378 lines
11 KiB
TeX
378 lines
11 KiB
TeX
%
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% PACKAGES
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%
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% Standard Packages
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\usepackage{babel}
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\usepackage{enumitem}
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\usepackage{hyperref}
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\usepackage[utf8]{inputenc}
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\usepackage{newunicodechar}
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\usepackage{mathtools}
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\usepackage{varioref}
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\usepackage[arrow,curve,matrix]{xy}
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% Graphics Packages
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\usepackage{colortbl}
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\usepackage{graphicx}
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\usepackage{tikz}
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% Font packages
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\usepackage{mathrsfs}
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%
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% GENERAL TYPESETTING
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%
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% Colours for hyperlinks
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\definecolor{linkred}{rgb}{0.7,0.2,0.2}
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\definecolor{linkblue}{rgb}{0,0.2,0.6}
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% Limit table of contents to section titles
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\setcounter{tocdepth}{1}
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% Numbering of figures (see below for numbering of equations)
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\numberwithin{figure}{section}
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% Add an uparrow to the bibliography entries, just before the back-list of references
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\usepackage[hyperpageref]{backref}
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\renewcommand{\backref}[1]{$\uparrow$~#1}
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% Numbering of parts in roman numbers
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\renewcommand\thepart{\rm \Roman{part}}
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% Sloppy formatting -- often looks better
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\sloppy
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% Changes the layout of descriptions and itemized lists. The indent specified in
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% the original amsart style is too much for my taste.
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\setdescription{labelindent=\parindent, leftmargin=2\parindent}
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\setitemize[1]{labelindent=\parindent, leftmargin=2\parindent}
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\setenumerate[1]{labelindent=0cm, leftmargin=*, widest=iiii}
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%
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% Input characters
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%
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\newunicodechar{א}{\ensuremath{\aleph}}
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\newunicodechar{α}{\ensuremath{\alpha}}
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\newunicodechar{β}{\ensuremath{\beta}}
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\newunicodechar{χ}{\ensuremath{\chi}}
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\newunicodechar{δ}{\ensuremath{\delta}}
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\newunicodechar{ε}{\ensuremath{\varepsilon}}
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\newunicodechar{Δ}{\ensuremath{\Delta}}
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\newunicodechar{η}{\ensuremath{\eta}}
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\newunicodechar{γ}{\ensuremath{\gamma}}
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\newunicodechar{Γ}{\ensuremath{\Gamma}}
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\newunicodechar{ι}{\ensuremath{\iota}}
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\newunicodechar{κ}{\ensuremath{\kappa}}
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\newunicodechar{λ}{\ensuremath{\lambda}}
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\newunicodechar{Λ}{\ensuremath{\Lambda}}
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\newunicodechar{ν}{\ensuremath{\nu}}
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\newunicodechar{μ}{\ensuremath{\mu}}
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\newunicodechar{ω}{\ensuremath{\omega}}
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\newunicodechar{Ω}{\ensuremath{\Omega}}
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\newunicodechar{π}{\ensuremath{\pi}}
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\newunicodechar{Π}{\ensuremath{\Pi}}
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\newunicodechar{φ}{\ensuremath{\phi}}
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\newunicodechar{Φ}{\ensuremath{\Phi}}
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\newunicodechar{ψ}{\ensuremath{\psi}}
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\newunicodechar{Ψ}{\ensuremath{\Psi}}
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\newunicodechar{ρ}{\ensuremath{\rho}}
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\newunicodechar{σ}{\ensuremath{\sigma}}
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\newunicodechar{Σ}{\ensuremath{\Sigma}}
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\newunicodechar{τ}{\ensuremath{\tau}}
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\newunicodechar{θ}{\ensuremath{\theta}}
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\newunicodechar{Θ}{\ensuremath{\Theta}}
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\newunicodechar{ξ}{\ensuremath{\xi}}
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\newunicodechar{Ξ}{\ensuremath{\Xi}}
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\newunicodechar{ζ}{\ensuremath{\zeta}}
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\newunicodechar{ℓ}{\ensuremath{\ell}}
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\newunicodechar{ï}{\"{\i}}
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\newunicodechar{𝔸}{\ensuremath{\bA}}
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\newunicodechar{𝔹}{\ensuremath{\bB}}
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\newunicodechar{ℂ}{\ensuremath{\bC}}
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\newunicodechar{𝔻}{\ensuremath{\bD}}
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\newunicodechar{𝔼}{\ensuremath{\bE}}
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\newunicodechar{𝔽}{\ensuremath{\bF}}
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\newunicodechar{𝔾}{\ensuremath{\bG}}
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\newunicodechar{ℕ}{\ensuremath{\bN}}
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\newunicodechar{ℙ}{\ensuremath{\bP}}
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\newunicodechar{ℚ}{\ensuremath{\bQ}}
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\newunicodechar{ℝ}{\ensuremath{\bR}}
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\newunicodechar{𝕏}{\ensuremath{\bX}}
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\newunicodechar{ℤ}{\ensuremath{\bZ}}
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\newunicodechar{𝒜}{\ensuremath{\sA}}
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\newunicodechar{ℬ}{\ensuremath{\sB}}
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\newunicodechar{𝒞}{\ensuremath{\sC}}
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\newunicodechar{𝒟}{\ensuremath{\sD}}
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\newunicodechar{ℰ}{\ensuremath{\sE}}
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\newunicodechar{ℱ}{\ensuremath{\sF}}
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\newunicodechar{𝒢}{\ensuremath{\sG}}
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\newunicodechar{ℋ}{\ensuremath{\sH}}
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\newunicodechar{𝒥}{\ensuremath{\sJ}}
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\newunicodechar{ℒ}{\ensuremath{\sL}}
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\newunicodechar{ℳ}{\ensuremath{\sM}}
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\newunicodechar{𝒪}{\ensuremath{\sO}}
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\newunicodechar{𝒬}{\ensuremath{\sQ}}
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\newunicodechar{𝒮}{\ensuremath{\sS}}
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\newunicodechar{𝒯}{\ensuremath{\sT}}
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\newunicodechar{𝒲}{\ensuremath{\sW}}
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\newunicodechar{∂}{\ensuremath{\partial}}
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\newunicodechar{∇}{\ensuremath{\nabla}}
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\newunicodechar{↺}{\ensuremath{\circlearrowleft}}
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\newunicodechar{∞}{\ensuremath{\infty}}
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\newunicodechar{⊕}{\ensuremath{\oplus}}
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\newunicodechar{⊗}{\ensuremath{\otimes}}
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\newunicodechar{•}{\ensuremath{\bullet}}
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\newunicodechar{Λ}{\ensuremath{\wedge}}
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\newunicodechar{↪}{\ensuremath{\into}}
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\newunicodechar{→}{\ensuremath{\to}}
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\newunicodechar{↦}{\ensuremath{\mapsto}}
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\newunicodechar{⨯}{\ensuremath{\times}}
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\newunicodechar{∪}{\ensuremath{\cup}}
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\newunicodechar{∩}{\ensuremath{\cap}}
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\newunicodechar{⊋}{\ensuremath{\supsetneq}}
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\newunicodechar{⊇}{\ensuremath{\supseteq}}
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\newunicodechar{⊃}{\ensuremath{\supset}}
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\newunicodechar{⊊}{\ensuremath{\subsetneq}}
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\newunicodechar{⊆}{\ensuremath{\subseteq}}
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\newunicodechar{⊂}{\ensuremath{\subset}}
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\newunicodechar{⊄}{\ensuremath{\not \subset}}
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\newunicodechar{≥}{\ensuremath{\geq}}
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\newunicodechar{≠}{\ensuremath{\neq}}
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\newunicodechar{≫}{\ensuremath{\gg}}
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\newunicodechar{≪}{\ensuremath{\ll}}
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\newunicodechar{≤}{\ensuremath{\leq}}
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\newunicodechar{∈}{\ensuremath{\in}}
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\newunicodechar{∉}{\ensuremath{\not \in}}
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\newunicodechar{∖}{\ensuremath{\setminus}}
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\newunicodechar{◦}{\ensuremath{\circ}}
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\newunicodechar{°}{\ensuremath{^\circ}}
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\newunicodechar{…}{\ifmmode\mathellipsis\else\textellipsis\fi}
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\newunicodechar{·}{\ensuremath{\cdot}}
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\newunicodechar{⋯}{\ensuremath{\cdots}}
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\newunicodechar{∅}{\ensuremath{\emptyset}}
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\newunicodechar{⇒}{\ensuremath{\Rightarrow}}
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\newunicodechar{⁰}{\ensuremath{^0}}
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\newunicodechar{¹}{\ensuremath{^1}}
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\newunicodechar{²}{\ensuremath{^2}}
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\newunicodechar{³}{\ensuremath{^3}}
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\newunicodechar{⁴}{\ensuremath{^4}}
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\newunicodechar{⁵}{\ensuremath{^5}}
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\newunicodechar{⁶}{\ensuremath{^6}}
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\newunicodechar{⁷}{\ensuremath{^7}}
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\newunicodechar{⁸}{\ensuremath{^8}}
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\newunicodechar{⁹}{\ensuremath{^9}}
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\newunicodechar{ⁱ}{\ensuremath{^i}}
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\newunicodechar{⌈}{\ensuremath{\lceil}}
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\newunicodechar{⌉}{\ensuremath{\rceil}}
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\newunicodechar{⌊}{\ensuremath{\lfloor}}
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\newunicodechar{⌋}{\ensuremath{\rfloor}}
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\newunicodechar{≅}{\ensuremath{\cong}}
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\newunicodechar{⇔}{\ensuremath{\Leftrightarrow}}
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\newunicodechar{∃}{\ensuremath{\exists}}
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\newunicodechar{±}{\ensuremath{\pm}}
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%
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% FONT DEFINTIONS
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%
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% Script Font used for sheaves
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\DeclareFontFamily{OMS}{rsfs}{\skewchar\font'60}
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\DeclareFontShape{OMS}{rsfs}{m}{n}{<-5>rsfs5 <5-7>rsfs7 <7->rsfs10 }{}
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\DeclareSymbolFont{rsfs}{OMS}{rsfs}{m}{n}
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\DeclareSymbolFontAlphabet{\scr}{rsfs}
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\DeclareSymbolFontAlphabet{\scr}{rsfs}
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% Code from mathabx.sty and mathabx.dcl, define macro \wcheck
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\DeclareFontFamily{U}{mathx}{\hyphenchar\font45}
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\DeclareFontShape{U}{mathx}{m}{n}{
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<5> <6> <7> <8> <9> <10>
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<10.95> <12> <14.4> <17.28> <20.74> <24.88>
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mathx10
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}{}
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\DeclareSymbolFont{mathx}{U}{mathx}{m}{n}
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\DeclareFontSubstitution{U}{mathx}{m}{n}
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\DeclareMathAccent{\wcheck}{0}{mathx}{"71}
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%
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% MATHEMATICS DEFINITIONS
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%
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% Operators
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\DeclareMathOperator{\Aut}{Aut}
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\DeclareMathOperator{\codim}{codim}
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\DeclareMathOperator{\coker}{coker}
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\DeclareMathOperator{\const}{const}
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\DeclareMathOperator{\Ext}{Ext}
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\DeclareMathOperator{\Hom}{Hom}
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\DeclareMathOperator{\Id}{Id}
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\DeclareMathOperator{\Image}{Image}
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\DeclareMathOperator{\img}{img}
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\DeclareMathOperator{\Pic}{Pic}
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\DeclareMathOperator{\rank}{rank}
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\DeclareMathOperator{\Ramification}{Ramification}
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\DeclareMathOperator{\red}{red}
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\DeclareMathOperator{\reg}{reg}
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\DeclareMathOperator{\sat}{sat}
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\DeclareMathOperator{\sEnd}{\sE\negthinspace \mathit{nd}}
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\DeclareMathOperator{\sing}{sing}
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\DeclareMathOperator{\Spec}{Spec}
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\DeclareMathOperator{\Sym}{Sym}
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\DeclareMathOperator{\supp}{supp}
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\DeclareMathOperator{\tor}{tor}
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\DeclareMathOperator{\Tor}{Tor}
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\DeclareMathOperator{\Frob}{Frob}
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% Sheaves
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\newcommand{\sA}{\scr{A}}
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\newcommand{\sB}{\scr{B}}
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\newcommand{\sC}{\scr{C}}
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\newcommand{\sD}{\scr{D}}
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\newcommand{\sE}{\scr{E}}
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\newcommand{\sF}{\scr{F}}
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\newcommand{\sG}{\scr{G}}
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\newcommand{\sH}{\scr{H}}
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\newcommand{\sHom}{\scr{H}\negthinspace om}
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\newcommand{\sI}{\scr{I}}
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\newcommand{\sJ}{\scr{J}}
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\newcommand{\sK}{\scr{K}}
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\newcommand{\sL}{\scr{L}}
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\newcommand{\sM}{\scr{M}}
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\newcommand{\sN}{\scr{N}}
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\newcommand{\sO}{\scr{O}}
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\newcommand{\sP}{\scr{P}}
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\newcommand{\sQ}{\scr{Q}}
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\newcommand{\sR}{\scr{R}}
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\newcommand{\sS}{\scr{S}}
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\newcommand{\sT}{\scr{T}}
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\newcommand{\sU}{\scr{U}}
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\newcommand{\sV}{\scr{V}}
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\newcommand{\sW}{\scr{W}}
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\newcommand{\sX}{\scr{X}}
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\newcommand{\sY}{\scr{Y}}
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\newcommand{\sZ}{\scr{Z}}
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% C-infty sheaves
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\newcommand{\cA}{\mathcal A}
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\newcommand{\cC}{\mathcal C}
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\newcommand{\cD}{\mathcal D}
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\newcommand{\cE}{\mathcal E}
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\newcommand{\cM}{\mathcal M}
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\newcommand{\cN}{\mathcal N}
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\newcommand{\cV}{\mathcal V}
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% Blackboard Bold Symbols
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\newcommand{\bA}{\mathbb{A}}
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\newcommand{\bB}{\mathbb{B}}
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\newcommand{\bC}{\mathbb{C}}
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\newcommand{\bD}{\mathbb{D}}
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\newcommand{\bE}{\mathbb{E}}
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\newcommand{\bF}{\mathbb{F}}
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\newcommand{\bG}{\mathbb{G}}
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\newcommand{\bH}{\mathbb{H}}
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\newcommand{\bI}{\mathbb{I}}
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\newcommand{\bJ}{\mathbb{J}}
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\newcommand{\bK}{\mathbb{K}}
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\newcommand{\bL}{\mathbb{L}}
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\newcommand{\bM}{\mathbb{M}}
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\newcommand{\bN}{\mathbb{N}}
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\newcommand{\bO}{\mathbb{O}}
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\newcommand{\bP}{\mathbb{P}}
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\newcommand{\bQ}{\mathbb{Q}}
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\newcommand{\bR}{\mathbb{R}}
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\newcommand{\bS}{\mathbb{S}}
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\newcommand{\bT}{\mathbb{T}}
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\newcommand{\bU}{\mathbb{U}}
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\newcommand{\bV}{\mathbb{V}}
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\newcommand{\bW}{\mathbb{W}}
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\newcommand{\bX}{\mathbb{X}}
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\newcommand{\bY}{\mathbb{Y}}
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\newcommand{\bZ}{\mathbb{Z}}
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% Sans serif symbols
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\newcommand{\aB}{{\sf B}}
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\newcommand{\aD}{{\sf D}}
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\newcommand{\aE}{{\sf E}}
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\newcommand{\aF}{{\sf F}}
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% Theorem type environments
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\theoremstyle{plain}
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\newtheorem{thm}{Theorem}[section]
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\newtheorem{aassumption}[thm]{Additional Assumption}
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\newtheorem{conjecture}[thm]{Conjecture}
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\newtheorem{cor}[thm]{Corollary}
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\newtheorem{defn}[thm]{Definition}
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\newtheorem{fact}[thm]{Fact}
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\newtheorem{lem}[thm]{Lemma}
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\newtheorem{lemDef}[thm]{Lemma and Definition}
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\newtheorem{lemNot}[thm]{Lemma and Notation}
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\newtheorem{problem}[thm]{Problem}
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\newtheorem{prop}[thm]{Proposition}
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\newtheorem{setup}[thm]{Setup}
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\newtheorem{subthm}[thm]{Sub-Theorem}
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\newtheorem{summary}[thm]{Summary}
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\theoremstyle{remark}
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\newtheorem{assumption}[thm]{Assumption}
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\newtheorem{asswlog}[thm]{Assumption w.l.o.g.}
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\newtheorem{claim}[thm]{Claim}
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\newtheorem{c-n-d}[thm]{Claim and Definition}
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\newtheorem{consequence}[thm]{Consequence}
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\newtheorem{construction}[thm]{Construction}
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\newtheorem{computation}[thm]{Computation}
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\newtheorem{example}[thm]{Example}
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\newtheorem{explanation}[thm]{Explanation}
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\newtheorem{notation}[thm]{Notation}
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\newtheorem{obs}[thm]{Observation}
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\newtheorem{rem}[thm]{Remark}
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\newtheorem{question}[thm]{Question}
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\newtheorem*{rem-nonumber}{Remark}
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\newtheorem{setting}[thm]{Setting}
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\newtheorem{warning}[thm]{Warning}
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% Numbering of equations. Number equation subordniate to theorems.
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\numberwithin{equation}{thm}
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% Style for enumerated lists. The following makes sure that enumerated lists are
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% numbered in the same way as equations are.
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\setlist[enumerate]{label=(\thethm.\arabic*), before={\setcounter{enumi}{\value{equation}}}, after={\setcounter{equation}{\value{enumi}}}}
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% Shorthand notations
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\newcommand{\into}{\hookrightarrow}
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\newcommand{\onto}{\twoheadrightarrow}
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\newcommand{\wtilde}{\widetilde}
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\newcommand{\what}{\widehat}
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%
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% HYPENTATION
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%
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\hyphenation{com-po-nents}
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\hyphenation{pos-i-tive}
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\hyphenation{Theo-rem}
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\hyphenation{Vojta}
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%
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% SPECIALIZED MACROS
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%
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% CounterStep - increases equation counter
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\newcommand\CounterStep{\addtocounter{thm}{1}\setcounter{equation}{0}}
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% factor - quotient groups
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\newcommand{\factor}[2]{\left. \raise 2pt\hbox{$#1$} \right/\hskip -2pt\raise -2pt\hbox{$#2$}}
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