From 94e6bac00ac674b8722e3b80ece83a94a94d0f45 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Thu, 13 Jan 2022 12:46:06 +0100
Subject: =?UTF-8?q?Wst=C4=99pny=20szkielet=20pracy.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 lic_malinka.tex | 208 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 208 insertions(+)
 create mode 100644 lic_malinka.tex

(limited to 'lic_malinka.tex')

diff --git a/lic_malinka.tex b/lic_malinka.tex
new file mode 100644
index 0000000..7abd37d
--- /dev/null
+++ b/lic_malinka.tex
@@ -0,0 +1,208 @@
+\documentclass[11pt, a4paper, final]{amsart}
+\setlength{\emergencystretch}{2em}
+
+
+\usepackage[T1]{fontenc}
+\usepackage{mathtools}
+\usepackage[activate={true,nocompatibility},final,tracking=true,kerning=true,spacing=true,stretch=10,shrink=10]{microtype}
+\microtypecontext{spacing=nonfrench}
+\usepackage[utf8]{inputenc}
+
+\usepackage{amsfonts}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{amsthm}
+\usepackage{XCharter}
+\usepackage[charter,expert, greekuppercase=italicized, greekfamily=didot]{mathdesign}
+\usepackage{mathtools}
+\usepackage{enumitem}
+\usepackage[utf8]{inputenc}
+\usepackage{tikz-cd}
+\usepackage{tikz}
+
+\usepackage{etoolbox}
+
+
+\usepackage{xcolor}
+\definecolor{green}{RGB}{0,127,0}
+\definecolor{redd}{RGB}{191,0,0}
+\definecolor{red}{RGB}{105,89,205}
+\usepackage[colorlinks=true]{hyperref}
+
+\usepackage[notref, notcite]{showkeys}
+\usepackage[cmtip,arrow]{xy}
+%\usepackage[backend=biber,
+%url=false,
+%isbn=false,
+%backref=true,
+%citestyle=alphabetic,
+%bibstyle=alphabetic,
+%autocite=inline,
+%maxnames=99,
+%minalphanames=4,
+%maxalphanames=4,
+%sorting=nyt,]{biblatex}
+%\addbibresource{linear_strucures.bib}
+
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\Stab}{Stab}
+\DeclareMathOperator{\st}{st}
+\DeclareMathOperator{\Flim}{FLim}
+
+\newcommand{\cupdot}{\mathbin{\mathaccent\cdot\cup}}
+\newcommand{\cC}{\mathcal C}
+\newcommand{\bN}{\mathbf N}
+\newcommand{\bR}{\mathbf R}
+\newcommand{\bZ}{\mathbf Z}
+
+\DeclareMathOperator{\im}{{Im}}
+\DeclareMathOperator{\lin}{{Lin}}
+\DeclareMathOperator{\Th}{{Th}}
+
+
+\newtheorem{theorem}{Theorem}
+\numberwithin{theorem}{section}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{claim}[theorem]{Claim}
+\newtheorem{fact}[theorem]{Fact}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{conjecture}[theorem]{Conjecture}
+\newtheorem{axiom}[theorem]{Axiom}
+\newtheorem{question}[theorem]{Question}
+\newtheorem{corollary}[theorem]{Corollary}
+\newtheorem*{theorem2}{Theorem}
+\newtheorem*{claim2}{Claim}
+\newtheorem*{corollary2}{Corollary}
+\newtheorem*{question2}{Question}
+\newtheorem*{conjecture2}{Conjecture}
+
+
+\newtheorem{clm}{Claim}
+\newtheorem*{clm*}{Claim}
+
+
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem*{definition2}{Definition}
+\newtheorem{example}[theorem]{Example}
+
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newtheorem*{remark2}{Remark}
+
+
+\AtEndEnvironment{proof}{\setcounter{clm}{0}}
+\newenvironment{clmproof}[1][\proofname]{\proof[#1]\renewcommand{\qedsymbol}{$\square$(claim)}}{\endproof}
+
+\newcommand{\xqed}[1]{%
+    \leavevmode\unskip\penalty9999 \hbox{}\nobreak\hfill
+    \quad\hbox{\ensuremath{#1}}}
+
+
+\title{Tytuł}
+\author{Franciszek Malinka}
+
+\begin{document}
+
+    \begin{abstract}
+        Abstract
+    \end{abstract}
+    \section{Introduction}
+
+    \section{Preliminaries}
+    \subsection{Descriptive set theory}
+    \begin{definition}
+        Suppose $X$ is a topological space and $A\subseteq X$. We say that $A$ is \emph{meagre} in $X$ if... Wea say that $A$ is \emph{comeagre} in $X$ if... .
+    \end{definition}
+
+    \begin{definition}
+        We say that a topological space $X$ is a \emph{Baire space} if every comeagre subset of $X$ is dense in $X$ (equivalently, every meagre set has empty interior).
+    \end{definition}
+
+    \begin{definition}
+        Suppose $X$ is a Baire space. We say that a property $P$ holds generically for a point in $x\in X$ if $\{x\in X\mid P\textrm{ holds for }x\}$ is comeagre in $X$.
+    \end{definition}
+
+    \begin{example}
+        content
+    \end{example}
+
+    \subsection{Fraïssé classes}
+    \begin{fact}[Fraïssé theorem]
+        \label{fact:fraisse_thm}
+        Suppose $\cC$ is a class of finitely generated $L$-structures such that...
+
+        Then there exists a unique up to isomorphism counable $L$-structure $M$ such that...
+    \end{fact}
+
+
+    \begin{definition}
+        For $\cC$, $M$ as in Fact~\ref{fact:fraisse_thm}, we write $\Flim(\cC)\coloneqq M$.
+    \end{definition}
+
+    \begin{fact}
+        If $\cC$ is a uniformly locally finite Fraïssé class, then $\Flim(\cC)$ is $\aleph_0$-categorical and has quantifier elimination.
+    \end{fact}
+
+    \section{Conjugacy classes in automorphism groups}
+
+    \subsection{Prototype: pure set}
+    In this section, $M=(M,=)$ is an infinite countable set (with no structure beyond equality).
+    \begin{proposition}
+        If $f_1,f_2\in \Aut(M)$, then $f_1$ and $f_2$ are conjugate if and only if for each $n\in \bN\cup \{\aleph_0\}$, $f_1$ and $f_2$ have the same number of orbits of size $n$.
+    \end{proposition}
+
+    \begin{proposition}
+        The conjugacy class of $f\in \Aut(M)$ is dense if and only if...
+    \end{proposition}
+    \begin{proposition}
+        If $f\in \Aut(M)$ has an infinite orbit, then the conjugacy class of $f$ is meagre.
+    \end{proposition}
+
+    \begin{proposition}
+        An automorphism $f$ of $M$ is generic if and only if...
+    \end{proposition}
+
+    \begin{proof}
+
+    \end{proof}
+
+    \subsection{More general structures}
+
+
+    \begin{proposition}
+        Suppose $M$ is an arbitrary structure and $f_1,f_2\in \Aut(M)$. Then $f_1$ and $f_2$ are conjugate if and only if $(M,f_1)\cong (M,f_2)$.
+    \end{proposition}
+
+    \begin{definition}
+        We say that a Fraïssé class $\cC$ has \emph{weak Hrushovski property} (\emph{WHP}) if for every $A\in \cC$ and partial automorphism $p\colon A\to A$, there is some $B\in \cC$ such that $p$ can be extended to an automorphism of $B$, i.e.\ there is an embedding $i\colon A\to B$ and a $\bar p\in \Aut(B)$ such that the following diagram commutes:
+        \begin{center}
+            \begin{tikzcd}
+                B\ar[r,"\bar p"]&B\\
+                A\ar[u,"i"]\ar[r,"p"]&A\ar[u,"i"]
+            \end{tikzcd}
+        \end{center}
+    \end{definition}
+
+    \begin{proposition}
+        Suppose $\cC$ is a Fraïssé class in a relational language with WHP. Then generically, for an $f\in \Aut(\Flim(\cC))$, all orbits of $f$ are finite.
+    \end{proposition}
+    \begin{proposition}
+        Suppose $\cC$ is a Fraïssé class in an arbitrary countable language with WHP. Then generically, for an $f\in \Aut(\Flim(\cC))$ ...
+    \end{proposition}
+
+    \subsection{Random graph}
+    \begin{definition}
+        The \emph{random graph} is...
+    \end{definition}
+
+    \begin{fact}
+        The
+    \end{fact}
+
+    \begin{proposition}
+        Generically, the set of fixed points of $f\in \Aut(M)$ is isomorphic to $M$ (as a graph).
+    \end{proposition}
+
+\end{document}
-- 
cgit v1.2.3

