watml-math.tex

\newcommand\hmmax{0}
\newcommand\bmmax{0}

\RequirePackage{latexsym}
\RequirePackage{amsmath}
\RequirePackage{empheq}
\RequirePackage{dsfont}
\RequirePackage{amssymb}
\RequirePackage{amsfonts}
\RequirePackage{bm}
\RequirePackage{xspace}
\RequirePackage{mathrsfs}
\RequirePackage[most,breakable]{tcolorbox}
\usepackage{derivative}

%%%%%%%%%%%%%%% exotic %%%%%%%%%%%%%%%
\newcommand\barbelow[1]{\stackunder[1.2pt]{$#1$}{\rule{1ex}{.075ex}}}
\newcommand*\circled[1]{\tikz[baseline=(char.base)]{
	\node[shape=circle,draw,inner sep=2pt] (char) {#1};}}
\makeatletter
\newcommand*{\indep}{%
	\mathbin{%
		\mathpalette{\@indep}{}%
	}%
}
\newcommand*{\nindep}{%
	\mathbin{% The final symbol is a binary math operator
		\mathpalette{\@indep}{\not}% \mathpalette helps for the adaptation
		% of the symbol to the different math styles.
	}%
}
\newcommand*{\@indep}[2]{%
% #1: math style
% #2: empty or \not
	\sbox0{$#1\perp\m@th$}%        box 0 contains \perp symbol
	\sbox2{$#1=$}%                 box 2 for the height of =
	\sbox4{$#1\vcenter{}$}%        box 4 for the height of the math axis
	\rlap{\copy0}%                 first \perp
	\dimen@=\dimexpr\ht2-\ht4-.2pt\relax
	% The equals symbol is centered around the math axis.
	% The following equations are used to calculate the
	% right shift of the second \perp:
	% [1] ht(equals) - ht(math_axis) = line_width + 0.5 gap
	% [2] right_shift(second_perp) = line_width + gap
	% The line width is approximated by the default line width of 0.4pt
	\kern\dimen@{#2}%
	% {\not} in case of \nindep;
	% the braces convert the relational symbol \not to an ordinary
	% math object without additional horizontal spacing.
	\kern\dimen@
	\copy0 %                       second \perp
} 
\makeatother

\newcommand{\mydot}[1]{\underset{\cdot}{#1}}
%%%%%%%%%%%%%%% boxed color equations %%%%%%%%%%%%%%%
% https://tex.stackexchange.com/questions/20575/attractive-boxed-equations
% https://tex.stackexchange.com/questions/419132/newenvironment-with-empheq-inside-tcolorbox
\newtcbox{\mymath}[1][red]{%
    nobeforeafter, math upper, tcbox raise base,
    enhanced, colframe=#1!30!black,
    colback=#1!30, boxrule=1pt%, top=1pt, bottom=1pt
}
\newenvironment{myboxeq}[1][red]{\empheq[box={\mymath[#1]}]{align}}{\endempheq}

%%%%%%%%%%%%%%% Widely accepted Sets and Symbols %%%%%%%%%%%%%%%
\newcommand{\twoheadmapsto}{\mapsto\!\!\!\!\to}
\newcommand{\CC}{\mathds{C}} % Complex numbers
\newcommand{\CE}{\text{\Cross}} % Cross Entropy
\newcommand{\EE}{\mathds{E}} % Expectation
\newcommand{\Ent}{\mathds{H}}% Entropy 
\newcommand{\FF}{\mathds{F}} % Arbitrary field
\newcommand{\HH}{\mathds{H}} % Entropy
\newcommand{\II}{\mathds{I}} % Mutual Information
\newcommand{\KK}{\mathds{K}} % Arbitrary field
\newcommand{\MM}{\mathsf{M}} % Median
\newcommand{\MI}{\mathds{I}} % Mutual Information
\newcommand{\NN}{\mathds{N}} % Natural numbers
\newcommand{\PP}{\mathds{P}} % Probability
\newcommand{\prob}{\mathrm{pr}} % Probability
\newcommand{\pleft}[1][p]{\overset{\leftarrow}{#1}}
\newcommand{\pright}[1][p]{\overset{\rightarrow}{#1}}
\newcommand{\pp}{\mathsf{p}} % density
\newcommand{\qq}{\mathsf{q}} % density
\newcommand{\QQ}{\mathds{Q}} % Rationals
\newcommand{\RR}{\mathds{R}} % Real numbers
\newcommand{\RRbar}{\overline{\RR}} % Real numbers
\newcommand{\RRe}{[-\infty,\infty]} % Real numbers
\newcommand{\RRi}{(-\infty,\infty]} % Real numbers
\newcommand{\RRinf}{\RR\cup\{\infty\}} % Real numbers
\newcommand{\SP}{\mathds{S}} % Symmetric positive semidefinite
\newcommand{\VV}{\mathsf{Var}} % Variance
\newcommand{\ZZ}{\mathds{Z}} % Integers
\newcommand{\Am}{\mathsf{A}} % operator A
\newcommand{\Bm}{\mathsf{B}} % operator B
\newcommand{\Fm}{\mathsf{F}} % operator F
\newcommand{\Rm}{\mathsf{R}}
\newcommand{\Sm}{\mathsf{S}} % operator S
\newcommand{\Tm}{\mathsf{T}} % operator T
\newcommand{\Zm}{\mathsf{Z}}
\newcommand{\Lzero}{\Lsf_0}
\newcommand{\Lone}{\Lsf_1}
\newcommand{\Ltwo}{\Lsf_2}
\newcommand{\LL}{\ensuremath{\mathfrak{L}}}
\newcommand{\psum}[1][$L$]{\textcircled{#1}}
\newcommand{\Di}{\mathsf{D}_{+}}
\newcommand{\He}{\mathsf{H}}
\newcommand{\Hi}{\mathsf{H}_{+}}
\newcommand{\Hs}{\ensuremath{\mathsf{H}}}
\newcommand{\K}[1][]{\mathsf{K}_{\mathsf{#1}}}
\newcommand{\h}{h}
\newcommand{\D}{\mathsf{D}}
\newcommand{\Af}{\mathfrak{A}}
\newcommand{\Bf}{\mathfrak{B}}
\newcommand{\Ff}{\mathfrak{F}}
\newcommand{\Jf}{\mathfrak{J}}
\newcommand{\Mf}{\mathfrak{M}}
\newcommand{\Nf}{\mathfrak{N}}
\newcommand{\Wf}{\mathfrak{W}}
\newcommand{\Gs}{\Omega}
\newcommand{\gs}{\omega}
\newcommand{\TNN}{\Nsf\!\!\!\!\!\Nsf}
\newcommand{\DNN}{\Dsf\Nsf}
%%%%%%%%%%%%%%% graph terminology  %%%%%%%%%%%%%%%%%
\newcommand{\gpa}{\mathrm{pa}}
\newcommand{\gch}{\mathrm{ch}}
\newcommand{\gde}{\mathrm{de}}
\newcommand{\gnd}{\mathrm{nd}}
\newcommand{\gan}{\mathrm{an}}
\newcommand{\pre}{\mathfrak{P}}
%%%%%%%%%%%%%%% standard abbreviations %%%%%%%%%%%%%%%
\newcommand{\ans}[1]{{\color{red} Ans: #1}}
\newcommand{\att}{\texttt{att}\xspace}
\newcommand{\Breg}{\mathrm{D}}
\newcommand{\emp}{\mathop{\mathrm{emp}}\xspace}
\newcommand{\err}{{\sf err}\xspace}
\newcommand{\Fix}{\ensuremath{\mathsf{Fix}}\xspace}
\newcommand{\fl}{\mathsf{FL}\xspace} 
\newcommand{\gl}{\mathsf{GL}\xspace}
\newcommand{\hinge}{\mathop{\mathrm{hinge}}\xspace}
\newcommand{\iter}{\texttt{iter}\xspace}
\newcommand{\logit}{\ensuremath{\mathtt{lgt}}\xspace}
\newcommand{\JS}{\ensuremath{\mathtt{JS}}\xspace}
\newcommand{\KL}{\ensuremath{\mathsf{KL}}\xspace}
\newcommand{\KW}{\ensuremath{\mathsf{W}}\xspace}
\newcommand{\len}{\mathsf{len}\xspace}
\newcommand{\link}{\mathtt{link}\xspace}
\newcommand{\loss}{\mathop{\mathrm{loss}}\xspace}
\newcommand{\Ltwobar}{\overline{M^2}}
\newcommand{\MAP}{\ensuremath{\mathsf{MAP}}\xspace}
\newcommand{\ML}{\ensuremath{\mathsf{MLE}}\xspace}
\newcommand{\nil}{\textsc{Nil}\xspace}
\newcommand{\NLL}{\ensuremath{\mathtt{NLL}}\xspace}
\newcommand{\perf}{\mathsf{perf}\xspace}
\newcommand{\predn}{\mathsf{pred}\xspace}
\newcommand{\qvar}{\mathrm{QV}\xspace} % quadratic variation
\newcommand{\reg}{\mathop{\mathrm{reg}}\xspace}
\newcommand{\RKL}{\ensuremath{\mathsf{LK}}\xspace}
\newcommand{\relu}{\mathtt{relu}\xspace}
\newcommand{\sgm}{\ensuremath{\mathtt{sgm}\xspace}}
\newcommand{\sm}{\ensuremath{\mathtt{softmax}\xspace}}
\newcommand{\spm}{\ensuremath{\mathtt{sparsemax}\xspace}}
\newcommand{\softmax}{\ensuremath{\mathtt{softmax}\xspace}}
\newcommand{\SC}[1][\eta]{\mathsf{SC}_{#1}}  % strong convexity
\renewcommand{\SS}[1][\eta]{\mathsf{SS}_{1/#1}} % strong smoothness
\newcommand{\TVd}{\ensuremath{\mathsf{TV}}\xspace}
\newcommand{\UKL}{\ensuremath{\widetilde{\KL}}\xspace}
%%%%%%%%%%%%%%% Mathematical Operations %%%%%%%%%%%%%%%
\newcommand{\adj}{\mathop{\mathrm{adj}}}
\newcommand{\arginf}{\mathop{\mathrm{arginf}}}
\newcommand{\argmax}{\mathop{\mathrm{argmax}}}
\newcommand{\argmin}{\mathop{\mathrm{argmin}}}
\newcommand{\argsup}{\mathop{\mathrm{argsup}}}
\newcommand{\Ball}[1][]{\mathcal{B}_{#1}}
\newcommand{\baru}[1]{\underset{\bar{}}{#1}}
\newcommand{\bd}{\mathop{\mathrm{bd}}}
\newcommand{\card}[1]{\abs{#1}}
\newcommand{\cc}[1]{\mathrm{\overline{conv}}#1}
\newcommand{\ccp}{\Gamma_0} % closed convex proper
\def\ci{\perp\!\!\!\perp} % conditional independence
\newcommand{\cl}{\mathop{\mathrm{cl}}}
\newcommand{\co}{\mathop{\mathrm{co}}}
\newcommand{\cone}{\mathop{\mathrm{cone}}}
\newcommand{\const}{\mathop{\mathrm{constant}}}
\newcommand{\core}{\mathop{\mathrm{core}}}
\newcommand{\conv}{\mathop{\mathrm{conv}}}
\newcommand{\Cov}{\mathop{\mathrm{Cov}}}
\newcommand{\dep}{\mathsf{dep}} % dependence
\newcommand{\diag}{\mathop{\mathrm{diag}}}
\newcommand{\Diag}{\mathop{\mathrm{Diag}}}
\newcommand{\diam}{\mathop{\textrm{diam}}}
\newcommand{\dist}{\ensuremath{\mathrm{dist}}\xspace}
\newcommand{\dom}{\mathop{\mathrm{dom}}}
\newcommand{\env}[2][\eta]{\mathrm{M}^{#1}_{#2}}
\newcommand{\epi}{\mathop{\mathrm{epi}}}
\newcommand{\eproof}{$\null\hfill\blacksquare$}
\newcommand{\gph}{\mathop{\mathrm{gph}}}
\newcommand{\grad}{{\nabla}}
\newcommand{\gradtil}{{\tilde{\grad}}}
\newcommand{\id}{\ensuremath{\mathrm{Id}}\xspace} % Identity
\newcommand{\Img}{\ensuremath{\mathrm{Im}}}
\newcommand{\ind}{\mathcal{I}}
\newcommand{\intr}{\mathop{\mathrm{int}}}
\newcommand{\Iver}[1]{[\![#1]\!]} % Iverson predicate
\newcommand{\Ker}{\ensuremath{\mathrm{Ker}}}
\newcommand{\lhs}{\mathrm{LHS}}
\usepackage{actuarialsymbol}
\newcommand{\Lsub}[2][]{\actsymb[][#1]{#2}{}{}} % Lipschitz constant
\newcommand{\mat}{\mathfrak{M}}
\newcommand{\maxi}{\mathop{\mathrm{maximize}}}
\newcommand{\mini}{\mathop{\mathrm{minimize}}}
\newcommand{\med}{\mathop{\mathrm{med}}}
\newcommand{\MED}{{\text{MED}}}
\newcommand{\mf}[1][f]{\mbox{\textcolor{red}{-}}#1} % minus function
\providecommand*{\mgrad}{\mathbin{\overline{\vee}\!}} % metric gradient
\newcommand{\ncone}{\mathcal{N}} % normal cone
\newcommand{\nnz}{\mathop{\mathrm{nnz}}}
\newcommand{\pa}[1][\eta]{\mathrm{A}^{#1}} % prox-avg
\newcommand{\pos}{\mathop{\mathrm{pos}}}
\newcommand{\pred}[1]{\left\llbracket#1\right\rrbracket}
\newcommand{\proj}{\mathrm{P}}
\newcommand{\prox}[2][\eta]{\mathrm{P}^{#1}_{#2}}
\newcommand{\qcc}[1]{\mathrm{\overline{qconv}}#1}
%\newcommand{\ran}{\mathrm{ran}}
\newcommand{\oeps}{\overline{\epsilon}} 
\newcommand{\of}[1][f]{\overline{#1}} % overline function
\newcommand{\opt}{\mathrm{opt}}
\newcommand{\ran}{\mathop{\mathrm{rge}}}
\newcommand{\rank}{\mathop{\mathrm{rank}}}
\newcommand{\rcore}{\mathop{\mathrm{rcore}}}
\def\restrict#1{\raise-.5ex\hbox{\ensuremath|}_{#1}}
\newcommand{\rhs}{\mathrm{RHS}}
\newcommand{\ri}{\mathop{\mathrm{ri}}}
\newcommand{\rint}{\mathop{\mathrm{rint}}}
\newcommand{\sat}{\mathsf{sat}}
\newcommand{\sgn}{\mathop{\mathrm{sign}}}
\newcommand{\sign}{\mathop{\mathrm{sign}}}
\DeclareMathOperator*{\sol}{\mathrm{SOL}}
\newcommand{\spe}{\ensuremath{\mathrm{sp}}}
\newcommand{\srank}{\mathop{\mathrm{srank}}}
\newcommand{\sri}{\mathop{\mathrm{sri}}}
\newcommand{\st}{~\mathrm{s.t.}~}
\newcommand{\supp}{\mathop{\mathrm{supp}}}
\newcommand{\tn}[1]{|\!\!|\!\!|#1|\!\!|\!\!|} % triple norm
\newcommand{\tr}{\mathop{\mathrm{tr}}}
\newcommand{\traj}{\mathop{\mathrm{Traj}}}
\newcommand{\tvn}{\ensuremath{\mathrm{tv}}\xspace} % total variation norm
\newcommand{\ueps}{\underline{\epsilon}}
\newcommand{\uf}[1][f]{\underline{#1}} % underline function
\newcommand{\var}{\mathop{\mathrm{Var}}}
\newcommand{\vect}{\mathop{\mathrm{vec}}}
\DeclareMathOperator*{\vi}{\mathrm{VI}}
\newcommand{\where}{{\quad \text{where} \quad}}
%\newcommand{\ln}{\mathop{\mathrm{ln}}}
%%%%%%%%%%%%%%% trigonometric %%%%%%%%%%%%%%%
%\DeclareMathOperator{\arcsin}{arcsin}   % already defined
%\DeclareMathOperator{\arccos}{arccos}% already defined
%\DeclareMathOperator{\arctan}{arctan} % already defined
\DeclareMathOperator{\arccot}{arccot}
\DeclareMathOperator{\arcsinh}{arcsinh}
\DeclareMathOperator{\arccosh}{arccosh}
\DeclareMathOperator{\arctanh}{arctanh}
\DeclareMathOperator{\arccoth}{arccoth}
%%%%%%%%%%%%%%% Utility functions %%%%%%%%%%%%%%%
\newcommand{\mymatrix}[2]{\left[\begin{array}{#1} #2 \end{array}\right]}
\newcommand{\mychoose}[2]{\left(\begin{array}{c} #1 \\ #2 \end{array}\right)}
\newcommand{\mydet}[1]{\det\left[ #1 \right]}
\newcommand{\myspan}[1]{\mathrm{span}\cbr{#1}}
\newcommand{\smallfrac}[2]{{\textstyle \frac{#1}{#2}}}
\newcommand{\pwrt}[1]{\frac{\partial}{\partial #1}}
\newcommand{\ppwrt}[1]{\frac{\partial^2}{(\partial #1)^2}}
\newcommand{\aleq}{\preccurlyeq}
\newcommand{\ageq}{\succcurlyeq}
\newcommand{\vardef}{\stackrel{\triangle}{=}}
%%%%%%%%%%%%%%% Short Forms %%%%%%%%%%%%%%%
\newcommand{\aka}{{a.k.a.}\xspace}
\newcommand{\cf}{{cf.}\xspace}
\newcommand{\ea}{{et al.}\xspace}
\newcommand{\eg}{{e.g.}\xspace}
\newcommand{\etc}{{etc.}\xspace}
\newcommand{\ie}{{i.e.}\xspace}
\newcommand{\iid}{{i.i.d.}\xspace}
\newcommand{\wrt}{{w.r.t.}\xspace}
\newcommand{\vs}{{vs.}\xspace}
%%%%%%%%%%%%%%% Brackets %%%%%%%%%%%%%%%
\newcommand{\abr}[1]{\left|#1\right|}
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\cbr}[1]{\left\{#1\right\}}
\newcommand{\ccc}[1]{|\!|\!|#1|\!|\!|}
\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}
\newcommand{\dual}[2]{\left\langle #1;#2 \right\rangle}
\newcommand{\dualnorm}[1]{\left\|#1\right\|_{\circ}}
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
\newcommand{\inner}[2]{\left\langle #1,#2 \right\rangle}
\newcommand{\nbr}[1]{\left\|#1\right\|}
\newcommand{\norm}[1]{\left\|#1\right\|}
\newcommand{\rbr}[1]{\left(#1\right)}
\newcommand{\sbr}[1]{\left[#1\right]}
\newcommand{\sembrack}[1]{[\![#1]\!]}
\newcommand{\sip}[2]{[#1, #2]}
%%%%%%%%%%%%%%%  Mathematical Constants  %%%%%%%%%%%%%%%
\newcommand{\face}[1][F]{\mathfrak{#1}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\invsqrttwo}{\frac{1}{\sqrt{2}}}
\newcommand{\one}{\mathbf{1}}  % Identity
\newcommand{\poly}[1][P]{\mathfrak{#1}}
\newcommand{\rmd}{\mathop{}\!\mathrm{d}}
\newcommand{\shalf}{\textstyle\frac{1}{2}}
\newcommand{\sqrttwo}{\sqrt{2}}
\newcommand{\zero}{\mathbf{0}} % Zero


%%%%%%%%%%%%%%% Theorems and Friends %%%%%%%%%%%%%%%
%% Some style files might actually define these variables.
%% So don't mess with them if they are already defined

\ifx\BlackBox\undefined
\newcommand{\BlackBox}{\rule{1.5ex}{1.5ex}}  % end of proof
\fi

\ifx\QED\undefined
\def\QED{~\rule[-1pt]{5pt}{5pt}\par\medskip}
\fi

\ifx\proof\undefined
\newenvironment{proof}{\par\noindent{\em Proof:\ }}{\hfill\BlackBox\\[.0mm]}
\fi
%\newenvironment{proof}{\par\noindent{\bf Proof\ }}{\hfill\BlackBox\\[2mm]}

\ifx\theorem\undefined
\newtheorem{theorem}{Theorem}
\fi

\ifx\example\undefined
\newtheorem{example}{Example}
\fi

\ifx\property\undefined
\newtheorem{property}{Property}
\fi

\ifx\lemma\undefined
\newtheorem{lemma}{Lemma}
\fi

\ifx\proposition\undefined
\newtheorem{proposition}{Proposition}
\fi

\ifx\remark\undefined
\newtheorem{remark}{Remark}
\fi

\ifx\corollary\undefined
\newtheorem{corollary}{Corollary}
\fi

\ifx\definition\undefined
\newtheorem{definition}{Definition}
\fi

\ifx\conjecture\undefined
\newtheorem{conjecture}{Conjecture}
\fi

\ifx\axiom\undefined
\newtheorem{axiom}[theorem]{Axiom}
\fi

\ifx\claim\undefined
\newtheorem{claim}[theorem]{Claim}
\fi

\ifx\assumption\undefined
\newtheorem{assumption}{Assumption}
\fi

\providecommand{\qedsymbol}{$\blacksquare$}


%%%%%%%%%%%%%%% Stock standard definitions %%%%%%%%%%%%%%%
\newcommand{\Abb}{\mathbb{A}}
%\newcommand{\Bbb}{\mathbb{B}} % conflict
\newcommand{\Cbb}{\mathbb{C}}
\newcommand{\Dbb}{\mathbb{D}}
\newcommand{\Ebb}{\mathbb{E}}
\newcommand{\Fbb}{\mathbb{F}}
\newcommand{\Gbb}{\mathbb{G}}
\newcommand{\Hbb}{\mathbb{H}}
\newcommand{\Ibb}{\mathbb{I}}
\newcommand{\Jbb}{\mathbb{J}}
\newcommand{\Kbb}{\mathbb{K}}
\newcommand{\Lbb}{\mathbb{L}}
\newcommand{\Mbb}{\mathbb{M}}
\newcommand{\Nbb}{\mathbb{N}}
\newcommand{\Obb}{\mathbb{O}}
\newcommand{\Pbb}{\mathbb{P}}
\newcommand{\Qbb}{\mathbb{Q}}
\newcommand{\Rbb}{\mathbb{R}}
\newcommand{\Sbb}{\mathbb{S}}
\newcommand{\Tbb}{\mathbb{T}}
\newcommand{\Ubb}{\mathbb{U}}
\newcommand{\Vbb}{\mathbb{V}}
\newcommand{\Wbb}{\mathbb{W}}
\newcommand{\Xbb}{\mathbb{X}}
\newcommand{\Ybb}{\mathbb{Y}}
\newcommand{\Zbb}{\mathbb{Z}}

\newcommand{\abf}{\mathbf{a}}
\newcommand{\bbf}{\mathbf{b}}
\newcommand{\cbf}{\mathbf{c}}
\newcommand{\dbf}{\mathbf{d}}
\newcommand{\ebf}{\mathbf{e}}
\newcommand{\fbf}{\mathbf{f}}
\newcommand{\gbf}{\mathbf{g}}
\newcommand{\hbf}{\mathbf{h}}
\newcommand{\ibf}{\mathbf{i}}
\newcommand{\jbf}{\mathbf{j}}
\newcommand{\kbf}{\mathbf{k}}
\newcommand{\lbf}{\mathbf{l}}
\newcommand{\mbf}{\mathbf{m}}
\newcommand{\nbf}{\mathbf{n}}
\newcommand{\obf}{\mathbf{o}}
\newcommand{\pbf}{\mathbf{p}}
\newcommand{\qbf}{\mathbf{q}}
\newcommand{\rbf}{\mathbf{r}}
\newcommand{\sbf}{\mathbf{s}}
\newcommand{\tbf}{\mathbf{t}}
\newcommand{\ubf}{\mathbf{u}}
\newcommand{\vbf}{\mathbf{v}}
\newcommand{\wbf}{\mathbf{w}}
\newcommand{\xbf}{\mathbf{x}}
\newcommand{\ybf}{\mathbf{y}}
\newcommand{\zbf}{\mathbf{z}}

\newcommand{\Abf}{\mathbf{A}}
\newcommand{\Bbf}{\mathbf{B}}
\newcommand{\Cbf}{\mathbf{C}}
\newcommand{\Dbf}{\mathbf{D}}
\newcommand{\Ebf}{\mathbf{E}}
\newcommand{\Fbf}{\mathbf{F}}
\newcommand{\Gbf}{\mathbf{G}}
\newcommand{\Hbf}{\mathbf{H}}
\newcommand{\Ibf}{\mathbf{I}}
\newcommand{\Jbf}{\mathbf{J}}
\newcommand{\Kbf}{\mathbf{K}}
\newcommand{\Lbf}{\mathbf{L}}
\newcommand{\Mbf}{\mathbf{M}}
\newcommand{\Nbf}{\mathbf{N}}
\newcommand{\Obf}{\mathbf{O}}
\newcommand{\Pbf}{\mathbf{P}}
\newcommand{\Qbf}{\mathbf{Q}}
\newcommand{\Rbf}{\mathbf{R}}
\newcommand{\Sbf}{\mathbf{S}}
\newcommand{\Tbf}{\mathbf{T}}
\newcommand{\Ubf}{\mathbf{U}}
\newcommand{\Vbf}{\mathbf{V}}
\newcommand{\Wbf}{\mathbf{W}}
\newcommand{\Xbf}{\mathbf{X}}
\newcommand{\Ybf}{\mathbf{Y}}
\newcommand{\Zbf}{\mathbf{Z}}

%\newcommand{\abs}{\bm{\mathsf{a}}} % conflict
\newcommand{\bbs}{\bm{\mathsf{b}}}
\newcommand{\cbs}{\bm{\mathsf{c}}}
\newcommand{\dbs}{\bm{\mathsf{d}}}
\newcommand{\ebs}{\bm{\mathsf{e}}}
\newcommand{\fbs}{\bm{\mathsf{f}}}
\newcommand{\gbs}{\bm{\mathsf{g}}}
\newcommand{\hbs}{\bm{\mathsf{h}}}
\newcommand{\ibs}{\bm{\mathsf{i}}}
\newcommand{\jbs}{\bm{\mathsf{j}}}
\newcommand{\kbs}{\bm{\mathsf{k}}}
\newcommand{\lbs}{\bm{\mathsf{l}}}
\newcommand{\mbs}{\bm{\mathsf{m}}}
\newcommand{\nbs}{\bm{\mathsf{n}}}
\newcommand{\obs}{\bm{\mathsf{o}}}
\newcommand{\pbs}{\bm{\mathsf{p}}}
\newcommand{\qbs}{\bm{\mathsf{q}}}
\newcommand{\rbs}{\bm{\mathsf{r}}}
\newcommand{\sbs}{\bm{\mathsf{s}}}
\newcommand{\tbs}{\bm{\mathsf{t}}}
\newcommand{\ubs}{\bm{\mathsf{u}}}
\newcommand{\vbs}{\bm{\mathsf{v}}}
\newcommand{\wbs}{\bm{\mathsf{w}}}
\newcommand{\xbs}{\bm{\mathsf{x}}}
\newcommand{\ybs}{\bm{\mathsf{y}}}
\newcommand{\zbs}{\bm{\mathsf{z}}}

%\newcommand{\Abs}{\bm{\mathsf{A}}} % conflict
\newcommand{\Bbs}{\bm{\mathsf{B}}}
\newcommand{\Cbs}{\bm{\mathsf{C}}}
\newcommand{\Dbs}{\bm{\mathsf{D}}}
\newcommand{\Ebs}{\bm{\mathsf{E}}}
\newcommand{\Fbs}{\bm{\mathsf{F}}}
\newcommand{\Gbs}{\bm{\mathsf{G}}}
\newcommand{\Hbs}{\bm{\mathsf{H}}}
\newcommand{\Ibs}{\bm{\mathsf{I}}}
\newcommand{\Jbs}{\bm{\mathsf{J}}}
\newcommand{\Kbs}{\bm{\mathsf{K}}}
\newcommand{\Lbs}{\bm{\mathsf{L}}}
\newcommand{\Mbs}{\bm{\mathsf{M}}}
\newcommand{\Nbs}{\bm{\mathsf{N}}}
\newcommand{\Obs}{\bm{\mathsf{O}}}
\newcommand{\Pbs}{\bm{\mathsf{P}}}
\newcommand{\Qbs}{\bm{\mathsf{Q}}}
\newcommand{\Rbs}{\bm{\mathsf{R}}}
\newcommand{\Sbs}{\bm{\mathsf{S}}}
\newcommand{\Tbs}{\bm{\mathsf{T}}}
\newcommand{\Ubs}{\bm{\mathsf{U}}}
\newcommand{\Vbs}{\bm{\mathsf{V}}}
\newcommand{\Wbs}{\bm{\mathsf{W}}}
\newcommand{\Xbs}{\bm{\mathsf{X}}}
\newcommand{\Ybs}{\bm{\mathsf{Y}}}
\newcommand{\Zbs}{\bm{\mathsf{Z}}}

\newcommand{\bsa}{\bm{\mathsf{a}}}
\newcommand{\bsb}{\bm{\mathsf{b}}}
\newcommand{\bsc}{\bm{\mathsf{c}}}
\newcommand{\bsd}{\bm{\mathsf{d}}}
\newcommand{\bse}{\bm{\mathsf{e}}}
\newcommand{\bsf}{\bm{\mathsf{f}}}
\newcommand{\bsg}{\bm{\mathsf{g}}}
\newcommand{\bsh}{\bm{\mathsf{h}}}
\newcommand{\bsi}{\bm{\mathsf{i}}}
\newcommand{\bsj}{\bm{\mathsf{j}}}
\newcommand{\bsk}{\bm{\mathsf{k}}}
\newcommand{\bsl}{\bm{\mathsf{l}}}
\newcommand{\bsm}{\bm{\mathsf{m}}}
\newcommand{\bsn}{\bm{\mathsf{n}}}
\newcommand{\bso}{\bm{\mathsf{o}}}
\newcommand{\bsp}{\bm{\mathsf{p}}}
\newcommand{\bsq}{\bm{\mathsf{q}}}
\newcommand{\bsr}{\bm{\mathsf{r}}}
\newcommand{\bss}{\bm{\mathsf{s}}}
\newcommand{\bst}{\bm{\mathsf{t}}}
\newcommand{\bsu}{\bm{\mathsf{u}}}
\newcommand{\bsv}{\bm{\mathsf{v}}}
\newcommand{\bsw}{\bm{\mathsf{w}}}
\newcommand{\bsx}{\bm{\mathsf{x}}}
\newcommand{\bsy}{\bm{\mathsf{y}}}
\newcommand{\bsz}{\bm{\mathsf{z}}}

\newcommand{\bsA}{\bm{\mathsf{A}}}
\newcommand{\bsB}{\bm{\mathsf{B}}}
\newcommand{\bsC}{\bm{\mathsf{C}}}
\newcommand{\bsD}{\bm{\mathsf{D}}}
\newcommand{\bsE}{\bm{\mathsf{E}}}
\newcommand{\bsF}{\bm{\mathsf{F}}}
\newcommand{\bsG}{\bm{\mathsf{G}}}
\newcommand{\bsH}{\bm{\mathsf{H}}}
\newcommand{\bsI}{\bm{\mathsf{I}}}
\newcommand{\bsJ}{\bm{\mathsf{J}}}
\newcommand{\bsK}{\bm{\mathsf{K}}}
\newcommand{\bsL}{\bm{\mathsf{L}}}
\newcommand{\bsM}{\bm{\mathsf{M}}}
\newcommand{\bsN}{\bm{\mathsf{N}}}
\newcommand{\bsO}{\bm{\mathsf{O}}}
\newcommand{\bsP}{\bm{\mathsf{P}}}
\newcommand{\bsQ}{\bm{\mathsf{Q}}}
\newcommand{\bsR}{\bm{\mathsf{R}}}
\newcommand{\bsS}{\bm{\mathsf{S}}}
\newcommand{\bsT}{\bm{\mathsf{T}}}
\newcommand{\bsU}{\bm{\mathsf{U}}}
\newcommand{\bsV}{\bm{\mathsf{V}}}
\newcommand{\bsW}{\bm{\mathsf{W}}}
\newcommand{\bsX}{\bm{\mathsf{X}}}
\newcommand{\bsY}{\bm{\mathsf{Y}}}
\newcommand{\bsZ}{\bm{\mathsf{Z}}}

\newcommand{\Acal}{\mathcal{A}}
\newcommand{\Bcal}{\mathcal{B}}
\newcommand{\Ccal}{\mathcal{C}}
\newcommand{\Dcal}{\mathcal{D}}
\newcommand{\Ecal}{\mathcal{E}}
\newcommand{\Fcal}{\mathcal{F}}
\newcommand{\Gcal}{\mathcal{G}}
\newcommand{\Hcal}{\mathcal{H}}
\newcommand{\Ical}{\mathcal{I}}
\newcommand{\Jcal}{\mathcal{J}}
\newcommand{\Kcal}{\mathcal{K}}
\newcommand{\Lcal}{\mathcal{L}}
\newcommand{\Mcal}{\mathcal{M}}
\newcommand{\Ncal}{\mathcal{N}}
\newcommand{\Ocal}{\mathcal{O}}
\newcommand{\Pcal}{\mathcal{P}}
\newcommand{\Qcal}{\mathcal{Q}}
\newcommand{\Rcal}{\mathcal{R}}
\newcommand{\Scal}{\mathcal{S}}
\newcommand{\Tcal}{\mathcal{T}}
\newcommand{\Ucal}{\mathcal{U}}
\newcommand{\Vcal}{\mathcal{V}}
\newcommand{\Wcal}{\mathcal{W}}
\newcommand{\Xcal}{\mathcal{X}}
\newcommand{\Ycal}{\mathcal{Y}}
\newcommand{\Zcal}{\mathcal{Z}}

\newcommand{\Ac}{\mathcal{A}}
\newcommand{\Bc}{\mathcal{B}}
\newcommand{\Cc}{\mathcal{C}}
\newcommand{\Dc}{\mathcal{D}}
\newcommand{\Ec}{\mathcal{E}}
\newcommand{\Fc}{\mathcal{F}}
\newcommand{\Gc}{\mathcal{G}}
\newcommand{\Hc}{\mathcal{H}}
\newcommand{\Ic}{\mathcal{I}}
\newcommand{\Jc}{\mathcal{J}}
\newcommand{\Kc}{\mathcal{K}}
\newcommand{\Lc}{\mathcal{L}}
\newcommand{\Mc}{\mathcal{M}}
\newcommand{\Nc}{\mathcal{N}}
\newcommand{\Oc}{\mathcal{O}}
\newcommand{\Pc}{\mathcal{P}}
\newcommand{\Qc}{\mathcal{Q}}
\newcommand{\Rc}{\mathcal{R}}
\newcommand{\Sc}{\mathcal{S}}
\newcommand{\Tc}{\mathcal{T}}
\newcommand{\Uc}{\mathcal{U}}
\newcommand{\Vc}{\mathcal{V}}
\newcommand{\Wc}{\mathcal{W}}
\newcommand{\Xc}{\mathcal{X}}
\newcommand{\Yc}{\mathcal{Y}}
\newcommand{\Zc}{\mathcal{Z}}

\newcommand{\Ads}{\mathds{A}}
\newcommand{\Bds}{\mathds{B}}
\newcommand{\Cds}{\mathds{C}}
\newcommand{\Dds}{\mathds{D}}
\newcommand{\Eds}{\mathds{E}}
\newcommand{\Fds}{\mathds{F}}
\newcommand{\Gds}{\mathds{G}}
\newcommand{\Hds}{\mathds{H}}
\newcommand{\Ids}{\mathds{I}}
\newcommand{\Jds}{\mathds{J}}
\newcommand{\Kds}{\mathds{K}}
\newcommand{\Lds}{\mathds{L}}
\newcommand{\Mds}{\mathds{M}}
\newcommand{\Nds}{\mathds{N}}
\newcommand{\Ods}{\mathds{O}}
\newcommand{\Pds}{\mathds{P}}
\newcommand{\Qds}{\mathds{Q}}
\newcommand{\Rds}{\mathds{R}}
\newcommand{\Sds}{\mathds{S}}
\newcommand{\Tds}{\mathds{T}}
\newcommand{\Uds}{\mathds{U}}
\newcommand{\Vds}{\mathds{V}}
\newcommand{\Wds}{\mathds{W}}
\newcommand{\Xds}{\mathds{X}}
\newcommand{\Yds}{\mathds{Y}}
\newcommand{\Zds}{\mathds{Z}}

\newcommand{\afk}{\mathfrak{a}}
\newcommand{\bfk}{\mathfrak{b}}
\newcommand{\cfk}{\mathfrak{c}}
\newcommand{\dfk}{\mathfrak{d}}
\newcommand{\efk}{\mathfrak{e}}
\newcommand{\ffk}{\mathfrak{f}}
\newcommand{\gfk}{\mathfrak{g}}
\newcommand{\hfk}{\mathfrak{h}}
\newcommand{\ifk}{\mathfrak{i}}
\newcommand{\jfk}{\mathfrak{j}}
\newcommand{\kfk}{\mathfrak{k}}
\newcommand{\lfk}{\mathfrak{l}}
\newcommand{\mfk}{\mathfrak{m}}
\newcommand{\nfk}{\mathfrak{n}}
\newcommand{\ofk}{\mathfrak{o}}
\newcommand{\pfk}{\mathfrak{p}}
\newcommand{\qfk}{\mathfrak{q}}
\newcommand{\rfk}{\mathfrak{r}}
\newcommand{\sfk}{\mathfrak{s}}
\newcommand{\tfk}{\mathfrak{t}}
\newcommand{\ufk}{\mathfrak{u}}
\newcommand{\vfk}{\mathfrak{v}}
\newcommand{\wfk}{\mathfrak{w}}
\newcommand{\xfk}{\mathfrak{x}}
\newcommand{\yfk}{\mathfrak{y}}
\newcommand{\zfk}{\mathfrak{z}}

\newcommand{\Afk}{\mathfrak{A}}
\newcommand{\Bfk}{\mathfrak{B}}
\newcommand{\Cfk}{\mathfrak{C}}
\newcommand{\Dfk}{\mathfrak{D}}
\newcommand{\Efk}{\mathfrak{E}}
\newcommand{\Ffk}{\mathfrak{F}}
\newcommand{\Gfk}{\mathfrak{G}}
\newcommand{\Hfk}{\mathfrak{H}}
\newcommand{\Ifk}{\mathfrak{I}}
\newcommand{\Jfk}{\mathfrak{J}}
\newcommand{\Kfk}{\mathfrak{K}}
\newcommand{\Lfk}{\mathfrak{L}}
\newcommand{\Mfk}{\mathfrak{M}}
\newcommand{\Nfk}{\mathfrak{N}}
\newcommand{\Ofk}{\mathfrak{O}}
\newcommand{\Pfk}{\mathfrak{P}}
\newcommand{\Qfk}{\mathfrak{Q}}
\newcommand{\Rfk}{\mathfrak{R}}
\newcommand{\Sfk}{\mathfrak{S}}
\newcommand{\Tfk}{\mathfrak{T}}
\newcommand{\Ufk}{\mathfrak{U}}
\newcommand{\Vfk}{\mathfrak{V}}
\newcommand{\Wfk}{\mathfrak{W}}
\newcommand{\Xfk}{\mathfrak{X}}
\newcommand{\Yfk}{\mathfrak{Y}}
\newcommand{\Zfk}{\mathfrak{Z}}

\newcommand{\Ascr}{\mathscr{A}}
\newcommand{\Bscr}{\mathscr{B}}
\newcommand{\Cscr}{\mathscr{C}}
\newcommand{\Dscr}{\mathscr{D}}
\newcommand{\Escr}{\mathscr{E}}
\newcommand{\Fscr}{\mathscr{F}}
\newcommand{\Gscr}{\mathscr{G}}
\newcommand{\Hscr}{\mathscr{H}}
\newcommand{\Iscr}{\mathscr{I}}
\newcommand{\Jscr}{\mathscr{J}}
\newcommand{\Kscr}{\mathscr{K}}
\newcommand{\Lscr}{\mathscr{L}}
\newcommand{\Mscr}{\mathscr{M}}
\newcommand{\Nscr}{\mathscr{N}}
\newcommand{\Oscr}{\mathscr{O}}
\newcommand{\Pscr}{\mathscr{P}}
\newcommand{\Qscr}{\mathscr{Q}}
\newcommand{\Rscr}{\mathscr{R}}
\newcommand{\Sscr}{\mathscr{S}}
\newcommand{\Tscr}{\mathscr{T}}
\newcommand{\Uscr}{\mathscr{U}}
\newcommand{\Vscr}{\mathscr{V}}
\newcommand{\Wscr}{\mathscr{W}}
\newcommand{\Xscr}{\mathscr{X}}
\newcommand{\Yscr}{\mathscr{Y}}
\newcommand{\Zscr}{\mathscr{Z}}

\newcommand{\asf}{\mathsf{a}}
%\newcommand{\bsf}{\mathsf{b}} % conflict
\newcommand{\csf}{\mathsf{c}}
\newcommand{\dsf}{\mathsf{d}}
\newcommand{\esf}{\mathsf{e}}
\newcommand{\fsf}{\mathsf{f}}
\newcommand{\gsf}{\mathsf{g}}
\newcommand{\hsf}{\mathsf{h}}
\newcommand{\isf}{\mathsf{i}}
\newcommand{\jsf}{\mathsf{j}}
\newcommand{\ksf}{\mathsf{k}}
\newcommand{\lsf}{\mathsf{l}}
\newcommand{\msf}{\mathsf{m}}
\newcommand{\nsf}{\mathsf{n}}
\newcommand{\osf}{\mathsf{o}}
\newcommand{\psf}{\mathsf{p}}
\newcommand{\qsf}{\mathsf{q}}
\newcommand{\rsf}{\mathsf{r}}
\newcommand{\ssf}{\mathsf{s}}
\newcommand{\tsf}{\mathsf{t}}
\newcommand{\usf}{\mathsf{u}}
\newcommand{\vsf}{\mathsf{v}}
\newcommand{\wsf}{\mathsf{w}}
\newcommand{\xsf}{\mathsf{x}}
\newcommand{\ysf}{\mathsf{y}}
\newcommand{\zsf}{\mathsf{z}}

\newcommand{\Asf}{\mathsf{A}}
\newcommand{\Bsf}{\mathsf{B}}
\newcommand{\Csf}{\mathsf{C}}
\newcommand{\Dsf}{\mathsf{D}}
\newcommand{\Esf}{\mathsf{E}}
\newcommand{\Fsf}{\mathsf{F}}
\newcommand{\Gsf}{\mathsf{G}}
\newcommand{\Hsf}{\mathsf{H}}
\newcommand{\Isf}{\mathsf{I}}
\newcommand{\Jsf}{\mathsf{J}}
\newcommand{\Ksf}{\mathsf{K}}
\newcommand{\Lsf}{\mathsf{L}}
\newcommand{\Msf}{\mathsf{M}}
\newcommand{\Nsf}{\mathsf{N}}
\newcommand{\Osf}{\mathsf{O}}
\newcommand{\Psf}{\mathsf{P}}
\newcommand{\Qsf}{\mathsf{Q}}
\newcommand{\Rsf}{\mathsf{R}}
\newcommand{\Ssf}{\mathsf{S}}
\newcommand{\Tsf}{\mathsf{T}}
\newcommand{\Usf}{\mathsf{U}}
\newcommand{\Vsf}{\mathsf{V}}
\newcommand{\Wsf}{\mathsf{W}}
\newcommand{\Xsf}{\mathsf{X}}
\newcommand{\Ysf}{\mathsf{Y}}
\newcommand{\Zsf}{\mathsf{Z}}

\newcommand{\Att}{\mathtt{A}}
\newcommand{\Btt}{\mathtt{B}}
\newcommand{\Ctt}{\mathtt{C}}
\newcommand{\Dtt}{\mathtt{D}}
\newcommand{\Ett}{\mathtt{E}}
\newcommand{\Ftt}{\mathtt{F}}
\newcommand{\Gtt}{\mathtt{G}}
\newcommand{\Htt}{\mathtt{H}}
\newcommand{\Itt}{\mathtt{I}}
\newcommand{\Jtt}{\mathtt{J}}
\newcommand{\Ktt}{\mathtt{K}}
\newcommand{\Ltt}{\mathtt{L}}
\newcommand{\Mtt}{\mathtt{M}}
\newcommand{\Ntt}{\mathtt{N}}
\newcommand{\Ott}{\mathtt{O}}
\newcommand{\Ptt}{\mathtt{P}}
\newcommand{\Qtt}{\mathtt{Q}}
\newcommand{\Rtt}{\mathtt{R}}
\newcommand{\Stt}{\mathtt{S}}
\newcommand{\Ttt}{\mathtt{T}}
\newcommand{\Utt}{\mathtt{U}}
\newcommand{\Vtt}{\mathtt{V}}
\newcommand{\Wtt}{\mathtt{W}}
\newcommand{\Xtt}{\mathtt{X}}
\newcommand{\Ytt}{\mathtt{Y}}
\newcommand{\Ztt}{\mathtt{Z}}

% mathbf lowercase
\newcommand{\av}{\mathbf{a}}
\newcommand{\bv}{\mathbf{b}}
\newcommand{\cv}{\mathbf{c}}
\newcommand{\dv}{\mathbf{d}}
\newcommand{\ev}{\mathbf{e}}
\newcommand{\fv}{\mathbf{f}}
\newcommand{\gv}{\mathbf{g}}
\newcommand{\hv}{\mathbf{h}}
\newcommand{\iv}{\mathbf{i}}
\newcommand{\jv}{\mathbf{j}}
\newcommand{\kv}{\mathbf{k}}
\newcommand{\lv}{\mathbf{l}}
\newcommand{\mv}{\mathbf{m}}
\newcommand{\nv}{\mathbf{n}}
\newcommand{\ov}{\mathbf{o}}
\newcommand{\pv}{\mathbf{p}}
\newcommand{\qv}{\mathbf{q}}
\newcommand{\rv}{\mathbf{r}}
\newcommand{\sv}{\mathbf{s}}
\newcommand{\tv}{\mathbf{t}}
\newcommand{\uv}{\mathbf{u}}
\newcommand{\vv}{\mathbf{v}}
\newcommand{\wv}{\mathbf{w}}
\newcommand{\xv}{\mathbf{x}}
\newcommand{\yv}{\mathbf{y}}
\newcommand{\zv}{\mathbf{z}}

% mathbf uppercase
\newcommand{\Av}{\mathbf{A}}
\newcommand{\Bv}{\mathbf{B}}
\newcommand{\Cv}{\mathbf{C}}
\newcommand{\Dv}{\mathbf{D}}
\newcommand{\Ev}{\mathbf{E}}
\newcommand{\Fv}{\mathbf{F}}
\newcommand{\Gv}{\mathbf{G}}
\newcommand{\Hv}{\mathbf{H}}
\newcommand{\Iv}{\mathbf{I}}
\newcommand{\Jv}{\mathbf{J}}
\newcommand{\Kv}{\mathbf{K}}
\newcommand{\Lv}{\mathbf{L}}
\newcommand{\Mv}{\mathbf{M}}
\newcommand{\Nv}{\mathbf{N}}
\newcommand{\Ov}{\mathbf{O}}
\newcommand{\Pv}{\mathbf{P}}
\newcommand{\Qv}{\mathbf{Q}}
\newcommand{\Rv}{\mathbf{R}}
\newcommand{\Sv}{\mathbf{S}}
\newcommand{\Tv}{\mathbf{T}}
\newcommand{\Uv}{\mathbf{U}}
\newcommand{\Vv}{\mathbf{V}}
\newcommand{\Wv}{\mathbf{W}}
\newcommand{\Xv}{\mathbf{X}}
\newcommand{\Yv}{\mathbf{Y}}
\newcommand{\Zv}{\mathbf{Z}}

\newcommand{\abar}{\bar{a}}
\newcommand{\bbar}{\bar{b}}
\newcommand{\cbar}{\bar{c}}
\newcommand{\dbar}{\bar{d}}
\newcommand{\ebar}{\bar{e}}
\newcommand{\fbar}{\bar{f}}
\newcommand{\gbar}{\bar{g}}
%\newcommand{\hbar}{\bar{h}} % conflict
\newcommand{\ibar}{\bar{i}}
\newcommand{\jbar}{\bar{j}}
\newcommand{\kbar}{\bar{k}}
\newcommand{\lbar}{\bar{l}}
\newcommand{\mbar}{\bar{m}}
\newcommand{\nbar}{\bar{n}}
%\newcommand{\obar}{\bar{o}} % conflict
\newcommand{\pbar}{\bar{p}}
\newcommand{\qbar}{\bar{q}}
\newcommand{\rbar}{\bar{r}}
\newcommand{\sbar}{\bar{s}}
\newcommand{\tbar}{\bar{t}}
\newcommand{\ubar}{\bar{u}}
\newcommand{\vbar}{\bar{v}}
\newcommand{\wbar}{\bar{w}}
\newcommand{\xbar}{\bar{x}}
\newcommand{\ybar}{\bar{y}}
\newcommand{\zbar}{\bar{z}}

\newcommand{\Abar}{\bar{A}}
\newcommand{\Bbar}{\bar{B}}
\newcommand{\Cbar}{\bar{C}}
\newcommand{\Dbar}{\bar{D}}
\newcommand{\Ebar}{\bar{E}}
\newcommand{\Fbar}{\bar{F}}
\newcommand{\Gbar}{\bar{G}}
\newcommand{\Hbar}{\bar{H}}
\newcommand{\Ibar}{\bar{I}}
\newcommand{\Jbar}{\bar{J}}
\newcommand{\Kbar}{\bar{K}}
\newcommand{\Lbar}{\bar{L}}
\newcommand{\Mbar}{\bar{M}}
\newcommand{\Nbar}{\bar{N}}
\newcommand{\Obar}{\bar{O}}
\newcommand{\Pbar}{\bar{P}}
\newcommand{\Qbar}{\bar{Q}}
\newcommand{\Rbar}{\bar{R}}
\newcommand{\Sbar}{\bar{S}}
\newcommand{\Tbar}{\bar{T}}
\newcommand{\Ubar}{\bar{U}}
\newcommand{\Vbar}{\bar{V}}
\newcommand{\Wbar}{\bar{W}}
\newcommand{\Xbar}{\bar{X}}
\newcommand{\Ybar}{\bar{Y}}
\newcommand{\Zbar}{\bar{Z}}

\newcommand{\ahat}{\hat{a}}
\newcommand{\bhat}{\hat{b}}
\newcommand{\chat}{\hat{c}}
\newcommand{\dhat}{\hat{d}}
\newcommand{\ehat}{\hat{e}}
\newcommand{\fhat}{\hat{f}}
\newcommand{\ghat}{\hat{g}}
\newcommand{\hhat}{\hat{h}}
\newcommand{\ihat}{\hat{i}}
\newcommand{\jhat}{\hat{j}}
\newcommand{\khat}{\hat{k}}
\newcommand{\lhat}{\hat{l}}
\newcommand{\mhat}{\hat{m}}
\newcommand{\nhat}{\hat{n}}
\newcommand{\ohat}{\hat{o}}
\newcommand{\phat}{\hat{p}}
\newcommand{\qhat}{\hat{q}}
\newcommand{\rhat}{\hat{r}}
\newcommand{\shat}{\hat{s}}
\newcommand{\that}{\hat{t}}
\newcommand{\uhat}{\hat{u}}
\newcommand{\vhat}{\hat{v}}
\newcommand{\what}{\hat{w}}
\newcommand{\xhat}{\hat{x}}
\newcommand{\yhat}{\hat{y}}
\newcommand{\zhat}{\hat{z}}

\newcommand{\Ahat}{\hat{A}}
\newcommand{\Bhat}{\hat{B}}
\newcommand{\Chat}{\hat{C}}
\newcommand{\Dhat}{\hat{D}}
\newcommand{\Ehat}{\hat{E}}
\newcommand{\Fhat}{\hat{F}}
\newcommand{\Ghat}{\hat{G}}
\newcommand{\Hhat}{\hat{H}}
\newcommand{\Ihat}{\hat{I}}
\newcommand{\Jhat}{\hat{J}}
\newcommand{\Khat}{\hat{K}}
\newcommand{\Lhat}{\hat{L}}
\newcommand{\Mhat}{\hat{M}}
\newcommand{\Nhat}{\hat{N}}
\newcommand{\Ohat}{\hat{O}}
\newcommand{\Phat}{\hat{P}}
\newcommand{\Qhat}{\hat{Q}}
\newcommand{\Rhat}{\hat{R}}
\newcommand{\Shat}{\hat{S}}
\newcommand{\That}{\hat{T}}
\newcommand{\Uhat}{\hat{U}}
\newcommand{\Vhat}{\hat{V}}
\newcommand{\What}{\hat{W}}
\newcommand{\Xhat}{\hat{X}}
\newcommand{\Yhat}{\hat{Y}}
\newcommand{\Zhat}{\hat{Z}}

\newcommand{\atil}{\tilde{a}}
\newcommand{\btil}{\tilde{b}}
\newcommand{\ctil}{\tilde{c}}
\newcommand{\dtil}{\tilde{d}}
\newcommand{\etil}{\tilde{e}}
\newcommand{\ftil}{\tilde{f}}
\newcommand{\gtil}{\tilde{g}}
\newcommand{\htil}{\tilde{h}}
\newcommand{\itil}{\tilde{i}}
\newcommand{\jtil}{\tilde{j}}
\newcommand{\ktil}{\tilde{k}}
\newcommand{\ltil}{\tilde{l}}
\newcommand{\mtil}{\tilde{m}}
\newcommand{\ntil}{\tilde{n}}
\newcommand{\otil}{\tilde{o}}
\newcommand{\ptil}{\tilde{p}}
\newcommand{\qtil}{\tilde{q}}
\newcommand{\rtil}{\tilde{r}}
\newcommand{\stil}{\tilde{s}}
\newcommand{\ttil}{\tilde{t}}
\newcommand{\util}{\tilde{u}}
\newcommand{\vtil}{\tilde{v}}
\newcommand{\wtil}{\tilde{w}}
\newcommand{\xtil}{\tilde{x}}
\newcommand{\ytil}{\tilde{y}}
\newcommand{\ztil}{\tilde{z}}

\newcommand{\Atil}{\tilde{A}}
\newcommand{\Btil}{\tilde{B}}
\newcommand{\Ctil}{\tilde{C}}
\newcommand{\Dtil}{\tilde{D}}
\newcommand{\Etil}{\tilde{E}}
\newcommand{\Ftil}{\tilde{F}}
\newcommand{\Gtil}{\tilde{G}}
\newcommand{\Htil}{\tilde{H}}
\newcommand{\Itil}{\tilde{I}}
\newcommand{\Jtil}{\tilde{J}}
\newcommand{\Ktil}{\tilde{K}}
\newcommand{\Ltil}{\tilde{L}}
\newcommand{\Mtil}{\tilde{M}}
\newcommand{\Ntil}{\tilde{N}}
\newcommand{\Otil}{\tilde{O}}
\newcommand{\Ptil}{\tilde{P}}
\newcommand{\Qtil}{\tilde{Q}}
\newcommand{\Rtil}{\tilde{R}}
\newcommand{\Stil}{\tilde{S}}
\newcommand{\Ttil}{\tilde{T}}
\newcommand{\Util}{\tilde{U}}
\newcommand{\Vtil}{\tilde{V}}
\newcommand{\Wtil}{\tilde{W}}
\newcommand{\Xtil}{\tilde{X}}
\newcommand{\Ytil}{\tilde{Y}}
\newcommand{\Ztil}{\tilde{Z}}

\newcommand{\Atilde}{\widetilde{A}}
\newcommand{\Btilde}{\widetilde{B}}
\newcommand{\Ctilde}{\widetilde{C}}
\newcommand{\Dtilde}{\widetilde{D}}
\newcommand{\Etilde}{\widetilde{E}}
\newcommand{\Ftilde}{\widetilde{F}}
\newcommand{\Gtilde}{\widetilde{G}}
\newcommand{\Htilde}{\widetilde{H}}
\newcommand{\Itilde}{\widetilde{I}}
\newcommand{\Jtilde}{\widetilde{J}}
\newcommand{\Ktilde}{\widetilde{K}}
\newcommand{\Ltilde}{\widetilde{L}}
\newcommand{\Mtilde}{\widetilde{M}}
\newcommand{\Ntilde}{\widetilde{N}}
\newcommand{\Otilde}{\widetilde{O}}
\newcommand{\Ptilde}{\widetilde{P}}
\newcommand{\Qtilde}{\widetilde{Q}}
\newcommand{\Rtilde}{\widetilde{R}}
\newcommand{\Stilde}{\widetilde{S}}
\newcommand{\Ttilde}{\widetilde{T}}
\newcommand{\Utilde}{\widetilde{U}}
\newcommand{\Vtilde}{\widetilde{V}}
\newcommand{\Wtilde}{\widetilde{W}}
\newcommand{\Xtilde}{\widetilde{X}}
\newcommand{\Ytilde}{\widetilde{Y}}
\newcommand{\Ztilde}{\widetilde{Z}}

\newcommand{\abbar}{\bar{\abf}}
\newcommand{\bbbar}{\bar{\bbf}}
\newcommand{\cbbar}{\bar{\cbf}}
\newcommand{\dbbar}{\bar{\dbf}}
\newcommand{\ebbar}{\bar{\ebf}}
\newcommand{\fbbar}{\bar{\fbf}}
\newcommand{\gbbar}{\bar{\gbf}}
\newcommand{\hbbar}{\bar{\hbf}}
\newcommand{\ibbar}{\bar{\ibf}}
\newcommand{\jbbar}{\bar{\jbf}}
\newcommand{\kbbar}{\bar{\kbf}}
\newcommand{\lbbar}{\bar{\lbf}}
\newcommand{\mbbar}{\bar{\mbf}}
\newcommand{\nbbar}{\bar{\nbf}}
\newcommand{\obbar}{\bar{\obf}}
\newcommand{\pbbar}{\bar{\pbf}}
\newcommand{\qbbar}{\bar{\qbf}}
\newcommand{\rbbar}{\bar{\rbf}}
\newcommand{\sbbar}{\bar{\sbf}}
\newcommand{\tbbar}{\bar{\tbf}}
\newcommand{\ubbar}{\bar{\ubf}}
\newcommand{\vbbar}{\bar{\vbf}}
\newcommand{\wbbar}{\bar{\wbf}}
\newcommand{\xbbar}{\bar{\xbf}}
\newcommand{\ybbar}{\bar{\ybf}}
\newcommand{\zbbar}{\bar{\zbf}}

\newcommand{\Abbar}{\bar{\Abf}}
\newcommand{\Bbbar}{\bar{\Bbf}}
\newcommand{\Cbbar}{\bar{\Cbf}}
\newcommand{\Dbbar}{\bar{\Dbf}}
\newcommand{\Ebbar}{\bar{\Ebf}}
\newcommand{\Fbbar}{\bar{\Fbf}}
\newcommand{\Gbbar}{\bar{\Gbf}}
\newcommand{\Hbbar}{\bar{\Hbf}}
\newcommand{\Ibbar}{\bar{\Ibf}}
\newcommand{\Jbbar}{\bar{\Jbf}}
\newcommand{\Kbbar}{\bar{\Kbf}}
\newcommand{\Lbbar}{\bar{\Lbf}}
\newcommand{\Mbbar}{\bar{\Mbf}}
\newcommand{\Nbbar}{\bar{\Nbf}}
\newcommand{\Obbar}{\bar{\Obf}}
\newcommand{\Pbbar}{\bar{\Pbf}}
\newcommand{\Qbbar}{\bar{\Qbf}}
\newcommand{\Rbbar}{\bar{\Rbf}}
\newcommand{\Sbbar}{\bar{\Sbf}}
\newcommand{\Tbbar}{\bar{\Tbf}}
\newcommand{\Ubbar}{\bar{\Ubf}}
\newcommand{\Vbbar}{\bar{\Vbf}}
\newcommand{\Wbbar}{\bar{\Wbf}}
\newcommand{\Xbbar}{\bar{\Xbf}}
\newcommand{\Ybbar}{\bar{\Ybf}}
\newcommand{\Zbbar}{\bar{\Zbf}}

\newcommand{\abhat}{\hat{\abf}}
\newcommand{\bbhat}{\hat{\bbf}}
\newcommand{\cbhat}{\hat{\cbf}}
\newcommand{\dbhat}{\hat{\dbf}}
\newcommand{\ebhat}{\hat{\ebf}}
\newcommand{\fbhat}{\hat{\fbf}}
\newcommand{\gbhat}{\hat{\gbf}}
\newcommand{\hbhat}{\hat{\hbf}}
\newcommand{\ibhat}{\hat{\ibf}}
\newcommand{\jbhat}{\hat{\jbf}}
\newcommand{\kbhat}{\hat{\kbf}}
\newcommand{\lbhat}{\hat{\lbf}}
\newcommand{\mbhat}{\hat{\mbf}}
\newcommand{\nbhat}{\hat{\nbf}}
\newcommand{\obhat}{\hat{\obf}}
\newcommand{\pbhat}{\hat{\pbf}}
\newcommand{\qbhat}{\hat{\qbf}}
\newcommand{\rbhat}{\hat{\rbf}}
\newcommand{\sbhat}{\hat{\sbf}}
\newcommand{\tbhat}{\hat{\tbf}}
\newcommand{\ubhat}{\hat{\ubf}}
\newcommand{\vbhat}{\hat{\vbf}}
\newcommand{\wbhat}{\hat{\wbf}}
\newcommand{\xbhat}{\hat{\xbf}}
\newcommand{\ybhat}{\hat{\ybf}}
\newcommand{\zbhat}{\hat{\zbf}}

\newcommand{\Abhat}{\hat{\Abf}}
\newcommand{\Bbhat}{\hat{\Bbf}}
\newcommand{\Cbhat}{\hat{\Cbf}}
\newcommand{\Dbhat}{\hat{\Dbf}}
\newcommand{\Ebhat}{\hat{\Ebf}}
\newcommand{\Fbhat}{\hat{\Fbf}}
\newcommand{\Gbhat}{\hat{\Gbf}}
\newcommand{\Hbhat}{\hat{\Hbf}}
\newcommand{\Ibhat}{\hat{\Ibf}}
\newcommand{\Jbhat}{\hat{\Jbf}}
\newcommand{\Kbhat}{\hat{\Kbf}}
\newcommand{\Lbhat}{\hat{\Lbf}}
\newcommand{\Mbhat}{\hat{\Mbf}}
\newcommand{\Nbhat}{\hat{\Nbf}}
\newcommand{\Obhat}{\hat{\Obf}}
\newcommand{\Pbhat}{\hat{\Pbf}}
\newcommand{\Qbhat}{\hat{\Qbf}}
\newcommand{\Rbhat}{\hat{\Rbf}}
\newcommand{\Sbhat}{\hat{\Sbf}}
\newcommand{\Tbhat}{\hat{\Tbf}}
\newcommand{\Ubhat}{\hat{\Ubf}}
\newcommand{\Vbhat}{\hat{\Vbf}}
\newcommand{\Wbhat}{\hat{\Wbf}}
\newcommand{\Xbhat}{\hat{\Xbf}}
\newcommand{\Ybhat}{\hat{\Ybf}}
\newcommand{\Zbhat}{\hat{\Zbf}}

\newcommand{\abtil}{\tilde{\abf}}
\newcommand{\bbtil}{\tilde{\bbf}}
\newcommand{\cbtil}{\tilde{\cbf}}
\newcommand{\dbtil}{\tilde{\dbf}}
\newcommand{\ebtil}{\tilde{\ebf}}
\newcommand{\fbtil}{\tilde{\fbf}}
\newcommand{\gbtil}{\tilde{\gbf}}
\newcommand{\hbtil}{\tilde{\hbf}}
\newcommand{\ibtil}{\tilde{\ibf}}
\newcommand{\jbtil}{\tilde{\jbf}}
\newcommand{\kbtil}{\tilde{\kbf}}
\newcommand{\lbtil}{\tilde{\lbf}}
\newcommand{\mbtil}{\tilde{\mbf}}
\newcommand{\nbtil}{\tilde{\nbf}}
\newcommand{\obtil}{\tilde{\obf}}
\newcommand{\pbtil}{\tilde{\pbf}}
\newcommand{\qbtil}{\tilde{\qbf}}
\newcommand{\rbtil}{\tilde{\rbf}}
\newcommand{\sbtil}{\tilde{\sbf}}
\newcommand{\tbtil}{\tilde{\tbf}}
\newcommand{\ubtil}{\tilde{\ubf}}
\newcommand{\vbtil}{\tilde{\vbf}}
\newcommand{\wbtil}{\tilde{\wbf}}
\newcommand{\xbtil}{\tilde{\xbf}}
\newcommand{\ybtil}{\tilde{\ybf}}
\newcommand{\zbtil}{\tilde{\zbf}}

\newcommand{\Abtil}{\tilde{\Abf}}
\newcommand{\Bbtil}{\tilde{\Bbf}}
\newcommand{\Cbtil}{\tilde{\Cbf}}
\newcommand{\Dbtil}{\tilde{\Dbf}}
\newcommand{\Ebtil}{\tilde{\Ebf}}
\newcommand{\Fbtil}{\tilde{\Fbf}}
\newcommand{\Gbtil}{\tilde{\Gbf}}
\newcommand{\Hbtil}{\tilde{\Hbf}}
\newcommand{\Ibtil}{\tilde{\Ibf}}
\newcommand{\Jbtil}{\tilde{\Jbf}}
\newcommand{\Kbtil}{\tilde{\Kbf}}
\newcommand{\Lbtil}{\tilde{\Lbf}}
\newcommand{\Mbtil}{\tilde{\Mbf}}
\newcommand{\Nbtil}{\tilde{\Nbf}}
\newcommand{\Obtil}{\tilde{\Obf}}
\newcommand{\Pbtil}{\tilde{\Pbf}}
\newcommand{\Qbtil}{\tilde{\Qbf}}
\newcommand{\Rbtil}{\tilde{\Rbf}}
\newcommand{\Sbtil}{\tilde{\Sbf}}
\newcommand{\Tbtil}{\tilde{\Tbf}}
\newcommand{\Ubtil}{\tilde{\Ubf}}
\newcommand{\Vbtil}{\tilde{\Vbf}}
\newcommand{\Wbtil}{\tilde{\Wbf}}
\newcommand{\Xbtil}{\tilde{\Xbf}}
\newcommand{\Ybtil}{\tilde{\Ybf}}
\newcommand{\Zbtil}{\tilde{\Zbf}}

\renewcommand{\vec}[1]{\mathbf{\boldsymbol{#1}}}

\newcommand{\avec}{\vec{a}}
\newcommand{\bvec}{\vec{b}}
\newcommand{\cvec}{\vec{c}}
\newcommand{\dvec}{\vec{d}}
\newcommand{\evec}{\vec{e}}
\newcommand{\fvec}{\vec{f}}
\newcommand{\gvec}{\vec{g}}
\newcommand{\hvec}{\vec{h}}
\newcommand{\ivec}{\vec{i}}
\newcommand{\jvec}{\vec{j}}
\newcommand{\kvec}{\vec{k}}
\newcommand{\lvec}{\vec{l}}
\newcommand{\mvec}{\vec{m}}
\newcommand{\nvec}{\vec{n}}
\newcommand{\ovec}{\vec{o}}
\newcommand{\pvec}{\vec{p}}
\newcommand{\qvec}{\vec{q}}
\newcommand{\rvec}{\vec{r}}
\newcommand{\svec}{\vec{s}}
\newcommand{\tvec}{\vec{t}}
\newcommand{\uvec}{\vec{u}}
\newcommand{\vvec}{\vec{v}}
\newcommand{\wvec}{\vec{w}}
\newcommand{\xvec}{\vec{x}}
\newcommand{\yvec}{\vec{y}}
\newcommand{\zvec}{\vec{z}}

\newcommand{\Avec}{\vec{A}}
\newcommand{\Bvec}{\vec{B}}
\newcommand{\Cvec}{\vec{C}}
\newcommand{\Dvec}{\vec{D}}
\newcommand{\Evec}{\vec{E}}
\newcommand{\Fvec}{\vec{F}}
\newcommand{\Gvec}{\vec{G}}
\newcommand{\Hvec}{\vec{H}}
\newcommand{\Ivec}{\vec{I}}
\newcommand{\Jvec}{\vec{J}}
\newcommand{\Kvec}{\vec{K}}
\newcommand{\Lvec}{\vec{L}}
\newcommand{\Mvec}{\vec{M}}
\newcommand{\Nvec}{\vec{N}}
\newcommand{\Ovec}{\vec{O}}
\newcommand{\Pvec}{\vec{P}}
\newcommand{\Qvec}{\vec{Q}}
\newcommand{\Rvec}{\vec{R}}
\newcommand{\Svec}{\vec{S}}
\newcommand{\Tvec}{\vec{T}}
\newcommand{\Uvec}{\vec{U}}
\newcommand{\Vvec}{\vec{V}}
\newcommand{\Wvec}{\vec{W}}
\newcommand{\Xvec}{\vec{X}}
\newcommand{\Yvec}{\vec{Y}}
\newcommand{\Zvec}{\vec{Z}}

%%%%%%%%%%%%%%% bold greek lowercase %%%%%%%%%%%%%%%
\newcommand{\alphav     }{\boldsymbol \alpha     }
\newcommand{\betav      }{\boldsymbol \beta      }
\newcommand{\gammav     }{\boldsymbol \gamma     }
\newcommand{\deltav     }{\boldsymbol \delta     }
\newcommand{\epsilonv   }{\boldsymbol \epsilon   }
\newcommand{\varepsilonv}{\boldsymbol \varepsilon}
\newcommand{\zetav      }{\boldsymbol \zeta      }
\newcommand{\etav       }{\boldsymbol \eta       }
\newcommand{\thetav     }{\boldsymbol \theta     }
\newcommand{\varthetav  }{\boldsymbol \vartheta  }
\newcommand{\iotav      }{\boldsymbol \iota      }
\newcommand{\kappav     }{\boldsymbol \kappa     }
\newcommand{\varkappav  }{\boldsymbol \varkappa  }
\newcommand{\lambdav    }{\boldsymbol \lambda    }
\newcommand{\muv        }{\boldsymbol \mu        }
\newcommand{\nuv        }{\boldsymbol \nu        }
\newcommand{\xiv        }{\boldsymbol \xi        }
\newcommand{\omicronv   }{\boldsymbol \omicron   }
\newcommand{\piv        }{\boldsymbol \pi        }
\newcommand{\varpiv     }{\boldsymbol \varpi     }
\newcommand{\rhov       }{\boldsymbol \rho       }
\newcommand{\varrhov    }{\boldsymbol \varrho    }
\newcommand{\sigmav     }{\boldsymbol \sigma     }
\newcommand{\varsigmav  }{\boldsymbol \varsigma  }
\newcommand{\tauv       }{\boldsymbol \tau       }
\newcommand{\upsilonv   }{\boldsymbol \upsilon   }
\newcommand{\phiv       }{\boldsymbol \phi       }
\newcommand{\varphiv    }{\boldsymbol \varphi    }
\newcommand{\chiv       }{\boldsymbol \chi       }
\newcommand{\psiv       }{\boldsymbol \psi       }
\newcommand{\omegav     }{\boldsymbol \omega     }
\newcommand{\ellv       }{\boldsymbol \ell       }

% bold greek uppercase
\newcommand{\Gammav     }{\boldsymbol \Gamma     }
\newcommand{\Deltav     }{\boldsymbol \Delta     }
\newcommand{\Thetav     }{\boldsymbol \Theta     }
\newcommand{\Lambdav    }{\boldsymbol \Lambda    }
\newcommand{\Xiv        }{\boldsymbol \Xi        }
\newcommand{\Piv        }{\boldsymbol \Pi        }
\newcommand{\Sigmav     }{\boldsymbol \Sigma     }
\newcommand{\Upsilonv   }{\boldsymbol \Upsilon   }
\newcommand{\Phiv       }{\boldsymbol \Phi       }
\newcommand{\Psiv       }{\boldsymbol \Psi       }
\newcommand{\Omegav     }{\boldsymbol \Omega     }

\DeclareMathSymbol{\Alpha}{\mathalpha}{operators}{"41}
\DeclareMathSymbol{\Beta}{\mathalpha}{operators}{"42}
\DeclareMathSymbol{\Epsilon}{\mathalpha}{operators}{"45}
\DeclareMathSymbol{\Zeta}{\mathalpha}{operators}{"5A}
\DeclareMathSymbol{\Eta}{\mathalpha}{operators}{"48}
\DeclareMathSymbol{\Iota}{\mathalpha}{operators}{"49}
\DeclareMathSymbol{\Kappa}{\mathalpha}{operators}{"4B}
\DeclareMathSymbol{\Mu}{\mathalpha}{operators}{"4D}
\DeclareMathSymbol{\Nu}{\mathalpha}{operators}{"4E}
\DeclareMathSymbol{\Omicron}{\mathalpha}{operators}{"4F}
\DeclareMathSymbol{\Rho}{\mathalpha}{operators}{"50}
\DeclareMathSymbol{\Tau}{\mathalpha}{operators}{"54}
\DeclareMathSymbol{\Chi}{\mathalpha}{operators}{"58}
\DeclareMathSymbol{\omicron}{\mathord}{letters}{"6F}