commit 0a0cbb5730039680562f1a6c1f0b7a0c712536d8
parent 3041a37a31d5b07d7d6ff27ee6d41ed9ee0bf096
Author: miksa234 <milutin@popovic.xyz>
Date: Mon, 26 Jul 2021 13:02:14 +0200
checkpoint
Diffstat:
9 files changed, 5604 insertions(+), 200 deletions(-)
diff --git a/src/thesis/chapters/basics.tex b/src/thesis/chapters/basics.tex
@@ -1,267 +1,210 @@
-
\subsection{Noncommutative Geometric Spaces}
-\subsubsection{Matrix Algebras and Finite Spaces}
\subsubsection{$*$-Algebra}
+To grasp the idea of encoding geometrical data into a spectral triple we
+introduce the first ingredient of a spectral triple, an unital $C^*$ algebra.
\begin{definition}
A \textit{vector space} $A$ over $\mathbb{C}$ is called a \textit{complex, unital Algebra} if, \\
$\forall a,b \in A$ :
\begin{enumerate}
\item
- $A \times A \rightarrow A$ \hspace{0.1\textwidth} \textit{bilinear} \\
- $(a, b)\ \mapsto \ a\cdot b$
- \item
- $1a = a1 =a$ \hspace{0.08\textwidth} \textit{unital} \\
+ $A \times A \rightarrow A$,
+ $(a, b)\ \mapsto \ a\cdot b$,
+ \item with an identity element
+ $1a = a1 =a$.
\end{enumerate}
-\end{definition}
-
-\begin{definition}
- A $*$-algebra is an algebra $A$ with a \textit{conjugate linear map (involution)} $*:A\ \rightarrow A$,
+ Extending the definition, a $*$-algebra is an algebra $A$ with a \textit{conjugate linear map (involution)} $*:A\ \rightarrow A$,
$\forall a, b \in A$ satisfying:
\begin{enumerate}
\item
- $(ab)^* = b^*a^*$ \hspace{0.05\textwidth} \textit{antidistributive}
+ $(ab)^* = b^*a^*$,
\item
- $(a^*)^* = a$ \hspace{0.1\textwidth} \textit{closure}
+ $(a^*)^* = a$.
\end{enumerate}
\end{definition}
In the following all unital algebras are referred to as algebras.
-\subsubsection{Functions on Discrete Spaces}
-Let $X$ be a \textit{discretized topological} space with $N$ points.
-Consider functions of a continuous $*$-algebra $C(X)$ assigning values to $\mathbb{C}$, for $f, g \in C(X)$,
-$\lambda \in \mathbb{C}$ and $x \in X$ they provide the following structures:
-
+\subsubsection{Finite Discrete Space}
+Let us consider an example of an $*$-algebra of continuous functions $C(X)$
+on a discrete topological space $X$ with $N$ points. Functions of a
+continuous $*$-algebra $C(X)$ assign values to $\mathbb{C}$, thus $f, g \in
+C(X)$, $\lambda \in \mathbb{C}$ and $x \in X$ they provide the following structure:
\begin{itemize}
\item \textit{pointwise linear} \\
- $(f + g)(x) = f(x) + g(x)$\\
- $(\lambda f)(x) = \lambda (f(x))$
+ $(f + g)(x) = f(x) + g(x)$,\\
+ $(\lambda f)(x) = \lambda (f(x)),$
\item \textit{pointwise multiplication} \\
- $fg(x) = f(x)g(x)$ \hspace{0.1\textwidth} same as $(fg)(x) = f(x)g(x))$?
+ $fg(x) = f(x)g(x)$,
\item \textit{pointwise involution} \\
- $f^*(x) = \overline{f(x)}$
+ $f^*(x) = \overline{f(x)}.$
\end{itemize}
+The $*$-algebra $C(X)$ is \textit{isomorphic} to a $*$-algebra $\mathbb{C}^N$
+with involution ($N$ number of points in $X$), we write $C(X) \simeq
+\mathbb{C}^N$. Isomorphisms are bijective maps that preserve structure and
+don't lose physical information. A function $f:X\ \rightarrow\ \mathbb{C}$
+can be represented with $N \times N$ diagonal matrices, where each diagonal
+value represents the function value at the corresponding $i$-th point for $i
+= 1,...,N$. Because of matrix multiplication and hermitian conjugate of
+matrices we have a preserving structure.
-\begin{question}
- Mathematical difference between Topological Discreet Spaces and just Discreet Spaces?
-\end{question}
-
-The author indicates that $\mathbb{C}$-valued functions on $X$ are automatically continuous.
-\begin{idea}
- CAN WE USE THE METRIC? NO!
- We know that $X$ is a \textit{finite discrete space}, meaning in an $\epsilon$-$\delta$ approach
- for each $x \in X$ the only $y \in X$, that is small enough is $x$ by itself, which implies
- $\epsilon$ is always bigger than zero, thus every function $f:X\ \rightarrow\ \mathbb{C}$ is continuous.
-\end{idea}
-
-\subsubsection{Isomorphism Property}
-Furthermore $C(X)$ $*$-algebra is \textit{isomorphic} to a $*$-algebra $\mathbb{C}^N$ with involution
-($N$ number of points in $X$), written as $C(X) \simeq \mathbb{C}^N$.
-A function $f:X\ \rightarrow\ \mathbb{C}$ can be represented with $N \times N$ diagonal matrices,
-where the value $(ii)$ is the value of the function at the corresponding
-$i$-th point ($i = 1,...,N$). The structure is preserved because of the definitions of
-matrix multiplication and the hermitian conjugate of matrices.
-
-\begin{question}
- Can isomorphisms between $C(X)$ and $\mathbb{C}^N$ be shown with matrix factorization?
-\end{question}
- Isomorphisms are bijective preserve structure and don't lose physical information/
-
-\subsubsection{Mapping Finite Discrete Spaces}
-
-\begin{definition}
- A \textit{map} between finite discrete spaces $X_1$ and $X_2$ is a function $\phi:\ X_1 \rightarrow\ X_2$
-\end{definition}
-
-For every map between finite discrete spaces there exists a corresponding map \\
-$\phi ^*:C(X_2)\ \rightarrow C(X_1)$, which `pulls back' values even if $\phi$ is not bijective.
+Moreover we can \textit{map} between finite discrete spaces $X_1$ and $X_2$ with a
+function
+\begin{align}
+ \phi:\ X_1 \rightarrow\ X_2.
+\end{align}
+For every such map there exists a corresponding map
+\begin{align}
+ \phi ^*:C(X_2)\ \rightarrow C(X_1),
+\end{align}
+which `pulls back' values even if $\phi$ is not bijective.
Note that the pullback doesn't map points back, but maps functions on an $*$-algebra $C(X)$.
-
-
-This map is called a pullback (or a $*$-homomorphism or a $*$-algebra map under pointwise product).
-Under the pointwise product:
+The pullback, in literature often called a $*$-homomorphism or a $*$-algebra map under
+pointwise product has the following properties
\begin{itemize}
- \item $\phi ^*(fg) = \phi ^*(f) \phi ^*(g)$
- \item $\phi ^*(\overline{f}) = \overline{\phi ^*(f)}$
- \item $\phi ^*(\lambda f + g) = \lambda \phi ^*(f) + \phi ^*(g)$
+ \item $\phi ^*(fg) = \phi ^*(f) \phi ^*(g)$,
+ \item $\phi ^*(\overline{f}) = \overline{\phi ^*(f)}$,
+ \item $\phi ^*(\lambda f + g) = \lambda \phi ^*(f) + \phi ^*(g)$.
\end{itemize}
-
-\begin{question}
- $\phi$ is in most cases not bijective, so how can we prove that there exists such a
- pullback for every map between discrete spaces which preserves information? For bijective
- it is given by its inverse, which by definition exists because $\phi$ is a map.
- Or I didn't understand this correctly?
-\end{question}
-
-\begin{MyExercise}
- \textbf{
- Show that $\phi :X_1\ \rightarrow \ X_2$ is injective (surjective) map of finite spaces iff
- $\phi ^* :C(X_2)\ \rightarrow \ C(X_1)$ is surjective (injective).
-}\newline
-
- Consider $X_1$ with $n$ points and $X_2$ with $m$ points. Then there are three cases:
- \begin{enumerate}
- \item $n=m$ \\
- Obviously $\phi$ is bijective and $\phi ^*$ too.
- \item $n \rangle m$ \\
- $\phi$ assigns $n$ points to $m$ points when $n \rangle m$,
- which is by definition surjective. \\
- $\phi ^*$ assigns $m$ points to $n$ points when $n \rangle m$,
- which is by definition injective. \\
- \item $n \langle m $ \\
- analogous
- \end{enumerate}
-\end{MyExercise}
+%------------ Exercise
+ The map $\phi :X_1\ \rightarrow \ X_2$ is an injective (surjective) map,
+ if only if the corresponding pullback $\phi ^* :C(X_2)\ \rightarrow \
+ C(X_1)$ is surjective (injective). Let us say, that $X_1$ has $n$ points and
+ $X_2$ with $m$ points. Then there are three different cases, first $n=m$ and
+ obviously $\phi$ is bijective and $\phi ^*$ too. Then $n > m$, in this case
+ $\phi$ assigns $n$ points to $m$ points when $n > m$, which is by definition
+ surjective. On the other hand $\phi ^*$ assigns $m$ points to $n$ points when
+ $n > m$, which is by definition injective. Lastly $n < m $, which is
+ completely analogous to the case $n > m$.
+%------------ Exercise
\subsubsection{Matrix Algebras}
\begin{definition}
- A \textit{(complex) matrix algebra} A is a direct sum, for $n_i, N \in \mathbb{N}$.
+ A \textit{(complex) matrix algebra} A is a direct sum, for $n_i, N \in
+ \mathbb{N}$
\begin{align}
- A = \bigoplus _{i=1}^{N} M_{n_i}(\mathbb{C})
+ A = \bigoplus _{i=1}^{N} M_{n_i}(\mathbb{C}).
\end{align}
The involution is the hermitian conjugate, a $*$ algebra with involution is referred to as
a matrix algebra
\end{definition}
+From a topological discrete space $X$, we can construct a $*$-algebra
+$C(X)$ which is isomorphic to a matrix algebra $A$. Then the question instantly
+arises, if we can construct $X$ given $A$? For a matrix algebra $A$,
+which in most cases is not commutative, the answer is generally no.
-So from a topological discrete space $X$, we can construct a $*$-algebra $C(X)$ which is isomorphic
-to a matrix algebra $A$. The question is can we construct $X$ given $A$? $A$ is a matrix algebra,
-which are in most cases is not commutative, so the answer is generally no.
-
-There are two options. We can restrict ourselves to commutative matrix algebras,
+Thus there are two options. We can restrict ourselves to commutative matrix algebras,
which are the vast minority and not physically interesting.
-Or we can allow more morphisms(isomorphisms) between matrix algebras.
-
-\begin{question}
- Why are non-commutative algebras not physically interesting?
- Maybe too far fetched,but because physical observables (QM-Operators) are not commutative?
-\end{question}
-Exactly.
+Or we can allow more morphisms (isomorphisms) between matrix algebras.
\subsubsection{Finite Inner Product Spaces and Representations}
-Until now we looked at a finite topological discrete space, moreover we can consider a
+Until now we looked at finite topological discrete spaces, moreover we can consider a
finite dimensional inner product space $H$ (finite Hilbert-spaces), with inner product
-$(\cdot,\cdot)\rightarrow \mathbb{C}$. $L(H)$ is the $*$-algebra of operators on $H$
-with product given by composition and involution given by the adjoint, $T \mapsto T^*$.
-$L(H)$ is a \textit{normed vector space} with
+$(\cdot,\cdot)\rightarrow \mathbb{C}$. We denote $L(H)$ as the $*$-algebra of operators on $H$
+equipped with a product given by composition and involution of the adjoint, $T \mapsto T^*$.
+Then $L(H)$ is a \textit{normed vector space} with
\begin{align}
&\|T\|^2 = \text{sup}_{h \in H}\{(Th,Th): (h,h) \leq 1\} \hspace{0.1\textwidth} T \in L(H) \\
&\|T\| = \text{sup}\{\sqrt{\lambda}: \lambda \text{ eigenvalue of } T\}
\end{align}
-
-
+This allows us to define representations of $*$-algebras.
\begin{definition}
- The \textit{representation} of a finite dimensional $*$-algebra A is a pair $(H, \pi)$.
- $H$ is a finite, dimensional inner product space and $\pi$ is a $*$-\textit{algebra map}
+ The \textit{representation} of a finite dimensional $*$-algebra $A$ is a
+ pair $(H, \pi)$, where $H$ is a finite dimensional inner product space
+ and $\pi$ is a $*$-\textit{algebra map}
\begin{align}
- \pi:A\ \rightarrow \ L(H)
+ \pi:A\ \rightarrow \ L(H).
\end{align}
-\end{definition}
-\begin{definition}
- $(H, \pi)$ is called \textit{irreducible} if:
+ We call the representation $(H, \pi)$ \textit{irreducible} if
\begin{itemize}
- \item $H \neq \emptyset$
- \item only $\emptyset$ or $H$ is invariant under the action of $A$ on $H$
+ \item $H \neq \emptyset$,
+ \item only $\emptyset$ or $H$ is invariant under the action of $A$ on
+ $H$.
\end{itemize}
\end{definition}
-
-Examples for reducible and irreducible representations
+Here are some examples of reducible and irreducible representations
\begin{itemize}
- \item $A = M_n(\mathbb{C})$, representation $H=\mathbb{C}^n$, $A$ acts as matrix multiplication\\
+ \item For $A = M_n(\mathbb{C})$ the representation $H=\mathbb{C}^n$, $A$ acts as matrix multiplication\\
$H$ is irreducible.
- \item $A = M_n(\mathbb{C})$, representation $H=\mathbb{C}^n\oplus \mathbb{C}^n$, with $a \in A$ acting
+ \item For $A = M_n(\mathbb{C})$ the representation $H=\mathbb{C}^n\oplus \mathbb{C}^n$, with $a \in A$ acting
in block form \\ $\pi: a \mapsto \big(\begin{smallmatrix} a & 0\\ 0 & a \end{smallmatrix}\big)$ is
reducible.
\end{itemize}
-
+Naturally there are also certain equivalences between different
+representations.
\begin{definition}
- Let $(H_1, \pi _1)$ and $(H_2, \pi _2)$ be representations of a $*$-algebra $A$. They are called
- \textit{unitary equivalent} if there exists a map $U: H_1 \rightarrow H_2$ such that.
+Two representations of a $*$-algebra $A$, $(H_1, \pi _1)$ and
+$(H_2, \pi _2)$ are called \textit{unitary equivalent} if there exists a map
+$U: H_1 \rightarrow H_2$ such that.
\begin{align}
\pi _1(a) = U^* \pi _2(a) U
\end{align}
\end{definition}
-\begin{question}
- In matrix representation this is diagonalization condition? (unitary diagonalization)
-\end{question}
-Yes
-
+Furthermore we define a mathematical structure called the structure space,
+which will later become important, when speaking of the duality between a spectral
+triple and a space.
\begin{definition}
- $A$ a $*$-algebra then, $\hat{A}$ is called the structure space of all \textit{unitary equivalence classes
+ Let $A$ a $*$-algebra then, $\hat{A}$ is called the structure space of all \textit{unitary equivalence classes
of irreducible representations of A}
\end{definition}
-
-\begin{question}
- Gelfand duality and the spectrum of $\hat{A}$, examples Fourier-Transform and Laplace-Transform
- for simple spaces.
-\end{question}
-More on that in later chapters.
-
-\begin{MyExercise}
- \textbf{
- Given $(H, \pi)$ of a $*$-algebra $A$, the \textbf{commutant} $\pi (A)'$ of $\pi (A)$ is defined as a set
+%------------- EXERCISE
+ Given a representation $(H, \pi)$ of a $*$-algebra $A$, the \textbf{commutant} $\pi (A)'$ of $\pi (A)$ is defined as a set
of operators in $L(H)$ that commute with all $\pi (a)$
\begin{align}
\pi (A)' = \{T \in L(H):\pi (a)T = T\pi (a) \;\;\; \forall a\in A\}
\end{align}
- \begin{enumerate}
- \item Show that $\pi (A)'$ is a $*$-algebra.
- \item Show that a representation $(H, \pi)$ of $A$ is irreducible iff the commutant $\pi (A)'$
- consists of multiples of the identity
- \end{enumerate}
-}
+ The commutant $\pi (A)'$ is also a $*$-algebra, because it has unital,
+ associative and involutive properties.
+ We note that $\pi (a) \in L(H)\ \forall a \in A$, unitary property is given
+ by the unital operator of the $*$-algebra of operators $L(H)$, which exists
+ by definition because H is a inner product space. Associativity is given by
+ the $*$-algebra of $L(H)$, where $L(H) \times L(H) \mapsto L(H)$, which is
+ associative by definition. The involutive property is also given by the $*$-algebra $L(H)$
+ with a map $*: L(H) \mapsto L(H)$ only for a $T$ that commutes with $\pi (a)$.
+%------------- EXERCISE
- 1. To show that $\pi (A)'$ is a $*$-algebra we have to show that it is unital, associative and involute.
- And note that $\pi (a) \in L(H)\ \forall a \in A$.
- Unitarity is given by the unital operator of the $*$-algebra of operators $L(H)$, which exists by definition
- because H is a inner product space. Associativity is given by $*$-algebra of $L(H)$, $L(H) \times L(H) \mapsto L(H)$,
- which is associative by definition. Involutnes is also given by the $*$-algebra $L(H)$
- with a map $*: L(H) \mapsto L(H)$ only for $T$ that commute with $\pi (a)$.
- \\
- 2.?
-\end{MyExercise}
+%------------- EXERCISE
+ For a unital algebra $*$-algebra $A$, the matrices $M_n(A)$ with entries
+ in $A$ form a unital $*$-algebra, because unitary operation in $M_n(A)$ is given by the identity Matrix, which
+ has to exists in every entry in $M_n(A)$, and behaves like in $A$. Associativity is given by
+ matrix multiplication. Lastly involution is given by the conjugate transpose.
-\begin{MyExercise}
- \textbf{
- \begin{enumerate}
- \item If $A$ is a unital $*$-algebra, show that the $n \times n$ matrices $M_n(A)$ with entries
- in $A$ form a unital $*$-algebra.
- \item Let $\pi :A\ \rightarrow \ L(H)$ be a representation of a $*$-algebra $A$ and set
- $H^n = H \oplus ... \oplus H$, $n$ times. Show that $\tilde{\pi}:M_n(A) \rightarrow L(H^n)$
- of $M_n(A)$ with\\
- $\tilde{\pi}((a_{ij})) = (\tilde{\pi}(a_{ij})) \in M_n(A)$.
- \item Let $\tilde{\pi}:M_n(A) \rightarrow L(H^n)$ be a $*$ algebra representation of $M_n(A)$.
- Show that $\pi: A \rightarrow L(H^n)$ is a representation of $A$.
- \end{enumerate}
-}
- 1. We know $A$ is a $*$ algebra. Unitary operaton in $M_n(A)$ is given by the identity Matrix, which
- has to exists because every entry in $M_n(A)$ has to behave like in $A$. Associativity is given by
- matrix multiplication. Involutnes is given by the conjugate transpose.\\
- 2. $A \simeq M_n(A)$ and $H \simeq H^n$ meaning $\tilde{\pi}$ is a valid reducible representation.\\
- 3. $\tilde{\pi}$ and $\pi$ are unitary equivalent, there is a map $U: H^n \rightarrow H^n$ given by
- $U=\mathbbm{1}_n$:\\
- $\pi (a) = \mathbbm{1}_n^*\ \tilde{\pi}((a_{ij}))\ \mathbbm{1}_n = \tilde{\pi}((a_{ij})) = \pi (a_{ij})
- \Rightarrow a_{ij} = a\mathbbm{1}_n$.
-\end{MyExercise}
+ A representation $\pi :A\ \rightarrow \ L(H)$ of a $*$-algebra $A$, for
+ $H^n = H \oplus ... \oplus H$, $n$ times. Then we have the following
+ representation $\tilde{\pi}:M_n(A) \rightarrow L(H^n)$ for the Matrix
+ Algebra with $\tilde{\pi}((a_{ij})) = (\tilde{\pi}(a_{ij})) \in M_n(A)$.
+ We have direct isomorphisms of $A \simeq M_n(A)$ and $H \simeq H^n$
+ meaning $\tilde{\pi}$ is a valid reducible representation.
-\subsubsection{Commutative Matrix Algebras}
-\begin{itemize}
- \item Commutative matrix algebras can be used to reconstruct a discrete space given
- a matrix \textit{commutative} matrix algebra.
- \item The structure space $\hat{A}$ is used for this. Because $A \simeq \mathbb{C}^N$ we all any
- irreducible representation are of the form
- $\pi _i:(\lambda_1,...,\lambda_N)\in \mathbb{C}^N \mapsto \lambda_i \in \mathbb{C}$ \\
- for $i = 1,...,N \Rightarrow \hat{A} \simeq \{1,...,N\}.$
- \item Conclusion is that there is a duality between discrete spaces and commutative matrix algebra
- this duality is called the \textit{finite dimensional Gelfand duality}
-\end{itemize}
+ Let $\tilde{\pi}:M_n(A) \rightarrow L(H^n)$ be a $*$ algebra
+ representation of $M_n(A)$, then $\pi: A \rightarrow L(H^n)$ is a representation of $A$.
+ The fact that $\tilde{\pi}$ and $\pi$ are unitary equivalent, there is
+ a map $U: H^n \rightarrow H^n$ given by $U=\mathbbm{1}_n$, thus
+ \begin{align}
+ \pi (a) &= \mathbbm{1}_n^*\ \tilde{\pi}((a_{ij})), \\
+ \mathbbm{1}_n &= \tilde{\pi}((a_{ij})) = \pi (a_{ij})
+ \Rightarrow a_{ij} = a\mathbbm{1}_n.
+ \end{align}
+%------------- EXERCISE
-\subsubsection{Noncommutative Matrix Algebras}
-Aim is to construct duality between finite dimensional spaces and \textit{equivalence classes}
-of matrix algebras, to preserve general non-commutivity of matrices.
-\begin{itemize}
- \item Equivalence classes are described by a generalized notion of ispomorphisms between matrix
- algebras (\textit{Morita Equivalence})
-\end{itemize}
+
+A commutative matrix algebra can be used to reconstruct a discrete space.
+The structure space $\hat{A}$ is used for this. Because $A \simeq
+\mathbb{C}^N$ all
+irreducible representation are of the form
+\begin{align}
+ \pi _i:(\lambda_1,...,\lambda_N)\in \mathbb{C}^N \mapsto \lambda_i \in
+ \mathbb{C}
+\end{align}
+for $i = 1,...,N$ and thus $\hat{A} \simeq \{1,...,N\}$.
+The conclusion is that, there is a duality between discrete spaces and
+commutative matrix algebra this duality is called the \textit{finite
+dimensional Gelfand duality}
+
+Our aim is to construct a duality between finite dimensional spaces and
+\textit{equivalence classes} of matrix algebras, to preserve general
+non-commutativity of matrices. Equivalence classes are described by a
+generalized notion of isomorphisms between matrix algebras (\textit{Morita
+Equivalence})
\subsubsection{Algebraic Modules}
\begin{definition}
@@ -323,3 +266,1314 @@ Remark on the notation
$\gamma: A\times A\times A \rightarrow A$ which is given by the inner product of the $*$-algebra.
\end{MyExercise}
+
+%\subsubsection{Balanced Tensor Product and Hilbert Bimodules}
+%
+%\begin{definition}
+% Let $A$ be an algebra, $E$ be a \textit{right} $A$-module and $F$ be a \textit{left} $A$-module.
+% The \textit{balanced tensor product} of $E$ and $F$ forms a $A$-bimodule.
+% \begin{align}
+% E \otimes _A F := E \otimes F / \left\{\sum _i e_i a_i \otimes f_i - e_i \otimes a_i f_i : \;\;\;
+% a_i \in A,\ e_i \in E,\ f_i \in F \right\}
+% \end{align}
+%\end{definition}
+%Note $/$ denotes the quotient space. So $\otimes _A$ takes two left/right modules and makes a
+%bimodule with the help the tensor product of the two modules and the quotient space that takes
+%out all the elements from the tensor product that dont preserver the left/right representation and that
+%are duplicates.
+%\begin{definition}
+% Let $A$, $B$ be \textit{matrix algebras}. The \textit{Hilbert bimodule} for $(A, B)$ is given by
+% \begin{itemize}
+% \item $E$, an $A$-$B$-bimodue $E$ and by
+% \item an $B$-valued \textit{inner product} $\langle \cdot,\cdot\rangle_E: E\times E \rightarrow B$
+% \end{itemize}
+%$\langle \cdot,\cdot\rangle_E$ needs to satisfy the following for $e, e_1, e_2 \in E,\ a \in A$ and $b \in B$.
+%\begin{align}
+% \langle e_1, a\cdot e_2\rangle_E &= \langle a^*\cdot e_1, e_2\rangle_E \;\;\;\; & \text{sesquilinear in $A$}\\
+% \langle e_1, e_2 \cdot b\rangle_E &= \langle e_1, e_2\rangle_E b \;\;\;\; & \text{scalar in $B$} \\
+% \langle e_1, e_2\rangle_E &= \langle e_2,e_1\rangle^*_E \;\;\;\; & \text{hermitian} \\
+% \langle e, e\rangle_E &\ge 0 \;\;\;\; & \text{equality holds iff $e=0$}
+%\end{align}
+%
+%\end{definition}
+%
+%We denote $KK_f(A,B)$ the set of all \textit{Hilbert bimodules} of $(A,B)$.
+%
+%%\begin{MyExercise}
+%% \textbf{
+%% Check that a representation $\pi:\ A \ \rightarrow L(H)$ of a matrix algebra $A$ turns $H$ into
+%% a Hilbert bimodule for $(A, \mathbb{C})$.
+%% \label{ex: bimodule}
+%%}\newline
+%%
+%%
+%% We check if the representation of $a \in A$, $\pi(a)=T \in L(H)$ fulfills
+%% the conditions on the $\mathbb{C}$-valued inner product for $h_1, h_2 \in H$:
+%% \begin{itemize}
+%% \item $\langle h_1, \pi(a) h)2\rangle _\mathbb{C} = \langle h_1, T h_2\rangle _\mathbb{C} =
+%% \langle T^* h_1, h_2\rangle _\mathbb{C}$, $T^*$ given by the adjoint
+%% \item $\langle h_1, h_2 \pi(a)\rangle _\mathbb{C} = \langle h_1, h_2 T\rangle _\mathbb{C} = \langle h_1, h_2\rangle _\mathbb{C}$, $T$ acts from the left
+%% \item $\langle h_1, h_2\rangle _\mathbb{C}^* = \langle h_2,h_1\rangle _\mathbb{C}$, hermitian because of the
+%% $\mathbb{C}$-valued inner product
+%% \item $\langle h_1, h_2\rangle \ge 0$, $\mathbb{C}$-valued inner product.
+%% \end{itemize}
+%%\end{MyExercise}
+%
+%%\begin{MyExercise}
+%% \textbf{
+%% Show that the $A-A$ bimodule given by $A$ is in $KK_f(A,A)$ by taking the following inner product
+%% $\langle \cdot,\cdot\rangle_A:A \times A \rightarrow A$:
+%% \begin{align}
+%% \langle a, a\rangle_A = a^*a' \;\;\;\; a,a'\in A
+%% \end{align}
+%% \label{exercise: inner-product}
+%%}\newline
+%%
+%%
+%% We check again the conditions on $\langle \cdot, \cdot\rangle _A$, let $a, a_1, a_2 \in A$:
+%% \begin{itemize}
+%% \item $\langle a_1, a\cdot a_2\rangle _A = a^*\ a\cdot a_2 = (a^*a_1)^* a_2 = \langle a^*a_1, a_2\rangle $
+%% \item $\langle a_1, a_2 \cdot a\rangle _A = a^*_1 (a_2\cdot a) = (a^*a_2)\cdot a = \langle a_1, a_2\rangle _A a$
+%% \item $\langle a_1, a_2\rangle _A^* = (a_1^* a_2)^* = a_2^*(a_1^*)^* = a_2^* a_1 = \langle a_2, a_1\rangle $
+%% \end{itemize}
+%%\end{MyExercise}
+%
+%\begin{example}
+% Consider a $*$ homomorphism between two matrix algebras $\phi:A\rightarrow B$.
+% From it we can construct a Hilbert bimodule $E_{\phi} \in KK_f(A, B)$ in the following way.
+% We let $E_{\phi}$ be $B$ in the vector space sense and an inner product from the above
+% Exercise \ref{exercise: inner-product}, with $A$ acting on the left with $\phi$.
+% \begin{align}
+% a\cdot b = \phi(a)b \;\;\;\; a\in A, b\in E_{\phi}
+% \end{align}
+%\end{example}
+%
+%
+%\subsubsection{Kasparov Product and Morita Equivalence}
+%\begin{definition}
+% Let $E \in KK_f(A, B)$ and $F \in KK_F(B, D)$ the \textit{Kasparov product} is defined as
+% with the balanced tensor product
+% \begin{align}
+% F \circ E := E \otimes _B F
+% \end{align}
+% Such that $F\circ E \in KK_f(A,D)$ with a $D$-valued inner product.
+% \begin{align}
+% \langle e_1 \otimes f_1, e_2 \otimes f_2\rangle _{E\otimes _B F} = \langle f_1,\langle e_1, e_2\rangle _E f_2\rangle _F
+% \end{align}
+%\end{definition}
+%
+%\begin{question}
+% How do we go from $\langle e_1 \otimes f_1, e_2 \otimes f_2\rangle _{E\otimes _B F}$ to $
+% \langle f_1,\langle e_1, e_2\rangle _E f_2\rangle _F$ \label{q: tensorproduct}\\
+% This statement is still in the definition.
+%\end{question}
+%
+%%\begin{question}
+%%What is the meaning of `associative up to isomorphism'? Isomorphism of $F \circ E$ or of $A, B$ or $D$?
+%%\end{question}
+%
+%% \begin{MyExercise}
+%% \textbf{
+%% Show that the association $\phi \leadsto E_\phi$ (from the previous Example) is natural
+%% in the sense
+%% \begin{enumerate}
+%% \item $E_{\text{id}_A} \simeq A \in KK_f(A,A)$
+%% \item for $*$-algebra homomorphism $\phi: A \rightarrow B$ and $\psi: B \rightarrow C$ we have
+%% an isomorphism
+%% \begin{align}
+%% E_{\psi} \circ E_{\phi}\ \equiv\ E_{\phi} \otimes _B E_{\psi}\ \simeq\
+%% E_{\psi \circ \phi} \in KK_f(A,C)
+%% \end{align}
+%% \end{enumerate}
+%%}
+%% \begin{enumerate}
+%% \item $\text{id}_A: A \rightarrow A$.\\
+%% To construct $E_{\phi}\in KK_f(A,A)$, we let $E_{\phi}$ be $A$ with a natural right
+%% representation, so $\Rightarrow E_{\phi}\simeq A$.\\
+%% With an inner product, acting on $A$ from the left with $\phi$, $a', a\in A$\\
+%% $a'a = (\phi(a') a) \in A $, which is satisfied by $\text{id}_A$, so $\phi = \text{id}_A$.
+%% \item $a \cdot b \cdot c = \psi(\phi (a) \cdot b) \cdot c$ for $a \in A$, $b\in B$, and $c\in C$
+%% which is $\psi \circ \phi$
+%% \end{enumerate}
+%%\end{MyExercise}
+%
+%%\begin{MyExercise}
+%% \textbf{
+%% In the definition of Morita equivalence:
+%% \begin{enumerate}
+%% \item Check that $E \otimes _B F$ is a $A-D$ bimodule
+%% \item Check that $\langle \cdot,\cdot\rangle _{E\oplus _B F}$ defines a $D$ valued inner product
+%% \item Check that $\langle a^*(e_1 \otimes f_1), e_2 \otimes f_2\rangle _{E \otimes _B F} = \langle e_1 \otimes f_1, a(e_2 \otimes f_2)\rangle _{E \otimes _B F}$.
+%% \end{enumerate}
+%%}
+%% \begin{enumerate}
+%% \item $E \otimes _B F = E \otimes F / \{\sum_i e_i b_i \otimes f_i - e_i \otimes b_i f_i;
+%% e_i \in E_i, b_i \in B, f_i \in F\}$ the last part takes out all tensor product elements of
+%% $E$ and $F$ that don't preserver the left/right representation and that are duplicates.
+%% \item $\langle e_1, e_2\rangle _E \in B$ and $\langle f_1, f_2\rangle _F \in C$ by definition. So let $\langle e_1, e_2\rangle _E =b$. \\
+%% Then $\langle e_1 \otimes f_1, e_2 \otimes f_2\rangle _{E\otimes _B F} = \langle f_1, \langle e_1, e_2\rangle _E f_2\rangle _F =
+%% \langle f_1, b f_2\rangle _F \in C$
+%% \item Check Question \ref{q: tensorproduct}.\\
+%% But let $G := E\otimes _B F \in KK_f(A,C)$ then $\forall g_1, g_2 \in G$ and $a \in A$ we need
+%% by definition $\langle g_1, ag_2\rangle _G = \langle a^*g_1, g_2\rangle _G$ and we set $g_1 = e_1 \otimes f_1$ and
+%% $g_2 = e_2 \otimes f_2$ for some $e_1, e_2 \in E$ and $f_1, f_2 \in F$, or else
+%% $G \notin KK_f(A,C)$ which would violate the Kasparov product
+%% \end{enumerate}
+%% \end{MyExercise}
+%
+%\begin{definition}
+% Let $A$, $B$ be \textit{matrix algebras}. They are called \textit{Morita equivalent} if there
+% exists an $E \in KK_f(A, B)$ and an $F \in KK_f(B, A)$ such that:
+% \begin{align}
+% E \otimes _B F \simeq A \;\;\; \text{and} \;\;\; F \otimes _A E \simeq B
+% \end{align}
+% Where $\simeq$ denotes the isomorphism between Hilbert bimodules, note that $A$ or $B$ is a bimodule by
+% itself.
+%\end{definition}
+%
+%\begin{question}
+% Why are $E$ and $F$ each others inverse in the Kasparov Product? \\
+% They are each others inverse with respect to the Kasparov Product because we land in the same space as we started.
+% In the definition we have $E \in KK_f(A, B)$ we start from $A$ and $E \otimes _B F$ lands in $A$.\\
+% On the other hand we have $F \in KK_f(B, D)$ we start from $B$ and $F \otimes _A E$ lands in $B$.
+%\end{question}
+%
+%\begin{example}
+% \
+% \begin{itemize}
+% \item Hilber bimodule of $(A,A)$ is $A$
+% \item Let $E \in KK_f(A,B)$, we take $E \circ A = A\oplus _A E \simeq E$
+% \item we conclude, that $_A A_A$ is the identity in the Kasparov product (up to isomorphism)
+% \end{itemize}
+%\end{example}
+%
+%\begin{example}
+% Let $E = \mathbb{C}^n$, which is a $(M_n(\mathbb{C}), \mathbb{C})$ Hilbert bimodule with the
+% standard $\mathbb{C}$ inner product.\\
+% On the other hand let $F = \mathbb{C}^n$, which is a $(\mathbb{C}, M_n(\mathbb{C}))$ Hilbert
+% bimodule by right matrix multiplication with $M_n(\mathbb{C})$ valued inner product:
+% \begin{align}
+% \langle v_1, v_2\rangle =\bar{v_1}v_2^t \;\; \in M_n(\mathbb{C})
+% \end{align}
+% Now we take the Kasparov product of $E$ and $F$:
+% \begin{itemize}
+% \item $F\circ E\ =\ E\otimes _{\mathbb{C}}F\ \;\;\;\;\;\; \simeq \ M_n(\mathbb{C})$
+% \item $E\circ F\ =\ F\otimes _{M_n(\mathbb{C})}E\ \simeq\ \mathbb{C}$
+% \end{itemize}
+% $M_n(\mathbb{C})$ and $\mathbb{C}$ are Morita equivalent
+%\end{example}
+%
+%\begin{theorem}
+% Two matrix algebras are Morita Equivalent iff their their Structure spaces
+% are isomorphic as discreet spaces (have the same cardinality / same number of elements)
+%\end{theorem}
+%\begin{proof}
+% Let $A$, $B$ be \textit{Morita equivalent}. So there exists $_A E_B$ and $_B F_A$ with
+% \begin{align}
+% E \otimes _B F \simeq A \;\;\; \text{and} \;\;\; F \otimes _A E \simeq B
+% \end{align}
+% Consider $[(\pi _B, H)] \in \hat{B}$ than we construct a representation of $A$,
+% \begin{align}
+% \pi _A \rightarrow L(E \otimes _B H)\;\;\; \text{with} \;\;\; \pi _A(a) (e \otimes v) = a e \otimes w
+% \end{align}
+% \begin{question}
+% Is $E \simeq H$ and $F \simeq W$? \\
+% Not in particular, there is a theorem that all infinite dimensional Hilbert spaces are isomorphic.
+% Here we are looking at finite dimensional Hilbert spaces.\\
+% Another thing to is that $[\pi _B, H] \in \hat{B}$ and looking at Exercise \ref{ex: bimodule}
+% we know that $H$ is a bimodule of $B$, hence $E \otimes _B H\simeq A$, and for $[\pi _A, W]$
+% the same.
+% \end{question}
+% \textit{vice versa}, consider $[(\pi _A, W)] \in \hat{A}$ we can construct $\pi _B$
+% \begin{align}
+% \pi _B: B \rightarrow L(F \otimes _A W) \;\;\; \text{and}\;\;\; \pi _B(b) (f\otimes w) = bf\otimes w
+% \end{align}
+% These maps are each others inverses, thus $\hat{A} \simeq \hat{B}$
+%\end{proof}
+%
+%%\begin{MyExercise}
+%% \textbf{
+%% Fill in the gaps in the above proof:
+%% \begin{enumerate}
+%% \item show that the representation of $\pi _A$ defined is irreducible iff $\pi _B$ is.
+%% \item Show that the association of the class $[\pi _A]$ to $[\pi _B]$ is independent
+%% of the choice of representatives $\pi _A$ and $\pi _B$
+%% \end{enumerate}
+%%}
+%%
+%% \begin{enumerate}
+%% \item $(\pi _B, H)$ is irreducible means $H \neq \emptyset$ and only $\emptyset$ or $H$
+%% is invariant under the Action of $B$ on $H$.
+%% Than $E\otimes _B H$ cannot be empty, because also $E$ preserves left representation of $A$
+%% and also $E\otimes _B H \simeq A$.
+%% \item The important thing is that $[\pi _A] \in \hat{A}$ respectively $[\pi _B] \in \hat{B}$,
+%% hence any choice of representation is irreducible, because the structure space denotes all unitary
+%% equivalence classes of irreducible representations.
+%% \end{enumerate}
+%%\end{MyExercise}
+%
+% \begin{lemma}
+% The matrix algebra $M_n(\mathbb{C})$ has a unique irreducible representation (up to isomorphism)
+% given by the defining representation on $\mathbb{C}^n$.
+%\end{lemma}
+%\begin{proof}
+% We know $\mathbb{C}^n$ is a irreducible representation of $A= M_n(\mathbb{C})$. Let $H$ be irreducible
+% and of dimension $k$, then we define a map
+% \begin{align}
+% \phi : A\oplus...\oplus A &\rightarrow H^* \\
+% (a_1,...,a_k) &\mapsto e^1\circ a_1^t+...+e^k\circ a_k^t
+% \end{align}
+% With $\{e^1,...,e^k\}$ being the basis of the dual space $H^*$ and $(\circ)$ being the pre-composition
+% of elements in $H^*$ and $A$ acting on $H$. This forms a morphism of $M_n(\mathbb{C})$ modules,
+% provided a matrix $a \in A$ acts on $H^*$ with $v\mapsto v\circ a^t$ ($v\in H^*$).
+% Furthermore this morphism is surjective, thus making the pullback $\phi ^*:H\mapsto (A^k)^*$ injective.
+% Now identify $(A^k)^*$ with $A^k$ as a $A$-module and note that
+% $A=M_n(\mathbb{C}) \simeq \oplus ^n \mathbb{C}^n$ as a n A module.
+% It follows that $H$ is a submodule of $A^k \simeq \oplus ^{nk}\mathbb{C}$. By irreducibility
+% $H \simeq \mathbb{C}$.
+%\end{proof}
+%
+%\begin{example}
+% Consider two matrix algebras $A$, and $B$.
+% \begin{align}
+% A = \bigoplus ^N_{i=1} M_{n_i}(\mathbb{C}) \;\;\; B = \bigoplus ^M_{j=1} M_{m_j}(\mathbb{C})
+% \end{align}
+% Let $\hat{A} \simeq \hat{B}$ that implies $N=M$ and define $E$ with $A$ acting by block-diagonal
+% matrices on the first tensor and B acting in the same way on the second tensor. Define $F$ vice versa.
+% \begin{align}
+% E:= \bigoplus _{i=1}^N \mathbb{C}^{n_i} \otimes \mathbb{C}^{m_i} \;\;\;
+% F:= \bigoplus _{i=1}^N \mathbb{C}^{m_i} \otimes \mathbb{C}^{n_i}
+% \end{align}
+% Then we calculate the Kasparov product.
+% \begin{align}
+% E \otimes _B F &\simeq \bigoplus _{i=1}^N (\mathbb{C}^{n_i}\otimes\mathbb{C}^{m_i})
+% \otimes _{M_{m_i}(\mathbb{C})} (\mathbb{C}^{m_i}\otimes\mathbb{C}^{n_i}) \\
+% &\simeq \bigoplus _{i=1}^N \mathbb{C}^{n_i}\otimes
+% \left(\mathbb{C}^{m_i}\otimes _{M_{m_i}(\mathbb{C})}\mathbb{C}^{m_i}\right)
+% \oplus \mathbb{C}^{n_i} \\
+% &\simeq \bigoplus _{i=1}^N \mathbb{C}^{m_i}\otimes\mathbb{C}^{n_i} \simeq A
+% \end{align}
+% and from $F \otimes _A E \simeq B$.
+%\end{example}
+%
+%We conclude that.
+%\begin{itemize}
+% \item There is a duality between finite spaces and Morita equivalence classes of matrix algebras.
+% \item By replacing $*$-homomorphism $A\rightarrow B$ with Hilbert bimodules $(A,B)$ we introduce
+% a richer structure of morphism between matrix algebras.
+%\end{itemize}
+%
+%\subsection{Noncommutative Geometric Spaces }
+%\subsubsection{Exercises}
+%
+%%\begin{MyExercise}
+%%\textbf{
+%% Make the proof of the last theorem (see week4.pdf) explicit for $N=3$.
+%%}\newline
+%%
+%% For the C* algebra we have $A=\mathbb{C}^3$
+%% For $H$ we have $H = (\mathbb{C}^2)^{\oplus 3} = H_2 \oplus H_2^1 \oplus H_2^2$.
+%% The symmetric operator $D$ acting on $H$ and the representation $\pi (a)$:
+%% \begin{align}
+%% \pi((a(1), a(2), a(3)) &=
+%% \begin{pmatrix}
+%% a(1) & 0 \\ 0 & a(2)
+%% \end{pmatrix} \oplus
+%% \begin{pmatrix}
+%% a(1) & 0 \\ 0 & a(3)
+%% \end{pmatrix} \oplus
+%% \begin{pmatrix}
+%% a(2) & 0 \\ 0 & a(2)
+%% \end{pmatrix} \nonumber \\
+%% & =
+%% \begin{pmatrix}
+%% a(1) & 0 & 0 & 0 & 0 & 0 \\
+%% 0 & a(2) & 0 & 0 & 0 & 0 \\
+%% 0 & 0 & a(1) & 0 & 0 & 0 \\
+%% 0 & 0 & 0 & a(3) & 0 & 0 \\
+%% 0 & 0 & 0 & 0 & a(2) & 0 \\
+%% 0 & 0 & 0 & 0 & 0 & a(3)
+%% \end{pmatrix} \\
+%% D &=
+%% \begin{pmatrix}
+%% 0 & x_1 \\ x_1 & 0
+%% \end{pmatrix} \oplus
+%% \begin{pmatrix}
+%% 0 & x_2 \\ x_2 & 0
+%% \end{pmatrix} \oplus
+%% \begin{pmatrix}
+%% 0 & x_3 \\ x_3 & 0
+%% \end{pmatrix} \nonumber \\
+%% &=
+%% \begin{pmatrix}
+%% 0 & x_1 & 0 & 0 & 0 & 0 \\
+%% x_1 & 0 & 0 & 0 & 0 & 0 \\
+%% 0 & 0 & 0 & x_2 & 0 & 0 \\
+%% 0 & 0 & x_2 & 0 & 0 & 0 \\
+%% 0 & 0 & 0 & 0 & 0 & x_3 \\
+%% 0 & 0 & 0 & 0 & x_3 & 0 \\
+%% \end{pmatrix} \\
+%% \end{align}
+%% Then the norm of the commutator would be the largest eigenvalue
+%% \begin{align}
+%% &||[D, \pi(a)]|| = ||D\pi(a) - \pi(a)D||\nonumber\\
+%% &=
+%% \left|\left|
+%% \setlength{\arraycolsep}{0.1cm}
+%% \renewcommand{\arraystretch}{0.1}
+%% \begin{pmatrix}
+%% 0 & x_1(a(2)-a(1)) & 0 & 0 & 0 & 0 \\
+%% -x_1(a(2)-a(1)) & 0 & 0 & 0 & 0 & 0 \\
+%% 0 & 0 & 0 & x_2(a(3)-a(1)) & 0 & 0 \\
+%% 0 & 0 & -x_2(a(3)-a(1)) & 0 & 0 & 0 \\
+%% 0 & 0 & 0 & 0 & 0 & x_3(a(3)-a(2)) \\
+%% 0 & 0 & 0 & 0 & -x_3(a(2)-a(3)) & 0 \\
+%% \end{pmatrix}\right|\right| \label{skew matrix}
+%% \end{align}
+%%The matrix in Equation \ref{shew matrix} is a skew symmetric matrix its eigenvalues
+%%are $i\lambda_1, i\lambda_2, i\lambda_3, i\lambda_4$, where the $\lambda$'s are on the
+%%upper and lower diagonal check \url{https://en.wikipedia.org/wiki/Skew-symmetric_
+%%matrix#Skew-symmetrizable_matrix}. The matrix norm of would be the maximum of the norm of
+%%the larges eigenvalues:
+%%\begin{align}
+%% ||[D, \pi(a)]|| = \max_{a\in A}\{&x_1|a(2)-a(1)|,\\
+%% &x_2|(a(3)-a(1))|,\nonumber\\
+%% &x_3|(a(3)-a(2))|,\}\nonumber
+%%\end{align}
+%%The metric is then:
+%%\begin{align}
+%% d =
+%% \begin{pmatrix}
+%% 0 & a(1)-a(2) & a(1)-a(3)\\
+%% a(2)-a(1) & 0 & a(2)-a(3)\\
+%% a(3)-a(1) & a(3)-a(2) & 0
+%% \end{pmatrix}
+%%\end{align}
+%%\end{MyExercise}
+%
+%%\begin{MyExercise}
+%% \textbf{
+%% Compute the metric on the space of three points given by $d_{ij} =
+%% \sup_{a\in A}\{|a(i) - a(j)|: ||[D, \pi(a)]|| \leq 1\}$ for the set of data
+%% $A = \mathbb{C}^3$ acting in the defining representation $H = \mathbb{C}^3$, and
+%% \begin{align}
+%% D =
+%% \begin{pmatrix}
+%% 0 & d^{-1} & 0 \\
+%% d^{-1} & 0 & 0 \\
+%% 0 & 0 & 0
+%% \end{pmatrix}
+%% \end{align}
+%% for some $d \in \mathbb{R}$
+%%}\newline
+%%
+%% We have $A=\mathbb{C}^3$, $H=\mathbb{C}^3$ and $D$ from above, then
+%%
+%% \begin{align}
+%% ||[D, \pi(a)]|| &= d^{-1}\left|\left|
+%% \begin{pmatrix}
+%% 0 & a(2)-a(1) & 0 \\
+%% -(a(2)-a(1)) & 0 & 0 \\
+%% 0 & 0 & 0
+%% \end{pmatrix} \right|\right|
+%% \end{align}
+%% The metric is then
+%% \begin{align}
+%% d =
+%% \begin{pmatrix}
+%% 0 & a(1)-a(2) & a(1) \\
+%% a(2)-a(1) & 0 & a(2) \\
+%% -a(1) & -a(2) & 0
+%% \end{pmatrix}
+%% \end{align}
+%%\end{MyExercise}
+%
+%%\begin{MyExercise}
+%% \textbf{
+%% Show that $d_{ij}$ from Equation \ref{ext metric} is a metric on $\hat{A}$ by
+%% establishing that:
+%% \begin{align}
+%% d_{ij} &= 0\;\;\; \Leftrightarrow \;\;\; i=j \label{metric 1} \\
+%% d_{ij} &= d_{ji} \label{metric 2}\\
+%% d_{ij} &\leq d_{ik} + d_{kj} \label{metric 3}
+%% \end{align}
+%% \begin{equation} \label{ext metric}
+%% d_{ij} = \sup_{a\in A}\big\{|\text{Tr}(a(i)) - \text{Tr}((a(j))|: ||[D, a]|| \leq 1\big\}
+%% \end{equation}
+%%}\newline
+%%
+%%For Equation \ref{metric 1} set $i=j$ in \ref{ext metric}.
+%%\begin{align}
+%% d_{ii} &= \sup_{a \in A}\{|\text{Tr}(a(i)) - \text{Tr}((a(i))|: ||[D, a]|| \leq
+%% 1\big\} \\
+%% &= \sup_{a \in A}\{0: ||[D, a]|| \leq 1\big\} = 0
+%%\end{align}
+%%For Equation \ref{metric 2} obviously we have the commuting property of
+%%addition.
+%%\newline
+%%For Equation \ref{metric 3}, for $k=j$ then $d_{kj} = 0$ and the equality
+%%holds. For $i = k$ then $d_{ik} = 0$ and equality holds. Else set $d_{ik} =
+%%1$ and $d_{kj} = 1$ then $d_{ij} = 1 \leq d_{ik} + d_{kj} = 2$
+%%\end{MyExercise}
+%
+%\subsubsection{Properties of Matrix Algebras}
+%\begin{lemma}
+% If $A$ is a unital C* algebra that acts faithfully on a finite
+% dimensional Hilbert space, then $A$ is a matrix algebra of the Form:
+% \begin{equation}
+% A \simeq \bigoplus _{i=1}^N M_{n_i}(\mathbb{C})
+% \end{equation}
+%\end{lemma}
+%\begin{proof}
+% Since $A$ acts faithfully on a Hilbert space, then $A$ is a C*
+% subalgebra of a matrix algebra $L(H) = M_{\dim (H)}(\mathbb{C}
+% \Rightarrow A \simeq \text{Matrix algebra}$.
+%\end{proof}
+%
+%\begin{question}
+% What does the author mean when he sais 'acts faithfully on a
+% Hilbertspace`? Then the representation is fully reducible, or that the
+% presentation is irreducible?
+% \newline
+%
+% For a *-representation 'faithful` if it is injective. For a
+% *-homomorphism 'faithful` means one-to-one correspondance
+%\end{question}
+%
+%\begin{example}
+% $A = M_n(\mathbb{C})$ and $H=\mathbb{C}^n$, $A$ acts on $H$ with matrix
+% multiplication and standard inner product. $D$ on $H$ is a hermitian
+% matrix $n\times n$ matrix.
+%\end{example}
+%
+%$D$ is referred to as a finite Dirac operator as in as its $\infty$
+%dimensional on Riemannian Spin manifolds coming in Chapter 4.
+%\newline
+%
+%Now can introduce a 'differential 'geometric structure` on the finite space X
+%with the \textbf{devided difference}
+%\begin{equation}
+% \frac{a(i)-a(j)}{d_{ij}}
+%\end{equation}
+%for each pair $i$, $j$ $\in X$ the finite dimensional discrete space $X$.
+%This appears in the entries in the commutator $[D, a]$ in the above
+%exercises.
+%
+%\begin{definition}
+% Given an finite spectral triple $(A, H, D)$, the $A$-bimodule of
+% Connes' differential one-forms is:
+% \begin{equation}
+% \Omega _D ^1 (A) := \left\{ \sum _k a_k[D, b_k]: a_k, b_k \in A \right\}
+% \end{equation}
+%\end{definition}
+%
+%\begin{question}
+% Is the Conne's differential one form the set of all '1st order
+% differential operators` given $A$, that act on $H$?
+%\end{question}
+%Then there is a map $d:A\rightarrow \Omega _D ^1 (A)$, $d = [D, \cdot]$.
+%%\begin{MyExercise}
+%% \textbf{
+%% Verify that 'd` is a derivation of the C* algebra
+%% \begin{align}
+%% d(ab) = d(a)b + ad(b) \\
+%% d(a^*) = -d(a)^*
+%% \end{align}
+%%}\newline
+%%
+%% For the record $d(\cdot) = [D, \cdot]$, then we have
+%% \begin{enumerate}
+%% \item
+%% \begin{align}
+%% d(ab) &= [D, ab] = [D, a]b + a[D,b]\\
+%% &= d(a)b + ad(b)
+%% \end{align}
+%% \item
+%% \begin{align}
+%% d(a^*) &= [D, a^*] = Da^* - a^*D \\
+%% &=-(D^*a - aD^*) = -[D^*, a] \\
+%% &= -d(a)^*
+%% \end{align}
+%% \end{enumerate}
+%%\end{MyExercise}
+%%\begin{MyExercise}
+%% \textbf{
+%% Verify that $\Omega _D^1 (A)$ is an $A$-bimodule by rewriting
+%% }
+%% \begin{align}
+%% a(a_k[D, b_k])b = \sum_k a'_k[D, b'_k] \;\;\;\; a'_k, b'_k \in A
+%% \end{align}
+%% \newline
+%%
+%% Begin
+%% \begin{align}
+%% a(a_k[D, b_k])b &= aa_k(Db_k - b_k D) b = \\
+%% &= aa_k(Db_k b - b_k D b) = aa_k(Db_k b - b_k Db - b_kbD +b_kbD)=\\
+%% &= aa_k(Db_kb-b_kbD + b_k b D - b_k D b) = \\
+%% &= aa_k [D, b_kb] + aa_k b [D, b]=\\
+%% &= \sum _k a_k' [D, b_k']
+%% \end{align}
+%%
+%%\end{MyExercise}
+%
+%\begin{lemma}
+% Let $(A, H, D) = (M_n(\mathbb{C}, \mathbb{C}^n, D)$, with $D$ a hermitian
+% $n\times n$ matrix. If $D$ is not a multiple of the identity then:
+% \begin{equation}
+% \Omega _D ^1 (A) \simeq M_n(\mathbb{C}) = A
+% \end{equation}
+%\end{lemma}
+%
+%\begin{proof}
+% Assume $D = \sum _i \lambda _i e_{ii}$ (diagonal), $\lambda _i \in \mathbb{R}$ and
+% $\{e_{ij}\}$ the basis of $M_n(\mathbb{C}$. For fixed $i$, $j$ choose $k$
+% such that $\lambda _k \neq \lambda _j$ then
+% \begin{align} \label{basis}
+% \left(\frac{1}{\lambda _k - \lambda _j} e_{ik}\right) [D, e_{kj}] =
+% e_{ij}
+% \end{align}
+% $e_{ij}\in \Omega _D ^1 (A)$ by the above definition. And $\Omega _D ^1
+% (A) \subset L(\mathbb{C}^n) = H \simeq M_n(\mathbb{C}) = A$
+%\end{proof}
+%
+%%\begin{MyExercise}
+%% \textbf{
+%% Consider $(A=\mathbb{C}^2, H=\mathbb{C}^2,
+%% D = \begin{pmatrix} 0 & \lambda \\ \bar{\lambda} & 0
+%% \end{pmatrix})$ with $\lambda \neq 0$. Show that $\Omega _D^1(A)
+%% \simeq M_2(\mathbb{C})$
+%% }
+%%\newline
+%%
+%% Because of the Hilbert Basis $D$ can be extended in terms of
+%% the basis of $M_2(\mathbb{C})$, plugging this into Equation
+%% \ref{basis} will get us the same cyclic result, thus
+%% $\Omega _D^1(A) \simeq M_2(\mathbb{C})$
+%%\
+%%\end{MyExercise}
+%
+%\subsubsection{Morphisms Between Finite Spectral Triples}
+%\begin{definition}
+% two finite spectral tripes $(A_1, H_1, D_1)$ and $(A_2, H_2, D_2)$ are
+% called unitarily equivalent if
+% \begin{itemize}
+% \item $A_1 = A_2$
+% \item $\exists \;\; U: H_1 \rightarrow H_2$, unitary with
+% \begin{enumerate}
+% \item $U\pi_1(a)U^* = \pi_2(a)$ with $a \in A_1$
+% \item $UD_1 U^* = D_2$
+% \end{enumerate}
+% \end{itemize}
+%\end{definition}
+%
+%Some remarks
+%\begin{itemize}
+% \item the above is an equivalence relation
+% \item spectral unitary equivalence is given by the unitaries of the
+% matrix algebra itself
+% \item for any such $U$ then $(A, H, D) \sim (A, H, UDU^*)$
+% \item $UDU^* = D + U[D, U^*]$ of the form of elements in
+% $\Omega _D^1 (A)$.
+%\end{itemize}
+%
+%%\begin{MyExercise}
+%% \textbf{
+%% Show that the unitary equivalence between finite spectral
+%% triples is a equivalence relation
+%%}\newline
+%%
+%% An equivalence relation needs to satisfy reflexivity, symmetry
+%% transitivity.
+%% Let $(A_1, H_1, D_1)$, $(A_2, H_2, D_2)$ and $(A_3, H_3, D_3)$
+%% be three finite spectral triples.
+%% \newline
+%%
+%% For reflexivity $(A_1, H_1, D_1) \sim (A_1, H_1, D_1)$. So there
+%% exists a $U: H_1 \rightarrow H_1$ unitary, which is the identity
+%% and always exists.
+%% \newline
+%%
+%% For symmetry we need
+%% \begin{align}
+%% (A_1, H_1, D_1) \sim (A_2, H_2, D_2) \Leftrightarrow
+%% (A_2, H_2, D_2) \sim (A_1, H_1, D_1)
+%% \end{align}
+%% because $U$ is unitary:
+%% \begin{align}
+%% &U\pi_1(a)U^* = \pi_2(a) \;\;\; | \cdot U^*\boxdot U \\
+%% &U^*U\pi_1(a)U^*U = \pi_1(a) = U^*\pi_2(a)U \\
+%% \end{align}
+%% The same with the symmetric operator $D$.
+%% \newline
+%%
+%% For transitivity we need
+%% \begin{align}
+%% (A_1, H_1, D_1) &\sim (A_2, H_2, D_2) \;\;\; \text{and} \;\;\;
+%% (A_2, H_2, D_2) \sim (A_3, H_3, D_3) \\
+%% &\Rightarrow (A_1, H_1, D_1) \sim (A_3, H_3, D_3)
+%% \end{align}
+%% There are two unitary maps $U_{12}:H_1 \rightarrow H_2$ and
+%% $U_{23}: H_2 \rightarrow H_3$ then
+%% \begin{align}
+%% U_{23}U_{12} \pi_1(a) U^*_{12}U^*_{23} &= U_{23}
+%% \pi_2(a) U_23^* \\
+%% &= \pi_3(a) \\
+%% U_{23}U_{12} D_1U^*_{12}U^*_{23} &= U_{23}
+%% D_2 U_23^* \\
+%% &= D_3
+%% \end{align}
+%%\end{MyExercise}
+%
+%Extending the this relation we look again at the notion of equivalence from
+%Morita equivalence of Matrix Algebras.
+%\newline
+%
+%\begin{definition}
+% Let $A$ be an algebra. We say that $I \subset A$, as a vector space, is a
+% right(left) ideal if $ab \in I$ for $a \in A$ and $b\in I$ (or $ba \in
+% I$, $b\in I$, $a\in A$). We call a left-right ideal simply an ideal.
+%\end{definition}
+%
+%Given a Hilbert bimodule $E \in KK_f(B, A)$ and $(A, H, D)$ we construct
+%a finite spectral triple on $B$, $(B, H', D')$
+%\begin{equation}
+% H' = E \otimes _A H
+%\end{equation}
+%We might define $D'$ with $D'(e \otimes \xi) = e\otimes D\xi$, thought this
+%would not satisfy the ideal defining the balanced tensor product over $A$,
+%which is generated by elements of the form
+%\begin{align}
+% e a \otimes \xi - e\otimes a \xi ;\;\;\;\; e\in E, a\in A, \xi \in H
+%\end{align}
+%This inherits the left action on $B$ from $E$ and has a $\mathbb{C}$
+%valued inner product space. $B$ also satisfies the ideal.
+%\begin{equation}
+% D'(e\otimes \xi) = e \otimes D \xi + \nabla (e) \xi \;\;\;\; e\in
+% E, a\in A
+%\end{equation}
+%Where $\nabla$ is called the \textit{connection on the right A-module E}
+%associated with the derivation $d=[D, \cdot]$ and satisfying the
+%\textit{Leibnitz Rule} which is
+%\begin{equation}
+% \nabla(ae) = \nabla(e)a + e \otimes [D, a] \;\;\;\;\; e\in E,\; a\in A
+%\end{equation}
+%Then $D'$ is well defined on $E \otimes _A H$:
+%\begin{align}
+% D'(ea \otimes \xi - e \otimes a \xi) &= D'(ea \otimes \xi) - D'(e
+% \otimes \xi) \\
+% &= ea\otimes D\xi + \nabla(ae) \xi - e \otimes D(a\xi ) - \nabla (e)a
+% \xi \\
+% &= 0.
+%\end{align}
+%With the information thus far we can prove the following theorem
+%\begin{theorem}
+% If $(A, H, D)$ a finite spectral triple, $E \in KK_f(B, A)$.
+% Then $(V, E\otimes _A H, D')$ is a finite spectral triple, provided that
+% $\nabla$ satisfies the compatibility condition
+% \begin{equation}
+% \langle e_1, \nabla e_2 \rangle _E - \langle \nabla e_1, e_2
+% \rangle _E = d\langle e_1, e_2 \rangle _E \;\;\;\; e_1, e_2 \in E
+% \end{equation}
+%\end{theorem}
+%\begin{proof}
+% $E\otimes _A H$ was shown in the previous subsection (text before the
+% theorem). The only thing left is to show that $D'$ is a symmetric
+% operator, this we can just compute. Let $e_1, e_2 \in E$ and $\xi _1,
+% \xi _2 \in H$ then
+% \begin{align}
+% \langle e_1 \otimes \xi _1, D'(e_2 \otimes \xi_2)\rangle _{E\otimes _A H} &=
+% \langle \xi _1, \langle e_1, \nabla e_2\rangle _E \xi _2\rangle + \langle \xi _1 , \langle e_1, e_2\rangle _E D\xi
+% _2\rangle _H \\
+% &= \langle \xi _1, \langle \nabla e_1, e_2\rangle _E \xi _2\rangle _H + \langle \xi _1, d\langle e_1, e_2\rangle _E
+% \xi _2\rangle _H \\
+% &+ \langle D\xi _1,\langle e_1, e_2\rangle _E \xi _2\rangle _H - \langle \xi _1, [D, \langle e_1, e_2\rangle _E] \xi
+% _2 \rangle _H \\
+% &= \langle D'(e_1 \otimes \xi _1), e_2 \otimes \xi _2\rangle _{E \otimes _A H}
+% \end{align}
+%\end{proof}
+%
+%%\begin{MyExercise}
+%% \textbf{
+%% Let $\nabla$ and $\nabla'$ be two connections on a right $A$-module
+%% $E$. Show that $\nabla - \nabla'$ is a right $A$-linear map
+%% $E \rightarrow E\otimes _A \Omega _D^1(A)$
+%%}\newline
+%%
+%% Both $\nabla$ and $\nabla'$ need to satisfy the Leiblitz rule, so
+%% let's see if $\nabla - \nabla'$ does.
+%%
+%% \begin{align}
+%% \nabla(ea)-\nabla'(ea)&=\nabla(e) + e\otimes[D, a]\\
+%% &-(\nabla'(e)a + e\otimes[D',a])\\
+%% &=\bar{\nabla}a + e\otimes(Da-aD-D'a+aD')\\
+%% &=\bar{\nabla}a + e\otimes((D-D')a-a(D-D'))\\
+%% &=\bar{\nabla}a + e\otimes[D', a]\\
+%% &=\bar{\nabla}(ea)
+%% \end{align}
+%% Therefore $\nabla-\nabla'$ is a linear map.
+%%\end{MyExercise}
+%
+%%\begin{MyExercise}
+%% \textbf{
+%% Construct a finite spectral triple $(A, H', D')$ from $(A, H, D)$
+%% \begin{enumerate}
+%% \item show that the derivation $d(\cdot):A \rightarrow A\otimes _A
+%% \Omega_D^1(A)=\Omega_D^1(A)$ is a connection on $A$
+%% considered a right $A$-module
+%% \item Upon identifying $A\otimes_A H\simeq H$, what is $D'$
+%% when the connection is $d(\cdot)$.
+%% \item Use 1) and 2) to show that any connection $\nabla:
+%% A\rightarrow A\otimes_A \Omega_D^1(A)$ is given by
+%% \begin{align}
+%% \nabla = d + \omega
+%% \end{align}
+%% where $\omega \in \Omega_D^1(A)$
+%% \item Upon identifying $A\otimes_A H \simeq H$, what is the
+%% difference operator $D'$ with the connection on $A$ given by
+%% $\nabla = d + \omega$
+%% \end{enumerate}
+%%}
+%% \begin{enumerate}
+%% \item $\nabla(e \cdot a) = d(a)$
+%% \item
+%% $D'(a\xi) = a(D\xi) + (\nabla a) \xi = D(a\xi)$
+%% \item Use the identity element $e \in A$\\
+%% $\nabla (e\cdot a) = \nabla(e) a + 1 \otimes d(a)=d(a)
+%% \nabla(e) a$
+%% \item $D'(a\otimes \xi) = D'(a \xi) = a(D\xi) + (\nabla a)\xi =
+%% a(D\xi) + \nabla(e \cdot a) \xi \\
+%% = D(a\xi) + \nabla(e) (a\xi)$
+%% \end{enumerate}
+%%\end{MyExercise}
+%
+%\subsubsection{Graphing Finite Spectral Triples}
+%\begin{definition}
+% A \textit{graph} is a ordered pair $(\Gamma ^{(0)}, \Gamma ^{(1)})$.
+% Where $\Gamma ^{(0)}$ is the set of vertices (nodes) and $\Gamma ^{(1)}$
+% a set of pairs of vertices (edges)
+%\end{definition}
+%\begin{figure}[h!]
+% \centering
+%\begin{tikzpicture}[
+% mass/.style = {draw,circle, minimum size=0.2cm, inner sep=0pt, thick},
+% spring/.style = {decorate,decoration={zigzag, pre length=1cm,post length=1cm,segment length=5pt}},]
+% \node[mass] (m1) at (1,1.5) {};
+% \node[mass] (m2) at (-1,1.5) {};
+% \node[mass] (m3) at (0,0) {};
+%
+% \draw (m1) -- (m2);
+% \draw (m1) -- (m3);
+% \draw (m2) -- (m3);
+% \end{tikzpicture}
+% \caption{A simple graph with three vertices and three edges}
+%\end{figure}
+%%\begin{MyExercise}
+%% \textbf{
+%% Show that any finite-dimensional faithful representation $H$ of a matrix
+%% algebra $A$ is completely reducible. To do that show that the complement
+%% $W^{\perp}$ of an $A$-submodule $W\subset H$ is also an $A$-submodule
+%% of $H$.
+%%}\newline
+%%
+%% $A\simeq \bigoplus_{i=1}^N M_{n_i}(\mathbb{C})$ is the matrix algebra
+%% then $H$ is a Hilbert $A$-bimodule and $W$ a submodule of $A$.
+%% Because we have $H = W \cup W^{\perp}$, then $W^{\perp}$ is naturally a
+%% $A$-submodule, because elements in $W^{\perp}$ need to satisfy the
+%% bimodularity.
+%%\end{MyExercise}
+%\begin{definition}
+% A $\Lambda$-decorated graph is given by an ordered pair $(\Gamma,
+% \Lambda)$ of a finite graph $\Gamma$ and a set of positive integers
+% $\Lambda$ with the labeling
+% \begin{itemize}
+% \item of the vetices $v\in \Gamma ^{(0)}$ given by $n(\nu) \in
+% \Lambda$
+% \item of the edges $e = (\nu _1, \nu _2) \in \Gamma ^{(1)}$ by
+% operators
+% \begin{itemize}
+% \item $D_e: \mathbb{C}^{n(\nu _1)} \rightarrow
+% \mathbb{C}^{n(\nu _2)}$
+% \item and $D_e^*: \mathbb{C}^{n(\nu _2)} \rightarrow
+% \mathbb{C}^{n(\nu _1)}$ its conjugate traspose
+% (pullback?)
+% \end{itemize}
+% \end{itemize}
+% such that
+% \begin{equation}
+% n(\Gamma ^{(0)}) = \Lambda
+% \end{equation}
+%\end{definition}
+%\begin{question}
+% Would then $D_e$ be the pullback?
+%\end{question}
+%\begin{question}
+% These graphs are important in the next chapter I should look
+% into it more, I don't understand much here, specific
+% how to construct them with the abstraction of a spectral triple...
+%\end{question}
+%
+%The operator $D_e$ between $\textbf{n}_i$ and $\textbf{n}_j$ add up to
+%$D_{ij}$
+%\begin{align}
+% D_{ij} = \sum\limits_{\substack{e = (\nu _1, \nu _2) \\ n(\nu _1) =
+% \textbf{n}_i \\ n(\nu _2) = \textbf{n}_j}} D_e
+%\end{align}
+%
+%\begin{theorem}
+% There is a on to one correspondence between finite spectral triples
+% modulo unitary equivalence and $\Lambda$-decorated graphs, given by
+% associating a finite spectral triples $(A, H, D)$ to a $\Lambda$ decorated
+% graph $(\Gamma, \Lambda)$ in the following way:
+% \begin{equation}
+% A = \bigoplus _{n\in \Lambda} M_n(\mathbb{C}); \;\;\;
+% H = \bigoplus _{\nu \in \Gamma ^{(0)}} \mathbb{C}^{n(\nu)}; \;\;\;
+% D = \sum _{e \in \Gamma ^{(1)}} D_e + D_e^*
+% \end{equation}
+%\end{theorem}
+% \begin{figure}[h!]
+% \centering
+% \begin{tikzpicture}[
+% mass/.style = {draw,circle, minimum size=0.3cm, inner sep=0pt, thick},
+% ]
+%
+% \node[mass, label={\textbf{n}}] (m1) at (1,0) {};
+% \draw (m1) to [out=330, in=210, looseness=25] node[above] {$D_e$} (m1);
+% \end{tikzpicture}
+% \caption{A $\Lambda$-decorated Graph of $(M_n(\mathbb{C}), \mathbb{C}^n,
+% D = D_e + D_e^*)$}
+%\end{figure}
+%
+%%\begin{MyExercise}
+%% \textbf{
+%% Draw a $\Lambda$ decorated graph corresponding to the spectral triple
+%% $(A=\mathbb{C}^3, H=\mathbb{C}^3, D=\begin{pmatrix}0 & \lambda & 0\\
+%% \bar{\lambda} &0 &0 \\ 0&0&0\end{pmatrix})$
+%%}\newline
+%%
+%%\centering
+%%\begin{tikzpicture}[
+%% mass/.style = {draw,circle, minimum size=0.4cm, inner sep=0pt, thick},
+%% spring/.style = {decorate,decoration={zigzag, pre length=1cm,post length=1cm,segment length=5pt}},]
+%% \node[mass] (m1) at (-1,1.5) {\textbf{1}};
+%% \node[mass] (m2) at (1,1.5) {\textbf{2}};
+%% \node[mass] (m3) at (3,1.5) {\textbf{3}};
+%%
+%% \draw[style=thick, -] (1.1,1.7) -- (-1.1,1.7);
+%% \draw[style=thick, -] (1.1,1.3) -- (-1.1,1.3);
+%% \end{tikzpicture}
+%% % \captionof{figure}{Solution}
+%%\end{MyExercise}
+%%\begin{MyExercise}
+%% \textbf{
+%% Use $\Lambda$-decorated graphs to classify all finite spectral triples
+%% (modulo unitary equivalence) on the matrix algebra
+%% $A=\mathbb{C}\oplus M_2(\mathbb{C})$
+%%}\newline
+%%
+%% \centering
+%%\begin{tikzpicture}[
+%% mass/.style = {draw,circle, minimum size=0.4cm, inner sep=0pt, thick},
+%% spring/.style = {decorate,decoration={zigzag, pre length=1cm,post length=1cm,segment length=5pt}},]
+%% \node[mass] (m1) at (-1,1) {\textbf{1}};
+%% \node[mass] (m2) at (1,1) {\textbf{2}};
+%% \node[mass] (m3) at (3,1) {\textbf{3}};
+%%
+%% \node[mass] (m4) at (-1,0) {\textbf{1}};
+%% \node[mass] (m5) at (1,0) {\textbf{2}};
+%% \node[mass] (m6) at (3,0) {\textbf{3}};
+%%
+%% \node[mass] (m7) at (-1,-1) {\textbf{1}};
+%% \node[mass] (m8) at (1,-1) {\textbf{2}};
+%% \node[mass] (m9) at (3,-1) {\textbf{3}};
+%%
+%% \node[mass] (m10) at (-1,-2) {\textbf{1}};
+%% \node[mass] (m11) at (1,-2) {\textbf{2}};
+%% \node[mass] (m12) at (3,-2) {\textbf{3}};
+%%
+%% \draw[style=thick, -] (1.1,0.2) -- (-1.1,0.2);
+%% \draw[style=thick, -] (1.1,-0.2) -- (-1.1,-0.2);
+%% \draw[style=thick, -] (m7) to [out=330, in=210, looseness=10] node[above] {} (m7);
+%% \draw[style=thick, -] (m10) -- (m11) ;
+%%
+%%\end{tikzpicture}
+%%% \captionof{figure}{Solution $A=M_3(\mathbb{C})$}
+%%\end{MyExercise}
+%\subsubsection{Graph Construction of Finite Spectral Triples}
+%\textbf{Algebra:}We know if a acts on a finite dimensional Hilbert space then
+%this C* algebra is isomorphic to a matrix algebra so $A \simeq
+%\bigoplus_{i=1}^{N}M_{n_i}(\mathbb{C})$. Where $i\in
+%\hat{A}$ represents an equivalence class and runs from $1$ to $N$,
+%thus $\hat{A}\simeq\{1,\dots, N\}$. We label equivalence classes by
+%$\textbf{n}_i$, then $\hat{A}\simeq\{\textbf{n}_1,\dots,\textbf{n}_N\}$.
+%\newline
+%
+%\textbf{Hilbert Space:} Since every Hilbert space that acts faithfully on a
+%C* algebra is completely reducible, it is isomorphic to the composition
+%of irreducible representations. $H \simeq \bigoplus_{i=1}^N\mathbb{C}^{n_i}
+%\otimes V_i$. Where all $V_i$'s are Vector spaces, their dimension is the
+%multiplicity of the representation landed by $\textbf{n}_i$ to $V_i$ itself
+%by the multiplicity space.
+%\newline
+%
+%\textbf{Finite Dirac Operator:} $D_{ij}$ is connecting nodes $\textbf{n}_i$
+%and $\textbf{n}_j$, with a symmetric map $D_{ij}:\mathbb{C}^{n_i}\otimes V_i
+%\rightarrow \mathbb{C}^{n_j}\otimes V_j$
+%\newline
+%
+%To draw a graph, draw nodes in position $\textbf{n}_i\in \hat{A}$.
+%Multiple nodes at the same position represent multiplicities in $H$.
+%Draw lines between nodes to represent $D_{ij}$.
+%
+%\begin{figure}[h!]
+% \centering
+%\begin{tikzpicture}
+% \node[draw, label=above:{$\textbf{n}_1$},circle, thick] at (-3,0) {};
+% \node[label=above:{$\dots$}] at (-2,0) {};
+% \node[draw, label=above:{$\textbf{n}_i$},circle, thick] at (-1,0) {};
+% \node[label=above:{$\dots$}] at (0,0) {};
+% \node[draw, label=above:{$\textbf{n}_j$},circle, thick] at (1,0) {};
+% \node[draw, label=above:{},circle, thick, inner sep=0cm, minimum
+% size=0.2cm] at (1,0) {};
+% \node[label=above:{$\dots$}] at (2,0) {};
+% \node[draw, label=above:{$\textbf{n}_N$},circle, thick] at (3,0) {};
+%
+% \draw[style=thick, -] (-1,-0.2) -- (1,-0.2);
+% \draw[style=thick, -] (-1,0.2) -- (1,0.2);
+% \path[style=thick, -] (-1,-0.2) edge[bend right=15]
+% node[pos=0.5,below] {} (3,-0.2);
+% \end{tikzpicture}
+% \caption{Example}
+%\end{figure}
+%
+%\subsection{Finite Real Noncommutative Spaces}
+%\subsubsection{Finite Real Spectral Triples}
+%Add on to finite real spectral triples a \textit{real structure}. The
+%requirement is that $H$ is a $A$-$A$-bimodule (before only a $A$-left
+%module).
+%\newline
+%
+%For this we introduce a $\mathbb{Z}_2$-grading $\gamma$ with
+%\begin{align}
+% &\gamma ^* = \gamma \\
+% &\gamma ^2 = 1 \\
+% &\gamma D = - D \gamma\\
+% &\gamma a = a \gamma \;\;\;\; a\in A
+%\end{align}
+%
+%\begin{definition}
+% A \textit{finite real spectral triple} is given by a finite spectral
+% triple $(A, H, D)$ and a anti-unitary operator $J:H\rightarrow H$ called
+% the \textit{real structure}, such that
+% \begin{align}
+% a^\circ := J a^* J^{-1}
+% \end{align}
+% is a right representation of $A$ on $H$, that is $(ab)^\circ = b^\circ
+% a^\circ$. With two requirements
+% \begin{align}
+% &[a, b^\circ] = 0\\
+% &[[D, a],b^\circ] = 0.
+% \end{align}
+% They are called the \textit{commutant property}, and mean that the left
+% action of an element in $A$ and $\Omega _D^1(A)$ commutes with the right
+% action on $A$.
+%\end{definition}
+%\begin{definition}
+% The $KO$-dimension of a real spectral triple is determined by the sings
+% $\epsilon, \epsilon ' ,\epsilon '' \in \{-1, 1\}$ appearing in
+% \begin{align}
+% &J^2 = \epsilon \\
+% &JD = \epsilon \ DJ\\
+% &J\gamma = \epsilon '' \gamma J.
+% \end{align}
+%\end{definition}
+%\begin{table}[h!]
+% \centering
+% \caption{$KO$-dimension $k$ modulo $8$ of a real spectral triple}
+% \begin{tabular}{ c | c c c c c c c c}
+% \hline
+% $k$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
+% \hline
+% $\epsilon$ & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
+% $\epsilon '$ & 1 & -1 & 1 & 1 & 1 & -1 & 1 & 1 \\
+% $\epsilon ''$ & 1 & & -1 & & 1 & & -1 & \\
+% \hline
+% \end{tabular}
+%\end{table}
+%
+%
+%\begin{definition}
+%An opposite-algebra $A^\circ$ of a $A$ is defined to be equal to $A$ as a
+%vector space with the opposite product
+%\begin{align}
+% &a\circ b := ba\\
+% &\Rightarrow a^\circ = Ja^* J^{-1} \;\;\; \text{defines the left
+% representation of $A^\circ$ on $H$}
+%\end{align}
+%\end{definition}
+%
+%
+%\begin{example}
+% Matrix algebra $M_N(\mathbb{C})$ acting on $H=M_N(\mathbb{C})$ by left
+% matrix multiplication with the Hilbert Schmidt inner product.
+% \begin{align}
+% \langle a , b \rangle = \text{Tr}(a^* b)
+% \end{align}
+% Then we define $\gamma (a) = a$ and $J(a) = a^*$ with $a\in H$.
+% Since $D$ mus be odd with respect to $\gamma$ it vanishes identically.
+%\end{example}
+%
+%\begin{definition}
+% We call $\xi \in H$ \textbf{cyclic vector} in $A$ if:
+% \begin{align}
+% A\xi := { a\xi:\;\; a\in A} = H
+% \end{align}
+%
+% We call $\xi \in H$ \textbf{separating vector} in $A$ if:
+% \begin{align}
+% a\xi = 0\;\; \Rightarrow \;\; a=0;\;\;\; a\in A
+% \end{align}
+%\end{definition}
+%
+%%\begin{MyExercise}
+%% \textbf{
+%% In the previous example, show that the right action on $M_N(\mathbb{C})$
+%% on $H = M_N(\mathbb{C})$ as defined by $a \mapsto a^\circ$
+%% is given by right matrix multiplication.
+%%}\newline
+%%
+%% \begin{align}
+%% a^\circ \xi = J a^* J^{-1}\xi = Ja^* \xi^* = J\xi a=\xi^* a
+%% \end{align}
+%%\end{MyExercise}
+%%\begin{MyExercise}
+%% \textbf{
+%% Let $A= \bigoplus _i M_{n_i}(\mathbb{C})$, represented on $H = \bigoplus_i \mathbb{C}^{n_i}
+%% \otimes \mathbb{C}^{m_i}$, meaning that the irreducible representation $\textbf{n}_i$ has
+%% multiplicity $m_i$.
+%% \begin{enumerate}
+%% \item Show that the commutant $A'$ of $A$ is $A'\simeq \bigoplus_i M_{m_i} (\mathbb{C})$. As a consequence show $A'' \simeq A$.
+%% \item Show that if $\xi$ is a separating vector for $A$ than it is cyclic for $A'$.
+%% \end{enumerate}
+%% }
+%%
+%%
+%% \begin{enumerate}
+%% \item We know the multiplicity space is $V_i = \mathbb{C}^{m_i}$. We know that
+%% for $T\in H$ and
+%% $a\in A'$ to work we need $aT=Ta$ by laws of matrix multiplication we need
+%% $A' \simeq \oplus _i M_{m_i}(\mathbb{C})$ for this to work since $H = \bigoplus_i
+%% \mathbb{C}^{n_i}
+%% \otimes \mathbb{C}^{m_i}$
+%%
+%% \item Suppose $\xi$ is cyclic for $A$ then $A'\xi = \{0\}$. Under the action of $A$ we
+%% then have $A'A\xi = AA' \xi = 0 \Rightarrow A' = 0$.\\
+%% Suppose now $\xi$ is separating for $A'$, we have $A'\xi = \{0\}$. We can define a
+%% projection in $A'$, $A\xi = P'$. With this projection we have $(1-P')\xi = 0
+%% \Rightarrow 1-P' = 0 \Rightarrow A\xi = H$.
+%% \end{enumerate}
+%%\end{MyExercise}
+%%\begin{MyExercise}
+%% \textbf{ Suppose $(A, H, D = 0)$ is a finite spectral triple such that $H$ possesses a
+%% cyclic and separating vector for $A$.
+%% \begin{enumerate}
+%% \item Show that the formula $S(a \xi) = a* \xi$ defines a anti-linear operator\\
+%% $S: H \rightarrow H$.
+%% \item Show that $S$ is invertible
+%% \item Let $J: H \rightarrow H$ be the operator in $S = J \Delta ^{1/2}$ with
+%% $\Delta = S^*S$. Show that $J$ is anti-unitary
+%% \end{enumerate}
+%% }
+%%
+%%
+%% \begin{enumerate}
+%% \item By composition $S(a\xi) = a*\xi$ this is literally anti-linearity. Does this mean
+%% $S\xi = \xi$?
+%% \item Let $\xi \in H$ be cyclic then: $S(A\xi) = A^*\xi = A\xi = H$. The same has to work
+%% for $S^{-1}$ if not then $\xi$ wouldn't exist. $S^{-1}(A^*\xi) = S^{-1}(H) = H$.
+%% \item Since $S$ is bijective then $\Delta ^{1/2}$ and $J$ need to be bijective.
+%% We also have $J = S \Delta^{-1/2}$ and $\Delta^* = \Delta$\\
+%% Now let $\xi _1 , \xi _2 \in H$ \begin{align}
+%% <J \xi _1 , J \xi _2 > &= < J^*J\xi_1 , \xi_2>^* =\\
+%% &= <(\Delta ^{-1/2})^* S^* S \Delta ^{-1/2} \xi_1, \xi_2>^* = \\
+%% &= <(\Delta^{-1/2})^* \Delta \Delta^{-1/2} \xi_1, \xi_2>^* =\\
+%% &= <\Delta^{-1/2} \Delta^{1/2}\Delta^{1/2} \Delta^{-1/2} \xi_1, \xi_2>^* =\\
+%% &= <\xi _1, \xi_2>^* = <\xi_2 , \xi_1>.
+%% \end{align}
+%% \end{enumerate}
+%%\end{MyExercise}
+%\subsubsection{Morphisms Between Finite Real Spectral Triples}
+%Extend unitary equivalence of finite spectral triples to real ones (with $J$
+%and $\gamma$)
+%
+%\begin{definition}
+% We call two finite real spectral triples $(A_1, H_1 ,D_1 ; J_1 , \gamma
+% _1)$ and $(A_2, H_2, D_2; J_2, \gamma _2)$ unitarily equivalent if $A_1 =
+% A_2$ and if there exists a unitary operator $U: H_1 \rightarrow H_2$ such
+% that
+% \begin{align}
+% &U\pi_1(a) U^* = \pi _2(a)\\
+% &UD_1U^*=D_2\\
+% &U\gamma _1 U^* = \gamma _2\\
+% &UJ_1 U^* = J_2
+% \end{align}
+%\end{definition}
+%\begin{definition}
+% Let $E$ be a $B$-$A$ bimodule. The \textit{conjugate Module} $E^\circ$ is
+% given by the $A$-$B$-bimodule.
+% \begin{align}
+% E^\circ = \{\bar{e} : e\in E\}
+% \end{align}
+% with
+% \begin{align}
+% a \cdot \bar{e} \cdot b = b^* \bar{e} a^* \;\;\;\; \forall a\in A, b \in
+% B
+% \end{align}
+%\end{definition}
+%$E^\circ$ is not a Hilbert bimodule for $(A, B)$ because it doesn't have a
+%natural $B$-valued inner product. But there is a $A$-valued inner product on
+%the left $A$-module $E^\circ$ with
+%\begin{align}
+% \langle \bar{e}_1, \bar{e}_2 \rangle = \langle e_2 , e_1 \rangle
+% \;\;\;\; e_1, e_2 \in E
+%\end{align}
+%and linearity in $A$:
+%\begin{align}
+% \langle a \bar{e}_1, \bar{e}_2 \rangle = a \langle \bar{e}_1, \bar{e}_2
+% \rangle \;\;\;\; \forall a \in A.
+%\end{align}
+%
+%%\begin{MyExercise}
+%% \textbf{Show that $E^\circ$ is a Hilbert bimodule $(B^{\circ}, A^{\circ})$
+%% }\newline
+%%
+%%
+%% Straightforward show properties of the Hilbert bimodule and its $B^{\circ}$
+%% valued inner product. Let $\bar{e}_1, \bar{e}_2 \in E^{\circ}$ and $a^\circ \in A,
+%% b^\circ \in B$. \\
+%% \begin{align}
+%% <\bar{e}_1, a^\circ \bar{e}_2> &= <\bar{e}_1, Ja^*J^{-1} \bar{e}_2>=\\
+%% &= <\bar{e}_1 , J a^* e_2> = \\
+%% &= <J^{-1} e_1, a^* e_2> =\\
+%% & = <a^* e_1, e_2>= <J^{-1}(a^\circ)^* J e_1, e_2> = \\
+%% & = <J^{-1} (a^\circ)^* \bar{e}_1, e_2> =\\
+%% & = <(a^\circ)^* \bar{e}_1 , \bar{e}_2>.
+%% \end{align}
+%%
+%% Next $<\bar{e}_1, \bar{e}_2 b^\circ> = <\bar{e}_1, \bar{e_2}> b^\circ$.
+%% \begin{align}
+%% <\bar{e}_1, \bar{e}_2 b^\circ> &= <\bar{e}_1, \bar{e}_2 Jb^*J^{-1}> =\\
+%% &= <\bar{e}_1, \bar{e_2}> Jb^*J^{-1} = \\
+%% &= <\bar{e}_1, \bar{e}_2> b^\circ.
+%% \end{align}
+%% Then:
+%% \begin{align}
+%% (<\bar{e}_1, \bar{e}_2)>_{E^\circ})^* &= (<e_2, e_1>_E)^* =\\
+%% &= <e_1, e_2>_E^* = <\bar{e}_2, \bar{e}_2>_{E^\circ}
+%% \end{align}
+%% And of course $<\bar{e}, \bar{e}> = <e, e> \geq 0$
+%%\end{MyExercise}
+%
+%\subsubsection{Construction of a Finite Real Spectral Triple from a Finite
+%Real Spectral Triple}
+%Given a Hilbert bimodule $E$ for $(B, A)$ we construct a spectral triple
+%$(B, H', D'; J', \gamma ')$ from $(A, H, D; J, \gamma)$
+%
+%For the $H'$ we make a $\mathbb{C}$-valued inner product on $H'$ by combining
+%the $A$ valued inner product on $E$ and $E^\circ$ with the
+%$\mathbb{C}$-valued inner product on $H$.
+%\begin{align}
+% H' := E\otimes _A H \otimes _A E^\circ
+%\end{align}
+%
+%Then the action of $B$ on $H'$ is:
+%\begin{align}
+% b(e_2 \otimes \xi \otimes \bar{e}_2 ) = (be_1) \otimes \xi \otimes
+% \bar{e}_2
+%\end{align}
+%The right action of $B$ on $H'$ defined by action on the right component
+%$E^\circ$
+%\begin{align}
+% J'(e_1 \otimes \xi \otimes \bar{e}_2) = e_2 \otimes J \xi \otimes
+% \bar{e}_1
+%\end{align}
+%with $b^\circ = J' b^* (J')^{-1}$, $b^* \in B$ action on $H'$.
+%\newline
+%
+%
+%\newpage
+%%\begin{MyExercise}
+%% \textbf{ Let $\nabla : E \Rightarrow E \otimes _A \Omega _d^1 (A)$ be a right connection on $E$
+%% consider the following anti-linear map:
+%% \begin{align}
+%% \tau : E \otimes_A \Omega _D^1 (A) &\rightarrow \Omega _D^1 (A) \otimes_A E^\circ\\
+%% e \otimes \omega &\mapsto -\omega ^* \otimes \bar{e}
+%% \end{align}
+%% Show that the map $\bar{\nabla} : E^\circ \rightarrow \Omega _D^1(A) \otimes E^\circ$
+%% with $\bar{\nabla}(\bar{e}) = \tau \circ \nabla(e)$ is a left connection, that means
+%% show that it satisfied the left Leibniz rule:
+%% \begin{equation}
+%% \bar{\nabla}(a\bar{e}) = [D, a] \otimes \bar{e} + a \bar{\nabla}(\bar{e})
+%% \end{equation}
+%% }
+%%
+%% Hagime:
+%% \begin{align}
+%% &\text{For one:}\\
+%% &\tau \circ \nabla(ae) = \bar{\nabla}(a\bar{e}) = \bar{\nabla}(a^* \bar{e})\\
+%% &\text{For two:}\\
+%% &\tau \circ \nabla(ae) = \tau(\nabla(e)a) + \tau \circ(e \otimes d(a))=\\
+%% &=a^*\bar{\nabla}(\bar{e}) - d(a)^* \otimes \bar{e}. \\
+%% &= a^*\bar{\nabla}(\bar{e}) + d(a^*) \otimes \bar{e}.
+%% \end{align}
+%%\end{MyExercise}
+%Then the connections
+%\begin{align}
+% &\nabla: E \rightarrow E\otimes _A \Omega _D ^1(A) \\
+% &\bar{\nabla}:E^\circ \rightarrow \Omega _D^1(A) \otimes _A E^\circ
+%\end{align}
+%give us the Dirac operator on $H' = E \otimes _A H \otimes _A E^\circ$
+%\begin{align}
+% D'(e_1 \otimes \xi \otimes \bar{e}_2) = (\nabla e_1) \xi \otimes
+% \bar{e_2}+ e_1 \otimes D\xi \otimes \bar{e}_2 + e_1 \otimes
+% \xi(\bar{\nabla}\bar{e}_2)
+%\end{align}
+%
+%And the right action of $\omega \in \Omega _D ^1(A)$ on $\xi \in H$ is
+%defined by
+%\begin{align}
+% \xi \mapsto \epsilon' J \omega ^* J^{-1}\xi
+%\end{align}
+%
+%Finally for the grading
+%\begin{align}
+% \gamma ' = 1 \otimes \gamma \otimes 1
+%\end{align}
+%
+%\begin{theorem}
+% Suppose $(A, H, D; J, \gamma)$ is a finite spectral triple of
+% $KO$-dimension $k$, let $\nabla$ be like above satisfying the
+% compatibility condition (like with finite spectral triples).
+%
+% Then $(B, H',D'; J', \gamma')$ is a finite spectral triple of
+% $KO$-Dimension $k$. ($H', D', J', \gamma'$ like above)
+%\end{theorem}
+%
+%\begin{proof}
+% The only thing left is to check if the $KO$-dimension is preserved,
+% for this we check if the $\epsilon$'s are the same.
+% \begin{align}
+% &(J')^2 = 1 \otimes J^2 \otimes 1 = \epsilon\\
+% &J' \gamma '= \epsilon ''\gamma'J'
+% \end{align}
+% and for $\epsilon '$
+% \begin{align}
+% J'D'(e_1 \otimes \xi \otimes \bar{e}_2)&=J'((\nabla e_1) \xi \otimes
+% \bar{e_2} + e_1 \otimes D\xi \otimes \bar{e}_2 + e_1 \otimes \xi (\tau
+% \nabla e_2))\\
+% &= \epsilon' D'(e_2 \otimes J\xi \otimes \bar{e}_2)\\
+% &= \epsilon' D'J'(e_1 \otimes \xi \bar{e}_2)
+% \end{align}
+%\end{proof}
+%
+%
+%\end{document}
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+ </bcf:options>
+ <!-- BIBLATEX OPTIONS -->
+ <!-- GLOBAL -->
+ <bcf:options component="biblatex" type="global">
+ <bcf:option type="singlevalued">
+ <bcf:key>alphaothers</bcf:key>
+ <bcf:value>+</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>labelalpha</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="multivalued">
+ <bcf:key>labelnamespec</bcf:key>
+ <bcf:value order="1">shortauthor</bcf:value>
+ <bcf:value order="2">author</bcf:value>
+ <bcf:value order="3">shorteditor</bcf:value>
+ <bcf:value order="4">editor</bcf:value>
+ <bcf:value order="5">translator</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>labeltitle</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="multivalued">
+ <bcf:key>labeltitlespec</bcf:key>
+ <bcf:value order="1">shorttitle</bcf:value>
+ <bcf:value order="2">title</bcf:value>
+ <bcf:value order="3">maintitle</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>labeltitleyear</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>labeldateparts</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="multivalued">
+ <bcf:key>labeldatespec</bcf:key>
+ <bcf:value order="1" type="field">date</bcf:value>
+ <bcf:value order="2" type="field">year</bcf:value>
+ <bcf:value order="3" type="field">eventdate</bcf:value>
+ <bcf:value order="4" type="field">origdate</bcf:value>
+ <bcf:value order="5" type="field">urldate</bcf:value>
+ <bcf:value order="6" type="string">nodate</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>julian</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>gregorianstart</bcf:key>
+ <bcf:value>1582-10-15</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>maxalphanames</bcf:key>
+ <bcf:value>3</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>maxbibnames</bcf:key>
+ <bcf:value>3</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>maxcitenames</bcf:key>
+ <bcf:value>3</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>maxsortnames</bcf:key>
+ <bcf:value>3</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>maxitems</bcf:key>
+ <bcf:value>3</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>minalphanames</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>minbibnames</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>mincitenames</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>minsortnames</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>minitems</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>nohashothers</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>noroman</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>nosortothers</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>singletitle</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>skipbib</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>skipbiblist</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>skiplab</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>sortalphaothers</bcf:key>
+ <bcf:value>+</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>sortlocale</bcf:key>
+ <bcf:value>english</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>sortingtemplatename</bcf:key>
+ <bcf:value>none</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>sortsets</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>uniquelist</bcf:key>
+ <bcf:value>false</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>uniquename</bcf:key>
+ <bcf:value>false</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>uniqueprimaryauthor</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>uniquetitle</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>uniquebaretitle</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>uniquework</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useprefix</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useafterword</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useannotator</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useauthor</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>usebookauthor</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>usecommentator</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useeditor</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useeditora</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useeditorb</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useeditorc</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useforeword</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useholder</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useintroduction</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>usenamea</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>usenameb</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>usenamec</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>usetranslator</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useshortauthor</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useshorteditor</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ </bcf:options>
+ <!-- online -->
+ <bcf:options component="biblatex" type="online">
+ <bcf:option type="singlevalued">
+ <bcf:key>labelalpha</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="multivalued">
+ <bcf:key>labelnamespec</bcf:key>
+ <bcf:value order="1">shortauthor</bcf:value>
+ <bcf:value order="2">author</bcf:value>
+ <bcf:value order="3">shorteditor</bcf:value>
+ <bcf:value order="4">editor</bcf:value>
+ <bcf:value order="5">translator</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>labeltitle</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="multivalued">
+ <bcf:key>labeltitlespec</bcf:key>
+ <bcf:value order="1">shorttitle</bcf:value>
+ <bcf:value order="2">title</bcf:value>
+ <bcf:value order="3">maintitle</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>labeltitleyear</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>labeldateparts</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="multivalued">
+ <bcf:key>labeldatespec</bcf:key>
+ <bcf:value order="1" type="field">date</bcf:value>
+ <bcf:value order="2" type="field">year</bcf:value>
+ <bcf:value order="3" type="field">eventdate</bcf:value>
+ <bcf:value order="4" type="field">origdate</bcf:value>
+ <bcf:value order="5" type="field">urldate</bcf:value>
+ <bcf:value order="6" type="string">nodate</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>maxalphanames</bcf:key>
+ <bcf:value>3</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>maxbibnames</bcf:key>
+ <bcf:value>3</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>maxcitenames</bcf:key>
+ <bcf:value>3</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>maxsortnames</bcf:key>
+ <bcf:value>3</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>maxitems</bcf:key>
+ <bcf:value>3</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>minalphanames</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>minbibnames</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>mincitenames</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>minsortnames</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>minitems</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>nohashothers</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>noroman</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>nosortothers</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>singletitle</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>skipbib</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>skiplab</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>skipbiblist</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>uniquelist</bcf:key>
+ <bcf:value>false</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>uniquename</bcf:key>
+ <bcf:value>false</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>uniqueprimaryauthor</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>uniquetitle</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>uniquebaretitle</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>uniquework</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useprefix</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useafterword</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useannotator</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useauthor</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>usebookauthor</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>usecommentator</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useeditor</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useeditora</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useeditorb</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useeditorc</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useforeword</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useholder</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useintroduction</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>usenamea</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>usenameb</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>usenamec</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>usetranslator</bcf:key>
+ <bcf:value>0</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useshortauthor</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ <bcf:option type="singlevalued">
+ <bcf:key>useshorteditor</bcf:key>
+ <bcf:value>1</bcf:value>
+ </bcf:option>
+ </bcf:options>
+ <!-- BIBLATEX OPTION SCOPE -->
+ <bcf:optionscope type="GLOBAL">
+ <bcf:option datatype="xml">datamodel</bcf:option>
+ <bcf:option datatype="xml">labelalphanametemplate</bcf:option>
+ <bcf:option datatype="xml">labelalphatemplate</bcf:option>
+ <bcf:option datatype="xml">inheritance</bcf:option>
+ <bcf:option datatype="xml">translit</bcf:option>
+ <bcf:option datatype="xml">uniquenametemplate</bcf:option>
+ <bcf:option datatype="xml">sortingnamekeytemplate</bcf:option>
+ <bcf:option datatype="xml">sortingtemplate</bcf:option>
+ <bcf:option datatype="xml">extradatespec</bcf:option>
+ <bcf:option datatype="xml">labelnamespec</bcf:option>
+ <bcf:option datatype="xml">labeltitlespec</bcf:option>
+ <bcf:option datatype="xml">labeldatespec</bcf:option>
+ <bcf:option datatype="string">controlversion</bcf:option>
+ <bcf:option datatype="string">alphaothers</bcf:option>
+ <bcf:option datatype="string">sortalphaothers</bcf:option>
+ <bcf:option datatype="string">presort</bcf:option>
+ <bcf:option datatype="string">texencoding</bcf:option>
+ <bcf:option datatype="string">bibencoding</bcf:option>
+ <bcf:option datatype="string">sortingtemplatename</bcf:option>
+ <bcf:option datatype="string">sortlocale</bcf:option>
+ <bcf:option datatype="string">language</bcf:option>
+ <bcf:option datatype="string">autolang</bcf:option>
+ <bcf:option datatype="string">langhook</bcf:option>
+ <bcf:option datatype="string">indexing</bcf:option>
+ <bcf:option datatype="string">hyperref</bcf:option>
+ <bcf:option datatype="string">backrefsetstyle</bcf:option>
+ <bcf:option datatype="string">block</bcf:option>
+ <bcf:option datatype="string">pagetracker</bcf:option>
+ <bcf:option datatype="string">citecounter</bcf:option>
+ <bcf:option datatype="string">citetracker</bcf:option>
+ <bcf:option datatype="string">ibidtracker</bcf:option>
+ <bcf:option datatype="string">idemtracker</bcf:option>
+ <bcf:option datatype="string">opcittracker</bcf:option>
+ <bcf:option datatype="string">loccittracker</bcf:option>
+ <bcf:option datatype="string">labeldate</bcf:option>
+ <bcf:option datatype="string">labeltime</bcf:option>
+ <bcf:option datatype="string">dateera</bcf:option>
+ <bcf:option datatype="string">date</bcf:option>
+ <bcf:option datatype="string">time</bcf:option>
+ <bcf:option datatype="string">eventdate</bcf:option>
+ <bcf:option datatype="string">eventtime</bcf:option>
+ <bcf:option datatype="string">origdate</bcf:option>
+ <bcf:option datatype="string">origtime</bcf:option>
+ <bcf:option datatype="string">urldate</bcf:option>
+ <bcf:option datatype="string">urltime</bcf:option>
+ <bcf:option datatype="string">alldatesusetime</bcf:option>
+ <bcf:option datatype="string">alldates</bcf:option>
+ <bcf:option datatype="string">alltimes</bcf:option>
+ <bcf:option datatype="string">gregorianstart</bcf:option>
+ <bcf:option datatype="string">autocite</bcf:option>
+ <bcf:option datatype="string">notetype</bcf:option>
+ <bcf:option datatype="string">uniquelist</bcf:option>
+ <bcf:option datatype="string">uniquename</bcf:option>
+ <bcf:option datatype="string">refsection</bcf:option>
+ <bcf:option datatype="string">refsegment</bcf:option>
+ <bcf:option datatype="string">citereset</bcf:option>
+ <bcf:option datatype="string">sortlos</bcf:option>
+ <bcf:option datatype="string">babel</bcf:option>
+ <bcf:option datatype="string">datelabel</bcf:option>
+ <bcf:option datatype="string">backrefstyle</bcf:option>
+ <bcf:option datatype="string">arxiv</bcf:option>
+ <bcf:option datatype="boolean">familyinits</bcf:option>
+ <bcf:option datatype="boolean">giveninits</bcf:option>
+ <bcf:option datatype="boolean">prefixinits</bcf:option>
+ <bcf:option datatype="boolean">suffixinits</bcf:option>
+ <bcf:option datatype="boolean">useafterword</bcf:option>
+ <bcf:option datatype="boolean">useannotator</bcf:option>
+ <bcf:option datatype="boolean">useauthor</bcf:option>
+ <bcf:option datatype="boolean">usebookauthor</bcf:option>
+ <bcf:option datatype="boolean">usecommentator</bcf:option>
+ <bcf:option datatype="boolean">useeditor</bcf:option>
+ <bcf:option datatype="boolean">useeditora</bcf:option>
+ <bcf:option datatype="boolean">useeditorb</bcf:option>
+ <bcf:option datatype="boolean">useeditorc</bcf:option>
+ <bcf:option datatype="boolean">useforeword</bcf:option>
+ <bcf:option datatype="boolean">useholder</bcf:option>
+ <bcf:option datatype="boolean">useintroduction</bcf:option>
+ <bcf:option datatype="boolean">usenamea</bcf:option>
+ <bcf:option datatype="boolean">usenameb</bcf:option>
+ <bcf:option datatype="boolean">usenamec</bcf:option>
+ <bcf:option datatype="boolean">usetranslator</bcf:option>
+ <bcf:option datatype="boolean">useshortauthor</bcf:option>
+ <bcf:option datatype="boolean">useshorteditor</bcf:option>
+ <bcf:option datatype="boolean">debug</bcf:option>
+ <bcf:option datatype="boolean">loadfiles</bcf:option>
+ <bcf:option datatype="boolean">safeinputenc</bcf:option>
+ <bcf:option datatype="boolean">sortcase</bcf:option>
+ <bcf:option datatype="boolean">sortupper</bcf:option>
+ <bcf:option datatype="boolean">terseinits</bcf:option>
+ <bcf:option datatype="boolean">abbreviate</bcf:option>
+ <bcf:option datatype="boolean">dateabbrev</bcf:option>
+ <bcf:option datatype="boolean">clearlang</bcf:option>
+ <bcf:option datatype="boolean">sortcites</bcf:option>
+ <bcf:option datatype="boolean">sortsets</bcf:option>
+ <bcf:option datatype="boolean">backref</bcf:option>
+ <bcf:option datatype="boolean">backreffloats</bcf:option>
+ <bcf:option datatype="boolean">trackfloats</bcf:option>
+ <bcf:option datatype="boolean">parentracker</bcf:option>
+ <bcf:option datatype="boolean">labeldateusetime</bcf:option>
+ <bcf:option datatype="boolean">datecirca</bcf:option>
+ <bcf:option datatype="boolean">dateuncertain</bcf:option>
+ <bcf:option datatype="boolean">dateusetime</bcf:option>
+ <bcf:option datatype="boolean">eventdateusetime</bcf:option>
+ <bcf:option datatype="boolean">origdateusetime</bcf:option>
+ <bcf:option datatype="boolean">urldateusetime</bcf:option>
+ <bcf:option datatype="boolean">julian</bcf:option>
+ <bcf:option datatype="boolean">datezeros</bcf:option>
+ <bcf:option datatype="boolean">timezeros</bcf:option>
+ <bcf:option datatype="boolean">timezones</bcf:option>
+ <bcf:option datatype="boolean">seconds</bcf:option>
+ <bcf:option datatype="boolean">autopunct</bcf:option>
+ <bcf:option datatype="boolean">punctfont</bcf:option>
+ <bcf:option datatype="boolean">labelnumber</bcf:option>
+ <bcf:option datatype="boolean">labelalpha</bcf:option>
+ <bcf:option datatype="boolean">labeltitle</bcf:option>
+ <bcf:option datatype="boolean">labeltitleyear</bcf:option>
+ <bcf:option datatype="boolean">labeldateparts</bcf:option>
+ <bcf:option datatype="boolean">nohashothers</bcf:option>
+ <bcf:option datatype="boolean">nosortothers</bcf:option>
+ <bcf:option datatype="boolean">noroman</bcf:option>
+ <bcf:option datatype="boolean">singletitle</bcf:option>
+ <bcf:option datatype="boolean">uniquetitle</bcf:option>
+ <bcf:option datatype="boolean">uniquebaretitle</bcf:option>
+ <bcf:option datatype="boolean">uniquework</bcf:option>
+ <bcf:option datatype="boolean">uniqueprimaryauthor</bcf:option>
+ <bcf:option datatype="boolean">defernumbers</bcf:option>
+ <bcf:option datatype="boolean">locallabelwidth</bcf:option>
+ <bcf:option datatype="boolean">bibwarn</bcf:option>
+ <bcf:option datatype="boolean">useprefix</bcf:option>
+ <bcf:option datatype="boolean">skipbib</bcf:option>
+ <bcf:option datatype="boolean">skipbiblist</bcf:option>
+ <bcf:option datatype="boolean">skiplab</bcf:option>
+ <bcf:option datatype="boolean">dataonly</bcf:option>
+ <bcf:option datatype="boolean">defernums</bcf:option>
+ <bcf:option datatype="boolean">firstinits</bcf:option>
+ <bcf:option datatype="boolean">sortfirstinits</bcf:option>
+ <bcf:option datatype="boolean">sortgiveninits</bcf:option>
+ <bcf:option datatype="boolean">labelyear</bcf:option>
+ <bcf:option datatype="boolean">isbn</bcf:option>
+ <bcf:option datatype="boolean">url</bcf:option>
+ <bcf:option datatype="boolean">doi</bcf:option>
+ <bcf:option datatype="boolean">eprint</bcf:option>
+ <bcf:option datatype="boolean">related</bcf:option>
+ <bcf:option datatype="boolean">subentry</bcf:option>
+ <bcf:option datatype="boolean">bibtexcaseprotection</bcf:option>
+ <bcf:option datatype="integer">mincrossrefs</bcf:option>
+ <bcf:option datatype="integer">minxrefs</bcf:option>
+ <bcf:option datatype="integer">maxnames</bcf:option>
+ <bcf:option datatype="integer">minnames</bcf:option>
+ <bcf:option datatype="integer">maxbibnames</bcf:option>
+ <bcf:option datatype="integer">minbibnames</bcf:option>
+ <bcf:option datatype="integer">maxcitenames</bcf:option>
+ <bcf:option datatype="integer">mincitenames</bcf:option>
+ <bcf:option datatype="integer">maxsortnames</bcf:option>
+ <bcf:option datatype="integer">minsortnames</bcf:option>
+ <bcf:option datatype="integer">maxitems</bcf:option>
+ <bcf:option datatype="integer">minitems</bcf:option>
+ <bcf:option datatype="integer">maxalphanames</bcf:option>
+ <bcf:option datatype="integer">minalphanames</bcf:option>
+ <bcf:option datatype="integer">maxparens</bcf:option>
+ <bcf:option datatype="integer">dateeraauto</bcf:option>
+ </bcf:optionscope>
+ <bcf:optionscope type="ENTRYTYPE">
+ <bcf:option datatype="string">alphaothers</bcf:option>
+ <bcf:option datatype="string">sortalphaothers</bcf:option>
+ <bcf:option datatype="string">presort</bcf:option>
+ <bcf:option datatype="string">indexing</bcf:option>
+ <bcf:option datatype="string">citetracker</bcf:option>
+ <bcf:option datatype="string">ibidtracker</bcf:option>
+ <bcf:option datatype="string">idemtracker</bcf:option>
+ <bcf:option datatype="string">opcittracker</bcf:option>
+ <bcf:option datatype="string">loccittracker</bcf:option>
+ <bcf:option datatype="string">uniquelist</bcf:option>
+ <bcf:option datatype="string">uniquename</bcf:option>
+ <bcf:option datatype="boolean">familyinits</bcf:option>
+ <bcf:option datatype="boolean">giveninits</bcf:option>
+ <bcf:option datatype="boolean">prefixinits</bcf:option>
+ <bcf:option datatype="boolean">suffixinits</bcf:option>
+ <bcf:option datatype="boolean">useafterword</bcf:option>
+ <bcf:option datatype="boolean">useannotator</bcf:option>
+ <bcf:option datatype="boolean">useauthor</bcf:option>
+ <bcf:option datatype="boolean">usebookauthor</bcf:option>
+ <bcf:option datatype="boolean">usecommentator</bcf:option>
+ <bcf:option datatype="boolean">useeditor</bcf:option>
+ <bcf:option datatype="boolean">useeditora</bcf:option>
+ <bcf:option datatype="boolean">useeditorb</bcf:option>
+ <bcf:option datatype="boolean">useeditorc</bcf:option>
+ <bcf:option datatype="boolean">useforeword</bcf:option>
+ <bcf:option datatype="boolean">useholder</bcf:option>
+ <bcf:option datatype="boolean">useintroduction</bcf:option>
+ <bcf:option datatype="boolean">usenamea</bcf:option>
+ <bcf:option datatype="boolean">usenameb</bcf:option>
+ <bcf:option datatype="boolean">usenamec</bcf:option>
+ <bcf:option datatype="boolean">usetranslator</bcf:option>
+ <bcf:option datatype="boolean">useshortauthor</bcf:option>
+ <bcf:option datatype="boolean">useshorteditor</bcf:option>
+ <bcf:option datatype="boolean">terseinits</bcf:option>
+ <bcf:option datatype="boolean">abbreviate</bcf:option>
+ <bcf:option datatype="boolean">dateabbrev</bcf:option>
+ <bcf:option datatype="boolean">clearlang</bcf:option>
+ <bcf:option datatype="boolean">labelnumber</bcf:option>
+ <bcf:option datatype="boolean">labelalpha</bcf:option>
+ <bcf:option datatype="boolean">labeltitle</bcf:option>
+ <bcf:option datatype="boolean">labeltitleyear</bcf:option>
+ <bcf:option datatype="boolean">labeldateparts</bcf:option>
+ <bcf:option datatype="boolean">nohashothers</bcf:option>
+ <bcf:option datatype="boolean">nosortothers</bcf:option>
+ <bcf:option datatype="boolean">noroman</bcf:option>
+ <bcf:option datatype="boolean">singletitle</bcf:option>
+ <bcf:option datatype="boolean">uniquetitle</bcf:option>
+ <bcf:option datatype="boolean">uniquebaretitle</bcf:option>
+ <bcf:option datatype="boolean">uniquework</bcf:option>
+ <bcf:option datatype="boolean">uniqueprimaryauthor</bcf:option>
+ <bcf:option datatype="boolean">useprefix</bcf:option>
+ <bcf:option datatype="boolean">skipbib</bcf:option>
+ <bcf:option datatype="boolean">skipbiblist</bcf:option>
+ <bcf:option datatype="boolean">skiplab</bcf:option>
+ <bcf:option datatype="boolean">dataonly</bcf:option>
+ <bcf:option datatype="boolean">skiplos</bcf:option>
+ <bcf:option datatype="boolean">labelyear</bcf:option>
+ <bcf:option datatype="boolean">isbn</bcf:option>
+ <bcf:option datatype="boolean">url</bcf:option>
+ <bcf:option datatype="boolean">doi</bcf:option>
+ <bcf:option datatype="boolean">eprint</bcf:option>
+ <bcf:option datatype="boolean">related</bcf:option>
+ <bcf:option datatype="boolean">subentry</bcf:option>
+ <bcf:option datatype="boolean">bibtexcaseprotection</bcf:option>
+ <bcf:option datatype="xml">labelalphatemplate</bcf:option>
+ <bcf:option datatype="xml">translit</bcf:option>
+ <bcf:option datatype="xml">sortexclusion</bcf:option>
+ <bcf:option datatype="xml">sortinclusion</bcf:option>
+ <bcf:option datatype="xml">labelnamespec</bcf:option>
+ <bcf:option datatype="xml">labeltitlespec</bcf:option>
+ <bcf:option datatype="xml">labeldatespec</bcf:option>
+ <bcf:option datatype="integer">maxnames</bcf:option>
+ <bcf:option datatype="integer">minnames</bcf:option>
+ <bcf:option datatype="integer">maxbibnames</bcf:option>
+ <bcf:option datatype="integer">minbibnames</bcf:option>
+ <bcf:option datatype="integer">maxcitenames</bcf:option>
+ <bcf:option datatype="integer">mincitenames</bcf:option>
+ <bcf:option datatype="integer">maxsortnames</bcf:option>
+ <bcf:option datatype="integer">minsortnames</bcf:option>
+ <bcf:option datatype="integer">maxitems</bcf:option>
+ <bcf:option datatype="integer">minitems</bcf:option>
+ <bcf:option datatype="integer">maxalphanames</bcf:option>
+ <bcf:option datatype="integer">minalphanames</bcf:option>
+ </bcf:optionscope>
+ <bcf:optionscope type="ENTRY">
+ <bcf:option datatype="string">noinherit</bcf:option>
+ <bcf:option datatype="string" backendin="sortingnamekeytemplatename,uniquenametemplatename,labelalphanametemplatename">nametemplates</bcf:option>
+ <bcf:option datatype="string" backendout="1">labelalphanametemplatename</bcf:option>
+ <bcf:option datatype="string" backendout="1">uniquenametemplatename</bcf:option>
+ <bcf:option datatype="string" backendout="1">sortingnamekeytemplatename</bcf:option>
+ <bcf:option datatype="string">presort</bcf:option>
+ <bcf:option datatype="string" backendout="1">indexing</bcf:option>
+ <bcf:option datatype="string" backendout="1">citetracker</bcf:option>
+ <bcf:option datatype="string" backendout="1">ibidtracker</bcf:option>
+ <bcf:option datatype="string" backendout="1">idemtracker</bcf:option>
+ <bcf:option datatype="string" backendout="1">opcittracker</bcf:option>
+ <bcf:option datatype="string" backendout="1">loccittracker</bcf:option>
+ <bcf:option datatype="string">uniquelist</bcf:option>
+ <bcf:option datatype="string">uniquename</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">familyinits</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">giveninits</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">prefixinits</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">suffixinits</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useafterword</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useannotator</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useauthor</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">usebookauthor</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">usecommentator</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useeditor</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useeditora</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useeditorb</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useeditorc</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useforeword</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useholder</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useintroduction</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">usenamea</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">usenameb</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">usenamec</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">usetranslator</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useshortauthor</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useshorteditor</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">terseinits</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">abbreviate</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">dateabbrev</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">clearlang</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">labelnumber</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">labelalpha</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">labeltitle</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">labeltitleyear</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">labeldateparts</bcf:option>
+ <bcf:option datatype="boolean">nohashothers</bcf:option>
+ <bcf:option datatype="boolean">nosortothers</bcf:option>
+ <bcf:option datatype="boolean">noroman</bcf:option>
+ <bcf:option datatype="boolean">singletitle</bcf:option>
+ <bcf:option datatype="boolean">uniquetitle</bcf:option>
+ <bcf:option datatype="boolean">uniquebaretitle</bcf:option>
+ <bcf:option datatype="boolean">uniquework</bcf:option>
+ <bcf:option datatype="boolean">uniqueprimaryauthor</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useprefix</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">skipbib</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">skipbiblist</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">skiplab</bcf:option>
+ <bcf:option datatype="boolean" backendin="uniquename=false,uniquelist=false,skipbib=true,skipbiblist=true,skiplab=true">dataonly</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">skiplos</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">isbn</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">url</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">doi</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">eprint</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">related</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">subentry</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">bibtexcaseprotection</bcf:option>
+ <bcf:option datatype="integer" backendin="maxcitenames,maxbibnames,maxsortnames">maxnames</bcf:option>
+ <bcf:option datatype="integer" backendin="mincitenames,minbibnames,minsortnames">minnames</bcf:option>
+ <bcf:option datatype="integer" backendout="1">maxbibnames</bcf:option>
+ <bcf:option datatype="integer" backendout="1">minbibnames</bcf:option>
+ <bcf:option datatype="integer" backendout="1">maxcitenames</bcf:option>
+ <bcf:option datatype="integer" backendout="1">mincitenames</bcf:option>
+ <bcf:option datatype="integer" backendout="1">maxsortnames</bcf:option>
+ <bcf:option datatype="integer" backendout="1">minsortnames</bcf:option>
+ <bcf:option datatype="integer" backendout="1">maxitems</bcf:option>
+ <bcf:option datatype="integer" backendout="1">minitems</bcf:option>
+ <bcf:option datatype="integer" backendout="1">maxalphanames</bcf:option>
+ <bcf:option datatype="integer" backendout="1">minalphanames</bcf:option>
+ </bcf:optionscope>
+ <bcf:optionscope type="NAMELIST">
+ <bcf:option datatype="string" backendin="sortingnamekeytemplatename,uniquenametemplatename,labelalphanametemplatename">nametemplates</bcf:option>
+ <bcf:option datatype="string" backendout="1">labelalphanametemplatename</bcf:option>
+ <bcf:option datatype="string" backendout="1">uniquenametemplatename</bcf:option>
+ <bcf:option datatype="string" backendout="1">sortingnamekeytemplatename</bcf:option>
+ <bcf:option datatype="string">uniquelist</bcf:option>
+ <bcf:option datatype="string">uniquename</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">familyinits</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">giveninits</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">prefixinits</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">suffixinits</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">terseinits</bcf:option>
+ <bcf:option datatype="boolean">nohashothers</bcf:option>
+ <bcf:option datatype="boolean">nosortothers</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useprefix</bcf:option>
+ </bcf:optionscope>
+ <bcf:optionscope type="NAME">
+ <bcf:option datatype="string" backendin="sortingnamekeytemplatename,uniquenametemplatename,labelalphanametemplatename">nametemplates</bcf:option>
+ <bcf:option datatype="string" backendout="1">labelalphanametemplatename</bcf:option>
+ <bcf:option datatype="string" backendout="1">uniquenametemplatename</bcf:option>
+ <bcf:option datatype="string" backendout="1">sortingnamekeytemplatename</bcf:option>
+ <bcf:option datatype="string">uniquename</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">familyinits</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">giveninits</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">prefixinits</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">suffixinits</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">terseinits</bcf:option>
+ <bcf:option datatype="boolean" backendout="1">useprefix</bcf:option>
+ </bcf:optionscope>
+ <!-- DATAFIELDSETS -->
+ <bcf:datafieldset name="setnames">
+ <bcf:member datatype="name" fieldtype="list"/>
+ </bcf:datafieldset>
+ <bcf:datafieldset name="settitles">
+ <bcf:member field="title"/>
+ <bcf:member field="booktitle"/>
+ <bcf:member field="eventtitle"/>
+ <bcf:member field="issuetitle"/>
+ <bcf:member field="journaltitle"/>
+ <bcf:member field="maintitle"/>
+ <bcf:member field="origtitle"/>
+ </bcf:datafieldset>
+ <!-- SOURCEMAP -->
+ <bcf:sourcemap>
+ <bcf:maps datatype="bibtex" level="driver">
+ <bcf:map>
+ <bcf:map_step map_field_set="day" map_null="1"/>
+ </bcf:map>
+ <bcf:map>
+ <bcf:map_step map_type_source="conference" map_type_target="inproceedings"/>
+ <bcf:map_step map_type_source="electronic" map_type_target="online"/>
+ <bcf:map_step map_type_source="www" map_type_target="online"/>
+ </bcf:map>
+ <bcf:map>
+ <bcf:map_step map_type_source="mastersthesis" map_type_target="thesis" map_final="1"/>
+ <bcf:map_step map_field_set="type" map_field_value="mathesis"/>
+ </bcf:map>
+ <bcf:map>
+ <bcf:map_step map_type_source="phdthesis" map_type_target="thesis" map_final="1"/>
+ <bcf:map_step map_field_set="type" map_field_value="phdthesis"/>
+ </bcf:map>
+ <bcf:map>
+ <bcf:map_step map_type_source="techreport" map_type_target="report" map_final="1"/>
+ <bcf:map_step map_field_set="type" map_field_value="techreport"/>
+ </bcf:map>
+ <bcf:map>
+ <bcf:map_step map_field_source="hyphenation" map_field_target="langid"/>
+ <bcf:map_step map_field_source="address" map_field_target="location"/>
+ <bcf:map_step map_field_source="school" map_field_target="institution"/>
+ <bcf:map_step map_field_source="annote" map_field_target="annotation"/>
+ <bcf:map_step map_field_source="archiveprefix" map_field_target="eprinttype"/>
+ <bcf:map_step map_field_source="journal" map_field_target="journaltitle"/>
+ <bcf:map_step map_field_source="primaryclass" map_field_target="eprintclass"/>
+ <bcf:map_step map_field_source="key" map_field_target="sortkey"/>
+ <bcf:map_step map_field_source="pdf" map_field_target="file"/>
+ </bcf:map>
+ </bcf:maps>
+ </bcf:sourcemap>
+ <!-- LABELALPHA NAME TEMPLATE -->
+ <bcf:labelalphanametemplate name="global">
+ <bcf:namepart order="1" use="1" pre="1" substring_width="1" substring_compound="1">prefix</bcf:namepart>
+ <bcf:namepart order="2">family</bcf:namepart>
+ </bcf:labelalphanametemplate>
+ <!-- LABELALPHA TEMPLATE -->
+ <bcf:labelalphatemplate type="global">
+ <bcf:labelelement order="1">
+ <bcf:labelpart final="1">shorthand</bcf:labelpart>
+ <bcf:labelpart>label</bcf:labelpart>
+ <bcf:labelpart substring_width="3" substring_side="left" ifnames="1">labelname</bcf:labelpart>
+ <bcf:labelpart substring_width="1" substring_side="left">labelname</bcf:labelpart>
+ </bcf:labelelement>
+ <bcf:labelelement order="2">
+ <bcf:labelpart substring_width="2" substring_side="right">year</bcf:labelpart>
+ </bcf:labelelement>
+ </bcf:labelalphatemplate>
+ <!-- EXTRADATE -->
+ <bcf:extradatespec>
+ <bcf:scope>
+ <bcf:field order="1">labelyear</bcf:field>
+ <bcf:field order="2">year</bcf:field>
+ </bcf:scope>
+ </bcf:extradatespec>
+ <!-- INHERITANCE -->
+ <bcf:inheritance>
+ <bcf:defaults inherit_all="true" override_target="false">
+ </bcf:defaults>
+ <bcf:inherit>
+ <bcf:type_pair source="mvbook" target="inbook"/>
+ <bcf:type_pair source="mvbook" target="bookinbook"/>
+ <bcf:type_pair source="mvbook" target="suppbook"/>
+ <bcf:type_pair source="book" target="inbook"/>
+ <bcf:type_pair source="book" target="bookinbook"/>
+ <bcf:type_pair source="book" target="suppbook"/>
+ <bcf:field source="author" target="author"/>
+ <bcf:field source="author" target="bookauthor"/>
+ </bcf:inherit>
+ <bcf:inherit>
+ <bcf:type_pair source="mvbook" target="book"/>
+ <bcf:type_pair source="mvbook" target="inbook"/>
+ <bcf:type_pair source="mvbook" target="bookinbook"/>
+ <bcf:type_pair source="mvbook" target="suppbook"/>
+ <bcf:field source="title" target="maintitle"/>
+ <bcf:field source="subtitle" target="mainsubtitle"/>
+ <bcf:field source="titleaddon" target="maintitleaddon"/>
+ <bcf:field source="shorttitle" skip="true"/>
+ <bcf:field source="sorttitle" skip="true"/>
+ <bcf:field source="indextitle" skip="true"/>
+ <bcf:field source="indexsorttitle" skip="true"/>
+ </bcf:inherit>
+ <bcf:inherit>
+ <bcf:type_pair source="mvcollection" target="collection"/>
+ <bcf:type_pair source="mvcollection" target="reference"/>
+ <bcf:type_pair source="mvcollection" target="incollection"/>
+ <bcf:type_pair source="mvcollection" target="inreference"/>
+ <bcf:type_pair source="mvcollection" target="suppcollection"/>
+ <bcf:type_pair source="mvreference" target="collection"/>
+ <bcf:type_pair source="mvreference" target="reference"/>
+ <bcf:type_pair source="mvreference" target="incollection"/>
+ <bcf:type_pair source="mvreference" target="inreference"/>
+ <bcf:type_pair source="mvreference" target="suppcollection"/>
+ <bcf:field source="title" target="maintitle"/>
+ <bcf:field source="subtitle" target="mainsubtitle"/>
+ <bcf:field source="titleaddon" target="maintitleaddon"/>
+ <bcf:field source="shorttitle" skip="true"/>
+ <bcf:field source="sorttitle" skip="true"/>
+ <bcf:field source="indextitle" skip="true"/>
+ <bcf:field source="indexsorttitle" skip="true"/>
+ </bcf:inherit>
+ <bcf:inherit>
+ <bcf:type_pair source="mvproceedings" target="proceedings"/>
+ <bcf:type_pair source="mvproceedings" target="inproceedings"/>
+ <bcf:field source="title" target="maintitle"/>
+ <bcf:field source="subtitle" target="mainsubtitle"/>
+ <bcf:field source="titleaddon" target="maintitleaddon"/>
+ <bcf:field source="shorttitle" skip="true"/>
+ <bcf:field source="sorttitle" skip="true"/>
+ <bcf:field source="indextitle" skip="true"/>
+ <bcf:field source="indexsorttitle" skip="true"/>
+ </bcf:inherit>
+ <bcf:inherit>
+ <bcf:type_pair source="book" target="inbook"/>
+ <bcf:type_pair source="book" target="bookinbook"/>
+ <bcf:type_pair source="book" target="suppbook"/>
+ <bcf:field source="title" target="booktitle"/>
+ <bcf:field source="subtitle" target="booksubtitle"/>
+ <bcf:field source="titleaddon" target="booktitleaddon"/>
+ <bcf:field source="shorttitle" skip="true"/>
+ <bcf:field source="sorttitle" skip="true"/>
+ <bcf:field source="indextitle" skip="true"/>
+ <bcf:field source="indexsorttitle" skip="true"/>
+ </bcf:inherit>
+ <bcf:inherit>
+ <bcf:type_pair source="collection" target="incollection"/>
+ <bcf:type_pair source="collection" target="inreference"/>
+ <bcf:type_pair source="collection" target="suppcollection"/>
+ <bcf:type_pair source="reference" target="incollection"/>
+ <bcf:type_pair source="reference" target="inreference"/>
+ <bcf:type_pair source="reference" target="suppcollection"/>
+ <bcf:field source="title" target="booktitle"/>
+ <bcf:field source="subtitle" target="booksubtitle"/>
+ <bcf:field source="titleaddon" target="booktitleaddon"/>
+ <bcf:field source="shorttitle" skip="true"/>
+ <bcf:field source="sorttitle" skip="true"/>
+ <bcf:field source="indextitle" skip="true"/>
+ <bcf:field source="indexsorttitle" skip="true"/>
+ </bcf:inherit>
+ <bcf:inherit>
+ <bcf:type_pair source="proceedings" target="inproceedings"/>
+ <bcf:field source="title" target="booktitle"/>
+ <bcf:field source="subtitle" target="booksubtitle"/>
+ <bcf:field source="titleaddon" target="booktitleaddon"/>
+ <bcf:field source="shorttitle" skip="true"/>
+ <bcf:field source="sorttitle" skip="true"/>
+ <bcf:field source="indextitle" skip="true"/>
+ <bcf:field source="indexsorttitle" skip="true"/>
+ </bcf:inherit>
+ <bcf:inherit>
+ <bcf:type_pair source="periodical" target="article"/>
+ <bcf:type_pair source="periodical" target="suppperiodical"/>
+ <bcf:field source="title" target="journaltitle"/>
+ <bcf:field source="subtitle" target="journalsubtitle"/>
+ <bcf:field source="titleaddon" target="journaltitleaddon"/>
+ <bcf:field source="shorttitle" skip="true"/>
+ <bcf:field source="sorttitle" skip="true"/>
+ <bcf:field source="indextitle" skip="true"/>
+ <bcf:field source="indexsorttitle" skip="true"/>
+ </bcf:inherit>
+ <bcf:inherit>
+ <bcf:type_pair source="*" target="*"/>
+ <bcf:field source="ids" skip="true"/>
+ <bcf:field source="crossref" skip="true"/>
+ <bcf:field source="xref" skip="true"/>
+ <bcf:field source="entryset" skip="true"/>
+ <bcf:field source="entrysubtype" skip="true"/>
+ <bcf:field source="execute" skip="true"/>
+ <bcf:field source="label" skip="true"/>
+ <bcf:field source="options" skip="true"/>
+ <bcf:field source="presort" skip="true"/>
+ <bcf:field source="related" skip="true"/>
+ <bcf:field source="relatedoptions" skip="true"/>
+ <bcf:field source="relatedstring" skip="true"/>
+ <bcf:field source="relatedtype" skip="true"/>
+ <bcf:field source="shorthand" skip="true"/>
+ <bcf:field source="shorthandintro" skip="true"/>
+ <bcf:field source="sortkey" skip="true"/>
+ </bcf:inherit>
+ </bcf:inheritance>
+ <!-- UNIQUENAME TEMPLATES -->
+ <bcf:uniquenametemplate name="global">
+ <bcf:namepart order="1" use="1" base="1">prefix</bcf:namepart>
+ <bcf:namepart order="2" base="1">family</bcf:namepart>
+ <bcf:namepart order="3">given</bcf:namepart>
+ </bcf:uniquenametemplate>
+ <!-- SORTING NAME KEY TEMPLATES -->
+ <bcf:sortingnamekeytemplate name="global">
+ <bcf:keypart order="1">
+ <bcf:part type="namepart" order="1" use="1">prefix</bcf:part>
+ <bcf:part type="namepart" order="2">family</bcf:part>
+ </bcf:keypart>
+ <bcf:keypart order="2">
+ <bcf:part type="namepart" order="1">given</bcf:part>
+ </bcf:keypart>
+ <bcf:keypart order="3">
+ <bcf:part type="namepart" order="1">suffix</bcf:part>
+ </bcf:keypart>
+ <bcf:keypart order="4">
+ <bcf:part type="namepart" order="1" use="0">prefix</bcf:part>
+ </bcf:keypart>
+ </bcf:sortingnamekeytemplate>
+ <bcf:presort>mm</bcf:presort>
+ <!-- DATA MODEL -->
+ <bcf:datamodel>
+ <bcf:constants>
+ <bcf:constant type="list" name="gender">sf,sm,sn,pf,pm,pn,pp</bcf:constant>
+ <bcf:constant type="list" name="nameparts">family,given,prefix,suffix</bcf:constant>
+ <bcf:constant type="list" name="optiondatatypes">boolean,integer,string,xml</bcf:constant>
+ <bcf:constant type="list" name="multiscriptforms">default,transliteration,transcription,translation</bcf:constant>
+ </bcf:constants>
+ <bcf:entrytypes>
+ <bcf:entrytype>article</bcf:entrytype>
+ <bcf:entrytype>artwork</bcf:entrytype>
+ <bcf:entrytype>audio</bcf:entrytype>
+ <bcf:entrytype>bibnote</bcf:entrytype>
+ <bcf:entrytype>book</bcf:entrytype>
+ <bcf:entrytype>bookinbook</bcf:entrytype>
+ <bcf:entrytype>booklet</bcf:entrytype>
+ <bcf:entrytype>collection</bcf:entrytype>
+ <bcf:entrytype>commentary</bcf:entrytype>
+ <bcf:entrytype>customa</bcf:entrytype>
+ <bcf:entrytype>customb</bcf:entrytype>
+ <bcf:entrytype>customc</bcf:entrytype>
+ <bcf:entrytype>customd</bcf:entrytype>
+ <bcf:entrytype>custome</bcf:entrytype>
+ <bcf:entrytype>customf</bcf:entrytype>
+ <bcf:entrytype>dataset</bcf:entrytype>
+ <bcf:entrytype>inbook</bcf:entrytype>
+ <bcf:entrytype>incollection</bcf:entrytype>
+ <bcf:entrytype>inproceedings</bcf:entrytype>
+ <bcf:entrytype>inreference</bcf:entrytype>
+ <bcf:entrytype>image</bcf:entrytype>
+ <bcf:entrytype>jurisdiction</bcf:entrytype>
+ <bcf:entrytype>legal</bcf:entrytype>
+ <bcf:entrytype>legislation</bcf:entrytype>
+ <bcf:entrytype>letter</bcf:entrytype>
+ <bcf:entrytype>manual</bcf:entrytype>
+ <bcf:entrytype>misc</bcf:entrytype>
+ <bcf:entrytype>movie</bcf:entrytype>
+ <bcf:entrytype>music</bcf:entrytype>
+ <bcf:entrytype>mvcollection</bcf:entrytype>
+ <bcf:entrytype>mvreference</bcf:entrytype>
+ <bcf:entrytype>mvproceedings</bcf:entrytype>
+ <bcf:entrytype>mvbook</bcf:entrytype>
+ <bcf:entrytype>online</bcf:entrytype>
+ <bcf:entrytype>patent</bcf:entrytype>
+ <bcf:entrytype>performance</bcf:entrytype>
+ <bcf:entrytype>periodical</bcf:entrytype>
+ <bcf:entrytype>proceedings</bcf:entrytype>
+ <bcf:entrytype>reference</bcf:entrytype>
+ <bcf:entrytype>report</bcf:entrytype>
+ <bcf:entrytype>review</bcf:entrytype>
+ <bcf:entrytype>set</bcf:entrytype>
+ <bcf:entrytype>software</bcf:entrytype>
+ <bcf:entrytype>standard</bcf:entrytype>
+ <bcf:entrytype>suppbook</bcf:entrytype>
+ <bcf:entrytype>suppcollection</bcf:entrytype>
+ <bcf:entrytype>suppperiodical</bcf:entrytype>
+ <bcf:entrytype>thesis</bcf:entrytype>
+ <bcf:entrytype>unpublished</bcf:entrytype>
+ <bcf:entrytype>video</bcf:entrytype>
+ <bcf:entrytype skip_output="true">xdata</bcf:entrytype>
+ </bcf:entrytypes>
+ <bcf:fields>
+ <bcf:field fieldtype="field" datatype="integer">sortyear</bcf:field>
+ <bcf:field fieldtype="field" datatype="integer">volume</bcf:field>
+ <bcf:field fieldtype="field" datatype="integer">volumes</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">abstract</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">addendum</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">annotation</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">booksubtitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">booktitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">booktitleaddon</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">chapter</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">edition</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">eid</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">entrysubtype</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">eprintclass</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">eprinttype</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">eventtitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">eventtitleaddon</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">gender</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">howpublished</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">indexsorttitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">indextitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">isan</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">isbn</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">ismn</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">isrn</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">issn</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">issue</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">issuesubtitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">issuetitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">issuetitleaddon</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">iswc</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">journalsubtitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">journaltitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">journaltitleaddon</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">label</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">langid</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">langidopts</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">library</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">mainsubtitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">maintitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">maintitleaddon</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">nameaddon</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">note</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">number</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">origtitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">pagetotal</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">part</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">relatedstring</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">relatedtype</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">reprinttitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">series</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">shorthandintro</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">subtitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">title</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">titleaddon</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">usera</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">userb</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">userc</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">userd</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">usere</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">userf</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">venue</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal">version</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal" label="true">shorthand</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal" label="true">shortjournal</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal" label="true">shortseries</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal" label="true">shorttitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal" skip_output="true">sorttitle</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal" skip_output="true">sortshorthand</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal" skip_output="true">sortkey</bcf:field>
+ <bcf:field fieldtype="field" datatype="literal" skip_output="true">presort</bcf:field>
+ <bcf:field fieldtype="list" datatype="literal">institution</bcf:field>
+ <bcf:field fieldtype="list" datatype="literal">lista</bcf:field>
+ <bcf:field fieldtype="list" datatype="literal">listb</bcf:field>
+ <bcf:field fieldtype="list" datatype="literal">listc</bcf:field>
+ <bcf:field fieldtype="list" datatype="literal">listd</bcf:field>
+ <bcf:field fieldtype="list" datatype="literal">liste</bcf:field>
+ <bcf:field fieldtype="list" datatype="literal">listf</bcf:field>
+ <bcf:field fieldtype="list" datatype="literal">location</bcf:field>
+ <bcf:field fieldtype="list" datatype="literal">organization</bcf:field>
+ <bcf:field fieldtype="list" datatype="literal">origlocation</bcf:field>
+ <bcf:field fieldtype="list" datatype="literal">origpublisher</bcf:field>
+ <bcf:field fieldtype="list" datatype="literal">publisher</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">afterword</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">annotator</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">author</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">bookauthor</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">commentator</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">editor</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">editora</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">editorb</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">editorc</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">foreword</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">holder</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">introduction</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">namea</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">nameb</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">namec</bcf:field>
+ <bcf:field fieldtype="list" datatype="name">translator</bcf:field>
+ <bcf:field fieldtype="list" datatype="name" label="true">shortauthor</bcf:field>
+ <bcf:field fieldtype="list" datatype="name" label="true">shorteditor</bcf:field>
+ <bcf:field fieldtype="list" datatype="name" skip_output="true">sortname</bcf:field>
+ <bcf:field fieldtype="field" datatype="key">authortype</bcf:field>
+ <bcf:field fieldtype="field" datatype="key">editoratype</bcf:field>
+ <bcf:field fieldtype="field" datatype="key">editorbtype</bcf:field>
+ <bcf:field fieldtype="field" datatype="key">editorctype</bcf:field>
+ <bcf:field fieldtype="field" datatype="key">editortype</bcf:field>
+ <bcf:field fieldtype="field" datatype="key">bookpagination</bcf:field>
+ <bcf:field fieldtype="field" datatype="key">nameatype</bcf:field>
+ <bcf:field fieldtype="field" datatype="key">namebtype</bcf:field>
+ <bcf:field fieldtype="field" datatype="key">namectype</bcf:field>
+ <bcf:field fieldtype="field" datatype="key">pagination</bcf:field>
+ <bcf:field fieldtype="field" datatype="key">pubstate</bcf:field>
+ <bcf:field fieldtype="field" datatype="key">type</bcf:field>
+ <bcf:field fieldtype="list" datatype="key">language</bcf:field>
+ <bcf:field fieldtype="list" datatype="key">origlanguage</bcf:field>
+ <bcf:field fieldtype="field" datatype="entrykey">crossref</bcf:field>
+ <bcf:field fieldtype="field" datatype="entrykey">xref</bcf:field>
+ <bcf:field fieldtype="field" datatype="date" skip_output="true">date</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart" nullok="true">endyear</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart" nullok="true">year</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">month</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">day</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">hour</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">minute</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">second</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">timezone</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">season</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">endmonth</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">endday</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">endhour</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">endminute</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">endsecond</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">endtimezone</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">endseason</bcf:field>
+ <bcf:field fieldtype="field" datatype="date" skip_output="true">eventdate</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart" nullok="true">eventendyear</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart" nullok="true">eventyear</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">eventmonth</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">eventday</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">eventhour</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">eventminute</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">eventsecond</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">eventtimezone</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">eventseason</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">eventendmonth</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">eventendday</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">eventendhour</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">eventendminute</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">eventendsecond</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">eventendtimezone</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">eventendseason</bcf:field>
+ <bcf:field fieldtype="field" datatype="date" skip_output="true">origdate</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart" nullok="true">origendyear</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart" nullok="true">origyear</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">origmonth</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">origday</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">orighour</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">origminute</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">origsecond</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">origtimezone</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">origseason</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">origendmonth</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">origendday</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">origendhour</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">origendminute</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">origendsecond</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">origendtimezone</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">origendseason</bcf:field>
+ <bcf:field fieldtype="field" datatype="date" skip_output="true">urldate</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart" nullok="true">urlendyear</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart" nullok="true">urlyear</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">urlmonth</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">urlday</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">urlhour</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">urlminute</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">urlsecond</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">urltimezone</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">urlseason</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">urlendmonth</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">urlendday</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">urlendhour</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">urlendminute</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">urlendsecond</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">urlendtimezone</bcf:field>
+ <bcf:field fieldtype="field" datatype="datepart">urlendseason</bcf:field>
+ <bcf:field fieldtype="field" datatype="verbatim">doi</bcf:field>
+ <bcf:field fieldtype="field" datatype="verbatim">eprint</bcf:field>
+ <bcf:field fieldtype="field" datatype="verbatim">file</bcf:field>
+ <bcf:field fieldtype="field" datatype="verbatim">verba</bcf:field>
+ <bcf:field fieldtype="field" datatype="verbatim">verbb</bcf:field>
+ <bcf:field fieldtype="field" datatype="verbatim">verbc</bcf:field>
+ <bcf:field fieldtype="field" datatype="uri">url</bcf:field>
+ <bcf:field fieldtype="field" format="xsv" datatype="entrykey" skip_output="true">xdata</bcf:field>
+ <bcf:field fieldtype="field" format="xsv" datatype="entrykey" skip_output="true">ids</bcf:field>
+ <bcf:field fieldtype="field" format="xsv" datatype="entrykey" skip_output="true">entryset</bcf:field>
+ <bcf:field fieldtype="field" format="xsv" datatype="entrykey">related</bcf:field>
+ <bcf:field fieldtype="field" format="xsv" datatype="keyword">keywords</bcf:field>
+ <bcf:field fieldtype="field" format="xsv" datatype="option" skip_output="true">options</bcf:field>
+ <bcf:field fieldtype="field" format="xsv" datatype="option" skip_output="true">relatedoptions</bcf:field>
+ <bcf:field fieldtype="field" datatype="range">pages</bcf:field>
+ <bcf:field fieldtype="field" datatype="code">execute</bcf:field>
+ </bcf:fields>
+ <bcf:entryfields>
+ <bcf:field>abstract</bcf:field>
+ <bcf:field>annotation</bcf:field>
+ <bcf:field>authortype</bcf:field>
+ <bcf:field>bookpagination</bcf:field>
+ <bcf:field>crossref</bcf:field>
+ <bcf:field>day</bcf:field>
+ <bcf:field>doi</bcf:field>
+ <bcf:field>eprint</bcf:field>
+ <bcf:field>eprintclass</bcf:field>
+ <bcf:field>eprinttype</bcf:field>
+ <bcf:field>endday</bcf:field>
+ <bcf:field>endhour</bcf:field>
+ <bcf:field>endminute</bcf:field>
+ <bcf:field>endmonth</bcf:field>
+ <bcf:field>endseason</bcf:field>
+ <bcf:field>endsecond</bcf:field>
+ <bcf:field>endtimezone</bcf:field>
+ <bcf:field>endyear</bcf:field>
+ <bcf:field>entryset</bcf:field>
+ <bcf:field>entrysubtype</bcf:field>
+ <bcf:field>execute</bcf:field>
+ <bcf:field>file</bcf:field>
+ <bcf:field>gender</bcf:field>
+ <bcf:field>hour</bcf:field>
+ <bcf:field>ids</bcf:field>
+ <bcf:field>indextitle</bcf:field>
+ <bcf:field>indexsorttitle</bcf:field>
+ <bcf:field>isan</bcf:field>
+ <bcf:field>ismn</bcf:field>
+ <bcf:field>iswc</bcf:field>
+ <bcf:field>keywords</bcf:field>
+ <bcf:field>label</bcf:field>
+ <bcf:field>langid</bcf:field>
+ <bcf:field>langidopts</bcf:field>
+ <bcf:field>library</bcf:field>
+ <bcf:field>lista</bcf:field>
+ <bcf:field>listb</bcf:field>
+ <bcf:field>listc</bcf:field>
+ <bcf:field>listd</bcf:field>
+ <bcf:field>liste</bcf:field>
+ <bcf:field>listf</bcf:field>
+ <bcf:field>minute</bcf:field>
+ <bcf:field>month</bcf:field>
+ <bcf:field>namea</bcf:field>
+ <bcf:field>nameb</bcf:field>
+ <bcf:field>namec</bcf:field>
+ <bcf:field>nameatype</bcf:field>
+ <bcf:field>namebtype</bcf:field>
+ <bcf:field>namectype</bcf:field>
+ <bcf:field>nameaddon</bcf:field>
+ <bcf:field>options</bcf:field>
+ <bcf:field>origday</bcf:field>
+ <bcf:field>origendday</bcf:field>
+ <bcf:field>origendhour</bcf:field>
+ <bcf:field>origendminute</bcf:field>
+ <bcf:field>origendmonth</bcf:field>
+ <bcf:field>origendseason</bcf:field>
+ <bcf:field>origendsecond</bcf:field>
+ <bcf:field>origendtimezone</bcf:field>
+ <bcf:field>origendyear</bcf:field>
+ <bcf:field>orighour</bcf:field>
+ <bcf:field>origminute</bcf:field>
+ <bcf:field>origmonth</bcf:field>
+ <bcf:field>origseason</bcf:field>
+ <bcf:field>origsecond</bcf:field>
+ <bcf:field>origtimezone</bcf:field>
+ <bcf:field>origyear</bcf:field>
+ <bcf:field>origlocation</bcf:field>
+ <bcf:field>origpublisher</bcf:field>
+ <bcf:field>origtitle</bcf:field>
+ <bcf:field>pagination</bcf:field>
+ <bcf:field>presort</bcf:field>
+ <bcf:field>related</bcf:field>
+ <bcf:field>relatedoptions</bcf:field>
+ <bcf:field>relatedstring</bcf:field>
+ <bcf:field>relatedtype</bcf:field>
+ <bcf:field>season</bcf:field>
+ <bcf:field>second</bcf:field>
+ <bcf:field>shortauthor</bcf:field>
+ <bcf:field>shorteditor</bcf:field>
+ <bcf:field>shorthand</bcf:field>
+ <bcf:field>shorthandintro</bcf:field>
+ <bcf:field>shortjournal</bcf:field>
+ <bcf:field>shortseries</bcf:field>
+ <bcf:field>shorttitle</bcf:field>
+ <bcf:field>sortkey</bcf:field>
+ <bcf:field>sortname</bcf:field>
+ <bcf:field>sortshorthand</bcf:field>
+ <bcf:field>sorttitle</bcf:field>
+ <bcf:field>sortyear</bcf:field>
+ <bcf:field>timezone</bcf:field>
+ <bcf:field>url</bcf:field>
+ <bcf:field>urlday</bcf:field>
+ <bcf:field>urlendday</bcf:field>
+ <bcf:field>urlendhour</bcf:field>
+ <bcf:field>urlendminute</bcf:field>
+ <bcf:field>urlendmonth</bcf:field>
+ <bcf:field>urlendsecond</bcf:field>
+ <bcf:field>urlendtimezone</bcf:field>
+ <bcf:field>urlendyear</bcf:field>
+ <bcf:field>urlhour</bcf:field>
+ <bcf:field>urlminute</bcf:field>
+ <bcf:field>urlmonth</bcf:field>
+ <bcf:field>urlsecond</bcf:field>
+ <bcf:field>urltimezone</bcf:field>
+ <bcf:field>urlyear</bcf:field>
+ <bcf:field>usera</bcf:field>
+ <bcf:field>userb</bcf:field>
+ <bcf:field>userc</bcf:field>
+ <bcf:field>userd</bcf:field>
+ <bcf:field>usere</bcf:field>
+ <bcf:field>userf</bcf:field>
+ <bcf:field>verba</bcf:field>
+ <bcf:field>verbb</bcf:field>
+ <bcf:field>verbc</bcf:field>
+ <bcf:field>xdata</bcf:field>
+ <bcf:field>xref</bcf:field>
+ <bcf:field>year</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>set</bcf:entrytype>
+ <bcf:field>entryset</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>article</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>annotator</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>commentator</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editora</bcf:field>
+ <bcf:field>editorb</bcf:field>
+ <bcf:field>editorc</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>editoratype</bcf:field>
+ <bcf:field>editorbtype</bcf:field>
+ <bcf:field>editorctype</bcf:field>
+ <bcf:field>eid</bcf:field>
+ <bcf:field>issn</bcf:field>
+ <bcf:field>issue</bcf:field>
+ <bcf:field>issuetitle</bcf:field>
+ <bcf:field>issuesubtitle</bcf:field>
+ <bcf:field>issuetitleaddon</bcf:field>
+ <bcf:field>journalsubtitle</bcf:field>
+ <bcf:field>journaltitle</bcf:field>
+ <bcf:field>journaltitleaddon</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>origlanguage</bcf:field>
+ <bcf:field>pages</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>series</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>translator</bcf:field>
+ <bcf:field>version</bcf:field>
+ <bcf:field>volume</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>bibnote</bcf:entrytype>
+ <bcf:field>note</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>book</bcf:entrytype>
+ <bcf:field>author</bcf:field>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>afterword</bcf:field>
+ <bcf:field>annotator</bcf:field>
+ <bcf:field>chapter</bcf:field>
+ <bcf:field>commentator</bcf:field>
+ <bcf:field>edition</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editora</bcf:field>
+ <bcf:field>editorb</bcf:field>
+ <bcf:field>editorc</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>editoratype</bcf:field>
+ <bcf:field>editorbtype</bcf:field>
+ <bcf:field>editorctype</bcf:field>
+ <bcf:field>eid</bcf:field>
+ <bcf:field>foreword</bcf:field>
+ <bcf:field>introduction</bcf:field>
+ <bcf:field>isbn</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>maintitle</bcf:field>
+ <bcf:field>maintitleaddon</bcf:field>
+ <bcf:field>mainsubtitle</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>origlanguage</bcf:field>
+ <bcf:field>pages</bcf:field>
+ <bcf:field>pagetotal</bcf:field>
+ <bcf:field>part</bcf:field>
+ <bcf:field>publisher</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>series</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>translator</bcf:field>
+ <bcf:field>volume</bcf:field>
+ <bcf:field>volumes</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>mvbook</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>afterword</bcf:field>
+ <bcf:field>annotator</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>commentator</bcf:field>
+ <bcf:field>edition</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editora</bcf:field>
+ <bcf:field>editorb</bcf:field>
+ <bcf:field>editorc</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>editoratype</bcf:field>
+ <bcf:field>editorbtype</bcf:field>
+ <bcf:field>editorctype</bcf:field>
+ <bcf:field>foreword</bcf:field>
+ <bcf:field>introduction</bcf:field>
+ <bcf:field>isbn</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>origlanguage</bcf:field>
+ <bcf:field>pagetotal</bcf:field>
+ <bcf:field>publisher</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>series</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>translator</bcf:field>
+ <bcf:field>volume</bcf:field>
+ <bcf:field>volumes</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>inbook</bcf:entrytype>
+ <bcf:entrytype>bookinbook</bcf:entrytype>
+ <bcf:entrytype>suppbook</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>afterword</bcf:field>
+ <bcf:field>annotator</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>booktitle</bcf:field>
+ <bcf:field>bookauthor</bcf:field>
+ <bcf:field>booksubtitle</bcf:field>
+ <bcf:field>booktitleaddon</bcf:field>
+ <bcf:field>chapter</bcf:field>
+ <bcf:field>commentator</bcf:field>
+ <bcf:field>edition</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editora</bcf:field>
+ <bcf:field>editorb</bcf:field>
+ <bcf:field>editorc</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>editoratype</bcf:field>
+ <bcf:field>editorbtype</bcf:field>
+ <bcf:field>editorctype</bcf:field>
+ <bcf:field>eid</bcf:field>
+ <bcf:field>foreword</bcf:field>
+ <bcf:field>introduction</bcf:field>
+ <bcf:field>isbn</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>mainsubtitle</bcf:field>
+ <bcf:field>maintitle</bcf:field>
+ <bcf:field>maintitleaddon</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>origlanguage</bcf:field>
+ <bcf:field>part</bcf:field>
+ <bcf:field>publisher</bcf:field>
+ <bcf:field>pages</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>series</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>translator</bcf:field>
+ <bcf:field>volume</bcf:field>
+ <bcf:field>volumes</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>booklet</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>chapter</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>eid</bcf:field>
+ <bcf:field>howpublished</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>pages</bcf:field>
+ <bcf:field>pagetotal</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>type</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>collection</bcf:entrytype>
+ <bcf:entrytype>reference</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>afterword</bcf:field>
+ <bcf:field>annotator</bcf:field>
+ <bcf:field>chapter</bcf:field>
+ <bcf:field>commentator</bcf:field>
+ <bcf:field>edition</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editora</bcf:field>
+ <bcf:field>editorb</bcf:field>
+ <bcf:field>editorc</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>editoratype</bcf:field>
+ <bcf:field>editorbtype</bcf:field>
+ <bcf:field>editorctype</bcf:field>
+ <bcf:field>eid</bcf:field>
+ <bcf:field>foreword</bcf:field>
+ <bcf:field>introduction</bcf:field>
+ <bcf:field>isbn</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>mainsubtitle</bcf:field>
+ <bcf:field>maintitle</bcf:field>
+ <bcf:field>maintitleaddon</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>origlanguage</bcf:field>
+ <bcf:field>pages</bcf:field>
+ <bcf:field>pagetotal</bcf:field>
+ <bcf:field>part</bcf:field>
+ <bcf:field>publisher</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>series</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>translator</bcf:field>
+ <bcf:field>volume</bcf:field>
+ <bcf:field>volumes</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>mvcollection</bcf:entrytype>
+ <bcf:entrytype>mvreference</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>afterword</bcf:field>
+ <bcf:field>annotator</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>commentator</bcf:field>
+ <bcf:field>edition</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editora</bcf:field>
+ <bcf:field>editorb</bcf:field>
+ <bcf:field>editorc</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>editoratype</bcf:field>
+ <bcf:field>editorbtype</bcf:field>
+ <bcf:field>editorctype</bcf:field>
+ <bcf:field>foreword</bcf:field>
+ <bcf:field>introduction</bcf:field>
+ <bcf:field>isbn</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>origlanguage</bcf:field>
+ <bcf:field>publisher</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>translator</bcf:field>
+ <bcf:field>volume</bcf:field>
+ <bcf:field>volumes</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>incollection</bcf:entrytype>
+ <bcf:entrytype>suppcollection</bcf:entrytype>
+ <bcf:entrytype>inreference</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>afterword</bcf:field>
+ <bcf:field>annotator</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>booksubtitle</bcf:field>
+ <bcf:field>booktitle</bcf:field>
+ <bcf:field>booktitleaddon</bcf:field>
+ <bcf:field>chapter</bcf:field>
+ <bcf:field>commentator</bcf:field>
+ <bcf:field>edition</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editora</bcf:field>
+ <bcf:field>editorb</bcf:field>
+ <bcf:field>editorc</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>editoratype</bcf:field>
+ <bcf:field>editorbtype</bcf:field>
+ <bcf:field>editorctype</bcf:field>
+ <bcf:field>eid</bcf:field>
+ <bcf:field>foreword</bcf:field>
+ <bcf:field>introduction</bcf:field>
+ <bcf:field>isbn</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>mainsubtitle</bcf:field>
+ <bcf:field>maintitle</bcf:field>
+ <bcf:field>maintitleaddon</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>origlanguage</bcf:field>
+ <bcf:field>pages</bcf:field>
+ <bcf:field>part</bcf:field>
+ <bcf:field>publisher</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>series</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>translator</bcf:field>
+ <bcf:field>volume</bcf:field>
+ <bcf:field>volumes</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>dataset</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>edition</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>organization</bcf:field>
+ <bcf:field>publisher</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>series</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>type</bcf:field>
+ <bcf:field>version</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>manual</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>chapter</bcf:field>
+ <bcf:field>edition</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>eid</bcf:field>
+ <bcf:field>isbn</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>organization</bcf:field>
+ <bcf:field>pages</bcf:field>
+ <bcf:field>pagetotal</bcf:field>
+ <bcf:field>publisher</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>series</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>type</bcf:field>
+ <bcf:field>version</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>misc</bcf:entrytype>
+ <bcf:entrytype>software</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>howpublished</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>organization</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>type</bcf:field>
+ <bcf:field>version</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>online</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>organization</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>version</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>patent</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>holder</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>type</bcf:field>
+ <bcf:field>version</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>periodical</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editora</bcf:field>
+ <bcf:field>editorb</bcf:field>
+ <bcf:field>editorc</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>editoratype</bcf:field>
+ <bcf:field>editorbtype</bcf:field>
+ <bcf:field>editorctype</bcf:field>
+ <bcf:field>issn</bcf:field>
+ <bcf:field>issue</bcf:field>
+ <bcf:field>issuesubtitle</bcf:field>
+ <bcf:field>issuetitle</bcf:field>
+ <bcf:field>issuetitleaddon</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>season</bcf:field>
+ <bcf:field>series</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>volume</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>mvproceedings</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>eventday</bcf:field>
+ <bcf:field>eventendday</bcf:field>
+ <bcf:field>eventendhour</bcf:field>
+ <bcf:field>eventendminute</bcf:field>
+ <bcf:field>eventendmonth</bcf:field>
+ <bcf:field>eventendseason</bcf:field>
+ <bcf:field>eventendsecond</bcf:field>
+ <bcf:field>eventendtimezone</bcf:field>
+ <bcf:field>eventendyear</bcf:field>
+ <bcf:field>eventhour</bcf:field>
+ <bcf:field>eventminute</bcf:field>
+ <bcf:field>eventmonth</bcf:field>
+ <bcf:field>eventseason</bcf:field>
+ <bcf:field>eventsecond</bcf:field>
+ <bcf:field>eventtimezone</bcf:field>
+ <bcf:field>eventyear</bcf:field>
+ <bcf:field>eventtitle</bcf:field>
+ <bcf:field>eventtitleaddon</bcf:field>
+ <bcf:field>isbn</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>organization</bcf:field>
+ <bcf:field>pagetotal</bcf:field>
+ <bcf:field>publisher</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>series</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>venue</bcf:field>
+ <bcf:field>volumes</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>proceedings</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>chapter</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>eid</bcf:field>
+ <bcf:field>eventday</bcf:field>
+ <bcf:field>eventendday</bcf:field>
+ <bcf:field>eventendhour</bcf:field>
+ <bcf:field>eventendminute</bcf:field>
+ <bcf:field>eventendmonth</bcf:field>
+ <bcf:field>eventendseason</bcf:field>
+ <bcf:field>eventendsecond</bcf:field>
+ <bcf:field>eventendtimezone</bcf:field>
+ <bcf:field>eventendyear</bcf:field>
+ <bcf:field>eventhour</bcf:field>
+ <bcf:field>eventminute</bcf:field>
+ <bcf:field>eventmonth</bcf:field>
+ <bcf:field>eventseason</bcf:field>
+ <bcf:field>eventsecond</bcf:field>
+ <bcf:field>eventtimezone</bcf:field>
+ <bcf:field>eventyear</bcf:field>
+ <bcf:field>eventtitle</bcf:field>
+ <bcf:field>eventtitleaddon</bcf:field>
+ <bcf:field>isbn</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>mainsubtitle</bcf:field>
+ <bcf:field>maintitle</bcf:field>
+ <bcf:field>maintitleaddon</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>organization</bcf:field>
+ <bcf:field>pages</bcf:field>
+ <bcf:field>pagetotal</bcf:field>
+ <bcf:field>part</bcf:field>
+ <bcf:field>publisher</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>series</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>venue</bcf:field>
+ <bcf:field>volume</bcf:field>
+ <bcf:field>volumes</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>inproceedings</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>booksubtitle</bcf:field>
+ <bcf:field>booktitle</bcf:field>
+ <bcf:field>booktitleaddon</bcf:field>
+ <bcf:field>chapter</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editortype</bcf:field>
+ <bcf:field>eid</bcf:field>
+ <bcf:field>eventday</bcf:field>
+ <bcf:field>eventendday</bcf:field>
+ <bcf:field>eventendhour</bcf:field>
+ <bcf:field>eventendminute</bcf:field>
+ <bcf:field>eventendmonth</bcf:field>
+ <bcf:field>eventendseason</bcf:field>
+ <bcf:field>eventendsecond</bcf:field>
+ <bcf:field>eventendtimezone</bcf:field>
+ <bcf:field>eventendyear</bcf:field>
+ <bcf:field>eventhour</bcf:field>
+ <bcf:field>eventminute</bcf:field>
+ <bcf:field>eventmonth</bcf:field>
+ <bcf:field>eventseason</bcf:field>
+ <bcf:field>eventsecond</bcf:field>
+ <bcf:field>eventtimezone</bcf:field>
+ <bcf:field>eventyear</bcf:field>
+ <bcf:field>eventtitle</bcf:field>
+ <bcf:field>eventtitleaddon</bcf:field>
+ <bcf:field>isbn</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>mainsubtitle</bcf:field>
+ <bcf:field>maintitle</bcf:field>
+ <bcf:field>maintitleaddon</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>organization</bcf:field>
+ <bcf:field>pages</bcf:field>
+ <bcf:field>part</bcf:field>
+ <bcf:field>publisher</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>series</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>venue</bcf:field>
+ <bcf:field>volume</bcf:field>
+ <bcf:field>volumes</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>report</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>chapter</bcf:field>
+ <bcf:field>eid</bcf:field>
+ <bcf:field>institution</bcf:field>
+ <bcf:field>isrn</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>number</bcf:field>
+ <bcf:field>pages</bcf:field>
+ <bcf:field>pagetotal</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>type</bcf:field>
+ <bcf:field>version</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>thesis</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>chapter</bcf:field>
+ <bcf:field>eid</bcf:field>
+ <bcf:field>institution</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>pages</bcf:field>
+ <bcf:field>pagetotal</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>type</bcf:field>
+ </bcf:entryfields>
+ <bcf:entryfields>
+ <bcf:entrytype>unpublished</bcf:entrytype>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>eventday</bcf:field>
+ <bcf:field>eventendday</bcf:field>
+ <bcf:field>eventendhour</bcf:field>
+ <bcf:field>eventendminute</bcf:field>
+ <bcf:field>eventendmonth</bcf:field>
+ <bcf:field>eventendseason</bcf:field>
+ <bcf:field>eventendsecond</bcf:field>
+ <bcf:field>eventendtimezone</bcf:field>
+ <bcf:field>eventendyear</bcf:field>
+ <bcf:field>eventhour</bcf:field>
+ <bcf:field>eventminute</bcf:field>
+ <bcf:field>eventmonth</bcf:field>
+ <bcf:field>eventseason</bcf:field>
+ <bcf:field>eventsecond</bcf:field>
+ <bcf:field>eventtimezone</bcf:field>
+ <bcf:field>eventyear</bcf:field>
+ <bcf:field>eventtitle</bcf:field>
+ <bcf:field>eventtitleaddon</bcf:field>
+ <bcf:field>howpublished</bcf:field>
+ <bcf:field>language</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>pubstate</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>type</bcf:field>
+ <bcf:field>venue</bcf:field>
+ </bcf:entryfields>
+ <bcf:multiscriptfields>
+ <bcf:field>abstract</bcf:field>
+ <bcf:field>addendum</bcf:field>
+ <bcf:field>afterword</bcf:field>
+ <bcf:field>annotator</bcf:field>
+ <bcf:field>author</bcf:field>
+ <bcf:field>bookauthor</bcf:field>
+ <bcf:field>booksubtitle</bcf:field>
+ <bcf:field>booktitle</bcf:field>
+ <bcf:field>booktitleaddon</bcf:field>
+ <bcf:field>chapter</bcf:field>
+ <bcf:field>commentator</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>editora</bcf:field>
+ <bcf:field>editorb</bcf:field>
+ <bcf:field>editorc</bcf:field>
+ <bcf:field>foreword</bcf:field>
+ <bcf:field>holder</bcf:field>
+ <bcf:field>institution</bcf:field>
+ <bcf:field>introduction</bcf:field>
+ <bcf:field>issuesubtitle</bcf:field>
+ <bcf:field>issuetitle</bcf:field>
+ <bcf:field>issuetitleaddon</bcf:field>
+ <bcf:field>journalsubtitle</bcf:field>
+ <bcf:field>journaltitle</bcf:field>
+ <bcf:field>journaltitleaddon</bcf:field>
+ <bcf:field>location</bcf:field>
+ <bcf:field>mainsubtitle</bcf:field>
+ <bcf:field>maintitle</bcf:field>
+ <bcf:field>maintitleaddon</bcf:field>
+ <bcf:field>nameaddon</bcf:field>
+ <bcf:field>note</bcf:field>
+ <bcf:field>organization</bcf:field>
+ <bcf:field>origlanguage</bcf:field>
+ <bcf:field>origlocation</bcf:field>
+ <bcf:field>origpublisher</bcf:field>
+ <bcf:field>origtitle</bcf:field>
+ <bcf:field>part</bcf:field>
+ <bcf:field>publisher</bcf:field>
+ <bcf:field>relatedstring</bcf:field>
+ <bcf:field>series</bcf:field>
+ <bcf:field>shortauthor</bcf:field>
+ <bcf:field>shorteditor</bcf:field>
+ <bcf:field>shorthand</bcf:field>
+ <bcf:field>shortjournal</bcf:field>
+ <bcf:field>shortseries</bcf:field>
+ <bcf:field>shorttitle</bcf:field>
+ <bcf:field>sortname</bcf:field>
+ <bcf:field>sortshorthand</bcf:field>
+ <bcf:field>sorttitle</bcf:field>
+ <bcf:field>subtitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>titleaddon</bcf:field>
+ <bcf:field>translator</bcf:field>
+ <bcf:field>venue</bcf:field>
+ </bcf:multiscriptfields>
+ <bcf:constraints>
+ <bcf:entrytype>article</bcf:entrytype>
+ <bcf:entrytype>book</bcf:entrytype>
+ <bcf:entrytype>inbook</bcf:entrytype>
+ <bcf:entrytype>bookinbook</bcf:entrytype>
+ <bcf:entrytype>suppbook</bcf:entrytype>
+ <bcf:entrytype>booklet</bcf:entrytype>
+ <bcf:entrytype>collection</bcf:entrytype>
+ <bcf:entrytype>incollection</bcf:entrytype>
+ <bcf:entrytype>suppcollection</bcf:entrytype>
+ <bcf:entrytype>manual</bcf:entrytype>
+ <bcf:entrytype>misc</bcf:entrytype>
+ <bcf:entrytype>mvbook</bcf:entrytype>
+ <bcf:entrytype>mvcollection</bcf:entrytype>
+ <bcf:entrytype>online</bcf:entrytype>
+ <bcf:entrytype>patent</bcf:entrytype>
+ <bcf:entrytype>periodical</bcf:entrytype>
+ <bcf:entrytype>suppperiodical</bcf:entrytype>
+ <bcf:entrytype>proceedings</bcf:entrytype>
+ <bcf:entrytype>inproceedings</bcf:entrytype>
+ <bcf:entrytype>reference</bcf:entrytype>
+ <bcf:entrytype>inreference</bcf:entrytype>
+ <bcf:entrytype>report</bcf:entrytype>
+ <bcf:entrytype>set</bcf:entrytype>
+ <bcf:entrytype>thesis</bcf:entrytype>
+ <bcf:entrytype>unpublished</bcf:entrytype>
+ <bcf:constraint type="mandatory">
+ <bcf:fieldxor>
+ <bcf:field>date</bcf:field>
+ <bcf:field>year</bcf:field>
+ </bcf:fieldxor>
+ </bcf:constraint>
+ </bcf:constraints>
+ <bcf:constraints>
+ <bcf:entrytype>set</bcf:entrytype>
+ <bcf:constraint type="mandatory">
+ <bcf:field>entryset</bcf:field>
+ </bcf:constraint>
+ </bcf:constraints>
+ <bcf:constraints>
+ <bcf:entrytype>article</bcf:entrytype>
+ <bcf:constraint type="mandatory">
+ <bcf:field>author</bcf:field>
+ <bcf:field>journaltitle</bcf:field>
+ <bcf:field>title</bcf:field>
+ </bcf:constraint>
+ </bcf:constraints>
+ <bcf:constraints>
+ <bcf:entrytype>book</bcf:entrytype>
+ <bcf:entrytype>mvbook</bcf:entrytype>
+ <bcf:constraint type="mandatory">
+ <bcf:field>author</bcf:field>
+ <bcf:field>title</bcf:field>
+ </bcf:constraint>
+ </bcf:constraints>
+ <bcf:constraints>
+ <bcf:entrytype>inbook</bcf:entrytype>
+ <bcf:entrytype>bookinbook</bcf:entrytype>
+ <bcf:entrytype>suppbook</bcf:entrytype>
+ <bcf:constraint type="mandatory">
+ <bcf:field>author</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>booktitle</bcf:field>
+ </bcf:constraint>
+ </bcf:constraints>
+ <bcf:constraints>
+ <bcf:entrytype>booklet</bcf:entrytype>
+ <bcf:constraint type="mandatory">
+ <bcf:fieldor>
+ <bcf:field>author</bcf:field>
+ <bcf:field>editor</bcf:field>
+ </bcf:fieldor>
+ <bcf:field>title</bcf:field>
+ </bcf:constraint>
+ </bcf:constraints>
+ <bcf:constraints>
+ <bcf:entrytype>collection</bcf:entrytype>
+ <bcf:entrytype>reference</bcf:entrytype>
+ <bcf:entrytype>mvcollection</bcf:entrytype>
+ <bcf:entrytype>mvreference</bcf:entrytype>
+ <bcf:constraint type="mandatory">
+ <bcf:field>editor</bcf:field>
+ <bcf:field>title</bcf:field>
+ </bcf:constraint>
+ </bcf:constraints>
+ <bcf:constraints>
+ <bcf:entrytype>incollection</bcf:entrytype>
+ <bcf:entrytype>suppcollection</bcf:entrytype>
+ <bcf:entrytype>inreference</bcf:entrytype>
+ <bcf:constraint type="mandatory">
+ <bcf:field>author</bcf:field>
+ <bcf:field>editor</bcf:field>
+ <bcf:field>title</bcf:field>
+ <bcf:field>booktitle</bcf:field>
+ </bcf:constraint>
+ </bcf:constraints>
+ <bcf:constraints>
+ <bcf:entrytype>dataset</bcf:entrytype>
+ <bcf:constraint type="mandatory">
+ <bcf:field>title</bcf:field>
+ </bcf:constraint>
+ </bcf:constraints>
+ <bcf:constraints>
+ <bcf:entrytype>manual</bcf:entrytype>
+ <bcf:constraint type="mandatory">
+ <bcf:field>title</bcf:field>
+ </bcf:constraint>
+ </bcf:constraints>
+ <bcf:constraints>
+ <bcf:entrytype>misc</bcf:entrytype>
+ <bcf:entrytype>software</bcf:entrytype>
+ <bcf:constraint type="mandatory">
+ <bcf:field>title</bcf:field>
+ </bcf:constraint>
+ </bcf:constraints>
+ <bcf:constraints>
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+ <bcf:citekey order="1">heatkernel</bcf:citekey>
+ <bcf:citekey order="2">ncgwalter</bcf:citekey>
+ <bcf:citekey order="3" nocite="1">electrodynamics</bcf:citekey>
+ <bcf:citekey order="4">ncgwalter</bcf:citekey>
+ <bcf:citekey order="5" nocite="1">liealgebra</bcf:citekey>
+ <bcf:citekey order="6" nocite="1">ncg4pages</bcf:citekey>
+ <bcf:citekey order="7" nocite="1">ncgshort</bcf:citekey>
+ <bcf:citekey order="8">heatkernel</bcf:citekey>
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diff --git a/src/thesis/main.tex b/src/thesis/main.tex
@@ -24,13 +24,13 @@
\input{chapters/main_sec}
-%\input{chapters/basics}
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-\input{chapters/heatkernel}
-
-\input{chapters/twopointspace}
-
-\input{chapters/electroncg}
+%\input{chapters/heatkernel}
+%
+%\input{chapters/twopointspace}
+%
+%\input{chapters/electroncg}
%------------------ OUTRO -------------------------
diff --git a/src/thesis/main.toc b/src/thesis/main.toc
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+\defcounter {refsection}{0}\relax
+\contentsline {section}{\numberline {3}Conclusion}{20}{section.3}%
+\defcounter {refsection}{0}\relax
+\contentsline {section}{\numberline {4}Acknowledgment}{21}{section.4}%
