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28 pages, 2 figures
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covering manifolds, spectral gaps, residually finite groups,
min-max principle
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% mangletex (24 Nov 1995) run at 18:24 BST Monday 13 May 2002
\message{<Paul Taylor's commutative diagrams, version 3.89, March 2002>}%
%% This code runs the LaTeX \ProvidesPackage command iff it is defined.
%% included at the request of Michael Downes <mjd@ams.org> March 2002
\expandafter\ifx\csname ProvidesPackage\endcsname\relax\toks0=\expandafter{%
\fi\ProvidesPackage{diagrams}[2002/03/15 v3.89 Paul Taylor's commutative
diagrams]%%
\toks0=\bgroup}%%
%%======================================================================%
%%      TeX  macros for drawing category-theoretic diagrams             %
%%                                                                      %
%%                              Paul  Taylor                            %
%%                                                                      %
%%          www.ctan.org/tex-archive/macros/generic/diagrams/taylor     %
%%                      www.dcs.qmul.ac.uk/~pt/diagrams                 %
%%                                                                      %
%%                           PLEASE READ THE MANUAL!                    %
%%                                                                      %
%%      Please ensure that you are registered with me as a user so that %
%%      you can be informed of future releases.  Any electronic mail    %
%%      message with "commutative" or "diagram" in the subject line     %
%%      (such as your request for the package, a question about it, or  %
%%      even an otherwise blank message) automatically registers you.   %
%%                                                                      %
%%                                                                      %
%% CONTENTS:                                                            %
%%  (O) corruption-sensitive hacks    (to approx line 339)              %
%%              Arrow components & commands - starts approx line 1256   %
%% (22) auxillary macros for adjustment of components                   %
%% (23) bits of arrows  (\rhvee etc)                                    %
%% (24) arrow commands  (\rTo etc)                                      %
%% (25) miscellaneous                                                   %
%% Apart from these five sections, the rest is intended to be totally   %
%% meaningless in the undocumented version, which is approximately 1920 %
%% lines long. Please do not waste trees by printing it out.            %
%%                                                                      %
%% COPYRIGHT NOTICE:                                                    %
%%      This package may be copied and used freely for any academic     %
%%      (not commercial or military) purpose, on condition that it      %
%%      is not altered in any way, and that an acknowledgement is       %
%%      included in any published work making substantial use of it.    %
%%                                                                      %
%%      IT IS SUPPLIED "AS IS", WITHOUT WARRANTY, EXPRESS OR IMPLIED.   %
%%                                                                      %
%%      If you are doing something where mistakes cost money (or where  %
%%      success brings financial profit) then you must use commercial   %
%%      software, not this package. In any case, please remember to     %
%%      keep several backup copies of all files, and check everything   %
%%      visually before sending final copy to the publishers.           %
%%                                                                      %
%%      You may use this package as a (substantial) aid to writing an   %
%%      academic research or text book on condition that                %
%%       (i) you contact me at a suitable time to ensure that you have  %
%%           an up-to-date version (and any infelicities can be fixed), %
%%      (ii) you send me a copy of the book when it's published.        %
%%                                                                      %
%% HISTORY                                                              %
%% 3.89                                                                 %
%%      Added support for pdftex, which is recognised automatically.    %
%% 3.88 Released 1 September 2000                                       %
%%      Square hook tail: \newarrow{SquareInto}{sqhook}--->             %
%% 3.87 Released 1 September 1999                                       %
%%      This version was used for the final 1200dpi PS copy of my book  %
%%      ``Practical Foundations of Mathematics'' (Cambridge Univ Press) %
%%      see   http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations        %
%% 3.86 Released 1 September 1998                                       %
%%      New options hug and nohug in PostScript mode: [PS,nohug] uses   %
%%      PS for the arrows without rotating the labels, but the way of   %
%%      calculating the actual position of these horizontal labels on   %
%%      will remain subject to alteration for some period of time ---   %
%%      please send me examples if you feel that adjustment is needed.  %
%%                                                                      %
%%      Suppress indentation of following text (due to LaTeX change)    %
%%      Mangle \left@label and \right@label (conflict with Elsevier).   %
%%      midvshaft and snake for vertical arrows                         %
%%      New option [gap=width] (default=shortfall) to use instead of    %
%%      ~{\;} on horizontals and PS diagonals, as this caused ^ and _   %
%%      labels to be moved too far away from the shaft.                 %
%%      Added >-> and <-< heads and tails, same as >> and << but the    %
%%      shaft goes *through* the extra arrowhead.                       %
%% 3.85 Released 20 August 1997                                         %
%%      New option [crab=distance] shifts horizontals and PS diagonals  %
%%      transversally by the specified distance (positive=upward).      %
%%      New option [snake=distance] shifts midshaft horizontals and     %
%%      PS diagonals longitudinally by the specified distance.          %
%%      New option [leftflush], like [flushleft] but reckons alignment  %
%%      from multiple verticals, or from text if there's no vertical.   %
%%      Most of the history has been suppressed from the user version.  %
%% 3.83 Released 18 May 1995                                            %
%%      "dotted" option (set dot filler on maps)                        %
%%      Parallel maps (\pile) outside diagrams stretch correctly.       %
%%      Option "LaTeXeqno" uses LaTeX's equation number and style       %
%%      for "eqno";  LaTeX's \label command picks this up.              %
%%      Suppress warnings & 2nd pass errors with "silent" option.       %
%% 3.81 Second alpha release 18 July 1994                               %
%%      \overprint{text} sets text in maths and overprints it in the    %
%%      current cell, centered in the column irrespective of other stuff%
%%      "repositionpullbacks" option uses this for \SEpbk etc           %
%%      \newdiagramgrid declaration, grid option and pentagon grid.     %
%% 3.80 Alpha release for adjusted diagonals 15 July 1994.              %
%%      Introduced landscape and portrait options.                      %
%%      Diagonals adjusted to meet their endpoints, at last!!!!         %
%% 3.28 Released 30 November 1993                                       %
%%      Peter Freyd's \puncture symbol provided.                        %
%% 3.25 Released 30 January 1993                                        %
%%      LaTeX heads made default (unless \tenln undefined, when vee)    %
%%      Circle, cross, little vee, little black triangle heads.         %
%%      First-use warning when defaulted diagonal components are used.  %
%%      AMSTEX emulation - works at least when amstex not present.      %
%% 3.24 Release 7 Sept 1992 advertised to users.                        %
%%      PostScript option introduced:                                   %
%%      LaTeX, vee, curlyvee, triangle & blacktriangle heads & tails    %
%%      TPIC option introduced as an alternative to \LaTeX@make@line.   %
%%      Postscript arrows (basic code).                                 %
%%      Implemented \newarrow \newarrowhead etc.                        %
%%      Horizontal arrows outside diagram can stretch by wordspacing.   %
%%      Added < and > for labels on left and right of arrow;            %
%% 3.16 (20.7.90) as mass mailed; only have mangled version             %
%% -- all following version numbers are post-facto --                   %
%%  3   (Jan 90) stretching vertical arrows                             %
%%  2   (Sept 89) horizontals stretch to objects; "superscript" labels  %
%%  1   (1987) horizontal arrows stretch to edge of cell                %
%%  0   (1986) implementation of Knuth's TeXercise 18.46                %
%%======================================================================%

%%======================================================================%
%%                                                                      %
%%      (1) CORRUPTION-SENSITIVE HACKS                                  %
%%                                                                      %
%%======================================================================%

%%                      CORRUPTION & \catcode WARNING

%% BITNET (IBM) machines may corrupt certain important characters
%% in transmission by electronic mail:
%%        0123456789=digits, abcdefghijklmnopqrstuvwxyz=lowers,
%%        ABCDEFGHIJKLMNOPQRSTUVWXYZ=uppers, @=at (internal names),
%%        {}=curly braces (grouping), \=backslash (keywords),
%%        %=percent (comment), #=hash/sharp (argument), +=plus, -=minus,
%%        <>=angle brackets (relations \ifnum,\ifdim), ==equals,
%%        ,=comma, .=dot, :=colon, ;=semicolon,  =space
%% $=dollar (maths) is only used in the "bits of maps" section

%% The following characters are marked by a comment including the word "ASCII",
%% except in comments and messages:
%%        &=and (alignment), ~=tilde, |=vertical, []=square brackets,
%%        ^=caret (superscript), _=underline (subscript),
%%        `=grave/backquote (catcodes), '=acute/single quote (octal),
%%        "=double quote (hex), ()=round brackets,
%%        /=slash, ?=query, !=pling/bang, 

%% The \catcode's marked * are assumed for reading this file:
%%          \=0* {=1* }=2* $=3 &=4 return=5* #=6 ^=7 _=8 ignored=9*
%%          space=10* letter=11* other=12 active=13 %=14* invalid=15
%% If you want to load this package inside Stallman's "texinfo", you must do
%%% @catcode`@\=0 \catcode`\%=14 \input diagrams \catcode`\%=12 \catcode`\\=13
%% and then use @diagram @rTo @\ @enddiagram etc. (braces {} stay the same).
%% Also need @catcode`@&=4.

%%*** You *MUST NOT* use the internal commands (with names beginning \CD@)****
% will substitute \CD@A for \across@cells
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% will substitute \CD@KC for \deepen@donations
% will substitute \CD@LC for \default@args
% will substitute \CD@MC for \default@diagonal
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% will substitute \CD@EF for \if@use@TPIC@
% will substitute \CD@FF for \if@was@pair@
% will substitute \CD@GF for \info@at@end
% will substitute \CD@HF for \@in@matrix@false
% will substitute \CD@IF for \@in@matrix@true
% will substitute \CD@JF for \inner@environment
% will substitute \CD@KF for \inner@pile
% will substitute \CD@LF for \intended@breadth
% will substitute \CD@MF for \interim@shortfall
% will substitute \CD@NF for \inter@row
% will substitute \CD@OF for \iterate@
% will substitute \CD@PF for \iterate@@
% will substitute \CD@QF for \iterate@@@
% will substitute \CD@RF for \@junk
% will substitute \CD@SF for \@l
% will substitute \CD@TF for \label@eat@space
% will substitute \CD@UF for \label@egroup
% will substitute \CD@VF for \label@egroups
% will substitute \CD@WF for \label@horiz
% will substitute \CD@XF for \label@optional
% will substitute \CD@YF for \last@map@type
% will substitute \CD@ZF for \LaTeX@arrow@char
% will substitute \CD@aF for \@@LaTeX@diagonal
% will substitute \CD@bF for \LaTeX@dot@filler
% will substitute \CD@cF for \LaTeX@line@char
% will substitute \CD@dF for \LaTeX@make@line
% will substitute \CD@eF for \LaTeX@name
% will substitute \CD@fF for \LaTeX@NE
% will substitute \CD@gF for \LaTeX@SW
% will substitute \CD@hF for \left@donated
% will substitute \CD@iF for \left@label
% will substitute \CD@jF for \left@LaTeX@tail
% will substitute \CD@kF for \left@over
% will substitute \CD@lF for \left@part
% will substitute \CD@mF for \left@width
% will substitute \CD@nF for \let@names
% will substitute \CD@oF for \lift@map@part
% will substitute \CD@pF for \lift@map@parts
% will substitute \CD@qF for \line@char
% will substitute \CD@rF for \line@font
% will substitute \CD@sF for \loop@on@cells
% will substitute \CD@tF for \loop@on@maps
% will substitute \CD@uF for \loop@on@rows
% will substitute \CD@vF for \make@diag@part
% will substitute \CD@wF for \@make@double@head
% will substitute \CD@xF for \make@double@head
% will substitute \CD@yF for \@make@double@through@head
% will substitute \CD@zF for \make@end@at@line
% will substitute \CD@@G for \make@filler
% will substitute \CD@AG for \make@found@end@diagram
% will substitute \CD@BG for \make@horiz@part
% will substitute \CD@CG for \make@line
% will substitute \CD@DG for \makeline
% will substitute \CD@EG for \make@old@fill
% will substitute \CD@FG for \make@profile
% will substitute \CD@GG for \make@vert@part
% will substitute \CD@HG for \math@off
% will substitute \CD@IG for \math@off@box
% will substitute \CD@JG for \math@on
% will substitute \CD@KG for \math@on@box
% will substitute \CD@LG for \maths@display
% will substitute \CD@MG for \matrix@row@to@buffer
% will substitute \CD@NG for \matrix@to@buffer
% will substitute \CD@OG for \max@fraction
% will substitute \CD@PG for \message@at@end@doc
% will substitute \CD@QG for \@middle
% will substitute \CD@RG for \min@map@length
% will substitute \CD@SG for \misplaced@enddiagram
% will substitute \CD@TG for \missive@diagonal
% will substitute \CD@UG for \missive@outside
% will substitute \CD@VG for \missive@to@cell
% will substitute \CD@WG for \missive@to@row
% will substitute \CD@XG for \mk@new@donat
% will substitute \CD@YG for \modify@cell
% will substitute \CD@ZG for \monomial@
% will substitute \CD@aG for \more@dim
% will substitute \CD@bG for \more@donations
% will substitute \CD@cG for \@morse
% will substitute \CD@dG for \multiply@
% will substitute \CD@eG for \multiply@@
% will substitute \CD@fG for \name@prefix
% will substitute \CD@gG for \need@extra
% will substitute \CD@hG for \@need@PS@lib@false
% will substitute \CD@iG for \@need@PS@lib@true
% will substitute \CD@jG for \@new@arrow
% will substitute \CD@kG for \newarrow@mmode
% will substitute \CD@lG for \newarrow@name
% will substitute \CD@mG for \new@arrow@part
% will substitute \CD@nG for \new@diagonal
% will substitute \CD@oG for \new@five@map
% will substitute \CD@pG for \new@half@rpt@map
% will substitute \CD@qG for \new@half@rule@map
% will substitute \CD@rG for \new@help
% will substitute \CD@sG for \new@horiz@map
% will substitute \CD@tG for \new@if
% will substitute \CD@uG for \new@line
% will substitute \CD@vG for \new@map
% will substitute \CD@wG for \new@rule@map
% will substitute \CD@xG for \new@simple@map
% will substitute \CD@yG for \new@slant@map
% will substitute \CD@zG for \new@slope@map
% will substitute \CD@@H for \new@thru@map
% will substitute \CD@AH for \new@vector@map
% will substitute \CD@BH for \new@vert@map
% will substitute \CD@CH for \next
% will substitute \CD@DH for \next@opt@arg
% will substitute \CD@EH for \next@token
% will substitute \CD@FH for \no@check@vert
% will substitute \CD@GH for \no@error@context
% will substitute \CD@HH for \nohcheck
% will substitute \CD@IH for \no@make@line
% will substitute \CD@JH for \@noptcmd
% will substitute \CD@KH for \no@tenln
% will substitute \CD@LH for \numerator@
% will substitute \CD@MH for \object@profile
% will substitute \CD@NH for \obs@across
% will substitute \CD@OH for \@obs@countq
% will substitute \CD@PH for \obs@display
% will substitute \CD@QH for \obs@positional
% will substitute \CD@RH for \odef
% will substitute \CD@SH for \old@horiz@fill
% will substitute \CD@TH for \old@horiz@map
% will substitute \CD@UH for \old@horiz@part
% will substitute \CD@VH for \old@vert@fill
% will substitute \CD@WH for \old@vert@map
% will substitute \CD@XH for \one@sp
% will substitute \CD@YH for \@optcmd
% will substitute \CD@ZH for \outer@empty
% will substitute \CD@aH for \outer@environment
% will substitute \CD@bH for \output@cell
% will substitute \CD@cH for \output@diagram
% will substitute \CD@dH for \outside@diagram
% will substitute \CD@eH for \over@print
% will substitute \CD@fH for \overprint@pbk
% will substitute \CD@gH for \@p
% will substitute \CD@hH for \pair@diagram@end
% will substitute \CD@iH for \@par@enddiagram
% will substitute \CD@jH for \par@enddiagram
% will substitute \CD@kH for \par@name
% will substitute \CD@lH for \par@not@allowed
% will substitute \CD@mH for \@parse@opt@arg
% will substitute \CD@nH for \parse@opt@arg
% will substitute \CD@oH for \@parse@pair
% will substitute \CD@pH for \pbk@osh
% will substitute \CD@qH for \pbk@sh
% will substitute \CD@rH for \pbk@shsh
% will substitute \CD@sH for \pending@left
% will substitute \CD@tH for \pending@right
% will substitute \CD@uH for \@pile
% will substitute \CD@vH for \pile@cr
% will substitute \CD@wH for \pile@empty@row
% will substitute \CD@xH for \pile@junk
% will substitute \CD@yH for \pile@state
% will substitute \CD@zH for \pixel@round
% will substitute \CD@@I for \pixel@size
% will substitute \CD@AI for \polar@to@rect
% will substitute \CD@BI for \PositiveGradientfalse
% will substitute \CD@CI for \PositiveGradienttrue
% will substitute \CD@DI for \ps@prog
% will substitute \CD@EI for \push@queue
% will substitute \CD@FI for \push@stack
% will substitute \CD@GI for \put@cd@box
% will substitute \CD@HI for \put@horiz
% will substitute \CD@II for \pyth@add@term
% will substitute \CD@JI for \Pythagorean@sum
% will substitute \CD@KI for \@pyth@sum
% will substitute \CD@LI for \@r
% will substitute \CD@MI for \reassign@filler
% will substitute \CD@NI for \reassign@parts
% will substitute \CD@OI for \receive@cell@type
% will substitute \CD@PI for \reformat@cell
% will substitute \CD@QI for \reformat@complex
% will substitute \CD@RI for \reformat@empty
% will substitute \CD@SI for \reformat@HmeetV
% will substitute \CD@TI for \reformat@HonV
% will substitute \CD@UI for \@reformat@horiz
% will substitute \CD@VI for \reformat@horiz
% will substitute \CD@WI for \reformat@matrix
% will substitute \CD@XI for \reformat@object
% will substitute \CD@YI for \reformat@pile
% will substitute \CD@ZI for \reformat@row
% will substitute \CD@aI for \@reformatting@false
% will substitute \CD@bI for \@reformatting@true
% will substitute \CD@cI for \reformat@vert
% will substitute \CD@dI for \reformat@VonH
% will substitute \CD@eI for \reoutput@cell
% will substitute \CD@fI for \resend@missive
% will substitute \CD@gI for \reset@map@breadth
% will substitute \CD@hI for \right@donated
% will substitute \CD@iI for \right@label
% will substitute \CD@jI for \right@LaTeX@tail
% will substitute \CD@kI for \right@over
% will substitute \CD@lI for \right@part
% will substitute \CD@mI for \right@width
% will substitute \CD@nI for \Rotated@Horizontal
% will substitute \CD@oI for \rounded@breadth
% will substitute \CD@pI for \row@depth
% will substitute \CD@qI for \row@height
% will substitute \CD@rI for \row@to@buffer
% will substitute \CD@sI for \rule@depth
% will substitute \CD@tI for \rule@height
% will substitute \CD@uI for \save@begin@at@line
% will substitute \CD@vI for \save@cell@at@line
% will substitute \CD@wI for \save@row@no
% will substitute \CD@xI for \see@above
% will substitute \CD@yI for \send@cell@type
% will substitute \CD@zI for \separate@verticals
% will substitute \CD@@J for \set@arrowhead
% will substitute \CD@AJ for \set@axis
% will substitute \CD@BJ for \set@cmex@axis
% will substitute \CD@CJ for \set@cmex@offset@axis
% will substitute \CD@DJ for \set@col@width
% will substitute \CD@EJ for \set@currentlabel
% will substitute \CD@FJ for \set@currentlabel@to@equation
% will substitute \CD@GJ for \@set@default@part
% will substitute \CD@HJ for \set@default@part
% will substitute \CD@IJ for \set@default@parts
% will substitute \CD@JJ for \set@dot@filler
% will substitute \CD@KJ for \set@fraction
% will substitute \CD@LJ for \set@h@grid
% will substitute \CD@MJ for \set@map@parts
% will substitute \CD@NJ for \set@max@fraction
% will substitute \CD@OJ for \set@mil
% will substitute \CD@PJ for \set@new@display
% will substitute \CD@QJ for \set@offset@axis
% will substitute \CD@RJ for \set@resolution
% will substitute \CD@SJ for \set@resultion
% will substitute \CD@TJ for \@setup@horiz
% will substitute \CD@UJ for \setup@pile
% will substitute \CD@VJ for \set@v@grid
% will substitute \CD@WJ for \showboxname
% will substitute \CD@XJ for \showname
% will substitute \CD@YJ for \show@prevgraf
% will substitute \CD@ZJ for \showthename
% will substitute \CD@aJ for \@signature
% will substitute \CD@bJ for \signature@countdown
% will substitute \CD@cJ for \size@dependencies
% will substitute \CD@dJ for \skip@case
% will substitute \CD@eJ for \snake@
% will substitute \CD@fJ for \spAn
% will substitute \CD@gJ for \split@pile@row
% will substitute \CD@hJ for \square@fraction
% will substitute \CD@iJ for \start@bomb@vlist
% will substitute \CD@jJ for \stretch@PS@diagonal
% will substitute \CD@kJ for \stretch@vert
% will substitute \CD@lJ for \@sub
% will substitute \CD@mJ for \switch@label@arg
% will substitute \CD@nJ for \@szdep
% will substitute \CD@oJ for \@tail
% will substitute \CD@pJ for \test@loose@diagonals
% will substitute \CD@qJ for \test@signature
% will substitute \CD@rJ for \the@bomb
% will substitute \CD@sJ for \the@default@head
% will substitute \CD@tJ for \then
% will substitute \CD@uJ for \the@signature
% will substitute \CD@vJ for \this@map@type
% will substitute \CD@wJ for \total@donated
% will substitute \CD@xJ for \tpic@make@line
% will substitute \CD@yJ for \@u
% will substitute \CD@zJ for \unbraced@label
% will substitute \CD@@K for \use@line@char
% will substitute \CD@AK for \use@name
% will substitute \CD@BK for \use@newarrow
% will substitute \CD@CK for \@use@TPIC@false
% will substitute \CD@DK for \@use@TPIC@true
% will substitute \CD@EK for \using@ps
% will substitute \CD@FK for \@v
% will substitute \CD@GK for \@value
% will substitute \CD@HK for \@vector@setup
% will substitute \CD@IK for \verbatim@ps@special
% will substitute \CD@JK for \vert@extra
% will substitute \CD@KK for \vertical@name
% will substitute \CD@LK for \vert@outside
% will substitute \CD@MK for \vert@repeater
% will substitute \CD@NK for \vert@wd@in@leftmost
% will substitute \CD@OK for \@VonH
% will substitute \CD@PK for \@was@pair@false
% will substitute \CD@QK for \@was@pair@true
% will substitute \CD@RK for \widen@hbox
% will substitute \CD@SK for \widen@horiz
% will substitute \CD@TK for \widen@pile
% will substitute \CD@UK for \word@shrink
% will substitute \CD@VK for \word@stretch
% will substitute \CD@WK for \x@coord
% will substitute \CD@XK for \xlc@ii
% will substitute \CD@YK for \xlc@iv
% will substitute \CD@ZK for \x@offset
% will substitute \CD@aK for \y@coord
% will substitute \CD@bK for \y@offset

% will substitute \CD@cK for \@pdf@false
% will substitute \CD@dK for \@pdf@true
% will substitute \CD@eK for \denom@@
% will substitute \CD@fK for \div@@
% will substitute \CD@gK for \do@eturn
% will substitute \CD@hK for \if@pdf@
% will substitute \CD@iK for \make@division
% will substitute \CD@jK for \mk@@div
% will substitute \CD@kK for \mnum@@
% will substitute \CD@lK for \num@@
% will substitute \CD@mK for \rotate@box@z
%% don't load me twice!
\ifx\diagram\undefined\else\message{WARNING: the \string\diagram\space command
is already defined and will not be loaded again}\expandafter\endinput\fi

%% make @ letter, saving its old code to restore at the end of this file
%%% look for this on the last line of the file if you think something's missing!
%% the other \catcode assignments are to make it work with texinfo.
\edef\cdrestoreat{%%
\noexpand\catcode\lq\noexpand\@=\the\catcode\lq\@%%
\noexpand\catcode\lq\noexpand\#=\the\catcode\lq\#%%
\noexpand\catcode\lq\noexpand\$=\the\catcode\lq\$%%
\noexpand\catcode\lq\noexpand\<=\the\catcode\lq\<%%
\noexpand\catcode\lq\noexpand\>=\the\catcode\lq\>%%
\noexpand\catcode\lq\noexpand\:=\the\catcode\lq\:%% Johannes L. Braams's
\noexpand\catcode\lq\noexpand\;=\the\catcode\lq\;%% Babel languages package
\noexpand\catcode\lq\noexpand\!=\the\catcode\lq\!%% makes these \active.
\noexpand\catcode\lq\noexpand\?=\the\catcode\lq\?%%
\noexpand\catcode\lq\noexpand\+=\the\catcode\rq53%
%% texinfo @+ is @outer@active
}\catcode\lq\@=11 \catcode\lq\#=6 \catcode\lq\<=12 \catcode\lq\>=12 \catcode
\rq53=12 \catcode\lq\:=12 \catcode\lq\;=12 \catcode\lq\!=12 \catcode\lq\?=12

%% Change y to n if pool_size in your implementation of TeX is small.
%% This is reasonable if you have a small ("personal") computer, but if you
%%% have a sun, dec, hp, ... workstation or a mainframe, complain to your local
%% system manager and get him/her to install a version of TeX with bigger
%% parameters. The "hash size" (number of command names) gets you next.
\ifx\diagram@help@messages\undefined\let\diagram@help@messages y\fi

%% The following macro is used to include literal PostScript commands in the
%% DVI file for rotation, etc.  Since this goes beyond standard TeX, it is
%%% dependent on the convention used by your local DVI-to-PostScript translator.
%% Choose whichever line applies to the program used at your site, or, if
%% yours is not listed, consult the manual, experiment with this macro and
%% (please) tell me what is needed to make it work.
%%
%%
%%% dvips (Tomas Rokicki, Radical Eye) labrea.stanford.edu /pub/dvips9999.tar.Z
%% CTAN: dviware/dvips
\def\cdps@Rokicki#1{\special{ps:#1}}\let\cdps@dvips\cdps@Rokicki\let
\cdps@RadicalEye\cdps@Rokicki\let\CD@IK\cdps@Rokicki\let\CD@HB\cdps@Rokicki
%%
%% I'm not sure that the rest work.
%%
%% dvitps (Stephan Bechtolsheim, Integrated Computer Systems)
%% arthur.cs.purdue.edu /pub/TeXPS-9.99.tar.Z
\def\cdps@Bechtolsheim#1{\special{dvitps: Literal "#1"}}%
%% ASCII two dbl quotes
\let\cdps@dvitps\cdps@Bechtolsheim\let\cdps@IntegratedComputerSystems
\cdps@Bechtolsheim%%
%% dvitops (James Clark)
%% CTAN: dviware/dvitops
\def\cdps@Clark#1{\special{dvitops: inline #1}}%%
\let\cdps@dvitops\cdps@Clark%%
%% OzTeX (Andrew Trevorrow) cannot be used
\let\cdps@OzTeX\empty\let\cdps@oztex\empty\let\cdps@Trevorrow\empty%%
%% dvi3ps (Kevin Coombes)
%% CTAN: dviware/dvi2ps/dvi3ps
\def\cdps@Coombes#1{\special{ps-string #1}}%%
%% psprint (Trevorrow) CTAN: dviware/psprint
%% dvi2ps (Senn) CTAN: dviware/dvi2ps
%% psdvi (Elwell) CTAN: dviware/dvi2ps/psdvi

\count@=\year\multiply\count@12 \advance\count@\month%%
\ifnum\count@>34049 %% changed (January 2004) It's changing very fast at the
                    %% moment! 
\message{***********************************************************}%%ascii
\message{! YOU HAVE AN OUT OF DATE VERSION OF COMMUTATIVE DIAGRAMS! *}%%
\message{! it expired in December 2002 and is time-bombed for January *}%%
\message{! You may get an up to date version of this package from *}%%ascii
\message{! ftp://ftp.dcs.qmw.ac.uk/pub/tex/contrib/pt/diagrams/ *}%%
\message{***********************************************************}%%ascii
\ifnum\count@>24052 %% (April 2004)
\errhelp{You may press RETURN and carry on for the time being.}\message{! It
is embarrassing to see papers in conference proceedings}\message{! and
journals containing bugs which I had fixed years before.}\message{! It is easy
to obtain and install a new version, which will}\errmessage{! remain
compatible with your files. Please get it NOW.}\fi\fi

\def\CD@DE{\global\let}\def\CD@RH{\outer\def}

%% safe names for braces, tab (&) and maths ($), as commands and for messages
{\escapechar\m@ne\xdef\CD@o{\string\{}\xdef\CD@yC{\string\}}%%
%%
%% three ASCII ampersands (ands) (&&&) appear on the next line
\catcode\lq\&=4 \CD@DE\CD@Q=&\xdef\CD@S{\string\&}%%ascii three ands
%%
%% ASCII ^ and _ each appear twice on next line
%% six ASCII dollars ($$$$$$) appear on the next two lines.
\catcode\lq\$=3 \CD@DE\CD@k=$\CD@DE\CD@ND=$%%ascii three dollars
\xdef\CD@nC{\string\$}\gdef\CD@LG{$$}%%ascii three dollars
%%
%% two ASCII underlines (__) appear on the next line.
\catcode\lq\_=8 \CD@DE\CD@lJ=_%%ascii two underlines
%%
%% eight ASCII carets (^^^^^^^^) appear on the next three lines.
\obeylines\catcode\lq\^=7 \CD@DE\@super=^%%ascii two carets
\ifnum\newlinechar=10 \gdef\CD@uG{^^J}%%ascii two carets
\else\ifnum\newlinechar=13 \gdef\CD@uG{^^M}%%ascii two carets
\else\ifnum\newlinechar=-1 \gdef\CD@uG{^^J}%%ascii two carets
\else\CD@DE\CD@uG\space\expandafter\message{! input error: \noexpand
\newlinechar\space is ASCII \the\newlinechar, not LF=10 or CR=13.}%%
\fi\fi\fi}%%

%% avoid using <> (because I personally re-define them to mean \langle\rangle)
\mathchardef\lessthan=\rq30474 \mathchardef\greaterthan=\rq30476

%% LaTeX line and arrowhead font
%% the "hit return" comments show up if the font is missing.
\ifx\tenln\undefined%%
\font\tenln=line10\relax%% Hit return - who needs diagonals?
\fi\ifx\tenlnw\undefined\ifx\tenln\nullfont\let\tenlnw\nullfont\else%%
\font\tenlnw=linew10\relax%% Hit return - who needs diagonals?
\fi\fi%%

%% report line numbers in TeX3 only
\ifx\inputlineno\undefined\csname newcount\endcsname\inputlineno\inputlineno
\m@ne\message{***************************************************}\message{!
Obsolete TeX (version 2). You should upgrade to *}\message{! version 3, which
has been available since 1990. *}\message{***********************************%
****************}\fi

\def\cd@shouldnt#1{\CD@KB{* THIS (#1) SHOULD NEVER HAPPEN! *}}

%% turn round- and square-bracketed arguments into curly-bracketed
\def\get@round@pair#1(#2,#3){#1{#2}{#3}}%%ascii round brackets ()
\def\get@square@arg#1[#2]{#1{#2}}%%ascii square brackets []
\def\CD@AE#1{\CD@PK\let\CD@DH\CD@@E\CD@@E#1,],}%%ascii sq brackets
\def\CD@m{[}\def\CD@RD{]}\def\commdiag#1{{\let\enddiagram\relax\diagram[]#1%
\enddiagram}}

%% ASCII open square bracket occurs on next line
\def\CD@BF{{\ifx\CD@EH[\aftergroup\get@square@arg\aftergroup\CD@YH\else
\aftergroup\CD@JH\fi}}%%
\def\CD@CF#1#2{\def\CD@YH{#1}\def\CD@JH{#2}\futurelet\CD@EH\CD@BF}

%% ASCII vertical bar (|) occurs on the next line
\def\CD@KK{|}

\def\CD@PB{%% arguments to maps inside diagrams
\tokcase\CD@DD:\CD@y\break@args;\catcase\@super:\upper@label;\catcase\CD@lJ:%
\lower@label;\tokcase{~}:\middle@label;%%ascii tilde
\tokcase<:\CD@iF;%%ascii less-than
\tokcase>:\CD@iI;%%ascii greater-than
\tokcase(:\CD@BC;%%)%ascii open round bracket
\tokcase[:\optional@;%%]%ascii open square bracket
\tokcase.:\CD@JJ;%%ascii dot 12.7.94
\catcase\space:\eat@space;\catcase\bgroup:\positional@;\default:\CD@@A
\break@args;\endswitch}

\def\switch@arg{%% arguments to horizontal maps outside diagrams
\catcase\@super:\upper@label;\catcase\CD@lJ:\lower@label;\tokcase[:\optional@
;%%]%ascii open square bracket
\tokcase.:\CD@JJ;%%ascii dot 12.7.94 % ; was : before 15.6.97
\catcase\space:\eat@space;\catcase\bgroup:\positional@;\tokcase{~}:%
\middle@label;%%ascii tilde (questionable!)
\default:\CD@y\break@args;\endswitch}

%% That's as much as you get to read "in clear" - the rest is private!

\let\CD@tJ\relax\ifx\protect\undefined\let\protect\relax\fi\ifx\AtEndDocument
\undefined\def\CD@PG{\CD@gB}\def\CD@GF#1#2{}\else\def\CD@PG#1{\edef\CD@CH{#1}%
\expandafter\CD@oC\CD@CH\CD@OD}\def\CD@oC#1\CD@OD{\AtEndDocument{\typeout{%
\CD@tA: #1}}}\def\CD@GF#1#2{\gdef#1{#2}\AtEndDocument{#1}}\fi\def\CD@ZA#1#2{%
\def#1{\CD@PG{#2\CD@mD\CD@W}\CD@DE#1\relax}}\def\CD@uF#1\repeat{\def\CD@p{#1}%
\CD@OF}\def\CD@OF{\CD@p\relax\expandafter\CD@OF\fi}\def\CD@sF#1\repeat{\def
\CD@q{#1}\CD@PF}\def\CD@PF{\CD@q\relax\expandafter\CD@PF\fi}\def\CD@tF#1%
\repeat{\def\CD@r{#1}\CD@QF}\def\CD@QF{\CD@r\relax\expandafter\CD@QF\fi}\def
\CD@tG#1#2#3{\def#2{\let#1\iftrue}\def#3{\let#1\iffalse}#3}\if y%
\diagram@help@messages\def\CD@rG#1#2{\csname newtoks\endcsname#1#1=%
\expandafter{\csname#2\endcsname}}\else\csname newtoks\endcsname\no@cd@help
\no@cd@help{See the manual}\def\CD@rG#1#2{\let#1\no@cd@help}\fi\chardef\CD@lF
=1 \chardef\CD@lI=2 \chardef\CD@MH=5 \chardef\CD@tH=6 \chardef\CD@sH=7
\chardef\CD@PC=9 \dimendef\CD@hI=2 \dimendef\CD@hF=3 \dimendef\CD@mF=4
\dimendef\CD@mI=5 \dimendef\CD@wJ=6 \dimendef\CD@tI=8 \dimendef\CD@sI=9
\skipdef\CD@uB=1 \skipdef\CD@NF=2 \skipdef\CD@tB=3 \skipdef\CD@ZE=4 \skipdef
\CD@JK=5 \skipdef\CD@kI=6 \skipdef\CD@kF=7 \skipdef\CD@qI=8 \skipdef\CD@pI=9
\countdef\CD@JC=9 \countdef\CD@gD=8 \countdef\CD@A=7 \def\sdef#1#2{\def#1{#2}%
}\def\CD@L#1{\expandafter\aftergroup\csname#1\endcsname}\def\CD@RC#1{%
\expandafter\def\csname#1\endcsname}\def\CD@sD#1{\expandafter\gdef\csname#1%
\endcsname}\def\CD@vC#1{\expandafter\edef\csname#1\endcsname}\def\CD@nF#1#2{%
\expandafter\let\csname#1\expandafter\endcsname\csname#2\endcsname}\def\CD@EE
#1#2{\expandafter\CD@DE\csname#1\expandafter\endcsname\csname#2\endcsname}%
\def\CD@AK#1{\csname#1\endcsname}\def\CD@XJ#1{\expandafter\show\csname#1%
\endcsname}\def\CD@ZJ#1{\expandafter\showthe\csname#1\endcsname}\def\CD@WJ#1{%
\expandafter\showbox\csname#1\endcsname}\def\CD@tA{Commutative Diagram}\edef
\CD@kH{\string\par}\edef\CD@dC{\string\diagram}\edef\CD@HD{\string\enddiagram
}\edef\CD@EC{\string\\}\def\CD@eF{LaTeX}\ifx\@ignoretrue\undefined
\expandafter\CD@tG\csname if@ignore\endcsname\ignore@true\ignore@false\def
\@ignoretrue{\global\ignore@true}\def\@ignorefalse{\global\ignore@false}\fi

\def\CD@g{{\ifnum0=\lq}\fi}\def\CD@wC{\ifnum0=\lq{\fi}}\def\catcase#1:{\ifcat
\noexpand\CD@EH#1\CD@tJ\expandafter\CD@kC\else\expandafter\CD@dJ\fi}\def
\tokcase#1:{\ifx\CD@EH#1\CD@tJ\expandafter\CD@kC\else\expandafter\CD@dJ\fi}%
\def\CD@kC#1;#2\endswitch{#1}\def\CD@dJ#1;{}\let\endswitch\relax\def\default:%
#1;#2\endswitch{#1}\ifx\at@\undefined\def\at@{@}\fi\edef\CD@P{\CD@o pt\CD@yC}%
\CD@RC{\CD@P>}#1>#2>{\CD@z\rTo\sp{#1}\sb{#2}\CD@z}\CD@RC{\CD@P<}#1<#2<{\CD@z
\lTo\sp{#1}\sb{#2}\CD@z}\CD@RC{\CD@P)}#1)#2){\CD@z\rTo\sp{#1}\sb{#2}\CD@z}%
%%ascii round
\CD@RC{\CD@P(}#1(#2({\CD@z\lTo\sp{#1}\sb{#2}\CD@z}%%ascii brack
\def\CD@O{\def\endCD{\enddiagram}\CD@RC{\CD@P A}##1A##2A{\uTo<{##1}>{##2}%
\CD@z\CD@z}\CD@RC{\CD@P V}##1V##2V{\dTo<{##1}>{##2}\CD@z\CD@z}\CD@RC{\CD@P=}{%
\CD@z\hEq\CD@z}\CD@RC{\CD@P\CD@KK}{\vEq\CD@z\CD@z}\CD@RC{\CD@P\string\vert}{%
\vEq\CD@z\CD@z}\CD@RC{\CD@P.}{\CD@z\CD@z}\let\CD@z\CD@Q}\def\CD@IE{\let\tmp
\CD@JE\ifcat A\noexpand\CD@CH\else\ifcat=\noexpand\CD@CH\else\ifcat\relax
\noexpand\CD@CH\else\let\tmp\at@\fi\fi\fi\tmp}\def\CD@JE#1{\CD@nF{tmp}{\CD@P
\string#1}\ifx\tmp\relax\def\tmp{\at@#1}\fi\tmp}\def\CD@z{}\begingroup
\aftergroup\def\aftergroup\CD@T\aftergroup{\aftergroup\def\catcode\lq\@%
\active\aftergroup @\endgroup{\futurelet\CD@CH\CD@IE}}\newcount\CD@uA
\newcount\CD@vA\newcount\CD@wA\newcount\CD@xA\newdimen\CD@OA\newdimen\CD@PA
\CD@tG\CD@gE\CD@@A\CD@y\CD@tG\CD@hE\CD@EA\CD@BA\newdimen\CD@RA\newdimen\CD@SA
\newcount\CD@yA\newcount\CD@zA\newdimen\CD@QA\newbox\CD@DA\CD@tG\CD@lE\CD@dA
\CD@bA\newcount\CD@LH\newcount\CD@TC\def\CD@V#1#2{\ifdim#1<#2\relax#1=#2%
\relax\fi}\def\CD@X#1#2{\ifdim#1>#2\relax#1=#2\relax\fi}\newdimen\CD@XH\CD@XH
=1sp \newdimen\CD@zC\CD@zC\z@\def\CD@cJ{\ifdim\CD@zC=1em\else\CD@nJ\fi}\def
\CD@nJ{\CD@zC1em\def\CD@NC{\fontdimen8\textfont3 }\CD@@J\CD@NJ\setbox0=\vbox{%
\CD@t\noindent\CD@k\null\penalty-9993\null\CD@ND\null\endgraf\setbox0=%
\lastbox\unskip\unpenalty\setbox1=\lastbox\global\setbox\CD@IG=\hbox{\unhbox0%
\unskip\unskip\unpenalty\setbox0=\lastbox}\global\setbox\CD@KG=\hbox{\unhbox1%
\unskip\unpenalty\setbox1=\lastbox}}}\newdimen\CD@@I\CD@@I=1true in \divide
\CD@@I300 \def\CD@zH#1{\multiply#1\tw@\advance#1\ifnum#1<\z@-\else+\fi\CD@@I
\divide#1\tw@\divide#1\CD@@I\multiply#1\CD@@I}\def\MapBreadth{%
\afterassignment\CD@gI\CD@LF}\newdimen\CD@LF\newdimen\CD@oI\def\CD@gI{\CD@oI
\CD@LF\CD@V\CD@@I{4\CD@XH}\CD@X\CD@@I\p@\CD@zH\CD@oI\ifdim\CD@LF>\z@\CD@V
\CD@oI\CD@@I\fi\CD@cJ}\def\CD@RJ#1{\CD@zD\count@\CD@@I#1\ifnum\count@>\z@
\divide\CD@@I\count@\fi\CD@gI\CD@NJ}\def\CD@NJ{\dimen@\CD@QC\count@\dimen@
\divide\count@5\divide\count@\CD@@I\edef\CD@OC{\the\count@}}\def\CD@AJ{\CD@QJ
\z@}\def\CD@QJ#1{\CD@tI\axisheight\advance\CD@tI#1\relax\advance\CD@tI-.5%
\CD@oI\CD@zH\CD@tI\CD@sI-\CD@tI\advance\CD@tI\CD@LF}\newdimen\CD@DC\CD@DC\z@
\newdimen\CD@eJ\CD@eJ\z@\def\CD@CJ#1{\CD@sI#1\relax\CD@tI\CD@sI\advance\CD@tI
\CD@LF\relax}\def\horizhtdp{height\CD@tI depth\CD@sI}\def\axisheight{%
\fontdimen22\the\textfont\tw@}\def\script@axisheight{\fontdimen22\the
\scriptfont\tw@}\def\ss@axisheight{\fontdimen22\the\scriptscriptfont\tw@}\def
\CD@NC{0.4pt}\def\CD@VK{\fontdimen3\textfont\z@}\def\CD@UK{\fontdimen3%
\textfont\z@}\newdimen\PileSpacing\newdimen\CD@nA\CD@nA\z@\def\CD@RG{%
\ifincommdiag1.3em\else2em\fi}\newdimen\CD@YB\def\CellSize{\afterassignment
\CD@kB\DiagramCellHeight}\newdimen\DiagramCellHeight\DiagramCellHeight-%
\maxdimen\newdimen\DiagramCellWidth\DiagramCellWidth-\maxdimen\def\CD@kB{%
\DiagramCellWidth\DiagramCellHeight}\def\CD@QC{3em}\newdimen\MapShortFall\def
\MapsAbut{\MapShortFall\z@\objectheight\z@\objectwidth\z@}\newdimen\CD@iA
\CD@iA\z@\def\newarrowhead{\CD@mG h\CD@BG\CD@GG>}\def\newarrowtail{\CD@mG t%
\CD@BG\CD@GG>}\def\newarrowmiddle{\CD@mG m\CD@BG\hbox@maths\empty}\def
\newarrowfiller{\CD@mG f\CD@bE\CD@MK-}\def\CD@mG#1#2#3#4#5#6#7#8#9{\CD@RC{r#1%
:#5}{#2{#6}}\CD@RC{l#1:#5}{#2{#7}}\CD@RC{d#1:#5}{#3{#8}}\CD@RC{u#1:#5}{#3{#9}%
}\CD@vC{-#1:#5}{\expandafter\noexpand\csname-#1:#4\endcsname\noexpand\CD@MC}%
\CD@vC{+#1:#5}{\expandafter\noexpand\csname+#1:#4\endcsname\noexpand\CD@MC}}%
\CD@ZA\CD@MC{\CD@eF\space diagonals are used unless PostScript is set}\def
\defaultarrowhead#1{\edef\CD@sJ{#1}\CD@@J}\def\CD@@J{\CD@IJ\CD@sJ<>ht\CD@IJ
\CD@sJ<>th}\def\CD@IJ#1#2#3#4#5{\CD@HJ{r#4}{#3}{l#5}{#2}{r#4:#1}\CD@HJ{r#5}{#%
2}{l#4}{#3}{l#4:#1}\CD@HJ{d#4}{#3}{u#5}{#2}{d#4:#1}\CD@HJ{d#5}{#2}{u#4}{#3}{u%
#4:#1}}\def\CD@HJ#1#2#3#4#5{\begingroup\aftergroup\CD@GJ\CD@L{#1+:#2}\CD@L{#1%
:#2}\CD@L{#3:#4}\CD@L{#5}\endgroup}\def\CD@GJ#1#2#3#4{\csname newbox%
\endcsname#1\def#2{\copy#1}\def#3{\copy#1}\setbox#1=\box\voidb@x}\def\CD@sJ{}%
\CD@@J\def\CD@GJ#1#2#3#4{\setbox#1=#4}\ifx\tenln\nullfont\def\CD@sJ{vee}\else
\let\CD@sJ\CD@eF\fi\def\CD@xF#1#2#3{\begingroup\aftergroup\CD@wF\CD@L{#1#2:#3%
#3}\CD@L{#1#2:#3}\aftergroup\CD@yF\CD@L{#1#2:#3-#3}\CD@L{#1#2:#3}\endgroup}%
\def\CD@wF#1#2{\def#1{\hbox{\rlap{#2}\kern.4\CD@zC#2}}}\def\CD@yF#1#2{\def#1{%
\hbox{\rlap{#2}\kern.4\CD@zC#2\kern-.4\CD@zC}}}\CD@xF lh>\CD@xF rt>\CD@xF rh<%
\CD@xF rt<\def\CD@yF#1#2{\def#1{\hbox{\kern-.4\CD@zC\rlap{#2}\kern.4\CD@zC#2}%
}}\CD@xF rh>\CD@xF lh<\CD@xF lt>\CD@xF lt<\def\CD@wF#1#2{\def#1{\vbox{\vbox to%
\z@{#2\vss}\nointerlineskip\kern.4\CD@zC#2}}}\def\CD@yF#1#2{\def#1{\vbox{%
\vbox to\z@{#2\vss}\nointerlineskip\kern.4\CD@zC#2\kern-.4\CD@zC}}}\CD@xF uh>%
\CD@xF dt>\CD@xF dh<\CD@xF dt<\def\CD@yF#1#2{\def#1{\vbox{\kern-.4\CD@zC\vbox
to\z@{#2\vss}\nointerlineskip\kern.4\CD@zC#2}}}\CD@xF dh>\CD@xF ut>\CD@xF uh<%
\CD@xF ut<\def\CD@BG#1{\hbox{\mathsurround\z@\offinterlineskip\CD@k\mkern-1.5%
mu{#1}\mkern-1.5mu\CD@ND}}\def\hbox@maths#1{\hbox{\CD@k#1\CD@ND}}\def\CD@GG#1%
{\hbox to\CD@LF{\setbox0=\hbox{\offinterlineskip\mathsurround\z@\CD@k{#1}%
\CD@ND}\dimen0.5\wd0\advance\dimen0-.5\CD@oI\CD@zH{\dimen0}\kern-\dimen0%
\unhbox0\hss}}\def\CD@sB#1{\hbox to2\CD@LF{\hss\offinterlineskip\mathsurround
\z@\CD@k{#1}\CD@ND\hss}}\def\CD@vF#1{\hbox{\mathsurround\z@\CD@k{#1}\CD@ND}}%
\def\CD@bE#1{\hbox{\kern-.15\CD@zC\CD@k{#1}\CD@ND\kern-.15\CD@zC}}\def\CD@MK#%
1{\vbox{\offinterlineskip\kern-.2ex\CD@GG{#1}\kern-.2ex}}\def\@fillh{%
\xleaders\vrule\horizhtdp}\def\@fillv{\xleaders\hrule width\CD@LF}\CD@nF{rf:-%
}{@fillh}\CD@nF{lf:-}{@fillh}\CD@nF{df:-}{@fillv}\CD@nF{uf:-}{@fillv}\CD@nF{%
rh:}{null}\CD@nF{rm:}{null}\CD@nF{rt:}{null}\CD@nF{lh:}{null}\CD@nF{lm:}{null%
}\CD@nF{lt:}{null}\CD@nF{dh:}{null}\CD@nF{dm:}{null}\CD@nF{dt:}{null}\CD@nF{%
uh:}{null}\CD@nF{um:}{null}\CD@nF{ut:}{null}\CD@nF{+h:}{null}\CD@nF{+m:}{null%
}\CD@nF{+t:}{null}\CD@nF{-h:}{null}\CD@nF{-m:}{null}\CD@nF{-t:}{null}\def
\CD@@D{\hbox{\vrule height 1pt depth-1pt width 1pt}}\CD@RC{rf:}{\CD@@D}\CD@nF
{lf:}{rf:}\CD@nF{+f:}{rf:}\CD@RC{df:}{\CD@@D}\CD@nF{uf:}{df:}\CD@nF{-f:}{df:}%
\def\CD@BD{\CD@U\null\CD@@D\null\CD@@D\null}\edef\CD@lG{\string\newarrow}\def
\newarrow#1#2#3#4#5#6{\begingroup\edef\@name{#1}\edef\CD@oJ{#2}\edef\CD@iD{#3%
}\edef\CD@QG{#4}\edef\CD@jD{#5}\edef\CD@LE{#6}\let\CD@HE\CD@sG\let\CD@FK
\CD@BH\let\@x\CD@AH\ifx\CD@oJ\CD@iD\let\CD@oJ\empty\fi\ifx\CD@LE\CD@jD\let
\CD@LE\empty\fi\def\CD@LI{r}\def\CD@SF{l}\def\CD@IC{d}\def\CD@yJ{u}\def\CD@gH
{+}\def\@m{-}\ifx\CD@iD\CD@jD\ifx\CD@QG\CD@iD\let\CD@QG\empty\fi\ifx\CD@LE
\empty\ifx\CD@iD\CD@aE\let\@x\CD@yG\else\let\@x\CD@zG\fi\fi\else\edef\CD@a{%
\CD@iD\CD@oJ}\ifx\CD@a\empty\ifx\CD@QG\CD@jD\let\CD@QG\empty\fi\fi\fi\ifmmode
\aftergroup\CD@kG\else\CD@@A\CD@oB rh{head\space\space}\CD@LE\CD@oB rf{filler%
}\CD@iD\CD@oB rm{middle}\CD@QG\ifx\CD@jD\CD@iD\else\CD@oB rf{filler}\CD@jD\fi
\CD@oB rt{tail\space\space}\CD@oJ\CD@gE\CD@HE\CD@FK\@x\CD@nG l-2+2{lu}{nw}%
\NorthWest\CD@nG r+2+2{ru}{ne}\NorthEast\CD@nG l-2-2{ld}{sw}\SouthWest\CD@nG r%
+2-2{rd}{se}\SouthEast\else\aftergroup\CD@b\CD@L{r\@name}\fi\fi\endgroup}\def
\CD@sG{\CD@vG\CD@LI\CD@SF rl\Horizontal@Map}\def\CD@BH{\CD@vG\CD@IC\CD@yJ du%
\Vertical@Map}\def\CD@AH{\CD@vG\CD@gH\@m+-\Vector@Map}\def\CD@yG{\CD@vG\CD@gH
\@m+-\Slant@Map}\def\CD@zG{\CD@vG\CD@gH\@m+-\Slope@Map}\catcode\lq\/=\active
\def\CD@vG#1#2#3#4#5{\CD@jG#1#3#5t:\CD@oJ/f:\CD@iD/m:\CD@QG/f:\CD@jD/h:\CD@LE
//\CD@jG#2#4#5h:\CD@LE/f:\CD@jD/m:\CD@QG/f:\CD@iD/t:\CD@oJ//}\def\CD@jG#1#2#3%
#4//{\edef\CD@fG{#2}\aftergroup\sdef\CD@L{#1\@name}\aftergroup{\aftergroup#3%
\CD@M#4//\aftergroup}}\def\CD@M#1/{\edef\CD@EH{#1}\ifx\CD@EH\empty\else\CD@L{%
\CD@fG#1}\expandafter\CD@M\fi}\catcode\lq\/=12 \def\CD@nG#1#2#3#4#5#6#7#8{%
\aftergroup\sdef\CD@L{#6\@name}\aftergroup{\CD@L{#2\@name}\if#2#4\aftergroup
\CD@CI\else\aftergroup\CD@BI\fi\CD@L{#1\@name}%
%% ASCII round brackets and comma (,) appear on the next line
\aftergroup(\aftergroup#3\aftergroup,\aftergroup#5\aftergroup)\aftergroup}}%
\def\CD@oB#1#2#3#4{\expandafter\ifx\csname#1#2:#4\endcsname\relax\CD@y\CD@gB{%
arrow#3 "#4" undefined}\fi}\CD@rG\CD@VE{All five components must be defined
before an arrow.}\CD@rG\CD@SE{\CD@lG, unlike \string\HorizontalMap, is a
declaration.}\def\CD@b#1{\CD@YA{Arrows \string#1 etc could not be defined}%
\CD@VE}\def\CD@kG{\CD@YA{misplaced \CD@lG}\CD@SE}\def\newdiagramgrid#1#2#3{%
\CD@RC{cdgh@#1}{#2,],}%% ASCII close square bracket
\CD@RC{cdgv@#1}{#3,],}}%% ASCII close square bracket
\CD@tG\ifincommdiag\incommdiagtrue\incommdiagfalse\CD@tG\CD@@F\CD@IF\CD@HF
\newcount\CD@VA\CD@VA=0 \def\CD@yH{\CD@VA6 }\def\CD@OB{\CD@VA1 \global\CD@yA1
\CD@DE\CD@YF\empty}\def\CD@YF{}\def\CD@nB#1{\relax\CD@MD\edef\CD@vJ{#1}%
\begingroup\CD@rE\else\ifcase\CD@VA\ifmmode\else\CD@YG\CD@E0\fi\or\CD@cE5\or
\CD@YG\CD@F5\or\CD@YG\CD@B5\or\CD@YG\CD@B5\or\CD@YG\CD@C5\or\CD@cE7\or\CD@YG
\CD@D7\fi\fi\endgroup\xdef\CD@YF{#1}}\def\CD@pB#1#2#3#4#5{\relax\CD@MD\xdef
\CD@vJ{#4}\begingroup\ifnum\CD@VA<#1 \expandafter\CD@cE\ifcase\CD@VA0\or#2\or
#3\else#2\fi\else\ifnum\CD@VA<6 \CD@tJ\CD@YG\CD@B#2\else\CD@YG\CD@G#2\fi\fi
\endgroup\CD@DE\CD@YF\CD@vJ\ifincommdiag\let\CD@ZD#5\else\let\CD@ZD\CD@LK\fi}%
\def\CD@yI{\global\CD@yA=\ifnum\CD@VA<5 1\else2\fi\relax}\def\CD@OI{\CD@VA
\CD@yA}\def\CD@cE#1{\aftergroup\CD@VA\aftergroup#1\aftergroup\relax}\def
\CD@HH{\def\CD@nB##1{\relax}\let\CD@pB\CD@FH\let\CD@yH\relax\let\CD@OB\relax
\let\CD@yI\relax\let\CD@OI\relax}\def\CD@FH#1#2#3#4#5{\ifincommdiag\let\CD@ZD
#5\else\xdef\CD@vJ{#4}\let\CD@ZD\CD@LK\fi}\def\CD@YG#1{\aftergroup#1%
\aftergroup\relax\CD@cE}\def\CD@B{\CD@YE\CD@S\CD@ME\CD@Q}\def\CD@G{\CD@YE{%
\CD@yC\CD@S}\CD@XE\CD@QD\CD@Q}\def\CD@F{\CD@YE{*\CD@S}\CD@RE\clubsuit\CD@Q}%
\def\CD@C{\CD@YE{\CD@S*\CD@S}\CD@RE\CD@Q\clubsuit\CD@Q}\def\CD@D{\CD@YE\CD@EC
\CD@TE\\}\def\CD@E{\CD@YE\CD@nC\CD@QE\CD@k}\def\CD@LK{\CD@YA{\CD@vJ\space
ignored \CD@dH}\CD@WE}\def\CD@FE{}\def\CD@d{\CD@YA{maps must never be enclosed
in braces}\CD@OE}\def\CD@dH{outside diagram}\def\CD@FC{\string\HonV, \string
\VonH\space and \string\HmeetV}\CD@rG\CD@ME{The way that horizontal and
vertical arrows are terminated implicitly means\CD@uG that they cannot be
mixed with each other or with \CD@FC.}\CD@rG\CD@XE{\string\pile\space is for
parallel horizontal arrows; verticals can just be put together in\CD@uG a cell%
. \CD@FC\space are not meaningful in a \string\pile.}\CD@rG\CD@RE{The
horizontal maps must point to an object, not each other (I've put in\CD@uG one
which you're unlikely to want). Use \string\pile\space if you want them
parallel.}\CD@rG\CD@TE{Parallel horizontal arrows must be in separate layers
of a \string\pile.}\CD@rG\CD@QE{Horizontal arrows may be used \CD@dH s, but
must still be in maths.}\CD@rG\CD@WE{Vertical arrows, \CD@FC\space\CD@dH s don%
't know where\CD@uG where to terminate.}\CD@rG\CD@OE{This prevents them from
stretching correctly.}\def\CD@YE#1{\CD@YA{"#1" inserted \ifx\CD@YF\empty
before \CD@vJ\else between \CD@YF\ifx\CD@YF\CD@vJ s\else\space and \CD@vJ\fi
\fi}}\count@=\year\multiply\count@12 \advance\count@\month\ifnum\count@>60000
\message{because this one expired in December 2003!}\expandafter\endinput\fi
\def\Horizontal@Map{\CD@nB{horizontal map}\CD@LC\CD@TJ\CD@qD}\def\CD@TJ{%
\CD@GB-9999 \let\CD@ZD\CD@XD\ifincommdiag\else\CD@cJ\ifinpile\else\skip2\z@
plus 1.5\CD@VK minus .5\CD@UK\skip4\skip2 \fi\fi\let\CD@kD\@fillh\CD@nF{%
fill@dot}{rf:.}}\def\Vector@Map{\CD@HK4}\def\Slant@Map{\CD@HK{\CD@EF255\else6%
\fi}}\def\Slope@Map{\CD@HK\CD@OC}\def\CD@HK#1#2#3#4#5#6{\CD@LC\def\CD@WK{2}%
\def\CD@aK{2}\def\CD@ZK{1}\def\CD@bK{1}\let\Horizontal@Map\CD@nI\def\CD@OG{#1%
}\def\CD@NI{\CD@U#2#3#4#5#6}}\def\CD@nI{\CD@TJ\CD@JB\let\CD@ZD\CD@TD\CD@qD}%
\CD@tG\CD@pE\CD@rA\CD@qA\CD@rA\def\cds@missives{\CD@rA}\def\CD@TD{\CD@vE\let
\CD@OG\CD@OC\CD@x\CD@zE\CD@WF\fi\setbox0\hbox{\incommdiagfalse\CD@HI}\CD@pE
\CD@aD\else\global\CD@YC\CD@bD\fi\ifvoid6 \ifvoid7 \CD@eE\fi\fi\CD@zE\else
\CD@BD\global\CD@YC\let\CD@CG\CD@IH\CD@YD\fi\else\CD@NI\CD@MI\global\CD@YC
\CD@YD\fi}\def\CD@LC{\begingroup\dimen1=\MapShortFall\dimen2=\CD@RG\dimen5=%
\MapShortFall\setbox3=\box\voidb@x\setbox6=\box\voidb@x\setbox7=\box\voidb@x
\CD@pD\mathsurround\z@\skip2\z@ plus1fill minus 1000pt\skip4\skip2 \CD@TB}%
\CD@tG\CD@tE\CD@UB\CD@TB\def\CD@U#1#2#3#4#5{\let\CD@oJ#1\let\CD@iD#2\let
\CD@QG#3\let\CD@jD#4\let\CD@LE#5\CD@TB\ifx\CD@iD\CD@jD\CD@UB\fi}\def\CD@qD#1#%
2#3#4#5{\CD@U#1#2#3#4#5\CD@tD}\def\Vertical@Map{\CD@pB433{vertical map}\CD@cD
\CD@LC\CD@GB-9995 \let\CD@kD\@fillv\CD@nF{fill@dot}{df:.}\CD@qD}\def
\break@args{\def\CD@tD{\CD@ZD}\CD@ZD\endgroup\aftergroup\CD@FE}\def\CD@MJ{%
\setbox1=\CD@oJ\setbox5=\CD@LE\ifvoid3 \ifx\CD@QG\null\else\setbox3=\CD@QG\fi
\fi\CD@@G2\CD@iD\CD@@G4\CD@jD}\def\CD@pF#1{\ifvoid1\else\CD@oF1#1\fi\ifvoid2%
\else\CD@oF2#1\fi\ifvoid3\else\CD@oF3#1\fi\ifvoid4\else\CD@oF4#1\fi\ifvoid5%
\else\CD@oF5#1\fi} \def\CD@oF#1#2{\setbox#1\vbox{\offinterlineskip\box#1%
\dimen@\prevdepth\advance\dimen@-#2\relax\setbox0\null\dp0\dimen@\ht0-\dimen@
\box0}}\def\CD@@G#1#2{\ifx#2\CD@kD\setbox#1=\box\voidb@x\else\setbox#1=#2\def
#2{\xleaders\box#1}\fi}\CD@ZA\CD@BK{\string\HorizontalMap, \string
\VerticalMap\space and \string\DiagonalMap\CD@uG are obsolete - use \CD@lG
\space to pre-define maps}\def\HorizontalMap#1#2#3#4#5{\CD@BK\CD@nB{old
horizontal map}\CD@LC\CD@TJ\def\CD@oJ{\CD@UH{#1}}\CD@SH\CD@iD{#2}\def\CD@QG{%
\CD@UH{#3}}\CD@SH\CD@jD{#4}\def\CD@LE{\CD@UH{#5}}\CD@tD}\def\VerticalMap#1#2#%
3#4#5{\CD@BK\CD@pB433{vertical map}\CD@cD\CD@LC\CD@GB-9995 \let\CD@kD\@fillv
\def\CD@oJ{\CD@GG{#1}}\CD@VH\CD@iD{#2}\def\CD@QG{\CD@GG{#3}}\CD@VH\CD@jD{#4}%
\def\CD@LE{\CD@GG{#5}}\CD@tD}\def\DiagonalMap#1#2#3#4#5{\CD@BK\CD@LC\def
\CD@OG{4}\let\CD@kD\undefined\let\CD@ZD\CD@YD\def\CD@WK{2}\def\CD@aK{2}\def
\CD@ZK{1}\def\CD@bK{1}\def\CD@QG{\CD@vF{#3}}\ifPositiveGradient\let\mv\raise
\def\CD@oJ{\CD@vF{#5}}\def\CD@iD{\CD@vF{#4}}\def\CD@jD{\CD@vF{#2}}\def\CD@LE{%
\CD@vF{#1}}\else\let\mv\lower\def\CD@oJ{\CD@vF{#1}}\def\CD@iD{\CD@vF{#2}}\def
\CD@jD{\CD@vF{#4}}\def\CD@LE{\CD@vF{#5}}\fi\CD@tD}\def\CD@aE{-}\def\CD@AD{%
\empty}\def\CD@SH{\CD@EG\CD@bE\CD@aE\@fillh}\def\CD@VH{\CD@EG\CD@MK\CD@KK
\@fillv}\def\CD@EG#1#2#3#4#5{\def\CD@CH{#5}\ifx\CD@CH#2\let#4#3\else\let#4%
\null\ifx\CD@CH\empty\else\ifx\CD@CH\CD@AD\else\let#4\CD@CH\fi\fi\fi}\def
\CD@UH#1{\hbox{\mathsurround\z@\offinterlineskip\def\CD@CH{#1}\ifx\CD@CH
\empty\else\ifx\CD@CH\CD@AD\else\CD@k\mkern-1.5mu{\CD@CH}\mkern-1.5mu\CD@ND
\fi\fi}}\def\CD@yD#1#2{\setbox#1=\hbox\bgroup\setbox0=\hbox{\CD@k\labelstyle(%
)\CD@ND}%% ASCII round brackets
\setbox1=\null\ht1\ht0\dp1\dp0\box1 \kern.1\CD@zC\CD@k\bgroup\labelstyle
\aftergroup\CD@LD\CD@xD}\def\CD@LD{\CD@ND\kern.1\CD@zC\egroup\CD@tD}\def
\CD@xD{\futurelet\CD@EH\CD@mJ}\def\CD@mJ{%% qualifiers on label arguments
\catcase\bgroup:\CD@v;\catcase\egroup:\missing@label;\catcase\space:\CD@TF;%
\tokcase[:\CD@XF;%%]%ascii close square bracket 
\default:\CD@zJ;\endswitch}\def\CD@v{\let\CD@MD\CD@c\let\CD@CH}\def\CD@zJ#1{%
\let\CD@UF\egroup{\let\actually@braces@missing@around@macro@in@label\CD@ZH
\let\CD@MD\CD@xC\let\CD@UF\CD@VF#1%
\actually@braces@missing@around@macro@in@label}\CD@UF}\def
\actually@braces@missing@around@macro@in@label{\let\CD@CH=}\def\missing@label
{\egroup\CD@YA{missing label}\CD@PE}\def\CD@xC{\egroup\missing@label}\outer
\def\CD@ZH{}\def\CD@UF{}\def\CD@VF{\CD@wC\CD@UF}\def\CD@MD{}\def\CD@XF{\let
\CD@N\CD@xD\get@square@arg\CD@AE}\CD@rG\CD@PE{The text which has just been
read is not allowed within map labels.}\def\CD@c{\egroup\CD@YA{missing \CD@yC
\space inserted after label}\CD@PE}\def\upper@label{\CD@oD\CD@yD6}\def
\lower@label{\def\positional@{\CD@@A\break@args}\CD@yD7}\def\middle@label{%
\CD@yD3}\CD@tG\CD@yE\CD@pD\CD@oD\def\CD@iF{\ifPositiveGradient\CD@tJ
\expandafter\upper@label\else\expandafter\lower@label\fi}\def\CD@iI{%
\ifPositiveGradient\CD@tJ\expandafter\lower@label\else\expandafter
\upper@label\fi}\CD@ZA\CD@QH{labels as positional arguments are obsolete}\def
\positional@{\CD@QH\CD@yE\CD@tJ\expandafter\upper@label\else\expandafter
\lower@label\fi-}\def\CD@tD{\futurelet\CD@EH\switch@arg}\def\eat@space{%
\afterassignment\CD@tD\let\CD@EH= }\def\CD@TF{\afterassignment\CD@xD\let
\CD@EH= }\def\CD@BC{\get@round@pair\CD@uD}\def\CD@uD#1#2{\def\CD@WK{#1}\def
\CD@aK{#2}\CD@tD}\def\optional@{\let\CD@N\CD@tD\get@square@arg\CD@AE}\def
\CD@JJ.{\CD@sC\CD@tD}\def\CD@sC{\let\CD@iD\fill@dot\let\CD@jD\fill@dot\def
\CD@MI{\let\CD@iD\dfdot\let\CD@jD\dfdot}}\def\CD@MI{}\def\CD@@E#1,{\CD@nH#1,%
\begingroup\ifx\@name\CD@RD\CD@FF\aftergroup\CD@e\fi\aftergroup\CD@jC\else
\expandafter\def\expandafter\CD@RF\expandafter{\csname\@name\endcsname}%
\expandafter\CD@vD\CD@RF\CD@KD\ifx\CD@RF\empty\aftergroup\CD@pC\expandafter
\aftergroup\csname\CD@FB\@name\endcsname\expandafter\aftergroup\csname\CD@FB @%
\@name\endcsname\else\gdef\CD@GE{#1}\CD@gB{\string\relax\space inserted before
`[\CD@GE'}\message{(I was trying to read this as a \CD@tA\ option.)}%
\aftergroup\CD@H\fi\fi\endgroup}\def\CD@vD#1#2\CD@KD{\def\CD@RF{#2}}\def
\CD@jC{\let\CD@CH\CD@N\let\CD@N\relax\CD@CH}\def\CD@H#1],{%
%% ASCII close square bracket
\CD@jC\relax\def\CD@RF{#1}\ifx\CD@RF\empty\def\CD@RF{[\CD@GE]}%
%% ASCII open and close square bracket
\else\def\CD@RF{[\CD@GE,#1]}%% ASCII open and close square bracket
\fi\CD@RF}\def\CD@pC#1#2{\ifx#2\undefined\ifx#1\undefined\CD@gB{option `%
\@name' undefined}\else#1\fi\else\CD@FF\expandafter#2\CD@GK\CD@PK\else\CD@QK
\fi\fi\CD@DH}\CD@tG\CD@FF\CD@QK\CD@PK\def\CD@nH#1,{\CD@FF\ifx\CD@GK\undefined
\CD@e\else\expandafter\CD@oH\CD@GK,#1,(,),(,)[]%
%%ASCII 5commas two pairs round, pair square
\fi\fi\CD@FF\else\CD@mH#1==,\fi}\def\CD@e{\CD@gB{option `\@name' needs (x,y)
value}\CD@PK\let\@name\empty}\def\CD@mH#1=#2=#3,{\def\@name{#1}\def\CD@GK{#2}%
\def\CD@RF{#3}\ifx\CD@RF\empty\let\CD@GK\undefined\fi}%
%% ASCII 2commas 2pair round, pair square on next line
\def\CD@oH#1(#2,#3)#4,(#5,#6)#7[]{\def\CD@GK{{#2}{#3}}\def\CD@RF{#1#4#5#6}%
\ifx\CD@RF\empty\def\CD@RF{#7}\ifx\CD@RF\empty\CD@e\fi\else\CD@e\fi}\def
\CD@FB{cds@}\let\CD@N\relax\def\CD@zD#1{\ifx\CD@GK\undefined\CD@gB{option `%
\@name' needs a value}\else#1\CD@GK\relax\fi}\def\CD@BE#1#2{\ifx\CD@GK
\undefined#1#2\relax\else#1\CD@GK\relax\fi}\def\cds@@showpair#1#2{\message{x=%
#1,y=#2}}\def\cds@@diagonalbase#1#2{\edef\CD@ZK{#1}\edef\CD@bK{#2}}\def\CD@DI
#1{\CD@nF{@x}{cdps@#1}\ifx\@x\relax\CD@f{unknown}\else\ifx\@x\empty\CD@f{%
cannot be used}\else\let\CD@IK\@x\fi\fi}\def\CD@f#1{\CD@gB{PostScript
translator `\CD@GK' #1}\CD@ZB\let\cds@PS\empty\let\cds@noPS\empty}\def\CD@PH{%
}\def\CD@PJ{\CD@fA\edef\CD@PH{\noexpand\CD@KB{\@name\space ignored within
maths}}}\def\diagramstyle{\CD@cJ\let\CD@N\relax\CD@CF\CD@AE\CD@AE}\let
\diagramsstyle\diagramstyle\CD@tG\CD@sE\CD@SB\CD@RB\CD@tG\CD@qE\CD@EB\CD@DB
\CD@tG\CD@oE\CD@pA\CD@oA\CD@tG\CD@iE\CD@HA\CD@GA\CD@HA\CD@tG\CD@jE\CD@JA
\CD@IA\CD@tG\CD@kE\CD@LA\CD@KA\CD@tG\CD@vE\CD@aB\CD@ZB\CD@tG\CD@EF\CD@DK
\CD@CK\CD@tG\CD@rE\CD@JB\CD@IB\CD@tG\CD@mE\CD@gA\CD@fA\CD@tG\CD@nE\CD@kA
\CD@jA\CD@tG\CD@AF\CD@iG\CD@hG\CD@RC{cds@ }{}\CD@RC{cds@}{}\CD@RC{cds@1em}{%
\CellSize1\CD@zC}\CD@RC{cds@1.5em}{\CellSize1.5\CD@zC}\CD@RC{cds@2em}{%
\CellSize2\CD@zC}\CD@RC{cds@2.5em}{\CellSize2.5\CD@zC}\CD@RC{cds@3em}{%
\CellSize3\CD@zC}\CD@RC{cds@3.5em}{\CellSize3.5\CD@zC}\CD@RC{cds@4em}{%
\CellSize4\CD@zC}\CD@RC{cds@4.5em}{\CellSize4.5\CD@zC}\CD@RC{cds@5em}{%
\CellSize5\CD@zC}\CD@RC{cds@6em}{\CellSize6\CD@zC}\CD@RC{cds@7em}{\CellSize7%
\CD@zC}\CD@RC{cds@8em}{\CellSize8\CD@zC}\def\cds@abut{\MapsAbut\dimen1\z@
\dimen5\z@}\def\cds@alignlabels{\CD@IA\CD@KA}\def\cds@amstex{\ifincommdiag
\CD@O\else\def\CD{\diagram[amstex]}%%ascii square brackets []
\fi\CD@T\catcode\lq\@\active}\def\cds@b{\let\CD@dB\CD@bB}\def\cds@balance{%
\let\CD@hA\CD@AA}\let\cds@bottom\cds@b\def\cds@center{\cds@vcentre
\cds@nobalance}\let\cds@centre\cds@center\def\cds@centerdisplay{\CD@HA\CD@PJ
\cds@balance}\let\cds@centredisplay\cds@centerdisplay\def\cds@crab{\CD@BE
\CD@DC{.5\PileSpacing}}\CD@RC{cds@crab-}{\CD@DC-.5\PileSpacing}\CD@RC{%
cds@crab+}{\CD@DC.5\PileSpacing}\CD@RC{cds@crab++}{\CD@DC1.5\PileSpacing}%
\CD@RC{cds@crab--}{\CD@DC-1.5\PileSpacing}\def\cds@defaultsize{\CD@BE{\let
\CD@QC}{3em}\CD@NJ}\def\cds@displayoneliner{\CD@DB}\let\cds@dotted\CD@sC\def
\cds@dpi{\CD@RJ{1truein}}\def\cds@dpm{\CD@RJ{100truecm}}\let\CD@XA\undefined
\def\cds@eqno{\let\CD@XA\CD@GK\let\CD@EJ\empty}\def\cds@fixed{\CD@qA}\CD@tG
\CD@fE\CD@J\CD@I\def\cds@flushleft{\CD@I\CD@GA\CD@PJ\cds@nobalance\CD@BE
\CD@nA\CD@nA}\def\cds@gap{\CD@AJ\setbox3=\null\ht3=\CD@tI\dp3=\CD@sI\CD@BE{%
\wd3=}\MapShortFall} \def\cds@grid{\ifx\CD@GK\undefined\let\h@grid\relax\let
\v@grid\relax\else\CD@nF{h@grid}{cdgh@\CD@GK}\CD@nF{v@grid}{cdgv@\CD@GK}\ifx
\h@grid\relax\CD@gB{unknown grid `\CD@GK'}\else\CD@WB\fi\fi}\let\h@grid\relax
\let\v@grid\relax\def\cds@gridx{\ifx\CD@GK\undefined\else\cds@grid\fi\let
\CD@CH\h@grid\let\h@grid\v@grid\let\v@grid\CD@CH}\def\cds@h{\CD@zD
\DiagramCellHeight}\def\cds@hcenter{\let\CD@hA\CD@aA}\let\cds@hcentre
\cds@hcenter\def\cds@heads{\CD@BE{\let\CD@sJ}\CD@sJ\CD@@J\CD@vE\else\ifx
\CD@sJ\CD@eF\else\CD@MC\fi\fi}\let\cds@height\cds@h\let\cds@hmiddle
\cds@balance\def\cds@htriangleheight{\CD@BE\DiagramCellHeight
\DiagramCellHeight\DiagramCellWidth1.73205\DiagramCellHeight}\def
\cds@htrianglewidth{\CD@BE\DiagramCellWidth\DiagramCellWidth
\DiagramCellHeight.57735\DiagramCellWidth}\CD@tG\CD@zE\CD@eE\CD@dE\CD@eE\def
\cds@hug{\CD@eE} \def\cds@inline{\CD@gA\let\CD@PH\empty}\def
\cds@inlineoneliner{\CD@EB}\CD@RC{cds@l>}{\CD@zD{\let\CD@RG}\dimen2=\CD@RG}%
\def\cds@labelstyle{\CD@zD{\let\labelstyle}}\def\cds@landscape{\CD@kA}\def
\cds@large{\CellSize5\CD@zC}\let\CD@EJ\empty\def\CD@FJ{\refstepcounter{%
equation}\def\CD@XA{\hbox{\@eqnnum}}}\def\cds@LaTeXeqno{\let\CD@EJ\CD@FJ}\def
\cds@lefteqno{\CD@pA}\def\cds@leftflush{\cds@flushleft\CD@J}\def
\cds@leftshortfall{\CD@zD{\dimen1 }}\def\cds@lowershortfall{%
\ifPositiveGradient\cds@leftshortfall\else\cds@rightshortfall\fi}\def
\cds@loose{\CD@VB}\def\cds@midhshaft{\CD@JA}\def\cds@midshaft{\CD@JA}\def
\cds@midvshaft{\CD@LA}\def\cds@moreoptions{\CD@@A}\let\cds@nobalance
\cds@hcenter\def\cds@nohcheck{\CD@HH}\def\cds@nohug{\CD@dE} \def
\cds@nooptions{\def\CD@aC{\CD@WD}}\let\cds@noorigin\cds@nobalance\def
\cds@nopixel{\CD@@I4\CD@XH\CD@cJ}\def\cds@noPostScript{\global\let\CD@n\empty
\CD@BE\CD@DI\empty\CD@ZB\let\cds@PS\empty\let\cds@noPS\empty}\def\cds@noPS{%
\CD@ZB\global\let\CD@n\empty}\def\cds@notextflow{\CD@RB}\def\cds@noTPIC{%
\CD@CK}\def\cds@objectstyle{\CD@zD{\let\objectstyle}}\def\cds@origin{\let
\CD@hA\CD@iB}\def\cds@p{\CD@zD\PileSpacing}\CD@tG\CD@hK\CD@dK\CD@cK\CD@cK\let
\cds@pilespacing\cds@p\def\cds@pixelsize{\CD@zD\CD@@I\CD@gI}\def\cds@portrait
{\CD@jA}\def\cds@PostScript{\CD@aB\global\let\CD@n\empty\let\cds@PS\CD@aB\let
\cds@noPS\CD@ZB\CD@BE\CD@DI\empty}\def\cds@PS{\CD@aB\global\let\CD@n\empty}%
\CD@GF\CD@n{\typeout{\CD@tA: try the PostScript option for better results}}%
\def\cds@repositionpullbacks{\let\make@pbk\CD@fH\let\CD@qH\CD@pH}\def
\cds@righteqno{\CD@oA}\def\cds@rightshortfall{\CD@zD{\dimen5 }}\def
\cds@ruleaxis{\CD@zD{\let\axisheight}}\def\cds@cmex{\let\CD@GG\CD@sB\let
\CD@QJ\CD@CJ}\def\cds@s{\cds@height\DiagramCellWidth\DiagramCellHeight}\def
\cds@scriptlabels{\let\labelstyle\scriptstyle}\def\cds@shortfall{\CD@zD
\MapShortFall\dimen1\MapShortFall\dimen5\MapShortFall}\def\cds@showfirstpass{%
\CD@BE{\let\CD@nD}\z@}\def\cds@silent{\def\CD@KB##1{}\def\CD@gB##1{}}\let
\cds@size\cds@s\def\cds@small{\CellSize2\CD@zC}\def\cds@snake{\CD@BE\CD@eJ\z@
}\def\cds@t{\let\CD@dB\CD@fB}\def\cds@textflow{\CD@SB\CD@PJ}\def\cds@thick{%
\let\CD@rF\tenlnw\CD@LF\CD@NC\CD@BE\MapBreadth{2\CD@LF}\CD@@J}\def\cds@thin{%
\let\CD@rF\tenln\CD@BE\MapBreadth{\CD@NC}\CD@@J}\def\cds@tight{\CD@WB}\let
\cds@top\cds@t\def\cds@TPIC{\CD@DK}\def\cds@uppershortfall{%
\ifPositiveGradient\cds@rightshortfall\else\cds@leftshortfall\fi}\def
\cds@vcenter{\let\CD@dB\CD@cB}\let\cds@vcentre\cds@vcenter\def
\cds@vtriangleheight{\CD@BE\DiagramCellHeight\DiagramCellHeight
\DiagramCellWidth.577035\DiagramCellHeight}\def\cds@vtrianglewidth{\CD@BE
\DiagramCellWidth\DiagramCellWidth\DiagramCellHeight1.73205\DiagramCellWidth}%
\def\cds@vmiddle{\let\CD@dB\CD@eB}\def\cds@w{\CD@zD\DiagramCellWidth}\let
\cds@width\cds@w\def\diagram{\relax\protect\CD@bC}\def\enddiagram{\protect
\CD@SG}\def\CD@bC{\CD@g\CD@uI\incommdiagtrue\edef\CD@wI{\the\CD@NB}\global
\CD@NB\z@\boxmaxdepth\maxdimen\everycr{}\ifx\pdfoutput\undefined\else\ifnum
\pdfoutput>\z@\CD@dK\CD@aB\global\let\CD@n\empty\fi\fi\CD@aC}\def\CD@aC{\CD@y
\let\CD@N\CD@ZC\CD@CF\CD@AE\CD@WD}\def\CD@ZC{\CD@gE\expandafter\CD@aC\else
\expandafter\CD@WD\fi}\def\CD@WD{\let\CD@EH\relax\CD@nE\CD@vE\else\CD@hK\else
\CD@KB{landscape ignored without PostScript}\CD@jA\fi\fi\fi\CD@EJ\setbox2=%
\vbox\bgroup\CD@JF\CD@VD}\def\CD@cH{\CD@nE\CD@fB\else\CD@dB\fi\CD@hA
\nointerlineskip\setbox0=\null\ht0-\CD@pI\dp0\CD@pI\wd0\CD@kI\box0 \global
\CD@QA\CD@kF\global\CD@yA\CD@XB\ifx\CD@NK\undefined\global\CD@RA\CD@kF\else
\global\CD@RA\CD@NK\fi\egroup\CD@zF\CD@nE\setbox2=\hbox to\dp2{\vrule height%
\wd2 depth\CD@QA width\z@\global\CD@QA\ht2\ht2\z@\dp2\z@\wd2\z@\CD@hK
\pdfliteral{q 0 1 -1 0 0 0 cm}\else\global\CD@iG\CD@IK{0 1 bturn}\fi\box2%
\CD@gK\hss}\CD@DB\fi\ifnum\CD@yA=1 \else\CD@DB\fi\global\@ignorefalse\CD@mE
\leavevmode\fi\ifvmode\CD@TA\else\ifmmode\CD@PH\CD@GI\else\CD@qE\CD@gA\fi
\ifinner\CD@gA\fi\CD@mE\CD@GI\else\CD@sE\CD@QB\else\CD@TA\fi\fi\fi\fi\CD@dD}%
\def\CD@dD{\global\CD@NB\CD@wI\relax\CD@xE\global\CD@ID\else\aftergroup\CD@mC
\fi\if@ignore\aftergroup\ignorespaces\fi\CD@wC\ignorespaces}\def\CD@fB{%
\advance\CD@pI\dimen1\relax}\def\CD@eB{\advance\CD@pI.5\dimen1\relax}\def
\CD@bB{}\def\CD@cB{\CD@fB\advance\CD@pI\CD@YB\divide\CD@pI2 \advance\CD@pI-%
\axisheight\relax}\def\CD@aA{}\def\CD@iB{\CD@kF\z@}\def\CD@AA{\ifdim\dimen2>%
\CD@kF\CD@kF\dimen2 \else\dimen2\CD@kF\CD@kI\dimen0 \advance\CD@kI\dimen2 \fi
}\def\CD@QB{\skip0\z@\relax\loop\skip1\lastskip\ifdim\skip1>\z@\unskip
\advance\skip0\skip1 \repeat\vadjust{\prevdepth\dp\strutbox\penalty
\predisplaypenalty\vskip\abovedisplayskip\CD@UA\penalty\postdisplaypenalty
\vskip\belowdisplayskip}\ifdim\skip0=\z@\else\hskip\skip0 \global\@ignoretrue
\fi}\def\CD@TA{\CD@LG\kern-\displayindent\CD@UA\CD@LG\global\@ignoretrue}\def
\CD@UA{\hbox to\hsize{\CD@fE\ifdim\CD@RA=\z@\else\advance\CD@QA-\CD@RA\setbox
2=\hbox{\kern\CD@RA\box2}\fi\fi\setbox1=\hbox{\ifx\CD@XA\undefined\else\CD@k
\CD@XA\CD@ND\fi}\CD@oE\CD@iE\else\advance\CD@QA\wd1 \fi\wd1\z@\box1 \fi\dimen
0\wd2 \advance\dimen0\wd1 \advance\dimen0-\hsize\ifdim\dimen0>-\CD@nA\CD@HA
\fi\advance\dimen0\CD@QA\ifdim\dimen0>\z@\CD@KB{wider than the page by \the
\dimen0 }\CD@HA\fi\CD@iE\hss\else\CD@V\CD@QA\CD@nA\fi\CD@GI\hss\kern-\wd1\box
1 }}\def\CD@GI{\CD@AF\CD@@F\else\CD@SC\global\CD@hG\fi\fi\kern\CD@QA\box2 }%
\CD@tG\CD@wE\CD@YC\CD@XC\def\CD@JF{\CD@cJ\ifdim\DiagramCellHeight=-\maxdimen
\DiagramCellHeight\CD@QC\fi\ifdim\DiagramCellWidth=-\maxdimen
\DiagramCellWidth\CD@QC\fi\global\CD@XC\CD@IF\let\CD@FE\empty\let\CD@z\CD@Q
\let\overprint\CD@eH\let\CD@s\CD@rJ\let\enddiagram\CD@ED\let\\\CD@cC\let\par
\CD@jH\let\CD@MD\empty\let\switch@arg\CD@PB\let\shift\CD@iA\baselineskip
\DiagramCellHeight\lineskip\z@\lineskiplimit\z@\mathsurround\z@\tabskip\z@
\CD@OB}\def\CD@VD{\penalty-123 \begingroup\CD@jA\aftergroup\CD@K\halign
\bgroup\global\advance\CD@NB1 \vadjust{\penalty1}\global\CD@FA\z@\CD@OB\CD@j#%
#\CD@DD\CD@Q\CD@Q\CD@OI\CD@j##\CD@DD\cr}\def\CD@ED{\CD@MD\CD@GD\crcr\egroup
\global\CD@JD\endgroup}\def\CD@j{\global\advance\CD@FA1 \futurelet\CD@EH\CD@i
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\CD@aK\DiagramCellHeight\dimen7 \CD@WK\DiagramCellWidth\CD@jJ
\ifPositiveGradient\advance\dimen7-\CD@ZK\DiagramCellWidth\else\dimen7 \CD@ZK
\DiagramCellWidth\dimen6\z@\fi\advance\dimen6-\CD@bK\DiagramCellHeight\CD@mK
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\axisheight\box0 }\def\CD@mK{\setbox0\hbox{\ht\z@\z@\dp\z@\z@\wd\z@\z@\CD@hK
\expandafter\pdfliteral{q \CD@eK\space\CD@lK\space\CD@kK\space\CD@eK\space0 0
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\CD@hF\advance\CD@wJ\CD@hI\ifvoid\CD@sH\ifdim\CD@wJ<.1em\ifnum\CD@gD=\@m\else
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\CD@SK\else\CD@TK\fi\advance\CD@wJ\CD@mI\CD@bH{-\CD@mI}{\box\CD@sH}{\CD@wJ}%
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\CD@hI\fi\CD@tE\kern-\CD@hI\fi\hbox to\CD@hI{\unhbox\CD@tH}\CD@HG}}\CD@tG
\ifinpile\inpiletrue\inpilefalse\inpilefalse\def\pile{\protect\CD@UJ\protect
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\CD@rG\CD@UE{you left it out!}\def\CD@R{\CD@QD\CD@Q\relax\CD@YA{missing \CD@yC
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\CD@sH=\hbox to\CD@wJ{\hss\unhbox\CD@tH\hss}\else\setbox\CD@sH=\hbox to\CD@wJ
{\hss\kern\CD@hF\unhbox\CD@tH\kern\CD@hI\hss}\fi\or\setbox\CD@sH=\lastbox
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\setbox3\hbox{\ifvoid6\else\kern-\dimen0\unhbox6\fi\unhbox3 \ifvoid7\else
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\advance\dimen4\dp5 \advance\dimen4\dimen1 \let\CD@jD\empty\else\dimen4\ht3
\fi\setbox0\null\ht0\dimen4 \offinterlineskip\setbox8=\vbox spread2ex{\kern
\dimen5 \box1 \CD@iD\vfill\CD@tE\else\kern\CD@eJ\fi\box0}\ht8=\z@\setbox9=%
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\CD@GB\null\kern\dimen3}\def\CD@cI{\ifhbox\CD@VA\CD@KB{clashing verticals}\ht
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\advance\CD@xA\m@ne\setbox\CD@VA=\hbox{\box\CD@lF\unhbox\CD@VA}\unskip\setbox
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\setbox0=\hbox to\wd0\bgroup\unhbox0 \unskip\unpenalty\dimen7\lastkern\unkern
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\multiply\count@12 \advance\count@\month\ifnum\count@>60000 \loop\iftrue
\message{gone July 2004!}\repeat\fi\def\CD@wB{\CD@qI-\CD@NF\CD@pI\CD@NF
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\setbox\CD@FA=\box\voidb@x\CD@yB\repeat\CD@w\ht\CD@NB\CD@qI\dp\CD@NB\CD@pI}%
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\CD@NB\CD@sF\unskip\setbox0=\lastbox\ifhbox0 \advance\CD@FA\m@ne\setbox\CD@FA
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\count@\CD@sF\ifnum\CD@FA<\CD@LB\CD@DJ\CD@XG\dimen0\wd\CD@FA\advance\CD@FA1
\repeat\fi\CD@sF\CD@GB\lastpenalty\unpenalty\ifnum\CD@GB>\z@\CD@vA
\lastpenalty\unpenalty\CD@VG\repeat\endgraf\unskip\ifnum\lastpenalty=4
\unpenalty\else\cd@shouldnt S\fi}\def\CD@VG{\advance\CD@vA\CD@lA\advance
\CD@vA\m@ne\setbox0=\lastbox\ifnum\CD@vA<\CD@LB\setbox\CD@vA\hbox{\box0%
\penalty\CD@GB\unhbox\CD@vA}\else\CD@UG\fi}\def\CD@bG{}\CD@tG\CD@uE\CD@WB
\CD@VB\def\CD@DJ{\advance\dimen0\wd\CD@FA\divide\dimen0\tw@\CD@uE\dimen0%
\DiagramCellWidth\else\CD@V{\dimen0}\DiagramCellWidth\CD@pJ\fi\advance\CD@tB
\dimen0 }\def\CD@XG{\setbox\CD@MB=\vbox{}\dp\CD@MB=\CD@uB\wd\CD@MB\CD@tB
\advance\CD@MB1 }\def\CD@LJ#1,{\def\CD@GK{#1}\ifx\CD@GK\CD@RD\else\advance
\CD@tB\CD@GK\DiagramCellWidth\CD@XG\expandafter\CD@LJ\fi}\def\CD@VJ#1,{\def
\CD@GK{#1}\ifx\CD@GK\CD@RD\else\ifnum\CD@NB>\CD@CA\CD@NF\CD@GK
\DiagramCellHeight\advance\CD@NF-\dp\CD@NB\advance\CD@NB\m@ne\ht\CD@NB\CD@NF
\fi\expandafter\CD@VJ\fi}\def\CD@pJ{\CD@wE\CD@OA\dimen0 \advance\CD@OA-%
\DiagramCellWidth\ifdim\CD@OA>2\MapShortFall\CD@KB{badly drawn diagonals (see
manual)}\let\CD@pJ\empty\fi\else\let\CD@pJ\empty\fi}\def\CD@KC{\CD@VA\CD@mA
\CD@sF\ifnum\CD@VA<\CD@MB\dimen0\dp\CD@VA\advance\dimen0\CD@NF\dp\CD@VA\dimen
0 \advance\CD@VA1 \repeat}\def\CD@bH#1#2#3#4{\ifnum\CD@FA<\CD@LB\CD@OA=#1%
\relax\setbox\CD@FA=\hbox{\setbox0=#2\dimen7=#4\relax\dimen8=#3\relax\ifhbox
\CD@FA\unhbox\CD@FA\advance\CD@OA-\lastkern\unkern\fi\ifdim\CD@OA=\z@\else
\kern-\CD@OA\fi\raise\dimen7\box0 \kern-\dimen8 }\ifnum\CD@FA=\CD@lA\CD@V
\CD@kF\CD@OA\fi\else\cd@shouldnt O\fi}\def\CD@w{\setbox\CD@NB=\hbox{\CD@FA
\CD@lA\CD@VA\CD@mA\CD@PA\z@\relax\CD@sF\ifnum\CD@FA<\CD@LB\CD@tB\wd\CD@VA
\relax\CD@eI\advance\CD@FA1 \advance\CD@VA1 \repeat}\CD@V\CD@kI{\wd\CD@NB}\wd
\CD@NB\z@}\def\CD@eI{\ifhbox\CD@FA\CD@OA\CD@tB\relax\advance\CD@OA-\CD@PA
\relax\ifdim\CD@OA=\z@\else\kern\CD@OA\fi\CD@PA\CD@tB\advance\CD@PA\wd\CD@FA
\relax\unhbox\CD@FA\advance\CD@PA-\lastkern\unkern\fi}\def\CD@ZI{\setbox
\CD@sH=\box\voidb@x\CD@VA=\CD@MB\CD@FA\CD@LB\CD@VA\CD@mA\advance\CD@VA\CD@FA
\advance\CD@VA-\CD@lA\advance\CD@VA\m@ne\CD@tB\wd\CD@VA\count@\CD@LB\advance
\count@\m@ne\CD@hF.5\wd\count@\advance\CD@hF\CD@tB\CD@A\m@ne\CD@gD\@m\CD@sF
\ifnum\CD@FA>\CD@lA\advance\CD@FA\m@ne\advance\CD@hF-\CD@tB\CD@PI\wd\CD@VA
\CD@tB\advance\CD@hF\CD@tB\advance\CD@VA\m@ne\CD@tB\wd\CD@VA\repeat\CD@mF
\CD@kF\CD@mI-\CD@mF\CD@vB}\newcount\CD@GB\def\CD@s{}\def\CD@t{\mathsurround
\z@\hsize\z@\rightskip\z@ plus1fil minus\maxdimen\parfillskip\z@\linepenalty
9000 \looseness0 \hfuzz\maxdimen\hbadness10000 \clubpenalty0 \widowpenalty0
\displaywidowpenalty0 \interlinepenalty0 \predisplaypenalty0
\postdisplaypenalty0 \interdisplaylinepenalty0 \interfootnotelinepenalty0
\floatingpenalty0 \brokenpenalty0 \everypar{}\leftskip\z@\parskip\z@
\parindent\z@\pretolerance10000 \tolerance10000 \hyphenpenalty10000
\exhyphenpenalty10000 \binoppenalty10000 \relpenalty10000 \adjdemerits0
\doublehyphendemerits0 \finalhyphendemerits0 \CD@IA\prevdepth\z@}\newbox
\CD@KG\newbox\CD@IG\def\CD@JG{\unhcopy\CD@KG}\def\CD@HG{\unhcopy\CD@IG}\def
\CD@iJ{\hbox{}\penalty1\nointerlineskip}\def\CD@PI{\penalty5 \noindent\setbox
\CD@MH=\null\CD@mF\z@\CD@mI\z@\ifnum\CD@FA<\CD@LB\ht\CD@MH\ht\CD@FA\dp\CD@MH
\dp\CD@FA\unhbox\CD@FA\skip0=\lastskip\unskip\else\CD@OK\skip0=\z@\fi\endgraf
\ifcase\prevgraf\cd@shouldnt Y \or\cd@shouldnt Z \or\CD@RI\or\CD@XI\else
\CD@QI\fi\unskip\setbox0=\lastbox\unskip\unskip\unpenalty\noindent\unhbox0%
\setbox0\lastbox\unpenalty\unskip\unskip\unpenalty\setbox0\lastbox\CD@tF
\CD@GB\lastpenalty\unpenalty\ifnum\CD@GB>\z@\setbox\z@\lastbox\CD@lB\repeat
\endgraf\unskip\unskip\unpenalty}\def\CD@YJ{\CD@uA\CD@XB\advance\CD@uA-\CD@NB
\CD@vA\CD@FA\advance\CD@vA-\CD@lA\advance\CD@vA1 \expandafter\message{%
prevgraf=\the\prevgraf at (\the\CD@uA,\the\CD@vA)}}\def\CD@XI{\CD@CE\setbox
\CD@lI=\lastbox\setbox\CD@lI=\hbox{\unhbox\CD@lI\unskip\unpenalty}\unskip
\ifdim\ht\CD@lI>\ht\CD@PC\setbox\CD@MH=\copy\CD@lI\else\ifdim\dp\CD@lI>\dp
\CD@PC\setbox\CD@MH=\copy\CD@lI\else\CD@FG\CD@lI\fi\fi\advance\CD@mF.5\wd
\CD@lI\advance\CD@mI.5\wd\CD@lI\setbox\CD@lI=\hbox{\unhbox\CD@lI\CD@HG}\CD@bH
\CD@mF{\box\CD@lI}\CD@mI\z@\CD@yB\CD@vB}\def\CD@CE{\ifnum\CD@A>0 \advance
\dimen0-\CD@tB\CD@iA-.5\dimen0 \CD@A-\CD@A\else\CD@A0 \CD@iA\z@\fi\setbox
\CD@MH=\lastbox\setbox\CD@MH=\hbox{\unhbox\CD@MH\unskip\unskip\unpenalty
\setbox0=\lastbox\global\CD@QA\lastkern\unkern}\advance\CD@iA-.5\CD@QA\unskip
\setbox\CD@MH=\null\CD@mI\CD@iA\CD@mF-\CD@iA}\def\CD@Z{\ht\CD@MH\CD@tI\dp
\CD@MH\CD@sI}\def\CD@FG#1{\setbox\CD@MH=\hbox{\CD@V{\ht\CD@MH}{\ht#1}\CD@V{%
\dp\CD@MH}{\dp#1}\CD@V{\wd\CD@MH}{\wd#1}\vrule height\ht\CD@MH depth\dp\CD@MH
width\wd\CD@MH}}\def\CD@QI{\CD@CE\CD@Z\setbox\CD@lI=\lastbox\unskip\setbox
\CD@lF=\lastbox\unskip\setbox\CD@lF=\hbox{\unhbox\CD@lF\unskip\global\CD@yA
\lastpenalty\unpenalty}\advance\CD@yA9999 \ifcase\CD@yA\CD@VI\or\CD@YI\or
\CD@TI\or\CD@dI\or\CD@cI\or\CD@SI\else\cd@shouldnt9\fi}\def\CD@VI{\CD@FG
\CD@lI\CD@UI\setbox\CD@sH=\box\CD@lF\setbox\CD@tH=\box\CD@lI}\def\CD@YI{%
\CD@FG\CD@lF\setbox\CD@lI\hbox{\penalty8 \unhbox\CD@lI\unskip\unpenalty\ifnum
\lastpenalty=8 \else\CD@xH\fi}\CD@UI\setbox\CD@lF=\hbox{\unhbox\CD@lF\unskip
\unpenalty\global\setbox\CD@DA=\lastbox}\ifdim\wd\CD@lF=\z@\else\CD@xH\fi
\setbox\CD@sH=\box\CD@DA}\def\CD@xH{\CD@KB{extra material in \string\pile
\space cell (lost)}}\def\CD@UI{\CD@yB\ifvoid\CD@sH\else\CD@KB{Clashing
horizontal arrows}\CD@mI.5\CD@hF\CD@mF-\CD@mI\CD@vB\CD@mI\z@\CD@mF\z@\fi
\CD@hI\CD@hF\advance\CD@hI-\CD@mI\CD@hF-\CD@mF\CD@JC\CD@FA}\def\CD@RI{\setbox
0\lastbox\unskip\CD@iA\z@\CD@Z\ifdim\skip0>\z@\CD@tJ\CD@A0 \else\ifnum\CD@A<1
\CD@A0 \dimen0\CD@tB\fi\advance\CD@A1 \fi}\def\VonH{\CD@MA46\VonH{.5\CD@LF}}%
\def\HonV{\CD@MA57\HonV{.5\CD@LF}}\def\HmeetV{\CD@MA44\HmeetV{-\MapShortFall}%
}\def\CD@MA#1#2#3#4{\CD@pB34#1{\string#3}\CD@SD\CD@GB-999#2 \dimen0=#4\CD@tI
\dimen0\advance\CD@tI\axisheight\CD@sI\dimen0\advance\CD@sI-\axisheight\CD@CF
\CD@HC\CD@ZD}\def\CD@HC#1{\setbox0=\hbox{\CD@k#1\CD@ND}\dimen0.5\wd0 \CD@tI
\ht0 \CD@sI\dp0 \CD@ZD}\def\CD@SD{\setbox0=\null\ht0=\CD@tI\dp0=\CD@sI\wd0=%
\dimen0 \copy0\penalty\CD@GB\box0 }\def\CD@TI{\CD@GC\CD@yB}\def\CD@dI{\CD@GC
\CD@vB}\def\CD@SI{\CD@GC\CD@yB\CD@vB}\def\CD@GC{\setbox\CD@lI=\hbox{\unhbox
\CD@lI}\setbox\CD@lF=\hbox{\unhbox\CD@lF\global\setbox\CD@DA=\lastbox}\ht
\CD@MH\ht\CD@DA\dp\CD@MH\dp\CD@DA\advance\CD@mF\wd\CD@DA\advance\CD@mI\wd
\CD@lI}\CD@tG\ifPositiveGradient\CD@CI\CD@BI\CD@CI\CD@tG\ifClimbing\CD@rB
\CD@qB\CD@rB\newcount\DiagonalChoice\DiagonalChoice\m@ne\ifx\tenln\nullfont
\CD@tJ\def\CD@qF{\CD@KH\ifPositiveGradient/\else\CD@k\backslash\CD@ND\fi}%
\else\def\CD@qF{\CD@rF\char\count@}\fi\let\CD@rF\tenln\def\Use@line@char#1{%
\hbox{#1\CD@rF\ifPositiveGradient\else\advance\count@64 \fi\char\count@}}\def
\CD@cF{\Use@line@char{\count@\CD@TC\multiply\count@8\advance\count@-9\advance
\count@\CD@LH}}\def\CD@ZF{\Use@line@char{\ifcase\DiagonalChoice\CD@gF\or
\CD@fF\or\CD@fF\else\CD@gF\fi}}\def\CD@gF{\ifnum\CD@TC=\z@\count@\rq33 \else
\count@\CD@TC\multiply\count@\sixt@@n\advance\count@-9\advance\count@\CD@LH
\advance\count@\CD@LH\fi}\def\CD@fF{\count@\rq\ifcase\CD@LH55\or\ifcase\CD@TC
66\or22\or52\or61\or72\fi\or\ifcase\CD@TC66\or25\or22\or63\or52\fi\or\ifcase
\CD@TC66\or16\or36\or22\or76\fi\or\ifcase\CD@TC66\or27\or25\or67\or22\fi\fi
\relax}\def\CD@uC#1{\hbox{#1\setbox0=\Use@line@char{#1}\ifPositiveGradient
\else\raise.3\ht0\fi\copy0 \kern-.7\wd0 \ifPositiveGradient\raise.3\ht0\fi
\box0}}\def\CD@jF#1{\hbox{\setbox0=#1\kern-.75\wd0 \vbox to.25\ht0{%
\ifPositiveGradient\else\vss\fi\box0 \ifPositiveGradient\vss\fi}}}\def\CD@jI#%
1{\hbox{\setbox0=#1\dimen0=\wd0 \vbox to.25\ht0{\ifPositiveGradient\vss\fi
\box0 \ifPositiveGradient\else\vss\fi}\kern-.75\dimen0 }}\CD@RC{+h:>}{%
\Use@line@char\CD@fF}\CD@RC{-h:>}{\Use@line@char\CD@gF}\CD@nF{+t:<}{-h:>}%
\CD@nF{-t:<}{+h:>}\CD@RC{+t:>}{\CD@jF{\Use@line@char\CD@fF}}\CD@RC{-t:>}{%
\CD@jI{\Use@line@char\CD@gF}}\CD@nF{+h:<}{-t:>}\CD@nF{-h:<}{+t:>}\CD@RC{+h:>>%
}{\CD@uC\CD@fF}\CD@RC{-h:>>}{\CD@uC\CD@gF}\CD@nF{+t:<<}{-h:>>}\CD@nF{-t:<<}{+%
h:>>}\CD@nF{+h:>->}{+h:>>}\CD@nF{-h:>->}{-h:>>}\CD@nF{+t:<-<}{-h:>>}\CD@nF{-t%
:<-<}{+h:>>}\CD@RC{+t:>>}{\CD@jF{\CD@uC\CD@fF}}\CD@RC{-t:>>}{\CD@jI{\CD@uC
\CD@gF}}\CD@nF{+h:<<}{-t:>>}\CD@nF{-h:<<}{+t:>>}\CD@nF{+t:>->}{+t:>>}\CD@nF{-%
t:>->}{-t:>>}\CD@nF{+h:<-<}{-t:>>}\CD@nF{-h:<-<}{+t:>>}\CD@RC{+f:-}{\CD@EF
\null\else\CD@cF\fi}\CD@nF{-f:-}{+f:-}\def\CD@tC#1#2{\vbox to#1{\vss\hbox to#%
2{\hss.\hss}\vss}}\def\hfdot{\CD@tC{2\axisheight}{.5em}}%
%% % .7em until 29.7.98
\def\vfdot{\CD@tC{1ex}\z@}%% % 1.46ex until 29.7.98
\def\CD@bF{\hbox{\dimen0=.3\CD@zC\dimen1\dimen0 \ifnum\CD@LH>\CD@TC\CD@iC{%
\dimen1}\else\CD@dG{\dimen0}\fi\CD@tC{\dimen0}{\dimen1}}}\newarrowfiller{.}%
\hfdot\hfdot\vfdot\vfdot\def\dfdot{\CD@bF\CD@CK}\CD@RC{+f:.}{\dfdot}\CD@RC{-f%
:.}{\dfdot}\def\CD@@K#1{\hbox\bgroup\def\CD@CH{#1\egroup}\afterassignment
\CD@CH%%
\count@=\rq}\def\lnchar{\CD@@K\CD@qF}\let\laf\lnchar\let\lah\lnchar\def\lad{%
\CD@@K\xlad}\def\xlad{\setbox2=\hbox{\CD@qF}\setbox0=\hbox to.3\wd2{\hss.\hss
}\dimen0=\ht0 \advance\dimen0-\dp0 \dimen1=.3\ht2 \advance\dimen1-\dimen0 \dp
0=.5\dimen1 \dimen1=.3\ht2 \advance\dimen1\dimen0 \ht0=.5\dimen1 \raise\dp0%
\box0}\def\lahh{\CD@@K\xlahh}\def\lat{\CD@@K\xlat}\def\xlat{\setbox0=\hbox{%
\CD@qF}\dimen0=\ht0 \setbox1=\hbox to.25\wd0{\ifcase\DiagonalChoice\box0\hss
\or\hss\box0 \or\hss\box0 \or\box0\hss\fi}\vbox to.25\dimen0{\ifClimbing\box1%
\vss\else\vss\box1\fi\kern\z@}}\def\xlahh{\setbox0=\hbox{\CD@qF}%
\ifPositiveGradient\CD@tJ\copy0 \kern-.7\wd0 \mv.3\ht0\box0 \else\ifClimbing
\CD@tJ\copy0 \kern-.7\wd0 \mv.3\ht0\box0 \else\mv-.3\ht0\copy0 \kern-.7\wd0
\box0 \fi\fi}\def\CD@dF#1{\setbox#1=\hbox{\dimen5\dimen#1 \setbox8=\box#1
\dimen1\wd8 \count@\dimen5 \divide\count@\dimen1 \ifnum\count@=0 \box8 \ifdim
\dimen5<.95\dimen1 \CD@gB{diagonal line too short}\fi\else\dimen3=\dimen5
\advance\dimen3-\dimen1 \divide\dimen3\count@\dimen4\dimen3 \CD@dG{\dimen4}%
\ifPositiveGradient\multiply\dimen4\m@ne\fi\dimen6\dimen1 \advance\dimen6-%
\dimen3 \loop\raise\count@\dimen4\copy8 \ifnum\count@>0 \kern-\dimen6 \advance
\count@\m@ne\repeat\fi}}\def\CD@CG#1{\CD@EF\CD@xJ{#1}\else\CD@dF{#1}\fi}\def
\CD@IH#1{}\newdimen\objectheight\objectheight1.8ex \newdimen\objectwidth
\objectwidth1em \def\CD@YD{\dimen6=\CD@aK\DiagramCellHeight\dimen7=\CD@WK
\DiagramCellWidth\CD@KJ\ifnum\CD@LH>0 \ifnum\CD@TC>0 \CD@aF\else\aftergroup
\CD@VC\fi\else\aftergroup\CD@UC\fi}\def\CD@VC{\CD@YA{diagonal map is nearly
vertical}\CD@NA}\def\CD@UC{\CD@YA{diagonal map is nearly horizontal}\CD@NA}%
\CD@rG\CD@NA{Use an orthogonal map instead}\def\CD@aF{\CD@MJ\dimen3\dimen7%
\dimen7\dimen6\CD@iC{\dimen7}\advance\dimen3-\dimen7 \CD@MF\ifnum\CD@LH>%
\CD@TC\advance\dimen6-\dimen1\advance\dimen6-\dimen5 \CD@iC{\dimen1}\CD@iC{%
\dimen5}\else\dimen0\dimen1\advance\dimen0\dimen5\CD@dG{\dimen0}\advance
\dimen6-\dimen0 \fi\dimen2.5\dimen7\advance\dimen2-\dimen1 \dimen4.5\dimen7%
\advance\dimen4-\dimen5 \ifPositiveGradient\dimen0\dimen5 \advance\dimen1-%
\CD@WK\DiagramCellWidth\advance\dimen1 \CD@ZK\DiagramCellWidth\setbox6=\llap{%
\unhbox6\kern.1\ht2}\setbox7=\rlap{\kern.1\ht2\unhbox7}\else\dimen0\dimen1
\advance\dimen1-\CD@ZK\DiagramCellWidth\setbox7=\llap{\unhbox7\kern.1\ht2}%
\setbox6=\rlap{\kern.1\ht2\unhbox6}\fi\setbox6=\vbox{\box6\kern.1\wd2}\setbox
7=\vtop{\kern.1\wd2\box7}\CD@dG{\dimen0}\advance\dimen0-\axisheight\advance
\dimen0-\CD@bK\DiagramCellHeight\dimen5-\dimen0 \advance\dimen0\dimen6
\advance\dimen1.5\dimen3 \ifdim\wd3>\z@\ifdim\ht3>-\dp3\CD@TB\fi\fi\dimen3%
\dimen2 \dimen7\dimen2\advance\dimen7\dimen4 \ifvoid3 \else\CD@tE\else\dimen8%
\ht3\advance\dimen8-\axisheight\CD@iC{\dimen8}\CD@X{\dimen8}{.5\wd3}\dimen9%
\dp3\advance\dimen9\axisheight\CD@iC{\dimen9}\CD@X{\dimen9}{.5\wd3}%
\ifPositiveGradient\advance\dimen2-\dimen9\advance\dimen4-\dimen8 \else
\advance\dimen4-\dimen9\advance\dimen2-\dimen8 \fi\fi\advance\dimen3-.5\wd3
\fi\dimen9=\CD@aK\DiagramCellHeight\advance\dimen9-2\DiagramCellHeight\CD@tE
\advance\dimen2\dimen4 \CD@CG{2}\dimen2-\dimen0\advance\dimen2\dp2 \else
\CD@CG{2}\CD@CG{4}\ifPositiveGradient\dimen2-\dimen0\advance\dimen2\dp2 \dimen
4\dimen5\advance\dimen4-\ht4 \else\dimen4-\dimen0\advance\dimen4\dp4 \dimen2%
\dimen5\advance\dimen2-\ht2 \fi\fi\setbox0=\hbox to\z@{\kern\dimen1 \ifvoid1
\else\ifPositiveGradient\advance\dimen0-\dp1 \lower\dimen0 \else\advance
\dimen5-\ht1 \raise\dimen5 \fi\rlap{\unhbox1}\fi\raise\dimen2\rlap{\unhbox2}%
\ifvoid3 \else\lower.5\dimen9\rlap{\kern\dimen3\unhbox3}\fi\kern.5\dimen7
\lower.5\dimen9\box6 \lower.5\dimen9\box7 \kern.5\dimen7 \CD@tE\else\raise
\dimen4\llap{\unhbox4}\fi\ifvoid5 \else\ifPositiveGradient\advance\dimen5-\ht
5 \raise\dimen5 \else\advance\dimen0-\dp5 \lower\dimen0 \fi\llap{\unhbox5}\fi
\hss}\ht0=\axisheight\dp0=-\ht0\box0 }\def\NorthWest{\CD@BI\CD@rB
\DiagonalChoice0 }\def\NorthEast{\CD@CI\CD@rB\DiagonalChoice1 }\def\SouthWest
{\CD@CI\CD@qB\DiagonalChoice3 }\def\SouthEast{\CD@BI\CD@qB\DiagonalChoice2 }%
\def\CD@aD{\vadjust{\CD@uA\CD@FA\advance\CD@uA\ifPositiveGradient\else-\fi
\CD@ZK\relax\CD@vA\CD@NB\advance\CD@vA-\CD@bK\relax\hbox{\advance\CD@uA
\ifPositiveGradient-\fi\CD@WK\advance\CD@vA\CD@aK\hbox{\box6 \kern\CD@DC\kern
\CD@eJ\penalty1 \box7 \box\z@}\penalty\CD@uA\penalty\CD@vA}\penalty\CD@uA
\penalty\CD@vA\penalty104}}\def\CD@eH#1{\relax\vadjust{\hbox@maths{#1}%
\penalty\CD@FA\penalty\CD@NB\penalty\tw@}}\def\CD@lB{\ifcase\CD@GB\or\or
\CD@bH{.5\wd0}{\box0}{.5\wd0}\z@\or\unhbox\z@\setbox\z@\lastbox\CD@bH{.5\wd0}%
{\box0}{.5\wd0}\z@\unpenalty\unpenalty\setbox\z@\lastbox\or\CD@TG\else
\advance\CD@GB-100 \ifnum\CD@GB<\z@\cd@shouldnt B\fi\setbox\z@\hbox{\kern
\CD@mF\copy\CD@MH\kern\CD@mI\CD@uA\CD@XB\advance\CD@uA-\CD@NB\penalty\CD@uA
\CD@uA\CD@FA\advance\CD@uA-\CD@lA\penalty\CD@uA\unhbox\z@\global\CD@yA
\lastpenalty\unpenalty\global\CD@zA\lastpenalty\unpenalty}\CD@uA-\CD@yA\CD@vA
\CD@zA\CD@fI\fi}\def\CD@TG{\unhbox\z@\setbox\z@\lastbox\CD@uA\lastpenalty
\unpenalty\advance\CD@uA\CD@mA\CD@vA\CD@XB\advance\CD@vA-\lastpenalty
\unpenalty\dimen1\lastkern\unkern\setbox3\lastbox\dimen0\lastkern\unkern
\setbox0=\hbox to\z@{\unhbox0\setbox0\lastbox\setbox7\lastbox\unpenalty\CD@eJ
\lastkern\unkern\CD@DC\lastkern\unkern\setbox6\lastbox\dimen7\CD@tB\advance
\dimen7-\wd\CD@uA\ifdim\dimen7<\z@\CD@CI\multiply\dimen7\m@ne\let\mv\empty
\else\CD@BI\def\mv{\raise\ht1}\kern-\dimen7 \fi\ifnum\CD@vA>\CD@NB\dimen6%
\CD@uB\advance\dimen6-\ht\CD@vA\else\dimen6\z@\fi\CD@jJ\CD@mK\setbox1\null\ht
1\dimen6\wd1\dimen7 \dimen7\dimen2 \dimen6\wd1 \CD@KJ\CD@uA\CD@LH\CD@vA\CD@TC
\dimen6\ht1 \CD@KJ\setbox2\null\divide\dimen2\tw@\advance\dimen2\CD@eJ\CD@eG{%
\dimen2}\wd2\dimen2 \dimen0.5\dimen7 \advance\dimen0\ifPositiveGradient\else-%
\fi\CD@eJ\CD@dG{\dimen0}\advance\dimen0-\axisheight\ht2\dimen0 \dimen0\CD@DC
\CD@eG{\dimen0}\advance\dimen0\ht2\ht2\dimen0 \dimen0\ifPositiveGradient-\fi
\CD@DC\CD@dG{\dimen0}\advance\dimen0\wd2\wd2\dimen0 \setbox4\null\dimen0 .6%
\CD@zC\CD@eG{\dimen0}\ht4\dimen0 \dimen0 .2\CD@zC\CD@dG{\dimen0}\wd4\dimen0
\dimen0\wd2 \ifvoid6\else\dimen1\ht4 \advance\dimen1\ht2 \CD@CC6+-\raise
\dimen1\rlap{\ifPositiveGradient\advance\dimen0-\wd6\advance\dimen0-\wd4 \else
\advance\dimen0\wd4 \fi\kern\dimen0\box6}\fi\dimen0\wd2 \ifvoid7\else\dimen1%
\ht4 \advance\dimen1-\ht2 \CD@CC7-+\lower\dimen1\rlap{\ifPositiveGradient
\advance\dimen0\wd4 \else\advance\dimen0-\wd7\advance\dimen0-\wd4 \fi\kern
\dimen0\box7}\fi\mv\box0\hss}\ht0\z@\dp0\z@\CD@bH{\z@}{\box\z@}{\z@}{%
\axisheight}}\def\CD@CC#1#2#3{\dimen4.5\wd#1 \ifdim\dimen4>.25\dimen7\dimen4=%
.25\dimen7\fi\ifdim\dimen4>\CD@zC\dimen4.4\dimen4 \advance\dimen4.6\CD@zC\fi
\CD@eG{\dimen4}\dimen5\axisheight\CD@dG{\dimen5}\advance\dimen4-\dimen5 \dimen
5\dimen4\CD@eG{\dimen5}\advance\dimen0\ifPositiveGradient#2\else#3\fi\dimen5
\CD@dG{\dimen4}\advance\dimen1\dimen4 } \def\CD@eD#1{\expandafter\CD@IK{#1}}%
\CD@ZA\CD@EK{output is PostScript dependent}\def\CD@SC{\CD@IK{/bturn {gsave
currentpoint currentpoint translate 4 2 roll neg exch atan rotate neg exch neg
exch translate } def /eturn {currentpoint grestore moveto} def}}\def\CD@gK{%
\relax\CD@hK\pdfliteral{Q}\else\CD@IK{eturn}\fi} \def\CD@OJ#1{\count@#1\relax
\multiply\count@7\advance\count@16577\divide\count@33154 }\def\CD@fD#1{%
\expandafter\special{#1}} \def\CD@xJ#1{\setbox#1=\hbox{\dimen0\dimen#1\CD@dG{%
\dimen0}\CD@OJ{\dimen0}\setbox0=\null\ifPositiveGradient\count@-\count@\ht0%
\dimen0 \else\dp0\dimen0 \fi\box0 \CD@uA\count@\CD@OJ\CD@LF\CD@fD{pn \the
\count@}\CD@fD{pa 0 0}\CD@OJ{\dimen#1}\CD@fD{pa \the\count@\space\the\CD@uA}%
\CD@fD{fp}\kern\dimen#1}}\def\CD@JI{\CD@KJ\begingroup\ifdim\dimen7<\dimen6
\dimen2=\dimen6 \dimen6=\dimen7 \dimen7=\dimen2 \count@\CD@LH\CD@LH\CD@TC
\CD@TC\count@\else\dimen2=\dimen7 \fi\ifdim\dimen6>.01\p@\CD@KI\global\CD@QA
\dimen0 \else\global\CD@QA\dimen7 \fi\endgroup\dimen2\CD@QA\CD@iK\CD@lK{%
\ifPositiveGradient\else-\fi\dimen6}\CD@iK\CD@kK{\ifPositiveGradient-\fi
\dimen6}\CD@iK\CD@eK{\dimen7}}\def\CD@KI{\CD@hJ\ifdim\dimen7>1.73\dimen6
\divide\dimen2 4 \multiply\CD@TC2 \else\dimen2=0.353553\dimen2 \advance\CD@LH
-\CD@TC\multiply\CD@TC4 \fi\dimen0=4\dimen2 \CD@ZG4\CD@ZG{-2}\CD@ZG2\CD@ZG{-2%
.5}}\def\CD@AI{\begingroup\count@\dimen0 \dimen2 45pt \divide\count@\dimen2
\ifdim\dimen0<\z@\advance\count@\m@ne\fi\ifodd\count@\advance\count@1\CD@@A
\else\CD@y\fi\advance\dimen0-\count@\dimen2 \CD@gE\multiply\dimen0\m@ne\fi
\ifnum\count@<0 \multiply\count@-7 \fi\dimen3\dimen1 \dimen6\dimen0 \dimen7
3754936sp \ifdim\dimen0<6\p@\def\CD@OG{4000}\fi\CD@KJ\dimen2\dimen3\CD@dG{%
\dimen2}\CD@hJ\multiply\CD@TC-6 \dimen0\dimen2 \CD@ZG1\CD@ZG{0.3}\dimen1%
\dimen0 \dimen2\dimen3 \dimen0\dimen3 \CD@ZG3\CD@ZG{1.5}\CD@ZG{0.3}\divide
\count@2 \CD@gE\multiply\dimen1\m@ne\fi\ifodd\count@\dimen2\dimen1\dimen1%
\dimen0\dimen0-\dimen2 \fi\divide\count@2 \ifodd\count@\multiply\dimen0\m@ne
\multiply\dimen1\m@ne\fi\global\CD@QA\dimen0\global\CD@RA\dimen1\endgroup
\dimen6\CD@QA\dimen7\CD@RA}\def\CD@OC{255}\let\CD@OG\CD@OC\def\CD@KJ{%
\begingroup\ifdim\dimen7<\dimen6 \dimen9\dimen7\dimen7\dimen6\dimen6\dimen9%
\CD@@A\else\CD@y\fi\dimen2\z@\dimen3\CD@XH\dimen4\CD@XH\dimen0\z@\dimen8=%
\CD@OG\CD@XH\CD@lC\global\CD@yA\dimen\CD@gE0\else3\fi\global\CD@zA\dimen
\CD@gE3\else0\fi\endgroup\CD@LH\CD@yA\CD@TC\CD@zA}\def\CD@lC{\count@\dimen6
\divide\count@\dimen7 \advance\dimen6-\count@\dimen7 \dimen9\dimen4 \advance
\dimen9\count@\dimen0 \ifdim\dimen9>\dimen8 \CD@@C\else\CD@AC\ifdim\dimen6>%
\z@\dimen9\dimen6 \dimen6\dimen7 \dimen7\dimen9 \expandafter\expandafter
\expandafter\CD@lC\fi\fi}\def\CD@@C{\ifdim\dimen0=\z@\ifdim\dimen9<2\dimen8
\dimen0\dimen8 \fi\else\advance\dimen8-\dimen4 \divide\dimen8\dimen0 \ifdim
\count@\CD@XH<2\dimen8 \count@\dimen8 \dimen9\dimen4 \advance\dimen9\count@
\dimen0 \CD@AC\fi\fi}\def\CD@AC{\dimen4\dimen0 \dimen0\dimen9 \advance\dimen2%
\count@\dimen3 \dimen9\dimen2 \dimen2\dimen3 \dimen3\dimen9 }\def\CD@ZG#1{%
\CD@dG{\dimen2}\advance\dimen0 #1\dimen2 }\def\CD@dG#1{\divide#1\CD@TC
\multiply#1\CD@LH}\def\CD@eG#1{\divide#1\CD@vA\multiply#1\CD@uA}\def\CD@iC#1{%
\divide#1\CD@LH\multiply#1\CD@TC}\def\CD@hJ{\dimen6\CD@LH\CD@XH\multiply
\dimen6\CD@LH\dimen7\CD@TC\CD@XH\multiply\dimen7\CD@TC\CD@KJ}\def\CD@iK#1#2{%
\begingroup\dimen@#2\relax\loop\ifdim\dimen2<.4\maxdimen\multiply\dimen2\tw@
\multiply\dimen@\tw@\repeat\divide\dimen2\@cclvi\divide\dimen@\dimen2\relax
\multiply\dimen@\@cclvi\expandafter\CD@jK\the\dimen@\endgroup\let#1\CD@fK}{%
\catcode\lq p=12 \catcode\lq0=12 \catcode\lq.=12 \catcode\lq t=12 \gdef\CD@jK
#1pt{\gdef\CD@fK{#1}}}\ifx\errorcontextlines\undefined\CD@tJ\let\CD@GH\relax
\else\def\CD@GH{\errorcontextlines\m@ne}\fi\ifnum\inputlineno<0 \let\CD@CD
\empty\let\CD@W\empty\let\CD@mD\relax\let\CD@uI\relax\let\CD@vI\relax\let
\CD@zF\relax\message{! Why not upgrade to TeX version 3? (available since 1990%
)}\else\def\CD@W{ at line \number\inputlineno}\def\CD@mD{ - first occurred}%
\def\CD@uI{\edef\CD@h{\the\inputlineno}\global\let\CD@jB\CD@h}\def\CD@h{9999}%
\def\CD@vI{\xdef\CD@jB{\the\inputlineno}}\def\CD@jB{\CD@h}\def\CD@zF{\ifnum
\CD@h<\inputlineno\edef\CD@CD{\space at lines \CD@h--\the\inputlineno}\else
\edef\CD@CD{\CD@W}\fi}\fi\let\CD@CD\empty\def\CD@YA#1#2{\CD@GH\errhelp=#2%
\expandafter\errmessage{\CD@tA: #1}}\def\CD@KB#1{\begingroup\expandafter
\message{! \CD@tA: #1\CD@CD}\ifnum\CD@XB>\CD@NB\ifnum\CD@CA>\CD@NB\else\ifnum
\CD@lA>\CD@FA\else\ifnum\CD@LB>\CD@FA\advance\CD@XB-\CD@NB\advance\CD@FA-%
\CD@lA\advance\CD@FA1\relax\expandafter\message{! (error detected at row \the
\CD@XB, column \the\CD@FA, but probably caused elsewhere)}\fi\fi\fi\fi
\endgroup}\def\CD@gB#1{{\expandafter\message{\CD@tA\space Warning: #1\CD@W}}}%
\def\CD@CB#1#2{\CD@gB{#1 \string#2 is obsolete\CD@mD}}\def\CD@AB#1{\CD@CB{%
Dimension}{#1}\CD@DE#1\CD@BB\CD@BB}\def\CD@BB{\CD@OA=}\def\CD@@B#1{\CD@CB{%
Count}{#1}\CD@DE#1\CD@OH\CD@OH}\def\CD@OH{\count@=}\def\HorizontalMapLength{%
\CD@AB\HorizontalMapLength}\def\VerticalMapHeight{\CD@AB\VerticalMapHeight}%
\def\VerticalMapDepth{\CD@AB\VerticalMapDepth}\def\VerticalMapExtraHeight{%
\CD@AB\VerticalMapExtraHeight}\def\VerticalMapExtraDepth{\CD@AB
\VerticalMapExtraDepth}\def\DiagonalLineSegments{\CD@@B\DiagonalLineSegments}%
\ifx\tenln\nullfont\CD@ZA\CD@KH{\CD@eF\space diagonal line and arrow font not
available}\else\let\CD@KH\relax\fi\def\CD@aG#1#2<#3:#4:#5#6{\begingroup\CD@PA
#3\relax\advance\CD@PA-#2\relax\ifdim.1em<\CD@PA\CD@uA#5\relax\CD@vA#6\relax
\ifnum\CD@uA<\CD@vA\count@\CD@vA\advance\count@-\CD@uA\CD@KB{#4 by \the\CD@PA
}\if#1v\let\CD@CH\CD@JK\edef\tmp{\the\CD@uA--\the\CD@vA,\the\CD@FA}\else
\advance\count@\count@\if#1l\advance\count@-\CD@A\else\if#1r\advance\count@
\CD@A\fi\fi\advance\CD@PA\CD@PA\let\CD@CH\CD@ZE\edef\tmp{\the\CD@NB,\the
\CD@uA--\the\CD@vA}\fi\divide\CD@PA\count@\ifdim\CD@CH<\CD@PA\global\CD@CH
\CD@PA\fi\fi\fi\endgroup}\CD@tG\CD@xE\CD@JD\CD@ID\CD@rG\CD@xI{See the message
above.}\CD@rG\CD@lH{Perhaps you've forgotten to end the diagram before
resuming the text, in\CD@uG which case some garbage may be added to the
diagram, but we should be ok now.\CD@uG Alternatively you've left a blank line
in the middle - TeX will now complain\CD@uG that the remaining \CD@S s are
misplaced - so please use comments for layout.}\CD@rG\CD@hD{You have already
closed too many brace pairs or environments; an \CD@HD\CD@uG command was (%
over)due.}\CD@rG\CD@hH{\CD@dC\space and \CD@HD\space commands must match.}%
\def\CD@jH{\ifnum\inputlineno=0 \else\expandafter\CD@iH\fi}\def\CD@iH{\CD@MD
\CD@GD\crcr\CD@YA{missing \CD@HD\space inserted before \CD@kH- type "h"}%
\CD@lH\enddiagram\CD@AG\CD@kH\par}\def\CD@AG#1{\edef\enddiagram{\noexpand
\CD@rD{#1\CD@W}}}\def\CD@rD#1{\CD@YA{\CD@HD\space(anticipated by #1) ignored}%
\CD@xI\let\enddiagram\CD@SG}\def\CD@SG{\CD@YA{misplaced \CD@HD\space ignored}%
\CD@hH}\def\CD@mC{\CD@YA{missing \CD@HD\space inserted.}\CD@hD\CD@AG{closing
group}}\ifx\DeclareOption\undefined\else\ifx\DeclareOption\@notprerr\else
\DeclareOption*{\let\CD@N\relax\let\CD@DH\relax\expandafter\CD@@E
\CurrentOption,}\fi\fi
%%======================================================================%
%%                                                                      %
%%      (22) AUXILLARY MACROS FOR ADJUSTMENT OF COMPONENTS              %
%%                                                                      %
%%======================================================================%

%% NOTE: The recommended way of defining arrow commands is now
%%      \newarrow{Name}{tail}{filler}{middle}{filler}{head}
%% which defines \rName, \lName, \dName and \uName using arrow parts which
%% have themselves previously been defined using the commands
%% \newarrowtail, \newarrowfiller, \newarrowmiddle and \newarrowhead.

%% The components \rhvee etc have been retained for the time being, as an
%% intermediate stage and to continue to support the old \HorizontalMap and
%% \VerticalMap commands, but you should not rely on the continued existence
%% of these macros.

%% The various components usually need some correction
%% - longitudinally, ie to prevent gaps and overprints with the shaft,
%% - transversally,  ie to prevent "steps" in the junction with the shaft.
%% The former can be done safely ad hoc, eg with \mkern1mu.
%% The latter are now done with the macros \scriptaxis, \boldscriptaxis,
%% \shifthook and \raisehook, which include pixel corrections.

%% Please note that these and the other auxillary macros which follow are
%% interim. When it becomes clear exactly what kinds of adjustments are
%% needed for characters, this job will be done by a suitable extension
%% to the language of \newarrowhead, etc. If you have any other ideas for
%% transformations of general use please tell me.

%% By all means experiment with other characters for arrowheads, but
%% please, in your own interests, do not rely on macros like \rhvee,
%% send me a copy of your definitions for distribution to other users
%% in this file, and keep track of where your efforts get copied so
%% that they can be replaced with the "official" version when it is
%% incorporated.

%% ***** DONT use macros with mangled names like \Cd@gH. *****

\catcode\lq\$=3 %% make sure that $ means maths-shift
\def\vboxtoz{\vbox to\z@}%% \z@ is in plain TeX and means 0pt

%% print #1 in \scriptstyle, adjusting for the maths axis height
\def\scriptaxis#1{\@scriptaxis{$\scriptstyle#1$}}%%
\def\ssaxis#1{\@ssaxis{$\scriptscriptstyle#1$}}%%
\def\@scriptaxis#1{\dimen0\axisheight\advance\dimen0-\ss@axisheight\raise
\dimen0\hbox{#1}}\def\@ssaxis#1{\dimen0\axisheight\advance\dimen0-%
\ss@axisheight\raise\dimen0\hbox{#1}}

%% Some of the characters would look better in bold since they're
%% taken from sub/superscript fonts; we use LaTeX's \boldmath to
%% do this, defining this to do nothing if it doesn't exist.
%% With the old LaTeX font selection at other than 10pt you may still
%% get nothing happenning.  Also, PK fonts may be missing.
%% If you have problems, DONT use boldhook or boldlittlevee.
\ifx\boldmath\undefined%%
\let\boldscriptaxis\scriptaxis%%
\def\boldscript#1{\hbox{$\scriptstyle#1$}}%%
\def\boldscriptscript#1{\hbox{$\scriptscriptstyle#1$}}%%
\else\def\boldscriptaxis#1{\@scriptaxis{\boldmath$\scriptstyle#1$}}%%
\def\boldscript#1{\hbox{\boldmath$\scriptstyle#1$}}%%
\def\boldscriptscript#1{\hbox{\boldmath$\scriptscriptstyle#1$}}%%
\fi

%%  #1= {} or \boldmath; #2= + or -; #3=\subset or \supset
\def\raisehook#1#2#3{\hbox{\setbox3=\hbox{#1$\scriptscriptstyle#3$}%
%% the character to use
\dimen0\ss@axisheight%% \scriptscriptstyle axis height
\dimen1\axisheight\advance\dimen1-\dimen0%% difference in axis heights
\dimen2\ht3\advance\dimen2-\dimen0%
%%  height of char above axis (half spread)
\advance\dimen2-0.021em\advance\dimen1 #2\dimen2%
%% shift = axis_difference +/- half_spread
\raise\dimen1\box3}}%% print the character
%% Mark Dawson suggested using the width
\def\shifthook#1#2#3{\setbox1=\hbox{#1$\scriptscriptstyle#3$}\dimen0\wd1%
\divide\dimen0 12\CD@zH{\dimen0}%%  "u"
\dimen1\wd1\advance\dimen1-2\dimen0 \advance\dimen1-2\CD@oI\CD@zH{\dimen1}%
\kern#2\dimen1\box1}%% print

%% use the extension font (cmex) for double vertical arrows
\def\@cmex{\mathchar"03}%%ascii double quote

%%      ************* P U L L B A C K S ************

%% These will probably be replaced by something less ad hoc
%% in a future version.

\def\make@pbk#1{\setbox\tw@\hbox to\z@{#1}\ht\tw@\z@\dp\tw@\z@\box\tw@}\def
\CD@fH#1{\overprint{\hbox to\z@{#1}}}\def\CD@qH{\kern0.11em}\def\CD@pH{\kern0%
.35em}

%% This is a hack for my book ``Practical Foundations of Mathematics''
%% and WILL NOT BE SUPPORTED --- DO NOT USE IT!
\def\dblvert{\def\CD@rH{\kern.5\PileSpacing}}\def\CD@rH{}

\def\SEpbk{\make@pbk{\CD@qH\CD@rH\vrule depth 2.87ex height -2.75ex width 0.%
95em \vrule height -0.66ex depth 2.87ex width 0.05em \hss}}

\def\SWpbk{\make@pbk{\hss\vrule height -0.66ex depth 2.87ex width 0.05em
\vrule depth 2.87ex height -2.75ex width 0.95em \CD@qH\CD@rH}}

\def\NEpbk{\make@pbk{\CD@qH\CD@rH\vrule depth -3.81ex height 4.00ex width 0.%
95em \vrule height 4.00ex depth -1.72ex width 0.05em \hss}}

\def\NWpbk{\make@pbk{\hss\vrule height 4.00ex depth -1.72ex width 0.05em
\vrule depth -3.81ex height 4.00ex width 0.95em \CD@qH\CD@rH}}

%%  Freyd & Scedrov puncture symbol for non-commuting polygon
\def\puncture{{\setbox0\hbox{A}\vrule height.53\ht0 depth-.47\ht0 width.35\ht
0 \kern.12\ht0 \vrule height\ht0 depth-.65\ht0 width.06\ht0 \kern-.06\ht0
\vrule height.35\ht0 depth0pt width.06\ht0 \kern.12\ht0 \vrule height.53\ht0
depth-.47\ht0 width.35\ht0 }}

%% 2-cells: (24.11.95)
%%% \NEclck puts a clockwise (ie southeast) arrow to the northwest of cell etc
\def\NEclck{\overprint{\raise2.5ex\rlap{ \CD@rH$\scriptstyle\searrow$}}}%%
\def\NEanti{\overprint{\raise2.5ex\rlap{ \CD@rH$\scriptstyle\nwarrow$}}}%%
\def\NWclck{\overprint{\raise2.5ex\llap{$\scriptstyle\nearrow$ \CD@rH}}}%%
\def\NWanti{\overprint{\raise2.5ex\llap{$\scriptstyle\swarrow$ \CD@rH}}}%%
\def\SEclck{\overprint{\lower1ex\rlap{ \CD@rH$\scriptstyle\swarrow$}}}%%
\def\SEanti{\overprint{\lower1ex\rlap{ \CD@rH$\scriptstyle\nearrow$}}}%%
\def\SWclck{\overprint{\lower1ex\llap{$\scriptstyle\nwarrow$ \CD@rH}}}%%
\def\SWanti{\overprint{\lower1ex\llap{$\scriptstyle\searrow$ \CD@rH}}}

%%======================================================================%
%%                                                                      %
%%      (23) BITS OF ARROWS                                             %
%%                                                                      %
%%======================================================================%

%%       **********  H E A D S ***********

%% \diagramstyle[heads=xxx] defines {>} as {xxx} where xxx
%% has been defined by \newarrowhead{xxx} and \newarrowtail{xxx}

%% vee head
\def\rhvee{\mkern-10mu\greaterthan}%%
\def\lhvee{\lessthan\mkern-10mu}%%
\def\dhvee{\vboxtoz{\vss\hbox{$\vee$}\kern0pt}}%%
\def\uhvee{\vboxtoz{\hbox{$\wedge$}\vss}}%%
\newarrowhead{vee}\rhvee\lhvee\dhvee\uhvee

%% little vee head
\def\dhlvee{\vboxtoz{\vss\hbox{$\scriptstyle\vee$}\kern0pt}}%%
\def\uhlvee{\vboxtoz{\hbox{$\scriptstyle\wedge$}\vss}}%%
\newarrowhead{littlevee}{\mkern1mu\scriptaxis\rhvee}{\scriptaxis\lhvee}%
\dhlvee\uhlvee\ifx\boldmath\undefined%%
\newarrowhead{boldlittlevee}{\mkern1mu\scriptaxis\rhvee}{\scriptaxis\lhvee}%
\dhlvee\uhlvee\else%%
\def\dhblvee{\vboxtoz{\vss\boldscript\vee\kern0pt}}%%
\def\uhblvee{\vboxtoz{\boldscript\wedge\vss}}%%
\newarrowhead{boldlittlevee}{\mkern1mu\boldscriptaxis\rhvee}{\boldscriptaxis
\lhvee}\dhblvee\uhblvee%%
\fi

%% curly vee head (uses AMS symbols fonts)
\def\rhcvee{\mkern-10mu\succ}%%
\def\lhcvee{\prec\mkern-10mu}%%
\def\dhcvee{\vboxtoz{\vss\hbox{$\curlyvee$}\kern0pt}}%%
\def\uhcvee{\vboxtoz{\hbox{$\curlywedge$}\vss}}%%
\newarrowhead{curlyvee}\rhcvee\lhcvee\dhcvee\uhcvee

%% double vee head %% will probably be withdrawn later
\def\rhvvee{\mkern-13mu\gg}%% 24.8.92 changed 10mu to 13mu
\def\lhvvee{\ll\mkern-13mu}%% to make rule go through
\def\dhvvee{\vboxtoz{\vss\hbox{$\vee$}\kern-.6ex\hbox{$\vee$}\kern0pt}}%%
\def\uhvvee{\vboxtoz{\hbox{$\wedge$}\kern-.6ex \hbox{$\wedge$}\vss}}%%
\newarrowhead{doublevee}\rhvvee\lhvvee\dhvvee\uhvvee

%% open and closed triangles (uses AMS symbols fonts)
\def\triangleup{{\scriptscriptstyle\bigtriangleup}}%%
\def\littletriangledown{{\scriptscriptstyle\triangledown}}%% AMS
\def\rhtriangle{\triangleright\mkern1.2mu}%% 29.1.93
\def\lhtriangle{\triangleleft\mkern.8mu}%%
\def\uhtriangle{\vbox{\kern-.2ex \hbox{$\scriptscriptstyle\bigtriangleup$}%
\kern-.25ex}}%%
%% Changed \scriptstyle\triangledown to \scriptscriptstyle\bigtriangledown
%% at the suggestion of Martin Hofmann (25.11.92) to avoid using AMS symbols
%% and also for compatibility with upward arrow.
\def\dhtriangle{\vbox{\kern-.28ex \hbox{$\scriptscriptstyle\bigtriangledown$}%
\kern-.1ex}}%% 15.1.93 from -.25ex
\def\dhblack{\vbox{\kern-.25ex\nointerlineskip\hbox{$\blacktriangledown$}}}%
%% AMS
\def\uhblack{\vbox{\kern-.25ex\nointerlineskip\hbox{$\blacktriangle$}}}%
%% AMS
\def\dhlblack{\vbox{\kern-.25ex\nointerlineskip\hbox{$\scriptstyle
\blacktriangledown$}}}%% AMS
\def\uhlblack{\vbox{\kern-.25ex\nointerlineskip\hbox{$\scriptstyle
\blacktriangle$}}}%% AMS
\newarrowhead{triangle}\rhtriangle\lhtriangle\dhtriangle\uhtriangle
\newarrowhead{blacktriangle}{\mkern-1mu\blacktriangleright\mkern.4mu}{%
\blacktriangleleft}\dhblack\uhblack\newarrowhead{littleblack}{\mkern-1mu%
\scriptaxis\blacktriangleright}{\scriptaxis\blacktriangleleft\mkern-2mu}%
\dhlblack\uhlblack

%% LaTeX arrowheads
\def\rhla{\hbox{\setbox0=\lnchar55\dimen0=\wd0\kern-.6\dimen0\ht0\z@\raise
\axisheight\box0\kern.1\dimen0}}%%
\def\lhla{\hbox{\setbox0=\lnchar33\dimen0=\wd0\kern.05\dimen0\ht0\z@\raise
\axisheight\box0\kern-.5\dimen0}}%%
\def\dhla{\vboxtoz{\vss\rlap{\lnchar77}}}%%
\def\uhla{\vboxtoz{\setbox0=\lnchar66 \wd0\z@\kern-.15\ht0\box0\vss}}%% 1/93
\newarrowhead{LaTeX}\rhla\lhla\dhla\uhla

%% double LaTeX arrowheads %% will probably be withdrawn later
\def\lhlala{\lhla\kern.3em\lhla}%%
\def\rhlala{\rhla\kern.3em\rhla}%%
\def\uhlala{\hbox{\uhla\raise-.6ex\uhla}}%%
\def\dhlala{\hbox{\dhla\lower-.6ex\dhla}}%%
\newarrowhead{doubleLaTeX}\rhlala\lhlala\dhlala\uhlala

%% circles % \rho is a Greek letter!
\def\hhO{\scriptaxis\bigcirc\mkern.4mu} \def\hho{{\circ}\mkern1.2mu}%
\newarrowhead{o}\hho\hho\circ\circ%%
\newarrowhead{O}\hhO\hhO{\scriptstyle\bigcirc}{\scriptstyle\bigcirc}%%

%% crosses
\def\rhtimes{\mkern-5mu{\times}\mkern-.8mu}\def\lhtimes{\mkern-.8mu{\times}%
\mkern-5mu}\def\uhtimes{\setbox0=\hbox{$\times$}\ht0\axisheight\dp0-\ht0%
\lower\ht0\box0 }\def\dhtimes{\setbox0=\hbox{$\times$}\ht0\axisheight\box0 }%
\newarrowhead{X}\rhtimes\lhtimes\dhtimes\uhtimes\newarrowhead+++++

%% empty head {} is also available

%% Y from stmaryrd (vertical ones still need large adjustment)
\newarrowhead{Y}{\mkern-3mu\Yright}{\Yleft\mkern-3mu}\Yup\Ydown

%%       **********  H E A D S  with  S H A F T S  ***********

%% little arrow with shaft
\newarrowhead{->}\rightarrow\leftarrow\downarrow\uparrow

%% arrow with double shaft
%%\newarrowhead{=>}\Rightarrow\Leftarrow\Downarrow\Uparrow
\newarrowhead{=>}\Rightarrow\Leftarrow{\@cmex7F}{\@cmex7E}

%% harpoon with shaft (trailing up/left can be changed to down/right)
\newarrowhead{harpoon}\rightharpoonup\leftharpoonup\downharpoonleft
\upharpoonleft

%% little double-headed arrow with shaft (uses AMS symbols fonts)
\def\twoheaddownarrow{\rlap{$\downarrow$}\raise-.5ex\hbox{$\downarrow$}}%%
\def\twoheaduparrow{\rlap{$\uparrow$}\raise.5ex\hbox{$\uparrow$}}%%
\newarrowhead{->>}\twoheadrightarrow\twoheadleftarrow\twoheaddownarrow
\twoheaduparrow

%%       **********  T A I L S ***********

%% vee tail
\def\rtvee{\greaterthan}%%
\def\ltvee{\mkern-1mu{\lessthan}\mkern.4mu}%% \mkern added 15.1.93
\def\dtvee{\vee}%%
\def\utvee{\wedge}%%
\newarrowtail{vee}\greaterthan\ltvee\vee\wedge

%% little vee tail
\newarrowtail{littlevee}{\scriptaxis\greaterthan}{\mkern-1mu\scriptaxis
\lessthan}{\scriptstyle\vee}{\scriptstyle\wedge}\ifx\boldmath\undefined
\newarrowtail{boldlittlevee}{\scriptaxis\greaterthan}{\mkern-1mu\scriptaxis
\lessthan}{\scriptstyle\vee}{\scriptstyle\wedge}\else\newarrowtail{%
boldlittlevee}{\boldscriptaxis\greaterthan}{\mkern-1mu\boldscriptaxis
\lessthan}{\boldscript\vee}{\boldscript\wedge}\fi

%% curly vee tail (uses AMS symbols fonts)
\newarrowtail{curlyvee}\succ{\mkern-1mu{\prec}\mkern.4mu}\curlyvee\curlywedge

%% open and closed triangle tails (uses AMS symbols fonts)
\def\rttriangle{\mkern1.2mu\triangleright}%% 29.1.93
\newarrowtail{triangle}\rttriangle\lhtriangle\dhtriangle\uhtriangle
\newarrowtail{blacktriangle}\blacktriangleright{\mkern-1mu\blacktriangleleft
\mkern.4mu}\dhblack\uhblack\newarrowtail{littleblack}{\scriptaxis
\blacktriangleright\mkern-2mu}{\mkern-1mu\scriptaxis\blacktriangleleft}%
\dhlblack\uhlblack

%% LaTeX tails
\def\rtla{\hbox{\setbox0=\lnchar55\dimen0=\wd0\kern-.5\dimen0\ht0\z@\raise
\axisheight\box0\kern-.2\dimen0}}%%
\def\ltla{\hbox{\setbox0=\lnchar33\dimen0=\wd0\kern-.15\dimen0\ht0\z@\raise
\axisheight\box0\kern-.5\dimen0}}%%
\def\dtla{\vbox{\setbox0=\rlap{\lnchar77}\dimen0=\ht0\kern-.7\dimen0\box0%
\kern-.1\dimen0}}%% 15.1.93 from -.6
\def\utla{\vbox{\setbox0=\rlap{\lnchar66}\dimen0=\ht0\kern-.1\dimen0\box0%
\kern-.6\dimen0}}%%
\newarrowtail{LaTeX}\rtla\ltla\dtla\utla

%% double vee tail %% will probably be withdrawn later
\def\rtvvee{\gg\mkern-3mu}%%
\def\ltvvee{\mkern-3mu\ll}%%
\def\dtvvee{\vbox{\hbox{$\vee$}\kern-.6ex \hbox{$\vee$}\vss}}%%
\def\utvvee{\vbox{\vss\hbox{$\wedge$}\kern-.6ex \hbox{$\wedge$}\kern\z@}}%%
\newarrowtail{doublevee}\rtvvee\ltvvee\dtvvee\utvvee

%% double LaTeX tails %% will probably be withdrawn later
\def\ltlala{\ltla\kern.3em\ltla}%%
\def\rtlala{\rtla\kern.3em\rtla}%%
\def\utlala{\hbox{\utla\raise-.6ex\utla}}%%
\def\dtlala{\hbox{\dtla\lower-.6ex\dtla}}%%
\newarrowtail{doubleLaTeX}\rtlala\ltlala\dtlala\utlala

%% bar (as in \mapsto)
\def\utbar{\vrule height 0.093ex depth0pt width 0.4em}%%
\let\dtbar\utbar%%
\def\rtbar{\mkern1.5mu\vrule height 1.1ex depth.06ex width .04em\mkern1.5mu}%
%%
\let\ltbar\rtbar%%
\newarrowtail{mapsto}\rtbar\ltbar\dtbar\utbar%%
\newarrowtail{|}\rtbar\ltbar\dtbar\utbar%%ascii vertical bar (|)

%% hooks (as in \into): choice of after/above and before/below

\def\rthooka{\raisehook{}+\subset\mkern-1mu}%%
\def\lthooka{\mkern-1mu\raisehook{}+\supset}%%
\def\rthookb{\raisehook{}-\subset\mkern-2mu}%%
\def\lthookb{\mkern-1mu\raisehook{}-\supset}%%

\def\dthooka{\shifthook{}+\cap}%%
\def\dthookb{\shifthook{}-\cap}%%
\def\uthooka{\shifthook{}+\cup}%%
\def\uthookb{\shifthook{}-\cup}%%

\newarrowtail{hooka}\rthooka\lthooka\dthooka\uthooka\newarrowtail{hookb}%
\rthookb\lthookb\dthookb\uthookb

\ifx\boldmath\undefined\newarrowtail{boldhooka}\rthooka\lthooka\dthooka
\uthooka\newarrowtail{boldhookb}\rthookb\lthookb\dthookb\uthookb\newarrowtail
{boldhook}\rthooka\lthooka\dthookb\uthooka\else\def\rtbhooka{\raisehook
\boldmath+\subset\mkern-1mu}%%
\def\ltbhooka{\mkern-1mu\raisehook\boldmath+\supset}%%
\def\rtbhookb{\raisehook\boldmath-\subset\mkern-2mu}%%
\def\ltbhookb{\mkern-1mu\raisehook\boldmath-\supset}%%
\def\dtbhooka{\shifthook\boldmath+\cap}%%
\def\dtbhookb{\shifthook\boldmath-\cap}%%
\def\utbhooka{\shifthook\boldmath+\cup}%%
\def\utbhookb{\shifthook\boldmath-\cup}%%
\newarrowtail{boldhooka}\rtbhooka\ltbhooka\dtbhooka\utbhooka\newarrowtail{%
boldhookb}\rtbhookb\ltbhookb\dtbhookb\utbhookb\newarrowtail{boldhook}%
\rtbhooka\ltbhooka\dtbhooka\utbhooka\fi

%% square-ended hooks (used for closed subsets in ``lifting and gluing'')
\def\dtsqhooka{\shifthook{}+\sqcap}%%
\def\dtsqhookb{\shifthook{}-\sqcap}%%
\def\ltsqhooka{\mkern-1mu\raisehook{}+\sqsupset}%%
\def\ltsqhookb{\mkern-1mu\raisehook{}-\sqsupset}%%
\def\rtsqhooka{\raisehook{}+\sqsubset\mkern-1mu}%%
\def\rtsqhookb{\raisehook{}-\sqsubset\mkern-2mu}%%
\def\utsqhooka{\shifthook{}+\sqcup}%%
\def\utsqhookb{\shifthook{}-\sqcup}%%
\newarrowtail{sqhook}\rtsqhooka\ltsqhooka\dtsqhooka\utsqhooka

%% the following seem the better choices at 10pt & 300dpi
\newarrowtail{hook}\rthooka\lthookb\dthooka\uthooka\newarrowtail{C}\rthooka
\lthookb\dthooka\uthooka

%% circles
\newarrowtail{o}\hho\hho\circ\circ%%
\newarrowtail{O}\hhO\hhO{\scriptstyle\bigcirc}{\scriptstyle\bigcirc}%%

%% crosses
\newarrowtail{X}\lhtimes\rhtimes\uhtimes\dhtimes\newarrowtail+++++

%% empty tail {} is also available

%% Y from stmaryrd (vertical ones still need adjustment)
\newarrowtail{Y}\Yright\Yleft\Yup\Ydown

%% harpoon with shaft (trailing up/left can be changed to down/right)
\newarrowtail{harpoon}\leftharpoondown\rightharpoondown\upharpoonright
\downharpoonright

%%      **********  F I L L E R S ***********

%% shortening is up to 0.15em=2.7mu horiz and 0.35ex vertically at each end.

%% dot {.}, single rule {-} and empty {} fillers are also available

%% double and triple lines
%%\newarrowfiller{=}==\Vert\Vert%%
\newarrowfiller{=}=={\@cmex77}{\@cmex77}%% 16.1.93
\def\vfthree{\mid\!\!\!\mid\!\!\!\mid}%%ascii
\newarrowfiller{3}\equiv\equiv\vfthree\vfthree

%% dashed line
\def\vfdashstrut{\vrule width0pt height1.3ex depth0.7ex}%%
\def\vfthedash{\vrule width\CD@LF height0.6ex depth 0pt}%%
\def\hfthedash{\CD@AJ\vrule\horizhtdp width 0.26em}%%
\def\hfdash{\mkern5.5mu\hfthedash\mkern5.5mu}%%
\def\vfdash{\vfdashstrut\vfthedash}%%
\newarrowfiller{dash}\hfdash\hfdash\vfdash\vfdash

%%      ************* M I D D L E S ************

%% plus
\newarrowmiddle+++++

%%      ************* D I A G O N A L S ************

%% simple arrow heads
%%\def\nwhTO{\nwarrow\mkern-1mu}%%
%%\def\nehTO{\mkern-.1mu\nearrow}%%
%%\def\sehTO{\searrow\mkern-.02mu}%%
%%\def\swhTO{\mkern-.8mu\swarrow}%%

%%======================================================================%
%%                                                                      %
%%      (24) ARROW COMMANDS                                             %
%%                                                                      %
%%======================================================================%

%% change to \iftrue to get mixed heads
\iffalse%%
\newarrow{To}----{vee}%%
\newarrow{Arr}----{LaTeX}%%
\newarrow{Dotsto}....{vee}%%
\newarrow{Dotsarr}....{LaTeX}%%
\newarrow{Dashto}{}{dash}{}{dash}{vee}%%
\newarrow{Dasharr}{}{dash}{}{dash}{LaTeX}%%
\newarrow{Mapsto}{mapsto}---{vee}%%
\newarrow{Mapsarr}{mapsto}---{LaTeX}%%
\newarrow{IntoA}{hooka}---{vee}%%
\newarrow{IntoB}{hookb}---{vee}%%
\newarrow{Embed}{vee}---{vee}%%
\newarrow{Emarr}{LaTeX}---{LaTeX}%%
\newarrow{Onto}----{doublevee}%%
\newarrow{Dotsonarr}....{doubleLaTeX}%%
\newarrow{Dotsonto}....{doublevee}%%
\newarrow{Dotsonarr}....{doubleLaTeX}%%
\else%%
\newarrow{To}---->%%
\newarrow{Arr}---->%%
\newarrow{Dotsto}....>%%
\newarrow{Dotsarr}....>%%
\newarrow{Dashto}{}{dash}{}{dash}>%%
\newarrow{Dasharr}{}{dash}{}{dash}>%%
\newarrow{Mapsto}{mapsto}--->%%
\newarrow{Mapsarr}{mapsto}--->%%
\newarrow{IntoA}{hooka}--->%%
\newarrow{IntoB}{hookb}--->%%
\newarrow{Embed}>--->%%
\newarrow{Emarr}>--->%%
\newarrow{Onto}----{>>}%%
\newarrow{Dotsonarr}....{>>}%%
\newarrow{Dotsonto}....{>>}%%
\newarrow{Dotsonarr}....{>>}%%
\fi%%

\newarrow{Implies}===={=>}%% minimum cell height 9.5pt
\newarrow{Project}----{triangle}%%
\newarrow{Pto}----{harpoon}%% partial function
\newarrow{Relto}{harpoon}---{harpoon}%% binary relation

\newarrow{Eq}=====%%
\newarrow{Line}-----%%
\newarrow{Dots}.....%%
\newarrow{Dashes}{}{dash}{}{dash}{}%%

%% square hooked arrow (used in my ``gluing and lifting'' paper)
\newarrow{SquareInto}{sqhook}--->

%% braces and parentheses
%% \newarrow gives inappropriate directions, so we change the names
%% the vertical filler is too far to the right; horizontal too high
%% the vertical middles are too low with midvshaft
%% maybe we'll add square brackets and the integral sign one day
\newarrowhead{cmexbra}{\@cmex7B}{\@cmex7C}{\@cmex3B}{\@cmex38}%%
\newarrowtail{cmexbra}{\@cmex7A}{\@cmex7D}{\@cmex39}{\@cmex3A}%%
\newarrowmiddle{cmexbra}{\braceru\bracelu}{\bracerd\braceld}{\vcenter{%
\hbox@maths{\@cmex3D\mkern-2mu}}}%% right
{\vcenter{\hbox@maths{\mkern2mu\@cmex3C}}}%% left
\newarrow{@brace}{cmexbra}-{cmexbra}-{cmexbra}%% braces
\newarrow{@parenth}{cmexbra}---{cmexbra}%% straight parentheses
\def\rightBrace{\d@brace[thick,cmex]}%%ASCII square brackets []
\def\leftBrace{\u@brace[thick,cmex]}%%ASCII square brackets []
\def\upperBrace{\r@brace[thick,cmex]}%%ASCII square brackets []
\def\lowerBrace{\l@brace[thick,cmex]}%%ASCII square brackets []
\def\rightParenth{\d@parenth[thick,cmex]}%%ASCII square brackets []
\def\leftParenth{\u@parenth[thick,cmex]}%%ASCII square brackets []
\def\upperParenth{\r@parenth[thick,cmex]}%%ASCII square brackets []
\def\lowerParenth{\l@parenth[thick,cmex]}%%ASCII square brackets []

%% synonyms for reverse compatibility

\let\uFrom\uTo%%
\let\lFrom\lTo%%
\let\uDotsfrom\uDotsto%%
\let\lDotsfrom\lDotsto%%
\let\uDashfrom\uDashto%%
\let\lDashfrom\lDashto%%
\let\uImpliedby\uImplies%%
\let\lImpliedby\lImplies%%
\let\uMapsfrom\uMapsto%%
\let\lMapsfrom\lMapsto%%
\let\lOnfrom\lOnto%%
\let\uOnfrom\uOnto%%
\let\lPfrom\lPto%%
\let\uPfrom\uPto%%

\let\uInfromA\uIntoA%%
\let\uInfromB\uIntoB%%
\let\lInfromA\lIntoA%%
\let\lInfromB\lIntoB%%
\let\rInto\rIntoA%%
\let\lInto\lIntoA%%
\let\dInto\dIntoB%%
\let\uInto\uIntoA%%
\let\ruInto\ruIntoA%%
\let\luInto\luIntoA%%
\let\rdInto\rdIntoA%%
\let\ldInto\ldIntoA%%
%%
\let\hEq\rEq%%
\let\vEq\uEq%%
\let\hLine\rLine%%
\let\vLine\uLine%%
\let\hDots\rDots%%
\let\vDots\uDots%%
\let\hDashes\rDashes%%
\let\vDashes\uDashes%%

%%=========================================================================%
%% The following are included for reverse compatibility only.
%%=========================================================================%
\let\NW\luTo\let\NE\ruTo\let\SW\ldTo\let\SE\rdTo\def\nNW{\luTo(2,3)}\def\nNE{%
\ruTo(2,3)}%%ascii
\def\sSW{\ldTo(2,3)}\def\sSE{\rdTo(2,3)}%%ascii
\def\wNW{\luTo(3,2)}\def\eNE{\ruTo(3,2)}%%ascii
\def\wSW{\ldTo(3,2)}\def\eSE{\rdTo(3,2)}%%ascii
\def\NNW{\luTo(2,4)}\def\NNE{\ruTo(2,4)}%%ascii
\def\SSW{\ldTo(2,4)}\def\SSE{\rdTo(2,4)}%%ascii
\def\WNW{\luTo(4,2)}\def\ENE{\ruTo(4,2)}%%ascii
\def\WSW{\ldTo(4,2)}\def\ESE{\rdTo(4,2)}%%ascii
\def\NNNW{\luTo(2,6)}\def\NNNE{\ruTo(2,6)}%%ascii
\def\SSSW{\ldTo(2,6)}\def\SSSE{\rdTo(2,6)}%%ascii
\def\WWNW{\luTo(6,2)}\def\EENE{\ruTo(6,2)}%%ascii
\def\WWSW{\ldTo(6,2)}\def\EESE{\rdTo(6,2)}%%ascii

\let\NWd\luDotsto\let\NEd\ruDotsto\let\SWd\ldDotsto\let\SEd\rdDotsto\def\nNWd
{\luDotsto(2,3)}\def\nNEd{\ruDotsto(2,3)}%%ascii
\def\sSWd{\ldDotsto(2,3)}\def\sSEd{\rdDotsto(2,3)}%%ascii
\def\wNWd{\luDotsto(3,2)}\def\eNEd{\ruDotsto(3,2)}%%ascii
\def\wSWd{\ldDotsto(3,2)}\def\eSEd{\rdDotsto(3,2)}%%ascii
\def\NNWd{\luDotsto(2,4)}\def\NNEd{\ruDotsto(2,4)}%%ascii
\def\SSWd{\ldDotsto(2,4)}\def\SSEd{\rdDotsto(2,4)}%%ascii
\def\WNWd{\luDotsto(4,2)}\def\ENEd{\ruDotsto(4,2)}%%ascii
\def\WSWd{\ldDotsto(4,2)}\def\ESEd{\rdDotsto(4,2)}%%ascii
\def\NNNWd{\luDotsto(2,6)}\def\NNNEd{\ruDotsto(2,6)}%%ascii
\def\SSSWd{\ldDotsto(2,6)}\def\SSSEd{\rdDotsto(2,6)}%%ascii
\def\WWNWd{\luDotsto(6,2)}\def\EENEd{\ruDotsto(6,2)}%%ascii
\def\WWSWd{\ldDotsto(6,2)}\def\EESEd{\rdDotsto(6,2)}%%ascii

\let\NWl\luLine\let\NEl\ruLine\let\SWl\ldLine\let\SEl\rdLine\def\nNWl{\luLine
(2,3)}\def\nNEl{\ruLine(2,3)}%%ascii
\def\sSWl{\ldLine(2,3)}\def\sSEl{\rdLine(2,3)}%%ascii
\def\wNWl{\luLine(3,2)}\def\eNEl{\ruLine(3,2)}%%ascii
\def\wSWl{\ldLine(3,2)}\def\eSEl{\rdLine(3,2)}%%ascii
\def\NNWl{\luLine(2,4)}\def\NNEl{\ruLine(2,4)}%%ascii
\def\SSWl{\ldLine(2,4)}\def\SSEl{\rdLine(2,4)}%%ascii
\def\WNWl{\luLine(4,2)}\def\ENEl{\ruLine(4,2)}%%ascii
\def\WSWl{\ldLine(4,2)}\def\ESEl{\rdLine(4,2)}%%ascii
\def\NNNWl{\luLine(2,6)}\def\NNNEl{\ruLine(2,6)}%%ascii
\def\SSSWl{\ldLine(2,6)}\def\SSSEl{\rdLine(2,6)}%%ascii
\def\WWNWl{\luLine(6,2)}\def\EENEl{\ruLine(6,2)}%%ascii
\def\WWSWl{\ldLine(6,2)}\def\EESEl{\rdLine(6,2)}%%ascii

\let\NWld\luDots\let\NEld\ruDots\let\SWld\ldDots\let\SEld\rdDots\def\nNWld{%
\luDots(2,3)}\def\nNEld{\ruDots(2,3)}%%ascii
\def\sSWld{\ldDots(2,3)}\def\sSEld{\rdDots(2,3)}%%ascii
\def\wNWld{\luDots(3,2)}\def\eNEld{\ruDots(3,2)}%%ascii
\def\wSWld{\ldDots(3,2)}\def\eSEld{\rdDots(3,2)}%%ascii
\def\NNWld{\luDots(2,4)}\def\NNEld{\ruDots(2,4)}%%ascii
\def\SSWld{\ldDots(2,4)}\def\SSEld{\rdDots(2,4)}%%ascii
\def\WNWld{\luDots(4,2)}\def\ENEld{\ruDots(4,2)}%%ascii
\def\WSWld{\ldDots(4,2)}\def\ESEld{\rdDots(4,2)}%%ascii
\def\NNNWld{\luDots(2,6)}\def\NNNEld{\ruDots(2,6)}%%ascii
\def\SSSWld{\ldDots(2,6)}\def\SSSEld{\rdDots(2,6)}%%ascii
\def\WWNWld{\luDots(6,2)}\def\EENEld{\ruDots(6,2)}%%ascii
\def\WWSWld{\ldDots(6,2)}\def\EESEld{\rdDots(6,2)}%%ascii

\let\NWe\luEmbed\let\NEe\ruEmbed\let\SWe\ldEmbed\let\SEe\rdEmbed\def\nNWe{%
\luEmbed(2,3)}\def\nNEe{\ruEmbed(2,3)}%%ascii
\def\sSWe{\ldEmbed(2,3)}\def\sSEe{\rdEmbed(2,3)}%%ascii
\def\wNWe{\luEmbed(3,2)}\def\eNEe{\ruEmbed(3,2)}%%ascii
\def\wSWe{\ldEmbed(3,2)}\def\eSEe{\rdEmbed(3,2)}%%ascii
\def\NNWe{\luEmbed(2,4)}\def\NNEe{\ruEmbed(2,4)}%%ascii
\def\SSWe{\ldEmbed(2,4)}\def\SSEe{\rdEmbed(2,4)}%%ascii
\def\WNWe{\luEmbed(4,2)}\def\ENEe{\ruEmbed(4,2)}%%ascii
\def\WSWe{\ldEmbed(4,2)}\def\ESEe{\rdEmbed(4,2)}%%ascii
\def\NNNWe{\luEmbed(2,6)}\def\NNNEe{\ruEmbed(2,6)}%%ascii
\def\SSSWe{\ldEmbed(2,6)}\def\SSSEe{\rdEmbed(2,6)}%%ascii
\def\WWNWe{\luEmbed(6,2)}\def\EENEe{\ruEmbed(6,2)}%%ascii
\def\WWSWe{\ldEmbed(6,2)}\def\EESEe{\rdEmbed(6,2)}%%ascii

\let\NWo\luOnto\let\NEo\ruOnto\let\SWo\ldOnto\let\SEo\rdOnto\def\nNWo{\luOnto
(2,3)}\def\nNEo{\ruOnto(2,3)}%%ascii
\def\sSWo{\ldOnto(2,3)}\def\sSEo{\rdOnto(2,3)}%%ascii
\def\wNWo{\luOnto(3,2)}\def\eNEo{\ruOnto(3,2)}%%ascii
\def\wSWo{\ldOnto(3,2)}\def\eSEo{\rdOnto(3,2)}%%ascii
\def\NNWo{\luOnto(2,4)}\def\NNEo{\ruOnto(2,4)}%%ascii
\def\SSWo{\ldOnto(2,4)}\def\SSEo{\rdOnto(2,4)}%%ascii
\def\WNWo{\luOnto(4,2)}\def\ENEo{\ruOnto(4,2)}%%ascii
\def\WSWo{\ldOnto(4,2)}\def\ESEo{\rdOnto(4,2)}%%ascii
\def\NNNWo{\luOnto(2,6)}\def\NNNEo{\ruOnto(2,6)}%%ascii
\def\SSSWo{\ldOnto(2,6)}\def\SSSEo{\rdOnto(2,6)}%%ascii
\def\WWNWo{\luOnto(6,2)}\def\EENEo{\ruOnto(6,2)}%%ascii
\def\WWSWo{\ldOnto(6,2)}\def\EESEo{\rdOnto(6,2)}%%ascii

\let\NWod\luDotsonto\let\NEod\ruDotsonto\let\SWod\ldDotsonto\let\SEod
\rdDotsonto\def\nNWod{\luDotsonto(2,3)}\def\nNEod{\ruDotsonto(2,3)}%%ascii
\def\sSWod{\ldDotsonto(2,3)}\def\sSEod{\rdDotsonto(2,3)}%%ascii
\def\wNWod{\luDotsonto(3,2)}\def\eNEod{\ruDotsonto(3,2)}%%ascii
\def\wSWod{\ldDotsonto(3,2)}\def\eSEod{\rdDotsonto(3,2)}%%ascii
\def\NNWod{\luDotsonto(2,4)}\def\NNEod{\ruDotsonto(2,4)}%%ascii
\def\SSWod{\ldDotsonto(2,4)}\def\SSEod{\rdDotsonto(2,4)}%%ascii
\def\WNWod{\luDotsonto(4,2)}\def\ENEod{\ruDotsonto(4,2)}%%ascii
\def\WSWod{\ldDotsonto(4,2)}\def\ESEod{\rdDotsonto(4,2)}%%ascii
\def\NNNWod{\luDotsonto(2,6)}\def\NNNEod{\ruDotsonto(2,6)}%%ascii
\def\SSSWod{\ldDotsonto(2,6)}\def\SSSEod{\rdDotsonto(2,6)}%%ascii
\def\WWNWod{\luDotsonto(6,2)}\def\EENEod{\ruDotsonto(6,2)}%%ascii
\def\WWSWod{\ldDotsonto(6,2)}\def\EESEod{\rdDotsonto(6,2)}%%ascii

%%======================================================================%
%%                                                                      %
%%      (25) MISCELLANEOUS                                              %
%%                                                                      %
%%======================================================================%

\def\labelstyle{%%
\ifincommdiag%%
\textstyle%%
\else%%
\scriptstyle%%
\fi}%%
\let\objectstyle\displaystyle

\newdiagramgrid{pentagon}{0.618034,0.618034,1,1,1,1,0.618034,0.618034}{1.%
17557,1.17557,1.902113,1.902113}

\newdiagramgrid{perspective}{0.75,0.75,1.1,1.1,0.9,0.9,0.95,0.95,0.75,0.75}{0%
.75,0.75,1.1,1.1,0.9,0.9}

\diagramstyle[%%ascii open square bracket
dpi=300,%%              office laserwriters are usually 300 dots per inch
vmiddle,nobalance,%%    vertical and horizontal positioning
loose,%%                allow rows and columns to stretch
thin,%%                 line10 arrows; default rule thickness (TeXbook p447)
pilespacing=10pt,%
%%     parallel vertical separation (horizontals: half this)
shortfall=4pt,%%        distance between arrowheads and their targets
%% The following are defaulted on entry to the diagram itself.
%% l>=2em               minimum length of horizontal arrow shafts in text
%% l>=1em               ditto in diagrams
%% size=3em             cell size
%% heads=LaTeX          arrowheads
]%%ascii close square bracket

%% process options to LaTeX2e's \usepackage[options]{diagrams}
\ifx\ProcessOptions\undefined\else\CD@PK\ProcessOptions\relax\CD@FF\CD@e\fi
\fi

\cdrestoreat%% restore old category code for @ etc
%%============================== THE END ==============================

\dimen0 200pt \dimen1 210pt \dimen2 220pt \dimen3 230pt \dimen4 240pt \dimen5
250pt \dimen6 260pt \dimen7 270pt \dimen8 280pt \dimen9 290pt

%% This is the end of Paul Taylor's commutative diagrams package.


---------------0507150823876
Content-Type: application/x-tex; name="nc-floquet-v2.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="nc-floquet-v2.tex"

%-------------------------------------------------------------------
% nc-floquet-v2.tex Non-commutative Floquet Theory and Spectral gaps
%                   Fernando Lled'o, Olaf Post
% Comment:          Some errors corrected (Sec. 4.3)
% 2003-04-28        Institut f"ur Reine und Angewandte Mathematik
%                   RWTH Aachen, Germany
% last change       email: lledo@iram.rwth-aachen.de, post@iram.rwth-aachen.de
% 
% 2005-07-13, 2005-03-03, 2004-11-11, 2004-10-10, 2004-08-26, 2004-06-24, 
% 2004-03-01, 2003-10-24, 2003-04-28
%--------------------------------------------------------------------
\documentclass[12pt,reqno]{amsart}  % leqno: eq-numbers on right side

\listfiles            % file list in .log file
\usepackage{a4}
\usepackage[german,english]{babel}
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% begin of the document
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\begin{document}

\title[Existence of spectral gaps and residually finite groups]{Existence 
       of spectral gaps, covering manifolds and residually finite groups}

\author{Fernando Lled\'o}
\address{Institut f\"ur Reine und Angewandte Mathematik,
       Rheinisch-Westf\"alische Technische Hochschule Aachen,
       Templergraben 55,
       D-52062 Aachen,
       Germany}
\email{lledo@iram.rwth-aachen.de}\email{post@iram.rwth-aachen.de}
\author{Olaf Post}
% \address{Institut f\"ur Reine und Angewandte Mathematik,
%        Rheinisch-Westf\"alische Technische Hochschule Aachen,
%        Templergraben 55,
%        52062 Aachen,
%        Germany}
%\email{post@iram.rwth-aachen.de}
\date{\today}

%------------------------------------------------------------
% Subject classifications
%------------------------------------------------------------
%Subjclass contains the Classification of the paper following the 1991
%Mathematics Subject Classifications

\subjclass{58J50, 35P15, 20E26, 57M10}
\keywords{covering manifolds, spectral gaps, residually finite groups,
          min-max principle}

%------------------------------------------------------------
% Abstract.
%------------------------------------------------------------

\begin{abstract}
  In this paper we present two construction procedures of covering
  manifolds $X\to M$ with residually finite covering transformation
  group $\Gamma$ such that the spectrum of the Laplacian $\laplacian
  X$ has at least a prescribed number $n$ of spectral gaps, $n \in
  \N$.  If $\Gamma$ has a positive Kadison constant, then we can apply
  results by Br\"uning and Sunada to the manifolds constructed here.
  In this case $\spec \laplacian X$ has, in addition, band structure
  and there is an asymptotic estimate for the number $\mathcal
  N(\lambda)$ of components of $\spec \laplacian X$ that intersect the
  interval $[0,\lambda]$. Finally, we present several classes of
  examples of residually finite groups that fit with our construction
  procedure and study their mutual relations.
\end{abstract}

\maketitle

%------------------------------------------------------------
% The main part
%------------------------------------------------------------

%------------------------------------------------------------
%------------------------------------------------------------
%
\section{Introduction}
%
%------------------------------------------------------------
%------------------------------------------------------------
\label{sec:intro}

The analysis of spectral properties of a Schr\"odinger operator
$H=-\Delta+V$ on $\R^d$, $d \ge 2$, is radically simplified if there
is an action of a discrete \emph{abelian} group $\Gamma_{\mathrm
  {ab}}=\Z^d$ on $\R^d$ commuting with $H$.  In this case, the
potential is completely determined on a (relatively compact)
fundamental domain $D$ and one can decompose $H$ as a direct integral
over the character group $\widehat \Gamma_{\mathrm {ab}}$. If the
potential $V$ has high barriers near the boundary $\bd D$, then gaps
appear in the spectrum $\spec H$. Here, a \emph{spectral gap} means an
interval $(\alpha, \beta)$ with $(\alpha, \beta) \cap \spec H =
\emptyset$ and $\alpha, \beta \in \spec H$.  For a review on high
contrast media see~\cite{hempel-post:03}.

The Laplace operator $\laplacian X$ on a Riemannian manifold $(X,g)$
has a formal analogy with the Sch\"odinger operator due to the
influence of the non-trivial geometry. Therefore, it is natural to ask
whether one can generate some effects of the periodic potential $V$ by
purely geometrical means. Specifically, can we generate spectral gaps
in $\spec \laplacian X$ for a covering manifold $X$ with
transformation group $\Gamma$ by deforming appropriately the metric
$g$?

The positive answer to this question is based on the following
intuitive idea. The effect of high barriers of the potential on the
spectrum is replaced by the following geometrical picture: One
decouples in this case neighbouring translates of a the fundamental
domain by means of a metric $g_\eps$ on $X$, where the junctions
between different translates of a fundamental domain are small enough
(of order $\eps$, see Fig.~\ref{fig:per-mfd}).
\begin{figure}[h]
  \begin{center}
%----------------------------------------------------------------------
%    \input{per-mfd.pstex_t}
%----------------------------------------------------------------------
\begin{picture}(0,0)
  \includegraphics{per-mfd.eps}
\end{picture}%
\setlength{\unitlength}{4144sp}
\begin{picture}(5244,1501)(259,-695)
  \put(811,-556){$X$}
  \put(4501,-556){$D$}
  \put(2791,-646){$\eps$}
\end{picture}
%---------------------------------------------------------------------
    \caption{A covering manifold $X$ with fundamental domain $D$. The
      junctions between different translates of $D$ are of order
      $\eps$.}
    \label{fig:per-mfd}
  \end{center}
\end{figure}
The structure of the discrete group $\Gamma$ enters crucially when we
implement mathematically the previous idea.  In fact, for abelian
groups the existence of covering manifolds with a prescribed number $n
\in \N$ of spectral gaps was shown in~\cite{post:03a} (cf.~also the
references therein and~\cite{beg:03}). The methods presented in the
former paper (based on Floquet theory) can be extended to
(non-abelian) type~I groups (cf.~\cite{lledo-post:pre04a} and
Section~\ref{sec:type.I}). The reason for this relies on the following
facts: first, the dual $\hG$ (i.e., the set of all equivalence classes
of continuous irreducible representations) of a type~I group is a nice
measure space (\emph{smooth} in the terminology
of~\cite[Ch.~2]{mackey:76}) and, second, the fibres in the direct
integral decomposition over $\hG$ of the right regular representation
of $\Gamma$ are finite-dimensional.

In this paper we will present a new method to show the existence of a
prescribed number of spectral gaps of $\laplacian X$ for covering
manifolds $X \to M$ with residually finite covering transformation
groups. This is a much wider class of groups including type~I groups,
but also free groups or Fuchsian groups.  Roughly speaking, a
residually finite group has a lot of normal subgroups with finite
index. Geometrically, this is equivalent to the fact that one can
approximate the covering $X \to M$ by a sequence of finite coverings
$M_i \to M$ (cf.\ Section~\ref{sec:res.fin} for further details); such
coverings are also called a \emph{tower of coverings}.  The relation
between the spectrum of $\laplacian X$ and the spectra of $\laplacian
{M_i}$ was established in~\cite{ass:94,adachi:95}.  Moreover, if the
group $\Gamma$ is amenable, then $\spec \laplacian X$ is completely
determined by $\spec \laplacian {M_i}$, $i\in\N$
(cf.~Theorem~\ref{thm:res.fin}).

Under a further assumption, namely, that $\Gamma$ has a positive
Kadison constant (cf.\ Section~\ref{sec:kadison}), we can apply
results by Sunada and Br\"uning \cite{sunada:92,bruening-sunada:92} to
the manifolds $(X,g_\varepsilon)$ constructed here: We obtain in this
case, that $\spec \laplacian X$ has band structure and that there is
an asymptotic upper \emph{and} lower estimate for the number $\al
N(\lambda)$ of components of the spectrum that intersect the interval
$[0,\lambda]$.

To prove the existence of spectral gaps for $\laplacian X$, where
$(X,g_\eps)$ is the covering manifold mentioned before, one begins by
studying the spectral behaviour of the Dirichlet, respectively, Neumann
Laplacian $\laplacian D$ on the fundamental domain $D$. Since their
spectrum is purely discrete we can introduce the following intervals
\begin{displaymath}
  I_k(\eps) := [\EWN k (\eps), \EWD k (\eps)]
\end{displaymath}
which are non-empty due to the min-max principle. Here,
$\lambda_k^{+/-}(\eps)$ denotes the $k$-th Dirichlet, resp., Neumann
eigenvalue written in ascending order and counted according to
multiplicity. One can show that any of these intervals intersects
(non-trivially) $\spec \laplacian X$ (cf.~Theorem~\ref{thm:spectrum}).
Moreover, by tuning appropriately the parameter $\eps$ it is possible
to show that the union
\begin{displaymath}
  I(\eps) := \bigcup_{k \in \N} I_k(\eps)
\end{displaymath}
is a closed set having at least $n$ gaps, i.e., $n+1$ components as a
subset of $[0, \infty)$. In a second step, and using either the fact
that $\Gamma$ is type~I or residually finite, we can show that
\begin{displaymath}
  \spec \laplacian {(X, g_\eps)} \subset I(\eps)
  \quad \text{and} \quad
  I_k(\eps) \cap \spec \laplacian {(X, g_\eps)} \ne \emptyset
\end{displaymath}
for each $k$, hence $\spec \laplacian {(X, g_\eps)}$ has at least a
finite number of spectral gaps.

Note that the manifolds constructed in this paper are, in a sense,
complementary to those considered e.g.\ by Mazzeo or
Friedlander~\cite{friedlander:91,mazzeo:91}. In fact, these authors
proved that $\EWN {k+1}(D) \le \EWD k(D)$ for all $k \in \N$, i.e,
that $I_k \cap I_{k+1} \ne \emptyset$ for all $k \in \N$ provided $D$
is an open subset of $\R^n$ or a Riemannian symmetric space of
non-compact type. On such a space, we have a priory no information on
the existence of gaps. The authors use the Dirichlet-to-Neumann
operator and appropriate test functions to show that the Neumann
eigenvalue counting function is strictly greater than the
corresponding Dirichlet one.

We consider in the present paper $\laplacian X$ as a prototype of an
elliptic operator and avoid the use of a potential $V$ in order to
analyse only the effect of geometry on $\spec \laplacian X$. Of
course, our methods and results may also be extended to more general
periodic structures that have a ``reasonable'' Neumann Laplacian as a
lower bound and satisfy the spectral ``localisation'' result in
Theorem~\ref{thm:spec.incl}. For example, one can also study periodic
operators like $\laplacian X + V$, operators on quantum wave guides,
more general periodic elliptic operators or operators on metric graphs
(cf.~e.g.~\cite{exner-post:05} for examples of periodic metric graphs
with spectral gaps).

It was already observed by Kobayashi, Ono and Sunada
\cite{sunada:90,kos:89} that many properties of the spectrum of a
periodic Schr\"odinger operator (e.g.~band structure, Bloch's property
etc.)  generalise to the context of \emph{abelian} coverings.  An
exception to this is the existence of $L_2$-eigenvalues.  Some further
interesting results were shown by Brooks~\cite{brooks:81}, e.g.\ that
$\Gamma$ is amenable iff $0 \in \spec \laplacian X$.  Moreover,
in~\cite{brooks:86} there is a combinatorial criterion whether the
second eigenvalue of $\laplacian {M_i}$ is bounded from below as $i
\to \infty$, where $M_i \to M$ is a tower of coverings.

The paper is organised as follows: In the following section we set up
the problem, present the geometrical context and state some results
and conventions that will be needed later. In
Section~\ref{sec:construye} we present in detail two construction
procedures for covering manifolds, such that the corresponding
Laplacian will have a prescribed finite number of spectral gaps.  Each
method is well adapted to a given initial geometrical context
(cf.~Remark~\ref{ExplainMethods} as well as
Examples~\ref{ex:fund.group} and \ref{ex:heisenberg}).  In
Subsection~\ref{ssec:ex.spec} we prove a technical result that assures
the existence of spectrum of $\laplacian X$ outside the gaps.  In
Section~\ref{sec:floquet} we review briefly the Floquet theory for
non-abelian groups which, roughly speaking, consists of a Fourier
transformation of the group part $\lsqr \Gamma$ of $L_2(X)\cong\lsqr
\Gamma\otimes L_2(D)$.  The Floquet theory is applied in
Section~\ref{sec:type.I} for coverings with type~I groups.
Theorem~\ref{thm:gaps.type.I} shows the existence of spectral gaps in
this case.  In Section~\ref{sec:res.fin} we study a class of covering
manifolds with residually finite groups.  That is we approximate the
covering $X \to M$ by a sequence of finite coverings $M_i \to M$,
where we apply again the Floquet theory. The existence of a prescribed
number of spectral gaps for $\laplacian X$ for this class of groups is
stated in Theorem~\ref{thm:gaps.res.fin}. In Section~\ref{sec:kadison}
we study residually finite groups $\Gamma$ that in addition have a
positive Kadison constant (e.g., free groups).  In this case we give a
more complete description of the spectrum outside the gaps.  In
Section~\ref{sec:examples} we illustrate the results obtained with
some classes of examples and point out their mutual relations.
Finally, Subsection~\ref{OpenQuestion} contains an interesting example
of a covering with an amenable, \emph{not} residually finite group
which cannot be treated with our methods.  We expect though that in
this case one can still generate spectral gaps by the construction
presented in Section~\ref{sec:construye}.



%------------------------------------------------------------
%------------------------------------------------------------
%
\section{Geometrical preliminaries}
%
%------------------------------------------------------------
%------------------------------------------------------------
\label{sec:prelim}


%------------------------------------------------------------
\subsection{Covering manifolds and Laplacians}
%------------------------------------------------------------
\label{ssec:per.mfd}


We begin fixing our geometrical context and recalling some results
that will be useful later on. We denote by $X$ a \emph{non-compact}
Riemannian manifold of dimension $d \ge 2$ and metric $g$.  We also
assume the existence of a finitely generated (infinite) discrete group
$\G$ of isometries acting \emph{properly discontinuously} and
\emph{cocompactly} on $X$, i.e., for each $x \in X$ there is a
neighbourhood $U$ of $x$ such that the sets $\g U$ and $\g'U$ are
disjoint if $\g \ne \g'$ and $M:=X/\G$ is compact. Moreover, the
quotient $M$ is a Riemannian manifold which also has dimension $d$ and
is locally isometric to $X$.  In other words, $\map \pi X M$ is a
\emph{Riemannian covering space} with covering transformation group
$\G$.  We call such a manifold \emph{$\G$-periodic} or simply
\emph{periodic}. All groups $\G$ appearing in this paper will satisfy
the preceding properties.

We also fix a \emph{fundamental domain} $D$, i.e., an open set $D
\subset X$ such that $\g D$ and $\g' D$ are disjoint for all $\g \ne
\g'$ and $\bigcup_{\g \in \G} \g \overline D = X$. We always assume
that $\overline D$ is compact and that $\bd D$ is piecewise smooth. If
not otherwise stated we also assume that $D$ is connected.  Note that
we can embed $D\subset X$ isometrically into the quotient $M$.  In the
sequel, we will not always distinguish between $D$ as a subset of $X$
or $M$ since they are isometric.  For details we refer
to~\cite[\S6.5]{ratcliffe:94}.

As a prototype for an elliptic operator we consider the Laplacian
$\laplacian X$ on a Riemannian manifold $(X,g)$ acting on a dense
subspace of the Hilbert space $\Lsqr X$ with norm $\norm[X] \cdot$.
For the formulation of the Theorems~\ref{thm:gaps.type.I} and
\ref{thm:gaps.res.fin} and at other places, it is useful to denote
explicitly the dependence on the metric, since we deform the manifold
by changing the metric. In this case we will write $\laplacian
{(X,g)}$ for $\laplacian X$ or $\Lsqr {X,g}$ for $\Lsqr X$.

The positive self-adjoint operator 
$\laplacian X$ can be defined in terms of a
suitable quadratic form $q_X$ (see e.g.~\cite[Chapter~VI]{kato:95},
\cite{reed-simon-1} or \cite{davies:96}). Concretely we have
\begin{equation}
\label{def:quad.form}
  q_X(u):=\normsqr[X] {d u} = \Dint X {|d u|^2} X,\quad
          u \in \Cci X .
\end{equation}
In coordinates we write the pointwise norm of the $1$-form $d u$ as
\begin{displaymath}
  |d u |^2= \sum_{i,j} g^{ij} \partial_i u \, \partial_j\overline u ,
\end{displaymath}
where $(g^{ij})$ is the inverse of the metric tensor $(g_{ij})$ in a
chart. Taking the closure of the quadratic form we can extend $q_X$ onto
the Sobolev space
\begin{displaymath}
  \Sob X = \Sob {X,g} = \set{ u \in \Lsqr X}{ q_X(u) < \infty}.
\end{displaymath}
As usual the operator $\laplacian X$ is related with the quadratic form by the
formula $\iprod {\laplacian X u} u = q_X(u)$, $u \in \Cci X$. Since the metric
on $X$ is $\G$-invariant, the Laplacian $\laplacian X$ (i.e., its
resolvent) commutes with the translation on $X$ given by
\begin{equation}
  \label{eq:transl}
   (T_\g u )(x) := u(\g^{-1}x), \quad u \in \Lsqr X, \g \in \G.
\end{equation}
Operators with this property are called \emph{periodic}.

For an open, relatively compact subset $D \subset X$ with sufficiently smooth
boundary $\bd D$ (e.g.~Lipschitz) we define the Dirichlet (respectively,
Neumann) Laplacian $\laplacianD D$ (resp., $\laplacianN D$) via its quadratic
form $q_D^+$ (resp., $q_D^-$) associated to the closure of $q_D$ on $\Cci D$,
the space of smooth functions with compact support, (resp., $\Ci {\overline
  D}$, the space of smooth functions with continuous derivatives up to the
boundary).  We also use the notation $\Sobn D = \dom q_D^+$ (resp., $\Sob D =
\dom q_D^-$). Note that the usual boundary condition of the Neumann Laplacian
occurs only in the \emph{operator} domain via the Gau{\ss}-Green formula.
Since $\overline D$ is compact, $\laplacianD D$ has purely discrete spectrum
$\EWD k$, $k \in \N$.  It is written in ascending order and repeated according
to multiplicity.  The same is true for the Neumann Laplacian and we denote the
corresponding purely discrete spectrum by $\EWN k$, $k\in\N$.

One of the advantages of the quadratic form approach is that 
one can easily read off from the inclusion of domains an order relation 
for the eigenvalues. In fact, by the 
the \emph{min-max principle} we have
\begin{equation}
  \label{eq:min.max}
  \EWDN k =
  \inf_{L_k} \sup_{u \in L_k \setminus \{0\} }
      \frac {q_D^\pm(u)}{\normsqr u},
\end{equation}
where the infimum is taken over all $k$-dimensional subspaces $L_k$ of the
corresponding \emph{quadratic} form domain $\dom q_D^\pm$,
cf.~e.g.~\cite{davies:96}. Then the inclusion
\begin{equation}
  \label{eq:dom.mono}
   \dom q_D^+ = \Sobn D  \subset \Sob D = \dom q_D^- 
\end{equation}
implies the following important relation between the corresponding
eigenvalues
\begin{equation}
  \label{eq:ew.mono}
   \EWD k \ge \EWN k .
\end{equation}
This means, that the Dirichlet $k$-th eigenvalue is in general
larger than the $k$-th Neumann eigenvalue and this justifies 
the choice of the labels $+$, respectively, $-$.


%------------------------------------------------------------
%------------------------------------------------------------
%
\section{Construction of periodic manifolds}
%
%------------------------------------------------------------
%------------------------------------------------------------
\label{sec:construye}
In the present section we will give two different construction
procedures (labelled by the letters `A' and `B') for covering manifolds,
such that the corresponding Laplacian will have a prescribed finite
number of spectral gaps. In contrast with \cite{post:03a} (where only
abelian groups were considered) we will base the construction on the
specification of the quotient space $M=X/\Gamma$. By doing this, the
spectral convergence result in Theorem~\ref{thm:mfd.conv} becomes
manifestly independent of the fact whether $\Gamma$ is abelian or not.

Both constructions are done in two steps: first, we specify in two
ways the quotient $M$ together with a family of metrics $g_\eps$.
Second, we construct in either case the covering manifold with
covering transformation group $\Gamma$ which has $r$ generators. In
the last section we will localise the spectrum of the covering
Laplacian in certain intervals given by an associated Dirichlet,
respectively, Neumann eigenvalue problem.  Some reasons for presenting
two different methods~(A) and~(B) are formulated in a final remark of
this section.


%------------------------------------------------------------
\subsection{Construction of the quotient}
\label{ssec:quotient}
%------------------------------------------------------------


In the following two methods we define a family of Riemannian
manifolds $(M,g_\eps)$ that converge to a Riemannian manifold $(N,g)$
of the same dimension (cf.~Figure~\ref{fig:constr-mfd}). In each case
we will also specify a domain $D\subset M$ (in the following section
$D$ will become a fundamental domain of the corresponding covering):
\begin{enumerate}
\item[(1A)] \textbf{Attaching $r$ handles:} We construct the manifold
  $M$ by attaching $r$ handles diffeomorphic with $C := (0,1) \times
  \Sphere^{d-1}$ to a given $d$-dimensional compact orientable
  manifold $N$ with metric $g$.  For simplicity we assume that $N$ has
  no boundary.  Concretely, for each handle we remove two small discs
  of radius $\eps>0$ from $N$, denote the remaining set by $R_\eps$
  and identify $\{0\} \times \Sphere^{d-1}$ with the boundary of the
  first hole and $\{1\} \times \Sphere^{d-1}$ with the boundary of the
  second hole.  We denote by $D$ the open subset of $M$ where the mid
  section $\{1/2\} \times \Sphere^{d-1}$ of each handle is removed.
 
  One can finally define a family of metrics $(g_\eps)_\eps$,
  $\eps>0$, on $M$ such that the diameter and length of the handle is
  of order $\eps$ (see e.g.~\cite{post:03a,chavel-feldman:81}). In
  this situation the handles shrink to a point as $\eps\to 0$. Note
  that $(R_\eps, g)$ can be embedded isometrically into $(N,g)$, resp., 
  $(M, g_\eps)$. This fact will we useful for proving
  Theorem~\ref{thm:spectrum}.
  

\item[(1B)] \textbf{Conformal change of metric:} In the second
  construction, we start with an arbitrary compact $d$-dimensional
  Riemannian manifold $M$ with metric $g$. We consider only the case
  $d \ge 3$ (for a discussion of some two-dimensional examples
  see~\cite{post:03a}). Moreover, we assume that $N$ and $D$ are two
  open subsets of $M$ such that (i)~$\bd N$ is smooth, (ii)~$\overline
  N \subset D$, (iii)~$\overline D = M$ and (iv)~$D \setminus N$ can
  completely be described by Fermi coordinates (i.e., coordinates
  $(r,y)$, $r$ being the distance from $N$ and $y \in \bd N$) up to a
  set of measure $0$ (cf.~Figure~\ref{fig:constr-mfd}~(B)). The last
  assumption assures that $N$ is in some sense large in $D$.
  
  Suppose in addition, that $\map {\rho_\eps} M {(0,1]}$, $\eps>0$, is
  a family of smooth functions such that $\rho_\eps(x)=1$ if $x \in N$
  and $\rho_\eps(x)=\eps$ if $x \in M \setminus N$ and $\dist(x,\bd N)
  \ge \eps^d$. Then $\rho_\eps$ converges pointwise to the
  characteristic function of $N$. Furthermore, the Riemannian manifold
  $(M, g_\eps)$ with $g_\eps:= \rho_\eps^2 g$ converges to $(N,g)$ in
  the sense that $M \setminus N$ shrinks to a point in the metric
  $g_\eps$.
\end{enumerate}
\begin{figure}[h]
 \begin{center}
%----------------------------------------------------------------------
%    \input{constr-mfd.pstex_t}
%----------------------------------------------------------------------
\begin{picture}(0,0)
  \includegraphics{constr-mfd.eps}
\end{picture}
\setlength{\unitlength}{4144sp}
\begin{picture}(5335,2429)(361,-1861)
  \put(3601,-1800){(B)}
  \put(360,-1800){(A)}
  \put(3550,460){$r$}
  \put(4394,-44){$y$}
  \put(1391, 59){$R_\eps$}
  \put(360,-1550){$C=(0,1)\times\Sphere^{d-1}$}
  \put(360,250){$(N,g)$}
  \put(2000,-1650){$\beta_1$ (mid section)}
  \put(860,-400){$\alpha_1$}
  \put(4494,-854){$N$}
  \put(4970,178){$D \setminus N$}
  \put(4000,-1489){$\rho_\eps(x)=O(\eps)$}
  \put(4150,-1100){$\rho_\eps(x)=1$}
\end{picture}%
%----------------------------------------------------------------------
   \caption{Two constructions of a family of manifold $(M, g_\eps$), $\eps >
     0$: In both cases, the grey area has a length scale of order
     $\eps$ in all directions. (A)~We attach $r$ handles (here $r=1$)
     of diameter and length of order $\eps$ to the manifold $(N,g)$.
     We also denoted the two cycles $\alpha_1$ and $\beta_1$. (B)~We
     change the metric conformally to $g_\eps = \rho_\eps^2 g$. The
     grey area $D \setminus N$ (with Fermi coordinates in the upper
     left corner) shrinks conformally to a point as $\eps \to 0$
     whereas $N$ remains fixed. Note that the opposite sides of the
     square are identified (to obtain a torus as manifold $M$).}
   \label{fig:constr-mfd}
 \end{center}
\end{figure}

Now we can formulate the following spectral convergence result which was
proven in~\cite{post:03a}:
\begin{theorem}
  \label{thm:mfd.conv}
  Suppose $(M,g_\eps)$ and $D \subset M$ are constructed as in
  parts~(1A) or~(1B) above. In Case~(1B) we assume in addition that $d
  \ge 3$. Then
  \begin{displaymath}
    \EWDN k (\eps) \to \EW k (0)
  \end{displaymath}
  as $\eps \to 0$ for each $k$. Here, $\EWDN k (\eps)$ denotes the
  $k$-th Dirichlet, resp., Neumann eigenvalue of the Laplacian on
  $(D,g_\eps)$ whereas $\EW k (0)$ is the $k$-th eigenvalue of $(N,g)$
  (with Neumann boundary conditions at $\bd N$ in Case~(1B)).
\end{theorem}

%------------------------------------------------------------
\subsection{Construction of the covering spaces}
%------------------------------------------------------------
\label{ssec:constr.cov.sp}

Given $(M,g_\eps)$ and $D$ as in the previous subsection, we will
associate a Riemannian covering $\map \pi {(X,g_\eps)} {(M, g_\eps)}$
with covering transformation group $\G$ such that $D$ is a fundamental
domain. Note that we identify $D \subset M$ with a component of the
lift $\widetilde D := \pi^{-1} (D)$. Moreover, $\Gamma$ is isomorphic
to a normal subgroup of the fundamental group $\pi_1(M)$.
\begin{enumerate}
\item[(2A)] Suppose that $\G$ is a discrete group with $r$ generators
  $\gamma_1, \dots, \gamma_r$. We will construct a $\G$-covering
  $(X,g_\eps) \to (M, g_\eps)$ with fundamental domain $D$ where $D$
  and $(M,g_\eps)$ are given as in Part~(1A) of the previous
  subsection. Roughly speaking, we glue together $\Gamma$ copies of
  $D$ along the handles according to the Cayley graph of $\Gamma$
  w.r.t.\ the generators $\gamma_1, \dots, \gamma_r$. For convenience
  of the reader, we specify the construction:
  
  The fundamental group of $M$ is given by $\pi_1(M) = \pi_1(N) *
  \Z^{*r}$ in the case $d \ge 3$. Here, $G_1*G_2$ denotes the free
  product of $G_1$ and $G_2$, and $\Z^{*r}$ is the free group in $r$
  generators $\alpha_1, \dots, \alpha_r$. If $d=2$ we know from the
  classification result for $2$-dimensional orientable manifolds that
  $N$ is diffeomorphic to an $s$-holed torus. In this case the
  fundamental group is given by
  \begin{equation}
    \label{eq:fund.group}
    \phantom{4ex}
    \pi_1(M) = \langle \alpha_1, \beta_1, \dots,
  \alpha_{r+s}, \beta_{r+s} \mid [\alpha_1, \beta_1] \cdot \ldots \cdot
  [\alpha_{r+s},\beta_{r+s}] = e \rangle,
  \end{equation}
  where $[\alpha,\beta]:=\alpha \beta \alpha^{-1}\beta^{-1}$ is the usual
  commutator.  We may assume that $\alpha_i$
  represents the homotopy class of the cycle \emph{transversal} to the section
  of the $i$-th handle and that $\beta_i$ represents the section itself ($i=1,
  \dots, r$) (cf.~Figure~\ref{fig:constr-mfd}~(A)).
 
  One easily sees that there exists an epimorphism $\map \phi {\pi_1(M)} \G$
  which maps $\alpha_i \in \pi_1(M)$ to $\gamma_i \in \G$ ($i=1, \dots, r$) and
  all other generators to the unit element $e \in \G$. 
  Note that this map is also well-defined in the case $d=2$,
  since the relation in~\eqref{eq:fund.group} is trivially satisfied in the
  case when the $\beta_i$'s are mapped to $e$.

  Finally, $\G \cong \pi_1(M) / \ker \phi$, and 
  $X \to M$ is the associated covering with respect to the universal
  covering $\widetilde M \to M$ 
  (considered as a principal bundle with discrete fibre $\G$) and
  the natural action of $\Gamma$ on $\pi_1(M)$.

  Then $X \to M$ is a normal $\G$-covering with fundamental domain $D$
  constructed as in~(1A) of the preceding subsection. Here we use the
  fact that $\alpha_i$ is \emph{transversal} to the section of the
  handle in dimension~$2$.
  
\item[(2B)] Suppose $(X,g) \to (M,g)$ is a Riemannian covering 
  with fundamental
  domain $D$ such that $\bd D$ is piecewise smooth. Then $\overline D = M$,
  where we have embedded $D$ into the quotient,
  cf.~\cite[Theorem~6.5.8]{ratcliffe:94}. 
  According to~(1B) we can conformally change the metric on $M$, to produce
  a new covering $(X,g_\eps) \to (M,g_\eps)$ that satisfies the required 
  properties.
  

\end{enumerate}
In both cases, we lift for each $\eps >0$ the metric $g_\eps$ from $M$
to $X$ and obtain a Riemannian covering $(X,g_\eps) \to (M, g_\eps)$.
Note that the set $D$ specified in the first step of the previous
construction becomes a fundamental domain after the specification of
the covering in the second step.

The following statement is a direct consequence of the spectral convergence
result in Theorem~\ref{thm:mfd.conv}:
\begin{theorem}
  \label{thm:gaps}
  Suppose $(X,g_\eps) \to (M, g_\eps)$ ($\eps>0$) is a family of Riemannian
  coverings with fundamental domain $D$ constructed as in the 
  previous parts~(2A) or~(2B).
  Then for each $n \in \N$ there exists $\eps = \eps_n > 0$ such that
  \begin{equation}
    \label{eq:gaps}
    I(\eps):=\bigcup_{k \in \N} I_k(\eps), \qquad \text{with} \qquad
    I_k(\eps) := [\EWN k (\eps), \EWD k (\eps)],
  \end{equation}
  is a closed set having at least $n$ gaps, i.e., $n+1$ components as subset
  of $[0,\infty)$. Here, $\EWDN k (\eps)$ denotes the $k$-th Dirichlet, 
  resp., Neumann eigenvalue of the Laplacian on $(D,g_\eps)$.
\end{theorem}
\begin{proof}
  First, note that $\set{\EWDN k (\eps)}{ k \in \N}$, $\eps\geq 0$,
  has no finite accumulation point, since the spectrum is discrete.
  Second, Theorem~\ref{thm:mfd.conv} 
  shows that the intervals $I_k(\eps)$ reduce
  to the point $\{\EW k (0) \}$ as $\eps \to 0$.
  Therefore, $I(\eps)$ is a locally finite
  union of compact intervals, hence closed.
\end{proof}

%------------------------------------------------------------
\subsection{Existence of spectrum outside the gaps}
%------------------------------------------------------------
\label{ssec:ex.spec}

In the following subsection we will assure that each Neumann-Dirichlet
interval $I_k(\eps)$ contains at least one point of $\spec \laplacian
{(X,g_\eps)}$ provided $\eps$ is small enough. In our general setting
described below (cf.\ Theorems~\ref{thm:gaps.type.I}
and~\ref{thm:gaps.res.fin}) we will show the inclusion
\begin{equation}
  \label{eq:spec.incl}
  \spec \laplacian {(X,g_\eps)} \subset \bigcup_{k \in \N} I_k(\eps).
\end{equation}
It is a priori not clear that each $I_k(\eps)$ intersects the spectrum
of the Laplacian on $(X,g_\eps)$, i.e., that gaps in $\bigcup_{k \in
  \N} I_k(\eps)$ are also gaps in $\spec \laplacian {(X,g_\eps)}$. If
the covering group is amenable, the $k$-th eigenvalue of the Laplacian
on the quotient $(M,g_\eps)$ is always an element of $I_k(\eps) \cap
\spec (\laplacian X, g_\eps)$ (cf.~the argument in the proof of
Theorem~\ref{thm:gaps.type.I}). In general, this need not to be true.
Therefore, we need the following theorem which will be used in
Theorems~\ref{thm:gaps.res.fin} and~\ref{thm:band}:

\begin{theorem}
  \label{thm:spectrum}
  With the notation of the previous theorem, we have
  \begin{equation}
    \label{eq:spectrum}
    I_k(\eps) \cap \spec \laplacian {(X,g_\eps)} \ne \emptyset
  \end{equation}
  for all $k \in \N$.
\end{theorem}
We begin with a general criterion which will be useful to detect
points in the spectra of a parameter-dependent family of operators
using only its sesquilinear form. A similar result is also stated
in~\cite[Lemma~5.1]{krejcirik-kriz:pre03}.

Suppose that $H_\eps$ is a self-adjoint, non-negative, unbounded
operator in a Hilbert space $\HS_\eps$ for each $\eps>0$.  Denote by
$\HS_\eps^1 := \dom h_\eps$ the Hilbert space of the corresponding
quadratic form $h_\eps$ associated to $H_\eps$ with norm $\norm[1] u
:= (h_\eps(u) + \norm[\HS_\eps] u)^{1/2}$ and by $\HS_\eps^{-1}$ the
dual of $\HS_\eps^1$. Note that
$\map{H_\eps}{\HS_\eps^1}{\HS_\eps^{-1}}$ is continuous. In the next
lemma we characterise for each $\eps$ certain spectral points of
$H_\eps$.

\begin{lemma}
  \label{lem:char.spec}
  Suppose there exist a family $(u_\eps) \subset \HS_\eps^1$ and
  constants $\lambda \ge 0$, $c>0$ such that
  \begin{equation}
    \label{eq:char.spec}
    \norm[-1] {(H_\eps-\lambda)u_\eps} \to 0 \qquad \text{as} \qquad
    \eps \to 0
  \end{equation}
  and $\norm {u_\eps} \ge c > 0$ for all $\eps>0$, then there exists
  $\delta=\delta(\eps) \to 0$ as $\eps \to 0$ such that
  \begin{displaymath}
    \lambda + \delta(\eps) \in \spec H_\eps.
  \end{displaymath}
\end{lemma}
\begin{proof}
  Suppose that the conclusion is false. Then there exist a sequence
  $\eps_n \to 0$ and a constant $\delta_0>0$ such that
  \begin{displaymath}
    I_\lambda \cap \spec H_{\eps_n} = \emptyset \qquad \text{with} \qquad
    I_\lambda := (\lambda - \delta_0, \lambda + \delta_0)
  \end{displaymath}
  for all $n \in \N$.  Denote by $E_t$ the spectral resolution of
  $H_\eps$. Then
  \begin{multline*}
    \normsqr[-1] {(H_\eps - \lambda)u_\eps} =
    \Dint {\R_+ \setminus I_\lambda} {\frac{(t-\lambda)^2} {(t+1)}} 
                   {\iprod {E_tu_\eps} {u_\eps}} \\ \ge
    \frac {\delta_0^2} {\lambda + \delta_0 + 1}
    \Dint {\R_+ \setminus I_\lambda} 
          {} {\iprod {E_tu_\eps} {u_\eps}} \ge
    \frac {c \delta_0^2} {\lambda + \delta_0 + 1}
  \end{multline*}
  since $I_\lambda$ does not lie in the support of the spectral
  measure. But this inequality contradicts~\eqref{eq:char.spec}.
\end{proof}


\begin{remark}
  Eq.~\eqref{eq:char.spec} is equivalent to the inequality
  \begin{equation}
    \label{eq:char.spec2}
    |h_\eps(u_\eps, v_\eps) - \lambda \iprod {u_\eps} {v_\eps} | \le
    o(1) \norm[1] {v_\eps} \qquad \text{for all $v_\eps \in \HS_\eps^1$}
  \end{equation}
  as $\eps \to 0$. Note that $o(1)$ could depend on $u_\eps$.  The
  advantage of the criterion in the previous lemma is that one only
  needs to find a family $(u_\eps)$ in the domain of the quadratic
  form $h_\eps$.
\end{remark}


We will need the following lemma in order to define a cut-off function
with convergent $L_2$-integral of its derivative. Its proof is
straightforward.
\begin{lemma}
  \label{lem:cut-off}
  Denote by $h(r):= r^{-d+2}$ if $d \ge 3$ and $h(r) = \ln r$ if
  $d=2$. For $\eps \in (0,1)$ define
  \begin{equation}
  \label{eq:cut-off}
    \chi_\eps(r):=
    \begin{cases}
      0, & 0 < r \le \eps\\
      \frac{h(r)-h(\eps)}{h(\sqrt \eps) - h(\eps)}, & 
                        \eps \le r \le \sqrt \eps\\
      1, & \sqrt \eps \le r
    \end{cases}
  \end{equation}
  then $\chi_\eps \in \Sob{(0,1)}$ and
  \begin{displaymath}
    \normsqr {\chi_\eps'} := \Dint[1] 0 {|\chi_\eps'(r)|^2 r^{d-1}} r = o(1). 
  \end{displaymath}
\end{lemma}

Remember that $(N,g)$ is the unperturbed manifold as in
Fig.~\ref{fig:constr-mfd}. In Case~A of
Subsection~\ref{ssec:quotient}, we denoted by $R_\eps$ the manifold
$N$ with a closed ball of radius $\eps$ removed around each point
where the handles have been attached (note that $R_\eps$ is also
contained in $D$) and denote by $(r,y)$ the polar coordinates around
such a point ($r=\eps$ corresponds to a component of $\bd R_\eps$).

\begin{proof}[Proof of Theorem~\ref{thm:spectrum}]
  Let $\phi$ be the $k$-th eigenfunction of the limit operator
  $\laplacian N$ with eigenvalue $\lambda=\lambda_k(0)$. We will treat
  Case~A and~B of Subsection~\ref{ssec:quotient} separately.

  \noindent
  (3A) Set $u_\eps(r,y) := \chi_\eps(r) \phi(r,y)$ in the polar
  coordinates described above and $u_\eps := \phi$ on $R_{\sqrt
    \eps}$. Now, $\normsqr[R_{\sqrt \eps}] {\phi} \ge c$ since
  $\normsqr[R_{\sqrt \eps}] \phi \to \normsqr[N] \phi>0$ as $\eps \to
  0$. In addition, $u_\eps \in \Sobn {R_\eps} \subset \Sob {X,g_\eps}$
  and
    \begin{multline*}
      |\iprod {du_\eps} {dv_\eps} - \lambda \iprod {u_\eps} {v_\eps}| \\ =
      \Bigl| \int_{R_\eps} \bigl[\iprod {d\phi} {d(\chi_\eps v_\eps)}
                 -\lambda \phi \overline{\chi_\eps v} \bigr] +
        \int_{R_\eps} \phi \iprod {d\chi_\eps}{dv_\eps} -
        \int_{R_\eps} \overline v \iprod {d\phi}{d\chi_\eps}
      \Bigr|
    \end{multline*}
    for all $v_\eps \in \Sob {D_\eps}$.  Now the first integral
    vanishes since $\phi$ is the eigenfunction with eigenvalue
    $\lambda$ on $N$.  Note that $\chi_\eps v \in \Sobn {R_\eps}$ can
    be interpreted as function in $\Sob N$. The second and third
    integral can be estimated from above by
    \begin{displaymath}
     \sup_{x \in N} \bigl[|\phi(x)| + |d\phi(x)| \bigr]
       \norm {\chi_\eps'} \norm[1] {v_\eps} = o(1) \norm[1] {v_\eps}
    \end{displaymath}
    since $\phi$ is a smooth function on an $\eps$-independent space
    and due to Lemma~\ref{lem:cut-off}.
  
   \noindent
   (3B) Set $u_\eps := \phi$ on $N$ and $u_\eps(r,y):=\widetilde
   \chi_\eps(r) \phi(0,y)$, $r>0$, i.e., on $D \setminus N$ with
   $\widetilde \chi_\eps(r):=\chi_\eps(\sqrt \eps + \eps^d - r)$,
   where $\chi_\eps$ is defined in~\eqref{eq:cut-off} with $d=2$. Note
   that $\widetilde \chi_\eps'(r) \ne 0$ only for those $r=\dist(x,\bd
   N)$ where the conformal factor $\rho_\eps(x)=\eps$. Now, $u_\eps
   \in \Sobn {D,g_\eps} \subset \Sob {X,g_\eps}$. Furthermore, for
   $v_\eps \in \Sob {D,g_\eps}$ we have
    \begin{multline*}
      |\iprod {du_\eps} {dv_\eps} - \lambda \iprod {u_\eps} {v_\eps}| \le
      \int_{D \setminus N}
      \Bigl[ 
         \bigl|
             \widetilde \chi_\eps'(r)\phi(0,y) \partial_r v_\eps
         \bigr| \rho_\eps^{d-2} \\ +
         \bigl|
             \widetilde \chi_\eps (r) \iprod {d_y\phi(0,y)} {d_y v_\eps}
         \bigr| \rho_\eps^{d-2} +
         \lambda \widetilde \chi_\eps(r) |\phi(0,y) v_\eps| \rho_\eps^d
      \Bigr] \, \mathrm dr \, \mathrm dy \\ \le
       C
       \Bigl[
       \Bigl( 
         \int\limits_{\eps^d}^{\sqrt \eps + \eps^d - \eps} 
               |\widetilde \chi_\eps'(r)|^2 \eps^{d-2} \,\mathrm dr 
       \Bigr)^{\frac 12} \\+
       \Bigl( 
         \int\limits_0^{\sqrt \eps} 
               |\widetilde \chi_\eps(r)|^2 \rho_\eps^{d-2} \, \mathrm dr 
       \Bigr)^{\frac 12} +
       \Bigl( 
         \int\limits_0^{\sqrt \eps} 
               |\widetilde \chi_\eps(r)|^2 \rho_\eps^d \, \mathrm dr 
       \Bigr)^{\frac 12} 
       \Bigr] \norm[1]{v_\eps}
     \end{multline*}
     where we have used that $\phi$ is the Neumann eigenfunction on
     $N$. Furthermore, $C$ depends on the supremum of $\phi$ and
     $d\phi$ and on $\lambda$. Note that the conformal factor
     $\rho_\eps$ equals $\eps$ on the support of $\widetilde
     \chi_\eps'$, therefore, he first integral converges to $0$ since
     $d \ge 3$. Finally, estimating $\widetilde \chi_\eps$ and
     $\rho_\eps$ by $1$, the second and third integral are bounded by
     $\eps^{1/4}$.
\end{proof}


\begin{remark}
\label{ExplainMethods}
In the present section we have specified two construction procedures
for covering manifolds, such that the corresponding Laplacians will
have at least a prescribed number of spectral gaps
(cf.~Sections~\ref{sec:type.I} and~\ref{sec:res.fin}).  The
construction specified in method~(A) is feasible for every given
covering transformation group $\Gamma$ with $r$ generators. Note that
this method produces fundamental domains that have smooth boundaries
(see e.g.~Example~\ref{ex:fund.group} below).

  The construction in~(B) applies for every given Riemannian
  covering $(X,g) \to (M,g)$, since, by the procedure described,
  one can modify conformally this covering in order 
  to satisfy the spectral convergence
  result of Theorem~\ref{thm:mfd.conv}
  (cf.~Example~\ref{ex:heisenberg}).
\end{remark}

%------------------------------------------------------------
%------------------------------------------------------------
%
\section{Floquet theory for non-abelian groups}
%
%------------------------------------------------------------
%------------------------------------------------------------
\label{sec:floquet}

The aim of the present section is to state a spectral inclusion result
(cf.~Theorem~\ref{thm:spec.incl}) and the direct integral
decomposition of $\laplacian X$ (cf.~Theorem~\ref{thm:floquet}) for
certain \emph{non-abelian} discrete groups $\G$.  These results will
be used to prove the existence of spectral gaps in the situations
analysed in the next two sections.  A more detailed presentation of
the results in this section may be found in \cite{lledo-post:pre04a}.

%------------------------------------------------------------
\subsection{Equivariant Laplacians}
%------------------------------------------------------------
\label{ssec:equiv.lapl}
We will introduce next a new operator that lies ``between''
the Dirichlet and Neumann Laplacians and that will play an important
role in the following.
Suppose $\rho$ is a unitary representation of the discrete group $\G$ on the
Hilbert space $H$, i.e., $\map \rho \G {\Unitary H}$ is a
homomorphism. We fix a fundamental domain $D$ for the $\G$-covering $X \to M$.

We now introduce the following space of smooth $\rho$-equivariant
functions
\begin{equation}
  \label{def:equiv.fct}
  \CiR {D,H} := \set {h \restr D}
    {h \in \Ci {X, H}, \quad h(\g x) = \rho_\g h(x), \quad  
         \g \in \G, x \in X}.
\end{equation}  
This definition coincides with the usual one for abelian groups,
cf.~\cite{lledo-post:pre04a}. Note that we need \emph{vector-valued} functions
$\map h X H$ since the representation $\rho$ acts on the Hilbert space $H$,
which, in general, has dimension $\geq 2$.

We define next the so-called \emph{equivariant
  Laplacian} (w.r.t.\ the representation $\rho$) on $\Lsqr {D,H} \cong
\Lsqr D \otimes H$: Let a quadratic form be defined by
\begin{equation}
\label{eq:quad.form}
  \normsqr[D] {dh} := \Dint D {\normsqr[H] {dh(x)}} {X(x)}
\end{equation}
for $h \in \CiR {D,H}$, where the integrand is locally given by
\begin{displaymath}
    \normsqr[H] {dh(x)} = 
    \sum_{i,j} g^{ij}(x) \, \iprod[H] {\partial_i h(x)} {\partial_j h(x)},
     \qquad x \in D.
\end{displaymath}
This generalises Eq.~\eqref{def:quad.form} to the case of vector-valued
functions. We denote the closure by $q_D^\rho$ and its domain by 
$\SobR{D,H}$. The corresponding non-negative, 
self-adjoint operator on $\Lsqr {D,H}$, the
\emph{$\rho$-equivariant Laplacian}, will be denoted by $\laplacianR {D,H}$
(cf.~\cite[Chapter~VI]{kato:95}).


%------------------------------------------------------------
\subsection{Dirichlet-Neumann bracketing}
%------------------------------------------------------------
\label{ssec:dir-neu}
We study in this section the spectrum of a $\rho$-equivariant
Laplacian $\Delta^\rho$ associated with a finite-dimensional
representation $\rho$.  In particular, we show that $\spec
\Delta^\rho$ is contained in a suitable set determined by the spectrum
of the Dirichlet and Neumann Laplacians on $D$.  The key ingredient in
dealing with non-abelian groups is the observation that this set is
\emph{independent} of $\rho$.

We begin with the definition of certain operators acting in $\Lsqr {D,H}$ and
its eigenvalues. We denote by $\EWN m (H)$, $\EWR m (H)$, resp., $\EWD m (H)$
the $m$-th eigenvalue of the operator $\laplacianN {D,H}$, 
$\laplacianR {D,H}$, resp., $\laplacianD {D, H}$ corresponding to the quadratic
form~\eqref{eq:quad.form} on $\Sobn {D,H}$, $\SobR {D,H}$, resp., $\Sob {D,H}$.
Recall that $\Sobn {D,H}$ is the $\al H^1$-closure of the space of smooth
functions $\map h D H$ with support away from $\bd D$ and $\Sob {D,H}$ is the
closure of the space of smooth functions with derivatives continuous up to the
boundary.

The proof of the following lemma follows, as in the abelian case
(cf.~Eqs.~\eqref{eq:dom.mono} and~\eqref{eq:ew.mono}), from the reverse
inclusions of the quadratic form domains
\begin{equation}
  \label{eq:dom.mono.2}
  \Sob {D,H} \supset \SobR {D,H} \supset \Sobn {D,H}
\end{equation}
and the min-max principle~\eqref{eq:min.max}.

\begin{lemma}
  \label{lem:bracketing}
  We have
  \begin{displaymath}
   \EWN m (H)  \le \EWR m (H) \le \EWD m (H)
  \end{displaymath}
  for all $m \in \N$.
\end{lemma}

From the definition of the quadratic form in the Dirichlet, resp., Neumann
case we have that the corresponding
vector-valued Laplacians are a direct sum of the scalar operators. Therefore
the eigenvalues of the corresponding vector-valued Laplace operators consist
of repeated eigenvalues of the scalar Laplacian. We can 
arrange the former in the following way:
\begin{lemma}
  \label{lem:dn.scalar}
  If $n:=\dim H < \infty$ then
  \begin{displaymath}
    \EWDN m (H) = \EWDN k, \qquad
    m=(k-1)n+1, \dots, kn,
  \end{displaymath}
  where $\EWDN k$ denotes the (scalar) $k$-th Dirichet/Neumann eigenvalue on
  $D$.
\end{lemma}
\begin{proof}
  Note that $\laplacianDN {D, H}$ is unitarily equivalent to an $n$-fold
  direct sum of the scalar operator $\laplacianDN D$ on $\Lsqr D$ since there
  is no coupling between the components on the boundary.
\end{proof}

Recall the definition of the intervals $I_k := [\EWN k, \EWD k]$ in
Eq.~\eqref{eq:gaps} (for simplicity, we omit in the following the
index $\eps$). From the preceding two lemmas we may collect the $n$
eigenvalues of $\laplacianR {D,H}$ which lie in $I_k$:
\begin{equation}
  \label{eq:band.rho}
  B_k(\rho) := \set {\EWR m (H)} 
                  {m=(k-1)n+1, \dots, kn}
          \subset I_k,  \qquad n := \dim H.
\end{equation}
Therefore, we obtain the following spectral inclusion for equivariant
Laplacians. This result will be applied in 
Theorems~\ref{thm:gaps.type.I} and \ref{thm:gaps.res.fin}
below.
\begin{theorem}
  \label{thm:spec.incl}
  If $\rho$ is a unitary representation on a finite-dimensional
  Hilbert space $H$ then
  \begin{displaymath}
    \spec \laplacianR {D, H} =
    \bigcup_{k \in \N} B_k(\rho) \subseteq 
    \bigcup_{k \in \N} I_k %%=:I
  \end{displaymath}
  where $\laplacianR {D, H}$ denotes the $\rho$-equivariant Laplacian.
\end{theorem}



%------------------------------------------------------------
\subsection{Non-abelian Floquet transformation}
%------------------------------------------------------------
\label{ssec:floquet}

Consider first the right, respectively, left regular representation $R$,
resp., $L$ on the Hilbert space $\lsqr \G$:
\begin{equation}
  \label{def:reg.rep}
  (R_\g a)_\tg = a_{\tg \g}, \qquad
  (L_\g a)_\tg = a_{\g^{-1}\tg},  \quad\qquad a = (a_\g)_\g \in \lsqr \G, 
      \quad \g,\tg \in \G.
\end{equation}
Using standard results we introduce the following unitary map (see
e.g., \cite[Section~3 and the appendix]{lledo-post:pre04a} and
references cited therein)
\begin{equation}
  \label{eq:fourier}
  \map F {\lsqr \G}{\OintZ {H(z)}}
\end{equation}
for a suitable measure space $(Z, \mathrm dz)$.  The map $F$ is a
generalisation of the Fourier transformation in the abelian case.
Moreover, it transforms the right regular representation $R$ into the
following direct integral representation
\begin{equation}
  \label{eq:reg.rep.trafo}
    \widehat R_\g = F R_\g F^* = \OintZ {R_\g(z)}, \qquad \g \in \G.
\end{equation}

\begin{remark}
  \label{rem:meas.space}
  Let $\al R$ be the von Neumann algebra generated by all unitaries
  $R_\g$, $\g \in \G$, i.e.,
  \begin{equation}
    \label{eq:gen.vn.algebra}
    \al R = \set {R_\g} {\g \in \G}'',
  \end{equation}
  where $\al R'$ denotes the commutant of $\al R$ in $\End {\lsqr
    \G}$. Then we decompose $\al R$ with respect to a maximal abelian
  von Neumann subalgebra $\al A\subset \al R'$ (for a concrete example
  see Example~\ref{ex:dir.int}). The space $Z$ is the compact
  Hausdorff space associated, by Gelfand's isomorphism, to a
  \emph{separable} $C^*$-algebra $\al C$, which is strongly dense in
  $\al A$.  Furthermore, $\mathrm d z$ is a regular Borel measure on
  $Z$.  We may identify the algebra $\al A$ with $\Linfty {Z,\mathrm
    dz}$ and since it is maximal abelian, the fibre representations
  $R(z)$ are irreducible a.e.\ (see
  \cite[Section~14.8~ff.]{wallach:92}).
\end{remark}

The generalised Fourier transformation introduced in
Eq.~\eqref{eq:fourier} can be used to decompose $\Lsqr X$ into a
direct integral. In particular, we define for a.e.~$z \in Z$:
\begin{equation}
  \label{eq:floquet.short}
  (Uu)(z)(x) := \sum_{\g \in \G} \,u(\g x)  R_{\gamma^{-1}}(z) v(z)  ,
\end{equation}
where $v:=F \delta_e \in \lsqr \G$, $u \in \Cci X$ and $x \in D$.   
The map $U$ extends to a unitary map
\begin{displaymath}
  \map U {\Lsqr X} {\OintZ 
      {\Lsqr {D,H(z)}} \cong \OintZ {H(z)} \otimes \Lsqr D},
\end{displaymath}
the so-called \emph{Floquet} or \emph{partial Fourier transformation}.
Moreover, operators commuting with the translation $T$ on $\Lsqr X$
are decomposable, in particular, we can decompose $\laplacian X$ since
its resolvent commutes with all translations~\eqref{eq:transl}. One
can show in this context (cf.~\cite{sunada:88,lledo-post:pre04a}):
\begin{theorem}
  \label{thm:floquet}
   The operator $U$ maps $\Cci X$ into
   $\OintZ {\CiEq {D,H(z)}}$. Moreover, $\laplacian X$ is unitary
   equivalent to $\OintZ {\laplacianZ D}$ and
  \begin{equation}
  \label{eq:spec.dir.int}
    \spec \laplacian X \subseteq
    \overline {\bigcup_{z \in Z} \spec \laplacianZ D}.
  \end{equation}
    We denote by $\CiEq {D,
    H(z)}$ the set of smooth $R(z)$-equivariant functions defined
  in~\eqref{def:equiv.fct} and $\laplacianZ D$ is the
  $R(z)$-equivariant Laplacian in $\Lsqr {D,H(z)}$.

  If $\Gamma$ is amenable (cf.\ Remark~\ref{rem:amenable}), then we
  have equality in \eqref{eq:spec.dir.int}.
\end{theorem}

\begin{example}
  \label{ex:dir.int}
  Let us illustrate the above direct integral decomposition in the
  case of the free group $\Gamma = \Z * \Z$ generated by $\alpha$ and
  $\beta$. Let $A \cong \Z$ be the cyclic subgroup generated by
  $\alpha$. We can decompose the algebra $\mathcal R$ given
  in~\eqref{eq:gen.vn.algebra} w.r.t.\ the abelian algebra $\mathcal A
  := \set{L_a \in \mathcal L(\lsqr \Gamma)}{a \in A} \subset \mathcal
  R'$, and, in this case, we have $Z = \Sphere^1$. Since the set
  $\set{a \gamma a^{-1}}{a \in A}$ is infinite provided $\gamma \notin
  A$, the algebra is \emph{maximal} abelian in $\mathcal R'$ (i.e.,
  $\mathcal A = \mathcal A' \cap \mathcal R'$), and therefore, each
  fibre representation $R(z)$ is irreducible in $\mathcal H(z)$.
  Moreover, since $L_a \in \mathcal A'$ ($a \in A$) we can also
  decompose these operators w.r.t\ the previous direct integral.
  
  We can give a more concrete realisation of the abstract Fourier
  transformation $F=F_\Gamma$ (see e.g.~\cite[Section~19]{robert:83}):
  We interprete $\Gamma \to A \setminus \Gamma$ as covering space with
  abelian covering group $A$ acting on $\Gamma$ from the left; the
  corresponding translation action $T_a$ on $\lsqr \Gamma$ coincides
  with the left regular representation $L_a$ ($a \in A$). The
  (abelian) Floquet transformation $U=U_A$ gives a direct integral
  decomposition
  \begin{equation*}
    \map{F_\Gamma = U_A} 
           {\lsqr \Gamma} {\Oint {\widehat A} {H(\chi)} \chi},
  \end{equation*}
  where $H(\chi) \cong \lsqr {A \setminus \Gamma}$ is the space of
  $\chi$-equivariant sequences in $\lsqr \Gamma$. Note that $H(\chi)$
  is infinite dimensional. A straightforward calculation shows that
  \begin{equation*}
    R_\gamma \cong \Oint {\widehat A} {R_\gamma(\chi)} \chi
      \quad \text{and} \quad
    L_a      \cong \Oint {\widehat A} {L_a     (\chi)} \chi,
  \end{equation*}
  where $R_\gamma(\chi) u(\tg) = u(\tg \gamma)$ and $L_a(\chi) u(\tg)=
  \overline \chi(a) u(\tg)$ for $u \in H(\chi)$. Furthermore, one sees
  that $v = (U \delta_e)(\chi)$ is the \emph{unique} normalised
  eigenvector of $R_a(\chi)$ with eigenvalue $\chi(a)$. This follows
  from the fact that the set of cosets $\set{A\gamma a}{a \in A}
  \subset A\setminus \Gamma$ is infinite provided $\gamma \notin A$.
  From the previous facts one can directly check that each $R(\chi)$
  is an irreducible representation of $\Gamma$ in $H(\chi)$ and that
  these representations are mutually inequivalent. Finally, $R(\chi)$
  is also inequivalent to any irreducible component of the direct
  integral decomposition obtained from a different maximal abelian
  subgroup $B \ne A$.
\end{example}


%------------------------------------------------------------
%------------------------------------------------------------
%
\section{Spectral gaps for type~I groups}
%
%------------------------------------------------------------
%------------------------------------------------------------
\label{sec:type.I}


We will present in this section the first method to show that the
Laplacian of the manifolds constructed in Section~\ref{sec:construye}
with (in general \emph{non-abelian}) type~I covering groups have an
arbitrary finite number of spectral gaps.  We begin recalling the
definition of type~I groups in the context of discrete groups.

\begin{definition}
  \label{def:type.I}
  A discrete group $\G$ is of \emph{type~I} if $\G$ is a finite
  extension of an abelian group, i.e., if there is an exact sequence
   \begin{displaymath}
     0 \longrightarrow 
     A \longrightarrow
     \G \longrightarrow
     \G_0 \longrightarrow 
     0 ,
   \end{displaymath}
   where $A \lhd \G$ is abelian and $\G_0 \cong \G / A$ is a finite
   group.
\end{definition}

\begin{remark}
  \label{rem:type.I}
  \begin{enumerate}
  \item 
    \label{rem:type.I.i}
    In the previous definition we have used a simple characterisation
    of countable, \emph{discrete} groups of type~I due to Thoma,
    cf.~\cite{thoma:64} (for uncountable groups see \cite{thoma:68}).
    Moreover, all irreducible representations of a type~I group $\G$
    are finite-dimensional and have a uniform bound on the dimension
    (see~\cite{thoma:64,moore:72}). Therefore, the following
    properties are all equivalent: (a)~there is a uniform bound on the
    dimensions of irreducible representations of $\G$, (b)~all
    irreducible representations of $\G$ are finite-dimensional, (c)
    $\G$ is a finite extension of an abelian group, (d)~$\G$ is CCR
    (completely continuous representation,
    cf.~\cite[Ch.~14]{wallach:92}), (e)~$\G$ is of type~I.  Recall
    also that $\Gamma$ is of type~I iff the von Neumann algebra $\al
    R$ generated by $\Gamma$ (cf.~Eq.~\eqref{eq:gen.vn.algebra}) is of
    \emph{type~I} (cf.~\cite{kaniuth:69}).
  
    Note that for our application it would be enough if $\Gamma$ has a
    decomposition over a measure space $(Z,\mathrm d z)$ as in
    Remark~\ref{rem:meas.space} such that \emph{almost} every
    representation $\rho(z)$ is finite-dimensional. But such a group
    is already of type~I: indeed, if the set $\set {z \in Z} {\dim
      H(z) = \infty}$ has measure $0$, then it follows from
    \cite[Section~II.3.5]{dixmier:81} that the von Neumann Algebra
    $\mathcal R$ (cf.~Eq.~\eqref{eq:gen.vn.algebra}) is of type~I.  By
    the above equivalent characterisation this implies that $\Gamma$
    is of type~I.
  \item
    \label{rem:type.I.ii}
    The following criterion (cf.~\cite{kaniuth:69,kallman:70}) will
    be used in Examples~\ref{ex:heisenberg} and \ref{ex:free.group} to
    decide that a group is not of type~I: The von Neumann algebra $\al
    R$ is of type~II$_1$ iff $\Gamma_{\mathrm{fcc}}$ has infinite
    index in $\Gamma$.  Here,
    \begin{equation}
      \label{eq:fcc}
      \Gamma_{\mathrm{fcc}} :=
      \set{\gamma \in \Gamma} {C_\gamma \text{ is finite}}
    \end{equation}
    is the set of elements $\gamma\in\Gamma$ having finite conjugacy
    class $C_\gamma$.  In particular such a group is not of type~I.
    Even worse: Almost all representations in the direct integral
    decomposition~\eqref{eq:reg.rep.trafo} are of type~II$_1$
    (\cite[Section~II.3.5]{dixmier:81}) and therefore
    infinite-dimensional (see e.g.~Example~\ref{ex:dir.int}).
  \end{enumerate}
\end{remark}

\begin{remark}
  \label{rem:amenable}
  The notion of amenable discrete group will be useful at different
  stages of our approach.  For a definition of \emph{amenability} of
  a discrete group $\Gamma$ see e.g.~\cite{day:57} or
  \cite{brooks:81}. We will only need the following equivalent
  characterisations: (a)~$\Gamma$ is amenable. (b)~$0 \in \spec
  \laplacian X$~\cite{brooks:81}.  (c)~$\spec \laplacian M \subset
  \spec \laplacian X$~\cite[Propositions~7--8]{sunada:88}. Here, $X
  \to M$ is a covering with covering group $\Gamma$. Note that
  discrete type~I groups are amenable since they are finite extensions
  of abelian groups (extensions of amenable groups are again amenable,
  cf.~\cite[Section~4]{day:57}).
  
  We want to stress that Theorem~\ref{thm:spectrum} is no
  contradiction to the fact that $\Gamma$ is amenable iff $0 \in \spec
  \laplacian {(X,g_\eps)}$ although the first interval
  $I_1(g_\eps)=[0,\EWD k(g_\eps)]$ tends to $0$ as $\eps \to 0$.  Note
  that we have only shown there that $I_1(g_\eps) \cap \spec
  \laplacian {(X,g_\eps)} \ne \emptyset$ and \emph{not} $0=\EW
  1(M,g_\eps) \in \spec \laplacian {(X,g_\eps)}$ which is only true in
  the amenable case.
\end{remark}

The \emph{dual of $\G$}, which we denote by $\hG$, is the set of
equivalence classes of unitary irreducible representations of $\G$. We
denote by $[\rho]$ the (unitary) equivalence class of a unitary
representation $\rho$ on $H$.  Note that the spectrum of a
$\rho$-equivariant Laplacian and $\dim H$ only depend on the
\emph{equivalence class} of $\rho$.


If $\G$ is of type~I, then the dual $\hG$ becomes a nice measure space
(``smooth'' in the terminology of \cite[Chapter~2]{mackey:76}).
Furthermore, we can use $\hG$ as measure space in the direct integral
decomposition defined in Subsection~\ref{ssec:floquet}. In particular,
combining the results of Section~\ref{sec:prelim}
and~\ref{sec:floquet} we obtain the main result for type~I
groups:
\begin{theorem}
  \label{thm:gaps.type.I}
  Suppose $X \to M$ is a Riemannian $\G$-covering with fundamental
  domain $D$, where $\G$ is a type~I group and denote by $g$ the
  Riemannian metric on $X$. Then
  \begin{displaymath}
    \spec \laplacian {(X,g)} \subset \bigcup_{k \in \N} I_k(g), 
    \qquad\mathrm{and}\qquad
    I_k(g) \cap \spec \laplacian{(X,g)} \ne \emptyset, \quad k \in \N,
  \end{displaymath}
  where $I_k(g):=[\EWN k (D,g), \EWD k (D,g)]$ is the
  Neumann-Dirichlet interval defined as in~\eqref{eq:gaps}.  In
  particular, for each $n \in \N$ there exists a metric $g=g_{\eps_n}$
  constructed as in Subsection~\ref{ssec:constr.cov.sp} such that
  $\spec \laplacian {(X,g)}$ has at least $n$ gaps, i.e., $n+1$
  components as subset of $[0, \infty)$.
\end{theorem}
\begin{proof}
  We have
  \begin{displaymath}
    \spec \laplacian X =
    \overline {\bigcup_{[\rho] \in \hG} \spec \laplacianR {D, H}}
    \subseteq  \overline {\bigcup_{k \in \N} I_k(g)}
    = \bigcup_{k \in \N} I_k(g),
  \end{displaymath}
  where we used the Theorem~\ref{thm:floquet} with $Z=\hG$ for the
  first equality and Theorem~\ref{thm:spec.incl} for the inclusion.
  Note that $\Gamma$ is amenable and that the latter theorem applies
  since all (equivalence classes of) irreducible representations of a
  type~I group are finite-dimensional
  (cf.~Remark~\ref{rem:type.I}~(\ref{rem:type.I.i})). The existence of
  gaps in $\bigcup_k I_k(g)$ follows from Theorem~\ref{thm:gaps}.
  
  Since $\Gamma$ is amenable, $\spec \laplacian M \subset \spec
  \laplacian X$ (cf.~(c) in Remark~\ref{rem:amenable}).  Moreover,
  from Eq.~\eqref{eq:band.rho} with $\rho$ the trivial representation
  on $\mathcal{H}=\C$, we have that $\lambda_k(M) \in I_k$. Note that
  functions on $M$ correspond to functions on $D$ with periodic
  boundary conditions. Therefore, we have shown that every gap of the
  union $\bigcup_k I_k(g)$ is also a gap of $\spec \laplacian X$.
\end{proof}




%------------------------------------------------------------
%------------------------------------------------------------
%
\section{Spectral gaps for residually finite groups}
%
%------------------------------------------------------------
%------------------------------------------------------------
\label{sec:res.fin}

In this section, we present a new method to prove the existence of a
finite number of spectral gaps of $\laplacian X$.  The present
approach is applicable to so-called residually finite groups $\Gamma$,
which is a much larger class of groups containing type~I groups
(cf.~Section~\ref{sec:examples}). Roughly speaking, residually finite
means that $\Gamma$ has a lot of normal subgroups with finite index.
Geometrically, this implies that one can approximate the covering
$\map \pi X M$ with covering transformation group $\Gamma$ by
\emph{finite} coverings $\map {p_i} {M_i} M$, where the $M_i$'s are
compact.

Since the present section is central to the paper we will also give
for completeness proofs of known results (for
Theorem~\ref{thm:res.fin} see also \cite{ass:94,adachi:95})

%------------------------------------------------------------
\subsection{Subcoverings and residually finite groups}
%------------------------------------------------------------
\label{ssec:sub.cov}
Suppose that $\map \pi X M$ is a covering with covering transformation group
$\G$ (as in Subsection~\ref{ssec:per.mfd}). Corresponding to a normal
subgroup $\G_i \lhd \G$ we associate a covering $\map {\pi_i} X {M_i}$
such that
\begin{equation}
  \label{eq:sub.cov}
  \begin{diagram}
          &                             & X          &                   &\\
          &  \ldTo(2,2)^{\pi_i}_{\G_i}  &            & \rdTo(2,2)^\pi_\G & \\
     M_i  &                             & \rTo^{p_i}_{\G/\G_i} &         & M
  \end{diagram}
\end{equation}
is a commutative diagram. The groups under the arrows denote the
corresponding covering transformation groups.

\begin{definition}
\label{def:res.fin}
  A (countable, infinite) discrete group $\G$ is residually finite if
  there exists a monotonous decreasing sequence of normal subgroups
  $\G_i \lhd \G$ such that
\begin{equation}
  \label{eq:sub.groups}
  \G=\G_0 \rhd \G_1 \rhd \dots \rhd \G_i \rhd \cdots,  \quad
  \bigcap_{i \in \N} \G_i = \{e\}  \quad \text{and} \quad
  \text{$\G/\G_i$ is finite.}
\end{equation}
Denote by $\mathfrak R \mathcal F$ the class of residually finite
groups.
\end{definition}
Suppose now that $\G$ is residually finite. Then there exists a
corresponding sequence of coverings $\map {\pi_i} X {M_i}$ such that
$\map {p_i} {M_i} M$ is a \emph{finite} covering
(cf.~Diagram~\eqref{eq:sub.cov}). Such a sequence of covering maps is
also called \emph{tower of coverings}.

\begin{remark}
We recall also the following equivalent definitions of residually 
finite groups (see e.g.~\cite{magnus:69} or~\cite[Section~2.3]{robinson:82}).
\begin{itemize}
\item[(i)]
  A group $\G$ is called \emph{residually finite} if for all 
  $\g \in \G \setminus \{e\}$ there is a group
  homomorphism $\map \Psi \G G$ such that $\Psi(\g) \ne e$ and $\Psi(\G)$
  is a \emph{finite} group.
\item[(ii)]
  Let $\al F$ denote the class of finite groups.
  $\G$ is residually finite, if the so-called 
  \emph{$\al F$-residual}
  \begin{equation}
    \label{eq:residual}
    \mathfrak{R}_{\al F}(\G) := 
     \bigcap_{ \substack{N \lhd \G\\\G/N \in \al F}} N
\end{equation}  
is trivial, i.e.,~$\mathfrak{R}_{\al F}(\G)=\{e\}$. 
\end{itemize}
\end{remark}

Next we give some examples for residually finite groups (cf.~the
survey article~\cite{magnus:69}):
\begin{example}
  \label{ex:res.fin}
  (i)~Abelian and finite groups are residually finite. (ii)~Free
  products of residually finite groups are residually finite, in
  particular, the free group in $r$ generators $\Z^{*r}$ is residually
  finite. (iii)~Finitely generated linear groups are residually finite
  (for a simple proof of this fact cf.~\cite{alperin:87}; a group is
  called \emph{linear} iff it is isomorphic to a subgroup of
  $\mathrm{GL}_n(\C)$ for some $n \in \N$.) In particular,
  $\mathrm{SL}_n(\Z)$ and fundamental groups of closed, orientable
  surfaces of genus $g$ are residually finite.
\end{example}

Next we need to introduce a metric on the discrete space $\Gamma$:
\begin{definition}
  \label{def:word.met}
  Let $E$ be a set which generates $\Gamma$. The \emph{word metric}
  $d=d_E$ on $\Gamma$ is defined as follows: $d(\g,e)$ is the minimal
  number of elements in $E$ needed to express $\g$ as a word in the
  alphabet $E$; $d(e,e):=0$ and $d(\g,\tg) := d(\g \tg^{-1}, e)$.
\end{definition}

Geometrically, residually finiteness means that, given any compact set
$K \subset X$, there exists a finite covering $\map {p_i} {M_i} M$ and
a covering $\map {\pi_i} X {M_i}$ which is injective on $K$
(cf.~\cite{brooks:86}). This idea is used in the following lemma:
\begin{lemma}
  \label{lem:seq.fund.dom}
  Fix a fundamental domain $D$ for the covering $\map \pi X M$. Then for each
  covering $\map {\pi_i} X {M_i}$ there is a fundamental domain $D_i$ (not
  necessarily connected) such that
  \begin{displaymath}
    D_0 := D \subset D_1 \subset \dots \subset D_i \subset \cdots  
      \qquad \text{and} \qquad
    \bigcup_{i \in \N} D_i = X.
  \end{displaymath}
\end{lemma}
\begin{proof}
  It is enough to show the existence of a family of representants $R_i
  \subset \G$ of $\G/\G_i$, $i\in\N$, satisfying
  \begin{displaymath}
    R_0 := \{e\} \subset R_1 \subset \dots \subset R_i \subset \cdots  
      \qquad \text{and} \qquad
    \bigcup_{i \in \N} R_i = \G.
  \end{displaymath}
  In this case the fundamental domains are given explicitly by
  \begin{displaymath}
    D_i := \intr \bigcup_{r \in R_i} r^{-1} \overline D,
  \end{displaymath}
  where $\intr$ denotes the topological interior.
  
  Let $d$ be the word metric on $\Gamma$ with respect to the set of
  generators $E := \set{\g \in \G}{\g\overline D \cap \overline D \ne
    \emptyset}$, which is naturally adapted to the fundamental domain
  $D$. Note that $E$ is finite and generates $\G$ since $\overline D$
  is compact (cf.~\cite[Thms.~6.5.10 and~6.5.11]{ratcliffe:94}).
  
  We choose a set of representants $R_i$ of $\Gamma/\Gamma_i$ that
  have minimal distance in the word metric to the neutral element,
  i.e.~if $r\in R_i$, then $d(r,e)\leq d(r\Gamma_i, e)$. Note that
  since $\Gamma_{i+1}\subset\Gamma_i$ we have $R_{i+1}\supset R_i$.
  To conclude the proof we have to show that every $\g \in \G$ is
  contained in some $R_i$, $i \in \N$.  Since $\G$ is finitely
  generated, there exists $n \in \N$ such that $\gamma\in B_{n} :=
  \{\g \in \G \mid d(\g,e)\le n\}$.  Moreover, since $B_{2n}$ is
  finite and $\Gamma$ residually finite we also have
  $B_{2n}\cap\Gamma_i=\{e\}$ for $i$ large enough.  Therefore, any
  other element $\tg=\g \g_i^{-1}$ in the class $\g \G_i$ with $\g_i
  \in \G_i \setminus \{e\}$ has a distance greater than $n$, since
  \begin{displaymath}
    d(\tg, e) = d(\g\g_i^{-1},e) = d(\g, \g_i) \ge
    d(e,\g_i) - d(\g,e) > 2n - n = n.
  \end{displaymath}
  This implies that $\gamma\in R_i$ 
  by the minimality condition in the choice of the representants.
\end{proof}

\begin{theorem}
  \label{thm:res.fin}
  Suppose $\G$ is residually finite with the associated sequence of coverings
  $\map {\pi_i} X {M_i}$ and $\map {p_i} {M_i} M$ as in~\eqref{eq:sub.cov}.
  Then
  \begin{displaymath}
    \spec \laplacian X \subseteq \overline {\bigcup_{i \in \N} \spec \laplacian
    {M_i}},
  \end{displaymath}
  and the Laplacian $\laplacian {M_i}$ w.r.t.\ the finite covering
  $\map {p_i} {M_i} M$ has discrete spectrum. Equality holds iff
  $\Gamma$ is amenable.
\end{theorem}
\begin{proof}
  (Cf.~\cite{adachi:95}) If $\lambda \in \spec \laplacian X$, then for each
  $\eps>0$ there exists $u \in \Cci X$ such that
  \begin{displaymath}
    \frac{\normsqr[X] {(\laplacian X - \lambda) u}} {\normsqr[X] u} < \eps.
  \end{displaymath}
  Applying Lemma~\ref{lem:seq.fund.dom} there is an $i=i(\eps)$ such that
  $\supp u \subset D_i$. Furthermore, since
  $D_i \hookrightarrow M_i=X/\G_i$ is an
  isometry, $u$ can be written as the lift of a smooth $f$ on $M_i$,
  i.e., $f \circ \pi_i = u$. Therefore,
  \begin{displaymath}
    \frac{\normsqr[M_i] {(\laplacian {M_i} - \lambda) f}} {\normsqr[M_i] f} =
    \frac{\normsqr[X] {(\laplacian X - \lambda) u}} {\normsqr[X] u} < \eps,
  \end{displaymath}
  which implies $\lambda \in \overline {\bigcup_{i \in \N} \spec
    \laplacian {M_i}}$. Finally, since $M_i \to M$ is a finite
  covering and $M$ is compact, $\spec\laplacian {M_i}$ is discrete.
  For the second assertion cf.~\cite{adachi:95} or~\cite{ass:94}. One
  basically uses the characterisation due to \cite{brooks:81} that
  $\Gamma$ is amenable iff $0 \in \spec \laplacian X$ (cf.\ 
  Remark~\ref{rem:amenable}).
\end{proof}


Next we analyse the spectrum of the finite covering $M_i \to M$. Note that 
$D$ is also isometric 
to a fundamental domain for \emph{each} finite covering $M_i \to M$,
$i \in \N$.
\begin{lemma}
  \label{lem:fin.group}
  We have
  \begin{displaymath}
    \spec \laplacian {M_i} = 
       \bigcup_{[\rho] \in \widehat{G_i}}
                       \spec \laplacianR {D,H(\rho)},
  \end{displaymath}
  where $\Delta^\rho$ is the equivariant Laplacian introduced in
  Subsection~\ref{ssec:equiv.lapl} and $G_i := \G/\G_i$ is a finite
  group and $\widehat{G_i}$ its dual.
\end{lemma}
\begin{proof}
  Applying the results of Subsection~\ref{ssec:floquet} to the finite group
  $G_i$ and the finite measure space $Z:=\widehat{G_i}$ with the counting
  measure all direct integrals become direct sums. By Peter-Weyl's theorem
  (see e.g.~\cite[\S27.49]{hewitt-ross-2}) we also have
  \begin{displaymath}
    \map F {\lsqr {G_i}} 
       {\bigoplus_{[\rho] \in \widehat{G_i}} n(\rho) H(\rho)},
  \end{displaymath}
  where each multiplicity satisfies $n(\rho)=\dim H(\rho)<\infty$. 
  Finally,
  \begin{displaymath}
    \laplacian {M_i} \cong 
      \bigoplus_{[\rho] \in \widehat{G_i}} \laplacianR {D, H(\rho)}
  \end{displaymath}
  and the result follows.
\end{proof}


We now can formulate the main result of this section:
\begin{theorem}
  \label{thm:gaps.res.fin} 
  Suppose $X \to M$ is a Riemannian $\G$-covering with fundamental
  domain $D$, where $\G$ is a residually finite group and denote by
  $g$ the Riemannian metric on $X$.  Then
  \begin{displaymath}
    \spec \laplacian {(X,g)} \subset \bigcup_{k \in \N} I_k(g), \qquad
    I_k(g) \cap \spec \laplacian{(X,g)} \ne \emptyset, \quad k \in \N,
  \end{displaymath}
  where $I_k(g):=[\EWN k (D,g), \EWD k (D,g)]$ is defined as
  in~\eqref{eq:gaps}.  In particular, for each $n \in \N$ there exists
  a metric $g=g_{\eps_n}$, constructed as in
  Subsection~\ref{ssec:constr.cov.sp}, such that $\spec \laplacian
  {(X,g)}$ has at least $n$ gaps, i.e., $n+1$ components as subset of
  $[0, \infty)$.
\end{theorem}
\begin{proof}
  We have
  \begin{displaymath}
    \spec \laplacian X \subseteq 
    \overline {\bigcup_{i \in \N} \spec \laplacian {M_i}} =
    \overline {\bigcup_{\substack{i \in \N\\ [\rho] \in \widehat {G_i}}}
           \spec \laplacianR {D, H(\rho)}}
    \subseteq  
    \overline {\bigcup_{k \in \N} I_k(g) } = \bigcup_{k \in \N} I_k(g),
  \end{displaymath}
  where we used Theorem~\ref{thm:res.fin}, Lemma~\ref{lem:fin.group}
  and Theorem~\ref{thm:spec.incl}. Note that the latter theorem
  applies since all (equivalence classes of) irreducible
  representations of the finite groups $G_i$, $i\in\N$, are
  finite-dimensional. The existence of gaps in $\bigcup_k I_k(g)$
  follows from Theorem~\ref{thm:gaps}. Finally, by
  Theorem~\ref{thm:spectrum}, a gap of $\bigcup_k I_k(g)$ is in fact a
  gap of $\spec \laplacian X$.
\end{proof}

%------------------------------------------------------------
%------------------------------------------------------------
%
\section{Kadison constant and asymptotic behaviour}
%
%------------------------------------------------------------
%------------------------------------------------------------
\label{sec:kadison}

In the present section we will combine our main result stated in
Theorem~\ref{thm:gaps.res.fin} with some results by Sunada and
Br\"uning (cf.~\cite[Theorem~1]{sunada:92} or
\cite{bruening-sunada:92}), to give a more complete description of the
spectrum of the Laplacian $\laplacian X$, where $X \to M$ is the
covering manifold constructed in Section~\ref{sec:construye}. For
this, we need a further definition:
\begin{definition}
  \label{def:kadison}
  The \emph{Kadison constant} of $\Gamma$ is defined as
  \begin{displaymath}
     C(\Gamma) := \inf \set{ \tr_\Gamma (P)} 
           {\text{$P$  non-trivial projection in 
                           $C^*_{\mathrm{red}}(\Gamma,\al K)$}},
  \end{displaymath}
  where $\tr \Gamma (\cdot)$ is the canonical trace on
  $C^*_{\mathrm{red}}(\Gamma,\al K)$ , the tensor product of the
  reduced group $C^*$-algebra of $\Gamma$ and the algebra $\al K$ of
  compact operators on a separable Hilbert space of infinite dimension
  (see \cite[Section~1]{sunada:92} for more details.)
\end{definition}
In this section, we assume that $\Gamma$ is is residually finite and
has a strictly positive Kadison constant, i.e.~$C(\Gamma)>0$.  For
example, the free product $\Z^{*r} * \Gamma_1 * \dots * \Gamma_a$ with
finite groups $\Gamma_i$ satisfies both properties
(cf.~e.g.~\cite{magnus:69}, \cite[Appendix]{sunada:92}). Another such
group is the fundamental group (cf.~Eq.~\eqref{eq:fund.group}) of a
(compact, orientable) surface of genus $g$
(see~\cite{marcolli-mathai:99}).

\begin{remark}
  Suppose that $K$ is an integral operator on $\Lsqr X$ commuting
  with the group action, having smooth kernel $k(x,y)$ and satisfying
  \begin{displaymath}
    k(x,y) = 0 \qquad \text{for all $x,y \in X$ with $d(x,y) \ge c$}
  \end{displaymath}
  for some constant $c>0$. Then $K$ can be interpreted as an element
  of $C_{\mathrm {red}}^*(\Gamma, \al K)$ and one can write the
  $\Gamma$-trace as
  \begin{displaymath}
    \tr_\Gamma K = \int_D k(x,x)\, dx
  \end{displaymath}
  (see \cite[Section~1]{sunada:92} as well as \cite{atiyah:76} for
  further details), where $D$ is a fundamental domain of $X \to M$.

  If we consider the spectral
  resolution of the Laplacian $\laplacian X \cong \Oint{} \lambda
  {E(\lambda)}$, then it follows that
  \begin{displaymath}
    E(\lambda_2) - E(\lambda_1) \in C_{\mathrm {red}}^*(\Gamma, \al K)
  \end{displaymath}
  if $\lambda_1 < \lambda_2$ and $\lambda_1, \lambda_2 \not\in \spec
  \laplacian X$ (cf.~\cite[Section~2]{sunada:92}).
\end{remark}

\begin{theorem}
  \label{thm:band}
  Suppose $(X,g) \to (M,g)$ is a Riemannian $\Gamma$-covering where
  $\Gamma$ is a residually finite group having positive Kadison
  constant, i.e., $C(\Gamma)>0$. Denote by $\al N(g,\lambda)$ the number
  of components of $\spec\laplacian {(X,g)}$ which intersect the
  interval $[0, \lambda]$. Then we have
\begin{itemize}
\item[(i)] For each $n \in \N$ there exists a metric $g=g_{\eps_n}$
  constructed as in Subsection~\ref{ssec:constr.cov.sp} such that
  $\spec \laplacian {(X,g)}$ has at least $n$ gaps, i.e., $\al
  N(g_{\eps_n},\infty) \ge n+1$.
\item[(ii)] $\spec \laplacian {(X,g)}$ has band structure, i.e.~it is
  a locally finite union of closed intervals. In other words, $\al
  N(g,\lambda) < \infty$ for all $\lambda \ge 0$.
\item[(iii)] The following asymptotic upper bound holds:
  \begin{displaymath}
  \limsup_{\lambda\to\infty} 
     \frac{\al N(g, \lambda)} {(2\pi)^{-d} \omega_d \vol (M,g) \lambda^{d/2}}
  \leq \frac 1 {C(\Gamma)}.
  \end{displaymath}
\item[(iv)] Suppose in addition that the limit manifold $(N,g)$ has simple
  spectrum, i.e., all eigenvalues $\lambda_k(0)$ have multiplicity $1$
  (cf.~Theorem~\ref{thm:mfd.conv}). Then for each $\lambda \ge 0$ there
  exists $\eps(\lambda)>0$ such that
  \begin{displaymath}
  \liminf_{\lambda\to\infty} 
     \frac{\al N (g_{\eps(\lambda)}, \lambda)}
       {(2\pi)^{-d} \omega_d \vol (N,g) \lambda^{d/2}}
  \geq 1.
  \end{displaymath}
  Here, $g_\eps$ denote the metric constructed in
  Section~\ref{sec:construye}.
\end{itemize}
\end{theorem}
\begin{proof}
  (i)~follows immediately from Theorem~\ref{thm:gaps.res.fin}.
  (ii)~and (iii)~have been shown in~\cite{sunada:92} or
  \cite{bruening-sunada:92}. (iv)~Suppose $\lambda \notin \spec
  \laplacian N$, then $\lambda_k(0) < \lambda < \lambda_{k+1}(0)$ for
  some $k\in\N$.  Let $\eps=\eps(\lambda) \in (0,1]$ be the largest
  number such that $\al N(\lambda,g_\eps)$ is (at least) $k$, in other
  words, $\al N(\lambda, g_\eps) \ge k = \al N(\lambda, \laplacian N)$
  where the latter number denotes the number of eigenvalues of
  $\laplacian N$ below $\lambda$.  By the Weyl theorem,
  \begin{displaymath}
  \lim_{\lambda\to\infty} 
     \frac{\al N (\lambda, \laplacian N)}
       {(2\pi)^{-d} \omega_d \vol (N,g) \lambda^{d/2}} = 1
  \end{displaymath}
  and the result follows.
\end{proof}

\begin{remark}
  \label{rem:bethe.sommer}
  Note that (iii)~only gives an \emph{asymptotic} upper bound on the
  number of components of $\spec \laplacian X \cap [0,\lambda]$, not
  on the \emph{whole} spectrum itself.  Therefore, we have no
  assertion about the so-called \emph{Bethe-Sommerfeld conjecture}
  stating that the number of spectral gaps for a periodic operator in
  dimensions $d \ge 2$ remains \emph{finite}.
\end{remark}

To conclude the section we remark that that generically, $\laplacian
{(N,g)}$ has simple spectrum (cf.~\cite{uhlenbeck:76}). The assumption
in (iv)~is natural since $\al N(g,\lambda)$ counts the components
without multiplicity.




%------------------------------------------------------------
%------------------------------------------------------------
%
\section{Examples}
%
%------------------------------------------------------------
%------------------------------------------------------------
\label{sec:examples}

%
%------------------------------------------------------------
%
\subsection{Relation between the approaches presented in 
  Sections~\ref{sec:type.I} and \ref{sec:res.fin}}
%
%------------------------------------------------------------
%
\label{Relation5.6} 

We begin comparing the two main approaches presented in this
paper which assure the existence of spectral gaps
(cf.~Sections~\ref{sec:type.I} and \ref{sec:res.fin}).

One easily sees from Definition~\ref{def:res.fin} that a \emph{finite}
extension of a residually finite group is again residually finite. In
particular, type~I groups are residually finite as finite extensions
of abelian groups (cf.\ Definition~\ref{def:type.I}).  Therefore, for
type~I groups one can also produce spectral gaps by the approximation
method with finite coverings introduced in Section~\ref{sec:res.fin}.
Nevertheless we believe that the direct integral method will be useful
when analysing further spectral properties:

\begin{example}
 \label{rem:ev.dep.cont}
 One of the advantages of the method described in
 Section~\ref{sec:type.I} is that one has more information about the
 bands.  Suppose $\Gamma$ is finitely generated and \emph{abelian},
 i.e.\ $\Gamma \cong \Z^r \oplus \Gamma_0$, where $\Gamma_0$ is the
 torsion subgroup of $\Gamma$. Then $\hG$ is the disjoint union of
 finitely many copies of $\Torus^r$.  From the continuity of the map
 $\rho \to \EWR k$ (cf.~\cite{bjr:99} or~\cite{sunada:90}), we can
 simplify the characterisation of the spectrum in
 Theorem~\ref{thm:floquet} and obtain
 \begin{equation}
   \label{eq:char.spec.ab}
   \spec \laplacian X = \bigcup_{k \in \N} B_k, \quad \text{where} \quad
   B_k := \set{\EWR k} {\rho \in \hG} \subseteq I_k,
 \end{equation}
 the $k$-th \emph{band}. Since $\hG$ is compact, $B_k$ is also
 compact, but in general, $B_k$ need not to be connected (recall that
 $\hG$ is connected iff $\G$ is torsion free, i.e., $\G = \Z^r$).
 Note also that $B_k$ has only finitely many components.  For
 non-abelian groups this approach may be generalised in the direction
 of Hilbert C*-modules (cf.~\cite{gruber:01}).
\end{example}


In principle one could also consider a combination of the methods of
Section~\ref{sec:type.I} and~\ref{sec:res.fin}: denote by $\al T_1$
the class of type~I groups and by $\mathfrak{R}\mathcal{T}_1$ the
class of \emph{residually type~I} groups, i.e.,
$\Gamma\in\mathfrak{R}\mathcal{T}_1$ iff the $\al T_1$-residual
$\mathfrak{R}_{\mathcal{T}_1}(\Gamma)$ is trivial
(cf.~Eq.~\eqref{eq:residual}).  Similarly we denote by
$\mathfrak{R}\mathcal{F}$ the class of residually finite groups
(cf.~Definition~\ref{def:res.fin}).  If we consider a covering with a
group $\Gamma\in\mathfrak{R}\mathcal{T}_1$, then instead of the
\emph{finite} covering $\map{p_i} {M_i} M$ considered in
Eq.~\eqref{eq:sub.cov} we would have a covering with a type~I group.
For these groups, we can replace Lemma~\ref{lem:fin.group} by the
direct integral decomposition of Theorem~\ref{thm:floquet}.
Nevertheless the following lemma shows that the class of residually
finite and residually type~I groups coincide.
  

\begin{lemma}
  \label{lem:res.class}
  From the inclusion
  $\mathcal{F}\subset\mathcal{T}_1\subset\mathfrak{R}\mathcal{F}$ it
  follows that the corresponding residuals for the group $\Gamma$
  coincide,i.e., $\mathfrak R_{\al F}(\Gamma)=\mathfrak R_{\al T_1}
  (\Gamma)$.  Moreover,
  $\mathfrak{R}\mathcal{F}=\mathfrak{R}\mathcal{T}_1$.
\end{lemma}
\begin{proof}
  From the inclusion $\mathcal{F}\subset\mathcal{T}_1$ it follows
  immediately that $\mathfrak R_{\al F}(\Gamma)\supset \mathfrak
  R_{\al T_1} (\Gamma)$.  To show the reverse inclusion one uses the
  following characterisation: a group is residually $\al F$ iff it is
  a subcartesian product of finite groups
  (cf.~\cite[\S~2.3.3]{robinson:82}).  Finally, from the equality of
  the residuals it follows that
  $\mathfrak{R}\mathcal{F}=\mathfrak{R}\mathcal{T}_1$.
\end{proof}



%
%------------------------------------------------------------
%
\subsection{Examples with residually finite groups}
%
%------------------------------------------------------------
%
\label{ResFinGroups}

In the rest of this subsection we present several examples 
of residually finite groups which are not type~I. They
show different aspects of our analysis.

For the next example recall the construction procedure~(A)
described in Section~\ref{sec:construye}.
\begin{example}[Fundamental groups of closed surfaces]
  \label{ex:fund.group}
  Suppose that $N:=\Sphere^2$ is the two-dimensional sphere with a
  metric such that $\laplacian N$ has simple spectrum
  (cf.~\cite{uhlenbeck:76} for the existence of such metrics).
  Suppose, in addition, that $M$ is obtained by adding $r$ handles to
  $N$ as described in Section~\ref{sec:construye}, Case~A.  The
  fundamental group $\Gamma$ of $M$ (cf.~Eq.~\eqref{eq:fund.group}
  with $s=0$) is residually finite (recall
  Example~\ref{ex:res.fin}~(iii)).  Moreover, from the proof of
  Proposition~2.16 in \cite{marcolli-mathai:99}, $\Gamma$ has a
  positive Kadison constant. Therefore, Theorem~\ref{thm:band} applies
  to the the universal cover $X :=\widetilde M \to M$ with the metric
  $g_\eps$ specified in Section~\ref{sec:construye}.
\end{example}

The following example uses the construction procedure~(B)
in Section~\ref{sec:construye}.
\begin{example}[Heisenberg group]
  \label{ex:heisenberg}
  Let $\Gamma := H_3(\Z)$ be the \emph{discrete Heisenberg group},
  where $H_3(R)$ denotes the set of matrices
  \begin{equation}
    \label{eq:heisenberg}
    A_{x,y,z} := \begin{pmatrix}
      1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1
    \end{pmatrix}
  \end{equation}
  with coefficients $x,y,z$ in the ring $R$. A covering with group
  $\Gamma$ is given e.g.~by $X:=H_3(\R)$ with compact quotient
  $M:=H_3(\R)/H_3(\Z)$. Note that $X$ is diffeomorphic to $\R^3$.
  Clearly, $\Gamma$ is a finitely generated linear group and therefore
  residually finite (cf.\ Example~\ref{ex:res.fin}~(iii)). Now, by
  Theorem~\ref{thm:gaps.res.fin} one can deform conformally a
  $\Gamma$-invariant metric $g$ as in Case~(B) of
  Section~\ref{sec:construye}, such that $\spec \laplacian X$ has at
  least $n$ spectral gaps, $n\in\N$.

  In this case, $\Gamma$ is also amenable as an extension of amenable groups
  (cf.\ Remark~\ref{rem:amenable}). In fact,
  $\Gamma$ is isomorphic to the semi-direct product 
  $\Z\ltimes \Z^2$, where $1 \in \Z$ acts on $\Z^2$ by the matrix
  \begin{displaymath}
    \begin{pmatrix}
      1 & 1 \\ 0 & 1
    \end{pmatrix}.
  \end{displaymath}
  Therefore, we have equality in the
  characterisation of $\spec \laplacian X$ in
  Theorems~\ref{thm:floquet} and~\ref{thm:res.fin}.
  
  Note finally that the group $\Gamma$ is not of type~I since
  $\Gamma_{\mathrm {fcc}} = \set{A_{0,y,0}}{y \in \Z}$ has infinite
  index in $\Gamma$ (cf.\ 
  Remark~\ref{rem:type.I}~(\ref{rem:type.I.ii})). Thus, our method in
  Section~\ref{sec:type.I} does not apply since the measure $\mathrm
  dz$ in~\eqref{eq:fourier} is supported only on infinite-dimensional
  Hilbert spaces.  Curiously, one can construct a \emph{finitely}
  additive measure on the group dual $\hG$ supported by the set of
  finite-dimensional representations of $\hG$ (cf.~\cite{pytlik:79}).
  The group dual $\hG$ is calculated
  e.g.~in~\cite[Beispiel~1]{kaniuth:68}.
\end{example}

\begin{example}[Free groups]
\label{ex:free.group}
  Let $\Gamma = \Z^{*r}$ be the free group with $r>1$
  generators. Then $\Gamma$ is residually finite 
  (recall Example~\ref{ex:res.fin}~(ii))
  and has positive Kadison constant (cf.~\cite[Appendix]{sunada:92}). 
  Therefore, Theorem~\ref{thm:band} applies to the $\Gamma$-coverings 
  $X \to M$ specified in Section~\ref{sec:construye}.
  
  Note that $\Gamma$ is not of type~I since $\Gamma_{\mathrm
    {fcc}}=\{e\}$ (cf.~Remark~\ref{rem:type.I}~(\ref{rem:type.I.ii})).
  Such groups are called \emph{ICC (infinite conjugacy class) groups}.
  Again, for any direct integral decomposition~\eqref{eq:fourier},
  almost all Hilbert spaces $H(z)$ are infinite-dimensional. Finally,
  $\Gamma$ is not amenable.
\end{example}

%
%------------------------------------------------------------
%
\subsection{An example with an amenable, non-residually finite group}
%
%------------------------------------------------------------
%
\label{OpenQuestion}

Kirchberg mentioned in \cite[Section~5]{kirchberg:94} an interesting
example of a finitely generated \emph{amenable} group which is not
residually finite: Denote by $S_0$ the group of permutations of $\Z$
which leave unpermuted all but a finite number of integers. We call
$A_0$ the normal subgroup of even permutations in $S_0$.  Let $\Z$ act
on $S_0$ as shift operator.  Then the semi-direct product $\Gamma: =\Z
\ltimes S_0$ is (finitely) generated by the shift $n \mapsto n+1$ and
the transposition interchanging $0$ and $1$.  Note that $\Gamma$ and
$S_0$ are ICC groups.
\begin{lemma}
  \label{S0amenable}
  The group $\Gamma$ is amenable. Moreover, $\mathfrak{R}_{\al
    F}(\Gamma)=A_0$, hence $\Gamma$ is not residually finite.
\end{lemma}
\begin{proof}
  The group $S_0$ is amenable as inductive limit of amenable groups;
  therefore, $\Gamma$ is amenable as semi-direct product of amenable
  groups (cf.~\cite[Section~4]{day:57}).
  
  The equality $\mathfrak R_{\mathcal F} (\Gamma) = A_0$ follows from
  the fact that $A_0$ is simple.
\end{proof}

\begin{proposition}
\label{prop:OpenFinRep}
Every finite-dimensional unitary representation $\rho$ of $\Gamma$
leaves $A_0$ elementwise invariant, i.e., $\rho(\gamma)=\1$ for all
$\gamma \in A_0$.
\end{proposition}
\begin{proof}
  Let $\al E$ be the class of countable subgroups of $\mathrm U(n)$,
  $n \in \N$, and $\al {FG}$ the class of finitely generated groups.
  Note that $\al F \subset \al E \cap \al {FG}$ and that finitely
  generated linear groups are residually finite
  (cf.~Example~\ref{ex:res.fin}~(iii)), i.e.~$\al E \cap \al {FG}
  \subset\mathfrak R \al F$.  Arguing as in the proof of
  Lemma~\ref{lem:res.class} we obtain from the inclusions $\al F
  \subset \al E \cap \al {FG} \subset\mathfrak R \al F$ that
  $\mathfrak R_{\al E \cap \al{FG}}(\Gamma) = \mathfrak R_{\al
    F}(\Gamma)$.  Now by Lemma~\ref{S0amenable} the $\al F$-residual
  of $\Gamma$ is $A_0$.  Finally, since $\Gamma$ itself is finitely
  generated (i.e.\ $\Gamma \in \al {FG}$), we have
  \begin{displaymath}
     \mathfrak R_{\al E}(\Gamma) =
     \mathfrak R_{\al E \cap \al{FG}}(\Gamma)=A_0.
  \end{displaymath}
  This concludes the proof since $\rho$ is a finite-dimensional
  unitary representation iff $\mathrm {im} (\rho) \cong \Gamma/\ker
  \rho \in \al E$, i.e., $\mathfrak R_{\al E}(\Gamma)$ is the
  intersection of all $\ker \rho$, where $\rho$ are the
  finite-dimensional, unitary representations of $\Gamma$.
\end{proof}

In conclusion, we cannot analyse the spectrum of $\laplacian X$ by
none of the above methods since $\Gamma$ is not residually finite (and
therefore neither of type~I). Nevertheless, equality holds
in~\eqref{eq:spec.dir.int}, but we would need infinite-dimensional
Hilbert spaces $H(z)$ in the direct integral decomposition in order to
describe the spectrum of the whole covering $X \to M$ and not only of
the subcovering $X/A_0 \to M$ (with covering group $\Z \times \Z_2$,
cf.\ Diagram~\eqref{eq:sub.cov}).


%------------------------------------------------------------
%------------------------------------------------------------
%
\section{Outlook}
%
%------------------------------------------------------------
%------------------------------------------------------------
\label{sec:outlook}

Coverings with transformation groups as in
Subsection~\ref{OpenQuestion} cannot be treated with the methods
developped in this paper. It seems though reasonable that even for
non-residually finite groups the construction specified in
Section~\ref{sec:construye} still produces at least $n$ spectral gaps,
$n\in\N$.  Nevertheless one probably needs to replace the techniques
of Section~\ref{sec:floquet} that use the min-max principle in order
to prove the existence of spectral gaps for these types of covering
manifolds.

A further interesting line of research would be to consider the
opposite situation as in the present paper; that means the use of
geometry to prevent the appearance of spectral gaps
(cf.~\cite{friedlander:91,mazzeo:91}).




%------------------------------------------------------------
%------------------------------------------------------------
% 
\section*{Acknowledgements}
% 
%------------------------------------------------------------
%------------------------------------------------------------

It is a pleasure to thank Mohamed Barakat for helpful discussions on
residually finite groups. We are also grateful to David
Krej{\v{c}}i{\v{r}}{\'\i}k and Norbert Peyerimhoff for useful
comments. Finally, we would like to thank Volker En{\ss} for valuable
remarks and suggestions on the manuscript.


%------------------------------------------------------------
% References
%------------------------------------------------------------

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%\input nc-floquet.bbl

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 5046 1113 l 5042 1104 l 5039 1096 l 5036 1088 l 5032 1080 l
 5028 1072 l 5024 1064 l 5018 1058 l 5012 1053 l 5004 1048 l
 4996 1045 l 4987 1043 l 4976 1041 l 4964 1039 l 4952 1038 l
 4939 1038 l 4927 1039 l 4915 1040 l 4904 1042 l 4895 1045 l
 4884 1050 l 4874 1056 l 4865 1063 l 4856 1070 l 4850 1078 l
 4844 1086 l 4840 1092 l 4836 1098 l 4835 1101 l 4833 1103 l
 4832 1106 l 4831 1108 l 4831 1111 l 4832 1114 l 4833 1117 l
 4835 1120 l 4837 1123 l 4840 1127 l 4843 1131 l 4847 1136 l
 4853 1143 l 4860 1151 l 4868 1160 l 4877 1170 l 4886 1180 l
 4895 1191 l 4905 1201 l 4913 1211 l 4921 1219 l
 cp gs col7 0.80 shd ef gr 
% Polyline
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 4594 618 l 4605 622 l 4615 626 l 4624 630 l 4632 636 l
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 4653 667 l 4650 670 l 4644 672 l 4636 675 l 4627 677 l
 4616 679 l 4605 680 l 4594 682 l 4583 683 l 4572 685 l
 4561 687 l 4549 689 l 4537 691 l 4526 693 l 4515 693 l
 4507 693 l 4499 693 l 4490 690 l 4482 685 l 4475 679 l
 4469 672 l 4465 666 l 4461 660 l 4460 657 l 4458 654 l
 4457 651 l 4456 648 l 4455 645 l 4455 643 l 4456 640 l
 4457 637 l 4459 635 l 4461 633 l 4465 629 l 4471 626 l
 4478 622 l 4486 618 l 4495 615 l 4504 612 l 4514 610 l

 cp gs col7 0.80 shd ef gr 
% here ends figure;
% 
% here starts figure with depth 100
% Polyline
2 slj
7.500 slw
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 909 867 l 913 861 l 918 857 l 923 862 l 927 869 l
 930 876 l 932 883 l 934 895 l
 cp gs col0 s gr 
% Polyline
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 1024 1015 l 1031 1022 l 1036 1027 l 1041 1032 l 1047 1039 l
 1054 1045 l 1060 1052 l 1067 1059 l 1074 1065 l 1081 1071 l
 1088 1076 l 1097 1082 l 1106 1088 l 1117 1094 l 1127 1100 l
 1138 1105 l 1149 1110 l 1160 1114 l 1169 1117 l 1179 1121 l
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 1245 1136 l 1256 1138 l 1267 1140 l 1277 1141 l 1288 1143 l
 1299 1144 l 1311 1146 l 1323 1147 l 1336 1148 l 1348 1149 l
 1359 1150 l 1371 1151 l 1382 1151 l 1392 1152 l 1404 1152 l
 1416 1153 l 1429 1153 l 1442 1153 l 1455 1153 l 1467 1153 l
 1479 1153 l 1491 1153 l 1502 1152 l 1513 1152 l 1525 1151 l
 1537 1150 l 1549 1149 l 1562 1148 l 1574 1147 l 1587 1146 l
 1598 1145 l 1609 1143 l 1620 1142 l 1630 1140 l 1641 1138 l
 1652 1137 l 1664 1134 l 1675 1132 l 1687 1130 l 1698 1127 l
 1708 1125 l 1717 1122 l 1726 1120 l 1736 1117 l 1746 1114 l
 1757 1110 l 1767 1106 l 1777 1102 l 1786 1098 l 1795 1094 l
 1802 1090 l 1810 1086 l 1817 1081 l 1824 1076 l 1832 1070 l
 1839 1065 l 1845 1059 l 1851 1054 l 1856 1049 l 1862 1043 l
 1868 1036 l 1874 1029 l 1880 1022 l 1886 1016 l 1892 1010 l
 1898 1005 l 1906 999 l 1914 993 l 1923 987 l 1931 982 l
 1939 978 l 1946 975 l 1953 972 l 1960 970 l 1966 968 l
 1972 967 l 1977 966 l 1982 965 l 1987 964 l 1993 964 l
 1998 964 l
 1999 964 l gs col0 s gr 
% Polyline
n 917 859 m 918 859 l 924 857 l 934 854 l 948 850 l 960 846 l
 970 843 l 980 839 l 989 836 l 997 832 l 1007 828 l
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 1064 796 l 1072 790 l 1079 785 l 1087 778 l 1095 772 l
 1103 766 l 1110 760 l 1117 754 l 1125 749 l 1132 744 l
 1141 739 l 1150 734 l 1160 729 l 1171 724 l 1182 719 l
 1192 715 l 1203 712 l 1212 709 l 1221 706 l 1231 703 l
 1242 701 l 1253 698 l 1265 696 l 1277 693 l 1288 691 l
 1300 690 l 1311 688 l 1323 686 l 1336 685 l 1349 684 l
 1363 682 l 1378 681 l 1393 680 l 1407 680 l 1421 679 l
 1434 679 l 1447 679 l 1459 679 l 1471 679 l 1484 680 l
 1498 680 l 1511 681 l 1524 682 l 1537 683 l 1549 685 l
 1561 686 l 1571 687 l 1582 689 l 1593 690 l 1604 692 l
 1616 694 l 1627 696 l 1639 698 l 1649 700 l 1660 702 l
 1669 704 l 1679 707 l 1690 710 l 1701 713 l 1713 716 l
 1725 721 l 1736 725 l 1748 730 l 1758 735 l 1767 740 l
 1777 745 l 1787 751 l 1797 758 l 1807 766 l 1817 773 l
 1827 780 l 1836 787 l 1845 793 l 1854 798 l 1863 804 l
 1872 809 l 1881 814 l 1890 819 l 1899 824 l 1906 828 l
 1913 831 l 1921 835 l 1929 839 l 1938 842 l 1945 845 l
 1952 848 l 1957 849 l 1964 851 l 1972 852 l 1981 853 l
 1988 854 l
 1990 854 l gs col0 s gr 
% Polyline
n 1207 884 m 1209 885 l 1214 887 l 1222 891 l 1234 896 l 1249 903 l
 1266 909 l 1284 916 l 1305 923 l 1327 929 l 1352 935 l
 1381 940 l 1412 944 l 1447 946 l 1478 946 l 1507 945 l
 1533 943 l 1558 940 l 1580 936 l 1600 932 l 1620 928 l
 1638 923 l 1654 918 l 1669 914 l 1681 910 l 1691 906 l
 1697 904 l 1701 903 l
 1703 902 l gs col0 s gr 
% Polyline
n 1243 900 m 1245 899 l 1250 898 l 1259 895 l 1271 892 l 1286 888 l
 1303 884 l 1321 880 l 1342 876 l 1363 872 l 1387 869 l
 1413 867 l 1441 865 l 1472 865 l 1502 866 l 1529 869 l
 1553 872 l 1574 876 l 1593 881 l 1610 886 l 1626 891 l
 1640 895 l 1652 900 l 1661 904 l 1668 907 l 1672 908 l

 1674 909 l gs col0 s gr 
% Polyline
n 1999 963 m 2001 963 l 2008 965 l 2018 967 l 2026 969 l 2032 971 l
 2038 973 l 2043 975 l 2050 978 l 2056 981 l 2062 984 l
 2069 988 l 2075 992 l 2082 997 l 2089 1002 l 2097 1009 l
 2104 1015 l 2111 1022 l 2116 1027 l 2122 1032 l 2127 1039 l
 2134 1045 l 2141 1052 l 2147 1059 l 2154 1065 l 2161 1071 l
 2169 1076 l 2177 1082 l 2187 1088 l 2197 1094 l 2208 1100 l
 2219 1105 l 2230 1110 l 2241 1114 l 2250 1117 l 2260 1121 l
 2270 1123 l 2281 1126 l 2292 1129 l 2304 1132 l 2315 1134 l
 2326 1136 l 2337 1138 l 2348 1140 l 2358 1141 l 2369 1143 l
 2380 1144 l 2392 1146 l 2404 1147 l 2417 1148 l 2429 1149 l
 2440 1150 l 2452 1151 l 2463 1151 l 2473 1152 l 2485 1152 l
 2497 1153 l 2510 1153 l 2523 1153 l 2536 1153 l 2548 1153 l
 2560 1153 l 2572 1153 l 2583 1152 l 2594 1152 l 2606 1151 l
 2618 1150 l 2630 1149 l 2643 1148 l 2655 1147 l 2667 1146 l
 2679 1145 l 2690 1143 l 2701 1142 l 2711 1140 l 2722 1138 l
 2733 1137 l 2744 1134 l 2756 1132 l 2767 1130 l 2778 1127 l
 2788 1125 l 2797 1122 l 2806 1120 l 2816 1117 l 2827 1114 l
 2837 1110 l 2847 1106 l 2857 1102 l 2866 1098 l 2875 1094 l
 2882 1090 l 2890 1086 l 2897 1081 l 2904 1076 l 2912 1070 l
 2919 1065 l 2925 1059 l 2931 1054 l 2936 1049 l 2942 1043 l
 2948 1036 l 2954 1029 l 2961 1022 l 2967 1016 l 2973 1010 l
 2979 1005 l 2987 999 l 2995 993 l 3004 987 l 3012 982 l
 3019 978 l 3026 975 l 3033 972 l 3040 970 l 3047 968 l
 3052 967 l 3057 966 l 3062 965 l 3067 964 l 3073 964 l
 3078 964 l
 3079 964 l gs col0 s gr 
% Polyline
n 1989 854 m 1990 854 l 1996 853 l 2009 850 l 2023 847 l 2037 844 l
 2049 841 l 2059 838 l 2068 835 l 2077 832 l 2087 828 l
 2098 823 l 2108 818 l 2119 812 l 2129 807 l 2137 801 l
 2145 796 l 2153 790 l 2160 785 l 2168 778 l 2176 772 l
 2184 766 l 2191 760 l 2198 754 l 2206 749 l 2213 744 l
 2221 739 l 2231 734 l 2241 729 l 2251 724 l 2262 719 l
 2273 715 l 2283 712 l 2292 709 l 2301 706 l 2311 703 l
 2322 701 l 2334 698 l 2345 696 l 2357 693 l 2369 691 l
 2381 690 l 2392 688 l 2404 686 l 2417 685 l 2430 684 l
 2444 682 l 2459 681 l 2474 680 l 2488 680 l 2502 679 l
 2515 679 l 2528 679 l 2540 679 l 2552 679 l 2565 680 l
 2579 680 l 2592 681 l 2605 682 l 2618 683 l 2630 685 l
 2641 686 l 2652 687 l 2663 689 l 2674 690 l 2685 692 l
 2696 694 l 2708 696 l 2719 698 l 2730 700 l 2740 702 l
 2750 704 l 2759 707 l 2770 710 l 2781 713 l 2793 716 l
 2805 721 l 2817 725 l 2828 730 l 2839 735 l 2848 740 l
 2858 745 l 2867 751 l 2878 758 l 2888 766 l 2898 773 l
 2908 780 l 2917 787 l 2926 793 l 2935 798 l 2944 804 l
 2953 809 l 2962 814 l 2971 819 l 2980 824 l 2987 828 l
 2994 831 l 3002 835 l 3011 839 l 3019 842 l 3027 845 l
 3033 848 l 3040 850 l 3046 851 l 3053 852 l 3063 853 l
 3072 854 l 3079 855 l
 3080 855 l gs col0 s gr 
% Polyline
n 2288 884 m 2290 885 l 2295 887 l 2303 891 l 2315 896 l 2329 903 l
 2346 909 l 2365 916 l 2385 923 l 2408 929 l 2432 935 l
 2461 940 l 2492 944 l 2527 946 l 2558 946 l 2587 945 l
 2613 943 l 2638 940 l 2660 936 l 2681 932 l 2700 928 l
 2718 923 l 2735 918 l 2749 914 l 2762 910 l 2772 906 l
 2778 904 l 2782 903 l
 2784 902 l gs col0 s gr 
% Polyline
n 2323 900 m 2325 899 l 2330 898 l 2339 895 l 2351 892 l 2366 888 l
 2383 884 l 2402 880 l 2422 876 l 2444 872 l 2468 869 l
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 2634 872 l 2656 876 l 2675 881 l 2692 886 l 2707 891 l
 2721 895 l 2733 900 l 2742 904 l 2749 907 l 2753 908 l

 2755 909 l gs col0 s gr 
% Polyline
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 3565 1152 l 3577 1153 l 3590 1153 l 3603 1153 l 3616 1153 l
 3628 1153 l 3640 1153 l 3652 1153 l 3663 1152 l 3674 1152 l
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 3748 1146 l 3759 1145 l 3770 1143 l 3781 1142 l 3791 1140 l
 3802 1138 l 3813 1137 l 3825 1134 l 3836 1132 l 3848 1130 l
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 3907 1114 l 3918 1110 l 3928 1106 l 3938 1102 l 3947 1098 l
 3956 1094 l 3963 1090 l 3971 1086 l 3978 1081 l 3985 1076 l
 3993 1070 l 4000 1065 l 4006 1059 l 4012 1054 l 4017 1049 l
 4023 1043 l 4029 1036 l 4035 1029 l 4041 1022 l 4047 1016 l
 4053 1010 l 4060 1005 l 4067 999 l 4076 993 l 4084 987 l
 4092 982 l 4100 978 l 4107 975 l 4114 972 l 4121 970 l
 4127 968 l 4133 967 l 4138 966 l 4143 965 l 4148 964 l
 4154 964 l 4159 964 l
 4160 964 l gs col0 s gr 
% Polyline
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 3133 841 l 3142 838 l 3151 835 l 3159 832 l 3169 828 l
 3179 823 l 3189 818 l 3199 812 l 3209 807 l 3218 801 l
 3226 796 l 3233 790 l 3240 785 l 3248 778 l 3256 772 l
 3264 766 l 3271 760 l 3278 754 l 3286 749 l 3293 744 l
 3302 739 l 3311 734 l 3321 729 l 3332 724 l 3343 719 l
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 3938 745 l 3948 751 l 3958 758 l 3968 766 l 3978 773 l
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 4033 809 l 4042 814 l 4051 819 l 4060 824 l 4067 828 l
 4074 831 l 4083 835 l 4091 839 l 4099 842 l 4107 845 l
 4114 848 l 4121 850 l 4127 851 l 4134 852 l 4144 853 l
 4153 854 l 4160 855 l
 4161 855 l gs col0 s gr 
% Polyline
n 3369 884 m 3371 885 l 3376 887 l 3384 891 l 3396 896 l 3410 903 l
 3427 909 l 3446 916 l 3466 923 l 3489 929 l 3514 935 l
 3542 940 l 3573 944 l 3608 946 l 3639 946 l 3668 945 l
 3694 943 l 3718 940 l 3741 936 l 3761 932 l 3780 928 l
 3798 923 l 3815 918 l 3830 914 l 3842 910 l 3852 906 l
 3858 904 l 3862 903 l
 3864 902 l gs col0 s gr 
% Polyline
n 3404 900 m 3406 899 l 3411 898 l 3420 895 l 3432 892 l 3447 888 l
 3464 884 l 3483 880 l 3503 876 l 3525 872 l 3549 869 l
 3575 867 l 3603 865 l 3634 865 l 3664 866 l 3691 869 l
 3715 872 l 3736 876 l 3755 881 l 3772 886 l 3787 891 l
 3801 895 l 3813 900 l 3823 904 l 3829 907 l 3833 908 l

 3835 909 l gs col0 s gr 
% Polyline
n 5244 854 m 5244 855 l 5246 859 l 5248 868 l 5251 879 l 5253 893 l
 5253 911 l 5252 929 l 5249 943 l 5246 955 l 5243 964 l
 5241 969 l
 5241 970 l gs col0 s gr 
% Polyline
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 4709 1153 l 4721 1153 l 4733 1153 l 4744 1152 l 4755 1152 l
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 4829 1146 l 4840 1145 l 4851 1143 l 4862 1142 l 4872 1140 l
 4883 1138 l 4894 1137 l 4906 1134 l 4917 1132 l 4929 1130 l
 4940 1127 l 4950 1125 l 4959 1122 l 4968 1120 l 4978 1117 l
 4988 1114 l 4999 1110 l 5009 1106 l 5019 1102 l 5028 1098 l
 5036 1094 l 5044 1090 l 5051 1086 l 5059 1081 l 5066 1076 l
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 5134 1010 l 5140 1005 l 5148 999 l 5156 993 l 5165 987 l
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 5208 968 l 5213 967 l 5218 966 l 5223 965 l 5229 964 l
 5235 964 l 5240 964 l
 5241 964 l gs col0 s gr 
% Polyline
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 5155 831 l 5163 835 l 5172 839 l 5180 842 l 5188 845 l
 5195 848 l 5201 850 l 5207 851 l 5215 852 l 5224 853 l
 5234 854 l 5241 855 l
 5242 855 l gs col0 s gr 
% Polyline
n 4449 884 m 4451 885 l 4456 887 l 4464 891 l 4476 896 l 4490 903 l
 4507 909 l 4526 916 l 4546 923 l 4569 929 l 4593 935 l
 4622 940 l 4653 944 l 4688 946 l 4719 946 l 4748 945 l
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 4879 923 l 4896 918 l 4910 914 l 4923 910 l 4933 906 l
 4939 904 l 4943 903 l
 4945 902 l gs col0 s gr 
% Polyline
n 4485 900 m 4487 899 l 4492 898 l 4501 895 l 4513 892 l 4528 888 l
 4545 884 l 4563 880 l 4584 876 l 4605 872 l 4629 869 l
 4655 867 l 4683 865 l 4714 865 l 4744 866 l 4771 869 l
 4795 872 l 4816 876 l 4835 881 l 4852 886 l 4868 891 l
 4882 895 l 4894 900 l 4903 904 l 4910 907 l 4914 908 l

 4916 909 l gs col0 s gr 
% Polyline
0 slj
 [15 15] 15 sd
n 675 900 m
 810 900 l gs col0 s gr  [] 0 sd
% Polyline
 [15 15] 15 sd
n 5355 900 m
 5490 900 l gs col0 s gr  [] 0 sd
% Polyline
2 slj
n 530 372 m 528 389 l 526 401 l 523 410 l 520 417 l 516 424 l
 511 428 l 506 423 l 502 415 l 500 407 l 498 399 l
 496 386 l 495 369 l 497 352 l 499 341 l 501 333 l
 504 327 l 508 321 l 513 317 l 518 322 l 522 329 l
 525 336 l 527 343 l 529 355 l
 cp gs col0 s gr 
% Polyline
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 1534 438 l 1541 435 l 1548 432 l 1555 430 l 1561 428 l
 1567 427 l 1572 426 l 1577 425 l 1582 424 l 1588 424 l
 1593 424 l
 1594 424 l gs col0 s gr 
% Polyline
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 565 303 l 575 299 l 584 296 l 592 292 l 602 288 l
 612 283 l 623 278 l 633 272 l 643 267 l 651 261 l
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 1372 205 l 1382 211 l 1392 218 l 1402 226 l 1412 233 l
 1422 240 l 1431 247 l 1440 253 l 1449 258 l 1458 264 l
 1467 269 l 1476 274 l 1485 279 l 1494 284 l 1501 288 l
 1508 291 l 1516 295 l 1524 299 l 1533 302 l 1540 305 l
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 1583 314 l
 1585 314 l gs col0 s gr 
% Polyline
n 802 344 m 804 345 l 809 347 l 817 351 l 829 356 l 844 363 l
 861 369 l 879 376 l 900 383 l 922 389 l 947 395 l
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 1128 403 l 1153 400 l 1175 396 l 1195 392 l 1215 388 l
 1233 383 l 1249 378 l 1264 374 l 1276 370 l 1286 366 l
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% Polyline
n 838 360 m 840 359 l 845 358 l 854 355 l 866 352 l 881 348 l
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 1148 332 l 1169 336 l 1188 341 l 1205 346 l 1221 351 l
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 1269 369 l gs col0 s gr 
% Polyline
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 1729 505 l 1736 512 l 1742 519 l 1749 525 l 1756 531 l
 1764 536 l 1772 542 l 1782 548 l 1792 554 l 1803 560 l
 1814 565 l 1825 570 l 1836 574 l 1845 577 l 1855 581 l
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 1921 596 l 1932 598 l 1943 600 l 1953 601 l 1964 603 l
 1975 604 l 1987 606 l 1999 607 l 2012 608 l 2024 609 l
 2035 610 l 2047 611 l 2058 611 l 2068 612 l 2080 612 l
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 2155 613 l 2167 613 l 2178 612 l 2189 612 l 2201 611 l
 2213 610 l 2225 609 l 2238 608 l 2250 607 l 2262 606 l
 2274 605 l 2285 603 l 2296 602 l 2306 600 l 2317 598 l
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 2383 585 l 2392 582 l 2401 580 l 2411 577 l 2422 574 l
 2432 570 l 2442 566 l 2452 562 l 2461 558 l 2470 554 l
 2477 550 l 2485 546 l 2492 541 l 2499 536 l 2507 530 l
 2514 525 l 2520 519 l 2526 514 l 2531 509 l 2537 503 l
 2543 496 l 2549 489 l 2556 482 l 2562 476 l 2568 470 l
 2574 465 l 2582 459 l 2590 453 l 2599 447 l 2607 442 l
 2614 438 l 2621 435 l 2628 432 l 2635 430 l 2642 428 l
 2647 427 l 2652 426 l 2657 425 l 2662 424 l 2668 424 l
 2673 424 l
 2674 424 l gs col0 s gr 
% Polyline
n 1584 314 m 1585 314 l 1591 313 l 1604 310 l 1618 307 l 1632 304 l
 1644 301 l 1654 298 l 1663 295 l 1672 292 l 1682 288 l
 1693 283 l 1703 278 l 1714 272 l 1724 267 l 1732 261 l
 1740 256 l 1748 250 l 1755 245 l 1763 238 l 1771 232 l
 1779 226 l 1786 220 l 1793 214 l 1801 209 l 1808 204 l
 1816 199 l 1826 194 l 1836 189 l 1846 184 l 1857 179 l
 1868 175 l 1878 172 l 1887 169 l 1896 166 l 1906 163 l
 1917 161 l 1929 158 l 1940 156 l 1952 153 l 1964 151 l
 1976 150 l 1987 148 l 1999 146 l 2012 145 l 2025 144 l
 2039 142 l 2054 141 l 2069 140 l 2083 140 l 2097 139 l
 2110 139 l 2123 139 l 2135 139 l 2147 139 l 2160 140 l
 2174 140 l 2187 141 l 2200 142 l 2213 143 l 2225 145 l
 2236 146 l 2247 147 l 2258 149 l 2269 150 l 2280 152 l
 2291 154 l 2303 156 l 2314 158 l 2325 160 l 2335 162 l
 2345 164 l 2354 167 l 2365 170 l 2376 173 l 2388 176 l
 2400 181 l 2412 185 l 2423 190 l 2434 195 l 2443 200 l
 2453 205 l 2462 211 l 2473 218 l 2483 226 l 2493 233 l
 2503 240 l 2512 247 l 2521 253 l 2530 258 l 2539 264 l
 2548 269 l 2557 274 l 2566 279 l 2575 284 l 2582 288 l
 2589 291 l 2597 295 l 2606 299 l 2614 302 l 2622 305 l
 2628 308 l 2635 310 l 2641 311 l 2648 312 l 2658 313 l
 2667 314 l 2674 315 l
 2675 315 l gs col0 s gr 
% Polyline
n 1883 344 m 1885 345 l 1890 347 l 1898 351 l 1910 356 l 1924 363 l
 1941 369 l 1960 376 l 1980 383 l 2003 389 l 2027 395 l
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 2313 383 l 2330 378 l 2344 374 l 2357 370 l 2367 366 l
 2373 364 l 2377 363 l
 2379 362 l gs col0 s gr 
% Polyline
n 1918 360 m 1920 359 l 1925 358 l 1934 355 l 1946 352 l 1961 348 l
 1978 344 l 1997 340 l 2017 336 l 2039 332 l 2063 329 l
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 2229 332 l 2251 336 l 2270 341 l 2287 346 l 2302 351 l
 2316 355 l 2328 360 l 2337 364 l 2344 367 l 2348 368 l

 2350 369 l gs col0 s gr 
% Polyline
n 2672 426 m 2673 426 l 2679 427 l 2688 428 l 2696 429 l 2702 431 l
 2708 432 l 2713 433 l 2719 436 l 2726 438 l 2732 441 l
 2738 444 l 2745 448 l 2751 452 l 2758 457 l 2765 462 l
 2773 469 l 2780 475 l 2787 482 l 2792 487 l 2798 492 l
 2803 499 l 2810 505 l 2817 512 l 2823 519 l 2830 525 l
 2837 531 l 2845 536 l 2853 542 l 2863 548 l 2873 554 l
 2884 560 l 2895 565 l 2906 570 l 2917 574 l 2926 577 l
 2935 581 l 2946 583 l 2956 586 l 2968 589 l 2979 592 l
 2991 594 l 3002 596 l 3012 598 l 3023 600 l 3033 601 l
 3044 603 l 3055 604 l 3067 606 l 3079 607 l 3092 608 l
 3104 609 l 3115 610 l 3127 611 l 3138 611 l 3148 612 l
 3160 612 l 3172 613 l 3185 613 l 3198 613 l 3211 613 l
 3223 613 l 3235 613 l 3247 613 l 3258 612 l 3269 612 l
 3281 611 l 3293 610 l 3305 609 l 3318 608 l 3330 607 l
 3343 606 l 3354 605 l 3365 603 l 3376 602 l 3386 600 l
 3397 598 l 3408 597 l 3420 594 l 3431 592 l 3443 590 l
 3454 587 l 3464 585 l 3473 582 l 3482 580 l 3492 577 l
 3502 574 l 3513 570 l 3523 566 l 3533 562 l 3542 558 l
 3551 554 l 3558 550 l 3566 546 l 3573 541 l 3580 536 l
 3588 530 l 3595 525 l 3601 519 l 3607 514 l 3612 509 l
 3618 503 l 3624 496 l 3630 489 l 3636 482 l 3642 476 l
 3648 470 l 3655 465 l 3662 459 l 3671 453 l 3679 447 l
 3687 442 l 3695 438 l 3702 435 l 3709 432 l 3716 430 l
 3722 428 l 3728 427 l 3733 426 l 3738 425 l 3743 424 l
 3749 424 l 3754 424 l
 3755 424 l gs col0 s gr 
% Polyline
n 2675 313 m 2676 313 l 2682 312 l 2692 309 l 2705 306 l 2717 304 l
 2728 301 l 2737 298 l 2746 295 l 2754 292 l 2764 288 l
 2774 283 l 2784 278 l 2794 272 l 2804 267 l 2813 261 l
 2821 256 l 2828 250 l 2835 245 l 2843 238 l 2851 232 l
 2859 226 l 2866 220 l 2873 214 l 2881 209 l 2888 204 l
 2897 199 l 2906 194 l 2916 189 l 2927 184 l 2938 179 l
 2948 175 l 2959 172 l 2968 169 l 2977 166 l 2987 163 l
 2998 161 l 3009 158 l 3021 156 l 3033 153 l 3044 151 l
 3056 150 l 3068 148 l 3079 146 l 3092 145 l 3105 144 l
 3120 142 l 3134 141 l 3149 140 l 3164 140 l 3178 139 l
 3191 139 l 3203 139 l 3215 139 l 3228 139 l 3241 140 l
 3254 140 l 3268 141 l 3281 142 l 3294 143 l 3305 145 l
 3317 146 l 3328 147 l 3338 149 l 3349 150 l 3360 152 l
 3372 154 l 3383 156 l 3395 158 l 3405 160 l 3416 162 l
 3425 164 l 3435 167 l 3446 170 l 3457 173 l 3469 176 l
 3481 181 l 3492 185 l 3504 190 l 3514 195 l 3523 200 l
 3533 205 l 3543 211 l 3553 218 l 3563 226 l 3573 233 l
 3583 240 l 3592 247 l 3601 253 l 3610 258 l 3619 264 l
 3628 269 l 3637 274 l 3646 279 l 3655 284 l 3662 288 l
 3669 291 l 3678 295 l 3686 299 l 3694 302 l 3702 305 l
 3709 308 l 3716 310 l 3722 311 l 3729 312 l 3739 313 l
 3748 314 l 3755 315 l
 3756 315 l gs col0 s gr 
% Polyline
n 2964 344 m 2966 345 l 2971 347 l 2979 351 l 2991 356 l 3005 363 l
 3022 369 l 3041 376 l 3061 383 l 3084 389 l 3109 395 l
 3137 400 l 3168 404 l 3203 406 l 3234 406 l 3263 405 l
 3289 403 l 3313 400 l 3336 396 l 3356 392 l 3375 388 l
 3393 383 l 3410 378 l 3425 374 l 3437 370 l 3447 366 l
 3453 364 l 3457 363 l
 3459 362 l gs col0 s gr 
% Polyline
n 2999 360 m 3001 359 l 3006 358 l 3015 355 l 3027 352 l 3042 348 l
 3059 344 l 3078 340 l 3098 336 l 3120 332 l 3144 329 l
 3170 327 l 3198 325 l 3229 325 l 3259 326 l 3286 329 l
 3310 332 l 3331 336 l 3350 341 l 3367 346 l 3382 351 l
 3396 355 l 3408 360 l 3418 364 l 3424 367 l 3428 368 l

 3430 369 l gs col0 s gr 
% Polyline
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 4848 371 l 4847 389 l 4844 403 l 4841 415 l 4838 424 l
 4836 429 l
 4836 430 l gs col0 s gr 
% Polyline
n 3751 421 m 3752 421 l 3758 423 l 3767 425 l 3776 427 l 3783 429 l
 3788 431 l 3794 433 l 3800 435 l 3806 438 l 3813 441 l
 3819 444 l 3825 448 l 3831 452 l 3838 457 l 3845 462 l
 3853 469 l 3860 475 l 3867 482 l 3872 487 l 3878 492 l
 3884 499 l 3890 505 l 3897 512 l 3904 519 l 3911 525 l
 3918 531 l 3925 536 l 3934 542 l 3943 548 l 3954 554 l
 3964 560 l 3975 565 l 3986 570 l 3997 574 l 4006 577 l
 4016 581 l 4026 583 l 4037 586 l 4048 589 l 4060 592 l
 4071 594 l 4082 596 l 4093 598 l 4104 600 l 4114 601 l
 4125 603 l 4136 604 l 4148 606 l 4160 607 l 4173 608 l
 4185 609 l 4196 610 l 4208 611 l 4219 611 l 4229 612 l
 4241 612 l 4253 613 l 4266 613 l 4279 613 l 4292 613 l
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 4424 606 l 4435 605 l 4446 603 l 4457 602 l 4467 600 l
 4478 598 l 4489 597 l 4501 594 l 4512 592 l 4524 590 l
 4535 587 l 4545 585 l 4554 582 l 4563 580 l 4573 577 l
 4583 574 l 4594 570 l 4604 566 l 4614 562 l 4623 558 l
 4631 554 l 4639 550 l 4646 546 l 4654 541 l 4661 536 l
 4668 530 l 4675 525 l 4682 519 l 4687 514 l 4692 509 l
 4698 503 l 4704 496 l 4710 489 l 4717 482 l 4723 476 l
 4729 470 l 4735 465 l 4743 459 l 4751 453 l 4760 447 l
 4768 442 l 4775 438 l 4782 435 l 4789 432 l 4796 430 l
 4803 428 l 4808 427 l 4813 426 l 4818 425 l 4824 424 l
 4830 424 l 4835 424 l
 4836 424 l gs col0 s gr 
% Polyline
n 3753 317 m 3754 317 l 3760 315 l 3771 312 l 3784 309 l 3796 305 l
 3807 302 l 3817 299 l 3825 296 l 3834 292 l 3844 288 l
 3854 283 l 3865 278 l 3875 272 l 3885 267 l 3893 261 l
 3901 256 l 3909 250 l 3916 245 l 3924 238 l 3932 232 l
 3940 226 l 3947 220 l 3954 214 l 3962 209 l 3969 204 l
 3977 199 l 3987 194 l 3997 189 l 4007 184 l 4018 179 l
 4029 175 l 4039 172 l 4048 169 l 4057 166 l 4067 163 l
 4078 161 l 4090 158 l 4101 156 l 4113 153 l 4125 151 l
 4137 150 l 4148 148 l 4160 146 l 4173 145 l 4186 144 l
 4200 142 l 4215 141 l 4230 140 l 4244 140 l 4258 139 l
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 4335 140 l 4348 141 l 4361 142 l 4374 143 l 4386 145 l
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 4453 154 l 4464 156 l 4476 158 l 4486 160 l 4497 162 l
 4506 164 l 4516 167 l 4527 170 l 4538 173 l 4550 176 l
 4562 181 l 4573 185 l 4585 190 l 4595 195 l 4604 200 l
 4614 205 l 4624 211 l 4634 218 l 4644 226 l 4654 233 l
 4664 240 l 4673 247 l 4682 253 l 4691 258 l 4700 264 l
 4709 269 l 4718 274 l 4727 279 l 4736 284 l 4743 288 l
 4750 291 l 4758 295 l 4767 299 l 4775 302 l 4783 305 l
 4790 308 l 4796 310 l 4802 311 l 4810 312 l 4819 313 l
 4829 314 l 4836 315 l
 4837 315 l gs col0 s gr 
% Polyline
n 4044 344 m 4046 345 l 4051 347 l 4059 351 l 4071 356 l 4085 363 l
 4102 369 l 4121 376 l 4141 383 l 4164 389 l 4188 395 l
 4217 400 l 4248 404 l 4283 406 l 4314 406 l 4343 405 l
 4369 403 l 4394 400 l 4416 396 l 4437 392 l 4456 388 l
 4474 383 l 4491 378 l 4505 374 l 4518 370 l 4528 366 l
 4534 364 l 4538 363 l
 4540 362 l gs col0 s gr 
% Polyline
n 4080 360 m 4082 359 l 4087 358 l 4096 355 l 4108 352 l 4123 348 l
 4140 344 l 4158 340 l 4179 336 l 4200 332 l 4224 329 l
 4250 327 l 4278 325 l 4309 325 l 4339 326 l 4366 329 l
 4390 332 l 4411 336 l 4430 341 l 4447 346 l 4463 351 l
 4477 355 l 4489 360 l 4498 364 l 4505 367 l 4509 368 l

 4511 369 l gs col0 s gr 
% Polyline
0 slj
 [15 15] 15 sd
n 270 360 m
 405 360 l gs col0 s gr  [] 0 sd
% Polyline
 [15 15] 15 sd
n 4950 360 m
 5085 360 l gs col0 s gr  [] 0 sd
% here ends figure;
% 
% here starts figure with depth 50
% Polyline
2 slj
7.500 slw
 [15 15] 15 sd
n 4172 855 m 4172 857 l 4175 868 l 4178 884 l 4180 898 l 4181 909 l
 4181 918 l 4181 927 l 4179 936 l 4175 947 l 4171 958 l
 4169 966 l
 4168 967 l gs col0 s gr  [] 0 sd
% Polyline
n 990 540 m 991 540 l 997 540 l 1010 541 l 1029 543 l 1049 545 l
 1068 548 l 1085 552 l 1099 557 l 1113 563 l 1125 570 l
 1134 576 l 1143 584 l 1153 593 l 1163 603 l 1175 616 l
 1187 630 l 1201 647 l 1216 664 l 1230 682 l 1242 697 l
 1251 709 l 1257 716 l
 1260 720 l gs col0 s gr 
% Polyline
n 1395 450 m 1394 451 l 1391 456 l 1384 467 l 1375 482 l 1365 499 l
 1357 515 l 1351 529 l 1347 541 l 1346 552 l 1347 562 l
 1350 571 l 1354 580 l 1360 589 l 1368 600 l 1378 612 l
 1390 626 l 1404 640 l 1417 653 l 1429 664 l 1436 671 l
 1439 674 l
 1440 675 l gs col0 s gr 
% Polyline
0.000 slw
n 1289 531 m 1297 533 l 1304 536 l 1311 539 l 1318 544 l 1324 549 l
 1330 555 l 1334 561 l 1338 568 l 1341 574 l 1342 581 l
 1343 587 l 1342 593 l 1340 598 l 1338 603 l 1332 609 l
 1323 615 l 1313 619 l 1300 623 l 1287 625 l 1272 627 l
 1258 628 l 1244 628 l 1231 627 l 1219 626 l 1212 625 l
 1207 624 l 1201 622 l 1196 620 l 1190 619 l 1186 616 l
 1181 614 l 1178 611 l 1174 609 l 1171 606 l 1169 603 l
 1168 599 l 1167 596 l 1167 593 l 1167 589 l 1168 586 l
 1170 582 l 1172 579 l 1178 572 l 1186 565 l 1196 558 l
 1208 551 l 1222 544 l 1236 539 l 1250 534 l 1264 531 l
 1277 530 l
 cp gs col7 1.00 shd ef gr 
% Polyline
7.500 slw
n 2115 540 m 2116 540 l 2122 540 l 2135 541 l 2154 543 l 2174 545 l
 2193 548 l 2210 552 l 2224 557 l 2238 563 l 2250 570 l
 2259 576 l 2268 584 l 2278 593 l 2288 603 l 2300 616 l
 2312 630 l 2326 647 l 2341 664 l 2355 682 l 2367 697 l
 2376 709 l 2382 716 l
 2385 720 l gs col0 s gr 
% Polyline
n 2520 450 m 2519 451 l 2516 456 l 2509 467 l 2500 482 l 2490 499 l
 2482 515 l 2476 529 l 2472 541 l 2471 552 l 2472 562 l
 2475 571 l 2479 580 l 2485 589 l 2493 600 l 2503 612 l
 2515 626 l 2529 640 l 2542 653 l 2554 664 l 2561 671 l
 2564 674 l
 2565 675 l gs col0 s gr 
% Polyline
0.000 slw
n 2414 531 m 2422 533 l 2429 536 l 2436 539 l 2443 544 l 2449 549 l
 2455 555 l 2459 561 l 2463 568 l 2466 574 l 2467 581 l
 2468 587 l 2467 593 l 2465 598 l 2463 603 l 2457 609 l
 2448 615 l 2438 619 l 2425 623 l 2412 625 l 2397 627 l
 2383 628 l 2369 628 l 2356 627 l 2344 626 l 2337 625 l
 2332 624 l 2326 622 l 2321 620 l 2315 619 l 2311 616 l
 2306 614 l 2303 611 l 2299 609 l 2296 606 l 2294 603 l
 2293 599 l 2292 596 l 2292 593 l 2292 589 l 2293 586 l
 2295 582 l 2297 579 l 2303 572 l 2311 565 l 2321 558 l
 2333 551 l 2347 544 l 2361 539 l 2375 534 l 2389 531 l
 2402 530 l
 cp gs col7 1.00 shd ef gr 
% Polyline
7.500 slw
n 3150 540 m 3151 540 l 3157 540 l 3170 541 l 3189 543 l 3209 545 l
 3228 548 l 3245 552 l 3259 557 l 3273 563 l 3285 570 l
 3294 576 l 3303 584 l 3313 593 l 3323 603 l 3335 616 l
 3347 630 l 3361 647 l 3376 664 l 3390 682 l 3402 697 l
 3411 709 l 3417 716 l
 3420 720 l gs col0 s gr 
% Polyline
n 3555 450 m 3554 451 l 3551 456 l 3544 467 l 3535 482 l 3525 499 l
 3517 515 l 3511 529 l 3507 541 l 3506 552 l 3507 562 l
 3510 571 l 3514 580 l 3520 589 l 3528 600 l 3538 612 l
 3550 626 l 3564 640 l 3577 653 l 3589 664 l 3596 671 l
 3599 674 l
 3600 675 l gs col0 s gr 
% Polyline
0.000 slw
n 3449 531 m 3457 533 l 3464 536 l 3471 539 l 3478 544 l 3484 549 l
 3490 555 l 3494 561 l 3498 568 l 3501 574 l 3502 581 l
 3503 587 l 3502 593 l 3500 598 l 3498 603 l 3492 609 l
 3483 615 l 3473 619 l 3460 623 l 3447 625 l 3432 627 l
 3418 628 l 3404 628 l 3391 627 l 3379 626 l 3372 625 l
 3367 624 l 3361 622 l 3356 620 l 3350 619 l 3346 616 l
 3341 614 l 3338 611 l 3334 609 l 3331 606 l 3329 603 l
 3328 599 l 3327 596 l 3327 593 l 3327 589 l 3328 586 l
 3330 582 l 3332 579 l 3338 572 l 3346 565 l 3356 558 l
 3368 551 l 3382 544 l 3396 539 l 3410 534 l 3424 531 l
 3437 530 l
 cp gs col7 1.00 shd ef gr 
% Polyline
7.500 slw
n 4230 540 m 4231 540 l 4237 540 l 4250 541 l 4269 543 l 4289 545 l
 4308 548 l 4325 552 l 4339 557 l 4353 563 l 4365 570 l
 4374 576 l 4383 584 l 4393 593 l 4403 603 l 4415 616 l
 4427 630 l 4441 647 l 4456 664 l 4470 682 l 4482 697 l
 4491 709 l 4497 716 l
 4500 720 l gs col0 s gr 
% Polyline
n 4635 450 m 4634 451 l 4631 456 l 4624 467 l 4615 482 l 4605 499 l
 4597 515 l 4591 529 l 4587 541 l 4586 552 l 4587 562 l
 4590 571 l 4594 580 l 4600 589 l 4608 600 l 4618 612 l
 4630 626 l 4644 640 l 4657 653 l 4669 664 l 4676 671 l
 4679 674 l
 4680 675 l gs col0 s gr 
% Polyline
0.000 slw
n 4529 531 m 4537 533 l 4544 536 l 4551 539 l 4558 544 l 4564 549 l
 4570 555 l 4574 561 l 4578 568 l 4581 574 l 4582 581 l
 4583 587 l 4582 593 l 4580 598 l 4578 603 l 4572 609 l
 4563 615 l 4553 619 l 4540 623 l 4527 625 l 4512 627 l
 4498 628 l 4484 628 l 4471 627 l 4459 626 l 4452 625 l
 4447 624 l 4441 622 l 4436 620 l 4430 619 l 4426 616 l
 4421 614 l 4418 611 l 4414 609 l 4411 606 l 4409 603 l
 4408 599 l 4407 596 l 4407 593 l 4407 589 l 4408 586 l
 4410 582 l 4412 579 l 4418 572 l 4426 565 l 4436 558 l
 4448 551 l 4462 544 l 4476 539 l 4490 534 l 4504 531 l
 4517 530 l
 cp gs col7 1.00 shd ef gr 
% Polyline
7.500 slw
 [15 15] 15 sd
n 1774 1157 m 1767 1159 l 1760 1161 l 1754 1165 l 1748 1170 l 1743 1175 l
 1738 1182 l 1734 1189 l 1731 1197 l 1728 1205 l 1726 1213 l
 1725 1221 l 1725 1229 l 1725 1237 l 1726 1245 l 1728 1253 l
 1730 1261 l 1734 1269 l 1739 1277 l 1745 1285 l 1751 1291 l
 1758 1297 l 1766 1301 l 1774 1305 l 1782 1307 l 1790 1308 l
 1798 1307 l 1806 1306 l 1813 1303 l 1821 1300 l 1828 1295 l
 1836 1290 l 1843 1284 l 1850 1277 l 1856 1270 l 1862 1263 l
 1866 1255 l 1869 1247 l 1871 1239 l 1872 1232 l 1871 1225 l
 1869 1216 l 1865 1207 l 1859 1199 l 1852 1191 l 1843 1183 l
 1833 1175 l 1823 1169 l 1812 1164 l 1802 1160 l 1792 1158 l
 1782 1157 l
 cp gs col0 s gr  [] 0 sd
% Polyline
n 1440 1080 m 1441 1080 l 1447 1080 l 1460 1081 l 1479 1083 l 1499 1085 l
 1518 1088 l 1535 1092 l 1549 1097 l 1563 1103 l 1575 1110 l
 1584 1116 l 1593 1124 l 1603 1133 l 1613 1143 l 1625 1156 l
 1637 1170 l 1651 1187 l 1666 1204 l 1680 1222 l 1692 1237 l
 1701 1249 l 1707 1256 l
 1710 1260 l gs col0 s gr 
% Polyline
n 1845 990 m 1844 991 l 1841 996 l 1834 1007 l 1825 1022 l 1815 1039 l
 1807 1055 l 1801 1069 l 1797 1081 l 1796 1092 l 1797 1102 l
 1800 1111 l 1804 1120 l 1810 1129 l 1818 1140 l 1828 1152 l
 1840 1166 l 1854 1180 l 1867 1193 l 1879 1204 l 1886 1211 l
 1889 1214 l
 1890 1215 l gs col0 s gr 
% Polyline
0.000 slw
n 1739 1071 m 1747 1073 l 1754 1076 l 1761 1079 l 1768 1084 l 1774 1089 l
 1780 1095 l 1784 1101 l 1788 1108 l 1791 1114 l 1792 1121 l
 1793 1127 l 1792 1133 l 1790 1138 l 1788 1143 l 1782 1149 l
 1773 1155 l 1763 1159 l 1750 1163 l 1737 1165 l 1722 1167 l
 1708 1168 l 1694 1168 l 1681 1167 l 1669 1166 l 1662 1165 l
 1657 1164 l 1651 1162 l 1646 1160 l 1640 1159 l 1636 1156 l
 1631 1154 l 1628 1151 l 1624 1149 l 1621 1146 l 1619 1143 l
 1618 1139 l 1617 1136 l 1617 1133 l 1617 1129 l 1618 1126 l
 1620 1122 l 1622 1119 l 1628 1112 l 1636 1105 l 1646 1098 l
 1658 1091 l 1672 1084 l 1686 1079 l 1700 1074 l 1714 1071 l
 1727 1070 l
 cp gs col7 1.00 shd ef gr 
% Polyline
7.500 slw
 [15 15] 15 sd
n 2854 1157 m 2847 1159 l 2840 1161 l 2834 1165 l 2828 1170 l 2823 1175 l
 2818 1182 l 2814 1189 l 2811 1197 l 2808 1205 l 2806 1213 l
 2805 1221 l 2805 1229 l 2805 1237 l 2806 1245 l 2808 1253 l
 2810 1261 l 2814 1269 l 2819 1277 l 2825 1285 l 2831 1291 l
 2838 1297 l 2846 1301 l 2854 1305 l 2862 1307 l 2870 1308 l
 2878 1307 l 2886 1306 l 2893 1303 l 2901 1300 l 2908 1295 l
 2916 1290 l 2923 1284 l 2930 1277 l 2936 1270 l 2942 1263 l
 2946 1255 l 2949 1247 l 2951 1239 l 2952 1232 l 2951 1225 l
 2949 1216 l 2945 1207 l 2939 1199 l 2932 1191 l 2923 1183 l
 2913 1175 l 2903 1169 l 2892 1164 l 2882 1160 l 2872 1158 l
 2862 1157 l
 cp gs col0 s gr  [] 0 sd
% Polyline
n 2520 1080 m 2521 1080 l 2527 1080 l 2540 1081 l 2559 1083 l 2579 1085 l
 2598 1088 l 2615 1092 l 2629 1097 l 2643 1103 l 2655 1110 l
 2664 1116 l 2673 1124 l 2683 1133 l 2693 1143 l 2705 1156 l
 2717 1170 l 2731 1187 l 2746 1204 l 2760 1222 l 2772 1237 l
 2781 1249 l 2787 1256 l
 2790 1260 l gs col0 s gr 
% Polyline
n 2925 990 m 2924 991 l 2921 996 l 2914 1007 l 2905 1022 l 2895 1039 l
 2887 1055 l 2881 1069 l 2877 1081 l 2876 1092 l 2877 1102 l
 2880 1111 l 2884 1120 l 2890 1129 l 2898 1140 l 2908 1152 l
 2920 1166 l 2934 1180 l 2947 1193 l 2959 1204 l 2966 1211 l
 2969 1214 l
 2970 1215 l gs col0 s gr 
% Polyline
0.000 slw
n 2819 1071 m 2827 1073 l 2834 1076 l 2841 1079 l 2848 1084 l 2854 1089 l
 2860 1095 l 2864 1101 l 2868 1108 l 2871 1114 l 2872 1121 l
 2873 1127 l 2872 1133 l 2870 1138 l 2868 1143 l 2862 1149 l
 2853 1155 l 2843 1159 l 2830 1163 l 2817 1165 l 2802 1167 l
 2788 1168 l 2774 1168 l 2761 1167 l 2749 1166 l 2742 1165 l
 2737 1164 l 2731 1162 l 2726 1160 l 2720 1159 l 2716 1156 l
 2711 1154 l 2708 1151 l 2704 1149 l 2701 1146 l 2699 1143 l
 2698 1139 l 2697 1136 l 2697 1133 l 2697 1129 l 2698 1126 l
 2700 1122 l 2702 1119 l 2708 1112 l 2716 1105 l 2726 1098 l
 2738 1091 l 2752 1084 l 2766 1079 l 2780 1074 l 2794 1071 l
 2807 1070 l
 cp gs col7 1.00 shd ef gr 
% Polyline
7.500 slw
 [15 15] 15 sd
n 3934 1157 m 3927 1159 l 3920 1161 l 3914 1165 l 3908 1170 l 3903 1175 l
 3898 1182 l 3894 1189 l 3891 1197 l 3888 1205 l 3886 1213 l
 3885 1221 l 3885 1229 l 3885 1237 l 3886 1245 l 3888 1253 l
 3890 1261 l 3894 1269 l 3899 1277 l 3905 1285 l 3911 1291 l
 3918 1297 l 3926 1301 l 3934 1305 l 3942 1307 l 3950 1308 l
 3958 1307 l 3966 1306 l 3973 1303 l 3981 1300 l 3988 1295 l
 3996 1290 l 4003 1284 l 4010 1277 l 4016 1270 l 4022 1263 l
 4026 1255 l 4029 1247 l 4031 1239 l 4032 1232 l 4031 1225 l
 4029 1216 l 4025 1207 l 4019 1199 l 4012 1191 l 4003 1183 l
 3993 1175 l 3983 1169 l 3972 1164 l 3962 1160 l 3952 1158 l
 3942 1157 l
 cp gs col0 s gr  [] 0 sd
% Polyline
n 3600 1080 m 3601 1080 l 3607 1080 l 3620 1081 l 3639 1083 l 3659 1085 l
 3678 1088 l 3695 1092 l 3709 1097 l 3723 1103 l 3735 1110 l
 3744 1116 l 3753 1124 l 3763 1133 l 3773 1143 l 3785 1156 l
 3797 1170 l 3811 1187 l 3826 1204 l 3840 1222 l 3852 1237 l
 3861 1249 l 3867 1256 l
 3870 1260 l gs col0 s gr 
% Polyline
n 4005 990 m 4004 991 l 4001 996 l 3994 1007 l 3985 1022 l 3975 1039 l
 3967 1055 l 3961 1069 l 3957 1081 l 3956 1092 l 3957 1102 l
 3960 1111 l 3964 1120 l 3970 1129 l 3978 1140 l 3988 1152 l
 4000 1166 l 4014 1180 l 4027 1193 l 4039 1204 l 4046 1211 l
 4049 1214 l
 4050 1215 l gs col0 s gr 
% Polyline
0.000 slw
n 3899 1071 m 3907 1073 l 3914 1076 l 3921 1079 l 3928 1084 l 3934 1089 l
 3940 1095 l 3944 1101 l 3948 1108 l 3951 1114 l 3952 1121 l
 3953 1127 l 3952 1133 l 3950 1138 l 3948 1143 l 3942 1149 l
 3933 1155 l 3923 1159 l 3910 1163 l 3897 1165 l 3882 1167 l
 3868 1168 l 3854 1168 l 3841 1167 l 3829 1166 l 3822 1165 l
 3817 1164 l 3811 1162 l 3806 1160 l 3800 1159 l 3796 1156 l
 3791 1154 l 3788 1151 l 3784 1149 l 3781 1146 l 3779 1143 l
 3778 1139 l 3777 1136 l 3777 1133 l 3777 1129 l 3778 1126 l
 3780 1122 l 3782 1119 l 3788 1112 l 3796 1105 l 3806 1098 l
 3818 1091 l 3832 1084 l 3846 1079 l 3860 1074 l 3874 1071 l
 3887 1070 l
 cp gs col7 1.00 shd ef gr 
% Polyline
7.500 slw
 [15 15] 15 sd
n 5014 1157 m 5007 1159 l 5000 1161 l 4994 1165 l 4988 1170 l 4983 1175 l
 4978 1182 l 4974 1189 l 4971 1197 l 4968 1205 l 4966 1213 l
 4965 1221 l 4965 1229 l 4965 1237 l 4966 1245 l 4968 1253 l
 4970 1261 l 4974 1269 l 4979 1277 l 4985 1285 l 4991 1291 l
 4998 1297 l 5006 1301 l 5014 1305 l 5022 1307 l 5030 1308 l
 5038 1307 l 5046 1306 l 5053 1303 l 5061 1300 l 5068 1295 l
 5076 1290 l 5083 1284 l 5090 1277 l 5096 1270 l 5102 1263 l
 5106 1255 l 5109 1247 l 5111 1239 l 5112 1232 l 5111 1225 l
 5109 1216 l 5105 1207 l 5099 1199 l 5092 1191 l 5083 1183 l
 5073 1175 l 5063 1169 l 5052 1164 l 5042 1160 l 5032 1158 l
 5022 1157 l
 cp gs col0 s gr  [] 0 sd
% Polyline
n 5085 990 m 5084 991 l 5081 996 l 5074 1007 l 5065 1022 l 5055 1039 l
 5047 1055 l 5041 1069 l 5037 1081 l 5036 1092 l 5037 1102 l
 5040 1111 l 5044 1120 l 5050 1129 l 5058 1140 l 5068 1152 l
 5080 1166 l 5094 1180 l 5107 1193 l 5119 1204 l 5126 1211 l
 5129 1214 l
 5130 1215 l gs col0 s gr 
% Polyline
n 4680 1080 m 4681 1080 l 4687 1080 l 4700 1081 l 4719 1083 l 4739 1085 l
 4758 1088 l 4775 1092 l 4789 1097 l 4803 1103 l 4815 1110 l
 4824 1116 l 4833 1124 l 4843 1133 l 4853 1143 l 4865 1156 l
 4877 1170 l 4891 1187 l 4906 1204 l 4920 1222 l 4932 1237 l
 4941 1249 l 4947 1256 l
 4950 1260 l gs col0 s gr 
% Polyline
0.000 slw
n 4979 1071 m 4987 1073 l 4994 1076 l 5001 1079 l 5008 1084 l 5014 1089 l
 5020 1095 l 5024 1101 l 5028 1108 l 5031 1114 l 5032 1121 l
 5033 1127 l 5032 1133 l 5030 1138 l 5028 1143 l 5022 1149 l
 5013 1155 l 5003 1159 l 4990 1163 l 4977 1165 l 4962 1167 l
 4948 1168 l 4934 1168 l 4921 1167 l 4909 1166 l 4902 1165 l
 4897 1164 l 4891 1162 l 4886 1160 l 4880 1159 l 4876 1156 l
 4871 1154 l 4868 1151 l 4864 1149 l 4861 1146 l 4859 1143 l
 4858 1139 l 4857 1136 l 4857 1133 l 4857 1129 l 4858 1126 l
 4860 1122 l 4862 1119 l 4868 1112 l 4876 1105 l 4886 1098 l
 4898 1091 l 4912 1084 l 4926 1079 l 4940 1074 l 4954 1071 l
 4967 1070 l
 cp gs 0.80 setgray ef gr 
% Arc
7.500 slw
gs  clippath
2233 1019 m 2283 985 l 2199 860 l 2241 977 l 2149 893 l cp
eoclip
 [30 14 15 10 15 14] 0 sd
n 1732.5 1256.8 582.2 -152.7 -27.3 arc
gs col0 s gr
 gr
 [] 0 sd
% arrowhead
0 slj
n 2149 893 m 2241 977 l 2199 860 l 2149 893 l  cp gs 0.00 setgray ef gr  col0 s
% Arc
gs  clippath
795 465 m 795 525 l 946 525 l 826 495 l 946 465 l cp
eoclip
 [30 14 15 10 15 14] 0 sd
n 878.3 852.3 363.8 22.2 -100.8 arcn
gs col0 s gr
 gr
 [] 0 sd
% arrowhead
n 946 465 m 826 495 l 946 525 l 946 465 l  cp gs 0.00 setgray ef gr  col0 s
% Ellipse
n 1222 1020 35 35 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr

% Ellipse
n 2295 1020 35 35 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr

% Ellipse
n 802 483 35 35 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr

% Polyline
 [15 15] 15 sd
n 1875 1312 m
 1965 1402 l gs col0 s gr  [] 0 sd
% Polyline
 [15 15] 15 sd
n 2970 1312 m
 3060 1402 l gs col0 s gr  [] 0 sd
% Polyline
 [15 15] 15 sd
n 4050 1312 m
 4140 1402 l gs col0 s gr  [] 0 sd
% Polyline
 [15 15] 15 sd
n 5130 1312 m
 5220 1402 l gs col0 s gr  [] 0 sd
% Polyline
 [15 15] 15 sd
n 810 45 m
 900 135 l gs col0 s gr  [] 0 sd
% Polyline
 [15 15] 15 sd
n 1935 45 m
 2025 135 l gs col0 s gr  [] 0 sd
% Polyline
 [15 15] 15 sd
n 3015 45 m
 3105 135 l gs col0 s gr  [] 0 sd
% Polyline
 [15 15] 15 sd
n 4095 45 m
 4185 135 l gs col0 s gr  [] 0 sd
% Polyline
gs  clippath
2805 1290 m 2805 1230 l 2654 1230 l 2774 1260 l 2654 1290 l cp
eoclip
n 2340 1260 m
 2790 1260 l gs col0 s gr gr

% arrowhead
n 2654 1290 m 2774 1260 l 2654 1230 l 2654 1290 l  cp gs col7 1.00 shd ef gr  col0 s
% Polyline
gs  clippath
2955 1230 m 2955 1290 l 3106 1290 l 2986 1260 l 3106 1230 l cp
eoclip
n 3420 1260 m
 2970 1260 l gs col0 s gr gr

% arrowhead
n 3106 1230 m 2986 1260 l 3106 1290 l 3106 1230 l  cp gs col7 1.00 shd ef gr  col0 s
% Polyline
2 slj
 [15 30] 30 sd
n 4455 646 m 4456 645 l 4461 642 l 4470 635 l 4482 628 l 4494 621 l
 4504 616 l 4513 613 l 4523 611 l 4531 611 l 4540 611 l
 4551 612 l 4564 614 l 4579 616 l 4595 619 l 4610 622 l
 4620 625 l 4625 626 l
 4626 626 l gs col0 s gr  [] 0 sd
% here ends figure;
$F2psEnd
rs
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