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% 2006.05.09:  I am commenting these:
%
%\usepackage{amssymb,amsmath,amsthm}
%\usepackage{amscd}     % commutative diagram
%\usepackage[mathscr]{euscript}
%\usepackage{url}
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%\usepackage{mathabx}
%\usepackage{theorem}

% Stylistic adjustments
%\renewcommand{\theequation}{\fnsymbol{equation}}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\newcommand{\defn}[1]{\textbf{#1}}   % for words being defined
\newcommand{\tech}[1]{\textsl{#1}}     % for technical terms
\newcommand{\ssp}{\,}                % space between \S and number
\newcommand{\modulo}{\emph{modulo}}
\newcommand{\ie}{\emph{i.e.}}
\newcommand{\opcit}{\emph{op.\ cit.}}
\newcommand{\qua}{\emph{qu\^a}\ }
\let\oldepsilon\epsilon
\renewcommand{\epsilon}{\varepsilon}
\newcommand{\pref}[1]{(\ref{#1})}
\newcommand{\Tau}{\mathrm{T}}


\newcommand{\mathconstant}[1]{\mathrm{#1}}
\newcommand{\me}{\mathconstant e}     % [already given above]
\newcommand{\mpi}{\mathconstant{\pi}}
\newcommand{\mi}{\mathconstant{i}}


% For boxes around math; adapted from The LaTeX Companion.  It's not
%  working properly yet.  

\newsavebox{\mathsavebox}
\newlength{\mathboxwidth}
\newenvironment{mathbox}[1][\linewidth]
 {\setlength{\mathboxwidth}{#1-2\fboxsep-2\fboxrule}%
    \begin{lrbox}{\mathsavebox}%
        \begin{minipage}{\mathboxwidth}}
         {\end{minipage}
      \end{lrbox}
  \begin{center}
        \fbox{\usebox{\mathsavebox}}
  \end{center}}

% Derivations:

%\DeclareMathOperator{\Der}{Der}    % space of derivations of...
\newcommand{\Der}[1]{\operatorname{Der}(#1)} % space of derivations of...
%\newcommand{\vs}{\mathcal{V}}      % subspace of this
\newcommand{\lrv}{E}               % the same: a Lie-ring of vectors

\DeclareMathOperator{\dee}{d}      % derivation into the dual of this
\newcommand{\Lie}[1]{\mathfrak #1} % the Lie algebra of #1
%\newcommand{\deriv}{\delta}        % arbitrary derivation
%\newcommand{\genvs}{\vs}           % its codomain

\newcommand{\glld}[4]{\left.\frac{\partial}{\partial
    x^{#1}_{#2}}#3\right|_{#4}} % On GL_n(\R), the partial
    % derivative, with respect to x^{#1}_{#2}, of #3, evaluated at #4
\newcommand{\gld}[2]{\frac{\partial}{\partial
    x^{#1}_{#2}}} % On GL_n(\R), the partial derivation with respect
    % to x^{#1}_{#2} 
\newcommand{\Tan}[2]{\operatorname{T}_{#1}(#2)}  % tangent space at #1
    % of #2


% (Differential) forms:

%\DeclareMathOperator{\A}{A} % space of alternating functions
\newcommand{\Aplain}[2]{\operatorname A^{#1}(#2)}
\newcommand{\A}[1]{\Aplain{#1}{\lrv}}
\newcommand{\forms}[3]{\Omega^{#1}_{#2/#3}}% this, with diff'ation:
\newcommand{\Forms}[1][1]{\forms{#1}{L}{\lrv}}% its most common instance:
\newcommand{\svs}{\mathcal{W}}     % a subspace of this
\DeclareMathOperator{\I}{I}        % ordinary ideal
\newcommand{\di}{\mathcal{D}}      % differential ideal

% Tuples

%\newcommand{\tuple}[1]{\mathbf{#1}}
%\newcommand{\tuple}[1]{\vec{#1}\;}
%\newcommand{\tuple}[1]{\vec{#1}\,}
%\newcommand{\tuple}[1]{\vec{#1}\rule{0ex}{3ex}}
%\newcommand{\tuple}[1]{\vec{#1}}
\newcommand{\tuple}[1]{\boldsymbol{#1}}
\newcommand{\gtuple}[1]{\boldsymbol{#1}}  % For bold Greek letters
\newcommand{\seqi}{\mathbf{i}}     % characteristic sequence of i
\newcommand{\concat}{\mathbin{\widehat{\ }}}


% [Commutative] algebra:

\newcommand{\categ}[1]{\mathbf{#1}}     % category
\newcommand{\op}{^{\mathrm{op}}}        % opposite [category]

\newcommand{\pairing}[1]{\langle#1\rangle} % pairing
\DeclareMathOperator{\Spec}{Spec}       % spectrum (of a ring)
\newcommand{\Var}[1]{\mathrm{V}(#1)}    % variety (set of prime ideals)
\DeclareMathOperator{\stone}{S}
\newcommand{\Stone}[1]{\stone(#1)}      % Stone space of #1
\newcommand{\Ideal}[1]{\mathfrak{I}(#1)} % set of submodules of #1
\newcommand{\stroke}{\mathrel{|}}       % stroke for semi-group example
\newcommand{\unit}[1]{{#1}^{\times}}    % group of units of a ring

\newcommand{\inv}{^{-1}}                % mult. inverse
% converse of a relation: see functions 
\newcommand{\integer}[1]{\mathbf{#1}}   % in getting integers from naturals

\newcommand{\norm}[1]{\left\Vert#1\right\Vert}

% Concerning permutations:

\DeclareMathOperator{\sh}{sh}      % shuffle permutations
\DeclareMathOperator{\sgn}{sgn}    % sign of a permutation
\newcommand{\Sym}[1]{\operatorname{Sym}(#1)}
%\DeclareMathOperator{\Sym}{S}      % symmetric group
\newcommand{\Alt}[1]{A_{#1}}       % alternating group
\newcommand{\Dih}[1]{D_{#1}}       % dihedral group

% More groups

\newcommand{\order}[1]{\left|#1\right|}
\newcommand{\gpgen}[1]{\left\langle#1\right\rangle}% subgroup generated by #1
\newcommand{\nsubgp}{\vartriangleleft}  % normal subgroup
\newcommand{\nsupgp}{\vartriangleright}  % normal supergroup
\newcommand{\weakprod}{\displaystyle\sideset{}{^{\mathrm{w}}}\prod}
\newcommand{\freeprod}{\displaystyle\sideset{}{^*}\prod}
\newcommand{\gpres}[2]{\gpgen{#1\mid#2}}% group on #1 with rel'ns #2
\newcommand{\centr}[1]{\operatorname{C}(#1)}  % center
\newcommand{\cseries}[2]{\operatorname{C}_{#1}(#2)} % central series
\newcommand{\centralizer}[2]{\operatorname{C}_{#2}(#1)} % centralizer
\newcommand{\normalizer}[2]{\operatorname{N}_{#2}(#1)}
\newcommand{\dsubgp}[2]{#2^{(#1)}}  % n-th derived subgroup of #2, where n=#1.
\newcommand{\tsubgp}[1]{#1_{\mathrm{t}}} % torsion sub-group

% matrices

\DeclareMathOperator{\rank}{rank}  % 
\newcommand{\transp}[1]{{#1}^{\mathrm t}} % transpose

% linear algebraic groups (see also derivations)

%\newcommand{\GL}[1]{\operatorname{GL}(#1)}
\newcommand{\GLn}[1]{\operatorname{GL}_n(#1)}
\newcommand{\GLnr}{\GLn{\R}}
%\newcommand{\gl}[1]{\operatorname{GL}(#1)}
\newcommand{\gln}[1]{\mathfrak{gl}_n(#1)}
\newcommand{\glnr}{\gln{\R}}
\newcommand{\Mnnr}{\operatorname{M}_{n\times n}(\R)}


% rings and modules

\newcommand{\pid}{\textsc{pid}}
\newcommand{\ufd}{\textsc{ufd}}
\newcommand{\ed}{\textsc{ed}}
\newcommand{\ri}{\mathfrak{O}}          % ring of integers
\DeclareMathOperator{\ord}{ord}

\newcommand{\ad}[1]{\operatorname{ad}_{#1}}  % adjoint representation on
				% a Lie-ring

% ideals:

\newcommand{\ideal}[1]{\mathfrak{#1}}   % ideals (possibly improper)
\newcommand{\fil}[1]{\mathfrak{#1}}     % filter
\newcommand{\primei}{\mathfrak{p}}      % a prime ideal
\newcommand{\maxi}{\mathfrak{m}}        % a maximal ideal
\DeclareMathOperator{\Maxid}{m}
\newcommand{\maxid}[1]{\Maxid(#1)}      % max ideal of a valuation-ring
\newcommand{\modu}[1]{/\mathnormal{#1}} % quotient by #1

\newcommand{\valr}{\mathfrak{O}}        % valuation-ring
\newcommand{\locr}{\mathfrak{O}}        % local ring

% fields:

\DeclareMathOperator{\trdeg}{tr-deg} % transcendence-degree
%\newcommand{\acl}[1]{#1^{\mathrm{alg}}}    % algebraic closure
\newcommand{\alg}[1]{#1^{\mathrm{alg}}}    % algebraic closure
\newcommand{\rc}{^\mathrm{rc}}          % real-closure
\newcommand{\scl}[1]{#1^\mathrm{sep}}        % separable closure
\newcommand{\ins}[1]{#1^{p^{-\infty}}}  % perfect closure


\newcommand{\abs}[1]{\left\lvert#1\right\rvert}  % absolute value
\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}% greatest-integer
\DeclareMathOperator{\Char}{char}       % characteristic

\DeclareMathOperator{\witt}{W}
\newcommand{\Witt}[1]{\witt[#1]}

\newcommand{\divides}{\mathrel{|}}
\newcommand{\ndivides}{\mathrel{\nmid}}
%\newcommand{\radix}[1][{\ }]{{}^{#1}\!\!\!\surd}
\newcommand{\radix}[1][{\ }]{{}^{#1}\!\!\!\sqrt{\rule{0ex}{1.5ex}}}
\renewcommand{\leq}{\leqslant}
\renewcommand{\geq}{\geqslant}
\newcommand{\pred}[1][]{\operatorname{pred}#1}
\newcommand{\suc}[1]{\mathrm{succ}(#1)}

\newcommand{\genus}[1]{\operatorname{g}(#1)}  % genus of curve

% Galois theory

\newcommand{\Fix}[1]{\operatorname{Fix}(#1)}
\newcommand{\sgm}[1]{#1^{\sigma}}

% algebraic geometry

\newcommand{\zs}[1]{\operatorname{Z}(#1)}      % zero-set of...
\newcommand{\idealof}[1]{\operatorname{I}(#1)} % ideal of...

% vector-spaces

\newcommand{\lspan}[2]{\langle{#1}\rangle_{#2}} %linear span of #1 over #2



% Standard structures
\newcommand{\stnd}[1]{\mathbb{#1}}
\newcommand{\N}{\stnd{N}}         % natural numbers
\newcommand{\varN}{\omega}        % my usual preference for this
\newcommand{\Zp}{\Z_{+}}          % positive integers
\newcommand{\Z}{\stnd{Z}}         % integers
\newcommand{\Q}{\stnd{Q}}         % rationals
\newcommand{\R}{\stnd{R}}         % reals
\newcommand{\C}{\stnd{C}}         % complex numbers
\newcommand{\Ham}{\stnd{H}}       % quaternions
\newcommand{\F}{\stnd{F}}         % a field (probably finite)
\newcommand{\Aff}{\stnd{A}}       % affine space
\newcommand{\Proj}{\stnd{P}}      % projective space
%\newcommand{\P}{\stnd{P}}         % projective space
\newcommand{\B}{\stnd{B}}         % the 2-element Boolean algebra
%\newcommand{\2}{\F_2}             % field with 2 elements
%    (incomp. with xypic)

\newcommand{\End}[1]{\operatorname{End}(#1)}
\newcommand{\Mult}[1]{\operatorname{Mult}(#1)}
%\newcommand{\mult}{\mathbin{\mathsf m}}
\newcommand{\invol}[1]{\mathbin{\widehat{#1}}}
\newcommand{\Aut}[1]{\operatorname{Aut}(#1)}
\newcommand{\Hom}[1]{\operatorname{Hom}(#1)}
\newcommand{\HomZ}[1]{\operatorname{Hom}_{\Z}(#1)}
\newcommand{\HomR}[1]{\operatorname{Hom}_{R}(#1)}

% Symbolic logic

\DeclareMathOperator{\arity}{arity}    %
\newcommand{\qsep}{\;}                 % follows a quantified variable
\newcommand{\Forall}[1]{\forall{#1}\qsep }
\newcommand{\Exists}[1]{\exists{#1}\qsep }
\newcommand{\Existsgeq}[2]{\exists^{{}\geq #1}{#2}\qsep }
\newcommand{\Existsgeqn}[1]{\Existsgeq n{#1}}
\newcommand{\Existsgeqk}[1]{\Existsgeq k{#1}}
\newcommand{\Existsleq}[2]{\exists^{{}\leq #1}{#2}\qsep }
\newcommand{\Existsleqn}[1]{\Existsleq n{#1}}
\newcommand{\Existseq}[2]{\exists^{{}= #1}{#2}\qsep }
\newcommand{\existsunique}{\exists!}
\newcommand{\Existsunique}[1]{\exists!\,{#1}\qsep }
\newcommand{\Frall}[2]{(\forall#1\in#2)\qsep }
\newcommand{\Exsts}[2]{(\exists#1\in#2)\qsep }
\newcommand{\Iff}{}
\let\Iff\iff
\renewcommand{\iff}{\leftrightarrow}
\newcommand{\eor}{\nleftrightarrow}
\newcommand{\amp}{\;{}\mathbin{\&}{}\;}
%\newcommand{\Enot}{\mathsf{not-}}
\newcommand{\Enot}{\text{\sf not-}}
\renewcommand{\land}{\mathrel{\wedge}} % (otherwise the spacing is wrong)
\newcommand{\lno}{{\lnot}}
\newcommand{\shstroke}{\mathrel{|}}    % Sheffer stroke
\newcommand{\psys}[1]{\mathcal{#1}}    % proof-system
\newcommand{\proves}[1][]{\vdash_{\psys{#1}}}
\newcommand{\nproves}[1][]{\nvdash_{\psys{#1}}}
\newcommand{\fcom}{,\;}                % comma in list of formulas

\newcommand{\false}{\bot}              %
\newcommand{\true}{\top}               %
\newcommand{\False}{\mathsf F}
\newcommand{\True}{\mathsf T}
\newcommand{\Bcon}{\mathrel{\square}}  % arbitrary bin. Bool. connective 
\newcommand{\quant}{\mathsf Q}         % arbitrary quantifier
\newcommand{\Quant}[1]{\quant{#1}\qsep } % 
\newcommand{\Quanti}[2]{\quant_{#1}{#2}\qsep } % indexed arb. quant.
\newcommand{\syntactic}[1]{\mathbf{#1}} % for symbols of the syntax language
\newcommand{\sF}{\syntactic F}
\newcommand{\sG}{\syntactic G}
\newcommand{\sH}{\syntactic H}
\newcommand{\sP}{\syntactic P}
\newcommand{\sQ}{\syntactic Q}
\newcommand{\sR}{\syntactic R}
\newcommand{\sA}{\syntactic A}
\newcommand{\sB}{\syntactic B}
\newcommand{\sX}{\syntactic X}

% Model theory

\newcommand{\PFm}{\mathrm{PF}}  % set of propositional formulas
\newcommand{\PVar}{V}          % set of propositional variables
\newcommand{\interpretation}{\mathscr I}% interpretation (part of a
    %structure)
\newcommand{\Tm}[2]{\operatorname{Tm}^{#1}_{#2}} % terms
%\newcommand{\Fm}[1]{\mathrm{Fm}^{#1}} % #1-ary formulas
\newcommand{\Fm}[2]{\operatorname{Fm}^{#1}(#2)} % #1-ary formulas
\newcommand{\Fmo}[1]{\mathrm{Fm}^{#1}_0} % #1-ary basic formulas
\newcommand{\Sn}[1][\lang]{\operatorname{Sn}_{#1}} % sentences
\newcommand{\fv}[1]{\operatorname{fv}(#1)}  % set of free variables in #1
\newcommand{\Cn}[2][\lang]{\operatorname{Con}_{#1}(#2)} % logical consequences
\newcommand{\conseq}[1]{\operatorname{Con}(#1)} % logical consequences
\newcommand{\LT}[2]{\operatorname{B}_{#1}(#2)}  % Lindenbaum--Tarski algebra
\newcommand{\ts}[2]{\stone_{#1}(#2)}   % type-space
\newcommand{\tp}[2]{\operatorname{tp}_{#1}(#2)}  % complete type
\renewcommand{\models}{\vDash}   % for parallelism with the following:
\newcommand{\nmodels}{\nvDash}
\newcommand{\acl}[1]{\operatorname{acl}(#1)}  % algebraic closure
\newcommand{\dcl}[1]{\operatorname{dcl}(#1)}  % definable closure
\newcommand{\cl}[1]{\operatorname{cl}(#1)}   % closure

%\newcommand{\free}{\mathbin{\underset{\textstyle\smash\smallsmile}{\smash\shortmid}}}

%\newcommand{\free}{\mathbin%
%    {\thickspace\mid\negthickspace\negthickspace\negthinspace\cup}} 

\newcommand{\free}{\mathbin{\text{\makebox[0.8em]{\makebox[0pt]{$\mid$}\makebox[0pt]{$\cup$}}}}}


\newcommand{\thy}[1]{\mathrm{#1}}      % for the following:

% Named theories
\newcommand{\ACF}{\thy{ACF}}           % algebraically closed fields
\newcommand{\SCF}{\thy{SCF}}           % separably closed fields
\newcommand{\DF}{\thy{DF}}             % differential fields
\newcommand{\DCF}{\thy{DCF}}           % differentially closed fields
\newcommand{\DPF}{\thy{DPF}}           % differentially perfect fields
\newcommand{\ODF}{\thy{ODF}}           % ordered differential fields
\newcommand{\CODF}{\thy{CODF}}         % closed ordered differential fields
\newcommand{\ACFA}{\thy{ACFA}}         % fields with generic autom.
\newcommand{\RCF}{\thy{RCF}}           % real-closed fields
\newcommand{\OF}{\thy{OF}}             % operator-fields
\newcommand{\DEF}{\thy{DEF}}           % differential endomorphism-fields
\newcommand{\LO}{\thy{LO}}             % linear orders
\newcommand{\TO}{\thy{TO}}             % total orders (yes, same as linear)
\newcommand{\ML}{\thy{ML}}             % modular Lie-rings
\newcommand{\VS}{\thy{VS}}             % vector-spaces
\newcommand{\VL}{\thy{VL}}             % vector Lie-rings (was modular)

\newcommand{\lang}{\mathcal{L}}        % a language or signature
\newcommand{\eq}{^\mathrm{eq}}         % 
\DeclareMathOperator{\theory}{Th}      %
\newcommand{\Th}[1]{\theory(#1)}       % the theory of...
\DeclareMathOperator{\modelclass}{Mod}
\newcommand{\Mod}[1]{\modelclass(#1)}  % the class of models of...
%\newcommand{\Mod}[1]{\modelclass^{\included}(#1)}  % the category of
				% models with embeddings
\newcommand{\Model}[1]{\modelclass^{\elsub}(#1)}  % the category of
				% models with elementary embeddings

\DeclareMathOperator{\diag}{diag}      % (Robinson) diagram
\newcommand{\eldiag}[1]{\Th{\str{#1}_{#1}}}
\newcommand{\str}[1]{\mathfrak{#1}}     % structure
\newcommand{\Substr}[1]{\mathrm{Ss}(#1)} % set of substructures
\newcommand{\elsub}{\preccurlyeq}      % is an elementary substructure of
%\newcommand{\elsubstr}{\preccurlyeq}   

\newcommand{\monster}{\mathfrak{C}}    % monster model

\newcommand{\rnk}{\operatorname R}    % rank (arbitrary)
\newcommand{\RCB}[1]{\rnk_{\mathrm{CB}}(#1)} % Cantor--Bendixson rank
\newcommand{\rRCB}[2]{\rnk^{#1}_{\mathrm{CB}}(#2)} % relative CB rank  
\newcommand{\dCB}[1]{\deg_{\mathrm{CB}}(#1)}
\newcommand{\Rdeg}[1]{\operatorname{R-deg}(#1)}
\newcommand{\RM}[1]{\operatorname{RM}(#1)}  % Morley rank
\newcommand{\dM}[1]{\operatorname{dM}(#1)}

%\newcommand{\modeled}{\mathrel{=|}}

\newcommand{\altequiv}{\mathrel{\mathord{\models}=\!\!\!|}}
%\newcommand{\altequiv}{\mathrel{\mathord{\vDash}\mathord{\Dashv}}}

%%%% SET THEORY

\newcommand{\universe}{\mathcal U}     

% set-theoretic operators:

% nullary:

\renewcommand{\emptyset}{\varnothing}  % [I like the other symbol better]

% unary:

\newcommand{\comp}{^{\mathrm{c}}}      % set-theoretic complement
\newcommand{\pow}[1]{\mathscr{P}(#1)}  % power set

% binary:
\renewcommand{\setminus}{\smallsetminus} % I don't like the standard symbol
\newcommand{\symdiff}{\vartriangle}    % symmetric difference

% [infinitary]:

\newcommand{\family}[1]{\mathcal{#1}}  % family (of sets)

% set-theoretic relations:

\newcommand{\included}{\subseteq}      % [the name suggests the meaning here]
\newcommand{\nincluded}{\not\subseteq} % not included
\newcommand{\pincluded}{\subset}       % proper inclusion    
\newcommand{\includes}{\supseteq}
\newcommand{\fsub}{\subset_\mathrm{f}}  % is a finite subset of [an
				% infinite set]
\newcommand{\fincluded}{\included_\mathrm{f}}  % is a finite subset of


% cardinality

\newcommand{\injects}{\preccurlyeq}    % injects in
\newcommand{\pinjects}{\prec}
\newcommand{\equip}{\approx}           % equipollent (is in bijection with)
\newcommand{\nequip}{\not\approx}      % not equipollent
\newcommand{\vscr}[1]{#1'}            % successor ordinal (v for von Neumann)
\newcommand{\size}[1]{\lvert#1\rvert}  % cardinality
\newcommand{\pl}{^{+}}                 % successor cardinal
\newcommand{\cntm}{\mathfrak c}        % cardinality of the continuum

\newcommand{\class}[1]{\mathbf{#1}}    % class
\newcommand{\Russell}{\class R}        % {x:x\notin x}

\newcommand{\on}{\class{ON}}           % the class of ordinals
\newcommand{\cn}{\class{CN}}           % the class of cardinals
\newcommand{\axioms}[1]{\mathbf{#1}}   % set-theoretic axiom or set of axioms
\newcommand{\zf}{\axioms{ZF}}          % Zermelo-Frankel axioms
\newcommand{\ac}{\axioms{AC}}	       % Axiom of Choice
\newcommand{\zfc}{\axioms{ZFC}}        % ZF + AC
\newcommand{\ch}{\axioms{CH}}          % Continuum Hypothesis
\newcommand{\pa}{\axioms{PA}}          % first-order Peano arithmetic

% relations and functions

\newcommand{\conv}[1]{\breve{#1}}       % converse of a relation
\newcommand{\To}{\longrightarrow}
\DeclareMathOperator{\dom}{dom}        % domain of a function
\newcommand{\setim}[1][]{''{#1}} % image of set under preceding function 
\newcommand{\setimb}[1]{[#1]}    % image of set under preceding function 
%\newcommand{\rest}[1]{\mathord{|}_{#1}}% restriction of function to #1
\newcommand{\rest}[1]{\restriction{#1}}% restriction of function to #1
\newcommand{\id}{\mathsf{id}}          % identity-map
\newcommand{\mapset}[2]{{}^{#1}#2}     % set of functions from #1 to #2
\newcommand{\chf}[1]{\chi_{#1}}        % characteristic function
\DeclareMathOperator{\im}{im}          % image
\DeclareMathOperator{\supp}{supp}      % support
\newcommand{\coordproj}[2]{\pi^{#1}_{#2}} % coordinate projection,
				% omitting entry {#2} from an {#1}-tuple

\newcommand{\pis}[1]{\overline{#1}}    % proper initial segment
\newcommand{\modsim}{/\mathord{\sim}}  % modulo the eq-ren \sim
\newcommand{\eqc}[1]{[#1]}             % equivalence-class
\newcommand{\length}[1]{\operatorname{\ell n}(#1)}  % length of a string

% Number theory
\newcommand{\scr}[1]{#1^{+}}           % successor
\DeclareMathOperator{\lcm}{lcm}
\newcommand{\parity}[1]{\operatorname{p}(#1)}
%\newcommand{\parity}[1]{\overline{#1}}
\newcommand{\rem}[2]{\operatorname{rem}(#1,#2)} % remainder on
				% dividing #1 by #2

%  Analysis

\newcommand{\cut}[1]{\mathfrak{#1}}    % Dedekind cut
\newcommand{\partit}{\mathcal{P}}      % set of partitions
\newcommand{\chng}{\mathord{\vartriangle}}       % change in...
\newcommand{\mn}[2]{\mathrm{m}_{#1}(#2)}  % minimum of f on...
\newcommand{\mx}[2]{\mathrm{M}_{#1}(#2)}  % maximum of f on...
\newcommand{\ls}[3]{\mathrm{L}(#1,#2,#3)} % lower sum
\newcommand{\us}[3]{\mathrm{U}(#1,#2,#3)} % upper sum
\newcommand{\nls}[2]{\mathrm{L}(#1,#2)} % ``normal'' lower sum
\newcommand{\nus}[2]{\mathrm{U}(#1,#2)} % ``normal'' upper sum
\newcommand{\upint}[3]{\overline{\int_{#1}^{#2}}#3} % upper integral
\newcommand{\lowint}[3]{\underline{\int_{#1}^{#2}}#3} % lower integral

\newcommand{\bndry}[3]{\left[#1\right]_{#2}^{#3}}  % `boundary'
\newcommand{\Lpt}[1]{\mathscr{L}\left\{#1\right\}} % Laplace transform
\newcommand{\Lpint}[1]{\int_0^{\infty}e^{-st}#1\dee t} % its value

\newcommand{\uhp}{\mathfrak H}   % upper half-plane
