diff --git a/data/10anhang/.50implementation.swp b/data/10anhang/.50implementation.swp
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diff --git a/data/10anhang/50implementation b/data/10anhang/50implementation
index bf3ced2c438706d25b5451611238e0abecbb3fd1..3d87bd040002014de96b6f33db3ff737f03a6a11 100644
--- a/data/10anhang/50implementation
+++ b/data/10anhang/50implementation
@@ -2,5 +2,12 @@
 
 
 <subsubsection title="overflow in angular differences, and how to solve it" label="aimplementationphi">
+Our input data contains a #phi# that is centered around its mean: so the most simple implementation would simply subtract the mean of #phi# from each #phi# value. This can lead to overflow problems, since #Eq(phi,2*pi)# is equivalent to #Eq(phi,0)# and thus a mean of about #6# could be sutracted from a tiny value<note or the inverse> resulting in weird phi distributions. Not solving this, results in a loss distribution with a small peak at very high losses
+<i f="none" wmode="True">loss distribution with angular overflow</i>
+So how to solve this? first we need the true mean value (#2*pi-0.1# and #0.1# have mean #0# and not #pi#), and then we use a modular operator to restrict every difference (the difference #2*pi-0.1# molulo #2*pi# equals the true difference of #-0.1#). And for finding this mean value we can cheat a little, but calculating the mean 4 vector, and finding out its #phi# value. For a more indepth look at out solution, you can also take a look at the actual implementation (ENTER LINK TO gpre5).
+
+
+
+
 
 
diff --git a/data/10anhang/60sorting b/data/10anhang/60sorting
index 08d37cd130890b83195c1022fa5788a9e1316c5a..e8c2a38f52b44dd4c5a9c6289d8aed3c7c6d34d8 100644
--- a/data/10anhang/60sorting
+++ b/data/10anhang/60sorting
@@ -1 +1,9 @@
 <subsection title="the consequences of sorting outputs by lpt" label="asort">
+
+Sorting nodes at the end of the Autoencoder breaks the permutation symmetry that we praised at the beginning of this thesis (Chapter <ref graphs>), and, since we use #lp_T# for this sorting, we artificially inflate the importance of the momentum variable compared to the other variables. That beeing said, turning of this sorting, does hurt the network performance: Given the two Networks
+<i f="none" f2="none" wmode="True">sorting in network plot comparison</i>
+The sorted Network reduces an AUC value (trained on qcd) of #0.6351# into #0.5788# and probably even more importantly, the training curve looks way worse
+<i f="none" f2="none" wmode="True">training curve comparison for sorting</i>
+You migth consider the not sortet curve more clean, but it also does not really improve any further at a fairly early epoch, result in the sorting network reaching a loss of about a factor 3 smaller. Seeing this, sorting seems like a clear choice for us, theoretical concerns dont fair well compared to practical results, and so basically all Networks in this Thesis are sorted. That beeing said, the original deficits of breaking permutations symmetry, could actually be interpreted to mean the opposite: while it is true, that switching each value, except the sorted one, would not result in the same loss, switching any whole node position, still would result in exactly the same loss. In fact, we can use this, to understand why not sorted Networks are so bad: The encoding includes a random<note actually not random, but you could see it like this, if you only see the initial and final node indices> node permutation, while the decoding does not, so after the autoencoder, the result is a random permuation of the Input features, which then are compared to the still initially ordered Input features. That this does not work that well should be clear: So either choose the momentum axis as sorting value<note which would not be what you would want, since locality in real space is much more important than similar energies, as Chapter <ref ametrikana> shows> or compare your predictions neirly randomly. In fact, you could argue, that this breaks permutation symmetry, as you impose a defined ordering on your node indices. Finally, if you would want to improve this, you migth look at two things: making the comparison variable learnable, would remove the artificial inflated importance of #lp_T#<note but maybe also make the training less stable, and add a less controlable importance to some other mixture of features> and making the decompression chance the node ordering, best case in a learnable way, would make this whole discussion moot, as the network could converge as good with, as without sorting. That beeing suggested, implementing this is not neccesarily easy, as you would not want any function to apply to all nodes, to make sure you dont break permutation symmetry, which for me looks like you restrict yourself to finding a variable to sort by and to reverse an initial sorting would in general not be easy at all, as the initial sorting could be completely random, but would result here probably in a network sorting the nodes by their transverse momentum, as this is the sorting of the initial Data, but this seems to us, as a more complicated implementation of our final sorting layer<note that could actually work less well, not only since it needs more calculcation time, but also since this sorting is done at deconstruction, meaning that later graph update layer wont have an effect on it><note you migth also ask yourself if you could not just remove the permutation from the encoding layer, but this is easier said than done: As it is true, that the sorting is generally just done for implementation, but as you combine 4 values into for example two, you could have situations, in which node 0 and 3, as well as node 1 and 2 are combined together into (0,3) and (1,2), and even without sorting, reconstruction this, would result in 0,3,1,2, so you would still either need some kind of permutation in the decompression, or some kind of shortcuts between the layers, that encodes the original position: This would not be bad style, as it could result in the network learning to misuse this information to encode arbitrary information, but would also not be very easy to implement, and migth require a nonpermutation invariant compression and decompression function to work well, which would obviously not be ideal, as keeping permutation invariance is the main reason for this chapter>.
+So finally: sorting seems to be the rigth choice for us, but a more advanced algorithm, migth still be useful: consider the data from Chapter <ref feyn>: sorting by one parameter is not that useful, when you only have boolean datapoints<note even though in this chapter no real sorting was used, and you could still work with our approach and multiple sorting layers fairly well, assuming tf.math.top\_k is stable (their documentation does not say so, but the implementation is, but this may chance since also tf.argsort is stable at the time of writing this, but they want to implement a not stable version later to improve the speed of this algorithm)>
+
diff --git a/out/label.json b/out/label.json
index d3a684cbf504794139aacb61560481be0e85d762..e20818945babfe53c3aa452284bd5426a0becec0 100644
--- a/out/label.json
+++ b/out/label.json
@@ -1510,6 +1510,15 @@
     "label": "aimplementationphi",
     "file": "..\\..\\write\\/data\\10anhang\\50implementation"
   },
+  {
+    "typ": "img",
+    "files": [
+      "../imgs/none"
+    ],
+    "label": "none",
+    "caption": "loss distribution with angular overflow",
+    "where": "..\\..\\write\\/data\\10anhang\\50implementation"
+  },
   {
     "typ": "subsection",
     "title": "misusing the Graph Structure as effective activation",
@@ -1522,6 +1531,26 @@
     "label": "asort",
     "file": "..\\..\\write\\/data\\10anhang\\60sorting"
   },
+  {
+    "typ": "img2",
+    "files": [
+      "../imgs/none",
+      "../imgs/none"
+    ],
+    "label": "none",
+    "caption": "sorting in network plot comparison",
+    "where": "..\\..\\write\\/data\\10anhang\\60sorting"
+  },
+  {
+    "typ": "img2",
+    "files": [
+      "../imgs/none",
+      "../imgs/none"
+    ],
+    "label": "none",
+    "caption": "training curve comparison for sorting",
+    "where": "..\\..\\write\\/data\\10anhang\\60sorting"
+  },
   {
     "typ": "subsection",
     "title": "Permutation Invariance good or bad?",
diff --git a/out/main.aux b/out/main.aux
index 2bea30bd8b4f3602c976780a708b77aa92c76dfb..6e6d7c698320bbeef60f01c03737a12cb034ced6 100644
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+++ b/out/main.aux
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+\newlabel{sec:ameans}{{9.15}{119}{Why Autoencoder reproduce mean Values}{subsection.9.15}{}}
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index f6cdfc096cbfd11e1ba984443fe3ae3c085e5144..c341bb304dce3d8f5ace516378e96800d1ee4aba 100644
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+++ b/out/main.log
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 %%EOF
diff --git a/out/main.tex b/out/main.tex
index 9f0e334a26f1f79eb326a8cd1958ae0532792a13..d3f7e64d06723749a444559ae5dd0a0da34e7d23 100644
--- a/out/main.tex
+++ b/out/main.tex
@@ -2179,6 +2179,20 @@ So how to solve this? one way would be to connect everything below a fixed dista
 
 \subsubsection{overflow in angular differences, and how to solve it}\label{sec:aimplementationphi}
 
+Our input data contains a $\phi$ that is centered around its mean: so the most simple implementation would simply subtract the mean of $\phi$ from each $\phi$ value. This can lead to overflow problems, since $\phi = 2 \cdot \pi$ is equivalent to $\phi = 0$ and thus a mean of about $6$ could be sutracted from a tiny value\footnote{or the inverse} resulting in weird phi distributions. Not solving this, results in a loss distribution with a small peak at very high losses
+\begin{figure}[H] 
+  \centering
+\includegraphics[width=0.9\textwidth]{../imgs/none}
+\caption{loss distribution with angular overflow}
+\label{fig:none}
+  \end{figure}
+
+
+So how to solve this? first we need the true mean value ($- 0.1 + 2 \cdot \pi$ and $0.1$ have mean $0$ and not $\pi$), and then we use a modular operator to restrict every difference (the difference $- 0.1 + 2 \cdot \pi$ molulo $2 \cdot \pi$ equals the true difference of $-0.1$). And for finding this mean value we can cheat a little, but calculating the mean 4 vector, and finding out its $\phi$ value. For a more indepth look at out solution, you can also take a look at the actual implementation (ENTER LINK TO gpre5).
+
+
+
+
 
 
 
@@ -2192,6 +2206,40 @@ So how to solve this? one way would be to connect everything below a fixed dista
 \subsection{the consequences of sorting outputs by lpt}\label{sec:asort}
 
 
+Sorting nodes at the end of the Autoencoder breaks the permutation symmetry that we praised at the beginning of this thesis (Chapter \ref{sec:graphs}), and, since we use $lp_{T}$ for this sorting, we artificially inflate the importance of the momentum variable compared to the other variables. That beeing said, turning of this sorting, does hurt the network performance: Given the two Networks
+\begin{figure}[H]
+\begin{center}$
+\begin{array}{c}
+\includegraphics[width=0.9\textwidth]{../imgs/none} \\
+\includegraphics[width=0.9\textwidth]{../imgs/none}
+\end{array}$
+\end{center}
+\caption{sorting in network plot comparison}
+\label{fig:none}
+\end{figure}
+
+
+
+
+The sorted Network reduces an AUC value (trained on qcd) of $0.6351$ into $0.5788$ and probably even more importantly, the training curve looks way worse
+\begin{figure}[H]
+\begin{center}$
+\begin{array}{c}
+\includegraphics[width=0.9\textwidth]{../imgs/none} \\
+\includegraphics[width=0.9\textwidth]{../imgs/none}
+\end{array}$
+\end{center}
+\caption{training curve comparison for sorting}
+\label{fig:none}
+\end{figure}
+
+
+
+
+You migth consider the not sortet curve more clean, but it also does not really improve any further at a fairly early epoch, result in the sorting network reaching a loss of about a factor 3 smaller. Seeing this, sorting seems like a clear choice for us, theoretical concerns dont fair well compared to practical results, and so basically all Networks in this Thesis are sorted. That beeing said, the original deficits of breaking permutations symmetry, could actually be interpreted to mean the opposite: while it is true, that switching each value, except the sorted one, would not result in the same loss, switching any whole node position, still would result in exactly the same loss. In fact, we can use this, to understand why not sorted Networks are so bad: The encoding includes a random\footnote{actually not random, but you could see it like this, if you only see the initial and final node indices} node permutation, while the decoding does not, so after the autoencoder, the result is a random permuation of the Input features, which then are compared to the still initially ordered Input features. That this does not work that well should be clear: So either choose the momentum axis as sorting value\footnote{which would not be what you would want, since locality in real space is much more important than similar energies, as Chapter \ref{sec:ametrikana} shows} or compare your predictions neirly randomly. In fact, you could argue, that this breaks permutation symmetry, as you impose a defined ordering on your node indices. Finally, if you would want to improve this, you migth look at two things: making the comparison variable learnable, would remove the artificial inflated importance of $lp_{T}$\footnote{but maybe also make the training less stable, and add a less controlable importance to some other mixture of features} and making the decompression chance the node ordering, best case in a learnable way, would make this whole discussion moot, as the network could converge as good with, as without sorting. That beeing suggested, implementing this is not neccesarily easy, as you would not want any function to apply to all nodes, to make sure you dont break permutation symmetry, which for me looks like you restrict yourself to finding a variable to sort by and to reverse an initial sorting would in general not be easy at all, as the initial sorting could be completely random, but would result here probably in a network sorting the nodes by their transverse momentum, as this is the sorting of the initial Data, but this seems to us, as a more complicated implementation of our final sorting layer\footnote{that could actually work less well, not only since it needs more calculcation time, but also since this sorting is done at deconstruction, meaning that later graph update layer wont have an effect on it}\footnote{you migth also ask yourself if you could not just remove the permutation from the encoding layer, but this is easier said than done: As it is true, that the sorting is generally just done for implementation, but as you combine 4 values into for example two, you could have situations, in which node 0 and 3, as well as node 1 and 2 are combined together into (0,3) and (1,2), and even without sorting, reconstruction this, would result in 0,3,1,2, so you would still either need some kind of permutation in the decompression, or some kind of shortcuts between the layers, that encodes the original position: This would not be bad style, as it could result in the network learning to misuse this information to encode arbitrary information, but would also not be very easy to implement, and migth require a nonpermutation invariant compression and decompression function to work well, which would obviously not be ideal, as keeping permutation invariance is the main reason for this chapter}.
+So finally: sorting seems to be the rigth choice for us, but a more advanced algorithm, migth still be useful: consider the data from Chapter \ref{sec:feyn}: sorting by one parameter is not that useful, when you only have boolean datapoints\footnote{even though in this chapter no real sorting was used, and you could still work with our approach and multiple sorting layers fairly well, assuming tf.math.top\_k is stable (their documentation does not say so, but the implementation is, but this may chance since also tf.argsort is stable at the time of writing this, but they want to implement a not stable version later to improve the speed of this algorithm)}
+
+
 
 %from file ..\..\write\/data\10anhang\65permutationprocon
 \subsection{Permutation Invariance good or bad?}\label{sec:aperminv}
diff --git a/out/main.toc b/out/main.toc
index 75aaadb3045921c95871a5cd191336bd04378d2f..d03f9081e1cfefc470a9c4a964dc731f5c980f6a 100644
--- a/out/main.toc
+++ b/out/main.toc
@@ -118,10 +118,10 @@
 \contentsline {subsubsection}{\numberline {9.6.1}overflow in angular differences, and how to solve it}{115}{subsubsection.9.6.1}%
 \contentsline {subsection}{\numberline {9.7}misusing the Graph Structure as effective activation}{115}{subsection.9.7}%
 \contentsline {subsection}{\numberline {9.8}the consequences of sorting outputs by lpt}{115}{subsection.9.8}%
-\contentsline {subsection}{\numberline {9.9}Permutation Invariance good or bad?}{115}{subsection.9.9}%
-\contentsline {subsection}{\numberline {9.10}The usage of a BatchNormalization Layer in the middle of the gae}{115}{subsection.9.10}%
-\contentsline {subsection}{\numberline {9.11}Uncertainity and reproducability of AUC values}{115}{subsection.9.11}%
-\contentsline {subsection}{\numberline {9.12}Is it a good idea to relearn the graph at each step}{115}{subsection.9.12}%
-\contentsline {subsection}{\numberline {9.13}Trainingsize, and why graph autoencoder dont care about it}{115}{subsection.9.13}%
-\contentsline {subsection}{\numberline {9.14}Graph Autoencoder as Autoencoder with some graph Layers in front}{115}{subsection.9.14}%
-\contentsline {subsection}{\numberline {9.15}Why Autoencoder reproduce mean Values}{115}{subsection.9.15}%
+\contentsline {subsection}{\numberline {9.9}Permutation Invariance good or bad?}{119}{subsection.9.9}%
+\contentsline {subsection}{\numberline {9.10}The usage of a BatchNormalization Layer in the middle of the gae}{119}{subsection.9.10}%
+\contentsline {subsection}{\numberline {9.11}Uncertainity and reproducability of AUC values}{119}{subsection.9.11}%
+\contentsline {subsection}{\numberline {9.12}Is it a good idea to relearn the graph at each step}{119}{subsection.9.12}%
+\contentsline {subsection}{\numberline {9.13}Trainingsize, and why graph autoencoder dont care about it}{119}{subsection.9.13}%
+\contentsline {subsection}{\numberline {9.14}Graph Autoencoder as Autoencoder with some graph Layers in front}{119}{subsection.9.14}%
+\contentsline {subsection}{\numberline {9.15}Why Autoencoder reproduce mean Values}{119}{subsection.9.15}%