From 458dd9ab1fad7a2628a6b5d6bd0462827d00a402 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Fri, 9 Feb 2024 20:23:05 +0100 Subject: [PATCH] fixed some typesetting problems --- bibliography/references.bib | 2 - inputs/lecture_10.tex | 2 +- inputs/lecture_13.tex | 1 + inputs/lecture_14.tex | 3 +- inputs/lecture_20.tex | 4 +- inputs/lecture_22.tex | 12 ++-- inputs/lecture_26.tex | 112 ++++++++++++++++++++++++++++++++++-- inputs/tutorial_04.tex | 15 +---- inputs/tutorial_12b.tex | 2 + 9 files changed, 122 insertions(+), 31 deletions(-) diff --git a/bibliography/references.bib b/bibliography/references.bib index f542bda..7d2ec24 100644 --- a/bibliography/references.bib +++ b/bibliography/references.bib @@ -48,8 +48,6 @@ year = {2012}, TITLE = {Embedding of countable linear orders into $\Bbb Q$ as topological spaces}, AUTHOR = {Eric Wofsey}, HOWPUBLISHED = {Mathematics Stack Exchange}, - NOTE = {URL:https://math.stackexchange.com/q/3722713 (version: 2020-06-16)}, - EPRINT = {https://math.stackexchange.com/q/3722713}, URL = {https://math.stackexchange.com/q/3722713} } diff --git a/inputs/lecture_10.tex b/inputs/lecture_10.tex index 514ee1f..4136e55 100644 --- a/inputs/lecture_10.tex +++ b/inputs/lecture_10.tex @@ -1,6 +1,6 @@ \subsection{The Lusin Separation Theorem} - \lecture{10}{2023-11-17}{} + \begin{theorem}[\vocab{Lusin separation theorem}] \yalabel{Lusin Separation Theorem}{Lusin Separation}{thm:lusinseparation} Let $X$ be Polish and $A,B \subseteq X$ disjoint analytic. diff --git a/inputs/lecture_13.tex b/inputs/lecture_13.tex index 51494d2..38b3180 100644 --- a/inputs/lecture_13.tex +++ b/inputs/lecture_13.tex @@ -143,6 +143,7 @@ with $(f^{-1}(\{1\}), <)$. \end{itemize} \end{definition} \begin{remark} + \leavevmode \begin{itemize} \item A prewellordering may not be a linear order since it is not necessarily antisymmetric. diff --git a/inputs/lecture_14.tex b/inputs/lecture_14.tex index d02c7be..3d48945 100644 --- a/inputs/lecture_14.tex +++ b/inputs/lecture_14.tex @@ -193,7 +193,8 @@ Let $\rho(\prec) \coloneqq \sup \{\rho_{\prec}(x) + 1 : x \in X\}$. Clearly $|W| \le \aleph_0$. Define $\prec^\ast$ on $W$ by setting - \[(s_0,u_1,s_1,\ldots, u_n,s_n) \succ^\ast (s_0',u_1', s_1', \ldots, u_m', s_m') :\iff\] + \[(s_0,u_1,s_1,\ldots, u_n,s_n) \succ^\ast (s_0',u_1', s_1', \ldots, u_m', s_m')\] + iff \begin{itemize} \item $n < m$ and \item $\forall i \le n.~s_i \subsetneq s_i' \land u_i \subsetneq u_i'$. diff --git a/inputs/lecture_20.tex b/inputs/lecture_20.tex index 48ffd08..895ddfa 100644 --- a/inputs/lecture_20.tex +++ b/inputs/lecture_20.tex @@ -95,7 +95,8 @@ coordinates. \] \gist{% \begin{claim} - $F(x_k, x') = \inf_m d(\sigma^m(w_k), 1)$, + It is + \[F(x_k, x') = \inf_m d(\sigma^m(w_k), 1),\] where $\sigma(\xi_1, \xi_2, \ldots) = (\xi_1, \xi_1\xi_2, \xi_2\xi_3, \ldots)$. \end{claim} \begin{subproof} @@ -111,7 +112,6 @@ coordinates. By minimality of $(X,T)$ for any $\epsilon >0$, there exists $m \in \Z$ such that $d(\sigma^m w_k, w^\ast) < \epsilon$. - % TODO Think about this Then \begin{IEEEeqnarray*}{rCl} \inf_m d(\sigma^m w_k, 1) &\le & \inf_m d(\sigma^m w_k, w^\ast) + d(w^\ast, 1)\\ diff --git a/inputs/lecture_22.tex b/inputs/lecture_22.tex index 2c3c431..6fde4d9 100644 --- a/inputs/lecture_22.tex +++ b/inputs/lecture_22.tex @@ -14,7 +14,7 @@ \arrow["h"', dashed, from=2-3, to=2-2] \arrow["{\overline{g}, \text{ max. isom.}}", curve={height=-12pt}, from=2-3, to=3-2] \end{tikzcd}\] - + We want to show that this tower is normal, i.e.~the isometric extensions are maximal isometric extension. \gist{% @@ -25,18 +25,18 @@ Then there are $x,x' \in X$ with $\pi'(x) \neq \pi'(x')$ but $\pi_n(x) = \pi_n(x') =t \in X_n$. Then $h^{-1}(t) \ni \pi'(x), \pi'(x')$. - + By a \yaref{lem:lec20:1} there is a sequence $(x_k)$ in $X$ with $\pi_{n-1}(x_k) = \pi_{n-1}(x) = \pi_{n-1}(x')$ for all $k$, such that $F(x_k, x) \to 0$ and $F(x_k, x') \to 0$. - + Let $\rho$ be a metric witnessing that $\overline{g}$ is an isometric extension, i.e.~ $\rho$ is defined on $\bigcup_{x \in X_{n-1}} (\overline{g}^{-1}(x))^2 \overset{\text{closed}}{\subseteq} Y \times Y$, continuous and $\rho(Ta, Tb) = \rho(a,b)$ for $\overline{g}(a) = \overline{g}(b)$. - + For $a,b \in X$ such that \[ \overline{g}(\pi'(a)) = \overline{g}(\pi'(b)) @@ -45,14 +45,14 @@ \[ R(a,b) \coloneqq \rho(\pi'(a), \pi'(b)). \] - + \begin{itemize} \item For any two out of $x,x',(x_k)$, $R$ is defined. \item $R(x,x_k) = R(\tau^m x, \tau^m x_k)$ for all $m$. \item $F(x,x_k) \xrightarrow{k\to \infty} 0$, so there is a sequence $(m_k)$ such that - $d(\tau^{m_k}x, \tau^{m_k} x_k) \xrightarrow{k \to \infty} 0$. + \[d(\tau^{m_k}x, \tau^{m_k} x_k) \xrightarrow{k \to \infty} 0.\] \end{itemize} By continuity of $\rho$, we have that $R(x,x_k) = R(\tau^{m_k} x, \tau^{m_k} x_k) \xrightarrow{k \to \infty} 0$, diff --git a/inputs/lecture_26.tex b/inputs/lecture_26.tex index 25e1a9a..6909bad 100644 --- a/inputs/lecture_26.tex +++ b/inputs/lecture_26.tex @@ -95,7 +95,109 @@ We do a second proof of \yaref{thm:hindman}: Consider $y(0)$. We will prove that this color works and construct a corresponding $H$. - \begin{itemize} +% def power(s): +% if len(s) == 0: +% return [[]] +% else: +% p = power(s[0:-1]) +% return [q + [s[-1]] for q in p] + [q for q in p] +% +% +% def draw(hs): +% s = "\\begin{tikzpicture}" +% s += "\n\t\\node at (-0.5,0.5) {$x$};"; +% for (i,h) in enumerate(hs): +% s += "\n\t\\node at (" + str(hs[i]) + ",0.5) {$h_"+str(i)+"$};"; +% +% for subset in power(range(0,len(hs))): +% if len(subset) <= 1: +% continue +% c = sum([hs[i] for i in subset]) +% c2 = 0.9 if 0 in subset else 0.7 +% s += "\n\t\\node[black!40!white] at (" + str(c) + ", " + str(c2) + ") {\\tiny{$" + " + ".join(map(lambda x : "h_{" + str(x) + "}", subset)) + "$}};" +% s += "\n\t\\draw[black!40!white, very thin] (" + str(c) + ", " + str(c2 - 0.1) + ") -- (" + str(c) + ",-0.1);" +% if subset != list(range(0,len(subset))): +% s += "\n\t\\node[blue!40!white] at (" + str(c) + ", " + str(c2-2) + "){\\tiny{$y(" + " + ".join(map(lambda x: "h_{" + str(x) + "}", subset[0:-1])) + ")$}};" +% s += "\n\t\\draw[blue!40!white, very thin] (" + str(c) + ", -0.1) -- (" + str(c) + ", " + str(c2-1.8) + ");" +% for (i,h) in enumerate(hs): +% hsum = sum(hs[0:i+1]) +% s += '\n\t\\draw[blue] ('\ +% + str(h) + ', 0) -- ('\ +% + str(h) + ', -0.2) -- ('\ +% + str(hsum) + ', -0.2) -- ('\ +% + str(hsum) + ', 0);' +% s += "\n\t\\node[blue] at (" + str(h) + ", -0.5) {$y(0)$};" +% if i > 0: +% s += "\n\t\\node[blue] at (" + str(hsum)\ +% + ", -0.5) {$y(" + ("\ +% + ".join(list(map(lambda x : "h_{" + str(x) + "}", range(0,i)))))\ +% + ")$};"; +% s += "\n\t\\draw[thick] (0,0) -- ("+ str(sum(hs) + 2) + ",0);" +% s += "\n\\end{tikzpicture}" +% return s +% print(draw(np.cumsum([0.8,1,2.4,4.5]))) + +\adjustbox{scale=0.7,center}{% + \begin{tikzpicture} + \node at (-0.5,0.5) {$x$}; + \node at (0.8,0.5) {$h_0$}; + \node at (1.8,0.5) {$h_1$}; + \node at (4.2,0.5) {$h_2$}; + \node at (8.7,0.5) {$h_3$}; + \node[black!40!white] at (15.5, 0.9) {\tiny{$h_{0} + h_{1} + h_{2} + h_{3}$}}; + \draw[black!40!white, very thin] (15.5, 0.8) -- (15.5,-0.1); + \node[black!40!white] at (14.7, 0.7) {\tiny{$h_{1} + h_{2} + h_{3}$}}; + \draw[black!40!white, very thin] (14.7, 0.6) -- (14.7,-0.1); + \node[blue!40!white] at (14.7, -1.3){\tiny{$y(h_{1} + h_{2})$}}; + \draw[blue!40!white, very thin] (14.7, -0.1) -- (14.7, -1.1); + \node[black!40!white] at (13.7, 0.9) {\tiny{$h_{0} + h_{2} + h_{3}$}}; + \draw[black!40!white, very thin] (13.7, 0.8) -- (13.7,-0.1); + \node[blue!40!white] at (13.7, -1.1){\tiny{$y(h_{0} + h_{2})$}}; + \draw[blue!40!white, very thin] (13.7, -0.1) -- (13.7, -0.9); + \node[black!40!white] at (12.899999999999999, 0.7) {\tiny{$h_{2} + h_{3}$}}; + \draw[black!40!white, very thin] (12.899999999999999, 0.6) -- (12.899999999999999,-0.1); + \node[blue!40!white] at (12.899999999999999, -1.3){\tiny{$y(h_{2})$}}; + \draw[blue!40!white, very thin] (12.899999999999999, -0.1) -- (12.899999999999999, -1.1); + \node[black!40!white] at (11.299999999999999, 0.9) {\tiny{$h_{0} + h_{1} + h_{3}$}}; + \draw[black!40!white, very thin] (11.299999999999999, 0.8) -- (11.299999999999999,-0.1); + \node[blue!40!white] at (11.299999999999999, -1.1){\tiny{$y(h_{0} + h_{1})$}}; + \draw[blue!40!white, very thin] (11.299999999999999, -0.1) -- (11.299999999999999, -0.9); + \node[black!40!white] at (10.5, 0.7) {\tiny{$h_{1} + h_{3}$}}; + \draw[black!40!white, very thin] (10.5, 0.6) -- (10.5,-0.1); + \node[blue!40!white] at (10.5, -1.3){\tiny{$y(h_{1})$}}; + \draw[blue!40!white, very thin] (10.5, -0.1) -- (10.5, -1.1); + \node[black!40!white] at (9.5, 0.9) {\tiny{$h_{0} + h_{3}$}}; + \draw[black!40!white, very thin] (9.5, 0.8) -- (9.5,-0.1); + \node[blue!40!white] at (9.5, -1.1){\tiny{$y(h_{0})$}}; + \draw[blue!40!white, very thin] (9.5, -0.1) -- (9.5, -0.9); + \node[black!40!white] at (6.800000000000001, 0.9) {\tiny{$h_{0} + h_{1} + h_{2}$}}; + \draw[black!40!white, very thin] (6.800000000000001, 0.8) -- (6.800000000000001,-0.1); + \node[black!40!white] at (6.0, 0.7) {\tiny{$h_{1} + h_{2}$}}; + \draw[black!40!white, very thin] (6.0, 0.6) -- (6.0,-0.1); + \node[blue!40!white] at (6.0, -1.3){\tiny{$y(h_{1})$}}; + \draw[blue!40!white, very thin] (6.0, -0.1) -- (6.0, -1.1); + \node[black!40!white] at (5.0, 0.9) {\tiny{$h_{0} + h_{2}$}}; + \draw[black!40!white, very thin] (5.0, 0.8) -- (5.0,-0.1); + \node[blue!40!white] at (5.0, -1.1){\tiny{$y(h_{0})$}}; + \draw[blue!40!white, very thin] (5.0, -0.1) -- (5.0, -0.9); + \node[black!40!white] at (2.6, 0.9) {\tiny{$h_{0} + h_{1}$}}; + \draw[black!40!white, very thin] (2.6, 0.8) -- (2.6,-0.1); + \draw[blue] (0.8, 0) -- (0.8, -0.2) -- (0.8, -0.2) -- (0.8, 0); + \node[blue] at (0.8, -0.5) {$y(0)$}; + \draw[blue] (1.8, 0) -- (1.8, -0.2) -- (2.6, -0.2) -- (2.6, 0); + \node[blue] at (1.8, -0.5) {$y(0)$}; + \node[blue] at (2.6, -0.5) {$y(h_{0})$}; + \draw[blue] (4.2, 0) -- (4.2, -0.2) -- (6.800000000000001, -0.2) -- (6.800000000000001, 0); + \node[blue] at (4.2, -0.5) {$y(0)$}; + \node[blue] at (6.800000000000001, -0.5) {$y(h_{0} + h_{1})$}; + \draw[blue] (8.7, 0) -- (8.7, -0.2) -- (15.5, -0.2) -- (15.5, 0); + \node[blue] at (8.7, -0.5) {$y(0)$}; + \node[blue] at (15.5, -0.5) {$y(h_{0} + h_{1} + h_{2})$}; + \draw[thick] (0,0) -- (17.5,0); + \end{tikzpicture} +} + + \begin{itemize} \item % Step 1 Let $G_0 \coloneqq [y(0)]$ and let $N_0$ be such that @@ -152,10 +254,10 @@ We do a second proof of \yaref{thm:hindman}: \item proximal $\leadsto$ $\forall N$.~$T^n(x)\defon{N} = T^n(y)\defon{N}$ for infinitely many $n$. \item uniform recurrence $\leadsto$ - \[ - \forall n .~\exists N.~\forall r.y\defon{\{r,\ldots,r+N-1\}} - \text{ contains } y\defon{\{0,\ldots,n\}} \text{ as a subsequence.} - \] + \begin{IEEEeqnarray*}{rl} + \forall n .~\exists N.~\forall r.~&y\defon{\{r,\ldots,r+N-1\}}\\ + &\text{ contains } y\defon{\{0,\ldots,n\}} \text{ as a subsequence.} + \end{IEEEeqnarray*} (consider neighbourhood $G_n = \{z \in X : z\defon{n} = y\defon{n} \}$). \end{itemize} \item Consider $c \coloneqq y(0)$. This color works: diff --git a/inputs/tutorial_04.tex b/inputs/tutorial_04.tex index ca0e1f4..914aef2 100644 --- a/inputs/tutorial_04.tex +++ b/inputs/tutorial_04.tex @@ -101,20 +101,7 @@ let \item Then $[T]$ is compact: \todo{TODO} - % Let $\langle s_n, n <\omega \rangle$ - % be a Cauchy sequence in $[T]$. - - % Then for every $m < \omega$ - % there exists an $N < \omega$ such that - % $s_n\defon{m} = s_{n'}\defon{m}$ - % for all $n, n' > N$. - % Thus there exists a pointwise limit $s$ of the $s_n$. - - % Since for all $m$ we have $s\defon{m} = s_n\defon{m} \in [T]$ - % for $m$ large enough, - % we get $s \in [T]$. - - % Hence $[T]$ is sequentially compact. + % https://alanmath.wordpress.com/2011/06/16/on-trees-compactness-and-finite-splitting/ \end{enumerate} \nr 2 diff --git a/inputs/tutorial_12b.tex b/inputs/tutorial_12b.tex index bddca3d..22ce81b 100644 --- a/inputs/tutorial_12b.tex +++ b/inputs/tutorial_12b.tex @@ -2,6 +2,8 @@ \subsection{Sheet 10} +\todo{Copy from Abdelrahman and Shiguma} + \nr 2 \todo{Def skew shift flow (on $(\R / \Z)^2$!)}