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https://ctan.math.washington.edu/tex-archive/info/examples/PSTricks_6_de/33-05-25.ltx | washington.edu | CC-MAIN-2022-27 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2022-27/segments/1656103941562.52/warc/CC-MAIN-20220701125452-20220701155452-00594.warc.gz | 256,885,412 | 1,050 | %%
%% Ein Beispiel der DANTE-Edition
%%
%%
%% Copyright (C) 2010 Herbert Voss
%%
%% It may be distributed and/or modified under the conditions
%% of the LaTeX Project Public License, either version 1.3
%% of this license or (at your option) any later version.
%%
%% See http://www.latex-project.org/lppl.txt for details.
%%
%%
%% ====
% Show page(s) 1
%%
\documentclass[]{exaarticle}
\pagestyle{empty}
\setlength\textwidth{190.324pt}
\setlength\parindent{0pt}
\StartShownPreambleCommands
\usepackage{psgo}
\StopShownPreambleCommands
\begin{document}
\psscalebox{0.7}{%
\begin{psgopartialboard}[9]{(4,1)(9,6)}
\stone{white}{c}{3} \stone{white}{e}{3}
\stone{white}{d}{2} \stone{white}{d}{4}
\stone{black}{f}{3} \stone{black}{e}{2}
\stone{black}{e}{4}
\end{psgopartialboard}}
\end{document}
|
http://arxmliv.kwarc.info/sty/aipproc.cls | kwarc.info | CC-MAIN-2018-47 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2018-47/segments/1542039742569.45/warc/CC-MAIN-20181115075207-20181115101207-00342.warc.gz | 29,043,822 | 9,697 | %%
%% This is file `aipproc.cls',
%% generated with the docstrip utility.
%%
%% The original source files were:
%%
%% aipproc.dtx (with options: `class')
%%
%% Class aipproc to use with LaTeX2e
%% (C) 1998,2000 American Institute of Physics and Frank Mittelbach
%% All rights reserved
%%
%% Class aipproc to use with LaTeX2e
%%
%% Copyright (C) 1998, 2000, 2001, 2002, 2004, 2005 Frank Mittelbach
%% Copyright (C) 1998, 2000, 2001, 2002, 2004, 2005 American Institute of Physics
%% All rights reserved.
%%
%% Development of this class was commissioned by American Institute of Physics.
%%
\NeedsTeXFormat{LaTeX2e}[1999/06/01]
\ProvidesClass{aipproc}
[2005/11/11 v1.5a AIP Proceedings (FMi)]
%% \CharacterTable
%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
%% Digits \0\1\2\3\4\5\6\7\8\9
%% Exclamation \! Double quote \" Hash (number) \#
%% Dollar \$ Percent \% Ampersand \&
%% Acute accent \' Left paren \( Right paren \)
%% Asterisk \* Plus \+ Comma \,
%% Minus \- Point \. Solidus \/
%% Colon \: Semicolon \; Less than \<
%% Equals \= Greater than \> Question mark \?
%% Commercial at \@ Left bracket \[ Backslash \\
%% Right bracket \] Circumflex \^ Underscore \_
%% Grave accent \` Left brace \{ Vertical bar \|
%% Right brace \} Tilde \~}
%%
\IfFileExists{fixltx2e.sty}
{\RequirePackage{fixltx2e}}
{\RequirePackage{fix2col}[1998/08/17]}
\@ifpackageloaded{fixltx2e}{%
\@ifpackagelater{fixltx2e}{1999/12/02}{}{%
\def\addpenalty#1{%
\ifvmode
\if@minipage
\else
\if@nobreak
\else
\ifdim\lastskip=\z@
\penalty#1\relax
\else
\@tempskipb\lastskip
\advance \@tempskipb
\ifdim\prevdepth>\maxdepth\maxdepth\else
\ifdim \prevdepth = -\@m\p@ \z@ \else \prevdepth \fi
\fi
\vskip -\@tempskipb
\penalty#1%
\vskip\@tempskipb
\fi
\fi
\fi
\else
\@noitemerr
\fi}
\def \@doclearpage {%
\ifvoid\footins
\setbox\@tempboxa\vsplit\@cclv to\z@ \unvbox\@tempboxa
\setbox\@tempboxa\box\@cclv
\xdef\@deferlist{\@toplist\@botlist\@deferlist}%
\global \let \@toplist \@empty
\global \let \@botlist \@empty
\global \@colroom \@colht
\ifx \@currlist\@empty
\else
\@latexerr{Float(s) lost}\@ehb
\global \let \@currlist \@empty
\fi
\@makefcolumn\@deferlist
\@whilesw\if@fcolmade \fi{\@opcol\@makefcolumn\@deferlist}%
\if@twocolumn
\if@firstcolumn
\xdef\@deferlist{\@dbltoplist\@deferlist}%
\global \let \@dbltoplist \@empty
\global \@colht \textheight
\begingroup
\@dblfloatplacement
\@makefcolumn\@deferlist
\@whilesw\if@fcolmade \fi{\@outputpage
\@makefcolumn\@deferlist}%
\endgroup
\else
\vbox{}\clearpage
\fi
\fi
\ifx\@deferlist\@empty \else\clearpage \fi
\else
\setbox\@cclv\vbox{\box\@cclv\vfil}%
\@makecol\@opcol
\clearpage
\fi
}
\def \@addtocurcol {%
\@insertfalse
\@setfloattypecounts
\ifnum \@fpstype=8
\else
\ifnum \@fpstype=24
\else
\@flsettextmin
\advance \@textmin \@textfloatsheight
\@reqcolroom \@pageht
\ifdim \@textmin>\@reqcolroom
\@reqcolroom \@textmin
\fi
\advance \@reqcolroom \ht\@currbox
\ifdim \@colroom>\@reqcolroom
\@flsetnum \@colnum
\ifnum \@colnum>\z@
\@bitor\@currtype\@deferlist
\@testwrongwidth\@currbox
\if@test
\else
\@bitor\@currtype\@botlist
\if@test
\@addtobot
\else
\ifodd \count\@currbox
\advance \@reqcolroom \intextsep
\ifdim \@colroom>\@reqcolroom
\global \advance \@colnum \m@ne
\global \advance \@textfloatsheight \ht\@currbox
\global \advance \@textfloatsheight 2\intextsep
\@cons \@midlist \@currbox
\if@nobreak
\nobreak
\@nobreakfalse
\everypar{}%
\else
\addpenalty \interlinepenalty
\fi
\vskip \intextsep
\box\@currbox
\penalty\interlinepenalty
\vskip\intextsep
\ifnum\outputpenalty <-\@Mii \vskip -\parskip\fi
\outputpenalty \z@
\@inserttrue
\fi
\fi
\if@insert
\else
\@addtotoporbot
\fi
\fi
\fi
\fi
\fi
\fi
\fi
\if@insert
\else
\@resethfps
\@cons\@deferlist\@currbox
\fi
}}}
{}
\RequirePackage{calc}
\RequirePackage{ifthen}
\RequirePackage[final]{graphicx}
\newif\if@load@natbib \@load@natbibtrue
\IfFileExists{url.sty}
{\RequirePackage{url}%
}
{\def\url##1{\texttt{##1}}%
\ClassWarningNoLine{aipproc}
{\noexpand\url command might fail with
this LaTeX \MessageBreak installation since url.sty
is missing}%
}
\IfFileExists{textcase.sty}
{\RequirePackage{textcase}%
}
{\global\let\MakeTextUppercase\MakeUppercase
\ClassWarningNoLine{aipproc}
{\noexpand\section commands should not contain math
as this on LaTeX \MessageBreak installation the textcase
package is missing}%
}
\newcommand\AIP@optionnotsupported[1]
{\ClassWarningNoLine{aipproc}%
{Option~ `#1'~ not~ supported~ ---~ request~ ignored}}
\newcommand\AIP@error{\ClassError{aipproc}}
\newcommand\AIP@cmdnotsupported[1]
{\def#1{\AIP@error{Command \noexpand#1not supported by
class}\@eha}}
\newcommand\AIP@natbibnotavailable[1]
{\def#1{\AIP@error{Command \noexpand#1not supported if natbib
not installed}\@eha}}
\newcommand*\DesignerError[1]{%
\AIP@error{#1}{Probably bug in class file.}}
\newcommand*\InformationError[1]{%
\AIP@error{#1}%
{Add the necessary information to the document.}}
\newcommand\MakeSpaceIgnore{%
\catcode`\~=10\relax
\catcode`\ = 9\relax
\catcode`\^^M = 9\relax
}
\newcommand\MakeSpaceNormal{%
\catcode`\~= 13\relax
\catcode`\ = 10\relax
\catcode`\^^M = 5\relax
}
\let\UnbreakableSpace~
\MakeSpaceIgnore
\DeclareOption{a5paper} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{b5paper} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{legalpaper} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{executivepaper}{\AIP@optionnotsupported\CurrentOption}
\DeclareOption{landscape} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{10pt} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{11pt} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{12pt} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{titlepage} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{notitlepage} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{oneside} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{twoside} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{onecolumn} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{twocolumn} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{leqno} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{fleqn} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{openbib} {\AIP@optionnotsupported\CurrentOption}
\DeclareOption{tnotealph} {\def\AIP@tnote@representation{\@alph}}
\DeclareOption{tnotesymbol}{\def\AIP@tnote@representation{\@fnsymbol}}
\newboolean{@cmrfonts}
\DeclareOption{cmfonts} {\setboolean{@cmrfonts}{true}
\def\AIP@mathfontsused{0}}
\DeclareOption{mathptm} {\def\AIP@mathfontsused{1}}
\DeclareOption{mathtime} {\def\AIP@mathfontsused{2}}
\DeclareOption{nomathfonts}{\def\AIP@mathfontsused{3}}
\DeclareOption{mathptmx} {\def\AIP@mathfontsused{4}}
\DeclareOption{mtpro} {\def\AIP@mathfontsused{5}}
\def\pageref{0}
\DeclareOption{varioref} {\def\pageref{1}}
\DeclareOption{nonvarioref} {\def\pageref{2}}
\DeclareOption{numcites} {\def\AIPcitestyleselect{num}}
\DeclareOption{bibliocites} {\def\AIPcitestyleselect{biblio}}
\DeclareOption{nonatbib} {\dont@load@natbibfalse}
\DeclareOption{numberedheadings}
{\AtEndOfClass{\setcounter{secnumdepth}{3}}}
\DeclareOption{unnumberedheadings}
{\AtEndOfClass{\setcounter{secnumdepth}{-\maxdimen}}}
\DeclareOption{draft}{\PassOptionsToClass{\CurrentOption}{article}%
\@drafttrue
\AtEndOfPackage{
\let\AIP@pagenumerror\@gobble
\def\@oddfoot{\reset@font
\AIPfoliofont
\AIPfolioformat\@shorttitle\@date\thepage
}}}
\newif\if@draft
\DeclareOption{final}{\PassOptionsToClass{\CurrentOption}{article}}
\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}}
\ExecuteOptions{mathptmx,tnotesymbol,numcites,unnumberedheadings,letterpaper}
\ProcessOptions\relax
\MakeSpaceNormal
\LoadClass{article}
\MakeSpaceIgnore
\def\layoutstyle#1{%
\expandafter\let\expandafter
\AIP@layoutstylename
\csname AIP@layout@style@#1 \endcsname
\ifx\AIP@layoutstylename\relax
\def\AIP@layoutstylename{#1}
\fi
\MakeSpaceIgnore
\makeatletter
\InputIfFileExists{aip-\AIP@layoutstylename.clo}
{\let\AIP@check@layoutstyle\relax}
{\AIP@error{The~ layout~ style~ `#1'~ is~ not~
known\MessageBreak or~ its~
support~ file~ can~ not~ be~ found}
{The~ \noexpand \layoutstyle command~ tried~ to~ load~ the~ file~
aip-\AIP@layoutstylename.clo~ without~ success!\MessageBreak
This~ might~ be~ due~ to~ misspelling~ the~ style~
name.\MessageBreak
Standard~ styles~ are~ `6x9',~ `8x11single',~
`8x11double',~ and~ `arlo',~ but\MessageBreak there~
might~ be~ others~ (see~ the~ class~
documentation).\MessageBreak
It~ could~ also~ be~ due~ to~ an~ incomplete~ installation~
of~ the~ class.
}
}
\MakeSpaceNormal
\makeatother
\ifdim\columnsep>\z@
\@twocolumntrue
\else
\@twocolumnfalse
\fi
}
\@onlypreamble\layoutstyle
\def\declare@layoutstyle#1#2{
\@namedef{AIP@layout@style@#1}{#2}
}
\@onlypreamble\declare@layoutstyle
\declare@layoutstyle{6x9}{6s}
\declare@layoutstyle{8x11single}{8s}
\declare@layoutstyle{8x11double}{8d}
\def\AIP@check@layoutstyle{
\AIP@error{No~ \noexpand\layoutstyle command~ seen}
{The~ class~ requires~ a~ \noexpand\layoutstyle{<name>}~
declaration~ in~ the~ preamble!\MessageBreak
Standard~ styles~ are~ `6x9',~ `8x11single',~
`8x11double',~ and~ `arlo',~ but\MessageBreak there~
might~ be~ others~ (see~ the~ class~
documentation).\MessageBreak
To~ be~ able~ to~ proceed~ the~ 6x9~ style~ is~ assumed.
}
\layoutstyle{6x9}
\@colht\textheight
\@colroom\textheight \vsize\textheight
\columnwidth\textwidth
\@clubpenalty\clubpenalty
\if@twocolumn
\advance\columnwidth -\columnsep
\divide\columnwidth\tw@ \hsize\columnwidth \@firstcolumntrue
\fi
\hsize\columnwidth \linewidth\hsize
}
\AtBeginDocument{\AIP@check@layoutstyle}
\newcommand*\SetInternalRegister[2]{#1=#2\relax}
\let\SetInternalCounter\count@assign
\newcommand*\DeclareParagraphLayout[9]{%
\@namedef{#1Para}{
\fontsize{#2}{#3}\selectfont
#9
\setlength\parindent {#4}
\setlength\leftskip {#5}
\setlength\rightskip {#6}
\@rightskip\rightskip
\setlength\parfillskip{#7}
\setlength\parskip {#8}
}
}
\@onlypreamble\DeclareParagraphLayout
\newcommand*\UseParagraphLayout[1]{
\@ifundefined{#1Para}
{\DesignerError{Paragraph~ layout~ '#1'~ undefined}}
{\@nameuse{#1Para}}
}
\newcommand*\DeclareParagraphLayoutAlias[2]{%
\@ifundefined{#2Para}
{\DesignerError{Paragraph~ layout~ '#2'~ undefined}}
{\expandafter\let
\csname#1Para\expandafter\endcsname
\csname#2Para\endcsname
}
}
\@onlypreamble\DeclareParagraphLayoutAlias
\newcommand*\UseBBskip[1]
{\ifvmode
\setlength\@tempskipa{#1 - \parskip - \baselineskip}
\vskip\@tempskipa
\else
\DesignerError{\protect\UseBBskip\space outside~ vmode}
\fi
}
\newcommand*\DeclarePagestyle[5]
{
\@namedef{ps@#1}
{
\def\@oddhead {#2}
\def\@oddfoot {#3}
\def\@evenhead{#4}
\def\@evenfoot{#5}
}
}
\newdimen\bodytextsize
\newdimen\bodytextbaselineskip
\newdimen\bodytextenspace
\newdimen\bodytextparindent
\pagestyle{empty}
\AIP@cmdnotsupported\pagestyle
\newcommand\AIP@pagenumerror[1]{%
\AIP@error{Command~ \string#1~ can't~ be~ used~
in~ production}%
{This~ command~ will~ produce~ page~ numbers~ which~ will~
be~ incorrect~ in~ the\MessageBreak final~ production.
It~ should~ therefore~ only~ be~ used~ while~ producing~
drafts.}}
\let\@@tableofcontents\tableofcontents
\let\@@listoffigures\listoffigures
\let\@@listoftables\listoftables
\renewcommand\tableofcontents{%
\AIP@pagenumerror\tableofcontents\@@tableofcontents}
\renewcommand\listoffigures{%
\AIP@pagenumerror\listoffigures\@@listoffigures}
\renewcommand\listoftables{%
\AIP@pagenumerror\listoftables\@@listoftables}
\RequirePackage{aipxfm}
\MakeSpaceIgnore
\def\AIP@startsection#1#2#3#4#5{
\@tempskipa#2\relax
\advance\@tempskipa-\parskip
\ifdim\@tempskipa<\z@
\DesignerError{#2~ -~ \protect\parskip needs~ to~ be~
non-negative}
\fi
\ifthenelse{\equal#1{true}}
\relax
{\@tempskipa-\@tempskipa}
\edef\AIP@preskip{\the\@tempskipa}
\@tempskipa#4\relax
\advance\@tempskipa-\parskip
\ifdim\@tempskipa<\z@
\DesignerError{#2~ -~ \protect\parskip needs~ to~ be~
non-negative}
\fi
\ifthenelse{\equal#3{true}}
{\@tempskipa-\@tempskipa}
\relax
\edef\AIP@postskip{\the\@tempskipa}
\@secpenalty#5\relax
\@startsection
}
\renewcommand\section
{\AIP@startsection
\AIPsectionafterindent\AIPsectionpreskip
\AIPsectionrunin\AIPsectionpostskip
\AIPsectionpenalty
{section}{1}{\AIPsectionindent}%
{\AIP@preskip}%
{\AIP@postskip}%
{\AIPsectionfont\AIPsectionformat}}
\renewcommand\subsection
{\AIP@startsection
\AIPsubsectionafterindent\AIPsubsectionpreskip
\AIPsubsectionrunin\AIPsubsectionpostskip
\AIPsubsectionpenalty
{subsection}{2}{\AIPsubsectionindent}%
{\AIP@preskip}%
{\AIP@postskip}%
{\AIPsubsectionfont\AIPsubsectionformat}}
\renewcommand\subsubsection
{\AIP@startsection
\AIPsubsubsectionafterindent\AIPsubsubsectionpreskip
\AIPsubsubsectionrunin\AIPsubsubsectionpostskip
\AIPsubsubsectionpenalty
{subsubsection}{3}{\AIPsubsubsectionindent}%
{\AIP@preskip}%
{\AIP@postskip}%
{\AIPsubsubsectionfont\AIPsubsubsectionformat}}
\renewcommand\paragraph
{\AIP@startsection
\AIPparagraphafterindent\AIPparagraphpreskip
\AIPparagraphrunin\AIPparagraphpostskip
\AIPparagraphpenalty
{paragraph}{4}{\AIPparagraphindent}%
{\AIP@preskip}%
{\AIP@postskip}%
{\AIPparagraphfont\AIPparagraphformat}}
\renewcommand\subparagraph
{\AIP@startsection
\AIPsubparagraphafterindent\AIPsubparagraphpreskip
\AIPsubparagraphrunin\AIPsubparagraphpostskip
\AIPsubparagraphpenalty
{subparagraph}{5}{\AIPsubparagraphindent}%
{\AIP@preskip}%
{\AIP@postskip}%
{\AIPsubparagraphfont\AIPsubparagraphformat}}
\newcommand\UseNoHyphens{\hyphenpenalty\@M\exhyphenpenalty\@M}
\ifcase \AIP@mathfontsused
% 0 use cm for everything
\or
\MakeSpaceNormal
\RequirePackage{mathptm} % 1
\MakeSpaceIgnore
\or
\MakeSpaceNormal
\RequirePackage{mathtime} % 2
\MakeSpaceIgnore
\or
% 3 use cm for math
\or
\MakeSpaceNormal
\RequirePackage{mathptmx} % 4
\MakeSpaceIgnore
\or
\MakeSpaceNormal
\RequirePackage{mtpro} % 5
\MakeSpaceIgnore
\fi
\ifnum \AIP@mathfontsused > 0
\RequirePackage{times}
\normalfont
\RequirePackage[T1]{fontenc}
\RequirePackage{textcomp}
\fi
\AtBeginDocument{\UseParagraphLayout{AIPbodytext}}
\renewcommand\footnoterule{
\setlength\skip@{\AIPfootnoteruleheight+\AIPfootnoterulepostskip}
\vskip-\skip@
\moveright \AIPfootnoteruleindent\vbox{%
\hrule \@width \AIPfootnoterulewidth
\@height \AIPfootnoteruleheight}%
\vskip \AIPfootnoterulepostskip \relax}
\AtBeginDocument{
\setlength{\skip\footins}{\AIPfootnoterulepreskip
+\AIPfootnoterulepostskip}}
\renewcommand\@makefntext[1]{
\UseParagraphLayout{AIPfootnote}
\noindent
\hbox{\AIPfootnotetextmarkerformat
{\AIPfootnotetextmarkerfont\@thefnmark}}%
\ignorespaces
#1}
\def\@makefnmark{\hbox{%
\AIPfootnotemarkerformat{\AIPfootnotemarkerfont\@thefnmark}}}
\def \@makecol {%
\setbox\@outputbox \box\@cclv
\@combinefloats
\ifvoid\footins
\else
\setbox\@outputbox \vbox {%
\boxmaxdepth \@maxdepth
\unvbox \@outputbox
\vskip \skip\footins
\color@begingroup
\normalcolor
\footnoterule
\unvbox \footins
\color@endgroup
}%
\fi
\xdef\@freelist{\@freelist\@midlist}%
\global \let \@midlist \@empty
\ifvbox\@kludgeins
\@makespecialcolbox
\else
\setbox\@outputbox \vbox to\@colht {%
\@texttop
\dimen@ \dp\@outputbox
\unvbox \@outputbox
\vskip -\dimen@
\@textbottom
}%
\fi
\global \maxdepth \@maxdepth
}
\def\@fnsymbol#1{\ensuremath{\ifcase#1\or *\or \dagger\or **\or
\ddagger\or \mathsection\or \mathparagraph\or \|\or \dagger\dagger
\or \ddagger\ddagger \or\mathsection\mathsection
\or \mathparagraph\mathparagraph \or *{*}*\or
\dagger{\dagger}\dagger \or\ddagger{\ddagger}\ddagger\or
\mathsection{\mathsection}\mathsection
\or \mathparagraph{\mathparagraph}\mathparagraph \else\@ctrerr\fi}}
\def\@alph#1{\ifcase#1\or a\or b\or c\or d\or e\or f\or g\or h\or i\or
j\or k\or l\or m\or n\or o\or p\or q\or r\or s\or t\or u\or v\or
w\or x\or y\or z\or aa\or bb\or cc\or dd\or ee\or ff\or gg\or hh\or
ii\or jj\or kk\or ll\or mm\or nn\or oo\or pp\or qq\or rr\or ss\or
tt\or uu\or vv\or ww\or xx\or yy\or zz\else\@ctrerr\fi}
\AtBeginDocument{%
\ifx\tagform@\@undefined
\def\eqref#1{\mbox{\AIPeqreffont\AIPeqrefformat{\ref{#1}}}}%
\else
\def\tagform@#1{\mbox{\AIPeqreffont
\AIPeqrefformat{\ignorespaces
#1\unskip\@@italiccorr}}}%
\fi
\def\@eqnnum{{\AIPeqfont\AIPeqformat\theequation}}
}
\ifnum\pageref>0
\MakeSpaceNormal
\RequirePackage{varioref}
\MakeSpaceIgnore
\renewcommand\reftextfaceafter {on~ the~ next~ page}
\renewcommand\reftextfacebefore{on~ the~ \reftextvario{previous}
{preceding}~ page}
\renewcommand\reftextafter {on~ the~ \reftextvario{following}
{next}~ page}
\renewcommand\reftextbefore {on~ the~ \reftextvario{preceding~ page}
{page~ before}}
\renewcommand\reftextcurrent {on~ \reftextvario{this}%
{the~ current}~ page}
\renewcommand\reftextfaraway[1]{%
\is@pos@number\@tempb
{\ifnum\@tempb<0\@tempa\relax
\reftextearlier
\else
\reftextlater
\fi}%
{\@setref\relax\relax{#1}}}
\newcommand\reftextearlier{\reftextvario{on~ an~ earlier~ page}
{earlier~ on}}
\newcommand\reftextlater {\reftextvario{later~ on}{further~ down}}
\ifnum\pageref=2
\def\reftextvario#1#2{#1}
\fi
\let\pageref\vpageref
\else
\renewcommand\pageref[1]
{\AIP@error{Page~ references~ not~ supported}
{This~ class~ does~ not~ support~ references~ to~ page~
numbers~ unless~ the~ varioref~ or~ the~ nonvarioref~ option~
is~ used,~ since~ it~ doesn't~ print~ page~ numbers.}}
\fi
\newcommand\AIP@maketablecaption[2]{%
\UseParagraphLayout{AIPtable-singlelinecaption}
\settowidth\@tempdima{%
\noindent
{\AIPtablecaptionheadfont\AIPtablecaptionheadformat{#1}}
\AIPtablecaptiontextfont\ignorespaces#2}
\ifdim\@tempdima>\hsize
\UseParagraphLayout{AIPtable-multilinecaption}
\fi
\noindent
{\AIPtablecaptionheadfont\AIPtablecaptionheadformat{#1}}
\AIPtablecaptiontextfont\ignorespaces#2\par
\vskip\AIPtablecaptionskip}
\newskip\AIPtablecaptionskip
\newcommand\AIP@makefigurecaption[2]{%
\UseParagraphLayout{AIPfigure-singlelinecaption}
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|
http://hendrikmaryns.name/Wiskunde/Kombinatoriek-Kans/bronbestanden/kansen%20intro.tex | hendrikmaryns.name | CC-MAIN-2017-47 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2017-47/segments/1510934805761.46/warc/CC-MAIN-20171119191646-20171119211646-00037.warc.gz | 132,578,369 | 2,308 | % preamble versie 2016-09-14
\documentclass[captions=tableabove,headinclude=false,footinclude=false,headsepline]{scrartcl}
% schreefloze lettertipes voor tekst en wiskunde, moet voor fontspec, of hoe anders oproepen?
\usepackage{cmbright}
\usepackage{fontspec}
% oplossing voor °. Deze beter niet als superskript gebruiken volgens
% Context-goeroes, bovendien bevat cmbright het Unicode-symbool niet.
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% standaartpakketten
% taalpakket voor Xe, beter (?) dan babel
\usepackage{polyglossia}
\setdefaultlanguage{dutch}
% standaart voor wiskunde
\usepackage{amsmath}
% komma als desimaalscheiding
\usepackage{icomma}
% allerhande verbeteringen aan enumerate en itemize
\usepackage{enumitem}
% handige dingen voor tabellen
\usepackage{array}
% slimme verwijzingen
\usepackage{cleveref}
% laden van afbeeldingen
\usepackage{graphicx,grffile}
% berekeningen
\usepackage{calc}
% hoofdingen
\usepackage{scrlayer-scrpage}
\pagestyle{scrheadings}
\ihead{}
\chead{\textbf{\Large Inleidende opgaven kansrekening}}
\ohead{}
\ifoot{Klas 10}
\cfoot{Combinatoriek en kansrekening: Opgaven – 4}
\ofoot{}
\usepackage{pstricks}
\usepackage{picins}
\begin{document}
\begin{enumerate}
\item
\begin{enumerate}
\item Hoeveel mogelijkheden zijn er om met 2 dobbelstenen een 1 en een 6 te
gooien?
\item Hoeveel mogelijkheden zijn er om met 2 dobbelstenen überhaupt te gooien?
\item Welke uitkomst zou je hierdoor verwachten voor ons dobbelexperiment?
\item Hoe exact is het resultaat van ons experiment voor deze uitkomst?
\end{enumerate}
\item Hoe groot is het aantal uitkomsten met 2 dobbelstenen, waarbij de som
van de ogen
\begin{enumerate}
\item 5 is;
\item 7 is;
\item 10 is.
\end{enumerate}
\item Hoe groot is de kans om met 2 zeskantige dobbelstenen
\begin{enumerate}
\item meer dan 9 te gooien als je de ogen optelt?
\item een dubbel te gooien?
\item minstens één 3 te gooien?
\item een even getal te gooien als je de ogen optelt?
\end{enumerate}
\item
\parpic[r]{\includegraphics[height=2cm]{D4.png}}
Hiernaast zie je een viervlaksdobbelsteen. Met zo’n dobbelsteen kun je
1, 2, 3 of 4 gooien. Je gooit met een gewone dobbelsteen en met een
viervlaksdobbelsteen. Bereken de kans dat:
\begin{enumerate}
\item de som minstens 7 is;
\item je met de viervlaksdobbelsteen meer gooit dan met de gewone dobbelsteen;
\item het aantal ogen met beide dobbelstenen even groot is;
\item het product vier is. (Vb.: Het product van 3 en 5 is $3 \times 5 = 15$.)
\end{enumerate}
\item Floor gooit met een octaëder (achtkantige dobbelsteen) en met een
viervlaksdobbelsteen. Op de zijvlakken van de octaëder staan de cijfers 1 tot en met 8. Bereken de kans op de gebeurtenis:
\begin{enumerate}
\item de aantallen ogen zijn gelijk;
\item de som van de ogen is 8;
\item het product van de ogen is meer dan 16.
\end{enumerate}
\item Yvette gooit met twee geldstukken. Bereken de kans dat:
\begin{enumerate}
\item Yvette twee keer kop gooit;
\item Yvette één keer kop gooit.
\end{enumerate}
\item * In een urn zitten ballen met de getallen 1 tot 5. Je haalt er in één
keer twee ballen uit. Wat is de kans dat de som een getal deelbaar door 3 is?
\newpage
\item Je gooit 5 keer met een geldstuk. Wat is de kans dat:
\begin{enumerate}
\item je precies 4 keer kop gooit?
\item je minder dan 2 keer kop gooit?
\item je 5 keer hetzelfde gooit?
\end{enumerate}
\item In het volgende stratenrooster rijdt een taxi van A naar B.
\begin{center}
\begin{pspicture}(0,0)(5,4.5)
\psline(0,0)(0,4)
\psline(1,0)(1,4)
\psline(2,0)(2,4)
\psline(3,0)(3,4)
\psline(4,0)(4,4)
\psline(5,0)(5,4)
\psline(0,4)(5,4)
\psline(0,3)(5,3)
\psline(0,2)(5,2)
\psline(0,1)(5,1)
\psline(0,0)(5,0)
\uput*[45](0,0){A}
\psdot(0,0)
\uput*[45](5,4){B}
\psdot(5,4)
\uput*[-135](3,2){C}
\psdot(3,2)
\end{pspicture}
\end{center}
\begin{enumerate}
\item Hoeveel routes zijn er van A naar B?
\item Hoeveel routes zijn er van A naar B via C?
\item Wat is dus de kans dat, als je bij elke kruising willekeurig kiest, je
via C komt?
\item Geef commentaar op deze opgave.
\end{enumerate}
\end{enumerate}
\end{document}
|
https://www.apmep.fr/IMG/tex/Centres_etrangers_ES_juin_2008.tex | apmep.fr | CC-MAIN-2019-30 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2019-30/segments/1563195524568.14/warc/CC-MAIN-20190716135748-20190716161748-00191.warc.gz | 617,261,984 | 6,033 | \documentclass[10pt]{article}
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\begin{document}
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\rhead{\textbf{A. P{}. M. E. P{}.}}
\lhead{\small Baccalauréat ES}
\lfoot{\small{Centres étrangers}}
\rfoot{\small{17 juin 2008}}
\pagestyle{fancy}
\thispagestyle{empty}
\begin{center}\textbf{Durée : 3 heures}
\vspace{0,5cm}
{\Large \textbf{\decofourleft~Baccalauréat ES Centres étrangers 17 juin 2008~\decofourright }}\end{center}
\vspace{0,25cm}
\textbf{\textsc{Exercice 1} \hfill 5 points}
\textbf{Commun à tous les candidats}
\medskip
Une association organise chaque année un séjour qui s'adresse à des adultes handicapés. À sa création en 1997, dix adultes handicapés sont partis durant cinq jours. Ainsi, on dira qu'en 1997 le nombre de \og journées participant \fg{} est de $5 \times 10$ soit $50$.
Le tableau suivant donne le nombre de \og journées participant \fg{} de 1997 à 2004. L'année 1997 a le rang~0.
\medskip
\begin{tabularx}{\linewidth}{|m{5.6cm}|*{8}{>{\centering \arraybackslash}X|}}\hline
Années&1997 &1998& 1999& 2000 &2001 &2002 &2003 &2004\\ \hline
Rang de l'année : $x_{i}$& 0&1&2&3&4&5&6&7\\ \hline
Nombre de \og journées participant \fg{} : $y_{i}$&50&130& 200 &240&250&280&300& 320\\ \hline
\end{tabularx}
\medskip
\begin{enumerate}
\item Calculer le pourcentage d'augmentation du nombre de \og journées participant \fg{} de 1997 à 2000, puis celui de 2000 à 2001
\item Ces données sont représentées par le nuage de points ci-joint.
\medskip
\begin{center}
\psset{xunit=1cm,yunit=0.02cm}
\begin{pspicture}(-1,-50)(10,400)
\psframe(-1,-50)(10,400)
\multido{\n=0+50}{8}{\psline[linewidth=0.25pt](0,\n)(8,\n)}
\psaxes[linewidth=1.5pt,Dx=2,Oy=0,Dy=50]{->}(0,0)(10,400)
\psdots[dotstyle=square*,dotangle=45,linecolor=red](0,50)(1,130)(2,200)(3,240)(4,250)(5,280)(6,300)(7,320)
\end{pspicture}
\end{center}
\medskip
On considère qu'un ajustement affine n'est pas pertinent.
L'allure du nuage suggère de chercher un ajustement de $y$ en $x$ de la forme $y = k\ln (ax + b)$ où $k,~a$ et $b$ sont trois nombres réels. Pour cela on pose : $z_{i} = \text{e}^{\frac{y_{i}}{100}}$.
\textbf{Dans cette question les calcul. seront effectués à la calculatrice. Aucune justification n'est demandée. Les résultats seront arrondis au centième.}
\begin{enumerate}
\item Recopier et compléter le tableau suivant :
\medskip
\hspace*{-1cm}\begin{tabularx}{1.1\linewidth}{|m{3.5cm}|*{8}{>{\centering \arraybackslash}X|}}\hline
Rang de l'année : $x_{i}$& 0& 1& 2& 3& 4& 5& 6 &7\\ \hline
Nombre de \og journées participant \fg{} : $y_{i}$& 50& 130& 200 &240& 250& 280& 300 &320\\ \hline
\rule[0.3cm]{0pt}{5pt}$z_{i} = \text{e}^{\frac{y_{i}}{100}}$& 1,65&&&&&&&\\ \hline
\end{tabularx}
\medskip
\item Représenter le nuage de points associé à la série $\left(x_{i}~;~z_{i}\right)$ dans un repère orthononnal (unités : 1~cm)
\item Donner les coordonnées du point moyen et placer ce point sur le graphique précédent.
\item Déterminer une équation de la droite (D) d'ajustement affine de $z$ en $x$ par la méthode des moindres carrés. Représenter la droite (D) sur le graphique précédent
\item Sachant que $z_{i} = \text{e}^{\frac{y_{i}}{100}}$ déterminer l'expression de $y$ en fonction de $x$.
\end{enumerate}
\item On suppose que l'évolution du nombre de \og journées participant \fg{} se poursuit dans un futur proche selon le modèle précédent.
\begin{enumerate}
\item Estimer, à l'unité près, quel serait le nombre de \og journées participant \fg{} prévu pour l'année 2007.
\item En réalité, le nombre de \og journées participant \fg{} en 2007 a été de 390. Si l'écart en valeur absolue entre la valeur estimée et la valeur réelle est inférieure à 10\,\% de la valeur réelle, on considère que le modèle est pertinent. Est-ce le cas ?
\end{enumerate}
\end{enumerate}
\vspace{0,25cm}
\textbf{\textsc{Exercice 2} \hfill 5 points}
\textbf{Candidats n 'ayant pas suivi l'enseignement de spécialité}
\medskip
Un magasin de sport propose à la location des skis de piste, des snowboards et des skis de randonnée.
Son matériel est constitué de 50\,\% de skis de piste, le reste étant également réparti entre les snowboards et les skis de randonnée.
Après la journée de location, le matériel est contrôlé et éventuellement réparé.
Il a été constaté que la moitié des skis de piste, deux tiers des snowboards et le quart des skis de randonnée nécessitent une réparation.
\medskip
Chaque paire de ski et chaque snowboard est répertorié sur une fiche qui précise son suivi. On tire au hasard une fiche. On considère les évènements suivants :
\setlength\parindent{5mm}
\begin{itemize}
\item[] $Sp$ : \og La fiche est celle d'une paire de skis de piste \fg{} ;
\item[] $Sn$ : \og La fiche est celle d'un snowboard \fg{} ;
\item[] $Sr$ : \og La fiche est celle d'une paire de skis de randonnée \fg{} ;
\item[] $R$ : \og Le matériel nécessite une réparation \fg{} ; $\overline{R}$ est son évènement contraire.
\end{itemize}
\setlength\parindent{0mm}
\medskip
\emph{Tous les résultats des quatre premières questions seront donnés sous forme de fractions irréductibles.}
\medskip
\begin{enumerate}
\item Traduire toutes les données de l'énoncé à l'aide d'un arbre pondéré (on ne demande aucune explication).
\item Calculer la probabilité que la fiche tirée concerne une paire de skis de piste ne nécessitant pas une réparation.
\item Calculer la probabilité que la fiche tirée concerne du matériel ne nécessitant pas une réparation.
\item La fiche tirée concerne du matériel ayant nécessité une réparation.
Quelle est la probabilité que cette fiche concerne un snowboard ?
\item Les paires de skis de piste, de randonnée, ainsi que les snowboards sont loués 30~\euro{} pour la journée.
Quelle est l'espérance de gain sur le matériel loué sachant que chaque réparation coûte 20~\euro{} au loueur ?
\end{enumerate}
\vspace{0,25cm}
\textbf{\textsc{Exercice 2} \hfill 5 points}
\textbf{Candidats ayant suivi l'enseignement de spécialité}
\medskip
Dans un village, l'association de gymnastique volontaire possédait $50$ adhérents en 2000.
Depuis cette date, la trésorière a remarqué que chaque année elle reçoit $18$ ~nouvelles adhésions et que
85\,\% des anciens inscrits renouvellent leur adhésion.
On note $a_{n}$ le nombre d'adhérents pour l'année $2000 + n$ ;
on a donc $a_{0} = 50$ et $a_{n+1} = 0,85 a_{n} + 18$
pour tout entier naturel $n$.
\medskip
\begin{enumerate}
\item Soit la suite $\left(u_{n}\right)$ définie par $u_{n} = a_{n} - 120$ pour tout $n \geqslant 0$.
\begin{enumerate}
\item Montrer que la suite $\left(u_{n}\right)$ est une suite géométrique dont on précisera la raison et le premier
terme.
\item Démontrer que, pour tout entier naturel $n,~ a_{n} = 120 - 70 \times 0, 85^n$.
\item Déterminer la limite de la suite $\left(a_{n}\right)$ quand $n$ tend vers l'infini. Interpréter ce résultat.
\end{enumerate}
\item Chaque semaine, 60\,\% des adhérents s'inscrivent pour une heure de gymnastique et 40\,\% pour deux heures de gymnastique.
\begin{enumerate}
\item Exprimer en fonction de $n$ le nombre d'heures de gymnastique à prevoir par semaine pour l'an $2000 + n$.
\item Une séance de gymnastique dure une heure et est limitée à 20 personnes. On veut déterminer à partir de quelle année l'association devra prévoir plus de 8 séances par semaine. Démontrer qu'alors $n$ doit vérifier l'inéquation $98 \times 0,85^n < 8$.
Résoudre cette inéquation et conclure.
\emph{ Dans cette question, toute trace de recherche, même incomplète, ou d'initiative même non fructueuse sera prise en compte dans l'évaluation.}
\end{enumerate}
\end{enumerate}
\vspace{0,25cm}
\textbf{\textsc{Exercice 3} \hfill 4 points}
\textbf{Commun à tous les candidats}
\medskip
\emph{Cet exercice est un Q. C. M. (Questionnaire à Choix Multiples).}
\emph{Chaque question admet une seule réponse exacte : a, b ou c. Pour chacune des questions indiquer sur la copie le numéro de la question et la lettre correspondant à la réponse choisie. Aucune justification n'est demandée.}
\emph{Barème : une bonne réponse rapporte $0,5$ point. Une mauvaise réponse enlève $0,25$ point. L'absence de réponse ne rapporte ni n'enlève de point. Si le total des points est négatif, la note globale attribuée à l'exercice est ramenée à $0$.}
\bigskip
{\small \begin{tabularx}{\linewidth}{|c|p{8cm}|X|}\hline
&QUESTIONS& RÉPONSES\\ \hline
\multirow{3}{0.5cm}{Q1}& \multirow{3}{8cm}{D'une année sur l'autre, un produit perd 10\,\% de sa valeur. Le produit a perdu au moins 70\,\% de sa valeur initiale au bout de :}&\textbf{a.}~~ 7 années\\
&&\textbf{b.}~~11 années\\
&&\textbf{c.}~~ 12 années\\ \hline
\multirow{3}{0.5cm}{Q2}& \multirow{3}{8cm}{Dans une expérience aléatoire, la probabilité d'un évènement A est égale à 0,4. On répète huit fois cette expérience de façon indépendante. La probabilité que l'évènement A se réalise au moins une fois est égale à :}&\textbf{a.}~~ $(0,4)^8$\\
&&\textbf{b.}~~ $(0,6)^8$\\
&&\textbf{c.}~~ $1 - (0,6)^8$\\
&& \\\hline
\multirow{3}{0.5cm}{Q3}& \multirow{3}{8cm}{$F$ est la primitive qui s'annule en 1 de la fonction $f$ définie sur $\R$ par $f(x) = x^2 + 1$. On a }&\textbf{a.}~~ $F(0) = 1$\\
&&\textbf{b.}~~ $F(0) = - \dfrac{4}{3}$\\
&&\textbf{c.}~~ $F(0) = \dfrac{4}{3}$\rule[-3mm]{0mm}{8mm}\\ \hline
\multirow{3}{0.5cm}{Q4}& \multirow{3}{8cm}{$f$ est la fonction définie sur $\R$ par $f(x) = \text{e}^{3x}$. On appelle ($\mathcal{C}$) la courbe représentative de $f$ dans un repère. La tangente ($\mathcal{T}$) à la courbe ($\mathcal{C}$) au point A d'abscisse $0$ a pour coefficient directeur :}& \textbf{a.}~~$0$\\
&&\textbf{b.}~~ $1$\\
&&\textbf{c.}~~ $3$\\
&&\\ \hline
\end{tabularx}}
\medskip
Pour toutes les questions suivantes, on donne ci-dessous le tableau de variations d'une fonction $f$ définie et dérivable sur $]- \infty~;~ - 3[$. On appelle $(\mathcal{C})$ sa courbe représentative dans un repère.\\
\medskip
\begin{center}
\psset{unit=1cm}
\begin{pspicture}(9,3.5)
\psframe(9,3.5) \psline(0,3)(9,3) \psline(1,0)(1,3.5)
\uput[u](0.5,3){$x$} \uput[u](1.45,3){$- \infty$} \uput[u](3,3){$-3$} \uput[u](5,3){$-2$}
\uput[u](7,3){$2$} \uput[u](8.8,3){$3$} \put(0.3,1.5){$f(x)$} \uput[d](1.45,3){$+ \infty$}
\uput[u](5,0){$-2$} \uput[d](8.6,3){$+ \infty$}
\psline{->}(1.7,2.5)(4.6,0.4) \psline{->}(5.4,0.4)(8.3,2.7)
\rput*(3,1.55){$0$} \rput*(7,1.65){0}
\end{pspicture}
\end{center}
\smallskip
{\small \begin{tabularx}{\linewidth}{|c|p{7cm}|X|}\hline
&QUESTIONS& RÉPONSES\\ \hline
\multirow{3}{0.5cm}{Q5}& \multirow{3}{7cm}{On peut affirmer que :}&\textbf{a.}~~ $f(0) < 0$\\
&&\textbf{b.}~~$f(0) = 0$\\
&&\textbf{c.}~~ $f(0) > 0$\\ \hline
\multirow{3}{0.5cm}{Q6}& \multirow{3}{7cm}{La courbe ($\mathcal{C}$) admet pour asymptote la droite d'équation :}&\textbf{a.}~~ $x = 0$\\
&&\textbf{b.}~~ $x = 3$\\
&&\textbf{c.}~~ $y = 3$\\
&& \\\hline
\multirow{3}{0.5cm}{Q7}& \multirow{3}{7cm}{$g$ est la fonction définie par $g(x) = \ln [f(x)]$ sur l'intervalle $]- \infty~;~-3[$. La limite de $g$ en $- \infty$ :}&\textbf{a.}~~ est $- \infty$\\
&&\textbf{b.}~~ est $+ \infty$\\
&&\textbf{c.}~~ n'existe pas\\ \hline
\multirow{3}{0.5cm}{Q8}& \multirow{3}{7cm}{$F$ désigne une primitive de $f$ sur $] - \infty~;~3[$. $F$ est : }&\textbf{a.}~~strictement décroissante sur $] - \infty~;~3[$\\
&&\textbf{b.}~~ strictement décroissante sur $]-3~;~2[$\\
&&\textbf{c.}~~ strictement croissante sur $]-2~;~3[$\\ \hline
\end{tabularx}}
\vspace{0,5cm}
\textbf{\textsc{Exercice 4} \hfill 6 points}
\textbf{Commun à tous les candidats}
\medskip
On considère la fonction $f$ définie sur $]-1~;~+\infty[$ par
\[f(x) = -3x + 4 + 8 \ln (x + 1).\]
On note $(\mathcal{C})$ sa courbe représentative dans un repère orthonormal.
\medskip
\begin{enumerate}
\item
\begin{enumerate}
\item Calculer la limite de $f$ en $-1$. Donner l'interprétation graphique du résultat obtenu.
\item Déterminer la limite de $f$ en $+ \infty$ (on pourra utiliser $\displaystyle\lim_{x \to + \infty} \dfrac{\ln (x + 1)}{x} = 0$).
\end{enumerate}
\item
\begin{enumerate}
\item On note $f'$ la dérivée de $f$ sur $]-1~;~+\infty[$. Démontrer que $f'(x) = \dfrac{5 - 3x}{x + 1}$.
\item Étudier le signe de $f'$ et dresser le tableau de variations de $f$. On donnera une valeur arrondie au
dixième du maximum de $f$ sur $]-1~;~+\infty[$.
\end{enumerate}
\item On se place dans l'intervalle $\left[\dfrac{5}{3}~;~+ \infty\right[$. Démontrer que dans cet intervalle, l'équation $f(x) = 0$
admet une solution unique notée $x_{0}$. Donner une valeur approchée de $x_{0}$ à $10^{-2}$ près.
\item
\begin{enumerate}
\item Vérifier que la fonction $F$ définie par
\[F(x) = - \dfrac{3}{2}x^2 - 4x + 8(x+1) \ln (x + 1)\]
est une primitive de $f$ sur $]-1~;~ +\infty[$.
\item Calculer l'aire, exprimée en unités d'aire, du domaine plan limité par la courbe $(\mathcal{C})$, l'axe des abscisses et les droites d'équations $x = 0$ et $x = 5$ (on donnera la valeur exacte de cette aire et une valeur approchée au dixième près).
\end{enumerate}
\end{enumerate}
\end{document} |
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https://doc.libelektra.org/api/0.9.2/latex/try__compile__zeromq_8c.tex | libelektra.org | CC-MAIN-2020-45 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-45/segments/1603107900200.97/warc/CC-MAIN-20201028162226-20201028192226-00443.warc.gz | 292,242,612 | 955 | \hypertarget{try__compile__zeromq_8c}{}\doxysection{try\+\_\+compile\+\_\+zeromq.\+c File Reference}
\label{try__compile__zeromq_8c}\index{try\_compile\_zeromq.c@{try\_compile\_zeromq.c}}
Compilation test for Zero\+MQ.
{\ttfamily \#include $<$zmq.\+h$>$}\newline
Include dependency graph for try\+\_\+compile\+\_\+zeromq.\+c\+:
\nopagebreak
\begin{figure}[H]
\begin{center}
\leavevmode
\includegraphics[width=205pt]{try__compile__zeromq_8c__incl}
\end{center}
\end{figure}
\doxysubsection{Detailed Description}
Compilation test for Zero\+MQ.
\begin{DoxyCopyright}{Copyright}
B\+SD License (see L\+I\+C\+E\+N\+S\+E.\+md or \href{https://www.libelektra.org}{\texttt{ https\+://www.\+libelektra.\+org}})
\end{DoxyCopyright}
|
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% This version: 30/V/2002
% Some changes by SL: 22-3/VII/2002
% Lecture Notes for a course on optimization at the
% Tenth EPSRC/LMS Numerical Analysis Summer School
% University of Durham, 15-19th July 2002
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%% title page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\paperauthor}{Nicholas I. M. Gould and Sven Leyffer}
\newcommand{\papertitle}{An introduction to algorithms for
nonlinear optimization}
\pagestyle{myheadings}
\markboth{\paperauthor}{\papertitle}
\newcommand{\theabstract}{We provide a concise introduction to
modern methods for solving nonlinear optimization problems.
We consider both linesearch and trust-region methods for
unconstrained minimization, interior-point methods for problems
involving inequality constraints, and SQP methods for those
involving equality constraints. Theoretical as well as practical
aspects are emphasised. We conclude by giving a personal view
of some of the most significant papers in the area,
and a brief guide to on-line resources.}
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{titlepage}
\begin{flushright} {\large \bf RAL-TR-2002-03 (revised)} \end{flushright}
\vspace*{0.2 cm}
{\LARGE \bf
\begin{center}
An introduction to algorithms \\
for nonlinear optimization$^{\mbox{\small 1,2}}$
\end{center}}
\vspace*{0.1 cm}
\begin{center}
\mbox{} \\
Nicholas I. M. Gould$^{3,4,5}$
and
Sven Leyffer$^{6,7,8}$
\\
\end{center}
\vspace{0.4cm}
\begin{center}
\parbox{\textwidth}{
{
{\bf ABSTRACT \newline}
\theabstract}
}
\end{center}
\vspace{0.2 cm}
\noindent \rule{\textwidth}{0.001in}
\vspace{0.1 cm}
{\small
\begin{description}
\item $^1$ These notes were written to accompany
a course on optimization we gave at the
10th EPSRC/LMS Numerical Analysis Summer School at the
University of Durham, 15-19th July 2002. They are heavily
influenced by a similar course given by the first author to
M.Sc. students at Oxford University in Trinity Term, 2001.
The notes appeared in 2003 as pages 109-197 of the book
``Frontiers in Numerical Analysis (Durham 2002)'',
which was edited by J. F. Blowey, A. W. Craig and T. Shardlow
and published by Springer Verlag.
\item $^2$ Some of this material appeared, in expanded form, in the
book ``Trust-region methods'' [SIAM, Philadelphia, 2000]
by Andy Conn, Philippe Toint and the first author.
We are grateful to both Andy and Philippe for their
contributions.
\item $^3$ Computational Science and Engineering Department,
Rutherford Appleton Laboratory, \\
Chilton, Oxfordshire, OX11 0QX, England, EU.
Email: n.gould@rl.ac.uk
\item $^4$ Current reports available from %\\
``http://www.numerical.rl.ac.uk/reports/reports.html''.
\item $^5$ This work was supported in part by the EPSRC grant GR/R46641
\item $^6$ Department of Mathematics, University of Dundee, DD1 4HN,
Scotland.\\
Email: sleyffer@maths.dundee.ac.uk
\item $^7$ Address from 1st September 2002:
Mathematics and Computer Science Division, \\
Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439, USA.
\\ Email: leyffer@mcs.anl.gov
\item $^8$ Current reports available from %\\
``http://www-unix.mcs.anl.gov/$\sim$leyffer/''.
\end{description}
}
\vspace{0.6 cm}
\noindent
Computational Science and Engineering Department
\\
Atlas Centre
\\
Rutherford Appleton Laboratory
\\
Oxfordshire OX11 0QX
\vspace{0.1 cm}
\noindent \today .
\end{titlepage}
% Put in a table of contents
\pagenumbering{roman}
\tableofcontents
\newpage
\pagenumbering{arabic}
% ====================== end of header ==================
%\setcounter{section}{-1}
\section*{INTRODUCTION}
%\section{INTRODUCTION}
\addcontentsline{toc}{section}{INTRODUCTION}
The solution of nonlinear optimization problems---that
is the minimization or maximization of an objective function
involving unknown parameters/variables in which the variables
may be restricted by constraints---is one of the core
components of computational mathematics. Nature (and man) loves
to optimize, and the world is far from linear. In his book
on Applied Mathematics, the eminent mathematician Gil Strang
opines that optimization, along with the solution of systems
of linear equations, and of (ordinary and partial) differential equations,
is one of the three cornerstones of modern applied mathematics.
It is strange, then, that despite fathering many of the pioneers
in the area, the United Kingdom (in particular, higher education)
has turned away from the subject in favour of Strang's other
key areas. Witness the numbers of lectures in the EPSRC summer
school over the past ten years (broadly 3 series of lectures
on numerical linear algebra, 4 on ODEs, 17 on PDEs, 4 in other areas,
but {\em none} in optimization in Summer Schools V-XI)
and the relative paucity of undergraduate or postgraduate
courses in the area.
It is timely, therefore, to be given the opportunity to be able
to review developments in nonlinear optimization. The past 10
years have shown an incredible growth in the power and
applicability of optimization techniques, fueled in part by
the ``interior-point revolution'' of the late 1980s.
We have purposely chosen not to consider linear or discrete
problems, nor to describe important applications
such as optimal control, partially because there are other experts
better able to review these fields, and partly because they would
all make exciting courses on their own. Indeed, the prospect of
simply describing nonlinear optimization in five lectures is
extremely daunting, and each of the subjects we shall describe
could easily fill the whole course! Of course, there is a strong
cross-fertilisation of ideas from discrete to linear to nonlinear
optimization, just as there are strong influences both to and from
other branches of numerical analysis and computational mathematics.
This article is partitioned in broadly the same way as the
course on which it is based. Optimality conditions play a vital
role in optimization, both in the identification of optima, and
in the design of algorithms to find them. We consider these in
Section 1. Sections 2 and 3 are concerned with the two main
techniques for solving unconstrained optimization problems.
Although it can be argued that such problems arise relatively
infrequently in practice (nonlinear fitting being a vital
exception), the underlying linesearch and trust-region ideas
are so important that it is best to understand them first in
their simplest setting. The remaining two sections cover the
problems we really wish to solve, those involving constraints.
We purposely consider inequality constraints (alone) in one
and equality constraints (alone) in the other, since then the
key ideas may be developed without the complication of treating
both kinds of constraints at once. Of course, real methods cope
with both, and suitable algorithms will be hybrids of the methods
we have considered.
We make no apologies for mixing theory in with algorithms, since
(most) good algorithms have good theoretical underpinnings.
So as not to disturb the development in the main text, the proofs of
stated theorems have been relegated to Appendix C. In addition, we
do not provide citations in the main text, but have devoted
Appendix A to an annotated bibliography of what we consider to be
essential references in nonlinear optimization. Such a list is, by
its nature, selective, but we believe that the given references
form a corpus of seminal work in the area, which should be read by
any student interested in pursuing a career in optimization.
Before we start, we feel that one key development during the last
five years has done more to promote the use of optimization than
possibly any other. This is NEOS, the Network Enabled Optimization
Server, at Argonne National Laboratory and Northwestern University
in Chicago, see {\tt http://www-neos.mcs.anl.gov/neos} . Here,
users are able to submit problems for remote solution, without charge,
by a large (and expanding) collection of the world's best optimization
solvers, many of them being only available otherwise commercially.
Further details of what may be found on the World-Wide-Web are given in
Appendix B.
%
% Lecture 1: optimality conditions
%
\setcounter{lecture}{1}
\section{OPTIMALITY CONDITIONS AND WHY THEY ARE IMPORTANT}
\setcounter{equation}{0}
\subsection{Optimization problems}
\label{op}
As we have said optimization is concerned with the
minimization or maximization of an objective function, say, $f(x)$.
Since
\disp{ \mbox{maximum} \; f(x) = - \; \mbox{minimum} \; ( - f(x) )}
there is no loss in generality in concentrating in this article
on minimization---throughout, minimization will take place
with respect to an $n$-vector, $x$, of real unknowns. A bit of terminology here:
the smallest value of $f$ gives its \defn{minimum}, while any
(there may be more than one) corresponding values of $x$ are a \defn{minimizer}.
There are a number of important subclasses of optimization
problems. The simplest is
\defn{unconstrained minimization}, where we aim to
\disp{\tmininx{} f(x)}
where the \defn{objective function} $f$: $\Re^n \longrightarrow \Re$.
One level up is
\defn{equality constrained minimization}, where now we try to
\disp{\tmininx{} f(x) \tim{subject to} c(x) = 0}
where the \defn{constraints} $c$: $\Re^n \longrightarrow \Re^m$.
For consistency we shall assume that $m \leq n$, for otherwise
it is unlikely (but not impossible) that there is an $x$ that
satisfies all of the equality constraints. Another important problem is
\defn{inequality constrained minimization}, in which we aim to
\disp{\tmininx{} f(x) \tim{subject to} c(x) \geq 0}
where $c$: $\Re^n \longrightarrow \Re^m$ and now $m$ may be larger than $n$.
The most general problem involves both equality and
inequality constraints---some inequalities may have upper as well as lower
bounds---and may be further sub-classified depending on the nature of the
constraints. For instance, some of the $c_i(x)$ may be linear
(that is $c_i(x) = a_i^T x - b_i$ for some vector $a_i$ and scalar
$b_i$), some may be simple bounds on individual components of $x$
(for example, $c_i(x) = x_i$), or some may result from a network (``flow in =
flow out'').
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\noindent
\subsection{Notation}
It is convenient to introduce our most common notation and terminology
at the outset. Suppose that $f(x)$ is at least twice continuously
differentiable ($f \in \calC^2$). We let $\nabla_x f(x)$ denote the
vector of first
partial derivatives, whose $i$-th component is $\partial f(x)/\partial x_i$.
Similarly, the $i,j$-th component of the (symmetric) matrix $\nabla_{xx} f(x)$
is the second partial derivative $\partial^2 f(x)/\partial x_i \partial x_j$.
We also write the usual Euclidean inner product between two $p$-vectors
$u$ and $v$ as $\ip{u}{v} \eqdef \sum_{i=1}^{p} u_i v_i$ (and mention,
for those who care, that some but not all of what we have to say remains
true in more general Hilbert spaces!). We denote the set of points
for which all the constraints are satisfied as $\calC$, and say that
any $x \in \calC$ (resp. $x \notin \calC$) is \defn{feasible} (resp.
\defn{infeasible}).
With this in mind we define the \defn{gradient} and \defn{Hessian} (matrix) of
the objective function $f$ to be $g(x) \eqdef \nabla_x f(x)$ and
$H(x) \eqdef \nabla_{xx} f(x)$, respectively.
Likewise, the gradient and Hessian of the $i$-th constraint are
$a_i(x) \eqdef \nabla_x c_i(x)$ and
$H_i(x) \eqdef \nabla_{xx}c_i(x)$.
The \defn{Jacobian} (matrix) is
\disp{A(x) \eqdef \nabla_{x} c(x) \equiv
\vect{a_1^T(x) \\ \cdots \\ a_m^T(x)}.}
Finally, if $y$ is a vector (of so-called \defn{Lagrange multipliers}), the
\defn{Lagrangian} (function) is
\disp{\ell(x,y) \eqdef f(x) - \ip{y}{c(x)},}
while its gradient and Hessian with
respect to $x$ are, respectively,
\disp{\arr{rcl}{g(x,y) & \eqdef & \nabla_{x}\ell(x,y) \equiv
g(x) - \bigsum_{i=1}^m y_i a_i(x) \equiv g(x) - A^T(x) y \tim{and} \\
H(x,y) & \eqdef & \nabla_{xx}\ell(x,y) \equiv
H(x) - \bigsum_{i=1}^m y_i H_i(x).}}
One last piece of notation:
$e_i$ is the $i$-th unit vector, while $e$
is the vector of ones, and $I$ is the (appropriately dimensioned)
identity matrix.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
%\noindent
\subsection{Lipschitz continuity and Taylor's theorem}
It might be argued that those who understand Taylor's theorem
and have a basic grasp of linear algebra have all the tools
they need to study continuous optimization---of course, this leaves aside
all the beautiful mathematics needed to fully appreciate optimization
in abstract settings, yet another future EPSRC summer school course,
we hope!
Taylor's theorem(s) can most easily be stated for functions
with Lipschitz continuous derivatives. Let
$\calX$ and $\calY$ open sets, let $F : \calX \rightarrow \calY$, and
let $\|\cdot\|_{\calX}^{ }$ and $\|\cdot\|_{\calY}^{ }$ be norms on
$\calX$ and $\calY$ respectively.
Then $F$ is \defn{Lipschitz continuous at} $x \in \calX$
if there exists a function $\gamma(x)$ such that
\disp{\| F(z) - F(x) \|_{\calY}^{ } \leq \gamma(x) \| z - x \|_{\calX}^{ }}
for all $z \in \calX$.
Moreover $F$ is \defn{Lipschitz continuous throughout/in}
$\calX$ if there exists a constant $\gamma$ such that
\disp{\| F(z) - F(x) \|_{\calY}^{ } \leq \gamma \| z - x \|_{\calX}^{ }}
for all $x$ and $z \in \calX$. Lipschitz continuity relates (either
locally or globally) the changes that occur in $F$ to those that are
permitted in $x$.
Armed with this, we have the following {\em Taylor} approximation results.
The first suggests how good (or bad) a first-order (linear)
or second-order (quadratic) Taylor series approximation to a scalar-valued
function may be.
\lthm{bao-first-order-Taylor-error}{Let $\calS$ be an open subset of $\Re^n$,
and suppose $f:\calS \rightarrow \Re$ is continuously differentiable throughout
$\calS$. Suppose further that $g(x)$ is Lipschitz continuous at $x$,
with Lipschitz constant $\gamma^L(x)$ in some appropriate vector norm.
Then, if the segment $x + \theta s \in \calS$ for all
$\theta \in [0,1]$,
\disp{| f(x + s) - m^L( x + s ) | \leq \half \gamma^L(x) \| s \|^2, \tim{where}}
\disp{m^L(x + s) = f(x) + \ip{g(x)}{s}.}
If $f$ is twice continuously differentiable
throughout $\calS$ and $H(x)$ is Lipschitz continuous at $x$,
with Lipschitz constant $\gamma^Q(x)$,
\disp{|f(x + s) - m^Q( x + s ) | \leq \sixth \gamma^Q(x) \| s \|^3, \tim{where}}
\disp{m^Q(x + s) = f(x) + \ip{g(x)}{s} +
\half \ip{s}{H(x)s}.}
}
\noindent
The second result is a variation on the theme of the first, and is
often refereed to as the {\em generalized mean-value} theorem.
\lthm{mean-value}{Let $\calS$ be an open subset of $\Re^n$,
and suppose $f:\calS \rightarrow \Re$ is twice continuously differentiable
throughout $\calS$. Suppose further that $s \neq 0$, and that
the interval $[x, x+s] \in \calS$. Then
\disp{f(x + s) = f(x) + \ip{g(x)}{s} + \half \ip{s}{H(z)s}}
for some $z \in (x,x+s)$.
}
\noindent
The third result compares how bad a first-order
Taylor series approximation to a vector valued function might be.
%\vspace{0.2in}
\lthm{bao-first-order-vector-Taylor-error}{Let $\calS$
be an open subset of $\Re^n$,
and suppose $F:\calS \rightarrow \Re^m$ is continuously differentiable
throughout $\calS$.
Suppose further that $\nabla_x F(x)$ is Lipschitz continuous at $x$,
with Lipschitz constant $\gamma^L(x)$
in some appropriate vector norm and its induced matrix norm.
Then, if the segment $x + \theta s \in \calS$ for all
$\theta \in [0,1]$,
\disp{\| F(x + s) - M^L( x + s ) \| \leq \half \gamma^L(x) \| s \|^2 ,}
where
\disp{M^L(x + s) = F(x) + \nabla_x F(x)s}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\noindent
\subsection{Optimality conditions}
Now is the time to come clean. It is very, very difficult
to say anything about the solutions to the optimization problems
given in Section~\ref{op}. This is almost entirely because we
are considering very general problems, for which there may be
many local, often non-global, minimizers.
There are two possible ways around this.
We might choose to restrict the class of problems we allow, so that
all local minimizers are global. But since this would rule out
the vast majority of nonlinear problems that arise in practice,
we instead choose to lower our sights, and only aim for local
minimizers---there are methods that offer some guarantee of global
optimality, but to date they are really restricted to small or very
specially structured problems.
Formally, we still need to define what we mean by a local minimizer.
A feasible point $x_*$ is a \defn{local} minimizer
of $f(x)$ if there is an
open neighbourhood $\calN$ of $x_*$ such that $f(x_*) \leq f(x)$ for all
$x \in \calC \bigcap \calN$.
If there is an open neighbourhood $\calN$ of $x_*$ such that
$f(x_*) < f(x)$ for all $x \neq x_* \in \calC \bigcap \calN$, it is
\emph{isolated}.
While such definitions agree with our intuition, they are of very little
use in themselves. What we really need are optimality conditions.
Optimality conditions are useful for three reasons.
Firstly, the provide a means of guaranteeing that a candidate solution
is indeed (locally) optimal---these are the so-called
\defn{sufficient conditions}.
Secondly, they indicate when a point is not optimal---these are the
\defn{necessary conditions}.
Finally they guide us in the design of algorithms, since lack of optimality
indicates when we may improve our objective. We now give details.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\subsection{Optimality conditions for unconstrained minimization}
We first consider what we might deduce if we were fortunate enough to
have found a local minimizer of $f(x)$. The following two results
provide first- and second-order necessary optimality conditions
(respectively).
\lthm{bao-unc-1stordernec}{Suppose that $f \in C^1$, and that
$x_*$ is a local minimizer of $f(x)$. Then
\disp{ g(x_*) = 0.}}
\vspace*{-5mm}
\lthm{bao-unc-2ndordernec}{Suppose that $f \in C^2$, and that
$x_*$ is a local minimizer of $f(x)$. Then $g(x_*) = 0$ and $H(x_*)$
is positive semi-definite, that is
\disp{ \ip{s}{H(x_*) s} \geq 0 \tim{for all} s \in \Re^n.}}
\noindent
But what if we have found a point that satisfies the above conditions?
Is it a local minimizer? Yes, an isolated one, provided the following
second-order sufficient optimality conditions are satisfied.
\lthm{bao-unc-2ndordersuf}{Suppose that $f \in C^2$, that
$x_*$ satisfies the condition $g(x_*) = 0$, and that additionally $H(x_*)$
is positive definite, that is
\disp{ \ip{s}{H(x_*) s} > 0 \tim{for all}
s \neq 0 \in \Re^n.}
Then $x_*$ is an isolated local minimizer of $f$.}
\noindent
Notice how slim is the difference between these
necessary and sufficient conditions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\subsection{Optimality conditions for constrained minimization}
When constraints are present, things get more complicated. In particular,
the geometry of the feasible region at (or near) to a minimizer plays
a very subtle role. Consider a suspected minimizer $x_*$. We shall
say that a constraint is \defn{active} at $x_*$ if and only if
$c_i(x_*) = 0$. By necessity, equality constraints will be active, while
determining which (if any) of the inequalities is active is probably the
overriding concern in constrained optimization.
In order to say anything about optimality, it
is unfortunately necessary to rule out ``nasty'' local minimizers such as cusps
on the constraint boundary. This requires that we have to ask that so-called
\defn{constraint qualifications} hold---essentially these say that linear
approximations to the constraints characterize all feasible perturbations
about $x_*$ and that perturbations which keep strongly active constraints
strongly active (a \defn{strongly} active constraint is one that will still
be active if the data, and hence minimizer, is slightly
perturbed) are completely characterized by their corresponding
linearizations being forced to be active.
Fortunately, such assumptions are automatically
satisfied if the constraints are linear, or if the constraints that are
active have independent gradients, and may actually be guaranteed in far weaker
circumstances than these.
\subsubsection{Optimality conditions for equality-constrained minimization}
Given constraint qualifications,
first- and second-order necessary optimality conditions for
problems involving equality constraints are (respectively) as follows.
\lthm{bao-econ-1stordernec}{Suppose that $f,\; c \in C^1$, and that
$x_*$ is a local minimizer of $f(x)$ subject to $c(x) = 0$. Then, so long as
a first-order constraint qualification
holds, there exist a vector of Lagrange multipliers $y_*$ such that
\disp{\arr{rl}{
c(x_*) = 0 & \mbox{(\defn{primal feasibility}) and} \\
g(x_*) - A^T(x_*) y_* = 0 & \mbox{(\defn{dual feasibility}).}
}}}
\vspace*{-5mm}
\lthm{bao-econ-2ndordernec}{Suppose that $f,\; c \in C^2$,
and that $x_*$ is a local minimizer of $f(x)$ subject to $c(x) = 0$.
Then, provided that first- and second-order constraint qualifications
hold, there exist a vector of Lagrange multipliers $y_*$ such that
\eqn{e2n}{\bip{s}{H(x_*,y_*) s}
\geq 0 \tim{for all} s \in \calN}
where
\disp{\calN = \left\{s \in \Re^n \; | \; A(x_*) s = 0 \right\}.}}
\noindent
Notice that there are two first-order optimality requirements:
primal feasibility (the constraints are satisfied), and dual feasibility
(the gradient of the objective function is expressible as a linear
combination of the gradients of the constraints).
It is not hard to anticipate that, just as in the unconstrained case,
sufficient conditions occur when the requirement \req{e2n} is strengthened
to $\bip{s}{H(x_*,y_*) s} > 0$ for all $s \in \calN$.
\subsubsection{Optimality conditions for inequality-constrained minimization}
Finally, when the problem involves inequality constraints, it is easy
to imagine that only the constraints that are active at $x_*$ play
a role---the inactive constraints play no part in defining the minimizer---and
indeed this is so. First- and second-order necessary optimality conditions
are (respectively) as follows.
\lthm{bao-con-1stordernec}{Suppose that $f,\; c \in C^1$, and that
$x_*$ is a local minimizer of $f(x)$ subject to $c(x) \geq 0$. Then, provided
that a first-order constraint qualification
holds, there exist a vector of Lagrange multipliers
$y_*$ such that
\eqn{fonci}{\arr{rl}{
c(x_*) \geq 0 & \mbox{(\defn{primal feasibility}),} \\
{g(x_*) - A^T(x_*) y_* = 0 \tim{and} y_* \geq 0} &
\mbox{(\defn{dual feasibility}) and} \\
c_i(x_*) [y_*]_i = 0 & \mbox{(\defn{complementary slackness}).}}}}
\vspace*{-5mm}
\lthm{bao-con-2ndordernec}{Suppose that $f,\; c \in C^2$,
and that $x_*$ is a local minimizer of $f(x)$ subject to $c(x) \geq 0$.
Then, provided that first- and second-order constraint qualifications
hold, there exist a vector of Lagrange multipliers $y_*$ for which
primal/dual feasibility and complementary slackness requirements hold as
well as
\disp{\bip{s}{H(x_*,y_*) s}
\geq 0 \tim{for all} s \in \calN_+}
where
\eqn{setn}{\calN_+ = \left\{s \in \Re^n \; \left|
\arr{c}{
\ip{s}{a_i(x_*)} = 0 \;\mbox{if} \;
c_i(x_*) = 0 \; \mbox{\&} \; [y_*]_i > 0 \; \mbox{and} \\ \;
\ip{s}{a_i(x_*)} \geq 0 \;\mbox{if}\;
c_i(x_*) = 0 \; \mbox{\&} \; [y_*]_i = 0 \;\;\;\;\;\;\; }\right.\!\!\right\}.}}
\noindent
See how dual feasibility now imposes an extra requirement, that the
Lagrange multipliers be non-negative, while as expected there is an
additional (complementary slackness) assumption that inactive constraints
necessarily have zero Lagrange multipliers.
Also notice that $\calN_+$, the set over which the Hessian of the Lagrangian is
required to be positive semi-definite, may now be the intersection of
a linear manifold and a cone, a particularly unpleasant set to work with.
The by-now obvious sufficient conditions also hold:
%\vspace*{-5mm}
\lthm{bao-con-2ndordersuf}{Suppose that $f,\; c \in C^2$,
and that $x_*$ and a vector of Lagrange multipliers $y_*$
satisfy \req{fonci} and
\disp{\bip{s}{H(x_*,y_*) s} > 0}
for all $s$ in the set $\calN_+$ given in \req{setn}.
Then $x_*$ is an isolated local
minimizer of $f(x)$ subject to $c(x) \geq 0$.}
%
% Lecture 2: linesearch methods
%
\newpage
\setcounter{lecture}{2}
\section[LINESEARCH METHODS FOR UNCONSTRAINED OPTIMIZATION]{LINESEARCH METHODS FOR UNCONSTRAINED \\ OPTIMIZATION}
\setcounter{equation}{0}
\setcounter{figure}{0}
\label{lsmfuo}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
%\oddsidemargin 20mm
%\evensidemargin 20mm
In this and the next sections, we shall concentrate on the
unconstrained minimization problem,
\noindent
\disp{\tmininx{} f(x),}
where the objective function $f$: $\Re^n \longrightarrow \Re$.
We shall assume that $f \in C^1$ (sometimes $C^2$) with Lipschitz
continuous derivatives. Often in practice this assumption is violated,
but nonetheless the methods converge (perhaps by good fortune) regardless.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Despite knowing how to characterise local minimizers of our problem,
in practice it is rather unusual for us to be able to provide or compute
an explicit minimizer.
Instead, we would normally expect to fall back on a suitable iterative
process. An {\em iteration} is simply a procedure whereby a
sequence of points \disp{\{x_k\},\;\; k = 1, 2, \ldots}
is generated, starting from some initial ``guess'' $x_0$, with the overall
aim of ensuring that (a subsequence) of the $\{x_k\}$ has favourable limiting
properties. These might include that any limit generated
satisfies first-order or, even better,
second-order necessary optimality conditions.
Notice that we will not be able to guarantee
that our iteration will converge to a global minimizer unless we know that
$f$ obeys very strong conditions, nor regrettably in general that
any limit point is even a local minimizer (unless by chance it
happens to satisfy second-order sufficiency conditions).
What we normally do try to ensure is that, at the very least, the iteration
is \defn{globally} convergent, that is that (for at least) a subsequence of
iterates $\{g(x_k)\}$ converges to zero. And our hope is that such
a sequence converges at a reasonably fast asymptotic rate. These two
preoccupations lie at the heart of computational optimization.
For brevity, in what follows, we shall write
$f_k = f(x_k)$, $g_k = g(x_k)$ and $H_k = H(x_k)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\subsection{Linesearch methods}
Generically, linesearch methods work as follows. Firstly,
a \defn{search direction} $p_k$ is calculated from $x_k$. This
direction is required to be a \defn{descent direction}, i.e.,
\disp{\ip{p_k}{g_k} < 0 \tim{if} g_k \neq 0,}
so that, for small steps along $p_k$, Taylor's theorem
(Theorem~\ref{bao-first-order-Taylor-error}) guarantees
that the objective function may be reduced. Secondly,
a suitable \defn{steplength} $\alpha_k > 0$ is calculated so that
\disp{ f(x_k + \alpha_k p_k ) < f_k.}
The computation of $\alpha_k$ is the \defn{linesearch},
and may itself be an iteration.
Finally, given both search direction and steplength, the iteration
concludes by setting
\disp{x_{k+1} = x_k + \alpha_k p_k.}
Such a scheme sounds both natural and simple. But as with most simple ideas,
it needs to be refined somewhat in order to become a viable technique.
What might go wrong? Firstly, consider the example in Figure~\ref{2_1}.
\begin{figure}[ht]
%\centerline{\psfig{figure=l2_1.ps,height=8.0cm,silent=}}
\centerline{\psfig{figure=l21.eps,height=8.0cm,silent=}}
\begin{picture}(375,1)(23,3.3)
\put(43,85.2){$\scriptstyle f(x)$}
\put(152,8.0){$\scriptstyle x$}
\put(67,70){${\scriptstyle (x_1,f(x_1))}$}
\put(132,51.5){${\scriptstyle (x_2,f(x_2))}$}
\put(58,44.5){${\scriptstyle (x_3,f(x_3))}$}
\put(128,41.5){${\scriptstyle (x_4,f(x_4))}$}
\put(60,40){${\scriptstyle (x_5,f(x_5))}$}
%\put(126,38){${\scriptstyle (x_6,f(x_6))}$}
\end{picture}
\caption{\label{2_1}The objective function $f(x) = x^2$ and the iterates
$x_{k+1} = x_k + \alpha_k p_k$
generated by the descent directions $p_k = (-1)^{k+1}$ and steps
$\alpha_k = 2 + 3 / 2^{k+1}$ from $x_0 = 2$.}
\end{figure}
%\noindent
%\fbox{\parbox{\linewidth}{Nick: Note that the annotation in Figures~\ref{2_1}
%and \ref{2_2} is not entirely correct. The points $x_i$ should be along the
%x-axis. Maybe, a 2D example would be better or just a few words to explain.}}
\noindent
Here the search direction gives a descent direction, and the iterates
oscillate from one side of the minimizer to the other. Unfortunately,
the decrease per iteration is ultimately so small that the iterates
converge to the pair $\pm 1$, neither of which is a stationary point.
What has gone wrong? Simply the steps are too long relative to the
amount of objective-function decrease that they provide.
Is this the only kind of failure? Unfortunately, no. For consider
the example in Figure~\ref{2_2}.
\begin{figure}[hbt]
%\centerline{\psfig{figure=l2_2.ps,height=8.0cm,silent=}}
\centerline{\psfig{figure=l22.eps,height=8.0cm,silent=}}
\begin{picture}(375,1)(23,3.3)
\put(43,85.2){$\scriptstyle f(x)$}
\put(152,8.0){$\scriptstyle x$}
\put(120,68){${\scriptstyle (x_1,f(x_1))}$}
\put(132,51.5){${\scriptstyle (x_2,f(x_2))}$}
\put(130,44.5){${\scriptstyle (x_3,f(x_3))}$}
\put(128,41.5){${\scriptstyle (x_4,f(x_4))}$}
%\put(60,40){${\scriptstyle (x_5,f(x_5))}$}
%\put(126,38){${\scriptstyle (x_6,f(x_6))}$}
\end{picture}
\caption{\label{2_2}The objective function $f(x) = x^2$ and the iterates
$x_{k+1} = x_k + \alpha_k p_k$
generated by the descent directions $p_k = -1$ and steps
$\alpha_k = 1 / 2^{k+1}$ from $x_0 = 2$.}
\end{figure}
\noindent
Now the iterates approach the minimizer from one side, but the
stepsizes are so small that each iterate falls woefully short of
the minimizer, and ultimately converge to the non-stationary value $1$.
So now we can see that a simple-minded linesearch method can fail
if the linesearch allows steps that are either too long or too short
relative to the amount of decrease that might be obtained with a
well-chosen step.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Practical linesearch methods}
In the early days, it was often suggested that $\alpha_k$ should
be chosen to minimize $f(x_k + \alpha p_k )$. This is known
as an \defn{exact} linesearch. In most cases, exact linesearches
prove to be both very expensive---they are essentially univariate
minimizations---and most definitely not cost effective,
and are consequently rarely used nowadays.
Modern linesearch methods prefer to use \defn{inexact} linesearches, which
are guaranteed to pick steps that are neither too long nor too short.
In addition, they aim to pick a ``useful'' initial ``guess'' for
each stepsize so as to ensure fast asymptotic convergence---we will return
to this when we discuss Newton's method. The main contenders amongst the
many possible inexact linesearches are the so-called
``backtracking- Armijo'' and the ``Armijo-Goldstein'' varieties. The former
are extremely easy to implement, and form the backbone of most Newton-like
linesearch methods. The latter are particularly important when using
secant quasi-Newton methods (see
Section~\ref{qnmethods}), but alas we do not have space to describe them here.
Here is a basic {\em backtracking} linesearch to find $\alpha_k$:
\vspace{0.2in}
\noindent
\centerline{\framebox[3.4in]{\hspace*{0.2in}\parbox{3.3in}{
\vspace*{0.2in}
Given $\alpha_{\mbox{init}} > 0$ (e.g., $\alpha_{\mbox{init}} = 1$), \\
let $\alpha^{(0)} = \alpha_{\mbox{init}}$ and $l = 0$. \\
Until $f( x_k + \alpha^{(l)} p_k ) < f_k$ \\
\hspace*{6mm} set $\alpha^{(l+1)} = \tau \alpha^{(l)}$, where
$\tau \in (0,1)$ (e.g., $\tau = \half$) \\
\hspace*{6mm} and increase $l$ by 1. \\
Set $\alpha_k = \alpha^{(l)}$.
\vspace{0.2in}
}}}
\vspace{0.2in}
\noindent
Notice that the backtracking strategy prevents the step from getting too
small, since the first allowable value stepsize of the form
$\alpha_{\mbox{init}} \tau^i$, $i = 0, 1, \ldots$ is accepted.
However, as it stands, there is still no mechanism for
preventing too large steps relative to decrease in $f$. What is needed
is a tighter requirement than simply that $f( x_k + \alpha^{(l)} p_k ) < f_k$.
Such a role is played by the Armijo condition.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
The \defn{Armijo condition} is that the steplength be asked to give
slightly more than simply decrease in $f$. The actual requirement is that
\disp{f( x_k + \alpha_k p_k )
\leq f( x_k) + \alpha_k \beta \ip{p_k}{g_k} }
for some $\beta \in (0,1)$ (e.g., $\beta = 0.1$ or even $\beta = 0.0001$)---this
requirement is often said to give \defn{sufficient decrease}.
Observe that, since $\ip{p_k}{g_k} < 0$, the longer the step, the larger the
required decrease in $f$. The range of permitted values for the stepsize is
illustrated in Figure~\ref{2_3}.
\begin{figure}[h]
%\centerline{\psfig{figure=l2_3.ps,height=8.0cm,silent=}}
\centerline{\psfig{figure=l23.eps,height=8.0cm,silent=}}
\begin{picture}(375,1)(23,3.3)
\put(152,8.0){$\scriptstyle \alpha$}
\put(120,31){${\scriptstyle f(x_k+\alpha p_k)}$}
\put(65,20){${\scriptstyle f(x_k)+\alpha \ip{g_k}{p_k}}$}
\put(105,64){${\scriptstyle f(x_k)+\alpha \beta \ip{g_k}{p_k}}$}
\end{picture}
\caption{\label{2_3} A steplength of anything up to 1.8 is permitted
for this example, in the case where $\beta = 0.2$.}
\end{figure}
The Armijo condition may then be inserted into our previous backtracking
scheme to give the aptly-named \defn{Backtracking-Armijo} linesearch:
\vspace{0.2in}
\noindent
\centerline{\framebox[3.4in]{\hspace*{0.2in}\parbox{3.3in}{
\vspace*{0.2in}
Given $\alpha_{\mbox{init}} > 0$ (e.g., $\alpha_{\mbox{init}} = 1$), \\
let $\alpha^{(0)} = \alpha_{\mbox{init}}$ and $l = 0$. \\
Until
$f( x_k + \alpha^{(l)} p_k )
\leq f( x_k) + \alpha^{(l)} \beta \ip{p_k}{g_k}$ \\
\hspace*{6mm} set $\alpha^{(l+1)} = \tau \alpha^{(l)}$, where
$\tau \in (0,1)$ (e.g., $\tau = \half$) \\
\hspace*{6mm} and increase $l$ by 1. \\
Set $\alpha_k = \alpha^{(l)}$. \vspace{0.2in}
}}}
\vspace{0.2in}
\noindent
Of course, it is one thing to provide likely-sounding rules to control
stepsize selection, but another to be sure that they have the
desired effect. Indeed, can we even be sure that there are points which
satisfy the Armijo condition? Yes, for we have
\lthm{Armijo}{Suppose that $f \in C^1$, that $g(x)$ is Lipschitz continuous
with Lipschitz constant $\gamma(x)$, that $\beta \in (0,1)$
and that $p$ is a descent direction at $x$. Then the Armijo condition
\disp{f(x + \alpha p) \leq f(x) + \alpha \beta \ip{p}{g(x)}}
is satisfied for all $\alpha \in [0, \alpha_{\max(x,p)}]$, where
\disp{\alpha_{\max}(x,p) =
\frac{2 ( \beta - 1 ) \ip{p}{g(x)}}{\gamma(x) \|p\|_2^2}.}}
\vspace*{0.1in}
\noindent
Note that since $\gamma(x)$ is rarely known, the theorem does not
provide a recipe for computing $\alpha_{\max}(x,p)$, merely a guarantee
that there is such a suitable value. The numerator in $\alpha_{\max}(x,p)$
corresponds to the slope and the denominator to the curvature term. It
can be interpreted as follows: If the curvature term is large, then the
admissible range of $\alpha$ is small. Similarly, if the projected
gradient along the search direction is large, then the range of admissible
$\alpha$ is larger.
It then follows that the Backtracking-Armijo linesearch can be
guaranteed to terminate with a suitably modest stepsize.
%\noindent
\lcor{Armijo-linesearch}{
Suppose that $f \in C^1$, that $g(x)$ is Lipschitz continuous
with Lipschitz constant $\gamma_k$ at $x_k$, that $\beta \in (0,1)$
and that $p_k$ is a descent direction at $x_k$. Then the stepsize
generated by the backtracking-Armijo linesearch terminates with
\disp{\alpha_k \geq \min \left( \alpha_{\mbox{init}},
\frac{2 \tau (\beta - 1) \ip{p_k}{g_k}}{\gamma_k \|p_k\|_2^2} \right).}}
\noindent
Again, since $\gamma_k$ is rarely known, the corollary does not
give a practical means for computing $\alpha_k$, just an assurance
that there is a suitable value. Notice that the stepsize is
certainly not too large, since it is bounded above by $\alpha_{\max}$,
and can only be small when $\ip{p}{g(x)} / \|p\|_2^2$ is. This will be the
key to the successful termination of generic linesearch methods.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\noindent
\subsection{Convergence of generic linesearch methods}
%\vspace{0.5in}
In order to tie all of the above together, we first need to state our
Generic Linesearch Method:
\vspace{0.2in}
\noindent
\centerline{\framebox[3.4in]{\hspace*{0.2in}\parbox{3.3in}{
\vspace*{0.1in}
Given an initial guess $x_0$, let $k = 0$ \\
Until convergence: \\
\hspace*{6mm} Find a descent direction $p_k$ at $x_k$. \\
\hspace*{6mm} Compute a stepsize $\alpha_k$ using a\\
\hspace*{12mm} backtracking-Armijo linesearch along $p_k$. \\
\hspace*{6mm} Set $x_{k+1} = x_k + \alpha_k p_k$, and increase $k$ by 1.
\vspace*{0.1in}
}}}
\vspace{0.2in}
\noindent
It is then quite straightforward to apply Corollary~\ref{Armijo-linesearch}
to deduce the following very general convergence result.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
%\noindent
%\subsection{Global convergence theorem}
\vspace*{0.2in}
\noindent
\lthm{global-convergence-ls}{
Suppose that $f \in C^1$ and that $g$ is Lipschitz continuous on $\Re^n$.
Then, for the iterates generated by the Generic Linesearch Method,
\begin{description}
\item{either} \disp{g_l = 0 \tim{for some} l \geq 0}
\item{or} \disp{\lim_{k \rightarrow \infty} f_k = - \infty}
\item{or} \disp{\lim_{k \rightarrow \infty}
\min \left ( |\ip{p_k}{g_k}|, \frac{|\ip{p_k}{g_k}|}{\| p_k \|_2} \right)= 0.}
\end{description}}
\vspace{0.2in}
\noindent
In words, either we find a first-order stationary point in a finite
number of iterations, or we encounter a sequence of iterates for which
the objective function is unbounded from below, or the slope (or a normalized
slope) along the search direction converges to zero. While the first two
of these possibilities are straightforward and acceptable consequences,
the latter is perhaps not. For one thing, it certainly does not say
that the gradient converges to zero, that is the iterates may not ultimately be
first-order critical, since it might equally occur if
the search direction and gradient tend to be mutually orthogonal. Thus we
see that simply requiring that $p_k$ be a descent direction is not a
sufficiently demanding requirement. We will return to this shortly, but
first we consider {\em the} archetypical globally convergent algorithm,
the method of steepest descent.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\noindent
\subsection{Method of steepest descent}
\noindent
We have just seen that the Generic Linesearch Method may not succeed
if the search direction becomes orthogonal to the gradient. Is there
a direction for which this is impossible? Yes, when the
search direction is the descent direction
\disp{p_k = - g_k,}
the so-called \defn{steepest-descent} direction---the epithet is appropriate
since this direction solves the problem
\disp{ \minin{p \in \smallRe^n} m_k^L(x_k + p) \eqdef f_k + \ip{p}{g_k}
\tim{subject to} \|p\|_2 = \|g_k\|_2,}
and thus gives the greatest possible reduction in a first-order model
of the objective function for a step whose length is specified.
Global convergence follows immediately from
Theorem~\ref{global-convergence-ls}.
\vspace*{0.2in}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
%\noindent
%\subsection{Global convergence for steepest descent}
\noindent
\lthm{global-convergence-sd}{
Suppose that $f \in C^1$ and that $g$ is Lipschitz continuous on $\Re^n$.
Then, for the iterates generated by the Generic Linesearch Method
using the steepest-descent direction,
\begin{description}
\item{either} \disp{g_l = 0 \tim{for some} l \geq 0}
\item{or} \disp{\lim_{k \rightarrow \infty} f_k = - \infty}
\item{or} \disp{\lim_{k \rightarrow \infty} g_k = 0.}
\end{description}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As we mentioned above, this theorem suggests that steepest descent
really is the archetypical globally convergent method, and in practice
many other methods resort to steepest descent when they run into trouble.
However, the method is not scale invariant, as re-scaling
variables can lead to widely different ``steepest-descent'' directions.
Even worse, as we can see in Figure~\ref{sd},
\begin{figure}[htb]
\centerline{\psfig{figure=l2_4.ps,height=8.0cm,silent=}}
\caption{\label{sd}Contours for the objective function
$f(x,y) = 10(y-x^2)^2 + (x-1)^2$, and the
iterates generated by the Generic Linesearch steepest-descent method.}
\end{figure}
convergence may be (and actually almost always is) very slow in theory, while
numerically convergence sometimes does not occur at all as the iteration
stagnates. In practice, steepest-descent is all but worthless in most cases.
The figure exhibits quite typical behaviour in which the iterates repeatedly
oscillate from one side of a objective function ``valley'' to the other.
All of these phenomena may be attributed to a lack of attention to problem
curvature when building the search direction. We now turn to methods
that try to avoid this defect.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\noindent
\subsection{More general descent methods}
%\vspace*{0.2in}
%\noindent
\subsubsection{Newton and Newton-like methods}
Let $B_k$ be a symmetric, positive definite matrix. Then it is
trivial to show that the search direction $p_k$ for which
\disp{B_k p_k = - g_k}
is a descent direction. In fact, this direction solves the
direction-finding problem
\eqn{nld}{ \minin{p \in \smallRe^n} m_k^Q(x_k + p)
\eqdef f_k + \ip{p}{g_k} + \half \ip{p}{B_k p},}
where $m^Q_k(x_k + p)$ is a quadratic approximation to the objective
function at $x_k$.
Of particular interest is the possibility that $B_k = H_k$,
for in this case $m_k^Q(x_k+p)$ gives a second-order Taylor's approximation
to $f(x_k+p)$. The resulting direction for which
\disp{H_k p_k = - g_k}
is known as the \defn{Newton} direction, and any method which uses
it is a Newton method. But notice that the Newton direction is
only guaranteed to be useful in a linesearch context if
the Hessian $H_k$ is positive definite, for otherwise $p_k$ might turn out
to be an ascent direction.
It is also worth saying that while one can
motivate such Newton-like methods from the prospective of minimizing
a local second-order model of the objective function, one could
equally argue that they aim to find a zero of a local first-order model
\disp{g(x_k + p) \approx g_k + B_k p_k}
of its gradient.
So long as $B_k$ remains ``sufficiently'' positive definite, we can make
precisely the same claims for these second-order methods as for those
based on steepest descent.
\vspace*{0.2in}
\noindent
\lthm{global-convergence-gm}{
Suppose that $f \in C^1$ and that $g$ is Lipschitz continuous on $\Re^n$.
Then, for the iterates generated by the Generic Linesearch Method
using the Newton or Newton-like direction,
\begin{description}
\item{either} \disp{g_l = 0 \tim{for some} l \geq 0}
\item{or} \disp{\lim_{k \rightarrow \infty} f_k = - \infty}
\item{or} \disp{\lim_{k \rightarrow \infty} g_k = 0}
\end{description}
provided that the eigenvalues of $B_k$ are uniformly bounded and bounded
away from zero.}
\vspace{0.2in}
\noindent
Indeed, one can regard such methods as ``scaled'' steepest descent,
but they have the advantage that they
can be made scale invariant for suitable $B_k$, and crucially,
as we see in Figure~\ref{nm},
\noindent
\begin{figure}[ht]
\centerline{\psfig{figure=l2_5.ps,height=8.0cm,silent=}}
\caption{\label{nm}Contours for the objective function
$f(x,y) = 10(y-x^2)^2 + (x-1)^2$, and the
iterates generated by the Generic Linesearch Newton method.}
\end{figure}
their convergence is often significantly faster than steepest descent.
In particular, in the case of the Newton direction, the Generic
Linesearch method will usually converge very rapidly indeed.
\vspace{0.2in}
\noindent
\lthm{local-convergence-nm}{
Suppose that $f \in C^2$ and that $H$ is Lipschitz continuous on $\Re^n$.
Then suppose that the iterates generated by the Generic Linesearch Method
with $\alpha_{\mbox{init}} = 1$ and $\beta < \half$,
in which the search direction is chosen to be the Newton direction
$p_k^{ } = -H_k^{-1} g_k^{ }$ whenever $H_k$ is positive definite, has a limit
point $x_*$ for which $H(x_*)$ is positive definite. Then
\begin{description}
\item (i) $\alpha_k = 1$ for all sufficiently large $k$,
\item (ii) the entire sequence $\{ x_k\}$ converges to $x_*$, and
\item (iii) the rate is Q-quadratic, i.e, there is a constant $\kappa \geq 0$.
\disp{\lim_{k \rightarrow \infty}
\frac{ \| x_{k+1} - x_* \|_2}{ \| x_k - x_* \|_2^2} \leq \kappa.}
\end{description}
}
\vspace{-0.2in}
\subsubsection{Modified-Newton methods}
Of course, away from a local minimizer there is no reason to believe
that $H_k$ will be positive definite, so precautions need to be
taken to ensure that Newton and Newton-like linesearch methods, for which
$B_k$ is (or is close to) $H_k$, satisfy the
assumptions of the global convergence Theorem~\ref{global-convergence-gm}.
If $H_k$ is indefinite, it is usual to solve instead
\disp{ (H_k + M_k) p_k \equiv B_k p_k = - g_k,}
where $M_k$ is chosen so that $B_k = H_k + M_k$ is ``sufficiently'' positive definite
and $M_k = 0$ when $H_k$ is itself ``sufficiently'' positive definite.
This may be achieved in a number of ways.
Firstly, if $H_k$ has the spectral (that is eigenvector-eigenvalue)
decomposition $H_k^{ } = Q_k^{ } D_k^{ } Q_k^T$, then $M_k$ may be chosen
so that
\disp{B_k \equiv H_k + M_k = Q_k^{ } \max ( \epsilon I , | D_k^{ }| ) Q_k^T}
for some ``small'' $\epsilon$''. This will shift all the insufficiently
positive eigenvalues by as little as possible as is needed to make the overall
matrix positive definite. While such a decomposition may be too expensive
to compute for larger problems, a second, cheaper alternative is to find (or
estimate) the smallest (necessarily real!) eigenvalue, $\lambda_{\min}(H_k)$,
of $H_k$, and to set
\disp{M_k = \max ( 0, \epsilon - \lambda_{\min}(H_k) ) I}
so as to shift {\em all} the eigenvalues by just enough as to make the
smallest ``sufficiently'' positive. While this is often tried in
practice, in the worst case it may have the effect of over-emphasising
one large, negative eigenvalue at the expense of the remaining small,
positive ones, and in producing a direction which is essentially
steepest descent. Finally, a good compromise is instead to attempt
a Cholesky factorization of $H_k$, and to alter the generated factors
if there is evidence that the factorization will otherwise fail.
There are a number of so-called \defn{Modified Cholesky} factorizations,
each of which will obtain
\disp{B_k \equiv H_k^{ } + M_k^{ } = L_k^{ } L_k^T,}
where $M_k$ is zero for sufficiently positive-definite $H_k$, and
``not-unreasonably large'' in all other cases.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Quasi-Newton methods}
\label{qnmethods}
It was fashionable in the 1960s and 1970s to attempts to build
suitable approximations $B_k$ to the Hessian, $H_k$. Activity in this area
has subsequently died down, possibly because people started to
realize that computing exact second derivatives was not as
onerous as they had previously contended, but these techniques
are still of interest particularly when gradients are awkward
to obtain (such as when the function values are simply given
as the result of some other, perhaps hidden, computation).
There are broadly two classes of what may be called quasi-Newton methods.
The first are simply based on estimating columns of $H_k$ by
\defn{finite differences}. For example, we might use the approximation
\disp{(H_k) e_i \approx h^{-1} ( g(x_k + h e_i) - g_k ) \eqdef (B_k) e_i}
for some ``small'' scalar $h>0$. The difficulty here is in choosing
an appropriate value for $h$: too large a value gives inaccurate approximations,
while a too small one leads to large numerical cancellation errors.
The second sort of quasi-Newton methods are known
as \defn{secant approximations}, and try to ensure the \defn{secant condition}
\disp{B_{k+1} s_k = y_k, \tim{where} s_k = x_{k+1} - x_k \tim{and}
y_k = g_{k+1} - g_k, }
that would be true if $H(x)$ were constant, is satisfied. The secant
condition gives a lot of flexibility, and among the many methods
that have been discovered, the \defn{Symmetric Rank-1} method, for which
\disp{B_{k+1} = B_k +
\frac{(y_k - B_k s_k) (y_k - B_k s_k)^T}{\ip{s_k}{y_k - B_k s_k}},}
and the \defn{BFGS} method, for which
\disp{B_{k+1} = B_k + \frac{y_k y_k^T}{\ip{s_k}{y_k} } -
\frac{B_k s_k s_k^T B_k}{\ip{s_k}{B_k s_k}}}
are the best known (and generally the best).
Note that the former may give indefinite approximations (or even fail),
while the latter is guaranteed to generate symmetric and positive definite
matrices so long as $B_0$ is positive definite and $\ip{s_k}{y_k} > 0$ (the
last condition may be ensured by an appropriate ``Goldstein'' linesearch).
Since both of these secant methods are based on low-rank updates, it
is possible to keep the per-iteration linear algebraic requirements
at a more modest level for such methods than is generally possible with
Newton or finite-difference methods.
\subsubsection{Conjugate-gradient and truncated-Newton methods}
And what if the problem is large and matrix factorization is out of the question?
We have already considered (and rejected) steepest-descent methods. Is there
something between the simplicity of steepest descent and the power (but
expense) of Newton-like methods? Fortunately, the answer is yes.
Suppose that instead of solving \req{nld}, we instead find our
search direction as
\disp{ p_k = \mbox{(approximate)} \argminover{p \in \smallRe^n} q(p)
= f_k + \ip{p}{g_k} + \half \ip{p}{B_k p},}
where we assume that $B_k$ is positive definite---the key word here is
{\em approximate}. Suppose that instead of minimizing $q$ over all
$p \in \Re^n$, we restrict $p$ to lie in a (much) smaller
subspace---of course if we
do this we will not (likely) obtain the optimal value of $q$, but we
might hope to obtain a good approximation with considerably less effort.
Let $D^i = (d^0 : \cdots : d^{i-1})$ be any collection of $i$ vectors, let
\disp{\calD^i = \{ p \st p = D^i p_d \tim{for some} p_d \in \Re^i\}}
be the subspace spanned by $D^i$, and suppose that we choose to pick
\disp{p^i = \argminover{p \in \calD^i} q(p).}
Then immediately $D^{i\;T} g^i = 0$, where $g^i = B_k p^i + g_k$
is the gradient of $q$ at $p^i$.
More revealingly, since $p^{i-1} \in \calD^i$, it follows that
$p^i = p^{i-1} + D^i p^i_d$, where
\disp{\arr{rl}{p^i_d = & \argminover{p_d \in \smallRe^i}
\ip{p_d}{D^{i\;T} g^{i-1}} +
\half \ip{p_d}{D^{i\;T} B_k D^i p_d} \\
= & - ( D^{i\;T} B_k D^i )^{-1} D^{i\;T} g^{i-1} =
- \ip{d^{i-1}}{g^{i-1}} ( D^{i\;T} B_k D^i )^{-1} e_i. }}
Hence
\eqn{pu}{p^i = p^{i-1} - \ip{d^{i-1}}{g^{i-1}}
D^i ( D^{i\;T} B_k D^i )^{-1} e_i.}
All of this is true regardless of $D^i$. But now suppose that
the members of $\calD^i$ are $B_k$\defn{-conjugate}, that is
to say that $\ip{d_i}{B_k d_j} = 0$ for all $i \neq j$. If this is so
\req{pu} becomes
\eqn{pu2}{p^i = p^{i-1} + \alpha^{i-1} d^{i-1}, \tim{where}
\alpha^{i-1} = - \bigfrac{\ip{d^{i-1}}{g^{i-1}}}{\ip{d^{i-1}}{B_k d^{i-1}}}.}
Thus so long as we can generate $B_k$-conjugate vectors, we can
build up successively improving approximations to the minimize of $q$
by solving a sequence of {\em one-dimensional} minimization problems---the
relationship \req{pu2} may be interpreted as finding $\alpha^{i-1}$
to minimize $q(p^{i-1} + \alpha d^{i-1})$. But can we find suitable
$B_k$-conjugate vectors?
Surprisingly perhaps, yes, it is easy.
Since $g^i$ is independent of $\calD^i$, let
\disp{d^i = - g^i + \bigsum_{j=0}^{i-1} \beta^{i j} d^j}
for some unknown $\beta^{i j}$. Then elementary manipulation (and a
cool head) shows that if we choose
$\beta^{i j}$ so that $d^i$ is $B$-conjugate to $\calD^i$, we obtain
the wonderful result that
\disp{\beta^{i j} = 0 \tim{for $j < i-1$, and}
\beta^{i \; i-1} \equiv \beta^{i-1} = \bigfrac{\|g_i\|_2^2}{\|g_{i-1}\|_2^2}.}
That is, almost all of the $\beta^{i j}$ are zero! Summing all of this
up, we arrive at the method of \defn{conjugate gradients} (CG):
\vspace*{0.2in}
\noindent
\centerline{\framebox[3.3in]{\hspace*{0.2in}\parbox{2.9in}{
\vspace*{0.2in}
Given $p^0 = 0$, set $g^0 = g_k$, $d^0 = - g_k$ and $i = 0$. \\
Until $g^i$ is ``small'', iterate: \\
\hspace*{6mm} $\alpha^i = \|g^i\|_2^2 / \ip{d^i}{Bd^i}$ \\
\hspace*{6mm} $p^{i+1} = p^i + \alpha^i d^i$ \\
\hspace*{6mm} $g^{i+1} = g^i + \alpha^i B_k d^i$ \\
\hspace*{6mm} $\beta^i = \|g^{i+1}\|_2^2 / \|g^{i} \|_2^2$ \\
\hspace*{6mm} $d^{i+1} = - g^{i+1} + \beta^i d^i$ \\
\hspace*{6mm} and increase $i$ by 1.
\vspace{0.2in}}}}
\vspace{0.2in}
\noindent
Important features are that
$\ip{d^{j}}{g^{i+1}} = 0$
and
$\ip{g^{j}}{g^{i+1}} = 0$
for all $j = 0, \ldots, i$,
and most particularly that
$\ip{p^i}{g_k} \leq \ip{p^{i-1}}{g_k} < 0$ for $i = 1, \ldots, n$,
from which we see that {\em any} $p_k = p^i$ is a descent direction.
In practice the above conjugate gradient iteration may be
seen to offer a compromise between the steepest-descent direction
(stopping when $i = 1$) and a Newton (-like) direction
(stopping when $i = n$). For this reason, using such
a curtailed conjugate gradient step within a linesearch (or trust-region)
framework is often known as a {\em truncated}-Newton method.
Frequently the size of $g^i$ relative
to $g_k$ is used as a stopping criteria, a particularly popular
rule being to stop the conjugate-gradient iteration when
\disp{\|g^i\| \leq \min( \|g_k\|^\omega , \eta ) \|g_k\|, }
where $\eta$ and $\omega \in (0,1)$, since then a
faster-than-linear
asymptotic convergence rate may be achieved if $B_k = H_k$.
%
% Lecture 3: trust-region methods
%
%\newpage
\setcounter{lecture}{3}
\section{TRUST-REGION METHODS FOR UNCONSTRAINED OPTIMIZATION}
\setcounter{equation}{0}
\setcounter{figure}{0}
\label{trmfuo}
In this section, we continue to concentrate on the unconstrained minimization
problem, and shall as before assume that
the objective function is $C^1$ (sometimes $C^2$)
with Lipschitz continuous derivatives.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\subsection{Linesearch vs. trust-region methods}
One might view linesearch methods as naturally ``optimistic''.
Fairly arbitrary search directions are permitted---essentially
50\% of all possible directions give descent from a given point---while
unruly behaviour is held in check via the linesearch. There is, however, another
possibility, that more control is taken when choosing the search direction,
with the hope that this will then lead to a higher probability that the (full)
step really is useful for reducing the objective.
This naturally ``conservative'' approach is the basis of trust-region methods.
As we have seen, linesearch methods pick a descent direction $p_k$,
then pick a stepsize $\alpha_k$ to ``reduce'' $f(x_k + \alpha p_k)$
and finally accept $x_{k+1} = x_k + \alpha_k p_k$. \defn{Trust-region}
methods, by contrast, pick the overall step
$s_k$ to reduce a ``model'' of $f(x_k + s)$, and accept
$x_{k+1} = x_k + s_k$ if the decrease predicted by the model is
realised by $f(x_k + s_k)$. Since there is no guarantee that this
will always be so, the fall-back mechanism is to set $x_{k+1} = x_k$,
and to ``refine'' the model when the existing model produces a poor
step. Thus, while a linesearch method recovers from a poor step
by retreating along a parametric (usually linear) curve, a trust-region
method recovers by reconsidering the whole step-finding procedure.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Trust-region models}
It is natural to build a model of $f(x_k+s)$ by considering Taylor series
approximations. Of particular interest are the
\defn{linear} model
\disp{m^L_k(s) = f_k + \ip{s}{g_k},}
and the \defn{quadratic} model
\disp{m^Q_k(s) = f_k + \ip{s}{g_k} + \half \ip{s}{B_k s},}
where $B_k$ is a symmetric approximation to the local Hessian matrix $H_k$.
However, such models are far from perfect. In particular, the
models are unlikely to resemble $f(x_k+s)$ if $s$ is large.
More seriously, the models may themselves be unbounded from below
so that any attempts to minimize them may result in a large step. This
defect will always occur for the linear model
(unless $g_k = 0$), and also for the quadratic model
if $B_k$ is indefinite (and possibly if $B_k$ is only positive semi-definite).
Thus simply using a Taylor-series model is fraught with danger.
There is, fortunately, a simple and effective way around this conundrum.
The idea is to prevent the model $m_k(s)$ from being unboundedness by imposing
a \defn{trust-region} constraint
\disp{\| s \| \leq \Delta_k,}
for some ``suitable'' scalar \defn{radius} $\Delta_k > 0$, on the step.
This is a natural idea, since we know from
Theorem~\ref{bao-first-order-Taylor-error} that we can improve the
approximation error $|f(x_k+s) - m_k(s)|$ by restricting the allowable step.
Thus our \defn{trust-region subproblem} is to
\disp{\mbox{approximately}\;\; \minin{s \in \smallRe^n} m_k(s) \tim{subject to}
\|s\| \leq \Delta_k,}
and we shall choose $s_k$ as approximate solution of this problem.
In theory, it does not depend on which norm $\| \cdot \|$ we use
(at least, in finite-dimensional spaces), but in practice it might!
For simplicity, we shall concentrate on the second-order (Newton-like) model
\disp{m_k(s) = m^Q_k(s) = f_k + \ip{s}{g_k} + \half \ip{s}{B_k s}}
and any (consistent) trust-region norm $\| \cdot \|$ for which
\disp{\kappa_s \| \cdot \| \leq \| \cdot \|_2 \leq \kappa_l \| \cdot \|}
for some $\kappa_l \geq \kappa_s > 0$.
Notice that the gradient of $m_k(s)$ at $s = 0$ coincides with the
gradient of $f$ at $x_k$, and also, unlike for linesearch methods, $B_k = H_k$
is always allowed.
The vast majority of models use the $\ell_1$, $\ell_2$ or $\ell_{\infty}$
norms on $\Re^n$, and for these we have
$\| \cdot \|_2 \leq \| \cdot \|_2 \leq \| \cdot \|_2$ (obviously!!),
$n^{-\half} \| \cdot \|_1 \leq \| \cdot \|_2 \leq \| \cdot \|_1$ and
$\| \cdot \|_{\infty} \leq \| \cdot \|_2 \leq n \| \cdot \|_{\infty}$.
%\disp{\arr{c}{
%\| \cdot \|_2 \leq \| \cdot \|_2 \leq \| \cdot \|_2 \tim{(obviously!!)} \\
%n^{-\half} \| \cdot \|_1 \leq \| \cdot \|_2 \leq \| \cdot \|_1 \tim{and} \\
%\| \cdot \|_{\infty} \leq \| \cdot \|_2 \leq n \| \cdot \|_{\infty}.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Basic trust-region method}
Having decided upon a suitable model, we now turn to the trust-region
algorithm itself. As we have suggested, we shall choose to ``accept''
$x_{k+1} = x_k + s_k$ whenever (a reasonable fraction of)
the predicted model decrease $f_k - m_k(s_k)$ is realized by the actual
decrease $f_k - f(x_k + s_k)$. We measure this by computing the ratio
\disp{\rho_k = \frac{f_k - f(x_k + s_k)}{f_k - m_k(s_k)}}
of actual to predicted decrease, and accepting the trust-region step
when $\rho_k$ is not unacceptably smaller than 1.0. If the ratio is
close to (or larger than) 1.0, there is good reason to believe that
future step computations may well benefit from an increase in the
trust-region radius, so we allow a radius increase in this case.
If, by contrast, there is poor agreement between the actual and predicted
decrease (and particularly, if $f$ actually increases), the current step
is poor and should be rejected. In this case, we reduce the trust-region
radius to encourage a more suitable step at the next iteration.
We may summarize the basic trust-region method as follows:
\vspace{0.3in}
\noindent
\centerline{\framebox[4.1in]{\hspace*{0.2in}\parbox{3.8in}{
\vspace*{0.2in}
Given $k = 0$, $\Delta_0 > 0$ and $x_0$, until ``convergence'' do: \\
\hspace*{6mm} Build the second-order model $m(s)$ of $f(x_k+s)$.\\
\hspace*{6mm} ``Solve'' the trust-region subproblem to find $s_k$ \\
\hspace*{6mm} for which $m(s_k)$ ``$<$'' $f_k$ and $\|s_k\| \leq \Delta_k$,
and define
\disp{\rho_k = \frac{f_k - f(x_k + s_k)}{f_k - m_k(s_k)}.\vspace*{-8mm}}
\hspace*{6mm} If $\rho_k \geq \eta_v$ [\defn{very successful}]
\hspace{0.96in} \fbox{$0 < \eta_v < 1$} \\
\hspace*{9mm} set $x_{k+1} = x_k + s_k$ and $\Delta_{k+1} = \gamma_i \Delta_k$.
\hspace{0.39in} \fbox{$\gamma_i \geq 1$} \\
\hspace*{6mm} Otherwise if $\rho_k \geq \eta_s$ then [\defn{successful}]
\hspace{0.0in} \fbox{$0 < \eta_s \leq \eta_v < 1$} \\
\hspace*{9mm} set $x_{k+1} = x_k + s_k$ and $\Delta_{k+1} = \Delta_k $. \\
\hspace*{6mm} Otherwise [\defn{unsuccessful}]\\
\hspace*{9mm} set $x_{k+1} = x_k$ and $\Delta_{k+1} = \gamma_d \Delta_k $.
\hspace{0.405in} \fbox{$0 < \gamma_d < 1$} \\
\hspace*{6mm} Increase $k$ by 1.
\vspace{0.2in}
}}}
\vspace{0.3in}
\noindent
Reasonable values might be
$\eta_v = 0.9$ or $0.99$,
$\eta_s = 0.1$ or $0.01$,
$\gamma_i = 2$,
and
$\gamma_d = 0.5$.
In practice, these parameters might even be allowed to vary (within
reasonable limits) from iteration to iteration. In particular,
there would seem to be little justification in increasing
the trust region radius following a very successful iteration
unless $\|s_k\| \approx \Delta_k$, nor in decreasing the radius
by less than is required to ``cut off'' an unsuccessful $s_k$.
In practice, the trust-region radius is {\em not\/} increased
for a very successful iterations, if the step is much shorter, say
less than half the trust-region radius. There exist various
schemes for choosing an initial trust-region radius. However,
if the problem is well scaled, then $\Delta_0 = O(1)$ is
reasonable. Poor scaling can affect the performance of trust-region
methods. In practice it often suffices that the variables
of the (scaled) problem have roughly the same order of magnitude.
It remains for us to decide what we mean by ``solving'' the
trust-region subproblem. We shall see in Section~\ref{trs}
that (at least in the $\ell_2$-trust-region norm case) it is possible
to find the (global) solution to the subproblem. However, since this
may result in a considerable amount of work, we first seek
``minimal'' conditions under which we can guarantee convergence
of the above algorithm to a first-order critical point.
We have already seen that steepest-descent linesearch methods have
very powerful (theoretical) convergence properties. The same is
true in the trust-region framework. Formally, at the very least,
we shall require that we achieve as much reduction in the model as we
would from an iteration of steepest descent. That is, if we define the
\defn{Cauchy} point as $s\s{C}_k = - \alpha\s{C}_k g_k^{ }$, where
\disp{\arr{rl}{
\alpha\s{C}_k = & \argminover{\alpha > 0} m_k^{ }( - \alpha g_k^{ })
\tim{subject to} \alpha \|g_k^{ }\| \leq \Delta_k^{ } \\
= & \argminover{0 < \alpha \leq \Delta_k / \|g_k\|} m_k^{ }(- \alpha g_k^{ })
},}
we shall require that our step $s_k$ satisfies
\eqn{bcp}{m_k^{ }(s_k^{ }) \leq m_k^{ }(s\s{C}_k) \tim{and} \|s_k^{ }\|
\leq \Delta_k^{ }.}
Notice that the Cauchy point is extremely easy to find, since it merely
requires that we minimize the quadratic model along a line segment.
In practice, we shall hope to---and can---do far better than this,
but for now \req{bcp} suffices.
Figure~\ref{F-TR} illustrates the trust-region problem in four different
situations. The contours of the original function are shown as dotted
lines, while the contours of the trust-region model appear as solid lines
with the $\ell_2$ trust-region ball in bold. Clockwise from top left, the
plots depict the following situations: first, a quadratic model
with positive definite Hessian, next a linear model about the same point,
the third plot shows a quadratic model with indefinite Hessian and the
final plot is a quadratic model with positive definite Hessian whose
minimizers lies outside the trust-region.
\begin{figure}[ht]
\centerline{\psfig{figure=tr.eps,height=8.0cm,silent=}}
\caption{\label{F-TR}Trust-region models of
$f(x) = x_1^4 + x_1^{ } x_2^{ } + (1+x_2^{ })^2$ about different points.}
\end{figure}
We now examine the convergence of this trust-region method.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Basic convergence of trust-region methods}
The first thing to note is that we can guarantee a reasonable
reduction in the model at the Cauchy point.
\lthm{thm1}{If $m_k(s)$ is the second-order model and $s\s{C}_k$
is its Cauchy point within the trust-region $\|s\| \leq \Delta_k$, then
\disp{f_k - m_k(s_k\s{C}) \geq \half \| g_k \|_2
\min\left[ \frac{\|g_k\|_2}{1 + \|B_k\|_2}, \kappa_s \Delta_k \right].}}
\vspace{0.1in}
\noindent
Observe that the guaranteed reduction depends on how large the
current gradient is, and is also affected by the size of both the trust-region
radius and the (inverse) of the Hessian.
Since our algorithm requires that the step does at least as well as
the Cauchy point, we then have the following immediate corollary.
\lcor{cor2}{If $m_k(s)$ is the second-order model, and $s_k$ is an
improvement on the Cauchy point within the trust-region $\|s\| \leq \Delta_k$,
\disp{f_k - m_k(s_k) \geq \half \| g_k \|_2
\min\left[ \frac{\|g_k\|_2}{1 + \|B_k\|_2}, \kappa_s \Delta_k \right].}}
\vspace{0.1in}
\noindent
This is a typical trust-region result, in that it relates the model reduction
to a measure of the distance to optimality, in this case measured in terms
of the norm of the gradient.
It is also necessary to say something about how much the
model and the objective can vary. Since we are using a second-order
model for which the first-two terms are exactly those from the
Taylor's approximation, it is not difficult to believe that the
difference between model and function will vary like the square of the
norm of $s_k$, and indeed this is so.
\llem{lem3}{Suppose that $f \in C^2$, and that the true and model Hessians
satisfy the bounds $\|H(x)\|_2 \leq \kappa_h$ for all $x$
and $\|B_k\|_2\leq \kappa_b$
for all $k$ and some $\kappa_h \geq 1$ and $\kappa_b \geq 0$. Then
\disp{|f(x_k^{ } +s_k^{ }) - m_k^{ }(s_k^{ })| \leq \kappa_d \Delta_k^2,}
where $\kappa_d = \half \kappa_l^2 ( \kappa_h^{ } + \kappa_b^{ } )$,
for all $k$.}
\vspace{0.1in}
\noindent
Actually the result is slightly weaker than necessary since, for our
purposes, we have
chosen to replace $\|s_k\|$ by its (trust-region) bound $\Delta_k$.
Moreover, rather than requiring a uniform bound on $H(x)$, all that
is actually needed is a similar bound for all $x$ between $x_k$ and
$x_k + s_k$.
Armed with these bounds, we now arrive at a crucial result, namely
that it will always be possible to make progress from a non-optimal
point ($g_k \neq 0$).
\llem{lem4}{Suppose that $f \in C^2$, that the true and model Hessians
satisfy the bounds $\|H_k\|_2 \leq \kappa_h$ and $\|B_k\|_2\leq \kappa_b$
for all $k$ and some $\kappa_h \geq 1$ and $\kappa_b \geq 0$, and that
$\kappa_d = \half \kappa_l^2 ( \kappa_h^{ } + \kappa_b^{ } )$.
Suppose furthermore that $g_k \neq 0$ and that
\disp{\Delta_k \leq \|g_k\|_2
\min \left( \frac{1}{ \kappa_s ( \kappa_h + \kappa_b )},
\frac{\kappa_s ( 1 - \eta_v ) }{2 \kappa_d} \right).}
Then iteration $k$ is very successful and
\disp{\Delta_{k+1} \geq \Delta_k.}}
\vspace{0.1in}
\noindent
This result is fairly intuitive, since when the radius shrinks
the model looks more and more like its first-order Taylor's expansion
(provided $B_k$ is bounded) and thus ultimately there must be good local
agreement between the model and objective functions.
The next result is a variation on its predecessor, and
says that the radius is uniformly bounded away from zero if the same
is true of the sequence of gradients, that is
the radius will not shrink to zero at non-optimal points.
\llem{lem5}{Suppose that $f \in C^2$, that the true and model Hessians
satisfy the bounds $\|H_k\|_2 \leq \kappa_h$ and $\|B_k\|_2\leq \kappa_b$
for all $k$ and some $\kappa_h \geq 1$ and $\kappa_b \geq 0$, and that
$\kappa_d = \half \kappa_l^2 ( \kappa_h^{ } + \kappa_b^{ } )$.
Suppose furthermore that there exists a
constant $\epsilon > 0$ such that
$\|g_k\|_2 \geq \epsilon$ for all $k$. Then
\disp{\Delta_k \geq \kappa_{\epsilon} \eqdef \epsilon \gamma_d
\min \left( \frac{1}{ \kappa_s ( \kappa_h + \kappa_b )},
\frac{\kappa_s ( 1 - \eta_v ) }{2 \kappa_d} \right)}
for all $k$.}
We may then deduce that if there are only a finite number of
successful iterations, the iterates must be first-order optimal
after the last of these.
\llem{lem6}{Suppose that $f \in C^2$, and that both the true and model Hessians
remain bounded for all $k$.
Suppose furthermore that there are only finitely many
successful iterations. Then $x_k = x_*$ for all sufficiently large $k$ and
$g(x_*) = 0$.}
Having ruled out this special (and highly unlikely) case,
we then have our first global convergence result, namely that otherwise
there is at least one sequence of gradients that converge to zero.
\lthm{thm7}{
Suppose that $f \in C^2$, and that both the true and model Hessians
remain bounded for all $k$. Then either
\disp{g_l = 0 \tim{for some} l \geq 0}
or
\disp{\lim_{k \rightarrow \infty} f_k = - \infty}
or
\disp{\liminf_{k \rightarrow \infty} \|g_k\| = 0.}}
Is this all we can show? Is it possible for a second sub-sequence of
gradients to stay bounded away from zero? Fortunately, no.
\lcor{cor8}{
Suppose that $f \in C^2$, and that both the true and model Hessians
remain bounded for all $k$. Then either
\disp{g_l = 0 \tim{for some} l \geq 0}
or
\disp{\lim_{k \rightarrow \infty} f_k = - \infty}
or
\disp{\lim_{k \rightarrow \infty} g_k = 0.}}
\vspace{0.1in}
\noindent
Thus we have the highly-satisfying result that the gradients of
the sequence $\{x_k\}$ generated by our algorithm converge to, or
are all ultimately, zero. This does not mean that
a subsequence of $\{x_k\}$ itself converges, but if it does, the limit
is first-order critical.
It is also possible to show that an enhanced version of our basic
algorithm converges to second-order critical points. To do so, we
need to ensure that the Hessian of the model converges to that of
the objective (as would obviously be the case if $B_k = H_k$),
and that the step $s_k$ has a significant component along the eigenvector
corresponding to the most negative eigenvalue of $B_k$ (if any).
It is also possible to show that if $B_k = H_k$, if $\{x_k\}$ has
a limit $x_*$ for which $H(x_*)$ is positive definite, and if
$s_k$ is chosen to
\eqn{trsp}{\minin{s \in \smallRe^n} m_k(s) \tim{subject to}
\|s\| \leq \Delta_k,}
the step $\Delta_k$ stays bounded away from zero, and thus the
iteration ultimately becomes Newton's method (c.f. \req{nld}).
In conclusion,
we have seen that trust-region methods have a very rich
underlying convergence theory.
But so much for theory. We now turn to the outstanding practical issue,
namely how one might hope to find a suitable step $s_k$. We will
consider two possibilities, one that aims to get a very good approximation
to \req{trsp}, and a second, perhaps less ambitious method that
is more geared towards large-scale computation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\subsection{Solving the trust-region subproblem}
\label{trs}
For brevity, we will temporarily drop the iteration subscript, and
consider the problem of
\eqn{tsp}{\mbox{(approximately)} \minin{s \in \smallRe^n} q(s) \equiv
\ip{s}{g} + \half \ip{s}{B s} \tim{subject to} \| s \| \leq \Delta.}
As we have already mentioned, our aim is to find $s_*$ so that
\disp{ q(s_*) \leq q(s\s{C}) \tim{and} \| s_* \| \leq \Delta,}
where $s\s{C}$ is the Cauchy point. We shall consider two approaches in
this section. The first aims to solve \req{tsp}
exactly, in which case our trust-region method will be akin to a
Newton-like method. The second aims for an approximate solution
using a conjugate-gradient like method. For simplicity, we shall only
consider the $\ell_2$-trust region $\|s\| \leq \Delta$,
mainly because there are very
powerful methods in this case, but of course other norms are possible
and are sometimes preferred in practice.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Solving the $\ell_2$-norm trust-region subproblem}
\label{etrs}
There is a really powerful solution characterisation result
for the $\ell_2$-norm trust-region subproblem.
\lthm{thm9}{Any {\em global} minimizer $s_*$ of
$q(s)$ subject to $\|s\|_2 \leq \Delta$ satisfies the equation
\disp{ (B + \lambda_* I ) s_* = - g ,}
where $B + \lambda_* I$ is positive semi-definite,
$\lambda_* \geq 0$ and $\lambda_* ( \|s_*\|_2 - \Delta ) = 0$.
If $B + \lambda_* I$ is positive definite, $s_*$ is unique.}
\vspace{0.1in}
\noindent
This result is extraordinary as it is very unusual to be able to
give necessary and sufficient {\em global} optimality conditions for a
non-convex optimization problem (that is, a problem which might have
a number of local minimizers). Even more extraordinary is the fact
that the necessary and sufficient conditions are identical. But
most crucially, these optimality conditions also suggest
how we might solve the problem.
There are two cases to consider. If
$B$ is positive definite and the solution $s$ to
\eqn{ns}{B s = - g}
satisfies $\|s\|_2 \leq \Delta$, then it immediately
follows that $s_* = s$ ($\lambda_* = 0$ in Theorem~\ref{thm9})---this
potential solution may simply be checked by seeing if
$B$ has Cholesky factors and, if so, using these factors to solve \req{ns}
$B s = - g$ and subsequently evaluate $\|s\|_2$. Otherwise, either
$B$ is positive definite but the solution to \req{ns} satisfies
$\|s\|_2 > \Delta$ or $B$ is singular or indefinite.
In these cases, Theorem~\ref{thm9} then says that $s_*$ satisfies
\eqn{nsl}{(B + \lambda I ) s = - g \tim{and} \ip{s}{s} = \Delta^2,}
which is a {\em nonlinear} (quadratic) system of algebraic equations in the
$n+1$ unknowns $s$ and $\lambda$. Thus, we now concentrate on methods
for solving this system.
Suppose $B$ has the spectral decomposition
\disp{ B = U^T \Lambda U; }
here $U$ is a matrix of (orthonormal) eigenvectors while
the diagonal matrix $\Lambda$ is made up of eigenvalues
$\lambda_1 \leq \lambda_2 \leq \ldots \leq \lambda_n$.
Theorem~\ref{thm9} requires that $B + \lambda I$ be positive semi-definite,
and so the solution $(s,\lambda)$ to \req{nsl} that we seek necessarily
satisfies $\lambda \geq - \lambda_1$. The first part of \req{nsl}
enables us to write $s$ explicitly in terms of $\lambda$, that is
\disp{ s(\lambda) = - (B+\lambda I)^{-1} g; }
we will temporarily disregard the possibility that the theorem permits a
singular $B+\lambda I$. Notice that once we have found $\lambda$,
\eqn{pns}{(B + \lambda I ) s = - g}
is a linear system. In this case, we may substitute $s(\lambda)$ into
the second part of \req{nsl} to reveal that
\eqn{pe}{ \psi(\lambda) \eqdef \|s(\lambda)\|_2^2
= \|U^T ( \Lambda + \lambda I )^{-1} Ug\|_2^2 =
\sum_{i=1}^n \frac{\gamma_i^2}{(\lambda_i + \lambda)^2} = \Delta^2, }
where $\gamma_i = \ip{e_i}{U g} = \ip{U^T e_i}{g}$.
Thus to solve the trust-region
subproblem, it appears that all we have to do is find a particular root
of a univariate nonlinear equation.
We illustrate this in Figures~\ref{sts-minsol1_fig}--\ref{sts-minsol3_fig}.
{\setlength{\unitlength}{0.8pt}
\begin{figure}[ht] \begin{center}
\centerline{\psfig{figure=tr1.eps,height=6.4cm,silent=}}
\begin{picture}(375,1)(23,3.3)
%
\put(42,240){$\scriptstyle \psi(\lambda)$}
\put(350,15){$\scriptstyle \lambda$}
\put(67,15){$\scriptstyle - 8$}
\put(107,15){$\scriptstyle - 6$}
\put(147,15){$\scriptstyle - 4$}
\put(187,15){$\scriptstyle -2$}
\put(230,15){$\scriptstyle 0$}
\put(270,15){$\scriptstyle 2$}
\put(310,15){$\scriptstyle 4$}
\put(58,30){$\scriptstyle 0$}
\put(58,82){$\scriptstyle 1$}
\put(58,137){$\scriptstyle 2$}
\put(58,191){$\scriptstyle 3$}
\put(70,245.5){\vector(0,1){2}}
\put(352,31){\vector(1,0){2}}
%
\put(360,202){$B = \mat{rrr}{ 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5}$}
\put(400,57){$g= \vect{ 1 \\ 1 \\ 1 }$}
\put(355,90){$\Delta^2 = 1.151$}
\put(290,150){\small solution curve as $\Delta$ varies}
\put(286,153){\vector(-1,1){52}}
\put(286,151){\vector(-1,-2){43}}
\put(320,147){\vector(0,-1){111}}
\end{picture}
\caption{\label{sts-minsol1_fig} A plot of $\psi(\lambda)$ as $\lambda$ varies
from $-8$ to $6$. Note the poles at the negatives of the eigenvalues of $H$.
The heavy curve plots $\lambda$ against $\Delta$; the dashed vertical component
corresponds to interior solutions which occur for all $\Delta^2$ larger than
roughly $1.15$,
while the remaining segment indicates boundary solutions.\vspace*{-0.5cm}}
\end{center} \end{figure}}
{\setlength{\unitlength}{0.8pt}
\begin{figure}[ht] \begin{center}
\centerline{\psfig{figure=tr2.eps,height=6.4cm,silent=}}
\begin{picture}(375,1)(23,3.3)
%
\put(42,240){$\scriptstyle \psi(\lambda)$}
\put(350,20){$\scriptstyle \lambda$}
\put(67,20){$\scriptstyle - 8$}
\put(107,20){$\scriptstyle - 6$}
\put(147,20){$\scriptstyle - 4$}
\put(187,20){$\scriptstyle -2$}
\put(230,20){$\scriptstyle 0$}
\put(270,20){$\scriptstyle 2$}
\put(310,20){$\scriptstyle 4$}
\put(58,30){$\scriptstyle 0$}
\put(58,117){$\scriptstyle 1$}
\put(58,200){$\scriptstyle 2$}
\put(70,245.5){\vector(0,1){2}}
\put(352,37){\vector(1,0){2}}
%
\put(284,160){\small minus leftmost eigenvalue}
\put(280,163){\vector(-1,0){26}}
\put(340,212){$B = \mat{rrr}{ - 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5}$}
\put(380,94){$g= \vect{ 1 \\ 1 \\ 1 }$}
\end{picture}
\caption{\label{sts-minsol2_fig} A plot of $\psi(\lambda)$ as $\lambda$
varies from
$-8$ to $6$. Again, note the poles at the negatives of the eigenvalues of $H$.}
\end{center} \end{figure}}
{\setlength{\unitlength}{0.8pt}
\begin{figure}[hbt] \begin{center}
\centerline{\psfig{figure=tr3.eps,height=6.4cm,silent=}}
\begin{picture}(375,1)(23,3.3)
%
\put(42,240){$\scriptstyle \psi(\lambda)$}
\put(350,20){$\scriptstyle \lambda$}
\put(67,20){$\scriptstyle - 8$}
\put(107,20){$\scriptstyle - 6$}
\put(147,20){$\scriptstyle - 4$}
\put(187,20){$\scriptstyle -2$}
\put(230,20){$\scriptstyle 0$}
\put(270,20){$\scriptstyle 2$}
\put(310,20){$\scriptstyle 4$}
\put(58,30){$\scriptstyle 0$}
\put(58,117){$\scriptstyle 1$}
\put(58,200){$\scriptstyle 2$}
\put(70,245.5){\vector(0,1){2}}
\put(352,37){\vector(1,0){2}}
%
\put(284,155){\small minus leftmost eigenvalue}
\put(280,158){\vector(-1,0){26}}
%\put(260,73){\small no root larger than 2}
%\put(270,40){\small root near 2.2}
%\put(266,43){\vector(-2,-1){18}}
\put(340,202){$B = \mat{rrr}{ - 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5}$}
\put(380,114){$g= \vect{ 0 \\ 1 \\ 1 }$}
\put(355,41){$\Delta^2 = 0.0903$}
\end{picture}
\caption{\label{sts-minsol3_fig} A plot of $\psi(\lambda)$ for the
modified model as
$\lambda$ varies from $-8$ to $6$. Note that there is no solution with
to the equation $\psi(\lambda) = \Delta^2$ with $\lambda \geq 1$ for
$\Delta^2$ larger than roughly $0.09$.}
\end{center} \end{figure}}
The first shows a convex example ($B$ positive definite).
For $\Delta^2$ larger than roughly 1.15, the solution to the problem
lies in the interior of the trust region, and may be found directly from
\req{ns}. When $\Delta$ is smaller than this, the solution lies on the
boundary of the trust region, and can be found as the right-most root
of \req{pe}. The second example is non-convex ($B$ indefinite).
Now the solution must lie on the boundary of the trust region
for all values of $\Delta$, and again can be found as the right-most root
of \req{pe}, to the right of $- \lambda_1$.
In both
Figures~\ref{sts-minsol1_fig} and \ref{sts-minsol2_fig} everything seems easy,
and at least a semblance of an algorithm is obvious. But now consider
the example in Figure~\ref{sts-minsol3_fig}. This example is especially
chosen so that the coefficient $\gamma_1$ in \req{pe} is zero, that is
$g$ is orthogonal to the eigenvector $u_1$ of $B$ corresponding to the
eigenvalue $\lambda_1 = - 2$. Remember that Theorem~\ref{thm9} tells us that
$\lambda \geq 2 = - \lambda_1$.
But Figure~\ref{sts-minsol3_fig} shows that there is no such root of
\req{pe} if $\Delta^2$ is larger than (roughly) 0.09.
This is an example of
what has become known as the \defn{hard} case, which always arises when
$\lambda_1 < 0$, $\ip{u_1}{g} = 0$ and $\Delta$ is too big. What is happening?
Quite simply, in the hard case $\lambda = - \lambda_1$ and \req{pns}
is a singular (but consistent) system---it is consistent precisely
because $\ip{u_1}{g} = 0$. But this system has other solutions
$s + \alpha u_1$ for any $\alpha$, because
\disp{(B + \lambda I ) ( s + \alpha u_1 ) = - g,}
and $u_1$ is an eigenvector of $B + \lambda I$.
The solution we require is that for which
$\|s + \alpha u_1\|_2^2 = \Delta^2$, which is a quadratic equation
for the unknown $\alpha$, and either root suffices.
In the easy (that is not ``hard'') case, it remains to see how
best to solve $| s(\lambda)\|_2 = \Delta$. The answer is blunt. Don't!
At least, not directly, since as the previous figures showed,
$\psi(\lambda)$ is an unappealing function with many poles. It is far
better to solve the equivalent \defn{secular} equation
\disp{ \phi(\lambda) \eqdef \frac{1}{\| s(\lambda)\|_2} - \frac{1}{\Delta} = 0,}
as this has no poles, indeed $\phi$ is an analytic function, and thus ideal
for Newton's method. We illustrate the secular equation in
Figure~\ref{sts-sets}.
{\setlength{\unitlength}{0.8pt}
\begin{figure}[ht] \begin{center}
\centerline{\psfig{figure=tr4.eps,height=6.4cm,silent=}}
\begin{picture}(375,60)(-57,-0.5)
\put(-24,127){$0$}
\put(-38,280){$\phi(\lambda)$}
\put(95,45){$0$}
\put(170,45){$- \lambda_1$}
\put(203,45){$\lambda_*$}
\put(264,45){$\lambda$}
\put(-7,289){\vector(0,1){2}}
\put(275,65.52){\vector(1,0){2}}
%\put(28,250){$\min - \quarter s_1^2 + \quarter s_2^2 + \half s_1^{} + s_2^{}$}
%\put(28,230){$\mbox{subject to} \| s \|_2 \leq 4$}
\end{picture}
\vspace*{-15mm}
\caption{\label{sts-sets}
A plot of $\phi(\lambda)$ against $\lambda$
for the problem of minimizing
$- \quarter s_1^2 + \quarter s_2^2 + \half s_1^{} + s_2^{}$ subject to
$\|s\|_2^{} \leq 4$.}
\end{center} \end{figure}}
\newpage
Without giving details (for these, see the appendix, page~\pageref{seceq}),
Newton's method for the secular equation
is as follows
\vspace{0.2in}
\noindent
\centerline{\framebox[2.8in]{\hspace*{0.2in}\parbox{2.7in}{
\vspace*{0.1in}
Let $\lambda > - \lambda_1$ and $\Delta > 0$ be given.\\
Until ``convergence'' do: \\
\hspace*{6mm} Factorize $B + \lambda I = L L^T$. \\
\hspace*{6mm} Solve $L L^T s = - g$. \\
\hspace*{6mm} Solve $ L w = s$. \\
\hspace*{6mm} Replace $\lambda$ by \\
\disp{\lambda + \left( \frac{\|s\|_2 - \Delta}{\Delta} \right)
\left( \frac{\|s\|_2^2}{\|w\|_2^2} \right).}
%\vspace{0.2in}
}}}
\vspace{0.1in}
\noindent
This is globally and ultimately quadratically convergent when started in the
interval $[- \lambda_1, \lambda_*]$ except in the hard case,
but needs to be safeguarded to make it robust for the hard and interior
solution cases. Notice that the main computational cost per iteration
is a Cholesky factorization of $B + \lambda I$, and while this may be
reasonable for small problems, it may prove unacceptably expensive when
the number of variables is large. We consider an alternative for this case
next.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Solving the large-scale problem}
Solving the large-scale trust-region subproblem using the above method
is likely out of the question in all but very special cases.
The obvious alternative is to use
an iterative method to approximate its solution. The simplest approximation
that is consistent with our fundamental requirement that we do as least as
well as we would at the Cauchy point is to use the Cauchy point itself.
Of course, this is simply the steepest descent method, and thus unlikely
to be a practical method. The obvious generalization is the conjugate-gradient
method, since the first step of CG is in the steepest-descent direction and,
as subsequent CG steps further reduce the model, any step generated by
the method is allowed by our theory. However, there are a number of
other issues we need to address first. In particular,
what about the interaction between conjugate gradients and the trust region?
And what if $B$ is indefinite?
The conjugate-gradient method to find an approximation to
a minimizer of $q(s)$ may be summarised as follows.
\vspace*{0.2in}
\noindent
\centerline{\framebox[3.0in]{\hspace*{0.2in}\parbox{2.9in}{
\vspace*{0.2in}
Given $s^0 = 0$, set $g^0 = g$, $d^0 = - g$ and $i = 0$. \\
Until ``breakdown'' or $g^i$ ``small'', iterate: \\
\hspace*{6mm} $\alpha^i = \|g^i\|_2^2 / \ip{d^{i}}{Bd^i}$ \\
\hspace*{6mm} $s^{i+1} = s^i + \alpha^i d^i$ \\
\hspace*{6mm} $g^{i+1} = g^i + \alpha^i B d^i$ \\
\hspace*{6mm} $\beta^i = \|g^{i+1}\|_2^2 / \|g^{i} \|_2^2$ \\
\hspace*{6mm} $d^{i+1} = - g^{i+1} + \beta^i d^i$ \\
\hspace*{6mm} and increase $i$ by 1.
\vspace{0.2in}}}}
\vspace{0.2in}
\noindent
Notice that we have inserted a termination statement concerning ``breakdown''.
This is intended to cover the fatal case when $\ip{d^{i}}{Bd^i} = 0$
(or, in practice, is close to zero), for which the iteration is undefined,
and the non-fatal case when
$\ip{d^{i}}{Bd^i} < 0$ for which $q(s)$ is unbounded from below along
the so-called \defn{direction of negative curvature} $d_i$.
But what of the trust-region constraint? Here we have a crucial result.
\lthm{thm10}{Suppose that the conjugate gradient method is applied
to minimize $q(s)$ starting from $s^0 = 0$,
and that $\ip{d^{i}}{B d^i} > 0$ for $0 \leq i \leq k$.
Then the iterates $s^j$ satisfy the inequalities
\disp{ \| s^j \|_2 < \| s^{j+1} \|_2}
for $0 \leq j \leq k - 1$.}
Simply put, since the norm of the approximate solution generated by
the conjugate gradients increases in norm at each iteration, if there
is an iteration for which $\| s^j \|_2 > \Delta$, it must be that
the solution to the trust-region subproblem lies on the trust-region
boundary. That is $\| s_* \|_2 = \Delta$. This then suggests that we
should apply the basic conjugate-gradient method above but
terminate at iteration $i$ if either (a)
$\ip{d^{i}}{Bd^i} \leq 0$, since this implies that $q(s)$
is unbounded along $d^i$,
or (b) $\|s^i + \alpha^i d^i\|_2 > \Delta$, since this implies that
the solution must lie on the trust-region boundary.
In both cases, the simplest strategy is to stop on the boundary at
$s = s^i + \alpha\s{B} d^i$, where
$\alpha\s{B}$ chosen as positive root of the quadratic equation
\disp{ \| s^i + \alpha\s{B} d^i \|^2_2 = \Delta^2.}
Crucially this $s$ satisfies
\disp{ q(s) \leq q(s\s{C}) \tim{and} \| s \|_2 \leq \Delta}
and thus Corollary~\ref{cor8} shows that the overall trust-region algorithm
converges to a first-order critical point.
How good is this truncated conjugate-gradient strategy?
In the convex case, it turns out to be very good. Indeed, no worse than
half optimal!
\lthm{thm11}{Suppose that the truncated conjugate gradient method is applied
to approximately minimize $q(s)$ within $\|s\|_2 \leq \Delta$,
and that $B$ is positive definite. Then the
computed and actual solutions to the problem, $s$ and $s_*$, satisfy the bound
$q(s) \leq \half q(s_*)$.}
\noindent
%That is to say that the truncated conjugate gradient method is guaranteed
%to get at least 50\% of the way to the solution to the convex trust-region
%sub-problem.
In the non-convex ($B_k$ indefinite) case, however, the strategy may be
rather poor. For example, if $g = 0$ and $B$ is indefinite, the above
truncated conjugate-gradient method will terminate at $s = 0$, while
the true solution lies on the trust-region boundary.
What can we do in the non-convex case? The answer is quite involved,
but one possibility is to recall that conjugate-gradients is trying to
solve the overall problem by successively solving the problem over
a sequence of nested subspaces. As we saw, the CG method uses
$B$-conjugate subspaces. But there is an equivalent method, the
\defn{Lanczos} method, that uses instead orthonormal bases. Essentially this
may be achieved by applying the Gram-Schmidt procedure to the
CG basis $\calD^i$ to build the equivalent basis
$\calQ^i = \{ s \st s = Q^i s_q \tim{for some} s_q \in \Re^i\}$. It is
easy to show that for this $Q^i$,
\disp{ Q^{i\;T} Q^i = I \tim{and} Q^{i\;T} B Q^i = T^i,}
where $T^i$ is tridiagonal, and $Q^{i\;T} g = \|g\|_2 \, e_1$,
and it is trivial to generate $Q^i$ from the CG $\calD^i$. In this case
the trust-region subproblem \req{tsp} may be rewritten as
\disp{s_q^i = \argminover{s_q \in \calR^i}
\|g\|_2\, \ip{e_1}{s_q} + \half \ip{s_q}{T^i s_q}
\tim{subject to} \|s_q\|_2 \leq \Delta,}
where $s^i = Q^i s_q^i$.
Since $T^i$ is tridiagonal, $T^i + \lambda I$ has
very sparse Cholesky factors, and thus we can afford to solve this
problem using the earlier secular equation approach. Moreover, since
we will need to solve a sequence of related problems over nested subspaces,
it is easy to imagine that one can use the solution for
one problem to initialize the next. In practice, since the approach
is equivalent to conjugate gradients, it is best to use CG until
the trust-region boundary is reached and then to switch to
the Lanczos method at that stage. Such a method has turned out to be
most effective in practice.
%
% Lecture 4: interior-point methods
%
%\newpage
\setcounter{lecture}{4}
\section{INTERIOR-POINT METHODS FOR INEQUALITY \\ CONSTRAINED OPTIMIZATION}
\setcounter{equation}{0}
\setcounter{figure}{0}
\label{ipm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Having given a break-neck description of methods for unconstrained
minimization, we now turn our attention to the real problems of interest,
namely those involving constraints. This section will focus on problems
involving inequality constraints, while its successor will be concerned with
equality constraints. But before we start, we need to discuss the
conflicting nature of constrained optimization problems, and how we might
deal with them.
Unconstrained minimization is ``simple'' because there is but one goal,
namely to minimize the objective. This is not so for constrained minimization
because there is now a conflict of requirements, the aforementioned
objective minimization but at the same time a requirement of feasibility
of the solution. While in some instances (such as for linear equality
constraints and, to a certain extent, all inequality constraints)
it may be possible to generate feasible iterates, and thus to regain
the advantages of having a single goal, this is not true for general
constrained optimization.
\subsection{Merit functions for constrained minimization}
Most (but not all, see Section~\ref{filter}) constrained optimization techniques
overcome this dichotomy by introducing a merit function to try to balance
the two conflicting requirements of minimization and feasibility.
Given parameters $p$, a composite function $\Phi(x,p)$ is a
\defn{merit function} if
(some) minimizers of $\Phi(x,p)$ with respect to $x$ approach those of $f(x)$
subject to the constraints as $p$ approaches some set $\calP$. Thus a
merit function combines both optimality requirements into a single
``artificial'' objective function. In principal, it then only remains to
use the best {\em unconstrained} minimization methods to solve the
constrained problem. If only life were that simple!
Consider the case of equality constrained minimization, that is finding $x_*$ to
\eqn{ecp}{\tmininx{} f(x) \tim{subject to} c(x) = 0.}
A suitable merit function in this case is the
\defn{quadratic penalty function}
\eqn{qpf}{\Phi(x,\mu) = f(x) + \frac{1}{2 \mu} \|c(x)\|_2^2,}
where $\mu$ is a positive scalar parameter.
It is easy to believe that if $\mu$ is small and we try to minimize
$\Phi(x,\mu)$ much of the effort will be concentrated on making the
second objective term $\frac{1}{2 \mu} \|c(x)\|_2^2$ small, that is
in forcing $c(x)$ to be small. But as $f$ has a slight presence in the
merit function, any remaining energy will be diverted to making $f(x)$ small
amongst all of the values for which $c(x)$ is. Formally, it is easy to
show that, under modest conditions, some minimizers of $\Phi(x,\mu)$
converge to solutions of \req{ecp} as $\mu$ approaches the set $\{0\}$
from above.
Unfortunately, it is possible that $\Phi(x,\mu)$ may have other
stationary points that are not solutions of \req{ecp}---indeed this must be
the case if $c(x) = 0$ are inconsistent. The quadratic penalty function is
but one of many merit functions for equality constrained minimization.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The logarithmic barrier function for inequality constraints}
For the inequality constrained problem
\eqn{icp}{\tmininx{} f(x) \tim{subject to} c(x) \geq 0}
the best known merit function is the
\defn{logarithmic barrier function}
\disp{\Phi(x,\mu) = f(x) - \mu \sum_{i=1}^m \log c_i(x),}
where $\mu$ is again a positive scalar \defn{barrier parameter}.
\begin{figure}[b]
\centerline{
\psfig{figure=barn10.eps,width=7.0cm,silent=}\hspace*{0.1in}
\psfig{figure=bar1.eps,width=7.0cm,silent=}}
\centerline{\hfill $\mu = 10$ \hfill \hfill $\mu = 1$ \hfill}
\vspace*{2mm}
\centerline{
\psfig{figure=barp1.eps,width=7.0cm,silent=}\hspace*{0.1in}
\psfig{figure=barp01.eps,width=7.0cm,silent=}}
\centerline{\hfill $\mu = 0.1$ \hfill \hfill $\mu = 0.01$ \hfill}
\caption{\label{lbfcont}The logarithmic barrier function for
$\min x_1^2 + x_2^2$ subject to $x_1 + x_2^2 \geq 1$.
The contours for $\mu = 0.01$ are visually indistinguishable from
$f(x)$ for feasible points.}
\end{figure}
Each logarithmic term $- \log c_i(x)$ becomes infinite as
$x$ approaches the boundary of the $i$-th inequality from
the feasible side, and is undefined (effectively infinite) beyond there.
The size of the logarithmic term is mitigated when $\mu$ is small, and
it is then possible to get close to the boundary of the feasible region
before its effect is felt, any minimization effort being directed towards
reducing the objective. Once again, it is easy to
show that, under modest conditions, some minimizers of $\Phi(x,\mu)$
converge to solutions of \req{icp} as $\mu$ approaches the set $\{0\}$
from above. And once again a possible defect is that $\Phi(x,\mu)$ may
have other, useless stationary points. The contours of a typical
example are shown in Figure~\ref{lbfcont}.
\subsection{A basic barrier-function algorithm}
The logarithmic barrier function is different in one vital aspect from
the quadratic penalty function in that it requires that there is a
{\em strictly} interior point. If we apply the obvious sequential minimization
algorithm to $\Phi(x,\mu)$, a strictly interior starting point is required, and
all subsequent iterates will be strictly interior. The obvious
``interior-point'' algorithm is as follows.
\vspace*{0.2in}
\noindent
\centerline{\framebox[3.3in]{\hspace*{0.2in}\parbox{3.2in}{
\vspace*{0.2in}
Given $\mu_0 > 0$, set $k = 0$. \\
Until ``convergence'', iterate: \\
\hspace*{6mm} Find $x\s{S}_k$ for which $c(x\s{S}_k) > 0$. \\
\hspace*{6mm} Starting from $x\s{S}_k$, use an unconstrained \\
\hspace*{9mm} minimization algorithm to find an \\
\hspace*{9mm} ``approximate'' minimizer $x_k$ of $\Phi(x,\mu_k).$ \\
\hspace*{6mm} Compute $\mu_{k+1}>0$ smaller than $\mu_k$ such \\
\hspace*{9mm} that $\lim_{k\rightarrow \infty} \mu_{k+1} = 0$.
%\\ \hspace*{9mm}
and increase $k$ by 1.
\vspace{0.2in}}}}
\vspace{0.2in}
\noindent
In practice it is common to choose
$\mu_{k+1} = 0.1 \mu_k$ or even $\mu_{k+1} = \mu_k^2$,
while perhaps the obvious choice for a subsequent starting point
is $x\s{S}_{k+1} = x_k^{ }$.
Fortunately, as we have hinted, basic convergence for the
algorithm is easily established. Recall that
the \defn{active set} $\calA(x)$ at a point $x$ is
$\calA(x) = \{ i \st c_i(x) = 0\}$. Then we have the following.
\lthm{thm3.1}{Suppose that $f$, $c \in \calC^2$,
that $(y_k)_i \eqdef \mu_k / c_i(x_k)$ for $i = 1, \ldots, m$, that
\disp{ \| \nabla_x \Phi(x_k,\mu_k) \| _2 \leq \epsilon_k}
where $\epsilon_k$ converges to zero as $k \rightarrow \infty$,
and that $x_k$ converges to $x_*$ for which
$\{ a_i(x_*) \}_{i \in \calA(x_*)}$ are linearly independent.
Then $x_*$ satisfies the first-order necessary optimality
conditions for the problem
\disp{\tmininx{} f(x) \tim{subject to} c(x) \geq 0}
and $\{y_k\}$ converge to the associated Lagrange multipliers $y_*$.}
\noindent
Notice here how the algorithm delivers something unexpected, namely
estimates of the Lagrange multipliers. Also see the role played by
the linearly independence of the active constraint gradients,
regrettably quite a strong constraint qualification.
\subsection{Potential difficulties}
As we now know that it suffices to (approximately) minimize
$\Phi(x,\mu)$, how should we proceed? As $\Phi(x,\mu)$ is a smooth
function, we can immediately appeal to the methods we discussed in
Sections~\ref{lsmfuo} and \ref{trmfuo}. But we need to be careful.
Very, very careful.
We could use a linesearch method. Of note here is the fact that
the barrier function has logarithmic singularities, indeed is
undefined for infeasible points. Thus it makes sense to design a
specialized linesearch to cope with the singularity of the $\log$.
Alternatively, we could use a trust-region method.
Here we need to be able to instantly reject candidate
steps for which $c(x_k+s_k) \not> 0$. More importantly,
while all (consistent) trust-region norms are equivalent,
(ideally) we should ``shape'' the trust region for any barrier-function
model to cope with the contours of the singularity. This implies that
the trust-region shape may vary considerably from iteration to iteration,
with its shape reflecting the eigenvalues arising from the singularity.
\subsubsection{Potential difficulty I: ill-conditioning of the barrier Hessian}
\label{pdi}
At the heart of both linesearch and trust-region methods is, of course,
the Newton (second-order) model and related Newton direction.
The computation of a Newton model/direction for the logarithmic
barrier function is vital, and the resulting equations have a lot
of (exploitable) structure. The gradient of the barrier function is
\disp{\nabla_x \Phi(x,\mu) = g(x) - \mu \sum_i a_i(x ) / c_i(x) =
g(x) - A^T(x) y(x) = g(x,y(x)),}
where $y_i(x) \eqdef \mu / c_i(x)$ and $g(x,y)$ is the gradient of the
Lagrangian function for \req{icp}. Likewise, the Hessian is
\disp{\nabla_{xx} \Phi(x,\mu) = H(x,y(x)) + \mu A^T(x) C^{-2}(x) A(x),}
where
$H(x,y(x)) = H(x) - \bigsum_{i=1}^m y_i(x) H_i(x)$ and
$C(x) = \mbox{diag}( c_1(x) , \ldots , c_m(x) )$, the diagonal matrix
whose entries are the $c_i(x)$. Thus the Newton correction
$s\s{P}$ from $x$ for the barrier function satisfies
\eqn{nbe}{ ( H(x,y(x)) + \mu A^T(x) C^{-2}(x) A(x) ) s\s{P} = - g(x,y(x)).}
Since $y(x) = \mu C^{-1}(x) e$, \req{nbe} is sometimes written as
\eqn{nbee}{ \left( H(x,y(x)) + A^T(x) C^{-1}(x) Y(x) A(x) \right)
s\s{P} = - g(x,y(x)),}
or
\eqn{nbef}{ \left( H(x,y(x)) + A^T(x) Y^2(x) A(x) / \mu \right)
s\s{P} = - g(x,y(x)),}
where $Y(x) = \mbox{diag}( y_1(x) , \ldots , y_m(x) ).$
This is where we need to be careful. For we have the following
estimates of the eigenvalues of the barrier function as we approach a
solution.
\lthm{thm46}{Suppose that the assumptions of Theorem~\ref{thm3.1}
are satisfied, that $A_{\calA}$ is the matrix whose rows are
$\{ a_i^T(x_*) \}_{i \in \calA(x_*)}$, that
$m_a = |\calA(x_*)|$, and that $x_*$ is \defn{non-degenerate}, that is
$(y_*)_i > 0$ for all $i \in \calA(x_*)$. Then
the Hessian matrix of the barrier function, $\nabla_{xx} \Phi(x_k,\mu_k)$,
has $m_a$ eigenvalues
\disp{\lambda_i( A_{\calA}^T Y_{\calA}^2 A_{\calA}^{ } ) / \mu_k + O(1)
\tim{for $i = 1, \ldots , m_a$}}
and the remaining $n-m_a$ eigenvalues
\disp{\lambda_i(N_{\calA}^T H(x_*,y_*) N_{\calA}^{ }) + O(\mu_k)
\tim{for $i = 1, \ldots , n-m_a$}}
as $k \rightarrow \infty$,
where $\lambda_i(.)$ denotes the $i$-th eigenvalue of its matrix argument,
$Y_{\calA}$ is the diagonal matrix of active Lagrange multipliers at $x_*$
and $N_{\calA} =$ is an orthogonal basis for the null-space of $A_{\calA}$.}
\noindent
This demonstrates that the condition number of
$\nabla_{xx} \Phi(x_k,\mu_k)$ is $O(1/\mu_k)$ as $\mu_k$ shrinks to zero,
and suggests that it may not be straightforward to find the minimizer
numerically.
Look at how the contours around $x_*$ in Figure~\ref{lbfcont} bunch together
as $\mu$ approaches zero.
\subsubsection{Potential difficulty II: poor starting points}
\label{pdii}
As if this potential defect isn't serious enough, there is a second
significant difficulty with the naive method we described earlier.
This is that $x\s{S}_{k+1} = x_k^{ }$ appears to be a very poor starting point
for a Newton step just after the (small) barrier parameter is reduced.
To see this suppose, as will be the case at the end of the minimization
for the $k$-th barrier subproblem, that
\disp{ 0 \approx \nabla_x \Phi(x_k, \mu_k ) =
g(x_k) - \mu_k A^T(x_k) C^{-1} (x_k) e \approx
g(x_k) - \mu_k A_{\calA}^T(x_k) C_{\calA}^{-1} (x_k) e,}
the approximation being true because the neglected terms involve
$y(x_k) = \mu_k / c_i(x_k)$ which converge to zero for inactive
constraints. Then in the non-degenerate case, again roughly speaking,
the Newton correction $s\s{P}$ for the new barrier parameter satisfies
\eqn{lns}{ \mu_{k+1}^{ } A_{\calA}^T(x_k^{ }) C_{\calA}^{-2}(x_k^{ })
A_{\calA}^{ }(x_k^{ }) s\s{P} \approx ( \mu_{k+1}^{ } - \mu_k^{ } )
A_{\calA}^T(x_k^{ }) C_{\calA}^{-1}(x_k^{ }) e }
since
\disp{\nabla_x \Phi(x_k, \mu_{k+1} ) \approx g(x_k)
- \mu_{k+1} A_{\calA}^T(x_k) C_{\calA}^{-1} (x_k) e
\approx ( \mu_{k+1}^{ } - \mu_k^{ } )
A_{\calA}^T(x_k^{ }) C_{\calA}^{-1}(x_k^{ }) e}
and the $ \mu_{k+1}^{ } A_{\calA}^T(x_k^{ }) C_{\calA}^{-2}(x_k^{ })
A_{\calA}^{ }(x_k^{ })$ term dominates $\nabla_{xx} \Phi(x_k, \mu_{k+1} )$.
If $A_{\calA}^{ }(x_k^{ })$ is full rank, then multiplying the approximation
\req{lns} from the left first by the generalized inverse,
$(A_{\calA} A_{\calA}^T)^{-1} A_{\calA}$ of $A_{\calA}$ and then by $C_{\calA}^2$
implies that
\disp{ A_{\calA}(x_k) s\s{P}
\approx \left( 1 - \frac{\mu_k}{\mu_{k+1}}\right ) c_{\calA}(x_k) }
from which a Taylor expansion of $c_{\calA}(x_k+s\s{P})$ reveals that
\disp{c_{\calA}(x_k+s\s{P}) \approx c_{\calA}(x_k) + A_{\calA}(x_k) s\s{P}
\approx \left( 2 - \frac{\mu_k}{\mu_{k+1}}\right ) c_{\calA}(x_k) < 0}
whenever $\mu_{k+1} < \half \mu_k$. Hence a
Newton step will asymptotically be infeasible for anything but the
most modest decrease in $\mu$, and thus the method
is unlikely to converge fast.
We will return to both of these issues shortly, but first we need to
examine barrier methods in a seemingly different light.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{A different perspective: perturbed optimality conditions}
We now consider what, superficially, appears to be a completely different
approach to inequality-constrained optimization. Recall from
Theorem~\req{bao-con-1stordernec} that the
first order optimality conditions for \req{icp} are that there
are Lagrange multipliers (or, as they are sometimes called, dual variables)
$y$ for which
\disp{\arr{rl}{g(x) - A^T(x) y = 0 & \hspace*{10mm} \mbox{(dual feasibility)} \\
C(x) y = 0 & \hspace*{10mm} \mbox{(complementary slackness)} \\
c(x) \geq 0 \tim{and} y \geq 0.}}
Now consider the ``perturbed'' problem
\disp{\arr{rl}{g(x) - A^T(x) y = 0 & \hspace*{10mm} \mbox{(dual feasibility)} \\
C(x) y = \mu e & \hspace*{10mm}
\mbox{(\defn{perturbed} complementary slackness)} \\
\hspace*{15mm} c(x) > 0 \tim{and} y > 0,}}
where $\mu > 0$.
\defn{Primal-dual path-following} methods aim to track solutions to
the system
\eqn{pfs}{g(x) - A^T(x) y = 0 \tim{and} C(x) y - \mu e = 0}
as $\mu$ shrinks to zero, while maintaining $c(x) > 0$ and $y > 0$.
This approach has been amazingly successful when applied to linear
programming problems, and has been extended to many other classes of
convex optimization problems.
Since \req{pfs} is simply a nonlinear system, an obvious (locally convergent)
way to solve the system is, as always, to use Newton's method.
It is easy to show that the Newton correction $(s\s{PD},w)$ to $(x,y)$ satisfies
\eqn{nbff}{ \mat{cc}{H(x,y) & - A^T(x) \\ Y A(x) & C(x) } \vect{s\s{PD}\\w} = -
\vect{g(x) - A^T(x) y \\ C(x) y - \mu e }.}
Using the second equation to eliminate $w$ gives that
\eqn{nbf}{ \left( H(x,y) + A^T(x) C^{-1}(x) Y A(x) \right) s\s{PD}
= - \left( g(x) - \mu A^T(x) C^{-1}(x) e \right) = g(x,y(x)),}
where, as before, $y(x) = \mu C^{-1}(x) e$.
But now compare this with the Newton barrier system \req{nbee}.
Amazingly, the only difference is that the (left-hand-side) coefficient
matrix in \req{nbee} mentions the specific $y(x)$ while that for \req{nbf}
uses a generic $y$. And it is this difference that turns out to be
crucial. The freedom to choose $y$ in
$H(x,y) + A^T(x) C^{-1}(x) Y A(x)$ for the primal-dual approach
proves to be vital. Making the primal choice $y(x) = \mu C^{-1}(x) e$
can be poor, while using a more flexible approach in which $y$ is
chosen by other means, such as through the primal-dual correction $y+w$
is often highly successful.
We now return to the potential difficulties with the primal approach
we identified in Sections~\ref{pdi} and \ref{pdii}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Potential difficulty II \ldots revisited}
We first show that, despite our reservations in Section~\ref{pdii},
the value $x\s{S}_{k+1} = x_k^{ }$ can be a good starting point.
The problem with the primal correction $s\s{P}$ is that the primal method
has to choose $y = y(x\s{S}_k) = \mu_{k+1} C^{-1}(x_k) e$, and this
is a factor $\mu_{k+1} / \mu_k$ too small to be a good Lagrange multiplier
estimate---recall that Theorem~\ref{thm3.1} shows that
$\mu_{k} C^{-1}(x_k) e$ converges to $y_*$.
But now suppose instead
that we use the primal-dual correction $s\s{PD}$ and choose the ``proper''
$y = \mu_k C^{-1}(x_k) e$ rather than $y(x\s{S}_k)$---we know that this
is a good choice insofar as this Newton step should decrease the
dual infeasibility and complementary slackness since $(x_k,\mu_k C^{-1}(x_k) e)$
are already good estimates.
In this case, arguing as before, in the non-degenerate case,
the correction $s\s{PD}$ satisfies
\disp{ \mu_k^{ } A_{\calA}^T(x_k^{ }) C_{\calA}^{-2}(x_k^{ })
A_{\calA}^{ }(x_k^{ }) s\s{PD} \approx ( \mu_{k+1}^{ } - \mu_k^{ } )
A_{\calA}^T(x_k^{ }) C_{\calA}^{-1}(x_k^{ }) e, }
and thus if $A_{\calA}(x_k)$ is full rank,
\disp{ A_{\calA}(x_k) s\s{PD}
\approx \left( \frac{\mu_{k+1}}{\mu_k} - 1 \right ) c_{\calA}(x_k).}
Then using a Taylor expansion of $c_{\calA}(x_k+s\s{PD})$ reveals that
\disp{c_{\calA}(x_k+s\s{PD}) \approx c_{\calA}(x_k) + A_{\calA}(x_k) s\s{PD}
\approx \frac{\mu_{k+1}}{\mu_k} c_{\calA}(x_k) > 0,}
and thus $x_k + s\s{PD}$ is feasible---the result is easy to show for
inactive constraints. Hence, simply by using a different model Hessian
we can compute a useful Newton correction from $x\s{S}_{k+1} = x_k$ that both
improves the violation of the optimality conditions (and
ultimately leads to fast convergence) {\em and} stays feasible.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\subsubsection{Primal-dual barrier methods}
In order to globalize the primal-dual iteration, we simply need to build
an appropriate model of the logarithmic barrier function within
either a linesearch or trust-region framework for minimizing $\Phi(x,\mu_k)$.
As we have already
pointed out the disadvantages of only allowing the (primal) Hessian
approximation $\nabla_{xx} \Phi(x_k,\mu_k)$, we instead prefer the
more flexible search-direction model problem to (approximately)
\eqn{mbf}{\minin{s \in \smallRe^n} \ip{s}{g(x,y(x))} + \half
\bip{s}{\left( H(x,y) + A^T(x) C^{-1}(x) Y A(x) \right) s},}
possibly subject to a trust-region constraint.
We have already noticed that the first-order term
$g(x,y(x)) = \nabla_x \Phi(x,\mu)$ as
$y(x) = \mu C^{-1}(x) e$, and thus the model gradient is that of
the barrier function as required by our global convergence analyses
of linesearch and trust-region methods. We have discounted always choosing
$y = y(x)$ in \req{mbf}, and have suggested that the choice
$y = ( \mu_{k-1} / \mu_k ) y(x)$ when changing the barrier parameter
results in good use of the starting point. Another possibility is to
use $y = y\s{OLD} + w\s{OLD}$, where $w\s{OLD}$ is the primal-dual
correction to the previous dual-variable estimates $y\s{OLD}$.
However, this needs to be used with care since there is no a priori
assurance that $y\s{OLD} + w\s{OLD} > 0$, and indeed it is usual
to prefer $y = \max(y\s{OLD} + w\s{OLD},\epsilon(\mu_k) e)$
for some ``small'' $\epsilon(\mu_k) > 0$. The choice
$\epsilon(\mu_k) = \mu_k^{1.5}$ leads to a realistic
primal-dual method, although other precautions need sometimes to be taken.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Potential difficulty I \ldots revisited}
We now return to the other perceived difficult with barrier
or primal-dual path-following methods,
namely that the inherent ill-conditioning in the barrier Hessian makes
it hard to generate accurate Newton steps when the barrier parameter is
small. Let $\calI$ be the set of inactive constraints at $x_*$, and
denote the active and inactive components of $c$ and $y$ with suffices
$\calA$ and $\calI$ respectively. Thus
$c_{\calA}(x_*) = 0$ and $c_{\calI}(x_*) > 0$,
while if the solution is non-degenerate,
$y_{\calA}(x_*) > 0$ and $y_{\calI}(x_*) = 0$.
As we have seen, the Newton correction $s\s{PD}$ satisfies \req{nbff},
while the equivalent system \req{nbf} clearly has a condition number that
approaches infinity as $x$ and $y$ reach their limits because
$c_{\calA}(x)$ approaches zero while $y_{\calA}(x)$ approaches
$y_{\calA}(x_*) > 0$.
But now suppose that we separate \req{nbff} into
\disp{ \mat{ccc}{H(x,y) & - A_{\calA}^T(x) & - A_{\calI}^T(x) \\
Y_{\calA} A_{\calA}(x) & C_{\calA}(x) & 0 \\
Y_{\calI} A_{\calA}(x) & 0 & C_{\calI}(x) }
\vect{s\s{PD}\\w_{\calA}\\w_{\calI}} = -
\vect{g(x) - A^T(x) y \\ C_{\calA}(x) y_{\calA} - \mu e \\
C_{\calI}(x) y_{\calI} - \mu e },}
and then eliminate the variables $w_{\calI}$, multiply the
second equation by $Y_{\calA}^{-1}$ and use $C_{\calI}(x) y_{\calI} = \mu e$,
we obtain
\eqn{pbs}{ \arr{c}{\mat{cc}{H(x,y) +
A_{\calI}^T(x) C_{\calI}(x)^{-1} Y_{\calI}^{ } A_{\calI}^{ }(x)
& - A_{\calA}^T(x) \\
A_{\calA}^{ }(x) & C_{\calA}^{ }(x) Y_{\calA}^{-1}}
\vect{s\s{PD} \\ w_{\calA}^{ } } \\ = -
\vect{g(x) - A_{\calA}^T(x) y_{\calA}^{ } -
\mu A_{\calI}^T(x) C_{\calI}^{-1}(x) e
\\ c_{\calA}^{ }(x) - \mu Y_{\calA}^{-1} e}.}}
But then we see that the terms involving inverses,
$C_{\calI}^{-1}(x)$ and $Y_{\calA}^{-1}$,
remain bounded, and indeed in the limit the system becomes
\disp{ \mat{cc}{H(x,y) & - A_{\calA}^T(x) \\
A_{\calA}^{ }(x) & 0} \vect{s\s{PD} \\ w_{\calA}^{ } } = -
\vect{g(x) - A_{\calA}^T(x) y_{\calA}^{ } -
\mu A_{\calI}^T(x) C_{\calI}^{-1}(x) e \\ 0 } }
which is well behaved. Thus just because
\req{nbf} is ill conditioned, this does not preclude us from
finding $s\s{PD}$ from an equivalent, perfectly well-behaved system
like \req{pbs}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{A practical primal-dual method}
Following on from the above, we now give the skeleton of a reasonable
primal-dual method.
\vspace*{0.2in}
\noindent
\centerline{\framebox[4.5in]{\hspace*{0.2in}\parbox{4.4in}{
\vspace*{0.2in}
Given $\mu_0 > 0$ and feasible $(x\s{S}_0, y\s{S}_0)$, set $k = 0$. \\
Until ``convergence'', iterate: \\
\hspace*{3mm} \defn{Inner minimization}: starting from $(x\s{S}_k, y\s{S}_k)$,
use an \\
\hspace*{6mm} unconstrained minimization algorithm to find $(x_k,y_k)$
for which \\
\hspace*{9mm} $\| C(x_k) y_k - \mu_k e \| \leq \mu_k $ and
%\\\hspace*{9mm}
$\| g(x_k) - A^T(x_k) y_k \| \leq \mu_k^{1.00005}$. \\
\hspace*{3mm} Set $\mu_{k+1}^{ } = \min( 0.1 \mu_k^{ }, \mu_{k}^{1.9999})$. \\
\hspace*{3mm} Find $(x\s{S}_{k+1}, y\s{S}_{k+1})$ using a
%\\ \hspace*{9mm}
primal-dual Newton step from $(x_k,y_k)$. \\
\hspace*{3mm} If $(x\s{S}_{k+1}, y\s{S}_{k+1})$ is infeasible, reset
$(x\s{S}_{k+1}, y\s{S}_{k+1})$ to $(x_k,y_k)$. \\
\hspace*{3mm} Increase $k$ by 1.
\vspace{0.2in}}}}
\vspace{0.2in}
\noindent
The inner minimization will be performed by either a linesearch or
trust-region method for minimizing $\Phi(x,\mu_k)$, the stopping
rules $\| C(x_k) y_k - \mu_k e \| \leq \mu_k$ and
$\| g(x_k) - A^T(x_k) y_k \| \leq \mu_k^{1.00005}$ certainly being
attainable as the first-order optimality condition for
minimizing $\Phi(x,\mu_k)$ is that $g(x) - A^T(x) y = 0$, where
$C(x) y = \mu_k e$. The extra step, in which the starting point
is computed by performing a primal-dual Newton step from $(x_k,y_k)$,
is simply included to generate a value that is already close to
first order critical, and the stopping tolerances are specially
chosen to encourage this. Indeed we have the following
asymptotic convergence result.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\lthm{thm2}{Suppose that $f$, $c \in \calC^2$,
that a subsequence $\{(x_k,y_k)\}$, $k \in \calK$,
of the practical primal-dual method converges to $(x_*,y_*)$ satisfying
second-order sufficiency conditions, that $A_{\calA}(x_*)$ is full-rank,
and that $(y_*)_{\calA} > 0$. Then the starting point satisfies the
inner-minimization termination test (i.e., $(x_k,y_k) = (x\s{S}_k, y\s{S}_k)$)
for all $k$ sufficiently large,
and the whole sequence $\{(x_k,y_k)\}$ converges to
$(x_*,y_*)$ at a superlinear rate (with a Q-factor at least 1.9998).}
\noindent
This is a highly acceptable result, the convergence being essentially
quadratic (which would correspond to a Q-factor of two---any sequence
$\{\sigma_k\}$ is said to converge to $\sigma_*$ with \defn{Q-factor}
at least q if
$|\sigma_{k+1} - \sigma_*| \leq \gamma |\sigma_{k} - \sigma_*|^q$ for
some $\gamma > 0$).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Primal-dual interior-point methods have the potential for both excellent
theoretical and practical behaviour. There are polynomial
interior-point algorithms for linear, (convex) quadratic and
semi-definite programming. While it is unlikely that this is
true for more general (nonconvex) problems, the barrier function
globalization is most effective in practice, and the asymptotic behaviour
is normally just as for the convex case.
From a global perspective, it is very important that
iterates are kept away from constraint boundary
until near to convergence, as otherwise very slow progress will be made---this
is certainly born out in practice. Finally, while the methods we
have discussed in this section have all required an interior starting point,
it is possible to find one (if there is one!) by solving the ``phase-one''
problem to
\disp{ \minin{(x,\gamma)} \gamma \tim{subject to} c(x) + \gamma e \geq 0;}
any feasible point $(x,\gamma)$ for this auxiliary problem for which
$\gamma < 0$ is suitable, for then $c(x) > 0$.
It is quite common in practice to replace the inequality $c_i(x) \geq 0$
by the equation $c_i(x) - s_i = 0$, and simple bound $s_i \geq 0$ on
the \defn{slack} variable $s_i$. This has the algebraic advantage that the
inequality constraints are then all simple bounds and thus
that barrier terms only appear on the diagonal of the Hessian model,
but arguably the disadvantages that the dimensionality of the problem has been
artificially increased, and that we now need to use some
means of coping with equality constraints. We consider this latter point next.
%
% Lecture 5: SQP methods
%
%\newpage
\setcounter{lecture}{5}
\section[SQP METHODS FOR EQUALITY CONSTRAINED OPTIMIZATION]{SQP METHODS FOR EQUALITY CONSTRAINED \\ OPTIMIZATION}
\setcounter{equation}{0}
\setcounter{figure}{0}
\label{sqp}
In this final section, having already investigated very good
methods for dealing with inequality constraints,
we now turn our attention to the problem \req{ecp}, in which there
are only equality constraints on the variables.
Of course in practice, there are frequently both equations
and inequalities, and composite methods using the barrier/interior-point
methods discussed in Section~\ref{ipm} and the SQP methods we shall
consider here are often used. Alternatively, SQP methods themselves
may easily be generalized to handle inequality constraints. For brevity
we shall not consider such extensions further here.
\subsection{Newton's method for first-order optimality}
\defn{Sequential Quadratic Programming} (SQP) methods (sometimes called
successive or recursive quadratic programming methods) are most naturally
derived by considering the first-order necessary conditions for
\req{ecp}---we will see where the names come from shortly.
Recall at optimality we expect to have
\eqn{fooc}{g(x,y) \equiv g(x) - A^T(x) y = 0 \tim{and} c(x) = 0.}
This is a system of nonlinear equations in the variables $x$ and the
Lagrange multipliers $y$. Notice that the system is actually linear in
$y$ so that if $x$ were known it would be straightforward to find $y$.
Suppose now that $(x,y)$ is an approximation
to a solution of \req{fooc}. Then, as always, we might apply Newton's
method to try to improve $(x,y)$, and this leads us to construct a
correction $(s,w)$ for which
\eqn{sqpe}{ \mat{cc}{H(x,y) & - A^T(x) \\ A(x) & 0 }
\vect{s \\ w } = - \vect{g(x,y) \\ c(x) }.}
Newton's method would then apply the same procedure to the
``improved'' estimate $(x_+,y_+) = (x+s,y+w)$.
There are a number of alternative formulations of \req{sqpe}. Firstly
\req{sqpe} may be written as the symmetric system of equations
\disp{ \mat{cc}{H(x,y) & A^T(x) \\ A(x) & 0} \vect{s \\ - w }
= - \vect{g(x,y) \\ c(x) };}
notice here that the coefficient matrix is indefinite because of its
zero 2,2 block. Secondly, on writing $y_+ = y + w$, the equation becomes
\disp{ \mat{cc}{H(x,y) & - A^T(x) \\ A(x) & 0 }
\vect{s \\ y_+ } = - \vect{g(x) \\ c(x) },}
or finally, in symmetric form,
\disp{ \mat{cc}{H(x,y) & A^T(x) \\ A(x) & 0 }
\vect{s \\ - y_+ } = - \vect{g(x) \\ c(x) }.}
In practice we might prefer to approximate $H(x,y)$ by some symmetric
$B$, and instead solve
\eqn{sqpsys}{ \mat{cc}{B & A^T(x) \\ A(x) & 0 }
\vect{s \\ - y_+ } = - \vect{g(x) \\ c(x) }
= \mat{cc}{B & - A^T(x) \\ A(x) & 0 }
\vect{s \\ y_+ }.}
One could imagine solving these related systems
by finding an LU factorization of the coefficient matrix in the
unsymmetric case, or a symmetric-indefinite (a generalization of
Cholesky) factorization in the symmetric case. Alternatively, if
$B$ is invertible, $s$ and $y_+$ might be found successively by solving
\disp{A(x)B^{-1}A(x)^T y = - c + A(x)B^{-1}g \tim{and then}
B s = A(x)^T y - g}
using symmetric factorizations of $B$ and $A(x)B^{-1}A(x)^T$.
For very large problems, iterative methods might be preferred, and
here GMRES(k) or QMR, for the unsymmetric case, or MINRES or
conjugate-gradients (restricted to the null-space of $A(x)$), for the
symmetric case, have all been suggested. Thus there are many ways to
solve the system(s) of linear equations that arise from SQP methods,
and there is currently much interest in exploiting the structure in such
systems to derive very efficient methods.
But where does the name ``sequential quadratic programming'' come from?
\subsection{The Sequential Quadratic Programming iteration}
A \defn{quadratic program} is a problem involving the optimization of a
quadratic function subject to a set of linear inequality and/or equality
constraints. Consider the quadratic programming problem
\eqn{eqp}{ \minin{s \in \smallRe^n} \ip{s}{g(x)} + \half \ip{s}{B s}
\tim{subject to} A(x) s = - c(x). }
Why this problem? Well, Theorem~\ref{bao-first-order-vector-Taylor-error}
indicates that $c(x) + A(x) s$ is a first-order (Taylor) approximation
to the constraint function $c(x+s)$, while
$ \ip{s}{g(x)} + \half \ip{s}{B s}$ is potentially a second-order model
of the decrease $f(x+s)-f(x)$. Thus one can argue that \req{eqp}
gives a suitable (at least first-order) model of \req{ecp}. An objection
might be that really we should be aiming for true second-order approximations
to all functions concerned, but this would lead to the
significantly-harder minimization of a quadratic function subject to
quadratic constraints---constraint curvature is a major obstacle.
The interesting feature of \req{eqp} is that it follows
immediately from Theorem~\ref{bao-econ-1stordernec} that
any first-order critical point of \req{eqp} is given by \req{sqpsys}.
Thus Newton-like methods for first-order optimality are equivalent
to the solution of a sequence of related quadratic programs. Hence
the name. Notice that if $B = H(x,y)$, solving \req{eqp} is actually
Newton's method for \req{fooc}, and this suggests that $B$ should be
an approximation to the Hessian of the Lagrangian function, not the
objective function. Clearly the constraint curvature that we would have
liked to have added to the linear approximations of the constraints
has worked its way into the objective function!
To summarize, the basic SQP iteration is as follows.
\vspace{0.2in}
\noindent
\centerline{\framebox[4.7in]{\hspace*{0.2in}\parbox{4.3in}{
\vspace*{0.1in}
Given $(x_0,y_0)$, set $k = 0$. \\
Until ``convergence'' iterate: \\
\hspace*{6mm} Compute a suitable symmetric $B_k$ using $(x_k,y_k)$. \\
\hspace*{6mm} Find
\eqn{sqps}{s_k = \argminover{s \in \smallRe^n} \ip{s}{g_k} +
\half \ip{s}{B_k s}
\tim{subject to} A_k s = - c_k }
\hspace*{6mm} along with associated Lagrange multiplier estimates $y_{k+1}$. \\
\hspace*{6mm} Set $x_{k+1} = x_k + s_k$
%\\ \hspace*{9mm}
and increase $k$ by 1.
\vspace{0.2in}}}}
\vspace{0.2in}
\noindent
The SQP method is both simple and fast. If $B_k = H(x_k,y_k)$, the
method is Newton's method for \req{fooc}, and thus is quadratically
convergent provided that $(x_0,y_0)$ is sufficiently close to
a first-order critical point $(x_*,y_*)$ of \req{ecp} for which
\disp{\mat{cc}{H(x_*,y_*) & A^T(x_*) \\ A(x_*) & 0 }}
is non-singular. Moreover, the method is
superlinearly convergent when $B_k$ is a ``good'' approximation
to $H(x_k,y_k)$, and there is even no necessity that this be so
for fast convergence. It should also be easy for the reader to believe that
had we wanted to solve the problem \req{icp} involving
inequality constraints, the suitable SQP subproblem would be
\disp{ \minin{s \in \smallRe^n} \ip{s}{g(x)} + \half \ip{s}{B s}
\tim{subject to} A(x) s \geq - c(x) }
in which the nonlinear inequalities have been linearized.
But, as the reader will already have guessed, this basic iteration
also has drawbacks, leading to a number of vital questions.
For a start it is a Newton-like iteration, and
thus may diverge from poor starting points.
So how do we globalize this iteration?
How should we pick $B_k$? What should we do if \req{eqp}
is unbounded from below? And precisely when is it unbounded?
The problem \req{eqp} only has a solution if the constraints $A(x)s = -c(x)$
are consistent. This is certainly the case if $A(x)$ is full rank, but
may not be so if $A(x)$ is rank deficient---we shall consider alternatives
that deal with this deficiency later. Applying
Theorem~\ref{bao-econ-2ndordernec} to \req{eqp}, we deduce that
any stationary point $(s,y_+)$ satisfying \req{sqpsys} solves \req{eqp}
only if $B$ is positive semi-definite on the manifold
$\{ s: A(x) s = 0\}$---if $B$ is positive definite on the manifold
$(s,y_+)$ is the unique solution to the problem. If the $m$ by $n$ matrix
$A(x)$ is full rank and the columns of $S(x)$ form a basis for the
null-space of $A(x)$,
it is easy to show that $B$ being positive (semi-)definite on the manifold
$\{ s: A(x) s = 0\}$ is equivalent to $S(x)^T B S(x)$ being
positive (semi-)definite which is in turn equivalent to the matrix
\disp{\mat{cc}{B & A^T(x) \\ A(x) & 0 }}
(being non-singular and) having $m$ negative eigenvalues.
If $B$ violates these assumptions, \req{eqp} is unbounded.
For the remainder of this section, we focus on methods to globalize
the SQP iteration. And it should not surprise the reader that we
shall do so by considering linesearch and trust-region schemes.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Linesearch SQP methods}
\label{ls-sqp}
The obvious way to embed the SQP step $s_k$ within a linesearch framework
is to pick $x_{k+1} = x_k + \alpha_k s_k$, where the step
$\alpha_k > 0$ is chosen so that
\eqn{lssqp}{ \Phi( x_k + \alpha_k s_k , p_k ) \tim{``$<$''} \Phi( x_k , p_k),}
and where $\Phi(x,p)$ is a ``suitable'' merit function depending on
parameters $p_k$. Of course it is then vital that
$s_k$ be a descent direction for $\Phi(x,p_k)$ at $x_k$, as otherwise
there may be no $\alpha_k$ for which \req{lssqp} is satisfied.
As always with linesearch methods, this limits the choice of $B_k$,
and it is usual to insist that $B_k$ be positive definite---the reader
may immediately object that this is imposing an unnatural requirement,
since $B_k$ is supposed to be approximating the (usually) indefinite
matrix $H(x_k,y_k)$, and we can only sympathise with such a view!
What might a suitable merit function be? One possibility is to
use the quadratic penalty function \req{qpf}. In this case, we have
the following result.
\lthm{thm4.1}{Suppose that $B_k$ is positive definite, and that $(s_k,y_{k+1})$
are the SQP search direction and its associated Lagrange multiplier
estimates for the problem
\disp{\tmininx{} f(x) \tim{subject to} c(x) = 0}
at $x_k$. Then if $x_k$ is not a first-order critical point,
$s_k$ is a descent direction for the
quadratic penalty function $\Phi(x,\mu_k)$ at $x_k$ whenever
\disp{ \mu_k \leq \frac{\|c(x_k)\|_2}{\|y_{k+1}\|_2}.}}
\noindent
We know that the parameter $\mu_k$ for the quadratic penalty function
needs to approach zero for its minimizers to converge to those of \req{ecp},
so Theorem~\ref{thm4.1} simply confirms this by suggesting how to adjust
the parameter.
The quadratic penalty function has another role to play if the constraints
are inconsistent. For consider the quadratic (Newton-like) model
\disp{ \minin{s \in \smallRe^n} \ip{s}{g_k^{ } + A_k^T c_k^{ } / \mu_k^{ } } +
\half \ip{s}{( B_k^{ } + 1/\mu_k^{ } A_k^T A_k^{ } ) s}}
that might be used to compute a step $s\s{Q}_k$ from $x_k$. Stationary
points of this model satisfy
\disp{( B_k^{ } + 1/\mu_k^{ } A_k^T A_k^{ } )s\s{Q}_k = -
( g_k + A_k^T c_k / \mu_k )}
or, on defining $y\s{Q}_k \eqdef - \mu_k^{-1} ( c_k + A_k s\s{Q}_k )$,
\eqn{qpfs}{ \mat{cc}{B_k^{ } & A^T_k \\ A_k^{ } & - \mu_k^{ } I }
\vect{s\s{Q}_k \\ - y\s{Q}_k } = - \vect{g_k^{ } \\ c_k^{ } }.}
But now compare this system with \req{sqpsys} that which defines the SQP step:
the only difference is the vanishingly small 2,2 block $-\mu_k I$ in the
coefficient matrix. While this indicates that Newton-like directions
for the quadratic penalty function will become increasingly good
approximations to SQP steps (and, incidentally, it can be shown that
a Newton iteration for \req{qpf} with well chosen $\mu_k$ converges
superlinearly under reasonable assumptions),
the main point of the alternative \req{qpfs} is that
rank-deficiency in $A_k$ is neutralised by the presence of
2,2 block term $-\mu_k I$.
Nevertheless, the quadratic penalty function is rarely used, its
place often being taken by non-differentiable exact penalty functions.
The \defn{non-differentiable exact penalty function} is given by
\eqn{ndpf}{\Phi(x,\rho) = f(x) + \rho \|c(x)\|}
for any norm $\|\cdot\|$ and scalar $\rho > 0$. Notice that
the function is non-differentiable particularly when $c(x) = 0$,
the very values we hope to attain! The following result helps
explain why such a function is considered so valuable.
\lthm{thm4.2}{Suppose that $f, c \in C^2$, and that $x_*$ is an
isolated local minimizer of $f(x)$ subject to $c(x) = 0$,
with corresponding Lagrange multipliers $y_*$.
Then $x_*$ is also an isolated local minimizer of $\Phi(x,\rho)$
provided that \vspace{-0.2in}
\disp{\rho > \|y_*\|_D,\vspace*{-0.3in}}
where the \defn{dual norm} \vspace{-0.2in}
\disp{\| y \|_D = \sup_{x \neq 0}\frac{\ip{y}{x}}{\|x\|}.}}
\noindent
Notice that the fact that $\rho$ merely needs to be larger than
some critical value for $\Phi(x,\rho)$ to be usable to try
to identify solutions to \req{ecp} is completely different
to the quadratic penalty function, for which the parameter
had to take on a limiting value.
More importantly, as we now see, $\Phi(x,\rho)$ may be used as
a merit function for the SQP step.
\lthm{thm3}{Suppose that $B_k$ is positive definite, and that $(s_k,y_{k+1})$
are the SQP search direction and its associated Lagrange multiplier
estimates for the problem
\disp{\tmininx{} f(x) \tim{subject to} c(x) = 0}
at $x_k$. Then if $x_k$ is not a first-order critical point,
$s_k$ is a descent direction for the
non-differentiable penalty function $\Phi(x,\rho_k)$ at $x_k$ whenever
$\rho_k \geq \|y_{k+1}\|_D$.}
\noindent
Once again, this theorem indicates how $\rho_k$ needs to be adjusted
for use within a linesearch SQP framework.
Thus far, everything looks perfect. We have methods for globalizing
the SQP iteration, an iteration that should ultimately converge very
fast. But unfortunately, it is not as simple as that. For consider
the example in Figure~\ref{maratos}.
\begin{figure}[htbf]
\centerline{
\psfig{figure=maratos.eps,width=7.0cm,height=7.0cm,silent=}}
\begin{picture}(120,0.1)(0,0)
\put(79,21){\small $x_k$}
\put(97,34){\small $s_k$}
\put(92,59){\small $x_*$}
\end{picture}
\vspace*{-5mm}
\caption{\label{maratos}$\ell_1$ non-differentiable exact penalty function
($\rho = 1$):
$f(x) = 2( x_1^2 + x_2^2 - 1 ) - x_1$ and
$c(x) = x_1^2 + x_2^2 - 1$. Solution: $x_* = (1,0)$, $y_* = \threehalves$.
The SQP direction using the optimal Hessian $H(x_*,y_*) = I$.
Notice how the merit function increases at the point $x_k + s_k$.}
\end{figure}
Here the current iterate lies close to (actually on) the constraint, the
SQP step moves tangentially from it, and thus moves away as the constraint
is nonlinear, but unfortunately, at the same time, the value of the
objective function rises. Thus any merit function like \req{qpf} or
\req{ndpf} composed simply from
positive combinations of the objective and (powers) of norms of constraint
violations will increase after such an SQP step, and thus necessarily
$\alpha_k \neq 1$ in \req{lssqp}---worse still, this behaviour can happen
arbitrarily close to the minimizer. This has the unfortunate side effect
that it may happen that the expected fast convergence achievable by
Newton-like methods will be thwarted by the merit function. That is, there
is a serious mismatch between the global and local convergence needs of the
SQP method. The fact that the merit function may
prevent acceptance of the full SQP step is known as the
\defn{Maratos effect}.
The Maratos effect occurs because the curvature of the
constraints is not adequately represented by linearization
in the SQP model. In particular,
\disp{c( x_k + s_k ) = O(\|s_k\|^2).}
This suggests that we need to correct for this curvature. We may
do this by computing a \defn{second-order correction} from $x_k + s_k$,
that is an extra step $s\s{C}_k$ for which
\eqn{soca}{c( x_k + s_k + s\s{C}_k ) = o(\|s_k\|^2).}
Since we do not want to destroy potential for fast convergence, we must
also insist that the correction is small relative to the SQP step,
and thus that
\eqn{socb}{s\s{C}_k = o(s_k). }
There are a number of ways to compute a second-order correction.
The first is simply to move back as quickly as possible towards the
constraints. This suggests we compute a
minimum ($\ell_2$-)norm solution to $c(x_k+s_k) + A(x_k+s_k) s\s{C}_k = 0$.
It is easy to check that the required solution satisfies
\disp{ \mat{cc}{I & A^T(x_k+s_k) \\ A(x_k+s_k) & 0 }
\vect{s\s{C}_k \\ - y\s{C}_{k+1} } = - \vect{0 \\ c(x_k+s_k) }.}
Since this requires that we re-evaluate the constraints {\em and}
their Jacobian at $x_k + s_k$, we might hope instead to find a
minimum norm solution to $c(x_k+s_k) + A(x_k) s\s{C}_k = 0$, and thus that
\disp{ \mat{cc}{I & A^T(x_k) \\ A(x_k) & 0 }
\vect{s\s{C}_k \\ - y\s{C}_{k+1} } = - \vect{0 \\ c(x_k+s_k) }.}
A third amongst many other possibilities is
to compute another SQP step from $x_k + s_k$,
that is to compute $s\s{C}_k$ so that
\disp{ \mat{cc}{B\s{C}_k & A^T(x_k+s_k) \\ A(x_k+s_k) & 0 }
\vect{s\s{C}_k \\ - y\s{C}_{k+1} } = - \vect{ g(x_k+s_k) \\ c(x_k+s_k) },}
where $B\s{C}_k$ is an approximation to $H( x_k+s_k, y_k^+ )$.
It can easily be shown that all of the above corrections
satisfy \req{soca}--\req{socb}. In Figure~\ref{soc},
we illustrate a second-order correction in action.
\begin{figure}[htbf]
\centerline{
\psfig{figure=soc.eps,width=7.0cm,height=7.0cm,silent=}}
\begin{picture}(120,0.1)(0,0)
\put(79,21){\small $x_k$}
\put(97,34){\small $s_k$}
\put(92,59){\small $x_*$}
\put(102,57){\small $\soc_k$}
\end{picture}
\vspace*{-5mm}
\caption{\label{soc}$\ell_1$ non-differentiable exact penalty function
($\rho = 1$):
$f(x) = 2( x_1^2 + x_2^2 - 1 ) - x_1$ and
$c(x) = x_1^2 + x_2^2 - 1$
solution: $x_* = (1,0)$, $y_* = \threehalves$.
See that the second-order correction $\soc_k$ helps avoid the
Maratos effect for the above problem with the
$\ell_1$-penalty function. Notice how $\soc_k$ more than compensates for the
increase in the merit function at the point $x_k + s_k$, and how
much closer $x_k+s_k+\soc_k$ is to $x_*$ than is $x_k$.}
\end{figure}
It is possible to show that, under reasonable assumptions,
any step $x_k + s_k + \soc_k$ made up from the SQP step $s_k$ and
a second-order correction $\soc_k$ satisfying \req{soca}--\req{socb}
will ultimately reduce \req{ndpf}.
So now we can have both global and very fast asymptotic convergence at the
expense of extra problem evaluations. Of course, we have stressed that
a second SQP step gives a second-order correction, so another way of viewing
this is to require that the merit function decreases at least every second
iteration, and to tolerate non-monotonic behaviour in the interim.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Trust-region SQP methods}
The main disadvantage of (at least naive) linesearch SQP methods is the
unnatural requirement that $B_k$ be positive definite. We saw the same
restriction in the unconstrained case, although at least then there
was some expectation that ultimately the true Hessian $H_k$
would be positive (semi-) definite.
In the unconstrained case, indefinite model Hessians were better handled
in a trust-region framework, and the same is true in the constrained case.
The obvious trust-region generalization of the basic SQP step-generation
subproblem \req{eqp} is to find
\eqn{sqptr}{s_k = \argminover{s \in \smallRe^n} \ip{s}{g_k} + \half
\ip{s}{B_k s} \tim{subject to} A_k s = - c_k \tim{and} \|s\| \leq \Delta_k.}
Since we do not require that $B_k$ be positive definite, this allows us
to use $B_k = H(x_k,y_k)$ if we so desire.
However a few moments reflection should make it
clear that such an approach has a serious flaw.
Let $\Delta\s{CRIT}$ be the least distance to the linearized constraints, i.e.
\disp{\Delta\s{CRIT} \eqdef \min \|s\| \tim{subject to} A_k s = - c_k.}
The difficulty is that if $\Delta_k < \Delta\s{CRIT}$, then
there is {\em no solution} to the trust-region subproblem \req{sqptr}.
This implies that unless $c_k = 0$, the subproblem is meaningless
for all sufficiently small trust-region radius (see Figure~\ref{sqp-infeas}).
\begin{figure}[htbf]
\centerline{
\psfig{figure=newfeas.eps,angle=270,width=6.0cm,height=6.0cm,silent=}
\hspace{0.75cm}
\psfig{figure=newinfeas.eps,angle=270,width=6.0cm,height=6.0cm,silent=}
}
\begin{picture}(120,0.1)(-10,0)
\put(39,58){\small The linearized constraint}
\put(38,59){\vector(-1,-1){8}}
\put(75.5,59){\vector(3,-1){19.5}}
\put(61,35){\small The trust region}
\put(60,36){\vector(-1,0){12}}
\put(85.5,36){\vector(1,0){10}}
\put(47,13){\small The nonlinear constraint}
\put(46,14){\vector(-1,2){4}}
\put(84,14){\vector(1,3){2.7}}
\end{picture}
\caption{\label{sqp-infeas}
The intersection between the linearization of a nonlinear constraint
and a spherical trust region. In the left figure, the trust-region radius
is sufficiently large for the trust region and the linearized constraint
to intersect. This is not so for the smaller trust region illustrated in the
right figure.}
\end{figure}
Thus we need to consider alternatives. In this section, we shall review
the S$\mathbf{\ell_p}$QP method of Fletcher, the
composite step SQP methods due to Vardi, to Byrd and Omojokun, and
to Celis, Dennis and-Tapia, and
the filter-SQP approach of Fletcher and Leyffer.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{The S$\mathbf{\ell_p}$QP method}
Our first trust-region approach is to try to minimize
the $\ell_p$-(exact) penalty function
\eqn{lpep}{\Phi(x,\rho) = f(x) + \rho \|c(x)\|_p}
for sufficiently large $\rho > 0$ and some $\ell_p$ norm
$(1 \leq p \leq \infty)$. We saw in Section~\ref{ls-sqp} that
feasible minimizers of \req{lpep} may be solutions to \req{ecp}
so long as $\rho > 0$ is large enough. Of course, as $\Phi(x,\rho)$
is non-differentiable, we cannot simply apply one of the unconstrained
trust-region methods discussed in Section~\ref{trmfuo}, but must instead
build a specialized method.
Since we are discussing trust-region methods, a suitable model problem is
the \defn{$\mathbf{\ell_p}$QP}
\disp{\minin{s \in \smallRe^n} f_k + \ip{s}{g_k} + \half \ip{s}{B_k s}
+ \rho \| c_k + A_k s \|_p \tim{subject to} \|s\| \leq \Delta_k.}
This has the major advantage that the model problem is always consistent,
since now the only constraint is the trust-region bound.
In addition, when $\rho$ and $\Delta_k$ are large enough, it can be
shown that the model minimizer {\em is} the SQP direction so long as
$A_k s = - c_k$ is consistent. Moreover, when the norms are polyhedral
(e.g., the $\ell_1$ or $\ell_{\infty}$ norms),
$\mathbf{\ell_p}$QP is equivalent to a quadratic program.
To see this, consider for example the
$\mathbf{\ell_1}$QP model problem with an $\ell_{\infty}$ trust region
\disp{\minin{s \in \smallRe^n} \ip{s}{g_k} + \half \ip{s}{B_k s}
+ \rho \| c_k + A_k s \|_1 \tim{subject to} \|s\|_{\infty} \leq \Delta_k.}
But we can always write
\disp{c_k + A_k s = u - v, \tim{where} (u , v) \geq 0.}
Hence the $\mathbf{\ell_1}$QP subproblem is equivalent to the quadratic program
\disp{\arr{rl}{
\minin{s \in \smallRe^n, \; u, v \in \smallRe^m} &
\ip{s}{g_k} + \half \ip{s}{B_k s}
+ \rho \ip{e}{u + v} \\
\tim{subject to} & A_k s - u + v = - c_k \\
& u \geq 0, \;\; v \geq 0 \\
\tim{and} & - \Delta_k e \leq s \leq \Delta_k e.}}
Notice that the QP involves inequality constraints, but there are
good methods (especially of the interior-point variety) for solving
such problems. In particular, it is
possible to exploit the structure of the $u$ and $v$ variables.
In order to develop a practical S$\mathbf{\ell_1}$QP method, it should
not surprise the reader that we need to ensure that every step
we generate achieves as much reduction in the model
$f_k + \ip{s}{g_k} + \half \ip{s}{B_k s} + \rho \| c_k + A_k s \|_p$
as would have been achieved at a Cauchy point. One such
Cauchy point requires the solution to $\mathbf{\ell_1}$LP model
\disp{\minin{s \in \smallRe^n} \ip{s}{g_k}
+ \rho \| c_k + A_k s \|_1 \tim{subject to} \|s\|_{\infty} \leq \Delta_k,}
which may be reformulated as a linear program. Fortunately
approximate solutions to both $\mathbf{\ell_1}$LP and
$\mathbf{\ell_1}$QP subproblems suffice. In practice it is also
important to adjust $\rho$ as the method progresses so as to ensure that
$\rho$ is larger than the (as yet unknown) $\|y_*\|_D$, and this may
be achieved by using the available Lagrange multiplier estimates $y_k$.
Such a scheme is globally convergent, but there is still a
need for a second-order correction to prevent the Maratos effect and
thus allow fast asymptotic convergence.
If $c(x) = 0$ are inconsistent, the method converges to (locally) least
value of the infeasibility $\|c(x)\|$ provided $\rho \rightarrow \infty$.
The alert reader will have noticed that in this section we have replaced
the $\ell_2$ trust-region of the unconstraint trust-region method by a box
or $\ell_{\infty}$ trust-region. The reason for this apparent lack of consistency
is that minimizing a quadratic subject to linear constraints {\em and\/}
an additional quadratic trust-region is too hard. On the other hand, adding
box-constraints does not increase the complexity of the
resulting (quadratic programming) trust-region subproblem.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Composite-step methods}
\label{csm}
Another approach to avoid the difficulties caused by inconsistent
QP subproblems is to separate the computation of the step into
two stages. The aim of a \defn{composite-step} method is to find
\disp{ s_k = n_k + t_k,}
where the \defn{normal step} $n_k$ moves towards feasibility of the
linearized constraints (within the trust region),
while the \defn{tangential step} $t_k$ reduces the
model objective function (again within the trust-region)
without sacrificing feasibility obtained from $n_k$.
Of course since the normal step is solely concerned with
feasibility, the model objective may get worse, and indeed
it may not recover during the tangential step.
The fact that the tangential step is required to maintain
any gains in (linearized) feasibility achieved during the
normal step implies that
\disp{ A_k (n_k + t_k) = A_k n_k \tim{and hence that} A_k t_k = 0.}
We illustrate possible normal and tangential steps
in Figure~\ref{sqp-vardi-fig}.
\begin{figure}[ht]
\centerline{
\psfig{figure=normalstep.eps,angle=270,width=6.0cm,height=6.0cm,silent=}
\hspace{0.75cm}
\psfig{figure=normalstepa.eps,angle=270,width=6.0cm,height=6.0cm,silent=}
}
%\begin{picture}(120,0.1)(0,0)
\begin{picture}(120,0.1)(-10,0)
\put(29,54){\small The linearized constraint}
\put(28,55){\vector(-1,-1){8}}
\put(65.5,55){\vector(3,-1){19.5}}
\put(50,20){\small The trust region}
\put(49,21){\vector(-1,0){4}}
\put(74.5,21){\vector(1,0){4}}
\put(29,45){\small Nearest point on linearized constraint}
\put(27,36){\small $n_k$}
\put(107,43){\small Close to nearest point}
\put(95,29){\small $n_k$}
\end{picture}
\caption{\label{sqp-vardi-fig}
Computing the normal step. The left-hand figure shows the largest
possible normal step. The right-hand figure illustrates a shorter normal
step $n$, and the freedom this then allows for the tangential step---any
point on the dotted line is a potential tangential step.}
\end{figure}
\vspace*{0.3cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
{\bf \ref{csm}.1 \,\, Constraint relaxation---Vardi's method}
\vspace*{0.3cm}
\noindent
Vardi's approach is an early composite-step method.
The normal step is found by relaxing the requirement
\disp{A_k s = - c_k \tim{and} \|s\| \leq \Delta_k}
to
\disp{A_k n = - \sigma_k c_k \tim{and} \|n\| \leq \Delta_k,}
where $\sigma_k \in [0,1]$ is small enough so that
there is a feasible $n_k$.
Clearly $s = 0$ is feasible if $\sigma_k = 0$, and the
largest possible $\sigma_{\max}$ may be found by computing
\disp{ \max_{\sigma \in (0, 1]}
\left[ \min_{\| s \| \leq \Delta_k} \| A_k s + \sigma c_k \| = 0 \right]. }
In practice, some value between zero and $\sigma_{\max}$ is
chosen, since this gives some ``elbow-room'' in which to
compute the tangential step. The main defect with the
approach is that there may be no normal step if the
linearized constraints are inconsistent.
Once a normal step has been determined, the
tangential step is computed as the
\disp{\arr{rl}{\mbox{(approximate)}
\argminover{t \in \smallRe^n} &
\ip{t}{g_k + B_k n_k} + \half \ip{t}{B_k t} \\
\tim{subject to} & A_k t = 0 \tim{and} \|n_k + t\| \leq \Delta_k.}}
Although of historical interest, the method has been effectively
superseded by the Byrd--Omojokun approach we describe next.
\vspace*{0.3cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
{\bf \ref{csm}.2 \,\, Constraint reduction---the Byrd--Omojokun method}
\vspace*{0.3cm}
\noindent
The Byrd--Omojokun method aims to cope with the inconsistency issue
that afflicts Vardi's approach. Rather than relaxing the constraints,
the normal step is now computed as
\disp{\mbox{approximately} \minin{}
\| A_k n + c_k \| \tim{subject to} \|n\| \leq \Delta_k,}
in order to achieve a reasonable improvement in linearized infeasibility
that is consistent with the trust-region. The
tangential step is then computed exactly as in Vardi's method.
An important aspect is that it is possible to use
the conjugate gradient method to solve both subproblems. This
provides Cauchy points in both cases and allows the method to
be used to solve large problems. The method has been shown
to be globally convergent (under reasonable assumptions)
using an $\ell_2$ merit function, and is the
basis of the successful {\tt KNITRO} software package.
\vspace*{0.3cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
{\bf \ref{csm}.3 \,\, Constraint lumping---the Celis--Dennis--Tapia method}
\vspace*{0.3cm}
\noindent
A third method which might be considered to be of the composite-step variety
is that due to Celis, Dennis and Tapia. In this approach, the
requirement that $A_k s = - c_k$ is replaced by requiring that
\disp{\| A_k s + c_k \| \leq \sigma_k}
for some $\sigma_k \in [0,\|c_k\|]$. The value of $\sigma_k$ is chosen so
that the normal step $n_k$ satisfies
\disp{\| A_k n + c_k \| \leq \sigma_k \tim{and} \|n\| \leq \Delta_k.}
Having found a suitable normal step, the
tangential step is found as an
\disp{\arr{rl}{\mbox{(approximate)}
\argminover{t \in \smallRe^n} &
\ip{t}{g_k + B_k n_k} + \half \ip{t}{B_k t} \\
\tim{subject to} & \| A_k t + A_k n_k + c_k \| \leq \sigma_k
\tim{and} \|t + n_k\| \leq \Delta_k.}}
While finding a suitable $\sigma_k$ is inconvenient, the real Achilles' heel
of this approach is that the tangential step subproblem is (much) harder than
those we have considered so far. If the $\ell_2$-norm is used for
the constraints, we need to find the minimizer of a quadratic objective
within the intersection of two ``spherical'' regions. Unlike the
case involving a single sphere (recall Section~\ref{etrs}),
it is not known if there is an efficient algorithm in the two-sphere
case. Alternatively, if polyhedral ($\ell_1$ or $\ell_{\infty}$)
norms are used and $B_k$ is indefinite, the subproblem becomes a non-convex
quadratic program for which there is unlikely to be an efficient
general-purpose algorithm---in the special case where $B_k$ is positive
semi-definite and the $\ell_{\infty}$ norm is used, the subproblem
is a convex QP.
For this reason, the Celis--Dennis--Tapia approach is rarely used in practice.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new page %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\subsubsection{Filter methods}
\label{filter}
The last SQP method we shall consider is the most recent.
The approach taken is quite radical in that, unlike all of the
methods we have considered so far, it makes no use of a
merit function to force global convergence. The main
objection to merit functions is that they depend, to a large degree,
on arbitrary or a-priori unknown parameters. A secondary objection is
that they tend to be overly conservative in accepting promising
potential iterates. But if we wish to avoid merit functions, we
need some other device to encourage convergence. The new idea is to
use a ``filter''
Let $\theta(x) = \|c(x)\|$ be some norm of the constraint violation at $x$.
A \defn{filter} is a set of pairs $\{(\theta_k,f_k)\}$
of violations and objective values such that no member
dominates another, i.e., it does not happen that
\disp{ \theta_i \mbox{``$<$''} \theta_j \tim{and} f_i \mbox{``$<$''} f_j }
for any pair of filter points $i \neq j$---the ``$<$'' here informally means
``very slightly smaller than''. We illustrate a filter in
Figure~\ref{filter-pic}.
{\setlength{\unitlength}{0.7mm}
%{\setlength{\unitlength}{1.0mm}
\begin{figure}
\begin{center}
\begin{picture}(150,135)(0,-135,)
\put(0,-131){\vector(0,1){134}}
\put(-6,-97){0}
\put(-15,-5){$f(x)$}
\put(-10,-90){\vector(1,0){167}}
\put(145,-97){$\theta(x)$}
\put(15,-30){\best{\circle*{1.5}}} \put(16,-28){\best{1}}
\put(35,-65){\best{\circle*{1.5}}} \put(36,-64){\best{4}}
\put(45,-100){\best{\circle*{1.5}}} \put(46,-99){\best{2}}
\put(90,-115){\best{\circle*{1.5}}} \put(91,-114){\best{3}}
\thicklines
\put(15,-30){\best{\line(1,0){135}}}
\put(15,-30){\best{\line(0,1){30}}}
\put(35,-65){\best{\line(1,0){115}}}
\put(35,-65){\best{\line(0,1){65}}}
\put(45,-100){\best{\line(1,0){105}}}
\put(45,-100){\best{\line(0,1){100}}}
\put(90,-115){\best{\line(1,0){60}}}
\put(90,-115){\best{\line(0,1){115}}}
\thinlines
\put(13.5,-31.5){\env{\dashbox{0.2}(18,0){}}}
\put(13.5,-31.5){\env{\dashbox{0.2}(0,31.5){}}}
\put(31.5,-68.5){\env{\dashbox{0.2}(9,0){}}}
\put(31.5,-68.5){\env{\dashbox{0.2}(0,37){}}}
\put(40.5,-104.5){\env{\dashbox{0.2}(40.5,0){}}}
\put(40.5,-104.5){\env{\dashbox{0.2}(0,36){}}}
\put(81,-124){\env{\dashbox{0.2}(69,0){}}}
\put(81,-124){\env{\dashbox{0.2}(0,19.5){}}}
\end{picture}
\caption{\label{filter-pic} A filter with four entries.
}
\end{center}
\end{figure}
}
A potential new entry to the ``north-east'' of any of the
existing filter entries would not be permitted, and the
forbidden region is the intersection of the solid horizontal
and vertical lines emanating to the right and above each
filter point. For theoretical
reasons (akin to requiring sufficient decrease), we
slightly enlarge the forbidden region by putting a small
margin around each filter point, and this is illustrated in
the figure by the dotted lines.
And now it is clear how to use a filter. Any potential
SQP (or other) iterate $x_k + s_k$ will immediately be rejected if
it lies in the forbidden filter region accumulated during the
previous $k$ iterations. This may be embedded in a trust-region
framework, and a typical iteration might be as follows:
\vspace{0.2in}
\noindent
\centerline{\framebox[5.3in]{\hspace*{0.2in}\parbox{4.9in}{
\vspace*{0.2in}
If possible find
\disp{s_k = \argminover{s \in \smallRe^n} \ip{s}{g_k} + \half \ip{s}{B_k s}
\tim{subject to} A_k s = - c_k \tim{and} \|s\| \leq \Delta_k,}
but otherwise, find $s_k$ such that
\disp{\theta(x_k+s_k) \mbox{``$<$''} \theta_i \tim{for all} i \leq k.}
If $x_k+s_k$ is ``acceptable'' for the filter, set $x_{k+1} = x_k+s_k$
\\ \hspace*{6mm}and possibly add $(f((x_k+s_k), \theta(x_k+s_k))$ to the filter,
\\ \hspace*{6mm} ``prune'' the filter, and increase $\Delta_k$. \\
Otherwise reduce $\Delta_k$ and try again.
\vspace*{0.2in}
}}}
\vspace{0.2in}
\noindent
A few words of explanation are needed.
The trust-region and linearized constraints will always be compatible if
$c_k$ is small enough so long as they are at $c(x) = 0$.
Thus if the trust-region subproblem is incompatible,
one remedy is simply to move closer to the constraints. This
is known as a \defn{restoration} step. By ``pruning'' the filter,
we mean that a new point may completely dominate one or more
existing filter points and, in this case, the dominated entry
may be removed without altering the filter. For example, if a new
entry were accepted to the ``south-west'' of point 4 in our figure,
point 4 would be pruned.
While the basic filter idea is rather simple, in
practice, it is significantly more complicated than this. In particular,
there are theoretical reasons why some points that are acceptable
to the filter should still be rejected if any decrease in the
SQP model of the objective function is far from
realized in practice.
\section*{CONCLUSION}
\addcontentsline{toc}{section}{CONCLUSION}
We hope we have conveyed the impression that research into the
design, convergence and implementation of algorithms
for nonlinear optimization is an exciting and expanding area.
We have only been able to outline the developments in the
field, and have made no attempt to survey
the vast literature that has built up over the last 50 years.
Current algorithms for specialized problems like linear
and quadratic programming and unconstrained optimization
are well capable of solving problems involving millions of
unknowns (and constraints), while those for generally
constrained optimization routinely solve problems in the
tens and, perhaps even, hundreds of thousands of unknowns and
constraints. The next big goal is to be able to design
algorithms that have some hope of finding global optima
for large problems, the current state-of-the-art being
for problems with tens or hundreds of unknowns. Clearly
closing the gap between local and global optimization
has some way to go!
\section*{APPENDIX A - SEMINAL BOOKS AND PAPERS}
\addcontentsline{toc}{section}{APPENDIX A - SEMINAL BOOKS AND PAPERS}
The following books and papers are classics in the field.
Although many of them
cover topics outside the material we have described,
they are all worth reading. This section constitutes a
personal view of the most significant papers in the area.
It is not meant to be a complete bibliography.
\vspace*{4mm}
\noindent
{\Large \bf General text books}
\vspace*{4mm}
\noindent
There are a large number of text books devoted to nonlinear (and
even more for linear) programming. Those we find most useful and
which emphasize practical methods are
\refs{J. Dennis and R. Schnabel,
``Numerical Methods for Unconstrained Optimization and Nonlinear Equations'',
(republished by) SIAM (Classics in Applied Mathematics 16) (1996),
\vspace*{2mm} \\
R. Fletcher,
``Practical Methods of Optimization'',
2nd edition Wiley (1987), (republished in paperback 2000),
\vspace*{2mm} \\
P. Gill, W. Murray and M. Wright,
``Practical Optimization'', Academic Press (1981), and
\vspace*{2mm} \\
J. Nocedal and S. Wright,
``Numerical Optimization'', Springer Verlag (1999).}
The first of these concentrates on unconstrained optimization, while
the remainder cover (continuous) optimization in general.
\vspace*{4mm}
\noindent
{\Large \bf Early quasi-Newton methods}
\vspace*{4mm}
\noindent
These methods were introduced by
\refs{
W. Davidon, ``Variable metric method for minimization'', manuscript (1958),
finally published {\em SIAM J. Optimization} {\bf 1} (1991) 1:17,
}
and championed by
\refs{
R. Fletcher and M. J. D. Powell, ``A rapidly convergent descent method for
minimization'', {\em Computer J.} (1963) 163:168.
}
Although the so-called DFP method has been superseded by the more reliable
BFGS method, it paved the way for a number of classes of important updates.
\vspace*{4mm}
\noindent
{\Large \bf More modern quasi-Newton methods}
\vspace*{4mm}
\noindent
Coincidentally, all of the papers
\refs{C. G. Broyden, ``The convergence of a class of double-rank minimization
algorithms'', {\em J. Inst. Math. Applcs.}, {\bf 6} (1970) 76:90,
\vspace*{2mm} \\
R. Fletcher, ``A new approach to variable metric algorithms'',
{\em Computer J.} (1970) {\bf 13} (1970) 317:322,
\vspace*{2mm} \\
D. Goldfarb, ``A family of variable metric methods derived by variational
means'', {\em Math. Computation} {\bf 24} (1970) 23:26, and
\vspace*{2mm} \\
D. F. Shanno, ``Conditioning of quasi-Newton methods for function
minimization'', {\em Math. Computation} {\bf 24} (1970) 647:657
}
appeared in the same year. The aptly-named BFGS method has stood the test of
time well, and is still regarded as possibly the best secant updating formula.
\vspace*{4mm}
\noindent
{\Large \bf Quasi-Newton methods for large problems}
\vspace*{4mm}
\noindent
Limited memory methods are secant-updating methods that discard old
information so as to reduce the amount of storage required when
solving large problems. The methods first appeared in
\refs{J. Nocedal, ``Updating quasi-Newton matrices with limited storage'',
{\em Math. Computation} {\bf 35} (1980) 773:782, and
\vspace*{2mm} \\
A. Buckley and A. Lenir, ``{QN}-like variable storage conjugate gradients'',
{\em Math. Programming} {\bf 27} (1983) 155:175.}
Secant updating formulae proved to be less useful for large-scale computation,
but a successful generalization, applicable to what are known as partially
separable functions, was pioneered by
\refs{A. Griewank and Ph. Toint, ``Partitioned variable metric updates
for large structured optimization problems'',
{\em Numerische Mathematik} {\bf 39} (1982) 119:137, see also 429:448,
as well as
\vspace*{2mm} \\
A. Griewank and Ph. Toint, ``On the unconstrained optimization of partially
separable functions'', in {\em Nonlinear Optimization 1981} (Powell, M.,
ed.) Academic Press (1982)}
\vspace*{4mm}
\noindent
{\Large \bf Conjugate gradient methods for large problems}
\vspace*{4mm}
\noindent
Generalizations of Conjugate Gradient methods for non-quadratic minimization
were originally proposed by
\refs{R. Fletcher and C. M. Reeves, ``Function minimization by conjugate
gradients'', {\em Computer J.} (1964) 149:154, and
\vspace*{2mm} \\
E. Polak and G. Ribi\'{e}re, ``Note sur la convergence
de m\'{e}thodes de directions conjugu\'{e}es'',
{\em Revue Fran\c{c}aise d'informatique
et de recherche op\'{e}rationelle} {\bf 16} (1969) 35:43.}
An alternative is to attempt to solve the (linear)
Newton system by a conjugate-gradient like method. Suitable methods
for terminating such a procedure while still maintaining fast convergence
were proposed by
\refs{R. S. Dembo and T. Steihaug, ``Truncated-Newton algorithms for
large-scale unconstrained optimization'',
{\em Math. Programming} {\bf 26} (1983) 190:212.}
\vspace*{4mm}
\noindent
{\Large \bf Non-monotone methods}
\vspace*{4mm}
\noindent
While it is usual to think of requiring that the objective function decreases
at every iteration, this is not actually necessary for convergence so long as
there is some overall downward trend. The first method along these lines
was by
\refs{ L. Grippo, F. Lampariello and S. Lucidi, ``A nonmonotone line search
technique for Newton's method'',
{\em SIAM J. Num. Anal.}, {\bf 23} (1986) 707:716.}
\vspace*{4mm}
\noindent
{\Large \bf Trust-region methods}
\vspace*{4mm}
\noindent
The earliest methods that might be regarded as trust-region methods
are those by
\refs{K. Levenberg, ``A method for the solution of certain problems in least
squares'', {\em Quarterly J. Appl. Maths}, {\bf 2}
(1944) 164:168, and
\vspace*{2mm} \\
D. Marquardt, ``An algorithm for least-squares estimation of nonlinear
parameters'' {\em SIAM J. Appl. Maths}, {\bf 11} (1963) 431:441}
for the solution of nonlinear least-squares problems, although they
are motivated from the perspective of modifying indefinite Hessians
rather than restricting the step. Probably the first ``modern'' interpretation
is by
\refs{S. Goldfeldt, R. Quandt and H. Trotter,
``Maximization by quadratic hill-climbing'', {\em Econometrica}, {\bf 34}
(1966) 541:551.}
Certainly, the earliest proofs of convergence are given by
\refs{M. Powell, ``A New Algorithm for Unconstrained Optimization'',
in {\em Nonlinear Programming},
(Rosen, J., Mangasarian, O., and Ritter, K., eds.) Academic Press (1970),}
while a good modern introduction is by
\refs{J. Mor\'{e}, ``Recent developments in algorithms and software for trust
region methods'', in {\em Mathematical Programming: The State of the Art},
(Bachem, A., Gr\"{o}tschel, M., and Korte, B., eds.) Springer Verlag (1983).}
You might want to see our book
\refs{A. Conn, N. Gould and Ph. Toint, ``Trust-region methods'', SIAM (2000)}
for a comprehensive history and review of the large variety of articles on
trust-region methods.
\vspace*{4mm}
\noindent
{\Large \bf Trust-region subproblems}
\vspace*{4mm}
\noindent
Almost all you need to know about solving small-scale trust-region subproblems
is contained in the paper
\refs{J. Mor\'{e} and D. Sorensen, ``Computing a trust region step'',
{\em SIAM J. Sci. Stat. Comp.} {\bf 4} (1983) 533:572.}
Likewise
\refs{T. Steihaug, ``The conjugate gradient method and trust regions in large
scale optimization'', {\em SIAM J. Num. Anal.} {\bf 20}
(1983) 626:637}
provides the basic truncated conjugate-gradient approach used so
successfully for large-scale problems. More recently\footnote{We would
hate to claim ``seminal'' status for one of our own papers!}
\refs{N. Gould, S. Lucidi, M. Roma and Ph. Toint, ``Solving the
trust-region subproblem using the Lanczos method'',
{\em SIAM J. Optimization} {\bf 9} (1999) 504:525}
show how to improve on Steihaug's approach by moving around the trust-region
boundary. A particularly nice new paper by
\refs{Y. Yuan, ``On the truncated conjugate-gradient method'',
{\em Math. Programming}, {\bf 87} (2000) 561:573}
proves that Steihaug's approximation gives at least 50\% of the
optimal function decrease when applied to convex problems.
\noindent
\vspace*{4mm}
\noindent
{\Large \bf The Symmetric Rank-One quasi-Newton approximation}
\vspace*{4mm}
\noindent
Since trust-region methods allow non-convex models, perhaps the simplest
of all Hessian approximation methods, the Symmetric Rank-One update,
is back in fashion. Although it is unclear who first suggested the method,
\refs{C. Broyden, ``Quasi-Newton methods and their application to
function minimization'', {\em Math. Computation} {\bf 21} (1967) 577:593}
is the earliest reference that we know of. Its revival in fortune
is due\footnote{See previous footnote \ldots} to
\refs{A. Conn, N. Gould and Ph. Toint, ``Convergence of quasi-Newton
matrices generated by the Symmetric Rank One update''
{\em Math. Programming}, {\bf 50} (1991) 177:196
(see also {\em Math. Comp.} {\bf 50} (1988) 399:430), and
\vspace*{2mm} \\
R. Byrd, H. Khalfan and R. Schnabel
``Analysis of a symmetric rank-one trust region method''
{\em SIAM J. Optimization} {\bf 6} (1996) 1025:1039,}
and it has now taken its place alongside the BFGS method as the
pre-eminent updating formula.
\noindent
\vspace*{4mm}
\noindent
{\Large \bf More non-monotone methods}
\vspace*{4mm}
\noindent
Non-monotone methods have also been proposed in the trust-region case.
The basic reference here is the paper by
\refs{Ph. Toint, ``A non-monotone trust-region algorithm for nonlinear
optimization subject to convex constraints'',
{\em Math. Programming}, {\bf 77} (1997) 69:94.}
\vspace*{4mm}
\noindent
{\Large \bf Barrier function methods}
\vspace*{4mm}
\noindent
Although they appear to have originated in a pair of unpublished University
of Oslo technical reports by K. Frisch in the mid 1950s, (logarithmic)
barrier function were popularized by
\refs{A. Fiacco and G. McCormick, ``The sequential unconstrained minimization
technique for nonlinear programming: a primal-dual method'',
{\em Management Science} {\bf 10} (1964) 360:366;
see also {\em ibid} (1964) 601:617.}
A full early history is given in the book
\refs{A. Fiacco and G. McCormick, ``Nonlinear programming: sequential
unconstrained minimization techniques'' (1968), republished as
{\em Classics in Applied Mathematics 4}, SIAM (1990).}
The worsening conditioning of the Hessian was first highlighted by
\refs{F. Lootsma, ``Hessian matrices of penalty functions for solving
constrained optimization problems'', {\em Philips Research Reports}, {\bf 24}
(1969) 322:331, and
\vspace*{2mm} \\
W. Murray, ``Analytical expressions for eigenvalues and eigenvectors of
the Hessian matrices of barrier and penalty functions'',
{\em J. Optimization Theory and Applications}, {\bf 7} (1971) 189:196,}
although recent work by
\refs{M. Wright, ``Ill-conditioning and computational error in interior
methods for nonlinear programming'',
{\em SIAM J. Optimization} {\bf 9} (1999) 84:111, and
\vspace*{2mm} \\
S. Wright, ``Effects of finite-precision arithmetic on interior-point
methods for nonlinear programming'',
{\em SIAM J. Optimization} {\bf 12} (2001) 36:78
}
demonstrates that this ``defect'' is far from fatal.
\vspace*{4mm}
\noindent
{\Large \bf Interior-point methods}
\vspace*{4mm}
\noindent
The interior-point revolution was started by
\refs{N. Karmarkar, ``A new polynomial-time algorithm for linear programming'',
{\em Combinatorica} {\bf 4} (1984) 373:395.}
It did not take long for
\refs{P. Gill, W. Murray, M. Saunders, J. Tomlin and M. Wright,
``On projected Newton barrier methods for linear
programming and an equivalence to Karmarkar's projective method'',
{\em Math. Programming}, {\bf 36} (1986) 183:209}
to realize that this radical ``new'' approach was actually something that
nonlinear programmers had tried (but, most unfortunately, discarded)
in the past.
\vspace*{4mm}
\noindent
{\Large \bf SQP methods}
\vspace*{4mm}
\noindent
The first SQP method was proposed in the overlooked 1963 Harvard Master's
thesis of R. Wilson. The generic linesearch SQP method is that of
\refs{B. Pschenichny, ``Algorithms for general problems of mathematical
programming'', {\em Kibernetica}, {\bf 6} (1970) 120:125,}
while there is a much larger variety of trust-region SQP methods, principally
because of the constraint incompatibility issue.
\vspace*{4mm}
\noindent
{\Large \bf Merit functions for SQP}
\vspace*{4mm}
\noindent
The first use of an exact penalty function to globalize the SQP method
was by
\refs{S. Han, ``A globally convergent method for nonlinear programming'',
{\em J. Optimization Theory and Applics}, {\bf 22} (1977) 297:309, and
\vspace*{2mm} \\
M. Powell, ``A fast algorithm for nonlinearly constrained optimization
calculations'', in
{\em Numerical Analysis, Dundee 1977} (G. Watson, ed) Springer Verlag (1978)
144:157.}
The fact that such a merit function may prevent full SQP steps was
observed N. Maratos in his 1978 U. of London Ph. D. thesis, while methods
for combating the Maratos effect were subsequently proposed by
\refs{R. Fletcher, ``Second-order corrections for non-differentiable
optimization'', in
{\em Numerical Analysis, Dundee 1981} (G. Watson, ed) Springer Verlag (1982)
85:114, and
\vspace*{2mm} \\
R. Chamberlain, M. Powell, C. Lemar\'{e}chal, and H. Pedersen,
``The watchdog technique for forcing convergence in
algorithms for constrained optimization'',
{\em Math. Programming Studies}, {\bf 16} (1982) 1:17.}
An SQP method that avoids the need for a merit function altogether
by staying feasible is given by %\newpage
\refs{E. Panier and A. Tits, ``On combining feasibility,
descent and superlinear convergence in inequality
constrained optimization'',{\em Mathematical Programming},
{\bf 59} (1992) 261;276.}
%\refs{J. Bonnans, E. Panier, A. Tits, and J. Zhou, ``Avoiding the Maratos
% effect by means of a nonmonotone linesearch {II}. Inequality constrained
% problems---feasible iterates'',
% {\em SIAM J. Num. Anal.}, {\bf 29} (1992) 1187:1202.}
\vspace*{4mm}
\noindent
{\Large \bf Hessian approximations}
\vspace*{4mm}
\noindent
There is a vast literature on suitable Hessian approximations for
use in SQP methods. Rather than point at individual papers, a good
place to start is
\refs{P. Boggs and J. Tolle, ``Sequential quadratic programming'',
{\em Acta Numerica} {\bf 4} (1995) 1:51,}
but see also our paper
\refs{N. Gould and Ph. Toint, ``SQP methods for large-scale nonlinear
programming'', in {\em System modelling and optimization,
methods, theory and applications} (M. Powell and S. Scholtes, eds.)
Kluwer (2000) 149:178.}
\vspace*{4mm}
\noindent
{\Large \bf Trust-region SQP methods}
\vspace*{4mm}
\noindent
Since the trust-region and the linearized constraints may be
incompatible, almost all trust-region SQP methods modify the basic
SQP method in some way. The S$\ell_1$QP method is due to
\refs{R. Fletcher, ``A model algorithm for composite non-differentiable
optimization problems'',
{\em Math. Programming Studies}, {\bf 17} (1982) 67:76.}
Methods that relax the constraints include those proposed by
\refs{A. Vardi, ``A trust region algorithm for equality constrained
minimization: convergence properties and implementation'',
{\em SIAM J. Num. Anal.}, {\bf 22} (1985) 575:591, and
\vspace*{2mm} \\
M. Celis, J. Dennis and R. Tapia, ``A trust region strategy for nonlinear
equality constrained optimization'', in
{\em Numerical Optimization 1984} (P. Boggs, R. Byrd and R. Schnabel, eds),
SIAM (1985) 71:82,}
as well as a method that appeared in the 1989 U. of Colorado at Boulder
Ph. D. thesis of E. Omojokun, supervised by R. Byrd. The Filter-SQP
approach may be
found in\footnote{Once again, see previous footnote \ldots}
\refs{R. Fletcher and S. Leyffer, ``Nonlinear programming without a
penalty function'', {\em Math. Programming}, {\bf 91} (2002) 239:269.}
\vspace*{4mm}
\noindent
{\Large \bf Modern methods for nonlinear programming}
\vspace*{4mm}
\noindent
Many modern methods for nonlinearly constrained optimization tend to be
SQP-interior-point hybrids. A good example is due to
\refs{R. Byrd, J. Gilbert and J. Nocedal, ``A trust region method based on
interior point techniques for nonlinear programming'',
{\em Math. Programming A} {\bf 89} (2000) 149:185,}
and forms the basis for the excellent KNITRO package.
\section*{APPENDIX B - OPTIMIZATION RESOURCES ON THE WORLD-WIDE-WEB}
\addcontentsline{toc}{section}{APPENDIX B - OPTIMIZATION RESOURCES ON THE WORLD-WIDE-WEB}
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\section*{\thesection \,\, Answering questions on the web}
A good starting point for finding out more about optimization are the
two lists of Frequently Asked Questions (FAQs) on optimization.
The Linear Programming FAQ,
\www{www-unix.mcs.anl.gov/otc/Guide/faq/linear-programming-faq.html}{ ,}
is dedicated to question on linear optimization problems as well as certain
aspects of mixed integer linear programming. The Nonlinear Programming FAQ,
\www{www-unix.mcs.anl.gov/otc/Guide/faq/nonlinear-programming-faq.html}{ ,}
offers a concise introduction to nonlinear optimization.
The NEOS guide,
\www{www-fp.mcs.anl.gov/otc/Guide}{ ,}
provides an overview of
optimization and the solvers available. It contains the optimization tree,
\www{www-fp.mcs.anl.gov/otc/Guide/OptWeb}{ ,}
a dichotomy of optimization problems.
Both sites are maintained by the Optimization Technology Center
\www{www.ece.nwu.edu/OTC}{ ,}
a loose collaboration between Argonne National
Laboratory and Northwestern University in the USA.
Hans Mittelmann of Arizona State University maintains a decision tree for
optimization software,
\www{plato.la.asu.edu/guide.html}{ ,}
and he also provides a useful set of benchmarks for
optimization software,
\www{plato.la.asu.edu/bench.html}{ .}
%\vspace*{-5mm}
Harvey Greenberg's Mathematical Programming Glossary,
\www{www.cudenver.edu/~hgreenbe/glossary/glossary.html}{ }
contains brief definitions of commonly used expressions
in optimization and operations research.
The usenet newsgroup
\www{sci.op-research}{ }
is dedicated to answering questions on
optimization and operations research. Brian Borchers edits a weekly
digest of postings to it. You can receive the digest by sending an
email to \www{listserv@listserv.okstate.edu}{ } with the
message
\www{SUBSCRIBE ORCS-L Your Name}{ .}
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\addtocounter{section}{1}
\section*{\thesection \,\, Solving optimization problems on the web}
%%%=========================================================================%%%
\addtocounter{subsection}{-3}
\subsection*{\thesubsection \,\, The NEOS server}
Probably the most important and useful optimization site on the web is the
NEOS server\footnote{J. Czyzyk, M. Mesnier and J. Mor\'{e}.
The {NEOS} server. {\em IEEE Journal on Computational Science and Engineering},
5:68--75, 1998.}
at
\www{www-neos.mcs.anl.gov/neos}{ }
which allows you to solve optimization problems
over the internet. NEOS handles several thousand (!) submissions
per week. The server provides a wide choice of state-of-the-art
optimization software which can be used remotely without the need to
install or maintain any software.
The problems should preferably be formulated in a modelling language such
as AMPL\footnote{R. Fourer, D. Gay and B. Kernighan.
{\em {AMPL}: A modelling Language for Mathematical Programming}.
Boyd \& Fraser Publishing Company, Massachusetts, 1993.}
or
GAMS\footnote{A. Brooke, D. Kendrick, A. Meeraus and R. Raman.
{\em {GAMS} A user's guide}.
{GAMS} Developments Corporation, 1217 Potomac Street, N.W.,
Washington DC 20007, USA, December 1998.}
(see Section~B.2.3).
However, some solvers also accept problem descriptions in other
formats such as C or fortran source code or the verbose
linear programming MPS format.
There are a number of solvers implementing algorithms for nonlinearly
constrained optimization problems. Most are hybrids, and thus capable
of handling both equality and inequality constraints.
There are at least three interior point solvers (see Section~\ref{ipm}).
\begin{description}
\item
KNITRO (with a silent ``K''), is a primal-dual interior-point
method which uses trust regions.
\item
LOQO is based on an infeasible primal-dual interior-point method.
It uses a linesearch and a version of a filter to enforce
global convergence.
\item
MOSEK can only be used to solve {\em convex\/}
large-scale smooth nonlinear optimization problems.
It does not work for {\em nonconvex\/} problems.
\end{description}
There are at least three solvers implementing SQP algorithms (see
Section~\ref{sqp}).
\begin{description}
\item
DONLP2 implements a linesearch SQP algorithm with an exact
non-differentiable $\ell_1$-penalty function as a merit function.
It uses dense linear algebra.
\item
FILTER implements a trust-region SQP algorithm which is suitable
for solving large nonlinearly constrained problems with small
degrees of freedom. It uses a filter (see Section~\ref{filter})
to promote global convergence.
\item
SNOPT implements a linesearch SQP algorithm which uses an
augmented Lagrangian as a merit function. It maintains a positive
definite limited memory approximation of the Hessian of the
Lagrangian.
\end{description}
There is also a range of other solvers not covered in this article.
\begin{description}
\item
CONOPT is a feasible path method based on the generalized
reduced gradient algorithm.
\item
LANCELOT implements an augmented Lagrangian algorithm. It uses
a trust-region to promote global convergence.
\item
MINOS implements a sequential linearly constrained algorithm.
Steplength control is heuristic (for want of a suitable merit
function), but superlinear convergence is often achieved.
\item
PATHNLP finds stationary points for the nonlinear problem by solving
the Karush-Kuhn-Tucker conditions (see
Theorems~\ref{bao-econ-1stordernec} and \ref{bao-con-1stordernec}),
% Exercise 1.1),
written as a mixed complementarity problem, using the PATH solver.
\end{description}
Consult the
NEOS guide (see Section~B.1) for appropriate contacts.
A wide range of other optimization problems can also be solved such as
semi-infinite optimization, mixed integer linear and nonlinear optimization,
semidefinite optimization, complementarity problems,
non-differentiable optimization, and unconstrained and
stochastic optimization problems. The fact that the server maintains
state-of-the-art optimization software makes is suitable for medium to
large scale applications.
Users with their own copy of the modelling systems AMPL or GAMS can even
invoke the NEOS solvers out of their local AMPL or GAMS session using KESTREL,
\www{www-neos.mcs.anl.gov/neos/kestrel.html}{ .}
This is very convenient as it makes it possible to post- or pre-process the
models using a local copy of the modelling tool.
%%%=========================================================================%%%
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\subsection*{\thesubsection \,\, Other online solvers}
The system www-Nimbus, from
\www{nimbus.mit.jyu.fi}{ ,}
is designed to solve (small) multi-objective optimization problems.
It consists of a sequence of
menus to input the multi-objective problem as well as some facilities
for displaying the solution. It requires the user to interactively
guide the optimization and requires some familiarity with
multi-objective
terminology. An online tutorial guides the user through the process.
Certain topology optimization problems can be solved at
\www{www.topopt.dtu.dk}{ }
The input is via a GUI and the solution is also display graphically.
The system Baron, \www{archimedes.scs.uiuc.edu/baron/availability.html}{ ,}
allows the solution of small global optimization problems online.
%%%=========================================================================%%%
\addtocounter{subsection}{1}
\subsection*{\thesubsection \,\, Useful sites for modelling problems prior to online solution}
AMPL (A Mathematical Programming Language)
\www{www.ampl.com}{ } is a modelling language for optimization problems.
The site lists extensions to the book, allows the solution of example
models and contains a list of available solvers.
Further AMPL models can be found at the following sites: \\
\hspace*{5mm} NLP models by Bob Vanderbei:
\www{www.sor.princeton.edu/$\sim$rvdb/ampl/nlmodels}{ .}
\hspace*{5mm} MINLP and MPEC models by Sven Leyffer:
\www{www.maths.dundee.ac.uk/$\sim$sleyffer/MacMINLP}{ and}
\vspace*{-7mm}
\www{www.maths.dundee.ac.uk/$\sim$sleyffer/MacMPEC}{ .}
\hspace*{5mm} The COPS collection of Jorge Mor\'{e}:
\www{www-unix.mcs.anl.gov/$\sim$more/cops}{ .}
These sites are especially useful to help with your own modelling exercises.
GAMS (the General Algebraic Modelling System)
\www{www.gams.com}{ } is another modelling language. The site
contains documentation on GAMS and some example models.
More GAMS models can be found on the GAMS-world pages. These are
sites, dedicated to important modelling areas, see
\www{www.gamsworld.org}{ .} It also offers a translation service
from one modelling language to another.
Recently, optimization solvers have also been interfaced to {\tt matlab} at
\www{tomlab.biz/}{ .}
%%%=========================================================================%%%
\addtocounter{subsection}{1}
\subsection*{\thesubsection \,\, Free optimization software}
An extension of MPS to nonlinear optimization, SIF (standard input format),
can be used to model optimization problems. The reference document can be found at
\www{www.numerical.rl.ac.uk/lancelot/sif/sifhtml.html}{ .} A collection of
optimization problems in SIF is available at
{\sf CUTEr} can be found via
\www{www.cse.clrc.ac.uk/Activity/CUTEr}{ .}
Two solvers, {\sf LANCELOT}
\www{www.cse.clrc.ac.uk/Activity/LANCELOT}{ } and {\sf GALAHAD}
\www{www.cse.clrc.ac.uk/Activity/GALAHAD}{ }
are available freely for non-commercial users.
AMPL and some solvers are also available freely in limited size student
versions,
which allow the solution of problems of up to 300 variables and constraints, see
\www{netlib.bell-labs.com/netlib/ampl/student/}{ .}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\addtocounter{section}{1}
\section*{\thesection \,\, Optimization reports on the web}
Optimization online, \www{www.optimization-online.org}{ ,}
is an e-print site for papers on optimization. It is sponsored by the
Mathematical Programming
Society. It allows you to search for preprints on certain subjects. A
monthly digest summarizes all monthly submissions.
The two main optimization journals,
Mathematical Programming and SIAM Journal on Optimization maintain
free sites with access to titles and abstracts, see
\www{link.springer.de/link/service/journals/10107/}{, }
and
\www{www.siam.org/journals/siopt/siopt.htm}{ .}
\section*{APPENDIX C - SKETCHES OF PROOFS}
\addcontentsline{toc}{section}{APPENDIX C - SKETCHES OF PROOFS}
\setcounter{lecture}{1}
\setcounter{section}{\thelecture}
\addtocounter{section}{-1}
\setcounter{equation}{0}
\renewcommand{\theequation}{C.\arabic{equation}}
\setcounter{figure}{0}
\renewcommand{\thefigure}{C.\arabic{figure}}
%
% proofs of results in lecture 1
%
%\section{Sketches of proofs for Section~\thelecture}
Theorems \thelecture.1---\thelecture.2 can be found in any good book
on analysis.
Theorems \thelecture.1 and \thelecture.2 follow
directly by considering the remainders of truncated Taylor expansions of
the univariate function $f(x+\alpha s)$ with $\alpha \in [0,1]$,
while Theorem \thelecture.2 uses the Newton formula
\disp{F(x+s) = F(x) + \int_0^1 \nabla_x F(x+\alpha s)s d \alpha.}
\subsection*{Proof of Theorem \thelecture.4}
Suppose otherwise, that $g(x_*) \neq 0$. A Taylor expansion in the direction
$-g(x_*)$ gives
\disp{f(x_* - \alpha g(x_*) ) = f(x_*) - \alpha \|g(x_*)\|^2 + O( \alpha^2).}
For sufficiently small $\alpha$, $\half \alpha \|g(x_*)\|^2 \geq O( \alpha^2)$,
and thus
\disp{f(x_* - \alpha g(x_*) ) \leq f(x_*) - \half \alpha \|g(x_*)\|^2 < f(x_*).}
This contradicts the hypothesis that $x_*$ is a local minimizer.
\subsection*{Proof of Theorem \thelecture.5}
Again, suppose otherwise that $\ip{s}{H(x_*) s} < 0$. A Taylor expansion in the
direction $s$ gives
\disp{f(x_*+\alpha s) = f(x_*) + \half \alpha^2 \ip{s}{H(x_*) s} + O(\alpha^3),}
since $g(x_*) = 0$.
For sufficiently small $\alpha$, $-\quarter \alpha^2 \ip{s}{H(x_*) s} \geq
O(\alpha^3)$, and thus
\disp{f(x_* + \alpha s ) \leq f(x_*) + \quarter \alpha^2 \ip{s}{H(x_*) s}
< f(x_*).}
Once again, this contradicts the hypothesis that $x_*$ is a local minimizer.
\subsection*{Proof of Theorem \thelecture.6}
By continuity $H(x)$ is positive definite for all $x$ in a open ball
$\calN$ around $x_*$. The generalized mean-value theorem then says that if
$x_* + s \in \calN$, there is a value $z$ between the points $x_*$
and $x_* + s$ for which
\disp{f(x_* + s ) = f(x_*) + \ip{s}{g(x_*)} + \half \ip{s}{H(z) s}
= f(x_*) + \half \ip{s}{H(z) s} > f(x_*)}
for all nonzero $s$, and thus $x_*$ is an isolated local minimizer.
\subsection*{Proof of Theorem \thelecture.7}
We consider feasible perturbations about $x_*$. Consider a vector valued $C^2$
($C^3$ for Theorem \thelecture.8) function $x(\alpha)$ of the scalar
$\alpha$ for which $x(0) = x_*$ and $c(x(\alpha)) = 0$.
(The constraint qualification is that all such feasible
perturbations are of this form). We may then write
\eqn{xalpha}{ x(\alpha) = x_* + \alpha s + \half \alpha^2 p + O(\alpha^3)}
and we require that
\disp{ \arr{rl}{
0 \;\; = & c_i(x(\alpha)) = c_i(x_* + \alpha s + \half \alpha^2 p + O(\alpha^3)) \\
= & c_i(x_*) + \ip{a_i(x_*)}{\alpha s + \half \alpha^2 p}
+ \half \alpha^2 \ip{s}{H_i(x_*) s} + O(\alpha^3) \\
= & \alpha \ip{a_i(x_*)}{s} + \half \alpha^2 \left(
\ip{a_i(x_*)}{p} + \ip{s}{H_i(x_*) s} \right ) + O(\alpha^3)}}
using Taylor's theorem. Matching similar asymptotic terms,
this implies that for such a feasible perturbation
\eqn{pert1}{A(x_*) s = 0}
and
\eqn{pert2}{\ip{a_i(x_*)}{p} + \ip{s}{H_i(x_*) s} = 0}
for all $i = 1, \ldots , m$.
Now consider the objective function
\eqn{fexpan}{\arr{rl}{f(x(\alpha)) \;\;
= & f(x_* + \alpha s + \half \alpha^2 p + O(\alpha^3) ) \\
= & f(x_*) + \ip{g(x_*)}{\alpha s + \half \alpha^2 p}
+ \half \alpha^2 \ip{s}{H(x_*) s} + O(\alpha^3) \\
= & f(x_*) + \alpha \ip{g(x_*)}{s} +
\half \alpha^2 \left( \ip{g(x_*)}{p} + \ip{s}{H(x_*) s} \right)
+ O(\alpha^3).}}
This function is unconstrained along $x(\alpha)$, so we may
deduce, as in Theorem \thelecture.4, that
\eqn{opt1}{ \ip{g(x_*)}{s} = 0 \tim{for all} s \tim{such that} A(x_*) s = 0.}
If we let $S$ be a basis for the null-space of $A(x_*)$, we may write
\eqn{opt1alt}{g(x_*) = A^T(x_*) y_* + S z_* }
for some $y_*$ and $z_*$. Since, by definition, $A(x_*) S = 0$,
and as it then follows from \req{opt1} that $g^T(x_*) S = 0$,
we have that
\disp{0 = S^T g(x_*) = S^T A^T(x_*) y_* + S^T S z_* = S^T S z_*.}
Hence $S^T S z_* = 0$ and thus $z_* = 0$ since $S$ is of full rank.
Thus \req{opt1alt} gives
\eqn{1stoo}{g(x_*) - A^T(x_*) y_* = 0.}
\subsection*{Proof of Theorem \thelecture.8}
We have shown that
\eqn{fexpan2}{f(x(\alpha)) = f(x_*) +
\half \alpha^2 \left( \ip{p}{g(x_*)} + \ip{s}{H(x_*) s} \right)
+ O(\alpha^3)}
for all $s$ satisfying $A(x_*)s = 0$, and that \req{1stoo} holds.
Hence, necessarily,
\eqn{2nd}{ \ip{p}{g(x_*)} + \ip{s}{H(x_*) s} \geq 0}
for all $s$ and $p$ satisfying \req{pert1} and \req{pert2}.
But \req{1stoo} and \req{pert2} combine to give
\disp{ \ip{p}{g(x_*)} = \sum_{i=1}^m (y_*)_i \ip{p}{a_i(x_*)}
= -\sum_{i=1}^m (y_*)_i \ip{s}{H_i(x_*) s}}
and thus \req{2nd} is equivalent to
\disp{ \bip{s}{\left( H(x_*) - \sum_{i=1}^m (y_*)_i H_i(x_*) \right) s}
\equiv \ip{s}{H(x_*,y_*) s} \geq 0}
for all $s$ satisfying \req{pert1}.
\subsection*{Proof of Theorem \thelecture.9}
As in the proof of Theorem \thelecture.7, we consider feasible
perturbations about $x_*$.
Since any constraint that is inactive at $x_*$ (i.e., $c_i(x_*) > 0$) will
remain inactive for small perturbations, we need only consider perturbations
that are constrained by the constraints active at $x_*$, (i.e., $c_i(x_*) = 0$).
Let $\calA$ denote the indices of the active constraints. We then
consider a vector valued $C^2$ ($C^3$ for Theorem \thelecture.10)
function $x(\alpha)$ of the scalar
$\alpha$ for which $x(0) = x_*$ and $c_i(x(\alpha)) \geq 0$ for $i \in \calA$.
In this case, assuming that $x(\alpha)$ may be expressed as \req{xalpha},
we require that
\disp{ \arr{rl}{
0 \; \leq & c_i(x(\alpha)) = c(x_* + \alpha s + \half \alpha^2 p +
O(\alpha^3)) \\
= & c_i(x_*) + \ip{a_i(x_*)}{\alpha s + \half \alpha^2 p}
+ \half \alpha^2 \ip{s}{H_i(x_*) s} + O(\alpha^3) \\
= & \alpha \ip{a_i(x_*)}{s} + \half \alpha^2
\left( \ip{a_i(x_*)}{p} + \ip{s}{H_i(x_*) s} \right ) + O(\alpha^3)}}
for all $i \in \calA$.
Thus
\eqn{pert1b}{\ip{s}{a_i(x_*)} \geq 0}
and
\eqn{pert2b}{\ip{p}{a_i(x_*)} + \ip{s}{H_i(x_*) s} \geq 0
\tim{when} \ip{s}{a_i(x_*)} = 0}
for all $i \in \calA$.
The expansion of $f(x(\alpha))$ \req{fexpan} then implies that $x_*$
can only be a local minimizer if
\disp{\calS =
\{ s \st \ip{s}{g(x_*)}< 0 \tim{and} \ip{s}{a_i(x_*)} \geq 0 \tim{for}
i \in \calA \} = \emptyset.}
But then the result follows directly from Farkas' Lemma---a proof of this
famous result is given, for example, as
Lemma 9.2.4 in
%\begin{center}
%\parbox{5.5in}{
R. Fletcher
``Practical Methods of Optimization'', Wiley
(1987, 2nd edition).
%}
%\end{center}
\vspace{\baselineskip}
\noindent\framebox[\textwidth]{\parbox{0.95\textwidth}{
\vspace{\baselineskip}
{\bf Farkas' Lemma.}
Given any vectors $g$ and $a_i$, $i \in \calA$,
the set
\disp{\calS = \{ s \st \ip{s}{g} < 0 \tim{and} \ip{s}{a_i} \geq 0 \tim{for}
i \in \calA \} }
is empty if and only if
%\eqn{lc}
\disp{ g = \sum_{i \in \calA} y_i a_i }
for some $y_i \geq 0$, $i \in \calA$
\vspace*{\baselineskip} }
} \vspace*{\baselineskip}
%\noindent
%{\bf Proof of Lemma.}
%Suppose that \req{lc} holds with $y_i \geq 0$ and that
%$\ip{s}{a_i} \geq 0$ for $i \in \calA$. Then
%\disp{ \ip{s}{g} = \sum_{i \in \calA} y_i \ip{s}{a_i} \geq 0.}
%Hence $\calS$ is empty, since $\ip{s}{g}$ cannot be negative.
%Conversely, suppose that $\calS$ is non-empty, and that
%\req{lc} holds. Then, again,
%\disp{ 0 > \ip{s}{g} = \sum_{i \in \calA} y_i \ip{s}{a_i}.}
%Since $\ip{s}{a_i} \geq 0$ for $i \in \calA$, it then must be
%that at least one of the $y_i < 0$.
%\noindent\fbox{\parbox{\linewidth}{Nick: Is that last part enough?
%What, if \req{lc} does not hold?}}
\subsection*{Proof of Theorem \thelecture.10}
The expansion \req{fexpan} for the change in
the objective function will be dominated by the first-order
term $\alpha \ip{s}{g(x_*)}$ for feasible perturbations unless
$\ip{s}{g(x_*)} = 0$, in which case the expansion \req{fexpan2}
is relevant. Thus we must have that \req{2nd} holds for
all feasible $s$ for which $\ip{s}{g(x_*)} = 0$. The latter
requirement gives that
\disp{ 0 = \ip{s}{g(x_*)} = \sum_{i \in \calA} y_i \ip{s}{a_i(x_*)} ,}
and hence that either $y_i = 0$ or $\ip{s}{a_i(x_*)} = 0$ (or both).
We now focus on the {\em subset} of all feasible arcs that ensure
$c_i(x(\alpha)) = 0$ if $y_i > 0$ and $c_i(x(\alpha)) \geq 0$ if $y_i = 0$
for $i \in \calA$. For those constraints for which $c_i(x(\alpha) = 0$,
we have that \req{pert1} and \req{pert2} hold, and thus for such perturbations
$s \in \calN_+$. In this case
\disp{\ip{p}{g(x_*)} = \sum_{i \in \calA} y_i \ip{p}{a_i (x_*)}
= \sum_{\stackrel{\scriptstyle i \in \calA}{y_i > 0}}
y_i \ip{p}{a_i (x_*)}
= -
\sum_{\stackrel{\scriptstyle i \in \calA}{y_i > 0}}
y_i \ip{s}{H_i(x_*) s}
= - \sum_{i \in \calA} y_i \ip{s}{H_i(x_*) s}}
This combines with \req{2nd} to give that
\disp{ \ip{s}{H(x_*,y_*) s} \equiv
\bip{s}{\left( H(x_*) - \sum_{i=1}^m (y_*)_i H_i(x_*) \right) s} =
\ip{p}{g(x_*)} + \ip{s}{H(x_*) s} \geq 0.}
for all $s \in \calN_+$, which is the required result.
\subsection*{Proof of Theorem \thelecture.11}
Consider any feasible arc $x(\alpha)$. We have seen that
\req{pert1b} and \req{pert2b} hold, and that first-order feasible perturbations
are characterized by $\calN_+$.
It then follows from \req{pert2b} that
\disp{\ip{p}{g(x_*)} = \sum_{i \in \calA} y_i \ip{p}{a_i (x_*)}
= \!\!\!\!\!\!\! \sum_{\stackrel{\scriptstyle i \in \calA}{\ip{s}{a_i (x_*)}
= 0}}
\!\!\!\!\!\!\! y_i \ip{p}{a_i (x_*)}
\geq - \!\!\!\!\!\!\!
\sum_{\stackrel{\scriptstyle i \in \calA}{\ip{s}{a_i (x_*)} = 0}}
\!\!\!\!\!\!\! y_i \ip{s}{H_i(x_*) s}
= - \sum_{i \in \calA} y_i \ip{s}{H_i(x_*) s},}
and hence by assumption that
\disp{
\ip{p}{g(x_*)} + \ip{s}{H(x_*) s} \geq
\bip{s}{\left( H(x_*) - \sum_{i=1}^m (y_*)_i H_i(x_*) \right) s}
\equiv \ip{s}{H(x_*,y_*) s} > 0}
for all $s \in \calN_+$. But this then combines with \req{fexpan} and
\req{pert1b} to show that $f(x(\alpha)) > f(x_*)$ for all sufficiently
small $\alpha$.
\setcounter{lecture}{2}
\setcounter{section}{\thelecture}
\addtocounter{section}{-1}
%
% proofs of results in lecture 2
%
\subsection*{Proof of Theorem \thelecture.1}
From Taylor's theorem (Theorem 1.1), and using the bound
\disp{\alpha \leq \frac{2( \beta - 1 ) \ip{p}{g(x)}}{\gamma(x) \|p\|_2^2},}
we have that
\disp{\arr{rl}{
f(x+\alpha p) \hspace*{-2mm}
& \leq f(x) + \alpha \ip{p}{g(x)} + \half \gamma(x) \alpha^2 \|p\|^2 \\
& \leq f(x) + \alpha \ip{p}{g(x)} + \alpha ( \beta - 1 ) \ip{p}{g(x)} \\
& = f(x) + \alpha \beta \ip{p}{g(x)}.}}
\subsection*{Proof of Corollary \thelecture.2}
Theorem \thelecture.1 shows that the linesearch will terminate
as soon as $\alpha^{(l)} \leq \alpha_{\max}$. There are two cases
to consider. Firstly, it may be that $\alpha_{\mbox{init}}$
satisfies the Armijo condition, in which case
$\alpha_k = \alpha_{\mbox{init}}$.
If not, there must be a last linesearch iteration, say the $l$th, for which
$\alpha^{(l)} > \alpha_{\max}$ (if the linesearch has not already terminated).
Then $\alpha_k \geq \alpha^{(l+1)} =
\tau \alpha^{(l)} > \tau \alpha_{\max}$. Combining these two cases gives
the required result.
\subsection*{Proof of Theorem \thelecture.3}
We shall suppose that $g_k \neq 0$ for all $k$ and that
\disp{\lim_{k \rightarrow \infty} f_k > - \infty.}
From the Armijo condition, we have that
\disp{f_{k+1} - f_k \leq \alpha_k \beta \ip{p_k}{g_k}}
for all $k$, and hence summing over the first j iterations
\disp{f_{j+1} - f_0 \leq
\sum_{k=0}^{j} \alpha_k \beta \ip{p_k}{g_k}.}
Since the left-hand side of this inequality is, by assumption, bounded
below, so is the sum on right-hand-side.
As this sum is composed of negative terms, we deduce that
\disp{\lim_{k \rightarrow \infty} \alpha_k \ip{p_k}{g_k} = 0.}
Now define the two sets
\disp{\calK_1 = \left \{k \st
\alpha_{\mbox{init}} >
\frac{2 \tau (\beta - 1) \ip{p_k}{g_k}}{\gamma \|p_k\|_2^2} \right \}}
and
\disp{\calK_2 = \left \{k \st
\alpha_{\mbox{init}} \leq
\frac{2 \tau (\beta - 1) \ip{p_k}{g_k}}{\gamma \|p_k\|_2^2} \right \},}
where $\gamma$ is the assumed uniform Lipschitz constant.
For $k \in \calK_1$,
\disp{\alpha_k \geq
\frac{2 \tau (\beta - 1) \ip{p_k}{g_k}}{\gamma \|p_k\|_2^2}}
in which case
\disp{\alpha_k \ip{p_k}{g_k}
\leq \frac{2 \tau (\beta - 1)}{\gamma}
\left( \frac{\ip{p_k}{g_k}}{\|p_k\|} \right)^2 < 0.}
Thus
\eqn{lim1}{\lim_{k \in \calK_1 \rightarrow \infty}
\frac{| \ip{p_k}{g_k} |}{\| p_k \|_2} = 0.}
For $k \in \calK_2$,
\disp{\alpha_k \geq \alpha_{\mbox{init}}}
in which case
\eqn{lim2}{\lim_{k \in \calK_2 \rightarrow \infty} | \ip{p_k}{g_k} | = 0.}
Combining \req{lim1} and \req{lim2} gives the required result.
\subsection*{Proof of Theorem \thelecture.4}
Follows immediately from Theorem \thelecture.3, since for $p_k = - g_k$,
\disp{\min \left ( | \ip{p_k}{g_k} |, | \ip{p_k}{g_k} | / \| p_k \|_2 \right)
= \|g_k\|_2^{} \min \left ( 1, \|g_k\|_2^{} \right)}
and thus \disp{\lim_{k \rightarrow \infty}
\min \left ( | \ip{p_k}{g_k} |, | \ip{p_k}{g_k} | / \| p_k \|_2 \right)= 0}
implies that $\lim_{k \rightarrow \infty} g_k = 0$.
\subsection*{Proof of Theorem \thelecture.5}
Let $\lambda^{\min}(B_k)$ and $\lambda^{\max}(B_k)$ be the smallest and largest
eigenvalues of $B_k$.
By assumption, there are bounds $\lambda^{\min} > 0$ and $\lambda^{\max}$
such that
\disp{\lambda^{\min} \leq \lambda^{\min}(B_k) \leq \frac{\ip{s}{B_k s}}{\|s\|^2}
\leq \lambda^{\max}(B_k) \leq \lambda^{\max}}
for any nonzero vector $s$.
Thus
\disp{|\ip{p_k}{g_k}| = | \ip{g_k}{B_k^{-1} g_k} |
\geq \lambda_{\min}(B_k^{-1}) \|g_k \|_2^2
= \frac{1}{\lambda_{\max}(B_k)} \|g_k \|_2^2
\geq \lambda_{\max}^{-1} \|g_k \|_2^2.}
In addition
\disp{ \|p_k\|_2^2 = \ip{g_k}{B_k^{-2} g_k} \leq
\lambda_{\max}(B_k^{-2}) \|g_k \|_2^2 =
\frac{1}{\lambda_{\min}(B_k^2)} \|g_k \|_2^2 \leq
\lambda^{-2}_{\min} \|g_k \|_2^2 ,}
and hence
\disp{ \|p_k\|_2 \leq \lambda^{-1}_{\min} \|g_k \|_2,}
which leads to
\disp{\frac{|\ip{p_k}{g_k}|}{\| p_k \|_2} \geq
\frac{\lambda_{\min}}{\lambda_{\max}} \|g_k \|_2.}
Thus
\disp{\min \left ( | \ip{p_k}{g_k} | , | \ip{p_k}{g_k} | / \| p_k \|_2 \right)
\geq \lambda^{-1}_{\max} \|g_k\|_2^{}
\min \left ( \|g_k\|_2^{} , \lambda_{\min} \right).}
and hence \disp{\lim_{k \rightarrow \infty}
\min \left ( | \ip{p_k}{g_k} |, |\ip{p_k}{g_k}| / \| p_k \|_2 \right)= 0}
implies, as before, that $\lim_{k \rightarrow \infty} g_k = 0$.
\subsection*{Proof of Theorem \thelecture.6}
Consider the sequence of iterates $x_k$, $k \in \calK$, whose limit is $x_*$.
By continuity, $H_k$ is positive definite for all such $k$ sufficiently large.
In particular, we have that there is a $k_0 \geq 0$ such that
\disp{ \ip{p_k}{H_k p_k} \geq \half \lambda_{\min}(H_*) \|p_k \|_2^2}
for all $k \in \calK \geq k_0$, where $\lambda_{\min}(H_*)$ is the smallest
eigenvalue of $H(x_*)$. We may then deduce that
\eqn{b1}{|\ip{p_k}{g_k}| = - \ip{p_k}{g_k} = \ip{p_k}{H_k p_k}
\geq \half \lambda_{\min}(H_*) {\|p_k\|_2^2}. }
for all such $k$, and also that
\disp{\lim_{k \in \calK \rightarrow \infty} p_k = 0} since
Theorem \thelecture.5 implies
that at least one of the left-hand sides of \req{b1} and
\disp{\frac{|\ip{p_k}{g_k}|}{\|p_k\|_2} =
-\frac{\ip{p_k}{g_k}}{\|p_k\|_2} \geq
\half \lambda_{\min}(H_*) {\|p_k\|_2} }
converges to zero for all such $k$.
From Taylor's theorem, there is a $z_k$ between $x_k$ and $x_k + p_k$
such that
\disp{f(x_k + p_k ) = f_k + \ip{p_k}{g_k} +
\half \ip{p_k}{H(z_k) p_k}.}
Thus, the Lipschitz continuity of $H$ gives that
\eqn{dec}{\arr{rl}{f(x_k + p_k ) - f_k - \half \ip{p_k}{g_k}
\; = & \half ( \ip{p_k}{g_k} + \ip{p_k}{H(z_k) p_k} ) \\
= & \half ( \ip{p_k}{g_k} + \ip{p_k}{H_k p_k} ) +
\half \ip{p_k}{( H(z_k) - H_k ) p_k} \\
\leq & \half \gamma \|z_k - x_k \|_2 \| p_k \|_2^2
\leq \half \gamma \| p_k \|_2^3 }}
since $H_k p_k + g_k = 0$. Now pick $k$ sufficiently large so that
\disp{\gamma \| p_k \|_2 \leq \lambda_{\min}(H_*) (1-2\beta).}
In this case, \req{b1} and \req{dec} give that
\disp{f(x_k + p_k ) - f_k \leq \half \ip{p_k}{g_k} +
\half \lambda_{\min}(H_*) (1-2\beta) \| p_k \|_2^2
\leq \half ( 1 - (1-2\beta) ) \ip{p_k}{g_k} = \beta \ip{p_k}{g_k},}
and thus that a unit stepsize satisfies the Armijo condition
for all sufficiently large $k \in \calK$.
Now note that
$\| H_k^{-1}\|_2^{ } \leq 2 / \lambda_{\min}(H_*)$ for all sufficiently large
$k \in \calK$. The iteration gives
\disp{x_{k+1} - x_* = x_k - x_* - H_k^{-1} g_k
= x_k - x_* - H_k^{-1} \left ( g_k - g(x_*) \right )
= H_k^{-1} \left ( g(x_*) - g_k - H_k ( x_* - x_k ) \right ).}
But Theorem 1.3 gives that
\disp{\| g(x_*) - g_k - H_k \left ( x_* - x_k \right ) \|_2^{ } \leq
\gamma \| x_* - x_k \|_2^2.}
Hence
\disp{ \| x_{k+1} - x_* \|_2^{ } \leq \gamma \| H_k^{-1} \|_2^{ }
\| x_* - x_k \|_2^2}
which is (iii) when $\kappa = 2 \gamma / \lambda_{\min}(H_*)$
for $k \in \calK$. Result
(ii) follows since once an iterate becomes sufficiently close to $x_*$
for sufficiently large $k \in \calK$, this implies $k+1 \in \calK$, and
hence $\calK = \Na$. Thus (i) and (iii) are true for all $k$
sufficiently large.
\subsection*{Conjugate Gradient methods (Section~\thelecture)}
All of the results given here are easy to verify,
and may be found in any of the books of suggested background
reading material. The result that any $p_k = p^i$ is a descent direction
follows immediately since the fact that
$p^i$ minimizes $q(p)$ in $\calD^i$ implies that
\disp{ p^i = p^{i-1} - \frac{\ip{g_k}{d^{i-1}}}{\ip{d^{i-1}}{ B_k d^{i-1}} d^{i-1}}.}
Thus
\disp{\ip{g_k}{p^i} = \ip{g_k}{p^{i-1}} -
\frac{(\ip{g_k}{d^{i-1}})^2}{\ip{d^{i-1}}{B_k d^{i-1}}},}
from which it follows that $\ip{g_k}{p^i} < \ip{g_k}{p^{i-1}}$. The result then
follows by induction, since
\disp{\ip{g_k}{p^1} = - \frac{\|g_k\|_2^4}{\ip{g_k}{B_k g_k}} < 0.}
\setcounter{lecture}{3}
\setcounter{section}{\thelecture}
\addtocounter{section}{-1}
%
% proofs of results in lecture 3
%
\subsection*{Proof of Theorem \thelecture.1}
Firstly note that, for all $\alpha\geq 0$,
\eqn{uct-C2d1}{
m_k(-\alpha g_k) = f_k - \alpha \| g_k \|_2^2 +
\half \alpha^2 \ip{g_k}{B_k g_k}.}
If $g_k$ is zero, the result is immediate. So suppose otherwise.
In this case, there are three possibilities:
\begin{description}
\item[{\rm (i)}] the curvature $\ip{g_k}{B_k g_k}$
is not strictly positive; in this case
$m_k(-\alpha g_k)$ is unbounded from below as $\alpha$ increases, and hence
the Cauchy point occurs on the trust-region boundary.
\item[{\rm (ii)}] the curvature $\ip{g_k}{B_k g_k} > 0$ and the
minimizer of $m_k(-\alpha g_k)$
occurs at or beyond the trust-region boundary; once again, the
the Cauchy point occurs on the trust-region boundary.
\item[{\rm (iii)}] the curvature $\ip{g_k}{B_k g_k} > 0$
and the minimizer of $m_k(-\alpha g_k)$,
and hence the Cauchy point, occurs before the trust-region is reached.
\end{description}
We consider each case in turn;
\noindent Case (i).
In this case, since $\ip{g_k}{B_k g_k} \leq 0$, \req{uct-C2d1} gives
\eqn{uct-C2d10}{m_k(-\alpha g_k)
= f_k - \alpha \| g_k \|_2^2 + \half \alpha^2 \ip{g_k}{B_k g_k}
\leq f_k - \alpha \| g_k \|_2^2}
for all $\alpha\geq0$.
Since the Cauchy point lies on the boundary of the trust region
\eqn{uct-C2d7}{\alpha_k\s{C} = \frac{\Delta_k^{ }}{\|g_k^{ }\|}.}
Substituting this value into \req{uct-C2d10} gives
\eqn{uct-C2d11}{f_k^{ } - m_k^{ }(s_k\s{C})
\geq \|g_k^{ } \|_2^2 \frac{\Delta_k}{\|g_k^{ }\|}
\geq \kappa_s^{ } \|g_k^{ }\|_2^{ } \Delta_k^{ }
\geq \half \kappa_s^{ } \|g_k^{ } \|_2^{ } \Delta_k^{ }}
since $\|g_k\|_2 \geq \kappa_s \|g_k\|$.
\noindent Case (ii).
In this case, let $\alpha_k^*$ be the unique minimizer of \req{uct-C2d1};
elementary calculus reveals that
\eqn{uct-C2d4}{\alpha_k^* = \frac{\|g_k\|_2^2}{\ip{g_k}{B_k g_k}}.}
Since this minimizer lies on or beyond the trust-region boundary
\req{uct-C2d7} and \req{uct-C2d4} together imply that
\disp{ \alpha_k\s{C} \ip{g_k}{B_k g_k} \leq \|g_k\|_2^2.}
Substituting this last inequality in \req{uct-C2d1}, and using \req{uct-C2d7}
and $\|g_k\|_2 \geq \kappa_s \|g_k\|$, it follows that
\disp{ f_k^{ }-m_k^{ }(s_k\s{C}) =
\alpha_k\s{C} \|g_k^{ }\|_2^2 - \half [\alpha_k\s{C}]^2 \ip{g_k}{B_k g_k}
\geq \half \alpha_k\s{C} \|g_k^{ }\|_2^2
= \half \|g_k^{ } \|_2^2 \frac{\Delta_k}{\|g_k^{ }\|}
\geq \half \kappa_s^{ } \|g_k^{ } \|_2^{ } \Delta_k^{ }.}
\noindent Case (iii).
In this case, $\alpha_k\s{C}=\alpha_k^*$, and \req{uct-C2d1} becomes
\disp{f_k - m_k(s_k\s{C})
= \frac{\|g_k\|_2^4}{\ip{g_k}{B_k g_k}}
- \half \frac{\|g_k\|_2^4}{\ip{g_k}{B_k g_k}}
=\half \frac{\|g_k\|_2^4}{\ip{g_k}{B_k g_k}}
\geq \half \frac{\|g_k\|_2^2}{1+\|B_k\|_2},}
where
\disp{|\ip{g_k}{B_k g_k}| \leq \|g_k\|_2^2 \| B_k\|_2
\leq \|g_k\|_2^2 ( 1 + \| B_k\|_2)}
because of the Cauchy-Schwarz inequality.
The result follows since it is true in each of the above three
possible cases. Note that the ``$1+$'' is only needed to cover case where
$B_k = 0$, and that in this case, the ``$\min$'' in the theorem might
actually be replaced by $\kappa_s \Delta_k$.
\subsection*{Proof of Corollary \thelecture.2}
Immediate from Theorem~\thelecture.1 and the requirement that
$m_k^{ }(s_k^{ }) \leq m_k^{ }(s\s{C}_k)$.
\subsection*{Proof of Lemma \thelecture.3}
The generalized mean-value theorem gives that
\disp{f(x_k+s_k)
= f(x_k) + \ip{s_k}{\nabla_x f(x_k)} + \half \ip{s_k}{\nabla_{xx} f(\xi_k) s_k}}
for some $\xi_k$ in the segment $[x_k, x_k+s_k]$. Thus
\disp{\begin{array}{lcl}
|f(x_k+s_k) - m_k(s_k)|
& = & \half |\ip{s_k}{H(\xi_k) s_k} - \ip{s_k}{B_k s_k} |
\leq \half |\ip{s_k}{H(\xi_k) s_k}| + \half |\ip{s_k}{B_k s_k}| \\
& \leq & \half ( \kappa_h + \kappa_b ) \|s_k\|_2^2
\leq \half \kappa_l^2 ( \kappa_h + \kappa_b ) \|s_k\|^2
\leq \kappa_d \Delta_k^2
\end{array}}
using the triangle and Cauchy-Schwarz inequalities.
\subsection*{Proof of Lemma \thelecture.4}
By definition,
\disp{ 1 + \|B_k\|_2 \leq \kappa_h + \kappa_b , }
and hence for any radius satisfying the given (first) bound,
\disp{\kappa_s \Delta_k \leq \frac{\|g_k\|_2}{ \kappa_h + \kappa_b }
\leq \frac{\|g_k\|_2}{ 1 + \|B_k\|_2}.}
As a consequence, Corollary~\thelecture.2 gives that
\eqn{uct-ls1}{f_k - m_k(s_k) \geq \half \| g_k \|_2
\min\left[ \frac{\|g_k\|_2}{1 + \|B_k\|_2}, \kappa_s \Delta_k \right]
= \half \kappa_s \| g_k \|_2 \Delta_k.}
But then Lemma~\thelecture.3 and the assumed (second)
bound on the radius gives that
\eqn{uct-ls1b}{| \rho_k - 1 |
= \left|\frac{f(x_k+s_k) - m_k(s_k)}{f_k - m_k(s_k)} \right|
\leq 2 \frac{\kappa_d \Delta_k^2}{ \kappa_s \| g_k \|_2 \Delta_k }
= \frac{2 \kappa_d}{\kappa_s} \frac{\Delta_k}{\| g_k \|_2}
\leq 1 - \eta_v.}
Therefore, $\rho_k \geq \eta_v$ and the iteration is very successful.
\subsection*{Proof of Lemma \thelecture.5}
Suppose otherwise that $\Delta_k$ can become arbitrarily small.
In particular, assume that iteration $k$ is the first such that
\eqn{uct-db1}{\Delta_{k+1}
\leq \kappa_{\epsilon}.}
Then since the radius for the previous iteration must have been larger, the
iteration was unsuccessful, and thus $\gamma_d \Delta_k \leq \Delta_{k+1}$.
Hence
\disp{ \Delta_k \leq \epsilon
\min \left( \frac{1}{ \kappa_s ( \kappa_h + \kappa_b )},
\frac{\kappa_s ( 1 - \eta_v ) }{2 \kappa_d} \right) \leq \|g_k\|
\min \left( \frac{1}{ \kappa_s ( \kappa_h + \kappa_b )},
\frac{\kappa_s ( 1 - \eta_v ) }{2 \kappa_d} \right).}
But this contradicts the assertion of Lemma~\thelecture.4 that the $k$-th
iteration must be very successful.
\subsection*{Proof of Lemma \thelecture.6}
The mechanism of the algorithm ensures that
$x_* = x_{k_0+1}= x_{k_0+j}$ for all $j>0$,
where $k_0$ is the index of the last successful iterate. Moreover, since all
iterations are unsuccessful for sufficiently large $k$,
the sequence $\{\Delta_k\}$ converges to zero.
If $\|g_{k_0+1}\| > 0$, Lemma~\thelecture.4 then implies that
there must be a successful iteration of index larger than $k_0$, which is
impossible. Hence $\|g_{k_0+1}\| =0$.
\subsection*{Proof of Theorem \thelecture.7}
Lemma~\thelecture.6 shows that the result is true when
there are only a finite number of successful iterations. So it remains to
consider the case where there are an infinite number of successful
iterations. Let $\calS$ be the index set of successful iterations.
Now suppose that
\eqn{uct-li1}{\|g_k\| \geq \epsilon}
for some $\epsilon > 0 $ and all $k$,
and consider a successful iteration of index $k$.
The fact that $k$ is successful, Corollary~\thelecture.2, Lemma~\thelecture.5,
and the assumption \req{uct-li1} give that
\eqn{uct-zas0}{f_k - f_{k+1} \geq \eta_s [ f_k - m_k(s_k) ]
\geq \delta_{\epsilon} \eqdef \half \eta_s \epsilon
\min\left[ \frac{\epsilon}{1 + \kappa_b}, \kappa_s \kappa_{\epsilon} \right].}
Summing now over all successful iterations from 0 to $k$, it follows that
\disp{ f_0 - f_{k+1} =
\sum_{\stackrel{j = 0}{{\scriptscriptstyle j \in \calS}}}^k [f_j - f_{j+1} ]
\geq \sigma_k \delta_{\epsilon},}
where $\sigma_k$ is the number of successful iterations up to iteration $k$.
But since there are infinitely many such iterations, it must be that
\disp{ \lim_{k \rightarrow \infty} \sigma_k = + \infty.}
Thus \req{uct-li1} can only be true if $f_{k+1}$ is unbounded from below,
and conversely, if $f_{k+1}$ is bounded from below, \req{uct-li1} must
be false, and there is a subsequence of the $\|g_k\|$ converging to zero.
\subsection*{Proof of Corollary \thelecture.8}
Suppose otherwise that $f_k$ is bounded from below,
and that there is a subsequence of successful iterates,
indexed by $\{t_i\} \subseteq {\cal S}$, such that
\eqn{uct-w1}{\|g_{t_i}\| \geq 2 \epsilon > 0}
for some $\epsilon > 0 $ and for all $i$.
Theorem~ \thelecture.7 ensures the existence, for
each $t_i$, of a first successful iteration $\ell_i > t_i$ such that
$\|g_{\ell_i}\| < \epsilon$. That is to say that
there is another subsequence of ${\cal S}$ indexed by
$\{\ell_i\}$ such that
\eqn{uct-w2}{\|g_k \| \geq \epsilon \tim {for} t_i \leq k < \ell_i
\tim{and} \|g_{\ell_i}\| < \epsilon.}
We now restrict our attention to the subsequence of successful iterations
whose indices are in the
set
\disp{ {\cal K} \eqdef \{ k \in {\cal S} \mid t_i \leq k < \ell_i \},}
where $t_i$ and $\ell_i$ belong to the two subsequences defined above.
The subsequences $\{t_i\}$, $\{\ell_i\}$
and ${\cal K}$ are all illustrated in Figure~\ref{uct-subseq_fig},
where, for simplicity, it is assumed
that all iterations are successful. In this figure, we have
marked position $j$ in each of the subsequences represented
in abscissa when $j$ belongs to that subsequence. Note in this example
that $\ell_0=\ell_1= \ell_2 = \ell_3 = \ell_4 = \ell_5=8$, which we indicated
by arrows from $t_0 = 0$, $t_1=1$, $t_2=2$, $t_3=3$, $t_4=4$ and $t_5=7$ to
$k=9$, and so on.
\vspace*{4mm}
\setlength{\unitlength}{0.75mm}
\begin{figure}[ht]
\begin{center}
\begin{picture}(170,95)(-18,-5)
\put(0,0){\vector(0,1){80}}
\put(2,80){$\|g_k\|$}
\put(0,40){\dashbox{1}(150,0){}}
\put(-5,39){$2\epsilon$}
\put(0,20){\dashbox{1}(150,0){}}
\put(-5,19){$\epsilon$}
\put(0,0){\vector(1,0){150}}
\put(152,0){$k$}
\put(-5,-1){$\cal S$}
\multiput(0,0)(5,0){30}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\put(0,-4){\vector(1,0){150}}
\put(-7,-5){$\scriptstyle \{t_i\}$}
\multiput(0,-4)(5,0){5}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\put(35,-4){\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(50,-4)(5,0){3}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(85,-4)(5,0){3}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(125,-4)(5,0){2}{\makebox(0,0){\rule{0.75mm}{2mm}}}
%\put(0,-8){\vector(1,0){150}}
\put(-7,-9){$\scriptstyle \{\ell_i\}$}
\multiput(0,-8)(5,0){5}{\line(0,1){2}}
\put(35,-8){\line(0,1){2}}
\put(0,-8){\vector(1,0){39}}
\put(40,-8){\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(50,-8)(5,0){3}{\line(0,1){2}}
\put(50,-8){\vector(1,0){24}}
\put(75,-8){\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(85,-8)(5,0){3}{\line(0,1){2}}
\put(85,-8){\vector(1,0){19}}
\put(105,-8){\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(125,-8)(5,0){2}{\line(0,1){2}}
\put(125,-8){\vector(1,0){14}}
\put(140,-8){\makebox(0,0){\rule{0.75mm}{2mm}}}
\put(0,-12){\vector(1,0){150}}
\put(-5,-13){$\cal K$}
\multiput(0,-12)(5,0){8}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(50,-12)(5,0){5}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(85,-12)(5,0){4}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\multiput(125,-12)(5,0){3}{\makebox(0,0){\rule{0.75mm}{2mm}}}
\put(0,75){\redcircle}
\put(5,60){\redcircle}
\put(10,65){\redcircle}
\put(15,50){\redcircle}
\put(20,52){\redcircle}
\put(25,37){\yellowcircle}
\put(30,34){\yellowcircle}
\put(35,43){\redcircle}
\put(40,18){\bluestar}
\put(45,30){\yellowcircle}
\put(50,41){\redcircle}
\put(55,46){\redcircle}
\put(60,42){\redcircle}
\put(65,24){\yellowcircle}
\put(70,31){\yellowcircle}
\put(75,15){\bluestar}
\put(80,27){\yellowcircle}
\put(85,51){\redcircle}
\put(90,47){\redcircle}
\put(95,41){\redcircle}
\put(100,29){\yellowcircle}
\put(105,13){\bluestar}
\put(110,17){\bluestar}
\put(115,23){\yellowcircle}
\put(120,22){\yellowcircle}
\put(125,46){\redcircle}
\put(130,41){\redcircle}
\put(135,21){\yellowcircle}
\put(140,16){\bluestar}
\put(145,8){\bluestar}
\end{picture}
\end{center}
\caption{\label{uct-subseq_fig}The subsequences of the proof of
Corollary~\thelecture.8}
\end{figure}
\setlength{\unitlength}{1mm}
As in the previous proof, it immediately follows that
\eqn{uct-w4}{f_k - f_{k+1}
\geq \eta_s [f_k - m_k(s_k)]
\geq \half \eta_s \epsilon
\min\left[ \frac{\epsilon}{1 + \kappa_b}, \kappa_s \Delta_k \right]}
holds for all $k \in {\cal K}$ because of \req{uct-w2}.
Hence, since $\{f_k\}$ is, by assumption, bounded from below,
the left-hand side of \req{uct-w4} must tend to zero when $k$ tends to
infinity, and thus that
\disp{ \lim_{\stackrel{k \rightarrow \infty}{k \in {\cal K}}} \Delta_k = 0.}
As a consequence, the second term dominates in the minimum of \req{uct-w4}
and it follows that, for $k \in {\cal K}$ sufficiently large,
\disp{
\Delta_k \leq \frac{2}{\epsilon \eta_s \kappa_s}[ f_k - f_{k+1} ].}
We then deduce from this bound that, for $i$ sufficiently large,
\eqn{uct-ww4}{\| x_{t_i} - x_{\ell_i} \|
\leq \sum_{\stackrel{j=t_i}{j \in {\cal K}}}^{\ell_i-1} \|x_j - x_{j+1}\|
\leq \sum_{\stackrel{j=t_i}{j \in {\cal K}}}^{\ell_i-1} \Delta_j
\leq \frac{2}{\epsilon \eta_s \kappa_s} [ f_{t_i} - f_{\ell_i} ].}
But, because $\{f_k\}$ is monotonic and, by assumption, bounded from below,
the right-hand side of \req{uct-ww4} must
converge to zero.
Thus $\| x_{t_i} - x_{\ell_i} \|$ tends to zero as $i$ tends to infinity, and
hence, by continuity, $\|g_{t_i}-g_{\ell_i}\|$ also tend to
zero. However this is impossible because of the definitions of $\{t_i\}$ and
$\{\ell_i\}$, which imply that $\|g_{t_i} - g_{\ell_i}\| \geq \epsilon$.
Hence, no subsequence satisfying \req{uct-w1} can exist.
\subsection*{Proof of Theorem \thelecture.9}
The constraint $\| s \|_2 \leq \Delta$ is equivalent to
\eqn{cons}{\half \Delta^2 - \half \ip{s}{s} \geq 0.}
Applying Theorem 1.9 to the problem of minimizing $q(s)$ subject to
\req{cons} gives
\eqn{grad}{g + B s_* = - \lambda_* s_*}
for some Lagrange multiplier $\lambda_* \geq 0$ for which either
$\lambda_* = 0$ or $\| s_* \|_2 = \Delta$ (or both). It remains to show
that $B + \lambda_* I$ is positive semi-definite.
If $s_*$ lies in the interior of the trust-region,
necessarily $\lambda_* = 0$, and Theorem 1.10
implies that $B + \lambda_* I = B$ must be positive semi-definite.
Likewise if $\| s_* \|_2 = \Delta$ and $\lambda_* = 0$, it follows
from Theorem 1.10 that necessarily $\ip{v}{B v} \geq 0$ for all
$v \in \calN_+ = \{ v | \ip{s_*}{v} \geq 0\}$. If $v \notin \calN_+$,
then $-v \in \calN_+$, and thus $\ip{v}{B v} \geq 0$ for all $v$.
Thus the only outstanding case is where
$\| s_* \|_2 = \Delta$ and $\lambda_* > 0$. In this case, Theorem 1.10 shows
that $\ip{v}{( B + \lambda_* I ) v} \geq 0$ for all
$v \in \calN_+ = \{ v | \ip{s_*}{v} = 0\}$, so it remains to consider
$\ip{v}{B v}$ when $\ip{s_*}{v} \neq 0$.
\begin{figure}[ht]
\centerline{\psfig{figure=theo3.9b.eps,height=8.0cm,silent=}}
%\centerline{\psfig{figure=theo3.9.eps,height=8.0cm,silent=}}
\begin{picture}(375,1)(23,3.3)
\put(78.,50){$w$}
\put(103.,70){\circle*{1.5}}
\put(105.,71.5){$s_*$}
\put(125.,63){$\calN_+$}
\put(57.,26){$s$}
\end{picture}
\caption{\label{F-constr}Construction of ``missing'' directions of positive
curvature.}
\end{figure}
Let $s$ be any point on the boundary of the trust-region, and
let $w = s - s_*$, as in Figure~\ref{F-constr}. Then
\eqn{wts}{- \ip{w}{s_*} = \ip{s_* - s}{s_*} = \half \ip{ s_* - s}{s_* - s}
= \half \ip{w}{w}}
since $\|s\|_2 = \Delta = \|s_*\|_2$. Combining this with \req{grad} gives
\eqn{dm}{q(s) - q(s_*) = \ip{w}{g + B s_*} + \half \ip{w}{B w}
= - \lambda_* \ip{w}{s_*} + \half \ip{w}{B w}
= \half \ip{w}{( B + \lambda_* I ) w} ,}
and thus necessarily $\ip{w}{( B + \lambda_* I ) w} \geq 0$ since $s_*$ is
a global minimizer. It is easy to show that
\disp{s = s_* - 2 \frac{\ip{s_*}{v}}{\ip{v}{v}} v}
lies on the trust-region boundary, and thus for this $s$, $w$ is parallel to
$v$ from which it follows that $\ip{v}{( B + \lambda_* I ) v} \geq 0$.
When $B + \lambda_* I$ is positive definite,
$s_* = - (B + \lambda_* I)^{-1} g$. If this point is on the trust-region
boundary, while $s$ is any value in the trust-region, \req{wts} and \req{dm}
become $- \ip{w}{s_*} \geq \half \ip{w^}{w}$ and
$q(s) \geq q(s_*) + \half \ip{w}{( B + \lambda_* I ) w}$ respectively.
Hence, $q(s) > q(s_*)$ for any $s \neq s_*$. If $s_*$ is interior,
$\lambda_* = 0$, $B$ is positive definite, and thus $s_*$ is the
unique unconstrained minimizer of $q(s)$.
\subsection*{Newton's method for the secular equation (Section~\thelecture)}
\label{seceq}
Recall that the Newton correction
at $\lambda$ is $- \phi(\lambda)/ \phi^{\prime}(\lambda)$. Since
\disp{\phi(\lambda) = \frac{1}{\| s(\lambda)\|_2} - \frac{1}{\Delta}
= \frac{1}{(\ip{s(\lambda)}{s(\lambda)})^{\half}} - \frac{1}{\Delta},}
it follows, on differentiating, that
\disp{
\phi^{\prime}(\lambda) = - \frac{\ip{s(\lambda)}{\nabla_{\lambda}
s(\lambda)}}{(\ip{s(\lambda)}{s(\lambda)})^{\threehalves}}
= - \frac{\ip{s(\lambda)}{\nabla_{\lambda} s(\lambda)}}{\|s(\lambda)\|_2^3}.}
In addition, on differentiating the defining equation
\disp{(B + \lambda I ) s(\lambda) = - g,}
it must be that
\disp{(B + \lambda I ) \nabla_{\lambda} s(\lambda) + s(\lambda) = 0.}
Notice that, rather than the value of
$\nabla_{\lambda} s(\lambda)$, merely the numerator
\disp{\ip{s(\lambda)}{\nabla_{\lambda} s(\lambda)} = -
\ip{s(\lambda)}{(B + \lambda I)(\lambda)^{-1} s(\lambda)}}
is required in the expression for $\phi^{\prime}(\lambda)$.
Given the factorization $B + \lambda I = L(\lambda) L^T(\lambda)$,
the simple relationship
\disp{\ip{s(\lambda)}{(B + \lambda I )^{-1} s(\lambda)}
= \ip{s(\lambda)}{ L^{-T}(\lambda) L^{-1}(\lambda) s(\lambda)}
= \ip{L^{-1}(\lambda) s(\lambda)}{L^{-1}(\lambda) s(\lambda)} =
\| w(\lambda) \|_2^2}
where $L(\lambda) w(\lambda) = s(\lambda)$ then justifies the Newton step.
\subsection*{Proof of Theorem \thelecture.10}
We first show that
\eqn{sts-pitmpj=0}{\ip{d^{i}}{d^j} = \frac{\|g^i\|_2^2}{\|g^j\|_2^2}
\|d^j\|_2^2 > 0}
for all $0 \leq j \leq i \leq k$. For any $i$, \req{sts-pitmpj=0} is trivially
true for $j = i$. Suppose it is also true for all $i \leq l$.
Then, the update for $d^{l+1}$ gives
\disp{d^{l+1} = - g^{l+1} + \frac{\|g^{l+1}\|_2^2}{\|g^l\|_2^2} d^l.}
Forming the inner product with $d^j$, and using the fact that
$\ip{d^{j}}{g^{l+1}} = 0$ for all $j = 0, \ldots, l$,
and \req{sts-pitmpj=0} when $j = l$, reveals
\disp{ \ip{d^{l+1}}{d^j} = - \ip{g^{l+1}}{d^j} +
\frac{\|g^{l+1}\|_2^2}{\|g^l\|_2^2} \ip{d^{l}}{d^j}
= \frac{\|g^{l+1}\|_2^2}{\|g^l\|_2^2}
\frac{\|g^{l}\|_2^2}{\|g^j\|_2^2} {\|d^j\|_2^2}
= \frac{\|g^{l+1}\|_2^2}{\|g^j\|_2^2} {\|d^j\|_2^2} > 0.}
Thus \req{sts-pitmpj=0} is true for $i \leq l + 1$, and hence for all
$0 \leq j \leq i \leq k$.
We now have from the algorithm that
\disp{s^i = s^0 + \sum_{j=0}^{i-1} \alpha^j d^j = \sum_{j=0}^{i-1} \alpha^j d^j}
as, by assumption, $s^0 = 0$. Hence
\eqn{sts-pms}{\ip{s^{i}}{d^i} = \bip{\sum_{j=0}^{i-1} \alpha^j d^{j}}{d^i}
= \sum_{j=0}^{i-1} \alpha^j \ip{d^{j}}{d^i} > 0}
as each $\alpha^j > 0$, which follows from the definition of $\alpha^j$,
since $\ip{d^{j}}{H d^j} > 0$, and from relationship \req{sts-pitmpj=0}.
Hence
\disp{\arr{rl}{ \| s^{i+1} \|_2^2
& = \ip{s^{i+1}}{s^{i+1}}
= \ip{s^i + \alpha^i d^i}{s^i + \alpha^i d^i} \\
& = \ip{s^{i}}{s^i} + 2 \alpha^i \ip{s^{i}}{ d^i} + \alpha^{i\;2}
\ip{d^{i}}{ d^i}
> \ip{s^{i}}{s^i} = \| s^i \|_2^2 }}
follows directly from \req{sts-pms} and $\alpha^i > 0$ which is the required
result.
\subsection*{Proof of Theorem \thelecture.11}
The proof is elementary but rather complicated. See
\begin{center}
\parbox{5.5in}{Y. Yuan, ``On the truncated conjugate-gradient method'',
{\em Mathematical Programming}, {\bf 87} (2000) 561:573}
\end{center}
for full details.
\setcounter{lecture}{4}
\setcounter{section}{\thelecture}
\addtocounter{section}{-1}
%
% proofs of results in lecture 4
%
\subsection*{Proof of Theorem \thelecture.1}
Let $\calA = \calA(x_*)$, and $\calI = \{1,\ldots,m\} \; \backslash \; \calA$
be the indices of constraints that are active and inactive at $x_*$.
Furthermore let subscripts ${}_\calA$ and ${}_\calI$ denote the rows
of matrices/vectors whose indices are indexed by these sets.
Denote the left generalized inverse of $A_{\calA}^T(x)$ by
\disp{A_{\calA}^+(x) =
\left(A_{\calA}(x) A_{\calA}^T(x)\right)^{-1} A_{\calA}(x)}
at any point for which $A_{\calA}(x)$ is full rank. Since, by assumption,
$A_{\calA}(x_*)$ is full rank, these generalized inverses
exists, and are bounded and continuous in some open neighbourhood of $x_*$.
Now let
\disp{(y_k)_i = \frac{\mu_k}{c_i(x_k)}}
for $i = 1, \ldots, m$, as well as
\disp{(y_*)_{\calA} = A_{\calA}^+(x_*) g(x_*)}
and $(y_*)_{\calI} = 0$.
If $ \calI \neq \emptyset$, then
\eqn{yI}{\| (y_k)_{\calI} \|_2 \leq
2 \mu_k \sqrt{| \calI |} / \min_{i \in \calI} |c_i(x_*)|}
for all sufficiently large $k$.
It then follows from the inner-iteration termination test that
\eqn{kt}{ \arr{rl}{ \| g(x_k^{ }) -
A_{\calA}^T(x_k) (y_k^{ })_{\calA}^{ } \|_2^{ } \leq &
\| g(x_k^{ }) - A^T(x_k) y_k^{ } \|_2^{ } +
\| A_{\calI}^T(x_k^{ }) (y_k^{ })_{\calI}^{ }\|_2^{ } \\
\leq & \bar{\epsilon}_k \eqdef \epsilon_k +
\mu_k \bigfrac{2 \sqrt{| \calI |}
\| A_{\calI} \|_2}{\min_{i \in \calI} |c_i(x_*)|}.}}
Hence
\disp{\| A_{\calA}^+(x_k^{ }) g(x_k^{ }) - (y_k^{ })_{\calA}^{ }\|_2^{ } =
\| A_{\calA}^+(x_k^{ }) ( g(x_k^{ }) - A_{\calA}^T(x_k)
(y_k^{ })_{\calA}^{ } ) \|_2 ^{ }
\leq 2 \| A_{\calA}^+(x_*) \|_2^{ } \bar{\epsilon}_k.}
Then
\disp{ \|(y_k^{ })_{\calA}^{ } - (y_*^{ })_{\calA}^{ }\|_2^{ } \leq
\| A_{\calA}^+(x_*) g(x_*^{ }) - A_{\calA}^+(x_k^{ }) g(x_k^{ }) \|_2^{ } +
\| A_{\calA}^+(x_k^{ }) g(x_k^{ }) - (y_k^{ })_{\calA} \|_2^{ }}
which, in combination with \req{yI} and convergence of $x_k$, implies that
$\{ y_k \}$ converges to $y_*$.
In addition, continuity of the gradients and \req{kt} implies that
\disp{g(x_*) - A^T(x_*) y_* = 0}
while the fact that $c(x_k) > 0$ for all $k$, the definition of
$y_k$ and $y_*$ (and the implication that $c_i(x_k) (y_k)_i = \mu_k$)
shows that $c(x_*) \geq 0$, $y_* \geq 0$ and $c_i(x_*) (y_*)_i = 0$.
Hence $(x_*, y_*)$ satisfies the first-order optimality conditions.
\subsection*{Proof of Theorem \thelecture.2}
A formal proof is given by
\refs{ W. Murray, ``Analytical expressions for eigenvalues and eigenvectors of
the Hessian matrices of barrier and penalty functions'',
{\em J. Optimization Theory and Applics}, {\bf 7} (1971) 189:196.}
By way of a sketch, let $Q(x)$ and $S(x)$ be orthonormal bases for the
range- and null-spaces of $A_{\calA(x_*)}(x)$, and let $A_{\calI}(x)$
be the matrix whose rows are $\{ a_i^T(x) \}_{i \notin \calA(x_*)}$.
As we have shown, the required Hessian may be expressed (in decreasing terms
of asymptotic dominance) as
\disp{\nabla_{xx} \Phi(x,\mu) =
A_{\calA}^T(x) Y_{\calA}^2(x,\mu) A_{\calA}^{ }(x) / \mu + H(x,y(x,\mu)) +
\mu A_{\calI}^T(x) C_{\calI}^{-2}(x) A_{\calI}^{ }(x).}
Since the eigenvalues of $\nabla_{xx} \Phi(x,\mu)$ are not affected
by orthonormal transformations, on pre- and post-multiplying
$\nabla_{xx} \Phi(x,\mu)$
by $(Q(x) \;\; S(x))$ and its transpose, we see that the required eigenvalues
are those of
\eqn{bm}{
\mat{cc}{ Q(x)^T A_{\calA}^T(x) Y_{\calA}^2(x,\mu) A_{\calA}^{ }(x) Q(x) / \mu
+ Q(x)^T H(x,y(x,\mu)) Q(x) & Q(x)^T H(x,y(x,\mu)) S(x) \\
S(x)^T H(x,y(x,\mu)) Q(x) & S(x)^T H(x,y(x,\mu)) S(x)} + O(\mu),}
where we have use the relationship $A(x) S(x) = 0$.
The dominant eigenvalues are those arising from the 1,1 block of \req{bm},
and these are those of
$Q(x)^T A_{\calA}^T(x) Y_{\calA}^2(x,\mu) A_{\calA}^{ }(x) Q(x) / \mu$
with an $O(1)$ error---these are the same as those of
\disp{Y_{\calA}^{ }(x,\mu) A_{\calA}^{ }(x) Q(x)
Q(x)^T A_{\calA}^T(x) Y_{\calA}^{ }(x,\mu)
/ \mu =
Y_{\calA}^{ }(x,\mu) A_{\calA}^{ }(x) A_{\calA}^T(x) Y_{\calA}^{ }(x,\mu) /\mu}
as $Q(x)Q^T(x) = I$ and because the non-zero eigenvalues of $B^T B$ and
$BB^T$ agree for any (rectangular or otherwise) matrix $B$.
Since the remaining eigenvalues must occur for eigenvectors orthogonal to
those giving the 1,1 block, they will asymptotically
be those of the 2,2 block, and thus those of $S(x)^T H(x,y(x,\mu)) S(x)$ with
an $O(\mu)$ term.
\subsection*{Proof of Theorem \thelecture.3}
The proof of this result is elementary, but rather long and involved. See
\begin{center}
\parbox{5.5in}{
N. Gould, D. Orban, A. Sartenaer and Ph. L. Toint, ``Superlinear
convergence of primal-dual interior point algorithms for nonlinear
programming'',
{\em SIAM J. Optimization}, {\bf 11}(4) (2001) 974:1002}
\end{center}
for full details.
\setcounter{lecture}{5}
\setcounter{section}{\thelecture}
\addtocounter{section}{-1}
%
% proofs of results in lecture 5
%
%\section{Sketches of proofs for Section~\thelecture}
\subsection*{Proof of Theorem \thelecture.1}
The SQP search direction $s_k$ and its associated Lagrange multiplier
estimates $y_{k+1}$ satisfy
\eqn{1}{ B_k^{ } s_k^{ } - A_k^T y_{k+1}^{ } = - g_k }
and
\eqn{2}{ A_k s_k = - c_k.}
Pre-multiplying \req{1} by $s_k$ and using \req{2} gives that
\eqn{3}{ \ip{s_k}{g_k} = - \ip{s_k}{B_k s_k} + \ip{s_k}{A_k^T y_{k+1}}
= - \ip{s_k}{B_k s_k} - \ip{c_k}{y_{k+1}}.}
Likewise \req{2} gives
\eqn{4}{ \frac{1}{\mu_k} \ip{s_k}{A_k^T c_k} =
- \frac{\| c_k^{ } \|_2^2}{\mu_k}.}
Combining \req{3} and \req{4}, and using the positive definiteness of $B_k$,
the Cauchy-Schwarz inequality and the fact that $s_k \neq 0$ if
$x_k$ is not critical, yields
\disp{\arr{rl}{\ip{s_k}{\nabla_x \Phi(x_k)} = &
\bip{s_k}{g_k + \bigfrac{1}{\mu_k} A_k^T c_k}
= - \ip{s_k}{B_k s_k} - \ip{c_k}{y_{k+1}}
- \bigfrac{\| c_k \|_2^2}{\mu_k} \\
< & - \| c_k \|_2 \left( \bigfrac{\| c_k \|_2}{\mu_k} -
\|y_{k+1}\|_2 \right) \leq 0}}
because of the required bound on $\mu_k$.
\subsection*{Proof of Theorem \thelecture.2}
The proof is slightly complicated as it uses the calculus of non-differentiable
functions. See Theorem 14.3.1 in
\begin{center}
\parbox{5.5in}{
R. Fletcher, ``Practical Methods of Optimization'', Wiley
(1987, 2nd edition),}
\end{center}
where the converse result, that if $x_*$ is
an isolated local minimizer of $\Phi(x,\rho)$ for which $c(x_*) = 0$
then $x_*$ solves the given nonlinear program so long as $\rho$ is
sufficiently large, is also given.
Moreover, Fletcher showns (Theorem 14.3.2) that
$x_*$ cannot be a local minimizer of $\Phi(x,\rho)$ when $\rho < \|y_*\|_D$.
\subsection*{Proof of Theorem \thelecture.3}
For small steps $\alpha$, Taylor's theorem applied separately to
$f$ and $c$, along with \req{2}, gives that
\disp{\arr{rl}{\Phi(x_k + \alpha s_k, \rho_k) -
\Phi(x_k, \rho_k) = &
\alpha \ip{s_k}{g_k} + \rho_k
\bigleft ( \| c_k + \alpha A_k s_k \| - \| c_k \| \right )
+ O( \alpha^2 )
\\ = & \alpha \ip{s_k}{g_k} + \rho_k
\bigleft ( \| (1 - \alpha ) c_k \| - \| c_k \| \right )
+ O( \alpha^2 )
\\ = & \alpha \left(
\ip{s_k}{g_k} - \rho_k \| c_k \| \right) + O\left( \alpha^2 \right).}}
Combining this with \req{3}, and once again
using the positive definiteness of $B_k$,
the H\"{o}lder inequality (that is that $\ip{u}{v} \leq \|u\| \|v\|_D$
for any $u$, $v$)
and the fact that $s_k \neq 0$ if
$x_k$ is not critical, yields
\disp{\arr{rl}{\Phi(x_k + \alpha s_k, \rho_k) -
\Phi(x_k, \rho_k) = & - \alpha \left(
\ip{s_k}{B_k s_k} + \ip{c_k}{y_{k+1}} + \rho_k \| c_k \| \right)
+ O( \alpha^2 ) \\
< & - \alpha \left ( - \|c_k| \|y_{k+1}\|_D
+ \rho_k \| c_k \| \right ) + O( \alpha^2 ) \\
= & - \alpha \| c_k \| \bigleft( \rho_k - \|y_{k+1}\|_D \right)
+ O( \alpha^2 ) < 0}}
because of the required bound on $\rho_k$, for sufficiently small $\alpha$.
Hence sufficiently small steps along $s_k$ from non-critical $x_k$ reduce
$\Phi(x, \rho_k)$.
\end{document}
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https://mirror.anarhija.net/usa.anarchistlibraries.net/mirror/j/jg/jason-godesky-5-common-objections-to-primitivism-and-why-they-re-wrong.tex | anarhija.net | CC-MAIN-2021-10 | application/octet-stream | application/x-tex | crawl-data/CC-MAIN-2021-10/segments/1614178375096.65/warc/CC-MAIN-20210306131539-20210306161539-00017.warc.gz | 454,128,530 | 11,207 | \documentclass[DIV=12,%
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\title{5 Common Objections to Primitivism}
\date{2005}
\author{Jason Godesky}
\subtitle{and Why They’re Wrong}
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\section{1. Isn’t it hypocritical of primitivists to use modern technology? If they want to live primitively so badly, why don’t they just run off into the woods already and do it?}
Not all primitivists are against technology in and of itself; only some. Many primitivists hold a view that technology is ambiguous. Technology is found among all “primitive” peoples to one extent or another, so obviously there is \emph{some} sustainable level of technology. There is great disagreement among primitivists as to where that level is, but all agree that it isn’t our current level. Yes, we would like to see a lower level of technology, but since we have no problem with technology itself, why would we abstain from the use of our current, unsustainable technologies while they remain? One does not need to believe that a hammer is the greatest achievement of mankind, a miracle that ennobles us above all other animals and justifies our dominion over the earth, in order to use it to drive in a nail, after all. Neither does a computer. One can value science highly and still not believe that it is the sole or highest arbiter of truth; these are not mutually exclusive. And one can use the internet to spread the message that “the internet” and the infrastructure that supports it, is not going to last.
So, the charge of hypocrisy only holds up if we extend the beliefs of \emph{some} primitivists to \emph{all} primitivists, or to primitivism itself. What of the second question — why don’t primitivists run off into the woods already?
There are two issues here; the first is education. We were all raised within civilization, which has a vested interest in ensuring its children have as little independent survival value as possible. The civilized cultural system has adapted well — it reinforces itself memetically in precisely those areas where individuals are closest to self-sufficiency, creating a feeling of dependence even where little actual dependence exists. Regardless, most primitivists no more possess the skills of survival than your average suburbanite — skills every six year old “primitive” would have. Most primitivists are working to remedy that situation, but in the same way that you wouldn’t tell a !Kung man with dreams of brokering stock to just go to Wall Street already, but to learn a thing or two about the stock market first, so we are learning the skills we will need before hanging our lives on such skills. “Running off into the woods already” \emph{is} a goal, ultimately, but one we must work towards, not one we can simply pick up and go with. If it \emph{were} that easy, well, you wouldn’t be reading this, I can tell you that.
Secondly, there is the issue of lands and laws. Civilization has precluded “running off into the woods” as an option fairly well. Hunting regulations pose serious encumberments, to say nothing of the fact that some meager income must be maintained to pay for hunting and fishing licenses, as well as taxes on land. Ultimately, such a “micro-collapse” is impossible so long as civilization still exists — the pressing needs of ever-increasing complexity will lead to our re-absorption, by force if necessary. There is the essential problem; if civilization were willing to coexist with us, we would be happy to return the favor. But ultimately, civilization is incapable of letting anything but itself exist. We’re happy to live alongside anyone who’s willing to live alongside us — but civilization is not. “Running off into the woods,” so long as civilization remains, merely ensures our eventual, violent destruction at civilizaton’s hands.
\section{2. We have a stable, abundant supply of food. Primitivists want us to spend our lives desperate as to where our next meal is coming from.}
Why, then, is it only agriculturalists who starve? In fact, civilization’s food supply has always been shaky and meager. It is only recently that industrialized nations have increased production sufficiently to reap the benefits of “affluent malnutrition.” That’s the key to the success of modern life. We still eat things that are terribly maladapted to our physiology, but we eat them in prodigious quantities, allowing us to stay alive (if constantly sickly and degenerative) for the normal human lifespan of about 70 years, surpassing the average lifespan of medieval European nobility, but still slightly shy of our Mesolithic ancestors.
As the elite of the world system, the industrialized world is able to enjoy this standard of living because the non-industrialized world suffers chronic malnutrition and starvation. By contrast, foragers are transhumant omnivores — as well as being some of the most adaptable creatures on the planet. Foragers make their home among the islands of Tierra del Fuego, the frozen wastes of the Arctic, the Kalahari desert, and the thick jungles of the Congo — among areas so remote and desolate no crop would ever grow. To starve out foragers would require the end of nearly all multicellular life on this planet in the kind of mass extinction never before seen. By contrast, to starve out a bunch of farmers requires a slightly dry summer.
The idea that agriculture provides an abundant, stable food supply is demonstrably false. It is a myth. Agriculturalists rely on a small number of domesticable species — and those species tend to be closely related to one another, as well. It’s the fallacy of “putting all of your eggs in one basket.” By comparison, foragers rely not only on a much larger number of species, but a much wider diversity of species, as well. So, in fact, primitivists are advocating that we give up a higly unreliable and meager supply of food, for a supply that is genuinely stable and abundant.
\section{3. Primitivism would mean a drastic reduction in quality of life — no more medicine, no more art or music. Instead, you get euthanasia, astronomical infant mortality, and a life expectancy of about 30.}
The “euthanasia” charge comes from the Inuit, who were once slandered as leaving their elderly to die on ice floes. In fact, it was a rare custom, but a form of voluntary self-sacrifice that elders sometimes engaged in for the good of their bands, despite the pleading protestations of the rest of the band. The Inuit are full of such exceptions that prove the rule, because even for a forager, the arctic is a harsh and unforgiving place.
The infant mortality has simply been completely misrepresented, though. Yes, infant mortality among foragers is high — but not for the reasons such a statement would seem to imply. It is not because of disease or malnutrition — quite the opposite, as these things are fairly peculiar to civilized societies. Rather, just as we argue whether life begins at conception or at birth, foragers believe that life does not begin until, usually, the age of two. Foragers look at infanticide much the same way we do abortion. Among the !Kung, a pregnant woman goes into labor, and walks off into the bush (I’m told that childbirth is significantly less an ordeal among those who are not malnourished — affluently or otherwise). Maybe she comes back with a child; maybe she doesn’t. Either way, no questions are asked. So, our calculations of forager lifespans are quite unfair — if we’re going to include their infanticide, then we must include our own abortions. To do otherwise would simply be ethnocentric. In fact, when we do that, we see that forager lifespans are as long as, and sometimes longer, than our own.
The charge on medicine is common, but utterly anthropocentric. In the anthropology of medicine, one refers to “ethnomedicine” — whatever a given culture considers to be “medicine.” Given the overlap of food-as-medicine, this can be as arbitrary as how a culture divides up the color spectrum. Western biomedicine is \emph{our} ethnomedicine. Every culture believes that \emph{their} ethnomedicine is the only valuable one, and all others are naught but silly superstition. This is simply ethnocentrism. At the root of the claim that primitivism precludes medicine is precisely this ethnocentrism. In fact, when we look at the actual efficacy of the various ethnomedicines in the world, there’s very little variation. Most ethnomedicines are quite effective, just like ours; most have one or more area where they fail utterly (ours tries to ignore placebo rather than use it; shamanism is the opposite, but has no conept of surgery, etc.), and all end up being roughly interchangeable if one is only concerned with efficacy. So, by no means does primitivism require the end of medicine — it merely means a radically different, but equally effective, form of medicine. In fact, if we attempt a syncretic type of medicine that seeks to combine the best of several ethnomedicines, we may actually come up with one of the first medical systems that actually \emph{is} more effective.
Finally, the charge that primitivism would mean the end of art and music is patently false. Art, music and the rest were universal among primitive peoples for 30,000 years before civilization even began. They have had these things for four times as long as civilization has even existed. The cave art as Lasceaux is easily comparable to Michelangelo, and the Pygmy tribes of the Congo sing songs with a polyphonic complexity that Europe did not match until the 14\textsuperscript{th} century. One can only claim that primitive peoples have no art or music if we ethnocentrically define “art” and “music” to mean, “it only counts if a white guy did it.” In \emph{Savages \& Civilization}, Jack Weatherford makes the case that the scientific, artistic, musical and philosophical achievements of civilization were all inspired by our contact with savages. Primitivists believe that, if it is at all possible to call any culture “superior,” then it must be that of the primitives — those who inspired all of our greatest achievements, and suffer none of our worst flaws.
\section{4. Primitivists are misanthropic.}
This charge requires a unique definition of “misanthropic,” but it is usually attached to the next objection, below. To make this statement, the speaker first conflates humanity and civilization with some mythology about civilization being mankind’s natural destiny, rather than the momentary abberation it truly is. In fact, domesticated \emph{Homo sapiens} exists in a pitiful state of captivity, bound to a moribund existence to which she is entirely maladapted. Humans in the wild experience a level of freedom and fullness of life that is incomprehensible to their domesticated brethren, just as Plato’s protagonist could not explain the outside world to those poor wretches chained to the wall in the allegory of the cave. The goal of primitivism is rewilding, that is, to return as many domesticated \emph{Homo sapiens} to that happy, natural state as possible.
To the primitivist, it is, in fact, the progressivist who is misanthropic. It is the progressivist who claims that the natural state of humanity is to labor for the benefit of others and to be subject to despots — at best, kind-hearted and duly-elected despots, but despots all the same. It is the progressivist who thinks that humanity is not sufficient in itself, but must be ennobled by Science and Reason, redeemed from his fallen state of primitive fear and violence by Technology. The progressivist sees nothing but misery in our past, a savage in our soul that must be denied and sublimated, and for our future, a cold, aloof godhood, an apotheosis by nanotechnology, and the alienation of dominion over the earth that precludes ever being \emph{part} of it. The progressivist takes a very dim view of the human being indeed: her passions must be denied, her nature is savage and must be sublimated, her natural state is a never-ending Hobbesian nightmare.
The primitivist knows all of this is so many fairy tales. We know that primitive societies live in no such nightmare, but are, in fact, as Marshal Sahlins put it, “the original affluent society.” We know that we are not the forgotten children of evolution, the only species of all the earth left without an easy adaptation to the world. We know that human nature is neither demonic, nor angelic. We do not see humanity as something fallen that must be fixed — whether by faith in some number of gods (whether many, one, or none at all), or by Reason, or by Technology. We believe that being human is a wonderful thing. We can also see that the progressivist agenda has shackled humanity, that civilization dehumanizes us and strips us of all those things that are so good about our species.
It was for this abiding faith in humanity and our conviction that humanity is most emphatically \emph{not} broken, and neither is it in need of us to “fix” it, that I chose the name “Anthropik” for our tribe. The term “humanist” might have done just as well, had it not been adopted (rather inappropriately, to my mind) by a particular camp of progressivists, but as it is, it plays well against the term “misanthropy.” Progressivists are misanthropic; it is primitivists who are anthropic.
\section{5. Primitivists are genocidal maniacs whose planned “utopia” requires them to orchestrate the mass murder of 99\% of the human population!}
I’ve saved the best for last. This is the single most common, and the single most powerful attack launched against primitivists by the progressivist camp.
It is undeniably true that the world’s population cannot be sustained without modern civilization. Of course, it is abundantly clear that modern civilization is not sustainable, either. Given those two facts, then some kind of massive die-off is inevitable. It might be through genocide, but since primitvists are a fringe of a fringe (and will always be so) it’s unlikely to come from us. There are many other parties with a much greater interest in genocide for its own sake, who are far closer to power than we will ever be. Ultimately, genocide might be the kindest method, just as it is kind to deliver a \emph{coup de grace} to a dying animal. The alternative is to waste away by hunger or disease. But ultimately, genocide on such a scale would be nigh impossible, and though die-off is guaranteed, it is almost as guaranteed \emph{not} to come by way of genocide.
Rather, collapse is more likely to occur as it always has. The diminishing returns of complexity lead to the breakdown of civilization, until some minor turbulence that might have been easily overcome in a former time, instead ends our civilization — the way an AIDS victim dies not of AIDS, but of some minor disease a healthy person would have easily shrugged off. Perhaps Peak Oil, perhaps global warming, whatever the proximate cause, our ability to produce food will be cut off. Starvation will lead to food riots, until, in the end, the survivors will turn to cannibalism. The cities will be killing fields, but those who can look at the wilderness and call it home, those who can find their food without having someone grow it for them — those who are rewilded — will have access to vast resources that no others will even think to exploit.
This is the way evolution has always worked. The “oxygen holocaust” was caused by the abundance of microbes that breathed carbon dioxide, and exhaled oxygen. Eventually, they changed the very composition of the atmosphere, and began to choke and die in the toxic environment. But those microbes that were adapted and could actually breathe the toxic oxygen emerged and proliferated, striking a balance with their forebears, the carbon dioxide breathing microbes, and beginning the oxygen cycle that regulates our atmosphere today. So, too, the collapse will permanently end civilization, and with it the dehumanizing domestication and captivity of \emph{Homo sapiens}, leaving only rewilded humans to inherit the earth.
The fanciful genocide scenario is embraced by some primitivists, but this is quite patently madness — and unspeakably wicked. As I said, for those who die, dying quickly of a gunshot may be preferable to dying slowly of hunger and disease, or living to see their cities torn apart by warring gangs of cannibals. However, there is an evolutionary elegance to the collapse that such an alternative violates. Every individual on earth will have a choice. They will be free to choose to remain part of their culture to the bitter end, and die with it; or, they wll have the choice to embrace a new culture, embrace their own humanity, and survive into a new world. An act of active genocide violates that. The one who perpetrates such an act elevates himself to the status of a god (as the progressivists would do, only without their silly, illogical, anthropocentric qualms distinguishing between humans and all other life on the planet), to dictate who should live and who should die. This is why I believe Ted Kazcinski is evil: besides the complete counter-effectiveness of his campaign of terror, he committed the ultimate sin, the sin of civilization itself. He placed himself in the role of a god, dictating life and death.
Most will choose to die; we cannot change that. It would be just as wrong to force them to choose life as it was for Kaczinski to force others to die. What we \emph{can} do is try as hard as we can to make sure everyone understands that it truly is a \emph{choice} they face.
When hearing this defense, many progressivists will claim that our willingness to “allow” such a thing to happen is characterized as monstrous. First, the hubris dripping from such a statement is absurd; we do not “allow” such things to happen any more than we “allow” the sun to shine or the rain to fall. By comparison, a progressivist tries to dream up ways to control the weather, while a primitivist makes an umbrella or some sun screen. There is the difference between us; progressivists aspire to such divine control, where primitivists accede and accept that they are \emph{part} of the world, not gods of it.
But, addressing the point of such an absurd statement — the idea that we have some moral obligation to try to stop collapse — consider a sickly child. Consider my brother. It is my earliest memory. The doctors insisted it was not meningitis, even though it matched all the symptoms — after all, how could it be? He had just a few days before had a large number of meningitis pathogens injected into his body, and, having been vaccinated, it couldn’t possibly be. That would mean that science and medcine had failed.
My mother told me not to watch, but I peeked, and the image was seared into my brain forever. My tiny brother’s body, screaming in agony, pinned down by my father and a doctor, as another took a needle nearly as long as my little brother’s entire body, and slipped it into his spine.
I cannot imagine my brother’s pain — or my father’s holding him down for such a thing. But he did the right thing — the hard thing. My brother very nearly died that night, but because my father could see that avoiding that passing agony would mean death, he survived. There was great pain, but once that pain passed, there was life.
That is very much the situation the human race is in now. Had our civilization collapsed in the Bronze Age, it would have killed millions and caused ecological devastation throughout the Mediterranean. It was avoided, and instead we had wars, empires, the decimation of the New World, and we have ushered in the single greatest mass extinction in the planet’s history. Now, we stand on the same precipice. Collapse now would involve the deaths of \emph{billions}, and we can look back and see that it would have been better if our civilization had \emph{not} survived the Bronze Age. But it did, for all the same pressures that push us forward now. If by some miracle we \emph{do} find another \emph{deus ex machina}, then we will only make it still worse — the deaths of trillions, and the very real poossibility of the extinction of our species, and all multicellular life on earth,
The cost of collapse is terrible. It should have been paid by our ancestors, and damn them for not paying it! The cost would have been so much less. Instead, the debt has fallen on us, and it is almost more than we can bear. Yet bear it — and pay it — we must. If we do, then humanity will be free once again. If we don’t, then our children will pay it, and then the cost \emph{will} be too much to bear — they will damn us as we damn our ancestors’ weakness, for because of \emph{our} weakness, there will be no bright, shining hope once the debt is paid. For them, the debt will be so great that it must be paid with the extinction of our entire species.
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Anarchist library
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Anti-Copyright
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Jason Godesky
5 Common Objections to Primitivism
and Why They’re Wrong
2005
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Retrieved on January 27, 2010 from \href{http://tobyspeople.com/anthropik/2005/10/5-common-objections-to-primitivism-and-why-theyre-wrong/}{tobyspeople.com}
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\textbf{en.anarchistlibraries.net}
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\item Pense à un nombre (par exemple 5). Ajoute 7 à ce
nombre. Multiplie le résultat par 3. Retranche 20 au
résultat. Retranche le triple du nombre auquel tu as pensé. Divise le
résultat par 2.\\Combien trouves-tu ?
\item Démontre que quel que soit le nombre que tu choisis au départ,
le résultat trouvé est le même (on pourra appeler $x$ le nombre du
départ).
\end{myenumerate} |
https://www.authorea.com/users/350598/articles/475403/download_latex | authorea.com | CC-MAIN-2021-39 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-39/segments/1631780057083.64/warc/CC-MAIN-20210920161518-20210920191518-00327.warc.gz | 678,404,569 | 8,323 | \documentclass[10pt]{article}
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\title{Prolonged remission using CD19 chimeric antigen receptor-T cell therapy
followed by haploidentical transplantation in a 5-month-old patient with
infantile acute lymphoblastic leukemia}
\author[1]{Tamara Hagoel}%
\author[2]{Bree Kramer}%
\author[1]{Joanne Becker}%
\author[1]{Kara Kelly}%
\author[1]{Meghan Higman}%
\affil[1]{Roswell Park Comprehensive Cancer Center}%
\affil[2]{University at Buffalo Jacobs School of Medicine and Biomedical Sciences}%
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\begin{abstract}
Tisagenlecleucel offers promise to children with relapsed/refractory
(r/r) acute lymphoblastic leukemia (ALL). However, there is limited
experience with and data supporting the use of tisagenlecleucel in
infants. We describe our successful experience using tisagenlecleucel
followed by a haploidentical donor hematopoietic stem cell
transplantation in an infant with r/r KMT2A-rearranged ALL, the youngest
infant to survive and achieve prolonged remission using this approach.%
\end{abstract}%
\sloppy
\textbf{INTRODUCTION}
Infants younger than 6-months of age diagnosed with precursor B-cell
acute lymphoblastic leukemia (ALL) have a rare and aggressive form of
leukemia associated with inferior outcomes, especially if associated
with KMT2A gene rearrangements (KMT2A-r).{[}1-4{]} Dismal outcomes are
observed despite intensive therapies including allogeneic hematopoietic
stem cell transplantation (HSCT).{[}1,5-9{]} While a therapeutic
advantage exists for early HSCT in a subset of high-risk infants with
KMT2A-r ALL transplanted in a first complete remission, early disease
recurrence, refractory or progressive disease often limits this
approach.{[}6,8-12{]} Those who do proceed to HSCT frequently relapse
without curative treatment options.{[}6, 13{]}
Tisagenlecleucel, an autologous anti-CD19 chimeric antigen receptor
T-cell (CAR-T) agent, offers promise to patients with relapsed or
refractory (r/r) ALL, demonstrating remission rates as high as 81\% in
both children and adolescents.{[}14-16{]} However, there is limited
experience utilizing CAR-T in infants resulting from the exclusion of
children \textless{}1 year of age from initial clinical trials.{[}1{]}
The approach has been limited due to an infant's smaller size and blood
volume; concerns include cellular collection safety, the timing of
autologous T-cell harvest during a patient's treatment course, the
optimal infusion dose, and the management of post-infusion
toxicities.{[}10,17-19{]} There are inherent difficulties in obtaining
sufficient autologous T-cells in infants to generate a suitable CAR-T
product.{[}17-21{]} Whether CAR-T therapy alone is satisfactory to
achieve durable long-term remissions remains under investigation for all
recipients with little information in the setting of aggressive
infantile ALL. Failure following CAR-T therapy is frequently secondary
to antigen escape.{[}14,22{]} Multiple reports demonstrate superior
outcomes and sustained remissions in pediatric and adult patients with
r/r ALL receiving CAR-T as a bridge to allogeneic HSCT;{[}10,14,15{]}
however, there is a dearth of data supporting the use of CAR-T therapy
in infants.{[}18{]}
Here we present our experience in a high-risk infant with r/r KMT2A-r
ALL who successfully received the CD19 CAR-T cell therapy,
tisagenlecleucel, in tandem with haploidentical donor HSCT. This is the
youngest infant to survive and achieve prolonged remission following
CAR-T therapy and subsequent HSCT.{[}18{]}
\textbf{CASE DESCRIPTION}
Our patient is an African American female diagnosed at 6-weeks of age
who presented for evaluation of persistent vomiting, lethargy and an
unusual skin rash after several weeks of concern for failure to thrive.
Initial white blood cell (WBC) count was 834x10\textsuperscript{9}/L
with 98\% circulating lymphoblasts. Flow cytometry performed on
peripheral blood revealed ALL with 98\% of total cells expressing CD19.
Cytogenetic analysis revealed a t(11;19)(q23.3;p13.3) translocation and
flouresent \emph{in situ} hybridization analysis confirmed a KMT2A-r
ALL. Cerebrospinal fluid (CSF) analysis revealed central nervous system
(CNS) involvement. The patient underwent leukoreduction and subsequently
initiated treatment as per the Interfant-06 standard protocol
(clinigaltrials.gov: NCT00550992).
End of induction assessment demonstrated clearance of the
extra-medullary disease and a bone marrow morphologic remission but flow
cytometric evidence of minimal residual disease of 0.36\%. Three months
after diagnosis she relapsed with confirmed ALL detected in the bone
marrow and CSF with chloromatous skin lesions following a period of
delayed blood cell count recovery from consolidation therapy. She
received high-dose methotrexate (HD-MTX) (3.3gm/m\textsuperscript{2})
with 6-mercaptopurine (6MP) and intrathecal methotrexate/hydrocortisone
(IT MTX/HC) that resulted in clearance of peripheral blood and CSF
lymphoblasts and resolution of chloromas.
She underwent autologous CD3 lymphocyte apheresis using the CMNC program
in the Spectra Optia automatic apheresis system for CAR-T generation
over a course of 3 collection days at the age of 5-months and weighing
6.8kg. The device was primed with whole blood which was reinfused to her
at the completion of the procedures. Absolute lymphocyte count (ALC) at
the start of collection was 0.88x10\textsuperscript{9}/L. On day 1 the
product output was poor with a total nuclear cell count (TNC) of
0.14x10\textsuperscript{9}/L and a CD3 count of
0.02x10\textsuperscript{9}/L cells.
Adjustments were made for collection on days 2 and 3. On day 2, the TNC
obtained was 1.4x10\textsuperscript{9} and the CD3 count was
0.42x10\textsuperscript{9} cells. Day 3 yielded
2.64x10\textsuperscript{9} TNC and 0.31x10\textsuperscript{9} CD3
lymphocytes. The procedure totaled 4.18x10\textsuperscript{9} TNC and
0.76x10\textsuperscript{9} CD3 cells over 3 days. Despite her small
size, the cellular goals for apheresis were the same as for an adult and
were appropriately met without any complications.
CNS and peripheral blood lymphoblasts re-emerged following chemotherapy
with high dose cytarabine and pegaspargase but cleared following HD-MTX
(3.75gm/m\textsuperscript{2}), 6MP, dexamethasone, vincristine and IT
MTX/HC. Lymphodepleting therapy with a 2-day course of cyclophosphamide
and fludarabine was administered 7 days prior to tisagenlecleucel
infusion. The total cell viability in the final CAR-T product was 85.4\%
with 2.7\% CAR positive viable cells, below the 3\% threshold required
by the Food and Drug Administration (FDA) for product release. However,
the overall number of chimeric antigen receptor (CAR) viable transduced
T-cells was 2.3x10\textsuperscript{6} cells/kg, meeting the clinical
specifications required for commercial use.{[}23{]}~As the final product
did not meet all the required specifications, she received the
manufactured cells on compassionate use exemption at 7-months of age.
Figure-1 summarizes her WBC count, lymphoblast count, C-reactive
protein, ferritin, fibrinogen, maximum daily temperature and clinical
course following CAR-T infusion. Her course was complicated by grade 2
cytokine release syndrome (CRS) in conjunction with the emergence of
confirmed CD19 lymphoblasts detected in peripheral blood on day 7. CRS
was manifested by persistent high fevers, tachycardia, tachypnea and
grunting. She received two doses of tocilizumab on days 7 and 10. She
developed tumor lysis syndrome (TLS) and subsequent acute kidney injury
manifested by an increase in creatinine from 0.32mg/dL to 0.61mg/dL. She
subsequently developed \emph{Klebsiella pneumoniae} bacteremia,
disseminated intravascular coagulation and unexplained hypoglycemia.
Full recovery from TLS and CRS occurred following the clearance of
peripheral lymphoblasts by day 13. She was discharged home on day 26 in
remission confirmed by morphologic and flow cytometric assessment of
bone marrow.
At 9-months old she proceeded with a planned myeloablative
(busulfan/cyclophosphamide) haploidentical allogeneic HSCT from her
14-year old sister. Tacrolimus and mycophenolate mofetil were used for
graft-vs-host-disease prophylaxis. She developed reversible
hyperbilirubinemia, transaminitis and hepatomegaly. Complications
included \emph{Klebsiella pneumoniae} bacteremia, mucositis,
hypertension and pericardial effusion. She was discharged on day 34 in
remission confirmed by CSF evaluation and morphologic and flow
cytometric analysis of her bone marrow. She is now 12-months post-HSCT
and remains in a remission.
\textbf{DISCUSSION}
This case highlights that apheresis for CAR-T cell generation can be
done safely and successfully in an infant. Secondly, use of a product
with modified CAR-T parameters can result in remission. Despite not
meeting the FDA's percent of CAR positive viable cell specifications,
our patient's manufactured CAR-T product was equally capable of
expansion and leukemia targeting with no evidence of greater safety
risk.
Lastly, our patient's ongoing remission status cannot be stressed
enough. The availability of tisagenlecleucel provided an opportunity to
control very chemotherapy resistant disease prior to HSCT.
Pre-transplant remission status following second-line therapy
significantly impacts survival following HSCT in patients with r/r
KMT2A-r ALL;{[}3,8{]} the added benefit of tisagenlecleucel facilitated
a deeper remission prior to receiving HSCT, thereby providing an
opportunity for maximal long-term~survival in an infant who previously
had a dismal prognosis. With her being approximately 12-months in
remission post-HSCT, she is now beyond the greatest risk period for
relapse.{[}3{]} Future clinical investigations using CAR-T therapy prior
to HSCT for infants with r/r KMT2A-r ALL offers the potential for
improved survival in a disease where little progress has been made in
treatment outcomes.
\textbf{Conflict of Interest Statement}
The authors have nothing to disclose.
\textbf{Acknowledgements}
The authors would like to acknowledge Bridget Newsom for facilitating
our constant communications with Novartis and her dedicated efforts in
helping obtain tisagenlecleucel on compassionate use protocol for our
patient. We would also like to acknowledge all the members of the
Novartis team who supported our medical team through the pre- and
post-collection challenges.
\textbf{Correspondence}
Meghan Higman, MD
Department of Pediatrics, Division of Pediatric Hematology/Oncology,
Roswell Park Comprehensive Cancer Center
Clinical Assistant Professor, Division of Pediatric Hematology/Oncology,
University at Buffalo Jacobs School of Medicine and Biomedical Sciences
Buffalo, New York 14203, USA
e-mail: meghan.higman@roswellpark.org
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9. Mann G, Attarbaschi A, Schrappe M, et al. Improved outcome with
hematopoietic stem cell transplantation in a poor prognostic subgroup of
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\emph{Blood.}2010;116(15):2644-2650.
10. Grupp SA. Advances in T-cell therapy for ALL. \emph{Best Pract Res
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11. Jacobsohn DA, Hewlett B, Morgan E, Tse W, Duerst RE, Kletzel M.
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hematopoietic stem cell transplantation. \emph{Biol Blood Marrow
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12. Koh K, Tomizawa D, Moriya Saito A, et al. Early use of allogeneic
hematopoietic stem cell transplantation for infants with MLL
gene-rearrangement-positive acute lymphoblastic
leukemia.\emph{Leukemia.} 2015;29(2):290-296.
13. Pulsipher MA, Wayne AS, Schultz KR. New frontiers in pediatric
Allo-SCT: novel approaches for children and adolescents with
ALL.\emph{Bone Marrow Transplant.} 2014;49(10):1259-1265.
14. Hucks G, Rheingold SR. The journey to CAR T cell therapy: the
pediatric and young adult experience with relapsed or refractory
B-ALL.\emph{Blood Cancer J.} 2019;9(2):10.
15. Maude SL, Frey N, Shaw PA, et al. Chimeric antigen receptor T cells
for sustained remissions in leukemia. \emph{N Engl J
Med.}2014;371(16):1507-1517.
16. O'Leary MC, Lu X, Huang Y, et al. FDA Approval Summary:
Tisagenlecleucel for Treatment of Patients with Relapsed or Refractory
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Res.}2019;25(4):1142-1146.
17. Michon B, Moghrabi A, Winikoff R, et al. Complications of apheresis
in children. \emph{Transfusion.} 2007;47(10):1837-1842.
18. Shyr DC, Homsombath AA, Chan PP, Boyer MW, Harris AC. Using CD19
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20. Mahadeo KM, Khazal SJ, Abdel-Azim H, et al. Management guidelines
for paediatric patients receiving chimeric antigen receptor T cell
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21. Pahys J, Fisher V, Carneval M, Yomtovian R, Sarode R, Nieder M.
Successful large volume leukapheresis on a small infant allogeneic
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22. Majzner RG, Mackall CL. Tumor Antigen Escape from CAR T-cell
Therapy. \emph{Cancer Discov.} 2018;8(10):1219-1226.
23. Maude SL, Laetsch TW, Buechner J, et al. Tisagenlecleucel in
Children and Young Adults with B-Cell Lymphoblastic Leukemia. \emph{N
Engl J Med.} 2018;378(5):439-448.
\textbf{Figure Legends}
\textbf{Figure 1:} A, Time course of patient's WBC count versus
lymphoblast count post-tisagenlecleucel infusion; B, Time course of
patient's CRP versus maximum daily temperature post-tisagenlecleucel
infusion; C, Time course of patient's ferritin versus maximum daily
temperature post-tisagenlecleucel infusion; D, Time course of patient's
fibrinogen versus maximum daily temperature post-tisagenlecleucel
infusion. CAR-T, chimeric antigen receptor-T cell; CRP, C-reactive
protein; WBC, white blood cell.\selectlanguage{english}
\begin{figure}[H]
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\end{center}
\end{figure}
\selectlanguage{english}
\FloatBarrier
\end{document}
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\begin{document}
{\centerline {\large {\bf 1996 Canadian Mathematical Olympiad}}}
\vspace{0.2 in}
\begin{enumerate}
\item If $\alpha$, $\beta$, and $\gamma$ are the roots of $x^3 - x - 1 = 0$,
compute $\d \frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} +
\frac{1+\gamma}{1-\gamma}$.
\item Find all real solutions to the following system of equations. Carefully
justify your answer.
$$\left\{
\begin{array}{c}
\d \frac{4x^2}{1+4x^2} = y \\
\\
\d \frac{4y^2}{1+4y^2} = z \\
\\
\d \frac{4z^2}{1+4z^2} = x
\end{array}
\right. $$
\item We denote an arbitrary permutation of the integers 1, 2, \ldots, $n$ by
$a_1$, $a_2$, \ldots, $a_n$. Let $f(n)$ denote the number of these
permutations such that:
\begin{enumerate}
\item[(i)] $a_1 = 1$;
\item[(ii)] $|a_i - a_{i+1}| \leq 2$, \ $i = 1, \ldots, n - 1$.
\end{enumerate}
Determine whether $f(1996)$ is divisible by 3.
\item Let triangle $ABC$ be an isosceles triangle with $AB = AC$. Suppose that
the angle bisector of $\angle B$ meets $AC$ at $D$ and that $BC = BD + AD$.
Determine $\angle A$.
\item Let $r_1$, $r_2$, \ldots, $r_m$ be a given set of $m$ positive rational
numbers such that $\d \sum_{k=1}^m r_k = 1$. Define the function $f$ by $f(n)
= n - \d \sum_{k=1}^m \: [r_k n]$ for each positive integer $n$. Determine the
minimum and maximum values of $f(n)$. Here $[x]$ denotes the greatest integer
less than or equal to $x$.
\end{enumerate}
\end{document}
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\bibitem{Jiang2005} Jiang, Z. (2005). \textit{Snow leopards in the Dulan International Hunting Ground, Qinghai, China}.
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https://fifthestate.anarchistlibraries.net/library/391-springsummer-2014-mutual-aid-saves-fifth-estate.tex | anarchistlibraries.net | CC-MAIN-2021-17 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-17/segments/1618038083007.51/warc/CC-MAIN-20210415035637-20210415065637-00376.warc.gz | 357,725,717 | 2,833 | \documentclass[DIV=12,%
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\title{Mutual Aid Saves \emph{Fifth Estate}}
\date{}
\author{Fifth Estate Collective}
\subtitle{}
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We came close to a disaster during preparation of this edition, but due to the incredible mutual aid offered by readers and supporters, we have come out stronger than ever. We were very close to our publication deadline when our computer crashed! We could have lost all of our work, plus, we needed to spend almost a thousand dollars on a new machine.
We sent out an email blast to our list about the situation and readers and supporters responded exceedingly generously with donations, further offers of technical assistance, and subscription renewals, all made in the spirit of mutual aid.
Fortunately, all of our data was backed up so we lost only several weeks time while good friends helped reinstall everything on our new machine.
If it hadn’t been for this support, it could have been the end of our almost 50-year-old publication. Thanks to all of you who helped, and the issue you are holding is our part of the circle.
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Fifth Estate Collective
Mutual Aid Saves \emph{Fifth Estate}
\bigskip
\href{https://www.fifthestate.org/archive/391-springsummer-2014/mutual-aid-saves-fifth-estate}{\texttt{https://www.fifthestate.org/archive/391-springsummer-2014/mutual-aid-saves-fifth-estate}}
Fifth Estate \#391, Spring\Slash{}Summer 2014 — Anarchy!
\bigskip
\textbf{fifthestate.anarchistlibraries.net}
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%\SetWatermarkText{Lycée Henri IV}
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{0pt}%␣retrait␣à␣gauche
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\begin{center}
{\Large \textbf{Corrigé du baccalauréat ES -- Polynésie~\decofourright\\[4pt]16 juin 2017}}
\end{center}
\medskip
\textbf{Exercice 1 \hfill 4 points}
\textbf{Commun à tous les candidats}
\medskip
\begin{minipage}{\linewidth}
\emph{Cet exercice est un QCM (questionnaire à choix multiples). Pour chacune des questions posées, une seule des quatre réponses est exacte. Recopier le numéro de la question et la réponse exacte. Aucune justification n'est demandée.\\
Une réponse exacte rapporte $1$ point, une réponse fausse ou l'absence de réponse ne rapporte ni n'enlève de point. Une réponse multiple ne rapporte aucun point.}
\end{minipage}
\medskip
\surj{\textbf{Les justifications n'étaient pas demandées, elles sont données ici à titre indicatif}}
\medskip
\begin{enumerate}
\item La solution exacte de l'équation $\left( \dfrac{1}{2} \right)^x = \dfrac{3}{10}$ est :
\medskip
\begin{enumerate}
\begin{tabularx}{\linewidth}{*{4}{X}}
\item 1,74 & \item \surj{$\dfrac{\ln 10-\ln 3}{\ln 2}$} & \item $-\dfrac{\ln 3}{\ln 5}$ & \item 0,5 \end{tabularx} \end{enumerate}
\medskip
\begin{solution}
$\left(\dfrac{1}{2} \right) ^x=\dfrac{3}{10} \equi 2^x =\dfrac{10}{3}$
$\hphantom{\left(\dfrac{1}{2} \right) ^x=\dfrac{3}{10}} \equi \ln\left(2^x \right)=\ln\left(\dfrac{10}{3} \right)$
$\hphantom{\left(\dfrac{1}{2} \right) ^x=\dfrac{3}{10}} \equi x\ln(2) =\ln(10)-\ln(3)$
$\hphantom{\left(\dfrac{1}{2} \right) ^x=\dfrac{3}{10}} \equi x =\dfrac{\ln(10)-\ln(3)}{\ln(2)}$
\end{solution}
\item $f$ est la fonction définie pour tout nombre réel $x$ par $f(x)=2x\mathrm{e}^{x^2}$.
La valeur exacte de l'intégrale $\displaystyle\int_{-2}^{2}f(x)\;\mathrm {d}x$ est :
\medskip
\begin{enumerate}
\begin{tabularx}{\linewidth}{*{4}{X}}
\item $4\mathrm{e}^{4}-4\mathrm{e}^{-4}$ & \item $4\left(\mathrm{e}^{4}+\mathrm{e}^{-4}\right)$ & \item \surj{0} & \item 1 \end{tabularx} \end{enumerate}
\medskip
\begin{solution}
\integ{-2}{2}{2x\text{e}^{x^2}}{x}=~$\left[\text{e}^{x^2} \right]_{_{-2}}^{^2}=\text{e}^{4}-\text{e}^{4}=0$
\end{solution}
\item $f$ est la fonction définie pour tout $x$ de l'intervalle $]~0;~+ \infty [$ par $f(x)=(2x+3)\ln x$.
On admet que la fonction $f$ est dérivable sur l'intervalle $]~0;~+ \infty [$ .
On rappelle que $f'$ désigne la fonction dérivée de la fonction $f$.
Pour tout nombre réel $x$ de 'intervalle $]~0;~+ \infty [$ on a :
\medskip
\begin{enumerate}
\begin{tabularx}{\linewidth}{*{2}{X}}
\item $f'(x)=\dfrac{2x+3}{x}$ & \item $f'(x)=\dfrac{2}{x}$ \\ \item \surj{$f'(x)=2\ln x + \dfrac{3}{x} +2$} & \item $f'(x)=2\ln x + \dfrac{3}{x}$ \end{tabularx} \end{enumerate}
\medskip
\begin{solution}
$f=uv \Longrightarrow f'=u'v+uv'$ avec $\begin{cases}
u(x)=2x+3\\v(x)=\ln(x)
\end{cases}~\Longrightarrow \begin{cases}
u'(x)=2\\v'(x)=\dfrac{1}{x}
\end{cases}$
$f'(x)=2\ln(x)+(2x+3)\times \dfrac{1}{x}=2\ln(x)+\dfrac{3}{x}+2$
\end{solution}
\item Une grandeur a été augmentée de 5\,\% la première année, puis de 7\,\% la deuxième année.
Sur ces deux années, le pourcentage global d'augmentation est égal à :
\medskip
\begin{enumerate}
\begin{tabularx}{\linewidth}{*{4}{X}}
\item 12\,\% & \item 35\,\% & \item 0,35\,\% & \item \surj{12,35\,\%} \end{tabularx} \end{enumerate}
\end{enumerate}
\begin{solution}
Le coefficient multiplicateur associé à une hausse de 5\% est 1,05 et celui associé à une hausse de 7\% est 1,07
Donc sur les deux ans, le coefficient est $1,05\times 1,07=1,1235$ qui correspond à une hausse de 12,35\%
\end{solution}
\vspace{0,5cm}
\textbf{Exercice 2 \hfill 5 points}
\textbf{Commun à tous les candidats}
\medskip
Les trois parties de cet exercice sont indépendantes.
\begin{center}
\textsf {\textbf{\textsc{partie a}}}
\end{center}
\begin{minipage}{\linewidth}
D'après le \og bilan des examens du permis de conduire \fg{} pour l'année 2014 publiée par le Ministère de l'Intérieur en novembre 2015, 20\,\% des personnes qui se sont présentées à l'épreuve pratique du permis de conduire avaient suivi la filière de l'apprentissage anticipé de la conduite (AAC). Parmi ces candidats, 75\,\% ont été reçus à l'examen. Pour les candidats n'ayant pas suivi la filière AAC, le taux de réussite à l'examen était seulement de 56,6\,\%.
On choisit au hasard l'un des candidats à l'épreuve pratique du permis de conduire en 2014.
On considère les évènements suivants :
\begin{itemize}
\item $A$ \og le candidat a suivi la filière AAC \fg{} ;
\item $R$ \og le candidat a été reçu à l'examen \fg .
\end{itemize}
On rappelle que si $E$ et $F$ sont deux évènements, la probabilité de l'évènement $E$ est notée $P(E)$ et celle de $E$ sachant $F$ est notée $P_F(E)$. De plus $\overline{E}$ désigne l'évènement contraire de $E$.
\end{minipage}
\medskip
\begin{enumerate}
\item
\begin{enumerate}
\item Donner les probabilités $P(A)$, $P_A(R)$ et $P_{\overline{A}}(R)$.
\begin{solution}
$P(A)=0,2$ car 20\% des candidats ont suivi la filière AAC
$P_A(R)=0,75$ car parmi les candidats ayant suivi la filière AAC, 75\% sont reçus
$P_{\overline{A}}(R)=0,566$ car parmi les candidats n'ayant pas suivi la filière AAC, 56,6\% sont reçus
\end{solution}
\item Traduire la situation par un arbre pondéré.
\begin{solution}
avec les données précédentes ont obtient l'arbre suivant:
\begin{center}
\psset{nodesepA=0mm,nodesepB=4pt,levelsep=20mm,treesep=10mm}
\pstree[treemode=R]{\Tdot}
{
\pstree
{\TR{$A~$}\taput{\small $0,2$}}
{
\TR{$R$}\taput{\small $0,75$}
\TR{$\overline{R}$}\tbput{\small $0,25$}
}
\pstree
{\TR{$\overline{A}~$}\tbput{\small $0,8$}}
{
\TR{$R$}\taput{\small $0,566$}
\TR{$\overline{R}$}\tbput{\small $0,434$}
}
}
\end{center}
\end{solution}
\end{enumerate}
\item
\begin{enumerate}
\item Calculer la probabilité $P\left( A \cap R\right)$.
\begin{solution}
$P\left( A \cap R\right)=P_A(R)\times P(A)=0,75\times0,2=0,15$
\end{solution}
\item Interpréter ce résultat dans le cadre de l'énoncé.
\begin{solution}
Il s'agit de la probabilité que la personne choisie ait réussi et ait suivi la filière AAC
\end{solution}
\end{enumerate}
\item Justifier que $P(R)= \np{0,6028}$.
\begin{solution}
$A$ et $\overline{A}$ forment une partition de l'univers donc d'après les probabilités totales on a
$P(R)=P\left( A \cap R\right)+P\left( \overline{A} \cap R\right)=0,15+P_{\overline{A}}(R)\times P\left( \overline{A} \right)=0,15+0,566\times 0,8=0,6028$
\end{solution}
\item Sachant que le candidat a été reçu à l'examen, calculer la probabilité qu'il ait suivi la filière AAC.
On donnera une valeur approchée à $10^{-4}$ près de cette probabilité.
\begin{solution}
On cherche $P_R(A)$
$P_R(A) = \dfrac{P\left( A \cap R\right)}{P(R)}=\dfrac{0,15}{\np{0,6028}}\approx \np{0,2488}$
\end{solution}
\end{enumerate}
\begin{center}
\textsf {\textbf{\textsc{partie b}}}
\end{center}
\begin{minipage}{\linewidth}
Un responsable d'auto-école affirme que pour l'année 2016, la probabilité d'être reçu à l'examen est égale à $0,62$.
Ayant des doutes sur cette affirmation, une association d'automobilistes décide d'interroger $400$ candidats à l'examen parmi ceux de 2016. Il s'avère que $220$ d'entre eux ont effectivement obtenu le permis de conduire.
\end{minipage}
\medskip
\begin{enumerate}
\item Déterminer un intervalle de fluctuation asymptotique au seuil de 95\,\% de la fréquence de candidats reçus dans un échantillon aléatoire de 400 candidats.
\begin{solution}
La taille de l'échantillon est $n=400$ et la proportion supposée de candidats reçus dans la population est $p=0,62$
On a $n\geqslant 30~,~np=80 \geqslant 5~\text{et}~n(1-p)=80 \geqslant 5$ on peut donc bâtir l'intervalle de fluctuation asymptotique au seuil de 95\%
$I=\left[p-1,96\dfrac{\sqrt{p(1-p)}}{\sqrt{n}}~;~p+1,96\dfrac{\sqrt{p(1-p)}}{\sqrt{n}} \right]\approx [0,572~;~0,668]$
\end{solution}
\item Peut-on émettre des doutes sur l'affirmation du responsable de cette auto-école ?
Justifier votre réponse.
\begin{solution}
la fréquence orbservée de reçus sur l'échantillon est $f=\dfrac{220}{400}=0,55 \notin I$
On peut donc remettre en doute l'affirmation du responsable au risque de 5\% de se tromper.
\end{solution}
\end{enumerate}
\begin{center}
\textsf {\textbf{\textsc{partie c}}}
\end{center}
\begin{minipage}{\linewidth}
Selon une enquête menée en 2013 par l'association \og Prévention Routière \fg, le coût moyen d'obtention du permis de conduire atteignait environ \np{1500}~\euro . On décide de modéliser le coût d'obtention du permis de conduire par une variable aléatoire $X$ qui suit la loi normale d'espérance $\mu= \np{1500}$ et d'écart-type $\sigma = 410$.
\end{minipage}
\medskip
\begin{enumerate}
\item Déterminer une valeur approchée à $10^{-2}$ près de la probabilité que le coût du permis de conduire soit compris entre \np{1090}~\euro{} et \np{1910}~\euro .
\begin{solution}
$P(\np{1090}\leqslant X \leqslant \np{1910})=P(\mu-\sigma \leqslant X \leqslant \mu+\sigma)\approx 0,68$
\end{solution}
\item Déterminer $P\left( X \leqslant \np{1155}\right)$.
On donnera le résultat sous forme approchée à $10^{-2}$ près.
\begin{solution}
$P(X \leqslant \np{1155})=0,5-P(\np{1155}\leqslant X \leqslant \np{1500})\approx 0,20$
\end{solution}
\item
\begin{enumerate}
\item Par la méthode de votre choix, estimer la valeur du nombre réel $a$ arrondi à l'unité, vérifiant $P\left(X \geqslant a\right)=0,2$.
\begin{solution}
La courbe de la fonction densité de la loi normale suivie par $X$ est symétrique par rapport à la droite d'équation $x=\np{1500}$
On a $P(X \leqslant \np{1155})=P(X \leqslant \np{1500}-\np{345})\approx 0,20$ donc par symétrie, on peut affirmer
$P(X \geqslant \np{1500}+\np{345})=P(X \geqslant \np{1845})\approx 0,20$
donc $a = \np{1845}$
\end{solution}
\item Interpréter ce résultat dans le cadre de l'énoncé.
\begin{solution}
La probabilité qu'un permis coûte plus de \np{1845} \euro est d'environ 0,2
\end{solution}
\end{enumerate}
\end{enumerate}
\vspace{0,5cm}
\textbf{Exercice 3 \hfill 5 points}
\textbf{Candidats ES n'ayant pas suivi l'enseignement de spécialité et candidats de la série L}
\medskip
\begin{minipage}{\linewidth}
En 2015, les forêts couvraient environ \np{4000} millions d'hectares sur terre. On estime que, chaque année, cette surface diminue de 0,4\,\%. Cette perte est en partie compensée par le reboisement, naturel ou volontaire, qui est estimé à $7,2$ millions d'hectares par an.
On considère la suite $\left(u_n\right)$ définie par $u_0= \np{4000}$ et, pour tout entier naturel $n$,
\mbox{$u_{n+1}= 0,996\times u_n +7,2$}.
\end{minipage}
\begin{enumerate}
\item Justifier que, pour tout entier naturel $n$, $u_n$ permet d'obtenir une estimation de la surface mondiale de forêt, en millions d'hectares l'année $2015 + n$.
\begin{solution}
Le coefficient multiplicateur associé à une baisse de 0,4\% est $1-\dfrac{0,4}{100}=0,996$
Pour trouver la surface de l'année de rang $(n+1)$, on multiplie la surface de l'année de rang $n$ par 0,996 puis on ajoute 7,2 millions d'hectares de reboisement
On a donc bien $u_{n+1}=0,996u_n+7,2$ de plus $u_0=\np{4000}$ car en 2015 il y avait \np{4000} millions d'hectares de forêts sur terre
\end{solution}
\item Recopier et compléter l'algorithme ci-dessous pour qu'il calcule et affiche la première année pour laquelle la surface totale de forêt couvre moins de \np{3500} millions d'hectares sur terre.
\begin{solution}
\begin{center}
\begin{tabularx}{.6\linewidth}{|l X|}\hline
Variables : & $N$ est un entier naturel\\
& $U$ est un nombre réel\\
Traitement :& $n$ prend la valeur 0\\
& {\red Tant que $U > \np{3500}$}\\
& ~~~{\red $U$ prend la valeur $0,996U + 7,2$}\\
& ~~~{\red $N$ prend la valeur $N + 1$}\\
& {\red Fin Tant que}\\
Sortie : & Afficher $N$\\ \hline
\end{tabularx}
\end{center}
\end{solution}
\item On considère la suite $\left(v_n\right)$ définie pour tout entier naturel $n$ par $v_n= u_n- \np{1800}$.
\begin{enumerate}
\item Démontrer que la suite $\left(v_n\right)$ est géométrique puis préciser son premier terme et sa raison.
\begin{solution}
$\forall n \in \N~,~v_{n+1}=u_{n+1}-\np{1800}$
$\hphantom{\forall n \in \N~,~v_{n+1}}=0,996u_n+7,2-\np{1800}$
$\hphantom{\forall n \in \N~,~v_{n+1}}=0,996u_n-\np{1792,8}$
$\hphantom{\forall n \in \N~,~v_{n+1}}=0,996\left(u_n-\np{1800} \right) $
$\hphantom{\forall n \in \N~,~v_{n+1}}=0,996v_n$
On en déduit que $\left( v_n\right)$ est géométrique de raison $q=0,996$ et de premier terme $v_0=u_0-\np{1800}=\np{2200}$
\end{solution}
\item En déduire que pour tout entier naturel $n$, on a : $u_n= 2200\times 0,996^n+ \np{1800}$.
\begin{solution}
$\forall n \in \N~,~v_n=v_0\times q^n=\np{2200}\times 0,996^n$
On en déduit que $\forall n \in \N~,~u_n=v_n+\np{1800} = \np{2200}\times 0,996^n+\np{1800}$
\end{solution}
\item Selon ce modèle et si le phénomène perdure, la surface des forêts sur terre va-t-elle finir par disparaître ? Justifier la réponse.
\begin{solution}
$\left|0,996 \right|<1$ donc \Lim{n}{+\infty}{0,996^n}=~0 et par opération sur les limites, \Lim{n}{+\infty}{u_n}=~\np{1800}
Donc selon ce modèle les forêts ne devraient pas disparaître mais leur surface devrait se stabiliser aux alentours de $\np{1800}$ millions d'hectares
\end{solution}
\end{enumerate}
\item Une étude montre que, pour compenser le nombre d'arbres détruits ces dix dernières années, il faudrait planter 140 milliards d'arbres en 10 ans.
En 2016 on estime que le nombre d'arbres plantés par l'Organisation des Nations unies (ONU) est de 7,3 milliards.
On suppose que le nombre d'arbres plantés par l'ONU augmente chaque année de 10\,\%.
L'ONU peut-elle réussir à replanter 140 milliards d'arbres de 2016 à 2025 ?
Justifier la réponse.
\begin{solution}
Soit $w_n$ le nombre d'arbres replantés en 2016 + $n$ en milliards.
$\left(w_n \right)$ est une suite géométrique de raison $q=1,1$ (coefficient multiplicateur associé à la hausse de 10\%) et de premier terme $w_0 = 7,3$.
On cherche à savoir si la somme $S = w_0 + w_1 + \cdots + w_9$ est supérieure à 140.
$S=w_0+w_1+ \cdots + w_9=w_0\left(\dfrac{q^{10}-1}{q-1} \right)=7,3\times \left(\dfrac{1,1^{10}-1}{0,1} \right)\approx 116$.
L'ONU n'atteindra donc pas l'objectif des 140 milliards d'arbres plantés entre 2016 et 2025
\end{solution}
\end{enumerate}
\newpage
\textbf{Exercice 3 \hfill 5 points}
\textbf{Candidats de la série ES ayant suivi l'enseignement de spécialité}
\medskip
Les deux parties de cet exercice sont indépendantes.
\begin{center}
\textsf {\textbf{\textsc{partie a}}}
\end{center}
\begin{minipage}{\linewidth}
Alex a téléchargé sur son smartphone un jeu lui permettant de combattre des animaux virtuels par localisation GPS. Le graphe pondéré représenté ci-dessous illustre le trajet qu'Alex doit suivre en marchant dans les rues de sa ville et le nombre d'animaux virtuels qu'il doit combattre sur la route suivie
\end{minipage}
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\end{center}
\begin{minipage}{\linewidth}
À l'aide d'un algorithme, déterminer le nombre minimal de créatures qu'Alex doit combattre s'il part du point O pour arriver au point F de la ville. Détailler les étapes de l'algorithme.
\end{minipage}
\begin{solution}
On applique l'algorithme de DIJKSTRA
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}\hline
Départ-Sommet & O & A & B & C & D & E & F \\ \hline
C & \surj{0} &2,O & 5,O & 4, O & $+\infty$ & $+\infty$ & $+\infty$ \\ \hline
A(O) & & \surj{2, 0} &\sout{5, 0} & 4,O & \sout{$+\infty$} & $+\infty$ & $+\infty$ \\
& & &4, A & &9, A & & \\\hline
B(A) & & &\surj{4, A} &4, O &9, A &\sout{$+\infty$} &$+\infty$\\
& & & &\sout{5, B} & 9, B & 7, B & \\ \hline
C(O) & & & &\surj{4, O} & 9, A-B & 7, B & $+\infty$ \\
& & & & & &\sout{8, C} & \\\hline
E(B) & & & & &\sout{9, A-B} &\surj{7, B} &\sout{$+\infty$} \\
& & & & & 8, E & & 15, E \\ \hline
D(E) & & & & &\surj{8,E} & & \sout{(15, E)} \\
& & & & & & &\surj{14, D} \\ \hline
\end{tabular}
\end{center}
\vspace{0,3cm}
Alex doit combattre au minimum 14 créatures en allant de O à F : \surj{ $\text{O} \longrightarrow \text{A} \longrightarrow \text{B} \longrightarrow \text{E} \longrightarrow \text{D}
\longrightarrow \text{F}$ }
\end{solution}
\begin{center}
\textsf {\textbf{\textsc{partie b}}}
\end{center}
\begin{minipage}{\linewidth}
Alex retrouve d'autres personnes, ayant le même jeu, dans le parc de la ville dans le but de comparer le nombre de créatures qu'ils ont combattues.
Le premier jour, 8 personnes se sont retrouvées dans le parc. Le second jour, on comptait 25 personnes et le troisième jour, 80 personnes se sont retrouvées dans le parc.
Soit $f$ la fonction définie par $f(x)= ax^2 + bx + c$, où $a$, $b$ et $c$ sont trois nombres réels et $x$ un nombre entier compris entre 1 et 10. On admet que la fonction $f$ modélise le nombre de personnes qui se retrouvent dans le parc le $x$-ième jour.
\end{minipage}
\begin{enumerate}
\item Traduire l'énoncé par un système de trois équations à trois inconnues $a$, $b$ et $c$.
\begin{solution}
D'après l'énoncé on a $f(1)=8~,~f(2)=25~\text{et}~f(3)=80$. on obtient le système suivant
\begin{center}
$(S)~:~\left\lbrace \begin{array}{rcccccr}
a&+&b&+&c&=&8\\4a&+&2b&+&c&=&10\\9a&+&3b&+&c&=&80
\end{array} \right.$
\end{center}
\end{solution}
\item Vérifier que ce système est équivalent à l'équation $AX=B$ avec :\[A=\begin{pmatrix}
1 & 1 & 1 \\
4 & 2 & 1 \\
9 & 3 & 1 \\
\end{pmatrix} \, \text{, }\: X=\begin{pmatrix}a \\b \\c \\\end{pmatrix}\:\text{ et }\: B=\begin{pmatrix}8 \\25\\80 \\\end{pmatrix} \]
\begin{solution}
$AX=\begin{pmatrix}
a+b+c\\4a+2b+c\\9a+3b+c
\end{pmatrix}$ donc on a bien $(S) \equi AX=B$
\end{solution}
\item Soit la matrice $M=\begin{pmatrix}0,5 & -1 & 0,5 \\-2,5 & 4 & -1,5 \\3 & -3 & 1 \\\end{pmatrix}$.
\begin{enumerate}
\item Calculer $M\times A$.
\begin{solution}
$MA=\begin{pmatrix}
1&0&0\\0&1&0\\0&0&1
\end{pmatrix}=I_3$ avec $I_3$ la matrice identité d'ordre 3
\end{solution}
\item Que représente la matrice $M$ pour la matrice $A$ ?
\begin{solution}
On a $MA=I_3$ et on montre facilement que $AM=I_3$
Donc $AM=MA=I_3$ ce qui signifie que $A$ est inversible et $A^{-1}=M$
\end{solution}
\end{enumerate}
\item A l'aide d'un calcul matriciel, déterminer les valeurs des nombres $a,~b~\text{et}~c$
\begin{solution}
$(S) \equi AX=B \equi MAX=MB \equi I_3X=MB \equi X=MB$
$MB=\begin{pmatrix}
19\\-40\\29
\end{pmatrix}$
~~donc~~ $\begin{cases}
a=19\\b=-40\\c=29
\end{cases}$
\end{solution}
\item Le parc de la ville a une capacité d'accueil de \np{2500} personnes.
Selon ce modèle, le parc risque-t-il de refuser d'accueillir des personnes un de ces dix jours ?
Justifier la réponse.
\begin{solution}
On peut faire un tableau des valeurs prises par $f$:
\begin{center}
\begin{tabularx}{0.8\linewidth}{|c|*{10}{>{\centering \arraybackslash}X|}}\hline
$x$ &1 &2 &3 &4 &5 &6 &7 &8 &9 &10\\ \hline
$f(x)$ &8 &25 &80 &173&304&473&680&925&\np{1208} &\np{1529}\\ \hline
\end{tabularx}
\end{center}
On voit donc que le parc ne rique pas de refuser d'acceuillir des personnes sur un de ces dix jours
\end{solution}
\end{enumerate}
\textbf{Exercice 4 \hfill 6 points}
\textbf{Commun à tous les candidats}
\medskip
Soit $f$ une fonction définie sur l'intervalle $[ 0 ; 5]$ par $f(x)=(ax-2)\mathrm{e}^{-x}$, où $a$ est un nombre réel.
On admet dans tout l'exercice que la fonction $f$ est deux fois dérivable sur l'intervalle $[ 0~;~5]$.
La courbe représentative $\mathcal{C}$ de la fonction $f$ est donnée ci-dessous dans un repère d'origine O.
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Les courbes $\mathcal{C}$ et $\mathcal{D}$ passent toutes les deux par le point $A(0 ; -2)$.
La droite $\mathcal{D}$ est tangente à la courbe $\mathcal{C}$ au point $A$ et admet pour équation $y=10x-2$.
On rappelle que $f'$ désigne la fonction dérivée de la fonction $f$.
\medskip
\begin{enumerate}
\item Donner, à l'aide des informations ci-dessus et sans justifier les valeurs de $f(0)$ et de $f'(0)$.
\begin{solution}
$f(0)=-2$ et $f'(0)=10$
\end{solution}
\item \begin{enumerate}
\item Montrer que pour tout réel $x$ de l'intervalle $[ 0~;~5]$ on a :\[f'(x) = (- ax + a + 2)\mathrm{e}^{- x}\]
\begin{solution}
$f=uv \Longrightarrow f'=u'v+uv'$ avec $\begin{cases}
u(x)=ax-2\\v(x)=\text{e}^{-x}
\end{cases}~\Longrightarrow \begin{cases}
u'(x)=a\\v'(x)=-\text{e}^{-x}
\end{cases}$
$f'(x)=a\text{e}^{-x}-(ax-2)\text{e}^{-x}=(-ax+a+2)\text{e}^{-x}$
\end{solution}
\item Déduire des questions précédentes que $a = 8$.
\begin{solution}
$f'(0)=10 \equi a+2=10 \equi a=8$
\end{solution}
\item Donner l'expression de $f'(x)$.
\begin{solution}
On en déduit que $f'(x)=(-8x+10)\text{e}^{-x}$
\end{solution}
\end{enumerate}
\item
\begin{enumerate}
\item Préciser le signe de $f'(x)$ sur l'intervalle $[ 0~;~5]$. On pourra faire un tableau.
\begin{solution}
$\text{e}^{-x}>0$ sur $[0~;~5]$ donc $f'(x)$ est du signe de $(-8x+10)$
\begin{center}
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\end{pspicture}
\end{center}
\end{solution}
\item En déduire le tableau des variations de la fonction $f$ sur ce même intervalle.
\begin{solution}
On obtient les variations de $f$ sur $[0~;~5]$
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\end{center}
\end{solution}
\item Résoudre sur l'intervalle $[ 0 ; 5]$ l'équation $f(x)=0$.
\begin{solution}
$f(x)=0\equi (8x-2)\text{e}^{-x}=0 \equi 8x-2 = 0 \equi x =0,25$
\end{solution}
\end{enumerate}
\item À l'aide d'un logiciel de calcul formel, on a obtenu les résultats suivants :
\begin{center}
\begin{tabularx}{0.6\linewidth}{|c|X|}\hline
\multirow{2}*{1}& \rule[-1ex]{0pt}{3ex} $g(x) := \left(- 8*x + 10\right) * \mathrm{exp} (- x)$\\
&$\to g(x) := \left(- 8x +10\right)\mathrm{e}^{- x}$\\ \hline
\multirow{2}*{2}& \rule[-1ex]{0pt}{3ex} Dériver $\left[g(x) , x\right]$\\
&$\to ( 8 * x -18) * \mathrm{exp} (- x)$\\ \hline
\multirow{2}*{3}& \rule[-1ex]{0pt}{3ex} Résoudre $\left[( 8 * x -18) * \mathrm{exp} (- x) >0 , x\right]$\\
&$\to x > 9/4$\\ \hline
\end{tabularx}
\end{center}
En utilisant ces résultats :
\smallskip
\begin{enumerate}
\item Donner l'expression de $f''$, fonction dérivée seconde de la fonction $f$.
\begin{solution}
La première ligne du logiciel donne l'expression de $f'(x)$ donc la deuxième ligne donne celle de $f''(x)$
On a donc $f''(x)=(8x-18)\text{e}^{-x}$
\end{solution}
\item Justifier que la courbe $\mathcal{C}$ admet un point d'inflexion dont on donnera la valeur exacte de l'abscisse.
\begin{solution}
La troisième ligne permet de déterminer le signe de $f''(x)$:
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\end{center}
Donc $f''(x)$ s'annule en $x = \dfrac{9}{4} = 2,25$ en changeant de signe, on peut alors affirmer que $\mathcal{C}$ admet un point d'inflexion au point d'abscisse $x = \dfrac{9}{4}$
\end{solution}
\end{enumerate}
\item Une entreprise fabrique des grille-pains. Après avoir fait une étude, son directeur constate que si l'entreprise fabrique chaque jour $x$ milliers de grille-pains (où $x$ est un nombre réel de l'intervalle $[ 0 ; 5]$), alors le bénéfice quotidien est donné, en centaine de milliers d'euros, par la fonction $f$ définie par : \[f(x)=(8x - 2)\mathrm{e}^{- x}\]
\begin{enumerate}
\item Quelle quantité de grille-pains l'entreprise doit-elle fabriquer afin de réaliser un bénéfice maximal ?
\begin{solution}
Le bénéfice sera maximal quand $f$ sera maximal. D'après la question 3, on peut déduire que le bénéfice sera maximal pour \np{1250} grille-pains fabriqués
\end{solution}
\item Quel est alors la valeur de ce bénéfice maximal ?
On donnera une valeur approchée du résultat à l'euro près.
\begin{solution}
Le bénéfice maximal est $f(1,25)=8\text{e}^{-1,25}\approx \np{2,29204}$ soit environ \np{229204} \euro.
\end{solution}
\end{enumerate}
\end{enumerate}
\end{document} |
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% (try to) forbid widows/orphans
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% given that we said footinclude=false, this should be safe
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\title{Version 2.2}
\date{2018-08-16}
\author{}
\subtitle{}
% https://groups.google.com/d/topic/comp.text.tex/6fYmcVMbSbQ/discussion
\hypersetup{%
pdfencoding=auto,
pdftitle={Version 2.2},%
pdfauthor={},%
pdfsubject={},%
pdfkeywords={releases}%
}
\begin{document}
\begin{titlepage}
\strut\vskip 2em
\begin{center}
{\usekomafont{title}{\huge Version 2.2\par}}%
\vskip 1em
\vskip 2em
\vskip 1.5em
\vfill
{\usekomafont{date}{2018-08-16\par}}%
\end{center}
\end{titlepage}
\cleardoublepage
\tableofcontents
% start a new right-handed page
\cleardoublepage
\section{2.226 2018-08-16}
\begin{itemize}
\item\relax
Expose the legacy links via \Slash{}api\Slash{}legacy-links (\#205)
\item\relax
Show the git log when pulling (\#207)
\end{itemize}
\section{2.225 2018-07-29}
\begin{itemize}
\item\relax
Improve category sorting (take numbers in account)
\end{itemize}
\section{2.224 2018-07-28}
\begin{itemize}
\item\relax
Dependencies bump with RTL support in the parser
\item\relax
Update the wordpress import script
\end{itemize}
\section{2.223 2018-07-20}
\begin{itemize}
\item\relax
New Bookbuilder\Slash{}custom format options: centerchapter, centersection,
continuefootnotes
\item\relax
Respect the default sorting setting in static indexes
\end{itemize}
\section{2.222 2018-07-04}
\begin{itemize}
\item\relax
Add Bahasa Indonesia support
\item\relax
AMW-Meta: RSS feeds, avoid use of the DB
\item\relax
Mirror script: port to perl and speed up
\end{itemize}
\section{2.221 2018-06-10}
\begin{itemize}
\item\relax
Add nl tranlations
\end{itemize}
\section{2.220 2018-05-13}
\begin{itemize}
\item\relax
Add empty localization for Turkish language
\item\relax
Add new imposition schema: 2x4x1 (via upgraded PDF::Imposition)
\item\relax
Provide a disabled and not yet documented application to provide
an aggregated search of a given set of amusewiki sites residing on
the same server
\end{itemize}
\section{2.210 2018-05-03}
\begin{itemize}
\item\relax
Various fixes and optimizations on mirror routes and Xapian
\end{itemize}
\section{2.203 2018-04-23}
\begin{itemize}
\item\relax
Bump dependency on Text::Amuse 1.10 with improved anchors and
restored compatibility with Emacs Muse
\item\relax
Improve anchor display on editing and preview
\end{itemize}
\section{2.202 2018-03-31}
\begin{itemize}
\item\relax
Bump dependency on Text::Amuse 1.01 and Text::Amuse::Compile 1.04
\item\relax
Do not list ignored files in \Slash{}mirror.txt
\item\relax
Debian: install a disabled fontconfig setting for woff fonts
\item\relax
Optimize static indexes
\item\relax
Use option restrict\_mirror to disable mirroring
\item\relax
Update ru i18n (thanks @labdsf)
\end{itemize}
\section{2.201 2018-03-27}
\begin{itemize}
\item\relax
Bump dependency on Text::Amuse 1.00 and Text::Amuse::Compile 1.03
\item\relax
Fix secondary footnotes CSS
\end{itemize}
With Text::Amuse::Compile 1.03, a relatively recent \texttt{bigfoot.sty}
version is needed for the LaTeX compiler. If your installation is
missing it (notably jessie, stretch is fine), you can install it as
the user running the amusewiki instance with the following commands:
\begin{alltt}
\$ cd /tmp/
\$ mkdir -p `kpsewhich -var-value TEXMFHOME`/tex/latex/bigfoot
\$ wget http://mirrors.ctan.org/macros/latex/contrib/bigfoot.zip
\$ unzip bigfoot.zip
\$ cd bigfoot
\$ make
\$ mv *.sty `kpsewhich -var-value TEXMFHOME`/tex/latex/bigfoot
\$ texhash `kpsewhich -var-value TEXMFHOME`
\end{alltt}
\section{2.200 2018-03-20}
\begin{itemize}
\item\relax
Refactored the search page, using facets now
\item\relax
Updated ru (thanks @labdsf), it and hr translations, add cs
preliminary support
\item\relax
serve a list of urls to mirror under \texttt{/mirror.txt} and \texttt{/mirror.ts.txt}
to feed wget with it. Plus provided a client in
\texttt{script/mirror-site.sh}
\end{itemize}
This version brings a refactored, faceted search page. To get the
facets working, you need a \texttt{Search::Xapian} module newer then 1.2.22.0.
Notably, Debian jessie has a 1.2.19.0. Another problem is that the
current 1.2.25.0 version on CPAN fails to install using Xapian system
libraries in the 1.2 branch (I believe the issue, a single test
failing, will be addressed in the next \texttt{Search::Xapian} release).
Please note that without satisfying this dependency, Amusewiki will
still work without problems. You are just going to miss a nice
feature.
There are 4 cases here:
\begin{itemize}
\item\relax
Debian stretch, Amusewiki installed with deb package: works out of
the box.
\item\relax
Recent distro, with Xapian (system library) > 1.4, and Amusewiki
installed from git. It could already work out of the box. If not
so, upgrading \texttt{Search::Xapian} from CPAN (e.g. \texttt{cpanm Search::Xapian})
will do.
\item\relax
Older distro (with Xapian system library 1.2) and Amusewiki
installed with git: you need to install \texttt{Search::Xapian} from CPAN.
If it refuses to install because of a single test failing, force
the installation skipping the tests \texttt{cpanm -f Search::Xapian}. I
asked the upstream, and they confirmed it's harmless.
\item\relax
Debian jessie, Amusewiki installed with deb package: you need to
build a recent \texttt{Search::Xapian.} You can install it from CPAN
system-wide, but it has the downside of making your system dirty,
so it's not recommended. Instead, you could build a deb package
with the following procedure (the \texttt{libsearch-xapian-perl} sources
are provided as courtesy for this case, incorporating the needed
patch) and install the resulting deb (as root) in the parent
directory.
\end{itemize}
\begin{alltt}
\# apt-get install libdevel-leak-perl libtest-pod-perl devscripts \textbackslash{}
build-essential fakeroot libxapian-dev
\$ git clone https://github.com/melmothx/amusewiki-debian-packages.git
\$ cd amusewiki-debian-packages/libsearch-xapian-perl-1.2.24.0/
\$ debuild -i -us -uc -b
\end{alltt}
% begin final page
\clearpage
% new page for the colophon
\thispagestyle{empty}
\begin{center}
\bigskip
\includegraphics[width=0.25\textwidth]{logo-amw.pdf}
\bigskip
\end{center}
\strut
\vfill
\begin{center}
Version 2.2
2018-08-16
\bigskip
\bigskip
\textbf{amusewiki.org}
\end{center}
% end final page with colophon
\end{document}
% No format ID passed.
|
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\left(\mu+n+\tfrac{1}{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\mu+\tfrac{%
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You can get started by \textbf{double clicking} this text block and begin editing. You can also click the \textbf{Text} button below to add new block elements. Or you can \textbf{drag and drop an image} right onto this text. Happy writing!
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https://bootlin.com/pub/conferences/2017/elce/brezillon-crypto-framework/src/beamerthemeFreeElectrons.sty | bootlin.com | CC-MAIN-2018-13 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2018-13/segments/1521257647251.74/warc/CC-MAIN-20180320013620-20180320033620-00782.warc.gz | 537,828,778 | 2,362 | % Beamer theme for Free-Electrons presentations
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Free Electrons. Kernel, drivers and embedded Linux - Development,
consulting, training and support.
~{\tt http://free-electrons.com} & \insertframenumber{}/\inserttotalframenumber{}
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© Copyright 2004-\the\year, Free Electrons.\\
Creative Commons BY-SA 3.0 license.\\
Latest update: \today.\\
\ifdefempty{\sessionurl}{}{
Electronic copies of these materials:\\
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Document updates and sources:\\
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Corrections, suggestions, contributions and translations are welcome!
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© Copyright 2004-\the\year, Free Electrons.\\
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Corrections, suggestions, contributions and translations are welcome!
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%% Create a slide announcing a lab, with the beautiful worker penguin
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Slides under CC-BY-SA 3.0\\
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\newcommand\todo[1]{
{\large \color{red}TODO: #1}
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https://ldn-fai.net/wp-content/uploads/2018/01/Cr_ago_ldn_270118.tex | ldn-fai.net | CC-MAIN-2021-43 | application/octet-stream | application/x-tex | crawl-data/CC-MAIN-2021-43/segments/1634323585424.97/warc/CC-MAIN-20211021133500-20211021163500-00211.warc.gz | 475,850,748 | 13,184 | \documentclass[10pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage[french]{babel}
\usepackage[T1]{fontenc}
\usepackage{calc, tikz, lmodern, marvosym, graphicx, geometry, eurosym, multicol, enumitem}
\usepackage[hyphens]{url}
\usepackage[hyperindex=true, colorlinks=true, breaklinks=true, linkcolor=blue]{hyperref}
\usepackage{fancyhdr, lastpage}
\geometry{hmargin=2cm, vmargin=3cm}
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\interfootnotelinepenalty=10000
\renewcommand{\ttdefault}{pcr}
\pagestyle{fancy}
\renewcommand{\headrulewidth}{1pt}
\lhead{\emph{AGO LDN 2017}}
\lfoot{contact \emph{(CHEZ)} ldn-fai.net}
\cfoot{\thepage{} / \pageref{LastPage}}
\rfoot{AGO 2017}
\begin{document}
\thispagestyle{empty}
\hspace{-1cm}
\begin{tabular}{ c c }
\begin{minipage}{5.5cm}
\vspace{-1cm}
\includegraphics[width=150pt]{logo_ldn.pdf}
\end{minipage} &
\begin{minipage}{7cm}
\vspace{-1cm}
{\footnotesize
Lorraine Data Network\\
C/O CCAN\\
69 rue de Mon-Désert\\
54000 Nancy\\
\Letter~contact \emph{(CHEZ)} ldn-fai.net\\
}
{\scriptsize
\textbf{SIRET :} 528 368 624 00029\\
\textbf{ARCEP :} 10-1320
\textbf{RNA : } W543005639
}
\end{minipage}
\end{tabular}
\vspace{3.7cm}
\begin{center}
\Huge{Assemblée générale ordinaire\\
Lorraine Data Network (2017)}
\vspace{2cm}
\large{Samedi 27 janvier 2018\\
14h00 - 18h00\\
~\\
Centre Culturel Autogéré de Nancy (CCAN)}
\end{center}
\newpage
\thispagestyle{empty}
\tableofcontents
\newpage
\thispagestyle{empty}
\listoftables
\newpage
\section{Assemblée}
\subsection{Présents}
\begin{itemize}
\item Vincent Me \textbf{(collège solidaire)}
\item Stéphane \textbf{(collège solidaire)}
\item Thomas \textbf{(collège solidaire)}
\item Alexandre \textbf{(collège solidaire)}
\item Sebastien \textbf{(collège solidaire)}
\item Emile
\item Tribu
\item Grégoire
\item Maxime
\item Philippe
\item Anne
\item Thibaut
\end{itemize}
\subsection{Présents par procuration}
\begin{itemize}
\item Julien V. (Sebastien B.)
\item Jean-Christophe (Sebastien B.)
\item Annie et Bernard (Vincent Me.)
\item Pascal L. (Vincent Me.)
\item Luc D. (Alexandre B.)
\end{itemize}
\subsection{Détails des votants}
Sur les 17 personnes présentes ou représentées :
\begin{itemize}
\item 15 ont pu voter ;
\item 2 n'étaient pas adhérent.e.s ;
\item Personne n'était pas adhérent.e.s depuis plus de 1 an (condition imposée par nos statuts\footnote{\url{http://ldn-fai.net/statuts/}} pour voter) ;
\end{itemize}
\newpage
\section{Bilan de cette septième année}
Principaux événements de cette année :
\begin{description}
\addtolength{\parskip}{4pt}
\item[31/01/2018] Conférence gesticulée « Informatique ou libertés ? » à la MJC des 3 maisons (ancienne école) à 20h.
\item[18/12/2017] Première prise de contact avec Midway's Network, nouveau FAI Belfortin (Jura).
\item[04/12/2017] Mise en place des sauvegardes mutualisées lors de l'AG de Grenode\footnote{\url{https://www.grenode.net/Services/Espaces_de_sauvegarde/}}.
\item[03/12/2017] Assemblée Générale de Grenode à Lyon, LDN représenté par deux membre de LDN.
\item[04/11/2017] Atelier Network\&Magic à Metz. \footnote{\url{http://www.graoulug.org/wordpress/?p=465}} \footnote{\url{https://code.ffdn.org/ljf/networkandmagic}}.
\item[20/10/2017] Ré-installation de nos machines physiques et migration de l’infra LDN chez Gitoyen à Paris.
\item[04-10/10/2017] MMMFest 2017 à Millemont, déploiement du réseau par Julien V.\footnote{\url{https://julien.vaubourg.com/files/cr_millemont.pdf}}.
\item[09/08/2017] Nouvelle offre de collecte xDSL, via Liazo (FFDN).
\item[02/07/2017] L’association a tenu un stand à la fête des associations à Vandœuvre-lès-Nancy.
\item[11/06/2017] La Brique Internet, c’est facile. Présenté par Petit, et produit par ARTE TV.
\item[03/06/2017] Assemblée Générale FFDN, dans l’Yonne.
\item[26/04/2017] Ajout de l’association sur l’annuaire Framalibre.
\item[30/03/2017] Les T-shirts LDN sont de nouveau disponibles.
\item[25/03/2017] OBC54 – Open Bidouille Camp. LDN n’a pas pu tenir de stand. Le Mirabellug était présent.
\item[09/03/2017] Amélioration des services DNS, avec l’intégration de DNS Privacy.
\end{description}
\newpage
\section{Rapport financier}
\subsection{Compte de résultat}
\begin{tabular}{|p{4cm}|p{2cm}|p{2cm}|}
\hline
\textbf{Charges} & \hfill \textbf{2017} & \hfill \textbf{2016} \\ \hline
\hline
Assurance & \hfill 110,35 & \hfill 41,23 \\ \hline
Frais bancaires & \hfill 205,70 & \hfill 198,00 \\ \hline
Local & \hfill 120,00 & \hfill 120,00 \\ \hline
Ressources Internet & \hfill 174,57 & \hfill 161,38 \\ \hline
Matériel serveurs & \hfill 719,36 & \hfill \\ \hline
ADSL & \hfill 3109,80 & \hfill 2873,40 \\ \hline
Hébergement serveurs & \hfill 2637,60 & \hfill 1197,60 \\ \hline
Achat briques & \hfill & \hfill 916,12 \\ \hline
Achat t-shirts & \hfill 555,00 & \hfill \\ \hline
Variation des stocks & \hfill 858,00 & \hfill 140,00 \\ \hline
Dotation aux provisions & & \hfill 752,40 \\ \hline
Charges exceptionnelles & & \hfill 1508,40 \\ \hline
\emph{Total} & \hfill \emph{8490,38} & \hfill \emph{7908,53} \\ \hline
\end{tabular}
\vspace{0.3cm}
\begin{tabular}{|p{4cm}|p{2cm}|p{2cm}|}
\hline
\textbf{Produits} & \hfill \textbf{2017} & \hfill \textbf{2016} \\ \hline
\hline
Cotisations & \hfill 910,00 & \hfill 834,00 \\ \hline
Dons & \hfill 48,80 & \hfill 758,60 \\ \hline
Dividendes & \hfill 0,62 & \hfill 0,69 \\ \hline
ADSL & \hfill 3008,00 & \hfill 3342,00 \\ \hline
VPN & \hfill 1297,00 & \hfill 964,00 \\ \hline
DCPBay & & \hfill 266,67 \\ \hline
Vente briques & \hfill 438,00 & \hfill 232,00 \\ \hline
Vente t-shirts & \hfill 430,00 & \hfill \\ \hline
Variation des stocks & \hfill 600,00 & \hfill 828,00 \\ \hline
Produits exceptionnels & & \hfill 4506,00 \\ \hline
Reprise de provision & \hfill 752,40 & \hfill \\ \hline
\emph{Total} & \hfill \emph{7484,82} & \hfill \emph{11731,96} \\ \hline
\end{tabular}
\vspace{0.3cm}
\begin{tabular}{|p{4cm}|p{2cm}|p{2cm}|}
\hline
\textbf{Résultat} & \hfill -1005,56 & \hfill 3823,43\\ \hline
\end{tabular}
\vspace{0.3cm}
Cette année, nous avons payé une cotisation (120~\euro) au CCAN pour
pouvoir disposer de leur local pour nos réunions mensuelles (la
cotisation n'était pas imposée, nous l'avons votée pour les
remercier).
Le bond dans les frais d'assurance est dû à la forte augmentation du
budget de l'année dernière.
L'association a acheté du matériel pour les serveurs: un CPU, des
disques SSD et des adaptateurs pour ces derniers.
En 2017, 37 adhérents ont payé une cotisation (contre 29 en 2016). La
cotisation moyenne est de 24,59~\euro.
Cette année, une ligne ADSL a été résiliée. Au 31/12/2017, nous avons
donc 7 lignes ADSL en marque blanche chez FDN qui nous rapportent
13,80~\euro{} par mois. La perte apparente de 101,80~\euro{} est dûe
au retard de facturation d'un mois de l'année dernière (13 mois ont
été facturés en 2017).
3 nouvelles lignes VPN ont été ouvertes. Au 31/12/2017, nous avons
donc 14 lignes VPN. Ces lignes nous rapportent 115~\euro{} par
mois.
Chaque mois, en dépenses récurrentes, nous avons 16,60~\euro{} de
frais bancaires, 226,20~\euro{} de lignes ADSL chez FDN, 156~\euro{}
d'hébergement de serveurs et environ 100~\euro{} de bande passante
chez Gitoyen. En recettes, nous recevons 355~\euro{} de la part des
abonnés aux lignes ADSL et VPN. Nous perdons donc environ
143,80~\euro{} par mois.
La dernière brique à base de LIME1 a été vendue (70~\euro), ainsi que
4 briques à base de LIME2 (92~\euro{} chacune). Il reste donc en stock
5 briques à base de LIME2.
LDN a fait une commande de 30 T-shirts à 555~\euro. L'association
compte les revendre à 20~\euro{} l'unité (25~\euro{} pour les
non-adhérents). LDN réalise donc (au moins) 45~\euro{} de bénéfices
sur cette opération. 21 T-shirts ont été vendus en 2017, il en reste
donc 9.
La provision qui avait été faite l'année dernière en prévision des
pertes récurrentes s'est avérée être insuffisante. Cela est dû à
l'augmentation de l'utilisation de bande passante. Le résultat étant
très déficitaire, il a été décidé de ne pas imputer de provision en
2017. Toutefois, le report à nouveau permet de compenser les pertes.
\subsection{Bilan}
\begin{tabular}{|p{4cm}|p{2cm}|p{2cm}|}
\hline
\textbf{Actif} & \hfill \textbf{2017} & \hfill \textbf{2016} \\ \hline
\hline
Crédit Coopératif & \hfill 2879,51 & \hfill 3749,85 \\ \hline
Espèces & \hfill 132,00 & \hfill 130,00 \\ \hline
Chèques à encaisser & \hfill 170,00 & \\ \hline
Parts sociales CC & \hfill 61,00 & \hfill 61,00 \\ \hline
Créances & \hfill 70,00 & \hfill 20,00 \\ \hline
Stock briques & \hfill 460,00 & \hfill 898,00 \\ \hline
Stock t-shirts & \hfill 180,00 & \\ \hline
\emph{Total} & \hfill \emph{3312,51} & \hfill \emph{4858,85} \\ \hline
\end{tabular}
\vspace{0.3cm}
\begin{tabular}{|p{4cm}|p{2cm}|p{2cm}|}
\hline
\textbf{Passif} & \hfill \textbf{2017} & \hfill \textbf{2016} \\ \hline
\hline
Report à nouveau & \hfill 3906,45 & \hfill 83,02 \\ \hline
Résultat de l'exercice & \hfill -1005,56 & \hfill 3823,43 \\ \hline
Provisions & & \hfill 752,40 \\ \hline
Dettes fournisseurs & \hfill 361,80 & \hfill \\ \hline
Dettes adhérents & \hfill 689,82 & \hfill 200,00 \\ \hline
\emph{Total} & \hfill \emph{3312,51} & \hfill \emph{4858,85} \\ \hline
\end{tabular}
\subsection{Budget prévisionnel}
\begin{tabular}{|p{4cm}|p{2cm}|p{2cm}|}
\hline
\textbf{Charges} & \hfill \textbf{2018} \\ \hline
\hline
Assurance & \hfill 100 \\ \hline
Frais bancaires & \hfill 200 \\ \hline
Local & \hfill 120 \\ \hline
Ressources Internet & \hfill 200 \\ \hline
ADSL & \hfill 2715 \\ \hline
Hébergement serveurs & \hfill 3000 \\ \hline
Variation des stocks & \hfill 640 \\ \hline
\emph{Total} & \hfill \emph{6975} \\ \hline
\end{tabular}
\vspace{0.3cm}
\begin{tabular}{|p{4cm}|p{2cm}|p{2cm}|}
\hline
\textbf{Produits} & \hfill \textbf{2018} \\ \hline
\hline
Cotisations et dons & \hfill 1000 \\ \hline
ADSL & \hfill 2880 \\ \hline
VPN & \hfill 1380 \\ \hline
Vente briques & \hfill 460 \\ \hline
Vente t-shirts & \hfill 180 \\ \hline
\emph{Total} & \hfill \emph{5900} \\ \hline
\end{tabular}
\vspace{0.3cm}
Ce budget prévisionnel est déficitaire de 1075~\euro{}, ce qui est du
même ordre de grandeur que le déficit 2017. L'association pourra
encaisser cette perte grâce au report à nouveau qui, après imputation
du résultat 2017, s'élève à 2900,89~\euro{}. Néanmoins, cela veut dire
que l'association est limitée dans ses investissements à hauteur de
1825~\euro{} pour 2018. De plus, la question de la viabilité à long
terme de l'association dans ces conditions se pose.
Pour information, il faudrait ouvrir entre 9 et 15 lignes VPN
supplémentaires, ou que la cotisation moyenne monte à 56~\euro{}, pour
couvrir les pertes prévues.
\vspace{0.3cm}
\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{Pours} & \textbf{Contres} & \textbf{Abstentions} & \textbf{Ne prend pas part au vote}\\
\hline
14 & 0 & 0 & 3 \\
\hline
\end{tabular}
\caption{Détail du vote \og\emph{Validation du budget}\fg}
\end{center}
\end{table}
\newpage
\section{La fédération (FFDN)}
\subsection{Membres}
La fédération FDN (FFDN) compte actuellement 27 associations membres. La fédération totalise 3185 membres (simple addition de tous les membres de chaque association) soit 1 association de moins et 1098 membres de plus depuis notre assemblée générale précédente. Pour information, toutes ces statistiques sont disponibles sur la base de données de la fédération\footnote{\url{https://db.ffdn.org/}} (cette base est mise à jour régulièrement par un programme qui agrège automatiquement les informations de chaque association).
Afin de faciliter le décompte via db.ffdn.org, et depuis l'arrêt du système de mise à jour avec munin (stats.ffdn.org), Sébastien a fait un petit script qui récupère le nombre d'adhérents directement sur db.ffdn.org. \footnote{Ce script est disponible sur github : \url{https://github.com/sbadia/stats.ffdn.org}}
\subsection{Correspondants}
Pour rappel, selon les statuts\footnote{\url{http://www.ffdn.org/fr/statuts}} de FFDN : \og\emph{Sont correspondants les associations, entreprises, personnes physiques ou morales, qui n'ont pas statutairement vocation à être membres de la Fédération, mais avec lesquelles il existe une volonté commune et réciproque de collaboration et d'échange en vue de satisfaire à l'objet de la Fédération}\fg.
Les correspondants actuels sont Grenode, Gitoyen, Absolight et Wan2Many.
\subsection{Nouvelles régulières}
Aussi appelé gazette, une nouvelle pratique de cette année a été la mise en place de messages de nouvelles de chaque FAI composant la Fédération, ces nouvelles sont ensuite agrégés dans un document
\footnote{Les derrnières nouvelles de LDN données à FFDN sont ici: \url{https://listes.ldn-fai.net/pipermail/benevoles/2017-December/003530.html} et les nouvelles agrégées au niveau FFDN se trouvent sur le wiki \url{https://www.ffdn.org/wiki/doku.php?id=travaux:nouvelles&s[]=gazette}}
. C'est un moyen assez pratique d'avoir l'info synthétisé de chaque FAI, ces nouvelles remplacent aussi les réunion que la fédération organisait sur IRC (mais qui était devenue compliqués du fait du nombre de personnes à participer).
\subsection{Groupes de travail}
Une des nouveautés de cette année au niveau Fédéral est aussi l'apparition de groupes de travail. On compte actuellement quelques groupes bien formées, comme les groupes FTTH, juridique, compta, newsletter, ….
\subsection{Assemblée Générale}
L'assemblée générale de FFDN a eu lieu à Seignelay (dans l'Yonne), France, du 3 au 5 juin 2017. C'est entre Troyes et Auxerre. Elle a été organisée par Scani\footnote{\url{http://www.scani.fr/}}. Cinq adhérents LDN ont fait le trajet.
Le bilan de cette assemblée générale est plutôt positif. L'AG, se déroulant sur plusieurs jours, a permis aux membres des différentes associations de se retrouver et d'avancer sur des projets communs.
\newpage
\section{Vie de l'association}
\subsection{La Brique Internet}
Le début de l'année 2017 fut dans la continuité de l'année 2016 : de l'avancement à la stabilisation des projets basés sur Yunohost\footnote{Prononcer Why you no host}.
Cependant, nous ne pouvons pas en dire autant sur la fin de l'année. En effet, nous constatons un essoufflement des membres des associations apportant un support au niveau de La Brique Internet. Les divers plantages aléatoires et autres problèmes difficiles à reproduire\footnote{Le fait que le fabricant ait sorti différentes révisions des cartes que nous utilisons n'aide pas au dépannage.}, en plus des cartes micro SD qui finissent par lâcher finit par provoquer un sentiment de lassitude (aussi observée au niveau Fédéral). Une discussion\footnote{\url{https://listes.labriqueinter.net/pipermail/discussions/2017-December/001773.html}} a eu lieu sur l'intérêt de poursuivre l'utilisation du matériel actuel, et si utiliser un autre matériel ne permettrait pas de résoudre ces problèmes. Cependant, si on ajoute d'autres cartes, cela ne ferait qu'éparpiller encore plus les ressources des différentes associations. Au lieu de ne supporter que 2 cartes, on se retrouverait à en supporter 36\footnote{Bon, OK, pas 36. Olimex Lime 1 et 2, Banana PI, Banana PI R1, R2 (routeurs), Raspberry PI et autres, sans oublier des cartes à base x86}, ce qui ne serait pas une bonne chose.
L'autre problème, c'est que les briques sont également distribuées à des personnes n'ayant pas forcément les compétences nécessaires pour faire elles-mêmes les dépannages. Dans ce cas, le soutien physique (et moral) des associations est à ne surtout pas négliger, bien au contraire.
Présentement, il nous reste 5 briques en stock.
\subsection{Réunions mensuelles}
Les réunions mensuelles ont continué en 2017. Elles ont toujours lieu le premier mercredi du mois\footnote{Excepté celle du mois d'août qui fut annulée} au Centre Culturel Autogéré de Nancy\footnote{\url{http://ccan.herbesfolles.org/}} (CCAN).
Après chaque réunion mensuelle, un compte-rendu est envoyé sur la liste de diffusion « bénévoles », et une copie est enregistré sur le wiki de l'association\footnote{\url{https://wiki.ldn-fai.net/wiki/Catégorie:Réunions}}.
Sur les 10 dernières réunions, nous avons constaté une participation moyenne de 8 participants (5 en physique et 3 à distance), avec un minimum de 5 et un maximum de 11. En 2016, ces chiffres étaient de 9,7 participants (6,3 en physique et 3,4 à distance), avec un minimum de 8 et un maximum de 12. En 2015, ces chiffres étaient de 10,9 participants (7,9 en physique et 3 à distance), avec un minimum de 6 et un maximum de 15.
La participation aux réunions, que ça soit physiquement ou à distance\footnote{via Mumble} est en baisse, comme constaté lors de la précédente AG.
Suite aux changements de statuts, validés lors de la précédente AGE, les décisions sont prises lors de ces réunions mensuelles. Ces décisions sont enregistrées dans le compte rendu de la réunion mensuelle et sont également annoncés sur la liste de diffusion bénévoles, dans un message à part.
\textit{A priori}, ce système semble fonctionner, vu que les décisions prises n'ont pas provoqué de levée de boucliers ni de contestation.
Les décisions prises lors des réunions mensuelles furent :
\textbf{au 1\ier{} mars 2017}
\begin{itemize}
\item Fin de l'attente de devis après d'Adista (abandon de cette piste pour l'hébergement des machines).
\item Installation de façon pérenne (le temps de relancer les services) chez Gitoyen.
\item Achat de 2 SSD, de 1 CPU.
\item Passage de la commande de T-shirts.
\item Constitution du collège solidaire.
\item Arrêt de l'utilisation du registre spécial, qui n'est plus légalement obligatoire\footnote{\url{https://www.service-public.fr/associations/vosdroits/F338}}.
\end{itemize}
\textbf{au 5 avril 2017}
\begin{itemize}
\item \textbf{Ajout d'un point CCAN dans les réunions mensuelles}.
\item Achat d'un processeur sur Ebay, pour remplacer celui de l'une des machines de l'infra.
\item Achat de deux disques durs SSD Crucial pour les machines de l'infra.
\end{itemize}
\textbf{au 5 juillet 2017}
\begin{itemize}
\item ``Achat'' du nom de domaine actricedu.net. (À priori, ce domaine n'a pas été pris par LDN et est toujours disponible).
\end{itemize}
\subsection{Secrétariat mensuel tournant}
L'idée du secrétariat mensuel tournant est de désigner une personne pour répondre aux courriels destinés à l'association, et on tourne tous les mois. Cela permet alors de réduire le temps de réponse, et de « responsabiliser » les réponses.
Cela a été mis en place en mars 2016. Lors de la réunion mensuelle suivante, un récapitulatif est effectué, et un nouveau membre volontaire est désigné pour répondre aux courriels.
Ce système de secrétariat tournant est plutôt efficace, et ce, sur plusieurs points. Les courriels sont traités rapidement, et si des courriels sont ratés, on le constate lors de la réunion mensuelle suivante, et le nouveau secrétaire mensuel s'en occupe.
Ce système est toujours en place, même si on constate qu'il s'agit régulièrement des mêmes personnes qui sont volontaires pour prendre en charge le secrétariat tournant.
\subsection{Ateliers sysadmin}
Les ateliers/sessions sysadmin ont été mis en place l'an dernier. Ces ateliers, qui avaient d'abord lieu le jeudi soir, et maintenant le lundi soir (au besoin, avec planification du sujet sur la liste bénévoles@), exceptionnellement le week-end, permettent aux bénévoles de pouvoir s'attaquer aux tâches d'administration système,et de former les membres aux technologies utilisées sur l'infrastructure (git, puppet, bgp…).
Actuellement, quelques sessions ont eu lieu, la migration des services a été effectuée lors de ces ateliers.
\subsection{Hébergement des machines}
\subsubsection{Paris}
Suite à la décision prise en 2015, sur la refonte de l'infrastructure, et à la séparation des services partagés avec ARN, de nouveaux serveurs ont été mis en place chez Gitoyen, à Paris, en octobre 2016.
Le déménagement de l'infrastructure a été effectuée essentiellement en décembre 2016, suivi de la migration progressive des services les semaines suivantes.
Les services de l'association (sites web, listes de diffusions), hébergées chez Online, via le compte d'un adhérent (Rémi J.), ont également été migré vers cette nouvelle infrastructure. Cela a donc enfin permis de rassembler tous les services de l'association sur des même serveurs avec un accès (gestion) uniforme.
Les machines de prêt d'ARN (installées chez Gitoyen) ont été remplacées en 2017 par nos machines précédement hébergées chez WideVoip. L'infrastructure de Lorraine Data Network fonctionne donc maintenant sur 2 machines physiques hébergées chez Gitoyen.
\subsection{Conférences}
Cette année à été très calme au niveau conférences et, plus généralement au niveau des messages politique véhiculés par l'association.
\subsubsection{La Brique Internet}
La Brique Internet, c'est facile. Présenté par Petit, réalisé par Arte\footnote{\url{http://tracks.arte.tv/fr/off-track-episode-2}}
\subsubsection{Invitations déclinées}
Nous n'avons pas participé cette année, faute de volontaires :
\begin{description}
\addtolength{\parskip}{4pt}
\item[11/11/2017] Conférence à Supelec/Federez à Metz
\end{description}
\subsection{Goodies}
\subsubsection{Cartes de visites}
Nous avons toujours des cartes de visites de l'association, en deux modèles : classique, et my little pony (édition limitée), ainsi que des cartes de visite La Brique Internet. Ces cartes de visite sont distribuées lors des événements.
\subsubsection{T-shirts}
Nous avons annoncé à quelques reprises sur la liste de diffusion que les t-shirts allaient être réimprimés. Nous avons, suite à ces annonces, pu atteindre le seuil des 30 t-shirts. De fait, une nouvelle commande de t-shirts a pu être effectuée le 14 mars 2017.
Lors des discussions sur le template (modèle utilisé par l'imprimeur), nous nous sommes rendus-compte que l'association n'imprimait que des modèles de t-shirts masculins\footnote{\url{https://listes.ldn-fai.net/pipermail/benevoles/2016-June/002774.html}}. La décision d'imprimer également des modèles féminins fut prise.
L'association propose désormais des t-shirts en différentes tailles, pour hommes comme pour femmes.\footnote{\url{https://ldn-fai.net/t-shirt/}}
Comme vu dans la partie bilan financier, il nous reste 9 t-shirts, répartis comme suit :
\begin{table}[!h]
\begin{center}
\begin{tabular}{|p{2cm}|p{2cm}|p{2cm}|p{2cm}|p{2cm}|p{2cm}|}
\hline
\textbf{} & \hfill \textbf{S} & \hfill \textbf{M} & \hfill \textbf{L} & \hfill \textbf{XL} & \hfill \textbf{XXL} \\ \hline
\hline
Homme & \hfill 0 & \hfill 1 & \hfill 1 & \hfill 2 & \hfill 1 \\ \hline
Femme & \hfill 0 & \hfill 1 & \hfill 2 & \hfill 1 & \hfill 0 \\ \hline
\emph{Total} & \hfill \emph{0} & \hfill \emph{2} & \hfill \emph{3} & \hfill \emph{3} & \hfill \emph{1} \\ \hline
\end{tabular}
\caption{Stock restant pour les t-shirts}
\end{center}
\end{table}
\newpage
\section{Infrastructure}
\subsection{VPN}
Nous avons actuellement 14 lignes VPN actives. Mis à part quelques soucis de routage qui sont apparus par moment, problèmes rapidement corrigés, ce service semble plutôt fiable, et est utilisé au quotidien par les membres.
Ce service VPN est également utilisé lors de certains événements, afin de fournir un accès Internet complet, indépendant des contraintes liées au média utilisé (par exemple, restrictions sur les connexions 4G).
\subsection{Refonte de l'architecture}
Suite à la réunion du 18 octobre 2015\footnote{\url{https://wiki.ldn-fai.net/wiki/Réunion_adminsys_du_18_octobre_2015}} nous avons mis en place un plan de migration pour déplacer nos services de nos machines du NetCenter de Strasbourg aux machines hébergés chez Gitoyen à Paris.
Cette migration à été faite en trois étapes grâce au don d'ARN de deux serveur. Nous avons donc déplacé les services de Strasbourg à Paris, puis les les services propres à l'association (site, listes de diffusion, etc), hébergée sur un serveur Online d'un adhérent (Rémi J.), on été migré sur la nouvelle infrastructure à la fin de l'année 2016, et enfin nous avons remplacé les deux serveurs de don par nos machines récupérées à Strasbourg en octobre 2017.
Nous profitons de cette migration pour refondre l'architecture de notre infrastructure, avec pour objectifs :
\begin{itemize}
\item simplifier l'architecture (avoir deux FAI, avec deux livraisons sur une même machine impliquait pas mal de complications (virtualisation, netns, …).
\item éviter la multiplication des solutions de virtualisation (nous utilisions précedement LXC dans des KVM ainsi que des netns) ;
\item mettre en place de la redondance entre les machines (DRDB, iBGP, Ganeti).
\end{itemize}
\subsection{Services mutualisés}
Quelques adhérents utilisaient les services mutualisés de l'infrastructure (proposés avec Yunohost), lors de la refonte de l'infrastructure, nous avons fait le choix d'arréter Yunohost afin d'utiliser le groupware zimbra pour la partie mail, calendrier et partage de documents, nous n'avons cependant pas encore mis cette partie en place\footnote{Nous avons quelques volontaires pour regarder ce point :)}.
La partie messagerie instantanée (via Jabber) a été remise en service.
\subsection{Documentation de l'infrastructure}
La documentation sur le wiki n'est plus forcément à jour, notamment sur les pages techniques. Un travail de relecture a commencé l'an dernier. Une partie des pages a été actualisée, cependant, le wiki étant relativement conséquent, il reste encore un certain nombre de pages à revoir.
\subsection{Backups mutualisés via Grenode}
Lors de l'assemblée générale de Grenode de décembre 2017, nous avons commencé la mise en place des sauvegardes de notre infrastructure ; ces sauvegardes sont faites via le logiciel borg et sont envoyées sur le serveur de sauvegardes de Grenode\footnote{Ce serveur est hébergé à Toulouse par Tetaneutral.net}.
Le logiciel borg permet de faire des sauvegardes incrémentales, de dédupliquer\footnote{Les fichiers sont découpés en différents morceaux, de manière à faire en sorte à ce qu'il n'y ait pas de fragment identique en double (ou plus), et ce, même si le fichier est déplacé, renommé ou copié plusieurs fois.} les données et de les chiffrer\footnote{On crypte les chaines de TV. Pour tout le reste, on chiffre.}.
La documentation de la mise en place chez LDN est disponible sur notre wiki.\footnote{\url{https://wiki.ldn-fai.net/wiki/Borg}}.
\subsection{Supervision}
\subsubsection{Supervision et alertes}
Afin de s'assurer que l'infrastructure de l'association fonctionne correctement, nous utilisons toujours le logiciel checkmk (logiciel pour générer de la configuration au format nagios).
Nous utilisons la machine virtuelle conrad (vm chiffrée) hébergé par grenode à Lyon pour effectuer cette supervision externe de notre infrastructure.
\subsubsection{Courbes}
Suite aux soucis de performance sur notre ancienne infrastructure (affaire des 800\% :)) un adhérent (Sébastien B.) avait mis en place sur un de ses serveurs personnel le logiciel munin afin de faire des graphiques sur les différentes métriques des serveurs.
Le logiciel munin\footnote{Munin est un outil de supervision système et réseau, générant de beaux graphiques et courbes, permettant de voir l'évolution de la consommation de la bande passante, de la charge CPU (800\%) et du niveau du café dans la cafetière.} est maintenant installé sur leela (une machine virtuelle de l'association). Toutes les machines de l'infrastructure sont maintenant supervisés.
\newpage
\section{Les copains avec qui on bosse}
\subsection{Les Petits Débrouillards de Lorraine}
Nous avons rencontré les membres de Petits Débrouillards de Lorraine en 2015, notamment lors de l'Open Bidouille Camp, et au Forum Social Mondial.
Les petits débrouillards de Lorraine est une association très dynamique qui dispose d'une grande visibilité et de beaucoup de moyens financiers. Il y a fort à parier que nous serons à nouveau amenés à travailler avec eux.
L'Open Bidouille Camp n'a pas pu avoir lieu en 2016, à cause du plan Vigipirate \footnote{Pas mal d'événements ont été annulés ou reportés suite aux attentats de 2015}. L'OBC a eu lieu les 25 et 26 mars 2017. Cependant, cette date entrant en conflit avec de nombreux événements, LDN n'a pu tenir de stand lors de cet événement.
\subsection{Midway's Network}
En décembre 2017, Midway's Network association naissante basé à Belfort ayant pour but de devenir FAI dans le Jura, nous a contacté pour que LDN les aident à se créer.
Cette aide se formaliserait par la fourniture d'une machine virtuelle pour qu'ils annoncent sur Internet leurs ressources IPv4/IPv6 via leur AS. La vm contiendrait aussi un VPN vers le reste de leur infrastructure.
Pour présenter leur association, ils ont pour but de promouvoir la neutralité du net et défendre la vie privé des internautes.
Ils comptent réaliser cela via trois axes:
\begin{itemize}
\item Ateliers et conférences sur tout sujet touchant de près ou de loin l'objet de l'association.
\item Ils sont en train de monter un fournisseur d'accès à internet associatif Belfortain où leurs adhérents serait les acteurs de ce FAI.
\item Fournir des services web alternatif axé sur l'auto hébergement et l'apprentissage de chacun.
\end{itemize}
\subsection{Participation au projet TDCPB (en sommeil)}
Nous avons rejoint le projet TDCPB\footnote{\url{http://tdcpb.org/}} en 2014.
Ce service a été interrompu pour faciliter la migration de l'infrastructure.
L'infra étant de nouveau en service, il sera possible de remettre en service par la suite.
\newpage
\section{Projets en cours}
\subsection{Collecte Wifi}
L'an dernier, nous avions trouvé un point haut à Nancy, bien placé, avec un accès Internet très haut débit (FTTLA). Cette année, nous n'avons pas avancé sur ce projet. Ce projet est en pause.
\subsection{Services de l'association}
La nouvelle vague de services qui devrait être proposée par l'association est :
\begin{itemize}
\item \textbf{VPN :} Le service est en production et déjà utilisé par 13 membres, pour 14 lignes. Il est intéressant de noter que ce service est aussi utilisé conjointement au projet de brique Internet.
\item \textbf{DNS :} Le service DNS récursif est ouvert, et accessible à partir de n'importe où dans le monde. Ce service est limité au niveau bande passante (à confirmer). Ce service est aussi accessible via TLS.
\item \textbf{Pack :} Nous avons actuellement deux adhérents qui utilisent les services de MX (mail) et DNS secondaires de l'association.
\item \textbf{Mastodon :} L'association ne propose pas d'instance Mastodon :) Mais si un ou plusieurs membres veulent une instance…
\item \textbf{Zimbra :} L'association pourra proposer à terme un ensemble de services (courriel, DKIM, antispam…) via la suite collaborative Zimbra. (les AMIP, c'est comme les AMAP (Association pour le Maintien d'un Internet Proche)
\item \textbf{Café citoyen :} Diffuser des idées, parler de l'association. Éducation populaire.
\item \textbf{Autres idées :} N'hésitez pas à proposer des nouvelles idées lors de cette AG.
\end{itemize}
Les détails des services et des prix sont disponibles sur le site\footnote{\url{http://ldn-fai.net/services-de-lassociation/}}.
\newpage
\section{Objectifs}
\subsection{Objectifs fixés pour 2017 lors de l'AG de 2016}
\begin{itemize}
\item Finir la refonte de l'infrastucture. Sur ce point, on peut considérer que c'est une réussite. La nouvelle infra est opérationnelle.
\item Communication sur LaBriqueInterNet.
\item Atteindre un équilibre financier qui permettrait de payer un hébergement et du transit sans cadeaux ; Il manquerait une dizaine de lignes VPN pour arriver à l'équilibre. Malheureusement, ce n'est pas une réussite.
\item Avancer sur la collecte Wifi. Ce point est actuellement en pause.
\item Créer et présenter de nouvelles conférences, avec de nouveaux adhérents conférenciers. Ce point n'était pas une priorité pour l'association. Nous n'avons pas de nouvelle conférences, ni de nouveaux conférenciers et conférencières. Cependant, des rencontres (comme Network\&Magic) ont permis de rencontrer des gens et de les sensibiliser aux problématiques de données personnelles sur Internet.
\end{itemize}
\subsection{Objectifs pour 2018}
\begin{itemize}
\item Atteindre un équilibre financier ;
\item Créer et présenter de nouvelles conférences, café citoyen, avec de nouvelles et nouveaux adhérents conférenciers ;
\item Proposer de nouveaux services, dont un service de courriel, …
\end{itemize}
\newpage
\section{Collège solidaire}
Lors de l'AGE de 2016\footnote{\url{https://ldn-fai.net/bureau}}, nous avons votés le changement des statuts. Nous sommes passés d'une organisation classique (président, vice-président, trésorier, secrétaire) à une organisation de type collège solidaire. Ce fonctionnement nous semble satisfaisant, il n'y a pas de raisons à l'heure actuelle de changer de mode de fonctionnement.
Membres du collège solidaire
\begin{itemize}
\item Jean-Christophe Bach
\item Sébastien Badia
\item Alexandre Bailly
\item Abdelkarim Berkane
\item Gabriel Corona
\item Thomas Girod
\item Stéphane Glondu
\item Sébastien Jean
\item Vincent Merlet
\item Vincent Mollimard
\item Emmanuel Monbronssou
\item Julien Vaubourg
\end{itemize}
\end{document} |
http://tug.ctan.org/tex-archive/macros/latex/contrib/aastex/widetab.tex | ctan.org | CC-MAIN-2018-51 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2018-51/segments/1544376825029.40/warc/CC-MAIN-20181213171808-20181213193308-00358.warc.gz | 291,562,425 | 3,569 | \begin{longrotatetable}
\begin{deluxetable}{lllrrrrrrll}
\tablecaption{Observable Characteristics of
Galactic/Magellanic Cloud novae with X-ray observations\label{chartable}}
\tablewidth{700pt}
\tabletypesize{\scriptsize}
\tablehead{
\colhead{Name} & \colhead{V$_{max}$} &
\colhead{Date} & \colhead{t$_2$} &
\colhead{FWHM} & \colhead{E(B-V)} &
\colhead{N$_H$} & \colhead{Period} &
\colhead{D} & \colhead{Dust?} & \colhead{RN?} \\
\colhead{} & \colhead{(mag)} & \colhead{(JD)} & \colhead{(d)} &
\colhead{(km s$^{-1}$)} & \colhead{(mag)} & \colhead{(cm$^{-2}$)} &
\colhead{(d)} & \colhead{(kpc)} & \colhead{} & \colhead{}
}
\startdata
CI Aql & 8.83 (1) & 2451665.5 (1) & 32 (2) & 2300 (3) & 0.8$\pm0.2$ (4) & 1.2e+22 & 0.62 (4) & 6.25$\pm5$ (4) & N & Y \\
{\bf CSS081007} & \nodata & 2454596.5 & \nodata & \nodata & 0.146 & 1.1e+21 & 1.77 (5) & 4.45$\pm1.95$ (6) & \nodata & \nodata \\
GQ Mus & 7.2 (7) & 2445352.5 (7) & 18 (7) & 1000 (8) & 0.45 (9) & 3.8e+21 & 0.059375 (10) & 4.8$\pm1$ (9) & N (7) & \nodata \\
IM Nor & 7.84 (11) & 2452289 (2) & 50 (2) & 1150 (12) & 0.8$\pm0.2$ (4) & 8e+21 & 0.102 (13) & 4.25$\pm3.4$ (4) & N & Y \\
{\bf KT Eri} & 5.42 (14) & 2455150.17 (14) & 6.6 (14) & 3000 (15) & 0.08 (15) & 5.5e+20 & \nodata & 6.5 (15) & N & M \\
{\bf LMC 1995} & 10.7 (16) & 2449778.5 (16) & 15$\pm2$ (17) & \nodata & 0.15 (203) & 7.8e+20 & \nodata & 50 & \nodata & \nodata \\
LMC 2000 & 11.45 (18) & 2451737.5 (18) & 9$\pm2$ (19) & 1700 (20) & 0.15 (203) & 7.8e+20 & \nodata & 50 & \nodata & \nodata \\
{\bf LMC 2005} & 11.5 (21) & 2453700.5 (21) & 63 (22) & 900 (23) & 0.15 (203) & 1e+21 & \nodata & 50 & M (24) & \nodata \\
{\bf LMC 2009a} & 10.6 (25) & 2454867.5 (25) & 4$\pm1$ & 3900 (25) & 0.15 (203) & 5.7e+20 & 1.19 (26) & 50 & N & Y \\
{\bf SMC 2005} & 10.4 (27) & 2453588.5 (27) & \nodata & 3200 (28) & \nodata & 5e+20 & \nodata & 61 & \nodata & \nodata \\
{\bf QY Mus} & 8.1 (29) & 2454739.90 (29) & 60: & \nodata & 0.71 (30) & 4.2e+21 & \nodata & \nodata & M & \nodata \\
{\bf RS Oph} & 4.5 (31) & 2453779.44 (14) & 7.9 (14) & 3930 (31) & 0.73 (32) & 2.25e+21 & 456 (33) & 1.6$\pm0.3$ (33) & N (34) & Y \\
{\bf U Sco} & 8.05 (35) & 2455224.94 (35) & 1.2 (36) & 7600 (37) & 0.2$\pm0.1$ (4) & 1.2e+21 & 1.23056 (36) & 12$\pm2$ (4) & N & Y \\
{\bf V1047 Cen} & 8.5 (38) & 2453614.5 (39) & 6 (40) & 840 (38) & \nodata & 1.4e+22 & \nodata & \nodata & \nodata & \nodata \\
{\bf V1065 Cen} & 8.2 (41) & 2454123.5 (41) & 11 (42) & 2700 (43) & 0.5$\pm0.1$ (42) & 3.75e+21 & \nodata & 9.05$\pm2.8$ (42) & Y (42) & \nodata \\
V1187 Sco & 7.4 (44) & 2453220.5 (44) & 7: (45) & 3000 (44) & 1.56 (44) & 8.0e+21 & \nodata & 4.9$\pm0.5$ (44) & N & \nodata \\
{\bf V1188 Sco} & 8.7 (46) & 2453577.5 (46) & 7 (40) & 1730 (47) & \nodata & 5.0e+21 & \nodata & 7.5 (39) & \nodata & \nodata \\
{\bf V1213 Cen} & 8.53 (48) & 2454959.5 (48) & 11$\pm2$ (49) & 2300 (50) & 2.07 (30) & 1.0e+22 & \nodata & \nodata & \nodata & \nodata \\
{\bf V1280 Sco} & 3.79 (51) & 2454147.65 (14) & 21 (52) & 640 (53) & 0.36 (54) & 1.6e+21 & \nodata & 1.6$\pm0.4$ (54) & Y (54) & \nodata \\
{\bf V1281 Sco} & 8.8 (55) & 2454152.21 (55) & 15:& 1800 (56) & 0.7 (57) & 3.2e+21 & \nodata & \nodata & N & \nodata \\
{\bf V1309 Sco} & 7.1 (58) & 2454714.5 (58) & 23$\pm2$ (59) & 670 (60) & 1.2 (30) & 4.0e+21 & \nodata & \nodata & \nodata & \nodata \\
{\bf V1494 Aql} & 3.8 (61) & 2451515.5 (61) & 6.6$\pm0.5$ (61) & 1200 (62) & 0.6 (63) & 3.6e+21 & 0.13467 (64) & 1.6$\pm0.1$ (63) & N & \nodata \\
{\bf V1663 Aql} & 10.5 (65) & 2453531.5 (65) & 17 (66) & 1900 (67) & 2: (68) & 1.6e+22 & \nodata & 8.9$\pm3.6$ (69) & N & \nodata \\
V1974 Cyg & 4.3 (70) & 2448654.5 (70) & 17 (71) & 2000 (19) & 0.36$\pm0.04$ (71) & 2.7e+21 & 0.081263 (70) & 1.8$\pm0.1$ (72) & N & \nodata \\
{\bf V2361 Cyg} & 9.3 (73) & 2453412.5 (73) & 6 (40) & 3200 (74) & 1.2: (75) & 7.0e+21 & \nodata & \nodata & Y (40) & \nodata \\
{\bf V2362 Cyg} & 7.8 (76) & 2453831.5 (76) & 9 (77) & 1850 (78) & 0.575$\pm0.015$ (79) & 4.4e+21 & 0.06577 (80) & 7.75$\pm3$ (77) & Y (81) & \nodata \\
{\bf V2467 Cyg} & 6.7 (82) & 2454176.27 (82) & 7 (83) & 950 (82) & 1.5 (84) & 1.4e+22 & 0.159 (85) & 3.1$\pm0.5$ (86) & M (87) & \nodata \\
{\bf V2468 Cyg} & 7.4 (88) & 2454534.2 (88) & 10: & 1000 (88) & 0.77 (89) & 1.0e+22 & 0.242 (90) & \nodata & N & \nodata \\
{\bf V2491 Cyg} & 7.54 (91) & 2454567.86 (91) & 4.6 (92) & 4860 (93) & 0.43 (94) & 4.7e+21 & 0.09580: (95) & 10.5 (96) & N & M \\
V2487 Oph & 9.5 (97) & 2450979.5 (97) & 6.3 (98) & 10000 (98) & 0.38$\pm0.08$ (98) & 2.0e+21 & \nodata & 27.5$\pm3$ (99) & N (100) & Y (101) \\
{\bf V2540 Oph} & 8.5 (102) & 2452295.5 (102) & \nodata & \nodata & \nodata & 2.3e+21 & 0.284781 (103) & 5.2$\pm0.8$ (103) & N & \nodata \\
V2575 Oph & 11.1 (104) & 2453778.8 (104) & 20: & 560 (104) & 1.4 (105) & 3.3e+21 & \nodata & \nodata & N (105) & \nodata \\
{\bf V2576 Oph} & 9.2 (106) & 2453832.5 (106) & 8: & 1470 (106) & 0.25 (107) & 2.6e+21 & \nodata & \nodata & N & \nodata \\
{\bf V2615 Oph} & 8.52 (108) & 2454187.5 (108) & 26.5 (108) & 800 (109) & 0.9 (108) & 3.1e+21 & \nodata & 3.7$\pm0.2$ (108) & Y (110) & \nodata \\
{\bf V2670 Oph} & 9.9 (111) & 2454613.11 (111) & 15: & 600 (112) & 1.3: (113) & 2.9e+21 & \nodata & \nodata & N (114) & \nodata \\
{\bf V2671 Oph} & 11.1 (115) & 2454617.5 (115) & 8: & 1210 (116) & 2.0 (117) & 3.3e+21 & \nodata & \nodata & M (117) & \nodata \\
{\bf V2672 Oph} & 10.0 (118) & 2455060.02 (118) & 2.3 (119) & 8000 (118) & 1.6$\pm0.1$ (119) & 4.0e+21 & \nodata & 19$\pm2$ (119) & \nodata & M \\
V351 Pup & 6.5 (120) & 2448617.5 (120) & 16 (121) & \nodata & 0.72$\pm0.1$ (122) & 6.2e+21 & 0.1182 (123) & 2.7$\pm0.7$ (122) & N & \nodata \\
{\bf V382 Nor} & 8.9 (124) & 2453447.5 (124) & 12 (40) & 1850 (23) & \nodata & 1.7e+22 & \nodata & \nodata & \nodata & \nodata \\
V382 Vel & 2.85 (125) & 2451320.5 (125) & 4.5 (126) & 2400 (126) & 0.05: (126) & 3.4e+21 & 0.146126 (127) & 1.68$\pm0.3$ (126) & N & \nodata \\
{\bf V407 Cyg} & 6.8 (128) & 2455266.314 (128) & 5.9 (129) & 2760 (129) & 0.5$\pm0.05$ (130) & 8.8e+21 & 15595 (131) & 2.7 (131) & \nodata & Y \\
{\bf V458 Vul} & 8.24 (132) & 2454322.39 (132) & 7 (133) & 1750 (134) & 0.6 (135) & 3.6e+21 & 0.06812255 (136) & 8.5$\pm1.8$ (133) & N (135) & \nodata \\
{\bf V459 Vul} & 7.57 (137) & 2454461.5 (137) & 18 (138) & 910 (139) & 1.0 (140) & 5.5e+21 & \nodata & 3.65$\pm1.35$ (138) & Y (140) & \nodata \\
V4633 Sgr & 7.8 (141) & 2450895.5 (141) & 19$\pm3$ (142) & 1700 (143) & 0.21 (142) & 1.4e+21 & 0.125576 (144) & 8.9$\pm2.5$ (142) & N & \nodata \\
{\bf V4643 Sgr} & 8.07 (145) & 2451965.867 (145) & 4.8 (146) & 4700 (147) & 1.67 (148) & 1.4e+22 & \nodata & 3 (148) & N & \nodata \\
{\bf V4743 Sgr} & 5.0 (149) & 2452537.5 (149) & 9 (150) & 2400 (149) & 0.25 (151) & 1.2e+21 & 0.281 (152) & 3.9$\pm0.3$ (151) & N & \nodata \\
{\bf V4745 Sgr} & 7.41 (153) & 2452747.5 (153) & 8.6 (154) & 1600 (155) & 0.1 (154) & 9.0e+20 & 0.20782 (156) & 14$\pm5$ (154) & \nodata & \nodata \\
{\bf V476 Sct} & 10.3 (157) & 2453643.5 (157) & 15 (158) & \nodata & 1.9 (158) & 1.2e+22 & \nodata & 4$\pm1$ (158) & M (159) & \nodata \\
{\bf V477 Sct} & 9.8 (160) & 2453655.5 (160) & 3 (160) & 2900 (161) & 1.2: (162) & 4e+21 & \nodata & \nodata & M (163) & \nodata \\
{\bf V5114 Sgr} & 8.38 (164) & 2453081.5 (164) & 11 (165) & 2000 (23) & \nodata & 1.5e+21 & \nodata & 7.7$\pm0.7$ (165) & N (166) & \nodata \\
{\bf V5115 Sgr} & 7.7 (167) & 2453459.5 (167) & 7 (40) & 1300 (168) & 0.53 (169) & 2.3e+21 & \nodata & \nodata & N (169) & \nodata \\
{\bf V5116 Sgr} & 8.15 (170) & 2453556.91 (170) & 6.5 (171) & 970 (172) & 0.25 (173) & 1.5e+21 & 0.1238 (171) & 11$\pm3$ (173) & N (174) & \nodata \\
{\bf V5558 Sgr} & 6.53 (175) & 2454291.5 (175) & 125 (176) & 1000 (177) & 0.80 (178) & 1.6e+22 & \nodata & 1.3$\pm0.3$ (176) & N (179) & \nodata \\
{\bf V5579 Sgr} & 5.56 (180) & 2454579.62 (180) & 7: & 1500 (23) & 1.2 (181) & 3.3e+21 & \nodata & \nodata & Y (181) & \nodata \\
{\bf V5583 Sgr} & 7.43 (182) & 2455051.07 (182) & 5: & 2300 (182) & 0.39 (30) & 2.0e+21 & \nodata & 10.5 & \nodata & \nodata \\
{\bf V574 Pup} & 6.93 (183) & 2453332.22 (183) & 13 (184) & 2800 (184) & 0.5$\pm0.1$ & 6.2e+21 & \nodata & 6.5$\pm1$ & M (185) & \nodata \\
{\bf V597 Pup} & 7.0 (186) & 2454418.75 (186) & 3: & 1800 (187) & 0.3 (188) & 5.0e+21 & 0.11119 (189) & \nodata & N (188) & \nodata \\
{\bf V598 Pup} & 3.46 (14) & 2454257.79 (14) & 9$\pm1$ (190) & \nodata & 0.16 (190) & 1.4e+21 & \nodata & 2.95$\pm0.8$ (190) & \nodata & \nodata \\
{\bf V679 Car} & 7.55 (191) & 2454797.77 (191) & 20: & \nodata & \nodata & 1.3e+22 & \nodata & \nodata & \nodata & \nodata \\
{\bf V723 Cas} & 7.1 (192) & 2450069.0 (192) & 263 (2) & 600 (193) & 0.5 (194) & 2.35e+21 & 0.69 (195) & 3.86$\pm0.23$ (196) & N & \nodata \\
V838 Her & 5 (197) & 2448340.5 (197) & 2 (198) & \nodata & 0.5$\pm0.1$ (198) & 2.6e+21 & 0.2975 (199) & 3$\pm1$ (198) & Y (200) & \nodata \\
{\bf XMMSL1 J06} & 12 (201) & 2453643.5 (202) & 8$\pm2$ (202) & \nodata & 0.15 (203) & 8.7e+20 & \nodata & 50 & \nodata & \nodata \\
\enddata
\end{deluxetable}
\end{longrotatetable}
|
https://ftp.rrzn.uni-hannover.de/tex-archive/fonts/ifsym/ifsym.sty | uni-hannover.de | CC-MAIN-2022-49 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2022-49/segments/1669446711108.34/warc/CC-MAIN-20221206124909-20221206154909-00713.warc.gz | 289,785,243 | 2,777 | %%%%%%%%%%%%%%%%%% ifsym.sty %%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\HalfSun}{\textweathersymbol{17}}
\newcommand{\NoSun}{\textweathersymbol{18}}
\newcommand{\Fog}{\textweathersymbol{19}}
\newcommand{\ThinFog}{\textweathersymbol{20}}
\newcommand{\Rain}{\textweathersymbol{21}}
\newcommand{\WeakRain}{\textweathersymbol{22}}
\newcommand{\Hail}{\textweathersymbol{23}}
\newcommand{\Sleet}{\textweathersymbol{24}}
\newcommand{\Snow}{\textweathersymbol{25}}
\newcommand{\Lightning}{\textweathersymbol{26}}
\newcommand{\Cloud}{\textweathersymbol{27}}
\newcommand{\RainCloud}{\textweathersymbol{28}}
\newcommand{\WeakRainCloud}{\textweathersymbol{29}}
\newcommand{\SunCloud}{\textweathersymbol{30}}
\newcommand{\SnowCloud}{\textweathersymbol{31}}
\newcommand{\FilledCloud}{\textweathersymbol{32}}
\newcommand{\FilledRainCloud}{\textweathersymbol{33}}
\newcommand{\FilledWeakRainCloud}{\textweathersymbol{34}}
\newcommand{\FilledSunCloud}{\textweathersymbol{35}}
\newcommand{\FilledSnowCloud}{\textweathersymbol{36}}
% \wind{bedeckung 0(sonne)-4}{richtung in grad}{staerke in km/h}
\newcommand{\wind}[3]
{\rotatebox{#2}{%
\makebox[0pt][c]{\textweathersymbol{#1}}
\setcounter{ifsymcnt}{48+#3/10}%
\makebox[0pt][c]{\textweathersymbol{\value{ifsymcnt}}}
}
}
% \thermo{0-6}
\newcommand{\Thermo}[1]
{\setcounter{ifsymcnt}{5+#1}\textweathersymbol{\value{ifsymcnt}}}
}
\ProcessOptions\relax
%%%%%%%%%%%%%%%%%% end of ifsym.sty %%%%%%%%%%%%%%%%%%%%%%% |
http://ida.darksky.org/search.php?sqlQuery=SELECT%20author%2C%20title%2C%20type%2C%20year%2C%20publication%2C%20abbrev_journal%2C%20volume%2C%20issue%2C%20pages%2C%20keywords%2C%20abstract%2C%20thesis%2C%20editor%2C%20publisher%2C%20place%2C%20abbrev_series_title%2C%20series_title%2C%20series_editor%2C%20series_volume%2C%20series_issue%2C%20edition%2C%20language%2C%20author_count%2C%20online_publication%2C%20online_citation%2C%20doi%2C%20serial%2C%20area%20FROM%20refs%20WHERE%20serial%20%3D%202714%20ORDER%20BY%20first_author%2C%20author_count%2C%20author%2C%20year%2C%20title&client=&formType=sqlSearch&submit=Cite&viewType=&showQuery=0&showLinks=1&showRows=5&rowOffset=&wrapResults=1&citeOrder=&citeStyle=APA&exportFormat=RIS&exportType=html&exportStylesheet=&citeType=LaTeX&headerMsg= | darksky.org | CC-MAIN-2020-45 | application/x-latex | application/x-latex | crawl-data/CC-MAIN-2020-45/segments/1603107880878.30/warc/CC-MAIN-20201023073305-20201023103305-00074.warc.gz | 53,005,200 | 1,319 | %&LaTeX
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\begin{document}
\begin{thebibliography}{1}
\bibitem{Abay+Amare2018} Abay, K. A., \& Amare, M. (2018). Night light intensity and women{\textquoteright}s body weight: Evidence from Nigeria. \textit{Econ Hum Biol}, \textit{31}, 238--248.
\end{thebibliography}
\end{document}
|
https://git.ucc.asn.au/?p=ipdf/sam.git;a=blob_plain;f=chapters/BackgroundHardware.tex;hb=198adc71cff9a08993e465710d71a9880d4ab43c | ucc.asn.au | CC-MAIN-2022-40 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-40/segments/1664030337631.84/warc/CC-MAIN-20221005140739-20221005170739-00419.warc.gz | 318,331,952 | 1,429 | \chapter{Physical Limitations on Precision}\label{BackgroundHardware}
This chapter will focus on the physical limitations on precision due to hardware. It will describe the traditional approaches to working with real numbers and the current state of the art.
NOTE: Depending on how much I get into the VHDL stuff, this chapter may seem less relevant to the actual research we are doing and it might be very short.
However I think even if I do not actually do any hardware designs, a literature review about precision of numbers would not be complete if it only mentioned software algorithms.
\section{Real Number Representations}
This section will discuss representations of real numbers in hardware, including the trivial: fixed point, IEEE floating points, and anything else I find that is interesting.
\section{Floating Point Units}
This section will give an overview of FPUs, focusing on IEEE.
\section{Graphical Processing Units}
This section will discuss anything relevant we can find on GPUs, but will probably be very short.
|
http://smlnj-gforge.cs.uchicago.edu/scm/viewvc.php/*checkout*/sml-mode/trunk/sml-mode.texi?revision=2817&root=smlnj&pathrev=3106 | uchicago.edu | CC-MAIN-2020-29 | application/x-texinfo | application/x-tex | crawl-data/CC-MAIN-2020-29/segments/1593655906934.51/warc/CC-MAIN-20200710082212-20200710112212-00475.warc.gz | 131,756,394 | 13,018 | \input texinfo @c -*-texinfo-*-
@comment "@(#)$Name$:$Id$"
@comment Documentation for the GNU Emacs SML mode.
@comment Copyright (C) 1997-1999 Matthew J.@: Morley
@comment This file is part of the sml-mode distribution.
@comment sml-mode is free software; you can redistribute it and/or modify
@comment it under the terms of the GNU General Public License as published by
@comment the Free Software Foundation; either version 3 of the License,
@comment or (at your option) any later version.
@comment sml-mode is distributed in the hope that it will be useful,
@comment but WITHOUT ANY WARRANTY; without even the implied warranty of
@comment MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
@comment GNU General Public License for more details.
@comment You should have received a copy of the GNU General Public License
@comment along with sml-mode; see the file COPYING. If not, write to
@comment the Free Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA.
@setfilename sml-mode.info
@settitle SML mode - The Emacs SML editing mode
@dircategory Emacs
@direntry
* sml:(sml-mode). Emacs mode for editing SML
@end direntry
@setchapternewpage on
@titlepage
@sp 5
@center @titlefont{Editing and Running Standard ML}
@center @titlefont{under GNU Emacs}
@sp 5
@center {SML mode, Version $Name$}
@center {August 1999}
@sp 2
@author Authors: Matthew J.@: Morley and Stefan Monnier
@page
@vskip 0pt plus 1filll
Copyright @copyright{} (Anon)
@sp 1
@noindent
GNU General Public License as published by the Free Software Foundation;
either version 3, or (at your option) any later version.
@sp 1
@noindent
SML mode is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
Public License for more details.
@sp 1
@noindent
You should have received a copy of the GNU General Public License along
with GNU Emacs; see the file COPYING. If not, write to the Free Software
Foundation, 675 Mass Ave, Cambridge, MA 02139, USA.
@end titlepage
@setchapternewpage off
@headings double
@c ============================================================ TOP NODE
@node Top, Copying, (dir), (dir)
@ifinfo
@chapter SML Mode Info
@c == Top, Copying, (dir), (dir) =======================================
@noindent
You are looking at the top node of the Info tree documenting
@sc{sml-mode} (Version $Name$). Not all functions are documented here, but
those that aren't you probably won't miss. All commands and settable
variables have built-in documentation, as per usual Emacs conventions.
@end ifinfo
@menu
* Copying:: You can copy SML mode
* Introduction:: Setting things up
* SML Mode:: Editing SML source
* Interaction Mode:: Running ML processes
* Configuration:: Menus, highlighting, setting defaults
Indexes
* Command Index:: Commands you can invoke
* Variable Index:: Variables you can set
* Key Index:: Default keybindings
Introduction
* Contributors:: Who did what
* Getting Started:: What to tell Emacs
* Getting Help:: How Emacs can help
SML Mode
* Basics:: On entering SML mode
* Indentation:: Prettying SML text
* Magic Insertion:: Templates and electric keys
* SML Mode Defaults:: Variables controlling indentation
Interaction Mode
* Running ML:: Commands to run the ML compiler in a buffer
* ML Interaction:: Sending program fragments to the compiler
* Tracking Errors:: Finding reported syntax errors
* Process Defaults:: Setting defaults for process interaction
Configuration
* Hooks:: Creating hooks
* Key Bindings:: Binding commands to keys
* Highlighting:: Syntax colouring
* Advanced Topics:: You may need to speak Emacs Lisp
@end menu
@c ============================================================= COPYING
@node Copying, Introduction, Top, Top
@ifinfo
@chapter Copying
@c == Copying, Introduction, Top, Top ==================================
@noindent
You can freely copy, modify and redistribute SML mode because it's
made available under the liberal terms of the GNU General Public
License.
GNU General Public License as published by the Free Software Foundation;
either version 3, or (at your option) any later version.
SML mode is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
Public License for more details.
You should have received a copy of the GNU General Public License along
with GNU Emacs; see the file COPYING. If not, write to the Free Software
Foundation, 675 Mass Ave, Cambridge, MA 02139, USA.
@end ifinfo
@c ======================================================== INTRODUCTION
@node Introduction, SML Mode, Copying, Top
@chapter Introduction
@c == Introduction, SML Mode, Copying, Top =============================
@noindent
SML mode is a major mode for Emacs for editing Standard ML. It has
some novel bugs, and some nice features:
@itemize @bullet
@item
Automatic indentation of sml code---a number of variables to customise
the indentation.
@item
Easy insertion for commonly used templates like let, local, signature,
and structure declarations, with minibuffer prompting for types and
expressions.
@item
Magic pipe insertion: @code{|} automatically determines if it is used
in a case or fun construct, and indents the next line as appropriate,
inserting @code{=>} or the name of the function.
@item
Inferior shell for running ML. There's no need to leave Emacs, just keep
on editing while the compiler runs in another window.
@item
Automatic ``use file'' in the inferior shell---you can send files,
buffers, or regions of code to the ML subprocess.
@item
Menus, and syntax and keyword highlighting supported for Emacs 19 and
derivatives.
@item
Parsing errors from the inferior shell, and repositioning the
source with next-error---just like in c-mode.
@item
SML mode can be easily configured to work with a number of Standard
ML compilers, and other SML based tools.
@end itemize
@menu
* Contributors:: Who did what
* Getting Started:: What to tell Emacs
* Getting Help:: How Emacs can help
@end menu
@c ======================================================== CONTRIBUTORS
@node Contributors, Getting Started, Introduction, Introduction
@section Contributors to the SML mode
@cindex Contributors
@cindex Authors
Contributions to the package are welcome. I have limited time to work
on this project, but I will gladly add any code that you contribute to
me to this package.
Although the history of sml-mode is obscure, it seems that
the following persons have made contributions to sml-mode:
@itemize @bullet
@item
Lars Bo Nielsen wrote the original version of the code, providing the
sml editing mode and the inferior-sml support.
@item
Olin Shivers (@samp{shivers@@ai.mit.edu}) hacked the inferior-sml support
to use comint and call the whole thing ml-mode.
@item
Steven Gilmore supposedly provided some early attempt at menubar support.
@item
Matthew J. Morley (@samp{matthew@@verisity.com}) was maintainer for
a long time (until version 3.4) and provided many additions and fixes in
all areas.
@item
Frederick Knabe (@samp{knabe@@ecrc.de}) provided the original code for
font-lock and hilite support as well as for proper handling of nested
comments and of all the string escape sequences.
@item
Matthias Blume (@samp{blume@@kurims.kyoto-u.ac.jp}) provided a sml-make
which was replaced by sml-compile.
@item
Monnier Stefan (@samp{monnier@@cs.yale.edu}) completely reworked the
indentation engine as well as most of the rest of the code and is
the current maintainer since after version 3.4.
@end itemize
@c ===================================================== GETTING STARTED
@node Getting Started, Getting Help, Contributors, Introduction
@section Getting started
@c == Getting Started, Getting Help, Contributors, Introduction ========
@noindent
With luck your system administrator will have installed SML mode
somewhere convenient, so it will just magically all work---you can
skip the rest of this getting started section. Otherwise you will need
to tell Emacs where to find all the SML mode @file{.el} files, and
when to use them. The where is addressed by locating the Lisp code on
your Emacs Lisp load path---you may have to create a directory for this,
say @file{/home/mjm/elisp}, and then insert the following lines in your
@file{/home/mjm/.emacs} file:
@lisp
(add-to-list 'load-path "/home/mjm/elisp")
(autoload 'sml-mode "sml-mode" "Major mode for editing SML." t)
(autoload 'run-sml "sml-proc" "Run an inferior SML process." t)
@end lisp
@noindent
The first line adjusts Emacs' internal search path so it can locate the
Lisp source you have copied to that directory; the second and third
lines tell Emacs to load the code automatically when it is needed. You
can then switch any Emacs buffer into SML mode by entering the command
@example
M-x sml-mode
@end example
@noindent
It is usually more convenient to have Emacs automatically place the
buffer in SML mode whenever you visit a file containing ML
programs. The simplest way of achieving this is to put something like
@lisp
(add-to-list 'auto-mode-alist '("\\.\\(sml\\|sig\\)\\'" . sml-mode))
@end lisp
@noindent
also in your @file{.emacs} file. Subsequently (after a restart), any
files with these extensions will be placed in SML mode buffers when
you visit them.
You may want to pre-compile the @file{sml-*.el} files (@kbd{M-x
byte-compile-file}) for greater speed---byte compiled code loads and
runs somewhat faster.
@c ======================================================== GETTING HELP
@node Getting Help, , Getting Started, Introduction
@section Help!
@c == Getting Help, , Getting Started, Introduction ====================
@noindent
You're reading it. Apart from the on-line info tree (@kbd{C-h i} is the
Emacs key to enter the @code{info} system---you should follow the brief
tutorial if this is unfamiliar), there are further details on specific
commands in their documentation strings. Only the most useful
SML mode commands are documented in the info tree: to find out more
use Emacs' help facilities.
Briefly, to get help on a specific function use @kbd{C-h f} and enter
the command name. All (almost all, then) SML mode commands begin
with @code{sml-}, so if you type this and press @key{TAB} (for
completion) you will get a list of all commands. Another way is to use
@kbd{C-h a} and enter the string @code{sml}. This is command apropos; it
will list all commands with that sub-string in their names, and any key
binding they may have in the current buffer. Command apropos gives a
one-line synopsis of what each command does.
Some commands are also variables---such things are allowed in Lisp, if
not in ML! @xref{Command Index}, for a list of (info) documented
functions. @xref{Variable Index}, for a list of user settable variables
to control the behaviour of SML mode.
Before accessing this information on-line from within Emacs you may have
to set the variable @code{sml-mode-info}. Put in your @file{.emacs} file
something like:
@vindex sml-mode-info
@findex sml-mode-info
@kindex @kbd{C-c C-i}
@lisp
(setq sml-mode-info "/home/mjm/info/sml-mode.info")
@end lisp
@noindent
When different from the default this variable should be a string giving
the absolute name of the @file{.info} file. Then @kbd{C-c C-i} in
SML mode (i.e., the command @kbd{M-x sml-mode-info}) will bring up
the manual. This help is also accessible from the menu. (Resetting this
variable will not be necessary if your site administrator has been kind
enough to install SML mode and its attendant documentation in the
Emacs hierarchy.)
@c ============================================================ SML MODE
@node SML Mode, Interaction Mode, Introduction, Top
@chapter Editing with SML Mode
@c == SML Mode, Interaction Mode, Introduction, Top ====================
@noindent
Now SML mode provides just a few additional editing commands. Most of
the work has gone into implementing the indentation algorithm which, if
you think about it, has to be complicated for a language like
ML. @xref{SML Mode Defaults,,Indentation Defaults}, for details on how
to control some of the behaviour of the indentation algorithm. Principal
goodies are the `electric pipe' feature, and the ability to insert
common SML forms (macros or templates).
@menu
* Basics:: On entering SML mode
* Indentation:: Prettying SML text
* Magic Insertion:: Templates and electric keys
* SML Mode Defaults:: Variables controlling indentation
@end menu
@c ============================================================== BASICS
@node Basics, Indentation, SML Mode, SML Mode
@section On entering SML mode
@c == Basics, Indentation, SML Mode, SML Mode ==========================
@noindent
@deffn Command sml-mode
This switches a buffer into SML mode. This is a @emph{major mode} in
Emacs. To get out of SML mode the buffer's major mode must be set to
something else, like @t{text-mode}. @xref{Getting Started}, for details
on how to set this up automatically when visiting an SML file.
@end deffn
Emacs is all hooks of course. A hook is a variable: if the variable is
non-nil it binds a list of Emacs Lisp functions to be run in some order
(usually left to right). You can customise SML mode with these
hooks:
@defvr Hook sml-mode-hook
Default: @code{nil}
This is run every time a new SML mode buffer is created (or if you
type @kbd{M-x sml-mode}). This is one place to put your preferred key
bindings. @xref{Configuration}, for some examples.
@end defvr
@c ========================================================= INDENTATION
@node Indentation, Magic Insertion, Basics, SML Mode
@section Automatic indentation
@c == Indentation, Magic Insertion, Basics, SML Mode ===================
@noindent
ML is a complicated language to parse, let alone compile. The
indentation algorithm is a little wooden (for some tastes), and the best
advice is not to fight it! There are several variables that can be
adjusted to control the indentation algorithm (@pxref{SML Mode
Defaults,,Customising SML Mode}, below).
@deffn Command indent-for-tab-command
Key: @key{TAB}
@kindex @key{TAB}
This command indents the current line. If you set the indentation of the
previous line by hand, @code{indent-for-tab-command} will indent relative to
this setting.
@end deffn
@deffn Command indent-region
Key: @kbd{C-M-\}
@kindex @kbd{C-M-\}
Indent the current region. Be patient if the region is large (like the
whole buffer).
@end deffn
@deffn Command sml-back-to-outer-indent
Key: @kbd{M-@key{TAB}}
@kindex @kbd{M-@key{TAB}}
Unindents the line to the next outer level of indentation.
@end deffn
Further indentation commands that Emacs provides (generically, for all
modes) that you may like to recall:
@itemize @minus
@item
@kbd{M-x newline-and-indent}
On @key{LFD} by default.
@kindex @key{LFD}
Insert a newline, then indent according to the major mode. @xref{Program
Indent,,Indentation for Programs,emacs,The Emacs Editor Manual}, for
details.
@item
@kbd{M-x indent-rigidly}
On @kbd{C-x @key{TAB}} by default.
@kindex @kbd{C-x @key{TAB}}
Moves all lines in the region right by its argument (left, for negative
arguments). @xref{Indentation,,,emacs,The Emacs Editor Manual}.
@item
@kbd{M-x indent-for-comment}
On @kbd{M-;} by default.
@kindex @kbd{M-;}
Indent this line's comment to comment column, or insert an empty
comment. @xref{Comment Commands,,,emacs,The Emacs Editor
Manual}.
@item
@kbd{M-x indent-new-comment-line}
On @kbd{M-@key{LFD}} by default.
@kindex @kbd{M-@key{LFD}}
Break line at point and indent, continuing comment if within one.
@xref{Multi-Line Comments,,,emacs,The Emacs Editor Manual}.
@end itemize
@kindex @kbd{C-x ;}
As with other language modes, @kbd{M-;} gives you a comment at the end
of the current line. The column where the comment starts is determined
by the variable @code{comment-column}---default is 40, but it can be
changed with @code{set-comment-column} (on @kbd{C-x ;} by default).
@c ===================================================== MAGIC INSERTION
@node Magic Insertion, SML Mode Defaults, Indentation, SML Mode
@section Electric features
@c == Magic Insertion, SML Mode Defaults, Indentation, SML Mode ========
@noindent
Electric keys are generally pretty irritating, so those provided by
SML mode are fairly muted. The only truly electric key is @kbd{;},
and this has to be enabled to take effect.
@deffn Command sml-electric-pipe
Key: @kbd{M-|}
@kindex @kbd{M-|}
When the point is in a `case' statement this opens a new line, indents
and inserts @code{| =>} leaving point just before the double arrow; if
the enclosing construct is a `fun' declaration, the newline is indented
and the function name copied at the appropriate column. Generally, try
it whenever a @code{|} is wanted---you'll like it!
@end deffn
@deffn Command sml-electric-space
Key: @kbd{M-SPC}
@kindex @kbd{M-SPC}
When the point is after a keyword like `let', this inserts the
corresponding predefined skeleton if one exists. Else it just inserts a
space. Another way to insert those skeletons is to use
@code{sml-insert-form}, described below.
@end deffn
@deffn Command sml-electric-semi
Key: @kbd{;}
@kindex @kbd{;}
Just inserts a semi-colon, usually. The behaviour of this command is
governed by the variable @code{sml-electric-semi-mode}.
@end deffn
@defvr Variable sml-electric-semi-mode
Default: @code{nil}
If this variable is @code{nil}, @code{sml-electric-semi} just inserts a
semi-colon, otherwise it inserts a semi-colon and a newline, and indents
the newline for SML.
@end defvr
@deffn Command sml-insert-form
Key: @kbd{C-c @key{RET}}
@kindex @kbd{C-c @key{RET}}
Interactive short-cut to insert common ML forms (a.k.a.@: macros, or
templates). Recognised forms are `let', `local', `case', `abstype',
`datatype', `signature', `structure', and `functor'. Except for `let'
and `local', these will prompt for appropriate parameters like functor
name and signature, etc.. This command prompts in the mini-buffer, with
completion.
By default @kbd{C-c @key{RET}} will insert at point, with the
indentation of the current column; if you give a prefix argument (i.e.,
@kbd{C-u C-c @key{RET}}) the command will insert a newline first,
indent, and then insert the template.
@end deffn
@code{sml-insert-form} is also extensible: see @ref{Configuration} for
further details.
@c ======================================================= MODE DEFAULTS
@node SML Mode Defaults, , Magic Insertion, SML Mode
@section Indentation defaults
@c == SML Mode Defaults, , Magic Insertion, SML Mode ===================
@noindent
Several variables try to control the indentation algorithm and other
features of SML mode. Most of them are still in flux so they are not
described here yet.
If the default values are not acceptable you can set these variables
permanently in your @file{.emacs} file. @xref{Configuration}, for
details and examples.
@defvr Variable sml-indent-level
@findex sml-indent-level
Default: @code{4}
This variable controls the block indentation level.
@end defvr
@c end vtable
@c ========================================================= INTERACTION
@node Interaction Mode, Configuration, SML Mode, Top
@chapter Running ML under Emacs
@c == Interaction Mode, Configuration, SML Mode, Top ===================
@noindent
The most useful feature of SML mode is that it provides a convenient
interface to the compiler. How serious users of ML put up with a
teletype interface to the compiler is beyond me@.@.@. but perhaps there
are other interfaces to compilers that require one to part with serious
money. Such remarks can quickly become dated---in this case, let's hope
so!
Anyway, SML mode provides an interaction mode,
@code{inferior-sml-mode}, where the compiler runs in a separate buffer
in a window or frame of its own. You can use this buffer just like a
terminal, but it's usually more convenient to mark some text in the
SML mode buffer and have Emacs communicate with the sub-process. The
features discussed below are syntax-independent, so they should work
with a wide range of ML-like tools and compilers. @xref{Process
Defaults}, for some hints.
@findex inferior-sml-mode
@code{inferior-sml-mode} is a specialisation of the @file{comint}
package that comes with Emacs and XEmacs.
@menu
* Running ML:: Commands to run the ML compiler in a buffer
* ML Interaction:: Sending program fragments to the compiler
* Tracking Errors:: Finding reported syntax errors
* Process Defaults:: Setting defaults for process interaction
@end menu
@c ========================================================== RUNNING ML
@node Running ML, ML Interaction, Interaction Mode, Interaction Mode
@section Starting the compiler
@c == Running ML, ML Interaction, Interaction Mode, Interaction Mode ==
@noindent
Start your favourite ML compiler with the command
@example
@kbd{M-x run-sml}
@end example
@noindent
This creates a process interaction buffer that inherits some key
bindings from SML mode and from @file{comint} (@pxref{Shell Mode, ,
, emacs, The Emacs Editor Manual}). Starting the ML compiler adds some
functions to SML mode buffers so that program text can be
communicated between editor and compiler (@pxref{ML Interaction}).
The name of the ML compiler is the first thing you should know how to
specify:
@defvar sml-program-name
Default: @code{"sml"}
The program to run as ML. You might need to specify the full path name
of the program.
@end defvar
@defvar sml-default-arg
Default: @code{""}
Useful for Poly/ML users who may supply a database file, or others who
have wrappers for setting various options around the command to run the
compiler. Moscow ML people might set this to @code{"-P full"}, etc..
@end defvar
The variable @code{sml-program-name} is a string holding the name
of the program @emph{as you would type it at the shell}. You
can always choose a program different to the default by invoking
@example
@kbd{C-u M-x run-sml}
@end example
@noindent
With the prefix argument Emacs will prompt for the command name and any
command line arguments to pass to the compiler. Thereafter Emacs will
use this new name as the default, but for a permanent change you should
set this in your @file{.emacs} with, e.g.:
@lisp
(setq sml-program-name "nj-sml")
@end lisp
@deffn Command run-sml
Launches ML as an inferior process in another buffer; if an ML process
already exists, just switch to the process buffer. A prefix argument
allows you to edit the command line to specify the program, and any
command line options.
@end deffn
@defvr Hook inferior-sml-mode-hook
Default: @code{nil}
@kbd{M-x run-sml} runs @code{comint-mode-hook} and
@code{inferior-sml-mode-hook} hooks in that order, but @emph{after} the
compiler is started. Use @code{inferior-sml-mode-hook} to set any
@code{comint} buffer-local configurations for SML mode you like.
@end defvr
@deffn Command switch-to-sml
Key: @kbd{C-c C-s}
@kindex @kbd{C-c C-s}
Switch from the SML buffer to the interaction buffer. By default point
will be placed at the end of the process buffer, but a prefix argument
will leave point wherever it was before. If you try @kbd{C-c C-s} before
an ML process has been started, you'll just get an error message to the
effect that there's no current process buffer.
@end deffn
@deffn Command sml-cd
When started, the ML compiler's default working directory is the
current buffer's default directory. This command allows the working
directory to be changed, if the compiler can do this. The variable
@code{sml-cd-command} specifies the compiler command to invoke
(@pxref{Process Defaults}).
@end deffn
@c ======================================================== SENDING TEXT
@node ML Interaction, Tracking Errors, Running ML, Interaction Mode
@section Speaking to the compiler
@c == ML Interaction, Tracking Errors, Running ML, Interaction Mode ====
@noindent
Several commands are defined for sending program fragments to the
running compiler. Each of the following commands takes a prefix argument
that will switch the input focus to the process buffer afterwards
(leaving point at the end of the buffer):
@deffn Command sml-load-file
Key: @kbd{C-c C-l}
@kindex @kbd{C-c C-l}
Send a `use file' command to the current ML process. The variable
@code{sml-use-command} is used to define the correct template for the
command to invoke (@pxref{Process Defaults}). The default file is the
file associated with the current buffer, or the last file loaded if you
are in the interaction buffer.
@end deffn
@deffn Command sml-send-region
@findex sml-send-region-and-go
Key: @kbd{C-c C-r}
@kindex @kbd{C-c C-r}
Send the current region of text in the SML buffer.
@code{sml-send-region-and-go} is a similar command for you to bind in
SML mode if you wish: it'll send the region and then switch-to-sml.
@end deffn
@c @deffn Command sml-send-function
@c @findex sml-send-function-and-go
@c Send the enclosing `function' definition. Contrary to the suggestive
@c name, this command @emph{does not} try to determine the extent of the
@c function definition because that is too difficult with ML. Instead
@c this just sends the enclosing @emph{paragraph} (delimited by blank
@c lines or form-feed characters).
@c @end deffn
@deffn Command sml-send-buffer
Key: @kbd{C-c C-b}
@kindex @kbd{C-c C-b}
Send the contents of the current buffer to ML.
@end deffn
@c ===================================================== TRACKING ERRORS
@node Tracking Errors, Process Defaults, ML Interaction, Interaction Mode
@section Finding errors
@c == Tracking Errors, Process Defaults, ML Interaction, Interaction Mode
@noindent
SML mode provides one customisable function for locating the source
position of errors reported by the compiler. This should work whether
you type @code{use "puzzle.sml";} into the interaction buffer, or use
one of the mechanisms provided for sending programs directly to the
compiler---@pxref{ML Interaction}.
@deffn Command next-error
@findex next-error
Key: @kbd{C-x`}
@kindex @kbd{C-x`}
Jump to the source location of the next error reported by the compiler.
All the usual error-navigation commands are available, see
@pxref{Compilation Mode, , , emacs, The Emacs Editor Manual}.
@end deffn
@c ==================================================== PROCESS DEFAULTS
@node Process Defaults, , Tracking Errors, Interaction Mode
@section Process defaults
@c == Process Defaults, , Tracking Errors, Interaction Mode ============
@noindent
The process interaction code is independent of the compiler used,
deliberately, so SML mode will work with a variety of ML compilers
and ML-based tools. There are therefore a number of variables that may
need to be set correctly before SML mode can speak to the compiler.
Things are by default set up for Standard ML of New Jersey, but
switching to a new system is quite easy.
@defvar sml-use-command
Default: @code{"use \"%s\""}
Use file command template. Emacs will replace the @code{%s} with a file
name. Note that Emacs requires double quote characters inside strings
to be quoted with a backslash.
@end defvar
@defvar sml-cd-command
Default: @code{"OS.FileSys.chDir \"%s\""}
Compiler command to change the working directory. Not all ML systems
support this feature (well, Edinburgh (core) ML didn't), but they
should.
@end defvar
@defvar sml-prompt-regexp
Default: @code{"^[-=>#] *"}
Matches the ML compiler's prompt: @file{comint} uses this for various
purposes.
@end defvar
To customise error reportage for different ML compilers you need to set
two further variables before @code{next-error} can be useful:
@defvar sml-error-regexp-alist
Alist that specifies how to match errors in compiler output.
Each elt has the form (REGEXP FILE-IDX LINE-IDX [COLUMN-IDX FILE-FORMAT...])
If REGEXP matches, the FILE-IDX'th subexpression gives the file name, and
the LINE-IDX'th subexpression gives the line number. If COLUMN-IDX is
given, the COLUMN-IDX'th subexpression gives the column number on that line.
If any FILE-FORMAT is given, each is a format string to produce a file name to
try; %s in the string is replaced by the text matching the FILE-IDX'th
subexpression.
@end defvar
@c A typical way of (re)setting these variables correctly is to put
@c something in your @file{.emacs} file that resembles
@c @example
@c (setq sml-use-command "PolyML.use \"%s\"")
@c (setq sml-prompt-regexp "^[>#] *")
@c @end example
@c ======================================================= CONFIGURATION
@node Configuration, , Interaction Mode, Top
@chapter Configuration Summary
@c @footnote{@url{http://www.ahl.co.uk/}}
@c @footnote{@url{http://www.dina.kvl.dk/~sestoft/mosml.html}}
@noindent
This (sort of pedagogic) section gives more information on how to
configure SML mode: menus, key bindings, hooks and highlighting are
discussed, along with a few other random topics.
@menu
* Hooks:: Creating them
* Key Bindings:: Binding commands to keys
* Highlighting:: Syntax colouring
* Advanced Topics:: You may need to speak Emacs Lisp
@end menu
@c =============================================================== HOOKS
@node Hooks, Key Bindings, Configuration, Configuration
@section Hooks
@c == Hooks, Key Bindings, Configuration, Configuration ================
@noindent
One way to set SML mode variables (@pxref{SML Mode
Defaults,,Indentation Defaults}), and other defaults, is through the
@code{sml-mode-hook} in your @file{.emacs}. A simple example:
@lisp
(defun my-sml-mode-hook () "Local defaults for SML mode"
(setq sml-indent-level 2) ; conserve on horizontal space
(setq words-include-escape t) ; \ loses word break status
(setq indent-tabs-mode nil)) ; never ever indent with tabs
(add-hook 'sml-mode-hook 'my-sml-mode-hook)
@end lisp
@noindent
The body of @code{my-sml-mode-hook} is a sequence of assignments. In this
case it is not really necessary to set @code{sml-indent-level} in a hook
because this variable is global (most SML mode variables are). With
similar effect:
@lisp
(setq sml-indent-level 2)
@end lisp
@noindent
anywhere in your @file{.emacs} file. The variable @code{indent-tabs-mode} is
automatically made local to the current buffer whenever it is set
explicitly, so it @emph{must} be set in a hook if you always want
SML mode to behave like this.
Another hook is @code{inferior-sml-mode-hook}. This can be used to
control the behaviour of the interaction buffer through various
variables meaningful to @file{comint}-based packages:
@lisp
(defun my-inf-sml-mode-hook () "Local defaults for inferior SML mode"
(add-hook 'comint-output-filter-functions 'comint-truncate-buffer)
(setq comint-scroll-show-maximum-output t)
(setq comint-input-autoexpand nil))
(add-hook 'inferior-sml-mode-hook 'my-inf-sml-mode-hook)
@end lisp
@noindent
Again, the body is a sequence of assignments. Unless you run several ML
compilers simultaneously under one Emacs, this hook will normally only
get run once. You might want to look up the documentation (@kbd{C-h v}
and @kbd{C-h f}) for these buffer-local @code{comint} things.
@c ======================================================== Key Bindings
@node Key Bindings, Highlighting, Hooks, Configuration
@section Key bindings
@noindent
Customisation (in Emacs) usually entails putting favourite commands on
easily remembered keys. Two `keymaps' are defined in SML mode: one
is effective in program text buffers (@code{sml-mode-map}) and the other
is effective in interaction buffers (@code{inferior-sml-mode-map}).
The initial design ensures that (many of) the default key bindings from
the former keymap will also be available in the latter (e.g.,
@kbd{C-c`}).
Type @kbd{C-h m} in an SML mode buffer to find the default key
bindings (and similarly in an ML interaction buffer), and use the hooks
provided to install your preferred key bindings. Given that the keymaps
are global (variables):
@lisp
(defun my-sml-mode-hook () "Global defaults for SML mode"
(define-key sml-mode-map "\C-cd" 'sml-cd))
(add-hook 'sml-mode-hook 'my-sml-mode-hook)
@end lisp
@noindent
This has the effect of binding @code{sml-cd} to the key @kbd{C-c d}.
If you want the same behaviour from @kbd{C-c d} in the ML buffer:
@lisp
(defun my-inf-sml-mode-hook () "Global defaults for inferior SML mode"
(define-key inferior-sml-mode-map "\C-cd" 'sml-cd)
;; NB. for SML/NJ '96
(setq sml-cd-command "OS.FileSys.chDir \"%s\""))
(add-hook 'inferior-sml-mode-hook 'my-inf-sml-mode-hook)
@end lisp
There is nothing to stop you rebuilding the entire keymap for
SML mode and the ML interaction buffer in your @file{.emacs} of
course: SML mode won't define @code{sml-mode-map} or
@code{inferior-sml-mode-map} if you have already done so.
@c ======================================================== Highlighting
@node Highlighting, Advanced Topics, Key Bindings, Configuration
@section Syntax colouring
@noindent
Highlighting is very handy for picking out keywords in the program text,
spotting misspelled kewyords, and, if you have Emacs' @file{ps-print}
package installed (you usually do these days), obtaining pretty, even
colourful code listings---quite properly for your colourful ML programs.
The indentation scheme (strangely enough) also relies on the
highlighting code to properly handle nested comments, which is yet
another reason to turn on highlighting. To turn on highlighting,
use either of:
@lisp
M-x font-lock-mode
(add-hook 'sml-mode-hook 'turn-on-font-lock)
(global-font-lock-mode 1)
@end lisp
The first will turn it on in the current buffer.
The second will turn it on in all sml-mode buffers.
The last will turn it on everywhere.
This is valid for Emacs but maybe not for XEmacs. Check font-lock
documentation if you encounter problems.
@c ===================================================== ADVANCED TOPICS
@node Advanced Topics, , Highlighting, Configuration
@section Advanced Topics
@flushright
@emph{These forms are bloody useless; can't we have better ones?}
@end flushright
@sp 1
@noindent
You can indeed. @code{sml-insert-form} is extensible so all you need to
do is create the macros yourself. Define a @emph{keybord macro}
(@kbd{C-x (} <something> @kbd{C-x )}) and give it a suitable name:
@code{sml-addto-forms-alist} prompts for a name, say @code{NAME}, and
binds the macro @code{sml-form-NAME}. Thereafter @kbd{C-c @key{RET}
NAME} will insert the macro at point, and @kbd{C-u C-c @key{RET} NAME}
will insert the macro after a @code{newline-and-indent}. If you want to
keep your macros from one editing session to the next, go to your
@file{.emacs} file and call @code{insert-kbd-macro}; you'll need
to add @code{NAME} to @code{sml-forms-alist} permanently yourself:
@lisp
(defun my-sml-mode-hook () "Global defaults for SML mode"
;; whatever else you do
(add-to-list 'sml-forms-alist '("NAME" . FUNCTION)))
@end lisp
If you want to create templates like `case' that prompt for parameters
you'll have to do some Lisp programming. The @code{skeleton} package is
a good stating point. Better yet, you can reuse the wrappers used by
sml-mode itself in your sml-mode-hook:
@lisp
(add-hook 'sml-mode-hook
(lambda ()
(sml-def-skeleton "case" "Case expr: "
str " of" \n _ " => ")))
@end lisp
This will redefine `case' in order to leave the `of' on the first line.
See the documentation of @code{skeleton-insert} to get a better
understanding of how this works.
@sp 1
@flushright
@emph{I hate that indentation algorithm; can't I tweak it?}
@end flushright
@sp 1
@noindent
Ah, yes, of course, but this manual will not tell you how.
@sp 1
@flushright
@emph{Can SML mode handle more than one compiler running at once?}
@end flushright
Sure, just rename the @samp{*sml*} buffer and then use @code{run-sml}
as usual.
@sp 1
@flushright
@emph{What needs to be done to support other ML compilers?}
@end flushright
@sp 1
@noindent
Not much really. Just add the right regular expressions to
@code{sml-error-regexp-alist} and that should be all.
@c ======================================================= COMMAND INDEX
@headings singleafter
@node Command Index, Variable Index, , Top
@unnumbered Command Index
@printindex fn
@c ====================================================== VARIABLE INDEX
@c node Variable Index, , Command Index, Top
@node Variable Index, Key Index, Command Index, Top
@unnumbered Variable Index
@c == Variable Index, Key Index, Command Index, Top ====================
@printindex vr
@c =========================================================== KEY INDEX
@node Key Index, , Variable Index, Top
@unnumbered Key Index
@c == Key Index, , Variable Index, Top =================================
@printindex ky
@contents
@bye
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https://mirror.anarhija.net/pl.anarchistlibraries.net/mirror/g/gd/gilles-deleuze-felix-guattari-1933-mikropolityka-i-segmentacja.tex | anarhija.net | CC-MAIN-2021-10 | application/octet-stream | application/x-tex | crawl-data/CC-MAIN-2021-10/segments/1614178375096.65/warc/CC-MAIN-20210306131539-20210306161539-00326.warc.gz | 443,585,355 | 38,689 | \documentclass[DIV=12,%
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\title{1933 - Mikropolityka i Segmentacja}
\date{1980}
\author{Gilles Deleuze, Félix Guattari}
\subtitle{}
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\includegraphics[keepaspectratio=true,height=0.75\textheight,width=\textwidth]{g-d-gilles-deleuze-felix-guattari-1933-mikropolity-2.jpg}
\caption[]{\noindent Segmentacje (zespół typów)}
\end{figure}
Segmentowanie dokonuje się zewsząd i w każdym kierunku. Człowiek to zwierzę segmentowe. Segmentacja jest własnością wszystkich warstw, które się na nas składają. Mieszkać, krążyć, pracować, bawić się: to, co przeżywane, jest segmentowane – przestrzennie i społecznie. Dom jest segmentowany wedle przeznaczenia jego części; ulice – wedle porządku miasta; fabryka – wedle charakteru prac i czynności. Jesteśmy segmentowani binarnie, wedle wielkich dwoistych opozycji: klasy społeczne, lecz także mężczyźni i kobiety, dorośli i dzieci itd. Jesteśmy segmentowani kołowo, w coraz szerszych kołach, coraz większych dyskach czy kręgach [couronnes], na sposób „listu” Joyce’a\footnote{Oryg. „»Lettre« de Joyce”. Zapytany przez brazylijską tłumaczkę o kontekst, w związku z dwuznacznością francuskiego słowa lettre, tu jeszcze wziętego w cudzysłów, Deleuze miał odpowiedzieć, że kontekstu nie pamięta i że można swobodnie pominąć tę frazę: Felix Guattari, Suely Rolnik, Micropolitiques, Les Empêcheurs de penser en rond 2007, s. 223. Tłumacząc termin circulaire jako „kołowy\Slash{}-a”, podążamy za rozstrzygnięciem Krzysztofa Pomiana: Claude Lévi-Strauss, Antropologia strukturalna, tłum. K. Pomian, Aletheia 2009, s. 136 i nast. (tam także propozycja tłumaczenia pojawiającego się niżej terminu concentrique jako „współśrodkowy”, którą zaznaczamy, decydując się wszakże na termin „koncentryczny”) [przyp. tłum.].}: moje sprawy, sprawy mojej dzielnicy, mojego miasta, mojego kraju, świata\dots{} Jesteśmy segmentowani linearnie, po linii prostej, liniach prostych, gdzie każdy segment przedstawia odcinek lub „proces”: ledwie skończyliśmy jeden proces, już zaczynamy inny, zawsze dbając o procedury i na zawsze procedurom poddani – rodzina, szkoła, wojsko, zawód. A szkoła mówi nam: „nie jesteś już w domu”, wojsko zaś – „nie jesteś już w szkole\dots{}”. Niekiedy różne segmenty odsyłają do różnych jednostek lub grup, kiedy indziej ta sama jednostka lub grupa przechodzi z jednego do drugiego segmentu. Ale te postacie segmentacji – binarna, kołowa, linearna – zawsze ujmowane są jedna w drugiej, nawet przechodzą jedna w drugą, przekształcają się w zależności od punktu widzenia. Poświadczają to już dzicy: Lizot wskazuje, że Dom wspólny jest zorganizowany kołowo – od wewnątrz na zewnątrz, w serię kręgów, w których wykonywane są różne rodzaje dających się umiejscowić działań (kult i obrzędy, dalej wymiana dóbr, dalej życie rodzinne, dalej odpadki i łajno); ale jednocześnie „każdy z tych kręgów sam jest podzielony poprzecznie, każdy segment jest przypisany osobnemu lineażowi i ulega dalszym podziałom między różne grupy rodowe”\footnote{Jacques Lizot, Le cercle des feux, Éd. du Seuil, s. 118.}. W ogólniejszym kontekście Lévi-Strauss wykazał, że dualistyczna organizacja ludów pierwotnych odsyła do formy kołowej i przechodzi także w formę linearną obejmującą „dowolną liczbę grup” (przynajmniej trzy)\footnote{Claude Lévi-Strauss, Antropologia strukturalna, tłum. K. Pomian, KR 2000, rozdz. VIII: „Czy istnieją organizacje dualistyczne?”, s. 152.}.
Czemu jednak wracać do ludów pierwotnych, gdy idzie o nasze życie? Jest faktem, że pojęcie segmentacji zostało skonstruowane przez etnologów opisujących tak zwane społeczeństwa pierwotne, społeczeństwa bez stałego centralnego aparatu państwa, bez globalnej władzy i bez wyspecjalizowanych instytucji politycznych. Segmenty społeczne wykazują tu pewną elastyczność – zależnie od zadań i sytuacji – między dwiema pozycjami skrajnymi, biegunami całkowitego złączenia i pełnego rozdzielenia. Stąd zdolność szerokiego łączenia części heterogenicznych – tak, że spojenie segmentu z innym może dokonać się na wiele sposobów. Dalej – lokalna konstrukcja, która wyklucza możliwość określenia z góry domeny bazy (ekonomicznej, politycznej, prawnej, artystycznej). Następnie – zachowanie zewnętrznych własności sytuacyjnych czy też relacyjnych, jako nieredukowalnych do wewnętrznych własności struktury. Wreszcie – ciągła aktywność, która sprawia, że segmentacja nie jest ujmowana niezależnie od dokonującego się właśnie segmentowania, oddziałującego przez napory, zerwania, ponowne związania. Segmentacja pierwotna jest zarazem segmentacją wielogłosowego kodu, ufundowaną na lineażach, ich zmiennych relacjach i usytuowaniach, i segmentacją ruchomej terytorialności, ufundowaną na splątanych lokalnych podziałach. Kody i terytoria, lineaże klanowe i terytorialności plemienne tworzą względnie elastyczną tkankę segmentacji\footnote{Por. dwa przykładowe studia w Meyer Fortes, Edward Evans-Pritchard (red.), African Political Systems, Oxford University Press 1978, autorstwa Meyera Fortesa o ludzie Talensi, oraz Edwarda Evans-Pritcharda o Nuerach.}.
Wydaje nam się jednak kłopotliwym twierdzenie, że społeczeństwa państwowe, czy wręcz nasze nowoczesne państwa, są mniej segmentowe. Klasyczna opozycja między segmentowym a scentralizowanym wcale nie wydaje się trafna\footnote{Georges Balandier analizuje sposoby, w jakie etnologowie i socjologowie definiowali tę opozycję: Anthropologie politique, PUF 1967, s. 161–169.}. Państwo nie tylko oddziałuje na segmenty, które utrzymuje, lub którym pozwala przetrwać, lecz samo w sobie ma własną segmentację – i ją narzuca. Być może ustanowiona przez socjologów opozycja segmentowego i centralnego ma praźródło biologiczne: pierścieniowaty robak i centralny system nerwowy. Ale centralny mózg sam jest robakiem, bardziej jeszcze segmentowanym niż inne, gdy wziąć pod uwagę całą jego plastyczność – i pomimo niej. Nie istnieje opozycja między centralnym a segmentowym. Współczesny system polityczny jest globalną całością, zjednoczoną i jednoczącą, lecz może nią być o tyle, o ile wymaga istnienia zespołu podsystemów zestawionych razem, zazębiających się i skoordynowanych – analiza podejmowanych decyzji wydobywa na światło dzienne wszelkie rodzaje przegród, procesów częściowych, które zachodzą na siebie nawzajem, czemu towarzyszą zawsze przesunięcia i przemieszczenia. Technokracja postępuje naprzód za sprawą segmentowego podziału pracy (także w międzynarodowym podziale pracy). Biurokracja istnieje tylko poprzez swoje wydzielone przepierzeniami biura, funkcjonuje tylko poprzez „przemieszczanie celów” i odpowiadające mu „dysfunkcje”. Hierarchia nie ma jedynie kształtu piramidy, biuro szefa znajduje się tyleż na końcu korytarza, co na szczycie wieży. Krótko mówiąc, życie współczesne nie porzuciło segmentacji, lecz przeciwnie, jedynie ją usztywniło.
Zamiast przeciwstawiać segmentowe i scentralizowane, należałoby więc rozróżnić dwa typy segmentacji, jeden – „pierwotny” i elastyczny, drugi – „nowoczesny” i sztywny. To rozróżnienie przecina każdą z wcześniej zarysowanych postaci segmentacji:
\begin{enumerate}[1.]
\item\relax
Opozycje binarne (mężczyźni – kobiety, ci na górze – ci na dole itd.), bardzo silne w społeczeństwach pierwotnych, powstają, jak się wydaje, w maszynach i złożeniach, które same binarne nie są. Binarność społeczna: kobiety – mężczyźni w danej grupie, uruchamia reguły, wedle których jedne i drudzy wybierają sobie odpowiednio małżonków w grupie innej niż ich własna (stąd przynajmniej trzy grupy). W tym właśnie sensie Lévi-Strauss mógł wykazać, że organizacja dualistyczna w takich społeczeństwach sama w sobie nigdy nie jest wystarczająca. Z kolei cechą społeczeństw nowoczesnych, czy raczej państwowych, jest przydanie wartości maszynom dualnym, które funkcjonują jako takie, działając w trybie symultanicznym przez relacje jedno-jednoznaczne i ujmując następstwa w zbinaryzowane wybory. Klasy i płcie idą dwójkami, a zjawiska trójdzielności wynikają raczej z przemieszczania tego, co podwójne, niż odwrotnie. Widzieliśmy to szczególnie wyraźnie w wypadku maszyny Twarzy, która różni się w tej mierze od maszyn pierwotnych głów. Wydaje się, że społeczeństwa współczesne wyniosły dwójkową segmentację na poziom organizacji samowystarczalnej. Nie chodzi więc o to, czy status kobiet lub tych na dole jest lepszy, czy gorszy, ale z jakiego typu organizacji ten status wynika.
\item\relax
Podobnie możemy zauważyć, że u ludów pierwotnych segmentacja kołowa nie wymaga koniecznie, by koła były koncentryczne, współśrodkowe. W reżimie elastycznym centra już działają – jako tyle węzłów, oczu lub też czarnych dziur – ale nie współdrgają wszystkie razem, nie schodzą się w tym samym punkcie, nie zbiegają się w tej samej centralnej czarnej dziurze. Istnieje wielość animistycznych oczu, która sprawia, że na każde z nich wpływa na przykład osobny duch zwierzęcy (duch-wąż, duch-dzięcioł, duch-kajman\dots{}). Każda czarna dziura zostaje zajęta przez inne zwierzęce oko. Zapewne widać już, że gdzieniegdzie zarysowują się operacje usztywniania i centralizacji: wszystkie centra muszą przejść przez jedno jedyne koło, które samo z kolei ma tylko jedno centrum. Szaman rysuje linie między wszystkimi punktami czy duchami, wykreśla konstelację, promienisty zespół korzeni, który odsyła do centralnego drzewa. Narodziny władzy scentralizowanej czy też systemu drzewiastego przyniosą zdyscyplinowanie wykwitów pierwotnego kłącza?\footnote{O inicjacji szamana i roli drzewa u Janomamów, por. Jacques Lizot, Le cercle des feux, s. 127–135: „Między jego stopami wykopuje się pospiesznie dziurę, w którą wkłada się dół masztu – tym samym tam zasadzonego. Turaewë rysuje na ziemi wyobrażone linie, promieniujące wszędzie dookoła. Mówi: To są korzenie”.} Drzewo funkcjonuje tu zarówno jako zasada dychotomiczna, binarna, jak i oś obrotu\dots{} Lecz władza szamana jest jeszcze w pełni zlokalizowana, ściśle zależna od poszczególnego segmentu, uwarunkowana przez narkotyki, każdy zaś punkt nadal wytwarza własne niezależne sekwencje. Nie da się tego powiedzieć o społeczeństwach nowoczesnych czy nawet o państwach. Oczywiście scentralizowane nie przeciwstawia się segmentowemu, a koła pozostają wyodrębnione. Stają się jednak koncentryczne, zdecydowanie udrzewione. Segmentacja usztywnia się do tego stopnia, że wszystkie centra współdrgają, wszystkie czarne dziury sprowadzone zostają do jednego punktu akumulacji, swoistego punktu przecięcia, znajdującego się gdzieś za wszystkimi oczami. Twarz ojca, twarz nauczyciela, twarz pułkownika, szefa, tworzą powtarzalny nadmiar [se mettent a redonder], odsyłają do centrum znaczenia, które przemierza różne koła i przenika wszystkie segmenty. Elastyczne mikrogłówki, zwierzęce utwarzowienia [visagéifications] zostają zastąpione przez makro-twarz, której centrum znajduje się wszędzie, obwód zaś – nigdzie. Nie ma już n oczu w niebie, w roślinnych i zwierzęcych stawaniach się, lecz tylko centralne zarządzające oko, które promieniuje na wszystkie dziedziny. Centralne państwo nie konstytuuje się przez zniesienie segmentacji kołowej, lecz poprzez współśrodkowość odrębnych kół lub też przez koordynację i współdrganie centrów. Już w społeczeństwach pierwotnych istnieje tyle centrów władzy lub, inaczej mówiąc, tyle centrów władzy istnieje jeszcze w społeczeństwach państwowych. Społeczeństwa państwowe funkcjonują wszakże jako aparatura rezonansowa, organizują współdrganie, podczas gdy społeczeństwa pierwotne je tłumią\footnote{Państwo nie definiuje się więc jedynie przez typ władz publicznych, lecz jako pudło rezonansowe władz zarówno prywatnych, jak publicznych. W tym sensie Althusser mógł powiedzieć: „Rozróżnienie publicznego i prywatnego jest wewnętrznym odróżnieniem prawa burżuazyjnego, ważnym w dziedzinach (podporządkowanych), w których prawo burżuazyjne sprawuje swoją władzę. Dziedzina państwa mu się wymyka, ponieważ jest ono poza Prawem (\dots{}), jest – przeciwnie – warunkiem wszelkiego rozróżnienia między publicznym a prywatnym” (Louis Althusser, Ideologie i aparaty ideologiczne państwa, tłum. A. Staroń, http:\Slash{}\Slash{}www.nowakrytyka.pl\Slash{}spip.php?article374 [data dostępu: 01.01.2015], oryg. „La Pensée”, czerwiec 1970).}.
\item\relax
Wreszcie, z punktu widzenia segmentacji linearnej, można powiedzieć, że każdy segment zostaje podkreślony, skorygowany, zhomogenizowany sam w sobie, lecz także w odniesieniu do innych. Nie tylko każdy ma swoją jednostkę miary, lecz owe jednostki są równoważne i przekładalne jedna na drugą. Korelatem centralnego oka jest przestrzeń, w której się ono przemieszcza i – w odniesieniu do swoich przemieszczeń – samo pozostaje niezmienne. Wraz z polis grecką i reformą Klejstenesa objawia się homogeniczna i izotopiczna przestrzeń polityczna, która dokonuje nadkodowania segmentów lineażu, jednocześnie zaś odrębne ogniska zaczynają współdrgać w centrum, które działa jako wspólny mianownik\footnote{Jean-Pierre Vernant, Mythe et pensée chez les Grecs, Maspero 1971–1974, t. I, cz. III („Stając się wspólnym, wznosząc się w otwartej i wspólnej przestrzeni agory, już nie zaś we wnętrzu mieszkań prywatnych [\dots{}], ognisko wyraża odtąd centrum, jako wspólny mianownik wszystkich domów tworzących polis”, s. 210).}. Paul Virilio pokazał, jak w czasach późniejszych imperium rzymskie ustanawia linearną (lub geometryczną) rację stanu, która mieści w sobie ogólny zarys obozów i miejsc umocnionych, uniwersalną sztukę „wyznaczania granic według planów” [borner par des tracés], zagospodarowanie terytoriów, zastąpienie miejsc i terytorialności przestrzenią, przekształcenie świata w miasto, krótko mówiąc – coraz sztywniejszą segmentację\footnote{Paul Virilio, L’insécurité du territoire, Stock 1975, s. 120, 174–175. O „obozometrii” [castramétration]: „geometria jest konieczną podstawą skalkulowanej ekspansji władzy państwa w czasie i przestrzeni; państwo posiada więc już w sobie samym postać wystarczającą, idealną pod warunkiem, że będzie idealnie geometryczna (\dots{}). Lecz Fénelon, przeciwstawiając się polityce państwa Ludwika XIV, wykrzyknie: opierajcie się czarodziejskim urokom i diabolicznym właściwościom geometrii!”.}. Segmenty, podkreślone lub nadkodowane, zdają się w ten sposób tracić zdolność pączkowania, swoją dynamiczną relację z segmentowaniami właśnie się dokonującymi, właśnie się tworzącymi i rozkładającymi. Jeśli istnieje geometria „pierwotna” (protogeometria), to jest ona geometrią działania, w której figur nie daje się nigdy oddzielić od ich afektów, linii – od ich stawania się, segmentów – od segmentowania: istnieją „kolistości”, ale nie koła; istnieją „uszeregowania” [alignements], ale nie linie proste itd. Z kolei geometria państwa, czy raczej więź państwa z geometrią, objawia się w prymacie elementu-twierdzenia, zastępującego elastyczne formacje morfologiczne stałymi, idealnymi esencjami, afekty – własnościami, dokonujące się właśnie segmentowanie – uprzednio określonymi segmentami. Geometria i arytmetyka przejmują moc skalpela. Własność prywatna wymaga przestrzeni nadkodowanej i podzielonej na kwadraty przez kataster. Nie tylko każda linia ma swoje segmenty, ale też segmenty jednej linii odpowiadają segmentom innej linii: na przykład reżim płacy roboczej dokona uzgodnienia i ustanowi wzajemną odpowiedniość segmentów pieniężnych, segmentów produkcji i segmentów dóbr konsumpcyjnych.
\end{enumerate}
Możemy podsumować główne różnice między segmentacją sztywną a segmentacją elastyczną. W trybie sztywnym segmentacja binarna ma wartość sama dla siebie i zależy od wielkich maszyn bezpośredniej binaryzacji, podczas gdy w drugim trybie binarności wynikają z „wielości n-wymiarowych”. Po drugie, segmentacja kołowa dąży do formy koncentrycznej, to znaczy sprawia, że wszystkie jej ogniska stykają się w jednym centrum, które nie przestaje się przemieszczać, lecz pozostaje niezmienne w swoich przemieszczeniach, odsyłając do maszyny rezonansowej. Wreszcie segmentacja linearna przechodzi przez maszynę nadkodowania, która wytwarza przestrzeń homogeniczną more geometrico i zakreśla segmenty o określonej substancji, formie i relacjach. Zauważmy, że za każdym razem ową usztywnioną segmentację wyraża Drzewo. Drzewo jest węzłem drzewiastości, zasadą dychotomii; jest osią obrotu, zapewniającą współśrodkowość; jest strukturą lub siatką dzielącą na kwadraty to, co możliwe. Lecz jeśli przeciwstawiamy w ten sposób segmentację udrzewioną kłączastemu segmentowaniu, to nie tylko po to, by wskazać dwa stany tego samego procesu, lecz także po to, by odsłonić dwa różne procesy. Albowiem społeczeństwa pierwotne postępują zasadniczo wedle kodów i terytorialności. Samo rozróżnienie tych dwóch elementów – plemiennego systemu terytorium i klanowego systemu lineaży – uniemożliwia współdrganie, rezonans\footnote{Meyer Fortes analizuje u Talensi różnicę między „strażnikami ziemi” a wodzami. Takie rozróżnienie władz jest dość powszechne w społeczeństwach pierwotnych, lecz to, co się liczy, to fakt, że zostało ono precyzyjnie zorganizowane w sposób, który miał uniemożliwić rezonans władz. Na przykład wedle analizy Bertha dotyczącej Baduj na Jawie władza strażnika ziemi jest z jednej strony uważana za bierną czy też kobiecą, z drugiej zaś – przyznawana pierworodnemu: to nie „wtargnięcie pokrewieństwa w porządek polityczny”, lecz odwrotnie – „wymóg porządku politycznego przełożony na pojęcia pokrewieństwa”, by uniemożliwić ustanowienie rezonansu, z którego wyłoniłaby się własność prywatna (por. Louis Berthe, Aînés et cadets, l’alliance et la hierarchie chez les Baduj, „L’Homme”, lipiec 1965).}. Tymczasem społeczeństwa nowoczesne, czy też państwowe, zastąpiły słabnące kody jednoznacznym nadkodowaniem, utracone zaś terytorialności – specyficzną reterytorializacją (która tworzy się właśnie w nadkodowanej przestrzeni geometrycznej). Segmentacja objawia się zawsze jako wytwór maszyny abstrakcyjnej, ale w sztywnym i w elastycznym nie działa ta sama maszyna abstrakcyjna.
Nie wystarczy więc przeciwstawienie scentralizowanego segmentowemu. Ale nie wystarczy też przeciwstawienie dwóch segmentacji, jednej elastycznej i pierwotnej, drugiej – nowoczesnej i usztywnionej. Obie bowiem wyraźnie się różnią, lecz są nierozdzielne, splątane jedna z drugą, jedna w drugiej. Społeczeństwa pierwotne mają już zarodki sztywności, udrzewienia, które w tej samej mierze zapowiadają państwo, co je odpychają. I na odwrót, nasze społeczeństwa wciąż zanurzone są w elastycznej tkance, bez której nie stwardniałyby segmenty sztywne. Nie można zastrzegać elastycznej segmentacji dla ludów pierwotnych. Segmentacja elastyczna nie jest też przeżytkiem dzikusa w nas, jest funkcją doskonale nowoczesną i nierozerwalnie związaną z drugą. Każde społeczeństwo, lecz także każda jednostka są więc przenikane przez dwie segmentacje jednocześnie: jedną molową, drugą – molekularną. Są one odrębne o tyle, o ile nie obejmują tych samych pojęć, tych samych relacji, tej samej natury, tego samego rodzaju wielości. Ale są też nierozerwalnie związane w tej mierze, w jakiej współistnieją, przechodzą jedna w drugą, podług kształtów różnych u ludów pierwotnych i u nas – lecz zawsze zakładając siebie nawzajem. Krótko mówiąc, wszystko jest polityczne, ale wszelka polityka jest zarazem makropolityką i mikropolityką. Weźmy zespoły typu postrzeżeniowego lub uczuciowego: ich molowa organizacja, ich sztywna segmentacja nie wyklucza całego świata nieświadomych mikropostrzeżeń, nieświadomych afektów, subtelnych segmentacji, które nie ujmują i nie doznają tych samych rzeczy, które inaczej się rozkładają, które działają inaczej. To mikropolityka postrzeżenia, sympatii, rozmowy itd. Jeśli rozważymy wielkie binarne zespoły, jak płcie czy klasy, zobaczymy, że i one przechodzą w molekularne złożenia innej natury i że istnieje tam podwójna wzajemna zależność. Dwie płcie odsyłają bowiem do wielorakich kombinacji molekularnych, które wprowadzają do gry nie tylko mężczyznę w kobiecie i kobietę w mężczyźnie, lecz także relację każdego w drugim ze zwierzęciem, rośliną itd.: tysiąc maleńkich płci. Natomiast klasy społeczne same odsyłają do „mas”, których ruch, repartycja, cele, wreszcie sposoby walki są zupełnie różne. Wysiłki, by odróżnić masę i klasę, zmierzają w istocie ku tej granicy: pojęcie masy jest pojęciem molekularnym, działającym poprzez ten rodzaj segmentowania, który jest niesprowadzalny do molowej segmentacji klasowej. Jednakże klasy są przecież wyodrębniane w masach, krystalizują je. A masy nie przestają wyciekać, przeciekać z klas. To, że wzajemnie się zakładają, nie usuwa różnicy punktów widzenia, natury, skali i funkcji (tak rozumiane pojęcie masy ma zupełnie inne znaczenie niż to zaproponowane przez Canettiego).
Nie wystarczy zdefiniowanie biurokracji przez sztywną segmentację, z przepierzeniami oddzielającymi kolejne sąsiadujące z sobą kancelarie, z szefem kancelarii w każdym segmencie i z centralizacją odpowiadającą końcowi korytarza lub szczytowi wieży. Jednocześnie bowiem istnieje całe segmentowanie biurokratyczne, elastyczność i łączność kancelarii, perwersja biurokracji, ciągła wynalazczość lub kreatywność, która objawia się nawet w spotkaniu z regulaminami administracyjnymi. Kafka jest najwybitniejszym teoretykiem biurokracji właśnie dlatego, że pokazał, jak na pewnym poziomie (ale którym? – nielokalizowalnym) bariery między kancelariami przestają być „określoną granicą” i zanurzają się w środowisko molekularne, które je rozpuszcza, rozmnażając jednocześnie szefa w mikropostacie, tych zaś nie sposób rozpoznać, zidentyfikować oraz trudniej odróżnić, niż scentralizować: to inny reżim – współistniejący z oddzieleniem oraz całościowaniem sztywnych segmentów\footnote{Franz Kafka, Zamek, tłum. K. Radziwiłł, K. Truchanowski, Zielona Sowa 2003; przede wszystkim rozdz. XV (wyznania Barnabasa), cytat na s. 145. Przypowieść o dwóch biurach – molowym i molekularnym – nie podlega więc tylko interpretacji fizycznej, jak ta Arthura Eddingtona, lecz także interpretacji ściśle biurokratycznej. [Na zbieżność fizycznego obrazu świata w The Nature of the Physical World Eddingtona i wizji Kafkowskiej zwrócił uwagę Walter Benjamin, por. list do G. Scholema z 12 czerwca 1938 – Walter Benjamin, Briefe, t. II, Suhrkamp 1978, s. 761 – dopisek tłum.]}. Powiemy wręcz, że faszyzm wymaga istnienia reżimu molekularnego, który nie miesza się ani z molowymi segmentami, ani z ich centralizacją. Niewątpliwie faszyzm wynalazł pojęcie państwa totalitarnego, ale nie ma powodu, by definiować faszyzm za pomocą pojęcia, które on sam wynalazł: istnieją państwa totalitarne – typu stalinowskiego lub typu dyktatury wojskowej – w których nie ma faszyzmu. Koncepcja państwa totalitarnego ma wartość jedynie w skali makropolitycznej – w odniesieniu do sztywnej segmentacji i specjalnego trybu całościowania i centralizacji. Ale faszyzm jest nierozerwalnie związany z ogniskami molekularnymi, które mnożą się i skaczą z jednego punktu na drugi, wzajemnie na siebie oddziałując, zanim zaczną wszystkie razem współdrgać w państwie narodowosocjalistycznym. Faszyzm wiejski i faszyzm miasta, dzielnicy, faszyzm młodzieżowy i faszyzm dawnych kombatantów, faszyzm lewicy i prawicy, pary, rodziny, szkoły czy biura: każdy faszyzm określony jest przez mikro-czarną dziurę, która ma wartość sama przez się i w łączności z innymi, zanim zacznie współdrgać w wielkiej, centralnej, uogólnionej czarnej dziurze\footnote{Siła książki Jean-Pierre’a Faye’a, Langages totalitaires (Hermann 1972) polega na ukazaniu wielości tych ognisk, praktycznych i semiotycznych, które stanowią punkt wyjścia dla ukonstytuowania się nazizmu. Dlatego Faye, który jako pierwszy dokonał rygorystycznej analizy pojęcia państwa totalitarnego (w jego włoskich i niemieckich początkach), jednocześnie odmówił definiowania faszyzmu włoskiego i nazizmu za pomocą tego pojęcia (które działa w innym planie niż „proces głęboki” [procès sous-jacent]). Ze wszystkich tych kwestii Faye wytłumaczył się w La critique du langage et son économie, Éd. Galilée 1973.}. Faszyzm istnieje wtedy, gdy maszyna wojenna zostaje zainstalowana w każdej dziurze, w każdej niszy. Nawet ustanowione już państwo narodowosocjalistyczne wymaga utrzymania tych mikrofaszyzmów, które dostarczają mu niezrównanego środka oddziaływania na „masy”. Daniel Guérin słusznie powiada, że jeśli Hitler zdobył raczej władzę niż kadry niemieckiego państwa, stało się tak dlatego, że wcześniej miał do dyspozycji formy mikroorganizacji, dzięki którym zyskał „niezrównany, niezastąpiony środek, pozwalający przeniknąć do każdej komórki społeczeństwa”\footnote{\emph{Daniel Guérin, Sur le fascisme II. Fascisme et grand capital, F. Maspero 1969, s. 277.}}, segmentację elastyczną i molekularną, przepływy zdolne wchłonąć każdy rodzaj komórek. Z kolei jeśli kapitalizm ostatecznie uznał doświadczenie faszystowskie za katastrofę, jeśli wolał sprzymierzyć się z totalitaryzmem stalinowskim – z jego punktu widzenia daleko mądrzejszym i przystępniejszym – to dlatego właśnie, że stalinizm opierał się na bardziej klasycznej, mniej płynnej segmentacji i takiejż centralizacji. To właśnie potęga mikropolityczna, molekularna czyni faszyzm tak niebezpiecznym, jest to bowiem ruch masowy: raczej zrakowaciałe ciało niż totalitarny organizm. Kino amerykańskie często pokazywało owe ogniska molekularne, faszyzm bandy, gangu, sekty, rodziny, miasteczka, dzielnicy, samochodu, faszyzm, który nie oszczędza nikogo. Jedynie mikrofaszyzm stanowi odpowiedź na globalne pytanie: dlaczego pragnienie pragnie własnego stłumienia, jak może pragnąć swojego stłumienia? Rzecz jasna, masy nie poddają się biernie władzy, nie „chcą” też – bo nie powoduje nimi żadnego rodzaju masochistyczna histeria – być dławione, co więcej, nie zostały zwiedzione przez ideologiczne złudy. Lecz pragnienie nigdy nie daje się oddzielić od skomplikowanych złożeń, nieuchronnie przenikających poziomy molekularne, mikroformacje, które już kształtują postawy, dyspozycje, postrzeżenia, oczekiwania, semiotyki itd. Pragnienie nie jest nigdy niezróżnicowaną energią popędową, lecz samo jest wytworem starannie opracowanego montażu, inżynierii wysokich interakcji: całej tej elastycznej segmentacji, która opracowuje energie molekularne i może ukształtować pragnienie jako już faszystowskie. Organizacje lewicowe nie są wcale najdalsze od wytwarzania własnych mikrofaszyzmów. To zbyt łatwe – być antyfaszystą na poziomie molowym, nie widząc faszysty, którym się jest, którego się podtrzymuje i karmi, którego się ceni i wielbi w jego molekularnościach, prywatnych i kolektywnych.
Chcemy uniknąć czterech błędów związanych z segmentacją elastyczną i molekularną. Pierwszy jest błędem aksjologicznym i polega na wierze, że wystarczy nieco elastyczności, by być „lepszym”. Tymczasem faszyzm jest daleko bardziej niebezpieczny za sprawą swych mikrofaszyzmów, a subtelne segmentowania – równie szkodliwe, co najbardziej usztywnione segmenty. Błąd drugi to błąd psychologiczny, jak gdyby molekularne przynależało do domeny wyobrażania i odsyłało wyłącznie do jednostki lub tego, co międzyjednostkowe. A przecież na jednej linii jest nie mniej Realnego-społecznego niż na drugiej. Po trzecie, dwie formy nie różnią się po prostu rozmiarami, jako formy małe i duże. Jakkolwiek jest prawdą, że molekularne operuje na szczególe i przenika małe grupy, to przecież rozciąga się na całe pole społeczne – w tej samej mierze, co organizacja molowa. Wreszcie, różnica jakościowa dwóch linii nie wyklucza, by mogły się pobudzać lub przeciąć, zawsze bowiem istnieje między nimi dwiema relacja proporcjonalności – albo są wprost proporcjonalne, albo odwrotnie proporcjonalne.
W pierwszym przypadku im silniejsza jest organizacja molowa, tym bardziej sama pobudza molekularyzację swoich elementów, relacji i podstawowych aparatów. Kiedy maszyna staje się planetarna lub kosmiczna, złożenia wykazują coraz silniejszą tendencję do miniaturyzacji – przekształcania się w mikrozłożenia. Wedle formuły Gorza, w światowym kapitalizmie istnieje tylko jeden element pracy – molekularna lub zmolekularyzowana jednostka, to znaczy człowiek „masowy”. Korelatem administrowania wielkim, zorganizowanym bezpieczeństwem molowym jest całe to mikrozarządzanie małymi lękami, cały ten nieustanny molekularny brak bezpieczeństwa – do tego stopnia, że maksyma ministerstw spraw wewnętrznych mogłaby brzmieć: makropolityka społeczeństwa dla i poprzez mikropolitykę braku bezpieczeństwa\footnote{O tej komplementarności „makropolityki bezpieczeństwa – mikropolityki lęku” por. Paul Virilio, L’insécurité du territoire, s. 96, 130, 228–235. Często zwracano uwagę – w odniesieniu do wielkich miast współczesnych – na ową mikroorganizację permanentnego „stresu”.}. Niemniej jednak drugi przypadek – relacja odwrotnie proporcjonalna – jest jeszcze bardziej istotny w tej mierze, w jakiej ruchy molekularne już nie uzupełniają wielkiej, światowej organizacji, lecz dziurawią ją i stawiają jej opór. Prezydent Giscard d’Estaing dał temu wyraz w swym wykładzie geografii politycznej i wojskowej: im większa równowaga między Zachodem a Wschodem, w maszynie dualistycznej, nadkodującej i przezbrojonej, tym więcej „destabilizacji” na innej linii, z Północy na Południe. Zawsze jest jakiś Palestyńczyk, lecz także Bask, Korsykanin, gotów dokonać „regionalnej destabilizacji bezpieczeństwa”\footnote{Valéry Giscard d’Estaing, przemówienie z 1 czerwca 1976 w Instytucie Studiów Wyższych Obrony Narodowej (L’Institut des hautes études de Défense nationale), pełny tekst w „Le Monde”, 4 czerwca 1976.}. A zatem dwa wielkie zespoły molowe na Wschodzie i na Zachodzie są nieustannie nękane przez molekularne segmentowanie, przez zygzakowate pęknięcie, dlatego też z trudem są w stanie utrzymać swe własne segmenty. Jak gdyby linia ujścia, nawet jeśli zaczyna się od maleńkiego strumyczka, zawsze przepływała między segmentami, wymykając się centralizacji, unikając całościowania. Głębokie ruchy, które wstrząsają społeczeństwem, przedstawiają się właśnie w ten sposób, nawet jeśli są nieuchronnie „reprezentowane” jako zderzenie segmentów molowych. Twierdzi się błędnie (szczególnie na gruncie marksizmu), że społeczeństwo definiuje się przez swe sprzeczności. Lecz jest to prawdą tylko w wielkiej skali. Z punktu widzenia mikropolityki społeczeństwo definiuje się przez swoje linie ujścia, które są molekularne. Zawsze coś płynie lub uchodzi, wymykając się binarnym organizacjom, aparatowi rezonansowemu, maszynie nadkodowania: składa się to na karb „przemian obyczajowych” – to młodzi, kobiety, szaleńcy itd. Maj ‘68 we Francji był molekularny, stąd z punktu widzenia makropolityki jego uwarunkowania pozostawały niedostrzegalne. Doszło więc do tego, że ludzie bardzo ograniczeni lub bardzo starzy zdołali uchwycić to wydarzenie lepiej niż najbardziej postępowi politycy – czy raczej ci, którzy się za takich mieli, przyjmując punkt widzenia organizacji. Jak zauważył Gabriel Tarde, trzeba by wiedzieć, którzy chłopi – i w których regionach Południa – jako pierwsi przestali kłaniać się okolicznym właścicielom. Najstarszy, należący do minionej epoki właściciel mógł w tym względzie oceniać sprawy trafniej niż człowiek nowoczesny. Z majem ‘68 jest tak samo: wszyscy, którzy formułowali sądy w terminach makropolityki, nic z tego wydarzenia nie zrozumieli, umknęło im bowiem coś, czego nie dało się ściśle przypisać. Politycy, partie, związki zawodowe, wielu ludzi lewicy odczuwało wobec niego wielką niechęć, przypominali przecież bez ustanku, że „warunki” nie dojrzały. I oto jakby chwilowo odebrano im całą tę dualistyczną maszynę, która czyniła z nich poważnych rozmówców. Co dziwne, de Gaulle i nawet Pompidou zrozumieli dużo więcej niż inni. Molekularny przepływ wymykał się, najpierw maleńki, później nabrzmiewający, i nieustannie pozostawał nieprzypisany\dots{} A przecież prawdą jest i teza przeciwna: ujścia i ruchy molekularne byłyby niczym, gdyby nie przeniknęły ponownie organizacji molowych, gdyby nie przekształciły ich segmentów, ich binarnego rozmieszczenia płci, klas, partii.
Rzecz w tym, że molowe i molekularne nie różni się jedynie wielkością, skalą czy rozmiarem, lecz naturą danego systemu odniesień. Należy więc, być może, zastrzec słowa „linia” i „segmenty” dla organizacji molowej i szukać innych słów, które lepiej pasowałyby do składu [composition] molekularnego. W istocie za każdym razem, kiedy możemy wyznaczyć linię precyzyjnie określonych segmentów, spostrzegamy, że przedłuża się ona pod inną formą – w przepływ kwantowy. I za każdym razem „centrum władzy”, które jesteśmy w stanie zlokalizować, znajduje się na granicy między nimi i może zostać zdefiniowane nie przez absolutne wykonywanie władzy w pewnej domenie, lecz przez względne dostosowania i konwersje, które przeprowadza między linią a przepływem. Weźmy linię pieniądza i jej segmenty. Segmenty owe mogą zostać określone z różnych punktów widzenia: np. z punktu widzenia budżetu przedsiębiorstwa – płace realne, zyski netto, płace kierownictwa, odsetki kapitałowe, rezerwy, inwestycje\dots{} itd. Lecz ta linia pieniądza-płacy odsyła do zupełnie innego aspektu, to znaczy do przepływu pieniądza-finansowania, który nie zawiera już w sobie segmentów, lecz bieguny, osobliwości i kwanty (bieguny przepływu to kreowanie i niszczenie pieniądza, osobliwości – nominalne aktywa płynne, kwanty – inflacja, deflacja, stagflacja itd.). W tym względzie można było mówić o „przepływie mutującym, konwulsyjnym, twórczym i okrężnym”, związanym z pragnieniem, zawsze poza i pod zwartą linią, poza i pod segmentami określającymi odsetki, podaż i popyt\footnote{O „przepływie władzy mutującej” i rozróżnieniu dwóch typów pieniądza zob. Bernard Schmitt, Monnaie, salaires et profits, Éd. Castella 1980, s. 236, 275–277. [Por. też uwagi Marksa o cyrkulacji: „jej bezpośredni byt jest więc czystym pozorem. Jest zewnętrznym przejawem procesu odbywającego się gdzie indziej” (w tłum. franc. procès sous-jacent) – Karol Marks, Zarys krytyki ekonomii politycznej, tłum. Z.J. Wyrozembski, Książka i Wiedza 1986, s. 184 – dopisek tłum.]}. W bilansie płatniczym odnajdujemy binarną segmentację, która rozróżnia na przykład transakcje zwane autonomicznymi i transakcje wyrównawcze; ale ruchy kapitału są właśnie tym, co nie daje się w ten sposób posegmentować, są bowiem „w najwyższym stopniu zdekomponowane, zgodnie ze swą naturą, swoim czasem trwania, osobowością wierzyciela lub dłużnika” – a zatem w odniesieniu do tego przepływu „już zupełnie nie wiadomo, gdzie wytyczyć linię”\footnote{Michel Lelart, Le dolar monnaie internationale, Éd. Albatros 1975, s. 57.}. Niemniej jednak istnieje ciągła korelacja obu aspektów, ponieważ właśnie w ujęciu w linii i segmentacji przepływ wyczerpuje się, lecz stąd także wyrasta nowe tworzenie. Kiedy mówi się o władzy bankowej, skupionej zwłaszcza w bankach centralnych, chodzi właśnie o tę względną władzę, która polega na regulowaniu „o tyle, o ile” to możliwe połączeń, konwersji, współdostosowań obu części okręgu. Dlatego centra władzy są określone daleko bardziej przez to, co im umyka, przez swoją niemoc, niż przez swoją strefę mocy. Krótko mówiąc, tego, co molekularne, mikroekonomii, mikropolityki, nie określają niewielkie rozmiary ich elementów, lecz natura ich „masy” – przepływ kwantowy odróżniony od linii molowych segmentów\footnote{Weźmy analizę Foucaulta i to, co w Nadzorować i karać nazywa on „mikrofizyką władzy”: po pierwsze, chodzi właśnie o zminiaturyzowane mechanizmy, o ogniska molekularne, które ujawniają się w szczególe, w tym, co nieskończenie małe, i które składają się na tyle „dyscyplin” w szkole, w armii, w fabryce, w więzieniu itd. (por. Michel Foucault, Nadzorować i karać. Narodziny więzienia, tłum. T. Komendant, KR 1998, s. 134 i nast.). Ale po drugie, same te segmenty, a także ogniska, które opracowują je w skali mikrofizycznej, występują jako osobliwości abstrakcyjnego „diagramu”, przylegające w całości do pola społecznego (s. 208), lub jako kwanty pobrane z jakiegokolwiek przepływu – jakiegokolwiek przepływu określonego przez „wielość jednostek” do kontrolowania (por. s. 200 i nast.).}. Sprawienie, by segmenty odpowiadały kwantom, odpowiednie dopasowanie segmentów do kwantów wymaga zmian rytmu i trybu – udaje się to raz lepiej, raz gorzej, nie ma tu bowiem wszechmocy i zawsze coś uchodzi.
Można podać inne przykłady. Gdy zatem mówimy o władzy Kościoła, staje się widoczne, że władza ta zawsze pozostawała w związku z tym sposobem administrowania grzechem, który zawiera w sobie silną segmentację: rodzaje grzechów (siedem grzechów głównych), jednostki miary (ile razy?), reguły równoważności i odkupienia (spowiedź, pokuta\dots{}). Lecz czymś zupełnie innym, jakkolwiek komplementarnym, jest to, co można nazwać molekularnym przepływem grzeszności – który obejmuje strefę linearną, podlega w niej negocjowaniu, lecz sam w sobie zawiera jedynie bieguny (grzech pierworodny – odkupienie lub łaska) i kwanty („grzech nieobecności świadomości grzechu”, grzech świadomości grzechu, grzech następstwa świadomości grzechu)\footnote{W kwestii „grzeszności określonej ilościowo”, kwantów i skoku jakościowego odwołujemy się do całej mikroteologii zbudowanej przez Kierkegaarda w Pojęciu lęku [tłum. A. Szwed, Antyk 2000, s. 38 i 44; cytat w tekście odnosi się do tytułu rozdz. III: we franc. tłum. Kierkegaarda (Le concept de l’angoisse, tłum. K. Ferlov, J.J. Gateau, Gallimard 1935): „L’angoisse, conséquence du péché de ne pas atteindre à la conscience du péché”, w tłum. pol. inaczej: „Lęk jako następstwo grzechu, które jest spowodowane nieobecnością świadomości grzechu” – dopisek tłum.].}. W tej mierze można by też mówić o przepływie przestępczości odróżnionym od linii molowej kodeksu prawnego i jego wyodrębnionych paragrafów. Z kolei kiedy mówimy o władzy wojskowej, o władzy armii, rozważamy linię – podzieloną na segmenty wedle typów wojny ściśle odpowiadających państwom toczącym wojnę i celom politycznym, które te państwa dla siebie ustanawiają (od wojny „ograniczonej” po wojnę „totalną”). Lecz, idąc za intuicją Clausewitza, czymś zupełnie innym jest maszyna wojenna, to znaczy przepływ wojny absolutnej, który płynie od bieguna ofensywnego do bieguna defensywnego i jest oznaczony jedynie kwantami (siły materialne i psychiczne, stanowiące rodzaj nominalnych aktywów płynnych wojny). O czystym przepływie da się powiedzieć, że jest abstrakcyjny, a jednak rzeczywisty; idealny, a jednak efektywny; absolutny, a jednak „zróżnicowany”. To prawda, że ująć przepływ i jego kwanty daje się tylko poprzez wskaźniki linii segmentów; lecz i odwrotnie – owe wskaźniki i linie mogą istnieć tylko poprzez przepływ, który je obmywa. W każdym przypadku widzimy, że linia z segmentami (makropolityka) zanurza się i przedłuża w przepływie z kwantami (mikropolityce), który nieustannie ją przekształca i pobudza jej segmenty:
\begin{figure}[htbp!]
\centering
\includegraphics[keepaspectratio=true,height=0.75\textheight,width=\textwidth]{g-d-gilles-deleuze-felix-guattari-1933-mikropolity-1.jpg}
\caption[]{\noindent A: przepływ i bieguny
\forcelinebreak
a: kwanty
\forcelinebreak
b: linia i segmenty
\forcelinebreak
B: centrum władzy
\forcelinebreak
(Zespół [ensemble] jest cyklem lub okresem)}
\end{figure}
W hołdzie Gabrielowi Tarde (1843–1904): jego dzieło, długo zapomniane, odzyskało aktualność pod wpływem socjologii amerykańskiej, zwłaszcza mikrosocjologii. Zostało zmiażdżone przez Durkheima i jego szkołę (w polemice równie ostrej i tego samego rodzaju, co spór Cuviera z Geoffroyem Saint-Hilaire). Durkheim wyróżnionym przedmiotem badań uczynił wielkie przedstawienia zbiorowe – zasadniczo binarne, współdrgające i nadkodowane\dots{} Tarde oponował, zauważając, że przedstawienia zbiorowe zakładają to, co trzeba wyjaśnić, to znaczy „podobieństwo milionów ludzi”. Dlatego Tarde’a interesuje raczej świat szczegółu czy też świat tego, co nieskończenie małe: małe naśladownictwa, opozycje i wynalazki, które tworzą całą materię pod-przedstawieniową. Najlepsze strony, jakie Tarde zapisał, zawierają analizy maleńkiej innowacji biurokratycznej lub językowej. Szkoła Durkheima odpowiadała, że mielibyśmy tu do czynienia nie z socjologią, lecz z psychologią lub interpsychologią\footnote{W tłumaczeniu tego terminu podążamy za: Gabriel Tarde, Opinia i tłum, tłum. K. Skrzyńska, Gebethner i Wolff 1904, s. 138–139. Termin interpsychologie pojawia się jako tytuł kilku późnych wystąpień Tarde’a, np. w: „Bulletin de l’institut général psychologique”, nr 2, 1903, s. 91–118; czy w „Archives d’anthropologie criminelle”, t. 19, 1904, s. 536–564 [przyp. tłum.].}. Ale to prawda tylko pozornie, w pierwszym przybliżeniu: mikronaśladownictwo wydaje się rzeczywiście przechodzić z jednostki na jednostkę. Jednocześnie jednak, i na głębszym poziomie, odnosi się do przepływu czy też fali, nie zaś do indywiduum. Naśladownictwo jest rozszerzaniem przepływu; opozycja to binaryzacja, binarne ujęcie przepływu; wynalazek zaś – to sprzężenie czy też połączenie różnych przepływów. Czym jest przepływ wedle Tarde’a? Wierzeniem lub pragnieniem (to dwa aspekty wszelkiego złożenia), przepływ jest zawsze przepływem wierzenia i pragnienia. Wierzenia i pragnienia stanowią podstawę każdego społeczeństwa, są bowiem przepływami, i jako takie są „kwantyfikowalne”. To prawdziwe Wielkości społeczne, podczas gdy wrażenia są jakościowe, przedstawienia zaś – to proste wypadkowe\footnote{Według Tarde’a psychologia jest nauką ilościową, lecz tylko w tej mierze, w jakiej studiuje komponenty pragnienia i wierzenia we wrażeniu. Natomiast logika jest nauką ilościową, gdy nie trzyma się form przedstawienia, lecz sięga stopni wierzenia i pragnienia oraz ich kombinacji: por. La logique sociale, Alcan 1893.}. Nieskończenie małe naśladownictwo, opozycja, wynalazek stają się więc niejako kwantami przepływu, zaznaczającymi rozszerzanie, binaryzację lub sprzężenie wierzeń i pragnień. Stąd znaczenie statystyki – pod warunkiem wszakże, że zajmuje się ona punktami, a nie tylko „nieruchomą” strefą przedstawień. Ostatecznie różnica w żadnym razie nie zachodzi między społecznym a indywidualnym (czy interindywidualnym), lecz między molową domeną przedstawień, zarówno zbiorowych, jak i jednostkowych, a molekularną domeną wierzeń i pragnień, w której rozróżnienie na społeczne i indywidualne traci wszelki sens – przepływy bowiem w tym samym stopniu dają się przypisać jednostkom, co podlegają nadkodowaniu przez wspólne znaczące. O ile przedstawienia definiują od razu wielkie zespoły czy też określone na linii segmenty, o tyle wierzenia i pragnienia są przepływami oznaczonymi przez kwanty. Przepływami, które tworzą się, wyczerpują i zmieniają, dodają się, odejmują i łączą. Tarde jest wynalazcą mikrosocjologii, której nadał zakres i zasięg, wskazując od razu błędy, jakich padnie ona ofiarą.
Oto jak można rozróżnić linię segmentów i przepływ kwantowy. Mutujący przepływ zawiera zawsze coś, co dąży do wymknięcia się kodom, uniknięcia kodów, kwanty zaś są właśnie znakami lub stopniami deterytorializacji na zdekodowanym przepływie. Natomiast sztywna linia wymaga nadkodowania, które zastępuje słabnące kody, segmenty zaś są niejako reterytorializacjami na nadkodującej i nadkodowanej linii. Wróćmy do przypadku grzechu pierworodnego: samo oddziaływanie przepływu wyznacza dekodowanie w odniesieniu do stworzenia (przy zachowaniu samotnej wysepki dla Dziewicy) i deterytorializację w odniesieniu do ziemi Adamowej; ale też jednocześnie przeprowadza ono nadkodowanie poprzez organizacje binarne i rezonansowe (Władze, Kościół, imperia, bogaci-biedni, mężczyźni-kobiety itd.), a także dokonuje komplementarnych reterytorializacji (na ziemi Kaina, w pracy, w płodzeniu, w pieniądzu\dots{}). A zatem: dwa systemy odniesień pozostają w stosunku odwróconym – w tym znaczeniu, że jeden wymyka się drugiemu, który go wstrzymuje, uniemożliwia mu dalsze ujście – jakkolwiek są wobec siebie ściśle komplementarne, współistnieją, bo jeden może istnieć tylko jako funkcja drugiego; zarazem jednak są różne i proporcjonalne\footnote{Stosunek odwrócony (raison inverse) i proporcjonalność (raison directe): por. Euklides, Elementy. Księgi V-VI, teoria proporcji i podobieństwa, tłum. P. Błaszczyk, K. Mrówka, Copernicus Center Press 2013, s. 18 (definicje 6 i 13) [przyp. tłum.].}, choć wyrażenia jednego nie odpowiadają wyrażeniom drugiego, drugi bowiem może skutecznie zatrzymać pierwszy jedynie na „płaszczyźnie” [plan], która nie jest już płaszczyzną pierwszego, pierwszy zaś kontynuuje swój pęd [élan] na swojej własnej płaszczyźnie.
Pole społeczne jest nieustannie pobudzane przez wszelkiego rodzaju ruchy dekodowania i deterytorializacji, które wpływają na „masy” podług różnych prędkości i szybkości. To nie sprzeczności, lecz ujścia. Na tym poziomie wszystko jest zagadnieniem masy. Na przykład, widzimy, jak gdzieś w X–XIV wieku przyspieszają czynniki dekodowania i rosną prędkości deterytorializacji: masy ostatnich najeźdźców wyłaniające się z północy, ze wschodu, z południa; masy zbrojne, które przekształcają się w łupieżcze bandy; masy kościelne wydane na łup niewiernych i heretyków, wyznaczające sobie coraz bardziej zdeterytorializowane cele; masy chłopskie opuszczające włości senioralne; masy senioralne, które same muszą znaleźć metody wyzysku daleko mniej terytorialne niż poddaństwo i pańszczyzna; masy miejskie, które oddzielają się od wiejskiego zaplecza i znajdują w miastach zaopatrzenie coraz mniej sterytorializowane; masy kobiece, zrywające z dawnym kodem namiętności i kodem małżeńskim; masy pieniężne, które przestają być przedmiotem tezauryzacji i zostają wstrzyknięte w wielkie obwody handlowe\footnote{O wszystkich tych punktach por. szczególnie Maurice Dobb, Studia o rozwoju kapitalizmu, tłum. H. Hagemejer, J. Zdanowicz, Państwowe Wydawnictwo Ekonomiczne 1964; Georges Duby, Guerriers et paysans, Gallimard 1973.}. Można przywołać krucjaty, widząc w nich operator takiego złączenia tych przepływów, że każdy uruchamia i przyspiesza inne (nawet przepływ kobiecości w princesse lointaine [„dalekiej księżniczce”]\footnote{„Daleka księżniczka” to postać pojawiająca się w średniowiecznych romansach rycerskich – dama serca bohaterów-rycerzy, często niedostępna, a nawet nieznana (ponieważ rycerz mógł zakochać się w damie, nigdy jej nie widząc, a tylko słysząc o jej niezwykłej piękności). La Princesse lointaine to także tytuł sztuki Edmonda Rostanda z 1895 roku [przyp. red.].}, nawet przepływ dzieci w krucjatach XIII wieku). Lecz w tym samym czasie i w ścisłym związku tworzą się nadkodowania i reterytorializacje. Krucjatom, nadkodowanym przez papieża, wyznaczone zostają cele terytorialne. Ziemia Święta, pokój boży, nowe typy zakonów, nowe postacie pieniądza, nowe formy wyzysku chłopów: dzierżawa i praca najemna (lub też powrót do niewolnictwa), reterytorializacje miasta itd. – wszystkie tworzą złożony system. Przyjmując ten punkt widzenia, musimy odtąd wprowadzić rozróżnienie między dwoma pojęciami: połączeniem [connexion] i sprzężeniem [conjugaison] przepływów. O ile „połączenie” oznacza sposób, w jaki zdekodowane i zdeterytorializowane przepływy uruchamiają się nawzajem, przyspieszają swe wspólne ujście i dodają czy też rozgrzewają swe kwanty, o tyle „sprzężenie” tych samych przepływów wskazuje raczej na ich względne zatrzymanie, jako na punkt akumulacji, zatykający, tamujący teraz linie ujścia – w tym punkcie dokonuje się ogólna reterytorializacja, a przepływy zostają poddane jednemu spośród nich, zdolnemu je nadkodować. Za każdym razem właśnie ten przepływ, który był – w pierwszym aspekcie, połączeniu – najbardziej zdeterytorializowany, przeprowadza akumulację czy koniunkcję procesów i – w drugim aspekcie, sprzężeniu – określa nadkodowanie oraz służy za podstawę reterytorializacji (zetknęliśmy się już z twierdzeniem, zgodnie z którym reterytorializacja dokonuje się zawsze w tym, co najbardziej zdeterytorializowane). A zatem miejska burżuazja handlowa sprzęgła czy też skapitalizowała wiedzę, technologię, złożenia i okręgi, w zależność od których wejdzie arystokracja, Kościół, rzemieślnicy i sami chłopi. Właśnie jako szpica deterytorializacji, prawdziwy akcelerator cząstek, burżuazja może również przeprowadzić reterytorializację zespołu.
Zadanie historyka polega na wyznaczeniu „okresu” współistnienia czy też równoczesności tych dwóch ruchów (z jednej strony dekodowanie-deterytorializacja, z drugiej nadkodowanie-reterytorializacja). Dopiero bowiem w ramach tego okresu można wyróżnić aspekt molekularny i aspekt molowy: z jednej strony masy lub przepływy, z ich mutacjami, ich kwantami deterytorializacji, ich połączeniami i przyspieszeniami; z drugiej zaś – klasy lub segmenty, z ich binarną organizacją, ich współdrganiem, koniunkcją czy też akumulacją, ich samouprzywilejowującą się linią nadkodowania\footnote{Problem różnic i związków między masami a klasami został postawiony przez Różę Luksemburg (Oeuvres I, Maspero 1969; tom zawiera dwa teksty: Reforma socjalna czy rewolucja? oraz Strajk masowy, partia i związki zawodowe, tłum. polskie w: Róża Luksemburg, Wybór pism, t. I, Książka i Wiedza 1959, s. 141–235, 492–588), lecz jeszcze z subiektywnego punktu widzenia: masy jako „instynktowa baza świadomości klasowej” (por. artykuł Nicolasa Boulte’a i Jacques’a Moiroux, Masses et parti, „Partisans” nr 45, 1969: Rosa Luxemburg vivante). Alain Badiou i François Balmès zaproponowali hipotezę bardziej obiektywną: masy byłyby „inwariantami”, które przeciwstawiają się wyzyskowi i formie-państwu w ogóle, natomiast klasy byłyby zmiennymi historycznymi, które określają konkretne państwo i – w wypadku proletariatu – możliwość skutecznego rozpuszczenia klas (De l’idéologie, Maspero 1976). Nie jest jednak jasne, z jednej strony – dlaczego same masy nie są zmiennymi historycznymi, z drugiej zaś – dlaczego pojęcie masy zostało zastrzeżone dla wyzyskiwanych („masa chłopsko-plebejska”), skoro słowo to równie dobrze odpowiada masom senioralnym, burżuazyjnym – a nawet pieniężnym.}. Różnica między makrohistorią a mikrohistorią w żadnym razie nie dotyczy długości rozważanego trwania, znacznej lub niewielkiej, lecz odrębnych systemów odniesień, podług których rozpatruje się albo nadkodowaną linię segmentów, albo też zmienny przepływ kwantowy. System sztywny nie zatrzymuje zmiennego: przepływ istnieje dalej pod linią, nieustannie mutując, gdy tymczasem linia całościuje. Masa i klasa nie mają tych samych konturów ani tej samej dynamiki, jakkolwiek obu znakom przypisana zostaje ta sama grupa. Burżuazja jako masa i jako klasa\dots{} Związek masy z innymi masami nie jest tym samym, co związek „odpowiadającej” jej klasy z innymi klasami. Oczywiście relacji siły i przemocy po jednej stronie jest nie mniej niż po drugiej. Ale właśnie ta sama walka przybiera dwa zupełnie różne aspekty, stąd zwycięstwa i klęski nie są tym samym. Ruchy mas przyspieszają i zastępują się wzajemnie (lub nikną na dłuższy czas w długotrwałych stuporach), ale i przeskakują z klasy na klasę, mutują, uwalniają lub emitują nowe kwanty, które przekształcą stosunki klasowe, podadzą w wątpliwość ich nadkodowanie i reterytorializację, sprawią, że gdzie indziej przebiegną nowe linie ujścia. Pod reprodukcją klas znajduje się zawsze zmienna mapa mas. Polityka działa poprzez makrodecyzje i binarne wybory, binarnie zorganizowane interesy – ale domena tego, co podlega decyzji, pozostaje wąska. Na dodatek decyzja polityczna nieuchronnie zanurza się w świat mikrodeterminacji, przyciągań i pragnień, które musi przeczuwać lub szacować w inny sposób. Oszacowanie przepływów i ich kwantów – pod linearnym pojmowaniem i segmentowymi decyzjami. Michelet w ciekawym fragmencie zarzuca Franciszkowi I złe oszacowanie przepływu emigracji, który pchał ku Francji wielu ludzi walczących z Kościołem: Franciszek I dostrzegł tu jedynie źródło potencjalnych żołnierzy, zamiast wyczuć molekularny przepływ masy, który Francja mogła obrócić na swoją korzyść, stając na czele reformacji innej niż ta, która się dokonała\footnote{Jules Michelet, Histoire de France, la Renaissance, Flammarion 1971.}. Problemy zawsze przedstawiają się w ten sposób. Dobre czy złe, polityka i jej osądy są zawsze molowe, lecz to właśnie molekularne i jego oszacowanie „robi” politykę.
Potrafimy teraz lepiej naszkicować mapę. Jeśli przywrócimy słowu „linia” sens najbardziej ogólny, zobaczymy, że ostatecznie istnieją trzy, nie zaś jedynie dwie linie: 1) względnie elastyczna linia splecionych kodów i terytorialności; dlatego wyszliśmy od segmentacji zwanej pierwotną, w którym to przypadku segmentowania terytoriów i lineaży tworzyły przestrzeń społeczną; 2) sztywna linia, która zmierza do dualnej organizacji segmentów, koncentryczności współdrgających kół i ogólnego nadkodowania: przestrzeń społeczna wymaga w tym wypadku aparatu państwa. To system inny niż system pierwotny, przede wszystkim dlatego, że nadkodowanie nie jest silniejszym kodem, lecz specyficzną procedurą, różną od procedury kodów (tak samo reterytorializacja nie jest dodatkowym terytorium, lecz dokonuje się w innej przestrzeni niż przestrzeń terytoriów, ściśle mówiąc, w nadkodowanej przestrzeni geometrycznej); 3) linia lub linie ujścia, oznaczone kwantami, określane przez dekodowanie i deterytorializację (na tych liniach zawsze funkcjonuje coś w rodzaju maszyny wojennej).
Powyższy wywód zawiera jeszcze pewną niedogodność, powstaje bowiem wrażenie, że społeczeństwa pierwotne były pierwsze. W istocie kody nigdy nie są odłączne od ruchu dekodowania, terytoria – od przeszywających je wektorów deterytorializacji. Nadkodowanie zaś i reterytorializacja nie nadchodzą później. Chodzi więc raczej o przestrzeń, w której współistnieją trzy rodzaje ściśle splątanych linii: plemiona, imperia i maszyny wojenne. Można więc równie dobrze powiedzieć, że pierwsze są linie ujścia albo już usztywnione segmenty, i że elastyczne segmentowania nieustannie oscylują między nimi. Weźmy taką tezę historyka Pirenne’a dotyczącą plemion barbarzyńskich: „W tym, że Barbarzyńcy rzucili się na Imperium nie było żadnej spontaniczności; zostali tam oni wypchnięci przez napór Hunów, który miał stać się przyczyną wszystkich kolejnych inwazji\dots{}”\footnote{Henri Pirenne, Mahomet et Charlemagne, PUF 1970, s. 7.}. Oto z jednej strony sztywna segmentacja imperium rzymskiego, z jego centrum rezonansowym i jego peryferiami, jego państwem, pax romana, geometrią, obozami, limesem. A z drugiej strony, na horyzoncie, zupełnie inna linia, linia koczowników, którzy wyruszają ze stepu, podejmują ujście aktywne i płynne, wszędzie niosą deterytorializację, puszczają w ruch przepływy, których kwanty rozgrzewają się porwane przez maszynę wojenną bez państwa. Barbarzyńscy migranci znajdują się właśnie pomiędzy: odchodzą i przychodzą, przekraczają raz po raz granice, plądrują lub żądają okupu, lecz także integrują się i reterytorializują. Już to wciskają się w imperium, którego segment sobie zastrzegają, stają się najemnikami lub sprzymierzeńcami, stabilizują się, zajmują ziemię lub wykrawają sobie państwa (mądrzy Wizygoci). Już to, przeciwnie, przechodzą na stronę nomadów, sprzymierzają się z nimi, stając się nieodróżnialnymi (bystrzy Ostrogoci). Być może dlatego, że Hunowie i Wizygoci nieustannie ich bili, Wandalowie, „Goci drugiej strefy”, wytyczyli linię ujścia, która uczyniła ich równie silnymi, co ich panowie; oto jedyna banda czy też masa, która przebyła Morze Śródziemne. Ale też to właśnie oni dokonali reterytorializacji najbardziej nieoczekiwanej: imperium w Afryce\footnote{Por. Émile Félix Gautier, Genséric, roi des Vandales, Payot 1935 („właśnie dlatego, że byli najsłabsi, wiecznie spychani, musieli iść najdalej”).}. Wydaje się więc, że trzy linie nie tylko współistnieją, lecz się wzajemnie przekształcają, przechodzą każda w każdą. Znów podaliśmy zwięzły przykład, w którym linie zostały zobrazowane przez różne grupy. Tym bardziej stosuje się to do jednej grupy czy do jednostki.
Lepiej byłoby więc odtąd rozważać równoczesne stany Maszyny abstrakcyjnej. Z jednej strony istnieje abstrakcyjna maszyna nadkodowania: to ona wyznacza sztywną segmentację, makrosegmentację, ponieważ produkuje czy raczej reprodukuje segmenty, zestawiając je w przeciwstawne pary, zapewniając współdrganie wszystkich centrów i rozpościerając podzielną, homogeniczną przestrzeń, wyżłobioną we wszystkich kierunkach. Taka maszyna abstrakcyjna odsyła do aparatu państwa. Nie mylimy jej wszelako z samym aparatem państwa. Możemy na przykład definiować maszynę abstrakcyjną more geometrico lub też, w innych warunkach, za pomocą „aksjomatyki”; natomiast aparat państwa nie jest ani geometrią, ani aksjomatyką, jest tylko złożeniem reterytorializacji, które – w tych granicach, w takich warunkach – uruchamia maszynę nadkodowania. Możemy jedynie powiedzieć, że aparat państwa wykazuje mniejszą lub większą skłonność do utożsamienia się z maszyną abstrakcyjną, którą wprawia w ruch. Właśnie stąd bierze swój sens pojęcie państwa totalitarnego: państwo staje się totalitarne, kiedy – zamiast uruchamiać we własnych granicach – światową maszynę nadkodowania, utożsamia się z nią, tworząc warunki „autarkii”, dokonując reterytorializacji „pod kloszem”, w sztucznej pustce (co nie jest nigdy operacją ideologiczną, lecz ekonomiczną i polityczną)\footnote{Totalitaryzmu nie określa znaczenie sektora publicznego, ponieważ gospodarka pozostaje w wielu przypadkach liberalna, lecz sztuczne wytwarzanie „kloszy”, szczególnie pieniężnych, a nawet przemysłowych. To przede wszystkim w tym sensie – jak pokazał Daniel Guérin (Fascisme et grand capital, Maspero 1965, rozdz. IX) – faszyzm włoski i nazizm niemiecki ustanawiają państwa totalitarne.}.
Z drugiej strony, na drugim biegunie znajduje się abstrakcyjna maszyna mutacji, która działa przez dekodowanie i deterytorializację. To ona wytycza linie ujścia: steruje przepływami kwantowymi, zapewnia tworzenie-połączenie przepływów, emituje nowe kwanty. Sama jest w stanie ujścia i ustawia maszyny wojenne na swych liniach. Drugi biegun tworzy zaś dlatego, że sztywne, molowe segmenty nieustannie tamują, zatykają, zagradzają linie ujścia, ona tymczasem nieprzerwanie sprawia, że linie te wyciekają – „między” sztywnymi segmentami i w innym, submolekularnym kierunku. Ale też między dwoma biegunami znajduje się cała domena negocjacji, przekładu, prawdziwie molekularnej transdukcji. Niektóre linie molowe zostały tam już przeorane szczelinami i rysami, z kolei niektóre linie ujścia zaginają się już ku czarnym dziurom, połączenia przepływów są zastępowane przez ograniczające koniunkcje, emisje kwantów podlegają konwersji w punkty-centra. I wszystko to naraz. Linie ujścia łączą się, przedłużają swoje intensywności i sprawiają, że z czarnych dziur tryskają znaki-cząstki [signes-particules], lecz jednocześnie owe linie wpadają w mikro-czarne dziury, gdzie zaczynają wirować, w molekularne koniunkcje, które je przerywają, a wreszcie wstępują w stabilne, zbinaryzowane segmenty, ześrodkowane, uszeregowane na centralnej czarnej dziurze i nadkodowane.
Splątanie wszystkich tych linii może ujawnić się w pytaniu: Czym jest centrum lub ognisko władzy? Mówi się o władzy armii, Kościoła, szkoły, o władzy publicznej lub prywatnej\dots{} Pojęcie centrów władzy dotyczy oczywiście sztywnych segmentów. Każdy segment molowy ma swoje centrum, centra. Można wysunąć obiekcję, że same te segmenty zakładają istnienie centrum władzy – czegoś, co je wyróżnia i ponownie łączy w jedno, co je sobie nawzajem przeciwstawia i sprawia, że współdrgają. Lecz nie ma żadnej sprzeczności między częściami segmentowymi i scentralizowanym aparatem. Z jednej strony najsztywniejsza segmentacja nie stoi na przeszkodzie centralizacji: centralny punkt wspólny nie funkcjonuje jako punkt, w którym zlewają się i łączą inne punkty, lecz jako punkt rezonansu znajdujący się na horyzoncie, za wszystkimi innymi punktami. Państwo nie jest punktem, który zbiera pod sobą inne, lecz pudłem rezonansowym dla wszystkich punktów. I nawet kiedy państwo jest totalitarne, nie zmienia się jego funkcja rezonansowa, zapewniająca współdrganie wyodrębnionych centrów i segmentów; tyle że realizuje się ona pod kloszem, co zwiększa wewnętrzną donośność lub też podwaja „rezonans” poprzez „ruch wymuszony”\footnote{Por. Arystoteles, O niebie III. 2, w: Dzieła wszystkie, t. II, tłum. P. Siwek, PWN 1990, s. 309 [przyp. tłum.].}. Z drugiej strony, i na odwrót, najściślejsza centralizacja nie eliminuje odrębności centrów, segmentów i kół. W istocie wytyczenie linii nadkodującej wymaga zapewnienia przewagi wybranego segmentu nad innym (w wypadku segmentacji binarnej), przekazania wybranemu centrum względnej władzy rezonansowej w odniesieniu do innych (w wypadku segmentacji okrężnej), podkreślenia dominującego segmentu, przez który przechodzi sama ta linia (w wypadku segmentacji linearnej). W tym sensie centralizacja jest zawsze hierarchiczna, ale hierarchia jest zawsze segmentowa.
Każde centrum władzy jest również molekularne, rozciąga się na tkankę mikrologiczną, gdzie istnieje już tylko jako rozsiane, rozproszone, pomnożone, zminiaturyzowane i nieustannie przemieszczane, działając poprzez subtelne segmentowania, które operują na szczególe i szczególe szczegółów. Przeprowadzona przez Foucaulta analiza „dyscyplin” czy mikrowładz (szkoły, armii, fabryki, szpitala itd.) poświadcza istnienie owych „ognisk niestabilności”, gdzie ścierają się przegrupowania i akumulacje, lecz także ucieczki i ujścia, i gdzie tworzą się odwrócenia\footnote{Michel Foucault, Nadzorować i karać, s. 28: „stosunki te sięgają daleko w głąb społeczeństwa, nie sprowadzają się do relacji państwa do obywateli lub do walki klasowej i nie zadowalają się odtwarzaniem (\dots{}) ogólnej formy prawa czy rządu (\dots{}) – mają niezliczone punkty konfrontacji, ogniska niestabilności, z których każde zawiera właściwe sobie ryzyko konfliktu, walki i przejściowego przynajmniej odwrócenia relacji sił”.}. Nie ma już w szkole Nauczyciela – jest nadzorca, prymus, nieuk, woźny itd. Już nie generał, lecz młodsi oficerowie, podoficerowie, żołnierz we mnie, także maruder – każdy ze swoimi skłonnościami, biegunami, konfliktami i relacjami siły. Przywołujemy adiutanta, woźnego jedynie po to, by wyjaśnić, że tak jak oni mają stronę molową i stronę molekularną, tak samo, oczywiście, dwie strony mają już także generał i właściciel. Można powiedzieć, że nazwa własna nie traci swej władzy, lecz odkrywa nową, gdy wkracza w te strefy nierozróżnialności. Żeby mówić jak Kafka, nie trzeba już urzędnika Klamma, lecz może jego sekretarza Momusa albo innych molekularnych Klammów, którzy różnią się od Klamma już tak bardzo, że nie sposób tej różnicy wskazać („[Urzędnicy] nie zatrzymują się nigdy długo przy jednej księdze, ksiąg wszakże nie przekładają, lecz zamieniają się miejscami (\dots{}) [i wówczas] muszą się przeciskać z powodu ciasnoty pomieszczenia\dots{}”, „Ten urzędnik jest do Klamma bardzo podobny. Gdyby siedział we własnym gabinecie i przy własnym biurku, a na drzwiach byłaby tabliczka z jego nazwiskiem, nie miałbym żadnych wątpliwości”\footnote{Franz Kafka, Zamek, s. 147 i 150 (rozdz. XV).}, powiada Barnabas, który śnił o segmentacji wyłącznie molowej, jakkolwiek by nie była sztywna i straszliwa, jako jedynej rękojmi pewności i bezpieczeństwa. Zmuszony był jednak spostrzec, że molowe segmenty nieuchronnie zanurzają się w ową molekularną zupę, służącą im za strawę i zaburzającą ich kontury). Nie istnieje centrum władzy, które nie miałoby tej mikrotkanki. To ona – a nie masochizm – wyjaśnia, jak prześladowany gotów jest zawsze wziąć czynny udział w systemie opresji: robotnicy z krajów bogatych aktywnie uczestniczą w wyzysku trzeciego świata, w zbrojeniu dyktatur, w zanieczyszczaniu atmosfery.
Nie ma w tym nic dziwnego, skoro owa tkanka znajduje się między linią nadkodowania, z jej sztywnymi segmentami, a linią ostateczną – kwantową. Nieustannie oscyluje między nimi dwiema i niekiedy spycha linię kwantową na linię segmentów, niekiedy zaś sprawia, że przepływy i kwanty z linii segmentów uchodzą. Tu właśnie ujawnia się trzeci aspekt centrów władzy lub ich granica. Centra te istnieją tylko po to, by – na ile potrafią – dokonywać przekładu kwantów przepływu na segmenty linii (tylko segmenty poddają się bowiem całościowaniu, w ten czy inny sposób). Lecz tu właśnie napotykają jednocześnie zasadę swej mocy i źródło swej niemocy. Dalekie od zaprzeczania sobie, moc i niemoc uzupełniają się i wzmacniają jedna drugą w swoiście fascynującym zadowoleniu, którego najwyższy stopień odnajdujemy u najbardziej przeciętnych mężów stanu, i które wyznacza ich „chwałę”. Albowiem chwałę czerpią oni ze swej krótkowzroczności, moc zaś ze swej niemocy stanowiącej potwierdzenie, że nie było wyboru. Jedynymi „wielkimi” mężami stanu są ci, którzy łączą się z przepływami niczym znaki-przewodnicy (signes-pilotes), znaki-cząstki, i emitują kwanty, przezwyciężając czarne dziury: nie przypadkiem ludzi tych spotyka się tylko na liniach ujścia, gdy je wytyczają, przeczuwają, podążają za nimi i je wyprzedzają, nawet gdy się mylą i upadają (Żyd Mojżesz, Wandal Genzeryk, Mongoł Dżyngis-Chan, Chińczyk Mao\dots{}). Nie istnieje natomiast Władza, która regulowałaby same przepływy. Nie sposób zapanować nawet nad przyrostem „masy pieniężnej”. Kiedy rzutujemy na granice wszechświata obraz pana, ideę państwa czy tajnego rządu – jakby panowanie oddziaływało na przepływy w nie mniejszej mierze i w ten sam sposób, co na segmenty – wówczas poddajemy się przedstawieniom śmiesznym i fikcyjnym. To giełda, nie zaś państwo, dostarcza obrazu przepływów i ich kwantów. Kapitaliści mogą zarządzać wartością dodatkową i jej rozdzielaniem, lecz nie panują nad przepływami, z których pochodzi wartość dodatkowa. Centra władzy oddziałują zatem w punktach, w których przepływy przekształcają się w segmenty: jako wymienniki, konwertery, oscylatory. Same segmenty nie zależą jednak od władzy decydowania. Przeciwnie, widzieliśmy, jak segmenty (na przykład klasy) tworzą się w koniunkcji mas i zdeterytorializowanych przepływów, gdzie najbardziej zdeterytorializowany przepływ określa dominujący segment: stąd dolar jako dominujący segment pieniądza, burżuazja jako dominujący segment kapitalizmu\dots{} itd. Same segmenty zależą więc od maszyny abstrakcyjnej. Tym, co zależy od centrów władzy, są jednak układy, które ową maszynę abstrakcyjną wprawiają w ruch, to znaczy nieustannie dostosowują zmienności masy i przepływu do segmentów sztywnej linii, w zależności od segmentu dominującego i segmentu podporządkowanego. Tym dostosowaniom towarzyszyć może niemało perwersyjnej wynalazczości.
W tym właśnie sensie mówi się na przykład o władzy bankowej (Bank Światowy, banki centralne, banki kredytowe): jeśli przepływ pieniądza-finansowania, pieniądza kredytowego odsyła do masy transakcji ekonomicznych, zadaniem banków jest przekształcenie tego kreowanego pieniądza kredytowego w pieniądz segmentowej płatności, pieniądz gotówkowy, metalowy pieniądz państwa, pozwalający nabywać dobra, które same są segmentowane (w tym względzie waga stóp procentowych). Zadaniem banków jest więc konwersja dwóch rodzajów pieniądza, dalej konwersja segmentów drugiego rodzaju pieniądza w homogeniczny zespół i wreszcie konwersja drugiego rodzaju pieniądza w jakiekolwiek dobra\footnote{O tych aspektach władzy bankowej por. Suzanne de Brunhoff, L’offre de monnaie, Maspero 1971, passim, przede wszystkim s. 102–131.}. To samo da się powiedzieć o wszystkich centrach władzy. Każde bowiem centrum władzy obejmuje następujące trzy aspekty czy też trzy strefy: 1) swoją strefę mocy, w odniesieniu do segmentów trwałej, sztywnej linii; 2) strefę nierozróżnialności, w odniesieniu do swego rozproszenia na tkance mikrofizycznej; 3) swoją strefę niemocy, w odniesieniu do przepływów i kwantów, które może jedynie konwertować, nie osiągając zdolności ich kontrolowania czy określania. A zatem każde centrum władzy zawsze czerpie swą moc ze źródła własnej niemocy: stąd bierze się jego zasadnicza niegodziwość i próżność. Być raczej maleńkim kwantem przepływu niż konwerterem, oscylatorem, dystrybutorem molowym! Wracając do przykładu pieniądza – pierwszą strefę reprezentują narodowe banki centralne; drugą – „nieokreślona seria prywatnych związków między bankami a pożyczkobiorcami”; trzecią – pragnący przepływ pieniądza, przepływ, którego kwanty określone są przez masę transakcji ekonomicznych. Jest prawdą, że identyczne problemy pojawiają się ponownie na poziomie samych tych transakcji, w związku z innymi centrami władzy. Ale we wszystkich przypadkach strefa pierwsza, strefa centrum władzy zostaje określona w aparacie państwa jako układ, który wprawia w ruch abstrakcyjną maszynę molowego nadkodowania; strefa druga określona jest w tkance molekularnej, w którą zanurza się ten układ ; trzecia wreszcie jest określona w abstrakcyjnej maszynie mutacji, przepływu i kwantów.
Nie możemy jednak powiedzieć, że któraś z tych trzech linii jest – ze swej natury i koniecznie – dobra, inna zaś zła. Badanie zagrożeń na każdej linii o tyle stanowi przedmiot pragmatyki czy też schizoanalizy, o ile nie chce reprezentować, interpretować ani symbolizować, lecz jedynie tworzyć mapy i wykreślać linie, zaznaczając, gdzie owe linie są splątane, gdzie zaś się wyodrębniają. Zaratustra Nietzschego, a także indiański Don Juan u Castanedy powiadają: istnieją trzy lub nawet cztery zagrożenia, najpierw Lęk, potem Jasność, dalej Władza i wreszcie wielkie Obrzydzenie [Degout], chęć zabijania i umierania, Pasja kresu [abolition]\footnote{Carlos Castaneda, Nauki don Juana. Wiedza Indian z plemienia Yaqui, tłum. A. Szostkiewicz, wyd. 2 popr., Rebis 1997, s. 91–97.}. Czym jest lęk – możemy się domyślić. Nieustannie boimy się stracić. Bezpieczeństwo; wielka organizacja molowa, która nas podtrzymuje; drzewiastości, których się chwytamy; maszyny binarne, zapewniające nam ściśle określony status; współdrgania, w które wchodzimy; panujący nad nami system nadkodowania – wszystkiego tego pragniemy. „Wartości, zasady moralne, ojczyzny, religie i osobiste pewności, których nasza próżność i nasze samozadowolenie szczodrze nam udzielają – tyle miejsc zamieszkania świat przysposabia tym, co sądzą, że w ten sposób spoczywają, stojąc pośród rzeczy niewzruszonych; nie wiedzą oni nic o bezmiernej ucieczce, w którą się puścili\dots{} ujściu przed ujściem”\footnote{Maurice Blanchot, L’amitié, Gallimard 1971, s. 232.}. Uchodzimy przed ujściem, usztywniamy nasze segmenty, oddajemy się binarnej logice; o tyle sztywniejsi w jednym segmencie, o ile sztywniej potraktowano nas w innym, reterytorializujemy się gdziekolwiek, nie znamy segmentacji innej niż molowa – zarówno na poziomie wielkich zespołów, do których przynależymy, jak i małych grup, do których dołączamy, także w tym, co dzieje się w nas najbardziej intymnie i prywatnie. To dotyczy wszystkiego: sposobu postrzegania, rodzaju działania, maniery, z jaką się poruszamy, trybu życia, reżimu semiotycznego. Mężczyzna, który wraca do domu i mówi: „Zupa gotowa?”, żona, która odpowiada: „Co się krzywisz? Masz zły humor?” – to skutek zderzenia pary sztywnych segmentów. Im bardziej sztywna segmentacja, tym więcej daje nam pewności. Oto czym jest lęk i jak spycha nas na pierwszą linię.
Drugie zagrożenie, Jasność, wydaje się mniej oczywiste. Jasność bowiem dotyczy tego, co molekularne. I tu także dotyczy wszystkiego – nawet postrzegania, nawet semiotyki – lecz na drugiej linii. Castaneda wskazuje na przykład, że istnieje postrzeganie molekularne, które otwiera nam narkotyk (lecz tyle rzeczy może służyć jako narkotyk): dochodzi się do mikropostrzeżenia dźwiękowego i wzrokowego, objawiającego przestrzenie i otchłanie, swoiste dziury w strukturze molowej. Oto właśnie – jasność: owe rozróżnienia zachodzące w czymś, co wydawało się pełne, owe dziury w zwartym i szczelnym, i na odwrót: tam, gdzie dopiero co widzieliśmy zakończenia wyraźnie odrębnych segmentów, teraz istnieją raczej niepewne pogranicza, rozmycia, nasunięcia, migracje, akty segmentowania niezbieżne ze sztywną segmentacją. Wszystko staje się wyraziście elastyczne – pustki pośród pełni, mgławicowe kształty, drgające zarysy. Wszystko uzyskało mikroskopową jasność. Sądzimy więc, że wszystko zrozumieliśmy i wyciągnęliśmy wnioski. Jesteśmy nowymi rycerzami, mamy nawet misję. Mikrofizyka migranta zajęła miejsce makrogeometrii tego, co osiadłe. Lecz ta elastyczność i jasność nie tylko niosą zagrożenie, lecz same są zagrożeniem. Po pierwsze dlatego, że elastyczna segmentacja grozi odtworzeniem w miniaturze afekcji i afektacji sztywnych segmentów: rodzinę zastąpi się komuną, małżeństwo – reżimem wymiany i migracji. Ale dzieje się jeszcze gorzej: powstają mikro-Edypowie, mikrofaszyzmy tworzą prawo, matka czuje się zobowiązana brandzlować dziecko, ojciec staje się mamą. Takim właśnie smutkiem emanuje mroczna jasność, której nie zsyła żadna gwiazda. Ruchoma segmentacja wypływa bezpośrednio z segmentacji najbardziej sztywnej, jest jej bezpośrednią kompensacją. Im bardziej zespoły stają się molowe, tym bardziej ich elementy i stosunki między tymi elementami stają się molekularne, człowiek molekularny dla molowej ludzkości. Dokonuje się deterytorializacji, tworzy się masę, lecz tylko po to, by spętać i unieważnić ruchy masy i deterytorializacji, by wynajdywać wszelkie marginalne reterytorializacje jeszcze gorsze od innych. Przede wszystkim jednak elastyczna segmentacja rodzi własne zagrożenia. Nie poprzestają one na odtwarzaniu w małej skali zagrożeń segmentacji molowej. Nie wystarczy już, że z nich wypływają albo je kompensują. Jak widzieliśmy, mikrofaszyzmy mają swoją specyfikę, którą mogą wykrystalizować w makrofaszyzmie, lecz mogą ją także na własny rachunek upłynnić na elastycznej linii i w ten sposób wchłaniać każdą małą komórkę. Wielość czarnych dziur z łatwością może uniknąć centralizacji i istniejąc na sposób wirusów, które przystosowują się do najróżniejszych sytuacji, wydrążyć pustkę w postrzeżeniach i molekularnych semiotykach. Wzajemne oddziaływania bez współdrgania. Zamiast paranoicznego wielkiego lęku – tysiąc małych monomanii, oczywistości i jasności, w które jesteśmy pochwyceni, które tryskają z każdej czarnej dziury. To już nie system, lecz szmer i szum, oślepiające światła, dzięki którym ten lub ów odkrywa powołanie sędziego, obrońcy sprawiedliwości, samozwańczego policjanta, gauleitera budynku czy mieszkania. Pokonaliśmy lęk, opuściliśmy brzegi bezpieczeństwa, lecz weszliśmy w system nie mniej skoncentrowany, nie mniej zorganizowany – system małych niepewności, który sprawia, że każdy znajdzie swą czarną dziurę i w tej dziurze stanie się zagrożeniem, dysponując jasnością co do swojej sprawy, swojej roli i swojej misji, jasnością bardziej niepokojącą niż pewniki pierwszej linii.
Władza jest trzecim zagrożeniem, ponieważ znajduje się na obu liniach jednocześnie. Przechodzi od sztywnych segmentów, ich nadkodowania i współdrgania, do finezyjnych segmentowań z ich rozproszeniem i wzajemnym oddziaływaniem – i z powrotem. Nie istnieje człowiek władzy, który nie przeskakiwałby z jednej linii na drugą, który nie wykorzystywałby na przemian stylu niskiego i wysokiego, stylu grubiańskiego i stylu Bossueta, demagogii kiosku i imperializmu wysokiego urzędnika. Lecz ten łańcuch i ta sieć władzy całe zanurzają się w świat, który im umyka, świat mutujących przepływów. I właśnie ta niemoc czyni władzę tak niebezpieczną. Człowiek władzy nieustannie chce zatrzymać linie ujścia i dlatego ciągle stara się pochwycić, umocować maszynę mutacji w maszynie nadkodowania. Może to wszakże uczynić jedynie wytwarzając pustkę, to znaczy stabilizując najpierw samą maszynę nadkodowania, umieszczając ją w lokalnym układzie, który ma ją wprawiać w ruch – krótko mówiąc, nadając układowi wymiary maszyny: to właśnie zachodzi w sztucznych warunkach totalitaryzmu, „pod kloszem”.
Lecz istnieje jeszcze czwarte zagrożenie i zapewne ono interesuje nas najbardziej, ponieważ dotyczy samych linii ujścia. Mogliśmy przedstawiać te linie jako rodzaj mutacji, tworzenia, jako wykreślające się nie w wyobraźni, lecz w samej tkance rzeczywistości społecznej, mogliśmy nadać im ruch strzały i absolutną prędkość – byłoby jednak uproszczeniem sądzić, że jedynym niebezpieczeństwem, którego się lękają i które napotykają jest to, że mimo wszystko dadzą się pochwycić, zatamować, podłączyć, zawiązać, zreterytorializować. One same wydzielają dziwną rozpacz, niczym zapach śmierci i ofiarowania, niczym stan wojny, z której wychodzi się złamanym; one same niosą własne zagrożenia, różne od poprzednich. Dokładnie to kazało Fitzgeraldowi powiedzieć: „Miałem poczucie, że stoję o zmierzchu na opuszczonej strzelnicy, pusta strzelba w ręku, cele strącone. Żadnego problemu do rozwiązania. Po prostu cisza i tylko szmer mojego własnego oddechu. (\dots{}) Ofiara, którą uczyniłem sam z siebie, była czymś mrocznym i wilgotnym”\footnote{Francis Scott Fitzgerald, La fêlure, Gallimard 1963, s. 350, 354 [The Crack-Up, New Directions 1945]; trzyczęściowy esej opublikowany oryginalnie w „Esquire” w lutym, marcu i kwietniu 1936 roku. Cytaty pochodzą z cz. 2 (Pasting It Together) oraz 3 (Handle with Care). Drugi cytat tłumaczymy z oryg. ang. „My self-immolation was something sodden-dark”, w Tysiącu plateau: „Mon immolation de moi-même était une fusée sombre et mouillée”.}. Dlaczego linia ujścia jest wojną, z której najpewniej wróci się rozbitym, zniszczonym, zniszczywszy wcześniej wszystko, co się dało? Oto właśnie czwarte zagrożenie: że linia ujścia przedrze się przez mur, wydostanie się z czarnych dziur, lecz zamiast łączyć się z innymi liniami i za każdym razem zwiększać swą wartościowość, zmieni się w zniszczenie, kres czysty i prosty, pasję kresu. Taką jest linia ujścia Kleista, dziwaczna wojna, którą prowadzi, i samobójstwo, podwójne samobójstwo jako wyjście, które z linii ujścia czyni linię śmierci.
Nie odwołujemy się tu do żadnego popędu śmierci. W pragnieniu nie ma wewnętrznego popędu, są tylko złożenia. Pragnienie jest zawsze złożone i złożenie określa je, by było tym, czym jest. Na poziomie linii ujścia złożenie, które je wytycza, jest typem maszyny wojennej. Mutacje odsyłają do tej maszyny, której przedmiotem z pewnością nie jest wojna, lecz emisja kwantów deterytorializacji, przechodzenie mutujących przepływów (w tym sensie wszelkie tworzenie przechodzi przez maszynę wojenną). Istnieje wiele argumentów pozwalających wykazać, że maszyna wojenna jest innym złożeniem niż aparat państwa, wywodzi się skądinąd. Będąc pochodzenia koczowniczego, kieruje się przeciwko temu aparatowi. Będzie to jeden z podstawowych problemów państwa: przywłaszczyć sobie ową maszynę wojenną, która jest mu obca, uczynić z niej fragment własnego aparatu, nadając jej kształt ustabilizowanej instytucji militarnej; w tej mierze państwo będzie zawsze napotykać na wielkie trudności. Ale właśnie w chwili, gdy maszyna wojenna ma za przedmiot już tylko wojnę, gdy zatem zastępuje mutację zniszczeniem, wówczas uwalnia najstraszliwszy ładunek. Mutacja w żadnym razie nie była przekształceniem wojny, na odwrót – to wojna istnieje jako upadek czy odpad mutacji, jedyny przedmiot, jaki pozostaje maszynie wojennej, gdy utraci ona swą moc przekształcania. A zatem o samej wojnie trzeba powiedzieć, że jest tylko ohydną pozostałością maszyny wojennej, która albo została już przywłaszczona przez aparat państwa, albo, co gorsza, zbudowała sobie aparat państwa zdolny jedynie do destrukcji. Tak oto maszyna wojenna nie wykreśla już mutujących linii ujścia, ale czystą i zimną linię kresu (będziemy chcieli niżej przedstawić hipotezę dotyczącą tej złożonej relacji między maszyną wojenną a wojną.).
Tu właśnie ponownie docieramy do paradoksu faszyzmu i różnicy między faszyzmem a totalitaryzmem. Totalitaryzm jest bowiem sprawą państwa: dotyczy w istocie związku państwa – jako złożenia zlokalizowanego – z abstrakcyjną maszyną nadkodowania, którą wprawia ono w ruch. Nawet w wypadku dyktatury wojskowej to armia państwa – nie zaś maszyna wojenna – przejmuje władzę i podnosi państwo do stadium totalitarnego. Totalitaryzm jest w najwyższym stopniu konserwatywny. Tymczasem w faszyzmie chodzi rzeczywiście o maszynę wojenną. I gdy faszyzm buduje sobie państwo totalitarne, to już nie w tym znaczeniu, że armia państwa przejmuje władzę, lecz przeciwnie – oznacza to, że maszyna wojenna opanowuje państwo. Dziwna uwaga Virilia naprowadza nas na właściwy trop: w faszyzmie państwo jest nie tyle totalitarne, ile samobójcze. Faszyzm jest urzeczywistnionym nihilizmem. To znaczy, że w odróżnieniu od państwa totalitarnego, które wytęża siły, by zatamować wszystkie możliwe linie ujścia, faszyzm buduje się na intensywnej linii ujścia, którą przekształca w linię czystego zniszczenia i kresu. Ciekawe, że naziści od początku ogłaszali Niemcom, co im przynoszą: jednocześnie zaślubiny i śmierć, w tym swoją własną śmierć i śmierć Niemców. Sądzili, że zginą, lecz ich przedsięwzięcie zostało przynajmniej rozpoczęte – dla Europy, świata, systemu planetarnego. A ludzie wznosili radosne okrzyki – nie dlatego, że nie rozumieli, lecz dlatego, że chcieli tej śmierci, która miała się dokonać przez śmierć innych. Jakby chcieli za każdym razem grać o wszystko, stawiać śmierć innych przeciw swojej i wszystko wymierzyć „deleometrami”\footnote{Deleo, -ere (łac.), niszczyć, burzyć, kłaść kres [przyp. tłum.].}. Powieść Klausa Manna Mefisto dostarcza próbek dyskursu czy może całkiem zwyczajnych rozmów nazistowskich: „W naszej egzystencji zanikał coraz bardziej patos bohaterski (\dots{}). W istocie nie maszerujemy teraz, ale gnamy naprzód. Nasz ukochany führer porywa nas w ciemność i nicość. Jakże moglibyśmy, my, poeci, którzy mamy szczególny stosunek do ciemności i do przepaści, jakże moglibyśmy go za to nie podziwiać? (\dots{}) Łuny na horyzoncie, rzeki krwi na wszystkich drogach i opętany taniec dokoła trupów, taniec tych, co przeżyli, tych, których jeszcze oszczędzono!”\footnote{Klaus Mann, Mefisto, tłum. J. Dmochowska, Książnica 1997, s. 239–241 (rozdz. IX). Tego rodzaju deklaracji jest pod dostatkiem nawet w chwili sukcesu nazistów. Por. słynne sformułowania Goebbelsa: „W świecie absolutnego fatalizmu, w którym porusza się Hitler, nic już nie ma sensu, ani dobro, ani zło, ani czas, ani przestrzeń, a to, co inni ludzie nazywają sukcesem, nie może stanowić kryterium. (\dots{}) Możliwe, że Hitler doprowadzi do katastrofy\dots{}” (Hitler parle à ses généraux, Albin Michel 1964) [przywołana książka dostarcza kolejnych ilustracji dla tez Deleuze’a i Guattariego, nie jest natomiast źródłem tego cytatu, który pochodzi ze wspomnień księcia Friedricha Christiana zu Schaumburg-Lippe, adiutanta Goebbelsa – dopisek tłum.]. Ten katastrofizm nie kłóci się ze znacznym zadowoleniem, czystym sumieniem i wygodnym spokojem, co widzimy również, w zupełnie innym kontekście, u niektórych samobójców. Istnieje cała biurokracja katastrofy. W kwestii faszyzmu włoskiego odnosimy się przede wszystkim do analizy Marii A. Macciocchi, Sexualité féminine dans l’idéologie fasciste, „Tel Quel” nr 66, 1976: kobiecy szwadron śmierci stworzony z wdów i matek w żałobie, hasło „Trumna i kołyski” [u Macciocchi: „istnieje tylko śmiertelna jedność, stała spójność, od kołyski po grób, czy raczej od trumny do kołyski” – dopisek tłum.].}. Samobójstwo nie jawi się jako kara, lecz jako ukoronowanie śmierci innych. Możemy zawsze powiedzieć, że to kwestia mglistego dyskursu i ideologii – tylko ideologii. Lecz to nieprawda; z faktu, że definicje polityczne i ekonomiczne faszyzmu są niewystarczające, nie wynika jedynie konieczność dołączenia do nich nieostrych określeń zwanych ideologicznymi. Wolimy podążać za J.-P. Fayem, gdy bada, jak dokładnie kształtowały się nazistowskie wypowiedzenia funkcjonujące zarówno w polityce, w ekonomii, jak i w najbardziej absurdalnej rozmowie. Odnajdujemy w tych wypowiedzeniach zawsze ów okrzyk „głupi i odpychający” – Niech żyje śmierć!, nawet na poziomie ekonomicznym, gdzie rozprzestrzenienie zbrojeń zastępuje wzrost konsumpcji i gdzie inwestycje przesuwają się ze środków produkcji na środki czystego zniszczenia. Głęboko słuszna wydaje nam się analiza Paula Virilio, kiedy definiuje on faszyzm nie przez pojęcie państwa totalitarnego, lecz państwa samobójczego: wojna nazwana totalną jawi się tu w mniejszym stopniu jako przedsięwzięcie państwa, w większym zaś – maszyny wojennej, która przywłaszczyła sobie państwo i sprawiła, że przenika je przepływ wojny absolutnej, zmierzający ku jednemu tylko wyjściu, jakim jest samobójstwo samego państwa. „Uruchomienie nieznanego procesu materialnego faktycznie bez granic i bez celu. (\dots{}) Raz uruchomiony, jego mechanizm nie może dążyć do pokoju, pośrednia strategia bowiem skutecznie umieszcza panującą władzę poza zwykle stosowanymi kategoriami przestrzeni i czasu. (\dots{}) To w przerażeniu codziennością i jej środowiskiem Hitler znalazł ostatecznie swój najpewniejszy środek rządzenia, uprawomocnienie swej polityki i strategii militarnej – i to do samego końca, skoro ruiny, koszmary, zbrodnie i chaos wojny totalnej, dalekie od uderzenia w odrażającą naturę jego władzy, zazwyczaj jedynie powiększały jej zasięg. Telegram 71: Jeżeli wojnę przegramy, zginie również naród, w którym Hitler zdecydował związać swoje wysiłki z wysiłkami wroga, by dokonać zniszczenia swojego własnego ludu poprzez unicestwienie ostatnich zasobów środowiska i cywilnych rezerw wszelkiego rodzaju (woda pitna, paliwo, żywność) jest naturalnym zwieńczeniem\dots{}”\footnote{Paul Virilio, L’insécurité du territoire, rozdz. I. Hannah Arendt, jakkolwiek utożsamiła nazizm i totalitaryzm, wydobyła tę zasadę nazistowskiego panowania: „Ich koncepcja panowania stawia wymagania, którym nie potrafi sprostać żadne państwo ani zwyczajny aparat przemocy, lecz tylko ruch społeczny utrzymywany w nieustannym napięciu” [w Tysiącu plateau: „mouvement constamment en mouvement” – ruch stale w ruchu – dopisek tłum.]; nawet wojna i ryzyko klęski wojennej włączają się jako akceleratory (Hannah Arendt, Korzenie totalitaryzmu, tłum. D. Grinberg, Wydawnictwa Akademickie i Profesjonalne 2014, s. 407–408). [Zdanie „Jeżeli wojnę przegramy, zginie również naród” pochodzi z: Albert Speer, Wspomnienia, tłum. M. Fijałkowski i in., Wyd. MON 1990, s. 525. O dwunastu dokumentach z marca–kwietnia 1945 roku, nakazujących dokonywanie zniszczeń na terenie Niemiec zob. tamże, s. 528–548. O państwie nazistowskim jako państwie samobójczym w kontekście „telegramu 71” zob. także Michel Foucault, Trzeba bronić społeczeństwa. Wykłady w Collège de France, 1976, tłum. M. Kowalska, KR 1998, s. 257 (wykład z 17 marca 1976) – dopisek tłum.]}. To obrócenie się linii ujścia w linię zniszczenia wystarcza, by pobudzić wszystkie molekularne ogniska faszyzmu i doprowadzić je raczej do współdziałania z maszyną wojenną niż do współdrgania w aparacie państwa. Z maszyną wojenną, która miała za przedmiot jedynie wojnę i która pogodziła się raczej z unicestwieniem własnych sług, niż z zatrzymaniem zniszczenia. Wszystkie zagrożenia związane z innymi liniami są mało istotne wobec tego niebezpieczeństwa.
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Anarcho-Biblioteka
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Gilles Deleuze, Félix Guattari
1933 - Mikropolityka i Segmentacja
1980
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\textbf{pl.anarchistlibraries.net}
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https://dlmf.nist.gov/13.20.E19.tex | nist.gov | CC-MAIN-2018-13 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2018-13/segments/1521257647406.46/warc/CC-MAIN-20180320111412-20180320131412-00492.warc.gz | 543,698,044 | 820 | \[M_{\kappa,\mu}\left(x\right)=\left(8\mu\right)^{\frac{1}{4}}\*\left(\frac{2\mu%
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\mathrm{env}\mskip-1.0mu U\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)O\left(\mu^%
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http://dec41.user.srcf.net/notes/III_M/algebraic_topology_iii.tex | srcf.net | CC-MAIN-2018-51 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2018-51/segments/1544376823322.49/warc/CC-MAIN-20181210101954-20181210123454-00432.warc.gz | 70,015,067 | 63,184 | \documentclass[a4paper]{article}
\def\npart {III}
\def\nterm {Michaelmas}
\def\nyear {2016}
\def\nlecturer {O.\ Randal-Williams}
\def\ncourse {Algebraic Topology}
\input{header}
\theoremstyle{definition}
\newtheorem{cclaim}{Claim}
\setcounter{cclaim}{-1}
\begin{document}
\maketitle
{\small
\setlength{\parindent}{0em}
\setlength{\parskip}{1em}
Algebraic Topology assigns algebraic invariants to topological spaces; it permeates modern pure mathematics. This course will focus on (co)homology, with an emphasis on applications to the topology of manifolds. We will cover singular homology and cohomology, vector bundles and the Thom Isomorphism theorem, and the cohomology of manifolds up to Poincar\'e duality. Time permitting, there will also be some discussion of characteristic classes and cobordism, and conceivably some homotopy theory.
\subsubsection*{Pre-requisites}
Basic topology: topological spaces, compactness and connectedness, at the level of Sutherland's book. The course will not assume any knowledge of Algebraic Topology, but will go quite fast in order to reach more interesting material, so some previous exposure to simplicial homology or the fundamental group would be helpful. The Part III Differential Geometry course will also contain useful, relevant material.
Hatcher's book is especially recommended for the course, but there are many other suitable texts.
}
\tableofcontents
\section{Homotopy}
In this course, the word ``map'' will always mean ``continuous function''.
In topology, we study spaces up to ``continuous deformation''. Famously, a coffee mug can be continuously deformed into a doughnut, and thus they are considered to be topologically the same. Now we also talk about maps between topological spaces. So a natural question is if it makes sense to talk about the continuous deformations of maps. It turns out it does, and the definition is sort-of the obvious one:
\begin{defi}[Homotopy]\index{homotopy}
Let $X, Y$ be topological spaces. A \emph{homotopy} between $f_0, f_1: X \to Y$ is a map $F: [0, 1] \times X \to Y$ such that $F(0, x) = f_0(x)$ and $F(1, x) = f_1(x)$. If such an $F$ exists, we say $f_0$ is \emph{homotopic} to $f_1$, and write $f_0 \simeq f_1$.
This $\simeq$ defines an equivalence relation on the set of maps from $X$ to $Y$.
\end{defi}
Just as concepts in topology are invariant under homeomorphisms, it turns out the theories we develop in algebraic topology are invariant under homotopies, i.e.\ homotopic maps ``do the same thing''. This will be made more precise later.
Under this premise, if we view homotopic functions as ``the same'', then we have to enlarge our notion of isomorphism to take this into account. To do so, we just write down the usual definition of isomorphism, but with equality replaced with homotopy.
\begin{defi}[Homotopy equivalence]\index{homotopy equivalence}
A map $f: X \to Y$ is a \emph{homotopy equivalence} if there is some $g: Y \to X$ such that $f \circ g \simeq \id_Y$ and $g \circ f \simeq \id_X$. We call $g$ a \emph{homotopy inverse}\index{homotopy inverse} to $f$.
\end{defi}
As always, homotopy equivalence is an equivalence relation. This relies on the following straightforward property of homotopy:
\begin{prop}
If $f_0 \simeq f_1: X \to Y$ and $g_0 \simeq g_1: Y \to Z$, then $g_0 \circ f_0 \simeq g_1 \circ f_1: X \to Z$.
\[
\begin{tikzcd}
X \ar[r, bend left, "f_0"] \ar[r, bend right, "f_1"'] & Y \ar[r, bend left, "g_0"] \ar[r, bend right, "g_1"'] & Z
\end{tikzcd}
\]
\end{prop}
\begin{eg}[Stupid example]
If $f: X \to Y$ is a homeomorphism, then it is a homotopy equivalence --- we take the actual inverse for the homotopy inverse, since equal functions are homotopic.
\end{eg}
\begin{eg}[Interesting example]
Let $i: \{0\} \to \R^n$ be the inclusion map. To show this is a homotopy equivalence, we have to find a homotopy inverse. Fortunately, there is only one map $\R^n \to \{0\}$, namely the constant function $0$. We call this $r: \R^n \to \{0\}$. The composition $r \circ i: \{0\} \to \{0\}$ is exactly the identity. So this is good.
In the other direction, the map $i \circ r: \R^n \to \R^n$ sends everything to $0$. We need to produce a homotopy to the identity. We let $F: [0, 1] \times \R^n \to \R^n$ be
\[
F(t, \mathbf{v}) = t\mathbf{v}.
\]
We have $F(0, \mathbf{v}) = 0$ and $F(1, \mathbf{v}) = \mathbf{v}$. So this is indeed a homotopy from $i \circ r$ to $\id_{\R^n}$.
\end{eg}
So from the point of view of homotopy, the one-point space $\{0\}$ is the same as $\R^n$! So dimension, or even cardinality, is now a meaningless concept.
\begin{eg}[Also interesting example]
Let $S^n \subseteq \R^{n + 1}$ be the unit sphere, and $i: S^n \hookrightarrow \R^{n + 1} \setminus \{0\}$. We show that this is a homotopy equivalence. We define $r: \R^{n + 1} \setminus \{0\} \to S^n$ by
\[
r(\mathbf{v}) = \frac{\mathbf{v}}{\|\mathbf{v}\|}.
\]
Again, we have $r \circ i = \id_{S^n}$. In the other direction, we need to construct a path from each $\mathbf{v}$ to $\frac{\mathbf{v}}{\|\mathbf{v}\|}$ in a continuous way. We could do so by
\begin{align*}
H: [0, 1] \times (\R^{n + 1} \setminus \{0\}) &\to \R^{n + 1} \setminus \{0\}\\
(t, \mathbf{v}) &\mapsto (1 - t) \mathbf{v} + t \frac{\mathbf{v}}{\|\mathbf{v}\|}.
\end{align*}
We can easily check that this is a homotopy from $\id_{\R^{n + 1}\setminus \{0\}}$ to $i \circ r$.
\end{eg}
Again, homotopy equivalence allowed us to squash down one dimension of $\R^{n + 1} \setminus \{0\}$ to get $S^n$. However, there is one thing we haven't gotten rid of --- the hole. It turns out what \emph{is} preserved by homotopy equivalence is exactly the holes.
Now we might ask ourselves --- can we distinguish holes of ``different dimension''? For $n \not= m$, is $S^n$ homotopy equivalent to $S^m$? If we try hard to construct homotopies, we will always fail, and then we would start to think that maybe they aren't homotopy equivalent. However, at this point, we do not have any tools we can use the prove this.
The solution is algebraic topology. The idea is that we assign algebraic objects to each topological space in a way that is homotopy-invariant. We then come up with tools to compute these algebraic invariants. Then if two spaces are assigned different algebraic objects, then we know they cannot be homotopy equivalent.
What is this good for? At this point, you might not be convinced that it is a good idea to consider spaces up to homotopy equivalence only. However, algebraic topology can also help us show that spaces are not homeomorphic. After all, homeomorphic spaces are also homotopy equivalent. For example, suppose we want to show that if $\R^n \cong \R^m$, then $n = m$. Algebraic topology doesn't help us directly, because both $\R^n$ and $\R^m$ are homotopy equivalent to a point. Instead, we do the slightly sneaky thing of removing a point. If $\R^n \cong \R^m$, then it must also be the case that $\R^n \setminus \{0\} \cong \R^m \setminus \{0\}$. Since these are homotopy equivalent to $S^{n - 1}$ and $S^{m - 1}$, this implies that $S^{n - 1} \simeq S^{m - 1}$ are homotopy equivalent. By algebraic topology, we will show that that this can only happen if $m = n$. So indeed we can recover the notion of dimension using algebraic topology!
\section{Singular (co)homology}
\subsection{Chain complexes}
This course is called algebraic topology. We've already talked about some topology, so let's do some algebra. We will just write down a bunch of definitions, which we will get to use in the next chapter to define something useful.
\begin{defi}[Chain complex]\index{chain complex}
A \emph{chain complex} is a sequence of abelian groups and homomorphisms
\[
\begin{tikzcd}
\cdots \ar[r] & C_3 \ar[r, "d_3"] & C_2 \ar[r, "d_2"] & C_1 \ar[r, "d_1"] & C_0 \ar[r, "d_0"] & 0
\end{tikzcd}
\]
such that
\[
d_i \circ d_{i + 1} = 0
\]
for all $i$.
\end{defi}
Very related to the notion of a chain complex is a \emph{co}chain complex, which is the same thing with the maps the other way.
\begin{defi}[Cochain complex]\index{cochain complex}
A \emph{cochain complex} is a sequence of abelian groups and homomorphisms
\[
\begin{tikzcd}
0 \ar[r] & C^0 \ar[r, "d^0"] & C^1 \ar[r, "d^1"] & C^2 \ar[r, "d^2"] & C^3 \ar[r] & \cdots
\end{tikzcd}
\]
such that
\[
d^{i + 1} \circ d^i = 0
\]
for all $i$.
\end{defi}
Note that each of the maps $d$ is indexed by the number of its domain.
\begin{defi}[Differentials]\index{differentials}
The maps $d^i$ and $d_i$ are known as \emph{differentials}.
\end{defi}
Eventually, we will get lazy and just write all the differentials as $d$.
Given a chain complex, the only thing we know is that the composition of any two maps is zero. In other words, we have $\im d_{i+1} \subseteq \ker d_i$. We can then ask how good this containment is. Is it that $\im d_{i + 1} = \ker d_i$, or perhaps that $\ker d_i$ is huge but $\im d_{i + 1}$ is trivial? The homology or cohomology measures what happens.
\begin{defi}[Homology]\index{homology}
The \emph{homology} of a chain complex $C_{\Cdot}$ is
\[
H_i(C_{\Cdot}) = \frac{\ker (d_i: C_i \to C_{i - 1})}{\im (d_{i + 1}: C_{i + 1} \to C_i)}.
\]
An element of $H_i(C_{\Cdot})$ is known as a \term{homology class}.
\end{defi}
Dually, we have
\begin{defi}[Cohomology]\index{cohomology}
The \emph{cohomology} of a cochain complex $C^{\Cdot}$ is
\[
H^i(C^{\Cdot}) = \frac{\ker (d^i: C^i \to C^{i + 1})}{\im (d^{i - 1}: C^{i - 1} \to C^i)}.
\]
An element of $H^i(C^{\Cdot})$ is known as a \term{cohomology class}.
\end{defi}
More names:
\begin{defi}[Cycles and cocycles]\index{cycle}\index{cocycle}
The elements of $\ker d_i$ are the \emph{cycles}, and the elements of $\ker d^i$ are the \emph{cocycles}.
\end{defi}
\begin{defi}[Boundaries and coboundaries]\index{boundary}\index{coboundary}
The elements of $\im d_i$ are the \emph{boundaries}, and the elements of $\im d^i$ are the \emph{coboundaries}.
\end{defi}
As always, we want to talk about maps between chain complexes. These are known as chain maps.
\begin{defi}[Chain map]\index{chain map}
If $(C_{\Cdot}, d_{\Cdot}^C)$ and $(D_{\Cdot}, d_{\Cdot}^D)$ are chain complexes, then a \emph{chain map} $C_{\Cdot} \to D_{\Cdot}$ is a collection of homomorphisms $f_n: C_n \to D_n$ such that $d_n^D \circ f_n = f_{n - 1} \circ d_n^C$. In other words, the following diagram has to commute for all $n$:
\[
\begin{tikzcd}
C_n \ar[r, "f_n"] \ar[d, "d_n^C"] & D_n \ar[d, "d_n^D"]\\
C_{n - 1} \ar[r, "f_{n - 1}"] & D_{n - 1}
\end{tikzcd}
\]
\end{defi}
There is an obvious analogous definition for \term{cochain maps} between cochain complexes.
\begin{lemma}
If $f_{\Cdot}: C_{\Cdot} \to D_{\Cdot}$ is a chain map, then $f_*: H_n(C_{\Cdot}) \to H_n(D_{\Cdot})$ given by $[x] \mapsto [f_n(x)]$ is a well-defined homomorphism, where $x \in C_n$ is any element representing the homology class $[x] \in H_n(C_{\Cdot})$.
\end{lemma}
\begin{proof}
Before we check if it is well-defined, we first need to check if it is defined at all! In other words, we need to check if $f_n(x)$ is a cycle. Suppose $x \in C_n$ is a cycle, i.e.\ $d_n^C(x) = 0$. Then we have
\[
d_n^D(f_n(x)) = f_{n - 1}(d_n^C(x)) = f_{n - 1}(0) = 0.
\]
So $f_n(x)$ is a cycle, and it does represent a homology class.
To check this is well-defined, if $[x] = [y] \in H_n(C_{\Cdot})$, then $x - y = d_{n + 1}^C(z)$ for some $z\in C_{n + 1}$. So $f_n(x) - f_n(y) = f_n (d_{n + 1}^C(z)) = d_{n + 1}^D (f_{n + 1}(z))$ is a boundary. So we have $[f_n(x)] = [f_n(y)] \in H_n(D_{\Cdot})$.
\end{proof}
\subsection{Singular (co)homology}
The idea now is that given any space $X$, we construct a chain complex $C_\Cdot(X)$, and for any map $f: X \to Y$, we construct a chain map $f_\Cdot: C_\Cdot(X) \to C_\Cdot(Y)$. We then take the homology of these, so that we get some homology groups $H_*(X)$ for each space $X$, and a map $f_*: H_*(X) \to H_*(Y)$ for each map $f: X \to Y$.
There are many ways we can construct a chain complex $C_\Cdot(X)$ from a space $X$. The way we are going to define the chain complex is via \emph{singular homology}. The advantage of this definition is that it is obviously a property just of the space $X$ itself, whereas in other definitions, we need to pick, say, a triangulation of the space, and then work hard to show that the homology does not depend on the choice of the triangulation.
The disadvantage of singular homology is that the chain complexes $C_\Cdot(X)$ will be \emph{huge} (and also a bit scary). Except for the case of a point (or perhaps a few points), it is impossible to actually write down what $C_\Cdot(X)$ looks like and use that to compute the homology groups. Instead, we are going to play around with the definition and prove some useful results that help us compute the homology groups indirectly.
Later, for a particular type of spaces known as \emph{CW complexes}, we will come up with a different homology theory where the $C_{\Cdot}(X)$ are actually nice and small, so that we can use it to compute the homology groups directly. We will prove that this is equivalent to singular homology, so that this also provides an alternative method of computing homology groups.
Everything we do can be dualized to talk about cohomology instead. Most of the time, we will just write down the result for singular homology, but analogous results hold for singular cohomology as well. However, later on, we will see that there are operations we can perform on cohomology groups only, which makes them better, and the interaction between homology and cohomology will become interesting when we talk about manifolds at the end of the course.
We start with definitions.
\begin{defi}[Standard $n$-simplex]\index{standard $n$-simplex}
The standard $n$-simplex is
\[
\Delta^n = \left\{(t_0, \cdots, t_n) \in \R^{n + 1} : t_i \geq 0, \sum t_i = 1\right\}.
\]
\end{defi}
\begin{center}
\begin{tikzpicture}
\draw [->] (0, 0) -- (3, 0);
\draw [->] (0, 0) -- (0, 3);
\draw [->] (0, 0) -- (-1.5, -1.5);
\draw [fill=mblue, fill opacity=0.5] (0, 2) -- (2, 0) -- (-.866, -0.866) -- cycle;
\end{tikzpicture}
\end{center}
We notice that $\Delta^n$ has $n + 1$ ``faces''.
\begin{defi}[Face of standard simplex]\index{face of standard simplex}
The \emph{$i$th face} of $\Delta^n$ is
\[
\Delta_i^n = \{(t_0, \cdots, t_n) \in \Delta^n : t_i = 0\}.
\]
\end{defi}
\begin{eg}
The faces of $\Delta^1$ are labelled as follows:
\begin{center}
\begin{tikzpicture}
\draw [->] (-1, 0) -- (3, 0) node [right] {$t_0$};
\draw [->] (0, -1) -- (0, 3) node [above] {$t_1$};
\draw (2, 0) -- (0, 2) node [pos=0.5, anchor = south west] {$\Delta_1$};
\node [circ] at (2, 0) {};
\node [circ] at (0, 2) {};
\node at (2, 0) [below] {$\Delta_1^1$};
\node at (0, 2) [left] {$\Delta_0^1$};
\end{tikzpicture}
\end{center}
\end{eg}
We see that the $i$th face of $\Delta^n$ looks like the standard $(n-1)$-simplex. Of course, it is just homeomorphic to it, with the map given by
\begin{align*}
\delta_i: \Delta^{n - 1} &\to \Delta^n\\
(t_0, \cdots, t_{n - 1}) &\mapsto (t_0, \cdots, t_{i - 1}, 0, t_i, \cdots, t_{n - 1})
\end{align*}
This is a homeomorphism onto $\Delta_i^n$. We will make use of these maps to define our chain complex.
The idea is that we will look at subspaces of $X$ that ``look like'' these standard $n$-simplices.
\begin{defi}[Singular $n$-simplex]\index{singular $n$-simplex}
Let $X$ be a space. Then a \emph{singular $n$-simplex} in $X$ is a map $\sigma: \Delta^n \to X$.
\end{defi}
\begin{eg}
The inclusion of the standard $n$-simplex into $\R^{n+1}$ is a singular simplex, but so is the constant map to any point in $X$. So the singular $n$-simplices can be stupid.
\end{eg}
Given a space $X$, we obtain a \emph{set} of singular $n$-simplices. To talk about homology, we need to have an abelian group. We do the least imaginative thing ever:
\begin{defi}[Singular chain complex]
We let $C_n(X)$ be the free abelian group on the set of singular $n$-simplices in $X$. More explicitly, we have
\[
C_n(X) = \left\{\sum n_\sigma \sigma: \sigma: \Delta^n \to X, n_\sigma \in \Z, \text{only finitely many $n_\sigma$ non-zero}\right\}.
\]
We define $d_n: C_n(X) \to C_{n - 1}(X)$ by
\[
\sigma \mapsto \sum_{i = 0}^n (-1)^i \sigma \circ \delta_i,
\]
and then extending linearly.
\end{defi}
Note that it is essential that we insert those funny negative signs. Indeed, if the signs weren't there, then all terms in $d(d(\sigma))$ would have positive coefficients, and there is no hope that they all cancel. Intuitively, we can think of these signs as specifying the ``orientation'' of the faces. For example, if we have a line
\begin{center}
\begin{tikzpicture}
\draw (0, 0) node [circ] {} -- (2, 0) node [circ] {};
\end{tikzpicture}
\end{center}
then after taking $d$, one vertex would have a positive sign, and the other would have a negative sign.
Now we actually check that this indeed gives us a chain complex. The key tool is the following unexciting result:
\begin{lemma}
If $i < j$, then $\delta_j \circ \delta_i = \delta_i \circ \delta_{j - 1} : \Delta^{n - 2} \to \Delta^n$.
\end{lemma}
\begin{proof}
Both send $(t_0, \cdots, t_{n - 2})$ to
\[
(t_0, \cdots, t_{i - 1}, 0, t_i, \cdots, t_{j - 2}, 0, t_{j - 1}, \cdots, t_{n - 2}).\qedhere
\]
\end{proof}
\begin{cor}
The homomorphism $d_{n - 1} \circ d_n: C_n(X) \to C_{n - 2}(X)$ vanishes.
\end{cor}
\begin{proof}
It suffices to check this on each basis element $\sigma: \Delta^n \to X$. We have
\begin{align*}
d_{n - 1} \circ d_n (\sigma) &= \sum_{i = 0}^{n - 1}(-1)^i \sum_{j = 0}^n (-1)^j \sigma \circ \delta_j \circ \delta_i.
\intertext{We use the previous lemma to split the sum up into $i < j$ and $i \geq j$:}
&= \sum_{i < j} (-1)^{i + j} \sigma \circ \delta_j \circ \delta_i + \sum_{i \geq j} (-1)^{i + j} \sigma \circ \delta_j \circ \delta_i\\
&= \sum_{i < j} (-1)^{i + j} \sigma \circ \delta_i \circ \delta_{j - 1} + \sum_{i \geq j} (-1)^{i + j} \sigma \circ \delta_j \circ \delta_i\\
&= \sum_{i \leq j} (-1)^{i + j + 1} \sigma \circ \delta_i \circ \delta_j + \sum_{i \geq j} (-1)^{i + j} \sigma \circ \delta_j \circ \delta_i\\
&= 0.\qedhere
\end{align*}
\end{proof}
So the data $d_n: C_n (X) \to C_{n - 1}(X)$ is indeed a chain complex. The only thing we can do to a chain complex is to take its homology!
\begin{defi}[Singular homology]\index{singular homology}
The \emph{singular homology} of a space $X$ is the homology of the chain complex $C_\Cdot(X)$:
\[
H_i(X) = H_i(C_{\Cdot}(X), d_{\Cdot}) = \frac{\ker (d_i: C_i(X) \to C_{i - 1}(X))}{ \im(d_{i + 1}: C_{i + 1}(X) \to C_i(X))}.
\]
\end{defi}
We will also talk about the ``dual'' version of this:
\begin{defi}[Singular cohomology]\index{singular cohomology}
We define the dual cochain complex by
\[
C^n(X) = \Hom(C_n(X), \Z).
\]
We let
\[
d^n: C^n(X) \to C^{n + 1}(X)
\]
be the adjoint to $d_{n + 1}$, i.e.
\[
(\varphi: C_n(X) \to \Z) \mapsto (\varphi \circ d_{n + 1}: C_{n + 1}(X) \to \Z).
\]
We observe that
\[
\begin{tikzcd}
0 \ar[r] & C^0(X) \ar[r, "d^0"] & C^1(X) \ar[r] & \cdots
\end{tikzcd}
\]
is indeed a cochain complex, since
\[
d^{n + 1}(d^n(\varphi)) = \varphi \circ d_{n + 1} \circ d_{n + 2} = \varphi \circ 0 = 0.
\]
The \emph{singular cohomology} of $X$ is the cohomology of this cochain complex, i.e.
\[
H^i(X) = H^i(C^{\Cdot}(X), d^{\Cdot}) = \frac{\ker(d^i: C^i(X) \to C^{i + 1}(X))}{\im(d^{i - 1}: C^{i - 1}(X) \to C^i (X))}.
\]
\end{defi}
Note that in general, it is \emph{not} true that $H^n(X) = \Hom(H_n(X), \Z)$. Thus, dualizing and taking homology do not commute with each other. However, we will later come up with a relation between the two objects.
The next thing to show is that maps of spaces induce chain maps, hence maps between homology groups.
\begin{prop}
If $f: X \to Y$ is a continuous map of topological spaces, then the maps
\begin{align*}
f_n: C_n(X) &\to C_n(Y)\\
(\sigma: \Delta^n \to X) &\mapsto (f \circ \sigma: \Delta^n \to Y)
\end{align*}
give a chain map. This induces a map on the homology (and cohomology).
\end{prop}
\begin{proof}
To see that the $f_n$ and $d_n$ commute, we just notice that $f_n$ acts by composing on the left, and $d_n$ acts by composing on the right, and these two operations commute by the associativity of functional composition.
\end{proof}
Now if in addition we have a map $g: Y \to Z$, then we obtain two maps $H_n(X) \to H_n(Z)$ by
\[
\begin{tikzcd}
H_n(X) \ar[rr, "(g \circ f)_*"] \ar[rd, "f_*"] & & H_n(Z)\\
& H_n(Y) \ar[ru, "g_*"]
\end{tikzcd}.
\]
By direct inspection of the formula, we see that this diagram commutes. In equation form, we have
\[
(g \circ f)_* = g_* \circ f_*.
\]
Moreover, we trivially have
\[
(\id_X)_* = \id_{H_n(X)}: H_n(X) \to H_n(X).
\]
Thus we deduce that
\begin{prop}
If $f: X \to Y$ is a homeomorphism, then $f_*: H_n(X) \to H_n(Y)$ is an isomorphism of abelian groups.
\end{prop}
\begin{proof}
If $g: Y \to X$ is an inverse to $f$, then $g_*$ is an inverse to $f_*$, as $f_* \circ g_* = (f \circ g)_* = (\id)_* = \id$, and similarly the other way round.
\end{proof}
If one is taking category theory, what we have shown is that $H_*$ is a functor, and the above proposition is just the usual proof that functors preserve isomorphisms.
This is not too exciting. We will later show that \emph{homotopy equivalences} induce isomorphisms of homology groups, which is much harder.
Again, we can dualize this to talk about cohomology. Applying $\Hom(\ph, \Z)$ to $f_{\Cdot}: C_{\Cdot}(X) \to C_{\Cdot}(Y)$ gives homomorphisms $f^n: C^n(Y) \to C^n(X)$ by mapping
\[
(\varphi: C_n(Y) \to \Z) \mapsto (\varphi \circ f_n: C_n(X) \to \Z).
\]
Note that this map goes the other way! Again, this is a cochain map, and induces maps $f^*: H^n(Y) \to H^n(X)$.
How should we think about singular homology? There are two objects involved --- cycles and boundaries. We will see that cycles in some sense ``detect holes'', and quotienting out by boundaries will help us identify cycles that detect the same hole.
\begin{eg}
We work with the space that looks like this:
\begin{center}
\begin{tikzpicture}
\draw [fill=morange, opacity=0.3] circle [radius=1.5];
\draw circle [radius=1.5];
\draw [fill=white] circle [radius=0.5];
\end{tikzpicture}
\end{center}
Suppose we have a single chain complex $\sigma: \Delta^1 \to X$. Then its boundary $d_1(\sigma) = \sigma(1) - \sigma(0)$. This is in general non-zero, unless we have $\sigma(0) = \sigma(1)$. In other words, this has to be a loop!
\begin{center}
\begin{tikzpicture}
\draw [fill=morange, opacity=0.3] circle [radius=1.5];
\draw circle [radius=1.5];
\draw [fill=white] circle [radius=0.5];
\draw [mblue, thick] (-1, 0) to [out=-80, in=270, looseness=1.2] (0.8, 0) node [right] {$\sigma$} to [out=90, in=80, looseness=1.2] (-1, 0) node [circ] {};
\end{tikzpicture}
\end{center}
So the $1$-cycles represented by just one element are exactly the loops.
How about more complicated cycles? We could have, say, four $1$-simplices $\sigma_1, \sigma_2, \sigma_3, \sigma_4$.
\begin{center}
\begin{tikzpicture}
\draw [fill=morange, opacity=0.3] circle [radius=1.5];
\draw circle [radius=1.5];
\draw [fill=white] circle [radius=0.5];
\draw [mblue, thick] (-1, 0) to [out=-80, in=180] (0, -0.65) node [circ] {};
\draw [mgreen, thick] (0, -0.65) to [out=0, in=270] (0.8, 0) node [circ] {};
\draw [mred, thick] (0.8, 0) to [out=90, in=0] (0, 0.65) node [circ] {};
\draw [morange, thick] (0, 0.65) to [out=180, in=80] (-1, 0) node [circ] {};
\node [circ] at (-1, 0) {};
\node [circ] at (0, -0.65) {};
\node [circ] at (0.8, 0) {};
\node [circ] at (0, 0.65) {};
\node [mblue] at (-0.8, -0.7) {$\sigma_1$};
\node [mgreen] at (0.8, -0.7) {$\sigma_2$};
\node [mred] at (0.8, 0.7) {$\sigma_3$};
\node [morange] at (-0.8, 0.7) {$\sigma_4$};
\end{tikzpicture}
\end{center}
In this case we have
\begin{align*}
\sigma_1(1) &= \sigma_2(0)\\
\sigma_2(1) &= \sigma_3(0)\\
\sigma_3(1) &= \sigma_4(0)\\
\sigma_4(1) &= \sigma_1(0)
\end{align*}
Thus, we have $\sigma_1 + \sigma_2 + \sigma_3 + \sigma_4 \in C_1(X)$. We can think of these cycles as detecting the holes by surrounding them.
However, there are some stupid cycles:
\begin{center}
\begin{tikzpicture}
\draw [fill=morange, opacity=0.3] circle [radius=1.5];
\draw circle [radius=1.5];
\draw [fill=white] circle [radius=0.5];
\draw [mblue, thick] (0.7, 0) to [out=-80, in=270, looseness=1.2] (1.3, 0) to [out=90, in=80, looseness=1.2] (0.7, 0) node [circ] {};
\end{tikzpicture}
\end{center}
These cycles don't really surround anything. The solution is that these cycles are actually boundaries, as it is the boundary of the $2$-simplex that fills the region bounded by the loop (details omitted).
Similarly, the boundaries also allow us to identify two cycles that surround the same hole, so that we don't double count:
\begin{center}
\begin{tikzpicture}
\draw [fill=morange, opacity=0.3] circle [radius=1.5];
\draw circle [radius=1.5];
\draw [fill=white] circle [radius=0.5];
\draw [mblue, thick] (-0.8, 0) to [out=-80, in=270, looseness=1.4] (0.6, 0) to [out=90, in=80, looseness=1.4] (-0.8, 0) node [circ] {};
\draw [mgreen, thick] (-1.3, 0) to [out=-80, in=270, looseness=1.2] (1.1, 0) to [out=90, in=80, looseness=1.2] (-1.3, 0) node [circ] {};
\end{tikzpicture}
\end{center}
This time, the \emph{difference} between the two cycles is the boundary of the $2$-simplex given by the region in between the two loops.
Of course, some work has to be done to actually find a $2$-simplex whose boundary is the difference of the two loops above, and in fact we will have to write the region as the sum of multiple $2$-simplices for this to work. However, this is just to provide intuition, not a formal proof.
\end{eg}
We now do some actual computations. If we want to compute the homology groups directly, we need to know what the $C_\Cdot(X)$ look like. In general, this is intractable, unless we have the very simple case of a point:
\begin{eg}
Consider the one-point space $\pt = \{*\}$. We claim that
\[
H^n(\pt) = H_n(\pt) =
\begin{cases}
\Z & n = 0\\
0 & n > 0
\end{cases}.
\]
To see this, we note that there is always a single singular $n$-simplex $\sigma_n: \Delta^n \to \pt$. So $C_n(\pt) = \Z$, and is generated by $\sigma_n$. Now note that
\[
d_n(\sigma_n) = \sum_{i = 0}^n (-1)^i \sigma_n \delta_i =
\begin{cases}
\sigma_{n - 1} & n\text{ even}\\
0 & n\text{ odd}
\end{cases}.
\]
So the singular chain complex looks like
\[
\begin{tikzcd}
\Z \ar[r, "1"] & \Z\ar[r, "0"] & \Z \ar[r, "1"] & \Z \ar[r, "0"] & \Z \ar[r] & 0.
\end{tikzcd}
\]
The homology groups then follow from direct computation. The cohomology groups are similar.
\end{eg}
This result is absolutely unexciting, and it is also almost the only space whose homology groups we can find at this point. However, we are still capable of proving the following general result:
\begin{eg}
If $X = \coprod_{\alpha \in I} X_\alpha$ is a disjoint union of path components, then each singular simplex must lie in some $X_\alpha$. This implies that
\[
H_n(X) \cong \bigoplus_{\alpha \in I} H_n(X_\alpha).
\]
\end{eg}
Now we know how to compute, say, the homology of three points. How exciting.
\begin{lemma}
If $X$ is path-connected and non-empty, then $H_0(X) \cong \Z$.
\end{lemma}
\begin{proof}
Define a homomorphism $\varepsilon: C_0(X) \to \Z$ given by
\[
\sum n_\sigma \sigma \mapsto \sum n_\sigma.
\]
Then this is surjective. We claim that the composition
\[
\begin{tikzcd}
C_1(X) \ar[r, "d"] & C_0(X) \ar[r, "\varepsilon"] & \Z
\end{tikzcd}
\]
is zero. Indeed, each simplex has two ends, and a $\sigma: \Delta^1 \to X$ is mapped to $\sigma \circ \delta_0 - \sigma \circ \delta_1$, which is mapped by $\varepsilon$ to $1 - 1 = 0$.
Thus, we know that $\varepsilon(\sigma) = \varepsilon (\sigma + d \tau)$ for any $\sigma \in C_0(X)$ and $\tau \in C_1(X)$. So we obtain a well-defined map $\varepsilon: H_0(X) \to \Z$ mapping $[x] \mapsto \varepsilon(x)$, and this is surjective as $X$ is non-empty.
So far, this is true for a general space. Now we use the path-connectedness condition to show that this map is indeed injective. Suppose $\sum n_\sigma \sigma \in C_0(X)$ lies in $\ker \varepsilon$. We choose an $x_0 \in X$. As $X$ is path-connected, for each of $\Delta^0 \to X$ we can choose a path $\tau_\sigma: \Delta^1 \to X$ with $\tau_\sigma \circ \delta_0 = \sigma$ and $\tau_\sigma \circ \delta_1 = x_0$.
Given these $1$-simplices, we can form a $1$-chain $\sum n_\sigma \tau_\sigma \in C_1(X)$, and
\[
d_1\left(\sum n_\sigma \tau_\sigma\right)= \sum n_\sigma(\sigma + x_0) = \sum n_\sigma \cdot \sigma - \left(\sum n_\sigma\right) x_0.
\]
Now we use the fact that $\sum n_\sigma = 0$. So $\sum n_\sigma \cdot \sigma$ is a boundary. So it is zero in $H_0(X)$.
\end{proof}
Combining with the coproduct formula, we have
\begin{prop}
For any space $X$, we have $H_0(X)$ is a free abelian group generated by the path components of $X$.
\end{prop}
These are pretty much the things we can do by hand. To do more things, we need to use some tools.
\section{Four major tools of (co)homology}
We now state four major properties of cohomology. At the end, we will use these properties to compute a lot of homology groups. The proofs are long and boring, so we postpone them until we have gone through the properties and enjoyed some applications of them.
\subsection{Homotopy invariance}
The first result is pretty easy to state, but it is a really powerful result that really drives what we are doing:
\begin{thm}[Homotopy invariance theorem]\index{homotopy invariance theorem}
Let $f \simeq g: X \to Y$ be homotopic maps. Then they induce the same maps on (co)homology, i.e.
\[
f_* = g_*: H_\Cdot(X) \to H_\Cdot(Y)
\]
and
\[
f^* = g^* : H^\Cdot(Y) \to H^\Cdot(X).
\]
\end{thm}
\begin{cor}
If $f: X \to Y$ is a homotopy equivalence, then $f_*: H_{\Cdot}(X) \to H_{\Cdot}(Y)$ and $f^*: H^{\Cdot}(Y) \to H^{\Cdot}(X)$ are isomorphisms.
\end{cor}
\begin{proof}
If $g: Y \to X$ is a homotopy inverse, then
\[
g_* \circ f_* = (g \circ f)_* = (\id_X)_* = \id_{H_{\Cdot}(X)}.
\]
Similarly, we have $f_* \circ g_* = (\id_Y)_* = \id_{H_{\Cdot}(Y)}$. So $f_*$ is an isomorphism with an inverse $g_*$.
The case for cohomology is similar.
\end{proof}
Since we know that $\R^n$ is homotopy equivalent to a point, it immediately follows that:
\begin{eg}
We have
\[
H^n(\R^n) = H_n(\R^n) =
\begin{cases}
\Z & n = 0\\
0 & n > 0
\end{cases}.
\]
\end{eg}
\subsection{Mayer-Vietoris}
The next tool is something that allows us to compute (co)homology by computing the (co)homology of smaller subspaces. Unfortunately, the result doesn't just say we can directly add the cohomologies, or anything like that. Instead, the information is encoded in an \emph{exact sequence}.
\begin{defi}[Exact sequence]\index{exact sequence}
We say a pair of homomorphisms
\[
\begin{tikzcd}
A \ar[r, "f"] & B \ar[r, "g"] & C
\end{tikzcd}
\]
is \emph{exact at $B$} if $\im(f) = \ker(g)$.
We say a sequence
\[
\begin{tikzcd}
\cdots \ar[r] & X_0 \ar[r] & X_1 \ar[r] & X_2 \ar[r] & \cdots
\end{tikzcd}
\]
is \emph{exact} if it is exact at each $X_n$.
\end{defi}
\begin{eg}
If
\[
\begin{tikzcd}
0 \ar[r] & A \ar[r, "f"] & B \ar[r] & 0
\end{tikzcd}.
\]
is exact, then $f$ is an isomorphism.
\end{eg}
\begin{defi}[Short exact sequence]\index{short exact sequence}
A \emph{short exact sequence} is a sequence of the form
\[
\begin{tikzcd}
0 \ar[r] & A \ar[r] & B \ar[r] & C \ar[r] & 0
\end{tikzcd}.
\]
\end{defi}
It is an easy consequence of the first isomorphism theorem that
\begin{lemma}
In a short exact sequence
\[
\begin{tikzcd}
0 \ar[r] & A \ar[r, "f"] & B \ar[r, "g"] & C \ar[r] & 0
\end{tikzcd},
\]
the map $f$ is injective; $g$ is surjective, and $C \cong B/A$.
\end{lemma}
\begin{eg}
Consider the sequence
\[
\begin{tikzcd}
0 \ar[r] & \Z/n\Z \ar[r] & A \ar[r] & \Z/m\Z \ar[r] & 0
\end{tikzcd}
\]
There are many possible values of $A$. A few obvious examples are $A = \Z/nm\Z$ and $A = \Z/n\Z \oplus \Z/m\Z$. If $n$ and $m$ are not coprime, then these are different groups.
\end{eg}
Many of the important theorems for computing homology groups tell us that the homology groups of certain spaces fall into some exact sequence, and we have to try figure out what the groups are.
\begin{thm}[Mayer-Vietoris theorem]\index{Mayer-Vietoris theorem}
Let $X = A \cup B$ be the union of two open subsets. We have inclusions
\[
\begin{tikzcd}
A \cap B \ar[r, "i_A", hook] \ar[d, "i_B", hook] & A \ar[d, hook, "j_A"]\\
B \ar[r, hook, "j_B"] & X
\end{tikzcd}.
\]
Then there are homomorphisms $\partial_{MV}: H_n(X) \to H_{n - 1}(A \cap B)$ such that the following sequence is exact:
\[
\begin{tikzcd}
\ar[r, "\partial_{MV}"] & H_n(A \cap B) \ar[r, "i_{A*} \oplus i_{B*}"] & H_n(A) \oplus H_n(B) \ar[r, "j_{A*} - j_{B*}"] & H_n(X)\ar[out=0, in=180, looseness=2, overlay, lld, "\partial_{MV}"']\\
& H_{n - 1}(A \cap B) \ar[r, "i_{A*} \oplus i_{B*}"] & H_{n - 1}(A) \oplus H_{n - 1}(B) \ar[r, "j_{A*} - j_{B*}"] & H_{n - 1}(X) \ar [r] & \cdots\\
&\cdots \ar[r] & H_0(A) \oplus H_0(B) \ar[r, "j_{A*} - j_{B*}"] & H_0(X) \ar [r] & 0
\end{tikzcd}
\]
Furthermore, the Mayer-Vietoris sequence is \emph{natural}, i.e.\ if $f: X = A\cup B \to Y = U \cup V$ satisfies $f(A) \subseteq U$ and $f(B) \subseteq V$, then the diagram
\[
\begin{tikzcd}
H_{n + 1}(X) \ar[r, "\partial_{MV}"] \ar[d, "f_*"] & H_{n}(A \cap B) \ar[r, "i_{A*} \oplus i_{B*}"] \ar[d, "f|_{A \cap B*}"] & H_n(A) \oplus H_n(B) \ar[r, "j_{A*} - j_{B*}"] \ar[d, "f|_{A*} \oplus f|_{B*}"] & H_n(X) \ar[d, "f_*"]\\
H_{n + 1}(Y) \ar[r, "\partial_{MV}"] & H_{n}(U \cap V) \ar[r, "i_{U*} \oplus i_{V*}"] & H_n(U) \oplus H_n(V) \ar[r, "j_{U*} - j_{V*}"] & H_n(Y)
\end{tikzcd}
\]
commutes.
\end{thm}
For certain elements of $H_n(X)$, we can easily specify what $\partial_{MV}$ does to it. The meat of the proof is to show that every element of $H_n(X)$ can be made to look like that. If $[a + b] \in H_n(X)$ is such that $a \in C_n(A)$ and $b \in C_n(B)$, then the map $\partial_{MV}$ is specified by
\[
\partial_{MV}([a + b]) = [d_n(a)] = [-d_n(b)] \in H_{n - 1}(A \cap B).
\]
To see this makes sense, note that we have $d_n(a + b) = 0$. So $d_n(a) = - d_n(b)$. Moreover, since $a \in C_n(A)$, we have $d_n(a) \in C_{n - 1}(A)$. Similarly, $d_n(b) \in C_{n - 1}(B)$. So $d_n(a) = - d_n(b) \in C_{n - 1}(A) \cap C_{n - 1}(B) = C_{n - 1}(A \cap B)$.
\subsection{Relative homology}
The next result again gives us another exact sequence. This time, the exact sequence involves another quantity known as \emph{relative homology}.
\begin{defi}[Relative homology]\index{relative homology}
Let $A \subseteq X$. We write $i: A \to X$ for the inclusion map. Then the map $i_n: C_n(A) \to C_n(X)$ is injective as well, and we write
\[
C_n(X, A) = \frac{C_n(X)}{C_n(A)}.
\]
The differential $d_n: C_n(X) \to C_{n - 1}(X)$ restricts to a map $C_n(A) \to C_{n - 1}(A)$, and thus gives a well-defined differential $d_n: C_n(X, A) \to C_{n - 1}(X, A)$, sending $[c] \mapsto [d_n(c)]$. The \emph{relative homology} is given by
\[
H_n(X, A) = H_n(C_{\Cdot}(X, A)).
\]
\end{defi}
We think of this as chains in $X$ where we ignore everything that happens in $A$.
\begin{thm}[Exact sequence for relative homology]\index{exact sequence for relative homology}
There are homomorphisms $\partial: H_n(X, A) \to H_{n - 1}(A)$ given by mapping
\[
\big[[c]\big] \mapsto [d_n c].
\]
This makes sense because if $c \in C_n(X)$, then $[c] \in C_n(X)/C_n(A)$. We know $[d_n c] = 0 \in C_{n - 1}(X)/C_{n - 1}(A)$. So $d_n c \in C_{n - 1}(A)$. So this notation makes sense.
Moreover, there is a long exact sequence
\[
\begin{tikzcd}
\cdots \ar[r, "\partial"] & H_n(A) \ar[r, "i_*"] & H_n(X) \ar[r, "q_*"] & H_n(X, A)\ar[out=0, in=180, looseness=2, overlay, lld, "\partial"']\\
& H_{n - 1}(A) \ar[r, "i_*"] & H_{n - 1}(X) \ar[r, "q_*"] & H_{n - 1}(X, A) \ar [r] & \cdots\\
&\cdots \ar[r] & H_0(X) \ar[r, "q_*"] & H_0(X, A) \ar [r] & 0
\end{tikzcd},
\]
where $i_*$ is induced by $i: C_{\Cdot}(A) \to C_{\Cdot}(X)$ and $q_*$ is induced by the quotient $q: C_{\Cdot}(X) \to C_{\Cdot}(X, A)$.
\end{thm}
This, again, is natural. To specify the naturality condition, we need the following definition:
\begin{defi}[Map of pairs]\index{map of pairs}
Let $(X, A)$ and $(Y, B)$ be topological spaces with $A \subseteq X$ and $B \subseteq Y$. A \emph{map of pairs} is a map $f: X \to Y$ such that $f(A) \subseteq B$.
\end{defi}
Such a map induces a map $f_*: H_n(X, A) \to H_n(Y, B)$, and the exact sequence for relative homology is natural for such maps.
\subsection{Excision theorem}
Now the previous result is absolutely useless, because we just introduced a new quantity $H_n(X, A)$ we have no idea how to compute again. The main point of relative homology is that we want to think of $H_n(X, A)$ as the homology of $X$ when we ignore $A$. Thus, one might expect that the relative homology does not depend on the things ``inside $A$''. However, it is not true in general that, say, $H_n(X, A) = H_n(X \setminus A)$. Instead, what we are allowed to do is to remove subspaces of $A$ that are ``not too big''. This is given by excision:
\begin{thm}[Excision theorem]\index{excision theorem}
Let $(X, A)$ be a pair of spaces, and $Z \subseteq A$ be such that $\overline{Z} \subseteq \mathring{A}$ (the closure is taken in $X$). Then the map
\[
H_n(X \setminus Z, A \setminus Z) \to H_n(X, A)
\]
is an isomorphism.
\end{thm}
\begin{center}
\begin{tikzpicture}
\draw [draw=black, fill=morange, opacity=0.5] (0, 0) rectangle (3, 2);
\node at (0.5, 0.5) {$X$};
\draw [draw=black, fill=mblue, opacity=0.6] (1.8, 1) circle [radius=0.7];
\node at (2.6, 1.7) {$A$};
\draw [draw=black, fill=mred, opacity=0.8, decorate, decoration={snake}] (1.75, 0.95) circle [radius=0.4] node [white] {$Z$};
\end{tikzpicture}
\end{center}
While we've only been talking about homology, everything so far holds analogously for cohomology too. It is again homotopy invariant, and there is a Mayer-Vietoris sequence (with maps $\partial_{MV}: H^n(A \cap B) \to H^{n + 1}(X)$). The relative cohomology is defined by $C^{\Cdot}(X, A) = \Hom(C_{\Cdot}(X, A), \Z)$ and so $H^*(X, A)$ is the cohomology of that. Similarly, excision holds.
\subsection{Applications}
We now use all these tools to do lots of computations. The first thing we compute will be the homology of spheres.
\begin{thm}
We have
\[
H_i(S^1) =
\begin{cases}
\Z & i = 0, 1\\
0 & \text{otherwise}
\end{cases}.
\]
\end{thm}
\begin{proof}
We can split $S^1$ up as
\begin{center}
\begin{tikzpicture}
\draw circle [radius=1];
\draw [red] (1.105, -0.4673) arc(-22.9:202.9:1.2);
\node [red, above] at (0, 1.2) {$A$};
\draw [blue] (1.279, 0.545) arc(22.9:-202.9:1.4);
\node [blue, below] at (0, -1.4) {$B$};
\node [circ] at (1, 0) {};
\node at (1, 0) [left] {$q$};
\node [circ] at (-1, 0) {};
\node at (-1, 0) [right] {$p$};
\end{tikzpicture}
\end{center}
We want to apply Mayer-Vietoris. We have
\[
A \cong B \cong \R \simeq *,\quad A \cap B \cong \R \coprod \R \simeq \{p\} \coprod \{q\}.
\]
We obtain
\[
\begin{tikzcd}[row sep=tiny]
& 0 \ar[d, equals] & 0 \ar[d, equals] \\
\cdots \ar[r] & H_1(A \cap B) \ar[r] & H_1(A) \oplus H_1(B) \ar[r] & H_1(S^1)\ar[out=0, in=180, looseness=2, overlay, lldd, "\partial"']\\
\vphantom{a}\\
& H_0(A \cap B) \ar[r, "i_{A*} \oplus i_{B*}"] & H_0(A) \oplus H_0(B) \ar[r] & H_0(S^1) \ar [r] & 0\\
& \Z \oplus \Z \ar[u, equals] & \Z \oplus Z \ar[u, equals] & \Z \ar[u, equals]
\end{tikzcd}
\]
Notice that the map into $H_1(S^1)$ is zero. So the kernel of $\partial$ is trivial, i.e.\ $\partial$ is an injection. So $H_1(S^1)$ is isomorphic to the image of $\partial$, which is, by exactness, the kernel of $i_{A*} \oplus i_{B*}$. So we want to know what this map does.
We know that $H_0(A \cap B) \cong \Z \oplus \Z$ is generated by $p$ and $q$, and the inclusion map sends each of $p$ and $q$ to the unique connected components of $A$ and $B$. So the homology classes are both sent to $(1, 1) \in H_0(A) \oplus H_0(B) \cong \Z \oplus \Z$. We then see that the kernel of $i_{A*} \oplus i_{B*}$ is generated by $(p - q)$, and is thus isomorphic to $\Z$. So $H_1(S^1) \cong \Z$.
By looking higher up the exact sequence, we see that all other homology groups vanish.
\end{proof}
We can do the sphere in general in the same way.
\begin{thm}
For any $n \geq 1$, we have
\[
H_i(S^n) =
\begin{cases}
\Z & i = 0, n\\
0 & \text{otherwise}
\end{cases}.
\]
\end{thm}
\begin{proof}
We again cut up $S^n$ as
\begin{align*}
A &= S^n \setminus \{N\} \cong \R^n \simeq *,\\
B &= S^n \setminus \{S\} \cong \R^n \simeq *,
\end{align*}
where $N$ and $S$ are the north and south poles. Moreover, we have
\[
A\cap B \cong \R \times S^{n - 1} \simeq S^{n - 1}
\]
So we can ``induct up'' using the Mayer-Vietoris sequence:
\[
\begin{tikzcd}
\cdots \ar[r] & H_i(S^{n - 1}) \ar[r] & H_i(*) \oplus H_i(*) \ar[r] & H_i(S^n)\ar[out=0, in=180, looseness=2, overlay, lld, "\partial"]\\
& H_{i - 1}(S^{n - 1}) \ar[r] & H_{i - 1}(*) \oplus H_{i - 1}(*) \ar[r] & H_{i - 1}(S^n) \ar [r] & \cdots
\end{tikzcd}
\]
Now suppose $n \geq 2$, as we already did $S^1$ already. If $i > 1$, then $H_i(*) = 0 = H_{i - 1}(*)$. So the Mayer-Vietoris map
\[
\begin{tikzcd}
H_i(S^n) \ar[r, "\partial"] & H_{i - 1}(S^{n - 1})
\end{tikzcd}
\]
is an isomorphism.
All that remains is to look at $i = 0, 1$. The $i = 0$ case is trivial. For $i = 1$, we look at
\[
\begin{tikzcd}[row sep=tiny]
& & 0 \ar[d, equals] \\
& \cdots \ar[r] & H_1(*) \oplus H_1(*) \ar[r] & H_1(S^n)\ar[out=0, in=180, looseness=2, overlay, lldd, "\partial"]\\
\vphantom{a} \\
& H_0(S^{n - 1}) \ar[r, "f"] & H_0(*) \oplus H_0(*) \ar[r] & H_0(S^n) \ar [r] & 0\\
& \Z \ar[u, equals] & \Z \oplus \Z \ar[u, equals] & \Z \ar[u, equals]
\end{tikzcd}
\]
To conclude that $H_1(S^n)$ is trivial, it suffices to show that the map $f$ is injective. By picking the canonical generators, it is given by $1 \mapsto (1, 1)$. So we are done.
\end{proof}
\begin{cor}
If $n \not= m$, then $S^{n - 1} \not\simeq S^{m - 1}$, since they have different homology groups.
\end{cor}
\begin{cor}
If $n \not= m$, then $\R^n \not \cong \R^m$.
\end{cor}
Now suppose we have a map $f: S^n \to S^n$. As always, it induces a map $f_*: H_n(S^n) \to H_n(S^n)$. Since $H_n (S^n) \cong \Z$, we know the map $f_*$ is given by multiplication by some integer. We call this the \emph{degree} of the map $f$.
\begin{defi}[Degree of a map]\index{degree of map}
Let $f: S^n \to S^n$ be a map. The \emph{degree} $\deg(f)$ is the unique integer such that under the identification $H_n(S^n) \cong \Z$, the map $f_*$ is given by multiplication by $\deg(f)$.
\end{defi}
In particular, for $n = 1$, the degree is the winding number.
Note that there are two ways we can identify $H_n(S^n)$ and $\Z$, which differ by a sign. For the degree to be well-defined, and not just defined up to a sign, we must ensure that we identify both $H_n(S^n)$ in the same way.
Using the degree, we can show that certain maps are \emph{not} homotopic to each other. We first note the following elementary properties:
\begin{prop}\leavevmode
\begin{enumerate}
\item $\deg(\id_{S^n}) = 1$.
\item If $f$ is not surjective, then $\deg(f) = 0$.
\item We have $\deg(f\circ g) = (\deg f)(\deg g)$.
\item Homotopic maps have equal degrees.
\end{enumerate}
\end{prop}
\begin{proof}\leavevmode
\begin{enumerate}
\item Obvious.
\item If $f$ is not surjective, then $f$ can be factored as
\[
\begin{tikzcd}
S^n \ar[r, "f"] & S^n \setminus \{p\} \ar[r, hook] & S^n
\end{tikzcd},
\]
where $p$ is some point not in the image of $f$. But $S^n \setminus \{p\}$ is contractible. So $f_*$ factors as
\[
\begin{tikzcd}
f_*: H_n(S^n) \ar[r] & H_n(*) = 0 \ar[r] & H_n(S^n)
\end{tikzcd}.
\]
So $f_*$ is the zero homomorphism, and is thus multiplication by $0$.
\item This follows from the functoriality of $H_n$.
\item Obvious as well.\qedhere
\end{enumerate}
\end{proof}
As a corollary, we obtain the renowned Brouwer's fixed point theorem, a highly non-constructive fixed point existence theorem proved by a constructivist.
\begin{cor}[Brouwer's fixed point theorem]\index{Brouwer's fixed point theorem}
Any map $f: D^n \to D^n$ has a fixed point.\index{fixed point}
\end{cor}
\begin{proof}
Suppose $f$ has no fixed point. Define $r: D^n \to S^{n - 1} = \partial D^n$ by taking the intersection of the ray from $f(x)$ through $x$ with $\partial D^n$. This is continuous.
\begin{center}
\begin{tikzpicture}
\draw circle [radius=2cm];
\node [circ] at (-0.4, -0.3) {};
\node at (-0.4, -0.3) [below] {$x$};
\node [circ] at (0.4, 0.3) {};
\node at (0.4, 0.3) [right] {$f(x)$};
\draw (0.4, 0.3) -- (-1.6, -1.2);
\node at (-1.6, -1.2) [circ] {};
\node at (-1.6, -1.2) [anchor = north east] {$r(x)$};
\end{tikzpicture}
\end{center}
Now if $x \in \partial D^n$, then $r(x) = x$. So we have a map
\[
\begin{tikzcd}
S^{n - 1} = \partial D^n \ar[r, "i"] & D^n \ar[r, "r"] & \partial D^n = S^{n - 1}
\end{tikzcd},
\]
and the composition is the identity. This is a contradiction --- contracting $D^n$ to a point, this gives a homotopy from the identity map $S^{n - 1} \to S^{n - 1}$ to the constant map at a point. This is impossible, as the two maps have different degrees.
\end{proof}
A more manual argument to show this would be to apply $H_{n - 1}$ to the chain of maps above to obtain a contradiction.
We know there is a map of degree $1$, namely the identity, and a map of degree $0$, namely the constant map. How about other degrees?
\begin{prop}
A reflection $r: S^n \to S^n$ about a hyperplane has degree $-1$. As before, we cover $S^n$ by
\begin{align*}
A &= S^n \setminus \{N\} \cong \R^n \simeq *,\\
B &= S^n \setminus \{S\} \cong \R^n \simeq *,
\end{align*}
where we suppose the north and south poles lie in the hyperplane of reflection. Then both $A$ and $B$ are invariant under the reflection. Consider the diagram
\[
\begin{tikzcd}
H_n(S^n) \ar[r, "\partial_{MV}", "\sim"'] \ar[d, "r_*"] & H_{n - 1}(A \cap B) \ar[d, "r_*"] & H_{n - 1}(S^{n - 1}) \ar[l, "\sim"] \ar[d, "r_*"]\\
H_n(S^n) \ar[r, "\partial_{MV}", "\sim"'] & H_{n - 1}(A \cap B) & H_{n - 1}(S^{n - 1}) \ar[l, "\sim"]
\end{tikzcd}
\]
where the $S^{n - 1}$ on the right most column is given by contracting $A \cap B$ to the equator. Note that $r$ restricts to a reflection on the equator. By tracing through the isomorphisms, we see that $\deg(r) = \deg(r|_{\mathrm{equator}})$. So by induction, we only have to consider the case when $n = 1$. Then we have maps
\[
\begin{tikzcd}
0 \ar[r] & H_1(S^1) \ar[r, "\partial_{MV}", "\sim"'] \ar[d, "r_*"] & H_0(A \cap B) \ar[d, "r_*"] \ar[r] & H_0(A) \oplus H_0(B) \ar[d, "r_* \oplus r_*"]\\
0 \ar[r] & H_1(S^1) \ar[r, "\partial_{MV}", "\sim"'] & H_0(A \cap B) \ar[r] & H_0(A) \oplus H_0(B)
\end{tikzcd}
\]
Now the middle vertical map sends $p \mapsto q$ and $q \mapsto p$. Since $H_1(S^1)$ is given by the kernel of $H_0(A \cap B) \to H_0(A) \oplus H_0(B)$, and is generated by $p - q$, we see that this sends the generator to its negation. So this is given by multiplication by $-1$. So the degree is $-1$.
\end{prop}
\begin{cor}
The antipodal map $a: S^n \to S^n$ given by
\[
a(x_1, \cdots, x_{n + 1}) = (-x_1, \cdots, -x_{n + 1})
\]
has degree $(-1)^{n + 1}$ because it is a composition of $(n + 1)$ reflections.
\end{cor}
\begin{cor}[Hairy ball theorem]\index{Hairy ball theorem}
$S^n$ has a nowhere $0$ vector field iff $n$ is odd. More precisely, viewing $S^n \subseteq \R^{n + 1}$, a vector field on $S^n$ is a map $v: S^n \to \R^{n + 1}$ such that $\bra v(x), x\ket = 0$, i.e.\ $v(x)$ is perpendicular to $x$.
\end{cor}
\begin{proof}
If $n$ is odd, say $n = 2k - 1$, then
\[
v(x_1, y_1, x_2, y_2, \cdots, x_k, y_k) = (y_1, -x_1, y_2, -x_2, \cdots, y_k, -x_k)
\]
works.
Conversely, if $v: S^n \to \R^{n + 1} \setminus \{0\}$ is a vector field, we let $w = \frac{v}{\abs{v}}: S^n \to S^n$. We can construct a homotopy from $w$ to the antipodal map by ``linear interpolation'', but in a way so that it stays on the sphere. We let
\begin{align*}
H: [0, \pi] \times S^n &\to S^n\\
(t, x) &\mapsto \cos(t) x + \sin(t) w(x)
\end{align*}
Since $w(x)$ and $x$ are perpendicular, it follows that this always has norm $1$, so indeed it stays on the sphere.
Now we have
\[
H(0, x) = x,\quad H(\pi, x) = -x.
\]
So this gives a homotopy from $\id$ to $a$, which is a contradiction since they have different degrees.
\end{proof}
So far, we've only been talking about spheres all the time. We now move on to something different.
\begin{eg}
Let $K$ be the Klein bottle. We cut them up by
\begin{center}
\begin{tikzpicture}
\draw [->-=0.55, mred] (0, 0) -- (3, 0);
\draw [->-=0.55, mred] (3, 3) -- (0, 3);
\draw [->-=0.55, mblue] (3, 0) -- (3, 3);
\draw [->-=0.55, mblue] (0, 0) -- (0, 3);
\fill [morange, opacity=0.5] (0, 0) rectangle (3, 1.2);
\fill [morange, opacity=0.5] (0, 1.8) rectangle (3, 3);
\node [right, morange] at (3, 2.5) {$A$};
\fill [mgreen, opacity=0.5] (0, 0.8) rectangle (3, 2.2);
\node [right, mgreen] at (3, 1.5) {$B$};
\foreach \n in {0,1,2,3,4,5,6,7,8,9,10,11,12,13} {
\begin{scope}[shift={(0.15 + 0.2 * \n, 0.9)}]
\fill (0, 0) -- (0.05, 0) -- (0.1, 0.1) -- (0.05, 0.2) -- (0, 0.2) -- (0.05, 0.1) -- cycle;
\end{scope}
\begin{scope}[shift={(0.15 + 0.2 * \n, 1.9)}]
\fill (0, 0) -- (0.05, 0) -- (0.1, 0.1) -- (0.05, 0.2) -- (0, 0.2) -- (0.05, 0.1) -- cycle;
\end{scope}
}
\end{tikzpicture}
\end{center}
Then both $A$ and $B$ are cylinders, and their intersection is two cylinders:
\begin{center}
\begin{tikzpicture}
\fill [morange, opacity=0.5] (0, 0) rectangle (3, 2.6);
\draw [->-=0.55, mblue] (3, 0) -- (3, 2.6);
\draw [->-=0.55, mblue] (0, 0) -- (0, 2.6);
\draw [->-=0.55, mred] (3, 1.3) -- (0, 1.3);
\fill [mgreen, opacity=0.5] (0, 0) rectangle (3, 0.4);
\fill [mgreen, opacity=0.5] (0, 2.2) rectangle (3, 2.6);
\foreach \n in {0,1,2,3,4,5,6,7,8,9,10,11,12,13} {
\begin{scope}[shift={(0.15 + 0.2 * \n, 0.1)}]
\fill (0, 0) -- (0.05, 0) -- (0.1, 0.1) -- (0.05, 0.2) -- (0, 0.2) -- (0.05, 0.1) -- cycle;
\end{scope}
\begin{scope}[shift={(0.2 + 0.2 * \n, 2.3)}]
\fill (0, 0) -- (0.05, 0) -- (0, 0.1) -- (0.05, 0.2) -- (0, 0.2) -- (-0.05, 0.1) -- cycle;
\end{scope}
}
\node at (1.5, -0.1) [morange, below] {$A$};
\fill [morange, opacity=0.5] (4, 0.6) rectangle (7, 1);
\fill [morange, opacity=0.5] (4, 1.6) rectangle (7, 2);
\fill [mgreen, opacity=0.5] (4, 0.6) rectangle (7, 2);
\draw [->-=0.57, mblue] (4, 0.6) -- (4, 2);
\draw [->-=0.57, mblue] (7, 0.6) -- (7, 2);
\foreach \n in {0,1,2,3,4,5,6,7,8,9,10,11,12,13} {
\begin{scope}[shift={(4.15 + 0.2 * \n, 0.7)}]
\fill (0, 0) -- (0.05, 0) -- (0.1, 0.1) -- (0.05, 0.2) -- (0, 0.2) -- (0.05, 0.1) -- cycle;
\end{scope}
\begin{scope}[shift={(4.15 + 0.2 * \n, 1.7)}]
\fill (0, 0) -- (0.05, 0) -- (0.1, 0.1) -- (0.05, 0.2) -- (0, 0.2) -- (0.05, 0.1) -- cycle;
\end{scope}
}
\node at (5.5, -0.1) [mgreen, below] {$B$};
\begin{scope}[shift={(4, 0)}]
\fill [morange, opacity=0.5] (4, 0.6) rectangle (7, 1);
\fill [morange, opacity=0.5] (4, 1.6) rectangle (7, 2);
\fill [mgreen, opacity=0.5] (4, 0.6) rectangle (7, 1);
\fill [mgreen, opacity=0.5] (4, 1.6) rectangle (7, 2);
\draw [->-=0.8, mblue] (4, 0.6) -- (4, 1) node [left, pos=0.5] {$s$};
\draw [->-=0.8, mblue] (7, 0.6) -- (7, 1);
\draw [->-=0.8, mblue] (4, 1.6) -- (4, 2) node [left, pos=0.5] {$t$};
\draw [->-=0.8, mblue] (7, 1.6) -- (7, 2);
\foreach \n in {0,1,2,3,4,5,6,7,8,9,10,11,12,13} {
\begin{scope}[shift={(4.15 + 0.2 * \n, 0.7)}]
\fill (0, 0) -- (0.05, 0) -- (0.1, 0.1) -- (0.05, 0.2) -- (0, 0.2) -- (0.05, 0.1) -- cycle;
\end{scope}
\begin{scope}[shift={(4.15 + 0.2 * \n, 1.7)}]
\fill (0, 0) -- (0.05, 0) -- (0.1, 0.1) -- (0.05, 0.2) -- (0, 0.2) -- (0.05, 0.1) -- cycle;
\end{scope}
}
\node at (5.5, -0.1) [mgreen!50!morange, below] {$A \cap B$};
\end{scope}
\end{tikzpicture}
\end{center}
We have a long exact sequence
\[
\begin{tikzcd}
0 \ar[r] & H_2(K) \ar[r] & H_1(A \cap B) \ar[r, "{(i_A, i_B)}"] & H_1(A) \oplus H_1(B) \ar[lld, "j_A - j_B"', out=0, in=180, looseness=2, overlay] & \vphantom{0}\\
& H_1(K) \ar[r] & H_0(A \cap B) \ar[r, "{(i_A, i_B)}"] & H_0(A) \oplus H_0(B) \ar[lld, "j_A - j_B"', out=0, in=180, looseness=2, overlay]\\
& H_0(K) \ar[r] & 0
\end{tikzcd}
\]
We plug in the numbers to get
\[
\begin{tikzcd}
0 \ar[r] & H_2(K) \ar[r] & \Z \oplus \Z \ar[r, "{(i_A, i_B)}"] & \Z \oplus \Z \ar[dll, "j_A - j_B"', out=0, in=180, looseness=2, overlay] & \vphantom{0}\\
& H_1(K) \ar[r] & \Z \oplus \Z \ar[r, "{(i_A, i_B)}"] & \Z \oplus \Z \ar[r, "j_A - j_B"] & \Z \ar[r] & 0
\end{tikzcd}
\]
Now let's look at the first $(i_A, i_B)$ map. We suppose each $H_1$ is generated by the left-to-right loops. Then we have
\begin{align*}
i_A(s) &= -1\\
i_A(t) &= 1\\
i_B(s) &= 1\\
i_B(t) &= 1
\end{align*}
So the matrix of $(i_A, i_B)$ is
\[
\begin{pmatrix}
-1 & 1\\
1 & 1
\end{pmatrix}.
\]
This has determinant $2$, so it is injective and has trivial kernel. So we must have $H_2(K) = 0$. We also have
\[
\frac{H_1(A) \oplus H_1(B)}{\im(i_A, i_B)} = \frac{\bra a, b\ket}{\bra -a + b, a + b\ket} = \frac{\bra a\ket}{\bra 2a\ket} \cong \Z/2\Z.
\]
Now the second $(i_A, i_B)$ is given by
\[
\begin{pmatrix}
1 & 1\\
1 & 1
\end{pmatrix},
\]
whose kernel is isomorphic to $\Z$. So we have
\[
\begin{tikzcd}
0 \ar[r] & \Z/2\Z \ar[r] & H_1(K) \ar[r] & \Z \ar[r] & 0
\end{tikzcd}.
\]
So we have $H_1(K) \cong \Z \oplus \Z/2\Z$.
\end{eg}
Finally, we look at some applications of relative homology.
\begin{lemma}
Let $M$ be a $d$-dimensional manifold (i.e.\ a Hausdorff, second-countable space locally homeomorphic to $\R^d$). Then
\[
H_n(M, M \setminus \{x\}) \cong
\begin{cases}
\Z & n = d\\
0 & \text{otherwise}
\end{cases}.
\]
This is known as the \term{local homology}.
\end{lemma}
This gives us a homological way to define the dimension of a space.
\begin{proof}
Let $U$ be an open neighbourhood isomorphic to $\R^d$ that maps $x \mapsto 0$. We let $Z = M\setminus U$. Then
\[
\overline{Z} = Z \subseteq M \setminus \{x\} = \mathring{(M \setminus \{x\})}.
\]
So we can apply excision, and get
\[
H_n(U, U \setminus \{x\}) = H_n(M\setminus Z, (M\setminus \{x\})\setminus \Z) \cong H_n(M, M\setminus \{x\}).
\]
So it suffices to do this in the case $M \cong \R^d$ and $x = 0$. The long exact sequence for relative homology gives
\[
\begin{tikzcd}
H_n(\R^d) \ar[r] & H_n(\R^d, \R^d \setminus \{0\}) \ar[r] & H_{n - 1}(\R^d \setminus \{0\}) \ar[r] & H_{n - 1}(\R^d)
\end{tikzcd}.
\]
Since $H_n(\R^d) = H_{n - 1}(\R^d) = 0$ for $n \geq 2$ large enough, it follows that
\[
H_n(\R^d, \R^d \setminus \{0\}) \cong H_{n - 1}(\R^d \setminus \{0\}) \cong H_{n - 1}(S^{d - 1}),
\]
and the result follows from our previous computation of the homology of $S^{d - 1}$. We will have to check the lower-degree terms manually, but we shall not.
\end{proof}
These relative homology groups are exactly the homology group of spheres. So we might be able to define a degree-like object for maps between manifolds. However, there is the subtlety that the identification with $\Z$ is not necessarily canonical, and this involves the problem of orientation. For now, we will just work with $S^n$.
\begin{defi}[Local degree]\index{local degree}
Let $f: S^d \to S^d$ be a map, and $x \in S^d$. Then $f$ induces a map
\[
f_*: H_d(S^d, S^d \setminus \{x\}) \to H_d(S^d, S^d \setminus \{f(x)\}).
\]
We identify $H_d(S^d, S^d \setminus \{x\}) \cong H_d(S^d) \cong \Z$ for any $x$ via the inclusion $H_d(S^d) \to H_d(S^d, S^d \setminus \{x\})$, which is an isomorphism by the long exact sequence. Then $f_*$ is given by multiplication by a constant $\deg(f)_x$, called the \emph{local degree} of $f$ at $x$.
\end{defi}
What makes the local degree useful is that we can in fact compute the degree of a map via local degrees.
\begin{thm}
Let $f: S^d \to S^d$ be a map. Suppose there is a $y \in S^d$ such that
\[
f^{-1}(y) = \{x_1, \cdots, x_k\}
\]
is finite. Then
\[
\deg (f) = \sum_{i = 1}^k \deg(f)_{x_i}.
\]
\end{thm}
\begin{proof}
Note that by excision, instead of computing the local degree at $x_i$ via $H_d(S^d, S^d \setminus \{x\})$, we can pick a neighbourhood $U_i$ of $x_i$ and a neighbourhood $V$ of $x_i$ such that $f(U_i) \subseteq V$, and then look at the map
\[
f_*: H_d(U_i, U_i \setminus \{x_i\}) \to H_d(V, V \setminus \{y\})
\]
instead. Moreover, since $S^d$ is Hausdorff, we can pick the $U_i$ such that they are disjoint. Consider the huge commutative diagram:
\[
\begin{tikzcd}[row sep=large]
H_d(S^d) \ar[r, "f_*"] \ar[d] & H_d(S^d) \ar[d, "\sim"]\\
H_d(S^d, S^d \setminus \{x_1, \cdots, x_k\}) \ar[r, "f_*"] & H_d(S^d, S^d \setminus \{y\})\\
H_d\left(\coprod U_i, \coprod (U_i \setminus x_i)\right) \ar[u, "\text{excision}"]\\
\bigoplus_{i = 1}^ k H_d(U_i, U_i \setminus x_i) \ar[r, "\bigoplus f_*"] \ar[u, "\sim"] & H_d(V, V\setminus \{y\}) \ar[uu, "\sim"]
\end{tikzcd}
\]
Consider the generator $1$ of $H_d(S^d)$. By definition, its image in $H_d(S^d)$ is $\deg(f)$. Also, its image in $\bigoplus H_d(U_i, U_i \setminus \{x_i\})$ is $(1, \cdots, 1)$. The bottom horizontal map sends this to $\sum \deg(f)_{x_i}$. So by the isomorphisms, it follows that
\[
\deg(f) = \sum_{i = 1}^k \deg(f)_x.\qedhere
\]
\end{proof}
\subsection{Repaying the technical debt}
Finally, we prove all those theorems we stated without proof.
\subsubsection*{Long exact sequence of relative homology}
We start with the least bad one. In relative homology, we had a short exact sequence of chain complexes
\[
\begin{tikzcd}
0 \ar[r] & C_\Cdot(A) \ar[r] & C_\Cdot(X) \ar[r] & C_\Cdot(X, A) \ar[r] & 0.
\end{tikzcd}
\]
The claim is that when we take the homology groups of this, we get a long exact sequence of homology groups. This is in fact a completely general theorem.
\begin{thm}[Snake lemma]\index{snake lemma}
Suppose we have a short exact sequence of complexes
\[
\begin{tikzcd}
0 \ar[r] & A_{\Cdot} \ar [r, "i_{\Cdot}"] & B_{\Cdot} \ar [r, "q_{\Cdot}"] & C_{\Cdot} \ar[r] & 0
\end{tikzcd}.
\]
Then there are maps
\[
\partial: H_n(C_{\Cdot}) \to H_{n - 1}(A_{\Cdot})
\]
such that there is a long exact sequence
\[
\begin{tikzcd}
\cdots \ar [r] & H_n(A) \ar[r, "i_*"] & H_n(B) \ar[r, "q_*"] & H_n(C) \ar[out=0, in=180, looseness=2, overlay, dll, "\partial_*"']\\
& H_{n - 1}(A) \ar[r, "i_*"] & H_{n - 1}(B) \ar[r, "q_*"] & H_{n - 1}(C) \ar [r] & \cdots
\end{tikzcd}.
\]
\end{thm}
The method of proving this is sometimes known as ``diagram chasing'', where we just ``chase'' around commutative diagrams to find the elements we need. The idea of the proof is as follows --- in the short exact sequence, we can think of $A$ as a subgroup of $B$, and $C$ as the quotient $B/A$, by the first isomorphism theorem. So any element of $C$ can be represented by an element of $B$. We apply the boundary map to this representative, and then exactness shows that this must come from some element of $A$. We then check carefully that these is well-defined, i.e.\ does not depend on the representatives chosen.
\begin{proof}
The proof of this is in general not hard. It just involves a lot of checking of the details, such as making sure the homomorphisms are well-defined, are actually homomorphisms, are exact at all the places etc. The only important and non-trivial part is just the construction of the map $\partial_*$.
First we look at the following commutative diagram:
\[
\begin{tikzcd}
0 \ar[r] & A_n \ar[r, "i_n"] \ar[d, "d_n"] & B_n \ar[r, "q_n"] \ar[d, "d_n"] & C_n \ar[r] \ar[d, "d_n"] & 0\\
0 \ar[r] & A_{n - 1} \ar[r, "i_{n - 1}"] & B_{n - 1} \ar[r, "q_{n - 1}"] & C_{n - 1} \ar[r] & 0
\end{tikzcd}
\]
To construct $\partial_*: H_n(C) \to H_{n - 1}(A)$, let $[x] \in H_n(C)$ be a class represented by $x \in Z_n(C)$. We need to find a cycle $z \in A_{n - 1}$. By exactness, we know the map $q_n: B_n \to C_n$ is surjective. So there is a $y \in B_n$ such that $q_n(y) = x$. Since our target is $A_{n - 1}$, we want to move down to the next level. So consider $d_n(y) \in B_{n - 1}$. We would be done if $d_n(y)$ is in the image of $i_{n - 1}$. By exactness, this is equivalent saying $d_n(y)$ is in the kernel of $q_{n -1 }$. Since the diagram is commutative, we know
\[
q_{n - 1}\circ d_n(y) = d_n \circ q_n (y) = d_n(x) = 0,
\]
using the fact that $x$ is a cycle. So $d_n (y) \in \ker q_{n - 1} = \im i_{n - 1}$. Moreover, by exactness again, $i_{n - 1}$ is injective. So there is a unique $z \in A_{n - 1}$ such that $i_{n - 1}(z) = d_n(y)$. We have now produced our $z$.
We are not done. We have $\partial_* [x] = [z]$ as our candidate definition, but we need to check many things:
\begin{enumerate}
\item We need to make sure $\partial_*$ is indeed a homomorphism.
\item We need $d_{n - 1}(z) = 0$ so that $[z] \in H_{n - 1}(A)$;
\item We need to check $[z]$ is well-defined, i.e.\ it does not depend on our choice of $y$ and $x$ for the homology class $[x]$.
\item We need to check the exactness of the resulting sequence.
\end{enumerate}
We now check them one by one:
\begin{enumerate}
\item Since everything involved in defining $\partial_*$ are homomorphisms, it follows that $\partial_*$ is also a homomorphism.
\item We check $d_{n - 1}(z) = 0$. To do so, we need to add an additional layer.
\[
\begin{tikzcd}
0 \ar[r] & A_n \ar[r, "i_n"] \ar[d, "d_n"] & B_n \ar[r, "q_n"] \ar[d, "d_n"] & C_n \ar[r] \ar[d, "d_n"] & 0\\
0 \ar[r] & A_{n - 1} \ar[r, "i_{n - 1}"] \ar[d, "d_{n - 1}"] & B_{n - 1} \ar[r, "q_{n - 1}"] \ar[d, "d_{n - 1}"] & C_{n - 1} \ar[r] \ar[d, "d_{n - 1}"] & 0\\
0 \ar[r] & A_{n - 2} \ar[r, "i_{n - 2}"] & B_{n - 2} \ar[r, "q_{n - 2}"] & C_{n - 2} \ar[r] & 0
\end{tikzcd}
\]
We want to check that $d_{n - 1}(z) = 0$. We will use the commutativity of the diagram. In particular, we know
\[
i_{n - 2} \circ d_{n - 1}(z) = d_{n - 1} \circ i_{n - 1} (z) = d_{n - 1} \circ d_n(y) = 0.
\]
By exactness at $A_{n - 2}$, we know $i_{n - 2}$ is injective. So we must have $d_{n - 1}(z) = 0$.
\item
\begin{enumerate}
\item First, in the proof, suppose we picked a different $y'$ such that $q_n(y') = q_n(y) = x$. Then $q_n(y' - y) = 0$. So $y' - y \in \ker q_n = \im i_n$. Let $a \in A_n$ be such that $i_n(a) = y' - y$. Then
\begin{align*}
d_n(y') &= d_n(y' - y) + d_n(y) \\
&= d_n \circ i_n (a) + d_n(y) \\
&= i_{n - 1} \circ d_n (a) + d_n(y).
\end{align*}
Hence when we pull back $d_n(y')$ and $d_n(y)$ to $A_{n - 1}$, the results differ by the boundary $d_n(a)$, and hence produce the same homology class.
\item Suppose $[x'] = [x]$. We want to show that $\partial_* [x] = \partial_*[x']$. This time, we add a layer above.
\[
\begin{tikzcd}
0 \ar[r] & A_{n + 1} \ar[r, "i_{n + 1}"] \ar[d, "d_{n + 1}"] & B_{n + 1} \ar[r, "q_{n + 1}"] \ar[d, "d_{n + 1}"] & C_{n + 1} \ar[r] \ar[d, "d_{n + 1}"] & 0\\
0 \ar[r] & A_n \ar[r, "i_n"] \ar[d, "d_n"] & B_n \ar[r, "q_n"] \ar[d, "d_n"] & C_n \ar[r] \ar[d, "d_n"] & 0\\
0 \ar[r] & A_{n - 1} \ar[r, "i_{n - 1}"] & B_{n - 1} \ar[r, "q_{n - 1}"] & C_{n - 1} \ar[r] & 0
\end{tikzcd}
\]
By definition, since $[x'] = [x]$, there is some $c \in C_{n + 1}$ such that
\[
x' = x + d_{n + 1} (c).
\]
By surjectivity of $q_{n + 1}$, we can write $c = q_{n + 1}(b)$ for some $b \in B_{n + 1}$. By commutativity of the squares, we know
\[
x' = x + q_n \circ d_{n + 1} (b).
\]
The next step of the proof is to find some $y$ such that $q_n (y) = x$. Then
\[
q_n(y + d_{n + 1} (b)) = x'.
\]
So the corresponding $y'$ is $y' = y + d_{n + 1}(b)$. So $d_n (y) = d_n(y')$, and hence $\partial_*[x] = \partial_* [x']$.
\end{enumerate}
\item This is yet another standard diagram chasing argument. When reading this, it is helpful to look at a diagram and see how the elements are chased along. It is even more beneficial to attempt to prove this yourself.
\begin{enumerate}
\item $\im i_* \subseteq \ker q_*$: This follows from the assumption that $i_n \circ q_n = 0$.
\item $\ker q_* \subseteq \im i_*$: Let $[b] \in H_n(B)$. Suppose $q_*([b]) = 0$. Then there is some $c \in C_{n + 1}$ such that $q_n(b) = d_{n + 1}(c)$. By surjectivity of $q_{n + 1}$, there is some $b' \in B_{n + 1}$ such that $q_{n + 1}(b') = c$. By commutativity, we know $q_n(b) = q_n \circ d_{n + 1}(b')$, i.e.
\[
q_n (b - d_{n + 1}(b')) = 0.
\]
By exactness of the sequence, we know there is some $a \in A_n$ such that
\[
i_n(a) = b - d_{n + 1}(b').
\]
Moreover,
\[
i_{n - 1} \circ d_n(a) = d_n \circ i_n (a) = d_n(b - d_{n + 1}(b')) = 0,
\]
using the fact that $b$ is a cycle. Since $i_{n - 1}$ is injective, it follows that $d_n(a) = 0$. So $[a] \in H_n(A)$. Then
\[
i_*([a]) = [b] - [d_{n + 1}(b')] = [b].
\]
So $[b] \in \im i_*$.
\item $\im q_* \subseteq \ker \partial_*$: Let $[b] \in H_n(B)$. To compute $\partial_*(q_*([b]))$, we first pull back $q_n(b)$ to $b \in B_n$. Then we compute $d_n(b)$ and then pull it back to $A_{n + 1}$. However, we know $d_n(b) = 0$ since $b$ is a cycle. So $\partial_*(q_*([b])) = 0$, i.e.\ $\partial_* \circ q_* = 0$.
\item $\ker \partial_* \subseteq \im q_*$: Let $[c] \in H_n(C)$ and suppose $\partial_*([c]) = 0$. Let $b \in B_n$ be such that $q_n(b) = c$, and $a \in A_{n - 1}$ such that $i_{n - 1}(a) = d_n(b)$. By assumption, $\partial_*([c]) = [a] = 0$. So we know $a$ is a boundary, say $a = d_n (a')$ for some $a' \in A_n$. Then by commutativity we know $d_n(b) = d_n \circ i_n (a')$. In other words,
\[
d_n(b - i_n(a')) = 0.
\]
So $[b - i_n(a')] \in H_n(B)$. Moreover,
\[
q_*([b - i_n(a')]) = [q_n(b) - q_n \circ i_n(a')] = [c].
\]
So $[c] \in \im q_*$.
\item $\im \partial_* \subseteq \ker i_*$: Let $[c] \in H_n(C)$. Let $b \in B_n$ be such that $q_n(b) = c$, and $a \in A_{n - 1}$ be such that $i_n(a) = d_n(b)$. Then $\partial_*([c]) = [a]$. Then
\[
i_*([a]) = [i_n(a)] = [d_n(b)] = 0.
\]
So $i_* \circ \partial_* = 0$.
\item $\ker i_* \subseteq \im \partial_*$: Let $[a] \in H_n(A)$ and suppose $i_*([a]) = 0$. So we can find some $b \in B_{n + 1}$ such that $i_n(a) = d_{n + 1}(b)$. Let $c = q_{n + 1}(b)$. Then
\[
d_{n + 1}(c) = d_{n + 1}\circ q_{n + 1} (b) = q_n \circ d_{n + 1}(b) = q_n \circ i_n (a) = 0.
\]
So $[c] \in H_n(C)$. Then $[a] = \partial_*([c])$ by definition of $\partial_*$. So $[a] \in \im \partial_*$.\qedhere
\end{enumerate}%\qedhere
\end{enumerate}
\end{proof}
Another piece of useful algebra is known as the $5$-lemma:
\begin{lemma}[Five lemma]
Consider the following commutative diagram:
\[
\begin{tikzcd}
A \ar[r, "f"] \ar[d, "\ell"] & B \ar[r, "g"] \ar[d, "m"] & C \ar[r, "h"] \ar[d, "n"] & D \ar[r, "j"] \ar[d, "p"] & E \ar[d, "q"]\\
A' \ar[r, "r"] & B' \ar[r, "s"] & C' \ar[r, "t"] & D' \ar[r, "u"] & E'
\end{tikzcd}
\]
If the two rows are exact, $m$ and $p$ are isomorphisms, $q$ is injective and $\ell$ is surjective, then $n$ is also an isomorphism.
\end{lemma}
\begin{proof}
The philosophy is exactly the same as last time.
We first show that $n$ is surjective. Let $c' \in C'$. Then we obtain $d' = t(c') \in D'$. Since $p$ is an isomorphism, we can find $d \in D$ such that $p(d) = d'$. Then we have
\[
q(j(d)) = u(p(d)) = u(f(c')) = 0.
\]
Since $q$ is injective, we know $j(d) = 0$. Since the sequence is exact, there is some $c \in C$ such that $h(c) = d$.
We are not yet done. We do not know that $n(c) = c'$. All we know is that $d(n(c)) = d(c')$. So $d(c' - n(c)) = 0$. By exactness at $C'$, we can find some $b'$ such that $s(b') = n(c) - c'$. Since $m$ was surjective, we can find $b \in B$ such that $m(b) = b'$. Then we have
\[
n(g(b)) = n(c) - c'.
\]
So we have
\[
n(c - g(b)) = c'.
\]
So $n$ is surjective.
Showing that $n$ is injective is similar.
% complete
\end{proof}
\begin{cor}
Let $f: (X, A) \to (Y, B)$ be a map of pairs, and that any two of $f_*: H_*(X, A) \to H_*(Y, B)$, $H_*(X) \to H_*(Y)$ and $H_*(A) \to H_*(B)$ are isomorphisms. Then the third is also an isomorphism.
\end{cor}
\begin{proof}
Follows from the long exact sequence and the five lemma.
\end{proof}
That wasn't too bad, as it is just pure algebra.
\subsubsection*{Proof of homotopy invariance}
The next goal is want to show that homotopy of continuous maps does not affect the induced map on the homology groups. We will do this by showing that homotopies of maps induce homotopies of chain complexes, and chain homotopic maps induce the same map on homology groups. To make sense of this, we need to know what it means to be a homotopy of chain complexes.
\begin{defi}[Chain homotopy]\index{chain homotopy}
A \emph{chain homotopy} between chain maps $f_{\Cdot}, g_{\Cdot}: C_{\Cdot} \to D_{\Cdot}$ is a collection of homomorphisms $F_n: C_n \to D_{n + 1}$ such that
\[
g_n - f_n = d_{n + 1}^D \circ F_n + F_{n - 1} \circ d_n^C: C_n \to D_n
\]
for all $n$.
\end{defi}
The idea of the chain homotopy is that $F_n(\sigma)$ gives us an $n + 1$ simplex whose boundary is $g_n - f_n$, plus some terms arising from the boundary of $c$ itself:
\begin{center}
\begin{tikzpicture}
\fill [morange, opacity=0.3] (0, 0) rectangle (1.5, 2);
\node at (0.75, 1) {$F_n(\sigma)$};
\draw (0, 0) node [circ] {} -- (1.5, 0) node [circ] {} node [pos=0.5, below] {$f_n(\sigma)$};
\draw (0, 2) node [circ] {} -- (1.5, 2) node [circ] {} node [pos=0.5, above] {$g_n(\sigma)$};
\node at (2,1) {$:$};
\draw (2.5, 0) node [circ] {} -- (4, 0) node [circ] {};
\draw (2.5, 2) node [circ] {} -- (4, 2) node [circ] {};
\draw (2.5, 0) -- (2.5, 2);
\draw (4, 0) -- (4, 2);
\node at (3.25, 1) {$\d F_n(\sigma)$};
\node at (4.5, 1) {$=$};
\draw (5, 0) node [circ] {} -- (6.5, 0) node [circ] {} node [pos=0.5, below] {$f_n(\sigma)$};
\draw (5, 2) node [circ] {} -- (6.5, 2) node [circ] {} node [pos=0.5, above] {$g_n(\sigma)$};
\node at (7, 1) {$+$};
\draw (8, 0) node [circ] {} -- (8, 2) node [circ] {} ;
\draw (9.5, 0) node [circ] {} -- (9.5, 2) node [circ] {} ;
\node at (8.75, 1) {$F_{n - 1}(\d \sigma)$};
\end{tikzpicture}
\end{center}
We will not attempt to justify the signs appearing in the definition; they are what are needed for it to work.
The relevance of this definition is the following result:
\begin{lemma}
If $f_{\Cdot}$ and $g_{\Cdot}$ are chain homotopic, then $f_* = g_*: H_*(C_{\Cdot}) \to H_*(D_{\Cdot})$.
\end{lemma}
\begin{proof}
Let $[c] \in H_n(C_{\Cdot})$. Then we have
\[
g_n(c) - f_n(c) = d_{n + 1}^DF_n(c) + F_{n - 1}(d_n^C(c)) = d_{n + 1}^DF_n(c),
\]
where the second term dies because $c$ is a cycle. So we have $[g_n(c)] = [f_n(c)]$.
\end{proof}
That was the easy part. What we need to do now is to show that homotopy of maps between spaces gives a chain homotopy between the corresponding chain maps.
We will change notation a bit.
\begin{notation}
From now on, we will just write $d$ for $d_n^C$.
For $f: X \to Y$, we will write $f_\#: C_n(X) \to C_n(Y)$ for the map $\sigma \mapsto f \circ \sigma$, i.e.\ what we used to call $f_n$.
\end{notation}
Now if $H: [0, 1] \times X \to Y$ is a homotopy from $f$ to $g$, and $\sigma: \Delta^n \to X$ is an $n$-chain, then we get a homotopy
\[
\begin{tikzcd}
\lbrack0, 1\rbrack \times \Delta^n \ar[r, "{[0, 1] \times \sigma}"] & \lbrack0, 1\rbrack \times X \ar[r, "H"] & Y
\end{tikzcd}
\]
from $f_\#(\sigma)$ to $g_\#(\sigma)$. Note that we write $[0, 1]$ for the identity map $[0, 1] \to [0, 1]$.
The idea is that we are going to cut up $[0, 1] \times \Delta^n$ into $n + 1$-simplices. Suppose we can find a collection of chains $P_n \in C_{n + 1}([0, 1] \times \Delta^n)$ for $n \geq 0$ such that
\[
d(P_n) = i_1 - i_0 - \sum_{j = 0}^n (-1)^j ([0, 1] \times \delta_j)_\#(P_{n - 1}),
\]
where
\begin{align*}
i_0: \delta^n &= \{0\} \times \Delta^n \hookrightarrow [0, 1] \times \Delta^n\\
i_1: \delta^n &= \{0\} \times \Delta^n \hookrightarrow [0, 1] \times \Delta^n
\end{align*}
and $\delta_j: \Delta^{n - 1} \to \Delta^n$ is the inclusion of the $j$th face. These are ``prisms'' connecting the top and bottom face. Intuitively, the prism $P_2$ looks like this:
\begin{center}
\begin{tikzpicture}
\draw (-1, 0) -- (1, 0) -- (0, 0.5) -- cycle;
\draw (-1, -2) -- (1, -2);
\draw [dashed] (-1, -2) -- (0, -1.5) -- (1, -2);
\draw [dashed] (0, -1.5) -- (0, 0.5);
\draw (-1, -2) -- (-1, 0);
\draw (1, -2) -- (1, 0);
\end{tikzpicture}
\end{center}
and the formula tells us its boundary is the top and bottom triangles, plus the side faces given by the prisms of the edges.
Suppose we managed to find such prisms. We can then define
\[
F_n: C_n(X) \to C_{n + 1}(Y)
\]
by sending
\[
(\sigma: \Delta^n \to X) \mapsto (H \circ ([0, 1] \times \sigma))_\#(P_n).
\]
We now calculate.
\begin{align*}
\d F_n(\sigma) ={}& d((H \circ (1 \times \Delta^n))_\#(P_n))\\
={}& (H \circ ([0, 1] \times \sigma))_\#(d(P_n))\\
={}& (H \circ ([0, 1] \times \sigma))_\# \left(i_1 - i_0 - \sum_{j = 0}^n (-1)^j ([0, 1] \times \delta_j)_\#(P_{n - 1})\right)\\
={}& H \circ ([0, 1] \times \sigma) \circ i_1 - H \circ ([0, 1] \times \sigma) \circ i_0 \\
&\quad- \sum_{j = 0}^n (-1)^j H_\# \circ (([0, 1] \times \sigma) \circ \delta_j)_\# (P_{n - 1})\\
={}& g \circ \sigma - f \circ \sigma - \sum_{j = 0}^n (-1)^j H_\# \circ (([0, 1] \times \sigma) \circ \delta_j)_\# (P_{n - 1})\\
={}& g \circ \sigma - f \circ \sigma - F_{n - 1}(\d \sigma)\\
={}& g_\#(\sigma) - f_\#(\sigma) - F_{n - 1}(d\sigma).
\end{align*}
So we just have to show that $P_n$ exists. We already had a picture of what it looks like, so we just need to find a formula that represents it. We view $[0, 1] \times \Delta^n \in \R \times \R^{n + 1}$. Write $\{v_0, v_1, \cdots, v_n\}$ for the vertices of $\{0\} \times \Delta^n \subseteq [1, 0] \times \Delta^n$, and $\{w_0, \cdots, w_n\}$ for the corresponding vertices of $\{1\} \times \Delta^n$.
Now if $\{x_0, x_1, \cdots, x_{n + 1}\} \subseteq \{v_0, \cdots, v_n\} \cup \{w_0, \cdots, w_n\}$, we let
\[
[x_0, \cdots, x_n]: \Delta^n \to [0, 1] \times \Delta^n
\]
by
\[
(t_0, \cdots, t_{n + 1}) = \sum t_i x_i.
\]
This is still in the space by convexity. We let
\[
P_n = \sum_{i = 0}^n (-1)^i [v_0, v_1 , \cdots, v_i, w_i, w_{i + 1}, \cdots, w_n] \in C_{n + 1}([0, 1] \times \Delta^n).
\]
It is a boring check that this actually works, and we shall not bore the reader with the details.
\subsubsection*{Proof of excision and Mayer-Vietoris}
Finally, we prove excision and Mayer-Vietoris together. It turns out both follow easily from what we call the ``small simplices theorem''.
\begin{defi}[$C_n^\mathcal{U}(X)$ and $H_n^\mathcal{U}(X)$]\index{$C_n^\mathcal{U} (X)$}\index{$H_n^\mathcal{U}(X)$}
We let $\mathcal{U} = \{U_\alpha\}_{\alpha \in I}$ be a collection of subspaces of $X$ such that their interiors cover $X$, i.e.
\[
\bigcup_{\alpha \in I} \mathring{U}_\alpha = X.
\]
Let $C_n^\mathcal{U}(X) \subseteq C_n(X)$ be the subgroup generated by those singular $n$-simplices $\sigma: \Delta^n \to X$ such that $\sigma(\Delta^n) \subseteq U_\alpha$ for some $\alpha$. It is clear that if $\sigma$ lies in $U_\alpha$, then so do its faces. So $C_n^{\mathcal{U}}(X)$ is a sub-chain complex of $C_\Cdot(X)$.
We write $H_n^\mathcal{U}(X) = H_n(C_\Cdot^\mathcal{U}(X))$.
\end{defi}
It would be annoying if each choice of open cover gives a different homology theory, because this would be too many homology theories to think about. The \emph{small simplices theorem} says that the natural map $H_*^\mathcal{U}(X) \to H_*(X)$ is an isomorphism.
\begin{thm}[Small simplices theorem]\index{small simplices theorem}
The natural map $H_*^\mathcal{U}(X) \to H_*(X)$ is an isomorphism.
\end{thm}
The idea is that we can cut up each simplex into smaller parts by barycentric subdivision, and if we do it enough, it will eventually lie in on of the open covers. We then go on to prove that cutting it up does not change homology.
Proving it is not hard, but technically annoying. So we first use this to deduce our theorems.
\begin{proof}[Proof of Mayer-Vietoris]
Let $X = A \cup B$, with $A, B$ open in $X$. We let $\mathcal{U} = \{A, B\}$, and write $C_\Cdot(A + B) = C_\Cdot^\mathcal{U} (X)$. Then we have a natural chain map
\[
\begin{tikzcd}
C_\Cdot(A) \oplus C_{\Cdot}(B) \ar[r, "j_A - j_B"] & C_\Cdot(A + B)
\end{tikzcd}
\]
that is surjective. The kernel consists of $(x, y)$ such that $j_A(x) - j_B(y) = 0$, i.e.\ $j_A(x) = j_B(y)$. But $j$ doesn't really do anything. It just forgets that the simplices lie in $A$ or $B$. So this means $y = x$ is a chain in $A \cap B$. We thus deduce that we have a short exact sequence of chain complexes
\[
\begin{tikzcd}
C_\Cdot(A \cap B) \ar[r, "{(i_A, i_B)}"] & C_\Cdot(A) \oplus C_{\Cdot}(B) \ar[r, "j_A - j_B"] & C_\Cdot(A + B).
\end{tikzcd}
\]
Then the snake lemma shows that we obtain a long exact sequence of homology groups. So we get a long exact sequence of homology groups
\[
\begin{tikzcd}
\cdots\ar[r] & H_n(A \cap B) \ar[r, "{(i_A, i_B)}"] & H_n(A) \oplus H_n(B) \ar[r, "j_A - j_B"] & H_n^\mathcal{U}(X) \ar[r] & \cdots
\end{tikzcd}.
\]
By the small simplices theorem, we can replace $H_n^\mathcal{U}(X)$ with $H_n(X)$. So we obtain Mayer-Vietoris.
\end{proof}
Now what does the boundary map $\partial: H_n(X) \to H_{n - 1}(A \cap B)$ do? Suppose we have $c \in H_n(X)$ represented by a cycle $a + b \in C_n^\mathcal{U}(X)$, with $a$ supported in $A$ and $b$ supported in $B$. By the small simplices theorem, such a representative always exists. Then the proof of the snake lemma says that $\partial([a + b])$ is given by tracing through
\[
\begin{tikzcd}
& C_n(A) \oplus C_n(B) \ar[r, "j_A - j_B"] \ar[d, "d"] & C_n(A + B)\\
C_{n - 1}(A \cap B) \ar[r, "{(i_A, i_B)}"] & C_{n - 1}(A) \oplus C_{n - 1}(B)
\end{tikzcd}
\]
We now pull back $a + b$ along $j_A - j_B$ to obtain $(a, -b)$, then apply $d$ to obtain $(da, -db)$. Then the required object is $[da] = [-db]$.
We now move on to prove excision.
\begin{proof}[Proof of excision]
Let $X \supseteq A \supseteq Z$ be such that $\overline{Z} \supseteq \mathring{A}$. Let $B = X \setminus Z$. Then again take
\[
\mathcal{U} = \{A, B\}.
\]
By assumption, their interiors cover $X$. We consider the short exact sequences
\[
\begin{tikzcd}
0 \ar[r] & C_\Cdot(A) \ar[d, equals] \ar[r] & C_\Cdot(A + B) \ar[r] \ar[d] & C_{\Cdot}(A + B)/C_\Cdot(A) \ar[r] \ar[d] & 0\\
0 \ar[r] & C_\Cdot(A) \ar[r] & C_\Cdot(X) \ar[r] & C_{\Cdot}(X, A) \ar[r] & 0
\end{tikzcd}
\]
Looking at the induced map between long exact sequences on homology, the middle and left terms induce isomorphisms, so the right term does too by the $5$-lemma.
On the other hand, the map
\[
\begin{tikzcd}
C_{\Cdot}(B)/C_\Cdot(A \cap B) \ar[r] & C_{\Cdot}(A + B)/C_\Cdot(A)
\end{tikzcd}
\]
is an isomorphism of chain complexes. Since their homologies are $H_\Cdot(B, A \cap B)$ and $H_\Cdot (X, A)$, we infer they the two are isomorphic. Recalling that $B = X \setminus \bar{Z}$, we have shown that
\[
H_*(X \setminus Z, A \setminus Z) \cong H_*(X, A).\qedhere
\]
\end{proof}
We now provide a sketch proof of the small simplices theorem. As mentioned, the idea is to cut our simplices up, and one method to do so is barycentric subdivision.
Given a $0$-simplex $\{v_0\}$, its \term{barycentric subdivision} is itself.
If $x = \{x_0, \cdots, x_n\} \subseteq \R^n$ spans an $n$-simplex $\sigma$, we let
\[
b_x = \frac{1}{n + 1} \sum_{i = 0}^n x_i
\]
be its \term{barycenter}.
If we have a $1$-simplex
\begin{center}
\begin{tikzpicture}
\node [circ] {};
\node [circ] at (2, 0) {};
\draw (0, 0) -- (2, 0);
\end{tikzpicture}
\end{center}
Then the barycentric subdivision is obtained as
\begin{center}
\begin{tikzpicture}
\node [circ] {};
\node [circ] at (2, 0) {};
\node [circ] at (1, 0) {};
\draw (0, 0) -- (2, 0);
\end{tikzpicture}
\end{center}
We can degenerately describe this as first barycentrically subdividing the boundary (which does nothing in this case), and then add the barycenter.
In the case of a $2$-simplex:
\begin{center}
\begin{tikzpicture}
\draw [fill=mblue, fill opacity=0.5] (0, 0) -- (2, 0) -- (1, 1.732) -- cycle;
\node [circ] at (0, 0) {};
\node [circ] at (2, 0) {};
\node [circ] at (1, 1.732) {};
\end{tikzpicture}
\end{center}
we first barycentrically subdivide the boundary:
\begin{center}
\begin{tikzpicture}
\draw [fill=mblue, fill opacity=0.5] (0, 0) -- (2, 0) -- (1, 1.732) -- cycle;
\node [circ] at (0, 0) {};
\node [circ] at (2, 0) {};
\node [circ] at (1, 1.732) {};
\node [circ] at (1, 0) {};
\node [circ] at (0.5, 0.866) {};
\node [circ] at (1.5, 0.866) {};
\node [circ] at (1, 0.5773) {};
\end{tikzpicture}
\end{center}
Then add the barycenter $b_x$, and for each standard simplex in the boundary, we ``cone it off'' towards $b_x$:
\begin{center}
\begin{tikzpicture}
\draw [fill=mblue, fill opacity=0.5] (0, 0) -- (2, 0) -- (1, 1.732) -- cycle;
\node [circ] at (0, 0) {};
\node [circ] at (2, 0) {};
\node [circ] at (1, 1.732) {};
\node [circ] at (1, 0) {};
\node [circ] at (0.5, 0.866) {};
\node [circ] at (1.5, 0.866) {};
\node [circ] at (1, 0.5773) {};
\draw (0, 0) -- (1, 0.5773) -- (2, 0);
\draw (1, 0) -- (1, 1.732);
\draw (0.5, 0.866) -- (1, 0.5773) -- (1.5, 0.866);
\end{tikzpicture}
\end{center}
More formally, in the standard $n$-simplex $\Delta^n \subseteq \R^{n + 1}$, we let $B_n$ be its barycenter. For each singular $i$-simplex $\sigma: \Delta^i \to \Delta^n$, we define
\[
\mathrm{Cone}^{\Delta^n}_i (\sigma): \Delta^{i + 1} \to \Delta^n
\]
by
\[
(t_0, t_1, \cdots, t_{i + 1}) \mapsto t_0 b_n + (1 - t_0) \cdot \sigma\left(\frac{(t_1, \cdots, t_{i + 1})}{1 - t_0}\right).
\]
We can then extend linearly to get a map $\mathrm{Cone}_i^{\Delta^n}: C_i(\Delta^n) \to C_{i + 1}(\Delta^n)$.
\begin{eg}
In the $2$-simplex
\begin{center}
\begin{tikzpicture}
\draw [fill=mblue, fill opacity=0.5] (0, 0) -- (2, 0) -- (1, 1.732) -- cycle;
\node [circ] at (0, 0) {};
\node [circ] at (2, 0) {};
\node [circ] at (1, 1.732) {};
\end{tikzpicture}
\end{center}
the cone of the bottom edge is the simplex in orange:
\begin{center}
\begin{tikzpicture}
\draw [fill=mblue, fill opacity=0.5] (0, 0) -- (2, 0) -- (1, 1.732) -- cycle;
\node [circ] at (0, 0) {};
\node [circ] at (2, 0) {};
\node [circ] at (1, 1.732) {};
\draw [fill=morange] (0, 0) -- (1, 0.5773) -- (2, 0);
\end{tikzpicture}
\end{center}
\end{eg}
Since this increases the dimension, we might think this is a chain map. Then for $i > 0$, we have
\begin{align*}
d \mathrm{Cone}_i^{\Delta^n}(\sigma) &= \sum_{j = 0}^{i + 1} \mathrm{Cone}_i^{\Delta^n}(\sigma) \circ \delta_j \\
&= \sigma + \sum_{j = 1}^{i + 1} (-1)^j \mathrm{Cone}_{i - 1}^{\Delta^n} (\sigma \circ \delta_{j - 1})\\
&= \sigma - \mathrm{Cone}_{i - 1}^{\Delta^n} (d \sigma).
\end{align*}
For $i = 0$, we get
\[
d \mathrm{Cone}_i^{\Delta^n}(\sigma) = \sigma - \varepsilon(\sigma) \cdot b_n,
\]
In total, we have
\[
d \mathrm{Cone}_i^{\Delta^n} + \mathrm{Cone}_{i - 1}^{\Delta^n} d = \mathrm{id} - c_{\Cdot},
\]
where $c_i = 0$ for $i > 0$, and $c_0(\sigma) = \varepsilon(\sigma) b_n$ is a map $C_\Cdot(\Delta^n) \to C_\Cdot(\Delta^n)$.
We now use this cone map to construct a barycentric subdivision map $\rho_n^X: C_n(X) \to C_n(X)$, and insist that it is natural that if $f: X \to Y$ is a map, then $f_\# \circ \rho_n^X = \rho_n^Y \circ f_\#$, i.e.\ the diagram
\[
\begin{tikzcd}
C_n(X) \ar[r, "\rho_n^X"] \ar[d, "f_\#"] & C_n(X) \ar[d, "f_\#"]\\
C_n(Y) \ar[r, "\rho_n^Y"] & C_n(Y)
\end{tikzcd}.
\]
So if $\sigma: \Delta^n \to X$, we let $\iota_n: \Delta^n \to \Delta^n \in C_n(\Delta^n)$ be the identity map. Then we must have
\[
\rho_n^X (\sigma) = \rho_n^X (\sigma_\# \iota_n) = \sigma_\# \rho_n^{\Delta^n}(\iota_n).
\]
So if know how to do barycentric subdivision for $\iota_n$ itself, then by naturality, we have defined it for all spaces! Naturality makes life easier for us, not harder!
So we define $\rho_n^X$ recursively on $n$, for all spaces $X$ at once, by
\begin{enumerate}
\item $\rho_0^X = \id_{C_0}(X)$
\item For $n > 0$, we define the barycentric subdivision of $\iota_n$ by
\[
\rho_n^{\Delta^n}(\iota_n) = \mathrm{Cone}_{n - 1}^{\Delta^n} (\rho_{n - 1}^{\Delta^n} (\d \iota_n)),
\]
and then extend by naturality.
\end{enumerate}
This has all the expected properties:
\begin{lemma}
$\rho_\Cdot^X$ is a natural chain map.
\end{lemma}
\begin{lemma}
$\rho_\Cdot^X$ is chain homotopic to the identity.
\end{lemma}
\begin{proof}
No one cares.
\end{proof}
\begin{lemma}
The diameter of each subdivided simplex in $(\rho_n^{\Delta^n})^k(\iota_n)$ is bounded by $\left(\frac{n}{n + 1}\right)^k \diam(\Delta^n)$.
\end{lemma}
\begin{proof}
Basic geometry.
\end{proof}
\begin{prop}
If $c \in C_n^\mathcal{U}(X)$, then $p^X(c) \in C_n^{\mathcal{U}}(X)$.
Moreover, if $c \in C_n(X)$, then there is some $k$ such that $(\rho_n^{X})^k(c) \in C_n^{\mathcal{U}}(X)$.
\end{prop}
\begin{proof}
The first part is clear. For the second part, note that every chain is a finite sum of simplices. So we only have to check it for single simplices. We let $\sigma$ be a simplex, and let
\[
\mathcal{V} = \{\sigma^{-1} \mathring{U}_\alpha\}
\]
be an open cover of $\Delta^n$. By the Lebesgue number lemma, there is some $\varepsilon > 0$ such that any set of diameter $< \varepsilon$ is contained in some $\sigma^{-1} \mathring{U}_\alpha$. So we can choose $k > 0$ such that $(\rho_n^{\Delta^n}) (\iota_n)$ is a sum of simplices which each has diameter $< \varepsilon$. So each lies in some $\sigma^{-1}\mathring{U}_\alpha$. So
\[
(\rho_n^{\Delta^n})^k (\iota_n) = C_n^{\mathcal{V}} (\Delta^n).
\]
So applying $\sigma$ tells us
\[
(\rho_n^{\Delta^n})^k (\sigma) \in C_n^\mathcal{U}(X).\qedhere
\]
\end{proof}
Finally, we get to the theorem.
\begin{thm}[Small simplices theorem]\index{small simplices theorem}
The natural map $U: H_*^\mathcal{U}(X) \to H_*(X)$ is an isomorphism.
\end{thm}
\begin{proof}
Let $[c] \in H_n(X)$. By the proposition, there is some $k > 0$ such that $(\rho_n^X)^k (c) \in C_n^{\mathcal{U}}(x)$. We know that $\rho_n^X$ is chain homotopic to the identity. Thus so is $(\rho_n^X)^k$. So $[(\rho_n^X)^k (c)] = [c]$. So the map $H_n^\mathcal{U}(X) \to H_n(X)$ is surjective.
To show that it is injective, we suppose $U([c]) = 0$. Then we can find some $z \in H_{n + 1}(X)$ such that $dz = c$. We can then similarly subdivide $z$ enough such that it lies in $C^\mathcal{U}_{n + 1}(X)$. So this shows that $[c] = 0 \in H_n^{\mathcal{U}}(X)$.
\end{proof}
That's it. We move on to (slightly) more interesting stuff. The next few sections will all be slightly short, as we touch on various different ideas.
\section{Reduced homology}
\begin{defi}[Reduced homology]\index{reduced homology}
Let $X$ be a space, and $x_0 \in X$ a basepoint. We define the \emph{reduced homology} to be $\tilde{H}_*(X) = H_*(X, \{x_0\})$.
\end{defi}
Note that by the long exact sequence of relative homology, we know that $\tilde{H}_n(X) \cong H_n(X)$ for $n \geq 1$. So what is the point of defining a new homology theory that only differs when $n = 0$, which we often don't care about?
It turns out there is an isomorphism between $H_*(X, A)$ and $\tilde{H}_*(X/A)$ for suitably ``good'' pairs $(X, A)$.
\begin{defi}[Good pair]\index{good pair}
We say a pair $(X, A)$ is \emph{good} if there is an open set $U$ containing $\bar{A}$ such that the inclusion $A \hookrightarrow U$ is a deformation retract, i.e.\ there exists a homotopy $H: [0, 1] \times U \to U$ such that
\begin{align*}
H(0, x) &= x\\
H(1, x) &\in A\\
H(t, a) &= a\text{ for all $a \in A, t \in [0, 1]$}.
\end{align*}
\end{defi}
\begin{thm}
If $(X, A)$ is good, then the natural map
\[
\begin{tikzcd}
H_*(X, A) \ar[r] & H_*(X/A, A/A) = \tilde{H}_*(X/A)
\end{tikzcd}
\]
is an isomorphism.
\end{thm}
\begin{proof}
As $i: A \hookrightarrow U$ is in particular a homotopy equivalence, the map
\[
\begin{tikzcd}
H_*(A) \ar[r] & H_*(U)
\end{tikzcd}
\]
is an isomorphism. So by the five lemma, the map on relative homology
\[
\begin{tikzcd}
H_*(X, A) \ar[r] & H_*(X, U)
\end{tikzcd}
\]
is an isomorphism as well.
As $i: A \hookrightarrow U$ is a deformation retraction with homotopy $H$, the inclusion
\[
\{*\} = A/A \hookrightarrow U/A
\]
is also a deformation retraction. So again by the five lemma, the map
\[
\begin{tikzcd}
H_*(X/A, A/A) \ar[r] & H_*(X/A, U/A)
\end{tikzcd}
\]
is also an isomorphism. Now we have
\[
\begin{tikzcd}[column sep=large]
H_n(X, A) \ar[d] \ar[r, "\sim"] & H_n(X, U) \ar[d] \ar[r, "\text{excise }A"] & H_n(X \setminus A, U \setminus A)\ar[d]\\
H_n(X/A, A/A) \ar[r, "\sim"] & H_n(X/A, U/A) \ar[r, "\text{excise }A/A"] & H_n\left(\frac{X}{A}\setminus\frac{A}{A}, \frac{U}{A}\setminus\frac{A}{A}\right)
\end{tikzcd}
\]
We now notice that $X\setminus A = \frac{X}{A} \setminus \frac{A}{A}$ and $U \setminus A = \frac{U}{A}\setminus \frac{A}{A}$. So the right-hand vertical map is actually an isomorphism. So the result follows.
\end{proof}
\section{Cell complexes}
So far, everything we've said is true for \emph{arbitrary} spaces. This includes, for example, the topological space with three points $a, b, c$, whose topology is $\{\emptyset, \{a\}, \{a, b, c\}\}$. However, these spaces are horrible. We want to restrict our attention to nice spaces. Spaces that do feel like actual, genuine spaces.
The best kinds of space we can imagine would be manifolds, but that is a bit too strong a condition. For example, the union of the two axes in $\R^2$ is not a manifold, but it is still a sensible space to talk about. Perhaps we can just impose conditions like Hausdorffness and maybe second countability, but we can still produce nasty spaces that satisfy these properties.
So the idea is to provide a method to build spaces, and then say we only consider spaces built this way. These are known as \emph{cell complexes}, or \emph{CW complexes}
\begin{defi}[Cell complex]\index{cell complex}\index{CW complexes}
A \emph{cell complex} is any space built out of the following procedure:
\begin{enumerate}
\item Start with a discrete space $X^0$. The set of points in $X^0$ are called $I_0$.
\item If $X^{n - 1}$ has been constructed, then we may choose a family of maps $\{\varphi_\alpha: S^{n - 1} \to X^{n - 1}\}_{\alpha \in I_n}$, and set
\[
X^n = \left(X^{n - 1} \amalg \left(\coprod_{\alpha \in I_n} D_\alpha^n\right) \right)/\{x \in \partial D_\alpha^n \sim \varphi_\alpha(x) \in X^{n - 1}\}.
\]
We call $X^n$ the \term{$n$-skeleton} of $X$ We call the image of $D_\alpha^n \setminus \partial D_\alpha^n$ in $X^n$ the \term{open cell} $e_\alpha$.
\item Finally, we define
\[
X = \bigcup_{n \geq 0} X^n
\]
with the \term{weak topology}, namely that $A \subseteq X$ is open if $A \cap X^n$ is open in $X^n$ for all $n$.
\end{enumerate}
\end{defi}
We write $\Phi_\alpha: D_\alpha^n \to X^n$ for the obvious inclusion map. This is called the \term{characteristic map} for the cell $e_\alpha$.
\begin{defi}[Finite-dimensional cell complex]\index{finite-dimensional cell complex}\index{cell complex!finite-dimensional}
If $X = X^n$ for some $n$, we say $X$ is \emph{finite-dimensional}.
\end{defi}
\begin{defi}[Finite cell complex]\index{finite cell complex}\index{cell complex!finite}
If $X$ is finite-dimensional and $I_n$ are all finite, then we say $X$ is \emph{finite}.
\end{defi}
\begin{defi}[Subcomplex]\index{subcomplex}\index{cell complex!subcomplex}
A subcomplex $A$ of $X$ is a simplex obtained by using a subset $I_n' \subseteq I_n$.
\end{defi}
Note that we cannot simply throw away some cells to get a subcomplex, as the higher cells might want to map into the cells you have thrown away, and you need to remove them as well.
We note the following technical result without proof:
\begin{lemma}
If $A \subseteq X$ is a subcomplex, then the pair $(X, A)$ is \emph{good}.
\end{lemma}
\begin{proof}
See Hatcher 0.16.
\end{proof}
\begin{cor}
If $A \subseteq X$ is a subcomplex, then
\[
\begin{tikzcd}
H_n(X, A) \ar[r, "\sim"] & \tilde{H}_n(X/A)
\end{tikzcd}
\]
is an isomorphism.
\end{cor}
We are next going to show that we can directly compute the cohomology of a cell complex by looking at the cell structures, instead of going through all the previous rather ad-hoc mess we've been through. We start with the following lemma:
\begin{lemma}
Let $X$ be a cell complex. Then
\begin{enumerate}
\item
\[
H_i(X^n, X^{n - 1}) =
\begin{cases}
0 & i \not= n\\
\bigoplus_{i \in I_n} \Z & i = n
\end{cases}.
\]
\item $H_i(X^n) = 0$ for all $i > n$.
\item $H_i(X^n) \to H_i(X)$ is an isomorphism for $i < n$.
\end{enumerate}
\end{lemma}
\begin{proof}\leavevmode
\begin{enumerate}
\item As $(X^n, X^{n - 1})$ is good, we have an isomorphism
\[
\begin{tikzcd}
H_i(X^n, X^{n - 1}) \ar[r, "\sim"] & \tilde{H}_i(X^n/X^{n - 1})
\end{tikzcd}.
\]
But we have
\[
X^n/X^{n - 1} \cong \bigvee_{\alpha \in I_n} S_\alpha^n,
\]
the space obtained from $Y = \coprod_{\alpha \in I_n}S_\alpha^n$ by collapsing down the subspace $Z = \{x_\alpha: \alpha \in I_n\}$, where each $x_\alpha$ is the south pole of the sphere. To compute the homology of the wedge $X^n/X^{n - 1}$, we then note that $(Y, Z)$ is good, and so we have a long exact sequence
\[
\begin{tikzcd}
H_i(Z) \ar[r] & H_i(Y) \ar[r] & \tilde{H}_i(Y/Z) \ar[r] & H_{i - 1}(Z) \ar[r] & H_{i - 1}(Y)
\end{tikzcd}.
\]
Since $H_i(Z)$ vanishes for $i \geq 1$, the result then follows from the homology of the spheres plus the fact that $H_i(\coprod X_\alpha) = \bigoplus H_i(X_\alpha)$.
\item This follows by induction on $n$. We have (part of) a long exact sequence
\[
\begin{tikzcd}
H_i(X^{n - 1}) \ar[r] & H_i(X^n) \ar[r] & H_i(X^n, X^{n - 1})
\end{tikzcd}
\]
We know the first term vanishes by induction, and the third term vanishes for $i > n$. So it follows that $H_i(X^n)$ vanishes.
\item To avoid doing too much point-set topology, we suppose $X$ is finite-dimensional, so $X = X^m$ for some $m$. Then we have a long exact sequence
\[
\begin{tikzcd}[column sep=small]
H_{i + 1} (X^{n + 1}, X^n) \ar[r] & H_i(X^n) \ar[r] & H_i(X^{n + 1}) \ar[r] & H_i(X^{n + 1}, X^n)
\end{tikzcd}
\]
Now if $i < n$, we know the first and last groups vanish. So we have $H_i(X^n) \cong H_i(X^{n + 1})$. By continuing, we know that
\[
H_i(X^n) \cong H_i(X^{n + 1}) \cong H_i(X^{n + 2}) \cong \cdots \cong H_i(X^m) = H_i(X).
\]
To prove this for the general case, we need to use the fact that any map from a compact space to a cell complex hits only finitely many cells, and then the result would follow from this special case.\qedhere
\end{enumerate}
\end{proof}
For a cell complex $X$, let
\[
C_n^{\mathrm{cell}}(X) = H_n(X^n, X^{n - 1}) \cong \bigoplus_{\alpha \in I_n} \Z.
\]
We define $d_n^{\mathrm{cell}}: C_n^{\mathrm{cell}}(X) \to C_{n - 1}^{\mathrm{cell}}(X)$ by the composition
\[
\begin{tikzcd}
H_n(X^n, X^{n - 1}) \ar[r, "\partial"] & H_{n - 1}(X^{n - 1}) \ar[r, "q"] & H_{n - 1}(X^{n - 1}, X^{n - 2})
\end{tikzcd}.
\]
We consider
\[
\begin{tikzcd}[row sep=large, column sep=-0.5em]
&&& 0\\
0 \ar[dr] && H_n(X^{n + 1}) \ar[ur]\\
&H_n(X^n) \ar[ur] \ar[dr, "q_n"]\\
H_{n + 1}(X^{n + 1}, X^n) \ar[ur, "\partial"] \ar[rr, "d_{n + 1}^{\mathrm{cell}}"] & & H_n(X^n, X^{n - 1}) \ar[rr, "d_n^{\mathrm{cell}}"] \ar[rd, "\partial"] && H_{n - 1}(X^{n - 1}, X^{n - 2})\\
& & & H_{n - 1}(X^{n - 1}) \ar[rd] \ar[ru, "q_{n - 1}"]\\
& & 0 \ar[ur] & & H_{n - 1}(X^n)
\end{tikzcd}
\]
Referring to the above diagram, we see that
\[
d_n^{\mathrm{cell}} \circ d_{n + 1}^{\mathrm{cell}} = q_{n - 1} \circ \partial \circ q_n \circ \partial = 0,
\]
since the middle $\partial \circ q_n$ is part of an exact sequence. So $(C_{\Cdot}^{\mathrm{cell}}(X), d_{\Cdot}^{\mathrm{cell}})$ is a chain complex, and the corresponding homology groups are known as the \term{cellular homology} of $X$, written $H_n^{\mathrm{cell}}(X)$.
\begin{thm}
\[
H_n^{\mathrm{cell}}(X) \cong H_n(X).
\]
\end{thm}
\begin{proof}
We have
\begin{align*}
H_n(X) &\cong H_n(X^{n + 1}) \\
&= H_n(X^n)/\im(\partial: H_{n + 1}(X^{n + 1}, X^n) \to H_n(X^n))\\
\intertext{Since $q_n$ is injective, we apply it to and bottom to get}
&= q_n(H_n(X^n)) / \im(d_{n + 1}^{\mathrm{cell}}: H_{n + 1}(X^{n + 1}, X^n) \to H_n(X^n, X^{n - 1}))\\
\intertext{By exactness, the image of $q_n$ is the kernel of $\partial$. So we have}
&= \ker(\partial: H_n(X^n, X^{n - 1}) \to H_{n - 1}(X^{n - 1})) / \im(d_{n + 1}^{\mathrm{cell}})\\
&= \ker(d_n^{\mathrm{cell}}) / \im(d_{n + 1}^{\mathrm{cell}})\\
&= H_n^{\mathrm{cell}}(X).\qedhere
\end{align*}
\end{proof}
\begin{cor}
If $X$ is a finite cell complex, then $H_n(X)$ is a finitely-generated abelian group for all $n$, generated by at most $|I_n|$ elements. In particular, if there are no $n$-cells, then $H_n(X)$ vanishes.
If $X$ has a cell-structure with cells in even-dimensional cells only, then $H_*(X)$ are all free.
\end{cor}
We can similarly define cellular cohomology.
\begin{defi}[Cellular cohomology]\index{cellular cohomology}
We define \emph{cellular cohomology} by
\[
C_{\mathrm{cell}}^n(X) = H^n(X^n, X^{n - 1})
\]
and let $d_{\mathrm{cell}}^n$ be the composition
\[
\begin{tikzcd}
H^n(X^n, X^{n - 1}) \ar[r, "q^*"] & H^n(X^n) \ar[r, "\partial"] & H^{n + 1}(X^{n + 1}, X^n).
\end{tikzcd}
\]
This defines a cochain complex $C_{\mathrm{cell}}^\Cdot(X)$ with cohomology $H^*_{\mathrm{cell}}(X)$, and we have
\[
H_{\mathrm{cell}}^*(X) \cong H^*(X).
\]
One can directly check that
\[
C_{\mathrm{cell}}^\Cdot(X) \cong \Hom (C_\Cdot^{\mathrm{cell}}(X), \Z).
\]
\end{defi}
This is all very good, because cellular homology is very simple and concrete. However, to actually use it, we need to understand what the map
\[
d_n^{\mathrm{cell}} : C_n^{\mathrm{cell}}(X) = \bigoplus_{\alpha \in I_n} \Z\{e_\alpha\} \to C_{n - 1}^{\mathrm{cell}}(X) = \bigoplus_{\beta \in I_{n - 1}}\Z\{e_\beta\}
\]
is. In particular, we want to find the coefficients $d_{\alpha\beta}$ such that
\[
d_n^{\mathrm{cell}}(e_\alpha) = \sum d_{\alpha\beta} e_\beta.
\]
It turn out this is pretty easy
\begin{lemma}
The coefficients $d_{\alpha\beta}$ are given by the degree of the map
\[
\begin{tikzcd}
S_\alpha^{n - 1} = \partial D_\alpha^n \ar[r, "\varphi_\alpha"] \ar[rrr, bend right, "f_{\alpha\beta}", looseness=0.5] & X^{n - 1} \ar[r] & X^{n - 1}/X^{n - 2} = \bigvee_{\gamma \in I_{n - 1}} S_\gamma^{n - 1} \ar[r] & S_\beta^{n - 1}
\end{tikzcd},
\]
where the final map is obtained by collapsing the other spheres in the wedge.
In the case of cohomology, the maps are given by the transposes of these.
\end{lemma}
This is easier in practice that in sounds. In practice, the map is given by ``the obvious one''.
\begin{proof}
Consider the diagram
\[
\begin{tikzcd}
H_n(D_\alpha^n, \partial D_\alpha^n) \ar[d, "(\Phi_\alpha)_*"] \ar[r, "\partial", "\sim"'] & H_{n - 1}(\partial D_\alpha^n)\ar[d, "(\varphi_\alpha)_*"] \ar[r, dashed] & \tilde{H}_{n - 1}(S^{n - 1}_\beta) \\
H_n(X^n, X^{n - 1}) \ar[r, "\partial"] \ar[rd, "d_n^\mathrm{cell}"] & H_{n - 1}(X^{n - 1}) \ar[d, "q"] & \tilde{H}_{n - 1}\left(\bigvee S_\gamma^{n - 1}\right)\ar[u, "\text{collapse}"]\\
& H_{n - 1}(X^{n - 1}, X^{n - 2}) \ar[r, "\text{excision}", "\sim"'] & \tilde{H}_{n - 1}(X^{n - 1}/X^{n - 2}) \ar[u, equals]
\end{tikzcd}
\]
By the long exact sequence, the top left horizontal map is an isomorphism.
Now let's try to trace through the diagram. We can find
\[
\begin{tikzcd}
1 \ar[d, maps to] \ar[r, "\text{isomorphism}"] & 1 \ar[r, maps to, "f_{\alpha\beta}"]& d_{\alpha\beta}\\
e_\alpha \ar[rd, maps to]\\
& \sum d_{\alpha\gamma}e_\gamma \ar[r, maps to] & \sum d_{\alpha\gamma} e_\gamma \ar[uu, maps to]
\end{tikzcd}
\]
So the degree of $f_{\alpha\beta}$ is indeed $d_{\alpha\beta}$.
\end{proof}
\begin{eg}
Let $K$ be the Klein bottle.
\begin{center}
\begin{tikzpicture}
\draw [->-=0.55, mred] (0, 0) -- (3, 0);
\draw [->-=0.55, mred] (3, 3) -- (0, 3);
\draw [->-=0.55, mblue] (3, 0) -- (3, 3);
\draw [->-=0.55, mblue] (0, 0) -- (0, 3);
\node [circ] at (0, 0) {};
\node [circ] at (3, 0) {};
\node [circ] at (0, 3) {};
\node [circ] at (3, 3) {};
\node [anchor = south east] {$v$};
\node [left] at (0, 1.5) {$a$};
\node [below] at (1.5, 3) {$b$};
\node at (1.5, 1.5) {\huge $\pi$};
\end{tikzpicture}
\end{center}
We give it a cell complex structure by
\begin{itemize}
\item $K^0 = \{v\}$. Note that all four vertices in the diagram are identified.
\begin{center}
\begin{tikzpicture}
\node [circ] at (0, 0) {};
\node [above] at (0, 0) {$v$};
\end{tikzpicture}
\end{center}
\item $K^1 = \{a, b\}$.
\begin{center}
\begin{tikzpicture}[scale=0.75]
\draw (-1, 0) circle [radius=1];
\draw (1, 0) circle [radius=1];
\node [circ] {};
\node [left] at (-2, 0) {$a$};
\node [right] at (2, 0) {$b$};
\draw [->] (2, 0.1) -- (2, 0.11);
\draw [->] (-2, 0.1) -- (-2, 0.11);
\node [right] {$x_0$};
\end{tikzpicture}
\end{center}
\item $K^2$ is the unique $2$-cell $\pi$ we see in the picture, where $\varphi_\pi: S^1 \to K^1$ given by $a b a^{-1}b$.
\end{itemize}
The cellular chain complex is given by
\[
\begin{tikzcd}[row sep=small]
0 \ar[r] & C_2^{\mathrm{cell}}(K) \ar[d, equals] \ar[r, "d_2^{\mathrm{cell}}"] & C_1^{\mathrm{cell}}(K) \ar[d, equals] \ar[r, "d_1^{\mathrm{cell}}"] & C_0^{\mathrm{cell}}(K) \ar[d, equals]\\
& \Z\pi & \Z a \oplus \Z b & \Z_v
\end{tikzcd}
\]
We can now compute the maps $d_i^{\mathrm{cell}}$. The $d_1$ map is easy. We have
\[
d_1(a) = d_1(b) = v - v = 0.
\]
In the $d_2$ map, we can figure it out by using local degrees. Locally, the attaching map is just like the identity map, up to an orientation flip, so the local degrees are $\pm 1$. Moreover, the inverse image of each point has two elements. If we think hard enough, we realize that for the attaching map of $a$, the two elements have opposite degree and cancel each other out; while in the case of $b$ they have the same sign and give a degree of $2$. So we have
\[
d_2(\pi) = 0a + 2b.
\]
So we have
\begin{align*}
H_0(K) &= \Z\\
H_1(K) &= \frac{\Z \oplus \Z}{\bra2b\ket} = \Z \oplus \Z/2\Z\\
H_2(K) &= 0
\end{align*}
We can similarly compute the cohomology. By dualizing, we have
\[
\begin{tikzcd}[row sep=small]
C_{\mathrm{cell}}^2(K)(K) \ar[d, equals] & C_{\mathrm{cell}}^1(K) \ar[d, equals] \ar[l, "(0\; 2)"] & C_{\mathrm{cell}}^0(K) \ar[d, equals] \ar[l, "(0)"] \\
\Z & \Z \oplus \Z & \Z
\end{tikzcd}
\]
So we have
\begin{align*}
H_0(K) &= \Z\\
H_1(K) &= \Z\\
H_2(K) &= \Z/2\Z.
\end{align*}
Note that the second cohomology is \emph{not} the dual of the second homology!
However, if we forget where each factor is, and just add all the homology groups together, we get $\Z \oplus \Z \oplus \Z/2\Z$. Also, if we forget all the torsion components $\Z/2\Z$, then they are the same!
This is a general phenomenon. For a cell complex, if we know what all the homology groups are, we can find the cohomologies by keeping the $\Z$'s unchanged and moving the torsion components up. The general statement will be given by the \emph{universal coefficient theorem}.
\end{eg}
\begin{eg}
Consider $\RP^n = S^n/(x \sim -x)$. We notice that for any point in $\RP^n$, if it is not in the equator, then it is represented by a unique element in the northern hemisphere. Otherwise, it is represented by two points. So we have $\RP^n \cong D^n/(x \sim -x\text{ for }x \in \partial D^n)$. This is a nice description, since if we throw out the interior of the disk, then we are left with an $S^{n - 1}$ with antipodal points identified, i.e.\ an $\RP^{n - 1}$! So we can immediately see that
\[
\RP^n = \RP^{n - 1} \cup_f D^n,
\]
for $f: S^{n - 1} \to \RP^{n - 1}$ given by
\[
f(x) = [x].
\]
So $\RP^n$ has a cell structure with one cell in every degree up to $n$. What are the boundary maps?
We write $e_i$ for the $i$-th degree cell. We know that $e_i$ is attached along the map $f$ described above. More concretely, we have
\[
\begin{tikzcd}
f: S_s^{i - 1}\ar[r, "\varphi_i"] & \RP^{i - 1} \ar[r] &\RP^{i - 1}/\RP^{i - 2} = S_t^{i - 1}
\end{tikzcd}.
\]
The open upper hemisphere and lower hemisphere of $S^{i - 1}_s$ are mapped homeomorphically to $S_t^{i - 1} \setminus \{*\}$. Furthermore,
\[
f|_{\mathrm{upper}} = f|_{\mathrm{lower}} \circ a,
\]
where $a$ is the antipodal map. But we know that $\deg(a) = (-1)^i$. So we have a zero map if $i$ is odd, and a $2$ map if $i$ is even. Then we have
\[
\begin{tikzcd}
\cdots \ar[r, "2"] & \Z e_3 \ar[r, "0"] & \Z e_2 \ar[r, "2"] & \Z e_1 \ar[r, "0"] & \Z e_0
\end{tikzcd}.
\]
What happens on the left end depends on whether $n$ is even or odd. So we have
\[
H_i(\RP^n) =
\begin{cases}
\Z & i = 0\\
\Z/2\Z & i\text{ odd}, i < n\\
0 & i\text{ even}, 0 < i < n\\
\Z & i = n\text{ is odd}\\
0 & \text{otherwise}
\end{cases}.
\]
We can immediately work out the cohomology too. We will just write out the answer:
\[
H^i(\RP^n) =
\begin{cases}
\Z & i = 0\\
0 & i\text{ odd}, i < n\\
\Z/2\Z & i\text{ even}, 0 < i \leq n\\
\Z & i = n\text{ is odd}\\
0 & \text{otherwise}
\end{cases}.
\]
\end{eg}
\section{(Co)homology with coefficients}
Recall that when we defined (co)homology, we constructed these free $\Z$-modules from our spaces. However, we did not actually use the fact that it was $\Z$, we might as well replace it with any abelian group $A$.
\begin{defi}[(Co)homology with coefficients]\index{homology!with coefficients}\index{cohomology!with coefficients}
Let $A$ be an abelian group, and $X$ be a topological space. We let
\[
C_{\Cdot}(X; A) = C_\Cdot(X) \otimes A
\]
with differentials $d \otimes \id_A$. In other words $C_{\Cdot}(X; A)$ is the abelian group obtained by taking the direct sum of many copies of $A$, one for each singular simplex.
We let
\[
H_n(X; A) = H_n(C_{\Cdot}(X; A), d \otimes \id_A).
\]
We can also define
\[
H_n^{\mathrm{cell}}(X; A) = H_n(C_{\Cdot}^{\mathrm{cell}}(X) \otimes A),
\]
and the same proof shows that $H_n^{\mathrm{cell}}(X; A) = H_n(X; A)$.
Similarly, we let
\[
C^{\Cdot}(X; A) = \Hom(C_{\Cdot}(X), A),
\]
with the usual (dual) differential. We again set
\[
H^n(X; A) = H^n(C^\Cdot(X; A)).
\]
We similarly define cellular cohomology.
If $A$ is in fact a commutative ring, then these are in fact $R$-modules.
\end{defi}
We call $A$ the ``coefficients'', since a general member of $C_{\Cdot}(X; A)$ looks like
\[
\sum n_\sigma s,\quad \text{where } n_\sigma \in A,\quad \sigma: \Delta^n \to X.
\]
We will usually take $A = \Z, \Z/n\Z$ or $\Q$. Everything we've proved for homology holds for these with exactly the same proof.
\begin{eg}
In the case of $C_{\Cdot}^{\mathrm{cell}}(\RP^n)$, the differentials are all $0$ or $2$. So in $C_\Cdot^{\mathrm{cell}}(\RP^n, \Z/2)$, all the differentials are $0$. So we have
\[
H_i(\RP^n, \Z/2) =
\begin{cases}
\Z/2 & 0 \leq i \leq n\\
0 & i > n
\end{cases}
\]
Similarly, the cohomology groups are the same.
On the other hand, if we take the coefficients to be $\Q$, then multiplication by $2$ is now an isomorphism. Then we get
\[
C_{\Cdot}^\mathrm{cell}(\RP^n, \Q)
\begin{cases}
\Q & n \text{ odd}\\
0 & n\text{ even}
\end{cases}
\]
for $n$ not too large.
\end{eg}
\section{Euler characteristic}
There are many ways to define the Euler characteristic, and they are all equivalent. So to define it, we pick a definition that makes it obvious it is a number.
\begin{defi}[Euler characteristic]\index{Euler characteristic}
Let $X$ be a cell complex. We let
\[
\chi(X) = \sum_n (-1)^n \text{ number of $n$-cells of $X$} \in \Z.
\]
\end{defi}
From this definition, it is not clear that this is a property of $X$ itself, rather than something about its cell decomposition.
We similarly define
\[
\chi_{\Z}(X) = \sum_n (-1)^n \rank H_n(X; \Z).
\]
For any field $\F$, we define
\[
\chi_{\F}(X) = \sum_n (-1)^n \dim_\F H_n(X; \F).
\]
\begin{thm}
We have
\[
\chi = \chi_\Z = \chi_\F.
\]
\end{thm}
\begin{proof}
First note that the number of $n$ cells of $X$ is the rank of $C_n^{\mathrm{cell}}(X)$, which we will just write as $C_n$. Let
\begin{align*}
Z_n &= \ker (d_n: C_n \to C_{n - 1})\\
B_n &= \im (d_{n + 1}: C_{n + 1} \to C_n).
\end{align*}
We are now going to write down two short exact sequences. By definition of homology, we have
\[
\begin{tikzcd}
0 \ar[r] & B_n \ar[r] & Z_n \ar[r] & H_n(X; \Z) \ar[r] & 0
\end{tikzcd}.
\]
Also, the definition of $Z_n$ and $B_n$ give us
\[
\begin{tikzcd}
0 \ar[r] & Z_n \ar[r] & C_n \ar[r] & B_{n - 1} \ar[r] & 0
\end{tikzcd}.
\]
We will now use the first isomorphism theorem to know that the rank of the middle term is the sum of ranks of the outer terms. So we have
\[
\chi_\Z(X) = \sum (-1)^n \rank H_n(X) = \sum(-1)^n (\rank Z_n - \rank B_n).
\]
We also have
\[
\rank B_n = \rank C_{n + 1} - \rank Z_{n + 1}.
\]
So we have
\begin{align*}
\chi_\Z(X) &= \sum_n (-1)^n (\rank Z_n - \rank C_{n + 1} + \rank Z_{n + 1}) \\
&= \sum_n (-1)^{n + 1} \rank C_{n + 1}\\
&= \chi(X).
\end{align*}
For $\chi_\F$, we use the fact that
\[
\rank C_n = \dim_{\F} C_n \otimes \F.\qedhere
\]
\end{proof}
\section{Cup product}
So far, homology and cohomology are somewhat similar. We computed them, saw they are not the same, but they seem to contain the same information nevertheless. However, cohomology is 10 times better, because we can define a ring structure on them, and rings are better than groups.
Just like the case of homology and cohomology, we will be able to write down the definition easily, but will struggle to compute it.
\begin{defi}[Cup product]\index{cup product}\index{$\smile$}
Let $R$ be a commutative ring, and $\phi \in C^k(X; R)$, $\psi \in C^\ell(X; R)$. Then $\phi \smile \psi \in C^{k + \ell}(X; R)$ is given by
\[
(\phi \smile \psi)(\sigma: \Delta^{k + \ell} \to X) = \phi(\sigma|_{[v_0, \ldots, v_k]}) \cdot \psi(\sigma|_{[v_k, \ldots, v_{k + \ell}]}).
\]
Here the multiplication is multiplication in $R$, and $v_0, \cdots, v_\ell$ are the vertices of $\Delta^{k + \ell} \subseteq \R^{k + \ell + 1}$, and the restriction is given by
\[
\sigma|_{[x_0, \ldots, x_i]} (t_0, \ldots, t_i) = \sigma\left(\sum t_j x_j\right).
\]
This is a bilinear map.
\end{defi}
\begin{notation}
We write
\[
H^*(X; R) = \bigoplus_{n \geq 0} H^n(X; R).
\]
\end{notation}
This is the definition. We can try to establish some of its basic properties. We want to know how this interacts with the differential $d$ with the cochains. The obvious answer $d(\phi \smile \psi) = (\d \phi) \smile (\d \psi)$ doesn't work, because the degrees are wrong. What we have is:
\begin{lemma}
If $\phi \in C^k(X; R)$ and $\psi \in C^\ell(X; R)$, then
\[
d (\phi \smile \psi) = (d \phi)\smile \psi + (-1)^k \phi\smile(d \psi).
\]
\end{lemma}
This is like the product rule with a sign.
\begin{proof}
This is a straightforward computation.
Let $\sigma: \Delta^{k + \ell + 1} \to X$ be a simplex. Then we have
\begin{align*}
((d \phi)\smile \psi)(\sigma) &= (d \phi)(\sigma|_{[v_0, \ldots, v_{k + 1}]}) \cdot \psi(\sigma|_{[v_{k + 1}, \ldots, v_{k + \ell + 1}]})\\
&= \phi\left(\sum_{i = 0}^{k + 1} (-1)^i \sigma|_{[v_0, \ldots, \hat{v}_i, \ldots, v_{k + 1}]}\right) \cdot \psi(\sigma|_{[v_{k + 1}, \ldots, v_{k + \ell + 1}]})\\
(\phi \smile (d \psi))(\sigma) &= \phi(\sigma|_{[v_0, \ldots, v_k]}) \cdot (d \psi)(\sigma|_{v_k,\ldots, v_{k + \ell + 1}]})\\
&=\phi(\sigma|_{[v_0, \ldots, v_k]}) \cdot \psi\left(\sum_{i = k}^{k + \ell + 1} (-1)^{i - k} \sigma|_{[v_k, \ldots, \hat{v}_i, \ldots, v_{k + \ell + 1}]}\right)\\
&=(-1)^k \phi(\sigma|_{[v_0, \ldots, v_k]}) \cdot \psi\left(\sum_{i = k}^{k + \ell + 1} (-1)^{i} \sigma|_{[v_k, \ldots, \hat{v}_i, \ldots, v_{k + \ell + 1}]}\right).
\end{align*}
We notice that the last term of the first expression, and the first term of the second expression are exactly the same, except the signs differ by $-1$. Then the remaining terms overlap in exactly 1 vertex, so we have
\[
((d \phi) \smile \psi)(\sigma) + (-1)^k \phi \smile (d \psi)(\sigma) = (\phi \smile \psi)(d \sigma) = (d (\phi \smile \psi))(\sigma)
\]
as required.
\end{proof}
This is the most interesting thing about these things, because it tells us this gives a well-defined map on cohomology.
\begin{cor}
The cup product induces a well-defined map
\[
\begin{tikzcd}[row sep = 0ex
,/tikz/column 1/.append style={anchor=base east}
,/tikz/column 2/.append style={anchor=base west}
]
\smile: H^k(X; R) \times H^k(X; R) \ar[r] & H^{k + \ell}(X; R)\\
([\phi], [\psi] ) \ar[r, maps to] & \lbrack\phi \smile \psi\rbrack
\end{tikzcd}
\]
\end{cor}
\begin{proof}
To see this is defined at all, as $d \phi = 0 = d \psi$, we have
\[
d (\phi \smile \psi) = (d \phi) \smile \psi \pm \phi \smile (d \psi) = 0.
\]
So $\phi \smile \psi$ is a cocycle, and represents the cohomology class. To see this is well-defined, if $\phi' = \phi + d\tau$, then
\[
\phi' \smile \psi = \phi \smile \psi + d \tau \smile \psi = \phi \smile \psi + d(\tau \smile \psi) \pm \tau \smile (d \psi).
\]
Using the fact that $d \psi = 0$, we know that $\phi' \smile \psi$ and $\phi \smile \psi$ differ by a boundary, so $[\phi' \smile \psi] = [\phi \smile \psi]$. The case where we change $\psi$ is similar.
\end{proof}
Note that the operation $\smile$ is associative on cochains, so associative on $H^*$ too.
Also, there is a map $1: C_0(X) \to R$ sending $\sigma \mapsto 1$ for all $\sigma$. Then we have
\[
[1] \smile [\phi] = [\phi].
\]
So we have
\begin{prop}
$(H^*(X; R), \smile, [1])$ is a unital ring.
\end{prop}
Note that this is not necessarily commutative! Instead, we have the following \term{graded commutative} condition.
\begin{prop}
Let $R$ be a commutative ring. If $\alpha \in H^k(X; R)$ and $\beta \in H^\ell(X; R)$, then we have
\[
\alpha \smile \beta = (-1)^{k\ell}\beta \smile \alpha
\]
\end{prop}
Note that this is only true for the cohomology classes. It is not true in general for the cochains. So we would expect that this is rather annoying to prove.
The proof relies on the following observation:
\begin{prop}
The cup product is natural, i.e.\ if $f: X \to Y$ is a map, and $\alpha, \beta \in H^*(Y; R)$, then
\[
f^*(\alpha \smile \beta) = f^*(\alpha) \smile f^*(\beta).
\]
So $f^*$ is a homomorphism of unital rings.
\end{prop}
\begin{proof}[Proof of previous proposition]
Let $\rho_n: C_n(X) \to C_n(x)$ be given by
\[
\sigma \mapsto (-1)^{n(n + 1)/2} \sigma|_{[v_n, v_{n - 1}, \ldots, v_0]}
\]
The $\sigma|_{[v_n, v_{n - 1}, \ldots, v_0]}$ tells us that we reverse the order of the vertices, and the factor of $(-1)^{n(n + 1)/2}$ is the sign of the permutation that reverses $0, \cdots, n$. For convenience, we write
\[
\varepsilon_n = (-1)^{n (n + 1)/2}.
\]
\begin{claim}
We claim that $\rho_{\Cdot}$ is a chain map, and is chain homotopic to the identity.
\end{claim}
We will prove this later.
Suppose the claim holds. We let $\phi \in C^k(X; R)$ represent $\alpha$ and $\psi \in C^\ell(X; R)$ represent $\beta$. Then we have
\begin{align*}
(\rho^* \phi \smile \rho^* \psi)(\sigma) &= (\rho^* \phi)(\sigma|_{[v_0, \ldots, v_k]} (\rho^* \psi)(\sigma|_{[v_k, \ldots, v_{k + \ell}]})\\
&= \phi(\varepsilon_k \cdot \sigma|_{[v_k, \ldots, v_0]}) \psi(\varepsilon_\ell \sigma|_{[v_{k + \ell}, \ldots, v_k]}).
\end{align*}
Thus, we can compute
\begin{align*}
\rho^*(\psi \smile \phi)(\sigma) &= (\psi \smile \phi)(\varepsilon_{k + \ell} \sigma|_{[v_{k + \ell}, \ldots, v_0]})\\
&= \varepsilon_{k + \ell} \psi(\sigma|_{[v_{k + \ell}, \ldots, v_k}]) \phi(\sigma|_{[v_k, \ldots, v_0]})\\
&= \varepsilon_{k + \ell}\varepsilon_k \varepsilon_\ell (\rho^* \phi \smile \rho^* \psi)(\sigma).
\end{align*}
By checking it directly, we can see that $\varepsilon_{n + \ell}\varepsilon_k \varepsilon_\ell = (-1)^{k\ell}$. So we have
\begin{align*}
\alpha \smile \beta &= [\phi \smile \psi] \\
&= [\rho^* \phi \smile \rho^* \psi] \\
&= (-1)^{k\ell}[\rho^*(\psi \smile \phi)] \\
&= (-1)^{k\ell}[\psi \smile \phi] \\
&= (-1)^{kl} \beta \smile \alpha.
\end{align*}
Now it remains to prove the claim. We have
\begin{align*}
\d \rho(\sigma) &= \varepsilon_n \sum_{i = 0}^n (-1)^j \sigma|_{[v_n, \ldots, \hat{v}_{n - i}, \ldots, v_0]}\\
\rho(\d \sigma) &= \rho\left(\sum_{i = 0}^n (-1)^i \sigma|_{[v_0, \ldots, \hat{v}_i, \ldots., v_n]}\right)\\
&= \varepsilon_{n - 1} \sum_{j = 0}^n (-1)^j \sigma|_{[v_n, \ldots, \hat{v}_j, v_0]}.
\end{align*}
We now notice that $\varepsilon_{n - 1}(-1)^{n - i} = \varepsilon_n (-1)^i$. So this is a chain map!
We now define a chain homotopy. This time, we need a ``twisted prism''. We let
\[
P_n = \sum_i (-1)^i \varepsilon_{n - i} [v_0, \cdots, v_i, w_n, \cdots, w_i] \in C_{n + 1}([0, 1] \times \Delta^n),
\]
where $v_0, \cdots, v_n$ are the vertices of $\{0\} \times \Delta^n$ and $w_0, \cdots, w_n$ are the vertices of $\{1\} \times \Delta^n$.
We let $\pi: [0, 1] \times \Delta^n \to \Delta^n$ be the projection, and let $F_n^X: C_n(X) \to C_{n + 1}(X)$ be given by
\[
\sigma \mapsto (\sigma \circ \pi)_\#(P_n).
\]
We calculate
\begin{align*}
\d F_n^X(\sigma) &= (\sigma \circ \pi)_\#(\d P_n) \\
&= (\sigma \circ \pi_\#)\left(\sum_i \left(\sum_{j \leq i}(-1)^j (-1)^i \varepsilon_{n - i}[v_0, \cdots, \hat{v}_j, \cdots, v_i, w_0, \cdots, w_i]\right)\right.\\
&\quad +\left.\left(\sum_{j \geq i} (-1)^{n + i + 1 - j}(-1)^i \varepsilon_{n - i}[v_0, \cdots, v_i, w_n, \cdots, \hat{w}_j, \cdots, v_i]\right)\right).
\end{align*}
The terms with $j = i$ give
\begin{align*}
&(\sigma \circ \pi)_\#\left(\sum_i \varepsilon_{n - i}[v_0, \cdots, v_{i - 1}, w_n, \cdots, w_i]\right. \\
&\quad+ \left.\sum_i (-1)^{n + 1}(-1)^i \varepsilon_{n - i}[v_0, \cdots, v_i, w_n,\cdots, w_{i + 1}]\right)\\
={}& (\sigma \circ \pi)_\#(\varepsilon_n[w_n,\cdots, w_0] - [v_0, \cdots, v_n])\\
={}& \rho(\sigma) - \sigma
\end{align*}
The terms with $j \not= i$ are precisely $-F_{n - 1}^X (\d \sigma)$ as required. It is easy to see that the terms are indeed the right terms, and we just have to check that the signs are right. I'm not doing that.
\end{proof}
There are some other products we can define. One example is the cross product:
\begin{defi}[Cross product]\index{cross product}
Let $\pi_X: X \times Y \to X$, $\pi_Y: X \times Y\to Y$ be the projection maps. Then we have a \emph{cross product}
\[
\begin{tikzcd}[row sep = 0ex
,/tikz/column 1/.append style={anchor=base east}
,/tikz/column 2/.append style={anchor=base west}
]
\times: H^k(X; R) \otimes_R H^\ell(Y; R) \ar[r] & H^{k + \ell}(X \times Y; R)\\
a \otimes b \ar[r, maps to] & (\pi_X^* a) \smile (\pi_Y^* b)
\end{tikzcd}.
\]
\end{defi}
Note that the diagonal map $\Delta: X \to X \times X$ given by $\Delta(x) = (x, x)$ satisfies
\[
\Delta^*(a \times b) = a \smile b
\]
for all $a, b \in H^*(X; R)$. So these two products determine each other.
There is also a \emph{relative cup} product
\[
\smile: H^k(X, A; R) \otimes H^k(X; R) \to H^{k + \ell}(X, A; R)
\]
given by the same formula. Indeed, to see this is properly defined, note that if $\phi \in C^k(X, A; R)$, then $\phi$ is a map
\[
\phi:C_k(X, A) = \frac{C_k(X)}{C_k(A)} \to R.
\]
In other words, it is a map $C_k(X) \to R$ that vanishes on $C_k(A)$. Then if $\sigma \in C_{k + \ell}(A)$ and $\psi \in C^\ell(X; R)$, then
\[
(\phi \smile \psi)(\sigma) = \phi(\sigma|_{[v_0, \ldots, v_k]}) \cdot \psi(\sigma|_{[v_k, \ldots, v_{k + \ell}]}).
\]
We now notice that $[v_0, \cdots, v_k] \in C_k(A)$. So $\phi$ kills it, and this vanishes. So this is a term in $H^{k + \ell}(X, A; R)$.
You might find it weird that the two factors of the cup product are in different things, but note that a relative cohomology class is in particular a cohomology class. So this restricts to a map
\[
\smile: H^k(X, A; R) \otimes H^k(X, A; R) \to H^{k + \ell}(X, A; R),
\]
but the result we gave is more general.
\begin{eg}
Suppose $X$ is a space such that the cohomology classes are given by
\begin{center}
\begin{tabular}{cccccccc}
$k$ & 0 & 1 & 2 & 3 & 4 & 5 & 6\\
$H^k(X, \Z)$ & $\Z$ & 0 & 0 & $\Z = \bra x\ket$ & 0 & 0 & $\Z = \bra y\ket$
\end{tabular}
\end{center}
What can $x \smile x$ be? By the graded commutativity property, we have
\[
x\smile x = - x \smile x.
\]
So we know $2(x \smile x) = 0 \in H^6(X, \Z) \cong \Z$. So we must have $x \smile x = 0$.
\end{eg}
\section{\texorpdfstring{K\"unneth}{Kunneth} theorem and universal coefficients theorem}
We are going to prove two theorems of similar flavour --- K\"unneth's theorem and the universal coefficients theorem. They are both fairly algebraic results that relate different homology and cohomology groups. They will be very useful when we prove things about (co)homologies in general.
In both cases, we will not prove the ``full'' theorem, as they require knowledge of certain objects known as $\mathrm{Tor}$ and $\Ext$. Instead, we will focus on a particular case where the $\mathrm{Tor}$ and $\Ext$ vanish, so that we can avoid mentioning them at all.
We start with K\"unneth's theorem.
\begin{thm}[K\"unneth's theorem]\index{K\"unneth's theorem}
Let $R$ be a commutative ring, and suppose that $H^n(Y; R)$ is a free $R$-module for each $n$. Then the cross product map
\[
\begin{tikzcd}
\displaystyle\bigoplus_{k + \ell = n}H^k(X; R) \otimes H^\ell(Y; R) \ar[r, "\times"] & H^n(X \times Y; R)
\end{tikzcd}
\]
is an isomorphism for every $n$, for every finite cell complex $X$.
It follows from the five lemma that the same holds if we have a relative complex $(Y, A)$ instead of just $Y$.
\end{thm}
For convenience, we will write $H^*(X; R) \otimes H^*(Y; R)$ for the graded $R$-module which in grade $n$ is given by
\[
\bigoplus_{k + \ell = n} H^k(X; R) \otimes H^\ell (Y; R).
\]
Then K\"unneth says the map given by
\[
\begin{tikzcd}
H^*(X; R) \otimes H^*(Y; R) \ar[r, "\times"] & H^*(X \times Y; R)
\end{tikzcd}
\]
is an isomorphism of graded rings.
\begin{proof}
Let
\[
F^n(-) = \bigoplus_{k + \ell = n} H^k(-; R) \otimes H^\ell(Y; R).
\]
We similarly define
\[
G^n(-) = H^n(-\times Y; R).
\]
We observe that for each $X$, the cross product gives a map $\times: F^n(X) \to G^n(X)$, and, crucially, we know $\times_*: F^n(*) \to G^n(*)$ is an isomorphism, since $F^n(*) \cong G^n(*) \cong H^n(Y; R)$.
The strategy is to show that $F^n(-)$ and $G^n(-)$ have the same formal structure as cohomology and agree on a point, and so must agree on all finite cell complexes.
\separator
It is clear that both $F^n$ and $G^n$ are homotopy invariant, because they are built out of homotopy invariant things.
\separator
We now want to define the cohomology of pairs. This is easy. We define
\begin{align*}
F^n(X, A) &= \bigoplus_{i + j = n} H^i(X, A; R) \otimes H^j(Y; R)\\
G^n(X, A) &= H^n(X \times Y, A \times Y; R).
\end{align*}
Again, the relative cup product gives us a relative cross product, which gives us a map $F^n(X, A) \to G^n(X, A)$.
It is immediate $G^n$ has a long exact sequence associated to $(X, A)$ given by the usual long exact sequence of $(X \times Y, A \times Y)$. We would like to say $F$ has a long exact sequence as well, and this is where our hypothesis comes in.
If $H^*(Y; R)$ is a free $R$-module, then we can take the long exact sequence of $(X, A)$
\[
\begin{tikzcd}[column sep=small]
\cdots \ar[r] & H^n(A; R) \ar[r, "\partial"] & H^n(X , A ; R) \ar[r] & H^n(X ; R) \ar[r] & H^n(A ; R) \ar[r] & \cdots
\end{tikzcd},
\]
and then tensor with $H^j(Y; R)$. This preserves exactness, since $H^j(Y; R) \cong R^k$ for some $k$, so tensoring with $H^j(Y; R)$ just takes $k$ copies of this long exact sequence. By adding the different long exact sequences for different $j$ (with appropriate translations), we get a long exact sequence for $F$.
\separator
We now want to prove K\"unneth by induction on the number of cells and the dimension at the same time. We are going to prove that if $X = X' \cup_f D^n$ for some $S^{n - 1} \to X'$, and $\times: F(X') \to G(X')$ is an isomorphism, then $\times: F(X) \to G(X)$ is also an isomorphism. In doing so, we will assume that the result is true for attaching \emph{any} cells of dimension less than $n$.
Suppose $X = X' \cup_f D^n$ for some $f:S^{n - 1} \to X'$. We get long exact sequences
\[
\begin{tikzcd}
F^{* - 1}(X') \ar[r] \ar[d, "\times", "\sim"'] & F^*(X, X') \ar[r] \ar[d, "\times"] & F^*(X) \ar[r] \ar[d, "\times"] & F^*(X') \ar[r] \ar[d, "\times", "\sim"'] & F^{* + 1}(X, X') \ar[d, "\times"]\\
G^{* - 1}(X') \ar[r] & G^*(X, X') \ar[r] & G^*(X) \ar[r] & G^*(X') \ar[r] & G^{* + 1}(X, X')
\end{tikzcd}
\]
Note that we need to manually check that the boundary maps $\partial$ commute with the cross product, since this is not induced by maps of spaces, but we will not do it here.
Now by the five lemma, it suffices to show that the maps on the relative cohomology $\times: F^n(X, X') \to G^n(X, X')$ is an isomorphism.
We now notice that $F^*(-)$ and $G^*(-)$ have excision. Since $(X, X')$ is a good pair, we have a commutative square
\[
\begin{tikzcd}
F^*(D^n, \partial D^n) \ar[d, "\times"] & F^*(X, X') \ar[l, "\sim"] \ar[d, "\times"] \\
G^*(D^n, \partial D^n) & G^*(X, X') \ar[l, "\sim"]
\end{tikzcd}
\]
So we now only need the left-hand map to be an isomorphism. We look at the long exact sequence for $(D^n, \partial D^n)$!
\[
\begin{tikzcd}[column sep=small]
F^{* - 1}(\partial D^n) \ar[r] \ar[d, "\times", "\sim"'] & F^*(D^n, \partial D^n) \ar[r] \ar[d, "\times", "\sim"'] & F^*(D^n) \ar[r] \ar[d, "\times"] & F^*(\partial D^n) \ar[r] \ar[d, "\times", "\sim"'] & F^{* + 1}(D^n, \partial D^n) \ar[d, "\times", "\times"']\\
G^{* - 1}(\partial D^n) \ar[r] & G^*(D^n, \partial D^n) \ar[r] & G^*(D^n) \ar[r] & G^*(\partial D^n) \ar[r] & G^{* + 1}(D^n, \partial D^n)
\end{tikzcd}
\]
But now we know the vertical maps for $D^n$ and $\partial D^n$ are isomorphisms --- the ones for $D^n$ are because they are contractible, and we have seen the result of $*$ already; whereas the result for $\partial D^n$ follows by induction.
So we are done.
\end{proof}
The conditions of the theorem require that $H^n(Y; R)$ is free. When will this hold? One important example is when $R$ is actually a field, in which case \emph{all} modules are free.
\begin{eg}
Consider $H^*(S^1, \Z)$. We know it is $\Z$ in $* = 0, 1$, and $0$ elsewhere. Let's call the generator of $H^0(S^1, \Z)$ ``$1$'', and the generator of $H^1(S^1, \Z)$ as $x$. Then we know that $x \smile x = 0$ since there isn't anything in degree $2$. So we know
\[
H^*(S^1, \Z) = \Z[x]/(x^2).
\]
Then K\"unneth's theorem tells us that
\[
H^*(T^n, \Z) \cong H^*(S^1; \Z)^{\otimes n},
\]
where $T^n = (S^1)^n$ is the $n$-torus, and this is an isomorphism of rings. So this is
\[
H^*(T^n, \Z) \cong \Z[x_1, \cdots, x_n]/(x_i^2, x_i x_j + x_j x_i),
\]
using the fact that $x_i, x_j$ have degree $1$ so anti-commute. Note that this has an interesting cup product! This ring is known as the \emph{exterior algebra in $n$ generators}.
\end{eg}
\begin{eg}
Let $f: S^n \to T^n$ be a map for $n > 1$. We claim that this induces the zero map on the $n$th cohomology.
We look at the induced map on cohomology:
\[
\begin{tikzcd}
f^*: H^n(T^n; \Z) \ar[r] & H^n(S^n, \Z)
\end{tikzcd}.
\]
Looking at the presentation above, we know that $H^n(T^n, \Z)$ is generated by $x_1 \smile \cdots \smile x_n$, and $f^*$ sends it to $(f^* x_1) \smile \cdots \smile (f^* x_n)$. But $f^*x_i \in H^1(S^n, \Z) = 0$ for all $n > 1$. So $f^* (x_1 \cdots x_n) = 0$.
\end{eg}
Note that the statement does not involve cup products at all, but it would be much more difficult to prove this without using cup products!
We are now going to prove another useful result.
\begin{thm}[Universal coefficients theorem for (co)homology]\index{universal coefficient theorem for (co)homology}
Let $R$ be a PID and $M$ an $R$-module. Then there is a natural map
\[
H_*(X; R)\otimes M \to H_*(X; M).
\]
If $H_*(X; R)$ is a free module for each $n$, then this is an isomorphism. Similarly, there is a natural map
\[
H^*(X; M) \to \Hom_R(H_*(X; R), M),,
\]
which is an isomorphism again if $H^*(X; R)$ is free.
\end{thm}
In particular, when $R = \Z$, then an $R$-module is just an abelian group, and this tells us how homology and cohomology with coefficients in an abelian group relate to the usual homology and cohomology theory.
\begin{proof}
Let $C_n$ be $C_n(X; R)$ and $Z_n \subseteq C_n$ be the cycles and $B_n \subseteq Z_n$ the boundaries. Then there is a short exact sequence
\[
\begin{tikzcd}
0 \ar[r] & Z_n \ar[r, "i"] & C_n \ar[r, "g"] & B_{n - 1} \ar[r] & 0
\end{tikzcd},
\]
and $B_{n - 1} \leq C_{n - 1}$ is a submodule of a free $R$-module, and is free, since $R$ is a PID. So by picking a basis, we can find a map $s: B_{n - 1} \to C_n$ such that $g \circ s = \id_{B_{n - 1}}$. This induces an isomorphism
\[
\begin{tikzcd}
i \oplus s: Z_n \oplus B_{n - 1} \ar[r, "\sim"] & C_n.
\end{tikzcd}
\]
Now tensoring with $M$, we obtain
\[
\begin{tikzcd}
0 \ar[r] & Z_n \otimes M \ar[r] & C_n \otimes M \ar[r] & B_{n - 1} \otimes M \ar[r] & 0
\end{tikzcd},
\]
which is exact because we have
\[
C_n \otimes M \cong (Z_n \oplus B_{n - 1}) \otimes M \cong (Z_n \otimes M) \oplus (B_{n - 1} \otimes M).
\]
So we obtain a short exact sequence of chain complexes
\[
\begin{tikzcd}[column sep=scriptsize]
0 \ar[r] & (Z_n \otimes M, 0) \ar[r] & (C_n \otimes M, d \otimes \id) \ar[r] & (B_{n - 1} \otimes M, 0) \ar[r] & 0
\end{tikzcd},
\]
which gives a long exact sequence in homology:
\[
\begin{tikzcd}[column sep=scriptsize]
\cdots \ar[r] & B_n \otimes M \ar[r] & Z_n \otimes M \ar[r] & H_n(X; M) \ar[r] & B_{n - 1} \otimes M \ar[r] & \cdots
\end{tikzcd}
\]
We'll leave this for a while, and look at another short exact sequence. By definition of homology, we have a long exact sequence
\[
\begin{tikzcd}
0 \ar[r] & B_n \ar[r] & Z_n \ar[r] & H_n(X; R) \ar[r] & 0
\end{tikzcd}.
\]
As $H_n(X; R)$ is free, we have a splitting $t: H_n(X; R) \to Z_n$, so as above, tensoring with $M$ preserves exactness, so we have
\[
\begin{tikzcd}
0 \ar[r] & B_n\otimes M \ar[r] & Z_n\otimes M \ar[r] & H_n(X; R)\otimes M \ar[r] & 0
\end{tikzcd}.
\]
Hence we know that $B_n \otimes M \to Z_n \otimes M$ is injective. So our previous long exact sequence breaks up to
\[
\begin{tikzcd}
0 \ar[r] & B_n \otimes M \ar[r] & Z_n \otimes M \ar[r] & H_n(X; M) \ar[r] & 0.
\end{tikzcd}
\]
Since we have two short exact sequence with first two terms equal, the last terms have to be equal as well.
The cohomology version is similar.
\end{proof}
\section{Vector bundles}
\subsection{Vector bundles}
We now end the series of random topics, and work on a more focused topic. We are going to look at vector bundles. Intuitively, a vector bundle over a space $X$ is a continuous assignment of a vector space to each point in $X$. In the first section, we are just going to look at vector bundles as topological spaces. In the next section, we are going to look at homological properties of vector bundles.
\begin{defi}[Vector bundle]\index{vector bundle}
Let $X$ be a space. A (real) \emph{vector bundle} of dimension $d$ over $X$ is a map $\pi:E \to X$, with a (real) vector space structure on each \term{fiber} $E_x = \pi^{-1}(\{x\})$, subject to the local triviality condition: for each $x \in X$, there is a neighbourhood $U$ of $x$ and a homeomorphism $\varphi:E|_U = \pi^{-1}(U) \to U \times \R^d$ such that the following diagram commutes
\[
\begin{tikzcd}[column sep=0em]
E|_U \ar[rr, "\varphi"] \ar[rd, "\pi"] & & U \times \R^d \ar[dl, "\pi_1"]\\
& U
\end{tikzcd},
\]
and for each $y \in U$, the restriction $\varphi|_{E_y}: E_y \to \{y\} \times \R^d$ is a \emph{linear} isomorphism for each $y \in U$. This maps is known as a \term{local trivialization}.
\end{defi}
We have an analogous definition for complex vector bundles.
\begin{defi}[Section]\index{section}
A \emph{section} of a vector bundle $\pi: E \to X$ is a map $s: X \to E$ such that $\pi \circ s = \id$. In other words, $s(x) \in E_x$ for each $x$.
\end{defi}
\begin{defi}[Zero section]\index{zero section}\index{section!zero}
The \emph{zero section} of a vector bundle is $s_0: X \to E$ given by $s_0(x) = 0 \in E_x$.
\end{defi}
Note that the composition
\[
\begin{tikzcd}
E \ar[r, "\pi"] & X \ar[r, "s_0"] & E
\end{tikzcd}
\]
is homotopic to the identity map on $\id_E$, since each $E_x$ is contractible.
One important operation we can do on vector bundles is \emph{pullback}:
\begin{defi}[Pullback of vector bundles]\index{pullback!vector bundle}\index{vector bundle!pullback}
Let $\pi: E \to X$ be a vector bundle, and $f: Y \to X$ a map. We define the \emph{pullback}
\[
f^* E = \{(y, e) \in Y \times E: f(y) = \pi(e)\}.
\]
This has a map $f^*\pi: f^*E \to Y$ given by projecting to the first coordinate. The vector space structure on each fiber is given by the identification $(f^*E)_y = E_{f(y)}$. It is a little exercise in topology to show that the local trivializations of $\pi: E \to X$ induce local trivializations of $f^*\pi: f^* E \to Y$.
\end{defi}
Everything we can do on vector spaces can be done on vector bundles, by doing it on each fiber.
\begin{defi}[Whitney sum of vector bundles]\index{vector bundle!Whitney sum}\index{Whitney sum of vector bundles}
Let $\pi: E \to F$ and $\rho: F \to X$ be vector bundles. The \emph{Whitney sum} is given by
\[
E \oplus F = \{(e, f)\in E \times F: \pi(e) = \rho(f)\}.
\]
This has a natural map $\pi \oplus \rho: E \oplus F \to X$ given by $(\pi \oplus \rho)(e, f) = \pi(e) = \rho(f)$. This is again a vector bundle, with $(E \oplus F)_x = E_x \oplus F_x$ and again local trivializations of $E$ and $F$ induce one for $E \oplus F$.
\end{defi}
Tensor products can be defined similarly.
Similarly, we have the notion of subbundles.
\begin{defi}[Vector subbundle]\index{vector sub-bundle}\index{vector bundle!subbundle}
Let $\pi: E \to X$ be a vector bundle, and $F \subseteq E$ a subspace such that for each $x \in X$ there is a local trivialization $(U, \varphi)$
\[
\begin{tikzcd}[column sep=0em]
E|_U \ar[rr, "\varphi"] \ar[rd, "\pi"] & & U \times \R^d \ar[dl, "\pi_1"]\\
& U
\end{tikzcd},
\]
such that $\varphi$ takes $F|_U$ to $U \times \R^k$, where $\R^k \subseteq \R^d$. Then we say $F$ is a \emph{vector sub-bundle}.
\end{defi}
\begin{defi}[Quotient bundle]\index{quotient bundle}\index{vector bundle!quotient}
Let $F$ be a sub-bundle of $E$. Then $E/F$, given by the fiberwise quotient, is a vector bundle and is given by the \emph{quotient bundle}.
\end{defi}
We now look at one of the most important example of a vector bundle. In some sense, this is the ``universal'' vector bundle, as we will see later.
\begin{eg}[Grassmannian manifold]\index{Grassmannian manifold}\index{$\Gr_k(\R^n)$}
We let
\[
X = \Gr_k(\R^n) = \{k\text{-dimensional linear subgroups of $\R^n$}\}.
\]
To topologize this space, we pick a fixed $V \in \Gr_k(\R^n)$. Then any $k$-dimensional subspace can be obtained by applying some linear map to $V$. So we obtain a surjection
\begin{align*}
\GL_n(\R) &\to \Gr_k(\R^n)\\
M &\mapsto M(V).
\end{align*}
So we can given $\Gr_k(\R^n)$ the quotient (final) topology. For example,
\[
\Gr_1(\R^{n + 1}) = \RP^n.
\]
Now to construct a vector bundle, we need to assign a vector space to each point in $X$. But a point in $\Gr_k(\R^n)$ \emph{is} a vector space, so we have an obvious definition
\[
E = \{(V, v) \in \Gr_k(\R^n) \times \R^n: v \in V\}.
\]
This has the evident projection $\pi: E \to X$ given by the first projection. We then have
\[
E_V = V.
\]
To see that this is a vector bundle, we have to check local triviality. We fix a $V \in \Gr_k(\R^n)$, and let
\[
U = \{W \in\Gr_k(\R^n): W \cap V^\perp = \{0\}\}.
\]
We now construct a map $\varphi: E|_U \to U \times V \cong U \times \R^k$ by mapping $(W, w)$ to $(W, \pr_V(w))$, where $\pr_V: \R^n \to V$ is the orthogonal projection.
Now if $w \in U$, then $\pr_V(w) \not= 0$ since $W \cap V^\perp = \{0\}$. So $\varphi$ is a homeomorphism. We call this bundle $\gamma_{k, n}^\R \to \Gr_k(\R^n)$.\index{$\gamma_{k, n}^\R$}
In the same way, we can get a canonical complex vector bundle $\gamma_{k, n}^\C \to \Gr_k(\C^n)$.
\end{eg}
\begin{eg}
Let $M$ be a smooth $d$-dimensional manifold, then it naturally has a $d$-dimensional \term{tangent bundle} $\pi: TM \to M$ with $(TM)|_x = T_x M$.
If $M \subseteq N$ is a smooth submanifold, with $i$ the inclusion map, then $TM$ is a subbundle of $i^* TN$. Note that we cannot say $TM$ is a smooth subbundle of $TN$, since they have different base space, and thus cannot be compared without pulling back.
The \term{normal bundle} of $M$ in $N$ is
\[
\nu_{M \subseteq N} = \frac{i^* TN}{TM}.
\]
\end{eg}
Here is a theorem we have to take on faith, because proving it will require some differential geometry.
\begin{thm}[Tubular neighbourhood theorem]\index{tubular neighbourhood theorem}
Let $M \subseteq N$ be a smooth submanifold. Then there is an open neighbourhood $U$ of $M$ and a homeomorphism $\nu_{M \subseteq N} \to U$, and moreover, this homeomorphism is the identity on $M$ (where we view $M$ as a submanifold of $\nu_{M \subseteq N}$ by the image of the zero section).
\end{thm}
This tells us that locally, the neighbourhood of $M$ in $N$ looks like $\nu_{M \subseteq N}$.
We will also need some results from point set topology:
\begin{defi}[Partition of unity]\index{partition of unity}
Let $X$ be a compact Hausdorff space, and $\{U_\alpha\}_{\alpha \in I}$ be an open cover. A \emph{partition of unity subordinate to $\{U_\alpha\}$} is a collection of functions $\lambda_\alpha: X \to [0, \infty)$ such that
\begin{enumerate}
\item $\supp(\lambda_\alpha) = \overline{\{x \in X: \lambda_\alpha(x) > 0\}} \subseteq U_\alpha$.
\item Each $x \in X$ lies in finitely many of these $\supp (\lambda_\alpha)$.
\item For each $x$, we have
\[
\sum_{\alpha \in I}\lambda_\alpha(x) = 1.
\]
\end{enumerate}
\end{defi}
\begin{prop}
Partitions of unity exist for any open cover.
\end{prop}
You might have seen this in differential geometry, but this is easier, since we do not require the partitions of unity to be smooth.
Using this, we can prove the following:
\begin{lemma}
Let $\pi: E \to X$ be a vector bundle over a compact Hausdorff space. Then there is a continuous family of inner products on $E$. In other words, there is a map $E \otimes E \to \R$ which restricts to an inner product on each $E_x$.
\end{lemma}
\begin{proof}
We notice that every trivial bundle has an inner product, and since every bundle is locally trivial, we can patch these up using partitions of unity.
Let $\{U_\alpha\}_{\alpha \in I}$ be an open cover of $X$ with local trivializations
\[
\varphi_\alpha: E|_{U_\alpha} \to U_\alpha \times \R^d.
\]
The inner product on $\R^d$ then gives us an inner product on $E|_{U_\alpha}$, say $\bra \ph, \ph\ket_\alpha$. We let $\lambda_\alpha$ be a partition of unity associated to $\{U_\alpha\}$. Then for $u\otimes v \in E \otimes E$, we define
\[
\bra u, v\ket = \sum_{\alpha \in I} \lambda_\alpha(\pi(u)) \bra u, v\ket_\alpha.
\]
Now if $\pi(u) = \pi(v)$ is not in $U_\alpha$, then we don't know what we mean by $\bra u, v\ket_\alpha$, but it doesn't matter, because $\lambda_\alpha(\pi(u)) = 0$. Also, since the partition of unity is locally finite, we know this is a finite sum.
It is then straightforward to see that this is indeed an inner product, since a positive linear combination of inner products is an inner product.
\end{proof}
Similarly, we have
\begin{lemma}
Let $\pi: E \to X$ be a vector bundle over a compact Hausdorff space. Then there is some $N$ such that $E$ is a vector subbundle of $X \times \R^N$.
\end{lemma}
\begin{proof}
Let $\{U_\alpha\}$ be a trivializing cover of $X$. Since $X$ is compact, we may wlog assume the cover is finite. Call them $U_1, \cdots, U_n$. We let
\[
\varphi_i: E|_{U_i} \to U_i \times \R^d.
\]
We note that on each patch, $E|_{U_i}$ embeds into a trivial bundle, because it \emph{is} a trivial bundle. So we can add all of these together. The trick is to use a partition of unity, again.
We define $f_i$ to be the composition
\[
\begin{tikzcd}
E|_{U_i} \ar[r, "\varphi_i"] & U_i \times \R^d \ar[r, "\pi_2"] & \R^d
\end{tikzcd}.
\]
Then given a partition of unity $\lambda_i$, we define
\begin{align*}
f: E &\to X \times (\R^d)^n\\
v&\mapsto (\pi(v), \lambda_1(\pi(v)) f_1(v), \lambda_2(\pi(v)) f_2(v), \cdots, \lambda_n(\pi(v)) f_n(v)).
\end{align*}
We see that this is injective. If $v, w$ belong to different fibers, then the first coordinate distinguishes them. If they are in the same fiber, then there is some $U_i$ with $\lambda_i(\pi(u)) \not= 0$. Then looking at the $i$th coordinate gives us distinguishes them. This then exhibits $E$ as a subbundle of $X \times \R^n$.
\end{proof}
\begin{cor}
Let $\pi: E \to X$ be a vector bundle over a compact Hausdorff space. Then there is some $p: F \to X$ such that $E \oplus F \cong X \times \R^n$. In particular, $E$ embeds as a subbundle of a trivial bundle.
\end{cor}
\begin{proof}
By above, we can assume $E$ is a subbundle of a trivial bundle. We can then take the orthogonal complement of $E$.
\end{proof}
Now suppose again we have a vector bundle $\pi: E\to X$ over a compact Hausdorff $X$. We can then choose an embedding $E \subseteq X \times \R^N$, and then we get a map $f_\pi: X \to \Gr_d (\R^N)$ sending $x$ to $E_x \subseteq \R^N$. Moreover, if we pull back the tautological bundle along $f_\pi$, then we have
\[
f_{\pi}^* \gamma_{k, N}^\R \cong E.
\]
So every vector bundle is the pullback of the canonical bundle $\gamma_{k, N}^\R$ over a Grassmannian. However, there is a slight problem. Different vector bundles will require different $N$'s. So we will have to overcome this problem if we want to make a statement of the sort ``a vector bundle is the same as a map to a Grassmannian''.
The solution is to construct some sort of $\Gr_d(\R^\infty)$. But infinite-dimensional vector spaces are weird. So instead we take the union of all $\Gr_d(\R^N)$. We note that for each $N$, there is an inclusion $\Gr_d(\R^N) \to \Gr_d(\R^{n + 1})$, which induces an inclusion of the canonical bundle. We can then take the union to obtain $\Gr_d(\R^\infty)$ with a canonical bundle $\gamma_d^\R$. Then the above result shows that each vector bundle over $X$ is the pullback of the canoncial bundle $\gamma_d^\R$ along some map $f: X \to \Gr_d(\R^\infty)$.
Note that a vector bundle over $X$ does not uniquely specify a map $X \to \Gr_d(\R^\infty)$, as there can be multiple embeddings of $X$ into the trivial bundle $X \times \R^N$. Indeed, if we wiggle the embedding a bit, then we can get a new bundle. So we don't have a coorespondence between vector bundles $\pi: E \to X$ and maps $f_\pi: X \to \Gr_d(\R^\infty)$.
The next best thing we can hope for is that the ``wiggle the embedding a bit'' is all we can do. More precisely, two maps $f, g: X \to \Gr_d(\R^\infty)$ pull back isomorphic vector bundles if and only if they are homotopic. This is indeed true:
\begin{thm}
There is a correspondence
\[
\begin{tikzcd}[/tikz/column 1/.append style={anchor=base east}
,/tikz/column 2/.append style={anchor=base west}
,row sep=tiny]
\left\{\parbox{3cm}{\centering homotopy classess of maps $f: X \to \Gr_d(\R^\infty)$}\right\} \ar[r, leftrightarrow] & \left\{\parbox{3cm}{\centering $d$-dimensional vector bundles $\pi: E \to X$}\right\}\\
\lbrack f\rbrack \ar[r, maps to] & f^* \gamma_d^\R\\
\lbrack f_\pi\rbrack & \pi \ar[l, maps to]
\end{tikzcd}
\]
\end{thm}
The proof is mostly technical, and is left as an exercise on the example sheet.
\subsection{Vector bundle orientations}
We are now going to do something that actually involves algebraic topology. Unfortunately, we have to stop working with arbitrary bundles, and focus on \emph{orientable} bundles instead. So the first thing to do is to define orientability.
What we are going to do is to come up with a rather refined notion of orientability. For each commutative ring $R$, we will have the notion of \emph{$R$-orientability}. The strength of this condition will depend on what $R$ is --- any vector bundle is $\F_2$ orientable, while $\Z$-orientability is the strongest --- if a vector bundle is $\Z$-orientable, then it is $R$-orientable for all $R$.
While there isn't really a good geometric way to think about general $R$-orientability, the reason for this more refined notion is that whenever we want things to be true for (co)homology with coefficients in $R$, then we need the bundle to be $R$-orientable.
Let's begin. Let $\pi: E \to X$ be a vector bundle of dimension $d$. We write
\[
E^\# = E \setminus s_0(X).
\]
We now look at the relative homology groups
\[
H^i(E_x, E_x^\#; R),
\]
where $E_x^\# = E_x \setminus \{0\}$.
We know $E_x$ is a $d$-dimensional vector space. So we can choose an isomorphism $E_x \to \R^d$. So after making this choice, we know that
\[
H^i(E_x, E_x^\#; R) \cong H^i(\R^d, \R^d \setminus \{0\}; R) =
\begin{cases}
R & i = d\\
0 & \text{otherwise}
\end{cases}
\]
However, there is no canonical generator of $H^d(E_x, E_x^\#; R)$ as an $R$-module, as we had to pick an isomorphism $E_x \cong \R^d$.
\begin{defi}[$R$-orientation]\index{$R$-orientation}\index{orientation}\index{local orientation}\index{local $R$-orientation}
A \term{local $R$-orientation} of $E$ at $x \in X$ is a choice of $R$-module generator $\varepsilon_x \in H^d(E_x, E_x^\#; R)$.
An \term{$R$-orientation} is a choice of local $R$-orientation $\{\varepsilon_x\}_{x \in X}$ which are compatible in the following way: if $U\subseteq X$ is open on which $E$ is trivial, and $x, y \in U$, then under the homeomorphisms (and in fact linear isomorphisms):
\[
\begin{tikzcd}
E_x \ar[rd, hook] \ar[rrrd, bend left, "h_x"'] \\
& E|_U \ar[r, "\varphi_\alpha", "\cong"']& U \times \R^d \ar[r, "\pi_2"] & \R^d\\
E_y \ar[ru, hook] \ar[rrru, bend right, "h_y"]
\end{tikzcd}
\]
the map
\[
h_y^* \circ (h_x^{-1})^*: H^d(E_x, E_x^\#; R) \to H^d(E_y, E_y^\#; R)
\]
sends $\varepsilon_x$ to $\varepsilon_y$. Note that this definition does not depend on the choice of $\varphi_U$, because we used it twice, and they cancel out.
\end{defi}
It seems pretty horrific to construct an orientation. However, it isn't really that bad. For example, we have
\begin{lemma}
Every vector bundle is $\F_2$-orientable.
\end{lemma}
\begin{proof}
There is only one possible choice of generator.
\end{proof}
In the interesting cases, we are usually going to use the following result to construct orientations:
\begin{lemma}
If $\{U_\alpha\}_{\alpha \in I}$ is a family of covers such that for each $\alpha, \beta \in I$, the homeomorphism
\[
\begin{tikzcd}
(U_\alpha \cap U_\beta) \times \R^d & E|_{U_\alpha \cap U_\beta} \ar[l, "\cong", "\varphi_\alpha"'] \ar[r, "\cong", "\varphi_\beta"'] & (U_\alpha \cap U_\beta) \times \R^d
\end{tikzcd}
\]
gives an orientation preserving map from $(U_\alpha \cap U_\beta) \times \R^d$ to itself, i.e.\ has a positive determinant on each fiber, then $E$ is orientable for any $R$.
\end{lemma}
Note that we don't have to check the determinant at each point on $U_\alpha \cap U_\beta$. By continuity, we only have to check it for one point.
\begin{proof}
Choose a generator $u \in H^d(\R^d, \R^d \setminus \{0\}; R)$. Then for $x \in U_\alpha$, we define $\varepsilon_x$ by pulling back $u$ along
\[
\begin{tikzcd}
E_x \ar[r, hook] & E|_{U_\alpha} \ar[r, "\varphi_\alpha"] & U_\alpha \times \R^d \ar[r, "\pi_2"] & \R^d
\end{tikzcd}.\tag{$\dagger_\alpha$}
\]
If $x \in U_\beta$ as well, then the analogous linear isomorphism $\dagger_\alpha$ differs from $\dagger_\beta$ by post-composition with a linear map $L: \R^d \to \R^d$ of \emph{positive} determinant. We now use the fact that any linear map of positive determinant is homotopic to the identity. Indeed, both $L$ and $\id$ lies in $\GL_d^+(\R)$, a connected group, and a path between them will give a homotopy between the maps they represent. So we know $(\dagger_\alpha)$ is homotopic to $(\dagger_\beta)$. So they induce the same maps on cohomology classes.
\end{proof}
Now if we don't know that the maps have positive determinant, then $(\dagger_\alpha)$ and $(\dagger_\beta)$ might differ by a sign. So in any ring $R$ where $2 = 0$, we know every vector bundle is $R$-orientable. This is a generalization of the previous result for $\F_2$ we had.
\subsection{The Thom isomorphism theorem}
We now get to the main theorem about vector bundles.
\begin{thm}[Thom isomorphism theorem]
Let $\pi: E \to X$ be a $d$-dimensional vector bundle, and $\{\varepsilon_x\}_{x \in X}$ be an $R$-orientation of $E$. Then
\begin{enumerate}
\item $H^i(E, E^\#; R) = 0$ for $i < d$.
\item There is a unique class $u_E \in H^d(E, E^\#; R)$ which restricts to $\varepsilon_x$ on each fiber. This is known as the \term{Thom class}.
\item The map $\Phi$ given by the composition
\[
\begin{tikzcd}[column sep=large]
H^i(X; R) \ar[r, "\pi^*"] & H^i(E; R) \ar[r, "-\smile u_E"] & H^{i + d}(E, E^\#; R)
\end{tikzcd}
\]
is an isomorphism.
\end{enumerate}
Note that (i) follows from (iii), since $H^i(X; R) = 0$ for $i < 0$.
\end{thm}
Before we go on and prove this, we talk about why it is useful.
\begin{defi}[Euler class]
Let $\pi: E \to X$ be a vector bundle. We define the \term{Euler class} $e(E) \in H^d(X; R)$ by the image of $u_E$ under the composition
\[
\begin{tikzcd}
H^d(E, E^\#; R) \ar[r] & H^d(E; R) \ar[r, "s_0^*"] & H^d(X; R)
\end{tikzcd}.
\]
\end{defi}
This is an example of a \term{characteristic class}, which is a cohomology class related to an oriented vector bundle that behaves nicely under pullback. More precisely, given a vector bundle $\pi: E \to X$ and a map $f: Y \to X$, we can form a pullback
\[
\begin{tikzcd}
f^*E \ar[r, "\hat{f}"] \ar[d, "f^*\pi"] & E \ar[d, "\pi"]\\
Y \ar[r, "f"] & X
\end{tikzcd}.
\]
Since we have a fiberwise isomorphism $(f^*E)_y \cong E_{f(y)}$, an $R$-orientation for $E$ induces one for $f^* E$, and we know $f^*(u_E) = u_{f^* E}$ by uniqueness of the Thom class. So we know
\[
e(f^*(E)) = f^* e(E) \in H^d(Y; R).
\]
Now the Euler class is a cohomology class of $X$ itself, so we can use the Euler class to compare and contrast different vector bundles.
How can we think of the Euler class? It turns out the Euler class gives us an obstruction to the vector bundle having a non-zero section.
\begin{thm}
If there is a section $s: X \to E$ which is nowhere zero, then $e(E) = 0 \in H^d(X; R)$.
\end{thm}
\begin{proof}
Notice that any two sections of $E \to X$ are homotopic. So we have $e \equiv s_0^* u_E = s^* u_E$. But since $u_E \in H^d(E, E^\#; R)$, and $s$ maps into $E^\#$, we have $s^* u_E$.
Perhaps more precisely, we look at the long exact sequence for the pair $(E, E^\#)$, giving the diagram
\[
\begin{tikzcd}
H^d(E, E^\#; R) \ar[r] & H^d(E; R) \ar[r] \ar[d, "s_0^*"] & H^d(E^\#; R) \ar[dl, "s^*"]\\
& H^d(X; R)
\end{tikzcd}
\]
Since $s$ and $s_0$ are homotopic, the diagram commutes. Also, the top row is exact. So $u_E \in H^d(E, E^\#; R)$ gets sent along the top row to $0 \in H^d(E^\#; R)$, and thus $s^*$ sends it to $0 \in H^d(X; R)$. But the image in $H^d(X; R)$ is exactly the Euler class. So the Euler class vanishes.
\end{proof}
Now cohomology is not only a bunch of groups, but also a ring. So we can ask what happens when we cup $u_E$ with itself.
\begin{thm}
We have
\[
u_E \smile u_E = \Phi(e(E)) = \pi^*(e(E)) \smile u_E \in H^*(E, E^\#; R).
\]
\end{thm}
This is just tracing through the definitions.
\begin{proof}
By construction, we know the following maps commute:
\[
\begin{tikzcd}
H^d(E, E^\#; R) \otimes H^d(E, E^\#; R) \ar[r, "\smile"] \ar[d, "q^* \otimes \id"] & H^{2d}(E, E^\#; R)\\
H^d(E; R) \otimes H^d(E, E^\#; R) \ar[ur, "\smile"']
\end{tikzcd}
\]
We claim that the Thom class $u_E \otimes u_E \in H^d(E, E^\#; R) \otimes H^d(E, E^\#; R)$ is sent to $\pi^*(e(E)) \otimes u_E \in H^d(E; R) \otimes H^d(E, E^\#; R)$.
By definition, this means we need
\[
q^* u_E = \pi^*(e(E)),
\]
and this is true because $\pi^*$ is homotopy inverse to $s_0^*$ and $e(E) = s_0^* q^* u_E$.
\end{proof}
So if we have two elements $\Phi(c), \Phi(d) \in H^*(E, E^\#; R)$, then we have
\begin{align*}
\Phi(c) \smile \Phi(d) &= \pi^*c \smile u_E \smile \pi^*d \smile u_E \\
&= \pm \pi^*c \smile \pi^*d \smile u_E \smile u_E \\
&= \pm \pi^*(c \smile d \smile e(E)) \smile u_E\\
&= \pm \Phi(c \smile d \smile e(E)).
\end{align*}
So $e(E)$ is precisely the information necessary to recover the cohomology \emph{ring} $H^*(E, E^\#; R)$ from $H^*(X; R)$.
\begin{lemma}
If $\pi: E \to X$ is a $d$-dimensional $R$-module vector bundle with $d$ odd, then $2e(E) = 0 \in H^d(X; R)$.
\end{lemma}
\begin{proof}
Consider the map $\alpha: E \to E$ given by negation on each fiber. This then gives an isomorphism
\[
\begin{tikzcd}
a^*: H^d(E, E^\#; R) \ar[r, "\cong"] & H^d(E, E^\#; R).
\end{tikzcd}
\]
This acts by negation on the Thom class, i.e.
\[
a^*(u_E) = - u_E,
\]
as on the fiber $E_x$, we know $a$ is given by an odd number of reflections, each of which acts on $H^d(E_x, E_x^\#; R)$ by $-1$ (by the analogous result on $S^n$). So we change $\varepsilon_x$ by a sign. We then lift this to a statement about $u_E$ by the fact that $u_E$ is the unique thing that restricts to $\varepsilon_x$ for each $x$.
But we also know
\[
a \circ s_0 = s_0,
\]
which implies
\[
s_0^*(a^*(u_E)) = s_0^*(u_E).
\]
Combining this with the result that $a^*(u_E) = -u_E$, we get that
\[
2 e(E) = 2 s_0^*(u_E) = 0.\qedhere
\]
\end{proof}
This is a disappointing result, because if we already know that $H^d(X; R)$ has no $2$-torsion, then $e(E) = 0$.
After all that fun, we prove the Thom isomorphism theorem.
\begin{proof}[Proof of Thom isomorphism theorem]
We will drop the ``$R$'' in all our diagrams for readability (and also so that it fits in the page).
We first consider the case where the bundle is trivial, so $E = X \times \R^d$. Then we note that
\[
H^*(\R^d, \R^d \setminus \{0\}) =
\begin{cases}
R & * = d\\
0 & * \not= d
\end{cases}.
\]
In particular, the modules are free, and (a relative version of) K\"unneth's theorem tells us the map
\[
\begin{tikzcd}
\times: H^*(X) \otimes H^*(\R^d, \R^d \setminus \{0\}) \ar[r, "\cong"] & H^* (X \times \R^d, X \times (\R^d\setminus \{0\}))
\end{tikzcd}
\]
is an isomorphism. Then the claims of the Thom isomorphism theorem follow immediately.
\begin{enumerate}
\item For $i < d$, all the summands corresponding to $H^i(X \times \R^d, X \times (\R^d \setminus\{0\}))$ vanish since the $H^*(\R^d, \R^d \setminus\{0\})$ term vanishes.
\item The only non-vanishing summand for $H^d(X \times \R^d, X \times (\R^d \setminus\{0\})$ is
\[
H^0(X) \otimes H^d(\R^d, \R^d \setminus \{0\}).
\]
Then the Thom class must be $1 \otimes u$, where $u$ is the object corresponding to $\varepsilon_x \in H^d(E_x, E_x^\#) = H^d(\R^d, \R^d \setminus\{0\})$, and this is unique. % explain more
\item We notice that $\Phi$ is just given by
\[
\Phi(x) = \pi^*(x) \smile u_E = x \times u_E,
\]
which is an isomorphism by K\"unneth.
\end{enumerate}
\separator
We now patch the result up for a general bundle. Suppose $\pi: E \to X$ is a bundle. Then it has an open cover of trivializations, and moreover if we assume our $X$ is compact, there are finitely many of them. So it suffices to show that if $U, V \subseteq X$ are open sets such that the Thom isomorphism the holds for $E$ restricted to $U, V, U \cap V$, then it also holds on $U \cup V$.
The relative Mayer-Vietoris sequence gives us
\[
\begin{tikzcd}
H^{d - 1}(E|_{U \cap V}, E^\#|_{U \cap V}) \ar[r, "\partial^{MV}"] & H^d(E|_{U\cup V}, E^\#|_{U\cup V}) \ar[out=0, in=180, looseness=2, overlay, dl]\\
H^d(E|_U, E^\#|_U) \oplus H^d(E|_V, E^\#|_V) \ar[r] & H^d(E|_{U \cap V}, E^\#|_{U \cap V}).
\end{tikzcd}
\]
We first construct the Thom class. We have
\[
u_{E|_V} \in H^d(E|_V, E^\#),\quad u_{E|_U} \in H^d(E|_U, E^\#).
\]
We claim that $(u_{E|_U}, u_{E|_V}) \in H^d(E|_U, E^\#|_U) \oplus H^d(E|_V, E^\#|_V)$ gets sent to $0$ by $i_U^* - i_V^*$. Indeed, both the restriction of $u_{E|_U}$ and $u_{E|_V}$ to $U \cap V$ are Thom classes, so they are equal by uniqueness, so the difference vanishes.
Then by exactness, there must be some $u_{E|_{U\cup V}} \in H^d(E|_{U \cup V}, E^\#|_{U \cup V})$ that restricts to $u_{E|_U}$ and $u_{E|_V}$ in $U$ and $V$ respectively. Then this must be a Thom class, since the property of being a Thom class is checked on each fiber. Moreover, we get uniqueness because $H^{d - 1}(E|_{U \cap V}, E^\#|_{U \cap V}) = 0$, so $u_{E|_U}$ and $u_{E|_V}$ must be the restriction of a unique thing.
The last part in the Thom isomorphism theorem come from a routine application of the five lemma, and the first part follows from the last as previously mentioned.
\end{proof}
\subsection{Gysin sequence}
Now we do something interesting with vector bundles. We will come up with a long exact sequence associated to a vector bundle, known as the \emph{Gysin sequence}. We will then use the Gysin sequence to deduce something about the \emph{base} space.
Suppose we have a $d$-dimensional vector bundle $\pi: E \to X$ that is $R$-oriented. We want to talk about the unit sphere in every fiber of $E$. But to do so, we need to have a notion of length, and to do that, we want an inner product. But luckily, we do have one, and we know that any two norms on a finite-dimensional vector space are equivalent. So we might as well arbitrarily choose one.
\begin{defi}[Sphere bundle]\index{sphere bundle}
Let $\pi: E \to X$ be a vector bundle, and let $\bra \ph, \ph\ket : E \otimes E \to \R$ be an inner product, and let
\[
S(E) = \{v \in E; \bra v, v\ket = 1\} \subseteq E.
\]
This is the \emph{sphere bundle} associated to $E$.
\end{defi}
Since the unit sphere is homotopy equivalent to $\R^d \setminus \{0\}$, we know the inclusion
\[
\begin{tikzcd}
j: S(E) \ar[r, hook] & E^\#
\end{tikzcd}
\]
is a homotopy equivalence, with inverse given by normalization.
The long exact sequence for the pair $(E, E^\#)$ gives (as before, we do not write the $R$):
\[
\begin{tikzcd}
H^{i + d}(E, E^\#) \ar[r] & H^{i + d}(E) \ar[d, "s_0^*", xshift=2] \ar[r] & H^{i + d} (E^\#) \ar[r] \ar[d, "j^*"] & H^{i + d + 1}(E, E^\#)\\
H^i(X) \ar[u, "\Phi"] \ar[r, "\ph \smile e(E)"] & H^{i + d}(X) \ar[u, xshift=-2, "\pi^*"] \ar[r, "p^*"] & H^{i + d}(S(E)) \ar[r, "p_!"] & H^{i + 1}(X)\ar[u, "\Phi"]
\end{tikzcd}
\]
where $p: S(E) \to E$ is the projection, and $p_!$ is whatever makes the diagram commutes (since $j^*$ and $\Phi$ are isomorphisms). The bottom sequence is the \term{Gysin sequence}, and it is exact because the top row is exact. This is in fact a long exact sequence of $H^*(X; R)$-modules, i.e.\ the maps commute with cup products.
\begin{eg}
Let $L = \gamma_{1, n + 1}^\C \to \CP^n = \Gr_1(\C^{n + 1})$ be the tautological $1$ complex dimensional vector bundle on $\Gr_1(\C^{n +1})$. This is $\Z$-oriented as any complex vector bundle is, because if we consider the inclusion
\[
\GL_1(\C) \hookrightarrow \GL_2(\R)
\]
obtained by pretending $\C$ is $\R^2$, we know $\GL_1(\C)$ is connected, so lands in the component of the identity, so has positive determinant. The sphere bundle consists of
\[
S(L) = \{(V, v) \in \CP^n \times \C^{n + 1}: v \in V, |v| = 1\} \cong \{v \in \C^{n + 1}: |v| = 1\} \cong S^{2n + 1},
\]
where the middle isomorphism is given by
\[
\begin{tikzcd}[row sep=tiny]
(V, v) \ar[r, maps to] & v\\
(\bra v\ket, v) & v \ar[l, maps to]
\end{tikzcd}
\]
The Gysin sequence is
\[
\begin{tikzcd}
H^{i + 1}(S^{2n + 1}) \ar[r, "p_!"] & H^i(\CP^n) \ar[r, "\smile e(L)"] & H^{i + 2}(\CP^n) \ar[r, "p^*"] & H^{i + 2}(S^{2n + 1})
\end{tikzcd}
\]
Now if $i \leq 2n - 2$, then both outer terms are $0$. So the maps in the middle are isomorphisms. Thus we get isomorphisms
\[
\begin{tikzcd}[row sep=tiny]
H^0(\CP^n) \ar[d, equals] \ar[r, "\smile e(L)"] & H^2(\CP^n) \ar[d, equals] \ar[r, "\smile e(L)"] & H^4(\CP^n) \ar[d, equals] \ar[r] & \cdots\\
\Z \cdot 1 & \Z \cdot e(L) & \Z \cdot e(L)^2
\end{tikzcd}
\]
Similarly, we know that the terms in the odd degree vanish.
Checking what happens at the end points carefully, the conclusion is that
\[
H^*(\CP^n) = \Z[e(L)] / (e(L)^{n + 1})
\]
as a ring.
\end{eg}
\begin{eg}
We do the real case of the above computation. We have
\[
K = \gamma_{1, n + 1}^\R \to \RP^n = \Gr_1(\R^{n + 1}).
\]
The previous trick doesn't work, and indeed this isn't $\Z$-orientable. However, it is $\F_2$-oriented as every vector bundle is, and by exactly the same argument, we know
\[
S(L) \cong S^n.
\]
So by the same argument as above, we will find that
\[
H^*(\RP^n, \F_2) \cong \F_2[e(L)]/(e(L)^{n + 1}).
\]
Note that this is different from the one we had before, because here $\deg e(L) = 1$, while the complex case had degree $2$.
\end{eg}
\section{Manifolds and \texorpdfstring{Poincar\'e}{Poincare} duality}
We are going to prove Poincar\'e duality, and then use it to prove a lot of things about manifolds. Poincar\'e duality tells us that for a compact oriented manifold $M$ of dimension $n$, we have
\[
H_d(M) \cong H^{n - d}(M).
\]
To prove this, we will want to induct over covers of $M$. However, given a compact manifold, the open covers are in general not compact. We get compactness only when we join all of them up. So we need to come up with a version of Poincar\'e duality that works for non-compact manifolds, which is less pretty and requires the notion of compactly supported cohomology.
\subsection{Compactly supported cohomology}
\begin{defi}[Support of cochain]\index{support!cochain}
Let $\varphi \in C^n(X)$ be a cochain. We say $\varphi$ has \term{support} in $S \subseteq X$ if whenever $\sigma: \Delta^n \hookrightarrow X \setminus S \subseteq X$, then $\varphi(\sigma) = 0$. In this case, $\d \varphi$ also has support in $S$.
\end{defi}
Note that this has a slight subtlety. The definition only requires that if $\sigma$ lies \emph{completely} outside $S$, then $\varphi(\sigma)$ vanishes. However, we can have simplices that extends very far and only touches $S$ slightly, and the support does not tell us anything about the value of $\sigma$. Later, we will get around this problem by doing sufficient barycentric subdivision.
\begin{defi}[Compactly-supported cochain]\index{compactly-supported cochain}
Let $C_c^\Cdot(X) \subseteq C^\Cdot(X)$ be the sub-chain complex consisting of these $\varphi$ which has support in \emph{some} compact $K \subseteq X$.
\end{defi}
Note that this makes sense --- we have seen that if $\varphi$ has support in $K$, then $\d \varphi$ has support in $K$. To see it is indeed a sub-chain complex, we need to show that $C_c^\Cdot(X)$ is a subgroup! Fortunately, if $\varphi$ has support on $K$, and $\psi$ has support in $L$, then $\varphi + \psi$ has support in $K \cup L$, which is compact.
\begin{defi}[Compactly-supported cohomology]\index{compactly-supported cohomology}
The \emph{compactly supported cohomology} of $X$ is
\[
H^*_c(X) = H^*(C_c^\Cdot(X)).
\]
\end{defi}
Note that we can write
\[
C_c^{\Cdot}(X) = \bigcup_{K\text{ compact}} C^\Cdot(X, X \setminus K) \subseteq C^\Cdot(X).
\]
We would like to say that the compactly supported \emph{cohomology} is also ``built out of'' those relative cohomology, but we cannot just take the union, because the relative cohomology is not a subgroup of $H^*(X)$. To do that, we need something more fancy.
\begin{defi}[Directed set]\index{directed set}
A \emph{directed set} is a partial order $(I, \leq)$ such that for all $i, j \in I$, there is some $k \in I$ such that $i \leq k$ and $j \leq k$.
\end{defi}
\begin{eg}
Any total order is a directed system.
\end{eg}
\begin{eg}
$\N$ with divisibility $\mid$ as the partial order is a directed system.
\end{eg}
\begin{defi}[Direct limit]\index{direct system}\index{direct limit}
Let $I$ be a directed set. An \emph{direct system} of abelian groups indexed by $I$ is a collection of abelian groups $G_i$ for each $i \in I$ and homomorphisms
\[
\rho_{ij}: G_i \to G_j
\]
for all $i, j \in I$ such that $i \leq j$, such that
\[
\rho_{ii} = \id_{G_i}
\]
and
\[
\rho_{ik} = \rho_{jk} \circ \rho_{ij}
\]
whenever $i \leq j \leq k$.
We define the \emph{direct limit} on the system $(G_i, \rho_{ij})$ to be
\[
\varinjlim_{i \in I} G_i = \left(\bigoplus_{i \in I} G_i\right)/\bra x - \rho_{ij}(x): x \in G_i\ket.
\]
The underlying set of it is
\[
\left(\coprod_{i \in I}G_i\right)/\{x \sim \rho_{ij}(x): x \in G_i\}.
\]
\end{defi}
In terms of the second description, the group operation is given as follows: given $x \in G_i$ and $y \in G_j$, we find some $k$ such that $i, j \leq k$. Then we can view $x, y$ as elements as $G_k$ and do the operation there. It is an exercise to show that these two descriptions are indeed the same.
Now observe that if $J \subseteq I$ is a sub-directed set such that for all $a \in I$, there is some $b \in J$ such that $a \leq b$. Then we have
\[
\varinjlim_{i \in J} G_i \cong \varinjlim_{i \in I} G_i.
\]
So our claim is now
\begin{thm}
For any space $X$, we let
\[
\mathcal{K}(X) = \{K \subseteq X: K\text{ is compact}\}.
\]
This is a directed set under inclusion, and the map
\[
K \mapsto H^n(X, X \setminus K)
\]
gives a direct system of abelian groups indexed by $\mathcal{K}(X)$, where the maps $\rho$ are given by restriction.
Then we have
\[
H^*_c(X) \cong \varinjlim_{\mathbf{K}(X)} H^n(X, X \setminus K).
\]
\end{thm}
\begin{proof}
We have
\[
C_c^n(X) \cong \varinjlim_{\mathcal{K}(X)} C^n(X, X \setminus K),
\]
where we have a map
\[
\varinjlim_{K(\alpha)} C^n(X, X \setminus K)\to C_c^n(X)
\]
given in each component of the direct limit by inclusion, and it is easy to see that this is well-defined and bijective.
It is then a general algebraic fact that $H^*$ commutes with inverse limits, and we will not prove it.
\end{proof}
\begin{lemma}
We have
\[
H_c^i(\R^d; R) \cong
\begin{cases}
R & i = d\\
0 & \text{otherwise}
\end{cases}.
\]
\end{lemma}
\begin{proof}
Let $\mathcal{B} \in \mathcal{K}(\R^d)$ be the balls, namely
\[
\mathcal{B} = \{n D^d, n = 0,1, 2, \cdots\}.
\]
Then since every compact set is contained in one of them, we have
\[
H^n_c(X) \cong \varinjlim_{K \in \mathcal{K}(\R^d)} H^n(\R^d, \R^d \setminus K; R) \cong \varinjlim_{nD^d \in \mathcal{B}} H^n(\R^d,\R^d \setminus nD^d; R)
\]
We can compute that directly. Since $\R^d$ is contractible, the connecting map
\[
H^i(\R^d, \R^d \setminus nD^d; R) \to H^{i - 1}(\R^d \setminus nD^d; R)
\]
in the long exact sequence is an isomorphism. Moreover, the following diagram commutes:
\[
\begin{tikzcd}
H^i(\R^d, \R^d \setminus nD^n; R) \ar[d, "\partial"] \ar[r, "\rho_{n, n + 1}"] & H^i(\R^d, \R^d \setminus (n + 1)D^d; R) \ar[d, "\partial"]\\
H^{i - 1}(\R^d \setminus nD^d; R) \ar[r] & H^{i - 1}(\R^d\setminus (n + 1)D^d; R)
\end{tikzcd}
\]
But all maps here are isomorphisms because the horizontal maps are homotopy equivalences. So we know
\[
\varinjlim H^i(\R^d, \R^d\setminus nD^d; R) \cong H^i(\R^d, \R^d \setminus \{0\}; R) \cong H^{i - 1}(\R^d \setminus \{0\}; R).
\]
So it follows that
\[
H^i(\R^d, \R^d \setminus \{0\}; R) =
\begin{cases}
\R & i = d\\
0 & \text{otherwise}
\end{cases}.\qedhere
\]
\end{proof}
In general, this is how we always compute compactly-supported cohomology --- we pick a suitable subset of $\mathcal{K}(X)$ and compute the limit of that instead.
Note that compactly-supported cohomology is \emph{not} homotopy-invariant! It knows about the dimension of $\R^d$, since the notion of compactness is not homotopy invariant. Even worse, in general, a map $f: X \to Y$ does not induce a map $f^*: H^*_c(Y) \to H^*_c(X)$. Indeed, the usual map does not work because the preimage of a compact set of a compact set is not necessarily compact.
\begin{defi}[Proper map]\index{proper map}
A map $f: X \to Y$ of spaces is \emph{proper} if the preimage of a compact space is compact.
\end{defi}
Now if $f$ is proper, then it does induce a map $H_c^* (\ph)$ by the usual construction.
From now on, we will assume all spaces are Hausdorff, so that all compact subsets are closed. This isn't too bad a restriction since we are ultimately interested in manifolds, which are by definition compact.
Let $i: U \to X$ be the inclusion of an open subspace. We let $K \subseteq U$ be compact. Then by excision, we have an isomorphism
\[
H^*(U, U \setminus K) \cong H^*(X, X \setminus K),
\]
since the thing we cut out, namely $X \setminus U$, is already closed, and $U \setminus K$ is open, since $K$ is closed.
So a compactly supported cohomology class on $U$ gives one on $X$. So we get a map
\[
i_*: H_c^*(U) \to H_c^*(X).
\]
We call this ``extension by zero''. Indeed, this is how the cohomology class works --- if you have a cohomology class $\phi$ on $U$ supported on $K \subseteq U$, then given any simplex in $X$, if it lies inside $U$, we know how to evaluate it. If it lies outside $K$, then we just send it to zero. Then by barycentric subdivision, we can assume every simplex is either inside $U$ or outside $K$, so we are done.
\begin{eg}
If $i: U \to \R^d$ is an open ball, then the map
\[
i_*: H_c^* \to H_c^*(\R^d)
\]
is an isomorphism. So each cohomology class is equivalent to something with a support as small as we like.
\end{eg}
Since it is annoying to write $H^n(X, X \setminus K)$ all the time, we write
\[
H^n(X\mid K; R) = H^n(X, X \setminus K; R).
\]
By excision, this depends only on a neighbourhood of $K$ in $X$. In the case where $K$ is a point, this is local cohomology at a point. So it makes sense to call this \term{local cohomology} \emph{near $K \subseteq X$}.
Our end goal is to produce a Mayer-Vietoris for compactly-supported cohomology. But before we do that, we first do it for local cohomology.
\begin{prop}
Let $K, L \subseteq X$ be compact. Then there is a long exact sequence
\[
\begin{tikzcd}[column sep=small]
H^n(X \mid K \cap L) \ar[r] & H^n(X \mid K) \oplus H^n(X \mid L) \ar[r] & H^n(X \mid K \cup L) \ar[out=0, in=180, overlay, lld, "\partial"]\\
H^{n + 1}(X \mid K \cap L) \ar[r] & H^{n + 1}(X \mid K) \oplus H^{n + 1}(X \mid L) \ar[r] & \cdots
\end{tikzcd},
\]
where the unlabelled maps are those induced by inclusion.
\end{prop}
We are going to prove this by relating it to a Mayer-Vietoris sequence of some sort.
\begin{proof}
We cover $X \setminus K \cap L$ by
\[
\mathcal{U} = \{X \setminus K, X \setminus L\}.
\]
We then draw a huge diagram (here $^*$ denotes the dual, i.e.\ $X^* = \Hom(X; R)$, and $C^\Cdot(X \mid K) = C^\Cdot(X, X \setminus K)$):
\[
\begin{tikzcd}[column sep=small]
& 0 \ar[d] & 0 \ar[d] & 0 \ar[d]\\
0 \ar[r] & \left(\frac{C_\Cdot(X)}{C_\Cdot^\mathcal{U}(X \setminus K \cap L)}\right)^* \ar[r] \ar[d] & C^\Cdot(X \mid K) \oplus C^\Cdot(X \mid L) \ar[r] \ar[d] & C^\Cdot(X \mid K \cup L) \ar[r] \ar[d] & 0\\
0 \ar[r] & C^\Cdot(X) \ar[r, "{(\id, -\id)}"] \ar[d] & C^\Cdot(X) \oplus C^\Cdot(X) \ar[d] \ar[r, "\id + \id"] & C^\Cdot(X) \ar[r] \ar[d] & 0\\
0 \ar[r] & C^\Cdot_{\mathcal{U}}(X\setminus K \cap L) \ar[r, "{(j_1^*, -j_2^*})"] \ar[d] & C^\Cdot(X \setminus K) \oplus C^\Cdot(X \setminus L) \ar[r, "i_1^* + i_2^*"] \ar[d] & C^\Cdot(X \setminus K \cup L) \ar[r] \ar[d]& 0\\
& 0 & 0 & 0
\end{tikzcd}
\]
This is a diagram. Certainly.
The bottom two rows and all columns are exact. By a diagram chase (the \term{nine lemma}), we know the top row is exact. Taking the long exact sequence almost gives what we want, except the first term is a funny thing.
We now analyze that object. We look at the left vertical column:
\[
\begin{tikzcd}[column sep=small]
0 \ar[r] & \Hom\left(\frac{C_\Cdot(x)}{C_\Cdot^\mathcal{U}(X \setminus K \cap L)}, R \right) \ar[r] & C^\Cdot(X) \ar[r] & \Hom(C_\Cdot^\mathcal{U}(X \setminus K \cap L), R) \ar[r]& 0
\end{tikzcd}
\]
Now by the small simplices theorem, the right hand object gives the same (co)homology as $C_\Cdot(X \setminus K \cap L; R)$. So we can produce another short exact sequence:
\[
\begin{tikzcd}[column sep=small]
0 \ar[r] & \Hom\left(\frac{C_\Cdot(x)}{C_\Cdot^\mathcal{U}(X \setminus (K \cap L))}, R\right) \ar[r] & C^\Cdot(X) \ar[r] & \Hom(C_\Cdot^\mathcal{U}(X \setminus K \cap L), R) \ar[r] & 0\\
0 \ar[r] & C^\Cdot(X , X \setminus K \cap L) \ar[r] \ar[u] & C^\Cdot(X) \ar[u, equals] \ar[r] & \Hom(C_{\Cdot}(X \setminus K \cap L), R) \ar[r] \ar[u] & 0
\end{tikzcd}
\]
Now the two right vertical arrows induce isomorphisms when we pass on to homology. So by taking the long exact sequence, the five lemma tells us the left hand map is an isomorphism on homology. So we know
\[
H_*\left(\Hom\left(\frac{C_\Cdot(x)}{C_\Cdot^\mathcal{U}(X \setminus (K \cap L))}, R\right)\right) \cong H^*(X \mid K \cap L).
\]
So the long exact of the top row gives what we want.
\end{proof}
\begin{cor}
Let $X$ be a manifold, and $X = A \cup B$, where $A, B$ are open sets. Then there is a long exact sequence
\[
\begin{tikzcd}
H^n_c(A \cap B) \ar[r] & H^n_c(A) \oplus H^n_c(B) \ar[r] & H_c^n(X)\ar[out=0, in=180, looseness=2, overlay, lld, "\partial"]\\
H_c^{n+1}(A \cap B) \ar[r] & H_c^{n+1}(A) \oplus H_c^{n+1}(B) \ar[r] & \cdots
\end{tikzcd}
\]
\end{cor}
Note that the arrows go in funny directions, which is different from both homology and cohomology versions!
\begin{proof}
Let $K \subseteq A$ and $L \subseteq B$ be compact sets. Then by excision, we have isomorphisms
\begin{align*}
H^n(X \mid K) &\cong H^n(A \mid K)\\
H^n(X \mid L) &\cong H^n(B \mid L)\\
H^n(X \mid K \cap L) &\cong H^n(A \cap B \mid K\cap L).
\end{align*}
So the long exact sequence from the previous proposition gives us
\[
\begin{tikzcd}[column sep=small]
H^n(A \cap B\mid K \cap L) \ar[r] & H^n(A \mid K) \oplus H^n(B \mid L) \ar[r] & H^n(X \mid K \cup L) \ar[lld, out=0, in=180, overlay, "\partial"]\\
H^{n + 1}(A \cap B \mid K \cap L) \ar[r] & H^{n + 1}(A \mid K) \oplus H^{n + 1}(B \mid L) \ar[r] & \cdots
\end{tikzcd}
\]
The next step is to take the direct limit over $K \in \mathcal{K}(A)$ and $L \in \mathcal{K}(B)$. We need to make sure that these do indeed give the right compactly supported cohomology. The terms $H^n(A \mid K) \oplus H^n (B \mid L)$ are exactly right, and the one for $H^n(A \cap B \mid K \cap L)$ also works because every compact set in $A \cap B$ is a compact set in $A$ intersect a compact set in $B$ (take those to be both the original compact set).
So we get a long exact sequence
\[
\begin{tikzcd}[column sep=tiny]
H^n_c(A \cap B) \ar[r] & H^n_c(A) \oplus H^n_c(B) \ar[r] & \displaystyle\varinjlim_{\substack{K \in \mathcal{K}(A)\\ L \in \mathcal{K}(B)}} H^n(X \mid K \cup L) \ar[r, "\partial"] & H^{n + 1}_c(A \cap B)
\end{tikzcd}
\]
To show that that funny direct limit is really what we want, we have to show that every compact set $C \in X$ lies inside some $K \cup L$, where $K \subseteq A$ and $L \subseteq B$ are compact.
Indeed, as $X$ is a manifold, and $C$ is compact, we can find a finite set of closed balls in $X$, each in $A$ or in $B$, such that their interiors cover $C$. So done. (In general, this will work for any locally compact space)
\end{proof}
This is all we have got to say about compactly supported cohomology. We now start to talk about manifolds.
\subsection{Orientation of manifolds}
Similar to the case of vector bundles, we will only work with manifolds with orientation. The definition of orientation of a manifold is somewhat similar to the definition of orientation of a vector bundle. Indeed, it is true that an orientation of a manifold is just an orientation of the tangent bundle, but we will not go down that route, because the description we use for orientation here will be useful later on. After defining orientation, we will prove a result similar to (the first two parts of) the Thom isomorphism theorem.
For a $d$-manifold $M$ and $x \in M$, we know that
\[
H_i(M\mid x; R) \cong
\begin{cases}
R & i = d\\
0 & i \not= d
\end{cases}.
\]
We can then define a \emph{local orientation} of a manifold to be a generator of this group.
\begin{defi}[Local $R$-orientation of manifold]\index{local $R$-orientation!manifold}\index{manifold!local $R$-orientation}
Fr a $d$-manifold $M$, a local $R$-orientation of $M$ at $x$ is an $R$-module generator $\mu_x = H_d(M\mid x; R)$.
\end{defi}
\begin{defi}[$R$-orientation]\index{$R$-orientation!manifold}\index{orientation!manifold}\index{manifold!$R$-orientation}
An \emph{$R$-orientation} of $M$ is a collection $\{\mu_x\}_{x \in M}$ of local $R$-orientations such that if
\[
\varphi: \R^d \to U \subseteq M
\]
is a chart of $M$, and $p, q \in \R^d$, then the composition of isomorphisms
\[
\begin{tikzcd}
H_d(M \mid \varphi(p)) \ar[r, "\sim"] & H_d(U \mid \varphi(p)) & H_d(\R^d \mid p) \ar[l, "\varphi_*", "\sim"'] \ar[d, "\sim"] \\
H_d(M \mid \varphi(q)) \ar[r, "\sim"] & H_d(U \mid \varphi(q)) & H_d(\R^d \mid q) \ar[l, "\varphi_*", "\sim"']
\end{tikzcd}
\]
sends $\mu_x$ to $\mu_y$, where the vertical isomorphism is induced by a translation of $\R^d$.
\end{defi}
\begin{defi}[Orientation-preserving homeomorphism]\index{orientation preserving homeomorphism}\index{homeomorphism!orientation preserving}
For a homomorphism $f: U \to V$ with $U, V \in \R^d$ open, we say $f$ is $R$-orientation-preserving if for each $x \in U$, and $y = f(x)$, the composition
\[
\begin{tikzcd}[column sep=large]
H_d(\R^d \mid 0; R) \ar[r, "\text{translation}"] & H_d(\R^d \mid x; R) \ar[r, "\text{excision}"] & H_d(U \mid x; R) \ar[d, "f_*"]\\
H_d(\R^d \mid 0; R) \ar[r, "\text{translation}"] & H_d(\R^d \mid y; R) \ar[r, "\text{excision}"] & H_d(V \mid y; R)
\end{tikzcd}
\]
is the identity $H_d(\R^d \mid 0; R) \to H_d(\R^d \mid 0; R)$.
\end{defi}
As before, we have the following lemma:
\begin{lemma}\leavevmode
\begin{enumerate}
\item If $R = \F_2$, then every manifold is $R$-orientable.
\item If $\{\varphi_\alpha: \R^d \to U_\alpha \subseteq M\}$ is an open cover of $M$ by Euclidean space such that each homeomorphism
\[
\begin{tikzcd}
\R^d \supseteq \varphi_\alpha^{-1}(U_\alpha \cap U_\beta) & U_\alpha \cap U_\beta \ar[l, "\varphi_\alpha^{-1}"'] \ar[r, "\varphi_\beta^{-1}"] & \varphi_\beta^{-1}(U_\alpha \cap U_\beta) \subseteq \R^d
\end{tikzcd}
\]
is orientation-preserving, then $M$ is $R$-orientable.
\end{enumerate}
\end{lemma}
\begin{proof}\leavevmode
\begin{enumerate}
\item $\F_2$ has a unique $\F_2$-module generator.
\item For $x \in U_\alpha$, we define $\mu_x$ to be the image of the standard orientation of $\R^d$ via
\[
\begin{tikzcd}
H_d(M \mid x) & H_\alpha(U_d\mid x) \ar[l, "\cong"] & H_d(\R^d \mid \varphi_\alpha^{-1}(x)) \ar[l, "(\varphi_\alpha)_*"] & \R_d(\R^d \mid 0) \ar[l, "\text{trans.}"]
\end{tikzcd}
\]
If this is well-defined, then it is obvious that this is compatible. However, we have to check it is well-defined, because to define this, we need to pick a chart.
If $x \in U_\beta$ as well, we need to look at the corresponding $\mu_x'$ defined using $U_\beta$ instead. But they have to agree by definition of orientation-preserving.\qedhere
\end{enumerate}
\end{proof}
Finally, we get to the theorem:
\begin{thm}
Let $M$ be an $R$-oriented manifold and $A \subseteq M$ be compact. Then
\begin{enumerate}
\item There is a unique class $\mu_A \in H_d(M \mid A; R)$ which restricts to $\mu_x \in H_d(M \mid x; R)$ for all $x \in A$.
\item $H_i(M\mid A; R) = 0$ for $i > d$.
\end{enumerate}
\end{thm}
\begin{proof}
Call a compact set $A$ ``good'' if it satisfies the conclusion of the theorem.
\begin{claim}
We first show that if $K, L$ and $K \cap L$ is good, then $K \cup L$ is good.
\end{claim}
This is analogous to the proof of the Thom isomorphism theorem, and we will omit this.
Now our strategy is to prove the following in order:
\begin{enumerate}
\item If $A \subseteq \R^d$ is convex, then $A$ is good.
\item If $A \subseteq \R^d$, then $A$ is good.
\item If $A \subseteq M$, then $A$ is good.
\end{enumerate}
\begin{claim}
If $A \subseteq \R^d$ is convex, then $A$ is good.
\end{claim}
Let $x \in A$. Then we have an inclusion
\[
\R^d \setminus A \hookrightarrow \R^d \setminus \{x\}.
\]
This is in fact a homotopy equivalence by scaling away from $x$. Thus the map
\[
H_i(\R^d \mid A) \to H_i(\R^d \mid x)
\]
is an isomorphism by the five lemma for all $i$. Then in degree $d$, there is some $\mu_A$ corresponding to $\mu_x$. This $\mu_A$ is then has the required property by definition of orientability. The second part of the theorem also follows by what we know about $H_i(\R^d \mid x)$.
\begin{claim}
If $A \subseteq \R^d$, then $A$ is good.
\end{claim}
For $A \subseteq \R^d$ compact, we can find a finite collection of closed balls $B_i$ such that
\[
A \subseteq \bigcup_{i = 1}^n \mathring{B}_i = B.
\]
Moreover, if $U \supseteq A$ for any open $U$, then we can in fact take $B_i \subseteq U$. By induction on the number of balls $n$, the first claim tells us that any $B$ of this form is good.
We now let
\[
\mathcal{G} = \{B \subseteq \R^d: A \subseteq \mathring{B}, B\text{ compact and good}\}.
\]
We claim that this is a directed set under inverse inclusion. To see this, for $B, B' \in \mathcal{G}$, we need to find a $B'' \in \mathcal{G}$ such that $B'' \subseteq B, B'$ and $B''$ is good and compact. But the above argument tells us we can find one contained in $\mathring{B}' \cup \mathring{B}''$. So we are safe.
Now consider the directed system of groups given by
\[
B \mapsto H_i(\R^d \mid B),
\]
and there is an induced map
\[
\varinjlim_{B \in \mathcal{G}} H_i(\R^d \mid B) \to H_i(\R^d \mid A),
\]
since each $H_i(\R^d \mid B)$ maps to $H_i(\R^d \mid A)$ by inclusion, and these maps are compatible. We claim that this is an isomorphism. We first show that this is surjective. Let $[c] \in H_i(\R^d \mid A)$. Then the boundary of $c \in C_i(\R^d)$ is a finite sum of simplices in $\R^d \setminus A$. So it is a sum of simplices in some compact $C \subseteq \R^d \setminus A$. But then $A \subseteq \R^d \setminus C$, and $\R^d \setminus C$ is an open neighbourhood of $A$. So we can find a good $B$ such that
\[
A \subseteq B \subseteq \R^d \setminus C.
\]
Then $c \in C_i(\R^d \mid B)$ is a cycle. So we know $[c] \in H_i(\R^d \mid B)$. So the map is surjective. Injectivity is obvious.
An immediate consequence of this is that for $i > d$, we have $H_i(\R^d \mid A) = 0$. Also, if $i = d$, we know that $\mu_A$ is given uniquely by the collection $\{\mu_B\}_{B \in \mathcal{G}}$ (uniqueness follows from injectivity).
\begin{claim}
If $A \subseteq M$, then $A$ is good.
\end{claim}
This follows from the fact that any compact $A \subseteq M$ can be written as a finite union of compact $A_\alpha$ with $A_\alpha \subseteq U_\alpha \cong \R^d$. So $A_\alpha$ and their intersections are good. So done.
\end{proof}
\begin{cor}
If $M$ is compact, then we get a unique class $[M] = \mu_M \in H_n(M; R)$ such that it restricts to $\mu_x$ for each $x \in M$. Moreover, $H^i(M; R) = 0$ for $i > d$.
\end{cor}
This is not too surprising, actually. If we have a triangulation of the manifold, then this $[M]$ is just the sum of all the triangles.
\begin{defi}[Fundamental class]\index{fundamental class}
The \emph{fundamental class} of an $R$-oriented manifold is the unique class $[M]$ that restricts to $\mu_x$ for each $x \in M$.
\end{defi}
\subsection{\texorpdfstring{Poincar\'e}{Poincare} duality}
We now get to the main theorem --- Poincar\'e duality:
\begin{thm}[Poincar\'e duality]
Let $M$ be a $d$-dimensional $R$-oriented manifold. Then there is a map
\[
D_M: H^k_c(M; R) \to H_{d - k}(M; R)
\]
that is an isomorphism.
\end{thm}
The majority of the work is in defining the map. Afterwards, proving it is an isomorphism is a routine exercise with Mayer-Vietoris and the five lemma.
What does this tell us? We know that $M$ has no homology or cohomology in negative dimensions. So by Poincar\'e duality, there is also no homology or cohomology in dimensions $> d$.
Moreover, if $M$ itself is compact, then we know $H^0_c(M; R)$ has a special element $1$. So we also get a canonical element of $H_d(M; R)$. But we know there is a special element of $H_d(M; R)$, namely the fundamental class. They are in fact the same, and this is no coincidence. This is in fact how we are going to produce the map.
To define the map $D_M$, we need the notion of the cap product
\begin{defi}[Cap product]\index{cap product}
The \emph{cap product} is defined by
\begin{align*}
\ph \frown \ph : C_k(X; R) \times C^\ell(X; R) &\to C_{k - \ell} (X; R)\\
(\sigma, \varphi) &\mapsto \varphi(\sigma|_{[v_0, \ldots, v_\ell]}) \sigma|_{[v_{\ell}, \ldots, v_k]}.
\end{align*}
\end{defi}
We want this to induce a map on homology. To do so, we need to know how it interacts with differentials.
\begin{lemma}
We have
\[
d(\sigma \frown \varphi) = (-1)^d ((d \sigma) \frown \varphi - \sigma \frown (d \varphi)).
\]
\end{lemma}
\begin{proof}
Write both sides out.
\end{proof}
As with the analogous formula for the cup product, this implies we get a well-defined map on homology and cohomology, i.e.\ We obtain a map
\[
H_k(X; R) \times H^\ell(X; R) \to H_{k - \ell}(X; R).
\]
As with the cup product, there are also relative versions
\[
\frown\;: H_k(X, A; R) \times H^\ell(X; R) \to H_{k - \ell}(X, A; R)
\]
and
\[
\frown\;: H_k(X, A; R) \times H^\ell(X, A; R) \to H_{k - \ell}(X; R).
\]
We would like to say that the cap product is natural, but since maps of spaces induce maps of homology and cohomology in opposite directions, this is rather tricky. What we manage to get is the following:
\begin{lemma}
If $f: X \to Y$ is a map, and $x \in H_k(X; R)$ And $y \in H^\ell(Y; R)$, then we have
\[
f_*(x) \frown y = f_*(x \frown f^*(y)) \in H_{k - \ell}(Y; R).
\]
In other words, the following diagram commutes:
\[
\begin{tikzcd}
& H_k(Y; R) \times H^\ell(Y; R) \ar[r, "\frown"] & H_{k - \ell}(Y; R)\\
H_k(X; R) \times H^\ell(Y; R) \ar[ur, "f_* \times \id"] \ar[dr, "\id \times f^*"] \\
& H_k(X; R) \times H^\ell(X; R) \ar[r, "\frown"] & H_{k - \ell}(X; R) \ar[uu, "f_*"]
\end{tikzcd}
\]
\end{lemma}
\begin{proof}
We check this on the cochain level. We let $x = \sigma: \Delta^k \to X$. Then we have
\begin{align*}
f_\#(\sigma \frown f^\# y) &= f_\# \left((f^\# y) (\sigma|_{[v_0, \ldots, v_\ell]}) \sigma|_{[v_\ell, \ldots, v_k]}\right)\\
&= y(f_\# (\sigma|_{[v_0, \ldots, v_\ell]})) f_\# (\sigma|_{[v_{\ell}, \ldots, v_k]})\\
&= y((f_\# \sigma)|_{[v_0, \ldots, v_\ell]}) (f_\# \sigma)|_{[v_{\ell}, \ldots, v_k]}\\
&= (f_\# \sigma) \frown y.
\end{align*}
So done.
\end{proof}
Now if $M$ is compact, then we simply define the duality map as
\[
D_M = [M] \frown \ph : H^d (M; R) \to H_{d - \ell}(M; R).
\]
If not, we note that $H_c^d(M; R)$ is the directed limit of $H^d(M, K; R)$ with $K$ compact, and each $H^d(M, L; R)$ has a fundamental class. So we can define the required map for each $K$, and then put them together.
More precisely, if $K \subseteq L \subseteq M$ are such that $K, L$ are compact, then we have an inclusion map
\[
(\id, \mathrm{inc}): (M, M \setminus L) \to (M, M \setminus K).
\]
Then we have an induced map
\[
(\id, \mathrm{inc}): H_d(M \mid L; R) \to H_d(M\mid K; R)
\]
that sends the fundamental class $\mu_L$ to $\mu_K$, by uniqueness of the fundamental class.
Then the relative version of our lemma tells us that the following map commutes:
\[
\begin{tikzcd}
H^\ell(M \mid K; R) \ar[r, "{(\id, \mathrm{inc})^*}"] \ar[d, "\mu_K \frown \ph"] & H^\ell(M \mid L; R) \ar[d, "\mu_L \frown \ph"]\\
H_{d - \ell}(M; R) \ar[r, "\id"] & H_{d - \ell}(M; R)
\end{tikzcd}
\]
Indeed, this is just saying that
\[
(\id)_* (\mu_L \frown (\id, \mathrm{inc})^* (\varphi)) = \mu_K \frown \varphi.
\]
So we get a duality map
\[
D_M = \varinjlim (\mu_K \frown \ph): \varinjlim H^\ell(M \mid K; R) \to H_{d - \ell}(M; R).
\]
Now we can prove Poincar\'e duality.
\begin{proof}
We say $M$ is ``good'' if the Poincar\'e duality theorem holds for $M$. We now do the most important step in the proof:
\begin{cclaim}
$\R^d$ is good.
\end{cclaim}
The only non-trivial degrees to check are $\ell = 0, d$, and $\ell = 0$ is straightforward.
For $\ell =d $, we have shown that the maps
\[
\begin{tikzcd}
H_c^d(\R^d; R) & H^d(\R^d\mid 0; R) \ar[l, "\sim"] \ar[r, "\text{UCT}"] & \Hom_R(H_d(\R^d \mid 0; R), R)
\end{tikzcd}
\]
are isomorphisms, where the last map is given by the universal coefficients theorem.
Under these isomorphisms, the map
\[
\begin{tikzcd}
H_c^d(\R^d; R) \ar[r, "D_{\R^d}"] & H_0(\R^d; R) \ar[r, "\varepsilon"] & R
\end{tikzcd}
\]
corresponds to the map $\Hom_K(H_d(\R^d \mid 0; R), R) \to R$ is given by evaluating a function at the fundamental class $\mu_0$. But as $\mu_0 \in H_d(\R^d \mid 0; R)$ is an $R$-module generator, this map is an isomorphism.
\begin{cclaim}
If $M = U \cup V$ and $U, V, U \cap V$ are good, then $M$ is good.
\end{cclaim}
Again, this is an application of the five lemma with the Mayer-Vietoris sequence. We have
\[
\begin{tikzcd}[column sep=small]
H_c^\ell (U \frown V) \ar[r] \ar[d, "D_{U \frown V}"] & H_c^\ell(U) \oplus H_c^d(V) \ar[r] \ar[d, "D_U \oplus D_V"] & H_c^\ell(M) \ar[r] \ar[d, "D_M"] & H_c^{\ell + 1}(U \frown V) \ar[d, "D_{U\frown V}"]\\
H_{d - \ell}(U \frown V) \ar[r] & H_{d - \ell}(U) \oplus H_{d - \ell}(V) \ar[r] & H_{d - \ell}(M) \ar[r] & H_{d - \ell - 1}(U \frown V)
\end{tikzcd}
\]
We are done by the five lemma if this commutes. But unfortunately, it doesn't. It only commutes up to a sign, but it is sufficient for the five lemma to apply if we trace through the proof of the five lemma.
\begin{cclaim}
If $U_1 \subseteq U_2 \subseteq \cdots$ with $M = \bigcup_n U_n$, and $U_i$ are all good, then $M$ is good.
\end{cclaim}
Any compact set in $M$ lies in some $U_n$, so the map
\[
\varinjlim H_c^\ell (U_n) \to H_c^\ell(U_n)
\]
is an isomorphism. Similarly, since simplices are compact, we also have
\[
H_{d - k}(M) = \varinjlim H_{d - k}(U_n).
\]
Since the direct limit of open sets is open, we are done.
\begin{cclaim}
Any open subset of $\R^d$ is good.
\end{cclaim}
Any $U$ is a countable union of open balls (something something rational points something something). For finite unions, we can use Claims 0 and 1 and induction. For countable unions, we use Claim 2.
\begin{cclaim}
If $M$ has a countable cover by $\R^d$'s it is good.
\end{cclaim}
Same argument as above, where we instead use Claim 3 instead of Claim 0 for the base case.
\begin{cclaim}
Any manifold $M$ is good.
\end{cclaim}
Any manifold is second-countable by definition, so has a countable open cover by $\R^d$.
\end{proof}
\begin{cor}
For any compact $d$-dimensional $R$-oriented manifold $M$, the map
\[
[M] \frown \ph: H^\ell(M; R) \to H_{d - \ell}(M; R)
\]
is an isomorphism.
\end{cor}
\begin{cor}
Let $M$ be an odd-dimensional compact manifold. Then the Euler characteristic $\chi(M) = 0$.
\end{cor}
\begin{proof}
Pick $R = \F_2$. Then $M$ is $\F_2$-oriented. Since we can compute Euler characteristics using coefficients in $\F_2$. We then have
\[
\chi(M) = \sum_{r = 0}^{2n + 1} (-1)^i \dim_{\F_2} H_i(M, \F_2).
\]
But we know
\[
H_i(M, \F_2) \cong H^{2n + 1 - i}(M, \F_2) \cong (H_{2n + 1 - i}(M, \F_2))^* \cong H_{2n + 1 - i}(M, \F_2)
\]
by Poincar\'e duality and the universal coefficients theorem.
But the dimensions of these show up in the sum above with opposite signs. So they cancel, and $\chi(M) = 0$.
\end{proof}
What is the relation between the cap product and the cup product? If $\sigma \in C_{k + \ell}(X; R)$, $\varphi \in C^k(X; R)$ and $\psi \in C^{\ell}(X; R)$, then
\begin{align*}
\psi(\sigma \frown \varphi) &= \psi(\varphi(\sigma|_{[v_0, \ldots, v_k]}) \sigma|_{[v_k, \ldots, v_{k + \ell}]}) \\
&= \varphi(\sigma |_{[v_0, \ldots, v_k]}) \psi(\sigma|_{[v_k, \ldots, v_{k + \ell}]})\\
&= (\varphi \smile \psi)(\sigma),
\end{align*}
Since we like diagrams, we can express this equality in terms of some diagram commuting. The map $h: H^k(X; R) \to \Hom_R(H_k(X; R), R)$ in the universal coefficients theorem is given by
\[
[\varphi] \mapsto ([\sigma] \mapsto \varphi(\sigma)).
\]
This map exists all the time, even if the hypothesis of the universal coefficients theorem does not hold. It's just that it need not be an isomorphism. The formula
\[
\psi(\sigma \frown \varphi) = (\varphi \smile \psi)(\sigma)
\]
then translates to the following diagram commuting:
\[
\begin{tikzcd}
H^\ell(X; R) \ar[d, "\varphi \smile \ph"] \ar[r, "h"] & \Hom_R(H_\ell(X; R), R) \ar[d, "(\ph \frown \varphi)^*"]\\
H^{k + \ell}(X; R) \ar[r, "h"] & \Hom_R(H_{\ell + k}(X; R), R)
\end{tikzcd}
\]
Now when the universal coefficient theorem applies, then the maps $h$ are isomorphisms. So the cup product and cap product determine each other, and contain the same information.
Now since Poincar\'e duality is expressed in terms of cap products, this correspondence gives us some information about cupping as well.
\begin{thm}
Let $M$ be a $d$-dimensional compact $R$-oriented manifold, and consider the following pairing:
\[
\begin{tikzcd}[cdmap]
\bra\ph, \ph \ket: H^k(M; R) \otimes H^{d - k}(M, R) \ar[r] & R\\
\lbrack\varphi\rbrack \otimes \lbrack\psi\rbrack \ar[r, maps to] & (\varphi \smile \psi)[M]
\end{tikzcd}.
\]
If $H_*(M; R)$ is free, then $\bra \ph, \ph\ket$ is non-singular, i.e.\ both adjoints are isomorphisms, i.e.\ both
\[
\begin{tikzcd}[cdmap]
H^k(M; R) \ar[r] & \Hom(H^{d - k}(M; R), R)\\
\lbrack\varphi\rbrack \ar[r, maps to] & (\lbrack\psi\rbrack \mapsto \bra \varphi, \psi\ket)
\end{tikzcd}
\]
and the other way round are isomorphisms.
\end{thm}
\begin{proof}
We have
\[
\bra \varphi, \psi\ket = (-1)^{|\varphi||\psi|} \bra \psi, \varphi\ket,
\]
as we know
\[
\varphi \smile \psi = (-1)^{|\varphi||\psi|} \psi \smile \varphi.
\]
So if one adjoint is an isomorphism, then so is the other.
To see that they are isomorphsims, we notice that we have an isomorphism
\[
\begin{tikzcd}[row sep=tiny]
H^k(M; R) \ar[r, "\mathrm{UCT}"] & \Hom_R(H_k(M; R), R) \ar[r, "D_m^*"] & \Hom_R(H^{d - k}(M; R), R)\\
\lbrack\varphi\rbrack \ar[r, maps to] & (\lbrack\sigma\rbrack \mapsto \varphi(\sigma)) \ar[r, maps to] & (\lbrack\psi\rbrack \mapsto \varphi([M] \frown \psi))
\end{tikzcd}.
\]
But we know
\[
\varphi([M]\frown \psi) = (\psi \smile \varphi)([M]) = \bra \psi, \varphi\ket.
\]
So this is just the adjoint. So the adjoint is an isomorphism.
\end{proof}
This is a very useful result. We have already seen examples where we can figure out if a cup product vanishes. But this result tells us that certain cup products are \emph{not} zero. This is something we haven't seen before.
\begin{eg}
Consider $\CP^n$. We have previously computed
\[
H_*(\CP^n, \Z) =
\begin{cases}
\Z & * = 2i,\quad 0 \leq i \leq n\\
0 & \text{otherwise}
\end{cases}
\]
Also, we know $\CP^n$ is $\Z$-oriented, because it is a complex manifold. Let's forget that we have computed the cohomology ring structure, and compute it again.
Suppose for induction, that we have found
\[
H^*(\CP^{n - 1}, \Z)= \Z[x]/(x^n).
\]
As $\CP^n$ is obtained from $\CP^{n - 1}$ by attaching a $2n$ cell, the map given by inclusion $i^*: H^2(\CP^n, \Z) \to H^2(\CP^{n - 1}, \Z)$ is an isomorphism. Then the generator $x \in H^2(\CP^{n - 1}, \Z)$ gives us a generator $y \in H^2(\CP^n, \Z)$.
Now if we can show that $y^k \in H^{2k}(\CP^n, \Z) \cong \Z$ is a generator for all $k$, then $H^*(\CP^n, \Z) \cong \Z[y]/(y^{n + 1})$.
But we know that $y^{n - 1}$ generates $H^{2n - 2}(\CP^n, \Z)$, since it pulls back under $i^*$ to $x^{n - 1}$, which is a generator. Finally, consider
\[
\begin{tikzcd}[cdmap]
H^2(\CP^2, \Z) \otimes H^{2n - 2} (\CP^2, \Z) \ar[r] & \Z\\
y \otimes y^{n - 1} \ar[r, maps to] & y^n [\CP^n].
\end{tikzcd}
\]
Since this is non-singular, we know $y^n \in H^{2n}(\CP^n, \Z)$ must be a generator.
\end{eg}
Of course, we can get $H^*(\RP^n, \F_2)$ similarly.
\subsection{Applications}
We go through two rather straightforward applications, before we move on to bigger things like submanifolds.
\subsubsection*{Signature}
We focus on the case where $d = 2n$ is even. Then we have, in particular, a non-degenerate bilinear form
\[
\bra \ph, \ph\ket : H^n(M; R) \otimes H^n(M; R) \to R.
\]
Also, we know
\[
\bra a, b\ket = (-1)^{n^2}\bra b, a\ket = (-1)^n \bra b, a \ket.
\]
So this is a symmetric form if $n$ is even, and skew-symmetric form if $n$ is odd. These are very different scenario. For example, we know a symmetric matrix is diagonalizable with real eigenvalues (if $R = \R$), but a skew-symmetric form does not have these properties.
So if $M$ is $4k$-dimensional, and $\Z$-oriented, then in particular $M$ is $\R$-oriented. Then the map $\bra \ph, \ph\ket : H^{2k}(M; \R) \otimes H^{2k}(M; \R) \to \R$ can be represented by a symmetric real matrix, which can be diagonalized. This has some real eigenvalues. The eigenvalues can be positive or negative.
\begin{defi}[Signature of manifold]\index{signature of manifold}\index{manifold!signature}
Let $M$ be a $4k$-dimensional $\Z$-oriented manifold. Then the signature is the number of positive eigenvalues of
\[
\bra \ph, \ph\ket: H^{2k}(M; R) \otimes H^{2k}(M; \R) \to \R
\]
minus the number of negative eigenvalues. We write this as $\sgn(M)$.
\end{defi}
By Sylvester's law of inertia, this is well-defined.
\begin{fact}
If $M = \partial W$ for some compact $4k + 1$-dimensional manifold $W$ with boundary, then $\sgn(M) = 0$.
\end{fact}
\begin{eg}
$\CP^2$ has $H^2(\CP^2; \R) \cong \R$, and the bilinear form is represented by the matrix $(1)$. So the signature is $1$. So $\CP^2$ is not the boundary of a manifold.
\end{eg}
\subsubsection*{Degree}
Recall we defined the degree of a map from a sphere to itself. But if we have a $\Z$-oriented space, we can have the fundamental class $[M]$, and then there is an obvious way to define the degree.
\begin{defi}[Degree of map]\index{degree of map}
If $M, N$ are $d$-dimensional compact connected $\Z$-oriented manifolds, and $f: M \to N$, then
\[
f_*([M]) \in H_d(N, \Z) \cong \Z[N].
\]
So $f_*([M]) = k \Z[N]$ for some $k$. This $k$ is called the \emph{degree} of $f$, written $\deg (f)$.
\end{defi}
If $N = M = S^n$ and we pick the same orientation for them, then this recovers our previous definition.
By exactly the same proof, we can compute this degree using local degrees, just as in the case of a sphere.
\begin{cor}
Let $f: M \to N$ be a map between manifolds. If $\F$ is a field and $\deg(f) \not= 0 \in \F$, then then the induced map
\[
f^*: H^*(N, \F) \to H^*(M, \F)
\]
is injective.
\end{cor}
This seems just like an amusement, but this is powerful. We know this is in fact a map of rings. So if we know how to compute cup products in $H^*(M; \F)$, then we can do it for $H^*(N; \F)$ as well.
\begin{proof}
Suppose not. Let $\alpha \in H^k(N, \F)$ be non-zero but $f^*(\alpha) = 0$. As
\[
\bra \ph, \ph \ket: H^k(N, \F) \otimes H^{d - k}(N, \F) \to \F
\]
is non-singular, we know there is some $\beta \in H^{d - k}(N, \F)$ such that
\[
\bra \alpha, \beta\ket = (\alpha \smile \beta)[N] = 1.
\]
Then we have
\begin{align*}
\deg(f) &= \deg (f) \cdot 1 \\
&= (\alpha \smile \beta)(\deg(f) [N]) \\
&= (\alpha \smile \beta) (f_*[M]) \\
&= (f^*(\alpha) \smile f^*(\beta))([M])\\
&= 0.
\end{align*}
This is a contradiction.
\end{proof}
\subsection{Submanifolds}
\subsubsection*{Normal bundles}
We restrict our attention to smooth manifolds, and then we can talk about the tangent bundle. Recall (from the example sheet) that an orientation of the manifold is the same as an orientation on the tangent bundle.
Let $M$ be a compact smooth $R$-oriented manifold, and $N \subseteq M$ be an $n$-dimensional $R$-oriented submanifold, $i: N \hookrightarrow M$ be the inclusion. Suppose $\dim M = d$ and $\dim N = n$. Then we obtain a canonical homology class
\[
i_*[N] \in H_n(M; R).
\]
We will abuse notation and write $[N]$ for $i_* [N]$. This may or may not be zero.
As before, we can arbitrarily pick a metric on $TM$, and then decompose
\[
i^*TM \cong TN \oplus \nu_{N \subseteq M},
\]
where $\nu_{N\subseteq M}$ is the normal bundle. Since $TM$ is oriented, we obtain an orientation on the pullback $i^*TM$. Similarly, $TN$ is also oriented by assumption. In general, we have the following result:
\begin{lemma}
Let $X$ be a space and $V$ a vector bundle over $X$. If $V = U \oplus W$, then orientations for any two of $U, W, V$ give an orientation for the third.
\end{lemma}
\begin{proof}
Say $\dim V = d$, $\dim U = n$, $\dim W = m$. Then at each point $x \in X$, by K\"unneth's theorem, we have an isomorphism
\[
H^d(V_x, V_x^\#; R) \cong H^n(U_x, U_x^\#; R) \otimes H^m(W_x, W_x^\#; R) \cong R.
\]
So any local $R$-orientation on any two induces one on the third, and it is straightforward to check the local compatibility condition.
\end{proof}
Can we find a more concrete description of this orientation on $\nu_{N \subseteq M}$? By the same argument as when we showed that $H^i(\R^n \mid \{0\}) \cong H_c^i(\R^d)$, we know
\[
H^i(\nu_{N \subseteq M}, \nu_{N \subseteq M}^\#; R) \cong H_c^i(\nu_{N \subseteq M}; R).
\]
Also, by the tubular neighbourhood theorem, we know $\nu_{N \subseteq M}$ is homeomorphic to an open neighbourhood $U$ of $N$ in $M$. So we get isomorphisms
\[
H^i_c(\nu_{N \subseteq M}; R) \cong H_c^i(U; R) \cong H_{d - i}(U; R) \cong H_{d - i}(N; R),
\]
where the last isomorphism comes from the fact that $N$ is homotopy-equivalent to $U$.
In total, we have an isomorphism
\[
H^i(\nu_{N \subseteq M}, \nu_{N \subseteq M}^\#; R) \cong H_{d - i}(N; R).
\]
Under this isomorphism, the fundamental class $[N] \in H_n(N; R)$ corresponds to some
\[
\mathcal{E}_{N \subseteq M} \in H^{d - n}(\nu_{N \subseteq M}, \nu_{N \subseteq M}^\#; R)
\]
But we know $\nu_{N \subseteq M}$ has dimension $d - n$. So $\mathcal{E}_{N \subseteq M}$ is in the right dimension to be a Thom class, and is a generator for $H^{d - n}(\nu_{N \subseteq M}, \nu_{N \subseteq M}^\#; R)$, because it is a generator for $H_{d - n}(N; R)$. One can check that this is indeed the Thom class.
How is this related to the other things we've had? We can draw the commutative diagram
\[
\begin{tikzcd}
H_n(N; R) \ar[r, "\sim"] & H_n(U; R) \ar[r, "\sim"] \ar[d, "i_*"] & H_c^{d - n}(U; R) \ar[d, "\text{extension by }0"] \\
& H_n(M; R) \ar[r, "\sim"] & H^{d - n}(M; R)
\end{tikzcd}
\]
The commutativity of the square is a straightforward naturality property of Poincar\'e duality.
Under the identification $H_c^{d - n}(U; R) \cong H^{d - n}(\nu_{N \subseteq M}, \nu_{N \subseteq M}^\#; R)$, the above says that the image of $[N] \in H_n(N; R)$ in $H^{d - n}_c(U; R)$ is the Thom class of the normal bundle $\nu_{N \subseteq M}$.
On the other hand, if we look at this composition via the bottom map, then $[N]$ gets sent to $D_M^{-1}([N])$. So we know that the Poincar\'e dual of a submanifold is the (extension by zero of the) normal Thom class.
\subsubsection*{Intersection product}
Now if we had two submanifolds $N, W \subseteq M$. Then the normal Thom classes give us two cohomology classes of $M$. We can then take the cup product, and see what we get. It turns out the answer is great. However, this happens only when the manifolds intersect nicely.
\begin{defi}[Transverse intersection]\index{transverse intersection}\index{intersect transversely}
We say two submanifolds $N, W \subseteq M$ \emph{intersect transversely} if for all $x \in N \cap W$, we have
\[
T_x N + T_x W = T_x M.
\]
\end{defi}
\begin{eg}
We allow intersections like
\begin{center}
\begin{tikzpicture}
\draw (-1.5, 0) -- (1.5, 0);
\draw (0, -1.5) -- (0, 1.5);
\end{tikzpicture}
\end{center}
but not this:
\begin{center}
\begin{tikzpicture}
\draw (-2, 0) -- (2, 0);
\draw (-2, 2) parabola bend (0, 0) (2, 2);
\end{tikzpicture}
\end{center}
\end{eg}
It is a fact that we can always ``wiggle'' the submanifolds a bit so that the intersection is transverse, so this is not too much of a restriction. We will neither make this precise nor prove it.
Whenever the intersection is transverse, the intersection $N \cap W$ will be a submanifold of $M$, and of $N$ and $W$ as well. Moreover,
\[
(\nu_{N \cap W \subseteq M})_x = (\nu_{N \subseteq M})_x \oplus (\nu_{W \subseteq M})_x.
\]
Now consider the inclusions
\begin{align*}
i_N: N \cap W &\hookrightarrow N\\
i_W: N \cap W &\hookrightarrow W.
\end{align*}
Then we have
\[
\nu_{N \cap W \subseteq M} = i_N^* (\nu_{N \subseteq M}) \oplus i_W^*(\nu_{W \subseteq M}).
\]
So with some abuse of notation, we can write
\[
i_N^* \mathcal{E}_{N \subseteq M} \smile i_W^* \mathcal{E}_{W \subseteq M} \in H^*(\nu_{N \cap W \subseteq M}, \nu_{N \cap W \subseteq M}^\#; R),
\]
and we can check this gives the Thom class. So we have
\[
D_M^{-1}([N]) \smile D_M^{-1}([W]) = D_M^{-1}([N \cap W]).
\]
The slogan is ``cup product is Poincar\'e dual to intersection''. One might be slightly worried about the graded commutativity of the cup product, because $N \cap W = W \cap N$ as manifolds, but in general,
\[
D_M^{-1}([N]) \smile D_M^{-1}([W]) \not= D_M^{-1}([W]) \smile D_M^{-1}([N]).
\]
The fix is to say that $N \cap W$ are $W \cap N$ are not the same as \emph{oriented} manifolds in general, and sometimes they differ by a sign, but we will not go into details.
More generally, we can define
\begin{defi}[Intersection product]\index{intersection product}
The \emph{intersection product} on the homology of a compact manifold is given by
\[
\begin{tikzcd}[cdmap]
H_{n - k}(M) \otimes H_{n - \ell}(M) \ar[r] & H_{n - k - \ell} (M)\\
(a, b) \ar[r, maps to] & a \cdot b = D_M(D_M^{-1}(a) \smile D_M^{-1}(b))
\end{tikzcd}
\]
\end{defi}
\begin{eg}
We know that
\[
H_{2k}(\CP^n, \Z) \cong
\begin{cases}
\Z & 0 \leq k \leq n\\
0 & \text{otherwise}
\end{cases}.
\]
Moreover, we know the generator for these when we computed it using cellular homology, namely
\[
[\CP^k] \equiv y_k \in H_{2k}(\CP^n, \Z).
\]
To compute $[\CP^\ell] \cdot [\CP^\ell]$, if we picked the canonical copies of $\CP^\ell, \CP^k \subseteq \CP^n$, then one would be contained in the other, and this is exactly the opposite of intersecting transversely.
Instead, we pick
\[
\CP^k = \{[z_0:z_1 \cdots z_k:0:\cdots:0]\},\quad \CP^\ell = \{[0:\cdots : 0 : w_0 : \cdots : w_\ell]\}.
\]
It is a fact that any two embeddings of $\CP^k \hookrightarrow \CP^n$ are homotopic, so we can choose these.
Now these two manifolds intersect transversely, and the intersection is
\[
\CP^k \cap \CP^\ell = \CP^{k + \ell - n}.
\]
So this says that
\[
y_k \cdot y_\ell = \pm y_{l + \ell - n},
\]
where there is some annoying sign we do not bother to figure out.
So if $x_k$ is Poincar\'e dual to $y_{n - k}$, then
\[
x_k \smile x_\ell = x_{k + \ell},
\]
which is what we have previously found out.
\end{eg}
\begin{eg}
Consider the manifold with three holes
\begin{center}
\begin{tikzpicture}[yscale=2, xscale=1.3]
\draw plot [smooth cycle, tension=0.8] coordinates {(-1.7, 0) (-1.4, -0.27) (-0.7, -0.35) (0, -0.23) (0.8, -0.35) (1.6, -0.23) (2.3, -0.35) (3.0, -0.27) (3.3, 0) (3.0, 0.27) (2.3, 0.35) (1.6, 0.23) (0.8, 0.35) (0, 0.23) (-0.7, 0.35) (-1.4, 0.27)};
\foreach \x in {2.4, 0.8, -0.8}{
\begin{scope}[shift={(\x, -0.03)}]
\draw[rounded corners=14pt] (-0.55, 0.05)--(0, -.15)--(0.55, 0.05);
\draw[rounded corners=12pt] (-0.45, 0)--(0, 0.2)--(0.45, 0);
\end{scope}
}
\draw [mred, thick] (-0.8, 0) ellipse (0.6 and 0.2);
\draw [mred, thick] (0.8, 0) ellipse (0.6 and 0.2);
\draw [mred, thick] (2.4, 0) ellipse (0.6 and 0.2);
\node [mred, above] at (-0.8, 0.2) {$b_1$};
\node [mred, above] at (0.8, 0.2) {$b_2$};
\node [mred, above] at (2.4, 0.2) {$b_3$};
\draw [mblue, thick] (-0.8, -0.361) node [below] {$a_1$} arc(-90:90:0.05 and 0.125);
\draw [mblue, thick, dashed] (-0.8, -0.361) arc(270:90:0.05 and 0.125);
\draw [mblue, thick] (0.8, -0.361) node [below] {$a_2$} arc(-90:90:0.05 and 0.125);
\draw [mblue, thick, dashed] (0.8, -0.361) arc(270:90:0.05 and 0.125);
\draw [mblue, thick] (2.4, -0.361) node [below] {$a_3$} arc(-90:90:0.05 and 0.125);
\draw [mblue, thick, dashed] (2.4, -0.361) arc(270:90:0.05 and 0.125);
\end{tikzpicture}
\end{center}
Then we have
\[
a_i \cdot b_i = \{\mathrm{pt}\},\quad a_i \cdot b_j = 0\text{ for }i \not= j.
\]
So we find the ring structure of $H_*(\mathcal{E}_g, \Z)$, hence the ring structure of $H^*(\mathcal{E}_g, \Z)$.
This is so much easier than everything else we've been doing.
\end{eg}
Here we know that the intersection product is well-defined, so we are free to pick our own nice representatives of the loop to perform the calculation.
Of course, this method is not completely general. For example, it would be difficult to visualize this when we work with manifolds with higher dimensions, and more severely, not all homology classes of a manifold have to come from submanifolds (e.g.\ twice the fundamental class)!
\subsection{The diagonal}
Again, let $M$ be a compact $\Q$-oriented $d$-dimensional manifold. Then $M \times M$ is a $2d$-dimensional manifold. We can try to orient it as follows --- K\"unneth gives us an isomorphism
\[
H^{d + d}(M \times M; \Q) \cong H^d(M; \Q) \otimes H^d(M; \Q),
\]
as $H^k(M; \Q) = 0$ for $k > d$. By the universal coefficients theorem, plus the fact that duals commute with tensor products, we have an isomorphism
\[
H_{d + d}(M \times M; \Q) \cong H_d(M; \Q) \otimes H_d(M; \Q).
\]
Thus the fundamental class $[M]$ gives us a fundamental class $[M \times M] = [M] \otimes [M]$.
We are going to do some magic that involves the diagonal map
\[
\begin{tikzcd}[cdmap]
\Delta: M \ar[r] & M \times M\\
x \ar[r, maps to] & (x, x).
\end{tikzcd}
\]
This gives us a cohomology class
\[
\delta = D_{M \times M}^{-1}(\Delta_*[M]) \in H^d(M \times M, \Q) \cong \bigoplus_{i + j = d}H^i(M, \Q) \otimes H^j(M, \Q).
\]
It turns out a lot of things we want to do with this $\delta$ can be helped a lot by doing the despicable thing called picking a basis.
We let $\{a_i\}$ be a basis for $H^*(M, \Q)$. On this vector space, we have a non-singular form
\[
\bra \ph, \ph\ket: H^*(M, \Q) \otimes H^*(M; \Q) \to \Q
\]
given by $\bra \varphi, \psi\ket \mapsto (\varphi \smile \psi)([M])$. Last time we were careful about the degrees of the cochains, but here we just say that if $\varphi \smile \psi$ does not have dimension $d$, then the result is $0$.
Now let $\{b_i\}$ be the dual basis to the $a_i$ using this form, i.e.
\[
\bra a_i, b_j\ket = \delta_{ij}.
\]
It turns out $\delta$ has a really nice form expressed using this basis:
\begin{thm}
We have
\[
\delta = \sum_i (-1)^{|a_i|} a_i \otimes b_i.
\]
\end{thm}
\begin{proof}
We can certainly write
\[
\delta = \sum_{i, } C_{ij} a_i \otimes b_j
\]
for some $C_{ij}$. So we need to figure out what the coefficients $C_{ij}$ are. We try to compute
\begin{align*}
((b_k \otimes a_\ell) \smile \delta)[M \times M] &= \sum C_{ij} (b_k \otimes a_\ell) \smile (a_i \otimes b_j) [M \times M]\\
&= \sum C_{ij} (-1)^{|a_\ell||a_i|} (b_k \smile a_i) \otimes (a_\ell \smile b_i) [M] \otimes [M]\\
&= \sum C_{ij} (-1)^{|a_\ell||a_i|} (\delta_{ik}(-1)^{|a_i||b_k|}) \delta_{j \ell}\\
&= (-1)^{|a_k||a_\ell| + |a_k| |b_k|} C_{k\ell}.
\end{align*}
But we can also compute this a different way, using the definition of $\delta$:
\begin{align*}
(b_k \otimes a_\ell \smile \delta)[M \times M] = (b_k \otimes a_\ell)(\Delta_*[M]) = (b_k \smile a_\ell)[M] = (-1)^{|a_\ell||b_k|} \delta_{k\ell}.
\end{align*}
So we see that
\[
C_{k\ell} = \delta_{k\ell}(-1)^{|a_\ell|}.\qedhere
\]
\end{proof}
\begin{cor}
We have
\[
\Delta^*(\delta)[M] = \chi(M),
\]
the Euler characteristic.
\end{cor}
\begin{proof}
We note that for $a \otimes b \in H^n(M \times M)$, we have
\[
\Delta^*(a \otimes b) = \Delta^*(\pi_1^* a \smile \pi_2^* b) = a \smile b
\]
because $\pi_i \circ \Delta = \id$. So we have
\[
\Delta^*(\delta) = \sum (-1)^{|a_i|} a_i \smile b_i.
\]
Thus
\[
\Delta^*(\delta)[M] = \sum_i (-1)^{|a_i|} = \sum_k (-1)^k \dim_\Q H^k(M; \Q) = \chi(M).\qedhere
\]
\end{proof}
So far everything works for an arbitrary manifold. Now we suppose further that $M$ is smooth. Then $\delta$ is the Thom class of the normal bundle $\nu_{M \subseteq M \times M}$ of
\[
M \hookrightarrow M \times M.
\]
By definition, pulling this back along $\Delta$ to $M$ gives the Euler class of the normal bundle. But we know $\nu_{M \subseteq M \times M} \cong TM$, because the fiber at $x$ is the cokernel of the map
\[
\begin{tikzcd}
T_x M \ar[r, "\Delta"] & T_x M \oplus T_x M
\end{tikzcd}
\]
and the inverse given by
\begin{align*}
T_x M \oplus T_x M &\to T_x M\\
(v, w) &\mapsto v - w
\end{align*}
gives us an isomorphism
\[
\frac{T_x M \oplus T_x M}{\Delta T_x M} \cong T_x M.
\]
\begin{cor}
We have
\[
e(TM)[M] = \chi(M).
\]
\end{cor}
\begin{cor}
If $M$ has a nowhere-zero vector field, then $\chi(M) = 0$.
\end{cor}
More generally, this tells us that $\chi(M)$ is the number of zeroes of a vector field $M \to TM$ (transverse to the zero section) counted with sign.
\begin{lemma}
Suppose we have $R$-oriented vector bundles $E \to X$ and $F \to X$ with Thom classes $u_E, u_F$. Then the Thom class for $E \oplus F \to X$ is $u_E \smile u_F$. Thus
\[
e(E \oplus F) = e(E) \smile e(F).
\]
\end{lemma}
\begin{proof}
More precisely, we have projection maps
\[
\begin{tikzcd}
& E \oplus F \ar[dl, "\pi_E"'] \ar[rd, "\pi_F"]\\
E & & F
\end{tikzcd}.
\]
We let $U = \pi_E^{-1}(E^\#)$ and $V = \pi_F^{-1}(F^\#)$. Now observe that
\[
U \cup V = (E \oplus F)^\#.
\]
So if $\dim E = e, \dim F = f$, then we have a map
\[
\begin{tikzcd}
H^e(E, E^\#) \otimes H^f(F, F^\#) \ar[r, "\pi_E^* \otimes \pi_F^*"] & H^e(E \oplus F, U) \otimes H^f(E \oplus F, V) \ar[d, "\smile"] \\
& H^{e + f}(E \oplus F, (E \oplus F)^\#)
\end{tikzcd},
\]
and it is easy to see that the image of $u_E \otimes u_F$ is the Thom class of $E \oplus F$ by checking the fibers.
\end{proof}
\begin{cor}
$TS^{2n}$ has no proper subbundles.
\end{cor}
\begin{proof}
We know $e(TS^{2n}) \not= 0$ as $e(TS^{2n})[S^2] = \chi(S^{2n}) = 2$. But it cannot be a proper cup product of two classes, since there is nothing in lower cohomology groups. So $TS^{2n}$ is not the sum of two subbundles. Hence $TS^{2n}$ cannot have a proper subbundle $E$ or else $TS^{2n} = E \oplus E^\perp$ (for any choice of inner product).
\end{proof}
\subsection{Lefschetz fixed point theorem}
Finally, we are going to prove the Lefschetz fixed point theorem. This is going to be better than the version you get in Part II, because this time we will know how many fixed points there are.
So let $M$ be a compact $d$-dimensional manifold that is $\Z$-oriented, and $f: M \to M$ be a map. Now if we want to count the number of fixed points, then we want to make sure the map is ``transverse'' in some sense, so that there aren't infinitely many fixed points.
It turns out the right condition is that the graph
\[
\Gamma_f = \{(x, f(x)) \in M \times M\} \subseteq M \times M
\]
has to be transverse to the diagonal. Since $\Gamma_f \cap \Delta$ is exactly the fixed points of $f$, this is equivalent to requiring that for each fixed point $x$, the map
\[
\D_x \Delta \oplus \D_x F: T_x M \oplus T_x M \to T_x M \oplus T_x M
\]
is an isomorphism. We can write the matrix of this map, which is
\[
\begin{pmatrix}
I & I\\
\D_x f & I
\end{pmatrix}.
\]
Doing a bunch of row and column operations, this is equivalent to requiring that
\[
\begin{pmatrix}
I & 0\\
\D_x f & I - \D_x f
\end{pmatrix}
\]
is invertible. Thus the condition is equivalent to requiring that $1$ is not an eigenvalue of $\D_x f$.
The claim is now the following:
\begin{thm}[Lefschetz fixed point theorem]\index{Lefschetz fixed point theorem}\index{fixed point theorem!Lefschetz}
Let $M$ be a compact $d$-dimensional $\Z$-oriented manifold, and let $f: M \to M$ be a map such that the graph $\Gamma_f$ and diagonal $\Delta$ intersect transversely. Then
Then we have
\[
\sum_{x \in \fix(f)} \sgn \det(I - \D_x f) = \sum_k (-1)^k \tr(f^*: H^i(M; \Q) \to H^k(M; \Q)).
\]
\end{thm}
\begin{proof}
We have
\[
[\Gamma_f] \cdot [\Delta (M)] \in H_0(M \times M; \Q).
\]
We now want to calculate $\varepsilon$ of this. By Poincar\'e duality, this is equal to
\[
(D_{M \times M}^{-1}[\Gamma_f] \smile D_{M \times M}^{-1}[\Delta(M)])[M \times M] \in \Q.
\]
This is the same as
\[
(D_{M \times M}^{-1} [\Delta(M)]) ([\Gamma_f]) = \delta(F_*[M]) = (F^* \delta)[M],
\]
where $F: M \to M \times M$ is given by
\[
F(x) = (x, f(x)).
\]
We now use the fact that
\[
\delta = \sum (-1)^{|a_i|} a_i \otimes b_i.
\]
So we have
\[
F^* \delta = \sum (-1)^{|a_i|} a_i \otimes f^* b_i.
\]
We write
\[
f^* b_i = \sum C_{ij} b_j.
\]
Then we have
\[
(F^* \delta)[M] = \sum_{i, j} (-1)^{|a_i|} C_{ij} (a_i \otimes b_j) [M] = \sum_i (-1)^{|a_i|} C_{ii},
\]
and $C_{ii}$ is just the trace of $f^*$.
We now compute this product in a different way. As $\Gamma_f$ and $\Delta (M)$ are transverse, we know $\Gamma_f \cap \Delta(M)$ is a $0$-manifold, and the orientation of $\Gamma_f$ and $\Delta(M)$ induces an orientation of it. So we have
\[
[\Gamma_f] \cdot [\Delta(m)] = [\Gamma_f \cap \Delta(M)] \in H_0(M \times M; \Q).
\]
We know this $\Gamma_f \cap \Delta(M)$ has $|\fix(f)|$ many points, so $[\Gamma_f \cap \Delta(M)]$ is the sum of $|\fix(f)|$ many things, which is what we've got on the left above. We have to figure out the sign of each term is actually $\sgn \det(I - \D_x f)$, and this is left as an exercise on the example sheet.
\end{proof}
\begin{eg}
Any map $f: \CP^{2n} \to \CP^{2n}$ has a fixed point. We can't prove this using the normal fixed point theorem, but we can exploit the ring structure of cohomology to do this. We must have
\[
f^*(x) = \lambda x \in H^2(\CP^2; \Q) = \Q x.
\]
for some $\lambda \in \Q$. So we must have
\[
f^*(x^i) = \lambda^i x^i.
\]
We can now very easily compute the right hand side of the fixed point theorem
\[
\sum_k (-1)^k \tr(f^*: H^k \to H^k) = 1 + \lambda + \lambda^2 + \cdots + \lambda^{2n}.
\]
and this cannot be zero.
\end{eg}
\printindex
\end{document}
|
https://web.evanchen.cc/twitch/Ep023-Iran-2010-2-6-Solution.tex | evanchen.cc | CC-MAIN-2022-40 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-40/segments/1664030335396.92/warc/CC-MAIN-20220929225326-20220930015326-00553.warc.gz | 617,656,309 | 2,023 | \documentclass[11pt]{scrartcl}
\usepackage{evan}
\begin{document}
\title{Iran 2010/2/6}
\subtitle{Evan Chen}
\author{Twitch Solves ISL}
\date{Episode 23}
\maketitle
\section*{Problem}
Let $g$ and $n$ be positive integers such that
$\gcd(g^2-g, n) = 1$.
Define $B$ as the set of possible remainders
when $g^k$ is divided by $n$, across all integers $k \ge 0$.
For each $i = 0, \dots, g-1$ define
$a_i$ as the number of elements of $B$
which lie in the interval
\[ \left[ \frac{ni}{g}, \frac{n(i+1)}{g} \right). \]
Show that $g-1$ divides $\sum_{i=0}^{g-1} i a_i$.
\section*{Video}
\href{https://www.youtube.com/watch?v=CC5w3OLl18A&list=PLi6h8GM1FA6yHh4gDk_ZYezmncU1EJUmZ}{\texttt{https://youtu.be/CC5w3OLl18A}}
\newpage
\section*{Solution}
Let $e > 0$ denote the order of $g$ modulo $n$.
Also, by $a \% n$ we mean the remainder
when $a$ is divided by $n$.
The main observation is that an element $b \in B$
will fall in the $\left\lfloor \frac{g \cdot b}{n} \right\rfloor$'th
interval, and contribute that amount to the
sum given in the problem.
This gives the first equality in the following calculation:
\begin{align*}
\sum i a_i
&= \sum_{b \in B}
\left\lfloor \frac{g \cdot b}{n} \right\rfloor \\
&= \sum_{k=0}^{e-1}
\left\lfloor \frac{g \cdot (g^k \% n)}{n} \right\rfloor \\
&= \sum_{k=0}^{e-1}
\frac{g \cdot (g^k \% n) - \left( g \cdot (g^k \% n) \right) \% n}{n} \\
&= \sum_{k=0}^{e-1}
\frac{g \cdot (g^k \% n) - (g^{k+1} \% n)}{n}
\end{align*}
We may now take modulo $g-1$, noting that $g \equiv 1 \pmod{g-1}$
and $n$ is relatively prime to $g-1$, hence
\begin{align*}
\sum i a_i
&= \sum_{k=0}^{e-1}
\frac{(g^k \% n) - \left( g^{k+1} \% n \right)}{n} \pmod{g-1} \\
&= 0
\end{align*}
as desired, with the sum telescoping.
\end{document}
|
https://www.pharmascope.org/index.php/ijrps/article/download/4823/13808 | pharmascope.org | CC-MAIN-2021-43 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-43/segments/1634323585737.45/warc/CC-MAIN-20211023162040-20211023192040-00427.warc.gz | 1,116,832,869 | 12,600 | \documentclass[twocolumn,twoside]{article}
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\begin{document}
\def\RunningAuthor{Arshvir Kaur et al.}
\firstPage{2132}
\articleType{Original Article}
\receivedDate{02 Jun 2021}
\acceptedDate{12 Jul 2021}
\revisedDate{08 Jul 2021}
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\title{\textbf{Psychological distress and its associated factors in the hypertensive patient (HTN-Pt) in Ludhiana, Punjab: A questionnaire-based retrospective study }}
\author{Maninder~Singh\textsuperscript{1},
Bikram~Singh\textsuperscript{1},
Arshvir~Kaur\textsuperscript{*}\textsuperscript{2}~\\[5pt]{\textsuperscript{1}Lovely School of Pharmaceutical Sciences, Lovely Professional University, Jalandhar-Delhi GT Road, Phagwara-144001, Punjab, India}~\\{\textsuperscript{2}Department of Pharmacology, Delhi Institute of Pharmaceutical Sciences and Research, Sector 3, MB Road, Pushp Vihar-110017, New Delhi, India}}
\begin{abstract}
An elevation in blood pressure is an important risk factor of cardiovascular disease and several factors that can contribute to hypertension induce psychological distress. This study was aimed at estimating the prevalence of psychological distress and to assess general awareness regarding disease, concomitant substance abuse, and use of herbal drugs among hypertensive patients (HTN-Pt) at Satguru Pratap Singh (SPS) Hospitals, Ludhiana. The psychological distress was assessed using the standard Kessler-10 scale along with face-to-face interview among 275 outpatient department (OPD) HTN-Pt on follow-up. 15.30\% (n=33) of total participants (n=213) had alcohol use disorders and 8.80\% (n=19) of them were addicted to smoking habits. K10 scale results in patients, showed 46.9\% (100) patients were suffering from psychological distress out of which 26\% (n=56) were having mild, 17\% (n=36) moderate and 4\% (n=8) patients were having severe psychological distress. Highest percentage (33.80\%) of patients with psychological distress were from age group 31-60 years of age (p value=0.003, COR= 0.240, 95\% CI 0.072, 0.584). Many HTN-Pt were consuming the herbal supplements out of which 92 \% of patients consuming grapes were found to have psychological distress (p value=0.034, COR= 0.380, 95\% CI 0.155, 0.930). The results of the study indicated that there was a high prevalence of psychological distress in HTN-Pt belonging to age group of 31-60 years of age and patients involved in the consumption of grapes. This study asks for supervision on the concomitant administration of herbal supplements with allopathic medicines in HTN-Pt to avoid psychological distress.
\end{abstract}\def\keywordstitle{Keywords}
\begin{keywords}Hypertension,\newline herbal formulations,\newline Psychological distress,\newline Kessler-10,\newline grapes
\end{keywords}
\twocolumn[ \maketitle {\printKwdAbsBox}]
\makeatletter\textsuperscript{*}Corresponding Author\par Name:\ Arshvir~Kaur~\\ Phone:\ +91-9855518759~\\ Email:\ archie.dhwal@gmail.com
\par\vspace*{-11pt}\hrulefill\par{\fontsize{12}{14}\selectfont ISSN: 0975-7538}\par%
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\section{Introduction}
Hypertension (HTN) (140/90 mmHg), now and then called vascular hypertension, is a ceaseless clinical condition in which the strain due to circulation in the veins is raised. At rest, normally, systolic blood pressure (BP) is within the range of 100{\textendash}140 mm Hg and 60{\textendash}90mmHg diastolic \unskip~\citep{1156208:22694541}. HTN is one of the most common prevailing diseases in adult aged groups of people \unskip~\citep{1156208:22694545}. The burden of various sorts of chronic diseases in India is found to be associated with HTN, the prevalence of HTN in India is increasing day by day and less awareness and control are identified reasons from various studies done across India \unskip~\citep{1156208:22694542}.
The significant prevalence in HTN patients is found to be associated with an increase in psychological distress and raised the level of systolic BP, more commonly in females than males \unskip~\citep{1156208:22694544,1156208:22694539}. A substantial body of evidence supports the role of psychosocial factors, and psychological distress are primary risk factors for HTN \unskip~\citep{1156208:22694553,1156208:22694549}. Psychological distress and hypertension share a significant association \unskip~\citep{1156208:22694543}. Obesity, high alcohol intakes, physical inactivity, tobacco use and emotional stress are some of the other factors which are said to be associated with HTN \unskip~\citep{1156208:22694552}. A survey conducted among 396 HTN-Pt which were on follow-up at Jimma University, Teaching Hospital, located in Ethiopia (2017) reported the 31.6\% prevalence of psychological distress among single patients more likely as compared to the married and in participants, who were illiterate than who were capable of reading and writing. Also, 7.8\% of them had disorders firmly associated with consumption of alcohol, and 19.9\% were consuming khat or indulge in substance abuse on a daily basis \unskip~\citep{1156208:22694538}. However, another study on herbal extracts by Wong \textit{et al}. 2016, included grape seed, green tea, and ginkgo and ingredients in Brain, suggested a significant association with psychological distress. In that study, 46.47\% out of 85\% of patients who were consuming grapes were suffering from psychological distress. \unskip~\citep{1156208:22694540}.
So, as Psychological status in HTN-Pt was found to be affected by various socio-demographic, clinical and lifestyle factors in previous studies done at other locations, there is a need to explore this association in the North-India region, too as there is a paucity on related literature. To address the lacuna, this study was designed to explore the association of various factors (socio-demographic factors, perception about disease, lifestyle, clinical background and usage of herbal supplements) affecting psychological status in HTN-Pt in Ludhiana, Punjab, India.
\section{Materials and Methods}
\textbf{Study Design}
The retrospective cohort study was conducted at SPS Hospital, Ludhiana, comprising of 213 patients satisfying the selection criteria:
\begin{enumerate}
\item \relax Patients with primary and secondary hypertension
\item \relax Age more than 18 years
\item \relax Both the gender
\item \relax Patient suffering from a disease (not less than 1 year)
\item \relax Both IPD and OPD patients who appeared for treatment of hypertension except the pregnant women and patients having genotype-specific hypertension.
\end{enumerate}
The study was carried out over a period of a 1-month period from June 2017 to July 2017. All participants provided written informed consent. Ethics approval under protocol number (SPS 01/2017) was obtained from the SPS Hospitals review committee.
\textbf{Measures}
The background information was collected using authentic sciences article search engines, like Google Scholar, Medline Plus, Google, PubMed and other journal sources covering the recent information on the topic till 2019. Patients' data was collected using a hard copy version questionnaire by face-to-face interview.
\textbf{Socio-Statistic Qualities}
A qualitative questionnaire was utilized to estimating socio-statistic qualities of members (age, sexual orientation, married status, educational status, occupation, religion, place of living arrangement), utilization of liquor or cigarette, patient's clinical information on and about hypertension, and self practices, i.e. a way of life factors (work out, salt use and utilization of herbal options).
\textbf{Psychological distress using Kessler 10-Scale}
A quantitative questionnaire, i.e. the Kessler Psychological Distress Scale (K10), is a basic tool to measure mental or psychological distress \unskip~\citep{1156208:22694538,1156208:22694540}. It comprises ten questions for which response is recorded at five different levels, i.e. the frequency of experiencing symptoms that a common person has encountered in the latest 4-week time frame.
The total score ranges from 10 to 50. The total score was interpreted as follow: well (10 to 19), mild (20 to 24), moderate (25 to 29) and severe psychological distress (30 to 50).
\bgroup
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\caption{\boldmath {Prevalence of psychological distress in HTN-Pt}}
\label{f-2e36f64866ce}
\end{figure}
\egroup
\begin{table*}[!htbp]
\caption{\boldmath {Socio-Demographic Characteristics } }
\label{tw-58985ed33cc1}
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\tbltoprule \rowcolor{kwdboxcolor}\cAlignHack Variable & \cAlignHack Frequency & \cAlignHack Percent (\%)\\
\tblmidrule
\cAlignHack \textbf{Gender} &
&
\\
\cAlignHack Male &
\cAlignHack 105 &
\cAlignHack 48.8\\
\cAlignHack Female &
\cAlignHack 109 &
\cAlignHack 50.7\\
&
&
\\
\cAlignHack \textbf{Residence} &
&
\\
\cAlignHack Rural &
\cAlignHack 74 &
\cAlignHack 34.4\\
\cAlignHack Urban &
\cAlignHack 140 &
\cAlignHack 65.1\\
&
&
\\
\cAlignHack \textbf{Occupation} &
&
\\
\cAlignHack Daily Labor &
\cAlignHack 22 &
\cAlignHack 10.2\\
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\cAlignHack 13 &
\cAlignHack 6.0\\
\cAlignHack Housewife &
\cAlignHack 74 &
\cAlignHack 34.4\\
\cAlignHack Merchant &
\cAlignHack 68 &
\cAlignHack 7.0\\
\cAlignHack Teacher &
\cAlignHack 15 &
\cAlignHack 99.5\\
\cAlignHack Retired &
\cAlignHack 7 &
\cAlignHack 3.3\\
\cAlignHack Other &
\cAlignHack 68 &
\cAlignHack 31.6\\
&
&
\\
\cAlignHack \textbf{Marital Status} &
&
\\
\cAlignHack Single &
\cAlignHack 14 &
\cAlignHack 6.5\\
\cAlignHack Married &
\cAlignHack 165 &
\cAlignHack 76.7\\
\cAlignHack Divorced &
\cAlignHack 5 &
\cAlignHack 2.3\\
\cAlignHack Widowed &
\cAlignHack 30 &
\cAlignHack 14.0\\
&
&
\\
\cAlignHack \textbf{Education Status} &
&
\\
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\cAlignHack 13 &
\cAlignHack 6.0\\
\cAlignHack Read \& Write &
\cAlignHack 22 &
\cAlignHack 10.6\\
\cAlignHack Undergraduate &
\cAlignHack 98 &
\cAlignHack 45.6\\
\cAlignHack Graduate &
\cAlignHack 81 &
\cAlignHack 37.7\\
&
&
\\
\cAlignHack \textbf{Religion} &
&
\\
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\cAlignHack 90 &
\cAlignHack 41.9\\
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\cAlignHack 11 &
\cAlignHack 5.1\\
\cAlignHack Sikh &
\cAlignHack 109 &
\cAlignHack 50.7\\
\cAlignHack Orthodox &
\cAlignHack 4 &
\cAlignHack 1.9\\
\tblbottomrule
\end{tabulary}\par
\end{table*}
\begin{table*}[!htbp]
\caption{\boldmath {Awareness and perception of patients regarding hypertension} }
\label{tw-7fcddd77877e}
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\tbltoprule \rowcolor{kwdboxcolor}\cAlignHack Variable & \cAlignHack Frequency & \cAlignHack Percent (\%)\\
\tblmidrule
\cAlignHack \textbf{Is HTN Curable? } &
\cAlignHack &
\cAlignHack \\
\cAlignHack Yes &
\cAlignHack 13.5 &
\cAlignHack 62.8\\
\cAlignHack No &
\cAlignHack 79 &
\cAlignHack 36.7\\
\cAlignHack \textbf{Is HTN Deadly?} &
&
\\
\cAlignHack Yes &
\cAlignHack 92 &
\cAlignHack 42.8\\
\cAlignHack No &
\cAlignHack 122 &
\cAlignHack 56.7\\
\cAlignHack \textbf{Do you smoke?} &
&
\\
\cAlignHack Yes &
\cAlignHack 19 &
\cAlignHack 8.8\\
\cAlignHack No &
\cAlignHack 195 &
\cAlignHack 90.7\\
\cAlignHack \textbf{Do you exercise?} &
&
\\
\cAlignHack Yes &
\cAlignHack 95 &
\cAlignHack 44.2\\
\cAlignHack No &
\cAlignHack 119 &
\cAlignHack 55.3\\
\cAlignHack \textbf{Do you have junk food?} &
&
\\
\cAlignHack Yes &
\cAlignHack 53 &
\cAlignHack 24.7\\
\cAlignHack No &
\cAlignHack 161 &
\cAlignHack 74.9\\
\cAlignHack \textbf{Do you follow salt restrictions?} &
&
\\
\cAlignHack Yes &
\cAlignHack 24 &
\cAlignHack 11.2\\
\cAlignHack No &
\cAlignHack 190 &
\cAlignHack 88.4\\
\cAlignHack \textbf{Do you practice yoga?} &
&
\\
\cAlignHack Yes &
\cAlignHack 45 &
\cAlignHack 20.9\\
\cAlignHack No &
\cAlignHack 169 &
\cAlignHack 78.6\\
\cAlignHack \textbf{Do you consume alcohol?} &
&
\\
\cAlignHack Yes &
\cAlignHack 33 &
\cAlignHack 15.3\\
\cAlignHack No &
\cAlignHack 181 &
\cAlignHack 84.2\\
\tblbottomrule
\end{tabulary}\par
\end{table*}
\begin{table*}[!htbp]
\caption{\boldmath {Association of use of herbal alternatives with the prevalence of psychological distress } }
\label{tw-df60dbadceaf}
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\tbltoprule \rowcolor{kwdboxcolor}\cAlignHack Variables & \multicolumn{2}{p{\dimexpr(.30051428571428573\linewidth-2\tabcolsep)}}{\cAlignHack Psychological Distress} & \cAlignHack p-value & \cAlignHack OR & \multicolumn{2}{p{\dimexpr(.26485714285714298\linewidth-2\tabcolsep)}}{\cAlignHack 95 \% CI}\\
\rowcolor{kwdboxcolor}\cAlignHack & \cAlignHack Yes & \cAlignHack No & \cAlignHack & \cAlignHack & \cAlignHack Lower & \cAlignHack Upper\\
\rowcolor{kwdboxcolor}\cAlignHack & \cAlignHack N (\%) & \cAlignHack N (\%) & \cAlignHack & \cAlignHack & \cAlignHack & \cAlignHack \\
\tblmidrule
\cAlignHack \textbf{L-arginine} &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack \\
\cAlignHack Yes &
\cAlignHack 40(44.9) &
\cAlignHack 49(55.1) &
\cAlignHack .889 &
\cAlignHack 1.043 &
\cAlignHack .578 &
\cAlignHack 1.884\\
\cAlignHack No &
\cAlignHack 60(48.4) &
\cAlignHack 64(51.6) &
\cAlignHack &
\multicolumn{2}{p{\dimexpr(.26705714285714294\linewidth-2\tabcolsep)}}{\cAlignHack Reference} &
\cAlignHack \\
&
&
&
&
&
&
\\
\cAlignHack \textbf{Garlic} &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack \\
\cAlignHack Yes &
\cAlignHack 88(48.4) &
\cAlignHack 94(51.6) &
\cAlignHack .615 &
\cAlignHack .809 &
\cAlignHack .353 &
\cAlignHack 1.851\\
\cAlignHack No &
\cAlignHack 12(38.7) &
\cAlignHack 19(61.3) &
\cAlignHack &
\multicolumn{2}{p{\dimexpr(.26705714285714294\linewidth-2\tabcolsep)}}{\cAlignHack Reference} &
\cAlignHack \\
&
&
&
&
&
&
\\
\cAlignHack \textbf{Grapes} &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack \\
\cAlignHack Yes &
\cAlignHack 92(50.5) &
\cAlignHack 90(49.5) &
\cAlignHack .034 &
\cAlignHack .380 &
\cAlignHack .155 &
\cAlignHack .930\\
\cAlignHack No &
\cAlignHack 8(25.8) &
\cAlignHack 23(74.2) &
\cAlignHack &
\multicolumn{2}{p{\dimexpr(.26705714285714294\linewidth-2\tabcolsep)}}{\cAlignHack Reference} &
\cAlignHack \\
&
&
&
&
&
&
\\
\cAlignHack \textbf{Olive} &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack \\
\cAlignHack Yes &
\cAlignHack 29(41.4) &
\cAlignHack 41(58.6) &
\cAlignHack 0.75 &
\cAlignHack 1.856 &
\cAlignHack .939 &
\cAlignHack 3.669\\
\cAlignHack No &
\cAlignHack 71(49.7) &
\cAlignHack 72(50.3) &
\cAlignHack &
\multicolumn{2}{p{\dimexpr(.26705714285714294\linewidth-2\tabcolsep)}}{\cAlignHack Reference} &
\cAlignHack \\
&
&
&
&
&
&
\\
\cAlignHack \textbf{Green Tea} &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack \\
\cAlignHack Yes &
\cAlignHack 50(53.2) &
\cAlignHack 44(46.8) &
\cAlignHack 0.91 &
\cAlignHack .583 &
\cAlignHack .311 &
\cAlignHack 1.091\\
\cAlignHack No &
\cAlignHack 50(42.0) &
\cAlignHack 69(58.0) &
\cAlignHack &
\multicolumn{2}{p{\dimexpr(.26705714285714294\linewidth-2\tabcolsep)}}{\cAlignHack Reference} &
\cAlignHack \\
&
&
&
&
&
&
\\
\cAlignHack \textbf{Vitamins} &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack \\
\cAlignHack Yes &
\cAlignHack 79(47.9) &
\cAlignHack 86(52.1) &
\cAlignHack .566 &
\cAlignHack .816 &
\cAlignHack .408 &
\cAlignHack 1.634\\
\cAlignHack No &
\cAlignHack 21(43.8) &
\cAlignHack 27(56.3) &
\cAlignHack &
\multicolumn{2}{p{\dimexpr(.26705714285714294\linewidth-2\tabcolsep)}}{\cAlignHack Reference} &
\cAlignHack \\
&
&
&
&
&
&
\\
\multicolumn{3}{p{\dimexpr(.4664714285714286\linewidth-2\tabcolsep)}}{\textbf{Do you use Herbal medicines?}} &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack \\
\cAlignHack Yes &
\cAlignHack 18(46.2) &
\cAlignHack 21(53.8) &
\cAlignHack .893 &
\cAlignHack 1.123 &
\cAlignHack .205 &
\cAlignHack 6.168\\
\cAlignHack No &
\cAlignHack 82(47.1) &
\cAlignHack 92(52.9) &
\cAlignHack &
\multicolumn{2}{p{\dimexpr(.26705714285714294\linewidth-2\tabcolsep)}}{\cAlignHack Reference} &
\cAlignHack \\
&
&
&
&
&
&
\\
\multicolumn{3}{p{\dimexpr(.4664714285714286\linewidth-2\tabcolsep)}}{\textbf{Because they are Cheap?}} &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack \\
\cAlignHack Do not use &
\cAlignHack 84(47.2) &
\cAlignHack 94(52.8) &
\cAlignHack .314 &
\cAlignHack 5.982 &
\cAlignHack .184 &
\cAlignHack 194.644\\
\cAlignHack Yes &
\cAlignHack 8(34.8) &
\cAlignHack 15(65.2) &
\cAlignHack .098 &
\cAlignHack 4.867 &
\cAlignHack .749 &
\cAlignHack 31.630\\
\cAlignHack No &
\cAlignHack 8(66.7) &
\cAlignHack 4(33.3) &
\cAlignHack &
\multicolumn{2}{p{\dimexpr(.26705714285714294\linewidth-2\tabcolsep)}}{\cAlignHack Reference} &
\cAlignHack \\
&
&
&
&
&
&
\\
\multicolumn{3}{p{\dimexpr(.4664714285714286\linewidth-2\tabcolsep)}}{\textbf{Because they are effective?}} &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack \\
\cAlignHack Do not use &
\cAlignHack 84(47.2) &
\cAlignHack 94(52.8) &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack \\
\cAlignHack Yes &
\cAlignHack 10(43.5) &
\cAlignHack 13(56.5) &
\cAlignHack .921 &
\cAlignHack .914 &
\cAlignHack .153 &
\cAlignHack 5.459\\
\cAlignHack No &
\cAlignHack 6(50.0) &
\cAlignHack 6(50.0) &
\cAlignHack &
\multicolumn{2}{p{\dimexpr(.26705714285714294\linewidth-2\tabcolsep)}}{\cAlignHack Reference} &
\cAlignHack \\
&
&
&
&
&
&
\\
\multicolumn{3}{p{\dimexpr(.4664714285714286\linewidth-2\tabcolsep)}}{\textbf{Because they have lesser side effects?}} &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack \\
\cAlignHack Do not use &
\cAlignHack 84(47.2) &
\cAlignHack 94(52.8) &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack \\
\cAlignHack Yes &
\cAlignHack 10(41.7) &
\cAlignHack 14(58.3) &
\cAlignHack .551 &
\cAlignHack 1.732 &
\cAlignHack .285 &
\cAlignHack 10.531\\
\cAlignHack No &
\cAlignHack 6(54.5) &
\cAlignHack 5(45.5) &
\cAlignHack &
\multicolumn{2}{p{\dimexpr(.26705714285714294\linewidth-2\tabcolsep)}}{\cAlignHack Reference} &
\cAlignHack \\
&
&
&
&
&
&
\\
\multicolumn{3}{p{\dimexpr(.4664714285714286\linewidth-2\tabcolsep)}}{\textbf{Because they are popular?}} &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack \\
\cAlignHack Do not use &
\cAlignHack 84(47.2) &
\cAlignHack 94(52.8) &
\cAlignHack &
\cAlignHack &
\cAlignHack &
\cAlignHack \\
\cAlignHack Yes &
\cAlignHack 10(47.6) &
\cAlignHack 11(52.4) &
\cAlignHack .861 &
\cAlignHack .851 &
\cAlignHack .141 &
\cAlignHack 5.140\\
\cAlignHack No &
\cAlignHack 6(42.9) &
\cAlignHack 8(57.1) &
\cAlignHack &
\multicolumn{2}{p{\dimexpr(.26705714285714294\linewidth-2\tabcolsep)}}{\cAlignHack Reference} &
\cAlignHack \\
\tblbottomrule
\end{tabulary}\par
\end{table*}
\textbf{Data collection procedures}
Information gathering was done after the surveys were pretested on a small sample (5\% of the aggregate sample) of the patients with HTN going to the cardiac OPD at SPS Hospital.
The patients included in the pre-test were excluded from the fundamental examination. Self-collection of data in the form of questionnaires and software \textit{Akhil Systems Pvt. ltd (health care IT partner), 1mg, MedPlus mart, Mims }was done.
\textbf{Data analysis}
Information was interpreted utilizing the Statistical Package for Social Sciences (SPSS) form 22 using bivariate logistic regression analysis, keeping in mind the end goal to appraise the quality of affiliation utilizing odds ratios (COR). All factors related with mental distress with a \textit{p value}{\textless}0.05 were considered as altogether relate. All factors related with psychological distress with a \textit{p value}{\textless}0.05 under 0.25 were analysed using multivariable logistic regression for adjusting potential confounders. Ages, hypersensitivity, utilization of grapes and patients' perception were investigated as constant factors.
\textbf{Ethical considerations}
Ethical approval was received from the Human Ethic board of trustees of SPS Hospitals, Ludhiana, Punjab. Informed consent was also retrieved from every participant before information accumulation. Confidentiality was maintained at all phase of information preparing and examination.
\section{Results and Discussion}
\textbf{Socio-demographic characteristics of Hypertensive patient:}
A total of 275 patients were taken into this study and 213 patients agreed to participate with a response rate of 77.45\%. The mean age of patients was 50.97\ensuremath{\pm}1.051 years and ranged from 18 to 86 years. Table~\ref{tw-58985ed33cc1} shows that out of total participants, female HTN-Pt were in the majority, majorly belonging to Sikh religion, followed by Hindus residing in an urban area, and were retired from occupation. Most of HTN-Pt were married and educationally sounded, i.e. graduate or undergraduates.
Table~\ref{tw-7fcddd77877e} shows the percentage responses of an unstructured questionnaire, which was framed to assess the perception and awareness of the patients regarding hypertension and its maintenance.
The prevalence of alcohol use in this study, 15.3\%, was higher than the similar study performed in Southwest Ethiopia (7.8\%) but lower than the findings from a community-based study done in Jimma town (62.4\%), Gurage Zone (21\%) \unskip~\citep{1156208:22694548}.
\textbf{Clinical status}
During the filling up of questionnaires, the vital of the patients were also measured to assess their current clinical status. The mean BMI of patients was 25.274\ensuremath{\pm}0.339 (Overweight) and ranged from 20 to 50. An analysis of mean systolic and diastolic pressures among the Urban Indian population in the age range between 40-49 years showed an increase of mean systolic pressure 120.4 mmHg and mean diastolic pressure 73.2 mm of Hg in 1942 to 128.7/84.2 mm of Hg in 1985, 128.8/83.2 mm of Hg in 1995 and 141/85 mm of Hg in 2005. A higher prevalence of hypertension is often indicated by a rise in mean systolic and diastolic pressure \unskip~\citep{1156208:22694551}.
In this study, the mean pulse rate of patients was 91.04/min (ranged from 70-100 beats/min), and the mean Blood Pressure of HTN-Pt was 93.74\ensuremath{\pm}7.4 mm of Hg (Diastolic)\textbf{/}137.05\ensuremath{\pm}7.14 mm of Hg (Systolic).
\textbf{Prevalence of Psychological distress}
The Kessler {\textendash} 10 ten items questionnaire was administered to patients to record the rate of psychological morbidity. As per the collected data, the commonness of psychological distress was found to be 46.9 \%. As per the indications, 53 \% of the patients were well, 26 \% were suffering from mild distress, 17\% from moderate psychological distress and 4\% were severely affected (Figure~\ref{f-2e36f64866ce}).
The present study findings revealed that the prevalence of psychological distress (46.9\%) was higher than the findings of community-based studies done among HTN-Pt in England (15.7\%) and former the Soviet Union (9.9\%) \unskip~\citep{1156208:22694547}. Psychological distress was found to be highly prevalent among HTN-Pt on follow-up at Jimma University Hospital.
Similarly, the prevalence of psychological morbidity found in this study was higher than the finding of a similar study done in Southwest Ethiopia (36.6\%) \unskip~\citep{1156208:22694538}. The discrepancy between the four studies might be due to the difference in the tools used to assess psychological morbidity (Kessler-10, Kessler-6 and GHQ-12).
\textbf{Association of socio-demographic and several other factors governing awareness, patients' perception, lifestyle and clinical background with psychological distress}
Data obtained using the questionnaire was entered stepwise and analyzed using the Statistical Package for Social Sciences (SPSS). The outcome and explanatory variables were entered into a bivariate logistic regression analysis, one by one, in order to estimate the strength of association using odds ratios (OR). The data with a p-value{\textless}0.05 were considered significant.
Highest percentage (33.80\%) of patients with psychological distress were associated with the age group 31-60 years of age \textit{(p value=0.003, COR= 0.240, 95\% CI 0.072, 0.584)}. Table~\ref{tw-df60dbadceaf} shows the response received from the patients regarding the use of herbal formulations and supplements, significant association was found with the use of grapes and the occurrence of psychological distress. Table~\ref{tw-df60dbadceaf} shows that patients having diet enriched with grapes has 0.380 times (62\%) more likely to have psychological distress compared to the patients not consuming grapes \textit{(p value=0.034, COR= 0.380, 95\% CI 0.155, 0.930)} or patients not eating grapes are 2.63 times more likely not to have psychological morbidity. However, there was no association between substance use and psychological morbidity.
Most of the herbal therapies used by the participants of this study are similar to those reported from other studies in the literature. Garlic, grapes, green tea, olives and some of the vitamins are natural foods recommended for healthy nutrition. However, they may be harmful when consumed in higher amounts and they may interact with drugs used for hypertension treatment \unskip~\citep{1156208:22694546}.
In this study, most of the hypertensive subjects perceived herbal therapies or formulations containing garlic, grapes, green tea and vitamins. It has been suggested in the literature that minerals and vitamins present in garlic, grape juice, green tea, etc., play a role in reducing blood pressure \unskip~\citep{1156208:22694550}.
However, another study on herbal extracts included grape seed, green tea, and ginkgo, ingredients in Brain Awake and Brain Support. These and several other herbs were found to inhibit sulfotransferase 1A3, a phase II detoxifying enzyme in the intestinal epithelium that modulates dopamine sulfation, thus increasing the bioavailability of dopaminergic drugs. With the effect of elevated dopamine levels in psychosis, the ability of these herbal interactions to modulate dopamine metabolism and reuptake may have contributed to Mr A's psychotic symptoms. In this study, 85\% of patients using herbal formulations were using grapes and out of which 50.50\% patients were suffering from psychological distress \unskip~\citep{1156208:22694540}.
\section{Conclusion}
The prevalence of psychological distress among HTN-Pt was 46.9\% of the total participants, incorporating those who had alcohol use disorders and were addicted to smoking habits. Patients with age (31-60 years) and those who were using herbal alternatives containing grapes were more likely to have psychological distress. However, no association with substance abuse, clinical background, and patient's perception and awareness was found. The effects of alternative herbal substances on blood pressure needs further investigation.
\section*{Acknowledgement}The authors greatly acknowledge the support and guidance provided by Dr. Shivani Tandon (Head, Pharmacology Department) and Shalloo Devi (Supervisor, Clinical Pharmacology Unit), Satguru Pratap Singh (SPS) Hospitals, Ludhiana for the successful completion of the project work.
\textbf{Conflict of Interest}
The authors declare that they have no conflict of interest.
\textbf{Funding Support}
The authors declare that they have no funding support for this study.
\bibliographystyle{pharmascope_apa-custom}
\bibliography{\jobname}
\end{document}
|
https://dlmf.nist.gov/22.18.E8a.tex | nist.gov | CC-MAIN-2018-26 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2018-26/segments/1529267864256.26/warc/CC-MAIN-20180621172638-20180621192638-00472.warc.gz | 589,108,730 | 699 | $x_{3}=\frac{x_{1}y_{2}+x_{2}y_{1}}{1-k^{2}x_{1}^{2}x_{2}^{2}},$ |
http://aas.org/archives/cdrom/volume5/volume5/apj/v450/p051/table2.tex | aas.org | CC-MAIN-2015-48 | text/x-tex | null | crawl-data/CC-MAIN-2015-48/segments/1448398450745.23/warc/CC-MAIN-20151124205410-00090-ip-10-71-132-137.ec2.internal.warc.gz | 805,554 | 1,055 | \documentstyle[apjpt]{article}
\begin{document}
\label{taoxbins}
\begin{centering}
\begin{tabular}{clccc}
\multicolumn{4}{c}{ TABLE 2} \\
\multicolumn{4}{c}{WOR REGRESSIONS of $\alpha_{ox}$(log$~l_{opt}$) }
\\ \hline \hline
Bin Parameters & No. of Bins & Slope & Intercept \\ \hline
& & & \\
\multicolumn{4}{c}{ Full LBQS/RASS Sample } \\
% & & & \\
$N_H$/log$~l_{opt}$ & ~~~1/ 20 & 0.09$\pm$0.02 & $-1.4\pm$0.7 \\
$N_H$/log$~l_{opt}$ & ~~~6/ 7 & 0.07$\pm$0.02 & $-0.6\pm$0.5 \\
$N_H$/log$~l_{opt}$ & ~~~5/ 5 & 0.08$\pm$0.02 & $-0.9\pm$0.7 \\
$N_H$/log$~l_{opt}$ & ~~~1/ 5 & 0.11$\pm$0.03 & $-1.8\pm$0.9 \\
% & & & \\
$N_H$/Redshift & ~~~1/ 17 & 0.11$\pm$0.01 & $-1.9\pm$0.4 \\
$N_H$/Redshift & ~~~5/ 5 & 0.11$\pm$0.02 & $-1.9\pm$0.7 \\
$N_H$/Redshift & ~~~1/ 4 & 0.15$\pm$0.01 & $-3.0\pm$0.2 \\
& & & \\
\multicolumn{4}{c}{ Radio Quiet Sample } \\
$N_H$/Redshift & ~~~1/ 4 &0.15$\pm$0.07 & $-3.1\pm$2.3 \\
& & & \\
\multicolumn{4}{c}{ Radio Loud Sample } \\
$N_H$/Redshift & ~~~1/ 4 & $0.00\pm$0.26 & $1.5\pm$8.1 \\ \hline
\end{tabular}
\end{centering}
\end{document}
|
http://unilab.gbb60166.jp/prekou/tex/2keta-hikizan.tex | gbb60166.jp | CC-MAIN-2017-30 | application/x-tex | text/x-matlab | crawl-data/CC-MAIN-2017-30/segments/1500549424564.72/warc/CC-MAIN-20170723142634-20170723162634-00159.warc.gz | 347,329,824 | 2,563 | %
% gbb60166@gmail.com http://unilab.gbb60166.jp/prekou/prekou.htm
%
% aspectratio= は 1610, 169, 149, 54, 32 の中から選べる（省略時は 43）
% C:\w32tex\share\texmf\tex\latex\beamer\beamer.cls
\documentclass[20pt,dvipdfmx,aspectratio=169]{beamer}
% pdfの栞の字化けを防ぐ
%\AtBeginDvi{\special{pdf:tounicode EUC-UCS2}}
% テーマ
\usetheme{Copenhagen}
% navi. symbolsは目立たないが，dvipdfmxを使うと機能しないので非表示に
\setbeamertemplate{navigation symbols}{}
%\usepackage{graphicx}
%\usepackage{amsmath}
%\usepackage{amssymb}
%\usepackage{tkokugo,furikana,tsayusen,shiika,sfkanbun,jdkintou,plext}
\usepackage{furikana,utf,bm,type1cm}
\usepackage{tikzsymbols}
%\usepackage[dvipdfmx]{graphicx}%
\def\pgfsysdriver{pgfsys-dvipdfmx.def}%（graphicxパッケージを使用しない場合はこの行を有効に）
%\def\pgfsysdriver{pgfsys-dvips.def}%デフォルト
\usepackage{tikz}%（これで、pgfとpgfforが読み込まれます。）
\usetikzlibrary{patterns,intersections}
\usetikzlibrary{calc,intersections,through,backgrounds}
%\PassOptionsToPackage{dvipdfmx}{graphicx}
%\PassOptionsToPackage{dvipdfmx}{color}
% beamer(dvipdfmオプション？) と emath を併用したときの
%「Option clash for package graphicx」を回避するための
%\PassOptionsToPackage{dvipdfmx}{graphicx}
%\PassOptionsToPackage{dvipdfmx}{color}
% フォントはお好みで
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\setbeamerfont{frametitle}{size=\normalsize,series=\bfseries}
\setbeamertemplate{frametitle}[default][center]
\usefonttheme{professionalfonts}
% 参考にしたURL
% http://windom.phys.hirosaki-u.ac.jp/fswiki/wiki.cgi?page=LaTeX+Beamer%A4%C7%A5%D7%A5%EC%A5%BC%A5%F3%A5%C6%A1%BC%A5%B7%A5%E7%A5%F3
\newcommand{\Slash}[1]{\ooalign{\hfil\kern-3pt／\hfil\crcr$#1$}}
\everymath{\displaystyle}
\def\maruwaku#1{\begin{tikzpicture}[scale=0.7, baseline={([yshift=-24pt] current bounding box.north)}]
\filldraw[color=blue, line width=1pt, rounded corners=2pt] (-0.1,0)--(2.1,0)--(2.1,1.1)--(-0.1,1.1)--cycle;
\draw(1,0.5) node[white]{#1};
\end{tikzpicture}
}
\begin{document}
\title{プレ小算数科}\author{gbb60166}
%■■■■■■■■■■■■■ テスト領域 ■■■■■■■■■■■■■■
%\end{document}
%■■■■■■■■■■■■■ 完成品 ■■■■■■■■■■■■■■
\begin{frame}[t]
\frametitle{$68-15$のひっさんのしかた}
\vspace*{-16pt}
\begin{tikzpicture}[scale=1.5, thick, baseline={([yshift=-36pt] current bounding box.north)}]
%\draw[help lines] (0,0) grid (3,4);
\visible<2->{\draw(2,4.2) node[green]{\fontsize{8}{8}\selectfont 十のくらい};
\draw(3,4.2) node[green]{\fontsize{8}{8}\selectfont 一のくらい};
\draw(0.9,2.4) node{\Huge$-$};
\draw[ultra thick](0.4,1.7)--++(3.2,0);
\draw[dashed,cyan,thick](1.5,0.5)--++(0,3.9);
\draw[dashed,cyan,thick](2.5,0.5)--++(0,3.9);
\draw[dashed,cyan,thick](3.5,0.5)--++(0,3.9);
}
\visible<3-4>{\draw[red, ultra thick, decorate, decoration=zigzag](1.85,5.2)--++(0.6,0);
\draw(2.15,5) node[red]{\tiny 上};
\draw[red, ultra thick, decorate, decoration=zigzag](3.05,5.2)--++(0.6,0);
\draw(3.35,5) node[red]{\tiny 下};
}
\visible<4->{\draw(2,3.6) node{\Huge$6$};
\draw(3,3.6) node{\Huge$8$};
\draw(2,2.4) node{\Huge$1$};
\draw(3,2.4) node{\Huge$5$};
}
\visible<7->{\draw(3,1) node{\Huge$3$};}
\visible<10->{\draw(2,1) node{\Huge$5$};}
\end{tikzpicture}
\hfill
\begin{minipage}[t]{0.5\textwidth}
\only<2-7>{\hspace*{-1zw}\ajMaru{1}くらいを たてに\\そろえて かく。\\[-12pt]}
\only<5-7>{\hspace*{-1zw}\ajMaru{2}\textcolor{green}{一のくらい}から \\けいさんする。}
\only<6-7>{\( 8-5=3 \)}
\only<8->{\hspace*{-1zw}\ajMaru{3}\textcolor{green}{十のくらい}を\\けいさんする。}
\only<9->{\( 6\textcolor{yellow}{0}-1\textcolor{yellow}{0}=5\textcolor{yellow}{0} \)\\}
\only<11->{こたえは $53$}
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\title{Remember the Fallen}
\date{1987}
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Kuwasi. Now he's gone.
And its like his last loving joke on me, his last gentle bursting of my egotistic bubble. Because i was all ready for a long, lingering heroic battle against disease and death. Ultimately lost, but great tragic courage and sharing. And he did it his way. Fast. Kuwasi was never cut out to be a tragic hero. Or any kind of hero. He hated it and fought it. And subverted many people's attempts to mould him into one or create a myth with which to bury his real self. Which was simple human.
Simply lustful of life and life's sensuous pleasures – food, people, wine and laughter. Lustful too, for battle against the enemy.
He hated hypocracy and I am writing this because i want to combat our own need for hypocracy, for myth. Let's not make him bigger than life – but simply human.
Let's not distort his ideology, but claim him as the anarchist he was, who allied with New Afrikan nationalists because it was the best way he saw to fight for the human rights and liberation of his people and all people.
Let's not bury those parts of him - his kinkiness, iconoclasm, individualism, because like it or not, they are part of what fed his courage, his idealism and willingness to make his life the revolution.
For Kuwasi, fighting for Freedom and living free were one and the same thing.
Maybe Kuwasi died so quickly because he got to that point, looked around at the party that was planned for him and it sure wasn't one of those wine and music and a million people rockers he loved and he escaped before he could be trapped off.
Some people might wish .that Kuwasi died a more properly “revolutionary” death, in combat against the enemy or at least from a more respectable disease than AIDS. But AIDS is a scourge of the people, oppressed people. Its endemic because those who suffer its wrath are mainly the dispossessed, the hated, the marginalized. So the system has refused to address it and has punished its victims. Many of our communities have disowned our own in the face of it.
In this prison, (Bedford) women with AIDS are isolated into a filthy ward mixed among other sick women whose germs will kill them. They are punished double, disowned, humiliated, feared and hated. i am glad that Kuwasi did not have to suffer that indignity, even though i greedily wanted him to live longer, because i was not ready to lose him.
Did Kuwasi get AIDS from his transvestite lover, who he persisted to love and insisted entrusting despite pressures and conflicts from the rest of us? i would like to say "from those others" in the revolutionary movements who hardly celebrated that part of his life. But having called for an honest accounting i have to look at my own bourgeois moralism, hypocracy and self-hating anti-gayness.
But Kuwasi was persistent and consistent in his own way. Kuwasi could love women and men fully, freely, lustfully and most of all with such generosity of spirit that it never felt exploitative.
He didn't live by the rules. Not society's or Christianity's or Islam's or feminism's or the New Afrikan Independence Movement's.
But he did have principles and integrity and honesty.
He'd fight like hell for his positions - but if you convinced him he'd change, and he realized that one's actions had to be consistent with one's principles.
We used to fight furiously about his love of pornography, i can still recall my fury at his exchanging short ice with one of the prison guards! Yet i felt more comfortable, intimate and freer with Kuwasi than almost any man i'’ve known. Comfortable enough to hug and kiss and massage and play through our legal meetings in the county jails. And .when he once said that for him making love could mean anything, could mean playing footsies, as long as it was fun and with love, as he sat there, gleefully massaging my bare foot in his lap, i .believed him and was delighted and thrilled.
i am fighting the allure of putting my own stamp on Kuwasi, as though it would be any more accurate than any others.
Only Kuwasi can define Kuwasi.
i hope people collect his poems and his theoretical writings, because that will be the truest reflection.
All i can do is speak for myself. That's one of the things Kuwasi taught me.
We are each ourselves and can only vouch for our own partial truths and when we ennoble that into dogmas, or try to enforce or assume collective assumptions through social pressure, we delude ourselves and will pay for it in the end.
Kuwasi believed that and clung to his own ideology and dreams as dogmatically and subjectively as the rest of us.
Which is to say, he was contradictory.
Like the rest of us. Human!!!
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The Anarchist Library
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Judy Clark
Remember the Fallen
1987
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http://ctan.pp.ua/macros/latex/contrib/ucdavisthesis/example/ucdavisthesis_example_Chap1.tex | pp.ua | CC-MAIN-2023-06 | application/octet-stream | application/x-tex | crawl-data/CC-MAIN-2023-06/segments/1674764500288.69/warc/CC-MAIN-20230205193202-20230205223202-00425.warc.gz | 11,640,312 | 1,199 | \chapter{Information on Preparation of a Dissertation or Thesis}
%
The text for this example dissertation is taken from the Graduate Studies web site\footnote{Please see \texttt{http://gradstudies.ucdavis.edu/students/filing.html} for the latest information.}. Please read through it carefully to make sure that your dissertation/thesis meets the university requirements.
For information and assistance in the preparation of your thesis or dissertation, please contact the Student Affairs Coordinator responsible for your program. You must make an appointment with the Student Affairs Coordinator in order to file your thesis or dissertation.
You are responsible for observing the filing dates and for preparing the thesis in the proper format. As part of the electronic filing process you will complete the PQIL Masters Thesis Agreement (Master's students) or the UMI Doctoral Dissertation Agreement (Ph.D. students) on which you indicate your willingness to have the University supply copies of your thesis to interested persons immediately, or the dissertation submitted to ProQuest.
You must have committee approval (title page signed off by committee) before submitting your thesis/dis |
http://ctan.math.utah.edu/ctan/tex-archive/macros/latex/contrib/schule/doc/changelog.tex | utah.edu | CC-MAIN-2020-34 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-34/segments/1596439737204.32/warc/CC-MAIN-20200807143225-20200807173225-00022.warc.gz | 24,467,787 | 1,249 | \section{Changelog}
Im Laufe der Jahre wurde das Paket immer wieder erweitert. Nicht nur die Anpassung an veränderte Anforderungen, etwa bei den Unterrichtsbesuchen, sondern auch neue Funktionalitäten fließen in das Paket ein.
Die folgende Liste bietet eine Übersicht über die letzten Änderungen.
\begin{itemize}
\item \textbf{0.8.1} -- 2018-08-22
\begin{itemize}
\item Umbau auf flache Verzeichnistiefe für die Anforderungen von TeXLive
\end{itemize}
\item \textbf{0.8.0} -- 2018-08-12
\begin{itemize}
\item Vollständiger Umbau von \pkg{exsheets} auf \pkg{xsim}
\item Modul \enquote{Bewertung} hinzugefügt
\item Dokumententyp \enquote{Leitprogramm} hinzugefügt
\item Dokumententyp \enquote{Folie} hinzugefügt
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\item \textbf{0.7.2} -- 2017-01-29
\begin{itemize}
\item Modul \enquote{Lizenzen} hinzugefügt
\item Fix: Bearbeitungshinweise konnten keine Makros enthalten
\end{itemize}
\item \textbf{0.7.1} -- 2017-01-08
\begin{itemize}
\item Dokumentklassen verworfen
\item Dokumenttypen als Module implementiert
\item Bearbeitungshinweise zu Aufgaben hinzugefügt
\item Deklaration von \cs{aufgabeMC} und \cs{aufgabeLueckentext} für Hinweise angepasst \textbf{Inkompatibel} mit der bisherigen Schnittstelle
\item Ungenauigkeiten in der Doku zu Erwartungen behoben
\end{itemize}
\item \textbf{0.7} -- 2016-09-01
\begin{itemize}
\item Vollständige Überarbeitung des Pakets
\end{itemize}
\end{itemize}
|
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\title{Interview with Individualists Tending toward the Wild}
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\emph{January 24, 2014, Chicomóztoc, México.}
\emph{1.- When did Individualists Tending towards the Wild emerge, what ideas motivated you as an affinity group and what strategies have you decided to pursue to give continuity to this antagonistic project?}
Before we begin to respond to this interview, ITS would like to clarify that while we do not share many of the ideas presented in this book, we see a chance to be able to explain our ideas in a more real way, and this is what we are doing. We do not want to emphasize membership with anyone, our ideas are our own, but now that they are out in the public light, it is necessary that they (or the important parts) are completely understood, as there seems to be much confusion in regards to various themes (including criticisms previously given in our communiques), which were not understandable to the reader or which were not accepted or assumed.
While we are not anarchists, we appreciate this space given by the Editorial Ácrata.
Now that we have clarified this, we will begin the interview:
Individualists tending towards the wild formed at the beginning of 2011, and was motivated by the reasoning acquired during a slow process of getting to know, questioning, and the rejection of all that encompasses leftism and the civilized, and accordingly, employing all the above, we deemed it necessary to carry out the direct attack against the Technoindustrial System. We think that the struggle against this is not only a stance of wanting to abandon Civilization, regressing to Nature, or in refuting the system’s values, without also, attacking it.
Our immediate objectives are very clear: injure or kill scientists and researchers (by the means of whatever violent act) who ensure the Technoindustrial System continues its course. As we have declared on various occasions, our concrete objective is not the destruction of the Technoindustrial system, it is the attack with all the necessary resources, lashing out at this system which threatens to close off all paths to the reaching of our Individual Freedom, putting into practice our defensive instinct.
Our position does not stop at putting into question that which many do not question (like the risk of the utilization and expansion of the Technological complex), but what’s more, we use violence (as we are human, we distinguish ourselves from our more distant, primitive, and wild ancestors) to attack that which intimidates the development of wild human Freedom and tends towards the artificiality of all that is potentially free. In short, we are the contrary part to the Technological System, we are the reaction before the action, resulting from coincidence; while some dedicate themselves to manipulate, destroy, and artificialize the natural, we respond to their aggression.
\emph{2.- On the 8\textsuperscript{th} of August 2011, ITS made headlines on the front pages of Mexican newspapers with the news of the explosive attack against the area of nanotech research at Monterrey Tech, State of Mexico campus, in which two of its scientists were injured: Armando Herrera Corral–who the “parcel-bomb” was addressed to–and his colleague, Alejandro Aceves López. The act provoked skepticism in sectors of the left which did not see the fight against new technologies as valid and therefore, do not include it in their accustomed catalog of “fronts.” We heard more than one accusatory discourse including this, classifying them as “terrorist” in the typical acceptance of Power’s lexicon. We would like to know what your opinion of these acts is, as well as your comments around the different positionings that have motivated your anti-technological action.}
The attack on Monterey Tech and its claim caused a big commotion nationally and internationally, we as ITS know that the aforementioned act struck hard in the police, political, social, and of course scientific spheres. The act was such, and as we had hoped, with a great magnitude of consequences. With this we knew that we wouldn’t only have months of making these acts a reality, but it would also lift the curtain, proving the existence of a radical tendency which speaks to the root of the problems we are faced with in this epoch, which is the most refined expression of domination: the entirety of Technology.
Continuing with the question, we also knew that our acts would not be well received by society nor the leftist sectors (left, center, and right politics). But all of these campaigns and designations don’t bother us, we don’t waste our energy in trying to make ourselves look like “good activists” to these people, as they are accustomed to seeing, because we are not.
They label us as a terrorists, because in fact, this treatment is always given to those individuals or groups, who hurt people for some incentive (whatever it is). This is also why, before we mentioned our motivations, we took the word and ITS was named as a terrorist group.
We are focused on attacking the scientists who perfect nanotechnology (this is a fact), since now science has advanced significantly in Mexico (apart from biotechnology and transgenetic genetic engineering) and this is perhaps why many have not put thought into what nanotechnology entails for the future (or more concretely, the Technological complex as well), in any case ITS has already addressed this previously and don’t have reason to revisit it: if you want to read more about this theme, we suggest reading the 1\textsuperscript{st}-4\textsuperscript{th} communiques (in which the theme of nanotechnology, notably, is focused on).
\emph{3.- What is the objective of ITS? Is it the destruction of the technoindustrial system?}
We would like to emphasize that ITS has never proposed the destruction of the technoindustrial system as a concrete objective, although we would want to and would declare that our objective is to completely destroy this rotten system, we would be lying to ourselves, and would be moving towards something that can not happen quickly, this is why we DO NOT claim this adventurous objective. ITS wants to see this entire system destroyed and collapsed, wants this to be the “slogan” that we defend, but it is not like that.
As we have said, ITS has from the beginning proposed the attack against the system as the objective, striving to make these kinds of ideas spread around the globe through extreme acts, in defense of Wild Nature, as we have done.
What we have done with these acts is put the proposal against Technology and Civilization on the table, creating tension, and we think that, with time, these attacks will be refined. We act through trial and error, learning from our mistakes, since we do not (as we have previously written) have the “secret formula.”
\emph{4.-Is it not very reductionist then, that your objective is only the attack and nothing more than that?}
It can sound very simple to focus on the Technoindustrial System as your only attack, but that is what exists for now. If we propose to destroy it we fall into fantasy, into utopia.
We attack this system from our individuality, not only with attacks, but also by rejecting the Technoindustrial Society along with its values and attempting to abandon Civilization, it serves nothing to attack the system and continue having its own values rooted in you (or vice versa).
\emph{5.- The movement named 15M in the Spanish state and its replication in other cities around the world has generated hope in sectors of the left which have begun to label it the Spanish Revolution. How does ITS view the development of this movement? What do you hope to have come from it or what critiques come up?}
The 15M movement is a movement that only proposes to reform the system, which it improves. The demands of the subjects who comprise this movement are based in political remands around austerity, the lack of employment, and a “better” economic strategy (among others); what this type of movement does, is that the people who are demanding that the government be accountable for the way in which they administer their economics, financial management, etc., it is erroneous, that if they do not want a strong state crisis (or in an extreme case civil war), they should apply some reforms in order for the system to continue its course, in short, the system digests these types of protest as proposals to strengthen itself; these types of people are called leftists (the term we have already used in different ITS communiques and is also explained as well in \emph{Industrial Society and its Future} by the Freedom Club), leftism becomes one of the many more ingenious functions of the Techno-industrial System. Thousands of people (or even a few) say they are going to rebel against it, when in reality they are only helping it realize its faults, to make them better, it regenerates, and self perpetuates.
\emph{6.- Continuing with the theme of leftism, in the public critique made by the editorial group “Anonymous with Caution” they said that your attacks only serve to make the system stronger, that many universities and institutions have redoubled security around nanotechnology engineers as well as the researchers that develop them- What is your position in light of this criticism?}
Look, the critique of this editorial group falls short of what we are now, you can read more about this in our \href{http://waronsociety.noblogs.org/?p=3162}{last communique} published on January 28\textsuperscript{th} of this year (2012).
Responding to your question, we do not think the system is made stronger with the type of actions we have carried out, and we have seen evidence of this.
Since what happened at the Tec, institutions, businesses, and universities that develop nanoscience, declared an immediate alert, principally as to what arrives in courier mail, well, of this there is no doubt.
Now, is the system made stronger when an explosive detonates in the hands of a professor and leaves his colleague wounded (as well)? Only in these moments does the system intensify security, but does not reinforce it in its totality, we remember that the system is not only nanotechnology, that it is comprised of other things, roots equally or perhaps more important than nanoscale science. So, you can’t say that the system has made itself immune to the attacks of our actions because, whats more than this, we have checked these boasts and they say it is strengthened when in reality it isn’t; this became very clear during the attack on the Polytechnic University in Pachuca en Hidalgo (December 8, 2011), our device (which arrived by mail courier in fact) left a professor wounded (we will say here as we said in the communique in which we claimed they attack, there was a mistake in the name of the researcher of nanotechnology that was our fault, his name was Villanueva not Villafaña), this act was evidence that the system had not fortified because even with the security protocols, another person was left newly wounded by ITS.
This is not only confirmed now by the ITS, but also as well in the past, Freedom Club equally proved this, 23 people wounded and 3 killed over 20 years, this is not a sign that the system became resistant to these types of acts.
In any regard, to say what this editorial group said is to exaggerate that which for now we have done, the attacks of ITS, yes, have not had destructive results stronger than material damages, paranoia, a few wounded, and a death, besides the fact that for some months we were the only public group who carried these kinds of ideas out in practice. For the system this is not sufficient for them to consider us a real threat, because we are merely beginning, its safe to say that individuals or groups in the future, taking into account our errors, will carry out more destructive acts against the Techno-industrial System; with this, we are not saying that we have faith in this happening, but only that it is logical that we will not be the only ones.
\emph{7. On November 8, 2011, only three months after the parcel bomb was sent to the nanotechnology researchers at Monterrey Tech, in which Herrera Corral and Aceves López were wounded, a researcher of the Institute of Biotechnology at UNAM (National Autonomous University of México), Ernesto Méndez Salinas was assassinated by a bullet to the head in the middle of Teopanzaolco avenue, in the city of Cuernavaca. This action provoked various new speculations as to the author of this action, putting the spotlight on ITS once again. Does ITS claim this attack? And–in the hypothetical case of being its perpetrators–why did you not claim responsibility through a public communique as has been your custom? Perhaps you decided (as other groups of anti-system action) to renounce these types of pronouncements and focus on propaganda of the deed?}
Concerning this action, we want to publicly declare that the group ITS takes responsibility for the attack.
The “prominent” investigator, Méndez Salinas, received a shot to the head which ended his life, from this extremist group, this is a fact.
The Federal District (D.F.) police very well know that ITS was responsible for this act. Around the middle of February 2011, we sent a letter with a claim of responsibility inside addressed to the director of the Institute of Physics of UNAM, to Dr. Manuel Torres Labansat. Inside there was a .380 caliber bullet, in addition to a note which practically said the researchers of aforementioned campus would end up the same as Salinas. The sending of the package with the bullet and the note we claimed in our \href{http://waronsociety.noblogs.org/?p=3162}{last communique} (January 28, 2012), only we did not mention this, for practical reasons.
We want to make it clear that the actions we carry out in practice, we claim in a prudent manner, if the situation is favorable, the claim will follow (as has been done with past attacks), but if there are things that are not so favorable or that we can “get more juice” from, we wait, and this is what we have done.
Practically, looking at the situation this year (2011), everything is in tension in respect to what we have begun to do, we knew beforehand that the police would not tell the media that we were responsible for the aforementioned attack. This is why we were saving it for a precise moment.
Honestly we do not know when this interview will be published, but supposing that it is delayed in its publication, we are thinking of claiming this act in a more detailed way, when we have executed some other attack in the not-so-distant future, well, it is also clear that this small part of the claiming of Méndez Salinas’s assassination will also be made very public when this book is released.
\emph{8. Want to say anything else?}
We hope that with this interview (dated April 28, 2012) our position has been made a little more clear to readers. And we are grateful to the editorial for this interview and to the portal \href{https://waronsociety.noblogs.org}{War on Society} for serving as the intermediary to make this exchange possible. That is all for now.
\begin{flushright}
\textbf{\emph{Individualists tending toward the wild (ITS).}}
\end{flushright}
% begin final page
\clearpage
% if we are on an odd page, add another one, otherwise when imposing
% the page would be odd on an even one.
\ifthispageodd{\strut\thispagestyle{empty}\clearpage}{}
% new page for the colophon
\thispagestyle{empty}
\begin{center}
The Anarchist Library
\smallskip
Anti-Copyright
\bigskip
\includegraphics[width=0.25\textwidth]{logo-en}
\bigskip
\end{center}
\strut
\vfill
\begin{center}
Individualists Tending toward the Wild
Interview with Individualists Tending toward the Wild
2014
\bigskip
Retrieved on February 13\textsuperscript{th}, 2014 from http:\Slash{}\Slash{}waronsociety.noblogs.org\Slash{}?p=8802
It’s been a few weeks since the publication of the book “May the night be set alight! Genesis, development, and rise of the Informal Anarchist Tendency”, in which an interview with the ITS is presented, and on our behalf we want to (newly), re-clarify our total gratitude to those who have made it possible for our words to be spread in this form. While recently, upon re-reading our interview, ITS has decided to make a few changes (albeit very small ones) to the original text that we sent on April 28, 2012, which we now present.
\bigskip
\textbf{theanarchistlibrary.org}
\end{center}
% end final page with colophon
\end{document}
|
https://mirrors.ibiblio.org/CTAN/macros/latex/contrib/statex/statex-example.tex | ibiblio.org | CC-MAIN-2023-14 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2023-14/segments/1679296949009.11/warc/CC-MAIN-20230329151629-20230329181629-00792.warc.gz | 451,420,949 | 2,403 | \documentclass[dvipsnames,usenames]{report}
\usepackage{statex}
\usepackage{shortvrb}
\MakeShortVerb{@}
% Examples
\begin{document}
Many accents have been re-defined
@ c \c{c} \pi \cpi@ $$ c \c{c} \pi \cpi$$ %upright constants like the speed of light and 3.14159...
@int \e{\im x} \d{x}@ $$\int \e{\im x} \d{x}$$ %\d{x}; also note new commands \e and \im
@\^{\beta_1}=b_1@ $$\^{\beta_1}=b_1$$
@\=x=\frac{1}{n}\sum x_i@ $$\=x=\frac{1}{n}\sum x_i$$ %also, \b{x}, but see \ol{x} below
@\b{x} = \frac{1}{n} \wrap[()]{x_1 +\.+ x_n}@ $$\b{x} = \frac{1}{n} \wrap[()]{x_1 +\.+ x_n}$$
Sometimes overline is better: @\b{x}\ vs.\ \ol{x}@ $$\b{x}\ vs.\ \ol{x}$$
And, underlines are nice too: @\ul{x}@ $$\ul{x}$$
A few other nice-to-haves:
@\Gamma[n+1]=n!@ $$\Gamma[n+1]=n!$$
@\binom{n}{x}@ $$\binom{n}{x}$$ %provided by amsmath package
@\e{x}@ $$\e{x}$$
%$\H_0: \mu_\ij=0$ vs. $\H_1: \mu_\ij \neq 0$ %\ijk too
@\logit \wrap{p} = \log \wrap{\frac{p}{1-p}}@ $$\logit \wrap{p} = \log \wrap{\frac{p}{1-p}}$$
\pagebreak
Common distributions along with other features follows:
Normal Distribution
@Z ~ \N{0}{1}, \where \E{Z}=0 \and \V{Z}=1@ $$Z ~ \N{0}{1}, \where \E{Z}=0 \and \V{Z}=1$$
@\P{|Z|>z_\ha}=\alpha@ $$\P{|Z|>z_\ha}=\alpha$$
@\pN[z]{0}{1}@ $$\pN[z]{0}{1}$$
or, in general
@\pN[z]{\mu}{\sd^2}@ $$\pN[z]{\mu}{\sd^2}$$
Sometimes, we subscript the following operations:
@\E[z]{Z}=0, \V[z]{Z}=1, \and \P[z]{|Z|>z_\ha}=\alpha@ $$\E[z]{Z}=0, \V[z]{Z}=1, \and \P[z]{|Z|>z_\ha}=\alpha$$
Multivariate Normal Distribution
@\bm{X} ~ \N[p]{\bm{\mu}}{\sfsl{\Sigma}}@ $$\bm{X} ~ \N[p]{\bm{\mu}}{\sfsl{\Sigma}}$$ %\bm provided by the bm package
Chi-square Distribution
@Z_i \iid \N{0}{1}, \where i=1 ,\., n@ $$Z_i \iid \N{0}{1}, \where i=1 ,\., n$$
@\chisq = \sum_i Z_i^2 ~ \Chi{n}@ $$\chisq = \sum_i Z_i^2 ~ \Chi{n}$$
@\pChi[z]{n}@ $$\pChi[z]{n}$$
t Distribution
@\frac{\N{0}{1}}{\sqrt{\frac{\Chisq{n}}{n}}} ~ \t{n}@ $$\frac{\N{0}{1}}{\sqrt{\frac{\Chisq{n}}{n}}} ~ \t{n}$$
\pagebreak
F Distribution
@X_i, Y_{\~i} \iid \N{0}{1} \where i=1 ,\., n; \~i=1 ,\., m \and \V{X_i, Y_{\~i}}=\sd_\xy=0@ $$X_i, Y_{\~i} \iid \N{0}{1} \where i=1 ,\., n; \~i=1 ,\., m \and \V{X_i, Y_{\~i}}=\sd_\xy=0$$%\XY too
@\chisq_x = \sum_i X_i^2 ~ \Chi{n}@ $$\chisq_x = \sum_i X_i^2 ~ \Chi{n}$$
@\chisq_y = \sum_{\~i} Y_{\~i}^2 ~ \Chi{m}@ $$\chisq_y = \sum_{\~i} Y_{\~i}^2 ~ \Chi{m}$$
@\frac{\chisq_x}{\chisq_y} ~ \F{n, m}@ $$\frac{\chisq_x}{\chisq_y} ~ \F{n, m}$$
Beta Distribution
@B=\frac{\frac{n}{m}F}{1+\frac{n}{m}F} ~ \Bet{\frac{n}{2}, \frac{m}{2}}@ $$B=\frac{\frac{n}{m}F}{1+\frac{n}{m}F} ~ \Bet{\frac{n}{2}, \frac{m}{2}}$$
@\pBet{\alpha}{\beta}@ $$\pBet{\alpha}{\beta}$$
Gamma Distribution
@G ~ \Gam{\alpha, \beta}@ $$G ~ \Gam{\alpha, \beta}$$
@\pGam{\alpha}{\beta}@ $$\pGam{\alpha}{\beta}$$
Cauchy Distribution
@C ~ \Cau{\theta, \nu}@ $$C ~ \Cau{\theta, \nu}$$
@\pCau{\theta}{\nu}@ $$\pCau{\theta}{\nu}$$
Uniform Distribution
@X ~ \U{0, 1}@ $$X ~ \U{0, 1}$$
@\pU{0}{1}@ $$\pU{0}{1}$$
or, in general
@\pU{a}{b}@ $$\pU{a}{b}$$
Exponential Distribution
@X ~ \Exp{\lambda}@ $$X ~ \Exp{\lambda}$$
@\pExp{\lambda}@ $$\pExp{\lambda}$$
Hotelling's $T^2$ Distribution
@X ~ \Tsq{\nu_1, \nu_2}@ $$X ~ \Tsq{\nu_1, \nu_2}$$
Inverse Chi-square Distribution
@X ~ \IC{\nu}@ $$X ~ \IC{\nu}$$
Inverse Gamma Distribution
@X ~ \IG{\alpha, \beta}@ $$X ~ \IG{\alpha, \beta}$$
Pareto Distribution
@X ~ \Par{\alpha, \beta}@ $$X ~ \Par{\alpha, \beta}$$
@\pPar{\alpha}{\beta}@ $$\pPar{\alpha}{\beta}$$
Wishart Distribution
@\sfsl{X} ~ \W{\nu, \sfsl{S}}@ $$\sfsl{X} ~ \W{\nu, \sfsl{S}}$$
Inverse Wishart Distribution
@\sfsl{X} ~ \IW{\nu, \sfsl{S^{-1}}}@ $$\sfsl{X} ~ \IW{\nu, \sfsl{S^{-1}}}$$
Binomial Distribution
@X ~ \Bin{n, p}@ $$X ~ \Bin{n, p}$$
@\pBin{n}{p}@ $$\pBin{n}{p}$$
Bernoulli Distribution
@X ~ \B{p}@ $$X ~ \B{p}$$
Beta-Binomial Distribution
@X ~ \BB{p}@ $$X ~ \BB{p}$$
@\pBB{n}{\alpha}{\beta}@ $$\pBB{n}{\alpha}{\beta}$$
Negative-Binomial Distribution
@X ~ \NB{n, p}@ $$X ~ \NB{n, p}$$
Hypergeometric Distribution
@X ~ \HG{n, M, N}@ $$X ~ \HG{n, M, N}$$
Poisson Distribution
@X ~ \Poi{\mu}@ $$X ~ \Poi{\mu}$$
@\pPoi{\mu}@ $$\pPoi{\mu}$$
Dirichlet Distribution
@\bm{X} ~ \Dir{\alpha_1 \. \alpha_k}@ $$\bm{X} ~ \Dir{\alpha_1 \. \alpha_k}$$
Multinomial Distribution
@\bm{X} ~ \M{n, \alpha_1 \. \alpha_k}@ $$\bm{X} ~ \M{n, \alpha_1 \. \alpha_k}$$
\pagebreak
To compute critical values for the Normal distribution, create the
NCRIT program for your TI-83 (or equivalent) calculator. At each step, the
calculator display is shown, followed by what you should do (\Rect\ is the
cursor):\\
\Rect\\
\Prgm\to@NEW@\to@1:Create New@\\
@Name=@\Rect\\
NCRIT\Enter\\
@:@\Rect\\
\Prgm\to@I/O@\to@2:Prompt@\\
@:Prompt@ \Rect\\
\Alpha[A],\Alpha[T]\Enter\\
@:@\Rect\\
\Distr\to@DISTR@\to@3:invNorm(@\\
@:invNorm(@\Rect\\
1-(\Alpha[A]$\div$\Alpha[T]))\Sto\Alpha[C]\Enter\\
@:@\Rect\\
\Prgm\to@I/O@\to@3:Disp@\\
@:Disp@ \Rect\\
\Alpha[C]\Enter\\
@:@\Rect\\
\Quit\\
Suppose @A@ is $\alpha$ and @T@ is the number of tails. To run the program:\\
\Rect\\
\Prgm\to@EXEC@\to@NCRIT@\\
@prgmNCRIT@\Rect\\
\Enter\\
@A=?@\Rect\\
0.05\Enter\\
@T=?@\Rect\\
2\Enter\\
@1.959963986@
\end{document}
|
https://www.gust.org.pl/projects/pearls/2006p/bernd-raichle/bachotex2006-bernd-raichle-pearl5.tex/at_download/file | gust.org.pl | CC-MAIN-2022-33 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2022-33/segments/1659882572163.61/warc/CC-MAIN-20220815085006-20220815115006-00276.warc.gz | 717,544,471 | 1,744 | %%% Bernd Raichle: global assignments done locally
% Sometimes it is necessary to define a macro under a changed input regime,
% e.g, using different category codes or another end line character. Usually
% this is done inside a group to keep the changes locally and the macro or the
% token is defined \global'ly.
%
% In plain.tex you can find the following examples:
\catcode`\@=11
% 1. Helper macro for \newif
{\uccode`1=`i \uccode`2=`f \uppercase{\gdef\if@12{}}} % `if' is required
% 2. Definitions of \obeylines and \obeyspaces:
{\catcode`\^^M=\active % these lines must end with %
\gdef\obeylines{\catcode`\^^M\active \let^^M\par}%
\global\let^^M\par} % this is in case ^^M appears in a \write
{\obeyspaces\global\let =\space}
% 3. Definition of \getf@ctor:
{\catcode`p=12 \catcode`t=12 \gdef\\#1pt{#1}} \let\getf@ctor=\\
% 4. Math definitions for primes and underscore:
{\catcode`\'=\active \gdef'{^\bgroup\prim@s}}
{\catcode`\_=\active \global\let_=\_} % _ in math is either subscript or \_
% By placing the begin and end of group tokens in a bit
% ``unusual'' way using a temporary token register assignment
% or a macro definition, all these assignments can be done
% locally:
% 1. Helper macro for \newif:
\begingroup
\uccode`1=`i \uccode`2=`f
\uppercase{\endgroup
\def\if@12{}% `i'+`f' as delimited arguments are required
}%
% 2. Definitions of \obeylines and \obeyspaces:
\begingroup
\endlinechar=-1
\catcode`\^^M=\active
\toks0={\endgroup
\def\obeylines{\catcode`\^^M\active \let^^M\par}%
\let^^M=\par % this is in case ^^M appears in a \write
}\the\toks0\relax
\begingroup \obeyspaces\def\x{\endgroup\let =\space}\x
% 3. Definition of \getf@ctor:
\begingroup
\catcode`P=12 \catcode`T=12
% \lccode`P=`p \lccode`T=`t % is default setting
\lowercase{\endgroup
\def\getf@ctor#1PT{#1}% `p'+`t' with catcode `other'
}%
% 4. Math definitions for primes and underscore:
\begingroup
\catcode`\'=\active
\def\x{\endgroup
\def'{^\bgroup\prim@s}%
}\x
\begingroup
\catcode`\_=\active
\def\x{\endgroup
\let_=\_% % _ in math is either subscript or \_
}\x
% There is no advantage of a local definition for the shown plain.tex cases but
% if you want to do similar definitions within a group, the shown technique can
% be very helpful.
%
% Note: If you want to define macros with arguments it is better to use a token
% register assignment because you have to double the hash mark as macro
% parameter character inside the macro definition text.
\end
|
http://ctan.sharelatex.com/tex-archive/info/examples/Math/11-2-11.ltx | sharelatex.com | CC-MAIN-2013-48 | text/x-tex | null | crawl-data/CC-MAIN-2013-48/segments/1386165000886/warc/CC-MAIN-20131204135000-00033-ip-10-33-133-15.ec2.internal.warc.gz | 44,417,024 | 994 | %%
%% Der Mathematiksatz mit LaTeX, 1. Auflage 2009
%%
%% Example 11-2-11 on page 248.
%%
%% Copyright (C) 2009 Herbert Voss
%%
%% It may be distributed and/or modified under the conditions
%% of the LaTeX Project Public License, either version 1.3
%% of this license or (at your option) any later version.
%%
%% See http://www.latex-project.org/lppl.txt for details.
%%
\documentclass[]{ttctminimal}
\pagestyle{empty}
\setcounter{page}{6}
\setlength\textwidth{355.65944pt}
\usepackage[utf8]{inputenc}
\usepackage[ngerman]{babel}
\setlength\parindent{0pt}
\StartShownPreambleCommands
\usepackage[T1]{fontenc} \usepackage{txfonts}
\StopShownPreambleCommands
\begin{document}
\input{fontDemo}
\end{document}
|
https://dlmf.nist.gov/28.2.E6.tex | nist.gov | CC-MAIN-2018-13 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2018-13/segments/1521257645248.22/warc/CC-MAIN-20180317155348-20180317175348-00056.warc.gz | 570,694,116 | 700 | \[\mathscr{W}\left\{w_{\mbox{\tiny I}},w_{\mbox{\tiny II}}\right\}=1,\] |
https://www.azimuthproject.org/mathbook/tex/Notes+-+Real+Analysis%2C+Serge+Lang | azimuthproject.org | CC-MAIN-2020-45 | text/plain | application/x-tex | crawl-data/CC-MAIN-2020-45/segments/1603107910204.90/warc/CC-MAIN-20201030093118-20201030123118-00528.warc.gz | 622,470,036 | 4,113 | \documentclass[12pt,titlepage]{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}
%----Macros----------
%
% Unresolved issues:
%
% \righttoleftarrow
% \lefttorightarrow
%
% \color{} with HTML colorspec
% \bgcolor
% \array with options (without options, it's equivalent to the matrix environment)
% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}
% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.
% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother
% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
\ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
\else \append@sup#2 \@suptrue \fi%
\expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
\reset@sup\mk@scripts#3_/}
\makeatother
% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
\@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
\copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother
% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
% http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}
% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}
% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}
% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}
% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
<-6> MnSymbolE5
<6-7> MnSymbolE6
<7-8> MnSymbolE7
<8-9> MnSymbolE8
<9-10> MnSymbolE9
<10-12> MnSymbolE10
<12-> MnSymbolE12}{}
% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
\let#1=\undefined
\DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother
% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
\if\relax\noexpand#1%
\let#1\undefined
\fi
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\begin{document}
%-------------------------------------------------------------------
\section*{Notes - Real Analysis, Serge Lang}
\begin{itemize}%
\item Serge Lang, Real Analysis, Second Edition, Addison-Wesley, 1983. Text for first year graduate course in analysis.
\end{itemize}
\hypertarget{chapter_7_hilbert_space}{}\subsubsection*{{Chapter 7, Hilbert Space}}\label{chapter_7_hilbert_space}
Let $E$ be vector space over the complex numbers.
A \emph{sequilinear form} is a function from $E \times E$ into $\mathbb{C}$, which is linear in the first argument and \emph{semi-linear} (``conjugate linear'') in the second argument, which means $f(x + y) = f(x) + f(y)$ and $f(c x) = \overline{c} f(x)$.
The form is written $\langle v, w \rangle$.
The form is called \emph{hermitian} if $\langle w,v \rangle = \overline{\langle v,w \rangle}$.
Hermitian implies that $\langle v,v \rangle = \overline{\langle v,v \rangle}$, so $\langle v,v \rangle$ is real.
A hermitian form is called positive if $\langle v,v \rangle$ is never negative, and positive definite if it is always positive.
\emph{Orthogonal} means that $\langle v,w \rangle = 0$.
\end{document}
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\bibitem{Aulsebrook_etal2018} Aulsebrook, A. E., Jones, T. M., Mulder, R. A., \& Lesku, J. A. (2018). Impacts of artificial light at night on sleep: A review and prospectus. \textit{J Exp Zool A Ecol Integr Physiol}, \textit{329}(8-9), 409--418.
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% vim: set spell formatoptions=aw spelllang=pl:
\begin{elementlit}
{Piotr Duchliński}
{\autor{Piotr \kapit{Duchliński}}
\afiliacja{\wydzf, \aik}}
{Heurystyczna rola obrazów świata}
{Heurystyczna rola obrazów świata w~przyjmowaniu
faktów filozoficznych}
{The heuristic role of world-pictures in the process
by which philosophical facts come to be accepted}
\index{Duchliński, P.}
\index{Wittgenstein, L.}
\index{Akwinata|see{Tomasz z~Akwinu, św.}}
\oDef{\oAkwinat}{Akwinat}{Akwinata} %Tomasz z Akwinu
\oDef{\oDuchlinski}{Duchliński}{Duchliński, P.}
\oDef{\oPutnam}{Putnam}{Putnam, H.} %Hilary
\oDef{\oFleck}{Fleck}{Fleck, L.} %Ludwik
\oDef{\oAbel}{Abel}{Abel, G.} %Günter
\oDef{\oMaritain}{Maritain}{Maritain, J.} %Jacques
\oDef{\oKlosak}{Kłósak}{Kłósak, K.} %Kazimierz
\oDef{\oHajduk}{Hajduk}{Hajduk, Z.} %Zygmunt
\oDef{\oKaminski}{Kamiński}{Kamiński, S.} %Stanisław
\oDef{\oKrapiec}{Krąpiec}{Krąpiec, M.A.} %Mieczysław Albert
\oDef{\oStepien}{Stępień}{Stępień, A.B.} %Antoni Bazyli
\oDef{\oArago}{Arago}{Arago, F.} %François
\oDef{\oFizeau}{Fizeau}{Fizeau, H.} %Hippolyte
\oDef{\oMichelson}{Michelson}{Michelson, A.A.} %Albert Abraham
\oDef{\oMorley}{Morley}{Morley, E.W.} %Edward Williams
\oDef{\oHeller}{Heller}{Heller, M.} %Michał
\oDef{\oLavoisier}{Lavoisier}{Lavoisier, A.L.} %Antoine Laurent
\oDef{\oJung}{Jung}{Jung, C.G.} %Carl Gustav
\oDef{\oMorse}{Morse}{Morse, S.F.B.} %Samuel Finley Breese
\oDef{\oMoore}{Moore}{Moore, G.E.} %George Edward
\streszczenie{
Celem artykułu jest wykazanie, że wpływ na akceptację okreś\-lonych
faktów filozoficznych ma przyjęcie obrazu świata, jako wiedzy tła,
na podstawie której rozstrzyga się, co istnieje, a~co nie istnieje,
co jest prawdą, a~co fałszem.
Omówione zostało stanowisko tomistów egzystencjalnych
\oWittgenstein[a] oraz \ios{Abla}{Abel, G.},
z~odwołaniem także do prac \oPutnam[a] i~\oFleck[a].
Wstępnie wykazano, że fakty filozoficzne są akceptowane na bazie
obrazu świata, który jest splotem faktów wartości i~teorii o~różnym
stopniu ogólności.
O~ich akceptacji nie decyduje zatem żadne bezpośrednie doświadczenie.
Nie ma żadnych neutralnych faktów filozoficznych.
Obraz świata sugeruje, jakie fakty przyjmujemy oraz jakie metody
stosujemy do ich wyjaśnienia.
Na tym właśnie polega heurystyczna rola obrazów świata
w~,,kreowaniu'' faktów filozoficznych.
}{
obraz świata ---
fakt ---
fakty filozoficzne ---
tomizm ---
filozofia analityczna ---
filozofia interpretacji ---
paradygmat ---
filozofia nauki
}
\tytul{Uwagi wstępne}
Uprawianie filozofii nigdy nie zaczyna się od zera.
Każdy filozof akceptuje mniej lub bardziej rozbudowane założenia.
Metodologowie filozofii uważają, że filozofowie przyjmują tzw.
fakty filozoficzne, które mają stanowić punkt wyjścia teorii,
jak również jej uzasadnienie.
Niektórzy nawet twierdzą, że mają one moc rozstrzygania,
która teoria filozoficzna jest prawdziwa, a~która nie.
W~polskiej metodologii filozofii faktom filozoficznym sporo
uwagi poświęcali metodologowie tomistyczni\footnote{
Por. \cite{Mazierski:Prolegomena};
\cite{Mazierski:Elementy};
\cite{Mazierski:CzyFilozofia}.
Analizę faktów filozoficznych znajdujemy też w~pracach
\oMaritain[a], \oKlosak[a] oraz \oHajduk[a], którzy omawiają
poszczególne koncepcje oraz oceniają je pod kątem przydatności
dla filozofii przyrody.
Z~pozycji tomizmu egzystencjalnego problematyka faktów pojawi się
przede wszystkim u~\oKaminski[ego], \ios{Krąpca}{Krąpiec, M.A.}
oraz \ios{Stępnia}{Stępień, A.B.}.
}.
W~swoich analizach poruszali nie tylko problem metodologicznej
determinacji faktów filozoficznych, ale też próbowali określić
ich stosunek do tzw. faktów naukowych\footnote{
Szereg ważnych uwag dotyczących faktów przedstawia \oFleck{}
w~swojej epistemologii poznania naukowego (por.
\cite{Fleck:Psychosocjologia}).
Na temat tej koncepcji por.
\cite{Sady:Fleck}, s.~11--89.
}.
Interesowała ich zwłaszcza kwestia filozoficznej interpretacji
faktów naukowych\footnote{
Tomiści, jak się wydaje, dlatego tak bardzo zwracali uwagę
na kwestię faktów filozoficznych, gdyż ich zamiarem było
uprawianie filozofii naukowej, oczywiście rozumianej
niescjentystycznie.
Wyróżniam dwie koncepcje filozofii nauki: scjentystyczną,
która była właściwa dla wczesnego pozytywizmu i~pozytywizmu
Koła wiedeńskiego i~która charakteryzowała się wiarą
w~nieograniczoną moc wyjaśniającą metody naukowej,
oraz filozofię nauki niescjentystyczną, która zwraca uwagę
na ograniczoność epistemologiczną metody naukowej,
a~tym samym wskazuje na takie obszary badawcze, które mogą być
przedmiotem analizy filozoficznej.
}.
Problem nie jest przebrzmiały ani też nie należy do lamusa historii.
Stale bowiem pojawiają się prace, w~których podejmuje się te
zagadnienia, nadając im nowe rozwiązania\footnote{
Por. \cite{Turek:Filozoficzne}.
}.
Na ich rolę w~determinacji faktów filozoficznych wskazywali,
mniej lub bardziej wyraźnie, niektórzy autorzy neotomistyczni,
np. \oKlosak.
Będziemy starać się wykazać, że filozofowie nie akceptują tzw.
faktów filozoficznych jako bezpośrednich raportów z~doświadczenia,
ale przyjmują je na podstawie akceptacji określonego obrazu świata;
może być to obraz filozoficzny lub naukowy\footnote{
W~artykule nie charakteryzujemy bliżej naukowego obrazu świata.
Taka charakterystyka w~odniesieniu do filozofii aporetycznej
została przedstawiona w pracy:
\cite{Duchlinski:Czynniki}.
}.
Fakty filozoficzne nie są czymś, co filozof zastaje w~świecie
lub odkrywa w~toku „odczytywania” obiektywnej struktury świata.
W~analizach pomijamy ogólną problematykę faktu, która w~literaturze
przedmiotu doczekała się szeregu opracowań\footnote{
Przegląd najnowszej problematyki dotyczącej faktów zawiera
zwięzłe opracowanie
\cite{Mulligan:Facts}.
Tam też znajduje się obszerna literatura.
W~artykule tym nie ma jednak mowy o~czymś takim,
jak fakty filozoficzne.
}.
Nasze analizy będą miały kilka kroków.
W~pierwszym scharakteryzujemy propozycję tzw. neutralnych faktów
filozoficznych, głoszoną obecnie przez współczesnych tomistów
egzystencjalnych.
W~drugim kroku przedstawimy, powołując się zwłaszcza na prace
\oWittgenstein[a] i~\ios{Abla}{Abel, G.}, czym jest obraz świata
oraz w~jaki sposób go przyjmujemy.
Natomiast w~trzecim kroku wykażemy, w~jaki sposób obraz świata,
który przyswajamy razem z~grą językową, wpływa na akceptację takich
a~nie innych tzw. faktów filozoficznych.
Przedłożona propozycja nie rozwiązuje ostatecznie statusu
epistemologicznego faktów filozoficznych\footnote{
Kwestii tej należałoby poświęcić osobne opracowanie.
}.
Poddaje tylko pod dyskusję pewną propozycję ich objaśnienia,
korzystającą z~ustaleń współczes\-nej filozofii analitycznej
i~hermeneutyki.
W~żadnym też wypadku w~artykule nie głosi się gotowej doktryny
na temat tzw. faktów filozoficznych, raczej dyskutuje się różne
stanowiska (tomistów, analityków, zwolenników hermeneutyki)
w~tej sprawie, pokazując, że jest to problematyka dosyć istotna
dla uprawiania filozofii i~wcale nie przedawniona.
Wstępny przegląd tych zagadnień pozwala na przyjęcie uzasadnionego
stanowiska, w~którym bardziej chcemy zwrócić uwagę na
wieloaspektowość zagadnienia związanego z~tzw. faktami
filozoficznymi niż dekretowanie ostatecznego rozwiązania
tego problemu.
Zasadniczo chcemy tylko zwrócić uwagę na heurystyczną rolę
obrazów świata w~przyjmowaniu określonych typów faktów
filozoficznych.
Implikacje poniższych analiz odnoszą się nie tylko do tomizmu,
z~którym podejmujemy tutaj polemikę, ale też do innych nurtów
filozoficznych np. fenomenologii.
\tytul{1. Świat realny jako podstawa akceptacji neutralnych \\
faktów filozoficznych}
We współczesnej filozofii nauki przyjmuje się powszechnie,
że fakty są na różny sposób uteoretyzowane\footnote{
Szerzej na ten temat por.
\cite{Brown:Perception}, s.~81--109,
a~także:
\cite{Hajduk:Metodologiczna};
\cite{Hajduk:Niedookreslenie};
\cite{Jodkowski:Obserwacja};
\cite{Wolenski:Dlaczego}.
}.
\cytuj{
Nie posiadamy żadnej wiedzy o~świecie wobec jakiegoś systemu
neutralnej.
Wiedzy takiej nie dostarczają nam dane zmysłów.
Dane zmysłów stają się źródłem informacji tylko wtedy,
gdy pełnią funkcję sygnalizacyjną względem przygotowanego
wcześniej systemu.
Dane doświadczenia trzeba przecież nazwać, pojęciowo
poklasyfikować.
Z~samych danych nie wynika żaden opis --- zdanie nie może
wynikać z~obrazu, a~jedynie z~innych zdań.
To nie doświadczenie nas uczy, ale my się uczymy na podstawie
doświadczenia --- dopasowując do doświadczeń modele i~w~oparciu
o~te modele wyniki doświadczeń odczytując.
Wychowani na teorii względności odczytujemy wyniki doświadczeń
Arago\index{Arago, F.},
Fizeau\index{Fizeau, H.},
Michelsona\index{Michelson, A.A.}--Morleya\index{Morley, E.W.}
i~innych jako świadectwa nieistnienia eteru --- fizycy jednak
wychowani na mechanice Newtona\index{Newton, I.}
te same doświadczenia odczytywali jako świadectwa pewnych
jego szczególnych własności.
Ten sam zmysłowy obraz może świadczyć o~różnych rzeczach\footnote{
\cite{Sady:CoToZnaczy}, s.~17.
}.
}
Nawet jeśli pojawiają się dyskusje wokół faktów i~teorii, dotyczą one
tylko stopnia uteoretyzowania, ale nigdy nie kwestionowania tego,
że fakty są obciążone teoretycznie.
Wielu współczesnych filozofów przychyla się do tego poglądu.
Dla niektórych jednak problem ten jest specyficznie naukowy
i~to, czy filozofia może skorzystać z~rozstrzygnięć, które przyjęto
w~nauce, zależy od tego, jak pojmujemy związki zachodzące między
nauką a~filozofią.
Na ogół jednak nie kwestionuje się, że w~nauce nie ma nagich faktów.
\oKrapiec{} stwierdza, że
\cytuj{
cała dyskusja o~niemożliwości „nagich faktów” w~nauce
dotyczy faktów danych nam w~jakimś języku, język zaś
jest już pierwszym pośrednikiem w~ich poznaniu i~stąd właśnie
wniosek, że „nagich faktów” nie ma.
Twierdzę jednak, że w~obszarze naszej wiedzy
--- zapośredniczonej poprzez kontekst znakowy ---
mamy możliwość stwierdzenia istnienia jako podstawy
rzeczywistości\footnote{
\cite{Krapiec:Zawsze}, s.~51.
}.
}
Autor ten nie przeczy, że w~nauce fakty mają rzeczywiście
uteo\-retyzowany charakter, co wymuszone jest specyfiką poznania
naukowego.
Czy jednak taka sama sytuacja musi zachodzić w~filozofii?
Dla tomistów nie jest to już takie oczywiste.
Przede wszystkim dlatego, że według nich filozofia ma swoją
odrębność poznawczą, włas\-ny przedmiot badań oraz metodę
rozstrzygania problemów.
Żadnego z~tych elementów nie musi zapożyczać z~nauki.
Poza tym nie wydaje się, aby poznanie naukowe było tym właśnie
podstawowym poznaniem, za pomocą którego kontaktujemy się
z~realną rzeczywistością.
Znacznie bardziej podstawowym i~bazowym poznaniem jest filozofia.
Ta bowiem wyrasta z~poznania potocznego, w~ramach którego
kontaktujemy się ze światem realnym.
Jeśli tak, to nie musimy przejmować z~nauki żadnych rozstrzygnięć.
Nauka, jako poznanie „niższego stopnia”\footnote{
Nie chodzi tu o~żadne deprecjonowanie nauki, a~jedynie
o~podkreślenie, że w~porównaniu do nauki filozofia dostarcza
bardziej ogólnego poznania, które spełnia nie tylko funkcje
naukowe, ale i~światopoglądowe.
Tomiści nigdy nie deprecjonowali nauki, nawet jeśli nie brali
jej wyników jako punktów wyjścia swoich analiz.
Dlatego błędne są zarzuty przypisujące im ignorowanie nauki.
Z~tego, że ktoś nie bierze czegoś pod uwagę, wcale nie wynika,
że to coś deprecjonuje.
Choć tomiści mają świadomość tego, co się dzieje w~nauce,
wiedzę tę pozostawiają specjalistom, sami zaś próbują stworzyć
pewien szeroki fundament teoretyczny, w~ramach którego mogła\-by
również odnaleźć się nauka.
Inna sprawa, że nauka współczesna rzadko kiedy chce się w~tym
właśnie proponowanym przez tomistów fundamencie odnajdywać.
}
niż filozofia, nie dostarcza w~przekonaniu tomistów tego najbardziej
bazowego i~realistycznego poznania świata.
Filozofia, tak jak nauka, też dysponuje odpowiednimi
faktami\footnote{
Na sposób kształtowania tzw. faktów naukowych zwraca uwagę
\oFleck{} (por. \cite{Fleck:Powstanie}).
}.
Aby bowiem zacząć filozofować, trzeba przyjąć jakiś punkt wyjścia,
od którego tak naprawdę wszystko zależy.
To bowiem, jaki punkt wyjścia przyjmiemy, wyznacza teorię
i~ostateczne rozstrzygnięcie stawianych problemów.
„W~filozofii bardzo wiele zależy od punktu wyjścia”\footnote{
\cite{Krapiec:Metafizyka}, s.~191.
}.
Jakie zatem fakty trzeba przyjąć w~punkcie wyjścia, aby filozofia,
zgodnie z~tym, czego chcą tomiści, mogła być realistycznym
wyjaśnieniem rzeczywistości, a~nie konstruowaniem mitycznych
ontologii, które zajmują się tylko analizą myśli?
Nie mogą być one żadnymi danymi naocznymi, które bezpośrednio
prezentowałyby się świadomości, jak chcieli tego fenomenologowie.
Mówili oni nie tyle o~faktach, ile raczej o~świadomości faktów.
Dla fenomenologów fakty byłyby uwikłane w~określoną teorię
świadomości, a~tym samym nie dotyczyłyby realnego świata.
Jeśli już fenomenologowie przyjmowali jakieś fakty, to tylko
aprioryczne, odnoszące się do idealnej dziedziny bytu.
Tymczasem w~filozofii, która zamierza być realistycznym wyjaśnieniem
świata, nie może chodzić tylko o~świadomość faktów, ale o~realne
fakty, do których mamy dostęp w~spontanicznym doświadczeniu.
Nie mogą to być też żadne fakty językowe, dotyczące np. użycia
okreś\-lonych wyrażeń.
Rola faktów jest bardzo ważna, gdyż stanowią one „materiał”
do wyjaśniania.
\cytuj{
Fakty same w~sobie to jakby punkty wyjścia naszego poznania
i~zarazem ostateczne kryteria, punkty dojścia w~ukazywaniu
realności uzasadnień poznawczych\footnote{
\cite{Krapiec:Metafizyka}, s.~195--196.
}.
}
Trzeba zatem wyjść od takich faktów, które nie są jeszcze
teoretycznie zbyt obciążone.
Chodzi tu o~takie fakty, które możemy odkryć bez pomocy nauki,
ta bowiem, jeśli można tak powiedzieć, odkrywa fakty wtórne,
które aby zachować swój realistyczny charakter, powinny bazować
na faktach podstawowych, czyli tych, które przyjmuje w~punkcie
wyjścia filozofia, a~dokładniej metafizyka.
Metafizyka odkrywa tzw. fakty bazowe, z~którymi powinno liczyć się
poznanie naukowe.
Fakty te powinny znaleźć się w~zewnętrznej bazie poznania naukowego.
Wtedy mogłyby one stanowić coś w~rodzaju kontroli zabezpieczającej
przed nadmiernym oddalaniem się nauki od poznania świata realnego.
Wszystko to trzeba jednak widzieć w~znacznie szerszym kontekście
teoretycznym.
Autorzy tomistyczni uważają bowiem, że teoria bytu odgrywa rolę
fundującą dla poznania naukowego, jak i~dla wszystkich innych
dziedzin ludzkiej aktywności.
Wielokrotnie pod adresem nauki formułują zarzut, że oddaliła się
ona od poznawania rzeczywistości (faktów), a~niebezpiecznie
zbliżyła się do mitologii.
Dla tomistów egzystencjalnych fakty filozoficzne, które przyjmujemy
w~punkcie wyjścia filozofii, muszą być:
(i)~czymś zastanym, nie mogą być konstruowane czy apriorycznie
narzucane na rzeczywistość,
(ii)~wyrażone w~jednolitym języku teoretycznym; jeśli są to fakty
należące do płaszczyzny filozofii, to powinny być wyrażone tylko
w~języku filozoficznym.
Wykluczone jest konstruowanie faktów za pomocą semantyki języka
naukowego\footnote{
Szerzej na temat faktów por. \cite{Stepien:Studia}.
}.
Gdyby tak było, wówczas mamy do czynienia z~pomieszaniem płaszczyzn
poznawczych.
A~tego tomiści chcą uniknąć i~dlatego przestrzegają przed
swobodnym przechodzeniem między różnymi płaszczyznami poznawczymi,
zwłaszcza podkreślają trudności towarzyszące wszelkim badaniom
interdyscyplinarnym.
Ponadto
(iii)~fakty są też ostateczną instancją, na którą powołujemy się
przy rozstrzyganiu teorii filozoficznych\footnote{
Por. \cite{Krapiec:Metafizyka}, s.~195--196.
}
oraz
(iv)~pojęcie faktu zakresowo utożsamia się z~pojęciem bytu.
„Wszystko, co w~świecie nas otaczającym można nazwać bytem,
można też nazwać <<faktem>>, danym nam do wyjaśnienia”\footnote{
\cite{Krapiec:Metafizyka}, s.~195.
}.
Byt --- czyli to, co jest --- to podstawowy fakt metafizyczny,
który w~sposób pierwotny i~jeszcze nieuwyraźniony konstatujemy
już w~poznaniu spontanicznym.
Jeśli w~nauce nie ma nagich faktów, bo wszystkie są mniej
lub bardziej obciążone teorią, trzeba znaleźć takie poznanie,
w~którym możemy skonstatować fakty w~minimalny sposób
uteoretyzowane, które nie są jeszcze obciążone całą historią
naszego poznania.
Trzeba znaleźć fakty, które nie pociągają za sobą żadnych mocnych
hipotez interpretacyjnych.
Jest to możliwe, twierdzą tomiści, tylko wtedy, kiedy
\cytuj{
zwrócimy uwagę na to, że istnieje zneutralizowany przedmiot
dociekań filozoficznych.
I~właśnie w~świetle owego zneutralizowanego przedmiotu,
w~świetle ujawnionych i~uzasadnionych pierwszych zasad bytu
i~myślenia możemy fakt i~obiektywnie widzieć, i~zauważony
wytłumaczyć ostatecznie, czyli metafizycznie.
A~możemy to uczynić, tym bardziej, że jesteśmy jeszcze wsparci
rozumieniem historii interpretacji danego faktu.
Historia bowiem interpretacji, jeśli jest rozumiana jako
historia problemu i~„żyjącej” myśli, pozwala wyeliminować
wiele pseudoproblemów\footnote{
\cite{Krapiec:Metafizyka}, s.~196.
}.
}
Przedmiot zneutralizowany to taki, który nie pociąga za sobą
zbytniego obciążenia teoretycznego.
Taki przedmiot gwarantuje realizm i~obiektywizm eksplanacji
faktów realnych.
Dlatego
„owe fakty staramy się ująć w~świetle właściwego przedmiotu
filozofii, czyli w~świetle bytu jako istniejącego''\footnote{
\cite{Krapiec:Metafizyka}, s.~196.
}.
Pierwszym i~podstawowym faktem filozoficznym, który już konstatujemy
w~poznaniu spontanicznym, jest „fakt istnienia świata”.
Jest to bazowy fakt egzystencjalny (bytowy).
Jego werbalizacja dokonuje się w~strukturze sądu egzystencjalnego.
W~sądzie tym stwierdzamy obiektywne fakty.
Stwierdzamy, że jakiś przedmiot po prostu jest.
Afirmacja faktu istnienia przedmiotu ustawia całe poznanie na torach
realizmu i~obiektywizmu, gwarantuje także intersubiektywność
poznania.
To właśnie poznanie potoczne, spontaniczne jest tym najbardziej
podstawowym, bazowym typem poznania, w~ramach którego konstatujemy
fakty, które później w~poznaniu metafizycznym zostają tylko
uwyraźnione i~opracowane.
Metafizyk, wydając sądy egzystencjalne, uwypukla tylko to, czego
każdy doświadcza potocznie, choć nie zawsze zwraca na to swoją
uwagę poznawczą.
\cytuj{
Fakty bowiem dostrzegamy przede wszystkim w~poznaniu
spontanicznym, a~więc gdy nie ma jeszcze refleksji „destylującej”;
skąd fakty, które się pojawiają, już są uwikłane w~historię
naszego poznania\footnote{
\cite{Krapiec:Metafizyka}, s.~196.
}.
}
Tomistom zależy, aby maksymalnie ograniczyć uteoretyzowanie faktów.
Poznanie potoczne dostarcza metafizyce „gotowych” faktów,
które teoretyk bytu przyjmuje jako wiążące dla procedury
ich eksplanacji.
Czy w~poznaniu potocznym są to „surowe fakty”?
Nie wydaje się.
Poznanie to jest bowiem również teoretycznie obciążone
przez pochodzące z~różnych dziedzin wiedzy przekonania.
Tomiści zdają sobie sprawę z~tego, że poznanie dorosłego człowieka
ma własną historię, uwarunkowane jest kulturą, nawykami
oraz całą masą uwarunkowań o~charakterze biologicznym
i~psychologicznym.
Jednakże w~tym wszystkim można wskazać na tak pierwotne fakty,
które właściwie nie mogą nasuwać żadnej interpretacji, po za tym,
że zostaną one stwierdzone bezpośrednio w~odpowiednich aktach
poznawczych.
Potocznie ludzie nie zwracają uwagi na „fakt istnienia świata”;
traktują go jako oczywisty, dlatego nie dokonują żadnej nad nim
refleksji.
Ta oczywistość uchodzi ich uwadze.
Fakt istnienia świata jest przez każdego normalnego człowieka
presuponowany w~jego działalności poznawczej i~praktycznej.
Bez tego faktu wszelkie racjonalne działanie nie mogłoby się
urzeczywistniać.
Oprócz stwierdzenia istnienia świata, do innych faktów
ważnych z~punktu widzenia tomistycznej filozofii bytu należą:
fakt stwierdzenia pluralizmu bytowego,
fakt zachodzenia określonych przemian o~charakterze substancjalnym
i~przypadłościowym,
fakt podzielności bytów na określone części
oraz szereg różnych innych faktów ontycznych,
które stanowią dane do wyjaśnienia rzeczywistości\footnote{
Por. \cite{Krapiec:Metafizyka}, s.~197.
}.
Fakty te bowiem prowokują do zadania pytania: dlaczego?
Tym samym domagają się one wskazania na ostateczne racje,
które je uniesprzeczniają.
Chodzi tu przede wszystkim o~racje realne, a~nie tylko myślne
czy możliwe, jak ma to miejsce w~innych ontologiach.
„Ostateczna racja bytu ma definitywnie uniesprzecznić istnienie
zauważonego faktu, czyli oddzielić byt od nie bytu”\footnote{
Por. \cite{Krapiec:Metafizyka}, s.~197.
}.
I~na tym zasadniczo ma polegać cel poznania metafizycznego ---
na ostatecznej eksplanacji faktów dostrzeżonych w~bezpośrednim
doświadczeniu.
Trzeba podkreś\-lić, że ów pierwotny fakt „istnienia świata”
można tylko stwierdzić za pomocą sądu egzystencjalnego.
Nie można tego faktu spojęciować, gdyż istnienie, jako prosty
akt bytu, podczas kontaktu poznawczego nie wywołuje w~poznającym
żadnej reprezentacji poznawczej.
Faktu istnienia nie można skonceptualizować;
można jedynie stwierdzić, że przedmiot jest.
Ważne jest to, że fakty te poznajemy i~stwierdzamy w~sposób
bezpośredni, bez udziału jakichkolwiek pośredników poznawczych
(np. pośrednictwa systemowego, dowodzenia, emocji) poza pojęciami,
które są pośrednikami przeźroczystymi i~nie zatrzymują na sobie
naszej uwagi poznawczej.
\oKrapiec{} uważa, że fakt po prostu widzimy.
Nawet \ios{Stępień}{Stępień, A.B.}, który korzystał z~koncepcji
fenomenologów i~analityków, stwierdza:
„Byt to coś, co widzimy, co zastajemy w~jego własnościach
i~jego pozycji bytowej, polegającej na transcendencji
radykalnej”\footnote{
\cite{Stepien:DwaWyklady}, s.~103.
}.
Czyż trzeba komuś specjalnie tłumaczyć, że istnieje świat?
Można by powiedzieć:
wstań, otwórz oczy i~zobacz, masz go pod ręką...
Chodzi tylko o~uświadomienie sobie czegoś, o~czym bardzo dobrze
wiemy, co traktujemy jako fundament i~bazę wszelkiej naszej
aktywności.
Nie trzeba przecież nikomu dowodzić faktu, że „świat istnieje”.
Gdyby trzeba było to robić, mielibyśmy do czynienia z~kuriozalną
sytuacją ocierającą się o~zaburzenie psychiczne.
Przecież żaden normalny człowiek nie dowodzi faktów najbardziej
podstawowych.
Faktów tych doświadczamy w~sposób bezpośredni i~właśnie to
doświadczenie bezpośrednie, ten styk z~istnieniem powinien być
dla nas ostateczną rękojmią akceptacji tych faktów.
Są one po prostu niedowodliwe.
To --- według tomistów --- jeszcze bardziej uwypukla brak
teoretycznego obciążenia tego faktu.
Jeśli bowiem nie można go ująć w~pojęcia, to wydaje się,
że nie pociąga on za sobą żadnej teorii i~hipotezy, które w~jakiś
sposób ustawiałyby nasze interpretacje w~określony sposób,
niekoniecznie realistyczny.
Fakt owego „jest” świata, choć nie podlega konceptualizacji,
posiada obok istnienia element składowy, czyli istotę, a~ta
już może być skonceptualizowana.
Dzięki temu, że byt posiada istotę, możliwe jest poznanie
o~charakterze intersubiektywnym.
Fakt istnienia można stwierdzić, pokazać, ale nie można go wyrazić
w~pojęciu, natomiast istotę (treść bytu) można poznawać stopniowo
za pomocą różnych reprezentacji pojęciowych, które mają charakter
aspektowy.
Nie można o~czymś takim powiedzieć „fakt istoty”.
Byt to fakt, na który składa się istnienie i~istota.
Nie można z~faktu istnienia wyodrębnić niezależnych faktów istoty
i~oddzielnie ich badać.
Gdyby dokonać takiego usamodzielnienia, wówczas mielibyśmy
do czynienia z~esencjalną ontologią, która zajmuje się faktami
istotnościowymi, czyli faktami możliwymi.
Takie fakty, po pierwsze, nie istnieją realnie,
a~po drugie, od takich faktów --- nawet jeśli przyjąć ich
„istnienie” --- nie można przejść do faktów realnych, które mają
stanowić punkt wyjścia dla eksplanacji.
Fakt istoty wyodrębnia się podczas analizy bytu, wskazując tylko,
że między istotą a~istnieniem zachodzi realna różnica ontyczna.
\tytul{2. Obraz świata, struktura, przyswajanie}
Termin „obraz świata” nie należy do jednoznacznych\footnote{
Szerzej poglądy \oWittgenstein[a] i~\ios{Abla}{Abel, G.}
odnośnie do roli obrazu świata w~ludzkim doświadczeniu por.
\cite{Duchlinski:Odslony}, Rozdział 6, s.~211--237.
W~tym paragrafie wykorzystuję niektóre zmodyfikowane fragmenty
tekstu z~tego rozdziału.
}.
Aby jednak nie wchodzić w~zbyt zawiłe analizy semiotyczne,
można na potrzeby dalszych analiz przyjąć za \ios{Ablem}{Abel, G.},
że obraz świata to pewne odziedziczone tło, fundament naszej wiedzy,
który nabywamy razem z~przyswajaniem języka\footnote{
Osobnej analizie należałoby poddać tzw. wiedzę tła, która jest
częścią obrazu świata.
Trzeba by tutaj ustalić zakresy tych pojęć.
Można tylko wskazać, że obraz świata zakresowo jest nazwą
szerszą niż wiedza tła, która obejmuje w~dużej mierze elementy
niepropozycjonalne i~obrazowe.
Tymczasem, jak twierdzi \oAbel, nie są to jedyne składniki
obrazu świata.
}.
Ten obraz świata jest podstawą każdej kultury, naszego poznania
oraz działalności praktycznej.
Zawiera szereg elementów normatywnych regulujących naszą codzienną
praktykę.
Obraz świata jest niepodważalny w~takim znaczeniu,
że jako wiedza „tła” stanowi rzeczywistą podstawę dla przeprowadzania
wszelkiego rodzaju rozumowań i~ustalania tego, co jest prawdziwe,
a~co fałszywe\footnote{
Por. \cite{Abel:Swiat}, s.~78--79.
}.
Obraz ten jest tak głęboko w~nas osadzony, że tak naprawdę
spełniając konkretne czynności kognitywne i~praktyczne,
nie zdajemy sobie sprawy z~jego podskórnego oddziaływania.
Nie musimy za każdym razem sprawdzać wchodzących w~jego skład
twierdzeń, gdyż to prowadziłoby do kuriozalnych sytuacji
komunikacyjnych.
Nie ma jednego obrazu świata.
Trzeba mieć na uwadze wielość historycznie i~kulturowo zmiennych
obrazów świata.
Tak przecież możemy wyróżnić klasyczny obraz świata, który bardzo
często identyfikuje się z~\oArystoteles[em];
można mówić o~heliocentrycznym lub \ios{newtonowskim}{Newton, I.}
obrazie świata czy też obrazie świata, jaki wyłania się
ze współczesnej fizyki kwantowej.
Obecnie, przy całej niejednoznaczności, mówi się bardzo często
o~tzw. naukowym obrazie świata, na bazie którego wielu filozofów
chce uprawiać swoją tzw. naukową filozofię (np. \oHeller{} i~jego
uczniowie).
\oAbel{} zwraca uwagę, że ten obraz świata dziedziczymy jako
pewną ramę interpretacyjną.
Rodzimy się i~wzrastamy zawsze w~jakimś obrazie świata,
na który składa się szereg przekonań (mniej lub bardziej sztywnych,
niepodatnych na zmianę, jak i~tych stale fluktuujących),
artykułowanych językowo, wyobrażeń, obrazów mających swoją nośność
zwłaszcza dla zrozumienia człowieka i~świata, w~którym żyje.
Choć każdy obraz świata jest wewnętrznie ustrukturowany,
to jednak nie posiada doskonałej spójności i~jednolitości\footnote{
Por. \cite{Abel:Swiat}, s.~92--93.
}.
\oAbel{} twierdzi, że im obraz ten jest mniej spójny i~jednolity,
tym bardziej jest funkcjonalny i~efektywny.
W~przekonaniu zwolennika interpretacjonistycznej filozofii nauki
\cytuj{
wymaganie nadmiernej jednorodności i~formalnej spójności
prowadziłoby tu do dysfunkcji.
Wtedy zaś systemy utraciłyby charakter płynnego funkcjonowania
i~popadłyby w~stagnację, wskutek czego nie byłyby w~stanie nadal
zadowalająco wykorzystywać swych zorientowanych na świat
codziennego życia funkcji\footnote{
\cite{Abel:Swiat}, s.~92.
}.
}
Brak ścisłej spójności umożliwia poprawną komunikację
między członkami tej samej formy życia.
\oWittgenstein, do którego \oAbel{} się odwołuje, twierdzi w~swoim
traktacie \textit{O~pewności}, że obraz świata przyswajamy razem
z~nauką języka\footnote{
Por. \cite{Wittgenstein:OPewnosci};
\cite{Soin:WKwestii}.
}.
Obraz świata, który ujawnia się razem z~wchodzeniem w~okreś\-loną
grę językową, stanowi odziedziczone tło, na podstawie którego
możliwe jest dokonywanie rozróżnienia między prawdą a~fałszem.
\cytuj{
Ale nie zyskałem swego obrazu świata, gdyż przekonałem się
o~jego poprawności, ani nie dlatego, że jestem przekonany
o~jego poprawności.
Lecz jest to oddziedziczone tło, na którym rozróżniam prawdę
od fałszu\footnote{
\cite{Wittgenstein:OPewnosci}, s.~35.
}.
}
Owo tło nie do końca ma charakter wiedzy wypowiedzianej.
Znaczy to, że w~konkretnej praktyce funkcjonuje w~sposób
nietematyczny, jako wiedza tła.
Jest ona podstawą doświadczania świata i~przeprowadzania
jakichkolwiek eksperymentów naukowych.
Oto następujący przykład:
\cytuj{
Pomyślmy o~badaniach chemicznych.
Lavoisier\index{Lavoisier, A.L.} w~swoim laboratorium
przeprowadza eksperymenty z~substancjami i~konkluduje,
że przy spalaniu zachodzi to a~to.
Nie twierdzi, że innym razem mogło zdarzyć się coś innego.
Trzyma się określonego obrazu świata, oczywiście nie wynalazł go,
lecz nauczył się go jako dziecko.
Mówię obraz świata a~nie hipoteza, jest to bowiem oczywista
podstawa jego badań, i~jako taki nie jest też wysłowiony\footnote{
\cite{Wittgenstein:OPewnosci}, s.~46.
}.
}
Skąd wiemy, pyta \oWittgenstein, że przed moim narodzeniem
istniała Ziemia, skąd wiemy, że posiadamy rodziców lub też,
że do tej pory nikogo nie było na Księżycu.
Tych zdań nie możemy sprawdzić empirycznie.
Nie możemy w~ich potwierdzeniu odwołać się do doświadczenia,
a~pomimo to wierzymy, że one są prawdziwe, że stanowią
niepodważalny korpus naszej wiedzy.
Jest tak dlatego, gdyż
„od dziecka uczyłem się tak sądzić.
To jest sądzenie”\footnote{
\cite{Wittgenstein:OPewnosci}, s.~49.
}.
To, że wiem, iż Ziemia istniała na długo przed moim narodzeniem,
iż nikt z~ludzi nie był do tej pory na Księżycu, to wszystko
stanowi wiedzę tła, którą przyswajam, wchodząc w~określoną grę
językową.
Tych zdań jako pewnych uczę się już jako dziecko.
\cytuj{
Jako dzieci uczymy się faktów, że np. każdy człowiek ma mózg,
i~przyjmujemy na wiarę.
Wierzę, że istnieje wyspa Australia, o~takim to a~takim kształcie
i~tak dalej, wierzę, że miałem pradziadków, że ludzie, którzy
podają się za moich rodziców, są naprawdę moimi rodzicami itd.
Te przekonania nigdy nie zostały wypowiedzialne, a~nawet myśl,
że tak jest, nigdy nie została pomyślana\footnote{
\cite{Wittgenstein:OPewnosci}, s.~45.
}.
}
Jako dzieci przyswajamy tę wiedzę od dorosłych, których traktujemy
jako nauczycieli i~autorytety.
Dlatego nigdy nie sprawdzamy empirycznie, czy mamy rodziców,
czy też nie, uznajemy to za pewnik, który nam przekazano,
a~o~którym wiemy na podstawie innej wiedzy, np. od innych dorosłych.
Doświadczenie nie stanowi bezpośredniej racji przyjęcia takich
lub innych faktów.
\oWittgenstein{} pyta, co to znaczy, że doświadczenie czegoś uczy,
jak należy rozumieć to, że wiedzę czerpiemy z~doświadczenia.
Odpowiadając, stwierdza, że to nie doświadczenie nas uczy, ale
\cytuj{
my możemy wyprowadzać coś z~doświadczenia, doświadczenie
nie kieruje nas, by coś z~niego wyprowadzić.
Jeśli to jest racją, by tak sądzić (a~nie tylko przyczyną),
to nadal nie dysponujemy racją, aby uznać to za rację\footnote{
\cite{Wittgenstein:OPewnosci}, s.~40.
}.
}
Wiedza stanowiąca zawartość tła nie jest zatem wiedzą opartą
na bezpośrednim doświadczeniu.
Jest to wiedza, na którą składają się także przeszłe doświadczenia
innych ludzi, uznawanych bardzo często za autorytety w~różnych
dziedzinach wiedzy empirycznej i~pozaempirycznej.
\cytuj{
Jeśli doświadczenie stanowi podstawę naszej pewności,
to chodzi tu oczywiście o~doświadczenie przeszłe.
I~nie jest to tylko moje doświadczenie, lecz i~innych,
od których przyjmuję wiedzę\footnote{
\cite{Wittgenstein:OPewnosci}, s.~61.
}.
}
To, w~co wierzymy i~co wiemy, nie zależy tylko od doświadczenia,
ale od „tego, czego się uczymy”\footnote{
\cite{Wittgenstein:OPewnosci}, s.~62.
}.
Uczymy się także, w~jaki sposób mamy wyprowadzać konkluzję
z~takich lub innych doświadczeń.
\cytuj{
Uczymy się bowiem nie tylko tego, że takie a~takie doświadczenia
dają takie a~takie wyniki, lecz również wyprowadzanych
z~nich konkluzji.
I~nie ma w~tym oczywiście nic złego.
Bo to [wyprowadzone] zdanie jest instrumentem określonego
użytku\footnote{
\cite{Wittgenstein:OPewnosci}, s.~64.
}.
}
Słusznie też \oWittgenstein{} zauważa, że przyswajanie gry językowej
nie zaczyna się od wątpliwości.
Stan wątpienia nie jest dla filozofa naturalnym stanem poznawczym
człowieka.
On może pojawić się znacznie później, kiedy dysponujemy pewną siecią
trwałych przekonań.
Najpierw trzeba przyjąć jakieś fakty, aby potem móc w~nie wątpić.
\cytuj{
Bo czyż dziecko mogłoby od razu wątpić w~to, co mu się wpaja?
Znaczyć by to mogło jedynie, że nie mogłoby ono nauczyć się
pewnych gier językowych\footnote{
\cite{Wittgenstein:OPewnosci}, s.~62.
}.
}
Zanim przyjdą wątpliwości, najpierw trzeba uwierzyć nauczycielom
oraz podręcznikom, trzeba zdać się na autorytety, które przekazują
nam pewien obraz świata.
„Dziecko uczy się dzięki temu, że wierzy dorosłym.
Wątpienie pojawia się po uwierzeniu”\footnote{
\cite{Wittgenstein:OPewnosci}, s.~45.
}.
Dlatego rozsądny człowiek, jak go nazywa \oWittgenstein,
nie ma wątpliwości co do tego, że Ziemia jest okrągła,
czy że istniała na długo przed jego urodzeniem.
To są podwaliny wszystkich przekonań, których nauczyliśmy się,
przejmując język i~skorelowany z~nim obraz świata.
To są pewniki, bez których niemożliwe byłoby realizowanie
tej formy życia, w~której obecnie jesteśmy.
Wątpienie jest możliwe dopiero wtedy, kiedy przyjmujemy,
że istnieje coś pewnego.
U~podłoża wątpienia znajduje się zawsze coś ugruntowanego,
co stanowi wiedzę tła, na bazie której rozgrywa się konkretny
akt wątpienia.
Nigdy zatem nie można zwątpić globalnie we wszystko.
W~kontekście problematyki wątpienia \oAbel{} przywołuje
interesujący przykład „niewiernego Tomasza”, którego niepokoi
pytanie o~prawdę akceptowanego obrazu świata.
Potraktowane na poważnie zakłada ono, że mamy możliwość
wydostania się poza obraz świata, w~którym funkcjonujemy.
Zwolennik interpretacjonistycznej filozofii nauki przeczy jednak
takiej możliwości.
Argumentuje w~ten sposób, że
\cytuj{
nikt jednak nie potrafi ulokować się poza własnym obrazem świata.
Jeśli istotnie komuś by się to udało, to my, istoty skończone,
w~określonej sytuacji poznawczej nie mielibyśmy żadnego powodu,
by temu komuś uwierzyć.
Nie moglibyśmy z~uzasadnioną pewnością wiedzieć, czy rzeczywiście
zdołał on przyjąć perspektywę spoza obrazu świata\footnote{
\cite{Abel:Swiat}, s.~96.
}.
}
Wyjście poza obraz świata zakładałoby, zdaniem \ios{Abla}{Abel, G.},
możliwość przyjęcia „boskiego punktu widzenia”.
Dla człowieka, uwikłanego w~różne poziomy interpretacyjnych
odniesień do świata, przyjęcie bezstronnego punktu widzenia
jest niemożliwe do osiągnięcia.
Dlatego nie jesteśmy w~stanie wykazać prawdziwości obrazu świata
jako tła naszego poznania i~działania poprzez porównanie go
z~jakimś światem samym w~sobie, który byłby od niego
niezależny\footnote{
Możliwość „boskiego punktu widzenia” odrzuca \oPutnam{} --- por.
\cite{Putnam:Pragmatyzm};
\cite{Putnam:Reason}.
}.
Obraz świata zakłada wspólną bazę przekonań, która jest podzielana
przez członków określonej wspólnoty kulturowej.
\oAbel{} --- za \oWittgenstein[em] --- woli nazwać ją formą życia.
Ona to właśnie umożliwia intersubiektywność komunikacji
i~argumentowania.
\oAbel{} przekonuje, podobnie jak autor traktatu \textit{O~pewności},
że obraz świata stanowi tło, na bazie którego ustalamy, co jest
prawdą, a~co fałszem.
Twierdzi, że
\cytuj{
obraz świata tworzy tło, na którym rozstrzyga się,
co jest prawdą, a~co fałszem, to w~konsekwencji cała
opozycja prawda --- fałsz i~predykat „jest prawdziwy”
zakładają już strukturę obrazu świata i~mają ją
za swoją podstawę.
Skuteczne zastosowanie predykatów „jest prawdziwy”
i~„jest fałszywy” nie opiera się koniec końców na
trywialności takiej aplikacji, ale na tym, że dany jest
pewien obraz świata, obraz, na tle którego rozróżnienie
prawdy i~fałszu jest możliwe i~sensowne.
W~tym znaczeniu opozycja prawda i~fałsz jest powiązana
z~obrazem świata i~się do niego odnosi\footnote{
\cite{Abel:Swiat}, s.~84.
}.
}
Prawda i~fałsz są zatem pochodne od tego, co uznajemy za
prawdziwe$/$fałszywe na mocy akceptacji obrazu świata.
To on w~ostateczności rozstrzyga o~tym, jaką metafizyczną
wykładnię świata przyjmujemy.
Obraz mówi, co istnieje, a~co nie istnieje, decyduje o~akceptacji
określonych kryteriów istnienia.
Wszelka klasyfikacja bytów jest na nim oparta.
Obrazy świata, które przyswajamy dzięki językowi w~procesie edukacji,
decydują o~tym, jakie fakty uznajemy za pewne i~oczywiste.
To, że istnieje Ziemia, że posiadamy rodziców, że mamy takie a~nie
inne części naszego ciała, to wszystko zawdzięczamy procesowi
edukacji, w~trakcie którego nabywamy wiedzy i~podstawowych
kompetencji posługiwania się grą językową.
Fakty, które akceptujemy razem z~obrazem świata, są pewne
i~nie dopuszczają zwątpienia.
Są też uteoretyzowane, bo razem z~nimi zawsze idzie pewna teoria,
która je tłumaczy, a~którą nie zawsze znamy, ani której nie musimy znać,
aby racjonalnie działać.
Wiemy, że Ziemia obraca się wokół Słońca, czy też, że jest ona
okrągła; za tymi faktami stoi określona teoria fizyczna, która
je wyjaśnia.
Dla praktyki naszego działania ważne jest tylko, abyśmy akceptowali
to zdanie, natomiast nie musimy już szerzej znać kontekstu
jego uzasadnienia.
\oAbel{} dokładnie wyjaśnia, że
\cytuj{
obrazowe założenie, że Ziemia jest okrągła, należy przeto
do warunków tego, że zdania w~danym języku, na przykład
wypowiadane podczas przeprowadzania rezerwacji w~biurze podróży,
są semantycznie treściwe i~że można np. rezerwować przelot
z~Berlina do Japonii, co znaczy, że przenikają sam akt
rezerwowania, w~przypadku podróży \textit{last minute}
wkraczając także w~bezzwłoczną jazdę na lotnisko, a~podróż
objąć może „pół ziemskiego globu”\footnote{
\cite{Abel:Swiat}, s.~93.
}.
}
To porozumienie jest możliwe tylko dlatego, że istniejemy
w~przestrzeni intersubiektywnej, w~której inni ludzie podzielają
ten sam obraz świata, co my.
Jesteśmy o~tym przekonani, dlatego nie dowodzimy, że Ziemia
jest okrągła, tylko uznajemy to za grunt wszelkiego pragmatycznego
porozumienia z~innymi.
Obraz warunkuje intersubiektywną przestrzeń porozumienia.
A~dokładnie rzecz ujmując, warunkuje ją język, poprzez który
mamy dostęp do tego obrazu i~zawartych w~nim treści.
Obrazy świata, jako wiedza tła, nie są w~całości wyrażalne
za pomocą wiedzy propozycjonalnej.
Można wyrazić tylko część najważniejszych przekonań;
reszta pozostaje ukryta i~oddziałuje podskórnie.
Niemożliwość językowej eksplikacji treściowej zawartości
obrazu świata wynika z~tego, że obrazy nie składają się tylko
z~przekonań o~charakterze propozycjonalnym, ale też obrazowym
(wizualnym), wyobrażeniowym, czyli przedjęzykowym\footnote{
Można by powiedzieć, że jest w~nich miejsce na \oJung[owskie]
archetypy.
}.
\oWittgenstein{} i~\oAbel{} pokazują, w~jaki sposób przyswajamy
tzw. potoczny obraz świata, na bazie którego kształtuje się
nasze poznanie i~działanie.
Obraz ten, jak wynika z~analiz, jest treściowo niejednorodny,
pełno w~nim przekonań pochodzących z~różnych źródeł.
Do owych źródeł można zaliczyć naukę, bo przecież wiedzę
o~podstawowych faktach naukowych dotyczących świata zdobywamy
właśnie poprzez edukację.
Kolejnym źródłem jest kultura, od której przejmujemy szereg
faktów dotyczących życia społecznego, działania innych ludzi;
fakty te w~mocniejszym znaczeniu niż np. fakty naukowe wchodzą
w~zakres naszego samozrozumienia.
W~żadnym jednak wypadku nie oznacza to deprecjonowania faktów
naukowych.
Również i~one odgrywają swoją ważną rolę w~procesie samozrozumienia.
Kolejnym źródłem przekonań propozycjonalnych i~obrazowych
jest religia.
Dostarcza ona całej masy faktów dotyczących świata, człowieka,
a~także rzeczywistości, która przekracza doświadczenie empiryczne
i~stosowne kryteria sprawdzania uzyskanej na jego podstawie wiedzy.
\tytul{3. Świat --- obraz świata a~fakty filozoficzne}
Co determinuje filozofa do tego, aby przyjmował takie, a~nie inne
fakty filozoficzne?
Chodzi tu o~pewien kontekst odkrycia, w~którym dokonuje się
akceptacja określonych faktów, które mogą być wykorzystywane
w~mniej lub bardziej maksymalistycznych celach poznawczych.
Czy o~akceptacji tych faktów, jak chcą tego autorzy tomistyczni,
decyduje świat realnie istniejący, czy też może rozstrzyga o~tym
obraz świata, który przyswajamy w~drodze naszej ontogenezy
poznawczej, tak mocno związanej z~nabywaniem języka?
Czy alternatywa: świat albo obraz świata jest w~ogóle zasadna?
Może powinniśmy bardziej skłaniać się do koniunkcji i~powiedzieć:
świat i~obraz świata?
Pojawia się tutaj szereg ważnych zagadnień, których nie będziemy
w~stanie ostatecznie rozstrzygnąć, poza udzieleniem szkicowych
odpowiedzi np. na pytanie, jaki jest stosunek między światem realnym
a~obrazem świata.
Wydaje się, że dla naszej dyskusji jest to sprawa kluczowa.
Dotyka bowiem istotnej kwestii, jaką jest realizm naszych
ujęć poznawczych.
Spróbujmy wyważyć argumenty za którymś z~przytoczonych powyżej
stanowisk.
Pierwsza problematyczna kwestia dotyczy tego, czy przyjmować
coś takiego, jak fakty filozoficzne.
Nawet jeśli zgodzimy się, że są fakty filozoficzne, problemem
pozostaje, jaki mają charakter i~ich akceptacja.
Zwolennicy fundacjonalizmu w~teorii wiedzy
--- a~do takich zaliczają się i~tomiści, i~fenomenologowie ---
uważają, że trzeba przyjąć określone fakty bazowe, jako uzasadnienie
dla całości wiedzy.
Przyjmują oni model wiedzy w~formie odwróconego stożka,
u~podstawy którego znajdują się najbardziej trwałe i~niepodatne
na refutację przekonania.
Mniej skłonni do takiego postępowania będą zwolennicy
koherencjonizmu, pragmatyzmu czy holizmu w~teorii wiedzy\footnote{
Por. \cite{Putnam:Pragmatyzm}, s.~82--114.
}.
Nawet jeśli akceptują tego typu fakty, to nie uważają,
aby w~punkcie wyjścia dyskursu filozoficznego
można było wyróżnić jakieś uprzywilejowane fakty.
Nie podzielają optymistycznego przekonania fundacjonalistów,
jakoby fakty te miały uzasadnić całość naszej wiedzy.
Przy holistycznym rozumieniu wiedzy jawi się ona jako
zbiór przekonań o~różnej treści i~różnych sposobach akceptacji.
Fakty, wartości i~teorie stanowią tu pewną całość,
której poszczególne elementy mogą być wyróżnione mocą decyzji,
ale nigdy nie oddzielone od całościowej formy życia\footnote{
„Sposobu życia nie można jednak podzielić na zbiór przekonań
dotyczących faktów oraz na wiązkę wartości”
(\cite{Putnam:Pragmatyzm}, s.~93).
}.
Według zwolenników fundacjonalizmu fakty przyjmuje się na podstawie
danych bezpośredniego doświadczenia (przeważnie percepcji zewnętrznej
lub introspekcji), natomiast zwolennicy holizmu skłonni będą zwracać
uwagę na element konwencji i~decyzji metodologicznej przy wyborze
takich, a~nie innych faktów.
Spór między fundacjonalizmem a~innymi koncepcjami relacji wiedzy
do faktów dotyczy właśnie sposobu ich akceptacji.
Czy podstawą dla tej akceptacji jest bezpośrednie doświadczenie,
czy może konwencja lub też obraz świata przyswojony wraz z~nauką języka jako pewne tło?
Autorzy tomistyczni argumentują, że fakty poznajemy w~sposób
bezpośredni, po prostu je widzimy, jak zapewniał nas o~tym cytowany
wcześniej autor.
Przeciw takiemu podejściu formułuje się różne kontrargumenty.
Na przykład analitycy argumentują, że nasze
\cytuj{
poczucie bezpośredniości, naturalności i~pierwotności pewnej
sytuacji poznawczej związane jest z~tym, że nie zdajemy sobie
sprawy z~szerszych, bardziej podstawowych założeń, jakie
warunkują owo poczucie bezpośredniości.
Gdy metafizyk uzasadnia swoje twierdzenia przez odwołanie się
do bezpośredniej intuicji czegoś, to w~istocie manifestuje
po prostu swój konserwatyzm, polegający na akceptacji obrazu
świata, w~którym pewne rzeczy jawią się jako bezpośrednio dane,
a~stąd oczywiste i~nie wymagające dodatkowych uzasadnień\footnote{
\cite{Gutowski:Metafizyka}, s.~125.
}.
}
Autor zwraca uwagę, że to właśnie akceptacja określonego obrazu
świata decyduje o~tym, co uznajemy za istniejące i~w~jaki sposób
to możemy poznać.
Co to jednak znaczy „manifestacja konserwatyzmu”?
Chodzi o~to, że przyjmuje się taki obraz świata, w~którym
ani nic dodać, ani nic ująć nie można.
Jest to obraz gotowy, wykończony.
Również \oPopper, filozof nauki, skłania się ku poglądowi,
że w~poznawaniu świata
\cytuj{
nie ma nic bezpośredniego czy prostego w~naszym doświadczeniu;
musimy się uczyć, że posiadamy jaźń trwającą w~czasie,
istniejącą nawet we śnie i~podczas całkowitej utraty przytomności,
oraz że musimy się uczyć własnych ciał i~ciał innych ludzi.
Wszystko to polega na dekodowaniu i~interpretacji.
Uczymy się dekodować tak doskonale, że wszystko nam się wydaje
bardzo „bezpośrednie” i~„proste”, tak samo jak wydaje się komuś,
kto opanował alfabet Morse’a\index{Morse, S.F.B.}, czy ---
biorąc nieco bliższy przykład --- komuś, kto nauczył się czytać;
książka mówi doń „bezpośrednio” i~„po prostu”.
Niemniej jednak wiemy, że w~procesach tych zachodzi skomplikowany
proces dekodowania; pozorna bezpośredniość i~prostota jest
rezultatem ćwiczeń, tak samo jak gra na pianinie czy prowadzenie
samochodu\footnote{
\cite{Popper:Droga}, s.~65.
Podobnie twierdzi \oFleck, który wskazuje na wyuczony charakter
obserwacji, zwłaszcza naukowej, ale też pokazuje, że nasze
bezpośrednie doświadczenie, oparte na widzeniu, zawsze dokonuje
się z~perspektywy pewnego stylu myślowego (por.
\cite{Fleck:OObserwacji}, s.~230--231).
}.
}
\oPopper{} zgodnie ze swoją epistemologią ewolucyjną sugeruje,
że poznanie bezpośrednie jest konsekwencją nauki i~treningu\footnote{
Por. \cite{Popper:Droga}, s.~65-66.
}.
Podobnie też uważał \oIngarden\footnote{
Por. \cite{Ingarden:Dazenia}.
}
oraz \oFleck\footnote{
Por. \cite{Fleck:OKryzysie}.
}.
Twierdzą zgodnie, choć dzieli ich spora różnica w~przedzałożeniach,
że trzeba wyrobić sobie technikę bezpośredniego dostrzegania rzeczy.
Możemy do tego stopnia się wytrenować, że przestajemy zwracać uwagę
na sam początek nauki; to, co z~początku było dla nas trudno
poznawalne, z~biegiem czasu staje się widoczne jak na dłoni.
Dla tomistów obie propozycje są wątpliwe w~założeniach, a~tym samym
trudne do zaakceptowania.
Pierwsza, gdyż ustanawia obraz świata jako pewne \textit{a~priori}
ludzkiego poznania w~stosunku do rzeczywistości,
druga dlatego, że źle odczuwa naturę ludzkiego poznania
i~ustanawia \textit{a~priori} teoriopoznawcze wobec bytu.
Poza tym opiera się na kontrowersyjnych danych z~zakresu
teorii ewolucji.
Tomiści także mówią na temat obrazów świata, ale przeważnie
w~kontekście historii, co ma oczywiście niebagatelne znaczenie
dla akceptacji ich własnej teorii.
Zwracali uwagę, że tzw. wizja filozoficzna jest zawsze zakładana
w~każdym typie argumentacji\footnote{
Por. \cite{Hajduk:Rozumowanie}.
}.
Np. \oKlosak, który proponował koncepcję implikacji redukcyjnych
typu ontologicznego, twierdził, że ich przeprowadzenie zawsze
uzależnione jest --- jak pisał --- od „podstawowej wizji
filozoficznej''\footnote{
Por. \cite{Klosak:ZTeorii}, s.~160.
Szerzej o~tych zagadnieniach mowa w:
\cite{Lemanska:ZagadnienieFaktow};
\cite{Lemanska:ZagadnieniePrzejscia}.
}.
Inni zwolennicy filozofii klasycznej podkreślają, że dla Greków,
np. \oArystoteles[a], obraz świata był tak skonstruowany,
że nie można było dostrzec istnienia, jako aktu konstytuującego
realność rzeczy\footnote{
Por. \cite{Krapiec:ORozumieniu}, s.~34.
}.
\ios{Tomasz z~Akwinu}{Tomasz z Akwinu, św.} dokonał tu prawdziwej
rewolucji, która polegała na zastąpieniu
\ios{Arystotelesowskiego}{Arystoteles} obrazu świata, nowym obrazem,
w~którym na czoło wysuwa się koncepcja bytu, ukonstytuowanego
przez istnienie.
Taka koncepcja rzeczywistości była punktem zwrotnym w~dziejach
filozofii, doprowadziła bowiem --- jak uważają autorzy tomistyczni
--- do nowego rozumienia przedmiotu filozofii\footnote{
Por. \cite{Krapiec:Filozofia}, s.~87--99.
}.
Dlatego proponowana przez nich koncepcja wyjaśnienia oparta
na metodzie intuicyjno-redukcyjnej możliwa jest do przeprowadzenia,
tylko jeśli przyjmiemy powyższy obraz świata.
Obraz świata decyduje o~tym, że np. metoda dedukcyjna oparta
na związkach formalnych nie nadaje się dla teorii bytu,
która bierze tylko pod uwagę związki treściowo-egzystencjalne
zachodzące między skutkiem a~przyczyną.
Tomiści uważają, że to właśnie \ios{Tomaszowy}{Tomasz z Akwinu, św.}
obraz świata jest najbardziej poprawny i~adekwatny.
Adekwatny do czego?
Mówi on bowiem o~tym, jaka rzeczywistość jest naprawdę.
Przyjmując ten obraz świata, tomiści mogą twierdzić, że tym,
co jest bezpośrednio poznawalne, jest istnienie, które decyduje
o~tym, że byt jest realny.
Zwolennicy tej opcji filozoficznej twierdzą, że ten świat w~swoich
zasadniczych podstawach jest prawdziwy i~nie zmienił się do tej pory.
Choć wielu teoretyków nauki zwraca uwagę, że został on
zakwestionowany na progu nowożytności, autorzy tomistyczni nie są
skłonni go odrzucać tylko dlatego, że np. fizyka kwantowa pokazuje,
iż pojęcie substancji, którym operujemy na makropoziomie,
nie sprawdza się na mikropoziomie.
Uważają, że naukowy obraz świata dotyczy bardziej aspektów
jakościowych, wyrażanych za pomocą języka matematyki,
ta zaś nie dotyczy realnego istnienia.
Natomiast bardziej podstawowy od tego obrazu jest ten,
który zwraca uwagę na rolę podstawowych struktur ontycznych,
które konstytuują samo „jądro” rzeczywistości.
Natomiast dla filozofa nauki takie podejście będzie przejawem
akceptacji metafizyki zdroworozsądkowej opartej na realizmie
naiwnym\footnote{
Por. \cite{Zycinski:Ucieczka}.
}.
Z~perspektywy analityków obraz świata przedstawiany przez
neotomistów mógłby grzeszyć naiwnością, wyrażającą się w~notorycznym
podkreślaniu, że obraz ten jest taki sam jak rzeczywistość.
Dla współczesnych filozofów nauki przyjęcie obrazu świata bazuje
na założeniach realizmu krytycznego\footnote{
Por. \cite{Barbur:Mity}, s.~41--65.
Szerzej na temat realizmu i~antyrealizmu we współczesnej filozofii
nauki por. \cite{Bird:Philosophy}, s.~121--161.
}.
Zwolennicy tego poglądu głoszą, że poznanie świata nie jest jego
kopiowaniem, tylko bardziej modelowaniem i~schematyzowaniem\footnote{
„Poznawanie nie jest bowiem ani pasywną kontemplacją, ani
nabyciem jedynie możliwego pojmowania w~tym, co dane jest
jako gotowe.
Poznawanie jest czynnym, żywym, nawiązywaniem relacji,
przeformowywaniem i~byciem przeformowywanym, krótko ---
tworzeniem.
Samodzielna rzeczywistość nie przysługuje ani <<podmiotowi>>,
ani <<przedmiotowi>>; każda egzystencja opiera się na wzajemnym
oddziaływaniu i~jest relatywna” (\cite{Fleck:OKryzysie}, s.~176).
}.
Obraz świata jest tylko mniej lub bardziej przybliżonym, ale tylko
modelem, wybranych aspektów świata rzeczywistego\footnote{
Szerzej na temat różnych funkcji modeli we współczesnej filozofii
nauki por. \cite{Furtak:Funkcje}, s.~29--69.
}.
W~przypadku filozofii modele te są bardzo niedokładne i~pozbawione
możliwości empirycznego testowania.
Dlatego nie można argumentować za tym, że obraz świata jest taki sam
jak rzeczywistość.
Uznanie, że istnieje tylko jeden właściwy obraz świata, na którym
bazuje nawet tzw. naukowy obraz świata, sprawia, że tomiści nie mają
problemu z~przyjmowaniem dosyć mocnych założeń bazowych, co prowadzi
do zarzutu fundamentalizmu.
Jeśli natomiast przyjmiemy, jak czyni to zdecydowana większość filozofów nauki, że nie ma uprzywilejowanego obrazu świata,
a~sama filozofia ma zdolność generowania różnych obrazów,
które pozostają w~różnych stosunkach względem naukowego obrazu
świata, to przyjmowanie różnych faktów w~punkcie wyjścia filozofii
może być tylko kwestią konwencji, która związana jest z~akceptacją
odmiennych wizji, do których dopiero później dorabia się, mniej lub
bardziej wyczuloną na spójność, logiczną argumentację.
Różnorodność konstatowanych faktów filozoficznych np. w~fenomenologii
czy tomizmie wynikałaby z~tego, że zwolennicy tych tradycji
badawczych akceptują zupełnie inny obraz świata;
pierwsi platoński, a~ci drudzy arystotelesowski w~zmodyfikowanej
przez \oAkwinat[ę] formie.
Z~punktu widzenia filozofii nauki fakty filozoficzne przyjmujemy
nie na podstawie bezpośredniego doświadczenia świata,
tylko na podstawie uprzedniej akceptacji określonej tradycji
badawczej (paradygmatu filozoficznego).
Ujęcie ludzkiego poznania z~punktu widzenia ontogenezy potwierdza
wiele intuicji \oWittgenstein[a] i~\ios{Abla}{Abel, G.}.
Człowiek rozwija swoje czynności poznawcze zawsze w~określonym
kontekście kulturowym i~historycznym.
Wiedzę zdobywa poprzez kulturową edukację.
A~do tego konieczne jest opanowanie języka, z~którym zawsze
związany jest obraz świata, czyli jakaś metafizyka.
Nie ma bowiem języka, który byłby wolny od założeń ontologicznych.
Ontologia zakładana przez język nie zawsze jest widoczna,
skrywa się jako tło, z~którym zapoznajemy się stopniowo,
przyswajając sobie coraz to bardziej abstrakcyjne terminy i~pojęcia.
Już potoczne doświadczenie świata realnego jest splotem różnych
elementów doświadczeniowych i~teoretycznych, bez których te pierwsze
w~ogóle nie miałyby żadnego sensu.
Problematyczna może być jednak postulowana neutralność faktów
filozoficznych, kiedy spojrzymy na nią z~perspektywy ustaleń
współczesnej metodologii nauk.
W~tym miejscu tomiści zaprotestują, że metodologię czyni się
kryterium oceny dla metafizyki, że jest to ustawienie pewnego
\textit{a~priori} w~stosunku do rzeczywistości.
Jeśli jednak przyjmujemy bardziej liberalny stosunek między
dziedzinami nauki i~filozofii, to metodologii obawiać się nie musimy.
Z~tego typu neutralnością, o~jaką chodzi autorom tomistycznym,
nie spotykamy się w~nauce, gdzie mowa jest o~stałym obciążeniu
faktów naukowych przez teorię (\textit{theory-laden}).
Tomiści twierdzą, że fakty dostępne w~poznaniu spontanicznym cechuje
taka właśnie neutralność względem teorii naukowych i~filozoficznych.
Bo jaka teoria może zakładać, że coś jest?
Żadna.
To jest fakt do tego stopnia podstawowy, że może on dopiero
konstytuować teorię, której zadaniem będzie wyjaśnienie tego faktu.
A~jednak, na co trzeba zwrócić uwagę, fakt ten wyrażony jest w~jakimś
języku, który już niesie ze sobą całe tło wiedzy.
Ponadto fakty filozoficzne wyrażone są w~języku naturalnym,
a~ten ma przecież charakter \textit{do-rzeczny}, czyli bytowy.
To byt determinuje język i~wszelkie kategorie semantyczne,
za pomocą których opisujemy wszystko\footnote{
Por. \cite{Krapiec:Jezyk}.
}.
Tomiści zdają się twierdzić, że można opisać świat w~języku
samego świata\footnote{
Por. \cite{Putnam:Pragmatyzm}, s.~48.
}.
Język bytowy byłby takim właśnie językiem świata.
Tylko przyjęciem takiego języka można uzasadnić neutralność
podstawowego faktu filozoficznego.
Czy jednak taki uprzywilejowany język istnieje?
Czy nie jest tak, że to my, ludzie, filozofowie, wymyślamy po prostu
różne języki, za pomocą których próbujemy opisać świat?\footnote{
Por. \cite{Putnam:Pragmatyzm}, s.~48--49.
}
Jeśli przyjmiemy takie założenie, to konstatacja faktu,
że coś istnieje, nie jest tej neutralności pozbawiona.
W~tym przypadku, podobnie jak to jest w nauce, można mówić
o~filozoficznym obciążeniu faktów filozoficznych
(\textit{philosophy- or metaphysics-laden}).
Stopień tego obciążenia w~przypadku filozofii możne być nawet
znacznie większy niż w~przypadku nauk empirycznych.
Stwierdzenie istnienia bytu nie może być neutralne nawet w~znaczeniu
\textit{philosophy-laden}, gdyż suponuje okreś\-loną koncepcję
istnienia jako prostego aktu bytu, który sprawia, że właśnie ów byt
jest czymś realnym.
Dla filozofa nauki stwierdzenie „neutralny fakt filozoficzny”
jest wewnętrznie sprzeczne.
Samo bowiem ujęcie faktu konotuje już określoną wiedzę,
nawet jeśli, jak mamy z~tym do czynienia w~poznaniu potocznym,
nie jest to wiedza rozległa i~precyzyjna, ale zawsze każdy fakt
pozostaje w~otoczce wiedzy, choćby minimalnej.
\cytuj{
[Fakt filozoficzny]
zwerbalizowany w~zdaniu stanowi odpowiedź na pytanie sformułowane
w~języku pewnego systemu.
Jest wtedy twierdzeniem tego systemu oraz wynikiem implikowanych
przez to pytanie i~odpowiedź interpretacji.
Już zatem wyjściowy stan badań fragmentu rzeczywistości zależy
od ingerencji struktur teoretycznych, czego nie należy traktować
jako wyrazu konwencjonalizmu lub konstruktywizmu.
Uwzględnia raczej na tej drodze udział narzędzi badawczych
w~determinowaniu obiektu wiedzy.
Ustalane fakty są rezultatem doświadczenia oraz elementu
teoretycznego\footnote{
\cite{Hajduk:Filozofia}, s.~126.
}.
}
Tzw. przedmiot formalny nauki jest właśnie takim teoretycznym
wyznaczeniem przedmiotu badań.
Jest on w~jakiś sposób skonstruowany, wyznaczony przez teorię
i~jej założenia.
Dotyczy to także istnienia, jako przedmiotu metafizyki.
Teoretyk bytu po prostu wybiera ten aspekt świata, który czyni
przedmiotem badania.
Wybór ten nie jest neutralny, gdyż jest uwarunkowany akceptowaną
teorią metafizyki, koncepcją istnienia.
Zakłada jako pewną wiedzę tła uwarunkowany historycznie obraz świata.
Jest to sytuacja naturalna, można powiedzieć --- normalna.
W~każdym procesie badawczym, nie tylko w~nauce, ale też w~filozofii,
przyjmujemy jakieś ogólne założenia (przedzałożenia), które
umożliwiają postawienie wstępnych pytań wyznaczających kierunek
odpowiedzi.
Zarzut aprioryzmu i~konstruktywizmu wydaje się w~tym wypadku
chybiony.
Filozofia nie jest aż tak radykalnie różna od nauki, jeśli chodzi
o~kwestię pewnych czynności wiedzotwórczych oraz ich przedzałożeń.
Można nawet zaryzykować tezę, że element aprioryczny jest
w~filozofii bardziej dominujący niż w~nauce.
Nie wszyscy jednak filozofowie chcą tę prawdę zaakceptować,
co też jest naturalnym stanem rzeczy w~tej dziedzinie wiedzy.
Stwierdzenie, że istnienie jest przedmiotem sądu, a~nie ujęć
abstrakcyjnych, bardzo wiele mówi o~jego naturze.
Zakłada już bowiem określoną i~bardzo skomplikowaną teorię
metafizyczną.
\cytuj{
Przecież aby powiedzieć, jak czyni to
o.~Krąpiec\index{Krąpiec, M.A.}, że zneutralizowana koncepcja
przedmiotu metafizyki wskazuje, iż owym przedmiotem jest byt
jako coś istniejącego --- aktualnie, realnie, a~nie np.
w~przeszłości, potencjalnie, czy idealnie, i~żeby zwrócić uwagę
na aspekt istnienia w~bycie, który jakimś sposobem ma gwarantować
neutralność względem wszelkich interpretacji --- czyli całkowitą
obiektywność, żeby to wszystko uczynić, to trzeba dysponować
całkiem pokaźną teorią\footnote{
\cite{Gutowski:Metafizyka}, s.~114.
}.
}
Chyba że uznamy, iż istnieje tylko jeden podstawowy obraz świata,
czyli ten nakreślony przez
\ios{Tomasza z~Akwinu}{Tomasz z Akwinu, św.},
za którym opowiadają się tomiści.
Obraz, który mimo wszystko przez wieki nie uległ żadnym przemianom.
Gdyby jednak przyjąć takie założenie, to nie mielibyśmy żadnego
uteoretyzowania, bylibyśmy w~sytuacji, kiedy rzeczywistość byłaby
dokładnie taka sama, jak obraz świata, który ją przedstawia.
Jeśli jednak do konkurencji staje kilka różnych filozoficznych
obrazów, a~każdy z~nich posiada własną koncepcję istnienia,
pojawia się pytanie, który jest obrazem prawdziwym, w~jaki sposób
można dokonać ich epistemologicznej waloryzacji, czyli chodzi o~to,
jakie wybrać kryteria ich oceny.
Czy obrazy świata przyjmujemy tylko na podstawie czysto racjonalnej,
czy też w~grę wchodzą określone aspekty perswazji?
Tomiści powiedzą, że to sama rzeczywistość decyduje o~tym, jaki mamy
obraz świata.
Musi on być po prostu zgodny z~realnymi faktami, inaczej jest czymś
apriorycznie zaprojektowanym i~narzuconym na rzeczywistość.
Kiedy ma to miejsce, rzeczywistość przeważnie „skrzeczy”.
Niewielu jednak jest przekonanych co do takiego rozwiązania sprawy.
Dla \oWittgenstein[a]
„uczeń wierzy nauczycielom i~podręcznikom”\footnote{
\cite{Wittgenstein:OPewnosci}, s.~59.
}
i~wiara ta jest rozsądna.
Czyli nie świat, ale ludzkie autorytety decydują o~przyjęciu takich,
a~nie innych przekonań.
Akceptacja obrazu świata ma zatem więcej wspólnego z~pewnego rodzaju
epistemiczną wiarą niż z~racjonalnym uzasadnieniem.
Wydaje się, że autor \textit{Dociekań filozoficznych} wykluczał
taką możliwość, by przyjęcie obrazu świata dokonywało się tylko
za pomocą samych środków racjonalnych.
Dlatego też twierdził, że u~kresu akceptacji pewnych przekonań
zdaje się stać perswazja.
\cytuj{
Ponieważ użytkownik innego systemu językowego nie rozumie słów,
jakich używamy, a~nawet gdy je rozumie, to tylko częściowo,
to nie możemy go przekonać o~zasadności naszych poglądów
podając mu argumenty --- bo jemu nasze „dobre” argumenty
jawić się będą jako niewystarczające lub głupie lub niezrozumiałe.
Słowa, jakich używamy, są dla niego niezrozumiałe lub rozumie je
on inaczej niż my.
Reguły wnioskowania, które są dla nas oczywiste, dla niego jawią
się jako błędne lub nieobowiązujące.
Możemy więc jedynie próbować od podstaw wpoić mu nasz system
językowy i~związany z~nim obraz świata, a~tym samym sprawić,
żeby zaczął tak samo postrzegać świat i~myśleć o~świecie,
jak my go postrzegamy i~o~nim myślimy\footnote{
\cite{Sady:JakaTeorie}, s.~9.
}.
}
Oto jedna z~przyczyn, dla której nie może być wspólnego obrazu
świata, choćby dla filozofii.
Tym samym otrzymujemy pewną odpowiedź na pytanie, dlaczego różni
filozofowie przyjmują tak rozbieżne fakty w~punkcie wyjścia swoich
teorii.
Co się dzieje, kiedy próbujemy przekazać komuś akceptowany przez nas
obraz świata?
Czy cała procedura dokonywałaby się tylko za pomocą rozumu?
Dokonałoby się to, jak podkreśla cytowany wcześniej \oWittgenstein,
tylko w~wyniku swoistej perswazji.
Ktoś, komu wpojono zupełnie inną grę językową, niż wpojono nam,
postrzega świat inaczej niż my; ma po prostu inny filozoficzny
obraz świata.
Dlatego też nie ma obiektywnie ważnych argumentów, które mogłyby
rozstrzygnąć, w~jaki sposób przekonać interlokutora do naszego
obrazu świata\footnote{
„Dowody mogą nas przekonać, iż ktoś jest w~takim a~takim
stanie duchowym; że nie udaje.
Ale są tu też dowody o~charakterze imponderabiliów”
(\cite{Wittgenstein:Dociekania}, s.~319).
}.
Wszelkie argumenty są zawsze zrelatywizowane do określonej
gry językowej, z~której czerpią swoją prawomocność.
Można powiedzieć tak, że wszelkie argumenty stosowane przez filozofów
są zrelatywizowane do określonej filozoficznej tradycji badawczej,
do której akces zgłaszamy na mocy okreś\-lonych decyzji, w~których
nie bierzemy pod uwagę tego, czy teoria zgadza się z~rzeczywistością,
gdyż nie wiemy, w~przypadku filozofii, na czym to miałoby polegać.
Dla kogoś spoza tej tradycji akceptowany przez nas obraz świata
i~generowana przez niego ontologia nie mają żadnego znaczenia.
Dlatego tam, gdzie fenomenolog będzie widział konieczność
przyjmowania bytów idealnych dostępnych w~poznaniu apriorycznym,
jako gwarantów uzyskania wiedzy koniecznej, tam tomista będzie
z~właściwym sobie radykalizmem kwestionował to,
jako uprzedmiotowienie sensów ogólnych.
Postulowane przez jednego i~drugiego fakty są radykalnie
niewspółmierne, choć mogą być poddane interpretacji w~ramach
różnych koncepcji.
Wiara w~autorytet przekazujący prawdy nigdy nie jest w~stu
procentach oparta na samych przesłankach rozumowych.
Zawsze wchodzi tu w~grę element osobistego zawierzenia
i~nie dotyczy to tylko prawd przekazywanych drogą wiary religijnej,
ale też prawd naukowych i~filozoficznych.
Żaden obraz świata nie jest na tyle zniewalający w~swojej prawdzie,
aby był podzielany przez wszystkich.
Dlatego też żaden paradygmat filozoficzny, który zakłada jakiś
określony obraz świata, nie jest powszechnie akceptowany.
Przyjęcie określonego obrazu świata, uznanie go za podstawę
dla wszystkich rozstrzygnięć teoretycznych i~praktycznych sprawia,
że po jakimś czasie przestaje on być dla nas widoczny.
Oddziałuje tylko podskórnie, właśnie jako wiedza tła, na bazie której
przeprowadzamy określone argumenty oraz dokonujemy waloryzacji
pod kątem prawdy lub fałszu.
Im bardziej wzrastamy i~zrastamy się z~danym obrazem świata,
tym bardziej kształtuje się w~nas takie poczucie bezpośredniości
świata, gdzie wszystko wydaje się, jakby leżało na dłoni.
Tak, jak zżywamy się z~potocznym obrazem świata,
tak i~oswajamy się z~obrazem filozoficznym lub naukowym.
Z~tym że naukowy obraz świata --- w~przeciwieństwie do
filozoficznego --- bardzo często jest elastyczniejszy
i~podatniejszy na korekty, o~których decyduje nieustanny rozwój
poznania naukowego.
Filozof może, ale nie musi brać pod uwagę danych naukowego obrazu
świata.
Nie jest to ani koniecznym obowiązkiem filozofa, ani też nie
przyczynia się do uściślenia prowadzonych przez niego poszukiwań
i~uzyskiwanych rezultatów badawczych.
Dlatego bez posądzeń o~anachroniczność filozof może zupełnie
swobodnie głosić, że nadal obowiązujący jest platoński
(jak w~przypadku fenomenologów) lub arystotelesowski (jak w~przypadku
tomistów) obraz świata.
Współcześnie tak tomiści, jak i~fenomenologowie, głosząc takie
ustalenia, pozostają jakby z~boku naukowego obrazu świata,
choć --- co trzeba podkreślić --- ani go nie lekceważą, ani też
nie kwestionują w~całości.
Czasem tylko tomistom zdarza się posądzać współczesne nauki
przyrodnicze o~to, że w~swoich hipotetycznych spekulacjach
ocierają się o~mit\footnote{
Por. \cite{Krapiec:Filozofia}, s.~124.
}.
Owo zżycie polega na tym, że dany obraz świata bardzo często
uważa się ostatecznie za prawdziwy i~bezpośrednio pokazujący,
jaka rzeczywistość jest naprawdę.
Faktycznie zaczynamy widzieć pewne przedmioty jako byty, substancje,
przypadłości, idee, czyste jakości idealne itp.
Dlatego nie dziwi to, iż niektórzy autorzy tomistyczni piszą,
że widzą byt i~fakty, a~fenomenologowie, że doświadczają widzenia
apriorycznych idealnych stanów rzeczy, że dysponują takim właśnie
widzeniem, do którego nie każdy jest predysponowany.
Obraz świata nakłada pewne swoiste \textit{a priori}, które w~ludzkim
poznaniu --- ze względu na jego kulturowe i~historyczne obciążenie
--- jest niemożliwe do uniknięcia.
Samo istnienie świata, o~którym też dowiadujemy się na podstawie
obrazu świata, staje się do tego stopnia dla nas oczywiste,
że przestajemy je zauważać.
Dlatego słusznie powiada \oWittgenstein, że najważniejszych
i~najprostszych rzeczy po prostu nie zauważamy, nie potrafimy
ich spostrzec, choć przecież stale mamy je przed oczyma\footnote{
Por. \cite{Wittgenstein:Dociekania}, nr~129 i~89.
}.
W~codziennym doświadczeniu bardzo często mamy do czynienia
z~tzw. ślepotą na rzeczy bliskie\footnote{
Por. \cite{Abel:Swiat}, s.~95.
}.
Taka „ślepota na rzeczy bliskie” dotyczy też istnienia.
Razem z~nauką języka i~obrazem świata akceptujemy fakt jego
istnienia.
Staje się ono dla nas do tego stopnia czymś oczywistym i~powszechnym,
że nie zwracamy na nie uwagi w~naszych codziennych życiowych
działaniach poznawczych i~praktycznych.
Dobrze pokazał ten problem \oWittgenstein{} w~dyskusji z~analitykiem
\oMoore['m].
Zdania tego typu jak: „mam dwie ręce”, „posiadam ciało”
również uzyskują status wiedzy na gruncie przyjmowanego razem
z~grą językową obrazu świata.
To, że coś wiem, nie jest kwestią doświadczenia, ale pierwotnie
akceptowanej wiedzy tła.
Nie potrzeba odwoływać się do intuicji zdrowego rozsądku
czy innych specjalnych typów poznań, np. apriorycznej intuicji,
jak chcieli fenomenologowie.
Odpowiedź na pytanie, skąd wiem, że to jest moja ręka, odsyła nas
do gry językowej.
Zdanie to czerpie swoją pewność właśnie z~niej, jako pewnej sieci
zdań wzajemnie ze sobą powiązanych.
Autor \textit{Dociekań filozoficznych} uzasadnia to w~ten sposób:
\cytuj{
uczymy dziecko „To jest twoja ręka”, a~nie „To być może
(lub prawdopodobnie) jest twoja ręka”.
Tak dziecko uczy się niezliczonych gier językowych dotyczących
jego ręki.
W~ogóle nie przychodzi mu do głowy, by badać lub by pytać
„czy to naprawdę jest ręka”.
Z~drugiej strony nie uczy się też, że wie, iż to jest
ręka\footnote{
\cite{Wittgenstein:OPewnosci}, s.~76.
}.
}
Dlatego tomista zgodzi się z~\oWittgenstein[em], że nasze przekonanie
o~istnieniu jest pewne i~nie podlega refutacji.
Nie możemy komuś powiedzieć, że świat realny istnieje tylko
prawdopodobnie\footnote{
Tomiści również zgodziliby się z~\oWittgenstein[em], że wątpienie
nie jest pierwotnym stanem poznawczym człowieka.
Najpierw jest afirmacja świata, dopiero na tej podstawie można
wątpić w~określone treściowe uposażenie, ale nigdy nie
w~istnienie.
Dla \oWittgenstein[a] pewność co do faktów czerpiemy
z~przyswojenia językowego obrazu świata; nie mamy bezpośredniego
i~nieprzysłoniętego dostępu do rzeczywistości samej w~sobie.
Nie czerpiemy tej pewności z~żadnego bezpośredniego doświadczenia.
}.
Ten fakt musi być od początku obrośnięty niekwestionowaną pewnością.
Tylko dla tomisty pewność ta płynie z~bezpośredniego doświadczenia,
natomiast dla \oWittgenstein[a] z~językowego obrazu świata.
Tego typu fakty przyswajamy razem z~nauką; po prostu uczymy się,
że istnieją nasze ręce, inni ludzi, przedmioty itp.
Nie potrzeba zatem odwoływać się do żadnego bezpośredniego
doświadczenia.
Zbędne jest także wypowiadanie sądów egzystencjalnych.
Wobec obrazu świata ich ranga jest wtórna.
Przecież samo doświadczenie nie pokaże nam niczego, dopóki
nie nauczymy się poprawnie je odczytywać.
Trzeba zatem posiąść umiejętność jego interpretacji, wyciągania
wniosków, a~to wszystko dokonuje się w~procesie kulturowej edukacji,
poprzez transfer wiedzy\footnote{
Por. \cite{Tomasello:Kulturowe}.
}.
Stąd też ludzie posiadają tak zróżnicowaną sprawność w~wyprowadzaniu
określonych wniosków na podstawie doświadczenia.
Jeśli przyjmujemy określony obraz świata to on nam podpowiada,
co mamy uznać za przesłanki, a~co za wnioski dowodzenia.
Jeśli np. akceptujemy, że istnienie jest aktem prostym,
to --- jako proste --- nie może być skonceptualizowane.
Wówczas pozostaje tylko poznanie sądowe, jako prawomocny typ poznania
bezpośrednio udostępniający ów akt istnienia.
Obraz świata przynosi zatem określone typy faktów, które akceptujemy
jako podstawę naszego poznania, działania i~intersubiektywnej komunikacji.
Dotychczasowa argumentacja skłania nas do przypuszczenia,
że rzeczywistość nie narzuca nam żadnego uprzywilejowanego,
w~znaczeniu filozoficznym, obrazu świata.
Gdyby rzeczywistość faktycznie posiadała wbudowaną w~siebie
jakąś jedną uniwersalną strukturę ontyczną, która na ludzkie
poznania oddziaływałaby w~sposób konieczny, wówczas prawdopodobnie
dysponowalibyśmy tylko jednym obrazem świata.
Tymczasem rzeczywistość nie narzuca filozofowi żadnej określnej
obiektywnej wykładni metafizycznej, która byłaby prawidłowa,
zaś wszystkie inne nieadekwatne.
Pojawia się pytanie, które ze względu na propedeutyczny charakter
tych analiz nie znajdzie rozstrzygnięcia:
co to znaczy, że fakty filozoficzne w~jakiś sposób weryfikują teorię
czy też decydują o~jej prawdziwości (jak tego chcą np. autorzy
tomistyczni) lub mocy eksplanacyjnej?
Jaki tak naprawdę jest stosunek faktów do teorii filozoficznej?
W powyższych analizach udało nam się wstępnie ustalić, że fakty
filozoficzne tego typu, jak
„istnieje świat”, „Jan jest bytem” itp.,
akceptujemy na podstawie obrazu świata, który przyswajamy
poprzez język, wchodząc w~określoną filozoficzną tradycję
badawczą$/$paradygmat na mocy osobistej decyzji uwarunkowanej
nie tylko racjami rozumowymi, ale też psychologicznymi
i socjologicznymi.
Fakty filozoficzne, trawestując \oPutnam[a], nie są jakimś
gotowym wyborem.
Nie są zatem czymś, co czeka na filozofa, aby je odkrył,
a~następnie poukładał w~spójną całość\footnote{
Por. \cite{Putnam:Dlaczego};
\cite{Feyerabend:JakByc}.
}.
Zawsze uwikłane są w~określone założenia teoretyczne
oraz aksjologiczne; fakty zakładają teorie i~wartości\footnote{
Por. \cite{Putnam:Pragmatyzm}, s.~30--31.
}.
Nie wystarczy tylko otworzyć oczy, aby zobaczyć tak doniosły
fakt filozoficzny, jakim jest istnienie świata, dlatego,
że za tym stwierdzeniem stoi bardzo skomplikowana teoria istnienia,
która w~już określony sposób ustawia naszą interpretację
rzeczywistości na określonych torach\footnote{
„Pomocne może też być uświadomienie sobie, że dostęp do wspólnej
rzeczywistości nie wymaga dostępu do czegoś przedpojęciowego.
Wymaga raczej umiejętności tworzenia pojęć, które podzielamy”
(\cite{Putnam:Pragmatyzm}, s.~41).
}.
Jeśli tak, to fakt filozoficzny nie jest tworem obiektywnym,
w~znaczeniu radykalnej obiektywności, jako coś gotowego,
co oczekuje na czynności poznawcze, które w~sposób jednoznaczny
by go odzwierciedlały.
Zawiera zawsze interpretację, która jest konsekwencją akceptowanego
obrazu świata\footnote{
Por. \cite{Brozek:Granice}, s.~202.
Interesujące uwagi na temat interpretacji w~kontekście realizmu
por. \cite{Lenk:Interpretacja}.
},
w~którym mieszczą się czynniki kulturowe, wartości oraz szereg
założeń ontologicznych i~gnoseologicznych\footnote{
Por. \cite{Buczkowska:KilkaUwag};
\cite{Buczkowska:Rola};
\cite{Such:ORodzajach}.
}.
W~kontekście powyższych ustaleń trafne wydają się słowa \oPopper[a]:
\cytuj{
Nie wiemy, gdzie i~jak rozpocząć analizę rzeczywistości.
Żadna mądrość nam tego nie powie.
Nawet tradycja naukowa jako taka nie wskaże nam tego.
Ale powie nam, że ludzie zbudowali już na tym świecie pewien
ramowy zarys jego teorii --- pewnie nie najlepszy, lecz mniej
więcej przydatny jako osnowa wyjaśnienia zjawisk.
Służy on jako pewnego rodzaju siatka, układ współrzędnych,
do którego odnosimy nasze spostrzeżenia.
Korzystamy z~tego układu wciąż, sprawdzając i~krytykując
w~całej rozciągłości.
I~w~ten sposób osiągamy postęp\footnote{
\cite{Popper:Krytycyzm}, s.~864.
}.
}
Słowa te odnoszą się nie tylko do nauki, ale też i~filozofii,
a~może w~szczególny sposób właśnie do niej.
Fakty filozoficzne nie mają charakteru empirycznego i~nie mogą być
sprawdzone w~żadnym doświadczeniu.
Dotyczy to tak pozytywnego, jak i~negatywnego aspektu sprawdzania.
Problem ten otwiera jednak kolejną dyskusję dotyczącą możliwości
weryfikacji$/$falsyfikacji teorii filozoficznych poprzez tzw.
fakty filozoficzne.
Wstępnie pytamy, co tak naprawdę decyduje o~tym, że wybieramy
taką lub inną teorię filozoficzną (np. tomizm czy fenomenologię).
\tytul{Podsumowanie}
Na podstawie przeprowadzonych analiz wstępnie możemy skłonić się
ku tezie, że fakty filozoficzne mają raczej bardzo nikłą zdolność
wpływania na akceptację danej teorii filozoficznej.
Raczej decyduje o~tym obraz świata, którego heurystyczna funkcja
polega właśnie na tym, że on te fakty sugeruje, jako czasem coś
mniej lub bardziej oczywistego.
Aby przyjąć jakieś fakty, filozof po prostu musi dużo się nauczyć
od tych, których uznaje za autorytety w~swojej dziedzinie.
Fakty filozoficzne nie są czymś, co zastajemy w~świecie.
Są one konsekwencją interpretacji, z~którą zapoznajemy się,
przyjmując określoną grę językową oraz niesiony wraz z~nią
obraz świata.
Jako puentę warto przytoczyć za \oFleck[iem] zdanie, na które
bardzo często powołują się autorzy tomistyczni:
„\textit{nihil est in intellectu quod non prius fuerit in sensu}
--- także w~jego znaczeniu odwrotnym, choć niekoniecznie już
akceptowanym przez nich --- \textit{nihil est in sensu, quod
non fuerit in intellectu}''\footnote{
\cite{Fleck:OKryzysie}, s.~175.
Por. też \cite{Rembierz:Okreslanie}.
}.
Kiedy bierzemy pod uwagę prawdę wyrażoną w~obydwu tych łacińskich
sentencjach, wówczas uzyskujemy bardziej wieloaspektowy punkt
widzenia na fakty filozoficzne oraz ich rolę w~strukturze teorii
filozoficznej.
\summary{
This article aims to show that our acceptance or non-acceptance
of certain facts is influenced by our adoption of a~philosophical
world-picture as a~kind of background knowledge on the basis
of which one decides what does or does not exist, and what is true
or false.
For this purpose, I discuss the positions of the existential
Thomists, as well as those of \oWittgenstein{} and \oAbel,
while also occasionally invoking the work of \oPutnam{} and \oFleck.
To begin with, it is demonstrated that philosophical facts
are accepted on the basis of a~world-picture that is itself
a~tangle of facts, values, and theories exhibiting varying degrees
of generality.
Whether we embrace them or not is thus not determined by any sort
of direct experience.
There are no neutral philosophical facts.
Our world-picture suggests what sort of facts we are prepared
to accept, and what methods we use to explain them.
It is on precisely this that the heuristic role played
by world-pictures in “creating” philosophical facts depends.
}{
world-picture ---
fact ---
philosophical facts ---
Thomism ---
analytic philosophy ---
philosophy of interpretation ---
paradigm ---
philosophy of science
}
\end{elementlit}
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\title{CS302 - Problem Set 11}
\author{}
\date{}
\parindent=.25in
\begin{document}
\maketitle
\vspace{-2cm}
\begin{enumerate}
\item We've been learning about frameworks for deciding when algorithms are problematic and mitigating their harm. However, these frameworks are fairly abstract, and not yet in common use, and don't necessarily deal with the real world experience of discovering that your company or another company's algorithm is causing harm, and then working to mitigate that harm. We will look at two case studies of people who attempted to alert companies to harm or potential harm due their algorithms, and the results of their efforts:
\begin{itemize}
\item Watch this \href{https://www.youtube.com/watch?v=eRUEVYndh9c}{Youtube video} on the research of Joy Buolamwini. Then for a recent update on this story, read the first few paragraphs of \href{https://www.npr.org/2020/06/09/873298837/ibm-abandons-facial-recognition-products-condemns-racially-biased-surveillance}{IBM Abandons Facial Recognition (NPR)}. Optional: for a more in-depth look at Dr. Buolamwini's research and story and facial recognition more broadly, you should watch the movie \href{https://midd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=ef45a5cb-6fd8-41f4-9a2b-acaf014888d4}{Coded Bias} (will need Midd credentials to log in.)
\item Read about \href{https://middlebury.instructure.com/files/1288885/download?download_frd=1}{Timnit Gebru's Firing from Google (Washington Post)}
\end{itemize}
Comment on Dr. Buolamwini's approach to interacting with companies she studied. Which of her strategies were more successful in accomplishing change, and why do you think that was? Comment on how different companies reacted differently to her research.
Comment on Google's treatment of Dr. Gebru, and their approach to ethical research more generally (at least as described in the article). What are your feelings/reactions? What are some of the benefits and harms of Google's (in)actions on ethics? (I'm assuming there are some benefits, or Google wouldn't act the way it does!)
\item
Recall the following definitions:
\begin{itemize}
\item $k$-\textsf{INDSET}: Given an undirected, unweighted graph $G=(V,E)$, output YES if there is a set $V'\subseteq V$ such that $|V'|\geq k$, and for all $v,u\in V'$, there is no edge $\{u,v\}\in E$. (We call such a set a $k$-independent set.) Otherwise output NO.
\item $k$-\textsf{CLIQUE}: Given an undirected, unweighted graph $G=(V,E)$, output YES if there is a set $V'\subseteq V$ such that $|V'|\geq k$, and for all $v,u\in V'$, there is an edge $\{u,v\}\in E$. (We call such a set a $k$-clique.) Otherwise output NO.
\item \textsf{DOUBLE-3-SAT}: Given a CNF formula with $n$ variables, where each clause involves at most $3$ of the variables $z_1,z_2,\dots,z_n$ (and their negations) output YES if there at least two \textit{different} satisfying assignments to the formula. Otherwise output NO. For example, $(z_1 \vee \neg z_1 \vee \neg z_2) \wedge (z_2 \vee z_3) \wedge (\neg z_3)$ is a YES instance because it has two valid assignments, $z_1 = 1, z_2 = 1, z_3 = 0$ and $z_1 = 0, z_2 = 1, z_3 = 0$.
\end{itemize}
In the following, you can assume the results you proved in the prior PSet, that $k$-\textsf{INDSET}, $k$-\textsf{CLIQUE}, and \textsf{DOUBLE-3-SAT} are all in \textsf{NP}.
\begin{enumerate}
\item Prove \textsf{DOUBLE-3-SAT} is \textsf{NP}-Complete.
{}
{}
\item In this problem, you'll prove that that $k$-{\textsf{INDSET}} and $k$-{\textsf{CLIQUE}} are \textsf{NP}-Complete.
\begin{enumerate}
\item Prove that there is a polynomial reduction from $k$-{\textsf{INDSET}} to $k$-{\textsf{CLIQUE}}.
\item Prove that there is a polynomial reduction from $3$-\textsf{SAT} to $m$-\textsf{INDSET}, where $m$ is the number of clauses in the $3$-\textsf{SAT}.
\item Use the previous two parts to prove that $k$-{\textsf{INDSET}} and $k$-{\textsf{CLIQUE}} are \textsf{NP}-Complete.
\end{enumerate}
{}
{
}
\item \textbf{Challenge:} Consider the problem of outputting the {\it second} largest value of an array $A$. $A$ is an unsorted array $A$ of unique integers of length $n$ (you may assume $n$ is a power of 2).
\begin{enumerate}
\item Describe an algorithm that uses exactly $n+\log_2n-2$ comparison operations, where a comparison is an operation that tests whether one element is less than another element.
\item Justify why your algorithm is correct. (You can do this as a proof, or as a slightly less formal explanation.)
\end{enumerate}
(Hint: the algorithm combines ideas from both divide and conquer and dynamic programming.)
{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{enumerate}
\end{enumerate}
\end{document} |
http://users.umiacs.umd.edu/~hal/courses/2010F_CL1/out/hw05.tex | umd.edu | CC-MAIN-2023-14 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2023-14/segments/1679296945030.59/warc/CC-MAIN-20230323065609-20230323095609-00330.warc.gz | 50,487,770 | 1,692 | \documentclass[fleqn]{article}
\usepackage{haldefs}
\usepackage{notes}
\usepackage{url}
\usepackage{pict2e}
\usepackage{qtree}
\begin{document}
\lecture{Computational Linguistics I}{HW05: Unification}{CMSC723, Fall 2010}
% IF YOU ARE USING THIS .TEX FILE AS A TEMPLATE, PLEASE REPLACE
% "CMSC723, Fall 2010" WITH YOUR NAME AND UID.
Hand in at: \url{http://www.cs.utah.edu/~hal/handin.pl?course=cmsc723}.
Remember that only PDF submissions are accepted. We encourage using
\LaTeX\ to produce your writeups. See \verb+hw00.tex+ for an example
of how to do so. You can make a \verb+.pdf+ out of the \verb+.tex+ by
running ``\verb+pdflatex hw00.tex+''.
\section{Non-Unification Grammar}
Start with the following, basic CFG:
\begin{verbatim}
S -> NP VP
NP -> Det Noun
NP -> Pro
VP -> Verb
VP -> Verb NP
Pro -> I | you | she | we | they | me | her | us | them
Noun -> sandwich | sandwiches | fruit | apple | apples
Det -> the | a | an | many
Verb -> eat | eats | ate | like | likes | liked
\end{verbatim}
There are some problems with this grammar, including (at least):
\begin{itemize}
\item Det/Noun number agreement (``a sandwiches'' or ``many apple'')
\item Subject/Verb number agreement (``they eats'')
\item Pronoun case (``us ate'' or ``eats I'')
\item Determiner spelling (``an sandwich'' or ``a apple'')
\item Subject/Verb person agreement (``I eats'')
\end{itemize}
Add sufficient features to this grammar to capture the above
phenomena. Namely, augment the lexical items with their feature
values (warning: some are underspecified, for instance ``fruit'' can
be singular or plural). Then, for each production, state which
features of constituents need to be unified (eg, ``for NP -> Det
Noun'' we need to unify the \emph{blah} feature from Det and Noun).
Based on your grammar, construct parse structures for the following
sentences, and list the features at each node in the tree (terminals,
pre-terminals, internal nodes and root). For those that don't parse,
say where unification fails.
\begin{itemize}
\item I eat many sandwiches
\item A sandwich likes the fruit
\item We ate the sandwich
\item *I ate a sandwiches
\item *Us likes a sandwich
\item *I like a apple
\end{itemize}
\end{document}
|
https://git.sr.ht/~reesmichael1/chantpointer/blob/fix_ci/psalm.tem | sr.ht | CC-MAIN-2020-29 | text/plain | application/x-tex | crawl-data/CC-MAIN-2020-29/segments/1593655887360.60/warc/CC-MAIN-20200705121829-20200705151829-00498.warc.gz | 331,434,693 | 873 | \documentclass[((.PointSize))pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage[T1]{fontenc}
\usepackage{graphicx}
\usepackage{psalm}
\pagestyle{empty}
\begin{document}
\begin{center}
{\Large \textbf{(( .Title ))}}\\[12pt]
\textit{((.Subtitle ))}\\[30pt]
\includegraphics[width=\textwidth]{(( .ScorePath ))}
\end{center}
\vspace{24pt}
\setlength\textamount{12cm}
\begin{center}
\begin{psalm}
((range $v := .Verses ))
\vs{(( if .IsSecond ))\second(( end ))}{(( $v.FirstPart ))}
{}{(( $v.SecondPart ))}
(( end ))
\end{psalm}
\end{center}
\end{document}
|
https://dlmf.nist.gov/14.3.E11.tex | nist.gov | CC-MAIN-2022-27 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-27/segments/1656103329963.19/warc/CC-MAIN-20220627073417-20220627103417-00622.warc.gz | 255,684,559 | 743 | \[\mathsf{P}^{\mu}_{\nu}\left(x\right)=\cos\left(\tfrac{1}{2}(\nu+\mu)\pi\right)%
w_{1}(\nu,\mu,x)+\sin\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{2}(\nu,\mu,x),\] |
https://svn.geocomp.uq.edu.au/escript/trunk/doc/install/intro.tex?r1=2547&sortdir=down&pathrev=2547&r2=2546&view=patch | uq.edu.au | CC-MAIN-2020-45 | text/plain | application/x-tex | crawl-data/CC-MAIN-2020-45/segments/1603107905777.48/warc/CC-MAIN-20201029184716-20201029214716-00434.warc.gz | 523,010,624 | 990 | --- trunk/doc/install/intro.tex 2009/07/20 05:35:08 2546
+++ trunk/doc/install/intro.tex 2009/07/20 05:50:46 2547
@@ -47,7 +47,7 @@
escript poission.py
\end{shellCode}
If this produces a VTK file called \filename{u.vtu} then you are likely to have a functional \esfinley installation.
-You can try and visualise the VTK data or delete the file.
-For visualisation we suggest using \filename{VisIt}\footnote{\url{https://wci.llnl.gov/codes/visit/}} or \filename{MayaVi}\footnote{\url{http://mayavi.sourceforge.net}} which are both freely available.
+You can try and visualize the VTK data or delete the file.
+For visualization we suggest using \filename{VisIt}\footnote{\url{https://wci.llnl.gov/codes/visit/}} or \filename{MayaVi}\footnote{\url{http://mayavi.sourceforge.net}} which are both freely available.
See the site \url{https://answers.launchpad.net/escript-finley} for online help.
|
https://dlmf.nist.gov/25.11.E23.tex | nist.gov | CC-MAIN-2020-34 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-34/segments/1596439737238.53/warc/CC-MAIN-20200808021257-20200808051257-00390.warc.gz | 284,003,331 | 847 | \[\zeta'\left(1-2n,\tfrac{1}{3}\right)=-\frac{\pi(9^{n}-1)B_{2n}}{8n\sqrt{3}(3^{%
2n-1}-1)}-\frac{B_{2n}\ln 3}{4n\cdot 3^{2n-1}}-\frac{(-1)^{n}{\psi^{(2n-1)}}%
\left(\frac{1}{3}\right)}{2\sqrt{3}(6\pi)^{2n-1}}-\frac{\left(3^{2n-1}-1\right%
)\zeta'\left(1-2n\right)}{2\cdot 3^{2n-1}},\] |
http://kohnlehome.de/java/mvc.tex | kohnlehome.de | CC-MAIN-2017-34 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2017-34/segments/1502886105922.73/warc/CC-MAIN-20170819201404-20170819221404-00089.warc.gz | 230,522,546 | 2,646 | %Teilweise Erzeugt mit dem LaTeX-Generator: http://latex.sehnot.de
%Schriftgröße, Layout, Papierformat, Art des Dokumentes
\documentclass[10pt,oneside,a4paper]{scrartcl}
%Einstellungen der Seitenränder
\usepackage[left=2cm,right=2cm,top=1.5cm,bottom=1.5cm,includeheadfoot]{geometry}
%neue Rechtschreibung
\usepackage[ngerman]{babel}
%Umlaute ermöglichen
\usepackage[utf8]{inputenc}
%Gesamtseitenzahl
\usepackage{lastpage}
% Kopf- und Fußzeile
\usepackage[automark]{scrpage2}
%Quellcode-Listings
\usepackage{listings}
%Hyperlinks
\usepackage{hyperref}
%Kopfzeile
\ihead{Java}
\chead{}
\ohead{http://kohnlehome.de/java/mvc.pdf}
\setheadsepline{0.5pt}
%Fußzeile
\setfootsepline{0.5pt}
\ifoot{Franz Kohnle}
\cfoot{Seite \thepage\ von \pageref{LastPage}}
\ofoot{\today}
\pagestyle{scrheadings}
\begin{document}
% Überschrift
\begin{center}
\LARGE % Schriftgröße
\bfseries % Fettdruck
\sffamily % Serifenlose Schrift
Design Pattern: Model-View-Controller
\end{center}
\section{Designprinzip}
\begin{itemize}
\item Model: verwaltet Daten, benachrichtigt View (Observer) bei Änderung
\item View: Beobachter von Model (Anzeige der Daten), verwendet Controller als Strategie (Benutzeraktionen)
\item Controller: verarbeitet Benutzeraktionen und leitet sie an Model weiter, steuert View (Steuerelemente deaktivieren)
\end{itemize}
\section{Java-Code}
\subsection{Modelschnittstelle}
\lstset{language=Java}
\begin{lstlisting}[frame=tlRB]
public interface ModelSchnittstelle {
// View (Beobachter) muss sich registrieren
void registrieren(ViewSchnittstelle view);
// Methoden zum Lesezugriff auf Daten fuer View
int getDaten();
// Methoden zum Schreibzugriff fuer den Controller
void aendern();
}
\end{lstlisting}
\subsection{Viewschnittstelle}
\lstset{language=Java}
\begin{lstlisting}[frame=tlRB]
public interface ViewSchnittstelle {
// Diese Methode ruft das Model (Subject) bei Aenderung der Daten auf
void update();
// Diese Methoden kann der Controller aufrufen
// z.B. GUI erstellen,
// Steuerelemente deaktivieren
void fensterErstellen();
}
\end{lstlisting}
\subsection{Controllerschnittstelle}
\lstset{language=Java}
\begin{lstlisting}[frame=tlRB]
public interface ControllerSchnittstelle {
// Diese Methoden kann die View aufrufen
// Benutzeraktionen werden von View an Controller uebermittelt
void aendern();
}
\end{lstlisting}
\subsection{Hauptprogramm}
\lstset{language=Java}
\begin{lstlisting}[frame=tlRB]
ModelSchnittstelle model = new KonkretesModel();
ControllerSchnittstelle controller = new KonkreterController(model);
\end{lstlisting}
\subsection{Konkretes Model}
\lstset{language=Java}
\begin{lstlisting}[frame=tlRB]
public class KonkretesModel implements ModelSchnittstelle {
// Model kennt nur Referenzen von allen Views (Observer)
// Model kennt Controller nicht
private ArrayList<ViewSchnittstelle> beobachterliste
= new ArrayList<ViewSchnittstelle>();
// Vom Model selbst verwaltete Daten
private int daten;
private Random random = new Random();
// Views (Observer) muessen sich registrieren
@Override
public void registrieren(ViewSchnittstelle view) {
beobachterliste.add(view);
}
// Views (Observer) werden ueber Aenderung der Daten benachrichtigt
private void broadcast(){
for(int i=0; i<beobachterliste.size(); i++){
beobachterliste.get(i).update();
}
}
// Lesezugriff von aussen auf Daten (fuer Views, Observer)
@Override
public int getDaten() {
return daten;
}
// Schreibzugriff von aussen auf Daten (fuer Controller)
@Override
public void aendern() {
daten = random.nextInt();
// bei Aenderung der Daten alle Views benachrichtigen
broadcast();
}
}
\end{lstlisting}
\pagebreak
\subsection{Konkrete View}
\lstset{language=Java}
\begin{lstlisting}[frame=tlRB]
public class View implements ViewSchnittstelle{
// View holt sich Daten vom Model
private ModelSchnittstelle model;
// View uebergibt Benutzeraktionen an Controller
private ControllerSchnittstelle controller;
// GUI-Elemente
private JButton button;
public View(ControllerSchnittstelle controller,
ModelSchnittstelle model) {
// View kennt Controller zur Uebergabe von Aktionen
this.controller = controller;
// View kennt Model (Subject) zum Empfangen von Aenderungsmeldungen
// und zum Holen der Daten
this.model = model;
model.registrieren(this);
}
// wird vom Model (Subject) aufgerufen
@Override
public void update() {
// View holt Daten selbst (PULL)
int daten = model.getDaten();
button.setText("Zahl: " + daten);
}
// GUI
@Override
public void fensterErstellen() {
JFrame frame = new JFrame();
frame.setSize(200,200);
frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
button = new JButton("Drueck");
frame.add(button);
button.addActionListener(new ActionListener() {
@Override
public void actionPerformed(ActionEvent arg0) {
// Benutzeraktion wird an Controller uebermittelt
controller.aendern();
}
});
frame.setVisible(true);
}
}
\end{lstlisting}
\pagebreak
\subsection{Konkreter Controller}
\lstset{language=Java}
\begin{lstlisting}[frame=tlRB]
public class KonkreterController implements ControllerSchnittstelle {
// Controller ist Vermittler zwischen View und Model
private ModelSchnittstelle model;
private View view;
// Controller erstellt View
public KonkreterController(ModelSchnittstelle model) {
this.model = model;
view = new View(this,model);
view.fensterErstellen();
}
// Controller verarbeitet Nachrichten von View
// und gibt sie an Model weiter
@Override
public void aendern() {
model.aendern();
}
}
\end{lstlisting}
\begin{thebibliography}{999}
\bibitem [Head First Design Patterns]{} \url{http://www.oreilly.de/catalog/9780596007126/}
\bibitem [Wikipedia]{} \url{http://de.wikipedia.org/wiki/Model_View_Controller}
\end{thebibliography}
\end{document}
|
http://www.umr-marbec.fr/r3fbas3/search.php?sqlQuery=SELECT%20author%2C%20title%2C%20type%2C%20year%2C%20publication%2C%20abbrev_journal%2C%20volume%2C%20issue%2C%20pages%2C%20keywords%2C%20abstract%2C%20thesis%2C%20editor%2C%20publisher%2C%20place%2C%20abbrev_series_title%2C%20series_title%2C%20series_editor%2C%20series_volume%2C%20series_issue%2C%20edition%2C%20language%2C%20author_count%2C%20online_publication%2C%20online_citation%2C%20doi%2C%20serial%2C%20area%20FROM%20refs%20WHERE%20serial%20%3D%202678%20ORDER%20BY%20year%20DESC%2C%20first_author%2C%20author_count%2C%20author%2C%20title&client=&formType=sqlSearch&submit=Cite&viewType=&showQuery=0&showLinks=1&showRows=5&rowOffset=&wrapResults=1&citeOrder=year&citeStyle=MLA&exportFormat=RIS&exportType=html&exportStylesheet=&citeType=LaTeX&headerMsg= | umr-marbec.fr | CC-MAIN-2021-04 | application/x-latex | application/x-latex | crawl-data/CC-MAIN-2021-04/segments/1610703550617.50/warc/CC-MAIN-20210124173052-20210124203052-00662.warc.gz | 178,504,223 | 1,374 | %&LaTeX
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\begin{document}
\section*{2019}
Pag{\`e}s, R., et al. "Changes in rivers inputs during the last decades significantly impacted the biogeochemistry of the eastern Mediterranean basin: a modelling study." \textit{Progress in Oceanography} (2019): 102242.
\end{document}
|
https://www.authorea.com/users/408903/articles/518723/download_latex | authorea.com | CC-MAIN-2021-31 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-31/segments/1627046153931.11/warc/CC-MAIN-20210730025356-20210730055356-00294.warc.gz | 667,440,788 | 6,379 | \documentclass[10pt]{article}
\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}
\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
linkcolor = blue,
urlcolor = blue,
citecolor = blue,
anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
\errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother
\usepackage{natbib}
\renewenvironment{abstract}
{{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
{\bigskip}
\titlespacing{\section}{0pt}{*3}{*1}
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\usepackage{authblk}
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\usepackage{longtable}
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\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
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\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{float}
\begin{document}
\title{Statin related musculoskeletal complications; Necrotizing Autoimmune
Myositis\ldots{} more than Myalgia.}
\author[1]{William Scheuing}%
\author[1]{Dadhania Dadhania}%
\author[2]{adegbenga Bankole}%
\affil[1]{Carilion Clinic}%
\affil[2]{VTCSOM}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\selectlanguage{english}
\begin{abstract}
Statins are widely prescribed and well tolerated with most side effects
now considered a nocebo effect. Occasionally, statins can be associated
with immune mediated necrotizing myositis that is both difficult to
diagnose and treat. Aggressive immunosuppressive therapy is the best
recognized method of treatment of this complication.%
\end{abstract}%
\sloppy
Statin related musculoskeletal complications; Necrotizing Autoimmune
Myositis\ldots{} more than Myalgia.
William J. Scheuing MD\textsuperscript{1}, Frany B. Dadhania
MD\textsuperscript{1}, and Adegbenga A. Bankole MD\textsuperscript{2}
1: William J. Scheuing MD \& Frany B. Dadhania MD. Department of
Internal Medicine, Virginia Tech Carilion School of Medicine, 2
Riverside Circle, Roanoke, VA 24016, USA.
2: Adegbenga A. Bankole MD. Department of Internal Medicine, Section of
Rheumatology, Virginia Tech Carilion School of Medicine, 2 Riverside
Circle, Roanoke, VA 24016, USA. Email: aabankole@vt.edu Tel: 540
266-4795 Fax: 540 526-1099 ORCID identifiers: 0000-0001-6464-5367
(Corresponding author)
\emph{}
\emph{Introduction:}
Statins significantly reduce the risk of cardiovascular disease and are
generally considered safe. On rare occasions, statins can cause muscle
disease, and most of these cases recover on discontinuation of the
statin. Even more infrequently, statins can cause statin-associated
necrotizing autoimmune myositis (SANAM) that is characterized by muscle
necrosis on biopsy in the presence of antibodies to
3-hydroxy-3-methylglutaryl coenzyme A (HMG-CoA) reductase. These
patients need to be treated with aggressive immunosuppressive therapy,
but the treatment response is often poor with a variable clinical
response. With the development of newer therapies for dyslipidemia, the
prevalence of SANAM as a disease entity will decrease, making it even
harder to diagnose and treat. We present a typical case of SANAM with a
poor response to aggressive therapy.
\emph{Key Clinical Message:}
SANAM is difficult to diagnose with some patients presenting after
discontinuation of statins. A high degree of suspicion, early referral
to rheumatology and aggressive immunosuppressive therapy is required.
\emph{Case History:}
A 72 year-old man presented to the Emergency Department with a 6-week
history of progressive proximal symmetric muscle weakness. He noted some
difficulty rising from a seated position, climbing stairs, and lifting
up his arms to ninety degrees independently. He had no difficulty
chewing, talking, swallowing, or opening and closing his eyes. He had
diffused joint pain including in his proximal muscle groups in both limb
girdles. He had no rash on his face, chest, back, hands, or on his
eyelids. One week prior to this presentation, he was diagnosed with left
lung basal pneumonia and treated with oral antibiotics. He had fatigue,
malaise, night sweats, and dyspnea on exertion. He did not have
abdominal pain, change in bowel habits, or black or bloody stools. He
did not have dysuria, difficulty voiding, or hematuria. At the time of
admission, he was on metoprolol succinate 50 mg daily, furosemide 20 mg
daily, and aspirin 81 mg daily. He had been on atorvastatin and
sacubitril-valsartan for a number of years, but these medications had
been discontinued at the onset of his muscle weakness.
He has paraoesophageal hiatal hernia, Grover's disease, dyslipidemia,
hypertension, coronary artery disease, heart failure, and atrial
fibrillation. His mother had been diagnosed with dermatomyositis at the
age of 72.
His initial vital signs were normal. He had muscle atrophy in shoulder
and hip muscles, but no atrophy was noted in finger flexors. No muscle
tremors or fasciculations were observed. His right upper extremity
muscle power was 3/5, and 2/5 strength in the left upper extremity. The
power in his left and right hip flexors was 2/5. He had 5/5 power in his
hands and fingers. His deep tendon reflexes were normal. The nail and
nailfold capillaroscopy examination was normal. His joint, pulmonary,
and abdominal examinations were normal.
\textbf{Differential diagnosis, investigations, and treatment:}
The results of his laboratory tests are in Table 1. His blood tests
confirmed an elevated creatine kinase (CK) level, as well as elevations
in other muscle enzymes also. A bilateral quadricep muscle magnetic
resonance imaging (MRI) study was performed (Figure 1), and an MRI
directed muscle biopsy was performed (Figure 2a-c). Based on the muscle
enzymes levels, positive 3-hydroxy-3-methylglutaryl-coenzyme A reductase
antibody (anti-HMGCR Ab) and the results of the muscle biopsy, he was
diagnosed with SANAM.
\textbf{Outcome and follow-up:}
He was started on Prednisone 60 mg per day (approximately 1 mg/kg),
Methotrexate 20 mg weekly, along with folic acid 1 mg daily, with only a
modest improvement in his muscle enzymes and weakness initially. His
muscle weakness worsened as the prednisone was tapered. He was
readmitted to an in-patient rehabilitation facility and his further
changes to his medications. He is currently on a combination of
Prednisone, Methotrexate and Rituximab with both improvement in his
muscle strength and muscle enzymes levels. He has since been discharged
and continues out-patient rehabilitation.
\textbf{Discussion:}
Statins are very commonly prescribed for dyslipidemia and coronary
artery disease. They have anti-inflammatory properties and other
properties that are beneficial in treatment of a wide range of
cardiovascular diseases (5). The side effect profile of statins is very
good, with only mild side effects in most cases. Musculoskeletal side
effects are among the more commonly reported side effects, but recent
studies have shown that most of these complications are a nocebo effect.
Statin use can be associated with SANAM, which is a much more serious
complication. Only a small proportion of patients with SANAM improve
spontaneously, and even with treatment the outcomes can be poor. With
SANAM, prompt and immediate discontinuation of the statin drug is
required if the patient is still being treated with it. Following
discontinuation of the statin, aggressive immunosuppressive treatment is
needed though a clinical response is not always noted. In some cases,
the patients continue to deteriorate. There have being no clinical
trials on treatment protocols for SANAM and clinical and therapeutic
decision is based on case reports, cohort studies and clinical
experience and expertise. Oral prednisone at a dose of 1 mg per kilogram
of body weight per day is usually the initial therapy, with
methotrexate, azathioprine, or mycophenolate mofetil being added as
steroid sparing agents (2). Other therapies such as intravenous immune
globulin or rituximab may be needed if there is persistent muscle enzyme
elevation. Continued muscle weakness may not indicate ongoing muscle
disease as fatty replacement of muscle tissue develops and can cause
ongoing weakness.
Our case is a typical presentation as statin-associated necrotizing
myopathy and also features its poor response to therapy. SANAM must be
considered in the right clinical scenario and if the patients do not
respond as expected. Given the uniqueness and specificity of the
Anti-HMGCR Ab (6), screening patients with this antibody test may
prevent the need for more invasive testing. SANAM is likely to become
less common following the development of monoclonal antibody therapies
that lower cholesterol. This newer class of medication is highly
effective in reduces LDL cholesterol and becoming more commonly
prescribed. Given the current ubiquitous use of statins, SANAM is a
disease entity that all physicians should be aware of, as early
diagnosis allows for early and aggressive treatment that improved the
likely outcomes for the patient.
\emph{Author Contributions:}
William J. Scheuing MD: Performed the literature search, wrote, read,
revised manuscript.
Frany B. Dadhania MD: Performed the literature search, wrote, read,
revised manuscript.
Adegbenga A. Bankole MD: Performed the literature search, wrote, read,
revised and approved the final manuscript.
\emph{Conflict of Interest:}
No conflict of interests were reported by the authors.
\emph{Acknowledgement:}
We thank James W. Mandell, M.D., Ph.D Department of Pathology,
University of Virginia School of Medicine, Charlottesville, VA. for
aiding with the interpretation of the biopsy slides in this case.
\emph{References:}
\begin{enumerate}
\tightlist
\item
Gupta A, Thompson D, Whitehouse A, et al. 2017. Adverse events
associated with unblinded, but not with blinded, statin therapy in the
Anglo-Scandinavian Cardiac Outcomes Trial--Lipid-Lowering Arm
(ASCOT-LLA): a randomised double-blind placebo-controlled trial and
its non-randomised non-blind extension phase. Lancet
2017;389:2473-2481.
\item
Mammen, Andrew L. 2016. Statin-Associated Autoimmune Myopathy N Engl J
Med 2016; 374:664-669
\item
Nazir S, Lohani S, Tachamo N, Poudel D, Donato A. 2017.
Statin-Associated Autoimmune Myopathy: A Systematic Review of 100
Cases. J Clin Rheumatol. 2017 Apr;23(3):149-154.
\item
Gu HM, Zhang DW. 2015. Hypercholesterolemia, low density lipoprotein
receptor and proprotein convertase subtilisin/kexin-type 9. \emph{J
Biomed Res} . 2015;29(5):356-361. doi:10.7555/JBR.29.20150067
\item
Oesterle A, Liao JK. 2019. The Pleiotropic Effects of Statins - From
Coronary Artery Disease and Stroke to Atrial Fibrillation and
Ventricular Tachyarrhythmia. \emph{Curr Vasc Pharmacol} .
2019;17(3):222-232. doi:10.2174/1570161116666180817155058
\item
Lucile Musset, Yves Allenbach, Olivier Benveniste, Olivier Boyer,
Xavier Bossuyt, et al. 2016. Anti-HMGCR antibodies as a biomarker for
immune-mediated necrotizing myopathies: A history of statins and
experience from a large international multi-center study. Autoimmunity
Reviews 15 (2016) 983--993
\end{enumerate}
\textbf{Hosted file}
\verb`Figure 1 MR bilateral lower extremities (femurs) with and without Intravenous Contrast.pdf` available at \url{https://authorea.com/users/408903/articles/518723-statin-related-musculoskeletal-complications-necrotizing-autoimmune-myositis-more-than-myalgia}
\textbf{Hosted file}
\verb`Figure 2A Hematoxylin and eosin stain.pdf` available at \url{https://authorea.com/users/408903/articles/518723-statin-related-musculoskeletal-complications-necrotizing-autoimmune-myositis-more-than-myalgia}
\textbf{Hosted file}
\verb`Figure 2b Immunostain for type 2 (fast) myosin.pdf` available at \url{https://authorea.com/users/408903/articles/518723-statin-related-musculoskeletal-complications-necrotizing-autoimmune-myositis-more-than-myalgia}
\textbf{Hosted file}
\verb`Figure 2c Esterase histochemical stain for esterase activity.pdf` available at \url{https://authorea.com/users/408903/articles/518723-statin-related-musculoskeletal-complications-necrotizing-autoimmune-myositis-more-than-myalgia}
\textbf{Hosted file}
\verb`Table 1.pdf` available at \url{https://authorea.com/users/408903/articles/518723-statin-related-musculoskeletal-complications-necrotizing-autoimmune-myositis-more-than-myalgia}
\selectlanguage{english}
\FloatBarrier
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\iteman{ZMATH 2008e.00251}
\itemau{Ennemoser, Marco; Krajewski, Kristin}
\itemti{Effects of measures to develop understanding of the whole and its parts in mathematically challenged first-graders. (Effekte der F\"orderung des Teil-Ganzes-Verst\"andnisses bei Erstkl\"asslern mit schwachen Mathematikleistungen.)}
\itemso{Vierteljahresschr. Heilp\"adag. Nachbargeb., No. 3, 228-240 (2007).}
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\itemut{dyscalculia; remedial teaching; grade 1; primary education Rechenschw\"ache; F\"orderunterricht; Primarbereich; Mathematik; Intervention}
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\usepackage{minted} % Syntax highlighting via Pygments
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\title{Comparing Floating-Point Numbers Is Tricky}
\author{Matt Kline}
\date{\today}
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\href{mailto:matt@bitbashing.io}{matt@bitbashing.io}
\small
March 30, 2017
\end{center}
\begin{multicols*}{2}
\section{Abstract}
Floating-point math is fraught with subtle gotchas,
and comparing values properly is no exception.
Here we discuss common pitfalls,
examine some possible solutions,
and try to beat Boost.
\section{Things you probably know about floats}
If you need to represent a non-integer in a mainstream programming
language, you'll probably end up using IEEE~754 floating-point values.
Since their standardization in 1985, they've become ubiquitous.
Nearly all modern CPUs---and many microprocessors---contain special hardware
(called \emph{floating-point units}, or FPUs) to handle them.
\begin{center}
\def\svgwidth{\columnwidth}
\tiny
\input{double.pdf_tex}
\footnotesize The layout of a 64-bit IEEE~754 float \\
(from \href{https://commons.wikimedia.org/wiki/File:IEEE_754_Double_Floating_Point_Format.svg}{Wikipedia})
\end{center}
Each float consists of a sign bit, some bits representing an exponent,
and bits representing a fraction, also called the \emph{mantissa}.
Under most circumstances, the value of a float is:
\[ (-1)^s \times 1.m \times 2^{e - c}\]
where $s$ is our sign bit, $m$ is some fraction represented by the mantissa bits,
$e$ is an unsigned integer represented by the exponent bits,
and $c$ is half the maximum value of $e$, i.e.,
127 for a 32-bit float and 1023 for a 64-bit float.
There are also some special cases.
For example, when all exponent bits are zero, the formula changes to:
\[ (-1)^s \times 0.m \times 2^{-c + 1} \]
Note the lack of an implicit 1 preceding the mantissa---this allows us to store
small values close to zero, called
\emph{denormal} or \emph{subnormal} values.
And when all exponent bits are one, certain mantissa values represent
$+\infty$, $-\infty$, and ``Not a Number'' (NaN), the result of undefined
or unrepresentable operations such as dividing by zero.
We'll make two observations that will prove themselves useful shortly:
\begin{enumerate}
\item Floats cannot store arbitrary real numbers,
or even arbitrary rational numbers.
They can only store numbers representable by the equations shown before.
For example, if I declare some variable,
\begin{minted}[autogobble, fontsize=\codesize]{cpp}
float f = 0.1f;
\end{minted}
\texttt{f} becomes 0.100000001490116119384765625, the closest 32-bit float
value to 0.1.
\item Since the equations are exponential, the distance on the number line
between adjacent values increases (exponentially!) as you move away
from zero.
The distance between 1.0 and the next possible value is about
$1.19 \times 10^{-7}$, but the distance between adjacent floats near
$6.022 \times 10^{23}$ is roughly $3.6 \times 10^{16}$.
This will prove to be our greatest challenge: when comparing floats,
we want to handle inputs close to zero as well as we handle ones
close to the Avogadro constant.
\end{enumerate}
\section{What is equality?}
Since the result of every floating-point operation must be rounded to the
nearest possible value, math doesn't behave like it does with real numbers.
Depending on your hardware, compiler, and compiler flags,
$0.1 \times 10$ may produce a different result than
$\sum_{n=1}^{10} 0.1$.\punckern\footnote{On my setup,
the former gives 1.0 and the latter gives 1.000000119209290.}
Whenever we compare calculated values to each other,
we should provide some leeway to account for this.
Comparing their exact values with \texttt{==} won't cut it.
Instead,
we should consider two distinct values $a$ and $b$ equal if
$|a-b| \leq \epsilon$ for some sufficiently small $\epsilon$.
As luck would have it, the C standard library contains a \monobox{FLT\_EPSILON}.
Let's try it out!
\begin{minted}[fontsize=\codesize]{cpp}
bool almostEqual(float a, float b)
{
return fabs(a - b) <= FLT_EPSILON;
}
\end{minted}
We would hope that we're done here, but we would be wrong.
A look at the language standards reveals that
\monobox{FLT\_EPSILON} is equal to
the difference between 1.0 and the value that follows it.
But as we noted before, float values aren't equidistant!
For values less than 1, \monobox{FLT\_EPSILON} quickly becomes too large
to be useful.
For values greater than 2, \monobox{FLT\_EPSILON} is smaller than the
distance between adjacent values, so
\mintinline{cpp}{fabs(a - b) <= FLT_EPSILON}
will always be false.
To address these problems,
what if we scaled $\epsilon$ proportionally to our inputs?
\begin{minted}[fontsize=\codesize]{cpp}
bool relativelyEqual(float a, float b,
float maxRelativeDiff = FLT_EPSILON)
{
const float difference = fabs(a - b);
// Scale to the largest value.
a = fabs(a);
b = fabs(b);
const float scaledEpsilon =
maxRelativeDiff * max(a, b);
return difference <= scaledEpsilon;
}
\end{minted}
This works better than our initial solution,
but it's not immediately obvious what values of
\monobox{maxRelativeDiff} we might want for different cases.
The fact that we scale it by arbitrary inputs also means it can fall prey
to the same rounding we're worried about in the first place.
\section{What about Boost?}
Boost, the popular collection of \cpp{} libraries,
provides functions for similar purposes.\punckern\footnote{See
\href{http://www.boost.org/doc/libs/1_63_0/libs/math/doc/html/math_toolkit/float_comparison.html}%
{\textit{Floating-point Comparison}} in the floating-point utilities section of
Boost's Math toolkit.}
After removing template boilerplate and edge case handling for $\pm\infty$ and NaNs,
they resemble:
\begin{minted}[fontsize=\codesize]{cpp}
float relative_difference(float a, float b)
{
return fabs((a - b) / min(a, b));
}
float epsilon_difference(float a, float b)
{
return relative_difference(a, b) /
FLT_EPSILON;
}
\end{minted}
Unfortunately, these functions don't seem to solve
our problems.\punckern\footnote{Boost libraries
are usually high-quality and thoroughly reviewed,
so please contact me if I've missed some critical observation.}
Since the division in \monobox{relative\_difference} often makes its result
quite small,\punckern\footnote{For example,
the \monobox{relative\_difference} between 42 and
the next float value is about $9.08 \times 10^{-8}$.}
how do we know what a good threshold might be?
By dividing that result by \monobox{FLT\_EPSILON},
\monobox{epsilon\_difference} attempts to give an easier value to reason about.
But we just saw the dangers of \monobox{FLT\_EPSILON}\,!
This scheme becomes increasingly questionable as inputs move away from one.
\section{What about ULPs?}
It would be nice to define comparisons in terms of
something more concrete than arbitrary thresholds.
Ideally, we would like to know the number of possible floating-point
values---sometimes called \emph{units of least precision}, or ULPs---between
inputs.
If I have some value $a$, and another value $b$ is only two or three ULPs away,
we can probably consider them equal, assuming some rounding error.
Most importantly, this is true regardless of the distance between
$a$ and $b$ on the number line.
Boost offers a function called \monobox{float\_distance} to get the distance
between values in ULPs,
but it's about an order of magnitude slower than the approaches
discussed so far.
With some bit-fiddling, we can do better.
Consider some positive float $x$ where every mantissa
bit is one.
$x + 1\text{\textsc{ulp}}$ must use the next largest exponent,
and all its mantissa bits must be zero.
As an example, consider 1.99999988 and 2:
\columnbreak
\begin{center}
\small
\begin{tabular}{c|c|c|c}
Value & Bits & Exponent & Mantissa bits\\
\hline
1.99999988 & \texttt{0x3FFFFFFF} & 127 & \texttt{0x7FFFFF} \\
\hline
2.0 & \texttt{0x40000000} & 128 & \texttt{0x000000} \\
\end{tabular}
\end{center}
The property holds for denormals, even though they have a different
value equation.
Consider the largest denormal value and the smallest normal one:
\begin{center}
\small
\renewcommand{\arraystretch}{1.2} % For taller table rows
\setlength{\tabcolsep}{5pt} % For less horizontal margin
\begin{tabular}{c|c|c|c}
Value & Bits & Exp. & Man.~bits\\
\hline
$1.1754942 \times 10^{-38}$ & \texttt{0x007FFFFF} & $-126$ & \texttt{0x7FFFFF} \\
\hline
$1.17549435 \times 10^{-38}$ & \texttt{0x00800000} & $-126$ & \texttt{0x000000} \\
\end{tabular}
\end{center}
Notice an interesting corollary:
adjacent floats (of the same sign)
have adjacent integer values when reinterpreted as such.
This reinterpretation is sometimes called \emph{type punning},
and we can use it to calculate the distance between values in ULPs.
Traditionally in C and \cpp{}, one used a union trick:
\begin{minted}[fontsize=\codesize]{cpp}
union FloatPun {
float f;
int32_t i;
};
FloatPun fp;
fp.f = 25.624f;
// Read the same value as an integer.
printf("%x", fp.i);
\end{minted}
This still works in C,
but can run afoul of strict aliasing rules in
\cpp{}.\punckern\footnote{See \url{http://stackoverflow.com/q/11639947}
and \url{http://stackoverflow.com/q/17789928}.}
A better approach is to use \monobox{memcpy}.
Given the usual use of the function, one might assume that it would be less
efficient, but
\begin{minted}[fontsize=\codesize]{cpp}
int32_t floatToInt(float f)
{
int32_t r;
memcpy(&r, &f, sizeof(float));
return r;
}
\end{minted}
compiles to a single instruction that moves the value from a floating-point
register to an integer one.
This is exactly what we want.
With that problem solved, calculating the ULPs between values
becomes quite straightforward:
\begin{minted}[fontsize=\codesize]{cpp}
int32_t ulpsDistance(const float a, const float b)
{
// Save work if the floats are equal.
// Also handles +0 == -0.
if (a == b) return 0;
const auto max =
std::numeric_limits<int32_t>::max();
// Max distance for NaN
if (isnan(a) || isnan(b)) return max;
// If one's infinite and they're not equal,
// max distance.
if (isinf(a) || isinf(b)) return max;
int32_t ia, ib;
memcpy(&ia, &a, sizeof(float));
memcpy(&ib, &b, sizeof(float));
// Don't compare differently-signed floats.
if ((ia < 0) != (ib < 0)) return max;
// Return the absolute value of
// the distance in ULPs.
int32_t distance = ia - ib;
if (distance < 0) distance = -distance;
return distance;
}
\end{minted}
This code is quite portable---it only assumes that the platform supports
32-bit integers and that floats are stored in accordance with
IEEE~754.\punckern\footnote{The format of \mintinline{cpp}{float} is
implementation-defined according to the \cpp{} standard,
and not necessarily adherent to IEEE~754.
(See \url{http://stackoverflow.com/a/24157568}.)
Perhaps this is why Boost's \monobox{float\_distance} is implemented the way it is.}
We avoid comparing differently-signed values for a few reasons:
\begin{enumerate}
\item ULPs are the wrong tool to compare values near or across zero,
as we'll see below.
\item Almost all modern CPUs use \href{https://en.wikipedia.org/wiki/Two%27s_complement}{two's complement}
arithmetic, while floats use
\href{https://en.wikipedia.org/wiki/Signed_number_representations#Signed_magnitude_representation}{signed magnitude}.
Converting one format to the other in order to meaningfully add or subtract
differently-signed values requires some extra work.
For the same reason, the sign of our result might not be what we expect,
so we take its absolute value.
We only care about the distance between our two inputs.
\item If the subtraction overflows or underflows, we get undefined behavior
with signed integers and modular arithmetic with unsigned ones.
Neither is desirable here.
\end{enumerate}
We calculate the absolute value ourselves instead of using \monobox{std::abs}
for two reaons.
First, the integer versions of \monobox{std::abs} only take types---such as
\mintinline{c}{int}, \mintinline{c}{long}, and \mintinline{c}{long long}---whose
sizes are platform-specific.
We want to avoid assumptions about implicit conversions
between those types and \mintinline{c}{int32_t}.\punckern\footnote{Granted,
this is borderline paranoia---\mintinline{c}{int32_t} is one of those types
on nearly every relevant platform.}
The second is a strange pitfall related to the placement of
\monobox{std::abs} overloads in the \cpp{} standard library.
If you include \monobox{<cmath>} but not \monobox{<cstdlib>},
only the floating-point versions of \monobox{std::abs} are provided.
Several toolchains I tested then promote the \mintinline{cpp}{int32_t}
value to a \mintinline{c}{double},
even if your target only has
a 32-bit FPU and must emulate \mintinline{c}{double}
using integer registers.
(As one might guess, this is \emph{terrible} for performance.)
Warning flags such as \monobox{-Wconversion} can help us notice this happening,
or we can just avoid all these gotchas by calculating the absolute value directly.
At any rate, this is a trivial detail.
\section{No silver bullets}
Relative epsilons---including ULPs-based ones---don't make sense around zero.
The exponential nature of floats means that many more values are
gathered there than anywhere else on the number line.
Despite being a fairly small value in the context of many calculations,
0.1 is over one billion ULPs away from zero!
Consequently, fixed epsilons are probably the best
choice when you expect the results to be small.
What particular $\epsilon$ you want is entirely dependent on the calculations
performed.
Armed with this knowledge,
you may be tempted to write some end-all comparison function along the lines of:
\begin{minted}[fontsize=\codesize]{cpp}
bool nearlyEqual(float a, float b,
float fixedEpsilon, int ulpsEpsilon)
{
// Handle the near-zero case.
const float difference = fabs(a - b);
if (difference <= fixedEpsilon) return true;
return ulpsDistance(a, b) <= ulpsEpsilon;
}
\end{minted}
But using it meaningfully is difficult without understanding the theory
we've discussed.
\section{Brief aside: Other ULPs-based functions}
We can use the same techniques to write other useful functions,
such as one that increments a float by some number of ULPs.
Boost offers a similar family of functions
(\monobox{float\_next}, \monobox{float\_advance}, etc.),
but like its \monobox{float\_distance}, they pay a performance cost to avoid
type punning.
One would hope we could simply get our ULPs, perform our addition,
and pun the result back, e.g.,
\begin{minted}[fontsize=\codesize]{cpp}
/// Increases f by the given number of ulps
float ulpsIncrement(float f, int32_t ulps)
{
if (isnan(f) || isinf(f)) return f;
int32_t i;
memcpy(&i, &f, sizeof(float));
i += ulps;
memcpy(&f, &i, sizeof(float));
return f;
}
\end{minted}
This naïve solution works for positive values,
but on most hardware, ``incrementing'' a negative float by a positive number of
ULPs will move us away from zero!
This is probably not what we want.
We mentioned before that floats use a signed magnitude scheme,
whereas most CPUs use two's complement.
So, to operate on negative values, we need to convert from the former
to the CPU's native integer format.\punckern\footnote{On esoteric hardware,
the native format may also be signed magnitude.
In those cases, we trust the compiler to elide the needless work we do here.}
\begin{minted}[fontsize=\codesize]{cpp}
static const int32_t int32SignBit =
(int32_t)1 << 31;
\end{minted}
\begin{minted}[fontsize=\codesize]{cpp}
int32_t floatToNativeSignedUlps(float f)
{
int32_t i;
memcpy(&i, &f, sizeof(float));
// Positive values are the same in both
// two's complement and signed magnitude.
// For negative values, remove the sign bit
// and negate the result (subtract from 0).
return i >= 0 ? i : -(i & ~int32SignBit);
}
\end{minted}
After operating on the ULPs,
we must convert back to signed magnitude:
\begin{minted}[fontsize=\codesize]{cpp}
float nativeSignedUlpsToFloat(int32_t ulps)
{
if (ulps < 0) {
ulps = -ulps;
ulps |= int32SignBit;
}
float f;
memcpy(&f, &ulps, sizeof(float));
return f;
}
\end{minted}
With those functions defined,
we can return to our goal:
\begin{minted}[fontsize=\codesize]{cpp}
float ulpsIncrement(float f, int32_t ulps)
{
if (isnan(f) || isinf(f)) return f;
int32_t i = floatToNativeSignedUlps(f);
i += ulps;
return nativeSignedUlpsToFloat(i);
}
\end{minted}
\section{Takeaways}
When comparing floating-point values, remember:
\begin{itemize}
\item \monobox{FLT\_EPSILON}\ldots{} isn't float epsilon,
except in the ranges $[-2, -1]$ and $[1, 2]$.
The distance between adjacent values depends on the values in question.
\item When comparing to some known value---especially zero or values near
it---use a fixed $\epsilon$ that makes sense for your calculations.
\item When comparing non-zero values, some ULPs-based comparison is probably the
best choice.
\item When values could be anywhere on the number line,
some hybrid of the two is needed.
Choose epsilons carefully based on expected outputs.
\end{itemize}
\section{Acknowledgments}
Much of this was adapted from Bruce Dawson's \emph{fantastic}
exploration of the topic on his blog,
\href{https://randomascii.wordpress.com}{Random ASCII}.
Thanks also to
coworkers Evan Thompson and Matt Drees for their input.
\end{multicols*}
\newpage
\section{Appendix: Performance concerns}
The relatively poor performance of \monobox{boost::float\_distance}
was a large motivation for implementing
our own \monobox{ulpsDistance}.
For the sake of completeness, the following is a benchmark
(using \href{https://github.com/google/benchmark}{Google's benchmark library})
comparing the two with a handful of inputs.
\rule{0.9\textwidth}{0.5pt}
\begin{minted}[fontsize=\codesize]{cpp}
#include <cstring> // For memcpy
#include <limits> // for numeric_limits<float>::infinity
#include <random>
#include <benchmark/benchmark.h>
#include <boost/math/special_functions/next.hpp>
#include <boost/math/special_functions/relative_difference.hpp>
using namespace std;
using namespace boost::math;
std::pair<float, float> pickInput()
{
static auto re = mt19937(random_device()());
static auto coinFlip = bernoulli_distribution(0.5);
static auto inputPicker = uniform_int_distribution<int>(1, 10);
const float infinity = numeric_limits<float>::infinity();
switch(inputPicker(re)) {
// Let's say there's a 5% chance our values are denormal.
// (This is probably more pessimal than our actual data.)
case 1:
if (coinFlip(re)) return {1e-38f, float_advance(1e-38f, 3)};
// Intentional fall-through
// Let's throw in some huge numbers
case 2:
case 3:
case 4:
case 5:
return {6.022e23f, 2.998e8f};
break;
// And so not-so-huge ones.
case 6:
case 7:
case 8:
case 9:
return {1.0f, 11.0f};
// Let's say there's a 5% chance we have NaNs
// and another 5% chance they're infinity
case 10:
if (coinFlip(re)) return {42, numeric_limits<float>::quiet_NaN()};
else return {42, infinity};
default: assert(0);
}
}
__attribute__((noinline)) // For visibility when benchmarking
int32_t ulpsDistance(const float a, const float b)
{
// We can skip all the following work if they're equal.
if (a == b) return 0;
const auto max = numeric_limits<int32_t>::max();
// We first check if the values are NaN.
// If this is the case, they're inherently unequal;
// return the maximum distance between the two.
if (isnan(a) || isnan(b)) return max;
// If one's infinite, and they're not equal,
// return the max distance between the two.
if (isinf(a) || isinf(b)) return max;
// At this point we know that the floating-point values aren't equal and
// aren't special values (infinity/NaN).
// Because of how IEEE754 floats are laid out
// (sign bit, then exponent, then mantissa), we can examine the bits
// as if they were integers to get the distance between them in units
// of least precision (ULPs).
static_assert(sizeof(float) == sizeof(int32_t), "What size is float?");
// memcpy to get around the strict aliasing rule.
// The compiler knows what we're doing and will just transfer the float
// values into integer registers.
int32_t ia, ib;
memcpy(&ia, &a, sizeof(float));
memcpy(&ib, &b, sizeof(float));
// If the signs of the two values aren't the same,
// return the maximum distance between the two.
// This is done to avoid integer overflow, and because the bit layout of
// floats is closer to sign-magnitude than it is to two's complement.
// This *also* means that if you're checking if a value is close to zero,
// you should probably just use a fixed epsilon instead of this function.
if ((ia < 0) != (ib < 0)) return max;
// If we've satisfied all our caveats above, just subtract the values.
// The result is the distance between the values in ULPs.
int32_t distance = ia - ib;
if (distance < 0) distance = -distance;
return distance;
}
void benchFloatDistance(benchmark::State& state)
{
while (state.KeepRunning()) {
state.PauseTiming();
float a, b;
std::tie(a, b) = pickInput();
state.ResumeTiming();
// float_distance can't handle NaN and Infs.
if (!isnan(a) && !isnan(b) && !isinf(a) && !isinf(b)) {
benchmark::DoNotOptimize(float_distance(a, b));
}
}
}
BENCHMARK(benchFloatDistance);
void benchUlps(benchmark::State& state)
{
while (state.KeepRunning()) {
state.PauseTiming();
float a, b;
std::tie(a, b) = pickInput();
state.ResumeTiming();
benchmark::DoNotOptimize(ulpsDistance(a, b));
}
}
BENCHMARK(benchUlps);
BENCHMARK_MAIN();
\end{minted}
\rule{0.9\textwidth}{0.5pt}
On my laptop (an Intel Core i7 Skylake), I get:
{\codesize\selectfont
\begin{verbatim}
Benchmark Time CPU Iterations
---------------------------------------------------------
benchFloatDistance 717 ns 836 ns 850424
benchUlps 157 ns 176 ns 3914780
\end{verbatim}
}
And on an ARMv6 board we use at work for embedded Linux platforms, I get:
{\codesize\selectfont
\begin{verbatim}
Benchmark Time CPU Iterations
---------------------------------------------------------
benchFloatDistance 43674 ns 42609 ns 16646
benchUlps 4748 ns 4602 ns 151382
\end{verbatim}
}
Actual timing values obviously depend on the type of inputs,
the uniformity of the inputs (which influences branch prediction),
and \emph{many} other factors,
but our function seems to outperform Boost alternatives in the general case.
\end{document}
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http://theanarchistlibrary.org/library/max-nettlau-an-anarchist-manifesto.tex | theanarchistlibrary.org | CC-MAIN-2019-30 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2019-30/segments/1563195527000.10/warc/CC-MAIN-20190721123414-20190721145414-00057.warc.gz | 147,800,321 | 15,941 | \documentclass[DIV=12,%
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\title{An Anarchist Manifesto}
\date{1\textsuperscript{st} May, 1895}
\author{Max Nettlau}
\subtitle{}
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\par
Fellow Workers,
We come before you as Anarchist Communists to explain our principles. We are aware that the minds of many of you have been poisoned by the lies which all parties have diligently spread about us. But surely the persecutions to which we have been and are subjected by the governing classes of all countries should open the eyes of those who love fair play. Thousands of our comrades are suffering in prison or are driven homeless from one country to the other. Free speech — almost the only part of British liberty that can be of any use to the people — is denied to us in many instances, as the events of the last few years have shown.
The misery around us is increasing year by year. And yet there was never so much talk about labor as there is now, — labor, for the welfare of which all professional politicians profess to work day and night. A very few sincere and honest but impracticable reformers, in company with a multitude of mere quacks, ambitious placehunters, etc., say they are able to benefit labor, if labor will only follow their useless advice. All this does not lessen the misery in the least : look at the unemployed, the victims of hunger and cold, who die every year in the streets of our rich cities, where wealth of every description is stored up.
Not only do they suffer who are actually out of work and starving, but every working man who is forced to go through the same dreary routine day by day — the slavery and toil in the factory or workshop — the cheerless home, if the places where they are forced to herd together can be called homes. Is this life worth living? What becomes of the intellectual faculties, the artistic inclinations, nay, the ordinary human feeling and dignity of the greatest part of the workers? All these are warped and wasted, without any chance of development, making the wretched worker nothing but a human tool to be exploited until more profitably replaced by some new invention or machine.
Is all this misery necessary? It is not if you, the wealth producers, knew that there is enough and to spare of food and of the necessaries of life for all, if all would work. But now, in order to keep the rich in idleness and luxury, all the workers must lead a life of perpetual misery and exploitation. As to these facts we are all agreed; but as to the remedy most of you, unfortunately, have not given up trust in Parliament and the State. We shall explain how the very nature of the State prevents anything good coming from it. What does the State do? It protects the rich and their ill-gotten wealth; it suppresses the attempts of the workers to recover their rights, if these attempts are thought dangerous to the rich. Thus idle electioneering, labor politics etc. are not suppressed, but any effective popular demonstration, vigorous strikes as at Featherstone and Hull, Anarchist propaganda, etc., are suppressed or fought against by the vilest means. Moreover, the State pretending thereby to alleviate the sufferings of the poor, grants Royal Commissions on the Sweating System, the Aged Poor, on Labor in general, or select Committees on the Unemployed — which produce heaps of Blue Books, and give an opportunity to the politicians and labor leaders, “to show themselves off.” And that is about all. If the workers demand more — there is the workhouse; and if not satisfied with that, the truncheons of the police and the bullets and bayonets of the soldiers face them: — not bread, but lead!
All political prisoners are of the same value: either they are not kept, even if it could be, or they involve social changes which can only be effected by a revolution, and not by mere votes cast in Parliament. This applies to the promises of Socialist candidates, even if it could be admitted that these candidates could remain uncorrupted by the demoralizing influence of Parliament.
There can be no true humanity, no true self-respect, without self- reliance. No one can help you if you do not help yourselves. We do not promise to do anything for you, we do not want anything from you, we only appeal to you to co-operate with us to bring about a state of society which will make freedom, well-being possible for all.
To do this efficiently, we must all be imbued with the spirit of freedom, and this — freedom, and freedom alone — is the fundamental principle of Anarchy.
Freedom is a necessary condition to, and the only guarantee of, the proper development of mankind. Nature is most beautiful when unfettered by the artificial interference of man. Wild animals are stronger and more harmoniously developed than their domesticated kind, which the exploiting mind of man makes mere instruments of profit by developing chiefly those parts of them which are of use to him. The same threatens to be the case with the human victims of exploitation, if an end is not put to the system which allows the rich and crafty exploiters to reduce the greater part of mankind to a position resembling that of domestic animals — working machines, only fit to do mechanically a certain kind of work, but becoming intellectually wrecked and ruined.
All who acknowledge this to be the great danger to human progress should carefully ponder over it, and if they believe that it is necessary to ensure by every means the free development of humanity, and to remove by all means every obstacle placed in its path, they should join us and adopt the principles of Anarchism.
Belief in and submission to authority is the root cause of all our misery. The remedy we recommend: — struggle unto death against all authority, whether it be that of physical force identical with the State or that of doctrine and theories, the product of ages of ignorance and superstition inculcated into the workers minds from their childhood — such as religion, patriotism, obedience to the law, belief in the State, submission to the rich and titled, etc., generally speaking, the absence of any critical spirit in face of all the humbugs who victimise the workers again and again. We can only deal here briefly with all these subjects, and must limit ourselves to touch only on the chief points.
Economic exploitation — the result of the monopolisation of the land, raw materials and means of production by the capitalists and landlords — is at the bottom of the present misery. But the system which produces it would have long ago broken down if it were not upheld on one hand by the State, with its armies of officials, soldiers and police — the whole machinery of government, in one word; and on the other hand by the workers themselves, who tamely submit to their own spoliation and degradation, because they think it right, owing to a superstitious belief in a divine providence inculcated by their masters, or because they desire, by sneaking means, to become exploiters themselves — an object which only one in a thousand can succeed in — or because they have not lost faith in political action or the capacity of the State to do for them that which they are too ignorant to do for themselves. Under these protections the rich classes are enjoying their spoil in safety and comfort.
It is evident that this system, if to be destroyed at all, must be attacked by the workers themselves, as we cannot expect those who profit by it to cut their own throats, so to say.
Many still consider the State a necessity. Is this so in reality? The State, being only a machine for the protection and preservation of property, can only obstruct freedom and free development, being bound to
keep up the law and every statute law is an obstacle to progress and freedom.
Laws are of two kinds. They are either simple formulae, derived from the obsevation of phenomena as the so-called laws of nature, the phrasing of which is open to revision with the progress of human know-ledge and the accumulation of fresh material to draw dedcutions from. No authority is required to enforce them, they exist; and every being arranges his conduct in conformity with his knowledge of their action. The phenomenon of fire burning is the result of such a natural law, and all pay attention to it though there is no policeman posted behind every match and fireplace. Here again Nature gives us an example of free development and Anarchy, and in a free society all social facts and necessities would be equally well recognised and acted upon.
But there is the other kind of law. That which is the expression of the will of an unsrupulous minority, who, owing to the apathy and ignorance of the majority, have been able to usurp the means of power and purport to represent the whole people at the time of the enaction of the laws.
The fact that a great number of persons is in favor of something is evidently no guarantee that it is right. Experience, on the contrary, shows that progress is usually brought about by individuals. New discoveries, new lines of human activity are first found and practised by a few, and only gradually adopted by the many. The majority that makes the laws or abides to them will almost always lag behind progress, and the laws made by it will be reactionary from the very beginning. How much more so as time proceeds and new progress is made!
Of course, progress itself laughs at the puny efforts of the usurpers of power to stop its triumphant march. But its apostles and advocates have to suffer much and severely for the enthusiasm and the hope that is within them. Prison and often death itself is their doom; the penalty for having raised the standard of revolt against authority and law, the embodiment of the spirit of oppression.
And the very makers of these laws are forced to admit that their work is useless. Is not the continuous manufacture of new laws going on in the Parliaments of all countries throughout the greater part of this century, and in England for many centuries, a proof of the fact that laws never satisfy anybody, not even those who make them. They know, however, that their legislating is mere mockery and hypocrisy, having no other object but to make the people believe that something is being done for them, and that the public interest is well looked after. The people obey all these laws, whilst the State, in the alleged interest of all, in reality in the interest of the property owners and of its own power, violates them all and commits numberless crimes — which are glorified as deeds of valor committed in the interest of civilisation.
This principle, kept in the background in time of peace, is paraded before the eyes of the people in time of war. A trading company acquiring so-called “rights” in some savage territory, plunders and provokes the natives until they return force by force. Then the State steps in, in the pretended interest of religion and civilisation, slaughters them and annexes their land. The greater the slaughter, the greater the glory for these “heroic” pioneers. Or it may be in a war on a greater scale with a European State, when the workers of one country are let loose against those of another, to murder, plunder and burn homes and villages, and perform such like patriotic deeds of valor and chivalry.
We Anarchists are internationalists, we acknowledge no distinction of nationality or color. The workers of all countries suffer as we do here, and our comrades have everywhere to fight the same battle for freedom and justice. The capitalists are internationally unanimous in persecuting the defenders of freedom and in fleecing the workers. Even England is brought more and more under the sway of a continental police system, the dangers of which the British masses do not see at present, as it is used chiefly against friendless foreign refugees. They are regardless fo the fact that it is but the forerunner of an attack on their own liberties.
The workers as a rule are filled with an unreasoning dislike to the workers of other countries, whom their masters have succeeded in representing to them as their natural enemies, and herein lies one of the main sources of the strength of the capitalist system; a strength which has no other foundation than the weakness and helplessness of the people. It is in the interests of all governments to uphold patriotism, to have their own people ready to fly at the throats of their fellow workers of other nationalities whenever it suits the interests of the employers to open up new markets, or draw the attention of the people away from the contemplation of their own misery, which might drive them to revolt.
Patriotism and religion have always been the first and last refuges and strongholds of scoudrels. The meek and lowly servants of the one blessing — in the name of their God — the infamies committed for the sake of the other, and cursing in the same name the deeds they just now blessed if committed by the enemy.
Religion is mankind’s greatest curse! It is absurd to expect that science, in the few years that the State and the priests have left it to a certain extent alone — the stakeor the prison has been too often the reward of its pioneers — should have discovered everything. It would not be worth living in a world where everything had been discovered, analysed and registered. One fact is certain: all so-called religions are the products of human ignorance, mere phantastical efforts of barbarous people to reason out matters which they could not possibly understand without some knowledge of science and scientific methods. The opinion of a savage on the power that works a steam engine, or produces the electric light, is evidently worthless and could be refuted by anyone possessing elementary knowledge. In the same worthless way our forefathers, savages also, reasoned about the phenomena of nature, and came to the naive conclusion that somebody behind the curtains of the sky pulled the strings. This supposed individual they called God and the organic force of man the soul, and endowed it with a separate entity, although that organic force does not possess any more separate entity than that working a clock or a steam hammer. A dim consciousness of this has permeated the mind of most in spite of the fact that religion has been bolstered up by all the forces of authority, because it teaches submission to the law, and as a reward gives cheques drawn on the bank of heaven, which are not more likely to be met than the politician’s promises of what he will do when he is returned for Parliament. Religion is the most deadly enemy to human progress. It has always been used to poison the mind and deaden the judgment of the young, thus making grown up people accept all its absurdities because they are familiarised with them in their youth.
Unfortunately, religion is not kept out of the labor movement. Priests and parsons, who should be a horror to mankind, as their presence adds an additional element of corruption, sneak into it, and labor politicians use their services as the Liberals and Tories do. There is actually in existence a body of persons who prostitute the noble word “Labor” by coupling it with the disgusting word “Church”, forming the “Labor Church”, which is looked upon favorably by most of the prominent labor leaders. Why not start a “Labor Police”?
We are Atheists\footnote{This open statement of our convictions does not imply any spirit of persecution on our part against those who believe in the absurdities of the different religions. Persecution is essential to authority and religion, and fatal to freedom; we should destroy the basis of our own hopes and ideals, if we were ever carried away by the spirit of persecution, bigotry and intolerance, which is so commonly raised against us.} and believe that man cannot be free if he does not shake off the fetters of the authority of the absurd as well as those of every other authority. Authority assumes numerous shapes and disguises, and it will take a long period of development under freedom to get rid of all. To do this two things are wanted, to rid ourselves of all superstition and to root out the stronghold of all authority, the State.
We shall be asked what we intend to put in place of the State. We reply, “Nothing whatever!” The State is simply an obstacle to progress; this obstacle once removed we do not want to erect a fresh obstruction.
In this we differ essentially from the various schools of State Socialists, who either want to transform the present State into a benevolent public-spirited institution (just as easy as to transform a wolf into a lamb), or to create a new centralised organisation for the regulation of all production and consumption, the so-called Socialist society. In reality this is only the old State in disguise, with enormously strengthened powers. It would interfere with everything and would be the essence of tyranny and slavery, if it could be brought about. But, thanks to the tendency of the ways and means of production — which will lead to Anarchy — it cannot.
But whilst State Socialism is impracticable as a system of real Socialism, it is indeed possible if its advocates had their way, that all matters of general interest and more and more of private interest too would pass under the control of the State; whether it be a little more democratised or not, it does not matter, for we reject Democracy as well as Absolutism. Authority is equally hateful to us whether exercised by many, or by few, or by one. The last remnant of free initiative and self-reliance would be crushed under the hells of the State, and the emancipation of the workers would be far off as ever. State Socialism has indeed strengthened the decaying faith in, and renewed the prestige of, the State.
All we Anarchists want is equal freedom for all. The workers to provide for their own affairs by voluntary arrangements amongst themselves. This leads us to a consideration of the economic basis of the state of things we desire to bring about, and here we avow ourselves Communists.
Everybody has different faculties and abilities for work, and different wants and desires for the various necessities of life and leisure. These inclinations and wants require full satisfaction, but can only receive it in a state of freedom. Everybody supposing his faculties to be properly developed can best judge what is best for himself. Rules and regulations would hinder and make him a fettered, incomplete being who necessarily finds no pleasure in work forced upon him. But under Anarchy he would associate voluntarily with others to do the work he is best fitted to do, and would satisfy his wants in proportion to his needs from the common stock, the result of their common labor.
Cut-throat competition for the bare necessities of life would be done away with, leaving many matters of a more individual, private and intimate character, in which the free man would find opportunity for peaceful and harmonious emulation, and thereby develop his faculties in the highest possible degree.
One of the stock objections against Anarchist Communism is that no one would work. We reply that to- day work is viewed with disfavor and neglected by all who can possibly exist without it because it has to be carried on under the most disadvantageous conditions and is, moreover, looked upon as degrading. The worker earning his food by hard labor and ceaseless toil is a pariah, the outcast of society, while the idler who never does an hours work in his life is admired and glorified, and spends his days in luxurious ease amongst pleasant surroundings. We believe that under Anarchism everybody would be willing to work; work being freed from the badge of dishonor now associated with it will have become a labor of love, and the free man will feel ashamed to eat food he has not earned. But as to some atavistic remnants of modern capitalist society that would only work if forced? Well, nobody would want us to retard the emancipation of the immense mass of mankind on account of these few unsocial beings who may or may not exist then. Left to themselves and scorned by everyone they would soon come to their senses and work.
We cannot further enter here into the arguments which show the tendency of a development into Free Communism, and we refer to our literature on the subject. (See Kropotkin’s “Anarchism: its Basis and Principles.” Freedom Pamphlets, No. 4, etc.)
Anarchist society will consist of a great number of groups devoted each to the production of certain commodities free of access to all, and in local and interlocal contact with other groups to agree and make arrangements for purposes of exchange. With regard to the first necessities of life, food, clothes, shelter, education, Free Communism would be carried out thoroughly. All secondary matters would be left to a mutual agreement in the most varied ways. There would remain in such a society full freedom for the Individualist as long as he did not develop any monopolistic tendencies.
These are our principles; let us consider the means to realise them.
Here we are met by the cry “Dynamiters”, “Assassins”, “Fiends”, etc. Let us see who chiefly utter these cries.
The same people who, by colliery disasters, the ensuring of rotten ships, fires in death-trap-houses, railway accidents caused by overwork, etc., daily massacre more people than the Anarchists of all countries ever killed. The same people who are ready at any moment to have the natives of any country slaughtered, simply to rob them, who are overjoyed at the butchery of the Chinese War, which will enable them to make fresh profit, who are slowly starving and killing the millions of workers, whose lives are shortened by overwork, adulterated food, and overcrowding slums. These people have, in our eyes, no voice when the question of humanity is considered. They may abuse and insult us just as they like. The worst thing that could happen to us, indeed, would be to win their approbation, to be petted by them as the respectable labor politicians are.
Some well-meaning, but rather weak-minded people too, are misled by these cries. To these we say come and study our movement and gain a knowledge of its history and personalities, and you will find that every act of revolt is but a reply to a hundred, nay, a thousand villianeous crimes committed by the governing classes against us and against the workers in general. You will find that those who did these acts were the very best, the most human, unselfish, self-sacrificing of our comrades, who threw their lives away, meeting death or imprisonment in the hope that their acts would sow the seed of revolt, that they might show the way and wake an echo, by their deeds of rebellion, in the victims of the present system.
With the specific mode of action of anyone we have nothing to do. Anarchists advocate the propagation of their ideas by all means that lead to that end, and everyone is the best judge of his own actions. No one is required to do anything that is against his own inclination. Experience is in this as in other matters the best teacher, and the necessary experience can only be gained through entire freedom of action.
Thus the means which we would adopt embrace all that furthers our cause, and exclude all that will damage it. The decision of what is good or harmful must be left to persons or groups who choose to work together.
Nothing is more contrary to the real spirit of Anarchy than uniformity and intolerance. Freedom of development implies difference of development, hence difference of ideas and actions. Every person is likely to be open to a different kind of argument, so propaganda cannot be diversified enough if we want to touch all. We want it to pervade and penetrate all the utterances of life, social and political, domestic and artistic, educational and recreational. There should be propaganda by word and action, the platform and the press, the street corner, the workshop, and the domestic circle, acts of revolt, and the example of our own lives as free men. Those who agree with each other may co-operate; otherwise they should prefer to work each on his own lines to trying to persuade one the other of the superiority of his own method.
Organisation arises from the conciousness that, for a certain purpose, the co-operation of several forces is necessary. When this purpose is achieved the necessity for co-operation has ceased, and each force reassumes its previous independence, ready for other co-operation and combination if necessary. This is organisation in the Anarchist sense — ever varying, or, if necessary, continuous combinations of the elements that are considered to be the most suitable for the particular purpose on hand, and refers not only to the economical and industrial relations between man and man, but also to the sexual relations between man and woman, without which a harmonious social life is impossible.
These views differ immensely from those held by the believers in authority, who advocate permanent organisations with chiefs or councils elected by the majority, and who put all their trust in these institutions. The more they centralise these organisations and introduce stringent rules and regulations to preserve order and discipline, the more they will fail to achieve their object. In such organisations we see only obstacles to the free initiative and action of individuals, hot-beds of ambition, self seeking and rotten beliefs in authority etc. That means, we see in them agents of reaction to keep the people in continued ignorance of their own interests.
We do not therefore discourage workingmen from organisation, but such organisations could only be free groups of men and women with the same aims for identical purposes, disbanding when the object in view is achieved.
This brings us to the question of the advisability of Anarchists to join Trade Unions, not the question of the membership of Unions which may be a necessity for them as the case stands, but the question of propaganda in them. Anarchists do not wish to isolate themselves and Unions may be useful as a place to meet their fellow workers. But whether Unions should be formed by Anarchists is entirely dependent on the particular case. For we do not consider Trades Unionism as at present constituted as a serious force to overthrow the system, but only as a means to get a little better provision for the workers under the present conditions. Therefore they cannot be carried on without dealing with immediate so-called practical questions, which are never settled without compromises, as all members are not Anarchists.
In Unions the General Strike might form a proper subject to start the propaganda, and such a strike, though in itself not effective as a remedy, would probably bring about revolutionary situations which would advance the march of events in an unprecedented way. To speak plainly, we advocate the General Strike as a means to set the ball rolling: who knows whether it may not lead to the Social Revolution, which we all desire as the only thing that can help us.
The Social Revolution, as we conceive it, would consist in the paralysation of all existing authoritarian institutions and organisations, the prevention of new organisations of this character, the expropriation of the present exploiters of labor, and in the rearrangement of relations between men on the basis of voluntary agreements. This will appear to some to be rather a large program, but logical thinking will convince them of the fact that every one of these points is the necessary consequence of the others, and that they can only be carried out altogether, or not at all. For what is really impracticable are not full measures, but those half- hearted measures — so-called reforms — which pretend to do away with a part of the existing misery, whilst the root remains intact and makes the whole reform futile and useless.
These then are our means of propaganda, and we trust they are manifold enough to allow everybody full scope for his energies who chooses his place amongst us. The leading idea of our propaganda must always be defiance and destruction of the principle of authority in all its forms and disguises — full scope for freedom, the basis and condition of all human development and progress.
In conclusion, let us consider briefly the remedies proposed by the other parties — useless as they are, as the ever-increasing misery around us abundantly shows.
The State Socialist parties, apart from a few Socialists pure and simple who, if they were true to the foundations of their opinions, would come over to us, have of late become entirely parties for advocating political action. They believe in sending the right man to Parliament, and we have the choice between the chosen of the I.L.P., of the Fabians, and of the S.D.F. We do not consider their minor differences: what is the principle of political action worth? — is the question we ask. It is intended to bring about these social changes. Some palliatives may be adopted, but the system will continue to exist; for these labor parties make the workers believe in constitutional means, in the leadership and worship of men; in short, they destroy their self-reliance and self-respect, and do for them that which religion does — make them expect everything from others, nothing from themselves. The history of the labor movement in Europe and America shows the greater these parties become the less advanced their leaders grow and the less is achieved by these bulky, cast-iron organisations with no room for freedom left in them.
We have no more belief in Trades Unions as such than in political action, yet we prefer those Unionists, who rely upon their own action to those who cry for State help. Our propaganda might sometimes use this question as a starting point.
The Co-operative movement can only benefit a few who remain unnoticed among the general misery. Productive Co-operation on a large scale would have to compete with capitalism, which ruthlessly cuts down wages and gets a supply of cheap labor from the unemployed. Co-operators would have to work on similar lines, those of the greatest possible exploitation of labor and that will be no remedy for the needs of labor, or they would be crushed by the capitalist competition, being in fact the first victims of a commercial crisis. Thus on a large scale Co-operation is impracticable, and those who take part in it in is present form are only too often estranged from the general labor movement. So we consider Co-operators as workers who are no essential factor in the coming struggle.
The meanest and most repulsive “friends” of the workers are the Teetotalers, Malthusianists, and the advocates of thrift and saving, who propound each his particular crochet as an infallible remedy for poverty. They want the workers to give up the small mites of, however adulterated and paltry, pleasure and enjoyment that are left to them. “Hypocrisy is the compliment vice pays to virtue”, the proverb says, and the other parties make at any rate promises of better things, but these want to make life still more dreary and cheerless. Economically they are utterly wrong. If all were content to live as Coolies do, on a handful of rice per day, wages would be lowered by competition to the level of Coolie wages — a few pence per day. We want the standard of the workers’ living raised, not lowered, and all the things to which these “friends” object to a real, full, human life.
We need not dwell on all the cranks who have cut and dried remedies like the Free Currency advocates, who ignore the principle of every society with private property: “No property, no credit”. To be benefited by money cheques, it would be necessary to possess some kind of portable or realisable property to be given in exchange for the cheques or to have them secured on. Nothing would be altered by them, they could simply perpetuate the worst evils of the present system in a more aggravated form. To the worker who has no property but his labor to dispatch of, in times when work is slack and labor therefore not in demand, they would offer no resource whatever, and he would still be obliged to suffer and to starve. To make the remedy proportionate to the evil proposed to be cured, it would be requisite to abolish all private property and make the land and all it contains, together with all the implements of production, common property — that is, to introduce Communism, where money and money cheques will have become equally useless.
As you will have seen, Anarchism does not preach anything contrary to the principles which have always inspired men to strive for freedom and right. It would indeed be absurd to try and impose something new upon mankind. No! Anarchism is nothing but the full acknowledgment of the realisation of the principle that freedom is at the root of sound natural development. Nature knows no outside laws, no external powers, and only follows her own inward forces of attraction or repulsion. Everything is the result of the existing forces and tendencies, and this result becomes again in turn the cause of the next thing following. In its childhood, humanity suffered from the ignorance of this cause, and suffers still by being trodden under the heel of imaginary celestial and human authority (both arising from the same sources — ignorance and the fear of the unknown). All progress has been made by fighting and defying authority. Great men in history — men who have done real work, that is, work useful for the progress of the human race by breaking and defying laws and regulations apparently made for everlasting time — showed mankind new roads, opened new ground. There were rebels, and the last in this series — those who wish not only to be free themselves but who saw that which before them men did not see so clearly, that to be free ourselves we must be surrounded by free men; that the slavery of the meanest human being is our own slavery. Those last rebels for freedom and progress are the Anarchists of all countries, and in solidarity with them we appeal to you.
Study our principles, our movement, and if they convince you join us in our struggle against authority and exploitation, for freedom and happiness for all.
\begin{quote}
London, May 1\textsuperscript{st}, 1895.
\end{quote}
% begin final page
\clearpage
% if we are on an odd page, add another one, otherwise when imposing
% the page would be odd on an even one.
\ifthispageodd{\strut\thispagestyle{empty}\clearpage}{}
% new page for the colophon
\thispagestyle{empty}
\begin{center}
The Anarchist Library
\smallskip
Anti-Copyright
\bigskip
\includegraphics[width=0.25\textwidth]{logo-en}
\bigskip
\end{center}
\strut
\vfill
\begin{center}
Max Nettlau
An Anarchist Manifesto
1\textsuperscript{st} May, 1895
\bigskip
Anarchy is Order CD
\bigskip
\textbf{theanarchistlibrary.org}
\end{center}
% end final page with colophon
\end{document}
|
https://www.zentralblatt-math.org/matheduc/en/?id=6739&type=tex | zentralblatt-math.org | CC-MAIN-2019-30 | text/plain | application/x-tex | crawl-data/CC-MAIN-2019-30/segments/1563195526324.57/warc/CC-MAIN-20190719161034-20190719183034-00366.warc.gz | 904,377,350 | 1,251 | \input zb-basic
\input zb-matheduc
\iteman{ZMATH 2016e.00459}
\itemau{Hord, Casey; Marita, Samantha}
\itemti{Students with learning disabilities tackle multistep problems.}
\itemso{Math. Teach. Middle Sch. 19, No. 9, 548-555 (2014).}
\itemab
From the text: Do you ever worry about how you can help students with learning disabilities pass the state test? Are you even more concerned now that the new state test items have many steps and are extremely long? Lengthy problems with a lot of steps are difficult for many students, especially for those with learning disabilities. The teaching strategies we recommend can help students learn how to think through these types of problems and can offer them a better chance of scoring well on their next state test.
\itemrv{~}
\itemcc{D53 D73}
\itemut{special education; problem-solving strategies; learning disabilities; learning problems; multistep problems; tables; table templates; multiple tables; cognitive mechanisms; working memory}
\itemli{}
\end |
https://ctan.mines-albi.fr/macros/generic/expkv-bundle/expkv-bundle.tex | mines-albi.fr | CC-MAIN-2023-14 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2023-14/segments/1679296948765.13/warc/CC-MAIN-20230328042424-20230328072424-00796.warc.gz | 217,950,041 | 3,722 | \PassOptionsToPackage{full}{textcomp}
\documentclass[exfoo=value, exbar, exfoo=\empty]{l3doc}
% preamble >>=
\makeatletter
\let\save@onlypreamble\@onlypreamble
\let\@onlypreamble\@gobble
\usepackage[all]{expkv}
\let\@onlypreamble\save@onlypreamble
\makeatother
\usepackage[oldstylenums,nott]{kpfonts}
\input{glyphtounicode}
\pdfgentounicode=1
\usepackage{xfp} % required for an example
\usepackage{booktabs}
\usepackage{array}
\usepackage{collcell}
\usepackage{siunitx}
\DeclareSIUnit\ops{ops}
\usepackage{xcolor}
\usepackage{caption}
\usepackage{microtype}
\usepackage{accsupp}
\usepackage{enumitem}
\usepackage{randtext}
\usepackage{tcolorbox}%>>=
\newtcolorbox{exresult}[2][]
{%
colback=ekvgrey!10!white%
,colframe=ekvgrey%
,fontupper=\small
,width={\dimexpr#2\relax}%
,#1%
}
\newtcbox\exres[1][]
{
colback=ekvgrey!10!white
,colframe=ekvgrey
,size=small
,nobeforeafter
,tcbox raise base
,fontupper=\small
,#1
}
%=<<
\usepackage{listings}%>>=
\input{preamble-lst.tex}
%=<<
\let\metaORIG\meta
\protected\def\meta #1{\texttt{\metaORIG{#1}}}
\input{preamble-examples.tex}
\input{preamble-logos.tex}
\makeatletter
% shortcuts >>=
\newcommand\Vkey{\texttt{Val}-\key}
\newcommand*\Nkey{\texttt{NoVal}-\key}
\newcommand\kv{\meta{key}=\penalty2000\meta{value}}
\newcommand\kvarg{\{\kv, \ldots\}}
\newcommand\key{\meta{key}}
\newcommand\val{\meta{value}}
\newcommand\set{\meta{set}}
\newcommand\prefix{\texorpdfstring{\textit{prefix}}{prefix}}
\newcommand\prefixes{\textit{prefixes}}
\newcommand\type{\texorpdfstring{\textit{type}}{type}}
\newcommand\types{\textit{types}}
\newcommand\tkn[2]{\texttt{\char`#1}\textsubscript{#2}}
\newcommand\expansion{\meta{expansion}}
%\newcommand\expnotation{}
%\edef\expnotation
%{\noexpand\texttt{exp\string|}\penalty\@M-\hskip\z@skip notation}
\newcommand\singlecs[1]
{%
The \meta{cs} should be a single control sequence, such as
\cs[no-index]{#1}. \ignorespaces
}
\newcommand\ekvdocsection[8]
{%
\clearpage
\chardef\ekvdoc@insection1
\section[{#7}]%
{%
#7%
\hfill
\begingroup\scriptsize\ttfamily
\begin{tabular}{@{}r@{}l@{}}%
#1{\string\input\{expkv#8\}} & \rlap{#2{\ \% plain}}\\
#3{\string\usepackage\{expkv#8\}} & \rlap{#4{\ \% LaTeX}}\\
#5{\string\usemodule[expkv#8]} & \rlap{#6{\ \% ConTeXt}}\\
\end{tabular}%
\endgroup
\label{sec:expkv#8}%
}%
\chardef\ekvdoc@insection0
}
\chardef\ekvdoc@insection0
\newcommand*\genericekv
{\ekvdocsection{}{\textcolor{gray}}{}{\textcolor{gray}}{}{\textcolor{gray}}}
\newcommand*\latexekv
{\ekvdocsection\phantom\phantom{}{\textcolor{gray}}\phantom\phantom}
%=<<
\hypersetup{linkcolor=red!80!black,urlcolor=purple!80!black}
\input{preamble-prefixes.tex}
\input{preamble-noidx.tex}
\input{preamble-enverb.tex}
% vissp >>=
\ExplSyntaxOn
\cs_new_protected:Npn \vissp #1
{
\group_begin:
\tl_set:Nn \l_tmpa_tl {#1}
\tl_replace_all:Nnn \l_tmpa_tl { ~ } { \asciispace }
\l_tmpa_tl
\group_end:
}
\ExplSyntaxOff
% =<<
\ekvcSplit\expkvorules% >>=
{
cd = \emph{nothing}
,cu = \emph{nothing}
,pd = \emph{nothing}
,pu = \emph{nothing}
}
{%
\begin{description}
\item[Class:]
\begin{description}
\item[defined] #1
\item[undefined] #2
\end{description}
\item[Package:]
\begin{description}
\item[defined] #3
\item[undefined] #4
\end{description}
\end{description}%
}
\ekvcSecondaryKeys\expkvorules
{
meta d = {cd={#1},pd={#1}}
,meta u = {cu={#1},pu={#1}}
}% =<<
\newcommand*\pmso[1] % poor man's strike out%>>=
{%
\leavevmode
\begingroup
\sbox0{#1}%
\rlap{\vrule height .6ex depth -.5ex width \wd0\relax}%
\usebox0\relax
\endgroup
}%=<<
\@ifdefinable\gobbledocstriptag{\def\gobbledocstriptag#1>{}}
\renewcommand*\partname{Part}
% \addsec and friends >>=
\newcommand*\addsec@[2]
{%
\c@secnumdepth=%
\expanded
{%
\m@ne
\unexpanded{#1{#2}}%
\c@secnumdepth=\the\c@secnumdepth\relax
}%
}
\newcommand*\addsec {\addsec@\section}
\newcommand*\addssec {\addsec@\subsection}
\newcommand*\addsssec{\addsec@\subsubsection}
% =<<
\newenvironment{syntaxexample}% >>=
{%
\quote
\ttfamily\small\frenchspacing
\parskip=\z@
\def\indent{\leavevmode\phantom{mm}}%
}
{\endquote}
\newenvironment{syntaxexample*}
{\syntaxexample\obeylines}
{\endsyntaxexample}% =<<
\newcommand\expkvdocPrintErrors[1][]
{%
\protected\long\def\expkvdoc@errfont##1%
{\texttt{\frenchspacing\textcolor{red!80!black}{##1}}}%
\protected\long\def\ekv@err@collect##1\par##2%
{\expkvdoc@errfont{! \detokenize{##2} Error: ##1}#1}%
\protected\long\def\expkvdoc@errm##1##2%
{\expkvdoc@errfont{! expkv##1 Error: ##2}#1}%
\def\ekv@errm{\expkvdoc@errm{}}%
\def\ekvc@errm{\expkvdoc@errm{-cs}}%
\def\ekvd@errm{\expkvdoc@errm{-def}}%
\def\ekvp@errm{\expkvdoc@errm{-pop}}%
}
\makeatother
\ExplSyntaxOn
\str_new:N \g__expkvdoc_module_str
\cs_new_protected:Npn \expkvdocfile #1% >>=
{
\expkvdoc_for_module:nn {#1}
{
\str_if_eq:nnTF {#1} {main}
{ \lstset{style=expkv} }
{ \lstset{style=expkv-#1} }
\addtocontents{exs}
{
\medskip
\noindent
\use:c { expkv \str_if_eq:nnF {#1} {main} { \str_head:n {#1} } }
\smallskip
}
\input{pkg-#1.tex}
}
}% =<<
\cs_new_protected:Npn \expkvdocdtx #1% >>=
{
\expkvdoc_for_module:nn {#1}
{
\str_if_eq:nnTF {#1} {main}
{ \DocInput{expkv.dtx} }
{ \DocInput{expkv-#1.dtx} }
}
}% =<<
\cs_new_protected:Npn \expkvdoc_for_module:nn #1#2% >>=
{
\use:e
{
\exp_not:n
{
\str_gset:Nn \g__expkvdoc_module_str {#1}
#2
\str_gset:Nn \g__expkvdoc_module_str
}
{ \g__expkvdoc_module_str }
}
}% =<<
\ExplSyntaxOff
\input{preamble-l3doctweaks.tex}
%=<<
% for thanking Niranjan
% code from https://tex.stackexchange.com/a/635125/117050
\def\DevnagVersion{2.17}
\usepackage{devanagari}
\newif\ifexpkvDocImplementation%\expkvDocImplementationtrue
\begin{document}
\title{\expkvbundle}
%\title% >>=
%{%
%\texorpdfstring
%{%
%\huge\expkvbundle
%\\[\medskipamount]
%\Large an {\expFormat}andable
%\meta{{\kvstyle k}\kern-.05em ey}=%
%\meta{{\kvstyle v}\kern-.05em alue}
%implementation and more%
%}
%{expkv-bundle - an expandable <key>=<value> implementation and more}%
%}% =<<
\author
{%
Jonathan P. Spratte%
\thanks
{%
\protect\randomize{jspratte@yahoo.de};
Special thanks to {\protect\dn Enr\2jn} (Niranjan) for valuable
suggestions and additions to this documentation.%
}%
}
\date{2023-01-23}
\begingroup
\renewcommand*\thefootnote{\fnsymbol{footnote}}
\maketitle
\endgroup
\begin{abstract}%>>=
\noindent\parfillskip=0pt
The \expkvbundle\ provides at its core a \emph{fully expandable} \kv\ parser,
that is \emph{safe} for active commas and equals signs, \emph{reliable} to
only strip one set of braces after spaces are stripped, and blazingly
\emph{fast}, as of writing this only \pkg{keyval} is faster.
\par
\bigskip
This parser gets augmented by a family of packages. \expkvc\ allows to easily
define expandable macros using a \kv\ interface, making the expandable parser
available to the masses. \expkvd\ provides a \kv\ interface to
define common \key-types. With \expkvo\ you can parse package
and class options. \expkvp\ enables you to design your own prefix
oriented parsers for interface definitions.
\end{abstract}%=<<
\tableofcontents
\clearpage
\begin{documentation}% >>=
\ifexpkvDocImplementation\part{Documentation}\fi
\input{introduction.tex}
\clearpage
\input{impatient.tex}
\expkvdocfile{main}
\expkvdocfile{cs}
\expkvdocfile{def}
\expkvdocfile{opt}
\expkvdocfile{pop}
\clearpage
\lstset{style=expkv-all}
\input{comparison.tex}
\end{documentation}% =<<
\clearpage
\listofexamples
\ifexpkvDocImplementation
\clearpage
\begin{implementation}% >>=
\part{Implementation}
\expkvdocdtx{main}
\clearpage
\expkvdocdtx{cs}
\clearpage
\expkvdocdtx{def}
\clearpage
\expkvdocdtx{opt}
\clearpage
\expkvdocdtx{pop}
\end{implementation}% =<<
\fi
\clearpage
\PrintIndex
\end{document}
|
https://fifthestate.anarchistlibraries.net/library/354-spring-2000-news-reviews.tex | anarchistlibraries.net | CC-MAIN-2021-39 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-39/segments/1631780055601.25/warc/CC-MAIN-20210917055515-20210917085515-00603.warc.gz | 289,020,699 | 8,706 | \documentclass[DIV=12,%
BCOR=0mm,%
headinclude=false,%
footinclude=false,open=any,%
fontsize=10pt,%
oneside,%
paper=210mm:11in]%
{scrbook}
\usepackage{microtype}
\usepackage{graphicx}
\usepackage{alltt}
\usepackage{verbatim}
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\title{News \& Reviews}
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\author{Fifth Estate Collective}
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\textbf{Bound Together Books} is presenting the fifth annual Bay Area Anarchist Book Fair, Saturday, April 15 at the San Francisco Hall of Flowers in the city’s Golden Gate Park.
The fair showcases anarchist publishers, distributors and activist groups with attendance at last year’s event having grown to 3,000. According to its organizers, the idea for the fair is to build an international radical community, exchange ideas and sell literature.
Also, there will be an installation by Art and Revolution, an anti-authoritarian art exhibit, and a vegan\Slash{}vegetarian café. Speakers will include Wobbly singer and storyteller Utah Philips, and prison activist, Christian Parenti. A video collective will show footage documenting recent Bay Area radical activity.
Tables are available for presenters at a nominal fee, but admission to the fair is free. Contact Bound Together Books, 1369 Haight St., San Francisco CA 94117, or email: seansul@mindspring.com for information.
\textbf{Book fairs abound.} Chicago’s Autonomous Zone is sponsoring Matches ‘N Mayhem, The Midwest Anarchist Book Fair, May 5, 6 and 7.
The mayhem won’t be from the city’s notorious anarchist-hating cops, however, but from scheduled anarchist soccer matches. Yes, there will be rules.
There’ll also be a film festival, a propaganda gallery, variety show, and workshops. The films will be Friday night, the fair on Saturday with displays from numerous publishers, and a variety show in the evening. Then, the mayhem on the soccer field on Sunday.
If you want to field a soccer team or display your publications, contact A-Zone, 1573 N. Milwaukee, \#420, Chicago IL 60622; 773-252-6019; azone@wwa.com. There are small fees for the events.
\textbf{The northern Ohio college town of Bowling Green} will be the site of a second Underground Publishing Conference, June 10–11.
Last year, two of us from the Fifth Estate attended the first gathering at the university bearing the town’s name and were witness to the best of the small, alternative publication milieu. Attendees at the conference, mostly youth of college age, came from across the country to compare notes and resources, and to trade papers. Although the subject matter of the zines ranged from anarchism to poetry, anti-authoritarian politics was the dominant theme.
Unfortunately, most of the publications are of limited circulation, some printing as few as a couple dozen and the largest only in the low hundreds. Unfortunate because of the wide range of subject matter, good writing, and thoughtful analysis. The spirit in the workshops seemed similar to the Underground Press Syndicate meetings of the 1960s, but on a smaller scale which is due more to the political climate than the quality of the magazines.
A good example is a one-person zine, The Secret Files of Captain Sissy \#3, published by Andy Cornell featuring personal musings on the nature of life in’ modern America—“Being non-political means being an activist for the enemy,” he argues in it. Many of his themes are anti-work and anti-suburban boredom.
Cornell and his friends have put together the Words As Weapons zine distro, P.O. Box 4493, Ann Arbor MI 48106 which carries an array of zine publications including his. Also, in attendance were people from the distributor, Tree of Knowledge Press, P.O. Box 252766, Little Rock AR 7225, who have a large catalog of publications as well as instructional texts on how to publish a zine. Both outfits will consider publications sent to them for possible distribution.
For information on the conference contact Jason, 216 S. Church, Bowling Green OH 43402; 419\Slash{}353-7035; of upcon2000@hotmail.com.
\textbf{Louise M. Gagneur was a best selling novelist} in the late 19\textsuperscript{th} century and a passionate advocate of radical and feminist causes. In her essays and ten popular novels, she not only demanded equal rights for women, but also an end to the ancient tyrannies based on social classes and authoritarian institutions. -Despised by the male publishing and academic establishment in both France and this country, her writings were long assigned to oblivion until the recent re-issue of her title, The Nihilist Princess, by III Publishing.
This Gagneur novel has all the grand sweep of 19\textsuperscript{th} century literature, and centers on a young, beautiful Russian aristocrat who is appalled by her father’s treatment of his serfs. She joins the radical, and sometimes violent, underground Nihilist movement ‘which is seeking radical changes in Russia including abolition of serfdom and the czar. Thrilling adventure marks almost every page as the princess risks her life for her principles.
\$12 plus postage from III Publishing, POB 1581, Gualala CA 95445, (707) 882–1818; or www.iiipublishing.com.
\textbf{In our last issue, a debate} took up the question of the propriety of this paper’s and anarchist support of the striking unions at the \emph{Detroit News} and \emph{Free Press}.
In Carlotta R. Anderson’s excellent biography, \emph{All-American Anarchist: Joseph A. Labadie and the Labor Movement} (Wayne State University, 1998), a section about a strike during the 1880s against one of the same newspapers shows no record of a similar controversy at the time.
“Labadie was another featured speaker [at a mass meeting of the Trades Council]. He urged the audience to join the typographical union’s boycott against the \emph{Detroit Free Press} for its twelve-year refusal to employ union printers and pay union wages. ‘Just as long as it boycotts us, we will boycott them,’ he roared out. Fifteen hundred copies of a ‘Black List’ circular were distributed, listing \emph{Free Press} advertisers who should also be boycotted.”
The paper capitulated and recognized the unions in 1886 during a wave of labor militancy and agitation for the eight-hour day.
\emph{\textbf{De Fabel van de illegaal}} (The Myth of Illegality), a Dutch language newspaper printed on high gloss paper, sends along an English summary with each edition. The current translations are of articles concerning the overtures by the New Right to those campaigning against the World Trade Organization (WTO) and globalization
With the disintegration of the European old Right, a crop of neo-fascists are taking up themes of many Left campaigns and the opposition to multinationals. This is nothing new, of course. Nazis and fascists, particularly in the 1920s and 30s, sometimes sounded close to the Left and labor, and even ecologists, on many issues.
Still, when the veneer of issue-oriented politics is removed, the Right’s core authoritarian politics of a dictatorial state, conquest by war, racism, patriarchy and hierarchy are revealed. De Fabel van de ilIegaal urges readers to be overtly anti-capitalist, thereby ending any confusion that national capitalism is a preferable state of affairs to globalization.
The magazine charges that English millionaire, Edward Goldsmith, editor of the London-based, long-publishing magazine, The Ecologist, typifies the raproachement between Left and Right which they fear. De Fabel reports that Goldsmith regularly addresses neo-fascist gatherings and that its work is highly regarded in right-wing circles, many of which are influenced by the thought of Francis Parker Yockey and his neo-fascist, pan-European opus, Imperium.
Sorting out European politics with its history of fascist triumph, collaboration, and ideology-switching personalities is always difficult, so it’s hard to evaluate the publication’s charges against Goldsmith, but the phenomenon they warn about certainly is real.
Reach \emph{De Fabel} at Koppenhinksteeg 2, 2312 HX Leiden, Netherlands; or www.dsl.nl\Slash{}media\Slash{}lokabaal.
\textbf{\emph{Class War} newspaper is publishing again.} England’s sassiest tabloid still delights in photos of bloodied cops, reporting on the latest riot and sabotage, and is filled with general working class opposition and mischief towards the rulers. Issue \#78 contains an anti-BNP Donor Card to be carried in our wallet which contains stipulations body part allocation after death. It states, “No part of my body may be used for the treatment of members of the British National Party [fascists], serving or retired police officers, members of the British Royal Family, or\dots{}the House of Lords.” Subscriptions are \$20 (U.S.), from Class War, BM Box 357, London WC1N 3XX, United Kingdom.
\textbf{\emph{Do or Die} \#8.} We at this newspaper usually think we have a fat issue when we reach 32-pages, but this edition of DoD is a book-sized 346!!
The contents cover the entire spectrum of anarchist, radical environmental, and anti-fascist politics with a list of articles taking up an entire pages itself. If you want a distillation of English anti-authoritarian politics, you’ve got it in this volume.
They don’t take subscriptions but you can get this issue or next for about \$12 (U.S.) from Do or Die, c\Slash{}o 6 Tilbury Place, Brighton, East Sussex, BN2 2GY, United Kingdom; or, wwW.eco-action.org\Slash{}dod.
\textbf{A book banned} by the Michigan Department of Corrections, \emph{The Celling of America: An Inside Look at the US Prison Industry}, must now be allowed into state prisons under terms of a recent out-of-court settlement.
Two inmates brought suit in federal court against the prison system after being denied the book in 1998. Not only did the department agree to allow the title into its prisons but also agreed to change its notification system when book’s are banned.
Previously, only the inmate was informed of the seizure of a publication. Under the newly signed agreement, not only will the sender be notified, but must also be told how to appeal the seizure. Paul Wright, editor of Prison Legal News (PLN) said, “This is a victory both for the right of publishers to send their message to prisoners and for prisoners to receive ideas from outside the prison.”
The state agreed to pay the book’s publisher and PLN each \$1,000 in damages and \$10,000 for the plaintiffs’ legal fees. \emph{The Celling of America} is available from Common Courage Press, Box 702, Monroe ME 04951; (207) 525–0900; or www.commoncouragepress.com.
\emph{FE note:} Without sounding a negative note, we hope that this will be more than a paper settlement. Our experience with prison systems in other states where our paper has been refused admittance, is that the appeals process is a joke.
In one Texas gulag, an issue of the FE was banned because it contained a famous photo of Vietnamese civilians hideously burned by U.S. napalm in which a naked young girl could be seen. The paper was barred because it would “encourage deviate sexual behavior.” The exclusion was upheld by the appeals board.
\textbf{\emph{Guinea Pig Zero}} is an occupational job zine for people who are used as medical or pharmaceutical research subjects. GPZ \#6 contains true tales of guinea pig adventure stories, discussions of bioethics, news and reviews, even poetry and fiction relating to the “disposability of plebeian life.”
This very quirky zine is edited by Philadelphia Wobbly, and occasional guinea pig, Bob Helms, who has done several TV interviews as a result of its publication and even been featured in People magazine.
The content runs from the humorous to the grotesque. Current issue contains a “report card” on two labs by Dishwasher Pete, a Butcher of the Month Club centerfold, and a translation of Octave Mirbeau’s delightful little story, “The Enema.”
Sample copies from GPZ, POB 42531, Philadelphia PA 19101.
\textbf{As our lives go by,} it’s easy to forget that Leonard Peltier is more than a worthy cause, that he’s an innocent man who has spent almost 25 years in prison as a scapegoat for a failed government raid at Pine Ridge, South Dakota in 1975.
Peltier was falsely convicted in the deaths of two FBI agents who were part of an invasion force assaulting the native reserve.
\emph{In the Spirit of Crazy Horse}, the official newsletter for the Leonard Peltier Defense Committee, Peltier writes from prison of “knowing the grim pleasure the FBI takes in seeing me suffer here year after year.”
The 20-page tabloid contains updates on legal efforts and direct action to free Leonard plus other news of native people’s struggles. Subscriptions are \$15 from LPDC, Box 583 KS 66044. Or, 785–8425774; www.freepeltier.org.
The South Chicago Anarchist Black Cross (ABC) Zine Distro, P.O. Box 721, Homewood IL 60430, distributes over 100 zines covering anarchist, feminist, anti-racist\Slash{}abolitionist and prisoner issues.
SCABC specializes in helping articulate activist prisoners get the word out Send for their 40-page catalog or view it on-line at http:\Slash{}\Slash{}members.xoom.com\Slash{} thought bombs.
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Fifth Estate Collective
News \& Reviews
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\href{https://www.fifthestate.org/archive/354-spring-2000/news-reviews}{\texttt{https://www.fifthestate.org/archive/354-spring-2000/news-reviews}}
Fifth Estate \#354, Spring, 2000
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\textbf{fifthestate.anarchistlibraries.net}
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\rhead{\textbf{A. P. M. E. P.}}
\lhead{\small Brevet de technicien supérieur }
\lfoot{\small{Groupement D}}
\rfoot{\small{14 mai 2013}}
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\begin{center} \Large \textbf{\decofourleft~Brevet de technicien supérieur~\decofourright\\ Groupement D session 2013}
\end{center}
Analyses de biologie médicale
Bio analyses et contrôles
Biotechnologie
Hygiène-propreté-environnement
Industries plastiques-europlastic-à référentiel commun européen
Métiers de l'eau
Peintures, encres et adhésifs
Qualité dans les industries alimentaires et les bio-industries
\vspace{0,5cm}
\textbf{EXERCICE 1 }\hfill 10 points
\medskip
Les bordures d'autoroute possèdent parfois des bassins de décantation dont le rôle est de recueillir les eaux pluviales ruisselant sur l'asphalte et les éléments polluants qu'elles peuvent drainer.
\medskip
À la suite d'un accident de la circulation. un camion~citerne déverse une partie de son contenu sur la chaussée d'une autoroute. La réglementation en vigueur impose l'isolation, par fermeture de vannes, du bassin de décantation proche de l'accident de façon à ce que la concentration en matières polluantes dans le bassin ne dépasse pas 15 $\mu$g/L. Cette concentration est de 1,3 $\mu$g/L au moment où les matières polluantes provenant du camion-citerne commencent à se déverser dans le bassin.
\textbf{Dans cet exercice, on cherche à prévoir au bout de combien de temps la concentration en matières polluantes dans le bassin atteindra 15 $\mu$g/L si on n'isole pas le bassin et à quel moment les capteurs installés dans le bassin déclencheront la fermeture des vannes.}
\medskip
On mesure en minute le temps $t$ écoulé à partir de l'instant où les matières polluantes provenant du camion-citerne- commencent à se déverser dans le bassin de décantation.
On admet que, tant que le bassin n'est pas isolé par fermeture des vannes, la concentration à l'instant $t$ en matières polluantes dans le bassin, exprimée en $\mu$g/L peut être modélisée par $f(t)$ où $f$ est solution de l'équation différentielle
\[(E)\::\quad y' + 0,03y = 0,75.\]
On a donc $f(0) = 1,3$.
\begin{center}
\textbf{Les parties A et B peuvent être traitées de fa\c{c}on indépendante}
\end{center}
\textbf{A. Résolution sur l'intervalle \boldmath$[0~;~ +\infty[$\unboldmath de l'équation différentielle }
\boldmath$(E) \:: y' + 0,03y = 0,75$\unboldmath
\medskip
\begin{enumerate}
\item On considère l'équation différentielle
\[\left(E_{0}\right)\: :\quad y' + 0,03 y = 0,\]
où $y$ est une fonction de la variable $t$, définie et dérivable sur l'intervalle $[0~;~+ \infty]$, et $y'$ la fonction dérivée de la fonction $y$.
Déterminer les solutions de l'équation différentielle $\left(E_{0}\right)$ sur l'intervalle $[0~;~+ \infty]$.
\item Soit $g$ la fonction définie sur l'intervalle $[0~;~+ \infty]$ par $g(t) = a$, où $a$ est une constante réelle.
Déterminer $a$ pour que la fonction $g$ soit une solution particulière de l'équation différentielle $(E)$.
\item En déduire l'ensemble des solutions de l'équation différentielle $(E)$.
\item Démontrer que la solution $f$ de l'équation différentielle $(E)$ qui vérifie la condition initiaJe $f(0) = 1,3$ est la fonction définie sur l'intervalle $[0~;~+ \infty]$ par : $f(t) = 25 - 23,7 \text{e}^{- 0,03t}$.
\end{enumerate}
\bigskip
\textbf{B. Étude de la fonction }\boldmath $f$ \unboldmath
\medskip
$f$ est définie sur l'intervalle $[0~;~+ \infty]$ par :
\[f(t) = 25 - 23,7 \text{e}^{- 0,03t}.\]
On note $C$ sa courbe représentative dans le plan muni d'un repère orthogonal.
\medskip
\begin{enumerate}
\item Déterminer la limite de la fonction quand $t$ tend vers $- \infty$.
\item On désigne par $f'$ la fonction dérivée de la fonction $f$.
\begin{enumerate}
\item Calculer $f'(t)$ pour tout $t$ de l'intervalle $[0~;~+ \infty]$.
Expliquer le signe de $f'(t)$ pour tout $t$ de l'intervalle $[0~;~+ \infty]$.
\end{enumerate}
\item Dresser le tableau de variations complet de la fonction $f$.
\item
\begin{enumerate}
\item Recopier et compléter le tableau de valeurs ci-dessous. Arrondir les résultats au dixième.
\medskip
\begin{tabularx}{\linewidth}{|*{8}{>{\centering \arraybackslash}X|}}\hline
$t$&0 &10 &20 &30 &40 &50 &60 \\ \hline
$f(t)$& 1,3 &&&&&& \\ \hline
\end{tabularx}
\medskip
\item Tracer la courbe $C$ sur la feuille de papier millimétré jointe au sujet ou visualiser cette courbe sur l'écran de la calculatrice et indiquer sur la copie les caractéristiques de la fenêtre utilisée (valeurs de X min, X max, Y min et Y max et des \og pas \fg) et l'allure de la courbe obtenue.
\end{enumerate}
\end{enumerate}
\bigskip
\textbf{C. Traitement de la problématique}
\medskip
On rappelle que $f(t)$ modélise la concentration (exprimée en $\mu$g/L) en matières polluantes dans le bassin à l'instant $t$ (exprimé en minute) tant que le bassin n'est pas isolé par fermeture des vannes.
\medskip
\begin{enumerate}
\item Si le bassin n'était pas équipé d'un dispositif d'isolation par fermeture de vannes, quelle serait la valeur autour de laquelle se stabiliserait la concentration en matières polluantes ? Justifier.
\item A l'aide de la courbe C obtenue à la question B 4b), sur papier millimétré ou sur écran de la calculatrice. déterminer graphiquement une valeur approchée à l'unité du temps $t_{0}$ (exprimé en minute) au bout duquel la concentration en matières polluantes dans le bassin atteindrait 15 $\mu$g/L si le bassin n'était pas isolé par fermeture de vannes. Expliquer la démarche.
\item La concentration en matières polluantes dans le bassin est relevée par un capteur dont les mesures sont légèrement instables.
Pour prendre en compte cette instabilité, on met en place un dispositif associant la fermeture des vannes à l'instant $t\: (t \geqslant 2$) à la valeur moyenne de la concentration en matières polluantes mesurée par le capteur entre les instants $t - 2$ et $t$.
La fermeture des vannes est déclenchée lorsque cette valeur moyenne atteint 14 $\mu$g/L.
La valeur moyenne de la concentration (exprimée en $\mu$g/L) en matières polluantes entre les instants $t - 2$ et $t$ est modélisée par :
\[V(t) = \dfrac{1}{2}\int_{t - 2}^t f(u)\:\text{d}u = \dfrac{1}{2}(F(t) - F(t - 2)),\]
où $F$ est une primitive de la fonction $f$.
\begin{enumerate}
\item Donner une primitive $F$ de la fonction $f$ sur l'intervalle $[0~;~+ \infty]$.
\item Calculer $V(t)$ et vérifier que : $V(t) = 25 + A \text{e}^{-0,03t}$ avec $A = - 24,4$.
\item Résoudre l'équation : $25 - 24,4\text{e}^{-0,03t} = 14$.
Donner une valeur approchée au dixième de la solution $T$ de cette équation.
\item Que représente $T$ dans le contexte de l'exercice ?
\end{enumerate}
\end{enumerate}
\vspace{0,5cm}
\textbf{EXERCICE 2}\hfill 10 points
\medskip
Une coopérative est spécialisée dans la récolte de la fleur de sel.
Elle utilise une machine automatique pour remplir des sachets de fleur de sel dont la masse théorique doit être de 250 grammes.
Un sachet est dit conforme si sa masse $m$, exprimée en gramme, vérifie :
$240 \leqslant m \leqslant 260$.
\medskip
\emph{Les probabilités demandées dans cet exercice peuvent être calculées en utilisant le formulaire joint au sujet ou la calculatrice. Quelle que soit l'option retenue on fera figurer sur la copie quelques étapes de la démarche suivie.}
\medskip
Les résultats des calculs de probabilité seront arrondis au millième.
Les parties A, B et C de cet exercice peuvent être traitées de façon indépendante.
\bigskip
\textbf{Loi normale}
\medskip
L'étude statistique de la production permet d'admettre que la variable aléatoire $M$ qui mesure, en gramme, la masse d'un sachet suit une loi normale de moyenne $\mu = 250$ et d'écart type $\sigma = 5,3$.
\begin{enumerate}
\item On choisit au hasard un sachet dans la production.
Calculer la probabilité que le sachet soit conforme,
\item
\begin{enumerate}
\item Calculer $P(M \geqslant 245)$.
\item Un gros client exigeant souhaite qu'au moins trois quarts des sachets qu'il achète aient une masse supérieure à 245~grammes. Sera-t-il satisfait ? Justifier.
\end{enumerate}
\end{enumerate}
\bigskip
\textbf{B. Loi binomiale et loi de Poisson}
\medskip
\textbf{On considère dans cette partie que la probabilité qu'un sachet ne soit pas conforme est :\:}\boldmath $p = 0,06$\unboldmath.
\medskip
La coopérative constitue des lots de 50~sachets pour la vente et étudie le nombre de sachets non conformes contenus dans un lot.
La production de la coopérative est suffisamment importante pour que l'on puisse assimiler la constitution d'un lot à un tirage au hasard et avec remise de 50~sachets.
On note $X$ la variable aléatoire qui associe a chaque lot de 50~sachets le nombre de sachets non conformes de ce lot.
\medskip
\begin{enumerate}
\item Justifier que la variable aléatoire $X$ suit une loi binomiale et préciser ses paramètres.
\item Que représente la probabilité $P(X = 1)$ dans le contexte de l'exercice ? Calculer $P(X = 1)$.
\item On approche la loi de probabilité de $X$ par une loi de Poisson.
\begin{enumerate}
\item Justifier que cette loi de Poisson a pour paramètre $\lambda = 3$.
\item On note $Y$ une variable aléatoire qui suit une loi de Poisson de paramètre $\lambda = 3$.
En utilisant la variable aléatoire $Y$, estimer la probabilité qu'il y ait au plus cinq sachets non conformes dans un lot de 50~sachets.
\end{enumerate}
\end{enumerate}
\bigskip
\textbf{C. Test d'hypothèse}
\medskip
Après la révision annuelle de la machine utilisée pour remplir les sachets de fleur de sel, le responsable qualité de la coopérative veut contrôler la valeur de la masse moyenne $m$ (exprimée en gramme) d'un sachet de fleur de sel.
Il construit pour cela un test d'hypothèse bilatéral au seuil de signification de 5\,\%.
L'hypothèse nulle $H_{0}$ est : $m = 250$.
L'hypothèse alternative $H_{1}$ est : $m \neq 250$.
On note $\overline{M}$ la variable aléatoire qui, à chaque échantillon aléatoire de 50~sachets prélevés dans la production de la coopérative, associe la masse moyenne (en gramme) d'un sachet de l'échantillon.
La production est suffisamment importante pour qu'on puisse assimiler la constitution d'un échantillon à un tirage au hasard et avec remise de 50~sachets.
On suppose que la variable aléatoire $\overline{M}$ suit une loi normale de moyenne $m$ et d'écart type $\dfrac{5,3}{\sqrt{50}}$.
\medskip
\begin{enumerate}
\item Sous l'hypothèse nulle $H_{0}$, déterminer le nombre réel positif $a$ tel que
\[P\left(250 - a \leqslant \overline{M} \leqslant 250 + a\right) = 0,95,.\]
Arrondir au centième,
\item Énoncer la règle de décision du test.
\item On prélève au hasard 50~sachets dans la production.
Les masses en gramme de ces sachets se répartissent de la façon suivante:
\medskip
\begin{tabularx}{\linewidth}{|m{1.4cm}|*{7}{>{\centering \arraybackslash \tiny}X|}}\hline
\footnotesize Masse en gramme&[236 ; 240[ &[240 ; 244[ &[244 ; 248[ &[248 ; 252[ &[252 ; 256[ &[256 ; 260[ &[260 ; 264[\\ \hline
\footnotesize Nombre de sachets&5 &6 &9 &13 &8 &7 &2\\ \hline
\end{tabularx}
\medskip
\begin{enumerate}
\item En utilisant les centres des intervalles, calculer une valeur approchée de la masse moyenne d'un sachet de cet échantillon.
\item Quelle va être la conclusion du responsable qualité ?
\end{enumerate}
\end{enumerate}
\end{document} |
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\section{Inner Themes, Outer Themes, and Templates}
\label{section-elements}
This section discusses the inner and outer themes that are available in \beamer. These themes install certain \emph{templates} for the different elements of a presentation. The template mechanism is explained at the end of the section.
Before we plunge into the details, let us agree on some terminology for this section. In \beamer, an \emph{element} is part of a presentation that is potentially typeset in some special way. Examples of elements are frame titles, the author's name, or the footnote sign. The appearance of every element is governed by a \emph{template} for this element. Appropriate templates are installed by inner and outer themes, where the \emph{inner} themes only install templates for elements that are typically ``inside the main text,'' while \emph{outer} themes install templates for elements ``around the main text.'' Thus, from the templates's point of view, there is no real difference between inner and outer themes.
\subsection{Inner Themes}
An inner theme installs templates that dictate how the following elements are typeset:
\begin{itemize}
\item Title and part pages.
\item Itemize environments.
\item Enumerate environments.
\item Description environments.
\item Block environments.
\item Theorem and proof environments.
\item Figures and tables.
\item Footnotes.
\item Bibliography entries.
\end{itemize}
In the following examples, the color themes |seahorse| and |rose| are used to show where and how background colors are honored. Furthermore, background colors have been specified for all elements that honor them in the default theme. In the default color theme, all of the large rectangular areas are transparent.
\begin{innerthemeexample}{default}
The default element theme is quite sober. The only extravagance is the fact that a little triangle is used in |itemize| environments instead of the usual dot.
In some cases the theme will honor background color specifications for elements. For example, if you set the background color for block titles to green, block titles will have a green background. The background specifications are currently honored for the following elements:
\begin{itemize}
\item Title, author, institute, and date fields in the title page.
\item Block environments, both for the title and for the body.
\end{itemize}
This list may increase in the future.
\end{innerthemeexample}
\begin{innerthemeexample}{circles}
In this theme, |itemize| and |enumerate| items start with a small circle. Likewise, entries in the table of contents start with circles.
\end{innerthemeexample}
\begin{innerthemeexample}{rectangles}
In this theme, |itemize| and |enumerate| items and table of contents entries start with small rectangles.
\end{innerthemeexample}
\begin{innerthemeexample}[\oarg{options}]{rounded}
In this theme, |itemize| and |enumerate| items and table of contents entries start with small balls. If a background is specified for blocks, then the corners of the background rectangles will be rounded off. The following \meta{options} may be given:
\begin{itemize}
\item \declare{|shadow|} adds a shadow to all blocks.
\end{itemize}
\end{innerthemeexample}
\begin{innerthemeexample}{inmargin}
The idea behind this theme is to have ``structuring'' information on the left and ``normal'' information on the right. To this end, blocks are redefined such that the block title is shown on the left and the block body is shown on the right.
The code used to place text in the margin is a bit fragile. You may often need to adjust the spacing ``by hand,'' so use at your own risk.
Itemize items are redefined such that they appear on the left. However, only the position is changed by changing some spacing parameters; the code used to draw the items is not changed otherwise. Because of this, you can load another inner theme first and then load this theme afterwards.
This theme is a ``dirty'' inner theme since it messes with things that an inner theme should not mess with. In particular, it changes the width of the left sidebar to a large value. However, you can still use it together with most outer themes.
Using columns inside this theme is problematic. Most of the time, the result will not be what you expect.
\end{innerthemeexample}
\subsection{Outer Themes}
An outer theme dictates (roughly) the overall layout of frames. It specifies where any navigational elements should go (like a mini table of contents or navigational mini frames) and what they should look like. Typically, an outer theme specifies how the following elements are rendered:
\begin{itemize}
\item The head- and footline.
\item The sidebars.
\item The logo.
\item The frame title.
\end{itemize}
An outer theme will not specify how things like |itemize| environments should be rendered---that is the job of an inner theme.
In the following examples the color theme |seahorse| is used. Since the default color theme leaves most backgrounds empty, most of the outer themes look too unstructured with the default color theme.
\begin{outerthemeexample}{default}
The default layout theme is the most sober and minimalistic theme around. It will flush left the frame title and it will not install any head- or footlines. However, even this theme honors the background color specified for the frame title. If a color is specified, a bar occupying the whole page width is put behind the frame title. A background color of the frame subtitle is ignored.
\end{outerthemeexample}
\begin{outerthemeexample}{infolines}
This theme installs a headline showing the current section and the current subsection. It installs a footline showing the author's name, the institution, the presentation's title, the current date, and a frame count. This theme uses only little space.
The colors used in the headline and footline are drawn from |palette primary|, |palette secondary|, and |primary tertiary| (see Section~\ref{section-colors} for details on how to change these).
\end{outerthemeexample}
\begin{outerthemeexample}[\oarg{options}]{miniframes}
This theme installs a headline in which a horizontal navigational bar is shown. This bar contains one entry for each section of the presentation. Below each section entry, small circles are shown that represent the different frames in the section. The frames are arranged subsection-wise, that is, there is a line of frames for each subsection. If the class option |compress| is given, the frames will instead be arranged in a single row for each section. The navigation bars draws its color from |section in head/foot|.
Below the navigation bar, a line is put showing the title of the current subsection. The color is drawn from |subsection in head/foot|.
At the bottom, two lines are put that contain information such as the author's name, the institution, or the paper's title. What is shown exactly is influenced by the \meta{options} given. The colors are drawn from the appropriate \beamer-colors like |author in head/foot|.
At the top and bottom of both the head- and footline and between the navigation bar and the subsection name, separation lines are drawn \emph{if} the background color of |separation line| is set. This separation line will have a height of 3pt. You can get even more fine-grained control over the colors of the separation lines by setting appropriate colors like |lower separation line head|.
The following \meta{options} can be given:
\begin{itemize}
\item \declare{|footline=empty|} suppresses the footline (default).
\item \declare{|footline=authorinstitute|} shows the author's name and the institute in the footline.
\item \declare{|footline=authortitle|} shows the author's name and the title in the footline.
\item \declare{|footline=institutetitle|} shows the institute and the title in the footline.
\item \declare{|footline=authorinstitutetitle|} shows the author's name, the institute, and the title in the footline.
\item \declare{|subsection=|\meta{true or false}} shows or suppresses line showing the subsection in the headline. It is shown by default. If the document does not use subsections, this option should be set |false|.
\end{itemize}
\end{outerthemeexample}
\begin{outerthemeexample}[\oarg{options}]{smoothbars}
This theme behaves very much like the |miniframes| theme, at least with respect to the headline. The only differences are that smooth transitions are installed between the background colors of the navigation bar, the (optional) bar for the subsection name, and the background of the frame title. No footline is created. You can get the footlines of the |miniframes| theme by first loading that theme and then loading the |smoothbars| theme.
The following \meta{options} can be given:
\begin{itemize}
\item \declare{|subsection=|\meta{true or false}} shows or suppresses line showing the subsection in the headline. It is shown by default.
\end{itemize}
\end{outerthemeexample}
\begin{outerthemeexample}[\oarg{options}]{sidebar}
In this layout, a sidebar is shown that contains a small table of contents with the current section, subsection, or subsection highlighted. The frame title is vertically centered in a rectangular area at the top that always occupies the same amount of space in all frames. Finally, the logo is shown in the ``corner'' resulting from the sidebar and the frame title rectangle.
There are several ways of modifying the layout using the \meta{options}. If you set the width of the sidebar to 0pt, it is not shown, giving you a layout in which the frame title does not ``wobble'' since it always occupies the same amount of space on all slides. Conversely, if you set the height of the frame title rectangle to 0pt, the rectangular area is not used and the frame title is inserted normally (occupying as much space as needed on each slide).
The background color of the sidebar is taken from |sidebar|, the background color of the frame title from |frametitle|, and the background color of the logo corner from |logo|.
The colors of the entries in the table of contents are drawn from the \beamer-color |section in sidebar| and |section in sidebar current| as well as the corresponding \beamer-colors for subsections. If an entry does not fit on a single line it is automatically ``linebroken.''
The following \meta{options} may be given:
\begin{itemize}
\item
\declare{|height=|\meta{dimension}} specifies the height of the frame title rectangle. If it is set to 0pt, no frame title rectangle is created. Instead, the frame title is inserted normally into the frame. The default is 2.5 base line heights of the frame title font. Thus, there is about enough space for a two-line frame title plus a one-line subtitle.
\item
\declare{|hideothersubsections|} causes all subsections except those of the current section to be suppressed in the table of contents. This is useful if you have lots of subsections.
\item
\declare{|hideallsubsections|} causes all subsections to be suppressed in the table of contents.
\item
\declare{|left|} puts the sidebar on the left side. Note that in a left-to-right reading culture this is the side people look first. Note also that this table of contents is usually \emph{not} the most important part of the frame, so you do not necessarily want people to look at it first. Nevertheless, it is the default.
\item
\declare{|right|} puts the sidebar of the right side.
\item
\declare{|width=|\meta{dimension}} specifies the width of the sidebar. If it is set to 0pt, it is completely suppressed. The default is 2.5 base line heights of the frame title font.
\end{itemize}
\end{outerthemeexample}
\begin{outerthemeexample}{split}
This theme installs a headline in which, on the left, the sections of the talk are shown and, on the right, the subsections of the current section. If the class option |compress| has been given, the sections and subsections will be put in one line; normally there is one line per section or subsection.
The footline shows the author on the left and the talk's title on the right.
The colors are taken from |palette primary| and |palette quarternary|.
\end{outerthemeexample}
\begin{outerthemeexample}{shadow}
This layout theme extends the |split| theme by putting a horizontal shading behind the frame title and adding a little ``shadow'' at the bottom of the headline.
\end{outerthemeexample}
\begin{outerthemeexample}[\oarg{options}]{tree}
In this layout, the headline contains three lines that show the title of the current talk, the current section in this talk, and the current subsection in the section. The colors are drawn from |title in head/foot|, |section in head/foot|, and |subsection in head/foot|.
In addition, separation lines of height 3pt are shown above and below the three lines \emph{if} the background of |separation line| is set. More fine-grained control of the colors of these lines can be gained by setting |upper separation line head| and |lower separation line head|.
The following \meta{options} may be given:
\begin{itemize}
\item
\declare{|hooks|} causes little ``hooks'' to be drawn in front of the section and subsection entries. These are supposed to increase the tree-like appearance.
\end{itemize}
\end{outerthemeexample}
\begin{outerthemeexample}{smoothtree}
This layout is similar to the |tree| layout. The main difference is that the background colors change smoothly.
\end{outerthemeexample}
\subsection{Changing the Templates Used for Different Elements of a Presentation}
\label{section-templates}
This section explains how \beamer's template management works.
\subsubsection{Overview of Beamer's Template Management}
If you only wish to modify the appearance of a single or few elements, you do not need to create a whole new inner or outer theme. Instead, you can modify the appropriate template.
A template specifies how an element of a presentation is typeset. For example, the |frametitle| template dictates where the frame title is put, which font is used, and so on.
As the name suggests, you specify a template by writing the exact \LaTeX\ code you would also use when typesetting a single frame title by hand. Only, instead of the actual title, you use the command |\insertframetitle|.
\example
Suppose we would like to have the frame title typeset in red, centered, and boldface. If we were to typeset a single frame title by hand, it might be done like this:
\begin{verbatim}
\begin{frame}
\begin{centering}
\color{red}
\textbf{The Title of This Frame.}
\par
\end{centering}
Blah, blah.
\end{frame}
\end{verbatim}
In order to typeset the frame title in this way on all slides, in the simplest case we can change the frame title template as follows:
\begin{verbatim}
\setbeamertemplate{frametitle}
{
\begin{centering}
\color{red}
\textbf{\insertframetitle}
\par
\end{centering}
}
\end{verbatim}
We can then use the following code to get the desired effect:
\begin{verbatim}
\begin{frame}
\frametitle{The Title of This Frame.}
Blah, blah.
\end{frame}
\end{verbatim}
When rendering the frame, \beamer\ will use the code of the frame title template to typeset the frame title and it will replace every occurrence of |\insertframetitle| by the current frame title.
We can take this example a step further. It would be nicer if we did not have to ``hardwire'' the color of the frametitle, but if this color could be specified independently of the code for the template. This way, a color theme could change this color. Since this is a problem that is common to most templates, \beamer\ will automatically setup the \beamer-color |frametitle| when the template |frametitle| is used. Thus, we can remove the |\color{red}| command if we set the \beamer-color |frametitle| to red at some point.
\begin{verbatim}
\setbeamercolor{frametitle}{fg=red}
\setbeamertemplate{frametitle}
{
\begin{centering}
\textbf{\insertframetitle}
\par
\end{centering}
}
\end{verbatim}
Next, we can also make the font ``themable.'' Just like the color, the \beamer-font |frametitle| is installed before the |frametitle| template is typeset. Thus, we should rewrite the code as follows:
\begin{verbatim}
\setbeamercolor{frametitle}{fg=red}
\setbeamerfont{frametitle}{series=\bfseries}
\setbeamertemplate{frametitle}
{
\begin{centering}
\insertframetitle\par
\end{centering}
}
\end{verbatim}
Users, themes, or whoever can now easily change the color or font of the frametitle without having to mess with the code used to typeset it.
\articlenote
In |article| mode, most of the template mechanism is switched off and has no effect. However, a few templates are also available. If this is the case, it is specially indicated.
\smallskip
Here are a few hints that might be helpful when you wish to set a template:
\begin{itemize}
\item
Usually, you might wish to copy code from an existing template. The code often takes care of some things that you may not yet have thought about. The default inner and outer themes might be useful starting points. Also, the file |beamerbaseauxtemplates.sty| contains interesting ``auxilliary'' templates.
\item
When copying code from another template and when inserting this code in the preamble of your document (not in another style file), you may have to ``switch on'' the at-character (|@|). To do so, add the command |\makeatletter| before the |\setbeamertemplate| command and the command |\makeatother| afterward.
\item
Most templates having to do with the frame components (headlines, sidebars, etc.)\ can only be changed in the preamble. Other templates can be changed during the document.
\item
The height of the headline and footline templates is calculated automatically. This is done by typesetting the templates and then ``having a look'' at their heights. This recalculation is done right at the beginning of the document, \emph{after} all packages have been loaded and even \emph{after} these have executed their |\AtBeginDocument| initialization.
\item
Getting the boxes right inside any template is often a bit of a hassle. You may wish to consult the \TeX\ book for the glorious details on ``Making Boxes.'' If your headline is simple, you might also try putting everything into a |pgfpicture| environment, which makes the placement easier.
\end{itemize}
\subsubsection{Using Beamer's Templates}
As a user of the \beamer\ class you typically do not ``use'' or ``invoke'' templates yourself, directly. For example, the frame title template is automatically invoked by \beamer\ somewhere deep inside the frame typesetting process. The same is true of most other templates. However, if, for whatever reason, you wish to invoke a template yourself, you can use the following command.
\begin{command}{\usebeamertemplate\opt{|***|}\marg{element name}}
If none of the stars is given, the text of the \meta{element name} is directly inserted at the current position. This text should previously have been specified using the |\setbeamertemplate| command. No text is inserted if this command has not been called before.
\example
\begin{verbatim}
\setbeamertemplate{my template}{correct}
...
Your answer is \usebeamertemplate{my template}.
\end{verbatim}
If you add one star, three things happen. First, the template is put inside a \TeX-group, thereby limiting most side effects of commands used inside the template. Second, inside this group the \beamer-color named \meta{element name} is used and the foreground color is selected. Third, the \beamer-font \meta{element name} is also used. This one-starred version is usually the best version to use.
If you add a second star, nearly the same happens as with only one star. However, in addition, the color is used with the command |\setbeamercolor*|. This causes the colors to be reset to the normal text color if no special foreground or background is specified by the \beamer-color \meta{element name}. Thus, in this twice-starred version, the color used for the template is guaranteed to be independent of the color that was currently in use when the template is used.
Finally, adding a third star will also cause a star to be added to the |\setbeamerfont*| command. This causes the font used for the template also to be reset to normal text, unless the \beamer-font \meta{element name} specifies things differently. This three-star version is the ``most protected'' version available.
\end{command}
\begin{command}{\ifbeamertemplateempty\marg{beamer template name}\marg{executed if empty}\marg{executed otherwise}}
This command checks whether a template is defined and set to a non-empty text. If the text is empty or the template is not defined at all, \meta{executed if empty} is executed. Otherwise, \meta{executed otherwise} is executed.
\end{command}
\begin{command}{\expandbeamertemplate\marg{beamer template name}}
This command does the same as |\usebeamertemplate|\marg{beamer template name}. The difference is that this command performs a direct expansion and does not scan for a star. This is important inside, for example, an |\edef|. If you don't know the difference between |\def| and |\edef|, you won't need this command.
\end{command}
\subsubsection{Setting Beamer's Templates}
To set a \beamer-template, you can use the following command:
\begin{command}{\setbeamertemplate\marg{element name}\oarg{predefined option}\meta{args}}
In the simplest case, if no \meta{predefined option} is given, the \meta{args} must be a single argument and the text of the template \meta{element name} is setup to be this text. Upon later invocation of the template by the command |\usebeamertemplate| this text is used.
\example
\begin{verbatim}
\setbeamertemplate{answer}{correct}
...
Your answer is \usebeamertemplate*{answer}.
\end{verbatim}
If you specify a \meta{predefined option}, this command behaves slightly differently. In this case, someone has used the command |\defbeamertemplate| to predefine a template for you. By giving the name of this predefined template as the optional parameter \meta{predefined option}, you cause the template \meta{element name} to be set to this template.
\example
|\setbeamertemplate{bibliography item}[book]| causes the bibliography items to become little book icons. This command causes a subsequent call of |\usebeamertemplate{bibliography item}| to insert the predefined code for inserting a book.
Some predefined template options take parameters themselves. In such a case, the parameters are given as \meta{args}.
\example
The \meta{predefined option} |grid| for the template |background| takes an optional argument:
\begin{verbatim}
\setbeamertemplate{background}[grid][step=1cm]
\end{verbatim}
In the example, the second argument in square brackets is the optional argument.
In the descriptions of elements, if there are possible \meta{predefined option}, the description shows how the \meta{predefined option} can be used together with its arguments, but the |\setbeamertemplate{xxxx}| is omitted. Thus, the above example would be documented in the description of the |background| element like this:
\begin{itemize}
\itemoption{grid}{\oarg{step options}} causes a light grid to be \dots
\end{itemize}
\end{command}
\begin{command}{\addtobeamertemplate\marg{element name}\marg{pre-text}\marg{post-text}}
This command adds the \meta{pre-text} before the text that is currently installed as the template \meta{element name} and the \meta{post-text} after it. This allows you a limited form of modification of existing templates.
\example
The following commands have the same effect:
\begin{verbatim}
\setbeamertemplate{my template}{Hello world!}
\setbeamertemplate{my template}{world}
\addtobeamertemplate{my template}{Hello }{!}
\end{verbatim}
If a new template is installed, any additions will be deleted. On the other hand, you can repeatedly use this command to add multiple things.
\end{command}
\begin{command}{\defbeamertemplate\sarg{mode specification}\opt{|*|}\marg{element name}\marg{predefined option}\\ \oarg{argument number}\oarg{default optional argument}\marg{predefined text}\\ \opt{|[action]|\marg{action command}}}
This command installs a \emph{predefined option} for the template \meta{element name}. Once this command has been used, users can access the predefined template using the |\setbeamertemplate| command.
\example
|\defbeamertemplate{itemize item}{double arrow}{$\Rightarrow$}|
After the above command has been invoked, the following two commands will have the same effect:
\begin{verbatim}
\setbeamertemplate{itemize item}{$\Rightarrow$}
\setbeamertemplate{itemize item}[double arrow]
\end{verbatim}
Sometimes, a predefined template needs to get an argument when it is installed. Suppose, for example, we want to define a predefined template that draws a square as the itemize item and we want to make this size of this square configurable. In this case, we can specify the \meta{argument number} of the predefined option the same way one does for the |\newcommand| command:
\begin{verbatim}
\defbeamertemplate{itemize item}{square}[1]{\hrule width #1 height #1}
%% The following have the same effect:
\setbeamertemplate{itemize item}[square]{3pt}
\setbeamertemplate{itemize item}{\hrule width 3pt height 3pt}
\end{verbatim}
As for the |\newcommand| command, you can also specify a \meta{default optional argument}:
\begin{verbatim}
\defbeamertemplate{itemize item}{square}[1][1ex]{\hrule width #1 height #1}
%% The following have the same effect:
\setbeamertemplate{itemize item}[square][3pt]
\setbeamertemplate{itemize item}{\hrule width 3pt height 3pt}
%% So do the following:
\setbeamertemplate{itemize item}[square]
\setbeamertemplate{itemize item}{\hrule width 1ex height 1ex}
\end{verbatim}
The starred version of the command installs the predefined template option, but then immediately calls |\setbeamertemplate| for this option. This is useful for the default templates. If there are any arguments necessary, these are set to |\relax|.
In certain cases, if a predefined template option is chosen, you do not only wish the template text to be installed, but certain extra ``actions'' must also be taken once. For example, a shading must be defined that should not be redefined every time the shading is used later on. To implement such ``actions,'' you can use the optional argument \meta{action} following the keyword |[action]|. Thus, after the normal use of the |\defbeamertemplate| you add the text |[action]| and then any commands that should be executed once when the \meta{predefined option} is selected by the |\setbeamertemplate| command.
\example
\begin{verbatim}
\defbeamertemplate{background canvas}{my shading}[2]
{
\pgfuseshading{myshading}% simple enough
}
[action]
{
\pgfdeclareverticalshading{myshading}{\the\paperwidth}
{color(0cm)=(#1); color(\the\paperheight)=(#2)}
}
...
\setbeamertemplate{background canvas}{myshading}{red!10}{blue!10}
%% Defines the shading myshading right here. Subsequent calls to
%% \usebeamertemplate{background canvas} will yield
%% ``\pgfuseshading{myshading}''.
\end{verbatim}
\articlenote
Normally, this command has no effect in |article| mode. However, if a \meta{mode specification} is given, this command is applied for the specified modes. Thus, this command behaves like the |\\| command, which also gets the implicit mode specification |<presentation>| if no other specification is given.
\example
|\defbeamertemplate{my template}{default}{something}| has no effect in |article| mode.
\example
|\defbeamertemplate<article>{my template}{default}{something}| has no effect in |presentation| modes, but has an effect in |article| mode.
\example
|\defbeamertemplate<all>{my template}{default}{something}| applies to all modes.
\end{command}
It is often useful to have access to the same template option via different names. For this, you can use the following command to create aliases:
\begin{command}{\defbeamertemplatealias\marg{element name}\marg{new predefined option name}\marg{existing predefined option name}}
Causes the two predefined options to have the same effect.
\end{command}
There is no inheritance relation among templates as there is for colors and fonts. This is due to the fact the templates for one element seldom make sense for another. However, sometimes certain elements ``behave similarly'' and one would like a |\setbeamertemplate| to apply to a whole set of templates via inheritance. For example, one might want that |\setbeamertemplate{items}[circle]| causes all items to use the |circle| option, though the effects for the |itemize item| as opposed to the |itemize subsubitem| as opposed to |enumerate item| must be slightly different.
The \beamer-template mechanism implements a simple form of inheritance via \emph{parent templates}. In element descriptions, parent templates are indicated via a check mark in parentheses.
\begin{command}{\defbeamertemplateparent\marg{parent template name}\oarg{predefined option name}\marg{child template list}\\ \oarg{argument number}\oarg{default optional argument}\marg{arguments for children}}
The effect of this command is that whenever someone calls |\setbeamertemplate{|\meta{parent template name}|}{|\meta{args}|}|, the command |\setbeamertemplate{|\meta{child template name}|}{|\meta{args}|}| is called for each \meta{child template name} in the \meta{child template list}.
The \meta{arguments for children} come into play if the |\setbeamertemplate| command is called with a predefined option name (not necessarily the same as the \meta{predefined option name}, we'll come to that). If |\setbeamertemplate| is called with some predefined option name, the children are called with the \meta{arguments for children} instead. Let's look at two examples:
\example
The following is the typical, simple usage:
\begin{verbatim}
\defbeamertemplateparent{itemize items}{itemize item,itemize subitem,itemize subsubitem}
{}
%% The following command has the same effect as the three commands below:
\setbeamertemplate{itemize items}[circle]
\setbeamertemplate{itemite item}[circle] % actually, the ``empty'' argument is added
\setbeamertemplate{itemize subitem}[circle]
\setbeamertemplate{itemize subsubitem}[circle]
\end{verbatim}
\example
In the following case, an argument is passed to the children:
\begin{verbatim}
\defbeamertemplateparent{sections/subsections in toc shaded}
{section in toc shaded,subsection in toc shaded}[1][20]
{[#1]}
%% The following command has the same effect as the two commands below:
\setbeamertemplate{sections/subsection in toc shaded}[default][35]
\setbeamertemplate{section in toc shaded}[default][35]
\setbeamertemplate{subsection in toc shaded}[default][35]
%% Again:
\setbeamertemplate{sections/subsection in toc shaded}[default]
\setbeamertemplate{section in toc shaded}[default][20]
\setbeamertemplate{subsection in toc shaded}[default][20]
\end{verbatim}
In detail, the following happens: When |\setbeamertemplate| is encountered for a parent template, \beamer\ first checks whether a predefined option follows. If not, a single argument is read and |\setbeamertemplate| is called for all children for this template. If there is a predefined template option set, \beamer\ evaluates the \meta{argument for children}. It may contain parameters like |#1| or |#2|. These parameters are filled with the arguments that follow the call of |\setbeamertemplate| for the parent template. The number of arguments must be the number given as \meta{argument number}. An optional argument can also be specified in the usual way. Once the \meta{arguments for the children} have been computed, |\setbeamertemplate| is called for all children for the predefined template and with the computed arguments.
You may wonder what happens when certain predefined options take a certain number of arguments, but another predefined option takes a different number of arguments. In this case, the above-described mechanism cannot differentiate between the predefined options and it is unclear which or even how many arguments should be contained in \meta{arguments for children}. For this reason, you can give the optional argument \meta{predefined option name} when calling |\defbeamertemplateparent|. If this optional argument is specified, the parenthood of the template applies only to this particular \meta{predefined option name}. Thus, if someone calls |\setbeamertemplate| for this \meta{predefined option name}, the given \meta{argument for children} is used. For other predefined option names a possibly different definition is used. You can imaging that leaving out the optional \meta{predefined option name} means ``this \meta{argument for children} applies to all predefined option names that have not been specially defined differently.''
\end{command}
|
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https://joshua.smcvt.edu/linearalgebra/MatrixArithmetic.tex | smcvt.edu | CC-MAIN-2021-39 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-39/segments/1631780057202.68/warc/CC-MAIN-20210921101319-20210921131319-00696.warc.gz | 362,600,216 | 13,984 | \documentclass[12pt]{article}
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\begin{document}
\begin{center}
Matrix Arithmetic Updated \\
Harold W. Ellingsen Jr. \\
SUNY Potsdam \\
ellinghw@potsdam.edu
\end{center}
This supplements \underline{Linear Algebra} by Hefferon, which is lacking a chapter of matrix arithmetic for its own sake. Instructors who wish to introduce these manipulations earlier and without the rigor of linear transformations may find this useful. Please be sure to cover the text's Chapter One first.
This material is Free, including the LaTeX source, under the Creative Commons Attribution-ShareAlike 2.5 License. The latest version should be the one on this site.
Any feedback on the note is appreciated. \\[.25in]
\noindent
{\Large Matrix Arithmetic Updated} \\
\noindent
In this note we explore matrix arithmetic for its own sake. The author introduces it in Chapter Three using linear transformations. While his approach is quite rigorous, matrix arithmetic can be studied after Chapter One. This note assumes that Chapter One has been completed.
For a shortcut notation instead of writing a matrix $A$ as $\mtrx{a}$, we will write $A = (a_{ij})_{m \times n}$ or just $A = (a_{ij})$ if the size of $A$ is understood. %We understand that $1 \leq i \leq m$ and $1 \leq j \leq n$.
When given a set of objects in mathematics, there are two basic questions we should ask: When are two objects equal? and How can we combine two objects to produce a third object? For the first question we have the following definition.
\begin{framed}
{\bf Definition 1} \\
Two matrices $A = (a_{ij})$ and $B = (b_{ij})$ are {\itshape equal}, denoted by $A=B$, provided they have the same size and their corresponding entries are equal, that is, their sizes are both $m \times n$ and for each $1 \leq i \leq m$ and $1 \leq j \leq n$, $a_{ij} = b_{ij}$.
\end{framed}
{\bf Example 1}
\be
\it Let $A = \lp \begin{array}{rrr} 1 & -9 & 7 \\ 0 & 1 & -5 \end{array} \rp$ and $B = \lp \begin{array}{rrr} 1 & -9 \\ 0 & 1 \\ 7 & -5 \end{array} \rp$. Are $A$ and $B$ equal? \\
Since the size of $A$ is $2 \times 3$ and that of $B$ is $3 \times 2$, $A \not= B$. Do note, however, that they have the same entries.
\it Find all values of $x$ and $y$ so that $\begin{pmatrix} x^2 & y-x \\ 0 & y^2 \end{pmatrix} = \begin{pmatrix} 1 & x-y \\ x+1 & 1 \end{pmatrix}$. \\
We see that the size of each matrix is $2 \times 2$. So we set their corresponding entries equal:
\begin{align*}
x^2 &= 1 & y-x &= x-y \\
0 &= x+1 & y^2 &= 1.
\end{align*}
\noindent
We see that $x = \pm 1$ and $y = \pm 1$. From $0 = x + 1$, we get that $x$ must be $-1$. From $y-x = x-y$, we get that $2y = 2x$ and so $x=y$. Thus $y$ must also be $-1$.
\ee
As for the second question, we have been doing this for quite a while now: Adding, subtracting, multiplying, and dividing(when possible) real numbers. So how can we add and subtract two matrices? Eventually we will multiply matrices, but for now we consider another multiplication. Here are the definitions.
\begin{framed}
{\bf Definition 2} \\
Let $A = (a_{ij})$ and $B = (b_{ij})$ be $m \times n$ matrices. We define their {\itshape sum}, denoted by $A + B$, and their {\itshape difference}, denoted by $A - B$, to be the respective matrices $(a_{ij} + b_{ij})$ and $(a_{ij} - b_{ij})$. We define {\itshape scalar multiplication} by for any $r \in \R$, $rA$ is the matrix $(ra_{ij})$.
\end{framed}
These definitions should appear quite natural: When two matrices have the same size, we just add or subtract their corresponding entries, and for the scalar multiplication, we just multiply each entry by the scalar. Just a note: Since multiplication of real numbers is commutative, we have that $rA = Ar$ for any real number $r$ and matrix $A$. Here are some examples.
{\bf Example 2} \\
Let
$A = \lp \begin{array}{rr} 2 & 3 \\ -1 & 2 \end{array} \rp$,
$B = \lp \begin{array}{rr} -1 & 2 \\ 6 & -2 \end{array} \rp$, and
$C = \lp \begin{array}{rrr} 1 & 2 & 3 \\ -1 & -2 & -3 \end{array} \rp$.
Compute each of the following, if possible. If a computation is not possible, explain why it is not.
\be
\it $A + B$. \\
Since $A$ and $B$ are both $2 \times 2$ matrices, we can add them. Here we go:
\begin{align*}
A + B &= \lp \begin{array}{rr} 2 & 3 \\ -1 & 2 \end{array} \rp +
\lp \begin{array}{rr} -1 & 2 \\ 6 & -2 \end{array} \rp =
\lp \begin{array}{cc} 2+(-1) & 3+2 \\ -1+6 & 2+(-2) \end{array} \rp = \lp \begin{array}{rr} 1 & 5 \\ 5 & 0 \end{array} \rp.
\end{align*}
\it $B - A$. \\
Since $A$ and $B$ are both $2 \times 2$ matrices, we can subtract them. Here we go:
\begin{align*}
B - A &= \lp \begin{array}{rr} -1 & 2 \\ 6 & -2 \end{array} \rp - \lp \begin{array}{rr} 2 & 3 \\ -1 & 2 \end{array} \rp = \lp \begin{array}{cc} -1-2 & 2-3 \\ 6-(-1) & -2-2 \end{array} \rp = \lp \begin{array}{rr} -3 & -1 \\ 7 & -4 \end{array} \rp.
\end{align*}
\it $B + C$. \\
No can do. $B$ and $C$ have different sizes: $B$ is $2 \times 2$ and $C$ is $2 \times 3$.
\it $4C$. \\
We just multiply each entry of $C$ by $4$:
\begin{align*}
4C &= 4 \lp \begin{array}{rrr} 1 & 2 & 3 \\ -1 & -2 & -3 \end{array} \rp = \lp \begin{array}{ccc} 4 (1) & 4 (2) & 4 (3) \\ 4 (-1) & 4 (-2) & 4 (-3) \end{array} \rp
= \lp \begin{array}{rrr} 4 & 8 & 24 \\ -4 & -8 & -24 \end{array} \rp.
\end{align*}
\it $2A - 3B$. \\
Since scalar multiplication does not affect the size of a matrix, the matrices $2A$ and $3B$ have the same size and so we can subtract them. We'll do the scalar multiplication first and then the subtraction. Here we go:
\begin{align*}
2A - 3B &=
2 \lp \begin{array}{rr} 2 & 3 \\ -1 & 2 \end{array} \rp -
3 \lp \begin{array}{rr} -1 & 2 \\ 6 & -2 \end{array} \rp =
\lp \begin{array}{rr} 4 & 6 \\ -2 & 4 \end{array} \rp -
\lp \begin{array}{rr} -3 & 6 \\ 18 & -6 \end{array} \rp =
\lp \begin{array}{rr} 7 & 0 \\ -20 & 10 \end{array} \rp.
\end{align*}
\ee
Matrix arithmetic has some of the same properties as real number arithmetic.
\begin{framed}
{\bf Properties of Matrix Arithmetic} \\
Let $A$, $B$, and $C$ be $m \times n$ matrices and $r, s \in \R$.
\bttt
1. $A + B = B + A$ \> Matrix addition is commutative. \> \\[.05in]
2. $A + (B + C) = (A + B) + C$ \> Matrix addition is associative. \> \\[.05in]
3. $r(A + B) = rA + rB$ \> Scalar multiplication distributes over matrix addition. \> \\[.05in]
4. $(r + s)A = rA + sA$ \> Real number addition distributes over scalar multiplication. \> \\[.05in]
5. $(rs)A = r(sA)$ \> An associativity for scalar multiplication. \> \\[.05in]
6. There is a unique $m \times n$ matrix $\Theta$ such that for any $m \times n$ matrix $M$, $M + \Theta = M$. \> \ \> \\[.05in]
7. For every $m \times n$ matrix $M$ there is a unique $m \times n$ matrix $N$ such that $M + N = \Theta$. \> \ \>
\etb
\end{framed}
The above $\Theta$ is suggestively called the $m \times n$ {\itshape zero matrix}.
The above $N$ is suggestively called the {\itshape negative} of $M$ and is so denoted by $-M$. Let's prove something. How about that real number addition distributes over matrix addition, there is a unique zero matrix, and each matrix has a unique negative? You should prove the rest at some point in your life.
{\bf Proof} \\
Let $A = (a_{ij})$ be an $m \times n$ matrix and $r, s \in \R$. By definition, $(r + s)A$ and $rA + sA$ have the same size. Now we must show that their corresponding entries are equal. Let $1 \leq i \leq m$ and $1 \leq j \leq n$. Then the $ij$-entry of $(r + s)A$ is $(r + s)a_{ij}$. Using the usual properties of real number arithmetic, we have $(r + s)(a_{ij}) = ra_{ij} + sa_{ij}$, which is the sum of the $ij$-entries of $rA$ and $sA$, that is, it's the $ij$-entry of $rA + sA$. Hence $(r + s)A = rA + sA$.
Let $M = (m_{ij})$ be an $m \times n$ matrix and let $\Theta$ be the $m \times n$ matrix all of whose entries are $0$. By assumption $M$, $\Theta$, and $M + \Theta$
have the same size. Notice that the $ij$-entry of $M + \Theta$ is $m_{ij} + 0$. This is exactly the $ij$-entry of $M$. Hence $M + \Theta = M$. For uniqueness, suppose that $\Psi$ is an $m \times n$ matrix with the property that for any $m \times n$ matrix $C$, $C + \Psi = C$. Then $\Theta = \Theta + \Psi$ by the property of $\Psi$. But by the property of $\Theta$, $\Psi = \Psi + \Theta$. Since matrix addition is commutative, we see that $\Theta = \Psi$. Hence $\Theta$ is unique.
Let $N = (-m_{ij})$. Now this makes sense as each $m_{ij}$ is a real number and so its negative is also a real number. Notice that $M$, $N$, $M + N$, and $\Theta$ all have the same size. Now the $ij$-entry of $M + N$ is $m_{ij} + (-m_{ij}) = 0$, the $ij$-entry of $\Theta$. Hence a desired $N$ exists. For uniqueness suppose that $P$ is an $m \times n$ matrix with the property that $M + P = \Theta$. Then
\begin{align*}
N &= N + \Theta && \text{as $\Theta$ is the zero matrix} \\
&= N + (M + P) && \text{as $M + P = \Theta$} \\
&= (N + M) + P && \text{by associativity of matrix addition} \\
&= \Theta + P && \text{as $N + M = \Theta$} \\
&= P && \text{as $\Theta$ is the zero matrix}.
\end{align*}
Hence this $N$ is unique.
Now we will multiply matrices, but not in the way you're thinking. We will NEVER just simply multiply the corresponding entries! What we do is an extension of the dot product of vectors. (If you're not familiar with this, don't worry about it.) First we will multiply a row by a column and the result will be a real number(or scalar).
\begin{framed}
{\bf Definition 3} \\
We take a row vector $\rowp{a}$ with $p$ entries and a column vector $\colp{b}$ with $p$ entries and define their {\itshape product}, denoted by
$\rowp{a} \colp{b}$, to be the real number $a_1 b_1 + a_2 b_2 + \cdots + a_p b_p$. Notice that we're just taking the sum of the products of the corresponding entries and that we may view a real number as a $1 \times 1$ matrix.
\end{framed}
Let's do a couple of examples.
{\bf Example 3} \\
Multiply the row and column vectors.
\be
\it $\rowth{2}{3}{4} \colth{3}{4}{5} = 2(3) + 3(4) + 4(5) = 38$.
\it $\rowf{-1}{2}{-2}{3} \colf{2}{-2}{-1}{2} = -1(2) + 2(-2) + (-2)(-1) + 3(2) = 2$.
\ee
Now we'll multiply a general matrix by a column. After all, we can view a matrix as several row vectors of the same size put together. To do such a multiplication, the number of entries in each row must be the number of entries in the column and then we multiply each row of the matrix by the column.
\begin{framed}
{\bf Definition 4} \\
Let $A$ be an $m \times p$ matrix and $\bar{b}$ a $p \times 1$ column vector. We define their {\itshape product}, denoted by $A\bar{b}$, to be the $m \times 1$ column vector whose $i$-th entry, $1 \leq i \leq m$, is the product of the $i$-th row of $A$ and $\bar{b}$.
\end{framed}
Here are a couple of examples.
{\bf Example 4} \\
Multiply the matrix by the column.
\be
\it $\lp \begin{array}{rrr} 1 & 2 & 3 \\ -2 & 1 & 2 \end{array} \rp \colth{1}{2}{-3} = \colt{1(1)+2(2)+3(-3)}{-2(1)+1(2)+2(-3)} = \colt{-4}{-6}$.
\it $\lp \begin{array}{rr} 2 & -2 \\ 0 & 3 \\ -1 & 4 \end{array} \rp \colt{5}{-1} = \begin{pmatrix} 2(5)+(-2)(-1) \\ 0(5)+3(-1) \\ -1(5)+4(-1) \end{pmatrix} = \colth{12}{-3}{-9}$.
\ee
We now extend this multiplication to appropriately sized arbitrary matrices. We can view a matrix as several column vectors of the same size put together. To multiply a row by a column, we must be sure that they have the same number of entries. This means that the number of columns of our first matrix must be the number of rows of the second.
\begin{framed}
{\bf Definition 5} \\
Let $A$ be an $m \times p$ matrix and $B$ a $p \times n$ matrix. We define their {\itshape product}, denoted by $AB$, to be the $m \times n$ matrix whose $ij$-entry, $1 \leq i \leq m$ and $1 \leq j \leq n$, is the product of the $i$-th row of $A$ and the $j$-th column of $B$.
\end{framed}
Here are a few examples.
{\bf Example 5} \\
Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$,
$B = \lp \begin{array}{rr} 4 & -3 \\ -2 & 1 \end{array} \rp$,
$C = \lp \begin{array}{rrr} 2 & 2 & 9 \\ -1 & 0 & 8 \end{array} \rp$, and
$D = \begin{pmatrix} 1 & 2 & 3 \\ 5 & 2 & 3 \end{pmatrix}$.
Compute each of the following, if possible. If a computation is not possible, explain why it is not.
\be
\it $AB$. \\
This computation is possible, Since the size of both matrices is $2 \times 2$, the number of columns of the first is the same as the number of rows of the second. Note that the size of the product is $2 \times 2$. Here we go:
\begin{align*}
AB &= \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \lp \begin{array}{rr} 4 & -3 \\ -2 & 1 \end{array} \rp \\[.1in]
&=
\begin{pmatrix} \mbox{$1^{\text{st}}$ row of $A$ times $1^{\text{st}}$ column of $B$} & \mbox{$1^{\text{st}}$ row of $A$ times $2^{\text{nd}}$ column of $B$} \\
\mbox{$2^{\text{nd}}$ row of $A$ times $1^{\text{st}}$ column of $B$} & \mbox{$2^{\text{nd}}$ row of $A$ times $2^{\text{nd}}$ column of $B$} \end{pmatrix} \\[.1in]
&=
\begin{pmatrix} 1(4)+2(-2) & 1(-3)+2(1) \\ 3(4)+4(-2) & 3(-3)+4(1) \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 4 & -5 \end{pmatrix}.
\end{align*}
\it $BA$. \\
Again, the size of both matrices is $2 \times 2$, so this computation is possible and the size of the product is $2 \times 2$. Here we go again:
\begin{align*}
BA &= \lp \begin{array}{rr} 4 & -3 \\ -2 & 1 \end{array} \rp \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \\[.1in]
&=
\begin{pmatrix} \mbox{$1^{\text{st}}$ row of $B$ times $1^{\text{st}}$ column of $A$} & \mbox{$1^{\text{st}}$ row of $B$ times $2^{\text{nd}}$ column of $A$} \\
\mbox{$2^{\text{nd}}$ row of $B$ times $1^{\text{st}}$ column of $A$} & \mbox{$2^{\text{nd}}$ row of $B$ times $2^{\text{nd}}$ column of $A$} \end{pmatrix} \\[.1in]
&=
\begin{pmatrix} 4(1)+(-3)(3) & 4(2)+(-3)(4) \\ -2(1)+1(3) & -2(2)+1(4) \end{pmatrix} =\lp \begin{array}{rr} -5 & -4 \\ 1 & 0 \end{array} \rp.
\end{align*}
Did you notice what just happened? We have that $AB \not= BA$! Yes, it's true: Matrix multiplication is not commutative.
\it $CD$. \\
No can do. The size of the first matrix is $2 \times 3$ and the size of the second is also $2 \times 3$. The number of columns of the first, 3, is not the same as the number of rows of the second, 2.
\it $BC$. \\
This computation is possible. The size of the first matrix is $2 \times 2$ and the size of the second is $2 \times 3$. The number of columns of the first, 2, is the same as the number of rows of the second, 2. Then the size of this product is $2 \times 3$. Here we go:
\begin{align*}
BC &= \lp \begin{array}{rr} 4 & -3 \\ -2 & 1 \end{array} \rp
\lp \begin{array}{rrr} 2 & 2 & 9 \\ -1 & 0 & 8 \end{array} \rp \\
&= \begin{pmatrix} 4(2) + (-3)(-1) & 4(2) + (-3)(0) & 4(9) + (-3)(8) \\
-2(2) + 1(-1) & -2(2) + 1(0) & -2(9) + 1(8)
\end{pmatrix} \\
&=
\lp \begin{array}{rrr} 11 & 8 & 12 \\ -5 & -4 & -10 \end{array} \rp.
\end{align*}
\it $CB$. \\
No can do. The size of the first matrix is $2 \times 3$ and the size of the second is $2 \times 2$. The number of columns of the first, 3, is not the same as the number of rows of the second, 2.
Did you notice what just happened here? We were able to find the product $BC$, but not the product $CB$. It's not that $BC \neq CB$, it's that $CB$ isn't even possible. This is what makes matrix multiplication sooooo not commutative.
\ee
Now we state some more nice, and natural, properties of matrix arithmetic. \\
\begin{framed}
{\bf More Properties of Matrix Arithmetic} \\
Let $A$, $B$, and $C$ be matrices of the appropriate sizes and $r \in \R$. Then
\bttt
%1. $A + B = B + A$ \> Matrix addition is commutative. \> \\[.05in]
1. $A(BC) = (AB)C$ \> Matrix multiplication is associative. \> \\[.05in]
2. $(rA)B = r(AB) = A(rB)$ \> Scalar multiplication commutes with matrix multiplication. \> \\[.05in]
3. $A(B + C) = AB + AC$ \> \ \> \\[.05in]
4. $(A + B)C = AC + BC$ \> Matrix multiplication distributes over matrix addition. \> \\[.05in]
5. There exists a unique $n \times n$ matrix $I$ such that for all $n \times n$ matrices $M$, $I M = M I = M$. \> \ \>
\etb
\end{framed}
The matrix $I$ is called the $n \times n$ {\itshape identity matrix}. A proof that matrix multiplication is associative would be quite messy at this point. We will just take it to be true. There is an elegant proof, but we need to learn some more linear algebra first, which is in Chapter Three of the text. Let's prove the first distributive property and existence of the identity matrix. You should prove the rest at some point in your life.
{\bf Proof} \\
Let $A$ be an $m \times p$ matrix and $B$ and $C$ $p \times n$ matrices. This is what we mean by appropriate sizes: $B$ and $C$ must be the same size in order to add them, and the number of columns in $A$ must be the number of rows in $B$ and $C$ in order to multiply them. We have that the two matrices on each side of the equals sign have the same size, namely, $m \times n$. Now we show their corresponding entries are equal. Let $1 \leq i \leq m$ and $1 \leq j \leq n$. For simplicity, let's write the $i$-th row of $A$ as $\begin{pmatrix} a_1 & a_2 & \cdots & a_p \end{pmatrix}$ and the $j$-th columns of $B$ and $C$ as
$\begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_p \end{pmatrix}$ and $\begin{pmatrix} c_1 \\ c_2 \\ \vdots \\ c_p \end{pmatrix}$, respectively. Then the $j$-th column of $B + C$ is $\begin{pmatrix} b_1 + c_1 \\ b_2 +c_2 \\ \vdots \\ b_p + c_2 \end{pmatrix}$. So the $ij$-entry of $A(B + C)$ is the product of the $i$-th row of $A$ and the $j$-th column of $B+C$. Multiplying and then using the usual properties of real number arithmetic, we have
\begin{align*}
\begin{pmatrix} a_1 & a_2 & \cdots & a_p \end{pmatrix} \begin{pmatrix} b_1 + c_1 \\ b_2 +c_2 \\ \vdots \\ b_p + c_2 \end{pmatrix} &=
a_1(b_1 + c_1) + a_2(b_2 + c_2) + \cdots + a_p(b_p + c_p) \\
&= a_1 b_1 + a_1 c_1 + a_2 b_2 + b_2 c_2 + \cdots + a_p b_p + a_p c_p \\[.05in]
&= (a_1 b_1 + a_2 b_2 + \cdots + a_p b_p) + (a_1 c_1 + a_2 c_2 + \cdots + a_p c_p).
%\\[.05in]
%&= \begin{pmatrix} a_1 & a_2 & \cdots & a_p \end{pmatrix} \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_p \end{pmatrix} +
%\begin{pmatrix} a_1 & a_2 & \cdots & a_p \end{pmatrix} \begin{pmatrix} c_1 \\ c_2 \\ \vdots \\ c_p \end{pmatrix}.
\end{align*}
\noindent
We see that the two expressions in parentheses are the products of the $i$-th row of $A$ with the $j$-th columns of $B$ and $C$, respectively. We know that the sum of these two is the $ij$-entry of $AB + AC$. And we're done.
Now we will prove that last statement, about this mysterious identity matrix. We need a definition first: The {\itshape main diagonal} of a matrix $A$ consists of its entries of the from $a_{ii}$. Let $M = (m_{ij})$ be an $n \times n$ matrix. Let $I$ be the $n \times n$ matrix whose main diagonal entries are all $1's$ and all of its other entries $0's$, that is,
\bc
$I =
\begin{pmatrix} 1 & 0 & 0 & 0 & \cdots & 0 \\
0 & 1 & 0 & 0 & \cdots & 0 \\
0 & 0 & 1 & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \ & \vdots \\
\vdots & \vdots & \vdots & \ & \ddots & \vdots \\
0 & 0 & 0 & 0 & \cdots & 1
\end{pmatrix}$.
\ec
Since the sizes of $M$ and $I$ are $n \times n$, the sizes of the products $IM$ and $MI$ are also $n \times n$. Let $1 \leq i \leq m$ and $1 \leq j \leq n$. Notice that the $i$-th row of $I$ is the row vector whose $i$-th entry is $1$ and all others $0$'s. So when we multiply this $i$-th row of $I$ by the $j$-th column of $M$, the only entry in the column that gets multiplied by the $1$ is the $i$-th, which is $m_{ij}$. Thus $IM = M$. Now notice that the $j$-th column of $I$ is the column vector whose $j$-th entry is a $1$ and all others $0$'s. So when we multiply the $i$-th row of $M$ by the $j$-th column of $I$, the only entry in the row that gets multiplied by the $1$ is $j$-th, which is just $m_{ij}$. Thus $MI = M$. The proof that $I$ is unique is quite similar to that of the zero matrix. And we're done. \\
Now we return to linear systems. Here's a generic one now:
\begin{align*}
a_{11}x_1 &+ a_{12}x_2 + \cdots + a_{1n}x_n = b_1 \\
a_{21}x_1 &+ a_{22}x_2 + \cdots + a_{2n}x_n = b_2 \\
\vdots \\
a_{m1}x_1 &+ a_{m2}x_2 + \cdots + a_{mn}x_n = b_m.
\end{align*}
We can express this system as a matrix equation $A\bar{x} = \bar{b}$. How?, you ask. Just look at each equation: We're multiplying $a$'s by $x$'s and adding them up. This is exactly how we multiply a row by a column. The matrix $A$ we need is the matrix of the coefficients in the system and the $\bar{x}$ is the column vector of the variables. So the $\bar{b}$ is the column vector of the constants. More explicitly, we have
\bc
$A = \mtrx{a}$, $\bar{x} = \coln{x}$, and $\bar{b} = \colm{b}$.
\ec
So our matrix equation $A\bar{x} = \bar{b}$ represents the system of linear equations, which is a much more concise way of writing the system. It also provides a more convenient way of determining whether or not $\bar{u} = \coln{u}$ is a solution to the system: Just check whether or not $A\bar{u} = \bar{b}$. Let's do an example. \\[.05in]
{\bf Example 6} \\
Consider the linear system
\bc
$\begin{linsys}{3}
2x &- &y &\ &\ &= &0 \\
x & &\ &+ &z &= &4 \\
x &+ &2y &- &2z &= &-1
\end{linsys}$.
\ec
\be
\it Write it as a matrix equation $A\bar{x} = \bar{b}$. \\
Following the above, we let $A$ be the matrix of coefficients, $\bar{x}$ the column vector of the variables, and $\bar{b}$ the column vector of the constants. We have
\bc
$A = \lp \begin{array}{rrr} 2 & -1 & 0 \\ 1 & 0 & 1 \\ 1 & 2 & -2 \end{array} \rp$, $\bar{x} = \colth{x}{y}{z}$, and
$\bar{b} = \lp \begin{array}{r} 0 \\ 4 \\ -1 \end{array} \rp$.
\ec
Then the equation is
\bc
$\lp \begin{array}{rrr} 2 & -1 & 0 \\ 1 & 0 & 1 \\ 1 & 2 & -2 \end{array} \rp \colth{x}{y}{z} = \lp \begin{array}{r} 0 \\ 4 \\ -1 \end{array} \rp$.
\ec
\it Determine whether or not $\colth{1}{2}{3}$ is a solution to the system. \\
Let $\bar{u} = \colth{1}{2}{3}$. Then multiplying, we have
\begin{align*}
A\bar{u} &=
\lp \begin{array}{rrr} 2 & -1 & 0 \\ 1 & 0 & 1 \\ 1 & 2 & -2 \end{array} \rp \colth{1}{2}{3} = \colth{2(1)-1(2)+0(3)}{1(1)+0(2)+1(3)}{1(1)+2(2)-2(3)} = \lp \begin{array}{r} 0 \\ 4 \\ -1 \end{array} \rp = \bar{b}.
\end{align*}
So $\colth{1}{2}{3}$ is a solution to the system.
\it Determine whether or not $\colth{2}{4}{2}$ is a solution. \\
Let $\bar{v} = \colth{2}{4}{2}$. Then multiplying, we have
\begin{align*}
A\bar{v} &= \lp \begin{array}{rrr} 2 & -1 & 0 \\ 1 & 0 & 1 \\ 1 & 2 & -2 \end{array} \rp \colth{2}{4}{2} = \colth{2(2)-1(4)+0(2)}{1(2)+0(4)+1(2)}{1(2)+2(4)-2(2)} = \colth{0}{4}{6}
\not= \bar{b}.
\end{align*}
So $\colth{2}{4}{2}$ is not a solution to the system.
\ee
We know that every nonzero real number $x$ has a multiplicative inverse, namely $x^{-1} = 1/x$, as $x x^{-1} = 1$. Is there an analogous inverse for matrices? That is, for any nonzero $n \times n$ matrix $A$, is there an $n
\times n$ matrix $B$ such that $AB = BA = I$, where $I$ is the $n \times n$ identity matrix? Consider $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$. Let's try to find its $B$. Write $B = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. Then $AB = \begin{pmatrix} a & b \\ 0 & 0 \end{pmatrix}$. Oh. The matrix $AB$ has a row of $0$'s, so it can never be the identity, which is $\Idd$. So the answer to our question is no. Here, then, is a definition:
\begin{framed}
{\bf Definition 6} \\
An $n \times n$ matrix $A$ is {\itshape invertible} provided that there is an $n \times n$ matrix $B$ for which $AB = BA = I$. This $B$ is called {\itshape an inverse} of $A$.
\end{framed}
Notice that $II = I$, so there is at least one invertible matrix for each possible size. Are there more? Why, yes, there are. Thanks for asking. Here's one now.
{\bf Example 7} \\
Let $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ and
$B = \lp \begin{array}{rr} 1 & -1 \\ -1 & 2 \end{array} \rp$.
Multiplying, we get that
\begin{align*}
AB &= \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} \lp \begin{array}{rr} 1 & -1 \\ -1 & 2 \end{array} \rp =
\begin{pmatrix} 2(1)+1(-1) & 2(-1)+1(2) \\ 1(1)+ 1(-1) & 1(-1)+1(2) \end{pmatrix} = \Idd && \text{and} \\[.05in]
BA &= \lp \begin{array}{rr} 1 & -1 \\ -1 & 2 \end{array} \rp \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 1(2)-1(1) & 1(1)-1(1) \\
-1(2)+2(1) & -1(1)+2(1) \end{pmatrix} = \Idd.
\end{align*}
So $A$ is invertible. Perhaps you've noticed that $B$ is also invertible.
As you may have guessed, there are lots of invertible matrices. But there are also lots of matrices that are not invertible. Before determining a method to check whether or not a matrix is invertible, let's state and prove some properties of invertible matrices.
\begin{framed}
{\bf Properties of Invertible Matrices}
\be
\it If an $n \times n$ matrix is invertible, then its inverse is unique.
\it The inverse of an invertible matrix is invertible. Furthermore, if $A$
is an $n \times n$ invertible matrix, then the inverse of the inverse of
$A$ is $A$.
\it The product of two invertible $n \times n$ matrices is invertible.
Moreover, if $A$ and $B$ are invertible $n \times n$ matrices, then
$(AB)^{-1} = B^{-1} A^{-1}$.
\ee
\end{framed}
Now we can refer to {\itshape the inverse} of a square matrix $A$ and we will write its inverse as $A^{-1}$ and read it as ``$A$ inverse". In this case
we have $AA^{-1} = A^{-1}A = I$.
{\bf Proof} \\
Let $A$, $C$, and $D$ be $n \times n$ matrices and $I$ the $n \times n$ identity matrix. Assume that $A$ is invertible and $C$ and $D$ are its inverses. So we have that $AC = CA = AD = DA = I$. Now $C = CI = C(AD) = (CA)D = ID = D$. Notice that we used the associativity of matrix multiplication here.
Now we have that $AA^{-1} = A^{-1}A = I$. So $A$ satisfies the definition for $A^{-1}$ being invertible. Thus the inverse of the inverse of $A$ is $A$, that is, $(A^{-1})^{-1} = A$.
Finally, let $B$ be $n \times n$ invertible matrix.
To show that $AB$ is invertible, we will just multiply, taking full advantage of the associativity of matrix multiplication:
\begin{align*}
(AB)(B^{-1}A^{-1}) &= A(BB^{-1})A^{-1} = AIA^{-1} = AA^{-1} = I &&\text{and} \\
(B^{-1}A^{-1})(AB) &= B^{-1}(A^{-1}A)B = B^{-1}IB = B^{-1}B = I.
\end{align*}
Hence $AB$ is invertible and its inverse is $B^{-1}A^{-1}$.
The proof of the following corollary is a nice exercise using mathematical induction.
\begin{framed}
{\bf Corollary 1} \\
If $A_1, A_2, \ldots, A_m$ are invertible $n \times n$ matrices, then $A_1 A_2 \cdots A_m$ is invertible and $(A_1 A_2 \cdots A_m)^{-1} = A_{m}^{-1} \cdots A_{2}^{-1} A_{1}^{-1}$.
\end{framed}
How do we know if a matrix is invertible or not? The following theorem tells us. All vectors in $\R^n$ will be written as columns.
\begin{framed}
{\bf The Invertible Matrix Theorem(IMT) Part I} \\
Let $A$ be an $n \times n$ matrix, $I$ the $n \times n$ identity matrix, and $\bar{\th}$ the vector in $\R^n$ all of whose entries are zero. Then the following are equivalent.
\be
\it $A$ is invertible.
\it The reduced echelon form of $A$ is $I$.
\it For any $\bar{b} \in \R^n$ the matrix equation $A\bar{x} = \bar{b}$ has exactly one solution.
\it The matrix equation $A\bar{x} = \bar{\th}$ has only $\bar{x} = \bar{\th}$ as its solution.
\ee
\end{framed}
What we mean by ``the following are equivalent'' is that if one of the statements is true, then so are the others and if one of the statements is false, then so are the others. The proof takes advantage of the fact that the logical connective ``implies'', denoted by $\Rightarrow$, is transitive, that is, for any statements $P$, $Q$, and $R$, if $P \Rightarrow Q$ and $Q \Rightarrow R$, then $P \Rightarrow R$. We will prove that $(2) \Rightarrow (1)$, $(1) \Rightarrow (3)$, $(3) \Rightarrow (4)$, and $(4) \Rightarrow (2)$.
{\bf Proof} \\
Let $A$ be an $n \times n$ matrix, $I$ the $n \times n$ identity matrix, and $\bar{\th}$ the vector in $\R^n$ all of whose entries are zero. Assume that the reduced echelon form of $A$ is $I$. We wish to find an $n \times n$ matrix $B$ so that $AB = BA = I$. Recall that we can view the multiplication of two matrices as the multiplication of a matrix by a sequence of columns. In this way finding a matrix $B$ for which $AB = I$ is the same as solving the $n$ systems of linear equations whose matrix equations are given by $A\bar{x}_j = \bar{e}_j$ where $\bar{e}_j$ is the $j$-th column of $I$ for $1 \leq j \leq n$. To solve each system, we must reduce the augmented matrix $(A | \bar{e}_j)$. Since the reduced echelon form of $A$ is $I$, each of these systems has a unique solution. Notice, however, that it is the
$A$ part of the augmented matrix that dictates the row operations we must use; each $\bar{e}_j$ is just along for the ride. This suggests that in practice we can reduce the giant augmented matrix $(A | I)$ until the $A$ part is in its reduced echelon form, which in this case we assumed to be $I$. Hence we can reduce $(A | I)$ to $(I | B)$ for some $n \times n$ matrix $B$. For each $1 \leq j \leq n$ the solution to $A\bar{x}_j = \bar{e}_j$ is the $j$-th column of $B$. Thus $AB = I$.
Now since matrix multiplication is not commutative, we must still show that $BA = I$. Since we have reduced that giant augmented matrix $(A | I)$ to $(I | B)$, we have in fact reduced $I$ to $B$. By Lemma 1.6 in Chapter One Section III, ``reduces to'' is an equivalence relation. Since we can reduce $I$ to $B$, we can reduce $B$ to $I$. In other words, the reduced echelon form of $B$ is $I$. The previous argument then shows that there is an $n \times n$ matrix $C$ for which $BC = I$. Then $A = AI = A(BC) = (AB)C = IC = C$. Hence $BA = I$. Thus $A$ is invertible.
Now we assume that $A$ is invertible. Let $\bar{b} \in \R^n$ and consider $A^{-1}\bar{b}$. Since $A^{-1}$ is $n \times n$ and $\bar{b} \in \R^n$, $A^{-1}\bar{b} \in \R^n$. Now $A(A^{-1}\bar{b}) = (AA^{-1})\bar{b} = I\bar{b} = \bar{b}$. Technically we showed that $I$ is the identity for square matrices, but since we can make a square matrix whose columns are all $\bar{b}$, we see that $I\bar{b}$ is indeed $\bar{b}$. We have just shown that the equation $A\bar{x} = \bar{b}$ has a solution. For uniqueness, suppose that $\bar{u} \in \R^n$ is another one. Then we have
\begin{align*}
A\bar{u} &= \bar{b} \\
A^{-1}(A\bar{u}) &= A^{-1}\bar{b} \\
(A^{-1}A)\bar{u} &= A^{-1}\bar{b} \\
I\bar{u} &= A^{-1}\bar{b} \\
\bar{u} &= A^{-1}\bar{b}.
\end{align*}
Hence the only solution is $\bar{x} = A^{-1}\bar{b}$.
Since $\bar{\th}$ is a particular vector in $\R^n$, we automatically have that if for any $\bar{b} \in \R^n$ the matrix equation $A\bar{x} = \bar{b}$ has exactly one solution, then the matrix equation $A\bar{x} = \bar{\th}$ has only $\bar{x} = \bar{\th}$ as its solution.
Now to complete the proof, we must show that if the matrix equation $A\bar{x} = \bar{\th}$ has only $\bar{x} = \bar{\th}$ as its solution, then the reduced echelon form of $A$ is $I$. We do so by contraposition. Assume that the reduced echelon form of $A$ is not $I$. Since $A$ is square, its reduced echelon form must contain a row of zeroes. In solving the homogeneous system of linear equations corresponding to $A\bar{x} = \bar{\th}$, the augmented matrix $(A | \bar{\th})$ will have an entire row of zeroes when $A$ has been reduced to its reduced echelon form. As the number of equations and unknowns are the same, the system must have a free variable. This means that the system has more than one solution(in fact it has infinitely many, but who's counting?). Hence the matrix equation $A\bar{x} = \bar{\th}$ has more than one solution. Thus by contraposition we have proved that if the matrix equation $A\bar{x} = \bar{\th}$ has only $\bar{x} = \bar{\th}$ as its solution, then the reduced echelon form of $A$ is $I$. Therefore we have proved the theorem.
{\bf Example 8}
\be
\it The first part of the proof provides a method for determining whether or not
a matrix is invertible and if so, finding its inverse: Given an $n \times n$
matrix $A$, we form the giant augmented matrix $(A | I)$ and reduce it until the $A$ part is in reduced echelon form. If this form is $I$, then we
know that $A$ is invertible and the matrix in the $I$ part is its inverse; if this form is not $I$, then $A$ is not invertible. Determine whether or not
the matrix is invertible and if so, to find its inverse.
\be
\it Let $A = \lp \begin{array}{rrcrr} 2 & 1 \\ 1 & -1 \end{array} \rp$. \\
As stated above, we form the giant augmented matrix $(A | I )$ and reduce:
\begin{align*}
\lp \begin{array}{rrcrr} 2 & 1 & | & 1 & 0 \\
1 & -1 & | & 0 & 1 \end{array} \rp \sim
\lp \begin{array}{rrcrr} 1 & -1 & | & 0 & 1 \\
2 & 1 & | & 1 & 0 \end{array} \rp \sim
\lp \begin{array}{rrcrr} 1 & -1 & | & 0 & 1 \\
0 & 3 & | & 1 & -2 \end{array} \rp \\
\sim
\lp \begin{array}{rrcrr} 1 & -1 & | & 0 & 1 \\
0 & 1 & | & 1/3 & -2/3 \end{array} \rp \sim
\lp \begin{array}{rrcrr} 1 & 0 & | & 1/3 & 1/3 \\
0 & 1 & | & 1/3 & -2/3 \end{array} \rp.
\end{align*}
So we see that the reduced echelon form of $A$ is the identity. Thus $A$ is invertible and $A^{-1} = \lp \begin{array}{rr} 1/3 & 1/3 \\ 1/3 & -2/3 \end{array} \rp$. We can rewrite this inverse a bit more nicely by factoring out the $1/3$: $A^{-1} = \Frac{1}{3} \lp \begin{array}{rr} 1 & 1 \\ 1 & -2 \end{array} \rp$.
\it Let $A = \lp \begin{array}{rrr} 1 & 0 & 2 \\ -1 & 1 & -2 \\ 2 & 2 & 1 \end{array} \rp$. \\
We form the giant augmented matrix $(A | I)$ and reduce:
\begin{align*}
\lp \begin{array}{rrrcrrr}
1 & 0 & 2 & | & 1 & 0 & 0 \\
-1 & 1 & -2 & | & 0 & 1 & 0 \\
2 & 2 & 1 & | & 0 & 0 & 1 \end{array} \rp \sim
\lp \begin{array}{rrrcrrr}
1 & 0 & 2 & | & 1 & 0 & 0 \\
0 & 1 & 0 & | & 1 & 1 & 0 \\
0 & 2 & -3 & | & -2 & 0 & 1 \end{array} \rp \sim
\lp \begin{array}{rrrcrrr}
1 & 0 & 2 & | & 1 & 0 & 0 \\
0 & 1 & 0 & | & 1 & 1 & 0 \\
0 & 0 & -3 & | & -4 & -2 & 1 \end{array} \rp \\ \sim
\lp \begin{array}{rrrcrrr}
1 & 0 & 2 & | & 1 & 0 & 0 \\
0 & 1 & 0 & | & 1 & 1 & 0 \\
0 & 0 & 1 & | & 4/3 & 2/3 & -1/3 \end{array} \rp \sim
\lp \begin{array}{rrrcrrr}
1 & 0 & 0 & | & -5/3 & -4/3 & 2/3 \\
0 & 1 & 0 & | & 1 & 1 & 0 \\
0 & 0 & 1 & | & 4/3 & 2/3 & -1/3 \end{array} \rp.
\end{align*}
So we see that $A$ is invertible and $A^{-1} = \lp \begin{array}{rrr} -5/3 & -4/3 & 2/3 \\ 1 & 1 & 0 \\ 4/3 & 2/3 & -1/3 \end{array} \rp$. Factoring out a $1/3$, we get $A^{-1} = \Frac{1}{3} \lp \begin{array}{rrr} -5 & -4 & 2 \\ 3 & 3 & 0 \\ 4 & 2 & -1 \end{array} \rp$.
\it Let $B = \lp \begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array} \rp$. \\
We form the giant augmented matrix $(B | I)$ and reduce:
\begin{align*}
\lp \begin{array}{rrcrr} 1 & -1 & | & 1 & 0 \\
-1 & 1 & | & 0 & 1 \end{array} \rp \sim
\lp \begin{array}{rrcrr} 1 & -1 & | & 1 & 0 \\
0 & 0 & | & 1 & 1 \end{array} \rp.
\end{align*}
Since the reduced echelon form of $B$ is not $I$, $B$ is not invertible.
\ee
\it As seen in the proof of the theorem, we can use the inverse of a matrix to solve a linear system with the same number of equations and unknowns.
Specifically, we express the system as a matrix equation $A\bar{x} = \bar{b}$, where $A$ is the matrix of the coefficients. If $A$ is invertible, then the solution is $\bar{x} = A^{-1}\bar{b}$. Solve the following linear system using the inverse of the matrix of coefficients:
\bc
$\begin{linsys}{3}
x & &\ &+ &2z &= &3 \\
-x &+ &y &- &2z &= &-3 \\
2x &+ &2y &+ &z &= &6
\end{linsys}$.
\ec
Notice that the coefficient matrix is, conveniently, the matrix $A$ from part (b) above, whose inverse we've already found. The matrix equation for this system is
$A\bar{x} = \bar{b}$ where $\bar{x} = \colth{x}{y}{z}$ and $\bar{b} = \colth{3}{-3}{6}$. Multiplying and using the fact that scalars commute with matrix multiplication, we get that
\begin{align*}
\bar{x} = A^{-1}\bar{b} &=
\Frac{1}{3} \lp \begin{array}{rrr} -5 & -4 & 2 \\ 3 & 3 & 0 \\ 4 & 2 & -1 \end{array} \rp \colth{3}{-3}{6} =
\lp \begin{array}{rrr} -5 & -4 & 2 \\ 3 & 3 & 0 \\ 4 & 2 & -1 \end{array} \rp \cdot \Frac{1}{3} \colth{3}{-3}{6} \\[.05in]
&= \lp \begin{array}{rrr} -5 & -4 & 2 \\ 3 & 3 & 0 \\ 4 & 2 & -1 \end{array} \rp
\colth{1}{-1}{2} = \colth{3}{0}{0}.
\end{align*}
So $x = 3$, $y = 0$, and $z = 0$.
\ee
The following theorem tells us that if the product of two square matrices is the identity, then they are in fact inverses of each other.
\begin{framed}
{\bf Theorem 2} \\
Let $A$ and $B$ be $n \times n$ matrices and $I$ the $n \times n$ identity matrix. If $AB = I$, then $A$ and $B$ are invertible and $A^{-1} = B$.
\end{framed}
{\bf Proof} \\
Let $A$ and $B$ be $n \times n$ matrices, $I$ the $n \times n$ identity matrix, and $\bar{\th}$ the vector in $\R^n$ all of whose entries are zero. Assume $AB = I$. We will use the IMT Part I to prove that $B$ is invertible first. Consider the matrix equation $B\bar{x} = \bar{\th}$ and let $\bar{u} \in \R^n$ be a solution. So we have
\begin{align*}
B\bar{u} &= \bar{\th} \\
A(B\bar{u}) &= A\bar{\th} \\
(AB)\bar{u} &= \bar{\th} \\
I\bar{u} &= \bar{\th} \\
\bar{u} &= \bar{\th}.
\end{align*}
The only solution to $B\bar{x} = \bar{\th}$ is $\bar{x} = \bar{\th}$. Hence by the IMT Part I, $B$ is invertible. So $B^{-1}$ exists. Then multiplying both sides of $AB = I$ on the right by $B^{-1}$ gives us that $A = B^{-1}$. Since $B^{-1}$ is invertible, $A$ is too and $A^{-1} = (B^{-1})^{-1} = B$.
We finish this note off with what's called the transpose of a matrix. Here's the definition.
\begin{framed}
{\bf Definition 7} \\
Let $A = (a_{ij})$ be an $m \times n$ matrix. The {\itshape transpose} of $A$, denoted by $A^T$, is the matrix whose $i$-th column is the $i$-th row of $A$, or equivalently, whose $j$-th row is the $j$-th column of $A$. Notice that $A^T$ is an $n \times m$ matrix. We will write $A^T = (a^{T}_{ji})$ where $a^{T}_{ji} = a_{ij}$.
\end{framed}
Notice that the $ji$-entry of $A^T$ is the $ij$-entry of $A$.
This tells us that the main diagonals of a matrix and its transpose are the same and that entries of $A^T$ are the entries of $A$ reflected about the main diagonal. Here are a couple of examples.
{\bf Example 9} \\
Find the transpose of each of the given matrices.
\be
\it $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$. \\
The first row of $A$ is $\begin{pmatrix} 1 & 2 & 3 \end{pmatrix}$ and the second row is $\begin{pmatrix} 4 & 5 & 6 \end{pmatrix}$. So these become the columns of $A^T$, that is, $A^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$. Alternatively, we see that the columns of $A$ are $\begin{pmatrix} 1 \\ 4 \end{pmatrix}$, $\begin{pmatrix} 2 \\ 5 \end{pmatrix}$, and $\begin{pmatrix} 3 \\ 6 \end{pmatrix}$. So these become the rows of $A^T$, as we can see above.
\it $B = \lp \begin{array}{rrr} -1 & 0 & 6 \\ -4 & 1 & 9 \\ 2 & 3 & 0 \end{array} \rp$. \\
We make the rows of $B$ the columns of $B^T$. Doing so, we get
$B^T = \lp \begin{array}{rrr} -1 & -4 & 2 \\ 0 & 1 & 3 \\ 6 & 9 & 0 \end{array} \rp$. Notice how the entries of $B^T$ are those of $B$ reflected about the main diagonal.
\ee
\begin{framed}
{\bf Properties of the Transpose} \\
Let $A$ and $B$ be appropriately sized matrices and $r \in \R$. Then
\be
\it $(A^T)^T = A$.
\it $(A + B)^T = A^T + B^T$.
\it $(rA)^T = rA^T$.
\it $(AB)^T = B^T A^T$.
\ee
\end{framed}
The first three properties seem perfectly natural. You should try to prove them some time. But what about fourth one? Does that seem natural? Maybe. Given how the inverse of the product of matrices works, maybe this is fine. Let's prove it.
{\bf Proof} \\
Let $A$ be an $m \times p$ matrix and $B$ a $p \times n$ matrix. Then $AB$ is an $m \times n$ matrix. So $(AB)^T$ is an $n \times m$ matrix. Then $B^T$ is an $n \times p$ matrix and $A^T $ is a $p \times m$ matrix. Thus multiplying $B^T A^T$ makes sense and its size is also $n \times m$. But what about their corresponding entries? Let $1 \leq i \leq m$ and $1 \leq j \leq n$. The $ji$-entry of $(AB)^T$ is the $ij$-entry of $AB$, which is the $i$-th row of $A$ times the $j$-th column of $B$. For simplicity, let $\rowp{a}$ be the $i$-th row of $A$ and $\colp{b}$ the $j$-th column of $B$. Then the $ij$-entry of $AB$ is $a_1 b_1 + a_2 b_2 + \cdots + a_p b_p$, but this is also equal to $b_1 a_1 + b_2 a_2 + \cdots b_p a_p$, which is the product of $\rowp{b}$ and $\colp{a}$. This is exactly the product of the $j$-th row of $B^T$ and the $i$-th column of $A^T$, which is the $ji$-entry of $B^T A^T$. Thus $ji$-entry of $(AB)^T$ is the $ji$-entry of $B^T A^T$. Therefore $(AB)^T = B^T A^T$. \\
{\large Here are some exercises. Enjoy.}
\be
\it Let
\bc
$A = \lp \begin{array}{rrr} 1 & -2 & 3 \\ 1 & -1 & 0 \end{array} \rp$,
$B = \lp \begin{array}{rr} 3 & 4 \\ 5 & -1 \\ 1 & -1 \end{array} \rp$,
$C = \lp \begin{array}{rrr} 4 & -1 & 2 \\ -1 & 5 & 1 \end{array} \rp$,
$D = \lp \begin{array}{rrr} -1 & 0 & 1 \\ 0 & 2 & 1 \end{array} \rp$,
$E = \lp \begin{array}{rr} 3 & 4 \\ -2 & 3 \\ 0 & 1 \end{array} \rp$,
$F = \lp \begin{array}{r} 2 \\ -3 \end{array} \rp$, and
$G = \lp \begin{array}{rr} 2 & -1 \end{array} \rp$.
\ec
Compute each of the following, if possible. If a computation is not possible, explain why it is not.
\bfourt
(a) $3C - 4D$ \> (b) $A - (D + 2C)$ \> (c) $A - E$ \> (d) $AE$ \\[.05in]
(e) $3BC - 4BD$ \> (f) $CB + D$ \> (g) $GC$ \> (h) $FG$ \\[.05in]
(h) $3A - 2E^T$ \> (i) $A^T B$ \> (j) $B^T A$ \>
(k) $3A^T - 2E$ \\[.05in]
(l) $C C^T + FG$ \> (m) $(F^T + G)D$ \> (o) $(B - E)A^T$ \>
(p) $D^T (F + G^T)$
\etb
\it Illustrate the associativity of matrix multiplication by computing $(AB)C$
and $A(BC)$ where $A$, $B$, and $C$ are matrices above.
\it Let $A$ be an $n \times n$ matrix. Let $m \in \N$. As you would expect, we
define $A^m$ to be the product of $A$ with itself $m$ times. Notice that this makes sense as matrix multiplication is associative.
\be
\it Compute $A^4$ for $A =
\lp \begin{array}{rr} 1 & -2 \\ 1 & -1 \end{array} \rp$.
\it Provide a counter-example to the statement:
For any $2 \times 2$ matrices $A$ and $B$, $(AB)^2 = A^2 B^2$.
\ee
\it Prove that for all $m \times n$ matrices $A$, $B$, and $C$,
if $A + C = B + C$, then $A = B$.
\it Let $\Theta$ be the $m \times n$ zero matrix.
\be
\it Prove that for any $m \times n$ matrix $A$, $-A = (-1)A$ and
$0A = \Theta$.
\it Prove that for any $r \in \R$ and $m \times n$ matrix $A$,
if $rA = \Theta$, then $r = 0$ or $A = \Theta$.
\ee
\it We have seen that matrix multiplication is sooooo not commutative. Take this a
step further by finding two matrices $A$ and $B$ for which $AB$ and $BA$ are defined, but have different sizes.
\it Let $\Theta$ be the $2 \times 2$ zero matrix and $I$ the $2 \times 2$
identity matrix. Provide a counter-example to each of the following statements:
\be
\it For any $2 \times 2$ matrices $A$ and $B$, if $AB = \Theta$,
then $A = \Theta$ or $B = \Theta$.
\it For any $2 \times 2$ matrices, $A$, $B$, and $C$, if $AB = AC$,
then $B = C$.
\it For any $2 \times 2$ matrix $A$, if $A^2 = A$,
then $A = \Theta$ or $A = I$.
\ee
\it Suppose that we have a homogeneous linear system whose matrix
equation is given by $A\bar{x} = \bar{\th}$ where $A$ is the $m \times n$ matrix of coefficients, $\bar{x}$ is the column matrix of the $n$ variables, and $\bar{\th}$ is the column matrix of $m$ zeroes. Use the properties of matrix arithmetic to show that for any solutions $\bar{u}$ and $\bar{v}$ to the system and $r \in \R$, $\bar{u} + \bar{v}$ and $r\bar{u}$ are also solutions.
\it Consider the linear system:
\bc
$\begin{linsys}{4}
x_1 &+ &x_2 &+ &x_3 &+ &x_4 &= &3 \\
x_1 &- &x_2 &+ &x_3 &+ &x_4 &= &5 \\
\ &\ &x_2 &- &x_3 &- &x_4 &= &-4 \\
x_1 &+ &x_2 &- &x_3 &- &x_4 &= &-3
\end{linsys}$.
\ec
\be
\it Express the system as a matrix equation $A\bar{x} = \bar{b}$.
\it Use matrix multiplication to determine whether or not
$\bar{u} = \lp \begin{array}{r} 1 \\ -1 \\ 1 \\ 2 \end{array} \rp$ and $\bar{v} = \lp \begin{array}{r} -1 \\ 2 \\ 0 \\ 3 \end{array} \rp$ are solutions to the system.
\ee
\it Determine whether or not each of the following matrices is invertible.
If so, find its inverse.
\btt
(a) $A = \lp \begin{array}{rrrr}
1 & -2 & 0 & -1 \\
2 & 3 & 3 & 8 \\
4 & -6 & -3 & -5 \\
7 & -5 & 0 & 2 \end{array} \rp$ \>
(b) $B = \lp \begin{array}{rrr}
2 & 1 & -1 \\ 2 & -1 & 2 \\
1 & 1 & -1 \end{array} \rp$ \\[.05in]
(c) $C = \lp \begin{array}{rr} 4 & 3 \\ 2 & 3 \end{array} \rp$ \>
(d) $D^T D$ where $D = \lp \begin{array}{rrr} -1 & 0 & 1 \\ 0 & 2 & 1
\end{array} \rp$
\etb
\newpage
\it Solve each linear system using the inverse of its coefficient matrix.
\btt
(a) \
$\begin{linsys}{3}
2x &+ &y &- &z &= &2 \\
2x &- &y &+ &2z &= &-1 \\
x &+ &y &- &z &= &3
\end{linsys}$ \>
(b) \
$\begin{linsys}{4}
x_1 &+ &2x_2 &+ &x_3 &\ &\ &= &2 \\
\ &\ &x_2 &- &x_3 &+ &x_4 &= &-2 \\
2x_1 &+ &4x_2 &+ &3x_3 &\ &\ &= &0 \\
x_1 &+ &2x_2 &- &x_3 &+ &2x_4 &= &-4
\end{linsys}$
\etb
\it Provide a counter-example to the statement: For any $2 \times 2$ invertible
matrices $A$ and $B$, $A + B$ is invertible.
\it Find an example of a $2 \times 2$ nonidentity matrix whose transpose is its
inverse.
\it Let $A$ be an $n \times n$ invertible matrix.
\be
\it Prove that for all $m \in \N$, $A^m$ is invertible and determine a
formula for its inverse in terms of $A^{-1}$. Hint: Use mathematical induction and the fact that $A^{m+1} = A^m A$.
\it Prove that $A^T$ is invertible and determine its inverse in terms of
$A^{-1}$. Hint: Use a property of the transpose.
\ee
\it Determine $(B^T)^{-1}$ for the matrix $B$ in Problem \#10.
\it Prove that for any $n \times n$ matrices $A$ and $B$, if $AB$ is invertible,
then so are $A$ and $B$.
\it Here are a couple of new definitions: An $n \times n$ matrix $A$ is
{\itshape symmetric} provided $A^T = A$ and {\itshape skew-symmetric} provided $A^T = -A$.
\be
\it Give examples of symmetric and skew-symmetric $2 \times 2$, $3 \times 3$,
and $4 \times 4$ matrices.
\it What can you say about the main diagonal of a skew-symmetric matrix?
\it Give an example of a matrix that is both symmetric and skew-symmetric.
\it Prove that for any $n \times n$ matrix $A$, $A + A^T$, $AA^T$, and
$A^T A$ are symmetric and $A - A^T$ is skew-symmetric.
\it Prove that any $n \times n$ can be written as the sum of a symmetric and
skew-symmetric matrices. Hint: Did you do part (d) yet?
\ee
\ee
\end{document}
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\bibitem{Pape_etal2013} Pape, K., Bjorngaard, J. H., De Ridder, K. A. A., Westin, S., Holmen, T. L., \& Krokstad, S. (2013). Medical benefits in young Norwegians and their parents, and the contribution of family health and socioeconomic status. The HUNT Study, Norway. \textit{Scand J Public Health}, \textit{41}(5), 455--462.
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\bibitem{Dono_etal2013} Dono, G., Cortignani, R., Doro, L., Giraldo, L., Ledda, L., Pasqui, M., et al. (2013). Adapting to uncertainty associated with short-term climate variability changes in irrigated Mediterranean farming systems. \textit{Agricultural Systems}, \textit{117}, 1--12.
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\title{Swiss: Marco Camenisch free! (10\Slash{}03\Slash{}2017)}
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{\usekomafont{title}{\huge Swiss: Marco Camenisch free! (10\Slash{}03\Slash{}2017)\par}}%
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March 10\textsuperscript{th}, 2017 — We learn from the Red Aid that — finally — Marco Camenisch has completed the process of “gradual release”.
The comrade Marco Camenisch is free!
\bigskip
\textbf{ŠVICARSKA: MARCO CAMENISCH SLOBODAN! (10.03.2017.)}
10. mart 2017. — Doznajemo od Crvene Pomoći da je — napokon — Marco Camenisch okončao proces “postupnog oslobođenja”.
Drug Marco Camenisch je slobodan!
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Anarhija.info
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Swiss: Marco Camenisch free! (10\Slash{}03\Slash{}2017)
\bigskip
\href{http://www.informa-azione.info/prigionieri\_svizzera\_marco\_camenisch\_\%C3\%A8\_libero}{www.informa-azione.info}
\bigskip
\textbf{anarhija.info}
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https://dlmf.nist.gov/12.7.E8.tex | nist.gov | CC-MAIN-2022-40 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2022-40/segments/1664030337415.12/warc/CC-MAIN-20221003101805-20221003131805-00474.warc.gz | 248,439,666 | 743 | \[U\left(-2,z\right)=\frac{z^{5/2}}{4\sqrt{2\pi}}\left(2K_{\frac{1}{4}}\left(%
\tfrac{1}{4}z^{2}\right)+3K_{\frac{3}{4}}\left(\tfrac{1}{4}z^{2}\right)-K_{%
\frac{5}{4}}\left(\tfrac{1}{4}z^{2}\right)\right),\] |
https://watana.be/ku/tex/1981s_3.tex | watana.be | CC-MAIN-2021-39 | application/x-tex | text/x-matlab | crawl-data/CC-MAIN-2021-39/segments/1631780057580.39/warc/CC-MAIN-20210924201616-20210924231616-00490.warc.gz | 645,657,775 | 1,894 | %2007/04/22 06:10
%Copyright 2007 WATANABE,Masayuki
\documentclass[a4j,10pt,fleqn]{jsarticle}
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\usepackage{amsmath,amssymb,graphicx}
\pagestyle{plain}
\def\dsum{\displaystyle\sum}
\def\dlim{\displaystyle\lim}
\def\dint{\displaystyle\int}
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\begin{document}
{\large 京都大学 1981年 入学試験 理系数学 問題3}\\
\\
{\large 問題}\\
\\
$k$ は定数，$m$ は一つの自然数とする．\\
$x>0$ のとき，つねに $x^m-1\leqq k(x^{m+1}-1)$ であるならば，
$k=\dfrac{m}{m+1}$ であることを示せ．
\ \\
{\large 解答}\\
\\
$x^m-1\leqq k(x^{m+1}-1)$ならば\\
$x=1$のとき\\
$x^m-1=0 \leqq 0=k(x^{m+1}-1)$となり任意の$k$で成立するので$x\neq 0$の時のみ考えればよい\\
$k\leqq 0$ならば$x>1$のとき$x^m-1 >0$かつ$k(x^{m+1}-1)<0$となり条件が成立しないので\\
$k$がすべての$x$で条件を満たすとすると少なくとも$k>0$でなければならない。\\
よって$k>0$とする。\\
$x<1$のとき\\
$(x^{m+1}-1)<0$なので
\begin{eqnarray*}
x^m-1 &\leqq& k(x^{m+1}-1)\\
\dfrac{x^m-1}{x^{m+1}-1} &\geqq& k\\
\dfrac{(x-1)(x^{m-1}+x^{m-2}+\cdots+1}{(x-1)(x^m+x^{m-1}+\cdots+1)} &\geqq& k\\
\dfrac{x^{m-1}+x^{m-2}+\cdots+1}{x^m+x^{m-1}+\cdots+1} &\geqq& k
\end{eqnarray*}
$k>0$なので\\
\begin{eqnarray*}
\dfrac{x^m+x^{m-1}+\cdots+1}{x^{m-1}+x^{m-2}+\cdots+1} &\leqq& \dfrac{1}{k}\\
x+\dfrac{1}{x^{m-1}+x^{m-2}+\cdots+1} &\leqq& \dfrac{1}{k}
\end{eqnarray*}
$x<1$より$x^k<1$なので$x^{m-1}+x^{m-1}+\cdots+1<m$\\
$x+\dfrac{1}{x^{m-1}+x^{m-1}+\cdots+1}>x+\dfrac{1}{m}$\\
よって\\
\begin{eqnarray*}
x+\dfrac{1}{x^{m-1}+x^{m-2}+\cdots+1} &\leqq& \dfrac{1}{k}\\
x+\dfrac{1}{m}&<& \dfrac{1}{k}\\
k&<& \dfrac{m}{mx+1}
\end{eqnarray*}
$x<1$より$\dfrac{m}{mx+1}>\dfrac{m}{m+1}$\\
よって\\
$k\leqq \dfrac{m}{m+1}$ならば条件を満たす。\\
\ \\
$x>1$のとき\\
$(x^{m+1}-1)>0$なので
\begin{eqnarray*}
x^m-1 &\leqq& k(x^{m+1}-1)\\
\dfrac{x^m-1}{x^{m+1}-1} &\leqq& k\\
\dfrac{(x-1)(x^{m-1}+x^{m-2}+\cdots+1}{(x-1)(x^m+x^{m-1}+\cdots+1)} &\leqq& k\\
\dfrac{x^{m-1}+x^{m-2}+\cdots+1}{x^m+x^{m-1}+\cdots+1} &\geqq& k
\end{eqnarray*}
$k>0$なので\\
\begin{eqnarray*}
\dfrac{x^m+x^{m-1}+\cdots+1}{x^{m-1}+x^{m-2}+\cdots+1} &\leqq& \dfrac{1}{k}\\
x+\dfrac{1}{x^{m-1}+x^{m-2}+\cdots+1} &\leqq& \dfrac{1}{k}
\end{eqnarray*}
$x>1$より$x^k>1$なので$x^{m-1}+x^{m-1}+\cdots+1>m$\\
よって\\
$x+\dfrac{1}{x^{m-1}+x^{m-1}+\cdots+1}<x+\dfrac{1}{m}$\\
したがって\\
\begin{eqnarray*}
x+\dfrac{1}{x^{m-1}+x^{m-2}+\cdots+1} &\geqq& \dfrac{1}{k}\\
x+\dfrac{1}{m}&>& \dfrac{1}{k}\\
k&>& \dfrac{m}{mx+1}
\end{eqnarray*}
$x>1$より$\dfrac{m}{mx+1}<\dfrac{m}{m+1}$\\
よって\\
$k\geqq \dfrac{m}{m+1}$ならば条件を満たす。\\
\ \\
以上の通り、$\dfrac{m}{m+1} \leqq k \leqq \dfrac{m}{m+1}$ならば条件を満たすので\\
$k =\dfrac{m}{m+1}$ならば任意の$x$について式を満たす。\\
また
$\dlim_{x \to 1}\dfrac{x^m-1}{x^{m+1}-1}=\dlim_{x \to 1}\dfrac{x^{m-1}+x^{m-2}+\cdots+1}{x^m+x^{m-1}+\cdots+1}=\dfrac{m}{m+1}$\\
より$k\neq \dfrac{m}{m+1}$ならば条件の成立しない$x$が存在する。\\
$$
\text{よって条件を満たすならば}
k=\dfrac{m}{m+1}
\text{でなければならない。}$$
証明終了
\end{document}
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http://literatesoftware.com/axiom-website/CATS/schaum18.input.pamphlet | literatesoftware.com | CC-MAIN-2023-14 | unk | application/x-tex | crawl-data/CC-MAIN-2023-14/segments/1679296949107.48/warc/CC-MAIN-20230330070451-20230330100451-00102.warc.gz | 26,087,350 | 8,653 | \documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/input schaum18.input}
\author{Timothy Daly}
\maketitle
\eject
\tableofcontents
\eject
\section{\cite{1}:14.369~~~~~$\displaystyle
\int{\cos ax ~dx}$}
$$\int{\cos ax}=
\frac{\sin{ax}}{a}
$$
<<*>>=
)spool schaum18.output
)set message test on
)set message auto off
)clear all
--S 1
aa:=integrate(cos(a*x),x)
--R
--R
--R sin(a x)
--R (1) --------
--R a
--R Type: Union(Expression Integer,...)
--E
--S 2
bb:=sin(a*x)/a
--R
--R sin(a x)
--R (2) --------
--R a
--R Type: Expression Integer
--E
--S 3 14:369 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.370~~~~~$\displaystyle
\int{x\cos{ax}~dx}$}
$$\int{x\cos{ax}}=
\frac{\cos{ax}}{a^2}+\frac{x\sin{ax}}{a}
$$
<<*>>=
)clear all
--S 4
aa:=integrate(x*cos(a*x),x)
--R
--R
--R a x sin(a x) + cos(a x)
--R (1) -----------------------
--R 2
--R a
--R Type: Union(Expression Integer,...)
--E
--S 5
bb:=cos(a*x)/a^2+(x*sin(a*x))/a
--R
--R a x sin(a x) + cos(a x)
--R (2) -----------------------
--R 2
--R a
--R Type: Expression Integer
--E
--S 6 14:370 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.371~~~~~$\displaystyle
\int{x^2\cos{ax}~dx}$}
$$\int{x^2\cos{ax}}=
\frac{2x}{a^2}\cos{ax}+\left(\frac{x^2}{a}-\frac{2}{a^3}\right)\sin{ax}
$$
<<*>>=
)clear all
--S 7
aa:=integrate(x^2*cos(a*x),x)
--R
--R
--R 2 2
--R (a x - 2)sin(a x) + 2a x cos(a x)
--R (1) ----------------------------------
--R 3
--R a
--R Type: Union(Expression Integer,...)
--E
--S 8
bb:=(2*x)/a^2*cos(a*x)+(x^2/a-2/a^3)*sin(a*x)
--R
--R 2 2
--R (a x - 2)sin(a x) + 2a x cos(a x)
--R (2) ----------------------------------
--R 3
--R a
--R Type: Expression Integer
--E
--S 9 14:371 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.372~~~~~$\displaystyle
\int{x^3\cos{ax}~dx}$}
$$\int{x^3\cos{ax}}=
\left(\frac{3x^2}{a^2}-\frac{6}{a^4}\right)\cos{ax}
+\left(\frac{x^3}{a}-\frac{6x}{a^3}\right)\sin{ax}
$$
<<*>>=
)clear all
--S 10
aa:=integrate(x^3*cos(a*x),x)
--R
--R
--R 3 3 2 2
--R (a x - 6a x)sin(a x) + (3a x - 6)cos(a x)
--R (1) -------------------------------------------
--R 4
--R a
--R Type: Union(Expression Integer,...)
--E
--S 11
bb:=((3*x^2)/a^2-6/a^4)*cos(a*x)+(x^3/a-(6*x)/a^3)*sin(a*x)
--R
--R 3 3 2 2
--R (a x - 6a x)sin(a x) + (3a x - 6)cos(a x)
--R (2) -------------------------------------------
--R 4
--R a
--R Type: Expression Integer
--E
--S 12 14:372 Schaums and Axiom agree
cc:=aa-bb
--R
--R (3) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.373~~~~~$\displaystyle
\int{\frac{\cos{ax}}{x}}~dx$}
$$\int{\frac{\cos{ax}}{x}}=
\ln{x}-\frac{(ax)^2}{2\cdot 2!}+\frac{(ax)^4}{4\cdot 4!}
-\frac{(ax)^6}{6\cdot 6!}+\cdots
$$
<<*>>=
)clear all
--S 13 14:373 Schaums and Axiom agree by definition
aa:=integrate(cos(x)/x,x)
--R
--R
--R (1) Ci(x)
--R Type: Union(Expression Integer,...)
--E
@
\section{\cite{1}:14.374~~~~~$\displaystyle
\int{\frac{\cos{ax}}{x^2}}~dx$}
$$\int{\frac{\cos{ax}}{x^2}}=
-\frac{\cos{ax}}{x}-a\int{\frac{\sin{ax}}{x}}
$$
<<*>>=
)clear all
--S 14 14:374 Axiom cannot compute this integral
aa:=integrate(cos(a*x)/x^2,x)
--R
--R
--R x
--I ++ cos(%I a)
--I (1) | --------- d%I
--R ++ 2
--I %I
--R Type: Union(Expression Integer,...)
--E
@
\section{\cite{1}:14.375~~~~~$\displaystyle
\int{\frac{dx}{\cos{ax}}}$}
$$\int{\frac{1}{\cos{ax}}}=
\frac{1}{a}\ln(\sec{ax}-\tan{ax})=
\frac{1}{a}\ln\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right)
$$
<<*>>=
)clear all
--S 15
aa:=integrate(1/cos(a*x),x)
--R
--R
--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1
--R log(-----------------------) - log(-----------------------)
--R cos(a x) + 1 cos(a x) + 1
--R (1) -----------------------------------------------------------
--R a
--R Type: Union(Expression Integer,...)
--E
--S 16
bb1:=1/a*log(sec(a*x)+tan(a*x))
--R
--R log(tan(a x) + sec(a x))
--R (2) ------------------------
--R a
--R Type: Expression Integer
--E
--S 17
bb2:=1/a*log(tan(%pi/4+(a*x)/2))
--R
--R 2a x + %pi
--R log(tan(----------))
--R 4
--R (3) --------------------
--R a
--R Type: Expression Integer
--E
--S 18
cc1:=aa-bb1
--R
--R (4)
--R sin(a x) + cos(a x) + 1
--R - log(tan(a x) + sec(a x)) + log(-----------------------)
--R cos(a x) + 1
--R +
--R sin(a x) - cos(a x) - 1
--R - log(-----------------------)
--R cos(a x) + 1
--R /
--R a
--R Type: Expression Integer
--E
--S 19
cc2:=aa-bb2
--R
--R (5)
--R 2a x + %pi sin(a x) + cos(a x) + 1
--R - log(tan(----------)) + log(-----------------------)
--R 4 cos(a x) + 1
--R +
--R sin(a x) - cos(a x) - 1
--R - log(-----------------------)
--R cos(a x) + 1
--R /
--R a
--R Type: Expression Integer
--E
--S 20 14:375 Schaums and Axiom differ by a constant
complexNormalize cc1
--R
--R log(- 1)
--R (6) --------
--R a
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.376~~~~~$\displaystyle
\int{\frac{x~dx}{\cos{ax}}}$}
$$\int{\frac{x}{\cos{ax}}}=
\frac{1}{a^2}\left\{
\frac{(ax)^2}{2}+\frac{(ax)^4}{8}+\frac{5(ax)^6}{144}+\cdots+
\frac{E_n(ax)^{2n+2}}{(2n+2)(2n)!}+\cdots
\right\}
$$
<<*>>=
)clear all
--S 21 14:376 Axiom cannot compute this integral
aa:=integrate(x/cos(a*x),x)
--R
--R
--R x
--I ++ %I
--I (1) | --------- d%I
--I ++ cos(%I a)
--R Type: Union(Expression Integer,...)
--E
@
\section{\cite{1}:14.377~~~~~$\displaystyle
\int{\cos^2{ax}}~dx$}
$$\int{\cos^2{ax}}=
\frac{x}{2}+\frac{\sin{2ax}}{4a}
$$
<<*>>=
)clear all
--S 22
aa:=integrate(cos(a*x)^2,x)
--R
--R
--R cos(a x)sin(a x) + a x
--R (1) ----------------------
--R 2a
--R Type: Union(Expression Integer,...)
--E
--S 23
bb:=x/2+sin(2*a*x)/(4*a)
--R
--R sin(2a x) + 2a x
--R (2) ----------------
--R 4a
--R Type: Expression Integer
--E
--S 24
cc:=aa-bb
--R
--R - sin(2a x) + 2cos(a x)sin(a x)
--R (3) -------------------------------
--R 4a
--R Type: Expression Integer
--E
--S 25
cossinrule:=rule(cos(b)*sin(a) == 1/2*(sin(a-b)+sin(a+b)))
--R
--R
--I %M sin(b + a) - %M sin(b - a)
--I (4) %M cos(b)sin(a) == -----------------------------
--R 2
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
--S 26 14:377 Schaums and Axiom agree
dd:=cossinrule cc
--R
--R (5) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.378~~~~~$\displaystyle
\int{x\cos^2{ax}}~dx$}
$$\int{x\cos^2{ax}}=
\frac{x^2}{4}+\frac{x\sin{2ax}}{4a}+\frac{\cos{2ax}}{8a^2}
$$
<<*>>=
)clear all
--S 27
aa:=integrate(x*cos(a*x)^2,x)
--R
--R
--R 2 2 2
--R 2a x cos(a x)sin(a x) + cos(a x) + a x
--R (1) ----------------------------------------
--R 2
--R 4a
--R Type: Union(Expression Integer,...)
--E
--S 28
bb:=x^2/4+(x*sin(2*a*x))/(4*a)+cos(2*a*x)/(8*a^2)
--R
--R 2 2
--R 2a x sin(2a x) + cos(2a x) + 2a x
--R (2) ----------------------------------
--R 2
--R 8a
--R Type: Expression Integer
--E
--S 29
cc:=aa-bb
--R
--R 2
--R - 2a x sin(2a x) + 4a x cos(a x)sin(a x) - cos(2a x) + 2cos(a x)
--R (3) -----------------------------------------------------------------
--R 2
--R 8a
--R Type: Expression Integer
--E
--S 30
cossinrule:=rule(cos(b)*sin(a) == 1/2*(sin(a-b)+sin(a+b)))
--R
--R
--I %N sin(b + a) - %N sin(b - a)
--I (4) %N cos(b)sin(a) == -----------------------------
--R 2
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
--S 31
dd:=cossinrule cc
--R
--R 2
--R - cos(2a x) + 2cos(a x)
--R (5) ------------------------
--R 2
--R 8a
--R Type: Expression Integer
--E
--S 32
coscosrule:=rule(cos(a)*cos(b) == 1/2*(cos(a-b)+cos(a+b)))
--R
--R
--I %O cos(b + a) + %O cos(b - a)
--I (6) %O cos(a)cos(b) == -----------------------------
--I 2
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
--S 33
ee:=coscosrule dd
--R
--R 2
--R - cos(2a x) + 2cos(a x)
--R (7) ------------------------
--R 2
--R 8a
--R Type: Expression Integer
--E
--S 34
cossqrrule1:=rule(cos(a)^2 == 1/2+1/2*cos(2*a))
--R
--R 2 cos(2a) + 1
--R (8) cos(a) == -----------
--R 2
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
--S 35 14:378 Schaums and Axiom differ by a constant
ff:=cossqrrule1 ee
--R
--R 1
--R (9) ---
--R 2
--R 8a
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.379~~~~~$\displaystyle
\int{\cos^3{ax}}~dx$}
$$\int{\cos^3{ax}}=
\frac{\sin{ax}}{a}-\frac{\sin^3{ax}}{3a}
$$
<<*>>=
)clear all
--S 36
aa:=integrate(cos(a*x)^3,x)
--R
--R
--R 2
--R (cos(a x) + 2)sin(a x)
--R (1) -----------------------
--R 3a
--R Type: Union(Expression Integer,...)
--E
--S 37
bb:=sin(a*x)/a-sin(a*x)^3/(3*a)
--R
--R 3
--R - sin(a x) + 3sin(a x)
--R (2) -----------------------
--R 3a
--R Type: Expression Integer
--E
--S 38
cc:=aa-bb
--R
--R 3 2
--R sin(a x) + (cos(a x) - 1)sin(a x)
--R (3) -----------------------------------
--R 3a
--R Type: Expression Integer
--E
--S 39
cossqrrule:=rule(cos(a)^2 == 1-sin(a)^2)
--R
--R 2 2
--R (4) cos(a) == - sin(a) + 1
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
--S 40 14:379 Schaums and Axiom agree
dd:=cossqrrule cc
--R
--R (5) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.380~~~~~$\displaystyle
\int{\cos^4{ax}}~dx$}
$$\int{\cos^4{ax}}=
\frac{3x}{8}+\frac{\sin{2ax}}{4a}+\frac{\sin{4ax}}{32a}
$$
<<*>>=
)clear all
--S 41
aa:=integrate(cos(a*x)^4,x)
--R
--R
--R 3
--R (2cos(a x) + 3cos(a x))sin(a x) + 3a x
--R (1) ---------------------------------------
--R 8a
--R Type: Union(Expression Integer,...)
--E
--S 42
bb:=(3*x)/8+sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a)
--R
--R sin(4a x) + 8sin(2a x) + 12a x
--R (2) ------------------------------
--R 32a
--R Type: Expression Integer
--E
--S 43
cc:=aa-bb
--R
--R 3
--R - sin(4a x) - 8sin(2a x) + (8cos(a x) + 12cos(a x))sin(a x)
--R (3) ------------------------------------------------------------
--R 32a
--R Type: Expression Integer
--E
--S 44 14:380 Schaums and Axiom agree
complexNormalize cc
--R
--R (4) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.381~~~~~$\displaystyle
\int{\frac{dx}{\cos^2{ax}}}$}
$$\int{\frac{1}{\cos^2{ax}}}=
\frac{\tan{ax}}{a}
$$
<<*>>=
)clear all
--S 45
aa:=integrate(1/cos(a*x)^2,x)
--R
--R
--R sin(a x)
--R (1) ----------
--R a cos(a x)
--R Type: Union(Expression Integer,...)
--E
--S 46
bb:=tan(a*x)/a
--R
--R tan(a x)
--R (2) --------
--R a
--R Type: Expression Integer
--E
--S 47
cc:=aa-bb
--R
--R - cos(a x)tan(a x) + sin(a x)
--R (3) -----------------------------
--R a cos(a x)
--R Type: Expression Integer
--E
--S 48
tanrule:=rule(tan(a) == sin(a)/cos(a))
--R
--R sin(a)
--R (4) tan(a) == ------
--R cos(a)
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
--S 49 14:381 Schaums and Axiom agree
dd:=tanrule cc
--R
--R (5) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.382~~~~~$\displaystyle
\int{\frac{dx}{\cos^3{ax}}}$}
$$\int{\frac{1}{\cos^3{ax}}}=
\frac{\sin{ax}}{2a\cos^2{ax}}
+\frac{1}{2a}\ln\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right)
$$
<<*>>=
)clear all
--S 50
aa:=integrate(1/cos(a*x)^3,x)
--R
--R
--R (1)
--R 2 sin(a x) + cos(a x) + 1
--R cos(a x) log(-----------------------)
--R cos(a x) + 1
--R +
--R 2 sin(a x) - cos(a x) - 1
--R - cos(a x) log(-----------------------) + sin(a x)
--R cos(a x) + 1
--R /
--R 2
--R 2a cos(a x)
--R Type: Union(Expression Integer,...)
--E
--S 51
bb:=sin(a*x)/(2*a*cos(a*x)^2)+1/(2*a)*log(tan(%pi/4+(a*x)/2))
--R
--R 2 2a x + %pi
--R cos(a x) log(tan(----------)) + sin(a x)
--R 4
--R (2) ----------------------------------------
--R 2
--R 2a cos(a x)
--R Type: Expression Integer
--E
--S 52
cc:=aa-bb
--R
--R (3)
--R 2a x + %pi sin(a x) + cos(a x) + 1
--R - log(tan(----------)) + log(-----------------------)
--R 4 cos(a x) + 1
--R +
--R sin(a x) - cos(a x) - 1
--R - log(-----------------------)
--R cos(a x) + 1
--R /
--R 2a
--R Type: Expression Integer
--E
--S 53 14:382 Schaums and Axiom differ by a constant
complexNormalize cc
--R
--R log(- 1)
--R (4) --------
--R 2a
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.383~~~~~$\displaystyle
\int{\cos{px}\cos{qx}}~dx$}
$$\int{\cos{ax}\cos{px}}=
\frac{\sin{(a-p)x}}{2(a-p)}+\frac{\sin{(a+p)x}}{2(a+p)}
$$
<<*>>=
)clear all
--S 54
aa:=integrate(cos(a*x)*cos(p*x),x)
--R
--R p cos(a x)sin(p x) - a cos(p x)sin(a x)
--R (1) ---------------------------------------
--R 2 2
--R p - a
--R Type: Union(Expression Integer,...)
--E
--S 55
bb:=(sin((a-p)*x))/(2*(a-p))+(sin((a+p)*x))/(2*(a+p))
--R
--R (p - a)sin((p + a)x) + (p + a)sin((p - a)x)
--R (2) -------------------------------------------
--R 2 2
--R 2p - 2a
--R Type: Expression Integer
--E
--S 56
cc:=aa-bb
--R
--R (3)
--R (- p + a)sin((p + a)x) + 2p cos(a x)sin(p x) + (- p - a)sin((p - a)x)
--R +
--R - 2a cos(p x)sin(a x)
--R /
--R 2 2
--R 2p - 2a
--R Type: Expression Integer
--E
--S 57 14:383 Schaums and Axiom agree
complexNormalize cc
--R
--R (4) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.384~~~~~$\displaystyle
\int{\frac{dx}{1-\cos{ax}}}$}
$$\int{\frac{1}{1-\cos{ax}}}=
-\frac{1}{a}\cot\frac{ax}{2}
$$
<<*>>=
)clear all
--S 58
aa:=integrate(1/(1-cos(a*x)),x)
--R
--R
--R - cos(a x) - 1
--R (1) --------------
--R a sin(a x)
--R Type: Union(Expression Integer,...)
--E
--S 59
bb:=-1/a*cot((a*x)/2)
--R
--R a x
--R cot(---)
--R 2
--R (2) - --------
--R a
--R Type: Expression Integer
--E
--S 60
cc:=aa-bb
--R
--R a x
--R cot(---)sin(a x) - cos(a x) - 1
--R 2
--R (3) -------------------------------
--R a sin(a x)
--R Type: Expression Integer
--E
--S 61 14:384 Schaums and Axiom agree
dd:=complexNormalize cc
--R
--R (4) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.385~~~~~$\displaystyle
\int{\frac{x~dx}{1-\cos{ax}}}$}
$$\int{\frac{x}{1-\cos{ax}}}=
-\frac{x}{a}\cot\frac{ax}{2}
+\frac{2}{a^2}\ln~\sin\frac{ax}{2}
$$
<<*>>=
)clear all
--S 62
aa:=integrate(x/(1-cos(a*x)),x)
--R
--R (1)
--R sin(a x) 2
--R 2sin(a x)log(------------) - sin(a x)log(------------) - a x cos(a x) - a x
--R cos(a x) + 1 cos(a x) + 1
--R ---------------------------------------------------------------------------
--R 2
--R a sin(a x)
--R Type: Union(Expression Integer,...)
--E
--S 63
bb:=-x/a*cot((a*x)/2)+2/a^2*log(sin((a*x)/2))
--R
--R a x a x
--R 2log(sin(---)) - a x cot(---)
--R 2 2
--R (2) -----------------------------
--R 2
--R a
--R Type: Expression Integer
--E
--S 64
cc:=aa-bb
--R
--R (3)
--R sin(a x) a x
--R 2sin(a x)log(------------) - 2sin(a x)log(sin(---))
--R cos(a x) + 1 2
--R +
--R 2 a x
--R - sin(a x)log(------------) + a x cot(---)sin(a x) - a x cos(a x) - a x
--R cos(a x) + 1 2
--R /
--R 2
--R a sin(a x)
--R Type: Expression Integer
--E
--S 65
cotrule:=rule(cot(a) == cos(a)/sin(a))
--R
--R cos(a)
--R (4) cot(a) == ------
--R sin(a)
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
--S 66
dd:=cotrule cc
--R
--R (5)
--R a x sin(a x) a x a x
--R 2sin(---)sin(a x)log(------------) - 2sin(---)sin(a x)log(sin(---))
--R 2 cos(a x) + 1 2 2
--R +
--R a x 2 a x
--R - sin(---)sin(a x)log(------------) + a x cos(---)sin(a x)
--R 2 cos(a x) + 1 2
--R +
--R a x
--R (- a x cos(a x) - a x)sin(---)
--R 2
--R /
--R 2 a x
--R a sin(---)sin(a x)
--R 2
--R Type: Expression Integer
--E
--S 67
ee:=expandLog dd
--R
--R (6)
--R a x a x a x
--R 2sin(---)sin(a x)log(sin(a x)) - 2sin(---)sin(a x)log(sin(---))
--R 2 2 2
--R +
--R a x
--R - sin(---)sin(a x)log(cos(a x) + 1)
--R 2
--R +
--R a x a x a x
--R (- log(2)sin(---) + a x cos(---))sin(a x) + (- a x cos(a x) - a x)sin(---)
--R 2 2 2
--R /
--R 2 a x
--R a sin(---)sin(a x)
--R 2
--R Type: Expression Integer
--E
--S 68 14:385 Schaums and Axiom agree
complexNormalize ee
--R
--R (7) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.386~~~~~$\displaystyle
\int{\frac{dx}{1+\cos{ax}}}$}
$$\int{\frac{1}{1+\cos{ax}}}=
\frac{1}{a}\tan\frac{ax}{2}
$$
<<*>>=
)clear all
--S 69
aa:=integrate(1/(1+cos(a*x)),x)
--R
--R sin(a x)
--R (1) --------------
--R a cos(a x) + a
--R Type: Union(Expression Integer,...)
--E
--S 70
bb:=1/a*tan((a*x)/2)
--R
--R a x
--R tan(---)
--R 2
--R (2) --------
--R a
--R Type: Expression Integer
--E
--S 71
cc:=aa-bb
--R
--R a x
--R (- cos(a x) - 1)tan(---) + sin(a x)
--R 2
--R (3) -----------------------------------
--R a cos(a x) + a
--R Type: Expression Integer
--E
--S 72 14:386 Schaums and Axiom agree
complexNormalize cc
--R
--R (4) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.387~~~~~$\displaystyle
\int{\frac{x~dx}{1+\cos{ax}}}$}
$$\int{\frac{x}{1+\cos{ax}}}=
\frac{x}{a}\tan\frac{ax}{2}
+\frac{2}{a^2}\ln~\cos\frac{ax}{2}
$$
<<*>>=
)clear all
--S 73
aa:=integrate(x/(1+cos(a*x)),x)
--R
--R
--R 2
--R (- cos(a x) - 1)log(------------) + a x sin(a x)
--R cos(a x) + 1
--R (1) ------------------------------------------------
--R 2 2
--R a cos(a x) + a
--R Type: Union(Expression Integer,...)
--E
--S 74
bb:=x/a*tan((a*x)/2)+2/a^2*log(cos((a*x)/2))
--R
--R a x a x
--R 2log(cos(---)) + a x tan(---)
--R 2 2
--R (2) -----------------------------
--R 2
--R a
--R Type: Expression Integer
--E
--S 75
cc:=aa-bb
--R
--R (3)
--R a x 2
--R (- 2cos(a x) - 2)log(cos(---)) + (- cos(a x) - 1)log(------------)
--R 2 cos(a x) + 1
--R +
--R a x
--R (- a x cos(a x) - a x)tan(---) + a x sin(a x)
--R 2
--R /
--R 2 2
--R a cos(a x) + a
--R Type: Expression Integer
--E
--S 76
dd:=expandLog cc
--R
--R (4)
--R a x
--R (cos(a x) + 1)log(cos(a x) + 1) + (- 2cos(a x) - 2)log(cos(---))
--R 2
--R +
--R a x
--R (- a x cos(a x) - a x)tan(---) + a x sin(a x) - log(2)cos(a x) - log(2)
--R 2
--R /
--R 2 2
--R a cos(a x) + a
--R Type: Expression Integer
--E
--S 77 14:387 Schaums and Axiom agree
complexNormalize dd
--R
--R (5) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.388~~~~~$\displaystyle
\int{\frac{dx}{(1-\cos{ax})^2}}$}
$$\int{\frac{1}{(1-\cos{ax})^2}}=
-\frac{1}{2a}\cot\frac{ax}{2}
-\frac{1}{6a}\cot^3\frac{ax}{2}
$$
<<*>>=
)clear all
--S 78
aa:=integrate(1/(1-cos(a*x))^2,x)
--R
--R
--R 2
--R - cos(a x) + cos(a x) + 2
--R (1) --------------------------
--R (3a cos(a x) - 3a)sin(a x)
--R Type: Union(Expression Integer,...)
--E
--S 79
bb:=-1/(2*a)*cot((a*x)/2)-1/(6*a)*cot((a*x)/2)^3
--R
--R a x 3 a x
--R - cot(---) - 3cot(---)
--R 2 2
--R (2) -----------------------
--R 6a
--R Type: Expression Integer
--E
--S 80
cc:=aa-bb
--R
--R (3)
--R a x 3 a x 2
--R ((cos(a x) - 1)cot(---) + (3cos(a x) - 3)cot(---))sin(a x) - 2cos(a x)
--R 2 2
--R +
--R 2cos(a x) + 4
--R /
--R (6a cos(a x) - 6a)sin(a x)
--R Type: Expression Integer
--E
--S 81 14:388 Schaums and Axiom agree
complexNormalize cc
--R
--R (4) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.389~~~~~$\displaystyle
\int{\frac{dx}{(1+\cos{ax})^2}}$}
$$\int{\frac{1}{(1+\cos{ax})^2}}=
\frac{1}{2a}\tan\frac{ax}{2}
+\frac{1}{6a}\tan^3\frac{ax}{2}
$$
<<*>>=
)clear all
--S 82
aa:=integrate(1/(1+cos(a*x))^2,x)
--R
--R
--R (cos(a x) + 2)sin(a x)
--R (1) -------------------------------
--R 2
--R 3a cos(a x) + 6a cos(a x) + 3a
--R Type: Union(Expression Integer,...)
--E
--S 83
bb:=1/(2*a)*tan((a*x)/2)+1/(6*a)*tan((a*x)/2)^3
--R
--R a x 3 a x
--R tan(---) + 3tan(---)
--R 2 2
--R (2) ---------------------
--R 6a
--R Type: Expression Integer
--E
--S 84
cc:=aa-bb
--R
--R (3)
--R 2 a x 3
--R (- cos(a x) - 2cos(a x) - 1)tan(---)
--R 2
--R +
--R 2 a x
--R (- 3cos(a x) - 6cos(a x) - 3)tan(---) + (2cos(a x) + 4)sin(a x)
--R 2
--R /
--R 2
--R 6a cos(a x) + 12a cos(a x) + 6a
--R Type: Expression Integer
--E
--S 85 14:389 Schaums and Axiom agree
complexNormalize cc
--R
--R (4) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.390~~~~~$\displaystyle
\int{\frac{dx}{p+q\cos{ax}}}$}
$$\int{\frac{1}{p+q\cos{ax}}}=
\left\{
\begin{array}{l}
\displaystyle
\frac{2}{a\sqrt{p^2-q^q}}
\tan^{-1}\sqrt{(p-q)/(p+q)}\tan\frac{1}{2}ax
\\
\displaystyle
\frac{1}{a\sqrt{q^2-p^2}}\ln\left(
\frac{\tan\frac{1}{2}ax+\sqrt{(q+p)/(q-p)}}
{\tan\frac{1}{2}ax-\sqrt{(q+p)(q-p)}}\right)
\end{array}
\right.
$$
<<*>>=
)clear all
--S 86
aa:=integrate(1/(p+q*cos(a*x)),x)
--R
--R (1)
--R +-------+
--R | 2 2 2 2
--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
--R log(--------------------------------------------------)
--R q cos(a x) + p
--R [-------------------------------------------------------,
--R +-------+
--R | 2 2
--R a\|q - p
--R +---------+
--R | 2 2
--R sin(a x)\|- q + p
--R 2atan(-----------------------)
--R (q + p)cos(a x) + q + p
--R ------------------------------]
--R +---------+
--R | 2 2
--R a\|- q + p
--R Type: Union(List Expression Integer,...)
--E
--S 87
bb1:=2/(a*sqrt(p^2-q^2))*atan(sqrt((p-q)/(p+q))*tan(1/2*a*x))
--R
--R
--R +-------+
--R a x |- q + p
--R 2atan(tan(---) |------- )
--R 2 \| q + p
--R (2) -------------------------
--R +---------+
--R | 2 2
--R a\|- q + p
--R Type: Expression Integer
--E
--S 88
bb2:=1/(a*sqrt(q^2-p^2))*log((tan(1/2*a*x)+sqrt((q+p)/(q-p)))/(tan(1/2*a*x)-sqrt((q+p)/(q-p))))
--R
--R +-----+
--R |q + p a x
--R - |----- - tan(---)
--R \|q - p 2
--R log(---------------------)
--R +-----+
--R |q + p a x
--R |----- - tan(---)
--R \|q - p 2
--R (3) --------------------------
--R +-------+
--R | 2 2
--R a\|q - p
--R Type: Expression Integer
--E
--S 89
cc1:=aa.1-bb1
--R
--R
--R (4)
--R +-------+
--R +---------+ | 2 2 2 2
--R | 2 2 (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
--R \|- q + p log(--------------------------------------------------)
--R q cos(a x) + p
--R +
--R +-------+ +-------+
--R | 2 2 a x |- q + p
--R - 2\|q - p atan(tan(---) |------- )
--R 2 \| q + p
--R /
--R +---------+ +-------+
--R | 2 2 | 2 2
--R a\|- q + p \|q - p
--R Type: Expression Integer
--E
--S 90
cc2:=aa.2-bb1
--R
--R
--R +---------+
--R +-------+ | 2 2
--R a x |- q + p sin(a x)\|- q + p
--R - 2atan(tan(---) |------- ) + 2atan(-----------------------)
--R 2 \| q + p (q + p)cos(a x) + q + p
--R (5) ------------------------------------------------------------
--R +---------+
--R | 2 2
--R a\|- q + p
--R Type: Expression Integer
--E
--S 91
cc3:=aa.1-bb2
--R
--R (6)
--R +-----+
--R |q + p a x
--R - |----- - tan(---)
--R \|q - p 2
--R - log(---------------------)
--R +-----+
--R |q + p a x
--R |----- - tan(---)
--R \|q - p 2
--R +
--R +-------+
--R | 2 2 2 2
--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
--R log(--------------------------------------------------)
--R q cos(a x) + p
--R /
--R +-------+
--R | 2 2
--R a\|q - p
--R Type: Expression Integer
--E
--S 92 14:390 Axiom cannot simplify these expressions
cc4:=aa.2-bb2
--R
--R (7)
--R +-----+
--R |q + p a x
--R +---------+ - |----- - tan(---)
--R | 2 2 \|q - p 2
--R - \|- q + p log(---------------------)
--R +-----+
--R |q + p a x
--R |----- - tan(---)
--R \|q - p 2
--R +
--R +---------+
--R +-------+ | 2 2
--R | 2 2 sin(a x)\|- q + p
--R 2\|q - p atan(-----------------------)
--R (q + p)cos(a x) + q + p
--R /
--R +---------+ +-------+
--R | 2 2 | 2 2
--R a\|- q + p \|q - p
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.391~~~~~$\displaystyle
\int{\frac{dx}{(p+q\cos{ax})^2}}$}
$$\int{\frac{1}{(p+q\cos{ax})^2}}=
\frac{q\sin{ax}}{a(q^2-p^2)(p+q\cos{ax})}
-\frac{p}{q^2-p^2}\int{\frac{1}{p+q\cos{ax}}}
$$
<<*>>=
)clear all
--S 93
aa:=integrate(1/(p+q*cos(a*x))^2,x)
--R
--R
--R (1)
--R [
--R 2
--R (p q cos(a x) + p )
--R *
--R +-------+
--R | 2 2 2 2
--R (- p cos(a x) - q)\|q - p + (q - p )sin(a x)
--R log(------------------------------------------------)
--R q cos(a x) + p
--R +
--R +-------+
--R | 2 2
--R q sin(a x)\|q - p
--R /
--R +-------+
--R 3 2 2 3 | 2 2
--R ((a q - a p q)cos(a x) + a p q - a p )\|q - p
--R ,
--R
--R +---------+
--R | 2 2
--R 2 sin(a x)\|- q + p
--R (- 2p q cos(a x) - 2p )atan(-----------------------)
--R (q + p)cos(a x) + q + p
--R +
--R +---------+
--R | 2 2
--R q sin(a x)\|- q + p
--R /
--R +---------+
--R 3 2 2 3 | 2 2
--R ((a q - a p q)cos(a x) + a p q - a p )\|- q + p
--R ]
--R Type: Union(List Expression Integer,...)
--E
--S 94
t1:=integrate(1/(p+q*cos(a*x)),x)
--R
--R (2)
--R +-------+
--R | 2 2 2 2
--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
--R log(--------------------------------------------------)
--R q cos(a x) + p
--R [-------------------------------------------------------,
--R +-------+
--R | 2 2
--R a\|q - p
--R +---------+
--R | 2 2
--R sin(a x)\|- q + p
--R 2atan(-----------------------)
--R (q + p)cos(a x) + q + p
--R ------------------------------]
--R +---------+
--R | 2 2
--R a\|- q + p
--R Type: Union(List Expression Integer,...)
--E
--S 95
bb1:=(q*sin(a*x))/(a*(q^2-p^2)*(p+q*cos(a*x)))-p/(q^2-p^2)*t1.1
--R
--R (3)
--R 2
--R (- p q cos(a x) - p )
--R *
--R +-------+
--R | 2 2 2 2
--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
--R log(--------------------------------------------------)
--R q cos(a x) + p
--R +
--R +-------+
--R | 2 2
--R q sin(a x)\|q - p
--R /
--R +-------+
--R 3 2 2 3 | 2 2
--R ((a q - a p q)cos(a x) + a p q - a p )\|q - p
--R Type: Expression Integer
--E
--S 96
bb2:=(q*sin(a*x))/(a*(q^2-p^2)*(p+q*cos(a*x)))-p/(q^2-p^2)*t1.2
--R
--R (4)
--R +---------+
--R | 2 2 +---------+
--R 2 sin(a x)\|- q + p | 2 2
--R (- 2p q cos(a x) - 2p )atan(-----------------------) + q sin(a x)\|- q + p
--R (q + p)cos(a x) + q + p
--R -----------------------------------------------------------------------------
--R +---------+
--R 3 2 2 3 | 2 2
--R ((a q - a p q)cos(a x) + a p q - a p )\|- q + p
--R Type: Expression Integer
--E
--S 97
cc1:=aa.1-bb1
--R
--R (5)
--R +-------+
--R | 2 2 2 2
--R (- p cos(a x) - q)\|q - p + (q - p )sin(a x)
--R p log(------------------------------------------------)
--R q cos(a x) + p
--R +
--R +-------+
--R | 2 2 2 2
--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
--R p log(--------------------------------------------------)
--R q cos(a x) + p
--R /
--R +-------+
--R 2 2 | 2 2
--R (a q - a p )\|q - p
--R Type: Expression Integer
--E
--S 98
cc2:=aa.2-bb1
--R
--R (6)
--R +-------+
--R +---------+ | 2 2 2 2
--R | 2 2 (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
--R p\|- q + p log(--------------------------------------------------)
--R q cos(a x) + p
--R +
--R +---------+
--R +-------+ | 2 2
--R | 2 2 sin(a x)\|- q + p
--R - 2p\|q - p atan(-----------------------)
--R (q + p)cos(a x) + q + p
--R /
--R +---------+ +-------+
--R 2 2 | 2 2 | 2 2
--R (a q - a p )\|- q + p \|q - p
--R Type: Expression Integer
--E
--S 99
cc3:=aa.1-bb2
--R
--R (7)
--R +-------+
--R +---------+ | 2 2 2 2
--R | 2 2 (- p cos(a x) - q)\|q - p + (q - p )sin(a x)
--R p\|- q + p log(------------------------------------------------)
--R q cos(a x) + p
--R +
--R +---------+
--R +-------+ | 2 2
--R | 2 2 sin(a x)\|- q + p
--R 2p\|q - p atan(-----------------------)
--R (q + p)cos(a x) + q + p
--R /
--R +---------+ +-------+
--R 2 2 | 2 2 | 2 2
--R (a q - a p )\|- q + p \|q - p
--R Type: Expression Integer
--E
--S 100 14:391 Schaums and Axiom agree
cc4:=aa.2-bb2
--R
--R (8) 0
--R Type: Expression Integer
--E
@
\section{\cite{1}:14.392~~~~~$\displaystyle
\int{\frac{dx}{p^2+q^2\cos^2{ax}}}$}
$$\int{\frac{1}{p^2+q^2\cos^2{ax}}}=
\frac{1}{ap\sqrt{p^2+q^2}}\tan^{-1}\frac{p\tan{ax}}{\sqrt{p^2+q^2}}
$$
<<*>>=
)clear all
--S 101
aa:=integrate(1/(p^2+q^2*cos(a*x)^2),x)
--R
--R
--R (1)
--R +-------+
--R | 2 2 2 2 2
--R sin(a x)\|q + p ((q - p )cos(a x) - 2p )sin(a x)
--R atan(------------------) - atan(-----------------------------------------)
--R 2p cos(a x) + 2p +-------+
--R 2 | 2 2
--R (p cos(a x) + 2p cos(a x) + p)\|q + p
--R --------------------------------------------------------------------------
--R +-------+
--R | 2 2
--R a p\|q + p
--R Type: Union(Expression Integer,...)
--E
--S 102
bb:=1/(a*p*sqrt(p^2+q^2))*atan((p*tan(a*x))/sqrt(p^2+q^2))
--R
--R p tan(a x)
--R atan(----------)
--R +-------+
--R | 2 2
--R \|q + p
--R (2) ----------------
--R +-------+
--R | 2 2
--R a p\|q + p
--R Type: Expression Integer
--E
--S 103
cc:=aa-bb
--R
--R (3)
--R +-------+
--R | 2 2
--R sin(a x)\|q + p p tan(a x)
--R atan(------------------) - atan(----------)
--R 2p cos(a x) + 2p +-------+
--R | 2 2
--R \|q + p
--R +
--R 2 2 2
--R ((q - p )cos(a x) - 2p )sin(a x)
--R - atan(-----------------------------------------)
--R +-------+
--R 2 | 2 2
--R (p cos(a x) + 2p cos(a x) + p)\|q + p
--R /
--R +-------+
--R | 2 2
--R a p\|q + p
--R Type: Expression Integer
--E
--S 104
dd:=ratDenom cc
--R
--R (4)
--R +-------+
--R +-------+ | 2 2
--R | 2 2 p tan(a x)\|q + p
--R - \|q + p atan(--------------------)
--R 2 2
--R q + p
--R +
--R -
--R +-------+
--R | 2 2
--R \|q + p
--R *
--R +-------+
--R 2 2 2 | 2 2
--R ((q - p )cos(a x) - 2p )sin(a x)\|q + p
--R atan(--------------------------------------------------------)
--R 2 3 2 2 3 2 3
--R (p q + p )cos(a x) + (2p q + 2p )cos(a x) + p q + p
--R +
--R +-------+
--R +-------+ | 2 2
--R | 2 2 sin(a x)\|q + p
--R \|q + p atan(------------------)
--R 2p cos(a x) + 2p
--R /
--R 2 3
--R a p q + a p
--R Type: Expression Integer
--E
--S 105
atanrule2:=rule(atan(x) == 1/2*%i*(log(1-%i*x)-log(1+%i*x)))
--R
--R 1 1
--R (5) atan(x) == - - %i log(%i x + 1) + - %i log(- %i x + 1)
--R 2 2
--RType: RewriteRule(Integer,Complex Fraction Integer,Expression Complex Fraction Integer)
--E
--S 106
ee:=atanrule2 dd
--R
--R (6)
--R +-------+
--R +-------+ | 2 2 2 2
--R 1 | 2 2 %i p tan(a x)\|q + p + q + p
--R - %i\|q + p log(---------------------------------)
--R 2 2 2
--R q + p
--R +
--R +-------+
--R 1 | 2 2
--R - %i\|q + p
--R 2
--R *
--R log
--R +-------+
--R 2 2 2 | 2 2
--R ((%i q - %i p )cos(a x) - 2%i p )sin(a x)\|q + p
--R +
--R 2 3 2 2 3 2 3
--R (p q + p )cos(a x) + (2p q + 2p )cos(a x) + p q + p
--R /
--R 2 3 2 2 3 2 3
--R (p q + p )cos(a x) + (2p q + 2p )cos(a x) + p q + p
--R +
--R +-------+
--R 1 | 2 2
--R +-------+ - %i sin(a x)\|q + p + p cos(a x) + p
--R 1 | 2 2 2
--R - - %i\|q + p log(----------------------------------------)
--R 2 p cos(a x) + p
--R +
--R +-------+
--R 1 | 2 2
--R +-------+ - - %i sin(a x)\|q + p + p cos(a x) + p
--R 1 | 2 2 2
--R - %i\|q + p log(------------------------------------------)
--R 2 p cos(a x) + p
--R +
--R -
--R +-------+
--R 1 | 2 2
--R - %i\|q + p
--R 2
--R *
--R log
--R +-------+
--R 2 2 2 | 2 2
--R ((- %i q + %i p )cos(a x) + 2%i p )sin(a x)\|q + p
--R +
--R 2 3 2 2 3 2 3
--R (p q + p )cos(a x) + (2p q + 2p )cos(a x) + p q + p
--R /
--R 2 3 2 2 3 2 3
--R (p q + p )cos(a x) + (2p q + 2p )cos(a x) + p q + p
--R +
--R +-------+
--R +-------+ | 2 2 2 2
--R 1 | 2 2 - %i p tan(a x)\|q + p + q + p
--R - - %i\|q + p log(-----------------------------------)
--R 2 2 2
--R q + p
--R /
--R 2 3
--R a p q + a p
--R Type: Expression Complex Fraction Integer
--E
--S 107
ff:=expandLog ee
--R
--R (7)
--R +-------+ +-------+
--R 1 | 2 2 | 2 2 2 2
--R - - %i\|q + p log(p tan(a x)\|q + p + %i q + %i p )
--R 2
--R +
--R +-------+ +-------+
--R 1 | 2 2 | 2 2 2 2
--R - %i\|q + p log(p tan(a x)\|q + p - %i q - %i p )
--R 2
--R +
--R -
--R +-------+
--R 1 | 2 2
--R - %i\|q + p
--R 2
--R *
--R log
--R +-------+
--R 2 2 2 | 2 2
--R ((q - p )cos(a x) - 2p )sin(a x)\|q + p
--R +
--R 2 3 2 2 3
--R (%i p q + %i p )cos(a x) + (2%i p q + 2%i p )cos(a x)
--R +
--R 2 3
--R %i p q + %i p
--R +
--R +-------+
--R 1 | 2 2
--R - %i\|q + p
--R 2
--R *
--R log
--R +-------+
--R 2 2 2 | 2 2
--R ((q - p )cos(a x) - 2p )sin(a x)\|q + p
--R +
--R 2 3 2 2 3
--R (- %i p q - %i p )cos(a x) + (- 2%i p q - 2%i p )cos(a x)
--R +
--R 2 3
--R - %i p q - %i p
--R +
--R +-------+ +-------+
--R 1 | 2 2 | 2 2
--R - %i\|q + p log(sin(a x)\|q + p + 2%i p cos(a x) + 2%i p)
--R 2
--R +
--R +-------+ +-------+
--R 1 | 2 2 | 2 2
--R - - %i\|q + p log(sin(a x)\|q + p - 2%i p cos(a x) - 2%i p)
--R 2
--R +
--R +-------+
--R 1 1 1 1 | 2 2
--R (%i log(%i) - - %i log(- %i) + - %i log(- - %i) - %i log(- %i))\|q + p
--R 2 2 2 2
--R /
--R 2 3
--R a p q + a p
--R Type: Expression Complex Fraction Integer
--E
--S 108 14:392 Schaums and Axiom differ by a constant
complexNormalize ff
--R
--R (8)
--R 1 1 1 1
--R %i log(%i) - - %i log(- %i) + - %i log(- - %i) - %i log(- %i)
--R 2 2 2 2
--R +
--R 1
--R - - %i log(- 1)
--R 2
--R *
--R +-------+
--R | 2 2
--R \|q + p
--R /
--R 2 3
--R a p q + a p
--R Type: Expression Complex Fraction Integer
--E
@
\section{\cite{1}:14.393~~~~~$\displaystyle
\int{\frac{dx}{p^2-q^2\cos^2{ax}}}$}
$$\int{\frac{1}{p^2-q^2\cos^2{ax}}}=
\left\{
\begin{array}{l}
\displaystyle
\frac{1}{ap\sqrt{p^2-q^2}}\tan^{-1}\frac{p\tan{ax}}{\sqrt{p^2-q^2}}\\
\\
\displaystyle
\frac{1}{2ap\sqrt{q^2-p^2}}\ln\left(\frac{p\tan{ax}-\sqrt{q^2-p^2}}
{p\tan{ax}+\sqrt{q^2-p^2}}\right)
\end{array}
\right.
$$
<<*>>=
)clear all
--S 109
aa:=integrate(1/(p^2-q^2*cos(a*x)^2),x)
--R
--R
--R (1)
--R +-------+
--R 2 2 2 2 | 2 2 2 3
--R ((q - 2p )cos(a x) + p )\|q - p + (- 2p q + 2p )cos(a x)sin(a x)
--R log(----------------------------------------------------------------------)
--R 2 2 2
--R q cos(a x) - p
--R [---------------------------------------------------------------------------,
--R +-------+
--R | 2 2
--R 2a p\|q - p
--R
--R +---------+
--R | 2 2
--R sin(a x)\|- q + p
--R atan(--------------------)
--R 2p cos(a x) + 2p
--R +
--R 2 2 2
--R ((q + p )cos(a x) + 2p )sin(a x)
--R atan(-------------------------------------------)
--R +---------+
--R 2 | 2 2
--R (p cos(a x) + 2p cos(a x) + p)\|- q + p
--R /
--R +---------+
--R | 2 2
--R a p\|- q + p
--R ]
--R Type: Union(List Expression Integer,...)
--E
--S 110
bb1:=1/(a*p*sqrt(p^2-q^2))*atan((p*tan(a*x))/sqrt(p^2-q^2))
--R
--R p tan(a x)
--R atan(------------)
--R +---------+
--R | 2 2
--R \|- q + p
--R (2) ------------------
--R +---------+
--R | 2 2
--R a p\|- q + p
--R Type: Expression Integer
--E
--S 111
bb2:=1/(2*a*p*sqrt(q^2-p^2))*log((p*tan(a*x)-sqrt(q^2-p^2))/(p*tan(a*x)+sqrt(q^2-p^2)))
--R
--R +-------+
--R | 2 2
--R - \|q - p + p tan(a x)
--R log(-------------------------)
--R +-------+
--R | 2 2
--R \|q - p + p tan(a x)
--R (3) ------------------------------
--R +-------+
--R | 2 2
--R 2a p\|q - p
--R Type: Expression Integer
--E
--S 112
cc1:=aa.1-bb1
--R
--R (4)
--R +---------+
--R | 2 2
--R \|- q + p
--R *
--R log
--R +-------+
--R 2 2 2 2 | 2 2
--R ((q - 2p )cos(a x) + p )\|q - p
--R +
--R 2 3
--R (- 2p q + 2p )cos(a x)sin(a x)
--R /
--R 2 2 2
--R q cos(a x) - p
--R +
--R +-------+
--R | 2 2 p tan(a x)
--R - 2\|q - p atan(------------)
--R +---------+
--R | 2 2
--R \|- q + p
--R /
--R +---------+ +-------+
--R | 2 2 | 2 2
--R 2a p\|- q + p \|q - p
--R Type: Expression Integer
--E
--S 113
cc2:=aa.2-bb1
--R
--R (5)
--R +---------+
--R | 2 2
--R sin(a x)\|- q + p p tan(a x)
--R atan(--------------------) - atan(------------)
--R 2p cos(a x) + 2p +---------+
--R | 2 2
--R \|- q + p
--R +
--R 2 2 2
--R ((q + p )cos(a x) + 2p )sin(a x)
--R atan(-------------------------------------------)
--R +---------+
--R 2 | 2 2
--R (p cos(a x) + 2p cos(a x) + p)\|- q + p
--R /
--R +---------+
--R | 2 2
--R a p\|- q + p
--R Type: Expression Integer
--E
--S 114
cc3:=aa.1-bb2
--R
--R (6)
--R log
--R +-------+
--R 2 2 2 2 | 2 2 2 3
--R ((q - 2p )cos(a x) + p )\|q - p + (- 2p q + 2p )cos(a x)sin(a x)
--R ----------------------------------------------------------------------
--R 2 2 2
--R q cos(a x) - p
--R +
--R +-------+
--R | 2 2
--R - \|q - p + p tan(a x)
--R - log(-------------------------)
--R +-------+
--R | 2 2
--R \|q - p + p tan(a x)
--R /
--R +-------+
--R | 2 2
--R 2a p\|q - p
--R Type: Expression Integer
--E
--S 115
cc4:=aa.2-bb2
--R
--R (7)
--R +-------+
--R +---------+ | 2 2
--R | 2 2 - \|q - p + p tan(a x)
--R - \|- q + p log(-------------------------)
--R +-------+
--R | 2 2
--R \|q - p + p tan(a x)
--R +
--R +---------+
--R +-------+ | 2 2
--R | 2 2 sin(a x)\|- q + p
--R 2\|q - p atan(--------------------)
--R 2p cos(a x) + 2p
--R +
--R +-------+ 2 2 2
--R | 2 2 ((q + p )cos(a x) + 2p )sin(a x)
--R 2\|q - p atan(-------------------------------------------)
--R +---------+
--R 2 | 2 2
--R (p cos(a x) + 2p cos(a x) + p)\|- q + p
--R /
--R +---------+ +-------+
--R | 2 2 | 2 2
--R 2a p\|- q + p \|q - p
--R Type: Expression Integer
--E
--S 116
dd2:=ratDenom cc2
--R
--R (8)
--R +---------+
--R +---------+ | 2 2
--R | 2 2 p tan(a x)\|- q + p
--R - \|- q + p atan(----------------------)
--R 2 2
--R q - p
--R +
--R +---------+
--R | 2 2
--R \|- q + p
--R *
--R +---------+
--R 2 2 2 | 2 2
--R ((q + p )cos(a x) + 2p )sin(a x)\|- q + p
--R atan(--------------------------------------------------------)
--R 2 3 2 2 3 2 3
--R (p q - p )cos(a x) + (2p q - 2p )cos(a x) + p q - p
--R +
--R +---------+
--R +---------+ | 2 2
--R | 2 2 sin(a x)\|- q + p
--R - \|- q + p atan(--------------------)
--R 2p cos(a x) + 2p
--R /
--R 2 3
--R a p q - a p
--R Type: Expression Integer
--E
--S 117
tanrule:=rule(tan(a) == sin(a)/cos(a))
--R
--R sin(a)
--R (9) tan(a) == ------
--R cos(a)
--R Type: RewriteRule(Integer,Integer,Expression Integer)
--E
--S 118
ee2:=tanrule dd2
--R
--R (10)
--R +---------+
--R | 2 2
--R \|- q + p
--R *
--R +---------+
--R 2 2 2 | 2 2
--R ((q + p )cos(a x) + 2p )sin(a x)\|- q + p
--R atan(--------------------------------------------------------)
--R 2 3 2 2 3 2 3
--R (p q - p )cos(a x) + (2p q - 2p )cos(a x) + p q - p
--R +
--R +---------+
--R +---------+ | 2 2
--R | 2 2 sin(a x)\|- q + p
--R - \|- q + p atan(--------------------)
--R 2p cos(a x) + 2p
--R +
--R +---------+
--R +---------+ | 2 2
--R | 2 2 p sin(a x)\|- q + p
--R - \|- q + p atan(----------------------)
--R 2 2
--R (q - p )cos(a x)
--R /
--R 2 3
--R a p q - a p
--R Type: Expression Integer
--E
--S 119
atanrule2:=rule(atan(x) == 1/2*%i*(log(1-%i*x)-log(1+%i*x)))
--R
--R 1 1
--R (11) atan(x) == - - %i log(%i x + 1) + - %i log(- %i x + 1)
--R 2 2
--RType: RewriteRule(Integer,Complex Fraction Integer,Expression Complex Fraction Integer)
--E
--S 120
ff2:=atanrule2 ee2
--R
--R (12)
--R -
--R +---------+
--R 1 | 2 2
--R - %i\|- q + p
--R 2
--R *
--R log
--R +---------+
--R 2 2 2 | 2 2
--R ((%i q + %i p )cos(a x) + 2%i p )sin(a x)\|- q + p
--R +
--R 2 3 2 2 3 2 3
--R (p q - p )cos(a x) + (2p q - 2p )cos(a x) + p q - p
--R /
--R 2 3 2 2 3 2 3
--R (p q - p )cos(a x) + (2p q - 2p )cos(a x) + p q - p
--R +
--R +---------+
--R 1 | 2 2
--R +---------+ - %i sin(a x)\|- q + p + p cos(a x) + p
--R 1 | 2 2 2
--R - %i\|- q + p log(------------------------------------------)
--R 2 p cos(a x) + p
--R +
--R +---------+
--R +---------+ | 2 2 2 2
--R 1 | 2 2 %i p sin(a x)\|- q + p + (q - p )cos(a x)
--R - %i\|- q + p log(---------------------------------------------)
--R 2 2 2
--R (q - p )cos(a x)
--R +
--R +---------+
--R +---------+ | 2 2 2 2
--R 1 | 2 2 - %i p sin(a x)\|- q + p + (q - p )cos(a x)
--R - - %i\|- q + p log(-----------------------------------------------)
--R 2 2 2
--R (q - p )cos(a x)
--R +
--R +---------+
--R 1 | 2 2
--R +---------+ - - %i sin(a x)\|- q + p + p cos(a x) + p
--R 1 | 2 2 2
--R - - %i\|- q + p log(--------------------------------------------)
--R 2 p cos(a x) + p
--R +
--R +---------+
--R 1 | 2 2
--R - %i\|- q + p
--R 2
--R *
--R log
--R +---------+
--R 2 2 2 | 2 2
--R ((- %i q - %i p )cos(a x) - 2%i p )sin(a x)\|- q + p
--R +
--R 2 3 2 2 3 2 3
--R (p q - p )cos(a x) + (2p q - 2p )cos(a x) + p q - p
--R /
--R 2 3 2 2 3 2 3
--R (p q - p )cos(a x) + (2p q - 2p )cos(a x) + p q - p
--R /
--R 2 3
--R a p q - a p
--R Type: Expression Complex Fraction Integer
--E
--S 121
gg2:=expandLog ff2
--R
--R (13)
--R +---------+
--R 1 | 2 2
--R - %i\|- q + p
--R 2
--R *
--R log
--R +---------+
--R 2 2 2 | 2 2
--R ((q + p )cos(a x) + 2p )sin(a x)\|- q + p
--R +
--R 2 3 2 2 3 2
--R (%i p q - %i p )cos(a x) + (2%i p q - 2%i p )cos(a x) + %i p q
--R +
--R 3
--R - %i p
--R +
--R -
--R +---------+
--R 1 | 2 2
--R - %i\|- q + p
--R 2
--R *
--R log
--R +---------+
--R 2 2 2 | 2 2
--R ((q + p )cos(a x) + 2p )sin(a x)\|- q + p
--R +
--R 2 3 2 2 3
--R (- %i p q + %i p )cos(a x) + (- 2%i p q + 2%i p )cos(a x)
--R +
--R 2 3
--R - %i p q + %i p
--R +
--R +---------+ +---------+
--R 1 | 2 2 | 2 2 2 2
--R - - %i\|- q + p log(p sin(a x)\|- q + p + (%i q - %i p )cos(a x))
--R 2
--R +
--R +---------+ +---------+
--R 1 | 2 2 | 2 2 2 2
--R - %i\|- q + p log(p sin(a x)\|- q + p + (- %i q + %i p )cos(a x))
--R 2
--R +
--R +---------+ +---------+
--R 1 | 2 2 | 2 2
--R - - %i\|- q + p log(sin(a x)\|- q + p + 2%i p cos(a x) + 2%i p)
--R 2
--R +
--R +---------+ +---------+
--R 1 | 2 2 | 2 2
--R - %i\|- q + p log(sin(a x)\|- q + p - 2%i p cos(a x) - 2%i p)
--R 2
--R +
--R +---------+
--R 1 1 1 1 | 2 2
--R (- %i log(- %i) - - %i log(- - %i))\|- q + p
--R 2 2 2 2
--R /
--R 2 3
--R a p q - a p
--R Type: Expression Complex Fraction Integer
--E
--S 122 14:393 Schaums and Axiom differ by a constant
hh2:=complexNormalize gg2
--R
--R (14)
--R 1 1 1 1 1 1
--R (- - %i log(%i) + - %i log(- %i) - - %i log(- - %i) + - %i log(- %i))
--R 2 2 2 2 2 2
--R *
--R +---------+
--R | 2 2
--R \|- q + p
--R /
--R 2 3
--R a p q - a p
--R Type: Expression Complex Fraction Integer
--E
@
\section{\cite{1}:14.394~~~~~$\displaystyle
\int{x^m\cos{ax}}~dx$}
$$\int{x^m\cos{ax}}=
\frac{x^m\sin{ax}}{a}+\frac{mx^{m-1}}{a^2}\cos{ax}
-\frac{m(m-1)}{a^2}\int{x^{m-2}\cos{ax}}
$$
<<*>>=
)clear all
--S 123 14:394 Axiom cannot compute this integral
aa:=integrate(x^m*cos(a*x),x)
--R
--R
--R x
--R ++ m
--I (1) | cos(%I a)%I d%I
--R ++
--R Type: Union(Expression Integer,...)
--E
@
\section{\cite{1}:14.395~~~~~$\displaystyle
\int{\frac{\cos{ax}}{x^n}}~dx$}
$$\int{\frac{\cos{ax}}{x^n}}=
-\frac{\cos{ax}}{(n-1)x^{n-1}}-\frac{a}{n-1}\int{\frac{\sin{ax}}{x^{n-1}}}
$$
<<*>>=
)clear all
--S 124 14:395 Axiom cannot compute this integral
aa:=integrate(cos(a*x)/x^n,x)
--R
--R
--R x
--I ++ cos(%I a)
--I (1) | --------- d%I
--R ++ n
--I %I
--R Type: Union(Expression Integer,...)
--E
@
\section{\cite{1}:14.396~~~~~$\displaystyle
\int{\cos^n{ax}}~dx$}
$$\int{\cos^n{ax}}=
\frac{\sin{ax}\cos^{n-1}{ax}}{an}+\frac{n-1}{n}\int{\cos^{n-2}{ax}}
$$
<<*>>=
)clear all
--S 125 14:396 Axiom cannot compute this integral
aa:=integrate(cos(a*x)^n,x)
--R
--R
--R x
--R ++ n
--I (1) | cos(%I a) d%I
--R ++
--R Type: Union(Expression Integer,...)
--E
@
\section{\cite{1}:14.397~~~~~$\displaystyle
\int{\frac{1}{\cos^n{ax}}}~dx$}
$$\int{\frac{1}{\cos^n{ax}}}=
\frac{\sin{ax}}{a(n-1)\cos^{n-1}{ax}}
+\frac{n-2}{n-1}\int{\frac{1}{\cos^{n-2}{ax}}}
$$
<<*>>=
)clear all
--S 126 14:397 Axiom cannot compute this integral
aa:=integrate(1/(cos(a*x))^n,x)
--R
--R
--R x
--R ++ 1
--I (1) | ---------- d%I
--R ++ n
--I cos(%I a)
--R Type: Union(Expression Integer,...)
--E
@
\section{\cite{1}:14.398~~~~~$\displaystyle
\int{\frac{x~dx}{cos^n{ax}}}$}
$$\int{\frac{x}{cos^n{ax}}}=
\frac{x\sin{ax}}{a(n-1)\cos^{n-1}{ax}}
-\frac{1}{a^2(n-1)(n-2)\cos^{n-2}{ax}}
+\frac{n-2}{n-1}\int{\frac{x}{\cos^{n-2}{ax}}}
$$
<<*>>=
)clear all
--S 127 14:398 Axiom cannot compute this integral
aa:=integrate(x/cos(a*x)^n,x)
--R
--R
--R x
--I ++ %I
--I (1) | ---------- d%I
--R ++ n
--I cos(%I a)
--R Type: Union(Expression Integer,...)
--E
)spool
)lisp (bye)
@
\eject
\begin{thebibliography}{99}
\bibitem{1} Spiegel, Murray R.
{\sl Mathematical Handbook of Formulas and Tables}\\
Schaum's Outline Series McGraw-Hill 1968 pp77-78
\end{thebibliography}
\end{document}
|
https://www.zentralblatt-math.org/matheduc/en/?id=28759&type=tex | zentralblatt-math.org | CC-MAIN-2019-39 | text/plain | application/x-tex | crawl-data/CC-MAIN-2019-39/segments/1568514576355.92/warc/CC-MAIN-20190923105314-20190923131314-00042.warc.gz | 1,075,717,661 | 1,491 | \input zb-basic
\input zb-matheduc
\iteman{ZMATH 2010d.00618}
\itemau{Rathgeber, Carsten}
\itemti{Entropy, information, and reality: On the basic understanding of the world of information technology. Pt. 1: Reflections on the concept of information. (Entropie, Information und Realit\"at: Zum Grundverst\"andnis der informationstechnischen Welt. T. 1: Betrachtungen zum Informationsbegriff.)}
\itemso{Log In 29, No. 160-161, 83-91 (2010).}
\itemab
Zusammenfassung: Information ist einer der zentralen Informatikbegriffe. Sie wird in der Technik als Entropiegr\"o\ss e verstanden, die (scheinbar) nicht im Materie- oder Energiebegriff aufgeht. Dies und weitere Konzepte zur Information werden im folgenden Beitrag mit Blick auf unser Verst\"andnis der Realit\"at dargelegt.
\itemrv{~}
\itemab
Summary (translation): Information is one of the central concepts in informatics. In technology it is understood as an entropy quantity that (apparently) does not work out in the concept of matter or energy. The article explains this and other concepts on information in view of our understanding of the reality.
\itemrv{~}
\itemcc{P20 K90}
\itemut{expected values; entropy; information theory; theoretical computer science; probability; channels; physics; thermodynamics; data transmission Erwartungswert; Entropie; Informationstheorie; theoretische Informatik; Wahrscheinlichkeit; Kanal; Physik; Thermodynamik; Daten\"ubertragung}
\itemli{}
\end |
https://kohnlehome.de/netz/routerkonfiguration-hsrp.tex | kohnlehome.de | CC-MAIN-2023-06 | text/x-tex | text/x-matlab | crawl-data/CC-MAIN-2023-06/segments/1674764499744.74/warc/CC-MAIN-20230129144110-20230129174110-00362.warc.gz | 366,493,124 | 1,647 | %Teilweise Erzeugt mit dem LaTeX-Generator: http://latex.sehnot.de
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\ihead{Cisco}
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\begin{document}
% Überschrift
\begin{center}
\LARGE % Schriftgröße
\bfseries % Fettdruck
\sffamily % Serifenlose Schrift
Routerkonfiguration: HSRP (Hot Standby Router Protocol)
\end{center}
Mehrere Router bilden eine gemeinsame virtuelle Schnittstelle mit einer virtuellen MAC-Adresse und einer virtuellen IP-Adresse, die von einem Endgerät als Gatewayadresse benutzt werden kann. Der Router mit dem Zustand 'Active' leitet die Pakete weiter. Fällt dieser aus, übernimmt der Router mit dem Zustand 'Standby'.
\subsection*{Routerschnittstelle konfigurieren}
\begin{tabular}{l l}
\texttt{Router(config-if)\# ip address 10.0.0.1 255.0.0.0 } & reale IP-Adresse\\
\texttt{Router(config-if)\# standby version 2} & Version 2 aktivieren\\
\texttt{Router(config-if)\# standby 1 ip 10.0.0.2} & virtuelle IP-Adresse in Gruppe 1 (default = 0)\\
\texttt{Router(config-if)\# standby 1 priority 120} & default=100, je höher, desto 'Active'\\
\texttt{Router(config-if)\# standby 1 preempt} & versucht, 'Active' zu werden
\end{tabular}
\subsection*{Diagnose}
\texttt{Router\# show standby}\\
\texttt{Router\# show standby brief}\\
\texttt{Router\# debug standby packets}
\end{document}
|
https://kskedlaya.org/putnam-archive/2006s.tex | kskedlaya.org | CC-MAIN-2020-50 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2020-50/segments/1606141737946.86/warc/CC-MAIN-20201204131750-20201204161750-00483.warc.gz | 352,712,538 | 13,061 | \documentclass[amssymb,twocolumn,pra,10pt,aps]{revtex4-1}
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\begin{document}
\title{Solutions to the 67th William Lowell Putnam Mathematical Competition \\
Saturday, December 2, 2006}
\author{Kiran Kedlaya and Lenny Ng}
\noaffiliation
\maketitle
\begin{itemize}
\item[A--1]
We change to cylindrical coordinates, i.e., we put $r = \sqrt{x^2 + y^2}$.
Then the given inequality is equivalent to
\[
r^2 + z^2 + 8 \leq 6r,
\]
or
\[
(r-3)^2 + z^2 \leq 1.
\]
This defines a solid of revolution (a solid torus); the area being rotated
is the disc $(x-3)^2 + z^2 \leq 1$ in the $xz$-plane. By Pappus's theorem,
the volume of this equals the area of this disc, which is $\pi$, times the
distance through which the center of mass is being rotated, which is $(2\pi)3$.
That is, the total volume is $6 \pi^2$.
\item[A--2]
Suppose on the contrary that the set $B$ of values of $n$ for which Bob
has a winning strategy is finite; for convenience, we include $n=0$ in $B$,
and write $B = \{b_1, \dots, b_m\}$.
Then for every nonnegative integer $n$ not
in $B$, Alice must have some move on a heap of $n$ stones leading to a
position in which the second player wins. That is, every nonnegative integer
not in $B$ can be written as $b + p - 1$ for some $b \in B$ and some prime
$p$. However, there are numerous ways to show that this cannot happen.
\textbf{First solution:}
Let $t$ be any integer bigger than all of the $b \in B$. Then it is easy to
write down $t$ consecutive composite integers, e.g., $(t+1)! + 2, \dots,
(t+1)! + t+1$. Take $n = (t+1)! + t$; then for each $b \in B$,
$n - b + 1$ is one of the composite integers we just wrote down.
\textbf{Second solution:}
Let $p_1, \dots, p_{2m}$ be
any prime numbers; then by the Chinese remainder theorem, there exists a
positive integer $x$ such that
\begin{align*}
x - b_1 &\equiv -1 \pmod{p_1 p_{m+1}} \\
\dots \\
x - b_n &\equiv -1 \pmod{p_m p_{2m}}.
\end{align*}
For each $b \in B$,
the unique integer $p$ such that $x=b+p-1$ is divisible
by at least two primes, and so cannot itself be prime.
\textbf{Third solution:} (by Catalin Zara)
Put $b_1 = 0$, and take $n = (b_2 - 1)\cdots(b_m - 1)$; then $n$ is
composite because $3, 8 \in B$, and for any nonzero $b \in B$,
$n - b_i + 1$ is divisible by but not equal to $b_i - 1$.
(One could also take $n = b_2 \cdots b_m - 1$, so that
$n-b_i+1$ is divisible by $b_i$.)
\item[A--3]
We first observe that given any sequence of integers
$x_1, x_2, \dots$ satisfying a recursion
\[
x_k = f(x_{k-1}, \dots, x_{k-n}) \qquad (k > n),
\]
where $n$ is fixed and $f$ is a fixed polynomial of $n$ variables with
integer coefficients, for any positive integer $N$, the sequence modulo $N$
is eventually periodic. This is simply because there are only finitely many
possible sequences of $n$ consecutive values modulo $N$, and once such
a sequence is repeated, every subsequent value is repeated as well.
We next observe that if one can rewrite the same recursion as
\[
x_{k-n} = g(x_{k-n+1}, \dots, x_k) \qquad (k > n),
\]
where $g$ is also a polynomial with integer coefficients, then
the sequence extends uniquely to a doubly infinite sequence $\dots,
x_{-1}, x_0, x_1, \dots$ which is fully periodic modulo any $N$.
That is the case in the
situation at hand, because we can rewrite the given recursion as
\[
x_{k-2005} = x_{k+1} - x_k.
\]
It thus suffices to find 2005 consecutive terms divisible by $N$ in the
doubly infinite sequence, for any fixed $N$ (so in particular for
$N = 2006$).
Running the recursion backwards, we easily find
\begin{gather*}
x_1 = x_0 = \cdots = x_{-2004} = 1 \\
x_{-2005} = \cdots = x_{-4009} = 0,
\end{gather*}
yielding the desired result.
\item[A--4]
\textbf{First solution:}
By the linearity of expectation, the average number of local maxima is equal
to the sum of the probability of having a local maximum at $k$ over
$k=1,\dots, n$.
For $k=1$, this probability is 1/2: given the pair
$\{\pi(1), \pi(2)\}$, it is equally likely that $\pi(1)$ or $\pi(2)$ is
bigger. Similarly, for $k=n$, the probability is 1/2. For $1 < k < n$,
the probability is 1/3: given the pair $\{\pi(k-1), \pi(k), \pi(k+1)\}$,
it is equally likely that any of the three is the largest.
Thus the average number of local maxima is
\[
2 \cdot \frac{1}{2} + (n-2) \cdot \frac{1}{3} =
\frac{n+1}{3}.
\]
\textbf{Second solution:}
Another way to apply the linearity of expectation is to compute the
probability that $i \in \{1, \dots, n\}$ occurs as a local maximum.
The most efficient way to do this is to imagine the permutation
as consisting of the symbols $1, \dots, n, *$ written in a circle in
some order. The number $i$ occurs as a local maximum if the two symbols
it is adjacent to both belong to the set $\{*, 1, \dots, i-1\}$. There are
$i(i-1)$ pairs of such symbols and $n(n-1)$ pairs in total, so the
probability of $i$ occurring as a local maximum is $i(i-1)/(n(n-1))$, and
the average number of local maxima is
\begin{align*}
\sum_{i=1}^n \frac{i(i-1)}{n(n-1)} &=
\frac{2}{n(n-1)} \sum_{i=1}^n \binom{i}{2} \\
&= \frac{2}{n(n-1)} \binom{n+1}{3} \\
&= \frac{n+1}{3}.
\end{align*}
One can obtain a similar (if slightly more intricate)
solution inductively, by removing the known
local maximum $n$ and splitting into two shorter sequences.
\textbf{Remark:}
The usual term for a local maximum in this sense is a \emph{peak}.
The complete distribution for the number of peaks is known;
Richard Stanley suggests the reference:
F. N. David and D. E. Barton, \textit{Combinatorial Chance}, Hafner, New York,
1962, p.\ 162 and subsequent.
\item[A--5]
Since the desired expression involves symmetric functions of $a_1,
\dots, a_n$, we start by finding a polynomial with $a_1, \dots, a_n$
as roots. Note that
\[
1 \pm i \tan \theta = e^{\pm i \theta} \sec \theta
\]
so that
\[
1 + i \tan \theta = e^{2 i \theta} (1 - i \tan \theta).
\]
Consequently, if we put $\omega = e^{2 i n \theta}$, then the polynomial
\[
Q_n(x) = (1 + ix)^n - \omega (1-ix)^n
\]
has among its roots $a_1, \dots, a_n$. Since these are distinct and
$Q_n$ has degree $n$, these must be exactly the roots.
If we write
\[
Q_n(x) = c_n x^n + \cdots + c_1 x + c_0,
\]
then $a_1 + \cdots + a_n = -c_{n-1}/c_n$ and $a_1\cdots a_n = -c_0/c_n$,
so the ratio we are seeking is $c_{n-1}/c_0$.
By inspection,
\begin{align*}
c_{n-1} &= n i^{n-1} - \omega n (-i)^{n-1} = n i^{n-1} (1-\omega)\\
c_0 &= 1 - \omega
\end{align*}
so
\[
\frac{a_1+ \cdots + a_n}{a_1 \cdots a_n}
= \begin{cases} n & n \equiv 1 \pmod{4} \\ -n & n \equiv 3 \pmod{4}.
\end{cases}
\]
\textbf{Remark:} The same argument shows that the ratio between
any two \emph{odd} elementary
symmetric functions of $a_1, \dots, a_n$ is independent
of $\theta$.
\item[A--6]
\textbf{First solution:}
(by Daniel Kane)
The probability is $1 - \frac{35}{12\pi^2}$.
We start with some notation and simplifications.
For simplicity, we
assume without loss of generality that the circle has radius 1.
Let $E$ denote the expected value of a random variable over all
choices of $P,Q,R$.
Write $[XYZ]$ for the area of triangle $XYZ$.
If $P,Q,R,S$ are the four points, we may ignore the case where three
of them are collinear, as this occurs with probability zero. Then the only
way they can fail to form the vertices of a convex quadrilateral is if one
of them lies inside the triangle formed by the other three. There are four
such configurations, depending on which point lies inside the triangle, and
they are mutually exclusive. Hence the desired probability is 1 minus
four times the probability that $S$ lies inside triangle $PQR$. That latter
probability is simply $E([PQR])$ divided by the area of
the disc.
Let $O$ denote the center of the circle,
and let $P',Q',R'$ be the projections of $P,Q,R$ onto the circle from $O$.
We can write
\[
[PQR] = \pm [OPQ] \pm [OQR] \pm [ORP]
\]
for a suitable choice of signs, determined as follows. If the points
$P',Q',R'$ lie on no semicircle, then all of the signs are positive.
If $P',Q',R'$ lie on a semicircle in that order and
$Q$ lies inside the triangle $OPR$, then the sign on $[OPR]$ is
positive and the others are negative.
If $P',Q',R'$ lie on a semicircle in that order and
$Q$ lies outside the triangle $OPR$, then the sign on $[OPR]$ is
negative and the others are positive.
We first calculate
\[
E([OPQ] + [OQR] + [ORP]) = 3 E([OPQ]).
\]
Write $r_1 = OP, r_2 = OQ, \theta = \angle POQ$, so that
\[
[OPQ] = \frac{1}{2} r_1 r_2 (\sin \theta).
\]
The distribution of $r_1$ is given by $2r_1$ on $[0,1]$
(e.g., by the change of variable formula to polar coordinates),
and similarly for $r_2$.
The distribution of $\theta$ is uniform on $[0,\pi]$.
These three distributions are independent; hence
\begin{align*}
& E([OPQ]) \\
&= \frac{1}{2} \left( \int_0^{1} 2r^2\,dr \right)^2
\left( \frac{1}{\pi} \int_0^\pi \sin (\theta)\,d\theta \right) \\
&= \frac{4}{9 \pi},
\end{align*}
and
\[
E([OPQ] + [OQR] + [ORP]) = \frac{4}{3 \pi}.
\]
We now treat the case where $P',Q',R'$ lie on a semicircle in
that order.
Put $\theta_1 = \angle POQ$ and $\theta_2 = \angle QOR$; then
the distribution of $\theta_1, \theta_2$ is uniform on the region
\[
0 \leq \theta_1, \quad 0 \leq \theta_2, \quad \theta_1 + \theta_2 \leq \pi.
\]
In particular, the distribution on $\theta = \theta_1 + \theta_2$
is $\frac{2\theta}{\pi^2}$ on $[0, \pi]$.
Put $r_P = OP, r_Q = OQ, r_R = OR$. Again, the distribution on $r_P$
is given by $2 r_P$ on $[0,1]$, and similarly for $r_Q, r_R$; these
are independent from each other and from the joint distribution
of $\theta_1,\theta_2$.
Write $E'(X)$ for the expectation of a random variable $X$
restricted to this part of the domain.
Let $\chi$ be the random variable with value 1 if $Q$ is inside
triangle $OPR$ and 0 otherwise.
We now compute
\begin{align*}
&E'([OPR]) \\
&=
\frac{1}{2} \left( \int_0^1 2r^2\,dr \right)^2
\left( \int_0^\pi \frac{2\theta}{\pi^2} \sin(\theta) \,d\theta \right)\\
&= \frac{4}{9 \pi} \\
& E'(\chi [OPR]) \\
&= E'(2 [OPR]^2 / \theta) \\
&=
\frac{1}{2} \left( \int_0^1 2r^3\,dr \right)^2
\left( \int_0^\pi \frac{2\theta}{\pi^2} \theta^{-1} \sin^2(\theta) \,d\theta \right)\\
&= \frac{1}{8\pi}.
\end{align*}
Also recall that given any triangle $XYZ$, if $T$ is chosen uniformly
at random inside $XYZ$, the expectation of $[TXY]$ is the area of
triangle bounded by $XY$ and the centroid of $XYZ$, namely
$\frac{1}{3} [XYZ]$.
Let $\chi$ be the random variable with value 1 if $Q$ is inside
triangle $OPR$ and 0 otherwise. Then
\begin{align*}
&E'([OPQ] + [OQR] + [ORP] - [PQR]) \\
&= 2 E'(\chi ([OPQ] + [OQR]) + 2 E'((1-\chi)[OPR]) \\
&= 2 E'(\frac{2}{3} \chi [OPR]) + 2 E'([OPR]) - 2 E'(\chi [OPR]) \\
&= 2E'([OPR]) - \frac{2}{3} E'(\chi [OPR]) = \frac{29}{36 \pi}.
\end{align*}
Finally, note that the case when $P',Q',R'$
lie on a semicircle in some order occurs with probability $3/4$.
(The case where they lie on a semicircle proceeding clockwise from $P'$
to its antipode has probability 1/4; this case and its two analogues are
exclusive and exhaustive.) Hence
\begin{align*}
&E([PQR]) \\
&= E([OPQ]+[OQR]+[ORP]) \\
&\quad - \frac{3}{4} E'([OPQ] + [OQR] + [ORP] - [PQR]) \\
&= \frac{4}{3 \pi} - \frac{29}{48 \pi} = \frac{35}{48 \pi},
\end{align*}
so the original probability is
\[
1 - \frac{4 E([PQR])}{\pi} = 1 - \frac{35}{12 \pi^2}.
\]
\textbf{Second solution:}
(by David Savitt)
As in the first solution, it suffices to check that for
$P,Q,R$ chosen uniformly at random in the disc, $E([PQR]) = \frac{35}{48 \pi}$.
Draw the lines $PQ, QR, RP$, which with probability 1 divide the interior
of the circle into seven regions. Put $a = [PQR]$, let $b_1,b_2,b_3$
denote the areas of the
three other regions sharing a side with the triangle, and let
$c_1,c_2,c_3$ denote the areas of the other three regions.
Put $A = E(a)$, $B = E(b_1)$, $C = E(c_1)$, so that
$A + 3B + 3C = \pi$.
Note that $c_1 + c_2 + c_3 + a$ is the area of the region in which we can
choose a fourth point $S$ so that the quadrilateral $PQRS$ fails to be
convex. By comparing expectations, we have $3C + A = 4A$,
so $A = C$ and $4A + 3B = \pi$.
We will compute $B + 2A = B + 2C$, which is the expected area of the part
of the circle cut off by a chord through two random points $D,E$, on the
side of the chord not containing a third random point $F$.
Let $h$ be the distance from the center $O$ of the circle to the line $DE$.
We now determine the distribution of $h$.
Put $r = OD$; the distribution of $r$ is $2r$ on $[0,1]$.
Without loss of generality, suppose $O$ is the origin and
$D$ lies on the positive $x$-axis.
For fixed $r$, the distribution of $h$ runs over $[0,r]$,
and can be computed as the area of the infinitesimal region in which
$E$ can be chosen so the chord through $DE$ has distance to $O$
between $h$ and $h+dh$, divided by $\pi$.
This region splits into two symmetric pieces, one of which lies
between chords making angles of $\arcsin(h/r)$ and
$\arcsin((h + dh)/r)$ with the $x$-axis.
The angle between these is $d\theta = dh/(r^2 - h^2)$.
Draw the chord through $D$ at distance $h$ to $O$, and let $L_1,L_2$ be the
lengths of the parts on opposite sides of $D$; then
the area we are looking for is $\frac{1}{2}(L_1^2 + L_2^2) d\theta$.
Since
\[
\{L_1, L_2 \} = \sqrt{1-h^2} \pm \sqrt{r^2 - h^2},
\]
the area we are seeking (after doubling) is
\[
2\frac{1 + r^2 - 2h^2}{\sqrt{r^2 - h^2}}.
\]
Dividing by $\pi$, then integrating over $r$, we compute the distribution
of $h$ to be
\begin{align*}
&\frac{1}{\pi} \int_h^1 2 \frac{1 + r^2 - 2h^2}{\sqrt{r^2 - h^2}} 2r\,dr \\
&= \frac{16}{3\pi} (1-h^2)^{3/2}.
\end{align*}
We now return to computing $B +2A$.
Let $A(h)$ denote the smaller of the two areas of the disc cut off
by a chord at distance $h$.
The chance that the third point is in the smaller (resp.\
larger) portion is $A(h)/\pi$ (resp.\ $1 - A(h)/\pi$),
and then the area we are trying to compute is $\pi - A(h)$
(resp.\ $A(h)$).
Using the distribution on $h$,
and the fact that
\begin{align*}
A(h) &= 2 \int_h^1 \sqrt{1-h^2}\,dh \\
&= \frac{\pi}{2} - \arcsin(h) - h \sqrt{1-h^2},
\end{align*}
we find
\begin{align*}
&B+2A \\
&= \frac{2}{\pi} \int_0^1 A(h) (\pi - A(h))\, \frac{16}{3\pi} (1-h^2)^{3/2}
\,dh \\
&= \frac{35 + 24 \pi^2}{72 \pi}.
\end{align*}
Since $4A + 3B = \pi$, we solve to obtain
$A = \frac{35}{48 \pi}$ as in the first solution.
\textbf{Third solution:} (by Noam Elkies)
Again, we reduce to computing the average area of a triangle formed by
three random points $A,B,C$ inside a unit circle.
Let $O$ be the center of the circle, and put $c = \max\{OA,OB,OC\}$;
then the probability that $c \leq r$ is $(r^2)^3$, so the distribution
of $c$ is $6c^5\,dc$ on $[0,1]$.
Given $c$, the expectation of $[ABC]$ is equal to $c^2$ times $X$, the expected
area of a triangle formed by two random points $P,Q$ in a circle and
a fixed point $R$ on the boundary. We introduce polar coordinates centered
at $R$, in which the circle is given by $r = 2 \sin \theta$ for
$\theta \in [0, \pi]$. The distribution of a random point in that circle is
$\frac{1}{\pi} r\,dr\,d\theta$ over $\theta \in [0,\pi]$ and
$r \in [0, 2 \sin \theta]$. If $(r,\theta)$ and $(r',\theta')$ are the
two random points, then the area is $\frac{1}{2} rr' \sin |\theta - \theta'|$.
Performing the integrals over $r$ and $r'$ first, we find
\begin{align*}
X &= \frac{32}{9 \pi^2} \int_0^\pi \int_0^\pi \sin^3 \theta \sin^3 \theta'
\sin |\theta-\theta'|\,d\theta'\,d\theta \\
&= \frac{64}{9 \pi^2} \int_0^\pi \int_0^\theta \sin^3 \theta \sin^3 \theta'
\sin (\theta-\theta') \,d\theta'\,d\theta.
\end{align*}
This integral is unpleasant but straightforward; it yields
$X = 35/(36 \pi)$, and
$E([PQR]) = \int_0^1 6c^7 X\,dc = 35/(48 \pi)$, giving the desired
result.
\textbf{Remark:}
This is one of the oldest problems in geometric probability; it is an instance
of Sylvester's four-point problem, which nowadays is usually solved using
a device known as Crofton's formula.
We defer to \texttt{http://mathworld.wolfram.com/} for
further discussion.
\item[B--1]
The ``curve'' $x^3 + 3xy + y^3 - 1 = 0$ is actually reducible, because the
left side factors as
\[
(x+y-1)(x^2 -xy + y^2 + x + y + 1).
\]
Moreover, the second factor is
\[
\frac{1}{2} ((x+1)^2 + (y+1)^2 + (x-y)^2),
\]
so it only vanishes at $(-1,-1)$. Thus the curve in question consists of the
single point $(-1,-1)$ together with the line $x+y=1$. To form a triangle
with three points on this curve, one of its vertices must be $(-1,-1)$.
The other two vertices lie on the line $x+y=1$, so the length of the altitude
from $(-1,-1)$ is the distance from $(-1,-1)$ to $(1/2,1/2)$, or
$3 \sqrt{2}/2$. The area of an equilateral triangle of height $h$ is
$h^2 \sqrt{3}/3$, so the desired area is
$3 \sqrt{3}/2$.
\textbf{Remark:} The factorization used above is a special case of the fact
that
\begin{align*}
&x^3 + y^3 + z^3 - 3xyz \\
&= (x+y+z)(x+\omega y + \omega^2 z)(x + \omega^2 y
+ \omega z),
\end{align*}
where $\omega$ denotes a primitive cube root of unity. That fact in turn follows
from the evaluation of the determinant of the \emph{circulant matrix}
\[
\begin{pmatrix} x & y & z \\ z & x & y \\ y & z & x
\end{pmatrix}
\]
by reading off the eigenvalues of the eigenvectors
$(1, \omega^i, \omega^{2i})$ for $i=0,1,2$.
\item[B--2]
Let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of $x$.
For $i=0,\dots, n$, put $s_i = x_1 + \cdots + x_i$ (so that $s_0 = 0$).
Sort the numbers $\{s_0\}, \dots, \{s_n\}$ into ascending order,
and call the result $t_0, \dots, t_n$. Since $0 = t_0 \leq \cdots \leq
t_n < 1$, the differences
\[
t_1 - t_0, \dots, t_n - t_{n-1}, 1 - t_n
\]
are nonnegative and add up to 1. Hence (as in the pigeonhole principle) one
of these differences
is no more than $1/(n+1)$; if it is anything other than $1 - t_n$,
it equals $\pm (\{s_i\} - \{s_j\})$ for some
$0 \leq i < j \leq n$. Put $S = \{x_{i+1}, \dots, x_j\}$ and
$m = \lfloor s_i \rfloor - \lfloor s_j \rfloor$; then
\begin{align*}
\left| m + \sum_{s \in S} s \right|
&= |m + s_j - s_i| \\
&= |\{s_j\} - \{s_i\}| \\
&\leq \frac{1}{n+1},
\end{align*}
as desired. In case $1 - t_n \leq 1 / (n+1)$, we take
$S = \{x_1, \dots, x_n\}$ and $m = -\lceil s_n \rceil$, and again obtain
the desired conclusion.
\item[B--3]
The maximum is $\binom{n}{2} + 1$, achieved for instance by a
convex $n$-gon: besides the trivial partition (in which all of the points
are in one part), each linear
partition occurs by drawing a line crossing a unique pair
of edges.
\textbf{First solution:}
We will prove that $L_S = \binom{n}{2} + 1$ in any configuration in which
no two of the lines joining points of $S$ are parallel. This suffices
to imply the maximum in all configurations: given a maximal configuration,
we may vary the points slightly to get another maximal configuration in which
our hypothesis is satisfied.
For convenience, we assume $n \geq 3$, as the cases $n=1,2$ are easy.
Let $P$ be the line at infinity in the real projective plane; i.e., $P$
is the set of possible directions of lines in the plane, viewed as a circle.
Remove the directions corresponding to lines through two points of $S$;
this leaves behind $\binom{n}{2}$ intervals.
Given a direction in one of the intervals, consider the set of linear
partitions achieved by lines parallel to that direction. Note that the
resulting collection of partitions depends only on the interval. Then
note that the collections associated to adjacent intervals differ in only
one element.
The trivial partition that puts all of $S$ on one side is in every such
collection. We now observe that for any other linear partition
$\{A,B\}$, the set of intervals to which $\{A,B\}$ is:
\begin{enumerate}
\item[(a)] a consecutive block of intervals, but
\item[(b)] not all of them.
\end{enumerate}
For (a), note that if $\ell_1, \ell_2$ are nonparallel lines achieving
the same partition, then we can rotate around their point of intersection
to achieve all of the intermediate directions on one side or the other.
For (b), the case $n=3$ is evident; to reduce the general case to this case,
take points $P,Q,R$ such that $P$ lies on the opposite side of
the partition from $Q$ and $R$.
It follows now that that each linear partition,
except for the trivial one, occurs in exactly one place as the partition
associated to some interval but not to its immediate counterclockwise neighbor.
In other words, the number of linear partitions is one more than the
number of intervals, or $\binom{n}{2} + 1$ as desired.
\textbf{Second solution:}
We prove the upper bound
by induction on $n$. Choose a point $P$ in the convex hull of $S$.
Put $S' = S \setminus \{P\}$;
by the induction hypothesis, there are at most $\binom{n-1}{2} + 1$
linear partitions of $S'$. Note that each linear partition of $S$ restricts
to a linear partition of $S'$. Moreover, if two linear partitions of $S$
restrict to the same linear partition of $S'$, then that partition of $S'$
is achieved by a line through $P$.
By rotating a line through $P$, we see that there are at most $n-1$
partitions of $S'$ achieved by lines through $P$: namely, the partition only
changes when the rotating line passes through one of the points of $S$.
This yields the desired result.
\textbf{Third solution:} (by Noam Elkies) We enlarge the plane to a projective
plane by adding a line at infinity, then apply the polar duality map
centered at one of the points $O \in S$. This turns the rest of $S$ into
a set $S'$ of $n-1$ lines in the dual projective plane. Let $O'$ be the
point in the dual plane corresponding to the original line at infinity;
it does not lie on any of the lines in $S'$.
Let $\ell$ be a line in the original plane, corresponding to a point $P$ in
the dual plane. If we form the linear partition induced by $\ell$, then
the points of $S \setminus \{O\}$ lying in the same part as $O$
correspond to the lines of $S'$ which cross the segment $O'P$.
If we consider the dual affine plane as being divided into regions by
the lines of $S'$, then the lines of $S'$ crossing the segment $O'P$
are determined by which region $P$ lies in.
Thus our original maximum is equal to the maximum number of regions into
which $n-1$ lines divide an affine plane. By induction on $n$, this number
is easily seen to be $1 + \binom{n}{2}$.
\textbf{Fourth solution:} (by Florian Herzig)
Say that an \emph{$S$-line} is a line that intersects $S$ in at least two points.
We claim that the nontrivial linear partitions of $S$ are in natural bijection with pairs
$(\ell, \{X,Y\})$ consisting of an $S$-line $\ell$ and a nontrivial linear partition $\{X,Y\}$ of $\ell \cap S$.
Since an $S$-line $\ell$ admits precisely $|\ell\cap S|-1 \le \binom{|\ell \cap S|}{2}$ nontrivial linear partitions,
the claim implies that $L_S \le \binom n2 + 1$ with equality iff no three points of $S$ are collinear.
Let $P$ be the line at infinity in the real projective plane. Given any nontrivial linear partition $\{A,B\}$ of $S$, the
set of lines inducing this partition is a proper, open, connected subset $I$ of $P$. (It is proper because it has to omit
directions of $S$-lines that pass through both parts of the partition and open because we can vary the separating line. It is
connected because if we have two such lines that aren't parallel, we can rotate through their point of intersection to
get all intermediate directions.) Among all $S$-lines that intersect both $A$ and $B$ choose a line $\ell$ whose direction is
minimal (in the clockwise direction) with respect to the interval $I$; also, pick an arbitrary line $\ell'$ that induces
$\{A,B\}$. By rotating $\ell'$ clockwise to $\ell$ about their point of intersection, we see that the direction
of $\ell$ is the least upper bound of $I$. (We can't hit any point of $S$ during the rotation because of the minimality
property of $\ell$.) The line $\ell$ is in fact unique because if the (parallel) lines $pq$ and $rs$ are two choices for $\ell$,
with $p$, $q \in A$; $r$, $s \in B$, then one of the diagonals $ps$, $qr$ would contradict the minimality property of
$\ell$. To define the above bijection we send $\{A,B\}$ to $(\ell, \{A \cap \ell, B \cap \ell\})$.
Conversely, suppose that we are given an $S$-line $\ell$ and a nontrivial linear partition $\{X,Y\}$ of $\ell \cap S$.
Pick any point $p \in \ell$ that induces the partition $\{X,Y\}$. If we rotate the line $\ell$ about $p$ in the counterclockwise
direction by a sufficiently small amount, we get a nontrivial linear partitition of $S$ that is independent of all choices.
(It is obtained from the partition of $S-\ell$ induced by $\ell$ by adjoining $X$ to one part and $Y$ to the other.) This
defines a map in the other direction.
By construction these two maps are inverse to each other, and this proves the claim.
\textbf{Remark:}
Given a finite set $S$ of points in $\mathbb{R}^n$, a \emph{non-Radon partition}
of $S$ is a pair $(A,B)$
of complementary subsets that can be separated by
a hyperplane. \emph{Radon's theorem} states that if $\#S\geq n+2$, then not
every $(A,B)$ is a non-Radon partition. The result of this problem
has been greatly
extended, especially within the context of matroid theory and oriented
matroid theory. Richard Stanley suggests the following references:
T. H. Brylawski, A combinatorial
perspective on the Radon convexity theorem, \emph{Geom. Ded.} \textbf{5}
(1976),
459-466; and T. Zaslavsky, Extremal arrangements of hyperplanes,
\emph{Ann. N. Y. Acad. Sci.} \textbf{440} (1985), 69-87.
\item[B--4]
The maximum is $2^k$, achieved for instance by the subspace
\[
\{(x_1, \dots, x_n) \in \mathbb{R}^n: x_1 = \cdots = x_{n-k} = 0\}.
\]
\textbf{First solution:}
More generally, we show that any affine $k$-dimensional plane in
$\mathbb{R}^n$ can contain at most $2^k$ points in $Z$. The proof is by
induction on $k+n$; the case $k=n=0$ is clearly true.
Suppose that $V$ is a $k$-plane in $\mathbb{R}^n$. Denote the
hyperplanes $\{x_n = 0\}$ and $\{x_n = 1\}$ by $V_0$ and $V_1$,
respectively. If $V\cap V_0$ and $V\cap V_1$ are each at most
$(k-1)$-dimensional, then $V\cap V_0\cap Z$ and $V\cap V_1 \cap Z$ each
have cardinality at most $2^{k-1}$ by the induction assumption, and
hence $V\cap Z$ has at most $2^k$ elements. Otherwise, if $V\cap V_0$ or
$V\cap V_1$ is $k$-dimensional, then $V \subset V_0$ or $V\subset V_1$;
now apply the induction hypothesis on $V$, viewed as a subset of
$\mathbb{R}^{n-1}$ by dropping the last coordinate.
\textbf{Second solution:}
Let $S$ be a subset of $Z$ contained in a $k$-dimensional subspace of $V$.
This is equivalent to asking that any $t_1, \dots, t_{k+1} \in S$
satisfy a nontrivial linear dependence $c_1 t_1 + \cdots + c_{k+1} t_{k+1} = 0$
with $c_1, \dots, c_{k+1} \in \mathbb{R}$. Since $t_1, \dots, t_{k+1} \in
\mathbb{Q}^n$, given such a dependence we can always find another one with
$c_1, \dots, c_{k+1} \in \mathbb{Q}$; then by clearing denominators, we
can find one with $c_1, \dots, c_{k+1} \in \mathbb{Z}$ and not all having a
common factor.
Let $\mathbb{F}_2$ denote the field of two elements, and let
$\overline{S} \subseteq \mathbb{F}_2^n$ be the reductions modulo 2 of the points of
$S$. Then any $t_1, \dots, t_{k+1} \in \overline{S}$ satisfy a nontrivial
linear dependence, because we can take the dependence from the end of
the previous paragraph and reduce modulo 2. Hence $\overline{S}$ is contained
in a $k$-dimensional subspace of $\mathbb{F}_{2^n}$, and the latter has cardinality
exactly $2^k$. Thus $\overline{S}$ has at most $2^k$ elements, as does
$S$.
Variant (suggested by David Savitt): if $\overline{S}$ contained $k+1$
linearly independent elements, the $(k+1) \times n$ matrix formed by these
would have a nonvanishing maximal minor. The lift of that minor back to $\RR$
would also not vanish, so $S$ would contain $k+1$ linearly independent
elements.
\textbf{Third solution:} (by Catalin Zara)
Let $V$ be a $k$-dimensional subspace. Form the matrix whose rows are the elements
of $V \cap Z$; by construction, it has row rank at most $k$. It thus also has
column rank at most $k$; in particular, we can choose $k$ coordinates such that
each point of $V \cap Z$ is determined by those $k$ of its coordinates. Since
each coordinate of a point in $Z$ can only take two values, $V \cap Z$ can have
at most $2^k$ elements.
\textbf{Remark:} The proposers probably did not realize that this problem appeared
online about three months before the exam, at
\texttt{http://www.artofproblemsolving.com/ Forum/viewtopic.php?t=105991}. (It
may very well have also appeared even earlier.)
\item[B--5]
The answer is $1/16$. We have
\begin{align*}
&\int_0^1 x^2 f (x)\,dx - \int_0^1 x f(x)^2\,dx \\
&= \int_0^1 (x^3/4 - x
( f(x)-x/2)^2)\,dx \\
&\leq \int_0^1 x^3/4\,dx = 1/16,
\end{align*}
with equality when $f(x) = x/2$.
\item[B--6]
\textbf{First solution:}
We start with some easy
upper and lower bounds on $a_n$.
We write $O(f(n))$ and $\Omega(f(n))$ for functions $g(n)$ such that
$f(n)/g(n)$ and $g(n)/f(n)$, respectively, are bounded above.
Since $a_n$ is a nondecreasing sequence, $a_{n+1}-a_n$ is bounded above,
so $a_n = O(n)$. That means $a_n^{-1/k} = \Omega(n^{-1/k})$, so
\[
a_n = \Omega \left( \sum_{i=1}^n i^{-1/k} \right)
= \Omega(n^{(k-1)/k}).
\]
In fact, all we will need is that $a_n \to \infty$ as $n \to \infty$.
By Taylor's theorem with remainder, for $1 < m < 2$ and $x>0$,
\[
|(1+x)^m - 1 - mx| \leq \frac{m(m-1)}{2}x^2.
\]
Taking $m = (k+1)/k$ and $x = a_{n+1}/a_n = 1 + a_n^{-(k+1)/k}$, we obtain
\[
\left| a_{n+1}^{(k+1)/k} - a_n^{(k+1)/k} - \frac{k+1}{k} \right|
\leq \frac{k+1}{2k^2} a_n^{-(k+1)/k}.
\]
In particular,
\[
\lim_{n \to \infty} a_{n+1}^{(k+1)/k} - a_n^{(k+1)/k} = \frac{k+1}{k}.
\]
In general, if $x_n$ is a sequence with $\lim_{n \to \infty} x_n = c$, then
also
\[
\lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^n x_i = c
\]
by Cesaro's lemma. Explicitly, for any $\epsilon > 0$, we can find $N$ such that
$|x_n - c| \leq \epsilon/2$ for $n \geq N$, and then
\[
\left| c - \frac{1}{n} \sum_{i=1}^n x_i \right|
\leq \frac{n-N}{n} \frac{\epsilon}{2} + \frac{N}{n} \left| \sum_{i=1}^N (c-x_i) \right|;
\]
for $n$ large, the right side is smaller than $\epsilon$.
In our case, we deduce that
\[
\lim_{n \to \infty} \frac{a_n^{(k+1)/k}}{n} = \frac{k+1}{k}
\]
and so
\[
\lim_{n \to \infty} \frac{a_n^{k+1}}{n^k} = \left(\frac{k+1}{k} \right)^k,
\]
as desired.
\textbf{Remark:}
The use of Cesaro's lemma above is the special case $b_n = n$
of the \emph{Cesaro-Stolz
theorem}: if $a_n,b_n$ are sequences such that $b_n$ is positive,
strictly increasing, and unbounded, and
\[
\lim_{n \to \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = L,
\]
then
\[
\lim_{n \to \infty} \frac{a_n}{b_n} = L.
\]
\textbf{Second solution:}
In this solution, rather than applying Taylor's theorem with remainder
to $(1+x)^m$ for $1 < m < 2$ and $x > 0$, we only apply convexity to deduce
that $(1+x)^m \geq 1 + mx$. This gives
\[
a_{n+1}^{(k+1)/k} - a_n^{(k+1)/k} \geq \frac{k+1}{k},
\]
and so
\[
a_n^{(k+1)/k} \geq \frac{k+1}{k} n + c
\]
for some $c \in \RR$. In particular,
\[
\liminf_{n \to \infty} \frac{a_n^{(k+1)/k}}{n} \geq \frac{k+1}{k}
\]
and so
\[
\liminf_{n \to \infty} \frac{a_n}{n^{k/(k+1)}} \geq \left(\frac{k+1}{k} \right)^{k/(k+1)}.
\]
But turning this around, the fact that
\begin{align*}
&a_{n+1} - a_n \\
&= a_n^{-1/k} \\
&\leq \left(\frac{k+1}{k} \right)^{-1/(k+1)} n^{-1/(k+1)}
(1 + o(1)),
\end{align*}
where $o(1)$ denotes a function tending to 0 as $n \to \infty$,
yields
\begin{align*}
&a_n \\
&\leq
\left(\frac{k+1}{k} \right)^{-1/(k+1)} \sum_{i=1}^n i^{-1/(k+1)} (1 + o(1)) \\
&= \frac{k+1}{k} \left(\frac{k+1}{k} \right)^{-1/(k+1)} n^{k/(k+1)}(1 + o(1)) \\
&= \left( \frac{k+1}{k} \right)^{k/(k+1)} n^{k/(k+1)}(1 + o(1)),
\end{align*}
so
\[
\limsup_{n \to \infty} \frac{a_n}{n^{k/(k+1)}} \leq \left( \frac{k+1}{k}
\right)^{k/(k+1)}
\]
and this completes the proof.
\textbf{Third solution:}
We argue that $a_n \to \infty$ as in the first solution.
Write $b_n = a_n - L n^{k/(k+1)}$, for a value of $L$ to be determined
later.
We have
\begin{align*}
&b_{n+1} \\
&= b_n + a_n^{-1/k} - L ((n+1)^{k/(k+1)} - n^{k/(k+1)}) \\
&= e_1 + e_2,
\end{align*}
where
\begin{align*}
e_1 &= b_n + a_n^{-1/k} - L^{-1/k} n^{-1/(k+1)} \\
e_2 &= L ((n+1)^{k/(k+1)} - n^{k/(k+1)}) \\
&\quad - L^{-1/k} n^{-1/(k+1)}.
\end{align*}
We first estimate $e_1$.
For $-1 < m < 0$, by the convexity of $(1+x)^m$
and $(1+x)^{1-m}$, we have
\begin{align*}
1 + mx &\leq (1+x)^m \\
&\leq 1 + mx (1+x)^{m-1}.
\end{align*}
Hence
\begin{align*}
-\frac{1}{k} L^{-(k+1)/k} n^{-1} b_n &\leq e_1 - b_n \\
&\leq
-\frac{1}{k} b_n a_n^{-(k+1)/k}.
\end{align*}
Note that both bounds have sign opposite to $b_n$; moreover,
by the bound $a_n = \Omega(n^{(k-1)/k})$, both bounds have absolutely
value strictly less than that of $b_n$ for $n$ sufficiently large. Consequently,
for $n$ large,
\[
|e_1| \leq |b_n|.
\]
We now work on $e_2$.
By Taylor's theorem
with remainder applied to $(1+x)^m$ for $x > 0$ and $0 < m < 1$,
\begin{align*}
1+mx &\geq (1+x)^m \\
&\geq 1 + mx + \frac{m(m-1)}{2} x^2.
\end{align*}
The ``main term'' of $L ((n+1)^{k/(k+1)} - n^{k/(k+1)})$
is $L \frac{k}{k+1} n^{-1/(k+1)}$. To make this coincide with
$L^{-1/k} n^{-1/(k+1)}$, we take
\[
L = \left( \frac{k+1}{k} \right)^{k/(k+1)}.
\]
We then find that
\[
|e_2| = O(n^{-2}),
\]
and because $b_{n+1} = e_1 + e_2$, we have
$|b_{n+1}| \leq |b_n| + |e_2|$. Hence
\[
|b_n| = O\left (\sum_{i=1}^n i^{-2} \right) = O(1),
\]
and so
\[
\lim_{n \to \infty} \frac{a_n^{k+1}}{n^k} = L^{k+1} = \left( \frac{k+1}{k} \right)^k.
\]
\textbf{Remark:}
The case $k=2$ appeared on the 2004 Romanian Olympiad (district level).
\textbf{Remark:}
One can make a similar argument for any sequence given by
$a_{n+1} = a_n + f(a_n)$, when $f$ is a \emph{decreasing} function.
\textbf{Remark:}
Richard Stanley suggests a heuristic for determining the asymptotic
behavior of sequences of this type: replace the given recursion
\[
a_{n+1} - a_n = a_n^{-1/k}
\]
by the differential equation
\[
y' = y^{-1/k}
\]
and determine the asymptotics of the latter.
\end{itemize}
\end{document}
|
http://www.foo.be/docs-free/autonomy/chap07.tex | foo.be | CC-MAIN-2021-43 | text/x-tex | application/x-tex | crawl-data/CC-MAIN-2021-43/segments/1634323585911.17/warc/CC-MAIN-20211024050128-20211024080128-00714.warc.gz | 108,674,770 | 4,142 | \chapter{Un jour au soleil}
\citer{Robert M. Pirsig, C.E. 1974}{Personne ne soutient fanatiquement que le soleil se lèvera demain.\\
Tout le monde le sait.}
\begin{verbatim}
Lundi 24 septembre 2057
(Metadate: 2.073-9:96:285 kD nouvel epoch)
Champaign, Illinois
\end{verbatim}
Kyle avait toujours appréhendé la douleur qui accompagnait un
rétro-chargement vers le monde physique. Il savait par expérience que
les procédures de trans-chargement et rétro-chargement n'étaient pas
en elles-mêmes la cause de ces désagréments physiques, ce qu'il
ressentait était les changements normaux de son corps physique, ces
maux subtiles qu'il avait ressenti et ignoré tout au long de sa vie
physique. Il gémit et enleva l'interface neuronale de son visage,
frotta ses yeux et se leva doucement, le lit craquant sous son poids.
La pièce sentait la poussière, l'air était sucré, froid, emplit d'air
conditionné et de saletés en suspension.
Même lorqusqu'il avait résidé dans un n\oe ud autonome de première
génération, avec son facteur d'accélération par rapport au monde
physique de seulement trente, le temps qu'il avait passé dans le monde
réel s'était élevé à seulement quelques petites heures pour chaque
dizaine de circadiens de temps subjectif. Maintenant qu'il opérait
sur un n\oe ud de seconde génération, composé de nano-éléments qu'il
avait lui-même conçus, des centaines de circadiens s'écoulaient entre
chaque rétro-chargement dans son corps. En d'autres termes, il
passait en moyenne entre deux et trois ans de vie digitale entre
différentes intrusions dans le monde physique.
Kyle redoutait en particulier ce jour-là. Si les choses se déroulaient
comme prévu, il passerait la majeure partie de la journée dans le
Physique, rencontrerait les trois amis qui l'avaient aidé à fonder la
Communauté et le service de production de masse de nano-composants par
catalyse. En échange de leur aide et de leur soutien continu dans ce
service si important, ses trois amis rejoindraient la Communauté
Autonome, et seraient libérés des contraintes physiques qui faisaient
maintenant gémir Kyle.
Si les choses tournaient mal, il serait contraint de resterait ici
bien longtemps. Cette perspective était loin d'être réjouissante,
chaque minute dans le réel lui coûtant presque 2 déciCircadiens
d'expérience subjective dans le Virtuel. Ses contacts sociaux, son
cercle d'amis, tout ceci changerait complètement en une journée. La
société qu'il réintégrerait en se trans-chargeant à nouveau aurait
au moins vieilli d'un ou deux ans, peut-être même plus, selon le
temps qu'il aurait besoin de passer dans le Physique. Kyle pensait
que les gens changeraient sans doute beaucoup en une année. Les amis
qu'il avait rencontré aujourd'hui avaient parlé de ses changements de
personnalité et de tempérament des derniers jours, un temps très court
pour eux mais qui représentait pour lui deux décades. Il regrettait le
temps qu'il avait perdu ici, tous les projets, toutes les fêtes et les
expériences qu'il avait manqué, et toutes ces amitiés qui déclinaient
petit à petit, peut-être pour finalement disparaître, pendant qu'il
était absent.
Il lui restait un choix. L'importance du projet du jour ne devait pas
être sous-estimée. La Communauté Autonome avait besoin d'un accès
rapide et facile au monde réel, sans les contraintes laborieuses et
onéreuses du rétro-chargement. Après tout, quelqu'un, quelque part,
devait fabriquer des n\oe uds autonomes pour ceux qui souhaitent faire
évoluer leur équipement de première génération vers des n\oe uds bien
plus rapides de seconde génération, ou pour les nouveaux membres de la
Communauté, dont le nombre croissait sans cesse. Toute la Communauté
n'était pas nécessaire à leur fabrication. Des liens de fibre optique
vers Internet étaient indispensables, tout comme des appareils
d'expérimentation et des laboratoires, voire même (si quelques-unes
des rumeurs s'avéraient fondées) des moyens de défense physique pour
leur corps, pour que l'impensable ne se produise pas, que les
autorités n'apprennent quelque chose et ne tentent de tuer la
Communauté en plein développement. On avait besoin de tout cela et
même de plus, chaque tâche coûtant à chacun le type de temps que Kyle
sacrifiait aujourd'hui.
Bien sûr, à chaque fois que Kyle avait une idée de projet utile pour
la Communauté, il y avait des milliers d'autres personnes qui en
avaient d'autres. C'était la beauté de l'approche ouverte de la
science et de la culture que la Communauté avait engendré~: aucune
vision ne limitait la portée ou les domaines de connaissance qui
pouvaient être explorés~: un millier d'esprits avec mille visions
peuvent réaliser bien plus qu'un seul esprit ou une seule vision. Les
avantages de la liberté collaborative étaient exponentiels, avec une
synergie, un effet multiplicatif qui engendre dans un effet boule de
neige de nouvelles découvertes et inventions. Les découvertes étaient
faites dans un torrent d'activité, une percée menant à d'autres dans
un souffle de vitesse où les scientifiques publient leurs pensées et
leurs démonstrations sous la forme d'engrammes de connaissance pour
qui veut l'assimiler et la comprendre. Cela rendait la frénésie
scientifique du siècle dernier ridicule en comparaison. Malgré tout,
si les membres de la Communauté pensaient que le taux de progrès
scientifique pendant cette nouvelle renaissance était élevé, il
attendaient tous que le rétro-chargement devienne obsolète et qu'il ne
soit plus nécessaire de ralentir leur esprit par un facteur de cinq
cents ou plus à chaque fois d'un objet physique aurait besoin d'être
manipulé ou assemblé. La recherche et le progrès scientifique seraient
un petit jeu dangereux en comparaison.
Après cinq décades de stagnation et de litiges en tout genre, il
serait à nouveau passionnant d'être un scientifique.
Kyle était stupéfié de tous le temps subjectif qui était passé depuis
le jour précédent, lorqu'il s'était rétro-chargé pour s'exercer et
garder son corps physique en bonne santé. Avait-il réellement vécu
sept cents Circadiens~? Toute sa routine matinale n'était plus
habituelle, puisqu'il n'était pas certain de pouvoir se rappeler de
toutes les tâches quotidiennes nécessaires au maintien de sa vie dans
le Physique. Plutôt que son instinct, Kyle devait s'appuyer sur une
checklist mental d'activités, qu'il complétait dans son esprit au fur
et à mesure. Il revint difficilement à lui, passa encore nu et humide
la porte de la douche, quand il réalisa à quel point il n'était peu à
sa place dans le monde physique. Il se força à s'asseoir et fit un
checklist écrit des choses essentielles à faire pour entretenir son
corps aussi bien qu'il le pourrait. Malheureusement, tous les
perfectionnements intellectuels qu'il avait prévu pour le Virtuel lui
faisait gravement défaut ; ici son esprit était faible et limité, et
véritablement apte à l'erreur~! En ajoutant à cela toutes les décades
subjectives passées en tant que particules électroniques dans un
univers virtuel où il avait été projeté en juste quelques heures
d'expérience physique, il n'y avait rien de merveilleux dans son corps
et sa propre vie lui était devenue presque étrangère.
Il s'allongea sur le canapé de son salon et attendit que ses amis arrivent,
en regardant les rayons de soleil qui passaient dans un petit trou des
rideaux, comme une lame de lumière découpant l'air. Il essaya d'oublier
un peu les douleurs qui le parcourait. Un courant d'air frais fit frissonner
sa peau : c'était la climatisation bas de gamme qui dans un
gémissement perpétuel, crachait sa fraîcheur dans la pièce. Ça allait
être une bien longue journée.
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\date{April 20, 2011\\(revised July 13, 2012)}
\title{On Some Questions Concerning Strong Compactness
\thanks{2010 Mathematics Subject Classifications:
03E25, 03E35, 03E45, 03E55.}
\thanks{Keywords: Supercompact cardinal,
strongly compact cardinal, GCH, symmetric inner model.}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}
\thanks{The author wishes to thank
Brent Cody for helpful conversations
on the subject matter of this paper.
The author also wishes to thank the
second referee for helpful corrections
and suggestions which were incorporated
into the current version of the paper.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/aapter\\
awapter@alum.mit.edu}
\begin{document}
\maketitle
\begin{abstract}
A question of Woodin asks
if $\gk$ is strongly compact and GCH holds
below $\gk$, then must GCH hold everywhere?
One variant of this question asks
%a question of Woodin asks
if $\gk$ is strongly compact and GCH
fails at every regular cardinal
$\gd < \gk$, then must GCH fail at
some regular cardinal $\gd \ge \gk$?
Another variant asks if it is possible
for GCH to fail at every limit cardinal
less than or equal to
a strongly compact cardinal $\gk$.
We get a negative answer to the first
of these questions and positive answers
to the second of these questions
for a supercompact cardinal $\gk$
in the context of the absence of the full
Axiom of Choice.
%In all of our results, $\gk$ is fully supercompact.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In \cite[22.22, page 310]{K}, the following
question is attributed to Woodin:
If $\gk$ is strongly compact and GCH holds
below $\gk$, then must GCH hold everywhere?
Assuming the Axiom of Choice, an easy reflection
argument yields that the answer to this question
must be yes if $\gk$ is supercompact.
However, when full
AC is false, things are very different. Specifically,
we have the following
theorem from \cite{A00}, which provides
a negative answer to Woodin's question in
the context of the absence of AC.
\begin{theorem}\label{t1}
Let $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
There is then a partial ordering $\FP \in V$ and
a symmetric inner model $N$, $V \subseteq N \subseteq V^\FP$,
such that
$N \models ``$ZF + $\forall \gd < \gk[DC_{\gd}]$ +
$\gk$ is a strong limit cardinal +
$\forall \gd < \gk[2^\gd = \gd^+]$ + $\gk$ is supercompact +
There is a sequence $\la A_\ga \mid \ga < \gk^{++} \ra$
of distinct subsets of $\gk$''.
\end{theorem}
Woodin's question may be inverted to produce
related questions concerning strongly compact
cardinals and GCH. In particular, one may ask if
$\gk$ is strongly compact and
GCH fails at every regular cardinal $\gd < \gk$,
then must GCH fail at some regular cardinal
$\gd \ge \gk$?
As in Woodin's original question, a simple reflection
argument yields that the answer to this
question must be yes if $\gk$ is supercompact.
On the other hand, it is also possible to ask
about the possibility of $\gk$ being
strongly compact and
GCH failing at every limit cardinal $\gd \le \gk$.
Of course, by Solovay's
celebrated theorem \cite{S}, GCH must always
hold at any singular strong limit cardinal
above a strongly compact cardinal $\gk$.
Consequently, a simple reflection argument now shows
that the answer to this question
must be no if $\gk$ is supercompact.
The purpose of this paper is to
provide answers to these questions
in the context of the absence of full AC
for a supercompact cardinal $\gk$.
%but where $\gk$ is fully supercompact.
We show that as in
\cite{A00}, it is possible to get a negative
answer to the first of the
above questions.
On the other hand, it is also possible
to get a positive answer to the second of
the above questions.
Specifically,
we prove the following two theorems, where
we adopt as our terminology that when
AC is false and $\gd$ is a cardinal\footnote{For
the purposes of this paper, all cardinals will
be well-ordered, i.e., will be alephs.},
``GCH holds at $\gd$'' means that there is an injection
$f : \gd^+ \to \wp(\gd)$, and for every
cardinal $\gl > \gd^+$, there is no injection
$f : \gl \to \wp(\gd)$.
Similarly, in a choiceless context,
``GCH fails at $\gd$'' means that for some
cardinal $\gl > \gd^+$, there is an injection
$f : \gl \to \wp(\gd)$.
\begin{theorem}\label{t2}
Let $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
There is then a partial ordering $\FP \in V$ and
a symmetric inner model $N$, $V \subseteq N \subseteq V^\FP$,
such that
$N \models ``$ZF + $\forall \gd < \gk[DC_{\gd}]$ +
$\gk$ is a strong limit cardinal +
$\gk$ is supercompact +
Every successor cardinal is regular +
$\forall \gd < \gk[$If $\gd$ is regular, then
$2^\gd = \gd^{++}$, but if $\gd$ is singular,
then $2^\gd = \gd^+]$ +
GCH holds at every (regular or singular)
cardinal $\gd \ge \gk$''.
\end{theorem}
\begin{theorem}\label{t3}
Let $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
There is then a partial ordering $\FP \in V$ and
a symmetric inner model $N$, $V \subseteq N \subseteq V^\FP$,
such that
$N \models ``$ZF + $\neg {AC}_\go$ +
$\gk$ is a limit cardinal +
$\gk$ is supercompact +
Every successor cardinal is regular +
GCH fails at every limit
cardinal $\gd \le \gk$ +
GCH holds at every (regular or singular)
cardinal $\gd > \gk$''.
\end{theorem}
We take this opportunity to make a few
brief remarks concerning Theorems \ref{t2} and \ref{t3}.
Note that in the absence of full AC, $\gk$ being
supercompact means that for every cardinal
$\gl \ge \gk$, $P_\gk(\gl)$ carries a
$\gk$-additive, fine, normal ultrafilter, and
$\gk$ being strongly compact means that for every cardinal
$\gl \ge \gk$, $P_\gk(\gl)$ carries a
$\gk$-additive, fine (not necessarily normal) ultrafilter.
Consequently, the conclusions of Theorems \ref{t2} and \ref{t3}
remain valid, with ``$\gk$ is strongly compact'' replacing
``$\gk$ is supercompact''.
%our results are true in a choiceless context
%for strongly compact cardinals as well.
Note also that
as \cite[Example 15.57, pages 259--260]{J} shows, when AC is false,
it is possible for successor cardinals to be singular.
Thus, the fact that every successor cardinal is
regular in the models witnessing the
conclusions of Theorems \ref{t2} and \ref{t3}
is especially significant.
In addition, in Theorem \ref{t2},
in direct analogy to Theorem \ref{t1}, it will
literally be the case that ``$\gk$ is a
strong limit cardinal'',
``For every regular cardinal $\gd < \gk$,
$2^\gd = \gd^{++}$'', and
``For every singular cardinal $\gd < \gk$,
$2^\gd = \gd^+$'' mean the same thing
as when AC is true.
In Theorem \ref{t3}, however,
this won't be the situation. More specifically, GCH
holding and failing will be in the weaker sense
described above, although as is the case when
AC is true, $\gk$ remains a limit cardinal.\footnote{As
\cite[Theorem 21.16, pages 404--406]{J} shows,
without the Axiom of Choice, it is possible for
large cardinals to be successor cardinals.}
Finally, as each of our theorems shows, when
AC is false, a supercompact cardinal need
not possess its full reflection properties.
We mention very briefly some preliminary information.
We assume a basic knowledge of set theoretic
terminology and large cardinals and
forcing, as provided, e.g., by \cite{J}.
In particular, when $\FP$ is our forcing
partial ordering and $G$ is $V$-generic over $\FP$,
we will abuse notation somewhat and use both
$V^\FP$ and $V[G]$ to denote the generic extension by $\FP$.
We will also frequently abuse notation by writing
$x$ instead of $\check x$ for ground model sets.
We note in addition that for $\gk$ a regular cardinal and
$\ga$ an ordinal, $\add(\gk, \ga)$ is the standard
partial ordering for adding $\ga$ many Cohen
subsets of $\gk$, i.e.,
$\add(\gk, \ga) = \{f : \gk \times \ga \to \{0, 1\}
\mid \card{\dom(f)} < \gk\}$, ordered by inclusion.
For $\gk$ a regular cardinal and $\gl > \gk$ an
inaccessible cardinal, ${\rm Coll}(\gk, {<} \gl)$
is the standard L\'evy collapse partial ordering
for collapsing $\gl$ to $\gk^+$, i.e.,
${\rm Coll}(\gk, {<} \gl) = \{f : \gk \times \gl
\to \gl \mid \card{\dom(f)} < \gk$, and for every
$\la \ga, \gb \ra \in \dom(f)$,
$f(\la \ga, \gb \ra) < \gb\}$, ordered by inclusion.
\section{The Proofs of Theorems \ref{t2} and \ref{t3}}\label{s2}
We turn now to the proof of Theorem \ref{t2}.
We will be constructing a symmetric model of
``ZF + $\forall \d < \k[{\rm DC}_\gd]$'' in which
$\k$ is supercompact, $\k$ is a strong limit cardinal,
every successor cardinal is regular,
GCH fails at every regular cardinal $\gd < \gk$, and
GCH holds at all other cardinals.
\begin{proof}
The proof of Theorem \ref{t2} will be similar to
the proof of Theorem \ref{t1} found in \cite{A00}.
We will therefore freely quote (sometimes verbatim
when appropriate) from \cite{A00}.
Let $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
Let $\la \gd_\ga \mid \ga < \gk \ra$ enumerate in
increasing order the regular cardinals less than $\gk$.
For each ordinal $\a < \gk$, let
$\FP_\a = \add(\d_\a, \d_\a^{++})$.
The partial ordering $\FP$ with which we force is
then the Easton support product $\prod_{\a < \gk} \FP_\a$.
Let $G$ be $V$-generic over $\FP$. The full generic extension
$V[G]$ is not our desired model $N$.
In order to define $N$, we first let $G_\a$
for any $\a < \gk$ be the
projection of $G$ onto $\prod_{\gb < \a} \FP_\gb = \FQ_\a$.
By the Product Lemma, $G_\a$ is $V$-generic over $\FQ_\a$.
We can now intuitively describe $N$ as the least
model of ZF extending $V$ which contains,
for each $\a < \gk$, the set $G_\a$.
In order to define $N$ more formally, let
${\cal L}_1$ be the ramified sublanguage of the forcing
language ${\cal L}$ with respect to $\FP$
which contains symbols $\check v$ for each
$v \in V$, a unary predicate symbol $\check V$
(to be interpreted $\check V(\check v)$ iff
$v \in V$), and symbols $\dot G_\a$ for each
ordinal $\a < \gk$. $N$ can then
be defined inside $V[G]$ as follows.
\bigskip
\setlength{\parindent}{1.5in}
$N_0 = \emptyset$.
$N_\gl = \bigcup_{\ga < \gl} N_\ga$ if $\gl$ is a limit ordinal.
$N_{\ga + 1} = \Bigl\{\,x \subseteq N_\ga\, \Bigm|\,
\raise6pt\vtop{\baselineskip=10pt\hbox{$x$ is definable over the
model ${\la N_\ga, \in, c \ra}_{c \in N_\ga}$} \hbox{via a term $\tau \in
{\cal L}_1$ of rank $\le \ga$}}\,\Bigr\}$.
$N = \bigcup_{\ga \in {\rm Ord}^V} N_\ga$.
\setlength{\parindent}{1.5em}\bigskip
\noindent Standard arguments show $N \models {\rm ZF}$.
\begin{lemma}\label{l1}
Let $\l$ be an ordinal. If $x \subseteq \l$,
$x \in N$, then $x \in V[G_\a]$ for some $\a < \k$.
\end{lemma}
\begin{proof}
%We follow the proof of \cite[Lemma 1]{A00}.
We slightly modify the proof of \cite[Lemma 1]{A00}.
Let $\tau$ be a term for $x$ such that
$p \forces ``\tau \subseteq \l$''. Without loss of generality,
by coding if necessary, we can assume $\tau$ mentions only
one term of the form $\dot G_\a$ for some $\a < \k$.
For $q \in \FP$, $q = \la q_\b \mid \b < \k \ra$, define
$q \rest \a = \la q^*_\b \mid \b < \k \ra$ by
$q^*_\b = q_\b$ if $\b < \a$ and $q^*_\b = 0$
(the trivial condition) otherwise.
We can now define a term $\sigma$ by
$q \forces ``\g \in \sigma$'' iff $q$ extends $p$ and
%$q \forces ``\g \in \tau$'', and
$q \rest \a \forces ``\g \in \tau$''. It is clear that
$p \forces ``\sigma \subseteq \tau$''. We show in addition that
$p \forces ``\tau \subseteq \sigma$''.
To see that this is true,
let $q$ extending $p$,
$q = \la q_\b \mid \b < \k \ra$ be such that
$q \forces ``\g \in \tau$'', and assume towards a contradiction that
$q \rest \a \not\forces ``\g \in \tau$''.
Let $r$ extending $q \rest \a$,
$r = \la r_\b \mid \b < \k \ra$ be such that
$r \forces ``\g \not\in \tau$''. If we define
$s = \la s_\b \mid \b < \k \ra$ by $s_\b = r_\b$ for $\b < \a$
and $s_\b = q_\b$ otherwise, then by definition, $s$ extends
$q$ and $s \forces ``\g \in \tau$''.
Let $r_\b$ and $s_\b$ be such that $r_\b$ and $s_\b$ are
incompatible. Since $r_\b, s_\b \in \FP_\b$ and
$\FP_\b = \add(\d_\b, \d_\b^{++})$,
there is an automorphism
$\psi_\b : \FP_\b \to \FP_\b$ generated by a permutation of
$\d_\b$ such that $\psi_\b(r_\b)$ is compatible with $s_\b$.
(Any $t \in \add(\d_\b, \d_\b^{++})$ is a collection of
ordered triples of the form
$\la \xi_0, \xi_1, \xi_2 \ra$,
where $\xi_0 < \d_\b$, $\xi_1 < \d_\b^{++}$, and
$\xi_2 \in \{0, 1\}$. This means that we can let
$t$'s first domain
$\dom_1(t) = \{\xi < \d_\b \mid
\exists \xi_1 < \d_\b^{++} \exists \xi_2 \in \{0, 1\}
[\la \xi, \xi_1, \xi_2 \ra \in t]\}$.
Let $\eta < \d_\b$ be an ordinal greater than
$\max(\sup(\dom_1(s_\b)), \sup(\dom_1(r_\b)))$.
$\eta$ exists since for any condition $t \in \FP_\b$,
$\card{\dom_1(t)} < \d_\b$ and $\d_\b$ is a regular cardinal.
If $\la \rho_i \mid i < \zeta \ra$ enumerates
$\dom_1(r_\b)$ and $\la \rho_i' \mid i < \zeta \ra$ enumerates
the first $\zeta$ ordinals greater than $\eta$, then
$\psi^*_\b : \d_\b \to \d_\b$ given by
$\psi^*_\b(\rho_i) = \rho_i'$,
$\psi^*_\b(\rho_i') = \rho_i$,
and $\psi^*_\b$ is the identity otherwise is the
desired permutation. The automorphism $\psi_\b$ is defined by
applying $\psi^*_\b$ to each element
of a condition's first domain, i.e.,
for $t \in \FP_\gb$, $\psi_\b(t) =
\{\la \psi^*_\b(\xi_0), \xi_1, \xi_2 \ra \mid
\la \xi_0, \xi_1, \xi_2 \ra \in t \}$.)
Thus, if $\pi = \la \pi_\b \mid \b < \k \ra$ is defined by
$\pi_\b = \psi_\b$ if $r_\b$ and $s_\b$ are incompatible and
$\psi_\b$ is as just described
and $\pi_\b$ is the identity otherwise, $\pi$ generates an
automorphism of $\FP$ such that
$\pi(r)$ is compatible with $s$.
Note now that $\pi_\b$ is the identity for $\b < \a$.
Since terms for ground model sets and terms mentioning only
$\dot G_\a$ can be assumed to be invariant under automorphisms
of $\FP$ not changing the value of $G_\a$,
$\pi(r) \forces ``\g \not\in \tau$'',
$\pi(r)$ is compatible with $s$, and
$s \forces ``\g \in \tau$''. This contradiction means that
$q \rest \a \forces ``\g \in \tau$'', i.e.,
$p \forces ``\tau \subseteq \sigma$'', i.e.,
$p \forces ``\tau = \sigma$''.
Since $\sigma$ can clearly be realized in $V[G_\a]$,
$x \in V[G_\a]$.
This completes the proof of Lemma \ref{l1}.
\end{proof}
\begin{lemma}\label{l2}
$N \models ``\forall \d < \k
[{\rm DC}_\d]$''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l2} is
identical to the proof of \cite[Lemma 2]{A00}.
For completeness, we present it here.
Fix $\d < \k$ a cardinal in $N$.
Recall that ${\rm DC}_\d$ is the statement that
whenever $X$ is a set and $R \subseteq
[X]^{< \gd} \times X$ is a relation
such that for all
$\vec y \in [X]^{< \gd}$, there is $z \in X$
such that $\vec y \ R \ z$, then there is a
$\gd$ sequence $\vec Y$ such that for all
$\ga < \gd$, $\vec Y \rest \ga \ R \ Y(\ga)$.
Consequently, working inductively, assume
$p \forces ``\dot X \in \dot N$ is a set,
$\dot R \in \dot N$, $\dot R \subseteq
{[\dot X]}^{< \d} \times \dot X$ is a relation,
$\la \tau_\a \mid \a < \b < \d \ra \in \dot N$ is a sequence of
elements of $\dot X$, and for
$\la \tau_\a \mid \a < \g < \b \ra$,
$\la \tau_\a \mid \a < \g \ra \ \dot R \ \tau_\g$''.
We show how to define $\tau_\b$. Work in $V$. Let
$\eta = \sup(\{\a \mid \exists \g < \b [\dot G_\a$
occurs in $\tau_\g]\})$.
Since each $\tau_\g$ for $\g < \b$ can be assumed to be
an element of ${\cal L}_1$, and since $\k$ is a regular
limit cardinal, $\eta < \k$, so
$\la \tau_\a \mid \a < \b \ra$ can be defined using only
$\dot G_\eta$ and hence is an element of ${\cal L}_1$.
By AC in $V$,
since $\FP$ is an Easton support product of the appropriate
Cohen partial orderings, $\FP$ is $\k$-c.c. Thus,
again by AC in $V$, there is
%let ${\cal B} \subseteq \FP$,
${\cal B}$ with $|{\cal B}| < \k$,
${\cal B} = \{\la p_\rho, \sigma_\rho \ra \mid
\rho < \g^* < \k \}$ such that
%and ${\cal B}$ has the property that
${\cal A} = \{p_\rho \mid \rho < \g^*\}$
forms a maximal antichain of conditions extending $p$ and
$p_\rho \forces ``\la \tau_\a \mid \a < \b \ra \ \dot R
\ \sigma_\rho$''. As before,
$\eta^* = \sup(\{\a \mid \exists \g < \g^*
[\dot G_\a$ occurs in $\sigma_\g]\})$ is such that
$\eta^* < \k$, meaning ${\cal B}$ can be used to define
a term $\tau_\b \in {\cal L}_1$ such that
$p \forces ``\la \tau_\a \mid \a < \b \ra \ \dot R \ \tau_\b$''.
Since $\d < \k$, as before,
$\la \tau_\a \mid \a < \d \ra \in {\cal L}_1$. By the fact
$\la \tau_\a \mid \a < \d \ra$ can be realized in $N$,
$\la \tau_\a \mid \a < \d \ra$ will denote in $N$ a
${\hbox{\rm DC}}_\d$ sequence for $\dot R$ and $\dot X$.
This completes the proof of Lemma \ref{l2}.
\end{proof}
\begin{lemma}\label{l3}
$N \models ``\gk$ is a limit cardinal +
Every successor cardinal is regular''.
\end{lemma}
\begin{proof}
Standard arguments (see \cite{J}) in
conjunction with the fact that $\FP$
is the Easton support product of
$\add(\d_\a, \d_\a^{++})$ where $\a < \k$
show that $V$ and $V[G]$ have the same cardinals
and cofinalities.
%By the definition of $\FP$, $V$ and $V[G]$ have
%the same cardinals and cofinalities.
Therefore, since $V \subseteq N \subseteq V[G]$,
$N$ also has the same cardinals and cofinalities
as do $V$ and $V[G]$. In particular,
$N \models ``\gk$ is a limit cardinal +
Every successor cardinal is regular''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
\begin{lemma}\label{l4}
$N \models ``\forall \d < \k[$If $\d$ is regular, then
$2^\d = \d^{++}$, but if $\d$ is singular, then
$2^\d = \d^+]$''.
\end{lemma}
\begin{proof}
%We use ideas from the proof of \cite[Lemma 4]{A00}.
Let $\d < \k$ be a (regular or singular) cardinal.
Let $\gl$ be the least inaccessible cardinal greater than $\d$.
%As in the proof of Lemma \ref{l3},
Write
$\FP = \FQ_\gl \times \FQ^\gl$ and $G = G_\gl \times G^\gl$,
where $\FQ^\gl = \prod_{\gl \le \ga < \gk} \FP_\ga$ and
$G^\gl$ is the projection of $G$ onto $\FQ^\gl$.
Since $V[G^\gl]$ and $V$ contain the same bounded
subsets of $\gl$ and $V[G_\gl] \subseteq N$, it suffices
to show that $V[G_\gl] \models ``2^\d = \d^{++}$ if
$\d$ is regular, but $2^\d = \d^+$ if $\d$ is singular''.
However, once again, standard arguments (see \cite{J})
in conjunction with the fact that $\FQ_\gl$ is the
Easton support product of $\add(\d_\a, \d_\a^{++})$ where
$\a < \l$ yield that
$V[G_\gl] \models ``2^\d = \d^{++}$ if $\d$ is regular,
but $2^\d = \d^+$ if $\d$ is singular''.
This completes the proof of Lemma \ref{l4}.
\end{proof}
We remark that Lemmas \ref{l3}
and \ref{l4} show $N \models ``\k$ is
a strong limit cardinal''.
Also, note that
$N \models \neg {\hbox{\rm AC}}_\k$.
To see this, we follow the remark found
after the proof of \cite[Lemma 4]{A00}.
Define in $N$ for each $\a < \k$ the set
$X_\a = \{x \subseteq \d_\a^{++} \mid x$ codes a $\d_\a^{++}$
sequence of subsets of $\d_\a\}$.\footnote{By the proof
of Lemma \ref{l3}, $\d_\a$ is regular in
$V$, $N$, and $V[G]$.}
Although
$\la X_\a \mid \a < \k \ra \in N$, $
({\prod_{\a < \k} X_\a})^N
= \emptyset$. This follows since an element $y$ of $
({\prod_{\a < \k} X_\a})^N$
may be thought of as a set of ordinals, so by Lemma \ref{l1},
$y \in V[G_\b]$ for some $\b < \k$.
This, however, is impossible, as $\card{\FQ_\b} < \k$,
so a final segment of the sequence
of regular cardinals below $\k$ satisfies GCH in $V[G_\b]$.
\begin{lemma}\label{l5}
$N \models ``$GCH holds at every
(regular or singular) cardinal $\gd \ge \gk$''.
\end{lemma}
\begin{proof}
Since $\card{\FP} = \k$ and $V \models {\rm GCH}$,
$V[G] \models ``$GCH holds at every
(regular or singular) cardinal $\gd \ge \gk$''.
The fact that $V \subseteq N \subseteq V[G]$
then immediately implies that
$N \models ``$For every (regular or singular) cardinal
$\gd \ge \gk$, there is an injection $f : \d^+ \to \wp(\d)$,
but for every (regular or singular) cardinal $\gd \ge \gk$
and every cardinal $\gl > \gd^+$, there is no injection
$f : \gl \to \wp(\gd)$''.
This completes the proof of Lemma \ref{l5}.
\end{proof}
\begin{lemma}\label{l6}
$N \models ``\k$ is supercompact''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l6} is virtually identical to the proof of
\cite[Lemma 5]{A00}.
As before, for completeness, we include it here.
Fix $\l \ge \k$ and ${\cal U}$
a $\k$-additive, fine, normal ultrafilter
over $P_\k(\l)$ in $V$. Working
in $N$, let ${\cal U}' = \{x \subseteq
{(P_\k(\l))}^N
\mid \exists y \in \U[y \subseteq x]\}$. We show that
$N \models ``{\cal U}'$ is a $\k$-additive,
fine, normal ultrafilter over
${(P_\k(\l))}^N$''.
To see this, fix $x \subseteq
{(P_\k(\l))}^N
$, $x \in N$, and let $\tau$ be a term for $x$ mentioning
only $\dot G_\a$. Contained in the proof of Lemma \ref{l1}
is the fact that $y = \{p \in
{(P_\k(\l))}^V
\mid p \in x\}$
is actually a set in $V[G_\a]$. This follows since
the proof of Lemma \ref{l1} really shows that for a term
$\tau^*$ as just described and an element
$z \in V$, the statement $``z \in \tau^*$'' is decidable in
$V[G_\a]$. Thus, since $|\FQ_\a| < \k$, the
L\'evy-Solovay arguments \cite{LS}
show that in $V[G_\a] \subseteq N$,
either $y$ or $(P_\k(\l))^V - y$ contains a set in $\U$.
%its complement contains a ${\cal U}$ measure 1 set.
Further, if
$N \models ``\la x_\b \mid \b < \g < \k \ra$ is a sequence
such that each $x_\b \in {\cal U}'$'', then
let $\tau_1$ be such that $\tau_1$ denotes
$\la x_\b \mid \b < \g < \k \ra$ and mentions only $\dot G_\a$.
The methods of \cite{LS} yield that
for every $\b < \g$, there is
%a condition $p_\b \in G_\a$ and
a set $y_\b \in {\cal U}$ definable in $V$ such that
$\forces_{\FQ_\a} ``y_\b \subseteq \{p \in
{(P_\k(\l))}^V \mid p \in \dot x_\b\}$''.
%and $y_\b \in {\cal U}$.
%it is possible to define in $V[G_\a]$ a sequence
%$\la y_\b \mid \b < \g < \k \ra$ such that for each $\b < \g$,
%The methods of \cite{LS} then imply that
%$\bigcap_{\b < \g} y_\b$ contains a
%${\cal U}$ measure 1 set in $V[G_\a]$, so
%$\bigcap_{\b < \g} x_\b$ contains a ${\cal U}$
%measure 1 set in $N$.
Since $y^* = \bigcap_{\b < \g} y_\b \in {\cal U}$,
$N \models ``\exists y \in \U[y \subseteq
\bigcap_{\b < \g} x_\b]$''.
%so because
%$V[G_\a] \subseteq N$, this same statement is true in $N$.
Finally, if $N \models ``f :
{(P_\k(\l))}^N
\to \l$ is a choice function'', then if $\dot f$
denotes $f$ and mentions only $\dot G_\a$,
it is possible to define in $V[G_\a] \subseteq N$ the function
$g = f \rest {(P_\k(\l))}^V$. Once more, the results of
\cite{LS} show that for some $x \in {\cal U}$,
$V[G_\a] \models ``g$ is constant on $x$''. Thus,
$N \models ``{\cal U}'$ is a $\k$-additive,
fine, normal ultrafilter over
${(P_\k(\l))}^N$''.
This completes the proof of Lemma \ref{l6}.
\end{proof}
Lemmas \ref{l1} -- \ref{l6} and the intervening remarks
complete the proof of Theorem \ref{t2}.
\end{proof}
Having completed the proof of Theorem \ref{t2},
we turn now to the proof of Theorem \ref{t3}.
We will be constructing a symmetric model of
``ZF + $\neg {\rm AC}_\go$'' in which
$\k$ is supercompact, $\k$ is a limit cardinal,
every successor cardinal is regular,
GCH fails at every limit cardinal $\gd \le \gk$, and
GCH holds at all cardinals above $\gk$.
\begin{proof}
Let $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
We define a partial ordering $\ov \FQ$ such that
$V^{\ov \FQ} = \ov V \models ``$ZFC + $\k$ is supercompact +
$2^\k = \k^{++}$ + $2^\d = \d^+$ for every cardinal
$\d \ge \k^+$ + There is a club $C \subseteq \k$
composed of inaccessible cardinals and their limits with
$2^\d = 2^{\d^+} = \d^{++}$ for every $\gd \in C$''.
To obtain $\ov \FQ$, let $\FQ_1$ be Laver's partial ordering
of \cite{L} which makes $\k$'s supercompactness indestructible
under $\k$-directed closed forcing.
Since $\FQ_1$ may be defined so that $\card{\FQ_1} = \k$,
it is then the case that
$V^{\FQ_1 \ast \dot \add(\k, \k^{++})} = V_2 \models ``$ZFC +
$\k$ is supercompact + $2^\k = \k^{++}$ + $2^\gd = \gd^+$ for
every cardinal $\gd \ge \k^+$''.
Let $\FQ_3 \in V_2$ be Radin forcing defined over
$\gk$ using one repeat point (see either \cite{G10} or
\cite{R} for the precise definition of $\FQ_3$).
Standard facts about Radin forcing (see \cite{A91},
\cite{G10}, and \cite{R}) then show that
$V^{\FQ_3}_2 = V^{\FQ_1 \ast \dot \add(\k, \k^{++}) \ast \dot \FQ_3} =
\ov V \models
``$ZFC + $\k$ is supercompact +
$2^\k = \k^{++}$ + $2^\d = \d^+$ for every cardinal
$\d \ge \k^+$ + There is a club $C \subseteq \k$
composed of inaccessible cardinals and their limits with
$2^\d = 2^{\d^+} = \d^{++}$ for every $\gd \in C$''.
With an abuse of notation, we now let $\ov V = V$.
Let $\la \gk_i \mid i < \gk \ra \in V$ be the
continuous, increasing enumeration of $C \cup \{\go\}$.
For $i < \gk$, let $\FP_i = {\rm Coll}(\gk^{++}_i, {<} \gk_{i + 1})$.
The partial ordering $\FP$ with which we force is then the
Easton support product $\FP = \prod_{i < \k} \FP_i$.
%We now define $\FP = \prod_{i < \k} \FP_i$ with Easton support.
%\footnote{As our proof will show, the suport used
%can actually be arbitrary.}.
Let $G$ be $V$-generic over $\FP$.
$V[G]$, being a model of AC, is once more not our
desired model $N$. In order to define $N$, we
first note that as before, by the Product Lemma,
for $i < \gk$, $G_i$, the projection of $G$ onto $\FP_i$,
is $V$-generic over $\FP_i$.
Again by the Product Lemma, $G_I = \prod_{i \in I} G_i$
is $V$-generic over $\FP_I = \prod_{i \in I} \FP_i$.
We can now intuitively describe $N$ as the least model of
ZF extending $V$ which contains, for each finite set of
ordinals $I \subseteq \gk$, the set $G_I$.
In order to define $N$ more formally, let
${\cal L}_1$ be the ramified sublanguage of the forcing
language ${\cal L}$ with respect to $\FP$
which contains symbols $\check v$ for each
$v \in V$, a unary predicate symbol $\check V$
(to be interpreted $\check V(\check v)$ iff
$v \in V$), and symbols $\dot G_I$ for each
finite set of ordinals $I \subseteq \gk$. $N$ can then
be defined inside $V[G]$ as follows.
\bigskip
\setlength{\parindent}{1.5in}
$N_0 = \emptyset$.
$N_\gl = \bigcup_{\ga < \gl} N_\ga$ if $\gl$ is a limit ordinal.
$N_{\ga + 1} = \Bigl\{\,x \subseteq N_\ga\, \Bigm|\,
\raise6pt\vtop{\baselineskip=10pt\hbox{$x$ is definable over the
model ${\la N_\ga, \in, c \ra}_{c \in N_\ga}$} \hbox{via a term $\tau \in
{\cal L}_1$ of rank $\le \ga$}}\,\Bigr\}$.
$N = \bigcup_{\ga \in {\rm Ord}^V} N_\ga$.
\setlength{\parindent}{1.5em}\bigskip
\noindent As in the proof of Theorem \ref{t2},
standard arguments show $N \models {\rm ZF}$.\footnote{Although
defining $N$ using $G_i$ for every $i < \k$
is equivalent to our presentation, it is not
as useful for the arguments we are about to give.}
%this definition of $N$
%using G_I$ for every finite $I \subseteq \k$
\begin{lemma}\label{l7}
Let $\l$ be an ordinal. If $x \subseteq \l$,
$x \in N$, then $x \in V[G_I]$ %V[\prod_{i \in I} G_i]$
for some finite set of ordinals $I \subseteq \gk$.
\end{lemma}
\begin{proof}
Suppose $i < \gk$.
It is a standard fact (see, e.g., \cite[Lemma 5.2]{B78})
that since $\FP_i$ is a version of the L\'evy collapse,
for any $p, q \in \FP_i$, there is an automorphism
$\pi_i : \FP_i \to \FP_i$ such that $\pi_i(p)$ is
compatible with $q$. The proof of Lemma \ref{l7} is now
essentially the same as the proof of Lemma \ref{l1}, with
each occurrence of ``$\ga$'' in Lemma \ref{l1} replaced
by an occurrence of ``$I$''.
This completes the proof of Lemma \ref{l7}.
\end{proof}
\begin{lemma}\label{l8}
%If $N \models ``\gd < \gk$ is a successor cardinal'', then
%either $\gd = \gk_i$, $\gd = \gk^+_i$, or $\gd = \gk^{++}_i$
%for some $i < \gk$.
$N \models ``$Every successor cardinal is regular''.
\end{lemma}
\begin{proof}
Suppose first that $N \models ``\gd > \gk$ is a successor cardinal''.
Since $\FP = \prod_{i < \k} \FP_i$ is an Easton support product,
$\FP$ is $\gk$-c.c. This means that $V$ and $V[G]$ have the same
cardinals and cofinalities at and above $\gk$. Therefore, since
$V \subseteq N \subseteq V[G]$, $V$, $N$, and $V[G]$ all have
the same cardinals and cofinalities at and above $\gk$. In particular,
$N \models ``\gd$ is regular''.
Suppose next that $N \models ``\gd < \gk$ is a
(successor or limit) cardinal''.
We claim that
either $\gd = \gk_i$,
$\gd = (\gk^+_i)^V$, or
$\gd = (\gk^{++}_i)^V$
for some $i < \gk$.
To see this, since $C$ is club in $\gk$, we can let
$k < \gk$ be such that $\gk_{k + 1}$ is the least
member of $C$ greater than $\gd$.
If the claim is false, then because
$\gd \neq \gk_k$, $\gd \neq (\gk^+_k)^V$, and
$\gd \neq (\gk^{++}_k)^V$, $\gd \in ((\gk^{++}_k)^V, \gk_{k + 1})$.
However, since $G_k$ is $V$-generic over
${\rm Coll}(\gk^{++}_k, {<} \gk_{k + 1})$,
$V[G_k] \models ``\gd$ is not a cardinal''.
Consequently, because $V[G_k] \subseteq N$,
$N \models ``\gd$ is not a cardinal'', a contradiction
to the assumption that
$\gd \neq \gk_k$, $\gd \neq (\gk^+_k)^V$, and
$\gd \neq (\gk^{++}_k)^V$.
%The proof of our claim will thus be complete
%once we have shown
We next claim that for any $i < \gk$,
$N \models ``\gk_i$, $(\gk^+_i)^V$, and $(\gk^{++}_i)^V$
are all cardinals''. However, since any collapse map $f$
would have to be coded by a set of ordinals, if this were false,
then by Lemma \ref{l7}, there would have to be some
finite set of ordinals $I \subseteq \gk$ such that
$f \in V[G_I] = V[\prod_{j \in I} G_j]$.
Because $\prod_{j \in I} G_j$ is $V$-generic over
$\prod_{j \in I} \FP_j =
\prod_{j \in I} {\rm Coll}(\gk^{++}_j, {<} \gk_{j + 1})$
and $I$ is finite, this is impossible.
We now know that if
$N \models ``\gd < \gk$ is a successor cardinal'',
then there must be some $i < \gk$ such that
either $\gd = \gk_i$, $\gd = (\gk^+_i)^V$,
or $\gd = (\gk^{++}_i)^V$.
Assume that $\gd = \gk_i$.
It must be the case that $i$ is a successor ordinal.
This is since
if $i$ were a limit ordinal, then
$N \models ``\gk_i = \sup_{k < i} \gk_k$ and $\gk_k$
for $k < i$ is a cardinal'', i.e.,
$N \models ``\gk_i$ is a limit cardinal''.
Consequently, because $i$ is a successor ordinal,
$\gk_i$ is a successor member of the Radin generic
club $C$. This means that
$V \models ``\gk_i$ is inaccessible'', so in particular,
$V \models ``\gk_i$ is a regular cardinal''. If
$N \models ``\gk_i$ is singular'', then let
$S \subseteq \gk_i$, $S \in N$ be a witness to this fact.
Again by Lemma \ref{l7}, there must be some
finite set of ordinals $I \subseteq \gk$ such that
$f \in V[G_I] = V[\prod_{j \in I} G_j]$. Once more,
because $\prod_{j \in I} G_j$ is $V$-generic over
$\prod_{j \in I} \FP_j =
\prod_{j \in I} {\rm Coll}(\gk^{++}_j, {<} \gk_{j + 1})$
and $I$ is finite, this is impossible.
Hence, $N \models ``\gd$ is a regular cardinal''.
Assume finally that
either $\gd = (\gk^+_i)^V$ or $\gd = (\gk^{++}_i)^V$.
Clearly, since $V \models {\rm ZFC}$, it is also true that
$V \models ``\gk^+_i$ and $\gk^{++}_i$ are regular cardinals''.
The same contradiction as obtained in the preceding paragraph
again yields that
$N \models ``\gd$ is a regular cardinal''.
This completes the proof of Lemma \ref{l8}.
\end{proof}
As has just been noted in the the proof of Lemma \ref{l8},
$N \models ``\gd < \gk$ is a cardinal'' iff either
$\gd = \gk_i$, $\gd = (\gk^+_i)^V$, or $\gd = (\gk^{++}_i)^V$
for some $i < \gk$.
%if $i < \gk$, then $N \models ``\gk_i$ is a cardinal''.
Therefore, since $\gk = \sup_{i < \gk} \gk_i$,
$N \models ``\gk$ is a limit cardinal''.
\begin{lemma}\label{l9}
$N \models ``$GCH fails at every limit cardinal $\gd \le \gk$''.
\end{lemma}
\begin{proof}
Suppose first that $\gd = \gk$.
As noted in the proof of Lemma \ref{l8},
$V$, $N$, and $V[G]$ all have the same cardinals
and cofinalities at and above $\gk$.
In addition, $V \models ``2^\gk = \gk^{++}$''.
%Further, since $\card{\FP} = \gk$ and
%$V \models ``2^\gk = \gk^{++}$'', $V[G] \models ``2^\gk = \gk^{++}$''.
Therefore, because $V \subseteq N$, %\subseteq V[G]$,
$N \models ``$There is an injection
$f : \gk^{++} \to \wp(\gk)$''.
Suppose now that $\gd < \gk$.
By the second paragraph of the proof of Lemma \ref{l8},
there must be some $i < \gk$ such that either
$\gd = \gk_i$, $\gd = (\gk^+_i)^V$, or $\gd = (\gk^{++}_i)^V$.
Since $V \subseteq N$, it cannot be the case that
either $\gd = (\gk^+_i)^V$ or $\gd = (\gk^{++}_i)^V$.
This means we can let $i < \gk$ be such that $\gd = \gk_i$.
As $\gd \in C$, $V \models ``2^\gd = \gd^{++}$''.
Consequently, since
$V \subseteq N$ and the third paragraph of the proof
of Lemma \ref{l8} implies that
both $\gd = (\gk^+_i)^V$ and $\gd = (\gk^{++}_i)^V$
remain cardinals in $N$,
$N \models ``$There is an injection $f : \gd^{++} \to \wp(\gd)$''.
This completes the proof of Lemma \ref{l9}.
\end{proof}
\begin{lemma}\label{l10}
$N \models ``$GCH holds at every (regular or singular)
cardinal $\gd > \gk$''.
\end{lemma}
\begin{proof}
%As has been observed,
Since $V$, $N$, and $V[G]$ all have the same cardinals
and cofinalities at and above $\gk$,
$\card{\FP} = \gk$, and
$V \models ``2^\gd = \gd^{+}$
for every cardinal $\gd \ge \gk^+$'',
$V[G] \models ``2^\gd = \gd^{+}$
for every cardinal $\gd \ge \gk^+$''.
Therefore, again because $V \subseteq N \subseteq V[G]$,
$N \models ``$For every (regular or singular)
cardinal $\gd > \gk$, there is an injection
$f : \gd^+ \to \wp(\gd)$, and for every cardinal
$\gl > \gd^{+}$, there is no injection
$f : \gl \to \wp(\gd)$''.
This completes the proof of Lemma \ref{l10}.
\end{proof}
\begin{lemma}\label{l11}
$N \models \neg AC_\go$.
\end{lemma}
\begin{proof}
We follow the remarks given after the proofs of
\cite[Lemma 4]{A00} and Lemma \ref{l4}, making
the appropriate modifications in proof.
Define in $N$ for each $n < \go$ the set
$X_n = \{x \subseteq (\gk^{++}_n)^V \mid x$
codes a well-ordering of $(\gk^{+ 3}_n)^V$ of
order type $(\gk^{++}_n)^V\}$.\footnote{Since
$(\gk^{++}_n)^V = (\gk^{++}_n)^N$, it is
also possible to define $X_n$ in $N$ as
$X_n = \{x \subseteq \gk^{++}_n \mid x$
codes a well-ordering of $(\gk^{+ 3}_n)^V$ of
order type $\gk^{++}_n\}$.}
Although each $X_n \neq \emptyset$ and
$\la X_n \mid n < \go \ra \in N$,
$(\prod_{n < \go} X_n)^N = \emptyset$.
This follows since an element $y$ of $(\prod_{n < \go} X_n)^N$
is a set of ordinals, so by Lemma \ref{l7},
$y \in V[G_I] = V[\prod_{i \in I} G_i]$ for some finite set of
ordinals $I \subseteq \gk$.
Let $m$ be the maximum integer which is an element of $I$.
Write $I = I_0 \cup I_1$, where
$I_0 = \{i \in I \mid i \le m\}$ and $I_1 = I - I_0 =
\{i \in I \mid i > m\} = \{i \in I \mid i \ge \go\}$.
By the closure properties of the L\'evy collapse,
each member of the sequence $\la (\gk^{+ 3}_n)^V \mid n < \go \ra$
remains a cardinal in $V[\prod_{i \in I_1} G_i]$. Since
$\prod_{i \in I_0} \FP_i =
\prod_{i \in I_0} {\rm Coll}(\gk^{++}_i, {<} \gk_{i + 1})$
and $I_0$ is finite, there is some $j < \go$ such that
for all $\ell \ge j$, $\card{\prod_{i \in I_0} \FP_i} < \gk_\ell$.
Thus, a final segment of the sequence
$\la (\gk^{+ 3}_n)^V \mid n < \go \ra$ remains a sequence
of cardinals in $V[\prod_{i \in I_1} G_i][\prod_{i \in I_0} G_i] =
V[G_I]$, which is impossible.
This completes the proof of Lemma \ref{l11}.
\end{proof}
The proof that $N \models ``\gk$ is supercompact''
is the same as in Lemma \ref{l6}, with each occurrence
of ``$\ga$'' replaced by an occurrence of ``$I$''.
Lemmas \ref{l7} -- \ref{l11} and the intervening remarks
consequently complete the proof of Theorem \ref{t3}.
\end{proof}
The proof of Lemma \ref{l9} indicates that for any $i < \gk$,
GCH fails at $\gk_i$.
Since it is possible to show that
any $\gk_i$ for $i$ a successor ordinal is a
successor cardinal in $N$ (its predecessor in $N$
must be $\gk^{++}_{i - 1}$), there are many successor
cardinals below $\gk$ violating GCH.
We remark that by slightly changing the definition of $N$,
it is possible to obtain a model of ZF + $\neg{\rm AC}_\go$
satisfying the conclusions of Theorem \ref{t3} in which GCH
holds at every successor cardinal $\gd < \gk$. Specifically,
we have the following theorem.
\begin{theorem}\label{t4}
Let $V \models ``$ZFC + GCH + $\gk$ is supercompact''.
There is then a partial ordering $\FP \in V$ and
a symmetric inner model $N$, $V \subseteq N \subseteq V^\FP$,
such that
$N \models ``$ZF + $\neg {AC}_\go$ +
$\gk$ is a limit cardinal +
$\gk$ is supercompact +
Every successor cardinal is regular +
GCH fails at every limit
cardinal $\gd \le \gk$ +
GCH holds at every (regular or singular)
cardinal $\gd > \gk$, as well as at
every successor cardinal $\gd < \gk$''.
\end{theorem}
\begin{sketch}
We suppose as in the proof of Theorem \ref{t3} that
$V \models ``$ZFC + $\gk$ is supercompact +
$2^\gk = \gk^{++}$ + $2^\gd = \gd^+$ for every
cardinal $\gd \ge \gk^+$ + There is a club
$C \subseteq \gk$ composed of inaccessible cardinals
and their limits with $2^\gd = 2^{\gd^+} = \gd^{++}$ for
every $\gd \in C$''. Once again, let
$\la \gk_i \mid i < \gk \ra \in V$ be the continuous,
increasing enumeration of $C \cup \{\go\}$.
Change the definition of
$\FP_i$ so that $\FP_i = {\rm Coll}(\gk_i, {<} \gk_{i + 1})$
if $i < \gk$ is either $0$ or a successor ordinal, but
$\FP_i = {\rm Coll}(\gk^{++}_i, {<} \gk_{i + 1})$ if
$i < \gk$ is a limit ordinal.
The remainder of the definition of $\FP$ is as
before, i.e., $\FP = \prod_{i < \gk} \FP_i$ with Easton support.
Let $G$ be $V$-generic over $\FP$, and for each
$i < \gk$, let $G_i$ be the projection of
$G$ onto $\FP_i$.
$N$ is then constructed
as in the proof of Theorem \ref{t3}.
The same argument as given in the first paragraph of the
proof of Lemma \ref{l8} shows that
$N \models ``$Every successor cardinal $\gd > \gk$ is regular''.
The proofs of the natural analogues of Lemmas \ref{l6},
\ref{l7}, and \ref{l10} are as before.
The proof of the natural analogue of Lemma \ref{l11}
is as before, with the definition of $X_n$ changed to
$X_n = \{x \subseteq \gk_n \mid x$ codes a well-ordering of
$(\gk^+_n)^V$ of order type $\gk_n\}$.
This shows that
$N \models ``$ZF + $\neg {\rm AC}_\go$ + GCH holds at every
(regular or singular) cardinal $\gd > \gk$ + $\gk$
is supercompact''.
%The natural analogue of the argument found in the second
%paragraph of the proof of Lemma \ref{l8} shows that
%for any $i < \gk$, $N \models ``\gk_i$ is a cardinal''.
The natural analogue of the argument found in the second
paragraph of the proof of Lemma \ref{l8} shows that if
$N \models ``\gd < \gk$ is a (successor or limit) cardinal'', then
either $\gd = \gk_i$ for some $i < \gk$, or
for some limit ordinal $i < \gk$, either
$\gd = (\gk^+_i)^V$ or $\gd = (\gk^{++}_i)^V$.
(As in the proof of Lemma \ref{l8}, let $k < \gk$
be such that $\gk_{k + 1}$ is the least member of
$C$ greater than $\gd$. If this is false, then
either $\gd \in ((\gk^{++}_k)^V, \gk_{k + 1})$ if $k$ is a
limit ordinal, or $\gd \in (\gk_k, \gk_{k + 1})$ if $k$ is
either a successor ordinal or $0$. In each case, in
$V[G_k] \subseteq N$
%(a submodel of $N$)
and $N$, $\gd$ is not a cardinal.)
The same argument as given in the proof of Lemma \ref{l8}
now shows that $\gk_i$ for any $i < \gk$ and
both $\gk^+_i$ and $\gk^{++}_i$ for $i < \gk$
a limit ordinal remain cardinals in $N$.
From this, we may infer as in the proofs of
Lemmas \ref{l8} and \ref{l9} and the intervening remark that
$N \models ``\gk$ is a limit cardinal + Every successor cardinal
$\gd < \gk$ is regular + If
$i < \gk$ is a limit ordinal, then
$\gk_i$ is a limit cardinal + GCH fails at every
limit cardinal $\gd \le \gk$''.
It remains to show that
$N \models ``$GCH holds at every successor cardinal $\gd < \gk$''.
To see this, suppose first that $\gd = \gk_i$
for some $i < \gk$. As we have already observed, $i$ must be a
successor ordinal. As a consequence, $G_i$ must be $V$-generic over
${\rm Coll}(\gk_i, {<} \gk_{i + 1})$, so since
$V[G_i] \subseteq N$ and $N \models ``\gk_{i + 1}$
is a cardinal'',
$N \models ``\gk_{i + 1} = \gk^+_i = \gd^+$, and
there is an injection $f : \gd^+ \to \wp(\gd)$''.
Because $G_{i + 1}$ is $V$-generic over
${\rm Coll}(\gk_{i + 1}, {<} \gk_{i + 2})$,
$V[G_{i + 1}] \subseteq N$, and
$N \models ``\gk_{i + 2}$ is a cardinal'',
$N \models ``\gk_{i + 2} =
\gk^+_{i + 1} = \gd^{++}$''.
Assume now that $N \models ``$There is an injection
$f : \gd^{++} \to \wp(\gd)$'', i.e., that
$N \models ``$There is an injection
$f : \gk_{i + 2} \to \wp(\gk_i)$''.
Since $N \subseteq V[G]$, it must therefore be
the case that
$V[G] \models ``$There is an injection
$f : \gk_{i + 2} \to \wp(\gk_i)$''.
By the properties of the L\'evy collapse, however,
this is impossible.
%, then we can let
%$A = \la A_j \mid j < \gk_{i + 2} \ra \in N$ be such that
%each $A_j \subseteq \gk_i$.
%Since $A$ may be coded as a set of ordinals,
%there must be some finite
Suppose finally that either $\gd = (\gk^+_i)^V$ or
$\gd = (\gk^{++}_i)^V$ where $i < \gk$ is a limit ordinal.
If $\gd = (\gk^+_i)^V$, then since
$V \models ``2^{\gk^+_i} = \gk^{++}_i$'' and
$(\gk^{++}_i)^V$ remains a cardinal in $N$,
$(\gk^{++}_i)^V = (\gd^+)^N$, and
$N \models ``$There is an injection $f : \gd^+ \to \wp(\gd)$''.
Because $G_i$ is $V$-generic over
${\rm Coll}(\gk^{++}_i, {<} \gk_{i + 1})$,
$V[G_i] \subseteq N$, and
$N \models ``\gk_{i + 1}$ is a cardinal'',
$N \models ``\gk_{i + 1} = ((\gk^{++}_i)^V)^+ = \gd^{++}$''.
If $\gd = (\gk^{++}_i)^V$, then since $G_i$ is $V$-generic over
${\rm Coll}(\gk^{++}_i, {<} \gk_{i + 1})$,
$V[G_i] \models ``((\gk^{++}_i)^V)^+ = \gk_{i + 1}$ and
$2^{(\gk^{++}_i)^V} = \gk_{i + 1}$'', i.e.,
$V[G_i] \models ``2^\gd = \gd^+$''.
Because $V[G_i] \subseteq N$ and
$N \models ``\gk_{i + 1}$ is a cardinal'',
$N \models ``$There is an injection $f : \gd^+ \to \wp(\gd)$''.
As $G_{i + 1}$ is $V$-generic over
${\rm Coll}(\gk_{i + 1}, {<} \gk_{i + 2})$,
$V[G_{i + 1}] \subseteq N$, and
$N \models ``\gk_{i + 2}$ is a cardinal'',
$N \models ``\gk_{i + 2} = \gk^+_{i + 1} = \gd^{++}$''.
In either case, if $N \models ``$There is an injection
$f : \gd^{++} \to \wp(\gd)$'', then since
$N \subseteq V[G]$, we obtain a contradiction as
before to the properties of the L\'evy collapse.
This completes the sketch of the proof of Theorem \ref{t4}.
\end{sketch}
\section{Concluding Remarks}\label{s3}
In conclusion to this paper, we note that it is possible to modify
%we make several remarks.
%We begin by noting that it is possible to modify
Theorem \ref{t2} and its proof so that the behavior of
the continuum function at regular cardinals below
$\gk$ is given by a fixed ground model Easton function.
We leave it to readers to fill in the details.
However, the methods of this paper do not seem to
allow us to use a ground model Easton function
%defined on all cardinals below $\gk$
to control the behavior of the continuum function
on all cardinals at and below $\gk$ (in either the strong
sense of Theorem \ref{t2} or the weaker sense of
Theorems \ref{t3} and \ref{t4}) while having
GCH hold above $\gk$.
We ask if this is possible.
In particular, is is possible to construct
a model analogous to the ones for either
Theorem \ref{t2} or Theorem \ref{t3}
in which GCH fails everywhere below $\gk$?
Since AC fails completely in the models
witnessing the conclusions of Theorems \ref{t3} and \ref{t4},
we ask if it is possible to construct analogues
of these models in which some weak version of AC holds.
More generally, we finish by asking if it is possible
to prove analogues of Theorems \ref{t2} -- \ref{t4},
or the generalizations to which we have just alluded,
in the context of the full Axiom of Choice.
\begin{thebibliography}{99}
\bibitem{A91} A.~Apter, ``A Note on Strong Compactness
and Supercompactness'', {\it Bulletin of the London
Mathematical Society 23}, 1991, 113--115.
\bibitem{A00} A.~Apter, ``On a Problem of Woodin'',
{\it Archive for Mathematical Logic 39}, 2000, 253--259.
\bibitem{B78} E.~Bull, ``Successive Large Cardinals'',
{\it Annals of Mathematical Logic 15}, 1978, 161--191.
\bibitem{G10} M.~Gitik, ``Prikry-type Forcings'', in:
{\bf Handbook of Set Theory}, Springer-Verlag,
Berlin and New York, 2010, 1351--1448.
\bibitem{J} T.~Jech, {\it Set Theory.
The Third Millennium Edition,
Revised and Expanded}, Springer-Verlag,
Berlin and New York, 2003.
\bibitem{K} A.~Kanamori, {\it The
Higher Infinite}, Springer-Verlag,
Berlin and New York, 1994.
\bibitem{L} R.~Laver, ``Making the Supercompactness of
$\gk$ Indestructible under $\gk$-Directed Closed Forcing'',
{\it Israel Journal of Mathematics 29}, 1978, 385--388.
\bibitem{LS} A.~L\'evy, R.~Solovay,
``Measurable Cardinals and the Continuum Hypothesis'',
{\it Israel Journal of Mathematics 5}, 1967, 234--248.
\bibitem{R} L.~Radin, ``Adding Closed Cofinal
Sequences to Large Cardinals'', {\it Annals of
Mathematical Logic 22}, 1982, 243--261.
\bibitem{S} R.~Solovay, ``Strongly Compact Cardinals
and the GCH'', in: {\it Proceedings of the Tarski
Symposium}, {\bf Proceedings of Symposia in Pure
Mathematics 25}, American Mathematical Society,
Providence, 1974, 365--372.
\end{thebibliography}
\end{document}
\begin{lemma}\label{l3}
$N \models ``\k$ is a limit cardinal''.
\end{lemma}
\begin{proof}
The proof of Lemma \ref{l3} is
identical to the proof of \cite[Lemma 3]{A00}.
Once again, for completeness, we present it here.
Since $\k$ is measurable in $V$, there is a normal measure
$\mu \in V$ over $\k$ such that
$\{\d < \k \mid \FQ_\d$ is an Easton support product and $\d$ is
Mahlo$\} \in \mu$. For any such $\d$, write
$\FP = \FQ_\d \times \FQ^\d$ and $G = G_\d \times G^\d$, where
$\FQ^\d = \prod_{\a \le \d} \FP_\a$ and $G^\d$
is the projection of $G$ onto $\FQ^\d$. By the definition of
each $\FP_\a$,
$V[G^\d] \models ``\d$ is Mahlo and $\FQ_\d$ is an
Easton support product''. Thus,
$V[G^\d] \models ``\FQ_\d$ is $\d$-c.c.'', so
$V[G^\d][G_\d] = V[G] \models ``\d$ is a cardinal''.
As $V \subseteq N \subseteq V[G]$,
$N \models ``\d$ is a cardinal'', so because there are
unboundedly many in $\k$ such cardinals,
$N \models ``\k$ is a limit cardinal''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
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%; Copyright (C) 1997
%; Associated Universities, Inc. Washington DC, USA.
%;
%; This program is free software; you can redistribute it and/or
%; modify it under the terms of the GNU General Public License as
%; published by the Free Software Foundation; either version 2 of
%; the License, or (at your option) any later version.
%;
%; This program is distributed in the hope that it will be useful,
%; but WITHOUT ANY WARRANTY; without even the implied warranty of
%; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
%; GNU General Public License for more details.
%;
%; You should have received a copy of the GNU General Public
%; License along with this program; if not, write to the Free
%; Software Foundation, Inc., 675 Massachusetts Ave, Cambridge,
%; MA 02139, USA.
%;
%; Correspondence concerning AIPS should be addressed as follows:
%; Internet email: aipsmail@nrao.edu.
%; Postal address: AIPS Project Office
%; National Radio Astronomy Observatory
%; 520 Edgemont Road
%; Charlottesville, VA 22903-2475 USA
%-----------------------------------------------------------------------
\documentstyle [twoside]{article}
%
\newcommand{\AIPS}{{$\cal AIPS\/$}}
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\newcommand{\memtit}{Test of Errors of the Fitting Parameters at Gaussian Fitting task JMFIT.}
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\title{
% \hphantom{Hello World} \\
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% \fbox{AIPS Memo \memnum} \\
\fbox{{\large\whatmem}} \\
\vskip 28pt
\memtit \\}
%\title{Test of Errors of the Fitting Parameters at Gaussian Fitting task JMFIT}
\vspace{2 mm}
\author{ L.~Kogan
\vspace{2 mm}\\
\small National Radio Astronomy Observatory, Socorro, New Mexico,
USA\\}
%\date{September~ 5,~1993}
\vspace{2mm}
\begin{document}
\maketitle
\vspace{5mm}
\begin{abstract}
Two-dimensional elliptical Gaussian fits are used in astronomy for accurate measurements of source parameters such as central position, peak flux density and angular size. The revised error analysis based on \cite{con} and \cite{kog} is implemented at the AIPS task JMFIT. A test of the errors of six parameters of fitted Gaussian into an image provided by JMFIT has been carried out. The test demonstrates a good agreement with the error predicted by JMFIT.
\end{abstract}
\section{Correlation of the noise at the image pixels}
Many years ago the AIPS task IMFIT was written for fitting Gaussians at the image. The task found six parameters of the Gaussian together with predicted errors of each of them. For some reason (probably because of bad errors analysis) the people were not satisfied by the task and the new one (very similar) JMFIT was written later. JMFIT used a different algorithm for evaluating the solution and mutual errors. Evaluation of the errors at the both tasks was done using the same formulae independent on the ratio of beam and Gaussian size that is incorrect. The new error analysis based on \cite{kog} uses different formulae depending on the ratio of beam and Gaussian size. Such a dependence takes a place because of difference in correlation of noises at the neighbor pixels of image. Let's show that.
The visibility $V(\vec{B_i})$ measured at the given baseline $\vec{B_i}$ in the presence of noise $N(\vec{B_i})$ can be represented by the following equation:
\begin{equation}
V(\vec{B_i}) = V_{id}(\vec{B_i}) + N(\vec{B_i})
\label{eq:v(uv)}
\end{equation}
\begin{tabbing}
where~
\= $V_{id}(\vec{B_i})$ is the ideal visibility which would be measured \\
\> in the absence of noise.\\
\end{tabbing}
Having had the visibilities the image $Im(\vec{e})$ can be found as the Fourier transform:
\begin{equation}
Im(\vec{e}) = \frac{1}{I} \sum_{i} (V_{id}(\vec{B_i}) + N(\vec{B_i})) \cdot
\exp(j 2 \pi \, \vec{B_i}\cdot \vec{e})
\label{eq:im}
\end{equation}
\begin{tabbing}
where~
\= $\vec{e}$ is a vector at the picture plane of the source \\
\end{tabbing}
Correlation of the noise at the two different directions
$\vec{e_1}, \vec{e_2}$
at the picture plane is determined by the following equations:
\begin{eqnarray}
Cor(\vec{e_1}, \vec{e_2}) & = &
\overline{( Im(\vec{e_1}) - \overline{Im(\vec{e_1})} ) \cdot
( Im(\vec{e_2}) - \overline{Im(\vec{e_2})} ) } = \nonumber\\
& = &
= \frac{1}{I^2} \sum_i \sum_k \overline{N(\vec{B_i}) \, N(\vec{B_k})}
\exp(j 2 \pi \, \vec{B_i}\cdot \vec{e_1} ) \,
\exp(-j 2 \pi \, \vec{B_k}\cdot \vec{e_2})
\label{eq:cor1}
\end{eqnarray}
Noises at the different baselines are independent usually i.e.
$\overline{N(\vec{B_i}) \, N(\vec{B_k})} = 0 $ for different $i,k$.\\
Therefore the previous equation can be simplified:
\begin{equation}
Cor(\vec{e_1}, \vec{e_2})
= \frac{1}{I^2} \sum_i \overline{ N^2(\vec{B_i})}
\exp(j 2 \pi \, \vec{B_i}\cdot (\vec{e_1} - \vec{e_2} ))
\label{eq:cor2}
\end{equation}
If the noises variances are identical for all baselines (VLA, VLBA),
then:
\begin{equation}
Cor(\vec{e_1}, \vec{e_2})
= \frac{\overline{N^2}}{I} \frac {1}{I} \sum_i
\exp(j 2 \pi \, \vec{B_i}\cdot (\vec{e_1} - \vec{e_2} )) =
VARIM \cdot BEAM(\Delta \vec{e})
\label{eq:cor3}
\end{equation}
\begin{tabbing}
where~
\= $VARIM = \frac{\overline{N^2}}{I}$ is the noise variance at the image; \\
\> $BEAM(\Delta \vec{e}) = \frac {1}{I} \sum_i
\exp(j 2 \pi \, \vec{B_i}\cdot (\vec{e_1} - \vec{e_2} ))$ is the beam shape.\\
\end{tabbing}
{\em Thus, the correlation of the noise at the image pixels is completely determined by the beam shape }\\
We can consider three types of sources: very expanded source (the image is much bigger than the beam), the point source (image coincides with the beam), and intermediate case of partially resolved source.\\
\underline {A very expanded source}. The group of pixels inside of the beam area can be averaged and the averaged data are not correlated. So for very expanded source we can partially use the simplest theory of Gaussian parameters errors estimation supposing the noises uncorrelated. It is clear from this analysis, that errors should be proportional to {\bf sqrt} of ratio of area under the beam and under the image. \\
\underline {A point source}.~ All pixels noise at the image are deeply
correlated. This case was partially analyzed by Condon \cite{con}\\
\underline {Intermediate case}. This case is very difficult to analyze especially when the beam is not circular. Interpolation between the two above cases was can be used.\\
The noise estimation has been installed at the AIPS task JMFIT recently.
Not all estimations are based on the strong theory. That is why the errors prediction given by JMFIT need in test. Beside of that the errors have not been ever tested experimentally and as a result some people do not trust them. The result of the test is given below.
\section{Test of the Errors}
\underline {A point source}.
The method of the test was offered by Dr. M. Goss. Cube of image with 64 frequency was used for the test. The given image included several point sources of different intensity. The task JMFIT was applied for 48 central frequencies to evaluate the rms of the measured parameters of the fitted Gaussian. This rms was compared with predicted rms given by JMFIT. Two sources with $\sim 10 \%$
and $\sim 25 \%$ of noise relatively the Gaussian peaks were selected. Combined image including all frequencies (so called CH0) was used to compare solutions.
The result of this test is given at tables 1 and 2. \\
\underline {An expanded source}. ~
To check the JMFIT error prediction for expanded sources UV data were simulated
using AIPS task UVMOD. The source was selected 8 times (by area) larger than the beam.
Two value of implemented noise were used. The sequence of AIPS tasks: UVMOD
$\Rightarrow$ IMAGR $\Rightarrow$ JMFIT was repeated 16 times to have 16 independent measurement of the source. The result of the test is represented at the tables 3, 4. \\
{\em The comparison of predicted and measured errors of Gaussian fitting given at the tables 1-4 shows a good agreement for both point and expanded sources.
It proves that JMFIT's error analysis is plausible.}
\begin{thebibliography}{99}
\bibitem{con} J.J. Condon, , PASP, 109, 1997, February
\bibitem{kog} L. Kogan, AIPS memo~ 92, 1996
\end{thebibliography}
\newpage
\begin{table}
\caption{The Point source with ratio of rms to Gaussian peak 0.1}
\label{tab:1}
\begin{center}
\begin{tabular}{|c| c c c c|c c|} \hline
& \multicolumn{4}{c|}{Measured} &\multicolumn{2}{c|}{Predicted}\\ \cline{2-7}
& Min& Max& Mean &rms & mean(ch0)& rms(JMFIT)\\
\hline
PEAK & 0.87 & 1.3 & 1.11 & 0.09 & 1.12 &0.07 $\rightarrow$ 0.1\\
INT & 0.97 &1.87 & 1.3 & 0.18 & 1.2 &0.13 $\rightarrow$ 0.23\\
X & 627.0 & 627.9 & 627.4 & 0.16 & 627.4 &0.1 $\rightarrow$ 0.2\\
Y & 519.8 & 520.6 & 520.2 & 0.16 & 520.2 &0.1 $\rightarrow$ 0.2\\
MAJ & 3.67 & 5.85 & 4.55 & 0.42 & 4.29 &0.25 $\rightarrow$ 0.5 \\
MIN & 2.70 & 4.82 & 3.36 & 0.35 & 3.18 &0.2 $\rightarrow$ 0.43\\
BPA & 13 &63 & 43 & 10 & 45 & 5 $\rightarrow$ 56\\
\hline
\end{tabular}
\end{center}
{\small
PEAK is the Gaussian peak value \\
INT is integral flux in the Gaussian \\
X,Y are the right ascension and declination position , in pixels \\
MAJ, MIN are the major and minor axes of the half cross ellipse, in pixels \\
BPA is position angle of the major axis, in degrees\\
}
\end{table}
\begin{table}
\caption{The Point source with ratio of rms to Gaussian peak 0.3}
\label{tab:2}
\begin{center}
\begin{tabular}{|c| c c c c|c c|} \hline
& \multicolumn{4}{c|}{Measured} &\multicolumn{2}{c|}{Predicted}\\ \cline{2-7}
& Min& Max& Mean &rms & mean(ch0)& rms(JMFIT)\\
\hline
PEAK & 0.14 & 0.46 & 0.29 & 0.07 & 0.25 &0.07 $\rightarrow$ 0.09\\
INT & 0.18 &1.9 & 0.7 & 0.37 & 0.36 &0.1 $\rightarrow$ 0.54\\
X & 643.5 & 647.6 & 645.5 & 0.8 & 645.8 &0.35 $\rightarrow$ 2.6\\
Y & 736.6 & 742.7 & 741.0 & 1.1 & 741.8 &0.35 $\rightarrow$ 2.1\\
MAJ & 3.7 & 18 & 7.7 & 3.4 & 4.7 &1 $\rightarrow$ 8 \\
MIN & 2 & 7.1 & 4.0 & 1.2 & 3.8 &0.5 $\rightarrow$ 2.5\\
BPA & 7 &180 & 77 & 54 & 23 & 10 $\rightarrow$ 90\\
\hline
\end{tabular}
\end{center}
{\small
PEAK is the Gaussian peak value \\
INT is integral flux in the Gaussian \\
X,Y are the right ascension and declination position , in pixels \\
MAJ, MIN are the major and minor axes of the half cross ellipse, in pixels \\
BPA is position angle of the major axis, in degrees\\
}
\end{table}
\begin{table}
\caption{The expanded source (source/beam = 8) with ratio of rms to Gaussian peak 0.03}
\label{tab:3}
\begin{center}
\begin{tabular}{|c| c c c c|c c|} \hline
& \multicolumn{4}{c|}{Measured} &\multicolumn{2}{c|}{Predicted}\\ \cline{2-7}
& Min& Max& Mean &rms &value& rms(JMFIT)\\
\hline
PEAK & 11.4 & 12.9 & 12 & 0.39 & 12.0 &0.28 $\rightarrow$ 0.45\\
INT & 72.2 &84.3 & 78.8 & 3.3 & 81.5 &2.2 $\rightarrow$ 3\\
X & 27.7 & 28.2 & 27.9 & 0.18 & 28 &0.16$\rightarrow$ 0.21\\
Y & 34.1 & 34.9 & 34.5 & 0.23 & 34.3 &0.18 $\rightarrow$ 0.27\\
MAJ & 18.1 & 20 & 19.1 & 0.52 & 20 &0.46 $\rightarrow$ 0.66 \\
MIN & 13 & 14.3 & 13.6 & 0.37 & 13.3 &0.34 $\rightarrow$ 0.48\\
BPA & 19 &34 & 27 & 4.8 & 25 & 2.9 $\rightarrow$ 5.5\\
\hline
\end{tabular}
\end{center}
{\small
PEAK is the Gaussian peak value \\
INT is integral flux in the Gaussian \\
X,Y are the right ascension and declination position , in pixels \\
MAJ, MIN are the major and minor axes of the half cross ellipse, in pixels \\
BPA is position angle of the major axis, in degrees\\
}
\end{table}
\begin{table}
\caption{The expanded source (source/beam = 8) with ratio of rms to Gaussian peak 0.2}
\label{tab:4}
\begin{center}
\begin{tabular}{|c| c c c c|c c|} \hline
& \multicolumn{4}{c|}{Measured} &\multicolumn{2}{c|}{Predicted}\\ \cline{2-7}
& Min& Max& Mean &rms & value& rms(JMFIT)\\
\hline
PEAK & 8 & 18 & 12.1 & 2.3 & 12.0 &1 $\rightarrow$ 2.3\\
INT & 51 &87 & 70 & 9.5 & 81.5 &6 $\rightarrow$ 18\\
X & 23.2 & 29 & 27.4 & 1.4 & 28 &0.5$\rightarrow$ 2.9\\
Y & 33.3 & 37 & 35 & 1.2 & 34.3 &0.5 $\rightarrow$ 2\\
MAJ & 15 & 29 & 20.5 & 4.2 & 20 &1.3 $\rightarrow$ 7.7\\
MIN & 9 & 16 & 11.6 & 2 & 13.3 &1 $\rightarrow$ 3.4\\
BPA & 3 &154 & 56 & 49 & 25 & 3 $\rightarrow$ 90\\
\hline
\end{tabular}
\end{center}
{\small
PEAK is the Gaussian peak value \\
INT is integral flux in the Gaussian \\
X,Y are the right ascension and declination position , in pixels \\
MAJ, MIN are the major and minor axes of the half cross ellipse, in pixels \\
BPA is position angle of the major axis, in degrees\\
}
\end{table}
\end{document}
|
https://www.uni-due.de/~bm0031/chs07-18-2004.tex | uni-due.de | CC-MAIN-2019-30 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2019-30/segments/1563195524290.60/warc/CC-MAIN-20190715235156-20190716021156-00449.warc.gz | 871,956,018 | 5,614 | \documentclass{amsart}
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\begin{document}
\bibliographystyle{plain}
\title{Vanishing for negative $K$-theory}
\author{G. C., C. H., M. S. }
\thanks{Thanks}
\address{{}}
\email{chh@math.uiuc.edu}
\date{\today}
%%\begin{abstract} % Abstract.
%%\end{abstract}
%%\subjclass[2000]{19D35, 19E08, 14E15}
%%\keywords{}
%%\maketitle % Produces the title page.
%\tableofcontents
\section{Intro, Notation}
The following conjecture is due to C. Weibel, see \cite{WeibelKA}.
\begin{conj}\label{conj:reg}
Let $X$ be a Noetherian scheme of dimension $d$. Then $K_n(X) = 0$
for $n < -d$ and $X$ is $K_{-d}$-regular.
\end{conj}
In this paper we give a proof of this conjecture for $X$
essentially of finite type over a field $F$ of characteristic $0$.
As a matter of notation, we write $\mathrm{Sch}/F$ for the
category of schemes essentially of finite type over a field $F$,
$\mathrm{Sm}/F$ for the subcategory of essentially smooth schemes,
$\underline{\mathrm{Sch}}$ for the category of Noetherian
$\Q$-schemes, $\underline{\mathrm{Alg}}$ for the category of
$\Q$-algebras, $\underline{\mathrm{Rings}}$ for the category of
unitary rings.
If $A$ is a ring, $I\subset A$ a two-sided ideal and $\mathcal{E}$
a functor from $\underline{\mathrm{Rings}}$ to spectra, we write
$\mathcal{E}(A,I)$ for the homotopy fiber of $\mathcal{E}(A) \to
\mathcal{E}(A/I)$. If moreover $f:A \to B$ is a ring homomorphism
mapping $I$ isomorphically to a two-sided ideal (also denoted $I$)
of $B$, then we write $\cE(A,B,I)$ for the homotopy fiber of the
natural map $\cE(A,I) \to \cE(B,I)$. We say that $\cE$ satisfies
excision provided for all $A$,$I$, $f: A\to B$ as above,
$\cE(A,B,I)\simeq 0$.
If $\cE$ is a presheaf of spectra on $\mathrm{Sch}/F$, then we
write $G_{cdh}\cE$ for a fibrant replacement of $\cE$ in the local
injective model structure on presheaves of spectra on
$(\mathrm{Sch}/F)_{cdh}$ (this can be done functorially in $\cE$);
there is a natural transformation $\cE \to G_{cdh}\cE$.
We say a presheaf $\cE$ of spectra on $\underline{\mathrm{Sch}}$
satisfies descent for a cartesian square
$$
\begin{CD}
Y' @>>> X' \\
@VVV @VVV \\
Y @>>> X
\end{CD}
$$
of schemes if applying $\cE$ to this square results in a homotopy
cartesian square of spectra. We say $\cE$ satisfies $cdh$-descent
of $\mathrm{Sch}/F$ if $\cE$ satisfies descent for all elementary
Nisnevich and abstract blow-up squares in $\mathrm{Sch}/F$;
equivalently, if $\cE \to G_{cdh}\cE$ is an objectwise weak
equivalence.
\section{Thomason's theorem for cyclic homology}
By "cyclic homology" we always mean "cyclic homology over $\Q$" in
this paper, all our categories are $\Q$-linear, etc..
We use Keller's definition (cf. \cite{Keller99}, \cite{Keller98})
of cyclic homology for a "localization pair" to define the cyclic
homology of a scheme $X\in \underline{\mathrm{Sch}}$ as that of
the localization pair consisting of the categories of perfect
complexes on $X$ and its subcategory of acyclic complexes. Let me
try to recall the definition of cyclic homology etc. from this
(note that \cite{Keller99} never actually defines cyclic homology
at all...), for my own benefit.
Given a localization pair $\mathcal{A} =
(\mathcal{A}_0,\mathcal{A}_1)$ Keller assigns to it a mixed
complex $C = C(\mathcal{A})$ (i.e., a DG-module over the
DG-algebra $\Lambda = \Q[\epsilon] = H_*(S^1,\Q)$). Then the
Hochschild homology of $\mathcal{A}$ is $HH_*(\mathcal{A}) =
H^{-*}(C)$, the cyclic homology is $HC_*(\mathcal{A}) =
H^{-*}(C\otimes^{\mathbf{L}}_\Lambda \Q)$, and the periodic and
negative cyclic homology of $\mathcal{A}$ can likewise be defined
as some homological functor to the mixed complex {\tt this, of
course, we need to make precise}. In particular, Hochschild,
cyclic, periodic and negative cyclic homology all factor through
homological functors on the triangulated category $\mathcal{DMix}$
of mixed complexes. The main result of Keller is
\begin{thm}[Keller, \cite{Keller99}]\label{thm:keller}
\begin{enumerate}
\item Let $P: \mathcal{A} \to \mathcal{B}$ be a functor of
localization pairs inducing an equivalence (up to idempotent
completion) on the associated triangulated categories. Then the
induced map of mixed complexes $C(\mathcal{A}) \to C(\mathcal{B})$
is a quasiisomorphism. \item Let $\mathcal{A} \to \mathcal{B} \to
\mathcal{C}$ be a sequence of localization pairs such that the
associated sequence of triangulated categories is exact. Then the
induced sequence $C(\mathcal{A}) \to C(\mathcal{B}) \to
C(\mathcal{C})$ of mixed complexes is a distinguished triangle.
\end{enumerate}
\end{thm}
In particular, weak equivalences of localization pairs induce
isomorphisms in Hochschild, cyclic,... homology, and exact
sequences of pairs induce long exact sequences. That is, the
approximation theorem and fibration theorem of \cite{TT} hold for
Hochschild, cyclic,... homology. From now on, by taking
Eilenberg-Maclane spectra, we consider Hochschild, cyclic,...
homology as functors to spectra.
As a consequence we obtain the following theorem, originally
proven by Geller and Weibel in \cite{??}.
\begin{thm}\label{thm:nis}
Hochschild, cyclic,.. homology satisfy Nisnevich descent.
\end{thm}
Analyzing the proof of Thomason's blow-up formula for $K$-theory,
\cite{ThomRD}, we see that we also get the same formula for
Hochschild, cyclic,... homology.
\begin{thm}\label{thm:HCregdesc}
Let $Y \subset X$ be a regular embedding of $\Q$-schemes, $X' \to
X$ the blow-up of $X$ along $Y$ and $Y'$ the exceptional divisor.
Then the presheaves of spectra defining Hochschild, cyclic,...
homology satisfy descent for the square
$$
\begin{CD}
Y' @>>> X' \\
@VVV @VVV \\
Y @>>> X
\end{CD}
$$
\end{thm}
\begin{proof}
The proof of Thomason's theorem uses only two properties of
$K$-theory, namely, the approximation and fibration theorems.
Hence it applies unchanged to prove the corresponding formula for
Hochschild, cyclic,... homology. This is not quite what we want,
but it implies the assertion, see \cite[Theorem 5]{GS}.
\end{proof}
\begin{rem}
Of course, this works over any field, but we only need it over
$\Q$.
\end{rem}
\section{Descent for infinitesimal K-theory}
\begin{defn}\label{def:infK}
Let $X$ be a $\Q$-scheme. We define the infinitesimal $K$-theory
of $X$, $\cK^{inf}(X)$, as the homotopy fiber of the Chern
character $\cK(X) \to \mathbf{HN}(X)$, where $\mathbf{HN}$ is the
presheaf of spectra defining negative cyclic homology.
\end{defn}
Cortinas proved the so-called KABI-conjecture of Geller-Weibel.
\begin{thm}[Cortinas]\label{thm:KABI}
$\cK^{inf}$ satisfies excision on the category of rings containing
$\Q$.
\end{thm}
From the last section, we also conclude:
\begin{thm}\label{thm:Kinfregdesc}
$K^{inf}$ satisfies descent for every blow-up along a regular
sequence.
\end{thm}
\begin{proof}
Both $K$-theory an negative cyclic homology do.
\end{proof}
Finally, we have the following result, due to Goodwillie.
\begin{thm}[Goodwillie]\label{thm:infext}
Let $A$ be a $\Q$-algebra and $I \subset A$ a nilpotent ideal.
Then $\cK^{inf}(A,I)$ is contractible.
\end{thm}
We say that $\cK^{inf}$ is invariant under infinitesimal
extension.
Using the methods of \cite{HKH}, we can now prove that $\cK^{inf}$
satisfies $cdh$-descent. Let $F$ be a field of characteristic $0$.
\begin{thm}\label{thm:cdhdesc}
Let $\cE$ be a presheaf of spectra on $\mathrm{Sch}/F$ such that
$\cE$ satisfies excision, is invariant under infinitesimal
extension, satisfies Nisnevich descent and satisfies descent for
every blow-up along a regular sequence. Then $\cE$ satisfies
$cdh$-descent.
\end{thm}
\begin{proof}
First, by resolution of singularities, we have $\cE(X) \to
G_{cdh}\cE(X)$ is a weak equivalence whenever $X$ is smooth. Now
apply the arguments of \cite{HKH} accordingly.
\end{proof}
\begin{cor}\label{cor:Kinfdesc}
The presheaf of spectra $\cK^{inf}$ satisfies $cdh$-descent.
\end{cor}
\section{Nil groups}
For a presheaf of spectra $\cE$ on $\mathrm{Sch}/F$, we write
$N^{\leq j}\cE = \mathrm{cofib}(\cE \to \cE(-\times\A^j))$. Note
that $N^{\leq j}\cE$ is a direct factor of $\cE(-\times\A^j)$. In
particular, if use the same notation for presheaves of abelian
groups, then $N^{\leq j}\pi_r\cE \cong \pi_r N^{\leq j}\cE$.
\begin{prop}\label{prop:nil1}
Locally in the Zariski topology, $N^{\leq j}\mathbf{HN}$ is
$0$-connected.
\end{prop}
\begin{proof}
For a $\Q$-algebra $A$, we have $\pi_r\mathbf{HN}(A) \cong
\pi_r\mathbf{HP}(A)$ for $r\leq 0$. Here $\mathbf{HP}$ is periodic
cyclic homology, which is known to be homotopy invariant.
\end{proof}
\begin{prop}\label{prop:nil2}
Locally in the $cdh$-topology, $N^{\leq j}\cK$ is contractible.
\end{prop}
\begin{proof}
Follows readily from resolution of singularities.
\end{proof}
\begin{prop}\label{prop:nil3}
The $cdh$-sheafification $a_{cdh}\pi_r N^{\leq j}\cK^{inf} = 0$
for $r < 0$.
\end{prop}
\begin{proof}
This follows from the definition of $\cK^{inf}$ and the previous
two results. Indeed, Proposition \ref{prop:nil2} implies an
isomorphism $a_{cdh}\pi_r N^{\leq j}\cK^{inf} \cong a_{cdh}
\pi_{r+1} N^{\leq j}\mathbf{HN}$. Now apply Proposition
\ref{prop:nil1}.
\end{proof}
\begin{prop}\label{prop:nil4}
Let $X$ be a Noetherian $\Q$-scheme of dimension $d$. Then $\pi_r
N^{\leq j}\cK^{inf}(X) \cong \pi_r N^{\leq j}\cK(X)$ for $r\ leq
-d-1$.
\end{prop}
\begin{proof}
Locally in the Zariski topology, the asserted isomorphism holds
for $r \leq -1$, by Proposition \ref{prop:nil1}. Now apply Zariski
descent spectral sequence.
\end{proof}
\begin{prop}\label{prop:nildesc}
The presheaves of spectra $N^{\leq j}\cK^{inf}$ satisfy
$cdh$-descent on $\mathrm{Sch}/F$ for any field $F$ of
characteristic $0$.
\end{prop}
\begin{proof}
By Corollary \ref{cor:Kinfdesc}, the presheaf $\cK^{inf}$
satisfies descent. Hence, so does $\cK^{inf}(-\times\A^j)$ (this
implication holds since to check for a presheaf $\cE$ that $\cE
\simeq G_{cdh}\cE$ is suffice to check descent for a class of
cartesian squares that is stable under $-\times \A^j$). Therefore,
$N^{\leq j}\cK^{inf}$ also satisfies $cdh$-descent.
\end{proof}
\begin{thm}\label{thm:main}
Let $F$ be a field of characteristic $0$ and $X/F$ be a scheme,
essentially of finite type over $F$ and such that $\mathrm{dim}(X)
= d$. Then $X$ is $K_{-d-1}$-regular (that is, $N^{\leq j}K_r(X) =
0$ for $j\geq 0$ and $r < -d$) and $K_r(X) = 0$ for $r < -d$.
\end{thm}
\begin{proof}
The first assertion follows from Proposition \ref{prop:nil3},
Proposition \ref{prop:nil4}, Proposition \ref{prop:nildesc} and
the resulting $cdh$-descent spectral sequence
\[H^p_{cdh}(X,a_{cdh}\pi_{-q}N^{\leq j}\cK^{inf}) \Longrightarrow
\pi_{-p-q}N^{\leq j}\cK^{inf}(X).\] The second assertion now
follows from \cite[Theorem 7.1]{HKH} and the Bousfield-Friedlander
spectral sequence.
\end{proof}
\bibliography{refs}
\end{document}
|
http://voodoo-circle.sourceforge.net/doc_for_developers/latex/c__opers_8cpp_source.tex | sourceforge.net | CC-MAIN-2018-05 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2018-05/segments/1516084890771.63/warc/CC-MAIN-20180121135825-20180121155825-00685.warc.gz | 373,600,319 | 1,654 | \hypertarget{c__opers_8cpp}{
\section{c\_\-opers.cpp}
\label{c__opers_8cpp}\index{c\_\-opers.cpp@{c\_\-opers.cpp}}
}
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00001 \textcolor{comment}{/***************************************************************************}
00002 \textcolor{comment}{ c\_opers.cpp - description}
00003 \textcolor{comment}{ -------------------}
00004 \textcolor{comment}{ begin : Thu May 10 2007}
00005 \textcolor{comment}{ copyright : (C) 2004-2010 Marian VooDooMan Meravy}
00006 \textcolor{comment}{ email : ghostvoodooman -a.T= users (dot) sourceforge.net}
00007 \textcolor{comment}{ ***************************************************************************/}
00008
00009 \textcolor{comment}{/***************************************************************************}
00010 \textcolor{comment}{}
00011 \textcolor{comment}{VooDoo cIRCle - an IRC (ro)bot}
00012 \textcolor{comment}{Copyright (C) 2004-2010 Marian VooDooMan Meravy}
00013 \textcolor{comment}{ (ghostvoodooman -a.T= users (dot) sourceforge.net)}
00014 \textcolor{comment}{}
00015 \textcolor{comment}{This program is free software; you can redistribute it and/or}
00016 \textcolor{comment}{modify it under the terms of the GNU General Public License}
00017 \textcolor{comment}{as published by the Free Software Foundation; either version 2}
00018 \textcolor{comment}{of the License, or (at your option) any later version.}
00019 \textcolor{comment}{}
00020 \textcolor{comment}{This program is distributed in the hope that it will be useful,}
00021 \textcolor{comment}{but WITHOUT ANY WARRANTY; without even the implied warranty of}
00022 \textcolor{comment}{MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the}
00023 \textcolor{comment}{GNU General Public License for more details.}
00024 \textcolor{comment}{}
00025 \textcolor{comment}{You should have received a copy of the GNU General Public License}
00026 \textcolor{comment}{along with this program; if not, write to the Free Software}
00027 \textcolor{comment}{Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.}
00028 \textcolor{comment}{}
00029 \textcolor{comment}{****************************************************************************/}
00030 \textcolor{comment}{}
00031 \textcolor{comment}{/*!}
00032 \textcolor{comment}{ \(\backslash\)file}
00033 \textcolor{comment}{ \(\backslash\)brief Class 'c\_opers'}
00034 \textcolor{comment}{*/}
00035
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00042 \textcolor{preprocessor}{#include "../utils/my\_assert.h"}
00043 \textcolor{preprocessor}{#include "../compat.h"}
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00045 \textcolor{preprocessor}{#include "\hyperlink{c__opers_8h}{c_opers.h}"}
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\begin{document}
{\centering
\bfseries
\Large
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
CRITICAL SETS OF SOLUTIONS TO ELLIPTIC EQUATIONS%
\footnote{Supported by Ministerium f\"ur Wissenschaft und Verkehr der
Republik \"Osterreich}%
\footnote{Work supported by the European Union TMR grant FMRX-CT 96-0001 }%
\footnote{Research partially supported by the NSF}\\[2\baselineskip]
\mdseries\scshape\small
R. Hardt$^1$\\
M. Hoffmann-Ostenhof$^2$ \\
T. Hoffmann-Ostenhof$^{3,4}$ \\
N. Nadirashvili$^5$\\[2\baselineskip]
\par
\upshape
Mathematics Department, Rice University, Houston$^1$\\
Institut f\"ur Mathematik, Universit\"at Wien$^2$\\
Institut f\"ur Theoretische Chemie, Universit\"at Wien$^3$\\
International Erwin Schr\"odinger Institute for Mathematical Physics$^4$\\
Department of Mathematics, MIT$^5$\\[2\baselineskip]
}
\begin{abstract}
Let $u\not\equiv\operatorname{const}$ satisfy an elliptic equation
$L_0u\equiv \sum a_{ij}D_{ij}u+\sum b_jD_ju=0$ with smooth coefficients
in a domain in $\mathbf R^n$. It is shown that the critical set
$|\nabla u|^{-1}\{0\}$ has locally finite $n-2$ dimensional Hausdorff
measure. This implies in particular that for a solution $u\not\equiv 0$
of $(L_0+c)u=0$, with $c\in C^\infty$, the critical zero set
$u^{-1}\{0\}\cap |\nabla u|^{-1}\{0\}$ has locally finite $n-2$ dimensional
Hausdorff measure.
\end{abstract}
\section*{1. Introduction and Main Results}
Let $\Omega$ be a domain in $\mathbf R^n$, $n\ge 3$ and let $u\not\equiv 0$
be a real-valued classical solution of the elliptic partial differential
equation
\begin{equation*}\tag{1.1}
{\cal L}u \equiv \sum_{i,j=1}^n a_{ij}D_{ij}u + \sum_{i=1}^n b_jD_ju+cu=0
\quad\text{in } \Omega
\end{equation*}
where the real valued coefficients $a_{ij}, b_j, c$ are $C^\infty$ functions
in $\Omega$.
We denote
\begin{equation*}
\Sigma(u) = |\nabla u|^{-1}\{0\}, \text{ and }\Sigma_0(u) =
\Sigma(u)\cap u^{-1}\{0\}.
\end{equation*}
In the following we shall show that locally the singular set $\Sigma_0$ of $u$
has finite $(n-2)$ - dimensional Hausdorff measure, i.e. $\mathcal H^{n-2}(\Sigma_0(u)\cap K)<\infty$ for all compact $K\subset\Omega$. The first author
and the remaining three authors independently wrote preprints proving
this result and the present paper is a combination of these two works.
For $n=2$, $\&(u)$ is well-known to consist of isolated points. For
$n\>3$ an elementary argument (see [HS], \sec 1.9), first given by
L. Caffarelli and A. Friedman [CF] for $\D u+f(x,u)=0$, shows that
$\&(u)$ is contained in a countable union of smooth $n-2$ dimensional
submanifolds. Q. Han [Hn] obtains similar structural results with much
weaker assumptions on the smoothness of the coefficients, In
particular, he proved that $\&(u)$ is essentially contained in a
countable union of $\c^{1,\a}$ graphs if the coefficients are
Lipschitz. But, even for smooth coefficients, the question remained
concerning the size of $\&(u)$. Last year it was shown in [HOHON] that for
$n=3$, $\Sigma_0(u)$ has locally finite 1-dimensional Hausdorff measure.
Here we generalize the result to $n\ge3$ dimensions. The result is obtained by
showing that the critical set $\Sigma$ of a solution of (1.1) with $c\equiv0$
has locally finite $n-2$ dimensional Hausdorff measure.
Recently there is a rather rich literature describing the `size' of the
zero set and in particular the singular set $\Sigma_0$ of solutions
to elliptic equations in terms of the appropriate Hausdorff measure,
respectively, Hausdorff dimension. See the list of references in the
introduction of [HOHON].
The size of the nodal set was considered in the conjecture of S.T. Yau
[Y] that $\H^{n-1}(u_\l^{-1}\{0\}) \sim c{\sqrt\l}$ for the
$\l$-eigenfunction $u_\l$ on a compact Riemannian manifold. This was
established for real analytic metrics by H. Donnelly and C. Fefferman
[DF1]. Note that, for real analytic coefficients, the local
finiteness, without estimates, of $\H^{n-1}(u^{-1}\{0\})$ (or of
$\H^{n-2}(\&(u))$) follows just from the real analyticity of $u$ [F],
3.4.8. For the nonanalytic case, R. Hardt and L. Simon [HS] proved
the local finiteness of $\H^{n-1}(u^{-1}\{0\})$ with the coefficients
being only Lipschitz smooth. However, for the Riemannian manifold
application, their upper estimate $C\l^{c\sqrt{\l}}$ is weaker than
Yau's conjecture. F.H. Lin and Q. Han [L], [HnL1], [HnL2] proved a
parabolic nodal set estimate (with time-independent coefficients),
simplified several arguments in [DF1] and [HS], and made estimates
involving the frequency (or order)
$N_R\==\[R\J_{\B_R}|\7u|^2\,dx\]/\[\J_{\6\B_R}u^2\,d\H^{n-1}]$. Lin
[L] also conjectured that
$$
\H^{n-1}(u^{-1}\{0\}\A\B_{R/2})\< CN_R\ \:and:\
\H^{n-2}(\&(u)\A\B_{R/2})\< CN_R^2\ .
$$
While more precise results are known in $2$ dimensions [AL, D, DF2, N],
the general Yau and Lin conjectures remain open. Two very recent preprints
give some nonexplicit bounds. [HHL1] treats coefficients with finite
differentiability, and [HHL2] treats higher order equations.
Basic for all these investigations is the asymptotic behaviour of $u(x)$ for
$x\rightarrow x_0$, where $u(x_0)=0$. Let ${\cal O}\in\Omega$, and let $u$ be a solution of (1.1), then it is well known (see e.g. [B]) that
\begin{equation*}\tag{1.2}
u(x)=p_M+O(|x|^{M+1})\text{ as }|x|\rightarrow0
\end{equation*}
where $p_M\not\equiv 0$ is a homogeneous polynomial of degree $M$ satisfying the
osculating equation
\begin{equation*}
\sum_{i,j}a_{ij}({\cal O})D_{ij}p_M=0.
\end{equation*}
Assume without loss of generality that $a_{ij}({\cal O})=\delta_{ij}$, then $p_M$ is
harmonic.
Therefore the investigations of the zero set respectively singular set of a
solution of (1.1) are motivated by the desire to understand to which extent
these sets can be described locally by the zero sets respectively critical
sets of
harmonic homogeneous polynomials. For a harmonic polynomial $P_M$ of degree $M$
in $n$ variables it is known (see e.g. [HS]) that for some $C(n)<\infty$
\begin{equation*}\tag{1.3}
\mathcal H^{n-2}(\Sigma(P_M)\cap B_1)\le C(n)M^2,
\end{equation*}
$B_1$ denoting a ball with radius $1$.
On the other hand there are examples showing that the singular
set of a solution of an elliptic equation can be rather wild. See [HOHON],
section 1. Conversations with L. Simon also led to the following simple
example: For any closed subset $K$ of $\mathbf R$, let $f$ be a nonnegative
smooth function vanishing exactly on $K$ with $|ff''|+|f'{}^2|<1/4$. Then
$u(x,y,z)=xy+f^2(z)$ satisfies the elliptic equation
$u_{xx}+u_{yy}+u_{zz}-(f^2)''(z)u_{xy}=0$, and has singular set
equaling $\{(0,0)\}\times K$.
To state now our main results we define the elliptic operator ${\cal L}_0$ by
\begin{equation*}
{\cal L}_0 = {\cal L} - c
\end{equation*}
with ${\cal L}$ and $c$ given according to (1.1).
{\bf Theorem 1.1.} {\it Let $u\not\equiv\operatorname{const}$ satisfy
\begin{equation*}\tag{1.1'}
{\cal L}_0u=0\quad\text{in }\Omega,\quad\Omega\subset\mathbf{R}^n.
\end{equation*}
Then for every compact subset $K$ of $\Omega$
\begin{equation*}
\mathcal H^{n-2}(\Sigma(u)\cap K) < \infty.
\end{equation*}}
{\bf Corollary 1.1.}{\it Let $u\not\equiv0$ satisfy equation (1.1). Then for every compact subset $K$ of $\Omega$
\begin{equation*}
\mathcal H^{n-2}(\Sigma_0(u)\cap K) < \infty.
\end{equation*}}
The Corollary is a rather immediate consequence of Theorem 1.1:
\textit{Proof of the Corollary.} Given $x_0\in\Omega$ there is a neighbourhood $U(x_0)$
and a $u_0\in C^{\infty}(U(x_0))$ with $u_0>0$ and ${\cal L}u_0 = 0$ in $U(x_0)$.
See e.g. [BJS], p.228. It is easily seen that $\mu\equiv uu_0^{-1}$ satisfies
in $U(x_0)$ an equation of type (1.1), so that by Theorem 1.1, $H^{n-2}(\Sigma(
\mu)\cap U') < \infty$ for every compact subset $U'$ of $U(x_0)$. Furthermore
the singular set of $u$ is a subset of the critical set of $\mu$.\qed
{\bf Remark:} That the assertion of Theorem 1.1 is false if ${\cal L}_0$ is
replaced by ${\cal L}$ can be seen from the following example: Let $v\in C^{
\infty}(B)$, $B\subset\mathbf{R}^n$, with $|v|<1$, then with $u=v^2+1$ and $c=(
\Delta v^2)(v^2+1)^{-1}$, $\Delta u+cu=0$ and $\Sigma(u)=v^{-1}\{0\}$. But
every closed subset of
$\mathbf R^n$ can be the zero set of a $C^\infty$-function (see e.g. [T])!
The structure of the proof of Theorem 1.1 is similar to the $3$-dimensional
case in [HOHON]. For this it was crucial to show (Theorem 3.1) that {\it in 3
dimensions, the complex dimension of the complex critical set of a
homogeneous real harmonic polynomial is at most one}. Here it is shown that
the complex critical set of a homogeneous harmonic polynomial $P$ with real
coefficients has at most complex dimension $n-2$ (Theorem 2.1). With this
result it can be proven that for suitable complex $2$-planes $\epsilon_{ij}$,
$1\le i<j\le n$, $P|_{\epsilon_{ij}}$ has an
isolated critical point in the origin of $\mathbf C^2$ for all $i,j$. Using
results from singularity theory, [AGV], this implies that the algebraic
multiplicity of the gradient map of $P|_{\epsilon_{ij}}$
at the
origin is finite. Further looking at the restriction of the solution $u$ to
affine 2-planes it follows via a $C^{\infty}$-perturbation argument that the
number of critical points of $u$ restricted to these affine 2-plane slices is
uniformly bounded in a small enough neighborhood of the origin. This estimate
together
with the countable rectifiability of $\Sigma(u)$ (which follows immediately
using the arguments from the proof of Lemma 1.9 in [HS], or see also [CF])
allows us to apply a geometric measure inequality of Federer [F] which yields the
desired result.
\section*{2. The critical set of a homogeneous polynomial}
A {\it homogeneous polynomial} of degree $k\>1$ on $\R^n$ is a nonzero
function in the form
$$
u(x)\ =\ \sum_{\|\a\|=k}a_\a x^\a
$$
where $a_\a \in\R$ and $x^\a = x_1^{\a_1}\dots x_n^{\a_n}$ for
$x=(x_1,\dots,x_n)\in\R^n$, $\a=(\a_1,\dots,\a_n)\in\{0,1,\dots\}^n$, and
$\|\a\|\== \a_1+\dots +\a_n$.
The critical set $\&(v)$ of a polynomial $v(x)=\sum_{\|\a\|\<k}b_\a
x^\a$ is a real algebraic variety that is a cone in case $v$ is
homogeneous. Extending $v$ to a complex-valued
polynomial, also denoted $v$, on $\C^n$ by replacing each
$x_i$ by $z_i$, we also have the {\it complex critical (zero) set}
$$
\&_\C(v)\ \==\ \{ z=(z_1,\dots,z_n)\in\C^n\ :\
v(0)=\\{\6 v}/{\6z_1}(z)=\dots=\\{\6 v}/{\6z_n}(z)=0\}\ ,
$$
which is a complex algebraic variety that satisfies
$$
\&_\C(v)\A\R^n =\ \&(v)\ .
$$
Analogously we denote $\Sigma_{0\mathbf C}(v)=\Sigma_{\mathbf C}(v)\cap v^{-1}
\{0\}$.
For a nonconstant polynomial $v$ on $\R^n$, one thus always has the
rough estimates
$$
\:dim:_\R\&(v)\ \<\ n-1\ \ \:and:\ \ \:dim:_\C\&_\C(v)\ \<\ n-1\ .
$$
Suppose now that $v$ is a nonconstant {\it harmonic} polynomial on
$\R^n$. From [CF], [HS], we know that
$$
\:dim:_\R \&(v)\ \<\ n-2 \ .
$$
[HOHON], Th.3.1, also shows that
$$
\:dim_\C:\&_\C(v)\ \<\ 1\ \ \:in\ case:\ \ n=3\ ,
$$
and $v$ is homogeneous.
For the proof of Theorem 1.1 we need the generalisation of this result to
$n$ dimensions, which is given below. Thereby we thank H. Kn\"orrer for
crucial remarks.
{\bf Theorem 2.1.}{\it Let $P$ be a harmonic homogeneous polynomial in
$\mathbf C^n$ with real coefficients, $P\not\equiv\operatorname{const}$,
then $\dim \Sigma_{\mathbf C}(P)\le n-2$.}
\textit{Proof of Theorem 2.1.} For an irreducible polynomial in $\mathbf C^n$ the
assertion is true. See e.g. [S, Chap. II, 1.4].
Let $P$ be reducible. We assume for contradiction that dim$\Sigma (P)=n-1$.
Then $P$ can be represented as
\begin{equation*}\tag{2.1}
P=p^2q, \text{where p and q are homogenous and p is irreducible.}
\end{equation*}
This can be seen as follows: Let $P=\prod _{j=1}^k q_j$, $q_j$ irreducible
$\forall j$, with $k\ge 2$, and denote $N_j=N(q_j)$. If for $i\ne j$,
$\dim N_i\cap N_j <n-1$, then clearly $\dim \Sigma (P)<n-1$. Without loss
we assume $\dim N_1 \cap N_2 =n-1.$ Since $q_1, q_2$ are irreducible this
implies (see e.g. [M] Lemma 2.5) $q_1 =\operatorname{const} q_2$, proving (2.1).
(2.1) implies
\begin {equation*} \tag{2.2}
P=\tilde p^2\tilde q, \text{where } \tilde p, \tilde q \text{ are homogeneous
polynomials with real coefficients and } \tilde p\not\equiv
\operatorname{const}.
\end {equation*}
This can be seen as follows: Let$\underline f$ denote the polynomial which
is obtained from the polynomial $f$ by complex conjugations of its
coefficients. Since $P$ has real coefficients we conclude from (2.1) that
$P=\underline p^2 \underline q = p^2q$. Since $p$ is irreducible
$q=\underline p^2 \tilde q$ follows for some homogeneous polynomial
$\tilde q$. Hence $P=(p \underline p)^2 \tilde q$ and $p\underline p$ has
real coefficients.
Finally we use:
{\bf Proposition 2.2.} {\it Let $P\not\equiv\operatorname{const}$ be a harmonic polynomial
in $\mathbf R^n$ given by $P=p^2q, p, q$ polynomials with real coefficients
and $p\not\equiv\operatorname{const}$, then $P\equiv 0$.}
{\it Proof of Proposition 2.2:} Let $M$ denote the degree of $P$, then for some
spherical harmonic
$Y_M(x|x|^{-1})$, $P(x)=|x|^MY_M$ in polar coordinates. Further we have
\begin{equation*}
q|_{S^{n-1}} = \sum_{j=1}^{M-1} a_jY_j,
\end{equation*}
where the $Y_j$'s are spherical harmonics of degree $\le M-1$, which can be
taken orthonormal on $S^{n-1}$ and $a_j\in\mathbf{R}$, $\forall j$. Hence
$\int_{S^{n-1}}Y_jY_Md\omega=0$ for $j\ne M$, and
\begin{equation*}
\int_{S^{n-1}}p^2q^2d\omega = \int_{S^{n-1}}Pqd\omega = 0
\end{equation*}
implying $p=q=0$.\qed
Together with (2.2) this gives $P\equiv 0$ which is a contradiction to
$P\not\equiv\operatorname{const}$ and finishes the proof of Theorem 2.1.\qed
\section*{3. Restriction to 2-plane slices}
Suppose now $p:\C^n\->\C$ is a complex {\it homogeneous} polynomial;
hence,
$$
\Sigma_{0\mathbf C}(p)=\Sigma_{\mathbf C}(p).
$$
For any complex 2-dimensional subspace $\epsilon\subset\C^n$, the restriction
$p|\epsilon$ is
essentially a complex homogeneous polynomial of two variables. Moreover,
{\it for $z\in\epsilon\sm\{0\}$,
$$
\7(p|_\epsilon)(z)=0
$$
if and only if either
$$
z\in\Sigma_{\mathbf C}(p)
$$
or}
$$
z\in p^{-1}\{0\}\sm\Sigma_\C(p)\ \ \:and:\ \ \epsilon\
\:is\ tangent\ to:\ p^{-1}\{0\}\ \:at:\ z\ . \;{3.1}
$$
For each pair $i,\, j$ of integers with $1\<i<j\<n$ and point
$(z_1,\dots,z_n)\in \C^n$, let
$$
\p_{ij}(z_1,\dots,z_n)\ =\
(z_1,\dots,z_{i-1},z_{i+1},\dots,z_{j-1},z_{j+1},\dots,z_n)\ \in\
\C^{n-2}\ .
$$
For each real rotation $\g\in O(n)$, let $\g:\C^n\->\C^n$ also denote
the complex linear extension of $\g$. Thus each set
$(\p_{ij}\@\g)^{-1}\{y\}$, for $y\in\C^{n-2}$, is a complex affine $2$ plane
in $\C^n$.
\bb
\!{3.1 Lemma} {\it For any nonconstant homogeneous polynomial $p:\C^n\->\C$
having $\:dim:_\C\Sigma_\C(p)\ \<\ n-2$, there exists a rotation
$\g\in O(n)$ so that, for all integers $1\<i<j\<n$,
$$
\Sigma_\C(p)\A(\p_{ij}\@\g)^{-1}\{0\}\ =\ \{0\}
$$
and each complex $2$-plane $(\p_{ij}\@\g)^{-1}\{0\}$ is transverse to
$p^{-1}\{0\}\sm \Sigma_\C(p)$, hence},
$$
\[\7p|_{(\p_{ij}\@\g)^{-1}\{0\}}\]^{-1}\{0\}\ =\ \{0\}\ .
$$
\sb
\pf For $a\in\S^{n-1}$, let $\p_a$ be the complex rank $n-1$ projection of
$\C^n$ corresponding to the orthogonal projection of $\R^n$ onto
$a^\T$. Thus, $\p_a(z)=z-(z\cdot a)a$, and $\p_a$ has kernel
$\p_a^{-1}\{0\}$
equaling the complex span of $a$ and image $\p_a(\C^n)$
equaling the complex span of $a^\T$ in $\C^n$.
We also define, for $y\in\p_a (\C^n)$,
$$
D_a(y)\ =\ \prod_{z\in\p_a^{-1}(y)\A p^{-1}\{0\}}\[a_1\\{\6p}/{\6
z_1}(z)+\dots+a_n\\{\6p}/{\6 z_n}(z)\]\ ,
$$
which is the the discriminant of $p$, with respect to the direction $a$.
Then $D_a$ is a polynomial that is homogeneous because, for
$0\neq\lambda\in\C$,
$$
z\in\p_a^{-1}(\lambda y)\A p^{-1}\{0\}\ \:if\ and\ only\ if:\
\lambda^{-1}z\in\p_a^{-1}(y)\A p^{-1}\{0\}
$$
and $\\{\6p}/{\6 z_i}(z)=\lambda^{k-1}\\{\6p}/{\6 z_i}(\lambda^{-1}z)$.
Moreover, $D_a\not\equiv 0$, because, otherwise, $\7p$ would vanish on
a complex $n-1$ dimensional stratum of $p^{-1}\{0\}$, contradicting that
$\:dim:_\C\Sigma_\C(p)\< n-2$.
For $b\in\S^{n-1}$ with $a\cdot b=0$, $\p_b^{-1}\{0\}\ss\p_a(\C^n)$ and
$$
\epsilon_{a,b}\ =\ \p_a^{-1}\[\p_b^{-1}\{0\}\]
$$
is the complex span of $\{a,b\}$ in $\C^n$. Note that this complex
$2$-plane is transverse to $p^{-1}\{0\}\sm\{0\}$ if $D_a(b)\neq 0$.
In fact, if $z\in\epsilon_{a,b}\A p^{-1}\{0\}\sm\{0\}$, then $\p_a(z)=\lambda b$
for some $0\neq\lambda\in\C$. Since $D_a(\lambda b)\neq 0$, $a\cdot(\7 p)(z)\neq
0$, and, by the complex implicit function theorem, $p^{-1}\{0\}$, is,
locally near $z$, a holomorphic graph over a domain in $\p_a(\C^n)$.
In particular, $p^{-1}\{0\}$ is transverse at $z$ to
$\p_a^{-1}\[\p_b^{-1}\{0\}\]=\epsilon_{a,b}$.
In the set of all pairs
$$
{\cal A}=\{ (a,b)\in\S^{n-1}\x\S^{n-1}\, : a\cdot b=0\}\ ,
$$
we are now interested in the ``bad'' set
$$
{\cal B}\ = \{ (a,b)\in {\cal A}\,:\,\:either:\ \Sigma_\C(p)\A
\epsilon_{a,b}\sm\{0\}\neq\0\ \:or: D_a(b)=0\}\ .
$$
Since ${\cal B}$ is a semi-algebraic set [BR], we may show that
$\:dim:_\R{\cal B}<\:dim:_\R {\cal A} = (n-1)(n-2)$ by simply verifying that
${\cal B}$ contains no nonempty open subset $U$ of ${\cal A}$.
Suppose, for contradiction, that there were such a $U$. Since
$p\not\equiv 0$, $p|_{\S^{n-1}}\not\equiv 0$, because the coefficients of
$p$ are determined by $p|_{\R^n}$. Thus,
$$
\:dim:_\R\(\S^{n-1}\A p^{-1}\{0\}\)\ \<\ n-2\ ,
$$
and we may chose a pair $(a,b)\in U$ with $p(a)\neq 0$. By
homogeneity,
$p(\lambda a)\neq 0$ for all $0\neq\lambda\in\C$, hence,
$$
p^{-1}\{0\}\A\p_a^{-1}\{0\}\ =\ \{0\}\ .
$$
The projection $\p_a(\Sigma_\C(p))$ is, by the Proper Mapping Theorem and
Chow's Theorem [GR], pp.162,170, a complex homogeneous algebraic
subvariety of $\p_a(\C^n)$ of complex dimension $\< n-2$. Moreover,
the discriminant locus $D_a^{-1}\{0\}$ is also a complex homogeneous
algebraic subvariety of $\p_a(\C^n)$ of complex dimension $\<
n-2$. Thus,
$$
\p_a(\Sigma_\C(p))\U D_a^{-1}\{0\}\ \ss\ q^{-1}\{0\}
$$
for some not-identically-zero complex homogeneous polynomial $q$ on
$\p_a(\C^n)$,
and we may similarly find a point $c\in\S^{n-1}\A a^\T$ near $b$ so
that $(a,c)\in U\ss{\cal B}$ and $q(c)\neq 0$, hence,
$q^{-1}\{0\}\A\p_c^{-1}\{0\} =\{0\}$. But then
\begin{align*}
&\Sigma_\C(p)\A\epsilon_{a,c}\ \ss\
p^{-1}\{0\}\A\p_a^{-1}\[\p_a\(\Sigma_\C(p)\)\]\A\p_a^{-1}\[\p_c^{-1}\{0\}\]\cr
&\ss\ p^{-1}\{0\}\A\p_a^{-1}\[q^{-1}\{0\}\A\p_c^{-1}\{0\}\]\
\ss\ p^{-1}\{0\}\A\p_a^{-1}\{0\}\ =\{0\}\ ,
\end{align*}
and $D_a(c)\neq 0$, contradicting that $(a,c)\in {\cal B}$.
Thus, $\:dim:_\R{\cal B}< (n-1)(n-2)$. For each pair of integers
$1\<i<j\<n$, we deduce that, in the space ${\cal C}$ of ordered
orthonormal bases of $\R^n$, the set of ordered bases $(a_1,\dots,a_n)$
with $(a_i,a_j)\in {\cal B}$ has dimension $< \:dim:_\R{\cal
C}=(n-1)!$. In particular we are able to choose a ``good'' basis
$(a_1,\dots,a_n)$ so that $(a_i,a_j)\not\in {\cal B}$ for every
$1\<i<j\<n$. Such a basis readily determines the desired rotation $\g\in O(n)$.
\qed
{\bf Corollary 3.2.}{\it Let $P:\mathbf{C}^n \rightarrow \mathbf C$ be a
nonconstant homogeneous
harmonic polynomial with real coefficients.
Then for some real rotation
$\gamma\in O(n)$,
$P|_{(\pi_{ij}\circ\gamma)^{-1}\{0\}}$
has an isolated critical zero in the origin of $\mathbf{C}^2$, $\forall i,j$
with $1\le i <j \le n$.}
{\it Proof of Corollary 3.2:}
Because of Theorem 2.1
\begin{equation*}
\text{dim}_{\mathbf{C}}\Sigma _{\mathbf{C}}(P) \le n-2.
\end{equation*}
Therefore Lemma 3.1 is applicable and yields the desired result.\qed
\section*{4. Stability under smooth perturbations}
Let $u\ne\operatorname{const}$ satisfy (1.1'), i.e
\begin {equation*}
{ \cal L}_0 = 0 \quad \text{in } \Omega, \Omega \subset \mathbf R^n.
\end {equation*}
Without loss we assume that ${\cal O} \in \Omega$,
\begin {equation*}
a_{ij}({\cal O}) =\delta _{ij}, \quad 1\le i<j\le n
\end{equation*}
and that $u$ has a critical zero in ${\cal O}$. Then due to (1.2)
\begin {equation*}\tag{4.1}
u(x)=P_M(x)+O(|x|^{M+1})\quad\text{for }|x|\rightarrow0
\end {equation*}
for some harmonic homogeneous polynomial $P_M\not\equiv 0$ of
degree $M\ge 2$.
Denoting the complexification of $P_M$ for simplicity again by $P_M$ it
follows from Corollary 3.2 that
\begin{equation*}\tag{4.2}
P_M|_{(\pi_{ij}\circ\gamma)^{-1}\{0\}}
\text{ has an isolated critical zero in the origin of }
\mathbf C^2,\forall i,j.
\end{equation*}
This will be essential to show
{\bf Lemma 4.1.} {\it There exists $R>0$ such that
\begin {equation*}\tag{4.3}
\operatorname{card}\Sigma(u)\cap (\pi_{ij}\circ\gamma)^{-1}\{y\}\cap B_R\le(M-1)^2
\end{equation*}
for all $y\in B_R^{(n-2)}$ and for all $i,j$ such that $1\le i<j \le n$.}
{\it Proof of Lemma 4.1:}
The proof is similar to that of Lemma 2.3 in [HOHON]. From there we
use Proposition 2.2, namely:
{\bf Proposition 4.2.}
{\it Let $p(z_1,z_2)$ be a homogeneous polynomial in $\mathbf C^2$ of degree
$k$ with real coefficients, and assume that $p$ has an isolated critical point
in the origin in $\mathbf C^2$. Let further $\phi \in C^\infty(D_r({\cal O})
)$, $D_r({\cal O})=\{y\in\mathbf R^2:|y|<r\}$, and $r>0$, with
\begin {equation*}
\phi(y) = p(y) +o(|y|^k) \quad \text{for } |y|\rightarrow 0
\end{equation*}
and let $\phi _t(y)\in C^\infty(D({\cal O})\times I)$ for $t\in I$ where $I=
[-t_0,t_0]$, with $\phi_0 = \phi$. Then there exists $\tilde r$,
$0<\tilde r <r$ such that for $|t|\le t_0,\quad t_0$ small enough, the
number of critical points of $\phi_t(.)$ in $D_{\tilde r}({\cal O})$ is
uniformly bounded by $(k-1)^2$.}
For the proof we used results in [AGV], namely: for a homogeneous polynomial
$p(z)$, $z\in \mathbf C^2$ of degree $k$, with an isolated critical point
in the origin ${\cal O}$ the algebraic multiplicity of the gradient
map of $p$ in ${\cal O}$ is $(k-1)^2$. This together with the subadditivity
of the algebraic multiplicity yields the result, which can be stated as
\begin {equation*}
\operatorname{card} (\Sigma (\phi _t)\cap D_r({\cal O})) \le (k-1)^2 \quad \forall t,
\quad |t|\le t_0.
\end {equation*}
We apply this to our case and identify $\forall i,j\quad 1\le i<j\le n$
\begin{equation*}
p=P_M|_{(\pi_{ij}\circ\gamma)^{-1}\{0\}}
\end{equation*}
and
\begin{equation*}
\phi_0 = u|_{(\pi_{ij}\circ\gamma)^{-1}\{0\}}
\end{equation*}
Let $\sigma\in C^\infty$ denote a curve in $\mathbf R^{n-2}$ passing
through the origin, parametrized such that $\sigma(0)=\mathcal O$.
We define for $t\in I$, $I$ an interval with $0\in I$,
\begin{equation*}
\phi_t=u|_{(\pi_{ij}\circ\gamma)^{-1}\{\sigma (t)\}}.
\end{equation*}
Due to Lemma 3.2 and (4.1) we can apply Proposition 4.2 and obtain for
some $\tilde r >0$
\begin{equation*}
\operatorname{card} (\Sigma (u|_{(\pi_{ij}\circ\gamma)^{-1}\{\sigma (t)\}})\cap D_{\tilde r}(\mathcal O))\le (M-1)^2
\end{equation*}
for $t$, $|t|\le t_0$, $t_0$ small enough. This implies further that for
some $\overline R >0$ and $\overline t >0$
\begin{equation*}\tag{4.4}
\operatorname{card}(\Sigma (u)\cap (\pi_{ij}\circ \gamma)^{-1}\{\sigma (t)\}\cap B_{\overline R})\le (M-1)^2\quad \forall t, |t|\le\overline t.
\end{equation*}
Suppose now for contradiction that Lemma 4.1 is false, then for some
$i,j$ there are sequences $\{R_k\}$ and $\{y^{(k)}\}$ with
$y^{(k)}\in\mathbf R^{n-2}$, $R_k\rightarrow 0$, $|y^{(k)}|\rightarrow 0$
for $k\rightarrow \infty$ such that
\begin{equation*}
\operatorname{card} (\Sigma (u)\cap(\pi_{ij}\circ\gamma)^{-1}\{y^{(k)}\}\cap B_{R_k})>(M-1)^2.
\end{equation*}
{\bf Proposition 4.3.} {\it
Let $\{y^{(k)}\}$ denote a sequence in $\mathbf R^n$ convergent to some
$\overline y$. Then there is a subsequence which is a subset of a
$C^\infty$-curve in $\mathbf R^n$.}
{\it Proof of Proposition 4.3:} We use a result of Kriegl [Kr] (see also Lemma
4.2.15 in [FK]):
Let $x_m\in\mathbf R^n$, $x_m\rightarrow\overline x$ for $m\rightarrow\infty$
and let $t_m\in\mathbf R$, $t_m\downarrow 0$ for $m\rightarrow\infty$. If
$\forall k$, $k\in\mathbf N$, $\{(x_m-x_{m+1})(t_m-t_{m+1})^{-k}\}$ is
bounded, then for some $C^\infty$-curve $\gamma$, $\gamma(t_m)=x_m$,
$\forall m$ and $\gamma^{(j)}(t_m)=0$, $\forall j\in \mathbf N$.
>From any convergent sequence $\{y^{(k)}\}$ it is easily seen that we can
pick a subsequence converging fast enough so that the assumptions above are
satisfied.\qed
Therefore we conclude: We can pick a subsequence of $\{y^{(k)}\}$
(again denoted by $\{y^{(k)}\}$) such that for some $\sigma\in C^\infty$,
$\sigma (t_k)=y^{(k)}$, $\forall k$ and
\begin{equation*}\tag{4.5}
\operatorname{card}(\Sigma (u)\cap (\pi_{ij}\circ\gamma)^{-1}\{\sigma (t_k)\}\cap B_{R_k})>(M-1)^2\quad \forall k.
\end{equation*}
On the other hand given $\sigma$, there are $\overline R,\overline t \>0$
such that (4.4) holds which is a contradiction to (4.5).
This verifies Lemma 4.1. \qed
\section*{5. Finiteness of the measure of the critical set.}
We first need
{\bf Lemma 5.1.}{\it Let $u\not\equiv\operatorname{const}$ satisfy (1.1') and $B$ be a ball with
$\overline B \subset \Omega$, then $\Sigma (u)\cap B$ decomposes into
to the countable union of subsets of a pairwise disjoint collection
of smooth $n-2$ dimensional submanifolds, i.e $\Sigma (u) \cap B$
is a countably $(n-2)$-rectifiable subset in the sense of Federer [F].}
{\it Proof of Lemma 5.1:} The proof is in principle the same as the
one of Lemma 1.9 in [HS]:
Thereby the argument is essentially that used by Cafarelli and Friedman
[CF]:
Let for $q=1,2,3,\dots $
\begin{equation*}
S_q=\{x|D^\alpha u(x)=0, \forall \alpha \text{ with } 0<|\alpha |\le q,
\quad D^{q+1}u(x)\ne 0\}.
\end{equation*}
Let $B_{2R}$ be a ball with $\overline B_{2R}\subset \Omega$, then
$\forall x_0\in \overline B_R$, $\mathcal L_0 (u-u(x_0))=0$.
Since the coefficients of the equation are smooth it follows via
unique continuation that $u-u(x_0)$ vanishes of finite order
$M(x_0)$ in $x_0$ and
\begin{equation*}
\sup_{x_0\in\overline B_{2R}}M(x_0)\equiv \overline M <\infty
\end{equation*}
Therefrom we obtain that $\forall a\in\Sigma(u)\cap B_R$
\begin{equation*}
B_R(a)\cap\{x|\nabla u(x)=0\} = B_R(a)\cap\bigcup _{q=1}^{\overline M}S_q.
\end{equation*}
The remaining part of the proof is the same as in e.g. [HS] or [CF].\qed
Due to Lemma 5.1 we have in particular
\begin{equation*}
\Sigma (u)\cap B_R =\cup_{m=1}^\infty E_m
\end{equation*}
where $E_1\subset E_2\subset\dots$ are Borel subsets of $\Sigma(u)$ of
finite $\mathcal H^{n-2}$-measure.
Without loss of generality we change coordinates to make $\gamma = \text{Id}$
in Lemma 4.1.
Then we use the integral geometric inequality 3.2.27 in [F] and Lemma 4.1 to
obtain the following estimate:
With $R>0$ given in Lemma 4.1
\begin{align*}
\mathcal H^{n-1}(\Sigma&(u)\cap \overline B_R)
=\lim_{m\rightarrow\infty}\mathcal H^{n-2}(E_m\cap\overline B_R)\\
&\le\limsup_{m\rightarrow\infty}\sum_{1\le i<j\le n}\int_{B_R^{n-2}}
\operatorname{card}[\pi_{ij}^{-1}\{y\}\cap E_m\cap\overline B_R]d\mathcal H^{n-2}y\\
&\le \binom n2 \mathcal H^{n-2}(B_R^{n-2})(M-1)^2.
\end{align*}
This finishes the proof of Theorem 1.1.\qed
\vspace{1.5\baselineskip}
\ct{\bf\large References}
\bb
\1{[AGV]} V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, {\sl Singularities
of differentiable maps}, Vol.I, Monographs in Math. Vol. 82, Birkh\"auser,
Berlin, 1985.
\sb
\1{[AL]} G. Allessandrini, {\sl Singular solutions and the determination of
conductivity by boundary measurements}, J. Diff. Equ. {\bf 84} (1990), 252-272.
\sb
\1{[B]} L. Bers, {\sl Local behaviour of solutions of general linear elliptic
equations}, Commun. Pure Appl. Math. {\bf 8} (1955), 473-496.
\sb
\1{[BJS]} L. Bers, F. John, Schechter, {\sl Partial differential equations},
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\sb
\1{[BR]} R. Benedetti, R. Risler, {\sl Real algebraic sets. Actualites
Math\'ematiques}, Current Mathematical Topics, Hermann, Paris, 1990.
\sb
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420--439.
\sb
\1{[D]} R.-T. Dong, {\sl Nodal sets of eigenfunctions on
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\sb
\1{[DF1]} H. Donnelly, Ch. Fefferman {\sl Nodal sets of
eigenfunctions on Riemannian manifolds\/}, Invent. Math. {\bf93}
(1988), 161--183.
\sb
\1{[DF2]} \_, {\sl Nodal sets for
eigenfunctions of the Laplacian on surfaces\/},
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\sb
\1{[F]} H.\ Federer, {\sl Geometric measure theory}, Springer-Verlag,
Berlin and New York, 1969.
\sb
\1{[FK]} A. Fr\"ohlicher, A. Kriegl, {\sl Linear spaces and differentiation
theory}, Wiley, 1988.
\sb
\1{[GR]} R. Gunning, H. Rossi, {\sl Analytic functions of several
complex variables\/}. Prentice Hall, Englewood Cliffs, N.J. 1965
\sb
\1{[HHL1]} R. Hardt, Q. Han, F. H. Lin, {\sl Geometric measure of singular sets
of elliptic equations}, Preprint.
\sb
\1{[HHL2]} R. Hardt, Q. Han, F. H. Lin, {\sl Singular sets of higher order
equations}, Preprint.
\sb
\1{[Hn]} Q. Han, {\sl Singular sets of solutions to elliptic
equations}. Indiana Univ. Math. J. {\bf43}
(1994), no. 3, 983--1002.
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\sb
\1{[HnL2]} \_, {\sl On the geometric measure of nodal sets of solutions}. J.
Partial Differential Equations {\bf7} (1994), no. 2, 111--131.
\sb
\1{[HOHON]} M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and N.
Nadirashvili, {\sl Critical sets of smooth solutions of
elliptic equations in dimension 3\/}, Indiana Univ. Math. J. {\bf45}
(1996), no. 1, 15--37.
\sb
\1{[HS]} R.\ Hardt and L.\ Simon, {\sl Nodal sets for solutions for
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\sb\def\=={\equiv}
\1{[KR]} A. Kriegl, {\sl Die richtigen R\"aume f\"ur Analysis im Unendlich-
Dimensionalen}, Monatshefte f. Math. {\bf 94} (1982), 109-124.
\sb
\1{[L]} F.H.\ Lin, {\sl Nodal sets of solutions of elliptic and
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(1991), 287--308.
\sb
\1{[M]} J. Milnor, {\sl Singular points of complex hypersurfaces}, Annals of
Math. Studies 61, Princeton University Press, Princeton, 1968.
\sb
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\1{[N]} N. Nadirashvili, {\sl Uniqueness and stability of the solution of an
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{\bf 40}(1986), 623-627.
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\1{[S]} I-R. Shafarevich, {\sl Basic algebraic geometry\/}, Springer-Verlag,
Berlin and New York, 1977.
\sb
\1{[T]} J-C. Tougeron, {\sl Id\'eaux de fonctions diff\'erentiables\/},
Ergebnisse der Mathematik und ihrer Grenzgebiete 71, Springer-Verlag, Berlin-
Heidelberg-New York 1972.
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\1{[Y]} S.T. Yau, {\sl Open problems in geometry\/}, Proc. Symp.
Pure Math. {\bf54,1} (1993), 1--28.
{\scshape
R. Hardt: Mathematics Department, Rice University, Houston, TX 777251-1892 USA}
{\slshape
E-mail address:} \texttt{hardt@math.rice.edu}
\vspace\baselineskip
{\scshape
M. Hoffmann-Ostenhof: Institut f\"ur Mathematik, Universit\"at Wien,
Strudlhofgasse 4, A-1090 Wien, Austria}
{\slshape
E-mail address:} \texttt{mho@nelly.mat.univie.ac.at}
\vspace\baselineskip
{\scshape
T. Hoffmann-Ostenhof: Institut f\"ur Theoretische Chemie, Universit\"at
Wien, W\"ahringer Strasse 17, A-1090 Wien, Austria and International Erwin
Schr\"odinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090
Wien, Austria}
{\slshape
E-mail address:} \texttt{thoffman@esi.ac.at}
\vspace\baselineskip
{\scshape
N. Nadirashvili: Institute for Problems of Information Transmission, Bol'shoi
Karetnyi 19, 101447 Moscow, Russia
Current address: Massachusetts Institute of Technology,
Department of Mathematics, 77 Massachusetts Avenue,
Cambridge, 02139-4307 MA, USA}
{\slshape
E-mail address:} \texttt{nicolai@math.mit.edu}
\end{document}
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%--------------------------------------------------------
% A. objavljene znanstvene monografije
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%\begin{skupina}{A}
%\disertacija
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%{\it Integral Equations Operator Theory}, 2015, 6 str.
%$[$COBISS.SI-ID 17449049$]$\\
{\crta, H.~Radjavi}
{Invariant and almost-invariant subspaces for pairs of idempotents}
{Integral Equations Operator Theory}
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%\v{s}kod v pozavarovalni\v{s}tvu : magistrsko delo}. Ljubljana: $[$M. Kavalir$]$, 2015. VI, 57 str., ilustr. $[$COBISS.SI-ID 17283929$]$\\
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\title{Green Scared?}
\date{February 22, 2008}
\author{CrimethInc.}
\subtitle{Lessons from the FBI Crackdown on Eco-Activists}
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For years, the FBI targeted ecological activists as their \#1 priority. This is one of the chief reasons environmental devastation has continued unchecked.
At the end of 2005, the FBI opened a new phase of its assault on earth and animal liberation movements—known as the Green Scare—with the arrests and indictments of a large number of activists. This offensive, dubbed Operation Backfire, was intended to obtain convictions for many of the unsolved Earth Liberation Front arsons of the preceding ten years—but more so, to have a chilling effect on all ecological direct action. In the following analysis, originally published in \emph{Rolling Thunder} in 2008, we review everything we can learn from the Operation Backfire cases, with the intention of passing on the lessons for the next generation of environmental activists.
\section{For Those Who Came in Late\dots{}}
Of those charged in Operation Backfire, nine ultimately cooperated with the government and informed on others in hopes of reduced sentences: Stanislas Meyerhoff, Kevin Tubbs, Chelsea Dawn Gerlach, Suzanne Savoie, Kendall Tankersley, Jennifer Kolar, Lacey Phillabaum, Darren Thurston, and, much later, Briana Waters. Four held out through a terrifying year, during which it seemed certain they would end up serving decades in prison, until they were able to broker plea deals in which they could claim responsibility for their actions without providing information about others: Daniel McGowan, Jonathan Paul, Exile (aka Nathan Block), and Sadie (aka Joyanna Zacher)\footnote{After this writing, it came to light that Sadie and Exile hold both racist and transphobic views. The anarchist community has parted ways with them.}. Another defendant, William Rodgers (aka Avalon), tragically passed away in an alleged suicide while in custody shortly after his arrest. Fugitive Justin Solondz was captured in China in 2009 and completed his sentence in January 2017; Rebecca Rubin turned herself in in 2012, after many years on the run, and was sentenced to five years in prison. Joseph Dibee was extradited from Cuba to the US in August 2018 to face charges. One more defendant has been charged but not found.
The months following the launch of Operation Backfire saw an unprecedented increase in government repression of anarchist environmental activists, which came to be known as the Green Scare. Longtime animal liberation activist Rod Coronado was charged with a felony for answering a question during a speaking appearance, and faced potentially decades in prison. Six animal rights activists associated with SHAC, the campaign against animal testing corporation Huntingdon Life Sciences, were sentenced to several years in prison, essentially for running a website. Animal liberationist Peter Young, who had spent seven years on the run from the FBI, had finally been captured and was being threatened with double jeopardy. Tre Arrow, famous for surviving a 100-foot fall when police and loggers forced him out of a forest occupation, was fighting extradition from Canada to the United States to face arson charges. Innumerable people were subpoenaed to grand juries,\footnote{In theory, the task of a grand jury is to examine the validity of an accusation before trial. In practice, grand juries are used to force information out of people: by granting an individual immunity regarding a specific case, a grand jury can compel him or her to answer questions or else go to prison for contempt of court.} and some did jail time for refusing to cooperate. Perhaps most ominously of all, three young people were set up by an agent provocateur and arrested on conspiracy charges without having actually done anything at all. Two of them, Zachary Jenson and Lauren Weiner, pled guilty and became government informants; the third, Eric McDavid, who has contracted life-threatening health problems as a consequence of being denied vegan food by his jailers, was recently found guilty and awaits sentencing.
This phase of the Green Scare seems to be drawing to a close. Most of those apprehended in Operation Backfire are now serving their sentences. The first of the SHAC defendants has been released from prison. Peter Young has been out of prison for a year and is doing speaking tours. Rod Coronado’s trial ended in a deadlock, and he took a plea in return for a short sentence when the government threatened to bring further charges against him. It’s been months now since a new high profile felony case was brought against an environmental activist, though federal agents have been poking around in the Midwest. It’s time to begin deriving lessons from the past two years of government repression, to equip the next generation that will take the front lines in the struggle to defend life on earth.
\section{Distinguishing between Perceived and Real Threats}
In some anarchist circles, the initial onset of the Green Scare was met with a panic that rivaled the response to the September 11 attacks. This, of course, was exactly what the government wanted: quite apart from bringing individual activists to “justice,” they hoped to intimidate all who see direct action as the most effective means of social change. Rather than aiding the government by making exaggerated assumptions about how dangerous it is to be an anarchist today, we must sort out what these cases show about the current capabilities and limits of government repression.
The purpose of this inquiry is not to advocate or sensationalize any particular tactic or approach. We should be careful not to glorify illegal activity—it’s important to note that most of even the staunchest non-cooperating defendants have expressed regrets about their choices, though this must be understood in the context of their court cases. At the same time, federal repression affects everyone involved in resistance, not just those who participate in illegal direct action; the Green Scare offers case studies of the situation we are all in, like it or not.
\section{Case Study in Repression: Eugene, Oregon}
Operation Backfire took place against a backdrop of government investigation, harassment, and profiling of presumed anarchists in the Pacific Northwest. It is no coincidence that Eugene, Oregon was a major focus of the Operation Backfire cases, as it has been a hotbed of dissent and radicalism over the past decade and a half—although repression and other problems have taken a toll in recent years. We can’t offer a definitive analysis of the internal dynamics of the Eugene anarchist community, but we can look at how the authorities went about repressing it.
One useful resource for this inquiry is “Anarchist Direct Actions: A Challenge for Law Enforcement,”
an article that appeared in Studies in Conflict \& Terrorism in 2005, authored by Randy Borum of the University of South Florida and Chuck Tilby of the Eugene Police Department. According to Jeff (“Free”) Luers, Tilby was one of the cops who surveilled Free and his co-defendant Critter on the night of their arrest in June 2000. Tilby has given presentations on the “criminal anarchist” movement to law enforcement groups, and was intimately involved in the Operation Backfire cases, even making statements to the media and providing a quote to the FBI press release at the end of the Oregon federal prosecution.
Surprisingly, the article does not explicitly reference Eugene, Oregon at all. Besides Tilby’s byline at the beginning, there’s no indication that the paper was co-written from Eugene. All the same, the article provides several important clues about how the government proceeded against the Oregon defendants and those who were perceived to support them.
The authors centralize the importance of intelligence and informants for repressing criminal “anarchists,” while acknowledging the difficulty of obtaining them. In the case of grand jury subpoenas, anarchists regularly fail to comply, and support groups are often set up for those targeted; one of the more recent examples of this was Jeff Hogg, who received a grand jury subpoena while the Backfire prosecutions were underway and was jailed for nearly six months in 2006 as a result. The authors warn that “investigators and law enforcement officers should be cautious during questioning not to divulge more to the subject about the case (via questions), than is learned through their testimony.” Indeed, questions asked by grand juries turned up more than once in the pages of the \emph{Earth First! Journal}, which was edited from Eugene for a time. It is extremely important to support those under investigation and keep abreast of investigators’ efforts. Some believe that the Backfire investigation only arrived at a position of real strength once such support started to weaken in Oregon.
Regarding infiltration, “Anarchist Direct Actions” advises that:
What we know of the early Backfire investigation points to a strategy of generalized monitoring and infiltration. While investigators used increasingly focused tools and strategies as the investigation gained steam—for example, sending “cooperating witnesses” wearing body wires to talk to specific targets—they started out by sifting through a whole demographic of counter-cultural types. Activist and punk houses as well as gathering spots such as bars were placed under surveillance—anarchists who drink should be careful about the way alcohol can loosen lips. Infiltrators and informants targeted not only the most visibly committed anarchists, but also bohemians who inhabited similar cultural and social spheres. Police accumulated tremendous amounts of background information even while failing to penetrate the circles in which direct action was organized. The approximately 30,000 pages of discovery in the Oregon cases contain a vast amount of gossip and background information on quite a few from the Eugene community.
A similar profiling methodology appears to have been used in nearby Portland, Oregon. In March 2001, for example, a large-scale police raid was carried out on a house party attended by Portland punk rockers. The attendees were photographed and questioned about the Earth and Animal Liberation Fronts. Some were arrested and charged with kidnapping and assault on an officer—a standard over-charging which eventually led to plea deals. The defendants from the raid were videotaped at their court appearances by officers later identified as Gang Enforcement Unit members. In the aftermath of this raid, cops routinely harassed punks on the street, demanding to be told whether they were anarchists.
In retrospect, it seems likely that such efforts were not meant simply to intimidate Portland’s punks, but to uncover information relevant to the anarchist and ALF\Slash{}ELF cases of the time. This may have been a wrong step in the Backfire investigation; right now there’s no way to know. We do know, however, that “wide net” approaches by the state can be effective at stifling socially aware subcultures, even when they uncover no real links to radical action. Fortunately, in Portland those affected by the raid came together in response, aiding each other, limiting the damage done, and taking advantage of the situation to draw attention to police activity.
Another point of speculation is the degree to which authorities fostered division and infighting within radical circles in Eugene. This was a common COINTELPRO\footnote{The FBI’s Counter-Intelligence Program (COINTELPRO) existed officially from 1956 to 1971 and probably continues to this day in some form. Aiming to “expose, disrupt, misdirect, discredit, or otherwise neutralize” the activities of groups like the Black Panther Party, the Program utilized a wide variety of dirty tricks. Houses and offices were searched and documents stolen without any warrants having been issued; rumors were spread in order to foster mistrust and even violence between different organizations or factions within them; group members were harassed through the courts or even wholly framed for crimes they did not commit; infiltrators and agent provocateurs were distributed within target constituencies; no act of psychological warfare or blatant violence was ruled out. The program was finally exposed when radicals broke into an FBI Office and seized documents relating to the secret program, circulating them to various sources under the name of the “Citizens’ Commission to Investigate the FBI.”} tactic, and is probably still in use. Borum and Tilby hint at this in the final section of their paper, “Law Enforcement Strategies\Slash{} Implications”:
For those familiar with Eugene radical circles, this brings to mind the heated conflicts over gender and feminism within that community. There is no concrete evidence that government operatives were involved in escalating such debates, and we should be careful not to jump to conclusions; such speculation can only assist the state by propagating paranoia. However, law enforcement from local to federal levels must have been aware of the vulnerabilities that opened up when real debates turned to groupthink and factionalism in Eugene. Tilby and his cohorts must have used such insights to their advantage as they devised anti-anarchist strategies. By the time Operation Backfire grand juries began following up on real leads in Eugene, many who could have come together to oppose them were no longer on speaking terms. While this does not justify the lack of integrity shown by those who assisted grand juries, it does offer some context for why the grand juries weren’t resisted more effectively.
Borum and Tilby close their paper by urging investigators to display “patience and persistence”—and indeed, patience and persistence ultimately paid off in Operation Backfire. This is not to lend credibility to the notion that “The FBI always get their man.” The investigation was riddled with errors and missteps; plenty of other actions will never be prosecuted, as the authorities got neither lucky breaks nor useful cooperation. But we must understand that repression, and resistance to it, are both long-term projects, stretching across years and decades.
According to some accounts, one of the most significant leads in Operation Backfire came from a naïve request for police reports at a Eugene police station. According to this version, the police deduced from this request that they should pay attention to Jacob Ferguson; Ferguson later became the major informant in these cases. It is less frequently mentioned that the police were accusing Ferguson of an arson he did not participate in! With Ferguson, the unlikely happened and it paid off for the authorities to be wrong. Later on, when agents made their first arrests and presented grand jury subpoenas on December 7, 2005, two of those subpoenaed were wrongly assumed to have been involved in attacks. Their subpoenas were eventually dropped, as the authorities gained the cooperation of more informants and eventually made moves to arrest Exile and Sadie instead.
The investigation was not as unstoppable and dynamic as the government would like us to think, although the prosecution gathered force as more individuals rolled on others. The authorities spent years stumbling around, and they continued to falter even when prosecution efforts were underway—but they were tenacious and kept at their efforts. Meanwhile, radical momentum was less consistent.
Let’s review the arc of radical activity in Eugene over the past decade. The anticapitalist riot of June 18, 1999 in Eugene led to jubilation on the part of anarchists, even if one participant spent seven years in prison as a result. The participants in the June 18 Day of Action had put up a fight and fucked up some symbols of misery in the town, catching the police unprepared. The pitched battles on the streets of Seattle later that year at the WTO meeting only reinforced the feeling that the whole world was up for grabs. Most of the active anarchists in Eugene had never lived through such a period before. Despite the paltry demands and muddled analysis of much of the official “antiglobalization” movement, there was a sense that deeper change could be fought for and won. Being an anarchist seemed like the coolest thing you could be, and this perception was magnified by the media attention that followed. The ELF was setting fires all over the region at the time.
A series of reversals followed. In June 2001, Free received his initial sentence of 22 years and eight months. The following month, Carlo Giuliani was murdered on the streets of Genoa during protests against the G8 summit in Italy. While both of these tragedies illustrated the risks of confronting the capitalist system, Free’s sentence hit home especially hard in Eugene. In the changed atmosphere, some began dropping away and “getting on with their lives”—not necessarily betraying their earlier principles, but shifting their focus and priorities. This attrition intensified when American flags appeared everywhere in the aftermath of September 11, 2001. Anarchist efforts did not cease, but a period of relative disorientation followed. A year and a half later, the invasion of Iraq provided another opportunity for radicals to mobilize, but some consistency had been lost in the Eugene area. And all the while, FBI employees and police kept their regular hours, day in and day out.
Law enforcement received its most significant breakthrough in the Backfire cases—even though it started as an incorrect hypothesis—just before Free’s sentencing, in the period between anarchist jubilation and the shift to the defensive. The same fires that were incorrectly linked to Ferguson were used to justify Free’s stiff sentence, which intimidated some anarchists out of action. There was not enough revaluation, learning, and sharpening of skills, nor enough efforts at conflict resolution; the retreat occurred by default. What would have happened if the Backfire investigation had continued under different circumstances, while radicals maintained their momentum? That would be another story. Its conclusion is unknown.
\section{Putting up a Fight}
Repression will exist as long as there are states and people who oppose them. Complete invulnerability is impossible, for governments as well as their opponents. All the infiltrators and informants of the Tsarist secret police were powerless to prevent the Russian revolution of 1917, just as the East German Stasi were unable to prevent the fall of the Berlin Wall even though they had files on six million people. Revolutionary struggles can succeed even in the face of massive repression; for our part, we can minimize the effects of that repression by preparing in advance.
For many years now anarchists have focused on developing security culture, but security consciousness alone is not enough. There are some points one can never emphasize too much—don’t gossip about sensitive matters, share delicate information on a need-to-know basis,\footnote{It does appear that Operation Backfire defendants could have done better at limiting the flow of information inside their circles. Rather than organizing in closed, consistent cells, the defendants seem to have worked in more fluid arrangements, with enough crossover that once a few key participants turned informant the government had information about everyone.} don’t surrender your rights if detained or arrested, don’t cooperate with grand juries, don’t sell other people out. But one can abide by all these dictums and still make crucial mistakes. If anti-repression strategies center only on what we should not talk about, we lose sight of the necessity of clear communication for communities in struggle.
State disruption of radical movements can be interpreted as a kind of “armed critique,” in the way that someone throwing a brick through a Starbucks window is a critique in action. That is to say, a successful use of force against us demonstrates that we had pre-existing vulnerabilities. This is not to argue that we should blame the victim in situations of repression, but we need to learn how and why efforts to destabilize our activities succeed. Our response should not start with jail support once someone has been arrested. Of course this is important, along with longer-term support of those serving sentences—but our efforts must begin long before, countering the small vulnerabilities that our enemy can exploit. Open discussion of problems—for example, gender roles being imposed in nominally radical spaces—can protect against unhealthy resentments and schisms. This is not to say that every split is unwarranted—sometimes the best thing is for people to go their separate ways; but that even if that is necessary, they should try to maintain mutual respect or at least a willingness to communicate when it counts.
Risk is relative. In some cases, it may indeed be a good idea to lay low; in other cases, maintaining public visibility is viewed as too risky, when in fact nothing could be more dangerous than withdrawing from the public eye and letting momentum die. When we think about risk, we often picture security cameras and prison cells, but there are many more insidious threats. The Operation Backfire defendants ended up with much shorter sentences than expected; as it turned out, the most serious risk they faced was not prison time, after all, but recantation and betrayal—a risk that proved all too real. Likewise, we can imagine Eric McDavid, who currently awaits sentencing on conspiracy charges, idly discussing the risk factor of a hypothetical action with his supposed friends—who turned out to be two potential informants and a federal agent provocateur. Unfortunately, the really risky thing was having those discussions with those people in the first place.
\section{Preparing for the Worst}
Conventional activist wisdom dictates that one must not mix public and clandestine activity, but Daniel McGowan’s case seems to contradict this. McGowan was not brought to trial as a result of investigations based on his public organizing, but rather because he had worked with Jacob Ferguson, who turned snitch under police pressure. Though the government was especially eager to convict him on account of his extensive prisoner support work and organizing against the Republican National Convention, McGowan received tremendous public support precisely because he had been so visible.\footnote{This is not to say that all visibility is good visibility. Media attention was a significant factor in the conflicts that wracked Eugene. Such visibility can divide communities from within by creating the appearance that spokespeople have more power than everyone else, which provokes jealousy and stokes ego-driven conflicts whether or not what’s on the screen reflects reality on the ground. Those who fall prey to believing the media hype about themselves become dependent upon this attention, pursuing it rather than the unmediated connections and healthy relationships essential for long-term revolutionary struggle; the most valuable visibility is anchored in enduring communities, not media spectacles. There are reasonable arguments for using the media at times, but one must be aware of the danger of being \emph{used by it.}} Had he simply hidden in obscurity, he might have ended up in the same situation without the support that enabled him to weather it as successfully as he did—and without making as many important contributions to the anarchist movement.
Considering how many years it took the FBI to put together Operation Backfire and the prominent role of informants in so many Green Scare cases, it seems like it is possible to get away with a lot, provided you are careful and make intelligent decisions about who to trust. McGowan’s direct action résumé, as it appears in the government arguments at his sentencing, reads like something out of an adventure novel. One can’t help but think—just seven years, for all that!
The other side of this coin is that, despite all their precautions, the Green Scare defendants did get caught. No matter how careful and intelligent you are, it doesn’t pay to count on not getting caught; you have to be prepared for the worst. Those who are considering risky direct action should start from the assumption that they will be caught and prosecuted; before doing anything, before even talking about it, they should ask themselves whether they could accept the worst possible consequences. At the same time, as the government may target anyone at any time regardless of what they have actually done, it is important for even the most law-abiding activists—not to mention their friends and relatives—to think through how to handle being investigated, subpoenaed, or charged.
The Green Scare cases show that cooperating with the government is never in a defendant’s best interest. On average, the non-cooperating defendants in Operation Backfire are actually serving less time in proportion to their original threatened sentences than the informants
despite the government engaging the entire repressive apparatus of the United States to make an example of them. Exile and Sadie were threatened with over a thousand years in prison apiece, and are serving less than eight; if every arrestee understood the difference between what the state threatens and what it can actually do, far fewer would give up without a fight.
In the United States legal system, a court case is essentially a game of chicken. The state starts by threatening the worst penalties it possibly can, in hope of intimidating the defendant into pleading guilty and informing. It is easier if the defendant pleads guilty immediately; this saves the state immense quantities of time and money, not to mention the potential embarrassment of losing a well-publicized trial. Defendants should not be intimidated by the initial charges brought against them; it often turns out that many of these will not hold up, and are only being pressed to give the state more bargaining power. Even if a defendant fears he won’t have a leg to stand on in court, he can obtain some bargaining power of his own by threatening to put the state through a costly, challenging, and unpredictable trial—to that end, it is essential to acquire the best possible legal representation. When a defendant agrees to cooperate, he loses all that leverage, throwing himself at the mercy of forces that don’t have an ounce of mercy to offer.
As grim as things looked for Sadie, Exile, McGowan, and Jonathan Paul through most of 2006, they looked up when McGowan’s lawyer demanded information about whether prosecutors had used illegal National Security Agency wiretaps to gather evidence against the defendants. The government was loath to answer this question, and for good reason: there had just been a public scandal about NSA wiretaps, and if the court found that wiretaps had been used unconstitutionally, the entire Operation Backfire case would have been thrown out. That’s exactly why so many members of the Weather Underground are professors today rather than convicts: the FBI botched that case so badly the courts had to let them go free.
No matter how hopeless things look, never underestimate the power of fighting it out. Until Stanislas Meyerhoff and others capitulated, the linchpin of the federal case in Operation Backfire was Jacob Ferguson, a heroin addict and serial arsonist. Had all besides Ferguson refused to cooperate and instead fought the charges together, Operation Backfire would surely have ended differently.
\section{On Informants}
If becoming an informant is always a bad idea, why do so many people do it? At least eleven high profile defendants in Green Scare cases have chosen to cooperate with the government against their former comrades, not including Peter Young’s partner, who informed on him back in 1999. These were all experienced activists who presumably had spent years considering how they would handle the pressure of interrogation and trial, who must have been familiar with all the reasons it doesn’t pay to cooperate with the state! What, if anything, can we conclude from how many of them became informants?
There has been quite a bit of opportunist speculation on this subject by pundits with little knowledge of the circumstances and even less personal experience. We are to take it for granted that arrestees became informants because they were privileged middle class kids; in fact, both the cooperating and non-cooperating defendants are split along class and gender lines. We are told that defendants snitched because they hadn’t been fighting for their own interests; what exactly are one’s “own interests,” if not to live in a world without slaughterhouses and global warming? Cheaper hamburgers and air conditioning, perhaps? It has even been suggested that it’s inevitable some will turn informant under pressure, so we must not blame those who do, and instead should avoid using tactics that provoke investigations and interrogations. This last aspersion is not worth dignifying with a response, except to point out that no crime need be committed for the government to initiate investigations and interrogations. Whether or not you support direct action of any kind, it is never acceptable to equip the state to do harm to other human beings.
Experienced radicals who have been snitched on themselves will tell you that there is no surefire formula for determining who will turn informant and who won’t. There have been informants in almost every resistance movement in living memory, including the Black Panther Party, the Black Liberation Army, the American Indian Movement, and the Puerto Rican independence movement; the Green Scare cases are not particularly unusual in this regard, though some of the defendants seem to have caved in more swiftly than their antecedents. It may be that the hullabaloo about how many eco-activists have turned informant is partly due to commentators’ ignorance of past struggles.
If anything discourages people from informing on each other, it is blood ties. Historically, the movements with the least snitching have been the ones most firmly grounded in longstanding communities. Arrestees in the national liberation movements of yesteryear didn’t cooperate because they wouldn’t be able to face their parents or children again if they did; likewise, when gangsters involved in illegal capitalist activity refuse to inform, it is because doing so would affect the entirety of their lives, from their prospects in their chosen careers to their social standing in prison as well as their neighborhoods. The stronger the ties that bind an individual to a community, the less likely it is he or she will inform against it. North American radicals from predominantly white demographics have always faced a difficult challenge in this regard, as most of the participants are involved in defiance of their families and social circles rather than because of them. When an ex-activist is facing potentially decades in prison for something that was essentially a hobby, with his parents begging him not to throw his life away and the system he fought against apparently dominating the entirety of his present and future, it takes a powerful sense of right and wrong to resist selling out.
In this light, it isn’t surprising that the one common thread that links the non-cooperating defendants is that practically all of them were still involved in either anarchist or at least countercultural communities. Daniel McGowan was ceaselessly active in many kinds of organizing right up to his arrest; Exile and Sadie were still committed to life against the grain, if not political activity—a witness who attended their sentencing described their supporters as an otherworldly troop of black metal fans with braided beards and facial piercings. Here we see again the necessity of forging powerful, long-term communities with a shared culture of resistance; dropouts must do this from scratch, swimming against the tide, but it is not impossible.
Healthy relationships are the backbone of such communities, not to mention secure direct action organizing. Again—unaddressed conflicts and resentments, unbalanced power dynamics, and lack of trust have been the Achilles heel of countless groups. The FBI keeps psychological profiles on its targets, with which to prey on their weaknesses and exploit potential interpersonal fissures. The oldest trick in the book is to tell arrestees that their comrades already snitched on them; to weather this intimidation, people must have no doubts about their comrades’ reliability.
“Snitches get stitches” posters notwithstanding, anarchists aren’t situated to enforce a no-informing code by violent means. It’s doubtful that we could do such a thing without compromising our principles, anyway—when it comes to coercion and fear, the state can always outdo us, and we shouldn’t aspire to compete with it. Instead, we should focus on demystifying snitching and building up the collective trust and power that discourage it. If being a part of the anarchist community is rewarding enough, no one will wish to exile themselves from it by turning informant. For this to work, of course, those who do inform on others must be excluded from our communities with absolute finality; in betraying others for personal advantage, they join the ranks of the police officers, prison guards, and executioners they assist.
Those who may participate in direct action together should first take time to get to know each other well, including each other’s families and friends, and to talk over their expectations, needs, and goals. You should know someone long enough to know what you like least about him or her before committing to secure activity together; you have to be certain you’ll be able to work through the most difficult conflicts and trust them in the most frightening situations up to a full decade later.
Judging from the lessons of the 1970s, drug addiction is another factor that tends to correlate with snitching, as it can be linked to deep-rooted personal problems. Indeed, Jacob Ferguson, the first informant in Operation Backfire, was a longtime heroin addict. Just as the Operation Backfire cases would have been a great deal more difficult for the government if no one besides Jake had cooperated, the FBI might never have been able to initiate the cases at all if others had not trusted Jake in the first place.
Prompt prisoner support is as important as public support for those facing grand juries. As one Green Scare defendant has pointed out, defendants often turn informant soon after arrest when they are off balance and uncertain what lies ahead. Jail is notorious for being a harsher environment than prison; recent arrestees may be asking themselves whether they can handle years of incarceration without a realistic sense of what that would entail. Supporters should bail defendants out of jail as quickly as possible, so they can be informed and level-headed as they make decisions about their defense strategy. To this end, it is ideal if funds are earmarked for legal support long before any arrests occur.
It cannot be emphasized enough that informing is \emph{always} a serious matter, whether it is a question of a high profile defendant snitching on his comrades or an acquaintance of law-abiding activists answering seemingly harmless questions. The primary goal of the government in any political case is not to put any one defendant in prison but to obtain information with which to map radical communities, with the ultimate goal of repressing and controlling those communities. The first deal the government offered Peter Young was for him to return to animal rights circles to report to them from within: not just on illegal activity, but on all activity. The most minor piece of trivia may serve to jeopardize a person’s life, whether or not they have ever broken any law. \emph{It is never acceptable to give information about any other person without his or her express consent.}
\section{Regaining the Initiative}
We must not conceptualize our response to government repression in purely reactive terms. It takes a lot of resources for the government to mount a massive operation like the Green Scare cases, and in doing so they create unforeseen situations and open up new vulnerabilities. Like in Judo, when the state makes a move, we can strike back with a countermove that catches them off balance. To take an example from mass mobilizations, the powers that be were eventually able to cripple the so-called anti-globalization movement by throwing tremendous numbers of police at it; but in the wake of lawsuits subsequently brought against them, the police in places like Washington, D.C. now have their hands tied when it comes to crowd control, as demonstrated by their extreme restraint at the IMF\Slash{}World Bank protests in October 2007. We’re in a long war with hierarchical power that cannot be won or lost in any single engagement; the question is always how to make the best of each development, seizing the initiative whenever we can and passing whatever gains we make on to those who will fight after us.
There must be a way to turn the legacy of the Green Scare to our advantage. One starting place is to use it as an opportunity to learn how the state investigates underground activity and make sure those lessons are shared with the next generation. Another is to find common cause with other targeted communities; a promising example of this is the recent connection between animal liberation activists in the Bay Area and supporters of the San Francisco Eight, ex-Black Panthers who are now being charged with the 1971 murder of a police officer.
\section{Postscript: Cowards\dots{}}
In reflecting on Judge Aiken’s sentencing, let us put aside, for the time being, the question of whether executives who profit from logging, animal exploitation, and genetic engineering are “doing what they need to do to survive.” Let’s allow to pass, as well, the suggestion that those who run these industries are \emph{more} likely to enter into a “real dialogue” with environmentalists if the latter limit themselves to purely legal activity. Let’s even reserve judgment on Aiken’s attempt to draw parallels between domestic violence and sarcastically worded communiqués—which parallels the prosecutors’ assertion that the ELF, despite having never injured a single human being, is no different from the Ku Klux Klan.
There is but one question we cannot help but ask, in reference to Judge Aiken’s rhetoric about cowardice: if she found herself in a situation that called for action to be taken outside the established channels of the legal system, would she be capable of it? Or would she still insist on due process of law, urging others to be patient as human beings were sold into slavery or the Nazis carted people off to Dachau? Is it fair for a person whose complicity in the status quo is rewarded with financial stability and social status to accuse someone who has risked everything to abide by his conscience\dots{} of \emph{cowardice}? Perhaps Aiken would also feel entitled to inform John Brown that he was a coward, or the Germans who attempted to assassinate Hitler?
Once this question is asked, another question inexorably follows: what qualifies as a situation that calls for action to be taken outside the established channels of the legal system, if not the current ecological crisis? Species are going extinct all over the planet, climate change is beginning to wreak serious havoc on human beings as well, and scientists are giving us a very short window of time to turn our act around—while the US government and its corporate puppeteers refuse to make even the insufficient changes called for by liberals. If the dystopian nightmare those scientists predict comes to pass, will the refugees of the future look back at this encounter between McGowan and Aiken and judge \emph{McGowan} the coward?
We live in a democracy, Aiken and her kind insist: bypassing the established channels and breaking the law is akin to attacking freedom, community, and dialogue themselves. That’s the same thing they said in 1859.
Those who consider obeying the law more important than abiding by one’s conscience always try to frame themselves as the responsible ones, but the essence of that attitude is the desire to evade responsibility. Society, as represented—however badly—by its entrenched institutions, is responsible for decreeing right and wrong; all one must do is brainlessly comply, arguing for a change when the results are not to one’s taste but never stepping out of line. That is the creed of cowards, if anything is. At the hearing to determine whether the defendants should be sentenced as terrorists, Aiken acknowledged with frustration that she had no control over what the Bureau of Prisons would do with them regardless of her recommendations—but washed her hands of the matter and gave McGowan and others terrorism enhancements anyway. Doubtless, Aiken feels that whatever shortcomings the system has are not her responsibility, even if she participates in forcing them on others. She’s just doing her job.
That’s the Nuremberg defense. Regardless of what she thinks of McGowan’s actions or the Bureau of Prisons, Aiken is personally responsible for sending him to prison. She is responsible for separating him from his wife, for preventing him from continuing his work supporting survivors of domestic violence. If he is beaten or raped while in prison, it is the same as if Aiken beat or raped him. And not just McGowan, or Paul, or Sadie or Exile, but \emph{every single person} Aiken has ever sent to prison.
But Aiken and her kind are responsible for a lot more than this. As the polar icecaps melt, rainforests are reduced to pulp, and climate change inflicts more and more terrible catastrophes around the planet, they are responsible for stopping all who would take direct action to avert these tragedies. They are responsible, in short, for forcing the wholesale destruction of the natural environment upon everyone else on earth.
Aiken might counter that the so-called democratic system is the most effective way to go about halting that destruction. It sure has worked so far, hasn’t it! On the contrary, it seems more likely that she cannot bring herself to honestly consider whether there could be a higher good than the maintenance of law and order. For people like her, obedience to the law is more precious than polar icecaps, rainforests, and cities like New Orleans. Any price is worth paying to avoid taking responsibility for their part in determining the fate of the planet. Talk about \emph{cowardice.}
\section{\dots{}and Heroes}
So—if McGowan and the other non-cooperating Green Scare defendants are not cowards, does that mean they are heroes?
We should be cautious not to unthinkingly adopt the inverse of Aiken’s judgment. In presenting the case for the government, Peifer described the Operation Backfire defendants’ exploits as “almost like Mission Impossible.” It serves the powers that be to present the defendants as superhuman—the more exceptional their deeds seem to be, the further out of reach such deeds will feel to everyone else.
Similarly, lionizing “heroes” can be a way for the rest of us to let ourselves off the hook: as we are obviously not heroes of their caliber, we need not hold ourselves up to the same standards of conduct. It is a disservice to glorify McGowan, Exile, Sadie, Peter Young, and others like them; in choosing anonymous action, they did not set out to be celebrated, but to privately do what they thought was necessary, just as all of us ought to. They are as normal as any of us—any normal person who takes responsibility for his or her actions is capable of tremendous things.
This is not to say we should all become arsonists. There are countless paths available to those who would take responsibility for themselves, and each person must choose the one that is most appropriate to his or her situation. Let the courage of the non-cooperating Green Scare defendants, who dared to act on their beliefs and refused to betray those convictions even when threatened with life in prison, serve as reminders of just how much normal people like us can accomplish.
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\begin{center}
The Anarchist Library
\smallskip
Anti-Copyright
\bigskip
\includegraphics[width=0.25\textwidth]{logo-en}
\bigskip
\end{center}
\strut
\vfill
\begin{center}
CrimethInc.
Green Scared?
Lessons from the FBI Crackdown on Eco-Activists
February 22, 2008
\bigskip
Retrieved on 8\textsuperscript{th} November 2020 from \href{https://crimethinc.com/2008/02/22/green-scared}{crimethinc.com}
\bigskip
\textbf{theanarchistlibrary.org}
\end{center}
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\end{document}
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%
% Copyright (C) 2016, 2017 by Paul Gaborit
%
% Tis file may be distributed and/or modified
%
% 1. under the LaTeX Project Public License and/or
% 2. under the GNU Public License.
%
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\usepackage[utf8]{inputenc}
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belowskip=5pt,
language=[LaTeX]TeX,
moretexcs={
%
abstract, addbibresource, iscramset, keywords, mainmatter,
maketitle, printbibliography, subsection, subsubsection, url,
urldef, href, includegraphics, ldots, parencite, citeauthor,
citeyear, citetitle, midrule, toprule, bottomrule
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style=common,
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%CoRe Paper 2017={Documentation pseudo-Track},
title={
International Conference on
Information Systems for Crisis
Response and Management\\
\LaTeX{} Class
%Publications Format
},
short title={ISCRAM \LaTeX{} Class},
author={
short name={Paul Gaborit},
full name=Paul Gaborit\thanks{corresponding author},
affiliation={
Centre Génie Industriel -- Mines Albi%
\thanks{\href{http://gind.mines-albi.fr}{\sitewebgind}
and \href{http://www.mines-albi.fr}{\sitewebminesalbi}}\\
\href{mailto:paul.gaborit@gmail.com}{\gmailpaulgaborit}
},
},
author={
full name=Sébastien Truptil,
affiliation={
Centre Génie Industriel -- Mines Albi
},
},
}
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\begin{document}
\maketitle
\makeatletter
{\centering\large\iscram@version{}\\\iscram@date\par}
\makeatother
\abstract{
In this document we describe the formatting requirements for the
Proceedings of ISCRAM papers.
\emph{Please review this document carefully: submissions must follow
the format presented here} and be sure to adhere to the formatting
requirements as this will ultimately be your camera-ready version,
delivered as pdf.
\emph{Please note several limitations on length:} (1) your abstract
should be no more than 150 words, (2) your entire paper should be
between \pgfmathprintnumber{4000} and \pgfmathprintnumber{8000} words
in length for \textbf{CoRe Papers} (presenting completed work
including a complete description of methods, results and validation),
including all materials and references. Or (2) your entire paper
should be between \pgfmathprintnumber{3000} to
\pgfmathprintnumber{6000} words in length for \textbf{WiPe Papers}
(presenting work in earlier stages, outlining and discussing concepts
and methods and presenting first results), including all materials and
references.
\emph{Please make sure that your initial submission does not include
any author identifying information: use the \option{anonymous} class
option.} Avoid identifying self-citations as your own work
(e.g. ``In our previous research (Author Year) we found
\ldots{}''). Instead simply say ``Previous research (Author Year)
found \ldots{}'' Keep the self-citations in the bibliography so that
reviewers may refer to them if necessary. This will ensure a proper
double-blind-review process. If your paper is accepted please remove
the \option{anonymous} option before the final upload. }
\keywords{Guides, Instructions, Conference Publications, ISCRAM \LaTeX{}
Class.}
\section{Documentation}
The figure~\ref{fig:exampledoc} (at the end of this document) shows an
example of use of the ISCRAM document class.
\subsection{How to load the \texttt{iscram} document class}
The \lstinline[style=TeX]{iscram} document class accepts some
\generic{options}. You may use:
\begin{pseudoTeX}
\ttfamily\macro\documentclass[\generic{options}]\{iscram\}
\end{pseudoTeX}
or:
\begin{pseudoTeX}
\ttfamily\macro\documentclass\{iscram\}\\
\macro\iscramset\{\generic{options}\}
\end{pseudoTeX}
\subsection{\texorpdfstring{All the available \generic{options}}{The <options>}}
\begin{options}
\item[\option{draft}]
if this boolean option is set, the class shows any overfull
boxes. \emph{Note:} don't use this option in your final submission.
\item[\option{anonymous}]
if this boolean option is set, the \texttt{iscram} class produces an
anonymous version of the paper (no author, no affiliation, no
e-mail). \textbf{Use this option to submit the first version of your
paper (an anonymous version)}.
\item[\option{first alone}]
if this boolean option is set, the first author has its own line in
the list of authors.
\item[\option{title}=\generic{title}]
defines \generic{title} as the main title of your paper. This title is
inserted in your document by \macro\maketitle.
\item[\option{short title}=\generic{short title}]
defines \generic{short title} (up to 8 words) as the short title of
your paper, used in the header.
\item[\option{author}=\{\option{short name}=\generic{short name},
\option{full name}=\generic{full name},
\option{affiliation}=\generic{affiliation}\}]
appends an author (with its affiliation) at the end of the list of
authors (inserted in your paper by \macro\maketitle). The
\emph{\generic{affiliation}} may contain several lines (separated by
\macro{\\}). To add several authors, this option can be use several
times. The \emph{\generic{short name}} (default value:
\emph{\generic{full name}}) of the first author is used in the header of
your paper.
\item[\option{footer/line 1}=\generic{text},
\option{footer/line 2}=\generic{text},
\option{footer/line 3}=\generic{text}]
define respectively \generic{text} as content of the
first line, second line and third line of the footer.
\end{options}
\subsubsection{Prefefined styles for 2017 edition}
\begin{options}
\item[\option{iscram 2017 footer}]
a predefined style that sets the two last lines of the footer for a
paper published in ISCRAM 2017.
\item[\option{WiPe Paper 2017}=\generic{track name}]
a predefined style that sets the three lines of the footer for a
\textbf{WiPe Paper} published in the track \emph{\generic{track name}}
in ISCRAM 2017 (choose the appropriate track or use ``Open Track'' if
you do not have a specific track in mind).
\item[\option{CoRe Paper 2017}=\generic{track name}]
a predefined style that sets the three lines of the footer for a
\textbf{CoRe Paper} published in the track \emph{\generic{track name}}
in ISCRAM 2017 (choose the appropriate track or use ``Open Track'' if
you do not have a specific track in mind).
\end{options}
\subsubsection{Prefefined styles for 2018 edition}
\begin{options}
\item[\option{iscram 2018 footer}]
a predefined style that sets the two last lines of the footer for a
paper published in ISCRAM 2018.
\item[\option{WiPe Paper 2018}=\generic{track name}]
a predefined style that sets the three lines of the footer for a
\textbf{WiPe Paper} published in the track \emph{\generic{track name}}
in ISCRAM 2018 (choose the appropriate track or use ``Open Track'' if
you do not have a specific track in mind).
\item[\option{CoRe Paper 2018}=\generic{track name}]
a predefined style that sets the three lines of the footer for a
\textbf{CoRe Paper} published in the track \emph{\generic{track name}}
in ISCRAM 2018 (choose the appropriate track or use ``Open Track'' if
you do not have a specific track in mind).
\end{options}
\subsection{Using packages}
In your preamble, you may use your prefered packages with, for example
(choose the appropriate \generic{inputcoding}):
\begin{pseudoTeX}
\ttfamily\macro\usepackage[\generic{inputcoding}]\{inputenc\}
\end{pseudoTeX}
\subsubsection{Packages loaded by the \texttt{iscram} class}
The \texttt{iscram} class requires (and loads) some packages:
\begin{multicols}{6}
\texttt{biblatex}\\
\texttt{booktabs}\\
\texttt{caption}\\
\texttt{etex}\\
\texttt{etoolbox}\\
\texttt{float}\\
\texttt{fontenc}\\
\texttt{geometry}\\
\texttt{hyperref}\\
\texttt{microtype}\\
\texttt{newtxmath}\\
\texttt{nowidow}\\
\texttt{newtxtext}\\
\texttt{pgfopts}\\
\texttt{titlesec}\\
\texttt{url}\\
\texttt{xcolor}
\end{multicols}
To pass additional \generic{options} to one of these
\generic{package}, you may call \macro\PassOptionsToPackage{} \emph{before} the call to
\macro\documentclass{}:
\begin{pseudoTeX}
\ttfamily\macro\PassOptionsToPackage\{\generic{options}\}\{\generic{package}\}\\
\macro\documentclass\{iscram\}
\end{pseudoTeX}
\subsection{Useful commands}
Here are described some useful commands in order of usage:
\begin{options}
\item[\macro\addbibresource\{<bibfile>\}]
call this command in your preamble to add a \emph{\texttt{bibfile}} as
a resource to find your bibliographic references.
\item[\macro\maketitle]
to create a new page with the title and the list of authors or your
paper (to specifiy \emph{title} and \emph{authors}, use class options
or use \macro\iscramset{}).
\item[\macro\abstract\{\generic{abstract}\}]
to insert an \generic{abstract} as a section of your paper.
\item[\macro\keywords\{\generic{keywords}\}]
to insert the list of \generic{keywords} as a subsection of your paper.
\item[\macro\section\{\generic{section title}\}]
to insert a new section (sans-serif font, uppercase,
bold, 10bp).
\item[\macro\subsection\{\generic{subsection title}\}]
to insert a new subsection (sans-serif font, bold, 10bp).
\item[\macro\subsubsection\{\generic{subsubsection title}\}]
to insert a new subsubsection (sans-serif font, italice, 10bp).
\item[\macro\cite\{\generic{key}\} or
\macro\cite\{\generic{key1},\generic{key2}\}]
to insert one or more bibliographic references, referenced by
\generic{key}, \generic{key1}, \generic{key2} \ldots{}
\item[\macro\citeauthor\{\generic{key}\}]
to insert the authors from the \generic{key} bibliographic reference.
\item[\macro\citeyear\{\generic{key}\}]
to insert the year of publication of the \generic{key} bibliographic
reference.
\item[\macro\citetitle\{\generic{key}\}]
to insert the title of the \generic{key} bibliographic reference.
\item[\macro\printbibliography]
to insert the list of the cited references. Compile your document with
\texttt{latexmk} or use \texttt{biber} (not \texttt{bibtex}) after a
first compilation to produce the correct bibliographic file
(\texttt{.bbl}) from your bibliographic sources (\texttt{.bib}).
\end{options}
\subsection{Compilation}
The better way to compile your document is to use the \texttt{latexmk}
tool:
\begin{lstlisting}[style=console]
latexmk -pdf my-paper.tex
\end{lstlisting}
You may use the traditional method:
\begin{lstlisting}[style=console]
pdflatex my-paper.tex
biber my-paper
pdflatex my-paper.tex
pdflatex my-paper.tex
\end{lstlisting}
\section{Tips and Tricks}
\subsection{Compatibility}
The \texttt{iscram} class requires recent \TeX{} distributions (MikTeX or
TeXLive 2016).
For any questions, problems, suggestions concerning the iscram class,
contact the authors by mail
(\href{mailto:paul.gaborit@gmail.com}{\gmailpaulgaborit}).
\subsection{Title and Authors}
\subsubsection{Add footnotes in title or authors descriptions}
Use \macro{\thanks} to add footnotes attached to your \option{title} or
to the \option{name} of an author or to its \option{affiliation}
(\emph{note:} don't use \macro{\thanks} into the \option{short title} or
\option{short name} options).
\subsubsection{Links to web sites and to e-mail address}
You may use \macro{\href}, \macro{\urldef} and
\macro{\url} to add links to web pages or to e-mail address.
\begin{minipage}{.32\linewidth}
e-mail: \href{mailto:j.doe@example.com}{\jdmail}\\
web site: \href{http://www.example.com/}{\sitex}
\end{minipage}
\hfill
\begin{minipage}{.65\linewidth}
In your peamble:
\begin{lstlisting}[style=TeX]
\urldef{\jdmail}\url{j.doe@example.com}
\urldef{\sitex}\url{www.example.com}
\end{lstlisting}
Then in your document:
\begin{lstlisting}[style=TeX]
e-mail: \href{mailto:j.doe@example.com}{\jdmail}\\
web site: \href{http://www.example.com/}{\sitex}
\end{lstlisting}
\end{minipage}
\subsubsection{First author is important}
Use the \option{first alone} option to emphasize the first author: with
this option, the first author (and its affiliation) is alone on its line
just below the title. The others authors are grouped two by two.
\subsection{Abstract and Keywords}
Every submission should begin with an \macro{\abstract} of no more than
\textbf{150 words}, followed by a set of \textbf{up to five keywords}
(coma separated values). The abstract should be a concise statement of
the problem, approach, and conclusions of the work described. It should
clearly state the paper's contribution to the field.
\subsection{Figures and Tables}
Figures and tables should be centered. The caption of a figure should be
\emph{below} the figure. The caption of a table should be \emph{above}
the table. Read the documentation of the \texttt{booktabs} package to
find useful advices about composition of tables.
The \texttt{iscram} class uses TeX Gyre Termes (similar to Times) as
serif font and TeX Gyre Heros (similar to Helvetica) as sans-serif
font. You should use the same fonts in your figures and tables.
\subsubsection{Examples of figures}
Here is the code of the figure~\ref{fig:HMI} (a simple figure).
\begin{lstlisting}[style=TeX]
\begin{figure}
\centering
\includegraphics[width=4cm]{HMI}
\caption{Human Computer Interaction}
\label{fig:HMI}
\end{figure}
\end{lstlisting}
\begin{figure}
\centering \includegraphics[width=4cm]{HMI}
\caption{Human Computer Interaction}
\label{fig:HMI}
\end{figure}
Here is the code of the figure~\ref{fig:HMI2} (a figure with a
description).
\begin{lstlisting}[style=TeX]
\begin{figure}
\centering
\begin{minipage}[c]{.6\linewidth}
Here, some text to describe the illustration on the right. This
figure combines two minipages.
\end{minipage}
\hfill
\begin{minipage}[c]{.35\linewidth}
\centering \includegraphics[width=4cm]{HMI}\par
\end{minipage}
\caption{Human Computer Interaction (with description)}
\label{fig:HMI2}
\end{figure}
\end{lstlisting}
\begin{figure}
\centering
\begin{minipage}[c]{.6\linewidth}
Here, some text to describe the illustration on the right. This
figure combines two minipages.
\end{minipage}
\hfill
\begin{minipage}[c]{.35\linewidth}
\centering \includegraphics[width=4cm]{HMI}\par
\end{minipage}
\caption{Human Computer Interaction (with description)}
\label{fig:HMI2}
\end{figure}
Here is the code of the figure~\ref{fig:HMI3} (an HERE figure: note the \option{[H]} option).
\begin{lstlisting}[style=TeX]
\begin{figure}[H]
\centering
\includegraphics[width=4cm]{HMI}
\includegraphics[width=4cm,angle=90]{HMI}
\caption{Human Computer Interaction (example of HERE figure)}
\label{fig:HMI3}
\end{figure}
\end{lstlisting}
\begin{figure}[H]
\centering
\includegraphics[width=4cm]{HMI}
\includegraphics[width=4cm,angle=90]{HMI}
\caption{Human Computer Interaction (example of HERE figure)}
\label{fig:HMI3}
\end{figure}
\subsubsection{Example of table}
Here is the code of the table~\ref{tab:treatments}:
\begin{table}
\caption{A very nice table}
\label{tab:treatments}
\centering
\begin{tabular}{rrr}
\toprule
& \textit{Treatment 1} & \textit{Treatment 2}
\\\midrule
\textit{Setting A} & 125 & 95
\\
\textit{Setting B} & 85 & 102
\\
\textit{Setting C} & 98 & 85
\\\bottomrule
\end{tabular}
\end{table}
\begin{lstlisting}[style=Tex]
\begin{table}
\caption{A very nice table}
\label{tab:treatments}
\centering
\begin{tabular}{rrr}
\toprule
& \textit{Treatment 1} & \textit{Treatment 2} \\\midrule
\textit{Setting A} & 125 & 95 \\
\textit{Setting B} & 85 & 102 \\
\textit{Setting C} & 98 & 85 \\\bottomrule
\end{tabular}
\end{table}
\end{lstlisting}
\subsection{Bibliography: References and Citations}
\emph{Your references should comprise only published material accessible
to the public. Proprietary information (such as internal reports) may
not be cited.}
The \macro{\parencite} macro called with one or more bibliographic keys
is the standard way to insert citations. You may add one or more
\texttt{.bib} files as bibliographic sources via the
\macro{\addbibresource} macro (in your preamble).
The \texttt{biblatex} package provides many other macros to cite
references. Here is an example:\par
\nobreak
\begin{minipage}{.5\linewidth}
\begin{lstlisting}[style=TeX,basicstyle=\small\ttfamily]
In \citeyear{Agarwal2000},
\citeauthor{Agarwal2000} wrote an article
titled \citetitle{Agarwal2000}
\parencite{Agarwal2000} \ldots{}
\end{lstlisting}
\end{minipage}
\hfill
\begin{minipage}{.475\linewidth}
In \citeyear{Agarwal2000}, \citeauthor{Agarwal2000} wrote an article
titled \citetitle{Agarwal2000} \parencite{Agarwal2000} \ldots{}
\end{minipage}
All our references: \parencite{pittir13924,
Tractinsky:1997:AAU:258549.258626, Shneiderman:1997:DUI:523237,
Ghani:1991:EFC:126686.150736, ajzen1988attitudes,
AJZEN1991179,Agarwal2000}.
At the end of your paper, you should call the \macro{\printbibliography}
macro to insert all the cited references.
\printbibliography[heading=bibliography]
\iscramshowframe
\section{Changes}
\begin{itemize}
\item \textbf{v1.1.0} 2018 edition update.
\item \textbf{v1.0.2} PDF documentation included into CTAN package.
\item \textbf{v1.0.1} Fix bibliography style for papers with same
authors. First release to CTAN.
\item \textbf{v1.0.0} First public release.
\end{itemize}
\begin{figure}
\centering
\begin{lstlisting}[style=TeX]
\documentclass{iscram}
\iscramset{
CoRe Paper 2018={Open Track},
title={Example of title},
short title={Example of short title},
author={
short name={J. Doe},
full name={John Doe},
affiliation={Affiliation\\j.doe@example.com},
},
}
\addbibresource{example.bib}
\begin{document}
\maketitle
\abstract{A short abstract \ldots{}}
\keywords{Some keywords}
\section{First section}
\subsection{First subsection}
\subsubsection{First subsubsection}
Bla bla \parencite{key} \ldots{}
\printbibliography
\end{document}
\end{lstlisting}
\caption{Example of usage of ISCRAM document class}
\label{fig:exampledoc}
\end{figure}
\end{document}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End:
|
http://czt.sourceforge.net/latex/circus/circus-guide.tex | sourceforge.net | CC-MAIN-2017-47 | application/x-tex | application/x-tex | crawl-data/CC-MAIN-2017-47/segments/1510934806316.80/warc/CC-MAIN-20171121040104-20171121060104-00100.warc.gz | 72,287,255 | 24,985 | \documentclass{article}
\usepackage{multicol} % needed for Appendix reference card
%\usepackage{circus} % this document describes czt.sty
\usepackage[colour,cntbysection]{circus} % to test lucida bright
%%%%%%%%%% FOR HERE ONLY %%%%%%%%%%%
% Circus toolkit files - extra
%\usepackage{xspace}
\newcommand{\emfile}[1]{\texttt{#1}}%\xspace}} % adds trailing space if needed
\newcommand{\circuspreludefile}{\emfile{circu\_prelude.tex}}
\newcommand{\circustkfile}{\emfile{circus\_toolkit.tex}}
\makeatletter
% Code from Mike Spivey's zed2e.tex for demonstrating \LaTeX{} markup
\def\demo{\par\vbox\bgroup\begingroup\quote}
\def\gives{\endquote\endgroup\egroup}
\def\enddemo{\global\@ignoretrue}
% Code from Paul King's ozguide.tex for generating Appendix reference card
\newbox\zstrutbox \setbox\zstrutbox=\copy\strutbox
\def\zstrut{\relax\ifmmode\copy\zstrutbox\else\unhcopy\zstrutbox\fi}
\def\symbol#1{#1 & \tt\string#1}
\def\symbols{\@ifnextchar[{\@symbols}{\@symbols[0]}}
\def\@symbols[#1]{\interzedlinepenalty=\interdisplaylinepenalty\@@zed
\openup #1\@jot
\halign\bgroup\zstrut\hbox to 3.8em{$##$\hss}\tabskip=0pt&##\hss\cr
\noalign{\vskip-#1\@jot}}% equal vspace above and below argue display
\let\endsymbols=\endzed
\makeatother
% list dependencies in log file
\listfiles
\begin{document}
\title{\Circus-\LaTeX{} style explained\\ Community Z Tools (CZT)}
\author{Leo Freitas \\ \texttt{leo@cs.york.ac.uk} \\ \\ Department of Computer Science \\ University of York, YO10 5DD \\ York, United Kingdom \\ }
\date{October 2008}
\maketitle
\tableofcontents
\listoftables
%\section{Introduction}\label{sec:intro}
%
%In this document, we present a guide to the \textit{Community Z Tools} (CZT)~\cite{czt}
%style file (\cztstylefile). It is used to typeset ISO Standard Z notation~\cite{isoz} that is
%machine readable by CZT tools.
%
%The guide present all the Standard Z characters, as provided in
%the \textit{Community Z Tools} (CZT)~\cite{czt} \emfile{zchar.xml} file (from
%the \texttt{corejava} project within the SVN distribution). It implements the Unicode
%rendering and lexis as given in~\cite[Ch.~6--7]{isoz}. In what follows, each
%section corresponds to the XML groups within this XML file. Before we start, let us
%introduce some context and design decisions within the CZT Standard Z style file (\cztstylefile).
%
%The structure presented in this guide follows the structure presented in Standard Z for
%lexing, markup directives processor, and parsing. More details about all these symbols
%and their \LaTeX{} rendering can be found in~\cite[Appendix~A]{isoz}. For easy of
%reference, we mention at each table caption which part of that appendix symbols are related to.
%We summarise them all in the end of this document.
%Furthermore, some characters listed come from the mathematical toolkits, as defined
%in~\cite[Appendix~B]{isoz}. We add reference to them and the toolkit files
%within CZT where they come from. Mathematical toolkit files for Standard Z can
%be found within the CZT distribution under the \texttt{parser} project~\cite{czt} in
%its \texttt{lib} directory.
%
%Also, CZT lexing/parsing strategy is so that all markup formats are translated
%to a Unicode stream, which is then lexed/parsed according to the Standard Z
%concrete syntax grammar~\cite[Ch.~8]{isoz}. This way, we only need to have one parser and various
%markup translators, which reduces the work considerably. Unicode is chosen as
%a target (among other reasons) because it is an international ISO Standard for lexing. Now, that
%decision implies in some differences in rendering, as one would expect. For
%instance, subscripting, which in \LaTeX{} is done with ``\verb|\_|'', is represented
%in Unicode with so called \textit{word glues}.
%
%Similarly, whitespace and hard space are also treated differently:~in \LaTeX{}
%hard spaces are typeset as ``\verb|~|'', whereas in Unicode they are just normal spaces. Thus,
%as this document is only concerned with \LaTeX{} markup, word glues and Unicode
%considerations will not be discussed. On the other hand, \LaTeX{} specific issues,
%such as hard spaces, will be explained.
%
%\subsection{Design decisions}\label{sec:intro-design}
%
%The main design decision behind this document follows CZT guideline that
%``what you type is what you model''. That is, the document ``as-is'' becomes
%the source Standard Z (\LaTeX) specification to be processed by tools. Other
%design decisions included:~i)~keep the style file as minimal, simple, and
%consistent as possible;~ii)~document and acknowledge macro definition choices
%and their origin (when different);~iii)~normalise definitions for
%consistency;~iv)~complete missing cases with either normative rules from
%the Standard or using common sense;~v)~keep the style file well documented,
%but not verbose;~and vi)~follow order of definitions from Z Standard document.
%
%As the \cztstylefile may be used by both language extensions and \LaTeX{} users,
%we also provided and explained a series of useful macros for \LaTeX{} rendering
%that bare no relation with the Standard or the tools. They are useful for \LaTeX{}
%typesetting only, and are explained in Section~\ref{sec:intro-cztopt}, and
%Section~\ref{sec:cztsty}.
%
%\subsection{\cztstylefile{} package options and few useful commands}\label{sec:intro-cztopt}
%
%The \cztstylefile has few options, which are described below:
%%
%\begin{enumerate}
% \item \texttt{mathit}:~Latin letters in italic shape when in math mode;
% \item \texttt{mathrm}:~Latin letters in roman shape when in math mode;
% \item \texttt{lucida}:~use Lucida Bright fonts (\textit{e.g.,}~\texttt{lucidabr.sty});
% \item \texttt{tkkeyword}:~make some toolkit names render as keywords;
% \item \texttt{color}:~typeset Z-\LaTeX{} using colours;
% \item \texttt{colour}:~synonym for \texttt{color}
%\end{enumerate}
%%
%The default option for when the \cztstylefile is loaded is \texttt{mathit}.
%To change it to have \texttt{colour}ful \texttt{lucida} fonts, you can load it with
%%
%\begin{verbatim}
%\usepackage[colour,lucida]{czt}
%\end{verbatim}
%%
%AMS fonts are used when Lucida Bright is not loaded.
%
%A few style parameters affect the way Z text is set out; they can be
%changed at any time if your taste doesn't match mine.
%
%Other useful macros might be used in order to change the various
%space adjustment registers. They are detailed below, and were
%inherited from Mike Spivey's \texttt{zed.sty}.
%%
%\begin{description}
%\item[\tt\string\zedindent] The indentation for mathematical text.
% By default, this is the same as \verb|\leftmargini|, the
% indentation used for list environments.
%\item[\tt\string\zedleftsep] The space between the vertical line on the left
% of schemas, etc., and the maths inside. The default is 1em.
%\item[\tt\string\zedtab] The unit of indentation used by \verb|\t|.
% The default is 2em.
%\item[\tt\string\zedbar] The length of the horizontal bar in the middle
% of a schema. The default is 6em.
%\item[\tt\string\zedskip] The vertical space inserted by \verb|\also|.
% By default, this is the same as that inserted by approximately
% \verb|0.5\baselineskip|.
%\end{description}
%%
%Finally, two other macros that might be frequently used are those
%for marking commands as either Z-words ($\zword{text}$, \verb|$\zword{text}$|)
%or Z-keywords ($\zkeyword{text}$, \verb|$\zkeyword{text}$|). They are useful
%in rendering user defined \LaTeX{} commands, usually present in Z-\LaTeX{}
%markup directives, as shown in many examples below. We also offer another
%\verb|\ztoolkit| command, which renders toolkit names, such as $\dom$ or $\ran$,
%wither as \verb|\zkeyword| or \verb|\zword| depending on the option passed.
%
%\subsection{Background}\label{sec:intro-background}
%
%This document depends on the style file containing all the definitions for Standard Z
%(\emfile{czt.sty}). It is inspired in the work of many others (see Section~\ref{sec:conclusions}.
%By design, the resulting \emfile{czt.sty} is to be minimal, yet encompassing of the
%whole normative \LaTeX{} markup from the Z Standard.
%
%Although all other style files available worked well with various Z tools,
%they included a considerable amount of code that seemed unrelated to the Standard itself. For instance,
%presumably for backward compatibility, there were many characters for \textit{Fuzz},
%Mike Spivey's Z typechecker at Oxford University~\cite{zrm}. Another example are formatting for
%special formulas within \verb|\mathinner| mathematical operator class (see~\cite[8.9]{latexcomp}).
%
%That meant these style files sometimes created conflicts when used with other (newer) \LaTeX{} packages.
%For instance, because \textit{Fuzz} uses rather old \LaTeX{} $2.09$ (\textit{e.g.,}~\LaTeX{} symbols
%font \texttt{lasy}), some conflicts arise when using \emfile{zed.sty} (Jim Davies' style file
%used in~\cite{usingz}) and AMS fonts. We hope that, with time, any particular backward compatibility
%issue get solved with a separate (extension) of the base \cztstylefile{} file.
%
%These additions may be useful for some specific Z tools or editors, or indeed for
%beautification of the \LaTeX{} document itself. Nevertheless, they cannot be parsed
%by the Standard Z lexis, hence would produce errors if processed by CZT tools. As
%\LaTeX{} documents are meant to be machine-readable, such extensions seem outside
%the scope of CZT's aim. Again, if required, they can be incorporated by the specific
%users of the feature whom does not observe this machine-readability restriction.
%
%\subsection{Document structure}\label{sec:intro-struct}
%
%We organise this document following the specific parts within the Z Standard it is related to.
%We divided sections according to the Z lexis and mathematical toolkits, with a few extra
%sections for varied material.
%
%We tried to present, as exhaustively as possible, the use of every one
%of such commands with \LaTeX{} markup typeset in verbatim mode for
%clarity and reference. We summarise them all in Appendix~\ref{app:ref-card}.
%More details can be found at the \texttt{czt.dvi} file generated with
%the \texttt{docstrip} utility on the \texttt{czt.dtx} document from
%the CZT distribution.
%
%\section{Digit}\label{sec:digit}
%
%Loaded automatically by \LaTeX{} ($0$--$9$) in whichever font selected, hence
%no extra work is needed here.
%
%\section{Letters}\label{sec:letters}
%
%The Z Standard enables users to instruct the parser to recognise new \LaTeX{}
%commands as part of the Z lexis via the use of markup directives~\cite[A.2.3]{isoz},
%They are typeset as special \LaTeX{} comments \verb|%%Zxxxchar| or \verb|%%Zxxxword|,
%where ``\verb|xxx|'' can be either:~\verb|pre| for prefix names;~\verb|pos| for
%posfix names;~\verb|in| for infix names;~and empty for nofix names. Their syntax
%(accepted by the parser) expects two arguments:~the first is the \LaTeX{} command
%it represents, whereas the second determines how this command is to be rendered in Unicode.
%Thus, in order to add mathematical symbols as markup directives, one needs to know its
%corresponding Unicode character (number), which can be found in the Unicode chars~\cite{unicode}.
%
%From \preludefile, the Standard Z file containing \LaTeX{} markup directives for
%Z keywords and basic declarations, all markup directives given as \verb|%%Zprechar|
%or \verb|%%Zposchar| have special spacing as a pre/posfix operator, which in \cztstylefile
%is typeset with the \verb|\zpreop| and \verb|\zpostop| macros, respectively.
%Also, all \verb|%%Zinchar| have special spacing as an infix operator, which can be
%spaced as either a binary relation with the \verb|\zbinop| macro, or as a relational predicate
%operator with the \verb|\zrelop| macro. Other \verb|%%Zchar| directives (\textit{e.g.,}~$\Delta$, $\Xi$)
%do not require special spacing---in the Standard hard spacing is treated differently for them
%(see~\cite[A.6.28.2]{isoz}). The \verb|%%Zxxxword| markup directives are treated similarly.
%
%\subsection{Latin}\label{sec:letters-latin}
%
%Usual letters (\texttt{A}--\texttt{Z}, \texttt{a}--\texttt{z}) are
%loaded automatically by \LaTeX{} in whichever font selected.
%Moreover, in mathematical mode, Latin letters are rendered with either
%italics or roman shape. This depends on the package option selected
%(see Section~\ref{sec:intro-cztopt}), where italic shape is the default.
%
%\subsection{Greek}\label{sec:letters-greek}
%
%The Greek letters used in Z are given in Table~\ref{tbl:letters-greek}.
%The last two columns show how characters are rendered with the given
%\LaTeX{} markup on its side. The last row contains a name convention for
%framing schemas used in Z promotion~\cite[Ch.~13]{usingz} and have no semantic meaning.
%The spacing for $\lambda$ and $\mu$ changed, as they are prefix keywords in Z
%for function abstraction and definite description, respectively.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|c|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Capital Delta & schema inclusion & $\Delta$ & \verb|\Delta| \\
% \hline
% Capital Xi & schema inclusion & $\Xi$ & \verb|\Xi| \\
% \hline
% Small theta & schema bindings & $\theta$ & \verb|\theta| \\
% \hline
% Small lambda & function abstraction & $\lambda$ & \verb|\lambda| \\
% \hline
% Small mu & definite description & $\mu$ & \verb|\mu| \\
% \hline
% Capital Phi & schema promotion & $\Phi$ & \verb|\Phi| \\
% \hline
%\end{tabular}
%\caption{Greek letters used in Z \smallcaption{A.2.4.1}}\label{tbl:letters-greek}
%\end{table}
%
%
%
%
%The \preludefile define a few other letters as markup directives~\cite[A.2.3]{isoz},
%hence can also be used as variable names that are recognised by the parser,
%as given in Table~\ref{tbl:letters-greek-small}.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|c|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Small alpha & ordinary name & $\alpha$ & \verb|\alpha| \\
% \hline
% Small beta & ordinary name & $\beta$ & \verb|\beta| \\
% \hline
% Small gamma & ordinary name & $\gamma$ & \verb|\gamma| \\
% \hline
% Small delta & ordinary name & $\delta$ & \verb|\delta| \\
% \hline
% Small epsilon & ordinary name & $\epsilon$ & \verb|\epsilon| \\
% \hline
% Small zeta & ordinary name & $\zeta$ & \verb|\zeta| \\
% \hline
% Small eta & ordinary name & $\eta$ & \verb|\eta| \\
% \hline
% Small iota & ordinary name & $\iota$ & \verb|\iota| \\
% \hline
% Small kappa & ordinary name & $\kappa$ & \verb|\kappa| \\
% \hline
% Small nu & ordinary name & $\nu$ & \verb|\nu| \\
% \hline
% Small xi & ordinary name & $\xi$ & \verb|\xi| \\
% \hline
% Small pi & ordinary name & $\pi$ & \verb|\pi| \\
% \hline
% Small rho & ordinary name & $\rho$ & \verb|\rho| \\
% \hline
% Small sigma & ordinary name & $\sigma$ & \verb|\sigma| \\
% \hline
% Small tau & ordinary name & $\tau$ & \verb|\tau| \\
% \hline
% Small upsilon & ordinary name & $\upsilon$ & \verb|\upsilon| \\
% \hline
% Small phi & ordinary name & $\phi$ & \verb|\phi| \\
% \hline
% Small chi & ordinary name & $\chi$ & \verb|\chi| \\
% \hline
% Small psi & ordinary name & $\psi$ & \verb|\psi| \\
% \hline
% Small omega & ordinary name & $\omega$ & \verb|\omega| \\
% \hline
%\end{tabular}
%\caption{Small Greek letters \smallcaption{B.2, \preludefile}}\label{tbl:letters-greek-small}
%\end{table}
%%
%Similarly, few capital Greek letters are defined and given in Table~\ref{tbl:letters-greek-capital}.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|c|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Capital Gamma & ordinary name & $\Gamma$ & \verb|\Gamma| \\
% \hline
% Capital Theta & ordinary name & $\Theta$ & \verb|\Theta| \\
% \hline
% Capital Lambda & ordinary name & $\Lambda$ & \verb|\Lambda| \\
% \hline
% Capital Pi & ordinary name & $\Pi$ & \verb|\Pi| \\
% \hline
% Capital Sigma & ordinary name & $\Sigma$ & \verb|\Sigma| \\
% \hline
% Capital Upsilon & ordinary name & $\Upsilon$ & \verb|\Upsilon| \\
% \hline
% Capital Phi & ordinary name & $\Phi$ & \verb|\Phi| \\
% \hline
% Capital Psi & ordinary name & $\Psi$ & \verb|\Psi| \\
% \hline
% Capital Omega & ordinary name & $\Omega$ & \verb|\Omega| \\
% \hline
%\end{tabular}
%\caption{Capital Greek letters \smallcaption{B.2, \preludefile}}\label{tbl:letters-greek-capital}
%\end{table}
%
%\subsection{Other letter}\label{sec:letters-other}
%
%The other letters used in Z are given in Table~\ref{tbl:letters-other}.
%Note \LaTeX{} subscripting markup has no word glues (see Section~\ref{sec:special-wordglue}).
%Also, as $\power$ is defined with the \verb|%%Zprechar| markup directive, it is rendered
%with appropriate spacing as a prefix keyword. The same applies for finite subsets
%($\finset$) and their non-empty ($1$-subscripted) versions (\textit{e.g.,}~$\power_1$, $\finset_1$).
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|c|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Blackboard bold A & base numbers & $\arithmos$ & \verb|\arithmos| \\
% \hline
% Blackboard bold N & naturals & $\nat$ & \verb|\nat| \\
% \hline
% Blackboard bold P & power set & $\power\varg$ & \verb|\power| \\
% \hline
% Blackboard bold F & finite power set & $\finset\varg$ & \verb|\finset| \\
% \hline
%\end{tabular}
%\caption{Other letters \smallcaption{A.2.4.2, B.3.6, \preludefile, \settkfile}}\label{tbl:letters-other}
%\end{table}
%%
%In \numtkfile (see Section~\ref{sec:symbol-toolkit-number}) and
%\settkfile (see Section~\ref{sec:symbol-toolkit-set}) a few other
%markup directives also require special \LaTeX{} markup
%as letters, and is given in Table~\ref{tbl:letters-other-extra}.
%We also add extra ones for rational and real numbers, as well as boolean values.
%As they are not part of any toolkit, they are not recognised by the parser.
%Nevertheless, to amend that one just needs to add the following markup directives
%with their corresponding Unicode character hex-numbers.
%%
%\begin{verbatim}
%%%Zchar \rat U+2119
%%%Zchar \real U+211A
%%%Zchar \bool U-0001D539
%\end{verbatim}
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|c|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Blackboard bold Q & rationals & $\rat$ & \verb|\rat| \\
% \hline
% Blackboard bold R & reals & $\real$ & \verb|\real| \\
% \hline
% Blackboard bold B & boolean & $\bool$ & \verb|\bool| \\
% \hline
%\end{tabular}
%\caption{Extra letters that may be used in Z}\label{tbl:letters-other-extra}
%\end{table}
%
%\section{Special}\label{sec:special}
%
%In this section, we present a list of special characters used in Z.
%As noted in~\cite[A.2.4.3]{isoz}, ``no space characters need to be
%present around special characters, but it may be rendered if desired.''
%
%\subsection{Stroke characters}\label{sec:special-strokes}
%
%Strokes are summarised in Table~\ref{tbl:special-strokes}.
%Note that $\prime$ (\verb|\prime|) is not used in \LaTeX{}
%and $'$ (\verb|'|) is used in variables representing after state
%instead, whereas in Unicode $\prime$ is the one to use! That has to
%do with backward compatibility and issues related to Unicode.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|c|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Prime & after var. & $'$ & \verb|'| \\
% \hline
% Shriek & outputs & $!$ & \verb|!| \\
% \hline
% Query & inputs & $?$ & \verb|?| \\
% \hline
%\end{tabular}
%\caption{Special characters \smallcaption{A.2.4.3}}\label{tbl:special-strokes}
%\end{table}
%
%\subsection{Word glues}\label{sec:special-wordglue}
%
%Differently from Unicode, in \LaTeX, sub and superscripting markup
%has no word glues (see~\cite[A.2.4.3]{isoz}). Instead, the usual
%\LaTeX{} symbols are used, and no special rendering is needed
%for super (\verb|^|) and subscripting (\verb|_|).
%
%%<char id="NE" hex="2197" description="north east arrow"/>
%%<char id="SW" hex="2199" description="south west arrow"/>
%%<char id="SE" hex="2198" description="south east arrow"/>
%%<char id="NW" hex="2196" description="north west arrow"/>
%%<char id="LL" hex="005F" description="low line"/>
%
%\subsection{Brackets}\label{sec:special-bracket}
%
%Table~\ref{tbl:special-bracket} shows all the brackets used in Standard Z.
%The first two, parenthesis and square brackets, follow the usual \LaTeX{}
%spacing, whereas the last two (binding and free type brackets) should be
%treated as \verb|\mathopen/close| \LaTeX{} math operators, hence having a
%hard space around them. In Z mode, the curly bracket should be treated as
%a \verb|\mathopen/close| as well, since it is part of set constructors.
%As curly braces are such low-level \TeX{}, I could not find a way to go
%around this and just suggest the user to add the hard spaces manually
%(\textit{e.g.,}~\verb|\{~| and \verb|~\}|) as needed. This has no semantic
%difference, and is just for (personal) aesthetic reasons. Strangely,
%underscore is grouped at this table in the Standard. It serves both as part of
%a Z name or as a variable argument (\verb|\varg|) in a definition. For variable
%arguments, both forms (\verb|\_| and \verb|\varg|) are acceptable by CZT tools.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|l|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Left parenthesis & grouping & $($ & \verb|(| \\
% \hline
% Right parenthesis & grouping & $)$ & \verb|)| \\
% \hline
% Left square bracket & various & $[$ & \verb|[| \\
% \hline
% Right square bracket & various & $]$ & \verb|]| \\
% \hline
% Left curly bracket & sets & $\{$\textvisiblespace & \verb|\{~| \\
% \hline
% Right curly bracket & sets & \textvisiblespace$\}$ & \verb|~\}| \\
% \hline
% Left binding bracket & sets & $\lblot$ & \verb|\lblot| \\
% \hline
% Right binding bracket & sets & $\rblot$ & \verb|\rblot| \\
% \hline
% Left double angle bracket & free types & $\ldata$& \verb|\ldata| \\
% \hline
% Right double angle bracket & free types & $\rdata$ & \verb|\rdata| \\
% \hline
% Underscore & var. names & $\Rightarrow\_\Leftarrow$ & \verb|\_| \\
% \hline
% Op. template & var. argument & $\Rightarrow\varg\Leftarrow$ & \verb|\varg| \\
% \hline
%\end{tabular}
%\caption{Bracket characters \smallcaption{A.2.4.3}}\label{tbl:special-bracket}
%\end{table}
%
%\subsection{Box drawing characters}\label{sec:special-box}
%
%Table~\ref{tbl:special-box} lists the box drawing characters used to
%render various Z paragraphs, such as axiomatic definitions, schemas,
%and their generic counterparts, as well as section headers. %The rendering
%%column displays the Unicode character associated with the rendering,
%%which in \LaTeX{} will materialise as specific-width line drawings.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|c|l|l|}
% \hline
% \textbf{Description}& \textbf{Role} & \textbf{Rend.} & \textbf{\LaTeX} & \textbf{Unicode}\\
% \hline
% Light horizontal & para boxes & --- & N/A & U+$2500$ \\
% \hline
% Light down & para boxes & $|$ & N/A & U+$2577$ \\
% \hline
% Light down right & para boxes & $\zboxulcorner$ & N/A & U+$250C$ \\
% \hline
% Double horizontal & genpara boxes & N/A & N/A & U+$2550$ \\
% \hline
% Vertical line & box rendering & $\mid$ & \verb|\mid| & U+$007C$ \\
% \hline
% Paragraph separator & para marker & $\zboxllcorner$ & N/A & U+$2514$ \\
% \hline
% Paragraph separator & para marker & $|$ & \verb|\where|, \verb'|' & U+$007C$ \\
% \hline
%\end{tabular}
%\caption{Boxing characters \smallcaption{A.2.6, A.2.7}}\label{tbl:special-box}
%\end{table}
%%
%These box drawings characters are used for rendering the various
%Z-\LaTeX{} environments, as given in Table~\ref{tbl:latex-env} at
%Section~\ref{sec:latex-env}.
%
%\subsection{Other special characters}\label{sec:special-other}
%
%The other special characters from the Z Standard are hard space and new line~\cite[A.2.2]{isoz}.
%As \LaTeX{} provide rather fine grained spacing control, various \LaTeX{} commands correspond
%to the \texttt{SPACE} Unicode markup, as summarised in Table~\ref{tbl:special-other-hardspace}.
%Also, note the difference between \LaTeX{} whitespace (\textit{i.e.,}~those used to separate
%\LaTeX{} tokens in math mode) and Z-\LaTeX{} white (or hard) spaces (\textit{i.e.,}~those used
%to separate Z tokens).
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|l|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Inter word space & hard space & $\Rightarrow ~ \Leftarrow$ & \verb|~| \\
% \hline
% Inter word space & hard space & $\Rightarrow\ \Leftarrow$ & \verb|\|\textvisiblespace \\
% \hline
% Thin space & hard space & $\Rightarrow \, \Leftarrow$ & \verb|\,| \\
% \hline
% Medium space & hard space & $\Rightarrow \: \Leftarrow$ & \verb|\:| \\
% \hline
% Thick space & hard space & $\Rightarrow \; \Leftarrow$ & \verb|\;| \\
% \hline
% Tab stop $1$ & hard space & $\Rightarrow \t1 \Leftarrow$ & \verb|\t1| \\
% \hline
% Tab stop $2 \ldots$ & hard space & $\Rightarrow \t2 \Leftarrow$ & \verb|\t2| \\
% \hline
%\end{tabular}
%\caption{Hard space characters \smallcaption{A.2.2}}\label{tbl:special-other-hardspace}
%\end{table}
%%
%Thus, ASCII characters for space, tab, and new line are ``soft'', render as nothing and
%are not converted to any Z character. On the other hand, Z-\LaTeX{} hard space markup renders as
%specific quantities of space and is converted according to Table~\ref{tbl:special-other-hardspace}.
%The tab stops counter goes up to $9$ (\textit{i.e.,}~\verb|\t1|~\ldots\verb|\t9|).
%
%From \LaTeX, such mathematical spacing is regulated by the commands and skip values
%defined in Table~\ref{tbl:special-other-muskip}.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|l|l|}
% \hline
% \textbf{Description} & \textbf{Skip counter} & \textbf{Space command} & \textbf{\LaTeX} \\
% \hline
% Thin space skip & \verb|\thinmuskip| & \verb|\thinspace| & \verb|\,| \\
% \hline
% Medium space skip & \verb|\medmuskip| & \verb|\medspace| & \verb|\:| \\
% \hline
% Thick space skip & \verb|\thickmuskip| & \verb|\thickspace| & \verb|\;| \\
% \hline
%\end{tabular}
%\caption{Fine control of skip amount for space characters}\label{tbl:special-other-muskip}
%\end{table}
%%
%To illustrate how to use these skip amount counters, we provide the following
%\LaTeX{} code, which expands the skip amounts and then restores then back to
%their default value.
%%
%%\begin{multicols}{2} see it from symbols.tex for symbols-a4.pdf
%\begin{demo}
%\begin{verbatim}
%% Save original spacing on new skip counter
%\newmuskip\savemuskip
%\savemuskip=\thinmuskip
%
%Formula with default spacing \hfill $ x \, y \, z $
%
%% Change original spacing
%\thinmuskip=20mu
%
%Formula with $20$mu skip \hfill $ x \, y \, z $
%
%% restore default spacing
%\thinmuskip=\savemuskip
%
%Formula with default spacing \hfill $ x \, y \, z $
%\end{verbatim}
%\gives
%\begin{quote}
%% Save original spacing on new skip
%\newmuskip\savemuskip
%\savemuskip=\thinmuskip
%
%Formula with default spacing \hfill $ x \, y \, z $
%
%% Change original spacing
%\thinmuskip=20mu
%
%Formula with $20$mu skip \hfill $ x \, y \, z $
%
%% restore default spacing
%\thinmuskip=\savemuskip
%
%Formula with default spacing \hfill $ x \, y \, z $
%\end{quote}
%\end{demo}
%
%Similarly, we also have various characters for new lines,
%and formulae and page breaks, as shown in Table~\ref{tbl:special-other-newline}.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|l|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Carriage return & new line & (not shown) & \verb|\\| \\
% \hline
% Small vertical space & new line & (not shown) & \verb|\also| \\
% \hline
% Med. vertical space & new line & (not shown) & \verb|\Also| \\
% \hline
% Big vertical space & new line & (not shown) & \verb|\ALSO| \\
% \hline
% Small formula break & vert. space & (not shown) & \verb|\zbreak| \\
% \hline
% Med. formula break & vert. space & (not shown) & \verb|\zBreak| \\
% \hline
% Big formula break & vert. space & (not shown) & \verb|\ZBREAK| \\
% \hline
% New page & new page & (not shown) & \verb|\znewpage| \\
% \hline
%\end{tabular}
%\caption{New line and break characters \smallcaption{A.2.2}}\label{tbl:special-other-newline}
%\end{table}
%
%\section{Symbols}\label{sec:symbol}
%
%List of symbol characters are divided in core and toolkit symbols.
%The former are related to basic characters and keywords, whereas
%the latter is related to the Z mathematical toolkit~\cite[Appendix~B]{isoz}.
%
%\subsection{Core symbols}\label{sec:symbol-core}
%
%Many of the core symbols in \LaTeX{} come directly from the currently
%selected font, whereas others have special commands. We list them all
%in Table~\ref{tbl:symbol-core}, where expected arguments and their rendering
%position are given with ``$\varg$'' (\verb|\varg|).
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|c|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Bullet & set/pred separator & $@$, $\spot$ & \verb|@|, \verb|\spot| \\
% \hline
% Ampersand & recursive free types & $\varg\&\varg$ & \verb|\&| \\
% \hline
% Right tack & conjecture & $\vdash\varg$ & \verb|\vdash| \\
% \hline
% Wedge & logical and & $\varg\land\varg$ & \verb|\land| \\
% \hline
% Vee & logical or & $\varg\lor\varg$ & \verb|\lor| \\
% \hline
% Right double arrow & logical implication & $\varg\implies\varg$ & \verb|\implies| \\
% \hline
% L/R double arrow & logical equivalence & $\varg\iff\varg$ & \verb|\iff| \\
% \hline
% Not sign & logical negation & $\lnot\varg$ & \verb|\lnot| \\
% \hline
% Inverted A & universal quant. & $\forall\varg @ \varg$ & \verb|\forall| \\
% \hline
% Reversed E & existential quant. & $\exists\varg@ \varg$ & \verb|\exists| \\
% \hline
% $\exists$ subscript $1$ & unique existence & $\exists_1\varg @ \varg$ & \verb|\exists_1| \\
% \hline
% Pertinence & set membership & $\varg\in\varg$ & \verb|\in| \\
% \hline
% Math. \verb|\times| & cartesian product & $\varg\cross\varg$ & \verb|\cross| \\
% \hline
% Inverted solidus & schema hiding & $\varg\hide\varg$ & \verb|\hide| \\
% \hline
% Upwards harpoon & schema projection & $\varg\project\varg$ & \verb|\project| \\
% \hline
% Big fat semicolon & schema composition & $\varg\semi\varg$ & \verb|\semi| \\
% \hline
% Double greater than & schema piping & $\varg\pipe\varg$ & \verb|\pipe| \\
% \hline
% Big fat colon & typechecked term & $\varg\typecolon\varg$ & \verb|\typecolon| \\
% \hline
%\end{tabular}
%\caption{Core symbols \smallcaption{A.2.4.4}}\label{tbl:symbol-core}
%\end{table}
%%
%The ampersand (\verb|\&|) is needed in (the not so used) mutually
%recursive free types. Its syntax is described in~\cite[8.2]{isoz},
%whereas its semantics is given in~\cite[14.2.3.1]{isoz}.
%The fat \verb|\spot| also makes \verb|@| active in math mode
%so that it gets the right \verb|\mathrel| spacing.
%Wedge and Vee are the AMS terms for the logical operators.
%
%Other core symbols, such as ``$/$'', ``$;$'', ``$:$'', ``$,$'',
%``$.$'', ``$+$'', ``$=$'', \textit{etc.}, are typeset and spaced
%just as in \LaTeX. The symbol for schema projection ($\project$,
%\verb|\project|) is reused for sequence filtering in the
%toolkit defined in \seqtkfile (see Table~\ref{tbl:symbol-toolkit-seq} in Section~\ref{sec:symbol-toolkit-seq}).
%Also, the symbol for schema composition ($\semi$, \LaTeX{} \verb|\semi|, and Unicode character U+$2A1F$)
%is very similar (but slightly bigger) than the symbol for relational
%composition ($\comp$,~\verb|\comp|,~U+$2A3E$). Type checked markup is
%usually given with a big fat colon beside it ($\typecolon$,~\verb|\typecolon|,~U+$2982$).
%It is a binary operator with the expression in one side and its type on the other.
%%
%%\hline
%%Solidus & substitution & $/$ & \verb|/| \\
%%\hline
%%Double horz. bars & equality & $=$ & \verb|=| \\
%%\hline
%%Colon & type expr. sep. & $:$ & \verb|:| \\
%%\hline
%%Semicolon & var decl. sep. & $;$ & \verb|;| \\
%%\hline
%%Comma & var list sep. & $,$ & \verb|,| \\
%%\hline
%%Dot & tuple/bind sel. & $.$ & \verb|.| \\
%%\hline
%%Plus & integer sum & $+$ & \verb|+| \\
%%\hline
%
%Note that spacing with \LaTeX{} infix binary mathematical operators are rendered
%differently in the presence of new lines in between them.
%%
%\begin{demo}
%\begin{verbatim}
%\begin{zed}
% A ~~==~~ S \cup T \\ % usual
% B ~~==~~ S \cup \\ \t2 T \\ % spacing after \cup symbol
% C ~~==~~ S \cup{} \\ \t2 T % correction
%\end{zed}
%\end{verbatim}
%\gives
%\begin{zed}
% A ~~==~~ S \cup T \\ % usual
% B ~~==~~ S \cup \\ \t2 T \\ % bad spacing after \cup symbol
% C ~~==~~ S \cup{} \\ \t2 T % correction
%\end{zed}
%\end{demo}
%%
%So, when breaking lines
%near such operators, one need to add the usual \LaTeX{} marker for such situations,
%as illustrated below (see~\cite[p.525, Table~8.7]{latexcomp} for more details), new
%lines may change the spacing behaviour of infix binary mathematical operators,
%as the example above shows.
%
%\subsection{Toolkit symbols}\label{sec:symbol-toolkit}
%
%This section introduces all the characters used within \stdtkfile, as mentioned
%in~\cite[Appendix~B]{isoz}. It has been divided in subsections according to the various
%Z sections defined in the Standard.
%
%\subsubsection{\preludefile{} and Z keywords}\label{sec:symbol-toolkit-prelude}
%
%The \texttt{prelude} section is an implicit parent of every other section.
%It assists in defining the meaning of number literal expressions~\cite[12.2.6.9]{isoz}
%and the list arguments of operator templates~\cite[12.2.12]{isoz} via syntactic transformation
%rules. In Table~\ref{tbl:symbol-toolkit-prelude}, we present the list of symbols
%and Z keywords (and their fixture) defined in the prelude with markup directives.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|l|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Z section marker & prefix keyword & $\SECTION\varg$ & \verb|\SECTION| \\
% \hline
% Z section parent & infix keyword & $\parents\varg$ & \verb|\parents| \\
% \hline
% Conditional & prefix keyword & $\IF\varg$ & \verb|\IF| \\
% \hline
% Conditional & infix keyword & $\varg\THEN\varg$ & \verb|\THEN| \\
% \hline
% Conditional & infix keyword & $\varg\ELSE\varg$ & \verb|\ELSE| \\
% \hline
% Let definition & prefix keyword & $\LET\varg==\varg @ \varg$ & \verb|\LET| \\
% \hline
% Application expr. & prefix op. template & $\function\varg~\varg$ & \verb|\function| \\
% \hline
% Relational pred. & prefix op. template & $\relation\varg~\varg$ & \verb|\relation| \\
% \hline
% Generic expr. inst. & prefix op. template & $\generic\varg~\varg$ & \verb|\generic| \\
% \hline
% Left associative & infix op. template & $\leftassoc$ & \verb|\leftassoc| \\
% \hline
% Right associative & infix op. template & $\rightassoc$ & \verb|\rightassoc| \\
% \hline
% Schema precondition & prefix keyword & $\pre\varg$ & \verb|\pre| \\
% \hline
% List of arguments & infix op. template & $\Rightarrow\listarg\Leftarrow$ & \verb|\listarg| \\
% \hline
% Variable argument & infix op. template & $\Rightarrow\varg\Leftarrow$ & \verb|\varg| \\
% \hline
% Boolean truth & ordinary name & $\true$ & \verb|\true| \\
% \hline
% Boolean falsehood & ordinary name & $\false$ & \verb|\false| \\
% \hline
%\end{tabular}
%\caption{\preludefile symbols \smallcaption{A.2.4, B.2}}\label{tbl:symbol-toolkit-prelude}
%\end{table}
%%
%Z sections enable the user to define self contained named modules with (non cyclic)
%parent relationships given as a (possibly empty) list of section names. Conditional
%($\IF-\THEN-\ELSE$) allows one to test a predicate which yields an expression depending whether
%the predicate is $true$ or $false$. Let definitions ($\LET$) allow local variable scoping for expressions.
%
%Operator templates~\cite[C.4.13]{isoz} have syntactic significance only:~they tell
%the reader how to interpret the template associativity, and how it is rendered
%as prefix, infix, posfix or nofix. There are three categories of operator templates the
%user can define:~\verb|\function|, for application expressions as \textit{e.g.,}
%\[ S \cup T = (\_~\cup\_)~(S, T) \]
%\verb|\relation|, for relational predicates as \textit{e.g.,}
%\[ S \subseteq T = (S,T) (\_~\subseteq\_) \]
%and \verb|\generic|, for generic instantiation of expressions as \textit{e.g.,}
%\[ X \rel Y,\t1 \emptyset[\nat] \]
%Application expressions (\verb|\function|) are used for both fixed (as pre, in, or pos fixed)
%function operator application (\textit{e.g.,}~infix $S \cup T$), and as its equivalent
%(\textit{e.g.,}~nofix $(\_~\cup\_)~(S, T)$) version.
%Relational (or membership) predicates (\verb|\relation|) are used for both set membership
%(\textit{e.g.,}~\mbox{$x \in S$}), equality (\textit{e.g.,}~\mbox{$S = T$}), and as an
%operator that is a predicate (\textit{e.g.,}~\mbox{$S \subseteq T$}).
%Generic instantiation expressions are used for generic operator application
%as in when building relation (\mbox{$X \rel Y$}) or function (\mbox{$X \fun Y$}) spaces.
%
%Furthermore, all infix \verb|\function| and \verb|\generic| operator templates must have two explicit
%declarations:~one for their binding power, which is as a natural number (the higher the number tighter the
%precedence);~and one for their (left or right) associativity. They are used to resolve operator
%precedences. For instance, \mbox{$S \cup T \cap R = (S \cup (T \cap R))$} because $\cap$ binds tighter
%than $\cup$ (see binding powers in Table~\ref{tbl:symbol-toolkit-set} below). Relational predicates
%and prefix, posfix and nofix function and generic operators do not have precedence or associativity
%explicitly given. Examples of this notation can be found in~\cite[Appendix~B]{isoz}, and are highlighted
%in the \textbf{Role} column in the tables below for each operator template defined in the standard toolkits.
%When the binding powers are the same, the given associativity is used to resolve the precedence.
%For instance, set intersection and set difference have the same binding power ($30$), but are
%both left-associative, hence $S \cap T \setminus R = (S \cap T) \setminus R$ as the left-associativeness
%of set intersection gives it priority over set difference. Finally, if within the same section (and
%all its parent sections) there are two operator templates with the same binding power (even if different
%kinds, say one \verb|\function| and one \verb|\generic|), but different associativity, a parsing error
%is raised since precedence cannot be decided. For instance, if we have a section with \texttt{set\_toolkit}
%as its parent, and we define a new \verb|\function| operator template with binding power $30$
%and as right associative, a parsing error is raised, as it is not possible to decide its
%precedence (\textit{i.e.,} it conflicts with the operator template definition for $\cup$).
%
%Note that generic operator templates, such as finite subsets $(\finset~\_)$ and total functions $(\_~\fun\_)$,
%are not to be confused with a generic reference expression instantiation, such as empty sets ($\emptyset[\nat]$),
%which is not given as an operator template, but rather a reference name. Moreover, when generic
%references are instantiated by the typechecker they are implicit ($\emptyset$), whereas when given by
%the user they are explicit ($\emptyset[\nat]$---the empty set of natural numbers).
%
%When defining operator templates, we could have single arguments (\verb|\varg|) as in the
%definition of set union ($\varg\cup\varg$, \verb|\varg \cup \varg|) at \settkfile, or variable/list
%arguments (\verb|\listarg|) as in the definition of sequence display ($\langle \listarg \rangle$,
%\verb|\langle \listarg \rangle|) at \seqtkfile.
%
%
%Other Z style packages allow room for a keyword \verb|\inrel|, which could be used for changing
%the fixture of relations that were not defined as operator templates. For instance, suppose
%$R \in X \rel Y$, $x \in X$, and $y \in Y$. As $R$ is not an operator template, the usual
%way of relating $x$ and $y$ to $R$ would be either ``\mbox{$(x,y) \in R$}'' or ``\mbox{$(x \mapsto y \in R)$}''.
%With the \verb|\inrel| keyword, one was allowed to say ``\mbox{$(x~\inrel{R}~y)$}'' (\verb|(x~\inrel{R}~y)|).
%Nevertheless, such feature is not part of the Z Standard, hence not amenable to parsing, and thus
%not supported in \cztstylefile.
%
%Additionally, we add two special ``keywords'' as $\true$ (\verb|\true|) and $\false$ (\verb|\false|)
%to represent boolean values at the level of the logic, rather than as predicates $true$ (\verb|$true$|)
%and $false$ (\verb|$false$|). This is used in the Z logic of the Z Standard. It can also be used in the
%definition of a boolean free type in a user toolkit. This serves to illustrate how one can make use of
%Z markup directives once again.
%%
%\begin{demo}
%\begin{verbatim}
%% AMS black board B
%% \bool is already defined in czt.sty just like
%% \newcommand{\bool}{\zordop{\mathbb B}}
%
%% Note the markup directives are needed for parsing
%% since they are not present in any standard toolkit.
%%%Zchar \bool U-0001D539
%%%Zword \true True
%%%Zword \false False
%\begin{zed}
% \bool ::= \false | \true
%\end{zed}
%\end{verbatim}
%\gives
%% AMS black board B
%% \bool is already defined in czt.sty just like
%% \newcommand{\bool}{\zordop{\mathbb B}}
%
%% Note the markup directives are needed for parsing
%% since they are not present in any standard toolkit.
%%%Zchar \bool U-0001D539
%%%Zword \true True
%%%Zword \false False
%\begin{zed}
% \bool ::= \false | \true
%\end{zed}
%\end{demo}
%%
%Apart from typesetting purposes, logic boolean values can be used, for instance,
%to use Z as a meta-language to specify the semantics of other languages~\cite{circussem}.
%
%\subsubsection{\settkfile}\label{sec:symbol-toolkit-set}
%
%The \texttt{set\_toolkit} defines symbols for what a relation is, and operators about sets and finite sets.
%In Table~\ref{tbl:symbol-toolkit-set}, we present the list its symbols.
%The \textbf{Role} column contains the details for each operator template, or ``\textit{XXX name}''
%when the symbol is not an operator but a name, where the \textit{XXX} determines its fixture.
%Infix function and generic operator templates have their binding power given as numbers, and
%associativity given as either LA (left-associative) or RA (right-associative). Non-infix operator templates have
%their type and fixture given. For ease of reference, we also add the \verb|\varg| arguments to the
%\LaTeX{} rendering column (but not the verbatim \LaTeX{} itself for clarity).
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|c|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Relation space & generic $5$ RA & $\varg\rel\varg$ & \verb|\rel| \\
% \hline
% Function space & generic $5$ RA & $\varg\fun\varg$ & \verb|\fun| \\
% \hline
% Not set member & infix relation & $\varg\notin\varg$ & \verb|\notin| \\
% \hline
% Inequality & infix relation & $\varg\neq\varg$ & \verb|\neq| \\
% \hline
% Empty set & nofix name & $\emptyset$ & \verb|\emptyset| \\
% \hline
% Subset & infix relation & $\varg\subseteq\varg$ & \verb|\subseteq| \\
% \hline
% Proper subset & infix relation & $\varg\subset\varg$ & \verb|\subset| \\
% \hline
% Non-empty sets & prefix name & $\power_1\varg$ & \verb|\power_1| \\
% \hline
% Set union & function $30$ LA & $\varg\cup\varg$ & \verb|\cup| \\
% \hline
% Set intersection & function $40$ LA & $\varg\cap\varg$ & \verb|\cap| \\
% \hline
% Set difference & function $30$ LA & $\varg\setminus\varg$ & \verb|\setminus| \\
% \hline
% Set symmetric diff. & function $25$ LA & $\varg\symdiff\varg$ & \verb|\symdiff| \\
% \hline
% Generalised union & prefix name & $\bigcup\varg$ & \verb|\bigcup| \\
% \hline
% Generalised intersection & prefix name & $\bigcap\varg$ & \verb|\bigcap| \\
% \hline
% Finite sets & prefix generic & $\finset\varg$ & \verb|\finset| \\
% \hline
% Non empty $\finset$ & prefix generic & $\finset_1\varg$ & \verb|\finset_1| \\
% \hline
%\end{tabular}
%\caption{\settkfile symbols \smallcaption{A.2.5.1, B.3, B.4}}\label{tbl:symbol-toolkit-set}
%\end{table}
%
%The empty set symbol within the usual \LaTeX{} distribution (as found in file
%\emfile{fontmath.ltx} with font encoding \verb|OMS/cmsy/m/n| and hex number \verb|"3B|)
%is slightly different from the mathematical empty set symbol, which is present in the AMS font.
%Because of this, when using \cztstylefile, one can access the original empty set symbol with
%\verb|\mathemptyset|, which is rendered in \LaTeX{} as $\mathemptyset$.
%
%\subsubsection{\reltkfile}\label{sec:symbol-toolkit-relation}
%
%The \texttt{relation\_toolkit} has \texttt{set\_toolkit} as its parent and
%defines symbols for:~maplets;~domain and range;~relational and functional
%composition;~domain and range restriction and substraction;~relational
%inversion and overriding;~and transitive and reflexive transitive closures over relations.
%In Table~\ref{tbl:symbol-toolkit-relation}, we present its symbols.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|c|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Binary tuple projection & ordinary name & $first$ & \verb|first~\varg| \\
% \hline
% Binary tuple projection & ordinary name & $second$ & \verb|second~\varg| \\
% \hline
% Relation maplet & function $10$ LA & $\varg\mapsto\varg$ & \verb|\mapsto| \\
% \hline
% Domain of relation & prefix name & $\dom\varg$ & \verb|\dom| \\
% \hline
% Range of relation & prefix name & $\ran\varg$ & \verb|\ran| \\
% \hline
% Identity relation & prefix generic & $\id\varg$ & \verb|\id| \\
% \hline
% Relational composition & function $40$ LA & $\varg\comp\varg$ & \verb|\comp| \\
% \hline
% Functional composition & function $40$ LA & $\varg\circ\varg$ & \verb|\circ| \\
% \hline
% Domain restriction & function $65$ LA & $\varg\dres\varg$ & \verb|\dres| \\
% \hline
% Range restriction & function $60$ LA & $\varg\rres\varg$ & \verb|\rres| \\
% \hline
% Domain subtraction & function $65$ LA & $\varg\ndres\varg$ & \verb|\ndres| \\
% \hline
% Range subtraction & function $60$ LA & $\varg\nrres\varg$ & \verb|\nrres| \\
% \hline
% Relational inversion & prefix function & $\varg\inv$ & \verb|\inv| \\
% \hline
% Relational image left & mixfix function & $\varg\limg$ & \verb|\limg| \\
% \hline
% Relational image right & mixfix function & $\varg\rimg$ & \verb|\rimg| \\
% \hline
% Overriding & function $50$ LA & $\varg\oplus\varg$ & \verb|\oplus| \\
% \hline
% Transitive closure & posfix function & $\varg\plus$ & \verb|\plus| \\
% \hline
% Reflexive $(\_~\plus)$ & posfix function & $\varg\star$ & \verb|\star| \\
% \hline
%\end{tabular}
%\caption{\reltkfile symbols \smallcaption{A.2.5.2, B.5}}\label{tbl:symbol-toolkit-relation}
%\end{table}
%
%This toolkit defines tuple projection functions that do not use markup directives and
%are not given as operator templates, hence have no special \LaTeX{} markup associated
%with them. Despite this fact, the usual \LaTeX{} rendering is (historically)
%given as if they were Z keywords. To achieve this effect, however, the user need define his own
%``special'' rendering for that markup. For instance, $first$ and $second$, which project the first and second elements
%of a given binary tuple, are defined with ordinary names (\textit{i.e.,}~no markup directive)
%in \reltkfile. So, some users prefer to have keyword-like typesetting, which can be done as \verb|\zkeyword{first}|
%($\zkeyword{first}$). Unfortunately, this is no longer parseable, since \verb|\zkeyword| is not part of the Z lexis,
%but rather a \LaTeX{} rendering markup. Nevertheless, if the user still wants to keep a nice
%\LaTeX{} rendering, she could just define the appropriate \LaTeX{} command as an alternative markup for the name in
%question through markup directives. For our example, to have ``$first$'' typeset like a keyword ($\zkeyword{first}$),
%one should add the following markup directive and new \LaTeX{} command:
%%
%\newcommand{\first}{\zpreop{\zkeyword{first}}}
%\begin{verbatim}
%\newcommand{\first}{\zpreop{\zkeyword{first}}}
%%%Zpreword \first first
%\end{verbatim}
%%
%The markup directive will tell the parser to treat the command \verb|\first| as
%the string \verb|first|, which is loaded from \reltkfile. Then \LaTeX{} can now
%render \verb|\first| as desired ($\zkeyword{first}$). Furthermore, if the user
%wants to keep both choices conditional to the package option \texttt{tkkeyword},
%one can just use the \verb|\ztoolkit{first}|, instead. With current option, it
%typesets as $\ztoolkit{first}$. Finally, note that, since $\first$ a prefix word,
%we also wrap it with a \verb|\zpreop|. This way proper spacing is added, and one
%does not need to typeset it as \verb|$\first~x$| ($\first~x$) but just
%\verb|$\first x$| ($\first x$), as opposed to the mandatory hard space in
%\verb|$first~x$| ($first~x$) to avoid the wrong spacing from \verb|$first x$| ($first x$).
%
%The Z Standard also leaves room for mixfix (mixed fixture) operator templates, although those
%are more rarely used. One such operator is used for the definition of relational imagine as
%%
%\begin{verbatim}
%%%Zinchar \limg U+2987
%%%Zpostchar \rimg U+2988
%\begin{zed}
%\function (\_ \limg \_ \rimg)
%\end{zed}
%\end{verbatim}
%%
%So, each bracketing symbol is treated with a different fixture. That is, $\limg$ is treated
%as an infix operator, whereas $\rimg$ is treated as a posfixed one. This combination makes
%the relational image mixfix operator template as defined above.
%
%The AMS/Lucida bright font(s) already define(s) the \verb|\star| symbol as ``$\mathstar$''
%(\textit{e.g.,}~\texttt{msam10}, hex-number \verb|"46|), rather than the ``$\star$'' we want.
%Because of this, when using \cztstylefile, one can access the original AMS/Lucida bright star
%symbol with the \verb|\mathstar| command, which is rendered as ``$\mathstar$''.
%
%For relational inverse (\verb|R\inv|), the Z Standard does not specify it with superscripting
%word glues~\cite[A.2.4.3]{isoz}. Thus, its rendering is ``$R\inv$'', and it should not be superscripted
%as in ``$R^{\inv}$'', despite this being more common. This may perhaps be a Z Standard typo.
%
%\subsubsection{\fcntkfile}\label{sec:symbol-toolkit-function}
%
%The \texttt{function\_toolkit} has \texttt{relation\_toolkit} as its parent and
%defines symbols for generic operator templates representing the various subsets
%of function spaces, and a few relational predicates for sets. In
%Table~\ref{tbl:symbol-toolkit-function}, we present its list of symbols.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|c|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Partial function & generic $5$ RA & $\varg\pfun\varg$ & \verb|\pfun| \\
% \hline
% Partial injection & generic $5$ RA & $\varg\pinj\varg$ & \verb|\pinj| \\
% \hline
% Injection & generic $5$ RA & $\varg\inj\varg$ & \verb|\inj| \\
% \hline
% Partial surjection & generic $5$ RA & $\varg\psurj\varg$ & \verb|\psurj| \\
% \hline
% Surjection & generic $5$ RA & $\varg\surj\varg$ & \verb|\surj| \\
% \hline
% Bijection & generic $5$ RA & $\varg\bij\varg$ & \verb|\bij| \\
% \hline
% Finite partial function & generic $5$ RA & $\varg\ffun\varg$ & \verb|\ffun| \\
% \hline
% Finite partial injection & generic $5$ RA & $\varg\finj\varg$ & \verb|\finj| \\
% \hline
% Disjoint sets & prefix relation & $\disjoint\varg$ & \verb|\disjoint| \\
% \hline
% Set partitioning & infix relation & $\varg\partition\varg$ & \verb|\partition| \\
% \hline
%\end{tabular}
%\caption{\fcntkfile symbols \smallcaption{A.2.5.3, B.6}}\label{tbl:symbol-toolkit-function}
%\end{table}
%%
%Lucida Bright fonts render some of these symbols differently, if (and when) loaded.
%
%Disjointness of a relation states that a set of sets has no overlapping elements
%(\textit{i.e.,}~their pairwise set intersection is empty), whereas partitioning
%represents a disjoint set of sets that covers the whole elements of the set's type
%(\textit{i.e.,}~the generalised union of all sets being disjoint represents the whole type).
%
%\subsubsection{\numtkfile}\label{sec:symbol-toolkit-number}
%
%The \texttt{number\_toolkit} defines symbols for integer arithmetic.
%In Table~\ref{tbl:symbol-toolkit-number}, we present its list symbols.
%Note that summation is defined as an operator template in \preludefile,
%but most of its properties are defined in \numtkfile, hence we left it here.
%Subtraction is defined in terms of summation and unary negation
%(\textit{e.g.,}~$\negate\varg$, \verb|\negate|).
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|c|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% $\nat$ successor function & prefix function & $succ\varg$ & \verb|succ \varg| \\
% \hline
% Integers & ordinary name & $\num$ & \verb|\num| \\
% \hline
% Arithmetic negation & prefix function & $\negate\varg$ & \verb|\negate| \\
% \hline
% Subtraction & function $30$ LA & $\varg-\varg$ & \verb|-| \\
% \hline
% Summation & function $30$ LA & $\varg+\varg$ & \verb|+| \\
% \hline
% Less-than equal-to & infix relation & $\varg\leq\varg$ & \verb|\leq| \\
% \hline
% Less-than & infix relation & $\varg<\varg$ & \verb|<| \\
% \hline
% Greater-than equal-to & infix relation & $\varg\geq\varg$ & \verb|\geq| \\
% \hline
% Greater-than & infix relation & $\varg>\varg$ & \verb|>| \\
% \hline
% Non empty $\nat$ & prefix name & $\nat_1$ & \verb'\nat_1' \\
% \hline
% Non empty $\num$ & prefix name & $\num_1$ & \verb'\num_1' \\
% \hline
% Multiplication & function $40$ LA & $\varg*\varg$ & \verb|*| \\
% \hline
% Integer division & function $40$ LA & $\varg\div\varg$ & \verb|\div| \\
% \hline
% Integer modulus & function $40$ LA & $\varg\mod\varg$ & \verb|\mod| \\
% \hline
%\end{tabular}
%\caption{\numtkfile symbols \smallcaption{A.2.5.4, B.7}}\label{tbl:symbol-toolkit-number}
%\end{table}
%
%Like what happened in \reltkfile, where definitions were given without markup
%directives, in \numtkfile, the successor function for natural numbers ($succ$)
%is also defined without markup directives, yet one may be familiar with its
%specialised rendering as $\ztoolkit{succ}$ (\verb|\ztoolkit{succ}|). This is slightly
%different from $first$ and $second$ from \reltkfile, as $succ$ is defined as an operator
%template in \numtkfile, hence the \verb|\varg| on its description in Table~\ref{tbl:symbol-toolkit-number}.
%
%The division symbol within the usual \LaTeX{} distribution (\emfile{fontmath.ltx}
%with font encoding \verb|OMS/cmsy/m/n| and hex value \verb|"04|) is different from the Z
%integer division symbol, which is given as a Z toolkit word (\verb|\ztoolkit{div}|) in \cztstylefile. To access
%the original definition, one should use \verb|\mathdiv| ($\mathdiv$), instead.
%
%\subsubsection{\seqtkfile}\label{sec:symbol-toolkit-seq}
%
%The \texttt{sequence\_toolkit} has \texttt{function\_toolkit} and \texttt{number\_toolkit}
%as its parents and defines range, relational iteration, set cardinality, min/max, and
%finite sequences and its operators. In Table~\ref{tbl:symbol-toolkit-seq}, we present
%its list of symbols.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|c|l|}
% \hline
% \textbf{Description} & \textbf{Role} & \textbf{Rendering} & \textbf{\LaTeX} \\
% \hline
% Number range & function $20$ LA & $\varg\upto\varg$ & \verb|\upto| \\
% \hline
% Iteration & ordinary name & $iter\varg\varg$ & \verb|iter| \\
% \hline
% Iteration & prefix function & $(\varg~^{~\varg~})$ & \verb|\varg~^{~\varg~}| \\
% \hline
% $\finset$ cardinality & prefix function & $\#\varg$ & \verb|\#| \\
% \hline
% Minimum & prefix function & $min\varg$ & \verb|min~\varg| \\
% \hline
% Maximum & prefix function & $max\varg$ & \verb|max~\varg| \\
% \hline
% Finite seq. & prefix generic & $\seq\varg$ & \verb|\seq| \\
% \hline
% Non empty $\seq$ & prefix name & $\seq_1\varg$ & \verb|\seq_1| \\
% \hline
% Injective seq. & prefix generic & $\iseq\varg$ & \verb|\iseq| \\
% \hline
% Seq. brackets & mixfix function & $\langle \listarg \rangle$ & {\footnotesize \verb|\langle \listarg \rangle|} \\
% \hline
% Concatenation & function $30$ LA & $\varg\cat\varg$ & \verb|\cat| \\
% \hline
% Seq. reverse & ordinary name & $rev\varg$ & \verb|rev~\varg| \\
% \hline
% Seq. head & ordinary name & $head\varg$ & \verb|head~\varg| \\
% \hline
% Seq. last & ordinary name & $last\varg$ & \verb|last~\varg| \\
% \hline
% Seq. tail & ordinary name & $tail\varg$ & \verb|tail~\varg| \\
% \hline
% Seq. front & ordinary name & $front\varg$ & \verb|front~\varg| \\
% \hline
% Seq. re-indexing & ordinary name & $squash\varg$ & \verb|squash~\varg| \\
% \hline
% Seq. extraction & function $45$ LA & $\varg\extract\varg$ & \verb|\extract| \\
% \hline
% Seq. filtering & function $40$ LA & $\varg\filter\varg$ & \verb|\filter| \\
% \hline
% Seq. prefix & prefix relation & $\varg\prefix\varg$ & \verb|\prefix| \\
% \hline
% Seq. suffix & prefix relation & $\varg\suffix\varg$ & \verb|\suffix| \\
% \hline
% Seq. infix & prefix relation & $\varg\infix\varg$ & \verb|\infix| \\
% \hline
% Dist. concat. & ordinary name & $\dcat$ & \verb|\dcat| \\
% \hline
%\end{tabular}
%\caption{\seqtkfile symbols \smallcaption{A.2.5.5, B.8}}\label{tbl:symbol-toolkit-seq}
%\end{table}
%
%In \seqtkfile, few ordinary names or operator templates without markup directive
%also are typeset as keywords. They are:~relation iteration ($\ztoolkit{iter}~R~i$)
%and its superscript version ($R~^{~i~}$);~minimum ($min$) and maximum ($max$) of a set of
%numbers;~sequence $rev$erse, $head$, $last$, $tail$, $front$, and $squash$;~and
%distributed concatenation ($\dcat$). It is questionable if some of them should be
%made prefix function operator templates in the Z Standard. Note that, as these are
%ordinary names, no special \LaTeX{} spacing scheme is in place. Thus, although
%not explicitly required by the CZT tools, to properly render these names,
%a hard space is required in order to separate them from their arguments (\textit{e.g.,}~``$rev~s$'', \verb|$rev~s$|).
%Otherwise, \LaTeX{} will typeset them as a single word (\textit{e.g.,}~``$rev s$'', \verb|$rev s$|).
%Again, if wanted, markup directives with corresponding \LaTeX{} macros as \verb|\ztoolkit| can be added.
%
%\subsubsection{\stdtkfile}\label{sec:symbol-toolkit-std}
%
%The \texttt{standard\_toolkit} has \texttt{sequence\_toolkit} as its parent
%and defines nothing. It is the Z section implicitly inherited if no $\SECTION$
%keyword is present within a given file. Such files have so-called ``implicit''
%sections, where the implicit section is named as the file (without its extension),
%where the \texttt{standard} \texttt{\_toolkit} is its parent~\cite[B.9]{isoz}.
%
%\section{Z-\LaTeX{} environments}\label{sec:latex-env}
%
%In Table~\ref{tbl:latex-env}, we describe all the Z-\LaTeX{} environments used to
%typeset the various Z paragraphs, such as:~Z section headers containing the section
%name and its (optional, possibly empty, list of) parents;~horizontal paragraphs like given sets,
%operator templates, free types, horizontal schemas, and unnamed conjectures;~named
%conjecture paragraphs;~axiomatic and generic axiomatic definitions;~and schema and generic
%schema definitions. In many of these paragraphs, the \verb|\where| keyword is used
%to separate the declaration part from the predicate part. The \texttt{ENDCHAR}
%is used to mark the end of all Z paragraphs within the Unicode character stream.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|l|}
% \hline
% \textbf{Description} & \textbf{Markup} & \textbf{\LaTeX} \\
% \hline
% Section header & \texttt{ZEDCHAR} & \verb|\begin{zsection}| \\
% \hline
% Horizonal paragraph & \texttt{ZEDCHAR} & \verb|\begin{zed}| \\
% \hline
% Named conjecture & \texttt{ZEDCHAR} & \verb|\begin{theorem}{thm}| \\
% \hline
% Axiomatic definition & \texttt{AXCHAR} & \verb|\begin{axdef}| \\
% \hline
% Generic axdef & \texttt{AXCHAR GENCHAR} & \verb|\begin{gendef}| \\
% \hline
% Schema definition & \texttt{SCHCHAR} & \verb|\begin{schema}{S}| \\
% \hline
% Generic schema & \texttt{SCHCHAR GENCHAR} & \verb|\begin{schema}{S}[X]| \\
% \hline
% Declaration separator & \verb'~|~', or \verb|~\mid~| & \verb|\where| \\
% \hline
% End of all Z paras & \texttt{ENDCHAR} & \verb|\end{XXX}| \\
% \hline
%\end{tabular}
%\caption{Z-\LaTeX{} environments \smallcaption{A.2.6, A.2.7}}\label{tbl:latex-env}
%\end{table}
%
%Only material within Z paragraphs and \LaTeX{} markup directives are treated by CZT tools
%as part of a formal Z specification. Insofar as tools are concerned, everything else
%(\textit{e.g.,}~plain text, \LaTeX{} comments, other \LaTeX{} commands, \textit{etc.})
%is treated as a Z narrative paragraph, which can contain arbitrary text.
%
%To illustrate these boxes, we introduce a few Z paragraphs below. They are inspired in
%Mike Spivey's guide to Z-\LaTeX{} markup (\textit{i.e.,}~\texttt{zed2e.tex}). Firstly,
%we add a series of horizonal paragraphs.
%%
%\begin{demo}
%\begin{verbatim}
%\begin{zed}
% % Hard spaces (~) are optional below. They were
% % added for (personal) aesthetic reasons.
% [Set]
% \also % small vertical space
% List ~~::=~~ leaf | const \ldata List \rdata \\
% Sch ~~==~~ [~ x, y: \nat | x > y ~] \\
% Sch2 ~~==~~ Sch \land [~ z: \num ~]
%\end{zed}
%\end{verbatim}
%\gives
%\begin{zed}
% [Set]
% \also % small vertical space
% List ::= leaf | const \ldata List \rdata \\
% Sch ~~==~~ [~ x, y: \nat | x > y ~] \\
% Sch2 ~~==~~ Sch \land [~ z: \num ~]
%\end{zed}
%\end{demo}
%%
%Next, we typeset an axiomatic definition.
%%
%\begin{demo}
%\begin{verbatim}
%\begin{axdef}
% f, g: \power~\nat \fun (\num \cross \seq~\arithmos)
%\where
%\zbreak % may not break, depends on page placement
% \forall S, T: \power~\nat | f~S \subseteq g~S @ \\
% \t1 first~(f~S) < \#~(g~S).2
%\end{axdef}
%\end{verbatim}
%\gives
%\begin{axdef}
% f, g: \power~\nat \fun (\num \cross \seq~\arithmos)
%\where
%\zbreak % may not break, depends on placement
%
% \forall S, T: \power~\nat | f~S \subseteq g~S @ \\
% \t1 \, \! \: \, first~(f~S) < \#~(g~S).2
%\end{axdef}
%\end{demo}
%After that, we have a simple vertical schema.
%%
%\begin{demo}
%\begin{verbatim}
%\begin{schema}{Test}
% x, y: \nat; S, T: \power_1~\nat
%\where
% x > y \\
% S \subset T \\
%\znewpage % certainly breaks
% x \neq y = 0
% \Also % medium vspace
% x \in S \land y \notin T
%\end{schema}
%\end{verbatim}
%\gives
%\begin{schema}{Test}
% x, y: \nat; S, T: \power_1~\nat
%\where
% x > y \\
% S \subset T \\
%\znewpage % certainly breaks
% x \neq y = 0
% \Also % medium vspace
% x \in S \land y \notin T
%\end{schema}
%\end{demo}
%%
%Below we typeset a generic axiomatic definition.
%%
%\begin{demo}
%\begin{verbatim}
%\begin{gendef}[X, Y]
% S, T: \power~(X \cross Y)
%\where
% S \subseteq T
% \ALSO % big vertical space
%
% \exists U: \power~(X \cross Y) \spot \\
% \t2 U \subset (S \cup T)
%\end{gendef}
%\end{verbatim}
%\gives
%\begin{gendef}[X, Y]
% S, T: \power~(X \cross Y)
%\where
% S \subseteq T
% \ALSO % big vertical space
%
% \exists U: \power~(X \cross Y) \spot U \subset (S \cup T)
%\end{gendef}
%\end{demo}
%%
%And finally, a generic schema.
%%
%\begin{demo}
%\begin{verbatim}
%\begin{schema}{GenTest}[X]
% a: X; b: \power~X
%\where
% a \in b
%\end{schema}
%\end{verbatim}
%\gives
%\begin{schema}{GenTest}[X]
% a: X; b: \power~X
%\where
% a \in b
%\end{schema}
%\end{demo}
%%
%For schemas without names, which are not recognised by the parser, one could
%use the \verb|\begin{plainschema}| environment.
%%
%\begin{demo}
%\begin{verbatim}
%\begin{plainschema}
% x, y: \nat
% \where
% x = y
%\end{plainschema}
%\end{verbatim}
%\gives
%\begin{plainschema}
% x, y: \nat
% \where
% x = y
%\end{plainschema}
%\end{demo}
%%
%Finally, stared versions of the usual Z environments can be used to
%typeset Z-\LaTeX, but having its text ignored by the CZT tools as a narrative paragraph.
%%
%\begin{demo}
%\begin{verbatim}
%\begin{zed*}
% [NotParsed]
%\end{zed*}
%\begin{axdef*}
% a : \arithmos
%\end{axdef*}
%\begin{gendef*}[X]
% x: X
%\end{gendef*}
%\begin{schema*}{NotParsed}[X]
% x, y: X
%\where
% x > y
%\end{schema*}
%\end{verbatim}
%\gives
%\begin{zed*}
% [NotParsed]
%\end{zed*}
%\begin{axdef*}
% a : \arithmos
%\end{axdef*}
%\begin{gendef*}[X]
% x: X
%\end{gendef*}
%\begin{schema*}{NotParsed}[X]
% x, y: X
%\where
% x > y
%\end{schema*}
%\end{demo}
%
%\newpage
%
%After we have done that, let us test trailing spaces after Z paragraph environments are not
%affecting L/R mode indentation spacing, a known problem in some old Z-\LaTeX{} style files.
%Say, let us define a new operator template as a prefix function. For that we also add,
%together with the operator definition, its (Z) \LaTeX{} markup directive and associated
%\LaTeX{} markup command as a \verb|\ztoolkit|.
%%
%\begin{demo}
%\begin{verbatim}
%% Unicode markup in markup directive is the "text" to use
%\newcommand{\test}{\ztoolkit{test}}
%
%\begin{zed}
% %%Zword \test test
% \function (\test \_)
%\end{zed}
%Now some text to see if paragraph mode indentation is right.
%\[ \forall x: \nat_1 @ x > 0 \]
%What about with math display environments?
%All seems okay.
%\end{verbatim}
%\gives
%% Unicode markup in markup directive is the "text" to use
%\newcommand{\test}{\ztoolkit{test}}
%
%\begin{zed}
% %%Zword \test test
% \function (\test \_)
%\end{zed}
%Now some text to see if paragraph mode indentation is right.
%\[ \forall x: \nat_1 @ x > 0 \]
%What about with math display environments? All seems okay.
%\end{demo}
%%
%Finally, let us test the named conjecture environment.
%%
%\begin{demo}
%\begin{verbatim}
%\begin{theorem}{Thm1}
% \forall x: \nat @ x \geq 0
%\end{theorem}
%\end{verbatim}
%\gives
%\begin{theorem}{Thm1}
% \forall x: \nat @ x \geq 0
%\end{theorem}
%\end{demo}
%%
%Unfortunately, I could not find a way to make named conjectures
%colourful, whenever colour is enabled in Z math mode.
%
%\section{Extra macros and commands from \cztstylefile}\label{sec:cztsty}
%
%There are a few extra macros the user may refer to when extending the
%\cztstylefile, or adding her own markup directives. They are summarised
%in Table~\ref{tbl:cztsty-extra}.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|}
% \hline
% \textbf{Description} & \textbf{\LaTeX} \\
% \hline
% \cztstylefile version & \verb|\fileversion| \\
% \hline
% \cztstylefile date & \verb|\filedate| \\
% \hline
% \cztstylefile description & \verb|\filedesc| \\
% \hline
% \cztstylefile file name & \verb|\cztstylefile| \\
% \hline
% Prefix operators & \verb|\zpreop{XXX}| \\
% \hline
% Posfix operators & \verb|\zpostop{XXX}| \\
% \hline
% Binary operators & \verb|\zbinop{XXX}| \\
% \hline
% Relational operators & \verb|\zrelop{XXX}| \\
% \hline
% Ordinary operators & \verb|\zordop{XXX}| \\
% \hline
% Big symbol & \verb|\zbig{XXX}| \\
% \hline
% Bigger symbol & \verb|\zBig{XXX}| \\
% \hline
% Even bigger symbol & \verb|\zBIG{XXX}| \\
% \hline
% Smaller symbol & \verb|\zSmall{XXX}| \\
% \hline
% Even smaller symbol & \verb|\zsmall{XXX}| \\
% \hline
% Partial symbol & \verb|\p{XXX}| \\
% \hline
% Finite symbol & \verb|\f{XXX}| \\
% \hline
% Block alignment env. & {\small \verb|\begin{zblock}\end{zblock}|} \\
% \hline
%\end{tabular}
%\caption{Extra \LaTeX{} macros in \smallcaption{\cztstylefile}}\label{tbl:cztsty-extra}
%\end{table}
%%
%File version, date, and description are simple strings with
%information about \cztstylefile. The various operator wrappers
%are used to tell \LaTeX{} how spaces for some particular markup
%should be treated. They follow the usual \LaTeX{} mathematical
%operators spacing rules (see~\cite[p.~525, Table~8.7]{latexcomp}).
%Some symbols can be increased or decreased relative to their base symbol.
%For instance, the symbol for schema composition ($\semi$) is the \verb|\zBig|
%version of the symbol for relational composition ($\comp$).
%Similarly, partial function spaces ($\pfun$) are just the \verb|\p|
%version of total functions ($\fun$). Finally, block alignment can be
%used so that the treatment of new line within the block adds extra
%spacing just after the new line.
%
%\section{Conclusions and acknowledgements}\label{sec:conclusions}
%
%In this document, we presented a guide to typesetting ISO Standard Z~\cite{isoz} in \LaTeX{}
%when typeset using the \cztstylefile. The document is divided to mirror the
%Standard as much as possible. This style file is the result of merging, filtering,
%and removing definitions from various other style files, such as \texttt{oz.sty},
%\texttt{soz.sty}, \texttt{zed-csp.sty}, \texttt{zed.sty}, \texttt{fuzz.sty},
%\texttt{z-eves.sty}, and so on.
%
%The main design decision behind this document follows CZT guideline that
%``what you type is what you model''. That is, the document ``as-is'' becomes
%the source Standard Z (\LaTeX) specification to be processed by tools. Other
%design decisions included:~i)~keep the style file as minimal, simple, and
%consistent as possible;~ii)~document and acknowledge macro definition choices
%and their origin (when different);~iii)~normalise definitions for
%consistency;~iv)~complete missing cases with either normative rules from
%the Standard or using common sense;~v)~keep the style file well documented,
%but not verbose;~and vi)~follow order of definitions from Z Standard document.
%
%As the \cztstylefile may be used by both language extensions and \LaTeX{} users,
%we also provided and explained a series of useful macros for \LaTeX{} rendering
%that bare no relation with the Standard or the tools. They are useful for \LaTeX{}
%typesetting only, and are explained in Section~\ref{sec:intro-cztopt}, and
%Section~\ref{sec:cztsty}.
%
%We tried to present, as exhaustively as possible, the use of every one
%of such commands with \LaTeX{} markup typeset in verbatim mode for
%clarity and reference. We summarise them all in an Appendix below.
%More details can be found at the \texttt{czt.dvi} file generated with
%the \texttt{docstrip} utility on the \texttt{czt.dtx} document from
%the CZT distribution.
%
%Finally, the author would like to thank \textit{QinetiQ Malvern} in the
%UK for its long term support for the development of formal verification
%tools here at York. Also, the work to prepare this document and its companion
%style file benefited immensely by the good work of previous package
%builders for Z, namely Sebastian Rahtz (Object Z, \emfile{oz.sty}),
%Mike Spivey (ZRM and Fuzz, \emfile{zed.sty, fuzz.sty}), Jim Davies
%(ZRM and CSP$_M$, \emfile{zed-csp.sty}), Ian Toyn (Standard Z Editor,
%\emfile{ltcadiz.sty, soz.sty}), and Mark Utting (original CZT style
%based on \emfile{oz.sty}, \emfile{czt.sty}). Moreover, I would like
%to thank all the people in the \texttt{czt-devel} mailing list for
%their helpful comments on my many questions. Finally, I need to
%thank my York colleagues Jim Woodcock and Juan Perna for many helpful
%discussions about tool design and \LaTeX{} typesetting.
%
%\section{Features left out}\label{sec:todo}
%
%There were several features left out from the various packages we got
%inspiration from which might be of good use in typesetting \LaTeX{}
%specifications, as shown below in Table~\ref{tbl:todo}.
%%
%\begin{table}[ht]
%\centering
%\begin{tabular}{|l|l|l|}
% \hline
% \textbf{Description} & \textbf{Source} & \textbf{\LaTeX} \\
% \hline
% Multiple column math mode & \texttt{oz.sty} & \verb|\begin{sidebyside}| \\
% \hline
% Comment in math mode & \texttt{oz.sty} & \verb|\comment{XXX}| \\
% \hline
% indented new lines alignment & \texttt{oz.sty} & various \\
% \hline
% Tabular alignment math mode & \texttt{zed.sty} & \verb|\begin{syntax}| \\
% \hline
% Hand written proofs & \texttt{zed.sty} & \verb|\begin{argue}| \\
% \hline
% Inference rules & \texttt{zed.sty} & \verb|\begin{infrule}| \\
% \hline
% Mechanical proof scripts & \texttt{z-eves.sty} & \verb|\begin{zproof}| \\
% \hline
% Labelled predicates & \texttt{z-eves.sty} & \verb|\Label{XXX}| \\
% \hline
% Various new line alignment & \texttt{z-eves.sty} & \verb|\+, \-, \\| \\
% \hline
%\end{tabular}
%\caption{Some \LaTeX{} macros left out from other style files}\label{tbl:todo}
%\end{table}
%\newpage % add here to avoid table in same page as reference card
%%
%Although some of them could be introduced without problem as \textit{e.g.,}
%\begin{verbatim}
% \begin{sidebyside}...\end{sidebyside}
%\end{verbatim}
%for most others the trouble is their presence within the Z-\LaTeX{} lexis.
%That is, their presence would be detected by the parser as an error, hence they were left out.
%
%Finally, note that the Z Standard does not define a toolkit for multi sets also known as bags.
%That is despite the fact most Z tools do, and the symbols are well known from Spivey's guide~\cite{zrm}.
%Eventually, we should have in CZT extra toolkits from either known sources and rigorous experiments.
* PARSER DOES NOT YET SUPPORT LOGICAL CONSTANTS
\newpage
\appendix
\section{Environments}\label{app:ref-env}
\subsection{Informal argument}
To typeset an informal argument, you write in \LaTeX\
\begin{verbatim}
\begin{argue}
S \dres (T \dres R) \\
\t1 = \id S \comp \id T \comp R \\
\t1 = \id (S \cap T) \comp R & law about $\id$ \\
\t1 = (S \cap T) \dres R.
\end{argue}
\end{verbatim}
%
which corresponds to
%
\begin{argue}
S \dres (T \dres R) \\
\t1 = \id S \comp \id T \comp R \\
\t1 = \id (S \cap T) \comp R & law about $\id$ \\
\t1 = (S \cap T) \dres R.
\end{argue}
\newpage
\section{Reference card}\label{app:ref-card}
\setcounter{secnumdepth}{2}
\zedindent=0.2\zedindent
\subsection{Special \Circus{} symbols}
\vspace*{-0.5ex}
\subsubsection{Refinement}
\vspace*{-2.5ex}
\begin{symbols}
n \circassertref S = I &\t2 \verb|n \circassertref S = I| \\
n \circassertref S \circsimulates I &\t2 \verb|n \circassertref S \circsimulates I| \\
n \circassertref S \circrefines I &\t2 \verb|n \circassertref S \circrefines I| \\
n \circassertref S \circrefines Tr~ I &\t2 \verb|n \circassertref S \circrefines Tr~ I| \\
n \circassertref S \circrefines SFl~ I &\t2 \verb|n \circassertref S \circrefines SFl~ I| \\
n \circassertref S \circrefines FlDv~ I &\t2 \verb|n \circassertref S \circrefines FlDv~ I| \\
\end{symbols}
\subsection{\Circus{} channels and name sets}
\vspace*{-0.5ex}
\begin{symbols}
\circchannel e &\t5 \verb|\circchannel e| \\
\circchannel c : T &\t5 \verb|\circchannel c : T| \\
\circchannel [X] c : X &\t5 \verb|\circchannel [X] c : X| \\
\circchannelfrom S &\t5 \verb|\circchannelfrom S| \\
\circchannelfrom [X] S[X] &\t5 \verb|\circchannelfrom [X] S[X]| \\
\circchannelset n == \lchanset c \rchanset &\t5 \verb|\circchannelset n == \lchanset c \rchanset| \\
\circchannelset n == CSRef &\t5 \verb|\circchannelset n == CS| \\
\circchannelset [X] n == CSRef &\t5 \verb|\circchannelset [X] n == CS| \\
\circnameset n == \{~ x ~\} &\t5 \verb|\circnameset n == \{~ x ~\}| \\
\circnameset n == NSRef &\t5 \verb|\circnameset n == NS|
\end{symbols}
\subsection{\Circus{} actions}
\vspace*{-0.5ex}
\subsubsection{Action definition}
\vspace*{-2.5ex}
\begin{symbols}
n \circdef A &\t4 \verb|n \circdef A| \\
n \circdef x: T \circspot A &\t4 \verb|n \circdef x: T \circspot A|
\end{symbols}
\begin{multicols}{2}
\subsubsection{Basic actions}
\vspace*{-2.5ex}
\begin{symbols}
\Skip &\t1 \verb|\Skip| \\
\Stop &\t1 \verb|\Stop| \\
\Chaos &\t1 \verb|\Chaos|
\end{symbols}
\subsubsection{Prefixing action}
\vspace*{-2.5ex}
\begin{symbols}
e \then A &\t2 \verb|e \then A| \\
c.0 \then A &\t2 \verb|c.0 \then A| \\
c!v \then A &\t2 \verb|c!v \then A|\\
c?x \then A &\t2 \verb|c?x \then A| \\
c?x!y?z \then A &\t2 \verb|c?x!y?z \then A| \\
\end{symbols}
\end{multicols}
\subsubsection{Prefixing action (extra)}
\vspace*{-2.5ex}
\begin{symbols}
c?x \prefixcolon (P)!(f~x) \then A & \\
\quad \verb|c?x \prefixcolon (P)!(f~x) \then A| & \\
c?x \prefixcolon (x > 1)!(f~x) \then A & \\
\quad \verb|c?x \prefixcolon (x > 1)!(f~x) \then A| & \\
c[\nat \cross \power~\nat]?x!(\dom~R) \then A & \\
\quad \verb|c[\nat \cross \power~\nat]?x!(\dom~R) \then A| &
\end{symbols}
\subsubsection{Unary actions}
\vspace*{-2.5ex}
\begin{symbols}
\lschexpract S \rschexpract &\t4 \verb|\lschexpract S \rschexpract| \\
\circmu X \circspot A &\t4 \verb|\circmu X \circspot A| \\
A \circhide CS &\t4 \verb|A \circhide CS| \\
\lcircguard P \rcircguard \circguard A &\t4 \verb|\lcircguard P \rcircguard \circguard A|
\end{symbols}
\subsubsection{Binary actions}
\vspace*{-2.5ex}
\begin{symbols}
A \linter NSa | NSb \rinter B &\t5 \verb'A \linter NSa | NSb \rinter B' \\
A \interleave B &\t5 \verb|A \interleave B| \\
A \lpar NSa | CS | NSb \rpar B &\t5 \verb'A \lpar NSa | CS | NSb \rpar B' \\
A \lpar CS \rpar B &\t5 \verb|A \lpar CS \rpar B| \\
A [ NSb | CSa | CSb | NSb ] B &\t5 \verb'A [ NSb | CSa | CSb | NSb ] B' \\
A [ CSa | CSb ] B &\t5 \verb'A [ CSa | CSb ] B' \\
A \intchoice B &\t5 \verb|A \intchoice B| \\
A \extchoice B &\t5 \verb|A \extchoice B| \\
A \circseq B &\t5 \verb|A \circseq B|\\
AName &\t5 \verb|AName| \\
AName(x, y) &\t5 \verb|AName(x, y)| \\
AName[new/old, x/y] &\t5 \verb|AName[new/old, x/y]|\\
\end{symbols}
\subsubsection{Replicated actions}
\vspace*{-2.5ex}
\begin{symbols}
\Interleave x: T \circspot A & \t4 \verb|\Interleave x: T \circspot A| \\
\Interleave x: T \linter NS \rinter \circspot A & \t4 \verb|\Interleave x: T \linter NS \rinter \circspot A| \\
\lpar CS \rpar x: T \circspot \lpar NS \rpar A & \t4 \verb|\lpar CS \rpar x: T \circspot \lpar NS \rpar A|\\
\lpar CS \rpar x: T \circspot A & \t4 \verb|\lpar CS \rpar x: T \circspot A|\\
\Intchoice x : T \circspot A & \t4 \verb|\Intchoice x : T \circspot A| \\
\Extchoice x : T \circspot A & \t4 \verb|\Extchoice x : T \circspot A| \\
\Semi x : T \circspot A & \t4 \verb|\Semi x : T \circspot A|
\end{symbols}
\subsubsection{Parenthesised actions}
\vspace*{-2.5ex}
\begin{symbols}
(A) &\t4 \verb|(A)| \\
(x : T \circspot A) &\t4 \verb|(x : T \circspot A)| \\
(\circval x : T \circspot A) &\t4 \verb|(\circval x : T \circspot A)| \\
(\circres x : T \circspot A) &\t4 \verb|(\circres x : T \circspot A)| \\
(\circvres x : T \circspot A) &\t4 \verb|(\circvres x : T \circspot A)| \\
(x : T \circspot A)(v) &\t4 \verb|(x : T \circspot A)(v)| \\
(\circval x : T \circspot A)(v) &\t4 \verb|(\circval x : T \circspot A)(v)| \\
(\circres x : T \circspot A)(v) &\t4 \verb|(\circres x : T \circspot A)(v)| \\
(\circvres x : T \circspot A)(v) &\t4 \verb|(\circvres x : T \circspot A)(v)| \\
(\circmu X \circspot x : T \circspot A)(v) &\t4 \verb|(\circmu X \circspot x : T \circspot A)(v)| \\
(\circmu X \circspot (x : T \circspot A))(v) &\t4 \verb|(\circmu X \circspot (x : T \circspot A))(v)| \\
(\circmu X \circspot \circval x : T \circspot A)(v) &\t4 \verb|(\circmu X \circspot \circval x : T \circspot A)(v)|\\
(\circmu X \circspot \circres x : T \circspot A)(v) &\t4 \verb|(\circmu X \circspot \circres x : T \circspot A)(v)|\\
(\circmu X \circspot \circvres x : T \circspot A)(v) &\t4 \verb|(\circmu X \circspot \circvres x : T \circspot A)(v)|
\end{symbols}
\subsection{\Circus{} command definitions}
\vspace*{-0.5ex}
\subsubsection{Guarded commands}
\vspace*{-2.5ex}
\begin{symbols}
x, y := v1, v2 &\t4 \verb|x, y := v1, v2| \\
x, y \prefixcolon [~ P, Q ~] &\t4 \verb|x, y \prefixcolon [~ P, Q ~]| \\
\prefixcolon [~ P, Q ~] &\t4 \verb|\prefixcolon [~ P, Q ~]| \\
\{~ P ~\} &\t4 \verb|\{~ P ~\}| \\
[~ P ~] &\t4 \verb|[~ P ~]| \\
\circif P \circthen A \circelse B \circfi &\t4 \verb|\circif P \circthen A \circelse B \circfi| \\
\circdo P \circthen A \circod &\t4 \verb|\circdo P \circthen A \circod| \\
\circcon X \circspot A &\t4 \verb|\circcon X \circspot A| \\
\circvar x : T \circspot A &\t4 \verb|\circvar x : T \circspot A|
\end{symbols}
\subsubsection{Parameterised commands}
\vspace*{-2.5ex}
\begin{symbols}
\circval x : T \circspot A &\t4 \verb|\circval x : T \circspot A| \\
\circres x : T \circspot A &\t4 \verb|\circres x : T \circspot A| \\
\circvres x : T \circspot A &\t4 \verb|\circvres x : T \circspot A|
\end{symbols}
\subsection{\Circus{} processes}
\vspace*{-0.5ex}
\subsubsection{Process definition}
\vspace*{-2.5ex}
\begin{symbols}
\circprocess n \circdef PD &\t4 \verb|\circprocess n \circdef PD| \\
\circprocess [X] n \circdef PD &\t4 \verb|\circprocess [X] n \circdef PD| \\
\end{symbols}
\subsubsection{Basic process}
\vspace*{-2.5ex}
\begin{symbols}
\circprocess n \circdef \circbegin \ldots BP \ldots \circend & \\
\t1 \verb|\circprocess n \circdef \circbegin \ldots BP \ldots \circend| & \\
\circstate n == S &\t4 \verb|\circstate n == S| \\
\circstate S &\t4 \verb|\circstate S|
\end{symbols}
\subsubsection{Unary processes}
\vspace*{-2.5ex}
\begin{symbols}
P \circhide CS &\t4 \verb|P \circhide CS| \\
PName &\t4 \verb|PName| \\
PName[\nat] &\t4 \verb|PName[\nat]| \\
PName(x, y) &\t4 \verb|PName(x, y)| \\
PName[\nat](x, y) &\t4 \verb|PName[\nat](x, y)| \\
PName \lcircindex x \rcircindex &\t4 \verb|PName \lcircindex x \rcircindex| \\
PName[\nat] \lcircindex x \rcircindex &\t4 \verb|PName[\nat] \lcircindex x \rcircindex| \\
PName \lcircrename c, d := e, f \rcircrename &\t4 \verb|PName \lcircrename c, d := e, f \rcircrename| \\
PName[\nat] \lcircrename c, d := e, f \rcircrename &\t4 \verb|PName[\nat] \lcircrename c, d := e, f \rcircrename|
\end{symbols}
\subsubsection{Binary processes}
\vspace*{-2.5ex}
\begin{symbols}
P \interleave Q &\t4 \verb|P \interleave Q| \\
P \lpar CS \rpar Q &\t4 \verb|P \lpar CS \rpar Q| \\
P \intchoice Q &\t4 \verb|P \intchoice Q| \\
P \extchoice Q &\t4 \verb|P \extchoice Q| \\
P \circseq Q &\t4 \verb|P \circseq Q|
\end{symbols}
\subsubsection{Parameterised and indexed processes}
\vspace*{-2.5ex}
\begin{symbols}
x : T \circspot P &\t4 \verb|x : T \circspot P| \\
x : T \circindex P &\t4 \verb|x : T \circindex P|
\end{symbols}
\subsubsection{Parenthesised processes}
\vspace*{-2.5ex}
\begin{symbols}
(P) &\t4 \verb|(P)| \\
(x : T \circspot P) &\t4 \verb|(x : T \circspot P)| \\
(x : T \circindex P) &\t4 \verb|(x : T \circindex P)| \\
(P) \lcircrename c := d \rcircrename &\t4 \verb|(P) \lcircrename c := d \rcircrename| \\
(x : T \circspot P) \lcircrename c := d \rcircrename &\t4 \verb|(x : T \circspot P) \lcircrename c := d \rcircrename | \\
(x : T \circindex P) \lcircrename c := d \rcircrename &\t4 \verb|(x : T \circindex P) \lcircrename c := d \rcircrename| \\
(x : T \circspot P)(v) &\t4 \verb|(x : T \circspot P)(v)| \\
(x : T \circindex P)(v) &\t4 \verb|(x : T \circindex P)(v)| \\
[X](x : X \circspot P)[\nat](1) &\t4 \verb|[X](x : X \circspot P)[\nat](1)| \\
[X](x : X \circindex P)[\nat](1) &\t4 \verb|[X](x : X \circindex P)[\nat](1)| \\
(\circmu X \circspot x : T \circspot P)(v) &\t4 \verb|(\circmu X \circspot x : T \circspot P)(v)| \\
(\circmu X \circspot (x : T \circspot P))(v) &\t4 \verb|(\circmu X \circspot (x : T \circspot P))(v)| \\
(\circmu X \circspot \circval x : T \circspot P)(v) &\t4 \verb|(\circmu X \circspot \circval x : T \circspot P)(v)|
\end{symbols}
\subsubsection{Replicated processes}
\vspace*{-2.5ex}
\begin{symbols}
\Interleave x: T \circspot P &\t4 \verb|\Interleave x: T \circspot P| \\
\Parallel x: T \lpar CS \rpar \circspot P &\t4 \verb|\Parallel x: T \lpar CS \rpar \circspot P| \\
\Intchoice x : T \circspot P &\t4 \verb|\Intchoice x : T \circspot P| \\
\Extchoice x : T \circspot P &\t4 \verb|\Extchoice x : T \circspot P| \\
\Semi x : T \circspot P &\t4 \verb|\Semi x : T \circspot P|
\end{symbols}
\begin{multicols}{2}
\subsection{Mathematical toolkits}
\vspace*{-0.5ex}
\subsubsection{\Circus{} prelude}
\vspace*{-3ex}
\begin{symbols}
\boolean &\t1 \verb|\boolean| \\
\universe &\t1 \verb|\universe| \\
\true &\t1 \verb|\true| \\
\false &\t1 \verb|\false|
\end{symbols}
\subsubsection{\Circus{} model checking toolkit}
\vspace*{-3ex}
\begin{symbols}
SS \gendj TT &\t1 \verb|SS \gendj TT|\\
\regions SS &\t1 \verb|\regions SS|\\
SS \dsetminus S &\t1 \verb|SS \dsetminus S|\\
SS \dcap S &\t1 \verb|SS \dcap S |
\end{symbols}
\subsubsection{\Circus{} Spivey's Z bag toolkit}
\vspace*{-2.5ex}
\begin{symbols}
\bag~X &\t1 \verb|\bag~X| \\
B \bcount n &\t1 \verb|B \bcount n| \\
n \otimes B &\t1 \verb|n \otimes B| \\
B \uplus C &\t1 \verb|B \uplus C| \\
B \uminus C &\t1 \verb|B \uminus C| \\
x \inbag B &\t1 \verb|x \inbag B| \\
B \subbageq C &\t1 \verb|B \subbageq C| \\
\lbag x, y \rbag &\t1 \verb|\lbag x, y \rbag|
\end{symbols}
\end{multicols}
%\newpage
%\bibliographystyle{plain}
%\bibliography{circus-guide}
\end{document} |
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http://www1.combinatorics.org/Volume_5/Texfiles/v5i1r6.tex | combinatorics.org | CC-MAIN-2013-20 | application/x-tex | null | crawl-data/CC-MAIN-2013-20/segments/1368699056351/warc/CC-MAIN-20130516101056-00043-ip-10-60-113-184.ec2.internal.warc.gz | 814,774,408 | 8,564 | % A LaTeX file for an 11 page document
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\markright{\sc the electronic journal of combinatorics 5 (1998)
\#R6\hfill}
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\begin{document}
\title{Permutations which are the union of an increasing and a decreasing
subsequence}
\author{M.D. Atkinson \\School of Mathematical and Computational Sciences\\
North Haugh, St Andrews, Fife KY16 9SS, UK\\
\texttt{mda@@dcs.st-and.ac.uk}
}
\date{}
\maketitle
\begin{abstract} It is shown that there are ${2n\choose
n}-\sum_{m=0}^{n-1}2^{n-m-1}{2m\choose m}$ permutations which are the union of
an increasing sequence and a decreasing sequence.
\end{abstract}
\smallskip
\centerline{1991 {\em Mathematics Subject Classification} 05A15}
\smallskip
\centerline{Submitted: December 1, 1997; Accepted: January 10, 1998}
\smallskip
\section{Introduction}
{\em Merge permutations}, permutations which are the union of two increasing
subsequences, have been
studied for many years \cite{Knuth}. It is known that they are
characterised by the property of having no decreasing subsequence of
length $3$ and that there are ${2n\choose n}/(n+1)$ such
permutations of length $n$.
Recently there has been some interest in permutations which are the
union of an increasing subsequence with a decreasing subsequence. We
call such permutations {\em skew-merged}. Stankova
\cite{Stankova} proved that a permutation is skew-merged if and only
if it has no subsequence $abcd$ ordered in the same way as $2143$ or
$3412$. In \cite{BK,KSW} the more general problem of partitioning a
permutation into given numbers of increasing and decreasing
subsequences is considered.
This paper solves the enumeration problem for skew-merged permutations.
The proof yields another proof of Stankova's result. Finally, a
corollary allows the skew-merged enumeration result to be compared
with the enumeration of merge permutations.
\section{Points and colours}
We shall consider sets of `points' $(a,b)$ in the $(x,y)$-plane. Our
point sets will always have the property that there is no duplicated
first coordinate or second coordinate. In view of that condition
there are two natural total orders on points.
If $P=(a,b)$ and $Q=(c,d)$ are points then we define
$P..Q$ if $a<c$ and $P<Q$ if $b<d$.
Two sets of points $S,T$ are said to be {\em order isomorphic} if
there is a one-to one correspondence between them which respects both
total orders. A set of $n$ points is called {\em normal} if its sets of
$x$-coordinates and $y$-coordinates are each $\{1,2,\ldots,n\}$.
It is easy to see that every set of points is order isomorphic to a
unique
normal set. Note that a point set corresponds to a poset of
dimension $2$.
Every permutation $\sigma=[s_{1},\ldots,s_{n}]$ defines and is defined
by a normal set $\{(i,s_{i})\}_{i=1}^{n}$ and we shall find it
helpful to depict a permutation by its set of points plotted in the
$(x,y)$-plane. If the permutation is skew-merged this graph looks
like Figure 1 where the increasing and decreasing subsequences are
clearly visible. In the terminology of dimension $2$ posets, an
increasing subsequence is a chain and a decreasing subsequence is an
anti-chain.
\begin{figure}[t]
\centerline{\epsffile{Figure1.epsf}}
\caption{The graph of a skew-merged permutation}
\end{figure}
There are three involutory symmetry operations $r,s,t$ on point sets and
permutations defined on the points $P=(a,b)$ of a permutation $\sigma$ of
length $n$
as follows
\begin{eqnarray*}
r(P)&=&(n+1-a,b)\\
s(P)&=&(a,n+1-b)\\
t(P)&=&(b,a)
\end{eqnarray*}
We let $r(\sigma),s(\sigma),t(\sigma)$ be the corresponding
permutations and note the following elementary result:
\begin{lemma}
\begin{enumerate}
\item[(i)]If $P,Q$ are points of the permutation
$\sigma$, then $P..Q$ if and only if $r(Q)..r(P)$ in $r(\sigma)$ if and
only if
$s(P)..s(Q)$ in $s(\sigma)$ if and only if $t(P)<t(Q)$ in $t(\sigma)$
\item[(ii)] $\sigma$ is skew-merged if and only if any or all of
$r(\sigma),s(\sigma),t(\sigma)$ are skew-merged.
\end{enumerate}
\end{lemma}
If $P$ is one of the points in the set ${\cal P}$ of points of
the permutation $\sigma$ then ${\cal P}\setminus P$ is a set of
points with no duplicate first or second coordinates and therefore it
corresponds to some permutation $\tau$ and we write
\[\tau=\sigma-P\]
This corresponds to removing one of the components of
$\sigma=[s_{1},\ldots,s_{n}]$ and relabelling and renumbering the
remaining components appropriately.
We divide the points of a permutation $\sigma$ into 5 classes Red,
Blue, Green, Yellow, and White by the following rules:
\begin{enumerate}
\item[(r)] if $P..Q..R$ and $Q<R<P$ then $P$ is red
\item[(b)] if $P..Q..R$ and $Q<P<R$ then $R$ is blue
\item[(g)] if $P..Q..R$ and $P<R<Q$ then $P$ is green
\item[(y)] if $P..Q..R$ and $R<P<Q$ then $R$ is yellow
\item[(w)] if $P$ is not red, blue, green or yellow, then $P$ is white
\end{enumerate}
There is a more convenient way of expressing these conditions. If
$P,Q,R$ are any $3$ points and $[a,b,c]$ is any permutation of $1,2,3$
then we write $PQR\sim abc$ if $P..Q..R$ and, with respect
to ``$<$'', $P,Q,R$ are ordered in the same way as $a,b,c$. The
premises of (r), (b), (g), (y) above are then $PQR\sim 312$, $PQR\sim
213$, $PQR\sim 132$, $PQR\sim 231$ respectively. We also adopt this
language for sets of $4$ or more points. For example, if $P,Q,R,S$
are $4$ points such that $P..Q..R..S$ and $Q<P<S<R$ we would write
$PQRS\sim 2143$.
The aim of this section is to prove the following
\begin{theorem}\label{graph} The graph of a skew-merged permutation
has the form shown in Figure 2. In this figure the vertical and
horizontal lines have the obvious separation meaning; for example, if
$R$ and $W$ are red and white points we would have $R..W$ and $W<R$.
Furthermore, the green and blue points are increasing, the red and yellow
points are decreasing, and the white points are either increasing or
decreasing.
\end{theorem}
\begin{figure}[t]
\centerline{\epsffile{Figure2.epsf}}
\caption{Disposition of colours}
\end{figure}
In proving this theorem we shall use only the property of skew-merged
permutations that they avoid $3412$ and $2143$; in our `point'
terminology this means that there do not exist $4$ points $P,Q,R,S$
with $PQRS\sim 3412$ or $PQRS\sim 2143$. Since a
permutation with a graph of the above form is evidently skew-merged we
obtain another proof of Stankova's result.
\begin{lemma}\label{symmetry} Let $P$ be a point of a permutation $\sigma$.
Then
\begin{enumerate}
\item[(i)] $P$ is red in $\sigma$ if and only if $r(P)$ is blue in
$r(\sigma)$
\item [(ii)] $P$ is green in $\sigma$ if and only if $r(P)$ is yellow in
$r(\sigma)$
\item [(iii)] $P$ is red in $\sigma$ if and only if $s(P)$ is green in
$s(\sigma)$
\item [(iv)] $P$ is blue in $\sigma$ if and only if $s(P)$ is yellow in
$s(\sigma)$
\item [(v)] $P$ is red in $\sigma$ if and only if $t(P)$ is yellow in
$t(\sigma)$
\item [(vi)] $P$ is green in $\sigma$ if and only if $t(P)$ is green in
$t(\sigma)$
\item [(vii)] $P$ is blue in $\sigma$ if and only if $t(P)$ is blue in
$t(\sigma)$
\item [(viii)] $P$ is white in $\sigma$ if and only if any or all of
$r(P),s(P),t(P)$ are white in $r(\sigma),s(\sigma),t(\sigma)$
respectively.
\end{enumerate}
\end{lemma}
\begin{proof} All the statements follow easily from the definitions.
For example, suppose that $P$ is red in $\sigma$ so that there are
points $Q,R$ with $PQR\sim 312$. Then $r(P),r(Q),r(R)$ are points of
$r(\sigma)$ with $r(R)r(Q)r(P)\sim 213$; therefore
$r(P)$ is blue in $r(\sigma)$
\end{proof}
\begin{lemma} In a skew-merged permutation $\sigma$ every point has
exactly one colour.
\end{lemma}
\begin{proof} Suppose first that a point $P$ is coloured both red and green
in $\sigma$. Then there are points $Q,R,S,T$ with
$P..Q..R$, $P..S..T$, $Q<R<P$, and $P<T<S$.
In particular, $Q<R<P<T<S$. If $Q..S$ then $PQST\sim 2143$, a
contradiction, and if $S..Q$ then $PSQR\sim 3412$, also a contradiction.
It now follows from the symmetry operations and Lemma \ref{symmetry}
that $P$ cannot be
coloured both blue and yellow (or $r(P)$ would be both red and green
in $r(\sigma)$); nor can $P$ be both yellow and green (or $t(P)$ would
be red and green in $t(\sigma)$); nor can $P$ be both red and blue (or
$t(s(P))$ would be both red and green in $t(s(\sigma))$).
Suppose next that $P$ is both red and yellow in $\sigma$. Then there
are points $Q,R,S,T$ with
$P..Q..R, S..T..P, Q<R<P$, and $P<S<T$
But then $STQR\sim 3412$. The remaining
possibility, that $P$ is both blue and green, can also be excluded by
symmetry ($r(P)$ would be red and yellow in $r(\sigma)$).
\end{proof}
\begin{lemma} If $\sigma$ is skew-merged and $P..S$ are points of
$\sigma$ then
\begin{enumerate}
\item[(i)] if $P,S$ are both red or both yellow then $S<P$
\item[(ii)] if $P,S$ are both blue or both green then $P<S$
\end{enumerate}
\end{lemma}
\begin{proof} Suppose that $P,S$ are both red. Then there are points
$Q,R,T,U$ with
$P..Q..R$, $S..T..U$, $Q<R<P$, and $T<U<S$.
Suppose, for a contradiction, that $P<S$. Then, since $P..S..T..U$
and $PSTU\not\sim 3412$ we must have $P<U$. Also, $S..Q$
is impossible, otherwise $PSQR\sim 3412$. But now $PQSU\sim 2143$ which
is the required contradiction.
Suppose next that $P,S$ are both yellow. Then, as $P..S$, $t(P)<t(S)$
in $t(\sigma)$ and $t(P),t(S)$ are red in $t(\sigma)$. But we have
just seen that this means $t(S)..t(P)$ in $t(\sigma)$ and therefore
$S<P$ in $\sigma$.
For part (ii), if $P,S$ are both blue (green) then $s(P),s(S)$ are
both red (yellow) and $s(P)..s(S)$. So, by part (i), $s(S)<s(P)$ and
therefore $P<S$.
\end{proof}
\begin{lemma} Suppose that $\sigma$ is skew-merged with points
$R,B,G,Y$ coloured red, blue, green, yellow respectively. Then
\begin{tabular}{llll}
(i) $R..Y$ & (ii) $G..B$ & (iii) $G<B$ & (iv) $Y<R$\\
(v) $R..B$ & (vi) $G..Y$ & (vii) $G<R$ & (viii) $Y<B$\\
\end{tabular}
%\[ R..Y, G..B, G<B, Y<R\]
\end{lemma}
\begin{proof} Since $R$ is red and $Y$ is yellow there are points
$S,T,U,V$ with
$R..S..T$, $U..V..Y$, $S<T<R$, $Y<U<V$.
Suppose that $Y..R$. Then either
\begin{enumerate}
\item[(a)] $T<U$ in which case $UVST\sim 3412$, or
\item[(b)] $U<T$ in which case $UYRT\sim 2143$
\end{enumerate}
This contradiction proves that $R..Y$.
Symmetry arguments prove relations (ii), (iii), and (iv). Thus, since
$s(G)$ is red and $s(B)$ is yellow in $s(\sigma)$, we have
$s(G)..s(B)$ and therefore $G..B$. Next, since
$t(R),t(B),t(G),t(Y)$ are yellow, blue, green, red respectively, we
have $t(Y)..t(R)$ and $t(G)..t(B)$ in $t(\sigma)$ and so $Y<R$ and
$G<B$ in $\sigma$.
To prove part (v) we consider two points $X, Z$ such that $XZB\sim 213$
which exist since $B$ is blue.
Suppose that $B..R$. Then either
\begin{enumerate}
\item[(a)] $T<X$ in which case $XBST\sim 3412$, or
\item[(b)] $X<T$ in which case $XZRT\sim 2143$
\end{enumerate}
This contradiction proves that $R..B$ and, as before,
symmetry arguments justify the other relations.
%$s(G),s(Y)$ are red, blue respectively in $s(\sigma)$ so
%$s(G)..s(Y)$ and G..Y
%$t(R),t(B),t(G),t(Y) are yellow, blue, green, red in $t(\sigma)$
%respectively so $t(Y)..t(B)$ and $t(G)..t(R)$; so $Y<B$ and $G<R$
\end{proof}
These lemmas have proved that the disposition of the red, blue, green,
and yellow points in the graph of a skew-merged permutation $\sigma$
is as given in Theorem \ref{graph}.
The next two lemmas show that the white points are disposed as claimed.
\begin{lemma} If $R,B,G,Y$ are points of colour red, blue, green,
yellow in a permutation, and $P$ is a white point, then
\begin{tabular}{llll}
(i) $R..P$ & (ii) $G..P$ & (iii) $P..B$ & (iv) $P..Y$\\
(v) $G<P$ & (vi) $Y<P$ & (vii) $P<R$ & (viii) $P<B$\\
\end{tabular}
\end{lemma}
\begin{proof} Since $R$ is red there are two points $S,T$ with
$RST\sim 312$. If $P..R$ then either $T<P$ in which case
$PST\sim 312$ so $P$ would be red, or $P<T$ in which case
$PRT\sim 132$ and $P$ would be green. All the other
statements follow by symmetry.
\end{proof}
To complete the proof of Theorem \ref{graph} we have
\begin{lemma} The white points of a
permutation are either increasing or decreasing.
\end{lemma}
\begin{proof} From the definitions of red, blue, green, yellow every
triple $A..B..C$ of white points must satisfy $ABC\sim 123$ or
$ABC\sim 321$. Since every triple is either increasing or decreasing
the lemma follows.
\end{proof}
\section{Enumeration}
In this section we use Theorem \ref{graph} to derive the following
theorem.
\begin{theorem} \label{main} The number of skew-merged permutations of
length $n$
is
\[{2n\choose n}-\sum_{m=0}^{n-1}2^{n-m-1}{2m\choose m}\]
\end{theorem}
To prove this we enumerate the skew-merged permutations according to their
number of white points. Let $t_{i}(n)$ denote the number of
skew-merged permutations of length $n$ with exactly $i$ white points.
\begin{lemma} \label{eq1} $\sum_{i=0}^{n}(i+1)t_{i}(n)={2n \choose n}$
\end{lemma}
\begin{proof} Suppose that $\sigma$ is skew-merged and let
$\alpha,\beta$ be a pair of increasing and decreasing subsequences
whose union is $\sigma$. Let ${\cal A},{\cal B}$ be the sets of points of
$\alpha,\beta$. Consider a red point $R$ and let $S,T$ be the
corresponding points satisfying $R..S..T$ and $S<T<R$. Suppose that
$R\in {\cal A}$. Then, since $R..S$ and $S<R$, $S$ cannot also belong
to ${\cal A}$. Therefore $S\in {\cal B}$ and, similarly, $T\in {\cal
B}$. However, $S..T$ and $S<T$ and this is a contradiction.
Therefore all red points belong to ${\cal B}$. By a similar argument
all yellow points belong to ${\cal B}$ also, and all blue and green
points belong to ${\cal A}$.
Suppose that $\sigma$ has $i$ white points which, without loss in
generality, we shall suppose are increasing. Then at most one of the
white points can belong to ${\cal B}$. It follows that there are at most
$i+1$ possibilities for the pair $(\alpha,\beta)$ and, by Theorem
\ref{graph}, all of these possibilities do indeed yield a
pair of increasing and decreasing subsequences whose union is $\sigma$.
The left-hand side of the equation in the lemma therefore counts the
number of increasing, decreasing pairs $(\alpha,\beta)$ whose union is
a skew-merged permutation. However, we can count these in another
way. If $|\alpha|=r$ we may choose the first components of the
points of ${\cal A}$
in ${n\choose r}$ ways and, independently, the second components in
${n\choose r}$ ways also. So the number of $(\alpha,\beta)$ pairs is
therefore
\[\sum_{r=0}^{n}{n\choose r}^{2}={2n \choose n}\]
\end{proof}
\begin{lemma} \label{eq2} $\sum_{i=1}^{n}t_{i}(n)={2n-2 \choose n-1}$
\end{lemma}
\begin{proof} The left-hand side of the equation in the lemma is the
number of skew-merged permutations with at least one white point.
These permutations are exactly those which are the union of an
increasing subsequence $\alpha$ and a decreasing subsequence $\beta$
that have a common point. In such a permutation ($\sigma$, say) $\alpha$ is a
maximal increasing subsequence and $\beta$ is a maximal decreasing
subsequence. It follows that the pair of Young tableaux which
correspond, in the Robinson-Schensted correspondence, to $\sigma$ are
shaped like
the one shown in Figure 3.
\begin{figure}[t]
\centerline{\epsffile{Figure3.epsf}}
\caption{Young tableau}
\end{figure}
Conversely, every such pair of Young tableaux corresponds to a
skew-merged permutation with at least one white point. By the hook
formula \cite{Knuth} \S5.1.4, the number of such tableaux with
exactly $r$ cells in the first row is
\[\frac{n!}{n(r-1)!(n-r)!}={n-1 \choose r-1}\]
and hence the number of tableaux pairs is
\[\sum_{r=1}^{n}{n-1 \choose r-1}^{2}={2n-2 \choose n-1}\]
\end{proof}
Note that skew-merged permutations with no white points
correspond to Young tableau pairs
with a shape similar to Figure 3 but with a cell in the $(2,2)$
position. Unfortunately, not every Young tableau pair of this form
gives a skew-merged permutation.
\begin{lemma} \label{i>2} For all $i>2$, $t_{i}(n)=t_{i-1}(n-1)$
\end{lemma}
\begin{proof} Suppose that $\sigma$ is a permutation of length $n$
with $i$ white points. Then, by Theorem \ref{graph}, $\sigma-W$ is
independent of $W$ provided
only that $W$ is a white point of $\sigma$. Since $i>2$ the deletion
of $W$ cannot change the colour of any point of $\sigma$ and so
$\sigma-W$ has $i-1$ white points.
Conversely, suppose that $\tau$ is of length $n-1$ with $i-1$ white
points which, since $i>2$ are either increasing or decreasing but not
both. Then there is a unique permutation $\sigma$ with $i$ white
points such that $\sigma-W=\tau$ for some white point $W$ of
$\sigma$. This one to one correspondence proves the lemma.
\end{proof}
\begin{lemma} Suppose that $\sigma$ is a skew-merged permutation with
either one white point (or no white points). Then there exist points
$R,B,G,Y$ with colours red, blue, green, yellow such that either
\begin{enumerate}
\item[(i)]$RGWBY\sim 41352$ (or $RGBY\sim 3142$) and $W$ is the only point
(or there is no point) in $G..W..B$, or
\item[(ii)]$GRWYB\sim 25314$ (or $GRYB\sim 2413$) and $W$ is the only point
(or there is no point) in $R..W..Y$
\end{enumerate}
\end{lemma}
\begin{proof} Choose red, blue, green, yellow points $R,B,G,Y$ so that
$R,G$ are largest of their colour with respect to ``$..$'' and $B,Y$
are smallest of their colour (equivalently, $G,Y$ are largest of
their colour under ``$<$'' and $R,B$ are smallest of their colour).
According to Theorem \ref{graph}, $R, B, G, Y$ are ordered
under ``$..$'' as $R..G..B..Y$, $R..G..Y..B$, $G..R..B..Y$, or
$G..R..Y..B$. Similarly there are four possible ways in which they
can be ordered under ``$<$''.
Consider the ordering $R..G..Y..B$. Since $G$ is green there are
points $P,Q$ with $G..P..Q$ and $G<Q<P$. Not both $P,Q$ can be blue
since the blue points are increasing, and nor, by hypothesis, can
both be white. Further, neither can be red or green since $R..G$ and
$R$ and $G$ are the maximal red and green points under ``$..$''. Thus at
least one of them, $P$ say is yellow. Since the yellow points are
decreasing and $Y$ is the smallest yellow point under ``$..$'' we
have $Y..P$ or $Y=P$, and so $G<P\leq Y$. By a similar argument based
on the condition that $Y$ is yellow we can deduce that $Y<G$, a
contradiction. Thus $R..G..Y..B$ is impossible.
The symmetry conditions now also exclude the possibilities
$G..R..B..Y$ and also $G<Y<B<R$, and $Y<G<R<B$.
Suppose next that $R..G..B..Y$ and $Y<G<B<R$. Since $G$ is green
there are points $G..P..Q$ with $G<Q<P$. Since $Y<G$, $P$ and $Q$
cannot be yellow and so they must be white or blue with not both
white; it follows that $P<Q$, a contradiction. Finally, symmetry
rules out the case $G..R..Y..B$, $G<Y<R<B$.
The remaining cases are
\begin{enumerate}
\item[(i)] $R..G..B..Y$ and $G<Y<R<B$
\item[(ii)] $G..R..Y..B$ and $Y<G<B<R$
\end{enumerate}
In case (i) a single white point $W$ must, by Theorem \ref{graph},
satisfy $G..W..B$ and $Y<W<R$ so that $RGWBY\sim 41352$ giving the
first alternative of the lemma. Case (ii) gives the second
alternative.
\end{proof}
\begin{lemma} \label{i=0} $t_{1}(n)=t_{0}(n-1)$
\end{lemma}
\begin{proof} Suppose $\sigma$ is of length $n$ and has exactly one
white point. Let $R,B,G,Y$ be the points guaranteed by the previous
lemma. If we remove the single white point we are left with $4$
points such that
\begin{enumerate}
\item[(i)] $RGBY\sim 3142$ with no point between $G..B$ or
\item[(ii)] $GRYB\sim 2413$ with no point between $R..Y$
\end{enumerate}
These points remain coloured as they were ($GBY\sim 132$ ensures $G$ is
green, $RGB\sim 213$ ensures $B$ is blue etc) and so we have obtained a
sequence with no white points. Conversely, the last lemma shows that
a sequence without white points has this form for
some $R,B,G,Y$. If
we insert a point $W$ satisfying $G..W..B$ and $Y<W<R$ in case (i) and
$R..W..B$ and $G<W<B$ in case (ii) this point must be white (as it is
part of both an increasing sequence and a decreasing sequence in a
skew-merged decomposition, see the proof of Lemma \ref{eq1}). We therefore
have a one-to-one
correspondence proving that $t_{1}(n)=t_{0}(n-1)$.
\end{proof}
Since $t_{n-i}(n)$ is independent of $n$ if $i\leq n-2$ (Lemma
\ref{i>2}) we may set $b_{i}=t_{n-i}(n)$ for all $i\leq n-2$. Also,
we let $a_{n}=t_{0}(n)$ from which, by Lemma \ref{i=0}, we find
$t_{1}(n)=t_{0}(n-1)=a_{n-1}$. Hence the number of skew-merged
permutations is
\begin{eqnarray*}
s_{n}&=&t_{0}(n)+t_{1}(n)+\ldots +t_{n}(n)\\
&=&a_{n}+a_{n-1}+b_{n-2}+\ldots +b_{0}
\end{eqnarray*}
Rewriting Lemmas \ref{eq1} and \ref{eq2} we obtain
\begin{eqnarray*}
a_{n}+2a_{n-1}+3b_{n-2}+4b_{n-3}+\ldots +(n+1)b_{0}&=&{2n\choose n}\\
a_{n-1}+b_{n-2}+b_{n-3}+\ldots +b_{0}={2n-2\choose n-1}
\end{eqnarray*}
These equations are easily solved. Differencing reduces them to a
pair of low order inhomogeneous linear recurrence equations to which the
method of
generating functions may be applied. We obtain the generating
function
\[\sum_{n=0}^{\infty}s_{n}x^{n}=\frac{(1-3x)}{(1-2x)\sqrt{1-4x}}\]
from which we find
\[s_{n}={2n\choose n}-\sum_{m=0}^{n-1}2^{n-m-1}{2m\choose m}\]
proving Theorem \ref{main}. Finally, we have an asymptotic result.
\begin{corollary} \[\frac{s_{n}}{{2n\choose n}}\rightarrow 1/2\mbox{ as }
n\rightarrow\infty\]
\end{corollary}
\begin{proof} From Theorem \ref{main} it follows that
\[s_{n}=2s_{n-1}+\frac{n-2}{n}{2n-2\choose n-1}\]
In this equation we divide through by ${2n\choose n}$, and put
$r_{n}=s_{n}/{2n\choose n}$ to obtain
\[r_{n}=\frac{n}{2n-1}r_{n-1}+\frac{n-2}{4n-2}\] and take the limit as
$n\rightarrow\infty$.
\end{proof}
\noindent {\bf Acknowledgement} I thank Dominic Tulley for several
useful observations during the course of this work.
\begin{thebibliography}{99}
\bibitem{BK} A. Brandst\"adt, D. Kratsch: On partitions of
permutations into increasing and decreasing subsequences. Elektron.
Informationsverarb. Kybernet. 22 (1986), 263--273
\bibitem{KSW}A.E. K\'{e}zdy, H.S. Snevily, C. Wang: Partitioning
permutations into increasing and decreasing subsequences, J. Combinatorial
Theory A 74 (1996), 353--359.
\bibitem{Knuth} D.E. Knuth: Sorting and Searching (The Art of Computer
Programming volume 3), Addison Wesley, Reading, MA, 1973.
\bibitem{Stankova} Z. E. Stankova: Forbidden subsequences, Discrete Math. 132
(1994), 291--316
\end{thebibliography}
\end{document}
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\title{Articles from “Canenero”}
\date{1994–1997}
\author{Various Authors}
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\chapter{Translator’s Introduction}
\emph{Canenero} was a weekly anarchist publication that came out in Italy between the end of 1994 and the beginning of 1997 with one break. This was when the Marini investigation against anarchists began to bear its rotten fruit in an attempt to imprison dozens of anarchists on charges of “subversive association” or membership in an “armed gang”.\footnote{Venomous Butterfly Publications has published a pamphlet of material dealing specifically with this investigation and trial called simply \emph{The Marini Trial}.} One of the ideas behind \emph{Canenero} was to provide a means for ongoing communication and discussion in the face of this repressive operation of the state. A substantial portion of the material in the paper dealt with the situation and the various anarchist responses to it.
But the editors of \emph{Canenero }were not willing to allow the repressive activity of the state to define the limits of the discussion in the paper they published, so along with information and analysis of that specific situation other significant questions and idea were raised in its pages. Thus, within its pages one could find pointed, but brief, theoretical articles, social and historical analyses and bitingly witty looks at the weeks news.
Of course, as is appropriate for a weekly publication, most of the articles are specific to the time they were written, intended for immediate use in the heat of the situation that was going on. But there were enough articles of more general interest that I considered it worth my while to translate a number of them for publication in this form. I have already mad some of this material available in \emph{More, Much More}, a collection of writings by Massimo Passamani whose ideas I find particularly thought-provoking, and \emph{The Fullness of a Struggle Without Adjectives}, texts originally intended to stimulate a discussion about armed struggle groups that appeared in the last few issues of \emph{Canenero}.
In this booklet, I have collected a number of articles that I find particularly stimulating. I am certainly not in agreement with every word here. But I have found all of it to be a stimulus to deepening my own thinking on the sorts of questions raised. If, for example, Mario Cacciuco’s description of relationships between people as that of “spheres that bounce of each other” and his consequent rejection of the very idea of love and friendship seem rather bleak to me, this is precisely why his article provokes me to examine the nature of everyday relationships more closely, particularly those that we call “love” and “friendship”. In fact, one of the things that stands out for me in these articles is the way in which they are able to raise significant questions, often about matters that we take for granted, in so few words.
I have chosen to print the material in chronological order. The first article was an introduction to the project and the last was the editors’ explanation for bringing the project to a close. In this last piece, the problems that confront any anarchist publishing project are made clear. As anarchists, hopefully, we do not publish just in order to have something to do. There has to be a purpose that relates to our broader life project of revolt. If we don’t want to be leaders or evangelists carrying a supposed revolutionary gospel to whatever imaginary “masses”, then it seems to me that the idea of developing relationships of affinity and complicity in which significant discussion plays a central part would be a primary reason for publishing. Without this, publishing seems to be a meaningless spewing forth of words playing into the degradation of language that this society imposes through its own one-way “communication”. And real discussion is not a mere taking of positions and defending them from the fortress of our various ideologies. It has to be the real encounter between various and conflicting ideas.
If, ultimately, the editors of \emph{Canenero} did not feel that it stimulated the sort of discussion they desired, it is my hope that in publishing these articles in English, discussions may be stimulated here. There is a lot to think about in these brief writings. Perhaps it will stir something up.
\begin{quote}
\emph{Wolfi Landstreicher}\forcelinebreak
\emph{February 2006}
\end{quote}
\chapter{Vagabond Destruction}
\textbf{Canenero.}
One word alongside another. A sound that is lost in the continuous deafening noise that they still call language. A word different from others. A hiss in the midst of shouts. A sigh from which to move in search of new meanings in world where everything has been \emph{said}.
A word \emph{against} others, an \emph{against} that is other with respect to words, that doesn’t inhabit the space of the opposition between concepts, but that of the silence that precedes and accompanies it.
A word, finally, that doesn’t refer to itself, but that causes us to sense that region in which, in the silence where thought can move freely, the meaning of our singularity and the desire for revolt against all that suffocates it grow.
A paper for all those who, in this civilization of collective identity and reciprocal belonging, want to affirm their nature as “strangers everywhere”, as refractories against every fatherland (the “entire world” included).
\textbf{Vagabond} like the thought of the cynics, the Greek philosophers who in their scorn toward the regal condition of a philosophy addressed to power symbolized themselves with the image of the dog (\emph{Kýon}, in Greek), as a sign of refusal of hierarchy, social obligation and the supposed necessity for laws. Repaid, as is fitting for all free spirits, with censure and mystification. In our language — that is passed off as neutral but cannot hide its christian nature — “cynicism” has become synonymous with voluptuous indifference to the suffering of others. Thus, the police of ideas which travels through the centuries underground has gotten rid of what utterly did not give a damn for gods or laws.
So that the desire to be outside does not became resigned mutilation, but arms itself , but arms itself against every form of authority and exploitation.
So that one passes from the Power of dialogue (with which one thinks everything can be resolved) and from the dialogue of Power (that invites everyone to reasonable negotiation) to a feeling of radical hostility toward the existent, to the destruction of every structure that alienates, exploits, programs and regiments the lives of individuals. The black of the dog (this animal which is general associated with the idea of submission, of servile meekness) is precisely the desire to come out from the herd of voluntary servitude and open to the joy of rebellion. Not the black in which all cows are equal (even if it is in their being \emph{against} or \emph{outside}), but rather that in which the boundaries between destruction and creation, between extreme defense of oneself and the construction of relationships of mutuality with others, disappear.
A paper — to piece together a mosaic of thousands of possible meanings — of vagabond destruction, meaning by this the possibility of passing to the attack against state and domination in all its manifestations without pledging allegiance, to use a well-known expression, to any flag or organization.
As individuals, always, even where the unshakeable desire for the other leads us to choose the path of union.
\chapter{The Technique of Certainty \emph{by Marco Beaco}}
\begin{quote}
\emph{“I was frightened to find myself}\forcelinebreak
\emph{in the void, I myself a void.}\forcelinebreak
\emph{I felt like I was suffocating, }\forcelinebreak
\emph{considering and feeling}\forcelinebreak
\emph{that everything is void, }\forcelinebreak
\emph{solid void.”}
— \emph{Giacomo Leopardi}
\end{quote}
The metaphor of “mental illness” dispossesses the individual of whatever is most unique and personal in her way of life, in his method of perceiving reality and herself in it; this is one of the most dangerous attacks against the singular, because through it the individual is always brought back to the social, the collective, the only “healthy” dimension in existence.
The behavioral norms that regulate the human mass become absolute, the “deviant” act that follows a different logic is tolerated only when stripped of its peculiar “meaning”, of the particular “rationality” that underlies it. Reasons connect only to collective acts, which can be brought back, if not to the codes of the dominant culture, to those of various ethnic, antagonist and criminal subcultures that exist. The sharing of meanings, symbols and interpretations of reality thus appears as the best antidote to madness.
Thus if one who suddenly kills his family is a lunatic, or better, a “monster”, one who sets fire to a refuge for foreigners appears as a xenophobe (at most, from the method, a bit hasty, but still within reason) and one who slaughters in the situation of a declared war is nothing but a “good soldier”.
Thus, according to the classifying generalization that makes them all alike, expropriating them of their lived singularity, lunatics are “ dangerous to society”. Truthfully, one can only agree with this, certainly not because of the supposed and pretextual aggressiveness and violence attributed to those who suffer psychiatric diagnosis (the psychiatrists and educators of every sort are undoubtedly much more dangerous), but because they have violated, knowingly or not, the essentially quantitative codes that constitute normality. What is surprising is that after long years of domestication there is anybody who does not respond to cultural stimuli, if not quite automatically, at least in a highly predictable manner. Unpredictability is the source of the greatest anxiety for every society and its guardians, since it is often the quality of the individual; no motive, no value, no purpose that is socially comprehensible, only an individual logic, necessarily abnormal.
Defense from this danger is entrusted to the proclamations of science. In other words, the “unhealthy” gesture, the creator of which is not responsible, remains as a consequence of an external misfortune that could strike and give rise to thousands of people like him. The mechanism is therefore well contrived, a gesture deprived of meaning, of an underlying will, becomes innocuous, and it is easy to neutralize it, along with its creator, behind the alibi, which is “social” as well, of the cure.
The psychiatric diagnosis comes down on the individual like an axe, amputating her language, his meaning, her life paths; it claims to eliminate them as irrational, senseless; the psychiatrist behaves before them with the liquidating attitude of one who transforms the experiences of life into malfunctions of the psyche, the emotions into a malignant tumor to be removed.
Psychiatrists, as technicians of certainty, are the most efficient police of the social order. Reality, like the meaning of existence, has clear and unequivocal boundaries for these priests in white shirts; their mission: to “return” those who have gotten lost venturing onto the winding paths of nonsense “to their senses”.
If the police are limited, as is claimed, to beating you, the psychiatrist demands to hear you say, “Thank you, I am well now” as well.
The focal point in the discussion is not in the four walls and the bars of the asylum, nor in the electroshock and constraint beds, nor in bad as opposed to good psychiatry, but in “psychiatric thought” itself, in the form of thinking of anyone who addresses himself to different subjects with the clinical eye of diagnosis, always looking for the symptoms of a pathology in them, in order to annul the difference with a “therapy” that brings them back to being more like us.
If the real purpose of the “new places” of psychiatry was that of stimulating creativity, individual growth, liberating communication and developing the capacity for relations, they would not be “psychiatric” or “therapeutic\Slash{}rehabilitative” places, but probably ideal places for everyone, places of freedom. The problem is that these places are nothing but ghettoes in which one does not find individuals interacting on the level of mutuality, but rather two “categories” of persons in asymmetrical positions: the professionals and the clients , the healthy and the diseased, those who help and those who are helped; in these places, the healthy try to persuade the diseased that what they did and thought up to that time was wrong, or rather “unhealthy”, and through the “joyful” method of the encounter group, of dance, theatre and music\dots{}lead them toward the binaries of normality.
The “autonomy” and “self-realization” about which these democratic operators flap their tongues are exclusively their own and, to them, it is necessary to conform in order to be able to leave the healing enclosure. Psychiatric medicine itself, as analgesic (anesthetic) for the mind, is the sign of the attempt to block every development, every pathway however painful at times, that an individual puts into action as a reaction to that which oppresses her. Without mystifying this process, this moment of “crisis”, that is not necessarily a pathway to liberation, the fact of the matter remains that the answer of power is generalized narcosis, collective stupefaction, that renders us static and tranquil, anchored to our placid misery.
\chapter{The Obscure Clarity of Words \emph{by Alfredo M. Bonanno}}
One who writes, perhaps even more than one who speaks, is called to clarify, to bring light. A problem is posed — the problem of something the one who writes should be concerned with since otherwise his respect would be deprived of meaning. This problem is illuminated by the use of words, by a specific use, capable of being organized within the shell of certain rules and in view of a perspective to be attained.
One who reads, perhaps even more than one who listens, does not catch the individual words but their meaning within the sphere of the rules that organize them and the perspective that they affirm they desire to reach.
However weak the meaning of what one writes (or says) might be, the one who reads (or listens) does not carry out the role of passive receiver. The relationship often takes on the appearance of conflict, within which two different universes clash with each other. But this clash is not based on any active intention on the part of the one writing (or speaking), and a passive one on the part of the one hearing (or reading). The two movements are only contrary only in appearance. The reader participates in the effort of the writer and the writer in that of the reader. Even if the two movements are separated from each other, they are not so in the fact, which has not been much considered, that the one who writes is always (simultaneously) a reader of the text she is writing, and the one who reads is also himself (simultaneously) the writer of the text that he is reading.
Here two errors are committed. The first is that in which one encounters the writer who thinks that by reading while he writes, he understands what she is writing, and doesn’t realize that often her comprehension is not due to the clarity of the text, but to the reader-writer connection that reaches the highest level in the precise act of organizing word according to a project. The second is that which happens to the reader who, imagining himself in the act of writing the text that he is reading, refuses to accept word choices that are unthinkable to her, and doesn’t realize that often the incomprehensibility of the text that she reads is not so much due to a lack of clarity as to the fact that he would have written it differently.
The thing that seems to escape this binary relationship is the third element, i.e., the topic that is being discussed. The reality examined with words is a barrier that, on the one hand, may help to organize the words in a certain way (accepting some and rejecting others), but, on the other hand, carries out a distorting process with regards to the employment of the accepted words. No word is neutral, but each one, being organized within concepts, contributes to transferring into the reader (and in still different ways, into the listener) a conception of the diffraction of the reality examined (of which one writes or speaks).
Thus, no word is clear or obscure as such; there is no possibility of definitively casting a pool of light on reality, clarifying it once and for all. Once the word is detached from the reality to which it refers and thus from the choice that the writer (or speaker) made on the basis of the suggestions of the reality examined, it no longer means anything. It vanishes, and its possibility for being anything, a means for thought or action, an element for uniting or dividing human beings, vanishes with it. The dictionary is like a warehouse of words. They are lined up there on the shelves, some used continuously, others only rarely, all equally available, but only a few of them able to be coordinated together according to the intentions of the one who chooses and the suggestions of the reality she wants to dress up in words.
It’s just that we can understand words, and thus decide if each of them is “clear” for us, on the condition of being conversant with this operation of dressing up. There are not words on one side, dead objects shut up in dictionaries, and reality on the other side where individual objects exist beside words that are also themselves objects, but all in a haphazard manner, without relationship. Flows of meaning exist, i.e., working procedures in the course of which the elements of reality (that here, for convenience, we can call “objects”). They receive meaning through us, putting on linguistic clothes. There is no chair separate from the word that means it, and the different words to which different languages have recourse reconfirm this endeavor as a flow of meaning, proposing philological nuances that through the history of the millennia often cause incredible routes, extraordinary adventures, to emerge.
Dressing reality is thus the primary activity of the human being, the condition for acting and itself an action, the essential form of action, insofar as thought itself is the process of clothing reality (a fact that is not much considered). What could we “do” without the capacity of “reading” reality. We would find ourselves before a dark mass of foreboding and fear. The most important question is not that of the greatest clarity (easiest words, dressed most modestly, linearity in the correspondences), but rather, and maybe contrarily, that of the greatest richness (different words contrasting the commonplaces, dressed in the liveliest colors, uncertainty of correspondence). The word is also enchantment, marvel, joyous invention, fancy, evocation of something other, not the seal of the already seen, the confirmation of one’s certainties.
The aim of speaking and writing is therefore not that of “clarifying”, but of “enriching” reality, of inviting the unexpected, the unpredictable. The one who communicates has no obligation to give us prescriptions for repair, panaceas for our fears, confirmations of our knowledge, but can even feel free to suggest difficult routes, to make uncertainty and danger flare.
And whoever wants to feel safe in his house is free to stop his reading or cover her ears.
\chapter{The Reverse Road \emph{by Alfredo M. Bonanno}}
Times of doubt and uncertainty have arrived. New and old fears spur the search for guarantees. In the market where human affairs are managed, new models of comfort are briskly haggled over. Madonnas weep, politicians make promises; everywhere war and misery, savagery and horror are rife, rendering us now unable to even feel outrage, let alone to rebel.
People have been quick to accustom themselves to blood. They scarcely smell the odor of the massacres, and every day something new and more incredible awaits them: Tokyo, Gaza, the changeless Bosnia, Burundi and still more places, remote, distant, and yet nearby. What they ask is to be left out of it. Being informed, even of the smallest household massacres, those of Saturday evening for example, which pattern dozens of deaths weekly, with no other purpose than that of knowing in order to forget.
In a world that is revealed to be increasingly weak in real meanings, in motivations that give content to life, in projects worthy of being lived, people give away freedom for specters that are in easy reach, specters that come out from the studios of power. Religion is one of these specters. Not any religion whatsoever, objectified in distant and crusty practices, governed by priests and simulations lacking sense, but a religion that can reach the emptiness of their minds, filling it with the future, that is with hope.
I know well that a religion of this sort does not exist, but there are many people who try hard to exploit the need that exists for it. Against this need, the rationalist claims made by Cartesian veterans of the victories through which they have conquered, and destroyed, the world are worthless. Their chatter of scientific certainty no longer charms anyone. No one, except for a small group of relentless intellectuals, is willing to believe in the capacity of science to solve all the problems of humanity, to give an answer to all the questions concerning the eternal fear of the unknown.
Now it occurs that even we anarchists allow ourselves to take on this extraordinary laceration, to which we should instead remain extraneous, if we want to find a path for action, a path capable of making us understand reality, and thus putting us in a position to transform it. Even we don’t quite know what to do.
On the one hand, we withdraw, horrified, in the face of always delirious and disgusting manifestations of faith in all its forms. Sometimes we have pity for the man that stoops, that suffers under pain, and thus accepts the image of the incredible specter, and hopes, and continues to suffer and hope. But we can have no more than this for him. Immediately afterwards, contempt takes over, and with contempt, refusal, distancing, rejection.
On the other hand, still looking carefully, what do we find? We find an equally contemptible misery, but one that knows how to dress itself well, with the garments of culture and fine speech. This latter misery believes in science and in the world that can be systematized, in the world that is moving toward its highest destinies. But it closes its eyes and covers its ears, waiting for the storm to die down, unconscious and pitiless in the face of the pain and misery of the rest of the world. This universe of specialists and respectable people also disgusts us, in many ways as much as or more than the other, that at least had ignorance and the passionate force of emotion on its side.
But us, what do we do? We don’t beat our chests, nor do we go around with a slide-rule in our pockets. We believe neither in god nor in science. Neither miracle workers nor wise men in white coats interest us. But are we then really beyond all this?
I don’t think so. Merely reflecting, we realize that we are still children of our times. But, being anarchists, we are so in a reversed manner. We naively think that it is enough to overturn the errors of others like a glove in order to have the beautiful truth dished out in shovelfuls. It isn’t so.
Therefore, refusing that of the obscure which exists in the times in which we live, we set our feet on the certainties of a different science, indeed, a science that we must build completely ourselves, from top to bottom, but that like the other one will be based on reason and will. And, at the same time, refusing what there is of the functional and utilitarian in science, we go in search of sensations and emotions, intuitions and desires from which we expect answers for all questions, answers that cannot come to the extent that these stimuli crumble in our excessively rough hands.
Thus, we reel, now in one direction, now in another. We don’t have the ideological certainties of a few decades ago, but the critiques we have developed are still not able to tell us with the least bit of trustworthiness what to do. Thinking that we are in a position to act beyond every value, every foundation, in the moment that we ask ourselves what to do, we don’t know how to give ourselves a certain answer.
In other times, we had less fear of ridicule, we were more obtuse in our stubborn and coherent doing, less worried about matters of style. I fear that we are too much in love with subtleties, with nuances. Continuing along this path, we might even lose the meaning of the whole that has never been lacking, the projectual sense that made us feel rooted in reality, part of something in the course of transformation, not mere monads, brilliant in our own light, but dark to each other.
\chapter{Streamlined Production \emph{by Alfredo M. Bonanno}}
Among the various characteristics of the last several years, the failure of global automation in the factories (understood in strict sense) must be pointed out, a failure caused by the failure of the prospects and, if you will, the dreams of mass production.
The meeting between the telematic and traditional fixed production (harsh assembly lines later automated up to a certain point with the introduction of robots) has not developed toward a perfecting of the lines of automation. This is not due to problems of a technical nature, but due to problems of an economic nature and of the market. The threshold of saturation for technologies that can replace manual labor has not been exceeded; on the contrary there are always new possibilities opening in this direction. Rather, the strategies of mass production have been surpassed, and have thus come to have little importance for the economic model of maximum profit.
The flexibility that the telematic guaranteed and has steadily made possible in the phase of the rise of post-industrial transformation at a certain point caused such profound changes in the order of the market, and thus of the demand, as to render the opening that the telematic itself had made possible or rather put within reach useless. Thus, the flexibility and ease of production is moved from the sphere of the factory into the sphere of the market, causing a standstill in the telematic development of automation, and a reflourishing of new prospects for an extremely diversified demand that was unthinkable until a few years ago.
If one reads the shareholders’ reports of some of the great industries, it becomes clear that automation is only sustainable at increasing costs that quickly be come anti-economical. Only the prospect of social disorder of a great intensity could still drive the financially burdensome path of global automation.
For this reason, the reduction of the costs of production is now entrusted not only to the cost of labor, as has occurred in the past several years as a consequence of massive telematic replacement, but also to a rational management of so-called productive redundancy. In short, a ruthless analysis of waste, from whatever point of view, and, first of all, from the perspective of production times. In this way, by a variety of means, productive pressure is exercised once again on the producer in flesh and blood, dismantling the ideology of containment on the basis of which an easing of the conditions of suffering and exploitation that have always been characteristic of wage labor was credited to telematic technology.
The reduction of waste thus becomes the new aim of streamlined production, in its time based on the flexibility of labor already consolidated and the productive potentiality guaranteed by the telematic coupling as its starting point. And this reduction of waste falls entirely on the back of the producer. In fact, the mathematical analysis realized through complex systems already in widespread use in the major industries can easily solve the technical problems of contractors, which is to say, those relative to the combination of raw materials and machinery, in view of maintenance. But the solution to these problems would remain a marginal matter to production as a whole if the use of production time were not also placed under a regime of control.
Thus, the old taylorism comes back into fashion, though now it is filtered through the new psychological and computing technologies. The comprehensive flexibility of large industry is based on a sectoral flexibility of various components, as well as on the flexibility of the small manufacturers that peripherally support the productive unity of command. Work time is thus the basic unity for the new production; its control, without waste but also without stupidly repressive irritations, remains the indispensable connection between the old and new productive models.
These new forms of control have a pervasive nature. In other words, they tend to penetrate into the mentality of the individual producer, to create general psychological conditions so that little by little external control through a timetable of production is replaced by self-control and self-regulation of productive times and rhythms as a function of the choice of objectives, which is still determined by the bodies that manage productive unity. But these decisions might later be submitted to a democratic decision from below, asking the opinion of individuals employed in the various production units with the aim of implanting the process of self-management.
We are speaking of “suitable synchronism”, not realized once and for all, but dealt with time and again, for single productive periods or specific production campaigns and programs, with the aim of creating a convergence of interest of interests between workers and employers, a convergence to be realized not only on the technical terrain of production, but also on the indirect plane of solicitation of some claim to the demand, which is to say, on the plane of the market.
In fact, it is really in the market that two movements within the new productive flexibility are joined together. The old factory looked to itself as the center of the productive world and its structures as the stable element from which to start in order to conquer ever-expanding sections of consumption to satisfy. This would indirectly have to produce a worker-centered ideology, managed through guidance by a party of the sort called proletarian. The decline of this ideological-practical perspective could not be more evident today, not so much because of the collapse of real socialism, and all the direct and indirect consequences that followed from this and continue to grow out of it, but in reality, due to the productive changes which we are discussing. There is thus no longer a distinction between the rigidity of production and the chaotic and unpredictable flexibility of the market. Both these aspects are now brought back under the common denominator of variability and streamlining. The greater ability to penetrate into consumption, whether foreseeing and soliciting it or restraining it, allows the old chaos of the market to be transformed into an acceptable, if not entirely predictable, flexibility. At the same time, the old rigidity of the world of production has change into the new productive speed. These two movements are coming together in a new unifying dimension on which the economic and social domination of tomorrow will be built.
\chapter{A Eulogy to Opinion \emph{by Alfredo M. Bonanno}}
Opinion is a vast merchandise that everyone possesses and uses. Its production involves a large portion of the economy, and its consumption takes up much of people’s time. Its main characteristic is clarity.
We hasten to point out that there is no such thing as an unclear opinion. Everything is either yes or no. Different levels of thought or doubt, contradiction and painful confessions of uncertainty are foreign to it. Hence the great strength that opinion gives to those who use it and consume it in making decisions or impose it on the decisions of others.>
In a world that is moving at high speed toward positive\Slash{}negative binary logic, from red button to black, this reduction is an important factor in the development of civil cohabitation itself. What would become of our future if we were to continue to support ourselves on the unresolved cruelty of doubt? How could we be used? How could we produce?
Clarity emerges when the possibility of real choice is reduced. Only those with clear ideas know what to do. But ideas are never clear, so there are those on the scene who clarify them for us, by supplying simple comprehensible instruments: not arguments but quizzes, not studies but alternative binaries. Simply day and night, no sunset or dawn. Thus they solicit us to pronounce ourselves in favor of this or that. They do not show us the various facets of the problem, merely a highly simplified construction. It is a simple affair to pronounce ourselves in favor of a yes or no, but this simplicity hides complexity instead of attempting to understand and explain it. No complexity, correctly comprehended, can in fact be explained except by referring to other complexities. There is no such thing as a solution to be encountered. Joys of the intellect and of the heart are cancelled by binary propositions, and are replaced with the utility of “correct” decisions.
But no one is stupid enough to believe that the world rests on two logical positive and negative binaries. Surely there is a place for understanding, a place where ideas again take over and knowledge regains lost ground. Therefore, the desire arises to delegate this all to others who seem to hold the answers to the elaboration of complexity because they suggest simple solutions to us. They portray this elaboration as something that has taken place elsewhere and therefore represent themselves as witnesses and depositories of science.
So the circle closes. The simplifiers present themselves as those who guarantee the validity of the opinions asked, and their continual correct production in binary form. They seem to be wary of the fact that once opinion — this manipulation of clarity — has destroyed all capacity to understand the intricate tissue that underlies it, the complex unfoldings of the problems of conscience, the fevered activity of symbols and meanings, references and institutions, it destroys the connective tissues of differences. It annihilates them in the binary universe of codification where reality only seems to have two possible solutions, the light on or the light off. The model sums up reality, cancels the nuances of the latter and displays it in pre-wrapped formulas ready for consumption. Life projects no longer exist. Instead symbols take the place of desires and duplicate dreams, making them dreams twice over.
The unlimited amount of information potentially available to us does not allow us to go beyond the sphere of opinion. Just as most of the goods in a market where every possible, useless variety of the same product does not mean wealth and abundance but merely mercantile waste, an increase in information does not produce a qualitative growth in opinion. It does not produce any real capacity to decide what is true or false, good or bad, beautiful or ugly. It merely reduces one of these aspects to a systematic representation of a dominant model.
In reality, there is no good on the one side or bad on the other. Rather there is a whole range of conditions, cases, situations, theories and practices which only a capacity to understand can grasp, a capacity to use the intellect with the necessary presence of sensibility and intuition. Culture is not a mass of information, but a living and often contradictory system, through which we gain knowledge of the world and ourselves. This is a process which is at times painful and hardly ever satisfying, with which we realize the relationships which constitute our life and our capacity to live.
By canceling out all of these nuances, we again find ourselves with a statistical curve in our hands, an illusory course of events produced by a mathematical model, not a fractured and overwhelming reality,
Opinion provides us with certainty on the one hand, but on the other it impoverishes us and deprives us of the capacity to struggle, because we end up convinced that the world is simpler than it is. This is totally in the interest of those who control us. A mass of satisfied subjects convinced that science is on their side, that is what they need in order to realize the projects of domination in the future.
\chapter{The Specter That Reassures as it Kills \emph{by Alfredo M. Bonanno}}
All authority comes from god, said the apostle, and he was right. But not in the sense of offering legitimacy to authority due to its divine origins, but in the sense of the impossibility of authority in the absence of the idea of god.
The very concept of supreme security, of something beyond the parts, and thence also the concept of the sacred and untouchable function of government and justice, comes from the idea of god. The “immutable”, dreamed up by people as protection against the fear of the future and of the unknown that are hidden in the mists within which this last is enveloped, is god, the specter that reassures as it kills.
But in order for authority to be exercised in the sphere of human matters, that is to say, to become state and government, to insinuate itself into every fiber from which society is composed, it doesn’t just need the support provided by the idea of god; it also needs force, real force, suitable to the times and conditions of the conflict with all those who, because they suffer the authority and pay the consequences for it in terms of repression and restrictions of freedom, oppose it.
And this force is made up of weapons and armies, governments and parliaments, cops and spies, priests and laws, judges and professors, in short, of the entire apparatus at the service of power without which it remains a dead letter.
But the force is based on wealth, that is on the possibility of accumulating money or of securing oneself control of the flows in which the circulation of money is realized. With the development of commerce and industry, passing from ancient times through those of the industrial revolution up into the epoch in which we live, at the beginning of the third millennium, when wealth bends into a spasmodic essentialization of itself, passing from the old and static form of accumulation to the new, dynamic form of flux and high velocity circulation, its function as the basis of authority has not changed.
So we can say that an authority without wealth is a contradiction. All the tyrants of the past, like all the political people of today that have managed and continue to manage the public thing, have had immense quantity of wealth in their hands.
A poor person can never exercise authority, which is why an authority lacking the wealth that could form it into institutions and guarantee it as such in the concrete exercise of its functions tends to weaken into authoritativeness, thence into something quite different. A poor person may be authoritative for her knowledge, his coherence, her accuracy, but he would never constitute an authority.
This is why they Church, aware of its historical task, passed through a theoretical and practical torment that lasted three centuries and carried it from the initial critique of wealth (carried out in all the texts of primitive christianity), to the justification and acceptance of wealth, and the time in which this voyage was completed corresponds precisely to the philosophical maturity of St. Augustine and the conquest of power through Constantine, nearly simultaneous events.
This is why in the encyclical \emph{Evangelium Vitae}, the pope confuses us, limiting himself to quoting only half of the citation and thus misappropriating it to justify (or rather establish) the “gospel of life” as he calls it.
The fable speaks of a young man who approached the Master and asked him what to do in order to obtain eternal life, and the Master told him to observe the commandments, going through a list that begins with “Thou shalt not kill”. It is from this that the pope draws his cue to establish the “gospel of life” carrying out an act of confusion rather than reasoning. In other words, the mixing of the order of the commandments put into play here in the gospel text, which places “Thou shalt not kill” in the first place, is the proof of the will to defend life as the primary essential good. But the text of the story in Matthew continues. In fact, it tells us that the rich young man responds by saying that he had followed all of these commandments, but wanted to know something more, and the response was quite precise: “If you want to be perfect, go, sell all that you own, give it to the poor and you will have treasure in heaven, then come follow me.” As if to say that wealth were an obstacle and that the Church cannot accept it.
But to refuse wealth would have meant that the Church would condemn itself to exclusion from power and invalidate its participation in earthly authority that it always considered as a provisional passage toward the total conquest of power and the domination of the world, realized, of course, for the greater glory of god.
This is why it never accepted this refusal, but always persecuted with violence and death, with fire and sword, all those who supported the necessity for the Church to be poor in order to speak to the poor and not converse with the rich over the topics that interest them relating to the management of power or to mutually contend with them for power. And this is why the Church has always considered all those who support the refusal of wealth and all those who intend to fight against the rich of the earth to be heretics.
If it had taken the concrete force that comes from wealth and from commerce with the powerful, the Church would have removed the possibility of acting as the practical foundation of authority from the idea of god and would have forced authority to become blatant tyranny, clear and visible to everyone.
\chapter{Continue to Speak to Me \emph{by Alfredo M. Bonanno}}
Facing the understanding of oneself and others, unsuspected aspects of awareness are frequently discovered. When we approach a problem about which we know little or a person whom we have never met before, we feel a sense of panic (or of pleasure, a subtle difference that is never completely clear). Will we manage to get to the bottom of it? We ask ourselves. And the answer is not always positive.
Most of the time we look at the “stranger” with suspicion, the suspicion that always exists of the difference that is not yet codified. Where will this “stranger” take us? Certainly toward new things, and what will these be like? They might be good or bad, but they upset our balance, the sleep (and dreams) that we often create between one harsh awakening and the next.
From this, it is all the more necessary not to reveal ourselves. Since our personal world, our own world, is what is at stake when we risk venturing into the unknown, we are disposed to defend it to the death; its boundaries harden and propose an interpretive scheme. The “stranger”, whether person or problem, is thus catalogued in the sphere of our schemes; we dilute the form in the structure, suppress it by force, expecting the other to conform itself to our needs. Thus, after having killed it in the ritual manner that we can and within the limits of our capacity as killers, we reproduce it, adapted to our aims, even continuing to feed our inclusive desires, dreams and sleep.
In this way, some of us, and certainly not the worst, wrap ourselves up in the cocoon of codification, judging or suspending judgment without being aware of it. But in daily practice, this suspension is always expressed in trusting the other to remain in the sphere of our perspective by itself, without our needing to do it violence. In these cases, the common sense of ridicule helps in finding tunings that would otherwise be revealed as nonexistent.
Please, no shouting your contempt for order; it is sufficient that you show me that your way of living follows a lively, dancing qualitative logic and not the obligation of the routine of quiet and the code. But show me this with logical, accurate connections. Please, tell me that you are crazy, just like me, but say it with clarity. Please, speak to me of the terrible shudder of darkness, but tell me about it in the light of the sun, so that I can see it, here and now, represented in the distinct speech in which I was educated.
Encourage me with your chants about destruction — they are sweet lullabies for my heart’s needs — but speak of them in an orderly manner so that I can understand them and thanks to them understand what destruction is. In short, I want the words to reach me in a well-organized form. Alas, if you start to shout, I will no longer listen. It is good to destroy, but with the order that logic imposes. Otherwise we go into the chaos of the unrepeatable, where everything fades into the incomprehensible. Yes, granted, something could reach me even through the perplexing shouts of an Algerian marketplace on a feast day, but I am not used to that life, to that unpredictable and fleeting dance, to the unforeseen appearance of the “stranger”. It is necessary that you put the code of habit before me, that the language be made full of immediateness. Speak to me, I beg you, so that the word becomes the umbilical cord between me and the world of what has already happened, so that nothing presents itself as being thrown suddenly into the dark dimension of chaos.
Speak to me of love, of your love, for me, of every possible love, even of the most remote and difficult to understand, of the violence that goes at it from the hip, of violence and death, but, in order to let me see it with the eyes of the mind, speak to me about it imprisoned, captured in the slimy and corruptible web of words. Speak to me about it carefully, I beg you, so that my heart can bear its repercussions. Then I will make a habit of it. And really, since you have spoken to me about it, the love will become familiar to me and I will carry it with me everywhere, like one carries a knife in one’s pocket, a heavy object that furnishes security. As to that other possibility, as to the “stranger” that presented herself suddenly before my eyes, like a thief in the night, no longer beckoning to me there, it abandons the high howl that could still speak to me in the night.
Speak to me of the future society, of anarchy, that in which you and I believe, describe its conditions of uncertainty to me, the unpredictability of relations between human beings finally freed of every constraint; with your calm, persuasive words, tell me of the ferment of the passions that break loose, the hatred and the desire for destruction that don’t disappear from one day to the next, the fear and the blood that don’t stop spreading and flowing in the veins of a society that is finally different from every nightmare of the past. Tell me, I beg you, but do it in a way that does not frighten me, Speak to me about it in an orderly manner, speak to me about what we do, you and I, and the others, and the comrades, and those who were never comrades, but who come to understand from one moment to the next, all together, building, a little here, a little there, bit by bit, while everything within life, I mean true life, begins to flourish again. But speak to me about it with intelligible logic. Don’t shout into my ear that which shouts within you, frightening me. Keep it to yourself. Keep the difficulty of coordinating your needs and ideas with mine to yourself. Keep the indomitable strength to yourself that leads you far from any acceptance of my will, your own being irrepressibly hostile to all codification just like mine, after all. Not telling me all these things, you would stop frightening me.
I beg you, don’t give me anything more to worry about.
\chapter{Anarchism and Criticism of the Existent \emph{by Benedetto Gallucci}}
In a historical context like the one in which we live (the collapse of ideological dogmas, institutional certainties, etc.) it is a matter of fact that more and more people are beginning to show an interest in anarchism and to take libertarian ideas into consideration. Anarchist groups and circles and libertarian collectives are growing.
At this point, I don’t think it would be untimely to talk about the difference between the individual comrade who discovers an anarchist awareness and therefore begins to spread her anarchist ideas and the classical militant of a political organization. As anarchists, we are focused on the critique of the existence that surrounds us, but we don’t forget to take time for individual self-criticism that serves to make us keep our feet quite firmly on the ground. But self-criticism is lacking among political militants, and this inevitably leads them to set themselves up on a pedestal of arrogance and presumption. By self-criticism, I mean the individual process of self-analysis that is a part of the life of every libertarian, through which they constantly bring their way of thinking, acting, speaking and relating with others into question.
It isn’t a question of merely examining one’s character or temperament. On the contrary, it’s a question of driving out all the shit that Power and the Church (as well as the current everyday consumer society) shoves into us from the moment we’re born. Certain internal mechanisms with which we were shaped from a most tender age are quite difficult to destroy even when one has the lucidity to recognize that they are in clear conflict with libertarian principles. One always tends to think, “after all, I am made this way\dots{}” It is safe to say that it is a bit humiliating to discover people who speak of self-determination, anarchy and revolution who are totally incapable of carrying out an internal revolution that is necessary for destroying authoritarianism in whatever form it manifests itself.
For every future collective project of liberation, an individual voyage to grasp hold of the awareness of anarchist ideas is essential, a project that cannot be separated from a profound critique of the pathogenic germs of Power present in everyone of us.
\chapter{A Yellow Rose \emph{by Alfredo M. Bonanno}}
But have we truly finished interpreting the world? I did not realize that anyone was transforming it. The absolutely “other” event does not stand out on the horizon, whereas the mechanisms of the market are organizing themselves on the old codes and reproduce themselves, justifying poverty and wealth, the absurd polarizations of “the world goes this way”.
In \emph{A Yellow Rose}, Borges makes us see how the poet Marino, prince of fine speech, seventeenth century Italian master of human letters, realized at the point of death that speaking (or doing, which is really the same thing) as reproduction and mirror of the world, as grand interpretive picture, is not possible. He concludes more modestly with doing (and thus also speaking) as excess, as superfluous addition to a composition that is already complete, even if, for us, it is unwelcome and intolerable.
Thought and action, like this and that, are never simply projected, i.e., they don’t have a meaning “merely” as a function of what they contribute to determining or what one could foresee them as determining. First of all, they are a previous history, i.e., they are themselves events, significant in their sort of autonomy, full of meaning and, thence, carriers of the marking that human activity has attached to them.
In other words, they are characterized messages, pieces in motion of the humans that have thought and done them, as thoughts and actions. As such, they have no neat counterparts in the goal that they intend to achieve, i.e., they are not exhausted in the purposes that have apparently determined them. The study of this “difference” leads directly to the interior of the absolutely “other”.
If we think and act with the sole aim of adapting ourselves to reality, maybe wildly tooting our own horn to make ourselves better heard, and more distant, we don’t have time for nuances, for the thing added in excess of which I am speaking here. We produce what is necessary because the world goes forward with out contributions as well, and the rules of the market impose the codes of this production on us. They tell us (along broad, but sufficiently clear, lines) what to do so as to never come out below, \emph{or above}, what is required for the project to be realized. And when we fail in the capitulation that is required of us, we feel precisely that we have failed, we are failures, and we look at our inefficient hands and weep despondently.
Perhaps we will have to weep hotter tears when success has come precisely through the great capacity for adapting what we do to the goals to be reached. Perhaps precisely in this instance, that the increasingly intense efficiency of modern techniques suggests to us every day, we have supplied our little contribution to the great constructions of power. And this even when the project assumed the particulars of revolution, of the subversion of institutions and values, customs and traditions.
In this case, in small and big things, we are set up as suppliers of the future executioner, we have concluded our efforts in the perfection of what we had thought. A greater number of final details that correspond with the starting hypothesis is always seen as a higher degree of success. Goals have been achieved, finish lines crossed, hopes satisfied. Now the people have their free rules, old tyrannies are dead, new freedoms are engraved on shiny new tablets. We can present the bill. We are the liberators: we are the creators of the project and its details. We have incubated high social meaning the way a peacock egg is incubated, and now we witness the shining of the sun’s golden feathers.
The force of the goal to achieve has killed the initial character of action and thought. And that character was the adherence of to the concrete activity of the one who thought and acted, a manifestation of strength that wanted to leave its sign, to affirm itself in the world, to transform the world, not with the mark of subordination to something external, but with its own exuberance, with the excess that this very thinking and acting produce. The concern of the one who acts and thinks, and who makes of her thought and action a single thing, is thus not that of finding a measure outside himself, in the efficiency with which the project has been realized, in the completeness of the result, but is rather that of finding within the project itself, which was and remains a moment of doing and thinking, all the superabundance of the absolutely “other”. What does this mean?
It means not waiting for the goals to give reasons to the choices, ideas and means in order to act. Not waiting for practical authorization or moral foundation to arrive from the outside, from others or from what one hopes to obtain. If the project is not clear within us, if we are therefore not willing to incur the risks that our ideas and actions entail, we cannot expect a mere positive result to furnish us with what we lack. By accepting this conception, we present ourselves as creditors; we want a concrete result but only for ourselves, precisely because we have always been aware of that initial lack and have always gone in search of a completeness.
If, however, we are sure of what we think and of the reasons that move us to act, we are complete from the start. And if we are complete, we can make a gift of ourselves to the other, we can make a gift of ourselves to the objective we want to achieve. And this gift of ourselves will appear immediately for what it is: the exchange of a gift between ourselves and the other, between ourselves and the reality that stands before us, unknown but desired, that we want to transform. Our gift is not remedial, it doesn’t equalize, it doesn’t bring justice, it doesn’t smooth out faults. It destroys and creates, adds the immeasurable excess beyond which all calculation becomes impossible. It fills our hearts beyond any economic calculation.
\chapter{The Persistent Refusal of Paradise \emph{by Penelope Nin}}
It is rumored that we (a “we” not well-defined whose lack of definition suits the rumor-mongers) have nothing to do with anarchism, being in reality nihilists disguised for the purpose of penetrating into the sanctuary of anarchy with bad intentions. It is noted that one who takes up the task of guarding the temple ends up seeing thieves everywhere, and maybe the hour has come to quiet “our” troubled detractors.
First of all, they must explain what they mean by nihilism. Personally, I view anyone who extols the joys of nihilism to me with suspicion because I consider nihilism, as the substantiation of nothing, to be a deception. When the incompleteness of all is cultivated with a feeling of fullness, it is difficult to resist the temptation to replace the old absolute with its most abstract moment in which nothing is immediately transformed into all and is therefore totalized. Ultimately, nihilism seems to me to be a crafty form of reasoning, that drives the whole structure of knowledge into the darkness of Nothingness only to receive, through this spectacular, radical negation, still more of the light of the All.
But probably the rumored “nihilism” consists of something much simpler, that is, of a supposed absence of proposals. In other words, one is nihilistic when one persistently refuses to promise a future earthly paradise, to foresee its functioning, to study its organization, to praise its perfection. One is nihilistic when, instead of taking and valuing all the moments of relative freedom offered by this society, one radically negates it, preferring the drastic conclusion that none of it is worth saving. Finally, one is nihilistic when, instead of proposing something constructive, one’s activity comes down to an “ obsessive exultation of the destruction of this world.” If this is the argument, it is, indeed a meager one.
To begin, anarchism — the Idea — is one thing, and the anarchist movement — the ensemble of men and women who support this Idea — is another. It makes no sense to me to say of the Idea what in reality only a few anarchists assert. The Idea of anarchism is the absolute incompatibility between freedom and authority. From this it follows that one can enjoy total freedom in the complete absence of Power. Because Power exists and has no intention of disappearing voluntarily, it will be necessary indeed to create a way to eliminate it. Correct me if I’m mistaken.
I don’t understand why such a premise, which no anarchist “nihilist” has ever dreamed of denying and suppressing, must lead necessarily to postulating new social regulations. I don’t understand why, in order to “be part” of the anarchist movement, one must first undergo a doctoral examination in the architecture of the new world, and why it isn’t enough to love freedom and hate every form of authority with all that entails. All this is not only absurd from the theoretical point of view, but also false from the historical point of view (and the anarchist rumor-mongers show so much fervor for History). One of the points about which Malatesta and Galleani clashed regularly was precisely the question of whether it was necessary to plan what would be created after the revolution or not. Malatesta argued that anarchists must begin immediately to develop ideas of how to organize social life because it doesn’t allow for interruption; Galleani, on the other hand, argued that the task of anarchists was the destruction of this society, and that future generations that are immune to the logic of domination will figure out how to rebuild. In spite of these differences, Malatesta did not accuse Galleani of being nihilist. To make such an accusation would have been gratuitous because their difference was only over the constructive aspect of the question; they agreed completely about the destructive aspect. Though this is omitted by many of his exegetes, Malatesta was, indeed, an insurrectionalist, a confirmed supporter of a violent insurrection capable of demolishing the state.
Today, however, one merely needs to point out that anyone who holds power does not give up their privileges voluntarily and draw the due conclusions to be accused of nihilism. Within the anarchist movement, as everywhere, times change. Whereas once the debate among anarchists dealt with the way of conceiving the revolution, today it seems that all discussion centers around the way to avoid it. What other purpose could all these disquisitions on self-government, libertarian municipalism, or the blessed utopia of good sense have? It is clear that once one rejects the insurrectional project as such, the destructive hypothesis begins to assume frightful contours. What was only an error to Malatesta — limiting oneself to the demolition of the social order — for many present-day anarchists represents a horror.
When pious souls hear the bark of a dog, they always think that a ferocious wolf is coming. For them the blowing of the wind becomes an approaching tornado. In the same way, to anyone who has entrusted the task of transforming the world to persuasion alone, the word destruction is upsetting to the mind, evoking painful and unpleasant images. These things make a bad impression on the people who, if they are to be converted and finally flock into the ranks of reason, must have a religion that promises an Eden of peace and brotherhood. Whether it deals with paradise, nirvana or anarchy is of little importance. And anyone who dares to place such a religion into question cannot be thought of as simply a non-believer. In the course of things, such a person must be presented as a dangerous blasphemer.
And this is why “we” (but who is this “we”?) are called “nihilists”. But the nihilism in all this, what is the point?
\chapter{Prisoners of a Single World \emph{by Gruppo Anarchico Insurrezionalista “E. Malatesta”}}
\begin{quote}
“\emph{The fact is that the state would not be so pernicious if those who wanted to were able to ignore it and live their lives in their own way together with those with whom they get along. But it has invaded every function of social life, standing over all the activities of our lives and we are even prevented from defending ourselves when we are attacked.}\forcelinebreak
“\emph{It is necessary to submit to it or bring it down.”}
— \emph{Errico Malatesta}
\end{quote}
If we were not deeply dissatisfied with this world, we would not write on this paper and you would not read this article. It is therefore useless to waste further words to confirm our aversion to Power and its manifestations. Rather, what seems useful to us is the attempt to determine whether a revolt that is not openly and resolutely against the state and power is possible.
The question should not seem odd. In fact, there are those who see in the struggle against the state nothing but a further confirmation of the extent to which it has penetrated into us, managing to determine our actions — even if only in the negative. With its cumbersome presence, the state would distract us from that which should be our true objective: living life our way. If we think of taking down the state, of obstructing it, of fighting it, we don’t have the time to reflect on what we want to do ourselves. Rather than trying to realize our dreams here and now, we follow the state wherever it goes, becoming its shadow and putting off the realization of our projects to infinity. In a frenzy to be antagonist, to be against, we end up no longer being protagonist, in favor of something. Thus, if we want to be ourselves, we should cease to oppose ourselves to the state and start to consider it not with hostility, but with indifference. Rather than giving ourselves to trying to destroy its world — the world of authority — it is better to build our own, that of freedom. It is necessary to stop thinking about the enemy, what it does, where it is found, what to do to strike it, and dedicate ourselves to ourselves, to our “daily life”, to our relationships, to our spaces that need to expand and improve more and more. Otherwise, we will never do anything but follow the inclinations of power.
The anarchist movement today is full of this sort of reasoning, the continual search for justifications disguised as theoretical analyses that excuse one’s absolute inaction. There are those who want to do nothing because they are skeptical, those who do not want to impose anything on anyone, those who consider power too strong for them and those who don’t want to follow its rhythms and times; every one of these excuses is good. But these anarchists, do they have a dream capable of setting their hearts aflame?
In order to clear the field of these miserable excuses, it is worth the effort to remember a few things. There are not two worlds, ours and theirs, and even if, to be absurd, they did exist, how could they be made to co-exist? There is a single world, the world of authority and money, of exploitation and obedience: the world in which we are all forced to live. It is impossible to pretend that we are outside. This is why we cannot allow ourselves to be indifferent, this is why we cannot manage to ignore it. If we oppose ourselves to the state, if we are always quick to seize the occasion to attack it, it is not because we are indirectly molded by it, it is not because we have sacrificed our desires on the altar of revolution, but because our desires cannot be realized as long as the state exists, as long as any Power exists. The revolution does not distract us from our dreams, but rather is the only possibility that allows the conditions for their realization. We want to overturn this world as quickly as possible here and now, because here and now there are only barracks, courts, banks, concrete, supermarkets, prisons. Here and now there is only exploitation, while freedom, as we understand it, does not really exist.
This does not mean that we give up on creating spaces of our own in which to experiment with the relationships that we prefer. It only means that these spaces, these relationships, do not represent the complete freedom that we desire for ourselves and for everyone. They are a step, but not the final one, much less the definitive one. A freedom that ends on the threshold of our occupied house, of our “free” commune, is not enough, it does not satisfy us. Such freedom is illusory, because it frees only as long as we stay at home and don’t leave the confines that are imposed on us. If we don’t consider the necessity of attacking the state (and there is much that we could say about this concept of “attack”), then, by definition, we can only do what it allows us to do at its convenience, forever, limiting ourselves to surviving in the little “happy isle” that we will build ourselves. Keeping our distance from the state means conserving life, confronting it means living.
Our capitulation is implicit in indifference toward the state. It is as if we were admitting that the state is stronger, is invincible, is beyond contestation, one might as well lay down one’s arms and consider cultivating one’s kitchen garden. Is it possible to call this revolt? It seems to us rather to be a completely inner attitude, circumscribed by a kind of diffidence, incompatibility with and disinterest in that which surrounds us. But resignation remains implicit in such an attitude. Contemptuous resignation if you will, but resignation nonetheless.
It is like throwing punches that are limited to warding off blows without ever trying to bring the adversary that one hates down. But our adversary does not give us any respite. We cannot merely leave the ring and go on making a laughing-stock of it. It is necessary to bring our adversary down; dodging and expressing our disappointment in it is not sufficient.
\chapter{Camomillo \emph{by Penelope Nin}}
At this time, a lot of anarchists from all over Italy are flooding into Rome.
A month ago, by the order of a public prosecutor who was looking for easy glory, about thirty enemies of authority were taken into custody and locked up in Rebibbia, a prison in the outlying suburbs. To protest against the arrogance and vengeful spirit of the judges who have decided to take away their freedom, one of them has begun a hunger and thirst strike to the death.
But last Saturday, these anarchists were not alone in breathing the air of the eternal city. Others joined them there, guests this time of the international bookshop, \emph{Il Manifesto}, where they went to chatter — together with communists, marxists and historians — about Camillo Berneri, “an anarchist between Gramsci and Gobetti”, as the title of the conference said. It was promoted by the daily newspaper of via Tomacelli\footnote{Also called \emph{Il Manifesto.} — translator.} by the libertarian studies center of Milan and by the Historical Review of Anarchism of Pisa, in collaboration with the Roman bookshop Anomolia.
It’s a good thing that there are anarchists willing to cleanse the good name of anarchy, washing away the awful reputation that a few hotheads would like to attach to it. In printing the news of the arrests a month ago, \emph{Il Manifesto} had already attentively made note of how the investigators “a bit too easily” granted “a single ideological-political motivation to actions that seem like those of a band of common criminals.” But a fine convention organized all together was the thing needed to dissipate the last doubts, to finally bring back a bit of serenity.
In response to this proposal, it was immediately said that a better subject could not have been chosen. What anarchist more than Camomillo Berneri could have brought anarchists and personages such as Valentino Parlato, Goffredo Fofi (who is publishing an anthology of Berneri’s writings), and Enzo Santarelli onto a common terrain? Figures of this sort certainly could not remain insensitive to the fascination exercised by the leading exponent of anarchist revisionism and by his unsettling definitions of Anarchy — “the society in which technical authority, stripped of every function of political domination, comes to form a hierarchy conceived and realized as a system of distribution of work” — and of freedom — “the power of obeying reason”.
“Anarchist \emph{sui generis}\footnote{“his own kind of anarchist”. — translator.}” — so he loved to describe himself — Berneri fought like a lion to bring anarchism out from the mists of utopia at blows with reality. “Better the present evil than something worse” was the battle cry that accompanied him throughout his life and to which he always remained faithful. This sense of measure led him to salute the Bolshevik regime in 1918, despise abstentionism\footnote{refusal to participate in the electoral process. — translator.} which he dismissed as “cretinism”, collaborate with liberals like Gobetti, and make sympathetic gestures toward a part of the Catholic world with which he shared the idea of woman as wife, procreator and ideal housekeeper. And the deep sense of duty — which Camomillo identified with God is what made him write words full of cautious common sense about the necessity of money and the inevitability of prison, with the consciousness that it is always necessary to reach a “compromise between the Idea and the fact, between tomorrow and today.”
Berneri was killed in Barcelona during the days of May 1937, in the heat of the Spanish revolution. His martyrdom earned him canonization by a part of the venerable anarchist Church. The fact that his murderers were precisely the communists who Parlato, Fofi and their comrades praised so highly up until recently is a particular that is utterly insignificant.
The fact remains only Camomillo Berneri — the anarchist who used to candidly maintain that “a minimum of authority is indispensable” — could have become the line of union between stalinists and anarchists, the unbelievers who — like Gobetti and Gramsci — do nothing but feed dogma with their heresy.
But, okay, let’s say it: as far as it goes, these judges are perfectly right. There are anarchists and “anarchists”. Some are bad and are rightly in prison. But others — among them, it is worthwhile to recall, a few of the proposers of this convention, Claudio Venza, Gianni Carrozza, Giampietro Berti — are good. So good that they can enjoy the esteem of all the respectable people of this world.
A toast therefore to Camomillo. And to hell with the “anarchists” in prison.
\chapter{He Jokes with Men \emph{by Penelope Nin}}
\begin{quote}
“\emph{But expropriations and violent actions that put the lives of people at risk, and more generally the theory and practice of illegalism at all costs are far from our anarchism. Such actions are in clear contrast with the anti-violent Malatestian spirit that we have made our own.” }
(from \emph{Germinal}, \# 71\Slash{}72, p. 26)
\end{quote}
The greatest misfortune that can befall a human being endowed with any quality is to be surrounded by followers. As long as he remains alive, he will be perpetually compelled to keep watch so that nothing stupid is said or done in his name, toil that will prove useless however when, after his death, the initiates quarrel over how to advance the path of his endeavor. The followers are never at the level of their “teacher”, since only those who lack their own ideas take on those of others — becoming, precisely, their followers. Thus, followers not only prove to be incapable of causing something that has already been started to advance, but since they lack the qualities of the one who came before them, they easily reach the point of distorting and betraying the ideas they claim to support.
The phenomenon, deprecable in itself, takes on ludicrous and even amusing features and directions, particularly when the unfortunate “teacher” is an anarchist, that is to say an individual hostile to all authority and therefore opposed in principle to the herd mentality. And yet who can deny that even within the anarchist movement such cases have occurred? To avoid going too far, it is enough to consider Errico Malatesta, the famous Italian anarchist.
All the friends and scholars of the thoughts of Malatesta have had to agree on one fact. His sole preoccupation, his sole desire, throughout his life was to make revolution. For Malatesta, there was no doubt: anarchists are such because they want anarchy and it is only possible to realize anarchy by making revolution, a revolution that would necessarily be violent, the first step of which is insurrection. It seems to be a banality, and indeed it is. And yet it is a banality from which many anarchists tend to distance themselves with a sense of disgust.
Luigi Fabbri wrote: “Insurrection is the necessary and inescapable event of every revolution, the concrete event through which it becomes reality for everyone. It is from this fact that Malatesta’s aversion for every theory and method that tends, directly or indirectly, to discredit it, to avert the attention of the masses and the activity of revolutionaries from it, to replace it with means that are apparently more convenient and peaceful grew.”
Not just revolutionary, since “anyone can call themselves revolutionary while using the prudence to postpone the desired transformation to far distant times (when the time is ripe, as they say),” Malatesta was above all an insurrectionist inasmuch as he wanted to make the revolution immediately — a revolution understood “in the sense of violent change carried out through force against the preserving powers; and it thus implies material struggle, armed insurrection, with the retinue of barricades, armed groups, the confiscation of goods from the class against which one fights, sabotage of the means of communications, etc.” — not in a distant and undefined future, but immediately, as quickly as possible, as soon as the occasion presented itself, an occasion that had to be created intentionally by anarchists if it did not come on its own through natural events.
Yes, I know; who is not familiar with certain critiques Malatesta made of violence and polemics that he wrote about Emile Henry or Paolo Schichi? Nevertheless, Malatesta did not deny the legitimacy and even the necessity of the use of violence as such; he only opposed a violence that “strikes blindly, without distinguishing between the guilty and the innocent.” It is no accident that the example of blind violence that he Usually gave was that of the bomb that exploded in Barcelona during a religious procession, causing forty deaths and numerous injuries. This is because he would have no critique to make in the face of rebellious actions against precise targets that have no consequence for extraneous people. In fact, in the course of one of his famous interviews with conceded to Le Figaro, in which the interviewer tried to press him to disapprove of Ravachol’s bombs, and of the attack at the boulevard Magenta, Malatesta answered: “Your conclusions are hasty. In the affair of rue Clichy, it seems quite clear to me that it was intended to blow up a judge; but I regret that it was carried out — quite involuntarily, I believe — in a way that brought injury to people whom he had not considered. As to the bomb of boulevard Magenta — oh! I have no reservations about that! Lherot and Very had become accomplices of the police and it was a fine act of struggle to blow them up.”
It seems clear that all the discussion and polemics that occurred in those distant years — that certain present-day anarchists run through again in order to sell us the image of an anti-violent Malatesta — were not in fact aimed at the use of violence in itself, but only the limits one could not exceed without placing the very principles of anarchism in question, or at most those limits suggested by considerations of a tactical order.
But let’s leave “the dark end of an earlier century” and the polemics that then raged in the anarchist movement, and return to the present. No explosive actions claimed by anarchists in recent years could be considered as being carried out in a “blind” and “insensitive” manner. Rather all could be said to have been directed against the structures of domination without putting “the lives of people at risk.” So how can one justify the repudiation of these actions on the part of certain anarchists? Certainly not by borrowing from the thoughts of Malatesta since saying that there is a limit to the use of violence is not the same thing as saying that one must never have recourse to it.
Having recourse to the dead does not serve to justify one’s indolence.
\chapter{The Link That Isn’t There \emph{by Mario Cacciucco}}
In addition to explaining, language in its function of allowing communication between individuals, situations and materiality is set the misguided task of enclosing emotions, mental states and relationships between individuals and others within syllables.
In my opinion, the mystification of relationships of love and friendship is spurious. Examples from lived experience would be a great help in explaining my reflection, but I want to try to clarify it by using, in my own way, the written word.
I start from the presupposition that every individual is different in her attitudes, aspirations, physical aspect, pleasures. The relationships that exist between individuals are like spheres that bounce off each other in a whirl of contacts, without causing any fusion. Modifications, but never fusions. I on the other, the other on me. In every instance, each sphere maintains its uniqueness. Starting from my own uniqueness, I thus decide to embark on an unlimited search for contacts and situations close to mine, in order to realize myself excessively by enjoying the differences of others. And I do so by affirming my will to preserve my decision-making abilities however and whenever. In general, I recognize the difference of others, I am attracted to it, like a child who sees a clown pirouette and is attracted by the novelty and likableness that it communicates to him. I recognize the charm of all that is external to me, the known, the less known and the unknown.
The contacts that I establish may be more or less lasting. Circumstance contribute to a large extent. But they always end with the option of reopening.
When I talk about seeking affinity, I speak of granting myself a series of contacts with other individuals, which do not cause harm to my capacity to act, but are rather capable of giving me new strength, new capacities, multiplying the bouncing of my sphere on those of others, something indispensable for the search for myself and my satisfaction. The common meanings of “love” and “friendship” thus leave me perplexed.
When relationships open, one cannot establish \emph{a priori} how they might extend or end themselves. Relationships are and that is all. The randomness of events and the manifestation of individual will contribute to creating a certain something. And when I say a certain something, I mean everything. From the most heated passions, to carnality, to crime, to sensory ecstasy, to esteem, to indifference, to annoyance.
Excluding is a bit like making laws, depriving oneself of possibilities for movement. Uniting different events can cause the sense of their originality and uniqueness to be lost. If for some a kiss is love, for me it is a sensation of the lips to experiment with each time.
The individuals with whom I share moments are profoundly different from one another. Each instance, having peculiar characteristics, has nothing to do with any other instance. There is no doubt.
So, what is love and what is friendship when one speaks of relationships? Are they oracles to which to prostrate ourselves of obstacles to everything? Who is the person that we can get take part in one of these categories with certainty? And wouldn’t this certainty be a misguided and misleading boldness? Wouldn’t it always be to small? If “the fragile cage of language” is what still creates these problems for us, why not enter a bit more into contact with oneself and do away with these oh so mysterious and intangible words that lead the fruit of our personal emotions and agreeableness back to something that doesn’t exist? Why make oneself the spokesperson of concepts aimed at defining, establishing, when an unconditioned eruption of our desired could cancel all this in order to lead it into the abyss of the possible, the conceivable? And why not clearly, decisively, forcefully destroy the relationship when it becomes hateful to us since the past is a thing that becomes extraneous to the extent that you can no longer put your hands on it. And memories are useful, more than anything else, to those who momentarily live far from their will.
Comrades, friends, lovers, for me dissolution unites all these descriptions. I love, I prefer, I choose in my own way, as a lawless one. I don’t know what love is, and I don’t know what friendship is, perhaps because they don’t exist or perhaps because I have no need to use these words, because a have a more or less clear idea of what the dynamic of knowing and standing together with others, in agreement or disagreement, is.
Relationships without the disquieting and unbearable presence of authority are the only ones that I put up with, and I rely on them to express my boundless I When one of these relationships tends to create a bit of restlessness or sacrifice or that smarmy thing known as tolerance, then I hold that the time has come to remove myself from it, to start over in another of the infinite situations that the existent proposes to me.
Starting again from a gratifying detachment.
\chapter{A Little, Little Giant \emph{by Il Panda}}
\emph{[There are moments when it seems that anything could open up, that all possibilities are in play. These are the moments we need to seize in order to realize our rebellious dreams. There are no guarantees in these moments, only possibilities. The following article was written in the midst of one such moment that occurred several years ago in France. — translator]}
It is not just a matter of proportions. We always appear so very little in the face of this world that overwhelms us and that not only seems incomprehensible — with its endless and intricate network of relationships and dependencies between endless causes and effects — but also unassailable.
Yes, of course, we’d like to turn this world upside down, we’d like to destroy these relationships, but we don’t know where to begin; everything seems useless to us, all our destructive fury seems to be reduced to an almost inoffensive tickle against an impassive giant. Our hearts are stirred to revolt, but how many times have we run up against the supposed immutability of the giant that oppresses us? The pot is boiling, we think; but we don’t know how to lift its lid, this blessed pot, we don’t understand is rhyme or reason. And even if the urgency of things always goads us into action, it doesn’t seem to us that this manages to prime the mechanism that could put the existent into a hard spot. Our continue clashes with the world don’t succeed in reproducing themselves, rousing the passions, the wild and collective feasts, the revolutions that we desire. And yet, as we know, the giant is neither so big nor so passive as we imagine it to be. The feast is always right around the corner, because if the paths of domination are infinite, so are the paths of revolt: the giant that we have in our heads is really a network of relations, enormous indeed, but quite concrete, and these relations use determined channels, determined paths. And these paths could, indeed, be blocked, priming, in time, unpredictable mechanisms.
Such an eventuality has been bringing difficult moments to life for the French for several weeks. Truck drivers — those wage-laborers who drive back and forth across France and Europe, transporting commodities for the profit of capital — are on strike. Not only are all these goods not being bought and sold, with all the consequent problems for French cities and the economy; in fact, by strike, the French truck drivers did not just mean a mere abstention from work. No, they park their semis at the entrances of cities, on the expressways and block traffic; or they surround refineries in order to prevent the resupplying of fuel.
Bordeaux is already completely blocked, like a consistent number of the cities of the west and the southeast, and in Paris, the siege is starting. Think, what can a blockade of this sort arouse: already, just a few short days after the start of the protest, a few factories are noticeably slowing down production. Without raw materials, industry can’t work since its products are not transported and sold. And along with the factories, offices and ministries are shaken.
What can happen in a blockaded city? Everything and nothing, it’s a question of \emph{time}. Cities are built around work and its time. The time of the city is scanned from the hands of a clock, the ticking of which rules our lives branding our days with fire. The office, the family, Sundays, evenings, survival doesn’t survive without the ticking of the clocks.
However, in a blockaded city, time might not have any more need for clock faces and hands. It is released from work; it can expand and contract improbably even to the point of vanishing.
This might be dangerous for the giant. You will see that, without time, strange ideas enter people’s minds, strange vices are born that unleash unpredictable mechanisms — to such an extent that the they displace the narrow limits of demands, beyond which it no longer matters what the truck drivers wanted to negotiate, whether wages, pensions or work hours, because what is at stake is something else entirely, something for everyone.
Or else nothing could happen in a blockaded city. It could be a huge, very sad Sunday.
The pot boils and the giant is never too big for us; it cannot even sleep peacefully. Its arteries — that are roads, electric wires and computer networks — are exposed and can be cut, generating an infinite and unpredictable series of possibilities.
\chapter{Beyond the Law \emph{by Penelope Nin}}
To tell the truth, I don’t quite understand what is meant today when people speak of “illegalism”. I thought this word was no longer in use, that it could not slip out of the history books of the anarchist movement any more, shut up forever with the equally ancient “propaganda of the deed”. When I have heard it talked about again in recent times in such shamelessly critical tones, I haven’t been able to hold back a sensation of astonishment. I begin to find this mania for dusting off old arguments in order to avoid dealing with new discussions intolerable, but there is so much of this.
One thing, however, seems clear to me. The illegalism that is spoken of (badly) today is not the concept that was debated with so much heart-felt animation by the anarchist movement at the beginning of the 20\textsuperscript{th} century. At that time this term was used to indicate all those practices prohibited by law that were useful for resolving the economic problems of comrades: robbery, theft, smuggling, counterfeiting money and so on. It seems to me that today some anarchists, lacking anything concrete to discuss, are tending much too easily to claim that illegalism means a refined glorification for its own sake of every behavior forbidden by law, not only of those dictated by the requirements of survival. In short, illegalism would become a kind of theoretical framework for erecting illegality as a system, a life value.
Some people push it even further, to the point of censuring a no better defined “illegalism at all costs”, yearning for comrades who would violate the law even when they could do otherwise simply to savor the thrill of the forbidden or perhaps in order to satisfy some ideological dogma. But I ask, where have these comrades run across this illegalism at all costs, who has spoken of it? Who would be such a fool as to challenge the severity of the law when she could do otherwise? Obviously, nobody.
But there is probably another point on which it would be useful to reflect. Can an anarchist avoid challenging the law? Certainly in many circumstances this is possible. For example, at the moment I am writing for a paper that is published legally; does this perhaps make me a legalist anarchist? On the other hand, if I were to go this evening to put up clandestine flyers, would this make me an illegalist anarchist? But then, what would ever distinguish these two categories of anarchists?
The question of the relationship between an anarchist and the law cannot be settled in such a hasty and misleading way. As I see it, the actions of an anarchist cannot be conditioned by the law in either the positive or the negative. I mean that it cannot be either the reverential respect for the guiding standards of the time or the pleasure of transgression as an end in itself that drives her, but rather his ideas and dreams united to her individual inclinations. In other words, an anarchist can only be an alegalist, an individual who proposes to do what most pleases him beyond the law, without basing herself on what the penal code allows or forbids.
Of course, the law exists and one cannot pretend not to see it. I am quite aware that there is always a bludgeon ready to attend to our desires along the way toward their realization, but this threat should not influence our decision about the means to use to realize that which is dearest to our hearts. If I consider it important to publish a paper — a thing that is considered legal — I can easily attempt to follow the provisions of the law about the press in order to avoid useless annoyance, since this does not change the contents of what I intend to communicate at all.
But, on the other hand, if I consider it important to carry an action considered illegal — like the attack against the structures and people of power — I will not change my mind simply because someone waves the red flag of the risks I will face before my eyes. If I acted otherwise, the penal code would be advising me about what my conduct should be, greatly limiting my possibilities to act and thus to express myself.
But if it is an absurdity to describe an anarchist as “illegalist”, it would be ridiculous to attribute the quality of “legalist” to her. How could an anarchist, an individual who desires a world without authority, expect to be able to realize his dream without ever breaking the law, which is the most immediate expression of authority, that is to say, without transgressing those norms that have been deliberately established and written in order to defend the social order? Anyone who intends to radically transform this world would necessarily have to place herself sooner or later against the law that aims to conserve it.
Unless\dots{}Unless the desire to change that world that still smolders in the hearts of these anarchists is in some way subordinated to the worries about the risks they might face, about being persecuted by the police, about being brought under investigation, about losing the appreciation of friends and relations. Unless the absolute freedom that means so much to anarchists is considered a great and beautiful thing, but mainly in the realm of theory — manifesting itself in the inoffensive banter exchanged fork the armchairs after a suffocating day of work — because from the practical point of view the strength of domination offers no hope. Then it is advisable to make utopia into something concrete, with its feet upon the ground, uniting it with good sense, because revolution could never be considered legal under any penal code.
Enough of dreaming the impossible; let’s try to obtain the tolerable. Here it is, the invective against the myth of illegalism coming from certain anarchists takes on a precise meaning, that of justifying their self-interested predisposition to conform to the dictates of the law, setting aside every foolish, immoderate aspiration.
In the name of realism, of course.
\chapter{The Rudiments of Terror}
The ruling order and its challenger face each other. The former has everything: an organization — the state — economic power, military power, control over the entire nation. The latter has little at its disposal. Only a specific number of people, full of desperation, with a few rudimentary weapons. But these few are inspired by a terrible propulsive force, the ambition for domination, that is great enough to move them to launch their challenge. They know that they are weaker than their adversary, so they must strike and run, strike and run. And when a power — even in embryo — must strike, it knows only one tool: terrorism, the use of intentionally blind and indiscriminate violence. Like that of December 3, 1996 in Paris which caused the death of two people and the wounding of fifty more, mangled by the explosion of a bomb that happened in a subway car.
Terrorism has returned — the mass media throughout the world has begun to scream it. It has returned? But when did it ever go away?
Of course, the terrorism of the challenging power is blatant and is immediately denounced as such by the media of its rival. But who will have the boldness to denounce the terrorism of the power in office, the terrorism of the state, particularly the powerful states that maintain the global order? The images of mangled bodies have traveled around the globe, rousing the horror of all, perhaps enough to make people forget that for those in power (and for those seeking it) the “common people” have always been thought of as cannon-fodder. Slaughtering them in a subway car or on a battlefield doesn’t really make any difference.
These deaths and injuries are just like the deaths and injuries caused by aerial bombing, like those that occur year-round at workplaces, in barracks, in police stations, in hospitals, in prisons. Like those brought about by the paving over of wild places, by nuclear power plants, by the adulteration of our food, by atmospheric pollution or by the psychosomatic illnesses caused by the way of life that is imposed on us in this world.
So here it is, the violence that strikes everyone in a blind and indiscriminate fashion. Here it is, the terrorism of the state.
\chapter{Poor Heroes}
\begin{quote}
“\emph{His death unleashed a frantic propaganda about the hero Durruti. Any discussion would end with the citation of his name. And each time he was named, a bit of his thought and work was killed.”}
— \emph{Abel Paz, “Buenaventura Durutti”}
\end{quote}
Durutti is probably the best known anarchist in the world. His name is linked to the Spanish revolution, to the summer of 1936, when the Iberian proletariat rose up, arms in hand, against power and attacked the military bases, burned the churches, occupied the factories. It is this struggle, where he fought on the front lines together with the people of his column, that every one remembers. This is the struggle in which he lost his life on the morning of November 20, 1936, and due to which he became a hero to all.
And a hero is always right. No one ever dares to bring his statements or his actions into question. No one. The dark sides of heroes need never be put on display; they are justified. And Durutti had his dark sides as every human being does. Of those linked to his character, such as his hatred for homosexuals, there is nothing more to say. Everyone is made as they are, and besides so much water has passed under the bridge since then. But what of those linked to his choices in life? What can be said about these? What, for example, can be said about his past as a bank robber? Something needs to be said about it today when there are anarchists in prison accused of robbing banks. Can one sing the praises of that distant anarchist robber, dedicate a fine commemorative book to him and keep silent about the anarchist robbers of our time? A response to this is necessary; the comparison is far too obvious. And, as usual, the response is found in his time, in his implacable raids, in his ability to “objectively” change contexts and situations. And then there is the man, Buenaventura Durutti. Wasn’t he, in fact, the one who said — and the word of a hero is sacred — that “then I followed that method because the circumstances were different from those of the present day”, and “Banditry, no. Collective expropriation, yes! Yesterday is surpassed by the road of history itself. And anyone who desires to revive it, taking refuge in ‘the right to live’ is free to do so, but outside of our ranks, renouncing the title of militant and accepting individual responsibility for his action without compromising the life of the movement or its prestige before the working class”? Yes, he really was the one who said this, and we all need to remember it. All of us.
Only in this way could one forget. Forget that these words were said in 1933, when there were, to quote Durutti again, “a million union members” and “ a population awaiting the propitious moment to carry out the great revolution.” Forget that, after the propitious moment when he urged collective action had passed, it would be the time for Sabate, Facerias and other anarchist proponents of individual action — who were maligned and disowned for this by other anarchists afraid that their organization might lose its good reputation — to take this struggle up again.
But today, are we in a moment propitious for revolution? And besides, don’t Durutti’s thoughts exclusively deal with members of the FAI\Slash{}CNT? Wasn’t it the militants of these organizations who were to renounce their “titles” if they decided to attack a bank? And what of those who have never been part of such organizations, aho have always strongly affirmed individual responsibility for their actions? Has Durutti’s meaning been erased in order to use his words against these people? Those who have something to say are only his self-interested interpreters, preoccupied with confirming for the millionth time that there is no salvation outside the church.
Poor Durutti. His name — when not used to christen an after-work bar for comrades — is reduced to a mere polemical tool.
\clearpage
\begin{quote}
The next four texts were printed in \emph{Canenero} in order to stimulate on ongoing discussion. Unfortunately, this discussion never went beyond what is printed here and a few very brief statements that merely amounted to taking sides rather than furthering the debate. Although I am quite aware that the specific detail of the situation in Italy in 1996–7 were quite different from our present situation, I, nonetheless, think that there are broader ideas presented in these texts worthy of discussion and debate in relationship to a real practice here as well. I hope that there are those who will be moved to further this discussion in terms of our situation here and now. — the translator
\end{quote}
\clearpage
\chapter{Communiqué From Prison}
On the day that the state-capital in its two-fold capacity of judge-oppressor will officiate its vindicatory trial (in the Occorsio hall of the court in Rome on December 10, 1996) against the anarchist movement — an archaic rite of insult and criminalization against the transgressors of bourgeois society — in the attempt to expunge every form of individual or organized revolutionary antagonism combating the exploitation of the human being, we fearlessly affirm combatant revolutionary action, without unrealistic aphorisms or anathemas we will claim our identity as an armed organization against the state.
In that hall-like place, formal representation of the legitimacy of bourgeois law, we will practice militant anarchist anti-judicialism by abstaining from the farce of the debate of the trial. We will not endorse the mythical “de jure”, judicial doctrine, age-old normative heritage of states that are developed on the age-old usurpations of slavery, torture and the exploitation of other people’s labor, that guarantees defense for those investigated, offering them the judicial tool of reply, a way of guaranteeing the “democratic” form of the prosecuting trial, a sharp, corrupt and deceptive way disguise \emph{a priori }the prejudice against the defendants who don’t appear in court. We will not recognize the judges!
Industrial civilization is the highest of the aspirations of progress to which state-capital society aims. It forces millions of people in the world to give up the ancient indigenous culture of the population in order to embrace the modern culture of the factory. With the great means that the bourgeois capitalist state uses, beyond being functional as the dominant means of production, are powerful organizers of culture, the culture that is summed up in the symbols of the commodity as mediations between production and consumption.
The globalization of exploitation now so extremely normal is intellectual. The cerebral flattening to the preordained schemas of intelligent machines, the homogenization of the cultures of peoples to the new languages of communications and production are the aim of the new imperialist colonialism. Cybernetic universalism, or multimedia communication, is a tool of the systematic and quantitative reorganization of the new world order, in the sectors of the market, of capital, of the institutional order and of the territorial infrastructure, of the repression of antagonists, refractory to the homogenization of the new scientism, intellectual standardizer.
Inspiring ourselves critically with the experiences of the antagonist armed movement of the 1970s and particularly with the anarchist heritage, with struggles for regional independence, stable references for our path of conflict with the state-capital aimed at extinguishing them through insurrectional means, therefore, on the basis of this historical heritage, we allude to constructing a communist society in anarchist production in the anti-legal sense, without courts or prisons, through struggle against every form of government and power that is realized through the efforts of the exploited; an iconoclastic society inspired by free cooperation among people and by free education.
We recognize in this court the fawning role of the servant of the state, in which, living like a courtier off the sweat of the productive labor of workers and peasants, it insures that the exploited populace continues its obsequious service to bourgeois justice.
Every revolutionary action against the state and bourgeois institutions will be claimed as the sign of a beginning and a continuation of a precise antagonistic path, called Combatant Revolutionary Action, for which we will assume all responsibility in front of power.
No claim at all — at least on our part — for actions against the state with the circle A, because this exposes the anarchist movement to continuous provocations, while it is right to form specific groups that assume political responsibility for their actions.
Our combatant path is the formation in the revolutionary sense of a combatant, internationalist, anti-imperialist anarchist organization, in relation with all revolutionary forces that intend to subvert the order of the bourgeois capitalist state in its phase of globalization, in order to introduce ourselves as a unique productive and organizational model for relations between human beings.
To the many-centered and camouflaged conformation of cybernetic-industrial power, we will respond with wide-spread and well-aimed actions to undermine it both on the territory and in the urban space in which the organizational and informational infrastructures of its domination are centered.
Living force to all revolutionary prisoners and to all combatants, for a new free, anarchist and communist anti-authoritarian society.
Let’s remember to avenge all the comrades struck by the fire of the repression of the state-capital.
Long live anarchy, long live armed struggle.
\begin{quote}
\emph{Rome, December 1, 1996}
\emph{Pippo Stasi, Karechin Cricorian}
\emph{(Garagin Gregorian)}
\end{quote}
\chapter{The Fullness of a Struggle Without Adjectives}
Recently a communiqué from prison was distributed that has probably disturbed quite a few comrades. We are reproducing it here. Though it has the tone of a proclamation and certain statements are ambiguous, it seems to us that we can rule out the idea that we are confronting the announcement of the formation of an anarchist armed organization. This would be illogical for various reasons. For example, because, throughout time, armed groups have been shrewd enough to explain themselves after they have acted, and it doesn’t appear to us as if the acronym “Combatant Revolutionary Action” has ever claimed anything. Furthermore, if the comrades who signed the communiqué had, indeed, formed an armed organization, their document would become an explicit self-denunciation before the court, and this even before having initiated hostilities. If such a thing were true, it would make no sense at all.
From this, we deduce that the text should be interpreted as a simple proposal. Unfortunately, the wretched linguistic style in which it was formulated risks provoking misunderstandings and incomprehension that it would be best for everyone to avoid. More simply, we believe that Pippo Stasi and Garagin Gregorian wish to invite the anarchist movement to reflect on the arguments that they set forth, like the necessity for a portion of anarchists to undertake a path of armed struggle and, therefore to create a specific armed struggle. And since these comrades have not hesitated to state what they think, assuming all responsibility, we assume that no one will take it badly if we do the same.
As we have often taken the opportunity to say in the columns of this paper, we are decidedly opposed to all armed organization, including an unlikely anarchist armed organization. Here it is not a question of a mere divergence of views, but of a substantial radical difference that goes well beyond any considerations of expediency or contingency. We are against any armed organization today, as we were yesterday and will be tomorrow. And we confirm that this aversion of ours is not limited to formal disagreement. Not only will we never support an armed organization, but we will oppose it with a harsh critique. We will oppose its formation and spread because we consider it hostile to us, insofar as it is not capable of generating prospects that we find desirable.
We think that the individual who rises up, the individual who rebels against this world that is too cramped to contain his dreams, has no interest in limiting their possibilities, but in extending them infinitely if possible. Thirsty for freedom, eager for experience, anyone who rebels is in continuous search for new affinities, for new tools with which to express herself, with which to go to the attack on the existent in order to subvert it from the foundations. This is why insurrectional struggle should find its stimulus and energy in our capacity for filling its arsenal with ever new weapons, beyond and against all reductive specialization. The experts in pistols are like the experts in books, or occupations, or whatever else. They are boring because they always and only speak about themselves and their favorite means. Precisely because we do not privilege one tool over any of the others, we love and support numberless actions, carried out through the most varied means, that occur daily against the ruling order and its structures. Because revolt is like poetry: to be such it must be made by all, not by one alone, particularly not an expert.
Now the specific armed organization is the negation of this insurrectionary struggle, the parasite poisoning the blood. Whereas insurrection encourages enjoyment and the realization of what we have at heart, armed organization only promises sacrifice and ideology. Whereas insurrection exalts the possibilities of individuals, armed organization only exalts the techniques of its soldiers. Whereas insurrection considers a gun or a stick of dynamite to be only one of the weapons available to it, the armed organization makes it the only weapon, the only tool to use (“Long live armed struggle”). Whereas insurrection aims to generalize itself and invites everyone to participate in its festival, the armed organization is closed by force of circumstance and — except for its few militants — nothing is left for others to do except to cheer it on. The subversion of life is a vast project that knows no limits, because it aims to disrupt the totality of society. Armed organization is only able to glimpse a marginal aspect of this struggle — the military conflict against the state — and mistakes it for the whole. And even this conflict, even the armed attack against the state, loses any liberatory meaning, any breath of life, when its entire impetus is reduced to the promotion of a program an acronym to spend at the political market.
It is rather in anonymity that all political calculation vanishes, leaving space for the thousands of individual tensions and vibrations, and for the possibility for them to meet, come together and abandon themselves in each other. And of what use are neon signs to those with no commodities to sell. As to the accusation against those actions claimed with a “circle A”, claiming that they expose the whole anarchist movement to police provocation, other anarchists, terrorized by the idea that someone might come knocking at their door. Unfortunately for them and for the comrades who signed the document, a possible acronym will certainly not resolve the situation. At most, instead of suspecting anarchists of having signed an action with a “circle A”, the police will suspect them of being part of a specific group.
It seems to us to be a bit hasty to claim that in the 1970s, the anarchist movement knew specific experiences of the combatant model, since the “Revolutionary Action” (AR) archipelago — to which we assume Stasi and Gregorian are referring — can only be described as “anarchist” at the cost of a huge ideological distortion. In fact, comrades of various origins came together in AR , animated at the beginning by a libertarian and anti-stalinist spirit that defined its experiment for a brief time as anarcho-communist, considered as the summation of the various positions of the comrades. But it has become clear to many anarchists that armed organizations, none of them excluded, contributed to the decline of social subversion in those years. And these critical reflections are not new, but have been expressed by various anarchists on many occasions since the 1970s.
We don’t know what reasons pushed Stasi and Gregorian to distribute this writing. To say it all, their proposal seems out of this world to us, a bit like the rhetoric used for the occasion, that seems to come directly from debates that raged in the 1970s, poisoning the atmosphere. But more than anything else, we don’t like to see comrades accept the ultimatum the power puts forth today (either reformism or armed struggle) allowing themselves to get drawn into the foolish game of upping the ante: since we are accused of belonging to an armed band that doesn’t exist, why not form a real one? Well, this temptation, this attraction toward the one-way mirror of the armed organization, has no grip on us, and we will never tire of criticizing it wherever it manifests itself. Insurrection has desires and reasons that no military logic could ever understand.
\chapter{A Missing Debate}
Three weeks ago, when we published Garagin Gregorian and Pippo Stasi’s communiqué from prison, we thought that it might be able to open an interesting and worthwhile discussion. That document could have generated an endless series of reflection on topics that are always relevant (specialization, specific armed organization, attack, justice) and on others that — having never really disappeared — have returned after many years to shake up our lives (the question of going on the lam, for example). In our opinion, all these topics should be faced in perspective. By this we mean that they should be confronted not just on the basis of the much too obvious logic of “\emph{comrades are grown-up, weaned and choose what to do for themselves}”. We’ve all reached this point, and it seems ridiculous to repeat it. It is not so necessary to say which conception seems to us to be more or less compatible with “anarchist ethics and tradition”, but which one seems like it could move in our perspective. An armed band could possibly be organized in a horizontal manner, but what does that have to do with our insurrection? In the article that accompanied the comrades’ communiqué, we did nothing more than reassert the basic banalities on the question of armed struggle, the important matters that \emph{Canenero} has always been fond of emphasizing. But so many other questions remain open, questions that need to be raised sooner or later.
An example for all: the police knock at our door with an arrest warrant. In the situation where we manage to give them the slip, what do we do? Take care, this is a serious problem because forced clandestinity should not cause the interruption of our projects. We should make ourselves capable of facing the new situation in a way that makes it possible for us to still attack the ruling order, and to continue to live fully and with passion in all the spaces that, despite everything, we are able to conquer. To do this, clear ideas and useable tools would be of service to us — before the arrest warrants — to makes sure that our life is not reduced to flight. These tools are also the new way for organizing with respect to the new situation, the new way of communicating with struggles in course and with comrades who are not being pursued. Everything with the same perspective of the complete overturning of life, sacrifice and the existent that animated us before we had to go on the lam. And what about this, what could it ever have to do with a specific combatant organization — even one that is horizontal, but still has acronyms, programs and the limits that follow from this?
In any case, we were wrong. The debate had a hard time getting off the ground and only one contribution to the discussion has reached us up to now [\dots{}]. All the rest have been collective communiqués and the taking of stands [\dots{}] that don’t deal with the topics in question with sufficient depth. On the contrary, it seems to us that they reveal, at least partially, some common flaws and push us to consider a few things. The first is that it is necessary to know how to read. By this we mean that if someone writes that the specific armed organization, even when it declares itself anarchist, is a structure that we consider our enemy — as we wrote in the last issue — because it prospects utterly opposed to those we hope for, one should not read that those who propose it or practice it are our enemies. If we were to state that the anarcho-syndicalist perspective, for example, is not just extraneous, but also hostile, to us, we are certain that no one would misunderstand our words. No one would think that we intended to wait outside the houses of comrades who share this perspective in order to do them in, or that we would refuse to give our solidarity if they were struck by repression. The thing that touches us is that in their vision there is a place ready for us as well, that we, however, do not want to occupy. And our critique originates from their project of enclosing us in that place and our firm intention not to be enclosed. And these two perspectives, ours and theirs, have everything to gain from a mutual, constant and heated critique, even harsh when necessary. Because only through critique can distances widen or be bridged and the method be found for making the clash of projects that are so different as to be hostile worthwhile.
Knowing how to read also means that when someone writes that an experience like Revolutionary Action (AR) can be described as anarchist only at the cost of a huge distortion, one should not read that there were no anarchist in the AR. There were many anarchists in the AR, but there were also many other respectable comrades who, and this is not our fault, were not anarchists. It is not without reason that we consider the debate about the AR more interesting than that about the Red Brigades or other combatant parties.
And then — to bring up another flaw — if the one who proposes certain perspectives has the misfortune of being in prison, we certainly cannot play the role of Red Cross nurses, accepting anything that comes to us from behind bars with a compliant smile or applause even when we consider it rubbish. As long as we consider comrades in prison as poor things who we must always consider right so as not to cause them pain, or as heroes who we consider right because prisoners are always right, the problem will be left unresolved, new situations will catch us unprepared yet again and — in turn — the comrades in prison will be left more and more isolated. It would be best to shake the guerrilla war or political myths of medals from our heads — the myths according to which the more time one has been or has to be in prison, the more revolutionary and, thus, the more correct they must be — and reason passionately on our problems, which are also the problems of the imprisoned who have their say as well. This is why \emph{Canenero} dedicates these pages to this topic [\dots{}]
Finally, one more thing shines through in some of the statements of position: the concern that \emph{Canenero} should or wants to be the representative paper of “an area”. \emph{Canenero} represents a small piece of the lives of those who publish it. So don’t think ill of us if we don’t consult all (all of who? which area?) before saying what we think about what comes to us, or if we are not so many experts to teach the doctrine, since we want to have nothing to do with doctrines.
\begin{quote}
— \emph{the editors of }Canenero
\end{quote}
\chapter{Letter on Specialization}
(Not putting one’s destiny into play unless one is willing to play with all of one’s possibilities)
Today I thought about how sad it is to fall into the habit of defining ourselves in terms of one of the many activities in which we realize ourselves, as if that activity alone described the totality of our existence. All this recalls the separations that the state and the economy inflict on our lives much too closely. Take work, for example. The reproduction of the conditions of existence (i.e., the activity of putting out the effort in order to eat, sleep, stay warm, etc.) should be completely one with discussion, play, the continuous transformation of the environment, loving relationships, conflict, in short with the thousands of expressions of our uniqueness. Instead, work has not only become the center of every concern, but confident in its independence, it also imposes its measure on free time, amusement, encounters and reflection. In short, it is presented as the measure of life itself. In fact, since this is their social identity, almost everyone is defined in terms of the job they carry out, i.e., in terms of misery.
I am referring particularly to the repercussions that the fragmentation that power imposes on everyone’s lives has on the theory and practice of subversives. For example, take arms. It seems obvious to me that a revolution without arms is impossible, but it is equally clear that arms are not enough. On the contrary, I believe that the more revolutionary a change is, the less armed conflict is its measure. The broader, more conscious and more joyous the transformation is, the greater is the condition of no return that is created in relationship to the past. If subversion is carried into every sphere of existence, the armed defense of one’s possibility for destroying becomes completely one with the creation of new relationships and new environments. Then, everyone would be armed. Otherwise, specialists come into being — future bosses and bureaucrats — who “defend” while everyone else demolishes and rebuilds\dots{} their own slavery.
This is especially important because it is not “military” defeats that set off the decline and the consequent triumph of the old world, but rather the dying away of autonomous action and enthusiasm that are smothered by the lie of the “harsh necessities of the transition” (sacrifice before happiness in communism, obedience to power before freedom in anarchy). And historically, the most brutal repression is always played out precisely in this decline, never in the moment of widespread and uncontainable insurrection. Paradoxically, anarchists should push, arm in hand, so that arms are needed as little as possible and so that they are never separated from the totality of revolt. Then I ask myself what “armed struggle” could ever mean. I understand it when a leninist is speaking about it, since he possesses nothing of revolution except the misery he sets up — the coup d’etat, the taking of the Winter Palace. But for an anti-authoritarian? Perhaps, in the face of the general refusal to attack the state and capital, it could have the significance of emphasizing the inoffensiveness of every partial opposition and the illusoriness of a liberation that tries to abolish the ruling order simply by “delegitimating it”, or self-managing one’s elsewhere. It could be. But if there is anything partial, it is precisely the guerrilla mythology, with its entire stock of slogans, ideologies and hierarchical separations. So one is harmless to power, when one accepts going down the paths known to it and, thus, helps to impede all those it does not know. As to illusions, what else can one call the thesis according to which daily life — with its roles, duties and passivity — is criticized through armed organization. I absolutely recall the thesis: the endeavor was to supply a libertarian and non-vanguardist alternative to the stalinist combatant organizations. The results were already written in the methods. As if to attack the state and capital, there would be need for acronyms, boring claims, unreadable communiqués and all the rest. And still we hear talk of “Armed Struggle” and “combatant” organizations. Remembering — in the midst of so much self-interested amnesia — that arms also make up a part of the struggle can only be positive. But what does this mean? That we should no longer publish journals, have debates, publicly call for the elimination of the pope, throw eggs at judges or yogurt at journalists, loot during marches, occupy spaces or blockade the editorial office of whatever newspaper? Or does it mean — exactly as some magistrates dream — that this “level” should be left to some so that others can become specialists of the “attack”? Furthermore, with the intention of sparing the useless involvement of the entire movement for the actions of a few, as if it were not separations that have always prepared the best terrain for repression.
It would be necessary to free the practices of attack from any “combatant” phraseology, in order to cause them to become the real meeting of all revolts. This is the best way to prevent them from falling into a rut. So much the more so, since the exploited themselves sometimes move to the attack without waiting for instruction from any organization whatsoever. Dissatisfaction arms itself against the terrorist spectacle of power, sometimes feeding the spectacle. And anarchists should not be the one’s to disarm it. In order to hide every sign of dissatisfaction, in order to show that no one — except the latest “terrorists” — rebels against democracy, the state tries to invent a clandestine anarchist organization to which it attributes thousands of expressions of revolt — a revolt that goes beyond any gang, armed or not — in order to negate them. This way, it manages silence and social consensus. Precisely because the masters would like to enclose our activities into a military structure, dividing them into different “levels”, it is necessary for us to expand and unite them as much as possible into a revolutionary project that surpasses the armed mythology through excess. Each one with her own aptitudes and desires. And more than this, carrying subversion into every sphere of existence. The arm that contains all arms is the will to live with all one’s possibilities, immediately.
And what of the thesis according to which it is necessary to take one’s responsibility in the face of power by claiming one’s actions? It seems clear to me that acronyms ready for sticking on inconvenient individuals make the police happy. So if responsibility is not to be a lie or a pretext for control, it must be individual. Each person is responsible to herself in her actions. The mutual recognition of responsibility only happens on a plane of mutuality. Therefore, there is no responsibility in the face of those who, by exploiting, place themselves against all mutuality. In the face of authority, there is no terrain — political or military conflict — of common recognition, but only hostility. What does it mean, then, to take one’s responsibility in the face of power? Could it maybe mean — in perfect leninist observance — being recognized by it as an organization? Here responsibility ends and its collective substitute, the spectacle of social war, begins.
The leftist democrat, respectful of the law, is the first one to become infatuated with guerrilla iconography (especially when it is exotic) and once the guerrilla has laid down his arms, he is the first one to return, gradually from the left, to law and democracy. From this point of view, the one who declares the insurrectional perspective closed in its entire range, adhering more or less directly to reformism, helps to reinforce the false need for combatant organizations — reversed projections of political impotence. Leftist militants are even able to use subcommandante Marcos to legitimate their role against right through the game of postponements. For his part, the subcommandante hopes for nothing more than to be able to act democratically for his fatherland.
Leaving behind the more or less modernized leninists, we come to the sphere of anarchists. Even here, among the specialists of debate, many clasped the “Chiapas insurgents” to their hearts, provided that insurrection — this infantile disorder of anarchism — is never talked about from our side\dots{} And as long as one takes the due distance from those who continue to talk about it.
Once at the very end of a meeting on self-managed spaces, a friend of mine told me that in the 1970s there was the firm belief that anyone who used a gun, for this reason alone, was right, while now it seems that reason has been transferred lock, stock and barrel to those who occupy spaces. Interchangeable specializations. In itself, occupying spaces is an important method of struggle, which contains the very possibility of all subversion in a nutshell: the determination to reach out a hand and take one’s space. This clearly doesn’t mean that such a method, by itself, could put an end to the world of constraints and commodities. As always, the ideas and desires of those who apply it make the difference. If anyone in the occupied spaces seeks the guarantee of survival in a slapdash way, she will find it there, just as — by putting the occupation itself into play — she could find the point of departure for his most boundless demands there. The same goes for books, explosives or love affairs. The most important thing is not to place limits — in one direction or the other — borrowed from the ruling criteria (law, the number, the fortune of success).
Personally, I don’t know “the insurrectionalists”; I only know individuals who support the necessity of insurrection, each with his own reasons or methods. A necessity, as one of our friends said, determined by the fact that within the present society it is only possible to propose different ways of responding to the existing questions (perhaps with direct democracy, citizens’ committees, etc.), whereas with insurrection the questions themselves change.
And if we refuse all specialization, why describe ourselves as “squatters”? Why describe ourselves through one practice alone? Is it maybe because we can speak publicly of this practice, because it can spread further than others and because it implies a collective dimension? Poor criteria, in my opinion. One can also speak publicly of sabotage, as long as there isn’t any need to say, “I did this” or “that guy did the other thing”, in order to discuss a question. Several people could also carry out an act of sabotage together, but if only one person were to put it into practice, this would not make the action lose its meaning. It seems to me that the question of the capacity for spreading in itself should be a reason for reflection, certainly not a unit of measure. If someone who loves breaking the windows of banks or shopping centers were to say to you, “Hi, I am a vandal,” it would make you laugh. It would be equally ridiculous if a subversive described himself as a “writer” because he doesn’t disdain publishing some book or article. I have never heard any anarchist present herself as a “saboteur”. If I ever heard this, I would think I was meeting a cretin. Furthermore, who has ever critiqued occupation as such? Who has ever said that dynamite is “more revolutionary” than crowbars? Making the struggle in all its form into an indivisible totality — this is the point. I would say this not of the struggle, but of my life. Without “propaganda” and “the arms of critique”, “armed struggle” and “the critique of arms”, “daily life” and “revolution”, “individual” and “organization”, “self-management” and “direct action”, and away with pigeonholing.
But without specific proposals (labor struggle, the occupation of spaces or something else), how do you create a broader involvement? Proposals are possible, even though it is necessary to agree on what and with whom. But such proposals are either instances of a theoretical critique and a global practice, or they are\dots{} accepted proposals.
Nonetheless, not everything is to be destroyed. The possibility of destruction must not be destroyed. This is not wordplay. Destruction is thought, desired, projected and organized. To do this, no useful contribution, whether theoretical or practical, is wasted, no method abandoned. It is certainly not with fine proclamations of subversion that we can go to the assault on the world. This way, one only becomes a retiree of revolt. The possibility of destruction is completely to be invented, and no one can say that that there has been much effort put into doing this. Often with the alibi that he doesn’t want to construct anything, someone will go deeply into reasonings, and equally often, she lacks the will to be as open-minded and quick as her ideas, to refuse to remain at the mercy of events. In short, the ability to know how to choose the occasion. “In the heart of the occasion, everything is a weapon for the man whose will is not disarmed.”
I say again: everything together or nothing. When one claims to subvert the world only with discussion, or occupations, or books, or arms, one ends up trying to direct assemblies, occupying hovels, writing badly and shooting worse. The fact is that by repeating these banalities that should be the foundation for starting to truly discuss, one becomes boring like the specialists of repetition. The worn-out dialogues change by changing the situation.
\begin{quote}
\emph{Massimo Passamani}
\end{quote}
\chapter{An Adventure Without Regrets}
Dear readers,
What you have in your hands is the last issue of \emph{Canenero}. Various reasons have moved us to decide to bring it to a close. They all refer back to what we said in the editorial of \#33, the first in the new series: “\emph{Canenero }is a wager that only has meaning if there is someone willing to play.” And so now, those who have been willing to gamble on this stake are no longer so.
We are no longer available to do \emph{Canenero} because its publication has come to take up too much of the time of our lives, preventing us not only from carrying out other projects that are close to our hearts, but also from being able to fully utilize the very instrument to which we gave life. If an anarchist weekly doesn’t want to have the aim of merely being an account, it mus necessarily be used, and paradoxically those who made this one didn’t have the opportunity to use it as we would have liked.
Besides the limited length for articles in a weekly conceived like this (the famous page and a half) very often at most allowed us to outline certain discussions, to then leave them unresolved. Since it is unthinkable that the subsequent deepening of the discussion could happen in a weekly of this sort, it could only have been brought back to other more suitable venues, which up to now nobody has thought of creating. In the end, this situation became intolerable to us, first of all because of the current absence of other tools, like magazines that come our less frequently or books of some interest to us.
Finally, we have realized that, particularly in times like these, a weekly manages is able to stimulate reflection and worthwhile debate only with great difficulty. Incredibly, precisely due to irs decision to put out questions to be confronted, \emph{Canenero} has ended up becoming an object of debate itself, and not one of those involved in debate. To speak clearly, a weekly is alive when it is able to involve as many individuals as possible, i.e., when the ideas expressed are able to trigger chain reactions, even violent ones if you will, provided that they occur in conditions of mutuality. Otherwise, the paper falls back on itself and the only thing left is for it to die, if it doesn’t want to survive as a pathetic monument to the idea. And so, this confrontation is lacking. Those who didn’t agree with our ideas didn’t contribute, only being able to send letters of insults and accusation, lacking the least bit of argumentation. And those who shared our ideas — even if only partially — didn’t contribute. Worse yet, we realized that a representative task had been entrusted to the weekly: being the voice of those who have none. And the only discussions that \emph{Canenero} seems to have been able to raise are those relating to its ability or lack thereof to perform a task that none of us ever desired. In this regard, the position-taking that appeared in the last issue, in its “stodgy supplement”, are an indicative example. A broad, interesting debate capable of expressing many imaginable facets and nuances was not born from the clash of two different perspectives. All that was born was a distressing series of declarations \emph{for} or \emph{against}. But for or against what, and why? Silence. Everyone keeps quiet.
A silence that reconfirms our doubts about the current validity of \emph{Canenero}, and only increases the need to abandon an analytical tool like a weekly that maybe due to its overly narrow time schedule does not allow a better settling of the ideas contained in it, limiting itself inevitably to piling up problems and questions that still remain open.
And for all of these reasons, we have decided to put an end to \emph{Canenero}.
Without regrets.
\begin{quote}
\emph{The editors}
\end{quote}
% begin final page
\clearpage
% new page for the colophon
\thispagestyle{empty}
\begin{center}
Library.Anarhija.Net
\bigskip
\includegraphics[width=0.25\textwidth]{logo-yu.pdf}
\bigskip
\end{center}
\strut
\vfill
\begin{center}
Various Authors
Articles from “Canenero”
1994–1997
\bigskip
Personal communication with the translator
\bigskip
\textbf{lib.anarhija.net}
\end{center}
% end final page with colophon
\end{document}
|
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\alpha+\beta\right)}\frac{cd}{c+d}\frac{\left(-c/d;q\right)_{\infty}\left(-d/c%
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http://ftp.t.ring.gr.jp/archives/text/CTAN/graphics/pgf/base/tex/pgfmoduleplot.code.tex | ring.gr.jp | CC-MAIN-2020-10 | application/x-tex | text/x-matlab | crawl-data/CC-MAIN-2020-10/segments/1581875141460.64/warc/CC-MAIN-20200217000519-20200217030519-00074.warc.gz | 66,097,917 | 4,810 | % Copyright 2006 by Till Tantau
%
% This file may be distributed and/or modified
%
% 1. under the LaTeX Project Public License and/or
% 2. under the GNU Public License.
%
% See the file doc/generic/pgf/licenses/LICENSE for more details.
\ProvidesFileRCS{pgfmoduleplot.code.tex}
% PGF's plotting interface works as follows:
%
% In order to plot something, two things need to be done. First, you
% need to provide the coordinates (obviously) of the points that
% should be plotted. The coordinates are given via a long stream of
% commands. These commands are \pgfplotstreamstart, which is
% given exactly once at the beginning, the commands
% \pgfplotstreampoint, \pgfplotstreampointoutlier,
% \pgfplotstreampointundefined, \pgfplotstreamnewdataset, of which
% there are numerous in the middle, the special \pgfplotstreamspecial,
% of which there may be numerous in the middle, and \pgfplotstreamend,
% which must be given at the end. Between these commands arbitrary
% other commands may be given. Here is an example:
%
% ...
% \pgfplotstreamstart
% \pgfplotstreampoint{\pgfpointxy{0}{0}}
% \pgfplotstreampoint{\pgfpointxy{1}{1}}
% \pgfplotstreamnewdataset
% \pgfplotstreampoint{\pgfpointxy{2}{4}}
% \relax
% \pgfplotstreampointoutlier{\pgfpointxy{3}{9}}
% \pgfplotstreamspecial{some handler-dependent special stuff}
% \pgfplotstreamend
%
% By themselves, the \pgfplotstreamxxxx commands do not do anything by
% default. Rather, to ``use'' such a stream, you must first install a
% stream handler. For example, the ``lineto'' handler will simply
% basically every \pgfplotstreampoint into a \pgfpathlineto.
%
% One special things is the handling of "jumps" in a stream. For
% instance, when a lineto handler encounters an "outlier" or a new
% data set, the current line should end and a new subpath should start
% (unless configured otherwise). For this, the special jump handler is
% important.
%
% Example:
%
% \pgfpathmoveto{\pgfpointorigin}
%
% \pgfplothandlerlineto
% \pgfplotstreamstart
% \pgfplotstreampoint{\pgfpointxy{0}{0}}
% \pgfplotstreampoint{\pgfpointxy{1}{1}}
% \pgfplotstreampoint{\pgfpointxy{2}{4}}
% \relax
% \pgfplotstreampoint{\pgfpointxy{3}{9}}
% \pgfplotstreamend
% The stream commands actually call their ``internal'' versions, which
% are set by the handlers:
\def\pgfplotstreamstart{\pgf@plotstreamstart}%
\def\pgfplotstreampoint#1{\gdef\pgfplotlastpoint{#1}\pgf@plotstreampoint{#1}}%
\def\pgfplotstreampointoutlier#1{\pgfplot@outliers{}{\pgf@plotstreamjump}{\pgfplotstreampoint{#1}}}%
\def\pgfplotstreampointundefined{\pgfplot@outliers{}{\pgf@plotstreamjump}{}}%
\def\pgfplotstreamnewdataset{\pgfplot@outliers{}{\pgf@plotstreamjump}{}}%
\def\pgfplotstreamspecial{\pgf@plotstreamspecial}%
\def\pgfplotstreamend{\pgf@plotstreamend}%
\def\pgfplot@ignorer#1#2#3{#1}%
\def\pgfplot@jumper#1#2#3{#2}%
\def\pgfplot@plotter#1#2#3{#3}%
\let\pgfplot@outliers\pgfplot@jumper
\let\pgfplot@undefined\pgfplot@jumper
\let\pgfplot@newdata\pgfplot@jumper
\pgfset{
handle outlier points in plots/.is choice,
handle outlier points in plots/ignore/.code=\let\pgfplot@outliers\pgfplot@ignorer,
handle outlier points in plots/jump/.code=\let\pgfplot@outliers\pgfplot@jumper,
handle outlier points in plots/plot/.code=\let\pgfplot@outliers\pgfplot@plotter,
handle undefined points in plots/.is choice,
handle undefined points in plots/ignore/.code=\let\pgfplot@undefined\pgfplot@ignorer,
handle undefined points in plots/jump/.code=\let\pgfplot@undefined\pgfplot@jumper,
handle new data sets in plots/.is choice,
handle new data sets in plots/ignore/.code=\let\pgfplot@newdata\pgfplot@ignorer,
handle new data sets in plots/jump/.code=\let\pgfplot@newdata\pgfplot@jumper,
}%
% Declares a new plot handler
%
% #1 = macro that should install the stream handler subsequently
% #2 = parameter list of said macro.
% #3 = keys as described below
%
% Description:
%
% When you declare a new plot handler, you provide, through keys,
% different "actions" that should be taken when the different stream
% commands are used:
%
% start = some code to be executed when \pgfplotstreamstart is called
% for this plothandler
% end = same for the end of the stream
% point = code to be executed for a point of the stream
% jump = code to be executed for a "jump" in the stream
% special = code to be executed for \pgfplotstreamspecial
% start macro = here and for the following keys, the to-be-executed
% code is stored in the given macro
% end macro = see above
% jump macro = see above
% point macro = see above
% special macro = see above
%
% Example:
%
% \pgfdeclareplothandler{\ultrasimplehandler}{}{
% point=\pgfpathlineto{##1}
% }
\def\pgfdeclareplothandler#1#2#3{%
\def#1#2{%
\pgfkeys{%
/pgf/plots/@handler options/.cd,
start=\relax,
end macro=\relax,
point macro=\pgfutil@gobble,
jump macro=\relax,
special macro=\pgfutil@gobble,%
#3%
}%
}%
}%
\pgfkeys{%
/pgf/plots/@handler options/.cd,
start/.code=%
\gdef\pgf@plotstreamstart{%
\global\pgf@plot@startedfalse%
\global\let\pgf@plotstreamend\pgf@plotstreamend@init%
\global\let\pgf@plotstreampoint\pgf@plotstreampoint@init%
\global\let\pgf@plotstreamjump\pgf@plotstreamjump@init%
\global\let\pgf@plotstreamspecial\pgf@plotstreamspecial@init%
#1%
},%
start macro/.code=%
\gdef\pgf@plotstreamstart{%
\global\pgf@plot@startedfalse%
\global\let\pgf@plotstreamend\pgf@plotstreamend@init%
\global\let\pgf@plotstreampoint\pgf@plotstreampoint@init%
\global\let\pgf@plotstreamjump\pgf@plotstreamjump@init%
\global\let\pgf@plotstreamspecial\pgf@plotstreamspecial@init%
#1%
},%
end/.code=\gdef\pgf@plotstreamend@init{#1},
end macro/.code=\global\let\pgf@plotstreamend@init#1,
point/.code=\gdef\pgf@plotstreampoint@init##1{#1},
point macro/.code=\global\let\pgf@plotstreampoint@init#1,
jump/.code=\gdef\pgf@plotstreamjump@init{#1},
jump macro/.code=\global\let\pgf@plotstreamjump@init#1,
special/.code=\gdef\pgf@plotstreamspecial@init##1{#1},
special macro/.code=\global\let\pgf@plotstreamspecial@init#1,
}%
\newif\ifpgf@plot@started
% Sets the action taken for the first point of a plot to a lineto.
%
% Description:
%
% For certain handlers it makes sense either the start a plot by
% moving to the first point of the plot or to do a lineto to that
% first point. Using this command this action can be set to a lineto.
%
% Example:
%
% \pgfsetlinetofirstplotpoint
\def\pgfsetlinetofirstplotpoint{\let\pgf@plot@first@action=\pgfpathlineto}%
% Sets the action taken for the first point of a plot to a moveto.
%
% Example:
%
% \pgfsetmovetofirstplotpoint
\def\pgfsetmovetofirstplotpoint{\let\pgf@plot@first@action=\pgfpathmoveto}%
\let\pgf@plot@first@action=\pgfpathmoveto
%
% Handlers
%
% This handler converts each plot stream command into a lineto
% command, except for the first, which is converted to the action that
% has previously been specified using \pgfsetlinetofirstplotpoint or
% \pgfsetmovetofirstplotpoint.
%
% Example:
%
% \pgfplothandlerlineto
% \pgfplotxyfile{mytable}
\pgfdeclareplothandler{\pgfplothandlerlineto}{}{
point macro=\pgf@plot@line@handler,
jump=\global\let\pgf@plotstreampoint\pgf@plot@line@handler@move%
}%
\def\pgf@plot@line@handler#1{%
\pgf@plot@first@action{#1}%
\global\let\pgf@plotstreampoint=\pgfpathlineto%
}%
\def\pgf@plot@line@handler@move#1{%
\pgfpathmoveto{#1}%
\global\let\pgf@plotstreampoint=\pgfpathlineto%
}%
% This handler turns creates a series of lineto commands, with the
% last command being a closepath, resulting in a closed path. If a
% jump is encountered, the current subpath is closed and a new subpath
% is started.
%
% Example:
%
% \pgfplothandlerpolygon
% \pgfplotxyfile{mytable}
\pgfdeclareplothandler{\pgfplothandlerpolygon}{}{%
point macro=\pgf@plot@line@handler@close,
jump macro=\pgf@plot@next@close@and@moveto,
end macro=\pgf@plot@polygon@stop
}%
\def\pgf@plot@line@handler@close#1{%
\pgfpathmoveto{#1}%
\global\pgf@plot@startedtrue%
\global\let\pgf@plotstreampoint=\pgfpathlineto%
}%
\def\pgf@plot@polygon@stop{%
\ifpgf@plot@started%
\pgfpathclose%
\fi%
\global\pgf@plot@startedfalse%
}%
\def\pgf@plot@next@close@and@moveto{%
\ifpgf@plot@started%
\pgfpathclose%
\fi%
\global\let\pgf@plotstreampoint\pgf@plot@line@handler@close%
}%
% More handlers are defined in pgflibraryplothandlers
% This handler discards the plot.
%
% Example:
%
% \pgfplothandlerdiscard
% \pgfplotxyfile{mytable}
\pgfdeclareplothandler{\pgfplothandlerdiscard}{}{}%
% This handler records each plot stream command to a macro. This is
% useful if plot commands are difficult to generate and need to be
% ``recycled'' later on.
%
% Example:
%
% \pgfplothandlerrecord{\myplot}
% \pgfplotxyfile{mytable} % stored in \myplot now
% \pgfplothandlerline
% \myplot
% \pgftransformxshift{1cm}
% \myplot
\pgfdeclareplothandler{\pgfplothandlerrecord}{#1}{%
start=\gdef#1{\pgfplotstreamstart},
point=\expandafter\gdef\expandafter#1\expandafter{#1\pgfplotstreampoint{##1}},
jump=\expandafter\gdef\expandafter#1\expandafter{#1\pgf@plotstreamjump},
special=\expandafter\gdef\expandafter#1\expandafter{#1\pgfplotstreamspecial{##1}},
end=\expandafter\gdef\expandafter#1\expandafter{#1\pgfplotstreamend},
}%
% Read a plot stream from a file and plot it.
%
% #1 = file from which to read things
%
% File format:
%
% Each line of the file should begin with two numbers separated by a
% space. Such a line with number #1 and #2 is converted to a
% \pgfplotstreampoint{\pgfpointxy{#1}{#2}}. Extra characters following
% on the line are ignored.
%
% Lines starting with ``%'' and ``#'' are ignored.
%
% Example:
%
% \pgfplotxyfile{tableformgnuplot.dat}
\def\pgfplotxyfile#1{%
\begingroup%
\def\b@pgfplotsxyfile@scanning@for@first{1}%
\pgfplotstreamstart%
\openin\r@pgf@reada=#1
\ifeof\r@pgf@reada
\pgfwarning{Plot data file `#1' not found.}
\else
\catcode`\#=14
\catcode`\^^M=5
\pgf@readxyfile%
\fi
\pgfplotstreamend%
\endgroup%
}%
\let\pgf@savedpar=\par
\def\pgf@partext{\par}%
\def\pgf@readxyfile{%
\pgfutil@read\r@pgf@reada to \pgf@temp%
\let\par=\pgf@savedpar%
\edef\pgf@temp{\pgf@temp}%
\ifx\pgf@temp\pgfutil@empty%
\if1\b@pgfplotsxyfile@scanning@for@first
\else
\ifeof\r@pgf@reada\else\pgfplotstreamnewdataset\fi%
\fi
\else\ifx\pgf@temp\pgf@partext%
\if1\b@pgfplotsxyfile@scanning@for@first
\else
\ifeof\r@pgf@reada\else\pgfplotstreamnewdataset\fi%
\fi
\else%
\expandafter\pgf@parsexyline\pgf@temp\pgf@stop%
\fi\fi%
\ifeof\r@pgf@reada\else\expandafter\pgf@readxyfile\fi%
}%
\def\pgf@parsexyline#1 #2 #3\pgf@stop{%
\def\b@pgfplotsxyfile@scanning@for@first{0}%
\edef\pgf@xyline@flag@val{#3}%
\ifx\pgf@xyline@flag@val\pgf@xyline@flag@undef%
\pgfplotstreampointundefined%
\else\ifx\pgf@xyline@flag@val\pgf@xyline@flag@out%
\pgfplotstreampointoutlier{\pgfpointxy{#1}{#2}}%
\else%
\pgfplotstreampoint{\pgfpointxy{#1}{#2}}%
\fi\fi%
}%
\edef\pgf@xyline@flag@out{o\space}%
\edef\pgf@xyline@flag@undef{u\space}%
% Read a plot stream from a file and plot it.
%
% #1 = file from which to read things
%
% File format:
%
% Like xy, except that each line contains three numbers, which are
% converted to xyz coordinates.
%
% Example:
%
% \pgfplotxyfile{tableformgnuplot.dat}
\def\pgfplotxyzfile#1{%
\begingroup%
\pgfplotstreamstart%
\openin\r@pgf@reada=#1
\ifeof\r@pgf@reada
\pgfwarning{Plot data file `#1' not found.}
\else
\catcode`\#=14
\catcode`\^^M=5
\pgf@readxyzfile%
\fi
\pgfplotstreamend%
\endgroup%
}%
\def\pgf@readxyzfile{%
\pgfutil@read\r@pgf@reada to \pgf@temp%
\ifx\pgf@temp\pgfutil@empty%
\if1\b@pgfplotsxyfile@scanning@for@first
\else
\ifeof\r@pgf@reada\else\pgfplotstreamnewdataset\fi%
\fi
\else\ifx\pgf@temp\pgf@partext%
\if1\b@pgfplotsxyfile@scanning@for@first
\else
\ifeof\r@pgf@reada\else\pgfplotstreamnewdataset\fi%
\fi
\else%
\expandafter\pgf@parsexyzline\pgf@temp\pgf@stop%
\fi\fi%
\ifeof\r@pgf@reada\else\expandafter\pgf@readxyzfile\fi%
}%
\def\pgf@parsexyzline#1 #2 #3 #4\pgf@stop{%
\def\b@pgfplotsxyfile@scanning@for@first{0}%
\edef\pgf@xyline@flag@val{#4}%
\ifx\pgf@xyline@flag@val\pgf@xyline@flag@undef%
\pgfplotstreampointundefined%
\else\ifx\pgf@xyline@flag@val\pgf@xyline@flag@out%
\pgfplotstreampointoutlier{\pgfpointxyz{#1}{#2}{#3}}%
\else%
\pgfplotstreampoint{\pgfpointxyz{#1}{#2}{#3}}%
\fi\fi%
}%
% Render a function using gnuplot.
%
% #1 = filename prefix for .gnuplot and .table files (optional,
% default is \jobname)
% #2 = gnuplot function text
%
% Description:
%
% This command will write a file called #1.gnuplot that sets up
% some gnuplot commands to write output to a file called
% #1.table. Then it calls gnuplot (using the \write18 mechanism)
% to execute the file. Then it reads #2.table using \pgfplotxyfile.
%
% Example:
%
% \pgfplothandlerlineto
% \pgfplotgnuplot[\jobname]{plot [x=0:5] x*sin(x)}
{%
\catcode`\%=12
\catcode`\"=12
\xdef\pgf@gnuplot@head{set table \noexpand\pgf@plottablefile@quoted; set format "%.5f"}
}%
\let\pgf@plotwrite=\w@pgf@writea
\newif\ifpgf@resample@plot
\pgfkeys{%
/pgf/plot/gnuplot call/.initial={gnuplot}}%
\def\pgfplotgnuplot{\pgfutil@ifnextchar[{\pgf@plotgnuplot}{\pgf@plotgnuplot[\jobname]}}%}%
\def\pgf@plotgnuplot[#1]#2{%
\pgf@resample@plottrue%
\pgfutilpreparefilename{#1.gnuplot}%
\let\pgf@plotgnuplotfile=\pgfretval
\pgfutilpreparefilename{#1.table}%
\let\pgf@plottablefile=\pgfretval
\let\pgf@plottablefile@quoted=\pgfretvalquoted
% Check, whether it is up-to-date
\openin\pgfutil@inputcheck=\pgf@plotgnuplotfile\relax
\ifeof\pgfutil@inputcheck%
\else%
\pgfutil@read\pgfutil@inputcheck to\pgf@temp% ignored
\pgfutil@read\pgfutil@inputcheck to\pgf@plot@line%
\closein\pgfutil@inputcheck
\edef\pgf@plot@code{#2\space}%
\ifx\pgf@plot@code\pgf@plot@line%
\openin\pgfutil@inputcheck=\pgfretval\relax
\ifeof\pgfutil@inputcheck%
\else%
\closein\pgfutil@inputcheck
\pgf@resample@plotfalse%
\fi%
\fi%
\fi
\ifpgf@resample@plot%
\immediate\openout\pgf@plotwrite=\pgf@plotgnuplotfile\relax
\immediate\pgfutil@write\pgf@plotwrite{\pgf@gnuplot@head}%
\immediate\pgfutil@write\pgf@plotwrite{#2}%
\immediate\closeout\pgf@plotwrite%
\pgfutil@shellescape{%
\pgfkeysvalueof{/pgf/plot/gnuplot call} \pgf@plotgnuplotfile}%
\fi%
% \let\pgf@savedparsexyline=\pgf@parsexyline%
% \let\pgf@parsexyline=\pgf@parsegnuplotxyline%
\pgfplotxyfile{\pgf@plottablefile}%
% \let\pgf@parsexyline=\pgf@savedparsexyline%
}%
% \def\pgf@parsegnuplotxyline#1 #2 #3\pgf@stop{%
% \edef\pgf@xyline@flag@val{#3}%
% \edef\pgf@xyline@flag@out{o\space}%
% \edef\pgf@xyline@flag@undef{u\space}%
% \ifx\pgf@xyline@flag@val\pgf@xyline@flag@undef%
% \pgfplotstreampointundefined%
% \else\ifx\pgf@xyline@flag@val\pgf@xyline@flag@out%
% \pgfplotstreampointoutlier{\pgfpointxy{#1}{#2}}%
% \else%
% \pgfplotstreampoint{\pgfpointxy{#1}{#2}}%
% \fi\fi%
% }%
% This producer handler plots a function using pgf's mathematical engine.
%
% #1 = variable
% #2 = domain for the variable
% #3 = point, typically defined in terms of the value of the variable
%
% Description:
%
% This producer will iterate the variable #1 over all variables in #2
% (using the \foreach statement). For each value, a plot coordinate
% #3 is created.
%
% Note that this command is pretty slow.
%
% Example:
%
% \pgfplothandlerlineto
% \pgfplotfunction{\x}{0,0.1,...,3.141}{\pgfpointxy{\x}{sin(\x)}}
\def\pgfplotfunction#1#2#3{%
\pgfplotstreamstart%
\foreach#1in{#2}%
{%
\pgf@process{#3}%
\edef\pgf@marshal{\noexpand\pgfplotstreampoint{\noexpand\pgfqpoint{\the\pgf@x}{\the\pgf@y}}}%
\pgf@marshal%
}
\pgfplotstreamend%
}%
\endinput
|
http://itex.coastal.cheswick.com/itex_server/latex/pg/2726/2726.tex | cheswick.com | CC-MAIN-2017-17 | application/x-tex | null | crawl-data/CC-MAIN-2017-17/segments/1492917121752.57/warc/CC-MAIN-20170423031201-00428-ip-10-145-167-34.ec2.internal.warc.gz | 184,083,842 | 158,717 | \documentclass[12pt]{book}
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%% \usepackage{mathptmx} % times roman
%%\usepackage{lucidabr} % lucida bright
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\usepackage[pdftex,bookmarks=true,bookmarksopen=true,
bookmarksnumbered=true,bookmarksopenlevel=3,
colorlinks,urlcolor=blue,linkcolor=blue,
pdftitle={Eight Cousins},
pdfauthor={Louisa M. Alcott},
citecolor=blue]{hyperref}
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\begin{document}
\pagenumbering{alph} % bogus, never shown, names don't collide with below
\title{Eight Cousins}
\author{Louisa M. Alcott}
\maketitle
\pagenumbering{roman}
\frontmatter
The Project Gutenberg EBook of Eight Cousins, by Louisa M. Alcott
This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever. You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org
Title: Eight Cousins
Author: Louisa M. Alcott
Posting Date: December 6, 2008 [EBook \#2726]
Release Date: July, 2001
Language: English
Character set encoding: ASCII
*** START OF THIS PROJECT GUTENBERG EBOOK EIGHT COUSINS ***
Produced by David Reed
EIGHT COUSINS
By Louisa M. Alcott
Preface
The Author is quite aware of the defects of this little story, many of
which were unavoidable, as it first appeared serially. But, as Uncle
Alec's experiment was intended to amuse the young folks, rather than
suggest educational improvements for the consideration of the elders,
she trusts that these shortcomings will be overlooked by the friends of
the Eight Cousins, and she will try to make amends in a second volume,
which shall attempt to show The Rose in Bloom.
L.M.A.
This text was converted to LaTeX by means of \textbf{GutenMark} software (version Jul 12 2014).
The text has been further processed by software
in the iTeX project, by Bill Cheswick.
\cleardoublepage
\tableofcontents
\cleardoublepage
\mainmatter
\pagenumbering{arabic}
\gutchapter{Chapter 1---Two Girls}
Rose sat all alone in the big best parlor, with her little handkerchief laid ready to catch the first tear, for she was thinking of her troubles, and a shower was expected. She had retired to this room as a good place in which to be miserable; for it was dark and still, full of ancient furniture, sombre curtains, and hung all around with portraits of solemn old gentlemen in wigs, severe-nosed ladies in top-heavy caps, and staring children in little bob-tailed coats or short-waisted frocks. It was an excellent place for woe; and the fitful spring rain that pattered on the window-pane seemed to sob, ``Cry away: I'm with you.''
Rose really did have some cause to be sad; for she had no mother, and had lately lost her father also, which left her no home but this with her great-aunts. She had been with them only a week, and, though the dear old ladies had tried their best to make her happy, they had not succeeded very well, for she was unlike any child they had ever seen, and they felt very much as if they had the care of a low-spirited butterfly.
They had given her the freedom of the house, and for a day or two she had amused herself roaming all over it, for it was a capital old mansion, and was full of all manner of odd nooks, charming rooms, and mysterious passages. Windows broke out in unexpected places, little balconies overhung the garden most romantically, and there was a long upper hall full of curiosities from all parts of the world; for the Campbells had been sea-captains for generations.
Aunt Plenty had even allowed Rose to rummage in her great china closet a spicy retreat, rich in all the ``goodies'' that children love; but Rose seemed to care little for these toothsome temptations; and when that hope failed, Aunt Plenty gave up in despair.
Gentle Aunt Peace had tried all sorts of pretty needle-work, and planned a doll's wardrobe that would have won the heart of even an older child. But Rose took little interest in pink satin hats and tiny hose, though she sewed dutifully till her aunt caught her wiping tears away with the train of a wedding-dress, and that discovery put an end to the sewing society.
Then both old ladies put their heads together and picked out the model child of the neighbourhood to come and play with their niece. But Ariadne Blish was the worst failure of all, for Rose could not bear the sight of her, and said she was so like a wax doll she longed to give her a pinch and see if she would squeak. So prim little Ariadne was sent home, and the exhausted aunties left Rose to her own devices for a day or two.
Bad weather and a cold kept her in-doors, and she spent most of her time in the library where her father's books were stored. Here she read a great deal, cried a little, and dreamed many of the innocent bright dreams in which imaginative children find such comfort and delight. This suited her better than anything else, but it was not good for her, and she grew pale, heavy-eyed and listless, though Aunt Plenty gave her iron enough to make a cooking-stove, and Aunt Peace petted her like a poodle.
Seeing this, the poor aunties racked their brains for a new amusement and determined to venture a bold stroke, though not very hopeful of its success. They said nothing to Rose about their plan for this Saturday afternoon, but let her alone till the time came for the grand surprise, little dreaming that the odd child would find pleasure for herself in a most unexpected quarter.
Before she had time to squeeze out a single tear a sound broke the stillness, making her prick up her ears. It was only the soft twitter of a bird, but it seemed to be a peculiarly gifted bird, for while she listened the soft twitter changed to a lively whistle, then a trill, a coo, a chirp, and ended in a musical mixture of all the notes, as if the bird burst out laughing. Rose laughed also, and, forgetting her woes, jumped up, saying eagerly,
``It is a mocking-bird. Where is it?''
Running down the long hall, she peeped out at both doors, but saw nothing feathered except a draggle-tailed chicken under a burdock leaf. She listened again, and the sound seemed to be in the house. Away she went, much excited by the chase, and following the changeful song, it led her to the china-closet door.
``In there? How funny!'' she said. But when she entered, not a bird appeared except the everlastingly kissing swallows on the Canton china that lined the shelves. All of a sudden Rose's face brightened, and, softly opening the slide, she peered into the kitchen. But the music had stopped, and all she saw was a girl in a blue apron scrubbing the hearth. Rose stared about her for a minute, and then asked abruptly,
``Did you hear that mocking-bird?''
``I should call it a phebe-bird,'' answered the girl, looking up with a twinkle in her black eyes.
``Where did it go?''
``It is here still.''
``Where?''
``In my throat. Do you want to hear it?''
``Oh, yes! I'll come in.'' And Rose crept through the slide to the wide shelf on the other side, being too hurried and puzzled to go round by the door.
The girl wiped her hands, crossed her feet on the little island of carpet where she was stranded in a sea of soap-suds, and then, sure enough, out of her slender throat came the swallow's twitter, the robin's whistle, the blue-jay's call, the thrush's song, the wood-dove's coo, and many another familiar note, all ending as before with the musical ecstacy of a bobolink singing and swinging among the meadow grass on a bright June day.
Rose was so astonished that she nearly fell off her perch, and when the little concert was over clapped her hands delightedly.
``Oh, it was lovely! Who taught you?''
``The birds,'' answered the girl, with a smile, as she fell to work again.
``It is very wonderful! I can sing, but nothing half so fine as that. What is your name, please?''
``Phebe Moore.''
``I've heard of phebe-birds; but I don't believe the real ones could do that,'' laughed Rose, adding, as she watched with interest the scattering of dabs of soft soap over the bricks, ``May I stay and see you work? It is very lonely in the parlor.''
``Yes, indeed, if you want to,'' answered Phebe, wringing out her cloth in a capable sort of way that impressed Rose very much.
``It must be fun to swash the water round and dig out the soap. I'd love to do it, only aunt wouldn't like it, I suppose,'' said Rose, quite taken with the new employment.
``You'd soon get tired, so you'd better keep tidy and look on.''
``I suppose you help your mother a good deal?''
``I haven't got any folks.''
``Why, where do you live, then?''
``I'm going to live here, I hope. Debby wants some one to help round, and I've come to try for a week.''
``I hope you will stay, for it is very dull,'' said Rose, who had taken a sudden fancy to this girl, who sung like a bird and worked like a woman.
``Hope I shall; for I'm fifteen now, and old enough to earn my own living. You have come to stay a spell, haven't you?'' asked Phebe, looking up at her guest and wondering how life could be dull to a girl who wore a silk frock, a daintily frilled apron, a pretty locket, and had her hair tied up with a velvet snood.
``Yes, I shall stay till my uncle comes. He is my guardian now, and I don't know what he will do with me. Have you a guardian?''
``My sakes, no! I was left on the poor-house steps a little mite of a baby, and Miss Rogers took a liking to me, so I've been there ever since. But she is dead now, and I take care of myself.''
``How interesting! It is like Arabella Montgomery in the `Gypsy's Child.' Did you ever read that sweet story?'' asked Rose, who was fond of tales of found-lings, and had read many.
``I don't have any books to read, and all the spare time I get I run off into the woods; that rests me better than stories,'' answered Phebe, as she finished one job and began on another.
Rose watched her as she got out a great pan of beans to look over, and wondered how it would seem to have life all work and no play. Presently Phebe seemed to think it was her turn to ask questions, and said, wistfully,
``You've had lots of schooling, I suppose?''
``Oh, dear me, yes! I've been at boarding school nearly a year, and I'm almost dead with lessons. The more I got, the more Miss Power gave me, and I was so miserable that I 'most cried my eyes out. Papa never gave me hard things to do, and he always taught me so pleasantly I loved to study. Oh, we were so happy and so fond of one another! But now he is gone, and I am left all alone.''
The tear that would not come when Rose sat waiting for it came now of its own accord two of them in fact and rolled down her cheeks, telling the tale of love and sorrow better than any words could do it.
For a minute there was no sound in the kitchen but the little daughter's sobbing and the sympathetic patter of the rain. Phebe stopped rattling her beans from one pan to another, and her eyes were full of pity as they rested on the curly head bent down on Rose's knee, for she saw that the heart under the pretty locket ached with its loss, and the dainty apron was used to dry sadder tears than any she had ever shed.
Somehow, she felt more contented with her brown calico gown and blue-checked pinafore; envy changed to compassion; and if she had dared she would have gone and hugged her afflicted guest.
Fearing that might not be considered proper, she said, in her cheery voice,
``I'm sure you ain't all alone with such a lot of folks belonging to you, and all so rich and clever. You'll be petted to pieces, Debby says, because you are the only girl in the family.''
Phebe's last words made Rose smile in spite of her tears, and she looked out from behind her apron with an April face, saying in a tone of comic distress,
``That's one of my troubles! I've got six aunts, and they all want me, and I don't know any of them very well. Papa named this place the Aunt-hill, and now I see why.''
Phebe laughed with her as she said encouragingly,
``Everyone calls it so, and it's a real good name, for all the Mrs. Campbells live handy by, and keep coming up to see the old ladies.''
``I could stand the aunts, but there are dozens of cousins, dreadful boys all of them, and I detest boys! Some of them came to see me last Wednesday, but I was lying down, and when auntie came to call me I went under the quilt and pretended to be asleep. I shall have to see them some time, but I do dread it so.'' And Rose gave a shudder, for, having lived alone with her invalid father, she knew nothing of boys, and considered them a species of wild animal.
``Oh! I guess you'll like 'em. I've seen 'em flying round when they come over from the Point, sometimes in their boats and sometimes on horseback. If you like boats and horses, you'll enjoy yourself first-rate.''
``But I don't! I'm afraid of horses, and boats make me ill, and I hate boys!'' And poor Rose wrung her hands at the awful prospect before her. One of these horrors alone she could have borne, but all together were too much for her, and she began to think of a speedy return to the detested school.
Phebe laughed at her woe till the beans danced in the pan, but tried to comfort her by suggesting a means of relief.
``Perhaps your uncle will take you away where there ain't any boys. Debby says he is a real kind man, and always bring heaps of nice things when he comes.''
``Yes, but you see that is another trouble, for I don't know Uncle Alec at all. He hardly ever came to see us, though he sent me pretty things very often. Now I belong to him, and shall have to mind him, till I am eighteen. I may not like him a bit, and I fret about it all the time.''
``Well, I wouldn't borrow trouble, but have a real good time. I'm sure I should think I was in clover if I had folks and money, and nothing to do but enjoy myself,'' began Phebe, but got no further, for a sudden rush and tumble outside made them both jump.
``It's thunder,'' said Phebe.
``It's a circus!'' cried Rose, who from her elevated perch had caught glimpses of a gay cart of some sort and several ponies with flying manes and tails.
The sound died away, and the girls were about to continue their confidences when old Debby appeared, looking rather cross and sleepy after her nap.
``You are wanted in the parlor, Miss Rose.''
``Has anybody come?''
``Little girls shouldn't ask questions, but do as they are bid,'' was all Debby would answer.
``I do hope it isn't Aunt Myra; she always scares me out of my wits asking how my cough is, and groaning over me as if I was going to die,'' said Rose, preparing to retire the way she came, for the slide, being cut for the admission of bouncing Christmas turkeys and puddings, was plenty large enough for a slender girl.
``Guess you'll wish it was Aunt Myra when you see who has come. Don't never let me catch you coming into my kitchen that way again, or I'll shut you up in the big b'iler,'' growled Debby, who thought it her duty to snub children on all occasions.
\gutchapter{Chapter 2---The Clan}
Rose scrambled into the china-closet as rapidly as possible, and there refreshed herself by making faces at Debby, while she settled her plumage and screwed up her courage. Then she crept softly down the hall and peeped into the parlor. No one appeared, and all was so still she felt sure the company was upstairs. So she skipped boldly through the half-open folding-doors, to behold on the other side a sight that nearly took her breath away.
Seven boys stood in a row all ages, all sizes, all yellow-haired and blue-eyed, all in full Scotch costume, and all smiling, nodding, and saying as with one voice, ``How are you, cousin?''
Rose gave a little gasp, and looked wildly about her as if ready to fly, for fear magnified the seven and the room seemed full of boys. Before she could run, however, the tallest lad stepped out of the line, saying pleasantly,
``Don't be frightened. This is the Clan come to welcome you; and I'm the chief, Archie, at your service.''
He held out his hand as he spoke, and Rose timidly put her own into a brown paw, which closed over the white morsel and held it as the chief continued his introductions.
``We came in full rig, for we always turn out in style on grand occasions. Hope you like it. Now I'll tell you who these chaps are, and then we shall be all right. This big one is Prince Charlie, Aunt Clara's boy. She has but one, so he is an extra good one. This old fellow is Mac, the bookworm, called Worm for short. This sweet creature is Steve the Dandy. Look at his gloves and top-knot, if you please. They are Aunt Jane's lads, and a precious pair you'd better believe. These are the Brats, my brothers, Geordie and Will, and Jamie the Baby. Now, my men, step out and show your manners.''
At this command, to Rose's great dismay, six more hands were offered, and it was evident that she was expected to shake them all. It was a trying moment to the bashful child; but, remembering that they were her kinsmen come to welcome her, she tried her best to return the greeting cordially.
This impressive ceremony being over, the Clan broke ranks, and both rooms instantly appeared to be pervaded with boys. Rose hastily retired to the shelter of a big chair and sat there watching the invaders and wondering when her aunt would come and rescue her.
As if bound to do their duty manfully, yet rather oppressed by it, each lad paused beside her chair in his wanderings, made a brief remark, received a still briefer answer, and then sheered off with a relieved expression.
Archie came first, and, leaning over the chair-back, observed in a paternal tone,
``I'm glad you've come, cousin, and I hope you'll find the Aunt-hill pretty jolly.''
``I think I shall.''
Mac shook his hair out of his eyes, stumbled over a stool, and asked abruptly,
``Did you bring any books with you?''
``Four boxes full. They are in the library.''
Mac vanished from the room, and Steve, striking an attitude which displayed his costume effectively, said with an affable smile,
``We were sorry not to see you last Wednesday. I hope your cold is better.''
``Yes, thank you.'' And a smile began to dimple about Rose's mouth, as she remembered her retreat under the bed-cover.
Feeling that he had been received with distinguished marks of attention, Steve strolled away with his topknot higher than ever, and Prince Charlie pranced across the room, saying in a free and easy tone,
``Mamma sent her love and hopes you will be well enough to come over for a day next week. It must be desperately dull here for a little thing like you.''
``I'm thirteen and a half, though I do look small,'' cried Rose, forgetting her shyness in indignation at this insult to her newly acquired teens.
``Beg pardon, ma'am; never should have guessed it.'' And Charlie went off with a laugh, glad to have struck a spark out of his meek cousin.
Geordie and Will came together, two sturdy eleven and twelve year olders, and, fixing their round blue eyes on Rose, fired off a question apiece, as if it was a shooting match and she the target.
``Did you bring your monkey?''
``No; he is dead.''
``Are you going to have a boat?''
``I hope not.''
Here the two, with a right-about-face movement, abruptly marched away, and little Jamie demanded with childish frankness,
``Did you bring me anything nice?''
``Yes, lots of candy,'' answered Rose, whereupon Jamie ascended into her lap with a sounding kiss and the announcement that he liked her very much.
This proceeding rather startled Rose, for the other lads looked and laughed, and in her confusion she said hastily to the young usurper,
``Did you see the circus go by?''
``When? Where?'' cried all the boys in great excitement at once.
``Just before you came. At least I thought it was a circus, for I saw a red and black sort of cart and ever so many little ponies, and---''
She got no farther, for a general shout made her pause suddenly, as Archie explained the joke by saying in the middle of his laugh,
``It was our new dog-cart and the Shetland ponies. You'll never hear the last of your circus, cousin.''
``But there were so many, and they went so fast, and the cart was so very red,'' began Rose, trying to explain her mistake.
``Come and see them all!'' cried the Prince. And before she knew what was happening, she was borne away to the barn and tumultuously introduced to three shaggy ponies and the gay new dog-cart.
She had never visited these regions before, and had her doubts as to the propriety of her being there now, but when she suggested that ``Auntie might not like it,'' there was a general cry of,
``She told us to amuse you, and we can do it ever so much better out here than poking round in the house.''
``I'm afraid I shall get cold without my sacque,'' began Rose, who wanted to stay, but felt rather out of her element.
``No, you won't! We'll fix you,'' cried the lads, as one clapped his cap on her head, another tied a rough jacket round her neck by the sleeves, a third neatly smothered her in a carriage blanket, and a fourth threw open the door of the old barouche that stood there, saying with a flourish,
``Step in, ma'am, and make yourself comfortable while we show you some fun.''
So Rose sat in state enjoying herself very much, for the lads proceeded to dance a Highland Fling with a spirit and skill that made her clap her hands and laugh as she had not done for weeks.
``How is that, my lassie?'' asked the Prince, coming up all flushed and breathless when the ballet was over.
``It was splendid! I never went to the theatre but once, and the dancing was not half so pretty as this. What clever boys you must be!'' said Rose, smiling upon her kinsmen like a little queen upon her subjects.
``Ah, we're a fine lot, and that is only the beginning of our larks. We haven't got the pipes here or we'd,
'Sing for you, play for you
A dulcy melody,'''
answered Charlie, looking much elated at her praise.
``I did not know we were Scotch; papa never said anything about it, or seemed to care about Scotland, except to have me sing the old ballads,'' said Rose, beginning to feel as if she had left America behind her somewhere.
``Neither did we till lately. We've been reading Scott's novels, and all of a sudden we remembered that our grandfather was a Scotchman. So we hunted up the old stories, got a bagpipe, put on our plaids, and went in, heart and soul, for the glory of the Clan. We've been at it some time now, and it's great fun. Our people like it, and I think we are a pretty canny set.''
Archie said this from the other coach-step, where he had perched, while the rest climbed up before and behind to join in the chat as they rested.
``I'm Fitzjames and he's Roderick Dhu, and we'll give you the broadsword combat some day. It's a great thing, you'd better believe,'' added the Prince.
``Yes, and you should hear Steve play the pipes. He makes 'em skirl like a good one,'' cried Will from the box, eager to air the accomplishments of his race.
``Mac's the fellow to hunt up the old stories and tell us how to dress right, and pick out rousing bits for us to speak and sing,'' put in Geordie, saying a good word for the absent Worm.
``And what do you and Will do?'' asked Rose of Jamie, who sat beside her as if bound to keep her in sight till the promised gift had been handed over.
``Oh, I'm the little foot-page, and do errands, and Will and Geordie are the troops when we march, and the stags when we hunt, and the traitors when we want to cut any heads off.''
``They are very obliging, I'm sure,'' said Rose, whereat the ``utility men'' beamed with modest pride and resolved to enact Wallace and Montrose as soon as possible for their cousin's special benefit.
``Let's have a game of tag,'' cried the Prince, swinging himself up to a beam with a sounding slap on Stevie's shoulder.
Regardless of his gloves, Dandy tore after him, and the rest swarmed in every direction as if bent on breaking their necks and dislocating their joints as rapidly as possible.
It was a new and astonishing spectacle to Rose, fresh from a prim boarding-school, and she watched the active lads with breathless interest, thinking their antics far superior to those of Mops, the dear departed monkey.
Will had just covered himself with glory by pitching off a high loft head first and coming up all right, when Phebe appeared with a cloak, hood, and rubbers, also a message from Aunt Plenty that ``Miss Rose was to come in directly.''
``All right; we'll bring her!'' answered Archie, issuing some mysterious order, which was so promptly obeyed that, before Rose could get out of the carriage, the boys had caught hold of the pole and rattled her out of the barn, round the oval and up to the front door with a cheer that brought two caps to an upper window, and caused Debby to cry aloud from the back porch,
``Them harum-scarum boys will certainly be the death of that delicate little creter!''
But the ``delicate little creter'' seemed all the better for her trip, and ran up the steps looking rosy, gay, and dishevelled, to be received with lamentation by Aunt Plenty, who begged her to go and lie down at once.
``Oh, please don't! We have come to tea with our cousin, and we'll be as good as gold if you'll let us stay, auntie,'' clamoured the boys, who not only approved of ``our cousin'' but had no mind to lose their tea, for Aunt Plenty's name but feebly expressed her bountiful nature.
``Well, dears, you can; only be quiet, and let Rose go and take her iron and be made tidy, and then we will see what we can find for supper,'' said the old lady as she trotted away, followed by a volley of directions for the approaching feast.
``Marmalade for me, auntie.''
``Plenty of plum-cake, please.''
``Tell Debby to trot out the baked pears.''
``I'm your man for lemon-pie, ma'am.''
``Do have fritters; Rose will like 'em.''
``She'd rather have tarts, I know.''
When Rose came down, fifteen minutes later, with every curl smoothed and her most beruffled apron on, she found the boys loafing about the long hall, and paused on the half-way landing to take an observation, for till now she had not really examined her new-found cousins.
There was a strong family resemblance among them, though some of the yellow heads were darker than others, some of the cheeks brown instead of rosy, and the ages varied all the way from sixteen-year-old Archie to Jamie, who was ten years younger. None of them were especially comely but the Prince, yet all were hearty, happy-looking lads, and Rose decided that boys were not as dreadful as she had expected to find them.
They were all so characteristically employed that she could not help smiling as she looked. Archie and Charlie, evidently great cronies, were pacing up and down, shoulder to shoulder, whistling ``Bonnie Dundee''; Mac was reading in a corner, with his book close to his near-sighted eyes; Dandy was arranging his hair before the oval glass in the hat-stand; Geordie and Will investigating the internal economy of the moon-faced clock; and Jamie lay kicking up his heels on the mat at the foot of the stairs, bent on demanding his sweeties the instant Rose appeared.
She guessed his intention, and forestalled his demand by dropping a handful of sugar-plums down upon him.
At his cry of rapture the other lads looked up and smiled involuntarily, for the little kinswoman standing there above was a winsome sight with her shy, soft eyes, bright hair, and laughing face. The black frock reminded them of her loss, and filled the boyish hearts with a kindly desire to be good to ``our cousin,'' who had no longer any home but this.
``There she is, as fine as you please,'' cried Steve, kissing his hand to her.
``Come on, Missy; tea is ready,'' added the Prince encouragingly.
``I shall take her in.'' And Archie offered his arm with great dignity, an honour that made Rose turn as red as a cherry and long to run upstairs again.
It was a merry supper, and the two elder boys added much to the fun by tormenting the rest with dark hints of some interesting event which was about to occur. Something uncommonly fine, they declared it was, but enveloped in the deepest mystery for the present.
``Did I ever see it?'' asked Jamie.
``Not to remember it; but Mac and Steve have, and liked it immensely,'' answered Archie, thereby causing the two mentioned to neglect Debby's delectable fritters for several minutes, while they cudgelled their brains.
``Who will have it first?'' asked Will, with his mouth full of marmalade.
``Aunt Plenty, I guess.''
``When will she have it?'' demanded Geordie, bouncing in his seat with impatience.
``Sometime on Monday.''
``Heart alive! what is the boy talking about?'' cried the old lady from behind the tall urn, which left little to be seen but the topmost bow of her cap.
``Doesn't auntie know?'' asked a chorus of voices.
``No; and that's the best of the joke, for she is desperately fond of it.''
``What colour is it?'' asked Rose, joining in the fun.
``Blue and brown.''
``Is it good to eat?'' asked Jamie.
``Some people think so, but I shouldn't like to try it,'' answered Charlie, laughing so he split his tea.
``Who does it belong to?'' put in Steve.
Archie and the Prince stared at one another rather blankly for a minute, then Archie answered with a twinkle of the eye that made Charlie explode again,
``To Grandfather Campbell.''
This was a poser, and they gave up the puzzle, though Jamie confided to Rose that he did not think he could live till Monday without knowing what this remarkable thing was.
Soon after tea the Clan departed, singing ``All the blue bonnets are over the border,'' at the tops of their voices.
``Well, dear, how do you like your cousins?'' asked Aunt Plenty, as the last pony frisked round the corner and the din died away.
``Pretty well, ma'am; but I like Phebe better.'' An answer which caused Aunt Plenty to hold up her hands in despair and trot away to tell sister Peace that she never should understand that child, and it was a mercy Alec was coming soon to take the responsibility off their hands.
Fatigued by the unusual exertions of the afternoon, Rose curled herself up in the sofa corner to rest and think about the great mystery, little guessing that she was to know it first of all.
Right in the middle of her meditations she fell asleep and dreamed she was at home again in her own little bed. She seemed to wake and see her father bending over her; to hear him say, ``My little Rose''; to answer, ``Yes, papa''; and then to feel him take her in his arms and kiss her tenderly. So sweet, so real was the dream, that she started up with a cry of joy to find herself in the arms of a brown, bearded man, who held her close, and whispered in a voice so like her father's that she clung to him involuntarily,
``This is my little girl, and I am Uncle Alec.''
\gutchapter{Chapter 3---Uncles}
When Rose woke next morning, she was not sure whether she had dreamed what occurred the night before, or it had actually happened. So she hopped up and dressed, although it was an hour earlier than she usually rose, for she could not sleep any more, being possessed with a strong desire to slip down and see if the big portmanteau and packing cases were really in the hall. She seemed to remember tumbling over them when she went to bed, for the aunts had sent her off very punctually, because they wanted their pet nephew all to themselves.
The sun was shining, and Rose opened her window to let in the soft May air fresh from the sea. As she leaned over her little balcony, watching an early bird get the worm, and wondering how she should like Uncle Alec, she saw a man leap the garden wall and come whistling up the path. At first she thought it was some trespasser, but a second look showed her that it was her uncle returning from an early dip into the sea. She had hardly dared to look at him the night before, because whenever she tried to do so she always found a pair of keen blue eyes looking at her. Now she could take a good stare at him as he lingered along, looking about him as if glad to see the old place again.
A brown, breezy man, in a blue jacket, with no hat on the curly head, which he shook now and then like a water dog; broad-shouldered, alert in his motions, and with a general air of strength and stability about him which pleased Rose, though she could not explain the feeling of comfort it gave her. She had just said to herself, with a sense of relief, ``I guess I shall like him, though he looks as if he made people mind,'' when he lifted his eyes to examine the budding horse-chestnut overhead, and saw the eager face peering down at him. He waved his hand to her, nodded, and called out in a bluff, cheery voice,
``You are on deck early, little niece.''
``I got up to see if you had really come, uncle.''
``Did you? Well, come down here and make sure of it.''
``I'm not allowed to go out before breakfast, sir.''
``Oh, indeed!'' with a shrug. ``Then I'll come aboard and salute,'' he added; and, to Rose's great amazement, Uncle Alec went up one of the pillars of the back piazza hand over hand, stepped across the roof, and swung himself into her balcony, saying, as he landed on the wide balustrade: ``Have you any doubts about me now, ma'am?''
Rose was so taken aback, she could only answer with a smile as she went to meet him.
``How does my girl do this morning?'' he asked, taking the little cold hand she gave him in both his big warm ones.
``Pretty well, thank you, sir.''
``Ah, but it should be very well. Why isn't it?''
``I always wake up with a headache, and feel tired.''
``Don't you sleep well?''
``I lie awake a long time, and then I dream, and my sleep does not seem to rest me much.''
``What do you do all day?''
``Oh, I read, and sew a little, and take naps, and sit with auntie.''
``No running about out of doors, or house-work, or riding, hey?''
``Aunt Plenty says I'm not strong enough for much exercise. I drive out with her sometimes, but I don't care for it.''
``I'm not surprised at that,'' said Uncle Alec, half to himself, adding, in his quick way: ``Who have you had to play with?''
``No one but Ariadne Blish, and she was such a goose I couldn't bear her. The boys came yesterday, and seemed rather nice; but, of course, I couldn't play with them.''
``Why not?''
``I'm too old to play with boys.''
``Not a bit of it; that's just what you need, for you've been molly-coddled too much. They are good lads, and you'll be mixed up with them more or less for years to come, so you may as well be friends and playmates at once. I will look you up some girls also, if I can find a sensible one who is not spoilt by her nonsensical education.''
``Phebe is sensible, I'm sure, and I like her, though I only saw her yesterday,'' cried Rose, waking up suddenly.
``And who is Phebe, if you please?''
Rose eagerly told all she knew, and Uncle Alec listened, with an odd smile lurking about his mouth, though his eyes were quite sober as he watched the face before him.
``I'm glad to see that you are not aristocratic in your tastes, but I don't quite make out why you like this young lady from the poor-house.''
``You may laugh at me, but I do. I can't tell why, only she seems so happy and busy, and sings so beautifully, and is strong enough to scrub and sweep, and hasn't any troubles to plague her,'' said Rose, making a funny jumble of reasons in her efforts to explain.
``How do you know that?''
``Oh, I was telling her about mine, and asked if she had any, and she said, 'No, only I'd like to go to school, and I mean to some day.''
``So she doesn't call desertion, poverty, and hard work, troubles? She's a brave little girl, and I shall be proud to know her.'' And Uncle Alec gave an approving nod, that made Rose wish she had been the one to earn it.
``But what are these troubles of yours, child?'' he asked, after a minute of silence.
``Please don't ask me, uncle.''
``Can't you tell them to me as well as to Phebe?''
Something in his tone made Rose feel that it would be better to speak out and be done with it, so she answered, with sudden colour and averted eyes,
``The greatest one was losing dear papa.''
As she said that, Uncle Alec's arm came gently round her, and he drew her to him, saying, in the voice so like papa's,
``That is a trouble which I cannot cure, my child; but I shall try to make you feel it less. What else, dear?''
``I am so tired and poorly all the time, I can't do anything I want to, and it makes me cross,'' sighed Rose, rubbing the aching head like a fretful child.
``That we can cure and we will,'' said her uncle, with a decided nod that made the curls bob on his head, to that Rose saw the gray ones underneath the brown.
``Aunt Myra says I have no constitution, and never shall be strong,'' observed Rose, in a pensive tone, as if it was rather a nice thing to be an invalid.
``Aunt Myra is a ahem! an excellent woman, but it is her hobby to believe that everyone is tottering on the brink of the grave; and, upon my life, I believe she is offended if people don't fall into it! We will show her how to make constitutions and turn pale-faced little ghosts into rosy, hearty girls. That's my business, you know,'' he added, more quietly, for his sudden outburst had rather startled Rose.
``I had forgotten you were a doctor. I'm glad of it, for I do want to be well, only I hope you won't give me much medicine, for I've taken quarts already, and it does me no good.''
As she spoke, Rose pointed to a little table just inside the window, on which appeared a regiment of bottles.
``Ah, ha! Now we'll see what mischief these blessed women have been at.'' And, making a long arm, Dr. Alec set the bottles on the wide railing before him, examined each carefully, smiled over some, frowned over others, and said, as he put down the last: ``Now I'll show you the best way to take these messes.'' And, as quick as a flash, he sent one after another smashing down into the posy-beds below.
``But Aunt Plenty won't like it; and Aunt Myra will be angry, for she sent most of them!'' cried Rose, half frightened and half pleased at such energetic measures.
``You are my patient now, and I'll take the responsibility. My way of giving physic is evidently the best, for you look better already,'' he said, laughing so infectiously that Rose followed suit, saying saucily,
``If I don't like your medicines any better than those, I shall throw them into the garden, and then what will you do?''
``When I prescribe such rubbish, I'll give you leave to pitch it overboard as soon as you like. Now what is the next trouble?''
``I hoped you would forget to ask.''
``But how can I help you if I don't know them? Come, let us have No. 3.''
``It is very wrong, I suppose, but I do sometimes wish I had not quite so many aunts. They are all very good to me, and I want to please them; but they are so different, I feel sort of pulled to pieces among them,'' said Rose, trying to express the emotions of a stray chicken with six hens all clucking over it at once.
Uncle Alec threw back his head and laughed like a boy, for he could entirely understand how the good ladies had each put in her oar and tried to paddle her own way, to the great disturbance of the waters and the entire bewilderment of poor Rose.
``I intend to try a course of uncles now, and see how that suits your constitution. I'm going to have you all to myself, and no one is to give a word of advice unless I ask it. There is no other way to keep order aboard, and I am captain of this little craft, for a time at least. What comes next?''
But Rose stuck there, and grew so red, her uncle guessed what that trouble was.
``I don't think I can tell this one. It wouldn't be polite, and I feel pretty sure that it isn't going to be a trouble any more.''
As she blushed and stammered over these words, Dr. Alec turned his eyes away to the distant sea, and said so seriously, so tenderly, that she felt every word and long remembered them,
``My child, I don't expect you to love and trust me all at once, but I do want you to believe that I shall give my whole heart to this new duty; and if I make mistakes, as I probably shall, no one will grieve over them more bitterly than I. It is my fault that I am a stranger to you, when I want to be your best friend. That is one of my mistakes, and I never repented it more deeply than I do now. Your father and I had a trouble once, and I thought I could never forgive him; so I kept away for years. Thank God, we made it all up the last time I saw him, and he told me then, that if he was forced to leave her he should bequeath his little girl to me as a token of his love. I can't fill his place, but I shall try to be a father to her; and if she learns to love me half as well as she did the good one she has lost, I shall be a proud and happy man. Will she believe this and try?''
Something in Uncle Alec's face touched Rose to the heart, and when he held out his hand with that anxious troubled look in his eyes, she was moved to put up her innocent lips and seal the contract with a confiding kiss. The strong arm held her close a minute, and she felt the broad chest heave once as if with a great sigh of relief; but not a word was spoken till a tap at the door made both start.
Rose popped her head through the window to say ``come in,'' while Dr. Alec hastily rubbed the sleeve of his jacket across his eyes and began to whistle again.
Phebe appeared with a cup of coffee.
``Debby told me to bring this and help you get up,'' she said, opening her black eyes wide, as if she wondered how on earth ``the sailor man'' got there.
``I'm all dressed, so I don't need any help. I hope that is good and strong,'' added Rose, eyeing the steaming cup with an eager look.
But she did not get it, for a brown hand took possession of it as her uncle said quickly,
``Hold hard, my lass, and let me overhaul that dose before you take it. Do you drink all this strong coffee every morning, Rose?''
``Yes, sir, and I like it. Auntie says it `tones' me up, and I always feel better after it.''
``This accounts for the sleepless nights, the flutter your heart gets into at the least start, and this is why that cheek of yours is pale yellow instead of rosy red. No more coffee for you, my dear, and by and by you'll see that I am right. Any new milk downstairs, Phebe?''
``Yes, sir, plenty right in from the barn.''
``That's the drink for my patient. Go bring me a pitcherful, and another cup; I want a draught myself. This won't hurt the honeysuckles, for they have no nerves to speak of.'' And, to Rose's great discomfort, the coffee went after the medicine.
Dr. Alec saw the injured look she put on, but took no notice, and presently banished it by saying pleasantly,
``I've got a capital little cup among my traps, and I'll give it to you to drink your milk in, as it is made of wood that is supposed to improve whatever is put into it something like a quassia cup. That reminds me; one of the boxes Phebe wanted to lug upstairs last night is for you. Knowing that I was coming home to find a ready-made daughter, I picked up all sorts of odd and pretty trifles along the way, hoping she would be able to find something she liked among them all. Early to-morrow we'll have a grand rummage. Here's our milk! I propose the health of Miss Rose Campbell and drink it with all my heart.''
It was impossible for Rose to pout with the prospect of a delightful boxful of gifts dancing before her eyes; so, in spite of herself, she smiled as she drank her own health, and found that fresh milk was not a hard dose to take.
``Now I must be off, before I am caught again with my wig in a toss,'' said Dr. Alec, preparing to descend the way he came.
``Do you always go in and out like a cat, uncle?'' asked Rose, much amused at his odd ways.
``I used to sneak out of my window when I was a boy, so I need not disturb the aunts, and now I rather like it, for it's the shortest road, and it keeps me limber when I have no rigging to climb. Good-bye till breakfast.'' And away he went down the water-spout, over the roof, and vanished among the budding honey-suckles below.
``Ain't he a funny guardeen?'' exclaimed Phebe, as she went off with the cups.
``He is a very kind one, I think,'' answered Rose, following, to prowl round the big boxes and try to guess which was hers.
When her uncle appeared at sound of the bell, he found her surveying with an anxious face a new dish that smoked upon the table.
``Got a fresh trouble, Rosy?'' he asked, stroking her smooth head.
``Uncle, are you going to make me eat oatmeal?'' asked Rose, in a tragic tone.
``Don't you like it?''
``I de-test it!'' answered Rose, with all the emphasis which a turned-up nose, a shudder, and a groan could give to the three words.
``You are not a true Scotchwoman, if you don't like the `parritch.' It's a pity, for I made it myself, and thought we'd have such a good time with all that cream to float it in. Well, never mind.'' And he sat down with a disappointed air.
Rose had made up her mind to be obstinate about it, because she did heartily ``detest'' the dish; but as Uncle Alec did not attempt to make her obey, she suddenly changed her mind and thought she would.
``I'll try to eat it to please you, uncle; but people are always saying how wholesome it is, and that makes me hate it,'' she said, half-ashamed at her silly excuse.
``I do want you to like it, because I wish my girl to be as well and strong as Jessie's boys, who are brought up on this in the good old fashion. No hot bread and fried stuff for them, and they are the biggest and bonniest lads of the lot. Bless you, auntie, and good morning!''
Dr. Alec turned to greet the old lady, and, with a firm resolve to eat or die in the attempt, Rose sat down.
In five minutes she forgot what she was eating, so interested was she in the chat that went on. It amused her very much to hear Aunt Plenty call her forty-year-old nephew ``my dear boy''; and Uncle Alec was so full of lively gossip about all creation in general, and the Aunt-hill in particular, that the detested porridge vanished without a murmur.
``You will go to church with us, I hope, Alec, if you are not too tired,'' said the old lady, when breakfast was over.
``I came all the way from Calcutta for that express purpose, ma'am. Only I must send the sisters word of my arrival, for they don't expect me till to-morrow, you know, and there will be a row in church if those boys see me without warning.''
``I'll send Ben up the hill, and you can step over to Myra's yourself; it will please her, and you will have plenty of time.''
Dr. Alec was off at once, and they saw no more of him till the old barouche was at the door, and Aunt Plenty just rustling downstairs in her Sunday best, with Rose like a little black shadow behind her.
Away they drove in state, and all the way Uncle Alec's hat was more off his head than on, for everyone they met smiled and bowed, and gave him as blithe a greeting as the day permitted.
It was evident that the warning had been a wise one, for, in spite of time and place, the lads were in such a ferment that their elders sat in momentary dread of an unseemly outbreak somewhere. It was simply impossible to keep those fourteen eyes off Uncle Alec, and the dreadful things that were done during sermon-time will hardly be believed.
Rose dared not look up after a while, for these bad boys vented their emotions upon her till she was ready to laugh and cry with mingled amusement and vexation. Charlie winked rapturously at her behind his mother's fan; Mac openly pointed to the tall figure beside her; Jamie stared fixedly over the back of his pew, till Rose thought his round eyes would drop out of his head; George fell over a stool and dropped three books in his excitement; Will drew sailors and Chinamen on his clean cuffs, and displayed them, to Rose's great tribulation; Steve nearly upset the whole party by burning his nose with salts, as he pretended to be overcome by his joy; even dignified Archie disgraced himself by writing in his hymn book, ``Isn't he blue and brown?'' and passing it politely to Rose.
Her only salvation was trying to fix her attention upon Uncle Mac a portly, placid gentleman, who seemed entirely unconscious of the iniquities of the Clan, and dozed peacefully in his pew corner. This was the only uncle Rose had met for years, for Uncle Jem and Uncle Steve, the husbands of Aunt Jessie and Aunt Clara, were at sea, and Aunt Myra was a widow. Uncle Mac was a merchant, very rich and busy, and as quiet as a mouse at home, for he was in such a minority among the women folk he dared not open his lips, and let his wife rule undisturbed.
Rose liked the big, kindly, silent man who came to her when papa died, was always sending her splendid boxes of goodies at school, and often invited her into his great warehouse, full of teas and spices, wines and all sorts of foreign fruits, there to eat and carry away whatever she liked. She had secretly regretted that he was not to be her guardian; but since she had seen Uncle Alec she felt better about it, for she did not particularly admire Aunt Jane.
When church was over, Dr. Alec got into the porch as quickly as possible, and there the young bears had a hug all round, while the sisters shook hands and welcomed him with bright faces and glad hearts. Rose was nearly crushed flat behind a door in that dangerous passage from pew to porch; but Uncle Mac rescued her, and put her into the carriage for safe keeping.
``Now, girls, I want you to come and dine with Alec; Mac also, of course. But I cannot ask the boys, for we did not expect this dear fellow till tomorrow, you know, so I made no preparations. Send the lads home, and let them wait till Monday, for really I was shocked at their behaviour in church,'' said Aunt Plenty, as she followed Rose.
In any other place the defrauded boys would have set up a howl; as it was, they growled and protested till Dr. Alec settled the matter by saying,
``Never mind, old chaps, I'll make it up to you to-morrow, if you sheer off quietly; if you don't, not a blessed thing shall you have out of my big boxes.''
\gutchapter{Chapter 4---Aunts}
All dinner-time Rose felt that she was going to be talked about, and afterward she was sure of it, for Aunt Plenty whispered to her as they went into the parlour,
``Run up and sit awhile with Sister Peace, my dear. She likes to have you read while she rests, and we are going to be busy.''
Rose obeyed, and the quiet rooms above were so like a church that she soon composed her ruffled feelings, and was unconsciously a little minister of happiness to the sweet old lady, who for years had sat there patiently waiting to be set free from pain.
Rose knew the sad romance of her life, and it gave a certain tender charm to this great-aunt of hers, whom she already loved. When Peace was twenty, she was about to be married; all was done, the wedding dress lay ready, the flowers were waiting to be put on, the happy hour at hand, when word came that the lover was dead. They thought that gentle Peace would die, too; but she bore it bravely, put away her bridal gear, took up her life afresh, and lived on a beautiful, meek woman, with hair as white as snow and cheeks that never bloomed again. She wore no black, but soft, pale colours, as if always ready for the marriage that had never come.
For thirty years she had lived on, fading slowly, but cheerful, busy, and full of interest in all that went on in the family; especially the joys and sorrows of the young girls growing up about her, and to them she was adviser, confidante, and friend in all their tender trials and delights. A truly beautiful old maiden, with her silvery hair, tranquil face, and an atmosphere of repose about her that soothed whoever came to her!
Aunt Plenty was utterly dissimilar, being a stout, brisk old lady, with a sharp eye, a lively tongue, and a face like a winter-apple. Always trotting, chatting, and bustling, she was a regular Martha, cumbered with the cares of this world and quite happy in them.
Rose was right; and while she softly read psalms to Aunt Peace, the other ladies were talking about her little self in the frankest manner.
``Well, Alec, how do you like your ward?'' began Aunt Jane, as they all settled down, and Uncle Mac deposited himself in a corner to finish his doze.
``I should like her better if I could have begun at the beginning, and so got a fair start. Poor George led such a solitary life that the child has suffered in many ways, and since he died she has been going on worse than ever, judging from the state I find her in.''
``My dear boy, we did what we thought best while waiting for you to wind up your affairs and get home. I always told George he was wrong to bring her up as he did; but he never took my advice, and now here we are with this poor dear child upon our hands. I, for one, freely confess that I don't know what to do with her any more than if she was one of those strange, outlandish birds you used to bring home from foreign parts.'' And Aunt Plenty gave a perplexed shake of the head which caused great commotion among the stiff loops of purple ribbon that bristled all over the cap like crocus buds.
``If my advice had been taken, she would have remained at the excellent school where I placed her. But our aunt thought best to remove her because she complained, and she has been dawdling about ever since she came. A most ruinous state of things for a morbid, spoilt girl like Rose,'' said Mrs. Jane, severely.
She had never forgiven the old ladies for yielding to Rose's pathetic petition that she might wait her guardian's arrival before beginning another term at the school, which was a regular Blimber hot-bed, and turned out many a feminine Toots.
``I never thought it the proper school for a child in good circumstances an heiress, in fact, as Rose is. It is all very well for girls who are to get their own living by teaching, and that sort of thing; but all she needs is a year or two at a fashionable finishing school, so that at eighteen she can come out with eclat,'' put in Aunt Clara, who had been a beauty and a belle, and was still a handsome woman.
``Dear, dear! how short-sighted you all are to be discussing education and plans for the future, when this unhappy child is so plainly marked for the tomb,'' sighed Aunt Myra, with a lugubrious sniff and a solemn wag of the funereal bonnet, which she refused to remove, being afflicted with a chronic catarrh.
``Now, it is my opinion that the dear thing only wants freedom, rest, and care. There is look in her eyes that goes to my heart, for it shows that she feels the need of what none of us can give her a mother,'' said Aunt Jessie, with tears in her own bright eyes at the thought of her boys being left, as Rose was, to the care of others.
Uncle Alec, who had listened silently as each spoke, turned quickly towards the last sister, and said, with a decided nod of approval,
``You've got it, Jessie; and, with you to help me, I hope to make the child feel that she is not quite fatherless and motherless.''
``I'll do my best, Alec; and I think you will need me, for, wise as you are, you cannot understand a tender, timid little creature like Rose as a woman can,'' said Mrs. Jessie, smiling back at him with a heart full of motherly goodwill.
``I cannot help feeling that I, who have had a daughter of my own, can best bring up a girl; and I am very much surprised that George did not entrust her to me,'' observed Aunt Myra, with an air of melancholy importance, for she was the only one who had given a daughter to the family, and she felt that she had distinguished herself, though ill-natured people said that she had dosed her darling to death.
``I never blamed him in the least, when I remember the perilous experiments you tried with poor Carrie,'' began Mrs. Jane, in her hard voice.
``Jane Campbell, I will not hear a word! My sainted Caroline is a sacred object,'' cried Aunt Myra, rising as if to leave the room.
Dr. Alec detained her, feeling that he must define his position at once, and maintain it manfully if he hoped to have any success in his new undertaking.
``Now, my dear souls, don't let us quarrel and make Rose a bone of contention though, upon my word, she is almost a bone, poor little lass! You have had her among you for a year, and done what you liked. I cannot say that your success is great, but that is owing to too many fingers in the pie. Now, I intend to try my way for a year, and if at the end of it she is not in better trim than now, I'll give up the case, and hand her over to someone else. That's fair, I think.''
``She will not be here a year hence, poor darling, so no one need dread future responsibility,'' said Aunt Myra, folding her black gloves as if all ready for the funeral.
``By Jupiter! Myra, you are enough to damp the ardour of a saint!'' cried Dr. Alec, with a sudden spark in his eyes. ``Your croaking will worry that child out of her wits, for she is an imaginative puss, and will fret and fancy untold horrors. You have put it into her head that she has no constitution, and she rather likes the idea. If she had not had a pretty good one, she would have been `marked for the tomb' by this time, at the rate you have been going on with her. I will not have any interference please understand that; so just wash your hands of her, and let me manage till I want help, then I'll ask for it.''
``Hear, hear!'' came from the corner where Uncle Mac was apparently wrapt in slumber.
``You were appointed guardian, so we can do nothing. But I predict that the girl will be spoilt, utterly spoilt,'' answered Mrs. Jane, grimly.
``Thank you, sister. I have an idea that if a woman can bring up two boys as perfectly as you do yours, a man, if he devotes his whole mind to it, may at least attempt as much with one girl,'' replied Dr. Alec, with a humorous look that tickled the others immensely, for it was a well-known fact in the family that Jane's boys were more indulged than all the other lads put together.
``I am quite easy, for I really do think that Alec will improve the child's health; and by the time his year is out, it will be quite soon enough for her to go to Madame Roccabella's and be finished off,'' said Aunt Clara, settling her rings, and thinking, with languid satisfaction, of the time when she could bring out a pretty and accomplished niece.
``I suppose you will stay here in the old place, unless you think of marrying, and it's high time you did,'' put in Mrs. Jane, much nettled at her brother's last hit.
``No, thank you. Come and have a cigar, Mac,'' said Dr. Alec, abruptly.
``Don't marry; women enough in the family already,'' muttered Uncle Mac; and then the gentlemen hastily fled.
``Aunt Peace would like to see you all, she says,'' was the message Rose brought before the ladies could begin again.
``Hectic, hectic! dear me, dear me!'' murmured Aunt Myra, as the shadow of her gloomy bonnet fell upon Rose, and the stiff tips of a black glove touched the cheek where the colour deepened under so many eyes.
``I am glad these pretty curls are natural; they will be invaluable by and by,'' said Aunt Clara, taking an observation with her head on one side.
``Now that your uncle has come, I no longer expect you to review the studies of the past year. I trust your time will not be entirely wasted in frivolous sports, however,'' added Aunt Jane, sailing out of the room with the air of a martyr.
Aunt Jessie said not a word, but kissed her little niece, with a look of tender sympathy that made Rose cling to her a minute, and follow her with grateful eyes as the door closed behind her.
After everybody had gone home, Dr. Alec paced up and down the lower hall in the twilight for an hour, thinking so intently that sometimes he frowned, sometimes he smiled, and more than once he stood still in a brown study. All of a sudden he said, half aloud, as if he had made up his mind,
``I might as well begin at once, and give the child something new to think about, for Myra's dismals and Jane's lectures have made her as blue as a little indigo bag.''
Diving into one of the trunks that stood in a corner, he brought up, after a brisk rummage, a silken cushion, prettily embroidered, and a quaint cup of dark carved wood.
``This will do for a start,'' he said, as he plumped up the cushion and dusted the cup. ``It won't do to begin too energetically, or Rose will be frightened. I must beguile her gently and pleasantly along till I've won her confidence, and then she will be ready for anything.''
Just then Phebe came out of the dining-room with a plate of brown bread, for Rose had been allowed no hot biscuit for tea.
``I'll relieve you of some of that,'' said Dr. Alec, and, helping himself to a generous slice, he retired to the study, leaving Phebe to wonder at his appetite.
She would have wondered still more if she had seen him making that brown bread into neat little pills, which he packed into an attractive ivory box, out of which he emptied his own bits of lovage.
``There! if they insist on medicine, I'll order these, and no harm will be done. I will have my own way, but I'll keep the peace, if possible, and confess the joke when my experiment has succeeded,'' he said to himself, looking very much like a mischievous boy, as he went on with his innocent prescriptions.
Rose was playing softly on the small organ that stood in the upper hall, so that Aunt Peace could enjoy it; and all the while he talked with the old ladies, Uncle Alec was listening to the fitful music of the child, and thinking of another Rose who used to play for him.
As the clock struck eight, he called out,
``Time for my girl to be abed, else she won't be up early, and I'm full of jolly plans for to-morrow. Come and see what I've found for you to begin upon.''
Rose ran in and listened with bright attentive face, while Dr. Alec said impressively,
``In my wanderings over the face of the earth, I have picked up some excellent remedies, and, as they are rather agreeable ones, I think you and I will try them. This is a herb-pillow, given to me by a wise old woman when I was ill in India. It is filled with saffron, poppies, and other soothing plants; so lay your little head on it to-night, sleep sweetly without a dream, and wake to-morrow without a pain.''
``Shall I really? How nice it smells.'' And Rose willingly received the pretty pillow, and stood enjoying its faint, sweet odour, as she listened to the doctor's next remedy.
``This is the cup I told you of. Its virtue depends, they say, on the drinker filling it himself; so you must learn to milk. I'll teach you.''
``I'm afraid I never can,'' said Rose; but she surveyed the cup with favour, for a funny little imp danced on the handle, as if all ready to take a header into the white sea below.
``Don't you think she ought to have something more strengthening than milk, Alec? I really shall feel anxious if she does not have a tonic of some sort,'' said Aunt Plenty, eyeing the new remedies suspiciously, for she had more faith in her old-fashioned doses than all the magic cups and poppy pillows of the East.
``Well, ma'am, I'm willing to give her a pill, if you think best. It is a very simple one, and very large quantities may be taken without harm. You know hasheesh is the extract of hemp? Well, this is a preparation of corn and rye, much used in old times, and I hope it will be again.''
``Dear me, how singular!'' said Aunt Plenty, bringing her spectacles to bear upon the pills, with a face so full of respectful interest that it was almost too much for Dr. Alec's gravity.
``Take one in the morning, and a good-night to you, my dear,'' he said, dismissing his patient with a hearty kiss.
Then, as she vanished, he put both hands into his hair, exclaiming, with a comical mixture of anxiety and amusement,
``When I think what I have undertaken, I declare to you, aunt, I feel like running away and not coming back till Rose is eighteen!''
\gutchapter{Chapter 5---A Belt and a Box}
When Rose came out of her chamber, cup in hand, next morning, the first person she saw was Uncle Alec standing on the threshold of the room opposite, which he appeared to be examining with care. When he heard her step, he turned about and began to sing,
``Where are you going, my pretty maid?''
``I'm going a-milking, sir, she said,'' answered Rose, waving the cup; and then they finished the verse together in fine style.
Before either spoke, a head, in a nightcap so large and beruffled that it looked like a cabbage, popped out of a room farther down the hall, and an astonished voice exclaimed,
``What in the world are you doing about so early?''
``Clearing our pipes for the day, ma'am. Look here, auntie, can I have this room?'' said Dr. Alec, making her a sailor's bow.
``Any room you like, except sister's.''
``Thanks. And may I go rummaging round in the garrets and glory-holes to furnish it as I like?''
``My dear boy, you may turn the house upside down if you will only stay in it.''
``That's a handsome offer, I'm sure. I'll stay, ma'am; here's my little anchor, so you will get more than you want of me this time.''
``That's impossible! Put on your jacket, Rose. Don't tire her out with antics, Alec. Yes, sister, I'm coming!'' and the cabbage vanished suddenly.
The first milking lesson was a droll one; but after several scares and many vain attempts, Rose at last managed to fill her cup, while Ben held Clover's tail so that it could not flap, and Dr. Alec kept her from turning to stare at the new milkmaid, who objected to both these proceedings very much.
``You look chilly in spite of all this laughing. Take a smart run round the garden and get up a glow,'' said the doctor, as they left the barn.
``I'm too old for running, uncle; Miss Power said it was not lady-like for girls in their teens,'' answered Rose, primly.
``I take the liberty of differing from Madame Prunes and Prisms, and, as your physician, I order you to run. Off with you!'' said Uncle Alec, with a look and a gesture that made Rose scurry away as fast as she could go.
Anxious to please him, she raced round the beds till she came back to the porch where he stood, and, dropping down upon the steps, she sat panting, with cheeks as rosy as the rigolette on her shoulders.
``Very well done, child; I see you have not lost the use of your limbs though you are in your teens. That belt is too tight; unfasten it, then you can take a long breath without panting so.''
``It isn't tight, sir; I can breathe perfectly well,'' began Rose, trying to compose herself.
Her uncle's only answer was to lift her up and unhook the new belt of which she was so proud. The moment the clasp was open the belt flew apart several inches, for it was impossible to restrain the involuntary sigh of relief that flatly contradicted her words.
``Why, I didn't know it was tight! it didn't feel so a bit. Of course it would open if I puff like this, but I never do, because I hardly ever run,'' explained Rose, rather discomfited by this discovery.
``I see you don't half fill your lungs, and so you can wear this absurd thing without feeling it. The idea of cramping a tender little waist in a stiff band of leather and steel just when it ought to be growing,'' said Dr. Alec, surveying the belt with great disfavour as he put the clasp forward several holes, to Rose's secret dismay, for she was proud of her slender figure, and daily rejoiced that she wasn't as stout as Luly Miller, a former schoolmate, who vainly tried to repress her plumpness.
``It will fall off if it is so loose,'' she said anxiously, as she stood watching him pull her precious belt about.
``Not if you keep taking long breaths to hold it on. That is what I want you to do, and when you have filled this out we will go on enlarging it till your waist is more like that of Hebe, goddess of health, and less like that of a fashion-plate the ugliest thing imaginable.''
``How it does look!'' and Rose gave a glance of scorn at the loose belt hanging round her trim little waist. ``It will be lost, and then I shall feel badly, for it cost ever so much, and is real steel and Russia leather. Just smell how nice.''
``If it is lost I'll give you a better one. A soft silken sash is much fitter for a pretty child like you than a plated harness like this; and I've got no end of Italian scarfs and Turkish sashes among my traps. Ah! that makes you feel better, doesn't it?'' and he pinched the cheek that had suddenly dimpled with a smile.
``It is very silly of me, but I can't help liking to know that'' here she stopped and blushed and held down her head, ashamed to add, ``you think I am pretty.''
Dr. Alec's eyed twinkled, but he said very soberly,
``Rose, are you vain?''
``I'm afraid I am,'' answered a very meek voice from behind the veil of hair that hid the red face.
``That is a sad fault.'' And he sighed as if grieved at the confession.
``I know it is, and I try not to be; but people praise me, and I can't help liking it, for I really don't think I am repulsive.''
The last word and the funny tone in which it was uttered were too much for Dr. Alec, and he laughed in spite of himself, to Rose's great relief.
``I quite agree with you; and in order that you may be still less repulsive, I want you to grow as fine a girl as Phebe.''
``Phebe!'' and Rose looked so amazed that her uncle nearly went off again.
``Yes, Phebe; for she has what you need health. If you dear little girls would only learn what real beauty is, and not pinch and starve and bleach yourselves out so, you'd save an immense deal of time and money and pain. A happy soul in a healthy body makes the best sort of beauty for man or woman. Do you understand that, my dear?''
``Yes, sir,'' answered Rose, much taken down by this comparison with the girl from the poor-house. It nettled her sadly, and she showed that it did by saying quickly,
``I suppose you would like to have me sweep and scrub, and wear an old brown dress, and go round with my sleeves rolled up, as Phebe does?''
``I should very much, if you could work as well as she does, and show as strong a pair of arms as she can. I haven't seen a prettier picture for some time than she made of herself this morning, up to the elbows in suds, singing like a blackbird whilst she scrubbed on the back stoop.''
``Well, I do think you are the queerest man that ever lived!'' was all Rose could find to say after this display of bad taste.
``I haven't begun to show you my oddities yet, so you must make up your mind to worse shocks than this,'' he said, with such a whimsical look that she was glad the sound of a bell prevented her showing more plainly what a blow her little vanities had already received.
``You will find your box all open up in auntie's parlor, and there you can amuse her and yourself by rummaging to your heart's content; I've got to be cruising round all the morning getting my room to rights,'' said Dr. Alec, as they rose from breakfast.
``Can't I help you, uncle?'' asked Rose, quite burning to be useful.
``No, thank you, I'm going to borrow Phebe for a while, if Aunt Plenty can spare her.''
``Anybody anything, Alec. You will want me, I know, so I'll give orders about dinner and be all ready to lend a hand''; and the old lady bustled away full of interest and good-will.
``Uncle will find that I can do some things that Phebe can't, so now!'' thought Rose, with a toss of the head as she flew to Aunt Peace and the long-desired box.
Every little girl can easily imagine what an extra good time she had diving into a sea of treasures and fishing up one pretty thing after another, till the air was full of the mingled odours of musk and sandalwood, the room gay with bright colours, and Rose in a rapture of delight. She began to forgive Dr. Alec for the oatmeal diet when she saw a lovely ivory workbox; became resigned to the state of her belt when she found a pile of rainbow-coloured sashes; and when she came to some distractingly pretty bottles of attar of rose, she felt that they almost atoned for the great sin of thinking Phebe the finer girl of the two.
Dr. Alec meanwhile had apparently taken Aunt Plenty at her word, and was turning the house upside down. A general revolution was evidently going on in the green-room, for the dark damask curtains were seen bundling away in Phebe's arms; the air-tight stove retiring to the cellar on Ben's shoulder; and the great bedstead going up garret in a fragmentary state, escorted by three bearers. Aunt Plenty was constantly on the trot among her store-rooms, camphor-chests, and linen-closets, looking as if the new order of things both amazed and amused her.
Half the peculiar performances of Dr. Alec cannot be revealed; but as Rose glanced up from her box now and then she caught glimpses of him striding by, bearing a bamboo chair, a pair of ancient andirons, a queer Japanese screen, a rug or two, and finally a large bathing-pan upon his head.
``What a curious room it will be,'' she said, as she sat resting and refreshing herself with ``Lumps of Delight,'' all the way from Cairo.
``I fancy you will like it, deary,'' answered Aunt Peace, looking up with a smile from some pretty trifle she was making with blue silk and white muslin.
Rose did not see the smile, for just at that moment her uncle paused at the door, and she sprang up to dance before him, saying, with a face full of childish happiness,
``Look at me! look at me! I'm splendid I don't know myself. I haven't put these things on right, I dare say, but I do like them so much!''
``You look as gay as a parrot in your fez and cabaja, and it does my heart good to see the little black shadow turned into a rainbow,'' said Uncle Alec, surveying the bright figure before him with great approbation.
He did not say it, but he thought she made a much prettier picture than Phebe at the wash-tub, for she had stuck a purple fez on her blonde head, tied several brilliant scarfs about her waist, and put on a truly gorgeous scarlet jacket with a golden sun embroidered on the back, a silver moon on the front, and stars of all sizes on the sleeves. A pair of Turkish slippers adorned her feet, and necklaces of amber, coral, and filigree hung about her neck, while one hand held a smelling-bottle, and the other the spicy box of oriental sweetmeats.
``I feel like a girl in the `Arabian Nights,' and expect to find a magic carpet or a wonderful talisman somewhere. Only I don't see how I ever can thank you for all these lovely things,'' she said, stopping her dance, as if suddenly oppressed with gratitude.
``I'll tell you how by leaving off the black clothes, that never should have been kept so long on such a child, and wearing the gay ones I've brought. It will do your spirits good, and cheer up this sober old house. Won't it, auntie?''
``I think you are right, Alec, and it is fortunate that we have not begun on her spring clothes yet, for Myra thought she ought not to wear anything brighter than violet, and she is too pale for that.''
``You just let me direct Miss Hemming how to make some of these things. You will be surprised to see how much I know about piping hems and gathering arm-holes and shirring biases,'' began Dr. Alec, patting a pile of muslin, cloth and silk with a knowing air.
Aunt Peace and Rose laughed so that he could not display his knowledge any farther, till they stopped, when he said good-naturedly,
``That will go a great way toward filling out the belt, so laugh away, Morgiana, and I'll go back to my work, or I never shall be done.''
``I couldn't help it, `shirred biases' were so very funny!'' Rose said, as she turned to her box after the splendid laugh. ``But really, auntie,'' she added soberly, ``I feel as if I ought not to have so many nice things. I suppose it wouldn't do to give Phebe some of them? Uncle might not like it.''
``He would not mind; but they are not suitable for Phebe. Some of the dresses you are done with would be more useful, if they can be made over to fit her,'' answered Aunt Peace in the prudent, moderate tone which is so trying to our feelings when we indulge in little fits of charitable enthusiasm.
``I'd rather give her new ones, for I think she is a little bit proud and might not like old things. If she was my sister it would do, because sisters don't mind, but she isn't, and that makes it bad, you see. I know how I can manage beautifully; I'll adopt her!'' and Rose looked quite radiant with this new idea.
``I'm afraid you could not do it legally till you are older, but you might see if she likes the plan, and at any rate you can be very kind to her, for in one sense we are all sisters, and should help one another.''
The sweet old face looked at her so kindly that Rose was fired with a desire to settle the matter at once, and rushed away to the kitchen, just as she was. Phebe was there, polishing up the antique andirons so busily that she started when a voice cried out: ``Smell that, taste this, and look at me!''
Phebe sniffed attar of rose, crunched the ``Lump of Delight'' tucked into her mouth, and stared with all her eyes at little Morgiana prancing about the room like a brilliant paroquet.
``My stars, ain't you splendid!'' was all she could say, holding up two dusty hands.
``I've got heaps of lovely things upstairs, and I'll show them all to you, and I'd go halves, only auntie thinks they wouldn't be useful, so I shall give you something else; and you won't mind, will you? because I want to adopt you as Arabella was in the story. Won't that be nice?''
``Why, Miss Rose, have you lost your wits?''
No wonder Phebe asked, for Rose talked very fast, and looked so odd in her new costume, and was so eager she could not stop to explain. Seeing Phebe's bewilderment, she quieted down and said, with a pretty air of earnestness,
``It isn't fair that I should have so much and you so little, and I want to be as good to you as if you were my sister, for Aunt Peace says we are all sisters really. I thought if I adopted you as much as I can now, it would be nicer. Will you let me, please?''
To Rose's great surprise, Phebe sat down on the floor and hid her face in her apron for a minute without answering a word.
``Oh, dear, now she's offended, and I don't know what to do,'' thought Rose, much discouraged by this reception of her offer.
``Please, forgive me; I didn't mean to hurt your feelings, and hope you won't think---'' she faltered presently, feeling that she must undo the mischief, if possible.
But Phebe gave her another surprise, by dropping the apron and showing a face all smiles, in spite of tears in the eyes, as she put both arms round Rose and said, with a laugh and sob,
``I think you are the dearest girl in the world, and I'll let you do anything you like with me.''
``Then you do like the plan? You didn't cry because I seemed to be kind of patronising? I truly didn't mean to be,'' cried Rose, delighted.
``I guess I do like it! and cried because no one was ever so good to me before, and I couldn't help it. As for patronising, you may walk on me if you want to, and I won't mind,'' said Phebe, in a burst of gratitude, for the words, ``we are sisters'' went straight to her lonely heart and nestled there.
``Well, now, we can play I'm a good sprite out of the box, or, what is better, a fairy godmother come down the chimney, and you are Cinderella, and must say what you want,'' said Rose, trying to put the question delicately.
Phebe understood that, for she had a good deal of natural refinement, though she did come from the poor-house.
``I don't feel as if I wanted anything now, Miss Rose, but to find some way of thanking you for all you've done,'' she said, rubbing off a tear that went rolling down the bridge of her nose in the most unromantic way.
``Why, I haven't done anything but given you a bit of candy! Here, have some more, and eat 'em while you work, and think what I can do. I must go and clear up, so good-bye, and don't forget I've adopted you.''
``You've given me sweeter things than candy, and I'm not likely to forget it.'' And carefully wiping off the brick-dust, Phebe pressed the little hand Rose offered warmly in both her hard ones, while the black eyes followed the departing visitor with a grateful look that made them very soft and bright.
\gutchapter{Chapter 6---Uncle Alec's Room}
Soon after dinner, and before she had got acquainted with half her new possessions, Dr. Alec proposed a drive, to carry round the first instalment of gifts to the aunts and cousins. Rose was quite ready to go, being anxious to try a certain soft burnous from the box, which not only possessed a most engaging little hood, but had funny tassels bobbing in all directions.
The big carriage was full of parcels, and even Ben's seat was loaded with Indian war clubs, a Chinese kite of immense size, and a pair of polished ox-horns from Africa. Uncle Alec, very blue as to his clothes, and very brown as to his face, sat bolt upright, surveying well known places with interest, while Rose, feeling unusually elegant and comfortable, leaned back folded in her soft mantle, and played she was an Eastern princess making a royal progress among her subjects.
At three of the places their calls were brief, for Aunt Myra's catarrh was unusually bad; Aunt Clara had a room full of company; and Aunt Jane showed such a tendency to discuss the population, productions, and politics of Europe, Asia and Africa, that even Dr. Alec was dismayed, and got away as soon as possible.
``Now we will have a good time! I do hope the boys will be at home,'' said Rose, with a sigh of relief, as they wound yet higher up the hill to Aunt Jessie's.
``I left this for the last call, so that we might find the lads just in from school. Yes, there is Jamie on the gate watching for us; now you'll see the Clan gather; they are always swarming about together.''
The instant Jamie saw the approaching guests he gave a shrill whistle, which was answered by echoes from meadow, house and barn, as the cousins came running from all directions, shouting, ``Hooray for Uncle Alec!'' They went at the carriage like highwaymen, robbed it of every parcel, took the occupants prisoners, and marched them into the house with great exultation.
``Little Mum! little Mum! here they are with lots of goodies! Come down and see the fun right away! Quick!'' bawled Will and Geordie amidst a general ripping off of papers and a reckless cutting of strings that soon turned the tidy room into a chaos.
Down came Aunt Jessie with her pretty cap half on, but such a beaming face below it that one rather thought the fly-away head-gear an improvement than otherwise. She had hardly time to greet Rose and the doctor before the boys were about her, each clamouring for her to see his gift and rejoice over it with him, for ``little Mum'' went halves in everything. The great horns skirmished about her as if to toss her to the ceiling; the war clubs hurtled over her head as if to annihilate her; an amazing medley from the four quarters of the globe filled her lap, and seven excited boys all talked to her at once.
But she liked it; oh dear, yes! and sat smiling, admiring, and explaining, quite untroubled by the din, which made Rose cover up her ears and Dr. Alec threaten instant flight if the riot was not quelled. That threat produced a lull, and while the uncle received thanks in one corner, the aunt had some little confidences made to her in the other.
``Well, dear, and how are things going with you now? Better, I hope, than they were a week ago.''
``Aunt Jessie, I think I'm going to be very happy, now uncle has come. He does the queerest things, but he is so good to me I can't help loving him''; and, nestling closer to little Mum, Rose told all that had happened, ending with a rapturous account of the splendid box.
``I am very glad, dear. But, Rose, I must warn you of one thing; don't let uncle spoil you.''
``But I like to be spoilt, auntie.''
``I don't doubt it; but if you turn out badly when the year is over he will be blamed, and his experiment prove a failure. That would be a pity, wouldn't it? when he wants to do so much for you, and can do it if his kind heart does not get in the way of his good judgment.''
``I never thought of that, and I'll try not to be spoilt. But how can I help it?'' asked Rose anxiously.
``By not complaining of the wholesome things he wants you to do; by giving him cheerful obedience as well as love; and even making some small sacrifices for his sake.''
``I will, I truly will! and when I get in a worry about things may I come to you? Uncle told me to, and I feel as if I shouldn't be afraid.''
``You may, darling; this is the place where little troubles are best cured, and this is what mothers are for, I fancy''; and Aunt Jessie drew the curly head to her shoulder with a tender look that proved how well she knew what medicine the child most needed.
It was so sweet and comfortable that Rose sat still enjoying it till a little voice said,
``Mamma, don't you think Pokey would like some of my shells? Rose gave Phebe some of her nice things, and it was very good of her. Can I?''
``Who is Pokey?'' asked Rose, popping up her head, attracted by the odd name.
``My dolly; do you want to see her?'' asked Jamie, who had been much impressed by the tale of adoption he had overheard.
``Yes; I'm fond of dollies, only don't tell the boys, or they will laugh at me.''
``They don't laugh at me, and they play with my dolly a great deal; but she likes me best''; and Jamie ran away to produce his pet.
``I brought my old doll, but I keep her hidden because I am too big to play with her, and yet I can't bear to throw her away, I'm so fond of her,'' said Rose, continuing her confidences in a whisper.
``You can come and play with Jamie's whenever you like, for we believe in dollies up here,'' began Aunt Jessie, smiling to herself as if something amused her.
Just then Jamie came back, and Rose understood the smile, for his dolly proved to be a pretty four-year-old little girl, who trotted in as fast as her fat legs would carry her, and making straight for the shells, scrambled up an armful, saying, with a laugh that showed her little white teeth,
``All for Dimmy and me, for Dimmy and me!''
``That's my dolly; isn't she a nice one?'' asked Jamie, proudly surveying his pet with his hands behind him and his short legs rather far apart a manly attitude copied from his brothers.
``She is a dear dolly. But why call her Pokey?'' asked Rose, charmed with the new plaything.
``She is such an inquisitive little body she is always poking that mite of a nose into everything; and as Paul Pry did not suit, the boys fell to calling her Pokey. Not a pretty name, but very expressive.''
It certainly was, for, having examined the shells, the busy tot laid hold of everything she could find, and continued her researches till Archie caught her sucking his carved ivory chessmen to see if they were not barley sugar. Rice paper pictures were also discovered crumpled up in her tiny pocket, and she nearly smashed Will's ostrich egg by trying to sit upon it.
``Here, Jim, take her away; she's worse than the puppies, and we can't have her round,'' commanded the elder brother, picking her up and handing her over to the little fellow, who received her with open arms and the warning remark,
``You'd better mind what you do, for I'm going to 'dopt Pokey like Rose did Phebe, and then you'll have to be very good to her, you big fellows.''
``'Dopt away, baby, and I'll give you a cage to keep her in, or you won't have her long, for she is getting worse than a monkey''; and Archie went back to his mates, while Aunt Jessie, foreseeing a crisis, proposed that Jamie should take his dolly home, as she was borrowed, and it was time her visit ended.
``My dolly is better than yours, isn't she? 'cause she can walk and talk and sing and dance, and yours can't do anything, can she?'' asked Jamie with pride, as he regarded his Pokey, who just then had been moved to execute a funny little jig and warble the well-known couplet,
```Puss-tat, puss-tat, where you been?'
`I been Lunnin, to saw a Tween."'
After which superb display she retired, escorted by Jamie, both making a fearful din blowing on conch shells.
``We must tear ourselves away, Rose, because I want to get you home before sunset. Will you come for a drive, Jessie?'' said Dr. Alec, as the music died away in the distance.
``No, thank you; but I see the boys want a scamper, so, if you don't mind, they may escort you home, but not go in. That is only allowed on holidays.''
The words were hardly out of Aunt Jessie's mouth when Archie said, in a tone of command,
``Pass the word, lads. Boot and saddle, and be quick about it.''
``All right!'' And in a moment not a vestige of boy remained but the litter on the floor.
The cavalcade went down the hill at a pace that made Rose cling to her uncle's arm, for the fat old horses got excited by the antics of the ponies careering all about them, and went as fast as they could pelt, with the gay dog-cart rattling in front, for Archie and Charlie scorned shelties since this magnificent equipage had been set up. Ben enjoyed the fun, and the lads cut up capers till Rose declared that ``circus'' was the proper name for them after all.
When they reached the house they dismounted, and stood, three on each side the steps, in martial attitudes, while her ladyship was handed out with great elegance by Uncle Alec. Then the Clan saluted, mounted at word of command, and with a wild whoop tore down the avenue in what they considered the true Arab style.
``That was splendid, now it is safely ended,'' said Rose, skipping up the steps with her head over her shoulder to watch the dear tassels bob about.
``I shall get you a pony as soon as you are a little stronger,'' said Dr. Alec, watching her with a smile.
``Oh, I couldn't ride one of those horrid, frisky little beasts! They roll their eyes and bounce about so, I should die of fright,'' cried Rose, clasping her hands tragically.
``Are you a coward?''
``About horses I am.''
``Never mind, then; come and see my new room''; and he led the way upstairs without another word.
As Rose followed she remembered her promise to Aunt Jessie, and was sorry she had objected so decidedly. She was a great deal more sorry five minutes later, and well she might be.
``Now, take a good look, and tell me what you think of it,'' said Dr. Alec, opening the door and letting her enter before him, while Phebe was seen whisking down the backstairs with a dust-pan.
Rose walked to the middle of the room, stood still, and gazed about her with eyes that brightened as they looked, for all was changed.
This chamber had been built out over the library to suit some fancy, and had been unused for years, except at Christmas times, when the old house overflowed. It had three windows one to the east, that overlooked the bay; one to the south, where the horse-chestnuts waved their green fans; and one to the west, towards the hill and the evening sky. A ruddy sunset burned there now, filling the room with an enchanted glow; the soft murmur of the sea was heard, and a robin chirped ``Good-night!'' among the budding trees.
Rose saw and heard these things first, and felt their beauty with a child's quick instinct; then her eye took in the altered aspect of the room, once so shrouded, still and solitary, now so full of light and warmth and simple luxury.
India matting covered the floor, with a gay rug here and there; the antique andirons shone on the wide hearth, where a cheery blaze dispelled the dampness of the long-closed room. Bamboo lounges and chairs stood about, and quaint little tables in cosy corners; one bearing a pretty basket, one a desk, and on a third lay several familiar-looking books. In a recess stood a narrow white bed, with a lovely Madonna hanging over it. The Japanese screen half-folded back showed a delicate toilet service of blue and white set forth on a marble slab, and near by was the great bath-pan, with Turkish towels and a sponge as big as Rose's head.
``Uncle must love cold water like a duck,'' she thought, with a shiver.
Then her eye went on to the tall cabinet, where a half-open door revealed a tempting array of the drawers, shelves and ``cubby holes,'' which so delight the hearts of children.
``What a grand place for my new things,'' she thought, wondering what her uncle kept in that cedar retreat.
``Oh me, what a sweet toilet table!'' was her next mental exclamation, as she approached this inviting spot.
A round old-fashioned mirror hung over it, with a gilt eagle a-top, holding in his beak the knot of blue ribbon that tied up a curtain of muslin falling on either side of the table, where appeared little ivory-handled brushes, two slender silver candle-sticks, a porcelain match-box, several pretty trays for small matters, and, most imposing of all, a plump blue silk cushion, coquettishly trimmed with lace, and pink rose-buds at the corners.
That cushion rather astonished Rose; in fact, the whole table did, and she was just thinking, with a sly smile,
``Uncle is a dandy, but I never should have guessed it,'' when he opened the door of a large closet, saying, with a careless wave of the hand,
``Men like plenty of room for their rattle-traps; don't you think that ought to satisfy me?''
Rose peeped in and gave a start, though all she saw was what one usually finds in closets clothes and boots, boxes and bags. Ah! but you see these clothes were small black and white frocks; the row of little boots that stood below had never been on Dr. Alec's feet; the green bandbox had a gray veil straying out of it, and yes! the bag hanging on the door was certainly her own piece-bag, with a hole in one corner. She gave a quick look round the room and understood now why it had seemed too dainty for a man, why her Testament and Prayer Book were on the table by the bed, and what those rose-buds meant on the blue cushion. It came upon her in one delicious burst that this little paradise was all for her, and, not knowing how else to express her gratitude, she caught Dr. Alec round the neck, saying impetuously,
``O uncle, you are too good to me! I'll do anything you ask me; ride wild horses and take freezing baths and eat bad-tasting messes, and let my clothes hang on me, to show how much I thank you for this dear, sweet, lovely room!''
``You like it, then? But why do you think it is yours, my lass?'' asked Dr. Alec, as he sat down looking well pleased, and drew his excited little niece to his knee.
``I don't think, I know it is for me; I see it in your face, and I feel as if I didn't half deserve it. Aunt Jessie said you would spoil me, and I must not let you. I'm afraid this looks like it, and perhaps oh me! perhaps I ought not to have this beautiful room after all!'' and Rose tried to look as if she could be heroic enough to give it up if it was best.
``I owe Mrs. Jessie one for that,'' said Dr. Alec, trying to frown, though in his secret soul he felt that she was quite right. Then he smiled that cordial smile, which was like sunshine on his brown face, as he said,
``This is part of the cure, Rose, and I put you here that you might take my three great remedies in the best and easiest way. Plenty of sun, fresh air, and cold water; also cheerful surroundings, and some work; for Phebe is to show you how to take care of this room, and be your little maid as well as friend and teacher. Does that sound hard and disagreeable to you, dear?''
``No, sir; very, very pleasant, and I'll do my best to be a good patient. But I really don't think anyone could be sick in this delightful room,'' she said, with a long sigh of happiness as her eye went from one pleasant object to another.
``Then you like my sort of medicine better than Aunt Myra's, and don't want to throw it out of the window, hey?''
\gutchapter{Chapter 7---A Trip to China}
``Come, little girl, I've got another dose for you. I fancy you won't take it as well as you did the last, but you will like it better after a while,'' said Dr. Alec, about a week after the grand surprise.
Rose was sitting in her pretty room, where she would gladly have spent all her time if it had been allowed; but she looked up with a smile, for she had ceased to fear her uncle's remedies, and was always ready to try a new one. The last had been a set of light gardening tools, with which she had helped him put the flower-beds in order, learning all sorts of new and pleasant things about the plants as she worked, for, though she had studied botany at school, it seemed very dry stuff compared with Uncle Alec's lively lesson.
``What is it now?'' she asked, shutting her work-box without a murmur.
``Salt-water.''
``How must I take it?''
``Put on the new suit Miss Hemming sent home yesterday, and come down to the beach; then I'll show you.''
``Yes, sir,'' answered Rose obediently, adding to herself, with a shiver, as he went off: ``It is too early for bathing, so I know it is something to do with a dreadful boat.''
Putting on the new suit of blue flannel, prettily trimmed with white, and the little sailor-hat with long streamers, diverted her mind from the approaching trial, till a shrill whistle reminded her that her uncle was waiting. Away she ran through the garden, down the sandy path, out upon the strip of beach that belonged to the house, and here she found Dr. Alec busy with a slender red and white boat that lay rocking on the rising tide.
``That is a dear little boat; and `Bonnie Belle' is a pretty name,'' she said, trying not to show how nervous she felt.
``It is for you; so sit in the stern and learn to steer, till you are ready to learn to row.''
``Do all boats wiggle about in that way?'' she asked, lingering as if to tie her hat more firmly.
``Oh, yes, pitch about like nutshells when the sea is a bit rough,'' answered her sailor uncle, never guessing her secret woe.
``Is it rough to-day?''
``Not very; it looks a trifle squally to the eastward, but we are all right till the wind changes. Come.''
``Can you swim, uncle?'' asked Rose, clutching at his arm as he took her hand.
``Like a fish. Now then.''
``Oh, please hold me very tight till I get there! Why do you have the stern so far away?'' and, stifling several squeaks of alarm in her passage, Rose crept to the distant seat, and sat there holding on with both hands and looking as if she expected every wave to bring a sudden shipwreck.
Uncle Alec took no notice of her fear, but patiently instructed her in the art of steering, till she was so absorbed in remembering which was starboard and which larboard, that she forgot to say ``\textit{Ow}!'' every time a big wave slapped against the boat.
``Now where shall we go?'' she asked, as the wind blew freshly in her face, and a few, long swift strokes sent them half across the little bay.
``Suppose we go to China?''
``Isn't that rather a long voyage?''
``Not as I go. Steer round the Point into the harbour, and I'll give you a glimpse of China in twenty minutes or so.''
``I should like that!'' and Rose sat wondering what he meant, while she enjoyed the new sights all about her.
Behind them the green Aunt-hill sloped gently upward to the grove at the top, and all along the seaward side stood familiar houses, stately, cosy, or picturesque. As they rounded the Point, the great bay opened before them full of shipping, and the city lay beyond, its spires rising above the tall masts with their gay streamers.
``Are we going there?'' she asked, for she had never seen this aspect of the rich and busy old city before.
``Yes. Uncle Mac has a ship just in from Hong Kong, and I thought you would like to go and see it.''
``Oh, I should. I love dearly to go poking about in the warehouses with Uncle Mac; everything is so curious and new to me; and I'm specially interested in China because you have been there.''
``I'll show you two genuine Chinamen who have just arrived. You will like to welcome Whang Lo and Fun See, I'm sure.''
``Don't ask me to speak to them, uncle; I shall be sure to laugh at the odd names and the pig-tails and the slanting eyes. Please let me just trot round after you; I like that best.''
``Very well; now steer toward the wharf where the big ship with the queer flag is. That's the `Rajah,' and we will go aboard if we can.''
In among the ships they went, by the wharves where the water was green and still, and queer barnacles grew on the slippery piles. Odd smells saluted her nose, and odd sights met her eyes, but Rose liked it all, and played she was really landing in Hong Kong when they glided up to the steps in the shadow of the tall ``Rajah.'' Boxes and bales were rising out of the hold and being carried into the warehouse by stout porters, who tugged and bawled and clattered about with small trucks, or worked cranes with iron claws that came down and clutched heavy weights, whisking them aloft to where wide doors like mouths swallowed them up.
Dr. Alec took her aboard the ship, and she had the satisfaction of poking her inquisitive little nose into every available corner, at the risk of being crushed, lost, or drowned.
``Well, child, how would you like to take a voyage round the world with me in a jolly old craft like this?'' asked her uncle, as they rested a minute in the captain's cabin.
``I should like to see the world, but not in such a small, untidy, smelly place as this. We would go in a yacht all clean and comfortable; Charlie says that is the proper way,'' answered Rose, surveying the close quarters with little favour.
``You are not a true Campbell if you don't like the smell of tar and salt-water, nor Charlie either, with his luxurious yacht. Now come ashore and chin-chin with the Celestials.''
After a delightful progress through the great warehouse, peeping and picking as they went, they found Uncle Mac and the yellow gentlemen in his private room, where samples, gifts, curiosities, and newly arrived treasures of all sorts were piled up in pleasing pro-fusion and con-fusion.
As soon as possible Rose retired to a corner, with a porcelain god on one side, a green dragon on the other, and, what was still more embarrassing, Fun See sat on a tea-chest in front, and stared at her with his beady black eyes till she did not know where to look.
Mr. Whang Lo was an elderly gentleman in American costume, with his pig-tail neatly wound round his head. He spoke English, and was talking busily with Uncle Mac in the most commonplace way so Rose considered him a failure. But Fun See was delightfully Chinese from his junk-like shoes to the button on his pagoda hat; for he had got himself up in style, and was a mass of silk jackets and slouchy trousers. He was short and fat, and waddled comically; his eyes were very ``slanting,'' as Rose said; his queue was long, so were his nails; his yellow face was plump and shiny, and he was altogether a highly satisfactory Chinaman.
Uncle Alec told her that Fun See had come out to be educated and could only speak a little pigeon English; so she must be kind to the poor fellow, for he was only a lad, though he looked nearly as old as Mr. Whang Lo. Rose said she would be kind; but had not the least idea how to entertain the queer guest, who looked as if he had walked out of one of the rice-paper landscapes on the wall, and sat nodding at her so like a toy Mandarin that she could hardly keep sober.
In the midst of her polite perplexity, Uncle Mac saw the two young people gazing wistfully at one another, and seemed to enjoy the joke of this making acquaintance under difficulties. Taking a box from his table, he gave it to Fun See, with an order that seemed to please him very much.
Descending from his perch, he fell to unpacking it with great neatness and despatch, while Rose watched him, wondering what was going to happen. Presently, out from the wrappings came a teapot, which caused her to clasp her hands with delight, for it was made in the likeness of a plump little Chinaman. His hat was the cover, his queue the handle, and his pipe the nose. It stood upon feet in shoes turned up at the toes, and the smile on the fat, sleepy face was so like that on Fun's when he displayed the teapot, that Rose couldn't help laughing, which pleased him much.
Two pretty cups with covers, and a fine scarlet tray completed the set, and made one long to have a ``dish of tea,'' even in Chinese style, without cream or sugar.
When he had arranged them on a little table before her, Fun signified in pantomime that they were hers, from her uncle. She returned her thanks in the same way, whereupon he returned to his tea-chest, and, having no other means of communication, they sat smiling and nodding at one another in an absurd sort of way till a new idea seemed to strike Fun. Tumbling off his seat, he waddled away as fast as his petticoats permitted, leaving Rose hoping that he had not gone to get a roasted rat, a stewed puppy, or any other foreign mess which civility would oblige her to eat.
While she waited for her funny new friend, she improved her mind in a way that would have charmed Aunt Jane. The gentlemen were talking over all sorts of things, and she listened attentively, storing up much of what she heard, for she had an excellent memory, and longed to distinguish herself by being able to produce some useful information when reproached with her ignorance.
She was just trying to impress upon her mind that Amoy was two hundred and eighty miles from Hong Kong, when Fun came scuffling back, bearing what she thought was a small sword, till he unfurled an immense fan, and presented it with a string of Chinese compliments, the meaning of which would have amused her even more than the sound, if she could have understood it.
She had never seen such an astonishing fan, and at once became absorbed in examining it. Of course, there was no perspective whatever, which only gave it a peculiar charm to Rose, for in one place a lovely lady, with blue knitting-needles in her hair, sat directly upon the spire of a stately pagoda. In another charming view a brook appeared to flow in at the front door of a stout gentleman's house, and out at his chimney. In a third a zig-zag wall went up into the sky like a flash of lightning, and a bird with two tails was apparently brooding over a fisherman whose boat was just going aground upon the moon.
It was altogether a fascinating thing, and she would have sat wafting it to and fro all the afternoon, to Fun's great satisfaction, if Dr. Alec's attention had not suddenly been called to her by a breeze from the big fan that blew his hair into his eyes, and reminded him that they must go. So the pretty china was repacked, Rose furled her fan, and with several parcels of choice teas for the old ladies stowed away in Dr. Alec's pockets, they took their leave, after Fun had saluted them with ``the three bendings and the nine knockings,'' as they salute the Emperor, or ``Son of Heaven,'' at home.
``I feel as if I had really been to China, and I'm sure I look so,'' said Rose, as they glided out of the shadow of the ``Rajah.''
She certainly did, for Mr. Whang Lo had given her a Chinese umbrella; Uncle Alec had got some lanterns to light up her balcony; the great fan lay in her lap, and the tea-set reposed at her feet.
``This is not a bad way to study geography, is it?'' asked her uncle, who had observed her attention to the talk.
``It is a very pleasant way, and I really think I have learned more about China to-day than in all the lessons I had at school, though I used to rattle off the answers as fast as I could go. No one explained anything to us, so all I remember is that tea and silk come from there, and the women have little bits of feet. I saw Fun looking at mine, and he must have thought them perfectly immense,'' answered Rose, surveying her stout boots with sudden contempt.
``We will have out the maps and the globe, and I'll show you some of my journeys, telling stories as we go. That will be next best to doing it actually.''
``You are so fond of travelling, I should think it would be very dull for you here, uncle. Do you know, Aunt Plenty says she is sure you will be off in a year or two.''
``Very likely.''
``Oh, me! what shall I do then?'' sighed Rose, in a tone of despair that made Uncle Alec's face brighten with a look of genuine pleasure as he said significantly,
``Next time I go I shall take my little anchor with me. How will that suit?''
``Really, uncle?''
``Really, niece.''
Rose gave a little bounce of rapture which caused the boat to ``wiggle'' in a way that speedily quieted her down. But she sat beaming joyfully and trying to think which of some hundred questions she would ask first, when Dr. Alec said, pointing to a boat that was coming up behind them in great style,
``How well those fellows row! Look at them, and take notes for your own use by and by.''
The ``Stormy Petrel'' was manned by half a dozen jaunty looking sailors, who made a fine display of blue shirts and shiny hats, with stars and anchors in every direction.
``How beautifully they go, and they are only boys. Why, I do believe they are our boys! Yes, I see Charlie laughing over his shoulder. Row, uncle, row! Oh, please do, and not let them catch up with us!'' cried Rose, in such a state of excitement that the new umbrella nearly went overboard.
``All right, here we go!'' and away they did go with a long steady sweep of the oars that carried the ``Bonnie Belle'' through the water with a rush.
The lads pulled their prettiest, but Dr. Alec would have reached the Point first, if Rose, in her flurry, had not retarded him by jerking the rudder ropes in a most unseamanlike way, and just as she got right again her hat blew off. That put an end to the race, and while they were still fishing for the hat the other boat came alongside, with all the oars in the air, and the jolly young tars ready for a frolic.
``Did you catch a crab, uncle?''
``No, a blue-fish,'' he answered, as the dripping hat was landed on a seat to dry.
``What have you been doing?''
``Seeing Fun.''
``Good for you, Rose! I know what you mean. We are going to have him up to show us how to fly the big kite, for we can't get the hang of it. Isn't he great fun, though?''
``No, little Fun.''
``Come, stop joking, and show us what you've got.''
``You'd better hoist that fan for a sail.''
``Lend Dandy your umbrella; he hates to burn his pretty nose.''
``I say, uncle, are you going to have a Feast of Lanterns?''
``No, I'm going to have a feast of bread and butter, for it's tea-time. If that black cloud doesn't lie, we shall have a gust before long, so you had better get home as soon as you can, or your mother will be anxious, Archie.''
``Ay, ay, skipper. Good-night, Rose; come out often, and we'll teach you all there is to know about rowing,'' was Charlie's modest invitation.
Then the boats parted company, and across the water from the ``Petrel's'' crew came a verse from one of the Nonsense songs in which the boys delighted.
``Oh, Timballoo! how happy we are,
We live in a sieve and a crockery jar!
And all night long, in the starlight pale,
We sail away, with a pea-green sail,
And whistle and warble a moony song
To the echoing sound of a coppery gong.
Far and few, far and few
Are the lands where the Jumblies live;
Their heads are green, and their hands are blue,
And they went to sea in a sieve.''
\gutchapter{Chapter 8---And what came of it}
``Uncle, could you lend me a ninepence? I'll return it as soon as I get my pocket-money,'' said Rose, coming into the library in a great hurry that evening.
``I think I could, and I won't charge any interest for it, so you need not be in any hurry to repay me. Come back here and help me settle these books if you have nothing pleasanter to do,'' answered Dr. Alec, handing out the money with that readiness which is so delightful when we ask small loans.
``I'll come in a minute; I've been longing to fix my books, but didn't dare to touch them, because you always shake your head when I read.''
``I shall shake my head when you write, if you don't do it better than you did in making out this catalogue.''
``I know it's bad, but I was in a hurry when I did it, and I am in one now.'' And away went Rose, glad to escape a lecture.
But she got it when she came back, for Uncle Alec was still knitting his brows over the list of books, and sternly demanded, pointing to a tipsy-looking title staggering down the page,
``Is that meant for `Pulverized Bones,' ma'am?''
``No, sir; it's `Paradise Lost.'''
``Well, I'm glad to know it, for I began to think you were planning to study surgery or farming. And what is this, if you please? `Babies' Aprons' is all I can make of it.''
Rose looked hard at the scrawl, and presently announced, with an air of superior wisdom,
``Oh, that's `Bacon's Essays.'''
``Miss Power did not teach anything so old-fashioned as writing, I see. Now look at this memorandum Aunt Plenty gave me, and see what a handsome plain hand that is. She went to a dame-school and learnt a few useful things well; that is better than a smattering of half a dozen so-called higher branches, I take the liberty of thinking.''
``Well, I'm sure I was considered a bright girl at school, and learned everything I was taught. Luly and me were the first in all our classes, and 'specially praised for our French and music and those sort of things,'' said Rose, rather offended at Uncle Alec's criticism.
``I dare say; but if your French grammar was no better than your English, I think the praise was not deserved, my dear.''
``Why, uncle, we did study English grammar, and I could parse beautifully. Miss Power used to have us up to show off when people came. I don't see but I talk as right as most girls.''
``I dare say you do, but we are all too careless about our English. Now, think a minute, and tell me if these expressions are correct 'Luly and me,' `those sort of things,' and `as right as most girls.'''
Rose pulled her pet curl and put up her lip, but had to own that she was wrong, and said meekly, after a pause which threatened to be sulky,
``I suppose I should have said `Luly and I,' in that case, and 'that sort of things' and `rightly,' though `correctly' would have been a better word, I guess.''
``Thank you; and if you will kindly drop `I guess,' I shall like my little Yankee all the better. Now, see here, Rosy, I don't pretend to set myself up for a model in anything, and you may come down on my grammar, manners or morals as often as you think I'm wrong, and I'll thank you. I've been knocking about the world for years, and have got careless, but I want my girl to be what I call well-educated, even if she studies nothing but the three `Rs' for a year to come. Let us be thorough, no matter how slowly we go.''
He spoke so earnestly and looked so sorry to have ruffled her that Rose went and sat on the arm of his chair, saying, with a pretty air of penitence,
``I'm sorry I was cross, uncle, when I ought to thank you for taking so much interest in me. I guess no, I think you are right about being thorough, for I used to understand a great deal better when papa taught me a few lessons than when Miss Power hurried me through so many. I declare my head used to be such a jumble of French and German, history and arithmetic, grammar and music, I used to feel sometimes as if it would split. I'm sure I don't wonder it ached.'' And she held on to it as if the mere memory of the ``jumble'' made it swim.
``Yet that is considered an excellent school, I find, and I dare say it would be if the benighted lady did not think it necessary to cram her pupils like Thanks-giving turkeys, instead of feeding them in a natural and wholesome way. It is the fault with most American schools, and the poor little heads will go on aching till we learn better.''
This was one of Dr. Alec's hobbies, and Rose was afraid he was off for a gallop, but he reined himself in and gave her thoughts a new turn by saying suddenly, as he pulled out a fat pocket-book,
``Uncle Mac has put all your affairs into my hands now, and here is your month's pocket money. You keep your own little accounts, I suppose?''
``Thank you. Yes, Uncle Mac gave me an account book when I went to school, and I used to put down my expenses, but I couldn't make them go very well, for figures are the one thing I am not at all clever about,'' said Rose, rummaging in her desk for a dilapidated little book, which she was ashamed to show when she found it.
``Well, as figures are rather important things to most of us, and you may have a good many accounts to keep some day, wouldn't it be wise to begin at once and learn to manage your pennies before the pounds come to perplex you?''
``I thought you would do all that fussy part and take care of the pounds, as you call them. Need I worry about it? I do hate sums, so!''
``I shall take care of things till you are of age, but I mean that you shall know how your property is managed, and do as much of it as you can by and by; then you won't be dependent on the honesty of other people.''
``Gracious me! as if I wouldn't trust you with millions of billions if I had them,'' cried Rose, scandalised at the mere suggestion.
``Ah, but I might be tempted; guardians are sometimes; so you'd better keep your eye on me, and in order to do that you must learn all about these affairs,'' answered Dr. Alec, as he made an entry in his own very neat account-book.
Rose peeped over his shoulder at it, and then turned to the arithmetical puzzle in her hand with a sigh of despair.
``Uncle, when you add up your expenses do you ever find you have got more money than you had in the beginning?''
``No; I usually find that I have a good deal less than I had in the beginning. Are you troubled in the peculiar way you mention?''
``Yes; it is very curious, but I never can make things come out square.''
``Perhaps I can help you,'' began Uncle Alec, in the most respectful tone.
``I think you had better, for if I have got to keep accounts I may as well begin in the right way. But please don't laugh! I know I'm very stupid, and my book is a disgrace, but I never could get it straight.'' And with great trepidation, Rose gave up her funny little accounts.
It really was good in Dr. Alec not to laugh, and Rose felt deeply grateful when he said in a mildly suggestive tone,
``The dollars and cents seem to be rather mixed, perhaps if I just straightened them out a bit we should find things all right.''
``Please do, and then show me on a fresh leaf how to make mine look nice and ship-shape as yours do.''
As Rose stood by him watching the ease with which he quickly brought order out of chaos, she privately resolved to hunt up her old arithmetic and perfect herself in the four first rules, with a good tug at fractions, before she read any more fairy tales.
``Am I a rich girl, uncle?'' she asked suddenly, as he was copying a column of figures.
``Rather a poor one, I should say, since you had to borrow a ninepence.''
``That was your fault, because you forgot my pocket-money. But, really, shall I be rich by and by?''
``I am afraid you will.''
``Why afraid, uncle?''
``Too much money is a bad thing.''
``But I can give it away, you know; that is always the pleasantest part of having it I think.''
``I'm glad you feel so, for you can do much good with your fortune if you know how to use it well.''
``You shall teach me, and when I am a woman we will set up a school where nothing but the three R's shall be taught, and all the children live on oatmeal, and the girls have waists a yard round,'' said Rose, with a sudden saucy smile dimpling her cheeks.
``You are an impertinent little baggage, to turn on me in that way right in the midst of my first attempt at teaching. Never mind, I'll have an extra bitter dose for you next time, miss.''
``I knew you wanted to laugh, so I gave you a chance. Now, I will be good, master, and do my lesson nicely.''
So Dr. Alec had his laugh, and then Rose sat down and took a lesson in accounts which she never forgot.
``Now come and read aloud to me; my eyes are tired, and it is pleasant to sit here by the fire while the rain pours outside and Aunt Jane lectures upstairs,'' said Uncle Alec, when last month's accounts had been put in good order and a fresh page neatly begun.
Rose liked to read aloud, and gladly gave him the chapter in ``Nicholas Nickleby'' where the Miss Kenwigses take their French lesson. She did her very best, feeling that she was being criticised, and hoping that she might not be found wanting in this as in other things.
``Shall I go on, sir?'' she asked very meekly, when the chapter ended.
``If you are not tired, dear. It is a pleasure to hear you, for you read remarkably well,'' was the answer that filled her heart with pride and pleasure.
``Do you really think so, uncle? I'm so glad! Papa taught me, and I read for hours to him, but I thought perhaps, he liked it because he was fond of me.''
``So am I; but you really do read unusually well, and I'm very glad of it, for it is a rare accomplishment, and one I value highly. Come here in this cosy, low chair; the light is better, and I can pull these curls if you go too fast. I see you are going to be a great comfort as well as a great credit to your old uncle, Rosy.'' And Dr. Alec drew her close beside him with such a fatherly look and tone that she felt it would be very easy to love and obey him, since he knew how to mix praise and blame so pleasantly together.
Another chapter was just finished, when the sound of a carriage warned them that Aunt Jane was about to depart. Before they could go to meet her, however, she appeared in the doorway looking like an unusually tall mummy in her waterproof, with her glasses shining like cat's eyes from the depths of the hood.
``Just as I thought! petting that child to death and letting her sit up late reading trash. I do hope you feel the weight of the responsibility you have taken upon yourself, Alec,'' she said, with a certain grim sort of satisfaction at seeing things go wrong.
``I think I have a very realising sense of it, sister Jane,'' answered Dr. Alec, with a comical shrug of the shoulders and a glance at Rose's bright face.
``It is sad to see a great girl wasting these precious hours so. Now, my boys have studied all day, and Mac is still at his books, I've no doubt, while you have not had a lesson since you came, I suspect.''
``I've had five to-day, ma'am,'' was Rose's very unexpected answer.
``I'm glad to hear it; and what were they, pray?'' Rose looked very demure as she replied,
``Navigation, geography, grammar, arithmetic, and keeping my temper.''
``Queer lessons, I fancy; and what have you learned from this remarkable mixture, I should like to know?''
A naughty sparkle came into Rose's eyes as she answered, with a droll look at her uncle,
``I can't tell you all, ma'am, but I have collected some useful information about China, which you may like, especially the teas. The best are Lapsing Souchong, Assam Pekoe, rare Ankoe, Flowery Pekoe, Howqua's mixture, Scented Caper, Padral tea, black Congou, and green Twankey. Shanghai is on the Woosung River. Hong Kong means 'Island of Sweet waters.' Singapore is `Lion's Town.' `Chops' are the boats they live in; and they drink tea out of little saucers. Principal productions are porcelain, tea, cinnamon, shawls, tin, tamarinds and opium. They have beautiful temples and queer gods; and in Canton is the Dwelling of the Holy Pigs, fourteen of them, very big, and all blind.''
The effect of this remarkable burst was immense, especially the fact last mentioned. It entirely took the wind out of Aunt Jane's sails; it was so sudden, so varied and unexpected, that she had not a word to say. The glasses remained fixed full upon Rose for a moment, and then, with a hasty ``Oh, indeed!'' the excellent lady bundled into her carriage and drove away, somewhat bewildered and very much disturbed.
She would have been more so if she had seen her reprehensible brother-in-law dancing a triumphal polka down the hall with Rose in honour of having silenced the enemy's battery for once.
\gutchapter{Chapter 9---Phebe's Secret}
``Why do you keep smiling to yourself, Phebe?'' asked Rose, as they were working together one morning, for Dr. Alec considered house-work the best sort of gymnastics for girls; so Rose took lessons of Phebe in sweeping, dusting and bed-making.
``I was thinking about a nice little secret I know, and couldn't help smiling.''
``Shall I know it, sometime?''
``Guess you will.''
``Shall I like it?''
``Oh, won't you, though!''
``Will it happen soon?''
``Sometime this week.''
``I know what it is! The boys are going to have fireworks on the fourth, and have got some surprise for me. Haven't they?''
``That's telling.''
``Well, I can wait; only tell me one thing is uncle in it?''
``Of course he is; there's never any fun without him.''
``Then it's all right, and sure to be nice.''
Rose went out on the balcony to shake the rugs, and, having given them a vigorous beating, hung them on the balustrade to air, while she took a look at her plants. Several tall vases and jars stood there, and a month of June sun and rain had worked wonders with the seeds and slips she had planted. Morning-glories and nasturtiums ran all over the bars, making haste to bloom. Scarlet beans and honeysuckles were climbing up from below to meet their pretty neighbours, and the woodbine was hanging its green festoons wherever it could cling.
The waters of the bay were dancing in the sunshine, a fresh wind stirred the chestnut-trees with a pleasant sound, and the garden below was full of roses, butterflies and bees. A great chirping and twittering went on among the birds, busy with their summer house-keeping, and, far away, the white-winged gulls were dipping and diving in the sea, where ships, like larger birds, went sailing to and fro.
``Oh, Phebe, it's such a lovely day, I do wish your fine secret was going to happen right away! I feel just like having a good time; don't you?'' said Rose, waving her arms as if she was going to fly.
``I often feel that way, but I have to wait for my good times, and don't stop working to wish for 'em. There, now you can finish as soon as the dust settles; I must go do my stairs,'' and Phebe trudged away with the broom, singing as she went.
Rose leaned where she was, and fell to thinking how many good times she had had lately, for the gardening had prospered finely, and she was learning to swim and row, and there were drives and walks, and quiet hours of reading and talk with Uncle Alec, and, best of all, the old pain and ennui seldom troubled her now. She could work and play all day, sleep sweetly all night, and enjoy life with the zest of a healthy, happy child. She was far from being as strong and hearty as Phebe, but she was getting on; the once pale cheeks had colour in them now, the hands were growing plump and brown, and the belt was not much too loose. No one talked to her about her health, and she forgot that she had ``no constitution.'' She took no medicine but Dr. Alec's three great remedies, and they seemed to suit her excellently. Aunt Plenty said it was the pills; but, as no second batch had ever followed the first, I think the old lady was mistaken.
Rose looked worthy of her name as she stood smiling to herself over a happier secret than any Phebe had a secret which she did not know herself till she found out, some years later, the magic of good health.
```Look only,' said the brownie,
'At the pretty gown of blue,
At the kerchief pinned about her head,
And at her little shoe,"'
said a voice from below, as a great cabbage-rose came flying against her cheek.
``What is the princess dreaming about up there in her hanging-garden?'' added Dr. Alec as she flung back a morning-glory.
``I was wishing I could do something pleasant this fine day; something very new and interesting, for the wind makes me feel frisky and gay.''
``Suppose we take a pull over to the Island? I intended to go this afternoon; but if you feel more like it now, we can be off at once.''
``I do! I do! I'll come in fifteen minutes, uncle. I must just scrabble my room to rights, for Phebe has got a great deal to do.''
Rose caught up the rugs and vanished as she spoke, while Dr. Alec went in, saying to himself, with an indulgent smile,
``It may upset things a trifle, but half a child's pleasure consists in having their fun when they want it.''
Never did duster flap more briskly than the one Rose used that day, and never was a room ``scrabbled'' to rights in such haste as hers. Tables and chairs flew into their places as if alive; curtains shook as if a gale was blowing; china rattled and small articles tumbled about as if a young earthquake was playing with them. The boating suit went on in a twinkling, and Rose was off with a hop and a skip, little dreaming how many hours it would be before she saw her pretty room again.
Uncle Alec was putting a large basket into the boat when she arrived, and before they were off Phebe came running down with a queer, knobby bundle done up in a water-proof.
``We can't eat half that luncheon, and I know we shall not need so many wraps. I wouldn't lumber the boat up so,'' said Rose, who still had secret scares when on the water.
``Couldn't you make a smaller parcel, Phebe?'' asked Dr. Alec, eyeing the bundle suspiciously.
``No, sir, not in such a hurry,'' and Phebe laughed as she gave a particularly large knob a good poke.
``Well, it will do for ballast. Don't forget the note to Mrs. Jessie, I beg of you.''
``No, sir. I'll send it right off,'' and Phebe ran up the bank as if she had wings to her feet.
``We'll take a look at the lighthouse first, for you have not been there yet, and it is worth seeing. By the time we have done that it will be pretty warm, and we will have lunch under the trees on the Island.''
Rose was ready for anything, and enjoyed her visit to the lighthouse on the Point very much, especially climbing up the narrow stairs and going inside the great lantern. They made a long stay, for Dr. Alec seemed in no hurry to go, and kept looking through his spy-glass as if he expected to discover something remarkable on sea or land. It was past twelve before they reached the Island, and Rose was ready for her lunch long before she got it.
``Now this is lovely! I do wish the boys were here. Won't it be nice to have them with us all their vacation? Why, it begins to-day, doesn't it? Oh, I wish I'd remembered it sooner, and perhaps they would have come with us,'' she said, as they lay luxuriously eating sandwiches under the old apple-tree.
``So we might. Next time we won't be in such a hurry. I expect the lads will take our heads off when they find us out,'' answered Dr. Alec, placidly drinking cold tea.
``Uncle, I smell a frying sort of a smell,'' Rose said, pausing suddenly as she was putting away the remains of the lunch half an hour later.
``So do I; it is fish, I think.''
For a moment they both sat with their noses in the air, sniffing like hounds; then Dr. Alec sprang up, saying with great decision,
``Now, this won't do! No one is permitted on this island without asking leave. I must see who dares to fry fish on my private property.''
Taking the basket on one arm and the bundle on the other, he strode away towards the traitorous smell, looking as fierce as a lion, while Rose marched behind under her umbrella.
``We are Robinson Crusoe and his man Friday going to see if the savages have come,'' she said presently, for her fancy was full of the dear old stories that all children love so well.
``And there they are! Two tents and two boats, as I live! These rascals mean to enjoy themselves, that's evident.''
``There ought to be more boats and no tents. I wonder where the prisoners are?''
``There are traces of them,'' and Dr. Alec pointed to the heads and tails of fishes strewn on the grass.
``And there are more,'' said Rose, laughing, as she pointed to a scarlet heap of what looked like lobsters.
``The savages are probably eating their victims now; don't you hear the knives rattle in that tent?''
``We ought to creep up and peep; Crusoe was cautious, you know, and Friday scared out of his wits,'' added Rose, still keeping up the joke.
``But this Crusoe is going to pounce upon them, regardless of consequences. If I am killed and eaten, you seize the basket and run for the boat; there are provisions enough for your voyage home.''
With that Uncle Alec slipped round to the front of the tent and, casting in the big bundle like a bomb-shell, roared out, in a voice of thunder,
``Pirates, surrender!''
A crash, a shout, a laugh, and out came the savages, brandishing knives and forks, chicken bones, and tin mugs, and all fell upon the intruder, pommelling him unmercifully as they cried,
``You came too soon! We are not half ready! You've spoilt it all! Where is Rose?''
``Here I am,'' answered a half-stifled voice, and Rose was discovered sitting on the pile of red flannel bathing clothes, which she had mistaken for lobsters, and where she had fallen in a fit of merriment when she discovered that the cannibals were her merry cousins.
``You good-for-nothing boys! You are always bursting out upon me in some ridiculous way, and I always get taken in because I'm not used to such pranks. Uncle is as bad as the rest, and it's great fun,'' she said, as the lads came round her, half scolding, half welcoming, and wholly enjoying the double surprise.
``You were not to come till afternoon, and mamma was to be here to receive you. Everything is in a mess now, except your tent; we got that in order the first thing, and you can sit there and see us work,'' said Archie, doing the honours as usual.
``Rose felt it in her bones, as Dolly says, that something was in the wind, and wanted to be off at once. So I let her come, and should have kept her away an hour longer if your fish had not betrayed you,'' explained Uncle Alec, subsiding from a ferocious Crusoe into his good-natured self again.
``As this seat is rather damp, I think I'll rise,'' said Rose, as the excitement lessened a little.
Several fishy hands helped her up, and Charlie said, as he scattered the scarlet garments over the grass with an oar,
``We had a jolly good swim before dinner, and I told the Brats to spread these to dry. Hope you brought your things, Rose, for you belong to the Lobsters, you know, and we can have no end of fun teaching you to dive and float and tread water.''
``I didn't bring anything---'' began Rose, but was interrupted by the Brats (otherwise Will and Geordie), who appeared bearing the big bundle, so much demoralised by its fall that a red flannel tunic trailed out at one end and a little blue dressing-gown at the other, while the knobs proved to be a toilet-case, rubbers, and a silver mug.
``Oh, that sly Phebe! This was the secret, and she bundled up those things after I went down to the boat,'' cried Rose, with sparkling eyes.
``Guess something is smashed inside, for a bit of glass fell out,'' observed Will, as they deposited the bundle at her feet.
``Catch a girl going anywhere without a looking-glass. We haven't got one among the whole lot of us,'' added Mac, with masculine scorn.
``Dandy has; I caught him touching up his wig behind the trees after our swim,'' cut in Geordie, wagging a derisive finger at Steve, who promptly silenced him by a smart rap on the head with the drum-stick he had just polished off.
``Come, come, you lazy lubbers, fall to work, or we shall not be ready for mamma. Take Rose's things to her tent, and tell her all about it, Prince. Mac and Steve, you cut away and bring up the rest of the straw; and you small chaps, clear off the table, if you have stuffed all you can. Please, uncle, I'd like your advice about the boundary lines and the best place for the kitchen.''
Everyone obeyed the chief, and Rose was escorted to her tent by Charlie, who devoted himself to her service. She was charmed with her quarters, and still more so with the programme which he unfolded before her as they worked.
``We always camp out somewhere in vacation, and this year we thought we'd try the Island. It is handy, and our fireworks will show off well from here.''
``Shall we stay over the Fourth? Three whole days! Oh, me! what a frolic it will be!''
``Bless your heart, we often camp for a week, we big fellows; but this year the small chaps wanted to come, so we let them. We have great larks, as you'll see; for we have a cave and play Captain Kidd, and have shipwrecks, and races, and all sorts of games. Arch and I are rather past that kind of thing now, but we do it to please the children,'' added Charlie, with a sudden recollection of his sixteen years.
``I had no idea boys had such good times. Their plays never seemed a bit interesting before. But I suppose that was because I never knew any boys very well, or perhaps you are unusually nice ones,'' observed Rose, with an artless air of appreciation that was very flattering.
``We are a pretty clever set, I fancy; but we have a good many advantages, you see. There are a tribe of us, to begin with; then our family has been here for ages, and we have plenty of `spondulics,' so we can rather lord it over the other fellows, and do as we like. There, ma'am, you can hang your smashed glass on that nail and do up your back hair as fine as you please. You can have a blue blanket or a red one, and a straw pillow or an air cushion for your head, whichever you like. You can trim up to any extent, and be as free and easy as squaws in a wigwam, for this corner is set apart for you ladies and we never cross the line uncle is drawing until we ask leave. Anything more I can do for you, cousin?''
``No, thank you. I think I'll leave the rest till auntie comes, and go and help you somewhere else, if I may.''
``Yes, indeed, come on and see to the kitchen. Can you cook?'' asked Charlie, as he led the way to the rocky nook where Archie was putting up a sail-cloth awning.
``I can make tea and toast bread.''
``Well, we'll shew you how to fry fish, and make chowder. Now you just set these pots and pans round tastefully, and sort of tidy up a bit, for Aunt Jessie insists on doing some of the work, and I want it to be decent here.''
By four o'clock the camp was in order, and the weary workers settled down on Lookout Rock to watch for Mrs. Jessie and Jamie, who was never far from mamma's apron string. They looked like a flock of blue-birds, all being in sailor rig, with blue ribbon enough flying from the seven hats to have set up a milliner. Very tuneful blue-birds they were, too, for all the lads sang, and the echo of their happy voices reached Mrs. Jessie long before she saw them.
The moment the boat hove in sight up went the Island flag, and the blue-jackets cheered lustily, as they did on every possible occasion, like true young Americans. This welcome was answered by the flapping of a handkerchief and the shrill ``Rah! Rah! Rah!'' of the one small tar who stood in the stern waving his hat manfully, while a maternal hand clutched him firmly in the rear.
Cleopatra landing from her golden galley never received a heartier greeting than ``Little Mum'' as she was borne to her tent by the young folk, for love of whom she smilingly resigned herself to three days of discomfort; while Jamie immediately attached himself to Rose, assuring her of his protection from the manifold perils which might assail them.
Taught by long experience that boys are always hungry, Aunt Jessie soon proposed supper, and proceeded to get it, enveloped in an immense apron, with an old hat of Archie's stuck atop of her cap. Rose helped, and tried to be as handy as Phebe, though the peculiar style of table she had to set made it no easy task. It was accomplished at last, and a very happy party lay about under the trees, eating and drinking out of anyone's plate and cup, and quite untroubled by the frequent appearance of ants and spiders in places which these interesting insects are not expected to adorn.
``I never thought I should like to wash dishes, but I do,'' said Rose, as she sat in a boat after supper lazily rinsing plates in the sea, and rocking luxuriously as she wiped them.
``Mum is mighty particular; we just give 'em a scrub with sand, and dust 'em off with a bit of paper. It's much the best way, I think,'' replied Geordie, who reposed in another boat alongside.
``How Phebe would like this! I wonder uncle did not have her come.''
``I believe he tried to, but Dolly was as cross as two sticks, and said she couldn't spare her. I'm sorry, for we all like the Phebe bird, and she'd chirp like a good one out here, wouldn't she?''
``She ought to have a holiday like the rest of us. It's too bad to leave her out.''
This thought came back to Rose several times that evening, for Phebe would have added much to the little concert they had in the moonlight, would have enjoyed the stories told, been quick at guessing the conundrums, and laughed with all her heart at the fun. The merry going to bed would have been the best of all, for Rose wanted someone to cuddle under the blue blanket with her, there to whisper and giggle and tell secrets, as girls delight to do.
Long after the rest were asleep, Rose lay wide awake, excited by the novelty of all about her, and a thought that had come into her mind. Far away she heard a city clock strike twelve; a large star like a mild eye peeped in at the opening of the tent, and the soft plash of the waves seemed calling her to come out. Aunt Jessie lay fast asleep, with Jamie rolled up like a kitten at her feet, and neither stirred as Rose in her wrapper crept out to see how the world looked at midnight.
She found it very lovely, and sat down on a cracker keg to enjoy it with a heart full of the innocent sentiment of her years. Fortunately, Dr. Alec saw her before she had time to catch cold, for coming out to tie back the door-flap of his tent for more air, he beheld the small figure perched in the moonlight. Having no fear of ghosts, he quietly approached, and, seeing that she was wide awake, said, with a hand on her shining hair,
``What is my girl doing here?''
``Having a good time,'' answered Rose, not at all startled.
``I wonder what she was thinking about with such a sober look.''
``The story you told of the brave sailor who gave up his place on the raft to the woman, and the last drop of water to the poor baby. People who make sacrifices are very much loved and admired, aren't they?'' she asked, earnestly.
``If the sacrifice is a true one. But many of the bravest never are known, and get no praise. That does not lessen their beauty, though perhaps it makes them harder, for we all like sympathy,'' and Dr. Alec sighed a patient sort of sigh.
``I suppose you have made a great many? Would you mind telling me one of them?'' asked Rose, arrested by the sigh.
``My last was to give up smoking,'' was the very unromantic answer to her pensive question.
``Why did you?''
``Bad example for the boys.''
``That was very good of you, uncle! Was it hard?''
``I'm ashamed to say it was. But as a wise old fellow once said, 'It is necessary to do right; it is not necessary to be happy.'''
Rose pondered over the saying as if it pleased her, and then said, with a clear, bright look,
``A real sacrifice is giving up something you want or enjoy very much, isn't it?''
``Yes.''
``Doing it one's own self because one loves another person very much and wants her to be happy?''
``Yes.''
``And doing it pleasantly, and being glad about it, and not minding the praise if it doesn't come?''
``Yes, dear, that is the true spirit of self-sacrifice; you seem to understand it, and I dare say you will have many chances in your life to try the real thing. I hope they won't be very hard ones.''
``I think they will,'' began Rose, and there stopped short.
``Well, make one now, and go to sleep, or my girl will be ill to-morrow, and then the aunts will say camping out was bad for her.''
``I'll go good night!'' and throwing him a kiss, the little ghost vanished, leaving Uncle Alec to pace the shore and think about some of the unsuspected sacrifices that had made him what he was.
\gutchapter{Chapter 10---Rose's Sacrifice}
There certainly were ``larks'' on Campbell's Island next day, as Charlie had foretold, and Rose took her part in them like one intent on enjoying every minute to the utmost. There was a merry breakfast, a successful fishing expedition, and then the lobsters came out in full force, for even Aunt Jessie appeared in red flannel. There was nothing Uncle Alec could not do in the water, and the boys tried their best to equal him in strength and skill, so there was a great diving and ducking, for every one was bent on distinguishing himself.
Rose swam out far beyond her depth, with uncle to float her back; Aunt Jessie splashed placidly in the shallow pools, with Jamie paddling near by like a little whale beside its mother; while the lads careered about, looking like a flock of distracted flamingoes, and acting like the famous dancing party in ``Alice's Adventures in Wonderland.''
Nothing but chowder would have lured them from their gambols in the briny deep; that time-honoured dish demanded the concentrated action of several mighty minds; so the ``Water Babies'' came ashore and fell to cooking.
It is unnecessary to say that, when done, it was the most remarkable chowder ever cooked, and the quantity eaten would have amazed the world if the secret had been divulged. After this exertion a siesta was considered the thing, and people lay about in tents or out as they pleased, the boys looking like warriors slumbering where they fell.
The elders had just settled to a comfortable nap when the youngsters rose, refreshed and ready for further exploits. A hint sent them all off to the cave, and there were discovered bows and arrows, battle clubs, old swords, and various relics of an interesting nature. Perched upon a commanding rock, with Jamie to ``splain'' things to her, Rose beheld a series of stirring scenes enacted with great vigour and historical accuracy by her gifted relatives.
Captain Cook was murdered by the natives of Owhyhee in the most thrilling manner. Captain Kidd buried untold wealth in the chowder kettle at the dead of night, and shot both the trusting villains who shared the secret of the hiding place. Sinbad came ashore there and had manifold adventures, and numberless wrecks bestrewed the sands.
Rose considered them by far the most exciting dramas she had ever witnessed; and when the performance closed with a grand ballet of Feejee Islanders, whose barbaric yells alarmed the gulls, she had no words in which to express her gratification.
Another swim at sunset, another merry evening on the rocks watching the lighted steamers pass seaward and the pleasure-boats come into port, ended the second day of the camping out, and sent everyone to bed early that they might be ready for the festivities of the morrow.
``Archie, didn't I hear uncle ask you to row home in the morning for fresh milk and things?''
``Yes, why?''
``Please, may I go too? I have something of great importance to arrange; you know I was carried off in a hurry,'' Rose said in a confidential whisper as she was bidding her cousins good night.
``I'm willing, and I guess Charlie won't mind.''
``Thank you; be sure you stand by me when I ask leave in the morning, and don't say anything till then, except to Charlie. Promise,'' urged Rose, so eagerly, that Archie struck an attitude and cried dramatically,
``By yonder moon I swear!''
``Hush! it's all right, go along''; and Rose departed as if satisfied.
``She's a queer little thing, isn't she, Prince?''
``Rather a nice little thing, I think. I'm quite fond of her.''
Rose's quick ears caught both remarks, and she retired to her tent, saying to herself with sleepy dignity,
``Little thing, indeed! Those boys talk as if I was a baby. They will treat me with more respect after to-morrow, I guess.''
Archie did stand by her in the morning, and her request was readily granted, as the lads were coming directly back. Off they went, and Rose waved her hand to the islanders with a somewhat pensive air, for an heroic purpose glowed within her, and the spirit of self-sacrifice was about to be illustrated in a new and touching manner.
While the boys got the milk Rose ran to Phebe, ordered her to leave her dishes, to put on her hat, and take a note back to Uncle Alec, which would explain this somewhat mysterious performance. Phebe obeyed, and when she went to the boat Rose accompanied her, telling the boys she was not ready to go yet, but they could, some of them, come for her when she hung a white signal on her balcony.
``But why not come now? What are you about, miss? Uncle won't like it,'' protested Charlie, in great amazement.
``Just do as I tell you, little boy; uncle will understand and explain. Obey, as Phebe does, and ask no questions. I can have secrets as well as other people''; and Rose walked off with an air of lofty independence that impressed her friends immensely.
``It's some plot between uncle and herself, so we won't meddle. All right, Phebe? Pull away, Prince''; and off they went to be received with much surprise by the islanders.
This was the note Phebe bore:
``Dear Uncle, I am going to take Phebe's place to-day, and let her have all the fun she can. Please don't mind what she says, but keep her, and tell the boys to be very good to her for my sake. Don't think it is easy to do this; it is very hard to give up the best day of all, but I feel so selfish to have all the pleasure and Phebe none, that I wish to make this sacrifice. Do let me, and don't laugh at it; I truly do not wish to be praised, and I truly want to do it. Love to all from,
``Rose.''
``Bless the little dear, what a generous heart she has! Shall we go after her, Jessie, or let her have her way?'' said Dr. Alec, after the first mingled amusement and astonishment had subsided.
``Let her alone, and don't spoil her little sacrifice. She means it, I know, and the best way in which we can show our respect for her effort is to give Phebe a pleasant day. I'm sure she has earned it''; and Mrs. Jessie made a sign to the boys to suppress their disappointment and exert themselves to please Rose's guest.
Phebe was with difficulty kept from going straight home, and declared that she should not enjoy herself one bit without Miss Rose.
``She won't hold out all day, and we shall see her paddling back before noon, I'll wager anything,'' said Charlie; and the rest so strongly inclined to his opinion that they resigned themselves to the loss of the little queen of the revels, sure that it would be only a temporary one.
But hour after hour passed, and no signal appeared on the balcony, though Phebe watched it hopefully. No passing boat brought the truant back, though more than one pair of eyes looked out for the bright hair under the round hat; and sunset came, bringing no Rose but the lovely colour in the western sky.
``I really did not think the child had it in her. I fancied it was a bit of sentiment, but I see she was in earnest, and means that her sacrifice shall be a true one. Dear little soul! I'll make it up to her a thousand times over, and beg her pardon for thinking it might be done for effect,'' Dr. Alec said remorsefully, as he strained his eyes through the dusk, fancying he saw a small figure sitting in the garden as it had sat on the keg the night before, laying the generous little plot that had cost more than he could guess.
``Well, she can't help seeing the fireworks, any way, unless she is goose enough to think she must hide in a dark closet and not look,'' said Archie, who was rather disgusted at Rose's seeming ingratitude.
``She will see ours capitally, but miss the big ones on the hill, unless papa has forgotten all about them,'' added Steve, cutting short the harangue Mac had begun upon the festivals of the ancients.
``I'm sure the sight of her will be better than the finest fireworks that ever went off,'' said Phebe, meditating an elopement with one of the boats if she could get a chance.
``Let things work; if she resists a brilliant invitation we give her she will be a heroine,'' added Uncle Alec, secretly hoping that she would not.
Meanwhile Rose had spent a quiet, busy day helping Dolly, waiting on Aunt Peace, and steadily resisting Aunt Plenty's attempts to send her back to the happy island. It had been hard in the morning to come in from the bright world outside, with flags flying, cannon booming, crackers popping, and everyone making ready for a holiday, and go to washing cups, while Dolly grumbled and the aunts lamented. It was very hard to see the day go by, knowing how gay each hour must have been across the water, and how a word from her would take her where she longed to be with all her heart. But it was hardest of all when evening came and Aunt Peace was asleep, Aunt Plenty seeing a gossip in the parlor, Dolly established in the porch to enjoy the show, and nothing left for the little maid to do but sit alone in her balcony and watch the gay rockets whizz up from island, hill, and city, while bands played and boats laden with happy people went to and fro in the fitful light.
Then it must be confessed that a tear or two dimmed the blue eyes, and once, when a very brilliant display illuminated the island for a moment, and she fancied she saw the tents, the curly head went down on the railing, and a wide-awake nasturtium heard a little whisper,
``I hope someone wishes I was there!''
The tears were all gone, however, and she was watching the hill and island answer each other with what Jamie called ``whizzers, whirligigs and busters,'' and smiling as she thought how hard the boys must be working to keep up such a steady fire, when Uncle Mac came walking in upon her, saying hurriedly,
``Come, child, put on your tippet, pelisse, or whatever you call it, and run off with me. I came to get Phebe, but aunt says she is gone, so I want you. I've got Fun down in the boat, and I want you to go with us and see my fireworks. Got them up for you, and you mustn't miss them, or I shall be disappointed.''
``But, uncle,'' began Rose, feeling as if she ought to refuse even a glimpse of bliss, ``perhaps---''
``I know, my dear, I know; aunt told me; but no one needs you now so much as I do, and I insist on your coming,'' said Uncle Mac, who seemed in a great hurry to be off, yet was unusually kind.
So Rose went and found the little Chinaman with a funny lantern waiting to help her in and convulse her with laughter trying to express his emotions in pigeon English. The city clocks were striking nine as they got out into the bay, and the island fireworks seemed to be over, for no rocket answered the last Roman candle that shone on the Aunt-hill.
``Ours are done, I see, but they are going up all round the city, and how pretty they are,'' said Rose, folding her mantle about her, and surveying the scene with pensive interest.
``Hope my fellows have not got into trouble up there,'' muttered Uncle Mac, adding with a satisfied chuckle, as a spark shone out, ``No; there it goes! Look, Rosy, and see how you like this one; it was ordered especially in honour of your coming.''
Rose looked with all her eyes, and saw the spark grow into the likeness of a golden vase, then green leaves came out, and then a crimson flower glowing on the darkness with a splendid lustre.
``Is it a rose, uncle?'' she asked, clasping her hands with delight as she recognised the handsome flower.
``Of course it is! Look again, and guess what those are,'' answered Uncle Mac, chuckling and enjoying it all like a boy.
A wreath of what looked at first like purple brooms appeared below the vase, but Rose guessed what they were meant for, and stood straight up, holding by his shoulder, and crying excitedly,
``Thistles, uncle, Scotch thistles! There are seven of them one for each boy! Oh, what a joke!'' and she laughed so that she plumped into the bottom of the boat and stayed there till the brilliant spectacle was quite gone.
``That was rather a neat thing, I flatter myself,'' said Uncle Mac, in high glee at the success of his illumination. ``Now, shall I leave you on the Island or take you home again, my good little girl?'' he added, lifting her up with such a tone of approbation in his voice that Rose kissed him on the spot.
``Home, please uncle; and I thank you very very much for the beautiful firework you got up for me. I'm so glad I saw it; and I know I shall dream about it,'' answered Rose steadily, though a wistful glance went toward the Island, now so near that she could smell powder and see shadowy figures flitting about.
Home they went; and Rose fell asleep saying to herself, ``It was harder than I thought, but I'm glad I did it, and I truly don't want any reward but Phebe's pleasure.''
\gutchapter{Chapter 11---Poor Mac}
Rose's sacrifice was a failure in one respect, for, though the elders loved her the better for it, and showed that they did, the boys were not inspired with the sudden respect which she had hoped for. In fact, her feelings were much hurt by overhearing Archie say that he couldn't see any sense in it; and the Prince added another blow by pronouncing her ``the queerest chicken ever seen.''
It is apt to be so, and it is hard to bear; for, though we do not want trumpets blown, we do like to have our little virtues appreciated, and cannot help feeling disappointed if they are not.
A time soon came, however, when Rose, quite unconsciously, won not only the respect of her cousins, but their gratitude and affection likewise.
Soon after the Island episode, Mac had a sunstroke, and was very ill for some time. It was so sudden that everyone was startled, and for some days the boy's life was in danger. He pulled through, however; and then, just as the family were rejoicing, a new trouble appeared which cast a gloom over them all.
Poor Mac's eyes gave out; and well they might, for he had abused them, and never being very strong, they suffered doubly now.
No one dared to tell him the dark predictions of the great oculist who came to look at them, and the boy tried to be patient, thinking that a few weeks of rest would repair the overwork of several years.
He was forbidden to look at a book, and as that was the one thing he most delighted in, it was a terrible affliction to the Worm. Everyone was very ready to read to him, and at first the lads contended for this honour. But as week after week went by, and Mac was still condemned to idleness and a darkened room, their zeal abated, and one after the other fell off. It was hard for the active fellows, right in the midst of their vacation; and nobody blamed them when they contented themselves with brief calls, running of errands, and warm expressions of sympathy.
The elders did their best, but Uncle Mac was a busy man, Aunt Jane's reading was of a funereal sort, impossible to listen to long, and the other aunties were all absorbed in their own cares, though they supplied the boy with every delicacy they could invent.
Uncle Alec was a host in himself, but he could not give all his time to the invalid; and if it had not been for Rose, the afflicted Worm would have fared ill. Her pleasant voice suited him, her patience was unfailing, her time of no apparent value, and her eager good-will was very comforting.
The womanly power of self-devotion was strong in the child, and she remained faithfully at her post when all the rest dropped away. Hour after hour she sat in the dusky room, with one ray of light on her book, reading to the boy, who lay with shaded eyes silently enjoying the only pleasure that lightened the weary days. Sometimes he was peevish and hard to please, sometimes he growled because his reader could not manage the dry books he wished to hear, and sometimes he was so despondent that her heart ached to see him. Through all these trials Rose persevered, using all her little arts to please him. When he fretted, she was patient; when he growled, she ploughed bravely through the hard pages not dry to her in one sense, for quiet tears dropped on them now and then; and when Mac fell into a despairing mood, she comforted him with every hopeful word she dared to offer.
He said little, but she knew he was grateful, for she suited him better than anyone else. If she was late, he was impatient; when she had to go, he seemed forlorn; and when the tired head ached worst, she could always soothe him to sleep, crooning the old songs her father used to love.
``I don't know what I should do without that child,'' Aunt Jane often said.
``She's worth all those racketing fellows put together,'' Mac would add, fumbling about to discover if the little chair was ready for her coming.
That was the sort of reward Rose liked, the thanks that cheered her; and whenever she grew very tired, one look at the green shade, the curly head so restless on the pillow, and the poor groping hands, touched her tender heart and put new spirit into the weary voice.
She did not know how much she was learning, both from the books she read and the daily sacrifices she made. Stories and poetry were her delight, but Mac did not care for them; and since his favourite Greeks and Romans were forbidden, he satisfied himself with travels, biographies, and the history of great inventions or discoveries. Rose despised this taste at first, but soon got interested in Livingstone's adventures, Hobson's stirring life in India, and the brave trials and triumphs of Watt and Arkwright, Fulton, and ``Palissy, the Potter.'' The true, strong books helped the dreamy girl; her faithful service and sweet patience touched and won the boy; and long afterward both learned to see how useful those seemingly hard and weary hours had been to them.
One bright morning, as Rose sat down to begin a fat volume entitled ``History of the French Revolution,'' expecting to come to great grief over the long names, Mac, who was lumbering about the room like a blind bear, stopped her by asking abruptly,
``What day of the month is it?''
``The seventh of August, I believe.''
``More than half my vacation gone, and I've only had a week of it! I call that hard,'' and he groaned dismally.
``So it is; but there is more to come, and you may be able to enjoy that.''
``May be able! I will be able! Does that old noodle think I'm going to stay stived up here much longer?''
``I guess he does, unless your eyes get on faster than they have yet.''
``Has he said anything more lately?''
``I haven't seen him, you know. Shall I begin? this looks rather nice.''
``Read away; it's all one to me.'' And Mac cast himself down upon the old lounge, where his heavy head felt easiest.
Rose began with great spirit, and kept on gallantly for a couple of chapters, getting over the unpronounceable names with unexpected success, she thought, for her listener did not correct her once, and lay so still she fancied he was deeply interested. All of a sudden she was arrested in the middle of a fine paragraph by Mac, who sat bolt upright, brought both feet down with a thump, and said, in a rough, excited tone,
``Stop! I don't hear a word, and you may as well save your breath to answer my question.''
``What is it?'' asked Rose, looking uneasy, for she had something on her mind, and feared that he suspected what it was. His next words proved that she was right.
``Now, look here, I want to know something, and you've got to tell me.''
``Please, don't---'' began Rose, beseechingly.
``You must, or I'll pull off this shade and stare at the sun as hard as ever I can stare. Come now!'' and he half rose, as if ready to execute the threat.
``I will! oh, I will tell, if I know! But don't be reckless and do anything so crazy as that,'' cried Rose, in great distress.
``Very well; then listen, and don't dodge, as everyone else does. Didn't the doctor think my eyes worse the last time he came? Mother won't say, but you shall.''
``I believe he did,'' faltered Rose.
``I thought so! Did he say I should be able to go to school when it begins?''
``No, Mac,'' very low.
``Ah!''
That was all, but Rose saw her cousin set his lips together and take a long breath, as if she had hit him hard. He bore the disappointment bravely, however, and asked quite steadily in a minute,
``How soon does he think I can study again?''
It was so hard to answer that! Yet Rose knew she must, for Aunt Jane had declared she could not do it, and Uncle Mac had begged her to break the truth to the poor lad.
``Not for a good many months.''
``How many?'' he asked with a pathetic sort of gruffness.
``A year, perhaps.''
``A whole year! Why, I expected to be ready for college by that time.'' And, pushing up the shade, Mac stared at her with startled eyes, that soon blinked and fell before the one ray of light.
``Plenty of time for that; you must be patient now, and get them thoroughly well, or they will trouble you again when it will be harder to spare them,'' she said, with tears in her own eyes.
``I won't do it! I will study and get through somehow. It's all humbug about taking care so long. These doctors like to keep hold of a fellow if they can. But I won't stand it I vow I won't!'' and he banged his fist down on the unoffending pillow as if he were pommelling the hard-hearted doctor.
``Now, Mac, listen to me,'' Rose said very earnestly, though her voice shook a little and her heart ached. ``You know you have hurt your eyes reading by fire-light and in the dusk, and sitting up late, and now you'll have to pay for it; the doctor said so. You must be careful, and do as he tells you, or you will be blind.''
``No!''
``Yes, it is true, and he wanted us to tell you that nothing but entire rest would cure you. I know it's dreadfully hard, but we'll all help you; I'll read all day long, and lead you, and wait upon you, and try to make it easier.''
She stopped there, for it was evident that he did not hear a sound; the word ``blind'' seemed to have knocked him down, for he had buried his face in the pillow, and lay so still that Rose was frightened. She sat motionless for many minutes, longing to comfort him, but not knowing how, and wishing Uncle Alec would come, for he had promised to tell Mac.
Presently, a sort of choking sound came out of the pillow, and went straight to her heart the most pathetic sob she ever heard, for, though it was the most natural means of relief, the poor fellow must not indulge in it because of the afflicted eyes. The ``French Revolution'' tumbled out of her lap, and, running to the sofa, she knelt down by it, saying, with the motherly sort of tenderness girls feel for any sorrowing creature,
``Oh, my dear, you mustn't cry! It is so bad for your poor eyes. Take your head out of that hot pillow, and let me cool it. I don't wonder you feel so, but please don't cry. I'll cry for you; it won't hurt me.''
As she spoke she pulled away the cushion with gentle force, and saw the green shade all crushed and stained with the few hot tears that told how bitter the disappointment had been. Mac felt her sympathy, but, being a boy, did not thank her for it; only sat up with a jerk, saying, as he tried to rub away the tell-tale drops with the sleeve of his jacket, ``Don't bother; weak eyes always water. I'm all right.''
But Rose cried out, and caught his arm, ``Don't touch them with that rough woollen stuff! Lie down and let me bathe them, there's a dear boy; then there will be no harm done.''
``They do smart confoundedly. I say, don't you tell the other fellows that I made a baby of myself, will you?'' he added, yielding with a sigh to the orders of his nurse, who had flown for the eye-wash and linen cambric handkerchief.
``Of course I won't; but anyone would be upset at the idea of being well troubled in this way. I'm sure you bear it splendidly, and you know it isn't half so bad when you get used to it. Besides, it is only for a time, and you can do lots of pleasant things if you can't study. You'll have to wear blue goggles, perhaps; won't that be funny?''
And while she was pouring out all the comfortable words she could think of, Rose was softly bathing the eyes and dabbing the hot forehead with lavender-water, as her patient lay quiet with a look on his face that grieved her sadly.
``Homer was blind, and so was Milton, and they did something to be remembered by, in spite of it,'' he said, as if to himself, in a solemn tone, for even the blue goggles did not bring a smile.
``Papa had a picture of Milton and his daughters writing for him. It was a very sweet picture, I thought,'' observed Rose in a serious voice, trying to meet the sufferer on his own ground.
``Perhaps I could study if someone read and did the eye part. Do you suppose I could, by and by?'' he asked, with a sudden ray of hope.
``I dare say, if your head is strong enough. This sunstroke, you know, is what upset you, and your brain needs rest, the doctor says.''
``I'll have a talk with the old fellow next time he comes, and find out just what I may do; then I shall know where I am. What a fool I was that day to be stewing my brains and letting the sun glare on my book till the letters danced before me! I see 'em now when I shut my eyes; black balls bobbing round, and stars and all sorts of queer things. Wonder if all blind people do?''
``Don't think about them; I'll go on reading, shall I? We shall come to the exciting part soon, and then you'll forget all this,'' suggested Rose.
``No, I never shall forget. Hang the old `Revolution'! I don't want to hear another word of it. My head aches, and I'm hot. Oh, wouldn't I like to go for a pull in the `Stormy Petrel!"' and poor Mac tossed about as if he did not know what to do with himself.
``Let me sing, and perhaps you'll drop off; then the day will seem shorter,'' said Rose, taking up a fan and sitting down beside him.
``Perhaps I shall; I didn't sleep much last night, and when I did I dreamed like fun. See here, you tell the people that I know, and it's all right, and I don't want them to talk about it or howl over me. That's all; now drone away, and I'll try to sleep. Wish I could for a year, and wake up cured.''
``Oh, I wish, I wish you could!''
Rose said it so fervently that Mac was moved to grope for her apron and hold on to a corner of it, as if it was comfortable to feel her near him. But all he said was,
``You are a good little soul, Rosy. Give us `The Birks'; that is a drowsy one that always sends me off.''
Quite contented with this small return for all her sympathy, Rose waved her fan and sang, in a dreamy tone, the pretty Scotch air, the burden of which is,
``Bonny lassie, will ye gang, will ye gang
To the Birks of Aberfeldie?''
Whether the lassie went or not I cannot say, but the laddie was off to the land of Nod, in about ten minutes, quite worn out with hearing the bad tidings and the effort to bear them manfully.
\gutchapter{Chapter 12---``The Other Fellows''}
Rose did tell ``the people'' what had passed, and no one ``howled'' over Mac, or said a word to trouble him. He had his talk with the doctor, and got very little comfort out of it, for he found that ``just what he might do'' was nothing at all; though the prospect of some study by and by, if all went well, gave him courage to bear the woes of the present. Having made up his mind to this, he behaved so well that everyone was astonished, never having suspected so much manliness in the quiet Worm.
The boys were much impressed, both by the greatness of the affliction which hung over him and by his way of bearing it. They were very good to him, but not always particularly wise in their attempts to cheer and amuse; and Rose often found him much downcast after a visit of condolence from the Clan. She still kept her place as head-nurse and chief-reader, though the boys did their best in an irregular sort of way. They were rather taken aback sometimes at finding Rose's services preferred to their's, and privately confided to one another that ``Old Mac was getting fond of being molly-coddled.'' But they could not help seeing how useful she was, and owning that she alone had remained faithful a fact which caused some of them much secret compunction now and then.
Rose felt that she ruled in that room, if nowhere else, for Aunt Jane left a great deal to her, finding that her experience with her invalid father fitted her for a nurse, and in a case like this, her youth was an advantage rather than a drawback. Mac soon came to think that no one could take care of him so well as Rose, and Rose soon grew fond of her patient, though at first she had considered this cousin the least attractive of the seven. He was not polite and sensible like Archie, nor gay and handsome like Prince Charlie, nor neat and obliging like Steve, nor amusing like the ``Brats,'' nor confiding and affectionate like little Jamie. He was rough, absent-minded, careless, and awkward, rather priggish, and not at all agreeable to a dainty, beauty-loving girl like Rose.
But when his trouble came upon him, she discovered many good things in this cousin of hers, and learned not only to pity but to respect and love the poor Worm, who tried to be patient, brave, and cheerful, and found it a harder task than anyone guessed, except the little nurse, who saw him in his gloomiest moods. She soon came to think that his friends did not appreciate him, and upon one occasion was moved to free her mind in a way that made a deep impression on the boys.
Vacation was almost over, and the time drawing near when Mac would be left outside the happy school-world which he so much enjoyed. This made him rather low in his mind, and his cousins exerted themselves to cheer him up, especially one afternoon when a spasm of devotion seemed to seize them all. Jamie trudged down the hill with a basket of blackberries which he had ``picked all his ownself,'' as his scratched fingers and stained lips plainly testified. Will and Geordie brought their puppies to beguile the weary hours, and the three elder lads called to discuss baseball, cricket, and kindred subjects, eminently fitted to remind the invalid of his privations.
Rose had gone to drive with Uncle Alec, who declared she was getting as pale as a potato sprout, living so much in a dark room. But her thoughts were with her boy all the while, and she ran up to him the moment she returned, to find things in a fine state of confusion.
With the best intentions in life, the lads had done more harm than good, and the spectacle that met Nurse Rose's eye was a trying one. The puppies were yelping, the small boys romping, and the big boys all talking at once; the curtains were up, the room close, berries scattered freely about, Mac's shade half off, his cheeks flushed, his temper ruffled, and his voice loudest of all as he disputed hotly with Steve about lending certain treasured books which he could no longer use.
Now Rose considered this her special kingdom, and came down upon the invaders with an energy which amazed them and quelled the riot at once. They had never seen her roused before, and the effect was tremendous; also comical, for she drove the whole flock of boys out of the room like an indignant little hen defending her brood. They all went as meekly as sheep; the small lads fled from the house precipitately, but the three elder ones only retired to the next room, and remained there hoping for a chance to explain and apologise, and so appease the irate young lady, who had suddenly turned the tables and clattered them about their ears.
As they waited, they observed her proceedings through the half-open door, and commented upon them briefly but expressively, feeling quite bowed down with remorse at the harm they had innocently done.
``She's put the room to rights in a jiffey. What jacks we were to let those dogs in and kick up such a row,'' observed Steve, after a prolonged peep.
``The poor old Worm turns as if she was treading on him instead of cuddling him like a pussy cat. Isn't he cross, though?'' added Charlie, as Mac was heard growling about his ``confounded head.''
``She will manage him; but it's mean in us to rumple him up and then leave her to smooth him down. I'd go and help, but I don't know how,'' said Archie, looking much depressed, for he was a conscientious fellow, and blamed himself for his want of thought.
``No, more do I. Odd, isn't it, what a knack women have for taking care of sick folks?'' and Charlie fell a-musing over this undeniable fact.
``She has been ever so good to Mac,'' began Steve, in a self-reproachful tone.
``Better than his own brother, hey?'' cut in Archie, finding relief for his own regret in the delinquencies of another.
``Well, you needn't preach; you didn't any of you do any more, and you might have, for Mac likes you better than he does me. I always fret him, he says, and it isn't my fault if I am a quiddle,'' protested Steve, in self-defence.
``We have all been selfish and neglected him, so we won't fight about it, but try and do better,'' said Archie, generously taking more than his share of blame, for he had been less inattentive than either of the others.
``Rose has stood by him like a good one, and it's no wonder he likes to have her round best. I should myself if I was down on my luck as he is,'' put in Charlie, feeling that he really had not done ``the little thing'' justice.
``I'll tell you what it is, boys we haven't been half good enough to Rose, and we've got to make it up to her somehow,'' said Archie, who had a very manly sense of honour about paying his debts, even to a girl.
``I'm awfully sorry I made fun of her doll when Jamie lugged it out; and I called her `baby bunting' when she cried over the dead kitten. Girls are such geese sometimes, I can't help it,'' said Steve, confessing his transgressions handsomely, and feeling quite ready to atone for them if he only knew how.
``I'll go down on my knees and beg her pardon for treating her as if she was a child. Don't it make her mad, though? Come to think of it, she's only two years or so younger than I am. But she is so small and pretty, she always seems like a dolly to me,'' and the Prince looked down from his lofty height of five feet five as if Rose was indeed a pygmy beside him.
``That dolly has got a real good little heart, and a bright mind of her own, you'd better believe. Mac says she understands some things quicker than he can, and mother thinks she is an uncommonly nice girl, though she don't know all creation. You needn't put on airs, Charlie, though you are a tall one, for Rose likes Archie better than you; she said she did because he treated her respectfully.''
``Steve looks as fierce as a game-cock; but don't you get excited, my son, for it won't do a bit of good. Of course, everybody likes the Chief best; they ought to, and I'll punch their heads if they don't. So calm yourself, Dandy, and mend your own manners before you come down on other people's.''
Thus the Prince with great dignity and perfect good nature, while Archie looked modestly gratified with the flattering opinions of his kinsfolk, and Steve subsided, feeling he had done his duty as a cousin and a brother. A pause ensued, during which Aunt Jane appeared in the other room, accompanied by a tea-tray sumptuously spread, and prepared to feed her big nestling, as that was a task she allowed no one to share with her.
``If you have a minute to spare before you go, child, I wish you'd just make Mac a fresh shade; this has got a berry stain on it, and he must be tidy, for he is to go out to-morrow if it is a cloudy day,'' said Mrs. Jane, spreading toast in a stately manner, while Mac slopped his tea about without receiving a word of reproof.
``Yes, aunt,'' answered Rose, so meekly that the boys could hardly believe it could be the same voice which had issued the stern command, ``Out of this room, every one of you!'' not very long ago.
They had not time to retire, without unseemly haste, before she walked into the parlour and sat down at the work-table without a word. It was funny to see the look the three tall lads cast at the little person sedately threading a needle with green silk. They all wanted to say something expressive of repentance, but no one knew how to begin, and it was evident, from the prim expression of Rose's face, that she intended to stand upon her dignity till they had properly abased themselves. The pause was becoming very awkward, when Charlie, who possessed all the persuasive arts of a born scapegrace, went slowly down upon his knees before her, beat his breast, and said, in a heart-broken tone,
``Please forgive me this time, and I'll never do so any more.''
It was very hard to keep sober, but Rose managed it and answered gravely,
``It is Mac's pardon you should ask, not mine, for you haven't hurt me, and I shouldn't wonder if you had him a great deal, with all that light and racket, and talk about things that only worry him.''
``Do you really think we've hurt him, cousin?'' asked Archie, with a troubled look, while Charlie settled down in a remorseful heap among the table legs.
``Yes, I do, for he has got a raging headache, and his eyes are as red as as this emery bag,'' answered Rose, solemnly plunging her needle into a fat flannel strawberry.
Steve tore his hair, metaphorically speaking, for he clutched his cherished top-knot, and wildly dishevelled it, as if that was the heaviest penance he could inflict upon himself at such short notice. Charlie laid himself out flat, melodramatically begging someone to take him away and hang him; but Archie, who felt worst of all, said nothing except to vow within himself that he would read to Mac till his own eyes were as red as a dozen emery bags combined.
Seeing the wholesome effects of her treatment upon these culprits, Rose felt that she might relent and allow them a gleam of hope. She found it impossible to help trampling upon the prostrate Prince a little, in words at least, for he had hurt her feelings oftener than he knew; so she gave him a thimble-pie on the top of his head, and said, with an air of an infinitely superior being,
``Don't be silly, but get up, and I'll tell you something much better to do than sprawling on the floor and getting all over lint.''
Charlie obediently sat himself upon a hassock at her feet; the other sinners drew near to catch the words of wisdom about to fall from her lips, and Rose, softened by this gratifying humility, addressed them in her most maternal tone.
``Now, boys, if you really want to be good to Mac, you can do it in this way. Don't keep talking about things he can't do, or go and tell what fun you have had batting your ridiculous balls about. Get some nice book and read quietly; cheer him up about school, and offer to help him study by and by; you can do that better than I, because I'm only a girl, and don't learn Greek and Latin and all sorts of headachy stuff.''
``Yes, but you can do heaps of things better than we can; you've proved that,'' said Archie, with an approving look that delighted Rose, though she could not resist giving Charlie one more rebuke, by saying, with a little bridling of the head, and a curl of the lip that wanted to smile instead,
``I'm glad you think so, though I am a `queer chicken."'
This scathing remark caused the Prince to hide his face for shame, and Steve to erect his head in the proud consciousness that this shot was not meant for him. Archie laughed, and Rose, seeing a merry blue eye winking at her from behind two brown hands, gave Charlie's ear a friendly tweak, and extended the olive-branch of peace.
``Now we'll all be good, and plan nice things for poor Mac,'' she said, smiling so graciously that the boys felt as if the sun had suddenly burst out from behind a heavy cloud and was shining with great brilliancy.
The storm had cleared the air, and quite a heavenly calm succeeded, during which plans of a most varied and surprising sort were laid, for everyone burned to make noble sacrifices upon the shrine of ``poor Mac,'' and Rose was the guiding star to whom the others looked with most gratifying submission. Of course, this elevated state of things could not endure long, but it was very nice while it lasted, and left an excellent effect upon the minds of all when the first ardour had subsided.
``There, that's ready for to-morrow, and I do hope it will be cloudy,'' said Rose, as she finished off the new shade, the progress of which the boys had watched with interest.
``I'd bespoken an extra sunny day, but I'll tell the clerk of the weather to change it. He's an obliging fellow, and he'll attend to it, so make yourself easy,'' said Charlie, who had become quite perky again.
``It is very easy for you to joke, but how would you like to wear a blinder like that for weeks and weeks, sir?'' and Rose quenched his rising spirits by slipping the shade over his eyes, as he still sat on the cushion at her feet.
``It's horrid! Take it off, take it off! I don't wonder the poor old boy has the blues with a thing like that on''; and Charlie sat looking at what seemed to him an instrument of torture, with such a sober face that Rose took it gently away, and went in to bid Mac good-night.
``I shall go home with her, for it is getting darkish, and she is rather timid,'' said Archie, forgetting that he had often laughed at this very timidity.
``I think I might, for she's taking care of my brother,'' put in Steve, asserting his rights.
``Let's all go, that will please her''; proposed Charlie, with a burst of gallantry which electrified his mates.
``We will!'' they said with one voice, and they did, to Rose's great surprise and secret contentment; though Archie had all the care of her, for the other two were leaping fences, running races, and having wrestling matches all the way down.
They composed themselves on reaching the door, however; shook hands cordially all round, made their best bows, and retired with great elegance and dignity, leaving Rose to say to herself, with girlish satisfaction, as she went in,
``Now, that is the way I like to be treated.''
\gutchapter{Chapter 13---Cosey Corner}
Vacation was over, the boys went back to school, and poor Mac was left lamenting. He was out of the darkened room now, and promoted to blue goggles, through which he took a gloomy view of life, as might have been expected; for there was nothing he could do but wander about, and try to amuse himself without using his eyes. Anyone who has ever been condemned to that sort of idleness knows how irksome it is, and can understand the state of mind which caused Mac to say to Rose in a desperate tone one day,
``Look here, if you don't invent some new employment or amusement for me, I shall knock myself on the head as sure as you live.''
Rose flew to Uncle Alec for advice, and he ordered both patient and nurse to the mountains for a month, with Aunt Jessie and Jamie as escort. Pokey and her mother joined the party, and one bright September morning six very happy-looking people were aboard the express train for Portland two smiling mammas, laden with luncheon baskets and wraps; a pretty young girl with a bag of books on her arm; a tall thin lad with his hat over his eyes; and two small children, who sat with their short legs straight out before them, and their chubby faces beaming with the first speechless delight of ``truly travelling.''
An especially splendid sunset seemed to have been prepared to welcome them when, after a long day's journey, they drove into a wide, green door-yard, where a white colt, a red cow, two cats, four kittens, many hens, and a dozen people, old and young, were gaily disporting themselves. Everyone nodded and smiled in the friendliest manner, and a lively old lady kissed the new-comers all round, as she said heartily,
``Well, now, I'm proper glad to see you! Come right in and rest, and we'll have tea in less than no time, for you must be tired. Lizzie, you show the folks upstairs; Kitty, you fly round and help father in with the trunks; and Jenny and I will have the table all ready by the time you come down. Bless the dears, they want to go see the pussies, and so they shall!''
The three pretty daughters did ``fly round,'' and everyone felt at home at once, all were so hospitable and kind. Aunt Jessie had raptures over the home-made carpets, quilts and quaint furniture; Rose could not keep away from the windows, for each framed a lovely picture; and the little folks made friends at once with the other children, who filled their arms with chickens and kittens, and did the honours handsomely.
The toot of a horn called all to supper, and a goodly party, including six children besides the Camp-bells, assembled in the long dining-room, armed with mountain appetites and the gayest spirits. It was impossible for anyone to be shy or sober, for such gales of merriment arose they blew the starch out of the stiffest, and made the saddest jolly. Mother Atkinson, as all called their hostess, was the merriest there, and the busiest; for she kept flying up to wait on the children, to bring out some new dish, or to banish the live stock, who were of such a social turn that the colt came into the entry and demanded sugar; the cats sat about in people's laps, winking suggestively at the food; and speckled hens cleared the kitchen floor of crumbs, as they joined in the chat with a cheerful clucking.
Everybody turned out after tea to watch the sunset till all the lovely red was gone, and mosquitoes wound their shrill horns to sound the retreat. The music of an organ surprised the new-comers, and in the parlor they found Father Atkinson playing sweetly on the little instrument made by himself. All the children gathered about him, and, led by the tuneful sisters, sang prettily till Pokey fell asleep behind the door, and Jamie gaped audibly right in the middle of his favourite,
``Coo,'' said the little doves: ``Coo,'' said she,
``All in the top of the old pine-tree.''
The older travellers, being tired, went to ``bye low'' at the same time, and slept like tops in home-spun sheets, on husk mattresses made by Mother Atkinson, who seemed to have put some soothing powder among them, so deep and sweet was the slumber that came.
Next day began the wholesome out-of-door life, which works such wonders with tired minds and feeble bodies. The weather was perfect, and the mountain air made the children as frisky as young lambs; while the elders went about smiling at one another, and saying, ``Isn't it splendid?'' Even Mac, the ``slow coach,'' was seen to leap over a fence as if he really could not help it; and when Rose ran after him with his broad-brimmed hat, he made the spirited proposal to go into the woods and hunt for a catamount.
Jamie and Pokey were at once enrolled in the Cosey Corner Light Infantry a truly superb company, composed entirely of officers, all wearing cocked hats, carrying flags, waving swords, or beating drums. It was a spectacle to stir the dullest soul when this gallant band marched out of the yard in full regimentals, with Captain Dove a solemn, big-headed boy of eleven issuing his orders with the gravity of a general, and his Falstaffian regiment obeying them with more docility than skill. The little Snow children did very well, and Lieutenant Jack Dove was fine to see; so was Drummer Frank, the errand-boy of the house, as he rub-a-dub-dubbed with all his heart and drumsticks. Jamie had ``trained'' before, and was made a colonel at once; but Pokey was the best of all, and called forth a spontaneous burst of applause from the spectators as she brought up the rear, her cocked hat all over one eye, her flag trailing over her shoulder, and her wooden sword straight up in the air; her face beaming and every curl bobbing with delight as her fat legs tottered in the vain attempt to keep step manfully.
Mac and Rose were picking blackberries in the bushes beside the road when the soldiers passed without seeing them, and they witnessed a sight that was both pretty and comical. A little farther on was one of the family burial spots so common in those parts, and just this side of it Captain Fred Dove ordered his company to halt, explaining his reason for so doing in the following words,
``That's a graveyard, and it's proper to muffle the drums and lower the flags as we go by, and we'd better take off our hats, too; it's more respectable, I think.''
``Isn't that cunning of the dears?'' whispered Rose, as the little troop marched slowly by to the muffled roll of the drums, every flag and sword held low, all the little heads uncovered, and the childish faces very sober as the leafy shadows flickered over them.
``Let's follow and see what they are after,'' proposed Mac, who found sitting on the wall and being fed with blackberries luxurious but tiresome.
So they followed and heard the music grow lively, saw the banners wave in the breeze again when the graveyard was passed, and watched the company file into the dilapidated old church that stood at the corner of three woodland roads. Presently the sound of singing made the outsiders quicken their steps, and, stealing up, they peeped in at one of the broken windows.
Captain Dove was up in the old wooden pulpit, gazing solemnly down upon his company, who, having stacked their arms in the porch, now sat in the bare pews singing a Sunday-school hymn with great vigour and relish.
``Let us pray,'' said Captain Dove, with as much reverence as an army chaplain; and, folding his hands, he repeated a prayer which he thought all would know an excellent little prayer, but not exactly appropriate to the morning, for it was,
``Now I lay me down to sleep.''
Everyone joined in saying it, and it was a pretty sight to see the little creatures bowing their curly heads and lisping out the words they knew so well. Tears came into Rose's eyes as she looked; Mac took his hat off involuntarily, and then clapped it on again as if ashamed of showing any feeling.
``Now I shall preach you a short sermon, and my text is, 'Little children, love one another.' I asked mamma to give me one, and she thought that would be good; so you all sit still and I'll preach it. You mustn't whisper, Marion, but hear me. It means that we should be good to each other, and play fair, and not quarrel as we did this very day about the wagon. Jack can't always drive, and needn't be mad because I like to go with Frank. Annette ought to be horse sometimes and not always driver; and Willie may as well make up his mind to let Marion build her house by his, for she will do it, and he needn't fuss about it. Jamie seems to be a good boy, but I shall preach to him if he isn't. No, Pokey, people don't kiss in church or put their hats on. Now you must all remember what I tell you, because I am the captain, and you should mind me.''
Here Lieutenant Jack spoke right out in meeting with the rebellious remark,
``Don't care if you are; you'd better mind yourself, and tell how you took away my strap, and kept the biggest doughnut, and didn't draw fair when we had the truck.''
``Yes, and you slapped Frank; I saw you!'' bawled Willie Snow, bobbing up in his pew.
``And you took my book away and hid it 'cause I wouldn't go and swing when you wanted me to,'' added Annette, the oldest of the Snow trio.
``I shan't build my house by Willie's if he don't want me to, so now!'' put in little Marion, joining the mutiny.
``I will tiss Dimmy! and I tored up my hat 'tause a pin picked me,'' shouted Pokey, regardless of Jamie's efforts to restrain her.
Captain Dove looked rather taken aback at this outbreak in the ranks; but, being a dignified and calm personage, he quelled the rising rebellion with great tact and skill, by saying, briefly,
``We'll sing the last hymn; `Sweet, sweet good-by' you all know that, so do it nicely, and then we will go and have luncheon.''
Peace was instantly restored, and a burst of melody drowned the suppressed giggles of Rose and Mac, who found it impossible to keep sober during the latter part of this somewhat remarkable service. Fifteen minutes of repose rendered it a physical impossibility for the company to march out as quietly as they had marched in. I grieve to state that the entire troop raced home as hard as they could pelt, and were soon skirmishing briskly over their lunch, utterly oblivious of what Jamie (who had been much impressed by the sermon) called ``the captain's beautiful teck.''
It was astonishing how much they all found to do at Cosey Corner; and Mac, instead of lying in a hammock and being read to, as he had expected, was busiest of all. He was invited to survey and lay out Skeeterville, a town which the children were getting up in a huckleberry pasture; and he found much amusement in planning little roads, staking off house-lots, attending to the water-works, and consulting with the ``selectmen'' about the best sites for public buildings; for Mac was a boy still, in spite of his fifteen years and his love of books.
Then he went fishing with a certain jovial gentleman from the West; and though they seldom caught anything but colds, they had great fun and exercise chasing the phantom trout they were bound to have. Mac also developed a geological mania, and went tapping about at rocks and stones, discoursing wisely of ``strata, periods, and fossil remains''; while Rose picked up leaves and lichens, and gave him lessons in botany in return for his lectures on geology.
They led a very merry life; for the Atkinson girls kept up a sort of perpetual picnic; and did it so capitally, that one was never tired of it. So their visitors throve finely, and long before the month was out it was evident that Dr. Alec had prescribed the right medicine for his patients.
\gutchapter{Chapter 14---A Happy Birthday}
The twelfth of October was Rose's birthday, but no one seemed to remember that interesting fact, and she felt delicate about mentioning it, so fell asleep the night before wondering if she would have any presents. That question was settled early the next morning, for she was awakened by a soft tap on her face, and opening her eyes she beheld a little black and white figure sitting on her pillow, staring at her with a pair of round eyes very like blueberries, while one downy paw patted her nose to attract her notice. It was Kitty Comet, the prettiest of all the pussies, and Comet evidently had a mission to perform, for a pink bow adorned her neck, and a bit of paper was pinned to it bearing the words, ``For Miss Rose, from Frank.''
That pleased her extremely, and that was only the beginning of the fun, for surprises and presents kept popping out in the most delightful manner all through the day, the Atkinson girls being famous jokers and Rose a favourite. But the best gift of all came on the way to Mount Windy-Top, where it was decided to picnic in honour of the great occasion. Three jolly loads set off soon after breakfast, for everybody went, and everybody seemed bound to have an extra good time, especially Mother Atkinson, who wore a hat as broad-brimmed as an umbrella, and took the dinner-horn to keep her flock from straying away.
``I'm going to drive auntie and a lot of the babies, so you must ride the pony. And please stay behind us a good bit when we go to the station, for a parcel is coming, and you are not to see it till dinner-time. You won't mind, will you?'' said Mac, in a confidential aside during the wild flurry of the start.
``Not a bit,'' answered Rose. ``It hurts my feelings very much to be told to keep out of the way at any other time, but birthdays and Christmas it is part of the fun to be blind and stupid, and poked into corners. I'll be ready as soon as you are, Giglamps.''
``Stop under the big maple till I call then you can't possibly see anything,'' added Mac, as he mounted her on the pony his father had sent up for his use. ``Barkis'' was so gentle and so ``willin','' however, that Rose was ashamed to be afraid to ride him; so she had learned, that she might surprise Dr. Alec when she got home; meantime she had many a fine canter ``over the hills and far away'' with Mac, who preferred Mr. Atkinson's old Sorrel.
Away they went, and, coming to the red maple, Rose obediently paused; but could not help stealing a glance in the forbidden direction before the call came. Yes, there was a hamper going under the seat, and then she caught sight of a tall man whom Mac seemed to be hustling into the carriage in a great hurry. One look was enough, and with a cry of delight, Rose was off down the road as fast as Barkis could go.
``Now I'll astonish uncle,'' she thought. ``I'll dash up in grand style, and show him that I am not a coward, after all.''
Fired by this ambition, she startled Barkis by a sharp cut, and still more bewildered him by leaving him to his own guidance down the steep, stony road. The approach would have been a fine success if, just as Rose was about to pull up and salute, two or three distracted hens had not scuttled across the road with a great squawking, which caused Barkis to shy and stop so suddenly that his careless rider landed in an ignominious heap just under old Sorrel's astonished nose.
Rose was up again before Dr. Alec was out of the carryall, and threw two dusty arms about his neck crying with a breathless voice,
``O uncle, I'm so glad to see you! It is better than a cart-load of goodies, and so dear of you to come!''
``But aren't you hurt, child! That was a rough tumble, and I'm afraid you must be damaged somewhere,'' answered the Doctor, full of fond anxiety, as he surveyed his girl with pride.
``My feelings are hurt, but my bones are all safe. It's too bad! I was going to do it so nicely, and those stupid hens spoilt it all,'' said Rose, quite crestfallen, as well as much shaken.
``I couldn't believe my eyes when I asked `Where is Rose?' and Mac pointed to the little Amazon pelting down the hill at such a rate. You couldn't have done anything that would please me more, and I'm delighted to see how well you ride. Now, will you mount again, or shall we turn Mac out and take you in?'' asked Dr. Alec, as Aunt Jessie proposed a start, for the others were beckoning them to follow.
``Pride goeth before a fall better not try to show off again, ma'am,'' said Mac, who would have been more than mortal if he had refrained from teasing when so good a chance offered.
``Pride does go before a fall, but I wonder if a sprained ankle always comes after it?'' thought Rose, bravely concealing her pain, as she answered, with great dignity,
``I prefer to ride. Come on, and see who will catch up first.''
She was up and away as she spoke, doing her best to efface the memory of her downfall by sitting very erect, elbows down, head well up, and taking the motion of the pony as Barkis cantered along as easily as a rocking-chair.
``You ought to see her go over a fence and race when we ride together. She can scud, too, like a deer when we play `Follow the leader,' and skip stones and bat balls almost as well as I can,'' said Mac, in reply to his uncle's praise of his pupil.
``I'm afraid you will think her a sad tomboy, Alec; but really she seems so well and happy, I have not the heart to check her. She has broken out in the most unexpected way, and frisks like a colt; for she says she feels so full of spirits she must run and shout whether it is proper or not,'' added Mrs. Jessie, who had been a pretty hoyden years ago herself.
``Good good! that's the best news you could tell me,'' and Dr. Alec rubbed his hands heartily. ``Let the girl run and shout as much as she will it is a sure sign of health, and as natural to a happy child as frisking is to any young animal full of life. Tomboys make strong women usually, and I had far rather find Rose playing football with Mac than puttering over bead-work like that affected midget, Ariadne Blish.''
``But she cannot go on playing football very long, and we must not forget that she has a woman's work to do by and by,'' began Mrs. Jessie.
``Neither will Mac play football much longer, but he will be all the better fitted for business, because of the health it gives him. Polish is easily added, if the foundations are strong; but no amount of gilding will be of use if your timber is not sound. I'm sure I'm right, Jessie; and if I can do as well by my girl during the next six months as I have the last, my experiment will succeed.''
``It certainly will; for when I contrast that bright, blooming face with the pale, listless one that made my heart ache a while ago, I can believe in almost any miracle,'' said Mrs. Jessie, as Rose looked round to point out a lovely view, with cheeks like the ruddy apples in the orchard near by, eyes clear as the autumn sky overhead, and vigour in every line of her girlish figure.
A general scramble among the rocks was followed by a regular gypsy lunch, which the young folks had the rapture of helping to prepare. Mother Atkinson put on her apron, turned up her sleeves, and fell to work as gaily as if in her own kitchen, boiling the kettle slung on three sticks, over a fire of cones and fir boughs; while the girls spread the mossy table with a feast of country goodies, and the children tumbled about in everyone's way till the toot of the horn made them settle down like a flock of hungry birds.
As soon as the merry meal and a brief interval of repose were over, it was unanimously voted to have some charades. A smooth, green spot between two stately pines was chosen for the stage; shawls hung up, properties collected, audience and actors separated, and a word quickly chosen.
The first scene discovered Mac in a despondent attitude and shabby dress, evidently much troubled in mind. To him entered a remarkable creature with a brown paper bag over its head. A little pink nose peeped through one hole in the middle, white teeth through another, and above two eyes glared fiercely. Spires of grass stuck in each side of the mouth seemed meant to represent whiskers; the upper corners of the bag were twisted like ears, and no one could doubt for a moment that the black scarf pinned on behind was a tail.
This singular animal seemed in pantomime to be comforting his master and offering advice, which was finally acted upon, for Mac pulled off his boots, helped the little beast into them, and gave him a bag; then, kissing his paw, with a hopeful gesture, the creature retired, purring so successfully that there was a general cry of ``Cat, puss, boots!''
``Cat is the word,'' replied a voice, and the curtain fell.
The next scene was a puzzler, for in came another animal, on all-fours this time, with a new sort of tail and long ears. A gray shawl concealed its face, but an inquisitive sunbeam betrayed the glitter as of goggles under the fringe. On its back rode a small gentleman in Eastern costume, who appeared to find some difficulty in keeping his seat as his steed jogged along. Suddenly a spirit appeared, all in white, with long newspaper wings upon its back and golden locks about its face. Singularly enough, the beast beheld this apparition and backed instantly, but the rider evidently saw nothing and whipped up unmercifully, also unsuccessfully, for the spirit stood directly in the path, and the amiable beast would not budge a foot. A lively skirmish followed, which ended in the Eastern gentleman being upset into a sweet-fern bush, while the better bred animal abased itself before the shining one.
The children were all in the dark till Mother Atkinson said, in an inquiring tone,
``If that isn't Balaam and the ass, I'd like to know what it is. Rose makes a sweet angel, doesn't she?''
``Ass'' was evidently the word, and the angel retired, smiling with mundane satisfaction over the compliment that reached her ears.
The next was a pretty little scene from the immortal story of ``Babes in the Wood.'' Jamie and Pokey came trotting in, hand in hand, and, having been through the parts many times before, acted with great ease and much fluency, audibly directing each other from time to time as they went along. The berries were picked, the way lost, tears shed, baby consolation administered, and then the little pair lay down among the brakes and died with their eyes wide open and the toes of their four little boots turned up to the daisies in the most pathetic manner.
``Now the wobins tum. You be twite dead, Dimmy, and I'll peep in and see 'em,'' one defunct innocent was heard to say.
``I hope they'll be quick, for I'm lying on a stone, and ants are walking up my leg like fury,'' murmured the other.
Here the robins came flapping in with red scarves over their breasts and leaves in their mouths, which they carefully laid upon the babes wherever they would show best. A prickly blackberry leaf placed directly over Pokey's nose caused her to sneeze so violently that her little legs flew into the air; Jamie gave a startled ``Ow!'' and the pitying fowls fled giggling.
After some discussion it was decided that the syllable must be ``strew or strow'' and then they waited to see if it was a good guess.
This scene discovered Annette Snow in bed, evidently very ill; Miss Jenny was her anxious mamma, and her merry conversation amused the audience till Mac came in as a physician, and made great fun with his big watch, pompous manner, and absurd questions. He prescribed one pellet with an unpronounceable name, and left after demanding twenty dollars for his brief visit.
The pellet was administered, and such awful agonies immediately set in that the distracted mamma bade a sympathetic neighbour run for Mother Know-all. The neighbour ran, and in came a brisk little old lady in cap and specs, with a bundle of herbs under her arm, which she at once applied in all sorts of funny ways, explaining their virtues as she clapped a plantain poultice here, put a pounded catnip plaster there, or tied a couple of mullein leaves round the sufferer's throat. Instant relief ensued, the dying child sat up and demanded baked beans. The grateful parent offered fifty dollars; but Mother Know-all indignantly refused it and went smiling away, declaring that a neighbourly turn needed no reward, and a doctor's fee was all a humbug.
The audience were in fits of laughter over this scene, for Rose imitated Mrs. Atkinson capitally, and the herb cure was a good hit at the excellent lady's belief that ``yarbs'' would save mankind if properly applied. No one enjoyed it more than herself, and the saucy children prepared for the grand finale in high feather.
This closing scene was brief but striking, for two trains of cars whizzed in from opposite sides, met with a terrible collision in the middle of the stage, and a general smash-up completed the word catastrophe.
``Now let us act a proverb. I've got one all ready,'' said Rose, who was dying to distinguish herself in some way before Uncle Alec.
So everyone but Mac, the gay Westerner, and Rose, took their places on the rocky seats and discussed the late beautiful and varied charade, in which Pokey frankly pronounced her own scene the ``bestest of all.''
In five minutes the curtain was lifted; nothing appeared but a very large sheet of brown paper pinned to a tree, and on it was drawn a clock-face, the hands pointing to four. A small note below informed the public that 4 A.M. was the time. Hardly had the audience grasped this important fact when a long waterproof serpent was seen uncoiling itself from behind a stump. An inch-worm, perhaps, would be a better description, for it travelled in the same humpy way as that pleasing reptile. Suddenly a very wide-awake and active fowl advanced, pecking, chirping, and scratching vigorously. A tuft of green leaves waved upon his crest, a larger tuft of brakes made an umbrageous tail, and a shawl of many colours formed his flapping wings. A truly noble bird, whose legs had the genuine strut, whose eyes shone watchfully, and whose voice had a ring that evidently struck terror into the catterpillar's soul, if it was a catterpillar. He squirmed, he wriggled, he humped as fast as he could, trying to escape; but all in vain. The tufted bird espied him, gave one warbling sort of crow, pounced upon him, and flapped triumphantly away.
``That early bird got such a big worm he could hardly carry him off,'' laughed Aunt Jessie, as the children shouted over the joke suggested by Mac's nickname.
``That is one of uncle's favourite proverbs, so I got it up for his especial benefit,'' said Rose, coming up with the two-legged worm beside her.
``Very clever; what next?'' asked Dr. Alec as she sat down beside him.
``The Dove boys are going to give us an 'Incident in the Life of Napoleon,' as they call it; the children think it very splendid, and the little fellows do it rather nicely,'' answered Mac with condescension.
A tent appeared, and pacing to and fro before it was a little sentinel, who, in a brief soliloquy, informed the observers that the elements were in a great state of confusion, that he had marched some hundred miles or so that day, and that he was dying for want of sleep. Then he paused, leaned upon his gun, and seemed to doze; dropped slowly down, overpowered with slumber, and finally lay flat, with his gun beside him, a faithless little sentinel. Enter Napoleon, cocked hat, gray coat, high boots, folded arms, grim mouth, and a melodramatic stride. Freddy Dove always covered himself with glory in this part, and ``took the stage'' with a Napoleonic attitude that brought down the house; for the big-headed boy, with solemn, dark eyes and square brow, was ``the very moral of that rascal, Boneyparty,'' Mother Atkinson said.
Some great scheme was evidently brewing in his mighty mind a trip across the Alps, a bonfire at Moscow, or a little skirmish at Waterloo perhaps, for he marched in silent majesty till suddenly a gentle snore disturbed the imperial reverie. He saw the sleeping soldier and glared upon him, saying in an awful tone,
``Ha! asleep at his post! Death is the penalty he must die!''
Picking up the musket, he is about to execute summary justice, as emperors are in the habit of doing, when something in the face of the weary sentinel appears to touch him. And well it might, for a most engaging little warrior was Jack as he lay with his shako half off, his childish face trying to keep sober, and a great black moustache over his rosy mouth. It would have softened the heart of any Napoleon, and the Little Corporal proved himself a man by relenting, and saying, with a lofty gesture of forgiveness,
``Brave fellow, he is worn out; I will let him sleep, and mount guard in his place.''
Then, shouldering the gun, this noble being strode to and fro with a dignity which thrilled the younger spectators. The sentinel awakes, sees what has happened, and gives himself up for lost. But the Emperor restores his weapon, and, with that smile which won all hearts, says, pointing to a high rock whereon a crow happens to be sitting, ``Be brave, be vigilant, and remember that from yonder Pyramid generations are beholding you,'' and with these memorable words he vanishes, leaving the grateful soldier bolt upright, with his hand at his temple and deathless devotion stamped upon his youthful countenance.
The applause which followed this superb piece had hardly subsided, when a sudden splash and a shrill cry caused a general rush toward the waterfall that went gambolling down the rocks, singing sweetly as it ran. Pokey had tried to gambol also, and had tumbled into a shallow pool, whither Jamie had gallantly followed, in a vain attempt to fish her out, and both were paddling about half frightened, half pleased with the unexpected bath.
This mishap made it necessary to get the dripping infants home as soon as possible; so the wagons were loaded up, and away they went, as merry as if the mountain air had really been ``Oxygenated Sweets not Bitters,'' as Dr. Alec suggested when Mac said he felt as jolly as if he had been drinking champagne instead of the current wine that came with a great frosted cake wreathed with sugar roses in Aunt Plenty's hamper of goodies.
Rose took part in all the fun, and never betrayed by look or word the twinges of pain she suffered in her ankle. She excused herself from the games in the evening, however, and sat talking to Uncle Alec in a lively way, that both amazed and delighted him; for she confided to him that she played horse with the children, drilled with the light infantry, climbed trees, and did other dreadful things that would have caused the aunts to cry aloud if they knew of them.
``I don't care a pin what they say if you don't mind, uncle,'' she answered, when he pictured the dismay of the good ladies.
``Ah, it's all very well to defy them, but you are getting so rampant, I'm afraid you will defy me next, and then where are we?''
``No, I won't! I shouldn't dare; because you are my guardian, and can put me in a strait-jacket if you like;'' and Rose laughed in his face, even while she nestled closer with a confiding gesture pleasant to see.
``Upon my word, Rosy, I begin to feel like the man who bought an elephant, and then didn't know what to do with him. I thought I had got a pet and plaything for years to come; but here you are growing up like a bean-stalk, and I shall find I've got a strong-minded little woman on my hands before I can turn round. There's predicament for a man and an uncle!''
Dr. Alec's comic distress was mercifully relieved for the time being by a dance of goblins on the lawn, where the children, with pumpkin lanterns on their heads, frisked about like will-o'-the-wisps, as a parting surprise.
When Rose went to bed, she found that Uncle Alec had not forgotten her; for on the table stood a delicate little easel, holding two miniatures set in velvet. She knew them both, and stood looking at them till her eyes brimmed over with tears that were both sweet and sad; for they were the faces of her father and mother, beautifully copied from portraits fast fading away.
Presently, she knelt down, and, putting her arms round the little shrine, kissed one after the other, saying with an earnest voice, ``I'll truly try to make them glad to see me by and by.''
And that was Rose's little prayer on the night of her fourteenth birthday.
Two days later the Campbells went home, a larger party than when they came; for Dr. Alec was escort and Kitty Comet was borne in state in a basket, with a bottle of milk, some tiny sandwiches, and a doll's dish to drink out of, as well as a bit of carpet to lie on in her palace car, out of which she kept popping her head in the most fascinating manner.
There was a great kissing and cuddling, waving of handkerchiefs, and last good-byes, as they went; and when they had started, Mother Atkinson came running after them, to tuck in some little pies, hot from the oven, ``for the dears, who might get tired of bread and butter during that long day's travel.''
Another start, and another halt; for the Snow children came shrieking up to demand the three kittens that Pokey was cooly carrying off in a travelling bag. The unhappy kits were rescued, half smothered, and restored to their lawful owners, amid dire lamentation from the little kidnapper, who declared that she only ``tooked um 'cause they'd want to go wid their sister Tomit.''
Start number three and stoppage number three, as Frank hailed them with the luncheon basket, which had been forgotten, after everyone had protested that it was safely in.
All went well after that, and the long journey was pleasantly beguiled by Pokey and Pussy, who played together so prettily that they were considered public benefactors.
``Rose doesn't want to go home, for she knows the aunts won't let her rampage as she did up at Cosey Corner,'' said Mac, as they approached the old house.
``I can't rampage if I want to for a time, at least; and I'll tell you why. I sprained my ankle when I tumbled off of Barkis, and it gets worse and worse; though I've done all I know to cure it and hide it, so it shouldn't trouble anyone,'' whispered Rose, knitting her brows with pain, as she prepared to descend, wishing her uncle would take her instead of her bundles.
How he did it, she never knew; but Mac had her up the steps and on the parlour sofa before she could put her foot to the ground.
``There you are right side up with care; and mind, now, if your ankle bothers you, and you are laid up with it, I am to be your footman. It's only fair, you know; for I don't forget how good you have been to me.'' And Mac went to call Phebe, so full of gratitude and good-will that his very goggles shone.
\gutchapter{Chapter 15---Ear-Rings}
Rose's sprain proved to be a serious one, owing to neglect, and Dr. Alec ordered her to lie on the sofa for a fortnight at least; whereat she groaned dismally, but dared not openly complain, lest the boys turn upon her with some of the wise little sermons on patience which she had delivered for their benefit.
It was Mac's turn now, and honourably did he repay his debt; for, as school was still forbidden, he had plenty of leisure, and devoted most of it to Rose. He took many steps for her, and even allowed her to teach him to knit, after assuring himself that many a brave Scotchman knew how to ``click the pricks.'' She was obliged to take a solemn vow of secrecy, however, before he would consent; for, though he did not mind being called ``Giglamps,'' ``Granny'' was more than his boyish soul could bear, and at the approach of any of the Clan his knitting vanished as if by magic, which frequent ``chucking'' out of sight did not improve the stripe he was doing for Rose's new afghan.
She was busy with this pretty work one bright October afternoon, all nicely established on her sofa in the upper hall, while Jamie and Pokey (lent for her amusement) were keeping house in a corner, with Comet and Rose's old doll for their ``childerns.''
Presently, Phebe appeared with a card. Rose read it, made a grimace, then laughed and said,
``I'll see Miss Blish,'' and immediately put on her company face, pulled out her locket, and settled her curls.
``You dear thing, how do you do? I've been trying to call every day since you got back, but I have so many engagements, I really couldn't manage it till to-day. So glad you are alone, for mamma said I could sit awhile, and I brought my lace-work to show you, for it's perfectly lovely.'' cried Miss Blish, greeting Rose with a kiss, which was not very warmly returned, though Rose politely thanked her for coming, and bid Phebe roll up the easy chair.
``How nice to have a maid!'' said Ariadne, as she settled herself with much commotion. ``Still, dear, you must be very lonely, and feel the need of a bosom friend.''
``I have my cousins,'' began Rose, with dignity, for her visitor's patronising manner ruffled her temper.
``Gracious, child! you don't make friends of those great boys, do you? Mamma says she really doesn't think it's proper for you to be with them so much.''
``They are like brothers, and my aunts do think it's proper,'' replied Rose, rather sharply, for it struck her that this was none of Miss Blish's business.
``I was merely going to say I should be glad to have you for my bosom friend, for Hatty Mason and I have had an awful quarrel, and don't speak. She is too mean to live, so I gave her up. Just think, she never paid back one of the caramels I've given her, and never invited me to her party. I could have forgiven the caramels, but to be left out in that rude way was more than I could bear, and I told her never to look at me again as long as she lived.''
``You are very kind, but I don't think I want a bosom friend, thank you,'' said Rose, as Ariadne stopped to bridle and shake her flaxen head over the delinquent Hatty Mason.
Now, in her heart Miss Blish thought Rose ``a stuck-up puss,'' but the other girls wanted to know her and couldn't, the old house was a charming place to visit, the lads were considered fine fellows, and the Campbells ``are one of our first families,'' mamma said. So Ariadne concealed her vexation at Rose's coolness, and changed the subject as fast as possible.
``Studying French, I see; who is your teacher?'' she asked, flitting over the leaves of ``Paul and Virginia,'' that lay on the table.
``I don't study it, for I read French as well as English, and uncle and I often speak it for hours. He talks like a native, and says I have a remarkably good accent.''
Rose really could not help this small display of superiority, for French was one of her strong points, and she was vain of it, though she usually managed to hide this weakness. She felt that Ariadne would be the better for a little crushing, and could not resist the temptation to patronise in her turn.
``Oh, indeed!'' said Miss Blish, rather blankly, for French was not her strong point by any means.
``I am to go abroad with uncle in a year or two, and he knows how important it is to understand the languages. Half the girls who leave school can't speak decent French, and when they go abroad they are so mortified. I shall be very glad to help you, if you like, for, of course, you have no one to talk with at home.''
Now Ariadne, though she looked like a wax doll, had feelings within her instead of sawdust, and these feelings were hurt by Rose's lofty tone. She thought her more ``stuck up'' than ever, but did not know how to bring her down, yet longed to do it, for she felt as if she had received a box on the ear, and involuntarily put her hand up to it. The touch of an ear-ring consoled her, and suggested a way of returning tit for tat in a telling manner.
``Thank you, dear; I don't need any help, for our teacher is from Paris, and of course he speaks better French than your uncle.'' Then she added, with a gesture of her head that set the little bells on her ears to tingling: ``How do you like my new ear-rings? Papa gave them to me last week, and everyone says they are lovely.''
Rose came down from her high horse with a rapidity that was comical, for Ariadne had the upper hand now. Rose adored pretty things, longed to wear them, and the desire of her girlish soul was to have her ears bored, only Dr. Alec thought it foolish, so she never had done it. She would gladly have given all the French she could jabber for a pair of golden bells with pearl-tipped tongues, like those Ariadne wore; and, clasping her hands, she answered, in a tone that went to the hearer's heart,
``They are too sweet for anything! If uncle would only let me wear some, I should be perfectly happy.''
``I wouldn't mind what he says. Papa laughed at me at first, but he likes them now, and says I shall have diamond solitaires when I am eighteen,'' said Ariadne, quite satisfied with her shot.
``I've got a pair now that were mamma's, and a beautiful little pair of pearl and turquoise ones, that I am dying to wear,'' sighed Rose.
``Then do it. I'll pierce your ears, and you must wear a bit of silk in them till they are well; your curls will hide them nicely; then, some day, slip in your smallest ear-rings, and see if your uncle don't like them.''
``I asked him if it wouldn't do my eyes good once when they were red, and he only laughed. People do cure weak eyes that way, don't they?''
``Yes, indeed, and yours are sort of red. Let me see. Yes, I really think you ought to do it before they get worse,'' said Ariadne, peering into the large clear eye offered for inspection.
``Does it hurt much?'' asked Rose, wavering.
``Oh dear, no; just a prick and a pull, and it's all over. I've done lots of ears, and know just how. Come, push up your hair and get a big needle.''
``I don't quite like to do it without asking uncle's leave,'' faltered Rose, when all was ready for the operation.
``Did he ever forbid it?'' demanded Ariadne, hovering over her prey like a vampire.
``No, never!''
``Then do it, unless you are afraid,'' cried Miss Blish, bent on accomplishing the deed.
That last word settled the matter, and, closing her eyes, Rose said ``Punch!'' in the tone of one giving the fatal order ``Fire!''
Ariadne punched, and the victim bore it in heroic silence, though she turned pale and her eyes were full of tears of anguish.
``There! Now pull the bits of silk often, and cold-cream your ears every night, and you'll soon be ready for the rings,'' said Ariadne, well pleased with her job, for the girl who spoke French with ``a fine accent'' lay flat upon the sofa, looking as exhausted as if she had had both ears cut off.
``It does hurt dreadfully, and I know uncle won't like it,'' sighed Rose, as remorse began to gnaw. ``Promise not to tell, or I shall be teased to death,'' she added, anxiously, entirely forgetting the two little pitchers gifted with eyes as well as ears, who had been watching the whole performance from afar.
``Never. Mercy me, what's that?'' and Ariadne started as a sudden sound of steps and voices came up from below.
``It's the boys! Hide the needle. Do my ears show? Don't breathe a word!'' whispered Rose, scrambling about to conceal all traces of their iniquity from the sharp eyes of the Clan.
Up they came, all in good order, laden with the proceeds of a nutting expedition, for they always reported to Rose and paid tribute to their queen in the handsomest manner.
``How many, and how big! We'll have a grand roasting frolic after tea, won't we?'' said Rose, plunging both hands into a bag of glossy brown nuts, while the Clan ``stood at ease'' and nodded to Ariadne.
``That lot was picked especially for you, Rosy. I got every one myself, and they are extra whackers,'' said Mac, presenting a bushel or so.
``You should have seen Giglamps when he was after them. He pitched out of the tree, and would have broken his blessed old neck if Arch had not caught him,'' observed Steve, as he lounged gracefully in the window seat.
``You needn't talk, Dandy, when you didn't know a chestnut from a beech, and kept on thrashing till I told you of it,'' retorted Mac, festooning himself over the back of the sofa, being a privileged boy.
``I don't make mistakes when I thrash you, old Worm, so you'd better mind what you are about,'' answered Steve, without a ray of proper respect for his elder brother.
``It is getting dark, and I must go, or mamma will be alarmed,'' said Ariadne, rising in sudden haste, though she hoped to be asked to remain to the nut-party.
No one invited her; and all the while she was putting on her things and chatting to Rose the boys were telegraphing to one another the sad fact that someone ought to escort the young lady home. Not a boy felt heroic enough to cast himself into the breach, however; even polite Archie shirked the duty, saying to Charlie, as they quietly slipped into an adjoining room,
``I'm not going to do all the gallivanting. Let Steve take that chit home and show his manners.''
``I'll be hanged if I do!'' answered Prince, who disliked Miss Blish because she tried to be coquettish with him.
``Then I will,'' and, to the dismay of both recreant lads, Dr. Alec walked out of the room to offer his services to the ``chit.''
He was too late, however, for Mac, obeying a look from Rose, had already made a victim of himself, and trudged meekly away, wishing the gentle Ariadne at the bottom of the Red Sea.
``Then I will take this lady down to tea, as the other one has found a gentleman to go home with her. I see the lamps are lighted below, and I smell a smell which tells me that auntie has something extra nice for us to-night.''
As he spoke, Dr. Alec was preparing to carry Rose downstairs as usual; but Archie and Prince rushed forward, begging with penitent eagerness for the honour of carrying her in an arm-chair. Rose consented, fearing that her uncle's keen eye would discover the fatal bits of silk; so the boys crossed hands, and, taking a good grip of each curly pate, she was borne down in state, while the others followed by way of the banisters.
Tea was ordered earlier than usual, so that Jamie and his dolly could have a taste, at least, of the holiday fun, for they were to stay till seven, and be allowed twelve roasted chestnuts apiece, which they were under bonds not to eat till next day.
Tea was despatched rapidly, therefore, and the party gathered round the wide hearth in the dining-room, where the nuts were soon dancing gaily on hot shovels or bouncing out among the company, thereby causing delightful panics among the little ones.
``Come, Rosy, tell us a story while we work, for you can't help much, and must amuse us as your share,'' proposed Mac, who sat in the shade pricking nuts, and who knew by experience what a capital little Scheherazade his cousin was.
``Yes, we poor monkeys can't burn our paws for nothing, so tell away, Pussy,'' added Charlie, as he threw several hot nuts into her lap and shook his fingers afterwards.
``Well, I happen to have a little story with a moral to it in my mind, and I will tell it, though it is intended for younger children than you,'' answered Rose, who was rather fond of telling instructive tales.
``Fire away,'' said Geordie, and she obeyed, little thinking what a disastrous story it would prove to herself.
``Well, once upon a time, a little girl went to see a young lady who was very fond of her. Now, the young lady happened to be lame, and had to have her foot bandaged up every day; so she kept a basketful of bandages, all nicely rolled and ready. The little girl liked to play with this basket, and one day, when she thought no one saw her, she took one of the rolls without asking leave, and put it in her pocket.''
Here Pokey, who had been peering lovingly down at the five warm nuts that lay at the bottom of her tiny pocket, suddenly looked up and said, ``Oh!'' in a startled tone, as if the moral tale had become intensely interesting all at once.
Rose heard and saw the innocent betrayal of the small sinner, and went on in a most impressive manner, while the boys nudged one another and winked as they caught the joke.
``But an eye did see this naughty little girl, and whose eye do you think it was?''
``Eye of Dod,'' murmured conscience-stricken Pokey, spreading two chubby little hands before the round face, which they were not half big enough to hide.
Rose was rather taken aback by this reply, but, feeling that she was producing a good effect, she added seriously,
``Yes, God saw her, and so did the young lady, but she did not say anything; she waited to see what the little girl would do about it. She had been very happy before she took the bandage, but when it was in her pocket she seemed troubled, and pretty soon stopped playing, and sat down in a corner looking very sober. She thought a few minutes, and then went and put back the roll very softly, and her face cleared up, and she was a happy child again. The young lady was glad to see that, and wondered what made the little girl put it back.''
``Tonscience p'icked her,'' murmured a contrite voice from behind the small hands pressed tightly over Pokey's red face.
``And why did she take it, do you suppose?'' asked Rose, in a school-marmish tone, feeling that all the listeners were interested in her tale and its unexpected application.
``It was so nice and wound, and she wanted it deffly,'' answered the little voice.
``Well, I'm glad she had such a good conscience. The moral is that people who steal don't enjoy what they take, and are not happy till they put it back. What makes that little girl hide her face?'' asked Rose, as she concluded.
``Me's so 'shamed of Pokey,'' sobbed the small culprit, quite overcome by remorse and confusion at this awful disclosure.
``Come, Rose, it's too bad to tell her little tricks before everyone, and preach at her in that way; you wouldn't like it yourself,'' began Dr. Alec, taking the weeper on his knee and administering consolation in the shape of kisses and nuts.
Before Rose could express her regret, Jamie, who had been reddening and ruffling like a little turkey-cock for several minutes, burst out indignantly, bent on avenging the wound given to his beloved dolly.
``I know something bad that you did, and I'm going to tell right out. You thought we didn't see you, but we did, and you said uncle wouldn't like it, and the boys would tease, and you made Ariadne promise not to tell, and she punched holes in your ears to put ear-rings in. So now! and that's much badder than to take an old piece of rag; and I hate you for making my Pokey cry.''
Jamie's somewhat incoherent explosion produced such an effect that Pokey's small sin was instantly forgotten, and Rose felt that her hour had come.
``What! what! what!'' cried the boys in a chorus, dropping their shovels and knives to gather round Rose, for a guilty clutching at her ears betrayed her, and with a feeble cry of ``Ariadne made me!'' she hid her head among the pillows like an absurd little ostrich.
``Now she'll go prancing round with bird cages and baskets and carts and pigs, for all I know, in her ears, as the other girls do, and won't she look like a goose?'' asked one tormentor, tweaking a curl that strayed out from the cushions.
``I didn't think she'd be so silly,'' said Mac, in a tone of disappointment that told Rose she had sunk in the esteem of her wise cousin.
``That Blish girl is a nuisance, and ought not to be allowed to come here with her nonsensical notions,'' said the Prince, feeling a strong desire to shake that young person as an angry dog might shake a mischievous kitten.
``How do you like it, uncle?'' asked Archie, who, being the head of a family himself, believed in preserving discipline at all costs.
``I am very much surprised; but I see she is a girl, after all, and must have her vanities like all the rest of them,'' answered Dr. Alec, with a sigh, as if he had expected to find Rose a sort of angel, above all earthly temptations.
``What shall you do about it, sir?'' inquired Geordie, wondering what punishment would be inflicted on a feminine culprit.
``As she is fond of ornaments, perhaps we had better give her a nose-ring also. I have one somewhere that a Fiji belle once wore; I'll look it up,'' and, leaving Pokey to Jamie's care, Dr. Alec rose as if to carry out his suggestion in earnest.
``Good! good! We'll do it right away! Here's a gimlet, so you hold her, boys, while I get her dear little nose all ready,'' cried Charlie, whisking away the pillow as the other boys danced about the sofa in true Fiji style.
It was a dreadful moment, for Rose could not run away she could only grasp her precious nose with one hand and extend the other, crying distractedly,
``O uncle, save me, save me!''
Of course he saved her; and when she was securely barricaded by his strong arm, she confessed her folly in such humiliation of spirit, that the lads, after a good laugh at her, decided to forgive her and lay all the blame on the tempter, Ariadne. Even Dr. Alec relented so far as to propose two gold rings for the ears instead of one copper one for the nose; a proceeding which proved that if Rose had all the weakness of her sex for jewellery, he had all the inconsistency of his in giving a pretty penitent exactly what she wanted, spite of his better judgment.
\gutchapter{Chapter 16---Bread and Button-Holes}
``What in the world is my girl thinking about all alone here, with such a solemn face?'' asked Dr. Alec, coming into the study, one November day, to find Rose sitting there with folded hands and a very thoughtful aspect.
``Uncle, I want to have some serious conversation with you, if you have time,'' she said, coming out of a brown study, as if she had not heard his question.
``I'm entirely at your service, and most happy to listen,'' he answered, in his politest manner, for when Rose put on her womanly little airs he always treated her with a playful sort of respect that pleased her very much.
Now, as he sat down beside her, she said, very soberly,
``I've been trying to decide what trade I would learn, and I want you to advise me.''
``Trade, my dear?'' and Dr. Alec looked so astonished that she hastened to explain.
``I forgot that you didn't hear the talk about it up at Cosey Corner. You see we used to sit under the pines and sew, and talk a great deal all the ladies, I mean and I liked it very much. Mother Atkinson thought that everyone should have a trade, or something to make a living out of, for rich people may grow poor, you know, and poor people have to work. Her girls were very clever, and could do ever so many things, and Aunt Jessie thought the old lady was right; so when I saw how happy and independent those young ladies were, I wanted to have a trade, and then it wouldn't matter about money, though I like to have it well enough.''
Dr. Alec listened to this explanation with a curious mixture of surprise, pleasure, and amusement in his face, and looked at his little niece as if she had suddenly changed into a young woman. She had grown a good deal in the last six months, and an amount of thinking had gone on in that young head which would have astonished him greatly could he have known it all, for Rose was one of the children who observe and meditate much, and now and then nonplus their friends by a wise or curious remark.
``I quite agree with the ladies, and shall be glad to help you decide on something if I can,'' said the Doctor seriously. ``What do you incline to? A natural taste or talent is a great help in choosing, you know.''
``I haven't any talent, or any especial taste that I can see, and that is why I can't decide, uncle. So, I think it would be a good plan to pick out some very useful business and learn it, because I don't do it for pleasure, you see, but as a part of my education, and to be ready in case I'm ever poor,'' answered Rose, looking as if she rather longed for a little poverty so that her useful gift might be exercised.
``Well, now, there is one very excellent, necessary, and womanly accomplishment that no girl should be without, for it is a help to rich and poor, and the comfort of families depends upon it. This fine talent is neglected nowadays, and considered old-fashioned, which is a sad mistake, and one that I don't mean to make in bringing up my girl. It should be a part of every girl's education, and I know of a most accomplished lady who will teach you in the best and pleasantest manner.''
``Oh, what is it?'' cried Rose eagerly, charmed to be met in this helpful and cordial way.
``Housekeeping!'' answered Dr. Alec.
``Is that an accomplishment?'' asked Rose, while her face fell, for she had indulged in all sorts of vague, delightful dreams.
``Yes; it is one of the most beautiful as well as useful of all the arts a woman can learn. Not so romantic, perhaps, as singing, painting, writing, or teaching, even; but one that makes many happy and comfortable, and home the sweetest place in the world. Yes, you may open your big eyes; but it is a fact that I had rather see you a good housekeeper than the greatest belle in the city. It need not interfere with any talent you may possess, but it is a necessary part of your training, and I hope that you will set about it at once, now that you are well and strong.''
``Who is the lady?'' asked Rose, rather impressed by her uncle's earnest speech.
``Aunt Plenty.''
``Is she accomplished?'' began Rose in a wondering tone, for this great-aunt of hers had seemed the least cultivated of them all.
``In the good old-fashioned way she is very accomplished, and has made this house a happy home to us all, ever since we can remember. She is not elegant, but genuinely good, and so beloved and respected that there will be universal mourning for her when her place is empty. No one can fill it, for the solid, homely virtues of the dear soul have gone out of fashion, as I say, and nothing new can be half so satisfactory, to me at least.''
``I should like to have people feel so about me. Can she teach me to do what she does, and to grow as good?'' asked Rose, with a little prick of remorse for even thinking that Aunt Plenty was a commonplace old lady.
``Yes, if you don't despise such simple lessons as she can give. I know it would fill her dear old heart with pride and pleasure to feel that anyone cared to learn of her, for she fancies her day gone by. Let her teach you how to be what she has been a skilful, frugal, cheerful housewife; the maker and the keeper of a happy home, and by and by you will see what a valuable lesson it is.''
``I will, uncle. But how shall I begin?''
``I'll speak to her about it, and she will make it all right with Dolly, for cooking is one of the main things, you know.''
``So it is! I don't mind that a bit, for I like to mess, and used to try at home; but I had no one to tell me, so I never did much but spoil my aprons. Pies are great fun, only Dolly is so cross, I don't believe she will ever let me do a thing in the kitchen.''
``Then we'll cook in the parlour. I fancy Aunt Plenty will manage her, so don't be troubled. Only mind this, I'd rather you learned how to make good bread than the best pies ever baked. When you bring me a handsome, wholesome loaf, entirely made by yourself, I shall be more pleased than if you offered me a pair of slippers embroidered in the very latest style. I don't wish to bribe you, but I'll give you my heartiest kiss, and promise to eat every crumb of the loaf myself.''
``It's a bargain! it's a bargain! Come and tell aunty all about it, for I'm in a hurry to begin,'' cried Rose, dancing before him toward the parlor, where Miss Plenty sat alone knitting contentedly, yet ready to run at the first call for help of any sort, from any quarter.
No need to tell how surprised and gratified she was at the invitation she received to teach the child the domestic arts which were her only accomplishments, nor to relate how energetically she set about her pleasant task. Dolly dared not grumble, for Miss Plenty was the one person whom she obeyed, and Phebe openly rejoiced, for these new lessons brought Rose nearer to her, and glorified the kitchen in the good girl's eyes.
To tell the truth, the elder aunts had sometimes felt that they did not have quite their share of the little niece who had won their hearts long ago, and was the sunshine of the house. They talked it over together sometimes, but always ended by saying that as Alec had all the responsibility, he should have the larger share of the dear girl's love and time, and they would be contented with such crumbs of comfort as they could get.
Dr. Alec had found out this little secret, and, after reproaching himself for being blind and selfish, was trying to devise some way of mending matters without troubling anyone, when Rose's new whim suggested an excellent method of weaning her a little from himself. He did not know how fond he was of her till he gave her up to the new teacher, and often could not resist peeping in at the door to see how she got on, or stealing sly looks through the slide when she was deep in dough, or listening intently to some impressive lecture from Aunt Plenty. They caught him at it now and then, and ordered him off the premises at the point of the rolling-pin; or, if unusually successful, and, therefore, in a milder mood, they lured him away with bribes of ginger-bread, a stray pickle, or a tart that was not quite symmetrical enough to suit their critical eyes.
Of course he made a point of partaking copiously of all the delectable messes that now appeared at table, for both the cooks were on their mettle, and he fared sumptuously every day. But an especial relish was given to any dish when, in reply to his honest praise of it, Rose coloured up with innocent pride, and said modestly,
``I made that, uncle, and I'm glad you like it.''
It was some time before the perfect loaf appeared, for bread-making is an art not easily learned, and Aunt Plenty was very thorough in her teaching; so Rose studied yeast first, and through various stages of cake and biscuit came at last to the crowning glory of the ``handsome, wholesome loaf.'' It appeared at tea-time, on a silver salver, proudly borne in by Phebe, who could not refrain from whispering, with a beaming face, as she set it down before Dr. Alec,
``Ain't it just lovely, sir?''
``It is a regularly splendid loaf! Did my girl make it all herself?'' he asked, surveying the shapely, sweet-smelling object with real interest and pleasure.
``Every particle herself, and never asked a bit of help or advice from anyone,'' answered Aunt Plenty, folding her hands with an air of unmitigated satisfaction, for her pupil certainly did her great credit.
``I've had so many failures and troubles that I really thought I never should be able to do it alone. Dolly let one splendid batch burn up because I forgot it. She was there and smelt it, but never did a thing, for she said, when I undertook to bake bread I must give my whole mind to it. Wasn't it hard? She might have called me at least,'' said Rose, recollecting, with a sigh, the anguish of that moment.
``She meant you should learn by experience, as Rosamond did in that little affair of the purple jar, you remember.''
``I always thought it very unfair in her mother not to warn the poor thing a little bit; and she was regularly mean when Rosamond asked for a bowl to put the purple stuff in, and she said, in such a provoking way, `I did not agree to lend you a bowl, but I will, my dear.' Ugh! I always want to shake that hateful woman, though she was a moral mamma.''
``Never mind her now, but tell me all about my loaf,'' said Dr. Alec, much amused at Rose's burst of indignation.
``There's nothing to tell, uncle, except that I did my best, gave my mind to it, and sat watching over it all the while it was in the oven till I was quite baked myself. Everything went right this time, and it came out a nice, round, crusty loaf, as you see. Now taste it, and tell me if it is good as well as handsome.''
``Must I cut it? Can't I put it under a glass cover and keep it in the parlor as they do wax flowers and fine works of that sort?''
``What an idea, uncle! It would mould and be spoilt. Besides, people would laugh at us, and make fun of my old-fashioned accomplishment. You promised to eat it, and you must; not all at once, but as soon as you can, so I can make you some more.''
Dr. Alec solemnly cut off his favourite crusty slice, and solemnly ate it; then wiped his lips, and brushing back Rose's hair, solemnly kissed her on the forehead, saying, heartily,
``My dear, it is perfect bread, and you are an honour to your teacher. When we have our model school I shall offer a prize for the best bread, and you will get it.''
``I've got it already, and I'm quite satisfied,'' said Rose, slipping into her seat, and trying to hide her right hand which had a burn on it.
But Dr. Alec saw it, guessed how it came there, and after tea insisted on easing the pain which she would hardly confess.
``Aunt Clara says I am spoiling my hands, but I don't care, for I've had such good times with Aunt Plenty, and I think she has enjoyed it as much as I have. Only one thing troubles me, uncle, and I want to ask you about it,'' said Rose, as they paced up and down the hall in the twilight, the bandaged hand very carefully laid on Dr. Alec's arm.
``More little confidences? I like them immensely, so tell away, my dear.''
``Well, you see I feel as if Aunt Peace would like to do something for me, and I've found out what it can be. You know she can't go about like Aunty Plen, and we are so busy nowadays that she is rather lonely, I'm afraid. So I want to take lessons in sewing of her. She works so beautifully, and it is a useful thing, you know, and I ought to be a good needlewoman as well as housekeeper, oughtn't I?''
``Bless your kind little heart, that is what I was thinking of the other day when Aunt Peace said she saw you very seldom now, you were so busy I wanted to speak of it, but fancied you had as much on your hands as you could manage. It would delight the dear woman to teach you all her delicate handicraft, especially button-holes, for I believe that is where young ladies fail; at least, I've heard them say so. So, do you devote your mind to button-holes; make 'em all over my clothes if you want something to practice on. I'll wear any quantity.''
Rose laughed at this reckless offer, but promised to attend to that important branch, though she confessed that darning was her weak point. Whereupon Uncle Alec engaged to supply her with socks in all stages of dilapidation, and to have a new set at once, so that she could run the heels for him as a pleasant beginning.
Then they went up to make their request in due form, to the great delight of gentle Aunt Peace, who got quite excited with the fun that went on while they would yarn, looked up darning needles, and fitted out a nice little mending basket for her pupil.
Very busy and very happy were Rose's days now, for in the morning she went about the house with Aunt Plenty attending to linen-closets and store-rooms, pickling and preserving, exploring garret and cellar to see that all was right, and learning, in the good old-fashioned manner, to look well after the ways of the household.
In the afternoon, after her walk or drive, she sat with Aunt Peace plying her needle, while Aunt Plenty, whose eyes were failing, knitted and chatted briskly, telling many a pleasant story of old times, till the three were moved to laugh and cry together, for the busy needles were embroidering all sorts of bright patterns on the lives of the workers, though they seemed to be only stitching cotton and darning hose.
It was a pretty sight to see the rosy-faced little maid sitting between the two old ladies, listening dutifully to their instructions, and cheering the lessons with her lively chatter and blithe laugh. If the kitchen had proved attractive to Dr. Alec when Rose was there at work, the sewing-room was quite irresistible, and he made himself so agreeable that no one had the heart to drive him away, especially when he read aloud or spun yarns.
``There! I've made you a new set of warm night-gowns with four button-holes in each. See if they are not neatly done,'' said Rose, one day, some weeks after the new lessons began.
``Even to a thread, and nice little bars across the end so I can't tear them when I twitch the buttons out. Most superior work, ma'am, and I'm deeply grateful; so much so, that I'll sew on these buttons myself, and save those tired fingers from another prick.''
``You sew them on?'' cried Rose, with her eyes wide open in amazement.
``Wait a bit till I get my sewing tackle, and then you shall see what I can do.''
``Can he, really?'' asked Rose of Aunt Peace, as Uncle Alec marched off with a comical air of importance.
``Oh, yes, I taught him years ago, before he went to sea; and I suppose he has had to do things for himself, more or less, ever since; so he has kept his hand in.''
He evidently had, for he was soon back with a funny little work-bag, out of which he produced a thimble without a top; and, having threaded his needle, he proceeded to sew on the buttons so handily that Rose was much impressed and amused.
``I wonder if there is anything in the world that you cannot do,'' she said, in a tone of respectful admiration.
``There are one or two things that I am not up to yet,'' he answered, with a laugh in the corner of his eye, as he waxed his thread with a flourish.
``I should like to know what?''
``Bread and button-holes, ma'am.''
\gutchapter{Chapter 17---Good Bargains}
It was a rainy Sunday afternoon, and four boys were trying to spend it quietly in the ``liberry,'' as Jamie called the room devoted to books and boys, at Aunt Jessie's. Will and Geordie were sprawling on the sofa, deep in the adventures of the scapegraces and ragamuffins whose histories are now the fashion. Archie lounged in the easy chair, surrounded by newspapers; Charlie stood upon the rug, in an Englishman's favourite attitude, and, I regret to say, both were smoking cigars.
``It is my opinion that this day will never come to an end,'' said Prince, with a yawn that nearly rent him asunder.
``Read and improve your mind, my son,'' answered Archie, peering solemnly over the paper behind which he had been dozing.
``Don't you preach, parson, but put on your boots and come out for a tramp, instead of mulling over the fire like a granny.''
``No, thank you, tramps in an easterly storm don't strike me as amusing.'' There Archie stopped and held up his hand, for a pleasant voice was heard saying outside,
``Are the boys in the library, auntie?''
``Yes, dear, and longing for sunshine; so run in and make it for them,'' answered Mrs. Jessie.
``It's Rose,'' and Archie threw his cigar into the fire.
``What's that for?'' asked Charlie.
``Gentlemen don't smoke before ladies.''
``True; but I'm not going to waste my weed,'' and Prince poked his into the empty inkstand that served them for an ash tray.
A gentle tap at the door was answered by a chorus of ``Come in,'' and Rose appeared, looking blooming and breezy with the chilly air.
``If I disturb you, say so, and I'll go away,'' she began, pausing on the threshold with modest hesitation, for something in the elder boys' faces excited her curiosity.
``You never disturb us, cousin,'' said the smokers, while the readers tore themselves from the heroes of the bar-room and gutter long enough to nod affably to their guest.
As Rose bent to warm her hands, one end of Archie's cigar stuck out of the ashes, smoking furiously and smelling strongly.
``Oh, you bad boys, how could you do it, to-day of all days?'' she said reproachfully.
``Where's the harm?'' asked Archie.
``You know as well as I do; your mother doesn't like it, and it's a bad habit, for it wastes money and does you no good.''
``Fiddlesticks! every man smokes, even Uncle Alec, whom you think so perfect,'' began Charlie, in his teasing way.
``No, he doesn't! He has given it up, and I know why,'' cried Rose eagerly.
``Now I think of it, I haven't seen the old meerschaum since he came home. Did he stop it on our account?'' asked Archie.
``Yes,'' and Rose told the little scene on the seashore in the camping-out time.
Archie seemed much impressed, and said manfully, ``He won't have done that in vain so far as I'm concerned. I don't care a pin about smoking, so can give it up as easy as not, and I promise you I will. I only do it now and then for fun.''
``You too?'' and Rose looked up at the bonny Prince, who never looked less bonny than at that moment, for he had resumed his cigar just to torment her.
Now Charlie cared as little as Archie about smoking, but it would not do to yield too soon: so he shook his head, gave a great puff, and said loftily,
``You women are always asking us to give up harmless little things just because you don't approve of them. How would you like it if we did the same by you, miss?''
``If I did harmful or silly things, I'd thank you for telling me of them, and I'd try to mend my ways,'' answered Rose heartily.
``Well, now, we'll see if you mean what you say. I'll give up smoking to please you, if you will give up something to please me,'' said Prince, seeing a good chance to lord it over the weaker vessel at small cost to himself.
``I'll agree if it is as foolish as cigars.''
``Oh, it's ever so much sillier.''
``Then I promise; what is it?'' and Rose quite trembled with anxiety to know which of her pet habits or possessions she must lose.
``Give up your ear-rings,'' and Charlie laughed wickedly, sure that she would never hold to that bargain.
Rose uttered a cry and clapped both hands to her ears where the gold rings hung.
``Oh, Charlie, wouldn't anything else do as well? I've been through so much teasing and trouble, I do want to enjoy my pretty ear-rings, for I can wear them now.''
``Wear as many as you like, and I'll smoke in peace,'' returned this bad boy.
``Will nothing else satisfy you?'' imploringly.
``Nothing,'' sternly.
Rose stood silent for a minute, thinking of something Aunt Jessie once said ``You have more influence over the boys than you know; use it for their good, and I shall thank you all my life.'' Here was a chance to do some good by sacrificing a little vanity of her own. She felt it was right to do it, yet found it very hard, and asked wistfully,
``Do you mean never wear them, Charlie?''
``Never, unless you want me to smoke.''
``I never do.''
``Then clinch the bargain.''
He had no idea she would do it, and was much surprised when she took the dear rings from her ears, with a quick gesture, and held them out to him, saying, in a tone that made the colour come up to his brown cheek, it was so full of sweet good will,
``I care more for my cousins than for my ear-rings, so I promise, and I'll keep my word.''
``For shame, Prince! let her wear her little danglers if she likes, and don't bargain about doing what you know is right,'' cried Archie, coming out of his grove of newspapers with an indignant bounce.
But Rose was bent on showing her aunt that she could use her influence for the boys' good, and said steadily,
``It is fair, and I want it to be so, then you will believe I'm in earnest. Here, each of you wear one of these on your watch-guard to remind you. I shall not forget, because very soon I cannot wear ear-rings if I want to.''
As she spoke, Rose offered a little ring to each cousin, and the boys, seeing how sincere she was, obeyed her. When the pledges were safe, Rose stretched a hand to each, and the lads gave hers a hearty grip, half pleased and half ashamed of their part in the compact.
Just at that moment Dr. Alec and Mrs. Jessie came in.
``What's this? Dancing Ladies' Triumph on Sunday?'' exclaimed Uncle Alec, surveying the trio with surprise.
``No, sir, it is the Anti-Tobacco League. Will you join?'' said Charlie, while Rose slipped away to her aunt, and Archie buried both cigars behind the back log.
When the mystery was explained, the elders were well pleased, and Rose received a vote of thanks, which made her feel as if she had done a service to her country, as she had, for every boy who grows up free from bad habits bids fair to make a good citizen.
``I wish Rose would drive a bargain with Will and Geordie also, for I think these books are as bad for the small boys as cigars for the large ones,'' said Mrs. Jessie, sitting down on the sofa between the readers, who politely curled up their legs to make room for her.
``I thought they were all the fashion,'' answered Dr. Alec, settling in the big chair with Rose.
``So is smoking, but it is harmful. The writers of these popular stories intend to do good, I have no doubt, but it seems to me they fail because their motto is, `Be smart, and you will be rich,' instead of 'Be honest, and you will be happy.' I do not judge hastily, Alec, for I have read a dozen, at least, of these stories, and, with much that is attractive to boys, I find a great deal to condemn in them, and other parents say the same when I ask them.''
``Now, Mum, that's too bad! I like 'em tip-top. This one is a regular screamer,'' cried Will.
``They're bully books, and I'd like to know where's the harm,'' added Geordie.
``You have just shown us one of the chief evils, and that is slang,'' answered their mother quickly.
``Must have it, ma'am. If these chaps talked all right, there'd be no fun in 'em,'' protested Will.
``A boot-black mustn't use good grammar, and a newsboy must swear a little, or he wouldn't be natural,'' explained Geordie, both boys ready to fight gallantly for their favourites.
``But my sons are neither boot-blacks nor newsboys, and I object to hearing them use such words as `screamer,' `bully,' and `buster.' In fact, I fail to see the advantage of writing books about such people unless it is done in a very different way. I cannot think they will help to refine the ragamuffins if they read them, and I'm sure they can do no good to the better class of boys, who through these books are introduced to police courts, counterfeiters' dens, gambling houses, drinking saloons, and all sorts of low life.''
``Some of them are about first-rate boys, mother; and they go to sea and study, and sail round the world, having great larks all the way.''
``I have read about them, Geordie, and though they are better than the others, I am not satisfied with these optical delusions, as I call them. Now, I put it to you, boys, is it natural for lads from fifteen to eighteen to command ships, defeat pirates, outwit smugglers, and so cover themselves with glory, that Admiral Farragut invites them to dinner, saying, `Noble boy, you are an honour to your country!' Or, if the hero is in the army, he has hair-breadth escapes and adventures enough in one small volume to turn his hair white, and in the end he goes to Washington at the express desire of the President or Commander-in-chief to be promoted to no end of stars and bars. Even if the hero is merely an honest boy trying to get his living, he is not permitted to do so in a natural way, by hard work and years of patient effort, but is suddenly adopted by a millionaire whose pocket-book he has returned; or a rich uncle appears from sea just in the nick of time; or the remarkable boy earns a few dollars, speculates in pea-nuts or neckties, and grows rich so rapidly that Sinbad in the diamond valley is a pauper compared to him. Isn't it so, boys?''
``Well, the fellows in these books are mighty lucky, and very smart, I must say,'' answered Will, surveying an illustration on the open page before him, where a small but virtuous youth is upsetting a tipsy giant in a bar-room, and under it the elegant inscription, ``Dick Dauntless punches the head of Sam Soaker.''
``It gives boys such wrong ideas of life and business; shows them so much evil and vulgarity that they need not know about, and makes the one success worth having a fortune, a lord's daughter, or some worldly honour, often not worth the time it takes to win. It does seem to me that some one might write stories that should be lively, natural and helpful tales in which the English should be good, the morals pure, and the characters such as we can love in spite of the faults that all may have. I can't bear to see such crowds of eager little fellows at the libraries reading such trash; weak, when it is not wicked, and totally unfit to feed the hungry minds that feast on it for want of something better. There! my lecture is done; now I should like to hear what you gentlemen have to say,'' and Aunt Jessie subsided with a pretty flush on the face that was full of motherly anxiety for her boys.
``Tom Brown just suits mother, and me too, so I wish Mr. Hughes would write another story as good,'' said Archie.
``You don't find things of this sort in Tom Brown; yet these books are all in the Sunday-school libraries'' and Mrs. Jessie read the following paragraph from the book she had taken from Will's hand,
``'In this place we saw a tooth of John the Baptist. Ben said he could see locust and wild honey sticking to it. I couldn't. Perhaps John used a piece of the true cross for a tooth-pick.'''
``A larky sort of a boy says that, Mum, and we skip the parts where they describe what they saw in the different countries,'' cried Will.
``And those descriptions, taken mostly from guidebooks, I fancy, are the only parts of any real worth. The scrapes of the bad boys make up the rest of the story, and it is for those you read these books, I think,'' answered his mother, stroking back the hair off the honest little face that looked rather abashed at this true statement of the case.
``Anyway, mother, the ship part is useful, for we learn how to sail her, and by and by that will all come handy when we go to sea,'' put in Geordie.
``Indeed, then you can explain this manoeuvre to me, of course,'' and Mrs. Jessie read from another page the following nautical paragraph,
``The wind is south-south-west, and we can have her up four points closer to the wind, and still be six points off the wind. As she luffs up we shall man the fore and main sheets, slack on the weather, and haul on the lee braces.''
``I guess I could, if I wasn't afraid of uncle. He knows so much more than I do, he'd laugh,'' began Geordie, evidently puzzled by the question.
``Ho, you know you can't, so why make believe? We don't understand half of the sea lingo, Mum, and I dare say it's all wrong,'' cried Will, suddenly going over to the enemy, to Geordie's great disgust.
``I do wish the boys wouldn't talk to me as if I was a ship,'' said Rose, bringing forward a private grievance. ``Coming home from church this morning, the wind blew me about, and Will called out, right in the street, 'Brail up the foresail, and take in the flying-jib, that will ease her.'''
The boys shouted at the plaintive tone in which Rose repeated the words that offended her, and Will vainly endeavoured to explain that he only meant to tell her to wrap her cloak closer, and tie a veil over the tempest-tossed feathers in her hat.
``To tell the truth, if the boys must have slang, I can bear the 'sea lingo,' as Will calls it, better than the other. It afflicts me less to hear my sons talk about `brailing up the foresail' than doing as they `darn please,' and `cut your cable' is decidedly preferable to 'let her rip.' I once made a rule that I would have no slang in the house. I give it up now, for I cannot keep it; but I will not have rubbishy books; so, Archie, please send these two after your cigars.''
Mrs. Jessie held both the small boys fast with an arm round each neck, and when she took this base advantage of them they could only squirm with dismay. ``Yes, right behind the back log,'' she continued, energetically. ``There, my hearties (you like sea slang, so I'll give you a bit) now, I want you to promise not to read any more stuff for a month, and I'll agree to supply you with wholesome fare.''
``Oh, mother, not a single one?'' cried Will.
``Couldn't we just finish those?'' pleaded Geordie.
``The boys threw away half-smoked cigars; and your books must go after them. Surely you would not be outdone by the `old fellows,' as you call them, or be less obedient to little Mum than they were to Rose.''
``Course not! Come on, Geordie,'' and Will took the vow like a hero. His brother sighed and obeyed, but privately resolved to finish his story the minute the month was over.
``You have laid out a hard task for yourself, Jessie, in trying to provide good reading for boys who have been living on sensation stories. It will be like going from raspberry tarts to plain bread and butter; but you will probably save them from a bilious fever,'' said Dr. Alec, much amused at the proceedings.
``I remember hearing grandpa say that a love for good books was one of the best safeguards a man could have,'' began Archie, staring thoughtfully at the fine library before him.
``Yes, but there's no time to read nowadays; a fellow has to keep scratching round to make money or he's nobody,'' cut in Charlie, trying to look worldly-wise.
``This love of money is the curse of America, and for the sake of it men will sell honour and honesty, till we don't know whom to trust, and it is only a genius like Agassiz who dares to say, 'I cannot waste my time in getting rich,''' said Mrs. Jessie sadly.
``Do you want us to be poor, mother?'' asked Archie, wondering.
``No, dear, and you never need be, while you can use your hands; but I am afraid of this thirst for wealth, and the temptations it brings. O, my boys! I tremble for the time when I must let you go, because I think it would break my heart to have you fail as so many fail. It would be far easier to see you dead if it could be said of you as of Sumner 'No man dared offer him a bribe.'''
Mrs. Jessie was so earnest in her motherly anxiety that her voice faltered over the last words, and she hugged the yellow heads closer in her arms, as if she feared to let them leave that safe harbour for the great sea where so many little boats go down. The younger lads nestled closer to her, and Archie said, in his quiet, resolute way,
``I cannot promise to be an Agassiz or a Sumner, mother; but I do promise to be an honest man, please God.''
``Then I'm satisfied!'' and holding fast the hand he gave her, she sealed his promise with a kiss that had all a mother's hope and faith in it.
``I don't see how they ever can be bad, she is so fond and proud of them,'' whispered Rose, quite touched by the little scene.
``You must help her make them what they should be. You have begun already, and when I see those rings where they are, my girl is prettier in my sight than if the biggest diamonds that ever twinkled shone in her ears,'' answered Dr. Alec, looking at her with approving eyes.
``I'm so glad you think I can do anything, for I perfectly ache to be useful; everyone is so good to me, especially Aunt Jessie.''
``I think you are in a fair way to pay your debts, Rosy, for when girls give up their little vanities, and boys their small vices, and try to strengthen each other in well-doing, matters are going as they ought. Work away, my dear, and help their mother keep these sons fit friends for an innocent creature like yourself; they will be the manlier men for it, I can assure you.''
\gutchapter{Chapter 18---Fashion and Physiology}
``Please, sir, I guess you'd better step up right away, or it will be too late, for I heard Miss Rose say she knew you wouldn't like it, and she'd never dare to let you see her.''
Phebe said this as she popped her head into the study, where Dr. Alec sat reading a new book.
``They are at it, are they?'' he said, looking up quickly, and giving himself a shake, as if ready for a battle of some sort.
``Yes, sir, as hard as they can talk, and Miss Rose don't seem to know what to do, for the things are ever so stylish, and she looks elegant in 'em; though I like her best in the old ones,'' answered Phebe.
``You are a girl of sense. I'll settle matters for Rosy, and you'll lend a hand. Is everything ready in her room, and are you sure you understand how they go?''
``Oh, yes, sir; but they are so funny! I know Miss Rose will think it's a joke,'' and Phebe laughed as if something tickled her immensely.
``Never mind what she thinks so long as she obeys. Tell her to do it for my sake, and she will find it the best joke she ever saw. I expect to have a tough time of it, but we'll win yet,'' said the Doctor, as he marched upstairs with the book in his hand, and an odd smile on his face.
There was such a clatter of tongues in the sewing-room that no one heard his tap at the door, so he pushed it open and took an observation. Aunt Plenty, Aunt Clara, and Aunt Jessie were all absorbed in gazing at Rose, who slowly revolved between them and the great mirror, in a full winter costume of the latest fashion.
``Bless my heart! worse even than I expected,'' thought the Doctor, with an inward groan, for, to his benighted eyes, the girl looked like a trussed fowl, and the fine new dress had neither grace, beauty, nor fitness to recommend it.
The suit was of two peculiar shades of blue, so arranged that patches of light and dark distracted the eye. The upper skirt was tied so lightly back that it was impossible to take a long step, and the under one was so loaded with plaited frills that it ``wobbled'' no other word will express it ungracefully, both fore and aft. A bunch of folds was gathered up just below the waist behind, and a great bow rode a-top. A small jacket of the same material was adorned with a high ruff at the back, and laid well open over the breast, to display some lace and a locket. Heavy fringes, bows, puffs, ruffles, and revers finished off the dress, making one's head ache to think of the amount of work wasted, for not a single graceful line struck the eye, and the beauty of the material was quite lost in the profusion of ornament.
A high velvet hat, audaciously turned up in front, with a bunch of pink roses and a sweeping plume, was cocked over one ear, and, with her curls braided into a club at the back of her neck, Rose's head looked more like that of a dashing young cavalier than a modest little girl's. High-heeled boots tilted her well forward, a tiny muff pinioned her arms, and a spotted veil, tied so closely over her face that her eyelashes were rumpled by it, gave the last touch of absurdity to her appearance.
``Now she looks like other girls, and as I like to see her,'' Mrs. Clara was saying, with an air of great satisfaction.
``She does look like a fashionable young lady, but somehow I miss my little Rose, for children dressed like children in my day,'' answered Aunt Plenty, peering through her glasses with a troubled look, for she could not imagine the creature before her ever sitting in her lap, running to wait upon her, or making the house gay with a child's blithe presence.
``Things have changed since your day, Aunt, and it takes time to get used to new ways. But you, Jessie, surely like this costume better than the dowdy things Rose has been wearing all summer. Now, be honest, and own you do,'' said Mrs. Clara, bent on being praised for her work.
``Well, dear to be quite honest, then, I think it is frightful,'' answered Mrs. Jessie, with a candour that caused revolving Rose to stop in dismay.
``Hear, hear,'' cried a deep voice, and with a general start the ladies became aware that the enemy was among them.
Rose blushed up to her hat brim, and stood, looking, as she felt, like a fool, while Mrs. Clara hastened to explain.
``Of course, I don't expect you to like it, Alec, but I don't consider you a judge of what is proper and becoming for a young lady. Therefore, I have taken the liberty of providing a pretty street suit for Rose. She need not wear it if you object, for I know we promised to let you do what you liked with the poor dear for a year.''
``It is a street costume, is it?'' asked the Doctor, mildly. ``Do you know, I never should have guessed that it was meant for winter weather and brisk locomotion. Take a turn, Rosy, and let me see all its beauties and advantages.''
Rose tried to walk off with her usual free tread, but the under-skirt got in her way, the over-skirt was so tight she could not take a long step, and her boots made it impossible to carry herself perfectly erect.
``I haven't got used to it yet,'' she said, petulantly, kicking at her train, as she turned to toddle back again.
``Suppose a mad dog or a runaway horse was after you, could you get out of the way without upsetting, Colonel,'' asked the Doctor, with a twinkle in the eyes that were fixed on the rakish hat.
``Don't think I could, but I'll try,'' and Rose made a rush across the room. Her boot-heels caught on a rug, several strings broke, her hat tipped over her eyes, and she plunged promiscuously into a chair, where she sat laughing so infectiously that all but Mrs. Clara joined in her mirth.
``I should say that a walking suit in which one could not walk, and a winter suit which exposes the throat, head, and feet to cold and damp, was rather a failure, Clara, especially as it has no beauty to reconcile one to its utter unfitness,'' said Dr. Alec, as he helped Rose undo her veil, adding, in a low tone, ``Nice thing for the eyes; you'll soon see spots when it's off as well as when it's on, and, by and by, be a case for an oculist.''
``No beauty!'' cried Mrs. Clara, warmly, ``Now, that is just a man's blindness. This is the best of silk and camel's hair, real ostrich feathers, and an expensive ermine muff. What could be in better taste, or more proper for a young girl?''
``I'll shew you, if Rose will go to her room and oblige me by putting on what she finds there,'' answered the Doctor, with unexpected readiness.
``Alec, if it is a Bloomer, I shall protest. I've been expecting it, but I know I cannot bear to see that pretty child sacrificed to your wild ideas of health. Tell me it isn't a Bloomer!'' and Mrs. Clara clasped her hands imploringly.
``It is not.''
``Thank Heaven!'' and she resigned herself with a sigh of relief, adding plaintively, ``I did hope you'd accept my suit, for poor Rose has been afflicted with frightful clothes long enough to spoil the taste of any girl.''
``You talk of my afflicting the child, and then make a helpless guy like that of her!'' answered the Doctor, pointing to the little fashion plate that was scuttling out of sight as fast as it could go.
He closed the door with a shrug, but before anyone could speak, his quick eye fell upon an object which caused him to frown, and demand in an indignant tone,
``After all I have said, were you really going to tempt my girl with those abominable things?''
``I thought we put them away when she wouldn't wear them,'' murmured Mrs. Clara, whisking a little pair of corsets out of sight with guilty haste. ``I only brought them to try, for Rose is growing stout, and will have no figure if it is not attended to soon,'' she added, with an air of calm conviction that roused the Doctor still more, for this was one of his especial abominations.
``Growing stout! Yes, thank Heaven, she is, and shall continue to do it, for Nature knows how to mould a woman better than any corset-maker, and I won't have her interfered with. My dear Clara, have you lost your senses that you can for a moment dream of putting a growing girl into an instrument of torture like this?'' and with a sudden gesture he plucked forth the offending corsets from under the sofa cushion, and held them out with the expression one would wear on beholding the thumbscrews or the rack of ancient times.
``Don't be absurd, Alec. There is no torture about it, for tight lacing is out of fashion, and we have nice, sensible things nowadays. Everyone wears them; even babies have stiffened waists to support their weak little backs,'' began Mrs. Clara, rushing to the defence of the pet delusion of most women.
``I know it, and so the poor little souls have weak backs all their days, as their mothers had before them. It is vain to argue the matter, and I won't try, but I wish to state, once for all, that if I ever see a pair of corsets near Rose, I'll put them in the fire, and you may send the bill to me.''
As he spoke the corsets were on their way to destruction, but Mrs. Jessie caught his arm, exclaiming merrily, ``Don't burn them, for mercy sake, Alec; they are full of whalebones, and will make a dreadful odour. Give them to me. I'll see that they do no harm.''
``Whalebones, indeed! A regular fence of them, and metal gate-posts in front. As if our own bones were not enough, if we'd give them a chance to do their duty,'' growled the Doctor, yielding up the bone of contention with a last shake of contempt. Then his face cleared suddenly, and he held up his finger, saying, with a smile, ``Hear those girls laugh; cramped lungs could not make hearty music like that.''
Peals of laughter issued from Rose's room, and smiles involuntarily touched the lips of those who listened to the happy sound.
``Some new prank of yours, Alec?'' asked Aunt Plenty, indulgently, for she had come to believe in most of her nephew's odd notions, because they seemed to work so well.
``Yes, ma'am, my last, and I hope you will like it. I discovered what Clara was at, and got my rival suit ready for to-day. I'm not going to `afflict' Rose, but let her choose, and if I'm not entirely mistaken, she will like my rig best. While we wait I'll explain, and then you will appreciate the general effect better. I got hold of this little book, and was struck with its good sense and good taste, for it suggests a way to clothe women both healthfully and handsomely, and that is a great point. It begins at the foundations, as you will see if you will look at these pictures, and I should think women would rejoice at this lightening of their burdens.''
As he spoke, the Doctor laid the book before Aunt Plenty, who obediently brought her spectacles to bear upon the illustrations, and after a long look exclaimed, with a scandalised face,
``Mercy on us, these things are like the night-drawers Jamie wears! You don't mean to say you want Rose to come out in this costume? It's not proper, and I won't consent to it!''
``I do mean it, and I'm sure my sensible aunt will consent when she understands that these well I'll call them by an Indian name, and say pajamas are for underwear, and Rose can have as pretty frocks as she likes outside. These two suits of flannel, each in one piece from head to foot, with a skirt or so hung on this easily-fitting waist, will keep the child warm without burdening her with belts, and gathers, and buckles, and bunches round the waist, and leave free the muscles that need plenty of room to work in. She shall never have the back-ache if I can help it, nor the long list of ills you dear women think you cannot escape.''
``I don't consider it modest, and I'm sure Rose will be shocked at it,'' began Mrs. Clara, but stopped suddenly, as Rose appeared in the doorway, not looking shocked a bit.
``Come on, my hygienic model, and let us see you,'' said her uncle, with an approving glance, as she walked in, looking so mischievously merry, that it was evident she enjoyed the joke.
``Well, I don't see anything remarkable. That is a neat, plain suit; the materials are good, and it's not unbecoming, if you want her to look like a little school-girl; but it has not a particle of style, and no one would ever give it a second glance,'' said Mrs. Clara, feeling that her last remark condemned the whole thing.
``Exactly what I want,'' answered the provoking Doctor, rubbing his hands with a satisfied air. ``Rosy looks now like what she is, a modest little girl, who does not want to be stared at. I think she would get a glance of approval, though, from people who like sense and simplicity rather than fuss and feathers. Revolve, my Hebe, and let me refresh my eyes by the sight of you.''
There was very little to see, however, only a pretty Gabrielle dress, of a soft warm shade of brown, coming to the tops of a trim pair of boots with low heels. A seal-skin sack, cap, and mittens, with a glimpse of scarlet at the throat, and the pretty curls tied up with a bright velvet of the same colour, completed the external adornment, making her look like a robin redbreast wintry, yet warm.
``How do you like it, Rosy?'' asked the Doctor, feeling that her opinion was more important to the success of his new idea than that of all the aunts on the hill.
``I feel very odd and light, but I'm warm as a toast, and nothing seems to be in my way,'' answered Rose, with a skip which displayed shapely gaiters on legs that now might be as free and active as a boy's under the modest skirts of the girl.
``You can run away from the mad dogs, and walk off at a smart pace without tumbling on your nose, now, I fancy?''
``Yes, uncle! suppose the dog coming, I just hop over a wall so and when I walk of a cold day, I go like this.''
Entering fully into the spirit of the thing, Rose swung herself over the high back of the sofa as easily as one of her cousins, and then went down the long hall as if her stout boots were related to the famous seven-leaguers.
``There! you see how it will be; dress her in that boyish way and she will act like a boy. I do hate all these inventions of strong-minded women!'' exclaimed Mrs. Clara, as Rose came back at a run.
``Ah, but you see some of these sensible inventions come from the brain of a fashionable modiste, who will make you more lovely, or what you value more `stylish' outside and comfortable within. Mrs. Van Tassel has been to Madame Stone, and is wearing a full suit of this sort. Van himself told me, when I asked how she was, that she had given up lying on the sofa, and was going about in a most astonishing way, considering her feeble health.''
``You don't say so! Let me see that book a moment,'' and Aunt Clara examined the new patterns with a more respectful air, for if the elegant Mrs. Van Tassel wore these ``dreadful things'' it would never do to be left behind, in spite of her prejudices.
Dr. Alec looked at Mrs. Jessie, and both smiled, for ``little Mum'' had been in the secret, and enjoyed it mightily.
``I thought that would settle it,'' he said with a nod.
``I didn't wait for Mrs. Van to lead the way, and for once in my life I have adopted a new fashion before Clara. My freedom suit is ordered, and you may see me playing tag with Rose and the boys before long,'' answered Mrs. Jessie, nodding back at him.
Meantime Aunt Plenty was examining Rose's costume, for the hat and sack were off, and the girl was eagerly explaining the new under-garments.
``See, auntie, all nice scarlet flannel, and a gay little petticoat, and long stockings, oh, so warm! Phebe and I nearly died laughing when I put this rig on, but I like it ever so much. The dress is so comfortable, and doesn't need any belt or sash, and I can sit without rumpling any trimming, that's such a comfort! I like to be tidy, and so, when I wear fussed-up things, I'm thinking of my clothes all the time, and that's tiresome. Do say you like it. I resolved I would, just to please uncle, for he does know more about health than anyone else, I'm sure, and I'd wear a bag if he asked me to do it.''
``I don't ask that, Rose, but I wish you'd weigh and compare the two suits, and then choose which seems best. I leave it to your own commonsense,'' answered Dr. Alec, feeling pretty sure he had won.
``Why, I take this one, of course, uncle. The other is fashionable, and yes I must say I think it's pretty but it's very heavy, and I should have to go round like a walking doll if I wore it. I'm much obliged to auntie, but I'll keep this, please.''
Rose spoke gently but decidedly, though there was a look of regret when her eye fell on the other suit which Phebe had brought in; and it was very natural to like to look as other girls did. Aunt Clara sighed; Uncle Alec smiled, and said heartily,
``Thank you, dear; now read this book and you will understand why I ask it of you. Then, if you like, I'll give you a new lesson; you asked for one yesterday, and this is more necessary than French or housekeeping.''
``Oh, what?'' and Rose caught up the book which Mrs. Clara had thrown down with a disgusted look.
Though Dr. Alec was forty, the boyish love of teasing was not yet dead in him, and, being much elated at his victory, he could not resist the temptation of shocking Mrs. Clara by suggesting dreadful possibilities, so he answered, half in earnest, half in jest, ``Physiology, Rose. Wouldn't you like to be a little medical student, with Uncle Doctor for teacher, and be ready to take up his practice when he has to stop? If you agree, I'll hunt up my old skeleton to-morrow.''
That was too much for Aunt Clara, and she hastily departed, with her mind in a sad state of perturbation about Mrs. Van Tassel's new costume and Rose's new study.
\gutchapter{Chapter 19---Brother Bones}
Rose accepted her uncle's offer, as Aunt Myra discovered two or three days later. Coming in for an early call, and hearing voices in the study, she opened the door, gave a cry and shut it quickly, looking a good deal startled. The Doctor appeared in a moment, and begged to know what the matter was.
``How can you ask when that long box looks so like a coffin I thought it was one, and that dreadful thing stared me in the face as I opened the door,'' answered Mrs. Myra, pointing to the skeleton that hung from the chandelier cheerfully grinning at all beholders.
``This is a medical college where women are freely admitted, so walk in, madam, and join the class if you'll do me the honour,'' said the Doctor, waving her forward with his politest bow.
``Do, auntie, it's perfectly splendid,'' cried Rose's voice, and Rose's blooming face was seen behind the ribs of the skeleton, smiling and nodding in the gayest possible manner.
``What are you doing, child?'' demanded Aunt Myra, dropping into a chair and staring about her.
``Oh, I'm learning bones to-day, and I like it so much. There are twelve ribs, you know, and the two lower ones are called floating ribs, because they are not fastened to the breastbone. That's why they go in so easily if you lace tight and squeeze the lungs and heart in the let me see, what was that big word oh, I know thoracic cavity,'' and Rose beamed with pride as she aired her little bit of knowledge.
``Do you think that is a good sort of thing for her to be poking over? She is a nervous child, and I'm afraid it will be bad for her,'' said Aunt Myra, watching Rose as she counted vertebrae, and waggled a hip-joint in its socket with an inquiring expression.
``An excellent study, for she enjoys it, and I mean to teach her how to manage her nerves so that they won't be a curse to her, as many a woman's become through ignorance or want of thought. To make a mystery or terror of these things is a mistake, and I mean Rose shall understand and respect her body so well that she won't dare to trifle with it as most women do.''
``And she really likes it?''
``Very much, auntie! It's all so wonderful, and so nicely planned, you can hardly believe what you see. Just think, there are 600,000,000 air cells in one pair of lungs, and 2,000 pores to a square inch of surface; so you see what quantities of air we must have, and what care we should take of our skin so all the little doors will open and shut right. And brains, auntie, you've no idea how curious they are; I haven't got to them yet, but I long to, and uncle is going to show me a manikin that you can take to pieces. Just think how nice it will be to see all the organs in their places; I only wish they could be made to work as ours do.''
It was funny to see Aunt Myra's face as Rose stood before her talking rapidly with one hand laid in the friendliest manner on the skeleton's shoulder. Every word both the Doctor and Rose uttered hit the good lady in her weakest spot, and as she looked and listened a long array of bottles and pill-boxes rose up before her, reproaching her with the ``ignorance and want of thought'' that made her what she was, a nervous, dyspeptic, unhappy old woman.
``Well, I don't know but you may be right, Alec, only I wouldn't carry it too far. Women don't need much of this sort of knowledge, and are not fit for it. I couldn't bear to touch that ugly thing, and it gives me the creeps to hear about `organs,''' said Aunt Myra, with a sigh and her hand on her side.
``Wouldn't it be a comfort to know that your liver was on the right side, auntie, and not on the left!'' asked Rose with a naughty laugh in her eyes, for she had lately learnt that Aunt Myra's liver complaint was not in the proper place.
``It's a dying world, child, and it don't much matter where the pain is, for sooner or later we all drop off and are seen no more,'' was Aunt Myra's cheerful reply.
``Well, I intend to know what kills me if I can, and meantime, I'm going to enjoy myself in spite of a dying world. I wish you'd do so too, and come and study with uncle, it would do you good, I'm sure,'' and Rose went back to counting vertebrae with such a happy face, that Aunt Myra had not the heart to say a word to dampen her ardour.
``Perhaps it's as well to let her do what she likes the little while she is with us. But pray be careful of her, Alec, and not allow her to overwork,'' she whispered as she went out.
``That's exactly what I'm trying to do, ma'am, and rather a hard job I find it,'' he added, as he shut the door, for the dear aunts were dreadfully in his way sometimes.
Half an hour later came another interruption in the shape of Mac, who announced his arrival by the brief but elegant remark,
``Hullo! what new game is this?''
Rose explained, Mac gave a long whistle of surprise, and then took a promenade round the skeleton, observing gravely,
``Brother Bones looks very jolly, but I can't say much for his beauty.''
``You mustn't make fun of him, for he's a good old fellow, and you'd be just as ugly if your flesh was off,'' said Rose, defending her new friend with warmth.
``I dare say, so I'll keep my flesh on, thank you. You are so busy you can't read to a fellow, I suppose?'' asked Mac, whose eyes were better, but still too weak for books.
``Don't you want to come and join my class? Uncle explains it all to us, and you can take a look at the plates as they come along. We'll give up bones today and have eyes instead; that will be more interesting to you,'' added Rose, seeing no ardent thirst for physiological information in his face.
``Rose, we must not fly about from one thing to another in this way,'' began Dr. Alec, but she whispered quickly, with a nod towards Mac, whose goggles were turned wistfully in the direction of the forbidden books,
``He's blue to-day, and we must amuse him; give a little lecture on eyes, and it will do him good. No matter about me, uncle.''
``Very well; the class will please be seated,'' and the Doctor gave a sounding rap on the table.
``Come, sit by me, dear, then we can both see the pictures; and if your head gets tired you can lie down,'' said Rose, generously opening her little college to a brother, and kindly providing for the weaknesses that all humanity is subject to.
Side by side they sat and listened to a very simple explanation of the mechanism of the eye, finding it as wonderful as a fairy tale, for fine plates illustrated it, and a very willing teacher did his best to make the lesson pleasant.
``Jove! if I'd known what mischief I was doing to that mighty delicate machine of mine, you wouldn't have caught me reading by firelight, or studying with a glare of sunshine on my book,'' said Mac, peering solemnly at a magnified eye-ball; then, pushing it away, he added indignantly, ``Why isn't a fellow taught all about his works, and how to manage 'em, and not left to go blundering into all sorts of worries? Telling him after he's down isn't much use, for then he's found it out himself and won't thank you.''
``Ah, Mac, that's just what I keep lecturing about, and people won't listen. You lads need that sort of knowledge so much, and fathers and mothers ought to be able to give it to you. Few of them are able, and so we all go blundering, as you say. Less Greek and Latin and more knowledge of the laws of health for my boys, if I had them. Mathematics are all very well, but morals are better, and I wish, how I wish that I could help teachers and parents to feel it as they ought.''
``Some do; Aunt Jessie and her boys have capital talks, and I wish we could; but mother's so busy with her housekeeping, and father with his business, there never seems to be any time for that sort of thing; even if there was, it don't seem as if it would be easy to talk to them, because we've never got into the way of it, you know.''
Poor Mac was right there, and expressed a want that many a boy and girl feels. Fathers and mothers are too absorbed in business and housekeeping to study their children, and cherish that sweet and natural confidence which is a child's surest safeguard, and a parent's subtlest power. So the young hearts hide trouble or temptation till the harm is done, and mutual regret comes too late. Happy the boys and girls who tell all things freely to father or mother, sure of pity, help, and pardon; and thrice happy the parents who, out of their own experience, and by their own virtues, can teach and uplift the souls for which they are responsible.
This longing stirred in the hearts of Rose and Mac, and by a natural impulse both turned to Dr. Alec, for in this queer world of ours, fatherly and motherly hearts often beat warm and wise in the breasts of bachelor uncles and maiden aunts; and it is my private opinion that these worthy creatures are a beautiful provision of nature for the cherishing of other people's children. They certainly get great comfort out of it, and receive much innocent affection that otherwise would be lost.
Dr. Alec was one of these, and his big heart had room for every one of the eight cousins, especially orphaned Rose and afflicted Mac; so, when the boy uttered that unconscious reproach to his parents, and Rose added with a sigh, ``It must be beautiful to have a mother!'' the good Doctor yearned over them, and, shutting his book with a decided slam, said in that cordial voice of his,
``Now, look here, children, you just come and tell me all your worries, and with God's help, I'll settle them for you. That is what I'm here for, I believe, and it will be a great happiness to me if you can trust me.''
``We can, uncle, and we will!'' both answered, with a heartiness that gratified him much.
``Good! now school is dismissed, and I advise you to go and refresh your 600,000,000 air cells by a brisk run in the garden. Come again whenever you like, Mac, and we'll teach you all we can about your `works,' as you call them, so you can keep them running smoothly.''
``We'll come, sir, much obliged,'' and the class in physiology went out to walk.
Mac did come again, glad to find something he could study in spite of his weak eyes, and learned much that was of more value than anything his school had ever taught him.
Of course, the other lads made great fun of the whole thing, and plagued Dr. Alec's students half out of their lives. But they kept on persistently, and one day something happened which made the other fellows behave themselves for ever after.
It was a holiday, and Rose up in her room thought she heard the voices of her cousins, so she ran down to welcome them, but found no one there.
``Never mind, they will be here soon, and then we'll have a frolic,'' she said to herself, and thinking she had been mistaken she went into the study to wait. She was lounging over the table looking at a map when an odd noise caught her ear. A gentle tapping somewhere, and following the sound it seemed to come from the inside of the long case in which the skeleton lived when not professionally engaged. This case stood upright in a niche between two book-cases at the back of the room, a darkish corner, where Brother Bones, as the boys would call him, was out of the way.
As Rose stood looking in that direction, and wondering if a rat had got shut in, the door of the case swung slowly open, and with a great start she saw a bony arm lifted, and a bony finger beckon to her. For a minute she was frightened, and ran to the study door with a fluttering heart, but just as she touched the handle a queer, stifled sort of giggle made her stop short and turn red with anger. She paused an instant to collect herself, and then went softly toward the bony beckoner. A nearer look revealed black threads tied to the arm and fingers, the ends of threads disappearing through holes bored in the back of the case. Peeping into the dark recess, she also caught sight of the tip of an elbow covered with a rough gray cloth which she knew very well.
Quick as a flash she understood the joke, her fear vanished, and with a wicked smile, she whipped out her scissors, cut the threads, and the bony arm dropped with a rattle. Before she could say, ``Come out, Charlie, and let my skeleton alone,'' a sudden irruption of boys, all in a high state of tickle, proclaimed to the hidden rogue that his joke was a failure.
``I told him not to do it, because it might give you a start,'' explained Archie, emerging from the closet.
``I had a smelling bottle all ready if she fainted away,'' added Steve, popping up from behind the great chair.
``It's too bad of you not to squawk and run; we depended on it, it's such fun to howl after you,'' said Will and Geordie, rolling out from under the sofa in a promiscuous heap.
``You are getting altogether too strong-minded, Rose; most girls would have been in a jolly twitter to see this old fellow waggling his finger at them,'' complained Charlie, squeezing out from his tight quarters, dusty and disgusted.
``I'm used to your pranks now, so I'm always on the watch and prepared. But I won't have Brother Bones made fun of. I know uncle wouldn't like it, so please don't,'' began Rose just as Dr. Alec came in, and, seeing the state of the case at a glance, he said quietly,
``Hear how I got that skeleton, and then I'm sure you will treat it with respect.''
The boys settled down at once on any article of furniture that was nearest and listened dutifully.
``Years ago, when I was in the hospital, a poor fellow was brought there with a rare and very painful disease. There was no hope for him, but we did our best, and he was so grateful that when he died he left us his body that we might discover the mysteries of his complaint, and so be able to help others afflicted in the same way. It did do good, and his brave patience made us remember him long after he was gone. He thought I had been kind to him, and said to a fellow-student of mine, 'Tell the Doctor I lave him me bones, for I've nothing else in the wide world, and I'll nos be wanting 'em at all, at all, when the great pain hat kilt me entirely.' So that is how they came to be mine, and why I've kept them carefully, for, though only a poor, ignorant fellow, Mike Nolan did what he could to help others, and prove his gratitude to those who tried to help him.''
As Dr. Alec paused, Archie closed the door of the case as respectfully as if the mummy of an Egyptian king was inside; Will and Geordie looked solemnly at one another, evidently much impressed, and Charlie pensively remarked from the coal-hod where he sat,
``I've often heard of a skeleton in the house, but I think few people have one as useful and as interesting as ours.''
\gutchapter{Chapter 20---Under The Mistletoe}
Rose made Phebe promise that she would bring her stocking into the ``Bower,'' as she called her pretty room, on Christmas morning, because that first delicious rummage loses half its charm if two little night-caps at least do not meet over the treasures, and two happy voices Oh and Ah together.
So when Rose opened her eyes that day they fell upon faithful Phebe, rolled up in a shawl, sitting on the rug before a blazing fire, with her untouched stocking laid beside her.
``Merry Christmas!'' cried the little mistress smiling gaily.
``Merry Christmas!'' answered the little maid, so heartily that it did one good to hear her.
``Bring the stockings right away, Phebe, and let's see what we've got,'' said Rose, sitting up among the pillows, and looking as eager as a child.
A pair of long knobby hose were laid out upon the coverlet, and their contents examined with delight, though each knew every blessed thing that had been put into the other's stocking.
Never mind what they were; it is evident that they were quite satisfactory, for as Rose leaned back, she said, with a luxurious sigh of satisfaction, ``Now, I believe I've got everything in the world that I want,'' and Phebe answered, smiling over a lapful of treasures, ``This is the most splendid Christmas I ever had since I was born.'' Then she added with an important air,
``Do wish for something else, because I happen to know of two more presents outside the door this minute.''
``Oh, me, what richness!'' cried Rose, much excited. ``I used to wish for a pair of glass slippers like Cinderella's, but as I can't have them, I really don't know what to ask for.''
Phebe clapped her hands as she skipped off the bed and ran to the door, saying merrily, ``One of them is for your feet, anyway. I don't know what you'll say to the other, but I think it's elegant.''
So did Rose, when a shining pair of skates and a fine sled appeared.
``Uncle sent those; I know he did; and, now I see them, I remember that I did want to skate and coast. Isn't it a beauty? See! they fit nicely,'' and, sitting on the new sled, Rose tried a skate on her little bare foot, while Phebe stood by admiring the pretty tableau.
``Now we must hurry and get dressed, for there is a deal to do to-day, and I want to get through in time to try my sled before dinner.''
``Gracious me, and I ought to be dusting my parlors this blessed minute!'' and mistress and maid separated with such happy faces that anyone would have known what day it was without being told.
``Birnam Wood has come to Dunsinane, Rosy,'' said Dr. Alec, as he left the breakfast table to open the door for a procession of holly, hemlock, and cedar boughs that came marching up the steps.
Snowballs and ``Merry Christmases!'' flew about pretty briskly for several minutes; then all fell to work trimming the old house, for the family always dined together there on that day.
``I rode miles and mileses, as Ben says, to get this fine bit, and I'm going to hang it there as the last touch to the rig-a-madooning,'' said Charlie, as he fastened a dull green branch to the chandelier in the front parlor.
``It isn't very pretty,'' said Rose, who was trimming the chimney-piece with glossy holly sprays.
``Never mind that, it's mistletoe, and anyone who stands under it will get kissed whether they like it or not. Now's your time, ladies,'' answered the saucy Prince, keeping his place and looking sentimentally at the girls, who retired precipitately from the dangerous spot.
``You won't catch me,'' said Rose, with great dignity.
``See if I don't!''
``I've got my eye on Phebe,'' observed Will, in a patronising tone that made them all laugh.
``Bless the dear; I shan't mind it a bit,'' answered Phebe, with such a maternal air that Will's budding gallantry was chilled to death.
``Oh, the mistletoe bough,'' sang Rose.
``Oh, the mistletoe bough!'' echoed all the boys, and the teasing ended in the plaintive ballad they all liked so well.
There was plenty of time to try the new skates before dinner, and then Rose took her first lesson on the little bay, which seemed to have frozen over for that express purpose. She found tumbling down and getting up again warm work for a time, but with six boys to teach her, she managed at last to stand alone; and, satisfied with that success, she refreshed herself with a dozen grand coasts on the Amazon, as her sled was called.
``Ah, that fatal colour! it breaks my heart to see it,'' croaked Aunt Myra, as Rose came down a little late, with cheeks almost as ruddy as the holly berries on the wall, and every curl as smooth as Phebe's careful hands could make it.
``I'm glad to see that Alec allows the poor child to make herself pretty in spite of his absurd notions,'' added Aunt Clara, taking infinite satisfaction in the fact that Rose's blue silk dress had three frills on it.
``She's a very intelligent child, and has a nice little manner of her own,'' observed Aunt Jane, with unusual affability; for Rose had just handed Mac a screen to guard his eyes from the brilliant fire.
``If I had a daughter like that to show my Jem when he gets home, I should be a very proud and happy woman,'' thought Aunt Jessie, and then reproached herself for not being perfectly satisfied with her four brave lads.
Aunt Plenty was too absorbed in the dinner to have an eye for anything else; if she had not been, she would have seen what an effect her new cap produced upon the boys. The good lady owned that she did ``love a dressy cap,'' and on this occasion her head gear was magnificent; for the towering structure of lace was adorned with buff ribbons to such an extent that it looked as if a flock of yellow butterflies had settled on her dear old head. When she trotted about the rooms the ruches quivered, the little bows all stood erect, and the streamers waved in the breeze so comically that it was absolutely necessary for Archie to smother the Brats in the curtains till they had had their first laugh out.
Uncle Mac had brought Fun See to dinner, and it was a mercy he did, for the elder lads found a vent for their merriment in joking the young Chinaman on his improved appearance. He was in American costume now, with a cropped head, and spoke remarkably good English after six months at school; but, for all that, his yellow face and beady eyes made a curious contrast to the blonde Campbells all about him. Will called him the ``Typhoon,'' meaning Tycoon, and the name stuck to him to his great disgust.
Aunt Peace was brought down and set in the chair of state at table, for she never failed to join the family on this day, and sat smiling at them all, ``like an embodiment of Peace on earth,'' Uncle Alec said, as he took his place beside her, while Uncle Mac supported Aunt Plenty at the other end.
``I ate hardly any breakfast, and I've done everything I know to make myself extra hungry, but I really don't think I can eat straight through, unless I burst my buttons off,'' whispered Geordie to Will, as he surveyed the bounteous stores before him with a hopeless sigh.
``A fellow never knows what he can do till he tries,'' answered Will, attacking his heaped-up plate with an evident intention of doing his duty like a man.
Everybody knows what a Christmas dinner is, so we need waste no words in describing this one, but hasten at once to tell what happened at the end of it. The end, by the way, was so long in coming that the gas was lighted before dessert was over, for a snow flurry had come on and the wintry daylight faded fast. But that only made it all the jollier in the warm, bright rooms, full of happy souls. Everyone was very merry, but Archie seemed particularly uplifted so much so, that Charlie confided to Rose that he was afraid the Chief had been at the decanters.
Rose indignantly denied the insinuation, for when healths were drunk in the good old-fashioned way to suit the elders, she had observed that Aunt Jessie's boys filled their glasses with water, and had done the same herself in spite of the Prince's jokes about ``the rosy.''
But Archie certainly was unusually excited, and when someone remembered that it was the anniversary of Uncle Jem's wedding, and wished he was there to make a speech, his son electrified the family by trying to do it for him. It was rather incoherent and flowery, as maiden speeches are apt to be, but the end was considered superb; for, turning to his mother with a queer little choke in his voice, he said that she ``deserved to be blessed with peace and plenty, to be crowned with roses and lads'-love, and to receive the cargo of happiness sailing home to her in spite of wind or tide to add another Jem to the family jewels.''
That allusion to the Captain, now on his return trip, made Mrs. Jessie sob in her napkin, and set the boys cheering. Then, as if that was not sensation enough, Archie suddenly dashed out of the room, as if he had lost his wits.
``Too bashful to stay and be praised,'' began Charlie, excusing the peculiarities of his chief as in duty bound.
``Phebe beckoned to him; I saw her,'' cried Rose, staring hard at the door.
``Is it more presents coming?'' asked Jamie, just as his brother re-appeared, looking more excited than ever.
``Yes; a present for mother, and here it is!'' roared Archie, flinging wide the door to let in a tall man, who cried out,
``Where's my little woman? The first kiss for her, then the rest may come on as fast as they like.''
Before the words were out of his mouth, Mrs. Jessie was half-hidden under his rough great-coat, and four boys were prancing about him clamouring for their turn.
Of course, there was a joyful tumult for a time, during which Rose slipped into the window recess and watched what went on, as if it were a chapter in a Christmas story. It was good to see bluff Uncle Jem look proudly at his tall son, and fondly hug the little ones. It was better still to see him shake his brothers' hands as if he would never leave off, and kiss all the sisters in a way that made even solemn Aunt Myra brighten up for a minute. But it was best of all to see him finally established in grandfather's chair, with his ``little woman'' beside him, his three youngest boys in his lap, and Archie hovering over him like a large-sized cherub. That really was, as Charlie said, ``A landscape to do one's heart good.''
``All hearty and all here, thank God!'' said Captain Jem in the first pause that came, as he looked about him with a grateful face.
``All but Rose,'' answered loyal little Jamie, remembering the absent.
``Faith, I forgot the child! Where is George's little girl?'' asked the Captain, who had not seen her since she was a baby.
``You'd better say Alec's great girl,'' said Uncle Mac, who professed to be madly jealous of his brother.
``Here I am, sir,'' and Rose appeared from behind the curtains, looking as if she had rather have stayed there.
``Saint George Germain, how the mite has grown!'' cried Captain Jem, as he tumbled the boys out of his lap, and rose to greet the tall girl, like a gentleman as he was. But, somehow, when he shook her hand it looked so small in his big one, and her face reminded him so strongly of his dead brother, that he was not satisfied with so cold a welcome, and with a sudden softening of the keen eyes he took her up in his arms, whispering, with a rough cheek against her smooth one,
``God bless you, child! forgive me if I forgot you for a minute, and be sure that not one of your kinsfolk is happier to see you here than Uncle Jem.''
That made it all right; and when he set her down, Rose's face was so bright it was evident that some spell had been used to banish the feeling of neglect that had kept her moping behind the curtain so long.
That everyone sat round and heard all about the voyage home how the Captain had set his heart on getting there in time to keep Christmas; how everything had conspired to thwart his plan; and how, at the very last minute, he had managed to do it, and had sent a telegram to Archie, bidding him keep the secret, and be ready for his father at any moment, for the ship got into another port, and he might be late.
Then Archie told how that telegram had burnt in his pocket all dinner-time; how he had to take Phebe into his confidence, and how clever she was to keep the Captain back till the speech was over and he could come in with effect.
The elders would have sat and talked all the evening, but the young folks were bent on having their usual Christmas frolic; so, after an hour of pleasant chat, they began to get restless, and having consulted together in dumb show, they devised a way to very effectually break up the family council.
Steve vanished, and, sooner than the boys imagined Dandy could get himself up, the skirl of the bag-pipe was heard in the hall, and the bonny piper came to lead Clan Campbell to the revel.
``Draw it mild, Stevie, my man; ye play unco weel, but ye mak a most infernal din,'' cried Uncle Jem, with his hands over his ears, for this accomplishment was new to him, and ``took him all aback,'' as he expressed it.
So Steve droned out a Highland reel as softly as he could, and the boys danced it to a circle of admiring relations. Captain Jem was a true sailor, however, and could not stand idle while anything lively was going on; so, when the piper's breath gave out, he cut a splendid pigeon-wing into the middle of the hall, saying, ``Who can dance a Fore and After?'' and, waiting for no reply, began to whistle the air so invitingly that Mrs Jessie ``set'' to him laughing like a girl; Rose and Charlie took their places behind, and away went the four with a spirit and skill that inspired all the rest to ``cut in'' as fast as they could.
That was a grand beginning, and they had many another dance before anyone would own they were tired. Even Fun See distinguished himself with Aunt Plenty, whom he greatly admired as the stoutest lady in the company; plumpness being considered a beauty in his country. The merry old soul professed herself immensely flattered by his admiration, and the boys declared she ``set her cap at him,'' else he would never have dared to catch her under the mistletoe, and, rising on the tips of his own toes, gallantly salute her fat cheek.
How they all laughed at her astonishment, and how Fun's little black eyes twinkled over this exploit! Charlie put him up to it, and Charlie was so bent on catching Rose, that he laid all sorts of pitfalls for her, and bribed the other lads to help him. But Rose was wide-awake, and escaped all his snares, professing great contempt for such foolish customs. Poor Phebe did not fare so well, and Archie was the only one who took a base advantage of her as she stood innocently offering tea to Aunt Myra, whom she happened to meet just under the fatal bough. If his father's arrival had not rather upset him, I doubt if the dignified Chief would have done it, for he apologized at once in the handsomest manner, and caught the tray that nearly dropped from Phebe's hands.
Jamie boldly invited all the ladies to come and salute him; and as for Uncle Jem, he behaved as if the entire room was a grove of mistletoe. Uncle Alec slyly laid a bit of it on Aunt Peace's cap, and then softly kissed her; which little joke seemed to please her very much, for she liked to have part in all the home pastimes, and Alec was her favourite nephew.
Charlie alone failed to catch his shy bird, and the oftener she escaped the more determined he was to ensnare her. When every other wile had been tried in vain, he got Archie to propose a game with forfeits.
``I understand that dodge,'' thought Rose, and was on her guard so carefully that not one among the pile soon collected belonged to her.
``Now let us redeem them and play something else,'' said Will, quite unconscious of the deeply-laid plots all about him.
``One more round and then we will,'' answered the Prince, who had now baited his trap anew.
Just as the question came to Rose, Jamie's voice was heard in the hall, crying distressfully, ``Oh, come quick, quick!'' Rose started up, missed the question, and was greeted with a general cry of ``Forfeit! forfeit!'' in which the little traitor came to join.
``Now I've got her,'' thought the young rascal, exulting in his fun-loving soul.
``Now I'm lost,'' thought Rose, as she gave up her pin-cushion with a sternly defiant look that would have daunted anyone but the reckless Prince. In fact, it made even him think twice, and resolve to ``let Rose off easy,'' she had been so clever.
``Here's a very pretty pawn, and what shall be done to redeem it?'' asked Steve, holding the pin-cushion over Charlie's head, for he had insisted on being judge, and kept that for the last.
``Fine or superfine?''
``Super.''
``Hum, well, she shall take old Mac under the mistletoe, and kiss him prettily. Won't he be mad, though?'' and this bad boy chuckled over the discomfort he had caused two harmless beings.
There was an impressive pause among the young folks in their corner, for they all knew that Mac would ``be mad,'' since he hated nonsense of this sort, and had gone to talk with the elders when the game began. At this moment he was standing before the fire, listening to a discussion between his uncles and his father, looking as wise as a young owl, and blissfully unconscious of the plots against him.
Charlie expected that Rose would say, ``I won't!'' therefore he was rather astonished, not to say gratified, when, after a look at the victim, she laughed suddenly, and, going up to the group of gentlemen, drew her uncle Mac under the mistletoe and surprised him with a hearty kiss.
``Thank you, my dear,'' said the innocent gentleman, looking much pleased at the unexpected honour.
``Oh, come; that's not fair,'' began Charlie. But Rose cut him short by saying, as she made him a fine courtesy,
``You said `Old Mac,' and though it was very disrespectful, I did it. That was your last chance, sir, and you've lost it.''
He certainly had, for, as he spoke, Rose pulled down the mistletoe and threw it into the fire, while the boys jeered at the crestfallen Prince, and exalted quick-witted Rose to the skies.
``What's the joke?'' asked young Mac, waked out of a brown study by the laughter, in which the elders joined.
But there was a regular shout when, the matter having been explained to him, Mac took a meditative stare at Rose through his goggles, and said in a philosophical tone, ``Well, I don't think I should have minded much if she had done it.''
That tickled the lads immensely, and nothing but the appearance of a slight refection would have induced them to stop chaffing the poor Worm, who could not see anything funny in the beautiful resignation he had shown on this trying occasion.
Soon after this, the discovery of Jamie curled up in the sofa corner, as sound asleep as a dormouse, suggested the propriety of going home, and a general move was made.
They were all standing about the hall lingering over the good-nights, when the sound of a voice softly singing ``Sweet Home,'' made them pause and listen. It was Phebe, poor little Phebe, who never had a home, never knew the love of father or mother, brother or sister; who stood all alone in the wide world, yet was not sad nor afraid, but took her bits of happiness gratefully, and sung over her work without a thought of discontent.
I fancy the happy family standing there together remembered this and felt the beauty of it, for when the solitary voice came to the burden of its song, other voices took it up and finished it so sweetly, that the old house seemed to echo the word ``Home'' in the ears of both the orphan girls, who had just spent their first Christmas under its hospitable roof.
\gutchapter{Chapter 21---A Scare}
``Brother Alec, you surely don't mean to allow that child to go out such a bitter cold day as this,'' said Mrs. Myra, looking into the study, where the Doctor sat reading his paper, one February morning.
``Why not? If a delicate invalid like yourself can bear it, surely my hearty girl can, especially as she is dressed for cold weather,'' answered Dr. Alec with provoking confidence.
``But you have no idea how sharp the wind is. I am chilled to the very marrow of my bones,'' answered Aunt Myra, chafing the end of her purple nose with her sombre glove.
``I don't doubt it, ma'am, if you will wear crape and silk instead of fur and flannel. Rosy goes out in all weathers, and will be none the worse for an hour's brisk skating.''
``Well, I warn you that you are trifling with the child's health, and depending too much on the seeming improvement she has made this year. She is a delicate creature for all that, and will drop away suddenly at the first serious attack, as her poor mother did,'' croaked Aunt Myra, with a despondent wag of the big bonnet.
``I'll risk it,'' answered Dr. Alec, knitting his brows, as he always did when any allusion was made to that other Rose.
``Mark my words, you will repent it,'' and with that awful prophecy, Aunt Myra departed like a black shadow.
Now it must be confessed that among the Doctor's failings and he had his share was a very masculine dislike of advice which was thrust upon him unasked. He always listened with respect to the great-aunts, and often consulted Mrs. Jessie; but the other three ladies tried his patience sorely, by constant warnings, complaints and counsels. Aunt Myra was an especial trial, and he always turned contrary the moment she began to talk. He could not help it, and often laughed about it with comic frankness. Here now was a sample of it, for he had just been thinking that Rose had better defer her run till the wind went down and the sun was warmer. But Aunt Myra spoke, and he could not resist the temptation to make light of her advice, and let Rose brave the cold. He had no fear of its harming her, for she went out every day, and it was a great satisfaction to him to see her run down the avenue a minute afterward, with her skates on her arm, looking like a rosy-faced Esquimaux in her seal-skin suit, as she smiled at Aunt Myra stalking along as solemnly as a crow.
``I hope the child won't stay out long, for this wind is enough to chill the marrow in younger bones than Myra's,'' thought Dr. Alec, half an hour later, as he drove toward the city to see the few patients he had consented to take for old acquaintance' sake.
The thought returned several times that morning, for it was truly a bitter day, and, in spite of his bear-skin coat, the Doctor shivered. But he had great faith in Rose's good sense, and it never occurred to him that she was making a little Casabianca of herself, with the difference of freezing instead of burning at her post.
You see, Mac had made an appointment to meet her at a certain spot, and have a grand skating bout as soon as the few lessons he was allowed were over. She had promised to wait for him, and did so with a faithfulness that cost her dear, because Mac forgot his appointment when the lessons were done, and became absorbed in a chemical experiment, till a general combustion of gases drove him out of his laboratory. Then he suddenly remembered Rose, and would gladly have hurried away to her, but his mother forbade his going out, for the sharp wind would hurt his eyes.
``She will wait and wait, mother, for she always keeps her word, and I told her to hold on till I came,'' explained Mac, with visions of a shivering little figure watching on the windy hill-top.
``Of course, your uncle won't let her go out such a day as this. If he does, she will have the sense to come here for you, or to go home again when you don't appear,'' said Aunt Jane, returning to her ``Watts on the Mind.''
``I wish Steve would just cut up and see if she's there, since I can't go,'' began Mac, anxiously.
``Steve won't stir a peg, thank you. He's got his own toes to thaw out, and wants his dinner,'' answered Dandy, just in from school, and wrestling impatiently with his boots.
So Mac resigned himself, and Rose waited dutifully till dinner-time assured her that her waiting was in vain. She had done her best to keep warm, had skated till she was tired and hot, then stood watching others till she was chilled; tried to get up a glow again by trotting up and down the road, but failed to do so, and finally cuddled disconsolately under a pine-tree to wait and watch. When she at length started for home, she was benumbed with cold, and could hardly make her way against the wind that buffeted the frost-bitten rose most unmercifully.
Dr. Alec was basking in the warmth of the study fire, after his drive, when the sound of a stifled sob made him hurry to the door and look anxiously into the hall. Rose lay in a shivering bunch near the register, with her things half off, wringing her hands, and trying not to cry with the pain returning warmth brought to her half-frozen fingers.
``My darling, what is it?'' and Uncle Alec had her in his arms in a minute.
``Mac didn't come I can't get warm the fire makes me ache!'' and with a long shiver Rose burst out crying, while her teeth chattered, and her poor little nose was so blue, it made one's heart ache to see it.
In less time than it takes to tell it, Dr. Alec had her on the sofa rolled up in the bear-skin coat, with Phebe rubbing her cold feet while he rubbed the aching hands, and Aunt Plenty made a comfortable hot drink, and Aunt Peace sent down her own foot-warmer and embroidered blanket ``for the dear.''
Full of remorseful tenderness, Uncle Alec worked over his new patient till she declared she was all right again. He would not let her get up to dinner, but fed her himself, and then forgot his own while he sat watching her fall into a drowse, for Aunt Plenty's cordial made her sleepy.
She lay so several hours for the drowse deepened into a heavy sleep, and Uncle Alec, still at his post, saw with growing anxiety that a feverish colour began to burn in her cheeks, that her breathing was quick and uneven, and now and then she gave a little moan, as if in pain. Suddenly she woke up with a start, and seeing Aunt Plenty bending over her, put out her arms like a sick child, saying wearily,
``Please, could I go to bed?''
``The best place for you, deary. Take her right up, Alec; I've got the hot water ready, and after a nice bath, she shall have a cup of my sage tea, and be rolled up in blankets to sleep off her cold,'' answered the old lady, cheerily, as she bustled away to give orders.
``Are you in pain, darling?'' asked Uncle Alec, as he carried her up.
``My side aches when I breathe, and I feel stiff and queer; but it isn't bad, so don't be troubled, uncle,'' whispered Rose, with a little hot hand against his cheek.
But the poor doctor did look troubled, and had cause to do so, for just then Rose tried to laugh at Dolly charging into the room with a warming-pan, but could not, for the sharp pain took her breath away and made her cry out.
``Pleurisy,'' sighed Aunt Plenty, from the depths of the bath-tub.
``Pewmonia!'' groaned Dolly, burrowing among the bedclothes with the long-handled pan, as if bent on fishing up that treacherous disease.
``Oh, is it bad?'' asked Phebe, nearly dropping a pail of hot water in her dismay, for she knew nothing of sickness, and Dolly's suggestion had a peculiarly dreadful sound to her.
``Hush!'' ordered the Doctor, in a tone that silenced all further predictions, and made everyone work with a will.
``Make her as comfortable as you can, and when she is in her little bed I'll come and say good-night,'' he added, when the bath was ready and the blankets browning nicely before the fire.
Then he went away to talk quite cheerfully to Aunt Peace about its being ``only a chill''; after which he tramped up and down the hall, pulling his beard and knitting his brows, sure signs of great inward perturbation.
``I thought it would be too good luck to get through the year without a downfall. Confound my perversity! Why couldn't I take Myra's advice and keep Rose at home. It's not fair that the poor child should suffer for my sinful over-confidence. She shall not suffer for it! Pneumonia, indeed! I defy it,'' and he shook his fist in the ugly face of an Indian idol that happened to be before him, as if that particularly hideous god had some spite against his own little goddess.
In spite of his defiance his heart sunk when he saw Rose again, for the pain was worse, and the bath and blankets, the warming-pan and piping-hot sage tea, were all in vain. For several hours there was no rest for the poor child, and all manner of gloomy forebodings haunted the minds of those who hovered about her with faces full of the tenderest anxiety.
In the midst of the worst paroxysm Charlie came to leave a message from his mother, and was met by Phebe coming despondently downstairs with a mustard plaster that had brought no relief.
``What the dickens is the matter? You look as dismal as a tombstone,'' he said, as she held up her hand to stop his lively whistling.
``Miss Rose is dreadful sick.''
``The deuce she is!''
``Don't swear, Mr. Charlie; she really is, and it's Mr. Mac's fault,'' and Phebe told the sad tale in a few sharp words, for she felt at war with the entire race of boys at that moment.
``I'll give it to him, make your mind easy about that,'' said Charlie, with an ominous doubling up of his fist. ``But Rose isn't dangerously ill, is she?'' he added anxiously, as Aunt Plenty was seen to trot across the upper hall, shaking a bottle violently as she went.
``Oh, but she is though. The Doctor don't say much, but he don't call it a `chill' any more. It's `pleurisy' now, and I'm so afraid it will be pewmonia to-morrow,'' answered Phebe, with a despairing glance at the plaster.
Charlie exploded into a stifled laugh at the new pronunciation of pneumonia, to Phebe's great indignation.
``How can you have the heart to do it, and she in such horrid pain? Hark to that, and then laugh if you darst,'' she said with a tragic gesture, and her black eyes full of fire.
Charlie listened and heard little moans that went to his heart and made his face as sober as Phebe's. ``O uncle, please stop the pain, and let me rest a minute! Don't tell the boys I wasn't brave. I try to bear it, but it's so sharp I can't help crying.''
Neither could Charlie, when he heard the broken voice say that; but, boy-like, he wouldn't own it, and said pettishly, as he rubbed his sleeve across his eyes,
``Don't hold that confounded thing right under my nose; the mustard makes my eyes smart.''
``Don't see how it can, when it hasn't any more strength in it than meal. The Doctor said so, and I'm going to get some better,'' began Phebe, not a bit ashamed of the great tears that were bedewing the condemned plaster.
``I'll go!'' and Charlie was off like a shot, glad of an excuse to get out of sight for a few minutes.
When he came back all inconvenient emotion had been disposed of, and, having delivered a box of the hottest mustard procurable for money, he departed to ``blow up'' Mac, that being his next duty in his opinion. He did it so energetically and thoroughly that the poor Worm was cast into the depths of remorseful despair, and went to bed that evening feeling that he was an outcast from among men, and bore the mark of Cain upon his brow.
Thanks to the skill of the Doctor, and the devotion of his helpers, Rose grew easier about midnight, and all hoped that the worst was over. Phebe was making tea by the study fire, for the Doctor had forgotten to eat and drink since Rose was ill, and Aunt Plenty insisted on his having a ``good cordial dish of tea'' after his exertions. A tap on the window startled Phebe, and, looking up, she saw a face peering in. She was not afraid, for a second look showed her that it was neither ghost nor burglar, but Mac, looking pale and wild in the wintry moonlight.
``Come and let a fellow in,'' he said in a low tone, and when he stood in the hall he clutched Phebe's arm, whispering gruffly, ``How is Rose?''
``Thanks be to goodness, she's better,'' answered Phebe, with a smile that was like broad sunshine to the poor lad's anxious heart.
``And she will be all right again to-morrow?''
``Oh, dear no! Dolly says she's sure to have rheumatic fever, if she don't have noo-monia!'' answered Phebe, careful to pronounce the word rightly this time.
Down went Mac's face, and remorse began to gnaw at him again as he gave a great sigh and said doubtfully,
``I suppose I couldn't see her?''
``Of course not at this time of night, when we want her to go to sleep!''
Mac opened his mouth to say something more, when a sneeze came upon him unawares, and a loud ``Ah rash hoo!'' awoke the echoes of the quiet house.
``Why didn't you stop it?'' said Phebe reproachfully. ``I dare say you've waked her up.''
``Didn't know it was coming. Just my luck!'' groaned Mac, turning to go before his unfortunate presence did more harm.
But a voice from the stair-head called softly, ``Mac, come up; Rose wants to see you.''
Up he went, and found his uncle waiting for him.
``What brings you here at this hour, my boy?'' asked the Doctor in a whisper.
``Charlie said it was all my fault, and if she died I'd killed her. I couldn't sleep, so I came to see how she was, and no one knows it but Steve,'' he said with such a troubled face and voice that the Doctor had not the heart to blame him.
Before he could say anything more a feeble voice called ``Mac!'' and with a hasty ``Stay a minute just to please her, and then slip away, for I want her to sleep,'' the Doctor led him into the room.
The face on the pillow looked very pale and childish, and the smile that welcomed Mac was very faint, for Rose was spent with pain, yet could not rest till she had said a word of comfort to her cousin.
``I knew your funny sneeze, and I guessed that you came to see how I did, though it is very late. Don't be worried, I'm better now, and it is my fault I was ill, not yours; for I needn't have been so silly as to wait in the cold just because I said I would.''
Mac hastened to explain, to load himself with reproaches, and to beg her not to die on any account, for Charlie's lecture had made a deep impression on the poor boy's mind.
``I didn't know there was any danger of my dying,'' and Rose looked up at him with a solemn expression in her great eyes.
``Oh, I hope not; but people do sometimes go suddenly, you know, and I couldn't rest till I'd asked you to forgive me,'' faltered Mac, thinking that Rose looked very like an angel already, with the golden hair loose on the pillow, and the meekness of suffering on her little white face.
``I don't think I shall die; uncle won't let me; but if I do, remember I forgave you.''
She looked at him with a tender light in her eyes, and, seeing how pathetic his dumb grief was, she added softly, drawing his head down, ``I wouldn't kiss you under the mistletoe, but I will now, for I want you to be sure I do forgive and love you just the same.''
That quite upset poor Mac; he could only murmur his thanks and get out of the room as fast as possible, to grope his way to the couch at the far end of the hall, and lie there till he fell asleep, worn out with trying not to ``make a baby'' of himself.
\gutchapter{Chapter 22---Something to do}
Whatever danger there might have been from the effects of that sudden chill, it was soon over, though, of course, Aunt Myra refused to believe it, and Dr. Alec cherished his girl with redoubled vigilance and tenderness for months afterward. Rose quite enjoyed being sick, because as soon as the pain ended the fun began, and for a week or two she led the life of a little princess secluded in the Bower, while every one served, amused, and watched over her in the most delightful manner. But the doctor was called away to see an old friend, who was dangerously ill, and then Rose felt like a young bird deprived of its mother's sheltering wing; especially on one afternoon when the aunts were taking their naps, and the house was very still within while snow fell softly without.
``I'll go and hunt up Phebe, she is always nice and busy, and likes to have me help her. If Dolly is out of the way we can make caramels and surprise the boys when they come,'' Rose said to herself, as she threw down her book and felt ready for society of some sort.
She took the precaution to peep through the slide before she entered the kitchen, for Dolly allowed no messing when she was round. But the coast was clear, and no one but Phebe appeared, sitting at the table with her head on her arms apparently asleep. Rose was just about to wake her with a ``Boo!'' when she lifted her head, dried her wet eyes with her blue apron, and fell to work with a resolute face on something she was evidently much interested in. Rose could not make out what it was, and her curiosity was greatly excited, for Phebe was writing with a sputtering pen on some bits of brown paper, apparently copying something from a little book.
``I must know what the dear thing is about, and why she cried, and then set her lips tight and went to work with all her might,'' thought Rose, forgetting all about the caramels, and, going round to the door, she entered the kitchen, saying pleasantly,
``Phebe, I want something to do. Can't you let me help you about anything, or shall I be in the way?''
``Oh, dear no, miss; I always love to have you round when things are tidy. What would you like to do?'' answered Phebe, opening a drawer as if about to sweep her own affairs out of sight; but Rose stopped her, exclaiming, like a curious child,
``Let me see! What is it? I won't tell if you'd rather not have Dolly know.''
``I'm only trying to study a bit; but I'm so stupid I don't get on much,'' answered the girl reluctantly, permitting her little mistress to examine the poor contrivances she was trying to work with.
A broken slate that had blown off the roof, an inch or two of pencil, an old almanac for a reader, several bits of brown or yellow paper ironed smoothly and sewn together for a copy-book, and the copies sundry receipts written in Aunt Plenty's neat hand. These, with a small bottle of ink and a rusty pen, made up Phebe's outfit, and it was little wonder that she did not ``get on'' in spite of the patient persistence that dried the desponding tears and drove along the sputtering pen with a will.
``You may laugh if you want to, Miss Rose, I know my things are queer, and that's why I hide 'em; but I don't mind since you've found me out, and I ain't a bit ashamed except of being so backward at my age,'' said Phebe humbly, though her cheeks grew redder as she washed out some crooked capitals with a tear or two not yet dried upon the slate.
``Laugh at you! I feel more like crying to think what a selfish girl I am, to have loads of books and things and never remember to give you some. Why didn't you come and ask me, and not go struggling along alone in this way? It was very wrong of you, Phebe, and I'll never forgive you if you do so again,'' answered Rose, with one hand on Phebe's shoulder, while the other gently turned the leaves of the poor little copy-book.
``I didn't like to ask for anything more when you are so good to me all the time, miss, dear,'' began Phebe, looking up with grateful eyes.
``O you proud thing! just as if it wasn't fun to give away, and I had the best of it. Now, see here, I've got a plan and you mustn't say no, or I shall scold. I want something to do, and I'm going to teach you all I know; it won't take long,'' and Rose laughed as she put her arm around Phebe's neck, and patted the smooth dark head with the kind little hand that so loved to give.
``It would be just heavenly!'' and Phebe's face shone at the mere idea; but fell again as she added wistfully, ``Only I'm afraid I ought not to let you do it, Miss Rose. It will take time, and maybe the Doctor wouldn't like it.''
``He didn't want me to study much, but he never said a word about teaching, and I don't believe he will mind a bit. Anyway, we can try it till he comes, so pack up your things and go right to my room and we'll begin this very day; I'd truly like to do it, and we'll have nice times, see if we don't!'' cried Rose eagerly.
It was a pretty sight to see Phebe bundle her humble outfit into her apron, and spring up as if the desire of her heart had suddenly been made a happy fact to her; it was a still prettier sight to see Rose run gaily on before, smiling like a good fairy as she beckoned to the other, singing as she went,
``The way into my parlour is up a winding stair,
And many are the curious things I'll show you when you're there.
Will you, will you walk in, Phebe dear?''
``Oh, won't I!'' answered Phebe fervently, adding, as they entered the Bower, ``You are the dearest spider that ever was, and I'm the happiest fly.''
``I'm going to be very strict, so sit down in that chair and don't say a word till school is ready to open,'' ordered Rose, delighted with the prospect of such a useful and pleasant ``something to do.''
So Phebe sat demurely in her place while her new teacher laid forth books and slates, a pretty inkstand and a little globe; hastily tore a bit off her big sponge, sharpened pencils with more energy than skill, and when all was ready gave a prance of satisfaction that set the pupil laughing.
``Now the school is open, and I shall hear you read, so that I may know in which class to put you, Miss Moore,'' began Rose with great dignity, as she laid a book before her scholar, and sat down in the easy chair with a long rule in her hand.
Phebe did pretty well, only tripping now and then over a hard word, and pronouncing identical ``identickle,'' in a sober way that tickled Rose, though never a smile betrayed her. The spelling lesson which followed was rather discouraging; Phebe's ideas of geography were very vague, and grammar was nowhere, though the pupil protested that she tried so hard to ``talk nice like educated folks'' that Dolly called her ``a stuck-up piece who didn't know her place.''
``Dolly's an old goose, so don't you mind her, for she will say `nater,' `vittles,' and `doos' as long as she lives, and insist that they are right. You do talk very nicely, Phebe, I've observed it, and grammar will help you, and show you some things are right and others ain't are not, I mean,'' added Rose, correcting herself, and feeling that she must mind her own parts of speech if she was to serve as an example for Phebe.
When the arithmetic came, the little teacher was surprised to find her scholar quicker in some things than herself, for Phebe had worked away at the columns in the butcher's and baker's books till she could add so quickly and correctly that Rose was amazed, and felt that in this branch the pupil would soon excel the teacher if she kept on at the same pace. Her praise cheered Phebe immensely, and they went bravely on, both getting so interested that time flew unheeded till Aunt Plenty appeared, exclaiming, as she stared at the two heads bent over one slate,
``Bless my heart, what is going on now?''
``School, aunty. I'm teaching Phebe, and it's great fun!'' cried Rose, looking up with a bright face.
But Phebe's was brighter, though she added with a wistful look,
``Maybe I ought to have asked leave first; only when Miss Rose proposed this, I was so happy I forgot to. Shall I stop, ma'am?''
``Of course not, child; I'm glad to see you fond of your book, and to find Rose helping you along. My blessed mother used to sit at work with her maids about her, teaching them many a useful thing in the good old fashion that's gone by now. Only don't neglect your work, dear, or let the books interfere with the duties.''
As Aunt Plenty spoke, with her kind old face beaming approvingly upon the girls, Phebe glanced at the clock, saw that it pointed to five, knew that Dolly would soon be down, expecting to find preparations for supper under way, and, hastily dropping her pencil, she jumped up, saying,
``Please, can I go? I'll clear up after I've done my chores.''
``School is dismissed,'' answered Rose, and with a grateful ``Thank you, heaps and heaps!'' Phebe ran away singing the multiplication table as she set the tea ditto.
That was the way it began, and for a week the class of one went on with great pleasure and profit to all concerned; for the pupil proved a bright one, and came to her lessons as to a feast, while the young teacher did her best to be worthy the high opinion held of her, for Phebe firmly believed that Miss Rose knew everything in the way of learning.
Of course the lads found out what was going on, and chaffed the girls about the ``Seminary,'' as they called the new enterprise; but they thought it a good thing on the whole, kindly offered to give lessons in Greek and Latin gratis, and decided among themselves that ``Rose was a little trump to give the Phebe-bird such a capital boost.''
Rose herself had some doubts as to how it would strike her uncle, and concocted a wheedlesome speech which should at once convince him that it was the most useful, wholesome, and delightful plan ever devised. But she got no chance to deliver her address, for Dr. Alec came upon her so unexpectedly that it went out of her head entirely. She was sitting on the floor in the library, poring over a big book laid open in her lap, and knew nothing of the long-desired arrival till two large, warm hands met under her chin and gently turned her head back, so that someone could kiss her heartily on either cheek, while a fatherly voice said, half reproachfully, ``Why is my girl brooding over a dusty Encyclopedia when she ought to be running to meet the old gentleman who couldn't get on another minute without her?''
``O uncle! I'm so glad! and so sorry! Why didn't you let us know what time you'd be here, or call out the minute you came? Haven't I been home-sick for you? and now I'm so happy to have you back I could hug your dear old curly head off,'' cried Rose, as the Encyclopedia went down with a bang, and she up with a spring that carried her into Dr. Alec's arms, to be kept there in the sort of embrace a man gives to the dearest creature the world holds for him.
Presently he was in his easy chair with Rose upon his knee smiling up in his face and talking as fast as her tongue could go, while he watched her with an expression of supreme content, as he stroked the smooth round cheek, or held the little hand in his, rejoicing to see how rosy was the one, how plump and strong the other.
``Have you had a good time? Did you save the poor lady? Aren't you glad to be home again with your girl to torment you?''
``Yes, to all those questions. Now tell me what you've been at, little sinner? Aunty Plen says you want to consult me about some new and remarkable project which you have dared to start in my absence.''
``She didn't tell you, I hope?''
``Not a word more expect that you were rather doubtful how I'd take it, and so wanted to `fess' yourself and get round me as you always try to do, though you don't often succeed. Now, then, own up and take the consequences.''
So Rose told about her school in her pretty, earnest way, dwelling on Phebe's hunger for knowledge, and the delight it was to help her, adding, with a wise nod,
``And it helps me too, uncle, for she is so quick and eager I have to do my best or she will get ahead of me in some things. To-day, now, she had the word `cotton' in a lesson and asked all about it, and I was ashamed to find I really knew so little that I could only say that it was a plant that grew down South in a kind of a pod, and was made into cloth. That's what I was reading up when you came, and to-morrow I shall tell her all about it, and indigo too. So you see it teaches me also, and is as good as a general review of what I've learned, in a pleasanter way than going over it alone.''
``You artful little baggage! that's the way you expect to get round me, is it? That's not studying, I suppose?''
``No, sir, it's teaching; and please, I like it much better than having a good time by myself. Besides, you know, I adopted Phebe and promised to be a sister to her, so I am bound to keep my word, am I not?'' answered Rose, looking both anxious and resolute as she waited for her sentence.
Dr. Alec was evidently already won, for Rose had described the old slate and brown paper copy-book with pathetic effect, and the excellent man had not only decided to send Phebe to school long before the story was done, but reproached himself for forgetting his duty to one little girl in his love for another. So when Rose tried to look meek and failed utterly, he laughed and pinched her cheek, and answered in that genial way which adds such warmth and grace to any favour,
``I haven't the slightest objection in the world. In fact, I was beginning to think I might let you go at your books again, moderately, since you are so well; and this is an excellent way to try your powers. Phebe is a brave, bright lass, and shall have a fair chance in the world, if we can give it to her, so that if she ever finds her friends they need not be ashamed of her.''
``I think she has found some already,'' began Rose eagerly.
``Hey? what? has anyone turned up since I've been gone?'' asked Dr. Alec quickly, for it was a firm belief in the family that Phebe would prove to be ``somebody'' sooner or later.
``No, her best friend turned up when you came home, uncle,'' answered Rose with an approving pat, adding gratefully, ``I can't half thank you for being so good to my girl, but she will, because I know she is going to make a woman to be proud of, she's so strong and true, and loving.''
``Bless your dear heart, I haven't begun to do anything yet, more shame to me! But I'm going at it now, and as soon as she gets on a bit, she shall go to school as long as she likes. How will that do for a beginning?''
``It will be `just heavenly,' as Phebe says, for it is the wish of her life to `get lots of schooling,' and she will be too happy when I tell her. May I, please? it will be so lovely to see the dear thing open her big eyes and clap her hands at the splendid news.''
``No one shall have a finger in this nice little pie; you shall do it all yourself, only don't go too fast, or make too many castles in the air, my dear; for time and patience must go into this pie of ours if it is to turn out well.''
``Yes, uncle, only when it is opened won't `the birds begin to sing?"' laughed Rose, taking a turn about the room as a vent for the joyful emotions that made her eyes shine. All of a sudden she stopped and asked soberly,
``If Phebe goes to school who will do her work? I'm willing, if I can.''
``Come here and I'll tell you a secret. Dolly's `bones' are getting so troublesome, and her dear old temper so bad, that the aunts have decided to pension her off and let her go and live with her daughter, who has married very well. I saw her this week, and she'd like to have her mother come, so in the spring we shall have a grand change, and get a new cook and chamber-girl if any can be found to suit our honoured relatives.''
``Oh, me! how can I ever get on without Phebe? Couldn't she stay, just so I could see her? I'd pay her board rather than have her go, I'm so fond of her.''
How Dr. Alec laughed at that proposal, and how satisfied Rose was when he explained that Phebe was still to be her maid, with no duties except such as she could easily perform between school-hours.
``She is a proud creature, for all her humble ways, and even from us would not take a favour if she did not earn it somewhere. So this arrangement makes it all square and comfortable, you see, and she will pay for the schooling by curling these goldilocks a dozen times a day if you let her.''
``Your plans are always so wise and kind! That's why they work so well, I suppose, and why people let you do what you like with them. I really don't see how other girls get along without an Uncle Alec!'' answered Rose, with a sigh of pity for those who had missed so great a blessing.
When Phebe was told the splendid news, she did not ``stand on her head with rapture,'' as Charlie prophesied she would, but took it quietly, because it was such a happy thing she had no words ``big and beautiful enough to thank them in,'' she said; but every hour of her day was brightened by this granted wish, and dedicated to the service of those who gave it.
Her heart was so full of content that if overflowed in music, and the sweet voice singing all about the house gave thanks so blithely that no other words were needed. Her willing feet were never tired of taking steps for those who had smoothed her way; her skilful hands were always busy in some labour of love for them, and on the face fast growing in comeliness there was an almost womanly expression of devotion, which proved how well Phebe had already learned one of life's great lessons gratitude.
\gutchapter{Chapter 23---Peace-Making}
``Steve, I want you to tell me something,'' said Rose to Dandy, who was making faces at himself in the glass, while he waited for an answer to the note he brought from his mother to Aunt Plenty.
``P'raps I will, and p'raps I won't. What is it?''
``Haven't Arch and Charlie quarrelled?''
``Dare say; we fellows are always having little rows, you know. I do believe a sty is coming on my star-board eye,'' and Steve affected to be absorbed in a survey of his yellow lashes.
``No, that won't do; I want to know all about it; for I'm sure something more serious than a `little row' is the matter. Come, please tell me, Stenie, there's a dear.''
``Botheration! you don't want me to turn telltale, do you?'' growled Steve, pulling his top-knot, as he always did when perplexed.
``Yes, I do,'' was Rose's decided answer for she saw from his manner that she was right, and determined to have the secret out of him if coaxing would do it. ``I don't wish you to tell things to everyone, of course, but to me you may, and you must, because I have a right to know. You boys need somebody to look after you, and I'm going to do it, for girls are nice peacemakers, and know how to manage people. Uncle said so, and he is never wrong.''
Steve was about to indulge in a derisive hoot at the idea of her looking after them, but a sudden thought restrained him, and suggested a way in which he could satisfy Rose, and better himself at the same time.
``What will you give me if I'll tell you every bit about it?'' he asked, with a sudden red in his cheeks and an uneasy look in his eyes, for he was half ashamed of the proposition.
``What do you want?'' and Rose looked up rather surprised at his question.
``I'd like to borrow some money. I shouldn't think of asking you, only Mac never has a cent. since he's set up his old chemical shop, where he'll blow himself to bits some day, and you and uncle will have the fun of putting him together again,'' and Steve tried to look as if the idea amused him.
``I'll lend it to you with pleasure, so tell away,'' said Rose, bound to get at the secret.
Evidently much relieved by the promise, Steve set his top-knot cheerfully erect again, and briefly stated the case.
``As you say, it's all right to tell you, but don't let the boys know I blabbed, or Prince will take my head off. You see, Archie don't like some of the fellows Charlie goes with, and cuts 'em. That makes Prince mad, and he holds on just to plague Arch, so they don't speak to one another, if they can help it, and that's the row.''
``Are those boys bad?'' asked Rose, anxiously.
``Guess not, only rather wild. They are older than our fellows, but they like Prince, he's such a jolly boy; sings so well, dances jigs and breakdowns, you know, and plays any game that's going. He beat Morse at billiards, and that's something to brag of, for Morse thinks he knows everything. I saw the match, and it was great fun!''
Steve got quite excited over the prowess of Charlie, whom he admired immensely, and tried to imitate. Rose did not know half the danger of such gifts and tastes as Charlie's, but felt instinctively that something must be wrong if Archie disapproved.
``If Prince likes any billiard-playing boy better than Archie, I don't think much of his sense,'' she said severely.
``Of course he doesn't; but, you see, Charlie and Arch are both as proud as they can be, and won't give in. I suppose Arch is right, but I don't blame Charlie a bit for liking to be with the others sometimes, they are such a jolly set,'' and Steve shook his head morally, even while his eye twinkled over the memory of some of the exploits of the ``jolly set.''
``Oh, dear me!'' sighed Rose, ``I don't see what I can do about it, but I wish the boys would make up, for Prince can't come to any harm with Archie, he's so good and sensible.''
``That's the trouble; Arch preaches, and Prince won't stand it. He told Arch he was a prig and a parson, and Arch told him he wasn't a gentleman. My boots! weren't they both mad, though! I thought for a minute they'd pitch into one another and have it out. Wish they had, and not gone stalking round stiff and glum ever since. Mac and I settle our rows with a bat or so over the head, and then we are all right.''
Rose couldn't help laughing as Steve sparred away at a fat sofa-pillow, to illustrate his meaning; and, having given it several scientific whacks, he pulled down his cuffs and smiled upon her with benign pity for her feminine ignorance of this summary way of settling a quarrel.
``What droll things boys are!'' she said, with a mixture of admiration and perplexity in her face, which Steve accepted as a compliment to his sex.
``We're a pretty clever invention, miss, and you can't get on without us,'' he answered, with his nose in the air. Then, taking a sudden plunge into business, he added, ``How about that bit of money you were going to lend me? I've told, now you pay up.''
``Of course I will! How much do you want?'' and Rose pulled out her purse.
``Could you spare five dollars? I want to pay a little debt of honour that is rather pressing,'' and Steve put on a mannish air that was comical to see.
``Aren't all debts honourable?'' asked innocent Rose.
``Yes, of course; but this is a bet I made, and it ought to be settled up at once,'' began Steve, finding it awkward to explain.
``Oh, don't bet, it's not right, and I know your father wouldn't like it. Promise you won't do so again; please promise!'' and Rose held fast the hand into which she had just put the money.
``Well, I won't. It's worried me a good deal, but I was joked into it. Much obliged, cousin, I'm all right now,'' and Steve departed hastily.
Having decided to be a peace-maker, Rose waited for an opportunity, and very soon it came.
She was spending the day with Aunt Clara, who had been entertaining some young guests, and invited Rose to meet them, for she thought it high time her niece conquered her bashfulness and saw a little of society. Dinner was over, and everyone had gone. Aunt Clara was resting before going out to an evening party, and Rose was waiting for Charlie to come and take her home.
She sat alone in the elegant drawing-room, feeling particularly nice and pretty, for she had her best frock on, a pair of gold bands her aunt had just given her, and a tea-rose bud in her sash, like the beautiful Miss Van Tassel, whom everyone admired. She had spread out her little skirts to the best advantage, and, leaning back in a luxurious chair, sat admiring her own feet in new slippers with rosettes almost as big as dahlias. Presently Charlie came lounging in, looking rather sleepy and queer, Rose thought. On seeing her, however, he roused up and said with a smile that ended in a gape,
``I thought you were with mother, so I took forty winks after I got those girls off. Now, I'm at your service, Rosamunda, whenever you like.''
``You look as if your head ached. If it does, don't mind me. I'm not afraid to run home alone, it's so early,'' answered Rose, observing the flushed cheeks and heavy eyes of her cousin.
``I think I see myself letting you do it. Champagne always makes my headache, but the air will set me up.''
``Why do you drink it, then?'' asked Rose, anxiously.
``Can't help it, when I'm host. Now, don't you begin to lecture; I've had enough of Archie's old-fashioned notions, and I don't want any more.''
Charlie's tone was decidedly cross, and his whole manner so unlike his usual merry good-nature, that Rose felt crushed, and answered meekly,
``I wasn't going to lecture, only when people like other people, they can't bear to see them suffer pain.''
That brought Charlie round at once, for Rose's lips trembled a little, though she tried to hide it by smelling the flower she pulled from her sash.
``I'm a regular bear, and I beg your pardon for being so cross, Rosy,'' he said in the old frank way that was so winning.
``I wish you'd beg Archie's too, and be good friends again. You never were cross when he was your chum,'' Rose said, looking up at him as he bent toward her from the low chimney-piece, where he had been leaning his elbows.
In an instant he stood as stiff and straight as a ramrod, and the heavy eyes kindled with an angry spark as he said, in his high and mighty manner,
``You'd better not meddle with what you don't understand, cousin.''
``But I do understand, and it troubles me very much to see you so cold and stiff to one another. You always used to be together, and now you hardly speak. You are so ready to beg my pardon I don't see why you can't beg Archie's, if you are in the wrong.''
``I'm not!'' this was so short and sharp that Rose started, and Charlie added in a calmer but still very haughty tone: ``A gentleman always begs pardon when he has been rude to a lady, but one man doesn't apologize to another man who has insulted him.''
``Oh, my heart, what a pepperpot!'' thought Rose, and, hoping to make him laugh, she added slyly: ``I was not talking about men, but boys, and one of them a Prince, who ought to set a good example to his subjects.''
But Charlie would not relent, and tried to turn the subject by saying gravely, as he unfastened the little gold ring from his watch-guard,
``I've broken my word, so I want to give this back and free you from the bargain. I'm sorry, but I think it a foolish promise, and don't intend to keep it. Choose a pair of ear-rings to suit yourself, as my forfeit. You have a right to wear them now.''
``No, I can only wear one, and that is no use, for Archie will keep his word I'm sure!'' Rose was so mortified and grieved at this downfall of her hopes that she spoke sharply, and would not take the ring the deserter offered her.
He shrugged his shoulders, and threw it into her lap, trying to look cool and careless, but failing entirely, for he was ashamed of himself, and out of sorts generally. Rose wanted to cry, but pride would not let her, and, being very angry, she relieved herself by talk instead of tears. Looking pale and excited, she rose out of her chair, cast away the ring, and said in a voice that she vainly tried to keep steady,
``You are not at all the boy I thought you were, and I don't respect you one bit. I've tried to help you be good, but you won't let me, and I shall not try any more. You talk a great deal about being a gentleman, but you are not, for you've broken your word, and I can never trust you again. I don't wish you to go home with me. I'd rather have Mary. Good-night.''
And with that last dreadful blow, Rose walked out of the room, leaving Charlie as much astonished as if one of his pet pigeons had flown in his face and pecked at him. She was so seldom angry, that when her temper did get the better of her it made a deep impression on the lads, for it was generally a righteous sort of indignation at some injustice or wrong-doing, not childish passion.
Her little thunderstorm cleared off in a sob or two as she put on her things in the entry-closet, and when she emerged she looked the brighter for the shower. A hasty good-night to Aunt Clara now under the hands of the hairdresser and then she crept down to find Mary the maid. But Mary was out, so was the man, and Rose slipped away by the back-door, flattering herself that she had escaped the awkwardness of having Charlie for escort.
There she was mistaken, however, for the gate had hardly closed behind her when a well-known tramp was heard, and the Prince was beside her, saying in a tone of penitent politeness that banished Rose's wrath like magic,
``You needn't speak to me if you don't choose, but I must see you safely home, cousin.''
She turned at once, put out her hand, and answered heartily,
``I was the cross one. Please forgive me, and let's be friends again.''
Now that was better than a dozen sermons on the beauty of forgiveness, and did Charlie more good, for it showed him how sweet humility was, and proved that Rose practised as she preached.
He shook the hand warmly, then drew it through his arm and said, as if anxious to recover the good opinion with the loss of which he had been threatened,
``Look here, Rosy, I've put the ring back, and I'm going to try again. But you don't know how hard it is to stand being laughed at.''
``Yes, I do! Ariadne plagues me every time I see her, because I don't wear ear-rings after all the trouble I had getting ready for them.''
``Ah, but her twaddle isn't half as bad as the chaffing I get. It takes a deal of pluck to hold out when you are told you are tied to an apron string, and all that sort of thing,'' sighed Charlie.
``I thought you had a `deal of pluck,' as you call it. The boys all say you are the bravest of the seven,'' said Rose.
``So I am about some things, but I cannot bear to be laughed at.''
``It is hard, but if one is right won't that make it easier?''
``Not to me; it might to a pious parson like Arch.''
``Please don't call him names! I guess he has what is called moral courage, and you physical courage. Uncle explained the difference to me, and moral is the best, though often it doesn't look so,'' said Rose thoughtfully.
Charlie didn't like that, and answered quickly, ``I don't believe he'd stand it any better than I do, if he had those fellows at him.''
``Perhaps that's why he keeps out of their way, and wants you to.''
Rose had him there, and Charlie felt it, but would not give in just yet, though he was going fast, for somehow, in the dark he seemed to see things clearer than in the light, and found it very easy to be confidential when it was ``only Rose.''
``If he was my brother, now, he'd have some right to interfere,'' began Charlie, in an injured tone.
``I wish he was!'' cried Rose.
``So do I,'' answered Charlie, and then they both laughed at his inconsistency.
The laugh did them good, and when Prince spoke again, it was in a different tone pensive, not proud nor perverse.
``You see, it's hard upon me that I have no brothers and sisters. The others are better off and needn't go abroad for chums if they don't like. I am all alone, and I'd be thankful even for a little sister.''
Rose thought that very pathetic, and, overlooking the uncomplimentary word ``even'' in that last sentence, she said, with a timid sort of earnestness that conquered her cousin at once,
``Play I was a little sister. I know I'm silly, but perhaps I'm better than nothing, and I'd dearly love to do it.''
``So should I! and we will, for you are not silly, my dear, but a very sensible girl, we all think, and I'm proud to have you for a sister. There, now!'' and Charlie looked down at the curly head bobbing along beside him with real affection in his face.
Rose gave a skip of pleasure, and laid one seal-skin mitten over the other on his arm, as she said happily,
``That's so nice of you! Now, you needn't be lonely any more, and I'll try to fill Archie's place till he comes back, for I know he will, as soon as you let him.''
``Well, I don't mind telling you that while he was my mate I never missed brothers and sisters, or wanted anyone else; but since he cast me off, I'll be hanged if I don't feel as forlorn as old Crusoe before Friday turned up.''
This burst of confidence confirmed Rose in her purpose of winning Charlie's Mentor back to him, but she said no more, contented to have done so well. They parted excellent friends, and Prince went home, wondering why ``a fellow didn't mind saying things to a girl or woman which they would die before they'd own to another fellow.''
Rose also had some sage reflections upon the subject, and fell asleep thinking that there were a great many curious things in this world, and feeling that she was beginning to find out some of them.
Next day she trudged up the hill to see Archie, and having told him as much as she thought best about her talk with Charlie, begged him to forget and forgive.
``I've been thinking that perhaps I ought to, though I am in the right. I'm no end fond of Charlie, and he's the best-hearted lad alive; but he can't say No, and that will play the mischief with him, if he does not take care,'' said Archie in his grave, kind way.
``While father was home, I was very busy with him, so Prince got into a set I don't like. They try to be fast, and think it's manly, and they flatter him, and lead him on to do all sorts of things play for money, and bet, and loaf about. I hate to have him do so, and tried to stop it, but went to work the wrong way, so we got into a mess.''
``He is all ready to make up if you don't say much, for he owned to me he was wrong; but I don't think he will own it to you, in words,'' began Rose.
``I don't care for that; if he'll just drop those row-dies and come back, I'll hold my tongue and not preach. I wonder if he owes those fellows money, and so doesn't like to break off till he can pay it. I hope not, but don't dare to ask; though, perhaps, Steve knows, he's always after Prince, more's the pity,'' and Archie looked anxious.
``I think Steve does know, for he talked about debts of honour the day I gave him---'' There Rose stopped short and turned scarlet.
But Archie ordered her to ``fess,'' and had the whole story in five minutes, for none dared disobey the Chief. He completed her affliction by putting a five-dollar bill into her pocket by main force, looking both indignant and resolute as he said,
``Never do so again; but send Steve to me, if he is afraid to go to his father. Charlie had nothing to do with that; he wouldn't borrow a penny of a girl, don't think it. But that's the harm he does Steve, who adores him, and tries to be like him in all things. Don't say a word; I'll make it all right, and no one shall blame you.''
``Oh me! I always make trouble by trying to help, and then letting out the wrong thing,'' sighed Rose, much depressed by her slip of the tongue.
Archie comforted her with the novel remark that it was always best to tell the truth, and made her quite cheerful by promising to heal the breach with Charlie as soon as possible.
He kept his word so well that the very next afternoon, as Rose looked out of the window, she beheld the joyful spectacle of Archie and Prince coming up the avenue, arm-in-arm, as of old, talking away as if to make up for the unhappy silence of the past weeks.
Rose dropped her work, hurried to the door, and, opening it wide, stood there smiling down upon them so happily, that the faces of the lads brightened as they ran up the steps eager to show that all was well with them.
``Here's our little peace-maker!'' said Archie, shaking hands with vigour.
But Charlie added, with a look that made Rose very proud and happy, ``And my little sister.''
\gutchapter{Chapter 24---Which?}
``Uncle, I have discovered what girls are made for,'' said Rose, the day after the reconciliation of Archie and the Prince.
``Well, my dear, what is it?'' asked Dr. Alec, who was ``planking the deck,'' as he called his daily promenade up and down the hall.
``To take care of boys,'' answered Rose, quite beaming with satisfaction as she spoke. ``Phebe laughed when I told her, and said she thought girls had better learn to take care of themselves first. But that's because she hasn't got seven boy-cousins as I have.''
``She is right, nevertheless, Rosy, and so are you, for the two things go together, and in helping seven lads you are unconsciously doing much to improve one lass,'' said Dr. Alec, stopping to nod and smile at the bright-faced figure resting on the old bamboo chair, after a lively game of battledore and shuttlecock, in place of a run which a storm prevented.
``Am I? I'm glad of that; but really, uncle, I do feel as if I must take care of the boys, for they come to me in all sorts of troubles, and ask advice, and I like it so much. Only I don't always know what to do, and I'm going to consult you privately and then surprise them with my wisdom.''
``All right, my dear; what's the first worry? I see you have something on your little mind, so come and tell uncle.''
Rose put her arm in his, and, pacing to and fro, told him all about Charlie, asking what she could do to keep him straight, and be a real sister to him.
``Could you make up your mind to go and stay with Aunt Clara a month?'' asked the Doctor, when she ended.
``Yes, sir; but I shouldn't like it. Do you really want me to go?''
``The best cure for Charlie is a daily dose of Rose water, or Rose and water, or Rose and water; will you go and see that he takes it?'' laughed Dr. Alec.
``You mean that if I'm there and try to make it pleasant, he will stay at home and keep out of mischief?''
``Exactly.''
``But could I make it pleasant? He would want the boys.''
``No danger but he'd have the boys, for they swarm after you like bees after their queen. Haven't you found that out?''
``Aunt Plen often says they never used to be here half so much before I came, but I never thought I made the difference, it seemed so natural to have them round.''
``Little modesty doesn't know what a magnet she is; but she will find it out some day,'' and the Doctor softly stroked the cheek that had grown rosy with pleasure at the thought of being so much loved. ``Now, you see, if I move the magnet to Aunt Clara's, the lads will go there as sure as iron to steel, and Charlie will be so happy at home he won't care for these mischievous mates of his I hope,'' added the Doctor, well knowing how hard it was to wean a seventeen-year-old boy from his first taste of what is called ``seeing life,'' which, alas! often ends in seeing death.
``I'll go, uncle, right away! Aunt Clara is always asking me, and will be glad to get me. I shall have to dress and dine late, and see lots of company, and be very fashionable, but I'll try not to let it hurt me; and if I get in a puzzle or worried about anything I can run to you,'' answered Rose, good-will conquering timidity.
So it was decided, and without saying much about the real reason for this visit, Rose was transplanted to Aunt Clara's, feeling that she had a work to do, and very eager to do it well.
Dr. Alec was right about the bees, for the boys did follow their queen, and astonished Mrs. Clara by their sudden assiduity in making calls, dropping in to dinner, and getting up evening frolics. Charlie was a devoted host, and tried to show his gratitude by being very kind to his ``little sister,'' for he guessed why she came, and his heart was touched by her artless endeavours to ``help him be good.''
Rose often longed to be back in the old house with the simpler pleasures and more useful duties of the life there; but, having made up her mind, in spite of Phebe, that ``girls were made to take care of boys,'' here motherly little soul found much to enjoy in the new task she had undertaken.
It was a pretty sight to see the one earnest, sweet-faced girl among the flock of tall lads, trying to understand, to help and please them with a patient affection that worked many a small miracle unperceived. Slang, rough manners, and careless habits were banished or bettered by the presence of a little gentlewoman; and all the manly virtues cropping up were encouraged by the hearty admiration bestowed upon them by one whose good opinion all valued more than they confessed; while Rose tried to imitate the good qualities she praised in them, to put away her girlish vanities and fears, to be strong and just, and frank and brave, as well as modest, kind, and beautiful.
This trial worked so well that when the month was over, Mac and Steve demanded a visit in their turn, and Rose went, feeling that she would like to hear grim Aunt Jane say, as Aunt Clara did at parting, ``I wish I could keep you all my life, dear.''
After Mac and Steve had had their turn, Archie and Company bore her away for some weeks; and with them she was so happy, she felt as if she would like to stay for ever, if she could have Uncle Alec also.
Of course, Aunt Myra could not be neglected, and, with secret despair, Rose went to the ``Mausoleum,'' as the boys called her gloomy abode. Fortunately, she was very near home, and Dr. Alec dropped in so often that her visit was far less dismal than she expected. Between them, they actually made Aunt Myra laugh heartily more than once; and Rose did her so much good by letting in the sunshine, singing about the silent house, cooking wholesome messes, and amusing the old lady with funny little lectures on physiology, that she forgot to take her pills and gave up ``Mum's Elixir,'' because she slept so well, after the long walks and drives she was beguiled into taking, that she needed no narcotic.
So the winter flew rapidly away, and it was May before Rose was fairly settled again at home. They called her the ``Monthly Rose,'' because she had spent a month with each of the aunts, and left such pleasant memories of bloom and fragrance behind her, that all wanted the family flower back again.
Dr. Alec rejoiced greatly over his recovered treasure; but as the time drew near when his year of experiment ended, he had many a secret fear that Rose might like to make her home for the next twelve month with Aunt Jessie, or even Aunt Clara, for Charlie's sake. He said nothing, but waited with much anxiety for the day when the matter should be decided; and while he waited he did his best to finish as far as possible the task he had begun so well.
Rose was very happy now, being out nearly all day enjoying the beautiful awakening of the world, for spring came bright and early, as if anxious to do its part. The old horse-chestnuts budded round her windows, green things sprung up like magic in the garden under her hands, hardy flowers bloomed as fast as they could, the birds sang blithely overhead, and every day a chorus of pleasant voices cried, ``Good morning, cousin, isn't it jolly weather?''
No one remembered the date of the eventful conversation which resulted in the Doctor's experiment (no one but himself at least); so when the aunts were invited to tea one Saturday they came quite unsuspiciously, and were all sitting together having a social chat, when Brother Alec entered with two photographs in his hand.
``Do you remember that?'' he said, showing one to Aunt Clara, who happened to be nearest.
``Yes, indeed; it is very like her when she came. Quite her sad, unchildlike expression, and thin little face, with the big dark eyes.''
The picture was passed round, and all agreed that ``it was very like Rose a year ago.'' This point being settled, the Doctor showed the second picture, which was received with great approbation, and pronounced a ``charming likeness.''
It certainly was, and a striking contrast to the first one, for it was a blooming, smiling face, full of girlish spirit and health, with no sign of melancholy, though the soft eyes were thoughtful, and the lines about the lips betrayed a sensitive nature.
Dr. Alec set both photographs on the chimneypiece, and, falling back a step or two, surveyed them with infinite satisfaction for several minutes, then wheeled round, saying briefly, as he pointed to the two faces,
``Time is up; how do you think my experiment has succeeded, ladies?''
``Bless me, so it is!'' cried Aunt Plenty, dropping a stitch in her surprise.
``Beautifully, dear,'' answered Aunt Peace, smiling entire approval.
``She certainly has improved, but appearances are deceitful, and she had no constitution to build upon,'' croaked Aunt Myra.
``I am willing to allow that, as far as mere health goes, the experiment is a success,'' graciously observed Aunt Jane, unable to forget Rose's kindness to her Mac.
``So am I; and I'll go farther, for I really do believe Alec has done wonders for the child; she will be a beauty in two or three years,'' added Aunt Clara, feeling that she could say nothing better than that.
``I always knew he would succeed, and I'm so glad you all allow it, for he deserves more credit than you know, and more praise than he will ever get,'' cried Aunt Jessie, clapping her hands with an enthusiasm that caused Jamie's little red stocking to wave like a triumphal banner in the air.
Dr. Alec made them a splendid bow, looking much gratified, and then said soberly,
``Thank you; now the question is, shall I go on? for this is only the beginning. None of you know the hindrances I've had, the mistakes I've made, the study I've given the case, and the anxiety I've often felt. Sister Myra is right is one thing Rose is a delicate creature, quick to flourish in the sunshine, and as quick to droop without it. She has no special weakness, but inherits her mother's sensitive nature, and needs the wisest, tenderest care, to keep a very ardent little soul from wearing out a finely organised little body. I think I have found the right treatment, and; with you to help me, I believe we may build up a lovely and a noble woman, who will be a pride and comfort to us all.''
There Dr. Alec stopped to get his breath, for he had spoken very earnestly, and his voice got a little husky over the last words. A gentle murmur from the aunts seemed to encourage him, and he went on with an engaging smile, for the good man was slyly trying to win all the ladies to vote for him when the time came.
``Now, I don't wish to be selfish or arbitrary, because I am her guardian, and I shall leave Rose free to choose for herself. We all want her, and if she likes to make her home with any of you rather than with me, she shall do so. In fact, I encouraged her visits last winter, that she might see what we can all offer her, and judge where she will be happiest. Is not that the fairest way? Will you agree to abide by her choice, as I do?''
``Yes, we will,'' said all the aunts, in quite a flutter of excitement at the prospect of having Rose for a whole year.
``Good! she will be here directly, and then we will settle the question for another year. A most important year, mind you, for she has got a good start, and will blossom rapidly now if all goes well with her. So I beg of you don't undo my work, but deal very wisely and gently with my little girl, for if any harm come to her, I think it would break my heart.''
As he spoke, Dr. Alec turned his back abruptly and affected to be examining the pictures again; but the aunts understood how dear the child was to the solitary man who had loved her mother years ago, and who now found his happiness in cherishing the little Rose who was so like her. The good ladies nodded and sighed, and telegraphed to one another that none of them would complain if not chosen, or ever try to rob Brother Alec of his ``Heart's Delight,'' as the boys called Rose.
Just then a pleasant sound of happy voices came up from the garden, and smiles broke out on all serious faces. Dr. Alec turned at once, saying, as he threw back his head, ``There she is; now for it!''
The cousins had been a-Maying, and soon came flocking in laden with the spoils.
``Here is our bonny Scotch rose with all her thorns about her,'' said Dr. Alec, surveying her with unusual pride and tenderness, as she went to show Aunt Peace her basket full of early flowers, fresh leaves, and curious lichens.
``Leave your clutter in the hall, boys, and sit quietly down if you choose to stop here, for we are busy,'' said Aunt Plenty, shaking her finger at the turbulent Clan, who were bubbling over with the jollity born of spring sunshine and healthy exercise.
``Of course, we choose to stay! Wouldn't miss our Saturday high tea for anything,'' said the Chief, as he restored order among his men with a nod, a word, and an occasional shake.
``What is up? a court-martial?'' asked Charlie, looking at the assembled ladies with affected awe and real curiosity, for these faces betrayed that some interesting business was afloat.
Dr. Alec explained in a few words, which he made as brief and calm as he could; but the effect was exciting, nevertheless, for each of the lads began at once to bribe, entice, and wheedle ``our cousin'' to choose his home.
``You really ought to come to us for mother's sake, as a relish, you know, for she must be perfectly satiated with boys,'' began Archie, using the strongest argument he could think of at the moment.
``Oh, do! we'll never slam, or bounce at you or call you `fraid cat,' if you only will,'' besought Geordie and Will, distorting their countenances in the attempt to smile with overpowering sweetness.
``And I'll always wash my hands 'fore I touch you, and you shall be my dolly, 'cause Pokey's gone away, and I'll love you \textit{hard},'' cried Jamie, clinging to her with his chubby face full of affection.
``Brothers and sister ought to live together; especially when the brother needs some one to make home pleasant for him,'' added Charlie, with the wheedlesome tone and look that Rose always found so difficult to resist.
``You had her longest, and it's our turn now; Mac needs her more than you do, Prince, for she's `the light of his eyes,' he says. Come, Rose, choose us, and I'll never use the musky pomade you hate again as long as I live,'' said Steve, with his most killing air, as he offered this noble sacrifice.
Mac peered wistfully over his goggles, saying in an unusually wide-awake and earnest way, ---
``Do, cousin, then we can study chemistry together. My experiments don't blow up very often now, and the gases aren't at all bad when you get used to them.''
Rose meantime had stood quite still, with the flowers dropping from her hands as her eyes went from one eager face to another, while smiles rippled over her own at the various enticements offered her. During the laugh that followed Mac's handsome proposition, she looked at her uncle, whose eyes were fixed on her with an expression of love and longing that went to her heart.
``Ah! yes,'' she thought, ``he wants me most! I've often longed to give him something that he wished for very much, and now I can.''
So, when, at a sudden gesture from Aunt Peace, silence fell, Rose said slowly, with a pretty colour in her cheeks, and a beseeching look about the room, as if asking pardon of the boys,
``It's very hard to choose when everybody is so fond of me; therefore I think I'd better go to the one who seems to need me most.''
``No, dear, the one you love the best and will be happiest with,'' said Dr. Alec quickly, as a doleful sniff from Aunt Myra, and a murmur of ``My sainted Caroline,'' made Rose pause and look that way.
``Take time, cousin; don't be in a hurry to make up your mind, and remember, `Codlin's your friend,''' added Charlie, hopeful still.
``I don't want any time! I know who I love best, who I'm happiest with, and I choose uncle. Will he have me?'' cried Rose, in a tone that produced a sympathetic thrill among the hearers, it was so full of tender confidence and love.
If she really had any doubt, the look in Dr. Alec's face banished it without a word, as he opened wide his arms, and she ran into them, feeling that home was there.
No one spoke for a minute, but there were signs of emotion among the aunts, which warned the boys to bestir themselves before the water-works began to play. So they took hands and began to prance about uncle and niece, singing, with sudden inspiration, the nursery rhyme,
``Ring around a Rosy!''
Of course that put an end to all sentiment, and Rose emerged laughing from Dr. Alec's bosom, with the mark of a waistcoat button nicely imprinted on her left cheek. He saw it, and said with a merry kiss that half effaced it, ``This is my ewe lamb, and I have set my mark on her, so no one can steal her away.''
That tickled the boys, and they set up a shout of,
``Uncle had a little lamb!''
But Rose hushed the noise by slipping into the circle, and making them dance prettily like lads and lasses round a May-pole; while Phebe, coming in with fresh water for the flowers, began to twitter, chirp, and coo, as if all the birds of the air had come to join in the spring revel of the eight cousins.
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http://www.cs.berkeley.edu/~russell/slides/algorithms/conditional-planning-agent-algorithm.tex | berkeley.edu | CC-MAIN-2016-18 | application/x-tex | null | crawl-data/CC-MAIN-2016-18/segments/1461861700245.92/warc/CC-MAIN-20160428164140-00134-ip-10-239-7-51.ec2.internal.warc.gz | 448,465,463 | 1,035 | \code{
\func{Conditional-Planning-Agent}{{\ts}\var{percept}}{an \var{action}}
\firststatic{KB}{a knowledge base (includes action descriptions)}
\static{p}{a plan, initially $NoPlan$}
\static{t}{a counter, initially 0, indicating time}
\static{G}{a goal}
\bodysep
\noprog{Tell}(\var{KB}{\ac}\prog{Make-Percept-Sentence}({\ts}\var{percept}{\ac}\var{t}))
\setq{\var{current}}{\prog{State-Description}(\var{KB}{\ac}\var{t})}
\key{if} $\var{p} = NoPlan$ \key{then} \setq{\var{p}}{\prog{CPOP}(\var{current}{\ac}\var{G}{\ac}\var{KB})}
\key{if} $\var{p} = NoPlan$ \key{or} \var{p} is empty \key{then} \setq{\var{action}}{$NoOp$}
\key{else}
\setq{\var{action}}{\prog{First}({\ts}\var{p})}
\key{while} \prog{Conditional?}(\var{action}) \key{do}
\key{if} \noprog{Ask}(\var{KB}{\ac}\prog{Condition-Part}[\var{action}]) \key{then} \setq{\var{p}}{\prog{Append}(\prog{Then-Part}[\var{action}]{\ac}\prog{Rest}({\ts}\var{p}))}
\key{else} \setq{\var{p}}{\prog{Append}(\prog{Else-Part}[\var{action}]{\ac}\prog{Rest}({\ts}\var{p}))}
\setq{\var{action}}{\prog{First}({\ts}\var{p})}
\key{end}
\setq{\var{p}}{\prog{Rest}({\ts}\var{p})}
\noprog{Tell}(\var{KB}{\ac}\prog{Make-Action-Sentence}(\var{action}{\ac}\var{t}))
\setq{\var{t}}{\var{t} + 1}
\key{return} \var{action}
}
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