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Diffstat (limited to 'Master/texmf-dist/source/eplain/base/test/commdiags.tex')
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diff --git a/Master/texmf-dist/source/eplain/base/test/commdiags.tex b/Master/texmf-dist/source/eplain/base/test/commdiags.tex new file mode 100644 index 00000000000..ae5ca6221d4 --- /dev/null +++ b/Master/texmf-dist/source/eplain/base/test/commdiags.tex @@ -0,0 +1,111 @@ +% This file is public domain. +% +% Examples of commutative diagrams from Brockett, {\it Finite +% Dimensional Linear Systems}. + +\input arrow + +\def\Real{\mathord{R}} +\def\Rn{\Real^n} +\def\A{{\bf A}} +\def\B{{\bf B}} +\def\C{{\bf C}} +\def\F{{\bf F}} +\def\G{{\bf G}} +\def\H{{\bf H}} +\def\I{{\bf I}} +\def\K{{\bf K}} +\def\L{{\bf L}} +\def\M{{\bf M}} +\def\N{{\bf N}} +\def\P{{\bf P}} +\def\Q{{\bf Q}} +\def\R{{\bf R}} +\def\S{{\bf S}} +\def\W{{\bf W}} +\def\bp{{\bf\Phi}} + + +Page 29, Fig.~3 +$$\commdiag{\Rn&\mapright^{\bp(t_1,t_0)}&\Rn\cr +&\arrow(3,-2)\lft{\bp(t,t_0)}&\mapdown\rt{\bp(t,t_1)}\cr &&\Rn\cr}$$ + +Page 48, Fig.~1: +$$\commdiag{\Rn&\mapright^{\bp(t,t_0)}&\Rn\cr +\mapdown\lft{\P(t_0)}&&\mapup\rt{\P^{-1}(t)}\cr +\Rn&\mapright^{e^{\R(t-t_0)}}&\Rn\cr}$$ + +Page 69, Fig.~1: +$$\commdiag{\Rn&\mapright^{\M_1}&\Real^m\cr +\mapdown\lft\Q&&\mapup\rt\P\cr +\Rn&\mapright^{\M_2}&\Real^m\cr}$$ + +Page 69, Fig.~2: +$$\commdiag{\Rn&\mapright^\A&\Rn\cr +\mapdown\lft\P&&\mapup\rt{\P'}\cr +\Rn&\mapright^\S&\Rn\cr}$$ + +Page 71, Fig.~3: +$$\commdiag{\Rn&\mapright^\A&\Rn\cr +\mapdown\lft\P&&\mapup\rt{\P^{-1}}\cr +\Rn&\mapright^\B&\Rn\cr}$$ + +Page 92, Fig.~1: +$$\harrowlength=40pt \varrowlength=24.7pt \sarrowlength=\harrowlength +\commdiag{C^m&\mapright^{\B(\sigma)}&\Rn&\mapright^{\bp_\A(t_0,\sigma)}& +\Rn&\mapright^{\int_{t_0}^t-d\sigma}&\Rn&\mapright^{\bp_\A(t,t_0)}& +\Rn&\mapright^{\C(t)}&C^q\cr +&\arrow(3,-2)\lft{\P(\sigma)\B(\sigma)}&\mapdown\rt{\P(\sigma)}&& +\mapdown\rt{\P^{-1}(t_0)}&&\mapdown\rt{\P(t_0)}&&\mapdown\lft{\P^{-1}(t)}& +\arrow(3,2)\rt{\C(t)\P^{-1}(t)}\cr +&&\Rn&\mapright_{\bp_{\A_1}(t_0,\sigma)}&\Rn&\mapright_{\int_{t_0}^t-d\sigma}& +\Rn&\mapright_{\bp_{\A_1}(t,t_0)}&\Rn}$$ + +Page 96, Fig.~2: +$$\harrowlength=32pt \varrowlength=\harrowlength \sarrowlength=\harrowlength +\def\olap#1{\harrowlength=118pt\hbox to0pt{\hss$#1$\hss}} +\commdiag{C^m&&&&&&&&&&C^q\cr \mapup\lft{\G(\sigma)} +&\hmorphposn=2pt\arrow(1,-1)\rt{\int_{t_0}^{t_1}-\G(\sigma)\,d\sigma} +&&&&&&&&\arrow(1,1)\lft{\H(\ )}&\mapdown\rt +{\int_{t_0}^{t_1}-\H(\sigma)\,d\sigma}\cr +\Rn&\mapleft_{\W(t_0,t_1)}&\Rn&&&\olap{\mapright^{\I}}&&& +\Rn&\mapleft_{\M(t_0,t_1)}&\Rn\cr +\mapdown\lft{\P'}&&\mapup\lft\P&\arrow(-1,1)\rt\P&&&&\arrow(-1,-1)\lft\Q& +\mapdown\rt\Q&&\mapup\rt{\Q'}\cr +\Rn&\mapright^{\S_1}&\Rn&\mapright^{\S_1}&\Rn&\harrowlength=20pt +\mapright^{\Q\P}&\Rn&\mapright^{\S_2}&\Rn&\mapright^{\S_2}&\Rn\cr +&&\mapdown\lft{\N_1}&&&&&&\mapup\rt{\N_2}\cr +&&\Rn&&&\olap{\mapright^{\I}}&&&\Rn\cr}$$ + +Page 114, Fig.~1: +$$\sarrowlength=.42\harrowlength +\commdiag{&\Real^m\cr &\arrow(-1,-1)\lft\B\quad \arrow(1,-1)\rt\G\cr +\Rn&\mapright^\P&\Rn\cr \mapdown\lft{e^{\A t}}&&\mapdown\rt{e^{\F t}}\cr +\Rn&\mapright^\P&\Rn\cr &\arrow(1,-1)\lft\C\quad \arrow(-1,-1)\rt\H\cr +&\Real^q\cr}$$ + +Page 127, Fig.~2: +$$\commdiag{Y&\mapright^{L^*}&X\cr &\adjarrow(3,-2)\lft{LL^*}\rt{(LL^*)^{-1}}& +\mapdown\rt{L}\cr &&Y\cr}$$ + +Page 135, Fig.~2: +$$\harrowlength=80pt \varrowlength=40pt \sarrowlength=50pt +\def\olap#1{\harrowlength=214pt\hbox to0pt{\hss$#1$\hss}} +\def\vlap#1{\varrowlength=93pt\setbox0=\hbox{$#1$}\ht0=0pt\dp0=0pt\box0} +\commdiag{\Rn&&&\olap{\mapright^{\dot\K+\A'\K+\K\A+\L}}&&&\Rn\cr +&\arrow(3,-1)\rt\K&&&&\arrow(-3,-1)\lft\K\cr \noalign{\vskip-3pt} +&&\Rn&\mapright^{\B\B'}&\Rn\cr +\vlap{\mapdown\lft\P}&&\mapup\lft{\P'}&&\mapdown\rt\P&&\vlap{\mapup\rt{\P'}}\cr +&&\Rn&\mapleft^{\P\B\B'\P'}&\Rn\cr \noalign{\vskip-3pt} +&\arrow(3,1)\lft{\K_1}&&&&\arrow(-3,1)\rt{\K_1}\cr +\Rn&&&\olap{\mapright_{\dot\K_1+(\P\A\P^{-1}+\dot\P\P^{-1})\K_1 ++\K_1(\P\A\P^{-1}+\dot\P\P^{-1})+\P'^{-1}\L\P^{-1}}}&&&\Rn\cr +\noalign{\medskip}}$$ + +Page 139, Fig.~2: +$$\harrowlength=80pt \varrowlength=.618\harrowlength +\sarrowlength=\harrowlength +\commdiag{\Rn&\mapright^{\B'(\sigma)\bp'(t_0,\sigma)}&C^m\cr +&\adjarrow(3,-2)\lft{\W(t_0,t_1)}\rt{\W^{-1}(t_0,t_1)}& +\mapdown\rt{\int_{t_0}^{t_1}\bp(t_0,\sigma)\B(\sigma)\,d\sigma}\cr +&&\Rn\cr}$$ |