Warning: this is an htmlized version!
The original is across this link,
and the conversion rules are here.
% (find-angg "LATEX/2009fractions.tex")
% (find-dn4ex "edrx08.sty")
% (find-angg ".emacs.templates" "s2008a")
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2009fractions.tex && latex    2009fractions.tex"))
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2009fractions.tex && pdflatex 2009fractions.tex"))
% (eev "cd ~/LATEX/ && Scp 2009fractions.{dvi,pdf} edrx@angg.twu.net:slow_html/LATEX/")
% (find-dvipage "~/LATEX/2009fractions.dvi")
% (find-pspage  "~/LATEX/2009fractions.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o 2009fractions.ps 2009fractions.dvi")
% (find-pspage  "~/LATEX/2009fractions.ps")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o tmp.ps tmp.dvi")
% (find-pspage  "~/LATEX/tmp.ps")
% (ee-cp "~/LATEX/2009fractions.pdf" (ee-twupfile "LATEX/2009fractions.pdf") 'over)
% (ee-cp "~/LATEX/2009fractions.pdf" (ee-twusfile "LATEX/2009fractions.pdf") 'over)

% (find-angg ".emacs.papers" "fritz")
% (find-angg ".emacs.papers" "johnstone")
% (find-angg "books/__cats/__cats.el" "borceux")
% (find-angg "books/__cats/__cats.el" "gabriel-zisman")

\documentclass[oneside]{book}
\usepackage[latin1]{inputenc}
\usepackage{edrx08}       % (find-dn4ex "edrx08.sty")
%L process "edrx08.sty"  -- (find-dn4ex "edrx08.sty")
\input edrxheadfoot.tex   % (find-dn4ex "edrxheadfoot.tex")
\begin{document}

\input 2009fractions.dnt

%*
% (eedn4-51-bounded)

Notes on categories of fractions.

\msk

Index of the slides:
\msk
% To update the list of slides uncomment this line:
%\makelos{tmp.los}
% then rerun LaTeX on this file, and insert the contents of "tmp.los"
% below, by hand (i.e., with "insert-file"):
% (find-fline "tmp.los")
% (insert-file "tmp.los")



\def\ptwo#1#2{\begin{pmatrix} #1 \\ #2 \end{pmatrix}}
\def\sm#1{\begin{smallmatrix}#1\end{smallmatrix}}
\def\scrE{{\mathscr{E}}}
\def\scrF{{\mathscr{F}}}

%D diagram filter-Phi
%D 2Dx        100   +15  +15      +20 +15  +15        +35 +20 +20  +28 +18 +18 
%D 2D  100    U                                       a		     
%D 2D	      v                                       v		     
%D 2D	      |                                       |		     
%D 2D	      v     u               L(u)              v		     
%D 2D  +30 1_\scrE --> _\scrE    1 ---> L(_\scrE)   b ----> c    d ----> e
%D 2D	                                     |                             |
%D 2D	                                     |                             |
%D 2D	                                     v                             v
%D 2D  +30                                _\scrF                          f
%D 2D	   								     
%D 2D  +20         \scrE ----------> \scrF                E ---------> F    
%D 2D	                     L                                                  
%D 2D  +20
%D 2D
%D (( U   1_\scrE  _\scrE
%D    @ 0 @ 1 >-> @ 1 @ 2 -> .plabel= a u
%D ))
%D (( 1 L(_\scrE) _\scrF
%D    @ 0 @ 1 -> .plabel= a L(u) @ 1 @ 2 -> .plabel= r \Phi
%D    @ 0 @ 2 .> .plabel= l t
%D ))
%D (( \scrE \scrF -> .plabel= b L
%D ))
%D (( a .tex= \ptwo{m\N}{\N}
%D    b .tex= \ptwo{\N}{\N}
%D    c .tex= \ptwo{\N{×}}{\N}
%D    @ 0 @ 1 >-> @ 1 @ 2 -> .plabel= a \sm{n.\\nm\N}
%D ))
%D (( d .tex= 1
%D    e .tex= (N{->})
%D    f .tex= 
%D    d e -> .plabel= a \sm{*.n.\\nm\N}
%D    e f -> .plabel= r \sm{P.m.\\k.P(mk)}
%D    d f .> .plabel= l {}
%D ))
%D (( E .tex= \Set/\N
%D    F .tex= \Set
%D    @ 0 @ 1 -> sl__ .plabel= b å_\N
%D    @ 0 @ 1 <-      .plabel= m K
%D    @ 0 @ 1 .> sl^^ .plabel= a Æ_\N
%D ))
%D enddiagram
%D
$$\diag{filter-Phi}$$



%D diagram ??
%D 2Dx        100        +30       +30        +55
%D 2D  100  U^*(X) <---| X         a <------| b
%D 2D         |          |         |          |
%D 2D   U^*(f)|          |f        |          |
%D 2D         v          v         v          v
%D 2D  +30  U^*(Y) <---| Y         c <------| d
%D 2D
%D 2D  +20 \scrE/\U <- \scrE   \Set/m\N <-- \Set/\N
%D 2D
%D (( U^*(X) X U^*(Y) Y
%D    @ 0 @ 1 <-|
%D    @ 0 @ 2 -> .plabel= l U^*(f)
%D    @ 1 @ 3 -> .plabel= r     f
%D    @ 2 @ 3 <-|
%D    @ 0 @ 3 harrownodes nil 20 nil <-|
%D ))
%D (( \scrE/\U \scrE <-
%D ))
%D (( a .tex= \ptwo{Ækm\N.X_k}{m\N}
%D    b .tex= \ptwo{Æn\N.X_n}{\N}
%D    c .tex= \ptwo{Ækm\N.Y_k}{m\N}
%D    d .tex= \ptwo{Æn\N.Y_n}{\N}
%D    @ 0 @ 1 <-|
%D    @ 0 @ 2 -> .plabel= l U^*(f)
%D    @ 1 @ 3 -> .plabel= r     f
%D    @ 2 @ 3 <-|
%D    @ 0 @ 3 harrownodes nil 20 nil <-|
%D ))
%D (( \Set/m\N \Set/\N <-
%D ))
%D enddiagram
%D
$$\diag{??}$$






%D diagram filter-Phi2
%D 2Dx                   100   +15  +15     +20 +15  +15
%D 2D  100               U                                  
%D 2D                    v                                   
%D 2D                    |                                   
%D 2D                    v     u                   L(u)      
%D 2D  +30            1_\scrE --> _\scrE   1 ---> L(_\scrE)
%D 2D                                                  |
%D 2D                                                  |
%D 2D                                                  v
%D 2D  +30      U^*(X) <-----| X                    _\scrF
%D 2D             |            |                                                                    
%D 2D       U^*(f)|            |f                                                                   
%D 2D             v            v                                                                    
%D 2D  +30      U^*(Y) <-----| Y  
%D 2D                                                                                             
%D 2D  +20     \scrE/\U <--- \scrE ---------> \scrF
%D 2D                    U^*            L                                                  
%D 2D  +20
%D 2D
%D (( U   1_\scrE  _\scrE
%D    @ 0 @ 1 >-> @ 1 @ 2 -> .plabel= a u
%D ))
%D (( 1 L(_\scrE) _\scrF
%D    @ 0 @ 1 -> .plabel= a L(u) @ 1 @ 2 -> .plabel= r \Phi
%D    @ 0 @ 2 .> .plabel= l t
%D ))
%D (( \scrE \scrF -> .plabel= b L
%D ))
%D enddiagram
%D
$$\diag{filter-Phi2}$$


%D diagram filter-Phi3
%D 2Dx       100     +30 +20 +20  +20 +15 +15
%D 2D  100           a              
%D 2D                v              
%D 2D                |              
%D 2D                v              
%D 2D  +30           b ----> c    d ----> e
%D 2D                                     |        
%D 2D                                     |        
%D 2D                                     v        
%D 2D  +30   g <-------| h                f
%D 2D        |           |             
%D 2D        |           |                              
%D 2D        v           v                              
%D 2D  +30   i <-------| j
%D 2D                                                 
%D 2D  +25 E/mN <------- E ---------> F    
%D 2D           U^*            L                                                  
%D 2D  +20
%D 2D
%D (( a .tex= \ptwo{m\N}{\N}
%D    b .tex= \ptwo{\N}{\N}
%D    c .tex= \ptwo{\N{×}}{\N}
%D    @ 0 @ 1 >-> @ 1 @ 2 -> .plabel= a \sm{n.\\nm\N}
%D ))
%D (( d .tex= 1
%D    e .tex= (N{->})
%D    f .tex= 
%D    d e -> .plabel= a \sm{*.n.\\nm\N}
%D    e f -> .plabel= r \sm{P.m.\\k.P(mk)}
%D    d f .> .plabel= l {}
%D ))
%D (( E/mN .tex= \Set/m\N
%D    E .tex= \Set/\N
%D    F .tex= \Set
%D    @ 0 @ 1 <- .plabel= m U^*
%D    @ 1 @ 2 -> sl__ .plabel= b å_\N
%D    @ 1 @ 2 <-      .plabel= m K
%D    @ 1 @ 2 .> sl^^ .plabel= a Æ_\N
%D ))
%D (( g .tex= \ptwo{Ækm\N.X_k}{m\N}
%D    h .tex= \ptwo{Æn\N.X_n}{\N}
%D    i .tex= \ptwo{Ækm\N.Y_k}{m\N}
%D    j .tex= \ptwo{Æn\N.Y_n}{\N}
%D    @ 0 @ 1 <-|
%D    @ 0 @ 2 -> .plabel= l U^*(f)
%D    @ 1 @ 3 -> .plabel= r     f
%D    @ 2 @ 3 <-|
%D    @ 0 @ 3 harrownodes nil 20 nil <-|
%D ))
%D enddiagram
%D
$$\diag{filter-Phi3}$$



%D diagram ??
%D 2Dx        100        +30       +30        +55
%D 2D  100  U^*(X) <---| X         a <------| b
%D 2D         |          |         |          |
%D 2D   U^*(f)|          |f        |          |
%D 2D         v          v         v          v
%D 2D  +30  U^*(Y) <---| Y         c <------| d
%D 2D                       
%D 2D  +20 \scrE/\U <- \scrE   \Set/m\N <-- \Set/\N
%D 2D
%D (( U^*(X) X U^*(Y) Y
%D    @ 0 @ 1 <-|
%D    @ 0 @ 2 -> .plabel= l U^*(f)
%D    @ 1 @ 3 -> .plabel= r     f
%D    @ 2 @ 3 <-|
%D    @ 0 @ 3 harrownodes nil 20 nil <-|
%D ))
%D (( \scrE/\U \scrE <-
%D ))
%D (( a .tex= \ptwo{Ækm\N.X_k}{m\N}
%D    b .tex= \ptwo{Æn\N.X_n}{\N}
%D    c .tex= \ptwo{Ækm\N.Y_k}{m\N}
%D    d .tex= \ptwo{Æn\N.Y_n}{\N}
%D    @ 0 @ 1 <-|
%D    @ 0 @ 2 -> .plabel= l U^*(f)
%D    @ 1 @ 3 -> .plabel= r     f
%D    @ 2 @ 3 <-|
%D    @ 0 @ 3 harrownodes nil 20 nil <-|
%D ))
%D (( \Set/m\N \Set/\N <-
%D ))
%D enddiagram
%D
$$\diag{??}$$





%*

\end{document}

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