From 9215ef72c48cd9ad7c25cc10fa54c648e1d46c78 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Sun, 28 Aug 2022 21:15:26 +0200
Subject: kugel: Applications introductory paragraph

---
 buch/papers/kugel/applications.tex | 10 +++++++++-
 1 file changed, 9 insertions(+), 1 deletion(-)

(limited to 'buch/papers/kugel/applications.tex')

diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex
index b2f227e..40d5291 100644
--- a/buch/papers/kugel/applications.tex
+++ b/buch/papers/kugel/applications.tex
@@ -1,9 +1,17 @@
-% vim:ts=2 sw=2 et spell:
+% vim:ts=2 sw=2 et spell tw=80:
 
 \section{Applications}
 
+As suggested in the previous section, the fact that it is possible to write a
+Fourier style expansion of any function on the surface of the sphere is very
+useful in many fields of physics and engineering. Here we will present a few of
+the most interesting applications we came across during our research.
+
 \subsection{Electroencephalography (EEG)}
 
+\begin{figure}
+\end{figure}
+
 \subsection{Measuring Gravitational Fields}
 
 \subsection{Quantisation of Angular Momentum}
-- 
cgit v1.2.1


From 8f80489e9efbadba522098247bd240cc5c59dc38 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Mon, 29 Aug 2022 01:32:55 +0200
Subject: kugel: EEG electrodynamics recap

---
 buch/papers/kugel/applications.tex | 50 +++++++++++++++++++++++++++++++++++++-
 1 file changed, 49 insertions(+), 1 deletion(-)

(limited to 'buch/papers/kugel/applications.tex')

diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex
index 40d5291..39e4cfc 100644
--- a/buch/papers/kugel/applications.tex
+++ b/buch/papers/kugel/applications.tex
@@ -7,11 +7,59 @@ Fourier style expansion of any function on the surface of the sphere is very
 useful in many fields of physics and engineering. Here we will present a few of
 the most interesting applications we came across during our research.
 
-\subsection{Electroencephalography (EEG)}
+\subsection{Electroencephalography}
 
 \begin{figure}
+  \centering
+  \subfigure[EEG Electrodes \label{kugel:fig:eeg-electrodes}]%
+    {\kugelplaceholderfig{.4\linewidth}{5cm}}
+  \qquad
+  \subfigure[Gauss' Law \label{kugel:fig:eeg-flux}]%
+    {\includegraphics[width=.4\linewidth]{papers/kugel/figures/flux}}
+  \caption{
+    \label{kugel:fig:eeg}
+  }
 \end{figure}
 
+To start, we will look at an application that is from the field of medicine:
+electroencephalography. The \emph{electroencephalogram} (EEG) is a measurement
+of the electrical field on the scalp, which shows the brain's activity, and is
+used in many fields of research such as neurology and cognitive psychology.  The
+measurement is done by wearing a cap that contains a number of evenly
+distributed electrodes, each of which measures the electric potential (voltage)
+at their location (figure \ref{kugel:fig:eeg-electrodes}).  To see how this will
+relate to the spherical harmonics, we will first quickly recap a bit of physics,
+electrodynamics to be precise.
+
+In section \ref{kugel:sec:construction:eigenvalue} we have shown that the
+spherical harmonics arise from the surface spherical Laplacian operator, whose
+origin we did not consider too much, which is how mathematicians do their work.
+On the contrary, physicists usually do the opposite and start by discussing what
+is happening, since variables and functions represent physical quantities. So,
+we will quickly remind that the Laplacian operator does the following to an
+electric potential $\phi$:
+\begin{equation*}
+  \nabla^2 \phi
+  = \nabla \cdot \nabla \phi
+  = \nabla \cdot \mathbf{E}
+  = \rho / \varepsilon
+  \iff
+  \iiint_\Omega \nabla \cdot \mathbf{E} \, dv
+  = \iint_{\partial \Omega} \mathbf{E} \cdot d\mathbf{s}
+  = \Phi / \varepsilon.
+\end{equation*}
+Put into words: on the left we have the differential form, where we recall that
+the Laplacian (which is a second derivative) is the divergence of the gradient.
+Unpacking the notation we first see that we have the gradient of the potential,
+which is just the electric field $\mathbf{E}$, and then the divergence of said
+electric field is proportional to the charge density $\rho$. So, the Laplacian
+of the electric potential is the charge density! For those that are more
+familiar with the integral form of Maxwell's equation, we have also included an
+additional step using the divergence theorem, which brings us to the electric
+Flux, which by Gauss' law (shown in the iconic\footnote{Every electrical
+engineer has seen this picture so many times that is probably burnt in their
+eyes.} figure \ref{kugel:fig:eeg-flux}) is the net electric charge.
+
 \subsection{Measuring Gravitational Fields}
 
 \subsection{Quantisation of Angular Momentum}
-- 
cgit v1.2.1


From ddfa6f1258287cbbe4ece43869e6431a655f664b Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Tue, 30 Aug 2022 13:19:13 +0200
Subject: kugel: Start EEG interpolation problem

---
 buch/papers/kugel/applications.tex | 42 ++++++++++++++++++++++++++++++++------
 1 file changed, 36 insertions(+), 6 deletions(-)

(limited to 'buch/papers/kugel/applications.tex')

diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex
index 39e4cfc..eb6e02f 100644
--- a/buch/papers/kugel/applications.tex
+++ b/buch/papers/kugel/applications.tex
@@ -35,15 +35,15 @@ In section \ref{kugel:sec:construction:eigenvalue} we have shown that the
 spherical harmonics arise from the surface spherical Laplacian operator, whose
 origin we did not consider too much, which is how mathematicians do their work.
 On the contrary, physicists usually do the opposite and start by discussing what
-is happening, since variables and functions represent physical quantities. So,
-we will quickly remind that the Laplacian operator does the following to an
-electric potential $\phi$:
+is happening in the real world, since variables represent physical quantities.
+So, we will quickly remind that the Laplacian operator does the following to an
+electric potential $\phi(x, y, z)$:
 \begin{equation*}
   \nabla^2 \phi
   = \nabla \cdot \nabla \phi
   = \nabla \cdot \mathbf{E}
-  = \rho / \varepsilon
-  \iff
+  = \rho / \varepsilon,
+  \quad \text{or} \quad
   \iiint_\Omega \nabla \cdot \mathbf{E} \, dv
   = \iint_{\partial \Omega} \mathbf{E} \cdot d\mathbf{s}
   = \Phi / \varepsilon.
@@ -58,7 +58,37 @@ familiar with the integral form of Maxwell's equation, we have also included an
 additional step using the divergence theorem, which brings us to the electric
 Flux, which by Gauss' law (shown in the iconic\footnote{Every electrical
 engineer has seen this picture so many times that is probably burnt in their
-eyes.} figure \ref{kugel:fig:eeg-flux}) is the net electric charge.
+eyes.} figure \ref{kugel:fig:eeg-flux}) equals the net electric charge.
+
+Now, an important observation is that if we switch to spherical coordinates, the
+physics does not change. So, the spherical Laplacian $\sphlaplacian$ of the
+electric potential $\phi(r, \vartheta, \varphi)$ is still the charge density (in
+spherical coordinates). And what about the surface spherical Laplacian
+$\surflaplacian$? To that case the physics is also indifferent, the only change
+is that the units result is a \emph{surface} charge density $\rho_s$. Thus, we
+are done with physics and finally arrive at the engineers' perspective: how can
+we use this fact to build something that reads the current flows on the surface
+of the brain?
+
+The details of how EEG actually works gets very complicated very quickly, but we
+will try our best to give an broad overview of the mathematical machinery that
+makes it possible to measure brain waves. See, the problem neither the physicist
+nor the mathematician considered is that we cannot measure the electric field in
+its entirety. As show in figure \ref{kugel:fig:eeg-electrodes} the electrodes
+give measurements that are only available at discrete locations, and we are thus
+quite a lot of missing data. In other words, we have an interpolation problem.
+And (at this point not so surprisingly) we will show that it can be solved using
+the spherical harmonics.
+
+To solve this new interpolation problem, we will start with a blatantly
+engineering assumption: the human head is a sphere of radius $R$, with the value
+of $R$ begin the radius of the average human head. So, we will assume that the
+potential distribution on the head can be written as a finite linear combination
+of spherical harmonics:
+\begin{equation*}
+  \phi(\vartheta, \varphi)
+    = \sum_{n=0}^N \sum_{m=-n}^n a_{m,n} Y^m_n(\vartheta, \varphi)
+\end{equation*}
 
 \subsection{Measuring Gravitational Fields}
 
-- 
cgit v1.2.1


From 3c2859e6ea73355eb583e74756ee3584e353ca69 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Tue, 30 Aug 2022 17:24:54 +0200
Subject: kugel: Electrodes images and finish interpolation

---
 buch/papers/kugel/applications.tex | 82 +++++++++++++++++++++++++++++++++++---
 1 file changed, 76 insertions(+), 6 deletions(-)

(limited to 'buch/papers/kugel/applications.tex')

diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex
index eb6e02f..32095c4 100644
--- a/buch/papers/kugel/applications.tex
+++ b/buch/papers/kugel/applications.tex
@@ -12,7 +12,8 @@ the most interesting applications we came across during our research.
 \begin{figure}
   \centering
   \subfigure[EEG Electrodes \label{kugel:fig:eeg-electrodes}]%
-    {\kugelplaceholderfig{.4\linewidth}{5cm}}
+    % {\kugelplaceholderfig{.4\linewidth}{5cm}}
+    {\includegraphics[width=.45\linewidth, frame]{papers/kugel/figures/electrodes}}
   \qquad
   \subfigure[Gauss' Law \label{kugel:fig:eeg-flux}]%
     {\includegraphics[width=.4\linewidth]{papers/kugel/figures/flux}}
@@ -31,6 +32,9 @@ at their location (figure \ref{kugel:fig:eeg-electrodes}).  To see how this will
 relate to the spherical harmonics, we will first quickly recap a bit of physics,
 electrodynamics to be precise.
 
+
+\subsubsection{Electrodynamics}
+
 In section \ref{kugel:sec:construction:eigenvalue} we have shown that the
 spherical harmonics arise from the surface spherical Laplacian operator, whose
 origin we did not consider too much, which is how mathematicians do their work.
@@ -70,6 +74,8 @@ are done with physics and finally arrive at the engineers' perspective: how can
 we use this fact to build something that reads the current flows on the surface
 of the brain?
 
+\subsubsection{EEG as Interpolation Problem}
+
 The details of how EEG actually works gets very complicated very quickly, but we
 will try our best to give an broad overview of the mathematical machinery that
 makes it possible to measure brain waves. See, the problem neither the physicist
@@ -82,13 +88,77 @@ the spherical harmonics.
 
 To solve this new interpolation problem, we will start with a blatantly
 engineering assumption: the human head is a sphere of radius $R$, with the value
-of $R$ begin the radius of the average human head. So, we will assume that the
-potential distribution on the head can be written as a finite linear combination
-of spherical harmonics:
+of $R$ begin the average radius of a human head (which is around 11 cm). So, we
+will assume that the potential distribution on the head can be written as a
+finite linear combination of spherical harmonics:
 \begin{equation*}
-  \phi(\vartheta, \varphi)
-    = \sum_{n=0}^N \sum_{m=-n}^n a_{m,n} Y^m_n(\vartheta, \varphi)
+  V(\vartheta, \varphi)
+    = \sum_{n=0}^N \sum_{m=-n}^n a_{m,n} Y^m_n(\vartheta, \varphi),
 \end{equation*}
+where the values $a_{m,n}$ are the unknowns of our interpolation problem. Now to
+the measurements: we let $\phi_1, \phi_2, \ldots, p_M$ be the measured voltages
+at points in space $p_1, p_2, \ldots, p_M$ (position of the electrodes). To
+simplify, we will assume that the electrodes are reasonably evenly distributed,
+which means that we have no points that are on top of each other or at wildly
+different radii from the origin. With that out of the way, we can now write a
+minimization problem:
+\begin{subequations}
+  \begin{align}
+    a_{m,n}^* &= \arg \min_{a_{m,n}}
+      \int_{\partial S} | \surflaplacian V |^2 \, ds 
+      = \int_0^{2\pi} \int_{0}^\pi | \surflaplacian V |^2
+        \sin \vartheta \, d\vartheta d\varphi, 
+        \label{kugel:eqn:eeg-min} \\
+    &\text{under the constraints} \quad V(p_j) = \phi_j
+      \quad \text{ for } \quad 1 < j < M.
+      \label{kugel:eqn:eeg-min-constraints}
+  \end{align}
+\end{subequations}
+Essentially, with \eqref{kugel:eqn:eeg-min} we are are asking for the solution
+to be smooth by minimizing the square of the total curvature (recall that the
+surface spherical Laplacian $\surflaplacian$ is a measure of curvature), while
+at the same time with \eqref{kugel:eqn:eeg-min-constraints}, we force the
+solution to go through the measured points. The latter is the reason why we
+needed to assumed that the measurements are at reasonable locations, something
+that (as every engineer show know) is not necessarily the case in the real
+world! Thus, to solve this problem, we will use the suspiciously convenient fact
+that (hint: eigenvalues)
+\begin{equation*}
+  \surflaplacian V(\vartheta, \varphi)
+    = \sum_{n=0}^N \sum_{m=-n}^n a_{m,n}
+      \surflaplacian Y^m_n(\vartheta, \varphi)
+    = \sum_{n=0}^N \sum_{m=-n}^n a_{m,n}
+      n(n+1) Y^m_n(\vartheta, \varphi).
+\end{equation*}
+So that when substituted into \eqref{kugel:eqn:eeg-min} results in
+\begin{align*}
+  \int_{\partial S} \left[
+    \sum_{n=0}^N \sum_{m=-n}^n n(n+1) |a_{m,n}|
+    Y^m_n(\vartheta, \varphi)
+  \right]^2 ds
+  = \sum_{m, m'} \sum_{n, n'} |a_{m',n'}| |a_{m,n}|
+    n'(n'+1) n(n+1)
+    \underbrace{\int_{\partial S} Y^{m'}_{n'} Y^m_n \, ds}_{
+      \langle Y^{m'}_{n'}, Y^m_n \rangle
+    },
+\end{align*}
+where we used a ``sloppy'' double sum notation to indicate that we have a bunch
+of terms of that form. We did not bother to properly expand the product of
+double sums, because we can see that at the end we end up with an inner product
+$\langle Y^{m'}_{n'}, Y^m_n \rangle$, which as we know equals $\delta_{m'm}
+\delta_{n'n}$, so all of the terms where $n' \neq n$ or $m' \neq m$ can be
+dropped and \eqref{kugel:eqn:eeg-min} simplifies down to
+\nocite{pascual-marqui_current_1988}
+\begin{equation}
+  a^*_{m,n} = \arg \min_{a_{m,n}} 
+    \sum_{n=0}^N \sum_{m=-n}^n n^2 (n+1)^2 |a_{m,n}|^2.
+\end{equation}
+
+At this point, we could continue solving for an analytical solution to the
+minimization problem, for example by differentiating with respect to some
+$a_{j,k}$, setting that to zero and so forth, but the job of the spherical
+harmonics ends here. So, we will not pursue this further, and instead discuss a
+few interesting problems that come with this approach.
 
 \subsection{Measuring Gravitational Fields}
 
-- 
cgit v1.2.1


From eeade76703ccfc055e28e8356df60d8b2a45a6d5 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Tue, 30 Aug 2022 19:55:36 +0200
Subject: kugel: EEG Nyquist

---
 buch/papers/kugel/applications.tex | 40 +++++++++++++++++++++++++++-----------
 1 file changed, 29 insertions(+), 11 deletions(-)

(limited to 'buch/papers/kugel/applications.tex')

diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex
index 32095c4..1af0018 100644
--- a/buch/papers/kugel/applications.tex
+++ b/buch/papers/kugel/applications.tex
@@ -93,7 +93,7 @@ will assume that the potential distribution on the head can be written as a
 finite linear combination of spherical harmonics:
 \begin{equation*}
   V(\vartheta, \varphi)
-    = \sum_{n=0}^N \sum_{m=-n}^n a_{m,n} Y^m_n(\vartheta, \varphi),
+    = \sum_{n=1}^N \sum_{m=-n}^n a_{m,n} Y^m_n(\vartheta, \varphi),
 \end{equation*}
 where the values $a_{m,n}$ are the unknowns of our interpolation problem. Now to
 the measurements: we let $\phi_1, \phi_2, \ldots, p_M$ be the measured voltages
@@ -125,20 +125,20 @@ world! Thus, to solve this problem, we will use the suspiciously convenient fact
 that (hint: eigenvalues)
 \begin{equation*}
   \surflaplacian V(\vartheta, \varphi)
-    = \sum_{n=0}^N \sum_{m=-n}^n a_{m,n}
+    = \sum_{n=1}^N \sum_{m=-n}^n a_{m,n}
       \surflaplacian Y^m_n(\vartheta, \varphi)
-    = \sum_{n=0}^N \sum_{m=-n}^n a_{m,n}
+    = \sum_{n=1}^N \sum_{m=-n}^n a_{m,n}
       n(n+1) Y^m_n(\vartheta, \varphi).
 \end{equation*}
 So that when substituted into \eqref{kugel:eqn:eeg-min} results in
 \begin{align*}
-  \int_{\partial S} \left[
-    \sum_{n=0}^N \sum_{m=-n}^n n(n+1) |a_{m,n}|
+  \int_{\partial S} \left|
+    \sum_{n=1}^N \sum_{m=-n}^n n(n+1) a_{m,n}
     Y^m_n(\vartheta, \varphi)
-  \right]^2 ds
-  = \sum_{m, m'} \sum_{n, n'} |a_{m',n'}| |a_{m,n}|
+  \right|^2 ds
+  = \sum_{m, m'} \sum_{n, n'} a_{m',n'} \overline{a_{m,n}}
     n'(n'+1) n(n+1)
-    \underbrace{\int_{\partial S} Y^{m'}_{n'} Y^m_n \, ds}_{
+    \underbrace{\int_{\partial S} Y^{m'}_{n'} \overline{Y^m_n} \, ds}_{
       \langle Y^{m'}_{n'}, Y^m_n \rangle
     },
 \end{align*}
@@ -151,14 +151,32 @@ dropped and \eqref{kugel:eqn:eeg-min} simplifies down to
 \nocite{pascual-marqui_current_1988}
 \begin{equation}
   a^*_{m,n} = \arg \min_{a_{m,n}} 
-    \sum_{n=0}^N \sum_{m=-n}^n n^2 (n+1)^2 |a_{m,n}|^2.
+    \sum_{n=1}^N \sum_{m=-n}^n n^2 (n+1)^2 |a_{m,n}|^2.
 \end{equation}
 
 At this point, we could continue solving for an analytical solution to the
 minimization problem, for example by differentiating with respect to some
 $a_{j,k}$, setting that to zero and so forth, but the job of the spherical
-harmonics ends here. So, we will not pursue this further, and instead discuss a
-few interesting problems that come with this approach.
+harmonics ends here. So, we will not pursue this further, and instead briefly
+discuss a few interesting implications and problems. 
+
+\subsubsection{Sampling, smoothness and problems}
+
+The most interesting perhaps unforeseen fact is that with this method we are
+getting a free (!) spectral analysis, since the coefficients $a_{m,n}$ are the
+spectrum of the interpolated electric field $V(\vartheta, \varphi)$. However,
+like in the non spherical Fourier transformation, we only get a \emph{finite}
+resolution since our measurement are spatially discrete. In fact, if we know the
+mean angular inter-electrode distance $\gamma$ we can actually formulate a
+Nyquist frequency limit just like in the usual Fourier theory:
+\begin{equation*}
+  f_N = \frac{\pi}{2T}
+  \iff
+  n_N = \left\lfloor \frac{\pi}{2\gamma} \right\rfloor.
+\end{equation*}
+
+\nocite{wingeier_spherical_2001}
+
 
 \subsection{Measuring Gravitational Fields}
 
-- 
cgit v1.2.1


From 5c62e1f41394736e371983d4d9f85502f312a9eb Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Tue, 30 Aug 2022 20:42:08 +0200
Subject: kugel: EEG end

---
 buch/papers/kugel/applications.tex | 49 ++++++++++++++++++++++++++++----------
 1 file changed, 36 insertions(+), 13 deletions(-)

(limited to 'buch/papers/kugel/applications.tex')

diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex
index 1af0018..15a57d5 100644
--- a/buch/papers/kugel/applications.tex
+++ b/buch/papers/kugel/applications.tex
@@ -78,13 +78,13 @@ of the brain?
 
 The details of how EEG actually works gets very complicated very quickly, but we
 will try our best to give an broad overview of the mathematical machinery that
-makes it possible to measure brain waves. See, the problem neither the physicist
-nor the mathematician considered is that we cannot measure the electric field in
-its entirety. As show in figure \ref{kugel:fig:eeg-electrodes} the electrodes
-give measurements that are only available at discrete locations, and we are thus
-quite a lot of missing data. In other words, we have an interpolation problem.
-And (at this point not so surprisingly) we will show that it can be solved using
-the spherical harmonics.
+makes it possible to measure brain waves. The problem neither the physicist nor
+the mathematician considered is that we cannot measure the electric field in its
+entirety. As show in figure \ref{kugel:fig:eeg-electrodes} the electrodes give
+measurements that are only available at discrete locations, and we are thus
+missing quite a lot of data. Or in other words, we have an interpolation
+problem, which (at this point not so surprisingly) we will show can be solved
+using the spherical harmonics.
 
 To solve this new interpolation problem, we will start with a blatantly
 engineering assumption: the human head is a sphere of radius $R$, with the value
@@ -160,7 +160,8 @@ $a_{j,k}$, setting that to zero and so forth, but the job of the spherical
 harmonics ends here. So, we will not pursue this further, and instead briefly
 discuss a few interesting implications and problems. 
 
-\subsubsection{Sampling, smoothness and problems}
+\subsubsection{Sampling, Smoothness and Problems}
+\nocite{wingeier_spherical_2001, ruffini_spherical_2002}
 
 The most interesting perhaps unforeseen fact is that with this method we are
 getting a free (!) spectral analysis, since the coefficients $a_{m,n}$ are the
@@ -168,15 +169,37 @@ spectrum of the interpolated electric field $V(\vartheta, \varphi)$. However,
 like in the non spherical Fourier transformation, we only get a \emph{finite}
 resolution since our measurement are spatially discrete. In fact, if we know the
 mean angular inter-electrode distance $\gamma$ we can actually formulate a
-Nyquist frequency limit just like in the usual Fourier theory:
-\begin{equation*}
+Nyquist frequency just like in the usual Fourier theory:
+\begin{equation}
   f_N = \frac{\pi}{2T}
   \iff
   n_N = \left\lfloor \frac{\pi}{2\gamma} \right\rfloor.
-\end{equation*}
-
-\nocite{wingeier_spherical_2001}
+\end{equation}
 
+Before concluding this overview of EEG, we should point out that in practice
+there are about a million problems with this oversimplified approach. We do not
+intend to give an in depth explanation (since the authors themselves are not
+experts in any of these fields), but there are a few problems that are too big
+to ignore, so we will very briefly discuss them now. The first important
+real-world problem is that the electrodes are not necessarily at a reasonable
+location, so the constraint \eqref{kugel:eqn:eeg-min-constraints} is a bit too
+strong, and may end up fitting some noise or disturbances in the measurement. A
+simple solution may for example be to introduce a smoothness factor $\lambda >
+0$ as follows:
+\begin{equation}
+  V(\vartheta, \varphi) = \sum_{n=1}^N \sum_{m=-n}^n 
+    \frac{a_{m,n}}{1 + \lambda n^2(n+1)^2} Y^m_n(\vartheta, \varphi).
+\end{equation}
+To find proper smoothness factor $\lambda$, is another problem of its own, thus
+we will not discuss it here, since this is getting too long already. Another
+important issue is that in the real world, we cannot ``evenly distribute'' the
+electrodes on our head. As shown in the image, most of the electrodes are on a
+cap, and then there are just a few on the face, and almost none near the jawline
+and chin. This not something that can be ignored, and in fact, makes the
+analysis much more difficult. Finally, the most obvious problem is that human
+heads are not perfect spheres. Here too, it is possible to account for this fact
+and model the head with a more complex shape at the cost of making the math
+quite unwieldy.
 
 \subsection{Measuring Gravitational Fields}
 
-- 
cgit v1.2.1


From 1bdb803ced744bcfe7cf81c89a740fcbcf6bdc70 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Tue, 30 Aug 2022 22:42:54 +0200
Subject: kugel: Minor corrections

---
 buch/papers/kugel/applications.tex | 7 +++----
 1 file changed, 3 insertions(+), 4 deletions(-)

(limited to 'buch/papers/kugel/applications.tex')

diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex
index 15a57d5..10bf153 100644
--- a/buch/papers/kugel/applications.tex
+++ b/buch/papers/kugel/applications.tex
@@ -32,7 +32,6 @@ at their location (figure \ref{kugel:fig:eeg-electrodes}).  To see how this will
 relate to the spherical harmonics, we will first quickly recap a bit of physics,
 electrodynamics to be precise.
 
-
 \subsubsection{Electrodynamics}
 
 In section \ref{kugel:sec:construction:eigenvalue} we have shown that the
@@ -88,7 +87,7 @@ using the spherical harmonics.
 
 To solve this new interpolation problem, we will start with a blatantly
 engineering assumption: the human head is a sphere of radius $R$, with the value
-of $R$ begin the average radius of a human head (which is around 11 cm). So, we
+of $R$ being the average radius of a human head (which is around 11 cm). So, we
 will assume that the potential distribution on the head can be written as a
 finite linear combination of spherical harmonics:
 \begin{equation*}
@@ -110,7 +109,7 @@ minimization problem:
         \sin \vartheta \, d\vartheta d\varphi, 
         \label{kugel:eqn:eeg-min} \\
     &\text{under the constraints} \quad V(p_j) = \phi_j
-      \quad \text{ for } \quad 1 < j < M.
+      \quad \text{ for } \quad 1 \leq j \leq M.
       \label{kugel:eqn:eeg-min-constraints}
   \end{align}
 \end{subequations}
@@ -203,4 +202,4 @@ quite unwieldy.
 
 \subsection{Measuring Gravitational Fields}
 
-\subsection{Quantisation of Angular Momentum}
+% \subsection{Quantisation of Angular Momentum}
-- 
cgit v1.2.1


From 21f9b99c67c52e9d164a9a42f46fd2fedd9860b0 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Tue, 30 Aug 2022 23:18:54 +0200
Subject: kugel: Move figure

---
 buch/papers/kugel/applications.tex | 1 +
 1 file changed, 1 insertion(+)

(limited to 'buch/papers/kugel/applications.tex')

diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex
index 10bf153..481a3a5 100644
--- a/buch/papers/kugel/applications.tex
+++ b/buch/papers/kugel/applications.tex
@@ -18,6 +18,7 @@ the most interesting applications we came across during our research.
   \subfigure[Gauss' Law \label{kugel:fig:eeg-flux}]%
     {\includegraphics[width=.4\linewidth]{papers/kugel/figures/flux}}
   \caption{
+    Electroencephalography.
     \label{kugel:fig:eeg}
   }
 \end{figure}
-- 
cgit v1.2.1

