preamble_math.sty

\usepackage{amsthm, amsmath, mathtools, amssymb, physics} % for math and physics commands and symbols.

%%%%%%% Math-commands %%%%%%%
%%% GENERAL 
\newcommand{\function}[5]{
  \if\relax\detokenize{#1}\relax % first argument empty
    \begin{array}{c@{\hspace{0.5\tabcolsep}}c@{\hspace{0.5\tabcolsep}}c}
      #2 & \longrightarrow & #3 \\
      #4 & \longmapsto     & #5
    \end{array}
  \else % first argument non-empty
    \begin{array}{r@{\hspace{0.5\tabcolsep}}c@{\hspace{0.5\tabcolsep}}c@{\hspace{0.5\tabcolsep}}c}
      #1: & #2 & \longrightarrow & #3 \\
          & #4 & \longmapsto     & #5
    \end{array}
  \fi
} % 1 - name of the function
% 2 - domain
% 3 - image
% 4 - element in the domain
% 5 - element in the image

\newcommand{\rrightarrow}{\mathrel{\mathrlap{\rightarrow}\mkern1mu\rightarrow}} % a larger version of \twoheadrightarrow
% Default \Re{} for real part of a number 
% Default \Im{} for imaginary part of a number
% Default \abs for absolute value. \newcommand{\abs}[1]{\lvert#1\rvert}

\newcommand{\floor}[1]{
  \left\lfloor #1 \right\rfloor
}

\newcommand{\ceil}[1]{
  \left\lceil #1 \right\rceil
}


%%% CALCULUS
% command \grad is by default \nabla in bold face
% \renewcommand{\grad}{\operatorname{\vf{grad}}} % gradient
\DeclareMathOperator{\rot}{\vf{rot}} % curl
\DeclareMathOperator{\rotp}{\vf{\nabla}\times} % curl for physics 
\renewcommand{\div}{\operatorname{\vf{div}}} % divergence
\DeclareMathOperator{\divp}{\vf{\nabla}\cdot} % divergence for physics
% command \laplacian is by default \nabla^2
%\newcommand{\laplacian}{\nabla^2} % laplacian
\renewcommand{\laplacian}{\Delta} % laplacian
\DeclareDocumentCommand\derivative{ s o m g d() }{
  % Total derivative
  % s: star for \flatfrac flat derivative
  % o: optional n for nth derivative
  % m: mandatory (x in df/dx)
  % g: optional (f in df/dx)
  % d: long-form d/dx(...)
  \IfBooleanTF{#1}
  {\let\fractype\flatfrac}
  {\let\fractype\frac}
  \IfNoValueTF{#4}
  {
    \IfNoValueTF{#5}
    {\fractype{\diffd \IfNoValueTF{#2}{}{^{#2}}}{\diffd #3\IfNoValueTF{#2}{}{^{#2}}}}
    {\fractype{\diffd \IfNoValueTF{#2}{}{^{#2}}}{\diffd #3\IfNoValueTF{#2}{}{^{#2}}} \argopen(#5\argclose)}
  }
  {\fractype{\diffd \IfNoValueTF{#2}{}{^{#2}} #3}{\diffd #4\IfNoValueTF{#2}{}{^{#2}}}\IfValueT{#5}{(#5)}}
} % differential operator
\DeclareDocumentCommand\partialderivative{ s o m g d() }{
  % Total derivative
  % s: star for \flatfrac flat derivative
  % o: optional n for nth derivative
  % m: mandatory (x in df/dx)
  % g: optional (f in df/dx)
  % d: long-form d/dx(...)
  \IfBooleanTF{#1}
  {\let\fractype\flatfrac}
  {\let\fractype\frac}
  \IfNoValueTF{#4}{
    \IfNoValueTF{#5}
    {\fractype{\partial \IfNoValueTF{#2}{}{^{#2}}}{\partial #3\IfNoValueTF{#2}{}{^{#2}}}}
    {\fractype{\partial \IfNoValueTF{#2}{}{^{#2}}}{\partial #3\IfNoValueTF{#2}{}{^{#2}}} \argopen(#5\argclose)}
  }
  {\fractype{\partial \IfNoValueTF{#2}{}{^{#2}} #3}{\partial #4\IfNoValueTF{#2}{}{^{#2}}}\IfValueT{#5}{(#5)}}
} % partial differential operator
\newcommand{\ii}{\mathrm{i}} % imaginary unit
\renewcommand{\exp}[1]{\mathrm{e}^{#1}} % exponential function
\newcommand{\expp}[1]{\mathrm{exp}\!\left(#1\right)} % exponential function 2.0
\newcommand{\upint}[2]{\overline{\int_{#1}^{#2}}} % upper integral
\newcommand{\lowint}[2]{\underline{\int_{#1}^{#2}}} % lower integral
\DeclareMathOperator{\domain}{dom} % domain of a function
\DeclareMathOperator{\graph}{graph} % graph of a function
\DeclareMathOperator{\supp}{supp} % support of a function
\DeclareMathOperator{\vol}{vol} % volume of a region
\DeclareMathOperator{\area}{area} % area of a region
\newcommand{\crossprod}{\crossproduct} % cross product


%%% ALGEBRA
\newcommand{\bigzero}{\makebox(0,0){\text{\Huge0}}} % 0 big zero in a matrix
\DeclareMathOperator{\lcm}{lcm} % least common multiple
\DeclareMathOperator{\im}{im} % Image of a function
\newcommand{\id}{\mathrm{id}} % identity function
\DeclareMathOperator{\ch}{char} % characteristic of a field
\DeclareMathOperator{\sig}{sig} % signature of a bilinear map
\DeclareMathOperator{\rad}{rad} % radical of a bilinear map
\DeclareMathOperator{\sign}{sgn} % sign of a permutation
\DeclareMathOperator{\ord}{ord} % order of an element in a group
\newcommand{\quot}[2]{
  \mathchoice
  {% \displaystyle 
    \text{\raise1ex\hbox{$#1$}\!\Big/\!\lower1ex\hbox{$#2$}}}
  {% \textstyle
    #1/#2}
  {% \scriptstyle
    #1/#2}
  {% \scriptscriptstyle  
    #1/#2}
}% quotient group. Usage A/B--->\quot{A}{B}.
\newcommand{\NN}{\ensuremath{\mathbb{N}}} % set of natural numbers
\newcommand{\ZZ}{\ensuremath{\mathbb{Z}}} % set of integers
\newcommand{\QQ}{\ensuremath{\mathbb{Q}}} % set of rationals
\newcommand{\RR}{\ensuremath{\mathbb{R}}} % set of real numbers
\newcommand{\CC}{\ensuremath{\mathbb{C}}} % set of complex numbers
\newcommand{\KK}{\ensuremath{\mathbb{K}}} % a general field
\newcommand{\TT}{\ensuremath{\mathbb{T}}} % time for stochastic processes
\renewcommand{\SS}{\ensuremath{\mathbb{S}}} % other things
\newcommand{\MM}{\ensuremath{\mathbb{M}}} % other things

%%% TOPOLOGY
\DeclareMathOperator{\Int}{Int} % interior set
\DeclareMathOperator{\Ext}{Ext} % exterior set
\DeclareMathOperator{\Cl}{Cl} % clousure set
\newcommand{\cl}[1]{\overline{#1}} % clousure set
\DeclareMathOperator{\Fr}{\partial} % boundary set
\newcommand{\topo}{\tau} % symbol for the topology. Feasible options are: \tau, \mathcal{T}...
\newcommand{\conn}{\mathrel{\#}} % connected sum. \mathrel gives the space of a relation (like +,-,...) while \mathbin gives the space of a binary operator (like =).
\renewcommand{\S}{S} % S of the S ^ n (n-th dimensional sphere)
\newcommand{\Homeo}{\mathrm{Homeo}} % set of homeomorphisms
\newcommand{\Homeoplus}{\mathrm{Homeo}_+} % set of orientation-preserving homeomorphisms
\newcommand{\Diff}{\mathrm{Diff}} % set of diffeomorphisms
\newcommand{\Diffplus}{\mathrm{Diff}_+} % set of orientation-preserving diffeomorphisms

%%% GALOIS THEORY
\newcommand{\FF}{\ensuremath{\mathbb{F}}} % finite fields
\DeclareMathOperator{\irr}{Irr} % irreducible polynomial 
\DeclareMathOperator{\Mor}{Mor} % set of morphisms
\DeclareMathOperator{\Iso}{Iso} % set of isomorphisms
\DeclareMathOperator{\Aut}{Aut} % set of automorphisms
\DeclareMathOperator{\Gal}{Gal} % Galois group
\DeclareMathOperator{\Disc}{Disc} % discriminant of a polynomial


%%% PROBABILITY
\newcommand{\Prob}{\ensuremath{\mathbb{P}}} % probability
\newcommand{\Exp}{\mathbb{E}} % expected value
\newcommand{\Var}{\mathrm{Var}} % variance
\newcommand{\cov}{\mathrm{Cov}} % covariance
\newcommand{\iid}{i.i.d.\ } % independent and identically distributed
\newcommand{\indi}[1]{\vf{1}_{#1}}


%%% STATISTICS
\DeclareMathOperator{\bias}{bias} % Bias
\DeclareMathOperator{\MSE}{MSE} % Mean Square Error
\DeclareMathOperator{\Mod}{Mod} % mode
\DeclareMathOperator{\Med}{Med} % median
\DeclareMathOperator*{\argmax}{arg\,max} % argument where the maximum is attained
\DeclareMathOperator*{\argmin}{arg\,min} % argument where the minimum is attained


%%% COMPLEX ANALYSIS
% \DeclareMathOperator{\arg}{arg} % argument. By default is arg(z).
\DeclareMathOperator{\Arg}{Arg} % principal argument
\DeclareMathOperator{\Ln}{Ln} % principal logarithm
\DeclareMathOperator{\Ind}{Ind} % index of a curve
% \DeclareMathOperator{\Res}{Res} % residue of a function at a point. It is already defined as "Res".
\newcommand{\homoleg}[1]{\mathbin{\underset{#1}{\approx}}} % homology. \mathrel gives the space of a relation (like +,-,...) while \mathbin gives the space of a binary operator (like =). 
\newcommand{\DD}{\ensuremath{\mathbb{D}}} % unit disk


%%% DIFFERENTIAL GEOMETRY
\newcommand{\T}[1]{{\vf{T}_{\vf{#1}}}} % tangent vector
\newcommand{\N}[1]{{\vf{N}_{\vf{#1}}}} % normal vector
\newcommand{\B}[1]{{\vf{B}_{\vf{#1}}}} % binormal vector
\newcommand{\ka}[1]{{\kappa_{\vf{#1}}}} % curvature
\newcommand{\ta}[1]{{\tau_{\vf{#1}}}} % torsion
\newcommand{\rh}[1]{{\rho_{\vf{#1}}}} % radius of curvature
\newcommand{\I}{\ensuremath{\mathrm{I}}} % 1st fundamental form
\newcommand{\II}{\ensuremath{\mathrm{I\!I}}} % 2nd fundamental form
\newcommand{\homotop}{\mathbin{\sim}} % homotopy. \mathrel gives the space of a relation (like +,-,...) while \mathbin gives the space of a binary operator (like =). 
\newcommand{\HH}{\ensuremath{\mathbb{H}}} % half upper plane


%%% REAL AND FUNCTIONAL ANALYSIS
\newcommand{\m}[1]{
  \if\relax\detokenize{#1}\relax % first argument empty
    \abs{\cdot}
  \else % first argument non-empty
    \abs{#1}
  \fi
} % measure
\newcommand{\om}[1]{
  \if\relax\detokenize{#1}\relax % first argument empty
    \abs{\cdot}^*
  \else % first argument non-empty
    \abs{#1}^*
  \fi
} % outer measure
\newcommand{\almoste}[1]{\overset{\text{a.e.}}{#1}} % almost everywhere
\newcommand{\dotp}[2]{{\left\langle#1,#2\right\rangle}} % inner product
\DeclareMathOperator{\diam}{diam} % diameter

%%% PARTIAL DIFFERENTIAL EQUATIONS
\newcommand{\matdv}[2]{\frac{\mathrm{D}#1}{\mathrm{D}#2}} % material derivative
% \newcommand{\Tr}{\mathrm{Tr}} % trace of an operator (already defined and it is equal to Tr, so perfect)

%%% LINEAR MODELS
\DeclareMathOperator{\MST}{MST} % Total Mean Square
\DeclareMathOperator{\MSR}{MSR} % Regression Mean Square
\DeclareMathOperator{\SSE}{SSE} % Sum Square Error
\DeclareMathOperator{\SST}{SST} % Total Sum of Squares
\DeclareMathOperator{\SSR}{SSR} % Regression Sum of Squares
\DeclareMathOperator{\SSPE}{SSPE}
\DeclareMathOperator{\MSPE}{MSPE}
\DeclareMathOperator{\SSLOF}{SSLOF}
\DeclareMathOperator{\MSLOF}{MSLOF}
\DeclareMathOperator{\logit}{logit} % logit
\DeclareMathOperator{\VIF}{VIF} % Variance inflation factor
\newcommand{\Ent}{\mathrm{H}} % Shannon entropy

%%% DYNAMICAL SYSTEMS
\renewcommand{\o}[1]{\mathrm{o}\left(#1\right)} % little O notation
\renewcommand{\O}[1]{\mathrm{O}\left(#1\right)} % big O notation
\newcommand{\lyapunov}[1]{{\mathrm{L}}_{#1}} % Lyapunov constants
\newcommand{\poincare}[1]{{\mathrm{P}}_{#1}} % Poincaré constants
\newcommand{\sharkgeq}{\mathbin{\geq_{\mathrm{s}}}} % Sharkovskii order. \mathrel gives the space of a relation (like +,-,...) while \mathbin gives the space of a binary operator (like =). 
\newcommand{\sharkleq}{\mathbin{\leq_{\mathrm{s}}}} % Sharkovskii order. \mathrel gives the space of a relation (like +,-,...) while \mathbin gives the space of a binary operator (like =). 
\DeclareMathOperator{\Per}{Per} % Periods of a function

%%% HARMONIC ANALYSIS
\newcommand{\F}{\mathcal{F}} % Fourier transformoperator
\renewcommand{\H}{\mathcal{H}} % Fourier transform operator
\renewcommand{\pv}{\mathrm{p.v.}} % Cauchy principal value

%%% JUMP PROCESSES
\newcommand{\CPP}{\mathrm{CPP}} % Compound Poisson Process
\newcommand{\RPM}{\mathrm{RPM}} % Random Poisson Measure

%%% ADVANCED DYNAMICAL SYSTEMS
\newcommand{\Fix}{\mathrm{Fix}} % Fixed points

%%% NONLINEAR PDES 
\newcommand{\Conv}{\mathrm{Conv}} % convex hull
\newcommand{\codim}{\mathrm{codim}} % codimension

%%% vectors and matrices
\newcommand{\vf}[1]{\boldsymbol{\mathrm{#1}}} % math style for vectors and matrices and vector-values functions (previously it was \*vb{#1} but this does not apply to greek letters)
\newcommand{\transpose}[1]{{#1}^{\mathrm{T}}} % math style for vectors and matrices
\DeclareMathOperator{\adjugate}{adj} % adjugate matrix
\DeclareMathOperator{\cofactor}{cof} % cofactor matrix
\DeclareMathOperator{\diag}{diag} % diagonal matrix
\newcommand{\GL}{\mathrm{GL}} % set of invertible matrices
\renewcommand{\norm}[1]{\left\|#1\right\|} % norm of a vector



%%% PHYSICS
\newcommand{\const}{\text{const.}}
\newcommand{\Rey}{\mathrm{Re}} % Reynolds number
\newcommand{\Ma}{\mathrm{Ma}} % Mach number

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%% for long-vector arrows %%%%%
\newcommand{\vect}[1]{%
  \vbox{\m@th \ialign {##\crcr\vectfill\crcr\noalign{\kern-\p@ \nointerlineskip}$\hfil\displaystyle{#1}\hfil$\crcr}}
}
\def\vectfill{%
  $\m@th\smash-\mkern-7mu%
    \cleaders\hbox{$\mkern-2mu\smash-\mkern-2mu$}\hfill
    \mkern-7mu\raisebox{-3.66pt}[\p@][\p@]{$\mathord\mathchar"017E$}$
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%