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aeg-lemma.tex
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aeg-lemma.tex
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\documentclass{article}
\usepackage{ax/arxiv}
\usepackage{styles/quiver}
\usepackage[utf8]{inputenc} % allow utf-8 input
\usepackage[T1]{fontenc} % use 8-bit T1 fonts
\usepackage{hyperref} % hyperlinks
\usepackage{url} % simple URL typesetting
\usepackage{booktabs} % professional-quality tables
\usepackage[english]{babel}
\usepackage{nicefrac} % compact symbols for 1/2, etc.
\usepackage{microtype} % microtypography
\usepackage{graphicx}
\usepackage{stmaryrd}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{mathtools}
\usepackage{tikz}
\usetikzlibrary{angles,fit,arrows,calc,math,intersections,through,backgrounds}
\usepackage{qtree}
\usepackage{listings}
\lstset{
basicstyle=\itshape,
xleftmargin=3em,
literate={->}{$\rightarrow$}{2}
{α}{$\alpha$}{1}
{δ}{$\delta$}{1}
}
\usepackage{xstring}
\usepackage{stmaryrd}
\usepackage{wasysym}
\usepackage{textcomp}
\usepackage{blindtext}
\usepackage{subfiles}
\newtheorem{definition}{Definition}
\numberwithin{definition}{section}
\newtheorem{lemma}{Lemma}
\numberwithin{lemma}{section}
\newtheorem{proposition}{Proposition}
\numberwithin{proposition}{section}
\newtheorem{corollary}{Corollary}
\numberwithin{corollary}{section}
\newtheorem{theorem}{Theorem}
\numberwithin{theorem}{section}
\DeclareMathSymbol{\mathinvertedexclamationmark}{\mathclose}{operators}{'074}
\DeclareMathSymbol{\mathexclamationmark}{\mathclose}{operators}{'041}
\makeatletter
\newcommand{\raisedmathinvertedexclamationmark}{%
\mathclose{\mathpalette\raised@mathinvertedexclamationmark\relax}%
}
\newcommand{\raised@mathinvertedexclamationmark}[2]{%
\raisebox{\depth}{$\m@th#1\mathinvertedexclamationmark$}%
}
\begingroup\lccode`~=`! \lowercase{\endgroup
\def~}{\@ifnextchar`{\raisedmathinvertedexclamationmark\@gobble}{\mathexclamationmark}}
\mathcode`!="8000
\makeatother
\newcommand{\intga}{\mathclap{\smash{\oplus}}{\int}}
\newcommand{\intgp}{\mathclap{\smash{\otimes}}{\int}}
\newcommand{\intgg}{\mathclap{\rightsquigarrow}\mathclap{\int}}
\DeclareMathOperator{\arcsinh}{arcsinh}
\title{Lemmas on arithmetic expression geometry}
%\date{Decmber 8, 2022} % Here you can change the date presented in the paper title
%\date{} % Or removing it
\author{
Mingli Yuan \\
Swarma Research\\
Beijing, 100083 \\
\texttt{[email protected]}
}
% Uncomment to remove the date
%\date{}
% Uncomment to override the `A preprint' in the header
\renewcommand{\headeright}{A preprint}
\renewcommand{\undertitle}{A preprint}
\begin{document}
\maketitle
\begin{abstract}
TODO
\end{abstract}
\keywords{arithmetic expressions, hyperbolic geometry}
\setcounter{tocdepth}{2}
\tableofcontents
\newpage
\section{Introduction}\label{sec:introduction}
\input{sections/sec01.tex}
\newpage
\section{Basic concepts}\label{sec:concepts}
\input{sections/sec02-00.tex}
\input{sections/sec02-01.tex}
\newpage
\section{Flow equation}\label{sec:flowequation}
\input{sections/sec03.tex}
\newpage
\section{The first kind arithmetic expression space $\mathfrak{E}_1$}\label{sec:firstkind}
\input{sections/sec04.tex}
\newpage
\section{Fundamental domains in $\mathfrak{E}_1$}\label{sec:fundamentaldomains}
\input{sections/sec05.tex}
\newpage
\section{On order-4 apeirogonal tiling}\label{sec:tiling}
\input{sections/sec06.tex}
\newpage
\section{The second and third kind of arithmetic expression space $\mathfrak{K}_2$ and $\mathfrak{K}_3$}\label{sec:morekinds}
\input{sections/sec07.tex}
\newpage
\section{Topological arithmetic expression space}\label{sec:topology}
\input{sections/sec08.tex}
\newpage
\section{From arithmetic torsion to curvature}\label{sec:curvature}
\input{sections/sec09.tex}
\newpage
\section{Tube structure, complexification and fibration}\label{sec:morekinds}
\input{sections/sec10.tex}
\newpage
\section{General discussion}\label{sec:function}
\input{sections/sec11.tex}
\newpage
\section{A glossary of unsolved problems}\label{sec:problems}
\input{sections/sec12.tex}
\newpage
\bibliographystyle{plain}
\bibliography{aeg-paper.bib}
\appendix
\section{Solution of the flow equation}\label{sec:directformalsolution}
\input{sections/sec13.tex}
\newpage
\section{Infinitesimal-discrete conformance}\label{sec:conformance}
\input{sections/sec14.tex}
\newpage
\section{Geometry calculation}\label{sec:geometrycalculationn}
\input{sections/sec15.tex}
\section{Grid calculation}\label{sec:gridcalculation}
\input{sections/sec16.tex}
\section{Arithmetic expression, combinators and transformation over trees}\label{sec:expressions}
\input{sections/sec17.tex}
\newpage
\end{document}