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+%%
+%% This is file `PJLmath.sty',
+%% generated with the docstrip utility.
+
+%% Copyright (C) 2021 by Jinwen XU
+%%
+%% This is part of the ProjLib Toolkit.
+%%
+%% This work may be distributed and/or modified under the conditions of the
+%% LaTeX Project Public License, either version 1.3c of this license or (at
+%% your option) any later version. The latest version of this license is in
+%%
+%% http://www.latex-project.org/lppl.txt
+%%
+%% and version 1.3c or later is part of all distributions of LaTeX version
+%% 2005/12/01 or later.
+%%
+\NeedsTeXFormat{LaTeX2e}[2020-10-01]
+\ProvidesPackage{PJLmath}
+ [2021/12/16 Useful math macros and shortcuts]
+\RequirePackage{kvoptions}
+\RequirePackage{etoolbox}
+\RequirePackage{mathtools}
+\RequirePackage{mathrsfs}
+\@ifpackageloaded{unicode-math}{}{\RequirePackage{amssymb}}
+
+%%================================
+%% Shorthands
+%%================================
+\providecommand{\N}{\ensuremath{\mathbb{N}}}
+\providecommand{\Z}{\ensuremath{\mathbb{Z}}}
+\providecommand{\Q}{\ensuremath{\mathbb{Q}}}
+\providecommand{\R}{\ensuremath{\mathbb{R}}}
+\providecommand{\C}{\ensuremath{\mathbb{C}}}
+\providecommand{\F}{\ensuremath{\mathbb{F}}}
+\providecommand{\A}{\ensuremath{\mathbb{A}}}
+
+\providecommand{\bb}[1]{\ensuremath{\mathbb{#1}}}
+\providecommand{\bbA}{\ensuremath{\mathbb{A}}}
+\providecommand{\bbB}{\ensuremath{\mathbb{B}}}
+\providecommand{\bbC}{\ensuremath{\mathbb{C}}}
+\providecommand{\bbD}{\ensuremath{\mathbb{D}}}
+\providecommand{\bbE}{\ensuremath{\mathbb{E}}}
+\providecommand{\bbF}{\ensuremath{\mathbb{F}}}
+\providecommand{\bbG}{\ensuremath{\mathbb{G}}}
+\providecommand{\bbH}{\ensuremath{\mathbb{H}}}
+\providecommand{\bbI}{\ensuremath{\mathbb{I}}}
+\providecommand{\bbJ}{\ensuremath{\mathbb{J}}}
+\providecommand{\bbK}{\ensuremath{\mathbb{K}}}
+\providecommand{\bbL}{\ensuremath{\mathbb{L}}}
+\providecommand{\bbM}{\ensuremath{\mathbb{M}}}
+\providecommand{\bbN}{\ensuremath{\mathbb{N}}}
+\providecommand{\bbO}{\ensuremath{\mathbb{O}}}
+\providecommand{\bbP}{\ensuremath{\mathbb{P}}}
+\providecommand{\bbQ}{\ensuremath{\mathbb{Q}}}
+\providecommand{\bbR}{\ensuremath{\mathbb{R}}}
+\providecommand{\bbS}{\ensuremath{\mathbb{S}}}
+\providecommand{\bbT}{\ensuremath{\mathbb{T}}}
+\providecommand{\bbU}{\ensuremath{\mathbb{U}}}
+\providecommand{\bbV}{\ensuremath{\mathbb{V}}}
+\providecommand{\bbW}{\ensuremath{\mathbb{W}}}
+\providecommand{\bbX}{\ensuremath{\mathbb{X}}}
+\providecommand{\bbY}{\ensuremath{\mathbb{Y}}}
+\providecommand{\bbZ}{\ensuremath{\mathbb{Z}}}
+
+\providecommand{\mc}[1]{\ensuremath{\mathcal{#1}}}
+\providecommand{\mcA}{\ensuremath{\mathcal{A}}}
+\providecommand{\mcB}{\ensuremath{\mathcal{B}}}
+\providecommand{\mcC}{\ensuremath{\mathcal{C}}}
+\providecommand{\mcD}{\ensuremath{\mathcal{D}}}
+\providecommand{\mcE}{\ensuremath{\mathcal{E}}}
+\providecommand{\mcF}{\ensuremath{\mathcal{F}}}
+\providecommand{\mcG}{\ensuremath{\mathcal{G}}}
+\providecommand{\mcH}{\ensuremath{\mathcal{H}}}
+\providecommand{\mcI}{\ensuremath{\mathcal{I}}}
+\providecommand{\mcJ}{\ensuremath{\mathcal{J}}}
+\providecommand{\mcK}{\ensuremath{\mathcal{K}}}
+\providecommand{\mcL}{\ensuremath{\mathcal{L}}}
+\providecommand{\mcM}{\ensuremath{\mathcal{M}}}
+\providecommand{\mcN}{\ensuremath{\mathcal{N}}}
+\providecommand{\mcO}{\ensuremath{\mathcal{O}}}
+\providecommand{\mcP}{\ensuremath{\mathcal{P}}}
+\providecommand{\mcQ}{\ensuremath{\mathcal{Q}}}
+\providecommand{\mcR}{\ensuremath{\mathcal{R}}}
+\providecommand{\mcS}{\ensuremath{\mathcal{S}}}
+\providecommand{\mcT}{\ensuremath{\mathcal{T}}}
+\providecommand{\mcU}{\ensuremath{\mathcal{U}}}
+\providecommand{\mcV}{\ensuremath{\mathcal{V}}}
+\providecommand{\mcW}{\ensuremath{\mathcal{W}}}
+\providecommand{\mcX}{\ensuremath{\mathcal{X}}}
+\providecommand{\mcY}{\ensuremath{\mathcal{Y}}}
+\providecommand{\mcZ}{\ensuremath{\mathcal{Z}}}
+
+\providecommand{\cal}[1]{\ensuremath{\mathcal{#1}}}
+\providecommand{\calA}{\ensuremath{\mathcal{A}}}
+\providecommand{\calB}{\ensuremath{\mathcal{B}}}
+\providecommand{\calC}{\ensuremath{\mathcal{C}}}
+\providecommand{\calD}{\ensuremath{\mathcal{D}}}
+\providecommand{\calE}{\ensuremath{\mathcal{E}}}
+\providecommand{\calF}{\ensuremath{\mathcal{F}}}
+\providecommand{\calG}{\ensuremath{\mathcal{G}}}
+\providecommand{\calH}{\ensuremath{\mathcal{H}}}
+\providecommand{\calI}{\ensuremath{\mathcal{I}}}
+\providecommand{\calJ}{\ensuremath{\mathcal{J}}}
+\providecommand{\calK}{\ensuremath{\mathcal{K}}}
+\providecommand{\calL}{\ensuremath{\mathcal{L}}}
+\providecommand{\calM}{\ensuremath{\mathcal{M}}}
+\providecommand{\calN}{\ensuremath{\mathcal{N}}}
+\providecommand{\calO}{\ensuremath{\mathcal{O}}}
+\providecommand{\calP}{\ensuremath{\mathcal{P}}}
+\providecommand{\calQ}{\ensuremath{\mathcal{Q}}}
+\providecommand{\calR}{\ensuremath{\mathcal{R}}}
+\providecommand{\calS}{\ensuremath{\mathcal{S}}}
+\providecommand{\calT}{\ensuremath{\mathcal{T}}}
+\providecommand{\calU}{\ensuremath{\mathcal{U}}}
+\providecommand{\calV}{\ensuremath{\mathcal{V}}}
+\providecommand{\calW}{\ensuremath{\mathcal{W}}}
+\providecommand{\calX}{\ensuremath{\mathcal{X}}}
+\providecommand{\calY}{\ensuremath{\mathcal{Y}}}
+\providecommand{\calZ}{\ensuremath{\mathcal{Z}}}
+
+\providecommand{\mf}[1]{\ensuremath{\mathfrak{#1}}}
+\providecommand{\mfa}{\ensuremath{\mathfrak{a}}}
+\providecommand{\mfb}{\ensuremath{\mathfrak{b}}}
+\providecommand{\mfc}{\ensuremath{\mathfrak{c}}}
+\providecommand{\mfd}{\ensuremath{\mathfrak{d}}}
+\providecommand{\mfe}{\ensuremath{\mathfrak{e}}}
+\providecommand{\mff}{\ensuremath{\mathfrak{f}}}
+\providecommand{\mfg}{\ensuremath{\mathfrak{g}}}
+\providecommand{\mfh}{\ensuremath{\mathfrak{h}}}
+\providecommand{\mfi}{\ensuremath{\mathfrak{i}}}
+\providecommand{\mfj}{\ensuremath{\mathfrak{j}}}
+\providecommand{\mfk}{\ensuremath{\mathfrak{k}}}
+\providecommand{\mfl}{\ensuremath{\mathfrak{l}}}
+\providecommand{\mfm}{\ensuremath{\mathfrak{m}}}
+\providecommand{\mfn}{\ensuremath{\mathfrak{n}}}
+\providecommand{\mfo}{\ensuremath{\mathfrak{o}}}
+\providecommand{\mfp}{\ensuremath{\mathfrak{p}}}
+\providecommand{\mfq}{\ensuremath{\mathfrak{q}}}
+\providecommand{\mfr}{\ensuremath{\mathfrak{r}}}
+\providecommand{\mfs}{\ensuremath{\mathfrak{s}}}
+\providecommand{\mft}{\ensuremath{\mathfrak{t}}}
+\providecommand{\mfu}{\ensuremath{\mathfrak{u}}}
+\providecommand{\mfv}{\ensuremath{\mathfrak{v}}}
+\providecommand{\mfw}{\ensuremath{\mathfrak{w}}}
+\providecommand{\mfx}{\ensuremath{\mathfrak{x}}}
+\providecommand{\mfy}{\ensuremath{\mathfrak{y}}}
+\providecommand{\mfz}{\ensuremath{\mathfrak{z}}}
+\providecommand{\mfA}{\ensuremath{\mathfrak{A}}}
+\providecommand{\mfB}{\ensuremath{\mathfrak{B}}}
+\providecommand{\mfC}{\ensuremath{\mathfrak{C}}}
+\providecommand{\mfD}{\ensuremath{\mathfrak{D}}}
+\providecommand{\mfE}{\ensuremath{\mathfrak{E}}}
+\providecommand{\mfF}{\ensuremath{\mathfrak{F}}}
+\providecommand{\mfG}{\ensuremath{\mathfrak{G}}}
+\providecommand{\mfH}{\ensuremath{\mathfrak{H}}}
+\providecommand{\mfI}{\ensuremath{\mathfrak{I}}}
+\providecommand{\mfJ}{\ensuremath{\mathfrak{J}}}
+\providecommand{\mfK}{\ensuremath{\mathfrak{K}}}
+\providecommand{\mfL}{\ensuremath{\mathfrak{L}}}
+\providecommand{\mfM}{\ensuremath{\mathfrak{M}}}
+\providecommand{\mfN}{\ensuremath{\mathfrak{N}}}
+\providecommand{\mfO}{\ensuremath{\mathfrak{O}}}
+\providecommand{\mfP}{\ensuremath{\mathfrak{P}}}
+\providecommand{\mfQ}{\ensuremath{\mathfrak{Q}}}
+\providecommand{\mfR}{\ensuremath{\mathfrak{R}}}
+\providecommand{\mfS}{\ensuremath{\mathfrak{S}}}
+\providecommand{\mfT}{\ensuremath{\mathfrak{T}}}
+\providecommand{\mfU}{\ensuremath{\mathfrak{U}}}
+\providecommand{\mfV}{\ensuremath{\mathfrak{V}}}
+\providecommand{\mfW}{\ensuremath{\mathfrak{W}}}
+\providecommand{\mfX}{\ensuremath{\mathfrak{X}}}
+\providecommand{\mfY}{\ensuremath{\mathfrak{Y}}}
+\providecommand{\mfZ}{\ensuremath{\mathfrak{Z}}}
+
+\providecommand{\frak}[1]{\ensuremath{\mathfrak{#1}}}
+\providecommand{\fraka}{\ensuremath{\mathfrak{a}}}
+\providecommand{\frakb}{\ensuremath{\mathfrak{b}}}
+\providecommand{\frakc}{\ensuremath{\mathfrak{c}}}
+\providecommand{\frakd}{\ensuremath{\mathfrak{d}}}
+\providecommand{\frake}{\ensuremath{\mathfrak{e}}}
+\providecommand{\frakf}{\ensuremath{\mathfrak{f}}}
+\providecommand{\frakg}{\ensuremath{\mathfrak{g}}}
+\providecommand{\frakh}{\ensuremath{\mathfrak{h}}}
+\providecommand{\fraki}{\ensuremath{\mathfrak{i}}}
+\providecommand{\frakj}{\ensuremath{\mathfrak{j}}}
+\providecommand{\frakk}{\ensuremath{\mathfrak{k}}}
+\providecommand{\frakl}{\ensuremath{\mathfrak{l}}}
+\providecommand{\frakm}{\ensuremath{\mathfrak{m}}}
+\providecommand{\frakn}{\ensuremath{\mathfrak{n}}}
+\providecommand{\frako}{\ensuremath{\mathfrak{o}}}
+\providecommand{\frakp}{\ensuremath{\mathfrak{p}}}
+\providecommand{\frakq}{\ensuremath{\mathfrak{q}}}
+\providecommand{\frakr}{\ensuremath{\mathfrak{r}}}
+\providecommand{\fraks}{\ensuremath{\mathfrak{s}}}
+\providecommand{\frakt}{\ensuremath{\mathfrak{t}}}
+\providecommand{\fraku}{\ensuremath{\mathfrak{u}}}
+\providecommand{\frakv}{\ensuremath{\mathfrak{v}}}
+\providecommand{\frakw}{\ensuremath{\mathfrak{w}}}
+\providecommand{\frakx}{\ensuremath{\mathfrak{x}}}
+\providecommand{\fraky}{\ensuremath{\mathfrak{y}}}
+\providecommand{\frakz}{\ensuremath{\mathfrak{z}}}
+\providecommand{\frakA}{\ensuremath{\mathfrak{A}}}
+\providecommand{\frakB}{\ensuremath{\mathfrak{B}}}
+\providecommand{\frakC}{\ensuremath{\mathfrak{C}}}
+\providecommand{\frakD}{\ensuremath{\mathfrak{D}}}
+\providecommand{\frakE}{\ensuremath{\mathfrak{E}}}
+\providecommand{\frakF}{\ensuremath{\mathfrak{F}}}
+\providecommand{\frakG}{\ensuremath{\mathfrak{G}}}
+\providecommand{\frakH}{\ensuremath{\mathfrak{H}}}
+\providecommand{\frakI}{\ensuremath{\mathfrak{I}}}
+\providecommand{\frakJ}{\ensuremath{\mathfrak{J}}}
+\providecommand{\frakK}{\ensuremath{\mathfrak{K}}}
+\providecommand{\frakL}{\ensuremath{\mathfrak{L}}}
+\providecommand{\frakM}{\ensuremath{\mathfrak{M}}}
+\providecommand{\frakN}{\ensuremath{\mathfrak{N}}}
+\providecommand{\frakO}{\ensuremath{\mathfrak{O}}}
+\providecommand{\frakP}{\ensuremath{\mathfrak{P}}}
+\providecommand{\frakQ}{\ensuremath{\mathfrak{Q}}}
+\providecommand{\frakR}{\ensuremath{\mathfrak{R}}}
+\providecommand{\frakS}{\ensuremath{\mathfrak{S}}}
+\providecommand{\frakT}{\ensuremath{\mathfrak{T}}}
+\providecommand{\frakU}{\ensuremath{\mathfrak{U}}}
+\providecommand{\frakV}{\ensuremath{\mathfrak{V}}}
+\providecommand{\frakW}{\ensuremath{\mathfrak{W}}}
+\providecommand{\frakX}{\ensuremath{\mathfrak{X}}}
+\providecommand{\frakY}{\ensuremath{\mathfrak{Y}}}
+\providecommand{\frakZ}{\ensuremath{\mathfrak{Z}}}
+
+\providecommand{\ms}[1]{\ensuremath{\mathscr{#1}}}
+\providecommand{\msA}{\ensuremath{\mathscr{A}}}
+\providecommand{\msB}{\ensuremath{\mathscr{B}}}
+\providecommand{\msC}{\ensuremath{\mathscr{C}}}
+\providecommand{\msD}{\ensuremath{\mathscr{D}}}
+\providecommand{\msE}{\ensuremath{\mathscr{E}}}
+\providecommand{\msF}{\ensuremath{\mathscr{F}}}
+\providecommand{\msG}{\ensuremath{\mathscr{G}}}
+\providecommand{\msH}{\ensuremath{\mathscr{H}}}
+\providecommand{\msI}{\ensuremath{\mathscr{I}}}
+\providecommand{\msJ}{\ensuremath{\mathscr{J}}}
+\providecommand{\msK}{\ensuremath{\mathscr{K}}}
+\providecommand{\msL}{\ensuremath{\mathscr{L}}}
+\providecommand{\msM}{\ensuremath{\mathscr{M}}}
+\providecommand{\msN}{\ensuremath{\mathscr{N}}}
+\providecommand{\msO}{\ensuremath{\mathscr{O}}}
+\providecommand{\msP}{\ensuremath{\mathscr{P}}}
+\providecommand{\msQ}{\ensuremath{\mathscr{Q}}}
+\providecommand{\msR}{\ensuremath{\mathscr{R}}}
+\providecommand{\msS}{\ensuremath{\mathscr{S}}}
+\providecommand{\msT}{\ensuremath{\mathscr{T}}}
+\providecommand{\msU}{\ensuremath{\mathscr{U}}}
+\providecommand{\msV}{\ensuremath{\mathscr{V}}}
+\providecommand{\msW}{\ensuremath{\mathscr{W}}}
+\providecommand{\msX}{\ensuremath{\mathscr{X}}}
+\providecommand{\msY}{\ensuremath{\mathscr{Y}}}
+\providecommand{\msZ}{\ensuremath{\mathscr{Z}}}
+
+\providecommand{\scr}[1]{\ensuremath{\mathscr{#1}}}
+\providecommand{\scrA}{\ensuremath{\mathscr{A}}}
+\providecommand{\scrB}{\ensuremath{\mathscr{B}}}
+\providecommand{\scrC}{\ensuremath{\mathscr{C}}}
+\providecommand{\scrD}{\ensuremath{\mathscr{D}}}
+\providecommand{\scrE}{\ensuremath{\mathscr{E}}}
+\providecommand{\scrF}{\ensuremath{\mathscr{F}}}
+\providecommand{\scrG}{\ensuremath{\mathscr{G}}}
+\providecommand{\scrH}{\ensuremath{\mathscr{H}}}
+\providecommand{\scrI}{\ensuremath{\mathscr{I}}}
+\providecommand{\scrJ}{\ensuremath{\mathscr{J}}}
+\providecommand{\scrK}{\ensuremath{\mathscr{K}}}
+\providecommand{\scrL}{\ensuremath{\mathscr{L}}}
+\providecommand{\scrM}{\ensuremath{\mathscr{M}}}
+\providecommand{\scrN}{\ensuremath{\mathscr{N}}}
+\providecommand{\scrO}{\ensuremath{\mathscr{O}}}
+\providecommand{\scrP}{\ensuremath{\mathscr{P}}}
+\providecommand{\scrQ}{\ensuremath{\mathscr{Q}}}
+\providecommand{\scrR}{\ensuremath{\mathscr{R}}}
+\providecommand{\scrS}{\ensuremath{\mathscr{S}}}
+\providecommand{\scrT}{\ensuremath{\mathscr{T}}}
+\providecommand{\scrU}{\ensuremath{\mathscr{U}}}
+\providecommand{\scrV}{\ensuremath{\mathscr{V}}}
+\providecommand{\scrW}{\ensuremath{\mathscr{W}}}
+\providecommand{\scrX}{\ensuremath{\mathscr{X}}}
+\providecommand{\scrY}{\ensuremath{\mathscr{Y}}}
+\providecommand{\scrZ}{\ensuremath{\mathscr{Z}}}
+
+%%================================
+%% General symbols
+%%================================
+\providecommand{\abs}[1]{\lvert#1\rvert}
+\providecommand{\norm}[1]{\lVert#1\rVert}
+
+\providecommand*{\injection}{\ensuremath{\hookrightarrow}}
+\providecommand*{\surjection}{\ensuremath{\twoheadrightarrow}}
+\providecommand*{\bijection}{\ensuremath{\xrightarrow{\sim}}}
+
+%%================================
+%% Symbols for number theory
+%%================================
+\ifdefined\@@pmod\else
+\let\@@pmod\pmod
+\DeclareRobustCommand{\pmod}{\@ifstar\@pmods\@@pmod}
+\def\@pmods#1{\mkern4mu({\operator@font mod}\mkern 6mu#1)}
+\fi
+
+\providecommand{\genlegendre}[4]{%
+ \genfrac{(}{)}{}{#1}{#3}{#4}%
+ \if\relax\detokenize{#2}\relax\else_{\!#2}\fi%
+}
+\providecommand{\legendre}[3][]{\genlegendre{}{#1}{#2}{#3}}
+\providecommand{\dlegendre}[3][]{\genlegendre{0}{#1}{#2}{#3}}
+\providecommand{\tlegendre}[3][]{\genlegendre{1}{#1}{#2}{#3}}
+
+\endinput
+%%
+%% End of file `PJLmath.sty'.