\usepackage[mathscr]{eucal}
\usepackage[cmex10]{amsmath}
\usepackage{epsfig,epsf,psfrag}
\usepackage{amssymb,amsmath,amsthm,amsfonts,latexsym}
\usepackage{amsmath,graphicx,bm,xcolor,url}
\usepackage[caption=false]{subfig} 
\usepackage{fixltx2e}%ordering of single and double column floats
\usepackage{array}%array and tabular environments
\usepackage{verbatim}
\usepackage{bm}
\usepackage{algorithmic}
\usepackage{algorithm}
\usepackage{verbatim}
\usepackage{textcomp}
\usepackage{mathrsfs}
%\usepackage{hyperref}
\usepackage{epstopdf}
%\usepackage{graphicx} 
%\usepackage{caption} 

%\usepackage{setspace}

 


 
%% To produce a tilde in url
\catcode`~=11 \def\UrlSpecials{\do\~{\kern -.15em\lower .7ex\hbox{~}\kern .04em}} \catcode`~=13 

\allowdisplaybreaks[3]
 
\newcommand{\nn}{\nonumber} 
\newcommand{\cequal}{\stackrel{\mathrm{c}}{=}}
\newcommand{\dequal}{\stackrel{\mathrm{d}}{=}}

% Calligraphic stuff
\newcommand{\calA}{\mathcal{A}}
\newcommand{\calB}{\mathcal{B}}
\newcommand{\calC}{\mathcal{C}}
\newcommand{\calD}{\mathcal{D}}
\newcommand{\calE}{\mathcal{E}}
\newcommand{\calF}{\mathcal{F}}
\newcommand{\calG}{\mathcal{G}}
\newcommand{\calH}{\mathcal{H}}
\newcommand{\calI}{\mathcal{I}}
\newcommand{\calJ}{\mathcal{J}}
\newcommand{\calK}{\mathcal{K}}
\newcommand{\calL}{\mathcal{L}}
\newcommand{\calM}{\mathcal{M}}
\newcommand{\calN}{\mathcal{N}}
\newcommand{\calO}{\mathcal{O}}
\newcommand{\calP}{\mathcal{P}}
\newcommand{\calQ}{\mathcal{Q}}
\newcommand{\calR}{\mathcal{R}}
\newcommand{\calS}{\mathcal{S}}
\newcommand{\calT}{\mathcal{T}}
\newcommand{\calU}{\mathcal{U}}
\newcommand{\calV}{\mathcal{V}}
\newcommand{\calW}{\mathcal{W}}
\newcommand{\calX}{\mathcal{X}}
\newcommand{\calY}{\mathcal{Y}}
\newcommand{\calZ}{\mathcal{Z}}

\newcommand{\hatcalA}{\hat{\calA}} 
\newcommand{\hatcalB}{\hat{\calB}} 
\newcommand{\hatcalC}{\hat{\calC}} 
\newcommand{\hatcalD}{\hat{\calD}} 
\newcommand{\hatcalE}{\hat{\calE}} 
\newcommand{\hatcalF}{\hat{\calF}} 
\newcommand{\hatcalG}{\hat{\calG}} 
\newcommand{\hatcalH}{\hat{\calH}} 
\newcommand{\hatcalI}{\hat{\calI}} 
\newcommand{\hatcalJ}{\hat{\calJ}} 
\newcommand{\hatcalK}{\hat{\calK}} 
\newcommand{\hatcalL}{\hat{\calL}} 
\newcommand{\hatcalM}{\hat{\calM}} 
\newcommand{\hatcalN}{\hat{\calN}} 
\newcommand{\hatcalO}{\hat{\calO}} 
\newcommand{\hatcalP}{\hat{\calP}} 
\newcommand{\hatcalQ}{\hat{\calQ}} 
\newcommand{\hatcalR}{\hat{\calR}} 
\newcommand{\hatcalS}{\hat{\calS}} 
\newcommand{\hatcalT}{\hat{\calT}} 
\newcommand{\hatcalU}{\hat{\calU}} 
\newcommand{\hatcalV}{\hat{\calV}} 
\newcommand{\hatcalW}{\hat{\calW}} 
\newcommand{\hatcalX}{\hat{\calX}} 
\newcommand{\hatcalY}{\hat{\calY}} 
\newcommand{\hatcalZ}{\hat{\calZ}} 

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

\newcommand{\dU}{\dot{\mathbf{U}}}
\newcommand{\dD}{\dot{\mathbf{D}}}

% Roman stuff
\newcommand{\rma}{\mathrm{a}}
\newcommand{\rmA}{\mathrm{A}}
\newcommand{\rmb}{\mathrm{b}}
\newcommand{\rmB}{\mathrm{B}}
\newcommand{\rmc}{\mathrm{c}}
\newcommand{\rmC}{\mathrm{C}}
\newcommand{\rmd}{\mathrm{d}}
\newcommand{\rmD}{\mathrm{D}}
\newcommand{\rme}{\mathrm{e}}
\newcommand{\rmE}{\mathrm{E}}
\newcommand{\rmf}{\mathrm{f}}
\newcommand{\rmF}{\mathrm{F}}
\newcommand{\rmg}{\mathrm{g}}
\newcommand{\rmG}{\mathrm{G}}
\newcommand{\rmh}{\mathrm{h}}
\newcommand{\rmH}{\mathrm{H}}
\newcommand{\rmi}{\mathrm{i}}
\newcommand{\rmI}{\mathrm{I}}
\newcommand{\rmj}{\mathrm{j}}
\newcommand{\rmJ}{\mathrm{J}}
\newcommand{\rmk}{\mathrm{k}}
\newcommand{\rmK}{\mathrm{K}}
\newcommand{\rml}{\mathrm{l}}
\newcommand{\rmL}{\mathrm{L}}
\newcommand{\rmm}{\mathrm{m}}
\newcommand{\rmM}{\mathrm{M}}
\newcommand{\rmn}{\mathrm{n}}
\newcommand{\rmN}{\mathrm{N}}
\newcommand{\rmo}{\mathrm{o}}
\newcommand{\rmO}{\mathrm{O}}
\newcommand{\rmp}{\mathrm{p}}
\newcommand{\rmP}{\mathrm{P}}
\newcommand{\rmq}{\mathrm{q}}
\newcommand{\rmQ}{\mathrm{Q}}
\newcommand{\rmr}{\mathrm{r}}
\newcommand{\rmR}{\mathrm{R}}
\newcommand{\rms}{\mathrm{s}}
\newcommand{\rmS}{\mathrm{S}}
\newcommand{\rmt}{\mathrm{t}}
\newcommand{\rmT}{\mathrm{T}}
\newcommand{\rmu}{\mathrm{u}}
\newcommand{\rmU}{\mathrm{U}}
\newcommand{\rmv}{\mathrm{v}}
\newcommand{\rmV}{\mathrm{V}}
\newcommand{\rmw}{\mathrm{w}}
\newcommand{\rmW}{\mathrm{W}}
\newcommand{\rmx}{\mathrm{x}}
\newcommand{\rmX}{\mathrm{X}}
\newcommand{\rmy}{\mathrm{y}}
\newcommand{\rmY}{\mathrm{Y}}
\newcommand{\rmz}{\mathrm{z}}
\newcommand{\rmZ}{\mathrm{Z}}


% Numbers bb font
\newcommand{\bbA}{\mathbb{A}}
\newcommand{\bbB}{\mathbb{B}}
\newcommand{\bbC}{\mathbb{C}}
\newcommand{\bbD}{\mathbb{D}}
\newcommand{\bbE}{\mathbb{E}}
\newcommand{\bbF}{\mathbb{F}}
\newcommand{\bbG}{\mathbb{G}}
\newcommand{\bbH}{\mathbb{H}}
\newcommand{\bbI}{\mathbb{I}}
\newcommand{\bbJ}{\mathbb{J}}
\newcommand{\bbK}{\mathbb{K}}
\newcommand{\bbL}{\mathbb{L}}
\newcommand{\bbM}{\mathbb{M}}
\newcommand{\bbN}{\mathbb{N}}
\newcommand{\bbO}{\mathbb{O}}
\newcommand{\bbP}{\mathbb{P}}
\newcommand{\bbQ}{\mathbb{Q}}
\newcommand{\bbR}{\mathbb{R}}
\newcommand{\bbS}{\mathbb{S}}
\newcommand{\bbT}{\mathbb{T}}
\newcommand{\bbU}{\mathbb{U}}
\newcommand{\bbV}{\mathbb{V}}
\newcommand{\bbW}{\mathbb{W}}
\newcommand{\bbX}{\mathbb{X}}
\newcommand{\bbY}{\mathbb{Y}}
\newcommand{\bbZ}{\mathbb{Z}}


% Mathfrak font
\newcommand{\frakA}{\mathfrak{A}}
\newcommand{\frakB}{\mathfrak{B}}
\newcommand{\frakC}{\mathfrak{C}}
\newcommand{\frakD}{\mathfrak{D}}
\newcommand{\frakE}{\mathfrak{E}}
\newcommand{\frakF}{\mathfrak{F}}
\newcommand{\frakG}{\mathfrak{G}}
\newcommand{\frakH}{\mathfrak{H}}
\newcommand{\frakI}{\mathfrak{I}}
\newcommand{\frakJ}{\mathfrak{J}}
\newcommand{\frakK}{\mathfrak{K}}
\newcommand{\frakL}{\mathfrak{L}}
\newcommand{\frakM}{\mathfrak{M}}
\newcommand{\frakN}{\mathfrak{N}}
\newcommand{\frakO}{\mathfrak{O}}
\newcommand{\frakP}{\mathfrak{P}}
\newcommand{\frakQ}{\mathfrak{Q}}
\newcommand{\frakR}{\mathfrak{R}}
\newcommand{\frakS}{\mathfrak{S}}
\newcommand{\frakT}{\mathfrak{T}}
\newcommand{\frakU}{\mathfrak{U}}
\newcommand{\frakV}{\mathfrak{V}}
\newcommand{\frakW}{\mathfrak{W}}
\newcommand{\frakX}{\mathfrak{X}}
\newcommand{\frakY}{\mathfrak{Y}}
\newcommand{\frakZ}{\mathfrak{Z}}


% Mathscr
\newcommand{\scA}{\mathscr{A}}
\newcommand{\scB}{\mathscr{B}}
\newcommand{\scC}{\mathscr{C}}
\newcommand{\scD}{\mathscr{D}}
\newcommand{\scE}{\mathscr{E}}
\newcommand{\scF}{\mathscr{F}}
\newcommand{\scG}{\mathscr{G}}
\newcommand{\scH}{\mathscr{H}}
\newcommand{\scI}{\mathscr{I}}
\newcommand{\scJ}{\mathscr{J}}
\newcommand{\scK}{\mathscr{K}}
\newcommand{\scL}{\mathscr{L}}
\newcommand{\scM}{\mathscr{M}}
\newcommand{\scN}{\mathscr{N}}
\newcommand{\scO}{\mathscr{O}}
\newcommand{\scP}{\mathscr{P}}
\newcommand{\scQ}{\mathscr{Q}}
\newcommand{\scR}{\mathscr{R}}
\newcommand{\scS}{\mathscr{S}}
\newcommand{\scT}{\mathscr{T}}
\newcommand{\scU}{\mathscr{U}}
\newcommand{\scV}{\mathscr{V}}
\newcommand{\scW}{\mathscr{W}}
\newcommand{\scX}{\mathscr{X}}
\newcommand{\scY}{\mathscr{Y}}
\newcommand{\scZ}{\mathscr{Z}}

% San serif font
\DeclareMathAlphabet{\mathbsf}{OT1}{cmss}{bx}{n}
\DeclareMathAlphabet{\mathssf}{OT1}{cmss}{m}{sl}% slanted sans serif
\newcommand{\rva}{\mathsf{a}}
\newcommand{\rvA}{\mathsf{A}}
\newcommand{\rvba}{\mathbsf{a}}
\newcommand{\rvbA}{\mathbsf{A}}
\newcommand{\rvb}{\mathsf{b}}
\newcommand{\rvB}{\mathsf{B}}
\newcommand{\rvbb}{\mathbsf{b}}
\newcommand{\rvbB}{\mathbsf{B}}
\newcommand{\rvc}{\mathsf{c}}
\newcommand{\rvC}{\mathsf{C}}
\newcommand{\rvbc}{\mathbsf{c}}
\newcommand{\rvbC}{\mathbsf{C}}
\newcommand{\rvd}{\mathsf{d}}
\newcommand{\rvD}{\mathsf{D}}
\newcommand{\rvbd}{\mathbsf{d}}
\newcommand{\rvbD}{\mathbsf{D}}
\newcommand{\rve}{\mathsf{e}}
\newcommand{\rvE}{\mathsf{E}}
\newcommand{\rvbe}{\mathbsf{e}}
\newcommand{\rvbE}{\mathbsf{E}}
\newcommand{\rvf}{\mathsf{f}}
\newcommand{\rvF}{\mathsf{F}}
\newcommand{\rvbf}{\mathbsf{f}}
\newcommand{\rvbF}{\mathbsf{F}}
\newcommand{\rvg}{\mathsf{g}}
\newcommand{\rvG}{\mathsf{G}}
\newcommand{\rvbg}{\mathbsf{g}}
\newcommand{\rvbG}{\mathbsf{G}}
\newcommand{\rvh}{\mathsf{h}}
\newcommand{\rvH}{\mathsf{H}}
\newcommand{\rvbh}{\mathbsf{h}}
\newcommand{\rvbH}{\mathbsf{H}}
\newcommand{\rvi}{\mathsf{i}}
\newcommand{\rvI}{\mathsf{I}}
\newcommand{\rvbi}{\mathbsf{i}}
\newcommand{\rvbI}{\mathbsf{I}}
\newcommand{\rvj}{\mathsf{j}}
\newcommand{\rvJ}{\mathsf{J}}
\newcommand{\rvbj}{\mathbsf{j}}
\newcommand{\rvbJ}{\mathbsf{J}}
\newcommand{\rvk}{\mathsf{k}}
\newcommand{\rvK}{\mathsf{K}}
\newcommand{\rvbk}{\mathbsf{k}}
\newcommand{\rvbK}{\mathbsf{K}}
\newcommand{\rvl}{\mathsf{l}}
\newcommand{\rvL}{\mathsf{L}}
\newcommand{\rvbl}{\mathbsf{l}}
\newcommand{\rvbL}{\mathbsf{L}}
\newcommand{\rvm}{\mathsf{m}}
\newcommand{\rvM}{\mathsf{M}}
\newcommand{\rvbm}{\mathbsf{m}}
\newcommand{\rvbM}{\mathbsf{M}}
\newcommand{\rvn}{\mathsf{n}}
\newcommand{\rvN}{\mathsf{N}}
\newcommand{\rvbn}{\mathbsf{n}}
\newcommand{\rvbN}{\mathbsf{N}}
\newcommand{\rvo}{\mathsf{o}}
\newcommand{\rvO}{\mathsf{O}}
\newcommand{\rvbo}{\mathbsf{o}}
\newcommand{\rvbO}{\mathbsf{O}}
\newcommand{\rvp}{\mathsf{p}}
\newcommand{\rvP}{\mathsf{P}}
\newcommand{\rvbp}{\mathbsf{p}}
\newcommand{\rvbP}{\mathbsf{P}}
\newcommand{\rvq}{\mathsf{q}}
\newcommand{\rvQ}{\mathsf{Q}}
\newcommand{\rvbq}{\mathbsf{q}}
\newcommand{\rvbQ}{\mathbsf{Q}}
\newcommand{\rvr}{\mathsf{r}}
\newcommand{\rvR}{\mathsf{R}}
\newcommand{\rvbr}{\mathbsf{r}}
\newcommand{\rvbR}{\mathbsf{R}}
\newcommand{\rvs}{\mathsf{s}}
\newcommand{\rvS}{\mathsf{S}}
\newcommand{\rvbs}{\mathbsf{s}}
\newcommand{\rvbS}{\mathbsf{S}}
\newcommand{\rvt}{\mathsf{t}}
\newcommand{\rvT}{\mathsf{T}}
\newcommand{\rvbt}{\mathbsf{t}}
\newcommand{\rvbT}{\mathbsf{T}}
\newcommand{\rvu}{\mathsf{u}}
\newcommand{\rvU}{\mathsf{U}}
\newcommand{\rvbu}{\mathbsf{u}}
\newcommand{\rvbU}{\mathbsf{U}}
\newcommand{\rvv}{\mathsf{v}}
\newcommand{\rvV}{\mathsf{V}}
\newcommand{\rvbv}{\mathbsf{v}}
\newcommand{\rvbV}{\mathbsf{V}}
\newcommand{\rvw}{\mathsf{w}}
\newcommand{\rvW}{\mathsf{W}}
\newcommand{\rvbw}{\mathbsf{w}}
\newcommand{\rvbW}{\mathbsf{W}}
\newcommand{\rvx}{\mathsf{x}}
\newcommand{\rvX}{\mathsf{X}}
\newcommand{\rvbx}{\mathbsf{x}}
\newcommand{\rvbX}{\mathbsf{X}}
\newcommand{\rvy}{\mathsf{y}}
\newcommand{\rvY}{\mathsf{Y}}
\newcommand{\rvby}{\mathbsf{y}}
\newcommand{\rvbY}{\mathbsf{Y}}
\newcommand{\rvz}{\mathsf{z}}
\newcommand{\rvZ}{\mathsf{Z}}
\newcommand{\rvbz}{\mathbsf{z}}
\newcommand{\rvbZ}{\mathbsf{Z}}

\newcommand{\rvTh}{\ssfTheta}
\newcommand{\svTh}{\Theta}
\newcommand{\rvbTh}{\bsfTheta}
\newcommand{\svbTh}{\boldsymbol{\Theta}}
\newcommand{\rvPh}{\ssfPhi}
\newcommand{\svPh}{\Phi}
\newcommand{\rvbPh}{\bsfPhi}
\newcommand{\svbPh}{\boldsymbol{\Phi}}

% define some useful uppercase Greek letters in regular and bold sf
\DeclareSymbolFont{bsfletters}{OT1}{cmss}{bx}{n}  
\DeclareSymbolFont{ssfletters}{OT1}{cmss}{m}{n}
\DeclareMathSymbol{\bsfGamma}{0}{bsfletters}{'000}
\DeclareMathSymbol{\ssfGamma}{0}{ssfletters}{'000}
\DeclareMathSymbol{\bsfDelta}{0}{bsfletters}{'001}
\DeclareMathSymbol{\ssfDelta}{0}{ssfletters}{'001}
\DeclareMathSymbol{\bsfTheta}{0}{bsfletters}{'002}
\DeclareMathSymbol{\ssfTheta}{0}{ssfletters}{'002}
\DeclareMathSymbol{\bsfLambda}{0}{bsfletters}{'003}
\DeclareMathSymbol{\ssfLambda}{0}{ssfletters}{'003}
\DeclareMathSymbol{\bsfXi}{0}{bsfletters}{'004}
\DeclareMathSymbol{\ssfXi}{0}{ssfletters}{'004}
\DeclareMathSymbol{\bsfPi}{0}{bsfletters}{'005}
\DeclareMathSymbol{\ssfPi}{0}{ssfletters}{'005}
\DeclareMathSymbol{\bsfSigma}{0}{bsfletters}{'006}
\DeclareMathSymbol{\ssfSigma}{0}{ssfletters}{'006}
\DeclareMathSymbol{\bsfUpsilon}{0}{bsfletters}{'007}
\DeclareMathSymbol{\ssfUpsilon}{0}{ssfletters}{'007}
\DeclareMathSymbol{\bsfPhi}{0}{bsfletters}{'010}
\DeclareMathSymbol{\ssfPhi}{0}{ssfletters}{'010}
\DeclareMathSymbol{\bsfPsi}{0}{bsfletters}{'011}
\DeclareMathSymbol{\ssfPsi}{0}{ssfletters}{'011}
\DeclareMathSymbol{\bsfOmega}{0}{bsfletters}{'012}
\DeclareMathSymbol{\ssfOmega}{0}{ssfletters}{'012}

% Hat and Tilde
\newcommand{\hata}{\hat{a}}
\newcommand{\hatA}{\hat{A}}
\newcommand{\tila}{\tilde{a}}
\newcommand{\tilA}{\tilde{A}}
\newcommand{\hatba}{\hat{\ba}}
\newcommand{\hatbA}{\hat{\bA}}
\newcommand{\tilba}{\tilde{\ba}}
\newcommand{\tilbA}{\tilde{\bA}}

\newcommand{\hatb}{\hat{b}}
\newcommand{\hatB}{\hat{B}}
\newcommand{\tilb}{\tilde{b}}
\newcommand{\tilB}{\tilde{B}}
\newcommand{\hatbb}{\hat{\bb}}
\newcommand{\hatbB}{\hat{\bB}}
\newcommand{\tilbb}{\tilde{\bb}}
\newcommand{\tilbB}{\tilde{\bB}}

\newcommand{\hatc}{\hat{c}}
\newcommand{\hatC}{\hat{C}}
\newcommand{\tilc}{\tilde{c}}
\newcommand{\tilC}{\tilde{C}}
\newcommand{\hatbc}{\hat{\bc}}
\newcommand{\hatbC}{\hat{\bC}}
\newcommand{\tilbc}{\tilde{\bc}}
\newcommand{\tilbC}{\tilde{\bC}}

\newcommand{\hatd}{\hat{d}}
\newcommand{\hatD}{\hat{D}}
\newcommand{\tild}{\tilde{d}}
\newcommand{\tilD}{\tilde{D}}
\newcommand{\hatbd}{\hat{\bd}}
\newcommand{\hatbD}{\hat{\bD}}
\newcommand{\tilbd}{\tilde{\bd}}
\newcommand{\tilbD}{\tilde{\bD}}

\newcommand{\hate}{\hat{e}}
\newcommand{\hatE}{\hat{E}}
\newcommand{\tile}{\tilde{e}}
\newcommand{\tilE}{\tilde{E}}
\newcommand{\hatbe}{\hat{\be}}
\newcommand{\hatbE}{\hat{\bE}}
\newcommand{\tilbe}{\tilde{\be}}
\newcommand{\tilbE}{\tilde{\bE}}

\newcommand{\hatf}{\hat{f}}
\newcommand{\hatF}{\hat{F}}
\newcommand{\tilf}{\tilde{f}}
\newcommand{\tilF}{\tilde{F}}
\newcommand{\hatbf}{\hat{\boldf}}
\newcommand{\hatbF}{\hat{\bF}}
\newcommand{\tilbf}{\tilde{\boldf}}
\newcommand{\tilbF}{\tilde{\bF}}

\newcommand{\hatg}{\hat{g}}
\newcommand{\hatG}{\hat{G}}
\newcommand{\tilg}{\tilde{g}}
\newcommand{\tilG}{\tilde{G}}
\newcommand{\hatbg}{\hat{\bg}}
\newcommand{\hatbG}{\hat{\bG}}
\newcommand{\tilbg}{\tilde{\bg}}
\newcommand{\tilbG}{\tilde{\bG}}

\newcommand{\hath}{\hat{h}}
\newcommand{\hatH}{\hat{H}}
\newcommand{\tilh}{\tilde{h}}
\newcommand{\tilH}{\tilde{H}}
\newcommand{\hatbh}{\hat{\bh}}
\newcommand{\hatbH}{\hat{\bH}}
\newcommand{\tilbh}{\tilde{\bh}}
\newcommand{\tilbH}{\tilde{\bH}}

\newcommand{\hati}{\hat{i}}
\newcommand{\hatI}{\hat{I}}
\newcommand{\tili}{\tilde{i}}
\newcommand{\tilI}{\tilde{I}}
\newcommand{\hatbi}{\hat{\bi}}
\newcommand{\hatbI}{\hat{\bI}}
\newcommand{\tilbi}{\tilde{\bi}}
\newcommand{\tilbI}{\tilde{\bI}}

\newcommand{\hatj}{\hat{j}}
\newcommand{\hatJ}{\hat{J}}
\newcommand{\tilj}{\tilde{j}}
\newcommand{\tilJ}{\tilde{J}}
\newcommand{\hatbj}{\hat{\bj}}
\newcommand{\hatbJ}{\hat{\bJ}}
\newcommand{\tilbj}{\tilde{\bj}}
\newcommand{\tilbJ}{\tilde{\bJ}}

\newcommand{\hatk}{\hat{k}}
\newcommand{\hatK}{\hat{K}}
\newcommand{\tilk}{\tilde{k}}
\newcommand{\tilK}{\tilde{K}}
\newcommand{\hatbk}{\hat{\bk}}
\newcommand{\hatbK}{\hat{\bK}}
\newcommand{\tilbk}{\tilde{\bk}}
\newcommand{\tilbK}{\tilde{\bK}}

\newcommand{\hatl}{\hat{l}}
\newcommand{\hatL}{\hat{L}}
\newcommand{\till}{\tilde{l}}
\newcommand{\tilL}{\tilde{L}}
\newcommand{\hatbl}{\hat{\bl}}
\newcommand{\hatbL}{\hat{\bL}}
\newcommand{\tilbl}{\tilde{\bl}}
\newcommand{\tilbL}{\tilde{\bL}}

\newcommand{\hatm}{\hat{m}}
\newcommand{\hatM}{\hat{M}}
\newcommand{\tilm}{\tilde{m}}
\newcommand{\tilM}{\tilde{M}}
\newcommand{\hatbm}{\hat{\boldm}}
\newcommand{\hatbM}{\hat{\bM}}
\newcommand{\tilbm}{\tilde{\boldm}}
\newcommand{\tilbM}{\tilde{\bM}}

\newcommand{\hatn}{\hat{n}}
\newcommand{\hatN}{\hat{N}}
\newcommand{\tiln}{\tilde{n}}
\newcommand{\tilN}{\tilde{N}}
\newcommand{\hatbn}{\hat{\bn}}
\newcommand{\hatbN}{\hat{\bN}}
\newcommand{\tilbn}{\tilde{\bn}}
\newcommand{\tilbN}{\tilde{\bN}}

\newcommand{\hato}{\hat{o}}
\newcommand{\hatO}{\hat{O}}
\newcommand{\tilo}{\tilde{o}}
\newcommand{\tilO}{\tilde{O}}
\newcommand{\hatbo}{\hat{\bo}}
\newcommand{\hatbO}{\hat{\bO}}
\newcommand{\tilbo}{\tilde{\bo}}
\newcommand{\tilbO}{\tilde{\bO}}

\newcommand{\hatp}{\hat{p}}
\newcommand{\hatP}{\hat{P}}
\newcommand{\tilp}{\tilde{p}}
\newcommand{\tilP}{\tilde{P}}
\newcommand{\hatbp}{\hat{\bp}}
\newcommand{\hatbP}{\hat{\bP}}
\newcommand{\tilbp}{\tilde{\bp}}
\newcommand{\tilbP}{\tilde{\bP}}

\newcommand{\hatq}{\hat{q}}
\newcommand{\hatQ}{\hat{Q}}
\newcommand{\tilq}{\tilde{q}}
\newcommand{\tilQ}{\tilde{Q}}
\newcommand{\hatbq}{\hat{\bq}}
\newcommand{\hatbQ}{\hat{\bQ}}
\newcommand{\tilbq}{\tilde{\bq}}
\newcommand{\tilbQ}{\tilde{\bQ}}

\newcommand{\hatr}{\hat{r}}
\newcommand{\hatR}{\hat{R}}
\newcommand{\tilr}{\tilde{r}}
\newcommand{\tilR}{\tilde{R}}
\newcommand{\hatbr}{\hat{\br}}
\newcommand{\hatbR}{\hat{\bR}}
\newcommand{\tilbr}{\tilde{\br}}
\newcommand{\tilbR}{\tilde{\bR}}

\newcommand{\hats}{\hat{s}}
\newcommand{\hatS}{\hat{S}}
\newcommand{\tils}{\tilde{s}}
\newcommand{\tilS}{\tilde{S}}
\newcommand{\hatbs}{\hat{\bs}}
\newcommand{\hatbS}{\hat{\bS}}
\newcommand{\tilbs}{\tilde{\bs}}
\newcommand{\tilbS}{\tilde{\bS}}

\newcommand{\hatt}{\hat{t}}
\newcommand{\hatT}{\hat{T}}
\newcommand{\tilt}{\tilde{t}}
\newcommand{\tilT}{\tilde{T}}
\newcommand{\hatbt}{\hat{\bt}}
\newcommand{\hatbT}{\hat{\bT}}
\newcommand{\tilbt}{\tilde{\bt}}
\newcommand{\tilbT}{\tilde{\bT}}

\newcommand{\hatu}{\hat{u}}
\newcommand{\hatU}{\hat{U}}
\newcommand{\tilu}{\tilde{u}}
\newcommand{\tilU}{\tilde{U}}
\newcommand{\hatbu}{\hat{\bu}}
\newcommand{\hatbU}{\hat{\bU}}
\newcommand{\tilbu}{\tilde{\bu}}
\newcommand{\tilbU}{\tilde{\bU}}

\newcommand{\hatv}{\hat{v}}
\newcommand{\hatV}{\hat{V}}
\newcommand{\tilv}{\tilde{v}}
\newcommand{\tilV}{\tilde{V}}
\newcommand{\hatbv}{\hat{\bv}}
\newcommand{\hatbV}{\hat{\bV}}
\newcommand{\tilbv}{\tilde{\bv}}
\newcommand{\tilbV}{\tilde{\bV}}

\newcommand{\hatw}{\hat{w}}
\newcommand{\hatW}{\hat{W}}
\newcommand{\tilw}{\tilde{w}}
\newcommand{\tilW}{\tilde{W}}
\newcommand{\hatbw}{\hat{\bw}}
\newcommand{\hatbW}{\hat{\bW}}
\newcommand{\tilbw}{\tilde{\bw}}
\newcommand{\tilbW}{\tilde{\bW}}

\newcommand{\hatx}{\hat{x}}
\newcommand{\hatX}{\hat{X}}
\newcommand{\tilx}{\tilde{x}}
\newcommand{\tilX}{\tilde{X}}
\newcommand{\hatbx}{\hat{\bx}}
\newcommand{\hatbX}{\hat{\bX}}
\newcommand{\tilbx}{\tilde{\bx}}
\newcommand{\tilbX}{\tilde{\bX}}

\newcommand{\haty}{\hat{y}}
\newcommand{\hatY}{\hat{Y}}
\newcommand{\tily}{\tilde{y}}
\newcommand{\tilY}{\tilde{Y}}
\newcommand{\hatby}{\hat{\by}}
\newcommand{\hatbY}{\hat{\bY}}
\newcommand{\tilby}{\tilde{\by}}
\newcommand{\tilbY}{\tilde{\bY}}

\newcommand{\hatz}{\hat{z}}
\newcommand{\hatZ}{\hat{Z}}
\newcommand{\tilz}{\tilde{z}}
\newcommand{\tilZ}{\tilde{Z}}
\newcommand{\hatbz}{\hat{\bz}}
\newcommand{\hatbZ}{\hat{\bZ}}
\newcommand{\tilbz}{\tilde{\bz}}
\newcommand{\tilbZ}{\tilde{\bZ}}

\newcommand{\bara}{\bar{a}}
\newcommand{\barb}{\bar{b}}
\newcommand{\barc}{\bar{c}}
\newcommand{\bard}{\bar{d}}
\newcommand{\bare}{\bar{e}}
\newcommand{\barf}{\bar{f}}
\newcommand{\barg}{\bar{g}}
\newcommand{\bah}{\bar{h}}
\newcommand{\bari}{\bar{i}}
\newcommand{\barj}{\bar{j}}
\newcommand{\bark}{\bar{k}}
\newcommand{\barl}{\bar{l}}
\newcommand{\barm}{\bar{m}}
\newcommand{\barn}{\bar{n}}
\newcommand{\baro}{\bar{o}}
\newcommand{\barp}{\bar{p}}
\newcommand{\barq}{\bar{q}}
\newcommand{\barr}{\bar{r}}
\newcommand{\bars}{\bar{s}}
\newcommand{\bart}{\bar{t}}
\newcommand{\baru}{\bar{u}}
\newcommand{\barv}{\bar{v}}
\newcommand{\barw}{\bar{w}}
\newcommand{\barx}{\bar{x}}
\newcommand{\bary}{\bar{y}}
\newcommand{\barz}{\bar{z}}

\newcommand{\barA}{\bar{A}}
\newcommand{\barB}{\bar{B}}
\newcommand{\barC}{\bar{C}}
\newcommand{\barD}{\bar{D}}
\newcommand{\barE}{\bar{E}}
\newcommand{\barF}{\bar{F}}
\newcommand{\barG}{\bar{G}}
\newcommand{\barh}{\bar{H}}
\newcommand{\barI}{\bar{I}}
\newcommand{\barJ}{\bar{J}}
\newcommand{\barK}{\bar{K}}
\newcommand{\barL}{\bar{L}}
\newcommand{\barM}{\bar{M}}
\newcommand{\barN}{\bar{N}}
\newcommand{\barO}{\bar{O}}
\newcommand{\barP}{\bar{P}}
\newcommand{\barQ}{\bar{Q}}
\newcommand{\barR}{\bar{R}}
\newcommand{\barS}{\bar{S}}
\newcommand{\barT}{\bar{T}}
\newcommand{\barU}{\bar{U}}
\newcommand{\barV}{\bar{V}}
\newcommand{\barW}{\bar{W}}
\newcommand{\barX}{\bar{X}}
\newcommand{\barY}{\bar{Y}}
\newcommand{\barZ}{\bar{Z}}

% Bold greek
\newcommand{\balpha}{\bm{\alpha}}
\newcommand{\bbeta}{\bm{\beta}}
\newcommand{\bgamma}{\bm{\gamma}}
\newcommand{\bdelta}{\bm{\delta}}
\newcommand{\btheta}{\bm{\theta}}
\newcommand{\btau}{\bm{\tau}}
\newcommand{\bpi}{\bm{\pi}}
\newcommand{\bepsilon}{\bm{\epsilon}}
\newcommand{\veps}{\varepsilon}
\newcommand{\bvarepsilon}{\bm{\varepsilon}}
\newcommand{\bsigma}{\bm{\sigma}}
\newcommand{\bzeta}{\bm{\zeta}}
\newcommand{\bmeta}{\bm{\eta}}
\newcommand{\bkappa}{\bm{\kappa}}
\newcommand{\bchi}{\bm{\chi}}
\newcommand{\bphi}{\bm{\phi}}
\newcommand{\bpsi}{\bm{\psi}}
\newcommand{\bomega}{\bm{\omega}}
\newcommand{\bxi}{\bm{\xi}}
\newcommand{\blambda}{\bm{\lambda}}
\newcommand{\brho}{\bm{\rho}}

\newcommand{\bGamma}{\bm{\Gamma}}
\newcommand{\bLambda}{\bm{\Lambda}}
\newcommand{\bSigma	}{\bm{\Sigma}}
\newcommand{\bPsi}{\bm{\Psi}}
\newcommand{\bDelta}{\bm{\Delta}}
\newcommand{\bXi}{\bm{\Xi}}
\newcommand{\bUpsilon}{\bm{\Upsilon}}
\newcommand{\bOmega}{\bm{\Omega}}
\newcommand{\bPhi}{\bm{\Phi}}
\newcommand{\bPi}{\bm{\Pi}}
\newcommand{\bTheta}{\bm{\Theta}}

\newcommand{\talpha}{\tilde{\alpha}}
\newcommand{\tbeta}{\tilde{\beta}}
\newcommand{\tgamma}{\tilde{\gamma}}
\newcommand{\tdelta}{\tilde{\delta}}
\newcommand{\ttheta}{\tilde{\theta}}
\newcommand{\ttau}{\tilde{\tau}}
\newcommand{\tpi}{\tilde{\pi}}
\newcommand{\tepsilon}{\tilde{\epsilon}}
\newcommand{\tvarepsilon}{\tilde{\varepsilon}}
\newcommand{\tsigma}{\tilde{\sigma}}
\newcommand{\tzeta}{\tilde{\zeta}}
\newcommand{\tmeta}{\tilde{\eta}}
\newcommand{\tkappa}{\tilde{\kappa}}
\newcommand{\tchi}{\tilde{\chi}}
\newcommand{\tphi}{\tilde{\phi}}
\newcommand{\tpsi}{\tilde{\psi}}
\newcommand{\tomega}{\tilde{\omega}}
\newcommand{\txi}{\tilde{\xi}}
\newcommand{\tlambda}{\tilde{\lambda}}
\newcommand{\trho}{\tilde{\rho}}

\newcommand{\halpha}{\hat{\alpha}}
\newcommand{\hbeta}{\hat{\beta}}
\newcommand{\hgamma}{\hat{\gamma}}
\newcommand{\hdelta}{\hat{\delta}}
\newcommand{\htheta}{\hat{\theta}}
\newcommand{\htau}{\hat{\tau}}
\newcommand{\hpi}{\hat{\pi}}
\newcommand{\hepsilon}{\hat{\epsilon}}
\newcommand{\hvarepsilon}{\hat{\varepsilon}}
\newcommand{\hsigma}{\hat{\sigma}}
\newcommand{\hzeta}{\hat{\zeta}}
\newcommand{\hmeta}{\hat{\eta}}
\newcommand{\hkappa}{\hat{\kappa}}
\newcommand{\hchi}{\hat{\chi}}
\newcommand{\hphi}{\hat{\phi}}
\newcommand{\hpsi}{\hat{\psi}}
\newcommand{\homega}{\hat{\omega}}
\newcommand{\hxi}{\hat{\xi}}
\newcommand{\hlambda}{\hat{\lambda}}
\newcommand{\hrho}{\hat{\rho}}


% functional dot

\def\fndot{\, \cdot \,}
\def\fndia{\, \diamond \,}
\def\fnstar{\, \star \,}

% exponential

\newcommand{\eexp}[1]{e^{#1}}

% iid

\newcommand{\iid}{i.i.d.\ }

% convergence

\newcommand{\convp}{\stackrel{\mathrm{p}}{\longrightarrow}}
\newcommand{\convwpone}{\stackrel{\mathrm{w.p.1}}{\longrightarrow}}
\newcommand{\convd}{\stackrel{\mathrm{d}}{\longrightarrow}}
\newcommand{\convD}{\stackrel{\mathrm{D}}{\longrightarrow}}

\newcommand{\ceil}[1]{\lceil{#1}\rceil}
\newcommand{\floor}[1]{\lfloor{#1}\rfloor}
\newcommand{\lrangle}[2]{\langle{#1},{#2}\rangle}


\newcommand{\dotleq}{\stackrel{.}{\leq}}
\newcommand{\dotlt}{\stackrel{.}{<}}
\newcommand{\dotgeq}{\stackrel{.}{\geq}}
\newcommand{\dotgt}{\stackrel{.}{>}}
\newcommand{\dotdoteq}{\stackrel{\,..}{=}}
% Inequalities
\newcommand{\eqa}{\stackrel{(a)}{=}}
\newcommand{\eqb}{\stackrel{(b)}{=}}
\newcommand{\eqc}{\stackrel{(c)}{=}}
\newcommand{\eqd}{\stackrel{(d)}{=}}
\newcommand{\eqe}{\stackrel{(e)}{=}}
\newcommand{\eqf}{\stackrel{(f)}{=}}
\newcommand{\eqg}{\stackrel{(g)}{=}}
\newcommand{\eqh}{\stackrel{(h)}{=}}
\newcommand{\lea}{\stackrel{(a)}{\le}}
\newcommand{\leb}{\stackrel{(b)}{\le}}
\newcommand{\lec}{\stackrel{(c)}{\le}}
\newcommand{\led}{\stackrel{(d)}{\le}}
\newcommand{\lee}{\stackrel{(e)}{\le}}
\newcommand{\lef}{\stackrel{(f)}{\le}}
\newcommand{\leg}{\stackrel{(g)}{\le}}
\newcommand{\leh}{\stackrel{(h)}{\le}}
\newcommand{\gea}{\stackrel{(a)}{\ge}}
\newcommand{\geb}{\stackrel{(b)}{\ge}}
\newcommand{\gec}{\stackrel{(c)}{\ge}}
\newcommand{\ged}{\stackrel{(d)}{\ge}}
\newcommand{\gee}{\stackrel{(e)}{\ge}}
\newcommand{\gef}{\stackrel{(f)}{\ge}}
\newcommand{\geg}{\stackrel{(g)}{\ge}}
\newcommand{\geh}{\stackrel{(h)}{\ge}}
 
\newcommand{\Pen}{P_{\mathrm{e}}^{(n)}}
\newcommand{\Penone}{P_{\mathrm{e}, 1}^{(n)}}
\newcommand{\Pentwo}{P_{\mathrm{e}, 2}^{(n)}}
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\argsup}{arg\,sup}
\DeclareMathOperator*{\arginf}{arg\,inf}
\DeclareMathOperator{\minimize}{minimize}
\DeclareMathOperator{\maximize}{maximize}
\DeclareMathOperator{\st}{subject\,\,to}
\DeclareMathOperator{\erfc}{erfc}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\cum}{cum}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\spn}{span}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\adj}{adj}
\DeclareMathOperator{\var}{\mathsf{Var}}
\DeclareMathOperator{\Vol}{Vol}
\DeclareMathOperator{\cov}{\mathsf{Cov}}
\DeclareMathOperator{\sech}{sech}
\DeclareMathOperator{\sinc}{sinc}
\DeclareMathOperator{\col}{col}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\poly}{poly}
\DeclareMathOperator{\vect}{vec}
\newcommand{\Hb}{H_{\mathrm{b}}}%binary entropy
\newcommand{\Ber}{\mathrm{Bern}}
\DeclareMathOperator*{\lms}{l.i.m.\,}
\newcommand{\varop}[1]{\var\left[{#1}\right]}
\newcommand{\covop}[2]{\cov\left({#1},{#2}\right)} 
\newcommand{\bzero}{\mathbf{0}}
\newcommand{\bone}{\mathbf{1}}

 \def\independenT#1#2{\mathrel{\rlap{$#1#2$}\mkern5mu{#1#2}}}
\newcommand\indep{\protect\mathpalette{\protect\independenT}{\perp}}
 

\newtheorem{theorem}{Theorem} 
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}{Definition} 
\newtheorem{example}{Example} 
\newtheorem{exercise}{Exercise} 
\newtheorem{remark}{Remark}
\newtheorem{fact}{Fact}
\newtheorem{assumption}{Assumption}
\newtheorem{data model}{Data Model}
\newtheorem{requirement}{Requirement}
%\newenvironment{proof}[1][Proof]{\begin{trivlist}
%\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
%\newenvironment{definition}[1][Definition]{\begin{trivlist}
%\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
%\newenvironment{example}[1][Example]{\begin{trivlist}
%\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
%\newenvironment{remark}[1][Remark]{\begin{trivlist}
%\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}

\newcommand{\qednew}{\nobreak \ifvmode \relax \else
      \ifdim\lastskip<1.5em \hskip-\lastskip
      \hskip1.5em plus0em minus0.5em \fi \nobreak
      \vrule height0.75em width0.5em depth0.25em\fi}
