% strikeout text
\def\textsk#1{\setbox1\hbox{#1}\rule[.5ex]{\wd1}{.4pt}\hspace*{-\wd1}{\box1}}

\newcommand{\ret}{\\[10 pt]}
\newcommand{\norm}[1]{\left\Vert#1\right\Vert}
\newcommand{\abs}[1]{\left\vert#1\right\vert}
\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\parens}[1]{\!\left(#1\right)}
\newcommand{\brac}[1]{\!\left[#1\right]}
\newcommand{\ang}[1]{\left<#1\right>}
\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
\newcommand{\prob}[1]{\mathrm{Pr}\set{#1}}%{\mathbb{P}\set{#1}}%{P\brac{#1}}%{\mathbf{P}\!\parens{#1}}%% define how we take the probability of a quantity
\newcommand{\expect}[1]{\mathbf{E}\brac{#1}}%{\mathbb{E}\set{#1}}% expectation operator
\newcommand{\var}[1]{\mathrm{var}\!\parens{#1}}%{\mathbb{E}\set{#1}}% variance operator
\newcommand{\cov}[1]{\mathrm{cov}\!\parens{#1}}%{\mathbb{E}\set{#1}}% variance operator
\newcommand{\chiof}[1]{\chi\parens{#1}}
\newcommand{\pdev}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\ppdev}[2]{\frac{\partial^2 #1}{\partial #2^2}}
\newcommand{\dev}[2]{\frac{d #1}{d #2}}
\newcommand{\ddev}[2]{\frac{d^2 #1}{d #2^2}}
\renewcommand{\a}{\alpha}
\renewcommand{\b}{\beta}
\newcommand{\g}{\gamma}
\newcommand{\G}{\Gamma}
\newcommand{\e}{\epsilon}
\newcommand{\la}{\lambda}
\newcommand{\m}{\mu}
\renewcommand{\r}{\rho}
\newcommand{\s}{\sigma}
\newcommand{\z}{\zeta}
\newcommand{\x}{\hat{x}}
\newcommand{\bfA}{\mathbf{A}}
\newcommand{\bfC}{\mathbf{C}}
\newcommand{\orr}{\cup}
\newcommand{\andd}{\cap}

\newcommand{\con}{\omega}

\newcommand{\definedas}{\triangleq}%%{\overset{\bigtriangleup}{=}}

\newcommand{\ints}{\mathbb{Z}}
\newcommand{\nats}{\mathbb{N}}
\newcommand{\whole}{\mathbb{W}}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\into}{\rightarrow}
\newcommand{\us}{\text{ $\mu$s}}
\newcommand{\ns}{\text{ ns}}
\newcommand{\W}{\text{ W}}
\newcommand{\dB}{\text{ dB}}
\newcommand{\dBm}{\text{ dBm}}
\newcommand{\dBi}{\text{ dBi}}
\newcommand{\dBW}{\text{ dBW}}
\newcommand{\mW}{\text{ mW}}
\newcommand{\km}{\text{ km}}
\newcommand{\Hz}{\text{ Hz}}
\newcommand{\kHz}{\text{ kHz}}
\newcommand{\MHz}{\text{ MHz}}
\newcommand{\GHz}{\text{ GHz}}
\newcommand{\bps}{\text{ bps}}
\newcommand{\kbps}{\text{ kbps}}
\newcommand{\Mbps}{\text{ Mbps}}
\newcommand{\Mcps}{\text{ Mcps}}
\newcommand{\Erlang}{\text{ Erlang}}
\newcommand{\Erlangs}{\text{ Erlangs}}
\newcommand{\Rmax}{R_{max}}
\newcommand{\Rmin}{R_{min}}
\newcommand{\Rminplus}{R_{min}^+}
\newcommand{\Pmax}{P_{max}}
\newcommand{\Pmin}{P_{min}}
\newcommand{\Pminplus}{P_{min}^+}
\newcommand{\pmax}{p_{max}}
\newcommand{\pmin}{p_{min}}
\newcommand{\Popt}{P_{opt}}
\newcommand{\popt}{p_{opt}}
\newcommand{\hmin}{h_{min}}
\newcommand{\hmax}{h_{max}}
\newcommand{\cond}{\,|\,}
\newcommand{\logten}{\log_{10}}
\newcommand{\cartprod}{\mathsf{X}}% can also use \times

\newcommand{\eg}{e.g.,\xspace}
\newcommand{\ie}{i.e.,\xspace}
\newcommand{\apriori}{\emph{a priori}\xspace}
\newcommand{\aposteriori}{\emph{a posteriori}\xspace}
\newcommand{\etal}{\emph{et al.}\xspace}
\newcommand{\modelling}{modeling\xspace}
\newcommand{\Modelling}{Modeling\xspace}
\newcommand{\modelled}{modeled\xspace}
\newcommand{\Modelled}{Modeled\xspace}
\newcommand{\matlab}{\textsc{Matlab}\xspace}
\newcommand{\pmf}{probability mass function\xspace}
\newcommand{\pdf}{probability density function\xspace}
\newcommand{\CDF}{cumulative distribution function\xspace}
\newcommand{\rv}{random variable\xspace}


%\newcommand{\BayesNashEq}{Bayes Nash equilibrium\xspace}
\newcommand{\BayesNashEq}{BNE\xspace}
\newcommand{\BNE}{BNE\xspace}

\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
    T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}

\DeclareMathOperator{\LambertW}{W}
\DeclareMathOperator{\erf}{erf} 
\DeclareMathOperator{\erfinv}{erfinv}
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\card}{card}
\DeclareMathOperator{\Area}{Area}


\DeclareMathOperator{\modulo}{mod}

%\theoremstyle{plain} % the most bold
\theoremstyle{definition}
%\theoremstyle{remark} % the lightest

\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}{Acknowledgement}
\newtheorem{algorithm}{Algorithm}
\newtheorem{axiom}{Axiom}
\newtheorem{case}{Case}
\newtheorem{claim}{Claim}
\newtheorem{conclusion}{Conclusion}
\newtheorem{condition}{Condition}
\newtheorem{conjecture}{Conjecture}
\newtheorem{corollary}{Corollary}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{exercise}{Exercise}
\newtheorem{lemma}{Lemma}
\newtheorem{notation}{Notation}
\newtheorem{problem}{Problem}
\newtheorem{proposition}{Proposition}
\newtheorem{question}{Question}
\newtheorem{remark}{Remark}
\newtheorem{solution}{Solution}
\newtheorem{summary}{Summary}