\newtheorem{lem}{Lemma}
% define a few physics shortcuts
-\newcommand{\U}[1]{\textnormal{\ \bareU{#1}}} % units shortcut
+\newcommand{\unit}[1]{\textnormal{\ \bareU{#1}}} % units shortcut
+\providecommand{\U}[1]{\unit{#1}}
\newcommand{\bareU}[1]{\textnormal{#1}} % units without preceding value
\newcommand{\E}[1]{\ensuremath{\cdot 10 ^{#1}}} % exponent shortcut
\newcommand{\dg}{\ensuremath{^{\circ}}} % degree symbol ^o