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-rw-r--r--buch/papers/zeta/analytic_continuation.tex11
-rw-r--r--buch/papers/zeta/einleitung.tex32
-rw-r--r--buch/papers/zeta/euler_product.tex11
-rw-r--r--buch/papers/zeta/fazit.tex28
-rw-r--r--buch/papers/zeta/images/continuation_overview.tikz.tex (renamed from buch/papers/zeta/continuation_overview.tikz.tex)0
-rw-r--r--buch/papers/zeta/images/primzahlfunktion.pgf (renamed from buch/papers/zeta/primzahlfunktion.pgf)0
-rw-r--r--buch/papers/zeta/images/primzahlfunktion_paper.pgf505
-rw-r--r--buch/papers/zeta/images/youtube_screenshot.png (renamed from buch/papers/zeta/presentation/youtube_screenshot.png)bin378662 -> 378662 bytes
-rw-r--r--buch/papers/zeta/images/zeta_re_-1_plot.pgf (renamed from buch/papers/zeta/zeta_re_-1_plot.pgf)0
-rw-r--r--buch/papers/zeta/images/zeta_re_0.5_plot.pgf (renamed from buch/papers/zeta/zeta_re_0.5_plot.pgf)0
-rw-r--r--buch/papers/zeta/images/zeta_re_0_plot.pgf (renamed from buch/papers/zeta/zeta_re_0_plot.pgf)0
-rw-r--r--buch/papers/zeta/main.tex2
-rw-r--r--buch/papers/zeta/presentation/presentation.tex12
-rw-r--r--buch/papers/zeta/references.bib57
-rw-r--r--buch/papers/zeta/zeta_color_plot-img0.pngbin0 -> 37362 bytes
-rw-r--r--buch/papers/zeta/zeta_color_plot.pgf402
16 files changed, 1019 insertions, 41 deletions
diff --git a/buch/papers/zeta/analytic_continuation.tex b/buch/papers/zeta/analytic_continuation.tex
index a45791e..4046bb7 100644
--- a/buch/papers/zeta/analytic_continuation.tex
+++ b/buch/papers/zeta/analytic_continuation.tex
@@ -4,7 +4,7 @@
Die analytische Fortsetzung der Riemannschen Zetafunktion ist äusserst interessant.
Sie ermöglicht die Berechnung von $\zeta(-1)$ und weiterer spannender Werte.
So liegen zum Beispiel unendlich viele Nullstellen der Zetafunktion bei $\Re(s) = \frac{1}{2}$.
-Diese sind relevant für die Primzahlverteilung und sind Gegenstand der Riemannschen Vermutung.
+Wie bereits erwähnt sind diese Gegenstand der Riemannschen Vermutung.
Es werden zwei verschiedene Fortsetzungen benötigt.
Die erste erweitert die Zetafunktion auf $\Re(s) > 0$.
@@ -12,7 +12,7 @@ Die zweite verwendet eine Spiegelung an der $\Re(s) = \frac{1}{2}$ Geraden und e
Eine grafische Darstellung dieses Plans ist in Abbildung \ref{zeta:fig:continuation_overview} zu sehen.
\begin{figure}
\centering
- \input{papers/zeta/continuation_overview.tikz.tex}
+ \input{papers/zeta/images/continuation_overview.tikz.tex}
\caption{
Die verschiedenen Abschnitte der Riemannschen Zetafunktion.
Die originale Definition von \eqref{zeta:equation1} ist im grünen Bereich gültig.
@@ -237,7 +237,7 @@ Eine ganz ähnliche Spiegelungseigenschaft wurde bereits in Kapitel \ref{buch:fu
Ziel dieses Abschnittes ist es, zu zeigen wie das Integral $I_1$ aus Gleichung \eqref{zeta:equation:integral2} durch ein neues Integral mit den Integrationsgrenzen $1$ und $\infty$ ersetzt werden kann.
Da dieser Schritt ziemlich aufwendig ist, wird er hier in einem eigenen Abschnitt behandelt.
-Zunächst wird die poissonsche Summenformel hergeleitet, da diese verwendet werden kann um $\psi(x)$ zu berechnen.
+Zunächst wird die poissonsche Summenformel hergeleitet \cite{zeta:online:poisson}, da diese verwendet werden kann um $\psi(x)$ zu berechnen.
Um die poissonsche Summenformel zu beweisen, berechnen wir zunächst die Fourierreihe der Dirac Delta Funktion.
@@ -330,7 +330,7 @@ Um die poissonsche Summenformel zu beweisen, berechnen wir zunächst die Fourier
\sum_{k=-\infty}^{\infty}
\delta(x + k).
\end{align}
- Wenn wir dies einsetzen und erhalten wir den gesuchten Beweis für die poissonsche Summenformel
+ Wenn wir dies einsetzen und erhalten wir
\begin{equation}
\sum_{k=-\infty}^{\infty}
F(k)
@@ -348,8 +348,9 @@ Um die poissonsche Summenformel zu beweisen, berechnen wir zunächst die Fourier
\, dx
=
\sum_{k=-\infty}^{\infty}
- f(k).
+ f(k),
\end{equation}
+ was der gesuchte Beweis für die poissonsche Summenformel ist.
\end{proof}
Erinnern wir uns nochmals an unser Integral aus Gleichung \eqref{zeta:equation:integral2}
diff --git a/buch/papers/zeta/einleitung.tex b/buch/papers/zeta/einleitung.tex
index 3b70531..ad87fec 100644
--- a/buch/papers/zeta/einleitung.tex
+++ b/buch/papers/zeta/einleitung.tex
@@ -1,11 +1,41 @@
\section{Einleitung} \label{zeta:section:einleitung}
\rhead{Einleitung}
-Die Riemannsche Zetafunktion ist für alle komplexe $s$ mit $\Re(s) > 1$ definiert als
+Die Riemannsche Zetafunktion $\zeta(s)$ ist für alle komplexe $s$ mit $\Re(s) > 1$ definiert als
\begin{equation}\label{zeta:equation1}
\zeta(s)
=
\sum_{n=1}^{\infty}
\frac{1}{n^s}.
\end{equation}
+Die Zetafunktion ist bekannt als Bestandteil der Riemannschen Vermutung, welche besagt das alle nichttrivialen Nullstellen der Zetafunktion einen Realteil von $\frac{1}{2}$ haben.
+Mithilfe dieser Vermutung kann eine gute Annäherung an die Primzahlfunktion gefunden werden.
+Die Primzahlfunktion steigt immer an, sobald eine Primzahl vorkommt.
+Eine Darstellung davon ist in Abbildung \ref{fig:zeta:primzahlfunktion} zu finden.
+Die Riemannsche Vermutung ist eines der ungelösten Millennium-Probleme der Mathematik, auf deren Lösung eine Belohnung von einer Million Doller ausgesetzt ist \cite{zeta:online:millennium}.
+Auf eine genauere Beschreibung der Riemannschen Vermutung wird im Rahmen dieses Papers nicht eingegangen.
+\begin{figure}
+ \centering
+ \input{papers/zeta/images/primzahlfunktion_paper.pgf}
+ \caption{Die Primzahlfunktion von $0$ bis $30$.}
+ \label{fig:zeta:primzahlfunktion}
+\end{figure}
+Der grundlegende Zusammenhang der Primzahlen und der Zetafunktion wird im ersten Abschnitt \ref{zeta:section:eulerprodukt} über das Eulerprodukt gezeigt.
+Danach folgt die Verbindung zur bereits bekannten Gammafunktion in Abschnitt \ref{zeta:section:zusammenhang_mit_gammafunktion}.
+Schlussendlich folgt die Beschreibung der analytischen Fortsetzung die gesamte komplexe Ebene in Abschnitt \ref{zeta:section:analytische_fortsetzung}.
+
+Diese analytische Fortsetzung wird für die Riemannsche Vermutung benötigt, ermöglicht aber auch andere interessante Aussagen.
+So findet sich zum Beispiel immer wieder die aberwitzige Behauptung, das die Summe aller natürlichen Zahlen
+\begin{equation*}
+ \sum{n=1}^{\infty} n
+ =
+ \sum_{n=1}^{\infty}
+ \frac{1}{n^{-1}}
+ =
+ -\frac{1}{12}
+\end{equation*}
+sei.
+Obwohl diese Behauptung offensichtlich Falsch ist, hat sie doch ihre Berechtigung, wie durch die analytische Fortsetzung gezeigt werden wird.
+
+Die folgenden mathematischen Herleitungen sind, sofern nicht anders gekennzeichnet, eigene Darstellungen basierend auf den überaus umfangreichen Wikipedia-Artikeln auf Deutsch \cite{zeta:online:wiki_de} und Englisch \cite{zeta:online:wiki_en} sowie einer Video-Playlist \cite{zeta:online:mryoumath}.
diff --git a/buch/papers/zeta/euler_product.tex b/buch/papers/zeta/euler_product.tex
index 5f4f5ca..7915c84 100644
--- a/buch/papers/zeta/euler_product.tex
+++ b/buch/papers/zeta/euler_product.tex
@@ -1,9 +1,9 @@
\section{Eulerprodukt} \label{zeta:section:eulerprodukt}
\rhead{Eulerprodukt}
-Das Eulerprodukt stellt die Verbindung der Zetafunktion und der Primzahlen her.
-Diese Verbindung ist sehr wichtig, da durch sie eine Aussage zur Primzahlverteilung gemacht werden kann.
-Die Verteilung der Primzahlen ist Gegenstand der Riemannschen Vermutung, welche eines der grössten ungelösten Probleme der Mathematik ist.
+Das Eulerprodukt stellt die gesuchte Verbindung der Zetafunktion und der Primzahlen her.
+Wie der Name bereits sagt, wurde das Eulerprodukt bereits 1727 von Euler entdeckt.
+Um daraus die Riemannsche Vermutung herzuleiten, wäre aber noch einiges mehr nötig.
\begin{satz}
Für alle Zahlen $s$ mit $\Re(s) > 1$ ist die Zetafunktion identisch mit dem unendlichen Eulerprodukt
@@ -65,7 +65,7 @@ Die Verteilung der Primzahlen ist Gegenstand der Riemannschen Vermutung, welche
n = \prod_i p_i^{k_i} \quad \forall \quad n \in \mathbb{N}.
\end{equation}
Jeder Summand der Summen in \eqref{zeta:equation:eulerprodukt2} ist somit der Kehrwert genau einer natürlichen Zahl $n \in \mathbb{N}$.
- Da die Summen alle möglichen Kombinationen von Exponenten und Primzahlen in \eqref{zeta:equation:eulerprodukt2} enthält haben wir
+ Da die Summen alle möglichen Kombinationen von Exponenten und Primzahlen in \eqref{zeta:equation:eulerprodukt2} enthält, haben wir
\begin{equation}
\sum_{k_1=0}^{\infty}
\sum_{k_2=0}^{\infty}
@@ -79,7 +79,8 @@ Die Verteilung der Primzahlen ist Gegenstand der Riemannschen Vermutung, welche
\sum_{n=1}^\infty
\frac{1}{n^s}
=
- \zeta(s)
+ \zeta(s),
\end{equation}
+ wodurch das Eulerprudukt bewiesen ist.
\end{proof}
diff --git a/buch/papers/zeta/fazit.tex b/buch/papers/zeta/fazit.tex
new file mode 100644
index 0000000..f696f83
--- /dev/null
+++ b/buch/papers/zeta/fazit.tex
@@ -0,0 +1,28 @@
+\section{Fazit} \label{zeta:section:fazit}
+\rhead{Fazit}
+
+Ganz zu Beginn dieses Papers wurde die Behauptung erwähnt, dass die Summe aller natürlichen Zahlen $-\frac{1}{12}$ sei.
+Diese Summe ist nichts anderes als die Zetafunktion am Wert $s=-1$.
+Da wir die analytische Fortsetzung mit der Funktionalgleichung \eqref{zeta:equation:functional} gefunden haben, können wir diese Behauptung prüfen.
+Zunächst berechnen wir $\zeta(1-s) = \zeta(2) = \frac{\pi^2}{6}$, welches im konvergenten Bereich der Reihe liegt und auch bekannt ist als das Basler Problem.
+Somit haben wir
+\begin{align*}
+ \zeta(s) = \zeta(-1)
+ &=
+ \frac{\Gamma \left( \frac{1-s}{2} \right)}{\pi^{\frac{1-s}{2}}}
+ \zeta(1-s)
+ \frac{\pi^{\frac{s}{2}}}{\Gamma \left( \frac{s}{2} \right)}
+ \\
+ &=
+ \frac{\Gamma(1)}{\pi}
+ \frac{\pi^2}{6}
+ \frac{\pi^{\frac{-1}{2}}}{\Gamma \left( \frac{-1}{2} \right)}
+ \\
+ &=
+ \frac{1}{\pi}
+ \frac{\pi^2}{6}
+ \frac{1}{\sqrt{\pi} (-2\sqrt{\pi})}
+ &=
+ -\frac{1}{12},
+\end{align*}
+wobei die Werte der Gammafunktion TODO berechnet werden.
diff --git a/buch/papers/zeta/continuation_overview.tikz.tex b/buch/papers/zeta/images/continuation_overview.tikz.tex
index 836ab1d..836ab1d 100644
--- a/buch/papers/zeta/continuation_overview.tikz.tex
+++ b/buch/papers/zeta/images/continuation_overview.tikz.tex
diff --git a/buch/papers/zeta/primzahlfunktion.pgf b/buch/papers/zeta/images/primzahlfunktion.pgf
index 7d4f4fc..7d4f4fc 100644
--- a/buch/papers/zeta/primzahlfunktion.pgf
+++ b/buch/papers/zeta/images/primzahlfunktion.pgf
diff --git a/buch/papers/zeta/images/primzahlfunktion_paper.pgf b/buch/papers/zeta/images/primzahlfunktion_paper.pgf
new file mode 100644
index 0000000..b9d67d3
--- /dev/null
+++ b/buch/papers/zeta/images/primzahlfunktion_paper.pgf
@@ -0,0 +1,505 @@
+%% Creator: Matplotlib, PGF backend
+%%
+%% To include the figure in your LaTeX document, write
+%% \input{<filename>.pgf}
+%%
+%% Make sure the required packages are loaded in your preamble
+%% \usepackage{pgf}
+%%
+%% and, on pdftex
+%% \usepackage[utf8]{inputenc}\DeclareUnicodeCharacter{2212}{-}
+%%
+%% or, on luatex and xetex
+%% \usepackage{unicode-math}
+%%
+%% Figures using additional raster images can only be included by \input if
+%% they are in the same directory as the main LaTeX file. For loading figures
+%% from other directories you can use the `import` package
+%% \usepackage{import}
+%%
+%% and then include the figures with
+%% \import{<path to file>}{<filename>.pgf}
+%%
+%% Matplotlib used the following preamble
+%%
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diff --git a/buch/papers/zeta/presentation/youtube_screenshot.png b/buch/papers/zeta/images/youtube_screenshot.png
index 434041b..434041b 100644
--- a/buch/papers/zeta/presentation/youtube_screenshot.png
+++ b/buch/papers/zeta/images/youtube_screenshot.png
Binary files differ
diff --git a/buch/papers/zeta/zeta_re_-1_plot.pgf b/buch/papers/zeta/images/zeta_re_-1_plot.pgf
index dd15ba1..dd15ba1 100644
--- a/buch/papers/zeta/zeta_re_-1_plot.pgf
+++ b/buch/papers/zeta/images/zeta_re_-1_plot.pgf
diff --git a/buch/papers/zeta/zeta_re_0.5_plot.pgf b/buch/papers/zeta/images/zeta_re_0.5_plot.pgf
index 3ac7df8..3ac7df8 100644
--- a/buch/papers/zeta/zeta_re_0.5_plot.pgf
+++ b/buch/papers/zeta/images/zeta_re_0.5_plot.pgf
diff --git a/buch/papers/zeta/zeta_re_0_plot.pgf b/buch/papers/zeta/images/zeta_re_0_plot.pgf
index 29a844e..29a844e 100644
--- a/buch/papers/zeta/zeta_re_0_plot.pgf
+++ b/buch/papers/zeta/images/zeta_re_0_plot.pgf
diff --git a/buch/papers/zeta/main.tex b/buch/papers/zeta/main.tex
index caddace..de297a0 100644
--- a/buch/papers/zeta/main.tex
+++ b/buch/papers/zeta/main.tex
@@ -8,12 +8,12 @@
\begin{refsection}
\chapterauthor{Raphael Unterer}
-%TODO Einleitung
\input{papers/zeta/einleitung.tex}
\input{papers/zeta/euler_product.tex}
\input{papers/zeta/zeta_gamma.tex}
\input{papers/zeta/analytic_continuation.tex}
+\input{papers/zeta/fazit}
\printbibliography[heading=subbibliography]
\end{refsection}
diff --git a/buch/papers/zeta/presentation/presentation.tex b/buch/papers/zeta/presentation/presentation.tex
index e106089..53fd305 100644
--- a/buch/papers/zeta/presentation/presentation.tex
+++ b/buch/papers/zeta/presentation/presentation.tex
@@ -129,7 +129,7 @@
\begin{frame}
\frametitle{Summe aller Natürlichen Zahlen}
\begin{center}
- \includegraphics[width=0.7\textwidth]{youtube_screenshot.png}
+ \includegraphics[width=0.7\textwidth]{../images/youtube_screenshot.png}
\end{center}
\end{frame}
\begin{frame}
@@ -168,7 +168,7 @@
\begin{frame}
\frametitle{Plan für die Analytische Fortsetzung von $\zeta(s)$}
\begin{center}
- \input{../continuation_overview.tikz.tex}
+ \input{../images/continuation_overview.tikz.tex}
\end{center}
\end{frame}
\begin{frame}
@@ -331,7 +331,7 @@
\begin{frame}
\frametitle{Primzahlfunktion}
\begin{center}
- \scalebox{0.5}{\input{../primzahlfunktion.pgf}}
+ \scalebox{0.5}{\input{../images/primzahlfunktion.pgf}}
\end{center}
\end{frame}
@@ -348,19 +348,19 @@
\begin{frame}
\frametitle{Konstanter Realteil $\Re(s)=-1$ und $\Im(s)=0\ldots40$}
\begin{center}
- \scalebox{0.6}{\input{../zeta_re_-1_plot.pgf}}
+ \scalebox{0.6}{\input{../images/zeta_re_-1_plot.pgf}}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Konstanter Realteil $\Re(s)=0$ und $\Im(s)=0\ldots40$}
\begin{center}
- \scalebox{0.6}{\input{../zeta_re_0_plot.pgf}}
+ \scalebox{0.6}{\input{../images/zeta_re_0_plot.pgf}}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Konstanter Realteil $\Re(s)=0.5$ und $\Im(s)=0\ldots40$}
\begin{center}
- \scalebox{0.6}{\input{../zeta_re_0.5_plot.pgf}}
+ \scalebox{0.6}{\input{../images/zeta_re_0.5_plot.pgf}}
\end{center}
\end{frame}
diff --git a/buch/papers/zeta/references.bib b/buch/papers/zeta/references.bib
index a4f2521..e8d6b22 100644
--- a/buch/papers/zeta/references.bib
+++ b/buch/papers/zeta/references.bib
@@ -4,32 +4,43 @@
% (c) 2020 Autor, Hochschule Rapperswil
%
-@online{zeta:bibtex,
- title = {BibTeX},
- url = {https://de.wikipedia.org/wiki/BibTeX},
- date = {2020-02-06},
- year = {2020},
- month = {2},
- day = {6}
+@online{zeta:online:millennium,
+ title = {The Millennium Prize Problems},
+ url = {https://www.claymath.org/millennium-problems/millennium-prize-problems},
+ year = {2022},
+ month = {8},
+ day = {4}
}
-@book{zeta:numerical-analysis,
- title = {Numerical Analysis},
- author = {David Kincaid and Ward Cheney},
- publisher = {American Mathematical Society},
- year = {2002},
- isbn = {978-8-8218-4788-6},
- inseries = {Pure and applied undegraduate texts},
- volume = {2}
+@online{zeta:online:wiki_en,
+ title = {Riemann zeta function},
+ url = {https://en.wikipedia.org/wiki/Riemann_zeta_function},
+ year = {2022},
+ month = {8},
+ day = {7}
+}
+@online{zeta:online:wiki_de,
+ title = {Riemannsche Zeta-Funktion},
+ url = {https://de.wikipedia.org/wiki/Riemannsche_Zeta-Funktion},
+ year = {2022},
+ month = {8},
+ day = {7}
}
-@article{zeta:mendezmueller,
- author = { Tabea Méndez and Andreas Müller },
- title = { Noncommutative harmonic analysis and image registration },
- journal = { Appl. Comput. Harmon. Anal.},
- year = 2019,
- volume = 47,
- pages = {607--627},
- url = {https://doi.org/10.1016/j.acha.2017.11.004}
+@online{zeta:online:poisson,
+ title = {Deriving the Poisson Summation Formula},
+ url = {https://www.youtube.com/watch?v=4Bex-4BFYWo},
+ author = {Physics and Math Lectures},
+ year = {2022},
+ month = {8},
+ day = {7}
}
+@online{zeta:online:mryoumath,
+ title = {Riemann Zeta Function Playlist},
+ url = {https://www.youtube.com/playlist?list=PL32446FDD4DA932C9},
+ author = {MrYouMath},
+ year = {2022},
+ month = {8},
+ day = {7}
+}
diff --git a/buch/papers/zeta/zeta_color_plot-img0.png b/buch/papers/zeta/zeta_color_plot-img0.png
new file mode 100644
index 0000000..b8c7298
--- /dev/null
+++ b/buch/papers/zeta/zeta_color_plot-img0.png
Binary files differ
diff --git a/buch/papers/zeta/zeta_color_plot.pgf b/buch/papers/zeta/zeta_color_plot.pgf
new file mode 100644
index 0000000..0fd7cb8
--- /dev/null
+++ b/buch/papers/zeta/zeta_color_plot.pgf
@@ -0,0 +1,402 @@
+%% Creator: Matplotlib, PGF backend
+%%
+%% To include the figure in your LaTeX document, write
+%% \input{<filename>.pgf}
+%%
+%% Make sure the required packages are loaded in your preamble
+%% \usepackage{pgf}
+%%
+%% and, on pdftex
+%% \usepackage[utf8]{inputenc}\DeclareUnicodeCharacter{2212}{-}
+%%
+%% or, on luatex and xetex
+%% \usepackage{unicode-math}
+%%
+%% Figures using additional raster images can only be included by \input if
+%% they are in the same directory as the main LaTeX file. For loading figures
+%% from other directories you can use the `import` package
+%% \usepackage{import}
+%%
+%% and then include the figures with
+%% \import{<path to file>}{<filename>.pgf}
+%%
+%% Matplotlib used the following preamble
+%%
+\begingroup%
+\makeatletter%
+\begin{pgfpicture}%
+\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{6.400000in}{4.800000in}}%
+\pgfusepath{use as bounding box, clip}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
+\pgfsetmiterjoin%
+\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}%
+\pgfsetfillcolor{currentfill}%
+\pgfsetlinewidth{0.000000pt}%
+\definecolor{currentstroke}{rgb}{1.000000,1.000000,1.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetdash{}{0pt}%
+\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{6.400000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{6.400000in}{4.800000in}}%
+\pgfpathlineto{\pgfqpoint{0.000000in}{4.800000in}}%
+\pgfpathclose%
+\pgfusepath{fill}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
+\pgfsetmiterjoin%
+\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}%
+\pgfsetfillcolor{currentfill}%
+\pgfsetlinewidth{0.000000pt}%
+\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetstrokeopacity{0.000000}%
+\pgfsetdash{}{0pt}%
+\pgfpathmoveto{\pgfqpoint{2.588156in}{0.528000in}}%
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+\pgfpathlineto{\pgfqpoint{3.971844in}{4.224000in}}%
+\pgfpathlineto{\pgfqpoint{2.588156in}{4.224000in}}%
+\pgfpathclose%
+\pgfusepath{fill}%
+\end{pgfscope}%
+\begin{pgfscope}%
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