
%\newtheorem{theorem}{Theorem}[section]
%\newtheorem{lemma}[theorem]{Lemma}
%\newtheorem{corollary}[theorem]{Corollary}
%\newtheorem{algorithm}[theorem]{Algorithm}
%\newtheorem{remark}[theorem]{Remark}
%\newenvironment{proof}{\noindent {\bf Proof:} }{\hfill $\Box$ \\[2ex] }
%\newenvironment{keywords}{\begin{quote} {\bf Key words} }
%                        {\end{quote} }
%\newenvironment{AMS}{\begin{quote} {\bf AMS subject classifications} }
%                        {\end{quote} }

%
%\newcommand{\eref}[1]{\mbox{\rm(\ref{#1})}}
%\newcommand{\tref}[1]{\mbox{\rm\ref{#1}}}
%\newcommand{\set}[2]{\left\{ #1 \; : \; #2 \right\} }
%\newcommand{\deq}{\raisebox{0pt}[1ex][0pt]{$\stackrel{\scriptscriptstyle{\rm def}}{{}={}}$}}

\newcommand {\DS} {\displaystyle}

\newcommand{\real}{\mathbb{R}}



\newcommand {\half} {\mbox{$\frac{1}{2}$}}
\newcommand{\force}{{\mathbf{f}}}
\newcommand{\strain}{{\boldsymbol{\varepsilon}}}
\newcommand{\stress}{{\boldsymbol{\sigma}}}
\renewcommand{\div}{{\boldsymbol{\nabla}}}

\newcommand {\cA} {{\cal A}}
\newcommand {\cB} {{\cal B}}
\newcommand {\cC} {{\cal C}}
\newcommand {\cD} {{\cal D}}
\newcommand {\cE} {{\cal E}}
\newcommand {\cI} {{\cal I}}
\newcommand {\cH} {{\cal H}}
\newcommand {\cL} {{\cal L}}
\newcommand {\cO} {{\cal O}}
\newcommand {\cP} {{\cal P}}
\newcommand {\cQ} {{\cal Q}}
\newcommand {\cR} {{\cal R}}
\newcommand {\cV} {{\cal V}}
\newcommand {\cW} {{\cal W}}
\newcommand {\cM} {{\cal M}}
\newcommand {\cN} {{\cal N}}
\newcommand {\CH} {{\cal H}}
\newcommand {\CS} {{\cal S}}
\newcommand {\cS} {{\cal S}}
\newcommand {\cY} {{\cal Y}}

\newcommand{\bzero}{\mathbf{0}}
\newcommand{\ba}{\mathbf{a}}
\newcommand{\bb}{\mathbf{b}}
%\newcommand{\bbb}{\mathbf{b}}
\newcommand{\bc}{\mathbf{c}}
\newcommand{\bd}{\mathbf{d}}
\newcommand{\be}{\mathbf{e}}
\newcommand{\bff}{\mathbf{f}}
\newcommand{\bfe}{\mathbf{e}}
\newcommand{\bg}{\mathbf{g}}
\newcommand{\bh}{\mathbf{h}}
\newcommand{\bk}{\mathbf{k}}
\newcommand{\bn}{\mathbf{n}}
\newcommand{\bp}{\mathbf{p}}
\newcommand{\bq}{\mathbf{q}}
\newcommand{\br}{\mathbf{r}}
\newcommand{\bs}{\mathbf{s}}
\newcommand{\bt}{\mathbf{t}}
\newcommand{\bu}{\mathbf{u}}
\newcommand{\bv}{\mathbf{v}}
\newcommand{\bw}{\mathbf{w}}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\by}{\mathbf{y}}
\newcommand{\bz}{\mathbf{z}}
\newcommand{\bA}{\mathbf{A}}
\newcommand{\bB}{\mathbf{B}}
\newcommand{\bC}{\mathbf{C}}
\newcommand{\bD}{\mathbf{D}}
\newcommand{\bE}{\mathbf{E}}
\newcommand{\bF}{\mathbf{F}}
\newcommand{\bG}{\mathbf{G}}
\newcommand{\bH}{\mathbf{H}}
\newcommand{\bI}{\mathbf{I}}
\newcommand{\bJ}{\mathbf{J}}
\newcommand{\bK}{\mathbf{K}}
\newcommand{\bL}{\mathbf{L}}
\newcommand{\bM}{\mathbf{M}}
\newcommand{\bN}{\mathbf{N}}
\newcommand{\bO}{\mathbf{O}}
\newcommand{\bP}{\mathbf{P}}
\newcommand{\bQ}{\mathbf{Q}}
\newcommand{\bR}{\mathbf{R}}
\newcommand{\bS}{\mathbf{S}}
\newcommand{\bT}{\mathbf{T}}
\newcommand{\bU}{\mathbf{U}}
\newcommand{\bV}{\mathbf{V}}
\newcommand{\bW}{\mathbf{W}}
\newcommand{\bX}{\mathbf{X}}
\newcommand{\bY}{\mathbf{Y}}
\newcommand{\bZ}{\mathbf{Z}}
\newcommand{\trn}{{^{T}}}
\newcommand{\mtrn}{{^{-T}}}

\newcommand{\bgamma}{{\boldsymbol{\gamma}}}
\newcommand{\bmu}{{\boldsymbol{\mu}}}
\newcommand{\blambda}{{\boldsymbol{\lambda}}}
\newcommand{\bLambda}{{\boldsymbol{\Lambda}}}
\newcommand{\bpi}{{\boldsymbol{\pi}}}
\newcommand{\bPi}{{\boldsymbol{\Pi}}}
\newcommand{\btheta}{{\boldsymbol{\theta}}}
\newcommand{\bTheta}{{\boldsymbol{\Theta}}}
\newcommand{\bSigma}{{\boldsymbol{\Sigma}}}

\newcommand{\ul}{\underline}
\newcommand{\ov}{\overline}
\newcommand{\hsq}{\hspace*{.25in}}
\newcommand{\hso}{\hspace*{1in}}
\newcommand{\hsh}{\hspace*{.5in}}
\newcommand{\hse}{\hspace*{.125in}}
\newcommand{\vsq}{\vspace{.25in}}
\newcommand{\vso}{\vspace{1in}}
\newcommand{\vsh}{\vspace{.5in}}
\newcommand{\vse}{\vspace{.125in}}


 \newcommand{\beq}{\begin{equation}}
 \newcommand{\eeq}{\end{equation}}
 \newcommand{\barray}{\begin{array}}
 \newcommand{\ea}{\end{array}}
 \newcommand{\bea}{\begin{eqnarray*}}
 \newcommand{\eea}{\end{eqnarray*}}
 \newcommand{\bean}{\begin{eqnarray}}
 \newcommand{\eean}{\end{eqnarray}}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newenvironment{proof}{\noindent {\bf Proof:} }{\hfill $\square$ \\[2ex] }
\newenvironment{keywords}{\begin{quote} {\bf Key words} }
                         {\end{quote} }
\newenvironment{AMS}{\begin{quote} {\bf AMS subject classifications} }
                         {\end{quote} }


\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{claim}[theorem]{Claim}
%\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{assumption}[theorem]{Assumption}

\newfont{\Bb}{msbm10 scaled\magstep0}
\def\IR{\mbox {\Bb R}}
\def\IC{\mbox {\Bb C}}
\def\IZ{\mbox {\Bb Z}}

\newcommand{\thbth}[9]{
  \left[ \begin{array}{rrr}
       #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9
       \end{array} \right] }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   Steve's Commands
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\def\muz{\sqrt{RG}} \def\mul{\mu\ell} \def\g{\gamma} \def\s{\sigma}
%\def\hq{\hskip .75in} \def\mun{\mu_n} \def\pn{\phi_n}
%\def\exp#1{{\rm e}^{#1}} \def\O{\Omega} \def\eps{\varepsilon}
%\def\cc{\check c} \def\cs{\check s} \def\ct{\check t}
%\def\ref{\hangindent=1pc\hangafter=1\noindent}
%\def\bm{\ov m} \def\bh{\ov h} \def\bn{\ov n}
%\def\alm{\alpha_m} \def\bem{\beta_m} \def\cE{{\cal E}} \def\cC{{\cal C}}
%\def\alh{\alpha_h} \def\beh{\beta_h}
%\def\aln{\alpha_n} \def\ben{\beta_n}
%\def\bam{\ov\alpha_m} \def\bbm{\ov\beta_m}
%\def\bah{\ov\alpha_h} \def\bbh{\ov\beta_h}
%\def\ban{\ov\alpha_n} \def\bbn{\ov\beta_n}
%\def\cT{{\cal T}} \def\gb{\grave\gamma}
%\def\tm{\tilde m} \def\th{\tilde h} \def\tn{\tilde n}
%\def\tv{\tilde v}
%\def\DA{\Delta_A} \def\DH{\Delta_H}
%\def\p{\partial}
%\def\frac#1#2{{#1\over #2}} \def\cL{{\cal L}}
%%\def\la{\lambda} \def\ov{\overline}
%\def\iot#1{\int_0^\infty #1\,dt} \def\mus{\sqrt{R(Cs+G)}}
%\def\fh{\hat f} \def\hi{\hat i} \def\hv{\hat v} \def\tz{\tilde z}
