mirror of
https://github.com/janishutz/eth-summaries.git
synced 2026-01-11 19:48:27 +00:00
Merge branch 'main' of https://github.com/janishutz/eth-summaries
This commit is contained in:
RobinB27
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@@ -1,2 +1,27 @@
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\newsectionNoPB
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\subsection{Higher derivatives}
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\shortdef $f$ is in class $C^1$ if $f$ is differentiable and all its partial derivatives are continuous.
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$f$ is of class $C^k$ if it is differentiable and each of its partial derivatives are in $C^{k - 1}$.
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If $f \in C^k(X; \R^m)$ for all $k \geq 1$, then $f \in C^\infty(X; \R^m)$\\
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% ────────────────────────────────────────────────────────────────────
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\setLabelNumber{all}{4}
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\compactproposition{Mixed derivatives commute} $\partial_{x, y} f = \partial_{y, x}$, as well as $\partial_{x, y, z} = \partial_{x, z, y} = \ldots$, etc (all mixed derivatives commute)
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Since we have symmetry, we can use the notation $\partial_{x_1^{m_1}, \ldots, x_n^{m_n}} f = \frac{\partial^k}{\partial x^m} f = D^m f = \partial^m f$,
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where $m = (m_1, \ldots, m_n)$ and $m_1 + \ldots + m_n = k$.
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There are ${n + k - 1 \choose k}$ possible values for $m$ and e.g. $(1, 1, 2)$ corresponds to the derivative $\frac{\partial^4 f}{\partial x \partial y \partial^2 z}$\\
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% ────────────────────────────────────────────────────────────────────
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\stepLabelNumber{all}
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\shortremark Due to linearity of the partial derivative $\partial_x^m(a f_1 + b f_2) = a \partial_x^m f_1 + b \partial_x^m f_2$\\
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% ────────────────────────────────────────────────────────────────────
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\stepLabelNumber{all}
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\compactex{Laplace operator} $f \in C^2(X)$, $\nabla f \in C_1(X; \R^n)$, so
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$\displaystyle \text{div}(\nabla f) = \sum_{i = 1}^{n} \frac{\partial}{\partial_{x_i}} \left( \frac{\partial f}{\partial_{x_i}} \right)
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= \sum_{i = 1}^{n} \frac{\partial^2 f}{\partial x^2_i}$ (called \bi{Laplacian}, $\Delta f$)\\
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% ────────────────────────────────────────────────────────────────────
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\compactdef{Hessian} $f : X \rightarrow \R$ in $C^2$. For $x \in X$, the \bi{Hessian matrix} of $f$ at $x$ is the symmetric square matrix
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\vspace{-0.75pc}
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\begin{align*}
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\text{Hess}_f(x) = (\partial_{x_i, x_j} f)_{1 \leq i, j \leq n} = H_f(x) \mediumhspace (\text{$i$-th row, $j$-th column})
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\end{align*}
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\drmvspace
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@@ -1,2 +1,5 @@
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\newsectionNoPB
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\subsection{Change of variable}
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The idea is to substitute variables for others that make the equation easier to solve.
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A common example is to switch to polar coordinates from cartesian coordinates, as already demonstrated with continuity checks
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% TODO: Add notes from TA notes
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@@ -1,2 +1,25 @@
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\newsectionNoPB
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\subsection{Taylor polynomials}
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\compactdef{Taylor polynomials}
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Let $f : X \rightarrow \R$ with $f \in C^k(X, \R)$ and $y \in X$. The Taylor-Polynomial of order $k$ of $f$ at $y$ is:
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\vspace{-0.75pc}
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\begin{align*}
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T_k f(y; x - y) = \sum_{|i| \leq k} \frac{\partial_i f(y)(x - y)^i}{i!}
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\end{align*}
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\drmvspace\rmvspace
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% TODO: Find out what the \partial_1 notation means (likely TA notes 09)
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where $i$ is a \textit{multi-index}, so:
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\drmvspace
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\begin{multicols}{3}
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\begin{itemize}[noitemsep]
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\item $i = (i_1, \ldots, i_n)$ (each $i_j \geq 0$)
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\item $|i| = i_1 + \ldots + i_n$
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\item $\partial_i = \partial_1^{i_1} \ldots \partial_n^{i_n}$
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\item $(x - y)^i = (x_1 - y_1)^{i_1} \cdot \ldots \cdot (x_n - y_n)^{i_n}$
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\item $i! = i_1! \cdot \ldots \cdot i_n!$
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\end{itemize}
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\end{multicols}
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\drmvspace\rmvspace
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The concept this formula uses is that we iterate through all possible partial derivatives of $f$ and assigns each a multi-index $i$
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@@ -3,7 +3,7 @@
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\input{~/projects/latex/dist/full.tex}
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\usepackage{lmodern}
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% \usepackage{pgfplots}
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\usepackage{pgfplots}
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\setFontType{sans}
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\renewcommand{\authorTitle}{Robin Bacher, Janis Hutz\\\url{https://github.com/janishutz/eth-summaries}}
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@@ -23,23 +23,53 @@
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% │ Title page │
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% ╰────────────────────────────────────────────────╯
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\vspace{2cm}
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\begin{Huge}
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\begin{center}
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TITLE PAGE COMING SOON
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\end{center}
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\end{Huge}
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\begin{center}
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\begin{tikzpicture}
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\begin{axis}[
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legend pos=outer north east,
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axis lines = box,
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xlabel = size $n$,
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ylabel = Time ($s$),
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variable = t,
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trig format plots = rad,
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]
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\addplot [
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domain=1:10,
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samples=100,
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color=blue,
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]
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{x^2};
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\addlegendentry{$t = n^2$}
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\addplot [
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domain=1:10,
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samples=100,
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color=green,
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]
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{5 * x};
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\addlegendentry{$t = 5n$}
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\addplot [
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domain=1:10,
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samples=100,
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color=red,
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]
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{2^x};
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\addlegendentry{$t = 2^n$}
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\end{axis}
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\end{tikzpicture}
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\end{center}
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\vspace{4cm}
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\vspace{3cm}
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\begin{center}
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\begin{Large}
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\quote{Wenn ich keine Lust habe, das zu berechnen, dann wende ich einfach Gewalt an}
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\end{Large}
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\hspace{3cm} - Vasile Gradinaru, 2025
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\hspace{3cm} - Prof. Dr. Vasile Gradinaru, 2025
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\end{center}
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\vspace{3cm}
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\vspace{2cm}
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\begin{center}
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HS2025, ETHZ\\[0.2cm]
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\begin{Large}
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@@ -65,15 +95,18 @@
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\newpage
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\setcounter{section}{-1}
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\section{Introduction}
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This summary is intended to give you a broad overview of the topics relevant for the exam and is not intended to serve as a full on replacement for the script.
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This summary is intended to give you a broad overview of the topics relevant for the exam.
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While it aims to serve as a full on replacement for the script, please do not fully rely on it, as there may be mistakes, inaccuracies and missing details as compared to the script.
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Furthermore, you will only have access to the script during the exam, so getting familiar with the script is a good idea.
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We have decided to write it in German, as is the new script and for some of the topics that are poorly explained in the script, we have added further explanations.
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The numbering should match the script's numbering exactly (apart from the cases where two definitions were combined due to being closely related and short), making it easier for you to look up the relevant definitions, theorems, etc in context in the script.
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The numbering should match the script's numbering exactly (apart from the cases where two definitions were combined due to being closely related and short),
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making it easier for you to look up the relevant definitions, theorems, etc in context in the script.
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Many of the figures in this summary were taken directly from the Script or Lecture notes created by Professor Vasile Gradinaru.
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We have also taken some explanations and code examples from the slides of our TA, Nils Müller, whose slides can be found
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\color{MidnightBlue}\fbox{\href{https://n.ethz.ch/~muellerni/courses/numcs25.php}{here}}\color{black}
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\hlhref{https://n.ethz.ch/~muellerni/courses/numcs25.php}{here}. (Link will be updated if we are to get a new website link from him, as n.ethz.ch is down now)
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% TODO: Update this when n.ethz is taken offline completely
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% ────────────────────────────────────────────────────────────────────
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@@ -98,6 +98,7 @@ für alle Funktionen zweimal stetig differenzierbaren Funktionen $g$, für welch
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||f^{(k)} - s^{(k)}||_{L^\infty} \leq \frac{5}{384} h^{4 - k} ||f^{(4)}||_{L^\infty}
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\end{align*}
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\end{theorem}
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Zur Erinnerung, $C^4$ ist die Klasse aller vier (oder mehr) mal ableitbaren Funktionen.
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\innumpy verwendet \texttt{scipy.interpolate.CubicSpline} aktuell die ``not-a-knot''-Bedingung.
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Es ist möglich mithilfe von \texttt{bc\_type} beim Instanziieren der Klasse die Art des Splines zu ändern.
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@@ -17,24 +17,59 @@
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\startDocument
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\usetcolorboxes
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\vspace{2cm}
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\begin{Huge}
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\begin{center}
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TITLE PAGE COMING SOON
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\end{center}
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\end{Huge}
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\vspace{1cm}
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\begin{center}
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\begin{tikzpicture}[node distance = 1cm and 2cm, >={Stealth[round]}]
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\node[state, initial left, accepting] (q0p0) {$q_0, p_0$};
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\node[state] (q0p1) [right=of q0p0] {$q_0, p_1$};
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\node[state] (q0p2) [right=of q0p1] {$q_0, p_2$};
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\node[state, accepting] (q1p0) [below=of q0p0] {$q_1, p_0$};
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\node[state] (q1p1) [right=of q1p0] {$q_1, p_1$};
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\node[state] (q1p2) [right=of q1p1] {$q_1, p_2$};
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\node[state, accepting] (q2p0) [below=of q1p0] {$q_2, p_0$};
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\node[state, accepting] (q2p1) [right=of q2p0] {$q_2, p_1$};
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\node[state, accepting] (q2p2) [right=of q2p1] {$q_2, p_2$};
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\path[->]
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% Level 0
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(q0p0) edge node [above] {a} (q0p1)
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(q0p1) edge node [above] {a} (q0p2)
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(q0p2) edge [bend right] node [above] {a} (q0p0)
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% Level 0 to level 1
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(q0p0) edge node [right] {b} (q1p0)
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(q0p1) edge node [right] {b} (q1p1)
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(q0p2) edge node [right] {b} (q1p2)
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% Level 1 to level 2
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(q1p0) edge node [above] {a} (q2p1)
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(q1p1) edge node [above] {a} (q2p2)
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(q1p2) edge node [right, xshift=0.3cm] {a} (q2p0)
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% Level 2 to level 1
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(q2p0) edge node [right] {b} (q1p0)
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(q2p1) edge node [above left, yshift=0.1cm] {b} (q1p1)
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(q2p2) edge node [right] {b} (q1p2)
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% Level 2
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(q2p0) edge node [above] {a} (q2p1)
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(q2p1) edge node [above] {a} (q2p2)
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(q2p2) edge [bend left] node [below] {a} (q2p0)
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% ────────────────────────────────────────────────────────────────────
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% Loops on level 1
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(q1p0) edge [loop left] node {b} ()
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(q1p1) edge [loop left] node {b} ()
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(q1p2) edge [loop left] node {b} ();
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\end{tikzpicture}
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\end{center}
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\vspace{4cm}
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\vspace{2.5cm}
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\begin{center}
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\begin{Large}
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``\textit{Sie können also alle C Programme in Kanonischer Ordnung aufzählen. Sollten Sie dies tun. Wahrscheinlich nicht. Was aber zählt ist, sie \textbf{können} es tun}''
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``\textit{Sie können also alle C Programme in Kanonischer Ordnung aufzählen. Sollten Sie dies tun? Wahrscheinlich nicht. Was aber zählt ist, sie \textbf{können} es tun}''
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\end{Large}
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\hspace{3cm} - Prof. Dr. Dennis Komm, 2025
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\end{center}
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\vspace{3cm}
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\vspace{2cm}
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\begin{center}
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HS2025, ETHZ\\[0.2cm]
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\begin{Large}
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@@ -2,6 +2,8 @@
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\section{Combinatorics}
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\label{sec:combinatorics}
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\subsection{Introduction}
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Please note: This section was not part of the book and is written in very simple terms (it is taken from a summary I wrote 4 years ago during Gymnasium)
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Combinatorics was developed from the willingness of humans to gamble and the fact that everybody wanted to win as much money as possible.
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\subsection{Simple counting operations}
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Binary file not shown.
@@ -12,25 +12,58 @@
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\newcommand{\lempty}{L_{\text{empty}}}
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\renewcommand{\tc}{\text{Time}}
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\newcommand{\spc}{\text{Space}}
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\usetikzlibrary{automata, positioning, arrows.meta}
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\begin{document}
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\startDocument
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\usetcolorboxes
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\vspace{2cm}
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\begin{Huge}
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\begin{center}
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TITLE PAGE COMING SOON
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\end{center}
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\end{Huge}
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\vspace{1cm}
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\begin{center}
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\begin{tikzpicture}[node distance = 1cm and 2cm, >={Stealth[round]}]
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||||
\node[state, initial left, accepting] (q0p0) {$q_0, p_0$};
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\node[state] (q0p1) [right=of q0p0] {$q_0, p_1$};
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\node[state] (q0p2) [right=of q0p1] {$q_0, p_2$};
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\node[state, accepting] (q1p0) [below=of q0p0] {$q_1, p_0$};
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||||
\node[state] (q1p1) [right=of q1p0] {$q_1, p_1$};
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\node[state] (q1p2) [right=of q1p1] {$q_1, p_2$};
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||||
\node[state, accepting] (q2p0) [below=of q1p0] {$q_2, p_0$};
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\node[state, accepting] (q2p1) [right=of q2p0] {$q_2, p_1$};
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\node[state, accepting] (q2p2) [right=of q2p1] {$q_2, p_2$};
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||||
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||||
\path[->]
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||||
% Level 0
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||||
(q0p0) edge node [above] {a} (q0p1)
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(q0p1) edge node [above] {a} (q0p2)
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||||
(q0p2) edge [bend right] node [above] {a} (q0p0)
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||||
% Level 0 to level 1
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||||
(q0p0) edge node [right] {b} (q1p0)
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||||
(q0p1) edge node [right] {b} (q1p1)
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||||
(q0p2) edge node [right] {b} (q1p2)
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||||
% Level 1 to level 2
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||||
(q1p0) edge node [above] {a} (q2p1)
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||||
(q1p1) edge node [above] {a} (q2p2)
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||||
(q1p2) edge node [right, xshift=0.3cm] {a} (q2p0)
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||||
% Level 2 to level 1
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||||
(q2p0) edge node [right] {b} (q1p0)
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||||
(q2p1) edge node [above left, yshift=0.1cm] {b} (q1p1)
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||||
(q2p2) edge node [right] {b} (q1p2)
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||||
% Level 2
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||||
(q2p0) edge node [above] {a} (q2p1)
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||||
(q2p1) edge node [above] {a} (q2p2)
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(q2p2) edge [bend left] node [below] {a} (q2p0)
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||||
% ────────────────────────────────────────────────────────────────────
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% Loops on level 1
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(q1p0) edge [loop left] node {b} ()
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(q1p1) edge [loop left] node {b} ()
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(q1p2) edge [loop left] node {b} ();
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||||
\end{tikzpicture}
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||||
\end{center}
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\vspace{4cm}
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\vspace{3cm}
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\begin{center}
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\begin{Large}
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``\textit{Wenn Sie die Turing-Maschine als Touring-Maschine in der Prüfung schreiben, dann macht mich das sehr traurig. Ich seh das jeweils. Teilweise sind das sehr elaborierte Trolle, manchmal Leute die nie in die Vorlesungen kommen} (2025-10-14T08:51Z+02:00``)
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||||
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||||
``\textit{Sie können also alle C Programme in Kanonischer Ordnung aufzählen. Sollten Sie dies tun. Wahrscheinlich nicht. Was aber zählt ist, sie \textbf{können} es tun}''
|
||||
``\textit{Sie können also alle C Programme in Kanonischer Ordnung aufzählen. Sollten Sie dies tun? Wahrscheinlich nicht. Was aber zählt ist, sie \textbf{können} es tun}''
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||||
\end{Large}
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||||
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||||
\hspace{3cm} - Prof. Dr. Dennis Komm, 2025
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@@ -52,7 +85,7 @@
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\begin{scriptsize}
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\begin{itemize}
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\item \textit{Note: Definitions, Lemmas, etc are often 1:1 copies from the book or paraphrased (as I did not find an easier way of stating them)}
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\item \textit{Note: In case I forgot to add the PDF page numbers, you can take the PDF page number is given by $P_{\text{PDF}} = P_{\text{Book}} + 15$}
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\item \textit{Note: In case I forgot to add the PDF page numbers, the PDF page number is given by $P_{\text{PDF}} = P_{\text{Book}} + 15$}
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\end{itemize}
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\end{scriptsize}
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Reference in New Issue
Block a user