%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% symbols C,N,Q,R,Z for sets
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\C}{\mathbb{C}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\D}{\mathbb{D}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Expectation, Variane, Covariance
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\E}[1]{\operatorname{\mathbb{E}}[#1]}
\newcommand{\wE}[1]{\operatorname{\widehat{\mathbb{E}}}[#1]}
\newcommand{\Var}[1]{\operatorname{\mathbb{V}}\left(#1\right)}
\newcommand{\Cov}[1]{\operatorname{Cov}\left(#1\right)}
\newcommand{\Corr}[1]{\operatorname{Corr}\left(#1\right)}
\renewcommand{\P}[1]{\operatorname{\mathbb{P}}\left(#1\right)}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% timestamps
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\monthword}[1]{\ifcase#1\or Jan\or Feb\or M\"ar\or Apr\or Mai\or Jun\or Jul\or Aug\or Sep\or Okt\or Nov\or Dez\fi}
\newcommand{\leadingzero}[1]{\ifnum #1<10 0\the#1\else\the#1\fi}             %%fügt bei 1 bis 9 eine führende Null an

\newcommand{\todayI}{\the\year"~\leadingzero{\month}"~\leadingzero{\day}}    	%%YYYY-MM-DD
\newcommand{\todayII}{\the\year\leadingzero{\month}\leadingzero{\day}}       	%%YYYYMMDD
\newcommand{\todayIII}{\leadingzero{\day}/\leadingzero{\month}/\the\year}    	%%DD/MM/YYYY
\newcommand{\todayIV}{\leadingzero{\day}.\leadingzero{\month}.\the\year}     	%%DD.MM.YYYY
\newcommand{\todayV}{\the\day.\the\month.\the\year}                          	%%D.M.YYYY
\newcommand{\todayVI}{\the\day.~\monthword{\the\month}. \the\year}           	%%D. MMM. YYYY
\newcommand{\todayVII}{\leadingzero{\day}.~\monthword{\the\month}. \the\year}	%%DD. MMM. YYYY
\newcommand{\todayVIII}{\monthword{\the\month}. \the\year}										%%DD. MMM. YYYY

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% calligraphic capital letters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}{\chi}
\newcommand{\calX}{\mathcal{X}}
\newcommand{\calY}{\mathcal{Y}}
\newcommand{\calZ}{\mathcal{Z}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% bold lowercase and capital letters for vectors (v) and matrices (m)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newcommand{\bm}[1]{{\boldmath #1}}

\newcommand{\mA}{\bm A}
\newcommand{\va}{\bm a}
\newcommand{\mB}{\bm B}
\newcommand{\vb}{\bm b}
\newcommand{\mC}{\bm C}
\newcommand{\vc}{\bm c}
\newcommand{\mD}{\bm D}
\newcommand{\vd}{\bm d}
\newcommand{\mE}{\bm E}
\newcommand{\ve}{\bm e}
\newcommand{\mF}{\bm F}
\newcommand{\vf}{\bm f}
\newcommand{\mG}{\bm G}
\newcommand{\vg}{\bm g}
\newcommand{\mH}{\bm H}
\newcommand{\vh}{\bm h}
\newcommand{\mI}{\bm I}
\newcommand{\vi}{\bm i}
\newcommand{\mJ}{\bm J}
\newcommand{\vj}{\bm j}
\newcommand{\mK}{\bm K}
\newcommand{\vk}{\bm k}
\newcommand{\mL}{\bm L}
\newcommand{\vl}{\bm l}
\newcommand{\mM}{\bm M}
\newcommand{\vm}{\bm m}
\newcommand{\mN}{\bm N}
\newcommand{\vn}{\bm n}
\newcommand{\mO}{\bm O}
\newcommand{\vo}{\bm o}
\newcommand{\mP}{\bm P}
\newcommand{\vp}{\bm p}
\newcommand{\mQ}{\bm Q}
\newcommand{\vq}{\bm q}
\newcommand{\mR}{\bm R}
\newcommand{\vr}{\bm r}
\newcommand{\mS}{\bm S}
\newcommand{\vs}{\bm s}
\newcommand{\mT}{\bm T}
\newcommand{\vt}{\bm t}
\newcommand{\mU}{\bm U}
\newcommand{\vu}{\bm u}
\newcommand{\mV}{\bm V}
\newcommand{\vv}{\bm v}
\newcommand{\mW}{\bm W}
\newcommand{\vw}{\bm w}
\newcommand{\mX}{\bm X}
\newcommand{\vX}{\mathbf X}
\newcommand{\vx}{\bm x}
\newcommand{\mY}{\bm Y}
\newcommand{\vY}{\mathbf Y}
\newcommand{\vy}{\bm y}
\newcommand{\mZ}{\bm Z}
\newcommand{\vz}{\bm z}

\newcommand{\vnull}{\bm 0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% beta und gamma as estimators
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\muh}{\hat{\mu}}
\newcommand{\mut}{\tilde{\mu}}
\newcommand{\uh}{\hat{\vu}}
\newcommand{\yh}{\hat{\vy}}

\newcommand{\hbeta}{\hat{\beta}}
\newcommand{\tbeta}{\tilde{\beta}}
\newcommand{\bbeta}{\mathbf{\beta}}
\newcommand{\hbbeta}{\hat{\bbeta}}

\newcommand{\hgamma}{\hat{\gamma}}
\newcommand{\tgamma}{\tilde{\gamma}}
\newcommand{\bgamma}{\mathbf{\gamma}}
\newcommand{\hbgamma}{\hat{\bgamma}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% bold Greek lowercase letters for vectors (v)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\valpha}{\bm \alpha}
\newcommand{\vbeta}{\bm \beta}
\newcommand{\hvbeta}{\hat{\vbeta}}
\newcommand{\tvbeta}{\tilde{\vbeta}}
\newcommand{\vgamma}{\bm \gamma}
\newcommand{\vdelta}{\bm \delta}
\newcommand{\vepsi}{\bm \epsi}
\newcommand{\vvarepsilon}{\bm \varepsilon}
\newcommand{\vzeta}{\bm \zeta}
\newcommand{\veta}{\bm \eta}
\newcommand{\vtheta}{\bm \theta}
\newcommand{\viota}{\bm \iota}
\newcommand{\vkappa}{\bm \kappa}
\newcommand{\vlambda}{\bm \lambda}
\newcommand{\vmu}{\bm \mu}
\newcommand{\vnu}{\bm \nu}
\newcommand{\vxi}{\bm \xi}
\newcommand{\vpi}{\bm \pi}
\newcommand{\vrho}{\bm \rho}
\newcommand{\vsigma}{\bm \sigma}
\newcommand{\vtau}{\bm \tau}
\newcommand{\vupsilon}{\bm \upsilon}
\newcommand{\vphi}{\bm \phi}
\newcommand{\vchi}{\bm \chi}
\newcommand{\vpsi}{\bm \psi}
\newcommand{\vomega}{\bm \omega}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% bold Greek capital letters for matrices (m)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\mGamma}{\bm \varGamma}
\newcommand{\mDelta}{\bm \varDelta}
\newcommand{\mTheta}{\bm \varTheta}
\newcommand{\mLambda}{\bm \varLambda}
\newcommand{\mXi}{\bm \varXi}
\newcommand{\mPi}{\bm \varPi}
\newcommand{\mSigma}{\bm \varSigma}
\newcommand{\mUpsilon}{\bm \varUpsilon}
\newcommand{\mPhi}{\bm \varPhi}
\newcommand{\mPsi}{\bm \varPsi}
\newcommand{\mOmega}{\bm \varOmega}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% roman letters in mathematics
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\rb}{\ensuremath{\mathrm{b}}}
\newcommand{\rB}{\ensuremath{\mathrm{B}}}
\newcommand{\rC}{\ensuremath{\mathrm{C}}}
\newcommand{\rD}{\ensuremath{\mathrm{D}}}
\newcommand{\rf}{\ensuremath{\mathrm{f}}}
\newcommand{\rF}{\ensuremath{\mathrm{F}}}
\newcommand{\rH}{\ensuremath{\mathrm{H}}}
\newcommand{\rL}{\ensuremath{\mathrm{L}}}
\newcommand{\rN}{\ensuremath{\mathrm{N}}}
\newcommand{\rt}{\ensuremath{\mathrm{t}}}
\newcommand{\rU}{\ensuremath{\mathrm{U}}}
\newcommand{\rGam}{\ensuremath{\mathrm{Gam}}}
\newcommand{\rBeta}{\ensuremath{\mathrm{Beta}}}
%
\newcommand{\Bin}{\ensuremath{\mathrm{Bin}}}
\newcommand{\eu}{\ensuremath{\mathrm{e}}}
\newcommand{\iu}{\ensuremath{\mathrm{i}}}
\newcommand{\LN}{\ensuremath{\mathrm{LN}}}
\newcommand{\IN}{\ensuremath{\mathrm{IN}}}

\newcommand{\Poi}{\ensuremath{\mathrm{Poi}}}
%
\newcommand{\ped}[1]{\ensuremath{_\mathrm{#1}}} %pedex
\newcommand{\ap}[1]{\ensuremath{^\mathrm{#1}}} %apex

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% fractal letters in mathematics
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\frakA}{\mathfrak{A}}
\newcommand{\frakB}{\mathfrak{B}}
\newcommand{\frakC}{\mathfrak{C}}
\newcommand{\frakE}{\mathfrak{E}}
\newcommand{\frakF}{\mathfrak{F}}
\newcommand{\frakO}{\mathfrak{O}}
\newcommand{\frakS}{\mathfrak{S}}
\newcommand{\frakZ}{\mathfrak{Z}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% letters for (partial) differentiation
%\newcommand{\rd}{\ensuremath{\mathrm{d}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\makeatletter
\newcommand{\rd}{\@ifnextchar^{\DIfF}{\DIfF^{}}}
\def\DIfF^#1{%
   \mathop{\mathrm{\mathstrut d}}%
   \nolimits^{#1}\gobblespace}
\def\gobblespace{\futurelet\diffarg\opspace}
\def\opspace{%
   \let\DiffSpace\!%
   \ifx\diffarg(%
   \let\DiffSpace\relax
   \else
   \ifx\diffarg[%
   \let\DiffSpace\relax
   \else
   \ifx\diffarg\{%
   \let\DiffSpace\relax
   \fi\fi\fi\DiffSpace}
\newcommand{\deriv}[3][]{\frac{\rd^{#1}#2}{\rd #3^{#1}}}
\newcommand{\pderiv}[3][]{\frac{\partial^{#1}#2}{\partial #3^{#1}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% operatornames
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\bias}{\operatorname{bias}}
\newcommand{\col}{\operatorname{col}}
\newcommand{\dg}{\operatorname{dg}}
\newcommand{\diag}{\operatorname{diag}}
\newcommand{\etr}{\operatorname{etr}}
\newcommand{\kur}{\operatorname{kur}}
\newcommand{\median}{\operatorname{med}}
\newcommand{\MSE}{\operatorname{MSE}}
\newcommand{\rk}{\operatorname{rk}}
\newcommand{\sgn}{\operatorname{sgn}}
\newcommand{\vech}{\operatorname{vech}}
\renewcommand{\vec}{\operatorname{vec}}
\newcommand{\id}{\operatorname{id}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\Mor}{\operatorname{Mor}}
\newcommand{\Abb}{\operatorname{Abb}}
\newcommand{\Lin}{\operatorname{Lin}}
\newcommand{\Real}{\operatorname{Re}}
\newcommand{\Image}{\operatorname{Im}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\ggT}{\operatorname{ggT}}
\newcommand{\Ker}{\operatorname{Ker}}
\newcommand{\coker}{\operatorname{coker}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Ann}{\operatorname{Ann}}
\newcommand{\Frac}{\operatorname{Frac}}
\newcommand{\Spec}{{\operatorname{Spec}\,}}
\newcommand{\Proj}{{\operatorname{Proj}\,}}
\newcommand{\codim}{\operatorname{codim}}
\newcommand{\kon}{\operatorname{kon}}
\newcommand{\res}{\operatorname{res}}
\newcommand{\kor}{\operatorname{kor}}
\newcommand{\ad}{\operatorname{ad}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% other definitions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\adistr}{\stackrel{a}{\distr}}
\newcommand{\diff}{\Delta}
\newcommand{\fdiff}{\diff_{\rf}}
\newcommand{\bdiff}{\diff_{\rb}}
\newcommand{\eps}{\epsilon}
\newcommand{\epsi}{\varepsilon}
%
\newcommand{\longto}{\longrightarrow}
\newcommand{\pto}{\stackrel{p}{\longrightarrow}}
\newcommand{\dto}{\stackrel{d}{\longrightarrow}}
\newcommand{\wto}{\stackrel{w}{\longrightarrow}}
%
\newcommand{\Infmat}{\bm\calI}
\newcommand{\Hesmat}{\bm\calH}
\newcommand{\bcdot}{\raisebox{1pt}{\textbf{\large .}}}
%
\newcommand{\vones}{\bm\imath}
\newcommand{\vzeros}{\boldsymbol{0}}
\newcommand{\mZeros}{\mathbf{O}}
%
% additional commands
\newcommand{\interior}[1]{\overset{\circ}{#1}}
\newcommand{\mzeros}{\mZeros}
\newcommand{\mply}{\cdot}
\newcommand{\widebar}{\overline}
\newcommand{\bin}{\Bin}
\newcommand{\Po}{\Poi}
\newcommand{\e}{\eu}% exponential e



\newcommand{\MAX}[1]{\max\left\{#1\right\}}
\newcommand{\MIN}[1]{\min\left\{#1\right\}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Own abbreviations and commands:
% Aufgaben als sections
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\aufgabe}[1]{\section{#1}
\vspace*{-0.4 cm}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Pfeile
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\ra}{\longrightarrow}
\newcommand{\Ra}{\Longrightarrow}
\newcommand{\la}{\longleftarrow}
\newcommand{\La}{\Longleftarrow}
\newcommand{\LR}{\Longleftrightarrow}
\newcommand{\lr}{\longleftrightarrow}
\newcommand{\PRa}{\overset{P}{\longrightarrow}}
\newcommand{\PLa}{\overset{P}{\longleftarrow}}
\newcommand{\DRa}{\overset{d}{\longrightarrow}}
\newcommand{\DLa}{\overset{d}{\longleftarrow}}
\newcommand{\ARa}{\overset{a.s.}{\longrightarrow}}
\newcommand{\ALa}{\overset{a.s.}{\longleftarrow}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Limiten, Summen andere Mengenoperatoren
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\lim}[1]{\underset{#1}{\operatorname{lim}} \;}
\newcommand{\plim}[1]{\underset{#1}{\operatorname{plim}} \;}
\renewcommand{\sup}[1]{\underset{#1}{sup} \;}
\renewcommand{\inf}[1]{\underset{#1}{inf} \;}
\renewcommand{\max}[1]{\underset{#1}{\operatorname{max}} \;}
\renewcommand{\min}[1]{\underset{#1}{min} \;}
\newcommand{\limni}{\lim{n \to \infty} \;}
\newcommand{\limti}{\lim{t \to \infty} \;}
\newcommand{\limhn}{\stackrel{\operatorname{lim}}{h \to 0} \;}
\newcommand{\limTi}{\lim{T \to \infty} \;}
\newcommand{\limxi}{\lim{x \to \infty} \;}
\newcommand{\limzi}{\lim{z \to \infty} \;}
\newcommand{\plimni}{\plim{n \to \infty} \;}
\newcommand{\plimti}{\plim{t \to \infty} \;}
\newcommand{\tsum}{\sum_{t=1}^T \;}
\newcommand{\nsum}{\sum_{t=1}^T \;}
\newcommand{\bb }[1]{\bold  {#1}}
\newcommand{\bs }[1]{\boldsymbol {#1}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Umgebungen
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% a) b) c) Zähler in jeder Section durchlaufend
\newcounter{abcd}[section]
\newcommand{\bitabc}{\vspace{-1.8ex}\begin{enumerate}
\renewcommand{\labelenumi}{\alph{abcd})}
\itemsep-1.4ex \partopsep-1.8ex}
\newcommand{\eitabc}{\end{enumerate}}
\newcommand{\itabc}{\addtocounter{abcd}{1}\item}

% Punkte
\newcommand{\bitp}{\vspace{-1ex}\begin{itemize}
\itemsep-0.6ex \partopsep-0.3ex}
\newcommand{\eitp}{\end{itemize}}

% Farbige Umgebungen für Hervorhebungen
\newenvironment{fshaded}{\def\FrameCommand{\fcolorbox{framecolor}{shadecolor}}\MakeFramed{\FrameRestore}}{\endMakeFramed}
%orange Umgebung
\newenvironment{orangeumgebungs}[1][]{\definecolor{shadecolor}{rgb}{1,.8,.8}\definecolor{framecolor}{rgb}{1,0,0}\begin{fshaded}\begin{equation*}#1}{\end{equation*}\end{fshaded}}
\newcommand{\bous}{\begin{orangeumgebungs}}
\newcommand{\eous}{\end{orangeumgebungs}}

%hellblaue Umgebung
\newenvironment{blaueumgebungs}[1][]{\definecolor{shadecolor}{rgb}{.9,.9,1}%
\definecolor{framecolor}{rgb}{.1,.0,.7}%
\begin{fshaded}\begin{equation*}#1}{\end{equation*}\end{fshaded}}
\newcommand{\bbus}{\begin{blaueumgebungs}}
\newcommand{\ebus}{\end{blaueumgebungs}}

\newenvironment{blaueumgebunge}[1][]{\definecolor{shadecolor}{rgb}{.9,.9,1}%
\definecolor{framecolor}{rgb}{.1,.0,.7}%
\begin{fshaded}\begin{enumerate}#1}{\end{enumerate}\end{fshaded}}
\newcommand{\bbue}{\begin{blaueumgebunge}}
\newcommand{\ebue}{\end{blaueumgebunge}}

\newcommand{\formel}[1]{\begin{equation}\boxed{#1}\end{equation}}
\newcommand{\formels}[1]{\begin{equation*}\boxed{#1}\end{equation*}}

\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqs}{\begin{equation*}}
\newcommand{\eeqs}{\end{equation*}}

\newcommand{\bal}{\begin{align}}
\newcommand{\eal}{\end{align}}
\newcommand{\bals}{\begin{align*}}
\newcommand{\eals}{\end{align*}}

\newcommand{\bmat}{\begin{pmatrix}}
\newcommand{\emat}{\end{pmatrix}}

\newcommand{\bit}{\begin{itemize}}
\newcommand{\eit}{\end{itemize}}
\newcommand{\biit}{\begin{itemize}}
\newcommand{\eiit}{\end{itemize}}

\newcommand{\ben}{\begin{enumerate}}
\newcommand{\een}{\end{enumerate}}
\newcommand{\been}{\begin{enumerate}}
\newcommand{\eeen}{\end{enumerate}}

\newcommand{\bca}{\begin{cases}}
\newcommand{\eca}{\end{cases}}

\newcommand{\bpa}{\begin{parts}}
\newcommand{\epa}{\end{parts}}

\newcommand{\bspa}{\begin{subparts}}
\newcommand{\espa}{\end{subparts}}

% Porbel mit der Umgebung
 \newenvironment{NewSolution}
    {\SetTotalwidth\begin{solution}}
    {\end{solution}}
\newcommand{\bso}{\begin{NewSolution}}
%\newcommand{\bso}{\begin{solution}}
\newcommand{\eso}{\end{NewSolution}}
		
%\newcommand{\bso}{\SetTotalwidth\begin{solution}}
%\newcommand{\bso}{\begin{solution}}
%\newcommand{\eso}{\end{solution}}

\newcommand{\argmax}[1]{\underset{#1}\operatorname{argmax}}
\newcommand{\argmin}[1]{\underset{#1}\operatorname{argmin}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% horizonalte Abstände
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\hd}{\hspace{3mm}}
\newcommand{\hf}{\hspace{5mm}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% vertikale Abstände
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\vmd}{\hspace{3mm}}
\newcommand{\vmf}{\hspace{5mm}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% roman numbers
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}