pf.tex

%begin      p. fischer
\def\ds{\displaystyle}
\def\d{\partial}
\def\dO{\partial \Omega}
\def\dOt{\tilde {\partial \Omega}}
\def\spn{ {\textstyle span} \{ }
\def\ee{{\hat {\underline e}}}
\def\uxh{{\hat {\underline x}}}
\def\eq{ \; = \; }
%\def\eqq{ \, \equiv \, }
\def\eqq{ \, := \, }
% \def\es{ \, = \, }
\def\es{ = }
\def\ea{ \; & = & \; }
\def\sp{ \; + \; }
\def\sm{ \; - \; }
\def\ds{ \Delta s }
\def\dt{ \Delta t }
\def\dx{ \Delta x }
\def\XX{\raisebox{0.3ex}{$\chi$}}
\def\cW{{\cal W}}
\def\cL{{\cal L}}
\def\cB{{\cal B}}
\def\cH{{\cal H}}
\def\cI{{\cal I}}
\def\cM{{\cal M}}
\def\cN{{\cal N}}
\def\No{{N_o}}
\def\cR{{\cal R}}
\def\scriptO{{{\it O}\kern -.42em {\it `}\kern + .20em}}
\def\RR{{{\rm l}\kern - .15em {\rm R} }}
\def\PP{{{\rm l}\kern - .15em {\rm P} }}
%\def\L2{{{\cal L}^2}}
%\def\H1{{{\cal H}^1}}
%\def\L2{{{\bf L}^2}}
%\def\H1{{{\bf H}^1}}
\def\L2{{{\sf L}^2}}
\def\H1{{{\sf H}^1}}
\def\PNN{{\PP_{N}-\PP_{N}}}
\def\PN2{{\PP_{N}-\PP_{N-2}}}
\def\PNK{{\PP_{N,K}(\Omega)}}
\def\complex{{{\rm C} \kern - .53em {\rm l} \kern + .38em}}
\def\eop{\hbox{\vrule width 6pt height 6pt depth 0pt}}

\def\am{{ | \lambda_{\max} |}}
\def\a1{{ | \lambda_{\min} |}}
\def\lm{{   \lambda_{\max}  }}
\def\l1{{   \lambda_{\min}  }}
\def\bhu{{\hat   {\bf u}}}

\def\bff{{\bf f}}
\def\tp {{\tilde p}}
\def\btu{{\tilde {\bf u}}}
\def\utU{{\underline {\tilde {\bf u}}}}
\def\uU{{{\underline {\bf u}}}}
\def\bhu{{\hat   {\bf u}}}
\def\bhn{{\hat   {\bf n}}}
\def\ugt{{\tilde {\underline g}}}
\def\uut{{\tilde {\underline u}}}
\def\uxt{{\tilde {\underline x}}}
\def\ubt{{\tilde {\underline b}}}
\def\bue{{\underline {\bf e}}}
\def\bur{{\underline {\bf r}}}
\def\bu0{{\underline {\bf 0}}}
\def\buu{{\underline {\bf u}}}
\def\buf{{\underline {\bf f}}}
\def\br{{\bf r}}
\def\bu{{\bf u}}
\def\bv{{\bf v}}
\def\bx{{\bf x}}
\def\by{{\bf y}}
\def\bxt{{\tilde {\bf x}}}
\def\bA{{\bf A}}
\def\bB{{\bf B}}
\def\bD{{\bf D}}
\def\bH{{\bf H}}
\def\bI{{\bf I}}
\def\bQ{{\bf Q}}

\def\At{{\tilde A}}
\def\Bt{{\tilde B}}
\def\Bb{{\bar B}}
\def\Dt{{\tilde D}}
\def\It{{\tilde I}}
\def\Ut{{\tilde U}}
\def\xt{{\tilde x}}
\def\yt{{\tilde y}}
\def\zt{{\tilde z}}

\def\Ah{{\hat A}}
\def\Fh{{\hat F}}
\def\ih{{\hat \imath}}
\def\jh{{\hat \jmath}}
\def\Bh{{\hat B}}
\def\Dh{{\hat D}}
\def\Eh{{\hat E}}
\def\Oh{{\hat \Omega}}
\def\Ot{{\tilde \Omega}}
\def\Ok{{\Omega^k}}
\def\Rh{{\hat R}}
\def\hu{{\hat u}}
\def\hv{{\hat v}}

\def\ua{{\underline a}}
\def\ub{{\underline b}}
\def\uc{{\underline c}}
\def\ud{{\underline d}}
\def\ue{{\underline e}}
\def\uf{{\underline f}}
\def\ug{{\underline g}}
\def\uh{{\underline h}}
\def\ui{{\underline i}}
\def\uj{{\underline j}}
\def\uk{{\underline k}}
\def\ul{{\underline l}}
\def\um{{\underline m}}
\def\un{{\underline n}}
\def\uo{{\underline o}}
\def\up{{\underline p}}
\def\uq{{\underline q}}
\def\ur{{\underline r}}
\def\us{{\underline s}}
\def\ut{{\underline t}}
\def\uu{{\underline u}}
\def\uv{{\underline v}}
\def\uw{{\underline w}}
\def\ux{{\underline x}}
\def\vx{{\vec x}}
\def\uy{{\underline y}}
\def\uz{{\underline z}}
\def\u0{{\underline 0}}

\def\uxb{{\bar {\underline x}}}


\def\inn{ {\textstyle \;  \; {\rm in} \;  }}
\def\onn{ {\textstyle \;  \; {\rm on} \;  }}
\def\for{ {\textstyle \;  \; {\rm for} \;  }}


\newcommand{\pp}[2]{\frac{\partial #1}{\partial #2} }
\newcommand{\dd}[2]{\frac{d #1}{d #2} }


%end        p. fischer