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Family Counselor Michael Gurian, MFA, shares advice for parents on the benefits of your teenage son having a good peer mentor and four ways to find one for your son
Raising Boys | Advice On How To FInd Your Son A Good Mentor
Home » TEEN » Parenting Teens » At Risk Youth
Four ways to find a mentor for your son
You may know, you may sense that role models and mentors are really important to your kids. And you may look at your sons, especially as they are growing into adolescence, and say wow, this guy needs a role model, usually it will be a male role model, that he needs. And you may look around you and you may say I am a single mom, let´s say, or we are a blended family or something but my son does not really connect with the stepdad. I still got to help him find role models, and you are absolutely right. If you had to pick one of three things for adolescent boys, it is going to be role models and mentors. Some things you can do: You can look into Boy Scouts or organizations like that. Obvously, Big Brothers, sometimes they have waiting lists but still look into that. You can go to faith communities. That is really big. Even if you have turned away from a religion, maybe look at it again just to see because there are men who want to mentor. You can also look at putting the boy to work because boys get mentors from work. So if he has some free time and he is spending two or three hours a day on video games, it is time for him to work. So obviously he has got to be 16 or older or if he is like 13, 14, work in someone´s shop with just hanging around. But do something that involves work and involves purpose, the mentors come through that. The last thing you can do is you can look at elders, nursing homes or anything where there are old people. They really want to mentor young people. And so, let them apprentice this son of yours in anything, even if it is building a model, that will form the relationship.
Video Categories: ALL PARENTS, Family Life, Gender Differences, ELEMENTARY, Kids Health Ages 5 to 12, Therapy and Support, TEEN, Parenting Teens, At Risk Youth
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Meet Michael Gurian, MFA, CMHC
About the Gurian Institute
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Team sports vs. individual sports
Testosterone vs. Oxytocin
The importance of the mother-son relationship
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Michael Gurian, MFA, CMHC
Family Counselor & Author
Michael Gurian is the New York Times bestselling author of 25 books published in 21 languages. He provides counseling services at the Marycliff Center, in Spokane, Washington. The Gurian Institute, which he co-founded, conducts research internationally, launches pilot programs and trains professionals. Michael has been called "the people's philosopher" for his ability to bring together people's ordinary lives and scientific ideas.
He has pioneered efforts to bring neuro-biology and brain research into homes, schools, corporations, and public policy. A number of his books have sparked national debate, including The Wonder of Girls, The Wonder of Boys, and Boys and Girls Learn Differently!, and The Minds of Boys.
Michael has served as a consultant to families, corporations, therapists, physicians, school districts, community agencies, churches, criminal justice personnel and other professionals, traveling to approximately 20 cities per year to keynote at conferences. His training videos (also available as DVDs) for parents and volunteers are used by Big Brother and Big Sister agencies in the U.S. and Canada.
As an educator, Michael previously taught at Gonzaga University, Eastern Washington University, and Ankara University. His speaking engagements include Harvard University, Johns Hopkins University, Stanford University, Macalester College, University of Colorado, University of Missouri-Kansas City, and UCLA. His philosophy reflects the diverse cultures (European, Asian, Middle Eastern and American) in which he has lived, worked and studied.
Michael's work has been featured in various media, including the New York Times, the Washington Post, USA Today, Newsweek, Time, People Magazine, Reader's Digest, the Wall Street Journal, Forbes Magazine, Parenting, Good Housekeeping, Redbook, and on the Today Show, Good Morning America, CNN, PBS and National Public Radio.
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\section{Introduction}\label{sec:intro}%
A \Defn{simplicial complex} $\Delta$ is a collection of subsets of a finite
ground set, say $[n] := \{1,\dots,n\}$, such that $\sigma \in \Delta$ and
$\tau \subseteq \sigma$ implies $\tau \in \Delta$. Simplicial complexes are
fundamental objects in algebraic, geometric, and topological combinatorics; see,
for example,~\cite{Stanley96,crt,bjorner}. A basic combinatorial statistic of $\Delta$ is
the \Defn{face vector} (or \Defn{$\boldsymbol f$-vector})
\[
f(\Delta) = (f_{-1},f_0,\dots,f_{d-1}) \, ,
\]
where $f_k = f_k(\Delta)$ records the number of faces $\sigma \in \Delta$ of
dimension $k$, where $\dim \sigma := |\sigma| - 1$ and $d - 1 = \dim \Delta :=
\max \{ \dim \sigma : \sigma \in \Delta\}$. Notice that we allow $\Delta =
\emptyset$, the \emph{void} complex, which is the only complex with
$f_k(\Delta) = 0$ for all $k \ge -1$. A \Defn{relative simplicial complex}
$\Psi$ on the ground set $[n]$ is the collection of sets $ \Delta \setminus
\Gamma = \{ \tau \in \Delta : \tau \not \in \Gamma \}$, where $\Gamma \subset
\Delta \subseteq 2^{[n]}$ are simplicial complexes. In general, the pair of
simplicial complexes $(\Delta,\Gamma)$ is not uniquely determined by $\Psi$,
and we call $\Psi = (\Delta, \Gamma)$ a \Defn{presentation} of $\Psi$. We set
$\dim \Psi := \max \{ \dim \sigma : \sigma \in \Delta \setminus \Gamma \}$.
Relative complexes were introduced by Stanley~\cite{stanley87} and made
prominent recent appearances in, for example,~\cite{AS16,DGKM16, MN, MNY}. The
$f$-vector of a relative complex is given by
\[
f(\Psi) \ := \ f(\Delta) - f(\Gamma) \, ,
\]
where we set $f_k(\Gamma) := 0$ for all $k > \dim \Gamma$. When $\Gamma =
\emptyset$, then $\Psi$ is simply a simplicial complex and we write $\Delta$
instead of $\Psi$. We call $\Psi$ a \Defn{proper} relative complex if $\Gamma \neq
\emptyset$ or, equivalently, if $f_{-1}(\Psi) = 0$.
In contrast to simplicial complexes, much less is known about the
combinatorics of relative simplicial complexes. The first goal of this paper
is to address the following basic question:
\begin{center}
\it Which vectors $f = (0,f_0,\dots,f_{d-1}) \in \Z^{d+1}_{\ge0}$ are
$f$-vectors of proper relative simplicial complexes?
\end{center}
For simplicial complexes, this question is beautifully answered by the
Kruskal--Katona theorem~\cite{kruskal,katona}. Bj\"orner and Kalai~\cite{BK}
characterized the pairs $(f(\Delta),\beta(\Delta))$ where $\Delta$ is a
simplicial complex and $\beta(\Delta)$ is the sequence of Betti numbers of
$\Delta$ (over a field $\kk$). Duval~\cite{duval} characterized the
pairs
$(f(\Delta),f(\Gamma))$ where $\Delta
\subseteq \Gamma$ but, as stated before, the presentation $\Psi = \Delta
\setminus \Gamma$ is generally not unique. Moreover, the following example
shows that a characterization of $f$-vectors of relative complexes is trivial
without further qualifications.
\begin{ex} \label{ex:all_vectors}%
If $\Delta = 2^{[k+1]}$ is a $k$-dimensional simplex and $\partial \Delta :=
\Delta \setminus \{[k+1]\}$ denotes its boundary complex, then
$f_i(\Delta,\partial\Delta) = 1$ if $i = k$ and is zero otherwise. Hence, by
observing that relative simplicial complexes are closed under
disjoint unions, any vector $f = (0, f_0,\dots,f_{d-1}) \in
\Z_{\ge0}^{d+1}$ can occur as the $f$-vector of a proper relative simplicial
complex.
\end{ex}
The main difference between $f$-vectors of complexes and relative complexes is
that $f_0(\Psi)$ does not reveal the size of the ground set and the
construction outlined in Example~\ref{ex:all_vectors} produces relative
complexes with given $f$-vectors on large ground sets. Restricting the size
of the ground set is the key to a meaningful treatment of $f$-vectors of
relative complexes.
Therefore, we are going to characterize the $f$-vectors of relative complexes
$\Psi = \Delta \setminus \Gamma$ with $\Gamma \subset \Delta \subseteq 2^{[n]}$
for fixed $n$.
To state our characterization, we need to recall the
binomial representation of a natural number: For any $r,k \in \Z_{\ge 0}$ with
$k > 0$, there are unique integers $ r_k > r_{k-1} > \cdots > r_1 \ge 0$ such
that
\begin{equation}\label{eqn:binomial}
r \ = \
\binom{r_{k}}{k} +
\binom{r_{k-1}}{k-1} +
\cdots +
\binom{r_{1}}{1} \, .
\end{equation}
We refer the reader to Greene--Kleitman's excellent article~\cite[Sect.~8]{GK}
for details and combinatorial motivations for this and the following
definition. For the representation given in~\eqref{eqn:binomial} we define
\[
\partial_k(r) \ := \
\binom{r_{k}}{k-1} +
\binom{r_{k-1}}{k-2} +
\cdots +
\binom{r_{1}}{0} \, .
\]
The Kruskal-Katona theorem characterizes $f$-vectors of simplicial complexes in terms of these $\partial_k(r)$, see Theorem~\ref{thm:KK}.
We prove the following characterization of $f$-vectors of proper relative
complexes in Section~\ref{sec:f-rel}.
\begin{thm}\label{thm:relKK}%
Let $f = (0,f_0,\dots,f_{d-1}) \in \Znn^{d+1}$ and $n > 0$ and define two
sequences $(a_0,\dots,a_{d-1})$ and $(b_0,\dots,b_{d-1})$ by
$a_{d-1} := f_{d-1}$ and $b_{d-1} :=0$ and continue recursively
\begin{align*}
%
%
a_{k-1} &\ := \ \max(\partial_{k+1}(a_{k}),
f_{k-1} + \partial_{k+1}(b_{k}) ) \\
%
b_{k-1} &\ := \ \max(\partial_{k+1}(b_{k}),
\partial_{k+1}(a_{k})-f_{k-1} )
\end{align*}
for $k \ge 0$.
Then there is a proper relative simplicial complex $\Psi$ on the ground
set $[n]$ with $f = f(\Psi)$ if and only if $a_0 \le n$.
\end{thm}
The two sequences $(1,a_0,\dots,a_{d-1})$ and $(1,b_0,\dots,b_{d-1})$ are the
componentwise-minimal $f$-vectors of simplicial complexes $\Delta$ and
$\Gamma$ such that $\Gamma \subseteq \Delta$ and $f_{k-1} = f_{k-1}(\Delta) -
f_{k-1}(\Gamma)$ for all $0 \le k < d$.
(Relative) simplicial complexes can be generalized to (relative)
\emph{multicomplexes} by replacing sets with multisets. The notion of an
$f$-vector of a multicomplex is immediate (by taking into account
multiplicities) and the question above carries over to relative multicomplexes
on a ground set of fixed size. Multicomplexes are more natural from an
algebraic perspective: If $S := \kk[x_1,\dots,x_n]$ is the polynomial ring
over a field $\kk$ with $n$ indeterminates and $I \subseteq S$ is a monomial
ideal, then the monomials outside $I$ form a (possibly infinite) multicomplex
on ground set $[n]$ and every multicomplex over $[n]$ arises this way. In
particular, the $f$-vector of a multicomplex is the Hilbert function of $S/I$.
By appealing to initial ideals it is easy to see that $f$-vectors of
(infinite) multicomplexes are exactly the Hilbert functions of standard graded
algebras, which were characterized by Macaulay~\cite{macaulay}. In
Section~\ref{sec:f-rel-mult} we give precise definitions and
Theorem~\ref{thm:relKKm} is the corresponding analogue of
Theorem~\ref{thm:relKK} for proper, possibly infinite, relative
multicomplexes. The corresponding algebraic statement characterizes Hilbert
functions of $I/J$ where $J \subset I \subseteq S$ are pairs of homogeneous
ideals; see Corollary~\ref{cor:relKKm}.
The \Defn{$\boldsymbol h$-vector} $h(\Psi) = (h_0,\dots,h_d)$ of a
$(d-1)$-dimensional relative complex $\Psi$ is defined through
\begin{equation}\label{eqn:h-vec}
\sum_{k=0}^d f_{k-1}(\Psi) t^{d-k} \ = \ \sum_{i=0}^d h_{i}(\Psi)
(t+1)^{d-i} \, .
\end{equation}
Note that if $\dim \Delta = \dim \Gamma$, then $h(\Psi) = h(\Delta) -
h(\Gamma)$.
The $h$-vector clearly carries the same information as the $f$-vector but it
has been amply demonstrated that $h$-vectors often times reveal more
structure; see~\cite{Stanley96} for example. In particular, if $\Delta$ is a
\Defn{Cohen--Macaulay} simplicial complex (or CM complex, for short) over some
field $\kk$, then $h_i(\Delta) \ge 0$ for all $i \ge 0$.
Stanley~\cite{StanleyCM} showed that Macaulay's theorem characterizing Hilbert
functions of standard graded algebras yields a characterization of $h$-vectors
of CM complexes akin to the Kruskal--Katona theorem. Stronger even, Bj\"orner,
Frankl, and Stanley~\cite{BFS} showed that all admissible $h$-vectors can be
realized by shellable simplicial complexes, a proper subset of CM complexes.
In Section~\ref{sec:macaulay}, we recall the definition of a Cohen--Macaulay
relative complex and we give a characterization of $h$-vectors of
\emph{fully} CM relative complexes. We call a relative complex $\Psi$
\Defn{fully Cohen--Macaulay} over a ground set $[n]$ if it has a presentation
$\Psi = (\Delta,\Gamma)$ with $\Gamma \subset \Delta \subseteq 2^{[n]}$,
$\dim \Gamma = \dim \Psi$, and $\Psi$ as well as $\Delta$ and $\Gamma$ are
Cohen--Macaulay.
For $r,k \in \Znn$ with $k > 0$, let $r_k > \dots >
r_1 \ge 0$ as defined by~\eqref{eqn:binomial}. We define
\newcommand\Partial{\widetilde{\partial}}%
\[
\Partial_k(r) \ := \
\binom{r_{k}-1}{k-1} +
\binom{r_{k-1}-1}{k-2} +
\cdots +
\binom{r_{1}-1}{0} \, .
\]
Note that $\Psi$ is proper if and only if $h_0(\Psi) = 0$. Our
characterization of $h$-vectors of fully CM complexes parallels that of CM
complexes in that it suffices to consider \emph{fully shellable} relative
complexes; see Section~\ref{sec:macaulay} for a definition.
\begin{thm}\label{thm:relM}
Let $h = (0,h_1,\dots,h_{d}) \in \Znn^{d+1}$ and $n > 0$. Then the
following are equivalent:
\begin{enumerate}[\rm (a)]
\item There is a fully CM relative complex $\Psi$ on ground set $[n]$
with $h = h(\Psi)$;
\item There is a fully shellable relative complex $\Psi$ on ground set
$[n]$ with $h = h(\Psi)$;
\item Let $(a_0,\dots,a_{d-1})$ and
$(b_0,\dots,b_{d-1})$ be the sequences defined through $a_{d-1} := h_{d}$ and
$b_{d-1} :=0$ and recursively continued
\begin{align*}
%
%
a_{i-1} &\ := \ \max(\Partial_{i+1}(a_{i}),
h_{i} + \Partial_{i+1}(b_{i}) ) \\
%
b_{i-1} &\ := \ \max(\Partial_{i+1}(b_{i}),
\Partial_{i+1}(a_{i})-h_{i} )
\end{align*}
for $i \ge 1$. Then $a_0 \le n-d$.
\end{enumerate}
\end{thm}
In Section~\ref{sec:fully}, we discuss the difference between CM and fully CM
relative complexes. In particular, we show in Theorem~\ref{thm:nice} that
every $(d-1)$-dimensional CM relative complex has a presentation as a fully CM
relative complex if we allow the ground set to grow by at most $d$ elements.
From this, we derive the following necessary condition on $h$-vectors of
proper CM relative complexes.
\begin{cor}\label{cor:necessary}
Let $h = (0,h_1,\dots,h_{d}) \in \Znn^{d+1}$ and $n > 0$. Further, let
$(a_0,\dots,a_{d-1})$ and $(b_0,\dots,b_{d-1})$ be the sequences defined
in Theorem~\ref{thm:relM}(c). If there exists a CM relative complex $\Psi$
on ground set $[n]$ with $h = h(\Psi)$, then $a_0 \leq n$.
\end{cor}
We conjecture that it actually suffices to extend the ground set by a single
new vertex. This would strengthen the bound of Corollary~\ref{cor:necessary}
to $n-d+1$.
Finally, Theorem~\ref{thm:bjorner} gives a characterization of $h$-vectors of
relative multicomplexes if the dimensions of the minimal faces of $\Psi =
\Delta \setminus \Gamma$ are given. This resolves a question of A.~Bj\"orner
stated in~\cite{stanley87}.
\bigskip
\textbf{Acknowledgments.} Research that led to this paper was supported by the
National Science Foundation under Grant No.~DMS-1440140 while the authors were
in residence at the Mathematical Sciences Research Institute in Berkeley,
California, during the Fall 2017 semester on \emph{Geometric and Topological
Combinatorics}. G.C.~was also supported by the Center for International
Cooperation at Freie Universit\"at Berlin and the Einstein Foundation Berlin.
L.K.~was also supported by the DFG, grant KA 4128/2-1. R.S.~was also
supported by the DFG Collaborative Research Center SFB/TR 109 ``Discretization
in Geometry and Dynamics''. We thank the two referees for helpful suggestions.
\section{\texorpdfstring{$f$}{f}-vectors of relative simplicial complexes}
\label{sec:f-rel}
\newcommand\f{\mathbf{f}}%
\newcommand\relBnd{\partial^\mathsf{rel}}%
\newcommand\F{\mathcal{F}}%
\newcommand\C[1]{\mathrm{C}{#1}}%
The proof of Theorem~\ref{thm:relKK} follows the same ideas as that of the classical Kruskal--Katona theorem given in~\cite[Sect.~8]{GK}.
A simplicial complex $\Delta \subset 2^{[n]}$ is called \Defn{compressed} if its set of $k$-faces forms an initial segment with respect to the reverse lexicographic order on the $(k+1)$-subsets of $[n]$, for each $k$.
Note that if $\Delta$ and $\Gamma$ are both compressed simplicial complexes and $f_k(\Gamma) \leq f_k(\Delta)$ for all $k$, then $\Gamma \subseteq \Delta$.
The Kruskal--Katona theorem now states that $f$ is the $f$-vector of a simplicial complex if and only if it is the $f$-vector of a compressed simplicial complex, which can be checked by numerical conditions.
\begin{thm}[{Kruskal~\cite{kruskal}, Katona~\cite{katona}}]\label{thm:KK}
For a vector $f = (1,f_0,\dots,f_{d-1}) \in \Znn^{d+1}$, the following conditions are equivalent:
\begin{enumerate}[\rm (a)]
\item $f$ is $f$-vector of a simplicial complex;
\item $f$ is $f$-vector of a compressed simplicial complex;
\item $\partial_{k+1}(f_{k}) \le f_{k-1}$ for all $k \ge 1$.
\end{enumerate}
\end{thm}
The shadow of a family of $k$-sets consists of all
$(k-1)$-subsets of the $k$-sets of the family. The
Kruskal-Katona theorem tells us that $\partial_{k+1}(r)$ is
the minimum size of the shadow of a family $k$-sets of size
$r$. Actually, this minimum is always achieved if the family
is compressed. Note that this implies in particular that the
functions $\partial_k$ are monotone.
With these preparations, we can now give the proof of our Theorem~\ref{thm:relKK}.
\begin{proof}[Proof of Theorem~\ref{thm:relKK}]
Let us recall the definition of the sequences $(a_0,\dots,a_{d-1})$ and $(b_0,\dots,b_{d-1})$.
We have that
$a_{d-1} = f_{d-1}$, $b_{d-1} =0$ and
\begin{align*}
a_{k-1} &\ = \ \max(\partial_{k+1}(a_{k}), f_{k-1} + \partial_{k+1}(b_{k}) ) &=&\ \partial_{k+1}(a_{k}) + \max(0, f_{k-1} - (\partial_{k+1}(a_{k})- \partial_{k+1}(b_{k})) ); \\
b_{k-1} &\ = \ \max(\partial_{k+1}(b_{k}), \partial_{k+1}(a_{k})-f_{k-1}
) &=&\ \partial_{k+1}(b_{k}) + \max(0 , (\partial_{k+1}(a_{k})-
\partial_{k+1}(b_{k})) - f_{k-1} ) ,
\end{align*}
for $1 \leq k \leq d-1$.
From the second expression for $a_{k-1}$ and $b_{k-1}$ it is easy to see that $a_{k-1} - b_{k-1} = f_{k-1}$.
In particular, we have that $a_k \geq b_k$ for $k \geq 0$.
We now show the sufficiency of the condition, so assume that $a_0 \leq n$. As
both sequences $(1,a_0,\dots,a_{d-1})$ and $(1,b_0,\dots,b_{d-1})$ satisfy the
condition of the Kruskal-Katona theorem (Theorem~\ref{thm:KK}), there exist
compressed simplicial complexes $\Gamma, \Delta \subset 2^{[n]}$ whose
respective $f$-vectors equal the two sequences. In particular, since both
complexes are compressed and $f_k(\Gamma) = b_k \leq a_k = f_k(\Delta)$, it
holds that $\Gamma \subset \Delta$, and the relative complex
$\Psi:=(\Delta,\Gamma)$ has $f$-vector $f$.
Now we turn to the necessity of our condition. Assume that we are given a
relative complex $\Psi = (\Delta, \Gamma)$ on the ground set $[n]$ with
$f(\Psi) = f$. We show by induction on $k$ that $a_k \leq f_k(\Delta)$ and
$b_k \leq f_k(\Gamma)$ for $k \geq 0$.
The base case $k = d-1$ is obvious. For the inductive step, it follows from
Theorem~\ref{thm:KK} that $f_{k-1}(\Delta) \geq \partial_{k+1}(f_{k}(\Delta))$,
and further $f_{k}(\Delta) \geq a_k$ implies that $\partial_{k+1}(f_{k}(\Delta))
\geq \partial_{k+1}(a_k)$. Similarly, it holds that $f_{k-1}(\Delta) = f_{k-1}
+ f_{k-1}(\Gamma) \geq f_{k-1} + \partial_{k+1}(f_{k}(\Gamma)) \geq f_{k-1} +
\partial_{k+1}(b_k)$. Together, this implies that
\[
f_{k-1}(\Delta) \ \geq \ \max(\partial_{k+1}(a_k), f_{k-1} +
\partial_{k+1}(b_k)) \ = \ a_{k-1} \, .
\]
Moreover, the last inequality together with the fact that $f_{k-1}(\Delta) -
f_{k-1}(\Gamma) = a_{k-1} - b_{k-1}$ implies that $f_{k-1}(\Gamma) \geq
b_{k-1}$. In particular, $a_0 \leq f_0(\Delta) \leq n$.
\end{proof}
\section{\texorpdfstring{$f$}{f}-vectors of relative multicomplexes}
\label{sec:f-rel-mult}%
\newcommand\tDelta{\widetilde{\Delta}}%
\newcommand\tGamma{\widetilde{\Gamma}}%
\newcommand\tPsi{\widetilde{\Psi}}%
\newcommand\tF{\widetilde{\F}}%
A \Defn{$\boldsymbol k$-multiset} is a set with repetitions
allowed.
A \Defn{multicomplex} $\tDelta$ is a collection of multisets closed under taking (multi-)subsets.
We denote a $k$-multisubset of $[n]$ by $F = \{s_1, s_2, \dots,s_k\}_\le$ where $1 \le s_1 \le s_2 \le\cdots \le s_k \le n$.
We say that the dimension of $F$ is $k-1$ and in the same way as for simplicial complexes, one defines $f$-vectors of multicomplexes.
Note that multicomplexes can be infinite, even if the ground set is finite.
The sequences which arise as $f$-vectors of multicomplexes are called
\Defn{$\boldsymbol M$-sequences} and they have a well-known classification due
to Macaulay. Namely, a sequence $(1, f_0, f_1, \dots)$ is an $M$-sequence if
and only if $f_{k-1} \geq \Partial_{k+1}(f_{k})$. Moreover, as in the
simplicial case, for each $M$-sequence $f$ there exists a unique
\emph{compressed} multicomplex $\tDelta$ with $f = f(\tDelta)$. Here, being
compressed is defined as in the simplicial case.
We refer the reader to \cite[Sect.~8]{GK} of \cite[Sect.~II.2]{Stanley96} for
details.
Using compressed multicomplexes and the characterization of $M$-sequences, the
same proof as for Theorem \ref{thm:relKK} also yields the following
characterization for $f$-vectors of finite proper relative multicomplexes
$\tPsi = (\tDelta,\tGamma)$.
\begin{thm}\label{thm:relKKm}
Let $f = (0,f_{0},\dots,f_{d-1}) \in \Znn^{d+1}$ and $n > 0$ and define two
sequences $(a_0,\dots,a_{d-1})$ and $(b_0,\dots,b_{d-1})$ by
$a_{d-1} := f_{d-1}$ and $b_{d-1} :=0$ and continue recursively
\begin{align*}
a_{k-1} &\ := \ \max(\Partial_{k+1}(a_{k}),
f_{k-1} + \Partial_{k+1}(b_{k}) ) \\
b_{k-1} &\ := \ \max(\Partial_{k+1}(b_{k}),
\Partial_{k+1}(a_{k})-f_{k-1} )
\end{align*}
for $k \ge 0$.
Then there is a proper (finite) relative multicomplex $\tPsi$ on the ground set $[n]$
with $f = f(\tPsi)$ if and only if $a_0 \le n$.
\end{thm}
Now we turn to the classification of $f$-vectors of not necessarily finite multicomplexes.
In the proof of Theorem \ref{thm:relKK}, it was crucial that relative simplicial complexes have bounded dimension, so that we could proceed by induction from the top dimension downwards.
For general relative multicomplexes, we will instead proceed from dimension $0$ upwards.
This requires some new notation.
For $r,k \in \Znn$ with $k > 0$, let $r_k > \dots >
r_1 \ge 0$ as defined by~\eqref{eqn:binomial}. We define
\[
\upshad{r}{k} \ := \
\binom{r_{k}+1}{k+1} +
\binom{r_{k-1}+1}{k+2} +
\cdots +
\binom{r_{1}+1}{2} \, .
\]
It is not difficult to see that $\Partial_{k+1}(\upshad{r}{k}) = r$ and $\upshad{\Partial_{k}(r)}{k-1} \geq r$.
Therefore, $M$-sequences can be equivalently characterized as those sequences $(f_{-1}, f_0, \dotsc)$ which satisfy $f_{k+1} \geq \upshad{f_k}{k+1}$ for all $k$.
\begin{thm}\label{thm:relKKmi}
Let $f = (0,f_{0},f_1,\dots)$ be a sequence of non-negative integers and
$n > 0$ and define two sequences $(a_0,a_1, \dots)$ and $(b_0,b_1, \dots)$
by $a_{0} := n$, $b_{0} := n - f_0$ and continue recursively
\begin{align*}
a_{k+1} &\ := \ \min(\upshad{a_k}{k+1}, f_{k+1} + \upshad{b_k}{k+1})\\
b_{k+1} &\ := \ \min(\upshad{b_k}{k+1}, \upshad{a_k}{k+1} - f_{k+1} )
\end{align*}
for $k \ge 0$. Then, there is a proper relative multicomplex $\tPsi$ on
the ground set $[n]$ with $f = f(\tPsi)$ if and only if $b_k \geq 0$ for
all $k \geq 0$.
\end{thm}
The proof is almost the same as the proof of Theorem~\ref{thm:relKK}, using the characterization of $M$-sequences in terms of $\tilde{\partial}^{k}$.
The only difference is that to prove necessity, one needs to start
the induction at $k=0$ and proceed in increasing order.
The classical theorem by Macaulay characterizes Hilbert functions of standard
graded algebras, and Theorem \ref{thm:relKKmi} has a similar interpretation.
We denote the Hilbert function of a finitely generated graded module $M$ over the polynomial ring $\kk[x_1,\dotsc,x_n]$ by $H(M, k) := \dim_\kk M_k$.
\begin{cor}[Macaulay for quotients of ideals]\label{cor:relKKm}
Let $H : \Znn \to \Znn$ with $H(0) = 0$ and $n \geq H(1)$. Furthermore, let
$(a_0,a_1, \dots)$ and $(b_0,b_1, \dots)$ be the two sequences of
Theorem~\ref{thm:relKKmi}, where we set $f_k = H(k+1)$. Then, there exist
two proper homogeneous ideals $J \subset I \subsetneq
\kk[x_1,\dotsc,x_n]$ with $H(k) = H(I/J,k)$ for all $k$, if and only if
$b_k \geq 0$ for all $k \geq 0$.
\end{cor}
\begin{proof}
Consider a homogeneous ideal $I \subseteq \kk[x_1,\dotsc,x_n]$.
For any fixed term order $\preceq$, the collection of standard monomials, that is, the monomials not contained in the initial ideal of $I$ with respect to $\preceq$, is naturally identified with a multicomplex $\tDelta$.
Since the standard monomials form a vector space basis of $\kk[x_1,\dotsc,x_n]/I$ that respects the grading, the $f$-vector of $\tDelta$ coincides with the Hilbert function of $\kk[x_1,\dotsc,x_n]/I$.
Moreover, if $J \subseteq I \subseteq \kk[x_1,\dotsc,x_n]$ are two homogeneous ideals, then passing to the initial ideals (with respect to $\preceq$) preserves the inclusion.
Therefore, any Hilbert function of a quotient of ideals also arises as $f$-vector of a relative multicomplex.
For the converse we associate to any multicomplex $\tDelta$ the monomial ideal corresponding to all multisets not in $\tDelta$.
\end{proof}
\section{\texorpdfstring{$h$}{h}-vectors of relative Cohen-Macaulay complexes}\label{sec:macaulay}
Let $\Psi = (\Delta,\Gamma)$ be a $(d-1)$-dimensional relative simplicial
complex and let $\sigma_1,\dots,\sigma_m$ be some ordering of the
inclusion-maximal faces (i.e., the facets) of $\Psi$. Define
\[
\Psi_j \ := \ \left( 2^{\sigma_1} \cup 2^{\sigma_2} \cup \dots \cup
2^{\sigma_j} \right) \cap (\Delta \setminus \Gamma)
\]
for $j \ge 1$ and set $\Psi_0 := \emptyset$. We call the ordering of the facets a \Defn{shelling
order} if $\Psi_{j} \setminus \Psi_{j-1}$ has a unique inclusion-minimal
element $R(\sigma_j)$ for all $j=1,\dots,m$. Consequently, $\Psi$ is
\Defn{shellable} if it has a shelling order. If $\Gamma = \emptyset$ and hence
$\Psi$ is a simplicial complex, this recovers the usual notion of
shellability. The $h$-vector $h(\Psi)$ of a shellable relative complex has a
particularly nice interpretation:
\[
h_i(\Psi) \ = \ |\{ j : |R(\sigma_j)| = i \}| \, ,
\]
for $0 \le i \le d$. It is shown in~\cite[Sect.~III.7]{Stanley96} that a shellable
relative complex is Cohen--Macaulay but the converse does not need to hold.
We will call a relative complex $\Psi$ \Defn{fully shellable} if it has a
presentation $\Psi = (\Delta,\Gamma)$ such that $\dim \Psi = \dim \Gamma$ and
$\Psi$ as well as $\Delta$ and $\Gamma$ are shellable. By the above remarks,
it is clear that fully shellable relative complexes are fully Cohen--Macaulay
and, again, the converse does not necessarily hold.
In light of Theorem~\ref{thm:relKKm}, condition (c) of
Theorem~\ref{thm:relM} states that $h$ is the $f$-vector of a proper relative
multicomplex. In order to prove the implication (c) $\Longrightarrow$ (b), we
will show that for every relative multicomplex on the ground set $[n-d]$ with given
$f$-vector $h = (0,h_1,\dots,h_d)$, there is a fully shellable relative
complex $\Psi$ with $h(\Psi) = h$.
Let $\tPsi = (\tDelta,\tGamma)$ be a proper relative
$(d-1)$-dimensional multicomplex on ground set $[n-d]$ and
assume that $\tDelta$ and $\tGamma$ are compressed. To turn
$\tPsi$ into a relative complex, we follow the construction
in~\cite{BFS}. Order the collection of multisets of size
$\leq d$ on the ground set $[n-d]$ by graded reverse
lexicographic order, and the collection of $d$-sets on $[n]$
by reverse lexicographic order. There is a unique bijection
$\Phi_d$ between these two collections which preserves the
given orders. Explicitly, the map is
\[
F = \{b_1, b_2, \dots, b_k \}_{\le} \ \mapsto \ \Phi_d(F) \ := \
\{1,2,\dots,d-k, b_1 + d-k+1, b_2 + d-k+2, \dots, b_k + d \} \, .
\]
We denote by $\Delta$ the simplicial complex with
facets $\{ \Phi_d(F) : F \in \tDelta\}$ and $\Gamma$ likewise. Since
$\tGamma$ is a submulticomplex of $\tDelta$, it follows that $\Gamma \subset
\Delta$ and $\Psi = (\Delta,\Gamma)$ is a relative complex with $\dim \Psi =
\dim \Delta = \dim \Gamma = d - 1$.
\begin{prop}\label{prop:BFSrelShell}
Let $\tPsi = (\tDelta,\tGamma)$ be a $(d-1)$-dimensional relative
multicomplex such that $\tDelta$ and $\tGamma$ are compressed. Let $\Psi =
(\Delta, \Gamma)$ be the corresponding relative simplicial complex
constructed above. Given an ordering $\prec$ of the faces of $\tDelta$
such that $F \prec F'$ whenever $|F| < |F'|$, the induced ordering on the
facets $\Phi_d(F)$ of $\Delta$ is a shelling order for $\Delta$, $\Gamma$,
and $\Psi$.
\end{prop}
\begin{proof}
It was shown in~\cite{BFS} that any such ordering gives a shelling order for
$\Delta$ with restriction sets
\[
R(\sigma) \ = \ \sigma \setminus \{1,2,\dots,d-k\} \ = \ \{s_1 +
d-k+1,\dots,s_k + d\}
\]
if $\sigma = \Phi_d(\{s_1,\dots,s_k\}_\le)$.
We are left to prove that restricting this order
to the facets of $\Delta \setminus \Gamma$ yields a shelling order for
$\Psi$. It suffices to show that if $\sigma$ is a facet of $\Psi$, i.e., a
facet of $\Delta$ not contained in $\Gamma$, then $R(\sigma) \not\in
\Gamma$.
Let $F = \{s_1, \dots, s_k\}_{\leq}$ be the face of $\tDelta$ such that
$\sigma = \Phi_d(F)$. We will show that any facet $\sigma'$ of $\Delta$
which contains $R := R(\sigma)$ does not belong to $\Gamma$. By
construction, the facets of $\Gamma$ are a subset of the facets of $\Delta$,
and thus $R \notin \Gamma$.
Let $\sigma'$ be a facet of $\Delta$ which contains $R$ and let $F'$ be
the corresponding element of $\tDelta$ with $\sigma' = \Phi_d(F')$.
Observe that either $\sigma' = \sigma$ or $t = |F'| > |F| = k$. Indeed, if $t<k$,
$\{1, 2, \dots, d-k+1\} \subseteq \sigma'$, and since $R \cap \{1, 2,
\dots, d-k+1\} = \emptyset$, $R$ cannot be a subset of $\sigma'$.
If $t=k$, then $\sigma' \supseteq R$ implies $\sigma' = \sigma$.
So, let us assume that $t > k$. Let $G = \{r_1, \dots, r_t\}_{\leq}$ be
the smallest $t$-multiset in $\tDelta$ in reverse lexicographic order such that
$\tau = \Phi_d(G) \supseteq R$. Now $\tau = \{1, \dots, d-t\} \cup S$,
with $S= \{d-t +1 +r_1, \dots, d+r_t\}$. As before, observe that $R \cap
\{1, \dots, d-t\} = \emptyset$. Since $\Phi_d$ preserves the reverse
lexicographic order on $t$-multisets, $S$ is also minimal with respect to
reverse lexicographic order. Therefore the elements of $R$ are the largest
elements in $S$ and
\[
G \ = \ \{\underbrace{1, \dots, 1}_{t-k}, s_1, \dots, s_k\}_{\leq}.
\]
Then $F = \{s_1, \dots, s_k\}_{\leq} \subseteq G$, and since $F\notin
\tGamma$ and $\tGamma$ is a multicomplex, it follows that $G \notin
\tGamma$. Since $\tGamma$ is compressed and $G$ is smaller than $F'$, $F'$
also does not belong to $\tGamma$. This implies $\sigma \not \in \Gamma$.
\end{proof}
\begin{proof}[Proof of
Theorem~\ref{thm:relM}: (c) $\Longrightarrow$ (b) $\Longrightarrow$ (a)]
By Theorem~\ref{thm:relKKm}, condition (c) guarantees the existence of a
proper relative multicomplex $\tPsi$ with $f$-vector $h$.
By Proposition~\ref{prop:BFSrelShell}, the construction
above yields a fully shellable relative simplicial complex $\Psi$
with $h = h(\Psi)$. This proves (c) $\Longrightarrow$ (b).
Theorem 2.5 for relative complexes in~\cite{Stanley96}
asserts that $\Psi$ is fully Cohen--Macaulay and hence proves
(b) $\Longrightarrow$ (a).
\end{proof}
In order to prove the implication (a) $\Longrightarrow$ (c), we make use of
the powerful machinery of Stanley--Reisner modules. Let $\kk$ be an infinite
field. For a fixed $n > 0$, let $S := \kk[x_1,\dots,x_n]$ be the polynomial
ring. For a simplicial complex $\Delta \subseteq 2^{[n]}$, its
\Defn{Stanley--Reisner ideal} is $I_\Delta := \langle \x^\tau : \tau \not\in
\Delta \rangle$ and we write $\kk[\Delta] := S/I_\Delta$ for its
\Defn{Stanley--Reisner ring}. If $\Gamma \subset \Delta$ is a pair of
simplicial complexes, then $\kk[\Delta] \twoheadrightarrow \kk[\Gamma]$ and the
\Defn{Stanley--Reisner module} of $\Psi = (\Delta,\Gamma)$ is
\newcommand\SRmod{\mathrm{M}}
\[
\SRmod[\Psi] \ := \ \ker( \kk[\Delta] \twoheadrightarrow \kk[\Gamma]) \ = \
I_\Gamma / I_\Delta \, .
\]
This is a graded $S$-module and $\Psi$ is a \Defn{Cohen--Macaulay} relative
complex if $\SRmod[\Psi]$ is a Cohen--Macaulay module over $S$. In
particular, any choice of generic linear forms $\theta_1,\dots,\theta_d \in S$
for \mbox{$d = \dim \Psi + 1$}
is a regular sequence for $\SRmod[\Psi]$ and
\[
\dim_\kk ( \SRmod[\Psi] / \langle \theta_1,\dots,\theta_d \rangle \SRmod[\Psi])_i \ = \
h_i(\Psi) \, ,
\]
for all $i \ge 0$.
\begin{proof}[Proof of Theorem~\ref{thm:relM}: (a) $\Longrightarrow$ (c)]
Let $(\Delta,\Gamma)$ be a presentation of $\Psi$ such that $\dim \Gamma =
\dim \Psi$ and $\Delta$ and $\Gamma$ are CM. Consider the short exact
sequence
\begin{equation}\label{eq:seq1}
0 \ \to \ \SRmod[\Psi] \ \to \ \kk[\Delta] \ \to \ \kk[\Gamma] \
\to \ 0
\end{equation}
of $S$-modules. Let $\theta \in S$ be a generic linear form.
Tensoring~\eqref{eq:seq1} with $S / \theta$ yields
\begin{equation}\label{eq:seq2}
\Tor_1^S(\kk[\Gamma], S/\theta) \ \to \ \SRmod[\Psi]/\theta\SRmod[\Psi]
\ \to \ \kk[\Delta]/ \theta\kk[\Delta] \ \to \ \kk[\Gamma]/
\theta\kk[\Gamma] \ \to \ 0
\end{equation}
By resolving $S/\theta$, it is easy to see that $\Tor_1^S(\kk[\Gamma],
S/\theta) = (0 :_{\kk[\Gamma]} \theta) = 0$, so \eqref{eq:seq2} is a short
exact sequence as well.
By our choice of presentation, $\kk[\Gamma]$ is Cohen--Macaulay and
we may repeat the process for a full regular sequence $\Theta = (\theta_1,
\dotsc, \theta_{d})$ to arrive at
\begin{equation}\label{eq:seq3}
0 \ \to \ \SRmod[\Psi]/\Theta\SRmod[\Psi] \ \to \ \kk[\Delta]/
\Theta\kk[\Delta] \ \to \ \kk[\Gamma]/ \Theta\kk[\Gamma] \ \to \ 0 \, .
\end{equation}
\newcommand\In{\mathrm{in}_\preceq}%
Since $\Psi$ is Cohen--Macaulay, the Hilbert function of
$\SRmod[\Psi]/\Theta\SRmod[\Psi]$ is exactly the $h$-vector of $\Psi$ and,
moreover, we can identify $\SRmod[\Psi]/\Theta\SRmod[\Psi]$ with a graded
ideal in $\kk[\Delta]/ \Theta\kk[\Delta]$. By a linear change of coordinates,
this yields a pair of homogeneous ideals $J_\Delta \subset J_\Gamma
\subset R := \kk[y_1,\dots,y_{n-d}]$ with difference of Hilbert functions
exactly $h(\Psi)$. For any fixed term order $\preceq$, we denote by
$\In(J_\Delta), \In(J_\Gamma)$ the corresponding initial ideals. The
passage to initial ideals leaves the Hilbert functions invariant and
$\In(J_\Delta) \subseteq \In(J_\Gamma)$; c.f.~\cite[Prop.~9.3.9]{CLOS}.
The corresponding collections of standard monomials are naturally
identified with a pair of multicomplexes $\tGamma \subset \tDelta$ with
$f$-vector $h$ and this completes the proof.
\end{proof}
\section{Cohen--Macaulay versus fully Cohen--Macaulay}
\label{sec:fully}%
Theorem~\ref{thm:relM} only addresses the characterization of $h$-vectors of
fully CM relative complexes. By definition, a relative simplicial complex
$\Psi$ is the set difference of a pair $\Gamma \subset \Delta \subseteq
2^{[n]}$ of simplicial complexes. This presentation is by no means unique and
it is natural to ask if in the case that $\Psi$ is Cohen--Macaulay, there are
always CM complexes $\Gamma' \subseteq \Delta' \subseteq 2^{[n]}$ of dimension
$\dim \Psi$ such that $\Psi = \Delta' \setminus \Gamma'$. The following
example shows that this is not the case.
\begin{figure}[t]
\begin{tikzpicture}[scale=1.5]
\newcommand{gray!40}{gray!40}
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
\begin{scope}
\coordinate (v1) at (0,0);
\coordinate (v2) at (1,0);
\coordinate (v3) at (1,-1);
\coordinate (v4) at (0,-1);
\draw (v1)--(v2)--(v3)--(v4)--cycle;
\draw[line width=1mm] (v1) -- (v3) (v2) -- (v4);
\foreach \p in {(v1),(v2),(v3),(v4)}
\draw[fill=black] \p circle (0.06);
\path (v1) node[anchor=south] {$1$};
\path (v2) node[anchor=south] {$2$};
\path (v3) node[anchor=north] {$3$};
\path (v4) node[anchor=north] {$4$};
\end{scope}
\begin{scope}[xshift=3cm]
\coordinate (v1) at (0,0);
\coordinate (v2) at (1,0);
\coordinate (v3) at (1,-1);
\coordinate (v4) at (0,-1);
\draw (v1)--(v2)--(v3)--(v4)--cycle;
\foreach \p in {(v1),(v2),(v3),(v4)}
\draw[fill=black] \p circle (0.06);
\path (v1) node[anchor=south] {$1$};
\path (v2) node[anchor=south] {$2$};
\path (v3) node[anchor=north] {$3$};
\path (v4) node[anchor=north] {$4$};
\end{scope}
\begin{scope}[xshift=6cm]
\coordinate (v1) at (0,0);
\coordinate (v2) at (1,0);
\coordinate (v3) at (1,-1);
\coordinate (v4) at (0,-1);
\coordinate (v5) at (0.5,-0.5);
\coordinate (v5p) at (0.5,-0.55);
\draw (v1)--(v2)--(v3)--(v4)--cycle;
\foreach \p in {(v1),(v2),(v3),(v4)}
\draw[line width=1mm] (v5) -- \p;
\foreach \p in {(v1),(v2),(v3),(v4)}
\draw[fill=black] \p circle (0.06);
\draw[fill=black] (v5) circle (0.07);
\path (v1) node[anchor=south] {$1$};
\path (v2) node[anchor=south] {$2$};
\path (v3) node[anchor=north] {$3$};
\path (v4) node[anchor=north] {$4$};
\path (v5p) node[anchor=north] {$5$};
\end{scope}
\end{tikzpicture}
\caption{The relative complexes of Example \ref{ex:1}, Example \ref{ex:2},
and Example \ref{ex:3}. In each case, $\Gamma$ is drawn in
bold.}\label{fig:examples}
\end{figure}
\begin{ex}\label{ex:1}
Let $\Delta \subset 2^{[4]}$ be the complete graph on $4$ vertices, that
is, the complex consisting of all subsets of $[4]$ of size at most $2$.
Let $\Gamma \subset \Delta$ be a perfect matching, see Figure \ref{fig:examples}. Then $\Delta \setminus
\Gamma$ is the relative complex consisting of $4$ \emph{open} edges. This is
a shellable relative complex. It is easy to check that on the fixed
ground set $[4]$, this is the only presentation with $\dim \Delta = \dim
\Gamma = 1$ and hence $\Psi$ is not fully Cohen--Macaulay.
\end{ex}
There are several possibilities to weaken the requirements on fully
Cohen--Macaulay, for example, the requirement that $\dim \Gamma = \dim \Psi$.
The next example, however, shows that the characterization of
Theorem~\ref{thm:relM} then ceases to hold.
\begin{ex}\label{ex:2}
Let $\Delta \subseteq 2^{[4]}$ be the $1$-dimensional complex with facets
$\{1,2\}, \{2,3\}, \{3,4\}, \{1,4\}$ and let $\Gamma$ be the complex
composed of the vertices of $\Delta$. Then $\Psi = (\Delta,\Gamma)$ is a
relative complex isomorphic to the relative complex of Example~\ref{ex:1}.
Both $\Delta$ and $\Gamma$ are Cohen--Macaulay but $\dim \Gamma < \dim
\Psi$. In particular, $\Psi$ is shellable with $h$-vector $h := h(\Psi) =
(0,0,4)$. However, $h$ is not the $f$-vector of a relative multicomplex on
ground set $[4-2]$, as any such (relative) multicomplex can have at most
$3$ faces of dimension $1$.
\end{ex}
Nevertheless, it is possible to remedy the problem illustrated in
Example~\ref{ex:1} by allowing more vertices.
\begin{ex}\label{ex:3}
Let $\Psi = (\Delta,\Gamma)$ be the relative complex of
Example~\ref{ex:1}. Let $\Delta' := \Delta \cup \{ \{i,5\} : i \in [4]\}$
be the graph-theoretic cone over $\Delta$ and define $\Gamma'$
accordingly. Then $\Delta \setminus \Gamma = \Delta' \setminus \Gamma'$
and, since $\Delta'$ and $\Gamma'$ are connected graphs and hence
Cohen--Macaulay, this shows that $\Psi$ is a fully Cohen--Macaulay relative
complex over the ground set $[5]$.
\end{ex}
The following result now shows that every Cohen--Macaulay relative complex is
fully Cohen--Macaulay if the ground set is sufficiently enlarged.
\begin{thm}\label{thm:nice}
Let $\Gamma \subset \Delta \subseteq 2^{[n]}$ be simplicial complexes,
such that $\Psi = (\Delta, \Gamma)$ is Cohen-Macaulay of dimension $d-1$.
Let $e$ be the depth of $\kk[\Gamma]$.
Then there exist $\Gamma' \subseteq \Delta' \subseteq 2^{[n+d-e]}$, such
that $\Delta' \setminus \Gamma' = \Delta \setminus \Gamma$, and both
$\Delta'$ and $\Gamma'$ are Cohen-Macaulay of dimension $d-1$.
\end{thm}
\begin{proof}
Let $\Gamma_1$ be the $(d-e)$-fold cone over $\Gamma$ and set $\Delta_1 :=
\Delta \cup \Gamma_1$. Then $\Delta_1 \setminus \Gamma_1 = \Delta
\setminus \Gamma$. Further note that $\kk[\Gamma_1] =
\kk[\Gamma][y_1,\dotsc, y_{d-e}]$, where the $y_i$ are new variables.
Thus, the depth of $\kk[\Gamma_1]$ is $d$. Finally, we define $\Delta'$ and
$\Gamma'$ to be the $(d-1)$-dimensional skeleta of $\Delta_1$ and
$\Gamma_1$, respectively. Again, $\Delta' \setminus \Gamma' = \Delta
\setminus \Gamma$ and thus $\Psi \cong (\Delta', \Gamma')$. By
\cite[Corollary 2.6]{Hibi}, $\Gamma'$ is Cohen-Macaulay. By assumption,
$\Psi = \Delta' \setminus \Gamma'$ is Cohen-Macaulay, and since $\dim \Psi
= \dim \Delta' = \dim \Gamma'$, it follows from~\cite[Prop
1.2.9]{Bruns-Herzog} that $\Delta'$ is also Cohen--Macaulay.
\end{proof}
In the construction given in the course of the proof, the complexes $\Delta$
and $\Gamma$ occur as induced subcomplexes. If we are to abandon this
requirement, then our computations suggest that it suffices to add a single new
vertex. Based on this evidence, we offer the following conjecture.
\begin{conj}\label{conj1}
Every Cohen--Macaulay relative complex $\Psi$ on ground set $[n]$ is a
fully Cohen--Macaulay relative complex on ground set $[n+1]$. That is,
for every $(d-1)$-dimensional Cohen--Macaulay
relative complex $\Psi = (\Delta,\Gamma)$ on ground set $[n]$, there are
Cohen--Macaulay simplicial complexes $\Gamma' \subseteq
\Delta' \subseteq 2^{[n+1]}$ of dimension $d-1$, such that
$\Delta \setminus \Gamma = \Delta' \setminus \Gamma'$.
\end{conj}
We also offer a more precise conjecture on how the complexes $\Gamma'
\subset \Delta'$ can be obtained.
\begin{conj}\label{conj2}
Let $\emptyset \neq \Gamma \subsetneq \Delta \subset 2^{[n]}$ be two
simplicial complexes, such that the relative complex $(\Delta,\Gamma)$ is
Cohen--Macaulay of dimension $d-1$ over some field $\kk$. If $\Delta$ and
$\Gamma$ have no common minimal non-faces, then the depth of $\kk[\Gamma]$
is at least $d-1$.
\end{conj}
To see that Conjecture~\ref{conj2} implies Conjecture~\ref{conj1}, let $\Psi =
(\Delta,\Gamma)$ be a given presentation. We can assume that $\Delta$ and
$\Gamma$ have no minimal non-faces in common. Conjecture~\ref{conj2} then
assures us that $\kk[\Gamma]$ has depth $d-1$ and Theorem~\ref{thm:nice}
yields Conjecture~\ref{conj1}.
Instead of fixing the ground set, we may
instead consider the dimensions of the minimal faces in $\Psi = (\Delta,
\Gamma)$.
For a sequence $\alpha = (\alpha_1, \alpha_2, \alpha_3,\dotsc)$ of numbers and
$i \ge 0$ we set
\[
E^i\alpha \ := \ (\underbrace{0,\dotsc,0}_{i},\alpha_1, \alpha_2,
\alpha_3,\dotsc) \, .
\]
\begin{thm}\label{thm:bjorner}
For a vector $h = (h_0, \dotsc, h_{d}) \in \Znn^{d+1}$ and
numbers $a_1, \dotsc, a_r \in \Znn$, the following are
equivalent:
\begin{enumerate}[\rm (i)]
\item $h = h(\Delta, \Gamma)$ for a shellable relative complex $(\Delta, \Gamma)$, whose minimal faces have cardinalities $a_1, \dotsc, a_r$;
\item $h = h(\Delta, \Gamma)$ for a Cohen-Macaulay relative complex $(\Delta, \Gamma)$, whose minimal faces have cardinalities $a_1, \dotsc, a_r$;
\item $h$ is the $h$-vector of a graded Cohen-Macaulay module (over some polynomial ring), whose generators have the degrees $a_1, \dotsc, a_r$.
\item There exist M-sequences $\nu_1, \dotsc, \nu_r$ such that
\[
h \ = \
E^{a_1} \nu_1 +
E^{a_2} \nu_2 +
\cdots +
E^{a_r} \nu_r \, .
\]
\end{enumerate}
\end{thm}
The implications (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii) are clear, and
(iii) $\Rightarrow$ (iv) is Proposition 5.2 of~\cite{stanley87}.
In \emph{loc.~cit.} Anders Bj\"orner asked if the implication (iv)
$\Rightarrow$ (iii) also holds.
\begin{proof}
We only need to show (iv) $\Rightarrow$ (i). For each $i$, we can find a
shellable simplicial complex $\Delta_i$ whose $h$-vector is $\nu_i$.
Further, let $v_{i1},\dots,v_{i a_i}$ be new vertices and let $\Psi_i$ be
the relative complex with faces $\{ F \cup \{v_{i1},\dots,v_{i a_i}\}
\colon F \in \Delta_i\}$. It is clear that any shelling order on
$\Delta_i$ yields a shelling on $\Psi_i$, and that $h(\Psi_i) = E^{a_i}
\nu_i$. Finally, by taking cones if necessary, we may assume that all the
$\Psi_i$ have the same dimension. Then the disjoint union of the $\Psi_i$
is the desired shellable relative complex.
\end{proof}
\bibliographystyle{amsalpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 14 |
Q: How to add header to checkout page on magento 2 LUMA theme I have headers on all the pages in the LUMA theme except the checkout one. For reference, screenshots are attached below.
As you can see in the attached images, we have a header on the product view page, and it's missing on the checkout page? Is there any default way of adding it without overriding the module?
A: remove layout="checkout" in page layout.xml in your theme
Default checkout page magento in luma theme doesn't show header
Example
<page xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" layout="checkout" xsi:noNamespaceSchemaLocation="urn:magento:framework:View/Layout/etc/page_configuration.xsd">
<body></body>
</page>
remove layout="checkout" then header will show up
A: You need to copy:
/vendor/magento/module-checkout/view/frontend/page_layout/checkout.xml
in your theme:
/app/design/frontend/**YourVendor**/**YourTheme**/Magento_Checkout/page_layout/checkout.xml
Checkout.xml need looks like:
<?xml version="1.0"?>
<layout xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:View/Layout/etc/page_layout.xsd">
<referenceContainer name="page.wrapper">
<container name="header.container" as="header_container" label="Page Header Container" htmlTag="header" htmlClass="page-header" before="main.content"/>
<container name="page.top" as="page_top" label="After Page Header" after="-" />
</referenceContainer>
</layout>
And then just move logo in your checkout_index_index.xml
/app/design/frontend/**YourVendor**/**YourTheme**/Magento_Checkout/layout/checkout_index_index.xml
Content of checkout_index_index.xml:
<?xml version="1.0"?>
<page xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" layout="1column" xsi:noNamespaceSchemaLocation="urn:magento:framework:View/Layout/etc/page_configuration.xsd">
<body>
<move element="logo" destination="header-wrapper"/>
</body>
</page>
Good luck and don't forget Flush Cache ;)
PS. Main Navigation is hidden with CSS by default Luma style. You need override CSS to.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 483 |
As innovation in technology progresses by leaps and bounds, industry leaders are seeking to streamline their daily processes using RPA, allowing their employees to take on more strategic, creative and customer-facing tasks, leaving RPA to do the heavy lifting.
The benefits of RPA implementation are plenty, however, the main differentiator lies in the cost comparison between a full-time employee and RPA – a whopping 65% less expensive! Check out the industries and their processes that have been impacted by the power of our RPA business solutions. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,484 |
def extractKazamaTranslation(item):
"""
'Kazama Translation'
"""
vol, chp, frag, postfix = extractVolChapterFragmentPostfix(item['title'])
if not (chp or vol or frag) or 'preview' in item['title'].lower():
return None
return False
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,163 |
\section{Introduction}
\label{intro}
The role of tensor multiplets in supergravity has seen in the last
years a revived interest, in connection with the study of flux
compactifications in superstring or M-theory.
Two-index antisymmetric tensors are 2-form gauge fields whose
field-strengths are invariant under the (tensor)-gauge transformation
$B \to B + {\rm d}\hskip -1pt \Lambda$, $\Lambda$ being any 1-form. A physical pattern to
introduce massive tensor fields is the anti-Higgs mechanism, where
the dynamics allows the tensor to take a mass by a suitable
coupling to some vector field. The mass term plays the role of
magnetic charge in the theory.
The investigation of the role of massive tensor fields was
particularly fruitful for the $N=2$ theory in 4 dimensions, where the
study of the coupling of {\em tensor-scalar} multiplets (obtained by
Hodge-dualizing scalars covered by derivatives in the hypermultiplet
sector) to $N=2$ supergravity was considered, both as a CY compactification \cite{Louis:2002ny} and at a purely four dimensional supergravity level \cite{deWit:1982na,Theis:2003jj}. When this model was extended, in \cite{D'Auria:2004yi,Dall'Agata:2003yr,Sommovigo:2004vj}, to include the coupling to gauge multiplets, it allowed to construct new gaugings
containing also magnetic charges, and
to find the electric/magnetic duality completion of the $N=2$ scalar
potential.
However, a general formulation of $N=2$, $D=4$ supergravity coupled
to {\em tensor-vector } multiplets (obtained by Hodge-dualizing
scalars in the vector multiplet sector) is still missing, even if
important steps in that direction appeared quite recently
\cite{Gunaydin:2005df,Gunaydin:2005bf}.
On the other hand, the situation appears more promising in five
dimensional supergravity. There, 2-index antisymmetric tensors appear in the gauge
sector, since the field-strengths of massless two-index tensors are
Hodge-dual to vector field-strengths, and they naturally appear in the compactification
of higher dimensional theories \footnote{For
example, the $N=8$ gauged theory requires that a subset of the gauge
vectors be dualized to tensors
\cite{Gunaydin:1984qu,Gunaydin:1985cu,Pernici:1985ju}.}.
Various approaches to construct a general
coupling to tensor multiplets in the $N=2$ theory have been given
\cite{Gunaydin:1983rk,Gunaydin:1983bi,Gunaydin:1984pf,Sierra:1985ax,Lukas:1998yy,Gunaydin:1999zx,Ceresole:2000jd,Bergshoeff:2004kh,Gunaydin:2005bf}.
Towards a general understanding of the four dimensional case, we adopted the strategy of first
looking at the five dimensional theory in a framework as general as possible. In particular,
an ingredient generally used for the construction of the couplings is the ``self-duality in
odd dimensions'' \cite{Townsend:1983xs} that allows to work with massive, self-dual tensors
from the very beginning. However, in this way much of the algebraic structure underlying the
theory is not manifest. To find the most general theory in five dimensions in a way which can
give insight into the algebraic structure also for the four dimensional case, we found
useful to examine first at the bosonic level and in full generality the algebraic structure which any theory coupled to tensors and gauge vectors is based on. This requires the extension
of the notion of gauge algebra to that of free differential algebra (FDA in the following)
that naturally accomodates in a general algebraic structure the presence of $p$-forms ($p>1$).
We have then devoted the first part of the paper, section \ref{generalities}, to the study of
the gauge properties of a general FDA involving gauge vectors (1-forms) and two-index
antisymmetric tensors (2-forms). The discussion will be completely general, and will not rely
on the dimensions of space-time (apart from the obvious request $D\geq 4$, in order to have
dynamical 2-forms) nor on supersymmetry. Our procedure allows the FDA structure to be further
generalized, for $D\geq 5$, by including also couplings to higher order forms, as is the case,
in general, for flux compactifications. This is left to a future investigation.
When applying our results to the case of $D=5$, $N=2$ supergravity, in
section \ref{susy}, we
find some possible generalizations with respect to the current
literature in the subject
\cite{Gunaydin:1983rk,Gunaydin:1983bi,Gunaydin:1984pf,Sierra:1985ax,Lukas:1998yy,Gunaydin:1999zx,Ceresole:2000jd,Bergshoeff:2004kh,Gunaydin:2005bf}.
Besides the fact, already pointed out in \cite{Bergshoeff:2004kh} and
\cite{deWit:2004nw}, that it is possible to include in the 3-form
field-strength a coupling of the type $d_{\Lambda\Sigma M}F^\Lambda \wedge A^\Sigma$
(where $\Lambda$ enumerates gauge fields and $M$ tensor fields), we find
that the mass matrix for the tensor fields, which in five dimensional
supergravity has to be antisymmetric ($m^{MN}=-m^{NM}$), is however
not necessarily proportional to the symplectic metric
$\Omega^{MN}=\pmatrix{0& \relax{\rm 1 \kern-.35em 1}\cr -\relax{\rm 1 \kern-.35em 1} &0}$, which is the case to
which the literature on the subject usually refers to. On the
contrary, any general antisymmetric matrix may be considered. This may
be understood, for example, by looking at the $D=5$ $N=2$ theory
obtained by Scherk--Schwarz dimensional reduction from six dimensions
\cite{Andrianopoli:2004xu}. In this case, indeed, the tensor
mass-matrix is the Scherk--Schwarz phase and has in general different
eigenvalues. Therefore for a general five dimensional $N=2$ theory
the generators of the gauge algebra are not necessarily in a
symplectic representation, and constraints from supersymmetry give
milder constraints on the gauging than the ones usually considered.
As a consequence of these generalizations, the scalar potential of the
theory has some differences with respect to the previous investigations.
The FDA approach allows to interpret the resulting structure in a
general group-theoretical way which is not evident with other
approaches. Our starting point is a general gauge algebra, which is
represented via generators with indices in the adjoint representation
of the gauge group. The building blocks of the FDA are then p-form
{\em potentials} (with, for our case, p = 1,2, that is
$A=A_\mu {\rm d}\hskip -1pt x^\mu$ and $B=B_{\mu\nu} {\rm d}\hskip -1pt x^\mu \wedge {\rm d}\hskip -1pt x^\nu$) and
their field-strengths.
Let us emphasize that in this way the fields are subject to {\em gauge
constraints}, and are therefore {\em massless}.
The mechanism for which the 2-forms become massive is left to the
dynamics of the Lagrangian (or alternatively, in the supersymmetric
case, also of the supersymmetric Bianchi identities). At the bosonic
level, this is implemented via the anti-Higgs mechanism, that is by
fixing the gauge invariance of the system $(A,B)$:
\begin{eqnarray}
\left\{\matrix{\delta B &=& {\rm d}\hskip -1pt \Lambda \hfill \cr
\delta A &=& {\rm d}\hskip -1pt \Theta -m \Lambda}\right.,
\end{eqnarray}
with field-strengths
\begin{eqnarray}
\left\{\matrix{H &=& {\rm d}\hskip -1pt B \hfill \cr
F &=& {\rm d}\hskip -1pt A +m B}\right.,
\end{eqnarray}
via the tensor-gauge fixing $\bar \Lambda = \frac 1m A$.
Since our analysis does not rely on the space-time dimension, we
expect to retrieve in particular, with our approach, also the results
already known for the $D=5$, $N=2$ theory. However, there is a subtle
point here, because it appears not evident how to reconcile the
anti-Higgs mechanism with the fact that supersymmetry constrains
massive tensors in $D=5$ supergravity to obey the self-duality
condition:
\begin{equation}
m\partial_{[\mu }B_{\nu\rho]}\propto
\epsilon_{\mu\nu\rho\sigma\lambda}B^{\sigma\lambda}\,,\qquad \mu ,\nu,\cdots =
0,1,\dots,4.
\end{equation}
The way out from this puzzle may be found by looking again at the
subclass of models obtained by Scherk--Schwarz dimensional reduction
from six dimensions. Indeed, the six-dimensional Lorentz algebra
admits as irreducible representations self-dual tensors, satisfying
\begin{equation}
\partial_{[\hat\mu} B_{\hat\nu\hat\rho]M} = \frac{1}{6}
\epsilon_{\hat\mu \hat\nu \hat\rho \hat\sigma \hat\lambda \hat\tau}
\partial^{\hat\sigma} B^{\hat\lambda \hat\tau}_M\,,\qquad \hat\mu ,\hat\nu,\cdots =
0,1,\dots,5 .\label{sd6}
\end{equation}
Since $N=2$ matter-coupled supergravity in six dimensions contains one antiself-dual and $n_T$
self-dual tensors in the vector representation of $SO(1,n_T)$, one can use the $SO(n_T)\subset
SO(1,n_T)$ global symmetry of the model to dimensionally reduce the theory on a circle down to
five dimensions \`a la Scherk--Schwarz \cite{Andrianopoli:2004xu}, with S-S phase $m^{MN}= -
m^{NM} \in SO(n_T)$:
\begin{equation}
B_{\hat\mu\hat\nu M} (x,y_5)=\Bigl({\exp}[m y_5]\Bigr)_M^{\phantom{M}N} \sum_n
B^{(n)}_{\hat\mu\hat\nu N}(x)\exp\left[\frac{{\rm i} n }{2\pi
R}y_5\right]\,.\label{ss}
\end{equation}
Applying \eq{ss} to the self-duality relation \eq{sd6}, we find
\begin{equation}
\partial_{[\mu} B_{\nu\rho]M} = \frac16
\epsilon_{\mu\nu\rho\sigma\lambda 5}\left(m_M{}^N B_N^{\sigma\lambda}
+ 2 F^{\lambda\sigma}_N\right) \,,\qquad \mu=0,1,\dots,4\label{sd5}
\end{equation}
where $F_{\lambda\sigma N}\equiv\partial_{[\sigma} B_{\lambda] 5 N}$.
Eq. \eq{sd5} expresses the self-duality obeyed by the tensors in five
dimensional supergravity. However, it also shows that the
field-strengths of the vectors $B_{\mu 5 N}$, that give mass to the
tensors $B_{\mu\nu M} $ via the anti-Higgs mechanism, are in fact the
Hodge-dual of the tensors $B_{\mu\nu M} $ themselves.
From our analysis applied to $N=2$ supergravity in five dimensions, we
find this to be a general fact, not necessarily related to theories admitting
a six dimensional uplift: in each case, the massive tensor fields
belong to short representations of supersymmetry, and the dynamical
interpretation of the mechanism giving mass to the tensors requires
the coupling of the massless tensors to gauge vectors which are the
Hodge-dual of the tensors themselves.
The paper is organized as follows: In section \ref{generalities} we
study the general FDA describing the coupling of two-index antisymmetric
tensor fields to non-abelian gauge vectors and show in detail, for the general case, how the anti-Higgs mechanism takes place. In section
\ref{susy}, we apply the formalism to the case of $N=2$ five
dimensional supergravity, using the geometric approach to find the
Lagrangian, supersymmetry transformations rules and constraints on
the scalar geometry and gauging. Our results are summarized in the
concluding section, while we left to the appendices some technical
details and the comparison of our notations with the ones of
\cite{Gunaydin:1983bi} and of \cite{Ceresole:2000jd}.
\section{A general bosonic theory with massive tensors and
non-abelian vectors}
\label{generalities}
In this section we are going to study the gauge structure of a general
theory with two-index antisymmetric tensor fields coupled to
gauge vectors. The discussion here will be general, with no need to
make reference to any particular dimension of space-time nor to any
possible supersymmetric extension of the model. Later, in section
\ref{susy}, we will consider the supersymmetrization of the model,
specifying the discussion to the case of $N=2$ five dimensional
supergravity coupled to vector, tensor and hyper multiplets. The corresponding four
dimensional case of $N=2$ supergravity coupled to vector-tensor
multiplets is under investigation, and is left to a future publication.
\subsection{FDA and the anti-Higgs mechanism}
\subsubsection{Abelian case}
The simplest case of a FDA including 1-form and 2-form potentials \footnote{0-forms will also
be included in section \ref{susy}, when considering a supersymmetric version of the theory}
is described by a set of abelian gauge vectors $A^M$ and of massless tensor two-forms $B_M$
($M=1,\dots n_T$.) interacting by a coupling $m^{MN}$. The field-strengths are:
\begin{eqnarray}
\left\{
\matrix{F^M&=& {\rm d}\hskip -1pt A^M + m^{MN} B_N\hfill\cr
H_M &=& {\rm d}\hskip -1pt B_M \hfill }\right.
\end{eqnarray}
and are invariant under the gauge transformations:
\begin{eqnarray}
\left\{
\matrix{\delta A^M &=& {\rm d}\hskip -1pt \Theta^M- m^{MN} \Lambda_N \hfill\cr
\delta B_M &=& {\rm d}\hskip -1pt \Lambda_M \hfill}\right.
\end{eqnarray}
with $\Theta^M$ parameters of
infinitesimal U(1) gauge transformations and $\Lambda_M$ one-form
parameters of infinitesimal tensor-gauge transformations of the
two-forms $B_M$. In this case the system undergoes the anti-Higgs
mechanism, and it is possible to fix the tensor-gauge so that:
\begin{eqnarray}
\left\{\matrix{A^M &\to & A'^M= - m^{MN} \bar\Lambda_N\hfill\cr
B_M & \to & B'_M = B_M + {\rm d}\hskip -1pt \bar \Lambda_M;} \right.
\end{eqnarray}
In this way the gauge vectors $A^M$ disappear from the spectrum
providing the degrees of freedom necessary for the tensors to
acquire a mass, since:
\begin{eqnarray}
\left\{\matrix{ F'^M &=& m^{MN} B_N\hfill\cr
H'_M & = & {\rm d}\hskip -1pt B_M.\hfill}\right.\label{antihiggs0}
\end{eqnarray}
\subsubsection{Coupling to a non-abelian algebra}
The model outlined above may be generalized by including the coupling of this
system to $n_V$ gauge vectors $A^\Lambda$ ($\Lambda = 1, \dots n_V$), with
gauge algebra $G_0$ (not necessarily semisimple), if the index $M$ of
the tensors $B_M$ and of the abelian vectors $A^M$ runs over a
representation of $G_0$. In this case the FDA becomes \footnote{We will generally assume, here and in the following, that the tensor mass-matrix $m^{MN}$ is invertible. In case it has some 0-eigenvalues, we will restrict to the submatrix with non-vanishing rank. This is not a restrictive assumption, because any tensor corresponding to a zero-eigenvalue of $m$ may be dualized to a gauge vector and so included in the set of $\{A^\Lambda\}$.}:
\begin{eqnarray}
\left\{\matrix{F^\Lambda &=& {\rm d}\hskip -1pt A^\Lambda + \frac{1}{2} f_{\Sigma\Gamma}{}^\Lambda A^\Sigma \wedge
A^\Gamma \hfill\cr
F^M &=& {\rm d}\hskip -1pt A^M - T_{\Lambda N}{}^M A^\Lambda \wedge A^N + m^{MN} B_N \hfill\cr
& \equiv & D A^M + m^{MN} B_N \hfill\cr
H_M &=& {\rm d}\hskip -1pt B_M + T_{\Lambda M}{}^N A^\Lambda \wedge B_N + d_{\Lambda NM} F^\Lambda\wedge
A^N \hfill\cr
&\equiv & D B_M + d_{\Lambda NM} F^\Lambda\wedge A^N \hfill}
\right. \label{couplings}
\end{eqnarray}
Here $f_{\Sigma\Gamma}{}^\Lambda$ are the structure constants of the gauge algebra
$G_0$ and $T_{\Lambda M}{}^N $, $d_{\Lambda MN}$ suitable couplings. The
closure of the FDA (${{\rm d}\hskip -1pt}^2 A^\Lambda = {{\rm d}\hskip -1pt}^2 A^M = {{\rm d}\hskip -1pt}^2 B_M = 0$)
gives the following constraints:
\begin{eqnarray}
f_{[\Lambda\Sigma}{}^\Delta f_{\Gamma]\Delta}{}^\Omega&=&0 \label{gaugealgebra1} \\
T_{[\Lambda | M}{}^P T_{\Sigma ] P}{}^N & = &\frac 12 f_{\Lambda \Sigma}{}^\Gamma T_{\Gamma M}{}^N\label{gaugealgebra2} \\
T_{\Lambda M}{}^N &=& -d_{\Lambda MP }m^{NP} =d_{\Lambda PM}m^{PN}\label{gaugealgebra3} \\
T_{\Lambda N}{}^M m^{NP} &=& -T_{\Lambda N}{}^P m^{MN}\label{gaugealgebra4} \\
T_{\Sigma M}{}^N d_{\Gamma PN}& +& T_{\Sigma P}{}^N d_{\Gamma NM} - f_{\Sigma\Gamma}{}^\Lambda
d_{\Lambda P M}{}^Q = 0. \label{gaugealgebra5}
\end{eqnarray}
Eq.s \eq{gaugealgebra1}, \eq{gaugealgebra2} show in particular that
the structure constants $f_{\Lambda\Sigma}{}^\Gamma$
do indeed close the algebra $G_0$ and that $T_{\Lambda M}{}^N$ are
generators of $G_0$ in the representation spanned by the tensor
fields. Eq.s \eq{gaugealgebra3} and \eq{gaugealgebra4} imply:
\begin{equation}
\matrix{m^{MN} &=& \mp m^{NM}\hfill\cr
d_{\Lambda MN} &=& \pm d_{\Lambda NM},\hfill}
\end{equation}
and \eq{gaugealgebra5} is a consistency condition that, when multiplied by
$m^{PQ}$, is equivalent to \eq{gaugealgebra2} (upon use of \eq{gaugealgebra4}).
When \eq{gaugealgebra1} - \eq{gaugealgebra5} are satisfied, the Bianchi identities read:
\begin{eqnarray}
\left\{\matrix{{\rm d}\hskip -1pt F^\Lambda + f_{\Sigma\Gamma}{}^\Lambda A^\Sigma \wedge F^\Gamma &=& 0 \hfill\cr
{\rm d}\hskip -1pt F^M - T_{\Lambda N}{}^M A^\Lambda \wedge F^N &=& m^{MN} H_N \hfill\cr
{\rm d}\hskip -1pt H_M + T_{\Lambda M}{}^N A^\Lambda \wedge H_N &=& d_{\Lambda MN} F^N \wedge F^\Lambda.
\hfill}\right. \label{bisimplified}
\end{eqnarray}
To see how the anti-Higgs mechanism works in this more general case,
let us give the gauge and tensor-gauge transformations of the fields
(including the non-abelian transformations belonging to $G_0$, with parameter
$\epsilon^\Lambda$). They become:
\begin{eqnarray}
\left\{\matrix{\delta A^\Lambda &=& {\rm d}\hskip -1pt \epsilon^\Lambda + f_{\Sigma \Gamma}{}^\Lambda A^\Sigma \epsilon^\Gamma
\equiv D \epsilon^\Lambda \hfill\cr
\delta A^M &=& {\rm d}\hskip -1pt \Theta^M - T_{\Lambda N}{}^M A^\Lambda \Theta^N + T_{\Lambda
N}{}^M A^N \epsilon^\Lambda - m^{MN} \Lambda_N \hfill\cr
& \equiv & D \Theta^M + T_{\Lambda N}{}^M A^N \epsilon^\Lambda - m^{MN} \Lambda_N \hfill\cr
\delta B_M &=& {\rm d}\hskip -1pt \Lambda_M + T_{\Lambda M}{}^N A^\Lambda \wedge \Lambda_N - d_{\Lambda MN}
F^\Lambda \Theta^N - T_{\Lambda M}{}^N B_N \epsilon^\Lambda \hfill\cr
& \equiv & D \Lambda_M - d_{\Lambda MN} A^\Lambda \wedge {\rm d}\hskip -1pt \Theta^N - T_{\Lambda M}{}^N
B_N \epsilon^\Lambda ,\hfill} \right. \label{gaugeinvar1}
\end{eqnarray}
with:
\begin{eqnarray}
\left\{\matrix{\delta F^\Lambda &=& f_{\Sigma \Gamma} {}^\Lambda F^\Sigma \epsilon^\Gamma \hfill\cr
\delta F^M &=& T_{\Lambda N}{}^M F^N \epsilon^\Lambda \hfill\cr
\delta H_M &=& - T_{\Lambda M}{}^N H_N \epsilon^\Lambda.\hfill}\right.
\end{eqnarray}
Fixing the gauge of the tensor-gauge transformation as:
\begin{eqnarray}
\left\{\matrix{ A^\Lambda &\to& A'^\Lambda =A^\Lambda \hfill\cr A^M &\to& A'^M= - m^{MN} \bar \Lambda_N \hfill\cr B_M &\to&
B'_M = B_M + D \bar \Lambda_M,\hfill}\right. \label{gaugefix1}
\end{eqnarray}
we find:
\begin{eqnarray}
\left\{\matrix{F'^\Lambda &=& F^\Lambda \hfill\cr
F'^M &=& m^{MN} B_N \hfill\cr
H'_M &=& D B_M \hfill}\right. \label{gaugefixed1}
\end{eqnarray}
When the tensor-gauge is fixed as in \eq{gaugefix1},\eq{gaugefixed1},
the vectors $A^M$ disappear from the spectrum while the tensors $B_M$
acquire a mass. As anticipated in the introduction, this is in
particular the starting point of the formulation adopted in the literature to describe
$D=5$, $N=2$ supergravity coupled to massive tensor multiplets
\cite{Gunaydin:1999zx,Ceresole:2000jd,Bergshoeff:2004kh}.
\bigskip
However, let us observe that in this more general case the abelian
gauge vectors $A^M$, providing the degrees of freedom needed to give a
mass to the tensors via the anti-Higgs mechanism, are charged under
the gauge algebra $G_0$. It is not possible to make the gauge
transformation of the vectors $A^M$ compatible with that of the $A^\Lambda$
unless all together the vectors $\{A^\Lambda , A^M\} \equiv A^{\tilde I} $
form the co-adjoint representation of some larger non semisimple gauge
algebra $G\supset G_0$.
The relations so far obtained may then be written with the collective
index ${\tilde I} =(\Lambda , M)$, in terms of structure constants
$f_{{\tilde J}{\tilde K}}{}^{\tilde I}$ restricted to the following
non vanishing entries:
\begin{equation}
f_{{\tilde J}{\tilde K}}{}^{\tilde I}= ( f_{\Lambda\Sigma}{}^\Gamma , f_{\Lambda
M}{}^N=-T_{\Lambda M}{}^N)\,,
\label{algebrasimplified}
\end{equation}
and of the couplings:
\begin{eqnarray}
m^{\tilde I M}\equiv \delta^{\tilde I}_N m^{NM}\,, \qquad d_{\tilde I
\tilde J M} \equiv \delta_{\tilde I}^\Lambda \delta_{\tilde J}^N d_{\Lambda
NM}.\label{restrictions}
\end{eqnarray}
In terms of the tilded quantities the FDA \eq{couplings} reads:
\begin{eqnarray}
\left\{\matrix{F^{\tilde I} &\equiv& {\rm d}\hskip -1pt A^{\tilde I} + \frac{1}{2}
f_{{\tilde J}{\tilde K}}{}^{\tilde I} A^{\tilde J} \wedge A^{\tilde
K} + m^{{\tilde I} M} B_M \hfill\cr
H_M &\equiv& {\rm d}\hskip -1pt B_M + T_{{\tilde I} M}{}^N A^{\tilde I} B_N +
d_{{\tilde I}{\tilde J} M} F^{\tilde I} \wedge A^{\tilde
J}}\right. \label{fda}
\end{eqnarray}
with Bianchi identities:
\begin{eqnarray}
\left\{
\matrix{
{\rm d}\hskip -1pt F^{\tilde I} + \left( f_{{\tilde J}{\tilde K}}{}^{\tilde I} +
m^{{\tilde I} M} d_{{\tilde K}{\tilde J} M} \right)A^{\tilde J} F^{\tilde
K} &= \; m^{{\tilde I} M} H_M \hfill\cr
{\rm d}\hskip -1pt H_M +\left( T_{{\tilde I} M}{}^N + m^{{\tilde J} N} d_{{\tilde
J}{\tilde I} M} \right)A^{\tilde I} H_N & = \; d_{{\tilde I}{\tilde
J} M} F^{\tilde I} F^{\tilde J} \hfill}\right. , \label{BI}
\end{eqnarray}
provided the following relations, equivalent to \eq{gaugealgebra1} - \eq{gaugealgebra5}, hold:
\begin{eqnarray}
\matrix{f_{[{\tilde I}{\tilde J}}{}^{\tilde L} f_{{\tilde K}]{\tilde
L}}{}^{\tilde M} &=& 0 \hfill\cr
\left[ T_{\tilde I} , T_{\tilde J} \right] &=& f_{{\tilde I}{\tilde
J}}{}^{\tilde K} T_{\tilde K} \hfill\cr
T_{{\tilde I} M}{}^{(N} m^{{\tilde I} |P)} &=& 0\hfill\cr
m^{{\tilde I} N} T_{{\tilde J} N}{}^M &=& f_{{\tilde J}{\tilde
K}}{}^{\tilde I} m^{{\tilde K} M} \hfill\cr
T_{{\tilde I} M}{}^N &=& d_{{\tilde I}{\tilde J} M} m^{{\tilde J} N}
\hfill\cr
T_{[{\tilde I} | M}{}^N d_{{\tilde K} | {\tilde J}] N} &-& (f_{ [
{\tilde I} | {\tilde K}}{}^{\tilde L} + m^{{\tilde L} N}
d_{{\tilde K} [ {\tilde I} N} ) d_{{\tilde L} | {\tilde J}] M} -
\frac{1}{2} f_{{\tilde I}{\tilde J}}{}^{\tilde L} d_{{\tilde K}{\tilde
L} M} = 0.}
\label{closure}
\end{eqnarray}
Subject to the constraints \eq{closure}, the system is covariant under
the gauge transformations:
\begin{eqnarray}
\left\{\matrix{\delta A^{\tilde I} &=& {\rm d}\hskip -1pt \epsilon^{\tilde I} + f_{{\tilde
J}{\tilde K}}{}^{\tilde I} A^{\tilde J} \epsilon^{\tilde K} -
m^{{\tilde I} M} \Lambda_M \hfill\cr
\delta B_M &=& {\rm d}\hskip -1pt \Lambda_M + T_{{\tilde I} M}{}^N A^{\tilde I} \Lambda_N -
d_{{\tilde I}{\tilde J} M} F^{\tilde I} \epsilon^{\tilde J} -
T_{{\tilde I} M}{}^N \epsilon^{\tilde I} B_N}\right.\label{gaugefin}
\end{eqnarray}
implying the gauge transformation of the field strengths:
\begin{eqnarray}
\left\{\matrix{\delta F^{\tilde I} &=& - \left( f_{{\tilde J}{\tilde
K}}{}^{\tilde I} + m^{{\tilde I} M} d_{{\tilde K}{\tilde J} M}
\right) \epsilon^{\tilde J} F^{\tilde K} \hfill\cr
\delta H_M &=& - \left( T_{{\tilde I} M}{}^N + m^{{\tilde J} N}
d_{{\tilde J}{\tilde I} M} \right) \epsilon^{\tilde I} H_N}\right.
\label{labfalfa}
\end{eqnarray}
\subsubsection{A general FDA}
We now observe that the restrictions on the couplings
\eq{algebrasimplified} and \eq{restrictions} have been set to exactly
reproduce eqs. \eq{couplings} while exhibiting the fact that
$A^{\tilde I}$ collectively belong to the adjoint of some algebra
$G\supset G_0$. Actually eq.s \eq{couplings} and \eq{closure} allow in
fact a more general gauge structure than the one declared in
\eq{algebrasimplified}, \eq{restrictions}. Let $T_{\tilde I}\in {\rm
Adj}\,G$ be the gauge generators dual to $A^{\tilde I}$. For the
case of \eq{algebrasimplified}, $G$ has the semisimple structure
$G=G_0\ltimes \mathbb{R}^{n_T}$, and the generators $T_\Lambda \in G_0$ may be
realized in a block-diagonal way (with entries $T_{\Lambda \Sigma}{}^\Gamma
=f_{\Lambda\Sigma}{}^\Gamma$, $T_{\Lambda M}{}^N = - f_{\Lambda M}{}^N$) while the $T_M$ are
off-diagonal (with entries $T_{M \Lambda}{}^N =f_{\Lambda M}{}^N$). However, any
gauge algebra $G$ with structure constants $f_{\tilde I \tilde
J}{}^{\tilde K}$ may in principle be considered, provided it
satisfies the constraints \eq{closure}. In the general case, to match
\eq{closure} one must also relax the restrictions on the couplings
\eq{algebrasimplified}, \eq{restrictions}, and allow for more general
$f_{\tilde I \tilde J}{}^{\tilde K}$ and $d_{{\tilde I}{\tilde J} M}$. This includes
in particular the case
\begin{equation}
f_{\Lambda\Sigma}{}^M \neq 0\,, \qquad d_{\Lambda\Sigma M} \neq 0 \label{flsm}
\end{equation}
which was considered in \cite{Bergshoeff:2004kh} and \cite{deWit:2004nw}. In this case, $G$
cannot be semisimple, and $G_0$ is not a subalgebra of $G$ \footnote{We acknowledge an enlightening
discussion
with Maria~A.~Lled\'o on this point.}. This implies that the vectors
$A^M$ do not decouple anymore at the level of gauge algebra, and this,
at first sight, would be an obstruction to implement the anti-Higgs
mechanism. However, this apparent obstruction may be simply overcome in the FDA framework, due to the freedom of
redefining the tensor fields as \cite{Dall'Agata:2005mj}:
\begin{equation}
B_M \to B_M + k_{{\tilde I}{\tilde J} M} A^{\tilde I} \wedge
A^{\tilde J} ,\label{red}
\end{equation}
for any $k_{{\tilde I}{\tilde J} M} $ antisymmetric in ${\tilde
I},{\tilde J}$. It is then possible to implement the anti-Higgs
mechanism with the tensor-gauge fixing (which includes a field
redefinition as in \eq{red}):
\begin{eqnarray}
\left\{\matrix{ A^\Lambda &\to& A'^\Lambda = A^\Lambda \hfill\cr
A^M &\to& A'^M = - m^{ MN} \bar \Lambda_N \hfill\cr
B_M &\to& B'_M = B_M - \frac 12 d_{\Lambda\Sigma M}A^\Lambda \wedge A^\Sigma + D \bar
\Lambda_M\hfill}\right. \label{gaugefix}
\end{eqnarray}
This still gives:
\begin{eqnarray}
\left\{\matrix{F'^\Lambda &=& F^\Lambda \hfill\cr
F'^M &=& m^{MN} B_N \hfill\cr
H'_M &=& D B_M \hfill}\right. \label{gaugefixed}
\end{eqnarray}
provided that:
\begin{equation}
m^{MN} d_{[\Lambda\Sigma] N} = f_{\Lambda\Sigma}{}^M.
\end{equation}
With this observation, we may now analyze in full generality which
non trivial structure constants may be turned on in \eq{fda} in a way
compatible with the anti-Higgs mechanism.
First of all, it is immediate to see that if:
\begin{equation}
f_{\tilde I M}{}^\Sigma \neq 0 \,,
\end{equation}
it is impossible to implement the anti-Higgs
mechanism, because they introduce a coupling to the gauge vectors
$A^M$ in the field-strengths $F^\Lambda$ which is not possible to
reabsorb by any field-redefinition.
Considering then the case:
\begin{equation}
f_{MN}{}^P \neq 0 \,, \qquad d_{MNP}\neq 0 \,.
\end{equation}
we see that $f_{MN}{}^P$ would introduce a non-abelian interactions
among the vectors $A^M$ and in particular, for the case $\Lambda =0$,
this would imply that the $A^M$ close a non-abelian gauge
algebra. This case may be treated in a way quite similar to the case
\eq{flsm}, since again we may use the freedom in \eq{red} to absorb
the non-abelian contribution to $F^M$ in a redefinition of $B_M$. The
anti-Higgs mechanism may then be implemented via the tensor-gauge
fixing:
\begin{eqnarray}
\left\{\matrix{
A^\Lambda &\to& A'^\Lambda = A^\Lambda \hfill\cr
A^M &\to& A'^M = - m^{ MN} \bar \Lambda_N \hfill\cr
B_M &\to& B'_M = B_M - \frac 12 d_{NP M}A^N \wedge A^P + D \bar
\Lambda_M\hfill}\right. \label{gaugefix2}
\end{eqnarray}
giving, as before:
\begin{eqnarray}
\left\{\matrix{F'^\Lambda &=& F^\Lambda \hfill\cr
F'^M &=& m^{MN} B_N \hfill\cr
H'_M &=& D B_M \hfill}\right. \label{gaugefixed2}
\end{eqnarray}
provided that:
\begin{equation}
m^{MQ} d_{[NP] Q} = f_{NP}{}^M. \label{dmnp}
\end{equation}
This shows that also non-abelian gauge vectors $A^M$ may be
considered, and still may decouple from
the gauge-fixed theory by giving mass to the tensors $B_M$. For this
case, however, the constraints \eq{closure}, together with \eq{dmnp},
give the following conditions on the couplings:
\begin{equation}
\left\{\matrix{d_{MNP} &=& d_{[MNP]} \hfill\cr m^{MN}&=&+m^{NM}\hfill}\right.
.\label{msymm}
\end{equation}
As we are going to discuss in the next section, for the $D=5$, $N=2$
theory the matrix $m^{MN}$ has to be antisymmetric, and this then
implies, for this theory, $f_{MN}{}^P =0$. We conclude that even if
the algebra \eq{fda} can have non trivial extensions with new
couplings, this is not the case for the $D=5$, $N=2$ theory we shall
be concerned with in section \ref{susy}, so that the couplings
$f_{MN}{}^P$ and $d_{MNP}$ will be set to zero.
\bigskip
\subsection{General properties of the FDA}
\label{fdageneralities}
A further observation concerns eq.s \eq{BI} and \eq{labfalfa}. In
these equations, as in all the relations involving the physical field
strengths $F^{\tilde I}$ and $H_M$, the following objects appear:
\begin{eqnarray}
\matrix{\hat{f}_{{\tilde J}{\tilde K}}{}^{\tilde I} &\equiv&
f_{{\tilde J}{\tilde K}}{}^{\tilde I} + m^{{\tilde I} M} d_{{\tilde
K}{\tilde J} M} \, ; \hfill\cr
\hat{T}_{{\tilde I} M}{}^N &\equiv& T_{{\tilde I} M}{}^N + m^{{\tilde
J} N} d_{{\tilde J}{\tilde I} M}= 2 d_{({\tilde I}{\tilde J})M}
m^{{\tilde J} N}. \hfill }\label{generalgenerat}
\end{eqnarray}
The generalized couplings $\hat{f}_{{\tilde J}{\tilde K}}{}^{\tilde
I}$ belong to a representation of the gauge algebra $G$ which is
not the adjoint, since they are not antisymmetric in the lower
indices. In particular we find :
\begin{eqnarray}
\matrix{\hat f_{{\tilde I}{\tilde J}}{}^{\tilde K} m^{{\tilde J} M}
&=& \hat T_{{\tilde I} N}{}^Mm^{{\tilde J} N} \hfill\cr
\hat f_{{\tilde I}{\tilde J}}{}^{\tilde K} m^{{\tilde I} M} &=&
0.\hfill}
\end{eqnarray}
However, the $\hat{f}_{{\tilde J}{\tilde K}}{}^{\tilde I}$ and $\hat T_{{\tilde I} N}{}^M$ can be understood as representations of generators $\hat f_{\tilde I}$ and $\hat
T_{\tilde I}$ that still generate the gauge algebra $G$. Indeed the following relations hold (subject to the constraints
\eq{closure}):
\begin{eqnarray}
\matrix{\left[ \hat{f}_{\tilde I} , \hat{f}_{\tilde J} \right] &=& -
f_{{\tilde I}{\tilde J}}{}^{\tilde K} \hat{f}_{\tilde K},\hfill\cr
\left[ \hat{T}_{\tilde I} , \hat{T}_{\tilde J} \right] &=& f_{{\tilde
I}{\tilde J}}{}^{\tilde K} \hat{T}_{\tilde K}.\hfill}
\label{bigalgebra}
\end{eqnarray}
The generalized couplings $\hat f$ and $\hat T$ express the
deformation of the gauge structure due to the presence of the tensor
fields. In particular, only the structure constants of $G_0$ are unchanged, corresponding to the fact that this is
the algebra realized exactly in the interacting theory
\eq{fda} after the anti-Higgs mechanism has taken place.
The rest of the gauge algebra $G$ is instead spontaneously
broken by the anti-Higgs mechanism (which requires, if $f_{\Lambda\Sigma}{}^M\neq 0$, also a tensor redefinition, as explained in \eq{gaugefix}).
However, the entire algebra $G$
is still realized, even if in a more subtle way, as eq.s
\eq{bigalgebra} show.
From a
physical point of view, this is expected by a counting of degrees
of freedom, since the degrees of freedom required to make a
two-index tensor massive are the ones of a gauge vector connection
\footnote{Indeed, the on-shell degrees of freedom of a massless
(2-index) tensor and of a vector in $D$ dimensions are
$(D-2)(D-3)/2$ and $(D-2)$ respectively, while the ones of a massive
tensor are $(D-1)(D-2)/2= (D-2)(D-3)/2 + (D-2)$.}, so that also the vectors $A^M$, besides the $A^\Lambda$, are expected to be massless gauge vectors.
This algebra indeed closes provided the Jacobi
identities $f_{[{\tilde I}{\tilde J}}{}^{\tilde L} f_{{\tilde
K}]{\tilde L} }{}^{\tilde M} =0 $ are satisfied. We find indeed, using \eq{bigalgebra}:
\begin{eqnarray}
\matrix{ \left[ \left[\hat f_{[{\tilde I}},\hat f_{\tilde
J}\right],\hat f_{{\tilde K} ]}\right]_{\tilde L}{}^{\tilde
P}&=&-f_{[{\tilde I}{\tilde J}}{}^{\tilde N} f_{{\tilde K}]{\tilde
N} }{}^{\tilde M} \hat f_{{\tilde M} {\tilde L} }{}^{\tilde P}
=0 \hfill\cr
\left[ \left[\hat T_{[{\tilde I}},\hat T_{\tilde J}\right],\hat
T_{{\tilde K} ]}\right]_M{}^N&=&-f_{[{\tilde I}{\tilde
J}}{}^{\tilde M} f_{{\tilde K}]{\tilde M}}{}^{\tilde L} \hat
T_{{\tilde L}M}{}^N =0\hfill} \label{closed}
\end{eqnarray}
The
hatted generators $\hat f$, $\hat T$ play the role of {\em physical couplings} when the gauge structure is extended to include charged tensors. They
have then to be considered as the appropriate generators of the free
differential structure. It may be useful to recast the theory in terms of all the couplings
appearing in the Bianchi identities \eq{BI}, that is the hatted
generators and the symmetric part $d_{({\tilde I}{\tilde J})M}$ of the
Chern--Simons-like coupling $d_{{\tilde I}{\tilde J} M}$. This is done
by the field redefinition:
\begin{equation}
B_M \to \tilde B_M = B_M + \frac 12 d_{[{\tilde I}{\tilde J}]M}
A^{\tilde I}\wedge A^{\tilde J} \label{bridef}
\end{equation}
so that the FDA takes the form:
\begin{eqnarray}
\left\{\matrix{F^{\tilde I} &\equiv& {\rm d}\hskip -1pt A^{\tilde I} + \frac{1}{2} \hat
f_{{\tilde J}{\tilde K}}{}^{\tilde I} A^{\tilde J} \wedge A^{\tilde
K} + m^{{\tilde I} M}\tilde B_M \hfill\cr
H_M &\equiv& {\rm d}\hskip -1pt \tilde B_M + \frac 12 \hat T_{{\tilde I} M}{}^N
A^{\tilde I} \tilde B_N + d_{({\tilde I}{\tilde J}) M} F^{\tilde I}
\wedge A^{\tilde J} + \mathcal{K}_{M {\tilde I}{\tilde J}{\tilde
K}}A^{\tilde I}\wedge A^{\tilde J} \wedge A^{\tilde
K}\hfill}\right. \label{fda2}
\end{eqnarray}
and the constraints \eq{closure} in the new formulation read, after
introducing $\tilde f_{{\tilde I}{\tilde J}}{}^{\tilde K} \equiv \hat
f_{[{\tilde I}{\tilde J}]}{}^{\tilde K}$:
\begin{eqnarray}
\matrix{\tilde f_{[{\tilde I}{\tilde J}}{}^{\tilde M} \tilde
f_{{\tilde K}]{\tilde M}}{}^{\tilde L} &=& 2 m^{{\tilde L} M}
\mathcal{K}_{M[{\tilde I}{\tilde J}{\tilde K} ]} \hfill\cr
\hat T_{[{\tilde I} M}{}^N \hat T_{{\tilde J} ]N}{}^P &=& \tilde
f_{{\tilde I}{\tilde J}}{}^{\tilde K} \hat T_{\tilde K} + 12
\mathcal{K}_{M{\tilde I}{\tilde J}{\tilde K}}m^{{\tilde K} P} \hfill\cr
\hat T_{{\tilde I} M}{}^{N} m^{{\tilde I} P} &=& 0\hfill\cr
\frac{1}{2} m^{{\tilde I} N} \hat T_{{\tilde J} N}{}^M &=&\tilde
f_{{\tilde J}{\tilde K}}{}^{\tilde I} m^{{\tilde K} M} \hfill\cr
\hat T_{{\tilde I} M}{}^N &=& 2 d_{({\tilde I}{\tilde J}) M}
m^{{\tilde J} N} \hfill\cr
\hat T_{[{\tilde I} | M}{}^N d_{( {\tilde J}]{\tilde K} )N} &-& 2\hat
f_{ [ {\tilde I} | {\tilde K}}{}^{\tilde L} d_{ ({\tilde
J}]{\tilde L}) M} -\tilde f_{{\tilde I}{\tilde J}}{}^{\tilde L}
d_{({\tilde K}{\tilde L}) M} = -6 \mathcal{K}_{M{\tilde I}{\tilde
J}{\tilde K}} \hfill\cr
\mathcal{K}_{N [{\tilde J}{\tilde K}{\tilde L}} \hat T_{{\tilde
I}]|M}{}^N&-&3 \mathcal{K}_{M {\tilde P} [{\tilde I}{\tilde
J}}\tilde f_{{\tilde K}{\tilde L}]}{}^{\tilde P} =0.\hfill}
\label{closure2}
\end{eqnarray}
In eq.s \eq{fda2} and \eq{closure2} we have introduced the definition:
\begin{equation}
\mathcal{K}_{M{\tilde I}{\tilde J}{\tilde K}}= \frac 12 \hat
f_{[{\tilde I}{\tilde J}}{}^{\tilde{L}} d_{({\tilde K}] {\tilde{L}})
M} +\frac{2}{3} d_{({\tilde{L}}[{\tilde J})M} \hat f_{{\tilde
I}]{\tilde K}}{}^{\tilde{L}}. \label{k}
\end{equation}
that could also be found by directly studying the closure of the FDA
\eq{fda2} without referring to its derivation from \eq{fda}.
Eq. \eq{fda2}, which is expressed in terms of the physical couplings
only, is completely equivalent to \eq{fda}. This is in fact the
formulation used in
\cite{deWit:2004nw}, for the study of $N=8$ supergravity in 5
dimensions. However, as eq.s \eq{closure2} shows, in the formulation
\eq{fda2} the gauge structure is not completely manifest, because for
the ``structure constants'' $\tilde f_{\tilde I\tilde J}{}^{ \tilde
K}$ the Jacobi identities fail to close.
Equation \eq{fda} (or, equivalently, \eq{fda2}) is the most general FDA
involving vectors and 2-index antisymmetric
tensors. Any other possible deformation of \eq{fda} is indeed trivial
(unless the system is also coupled to higher order forms) as we will
show in detail in Appendix \ref{deformedfda}.
\bigskip
As a final remark, let us observe that, given the definitions
\eq{fda}, the FDA still enjoys a scale invariance under the transformation, with parameter $\alpha$:
\begin{eqnarray}
\left\{\matrix{m^{MN} &\to& \alpha \, m^{MN} \hfill\cr
B_M &\to& \frac{1}{\alpha} B_M \hfill\cr
d_{{\tilde I}{\tilde J} M} &\to& \frac{1}{\alpha} d_{{\tilde I}{\tilde J}
M} \hfill}\right. \label{scaleinv}
\end{eqnarray}
As we will see in the following, for the $N=2$ theory in five
dimensions this freedom corresponds to the possibility of choosing an
overall normalization for the tensor contributions to the
Chern--Simons Lagrangian.
\section{$D=5$, $N=2$ supergravity revisited}
\label{susy}
\subsection{Generalities and differences from previous approaches}
In this section we are going to apply the general
analysis of section \ref{generalities} to the case of $N=2$
supergravity theory in five dimensions coupled to vector- and
tensor-multiplets.
The field content of the theory, in the absence of couplings, is
\begin{itemize}
\item the gravity supermultiplet
$$(V_\mu^a, \psi_\mu^A, A^0_\mu)\,,\qquad a =0,1,\dots 4\,, \quad\mu =0,1,\dots 4\,, \quad
A=1,2$$
where $V^a_\mu$ is the space-time vielbein (with $a$ tangent-space indices and $\mu$ world-indices), $\psi^A_\mu$ the gravitino, with R-symmetry index in the fundamental representation of $Sp(2,\mathbb{R})$, and $A^0$ the graviphoton;
\item $n_V$ gauge multiplets
$$(A^i_\mu, \lambda^{iA},\varphi^i)\,,\qquad i=1,\dots n_V\,$$
with $ \varphi^i,\lambda^{iA}$ the scalar partners of the gauge vectors $A^i$ and the $Sp(2,\mathbb{R})$-valued gaugini respectively.
Since the gauge vectors mix in the interacting theory, in the following we will introduce the index $\Lambda =(0,i)=0,1,\dots n_V$ running
over all the gauge-vector indices, that is: $A^\Lambda\equiv (A^0,A^i)$;
\item $n_T$ massless tensor multiplets
$$(B_{M|\mu\nu}, \lambda^{MA},\varphi^M)\,,\qquad i=1,\dots n_T\,$$
with $ \varphi^M,\lambda^{MA}$ the scalar and spinor partners respectively
of the tensors $B_M$;
\item $n_H$ hypermultiplets
$$(q^u, \zeta^{\alpha})\,, \qquad u=(A\alpha )=1,\dots 4n_H\,; \quad \alpha
=1,\dots 2n_H$$
where the scalars $q^u$ span a quaternionic manifold of quaternionic
dimension $n_H$ and their spin-1/2 partners $\zeta^\alpha$ are labeled
with an index in the fundamental representation of $Sp(2n_H,\mathbb{R})$.
\end{itemize}
Before entering in the explicit construction of the theory, let us
emphasize the differences of our approach with respect to the existing
literature on $D=5$, $N=2$ supergravity. Inspired by the analysis of
the previous section, we are interested in exploiting all the rich
gauge structure underlying the bosonic sector of the model, so we want
to retrieve and possibly to extend the results in the existing
literature by starting with massless tensors and letting them take
mass via the anti-Higgs mechanism. Let us discuss this point in some
more detail than what has already done in the introduction.
While the anti--Higgs mechanism is very well understood at the bosonic
level, to implement it within a supersymmetric theory is a non trivial
task. This is due to the fact that the supersymmetry constraints
require the vectors $A^M$ giving mass to the tensors (in the notations
of section \ref{generalities}) to be related to the tensors themselves
in a non local way, involving Hodge-duality. This relation is codified
in the so-called {\em ``self-duality-in-odd-dimensions"} condition to
which all the tensor fields in odd-dimensional supergravity theories
have to comply \cite{Townsend:1983xs}:
\begin{equation}
m^{MN}H_{N|abc} \propto \epsilon_{abcde} F^{M|de}.\label{selfodd}
\end{equation}
In particular, for the five dimensional case the tensors are further required to be complex.
In fact, in the approach currently adopted in the literature \cite{Gunaydin:1983rk,Gunaydin:1983bi,Gunaydin:1984pf,Sierra:1985ax,Lukas:1998yy,Gunaydin:1999zx,Ceresole:2000jd,Bergshoeff:2004kh,Gunaydin:2005bf}, the tensors $B_M$
in the tensor multiplets are taken to be massive (and constrained to satisfy \eq{selfodd}) from the very
beginning, without any tensor-gauge freedom.
Naively, to implement the anti-Higgs mechanism at the supersymmetric level one could think of directly supersymmetrizing the FDA \eq{fda}, and try to give mass to the whole tensor multiplets
by coupling them to $n_T$ extra abelian vector multiplets added to the theory:
\begin{equation}
(A^M_\mu, \chi^{M A},\phi^M),\label{false}
\end{equation}
where the vectors $A^M$ and the
tensors $B_M$ admit the couplings and gauge invariance as in \eq{fda}
and \eq{gaugefin}. If this would be the case, in the interacting theory the fields in the extra vector multiplets would
couple to the tensor multiplets and one would
end up with $n_T$ {\em long} massive multiplets.
We found, however, from explicit calculation that this is not the case, since supersymmetry transformations never relate the tensors $B_M$ to the spinors $\chi^{MA}$
nor to the scalars $\phi^M$ in \eq{false}. Then the only way compatible with supersymmetry to couple $N=2$
supergravity with $n_T$ massive tensors involves {\em short BPS}
tensor multiplets
$$(B_{M|\mu\nu}, \lambda^{MA},\varphi^M)$$
where the massive tensors $B_M$ (that are complex because of CPT
invariance of the BPS multiplet) have to satisfy \eq{selfodd} (see
eq. \eq{selfmass}).
This is evident for the models having a six dimensional uplift, as
discussed in the introduction, since for these cases the mass of the
tensors is the BPS central charge gauged by the graviphoton $g_{\mu
5}$.
Then, in order to understand the $N=2$ supergravity
theory in five dimensions coupled to tensor and vector multiplets as a supersymmetrization of
the FDA discussed in section \ref{generalities}, we will adopt the following strategy: we
start from the massless theory with field content as outlined at the beginning of this
section, but we also introduce $n_T$ extra auxiliary abelian vectors $A^M$ coupled to the
system. The closure of the supersymmetry algebra will then fix their field-strengths,
on-shell, to be the Hodge-dual of the field-strengths of the tensors $B_M$. When the theory
also includes non-abelian gauge multiplets gauging some algebra $G_0$, and the tensor
multiplets are charged under some representation of $G_0$, then the spin-one part $(B_M,
A^M,A^\Lambda)$ of the bosonic sector is coupled as in \eq{fda}. In this case the closure of the
supersymmetry algebra also involves the non abelian field-strengths and give the set of
constraints \eq{bi1} - \eq{bilast} below.
According to the discussion in section \ref{generalities}, to simplify the notation we will generally use the index ${\tilde I}
= (\Lambda,M)$, valued in a representation of a group $G\supset G_0$ in the notations of section \ref{generalities}, that runs over all the vectors (including the auxiliary
ones) $$A^{{\tilde I}}\equiv (A^\Lambda,A^M)$$ and over the scalar sections
$$X^{{\tilde I}}(\varphi^x)\equiv (X^\Lambda,X^M)$$
($G$-valued functions of the scalar fields $\varphi^x \equiv (\varphi^i,\varphi^M)$) which appear in the supersymmetry transformations of the vector and tensor fields.
The world-index $x=1,\dots ,n_V +n_T$ will collectively enumerate the
scalar fields $\varphi^x$ and the spinors $\lambda^{x A}$ both in the
tensor and vector multiplets.
\bigskip
With respect to the analysis of
\cite{Gunaydin:1999zx},
our discussion will be a bit more general as we will include a
non-zero Chern--Simons coupling $d_{\Lambda\Sigma M}$, as in \cite{Bergshoeff:2004kh}.
Another important point, not considered so far in this general
context, concerns the couplings $m^{MN}$. The closure of the FDA \eq{fda}
demands (recalling \eq{closure}) the generators $T_{\tilde{I}
M}{}^N$ to be related to the couplings $d_{\tilde{I}\tilde{J} M}$
and to the structure constants $f_{\tilde{J}\tilde{K}}{}^{\tilde{I}}$
respectively by
\begin{equation}
T_{\tilde{I} N}{}^M = m^{\tilde{J}M} \, d_{\tilde{I}\tilde{J}N},
\qquad m^{\tilde{I}N} T_{\tilde{J} N}{}^M =
f_{\tilde{J}\tilde{K}}{}^{\tilde{I}} m^{\tilde{K}M}.
\end{equation}
Setting in the second relation $\tilde{I}=P$ and $\tilde{J}=\Lambda$,
it is immediate to obtain the following:
\begin{equation}
d_{\Lambda MN} m^{MQ}\, m^{NP} = - d_{\Lambda NM} \, m^{PN}m^{MQ}. \label{cs}
\end{equation}
Eq. \eq{cs} in principle admits two different solutions: either $d_{\Lambda
MN}$ is symmetric and $m^{MN}$ antisymmetric or the opposite.
But since these couplings enter the Lagrangian of five dimensional
supergravity respectively in the kinetic term for the tensors $m^{MN}
B_M {\rm d}\hskip -1pt B_N$ (which for $m^{MN}$ symmetric is a total derivative) and in the Chern--Simons term $d_{\tilde I MN} A^{\tilde I} F^M F^N$ (which is zero if $d_{\tilde I MN}$ is antisymmetric in $M$ and $N$),
we are forced to consider only the former solution\footnote{This is
not necessarily true for other cases, like the four dimensional
theories, where the equation (\ref{cs}) also seems to allow the
alternative solution
\begin{equation}
d_{{\tilde I} MN}= - d_{{\tilde I} NM}; \quad m^{MN} = m^{NM}.
\end{equation}}.
Furthermore it should be noted that this same choice
forbids the presence of $d_{MNP}$ couplings, due to eq. \eq{msymm}.
We want to stress, however, that this constraint leaves the freedom for
the tensor mass-matrix $m=-m^T$ to have $n_T$ different eigenvalues
$\pm {\rm i} m_\ell$ ($\ell=1,\dots n_T/2$)\footnote{For $n_T$ odd there
is one extra zero-eigenvalue.
However, this case is excluded when
the theory is embedded in $N=2$ supergravity, since in this case, as
we already anticipated, the closure of the superalgebra requires a
self-duality condition \cite{Townsend:1983xs} which needs an even
number of tensors.}.
As anticipated in the introduction, this is the case, for example, of the five dimensional theory
obtained by Scherk--Schwarz generalized dimensional reduction
\cite{Scherk:1979zr,Cremmer:1979uq} from the $(2,0)$ theory in six
dimensions \cite{Andrianopoli:2004xu}. In this theory the mass matrix $m^{MN} $ is in fact
the S-S phase, in the Cartan subalgebra of the global symmetry $SO(n_T)
\subset SO(1,n_T)$, which is the isometry group of the scalar sector
of the tensor multiplets in the $D=6$ parent theory. The five
dimensional theory one obtains in this way is a gauged theory with
flat group given by the semidirect product $U(1) \ltimes R_V$, where
$R_V$ is an $n_V$-dimensional representation of $\mathrm{SO}(1,n_T)$, and the
$U(1)$ group is gauged by the vector coming from the metric in six
dimensions. As remarked in \cite{Andrianopoli:2004xu}, such a
situation was not considered in previous classifications.
On the other hand, if we take all the eigenvalues of the matrix
$m^{MN}$ equal, which is the case generally considered in the
literature
\cite{Gunaydin:1983rk,Gunaydin:1983bi,Gunaydin:1984pf,Sierra:1985ax,Lukas:1998yy,Gunaydin:1999zx,Ceresole:2000jd,Bergshoeff:2004kh,Gunaydin:2005bf},
then $m^{MN}$ may be set in the form $m^{MN}=m\, \Omega^{MN}$ where
$m$ is one constant real parameter and $\Omega$ the symplectic
metric. In this case the constraints \eq{cs} require the generators $T_{\Lambda
I}^{\phantom{ \Lambda I}J}$ to belong to a symplectic representation of the
gauge group $T_\Lambda^T \cdot \Omega + \Omega \cdot T_\Lambda =0$.
\subsection{The construction of the theory}
Since our approach involves some generalizations with respect to those
in the existing literature, as discussed above, we have rederived
right from the beginning the theory in its full generality. We have
used the superspace geometric approach as far as the solution of
Bianchi identities is concerned (from which the supersymmetry
transformation laws of the fields follow) and the superspace
rheonomic Lagrangian for the derivation of the Lagrangian on
space-time. We also tried to use, as much as possible, the notations
existing in the previous literature; however in some cases we found
useful to adopt different normalizations for the fields and couplings
with respect to the seminal papers on the subject
\cite{Gunaydin:1983bi,Gunaydin:1999zx,Ceresole:2000jd}.
A dictionary between our normalizations and those adopted in the
previous papers is given in appendix \ref{rosetta}.
\bigskip
Our starting point, for the construction of the theory, is the
generalization of the bosonic FDA of section \ref{generalities} to a
super-FDA in superspace. Consequently, we introduce the supergravity
one-forms $\Omega^a{}_b$, $V^a$ and $\Psi^A$ denoting respectively
the spin-connection, the vielbein and the gravitino in superspace
($V^a$ and $\Psi^A$ spanning a basis on
superspace), together with their ``supercurvatures'' two-forms
$\mathcal{R}^a{}_b$, $\mathcal{T}^a$ and $\rho^A$. We further introduce
zero-forms for the scalars $\varphi^x, q^u$ and spin 1/2 fields
$\lambda^{xA},\zeta^\alpha$ and their curvatures (covariant
derivatives) .
The $D=5$, $N=2$ super-FDA is:
\begin{eqnarray}
\mathcal{R}^a{}_b &=& {\rm d}\hskip -1pt \Omega^a{}_b - \Omega^a{}_c \wedge \Omega^c{}_b
\label{eq:curvature} \\
\mathcal{T}^a &=& {\rm d}\hskip -1pt V^a - \Omega^a{}_b V^b - \frac{{\rm i}}{2} \overline \Psi_A \Gamma^a
\Psi^A \label{eq:tors} \\
F^{\tilde I} &=& {\rm d}\hskip -1pt A^{\tilde I} + \frac{1}{2} f_{{\tilde J}{\tilde
K}}{}^{\tilde I} A^{\tilde J} \wedge A^{\tilde K} + m^{{\tilde
I}M} B_M + {\rm i} X^{\tilde I} \overline \Psi_A \Psi^A \label{eq:allA} \\
H_M &=& {\rm d}\hskip -1pt B_M + T_{\tilde{I} M}{}^N A^{\tilde{I}} \wedge B_N +
d_{{\tilde I}{\tilde J}M} \left( F^{\tilde I} - {\rm i} X^{\tilde I} \overline
\Psi_A \Psi^A \right) A^{\tilde J} + \nonumber \\
&& + {\rm i} X_M \overline \Psi_A \Gamma_a \Psi^A V^a \label{eq:HI} \\
D \varphi^x &=& {\rm d}\hskip -1pt \varphi^x + k^x_{\tilde{I}} A^{\tilde{I}}
\label{eq:xi} \\
D q^u &=& {\rm d}\hskip -1pt q^u + k^u_{\tilde{I}} A^{\tilde{I}} \label{eq:q} \\
\rho^A &=& {\rm d}\hskip -1pt \Psi^A - \frac{1}{4} \Omega_{ab} \Gamma^{ab} \Psi^A +\tilde
\omega^A{}_B \Psi^B \label{eq:rho} \\
\nabla \lambda^{{x} A} &=& {\rm d}\hskip -1pt \lambda^{x A} - \frac{1}{4} \Omega_{ab} \Gamma^{ab} \lambda^{x A} +
\tilde \Gamma^x{}_{y} \lambda^{y A} + \tilde\omega^{A}{}_B \lambda^{xB}
\label{eq:alllambda} \\
\nabla \zeta^{\alpha} &=& {\rm d}\hskip -1pt \zeta^{\alpha} - \frac{1}{4} \Omega_{ab} \Gamma^{ab}\zeta^{\alpha} +
\tilde\Delta^{\alpha}{}_{\beta} \zeta^{\beta}. \label{eq:zeta}
\end{eqnarray}
In \eq{eq:rho} - \eq{eq:zeta} the gauged connections on the scalar
$\sigma$-models $\mathcal{M}(\varphi)$ and $\mathcal{M}_H(q)$ appear, where $\mathcal{M}(\varphi)$ is parametrized by the scalars in the vector and tensor multiplets while $\mathcal{M}_H(q)$ is parametrized by the scalars of the hypermultiplets (the quaternionic sector is unaffected by the presence of tensor multiplets).
They are defined as:
\begin{eqnarray}
\matrix{\tilde\Gamma^x{}_y &=&\Gamma^x{}_y + A^{\tilde I} \partial_y
k^x_{\tilde I} \qquad\qquad \mbox{ $\sigma$-model connection for
gauge sector} \hfill\cr
\tilde\omega^{AB} &=&\omega^{AB}+ \frac{3}{2} A^{\tilde I}
\mathcal{P}_{\tilde I}{}^{AB}\quad\qquad \mbox{ $SU(2)$ connection}
\hfill\cr
\tilde\Delta^{\alpha}{}_{\beta} &=& \Delta^{\alpha}{}_{\beta} + A^{\tilde I} \partial_v
k^u_{\tilde I} \, \mathcal{U}_u{}^{{\alpha}A}\mathcal{U}^{v}{}_{{\beta}A}\qquad \mbox{
$Sp(2n_H)$ connection}.\hfill}\label{gaugedspin}
\end{eqnarray}
Here $\Gamma^x{}_y(\varphi)$ is the Christoffel connection one-form of $\mathcal{M}(\varphi)$, while $\omega^{AB}(q)$ and $\Delta^\alpha{}_\beta(q)$ are respectively the $Sp(2,\mathbb{R})$ R-symmetry and the $Sp(2n_H,\mathbb{R})$ connections on $\mathcal{M}_H(q)$. Furthermore
$k^x_\Lambda(\varphi)$ and $k^u_\Lambda(q)$ denote the Killing vectors on the two $\sigma$-models. In eq. \eq{gaugedspin} also appear the geometric quantities $\mathcal{U}_u{}^{A\alpha}$ and $ \mathcal{P}_{\tilde I}{}^{AB}$. They are the vielbein ($\mathcal{U}^{A\alpha} \equiv \mathcal{U}^{A\alpha}_u dq^u$) and prepotential on $\mathcal{M}_H$. For their definition and geometric properties, we refer the reader to the standard literature, in particular \cite{Andrianopoli:1996cm,D'Auria:2001kv} where the same notations are used.
We adopted the following conventions
for raising and lowering $Sp(2,\mathbb{R})$ and $Sp(2n_H)$ indices:
\begin{equation}
\xi_A = \epsilon_{AB}\xi^B \,;\quad \xi^A =
-\epsilon^{AB}\xi_B\,;\qquad \xi_{\alpha} = \relax\,\hbox{$\vrule height1.5ex width.4pt depth0pt\kern-.3em{\rm C}$}_{{\alpha}{\beta}}\xi^{\beta}
\,;\quad \xi^{\alpha} = - \relax\,\hbox{$\vrule height1.5ex width.4pt depth0pt\kern-.3em{\rm C}$}^{{\alpha}{\beta}}\xi_{\beta}
\end{equation}
while the flat space-time indices $a,b$ are raised or lowered with the metric
\begin{equation}
\eta_{ab} = {\mathrm{diag(+,-,-,-,-)}}.
\end{equation}
With these definitions, the explicit construction proceeds by first solving the super-Bianchi's following from \eq{eq:curvature} - \eq{eq:zeta}:
\begin{eqnarray}
\mathcal{R}^a{}_b V^b &=& {\rm i} \overline \Psi_A \Gamma^a \rho^A \label{eq:bitors} \\
D \mathcal{R}^a{}_b &=& 0 \label{eq:curv} \\
D F^{\tilde I} &=& m^{{\tilde I}M} \left( H_M - {\rm i} X_M \overline \Psi_A \Gamma_a
\Psi^A V^a \right) + {\rm i} D X^{\tilde I} \overline \Psi_A \Psi^A - 2 {\rm i} X^{\tilde
I} \overline \Psi_A \rho^A \label{eq:biF} \\
D H_M &=& d_{{\tilde I}{\tilde J}M} \left( F^{\tilde I} - {\rm i} X^{\tilde
I} \overline \Psi_A \Psi^A \right) \wedge \left( F^{\tilde J} - {\rm i} X^{\tilde J}
\overline \Psi_B \Psi^B \right) + \nonumber \\
&& + {\rm i} D X_M \overline \Psi_A \Gamma_a \Psi^A V^a - 2 {\rm i} X_M \overline \Psi_A \Gamma_a
\rho^A V^a - \frac 12 X_M\overline \Psi_A \Gamma_a \Psi^A\overline \Psi_B \Gamma^a \Psi^B
\label{eq:biHI} \\
D^2 \varphi^x &=& k^x_{\tilde I} \left( F^{\tilde I} - {\rm i} X^{\tilde I}
\overline \Psi_A \Psi^A \right) \label{eq:bix} \\
D^2 q^u &=& k^u_{\tilde I} \left( F^{\tilde I} - {\rm i} X^{\tilde I} \overline
\Psi_A \Psi^A \right) \label{eq:biq} \\
\nabla \rho^A &=& - \frac{1}{4} \mathcal{R}_{ab} \Gamma^{ab} \Psi^A -\tilde \mathcal{R}^A{}_B \Psi^B
\label{eq:birho} \\
\nabla^2 \lambda^{x A} &=& - \frac{1}{4} \mathcal{R}_{ab} \Gamma^{ab} \lambda^{xA} - \tilde\mathcal{R}^{x}{}_{y}
\lambda^{yA} - \mathcal{R}^A{}_B \lambda^{xB} \label{eq:bialllambda}\\
\nabla^2 \zeta^{\alpha} &=& - \frac{1}{4} \mathcal{R}_{ab} \Gamma^{ab} \zeta^{\alpha} -\tilde
\mathcal{R}^{\alpha}{}_{\beta} \zeta^{\beta} \label{eq:bizeta},
\end{eqnarray}
where we have defined:
\begin{eqnarray}
D F^{\tilde I} &\equiv & {\rm d}\hskip -1pt F^{\tilde I} + \hat f_{{\tilde J}{\tilde
K}}{}^{\tilde I} A^{\tilde J} F^{\tilde K} \\
D X^{\tilde I} &\equiv & {\rm d}\hskip -1pt X^{\tilde I} + \hat f_{{\tilde J}{\tilde
K}}{}^{\tilde I} A^{\tilde J} X^{\tilde K} \\
D H_M &\equiv & {\rm d}\hskip -1pt H_M + \hat T_{{\tilde I} M}{}^N A^{\tilde I} H_N\\
D X_M &\equiv & {\rm d}\hskip -1pt X_M + \hat T_{{\tilde I} M}{}^N A^{\tilde I} X_N
\end{eqnarray}
with the $ \hat f_{{\tilde J}{\tilde K}}{}^{\tilde I}$, $\hat
T_{{\tilde I} M}{}^N $ introduced in \eq{generalgenerat}, and
\begin{eqnarray}
\tilde\mathcal{R}^x{}_y &\equiv & {\rm d}\hskip -1pt \tilde\Gamma^x{}_y + \tilde\Gamma^x{}_z
\tilde\Gamma^z{}_y \\
\tilde\mathcal{R}^A{}_B &\equiv & {\rm d}\hskip -1pt \tilde\omega^A{}_B + \tilde\omega^A{}_C
\tilde\omega^C{}_B \\
\tilde\mathcal{R}^{\alpha}{}_{\beta} &\equiv & {\rm d}\hskip -1pt \tilde\Delta^{\alpha}{}_{\beta} +
\tilde\Delta^{\alpha}{}_{\gamma} \tilde\Delta^{\gamma}{}_{\beta}.
\end{eqnarray}
Eq.s \eq{eq:bitors} - \eq{eq:bizeta} are solved by parametrizating the supercurvatures on superspace (from which the supersymmetry transformation laws follow) as:
\begin{eqnarray}
\mathcal{T}^a &=& 0 \label{par:tors} \\
\mathcal{R}_{ab} &=& \hat\mathcal{R}_{abcd} V^c V^d - {\rm i} \overline \Psi_A \Gamma_{[a} \rho^A_{b]c} V^c - \frac{{\rm i}}{8}
X_{\tilde I} \hat F^{{\tilde I} |cd} \epsilon_{abcde} \overline \Psi_A \Gamma^e \Psi^A -
\frac{{\rm i}}{2} X_{\tilde I}
\hat F^{\tilde I}_{ab} \overline \Psi_A \Psi^A + \nonumber \\
&&+ \frac 14 g_{xy}\left( \overline \lambda^{xA} \Gamma^c \lambda^{yB} \overline \Psi_A \Gamma_{abc}
\Psi_B - \overline \lambda^{xA} \Gamma_{ab} \lambda^{yB} \overline \Psi_A \Psi_B - \frac{1}{2} \overline
\lambda^{xA} \Gamma_{abc} \lambda^{yB} \overline \Psi_A \Gamma^c \Psi_B\right) + \nonumber \\
&& + {\rm i} S^{AB} \overline \Psi_A \Gamma^{ab} \Psi_B - \frac{1}{4} \overline \zeta_\alpha
\Gamma_{abc} \zeta^\alpha \overline \Psi_A \Gamma^c \Psi^A \label{par:r}\\
F^{{\tilde I}} &=&\hat F^{\tilde I}_{ab} V^a V^b - 2 f^{{\tilde
I}}_x\overline\Psi_A \Gamma_a \lambda^{x A} V^a
\label{par:anyA} \\
H_M &=& \hat H_{M|abc} V^a V^b V^c - h_{Mx} \overline \Psi_A \Gamma_{ab} \lambda^{x A} V^a
V^b \label{par:HI} \\
D \varphi^{x} &=& \hat D_a\varphi^{x} V^a + \overline \Psi_A \lambda^{x A}
\label{par:anyX} \\
Dq^u &=& \hat D_a q^u V^a + \mathcal{U}^u{}_{A{\alpha}} \overline \Psi^A \zeta^{\alpha} \\
\rho^A &=& \rho^A_{ab} V^a V^b - \frac{1}{8} X_{{\tilde I}} \hat F^{{\tilde
I} |bc} \left( \Gamma_{abc} - 4 \eta_{a[b} \Gamma_{c]} \right) \Psi^A V^a +
S^{AB} \Gamma_a \Psi_B V^a + \nonumber \\
&&\hskip -2mm + \frac{\rm i} 4 g_{xy} \Bigl[ \overline \lambda^{xA} \Gamma^{b} \lambda^{yB}
\left(\Gamma_{bc} + 2 \eta_{bc} \right) \Psi_B + \frac{1}{4} \overline \lambda^{xA} \Gamma^{ab}
\lambda^{yB} \left( \Gamma_{abc} + 4 \Gamma_a \eta_{bc} \right) \Psi_B \Bigr] V^c +
\nonumber\\
&& - \frac{{\rm i}}{8} \overline \zeta_\alpha \Gamma_{abc} \zeta^\alpha \Gamma^{ab} \Psi^A V^c
\label{par:rho} \\
\nabla \lambda^{xA} &=& \hat\nabla_a \lambda^{xA} V^a + \frac{{\rm i}}{2} D_a \varphi^x \Gamma^a \Psi^A +
\frac{{\rm i}}{4} g^x_{{\tilde I}} \hat F^{{\tilde I}}_{ab} \Gamma^{ab} \Psi^A + {\rm i}
W^{xAB} \Psi_B + \nonumber \\
&& + \frac 14 T^x{}_{yz}\left(-3 \overline \lambda^{yA} \lambda^{zB} \Psi_B + \overline
\lambda^{yA} \Gamma_a \lambda^{zB} \Gamma^a \Psi_B + \frac{1}{2} \overline \lambda^{yA} \Gamma^{ab}
\lambda^{zB} \Gamma_{ab} \Psi_B\right) \label{par:anylambda} \\
\nabla \zeta^\alpha &=&\hat \nabla_a \zeta^\alpha V^a + {\rm i} \mathcal{U}_{uA}{}^\alpha D_a q^u \Gamma^a
\Psi^A + {\rm i} \mathcal{N}^\alpha_A \Psi^A ,\label{par:zeta}
\end{eqnarray}
in terms of a set of scalar-dependent quantities:
$$f^{\tilde I}_x\,,\quad h_{Mx}\,,\quad g_{xy}\,,\quad g^x_{\tilde I}\,,\quad T^x{}_{yz}$$
and of the fermion-shifts due to the gauging $S^{AB}$, $ W^{xAB} $, $\mathcal{N}^\alpha_A $. The `hat' on the field-strengths and covariant derivatives denotes the
supercovariant part.
Eq.s \eq{eq:bitors} - \eq{eq:bizeta} give a set of constraints among the quanitites appearing in the parametrizations \eq{par:r} - \eq{par:zeta}. Part of them are reported below:
\begin{eqnarray}
f^{\tilde I}_{x} &=& D_x X^{\tilde I} \\
h_{Mx} &=& - D_x X_M \\
D_{(y} f^{\tilde I}_{x)} &=& T^{z}{}_{xy} f^{\tilde I}_{z} + X^{\tilde I} g_{xy} \label{bi1}\\
T^z{}_{[xy]}&=& 0\label{t[]}\\
X_M &=& - 2 d_{{\tilde I}{\tilde J}M} X^{\tilde I} X^{\tilde J} \\
\hat H_{M|abc} &=& -\frac 16 a_{M{\tilde I}}\epsilon_{abcde} \hat F^{{\tilde I}|de}\label{selfmass}\\
X^{\tilde I} X_{\tilde J} + f^{\tilde I}_x g^x_{\tilde J} &=& \delta^{\tilde I}_{\tilde J}\\
f^{\tilde I}_x W^{x[AB]} &=& \frac{1}{2} m^{{\tilde I}M} X_M \epsilon^{AB} \\
S^{AB} &=& X^{\tilde I} \mathcal{P}_{\tilde I}{}^{AB} \, ; \qquad S^{[AB]} = 0 \\
2 X_M S^{(AB)} &=& h_{M x} W^{x (AB)} \\
\mathcal{P}_{\tilde I}{}^{AB} m^{{\tilde I}M} &=& k_{\tilde I}^u m^{{\tilde I}M} = k_{\tilde I}^x m^{{\tilde I}M} = 0 \\
\mathcal{P}_{[{\tilde I}}{}^{AC} \mathcal{P}_{{\tilde J}]C}{}^B &=& \frac{1}{3}
f_{{\tilde I}{\tilde J}}{}^{\tilde K} \mathcal{P}_{\tilde K}{}^{AB} \label{bilast} .
\end{eqnarray}
In eq. \eq{selfmass} we have introduced the matrix
\begin{eqnarray}
a_{{\tilde I}{\tilde J}}&\equiv& X_{\tilde I} X_{\tilde J} +
h_{{\tilde I} x} g^x_{\tilde J}\label{aij}
\end{eqnarray}
which will appear in the Lagrangian as kinetic matrix for the vector
field-strengths.
Eq. \eq{selfmass} expresses, at the supersymmetric level, the duality
relation among the $B$-field-strengths and the vector
field-strengths.
Since the analysis has been done only at 2-fermion level, these are
not the totality of the algebraic and geometric
constraints of the theory. Further constraints are more easily
evaluated from the equations of motion in superspace of the rheonomic
Lagrangian given in appendix \ref{rheonomic} and will be reported
in the next subsection.
\subsection{The Lagrangian}
\label{lagrangian}
Writing the action as:
\begin{equation}
S =\int \sqrt{-g} {{\rm d}\hskip -1pt}^5 x \mathcal{L} ,
\end{equation}
the Lagrangian of the theory is:
\begin{eqnarray}
\mathcal{L} &=& \mathcal{L}_{Grav}+\mathcal{L}_{Kin} + \mathcal{L}_{Pauli} + \mathcal{L}_{gauge} + \mathcal{L}_{CS}
+\mathcal{L}_{4f} \label{susylag}
\end{eqnarray}
with:
\begin{eqnarray}
\mathcal{L}_{Grav} &=& R + \frac{{\rm i}}{\sqrt{-g}}\overline \Psi_{A|\mu} \Gamma^{\mu\nu\rho} \rho^A_{\nu\rho}
\label{eq:lgrav} \\
\mathcal{L}_{Kin} &=& - \frac{3}{8} a_{{\tilde I}{\tilde J}}
\mathcal{F}^{\tilde I}_{\mu\nu} \mathcal{F}^{{\tilde J}|\mu\nu} + \frac3{16}\frac 1{\sqrt{-g}}\epsilon^{\mu\nu\rho\sigma\lambda} m^{MN}
B_{M|\mu\nu}D_\rho B_{N|\sigma\lambda}+ \nonumber\\
&&+\frac{3}{4} g_{xy} D_\mu \varphi^x D^\mu \varphi^y + g_{uv} D_\mu q^u
D^\mu q^v+ \nonumber\\
&&+\frac{3}{2} {\rm i} g_{xy} \overline \lambda^{x}_A \Gamma^\mu \nabla_\mu \lambda^{y A} + {\rm i} \overline
\zeta_{\alpha} \Gamma^\mu D_\mu \zeta^{\alpha} \label{eq:lkin} \\
\mathcal{L}_{Pauli} &=&- \frac{3}{8} {\rm i} X_{\tilde I} \mathcal{F}^{\tilde I}_{\mu\nu} \overline
\Psi_{A|\rho}\Gamma^\mu \Gamma^{\rho\sigma} \Gamma^\nu \Psi^A_\sigma + \frac{3}{4}
h_{{\tilde I} x} \mathcal{F}^{\tilde I}_{\mu\nu} \overline \Psi_{A|\rho}
\left(\Gamma^{\mu\nu\rho}-2\Gamma^{\nu}g^{\mu\rho}\right) \lambda^{xA} + \nonumber \\
&& + \frac{{\rm i}}{4} \phi_{{\tilde I} xy} \mathcal{F}^{\tilde I}_{\mu\nu} \overline
\lambda^x_A \Gamma^{\mu\nu} \lambda^{yA}- \frac{3}{2} g_{xy} D_\nu \varphi^x \overline
\Psi_{A|\mu}\Gamma^\nu\Gamma^\mu \lambda^{yA} +\nonumber\\
&& + \frac{3{\rm i}}{4} X_{\tilde{I}} \mathcal{F}^{\tilde{I}}_{\mu\nu} \overline \zeta_{\alpha}
\Gamma^{\mu\nu} \zeta^{\alpha} - 2 D_\nu q^u \overline \Psi_{A|\mu} \Gamma^\nu\Gamma^\mu
\zeta_{\alpha} \mathcal{U}_u{}^{A{\alpha}} \label{eq:lpauli} \\
\mathcal{L}_{gauge} &=& - 3 {\rm i} S^{AB} \overline \Psi_A^\mu \Gamma_{\mu\nu} \Psi_B^\nu - 3
g_{xy} W^{y AB} \overline \lambda_A^x \Gamma_\mu \Psi_B^\mu +2 \mathcal{N}^{A}_{\alpha} \overline
\Psi_{A|\mu} \Gamma^\mu \zeta^{\alpha} + \nonumber \\
&& - \frac{3}{2} {\rm i} M_{xy|AB} \overline \lambda^{xA} \lambda^{yB} - {\rm i} \mathcal{M}^{{\alpha}{\beta}}
\overline \zeta_{\alpha} \zeta_{\beta} - 2 {\rm i} \mathcal{M}^{A{\alpha}}_x \overline \zeta_{\alpha} \lambda^x_A
- V(\phi) \label{eq:gauge}\\
\mathcal{L}_{CS} &=& \frac3{16}\Bigl[ 2 m^{MN} B_{M\mu\nu} d_{{\tilde I}{\tilde J}
N} \left( F^{\tilde I}_{\rho\sigma} -{\rm i} X^{\tilde I}
\bar\Psi_{A|\rho} \Psi_\sigma^A \right) A_{\tau}^{\tilde J} + \nonumber\\
&& + \frac 13 t_{{\tilde I}{\tilde J}{\tilde K}} A^{\tilde I}_\mu \partial_\nu
A^{\tilde J}_\rho \partial_\sigma A^{\tilde K}_\tau + \frac 1{4}
\left(t_{{\tilde I}{\tilde L}{\tilde M}} f_{{\tilde J}{\tilde
K}}{}^{\tilde L} + 4d_{ {\tilde I} {\tilde J} M} m^{MN}
d_{{\tilde M} {\tilde K} N}\right) A_\mu^{\tilde I} A_\nu^{\tilde
J} A_\rho^{\tilde K} \partial_\sigma A_\tau^{\tilde M} + \nonumber \\
&&+\frac{1}{20}\left( t_{{\tilde I}{\tilde L}{\tilde M}} f_{{\tilde
J}{\tilde K}}{}^{\tilde L} + 4 d_{{\tilde I}{\tilde J}M} m^{MN}
d_{{\tilde M} {\tilde K} N} \right)f_{{\tilde N}{\tilde P}}{}^{\tilde
M} A_\mu^{\tilde I} A_\nu^{\tilde J} A_\rho^{\tilde K}
A_\sigma^{\tilde N} A_\tau^{\tilde P} \Bigr]
\frac{\epsilon^{\mu\nu\rho\sigma\tau}}{\sqrt{-g}}
\label{csbis}
\end{eqnarray}
where:
\begin{eqnarray}
M_{xy|AB} &=& (g_{yz} k^z_{\tilde I} f^{\tilde I}_x - \frac{1}{2} h_{Mx}
m^{MN} h_{Ny}) \epsilon^{AB} - 2f^{\tilde I}_z T^z{}_{xy}
\mathcal{P}_{\tilde I}{}^{AB}\\
\mathcal{M}^{{\alpha}{\beta}} &=& \frac{1}{2} \mathcal{U}_v{}^{A\alpha} \mathcal{U}_{uA}{}^{\beta} D^{[u}
k^{v]}_{\tilde{I}} X^{\tilde{I}} \\
\mathcal{M}^{A{\alpha}}_x &=& -2 \,\mathcal{U}_u{}^{A{\alpha}} k^u_{\tilde{I}} f^{\tilde{I}}_x
\end{eqnarray}
and:
\begin{eqnarray}
\left\{\matrix{S^{AB} &=& X^{\tilde I} \mathcal{P}_{\tilde I}{}^{AB}
\hfill\cr
W^{xAB} &=& g^{xy} (\frac 12 h_{{\tilde I} y} m^{{\tilde I}M} X_M
\epsilon^{AB} - 2 f^{\tilde I}_y \mathcal{P}_{\tilde I}{}^{AB} )\hfill\cr
\mathcal{N}^{A{\alpha}} &=& 2 \mathcal{U}_u{}^{A{\alpha}} k^u_{\tilde{I}} X^{\tilde{I}}
.\hfill}\right. \,. \label{shifts}
\end{eqnarray}
Finally, $t_{\tilde I\tilde J \tilde K}$ introduced in \eq{csbis} is a covariantly constant tensor.
In \eq{csbis}, the freedom under rescaling \eq{scaleinv} has been used
to fix the overall normalization. More details on the calculation are
given in appendix \ref{rheonomic}. The 4-fermions contributions to the Lagrangian, from \cite{Gunaydin:1983bi}, is reported in appendix \ref{4f}.
The scalar potential is
\begin{eqnarray}
V &=& -12 S^{AB} S_{AB} + \frac 32 g_{xy} W^{x AB} W^y_{AB} +
\mathcal{N}_{{\alpha}A}\mathcal{N}^{{\alpha}A}\nonumber\\
&=& 6 \mathcal{P}_{\tilde{I}}^{AB}\mathcal{P}_{{\tilde{J}} AB}
\left(f^{\tilde{I}}_x g^{xy}f^{\tilde{J}}_y - 2X^{\tilde{I}}
X^{\tilde{J}}\right) + \frac{3}{4} X_M X_N m^{MP}
m^{NL}h_{P x}g^{xy}h_{Ly} + \nonumber\\
&&+4 g_{uv}k^u_{\tilde{I}} k^v_{\tilde{J}} X^{\tilde{I}} X^{\tilde{J}}
\label{potential}.
\end{eqnarray}
The following Ward-identity on the gauging holds
\begin{equation}
V \delta^B_A = - 24 S^{BC} S_{CA} + 3 g_{xy} W^{x CB} W^y_{CA} +
2\mathcal{N}_{{\alpha}A}\mathcal{N}^{{\alpha}B}.\label{ward}
\end{equation}
Eq. \eq{ward} is identically satisfied, given eq.s \eq{shifts}, for
any $SU(2)$-valued $\mathcal{P}^{AB}_{\tilde{I}}
=\mathcal{P}^r_{\tilde{I}} \sigma_r^{AB}$.
The Lagrangian \eq{susylag} is left invariant by the supersymmetry transformation
rules (with supersymmetry parameter $\epsilon^A$):
\begin{eqnarray}
\matrix{ \delta V^a_\mu &=& - {\rm i} \overline\Psi_{A\mu}\Gamma^a \epsilon^a \hfill \cr
\delta A^{\tilde I}_\mu &=& 2{\rm i} X^{\tilde I} \overline\Psi_{A\mu} \epsilon^A - 2
f^{\tilde I}_x \overline\epsilon_A \Gamma_\mu \lambda^{xA} \hfill \cr
\delta B_{M \mu\nu} &=& 2{\rm i} d_{{\tilde I}{\tilde J}M} X^{\tilde I} A^{\tilde
J}_{[\mu}\overline\Psi_{\nu] A} \epsilon^A + 2{\rm i} X_M \overline\Psi_{A[\mu}\Gamma_{\nu]}\epsilon^A -
h_{Mx} \overline\epsilon_A \Gamma_{\mu\nu}\lambda^{xA} \hfill \cr
\delta \varphi^x &=& \overline\epsilon_A \lambda^{xA} \hfill \cr
\delta q^u &=& \mathcal{U}^u{}_{A{\alpha}} \overline \epsilon^A \zeta^{\alpha} \hfill\cr
\delta \Psi^A_\mu &=& D_\mu \Psi^A - \omega^A_{u|\,B}\mathcal{U}^u{}_{C{\alpha}} \overline \epsilon^C
\zeta^{\alpha}\Psi^B_\mu- \frac{1}{8} X_{{\tilde I}} F^{{\tilde I} |\nu\rho}
\left( \Gamma_{\mu\nu\rho} - 4 \eta_{\mu[\nu} \Gamma_{\rho]} \right) \epsilon^A + S^{AB} \Gamma_\mu \epsilon_B
+ \hfill\cr
&&\hskip -2mm + \frac{\rm i} 4 g_{xy} \Bigl[ \overline \lambda^{xA} \Gamma^{\nu} \lambda^{yB}
\left(\Gamma_{\nu\mu} + 2 \eta_{\nu\mu} \right) \epsilon_B + \frac{1}{4} \overline \lambda^{xA} \Gamma^{\nu\rho}
\lambda^{yB} \left( \Gamma_{\mu\nu\rho} + 4 \Gamma_\nu \eta_{\rho\mu} \right) \epsilon_B \Bigr]
\hfill\cr
&& - \frac{{\rm i}}{8} \overline \zeta_\alpha \Gamma_{\mu\nu\sigma} \zeta^\alpha
\Gamma^{\nu\sigma} \epsilon^A \hfill\cr
\delta \lambda^{xA} &=& -\omega^A_{u|\,B}\mathcal{U}^u{}_{C{\alpha}} \overline \epsilon^C \zeta^{\alpha}\lambda^{xB}
- \Gamma^x_{yz}\overline\epsilon_C\lambda^{zC}\, \lambda^{yA} + \frac{{\rm i}}{2} D_\mu \phi^x \Gamma^\mu \epsilon^A +
\frac{{\rm i}}{4} g^x_{{\tilde I}} F^{{\tilde I}}_{\mu\nu} \Gamma^{\mu\nu} \epsilon^A +
{\rm i} W^{xAB} \epsilon_B + \hfill\cr
&& + \frac 14 T^x{}_{yz}\left(-3 \overline \lambda^{yA} \lambda^{zB} \epsilon_B + \overline \lambda^{yA}
\Gamma^\mu \lambda^{zB} \Gamma_\mu \epsilon_B + \frac{1}{2} \overline \lambda^{yA} \Gamma^{\mu\nu}
\lambda^{zB} \Gamma_{\mu\nu} \epsilon_B\right) \hfill\cr
\delta \zeta^\alpha &=& -\Delta^{\alpha}_{u|\,\beta}\mathcal{U}^u{}_{A{\gamma}} \overline \epsilon^A \zeta^{\gamma}
\zeta^{\beta}+ {\rm i} D_\mu q^u \Gamma^\mu U_{uA}{}^\alpha \epsilon^A + {\rm i} \mathcal{N}^\alpha_A \epsilon^A\hfill
}\label{susyrules}
\end{eqnarray}
For our calculations, we used the geometrical (rheonomic) approach
which, as is well known, provides not only the space-time Lagrangian
and the superspace equations of motion, but also the value of the
generalized curvatures in superspace thus providing constraints on the
physical fields of the theory. This is of course equivalent to require
space-time supersymmetry.
The extra constraints we find besides those
already given by closure of the Bianchi identities
\eq{bi1}-\eq{bilast} are:
\begin{eqnarray}
t_{{\tilde I}{\tilde J}{\tilde K}} X^{\tilde I} X^{\tilde J} X^{\tilde
K} &=&1 \label{surface0}\\
X_{\tilde I} &=& t_{{\tilde I}{\tilde J}{\tilde K}} X^{\tilde J}
X^{\tilde K} = a_{{\tilde I}{\tilde J}} X^{\tilde J} \\
f^{\tilde I}_x &=& D_x X^{\tilde I}\\
g_{xy}&=&- 2 t_{{\tilde I}{\tilde J}{\tilde K}}X^{\tilde K} f^{\tilde
I}_x f^{\tilde J}_y =a_{{\tilde I}{\tilde J}}f^{\tilde I}_x
f^{\tilde J}_y\\
T^z_{xy} &=& t_{{\tilde I}{\tilde J}{\tilde K}} g^{zw} f^{\tilde I}_w
f^{\tilde J}_x f^{\tilde K}_y \\
h_{{\tilde I} x} &=& - D_x X_{\tilde I} = a_{{\tilde I}{\tilde J}}
f_x^{\tilde J} = g_{xy} g_{\tilde I}^y \\
t_{{\tilde I}{\tilde J}{\tilde K}} X^{\tilde K} &=&-\frac{1}{2}
(a_{{\tilde I}{\tilde J}}-3X_{\tilde I} X_{\tilde J})\\
X_{\tilde I} X^{\tilde I} &=& 1 \\
D_x h_{{\tilde I} y} &=& - ( h_{{\tilde I} z} T^z_{xy} + X_{\tilde
I} g_{xy}) \,; \qquad
D_{[x} h_{{\tilde I} | y]} = 0 \\
\phi_{{\tilde I} xy} &=& 3 t_{{\tilde I}{\tilde J}{\tilde K}}
f^{\tilde J}_x f^{\tilde K}_y + \frac 94 X_{\tilde I} g_{xy} \\
d_{({\tilde I}{\tilde J} )M} &=& -\frac 12 t_{{\tilde I}{\tilde J}M}\label{3.54}\\
h_{{\tilde I} z}T^z_{xy} &=& t_{{\tilde I}{\tilde J}{\tilde K}}
f^{\tilde J}_x f^{\tilde K}_y + \frac 12 X_{\tilde I} g_{xy}.
\end{eqnarray}
In particular, \eq{surface0} defines the equation of the surface generally carachterizing the scalar geometry of $D=5$, $N=2$ tensor and vector multiplet sector.
Furthermore, the above relations also imply the constraints on the curvature of $\mathcal{M}(\varphi)$ characterizing its geometry:
\begin{eqnarray}
R^x_{\ yzt}&=& \left( \delta^x_{[t}g_{z]y}+ T^x_{\ w[t}T^w_{\
z]y}\right)\label{curva}\end{eqnarray}
and a relation between the constant $t_{{\tilde I}{\tilde J}{\tilde K}}$ defining the surface and the scalar-dependent couplings:
\begin{eqnarray}
t_{{\tilde I}{\tilde J}{\tilde K}}&=&\frac 12 \left(5 X_{\tilde I}
X_{\tilde J} X_{\tilde K} -3 a_{({\tilde I}{\tilde J}} X_{{\tilde K}
)} + 2 T_{xyz}g^x_{\tilde I} g^y_{\tilde J} g^z_{\tilde K}\right).
\end{eqnarray}
Since the geometrical properties of the $\sigma$-model $\mathcal{M}(\varphi)$ have been discussed thoroughly in the original paper \cite{Gunaydin:1983bi}, we omit further comments on this point.
\subsection{Comments on the scalar potential}
The scalar potential that we find in eq. \eq{potential}:
\begin{eqnarray}
V &=& 6 \mathcal{P}_{\tilde{I}}^{AB}\mathcal{P}_{{\tilde{J}} AB}
\left(f^{\tilde{I}}_x g^{xy}f^{\tilde{J}}_y - 2X^{\tilde{I}}
X^{\tilde{J}}\right) + \frac{3}{4} X_M X_N m^{MP} m^{NL}h_{P
x}g^{xy}h_{Ly} + \nonumber \\
&&+4 g_{uv}k^u_{\tilde{I}} k^v_{\tilde{J}} X^{\tilde{I}} X^{\tilde{J}}
\label{potential2}
\end{eqnarray}
is formally the same as the one found in the literature
\cite{Ceresole:2000jd} (a rescaling of the fields is required for a
precise comparison; a map is given in Appendix
\ref{GST/ADS}). However, since we are considering more general
couplings and a non-trivial $m^{MN}$ matrix, a few comments are in
order.
First of all, it is already known that the presence of the tensor multiplets allows a non-zero
$W^{x[AB]}= \frac 12 g^{xy} h_{{\tilde I} y} m^{{\tilde I}M} X_M \epsilon^{AB}$ but that the contribution to the scalar potential coming from the tensors is always positive, so that Anti de Sitter solutions may only be accounted
for a non trivial (possibly constant) $\mathcal{P}_{\tilde{I}}^{AB}$,
giving a mass to the gravitino, while, in the case
$\mathcal{P}_{\tilde{I}}^{AB}=0$, only Minkowski vacua are attainable,
for
\begin{equation}h_{Mx} m^{MN} X_N =0 .\label{vac}
\end{equation}
This is in particular the case when one considers as $N=2$ model
the Scherk--Schwarz generalized dimensional reduction of a six
dimensional theory, as discussed in \cite{Andrianopoli:2004xu}.
For the case
of the S-S dimensionally reduced theory, to have a non
negative scalar potential a cancellation is needed between the gaugino and gravitino
contributions, proportional to the prepotential $\mathcal{P}_{\tilde
I}^{AB}$. This does not
appear instead to be necessary in more general, purely five dimensional,
cases, still allowing, however, a general antisymmetric matrix $m^{MN}$.
Then, when $m^{MN}$ has general skew-eigenvalues, eq. \eq{vac} may have solutions more general than the ``symplectic-orthogonality'' condition between $h_{Mx}$ and $X_N$.
Let us now see the implications of having $d_{\Lambda\Sigma M} \neq
0$. Eq. \eq{vac} has a solution for:
\begin{eqnarray}
X_M = t_{{\tilde I}{\tilde J}M}X^{\tilde I} X^{\tilde J} = -4 d_{\Lambda
NM}X^\Lambda X^N -2 d_{\Lambda\Sigma M}X^\Lambda X^\Sigma =0 \label{vacuumtens}
\end{eqnarray}
where we used the relation \eq{3.54}
Eq. \eq{vacuumtens} must be solved together with the defining equation
of the scalar geometry
\begin{eqnarray}
t_{{\tilde I}{\tilde J}{\tilde K}} X^{\tilde I} X^{\tilde J} X^{\tilde
K} = 1
\end{eqnarray}
that is:
\begin{eqnarray}
X^\Lambda \left(t_{\Lambda\Sigma\Gamma} X^\Sigma X^\Gamma +2 t_{\Lambda\Sigma M} X^\Sigma X^M + 2 t_{\Lambda MN}
X^M X^N \right)= 1 .\label{surface}
\end{eqnarray}
In \eq{surface} we used the fact that, for $m^{MN}$ invertible, $t_{MNP} =0$, as explicitly shown in appendix \ref{rheonomic}, eq. \eq{tmnp0}.
Eq. \eq{surface} requires $X^\Lambda \neq 0$ for at least one value of $\Lambda$
(e.g. $X^\Lambda|_{\mbox{vac}} \propto \delta^\Lambda_0$), and it then implies that
the v.e.v. of the scalars $X^M$ are now shifted from zero, since
eq. \eq{vacuumtens} is solved for
\begin{equation}
d_{\Lambda MN} X^N|_{\mbox{vac}} =-\frac 12 d_{\Lambda\Sigma M}
X^\Sigma|_{\mbox{vac}}\neq 0.
\end{equation}
\section{Conclusions and outlook}
In the present paper we have studied the $D=5$, $N=2$ theory coupled
to vector, tensor and hyper multiplets by including all possible
couplings compatible with gauge symmetry and supersymmetry. We paid
particular attention in analyzing the algebraic structure of the FDA
which underlies the theory, This allowed to relax some constraints on the couplings usually considered, and correspondingly to write-down a scalar potential a bit more general than usually considered. It would be interesting to analyze in detail the critical points of models exhibiting the features described here, as in particular a magnetic coupling $m^{MN}$ with arbitrary skew-eigenvalues. Models of this kind (an example of which is found by Scherk--Schwarz compactification from six dimensions \cite{Andrianopoli:2004xu}) should appear in general flux compactifications from superstring or M-theory.
Our investigation may now be extended in various directions.
At a group-theoretical point of view, it would be interesting to extend the FDA to include also higher order forms, as is the case, in general, in theories corresponding to compactifications from superstrings or M-theory.
We would also like to perform an analysis, on the same lines of the one presented here, for the $D=4$ $N=2$ theory coupled to vector-tensor
multiplets. These developments are under investigation and will be
discussed elsewhere.
\acknowledgments{It is a pleasure to acknowledge valuable discussions with Mario Trigiante all over the preparation of this paper, and useful observations and careful reading from Maria Antonia Lled\'o on the group theoretical part.
Work supported in part by the European Community's Human Potential
Program under contract MRTN-CT-2004-005104 `Constituents,
fundamental forces and symmetries of the universe', in which L.A., and
R.D'A. are associated to Torino University. Partial support from the Spanish Ministry of
Education and Science (FIS2005-02761) and EU FEDER funds}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 144 |
Edwards, K. I7586 whakapapaonline | The Bristowe Branch - Capt David Reeves Bristow (I2939) Ani Te Patu Rameka (I2938), The Priestley-Bristowe Branch.
Edwards, M.H. I7387 whakapapaonline | The Priestley Branch - Charles Priestley (I49),Hokokai Kawakawa ( I1792 ), Rangiuia (5198), The Priestley-Bristowe Branch.
Edwards, Patrick I7385 b. 1936 whakapapaonline | The Priestley Branch - Charles Priestley (I49),Hokokai Kawakawa ( I1792 ), Rangiuia (5198), The Priestley-Bristowe Branch. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,197 |
Second Lawsuit Challenges Controversial Rent Rules
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Western Florida
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Management Relations
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Foreclosure Epidemic Skips Big Apple...So Far You Can't Take Manhattan
By Keith Loria 2008 July Finance
Whether you own your own home or not, you can't help but know about the subprime mortgage crisis that is sweeping across America as foreclosures are claiming people's homes everywhere we look.
Well, almost everywhere. Even in light of the foreclosure epidemic and what appears to be a looming economic recession, it seems to be that New York City—especially Manhattan—is immune to the kinds of mass foreclosures and repossessions that are plaguing the rest of the country.
"It has impacted it in the way that it's a little more difficult to secure a loan, especially in a co-op, but that being said, the market is still strong here," says Colleen Dwinell, a sales agent at DJK Residential. "We are insulated here in Manhattan, and because there are so many co-ops and the co-op boards are so strict, the people who got in are very qualified and you don't really see that foreclosure monster anywhere near us."
In fact, condos and co-ops in Manhattan continue to thrive, despite a downturn on Wall Street and a declining housing market nearly every other part of the country.
"Manhattan has been fairly isolated from the subprime meltdown because very few subprime loans are issued to Manhattan borrowers," says Vicki Been, director of the Furman Center for Real Estate and Urban Policy. "In 2006, only 0.8 percent of new home purchase loans in Manhattan were subprime, compared to 19.7 percent citywide."
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Refinancing Your Underlying Permanent Mortgage Like a Banker
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HDFC Co-ops and Foreclosures: Resolution and Restoration?
Experts Weigh in on What Could Be Done
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The Pros & Cons
BrokerGal on Thursday, July 31, 2008 9:59 PM
I am so tired of hearing about how the banks "did it to the homeowners" with the "subprime" issue. People did it to themselves. I know many people walked away from me to do their loans with an easy taker, because the "homeowner" wanted to go stated, or lie, or do whatever they wanted just to get into a house. Maybe we should allow those homeowners to accept their own consequences, so that this won't happen again, and then people like me--the HONEST brokers--will have normal business, not be subjected to dishosest brokers AND Buyers! And by the way. The MEDIA started the trouble with their coersion with the builders....that's where it all started: someone ought to research that... BrokerGal
Tue, Jan 21, 9:00pm – 12:00am
NYC DOB: Homeowners Night
Tue, Jan 21, 9:00pm – 12:00am add to calendar 21-01-2020 21:00 22-01-2020 00:00 America/New_York NYC DOB: Homeowners Night If you are a homeowner, tenant, small business owner or building manager, we encourage you to visit your local borough office where Department of Buildings staff can answer your questions and provide the information you need.Permits, construction codes, zoning regulations, sign offs, certificates of occupancy, place of assembly, equipment installations, violations and civil penalties are just a few of the things we can assist you with. Our plan examiners, inspectors and staff are here to help you build and live safely in New York City – no appointment needed. For information, visit https://www1.nyc.gov/site/buildings/homeowner/homeowner.page. MM/DD/YYYY
If you are a homeowner, tenant, small business owner or building manager, we encourage you to visit your local borough office where Department of Buildings staff can answer your questions and provide the information you need.Permits, construction codes, zoning regulations, sign offs, certificates of occupancy, place of assembly, equipment installations, violations and civil penalties are just a few of the things we can assist you with. Our plan examiners, inspectors and staff are here to help you build and live safely in New York City – no appointment needed. For information, visit https://www1.nyc.gov/site/buildings/homeowner/homeowner.page.
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Tue, Feb 4, 9:00pm – 12:00am add to calendar 04-02-2020 21:00 05-02-2020 00:00 America/New_York NYC DOB: Homeowners Night If you are a homeowner, tenant, small business owner or building manager, we encourage you to visit your local borough office where Department of Buildings staff can answer your questions and provide the information you need.Permits, construction codes, zoning regulations, sign offs, certificates of occupancy, place of assembly, equipment installations, violations and civil penalties are just a few of the things we can assist you with. Our plan examiners, inspectors and staff are here to help you build and live safely in New York City – no appointment needed. For information, visit https://www1.nyc.gov/site/buildings/homeowner/homeowner.page. MM/DD/YYYY
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Tue, Feb 11, 9:00pm – 12:00am add to calendar 11-02-2020 21:00 12-02-2020 00:00 America/New_York NYC DOB: Homeowners Night If you are a homeowner, tenant, small business owner or building manager, we encourage you to visit your local borough office where Department of Buildings staff can answer your questions and provide the information you need.Permits, construction codes, zoning regulations, sign offs, certificates of occupancy, place of assembly, equipment installations, violations and civil penalties are just a few of the things we can assist you with. Our plan examiners, inspectors and staff are here to help you build and live safely in New York City – no appointment needed. For information, visit https://www1.nyc.gov/site/buildings/homeowner/homeowner.page. MM/DD/YYYY
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Tue, Mar 3, 9:00pm – 12:00am add to calendar 03-03-2020 21:00 04-03-2020 00:00 America/New_York NYC DOB: Homeowners Night If you are a homeowner, tenant, small business owner or building manager, we encourage you to visit your local borough office where Department of Buildings staff can answer your questions and provide the information you need.Permits, construction codes, zoning regulations, sign offs, certificates of occupancy, place of assembly, equipment installations, violations and civil penalties are just a few of the things we can assist you with. Our plan examiners, inspectors and staff are here to help you build and live safely in New York City – no appointment needed. For information, visit https://www1.nyc.gov/site/buildings/homeowner/homeowner.page. MM/DD/YYYY
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Tue, Mar 10, 8:00pm – 11:00pm add to calendar 10-03-2020 20:00 10-03-2020 23:00 America/New_York NYC DOB: Homeowners Night If you are a homeowner, tenant, small business owner or building manager, we encourage you to visit your local borough office where Department of Buildings staff can answer your questions and provide the information you need.Permits, construction codes, zoning regulations, sign offs, certificates of occupancy, place of assembly, equipment installations, violations and civil penalties are just a few of the things we can assist you with. Our plan examiners, inspectors and staff are here to help you build and live safely in New York City – no appointment needed. For information, visit https://www1.nyc.gov/site/buildings/homeowner/homeowner.page. MM/DD/YYYY
Tue, Apr 7, 8:00pm – 11:00pm
Tue, Apr 7, 8:00pm – 11:00pm add to calendar 07-04-2020 20:00 07-04-2020 23:00 America/New_York NYC DOB: Homeowners Night If you are a homeowner, tenant, small business owner or building manager, we encourage you to visit your local borough office where Department of Buildings staff can answer your questions and provide the information you need.Permits, construction codes, zoning regulations, sign offs, certificates of occupancy, place of assembly, equipment installations, violations and civil penalties are just a few of the things we can assist you with. Our plan examiners, inspectors and staff are here to help you build and live safely in New York City – no appointment needed. For information, visit https://www1.nyc.gov/site/buildings/homeowner/homeowner.page. MM/DD/YYYY
Tue, Apr 14, 8:00pm – 11:00pm
Tue, Apr 14, 8:00pm – 11:00pm add to calendar 14-04-2020 20:00 14-04-2020 23:00 America/New_York NYC DOB: Homeowners Night If you are a homeowner, tenant, small business owner or building manager, we encourage you to visit your local borough office where Department of Buildings staff can answer your questions and provide the information you need.Permits, construction codes, zoning regulations, sign offs, certificates of occupancy, place of assembly, equipment installations, violations and civil penalties are just a few of the things we can assist you with. Our plan examiners, inspectors and staff are here to help you build and live safely in New York City – no appointment needed. For information, visit https://www1.nyc.gov/site/buildings/homeowner/homeowner.page. MM/DD/YYYY
Tue, May 5, 8:00pm – 11:00pm
Tue, May 5, 8:00pm – 11:00pm add to calendar 05-05-2020 20:00 05-05-2020 23:00 America/New_York NYC DOB: Homeowners Night If you are a homeowner, tenant, small business owner or building manager, we encourage you to visit your local borough office where Department of Buildings staff can answer your questions and provide the information you need.Permits, construction codes, zoning regulations, sign offs, certificates of occupancy, place of assembly, equipment installations, violations and civil penalties are just a few of the things we can assist you with. Our plan examiners, inspectors and staff are here to help you build and live safely in New York City – no appointment needed. For information, visit https://www1.nyc.gov/site/buildings/homeowner/homeowner.page. MM/DD/YYYY
Tue, May 12, 8:00pm – 11:00pm
Tue, May 12, 8:00pm – 11:00pm add to calendar 12-05-2020 20:00 12-05-2020 23:00 America/New_York NYC DOB: Homeowners Night If you are a homeowner, tenant, small business owner or building manager, we encourage you to visit your local borough office where Department of Buildings staff can answer your questions and provide the information you need.Permits, construction codes, zoning regulations, sign offs, certificates of occupancy, place of assembly, equipment installations, violations and civil penalties are just a few of the things we can assist you with. Our plan examiners, inspectors and staff are here to help you build and live safely in New York City – no appointment needed. For information, visit https://www1.nyc.gov/site/buildings/homeowner/homeowner.page. MM/DD/YYYY
Tue, Jun 2, 8:00pm – 11:00pm
Tue, Jun 2, 8:00pm – 11:00pm add to calendar 02-06-2020 20:00 02-06-2020 23:00 America/New_York NYC DOB: Homeowners Night If you are a homeowner, tenant, small business owner or building manager, we encourage you to visit your local borough office where Department of Buildings staff can answer your questions and provide the information you need.Permits, construction codes, zoning regulations, sign offs, certificates of occupancy, place of assembly, equipment installations, violations and civil penalties are just a few of the things we can assist you with. Our plan examiners, inspectors and staff are here to help you build and live safely in New York City – no appointment needed. For information, visit https://www1.nyc.gov/site/buildings/homeowner/homeowner.page. MM/DD/YYYY
1/17/2020 How Motorized Window Treatments Help Fight Sun Glare—Residential Tech Today
It was a co-op apartment upgrade that sounded great on paper. Our 58-year-old building in New York City was finally getting a window overhaul: full-pane windows to replace legacy double-hung units that sometimes wouldn't open and other times wouldn't close. But as the installation date approached, we were so focused on the upheaval of the window frame removal that we didn't anticipate the shade situation.
1/17/2020 You're Never Too Old to Apply for a Mortgage—Mansion Global
Mary Babinski, a senior loan officer with Motto Mortgage Champions in Trinity, Fla., recently wrote a 30-year loan for a retiree buying a home in New Port Richey. He had no problem qualifying, but he was surprised he could nonetheless.
1/16/2020 Why Manhattan's Skyscrapers Are Empty—The Atlantic
In Manhattan, the homeless shelters are full, and the luxury skyscrapers are vacant.
1/16/2020 The woman influencing the face of luxury real estate—Thrive Global
Meet Janine Yorio, the CEO and founder of Compound, a new app that is reimagining how the world invests in urban residential real estate.
1/16/2020 Tour the Bespoke NYC Co-Op of Two Art-World Maestros—Architectural Digest
Manhattan-based Richard Rabel founded his interiors and art-advising firm eight years ago, but his passion for design first revealed itself decades prior, when, as a child, he would sketch stools and other furnishings.
1/15/2020 REBNY fighting to keep New York City strong—Real Estate Weekly
Throughout 2019, REBNY educated policymakers and the broader New York public about misguided City Council proposals to impose commercial rent control on the city's commercial real estate market, as well an idea by the Mayor to implement a vacancy tax.
1/15/2020 De Blasio administration, NYC Council poised to backtrack on penalties to affordable co-ops—New York Daily News
Mayor de Blasio's administration voiced support Monday for measures that would exempt affordable co-ops from a law that requires owners register them with the city.
1/15/2020 Yet Another Zoning Fight That's All about Protecting the Rich—New York Post
Not in my backyard — NIMBY — never dies. More than that, it doesn't even change its tune.
1/14/2020 New York Seeks Cause of Water-Main Break That Unleashed Flooding—The Wall Street Journal
Investigators were working to determine the cause of a water-main break in Manhattan that flooded streets and crippled subway service during the Monday morning rush hour, with the weekend's unseasonably warm weather being explored as a factor, New York City officials said.
1/14/2020 Unsold Manhattan Condos Get New Year's Makeovers—Crain's
For a Manhattan condo developer in a slowing sales market, the request from an international travel website was irresistible: Let us furnish 20 of your unsold units and offer our users a two-night stay.
1/14/2020 How the Rise and Fall of the Luxury Condo Building Boom Transformed Brooklyn—Brownstoner
In the last decade, more than 250 luxury condo buildings went up in Brooklyn, bringing thousands of units to the borough and dramatically altering its character.
1/10/2020 Jonathan Rose Companies acquires Astoria building from Goodwill—QNS.com
Johnathan Rose Companies acquired Goodwill Terrace Apartments, a 202-unit property in Astoria, for about $35 million, the company announced on Wednesday, Jan. 8.
1/10/2020 Est4te Four gets $74M inventory loan for Red Hook condo—The Real Deal
Italian developer Est4te Four closed a $74 million inventory loan for its stalled, 70-unit condominium in Red Hook.
1/10/2020 RealReal CEO Julie Wainwright buys $6.75M condo in Woolworth Building—NY Post
Julie Wainwright — the founder and CEO of The RealReal, an online consignment shop for the resale of luxury goods — has bought a condo at the Woolworth Tower Residences for $6.75 million.
1/10/2020 Trump Jr and Ivanka Trump 'knew they were lying' over ploy to sell condos, book claims—The Guardian
Donald Trump Jr and Ivanka Trump took part in a fraudulent scheme to sell units in a luxury New York condominium-hotel and "knew they were lying", according to a new book that explores how the current US president built his business empire.
12/23/2019 Mitchell-Lama applicants pay thousands, with 'virtually no chance' at getting apartments—Curbed
New Yorkers forked over hundreds of thousands in fees to get on affordable housing wait lists with practically no chance of actually scoring one of those coveted homes, according to an audit by state Comptroller Thomas DiNapoli.
12/19/2019 DUMBO set to be next front for development's battle against homelessness—Brooklyn Daily Eagle
A new bill expected to pass in the City Council on Thursday will require certain developers that receive government funding to set aside 15 percent of their new rental units for homeless New Yorkers. All told, the legislation is projected to create about 1,000 new apartments for those most in need of housing each year — in addition to the 1,300 apartments the city will develop on an annual basis.
12/17/2019 I'm A Freelancer & I Bought A Two-Bedroom In NYC — Here's How I Did It—Refinery29
The day I closed on my Manhattan apartment, I sent a text to a friend that read, "My New York City checklist is now complete."
12/17/2019 Try talking your way out of this one: Three years after swearing he was going straight, real estate exec charged as Gambino family cohort—Daily News
The West Side's uber-popular High Line neighborhood is apparently a big draw for organized crime, too. Gambino family capo Andrew Campos and fellow made man Richard Martino allegedly paid "hundreds of thousands of dollars in bribes and kickbacks" to employees of the HFZ Capital Group, the developer behind the The XI — a high-end, high-rise, two-tower residential complex just west of the Manhattan attraction, federal prosecutors alleged.
12/17/2019 How Christmas lights brought these Upper East Side residents together—New York Post
When a Grinch tried to steal Christmas from this Upper East Side block, it brought the community together in holiday spirit.
Robin on HDFC's Proposed Regulations Shake Up Co-op Shareholders:
I'd love the info on the Turin house too, thanks.
Anthony de Fex on Recovering Legal Fees:
A sample of a default case would be the co-op board denying a shareholder's right to an inspection of the books of account under the proprie…
Mark Godsey on Maintenance Charge Increases:
I am a member of our buildings newly formed finance committee, responsible for analyzing the costs of properly funding current and future ph…
Joseph S Desrosiers on Underfunded Reserves:
This subject is very important, since I moved in my co-op in 1988 at Sherwood Village B. I have been asking the same questions I am asking…
Robert Nordlund on Maintenance Charge Increases:
Mike - well said. As a national provider of Reserve Studies (over 50,000 prepared for clients in all 50 states), we see all too commonly tha…
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«Оскар» — () комедийный кинофильм 1991 года режиссёра Джона Лэндиса. Фильм иногда называют голливудским ремейком одноимённого фильма 1967 года, хотя на самом деле он снят на основе той же пьесы Клода Манье.
Фильм стал первой попыткой Сильвестра Сталлоне сняться в комедийной роли.
Несмотря на то, что фильм в течение двух недель был номером один в американском прокате, его кассовые сборы небольшие — лишь $23 500 000, что не позволило окупить бюджет в $35 000 000.
Сюжет
Гангстер «Снэпс» («Щелкун», кличка происходит от привычки Проволоне щёлкать пальцами) Анджело Проволоне (Сильвестр Сталлоне) обещал своему умирающему отцу (Кирк Дуглас), что он оставит криминальное прошлое и займётся честным бизнесом.
Спустя месяц он готовится принять у себя банкиров, чтобы купить долю в их бизнесе и войти в правление банка — заняться серьёзным бизнесом.
Однако утром судьбоносного дня в доме Анджело Проволоне появляется Энтони Розано (Винсент Спано), молодой бухгалтер Снэпса. Скромный служащий обращается к нему с просьбой увеличить ему зарплату. Деньги ему нужны потому, что он собирается жениться на дочери Анджело. После свадьбы он готов вернуть $50000, которые он украл у Проволоне.
Дочь Снэпса, Лиза, несколько месяцев назад имела исключительно платонический роман с шофером своего отца Оскаром. Потому, когда к ней врывается Снэпс с обвинениями, что она тайком встречается с «его служащим», Лиза думает о том, что её руки просит Оскар.
В течение фильма горничная Анджело пытается сказать ему, что собирается уволиться, но тот постоянно игнорирует её. В отместку она подговаривает Лизу, которая переживает, что отец откажется выдавать её замуж «за какого-то шофера», но мечтает при этом наконец-то вырваться из родительского дома, соврать отцу, что она беременна…
Тем временем в дом к Снэпсу приходит девушка по имени Тереза, которая признается, что это она встречалась с Энтони Розано, что она любит его, и, чтобы придать себе в глазах амбициозного бухгалтера побольше значимости, назвалась дочерью Снэпса Проволоне.
Снэпс понимает, что у него есть легкий способ получить назад свои $50000 долларов, но при этом не выдавать замуж Лизу…
Итак, Анджело решил вопрос со своими $50000, но у него новая проблема: беременная незамужняя дочь — позор семьи…
Однако и тут Снэпс находит выход: Торнтон Пул, преподаватель правильного произношения, нанятый Анджело, чтобы он «поработал» с его речью (ведь респектабельный бизнесмен должен изъясняться вовсе не так, как это делал гангстер) — вот прекрасная партия для его Лизы. Дело за малым — уговорить доктора Пула…
Однако успеть всё в один день не так-то просто. Приходит наниматься на работу новая горничная, которая при этом оказывается бывшей пассией самого Снэпса и по совместительству матерью Терезы, выдававшей себя за дочь Анджело Проволоне… И, как выясняется, она все-таки дочь Анджело. При этом жена Снэпса София успела переговорить с Энтони Розано, который поставил её в известность о том, что женится на её дочери, при этом Лиза говорит, что она выходит замуж за Оскара да ещё беременна от него, и Софии очень интересно, что это у её мужа за новая дочь, «которая не Лиза» и на которой женится Энтони…
Кроме того, за домом Снэпса следят сразу 2 группы — под предводительством бывшего конкурента Снэпса гангстера Вендетти, который не верит, что Снэпс ушёл на покой, и думает, что Снэпс, наоборот, «что-то затевает», и готовится нанести удар по «логову Проволоне»; и под предводительством лейтенанта чикагской полиции Туи, который также не верит в то, что Снэпс отошёл от дел, и так же подозревает, что Снэпс «что-то задумал», и поэтому, наблюдая суету в доме Снэпса, ждет малейшего повода, чтобы «повязать всю шайку».
Да и банкиры, с которыми намерен встретиться Снэпс, тоже не так просты — они нуждаются в деньгах Анджело, но не готовы пускать гангстера в свой респектабельный бизнес…
Ещё больше запутывают все и братья Фануччи — первоклассные портные, которые весь день крутятся в доме Проволоне с новым костюмом Анджело и мыслями о том, как бы получить побольше денег, и подручные Анджело, бывшие головорезы, ныне исполняющие должности камердинера, дворецкого и лакея — они все время не могут запомнить, что босса больше нельзя звать «босс», а в доме лучше не хранить оружие…
И, наконец, весь день в дом вносят и выносят некий чёрный саквояж…
В общем, тем незабвенным утром Анджело Проволоне придется крутиться как Фигаро, чтобы решить все проблемы, остаться в выигрыше, выдать замуж свою дочь…
В конце концов появится сам Оскар.
В ролях
Сильвестр Сталлоне — «Снэпс» Анджело Проволоне
Мариса Томей — Лиза Проволоне
Орнелла Мути — София Проволоне
Кирк Дуглас — Эдуардо Проволоне
Линда Грей — Роксанна
Питер Ригерт — Альдо
Чезз Палминтери — Конни
Джой Траволта — Эйс
Пол Греко — Шимер
Ричард Фороньи — Наки
Ивонн де Карло — Тётя Роза
Дон Амичи — Отец Клементо
Ричард Романус — Вендетти
Арлин Соркин — Маникюрша Вендетти
Эдди Брэкен — Чарли-Пять-Звёзд
Тони Манафо — Фрэнки-Разрушитель
Роберт Лессер — Офицер Кьюф
Арт Ла Флёр — Офицер Куин
Кертвуд Смит — Лейтенант Чикагской полиции Туми
Винсент Спано — Энтони Розано
Джойселин О'Брайен — Нора
Мартин Ферреро — Луиджи Финуччи
Гарри Ширер — Гвидо Финуччи
Уильям Атертон — Овертон
Марк Меткалф — Мильхус
Кен Говард — Кирквуд
Сэм Чью, мл. — Ван Лиланд
Элизабет Барондес — Тереза
Сэл Веччио — Вендетти Худ
Тим Карри — Доктор Торнтон Пул
Дэнни Голдстин — водитель кэба
Кай Вульф — шофёр Андервуда
Маршалл Бэлл, Том Грант, Луис Д'Альто — репортёры
Рик Авери — шофёр Вендетти
Джим Малхолланд — Оскар
Интересные факты
Доктор Пул определяет место рождения по акценту так же, как профессор Генри Хиггинс в фильме «Моя прекрасная леди».
В фильме множество отсылок к боксу (он снят сразу после «Рокки 5»).
Фильм имеет три номинации на Золотую малину.
Примечания
Ссылки
Кинокомедии США
Кинокомедии 1991 года
Фильмы-ремейки США
Фильмы США 1991 года | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,914 |
\section{\bf Introduction}
\lb{int}
In pioneer works, V. L. Ambartsumyan and G. S. Saakyan considered the question of
superdense stellar matter made of a degenerate gas of elementary particles, comprising
neutron, protons, hyperons and electrons, at zero temperature\,\cite{Amb1,Amb2}.
Investigations of internal structures of these compact configurations led to the
possibility of stellar transitions of explosive character, from a metastable state to a stable
state, with great amounts of liberated energy\,\cite{Amb3,Sak}.
These transitions were related to stars with negative (or anomalous) mass defects
characterized by energy excesses with respect to the energies they would have to be (stable)
bound systems.
Anomalous mass defects were interpreted in terms of a catastrophic additivity violation
of the internal energy due to the very intense gravitational fields in the interior of such
superdense stellar configurations\,\cite{Amb4,Amb5}.
An important aspect of these early works rests on the fact that they included the strange
baryons - hyperons - in the degenerate (strange) nuclear plasma together with neutrons,
protons and electrons.
The baryons being made of quarks, it appeared to be natural to expect unbound quarks to exist
in the interior of hyperdense stars.
Within this assumption, and in an epoch when the physics of the strong interactions was very
incomplete, N. Itoh considered the possibility of hypothetical compact stars made of pure quark
matter\,\cite{Ito}.
With the subsequent developments of the strong interactions theory, new interests came into
play connecting the strange nuclear plasma with the physics of quarks.
The nuclear interactions within the superdense stellar matter turned out to be described in
terms of the baryon constituent quarks.
In this context, the strange quark matter (SQM) concept appeared leading to the conjecture of
the absolute stability of nuclear matter.
The Bodmer-Terazawa-Witten conjecture\,\cite{Bod,Wit}, which says that the SQM should be the true
ground state of the nuclear matter, has attracted a great deal of attention.
SQM is a type of quark matter made of approximately equal amounts of u, d and s quarks with
a small admixture of electrons in order to maintain the charge neutrality.
Its energy per baryon might be lower than the one in ordinary nuclear matter.
The SQM properties are of great importance for nuclear physics and astrophysics.
In the nuclear physics context, E. Farhi and R. L. Jaffe studied, within the MIT Bag Model
(BM), the dependence of the SQM stability on the model parameters, namely, the bag constant
$B$, the strong interactions coupling constant $\alpha_c$, and the strange quark mass
$m_{\rm s}$\,\cite{Far}.
In the astrophysical context, the Bodmer-Terazawa-Witten conjecture has proved to be of great
significance for the physics of the strange stars.
Since the 1980's, the properties of strange stars have been considered within the BM\,
\cite{AFO,HZS,Var2,Koh}.
In the BM, the quarks enter the equation of state (EOS) as free particles, with the quark
confinement being represented by the bag constant $B$.
Another well known model is the Nambu-Jona-Lasinio model\,\cite{NJL1,NJL2} used to investigate
quark matter properties in compact stars in Refs.\,\cite{DPM1,DPM2}.
It exhibits chiral symmetry breaking, but the quark confinement is not explicitly included.
In alternative investigations, mass density dependent models were considered to represent
confinement in Refs.\,\cite{BeL,LuB,Day1,Day2}.
In a recent work, the Richardson potential\,\cite{Ric}, which incorporates the asymptotic
freedom and linear quark confinement, and also used in Refs.\,\cite{Day1,Day2}, was considered
to investigate the SQM in strong magnetic field by the authors of Ref.\,\cite{SHS}.
Due to the highly nonlinear character of the theory of strong interactions, it was difficult
to deal with a definite model of EOS naturally including the quark confinement in terms of the
interactions between quarks and antiquarks, and gluons.
Thanks to the developments of quantum chromodynamics (QCD), the fundamental theory of the
strong interactions, great advances have been made to derive an EOS, including
perturbative and/or nonperturbative effects of confinement, to describe quark matter at all
finite densities and temperatures.
Recently, Yu. A. Simonov derived, from the first principles, the nonperturbative equation of
state (NPEOS) of the quark-gluon plasma in the framework of the Field Correlator Method (FCM)
\cite{Si6}.
In the FCM (for a review see Ref.\,\cite{DiG} and references therein), the dynamics of
confinement is naturally included in terms of the color-electric and color-magnetic
correlators.
The main parameters of the model that enter the NPEOS are the gluon condensate $G_2$ and the
large distance static ${\rm Q\bar Q}$ potential $V_1$, at fixed quark masses and temperature.
The model covers the entire phase diagram plane, from the low $T$ and large $\mu$ regime to
the large $T$ and low $\mu$ regime.
By connecting the FCM and lattice simulations, at $\mu_c=0$\,, the critical temperature turned
out to be $T_c\sim170$ MeV for $G_2=0.00682\;{\rm GeV}^4$\,\cite{ST1,ST2}.
Very recently, V. D. Orlovsky and Yu. A. Simonov considered the quark-hadron thermodynamics in
the presence of the magnetic field within the FCM\,\cite{OS}.
Astrophysical applications of the FCM have been made in the study of neutron stars interiors
\cite{Bur,Bal,Plu} and in the early Universe cosmology\,\cite{Cas}.
The authors of Ref.\,\cite{Bom,Log1,Log2} also applied the FCM to the study of phase transitions
in neutron stars matter and to the investigation of the structural properties and stability of
hybrid stars.
Recently we applied the FCM to investigate the properties of the strange stars and the SQM
stability in Refs.\,\cite{Fla1,Fla2}.
Of particular significance is the gradual decrease of the widths of the SQM stability windows
with the increase of $V_1$\,, being zero at $V_1=0.5\,{\rm GeV}$\,, the value of $V_1$
determined from lattice calculations\,\cite{KaZ}.
This aspect is of great importance to investigate the existence of strangelets, mainly in the
case of exploding stars with the liberation of matter/energy into the free space, which we here
briefly consider (at the end of the present paper).
In the present work, we study the general aspects of the mass defects of strange
stars, without crust and magnetic field, within the framework of the FCM.
We do not consider the crust here by the same reasons we disregarded it in our previous paper
\cite{Fla1}.
Crust contributions to the masses of strange stars have been estimated to be
$M_{\rm cr}\simeq2.5\times10^{-5}\,M_\odot$\,\cite{AFO},
$M_{\rm cr}\simeq(0.9\,-\,2.3)\times10^{-5}\,M_\odot$\,\cite{Var4,Var5}, and
$M_{\rm cr}\simeq3.4\times10^{-6}\,M_\odot$\,\cite{TLu}.
As we shall see below, the mass defects magnitudes we obtained in the present work are of
the order of $10^{53}\,{\rm erg}\simeq0.056\,M_\odot$\,, corresponding to
$\sim2.2\times10^3\,M_{\rm cr}$\,, $\sim(2.4\,-\,2.6)\times10^3\,M_{\rm cr}$\,, and
$\sim1.64\times10^4\,M_{\rm cr}$\,, respectively.
Crusts are important to investigate compact stars glitches, but we assume here
that they are insignificant for the purposes of the present work.
On the other hand, in this first attempt to investigate the mass defects of the strange
stars, we are interested in the solutions not affected by preferred directions due to
magnetic fields, of particular importance to investigate magnetars and soft gamma-ray
repeaters\,\cite{Hur} as well.
Differently from our strategy adopted in Ref. \cite{Fla1}, we here also consider unstable
solutions of the hydrostatic equilibrium equations of Tolman-Oppenheimer-Volkov.
As a result, depending on the values of the model parameters, $G_2$ and $V_1$\,, solutions with
non-negative and/or negative mass defects along a given sequence of stellar configurations are
possible.
Our aim is to understand the effects of the nonperturbative dynamics of confinement on the binding
energies of the strange stars.
We give special attention to the anomalous mass defects and, at the end of the paper, briefly
comment the corresponding consequences for astrophysical phenomena, such as gamma-ray bursts,
supernovae neutrinos or quark-novae explosions.
The present paper is organized as follows.
In Sec.\,\ref{npeos} we show the main equations to be used in our calculation.
In Sec.\,\ref{stc} we present the equations to calculate the important quantities of the stellar
configurations.
In Sec.\,\ref{res} we show the results and in Sec. \ref{frmks} we give the final remarks.
\section{The NPEOS at zero temperature}
\lb{npeos}
In previous works, we outlined the main features of the FCM and showed the equations
used to investigate strange stars and strange quark matter properties\,\cite{Fla1,Fla2}.
So, we now write only the main equations we need here.
For constant $V_1$, the pressure, energy density and number density of a (one flavor)
quark gas at $T=0$ are given by
\bq
p_q^{SLA}=\frac{N_c}{3\pi^2}\Bigg\{\frac{k_q^3}{4}\sqrt{k_q^2+m_q^2}-
\frac{3}{8}\;m_q^2\bigg[k_q\sqrt{k_q^2+m_q^2}-m_q^2\;\ln\bigg(\frac{k_q+\sqrt{k_q^2+m_q^2}}{m_q}\bigg)
\bigg]\Bigg\}\;,
\lb{pqT0}
\eq
\bqn
\varepsilon_q^{SLA}&=&\frac{N_c}{\pi^2}\Bigg\{\frac{k_q^3}{4}\sqrt{k_q^2+m_q^2}+
\frac{m_q^2}{8}\;\bigg[k_q\sqrt{k_q^2+m_q^2}-m_q^2\;\ln\bigg(\frac{k_q+\sqrt{k_q^2+m_q^2}}{m_q}
\bigg)\bigg]\nonumber\\
&+&\frac{V_1}{2}\;\frac{k_q^3}{3}
\Bigg\}
\lb{eqT0}
\eqn
and
\bq
n_q^{SLA}=\frac{N_c}{\pi^2}\;\frac{k_q^3}{3}\;,
\lb{nqT0}
\eq
where
\bq
k_q=\sqrt{(\mu_q-V_1/2)^2-m_q^2}\;,\;\;\;\;(q=\rm{u,d,s})\;,
\lb{kq}
\eq
and $N_c=3$ is the color number; and $SLA $ indicates the single line approximation considered
in Ref.\,\cite{Si6}.
When $V_1=0$, the ordinary Fermi momentum $k_F$ of a free quark gas is recovered in Eq.\,(\ref{kq}).
The additional term $(V_1/2)k_q^3/3\;$ in Eq.(\ref{eqT0}) comes from the large distance
static $Q{\bar Q}$ potential $V_1$.
Inside a strange star, the weak interaction reactions $d\rightarrow u+\ex+{\bar\nu}_\ex$\,,
$\ex+u\rightarrow d+\nu_\ex$\,, and $s\rightarrow u+\ex+{\bar\nu}_\ex$\,,
$u+\ex\rightarrow s+\nu_\ex$\, imply weak equilibrium between quarks, whereas neutrinos and
anti-neutrinos leave the star without interaction and their chemical potentials can be set to zero.
In this case, the chemical equilibrium is given by
\bq
\mu_d=\mu_u+\mu_\ex\;\;\;{\rm and}\;\;\;\mu_s=\mu_d\,.
\lb{mud}
\eq
The overall charge neutrality requires that
\bq
\frac{1}{3}(2n^{SLA}_u-n^{SLA}_d-n^{SLA}_s)-n_\ex=0\,,
\lb{chn}
\eq
where $n_i$ is the number density of the particle $i=$u,d,s,e.
To calculate stellar configurations, with charge neutrality and chemical equilibrium,
the total pressure and energy density, including electrons are given by
\bq
p=\sum_{q=u,d,s}p^{SLA}_{q}-\Delta|\varepsilon_{\rm vac}|+p_{\rm e}\;,
\lb{pqgl}
\eq
\bq
\varepsilon=\sum_{q=u,d,s}\varepsilon^{SLA}_{q}+
\Delta|\varepsilon_{\rm vac}|+\varepsilon_{\rm e}\;,
\lb{eqgl}
\eq
where
\bq
\Delta|\varepsilon_{\rm vac}|=\frac{11-\frac{2}{3}N_f}{32}\Delta G_2\;,
\lb{dvac}
\eq
is the vacuum energy density difference between confined and deconfined phases and
$N_f$ is the number of flavors.
The difference between the values of the gluon condensate, as predicted by lattice
calculations, is $\Delta G_2=G_2(T<T_c)-G_2(T>T_c)\simeq \frac{1}{2}G_2$ \cite{ST1,ST2}.
In order to obtain the numerical correspondence between FCM and BM, we make the
identifications: $\Delta|\varepsilon_{\rm vac}|=B$ and $V_1=0$.
However, we here emphasize that $\Delta|\varepsilon_{\rm vac}|$ is essentially a
nonperturbative quantity.
For the quark masses, we use $m_u=5$ MeV, $m_d=7$ MeV and $m_s=150$ MeV.
The corresponding equations for the degenerate electron gas are similar to the ones above
and can be easily obtained by making the changes: $N_c\rightarrow1$, $V_1\rightarrow0$,
$\mu_q\rightarrow\mu_\ex$ and $m_q\rightarrow m_\ex$.
We use the same numerical strategy adopted in Ref.\,\cite{Fla1} to calculate strange
stars configurations.
\section{compact stars configurations}
\lb{stc}
Compact stars configurations are calculated by numerical integration of the
Tolman-Oppenheimer-Volkov hydrostatic equilibrium equations \cite{ShT,Gle,ZeN}.
Of particular importance here is the total gravitational mass of a compact star,
\bq
M=4\pi\int^R_0\varepsilon(r)\,r^2dr\;,
\lb{stm}
\eq
which is the mass that governs the Keplerian orbital motion of the distant gravitating
bodies around it, as measured by external observers.
The proper mass is given by
\bq
M_P=\int^R_0\varepsilon(r)\,dV(r)\;,
\lb{ma}
\eq
where $dV(r)=4\pi[1-2Gm(r)/r]^{-1/2}\,r^2\,dr$ and $m(r)$ is the mass within a sphere of radius $r$.
The proper mass is the sum of the mass elements $dm(r)=\varepsilon(r)\,dV(r)$ measured by a local
observer.
The baryonic mass (also called rest mass) of a star is $M_A=N_Am_A$, where
\bq
N_A=\int^R_0n_A(r)\,dV(r)
\lb{NA}
\eq
is the number of baryons within the star, $m_A$ is the mass of the baryonic specie $A$, and
\bq
n_A=\frac{1}{3}(n^{SLA}_u+n^{SLA}_d+n^{SLA}_s)
\lb{na}
\eq
is the baryon number density.
The baryonic mass has a simple interpretation: it is the mass that the star would have if
its baryon content were dispersed at infinity.
In the case of the strange stars (because of the quark confinement), $N_A$ is the equivalent
number of baryons (not quarks).
There is some freedom to choose the baryonic mass.
In earlier texts, the baryonic mass was taken as the mass $m_{\rm H}$ of the hydrogen atom\,
\cite{Amb3}; the $^{56}{\rm F_e}$ mass per baryon $m_0\equiv m(^{56}{\rm Fe})/56$\,
\cite{Var2,HTW,ZeN,Var1,Var3}; or the neutron mass $m_n$\,\cite{Sak,Amb4,Fla1,Gle}.
We here assume $m_A=m_n$, as in Ref.\,\cite{Fla1}.
Comparison of some results with respect to $m_0$ is also made below.
Let us now consider the mass defect of a compact stars we are concerned in the present work.
The {\it incomplete mass defect} or, for short, the {\it mass defect} is the difference
$\Delta_2M=M_A-M$ (which in our notation\footnote{We here follow the notation according
to Refs.\,\cite{Amb1,Amb2,Sak,Var1,Var3}.
} is minus the binding energy $E_b$ defined in Refs.\,
\cite{ZeN,Gle}).
It corresponds to the energy released to aggregate from infinity the dispersed baryonic matter.
A stellar configuration is stable if $\Delta_2M>0$ (normal mass defect) and unstable if
$\Delta_2M<0$ (anomalous mass defect).
\section{Results}
\lb{res}
We calculated sequences of strange star configurations for central densities in the range
$11<\log\rho_c<18$.
In general, the forms of the sequences and the respective mass defects of stellar
configurations strongly depend on the values of the model parameters.
A typical example is shown in Fig.\,\ref{mg2v11}, for $G_2=0.006\,\text{GeV}^4\,$\cite{Si6}
and $V_1=0$.
The stellar sequence present three branches delimited at the labeled points 1 and 2, where
the solid and dashed curves cross itself, and in which $M=M_A$, as shown in panel (a).
In the intermediate branch we have $M<M_A$, required by the stability conditions
against transition to diffuse matter.
We have $M>M_A$ at densities $\sim10^{15}{\rm g\,cm^{-3}}$ in the first branch\footnote{Not
well visible in the scales of panels (a) and (b), but visible in panel (c).
}, and $>5.5\times10^{16}{\rm g\,cm^{-3}}$ in the third branch.
An investigation of strange stars within BM, for the values of the bag constant in the range
$50\,{\rm MeV\,fm^{-3}}\leq B\leq 70\,{\rm MeV\,fm^{-3}}$, showed the absence of the anomalous
mass defect in strange stars \cite{Var3}.
Anomalous mass defects does not have been obtained because of the low used values of $B$, which
numerically correspond (in our calculation with $V_1=0$) to lower values of
$\Delta|\varepsilon_{\rm vac.}|$ (or $G_2\,$).
In fact, in the FCM, for $V_1=0$ and $G_2\gappl0.0043\,{\rm GeV}^4$, the stellar configurations
present normal mass defects.
To obtain, within the BM, stellar configurations with anomalous mass defects in the first branch
we need $B>78.7\,{\rm MeV\,fm^{-3}}$ (corresponding to $G_2>0.0043\,\text{GeV}^4\,$).
To obtain anomalous mass defects both in first and in third branches, we need
$B\gappr110\,{\rm MeV\,fm^{-3}}$ (corresponding to $G_2\gappr0.006\,\text{GeV}^4\,$).
Values of $B$ between $150\,{\rm MeV\,fm^{-3}}$ and $170\,{\rm MeV\,fm^{-3}}$ were considered
to explain the time elapsed between the transition from a metastable neutron star generated by a
supernova explosion and the new collapse generating the delayed gamma-ray burst by the authors
of Ref.\,\cite{Ber}.
Moreover, higher values of $B$ up to $337\,{\rm MeV\,fm^{-3}}$ and $353\,{\rm MeV\,fm^{-3}}$
were considered, but to calculate at nonzero temperatures the quark deconfinement in the cores
of protoneutron stars\,\cite{Lug}.
Of particular interest is the dependence of $M$ with the number of baryons $N_A$
shown in panel (b), with the labels 1 and 2 as in panel (a).
The cusp is at the maximum value of $M$, where $N_A$ is also maximum\footnote{See footnote 2
in Ref.\,\cite{Fla1}.
}.
Also shown is the $M_A$ plot with its upper ``endpoint''\footnote{In reality, it is not an
endpoint because, at the upper point, the plot comes back along the same straight line.
} at the maximum $M_A$\,.
In the upper part of the $M\,{\rm vs.}\,N_A$ plot (above 1) the situation is analogous to that of neutron
stars in that $dM/dN_A<m_A$ everywhere on the corresponding plot segments; $M<M_A$ in the second
branch and $M>M_A$ in the third branch\,\cite{ZeN}.
However, a fact that was not observed in earlier works (because of the EOS used) is that $M>M_A$
in the first branch (below 1).
Moreover, the slope starts with $dM/dN_A>m_A$\,, turns to $dM/dN_A=m_A$ at an intermediate point
and then to $dM/dN_A<m_A$ as $N_A$ grows.
In contrast with neutron star configurations, we have here a situation with $M>M_A$ and
$dM/dN_A>m_A$ apparently not obeying the $dM/dN_A=m_A\,[1-2GM/R]^{1/2}$ prescription\,\cite{HTW}.
This is a characteristic feature of the anomalous mass defects occurring in the first branch
making evident the role of the confinement effects.
Panel\,(c) shows the mass defect as function of $M/M_\odot$ with the delimiters
1 and 2 as in panels (a) and (b).
Differently from Refs.\,\cite{Amb1,Amb2,Sak}, the mass defects are also negative in the first
and third branches\footnote{From now on, we do not mention the value of $\Delta_2M$ at the
origin because it is obviously zero as $M$\,, $ M_A$ and $N_A\rightarrow0$\,, unless stated
otherwise.
}.
For the given values of the parameters $G_2$ and $V_1$, the maximum $\Delta_2M$ magnitude
is of the order of $\sim0.15\times10^{53}\,{\rm erg}$ at $M\sim0.26\,M_\odot$ in the first branch,
and $\sim0.3\times10^{53}\,{\rm erg}$ at the endpoint of the third branch at $M\sim0.9\,M_\odot$.
A variety of behaviors can be obtained by the variation of the model parameters.
Some typical examples are shown in Fig.\,\ref{mg2v12}.
Panels (a) and (b) show the case $M<M_A$ for which the mass defect has the normal sign
($\Delta_2M>0$) everywhere along the sequence of stellar configurations, as in panel (c).
Panels (d) and (e) correspond to the limit $M=M_A$ at the maximum mass, but with anomalous
mass defects at all the other points of the stellar sequence with $M>M_A$, as depicted in
panel (f).
Finally, panels (g) and (h) show the case $M>M_A$, so $\Delta_2M<0$ at all points along
the sequence, as in panel (i).
Notice the pronounced confinement effects on the stellar sequences in panels (d)-(f) and
(g)-(i).
This general overview shows us that anomalous mass defects can (in principle) be obtained
for arbitrary values of the model parameters.
If $G_2$ is low, $V_1$ must be increased in order to yield anomalous mass defect; if $V_1$
is low, $G_2$ must grow in order to produce the same effect.
As it was stated above, new results emerge with the use of the NPEOS provided by the FCM: for
$V_1$ in the range $0\leq V_1\leq0.5\,{\rm GeV}$, stellar configurations with anomalous mass
defects are possible not only in the first branch but also in the third branch,
at densities larger than the nuclear one.
Merely illustratively, we also show the proper mass $M_P$ which is greater than both $M$ and
$M_A$.
On account of the above features, in the energy range we are considering, the $G_2-V_1$
plane can be divided in three different regions according to the signs of
$\Delta_2M$ as shown in Fig.\;\ref{d2mg2v1}.
In doing so, we obtain three regions.
The first region, A, with $\Delta_2M>0$ everywhere on the sequence.
In the second region, B, with features analogous to those in Fig.\;\ref{mg2v11},
both normal and anomalous mass defects are present in the same stellar sequence.
Finally, the third region, C, with $\Delta_2M<0$ along all the sequence of stellar
configurations.
The idea of Fig.\;\ref{d2mg2v1} serves to predict values of $V_1$ and $G_2$ according
to the types of the stellar configurations and the respective mass defects we want.
A star with anomalous mass defect has an exceeding stored energy with respect to the
one needed to form a compact stable bound system.
In principle, in a given sequence of stellar configurations, any star with $\Delta_2M<0$
might explode or implode (for example, in the presence of certain perturbations) with a
liberation of an enormous amount of energy.
In the case of explosion, the scattered matter will have a nonzero kinetic energy at
infinity.
On the other hand, it is well known that stellar configurations obtained from
Tolman-Oppenheimer-Volkov equations are stable if $dM/d\rho_c>0$\,, corresponding to the
stars in the ascending branch of the stellar sequence, as in panel (a) of Fig.\,\ref{mg2v11}.
In the descending branch, where $dM/d\rho_c<0$, the stellar configurations are unstable
against gravitational collapse to black hole.
Then, a stable configuration pass from stability to instability at the peak of the sequence
where $M$, $M_A$ and $N_A$ attain their maximums (for a detailed analysis of stability,
see Refs.:\,\cite{Gle,HTW}).
Hence, the investigation of the mass defects in this transition limit may be of particular
interest.
Among the many possibilities, as the ones shown in Figs.\;\ref{mg2v11} and \ref{mg2v12},
let us now consider (both normal and anomalous) mass defects at the maximum masses of the
stellar configurations, as depicted in Fig.\,\ref{d2mg2v11}.
The plots cover a large area in the $G_2-\Delta_2M$ plane, as shown in panel (a).
It is evident that nonnegative values of $\Delta_2M$ occur for $G_2\gappl0.00927\,{\rm GeV}^4$\;,
which gives the vacuum energy density
$\Delta|\varepsilon_{\rm vac}|\gappl0.0013\,{\rm GeV}^4\simeq170\,{\rm MeV\,fm^{-3}}$.
Above these values of $G_2$ (or $\Delta|\varepsilon_{\rm vac}|$), $\Delta_2M$ is anomalous
whatever the values of $V_1$ may be between zero and 0.5\,GeV.
At $V_1=0.5\,{\rm GeV}\,$, the anomalous mass defect attains its maximum magnitude at
$\simeq6.27\times10^{53}\,{\rm erg}$\,, but for $G_2\simeq0.000625\,{\rm GeV}^4$\,,
corresponding to a strange star with $M\simeq1.59\,M_\odot$ and $M_A=1.24\,M_\odot$\,.
In the FCM framework, this is the maximum allowed energy to be liberated in a possible
explosion.
Such a star has a fraction of stored energy around $|\Delta_2M|/M\sim22\,\%\,$, but it is
not a maximum.
For instance, in the range $0\leq G_2\gappl0.0095\,{\rm GeV}^4$\,, the fractions of the mass
excess may be as large as
$\sim25\,\%$\, for $V_1=0.3\,{\rm GeV}$ and $G_2=0.006\,{\rm GeV}^4$\,;
$\sim34\,\%$\, for $V_1=0.4\,{\rm GeV}$\, and $G_2=0.007\,{\rm GeV}^4$\,; and
$\sim42\,\%$\, for $V_1=0.5\,{\rm GeV}$\, and $G_2=0.0095\,{\rm GeV}^4$\,.
Concerning the masses in the above range of the anomalous mass defects, along the
$V_1=0.5\,{\rm GeV}$ curve, they vary from $M\simeq 4.7\,M_\odot\,$ (maximum at the
$\Delta_2M=0$ limit) at $G_2=2.5\times10^{-5}\,{\rm GeV}^4$ to $M\simeq0.63\,M_\odot\,$ at
$G_2=0.0095\,{\rm GeV}^4$.
For other values of $V_1$ the masses assume intermediate values.
As $G_2$ increases, the apparent "convergence" of the curves led us (speculatively) to
extrapolate our calculations to $G_2=0.1\,{\rm GeV}^4$, beyond the limits of the analysis
made in Ref.\,\cite{Iof}, as shown in panel (b).
As a result we obtained a slightly ascending tail with
$\Delta_2M\simeq-2.7\times10^{53}\,{\rm erg}$ at the endpoint.
Along this tail the masses vary, for example,
from $M\simeq0.49\,M_\odot\,$ at $V_1=0$ and $G_2=0.05\,{\rm GeV}^4$
to $M\simeq0.25\,M_\odot\,$ at $V_1=0.5\,{\rm GeV}$ and $G_2=0.1\,{\rm GeV}^4$.
For the intermediate values of $V_1$ the masses are also intermediate.
The mass excess fractions may be as large as
$\sim33\,\%$\, for $V_1=0$ and $G_2=0.05\,{\rm GeV}^4$\,;
$\sim52\,\%$\, for $V_1=0.3\,{\rm GeV}$ and $G_2=0.08\,{\rm GeV}^4$\,; and
$\sim60\,\%$\, for $V_1=0.5\,{\rm GeV}$ and $G_2=0.1\,{\rm GeV}^4$\,.
Extending (now, arbitrarily speculatively) our extrapolation to $G_2=1\,{\rm GeV}^4$\,, the
anomalous mass defects along the tail are not greater than $-1.2\times10^{53}\,{\rm erg}$\,.
At $G_2=1\,{\rm GeV}^4\,$, the masses vary from $M\simeq0.11\,M_\odot\,$ at $V_1=0$ to
$M\simeq0.09\,M_\odot\,$ at $V_1=0.5\,{\rm GeV}\,$; the excess fractions changing from
$\sim68\%$ to $\sim74\%$\,, respectively.
For large values of $G_2$ (say, $G_2>0.07\,{\rm GeV}^4$)\,, the curves concentrate in a
narrow band.
Then, it should be very difficult (or a very accurate determination, as for instance
in gamma-ray bursts observations, should be required) to extract, from the $|\Delta_2M|$
measurements around $\sim3\times10^{53}\,{\rm erg}$\,, reasonable estimates for $V_1$ and/or
$G_2$.
By the way, as a general case in the $G_2-\Delta_2M$ plane, we need an additional
measurement to determine unambiguously the model parameters $G_2$ and $V_1$.
For the sake of comparison with the BM, the open circles along the $V_1=0$ curve correspond
to the values of $B$ used in Refs.\;\cite{AFO,HZS,Koh,Var1,Var2,Var3,Ber}.
Here, a curious fact comes from the supernova SN1987A.
The neutrino signals were detected at Kamiokande II\,\cite{Kah} and at IMB\,\cite{Arn}.
The total energy of the observed neutrinos was found to be $\sim3\times10^{53}\,{\rm erg}$.
It was pointed out that one signal might have been originated in the formation of a
neutron star after the supernova explosion and the other signal in a possible formation of a
strange star \cite{Var3}.
It is inquisitive that the values of $|\Delta_2M|$ along the tail in panel (b) of
Fig.\,\ref{d2mg2v11} are roughly coincident with the energy of the SN1987A neutrinos.
Also, of similar magnitudes are the gamma-ray bursts GRB970828 with
$\sim2.7\times10^{53}\,{\rm erg}$\,\cite{Ber} and GRB971214 with an inferred energy loss of
$\sim3\times10^{53}\,{\rm erg}$\,\cite{Kul} (assuming isotropic emissions).
Hence, measurements of $|\Delta_2M|$ around these energy values may be of particular importance.
Finally, within our freedom to choose the value of $m_{\rm A}$, we performed our investigation
assuming $m_{\rm A}=m_{\rm n}$.
Taking into account that $m_A$ enters the expression of $M_A$ as an external factor multiplying
$N_A$ in Eq.\,\ref{NA}, the conversion formula given $\Delta_2M$ in terms of
$m_0=m({\rm ^{56} F_{\rm e}})/56$ is
\bqn
\Delta_2M({\rm ^{56} F_{\rm e}})&=&\Delta_2M+(\eta-1)M_A\nonumber\\
&=&\Delta_2M-0.175\times10^{53}(M_A/M_\odot)\,,
\lb{d2MFe}
\eqn
where $\eta=m_0/m_n\simeq0.9902$ and
$M_\odot\simeq1.988\times10^{33}{\rm g}\simeq1.787\times10^{54}{\rm erg}$\,.
In Fig.\,\ref{d2mg2v1Fe}, the curves corresponding to $m_A=m_0$ are slightly shifted with
respect the ones for $m_A=m_{\rm n}$.
In the context of baryonic stars, interesting comments about the possibility of states
with $\Delta_2M<0$ be reached by the release of nuclear energy, for the case of rarefied
hydrogen and the case of rarefied iron vapor dispersed at infinity, are made in Ref.:\,\cite{ZeN}.
\section{Final remarks}
\lb{frmks}
Previous investigations showed that the FCM provides new possibilities for the investigation
of compact stellar configurations\,\cite{Bal,Bur,Plu,Bom,Log1,Log2,Fla1}.
In the present work, we considered the general aspects of the mass defects of strange stars
within the FCM without magnetic field, with special emphasis on the anomalous mass defects.
Concerning the magnitudes of the anomalous mass defects, our results are consistent with the
estimated electromagnetic energies of the gamma-ray bursts, varying from $0.07\times10^{53}\,{\rm erg}$\,
in GRB970508 to $5\times10^{53}\,{\rm erg}$\, in GRB011211 (assuming isotropic emission),
given in Table 1 of Ref.\,\cite{Ber}\,, and the theoretical energy predictions of quark-nova
explosions \cite{RaD}.
Since quarks are not freely observed, a strange star in an explosion process must liberate
its energy excess as neutrinos, gamma-rays, gravitational waves, or other forms of matter/energy.
If the ejected matter is made of hadrons, then a transformation to the hadron phase is needed
in order to disperse the hadronic content to infinity.
Another possibility would be the energy release in the form of strangelets \cite{Wit,Far,AFO}.
Strangelets are lumps of self-bound matter containing as few as a thousand of u, d and s quarks.
The question of strangelets was considered in Ref.\,\cite{Klu}.
It was argued that a disruption of a strange star would contaminate the Galaxy with an exceeding
density of strangelets with respect to that required to transform neutron stars into strange
stars\,\cite{Cal}.
However, in order to such a contamination takes place, it is expected that the strangelets
must survive a long time, hence the need to investigate the SQM stability.
Very recently, we considered for several values of $V_1$ the behavior of the stability
windows of the SQM (with respect to the $^{56}F_\ex$ nucleus) with chemical equilibrium
and charge neutrality (cf. panel (a) of Fig.\,3 in Ref.:\,\cite{Fla2}).
Within the same line, we determined here the value of $V_1$ in order to give a zero stability
window at the s-quark mass $m_s=0.15\,{\rm GeV}$\,.
As a result we obtained $V_1=0.33\,{\rm GeV}$, as shown in panel (a) of Fig.\,\ref{fewin},
which also includes (for comparison) the nonzero windows at $V_1=0$\,, 0.1 GeV and 0.2 GeV.
At the given s-quark mass (dashed horizontal line), the stability windows are zero in the range
$0.33\,{\rm GeV}\leq V_1\leq0.5\,{\rm GeV}$\,.
On the other hand, the largest window width occurs at $V_1=0$ for any s-quark mass $m_s\geq0$,
being maximum at $m_s=0$\,. Then, it should be interesting to express the SQM stability, at a
given $m_s$, in terms of a relative stability, which we crudely estimated by considering
three different possibilities.
If for each value of $V_1$\,, we compare the width of the stability window at
$m_s=0.15\,{\rm GeV}$ with the corresponding (maximum) value at $m_s=0$\,, we observe
that the SQM stability gradually decreases from about 53\,\% at $V_1=0$ to zero at
$V_1=0.33\,{\rm GeV}$, as shown in panel (b).
On the other hand, if for each value of $V_1$\,, we compare the width of the stability
window at $m_s=0.15\,{\rm GeV}$ with the one at $V_1=0$ (at the same $m_s=0.15\,{\rm GeV}$),
the SQM stability also decreases very rapidly from 100\,\% at $V_1=0$ to zero at
$V_1=0.33\,{\rm GeV}$\,(short dashed line).
Finally, if for each value of $V_1$\,, we compare the width of the stability window at
$m_s=0.15\,{\rm GeV}$ with the largest one (at $m_s=0$ and $V_1=0$)\, we observe the
fastest rate of stability decrease (long dashed line).
The relative stabilities are $\gappl40\%$ at $V_1\simeq0.1\,{\rm GeV}$\,; $\gappl20.5\%$
at $V_1\simeq0.2\,{\rm GeV}$\, and $\gappl2.5\%$\, at $V_1\simeq0.3\,{\rm GeV}$\,.
Then, in this aspect, for reasonable values $V_1$ it is unlikely that the strangelets
ejected from strange star explosions should survive so long (before they decay) to arrive
on Earth or other place of the Galaxy.
These results appeared to be in accordance with terrestrial experiments at RHIC which
have not confirmed the existence of the SQM nor proved that it does not exists
\,\cite{San,Abe,Ble}.
In an investigation relating the gamma-ray bursts to a second explosion after the (first)
supernova explosion, it was pointed out that the main difficulties of the model were to
explain the causes of the second explosion and the time elapsed between the first explosion
and second explosion\,\cite{Bom2}.
In this regard, another interesting aspect to be considered would be the connection between
the anomalous mass defects and stellar instabilities.
Merely speculatively, it appears to be reasonable to expect that the greater the magnitude of
the anomalous mass defect is, the greater might be the instability of a strange star configuration.
Correspondingly, the lower might be the time interval between the first (supernova) explosion
and the second (quark-nova) explosion.
However, such an investigation should require detailed studies of the internal structure as
well as internal processes of the strange stars, that merit to be considered elsewhere, in the
author's opinion.
Finally, although the main aim of the present work is the study of the mass defects of the
strange stars, with emphasis on the anomalous mass defects, let us here consider some general
aspects concerning the low-mass strange stars as well as their importance for further
investigations in the FCM.
In Sec.\,\ref{res}, we showed that low-mass strange stars with anomalous mass defects are
possible in the first branch, as in Fig.\,\ref{mg2v11}.
These stars are important for both theoretical and observational investigations.
For $M>M_\odot$, strange stars and neutron stars with the same mass present similar radii,
but for $M<M_\odot$ their radii are markedly different\,\cite{AFO,XDL1}.
Another feature is that the strange stars, being more compact, have lager surface redshifts
than the ones for neutron stars\,\cite{NS1}.
It would be possible to distinguish strange stars and neutron star by radii direct measurements
of low mass pulsar-like stars by observations from X-ray satellites\,\cite{Xu1}(and references
therein).
The identification of strange stars with $M\gappl0.1M_\odot$ can show us if they are bare
stars given that their radii are much lower than the ones for stars with crust, in the low mass
limit\,\cite{Xu2}.
Due to the fact that the quarks and anti-quarks are held together by the strong interaction forces,
the strange stars are self-bound systems, even in the absence of gravitation, whereas neutron
stars are not.
In the low-mass regime, the mass-radius relations of strange stars have been well represented
by the use of the approximated BM equation of state in the $m_q\rightarrow0$ (q=u,d,s) limit,
\bq
p=\frac{1}{3}(\varepsilon-4\,B)
\lb{lmBMp}
\eq
which was used to calculate mass-radius relations in Refs.\,\cite{Wit,AFO}.
The low-mass strange stars obey the $M\propto R^3$ dependence which can be easily explained
within the BM.
For $M\gappl0.3M_\odot$\,, the gravitational pull becomes small compared to the contribution of
the vacuum energy density represented by the bag constant $B$.
Inside the star, due to the high degree of incompressibility of the SQM, the energy density is
nearly constant (cf. Fig.\,1 in Ref.\,\cite{AFO} or Fig.\,8.6 in Ref.\,\cite{NS1}.
In this case, the Newtonian approximation suffices to calculate the mass of the strange star
as being $M\simeq(4\pi/3)R^3\,\varepsilon$\,.
In the FCM, the corresponding simplified NPEOS is (from Eqs.\,(\ref{pqT0})-(\ref{dvac})) given by
\bqn
p&=&\frac{1}{3}\bigg[\varepsilon - \frac{3}{2}V_1n_A - 4\Delta|\varepsilon_{\rm vac}| \bigg] \nonumber\\
&=&\frac{1}{3}\bigg[\varepsilon - \frac{3}{2}V_1n_A - \frac{9}{16}G_2 \bigg]\,.
\lb{lowmp}
\eqn
There is here an important point to be addressed.
Apart from the low-mass strange stars in the first branch, where the use of Eq.\,(\ref{lowmp})
is valid, we also have sequences of stellar configurations with maximum masses in the low-mass
region (as in panel (g) of Fig.\,\ref{mg2v12}) where Eq.\,(\ref{lowmp}) is not generally valid,
even when $M_{\rm max}\simeq0.3\,M_\odot$.
For instance, in the region of anomalous mass defects, corresponding to $0\leq V_1\leq0.5$\,GeV
and $G_2$ in the extrapolated tail ($0.02\,{\rm GeV}^4\gappl G_2\gappl0.1\,{\rm GeV}^4$) in
Fig.\,\ref{d2mg2v11}\,(panel (b)), the maximum masses are in the region
$0.2\,M_\odot<M_{\rm max}<0.5\,M_\odot$\,.
However, as a part of a next work, we say in advance that it is not generally true that
Eq.\,(\ref{lowmp}) is an appropriate approximation\,\cite{Fla4}.
According to our preliminary exact numerical calculations, depending on the values of $V_1$ and
$G_2$, inside a star with $M=M_{\rm max}$ in the low-mass region it is not mandatory for the energy
density $\varepsilon(r)$ be nearly constant in order to allow for the usage of the Newtonian
approximation.
In this case, it appears that the concept of low-mass strange stars must be taken as meaning
$M<M_{\rm max}$ rather than $M<M_\odot$\,.
Correspondingly, we expect that these features may imply interesting consequences to the anomalous
mass defects.
\vspace{.2in}
\centerline{\bf ACKNOWLEDGMENTS}
I would like to thank J. S. Alcaniz, R. Silva and A. P. Santos.
This work was done with the support provided by the Minist\'erio da Ci\^encia , Tecnologia
e Inova\c c\~ao (MCTI).
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,337 |
Q: Can we get maths in PDFs that can be copy-pasted and screen-read? Mathematics in PDFs can be cumbersome for any purpose but reading or printing. In particular, copying mathematics in order to use it in another document, and screen-reading for visually impaired consumers.
Both issues can be solved using the accsup package; see
*
*here for copy-pasting (use accsup to add detokenized math environment) and
*here for screen-reading (use accsup to add a natural language version).
Since both solutions use the same mechanism, it is not clear how to achieve both at the same time, or if that is even possible.
Can we annotate formulae in PDFs so that they can be copy-pasted nicely and read by screen-reading software?
A: I use my answer at the OP's cited link, In which way have fake spaces made it to actual use?, to embed the detokenized math for copy/pasting. In addition, here, I use the pdfcomment package to add textual comments with the \copypaste invocations. The Acrobat pdf accessibility reader will read out loud the contents of the sticky notes (though the order of reading may be a bit strange). However, the sticky notes will not affect the copy/paste of the file.
\documentclass{article}
\usepackage{accsupp,pdfcomment}
\newcommand\copypaste[2]{%
\BeginAccSupp{method=escape,ActualText={\detokenize{#1}}}%
\pdfcomment{#2}#1%
\EndAccSupp{}%
}
\begin{document}
What I have is \copypaste{$x^2 + y^2 = z^2$}{Pythagorean formula} and
\copypaste{$\frac{1}{2}$}{a fraction}.
Try to copy/paste me.
\copypaste{\[
y = \frac{\cos{x}}{1+\cos{x}}
\]}{A display style trigonometric equation}
\end{document}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,599 |
import httplib, urllib
import random
from time import localtime, strftime
import time
def doit():
# voltage
a= random.randint(210, 220) #v1
b= random.randint(210, 230) #v2
c= random.randint(210, 240) #v3
#current
d= random.randint(5,7) #c1
e= random.randint(5,17) #c2
f= random.randint(5,27) #c3
#speed
g= random.randint(1100,1200)
#temperature
h= random.randint(38, 40)
params = urllib.urlencode({'field1': a ,'field2': b,'field3': c ,'field4': d,'field5': e ,'field6': f,'field7': g ,'field8': h,'key':'ETWI0YZCDOHZA6BN'})
headers = {"Content-type": "application/x-www-form-urlencoded","Accept": "text/plain"} #HHTP HEADERS FOR THEIR SERVER
conn = httplib.HTTPConnection("api.thingspeak.com:80") #opening a connection in port 80
try:
conn.request("POST", "/update", params, headers)
response = conn.getresponse()
print a
print b
print c
print d
print e
print f
print g
print h
#print strftime("%a, %d %b %Y %H:%M:%S", localtime())
print response.status, response.reason
data = response.read()
conn.close()
except:
print "connection failed"
#sleep for 16 seconds (api limit of 15 secs)
if __name__ == "__main__":
while True:
doit()
time.sleep(15)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,452 |
{"url":"https:\/\/lists.gnu.org\/archive\/html\/lilypond-devel\/2009-07\/msg00544.html","text":"lilypond-devel\n[Top][All Lists]\n\n## \"anchors\" in the music stream?\n\n From: Kieren MacMillan Subject: \"anchors\" in the music stream? Date: Wed, 22 Jul 2009 12:05:42 -0400\n\nHello all,\n\n\nLike many Lilyponders, I break down my code into variables, e.g. global (for time signature changes, etc.), notes, dynamics, etc. The main irritation with this (IMO) is that each variable requires a complete set of skips in order to keep the timing accurate.\n\n\nWould it be technically feasible\/possible to establish a system of \"anchors\" instead? That is, could we declare an anchor point in the global setup, and then refer to it in other code? e.g.\n\nglobal =\n{\n\\time 4\/4 s4*4*10\n\\time 3\/4 s4*3*5\n\\time 7\/4 s4*7\n\\time 4\/4 \\anchor #'coda s4*4*10\n\\bar \"|.\"\n}\n\ntempoChanges =\n{\n\\tempo \\markup \"Opening tempo\"\n\\anchorTo #'coda \\tempo \\markup \"Extremely slow\"\n}\n\n??\n\n\nJust a brainstorm from someone who doesn't understand the internals enough to immediately see the stupidity of this idea... =)\n\nThanks,\nKieren.","date":"2019-05-19 18:08:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6916140913963318, \"perplexity\": 13995.718834392463}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232255071.27\/warc\/CC-MAIN-20190519161546-20190519183546-00213.warc.gz\"}"} | null | null |
\section{Introduction}
The 2D Navier-Stokes equation forced at intermediate scales favors the transfer of energy to larger scales, a phenomenon known as an inverse cascade~\cite{kraichnan1967inertial, leith1968diffusion, batchelor1969computation}. Already the first experiments~\cite{sommeria1986experimental, paret1998intermittency} and numerical simulations~\cite{smith1993bose, smithr1994finite, borue1994inverse} on two-dimensional turbulence showed that in a finite domain with low bottom friction, the inverse cascade leads to the accumulation of energy at the system size and formation of coherent vortex structures. Subsequent numerical~\cite{chertkov2007dynamics, laurie2014universal, frishman2018turbulence} and experimental~\cite{xia2009spectrally,orlov2018large} studies demonstrated that these vortices have well-defined isotropic mean profiles with a radial power-law decay of vorticity in the inner region. Analytical progress can be made if the condensate is strong compared to turbulence. Then the self-action of turbulent pulsations is small compared to the action of the mean flow, and a quasi-linear passive theory can be developed~\cite{kolokolov2016structure, kolokolov2016velocity, frishman2017culmination}. If turbulence is excited at asymptotically small scales, the treatment allows one to derive an explicit formula for the condensate vorticity profile, $\Omega(r) = (3 \epsilon/\alpha)^{1/2} r^{-1}$, where $\alpha$ is the bottom friction coefficient and $\epsilon$ is the inverse energy flux. Since the profile does not depend on the type of small-scale dissipation and small-scale forcing, it was called universal~\cite{laurie2014universal}. Note that a similar quasi-linear approach was also used to describe jets in rectangular periodic domains~\cite{frishman2017jets}, on a sphere~\cite{falkovich2016interaction}, and on a periodic beta plane~\cite{woillez2017theoretical, woillez2019barotropic}.
Theoretical predictions in the universal limit agree with the results of numerical studies~\cite{laurie2014universal, frishman2018turbulence}. In these works, the simulations were carried out in periodic domain with dimensions of $2 \pi \times 2 \pi$, and the forcing wave number was at least $k_f = 100$. In laboratory experiments, the length of the inertial interval for the inverse cascade is much shorter, so the question arises of how the vorticity profile will change as $k_f$ decreases. In earlier experiments, the dependence $\Omega(r) \propto r^{-1.25}$ was reported, although a rather large error in the determination of the exponent should be noted~\cite{xia2009spectrally}. Here we systematically study the issue and demonstrate that for a random forcing with spatially homogeneous statistics the vorticity profile becomes steeper than the universal limit, with the difference increasing with the pumping scale but decreasing with the Reynolds number at the forcing scale. We discuss our results in the context of the Reynolds equation for the mean polar velocity of the vortex and conclude that the main distinction from the universal limit is that the effective pumping of the coherent vortex falls off as $r$ increases, leading to the observed steeper behaviour. Comparison with the quasi-linear passive theory~\cite{kolokolov2016structure} shows that it overestimates the effective pumping of the coherent vortex, and we suppose that this is caused by the self-action of fluctuations, which we model by the eddy viscosity resulting in a reasonable agreement with the simulations.
Next, the following question naturally arises: is the self-organization of coherent vortices with a vorticity profile flatter than the universal limit possible? We show that the answer is positive, and to demonstrate this we consider the forcing that has been spatially localized at two small spots. It turns out that the external forcing has no pinning effect on the coherent vortices and most of the time they are far from the pumped regions. In this case, the effective pumping of vortices mostly occurs due to velocity fluctuations that they meet with their outer edges during their movement through the system. Therefore, the effective pumping of vortex has a maximum at a certain distance from its center and decreases as one moves towards the center of the vortex. This effective pumping profile results in a vorticity distribution that is flatter than the universal limit.
\begin{figure*}[t]
\centering{\includegraphics[width=\linewidth]{./figure1.pdf}}
\caption{The stationary state for spatially uniform pumping: (a) snapshot of the vorticity field ($512^2$) for DNS run $A_1$, (b,c) radial profile of the mean vorticity for DNS runs $A_1$-$D_1$ and $C_0$-$C_4$. The straight dashed line in Fig.~\ref{fig:1}b corresponds to the universal limit, $\Omega(r) = (3 \epsilon/\alpha)^{1/2} r^{-1}$.}
\label{fig:1}
\end{figure*}
\section{Numerical Methods}
We solve the incompressible forced Navier-Stokes equation with hyperviscous dissipation and linear bottom friction for a fluid with unit density in 2D:
\begin{equation}\label{eq:1}
\partial_t \bm v + (\bm v \nabla) \bm v = - \nabla p -\alpha \bm v - \nu (-\nabla^2)^{q} \bm v + \bm f,
\end{equation}
where $\bm v$ is 2D velocity, $p$ is the pressure, $\alpha$ is the friction coefficient, $\nu$ is the hyperviscosity, and $\bm f$ is a random forcing. The domain is a doubly periodic square box of size $L=2 \pi$. We work with an isotropic, shortly correlated in time forcing acting in a narrow ring in Fourier space centered on wave number $k_f$, with $\varepsilon = \langle \bm v \cdot \bm f \rangle$ the average energy injection rate, and angular brackets denote time-averaging. In an unbounded system, the inverse energy cascade is terminated by the bottom friction at the scale $L_{\alpha} \sim \varepsilon^{1/2} \alpha^{-3/2}$, where a balance between the energy flux and the bottom friction is achieved~\cite{boffetta2012two}. We assume that the friction coefficient $\alpha$ is small enough, so that $L<L_{\alpha}$, and then the energy, transferred to the domain size $L$ by the inverse cascade, is accumulated there, giving rise to a mean (coherent) flow.
\begin{table}[b]
\begin{center}
\begin{tabular}{c|c|c|c|c|c}
\hline \hline
run & grid & $k_f$ & $\nu$ & $\Re$ & $\epsilon$ \\ \hline
$A_1$ & 512 & 100 & $5\cdot 10^{-35}$ & 304 & $1.94\cdot 10^{-4}$ \\
$B_1$ & 512 & 50 & $2 \cdot 10^{-30}$ & 313 & $2.01 \cdot 10^{-4}$ \\
$C_1$ & 256 & 25 & $8 \cdot 10^{-26}$ & 323 & $2.10 \cdot 10^{-4}$ \\
$D_1$ & 256 & 12.5& $3.5 \cdot 10^{-21}$ & 305 & $2.17 \cdot 10^{-4}$ \\
$C_0$ & 256 & 25 & $2.4 \cdot 10^{-25}$ & 108 & $1.88\cdot 10^{-4}$ \\
$C_2$ & 256 & 25 & $2.4 \cdot 10^{-26}$ & 1078 & $2.30 \cdot 10^{-4}$ \\
$C_3$ & 256 & 25 & $0.8 \cdot 10^{-26}$ & 3235 & $2.46 \cdot 10^{-4}$ \\
$C_4$ & 256 & 25 & $0.8 \cdot 10^{-27}$ & 32348 & $2.73 \cdot 10^{-4}$ \\
\hline \hline
\end{tabular}
\caption{Parameters for the DNS runs.}
\label{tab:1}
\end{center}
\end{table}
DNS results are obtained by integrating (\ref{eq:1}) in the vorticity formulation using the GeophysicalFlows.jl pseudospectral code~\cite{GeophysicalFlowsJOSS}, at resolution $256^2$ and $512^2$, with parameters $\varepsilon = 3.5 \cdot 10^{-4}$, $\alpha = 10^{-4}$, and $q=8$. The pumping covariance spectrum is Gaussian with mean $k_f$ and standard deviation $\delta_f=1.5 \ll k_f$. The high degree of hyperviscosity allows simulations to be performed with relatively low spatial resolution~\cite{chertkov2007dynamics, laurie2014universal, frishman2018turbulence, frishman2017jets}. The initial condition in all our simulations is a state of rest, and each simulation is run until the system reaches a non-equilibrium stationary state, observed by the saturation of the total kinetic energy. After that, the simulation continued for some time, necessary to collect statistics. The time step is fixed for each simulation and it satisfies $\Delta t < c_0 \Delta x/v_{max}$, where $\Delta x$ is the grid spacing, $v_{max}$ is the maximum value of the velocity field projections on the axes of the Cartesian coordinate system, and $c_0$ is equal to $0.3-0.5$. The time step $\Delta t$ is also the correlation time of the exciting force $\bm f$.
\begin{figure}[b]
\centering{\includegraphics[width=0.9\linewidth]{./figure2.pdf}}
\caption{Energy spectra for DNS runs $A_1$-$D_1$.}
\label{fig:2}
\end{figure}
In the first set of simulations ($A_1$-$D_1$), we vary the pumping wave number in the range from $k_f=100$ to $12.5$, while adjusting the hyperviscosity $\nu$ so that the Reynolds number at the forcing scale $\Re = \varepsilon^{1/3}/(\nu k_f^{46/3})$ is about 300, see Table~\ref{tab:1}. In agreement with previous studies~\cite{laurie2014universal,frishman2017jets,frishman2018turbulence}, the flow reaches a condensate steady state, taking the form of a system-sized vortex dipole. The vortices drift slowly over time, with fast turbulent pulsations superimposed onto them (see Fig.~\ref{fig:1}a and video~\cite{SM}). We find that in all cases a significant part of the injected energy $\varepsilon$ is dissipated due to hyperviscosity at high wave numbers $k>k_f$. This is consistent with recent numerical studies~\cite{frishman2017jets,frishman2018turbulence}, but differs from the theoretical analysis of the universal limit~\cite{kolokolov2016structure, frishman2017culmination}, where it is assumed that all the energy is dissipated by bottom friction on large scales. To estimate the inverse energy flux from numerical data, we compute the energy dissipation rate by bottom friction during the steady state regime, $\epsilon = \alpha \langle \int dxdy \, \bm v^2/L^2 \rangle$. The corresponding values are given in Table~\ref{tab:1}, and henceforce they are used to normalize velocities by $(\epsilon/\alpha)^{1/2}$. This estimate is justified because the main contribution to the total energy of the system comes from large scales corresponding to the coherent vortex dipole, see Fig.~\ref{fig:2}. The slopes of the spectra are steeper than $-5/3$ due to the presence of condensate, in agreement with previous studies~\cite{chertkov2007dynamics, frishman2017jets, chan2012dynamics}. The relative fluctuations of the total energy of the system are small and do not exceed $1\%$ even for the case with the largest forcing scale under consideration.
\begin{figure}[t]
\centering{\includegraphics[width=0.9\linewidth]{./figure3.pdf}}
\caption{The dependence of $1-Q$ on the distance $r$ from the vortex center in log-log scale calculated by expression (\ref{eq:4}) (solid lines with markers), using the quasi-linear theory (\ref{eq:kolokolov}) (dotted lines), and using the quasi-linear theory with the turbulent diffusion correction (dashed lines).}
\label{fig:3}
\end{figure}
To obtain the mean vorticity distribution, we shift the individual snapshots describing the instantaneous vorticity field so that the center of one of the vortices (identified by the vorticity maximum) is always at the center of the domain, and then average over all snapshots corresponding to the non-equilibrium steady state. The resulting vorticity distribution $\Omega$ inside the vortex is highly isotropic, and it can be also described in terms of the mean polar velocity $U$ depending on the distance $r$ from the vortex center, $\Omega = (1/r) \partial_r (rU)$. The mean polar velocity satisfies the Reynolds equation~\cite{kolokolov2016structure}
\begin{equation}\label{eq:2}
\alpha U + \nu (-\nabla^{2})^{q} U = -\left( \partial_r + \dfrac{2}{r}\right) \langle u_r u_{\phi} \rangle,
\end{equation}
where $u_r$ and $u_{\phi}$ are radial and polar components of the turbulent velocity fluctuations, and angular brackets denote time-averaging. Therefore, the coherent vortex maintains its existence due to the Reynolds shear stress $\langle u_r u_{\phi} \rangle$ (right-hand side), which balances the dissipative terms inside the vortex (left-hand side). The hyperviscous term is significant inside the vortex core~\cite{kolokolov2016structure}, at distances $r \lesssim R_{\alpha} = (\nu/\alpha)^{1/2q}$, and can be neglected outside, where the mean vorticity profile reveals the behaviour close to power-law for small-scale pumping, see Fig.~\ref{fig:1}b. Note that the velocity profile inside the viscous core was analyzed in detail in Refs.~\cite{parfenyev2021influence, doludenko2021coherent} in the case of an ordinary ($q=1$) viscous dissipation. This regime may be of interest for experiments with thin soap films~\cite{kellay2002two} or freely suspended smectic films~\cite{parfenyev2016nonlinear, yablonskii2017acoustic}, where friction against the bottom is absent. In what follows, we will focus on distances beyond the vortex core. We will use the scale $R_{\alpha}$ to normalize all distances.
Fig.~\ref{fig:1}b shows that the mean vorticity profile of run $A_1$ is close to the universal limit, but as the forcing scale increases (runs $B_1$-$D_1$), the vorticity profiles become steeper. To study the dependence of the profile slope on the Reynolds number (runs $C_0-C_4$), we fix the pumping wave number $k_f=25$ and change the hyperviscosity according to Table~\ref{tab:1}. The resulting mean vorticity profiles are presented in Fig.~\ref{fig:1}c. It can be concluded that an increase in the Reynolds number at the forcing scale results in flatter vorticity profiles.
\section{Effective Pumping of the Vortex}
\begin{figure}[t]
\centering{\includegraphics[width=0.9\linewidth]{./figure4.pdf}}
\caption{The intensity of velocity fluctuations as a function of the distance $r$ to the vortex center.}
\label{fig:4}
\end{figure}
To discuss the obtained results, let us analyze how and why they differ from the universal limit. Recall that the universal limit formally follows from Eq.~(\ref{eq:2}), if relation $\langle u_r u_{\phi} \rangle = \varepsilon(1-Q)/\Sigma$ is used for the Reynolds stress component, where $\Sigma = r \partial_r (U/r)$ describes the mean-flow shear rate, and if one neglects the hyperviscous term, since the region $r \gtrsim R_{\alpha}$ is analyzed~\cite{kolokolov2016structure}:
\begin{equation}\label{eq:3}
\alpha U = -\left( \partial_r + \dfrac{2}{r}\right) \dfrac{\varepsilon}{\Sigma} (1-Q).
\end{equation}
Here $Q$ can be thought of as a phenomenological parameter that describes the effective pumping intensity $\varepsilon (1-Q)$ of the coherent vortex. Now let us try to find the power-law solution of this equation in the form $\Omega \propto r^{-\beta}$, $U \propto r^{1-\beta}$, etc., and we will immediately obtain that the value of $1-Q$ should depend on $r$ as $1-Q \propto r^{2(1-\beta)}$. The universal limit corresponds to $\beta=1$ and then the solution of this equation is $U=(3 \epsilon/\alpha)^{1/2}$ and, accordingly, $\Omega = (3 \epsilon/\alpha)^{1/2} r^{-1}$, where $\epsilon=\varepsilon(1-Q)$. However, if $\beta>1$, then the effective pumping intensity of the coherent vortex $\varepsilon(1-Q)$ should decrease with increasing $r$. Can we support this statement quantitatively?
\begin{figure*}[t]
\centering{\includegraphics[width=\linewidth]{./figure5.pdf}}
\caption{The stationary state for spatially localized pumping: (a) snapshot ($512^2$) of the vorticity field, (b) radial profile of the mean vorticity demonstrates behaviour flatter than the universal limit, (c) the dependence of $1-Q$ on the distance $r$ from the vortex center in log-log scale calculated by expression (\ref{eq:4}) based on DNS data.}
\label{fig:5}
\end{figure*}
Instead of power-law analysis of Eq.~(\ref{eq:3}), which is not well suited to our DNS runs because the intervals of linear behaviour on the log-log plots are quite short (see Fig.~\ref{fig:1}), we can integrate it directly
\begin{equation}\label{eq:4}
1-Q = - \dfrac{\alpha \Sigma(r)}{\varepsilon r^2} \int_0^r d\xi \, \xi^2 U(\xi).
\end{equation}
From numerical simulations, we know the dependencies $U(r)$ and $\Sigma(r)$, and thus can calculate the dependence of $1-Q$ on $r$. Fig.~\ref{fig:3} shows the results for parameters corresponding to runs $A_1$-$C_1$ and $C_4$, which are chosen for illustrative purposes (solid lines with markers). One can conclude that the effective pumping intensity $\varepsilon(1-Q)$ of the coherent vortex actually decreases with increasing distance $r$ from its center, and the decrease is faster for steeper vorticity profiles. Note also that the effective pumping increases with increasing Reynolds number if the forcing scale is fixed.
The value of $Q$ can be also found theoretically in the framework of a quasi-linear treatment of relatively weak turbulent pulsations against the background of a strong coherent vortex (see Ref.~\cite{kolokolov2016structure} and Appendix~\ref{app:A})
\begin{eqnarray}\label{eq:kolokolov}
\nonumber
&Q \simeq \displaystyle 2 \nu \int_0^{\infty} d \tau \int \dfrac{d^2 \bm k \, k^2 \chi(\bm k) }{(2 \pi)^2} [(k_1 - \Sigma \tau k_2)^2 + k_2^2]^{q-1}& \\
&\displaystyle \times \exp \left[ -2 \int_{0}^{\tau} d \tau' \, \Gamma \left( \sqrt{(k_1 - \Sigma \tau' k_2)^2 + k_2^2} \right) \right],&
\end{eqnarray}
where $\Gamma (k) = \alpha + \nu k^{2q}$ describes dissipation including both the bottom friction and the hyperviscosity term. For our parameters, the pumping covariance spectrum can be safely replaced by $\chi(\bm k) = 2 \pi \delta(k-k_f)/k_f$ and using the dependence $\Sigma(r)$ obtained from DNS, we can calculate $1-Q$, see dotted lines in Fig.~\ref{fig:3}. It turns out that the quasi-linear theory correctly describes the effective pumping of the coherent vortex far from the outer region, but overestimates it on the periphery, especially for relatively small values of $k_f$.
We suppose that this overestimation is due to the self-action of fluctuations, which was neglected in the quasi-linear theory. Fig.~\ref{fig:4} shows the intensity of velocity fluctuations, which weakly depends on the distance $r$ to the vortex center outside the viscous core. Based on these results, one can find that velocity fluctuations are indeed small compared to the mean flow, $\langle u_r^2 + u_{\phi}^2 \rangle^{1/2}/U \sim 0.1$. However, at the same time, the nonlinear
self-action of fluctuations is significant compared to the dissipative terms at the forcing scale, which determine the value of $Q$, i.e. $|(\bm u \nabla) \bm u| \gg |\Gamma (k_f) \bm u|$. To take into account the self-action of fluctuations phenomenologically, we propose to add an additional term $\nu_T k^2$ into $\Gamma(k)$, corresponding to eddy viscosity. Note that in this case there is also an additional contribution to the value of $Q$ proportional to the parameter $\nu_T$, see Appendix~\ref{app:B}. The correlation time of velocity fluctuations can be estimated as $1/|\Sigma(r)|$, and therefore the turbulent viscosity can be modelled as $\nu_T = \gamma/|\Sigma(r)|$, where $\gamma$ is a free parameter that is chosen to match the theory with the DNS. We found $\gamma \sim 4 \cdot 10^{-6}$ for run $A_1$, $\gamma \sim 2 \cdot 10^{-5} $ for run $B_1$, $\gamma \sim 10^{-4} $ for run $C_1$, and $\gamma \sim 1.5 \cdot 10^{-4}$ for run $C_4$. The results are shown in Fig.~\ref{fig:3} with dashed lines and they demonstrate a reasonable agreement with the simulations.
\section{Spatially Localized Forcing}
Finally, we would like to address the question --- is the self-organization of coherent vortices with a vorticity profile flatter than the universal limit possible? To the best of our knowledge, the observation of such vortices has not been reported in the literature until now. In accordance with the previous discussion, the existence of such vortices is feasible if the effective pumping intensity of the coherent vortex increases with distance from its center. To implement such situation, we consider a random forcing localized in space in two regions with radii $a=0.1$, small compared to the expected sizes of coherent vortices, but large compared to the pumping scale ($k_f=100$). The centers of the regions are located at points with coordinates $(\pm L/4, \pm L/4)$. In comparison with DNS run $A_1$, the amplitude of the external force was increased so that the power pumped into the system, averaged over the entire domain area, remained approximately the same.
After some time, a pair of coherent vortices rotating in opposite directions is formed in the system, see Fig.~\ref{fig:5}a and video~\cite{SM}. In the stationary state, vortices slowly move through the system in random directions. The external forcing has no pinning effect on the vortices, see Fig.~\ref{fig:6}. So, the vortices explore all space uniformly.
The pumping creates fuel (turbulent pulsations) for the vortex. In the case of uniform pumping, at each moment of time, an external force excites fluctuations inside the vortex, which feed it. Now fluctuations are excited only in small regions, and the vortex spends most of the time away from them. In the process of its wandering, the vortex is fed by fluctuations that are encountered on its way. Thus, fluctuations penetrate into the vortex mostly from the outer region. In this case, it is reasonable to expect that the effective pumping of the vortex will have a maximum at some distance from its center. According to the previous analysis, such profile of the effective pumping should result in a vorticity distribution that is flatter than the universal limit.
\begin{figure}[t]
\centering{\includegraphics[width=0.8\linewidth]{./figure6.pdf}}
\caption{Each point corresponds to the position of the (positive) vortex center. About 12000 snapshots were analyzed.}
\label{fig:6}
\end{figure}
The above qualitative reasoning can be supported by quantitative observations. Fig.~\ref{fig:5}b shows the radial profile of the mean vorticity distribution. It exhibits behavior flatter than the universal limit, in line with our expectations. Fig.~\ref{fig:5}c presents the dependence of the effective pumping intensity of the coherent vortex on distance from its center. The value of $1-Q$ increases with $r$ inside the vortex and has a maximum at some distance from the vortex center, in accordance with the above qualitative arguments.
\section{Conclusion}
To conclude, we performed hyperviscous numerical study of the vortex condensate in forced 2D turbulence. In the case of spatially uniform pumping, the vorticity profile is steeper than the universal limit, with the difference increasing with the pumping scale but decreasing with the Reynolds number at the forcing scale. An analysis of the Reynolds equation for the mean polar velocity of a coherent vortex led us to the conclusion that this behaviour corresponds to a decrease in the effective pumping intensity of the coherent vortex with distance from its center. Inspired by this observation, we performed an additional simulation with spatially localized forcing, in which the effective pumping intensity of the coherent vortex, on the contrary, increases with $r$ and show that in this case the vorticity profile becomes flatter than the universal limit. Our findings demonstrate that spatially inhomogeneous forcing opens up an additional degree of freedom for controlling the self-organization of coherent vortices.
To explore the possibilities of this additional degree of freedom in future studies, it is reasonable to replace the periodic boundary conditions with no-slip or stress-free walls. In such systems, the positions of coherent vortex structures are almost fixed due to geometric constraints~\cite{xia2009spectrally, molenaar2004angular, doludenko2021coherent, gallet2013two}, and therefore it will be relatively easy to design the desired spatial profile $\varepsilon(r)$ of pumping in the reference frame associated with the vortex.
Despite the relative simplicity of the system under study, the obtained results can be used to understand the processes of self-organization of large-scale vortex currents in the atmosphere and oceans. Since the pumping of such currents occurs in a non-uniform manner, further research in this direction looks promising. In addition to the mean velocity profile inside vortices, an important object of study is the velocity of a coherent vortex as a whole and its statistical properties. In the future, this will help to better understand the motion of hurricanes.
\acknowledgments
I wish to thank Vladimir Lebedev, Igor Kolokolov, and Sergey Vergeles for helpful discussions. The work was supported by the Russian Ministry of Science and Higher Education, project No. 075-15-2022-1099, and by the Foundation for the Advancement of Theoretical Physics and Mathematics ''BASIS''. Simulations were performed on the cluster of the Landau Institute.
| {
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Amidst the controversy surrounding Lil Wayne's Tha Carter V, YMCMB fans are still looking forward to Drake's Views from the 6.
These past months saw several leaks that were possibly set aside for Views from the 6. Tracks that feature Tinashe, Kendrick Lamar and Beyoncé all hinted that his fifth studio album would surpass his widely successful If You're Reading This, It's Too Late. To add more excitement, Drake just shared a photo of Willow Smith with the caption that the "young muse" is all over the album. Although we don't know at what capacity exactly, it's great news. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,000 |
The Via6 Towers are a pair of 24-story apartment buildings in the Belltown neighborhood of Seattle, Washington. Construction began in 2011 and the building topped out in June 2012. The complex opened February 2013 and includes 18,000 square feet of retail space at street level. The building was constructed to Leed Gold standards.
References
External links
Official website at Pine Street Group
Residential skyscrapers in Seattle
Belltown, Seattle | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,006 |
{"url":"https:\/\/mathoverflow.net\/questions\/320714\/a-relation-between-spec1i-1r-and-specr-j","text":"# A relation between $Spec((1+I)^{-1}R)$ and $Spec(R\/J)$\n\nLet $$R$$ be a commutative ring with identity and let $$I$$ and $$J$$ be two finitely generated ideals of $$R$$. Clearly $$1+I:=\\{1+i:i\\in I\\}$$ is a multiplicative closed subset of $$R$$. We can consider the following natural homeomorphisms $$Spec((1+I)^{-1}R)\\cong\\{p\\in Spec(R): 1+i\\not\\in p\\}$$ and $$Spec(R\/J)\\cong\\{p\\in Spec(R): J\\subseteq p\\}$$.\n\nI am looking for (necessary or(and) sufficient)conditions on $$I$$ and $$J$$ under which $$\\{p\\in Spec(R): 1+i\\not\\in p\\}\\subseteq \\{p\\in Spec(R): J\\subseteq p\\}.$$ Or under the above homeomorphisms $$Spec((1+I)^{-1}R) \\subseteq Spec(R\/J).$$\n\nNote that $$Spec(S)$$ is the set of all prime ideals of a ring $$S$$.","date":"2019-02-20 02:23:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 14, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9776175618171692, \"perplexity\": 47.24644307486943}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-09\/segments\/1550247494125.62\/warc\/CC-MAIN-20190220003821-20190220025821-00416.warc.gz\"}"} | null | null |
Q: How to calibrate multiple touchscreen connected on 2 different graphic cards? I have a sytem define as in the attached piture:
So 2 graphic cards:
*
*Screen 0 = 1 Touchscreen + Normal Screen
*Screen 1 = 4 Touchscreens.
I have tried to calibrate with xinput-calibrator but I failed.
Doesn't matter if I launch with the screen 0 or screen 1 I get each time the mis-click problem after the first or second clicks.
I have tried also with the xinput method to calibrate the Transformation matrix:
xinput set-prop "Device Name" --type=float "Coordinate Transformation Matrix" c0 0 c1 0 c2 c3 0 0 1
I have got some results but the problem is that it seems the 2 screens are not really separated. For instance if I calibrate the screen 1 (just taking the TS10 into account), when I touch the screen the mouse move on the normal screen even on another graphic card.
I don't know how to manage this problem... and I did not yet find another post with this kind of configuration with 2 graphic cards.
Many thanks in advance for any idea I could try.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,963 |
\section{Introduction and main results}
\quad In recent work \cite{PTargmin}, we considered the {\em argmin process} $(\alpha_t; t \geq 0)$ of Brownian motion, defined by
\begin{equation}
\alpha_t : = \sup\left\{s \in [0,1]: B_{t+s}=\min_{u \in [0,1]}B_{t+u} \right\} \quad \mbox{for all}~t \geq 0.
\end{equation}
It is easy to see that the process $\alpha$ is stationary, with arcsine distributed invariant measure. We proved that $(\alpha_t; t \geq 0)$ is a Markov process with the Feller property, and computed its transition kernel $Q_t(x,\cdot)$ for $t>0$ and $x \in [0,1]$.
\quad The purpose of this paper is to extend our previous results to random walks and L\'evy processes. Fix $N \geq 1$. We study the {\em argmin chain} $(A_N(n); n \geq 0)$ of a random walk $(S_n; n \geq 0)$, defined by
\begin{equation}
\label{argminchain}
A_N(n):=\sup \left\{1 \leq i \leq N; S_{n+i} = \min_{1\leq i \leq N} S_{n+i} \right\} \quad \mbox{for all}~n \geq 0,
\end{equation}
where $S_n:=\sum_{i=1}^n X_i$ is the $n^{th}$ partial sum of $(X_n; n \geq 0)$ (with convention $S_0:=0$), and $(X_n; n \geq 0)$ is a sequence of independent and identically distributed random variables with the cumulative distribution function $F$. This is the discrete analog of the argmin process of Brownian motion.
A similar argument as in the Brownian case shows that $(A_N(n); n \geq 0)$ is a Markov chain. For $n \geq 1$, let
$$p_n: = \mathbb{P}(S_1 \geq 0, \cdots, S_n \geq 0) \quad \mbox{and} \quad \widetilde{p}_n: = \mathbb{P}(S_1 > 0, \cdots, S_n > 0).$$
Theorem \ref{GF} below recalls the classical theory of how the two sequences of probabilities $p_n$ and $\widetilde{p}_n$ are determined by the sequences of probabilities $\mathbb{P}(S_n \ge 0 )$ and $\mathbb{P}( S_n > 0 )$.
We give the transition matrix of the argmin chain $A_N$ in terms of $(p_n; n \geq 1)$ and $(\widetilde{p}_n; n \geq 1$), which can be made explicit for special choices of $F$.
\begin{theorem}
\label{discreteth}
Whatever the common distribution $F$ of $(X_n; n \geq 0)$, the argmin chain $(A_N(n); n \geq 0)$ is a stationary and time-homogeneous Markov chain on $\{0,1, \ldots, N\}$. Let $\Pi_N(k)$, $k \in [0,N]$ be the stationary distribution, and $P_N(i,j)$, $i,j \in [0,N]$ be the transition probabilities of the argmin chain $(A_N(n); n \geq 0)$ on $[0,N]$. Then
\begin{equation}
\label{stationaryPi}
\Pi_N(k) = p_k \widetilde{p}_{N-k} \quad \mbox{for}~0 \leq k \leq N;
\end{equation}
\begin{equation}
\label{trans1}
P_N(i,N ) = 1 -\frac{\widetilde{p}_{N+1-i}}{\widetilde{p}_{N-i}} \quad \mbox{and} \quad P_N(i,i-1) = \frac{\widetilde{p}_{N+1-i}}{\widetilde{p}_{N-i}} \quad \mbox{for}~0<i \leq N;
\end{equation}
\begin{equation}
\label{trans2}
P_N(0,j ) = \frac{(p_j-p_{j+1}) \widetilde{p}_{N-j}}{\widetilde{p}_{N}} \quad \mbox{for}~0\leq j < N \quad \mbox{and} \quad P_N(0,N) = 1 - \sum_{j=0}^{N-1} P_N(0,j).
\end{equation}
In particular,
\begin{enumerate}
\item
If $(S_n; n \geq 0)$ is a random walk with continuous distribution and $\mathbb{P}(S_n>0) = \theta \in (0,1)$ for all $n \geq 1$. Let $(\theta)_{n \uparrow}: = \prod_{i=0}^{n-1} (\theta + i)$ be the Pochhammer symbol. Then
\begin{equation}
\label{disarcsine}
\Pi_{N}(k) = \frac{(\theta)_{k \uparrow} (\theta)_{N-k \uparrow}}{k! (N-k)!} \quad \mbox{for}~0 \leq k \leq N;
\end{equation}
\begin{equation}
\label{eq5354}
P_N(i, N) = \frac{1-\theta}{N+1-i} \quad \mbox{and} \quad P_N(i,i-1) = \frac{N+\theta-i}{N+1-i} \quad \mbox{for}~0<i \leq N;
\end{equation}
\begin{equation}
\label{eq55}
P_N(0, j) = \frac{1 - \theta}{j+1} \binom{N}{j} \frac{(\theta)_{j \uparrow} (\theta)_{N-j \uparrow}}{(\theta)_{N \uparrow}} \quad \mbox{for}~0 \leq j < N;
\end{equation}
and
\begin{equation}
\label{eq56}
P_N(0,N) = \frac{2(1-\theta)}{N+1} - \frac{(1 - 2 \theta)(2 \theta)_{N \uparrow}}{(N+1)(\theta)_{N \uparrow}}.
\end{equation}
\item
If $(S_n; n \geq 0)$ is a simple symmetric random walk. Let $\lfloor x \rfloor$ be the integer part of $x$. Then
\begin{equation}
\label{eq513}
\Pi_N(k) = \frac{\displaystyle \left(\frac{1}{2}\right)_{ \lfloor{\frac{k+1}{2}}\rfloor \uparrow} \left(\frac{1}{2}\right)_{\lfloor \frac{N-k}{2} \rfloor \uparrow} }{\displaystyle 2 \cdot \left \lfloor \frac{k+1}{2} \right \rfloor ! \left \lfloor \frac{N-k}{2} \right \rfloor !} \quad \mbox{for}~0 \leq k \leq N;
\end{equation}
For $0 < i \leq N$,
\begin{equation}
\label{eq514}
P_N(i,N) = \left\{ \begin{array}{ccl}
\frac{N-i}{N+1-i} & \mbox{if}~ N-i ~\mbox{is odd}; \\ [8 pt]
1 & \mbox{if} ~N-i ~\mbox{is even};
\end{array}\right.
\mbox{and} ~
P_N(i, i-1) = 1 - P_N(i,N);
\end{equation}
for $0 \leq j <N$,
\begin{equation}
\label{eq515}
P_N(0,j) = \left\{ \begin{array}{ccl}
0 & \mbox{if}~ j ~\mbox{is odd}; \\ [8 pt]
\frac{\displaystyle \binom{j}{\frac{j}{2}} \binom{2 \lfloor \frac{N}{2} \rfloor-j}{\lfloor \frac{N}{2} \rfloor-\frac{j}{2}}}{\displaystyle(j+2)\binom{2 \lfloor \frac{N}{2} \rfloor}{ \lfloor\frac{N}{2} \rfloor}} & \mbox{if} ~j ~\mbox{is even};
\end{array}\right.
\end{equation}
and
\begin{equation}
\label{eq516}
P_N(0,N) = \left\{ \begin{array}{ccl}
\frac{1}{N+1} & \mbox{if}~ N ~\mbox{is odd}; \\ [8 pt]
\frac{2}{N+2} & \mbox{if} ~N ~\mbox{is even}.
\end{array}\right.
\end{equation}
\end{enumerate}
\end{theorem}
\quad For the argmin chain $A_N$, the transition probability from $0$ to $N$ is given by \eqref{trans2} in the general case. But this probability is simplified to \eqref{eq56} and \eqref{eq516} in the two special cases. These identities are proved analytically by Lemmas \ref{lem11} and \ref{lem12}. We do not have a simple explanation, and leave combinatorial interpretations for interested readers.
\quad Let $(X_t; t \geq 0)$ be a real-valued L\'evy process. We consider the argmin process $(\alpha_t^X; t \geq 0)$ of $X$, defined by
\begin{equation}
\alpha_t^X : = \sup\left\{s \in [0,1]: X_{t+s}=\inf_{u \in [0,1]}X_{t+u} \right\} \quad \mbox{for all}~t \geq 0.
\end{equation}
We are particularly interested in the case where $X$ is a stable L\'evy process. We follow the notations in Bertoin \cite[Chapter VIII]{Bertoin}. Up to a multiple factor, a stable L\'evy process $X$ is entirely determined by a {\em scaling parameter} $\alpha \in (0.2]$, and a {\em skewness parameter} $\beta \in [-1,1]$. The characteristics exponent of a stable L\'evy process $X$ with parameters $(\alpha,\beta)$ is given by
\begin{equation*}
\Psi(\lambda) : = \left\{ \begin{array}{ccl}
|\lambda|^{\alpha} (1 - i \beta \sgn(\lambda) \tan(\pi \alpha/2)) & \mbox{for}~\alpha \neq 1, \\ [8pt]
|\lambda| (1 + i \frac{2 \beta}{\pi} \sgn(\lambda) \log|\lambda|) & \mbox{for}~\alpha = 1.
\end{array}\right.
\end{equation*}
where $\sgn$ is the sign function. Let $\rho : = \mathbb{P}(X_1>0)$ be the {\em positivity parameter}. Zolotarev \cite[Section 2.6]{Zolo} found that
\begin{equation}
\label{positivity}
\rho = \frac{1}{2} + (\pi \alpha)^{-1}\arctan(\beta \tan(\pi \alpha/2)) \quad \mbox{for}~\alpha \in (0,2].
\end{equation}
\quad If $X$ (resp. $-X$ ) is a subordinator, then almost surely $\alpha^X_t =0$ (resp. $\alpha^X_t = 1$) for all $t \geq 0$. Relying on the excursion theory, we generalize Pitman and Tang \cite[Theorem 1.2]{PTargmin}.
\begin{theorem}
\label{mainLevy}
~
\begin{enumerate}
\item
Let $(X_t; t \geq 0)$ be a L\'evy process. Then the argmin process $(\alpha^X_t; t \geq 0)$ of $X$ is a stationary and time-homogeneous Markov process.
\item
Let $(X_t; t \geq 0)$ be a stable L\'evy process with parameters $(\alpha,\beta)$, and assume that neither $X$ nor $-X$ is a subordinator.
Let $\rho$ be defined by \eqref{positivity}. Then the argmin process $(\alpha^X_t; t \geq 0)$ of $X$ is a stationary Markov process, with generalized arcsine distributed invariant measure whose density is
\begin{equation}
f(x) : = \frac{\sin \pi \rho}{\pi} x^{-\rho} (1-x)^{\rho-1} \quad \mbox{for}~0<x<1.
\end{equation}
and Feller transition semigroup $Q^X_t(x, \cdot)$, $t >0$ and $x \in [0,1]$ where
\begin{equation}
\label{Qtrans}
Q^X_t(x,dy) = \left\{ \begin{array}{ccl}
1_{\{0<y<1\}} \frac{\sin \pi \rho}{\pi} y^{-\rho} (1-y)^{\rho-1} dy & \mbox{for}~0 \leq x \leq 1 <t, \\ [8pt]
\left(\frac{1-x}{1-x+t}\right)^{1-\rho} \delta_{x-t}(dy) + \frac{\sin \pi \rho}{\pi} \cdot \frac{ (1-y)^{\rho-1} (y+t-1)_{+}^{\rho} }{(y+t-x)}dy & \mbox{for}~0< t \leq x \leq 1\\ [8pt]
\frac{\sin \pi \rho}{\pi (y+t-x)} y^{-\rho} (1-y)^{\rho-1} [(t-x)^{\rho}(1-x)^{1-\rho} + y^{\rho}(y+t-1)_{+}^{1-\rho}] dy & \mbox{for}~0 \leq x<t \leq 1.
\end{array}\right.
\end{equation}
\end{enumerate}
\end{theorem}
\vskip 12 pt
{\bf Organization of the paper:} The layout of the paper is as follows.
\begin{itemize}
\item
In Section \ref{s5}, we study the argmin chain $(A_N(n); n \geq 0)$ for random walks. There Theorem \ref{discreteth} is proved.
\item
In Section \ref{s3}, we consider the argmin process $(\alpha^X_t; t \geq 0)$ of L\'evy process, and prove Theorem \ref{mainLevy}.
\end{itemize}
\section{The argmin chain of random walks}
\label{s5}
\quad In this section, we prove Theorem \ref{discreteth}. Recall the definition of the argmin chain $(A_N(n); n \geq 0)$ from \eqref{argminchain}. Fix $N \geq 1$. Let $(\overset{\rightarrow}{X}_N(n); n \geq 0)$ be the moving window process of length $N$, defined by
$$\overset{\rightarrow}{X}_N(n):=(X_{n+1}, \ldots, X_{n+N}) \quad \mbox{for}~n \geq 0,$$
with associated partial sums $\overset{\rightarrow}{S^X_N}(n):=(0, X_{n+1}, X_{n+1}+X_{n+2}, \ldots, \sum_{i=1}^N X_{n+i})$. Similarly, let $(\overset{\leftarrow}{X}_N(n); n \geq 0)$ be the reversed moving window process of length $N$, defined by
$$\overset{\leftarrow}{X}_N(n):=(-X_{n}, \ldots, -X_{n-N+1}) \quad \mbox{for}~n \geq N,$$
with associated partial sums $\overset{\leftarrow}{S^X_N}(n):=(0, -X_{n}, -X_{n}-X_{n-1}, \ldots, -\sum_{i=1}^N X_{n+1-i})$.
Note that $n+A_N(n)$ is the last time at which the minimum of $(S_k; k \geq 0)$ on $[n+1,n+N]$ is attained. So $A_N(n)$ is a function of $\overset{\rightarrow}{S^X_N}(n)$ or $\overset{\leftarrow}{S^X_N}(n+N)$.
The following path decomposition is due to Denisov.
\begin{theorem} [Denisov's decomposition for random walks] \cite{Denisov}
\label{DenisovRW}
Let $S_n:= \sum_{i=1}^n X_i$, where $X_i$ are independent random variables. For $N \geq 1$, let
$$A_N:=\sup\left\{0 \leq i \leq N: S_i=\min_{1 \leq k \leq N} S_k\right\}$$
be the last time at which $(S_k; k \geq 0)$ attains its minimum on $[0,N]$. For each positive integer $a$ with $0 \le a \le N$, given the event $\{A_N = a\}$,
the random walk is decomposed into two conditionally independent pieces:
\begin{enumerate}[(a).]
\item
$(S_{a-k}-S_{a}; 0 \leq k \leq a)$ has the same distribution as $\overset{\leftarrow \quad}{S^X_{a}}(a)$ conditioned to stay non-negative;
\item
$(S_{a+k}-S_{a}; 0 \leq k \leq N - a)$ has the same distribution as $\overset{\rightarrow \quad \quad}{S^X_{N - a}} (a)$ conditioned to stay positive.
\end{enumerate}
\end{theorem}
\quad By Denisov's decomposition for random walks, it is easy to adapt the argument of Pitman and Tang \cite[Proposition 3.4]{PTargmin} to show that $(A_N(n); n \geq 0)$ is a time-homogeneous Markov chain on $\{0,1,\cdots, N\}$. Here the detail is omitted.
\quad Now we compute the invariant distribution $\Pi_N$, and the transition matrix $P_N$ of the argmin chain $(A_N(n); n \geq 0)$ on $\{0,1, \ldots, N\}$. To proceed further, we need the following result regarding the law of ladder epochs, originally due to Sparre Anderson \cite{SA}, Spitzer \cite{Spitzer} and Baxter \cite{Baxter}. It can be read from Feller \cite[Chapter XII$.7$]{Fellervol2}.
\begin{theorem} \cite{SA, Fellervol2}
\label{GF}
\begin{enumerate}
\item
Let $\tau_n:=\mathbb{P}(S_1 \geq 0, \ldots, S_{n-1} \geq 0, S_n<0)$ and $\tau(s): = \sum_{n=0}^{\infty} \tau_n s^n$. Then for $|s|<1$,
$$\log \frac{1}{1-\tau(s)} = \sum_{n=1}^{\infty} \frac{s^n}{n} \mathbb{P}(S_n<0).$$
\item
Let $p_n:= \mathbb{P}(S_1 \geq 0, \ldots, S_n \geq 0)$ and $p(s): = \sum_{n=0}^{\infty} p_n s^n$. Then for $|s|<1$,
$$p(s) = \exp\left(\sum_{n=1}^{\infty} \frac{s^n}{n} \mathbb{P}(S_n \geq 0)\right).$$
\item
Let $\widetilde{p}_n:= \mathbb{P}(S_1 > 0, \ldots, S_n > 0)$ and $\widetilde{p}(s): = \sum_{n=0}^{\infty} \widetilde{p}_n s^n$. Then for $|s|<1$,
$$\widetilde{p}(s) = \exp\left(\sum_{n=1}^{\infty} \frac{s^n}{n} \mathbb{P}(S_n > 0)\right).$$
\end{enumerate}
\end{theorem}
\quad In the sequel, let $T_{-}:=\inf\{n \geq 1; S_n<0\}$ and $\widetilde{T}_{-}:=\inf\{n \geq 1; S_n \leq 0\}$ so that $p_n = \mathbb{P}(T_{-} > n)$ and $\widetilde{p}_n = \mathbb{P}(\widetilde{T}_{-} > n)$.
\begin{proof}[Proof of Theorem \ref{discreteth}]
Observe that the distribution of the argmin of sums on $\{0,1,\cdots,N\}$ is the stationary distribution of the argmin chain. Following Feller \cite[Chapter XII.8]{Feller}, this is the discrete arcsine law
\begin{equation*}
\Pi_N(k) = p_k \widetilde{p}_{N-k} \quad \mbox{for}~0 \leq k \leq N.
\end{equation*}
Let $t_i:=\mathbb{P}(T_{-}=i) = p_{i-1} - p_i$ and $\widetilde{t}_i:=\mathbb{P}(\widetilde{T}_{-}=i) = \widetilde{p}_{i-1} - \widetilde{p}_i$ for $i >0$. Now we calculate the transition probabilities of the argmin chain. We distinguish two cases.
\vskip 6 pt
{\bf Case $1$.} The argmin chain starts at $0 < i \leq N$: $A_N(0) = i$. This implies that for all $k \in [1, i-1]$, $S_k \geq S_i$, and for all $k \in [i+1, N]$, $S_k > S_i$.
\begin{itemize}
\item
If $S_{N+1}>S_i$, then the last time at which $(S_n)_{1 \leq n \leq N+1}$ attains its minimum is $i$, meaning that $A_{N}(1) = i-1$.
\item
If $S_{N+1} = S_i$, the the last time at which $(S_n)_{1 \leq n \leq N+1}$ attains its minimum is $N+1$, meaning that $A_N(1) = N$.
\end{itemize}
If we look forward from time $i$, $N+1$ is the first time at which the chain enters $(-\infty, 0]$. Consequently, for $0 < i \leq N$,
\begin{equation}
P_N(i,N) = \frac{\widetilde{t}_{N+1-i}}{\widetilde{p}_{N-i}} \quad \mbox{and} \quad P_N(i,i-1) = 1 - P_N(i,N),
\end{equation}
which leads to \eqref{trans1}.
\vskip 6 pt
{\bf Case $2$.} The argmin chain starts at $i=0$: $A_N(0) = 0$. For $0 \leq j <N$, let $j+1$ be the last time at which the minimum on $[1,N]$ is attained.
\begin{itemize}
\item
If $S_{N+1}>S_{j+1}$, then the last time at which $(S_n)_{1 \leq n \leq N+1}$ attains its minimum is $j+1$, meaning that $A_N(1)=j$.
\item
If $S_{N+1} = S_{j+1}$, then the last time at which $(S_n)_{1 \leq n \leq N+1}$ attains its minimum is $N+1$, meaning that $A_N(1)=N$.
\end{itemize}
If we look backward from time $j+1$, the origin is the first time at which the reversed walk enters $(-\infty, 0)$. So for $0 \leq j <N$,
\begin{equation}
P_{N}(0,j) = \frac{t_{j+1} \widetilde{p}_{N-j}}{\widetilde{p}_N},
\end{equation}
which yields \eqref{trans2}. The above formula fails for $j=N$, but $P_N(0, N) = 1 -\sum_{j=0}^{N-1}P_N(0,j)$.
\end{proof}
\quad We complete the proof of Theorem \ref{discreteth} in the following two subsections. In Section \ref{s51}, we consider the non-lattice case, and in Section \ref{s52}, we deal with the lattice case.
\subsection{$F$ is continuous and $\mathbb{P}(S_n>0) = \theta \in (0,1)$}
\label{s51}
From Theorem \ref{GF}, we deduce the well known facts that
\begin{equation*}
\log p(s) = \theta \sum_{n=1}^{\infty} \frac{s^n}{n} \Longrightarrow p(s) = (1-s)^{-\theta} = 1+ \sum_{n=1}^{\infty} \frac{(\theta)_{n \uparrow}}{n!} s^n,
\end{equation*}
where $(\theta)_{n \uparrow}: = \prod_{i=0}^{n-1} (\theta+i)$ is the {\em Pochhammer symbol}. This implies that
\begin{equation}
\label{pcts}
p_n = \widetilde{p}_n = \frac{(\theta)_{n \uparrow}}{n!} \quad \mbox{for all}~n>0.
\end{equation}
\quad By injecting \eqref{pcts} into \eqref{stationaryPi}, \eqref{trans1} and \eqref{trans2}, we get \eqref{disarcsine}, \eqref{eq5354} and \eqref{eq55}. The formula \eqref{eq56} is obtained by the following lemma.
\begin{lemma}
\label{lem11}
\begin{equation*}
P_{N}(0,N) = \frac{2(1-\theta)}{N+1} - \frac{(1 - 2 \theta)(2 \theta)_{N \uparrow}}{(N+1)(\theta)_{N \uparrow}}.
\end{equation*}
\end{lemma}
\begin{proof}
Note that $P_N(0,N) = 1 - \sum_{j=0}^{N-1} P_N(0,j)$. Thus , it suffices to show that
\begin{equation}
\label{magind}
\sum_{j=0}^{N-1} p_j p_{N-j} - \sum_{j=0}^{N-1} p_{j+1} p_{N-j} = \frac{1}{(N+1) !} \Bigg[ (N-2 \theta-1)(\theta)_{N \uparrow} + (1-2 \theta) (2 \theta)_{N \uparrow}\Bigg].
\end{equation}
Furthermore, for $|s|<1$,
\begin{equation*}
(1-s)^{-2\theta} = \left(\sum_{j=0}^{\infty}p_j s^j \right)^2 = \sum_{N=0}^{\infty} \left(\sum_{j=0}^N p_j p_{N-j} \right)s^j.
\end{equation*}
By identifying the coefficients on both sides, we get
\begin{equation*}
\sum_{j=0}^N p_j p_{N-j} = \frac{(2 \theta)_{N \uparrow}}{N !} \quad \mbox{and} \quad \sum_{j=0}^{N+1} p_jp_{N+1-j} = \frac{(2 \theta)_{N+1 \uparrow}}{(N+1) !}.
\end{equation*}
Therefore,
\begin{equation*}
\sum_{j=0}^{N-1} p_j p_{N-j} - \sum_{j=0}^{N-1} p_{j+1} p_{N-j} = \left[ \frac{(2 \theta)_{N \uparrow}}{N !} - \frac{(\theta)_{N \uparrow}}{N !}\right] - \left[\frac{(2 \theta)_{N+1 \uparrow}}{(N+1) !} - \frac{(\theta)_{N+1 \uparrow}}{(N+1) !}\right],
\end{equation*}
which leads to \eqref{magind}.
\end{proof}
\quad When $F$ is symmetric and continuous, the above results can be simplified. In this case, $\mathbb{P}(S_n \geq 0) = \mathbb{P}(S_n>0) = \frac{1}{2}$.
\begin{corollary}
Assume that $F$ is symmetric and continuous. Then the stationary distribution of the argmin chain $(A_N(n); n \geq 0)$ is given by
\begin{equation}
\Pi_N(k) = \binom{2k}{k} \binom{2N-2k}{N-k} 2^{-2N} \quad \mbox{for}~0 \leq k \leq N.
\end{equation}
In addition, the transition probabilities are
\begin{equation}
P_N(i,N) = \frac{1}{2(N+1-i)} \quad \mbox{and} \quad P_N(i,i-1) = \frac{2N +1 -2i}{2(N+1-i)} \quad \mbox{for}~0<i \leq N;
\end{equation}
\begin{equation}
P_N(0,j) = \frac{\binom{N}{j}^2}{2(j+1)\binom{2N}{2j}} \quad \mbox{for}~0 \leq j <N \quad \mbox{and} \quad P_N(0,N) = \frac{1}{N+1}.
\end{equation}
\end{corollary}
\subsection{Simple symmetric random walks}
\label{s52}
In \cite[Chapter III.3]{Feller}, Feller found for a simple symmetric walk,
\begin{equation}
\label{ptil2}
\widetilde{p}_{2n} = \widetilde{p}_{2n+1} = \frac{(\frac{1}{2})_{n \uparrow}}{2 \cdot n!} \quad \mbox{for all}~n \geq 1,
\end{equation}
and
\begin{equation}
\label{p2}
p_{2n-1} = p_{2n} = \frac{(\frac{1}{2})_{n \uparrow}}{ n!} \quad \mbox{for all}~n \geq 1.
\end{equation}
\quad By injecting \eqref{ptil2} and \eqref{p2} into \eqref{stationaryPi}, \eqref{trans1} and \eqref{trans2}, we get \eqref{eq513}, \eqref{eq514} and \eqref{eq515}. The formula \eqref{eq516} is obtained by the following lemma.
\begin{lemma}
\label{lem12}
\begin{equation*}
P_N(0,N) = \left\{ \begin{array}{ccl}
\frac{1}{N+1} & \mbox{if}~ N ~\mbox{is odd}; \\ [8 pt]
\frac{2}{N+2} & \mbox{if} ~N ~\mbox{is even}.
\end{array}\right.
\end{equation*}
\end{lemma}
\begin{proof}
Note that $P_N(0,N) = 1 -\sum_{j=0}^{N-1} P_N(0,j)$. Thus, it suffices to show that
\begin{equation}
\label{magindbis}
\sum_{j=0}^{N-1} p_j p_{N-j} - \sum_{j=0}^{N-1} p_{j+1}p_{N-j} =
\left\{ \begin{array}{ccl}
\frac{N}{N+1}\widetilde{p}_N & \mbox{if}~ N ~\mbox{is odd}; \\
\frac{N}{N+2}\widetilde{p}_N & \mbox{if} ~N ~\mbox{is even}.
\end{array}\right.
\end{equation}
Furthermore, for $s<1$,
\begin{equation*}
\frac{1}{1-s} = \left(\sum_{j=0}^{\infty}p_j s^j \right) \left(\sum_{j=0}^{\infty}\widetilde{p}_j s^j \right) = \sum_{N=0}^{\infty} \left(\sum_{j=0}^N p_j \widetilde{p}_{N-j} \right)s^j.
\end{equation*}
By identifying the coefficients on both sides, we get
\begin{equation*}
\sum_{j=0}^N p_j \widetilde{p}_{N-j} = \sum_{j=0}^{N+1} p_j \widetilde{p}_{N+1-j} = 1.
\end{equation*}
Therefore,
\begin{align*}
\sum_{j=0}^{N-1} p_j p_{N-j} - \sum_{j=0}^{N-1} p_{j+1}p_{N-j} = (1-p_N) - (1-p_{N+1}-\widetilde{p}_{N+1}),
\end{align*}
which leads to \eqref{magindbis}.
\end{proof}
\section{The argmin process of L\'evy processes}
\label{s3}
\quad In this section, we consider the argmin process $(\alpha^X_t; t \geq 0)$ of a L\'evy process $(X_t; t \geq 0)$.
According to the L\'evy-Khintchine formula, the characteristic exponent of $(X_t; t \geq 0)$ is given by
$$\Psi_X (\theta) : = ia \theta + \frac{\sigma^2}{2} \theta^2 + \int_{\mathbb{R}} (1-e^{i\theta x} + i \theta x 1_{\{|x|<1\}}) \Pi(dx),$$
where $a \in \mathbb{R}$, $\sigma \geq 0$, and $\Pi(\cdot)$ is the L\'evy measure satisfying $\int_{\mathbb{R}} \min(1,x^2) \Pi(dx) < \infty$. The L\'evy process $X$ is a compound Poisson process if and only if
$\sigma = 0$ and $\Pi(\mathbb{R}) < \infty$.
In this case, the process $X$ has the following representation:
\begin{equation}
\label{CPP}
X_t = ct + \sum_{i=1}^{N_t} Y_i \quad \mbox{for all}~t>0,
\end{equation}
where
$c = -a - \int_{|x|<1} x \Pi(dx)$,
$(N_t; t \geq 0)$ is a Poisson process with rate $\lambda$, and $(Y_i; i \geq 1)$ are independent and identically distributed random variables with cumulative distribution function $F$, independent of $N$ and satisfying $\lambda F(dx) = \Pi(dx)$.
See Bertoin \cite{Bertoin} and Sato \cite{Sato} for further development on L\'evy processes.
\quad In Section \ref{s31}, we review Millar-Denisov's decomposition for L\'evy processes with continuous distribution. In Section \ref{s32}, we consider a path decomposition for compound Poisson processes. Finally in Section \ref{s33}, we explain how to adapt the arguments in Pitman and Tang \cite{PTargmin} to prove Theorem \ref{mainLevy}.
\subsection{Millar-Denisov's decomposition for L\'evy processes}
\label{s31}
For $A \in \mathcal{B}(\mathbb{R})$, let
$T_A : = \inf\{t>0: X_t \in A\}$
be the hitting time of $A$ by $(X_t; t \geq 0)$.
Recall that $0$ is regular for the set $A$ if $\mathbb{P}(T_A = 0) = 1$.
\quad Assume that $X$ is not a compound Poisson process with drift, which is equivalent to
\begin{itemize}
\item[(CD).]
For all $t>0$, $X_t$ has a continuous distribution; that is for all $x \in \mathbb{R}$,
$\mathbb{P}(X_t = x) = 0$.
\end{itemize}
See Sato \cite[Theorem 27.4]{Sato}.
According to Blumenthal's zero-one law, $0$ is regular for at least one of the half-lines $(-\infty,0)$ and $(0,\infty)$. There are three subcases:
\begin{itemize}
\item[(RB).]
$0$ is regular for both half-lines $(-\infty,0)$ and $(0,\infty)$;
\item[(R$+$).]
$0$ is regular for the positive half-line $(0,\infty)$ but not for the negative half-line $(-\infty,0)$;
\item[(R$-$).]
$0$ is regular for the negative half-line $(-\infty,0)$ but not for the positive half-line $(0,\infty)$.
\end{itemize}
\quad Millar \cite{Millar78} proved that almost surely $(X_t; 0 \leq t \leq 1)$ achieves its minimum at a unique time $A \in [0,1]$, and
\begin{itemize}
\item
under the assumption (RB), $X_{A-} = X_A = \inf_{t \in [0,1]}X_t$ almost surely;
\item
under the assumption (R$+$), $X_{A-} > X_A = \inf_{t \in [0,1]}X_t$ almost surely;
\item
under the assumption (R$-$), $X_A > X_{A-} = \inf_{t \in [0,1]}X_t$ almost surely.
\end{itemize}
The following result is a simple consequence of Millar \cite[Proposition 4.2]{Millar78}.
\begin{theorem} \cite{Millar78}
\label{M78}
Assume that $(X_t; 0 \leq t \leq 1)$ is not a compound Poisson process with drift. Let $A$ be the a.s. unique time such that
$
\inf_{t \in [0,1]} X_t = \min (X_{A-}, X_A).
$
Given $A$, the L\'evy path is decomposed into two conditionally independent pieces:
$$\left(X_{(A-t)-}-\inf_{u \in [0,1]}X_u; 0 \leq t \leq A\right) \quad \mbox{and} \quad \left(X_{A+t}-\inf_{u \in [0,1]}X_u; 0 \leq t \leq 1-A\right).$$
\end{theorem}
\quad In \cite{Millar78}, Millar provided the law of the post-$A$ process $\left(X_{A+t}-\inf_{t \in [0,1]}X_t; 0 \leq t \leq 1-A\right)$ but he did not mention the law of the pre-$A$ process $\left(X_{(A-t)-}-\inf_{t \in [0,1]}X_t; 0 \leq t \leq A\right)$. Relying on Chaumont-Doney's construction \cite{CD10} of L\'evy meanders, Uribe Bravo \cite{UB14} proved that if $(X_t; 0 \leq t \leq 1)$ is not a compound Poisson process with drift and satisfies the assumption (RB), then
\begin{itemize}
\item
$\left(X_{(A-t)-}-\inf_{u \in [0,1]}X_u; 0 \leq t \leq A\right)$ is a L\'evy meander of length $A$;
\item
$(X_{A+t}-\inf_{u \in [0,1]}X_u; 0 \leq t \leq 1-A)$ is a L\'evy meander of length $1-A$.
\end{itemize}
This result generalizes Denisov's decomposition to L\'evy processes with continuous distribution. See also Chaumont \cite{Chaumont97} for related results for stable L\'evy processes.
\subsection{A path decomposition for compound Poisson processes}
\label{s32}
Here we give a path decomposition for compound Poisson processes, which is left out in the literature. Since a compound Poisson process is a continuous-time random walk, our construction is based on Denisov's decomposition for random walks.
\quad Let $(X_t; 0 \leq t \leq 1)$ be a compound Poisson process defined by \eqref{CPP}. Let
\begin{equation*}
A : = \sup\left\{0 \leq s \leq 1: X_{s} = \inf_{u \in [0,1]} X_{u}\right\}
\end{equation*}
be the last time at which $X$ achieves its minimum on $[0,1]$. Let $\{\xi_1, \cdots, \xi_N\}$ be the jumping positions of $(N_t; 0 \leq t \leq 1)$, with $N:=N_1$ and $0=:\xi_0 < \xi_1<\cdots <\xi_{N} < \xi_{N+1}:=1$.
\quad Given $\xi_1,\cdots,\xi_{N}$ and $A = \xi_k$ for some $k \leq N$, we distinguish two cases:
\vskip 6 pt
{\bf Case $1$.} $c \leq 0$. Then $\inf_{t \in [0,1]} X_t = X_{A-}$. Define $Y^c_i:=Y_i + c(\xi_{i+1} - \xi_i)$ for $1 \leq i \leq N$.
Let $\overset{\rightarrow}{Z}$ be distributed as $\overset{\rightarrow \quad \quad}{S^{Y^c}_{N-k+1}}(k-1)$ conditioned to stay positive, and
$\overset{\leftarrow}{Z}$ be distributed as $\overset{\leftarrow \quad}{S^{Y^c}_{k-1}}(k-1)$ conditioned to stay non-negative. Define
$$
\overset{\leftarrow}{X}_t : = \left\{ \begin{array}{cl}
-ct & \mbox{for}~0 \leq t < A - \xi_{k-1}, \\
\overset{\leftarrow}{Z}_1-c(t-A + \xi_{k-1}) & \mbox{for}~ A - \xi_{k-1} \leq t < A - \xi_{k-2}, \\
\vdots & \vdots \\
\overset{\leftarrow}{Z}_{k-1} - c(t-A+\xi_1) & \mbox{for}~A - \xi_1\leq t \leq A.
\end{array}\right.
$$
and
$$
\overset{\rightarrow}{X}_t:= \left\{ \begin{array}{cl}
\overset{\rightarrow}{Z}_1- c(\xi_{k+1} - A - t) & \mbox{for}~0 \leq t < \xi_{k+1} - A, \\
\overset{\rightarrow}{Z}_2 - c(\xi_{k+2} - A - t)& \mbox{for}~ \xi_{k+1} - A \leq t < \xi_{k+2} - A, \\
\vdots & \vdots \\
\overset{\rightarrow}{Z}_{N-k+1}-c(1-A-t) & \mbox{for}~\xi_N - A \leq t \leq 1 - A.
\end{array}\right.
$$
{\bf Case $2$.} $c>0$. Then $\inf_{t \in [0,1]} X_t = X_{A}$. Define $Y^c_i:=Y_i + c(\xi_{i} - \xi_{i-1})$ for $1 \leq i \leq N$.
Let $\overset{\rightarrow}{Z}$ be distributed as $\overset{\rightarrow \quad}{S^{Y^c}_{N-k}}(k)$ conditioned to stay positive, and
$\overset{\leftarrow}{Z}$ be distributed as $\overset{\leftarrow \quad}{S^{Y^c}_{k}}(k)$ conditioned to stay non-negative. Define
$$
\overset{\leftarrow}{X}_t : = \left\{ \begin{array}{cl}
\overset{\leftarrow}{Z}_1 + c(A - \xi_{k-1} - t)& \mbox{for}~0 \leq t < A - \xi_{k-1}, \\
\overset{\leftarrow}{Z}_2 + c(A - \xi_{k-2} - t) & \mbox{for}~ A - \xi_{k-1} \leq t < A - \xi_{k-2}, \\
\vdots & \vdots \\
\overset{\leftarrow}{Z}_{k} + c(A - t) & \mbox{for}~A - \xi_1\leq t \leq A.
\end{array}\right.
$$
and
$$
\overset{\rightarrow}{X}_t:= \left\{ \begin{array}{cl}
ct & \mbox{for}~0 \leq t < \xi_{k+1} - A, \\
\overset{\rightarrow}{Z}_1 + c(t -\xi_{k+1} + A )& \mbox{for}~ \xi_{k+1} - A \leq t < \xi_{k+2} - A, \\
\vdots & \vdots \\
\overset{\rightarrow}{Z}_{N-k} + c(t-\xi_N + A) & \mbox{for}~\xi_N - A \leq t \leq 1 - A.
\end{array}\right.
$$
By Theorem \ref{DenisovRW},
the path of $X$ is decomposed into two conditionally independent pieces:
\begin{equation}
\left(X_{(A-t)-}-\inf_{u \in [0,1]}X_u; 0 \leq t \leq A\right) \stackrel{(d)}{=} (\overset{\leftarrow}{X}_t; 0 \leq t \leq A),
\end{equation}
and
\begin{equation}
\left(X_{A+t}-\inf_{u \in [0,1]}X_u; 0 \leq t \leq 1-A \right) \stackrel{(d)}{=} (\overset{\rightarrow}{X}_t; 0 \leq t \leq 1-A).
\end{equation}
\subsection{The Markov property of the argmin process}
\label{s33}
Combining the results in the last two sections, we have the following corollary which is a slight extension to Theorem \ref{M78}.
\begin{corollary}
\label{haha}
Let $(X_t; 0 \leq t \leq 1)$ be a real-valued L\'evy process. Let
$$A : = \sup\left\{0 \leq s \leq 1: X_{s} = \inf_{u \in [0,1]} X_{u}\right\}$$
be the last time at which $X$ achieves its minimum on $[0,1]$. Given $A$, the path of $X$ is decomposed into two conditionally independent pieces:
$$\left(X_{(A-t)-}-\inf_{u \in [0,1]}X_u; 0 \leq t \leq A\right) \quad \mbox{and} \quad \left(X_{A+t}-\inf_{u \in [0,1]}X_u; 0 \leq t \leq 1-A\right).$$
\end{corollary}
With Corollary \ref{haha}, it is easy to adapt the argument of Pitman and Tang \cite[Proposition 3.4]{PTargmin} to prove that $(\alpha^X_t; t \geq 0)$ is a time-homogeneous Markov process. Here the detail is omitted.
\quad Now we turn to the stable L\'evy process. Let $(X_t; t \geq 0)$ be a stable L\'evy process with parameters $(\alpha,\beta)$, and neither $X$ nor $-X$ is a subordinator. It is well known that $0$ is regular for the reflected process $X- \underline{X}$. So It\^o's excursion theory can be applied to the process $X- \underline{X}$, see Sharpe \cite{Sharpe} for background on excursion theory of Markov processes.
\quad Let ${\bf n}(d\epsilon)$ be the It\^o measure of excursions of $X-\underline{X}$ away from $0$. Monrad and Silverstein \cite{MS79} computed the law of lifetime $\zeta$ of excursions under ${\bf n}$:
\begin{equation}
{\bf n}(\zeta > t) = c \frac{t^{\rho-1}}{\Gamma (\rho)} \quad \mbox{and} \quad {\bf n}(\zeta \in dt) = c(1-\rho) \frac{t^{\rho-2}}{\Gamma(\rho)}
\end{equation}
for some contant $c>0$. Following the argument of Pitman and Tang \cite[Remark 3.9]{PTargmin}, we have:
\begin{proposition}
Let $(X_t; t \geq 0)$ be a stable L\'evy process with parameters $(\alpha,\beta)$, and neither $X$ nor $-X$ is a subordinator. Then the jump rate of the argmin process $\alpha^X$ per unit time from $x \in (0,1)$ to $1$ is given by
\begin{equation}
\mu^{\uparrow 1}(x) = \frac{1 - \rho}{1-x} \quad \mbox{for}~0 < x <1.
\end{equation}
\end{proposition}
\quad Finally, by doing similar calculations as in Pitman and Tang \cite[Section 3.4]{PTargmin}, we obtain the Feller transition semigroup \eqref{Qtrans} for $X$.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
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\subsection*{Acknowledgements}
The first two authors would like to thank their teacher {\bf Joseph
Bernstein} for their mathematical education. They cordially thank
{\bf Joseph Bernstein} and {\bf Eitan Sayag} for guiding them
through this project. They would also like to thank {\bf Yuval
Flicker}, {\bf Erez Lapid}, {\bf Omer Offen} and {\bf Yiannis
Sakellaridis} for useful remarks.
The first two authors worked on this project while participating in
the program \emph{Representation theory, complex analysis and
integral geometry} of the Hausdorff Institute of Mathematics (HIM)
at Bonn joint with Max Planck Institute fur Mathematik. They wish to
thank the organizers of the activity and the director of HIM for
inspiring environment and perfect working conditions.
Finally, the first two authors wish to thank {\bf Vladimir
Berkovich}, {\bf Stephen Gelbart}, {\bf Maria Gorelik} and {\bf
Sergei Yakovenko} from the Weizmann Institute of Science for their
encouragement, guidance and support.
The last author thanks the Math Research Institute of Ohio State
University in Columbus for several invitations which allowed him to
work with the third author.
\@startsection {section}{1{\hglue -16pt . Theorem 2(2') implies Theorem 1(1')}
A group of type lctd is a locally compact, totally discontinuous
group which is countable at infinity. We consider smooth
representations of such groups. If $(\pi ,E_\pi )$ is such a
representation then $(\pi ^*,E_\pi ^*)$ is the smooth
contragradient. Smooth induction is denoted by $Ind$ and compact
induction by $ind$. For any topological space $T$ of type lctd,
$\mathcal {S}(T)$ is the space of functions locally constant,
complex valued, defined on $T$ and with compact support. The space
$\mathcal {S'}(T)$ of distributions on $T$ is the dual space to
$\mathcal {S}(T)$.
\begin{proposition}
Let $M$ be a lctd group and $N$ a closed subgroup, both unimodular. Suppose that there exists an involutive anti-automorphism $\sigma $ of $M$ such that $\sigma (N)=N$ and such that any distribution on $M$, biinvariant under $N$, is fixed by $\sigma $. Then, for any irreducible admissible representation $\pi $ of $M$
\[ {\mathrm {dim}} \left ({\mathrm {Hom}} _ M(ind_ N^M(1),\pi )\right )\times {\mathrm {dim}} \left ({\mathrm {Hom}} _ M(ind_ N^M(1),\pi ^*)\right )\leq 1.\]
\end{proposition}
This is well known (see for example \cite{P2}).
{\bf Remark\unskip . --- \ignorespaces} There is a variant for the non unimodular case; we will not need it.
\begin{corollary}
Let $M$ be a lctd group and $N$ a closed subgroup, both unimodular. Suppose that there exists an involutive anti-automorphism $\sigma $ of $M$ such that $\sigma (N)=N$ and such that any distribution on $M$, invariant under the adjoint action of $N$, is fixed by $\sigma $. Then, for any irreducible admissible representation $\pi $ of $M$ and any irreducible admissible representation $\rho $ of $N$
\[ \dim\left ({\mathrm {Hom}}_ N(\pi _{|N},\rho ^*)\right )\times \dim\left ({\mathrm {Hom}}_ N((\pi ^*)_{|N},\rho )\right )\leq 1.\]
\end{corollary}
{\it Proof.} Let $M'=M\times N$ and $N'$ be the closed subgroup of
$M'$ which is the image of the homomorphism $n\mapsto (n,n)$ of
$N$ into $M$. The map $(m,n)\mapsto mn^{-1}$ of $M'$ onto $M$
defines a homeomorphism of $M'/N'$ onto $M$. The inverse map is
$m\mapsto (m,1)N'$. On $M'/N'$ left translations by $N'$
correspond to the adjoint action of $N$ onto $M$. We have a
bijection between the space of distributions $T$ on $M$ invariant
under the adjoint action of $N$ and the space of distributions $S$
on $M'$ which are biinvariant under $N'$. Explicitly
\[ \langle S,f(m,n)\rangle =\langle T,\int _Nf(mn,n)dn\rangle . \]
Suppose that $T$ is invariant under $\sigma $ and consider the involutive anti-automorphism $\sigma '$ of $M'$ given by $\sigma '(m,n)=(\sigma (m),\sigma (n))$. Then
\[ \langle S,f\circ \sigma '\rangle =\langle T,\int _Nf(\sigma (n)\sigma (m),\sigma (n))dn\rangle. \]
Using the invariance under $\sigma $ and for the adjoint action of $N$ we get
\begin{eqnarray*}
\langle T,\int _Nf(\sigma (n)\sigma (m),\sigma (n))dn\rangle &=&\langle T,\int _Nf(\sigma (n)m,\sigma (n))dn\rangle \\
&=&\langle T,\int _Nf(mn,n)dn\rangle \\
&=&\langle S,f\rangle .
\end{eqnarray*}
\overfullrule=0pt Hence $S$ is invariant under $\sigma '$.
Conversely if $S$ is invariant under $\sigma '$ the same
computation shows that $T$ is invariant under $\sigma $. Under the
assumption of the corollary we can now apply Proposition 1-1 and
we obtain the inequality
\[ {\mathrm {dim}} \left ({\mathrm {Hom}} _ {M'}(ind_ {N'}^{M'}(1),\pi \otimes \rho )\right )\times {\mathrm {dim}} \left ({\mathrm {Hom}} _ {M'}(ind_ {N'}^{M'}(1),\pi ^*\otimes \rho ^*)\right )\leq 1.\]
We know that $Ind_ {N'}^{M'}(1)$ is the smooth contragredient representation of $ind_{N'}^{M'}(1)$; hence
\[ {\mathrm {Hom}} _ {M'}(ind_ {N'}^{M'}(1),\pi ^*\otimes \rho ^*)\approx {\mathrm{ Hom}} _ {M'}(\pi \otimes \rho , Ind _ {N'}^{M'}(1)).\]
Frobenius reciprocity tells us that
\[ {\mathrm {Hom}} _ {M'}\bigl (\pi \otimes \rho , Ind _ {N'}^{M'}(1)\bigr )\approx {\mathrm {Hom}} _ {N'}\bigl ((\pi \otimes \rho )_{|N'},1\bigr ).\]
Clearly
\[ {\mathrm {Hom}} _ {N'}\bigl ((\pi \otimes \rho )_{|N'},1\bigr )\approx {\mathrm {Hom}} _ {N}\bigl (\rho ,(\pi _ {|N})^*\bigr )\approx {\mathrm {Hom}}_ N(\pi _{|N},\rho ^*).\]
Using again Frobenius reciprocity we get
\[ {\mathrm {Hom}} _ {N}\bigl (\rho ,(\pi _ {|N})^*\bigr )\approx {\mathrm {Hom}} _ {M}\bigl (ind_ N^M(\rho ),\pi ^*\bigr ).\]
In the above computations we may replace $\rho $ by $\rho ^*$ and $\pi $ by $\pi ^*$. Finally
\begin{eqnarray*}
{\mathrm {Hom}} _ {M'}(ind_ {N'}^{M'}(1),\pi ^*\otimes \rho ^*)&\approx& {\mathrm {Hom}} _ {N}(\rho ,(\pi _ {|N})^*)\\
&\approx& {\mathrm {Hom}}_ N(\pi _{|N},\rho ^*)\\
&\approx& {\mathrm {Hom}} _ {M}(ind_ N^M(\rho ),\pi ^*).\\
{\mathrm {Hom}} _ {M'}(ind_ {N'}^{M'}(1),\pi \otimes \rho )
&\approx&{\mathrm {Hom}} _ {N}(\rho ^*,((\pi ^*)_ {|N})^*)\\
&\approx& {\mathrm {Hom}}_ N((\pi ^*)_{|N},\rho )\\
&\approx& {\mathrm {Hom}} _ {M}(ind_ N^M(\rho ^*),\pi ).
\end{eqnarray*}\hfill$\Box$\smallskip
Going back to our situation and keeping the notations of the
introduction consider first the case of the general linear group.
We take $M=GL(W)$ and $N=GL(V)$. Let $E_\pi $ be the space of the
representation $\pi $ and let $E_\pi ^*$ be the smooth dual
(relative to the action of $GL(W))$. Let $E_\rho $ be the space of
$\rho $ and $E^*_\rho $ be the smooth dual for the action of
$GL(V)$. We know (\cite{BZ} section 7), that the contragredient
representation $\pi ^*$ in $E_\pi ^*$ is isomorphic to the
representation $g\mapsto \pi (^tg^{{-1}})$ in $E_\pi $. The same
is true for $\rho ^*$. Therefore an element of ${\mathrm {Hom}}_
N(\pi _{|N},\rho ^*)$ may be described as a linear map $A$ from
$E_\pi $ into $E_\rho $ such that, for $g\in N$
\[ A\pi (g)=\rho (^tg^{-1})A.\]
An element of ${\mathrm {Hom}}_ N((\pi ^*)_{|N},\rho )$ may be described as a linear map $A'$ from $E_\pi $ into $E_\rho $ such that, for $g\in N$
\[ A'\pi (^tg^{-1})=\rho (g)A'.\]
We have obtained the same set of linear maps:
\[ {\mathrm {Hom}}_ N((\pi ^*)_{|N},\rho )\approx {\mathrm {Hom}}_ N(\pi _{|N},\rho ^*).\]
We are left with 2 possibilities: either both spaces have dimension 0 or they both have dimension 1 which is exactly what we want.
From now on we forget Theorem 1 and prove Theorem 2.
Consider the orthogonal/unitary case, with the notations of the introduction.
In Chapter 4 of \cite{MVW} the following result is proved. Choose $\delta \in GL_{\mathbb F}(W)$ such that $\langle \delta w,\delta w'\rangle =\langle w',w\rangle $. If $\pi $ is an irreducible admissible representation of $M$, let $\pi ^*$ be its smooth contragredient and define $\pi ^\delta $ by
\[ \pi ^\delta (x)=\pi (\delta x\delta ^{-1}).\]
Then $\pi ^\delta $ and $\pi ^*$ are equivalent. We choose $\delta =1$ in the orthogonal case ${\mathbb D}={\mathbb F}$. In the unitary case, fix an orthogonal basis of $W$, say $e_1,\dots ,e_{n+1}$, such that $e_2,\dots ,e_{n+1}$ is a basis of $V$; put $\langle e_i,e_i\rangle =a_i$. Then
\[ \langle \sum x_ie_i,\sum y_je_j\rangle =\sum a_ix_i\overline{y_i}.\]
Define $\delta $ by
\[ \delta \left (\sum x_ie_i\right )=\sum \overline{x_i}e_i.\]
Note that $\delta ^2=1$.
Let $E_\pi $ be the space of $\pi $. Then, up to equivalence, $\pi ^*$ is the representation $m\mapsto \pi (\delta m\delta ^{-1})$. If $\rho $ is an admissible irreducible representation of $G$ in a vector space $E_\rho $ then an element $A$ of ${\mathrm {Hom}}\left (\pi ^*_{|G},\rho \right )$ is a linear map from $E_\pi $ into $E_\rho $ such that
\[ A\pi (\delta g\delta ^{-1})=\pi (g)A,\quad g\in G.\]
In turn the contragredient $\rho ^*$ of $\rho $ is equivalent to the representation $g\mapsto \rho (\delta g\delta ^{-1})$ in $E_\rho $. Then an element $B$ of ${\mathrm {Hom}}\left (\pi _{|G},\rho ^*\right )$ is a linear map from $E_\pi $ into $E_\rho $ such that
\[ B\pi (g)=\rho (\delta g\delta ^{-1})B,\quad g\in G.\]
As $\delta ^2=1$ the conditions on $A$ and $B$ are the same:
\[ {\mathrm {Hom}}\left (\pi ^*_{|G},\rho \right )\approx {\mathrm {Hom}}\left (\pi _{|G},\rho ^*\right ). \]
However, assuming Theorem 2', by Corollary 1-1 we have
\[ \dim\biggl ({\mathrm {Hom}}\left (\pi ^*_{|G},\rho \right )\biggr )\times \dim\biggl ({\mathrm {Hom}}\left (\pi _{|G},\rho ^*\right )\biggr )\leq 1.
\]
so that both dimensions are 0 or 1. Replacing $\rho $ by $\rho ^*$ we get Theorem 1'. From now on we forget about Theorem 1'.
\@startsection {section}{1{\hglue -16pt . Reduction to the singular set : the GL(n) case}
If $ H$ is a topological group of type lctd, acting continuously
on a topological space $ E$ of the same type and if $\chi $ is a
continuous character of $ H$ we denote by $ {\mathcal
S}'(E)^{H,\chi }$ the space of distributions $ T$ on $ E$ such
that $ \langle T,f(h^{-1}x)\rangle =\chi (h)\langle T,f\rangle $ for
any $ f\in {\mathcal S}(E)$ and any $ h\in H$.
Consider the case of the general linear group. From the decomposition $ W=V\oplus {\mathbb F}e$ we get, with obvious identifications
$$ {\mathrm {End}} (W)={\mathrm {End}} (V)\oplus V\oplus V^*\oplus {\mathbb F}.$$
Note that $ {\mathrm {End}} (V)$ is the Lie algebra $ {\mathfrak
g}$ of $ G$. The group $ G$ acts on $ {\mathrm {End}} (W)$ by
$ g(X,v,v^*,t)=(gXg^{-1},gv,{}^t g^{-1}v^*,t)$. As before choose a
basis $ (e_1,\dots ,e_n)$ of $ V$ and let $ (e_1^*,\dots
,e_n^*)$ be the dual basis of $ V^*$. Define an isomorphism $ u$
of $ V$ onto $ V^*$ by $ u(e_i)=e_i^*$. On $ GL(W)$ the
involution $ \sigma $ is $ h\mapsto u^{-1}{}^th^{-1}u$. It
depends upon the choice of the basis but the action on the space
of invariant distributions does not depend upon this choice.
It will be convenient to introduce an extension $ \widetilde G$ of $ G$. Let $ {\mathrm {Iso}}(V,V^*)$ be the set of isomorphisms of $ V$ onto $ V^*$. We define $ \widetilde G=G\cup \mathrm {Iso}(V,V^*)$. The group law, for $ g,g'\in G$ and $ u,u'\in \mathrm {Iso}(V,V^*)$ is
$$ g \times g'=gg',\,\,\, u \times g=ug,\,\,\, g \times u=^t\hskip -3 pt g^{-1}u,\,\,\, u \times u'=^t\hskip -3 pt u^{-1}u'.$$
Now from $ W=V\oplus {\mathbb F}e$ we obtain an identification
of the dual space $ W^*$ with $ V^*\oplus {\mathbb F}e^*$ with
$ \langle e^*,V\rangle =(0)$ and $ \langle e^*,e\rangle =1$. Any
$ u$ as above extends to an isomorphism of $ W$ onto $ W^*$ by
defining $ u(e)=e^*$. The group $ \widetilde G$ acts on $ GL(W)$
:
$$ h\mapsto ghg^{-1},\,\,\,\, h\mapsto ^t\hskip -3pt (uhu^{-1}),\quad g\in G,\, h\in GL(W),\,\, u\in \mathrm {Iso}(V,V^*)$$
and also on $ {\mathrm {End}} (W)$ with the same formulas.
Let $\chi $ be the character of $ \widetilde G$ which is 1 on $
G$ and $ -1$ on $ \mathrm {Iso}(V,V^*)$. Our goal is to prove
that $ {\mathcal S}'(GL(W))^{\widetilde G, \chi }=(0)$.
\begin{proposition}
If $ {\mathcal S}'({\mathfrak g}\oplus V\oplus V^*)^{\widetilde
G, \chi }=(0)$ then $ {\mathcal S}'(GL(W))^{\widetilde G, \chi
}=(0)$.
\end{proposition}
{\it Proof.} We have $ {\mathrm {End}}(W)=\bigl (\,\,{\mathrm {End}} (V)\oplus V\oplus V^*\bigr )\oplus {\mathbb F}$ and the action of $ \widetilde G$ on $ {\mathbb F}$ is trivial thus $ {\mathcal S}'({\mathfrak g}\oplus V\oplus V^*)^{\widetilde G, \chi }=(0)$ implies that $ {\mathcal S}'({\mathrm {End}} (W))^{\widetilde G, \chi }=(0).$
Let $ T\in {\mathcal S}'(GL(W))^{\widetilde G, \chi }$. Let $
h\in GL(W)$ and choose a compact open neighborhood $ K$ of $
{\mathrm {Det}}\,\, h$ such that $ 0\notin K$. For $ x\in
{\mathrm {End} } (W)$ define $ \varphi (x)=1$ if $ {\mathrm
{Det}} x\in K $ and $ \varphi (x)=0$ otherwise. Then $ \varphi
$ is a locally constant function. The distribution $ (\varphi
_{|GL(W)})T$ has a support which is closed in $ {\mathrm {End}}
(W)$ hence may be viewed as a distribution on $ {\mathrm {End} }
(W)$. This distribution belongs to $ {\mathcal S}'({\mathrm
{End}} (W))^{\widetilde G, \chi }$ so it must be equal to 0.
It follows that $ T$ is 0 in the neighborhood of $ h$. As $ h$
is arbitrary we conclude that $ T=0$.
\hfill$\Box$\smallskip
Our task is now to prove that $ {\mathcal S}'({\mathfrak g}\oplus
V\oplus V^*)^{\widetilde G, \chi }=(0)$. We shall use induction
on the dimension $ n$ of $ V$. The action of $ \widetilde G$
is, for $ X\in {\mathfrak g},\, v\in V,\, v^*\in V^*,\, g\in
G,\, u\in \mathrm {Iso}(V,V^*)$
$$ (X,v,v^*)\mapsto (gXg^{-1},gv,^t\hskip -3pt g^{-1}v^*),\,\,\, (X,v,v^*)\mapsto (^t\hskip -1pt (uXu^{-1}),^t\hskip -3pt u^{-1}v^*,uv).$$
The case $n=0$ is trivial.
We suppose that $ V$ is of dimension $ n\geq 1$, assuming the
result up to dimension $ n-1$ and for all $\mathbb F$. If $ T\in
{\mathcal S}'({\mathfrak g}\oplus V\oplus V^*)^{\widetilde G,
\chi }$ we are going to show that its support is contained in
the "singular set". This will be done in two stages.
On $ V\oplus V^*$ let $ \Gamma $ be the cone $ \langle
v^*,v\rangle =0$. It is stable under $ \widetilde G$.
\begin{lemma}
The support of $ T$ is contained in $ {\mathfrak g} \times \Gamma $.
\end{lemma}
{\it Proof.} For $ (X,v,v^*)\in {\mathfrak g}\oplus V\oplus V^*$
put $ q(X,v,v^*)=\langle v^*,v\rangle $. Let $ \Omega $ be the
open subset $ q\ne 0$. We have to show that $ {\mathcal S}'(\Omega
)^{\widetilde G, \chi }=(0)$. By Bernstein's localization
principle (Corollary 6-1) it is enough to prove that, for any fiber $ \Omega
_t=q^{-1}(t),\,\,\, t\ne 0$, one has $ {\mathcal S}'(\Omega
_t)^{\widetilde G, \chi }=(0)$.
$ G$ acts transitively on the quadric $ \langle v^*,v\rangle =t$. Fix a decomposition
$ V={\mathbb F}\varepsilon \oplus V_1$ and identify $ V^*={\mathbb F}\varepsilon ^*\oplus V_1^*$
with $ \langle \varepsilon ^*,\varepsilon \rangle =1$. Then $ (X,\varepsilon ,t\varepsilon ^*)\in \Omega _t$
and the isotropy subgroup of $ (\varepsilon ,t\varepsilon ^*)$ in $ \widetilde G$ is, with an obvious notation
$ \widetilde G_{n-1}$. By Frobenius descent (Theorem 6-2) there is a linear bijection between
$ {\mathcal S}'(\Omega _t)^{\widetilde G, \chi }$ and the space
$ {\mathcal S}'({\mathfrak g})^{\widetilde G_{n-1}, \chi }$ and this last space is $ (0)$ by induction.
\hfill$\Box$\smallskip
Let $ {\mathfrak z}$ be the center of $ {\mathfrak g}$ that is
to say the space of scalar matrices. Let $ {\mathcal N}\subset
[{\mathfrak g},{\mathfrak g}]$ be the nilpotent cone in $
{\mathfrak g}$.
\begin{lemma}
The support of $ T$ is contained in $ {\mathfrak z} \times {\mathcal N} \times \Gamma $.
\end{lemma}
{\it Proof.} We use Harish-Chandra's descent. For $ X\in
{\mathfrak g}$ let $ X=X_s+X_n$ be the Jordan decomposition of
$ X$ with $ X_s$ semisimple and $ X_n$ nilpotent. This
decomposition commutes with the action of $ \widetilde G$. The
centralizer $ Z_G(X)$ of an element $ X\in {\mathfrak g}$ is
unimodular (\cite{SS} page 235) and there exists an isomorphism $
u$ of $ V$ onto $ V^*$ such that $ ^tX=uXu^{-1}$ (any matrix
is conjugate to its transpose). It follows that the centralizer $
Z_{\widetilde G}(X)$ of $ X$ in $ \widetilde G$, a semi direct
product of $ Z_G(X)$ and $ S_2$ is also unimodular.
Let $ E$ be the vector space of monic polynomials, of degree $ n$
, with coefficients in $ {\mathbb F}$. For $ p\in E$, let $
{\mathfrak g}_p$ be the set of all $ X\in {\mathfrak g}$ with
characteristic polynomial $ p$. Note that $ {\mathfrak g}_p$ is
fixed by $ \widetilde G$. By Bernstein localization principle (Corollary 6-1) it
is enough to prove that if $ p$ is not $ (T-\lambda )^n$ for
some $ \lambda $ then $ {\mathcal S}'({\mathfrak g}_p \times V
\times V^*)^{\widetilde G, \chi }=(0)$.
Fix $ p$. We claim that the map $ X\mapsto X_s$ restricted to $
{\mathfrak g}_p$ is continuous. Indeed let $ \widetilde {\mathbb
F}$ be a finite Galois extension of $ {\mathbb F}$ containing
all the roots of $ p$. Let
$$ p(\xi )=\prod _1^s(\xi -\l_i)^{n_i}$$
be the decomposition of $ p$. Recall that if $ X\in {\mathfrak
g}_p$ and $ V_i=\mathrm{Ker } (X-\lambda _I)^{n_i}$ then $
V=\oplus V_i$ and the restriction of $ X_s$ to $ V_i$ is the
multiplication by $ \lambda _i$. Then choose a polynomial $ R$,
with coefficients in $ \widetilde {\mathbb F}$ such that for
all $ i$, $ R$ is congruent to $ \lambda _i$ modulo $ (\xi
-\lambda _i)^{n_i}$ and $ R(0)=0$ to respect the tradition.
Clearly $ X_s=R(X)$. As the Galois group of $ \widetilde {\mathbb
F}$ over $ {\mathbb F}$ permutes the $ \lambda _i$ we may
even choose $ R\in {\mathbb F}[\xi ]$. This implies the
required continuity.
There is only one semi-simple orbit $ \gamma _p$ in $ {\mathfrak
g}_p$ and it is closed. We use Frobenius descent for the map $
(X,v,v^*)\mapsto X_s$ from $ {\mathfrak g}_p \times V \times
V^*$ to $ \gamma _p$.
Fix $ a\in \gamma _p$ ; its fiber is the product of $ V\oplus
V^*$ by the set of nilpotent elements which commute with $ a$. It
is a closed subset of the centralizer $ {\mathfrak m}={\mathfrak
Z}_{{\mathfrak g}}(a)$ of $ a$ in $ {\mathfrak g}$. Let $
M=Z_G(a)$ and $ \widetilde M=Z_{\widetilde G}(a)$.
Following (\cite{SS} ) let us describe these centralizers.
Let $P$ be the minimal polynomial of $ a$ ; all its roots are simple. Let $P=P_1\dots P_r$ be
the decomposition of $P$ into irreducible factors, over ${\mathbb F}$. Then the $P_i$
are two by two relatively prime. If $V_i={\mathrm Ker} P_i(a)$, then $V=\oplus V_i$ and $V^*=\oplus V_i^*$. An element $x$ of $G$
which commutes with $a$ is given by a family $\{x_1,\dots ,x_r\}$
where each $x_i$ is a linear map from $V_i$ to $V_i$, commuting
with the restriction of $a$ to $V_i$. Now ${\mathbb F}[\xi ]$
acts on $V_i$, by specializing $\xi $ to $a_{|V_i}$ and $P_i$ acts
trivially so that, if ${\mathbb F}_i={\mathbb F}[\xi
]/(P_i)$, then $V_i$ becomes a vector space over ${\mathbb
F}_i$. The ${\mathbb F}-$linear map $x_i$ commutes with $a$ if
and only if it is ${\mathbb F}_i-$linear.\hfill\break
Fix $i$. Let $\ell$ be a non zero ${\mathbb F}-$linear form on
${\mathbb F}_i$. If $v_i\in V_i$ and $v'_i\in V_i^*$ then
$\lambda \mapsto \langle \lambda v_i,v'_i\rangle $ is an
${\mathbb F}-$linear form on ${\mathbb F}_i$, hence there
exists a unique element $S(v_i,v'_i)$ of ${\mathbb F}_i$ such
that $\langle \lambda v_i,v'_i\rangle =\ell\left (\lambda
S(v_i,v'_i)\right )$. One checks trivially that $S$ is ${\mathbb
F}_i-$linear with respect to each variable and defines a non
degenerate duality, over ${\mathbb F}_i$ between $V_i$ and
$V_i^*$. Here ${\mathbb F}_i$ acts on $V_i^*$ by transposition,
relative to the ${\mathbb F}-$duality $\langle .,.\rangle $,
of the action on $V_i$. Finally if $x_i\in \mathrm {End}
_{{\mathbb F}_i} V_i$, its transpose, relative to the duality
$S(.,.)$ is the same as its transpose relative to the duality
$\langle .,.\rangle $.
Thus $M$ is a product of linear groups and the situation $(M,V,V^*)$ is a composite case, each component being
a linear case (over various extensions of ${\mathbb F}$).
Let $ u$ be an isomorphism of $ V$ onto $ V^*$ such that $
^ta=uau^{-1}$ and that, for each $ i,\,\,\, u(V_i)=V_i^*$. Then $
u\in \widetilde M$ and $ \widetilde M=M\cup uM$.
Suppose that $ a$ does not belong to the center of $ {\mathfrak
g}$. Then each $ V_i$ has dimension strictly smaller than $ n$
and we can use the inductive assumption. Therefore $ {\mathcal
S}'({\mathfrak m}\oplus V\oplus V^*)^{\widetilde M, \chi
}=(0)$. However the nilpotent cone $ {\mathcal N}_{\mathfrak m}$
in $ {\mathfrak m}$ is a closed subset so $ {\mathcal
S}'({\mathcal N}_{\mathfrak m} \times V \times V^*)^{\widetilde
M, \chi }=(0)$ which is what we need.
\hfill$\Box$\smallskip
If $ a$ belongs to the center then $ \widetilde M=\widetilde G$
and the fiber is $ (a+{\mathcal N}) \times V \times V^*$.
Therefore we have proved the following Proposition:
\begin{proposition}
If $ T\in {\mathcal S}'({\mathfrak g}\oplus V\oplus V^*)^{\widetilde G, \chi }$ then the support of $T$ is contained in ${\mathfrak z}\times{\mathcal N}\times{\Gamma}$.\hfill\break
If $ {\mathcal S}'({\mathcal N} \times\Gamma)^{\widetilde G, \chi }=(0)$ then $ {\mathcal S}'({\mathfrak g}\oplus V\oplus V^*)^{\widetilde G, \chi }=(0)$.
\end{proposition}
{\textbf{Remark}\unskip . --- \ignorespaces}Strictly speaking the singular set is
defined as the set of all $(X,v,v^*)$ such that for any
polynomial $P$ invariant under $\widetilde G$ one has
$P(X,v,v^*)=P(0)$. So we should take care of the invariants
$P(X,v,v^*)=\langle v^*,X^pv\rangle$ for all $p$ and not only for
$p=0$. It can be proved, a priori, that the support of the
distribution $T$ has to satisfy these extra conditions. As this is
not needed in the sequel we omit the proof.
\@startsection {section}{1{\hglue -16pt . End of the proof for GL(n)}
In this section we consider a distribution $ T\in {\mathcal
S}'({\mathcal N} \times \Gamma )^{\widetilde G, \chi }$ and
prove that $ T=0$. The following observation will play a crucial
role.
Choose a non trivial additive character $\psi$ of $\mathbb F$. On $V\oplus V^*$ we have the bilinear form
$$\bigl ((v_1,v_1^*),(v_2,v_2^*)\bigr )\mapsto \langle v_1^*,v_2\rangle +\langle v_2^*,v_1\rangle.$$
Define the Fourier transform by
$$\widehat \varphi (v_2,v_2^*)=\int _{V\oplus V^*}\varphi (v_1,v_1^*)\, \psi (\langle v_1^*,v_2\rangle +\langle v_2^*,v_1\rangle )\, dv_1dv_1^*$$
with $dv_1dv_1^*$ is normalized so that there is no constant factor appearing in the inversion formula.
This Fourier transform commutes with the action of $\widetilde G$;
hence the (partial) Fourier transform $\widehat T$ of our
distribution $T$ has the same invariance properties and the same
support conditions as $T$ itself.
Let $ {\mathcal N}_i$ be the union of nilpotent orbits of
dimension at most $ i$. We will prove, by descending induction on
$ i$, that the support of any $(\widetilde{G},\chi)-$equivariant
distribution must be contained in $ {\mathcal N}_i \times \Gamma
$. Suppose we already know that, for some $ i$, the support must
be contained in $ {\mathcal N}_i \times \Gamma $. We must show
that, for any nilpotent orbit $ {\mathcal O}$ of dimension $ i$,
the restriction of the distribution to $ {\mathcal O} \times
\Gamma $ is 0.
If $ v\in V$ and $ v^*\in V^*$ we call $ X_{v,v^*}$ the rank
one map $ x\mapsto \langle v^*,x\rangle v$. Let
$$ \nu_\lambda (X,v,v^*)=(X+\lambda X_{v,v^*},v,v^*),\quad (X,v,v^*)\in {\mathfrak g} \times \Gamma ,\,\,\, \lambda \in {\mathbb F}.$$
Then $ \nu_\lambda $ is a one parameter group of homeomorphisms
of $ {\mathfrak g} \times \Gamma $ and note that $ [{\mathfrak
g},{\mathfrak g}] \times \Gamma $ is invariant. The key
observation is that {\it $ \nu_\lambda $ commutes with the
action of $ \widetilde G$ }. Therefore the image of $ T$ by $
\nu_\lambda $ transforms according to the character $\chi $ of $
\widetilde G$. Its support is contained in $ [{\mathfrak
g},{\mathfrak g}] \times \Gamma $ and hence must be contained in $
{\mathcal N} \times \Gamma $ and in fact in $ {\mathcal N}_i
\times \Gamma $. This means that if $ (X,v,v^*)$ belongs to the
support of $ T$ then, for all $ \lambda $, $ (X+\lambda
X_{v,v^*},v,v^*)$ must belong to $ {\mathcal N}_i \times \Gamma
$.
The orbit $ {\mathcal O}$ is open in $ {\mathcal N}_i$. Thus if
$ X\in {\mathcal O}$ the condition $ X+\lambda X_{v,v^*}\in
{\mathcal N}_i$ implies that, at least for $ |\lambda |$ small
enough, $ X+\lambda X_{v,v^*}\in {\mathcal O}$. It follows that
$ X_{v,v^*}$ belongs to the tangent space to $ {\mathcal O}$ at
the point $ X$ and this tangent space is the image of $
\mathrm{ad } X$.
Let us call $Q(X)$ the set of all pairs $(v,v^*)$ such $X_{v,v^*}\in \mathrm {Im\,\, ad}X$.
Therefore it is enough to prove the following Lemma:
\begin{lemma}
Let $ T\in {\mathcal S}'({\mathcal O} \times V \times V^*)^{\widetilde G, \chi }$. Suppose that the support of $ T$ and of $ \widehat T$ are contained in the set of triplets $ (X,v,v^*)$ such that $ (v,v^*)\in Q(X)$.
Then $ T=0$.
\end{lemma}
Note that the trace of $ X_{v,v^*}$ is $ \langle v^*,v\rangle $
and that $ X_{v,v^*}\in {\mathrm Im}\,\, \mathrm{ad } \, X$
implies that its trace is 0. Therefore $ Q(X)$ is contained in $
\Gamma $.
We proceed in three steps. First we transfer the problem to $
V\oplus V^*$ and a fixed nilpotent endomorphism $ X$. Then we
show that if Lemma 3-1 is true for $ (V_1,X_1)$ and $ (V_2,X_2)$
then it is true for the direct sum $ (V_1\oplus V_2,X_1\oplus
X_2)$. Finally using the decomposition of $ X$ in Jordan blocks
we are left with the case of a principal nilpotent element for
which we give a direct proof, using Weil representation.
Consider the map $ (X,v,v^*)\mapsto X$ from $ {\mathcal O}
\times V \times V^*$ onto $ {\mathcal O}$. Choose $X\in{\mathcal
O}$ and let $ C$ (resp $ \widetilde C$) be the stabilizer in $
G$ (resp. in $ \widetilde G$) of an element $ X$ of $ {\mathcal
O}$ ; both groups are unimodular, hence we may use Frobenius
descent (Theorem 6-2).
Now we have to deal with a distribution, which we still call $T$, which belongs to $ {\mathcal S}'(V\oplus V^*)^{\widetilde C, \chi }$ such that both $ T$ and its Fourier transform are supported by $Q(X)$. Let us say that $ X$ is nice if the only such distribution is 0. We want to prove that all nilpotent endomorphisms are nice.
\begin{lemma}
Suppose that we have a decomposition $ V=V_1\oplus V_2$ such that $ X(V_i)\subset V_i$. Let $ X_i$ be the restriction of $ X$ to $ V_i$. Then if $ X_1$ and $ X_2$ are nice, so is $ X$.
\end{lemma}
{\it Proof of Lemma 3.2.} Let $ Q(X)$ be the set of pairs $ (v,v^*)$ such that $ X_{v,v^*}$ belongs to the image of $ \mathrm{ad } X$. Let $ (v,v^*)\in Q(X)$ and choose $ A\in {\mathfrak g}$ such that $ X_{v,v^*}=[A,X]$. Decompose $ v=v_1+v_2,\,\, v^*=v_1^*+v_2^*$ and put
$$ A=\pmatrix{A_{1,1}&A_{1,2}\cr A_{2,1}&A_{2,2}\cr}.$$
Writing $ X_{v,v^*}$ as a 2 by 2 matrix and looking at the diagonal blocks one gets that $ X_{v_i,v_i^*}=[A_{i,i},X_i]$. This means that
$$ Q(X)\subset Q(X_1) \times Q(X_2).$$
For $ i=1,2$ let $ C_i$ be the centralizer of $ X_i$ in $ GL(V_i)$ and $ \widetilde C_i$ the corresponding extension by $ S_2$.
Let $ T$ be a distribution as above and let $ \varphi _2\in {\mathcal S}(V_2\oplus V_2^*)$. Let $ T_{1}$ be the distribution on $ V_1\oplus V_1^*$ defined by $ \varphi _1\mapsto \langle T,\varphi _1\otimes \varphi _2\rangle $. The support of $ T_{1}$ is contained in $ Q(X_1)$ and $ T_{1}$ is invariant under the action of $ C_1$. We have
$$ \langle \widehat T_{1},\varphi _1\rangle =\langle T_{1},\widehat \varphi _1\rangle =\langle T, \widehat {\varphi _1}\otimes \varphi _2\rangle =\langle \widehat T,\check {\varphi _1}\otimes \widehat{\varphi _2}\rangle .$$
Here $ \check \varphi _1(v_1,v_1^*)=\varphi _1(-v_1,-v_1^*)$. By assumption the support of $ \widehat T$ is contained in $ Q(X)$ so that the support of $ \widehat{T_1}$ is supported in $ -Q(X_1)=Q(X_1)$. Because $ (X_1)$ is nice this implies that $ T_1$ in invariant under $ \widetilde C_1$. Imbedding $ \widetilde C_1$ into $ \widetilde C$ we get that $ T$ is invariant under $ \widetilde C_1$. Similarly it is invariant under $ \widetilde C_2$. However the subgroup $ \widetilde C_1 \times \widetilde C_2$ of $ \widetilde C$ is not contained in $ C$ so that $ T$ must be invariant under $ \widetilde C$ and hence must be 0.
\hfill$\Box$\hfill
Decomposing $ X$ into Jordan blocks we still have to prove Lemma 3-1 for a principal nilpotent element. We need some preliminary results.
\begin{lemma}
The distribution $ T$ satisfies the following homogeneity condition:
$$ \langle T,f(tv,tv^*)\rangle =|t|^{-n}\langle T,f(v,v^*)\rangle .$$
\end{lemma}
{\it Proof of Lemma 3.3.} We use a particular case of Weil or oscillator representation. Let $ E$ be a vector space over $ {\mathbb F}$ of finite dimension $ m$. To simplify assume that $ m$ is even. Let $ q$ be a non degenerate quadratic form on $ E$ and let $ b$ be the bilinear form
$$ b(e,e')=q(e+e')-q(e)-q(e').$$
Fix a continuous non trivial additive character $ \psi $ of $
{\mathbb F}$. We define the Fourier transform on $ E$ by
$$ \widehat f(e')=\int _Ef(e)\psi (b(e,e'))de$$
where $ de$ is the self dual Haar measure.
There exists (\cite{RS2}) a representation $ \pi $ of $
SL(2,{\mathbb F})$ in $ {\mathcal S}(E)$ such that:
\begin{eqnarray*}
\pi \pmatrix{1&u\cr 0&1\cr}f(e)&=&\psi (uq(e))f(e)\\
\pi \pmatrix{t&0\cr 0&t^{-1}}f(e)&=&\frac{\gamma (q)}{\gamma (tq)}|t|^{m/2}f(te)\\
\pi \pmatrix{0&1\cr -1&0\cr}f(e)&=&\gamma (q)\widehat f(e).\\
\end{eqnarray*}
The $ \gamma (tq)$ are complex numbers of modulus 1. In
particular if $(E,q)$ is a sum of hyperbolic planes these numbers
are all equal to 1.
We have a contragredient action in the dual space $ {\mathcal
S'(E)}$.
Suppose that $ T$ is a distribution on $ E$ such that $ T$ and
$ \widehat T$ are supported by the isotropic cone $ q(e)=0$. This
means that
$$ \langle T,\pi \pmatrix{1&u\cr 0&1\cr}f\rangle =\langle T,f\rangle ,\quad \langle \widehat T,\pi \pmatrix{1&u\cr 0&1\cr}f\rangle =\langle \widehat T,f\rangle . $$
Using the relation
$$ \langle \widehat T,\varphi \rangle =\langle T,\overline {\gamma (q)}\pi \pmatrix{0&1\cr -1&0\cr}f\rangle $$
the second relation is equivalent to
$$ \langle T,\pi \pmatrix{1&0\cr -u&0\cr}f\rangle =\langle T,f\rangle .$$
The matrices
$$ \pmatrix{1&u\cr 0&1\cr},\quad {\mathrm {and}}\quad \pmatrix{1&0\cr u&1\cr}\quad u\in {\mathbb F}$$
generate the group $ SL(2,{\mathbb F})$. Therefore the
distribution $ T$ is invariant by $ SL(2,{\mathbb F)}$. In
particular
$$ \langle T,f(te)\rangle =\frac{\gamma (tq)}{\gamma (q)}|t|^{-m/2}\langle T,f\rangle $$
and $ T=\gamma (q)\widehat T$.
{\bf Remark\unskip . --- \ignorespaces } For $ m$ even $ \gamma (tq)/\gamma (q)$
is a character and there do exist non zero distributions invariant
under $ SL(2,{\mathbb F})$. In odd dimension we get a
representation of the 2-fold covering of $ SL(2,{\mathbb F})$
and we obtain the same homogeneity condition. However $ \gamma
(tq)/\gamma (q)$ is not a character; hence the distribution $
T$ must be 0.
In our situation we take $ E=V\oplus V^*$ and $ q(v,v^*)=\langle
v^*,v\rangle $. Then
$$ b\Bigl ((v_1,v_1^*),(v_2,v_2^*)\Bigr )=\langle v_1^*,v_2\rangle +\langle v_2^*,v_1\rangle.$$
The Fourier transform commutes with the action of $ \widetilde G$. Both $
T$ and $ \widehat T$ are supported by $ Q(X)$ which is
contained in $ \Gamma $. As $ \gamma (tq)=1$ this proves the
Lemma and also that $ T=\widehat T$. \hfill$\Box$\hfill
{\bf Remark\unskip . --- \ignorespaces} The same type of argument could have been
used for the quadratic form $ Tr(XY)$ on $ {\mathfrak
s}{\mathfrak l}(V)=[{\mathfrak g},{\mathfrak g}]$. This would
have given a short proof for even $ n$ and a homogeneity
condition for odd $ n$.
Now we find $ Q(X)$.
\begin{lemma}
If $ X$ is principal then $ Q(X)$ is the set of pairs $ (v,v^*)$ such that for $ 0\leq k<n$, $ \langle v^*,X^kv\rangle =0$.
\end{lemma}
{\it Proof of Lemma 3.4.} Choose a basis $ (e_1,\dots ,e_n)$ of $
V$ such that $ Xe_1=0$ and $ Xe_j=e_{j-1}$ for $ j\geq 2$.
Consider the map $ A\mapsto XA-AX$ from the space of n by n
matrices into itself. A simple computation shows that the kernel
of this map, that is to say the Lie algebra $ {\mathfrak c}$ of
the centralizer $ C$, is the space of polynomials (of degree at
most $ n-1$ ) in $ X$. It is of dimension $ n$. The image is of
codimension $ n$ and calling $ b_{i,j}$ the coefficients of an n
by n matrix, a set of independent equations for this image is
$$ \sum _{j=1}^{n-r}b_{j+r,j}=0,\quad r=0,\dots n-1.$$
Let $ (e_1^*,\dots ,e_n^*)$ be the dual basis. Call $ x_1,\dots
,x_n$ the coordinates of $ v$ and $ (x_1^*,\dots ,x_n^*)$ the
coordinates of $ v^*$. The matrix of $ X_{v,v^*}$ is then given
by $ b_{i,j}=x_ix_j^*$ and we get the lemma.
\hfill$\Box$\hfill
{\it End of the proof of Lemma 3.1. }For $X$ principal, we proceed
by induction on $ n$. Keep the above notations. The centralizer $
C$ of $ X$ is the space of polynomials (of degree at most $ n-1$
) in $ X$ with non zero constant term. In particular the orbit $
\Omega $ of $ e_n$ is the open subset $ x_n\ne 0$. We shall
prove that the restriction of $ T$ to $ \Omega \times V^*$ is 0.
Note that the centralizer of $ e_n$ in $ C$ is trivial. By
Frobenius descent (Theorem 6-2), to the restriction of $ T$ corresponds a
distribution $ R$ on $ V^*$ with support in the set of $ v^*$
such that $ (e_n,v^*)\in Q(X)$. By the last Lemma this means that
$ R$ is a multiple $ a\delta $ of the Dirac measure at the
origin. The distribution $ T$ satisfies the two conditions
$$ \langle T,f(v,v^*)\rangle =\langle T,f(tv,t^{-1}v^*)\rangle =|t|^n\langle T,f(tv,tv^*)\rangle .$$
therefore
$$ \langle T,f(v,t^2v^*)=|t|^{-n}\langle T,f(v,v^*)\rangle .$$
Now $ T$ is recovered from $ R$ by the formula
$$ \langle T,f(v,v^*)=\int _C\langle R,f(ce_n,^tc^{-1}v^*\rangle dc=a\int _Cf(ce_n,0)dc,\,\,\, f\in {\mathcal S}(\Omega \times V^*).$$
Unless $ a=0$ this is not compatible with this last homogeneity condition.
Exactly in the same way one proves that $ T$ is 0 on $ V \times
\Omega ^*$ where $ \Omega ^*$ is the open orbit $ x_1^*\ne 0$ of
$ C$ in $ V^*$. The same argument is valid for $ \widehat T$
(which is even equal to $ T$ \dots ).
If $ n=1$ then $ T$ is obviously 0. If $ n\geq 2$ then there exists a distribution $ T'$ on
$$ \bigoplus_{1<j<n}{\mathbb F}e_j\oplus{\mathbb F}e_j^*$$
such that,
$$ T=T'\otimes \delta _{x_n=0}\otimes dx_1\otimes \delta _{x_1^*=0}\otimes dx_n^*.$$
Let $ u$ be the isomorphism of $ V$ onto $ V^*$ given by $
u(e_j)=e_{n+1-j}^*$. Recall that it acts on $ {\mathfrak g}
\times V \times V^*$ by $ (X,v,v^*)\mapsto
(^t(uXu^{-1}),^tu^{-1}v^*,uv)$. It belongs to $ \widetilde C$ but
not to $ C$ so it must transform $ T$ into $ -T$.
The case $ n=1$ has just been settled. If $ n=2$ in the above
formula $ T'$ should be replaced by a constant. The constant must
be 0 if we want $ u(T)=-T$. If $ n>2$ let
$$V'=\Bigl (\oplus_1^{n-1}{\mathbb F}e_i\Bigr )/{\mathbb F}e_1$$
and let $X'$ be the nilpotent endomorphism of $V'$ defined by $X$.
We may consider $T'$ as a distribution on $V\oplus V^{'*}$ and one
easily checks that, with obvious notations, it transforms
according to the character $\chi$ of the the centralizer
$\widetilde C'$ of $X'$ in $\widetilde G'$. By induction $T'=0$,
hence $T=0$.
\hfill$\Box$\hfill
\@startsection {section}{1{\hglue -16pt. Reduction to the singular set: the orthogonal and unitary cases}
We now turn our attention to the unitary case. We keep the
notations of the introduction. In particular $W=V\oplus {\mathbb
D}e$ is a vector space over ${\mathbb D}$ of dimension $n+1$ with
a non degenerate hermitian form $\langle.,.\rangle $ such that $e$
is orthogonal to $V$. The unitary group $G$ of $V$ is embedded
into the unitary group $M$ of $W$.
Let $A$ be the set of all bijective maps $u$ from $V$ to $V$ such that
$$ u(v_1+v_2)=u(v_1)+u(v_2),\,\,\, u(\lambda v)=\overline\lambda u(v),\,\,\, \langle u(v_1),u(v_2)\rangle =\overline{\langle v_1,v_2\rangle }.$$
An example of such a map is obtained by choosing a basis $e_1,\dots ,e_n$ of $V$ such that $\langle e_i,e_j\rangle \in {\mathbb F}$ and defining
$$ u(\sum x_ie_i)=\sum \overline x_ie_i.$$
Any $u\in A$ is extended to $W$ by the rule $u(v+\lambda e)=u(v)+\overline\lambda e$ and we define an action on ${\mathrm GL}(W)$ by $m\mapsto um^{-1}u^{-1}$. The group $G$ acts on ${\mathrm GL}(W)$ by the adjoint action.
Let $\widetilde G$ be the group of bijections of ${\mathrm GL}(W)$ onto itself generated by the actions of $G$ and $A$. It is a semi direct product of $G$ and $S_2$. We identify $G$ to a subgroup of $\widetilde G$ and $A$ to a subset. When a confusion is possible we denote the product in $\widetilde G$ with a $\times$.
We define a character $\chi $ of $\widetilde G$ by $\chi (g)=1$ for $g\in G$ and $\chi (u)=-1$ for $u\in \widetilde G\setminus G$.
Our overall goal is to prove that ${\mathcal S}'(M)^{\widetilde G,\chi }=(0)$.
Let $\widetilde G$ act on $G\times V$ as follows:
$$ g(x,v)=(gxg^{-1},g(v)),\,\, u(x,v)=(ux^{-1}u^{-1},-u(v)),\quad g\in G,u\in A,x\in G,v\in V$$
Our first step is to replace $M$ by $G\times V$.
\begin{proposition}
Suppose that for any $V$ and any hermitian form ${\mathcal S}'(G\times V)^{\widetilde G,\chi }=(0)$, then ${\mathcal S}'(M)^{\widetilde G,\chi }=(0)$.
\end{proposition}
{\it Proof.} We have in particular ${\mathcal S}'(M\times
W)^{\widetilde M,\chi }=(0)$. Let $Y$ be the set of all $(m,w)$
such that $\langle w,w\rangle =\langle e,e\rangle $; it is a
closed subset, invariant under $\widetilde M$, hence ${\mathcal
S}'(Y)^{\widetilde M,\chi }=(0)$. By Witt's theorem $M$ acts
transitively on $\Gamma =\{w|\langle w,w\rangle =\langle
e,e\rangle \} $. We can apply Frobenius descent (Theorem 6-2) to the map
$(m,w)\mapsto w$ of $Y$ onto $\Gamma $. The centralizer of $e$ in
$\widetilde M$ is isomorphic to $\widetilde G$ acting as before on
the fiber $M\times \{e\}$. We have a linear bijection between
${\mathcal S}'(M)^{\widetilde G.\chi }$ and ${\mathcal
S}'(Y)^{\widetilde M,\chi }$; therefore ${\mathcal
S}'(M)^{\widetilde G.\chi }=(0)$.
\hfill$\Box$\hfill
The proof that ${\mathcal S}'(G\times V)^{\widetilde G,\chi }=(0)$ is by induction on $n$. If ${\mathfrak g}$ is the Lie algebra of $G$ we shall prove simultaneously that ${\mathcal S}'({\mathfrak g}\times V)^{\widetilde G,\chi }=(0)$. In this case $G$ acts on its Lie algebra by the adjoint action and for $u\in \widetilde G\setminus G$ one puts, for $X\in {\mathfrak g},\,\,$ $u(X)=-uXu^{-1}$.
The case $n=0$ is trivial so we may assume that $n\geq 1$. If $T\in {\mathcal S}'(G\times V)^{\widetilde G,\chi }$ in this section we will prove that the support of $T$ must be contained in the "singular set".
Let $Z$ (resp. {$\mathfrak z$}) be the center of $G$ (resp. {$\mathfrak g$}) and ${\mathcal U}$ (resp. {$\mathcal N$}) the (closed) set of all unipotent (resp. nilpotent) elements of $G$ (resp. {$\mathfrak g$}).
\begin{lemma}
If $T\in {\mathcal S}'(G\times V)^{\widetilde G,\chi }$ (resp. $T\in {\mathcal S}'({\mathfrak g}\times V)^{\widetilde G,\chi }$) then the support of $T$ is contained in $Z{\mathcal U}\times V$ (resp. ${\mathfrak z}\times {\mathcal N}\times V)$.
\end{lemma}
This is Harish-Chandra's descent. We first review some facts about
the centralizers of semi-simple elements, following \cite{SS}.
Let $a\in G$, semi-simple; we want to describe its centralizer $M$ (resp. $\widetilde M$) in $G$ (resp. in $\widetilde G$) and to show that ${\mathcal S}'(M\times
V)^{\widetilde M,\chi }=(0)$.
View $a$ as a ${\mathbb D}-$linear endomorphism of $V$ and call $P$ its minimal polynomial. Then, as $a$ is semi-simple, $P$ decomposes into irreducible factors $P=P_1\dots P_r$ two by two relatively prime. Let $V_i={\mathrm Ker } P_i(a)$ so that $V=\oplus V_i$. Any element $x$ which commutes with $a$ will satisfy $xV_i\subset V_i$ for each $i$.
For
$$ R(\xi )=d_0+\cdots +d_m\xi^m,\quad d_0d_m\ne 0$$
let
$$ R^*(\xi )=\overline{d_0}\xi ^m+\cdots +\overline{d_m}.$$
Then, from $aa^*=1$ we obtain, if $m$ is the degree of $P$
$$ \langle P(a)v,v'\rangle =\langle v,a^{-m}P^*(a)v'\rangle $$
(note that the constant term of $P$ can not be 0 because $a$ is
invertible). It follows that $P^*(a)=0$ so that $P^*$ is
proportional to $P$. Now $P^*=P_1^*\dots P_r^*$; hence there exists
a bijection $\tau $ from $\{1,2,\dots ,r\}$ onto itself such that
$P^*_i$ is proportional to $P_{\tau (i)}$. Let $m_i$ be the degree
of $P_i$. Then, for some non zero constant $c$
$$ 0=\langle P_i(a)v_i,v_j\rangle =\langle v_i,a^{-m_i}P_i^*(a)v_j\rangle =c\langle v_i,a^{-m_i}P_{\tau (i)}(a)v_j\rangle
,\quad v_i\in V_i,\,\, v_j\in V_j.$$
We have two possibilities.
{\bf Case 1:}$\,\, \tau (i)=i$. The space $V_i$ is orthogonal to
$V_j$ for $j\ne i$; the restriction of the hermitian form to
$V_i$ is non degenerate. Let ${\mathbb D}_i={\mathbb D}[\xi
]/(P_i)$ and consider $V_i$ as a vector space over ${\mathbb D}_i$
through the action $(R(\xi ),v)\mapsto R(a)v$. As $a_{|V_i}$ is
invertible, $\xi $ is invertible modulo $(P_i)$; choose $\eta$
such that $\xi \eta=1$ modulo $(P_i)$. Let $\sigma _i$ be the
semi-linear involution of ${\mathbb D}_i$, as an algebra over
${\mathbb D}$:
$$ \sum d_j\xi^j\mapsto \sum \overline{d_j}\eta^j\quad{\mathrm modulo}\,\, (P_i)$$
Let ${\mathbb F}_i$ be the subfield of fixed points for
$\sigma_i$. It is a finite extension of ${\mathbb F}$, and
${\mathbb D}_i$ is either a quadratic extension of ${\mathbb
F}_i$ or equal to ${\mathbb F}_i$. There exists a ${\mathbb
D}-$linear form $\ell\ne 0$ on ${\mathbb D}_i$ such that $\ell
(\sigma_i(d))=\sigma_i(\ell (d))$ for all $d\in {\mathbb D }_i$.
Then any ${\mathbb D}-$linear form $L$ on ${\mathbb D}_i$ may be
written as $d\mapsto \ell (\lambda d)$ for some unique $\lambda
\in {\mathbb D}_i$.
If $v,v'\in V_i$ then $d\mapsto \langle d(a)v,v'\rangle $ is ${\mathbb D}-$linear map on ${\mathbb D}_i$; hence there exists $S(v,v')\in {\mathbb D}_i$ such that
$$ \langle d(a)v,v'\rangle =\ell (dS(v,v')).$$
One checks that $S$ is a non degenerate hermitian form on $V_i$ as
a vector space over ${\mathbb D}_i$. Also a ${\mathbb D}-$linear
map $x_i$ from $V_i$ into itself commutes with $a_i$ if and only
if it is ${\mathbb D}_i$-linear and it is unitary with respect to
our original hermitian form if and only if it is unitary with
respect to $S$. So in this case we call $G_i$ the unitary group
of $S$. It does not depend upon the choice of $\ell$. As no
confusion may arise, for $\lambda \in {{\mathbb D}_i}$ we define
$\overline\lambda =\sigma_i(\lambda )$.
We choose an ${\mathbb F}_i-$linear map $u_i$ from $V_i$ onto
itself, such that $u_i(\lambda v)=\overline\lambda u(v)$ and
$S(u_i(v),u_i(v'))=\overline{S(v,v')}$. Then because of our
original choice of $\ell$ we also have $\langle
u_i(v),u_i(v')\rangle =\overline{\langle v,v'\rangle }$. Note that
$u(a_{V_i})^{-1}u^{-1}=a_{V_i}$.
{\bf Case 2.} Suppose now that $j=\tau (i)\ne i$. Then $V_i\oplus V_j$ is orthogonal to $V_k$ for $k\ne i,j$ and the restriction of the hermitian form to $V_i\oplus V_j$ is non degenerate, both $V_i$ and $V_j$ being totally isotropic subspaces. Choose an inverse $\eta$ of $\xi $ modulo $P_j$. Then for any $P\in {\mathbb D}[\xi ]$
$$ \langle P(a)v_i,v_j\rangle =\langle v_i,\overline P(\eta (a))v_j\rangle ,\quad v_i\in V_i,\,\, v_j\in V_j$$
where $\overline P$ is the polynomial deduced from $P$ by changing
its coefficients into their conjugate. This defines a map, which
we call $\sigma_i$ from ${\mathbb D}_i$ onto ${\mathbb D}_j$. In
a similar way we have a map $\sigma _j$ which is the inverse of
$\sigma_i$. Then, for $\lambda \in {\mathbb D}_i$ we have
$\langle \lambda v_i,v_j\rangle =\langle v_i,\sigma_i(\lambda
)v_j\rangle $.
View $V_i$ as a vector space over ${\mathbb D}_i$. The action
$$ (\lambda ,v_j)\mapsto
\sigma_i(\lambda )v_j$$
defines a structure of ${\mathbb D}_i$ vector space on $V_j$.
However note that for $\lambda \in {\mathbb D}$ we have
$\sigma_i(\lambda )=\overline\lambda $ so that $\sigma_i(\lambda
)v_j$ may be different from $\lambda v_j$. To avoid confusion we
shall write, for $\lambda \in {\mathbb D}_i$
$$\lambda v_i=\lambda *v_i\quad {\mathrm and}\quad \sigma_i(\lambda )v_j=\lambda *v_j.$$
As in the first case choose a non zero ${\mathbb D}-$linear form $\ell$ on ${\mathbb D}_i$. For $v_i\in V_i$ and $v_j\in V_j$ the map $\lambda \mapsto \langle \lambda *v_i,v_j\rangle $ is a ${\mathbb D}-$linear form on ${\mathbb D}_i$; hence there exists a unique element $S(v_i,v_j)\in {\mathbb D}_i$ such that, for all $\lambda $
$$ \langle \lambda *v_i,v_j\rangle =\ell(\lambda S(v_i,v_j)).$$
The form $S$ is ${\mathbb D}_i-$ bilinear and non degenerate so that we can view $V_j$ as the dual space over ${\mathbb D}_i$ of the ${\mathbb D}_i$ vector space $V_i$.
Let $(x_i,x_j)\in {\mathrm End}_{\mathbb D}(V_i)\times {\mathrm End}_{\mathbb D}(V_j)$. They commute with $(a_i,a_j)$ if and only if they are ${\mathbb D}_i$-linear. The original hermitian form will be preserved, if and only if $S(x_iv_i,x_jv_j)=S(v_i,v_j)$ for all $v_i,v_j$. This means that $x_j$ is the inverse of the transpose of $x_i$. In this situation we define $G_i$ as the linear group of the ${\mathbb D}_i-$vector space $V_i$.
Let $u_i$ be a ${\mathbb D}_i-$linear bijection of $V_i$ onto $V_j$. Then $u_i(av_i)=a^{-1}u_i(v_i)$ and $u_i^{-1}(av_j)=a^{-1}u_i^{-1}(v_j)$.
Recall that $M$ is the centralizer of $a$ in $G$. Then $(M,V)$ decomposes as a "product", each "factor" being either of type $(G_i,V_i)$ with $G_i$ a unitary group
(case 1) or $(G_i,V_i\times V_j)$ with $G_i$ a general linear group (case 2). Gluing together the $u_i$ (case 1) and the $(u_i,u_i^{-1})$ (case 2)
we get an element $u\in \widetilde G\setminus G$ such that $ua^{-1}u^{-1}=a$ which means that it belongs to the centralizer of $a$ in $\widetilde G$.
Finally if $\widetilde M$ is the centralizer of $a$ in $\widetilde G$ then $(\widetilde M,V)$ is imbedded into a product each "factor" being either
of type $(\widetilde G_i,V_i)$ with $G_i$ a unitary group (case 1) or $(\widetilde G_i,V_i\times V_j)$ with $G_i$ a general linear group (case 2).
If $a$ is not central then for each $i$ the dimension of $V_i$ is
strictly smaller than $n$ and from the result for the general
linear group and the inductive assumption in the orthogonal or
unitary case we conclude that ${\mathcal S}'(M\times
V)^{\widetilde M,\chi }=(0)$.
{\it Proof of Lemma 4.1.} in the group case. Consider the map
$g\mapsto P_g$ where $P_g$ is the characteristic polynomial of
$g$. It is a continuous map from $G$ into the set of polynomials
of degree at most $n$. Each non empty fiber ${\mathcal F}$ is
stable under $ G$ but also under $\widetilde G\setminus G$.
Bernstein's localization principle tells us that it is enough to
prove that ${\mathcal S}'({\mathcal F}\times V)^{\widetilde
G,\chi }=(0)$.
Now it follows from \cite{SS} chapter IV that ${\mathcal F}$
contains only a finite number of semi-simple orbits; in particular
the set of semi-simple elements ${\mathcal F}_s$ in ${\mathcal
F}$ is closed. Let us use the multiplicative Jordan decomposition
into a product of a semi-simple and a unipotent element. Consider
the map $\theta $ from ${\mathcal F}\times V$ onto ${\mathcal
F}_s$ which associates to $(g,v)$ the semi-simple part $g_s$ of
$g$. This map is continuous (see the corresponding proof for $GL$)
and commutes with the action of $\widetilde G$. In ${\mathcal
F}_s$ each orbit $\gamma $ is both open and closed therefore
$\theta ^{-1}(\gamma )$ is open and closed and invariant under
$\widetilde G$. It is enough to prove that for each such orbit
${\mathcal S}'(\theta ^{-1}(\gamma ))^{\widetilde G,\chi }=(0)$.
By Frobenius descent (Theorem 6-2), if $a\in \gamma $ and is not central, this
follows from the above considerations on the centralizer of such
an $a$ and the fact that $\theta ^{-1}(a)$ is a closed subset of
the centralizer of $a$ in $\widetilde G$, the product of the set
of unipotent element commuting with $a$ by $V$. Now $g_s$ is
central if and only if $g$ belongs to $Z{\mathcal U}$, hence the
Lemma. For the Lie algebra the proof is similar, using the
additive Jordan decomposition.
\hfill$\Box$\hfill
Going back to the group if $a$ is central we see that it suffices to prove that ${\mathcal S}'({\mathcal U}\times V)^{\widetilde G,\chi }=(0)$ and similarly for the Lie algebra it is enough to prove that ${\mathcal S}'({\mathcal N}\times V)^{\widetilde G,\chi }=(0)$.
Now the exponential map (or the Cayley transform) is a
homeomorphism of ${\mathcal N}$ onto ${\mathcal U}$ commuting
with the action of $\widetilde G$. Therefore it is enough to
consider the Lie algebra case.
We now turn our attention to $V$. Let
$$ \Gamma =\{v\in V | \langle v,v\rangle =0\}.$$
\begin{proposition}
If $T\in {\mathcal S}'({\mathcal N}\times V)^{\widetilde G,\chi }$ then the support of $T$ is contained in ${\mathcal N}\times \Gamma $.
\end{proposition}
{\it Proof.} Let
$$ \Gamma _t=\{v\in V\,|\, \langle v,v\rangle =0\}.$$
Each $\Gamma _t$ is stable by $\widetilde G$, hence, by
Bernstein's localization principle (Corollary 6-1), to prove that the support of
$T$ is contained in ${\mathcal N}\times \Gamma _0$ it is enough
to prove that, for $t\ne 0$, $\,\, {\mathcal S}'({\mathcal
N}\times \Gamma _t)^{\widetilde G,\chi }=(0)$.
By Witt's theorem the group $G$ acts transitively on $\Gamma _t$.
We can apply Frobenius descent to the projection from ${\mathcal
N}\times \Gamma _t$ onto $\Gamma _t$. Fix a point $v_0\in \Gamma
_t$. The fiber is ${\mathcal N}\times \{v_0\}$. Let $\widetilde
G_1$ be the centralizer of $v_0$ in $\widetilde G$. We have to
show that ${\mathcal S}'({\mathcal N})^{\widetilde G_1,\chi
}=(0)$ and it is enough to prove that ${\mathcal S}'({\mathfrak
g})^{\widetilde G_1,\chi }=(0)$.
The vector $v_0$ is not isotropic so we have an orthogonal decomposition
$$ V={\mathbb D}v_0\oplus V_1$$
with $V_1$ orthogonal to $v_0$. The restriction of the hermitian form to $V_1$ is non degenerate and $G_1$ is identified with the unitary group of this restriction, and $\widetilde G_1$ is the expected semi-direct product with $S_2$. As a $\widetilde G_1-$module the Lie algebra ${\mathfrak g}$ is isomorphic to a direct sum
$$ {\mathfrak g\approx {\mathfrak g}_1\oplus V_1\oplus W}$$
where ${\mathfrak g}_1$ is the Lie algebra of $G_1$ and $W$ a vector space over ${\mathbb F}$ of dimension 0 or 1 and on which the action of $\widetilde G_1$ is trivial. The action on ${\mathfrak g}_1\oplus V_1$ is the usual one so that, by induction, we know that ${\mathcal S}'({\mathfrak g}_1\oplus V_1)^{\widetilde G_1,\chi }=(0)$. This readily implies that ${\mathcal S}'({\mathfrak g})^{\widetilde G_1,\chi }=(0)$.
\hfill$\Box$\hfill
Summarizing: we have to prove that ${\mathcal S}'({\mathcal N}\times\Gamma )^{\widetilde G,\chi}=(0)$.
\@startsection {section}{1{\hglue -16pt. End of the proof in the orthogonal and unitary cases}
We keep our general notations. We have to show that a distribution on ${\mathcal N}\times \Gamma $ which is invariant under $G$ is invariant under $\widetilde G$. To some extent the proof will be similar to the one we gave for the general linear group.
In particular we will use the fact that if $T$ is such a distribution then its partial Fourier transform on $V$ is also invariant under $G$. The Fourier transform on $V$ is defined using the bilinear form
$$ (v_1,v_2)\mapsto \langle v_1,v_2\rangle +\langle v_2,v_1\rangle $$
which is invariant under $\widetilde G$.
For $v\in V$ put
$$ \varphi _v(x)=\langle x,v\rangle v,\quad x\in V.$$
It is a rank one endomorphism of $V$ and $\langle \varphi _v(x),y\rangle =\langle x,\varphi _v(y)\rangle $.
\begin{lemma}
i)In the unitary case, for $\lambda \in {\mathbb D}$ such that $\lambda =-\overline\lambda $ the map
$$ \nu_\lambda : \quad (X,v)\mapsto (X+\lambda \varphi _v,v)$$
is a homeomorphism of $[{\mathfrak g},{\mathfrak g}]\times \Gamma $ onto itself which commutes with $\widetilde G$.\hfill\break
ii) In the orthogonal case, for $\lambda \in {\mathbb F}$ the map
$$ \mu _\lambda :\quad (X,v)\mapsto (X+\lambda X\varphi _v+\lambda \varphi _vX,v)$$
is a homeomorphism of $[{\mathfrak g},{\mathfrak g}]\times \Gamma $ onto itself which commutes with $\widetilde G$.
\end{lemma}
The proof is a trivial verification.
We now use the stratification of ${\mathcal N}$. Let us first check that
an adjoint orbit is stable not only by $G$ but by $\widetilde G$.
Choose a basis $e_1,\dots ,e_n$ of $V$ such that $\langle e_i,e_j\rangle \in {\mathbb F}$; this gives a conjugation $u: v=\sum x_ie_i\mapsto \overline v=\sum \overline{x_i}e_i$ on $V$. If $A$ is any endomorphism of $V$ then $\overline A$ is the endomorphism $v\mapsto \overline{A(\overline v)}$. The conjugation $u$ is an element of $\widetilde G\setminus G$ and, as such, it acts on ${\mathfrak g}\times V$ by $(X,v)\mapsto (-uXu^{-1},-u(v))=(-\overline X,-\overline v)$. In \cite{MVW} Chapter 4 Proposition 1-2
it is shown that for $X\in {\mathfrak g}$ there exists an
${\mathbb F}-$linear automorphism $a$ of $V$ such that $\langle
a(x),a(y)\rangle =\overline{\langle x,y\rangle }$ (this implies
that $a(\lambda x)=\overline{\lambda }x$) and such that
$aXa^{-1}=-X$. Then $g=ua\in G$ and $gXg^{-1}=-\overline X$ so
that $-\overline X$ belongs to the adjoint orbit of $X$. Note that
$a\in \widetilde G\setminus G$ and as such acts as
$a(X,v)=(X,-a(v))$; it is an element of the centralizer of $X$ in
$\widetilde G\setminus G$.
{\textbf {Remark.}} We need to check this only for nilpotent
orbits and this will be done later in an explicit way, using the
canonical form of nilpotent matrices.
Let ${\mathcal N}_i$ be the union of all nilpotent orbits of dimension at most $i$. We shall prove, by descending induction on $i$, that the support of a distribution $T\in {\mathcal S}'({\mathcal N}\times \Gamma )^{\widetilde G,\chi }$ must be contained in ${\mathcal N}_i\times \Gamma $.
So now assume that $i\geq 0$ and that we already know that the support of any $T\in {\mathcal S}'({\mathcal N}\times \Gamma )^{\widetilde G,\chi }$ must be contained in ${\mathcal N}_i\times \Gamma $. Let ${\mathcal O}$ be a nilpotent orbit of dimension $i$; we have to show that the restriction of $T$ to ${\mathcal O}$ is 0.
In the unitary case fix $\lambda \in {\mathbb D}$ such that
$\lambda =-\overline\lambda $ and consider, for every $t\in
{\mathbb F}$ the homeomorphism $\nu_{t\lambda }$; the image of $T$
belongs to ${\mathcal S}'({\mathcal N}\times \Gamma
)^{\widetilde G,\chi }$ so that the image of the support of $T$
must be contained in ${\mathcal N}_i\times \Gamma $. If $(X,v)$
belongs to this support this means that $X+t\lambda \varphi _v\in
{\mathcal N}_i$.
If $i=0$ so that $\mathcal {N}_i=\{0\}$ this implies that $v=0$ so that $T$ must be a multiple of the Dirac measure at the point $(0,0)$ and hence is invariant under $\widetilde G$ so must be 0.
If $i>0$ and $X\in {\mathcal O}$ then as ${\mathcal O}$ is open in ${\mathcal N}_i$, we get that, at least for $|t|$ small enough, $X+t\lambda \varphi _v\in {\mathcal O}$ and therefore $\lambda \varphi _v$ belongs to the tangent space $\mathrm{Im}\,\mathrm{ad} (X)$ of ${\mathcal O}$ at the point $X$. Define
$$ Q(X)=\{v\in V|\varphi _v\in \mathrm {Im}\,\mathrm {ad } (X)\},\quad X\in {\mathcal N},\,\,\, {\mathrm {(unitary \,\, case)}}.$$
Then we know that the support of the restriction of $T$ to ${\mathcal O}$ is contained in
$$ \{(X,v)|X\in {\mathcal O,\, v\in Q(X)}\}$$
and the same is true for the partial Fourier transform of $T$ on $V$.
In the orthogonal case for $i=0$, the distribution $T$ is the product of the Dirac measure at the origin of $\mathfrak{g}$ by a distribution $T'$ on $V$. The distribution $T'$ is invariant under $G$ but the image of $\widetilde G$ in $\mathrm{End} (V)$ is the same as the image of $G$ so that $T'$ is invariant under $\widetilde G$ hence must be 0.
If $i>0$ we proceed as in the unitary case, using $\mu _\lambda $. We define
$$ Q(X)=\{v\in V|X\varphi _v+\varphi _vX\in {\mathrm Im}\,{\mathrm ad } (X)\},\quad X\in {\mathcal N},\,\,\, {\mathrm {(orthogonal \,\, case)}}$$
and we have the same conclusion.
In both cases, for $i>0$, fix $X\in {\mathcal O}$. We use
Frobenius descent for the projection map $(Y,v)\mapsto Y$ of
${\mathcal O}\times V$ onto ${\mathcal O}$. Let $C$ (resp.
$\widetilde C$) be the stabilizer of $X$ in $G$ (resp. $\widetilde
G$). We have a linear bijection of ${\mathcal S}'({\mathcal
O}\times \Gamma )^{\widetilde G,\chi }$ onto ${\mathcal
S}'(V)^{\widetilde C,\chi }$.
\begin{lemma}
Let $T\in {\mathcal S}'(V)^{\widetilde C,\chi }$. If $T$ and its Fourier transform are supported in $Q(X)$ then $T=0$.
\end{lemma}
Let us say that a nilpotent element $X$ is nice if the above Lemma is true.
Suppose that we have a direct sum decomposition $V=V_1\oplus V_2$
such that $V_1$ and $V_2$ are orthogonal. By restriction we get
non degenerate hermitian forms $\langle .,.\rangle _i$ on $V_i$.
We call $G_i$ the unitary group of $\langle .,.\rangle _i$,
${\mathfrak g}_i$ its Lie algebra and so on. Suppose that
$X(V_i)\subset V_i$ so that $X_i=X_{|V_i}$ is a nilpotent element
of ${\mathfrak g}_i$.
\begin{lemma}
If $X_1$ and $X_2$ are nice so is $X$.
\end{lemma}
{\it Proof of Lemma 5.3. }We claim that $Q(X)\subset Q(X_1)\times Q(X_2)$. Indeed if
$$ A=\pmatrix{A_{1,1}&A_{1,2}\cr A_{2,1}&A_{2,2}\cr}\in {\mathfrak g}$$
then from
$$ \langle A\pmatrix{x_1\cr x_2\cr},\pmatrix{y_1\cr y_2\cr}\rangle +\langle \pmatrix{x_1\cr x_2\cr},A\pmatrix{y_1\cr y_2\cr}\rangle =0$$
we get in particular
$$ \langle A_{i,i}x_i,y_i\rangle +\langle x_i,A_{i,i}y_i\rangle =0$$
so that $A_{i,i}\in {\mathfrak g}_i$. Note that
$$ [X,A]=\pmatrix{[X_1,A_{1,1}]&*\cr *&[X_2,A_{2,2}]\cr}.$$
If $v_i\in V_i$ and $v_j\in V_j$ we define $\varphi _{v_i,v_j}: V_i\mapsto V_j$ by $\varphi _{v_i,v_j}(x_i)=\langle x_i,v_i\rangle v_j$. Then, for $v=v_1+v_2$
$$ \varphi _{v}=\pmatrix{\varphi _{v_1,v_1}&\varphi _{v_2,v_1}\cr \varphi _{v_1,v_2}&\varphi _{v_1,v_2}\cr}.$$
Therefore if, for $A\in {\mathfrak g}$ we have $\varphi _v=[X,A]$ then $\varphi _{v_i,v_i}=[X_i,A_{i,i}]$. This proves the assertion for the unitary case. The orthogonal case is similar.
The end of the proof is the same as the end of the proof of Lemma 3-2.
\hfill$\Box$\hfill
Now in both orthogonal and unitary cases nilpotent elements have normal forms which are orthogonal direct sums of "simple" nilpotent matrices. This is precisely described in \cite{SS} IV 2-19 page 259. By the above Lemma it is enough to prove that each "simple" matrix is nice.
{\bf Unitary case.} There is only one type to consider. There
exists a basis $e_1,\dots ,e_n$ of $V$ such that $Xe_1=0$ and
$Xe_i=e_{i-1}, \, i\geq 2$. The hermitian form is given by
$$ \langle e_i,e_j\rangle =0\, \, {\mathrm if}\, i+j\ne n+1,\quad \langle e_i,e_{n+1-i}\rangle =(-1)^{n-i}\alpha$$
with $\alpha\ne 0$. Note that $\overline\alpha =(-1)^{n-1}\alpha
$. Suppose that $v\in Q(X)$; for some $A\in {\mathfrak g}$ we
have $\lambda \varphi _v=XA-AX$. For any integer $p\geq 0$
$${\mathrm Tr } (\lambda \varphi _vX^p)={\mathrm Tr} (XAX^{p}-AX^{p+1})=0.$$
Now
${\mathrm Tr} (\varphi _vX^p)=\langle X^pv,v\rangle $
Let $v=\sum x_ie_i$. Hence
$$ \langle X^pv,v\rangle =\sum _1^{n-p}x_{i+p}\langle e_i,v\rangle =\sum _1^{n-p}(-1)^{n-i}\alpha x_{i+p}\overline x_{n+1-i}=0.$$
For $p=n-1$ this gives $x_n\overline x_n=0$. For $p=n-2$ we get
nothing new but for $p=n-3$ we obtain $x_{n-1}=0$. Going on, by an
easy induction, we conclude that $x_i=0$ if $i\geq (n+1)/2$.
If $n=2p+1$ is odd put $V_1=\oplus _1^p{\mathbb D}e_i$,$\,\, V_0={\mathbb D}e_{p+1}$ and $V_2=\oplus _{p+2}^{2p+1}{\mathbb D}e_i$. If $n=2p$ is even put $V_1=\oplus _1^p{\mathbb D}e_i$, $V_0=(0)$ and $V_2=\oplus _{p+1}^{2p}{\mathbb D}e_i$. In both cases we have $V=V_1\oplus V_0\oplus V_2$. We use the notation $v=v_2+v_0+v_1$
The distribution $T$ is supported by $V_1$. Call $\delta _i$ the
Dirac measure at $0$ on $V_i$. Then we may write $T=U\otimes
\delta _0\otimes \delta _2$ with $U\in {\mathcal S'(V_1)}$. The
same thing must be true of the Fourier transform of $T$. Note that
$\widehat U$ is a distribution on $V_2$, that $\widehat\delta _2$
is a Haar measure $dv_1$ on $V_1$ and that, for $n$ odd
$\widehat\delta _0$ is a Haar measure $dv_0$ on $V_0$. So we have
$\widehat T= dv_1\otimes \widehat U$ if $n$ is even and $\widehat
T=dv_1\otimes dv_0\otimes \widehat U$ if $n$ is odd. In the odd
case this forces $T=0$. In the even case, up to a scalar multiple
the only possiblity is $T=dv_1\otimes \delta _2$.
Let $$ a: \sum x_ie_i\mapsto \sum (-1)^i\overline x_ie_i.$$ Then
$a\in \widetilde G\setminus G$. It acts on ${\mathfrak g}$ by
$Y\mapsto -aYa^{-1}$ and in particular $-aXa^{-1}=X$ so that
$a\in \widetilde C\setminus C$. The action on $V$ is given by
$v\mapsto -a(v)$. It is an involution. The subspace $V_1$ is
invariant and so $dv_1$ is invariant. This implies that $T$ is
invariant under $\widetilde C$ so it must be 0.
{\bf Orthogonal case.} There are two different types of "simple"
nilpotent matrices.
{\bf The first type} is the same as the unitary case, with
$\alpha=1$ and thus $n$ odd but now our condition is that
$X\varphi _v+\varphi _vX=[X,A]$
for some $A\in {\mathfrak g}$. As before this implies that ${\mathrm Tr} (\varphi _vX^q)=0$ but only for $q\geq 1$. Put $n=2p+1$; we get $x_j=0$ for $j>p+1$.
Decompose $V$ as before: $V=V_1\oplus V_0\oplus V_2$. Our
distribution $T$ is supported by the subspace $v_2=0$ so we write
it $T=U\otimes \delta _2$ with $U\in {\mathcal S}'(V_1\oplus
V_0)$. This is also true for the distribution $\widehat T$ so we
must have $U=dv_1\otimes R$ with $R$ a distribution on $V_0$.
Finally $T=dv_1\otimes R\otimes \delta _2$. Now $-{\mathrm Id}\in
C$ and $T$ is invariant under $C$ so that $R$ must be an even
distribution. On the other end the endomorphism $a$ of $V$ defined
by $a(e_i)=(-1)^{i-p-1}e_i$ belongs to $C$and $aXa^{-1}=-X$ and
$u: (X,v)\mapsto (-X,-v)$ belongs to $\widetilde G\setminus G$.
The product $a*u$ of $a$ and $u$ in $\widetilde G$ belongs to
$\widetilde C \setminus C$. Clearly $T$ is invariant under $a*u$
so that $T$ is invariant under $\widetilde C$ so it must be 0.
{\bf The second type} is as follows. We have $n=2m$, an even integer and a decomposition $V=E\oplus F$ with both $E$ and $F$ of dimension $m$. We have a basis $e_1,\dots ,e_m$ of $E$ and a basis $f_1,\dots ,f_m$ of $F$ such that
$$ \langle e_i,e_j\rangle =\langle f_i,f_j\rangle =0$$
and
$$ \langle e_i,f_j\rangle =0 \,{\mathrm if}\, i+j\ne m+1\quad {\mathrm and}\quad \langle e_i,f_{m+1-i}\rangle =(-1)^{m-i}.$$
Finally $X$ is such that $Xe_i=e_{i-1},\,\, Xf_i=f_{i-1}$.
Let $\xi $ be the matrix of the restriction of $X$ to $E$ or to $F$. Write an element $A\in {\mathfrak g}$ as 2 by 2 matrix $A=(a_{i,j})$. Then
$$[X,A]=\pmatrix{[\xi ,a_{1,1}]&[\xi ,a_{1,2}]\cr [\xi ,a_{2,1}]& [\xi ,a_{2,2}]\cr}.$$
Suppose that $v\in Q(X)$ and let
$$ v=e+f\,\,\, {\mathrm with}\,\,\,e=\sum x_ie_i,\,\,\,f=\sum y_if_i.$$
We get
$$ X\varphi _v+\varphi _vX=\pmatrix{\xi \varphi _{f,e}+\varphi _{f,e}\xi &\xi \varphi _{f,f}+\varphi _{f,f}\xi \cr\xi \varphi _{e,e}+\varphi _{e,e}\xi &\xi \varphi _{e,f}+\varphi _{e,f}\xi \cr}$$
where, for example $\varphi _{e,e}$ is the map $f'\mapsto \langle f',e\rangle e$ from $F$ into $E$. Thus, for some $A$,
$$\xi \varphi _{e,e}+ \varphi _{e,e}\xi =\xi a_{2,1}-a_{2,1}\xi$$
In this formula, using the basis $(e_i),\, (f_i)$ replace all the
maps by their matrices.
Then, as before, we have ${\mathrm Tr} (\varphi _{e,e}\xi ^q)=0$ for $1\leq q\leq m-1$. If $e'=\sum x_if_i$ (the $x_i$ are the coordinates of $e$),
then ${\mathrm Tr} (\xi^q\varphi _{e,e})$ is $\langle\xi ^qe,e'\rangle $. Thus, as in the other cases, we have $x_j=0$ for $j>n/2$ if $n$ is even and $j>(n+1)/2$ if $n$ is odd. The same thing is true for the $y_i$.
If $n=2p$ is even, let $V_1=\oplus _{i\leq p}({\mathbb F}e_i\oplus {\mathbb
F}f_i)$ and $V_2=\oplus _{i>p}({\mathbb F}e_i\oplus {\mathbb
F}f_i)$; write $v=v_1+v_2$ the corresponding decomposition of an
arbitrary element of $V$. Let $\delta _2$ be the Dirac measure at
the origin in $V_2$ and $dv_1$ a Haar measure on $V_1$. Then, as
in the unitary case, using the Fourier transform, we see that the
distribution $T$ must be a multiple of $dv_1\otimes \delta_2$.
The endomorphism $a$ of $V$ defined by $a(e_i)=(-1)^ie_i$ and $a(f_i)=(-1)^{i+1}f_i$ belongs to $G$ and $aXa^{-1}=-X$. The map $u: (Y,v)\mapsto (-Y,-v)$ belongs to $\widetilde G\setminus G$ so that the product $a\times u$ in $\widetilde G$ belongs to $\widetilde C\setminus C$. It clearly leaves $T$ invariant so that $T=0$.
Finally if $n=2p+1$ is odd we put $V_1=\oplus _{i\leq p}({\mathbb F}e_i\oplus {\mathbb F}f_i)$, $V_0={\mathbb F}e_{p+1}\oplus {\mathbb F}f_{p+1}$,
$V_2=\oplus _{i\geq p+2}({\mathbb F}e_i\oplus {\mathbb F}f_i)$.
As in the unitary case we find that $T=dv_1\otimes R\otimes \delta _2$ with $R$ a distribution on $V_0$. As $-{\mathrm Id}\in C$ we see that $R$ must be even.
Then again, define $a\in G$ by $a(e_i)=(-1)^ie_i$ and $a(f_i)=(-1)^if_i$ and consider $a*u$ with $u(Y,v)=(-Y,-v)$.
As before $a*u\in \widetilde C\setminus C$ and leaves $T$ invariant so we have to take $T=0$.
\hfill$\Box$\hfill
\@startsection {section}{1{Appendix}
We shall state two theorems which are systematically used in our
proof.
If $X$ is a Hausdorff totally disconnected locally compact
topological space (lctd space in short) we denote by ${\mathcal
S}(X)$ the vector space of locally constant applications with
compact support of $X$ into the field of complex numbers ${\mathbb
C}$. The dual space ${\mathcal S}'(X)$ of ${\mathcal S}(X)$ is the
space of distributions on $X$. All the lctd spaces we introduce
are countable at infinity.
If an lctd topological group $G$ acts continuously on a lctd
space $X$ then it acts on ${{\mathcal S}}(X)$ by \[
(gf)(x)=f(g^{-1}x)\] and on distributions by
\[ (gT)(f)=T(g^{-1}f)\]
The space of invariant distributions is denoted by ${\mathcal
S}'(X)^G$. More generally, if $\chi$ is a character of $G$ we
denote by ${\mathcal S}'(X)^{G,\chi}$ the space of distributions
$T$ which transform according to $\chi$ that is to say
$T(f(g^{-1}x))=\chi (g)T(f)$.
The following result is due to Bernstein \cite{Ber}, section 1.4.
\begin{theorem}[Localization principle] Let $q:Z \to T$ be a continuous map between two topological spaces of type lctd. Denote $Z_t:=
q^{-1}(t)$. Consider $\mathcal{S}'(Z)$ as $\mathcal{S}(T)$-module.
Let $M$ be a closed subspace of $\mathcal{S}'(Z)$ which is an
$\mathcal{S}(T)$-submodule. Then $M=\overline{\bigoplus_{t \in T}
(M \cap \mathcal{S}'(Z_t))}$.
\end{theorem}
\begin{corollary} Let $q:Z \to T$ be a continuous map between topological spaces of type lctd. Let an lctd group
$H$ act on $Z$ preserving the fibers of $q$. Let $\mu$ be a
character of $H$. Suppose that for any $t\in T$,
$\mathcal{S}'(q^{-1}(t))^{H,\mu}=0$. Then
$\mathcal{S}'(Z)^{H,\mu}=0$.
\end{corollary}
The second theorem is a variant of Frobenius reciprocity.
\begin{theorem}[Frobenius descent]
Let a unimodular lctd topological group $H$ act transitively on
an lctd topological space $Z$. Let $\varphi:E \to Z$ be an
$H$-equivariant map of lctd topological spaces. Let $x\in Z$.
Suppose that its stabilizer $Stab_H(x)$ is unimodular. Let $W$ be
the fiber of $x$.
Let $\chi$ be a character of $H$. Then\\
(i) There exists a canonical isomorphism $Fr:
\mathcal{S}'(E)^{H,\chi}
\to \mathcal{S}'(W)^{Stab_H(x),\chi}$.\\
(ii) For any distribution $\xi \in \mathcal{S}'(E)^{H,\chi}$,
$\mathrm{Supp}(Fr(\xi))=\mathrm{Supp}(\xi)\cap W$.\\
(iii) Frobenius descent commutes with Fourier transform.
Namely, let $W$ be a finite dimensional linear space over $\mathbb
F$ with a nondegenerate bilinear form $B$. Let $H$ act on $W$
linearly preserving $B$.
Then
for any $\xi \in \mathcal{S}'(Z\times W)^{H,\chi}$, we have
$\mathcal{F}_{B}(\mathrm{Fr}(\xi))=\mathrm{Fr}(\mathcal{F}_{B}(\xi))$
where $\mathrm{Fr}$ is taken with respect to the projection $Z
\times W \to Z$.
\end{theorem}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,750 |
A-Team Group Data Management Summit Virtual Hit the Hot Topics of Data Strategy, Regulatory Reporting, KYC and Onboarding
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Keynote – Deborah Lorenzen
Meantime, back to Deborah Lorenzen, who presented a fascinating and entertaining keynote on the pitfalls of focusing on data management tools and the potential of focusing on data and aligning data and business strategies to achieve better decisions. She said: "You need to drive home data strategy, data governance and documentation. With your data in order, you can drive insights and get back to the predictive model that was first introduced 10 years ago."
To achieve a successful data strategy, Lorenzen highlighted the need for executive support, full-time positions in data strategy, data architecture and governance, and a technology architecture to operationalise the data architecture. She added: "Data also needs to be organised by domain. It then becomes possible to step out of data silos and gain predictive insight from layering data, looking across product data, and pulling datasets together."
In a final caveat, Lorenzen warned against writing a data strategy for regulation, saying: "This would be dead on arrival. If that is what your organisation is doing, pull back and rethink. You need the executive team to support data, not just regulatory compliance."
Don't miss A-Team Group's next Data Management Summit that will be hosted in New York City on 30 September 2021
Data Management Summit Virtual – Day 2
The second day of the livestreamed summit was dedicated to leveraging data management capabilities to drive innovation in regulatory reporting, KYC and client onboarding. CUSIP Global Services kicked off with a keynote entitled 'Standing at the intersection of regulatory need and standards development – A standards practitioner's view'.
A second keynote was presented by Ted Datta, director of compliance solutions at Bureau van Dijk. He raised the question of 'how does data management help with beneficial ownership due diligence?' and discussed the latest regulatory initiatives shaping beneficial ownership due diligence expectations, operational and practical data management challenges, and how data and technology can help firms answer the question, resolve challenges, and ensure best practice beneficial ownership due diligence.
Diving into the detail of how to get your customer data right to support regulatory compliance and deliver a digital customer experience, Allie Harris, vice president and CDO, global banking and markets, at Scotiabank, joined Lorraine Waters, recently appointed CDO at Solidatus, for a fireside chat – we'll bring you more on this session next week.
Panel sessions supporting the topic of data management for regulatory reporting, KYC and onboarding covered how to establish a robust data driven regulatory reporting framework, and best practice data management to support a digitised and streamlined client onboarding and KYC strategy. A final data strategy spotlight discussed how to future proof data strategy in a complex regulatory environment.
Closing the summit, Delaney thanked all the sponsors and speakers, delegates who joined the event and the many audience members who took part in polls during sessions and posed questions for the speakers. If you missed this summit, be sure to sign up for A-Team Group's next Data Management Summit, which we hope will be live in New York City on 30 September 2021.
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RegTech Summit London
Now in its 6th year, the RegTech Summit in London will bring together the RegTech ecosystem to explore how the European capital markets financial industry can leverage technology to drive innovation, cut costs and support regulatory change. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,727 |
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caller: Function;
}
declare var Function:{
/**
* Creates a new function.
* @param args A list of arguments the function accepts.
*/
new (...args:string[]): Function;
(...args:string[]): Function;
prototype: Function;
}
interface IArguments {
[index: number]: any;
length: number;
callee: Function;
}
interface String {
/** Returns a string representation of a string. */
toString(): string;
/**
* Returns the character at the specified index.
* @param pos The zero-based index of the desired character.
*/
charAt(pos:number): string;
/**
* Returns the Unicode value of the character at the specified location.
* @param index The zero-based index of the desired character. If there is no character at the specified index, NaN is returned.
*/
charCodeAt(index:number): number;
/**
* Returns a string that contains the concatenation of two or more strings.
* @param strings The strings to append to the end of the string.
*/
concat(...strings:string[]): string;
/**
* Returns the position of the first occurrence of a substring.
* @param searchString The substring to search for in the string
* @param position The index at which to begin searching the String object. If omitted, search starts at the beginning of the string.
*/
indexOf(searchString:string, position?:number): number;
/**
* Returns the last occurrence of a substring in the string.
* @param searchString The substring to search for.
* @param position The index at which to begin searching. If omitted, the search begins at the end of the string.
*/
lastIndexOf(searchString:string, position?:number): number;
/**
* Determines whether two strings are equivalent in the current locale.
* @param that String to compare to target string
*/
localeCompare(that:string): number;
/**
* Matches a string with a regular expression, and returns an array containing the results of that search.
* @param regexp A variable name or string literal containing the regular expression pattern and flags.
*/
match(regexp:string): RegExpMatchArray;
/**
* Matches a string with a regular expression, and returns an array containing the results of that search.
* @param regexp A regular expression object that contains the regular expression pattern and applicable flags.
*/
match(regexp:RegExp): RegExpMatchArray;
/**
* Replaces text in a string, using a regular expression or search string.
* @param searchValue A String object or string literal that represents the regular expression
* @param replaceValue A String object or string literal containing the text to replace for every successful match of rgExp in stringObj.
*/
replace(searchValue:string, replaceValue:string): string;
/**
* Replaces text in a string, using a regular expression or search string.
* @param searchValue A String object or string literal that represents the regular expression
* @param replaceValue A function that returns the replacement text.
*/
replace(searchValue:string, replaceValue:(substring:string, ...args:any[]) => string): string;
/**
* Replaces text in a string, using a regular expression or search string.
* @param searchValue A Regular Expression object containing the regular expression pattern and applicable flags
* @param replaceValue A String object or string literal containing the text to replace for every successful match of rgExp in stringObj.
*/
replace(searchValue:RegExp, replaceValue:string): string;
/**
* Replaces text in a string, using a regular expression or search string.
* @param searchValue A Regular Expression object containing the regular expression pattern and applicable flags
* @param replaceValue A function that returns the replacement text.
*/
replace(searchValue:RegExp, replaceValue:(substring:string, ...args:any[]) => string): string;
/**
* Finds the first substring match in a regular expression search.
* @param regexp The regular expression pattern and applicable flags.
*/
search(regexp:string): number;
/**
* Finds the first substring match in a regular expression search.
* @param regexp The regular expression pattern and applicable flags.
*/
search(regexp:RegExp): number;
/**
* Returns a section of a string.
* @param start The index to the beginning of the specified portion of stringObj.
* @param end The index to the end of the specified portion of stringObj. The substring includes the characters up to, but not including, the character indicated by end.
* If this value is not specified, the substring continues to the end of stringObj.
*/
slice(start?:number, end?:number): string;
/**
* Split a string into substrings using the specified separator and return them as an array.
* @param separator A string that identifies character or characters to use in separating the string. If omitted, a single-element array containing the entire string is returned.
* @param limit A value used to limit the number of elements returned in the array.
*/
split(separator:string, limit?:number): string[];
/**
* Split a string into substrings using the specified separator and return them as an array.
* @param separator A Regular Express that identifies character or characters to use in separating the string. If omitted, a single-element array containing the entire string is returned.
* @param limit A value used to limit the number of elements returned in the array.
*/
split(separator:RegExp, limit?:number): string[];
/**
* Returns the substring at the specified location within a String object.
* @param start The zero-based index number indicating the beginning of the substring.
* @param end Zero-based index number indicating the end of the substring. The substring includes the characters up to, but not including, the character indicated by end.
* If end is omitted, the characters from start through the end of the original string are returned.
*/
substring(start:number, end?:number): string;
/** Converts all the alphabetic characters in a string to lowercase. */
toLowerCase(): string;
/** Converts all alphabetic characters to lowercase, taking into account the host environment's current locale. */
toLocaleLowerCase(): string;
/** Converts all the alphabetic characters in a string to uppercase. */
toUpperCase(): string;
/** Returns a string where all alphabetic characters have been converted to uppercase, taking into account the host environment's current locale. */
toLocaleUpperCase(): string;
/** Removes the leading and trailing white space and line terminator characters from a string. */
trim(): string;
/** Returns the length of a String object. */
length: number;
// IE extensions
/**
* Gets a substring beginning at the specified location and having the specified length.
* @param from The starting position of the desired substring. The index of the first character in the string is zero.
* @param length The number of characters to include in the returned substring.
*/
substr(from:number, length?:number): string;
[index: number]: string;
}
/**
* Allows manipulation and formatting of text strings and determination and location of substrings within strings.
*/
declare var String:{
new (value?:any): String;
(value?:any): string;
prototype: String;
fromCharCode(...codes:number[]): string;
}
interface Boolean {
}
declare var Boolean:{
new (value?:any): Boolean;
(value?:any): boolean;
prototype: Boolean;
}
interface Number {
/**
* Returns a string representation of an object.
* @param radix Specifies a radix for converting numeric values to strings. This value is only used for numbers.
*/
toString(radix?:number): string;
/**
* Returns a string representing a number in fixed-point notation.
* @param fractionDigits Number of digits after the decimal point. Must be in the range 0 - 20, inclusive.
*/
toFixed(fractionDigits?:number): string;
/**
* Returns a string containing a number represented in exponential notation.
* @param fractionDigits Number of digits after the decimal point. Must be in the range 0 - 20, inclusive.
*/
toExponential(fractionDigits?:number): string;
/**
* Returns a string containing a number represented either in exponential or fixed-point notation with a specified number of digits.
* @param precision Number of significant digits. Must be in the range 1 - 21, inclusive.
*/
toPrecision(precision?:number): string;
}
/** An object that represents a number of any kind. All JavaScript numbers are 64-bit floating-point numbers. */
declare var Number:{
new (value?:any): Number;
(value?:any): number;
prototype: Number;
/** The largest number that can be represented in JavaScript. Equal to approximately 1.79E+308. */
MAX_VALUE: number;
/** The closest number to zero that can be represented in JavaScript. Equal to approximately 5.00E-324. */
MIN_VALUE: number;
/**
* A value that is not a number.
* In equality comparisons, NaN does not equal any value, including itself. To test whether a value is equivalent to NaN, use the isNaN function.
*/
NaN: number;
/**
* A value that is less than the largest negative number that can be represented in JavaScript.
* JavaScript displays NEGATIVE_INFINITY values as -infinity.
*/
NEGATIVE_INFINITY: number;
/**
* A value greater than the largest number that can be represented in JavaScript.
* JavaScript displays POSITIVE_INFINITY values as infinity.
*/
POSITIVE_INFINITY: number;
}
interface Math {
/** The mathematical constant e. This is Euler's number, the base of natural logarithms. */
E: number;
/** The natural logarithm of 10. */
LN10: number;
/** The natural logarithm of 2. */
LN2: number;
/** The base-2 logarithm of e. */
LOG2E: number;
/** The base-10 logarithm of e. */
LOG10E: number;
/** Pi. This is the ratio of the circumference of a circle to its diameter. */
PI: number;
/** The square root of 0.5, or, equivalently, one divided by the square root of 2. */
SQRT1_2: number;
/** The square root of 2. */
SQRT2: number;
/**
* Returns the absolute value of a number (the value without regard to whether it is positive or negative).
* For example, the absolute value of -5 is the same as the absolute value of 5.
* @param x A numeric expression for which the absolute value is needed.
*/
abs(x:number): number;
/**
* Returns the arc cosine (or inverse cosine) of a number.
* @param x A numeric expression.
*/
acos(x:number): number;
/**
* Returns the arcsine of a number.
* @param x A numeric expression.
*/
asin(x:number): number;
/**
* Returns the arctangent of a number.
* @param x A numeric expression for which the arctangent is needed.
*/
atan(x:number): number;
/**
* Returns the angle (in radians) from the X axis to a point (y,x).
* @param y A numeric expression representing the cartesian y-coordinate.
* @param x A numeric expression representing the cartesian x-coordinate.
*/
atan2(y:number, x:number): number;
/**
* Returns the smallest number greater than or equal to its numeric argument.
* @param x A numeric expression.
*/
ceil(x:number): number;
/**
* Returns the cosine of a number.
* @param x A numeric expression that contains an angle measured in radians.
*/
cos(x:number): number;
/**
* Returns e (the base of natural logarithms) raised to a power.
* @param x A numeric expression representing the power of e.
*/
exp(x:number): number;
/**
* Returns the greatest number less than or equal to its numeric argument.
* @param x A numeric expression.
*/
floor(x:number): number;
/**
* Returns the natural logarithm (base e) of a number.
* @param x A numeric expression.
*/
log(x:number): number;
/**
* Returns the larger of a set of supplied numeric expressions.
* @param values Numeric expressions to be evaluated.
*/
max(...values:number[]): number;
/**
* Returns the smaller of a set of supplied numeric expressions.
* @param values Numeric expressions to be evaluated.
*/
min(...values:number[]): number;
/**
* Returns the value of a base expression taken to a specified power.
* @param x The base value of the expression.
* @param y The exponent value of the expression.
*/
pow(x:number, y:number): number;
/** Returns a pseudorandom number between 0 and 1. */
random(): number;
/**
* Returns a supplied numeric expression rounded to the nearest number.
* @param x The value to be rounded to the nearest number.
*/
round(x:number): number;
/**
* Returns the sine of a number.
* @param x A numeric expression that contains an angle measured in radians.
*/
sin(x:number): number;
/**
* Returns the square root of a number.
* @param x A numeric expression.
*/
sqrt(x:number): number;
/**
* Returns the tangent of a number.
* @param x A numeric expression that contains an angle measured in radians.
*/
tan(x:number): number;
}
/** An intrinsic object that provides basic mathematics functionality and constants. */
declare var Math:Math;
/** Enables basic storage and retrieval of dates and times. */
interface Date {
/** Returns a string representation of a date. The format of the string depends on the locale. */
toString(): string;
/** Returns a date as a string value. */
toDateString(): string;
/** Returns a time as a string value. */
toTimeString(): string;
/** Returns a value as a string value appropriate to the host environment's current locale. */
toLocaleString(): string;
/** Returns a date as a string value appropriate to the host environment's current locale. */
toLocaleDateString(): string;
/** Returns a time as a string value appropriate to the host environment's current locale. */
toLocaleTimeString(): string;
/** Returns the stored time value in milliseconds since midnight, January 1, 1970 UTC. */
valueOf(): number;
/** Gets the time value in milliseconds. */
getTime(): number;
/** Gets the year, using local time. */
getFullYear(): number;
/** Gets the year using Universal Coordinated Time (UTC). */
getUTCFullYear(): number;
/** Gets the month, using local time. */
getMonth(): number;
/** Gets the month of a Date object using Universal Coordinated Time (UTC). */
getUTCMonth(): number;
/** Gets the day-of-the-month, using local time. */
getDate(): number;
/** Gets the day-of-the-month, using Universal Coordinated Time (UTC). */
getUTCDate(): number;
/** Gets the day of the week, using local time. */
getDay(): number;
/** Gets the day of the week using Universal Coordinated Time (UTC). */
getUTCDay(): number;
/** Gets the hours in a date, using local time. */
getHours(): number;
/** Gets the hours value in a Date object using Universal Coordinated Time (UTC). */
getUTCHours(): number;
/** Gets the minutes of a Date object, using local time. */
getMinutes(): number;
/** Gets the minutes of a Date object using Universal Coordinated Time (UTC). */
getUTCMinutes(): number;
/** Gets the seconds of a Date object, using local time. */
getSeconds(): number;
/** Gets the seconds of a Date object using Universal Coordinated Time (UTC). */
getUTCSeconds(): number;
/** Gets the milliseconds of a Date, using local time. */
getMilliseconds(): number;
/** Gets the milliseconds of a Date object using Universal Coordinated Time (UTC). */
getUTCMilliseconds(): number;
/** Gets the difference in minutes between the time on the local computer and Universal Coordinated Time (UTC). */
getTimezoneOffset(): number;
/**
* Sets the date and time value in the Date object.
* @param time A numeric value representing the number of elapsed milliseconds since midnight, January 1, 1970 GMT.
*/
setTime(time:number): number;
/**
* Sets the milliseconds value in the Date object using local time.
* @param ms A numeric value equal to the millisecond value.
*/
setMilliseconds(ms:number): number;
/**
* Sets the milliseconds value in the Date object using Universal Coordinated Time (UTC).
* @param ms A numeric value equal to the millisecond value.
*/
setUTCMilliseconds(ms:number): number;
/**
* Sets the seconds value in the Date object using local time.
* @param sec A numeric value equal to the seconds value.
* @param ms A numeric value equal to the milliseconds value.
*/
setSeconds(sec:number, ms?:number): number;
/**
* Sets the seconds value in the Date object using Universal Coordinated Time (UTC).
* @param sec A numeric value equal to the seconds value.
* @param ms A numeric value equal to the milliseconds value.
*/
setUTCSeconds(sec:number, ms?:number): number;
/**
* Sets the minutes value in the Date object using local time.
* @param min A numeric value equal to the minutes value.
* @param sec A numeric value equal to the seconds value.
* @param ms A numeric value equal to the milliseconds value.
*/
setMinutes(min:number, sec?:number, ms?:number): number;
/**
* Sets the minutes value in the Date object using Universal Coordinated Time (UTC).
* @param min A numeric value equal to the minutes value.
* @param sec A numeric value equal to the seconds value.
* @param ms A numeric value equal to the milliseconds value.
*/
setUTCMinutes(min:number, sec?:number, ms?:number): number;
/**
* Sets the hour value in the Date object using local time.
* @param hours A numeric value equal to the hours value.
* @param min A numeric value equal to the minutes value.
* @param sec A numeric value equal to the seconds value.
* @param ms A numeric value equal to the milliseconds value.
*/
setHours(hours:number, min?:number, sec?:number, ms?:number): number;
/**
* Sets the hours value in the Date object using Universal Coordinated Time (UTC).
* @param hours A numeric value equal to the hours value.
* @param min A numeric value equal to the minutes value.
* @param sec A numeric value equal to the seconds value.
* @param ms A numeric value equal to the milliseconds value.
*/
setUTCHours(hours:number, min?:number, sec?:number, ms?:number): number;
/**
* Sets the numeric day-of-the-month value of the Date object using local time.
* @param date A numeric value equal to the day of the month.
*/
setDate(date:number): number;
/**
* Sets the numeric day of the month in the Date object using Universal Coordinated Time (UTC).
* @param date A numeric value equal to the day of the month.
*/
setUTCDate(date:number): number;
/**
* Sets the month value in the Date object using local time.
* @param month A numeric value equal to the month. The value for January is 0, and other month values follow consecutively.
* @param date A numeric value representing the day of the month. If this value is not supplied, the value from a call to the getDate method is used.
*/
setMonth(month:number, date?:number): number;
/**
* Sets the month value in the Date object using Universal Coordinated Time (UTC).
* @param month A numeric value equal to the month. The value for January is 0, and other month values follow consecutively.
* @param date A numeric value representing the day of the month. If it is not supplied, the value from a call to the getUTCDate method is used.
*/
setUTCMonth(month:number, date?:number): number;
/**
* Sets the year of the Date object using local time.
* @param year A numeric value for the year.
* @param month A zero-based numeric value for the month (0 for January, 11 for December). Must be specified if numDate is specified.
* @param date A numeric value equal for the day of the month.
*/
setFullYear(year:number, month?:number, date?:number): number;
/**
* Sets the year value in the Date object using Universal Coordinated Time (UTC).
* @param year A numeric value equal to the year.
* @param month A numeric value equal to the month. The value for January is 0, and other month values follow consecutively. Must be supplied if numDate is supplied.
* @param date A numeric value equal to the day of the month.
*/
setUTCFullYear(year:number, month?:number, date?:number): number;
/** Returns a date converted to a string using Universal Coordinated Time (UTC). */
toUTCString(): string;
/** Returns a date as a string value in ISO format. */
toISOString(): string;
/** Used by the JSON.stringify method to enable the transformation of an object's data for JavaScript Object Notation (JSON) serialization. */
toJSON(key?:any): string;
}
declare var Date:{
new (): Date;
new (value:number): Date;
new (value:string): Date;
new (year:number, month:number, date?:number, hours?:number, minutes?:number, seconds?:number, ms?:number): Date;
(): string;
prototype: Date;
/**
* Parses a string containing a date, and returns the number of milliseconds between that date and midnight, January 1, 1970.
* @param s A date string
*/
parse(s:string): number;
/**
* Returns the number of milliseconds between midnight, January 1, 1970 Universal Coordinated Time (UTC) (or GMT) and the specified date.
* @param year The full year designation is required for cross-century date accuracy. If year is between 0 and 99 is used, then year is assumed to be 1900 + year.
* @param month The month as an number between 0 and 11 (January to December).
* @param date The date as an number between 1 and 31.
* @param hours Must be supplied if minutes is supplied. An number from 0 to 23 (midnight to 11pm) that specifies the hour.
* @param minutes Must be supplied if seconds is supplied. An number from 0 to 59 that specifies the minutes.
* @param seconds Must be supplied if milliseconds is supplied. An number from 0 to 59 that specifies the seconds.
* @param ms An number from 0 to 999 that specifies the milliseconds.
*/
UTC(year:number, month:number, date?:number, hours?:number, minutes?:number, seconds?:number, ms?:number): number;
now(): number;
}
interface RegExpMatchArray extends Array<string> {
index?: number;
input?: string;
}
interface RegExpExecArray extends Array<string> {
index: number;
input: string;
}
interface RegExp {
/**
* Executes a search on a string using a regular expression pattern, and returns an array containing the results of that search.
* @param string The String object or string literal on which to perform the search.
*/
exec(string:string): RegExpExecArray;
/**
* Returns a Boolean value that indicates whether or not a pattern exists in a searched string.
* @param string String on which to perform the search.
*/
test(string:string): boolean;
/** Returns a copy of the text of the regular expression pattern. Read-only. The rgExp argument is a Regular expression object. It can be a variable name or a literal. */
source: string;
/** Returns a Boolean value indicating the state of the global flag (g) used with a regular expression. Default is false. Read-only. */
global: boolean;
/** Returns a Boolean value indicating the state of the ignoreCase flag (i) used with a regular expression. Default is false. Read-only. */
ignoreCase: boolean;
/** Returns a Boolean value indicating the state of the multiline flag (m) used with a regular expression. Default is false. Read-only. */
multiline: boolean;
lastIndex: number;
// Non-standard extensions
compile(): RegExp;
}
declare var RegExp:{
new (pattern:string, flags?:string): RegExp;
(pattern:string, flags?:string): RegExp;
// Non-standard extensions
$1: string;
$2: string;
$3: string;
$4: string;
$5: string;
$6: string;
$7: string;
$8: string;
$9: string;
lastMatch: string;
}
interface Error {
name: string;
message: string;
}
declare var Error:{
new (message?:string): Error;
(message?:string): Error;
prototype: Error;
}
interface EvalError extends Error {
}
declare var EvalError:{
new (message?:string): EvalError;
(message?:string): EvalError;
prototype: EvalError;
}
interface RangeError extends Error {
}
declare var RangeError:{
new (message?:string): RangeError;
(message?:string): RangeError;
prototype: RangeError;
}
interface ReferenceError extends Error {
}
declare var ReferenceError:{
new (message?:string): ReferenceError;
(message?:string): ReferenceError;
prototype: ReferenceError;
}
interface SyntaxError extends Error {
}
declare var SyntaxError:{
new (message?:string): SyntaxError;
(message?:string): SyntaxError;
prototype: SyntaxError;
}
interface TypeError extends Error {
}
declare var TypeError:{
new (message?:string): TypeError;
(message?:string): TypeError;
prototype: TypeError;
}
interface URIError extends Error {
}
declare var URIError:{
new (message?:string): URIError;
(message?:string): URIError;
prototype: URIError;
}
interface JSON {
/**
* Converts a JavaScript Object Notation (JSON) string into an object.
* @param text A valid JSON string.
* @param reviver A function that transforms the results. This function is called for each member of the object.
* If a member contains nested objects, the nested objects are transformed before the parent object is.
*/
parse(text:string, reviver?:(key:any, value:any) => any): any;
/**
* Converts a JavaScript value to a JavaScript Object Notation (JSON) string.
* @param value A JavaScript value, usually an object or array, to be converted.
*/
stringify(value:any): string;
/**
* Converts a JavaScript value to a JavaScript Object Notation (JSON) string.
* @param value A JavaScript value, usually an object or array, to be converted.
* @param replacer A function that transforms the results.
*/
stringify(value:any, replacer:(key:string, value:any) => any): string;
/**
* Converts a JavaScript value to a JavaScript Object Notation (JSON) string.
* @param value A JavaScript value, usually an object or array, to be converted.
* @param replacer Array that transforms the results.
*/
stringify(value:any, replacer:any[]): string;
/**
* Converts a JavaScript value to a JavaScript Object Notation (JSON) string.
* @param value A JavaScript value, usually an object or array, to be converted.
* @param replacer A function that transforms the results.
* @param space Adds indentation, white space, and line break characters to the return-value JSON text to make it easier to read.
*/
stringify(value:any, replacer:(key:string, value:any) => any, space:any): string;
/**
* Converts a JavaScript value to a JavaScript Object Notation (JSON) string.
* @param value A JavaScript value, usually an object or array, to be converted.
* @param replacer Array that transforms the results.
* @param space Adds indentation, white space, and line break characters to the return-value JSON text to make it easier to read.
*/
stringify(value:any, replacer:any[], space:any): string;
}
/**
* An intrinsic object that provides functions to convert JavaScript values to and from the JavaScript Object Notation (JSON) format.
*/
declare var JSON:JSON;
/////////////////////////////
/// ECMAScript Array API (specially handled by compiler)
/////////////////////////////
interface Array<T> {
/**
* Gets or sets the length of the array. This is a number one higher than the highest element defined in an array.
*/
length: number;
/**
* Returns a string representation of an array.
*/
toString(): string;
toLocaleString(): string;
/**
* Appends new elements to an array, and returns the new length of the array.
* @param items New elements of the Array.
*/
push(...items:T[]): number;
/**
* Removes the last element from an array and returns it.
*/
pop(): T;
/**
* Combines two or more arrays.
* @param items Additional items to add to the end of array1.
*/
concat<U extends T[]>(...items:U[]): T[];
/**
* Combines two or more arrays.
* @param items Additional items to add to the end of array1.
*/
concat(...items:T[]): T[];
/**
* Adds all the elements of an array separated by the specified separator string.
* @param separator A string used to separate one element of an array from the next in the resulting String. If omitted, the array elements are separated with a comma.
*/
join(separator?:string): string;
/**
* Reverses the elements in an Array.
*/
reverse(): T[];
/**
* Removes the first element from an array and returns it.
*/
shift(): T;
/**
* Returns a section of an array.
* @param start The beginning of the specified portion of the array.
* @param end The end of the specified portion of the array.
*/
slice(start?:number, end?:number): T[];
/**
* Sorts an array.
* @param compareFn The name of the function used to determine the order of the elements. If omitted, the elements are sorted in ascending, ASCII character order.
*/
sort(compareFn?:(a:T, b:T) => number): T[];
/**
* Removes elements from an array and, if necessary, inserts new elements in their place, returning the deleted elements.
* @param start The zero-based location in the array from which to start removing elements.
*/
splice(start:number): T[];
/**
* Removes elements from an array and, if necessary, inserts new elements in their place, returning the deleted elements.
* @param start The zero-based location in the array from which to start removing elements.
* @param deleteCount The number of elements to remove.
* @param items Elements to insert into the array in place of the deleted elements.
*/
splice(start:number, deleteCount:number, ...items:T[]): T[];
/**
* Inserts new elements at the start of an array.
* @param items Elements to insert at the start of the Array.
*/
unshift(...items:T[]): number;
/**
* Returns the index of the first occurrence of a value in an array.
* @param searchElement The value to locate in the array.
* @param fromIndex The array index at which to begin the search. If fromIndex is omitted, the search starts at index 0.
*/
indexOf(searchElement:T, fromIndex?:number): number;
/**
* Returns the index of the last occurrence of a specified value in an array.
* @param searchElement The value to locate in the array.
* @param fromIndex The array index at which to begin the search. If fromIndex is omitted, the search starts at the last index in the array.
*/
lastIndexOf(searchElement:T, fromIndex?:number): number;
/**
* Determines whether all the members of an array satisfy the specified test.
* @param callbackfn A function that accepts up to three arguments. The every method calls the callbackfn function for each element in array1 until the callbackfn returns false, or until the end of the array.
* @param thisArg An object to which the this keyword can refer in the callbackfn function. If thisArg is omitted, undefined is used as the this value.
*/
every(callbackfn:(value:T, index:number, array:T[]) => boolean, thisArg?:any): boolean;
/**
* Determines whether the specified callback function returns true for any element of an array.
* @param callbackfn A function that accepts up to three arguments. The some method calls the callbackfn function for each element in array1 until the callbackfn returns true, or until the end of the array.
* @param thisArg An object to which the this keyword can refer in the callbackfn function. If thisArg is omitted, undefined is used as the this value.
*/
some(callbackfn:(value:T, index:number, array:T[]) => boolean, thisArg?:any): boolean;
/**
* Performs the specified action for each element in an array.
* @param callbackfn A function that accepts up to three arguments. forEach calls the callbackfn function one time for each element in the array.
* @param thisArg An object to which the this keyword can refer in the callbackfn function. If thisArg is omitted, undefined is used as the this value.
*/
forEach(callbackfn:(value:T, index:number, array:T[]) => void, thisArg?:any): void;
/**
* Calls a defined callback function on each element of an array, and returns an array that contains the results.
* @param callbackfn A function that accepts up to three arguments. The map method calls the callbackfn function one time for each element in the array.
* @param thisArg An object to which the this keyword can refer in the callbackfn function. If thisArg is omitted, undefined is used as the this value.
*/
map<U>(callbackfn:(value:T, index:number, array:T[]) => U, thisArg?:any): U[];
/**
* Returns the elements of an array that meet the condition specified in a callback function.
* @param callbackfn A function that accepts up to three arguments. The filter method calls the callbackfn function one time for each element in the array.
* @param thisArg An object to which the this keyword can refer in the callbackfn function. If thisArg is omitted, undefined is used as the this value.
*/
filter(callbackfn:(value:T, index:number, array:T[]) => boolean, thisArg?:any): T[];
/**
* Calls the specified callback function for all the elements in an array. The return value of the callback function is the accumulated result, and is provided as an argument in the next call to the callback function.
* @param callbackfn A function that accepts up to four arguments. The reduce method calls the callbackfn function one time for each element in the array.
* @param initialValue If initialValue is specified, it is used as the initial value to start the accumulation. The first call to the callbackfn function provides this value as an argument instead of an array value.
*/
reduce(callbackfn:(previousValue:T, currentValue:T, currentIndex:number, array:T[]) => T, initialValue?:T): T;
/**
* Calls the specified callback function for all the elements in an array. The return value of the callback function is the accumulated result, and is provided as an argument in the next call to the callback function.
* @param callbackfn A function that accepts up to four arguments. The reduce method calls the callbackfn function one time for each element in the array.
* @param initialValue If initialValue is specified, it is used as the initial value to start the accumulation. The first call to the callbackfn function provides this value as an argument instead of an array value.
*/
reduce<U>(callbackfn:(previousValue:U, currentValue:T, currentIndex:number, array:T[]) => U, initialValue:U): U;
/**
* Calls the specified callback function for all the elements in an array, in descending order. The return value of the callback function is the accumulated result, and is provided as an argument in the next call to the callback function.
* @param callbackfn A function that accepts up to four arguments. The reduceRight method calls the callbackfn function one time for each element in the array.
* @param initialValue If initialValue is specified, it is used as the initial value to start the accumulation. The first call to the callbackfn function provides this value as an argument instead of an array value.
*/
reduceRight(callbackfn:(previousValue:T, currentValue:T, currentIndex:number, array:T[]) => T, initialValue?:T): T;
/**
* Calls the specified callback function for all the elements in an array, in descending order. The return value of the callback function is the accumulated result, and is provided as an argument in the next call to the callback function.
* @param callbackfn A function that accepts up to four arguments. The reduceRight method calls the callbackfn function one time for each element in the array.
* @param initialValue If initialValue is specified, it is used as the initial value to start the accumulation. The first call to the callbackfn function provides this value as an argument instead of an array value.
*/
reduceRight<U>(callbackfn:(previousValue:U, currentValue:T, currentIndex:number, array:T[]) => U, initialValue:U): U;
[n: number]: T;
}
declare var Array:{
new (arrayLength?:number): any[];
new <T>(arrayLength:number): T[];
new <T>(...items:T[]): T[];
(arrayLength?:number): any[];
<T>(arrayLength:number): T[];
<T>(...items:T[]): T[];
isArray(arg:any): boolean;
prototype: Array<any>;
}
/////////////////////////////
/// IE10 ECMAScript Extensions
/////////////////////////////
/**
* Represents a raw buffer of binary data, which is used to store data for the
* different typed arrays. ArrayBuffers cannot be read from or written to directly,
* but can be passed to a typed array or DataView Object to interpret the raw
* buffer as needed.
*/
interface ArrayBuffer {
/**
* Read-only. The length of the ArrayBuffer (in bytes).
*/
byteLength: number;
/**
* Returns a section of an ArrayBuffer.
*/
slice(begin:number, end?:number): ArrayBuffer;
}
declare var ArrayBuffer:{
prototype: ArrayBuffer;
new (byteLength:number): ArrayBuffer;
}
interface ArrayBufferView {
buffer: ArrayBuffer;
byteOffset: number;
byteLength: number;
}
/**
* A typed array of 8-bit integer values. The contents are initialized to 0. If the requested number of bytes could not be allocated an exception is raised.
*/
interface Int8Array extends ArrayBufferView {
/**
* The size in bytes of each element in the array.
*/
BYTES_PER_ELEMENT: number;
/**
* The length of the array.
*/
length: number;
[index: number]: number;
/**
* Gets the element at the specified index.
* @param index The index at which to get the element of the array.
*/
get(index:number): number;
/**
* Sets a value or an array of values.
* @param index The index of the location to set.
* @param value The value to set.
*/
set(index:number, value:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:Int8Array, offset?:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:number[], offset?:number): void;
/**
* Gets a new Int8Array view of the ArrayBuffer store for this array, referencing the elements at begin, inclusive, up to end, exclusive.
* @param begin The index of the beginning of the array.
* @param end The index of the end of the array.
*/
subarray(begin:number, end?:number): Int8Array;
}
declare var Int8Array:{
prototype: Int8Array;
new (length:number): Int8Array;
new (array:Int8Array): Int8Array;
new (array:number[]): Int8Array;
new (buffer:ArrayBuffer, byteOffset?:number, length?:number): Int8Array;
BYTES_PER_ELEMENT: number;
}
/**
* A typed array of 8-bit unsigned integer values. The contents are initialized to 0. If the requested number of bytes could not be allocated an exception is raised.
*/
interface Uint8Array extends ArrayBufferView {
/**
* The size in bytes of each element in the array.
*/
BYTES_PER_ELEMENT: number;
/**
* The length of the array.
*/
length: number;
[index: number]: number;
/**
* Gets the element at the specified index.
* @param index The index at which to get the element of the array.
*/
get(index:number): number;
/**
* Sets a value or an array of values.
* @param index The index of the location to set.
* @param value The value to set.
*/
set(index:number, value:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:Uint8Array, offset?:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:number[], offset?:number): void;
/**
* Gets a new Uint8Array view of the ArrayBuffer Object store for this array, specifying the first and last members of the subarray.
* @param begin The index of the beginning of the array.
* @param end The index of the end of the array.
*/
subarray(begin:number, end?:number): Uint8Array;
}
declare var Uint8Array:{
prototype: Uint8Array;
new (length:number): Uint8Array;
new (array:Uint8Array): Uint8Array;
new (array:number[]): Uint8Array;
new (buffer:ArrayBuffer, byteOffset?:number, length?:number): Uint8Array;
BYTES_PER_ELEMENT: number;
}
/**
* A typed array of 16-bit integer values. The contents are initialized to 0. If the requested number of bytes could not be allocated an exception is raised.
*/
interface Int16Array extends ArrayBufferView {
/**
* The size in bytes of each element in the array.
*/
BYTES_PER_ELEMENT: number;
/**
* The length of the array.
*/
length: number;
[index: number]: number;
/**
* Gets the element at the specified index.
* @param index The index at which to get the element of the array.
*/
get(index:number): number;
/**
* Sets a value or an array of values.
* @param index The index of the location to set.
* @param value The value to set.
*/
set(index:number, value:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:Int16Array, offset?:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:number[], offset?:number): void;
/**
* Gets a new Int16Array view of the ArrayBuffer Object store for this array, specifying the first and last members of the subarray.
* @param begin The index of the beginning of the array.
* @param end The index of the end of the array.
*/
subarray(begin:number, end?:number): Int16Array;
}
declare var Int16Array:{
prototype: Int16Array;
new (length:number): Int16Array;
new (array:Int16Array): Int16Array;
new (array:number[]): Int16Array;
new (buffer:ArrayBuffer, byteOffset?:number, length?:number): Int16Array;
BYTES_PER_ELEMENT: number;
}
/**
* A typed array of 16-bit unsigned integer values. The contents are initialized to 0. If the requested number of bytes could not be allocated an exception is raised.
*/
interface Uint16Array extends ArrayBufferView {
/**
* The size in bytes of each element in the array.
*/
BYTES_PER_ELEMENT: number;
/**
* The length of the array.
*/
length: number;
[index: number]: number;
/**
* Gets the element at the specified index.
* @param index The index at which to get the element of the array.
*/
get(index:number): number;
/**
* Sets a value or an array of values.
* @param index The index of the location to set.
* @param value The value to set.
*/
set(index:number, value:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:Uint16Array, offset?:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:number[], offset?:number): void;
/**
* Gets a new Uint16Array view of the ArrayBuffer Object store for this array, specifying the first and last members of the subarray.
* @param begin The index of the beginning of the array.
* @param end The index of the end of the array.
*/
subarray(begin:number, end?:number): Uint16Array;
}
declare var Uint16Array:{
prototype: Uint16Array;
new (length:number): Uint16Array;
new (array:Uint16Array): Uint16Array;
new (array:number[]): Uint16Array;
new (buffer:ArrayBuffer, byteOffset?:number, length?:number): Uint16Array;
BYTES_PER_ELEMENT: number;
}
/**
* A typed array of 32-bit integer values. The contents are initialized to 0. If the requested number of bytes could not be allocated an exception is raised.
*/
interface Int32Array extends ArrayBufferView {
/**
* The size in bytes of each element in the array.
*/
BYTES_PER_ELEMENT: number;
/**
* The length of the array.
*/
length: number;
[index: number]: number;
/**
* Gets the element at the specified index.
* @param index The index at which to get the element of the array.
*/
get(index:number): number;
/**
* Sets a value or an array of values.
* @param index The index of the location to set.
* @param value The value to set.
*/
set(index:number, value:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:Int32Array, offset?:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:number[], offset?:number): void;
/**
* Gets a new Int32Array view of the ArrayBuffer Object store for this array, specifying the first and last members of the subarray.
* @param begin The index of the beginning of the array.
* @param end The index of the end of the array.
*/
subarray(begin:number, end?:number): Int32Array;
}
declare var Int32Array:{
prototype: Int32Array;
new (length:number): Int32Array;
new (array:Int32Array): Int32Array;
new (array:number[]): Int32Array;
new (buffer:ArrayBuffer, byteOffset?:number, length?:number): Int32Array;
BYTES_PER_ELEMENT: number;
}
/**
* A typed array of 32-bit unsigned integer values. The contents are initialized to 0. If the requested number of bytes could not be allocated an exception is raised.
*/
interface Uint32Array extends ArrayBufferView {
/**
* The size in bytes of each element in the array.
*/
BYTES_PER_ELEMENT: number;
/**
* The length of the array.
*/
length: number;
[index: number]: number;
/**
* Gets the element at the specified index.
* @param index The index at which to get the element of the array.
*/
get(index:number): number;
/**
* Sets a value or an array of values.
* @param index The index of the location to set.
* @param value The value to set.
*/
set(index:number, value:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:Uint32Array, offset?:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:number[], offset?:number): void;
/**
* Gets a new Int8Array view of the ArrayBuffer Object store for this array, specifying the first and last members of the subarray.
* @param begin The index of the beginning of the array.
* @param end The index of the end of the array.
*/
subarray(begin:number, end?:number): Uint32Array;
}
declare var Uint32Array:{
prototype: Uint32Array;
new (length:number): Uint32Array;
new (array:Uint32Array): Uint32Array;
new (array:number[]): Uint32Array;
new (buffer:ArrayBuffer, byteOffset?:number, length?:number): Uint32Array;
BYTES_PER_ELEMENT: number;
}
/**
* A typed array of 32-bit float values. The contents are initialized to 0. If the requested number of bytes could not be allocated an exception is raised.
*/
interface Float32Array extends ArrayBufferView {
/**
* The size in bytes of each element in the array.
*/
BYTES_PER_ELEMENT: number;
/**
* The length of the array.
*/
length: number;
[index: number]: number;
/**
* Gets the element at the specified index.
* @param index The index at which to get the element of the array.
*/
get(index:number): number;
/**
* Sets a value or an array of values.
* @param index The index of the location to set.
* @param value The value to set.
*/
set(index:number, value:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:Float32Array, offset?:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:number[], offset?:number): void;
/**
* Gets a new Float32Array view of the ArrayBuffer Object store for this array, specifying the first and last members of the subarray.
* @param begin The index of the beginning of the array.
* @param end The index of the end of the array.
*/
subarray(begin:number, end?:number): Float32Array;
}
declare var Float32Array:{
prototype: Float32Array;
new (length:number): Float32Array;
new (array:Float32Array): Float32Array;
new (array:number[]): Float32Array;
new (buffer:ArrayBuffer, byteOffset?:number, length?:number): Float32Array;
BYTES_PER_ELEMENT: number;
}
/**
* A typed array of 64-bit float values. The contents are initialized to 0. If the requested number of bytes could not be allocated an exception is raised.
*/
interface Float64Array extends ArrayBufferView {
/**
* The size in bytes of each element in the array.
*/
BYTES_PER_ELEMENT: number;
/**
* The length of the array.
*/
length: number;
[index: number]: number;
/**
* Gets the element at the specified index.
* @param index The index at which to get the element of the array.
*/
get(index:number): number;
/**
* Sets a value or an array of values.
* @param index The index of the location to set.
* @param value The value to set.
*/
set(index:number, value:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:Float64Array, offset?:number): void;
/**
* Sets a value or an array of values.
* @param array A typed or untyped array of values to set.
* @param offset The index in the current array at which the values are to be written.
*/
set(array:number[], offset?:number): void;
/**
* Gets a new Float64Array view of the ArrayBuffer Object store for this array, specifying the first and last members of the subarray.
* @param begin The index of the beginning of the array.
* @param end The index of the end of the array.
*/
subarray(begin:number, end?:number): Float64Array;
}
declare var Float64Array:{
prototype: Float64Array;
new (length:number): Float64Array;
new (array:Float64Array): Float64Array;
new (array:number[]): Float64Array;
new (buffer:ArrayBuffer, byteOffset?:number, length?:number): Float64Array;
BYTES_PER_ELEMENT: number;
}
/**
* You can use a DataView object to read and write the different kinds of binary data to any location in the ArrayBuffer.
*/
interface DataView extends ArrayBufferView {
/**
* Gets the Int8 value at the specified byte offset from the start of the view. There is no alignment constraint; multi-byte values may be fetched from any offset.
* @param byteOffset The place in the buffer at which the value should be retrieved.
*/
getInt8(byteOffset:number): number;
/**
* Gets the Uint8 value at the specified byte offset from the start of the view. There is no alignment constraint; multi-byte values may be fetched from any offset.
* @param byteOffset The place in the buffer at which the value should be retrieved.
*/
getUint8(byteOffset:number): number;
/**
* Gets the Int16 value at the specified byte offset from the start of the view. There is no alignment constraint; multi-byte values may be fetched from any offset.
* @param byteOffset The place in the buffer at which the value should be retrieved.
*/
getInt16(byteOffset:number, littleEndian?:boolean): number;
/**
* Gets the Uint16 value at the specified byte offset from the start of the view. There is no alignment constraint; multi-byte values may be fetched from any offset.
* @param byteOffset The place in the buffer at which the value should be retrieved.
*/
getUint16(byteOffset:number, littleEndian?:boolean): number;
/**
* Gets the Int32 value at the specified byte offset from the start of the view. There is no alignment constraint; multi-byte values may be fetched from any offset.
* @param byteOffset The place in the buffer at which the value should be retrieved.
*/
getInt32(byteOffset:number, littleEndian?:boolean): number;
/**
* Gets the Uint32 value at the specified byte offset from the start of the view. There is no alignment constraint; multi-byte values may be fetched from any offset.
* @param byteOffset The place in the buffer at which the value should be retrieved.
*/
getUint32(byteOffset:number, littleEndian?:boolean): number;
/**
* Gets the Float32 value at the specified byte offset from the start of the view. There is no alignment constraint; multi-byte values may be fetched from any offset.
* @param byteOffset The place in the buffer at which the value should be retrieved.
*/
getFloat32(byteOffset:number, littleEndian?:boolean): number;
/**
* Gets the Float64 value at the specified byte offset from the start of the view. There is no alignment constraint; multi-byte values may be fetched from any offset.
* @param byteOffset The place in the buffer at which the value should be retrieved.
*/
getFloat64(byteOffset:number, littleEndian?:boolean): number;
/**
* Stores an Int8 value at the specified byte offset from the start of the view.
* @param byteOffset The place in the buffer at which the value should be set.
* @param value The value to set.
*/
setInt8(byteOffset:number, value:number): void;
/**
* Stores an Uint8 value at the specified byte offset from the start of the view.
* @param byteOffset The place in the buffer at which the value should be set.
* @param value The value to set.
*/
setUint8(byteOffset:number, value:number): void;
/**
* Stores an Int16 value at the specified byte offset from the start of the view.
* @param byteOffset The place in the buffer at which the value should be set.
* @param value The value to set.
* @param littleEndian If false or undefined, a big-endian value should be written, otherwise a little-endian value should be written.
*/
setInt16(byteOffset:number, value:number, littleEndian?:boolean): void;
/**
* Stores an Uint16 value at the specified byte offset from the start of the view.
* @param byteOffset The place in the buffer at which the value should be set.
* @param value The value to set.
* @param littleEndian If false or undefined, a big-endian value should be written, otherwise a little-endian value should be written.
*/
setUint16(byteOffset:number, value:number, littleEndian?:boolean): void;
/**
* Stores an Int32 value at the specified byte offset from the start of the view.
* @param byteOffset The place in the buffer at which the value should be set.
* @param value The value to set.
* @param littleEndian If false or undefined, a big-endian value should be written, otherwise a little-endian value should be written.
*/
setInt32(byteOffset:number, value:number, littleEndian?:boolean): void;
/**
* Stores an Uint32 value at the specified byte offset from the start of the view.
* @param byteOffset The place in the buffer at which the value should be set.
* @param value The value to set.
* @param littleEndian If false or undefined, a big-endian value should be written, otherwise a little-endian value should be written.
*/
setUint32(byteOffset:number, value:number, littleEndian?:boolean): void;
/**
* Stores an Float32 value at the specified byte offset from the start of the view.
* @param byteOffset The place in the buffer at which the value should be set.
* @param value The value to set.
* @param littleEndian If false or undefined, a big-endian value should be written, otherwise a little-endian value should be written.
*/
setFloat32(byteOffset:number, value:number, littleEndian?:boolean): void;
/**
* Stores an Float64 value at the specified byte offset from the start of the view.
* @param byteOffset The place in the buffer at which the value should be set.
* @param value The value to set.
* @param littleEndian If false or undefined, a big-endian value should be written, otherwise a little-endian value should be written.
*/
setFloat64(byteOffset:number, value:number, littleEndian?:boolean): void;
}
declare var DataView:{
prototype: DataView;
new (buffer:ArrayBuffer, byteOffset?:number, length?:number): DataView;
}
/////////////////////////////
/// IE11 ECMAScript Extensions
/////////////////////////////
interface Map<K, V> {
clear(): void;
delete(key:K): boolean;
forEach(callbackfn:(value:V, index:K, map:Map<K, V>) => void, thisArg?:any): void;
get(key:K): V;
has(key:K): boolean;
set(key:K, value:V): Map<K, V>;
size: number;
}
declare var Map:{
new <K, V>(): Map<K, V>;
}
interface WeakMap<K, V> {
clear(): void;
delete(key:K): boolean;
get(key:K): V;
has(key:K): boolean;
set(key:K, value:V): WeakMap<K, V>;
}
declare var WeakMap:{
new <K, V>(): WeakMap<K, V>;
}
interface Set<T> {
add(value:T): Set<T>;
clear(): void;
delete(value:T): boolean;
forEach(callbackfn:(value:T, index:T, set:Set<T>) => void, thisArg?:any): void;
has(value:T): boolean;
size: number;
}
declare var Set:{
new <T>(): Set<T>;
}
declare module Intl {
interface CollatorOptions {
usage?: string;
localeMatcher?: string;
numeric?: boolean;
caseFirst?: string;
sensitivity?: string;
ignorePunctuation?: boolean;
}
interface ResolvedCollatorOptions {
locale: string;
usage: string;
sensitivity: string;
ignorePunctuation: boolean;
collation: string;
caseFirst: string;
numeric: boolean;
}
interface Collator {
compare(x:string, y:string): number;
resolvedOptions(): ResolvedCollatorOptions;
}
var Collator:{
new (locales?:string[], options?:CollatorOptions): Collator;
new (locale?:string, options?:CollatorOptions): Collator;
(locales?:string[], options?:CollatorOptions): Collator;
(locale?:string, options?:CollatorOptions): Collator;
supportedLocalesOf(locales:string[], options?:CollatorOptions): string[];
supportedLocalesOf(locale:string, options?:CollatorOptions): string[];
}
interface NumberFormatOptions {
localeMatcher?: string;
style?: string;
currency?: string;
currencyDisplay?: string;
useGrouping?: boolean;
}
interface ResolvedNumberFormatOptions {
locale: string;
numberingSystem: string;
style: string;
currency?: string;
currencyDisplay?: string;
minimumintegerDigits: number;
minimumFractionDigits: number;
maximumFractionDigits: number;
minimumSignificantDigits?: number;
maximumSignificantDigits?: number;
useGrouping: boolean;
}
interface NumberFormat {
format(value:number): string;
resolvedOptions(): ResolvedNumberFormatOptions;
}
var NumberFormat:{
new (locales?:string[], options?:NumberFormatOptions): Collator;
new (locale?:string, options?:NumberFormatOptions): Collator;
(locales?:string[], options?:NumberFormatOptions): Collator;
(locale?:string, options?:NumberFormatOptions): Collator;
supportedLocalesOf(locales:string[], options?:NumberFormatOptions): string[];
supportedLocalesOf(locale:string, options?:NumberFormatOptions): string[];
}
interface DateTimeFormatOptions {
localeMatcher?: string;
weekday?: string;
era?: string;
year?: string;
month?: string;
day?: string;
hour?: string;
minute?: string;
second?: string;
timeZoneName?: string;
formatMatcher?: string;
hour12: boolean;
}
interface ResolvedDateTimeFormatOptions {
locale: string;
calendar: string;
numberingSystem: string;
timeZone: string;
hour12?: boolean;
weekday?: string;
era?: string;
year?: string;
month?: string;
day?: string;
hour?: string;
minute?: string;
second?: string;
timeZoneName?: string;
}
interface DateTimeFormat {
format(date:number): string;
resolvedOptions(): ResolvedDateTimeFormatOptions;
}
var DateTimeFormat:{
new (locales?:string[], options?:DateTimeFormatOptions): Collator;
new (locale?:string, options?:DateTimeFormatOptions): Collator;
(locales?:string[], options?:DateTimeFormatOptions): Collator;
(locale?:string, options?:DateTimeFormatOptions): Collator;
supportedLocalesOf(locales:string[], options?:DateTimeFormatOptions): string[];
supportedLocalesOf(locale:string, options?:DateTimeFormatOptions): string[];
}
}
interface String {
/**
* Determines whether two strings are equivalent in the current locale.
* @param that String to compare to target string
* @param locales An array of locale strings that contain one or more language or locale tags. If you include more than one locale string, list them in descending order of priority so that the first entry is the preferred locale. If you omit this parameter, the default locale of the JavaScript runtime is used. This parameter must conform to BCP 47 standards; see the Intl.Collator object for details.
* @param options An object that contains one or more properties that specify comparison options. see the Intl.Collator object for details.
*/
localeCompare(that:string, locales:string[], options?:Intl.CollatorOptions): number;
/**
* Determines whether two strings are equivalent in the current locale.
* @param that String to compare to target string
* @param locale Locale tag. If you omit this parameter, the default locale of the JavaScript runtime is used. This parameter must conform to BCP 47 standards; see the Intl.Collator object for details.
* @param options An object that contains one or more properties that specify comparison options. see the Intl.Collator object for details.
*/
localeCompare(that:string, locale:string, options?:Intl.CollatorOptions): number;
}
interface Number {
/**
* Converts a number to a string by using the current or specified locale.
* @param locales An array of locale strings that contain one or more language or locale tags. If you include more than one locale string, list them in descending order of priority so that the first entry is the preferred locale. If you omit this parameter, the default locale of the JavaScript runtime is used.
* @param options An object that contains one or more properties that specify comparison options.
*/
toLocaleString(locales?:string[], options?:Intl.NumberFormatOptions): string;
/**
* Converts a number to a string by using the current or specified locale.
* @param locale Locale tag. If you omit this parameter, the default locale of the JavaScript runtime is used.
* @param options An object that contains one or more properties that specify comparison options.
*/
toLocaleString(locale?:string, options?:Intl.NumberFormatOptions): string;
}
interface Date {
/**
* Converts a date to a string by using the current or specified locale.
* @param locales An array of locale strings that contain one or more language or locale tags. If you include more than one locale string, list them in descending order of priority so that the first entry is the preferred locale. If you omit this parameter, the default locale of the JavaScript runtime is used.
* @param options An object that contains one or more properties that specify comparison options.
*/
toLocaleString(locales?:string[], options?:Intl.DateTimeFormatOptions): string;
/**
* Converts a date to a string by using the current or specified locale.
* @param locale Locale tag. If you omit this parameter, the default locale of the JavaScript runtime is used.
* @param options An object that contains one or more properties that specify comparison options.
*/
toLocaleString(locale?:string, options?:Intl.DateTimeFormatOptions): string;
}
/////////////////////////////
/// IE DOM APIs
/////////////////////////////
interface PositionOptions {
enableHighAccuracy?: boolean;
timeout?: number;
maximumAge?: number;
}
interface ObjectURLOptions {
oneTimeOnly?: boolean;
}
interface StoreExceptionsInformation extends ExceptionInformation {
siteName?: string;
explanationString?: string;
detailURI?: string;
}
interface StoreSiteSpecificExceptionsInformation extends StoreExceptionsInformation {
arrayOfDomainStrings?: string[];
}
interface ConfirmSiteSpecificExceptionsInformation extends ExceptionInformation {
arrayOfDomainStrings?: string[];
}
interface AlgorithmParameters {
}
interface MutationObserverInit {
childList?: boolean;
attributes?: boolean;
characterData?: boolean;
subtree?: boolean;
attributeOldValue?: boolean;
characterDataOldValue?: boolean;
attributeFilter?: string[];
}
interface PointerEventInit extends MouseEventInit {
pointerId?: number;
width?: number;
height?: number;
pressure?: number;
tiltX?: number;
tiltY?: number;
pointerType?: string;
isPrimary?: boolean;
}
interface ExceptionInformation {
domain?: string;
}
interface DeviceAccelerationDict {
x?: number;
y?: number;
z?: number;
}
interface MsZoomToOptions {
contentX?: number;
contentY?: number;
viewportX?: string;
viewportY?: string;
scaleFactor?: number;
animate?: string;
}
interface DeviceRotationRateDict {
alpha?: number;
beta?: number;
gamma?: number;
}
interface Algorithm {
name?: string;
params?: AlgorithmParameters;
}
interface MouseEventInit {
bubbles?: boolean;
cancelable?: boolean;
view?: Window;
detail?: number;
screenX?: number;
screenY?: number;
clientX?: number;
clientY?: number;
ctrlKey?: boolean;
shiftKey?: boolean;
altKey?: boolean;
metaKey?: boolean;
button?: number;
buttons?: number;
relatedTarget?: EventTarget;
}
interface WebGLContextAttributes {
alpha?: boolean;
depth?: boolean;
stencil?: boolean;
antialias?: boolean;
premultipliedAlpha?: boolean;
preserveDrawingBuffer?: boolean;
}
interface NodeListOf<TNode extends Node> extends NodeList {
length: number;
item(index:number): TNode;
[index: number]: TNode;
}
interface HTMLElement extends Element, ElementCSSInlineStyle, MSEventAttachmentTarget, MSNodeExtensions {
hidden: any;
readyState: any;
onmouseleave: (ev:MouseEvent) => any;
onbeforecut: (ev:DragEvent) => any;
onkeydown: (ev:KeyboardEvent) => any;
onmove: (ev:MSEventObj) => any;
onkeyup: (ev:KeyboardEvent) => any;
onreset: (ev:Event) => any;
onhelp: (ev:Event) => any;
ondragleave: (ev:DragEvent) => any;
className: string;
onfocusin: (ev:FocusEvent) => any;
onseeked: (ev:Event) => any;
recordNumber: any;
title: string;
parentTextEdit: Element;
outerHTML: string;
ondurationchange: (ev:Event) => any;
offsetHeight: number;
all: HTMLCollection;
onblur: (ev:FocusEvent) => any;
dir: string;
onemptied: (ev:Event) => any;
onseeking: (ev:Event) => any;
oncanplay: (ev:Event) => any;
ondeactivate: (ev:UIEvent) => any;
ondatasetchanged: (ev:MSEventObj) => any;
onrowsdelete: (ev:MSEventObj) => any;
sourceIndex: number;
onloadstart: (ev:Event) => any;
onlosecapture: (ev:MSEventObj) => any;
ondragenter: (ev:DragEvent) => any;
oncontrolselect: (ev:MSEventObj) => any;
onsubmit: (ev:Event) => any;
behaviorUrns: MSBehaviorUrnsCollection;
scopeName: string;
onchange: (ev:Event) => any;
id: string;
onlayoutcomplete: (ev:MSEventObj) => any;
uniqueID: string;
onbeforeactivate: (ev:UIEvent) => any;
oncanplaythrough: (ev:Event) => any;
onbeforeupdate: (ev:MSEventObj) => any;
onfilterchange: (ev:MSEventObj) => any;
offsetParent: Element;
ondatasetcomplete: (ev:MSEventObj) => any;
onsuspend: (ev:Event) => any;
onmouseenter: (ev:MouseEvent) => any;
innerText: string;
onerrorupdate: (ev:MSEventObj) => any;
onmouseout: (ev:MouseEvent) => any;
parentElement: HTMLElement;
onmousewheel: (ev:MouseWheelEvent) => any;
onvolumechange: (ev:Event) => any;
oncellchange: (ev:MSEventObj) => any;
onrowexit: (ev:MSEventObj) => any;
onrowsinserted: (ev:MSEventObj) => any;
onpropertychange: (ev:MSEventObj) => any;
filters: any;
children: HTMLCollection;
ondragend: (ev:DragEvent) => any;
onbeforepaste: (ev:DragEvent) => any;
ondragover: (ev:DragEvent) => any;
offsetTop: number;
onmouseup: (ev:MouseEvent) => any;
ondragstart: (ev:DragEvent) => any;
onbeforecopy: (ev:DragEvent) => any;
ondrag: (ev:DragEvent) => any;
innerHTML: string;
onmouseover: (ev:MouseEvent) => any;
lang: string;
uniqueNumber: number;
onpause: (ev:Event) => any;
tagUrn: string;
onmousedown: (ev:MouseEvent) => any;
onclick: (ev:MouseEvent) => any;
onwaiting: (ev:Event) => any;
onresizestart: (ev:MSEventObj) => any;
offsetLeft: number;
isTextEdit: boolean;
isDisabled: boolean;
onpaste: (ev:DragEvent) => any;
canHaveHTML: boolean;
onmoveend: (ev:MSEventObj) => any;
language: string;
onstalled: (ev:Event) => any;
onmousemove: (ev:MouseEvent) => any;
style: MSStyleCSSProperties;
isContentEditable: boolean;
onbeforeeditfocus: (ev:MSEventObj) => any;
onratechange: (ev:Event) => any;
contentEditable: string;
tabIndex: number;
document: Document;
onprogress: (ev:ProgressEvent) => any;
ondblclick: (ev:MouseEvent) => any;
oncontextmenu: (ev:MouseEvent) => any;
onloadedmetadata: (ev:Event) => any;
onafterupdate: (ev:MSEventObj) => any;
onerror: (ev:ErrorEvent) => any;
onplay: (ev:Event) => any;
onresizeend: (ev:MSEventObj) => any;
onplaying: (ev:Event) => any;
isMultiLine: boolean;
onfocusout: (ev:FocusEvent) => any;
onabort: (ev:UIEvent) => any;
ondataavailable: (ev:MSEventObj) => any;
hideFocus: boolean;
onreadystatechange: (ev:Event) => any;
onkeypress: (ev:KeyboardEvent) => any;
onloadeddata: (ev:Event) => any;
onbeforedeactivate: (ev:UIEvent) => any;
outerText: string;
disabled: boolean;
onactivate: (ev:UIEvent) => any;
accessKey: string;
onmovestart: (ev:MSEventObj) => any;
onselectstart: (ev:Event) => any;
onfocus: (ev:FocusEvent) => any;
ontimeupdate: (ev:Event) => any;
onresize: (ev:UIEvent) => any;
oncut: (ev:DragEvent) => any;
onselect: (ev:UIEvent) => any;
ondrop: (ev:DragEvent) => any;
offsetWidth: number;
oncopy: (ev:DragEvent) => any;
onended: (ev:Event) => any;
onscroll: (ev:UIEvent) => any;
onrowenter: (ev:MSEventObj) => any;
onload: (ev:Event) => any;
canHaveChildren: boolean;
oninput: (ev:Event) => any;
onmscontentzoom: (ev:MSEventObj) => any;
oncuechange: (ev:Event) => any;
spellcheck: boolean;
classList: DOMTokenList;
onmsmanipulationstatechanged: (ev:any) => any;
draggable: boolean;
dataset: DOMStringMap;
dragDrop(): boolean;
scrollIntoView(top?:boolean): void;
addFilter(filter:any): void;
setCapture(containerCapture?:boolean): void;
focus(): void;
getAdjacentText(where:string): string;
insertAdjacentText(where:string, text:string): void;
getElementsByClassName(classNames:string): NodeList;
setActive(): void;
removeFilter(filter:any): void;
blur(): void;
clearAttributes(): void;
releaseCapture(): void;
createControlRange(): ControlRangeCollection;
removeBehavior(cookie:number): boolean;
contains(child:HTMLElement): boolean;
click(): void;
insertAdjacentElement(position:string, insertedElement:Element): Element;
mergeAttributes(source:HTMLElement, preserveIdentity?:boolean): void;
replaceAdjacentText(where:string, newText:string): string;
applyElement(apply:Element, where?:string): Element;
addBehavior(bstrUrl:string, factory?:any): number;
insertAdjacentHTML(where:string, html:string): void;
msGetInputContext(): MSInputMethodContext;
addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerdown", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgotpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturedoubletap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerhover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturehold", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointermove", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturechange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturestart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointercancel", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgestureend", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturetap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerout", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msinertiastart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mslostpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerup", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"lostpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerenter", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"gotpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerleave", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mouseleave", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"keydown", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"move", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"keyup", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"reset", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"help", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"dragleave", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusin", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"seeked", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"durationchange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"blur", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"emptied", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"seeking", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"canplay", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"deactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"datasetchanged", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsdelete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"loadstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"losecapture", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragenter", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"controlselect", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"submit", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"change", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"layoutcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"beforeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"canplaythrough", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"filterchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"datasetcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"suspend", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mouseenter", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"errorupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"mouseout", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mousewheel", listener:(ev:MouseWheelEvent) => any, useCapture?:boolean): void;
addEventListener(type:"volumechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"cellchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowexit", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsinserted", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"propertychange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragend", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforepaste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragover", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseup", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragstart", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecopy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drag", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseover", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pause", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousedown", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"click", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"waiting", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"paste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"moveend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"stalled", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousemove", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforeeditfocus", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"ratechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dblclick", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"contextmenu", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadedmetadata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"afterupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"play", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizeend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"playing", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focusout", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dataavailable", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"readystatechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"keypress", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadeddata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforedeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"activate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"movestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"selectstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focus", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"timeupdate", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resize", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"cut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"select", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drop", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"copy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"ended", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"scroll", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"rowenter", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"input", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mscontentzoom", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"cuechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"msmanipulationstatechanged", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var HTMLElement:{
prototype: HTMLElement;
new(): HTMLElement;
}
interface Document extends Node, NodeSelector, MSEventAttachmentTarget, DocumentEvent, MSResourceMetadata, MSNodeExtensions, MSDocumentExtensions, GlobalEventHandlers {
/**
* Gets a reference to the root node of the document.
*/
documentElement: HTMLElement;
/**
* Retrieves the collection of user agents and versions declared in the X-UA-Compatible
*/
compatible: MSCompatibleInfoCollection;
/**
* Fires when the user presses a key.
* @param ev The keyboard event
*/
onkeydown: (ev:KeyboardEvent) => any;
/**
* Fires when the user releases a key.
* @param ev The keyboard event
*/
onkeyup: (ev:KeyboardEvent) => any;
/**
* Gets the implementation object of the current document.
*/
implementation: DOMImplementation;
/**
* Fires when the user resets a form.
* @param ev The event.
*/
onreset: (ev:Event) => any;
/**
* Retrieves a collection of all script objects in the document.
*/
scripts: HTMLCollection;
/**
* Fires when the user presses the F1 key while the browser is the active window.
* @param ev The event.
*/
onhelp: (ev:Event) => any;
/**
* Fires on the target object when the user moves the mouse out of a valid drop target during a drag operation.
* @param ev The drag event.
*/
ondragleave: (ev:DragEvent) => any;
/**
* Gets or sets the character set used to encode the object.
*/
charset: string;
/**
* Fires for an element just prior to setting focus on that element.
* @param ev The focus event
*/
onfocusin: (ev:FocusEvent) => any;
/**
* Sets or gets the color of the links that the user has visited.
*/
vlinkColor: string;
/**
* Occurs when the seek operation ends.
* @param ev The event.
*/
onseeked: (ev:Event) => any;
security: string;
/**
* Contains the title of the document.
*/
title: string;
/**
* Retrieves a collection of namespace objects.
*/
namespaces: MSNamespaceInfoCollection;
/**
* Gets the default character set from the current regional language settings.
*/
defaultCharset: string;
/**
* Retrieves a collection of all embed objects in the document.
*/
embeds: HTMLCollection;
/**
* Retrieves a collection of styleSheet objects representing the style sheets that correspond to each instance of a link or style object in the document.
*/
styleSheets: StyleSheetList;
/**
* Retrieves a collection of all window objects defined by the given document or defined by the document associated with the given window.
*/
frames: Window;
/**
* Occurs when the duration attribute is updated.
* @param ev The event.
*/
ondurationchange: (ev:Event) => any;
/**
* Returns a reference to the collection of elements contained by the object.
*/
all: HTMLCollection;
/**
* Retrieves a collection, in source order, of all form objects in the document.
*/
forms: HTMLCollection;
/**
* Fires when the object loses the input focus.
* @param ev The focus event.
*/
onblur: (ev:FocusEvent) => any;
/**
* Sets or retrieves a value that indicates the reading order of the object.
*/
dir: string;
/**
* Occurs when the media element is reset to its initial state.
* @param ev The event.
*/
onemptied: (ev:Event) => any;
/**
* Sets or gets a value that indicates whether the document can be edited.
*/
designMode: string;
/**
* Occurs when the current playback position is moved.
* @param ev The event.
*/
onseeking: (ev:Event) => any;
/**
* Fires when the activeElement is changed from the current object to another object in the parent document.
* @param ev The UI Event
*/
ondeactivate: (ev:UIEvent) => any;
/**
* Occurs when playback is possible, but would require further buffering.
* @param ev The event.
*/
oncanplay: (ev:Event) => any;
/**
* Fires when the data set exposed by a data source object changes.
* @param ev The event.
*/
ondatasetchanged: (ev:MSEventObj) => any;
/**
* Fires when rows are about to be deleted from the recordset.
* @param ev The event
*/
onrowsdelete: (ev:MSEventObj) => any;
Script: MSScriptHost;
/**
* Occurs when Internet Explorer begins looking for media data.
* @param ev The event.
*/
onloadstart: (ev:Event) => any;
/**
* Gets the URL for the document, stripped of any character encoding.
*/
URLUnencoded: string;
defaultView: Window;
/**
* Fires when the user is about to make a control selection of the object.
* @param ev The event.
*/
oncontrolselect: (ev:MSEventObj) => any;
/**
* Fires on the target element when the user drags the object to a valid drop target.
* @param ev The drag event.
*/
ondragenter: (ev:DragEvent) => any;
onsubmit: (ev:Event) => any;
/**
* Returns the character encoding used to create the webpage that is loaded into the document object.
*/
inputEncoding: string;
/**
* Gets the object that has the focus when the parent document has focus.
*/
activeElement: Element;
/**
* Fires when the contents of the object or selection have changed.
* @param ev The event.
*/
onchange: (ev:Event) => any;
/**
* Retrieves a collection of all a objects that specify the href property and all area objects in the document.
*/
links: HTMLCollection;
/**
* Retrieves an autogenerated, unique identifier for the object.
*/
uniqueID: string;
/**
* Sets or gets the URL for the current document.
*/
URL: string;
/**
* Fires immediately before the object is set as the active element.
* @param ev The event.
*/
onbeforeactivate: (ev:UIEvent) => any;
head: HTMLHeadElement;
cookie: string;
xmlEncoding: string;
oncanplaythrough: (ev:Event) => any;
/**
* Retrieves the document compatibility mode of the document.
*/
documentMode: number;
characterSet: string;
/**
* Retrieves a collection of all a objects that have a name and/or id property. Objects in this collection are in HTML source order.
*/
anchors: HTMLCollection;
onbeforeupdate: (ev:MSEventObj) => any;
/**
* Fires to indicate that all data is available from the data source object.
* @param ev The event.
*/
ondatasetcomplete: (ev:MSEventObj) => any;
plugins: HTMLCollection;
/**
* Occurs if the load operation has been intentionally halted.
* @param ev The event.
*/
onsuspend: (ev:Event) => any;
/**
* Gets the root svg element in the document hierarchy.
*/
rootElement: SVGSVGElement;
/**
* Retrieves a value that indicates the current state of the object.
*/
readyState: string;
/**
* Gets the URL of the location that referred the user to the current page.
*/
referrer: string;
/**
* Sets or gets the color of all active links in the document.
*/
alinkColor: string;
/**
* Fires on a databound object when an error occurs while updating the associated data in the data source object.
* @param ev The event.
*/
onerrorupdate: (ev:MSEventObj) => any;
/**
* Gets a reference to the container object of the window.
*/
parentWindow: Window;
/**
* Fires when the user moves the mouse pointer outside the boundaries of the object.
* @param ev The mouse event.
*/
onmouseout: (ev:MouseEvent) => any;
/**
* Occurs when a user clicks a button in a Thumbnail Toolbar of a webpage running in Site Mode.
* @param ev The event.
*/
onmsthumbnailclick: (ev:MSSiteModeEvent) => any;
/**
* Fires when the wheel button is rotated.
* @param ev The mouse event
*/
onmousewheel: (ev:MouseWheelEvent) => any;
/**
* Occurs when the volume is changed, or playback is muted or unmuted.
* @param ev The event.
*/
onvolumechange: (ev:Event) => any;
/**
* Fires when data changes in the data provider.
* @param ev The event.
*/
oncellchange: (ev:MSEventObj) => any;
/**
* Fires just before the data source control changes the current row in the object.
* @param ev The event.
*/
onrowexit: (ev:MSEventObj) => any;
/**
* Fires just after new rows are inserted in the current recordset.
* @param ev The event.
*/
onrowsinserted: (ev:MSEventObj) => any;
/**
* Gets or sets the version attribute specified in the declaration of an XML document.
*/
xmlVersion: string;
msCapsLockWarningOff: boolean;
/**
* Fires when a property changes on the object.
* @param ev The event.
*/
onpropertychange: (ev:MSEventObj) => any;
/**
* Fires on the source object when the user releases the mouse at the close of a drag operation.
* @param ev The event.
*/
ondragend: (ev:DragEvent) => any;
/**
* Gets an object representing the document type declaration associated with the current document.
*/
doctype: DocumentType;
/**
* Fires on the target element continuously while the user drags the object over a valid drop target.
* @param ev The event.
*/
ondragover: (ev:DragEvent) => any;
/**
* Deprecated. Sets or retrieves a value that indicates the background color behind the object.
*/
bgColor: string;
/**
* Fires on the source object when the user starts to drag a text selection or selected object.
* @param ev The event.
*/
ondragstart: (ev:DragEvent) => any;
/**
* Fires when the user releases a mouse button while the mouse is over the object.
* @param ev The mouse event.
*/
onmouseup: (ev:MouseEvent) => any;
/**
* Fires on the source object continuously during a drag operation.
* @param ev The event.
*/
ondrag: (ev:DragEvent) => any;
/**
* Fires when the user moves the mouse pointer into the object.
* @param ev The mouse event.
*/
onmouseover: (ev:MouseEvent) => any;
/**
* Sets or gets the color of the document links.
*/
linkColor: string;
/**
* Occurs when playback is paused.
* @param ev The event.
*/
onpause: (ev:Event) => any;
/**
* Fires when the user clicks the object with either mouse button.
* @param ev The mouse event.
*/
onmousedown: (ev:MouseEvent) => any;
/**
* Fires when the user clicks the left mouse button on the object
* @param ev The mouse event.
*/
onclick: (ev:MouseEvent) => any;
/**
* Occurs when playback stops because the next frame of a video resource is not available.
* @param ev The event.
*/
onwaiting: (ev:Event) => any;
/**
* Fires when the user clicks the Stop button or leaves the Web page.
* @param ev The event.
*/
onstop: (ev:Event) => any;
/**
* Occurs when an item is removed from a Jump List of a webpage running in Site Mode.
* @param ev The event.
*/
onmssitemodejumplistitemremoved: (ev:MSSiteModeEvent) => any;
/**
* Retrieves a collection of all applet objects in the document.
*/
applets: HTMLCollection;
/**
* Specifies the beginning and end of the document body.
*/
body: HTMLElement;
/**
* Sets or gets the security domain of the document.
*/
domain: string;
xmlStandalone: boolean;
/**
* Represents the active selection, which is a highlighted block of text or other elements in the document that a user or a script can carry out some action on.
*/
selection: MSSelection;
/**
* Occurs when the download has stopped.
* @param ev The event.
*/
onstalled: (ev:Event) => any;
/**
* Fires when the user moves the mouse over the object.
* @param ev The mouse event.
*/
onmousemove: (ev:MouseEvent) => any;
/**
* Fires before an object contained in an editable element enters a UI-activated state or when an editable container object is control selected.
* @param ev The event.
*/
onbeforeeditfocus: (ev:MSEventObj) => any;
/**
* Occurs when the playback rate is increased or decreased.
* @param ev The event.
*/
onratechange: (ev:Event) => any;
/**
* Occurs to indicate progress while downloading media data.
* @param ev The event.
*/
onprogress: (ev:ProgressEvent) => any;
/**
* Fires when the user double-clicks the object.
* @param ev The mouse event.
*/
ondblclick: (ev:MouseEvent) => any;
/**
* Fires when the user clicks the right mouse button in the client area, opening the context menu.
* @param ev The mouse event.
*/
oncontextmenu: (ev:MouseEvent) => any;
/**
* Occurs when the duration and dimensions of the media have been determined.
* @param ev The event.
*/
onloadedmetadata: (ev:Event) => any;
media: string;
/**
* Fires when an error occurs during object loading.
* @param ev The event.
*/
onerror: (ev:ErrorEvent) => any;
/**
* Occurs when the play method is requested.
* @param ev The event.
*/
onplay: (ev:Event) => any;
onafterupdate: (ev:MSEventObj) => any;
/**
* Occurs when the audio or video has started playing.
* @param ev The event.
*/
onplaying: (ev:Event) => any;
/**
* Retrieves a collection, in source order, of img objects in the document.
*/
images: HTMLCollection;
/**
* Contains information about the current URL.
*/
location: Location;
/**
* Fires when the user aborts the download.
* @param ev The event.
*/
onabort: (ev:UIEvent) => any;
/**
* Fires for the current element with focus immediately after moving focus to another element.
* @param ev The event.
*/
onfocusout: (ev:FocusEvent) => any;
/**
* Fires when the selection state of a document changes.
* @param ev The event.
*/
onselectionchange: (ev:Event) => any;
/**
* Fires when a local DOM Storage area is written to disk.
* @param ev The event.
*/
onstoragecommit: (ev:StorageEvent) => any;
/**
* Fires periodically as data arrives from data source objects that asynchronously transmit their data.
* @param ev The event.
*/
ondataavailable: (ev:MSEventObj) => any;
/**
* Fires when the state of the object has changed.
* @param ev The event
*/
onreadystatechange: (ev:Event) => any;
/**
* Gets the date that the page was last modified, if the page supplies one.
*/
lastModified: string;
/**
* Fires when the user presses an alphanumeric key.
* @param ev The event.
*/
onkeypress: (ev:KeyboardEvent) => any;
/**
* Occurs when media data is loaded at the current playback position.
* @param ev The event.
*/
onloadeddata: (ev:Event) => any;
/**
* Fires immediately before the activeElement is changed from the current object to another object in the parent document.
* @param ev The event.
*/
onbeforedeactivate: (ev:UIEvent) => any;
/**
* Fires when the object is set as the active element.
* @param ev The event.
*/
onactivate: (ev:UIEvent) => any;
onselectstart: (ev:Event) => any;
/**
* Fires when the object receives focus.
* @param ev The event.
*/
onfocus: (ev:FocusEvent) => any;
/**
* Sets or gets the foreground (text) color of the document.
*/
fgColor: string;
/**
* Occurs to indicate the current playback position.
* @param ev The event.
*/
ontimeupdate: (ev:Event) => any;
/**
* Fires when the current selection changes.
* @param ev The event.
*/
onselect: (ev:UIEvent) => any;
ondrop: (ev:DragEvent) => any;
/**
* Occurs when the end of playback is reached.
* @param ev The event
*/
onended: (ev:Event) => any;
/**
* Gets a value that indicates whether standards-compliant mode is switched on for the object.
*/
compatMode: string;
/**
* Fires when the user repositions the scroll box in the scroll bar on the object.
* @param ev The event.
*/
onscroll: (ev:UIEvent) => any;
/**
* Fires to indicate that the current row has changed in the data source and new data values are available on the object.
* @param ev The event.
*/
onrowenter: (ev:MSEventObj) => any;
/**
* Fires immediately after the browser loads the object.
* @param ev The event.
*/
onload: (ev:Event) => any;
oninput: (ev:Event) => any;
onmspointerdown: (ev:any) => any;
msHidden: boolean;
msVisibilityState: string;
onmsgesturedoubletap: (ev:any) => any;
visibilityState: string;
onmsmanipulationstatechanged: (ev:any) => any;
onmspointerhover: (ev:any) => any;
onmscontentzoom: (ev:MSEventObj) => any;
onmspointermove: (ev:any) => any;
onmsgesturehold: (ev:any) => any;
onmsgesturechange: (ev:any) => any;
onmsgesturestart: (ev:any) => any;
onmspointercancel: (ev:any) => any;
onmsgestureend: (ev:any) => any;
onmsgesturetap: (ev:any) => any;
onmspointerout: (ev:any) => any;
onmsinertiastart: (ev:any) => any;
msCSSOMElementFloatMetrics: boolean;
onmspointerover: (ev:any) => any;
hidden: boolean;
onmspointerup: (ev:any) => any;
msFullscreenEnabled: boolean;
onmsfullscreenerror: (ev:any) => any;
onmspointerenter: (ev:any) => any;
msFullscreenElement: Element;
onmsfullscreenchange: (ev:any) => any;
onmspointerleave: (ev:any) => any;
/**
* Returns a reference to the first object with the specified value of the ID or NAME attribute.
* @param elementId String that specifies the ID value. Case-insensitive.
*/
getElementById(elementId:string): HTMLElement;
/**
* Returns the current value of the document, range, or current selection for the given command.
* @param commandId String that specifies a command identifier.
*/
queryCommandValue(commandId:string): string;
adoptNode(source:Node): Node;
/**
* Returns a Boolean value that indicates whether the specified command is in the indeterminate state.
* @param commandId String that specifies a command identifier.
*/
queryCommandIndeterm(commandId:string): boolean;
getElementsByTagNameNS(namespaceURI:string, localName:string): NodeList;
createProcessingInstruction(target:string, data:string): ProcessingInstruction;
/**
* Executes a command on the current document, current selection, or the given range.
* @param commandId String that specifies the command to execute. This command can be any of the command identifiers that can be executed in script.
* @param showUI Display the user interface, defaults to false.
* @param value Value to assign.
*/
execCommand(commandId:string, showUI?:boolean, value?:any): boolean;
/**
* Returns the element for the specified x coordinate and the specified y coordinate.
* @param x The x-offset
* @param y The y-offset
*/
elementFromPoint(x:number, y:number): Element;
createCDATASection(data:string): CDATASection;
/**
* Retrieves the string associated with a command.
* @param commandId String that contains the identifier of a command. This can be any command identifier given in the list of Command Identifiers.
*/
queryCommandText(commandId:string): string;
/**
* Writes one or more HTML expressions to a document in the specified window.
* @param content Specifies the text and HTML tags to write.
*/
write(...content:string[]): void;
/**
* Allows updating the print settings for the page.
*/
updateSettings(): void;
/**
* Creates an instance of the element for the specified tag.
* @param tagName The name of an element.
*/
createElement(tagName:"a"): HTMLAnchorElement;
createElement(tagName:"abbr"): HTMLPhraseElement;
createElement(tagName:"acronym"): HTMLPhraseElement;
createElement(tagName:"address"): HTMLBlockElement;
createElement(tagName:"applet"): HTMLAppletElement;
createElement(tagName:"area"): HTMLAreaElement;
createElement(tagName:"article"): HTMLElement;
createElement(tagName:"aside"): HTMLElement;
createElement(tagName:"audio"): HTMLAudioElement;
createElement(tagName:"b"): HTMLPhraseElement;
createElement(tagName:"base"): HTMLBaseElement;
createElement(tagName:"basefont"): HTMLBaseFontElement;
createElement(tagName:"bdo"): HTMLPhraseElement;
createElement(tagName:"bgsound"): HTMLBGSoundElement;
createElement(tagName:"big"): HTMLPhraseElement;
createElement(tagName:"blockquote"): HTMLBlockElement;
createElement(tagName:"body"): HTMLBodyElement;
createElement(tagName:"br"): HTMLBRElement;
createElement(tagName:"button"): HTMLButtonElement;
createElement(tagName:"canvas"): HTMLCanvasElement;
createElement(tagName:"caption"): HTMLTableCaptionElement;
createElement(tagName:"center"): HTMLBlockElement;
createElement(tagName:"cite"): HTMLPhraseElement;
createElement(tagName:"code"): HTMLPhraseElement;
createElement(tagName:"col"): HTMLTableColElement;
createElement(tagName:"colgroup"): HTMLTableColElement;
createElement(tagName:"datalist"): HTMLDataListElement;
createElement(tagName:"dd"): HTMLDDElement;
createElement(tagName:"del"): HTMLModElement;
createElement(tagName:"dfn"): HTMLPhraseElement;
createElement(tagName:"dir"): HTMLDirectoryElement;
createElement(tagName:"div"): HTMLDivElement;
createElement(tagName:"dl"): HTMLDListElement;
createElement(tagName:"dt"): HTMLDTElement;
createElement(tagName:"em"): HTMLPhraseElement;
createElement(tagName:"embed"): HTMLEmbedElement;
createElement(tagName:"fieldset"): HTMLFieldSetElement;
createElement(tagName:"figcaption"): HTMLElement;
createElement(tagName:"figure"): HTMLElement;
createElement(tagName:"font"): HTMLFontElement;
createElement(tagName:"footer"): HTMLElement;
createElement(tagName:"form"): HTMLFormElement;
createElement(tagName:"frame"): HTMLFrameElement;
createElement(tagName:"frameset"): HTMLFrameSetElement;
createElement(tagName:"h1"): HTMLHeadingElement;
createElement(tagName:"h2"): HTMLHeadingElement;
createElement(tagName:"h3"): HTMLHeadingElement;
createElement(tagName:"h4"): HTMLHeadingElement;
createElement(tagName:"h5"): HTMLHeadingElement;
createElement(tagName:"h6"): HTMLHeadingElement;
createElement(tagName:"head"): HTMLHeadElement;
createElement(tagName:"header"): HTMLElement;
createElement(tagName:"hgroup"): HTMLElement;
createElement(tagName:"hr"): HTMLHRElement;
createElement(tagName:"html"): HTMLHtmlElement;
createElement(tagName:"i"): HTMLPhraseElement;
createElement(tagName:"iframe"): HTMLIFrameElement;
createElement(tagName:"img"): HTMLImageElement;
createElement(tagName:"input"): HTMLInputElement;
createElement(tagName:"ins"): HTMLModElement;
createElement(tagName:"isindex"): HTMLIsIndexElement;
createElement(tagName:"kbd"): HTMLPhraseElement;
createElement(tagName:"keygen"): HTMLBlockElement;
createElement(tagName:"label"): HTMLLabelElement;
createElement(tagName:"legend"): HTMLLegendElement;
createElement(tagName:"li"): HTMLLIElement;
createElement(tagName:"link"): HTMLLinkElement;
createElement(tagName:"listing"): HTMLBlockElement;
createElement(tagName:"map"): HTMLMapElement;
createElement(tagName:"mark"): HTMLElement;
createElement(tagName:"marquee"): HTMLMarqueeElement;
createElement(tagName:"menu"): HTMLMenuElement;
createElement(tagName:"meta"): HTMLMetaElement;
createElement(tagName:"nav"): HTMLElement;
createElement(tagName:"nextid"): HTMLNextIdElement;
createElement(tagName:"nobr"): HTMLPhraseElement;
createElement(tagName:"noframes"): HTMLElement;
createElement(tagName:"noscript"): HTMLElement;
createElement(tagName:"object"): HTMLObjectElement;
createElement(tagName:"ol"): HTMLOListElement;
createElement(tagName:"optgroup"): HTMLOptGroupElement;
createElement(tagName:"option"): HTMLOptionElement;
createElement(tagName:"p"): HTMLParagraphElement;
createElement(tagName:"param"): HTMLParamElement;
createElement(tagName:"plaintext"): HTMLBlockElement;
createElement(tagName:"pre"): HTMLPreElement;
createElement(tagName:"progress"): HTMLProgressElement;
createElement(tagName:"q"): HTMLQuoteElement;
createElement(tagName:"rt"): HTMLPhraseElement;
createElement(tagName:"ruby"): HTMLPhraseElement;
createElement(tagName:"s"): HTMLPhraseElement;
createElement(tagName:"samp"): HTMLPhraseElement;
createElement(tagName:"script"): HTMLScriptElement;
createElement(tagName:"section"): HTMLElement;
createElement(tagName:"select"): HTMLSelectElement;
createElement(tagName:"small"): HTMLPhraseElement;
createElement(tagName:"SOURCE"): HTMLSourceElement;
createElement(tagName:"span"): HTMLSpanElement;
createElement(tagName:"strike"): HTMLPhraseElement;
createElement(tagName:"strong"): HTMLPhraseElement;
createElement(tagName:"style"): HTMLStyleElement;
createElement(tagName:"sub"): HTMLPhraseElement;
createElement(tagName:"sup"): HTMLPhraseElement;
createElement(tagName:"table"): HTMLTableElement;
createElement(tagName:"tbody"): HTMLTableSectionElement;
createElement(tagName:"td"): HTMLTableDataCellElement;
createElement(tagName:"textarea"): HTMLTextAreaElement;
createElement(tagName:"tfoot"): HTMLTableSectionElement;
createElement(tagName:"th"): HTMLTableHeaderCellElement;
createElement(tagName:"thead"): HTMLTableSectionElement;
createElement(tagName:"title"): HTMLTitleElement;
createElement(tagName:"tr"): HTMLTableRowElement;
createElement(tagName:"track"): HTMLTrackElement;
createElement(tagName:"tt"): HTMLPhraseElement;
createElement(tagName:"u"): HTMLPhraseElement;
createElement(tagName:"ul"): HTMLUListElement;
createElement(tagName:"var"): HTMLPhraseElement;
createElement(tagName:"video"): HTMLVideoElement;
createElement(tagName:"wbr"): HTMLElement;
createElement(tagName:"x-ms-webview"): MSHTMLWebViewElement;
createElement(tagName:"xmp"): HTMLBlockElement;
createElement(tagName:string): HTMLElement;
/**
* Removes mouse capture from the object in the current document.
*/
releaseCapture(): void;
/**
* Writes one or more HTML expressions, followed by a carriage return, to a document in the specified window.
* @param content The text and HTML tags to write.
*/
writeln(...content:string[]): void;
createElementNS(namespaceURI:string, qualifiedName:string): Element;
/**
* Opens a new window and loads a document specified by a given URL. Also, opens a new window that uses the url parameter and the name parameter to collect the output of the write method and the writeln method.
* @param url Specifies a MIME type for the document.
* @param name Specifies the name of the window. This name is used as the value for the TARGET attribute on a form or an anchor element.
* @param features Contains a list of items separated by commas. Each item consists of an option and a value, separated by an equals sign (for example, "fullscreen=yes, toolbar=yes"). The following values are supported.
* @param replace Specifies whether the existing entry for the document is replaced in the history list.
*/
open(url?:string, name?:string, features?:string, replace?:boolean): any;
/**
* Returns a Boolean value that indicates whether the current command is supported on the current range.
* @param commandId Specifies a command identifier.
*/
queryCommandSupported(commandId:string): boolean;
/**
* Creates a TreeWalker object that you can use to traverse filtered lists of nodes or elements in a document.
* @param root The root element or node to start traversing on.
* @param whatToShow The type of nodes or elements to appear in the node list. For more information, see whatToShow.
* @param filter A custom NodeFilter function to use.
* @param entityReferenceExpansion A flag that specifies whether entity reference nodes are expanded.
*/
createTreeWalker(root:Node, whatToShow:number, filter:NodeFilter, entityReferenceExpansion:boolean): TreeWalker;
createAttributeNS(namespaceURI:string, qualifiedName:string): Attr;
/**
* Returns a Boolean value that indicates whether a specified command can be successfully executed using execCommand, given the current state of the document.
* @param commandId Specifies a command identifier.
*/
queryCommandEnabled(commandId:string): boolean;
/**
* Causes the element to receive the focus and executes the code specified by the onfocus event.
*/
focus(): void;
/**
* Closes an output stream and forces the sent data to display.
*/
close(): void;
getElementsByClassName(classNames:string): NodeList;
importNode(importedNode:Node, deep:boolean): Node;
/**
* Returns an empty range object that has both of its boundary points positioned at the beginning of the document.
*/
createRange(): Range;
/**
* Fires a specified event on the object.
* @param eventName Specifies the name of the event to fire.
* @param eventObj Object that specifies the event object from which to obtain event object properties.
*/
fireEvent(eventName:string, eventObj?:any): boolean;
/**
* Creates a comment object with the specified data.
* @param data Sets the comment object's data.
*/
createComment(data:string): Comment;
/**
* Retrieves a collection of objects based on the specified element name.
* @param name Specifies the name of an element.
*/
getElementsByTagName(name:"a"): NodeListOf<HTMLAnchorElement>;
getElementsByTagName(name:"abbr"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"acronym"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"address"): NodeListOf<HTMLBlockElement>;
getElementsByTagName(name:"applet"): NodeListOf<HTMLAppletElement>;
getElementsByTagName(name:"area"): NodeListOf<HTMLAreaElement>;
getElementsByTagName(name:"article"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"aside"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"audio"): NodeListOf<HTMLAudioElement>;
getElementsByTagName(name:"b"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"base"): NodeListOf<HTMLBaseElement>;
getElementsByTagName(name:"basefont"): NodeListOf<HTMLBaseFontElement>;
getElementsByTagName(name:"bdo"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"bgsound"): NodeListOf<HTMLBGSoundElement>;
getElementsByTagName(name:"big"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"blockquote"): NodeListOf<HTMLBlockElement>;
getElementsByTagName(name:"body"): NodeListOf<HTMLBodyElement>;
getElementsByTagName(name:"br"): NodeListOf<HTMLBRElement>;
getElementsByTagName(name:"button"): NodeListOf<HTMLButtonElement>;
getElementsByTagName(name:"canvas"): NodeListOf<HTMLCanvasElement>;
getElementsByTagName(name:"caption"): NodeListOf<HTMLTableCaptionElement>;
getElementsByTagName(name:"center"): NodeListOf<HTMLBlockElement>;
getElementsByTagName(name:"cite"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"code"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"col"): NodeListOf<HTMLTableColElement>;
getElementsByTagName(name:"colgroup"): NodeListOf<HTMLTableColElement>;
getElementsByTagName(name:"datalist"): NodeListOf<HTMLDataListElement>;
getElementsByTagName(name:"dd"): NodeListOf<HTMLDDElement>;
getElementsByTagName(name:"del"): NodeListOf<HTMLModElement>;
getElementsByTagName(name:"dfn"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"dir"): NodeListOf<HTMLDirectoryElement>;
getElementsByTagName(name:"div"): NodeListOf<HTMLDivElement>;
getElementsByTagName(name:"dl"): NodeListOf<HTMLDListElement>;
getElementsByTagName(name:"dt"): NodeListOf<HTMLDTElement>;
getElementsByTagName(name:"em"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"embed"): NodeListOf<HTMLEmbedElement>;
getElementsByTagName(name:"fieldset"): NodeListOf<HTMLFieldSetElement>;
getElementsByTagName(name:"figcaption"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"figure"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"font"): NodeListOf<HTMLFontElement>;
getElementsByTagName(name:"footer"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"form"): NodeListOf<HTMLFormElement>;
getElementsByTagName(name:"frame"): NodeListOf<HTMLFrameElement>;
getElementsByTagName(name:"frameset"): NodeListOf<HTMLFrameSetElement>;
getElementsByTagName(name:"h1"): NodeListOf<HTMLHeadingElement>;
getElementsByTagName(name:"h2"): NodeListOf<HTMLHeadingElement>;
getElementsByTagName(name:"h3"): NodeListOf<HTMLHeadingElement>;
getElementsByTagName(name:"h4"): NodeListOf<HTMLHeadingElement>;
getElementsByTagName(name:"h5"): NodeListOf<HTMLHeadingElement>;
getElementsByTagName(name:"h6"): NodeListOf<HTMLHeadingElement>;
getElementsByTagName(name:"head"): NodeListOf<HTMLHeadElement>;
getElementsByTagName(name:"header"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"hgroup"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"hr"): NodeListOf<HTMLHRElement>;
getElementsByTagName(name:"html"): NodeListOf<HTMLHtmlElement>;
getElementsByTagName(name:"i"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"iframe"): NodeListOf<HTMLIFrameElement>;
getElementsByTagName(name:"img"): NodeListOf<HTMLImageElement>;
getElementsByTagName(name:"input"): NodeListOf<HTMLInputElement>;
getElementsByTagName(name:"ins"): NodeListOf<HTMLModElement>;
getElementsByTagName(name:"isindex"): NodeListOf<HTMLIsIndexElement>;
getElementsByTagName(name:"kbd"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"keygen"): NodeListOf<HTMLBlockElement>;
getElementsByTagName(name:"label"): NodeListOf<HTMLLabelElement>;
getElementsByTagName(name:"legend"): NodeListOf<HTMLLegendElement>;
getElementsByTagName(name:"li"): NodeListOf<HTMLLIElement>;
getElementsByTagName(name:"link"): NodeListOf<HTMLLinkElement>;
getElementsByTagName(name:"listing"): NodeListOf<HTMLBlockElement>;
getElementsByTagName(name:"map"): NodeListOf<HTMLMapElement>;
getElementsByTagName(name:"mark"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"marquee"): NodeListOf<HTMLMarqueeElement>;
getElementsByTagName(name:"menu"): NodeListOf<HTMLMenuElement>;
getElementsByTagName(name:"meta"): NodeListOf<HTMLMetaElement>;
getElementsByTagName(name:"nav"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"nextid"): NodeListOf<HTMLNextIdElement>;
getElementsByTagName(name:"nobr"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"noframes"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"noscript"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"object"): NodeListOf<HTMLObjectElement>;
getElementsByTagName(name:"ol"): NodeListOf<HTMLOListElement>;
getElementsByTagName(name:"optgroup"): NodeListOf<HTMLOptGroupElement>;
getElementsByTagName(name:"option"): NodeListOf<HTMLOptionElement>;
getElementsByTagName(name:"p"): NodeListOf<HTMLParagraphElement>;
getElementsByTagName(name:"param"): NodeListOf<HTMLParamElement>;
getElementsByTagName(name:"plaintext"): NodeListOf<HTMLBlockElement>;
getElementsByTagName(name:"pre"): NodeListOf<HTMLPreElement>;
getElementsByTagName(name:"progress"): NodeListOf<HTMLProgressElement>;
getElementsByTagName(name:"q"): NodeListOf<HTMLQuoteElement>;
getElementsByTagName(name:"rt"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"ruby"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"s"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"samp"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"script"): NodeListOf<HTMLScriptElement>;
getElementsByTagName(name:"section"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"select"): NodeListOf<HTMLSelectElement>;
getElementsByTagName(name:"small"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"SOURCE"): NodeListOf<HTMLSourceElement>;
getElementsByTagName(name:"span"): NodeListOf<HTMLSpanElement>;
getElementsByTagName(name:"strike"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"strong"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"style"): NodeListOf<HTMLStyleElement>;
getElementsByTagName(name:"sub"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"sup"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"table"): NodeListOf<HTMLTableElement>;
getElementsByTagName(name:"tbody"): NodeListOf<HTMLTableSectionElement>;
getElementsByTagName(name:"td"): NodeListOf<HTMLTableDataCellElement>;
getElementsByTagName(name:"textarea"): NodeListOf<HTMLTextAreaElement>;
getElementsByTagName(name:"tfoot"): NodeListOf<HTMLTableSectionElement>;
getElementsByTagName(name:"th"): NodeListOf<HTMLTableHeaderCellElement>;
getElementsByTagName(name:"thead"): NodeListOf<HTMLTableSectionElement>;
getElementsByTagName(name:"title"): NodeListOf<HTMLTitleElement>;
getElementsByTagName(name:"tr"): NodeListOf<HTMLTableRowElement>;
getElementsByTagName(name:"track"): NodeListOf<HTMLTrackElement>;
getElementsByTagName(name:"tt"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"u"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"ul"): NodeListOf<HTMLUListElement>;
getElementsByTagName(name:"var"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"video"): NodeListOf<HTMLVideoElement>;
getElementsByTagName(name:"wbr"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"x-ms-webview"): NodeListOf<MSHTMLWebViewElement>;
getElementsByTagName(name:"xmp"): NodeListOf<HTMLBlockElement>;
getElementsByTagName(name:string): NodeList;
/**
* Creates a new document.
*/
createDocumentFragment(): DocumentFragment;
/**
* Creates a style sheet for the document.
* @param href Specifies how to add the style sheet to the document. If a file name is specified for the URL, the style information is added as a link object. If the URL contains style information, it is added to the style object.
* @param index Specifies the index that indicates where the new style sheet is inserted in the styleSheets collection. The default is to insert the new style sheet at the end of the collection.
*/
createStyleSheet(href?:string, index?:number): CSSStyleSheet;
/**
* Gets a collection of objects based on the value of the NAME or ID attribute.
* @param elementName Gets a collection of objects based on the value of the NAME or ID attribute.
*/
getElementsByName(elementName:string): NodeList;
/**
* Returns a Boolean value that indicates the current state of the command.
* @param commandId String that specifies a command identifier.
*/
queryCommandState(commandId:string): boolean;
/**
* Gets a value indicating whether the object currently has focus.
*/
hasFocus(): boolean;
/**
* Displays help information for the given command identifier.
* @param commandId Displays help information for the given command identifier.
*/
execCommandShowHelp(commandId:string): boolean;
/**
* Creates an attribute object with a specified name.
* @param name String that sets the attribute object's name.
*/
createAttribute(name:string): Attr;
/**
* Creates a text string from the specified value.
* @param data String that specifies the nodeValue property of the text node.
*/
createTextNode(data:string): Text;
/**
* Creates a NodeIterator object that you can use to traverse filtered lists of nodes or elements in a document.
* @param root The root element or node to start traversing on.
* @param whatToShow The type of nodes or elements to appear in the node list
* @param filter A custom NodeFilter function to use. For more information, see filter. Use null for no filter.
* @param entityReferenceExpansion A flag that specifies whether entity reference nodes are expanded.
*/
createNodeIterator(root:Node, whatToShow:number, filter:NodeFilter, entityReferenceExpansion:boolean): NodeIterator;
/**
* Generates an event object to pass event context information when you use the fireEvent method.
* @param eventObj An object that specifies an existing event object on which to base the new object.
*/
createEventObject(eventObj?:any): MSEventObj;
/**
* Returns an object representing the current selection of the document that is loaded into the object displaying a webpage.
*/
getSelection(): Selection;
msElementsFromPoint(x:number, y:number): NodeList;
msElementsFromRect(left:number, top:number, width:number, height:number): NodeList;
clear(): void;
msExitFullscreen(): void;
addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"keydown", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"keyup", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"reset", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"help", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"dragleave", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusin", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"seeked", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"durationchange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"blur", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"emptied", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"seeking", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"deactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"canplay", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"datasetchanged", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsdelete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"loadstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"controlselect", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragenter", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"submit", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"change", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"canplaythrough", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"datasetcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"suspend", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"errorupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"mouseout", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"msthumbnailclick", listener:(ev:MSSiteModeEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mousewheel", listener:(ev:MouseWheelEvent) => any, useCapture?:boolean): void;
addEventListener(type:"volumechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"cellchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowexit", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsinserted", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"propertychange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragend", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragover", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragstart", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseup", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drag", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseover", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pause", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousedown", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"click", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"waiting", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"stop", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mssitemodejumplistitemremoved", listener:(ev:MSSiteModeEvent) => any, useCapture?:boolean): void;
addEventListener(type:"stalled", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousemove", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforeeditfocus", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"ratechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dblclick", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"contextmenu", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadedmetadata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"play", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"afterupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"playing", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusout", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"selectionchange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"storagecommit", listener:(ev:StorageEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dataavailable", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"readystatechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"keypress", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadeddata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforedeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"activate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"selectstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focus", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"timeupdate", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"select", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drop", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"ended", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"scroll", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"rowenter", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"input", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerdown", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturedoubletap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msmanipulationstatechanged", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerhover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mscontentzoom", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"mspointermove", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturehold", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturechange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturestart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointercancel", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgestureend", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturetap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerout", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msinertiastart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerup", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msfullscreenerror", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerenter", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msfullscreenchange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerleave", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var Document:{
prototype: Document;
new(): Document;
}
interface Console {
info(message?:any, ...optionalParams:any[]): void;
warn(message?:any, ...optionalParams:any[]): void;
error(message?:any, ...optionalParams:any[]): void;
log(message?:any, ...optionalParams:any[]): void;
profile(reportName?:string): void;
assert(test?:boolean, message?:string, ...optionalParams:any[]): void;
msIsIndependentlyComposed(element:Element): boolean;
clear(): void;
dir(value?:any, ...optionalParams:any[]): void;
profileEnd(): void;
count(countTitle?:string): void;
groupEnd(): void;
time(timerName?:string): void;
timeEnd(timerName?:string): void;
trace(): void;
group(groupTitle?:string): void;
dirxml(value:any): void;
debug(message?:string, ...optionalParams:any[]): void;
groupCollapsed(groupTitle?:string): void;
select(element:Element): void;
}
declare var Console:{
prototype: Console;
new(): Console;
}
interface MSEventObj extends Event {
nextPage: string;
keyCode: number;
toElement: Element;
returnValue: any;
dataFld: string;
y: number;
dataTransfer: DataTransfer;
propertyName: string;
url: string;
offsetX: number;
recordset: any;
screenX: number;
buttonID: number;
wheelDelta: number;
reason: number;
origin: string;
data: string;
srcFilter: any;
boundElements: HTMLCollection;
cancelBubble: boolean;
altLeft: boolean;
behaviorCookie: number;
bookmarks: BookmarkCollection;
type: string;
repeat: boolean;
srcElement: Element;
source: Window;
fromElement: Element;
offsetY: number;
x: number;
behaviorPart: number;
qualifier: string;
altKey: boolean;
ctrlKey: boolean;
clientY: number;
shiftKey: boolean;
shiftLeft: boolean;
contentOverflow: boolean;
screenY: number;
ctrlLeft: boolean;
button: number;
srcUrn: string;
clientX: number;
actionURL: string;
getAttribute(strAttributeName:string, lFlags?:number): any;
setAttribute(strAttributeName:string, AttributeValue:any, lFlags?:number): void;
removeAttribute(strAttributeName:string, lFlags?:number): boolean;
}
declare var MSEventObj:{
prototype: MSEventObj;
new(): MSEventObj;
}
interface HTMLCanvasElement extends HTMLElement {
/**
* Gets or sets the width of a canvas element on a document.
*/
width: number;
/**
* Gets or sets the height of a canvas element on a document.
*/
height: number;
/**
* Returns an object that provides methods and properties for drawing and manipulating images and graphics on a canvas element in a document. A context object includes information about colors, line widths, fonts, and other graphic parameters that can be drawn on a canvas.
* @param contextId The identifier (ID) of the type of canvas to create. Internet Explorer 9 and Internet Explorer 10 support only a 2-D context using canvas.getContext("2d"); IE11 Preview also supports 3-D or WebGL context using canvas.getContext("experimental-webgl");
*/
getContext(contextId:"2d"): CanvasRenderingContext2D;
/**
* Returns an object that provides methods and properties for drawing and manipulating images and graphics on a canvas element in a document. A context object includes information about colors, line widths, fonts, and other graphic parameters that can be drawn on a canvas.
* @param contextId The identifier (ID) of the type of canvas to create. Internet Explorer 9 and Internet Explorer 10 support only a 2-D context using canvas.getContext("2d"); IE11 Preview also supports 3-D or WebGL context using canvas.getContext("experimental-webgl");
*/
getContext(contextId:"experimental-webgl"): WebGLRenderingContext;
/**
* Returns an object that provides methods and properties for drawing and manipulating images and graphics on a canvas element in a document. A context object includes information about colors, line widths, fonts, and other graphic parameters that can be drawn on a canvas.
* @param contextId The identifier (ID) of the type of canvas to create. Internet Explorer 9 and Internet Explorer 10 support only a 2-D context using canvas.getContext("2d"); IE11 Preview also supports 3-D or WebGL context using canvas.getContext("experimental-webgl");
*/
getContext(contextId:string, ...args:any[]): any;
/**
* Returns the content of the current canvas as an image that you can use as a source for another canvas or an HTML element.
* @param type The standard MIME type for the image format to return. If you do not specify this parameter, the default value is a PNG format image.
*/
toDataURL(type?:string, ...args:any[]): string;
/**
* Returns a blob object encoded as a Portable Network Graphics (PNG) format from a canvas image or drawing.
*/
msToBlob(): Blob;
}
declare var HTMLCanvasElement:{
prototype: HTMLCanvasElement;
new(): HTMLCanvasElement;
}
interface Window extends EventTarget, MSEventAttachmentTarget, WindowLocalStorage, MSWindowExtensions, WindowSessionStorage, WindowTimers, WindowBase64, IDBEnvironment, WindowConsole, GlobalEventHandlers {
ondragend: (ev:DragEvent) => any;
onkeydown: (ev:KeyboardEvent) => any;
ondragover: (ev:DragEvent) => any;
onkeyup: (ev:KeyboardEvent) => any;
onreset: (ev:Event) => any;
onmouseup: (ev:MouseEvent) => any;
ondragstart: (ev:DragEvent) => any;
ondrag: (ev:DragEvent) => any;
screenX: number;
onmouseover: (ev:MouseEvent) => any;
ondragleave: (ev:DragEvent) => any;
history: History;
pageXOffset: number;
name: string;
onafterprint: (ev:Event) => any;
onpause: (ev:Event) => any;
onbeforeprint: (ev:Event) => any;
top: Window;
onmousedown: (ev:MouseEvent) => any;
onseeked: (ev:Event) => any;
opener: Window;
onclick: (ev:MouseEvent) => any;
innerHeight: number;
onwaiting: (ev:Event) => any;
ononline: (ev:Event) => any;
ondurationchange: (ev:Event) => any;
frames: Window;
onblur: (ev:FocusEvent) => any;
onemptied: (ev:Event) => any;
onseeking: (ev:Event) => any;
oncanplay: (ev:Event) => any;
outerWidth: number;
onstalled: (ev:Event) => any;
onmousemove: (ev:MouseEvent) => any;
innerWidth: number;
onoffline: (ev:Event) => any;
length: number;
screen: Screen;
onbeforeunload: (ev:BeforeUnloadEvent) => any;
onratechange: (ev:Event) => any;
onstorage: (ev:StorageEvent) => any;
onloadstart: (ev:Event) => any;
ondragenter: (ev:DragEvent) => any;
onsubmit: (ev:Event) => any;
self: Window;
document: Document;
onprogress: (ev:ProgressEvent) => any;
ondblclick: (ev:MouseEvent) => any;
pageYOffset: number;
oncontextmenu: (ev:MouseEvent) => any;
onchange: (ev:Event) => any;
onloadedmetadata: (ev:Event) => any;
onplay: (ev:Event) => any;
onerror: ErrorEventHandler;
onplaying: (ev:Event) => any;
parent: Window;
location: Location;
oncanplaythrough: (ev:Event) => any;
onabort: (ev:UIEvent) => any;
onreadystatechange: (ev:Event) => any;
outerHeight: number;
onkeypress: (ev:KeyboardEvent) => any;
frameElement: Element;
onloadeddata: (ev:Event) => any;
onsuspend: (ev:Event) => any;
window: Window;
onfocus: (ev:FocusEvent) => any;
onmessage: (ev:MessageEvent) => any;
ontimeupdate: (ev:Event) => any;
onresize: (ev:UIEvent) => any;
onselect: (ev:UIEvent) => any;
navigator: Navigator;
styleMedia: StyleMedia;
ondrop: (ev:DragEvent) => any;
onmouseout: (ev:MouseEvent) => any;
onended: (ev:Event) => any;
onhashchange: (ev:Event) => any;
onunload: (ev:Event) => any;
onscroll: (ev:UIEvent) => any;
screenY: number;
onmousewheel: (ev:MouseWheelEvent) => any;
onload: (ev:Event) => any;
onvolumechange: (ev:Event) => any;
oninput: (ev:Event) => any;
performance: Performance;
onmspointerdown: (ev:any) => any;
animationStartTime: number;
onmsgesturedoubletap: (ev:any) => any;
onmspointerhover: (ev:any) => any;
onmsgesturehold: (ev:any) => any;
onmspointermove: (ev:any) => any;
onmsgesturechange: (ev:any) => any;
onmsgesturestart: (ev:any) => any;
onmspointercancel: (ev:any) => any;
onmsgestureend: (ev:any) => any;
onmsgesturetap: (ev:any) => any;
onmspointerout: (ev:any) => any;
msAnimationStartTime: number;
applicationCache: ApplicationCache;
onmsinertiastart: (ev:any) => any;
onmspointerover: (ev:any) => any;
onpopstate: (ev:PopStateEvent) => any;
onmspointerup: (ev:any) => any;
onpageshow: (ev:PageTransitionEvent) => any;
ondevicemotion: (ev:DeviceMotionEvent) => any;
devicePixelRatio: number;
msCrypto: Crypto;
ondeviceorientation: (ev:DeviceOrientationEvent) => any;
doNotTrack: string;
onmspointerenter: (ev:any) => any;
onpagehide: (ev:PageTransitionEvent) => any;
onmspointerleave: (ev:any) => any;
alert(message?:any): void;
scroll(x?:number, y?:number): void;
focus(): void;
scrollTo(x?:number, y?:number): void;
print(): void;
prompt(message?:string, _default?:string): string;
toString(): string;
open(url?:string, target?:string, features?:string, replace?:boolean): Window;
scrollBy(x?:number, y?:number): void;
confirm(message?:string): boolean;
close(): void;
postMessage(message:any, targetOrigin:string, ports?:any): void;
showModalDialog(url?:string, argument?:any, options?:any): any;
blur(): void;
getSelection(): Selection;
getComputedStyle(elt:Element, pseudoElt?:string): CSSStyleDeclaration;
msCancelRequestAnimationFrame(handle:number): void;
matchMedia(mediaQuery:string): MediaQueryList;
cancelAnimationFrame(handle:number): void;
msIsStaticHTML(html:string): boolean;
msMatchMedia(mediaQuery:string): MediaQueryList;
requestAnimationFrame(callback:FrameRequestCallback): number;
msRequestAnimationFrame(callback:FrameRequestCallback): number;
addEventListener(type:"mouseleave", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseenter", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"help", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focusout", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusin", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragend", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"keydown", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragover", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"keyup", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"reset", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mouseup", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragstart", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drag", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseover", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragleave", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"afterprint", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"pause", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeprint", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousedown", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"seeked", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"click", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"waiting", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"online", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"durationchange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"blur", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"emptied", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"seeking", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"canplay", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"stalled", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousemove", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"offline", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeunload", listener:(ev:BeforeUnloadEvent) => any, useCapture?:boolean): void;
addEventListener(type:"ratechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"storage", listener:(ev:StorageEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"dragenter", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"submit", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dblclick", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"contextmenu", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"change", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"loadedmetadata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"play", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"playing", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"canplaythrough", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"readystatechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"keypress", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadeddata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"suspend", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focus", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"message", listener:(ev:MessageEvent) => any, useCapture?:boolean): void;
addEventListener(type:"timeupdate", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resize", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"select", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drop", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseout", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"ended", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"hashchange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"unload", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"scroll", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mousewheel", listener:(ev:MouseWheelEvent) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"volumechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"input", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerdown", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturedoubletap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerhover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturehold", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointermove", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturechange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturestart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointercancel", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgestureend", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturetap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerout", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msinertiastart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"popstate", listener:(ev:PopStateEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerup", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"pageshow", listener:(ev:PageTransitionEvent) => any, useCapture?:boolean): void;
addEventListener(type:"devicemotion", listener:(ev:DeviceMotionEvent) => any, useCapture?:boolean): void;
addEventListener(type:"deviceorientation", listener:(ev:DeviceOrientationEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerenter", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"pagehide", listener:(ev:PageTransitionEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerleave", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var Window:{
prototype: Window;
new(): Window;
}
interface HTMLCollection extends MSHTMLCollectionExtensions {
/**
* Sets or retrieves the number of objects in a collection.
*/
length: number;
/**
* Retrieves an object from various collections.
*/
item(nameOrIndex?:any, optionalIndex?:any): Element;
/**
* Retrieves a select object or an object from an options collection.
*/
namedItem(name:string): Element;
// [name: string]: Element;
[index: number]: Element;
}
declare var HTMLCollection:{
prototype: HTMLCollection;
new(): HTMLCollection;
}
interface BlobPropertyBag {
type?: string;
endings?: string;
}
interface Blob {
type: string;
size: number;
msDetachStream(): any;
slice(start?:number, end?:number, contentType?:string): Blob;
msClose(): void;
}
declare var Blob:{
prototype: Blob;
new (blobParts?:any[], options?:BlobPropertyBag): Blob;
}
interface NavigatorID {
appVersion: string;
appName: string;
userAgent: string;
platform: string;
product: string;
vendor: string;
}
interface HTMLTableElement extends HTMLElement, MSDataBindingTableExtensions, MSDataBindingExtensions, DOML2DeprecatedBackgroundStyle, DOML2DeprecatedBackgroundColorStyle {
/**
* Sets or retrieves the width of the object.
*/
width: string;
/**
* Sets or retrieves the color for one of the two colors used to draw the 3-D border of the object.
*/
borderColorLight: any;
/**
* Sets or retrieves the amount of space between cells in a table.
*/
cellSpacing: string;
/**
* Retrieves the tFoot object of the table.
*/
tFoot: HTMLTableSectionElement;
/**
* Sets or retrieves the way the border frame around the table is displayed.
*/
frame: string;
/**
* Sets or retrieves the border color of the object.
*/
borderColor: any;
/**
* Sets or retrieves the number of horizontal rows contained in the object.
*/
rows: HTMLCollection;
/**
* Sets or retrieves which dividing lines (inner borders) are displayed.
*/
rules: string;
/**
* Sets or retrieves the number of columns in the table.
*/
cols: number;
/**
* Sets or retrieves a description and/or structure of the object.
*/
summary: string;
/**
* Retrieves the caption object of a table.
*/
caption: HTMLTableCaptionElement;
/**
* Retrieves a collection of all tBody objects in the table. Objects in this collection are in source order.
*/
tBodies: HTMLCollection;
/**
* Retrieves the tHead object of the table.
*/
tHead: HTMLTableSectionElement;
/**
* Sets or retrieves a value that indicates the table alignment.
*/
align: string;
/**
* Retrieves a collection of all cells in the table row or in the entire table.
*/
cells: HTMLCollection;
/**
* Sets or retrieves the height of the object.
*/
height: any;
/**
* Sets or retrieves the amount of space between the border of the cell and the content of the cell.
*/
cellPadding: string;
/**
* Sets or retrieves the width of the border to draw around the object.
*/
border: string;
/**
* Sets or retrieves the color for one of the two colors used to draw the 3-D border of the object.
*/
borderColorDark: any;
/**
* Removes the specified row (tr) from the element and from the rows collection.
* @param index Number that specifies the zero-based position in the rows collection of the row to remove.
*/
deleteRow(index?:number): void;
/**
* Creates an empty tBody element in the table.
*/
createTBody(): HTMLElement;
/**
* Deletes the caption element and its contents from the table.
*/
deleteCaption(): void;
/**
* Creates a new row (tr) in the table, and adds the row to the rows collection.
* @param index Number that specifies where to insert the row in the rows collection. The default value is -1, which appends the new row to the end of the rows collection.
*/
insertRow(index?:number): HTMLElement;
/**
* Deletes the tFoot element and its contents from the table.
*/
deleteTFoot(): void;
/**
* Returns the tHead element object if successful, or null otherwise.
*/
createTHead(): HTMLElement;
/**
* Deletes the tHead element and its contents from the table.
*/
deleteTHead(): void;
/**
* Creates an empty caption element in the table.
*/
createCaption(): HTMLElement;
/**
* Moves a table row to a new position.
* @param indexFrom Number that specifies the index in the rows collection of the table row that is moved.
* @param indexTo Number that specifies where the row is moved within the rows collection.
*/
moveRow(indexFrom?:number, indexTo?:number): any;
/**
* Creates an empty tFoot element in the table.
*/
createTFoot(): HTMLElement;
}
declare var HTMLTableElement:{
prototype: HTMLTableElement;
new(): HTMLTableElement;
}
interface TreeWalker {
whatToShow: number;
filter: NodeFilter;
root: Node;
currentNode: Node;
expandEntityReferences: boolean;
previousSibling(): Node;
lastChild(): Node;
nextSibling(): Node;
nextNode(): Node;
parentNode(): Node;
firstChild(): Node;
previousNode(): Node;
}
declare var TreeWalker:{
prototype: TreeWalker;
new(): TreeWalker;
}
interface GetSVGDocument {
getSVGDocument(): Document;
}
interface SVGPathSegCurvetoQuadraticRel extends SVGPathSeg {
y: number;
y1: number;
x: number;
x1: number;
}
declare var SVGPathSegCurvetoQuadraticRel:{
prototype: SVGPathSegCurvetoQuadraticRel;
new(): SVGPathSegCurvetoQuadraticRel;
}
interface Performance {
navigation: PerformanceNavigation;
timing: PerformanceTiming;
getEntriesByType(entryType:string): any;
toJSON(): any;
getMeasures(measureName?:string): any;
clearMarks(markName?:string): void;
getMarks(markName?:string): any;
clearResourceTimings(): void;
mark(markName:string): void;
measure(measureName:string, startMarkName?:string, endMarkName?:string): void;
getEntriesByName(name:string, entryType?:string): any;
getEntries(): any;
clearMeasures(measureName?:string): void;
setResourceTimingBufferSize(maxSize:number): void;
now(): number;
}
declare var Performance:{
prototype: Performance;
new(): Performance;
}
interface MSDataBindingTableExtensions {
dataPageSize: number;
nextPage(): void;
firstPage(): void;
refresh(): void;
previousPage(): void;
lastPage(): void;
}
interface CompositionEvent extends UIEvent {
data: string;
locale: string;
initCompositionEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, viewArg:Window, dataArg:string, locale:string): void;
}
declare var CompositionEvent:{
prototype: CompositionEvent;
new(): CompositionEvent;
}
interface WindowTimers extends WindowTimersExtension {
clearTimeout(handle:number): void;
setTimeout(handler:any, timeout?:any, ...args:any[]): number;
clearInterval(handle:number): void;
setInterval(handler:any, timeout?:any, ...args:any[]): number;
}
interface SVGMarkerElement extends SVGElement, SVGStylable, SVGLangSpace, SVGFitToViewBox, SVGExternalResourcesRequired {
orientType: SVGAnimatedEnumeration;
markerUnits: SVGAnimatedEnumeration;
markerWidth: SVGAnimatedLength;
markerHeight: SVGAnimatedLength;
orientAngle: SVGAnimatedAngle;
refY: SVGAnimatedLength;
refX: SVGAnimatedLength;
setOrientToAngle(angle:SVGAngle): void;
setOrientToAuto(): void;
SVG_MARKER_ORIENT_UNKNOWN: number;
SVG_MARKER_ORIENT_ANGLE: number;
SVG_MARKERUNITS_UNKNOWN: number;
SVG_MARKERUNITS_STROKEWIDTH: number;
SVG_MARKER_ORIENT_AUTO: number;
SVG_MARKERUNITS_USERSPACEONUSE: number;
}
declare var SVGMarkerElement:{
prototype: SVGMarkerElement;
new(): SVGMarkerElement;
SVG_MARKER_ORIENT_UNKNOWN: number;
SVG_MARKER_ORIENT_ANGLE: number;
SVG_MARKERUNITS_UNKNOWN: number;
SVG_MARKERUNITS_STROKEWIDTH: number;
SVG_MARKER_ORIENT_AUTO: number;
SVG_MARKERUNITS_USERSPACEONUSE: number;
}
interface CSSStyleDeclaration {
backgroundAttachment: string;
visibility: string;
textAlignLast: string;
borderRightStyle: string;
counterIncrement: string;
orphans: string;
cssText: string;
borderStyle: string;
pointerEvents: string;
borderTopColor: string;
markerEnd: string;
textIndent: string;
listStyleImage: string;
cursor: string;
listStylePosition: string;
wordWrap: string;
borderTopStyle: string;
alignmentBaseline: string;
opacity: string;
direction: string;
strokeMiterlimit: string;
maxWidth: string;
color: string;
clip: string;
borderRightWidth: string;
verticalAlign: string;
overflow: string;
mask: string;
borderLeftStyle: string;
emptyCells: string;
stopOpacity: string;
paddingRight: string;
parentRule: CSSRule;
background: string;
boxSizing: string;
textJustify: string;
height: string;
paddingTop: string;
length: number;
right: string;
baselineShift: string;
borderLeft: string;
widows: string;
lineHeight: string;
left: string;
textUnderlinePosition: string;
glyphOrientationHorizontal: string;
display: string;
textAnchor: string;
cssFloat: string;
strokeDasharray: string;
rubyAlign: string;
fontSizeAdjust: string;
borderLeftColor: string;
backgroundImage: string;
listStyleType: string;
strokeWidth: string;
textOverflow: string;
fillRule: string;
borderBottomColor: string;
zIndex: string;
position: string;
listStyle: string;
msTransformOrigin: string;
dominantBaseline: string;
overflowY: string;
fill: string;
captionSide: string;
borderCollapse: string;
boxShadow: string;
quotes: string;
tableLayout: string;
unicodeBidi: string;
borderBottomWidth: string;
backgroundSize: string;
textDecoration: string;
strokeDashoffset: string;
fontSize: string;
border: string;
pageBreakBefore: string;
borderTopRightRadius: string;
msTransform: string;
borderBottomLeftRadius: string;
textTransform: string;
rubyPosition: string;
strokeLinejoin: string;
clipPath: string;
borderRightColor: string;
fontFamily: string;
clear: string;
content: string;
backgroundClip: string;
marginBottom: string;
counterReset: string;
outlineWidth: string;
marginRight: string;
paddingLeft: string;
borderBottom: string;
wordBreak: string;
marginTop: string;
top: string;
fontWeight: string;
borderRight: string;
width: string;
kerning: string;
pageBreakAfter: string;
borderBottomStyle: string;
fontStretch: string;
padding: string;
strokeOpacity: string;
markerStart: string;
bottom: string;
borderLeftWidth: string;
clipRule: string;
backgroundPosition: string;
backgroundColor: string;
pageBreakInside: string;
backgroundOrigin: string;
strokeLinecap: string;
borderTopWidth: string;
outlineStyle: string;
borderTop: string;
outlineColor: string;
paddingBottom: string;
marginLeft: string;
font: string;
outline: string;
wordSpacing: string;
maxHeight: string;
fillOpacity: string;
letterSpacing: string;
borderSpacing: string;
backgroundRepeat: string;
borderRadius: string;
borderWidth: string;
borderBottomRightRadius: string;
whiteSpace: string;
fontStyle: string;
minWidth: string;
stopColor: string;
borderTopLeftRadius: string;
borderColor: string;
marker: string;
glyphOrientationVertical: string;
markerMid: string;
fontVariant: string;
minHeight: string;
stroke: string;
rubyOverhang: string;
overflowX: string;
textAlign: string;
margin: string;
animationFillMode: string;
floodColor: string;
animationIterationCount: string;
textShadow: string;
backfaceVisibility: string;
msAnimationIterationCount: string;
animationDelay: string;
animationTimingFunction: string;
columnWidth: any;
msScrollSnapX: string;
columnRuleColor: any;
columnRuleWidth: any;
transitionDelay: string;
transition: string;
msFlowFrom: string;
msScrollSnapType: string;
msContentZoomSnapType: string;
msGridColumns: string;
msAnimationName: string;
msGridRowAlign: string;
msContentZoomChaining: string;
msGridColumn: any;
msHyphenateLimitZone: any;
msScrollRails: string;
msAnimationDelay: string;
enableBackground: string;
msWrapThrough: string;
columnRuleStyle: string;
msAnimation: string;
msFlexFlow: string;
msScrollSnapY: string;
msHyphenateLimitLines: any;
msTouchAction: string;
msScrollLimit: string;
animation: string;
transform: string;
filter: string;
colorInterpolationFilters: string;
transitionTimingFunction: string;
msBackfaceVisibility: string;
animationPlayState: string;
transformOrigin: string;
msScrollLimitYMin: any;
msFontFeatureSettings: string;
msContentZoomLimitMin: any;
columnGap: any;
transitionProperty: string;
msAnimationDuration: string;
msAnimationFillMode: string;
msFlexDirection: string;
msTransitionDuration: string;
fontFeatureSettings: string;
breakBefore: string;
msFlexWrap: string;
perspective: string;
msFlowInto: string;
msTransformStyle: string;
msScrollTranslation: string;
msTransitionProperty: string;
msUserSelect: string;
msOverflowStyle: string;
msScrollSnapPointsY: string;
animationDirection: string;
animationDuration: string;
msFlex: string;
msTransitionTimingFunction: string;
animationName: string;
columnRule: string;
msGridColumnSpan: any;
msFlexNegative: string;
columnFill: string;
msGridRow: any;
msFlexOrder: string;
msFlexItemAlign: string;
msFlexPositive: string;
msContentZoomLimitMax: any;
msScrollLimitYMax: any;
msGridColumnAlign: string;
perspectiveOrigin: string;
lightingColor: string;
columns: string;
msScrollChaining: string;
msHyphenateLimitChars: string;
msTouchSelect: string;
floodOpacity: string;
msAnimationDirection: string;
msAnimationPlayState: string;
columnSpan: string;
msContentZooming: string;
msPerspective: string;
msFlexPack: string;
msScrollSnapPointsX: string;
msContentZoomSnapPoints: string;
msGridRowSpan: any;
msContentZoomSnap: string;
msScrollLimitXMin: any;
breakInside: string;
msHighContrastAdjust: string;
msFlexLinePack: string;
msGridRows: string;
transitionDuration: string;
msHyphens: string;
breakAfter: string;
msTransition: string;
msPerspectiveOrigin: string;
msContentZoomLimit: string;
msScrollLimitXMax: any;
msFlexAlign: string;
msWrapMargin: any;
columnCount: any;
msAnimationTimingFunction: string;
msTransitionDelay: string;
transformStyle: string;
msWrapFlow: string;
msFlexPreferredSize: string;
alignItems: string;
borderImageSource: string;
flexBasis: string;
borderImageWidth: string;
borderImageRepeat: string;
order: string;
flex: string;
alignContent: string;
msImeAlign: string;
flexShrink: string;
flexGrow: string;
borderImageSlice: string;
flexWrap: string;
borderImageOutset: string;
flexDirection: string;
touchAction: string;
flexFlow: string;
borderImage: string;
justifyContent: string;
alignSelf: string;
msTextCombineHorizontal: string;
getPropertyPriority(propertyName:string): string;
getPropertyValue(propertyName:string): string;
removeProperty(propertyName:string): string;
item(index:number): string;
[index: number]: string;
setProperty(propertyName:string, value:string, priority?:string): void;
}
declare var CSSStyleDeclaration:{
prototype: CSSStyleDeclaration;
new(): CSSStyleDeclaration;
}
interface SVGGElement extends SVGElement, SVGStylable, SVGTransformable, SVGLangSpace, SVGTests, SVGExternalResourcesRequired {
}
declare var SVGGElement:{
prototype: SVGGElement;
new(): SVGGElement;
}
interface MSStyleCSSProperties extends MSCSSProperties {
pixelWidth: number;
posHeight: number;
posLeft: number;
pixelTop: number;
pixelBottom: number;
textDecorationNone: boolean;
pixelLeft: number;
posTop: number;
posBottom: number;
textDecorationOverline: boolean;
posWidth: number;
textDecorationLineThrough: boolean;
pixelHeight: number;
textDecorationBlink: boolean;
posRight: number;
pixelRight: number;
textDecorationUnderline: boolean;
}
declare var MSStyleCSSProperties:{
prototype: MSStyleCSSProperties;
new(): MSStyleCSSProperties;
}
interface Navigator extends NavigatorID, NavigatorOnLine, NavigatorContentUtils, MSNavigatorExtensions, NavigatorGeolocation, MSNavigatorDoNotTrack, NavigatorStorageUtils, MSFileSaver {
msMaxTouchPoints: number;
msPointerEnabled: boolean;
msManipulationViewsEnabled: boolean;
pointerEnabled: boolean;
maxTouchPoints: number;
msLaunchUri(uri:string, successCallback?:MSLaunchUriCallback, noHandlerCallback?:MSLaunchUriCallback): void;
}
declare var Navigator:{
prototype: Navigator;
new(): Navigator;
}
interface SVGPathSegCurvetoCubicSmoothAbs extends SVGPathSeg {
y: number;
x2: number;
x: number;
y2: number;
}
declare var SVGPathSegCurvetoCubicSmoothAbs:{
prototype: SVGPathSegCurvetoCubicSmoothAbs;
new(): SVGPathSegCurvetoCubicSmoothAbs;
}
interface SVGZoomEvent extends UIEvent {
zoomRectScreen: SVGRect;
previousScale: number;
newScale: number;
previousTranslate: SVGPoint;
newTranslate: SVGPoint;
}
declare var SVGZoomEvent:{
prototype: SVGZoomEvent;
new(): SVGZoomEvent;
}
interface NodeSelector {
querySelectorAll(selectors:string): NodeList;
querySelector(selectors:string): Element;
}
interface HTMLTableDataCellElement extends HTMLTableCellElement {
}
declare var HTMLTableDataCellElement:{
prototype: HTMLTableDataCellElement;
new(): HTMLTableDataCellElement;
}
interface HTMLBaseElement extends HTMLElement {
/**
* Sets or retrieves the window or frame at which to target content.
*/
target: string;
/**
* Gets or sets the baseline URL on which relative links are based.
*/
href: string;
}
declare var HTMLBaseElement:{
prototype: HTMLBaseElement;
new(): HTMLBaseElement;
}
interface ClientRect {
left: number;
width: number;
right: number;
top: number;
bottom: number;
height: number;
}
declare var ClientRect:{
prototype: ClientRect;
new(): ClientRect;
}
interface PositionErrorCallback {
(error:PositionError): void;
}
interface DOMImplementation {
createDocumentType(qualifiedName:string, publicId:string, systemId:string): DocumentType;
createDocument(namespaceURI:string, qualifiedName:string, doctype:DocumentType): Document;
hasFeature(feature:string, version?:string): boolean;
createHTMLDocument(title:string): Document;
}
declare var DOMImplementation:{
prototype: DOMImplementation;
new(): DOMImplementation;
}
interface SVGUnitTypes {
SVG_UNIT_TYPE_UNKNOWN: number;
SVG_UNIT_TYPE_OBJECTBOUNDINGBOX: number;
SVG_UNIT_TYPE_USERSPACEONUSE: number;
}
declare var SVGUnitTypes:SVGUnitTypes;
interface Element extends Node, NodeSelector, ElementTraversal, GlobalEventHandlers {
scrollTop: number;
clientLeft: number;
scrollLeft: number;
tagName: string;
clientWidth: number;
scrollWidth: number;
clientHeight: number;
clientTop: number;
scrollHeight: number;
msRegionOverflow: string;
onmspointerdown: (ev:any) => any;
onmsgotpointercapture: (ev:any) => any;
onmsgesturedoubletap: (ev:any) => any;
onmspointerhover: (ev:any) => any;
onmsgesturehold: (ev:any) => any;
onmspointermove: (ev:any) => any;
onmsgesturechange: (ev:any) => any;
onmsgesturestart: (ev:any) => any;
onmspointercancel: (ev:any) => any;
onmsgestureend: (ev:any) => any;
onmsgesturetap: (ev:any) => any;
onmspointerout: (ev:any) => any;
onmsinertiastart: (ev:any) => any;
onmslostpointercapture: (ev:any) => any;
onmspointerover: (ev:any) => any;
msContentZoomFactor: number;
onmspointerup: (ev:any) => any;
onlostpointercapture: (ev:PointerEvent) => any;
onmspointerenter: (ev:any) => any;
ongotpointercapture: (ev:PointerEvent) => any;
onmspointerleave: (ev:any) => any;
getAttribute(name?:string): string;
getElementsByTagNameNS(namespaceURI:string, localName:string): NodeList;
hasAttributeNS(namespaceURI:string, localName:string): boolean;
getBoundingClientRect(): ClientRect;
getAttributeNS(namespaceURI:string, localName:string): string;
getAttributeNodeNS(namespaceURI:string, localName:string): Attr;
setAttributeNodeNS(newAttr:Attr): Attr;
msMatchesSelector(selectors:string): boolean;
hasAttribute(name:string): boolean;
removeAttribute(name?:string): void;
setAttributeNS(namespaceURI:string, qualifiedName:string, value:string): void;
getAttributeNode(name:string): Attr;
fireEvent(eventName:string, eventObj?:any): boolean;
getElementsByTagName(name:"a"): NodeListOf<HTMLAnchorElement>;
getElementsByTagName(name:"abbr"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"acronym"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"address"): NodeListOf<HTMLBlockElement>;
getElementsByTagName(name:"applet"): NodeListOf<HTMLAppletElement>;
getElementsByTagName(name:"area"): NodeListOf<HTMLAreaElement>;
getElementsByTagName(name:"article"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"aside"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"audio"): NodeListOf<HTMLAudioElement>;
getElementsByTagName(name:"b"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"base"): NodeListOf<HTMLBaseElement>;
getElementsByTagName(name:"basefont"): NodeListOf<HTMLBaseFontElement>;
getElementsByTagName(name:"bdo"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"bgsound"): NodeListOf<HTMLBGSoundElement>;
getElementsByTagName(name:"big"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"blockquote"): NodeListOf<HTMLBlockElement>;
getElementsByTagName(name:"body"): NodeListOf<HTMLBodyElement>;
getElementsByTagName(name:"br"): NodeListOf<HTMLBRElement>;
getElementsByTagName(name:"button"): NodeListOf<HTMLButtonElement>;
getElementsByTagName(name:"canvas"): NodeListOf<HTMLCanvasElement>;
getElementsByTagName(name:"caption"): NodeListOf<HTMLTableCaptionElement>;
getElementsByTagName(name:"center"): NodeListOf<HTMLBlockElement>;
getElementsByTagName(name:"cite"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"code"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"col"): NodeListOf<HTMLTableColElement>;
getElementsByTagName(name:"colgroup"): NodeListOf<HTMLTableColElement>;
getElementsByTagName(name:"datalist"): NodeListOf<HTMLDataListElement>;
getElementsByTagName(name:"dd"): NodeListOf<HTMLDDElement>;
getElementsByTagName(name:"del"): NodeListOf<HTMLModElement>;
getElementsByTagName(name:"dfn"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"dir"): NodeListOf<HTMLDirectoryElement>;
getElementsByTagName(name:"div"): NodeListOf<HTMLDivElement>;
getElementsByTagName(name:"dl"): NodeListOf<HTMLDListElement>;
getElementsByTagName(name:"dt"): NodeListOf<HTMLDTElement>;
getElementsByTagName(name:"em"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"embed"): NodeListOf<HTMLEmbedElement>;
getElementsByTagName(name:"fieldset"): NodeListOf<HTMLFieldSetElement>;
getElementsByTagName(name:"figcaption"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"figure"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"font"): NodeListOf<HTMLFontElement>;
getElementsByTagName(name:"footer"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"form"): NodeListOf<HTMLFormElement>;
getElementsByTagName(name:"frame"): NodeListOf<HTMLFrameElement>;
getElementsByTagName(name:"frameset"): NodeListOf<HTMLFrameSetElement>;
getElementsByTagName(name:"h1"): NodeListOf<HTMLHeadingElement>;
getElementsByTagName(name:"h2"): NodeListOf<HTMLHeadingElement>;
getElementsByTagName(name:"h3"): NodeListOf<HTMLHeadingElement>;
getElementsByTagName(name:"h4"): NodeListOf<HTMLHeadingElement>;
getElementsByTagName(name:"h5"): NodeListOf<HTMLHeadingElement>;
getElementsByTagName(name:"h6"): NodeListOf<HTMLHeadingElement>;
getElementsByTagName(name:"head"): NodeListOf<HTMLHeadElement>;
getElementsByTagName(name:"header"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"hgroup"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"hr"): NodeListOf<HTMLHRElement>;
getElementsByTagName(name:"html"): NodeListOf<HTMLHtmlElement>;
getElementsByTagName(name:"i"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"iframe"): NodeListOf<HTMLIFrameElement>;
getElementsByTagName(name:"img"): NodeListOf<HTMLImageElement>;
getElementsByTagName(name:"input"): NodeListOf<HTMLInputElement>;
getElementsByTagName(name:"ins"): NodeListOf<HTMLModElement>;
getElementsByTagName(name:"isindex"): NodeListOf<HTMLIsIndexElement>;
getElementsByTagName(name:"kbd"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"keygen"): NodeListOf<HTMLBlockElement>;
getElementsByTagName(name:"label"): NodeListOf<HTMLLabelElement>;
getElementsByTagName(name:"legend"): NodeListOf<HTMLLegendElement>;
getElementsByTagName(name:"li"): NodeListOf<HTMLLIElement>;
getElementsByTagName(name:"link"): NodeListOf<HTMLLinkElement>;
getElementsByTagName(name:"listing"): NodeListOf<HTMLBlockElement>;
getElementsByTagName(name:"map"): NodeListOf<HTMLMapElement>;
getElementsByTagName(name:"mark"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"marquee"): NodeListOf<HTMLMarqueeElement>;
getElementsByTagName(name:"menu"): NodeListOf<HTMLMenuElement>;
getElementsByTagName(name:"meta"): NodeListOf<HTMLMetaElement>;
getElementsByTagName(name:"nav"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"nextid"): NodeListOf<HTMLNextIdElement>;
getElementsByTagName(name:"nobr"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"noframes"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"noscript"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"object"): NodeListOf<HTMLObjectElement>;
getElementsByTagName(name:"ol"): NodeListOf<HTMLOListElement>;
getElementsByTagName(name:"optgroup"): NodeListOf<HTMLOptGroupElement>;
getElementsByTagName(name:"option"): NodeListOf<HTMLOptionElement>;
getElementsByTagName(name:"p"): NodeListOf<HTMLParagraphElement>;
getElementsByTagName(name:"param"): NodeListOf<HTMLParamElement>;
getElementsByTagName(name:"plaintext"): NodeListOf<HTMLBlockElement>;
getElementsByTagName(name:"pre"): NodeListOf<HTMLPreElement>;
getElementsByTagName(name:"progress"): NodeListOf<HTMLProgressElement>;
getElementsByTagName(name:"q"): NodeListOf<HTMLQuoteElement>;
getElementsByTagName(name:"rt"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"ruby"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"s"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"samp"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"script"): NodeListOf<HTMLScriptElement>;
getElementsByTagName(name:"section"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"select"): NodeListOf<HTMLSelectElement>;
getElementsByTagName(name:"small"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"SOURCE"): NodeListOf<HTMLSourceElement>;
getElementsByTagName(name:"span"): NodeListOf<HTMLSpanElement>;
getElementsByTagName(name:"strike"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"strong"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"style"): NodeListOf<HTMLStyleElement>;
getElementsByTagName(name:"sub"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"sup"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"table"): NodeListOf<HTMLTableElement>;
getElementsByTagName(name:"tbody"): NodeListOf<HTMLTableSectionElement>;
getElementsByTagName(name:"td"): NodeListOf<HTMLTableDataCellElement>;
getElementsByTagName(name:"textarea"): NodeListOf<HTMLTextAreaElement>;
getElementsByTagName(name:"tfoot"): NodeListOf<HTMLTableSectionElement>;
getElementsByTagName(name:"th"): NodeListOf<HTMLTableHeaderCellElement>;
getElementsByTagName(name:"thead"): NodeListOf<HTMLTableSectionElement>;
getElementsByTagName(name:"title"): NodeListOf<HTMLTitleElement>;
getElementsByTagName(name:"tr"): NodeListOf<HTMLTableRowElement>;
getElementsByTagName(name:"track"): NodeListOf<HTMLTrackElement>;
getElementsByTagName(name:"tt"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"u"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"ul"): NodeListOf<HTMLUListElement>;
getElementsByTagName(name:"var"): NodeListOf<HTMLPhraseElement>;
getElementsByTagName(name:"video"): NodeListOf<HTMLVideoElement>;
getElementsByTagName(name:"wbr"): NodeListOf<HTMLElement>;
getElementsByTagName(name:"x-ms-webview"): NodeListOf<MSHTMLWebViewElement>;
getElementsByTagName(name:"xmp"): NodeListOf<HTMLBlockElement>;
getElementsByTagName(name:string): NodeList;
getClientRects(): ClientRectList;
setAttributeNode(newAttr:Attr): Attr;
removeAttributeNode(oldAttr:Attr): Attr;
setAttribute(name?:string, value?:string): void;
removeAttributeNS(namespaceURI:string, localName:string): void;
msGetRegionContent(): MSRangeCollection;
msReleasePointerCapture(pointerId:number): void;
msSetPointerCapture(pointerId:number): void;
msZoomTo(args:MsZoomToOptions): void;
setPointerCapture(pointerId:number): void;
msGetUntransformedBounds(): ClientRect;
releasePointerCapture(pointerId:number): void;
msRequestFullscreen(): void;
addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerdown", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgotpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturedoubletap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerhover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturehold", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointermove", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturechange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturestart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointercancel", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgestureend", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturetap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerout", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msinertiastart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mslostpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerup", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"lostpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerenter", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"gotpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerleave", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var Element:{
prototype: Element;
new(): Element;
}
interface HTMLNextIdElement extends HTMLElement {
n: string;
}
declare var HTMLNextIdElement:{
prototype: HTMLNextIdElement;
new(): HTMLNextIdElement;
}
interface SVGPathSegMovetoRel extends SVGPathSeg {
y: number;
x: number;
}
declare var SVGPathSegMovetoRel:{
prototype: SVGPathSegMovetoRel;
new(): SVGPathSegMovetoRel;
}
interface SVGLineElement extends SVGElement, SVGStylable, SVGTransformable, SVGLangSpace, SVGTests, SVGExternalResourcesRequired {
y1: SVGAnimatedLength;
x2: SVGAnimatedLength;
x1: SVGAnimatedLength;
y2: SVGAnimatedLength;
}
declare var SVGLineElement:{
prototype: SVGLineElement;
new(): SVGLineElement;
}
interface HTMLParagraphElement extends HTMLElement, DOML2DeprecatedTextFlowControl {
/**
* Sets or retrieves how the object is aligned with adjacent text.
*/
align: string;
}
declare var HTMLParagraphElement:{
prototype: HTMLParagraphElement;
new(): HTMLParagraphElement;
}
interface HTMLAreasCollection extends HTMLCollection {
/**
* Removes an element from the collection.
*/
remove(index?:number): void;
/**
* Adds an element to the areas, controlRange, or options collection.
*/
add(element:HTMLElement, before?:any): void;
}
declare var HTMLAreasCollection:{
prototype: HTMLAreasCollection;
new(): HTMLAreasCollection;
}
interface SVGDescElement extends SVGElement, SVGStylable, SVGLangSpace {
}
declare var SVGDescElement:{
prototype: SVGDescElement;
new(): SVGDescElement;
}
interface Node extends EventTarget {
nodeType: number;
previousSibling: Node;
localName: string;
namespaceURI: string;
textContent: string;
parentNode: Node;
nextSibling: Node;
nodeValue: string;
lastChild: Node;
childNodes: NodeList;
nodeName: string;
ownerDocument: Document;
attributes: NamedNodeMap;
firstChild: Node;
prefix: string;
removeChild(oldChild:Node): Node;
appendChild(newChild:Node): Node;
isSupported(feature:string, version:string): boolean;
isEqualNode(arg:Node): boolean;
lookupPrefix(namespaceURI:string): string;
isDefaultNamespace(namespaceURI:string): boolean;
compareDocumentPosition(other:Node): number;
normalize(): void;
isSameNode(other:Node): boolean;
hasAttributes(): boolean;
lookupNamespaceURI(prefix:string): string;
cloneNode(deep?:boolean): Node;
hasChildNodes(): boolean;
replaceChild(newChild:Node, oldChild:Node): Node;
insertBefore(newChild:Node, refChild?:Node): Node;
ENTITY_REFERENCE_NODE: number;
ATTRIBUTE_NODE: number;
DOCUMENT_FRAGMENT_NODE: number;
TEXT_NODE: number;
ELEMENT_NODE: number;
COMMENT_NODE: number;
DOCUMENT_POSITION_DISCONNECTED: number;
DOCUMENT_POSITION_CONTAINED_BY: number;
DOCUMENT_POSITION_CONTAINS: number;
DOCUMENT_TYPE_NODE: number;
DOCUMENT_POSITION_IMPLEMENTATION_SPECIFIC: number;
DOCUMENT_NODE: number;
ENTITY_NODE: number;
PROCESSING_INSTRUCTION_NODE: number;
CDATA_SECTION_NODE: number;
NOTATION_NODE: number;
DOCUMENT_POSITION_FOLLOWING: number;
DOCUMENT_POSITION_PRECEDING: number;
}
declare var Node:{
prototype: Node;
new(): Node;
ENTITY_REFERENCE_NODE: number;
ATTRIBUTE_NODE: number;
DOCUMENT_FRAGMENT_NODE: number;
TEXT_NODE: number;
ELEMENT_NODE: number;
COMMENT_NODE: number;
DOCUMENT_POSITION_DISCONNECTED: number;
DOCUMENT_POSITION_CONTAINED_BY: number;
DOCUMENT_POSITION_CONTAINS: number;
DOCUMENT_TYPE_NODE: number;
DOCUMENT_POSITION_IMPLEMENTATION_SPECIFIC: number;
DOCUMENT_NODE: number;
ENTITY_NODE: number;
PROCESSING_INSTRUCTION_NODE: number;
CDATA_SECTION_NODE: number;
NOTATION_NODE: number;
DOCUMENT_POSITION_FOLLOWING: number;
DOCUMENT_POSITION_PRECEDING: number;
}
interface SVGPathSegCurvetoQuadraticSmoothRel extends SVGPathSeg {
y: number;
x: number;
}
declare var SVGPathSegCurvetoQuadraticSmoothRel:{
prototype: SVGPathSegCurvetoQuadraticSmoothRel;
new(): SVGPathSegCurvetoQuadraticSmoothRel;
}
interface DOML2DeprecatedListSpaceReduction {
compact: boolean;
}
interface MSScriptHost {
}
declare var MSScriptHost:{
prototype: MSScriptHost;
new(): MSScriptHost;
}
interface SVGClipPathElement extends SVGElement, SVGUnitTypes, SVGStylable, SVGTransformable, SVGLangSpace, SVGTests, SVGExternalResourcesRequired {
clipPathUnits: SVGAnimatedEnumeration;
}
declare var SVGClipPathElement:{
prototype: SVGClipPathElement;
new(): SVGClipPathElement;
}
interface MouseEvent extends UIEvent {
toElement: Element;
layerY: number;
fromElement: Element;
which: number;
pageX: number;
offsetY: number;
x: number;
y: number;
metaKey: boolean;
altKey: boolean;
ctrlKey: boolean;
offsetX: number;
screenX: number;
clientY: number;
shiftKey: boolean;
layerX: number;
screenY: number;
relatedTarget: EventTarget;
button: number;
pageY: number;
buttons: number;
clientX: number;
initMouseEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, viewArg:Window, detailArg:number, screenXArg:number, screenYArg:number, clientXArg:number, clientYArg:number, ctrlKeyArg:boolean, altKeyArg:boolean, shiftKeyArg:boolean, metaKeyArg:boolean, buttonArg:number, relatedTargetArg:EventTarget): void;
getModifierState(keyArg:string): boolean;
}
declare var MouseEvent:{
prototype: MouseEvent;
new(): MouseEvent;
}
interface RangeException {
code: number;
message: string;
name: string;
toString(): string;
INVALID_NODE_TYPE_ERR: number;
BAD_BOUNDARYPOINTS_ERR: number;
}
declare var RangeException:{
prototype: RangeException;
new(): RangeException;
INVALID_NODE_TYPE_ERR: number;
BAD_BOUNDARYPOINTS_ERR: number;
}
interface SVGTextPositioningElement extends SVGTextContentElement {
y: SVGAnimatedLengthList;
rotate: SVGAnimatedNumberList;
dy: SVGAnimatedLengthList;
x: SVGAnimatedLengthList;
dx: SVGAnimatedLengthList;
}
declare var SVGTextPositioningElement:{
prototype: SVGTextPositioningElement;
new(): SVGTextPositioningElement;
}
interface HTMLAppletElement extends HTMLElement, DOML2DeprecatedMarginStyle, DOML2DeprecatedBorderStyle, DOML2DeprecatedAlignmentStyle, MSDataBindingExtensions, MSDataBindingRecordSetExtensions {
width: number;
/**
* Sets or retrieves the Internet media type for the code associated with the object.
*/
codeType: string;
object: string;
form: HTMLFormElement;
code: string;
/**
* Sets or retrieves a character string that can be used to implement your own archive functionality for the object.
*/
archive: string;
/**
* Sets or retrieves a text alternative to the graphic.
*/
alt: string;
/**
* Sets or retrieves a message to be displayed while an object is loading.
*/
standby: string;
/**
* Sets or retrieves the class identifier for the object.
*/
classid: string;
/**
* Sets or retrieves the shape of the object.
*/
name: string;
/**
* Sets or retrieves the URL, often with a bookmark extension (#name), to use as a client-side image map.
*/
useMap: string;
/**
* Sets or retrieves the URL that references the data of the object.
*/
data: string;
/**
* Sets or retrieves the height of the object.
*/
height: string;
/**
* Gets or sets the optional alternative HTML script to execute if the object fails to load.
*/
altHtml: string;
/**
* Address of a pointer to the document this page or frame contains. If there is no document, then null will be returned.
*/
contentDocument: Document;
/**
* Sets or retrieves the URL of the component.
*/
codeBase: string;
/**
* Sets or retrieves a character string that can be used to implement your own declare functionality for the object.
*/
declare: boolean;
/**
* Returns the content type of the object.
*/
type: string;
/**
* Retrieves a string of the URL where the object tag can be found. This is often the href of the document that the object is in, or the value set by a base element.
*/
BaseHref: string;
}
declare var HTMLAppletElement:{
prototype: HTMLAppletElement;
new(): HTMLAppletElement;
}
interface TextMetrics {
width: number;
}
declare var TextMetrics:{
prototype: TextMetrics;
new(): TextMetrics;
}
interface DocumentEvent {
createEvent(eventInterface:"AnimationEvent"): AnimationEvent;
createEvent(eventInterface:"CloseEvent"): CloseEvent;
createEvent(eventInterface:"CompositionEvent"): CompositionEvent;
createEvent(eventInterface:"CustomEvent"): CustomEvent;
createEvent(eventInterface:"DeviceMotionEvent"): DeviceMotionEvent;
createEvent(eventInterface:"DeviceOrientationEvent"): DeviceOrientationEvent;
createEvent(eventInterface:"DragEvent"): DragEvent;
createEvent(eventInterface:"ErrorEvent"): ErrorEvent;
createEvent(eventInterface:"Event"): Event;
createEvent(eventInterface:"Events"): Event;
createEvent(eventInterface:"FocusEvent"): FocusEvent;
createEvent(eventInterface:"HTMLEvents"): Event;
createEvent(eventInterface:"IDBVersionChangeEvent"): IDBVersionChangeEvent;
createEvent(eventInterface:"KeyboardEvent"): KeyboardEvent;
createEvent(eventInterface:"LongRunningScriptDetectedEvent"): LongRunningScriptDetectedEvent;
createEvent(eventInterface:"MessageEvent"): MessageEvent;
createEvent(eventInterface:"MouseEvent"): MouseEvent;
createEvent(eventInterface:"MouseEvents"): MouseEvent;
createEvent(eventInterface:"MouseWheelEvent"): MouseWheelEvent;
createEvent(eventInterface:"MSGestureEvent"): MSGestureEvent;
createEvent(eventInterface:"MSPointerEvent"): MSPointerEvent;
createEvent(eventInterface:"MutationEvent"): MutationEvent;
createEvent(eventInterface:"MutationEvents"): MutationEvent;
createEvent(eventInterface:"NavigationCompletedEvent"): NavigationCompletedEvent;
createEvent(eventInterface:"NavigationEvent"): NavigationEvent;
createEvent(eventInterface:"PageTransitionEvent"): PageTransitionEvent;
createEvent(eventInterface:"PointerEvent"): MSPointerEvent;
createEvent(eventInterface:"PopStateEvent"): PopStateEvent;
createEvent(eventInterface:"ProgressEvent"): ProgressEvent;
createEvent(eventInterface:"StorageEvent"): StorageEvent;
createEvent(eventInterface:"SVGZoomEvents"): SVGZoomEvent;
createEvent(eventInterface:"TextEvent"): TextEvent;
createEvent(eventInterface:"TrackEvent"): TrackEvent;
createEvent(eventInterface:"TransitionEvent"): TransitionEvent;
createEvent(eventInterface:"UIEvent"): UIEvent;
createEvent(eventInterface:"UIEvents"): UIEvent;
createEvent(eventInterface:"UnviewableContentIdentifiedEvent"): UnviewableContentIdentifiedEvent;
createEvent(eventInterface:"WebGLContextEvent"): WebGLContextEvent;
createEvent(eventInterface:"WheelEvent"): WheelEvent;
createEvent(eventInterface:string): Event;
}
interface HTMLOListElement extends HTMLElement, DOML2DeprecatedListSpaceReduction, DOML2DeprecatedListNumberingAndBulletStyle {
/**
* The starting number.
*/
start: number;
}
declare var HTMLOListElement:{
prototype: HTMLOListElement;
new(): HTMLOListElement;
}
interface SVGPathSegLinetoVerticalRel extends SVGPathSeg {
y: number;
}
declare var SVGPathSegLinetoVerticalRel:{
prototype: SVGPathSegLinetoVerticalRel;
new(): SVGPathSegLinetoVerticalRel;
}
interface SVGAnimatedString {
animVal: string;
baseVal: string;
}
declare var SVGAnimatedString:{
prototype: SVGAnimatedString;
new(): SVGAnimatedString;
}
interface CDATASection extends Text {
}
declare var CDATASection:{
prototype: CDATASection;
new(): CDATASection;
}
interface StyleMedia {
type: string;
matchMedium(mediaquery:string): boolean;
}
declare var StyleMedia:{
prototype: StyleMedia;
new(): StyleMedia;
}
interface HTMLSelectElement extends HTMLElement, MSHTMLCollectionExtensions, MSDataBindingExtensions {
options: HTMLSelectElement;
/**
* Sets or retrieves the value which is returned to the server when the form control is submitted.
*/
value: string;
/**
* Retrieves a reference to the form that the object is embedded in.
*/
form: HTMLFormElement;
/**
* Sets or retrieves the name of the object.
*/
name: string;
/**
* Sets or retrieves the number of rows in the list box.
*/
size: number;
/**
* Sets or retrieves the number of objects in a collection.
*/
length: number;
/**
* Sets or retrieves the index of the selected option in a select object.
*/
selectedIndex: number;
/**
* Sets or retrieves the Boolean value indicating whether multiple items can be selected from a list.
*/
multiple: boolean;
/**
* Retrieves the type of select control based on the value of the MULTIPLE attribute.
*/
type: string;
/**
* Returns the error message that would be displayed if the user submits the form, or an empty string if no error message. It also triggers the standard error message, such as "this is a required field". The result is that the user sees validation messages without actually submitting.
*/
validationMessage: string;
/**
* Provides a way to direct a user to a specific field when a document loads. This can provide both direction and convenience for a user, reducing the need to click or tab to a field when a page opens. This attribute is true when present on an element, and false when missing.
*/
autofocus: boolean;
/**
* Returns a ValidityState object that represents the validity states of an element.
*/
validity: ValidityState;
/**
* When present, marks an element that can't be submitted without a value.
*/
required: boolean;
/**
* Returns whether an element will successfully validate based on forms validation rules and constraints.
*/
willValidate: boolean;
/**
* Removes an element from the collection.
* @param index Number that specifies the zero-based index of the element to remove from the collection.
*/
remove(index?:number): void;
/**
* Adds an element to the areas, controlRange, or options collection.
* @param element Variant of type Number that specifies the index position in the collection where the element is placed. If no value is given, the method places the element at the end of the collection.
* @param before Variant of type Object that specifies an element to insert before, or null to append the object to the collection.
*/
add(element:HTMLElement, before?:any): void;
/**
* Retrieves a select object or an object from an options collection.
* @param name Variant of type Number or String that specifies the object or collection to retrieve. If this parameter is an integer, it is the zero-based index of the object. If this parameter is a string, all objects with matching name or id properties are retrieved, and a collection is returned if more than one match is made.
* @param index Variant of type Number that specifies the zero-based index of the object to retrieve when a collection is returned.
*/
item(name?:any, index?:any): any;
/**
* Retrieves a select object or an object from an options collection.
* @param namedItem A String that specifies the name or id property of the object to retrieve. A collection is returned if more than one match is made.
*/
namedItem(name:string): any;
[name: string]: any;
/**
* Returns whether a form will validate when it is submitted, without having to submit it.
*/
checkValidity(): boolean;
/**
* Sets a custom error message that is displayed when a form is submitted.
* @param error Sets a custom error message that is displayed when a form is submitted.
*/
setCustomValidity(error:string): void;
}
declare var HTMLSelectElement:{
prototype: HTMLSelectElement;
new(): HTMLSelectElement;
}
interface TextRange {
boundingLeft: number;
htmlText: string;
offsetLeft: number;
boundingWidth: number;
boundingHeight: number;
boundingTop: number;
text: string;
offsetTop: number;
moveToPoint(x:number, y:number): void;
queryCommandValue(cmdID:string): any;
getBookmark(): string;
move(unit:string, count?:number): number;
queryCommandIndeterm(cmdID:string): boolean;
scrollIntoView(fStart?:boolean): void;
findText(string:string, count?:number, flags?:number): boolean;
execCommand(cmdID:string, showUI?:boolean, value?:any): boolean;
getBoundingClientRect(): ClientRect;
moveToBookmark(bookmark:string): boolean;
isEqual(range:TextRange): boolean;
duplicate(): TextRange;
collapse(start?:boolean): void;
queryCommandText(cmdID:string): string;
select(): void;
pasteHTML(html:string): void;
inRange(range:TextRange): boolean;
moveEnd(unit:string, count?:number): number;
getClientRects(): ClientRectList;
moveStart(unit:string, count?:number): number;
parentElement(): Element;
queryCommandState(cmdID:string): boolean;
compareEndPoints(how:string, sourceRange:TextRange): number;
execCommandShowHelp(cmdID:string): boolean;
moveToElementText(element:Element): void;
expand(Unit:string): boolean;
queryCommandSupported(cmdID:string): boolean;
setEndPoint(how:string, SourceRange:TextRange): void;
queryCommandEnabled(cmdID:string): boolean;
}
declare var TextRange:{
prototype: TextRange;
new(): TextRange;
}
interface SVGTests {
requiredFeatures: SVGStringList;
requiredExtensions: SVGStringList;
systemLanguage: SVGStringList;
hasExtension(extension:string): boolean;
}
interface HTMLBlockElement extends HTMLElement, DOML2DeprecatedTextFlowControl {
/**
* Sets or retrieves the width of the object.
*/
width: number;
/**
* Sets or retrieves reference information about the object.
*/
cite: string;
}
declare var HTMLBlockElement:{
prototype: HTMLBlockElement;
new(): HTMLBlockElement;
}
interface CSSStyleSheet extends StyleSheet {
owningElement: Element;
imports: StyleSheetList;
isAlternate: boolean;
rules: MSCSSRuleList;
isPrefAlternate: boolean;
readOnly: boolean;
cssText: string;
ownerRule: CSSRule;
href: string;
cssRules: CSSRuleList;
id: string;
pages: StyleSheetPageList;
addImport(bstrURL:string, lIndex?:number): number;
addPageRule(bstrSelector:string, bstrStyle:string, lIndex?:number): number;
insertRule(rule:string, index?:number): number;
removeRule(lIndex:number): void;
deleteRule(index?:number): void;
addRule(bstrSelector:string, bstrStyle?:string, lIndex?:number): number;
removeImport(lIndex:number): void;
}
declare var CSSStyleSheet:{
prototype: CSSStyleSheet;
new(): CSSStyleSheet;
}
interface MSSelection {
type: string;
typeDetail: string;
createRange(): TextRange;
clear(): void;
createRangeCollection(): TextRangeCollection;
empty(): void;
}
declare var MSSelection:{
prototype: MSSelection;
new(): MSSelection;
}
interface HTMLMetaElement extends HTMLElement {
/**
* Gets or sets information used to bind the value of a content attribute of a meta element to an HTTP response header.
*/
httpEquiv: string;
/**
* Sets or retrieves the value specified in the content attribute of the meta object.
*/
name: string;
/**
* Gets or sets meta-information to associate with httpEquiv or name.
*/
content: string;
/**
* Sets or retrieves the URL property that will be loaded after the specified time has elapsed.
*/
url: string;
/**
* Sets or retrieves a scheme to be used in interpreting the value of a property specified for the object.
*/
scheme: string;
/**
* Sets or retrieves the character set used to encode the object.
*/
charset: string;
}
declare var HTMLMetaElement:{
prototype: HTMLMetaElement;
new(): HTMLMetaElement;
}
interface SVGPatternElement extends SVGElement, SVGUnitTypes, SVGStylable, SVGLangSpace, SVGTests, SVGFitToViewBox, SVGExternalResourcesRequired, SVGURIReference {
patternUnits: SVGAnimatedEnumeration;
y: SVGAnimatedLength;
width: SVGAnimatedLength;
x: SVGAnimatedLength;
patternContentUnits: SVGAnimatedEnumeration;
patternTransform: SVGAnimatedTransformList;
height: SVGAnimatedLength;
}
declare var SVGPatternElement:{
prototype: SVGPatternElement;
new(): SVGPatternElement;
}
interface SVGAnimatedAngle {
animVal: SVGAngle;
baseVal: SVGAngle;
}
declare var SVGAnimatedAngle:{
prototype: SVGAnimatedAngle;
new(): SVGAnimatedAngle;
}
interface Selection {
isCollapsed: boolean;
anchorNode: Node;
focusNode: Node;
anchorOffset: number;
focusOffset: number;
rangeCount: number;
addRange(range:Range): void;
collapseToEnd(): void;
toString(): string;
selectAllChildren(parentNode:Node): void;
getRangeAt(index:number): Range;
collapse(parentNode:Node, offset:number): void;
removeAllRanges(): void;
collapseToStart(): void;
deleteFromDocument(): void;
removeRange(range:Range): void;
}
declare var Selection:{
prototype: Selection;
new(): Selection;
}
interface SVGScriptElement extends SVGElement, SVGExternalResourcesRequired, SVGURIReference {
type: string;
}
declare var SVGScriptElement:{
prototype: SVGScriptElement;
new(): SVGScriptElement;
}
interface HTMLDDElement extends HTMLElement {
/**
* Sets or retrieves whether the browser automatically performs wordwrap.
*/
noWrap: boolean;
}
declare var HTMLDDElement:{
prototype: HTMLDDElement;
new(): HTMLDDElement;
}
interface MSDataBindingRecordSetReadonlyExtensions {
recordset: any;
namedRecordset(dataMember:string, hierarchy?:any): any;
}
interface CSSStyleRule extends CSSRule {
selectorText: string;
style: MSStyleCSSProperties;
readOnly: boolean;
}
declare var CSSStyleRule:{
prototype: CSSStyleRule;
new(): CSSStyleRule;
}
interface NodeIterator {
whatToShow: number;
filter: NodeFilter;
root: Node;
expandEntityReferences: boolean;
nextNode(): Node;
detach(): void;
previousNode(): Node;
}
declare var NodeIterator:{
prototype: NodeIterator;
new(): NodeIterator;
}
interface SVGViewElement extends SVGElement, SVGZoomAndPan, SVGFitToViewBox, SVGExternalResourcesRequired {
viewTarget: SVGStringList;
}
declare var SVGViewElement:{
prototype: SVGViewElement;
new(): SVGViewElement;
}
interface HTMLLinkElement extends HTMLElement, LinkStyle {
/**
* Sets or retrieves the relationship between the object and the destination of the link.
*/
rel: string;
/**
* Sets or retrieves the window or frame at which to target content.
*/
target: string;
/**
* Sets or retrieves a destination URL or an anchor point.
*/
href: string;
/**
* Sets or retrieves the media type.
*/
media: string;
/**
* Sets or retrieves the relationship between the object and the destination of the link.
*/
rev: string;
/**
* Sets or retrieves the MIME type of the object.
*/
type: string;
/**
* Sets or retrieves the character set used to encode the object.
*/
charset: string;
/**
* Sets or retrieves the language code of the object.
*/
hreflang: string;
}
declare var HTMLLinkElement:{
prototype: HTMLLinkElement;
new(): HTMLLinkElement;
}
interface SVGLocatable {
farthestViewportElement: SVGElement;
nearestViewportElement: SVGElement;
getBBox(): SVGRect;
getTransformToElement(element:SVGElement): SVGMatrix;
getCTM(): SVGMatrix;
getScreenCTM(): SVGMatrix;
}
interface HTMLFontElement extends HTMLElement, DOML2DeprecatedColorProperty, DOML2DeprecatedSizeProperty {
/**
* Sets or retrieves the current typeface family.
*/
face: string;
}
declare var HTMLFontElement:{
prototype: HTMLFontElement;
new(): HTMLFontElement;
}
interface SVGTitleElement extends SVGElement, SVGStylable, SVGLangSpace {
}
declare var SVGTitleElement:{
prototype: SVGTitleElement;
new(): SVGTitleElement;
}
interface ControlRangeCollection {
length: number;
queryCommandValue(cmdID:string): any;
remove(index:number): void;
add(item:Element): void;
queryCommandIndeterm(cmdID:string): boolean;
scrollIntoView(varargStart?:any): void;
item(index:number): Element;
[index: number]: Element;
execCommand(cmdID:string, showUI?:boolean, value?:any): boolean;
addElement(item:Element): void;
queryCommandState(cmdID:string): boolean;
queryCommandSupported(cmdID:string): boolean;
queryCommandEnabled(cmdID:string): boolean;
queryCommandText(cmdID:string): string;
select(): void;
}
declare var ControlRangeCollection:{
prototype: ControlRangeCollection;
new(): ControlRangeCollection;
}
interface MSNamespaceInfo extends MSEventAttachmentTarget {
urn: string;
onreadystatechange: (ev:Event) => any;
name: string;
readyState: string;
doImport(implementationUrl:string): void;
addEventListener(type:"readystatechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var MSNamespaceInfo:{
prototype: MSNamespaceInfo;
new(): MSNamespaceInfo;
}
interface WindowSessionStorage {
sessionStorage: Storage;
}
interface SVGAnimatedTransformList {
animVal: SVGTransformList;
baseVal: SVGTransformList;
}
declare var SVGAnimatedTransformList:{
prototype: SVGAnimatedTransformList;
new(): SVGAnimatedTransformList;
}
interface HTMLTableCaptionElement extends HTMLElement {
/**
* Sets or retrieves the alignment of the caption or legend.
*/
align: string;
/**
* Sets or retrieves whether the caption appears at the top or bottom of the table.
*/
vAlign: string;
}
declare var HTMLTableCaptionElement:{
prototype: HTMLTableCaptionElement;
new(): HTMLTableCaptionElement;
}
interface HTMLOptionElement extends HTMLElement, MSDataBindingExtensions {
/**
* Sets or retrieves the ordinal position of an option in a list box.
*/
index: number;
/**
* Sets or retrieves the status of an option.
*/
defaultSelected: boolean;
/**
* Sets or retrieves the value which is returned to the server when the form control is submitted.
*/
value: string;
/**
* Sets or retrieves the text string specified by the option tag.
*/
text: string;
/**
* Retrieves a reference to the form that the object is embedded in.
*/
form: HTMLFormElement;
/**
* Sets or retrieves a value that you can use to implement your own label functionality for the object.
*/
label: string;
/**
* Sets or retrieves whether the option in the list box is the default item.
*/
selected: boolean;
}
declare var HTMLOptionElement:{
prototype: HTMLOptionElement;
new(): HTMLOptionElement;
create(): HTMLOptionElement;
}
interface HTMLMapElement extends HTMLElement {
/**
* Sets or retrieves the name of the object.
*/
name: string;
/**
* Retrieves a collection of the area objects defined for the given map object.
*/
areas: HTMLAreasCollection;
}
declare var HTMLMapElement:{
prototype: HTMLMapElement;
new(): HTMLMapElement;
}
interface HTMLMenuElement extends HTMLElement, DOML2DeprecatedListSpaceReduction {
type: string;
}
declare var HTMLMenuElement:{
prototype: HTMLMenuElement;
new(): HTMLMenuElement;
}
interface MouseWheelEvent extends MouseEvent {
wheelDelta: number;
initMouseWheelEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, viewArg:Window, detailArg:number, screenXArg:number, screenYArg:number, clientXArg:number, clientYArg:number, buttonArg:number, relatedTargetArg:EventTarget, modifiersListArg:string, wheelDeltaArg:number): void;
}
declare var MouseWheelEvent:{
prototype: MouseWheelEvent;
new(): MouseWheelEvent;
}
interface SVGFitToViewBox {
viewBox: SVGAnimatedRect;
preserveAspectRatio: SVGAnimatedPreserveAspectRatio;
}
interface SVGPointList {
numberOfItems: number;
replaceItem(newItem:SVGPoint, index:number): SVGPoint;
getItem(index:number): SVGPoint;
clear(): void;
appendItem(newItem:SVGPoint): SVGPoint;
initialize(newItem:SVGPoint): SVGPoint;
removeItem(index:number): SVGPoint;
insertItemBefore(newItem:SVGPoint, index:number): SVGPoint;
}
declare var SVGPointList:{
prototype: SVGPointList;
new(): SVGPointList;
}
interface SVGAnimatedLengthList {
animVal: SVGLengthList;
baseVal: SVGLengthList;
}
declare var SVGAnimatedLengthList:{
prototype: SVGAnimatedLengthList;
new(): SVGAnimatedLengthList;
}
interface SVGAnimatedPreserveAspectRatio {
animVal: SVGPreserveAspectRatio;
baseVal: SVGPreserveAspectRatio;
}
declare var SVGAnimatedPreserveAspectRatio:{
prototype: SVGAnimatedPreserveAspectRatio;
new(): SVGAnimatedPreserveAspectRatio;
}
interface MSSiteModeEvent extends Event {
buttonID: number;
actionURL: string;
}
declare var MSSiteModeEvent:{
prototype: MSSiteModeEvent;
new(): MSSiteModeEvent;
}
interface DOML2DeprecatedTextFlowControl {
clear: string;
}
interface StyleSheetPageList {
length: number;
item(index:number): CSSPageRule;
[index: number]: CSSPageRule;
}
declare var StyleSheetPageList:{
prototype: StyleSheetPageList;
new(): StyleSheetPageList;
}
interface MSCSSProperties extends CSSStyleDeclaration {
scrollbarShadowColor: string;
scrollbarHighlightColor: string;
layoutGridChar: string;
layoutGridType: string;
textAutospace: string;
textKashidaSpace: string;
writingMode: string;
scrollbarFaceColor: string;
backgroundPositionY: string;
lineBreak: string;
imeMode: string;
msBlockProgression: string;
layoutGridLine: string;
scrollbarBaseColor: string;
layoutGrid: string;
layoutFlow: string;
textKashida: string;
filter: string;
zoom: string;
scrollbarArrowColor: string;
behavior: string;
backgroundPositionX: string;
accelerator: string;
layoutGridMode: string;
textJustifyTrim: string;
scrollbar3dLightColor: string;
msInterpolationMode: string;
scrollbarTrackColor: string;
scrollbarDarkShadowColor: string;
styleFloat: string;
getAttribute(attributeName:string, flags?:number): any;
setAttribute(attributeName:string, AttributeValue:any, flags?:number): void;
removeAttribute(attributeName:string, flags?:number): boolean;
}
declare var MSCSSProperties:{
prototype: MSCSSProperties;
new(): MSCSSProperties;
}
interface SVGExternalResourcesRequired {
externalResourcesRequired: SVGAnimatedBoolean;
}
interface HTMLImageElement extends HTMLElement, MSImageResourceExtensions, MSDataBindingExtensions, MSResourceMetadata {
/**
* Sets or retrieves the width of the object.
*/
width: number;
/**
* Sets or retrieves the vertical margin for the object.
*/
vspace: number;
/**
* The original height of the image resource before sizing.
*/
naturalHeight: number;
/**
* Sets or retrieves a text alternative to the graphic.
*/
alt: string;
/**
* Sets or retrieves how the object is aligned with adjacent text.
*/
align: string;
/**
* The address or URL of the a media resource that is to be considered.
*/
src: string;
/**
* Sets or retrieves the URL, often with a bookmark extension (#name), to use as a client-side image map.
*/
useMap: string;
/**
* The original width of the image resource before sizing.
*/
naturalWidth: number;
/**
* Sets or retrieves the name of the object.
*/
name: string;
/**
* Sets or retrieves the height of the object.
*/
height: number;
/**
* Specifies the properties of a border drawn around an object.
*/
border: string;
/**
* Sets or retrieves the width of the border to draw around the object.
*/
hspace: number;
/**
* Sets or retrieves a Uniform Resource Identifier (URI) to a long description of the object.
*/
longDesc: string;
/**
* Contains the hypertext reference (HREF) of the URL.
*/
href: string;
/**
* Sets or retrieves whether the image is a server-side image map.
*/
isMap: boolean;
/**
* Retrieves whether the object is fully loaded.
*/
complete: boolean;
/**
* Gets or sets the primary DLNA PlayTo device.
*/
msPlayToPrimary: boolean;
/**
* Gets or sets whether the DLNA PlayTo device is available.
*/
msPlayToDisabled: boolean;
/**
* Gets the source associated with the media element for use by the PlayToManager.
*/
msPlayToSource: any;
crossOrigin: string;
msPlayToPreferredSourceUri: string;
}
declare var HTMLImageElement:{
prototype: HTMLImageElement;
new(): HTMLImageElement;
create(): HTMLImageElement;
}
interface HTMLAreaElement extends HTMLElement {
/**
* Sets or retrieves the protocol portion of a URL.
*/
protocol: string;
/**
* Sets or retrieves the substring of the href property that follows the question mark.
*/
search: string;
/**
* Sets or retrieves a text alternative to the graphic.
*/
alt: string;
/**
* Sets or retrieves the coordinates of the object.
*/
coords: string;
/**
* Sets or retrieves the host name part of the location or URL.
*/
hostname: string;
/**
* Sets or retrieves the port number associated with a URL.
*/
port: string;
/**
* Sets or retrieves the file name or path specified by the object.
*/
pathname: string;
/**
* Sets or retrieves the hostname and port number of the location or URL.
*/
host: string;
/**
* Sets or retrieves the subsection of the href property that follows the number sign (#).
*/
hash: string;
/**
* Sets or retrieves the window or frame at which to target content.
*/
target: string;
/**
* Sets or retrieves a destination URL or an anchor point.
*/
href: string;
/**
* Sets or gets whether clicks in this region cause action.
*/
noHref: boolean;
/**
* Sets or retrieves the shape of the object.
*/
shape: string;
/**
* Returns a string representation of an object.
*/
toString(): string;
}
declare var HTMLAreaElement:{
prototype: HTMLAreaElement;
new(): HTMLAreaElement;
}
interface EventTarget {
removeEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
dispatchEvent(evt:Event): boolean;
}
interface SVGAngle {
valueAsString: string;
valueInSpecifiedUnits: number;
value: number;
unitType: number;
newValueSpecifiedUnits(unitType:number, valueInSpecifiedUnits:number): void;
convertToSpecifiedUnits(unitType:number): void;
SVG_ANGLETYPE_RAD: number;
SVG_ANGLETYPE_UNKNOWN: number;
SVG_ANGLETYPE_UNSPECIFIED: number;
SVG_ANGLETYPE_DEG: number;
SVG_ANGLETYPE_GRAD: number;
}
declare var SVGAngle:{
prototype: SVGAngle;
new(): SVGAngle;
SVG_ANGLETYPE_RAD: number;
SVG_ANGLETYPE_UNKNOWN: number;
SVG_ANGLETYPE_UNSPECIFIED: number;
SVG_ANGLETYPE_DEG: number;
SVG_ANGLETYPE_GRAD: number;
}
interface HTMLButtonElement extends HTMLElement, MSDataBindingExtensions {
/**
* Sets or retrieves the default or selected value of the control.
*/
value: string;
status: any;
/**
* Retrieves a reference to the form that the object is embedded in.
*/
form: HTMLFormElement;
/**
* Sets or retrieves the name of the object.
*/
name: string;
/**
* Gets the classification and default behavior of the button.
*/
type: string;
/**
* Returns the error message that would be displayed if the user submits the form, or an empty string if no error message. It also triggers the standard error message, such as "this is a required field". The result is that the user sees validation messages without actually submitting.
*/
validationMessage: string;
/**
* Overrides the target attribute on a form element.
*/
formTarget: string;
/**
* Returns whether an element will successfully validate based on forms validation rules and constraints.
*/
willValidate: boolean;
/**
* Overrides the action attribute (where the data on a form is sent) on the parent form element.
*/
formAction: string;
/**
* Provides a way to direct a user to a specific field when a document loads. This can provide both direction and convenience for a user, reducing the need to click or tab to a field when a page opens. This attribute is true when present on an element, and false when missing.
*/
autofocus: boolean;
/**
* Returns a ValidityState object that represents the validity states of an element.
*/
validity: ValidityState;
/**
* Overrides any validation or required attributes on a form or form elements to allow it to be submitted without validation. This can be used to create a "save draft"-type submit option.
*/
formNoValidate: string;
/**
* Used to override the encoding (formEnctype attribute) specified on the form element.
*/
formEnctype: string;
/**
* Overrides the submit method attribute previously specified on a form element.
*/
formMethod: string;
/**
* Creates a TextRange object for the element.
*/
createTextRange(): TextRange;
/**
* Returns whether a form will validate when it is submitted, without having to submit it.
*/
checkValidity(): boolean;
/**
* Sets a custom error message that is displayed when a form is submitted.
* @param error Sets a custom error message that is displayed when a form is submitted.
*/
setCustomValidity(error:string): void;
}
declare var HTMLButtonElement:{
prototype: HTMLButtonElement;
new(): HTMLButtonElement;
}
interface HTMLSourceElement extends HTMLElement {
/**
* The address or URL of the a media resource that is to be considered.
*/
src: string;
/**
* Gets or sets the intended media type of the media source.
*/
media: string;
/**
* Gets or sets the MIME type of a media resource.
*/
type: string;
msKeySystem: string;
}
declare var HTMLSourceElement:{
prototype: HTMLSourceElement;
new(): HTMLSourceElement;
}
interface CanvasGradient {
addColorStop(offset:number, color:string): void;
}
declare var CanvasGradient:{
prototype: CanvasGradient;
new(): CanvasGradient;
}
interface KeyboardEvent extends UIEvent {
location: number;
keyCode: number;
shiftKey: boolean;
which: number;
locale: string;
key: string;
altKey: boolean;
metaKey: boolean;
char: string;
ctrlKey: boolean;
repeat: boolean;
charCode: number;
getModifierState(keyArg:string): boolean;
initKeyboardEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, viewArg:Window, keyArg:string, locationArg:number, modifiersListArg:string, repeat:boolean, locale:string): void;
DOM_KEY_LOCATION_RIGHT: number;
DOM_KEY_LOCATION_STANDARD: number;
DOM_KEY_LOCATION_LEFT: number;
DOM_KEY_LOCATION_NUMPAD: number;
DOM_KEY_LOCATION_JOYSTICK: number;
DOM_KEY_LOCATION_MOBILE: number;
}
declare var KeyboardEvent:{
prototype: KeyboardEvent;
new(): KeyboardEvent;
DOM_KEY_LOCATION_RIGHT: number;
DOM_KEY_LOCATION_STANDARD: number;
DOM_KEY_LOCATION_LEFT: number;
DOM_KEY_LOCATION_NUMPAD: number;
DOM_KEY_LOCATION_JOYSTICK: number;
DOM_KEY_LOCATION_MOBILE: number;
}
interface MessageEvent extends Event {
source: Window;
origin: string;
data: any;
ports: any;
initMessageEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, dataArg:any, originArg:string, lastEventIdArg:string, sourceArg:Window): void;
}
declare var MessageEvent:{
prototype: MessageEvent;
new(): MessageEvent;
}
interface SVGElement extends Element {
onmouseover: (ev:MouseEvent) => any;
viewportElement: SVGElement;
onmousemove: (ev:MouseEvent) => any;
onmouseout: (ev:MouseEvent) => any;
ondblclick: (ev:MouseEvent) => any;
onfocusout: (ev:FocusEvent) => any;
onfocusin: (ev:FocusEvent) => any;
xmlbase: string;
onmousedown: (ev:MouseEvent) => any;
onload: (ev:Event) => any;
onmouseup: (ev:MouseEvent) => any;
onclick: (ev:MouseEvent) => any;
ownerSVGElement: SVGSVGElement;
id: string;
addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerdown", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgotpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturedoubletap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerhover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturehold", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointermove", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturechange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturestart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointercancel", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgestureend", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturetap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerout", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msinertiastart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mslostpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerup", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"lostpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerenter", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"gotpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerleave", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mouseover", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mousemove", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseout", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dblclick", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusout", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusin", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mousedown", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mouseup", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"click", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var SVGElement:{
prototype: SVGElement;
new(): SVGElement;
}
interface HTMLScriptElement extends HTMLElement {
/**
* Sets or retrieves the status of the script.
*/
defer: boolean;
/**
* Retrieves or sets the text of the object as a string.
*/
text: string;
/**
* Retrieves the URL to an external file that contains the source code or data.
*/
src: string;
/**
* Sets or retrieves the object that is bound to the event script.
*/
htmlFor: string;
/**
* Sets or retrieves the character set used to encode the object.
*/
charset: string;
/**
* Sets or retrieves the MIME type for the associated scripting engine.
*/
type: string;
/**
* Sets or retrieves the event for which the script is written.
*/
event: string;
async: boolean;
}
declare var HTMLScriptElement:{
prototype: HTMLScriptElement;
new(): HTMLScriptElement;
}
interface HTMLTableRowElement extends HTMLElement, HTMLTableAlignment, DOML2DeprecatedBackgroundColorStyle {
/**
* Retrieves the position of the object in the rows collection for the table.
*/
rowIndex: number;
/**
* Retrieves a collection of all cells in the table row.
*/
cells: HTMLCollection;
/**
* Sets or retrieves how the object is aligned with adjacent text.
*/
align: string;
/**
* Sets or retrieves the color for one of the two colors used to draw the 3-D border of the object.
*/
borderColorLight: any;
/**
* Retrieves the position of the object in the collection.
*/
sectionRowIndex: number;
/**
* Sets or retrieves the border color of the object.
*/
borderColor: any;
/**
* Sets or retrieves the height of the object.
*/
height: any;
/**
* Sets or retrieves the color for one of the two colors used to draw the 3-D border of the object.
*/
borderColorDark: any;
/**
* Removes the specified cell from the table row, as well as from the cells collection.
* @param index Number that specifies the zero-based position of the cell to remove from the table row. If no value is provided, the last cell in the cells collection is deleted.
*/
deleteCell(index?:number): void;
/**
* Creates a new cell in the table row, and adds the cell to the cells collection.
* @param index Number that specifies where to insert the cell in the tr. The default value is -1, which appends the new cell to the end of the cells collection.
*/
insertCell(index?:number): HTMLElement;
}
declare var HTMLTableRowElement:{
prototype: HTMLTableRowElement;
new(): HTMLTableRowElement;
}
interface CanvasRenderingContext2D {
miterLimit: number;
font: string;
globalCompositeOperation: string;
msFillRule: string;
lineCap: string;
msImageSmoothingEnabled: boolean;
lineDashOffset: number;
shadowColor: string;
lineJoin: string;
shadowOffsetX: number;
lineWidth: number;
canvas: HTMLCanvasElement;
strokeStyle: any;
globalAlpha: number;
shadowOffsetY: number;
fillStyle: any;
shadowBlur: number;
textAlign: string;
textBaseline: string;
restore(): void;
setTransform(m11:number, m12:number, m21:number, m22:number, dx:number, dy:number): void;
save(): void;
arc(x:number, y:number, radius:number, startAngle:number, endAngle:number, anticlockwise?:boolean): void;
measureText(text:string): TextMetrics;
isPointInPath(x:number, y:number, fillRule?:string): boolean;
quadraticCurveTo(cpx:number, cpy:number, x:number, y:number): void;
putImageData(imagedata:ImageData, dx:number, dy:number, dirtyX?:number, dirtyY?:number, dirtyWidth?:number, dirtyHeight?:number): void;
rotate(angle:number): void;
fillText(text:string, x:number, y:number, maxWidth?:number): void;
translate(x:number, y:number): void;
scale(x:number, y:number): void;
createRadialGradient(x0:number, y0:number, r0:number, x1:number, y1:number, r1:number): CanvasGradient;
lineTo(x:number, y:number): void;
getLineDash(): number[];
fill(fillRule?:string): void;
createImageData(imageDataOrSw:any, sh?:number): ImageData;
createPattern(image:HTMLElement, repetition:string): CanvasPattern;
closePath(): void;
rect(x:number, y:number, w:number, h:number): void;
clip(fillRule?:string): void;
clearRect(x:number, y:number, w:number, h:number): void;
moveTo(x:number, y:number): void;
getImageData(sx:number, sy:number, sw:number, sh:number): ImageData;
fillRect(x:number, y:number, w:number, h:number): void;
bezierCurveTo(cp1x:number, cp1y:number, cp2x:number, cp2y:number, x:number, y:number): void;
drawImage(image:HTMLElement, offsetX:number, offsetY:number, width?:number, height?:number, canvasOffsetX?:number, canvasOffsetY?:number, canvasImageWidth?:number, canvasImageHeight?:number): void;
transform(m11:number, m12:number, m21:number, m22:number, dx:number, dy:number): void;
stroke(): void;
strokeRect(x:number, y:number, w:number, h:number): void;
setLineDash(segments:number[]): void;
strokeText(text:string, x:number, y:number, maxWidth?:number): void;
beginPath(): void;
arcTo(x1:number, y1:number, x2:number, y2:number, radius:number): void;
createLinearGradient(x0:number, y0:number, x1:number, y1:number): CanvasGradient;
}
declare var CanvasRenderingContext2D:{
prototype: CanvasRenderingContext2D;
new(): CanvasRenderingContext2D;
}
interface MSCSSRuleList {
length: number;
item(index?:number): CSSStyleRule;
[index: number]: CSSStyleRule;
}
declare var MSCSSRuleList:{
prototype: MSCSSRuleList;
new(): MSCSSRuleList;
}
interface SVGPathSegLinetoHorizontalAbs extends SVGPathSeg {
x: number;
}
declare var SVGPathSegLinetoHorizontalAbs:{
prototype: SVGPathSegLinetoHorizontalAbs;
new(): SVGPathSegLinetoHorizontalAbs;
}
interface SVGPathSegArcAbs extends SVGPathSeg {
y: number;
sweepFlag: boolean;
r2: number;
x: number;
angle: number;
r1: number;
largeArcFlag: boolean;
}
declare var SVGPathSegArcAbs:{
prototype: SVGPathSegArcAbs;
new(): SVGPathSegArcAbs;
}
interface SVGTransformList {
numberOfItems: number;
getItem(index:number): SVGTransform;
consolidate(): SVGTransform;
clear(): void;
appendItem(newItem:SVGTransform): SVGTransform;
initialize(newItem:SVGTransform): SVGTransform;
removeItem(index:number): SVGTransform;
insertItemBefore(newItem:SVGTransform, index:number): SVGTransform;
replaceItem(newItem:SVGTransform, index:number): SVGTransform;
createSVGTransformFromMatrix(matrix:SVGMatrix): SVGTransform;
}
declare var SVGTransformList:{
prototype: SVGTransformList;
new(): SVGTransformList;
}
interface HTMLHtmlElement extends HTMLElement {
/**
* Sets or retrieves the DTD version that governs the current document.
*/
version: string;
}
declare var HTMLHtmlElement:{
prototype: HTMLHtmlElement;
new(): HTMLHtmlElement;
}
interface SVGPathSegClosePath extends SVGPathSeg {
}
declare var SVGPathSegClosePath:{
prototype: SVGPathSegClosePath;
new(): SVGPathSegClosePath;
}
interface HTMLFrameElement extends HTMLElement, GetSVGDocument, MSDataBindingExtensions {
/**
* Sets or retrieves the width of the object.
*/
width: any;
/**
* Sets or retrieves whether the frame can be scrolled.
*/
scrolling: string;
/**
* Sets or retrieves the top and bottom margin heights before displaying the text in a frame.
*/
marginHeight: string;
/**
* Sets or retrieves the left and right margin widths before displaying the text in a frame.
*/
marginWidth: string;
/**
* Sets or retrieves the border color of the object.
*/
borderColor: any;
/**
* Sets or retrieves the amount of additional space between the frames.
*/
frameSpacing: any;
/**
* Sets or retrieves whether to display a border for the frame.
*/
frameBorder: string;
/**
* Sets or retrieves whether the user can resize the frame.
*/
noResize: boolean;
/**
* Retrieves the object of the specified.
*/
contentWindow: Window;
/**
* Sets or retrieves a URL to be loaded by the object.
*/
src: string;
/**
* Sets or retrieves the frame name.
*/
name: string;
/**
* Sets or retrieves the height of the object.
*/
height: any;
/**
* Retrieves the document object of the page or frame.
*/
contentDocument: Document;
/**
* Specifies the properties of a border drawn around an object.
*/
border: string;
/**
* Sets or retrieves a URI to a long description of the object.
*/
longDesc: string;
/**
* Raised when the object has been completely received from the server.
*/
onload: (ev:Event) => any;
/**
* Sets the value indicating whether the source file of a frame or iframe has specific security restrictions applied.
*/
security: any;
addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerdown", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgotpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturedoubletap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerhover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturehold", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointermove", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturechange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturestart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointercancel", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgestureend", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturetap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerout", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msinertiastart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mslostpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerup", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"lostpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerenter", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"gotpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerleave", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mouseleave", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"keydown", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"move", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"keyup", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"reset", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"help", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"dragleave", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusin", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"seeked", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"durationchange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"blur", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"emptied", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"seeking", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"canplay", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"deactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"datasetchanged", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsdelete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"loadstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"losecapture", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragenter", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"controlselect", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"submit", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"change", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"layoutcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"beforeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"canplaythrough", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"filterchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"datasetcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"suspend", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mouseenter", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"errorupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"mouseout", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mousewheel", listener:(ev:MouseWheelEvent) => any, useCapture?:boolean): void;
addEventListener(type:"volumechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"cellchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowexit", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsinserted", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"propertychange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragend", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforepaste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragover", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseup", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragstart", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecopy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drag", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseover", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pause", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousedown", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"click", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"waiting", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"paste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"moveend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"stalled", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousemove", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforeeditfocus", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"ratechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dblclick", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"contextmenu", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadedmetadata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"afterupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"play", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizeend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"playing", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focusout", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dataavailable", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"readystatechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"keypress", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadeddata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforedeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"activate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"movestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"selectstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focus", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"timeupdate", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resize", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"cut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"select", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drop", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"copy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"ended", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"scroll", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"rowenter", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"input", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mscontentzoom", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"cuechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"msmanipulationstatechanged", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var HTMLFrameElement:{
prototype: HTMLFrameElement;
new(): HTMLFrameElement;
}
interface SVGAnimatedLength {
animVal: SVGLength;
baseVal: SVGLength;
}
declare var SVGAnimatedLength:{
prototype: SVGAnimatedLength;
new(): SVGAnimatedLength;
}
interface SVGAnimatedPoints {
points: SVGPointList;
animatedPoints: SVGPointList;
}
interface SVGDefsElement extends SVGElement, SVGStylable, SVGTransformable, SVGLangSpace, SVGTests, SVGExternalResourcesRequired {
}
declare var SVGDefsElement:{
prototype: SVGDefsElement;
new(): SVGDefsElement;
}
interface HTMLQuoteElement extends HTMLElement {
/**
* Sets or retrieves the date and time of a modification to the object.
*/
dateTime: string;
/**
* Sets or retrieves reference information about the object.
*/
cite: string;
}
declare var HTMLQuoteElement:{
prototype: HTMLQuoteElement;
new(): HTMLQuoteElement;
}
interface CSSMediaRule extends CSSRule {
media: MediaList;
cssRules: CSSRuleList;
insertRule(rule:string, index?:number): number;
deleteRule(index?:number): void;
}
declare var CSSMediaRule:{
prototype: CSSMediaRule;
new(): CSSMediaRule;
}
interface WindowModal {
dialogArguments: any;
returnValue: any;
}
interface XMLHttpRequest extends EventTarget {
responseBody: any;
status: number;
readyState: number;
responseText: string;
responseXML: any;
ontimeout: (ev:Event) => any;
statusText: string;
onreadystatechange: (ev:Event) => any;
timeout: number;
onload: (ev:Event) => any;
response: any;
withCredentials: boolean;
onprogress: (ev:ProgressEvent) => any;
onabort: (ev:UIEvent) => any;
responseType: string;
onloadend: (ev:ProgressEvent) => any;
upload: XMLHttpRequestEventTarget;
onerror: (ev:ErrorEvent) => any;
onloadstart: (ev:Event) => any;
msCaching: string;
open(method:string, url:string, async?:boolean, user?:string, password?:string): void;
send(data?:any): void;
abort(): void;
getAllResponseHeaders(): string;
setRequestHeader(header:string, value:string): void;
getResponseHeader(header:string): string;
msCachingEnabled(): boolean;
overrideMimeType(mime:string): void;
LOADING: number;
DONE: number;
UNSENT: number;
OPENED: number;
HEADERS_RECEIVED: number;
addEventListener(type:"timeout", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"readystatechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadend", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var XMLHttpRequest:{
prototype: XMLHttpRequest;
new(): XMLHttpRequest;
LOADING: number;
DONE: number;
UNSENT: number;
OPENED: number;
HEADERS_RECEIVED: number;
create(): XMLHttpRequest;
}
interface HTMLTableHeaderCellElement extends HTMLTableCellElement {
/**
* Sets or retrieves the group of cells in a table to which the object's information applies.
*/
scope: string;
}
declare var HTMLTableHeaderCellElement:{
prototype: HTMLTableHeaderCellElement;
new(): HTMLTableHeaderCellElement;
}
interface HTMLDListElement extends HTMLElement, DOML2DeprecatedListSpaceReduction {
}
declare var HTMLDListElement:{
prototype: HTMLDListElement;
new(): HTMLDListElement;
}
interface MSDataBindingExtensions {
dataSrc: string;
dataFormatAs: string;
dataFld: string;
}
interface SVGPathSegLinetoHorizontalRel extends SVGPathSeg {
x: number;
}
declare var SVGPathSegLinetoHorizontalRel:{
prototype: SVGPathSegLinetoHorizontalRel;
new(): SVGPathSegLinetoHorizontalRel;
}
interface SVGEllipseElement extends SVGElement, SVGStylable, SVGTransformable, SVGLangSpace, SVGTests, SVGExternalResourcesRequired {
ry: SVGAnimatedLength;
cx: SVGAnimatedLength;
rx: SVGAnimatedLength;
cy: SVGAnimatedLength;
}
declare var SVGEllipseElement:{
prototype: SVGEllipseElement;
new(): SVGEllipseElement;
}
interface SVGAElement extends SVGElement, SVGStylable, SVGTransformable, SVGLangSpace, SVGTests, SVGExternalResourcesRequired, SVGURIReference {
target: SVGAnimatedString;
}
declare var SVGAElement:{
prototype: SVGAElement;
new(): SVGAElement;
}
interface SVGStylable {
className: SVGAnimatedString;
style: CSSStyleDeclaration;
}
interface SVGTransformable extends SVGLocatable {
transform: SVGAnimatedTransformList;
}
interface HTMLFrameSetElement extends HTMLElement {
ononline: (ev:Event) => any;
/**
* Sets or retrieves the border color of the object.
*/
borderColor: any;
/**
* Sets or retrieves the frame heights of the object.
*/
rows: string;
/**
* Sets or retrieves the frame widths of the object.
*/
cols: string;
/**
* Fires when the object loses the input focus.
*/
onblur: (ev:FocusEvent) => any;
/**
* Sets or retrieves the amount of additional space between the frames.
*/
frameSpacing: any;
/**
* Fires when the object receives focus.
*/
onfocus: (ev:FocusEvent) => any;
onmessage: (ev:MessageEvent) => any;
onerror: (ev:ErrorEvent) => any;
/**
* Sets or retrieves whether to display a border for the frame.
*/
frameBorder: string;
onresize: (ev:UIEvent) => any;
name: string;
onafterprint: (ev:Event) => any;
onbeforeprint: (ev:Event) => any;
onoffline: (ev:Event) => any;
border: string;
onunload: (ev:Event) => any;
onhashchange: (ev:Event) => any;
onload: (ev:Event) => any;
onbeforeunload: (ev:BeforeUnloadEvent) => any;
onstorage: (ev:StorageEvent) => any;
onpageshow: (ev:PageTransitionEvent) => any;
onpagehide: (ev:PageTransitionEvent) => any;
addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerdown", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgotpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturedoubletap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerhover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturehold", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointermove", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturechange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturestart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointercancel", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgestureend", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturetap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerout", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msinertiastart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mslostpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerup", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"lostpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerenter", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"gotpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerleave", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mouseleave", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"keydown", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"move", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"keyup", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"reset", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"help", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"dragleave", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusin", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"seeked", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"durationchange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"blur", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"emptied", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"seeking", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"canplay", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"deactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"datasetchanged", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsdelete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"loadstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"losecapture", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragenter", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"controlselect", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"submit", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"change", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"layoutcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"beforeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"canplaythrough", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"filterchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"datasetcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"suspend", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mouseenter", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"errorupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"mouseout", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mousewheel", listener:(ev:MouseWheelEvent) => any, useCapture?:boolean): void;
addEventListener(type:"volumechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"cellchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowexit", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsinserted", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"propertychange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragend", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforepaste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragover", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseup", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragstart", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecopy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drag", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseover", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pause", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousedown", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"click", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"waiting", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"paste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"moveend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"stalled", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousemove", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforeeditfocus", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"ratechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dblclick", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"contextmenu", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadedmetadata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"afterupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"play", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizeend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"playing", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focusout", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dataavailable", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"readystatechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"keypress", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadeddata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforedeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"activate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"movestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"selectstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focus", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"timeupdate", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resize", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"cut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"select", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drop", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"copy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"ended", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"scroll", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"rowenter", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"input", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mscontentzoom", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"cuechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"msmanipulationstatechanged", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"online", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"message", listener:(ev:MessageEvent) => any, useCapture?:boolean): void;
addEventListener(type:"afterprint", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeprint", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"offline", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"unload", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"hashchange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeunload", listener:(ev:BeforeUnloadEvent) => any, useCapture?:boolean): void;
addEventListener(type:"storage", listener:(ev:StorageEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pageshow", listener:(ev:PageTransitionEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pagehide", listener:(ev:PageTransitionEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var HTMLFrameSetElement:{
prototype: HTMLFrameSetElement;
new(): HTMLFrameSetElement;
}
interface Screen extends EventTarget {
width: number;
deviceXDPI: number;
fontSmoothingEnabled: boolean;
bufferDepth: number;
logicalXDPI: number;
systemXDPI: number;
availHeight: number;
height: number;
logicalYDPI: number;
systemYDPI: number;
updateInterval: number;
colorDepth: number;
availWidth: number;
deviceYDPI: number;
pixelDepth: number;
msOrientation: string;
onmsorientationchange: (ev:any) => any;
msLockOrientation(orientation:string): boolean;
msLockOrientation(orientations:string[]): boolean;
msUnlockOrientation(): void;
addEventListener(type:"msorientationchange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var Screen:{
prototype: Screen;
new(): Screen;
}
interface Coordinates {
altitudeAccuracy: number;
longitude: number;
latitude: number;
speed: number;
heading: number;
altitude: number;
accuracy: number;
}
declare var Coordinates:{
prototype: Coordinates;
new(): Coordinates;
}
interface NavigatorGeolocation {
geolocation: Geolocation;
}
interface NavigatorContentUtils {
}
interface EventListener {
(evt:Event): void;
}
interface SVGLangSpace {
xmllang: string;
xmlspace: string;
}
interface DataTransfer {
effectAllowed: string;
dropEffect: string;
types: DOMStringList;
files: FileList;
clearData(format?:string): boolean;
setData(format:string, data:string): boolean;
getData(format:string): string;
}
declare var DataTransfer:{
prototype: DataTransfer;
new(): DataTransfer;
}
interface FocusEvent extends UIEvent {
relatedTarget: EventTarget;
initFocusEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, viewArg:Window, detailArg:number, relatedTargetArg:EventTarget): void;
}
declare var FocusEvent:{
prototype: FocusEvent;
new(): FocusEvent;
}
interface Range {
startOffset: number;
collapsed: boolean;
endOffset: number;
startContainer: Node;
endContainer: Node;
commonAncestorContainer: Node;
setStart(refNode:Node, offset:number): void;
setEndBefore(refNode:Node): void;
setStartBefore(refNode:Node): void;
selectNode(refNode:Node): void;
detach(): void;
getBoundingClientRect(): ClientRect;
toString(): string;
compareBoundaryPoints(how:number, sourceRange:Range): number;
insertNode(newNode:Node): void;
collapse(toStart:boolean): void;
selectNodeContents(refNode:Node): void;
cloneContents(): DocumentFragment;
setEnd(refNode:Node, offset:number): void;
cloneRange(): Range;
getClientRects(): ClientRectList;
surroundContents(newParent:Node): void;
deleteContents(): void;
setStartAfter(refNode:Node): void;
extractContents(): DocumentFragment;
setEndAfter(refNode:Node): void;
createContextualFragment(fragment:string): DocumentFragment;
END_TO_END: number;
START_TO_START: number;
START_TO_END: number;
END_TO_START: number;
}
declare var Range:{
prototype: Range;
new(): Range;
END_TO_END: number;
START_TO_START: number;
START_TO_END: number;
END_TO_START: number;
}
interface SVGPoint {
y: number;
x: number;
matrixTransform(matrix:SVGMatrix): SVGPoint;
}
declare var SVGPoint:{
prototype: SVGPoint;
new(): SVGPoint;
}
interface MSPluginsCollection {
length: number;
refresh(reload?:boolean): void;
}
declare var MSPluginsCollection:{
prototype: MSPluginsCollection;
new(): MSPluginsCollection;
}
interface SVGAnimatedNumberList {
animVal: SVGNumberList;
baseVal: SVGNumberList;
}
declare var SVGAnimatedNumberList:{
prototype: SVGAnimatedNumberList;
new(): SVGAnimatedNumberList;
}
interface SVGSVGElement extends SVGElement, SVGStylable, SVGZoomAndPan, DocumentEvent, SVGLangSpace, SVGLocatable, SVGTests, SVGFitToViewBox, SVGExternalResourcesRequired {
width: SVGAnimatedLength;
x: SVGAnimatedLength;
contentStyleType: string;
onzoom: (ev:any) => any;
y: SVGAnimatedLength;
viewport: SVGRect;
onerror: (ev:ErrorEvent) => any;
pixelUnitToMillimeterY: number;
onresize: (ev:UIEvent) => any;
screenPixelToMillimeterY: number;
height: SVGAnimatedLength;
onabort: (ev:UIEvent) => any;
contentScriptType: string;
pixelUnitToMillimeterX: number;
currentTranslate: SVGPoint;
onunload: (ev:Event) => any;
currentScale: number;
onscroll: (ev:UIEvent) => any;
screenPixelToMillimeterX: number;
setCurrentTime(seconds:number): void;
createSVGLength(): SVGLength;
getIntersectionList(rect:SVGRect, referenceElement:SVGElement): NodeList;
unpauseAnimations(): void;
createSVGRect(): SVGRect;
checkIntersection(element:SVGElement, rect:SVGRect): boolean;
unsuspendRedrawAll(): void;
pauseAnimations(): void;
suspendRedraw(maxWaitMilliseconds:number): number;
deselectAll(): void;
createSVGAngle(): SVGAngle;
getEnclosureList(rect:SVGRect, referenceElement:SVGElement): NodeList;
createSVGTransform(): SVGTransform;
unsuspendRedraw(suspendHandleID:number): void;
forceRedraw(): void;
getCurrentTime(): number;
checkEnclosure(element:SVGElement, rect:SVGRect): boolean;
createSVGMatrix(): SVGMatrix;
createSVGPoint(): SVGPoint;
createSVGNumber(): SVGNumber;
createSVGTransformFromMatrix(matrix:SVGMatrix): SVGTransform;
getComputedStyle(elt:Element, pseudoElt?:string): CSSStyleDeclaration;
getElementById(elementId:string): Element;
addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerdown", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgotpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturedoubletap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerhover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturehold", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointermove", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturechange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturestart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointercancel", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgestureend", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturetap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerout", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msinertiastart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mslostpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerup", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"lostpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerenter", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"gotpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerleave", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mouseover", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mousemove", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseout", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dblclick", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusout", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusin", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mousedown", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mouseup", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"click", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"zoom", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"resize", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"unload", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"scroll", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var SVGSVGElement:{
prototype: SVGSVGElement;
new(): SVGSVGElement;
}
interface HTMLLabelElement extends HTMLElement, MSDataBindingExtensions {
/**
* Sets or retrieves the object to which the given label object is assigned.
*/
htmlFor: string;
/**
* Retrieves a reference to the form that the object is embedded in.
*/
form: HTMLFormElement;
}
declare var HTMLLabelElement:{
prototype: HTMLLabelElement;
new(): HTMLLabelElement;
}
interface MSResourceMetadata {
protocol: string;
fileSize: string;
fileUpdatedDate: string;
nameProp: string;
fileCreatedDate: string;
fileModifiedDate: string;
mimeType: string;
}
interface HTMLLegendElement extends HTMLElement, MSDataBindingExtensions {
/**
* Retrieves a reference to the form that the object is embedded in.
*/
align: string;
/**
* Retrieves a reference to the form that the object is embedded in.
*/
form: HTMLFormElement;
}
declare var HTMLLegendElement:{
prototype: HTMLLegendElement;
new(): HTMLLegendElement;
}
interface HTMLDirectoryElement extends HTMLElement, DOML2DeprecatedListSpaceReduction, DOML2DeprecatedListNumberingAndBulletStyle {
}
declare var HTMLDirectoryElement:{
prototype: HTMLDirectoryElement;
new(): HTMLDirectoryElement;
}
interface SVGAnimatedInteger {
animVal: number;
baseVal: number;
}
declare var SVGAnimatedInteger:{
prototype: SVGAnimatedInteger;
new(): SVGAnimatedInteger;
}
interface SVGTextElement extends SVGTextPositioningElement, SVGTransformable {
}
declare var SVGTextElement:{
prototype: SVGTextElement;
new(): SVGTextElement;
}
interface SVGTSpanElement extends SVGTextPositioningElement {
}
declare var SVGTSpanElement:{
prototype: SVGTSpanElement;
new(): SVGTSpanElement;
}
interface HTMLLIElement extends HTMLElement, DOML2DeprecatedListNumberingAndBulletStyle {
/**
* Sets or retrieves the value of a list item.
*/
value: number;
}
declare var HTMLLIElement:{
prototype: HTMLLIElement;
new(): HTMLLIElement;
}
interface SVGPathSegLinetoVerticalAbs extends SVGPathSeg {
y: number;
}
declare var SVGPathSegLinetoVerticalAbs:{
prototype: SVGPathSegLinetoVerticalAbs;
new(): SVGPathSegLinetoVerticalAbs;
}
interface MSStorageExtensions {
remainingSpace: number;
}
interface SVGStyleElement extends SVGElement, SVGLangSpace {
media: string;
type: string;
title: string;
}
declare var SVGStyleElement:{
prototype: SVGStyleElement;
new(): SVGStyleElement;
}
interface MSCurrentStyleCSSProperties extends MSCSSProperties {
blockDirection: string;
clipBottom: string;
clipLeft: string;
clipRight: string;
clipTop: string;
hasLayout: string;
}
declare var MSCurrentStyleCSSProperties:{
prototype: MSCurrentStyleCSSProperties;
new(): MSCurrentStyleCSSProperties;
}
interface MSHTMLCollectionExtensions {
urns(urn:any): any;
tags(tagName:any): any;
}
interface Storage extends MSStorageExtensions {
length: number;
getItem(key:string): any;
[key: string]: any;
setItem(key:string, data:string): void;
clear(): void;
removeItem(key:string): void;
key(index:number): string;
[index: number]: string;
}
declare var Storage:{
prototype: Storage;
new(): Storage;
}
interface HTMLIFrameElement extends HTMLElement, GetSVGDocument, MSDataBindingExtensions {
/**
* Sets or retrieves the width of the object.
*/
width: string;
/**
* Sets or retrieves whether the frame can be scrolled.
*/
scrolling: string;
/**
* Sets or retrieves the top and bottom margin heights before displaying the text in a frame.
*/
marginHeight: string;
/**
* Sets or retrieves the left and right margin widths before displaying the text in a frame.
*/
marginWidth: string;
/**
* Sets or retrieves the amount of additional space between the frames.
*/
frameSpacing: any;
/**
* Sets or retrieves whether to display a border for the frame.
*/
frameBorder: string;
/**
* Sets or retrieves whether the user can resize the frame.
*/
noResize: boolean;
/**
* Sets or retrieves the vertical margin for the object.
*/
vspace: number;
/**
* Retrieves the object of the specified.
*/
contentWindow: Window;
/**
* Sets or retrieves how the object is aligned with adjacent text.
*/
align: string;
/**
* Sets or retrieves a URL to be loaded by the object.
*/
src: string;
/**
* Sets or retrieves the frame name.
*/
name: string;
/**
* Sets or retrieves the height of the object.
*/
height: string;
/**
* Specifies the properties of a border drawn around an object.
*/
border: string;
/**
* Retrieves the document object of the page or frame.
*/
contentDocument: Document;
/**
* Sets or retrieves the horizontal margin for the object.
*/
hspace: number;
/**
* Sets or retrieves a URI to a long description of the object.
*/
longDesc: string;
/**
* Sets the value indicating whether the source file of a frame or iframe has specific security restrictions applied.
*/
security: any;
/**
* Raised when the object has been completely received from the server.
*/
onload: (ev:Event) => any;
sandbox: DOMSettableTokenList;
addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerdown", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgotpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturedoubletap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerhover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturehold", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointermove", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturechange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturestart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointercancel", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgestureend", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturetap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerout", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msinertiastart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mslostpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerup", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"lostpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerenter", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"gotpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerleave", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mouseleave", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"keydown", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"move", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"keyup", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"reset", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"help", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"dragleave", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusin", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"seeked", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"durationchange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"blur", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"emptied", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"seeking", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"canplay", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"deactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"datasetchanged", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsdelete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"loadstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"losecapture", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragenter", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"controlselect", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"submit", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"change", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"layoutcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"beforeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"canplaythrough", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"filterchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"datasetcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"suspend", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mouseenter", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"errorupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"mouseout", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mousewheel", listener:(ev:MouseWheelEvent) => any, useCapture?:boolean): void;
addEventListener(type:"volumechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"cellchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowexit", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsinserted", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"propertychange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragend", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforepaste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragover", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseup", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragstart", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecopy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drag", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseover", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pause", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousedown", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"click", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"waiting", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"paste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"moveend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"stalled", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousemove", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforeeditfocus", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"ratechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dblclick", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"contextmenu", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadedmetadata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"afterupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"play", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizeend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"playing", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focusout", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dataavailable", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"readystatechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"keypress", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadeddata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforedeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"activate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"movestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"selectstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focus", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"timeupdate", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resize", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"cut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"select", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drop", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"copy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"ended", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"scroll", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"rowenter", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"input", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mscontentzoom", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"cuechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"msmanipulationstatechanged", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var HTMLIFrameElement:{
prototype: HTMLIFrameElement;
new(): HTMLIFrameElement;
}
interface TextRangeCollection {
length: number;
item(index:number): TextRange;
[index: number]: TextRange;
}
declare var TextRangeCollection:{
prototype: TextRangeCollection;
new(): TextRangeCollection;
}
interface HTMLBodyElement extends HTMLElement, DOML2DeprecatedBackgroundStyle, DOML2DeprecatedBackgroundColorStyle {
scroll: string;
ononline: (ev:Event) => any;
onblur: (ev:FocusEvent) => any;
noWrap: boolean;
onfocus: (ev:FocusEvent) => any;
onmessage: (ev:MessageEvent) => any;
text: any;
onerror: (ev:ErrorEvent) => any;
bgProperties: string;
onresize: (ev:UIEvent) => any;
link: any;
aLink: any;
bottomMargin: any;
topMargin: any;
onafterprint: (ev:Event) => any;
vLink: any;
onbeforeprint: (ev:Event) => any;
onoffline: (ev:Event) => any;
onunload: (ev:Event) => any;
onhashchange: (ev:Event) => any;
onload: (ev:Event) => any;
rightMargin: any;
onbeforeunload: (ev:BeforeUnloadEvent) => any;
leftMargin: any;
onstorage: (ev:StorageEvent) => any;
onpopstate: (ev:PopStateEvent) => any;
onpageshow: (ev:PageTransitionEvent) => any;
onpagehide: (ev:PageTransitionEvent) => any;
createTextRange(): TextRange;
addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerdown", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgotpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturedoubletap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerhover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturehold", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointermove", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturechange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturestart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointercancel", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgestureend", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturetap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerout", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msinertiastart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mslostpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerup", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"lostpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerenter", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"gotpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerleave", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mouseleave", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"keydown", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"move", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"keyup", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"reset", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"help", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"dragleave", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusin", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"seeked", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"durationchange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"blur", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"emptied", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"seeking", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"canplay", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"deactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"datasetchanged", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsdelete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"loadstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"losecapture", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragenter", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"controlselect", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"submit", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"change", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"layoutcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"beforeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"canplaythrough", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"filterchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"datasetcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"suspend", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mouseenter", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"errorupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"mouseout", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mousewheel", listener:(ev:MouseWheelEvent) => any, useCapture?:boolean): void;
addEventListener(type:"volumechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"cellchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowexit", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsinserted", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"propertychange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragend", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforepaste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragover", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseup", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragstart", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecopy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drag", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseover", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pause", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousedown", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"click", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"waiting", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"paste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"moveend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"stalled", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousemove", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforeeditfocus", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"ratechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dblclick", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"contextmenu", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadedmetadata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"afterupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"play", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizeend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"playing", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focusout", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dataavailable", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"readystatechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"keypress", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadeddata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforedeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"activate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"movestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"selectstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focus", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"timeupdate", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resize", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"cut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"select", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drop", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"copy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"ended", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"scroll", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"rowenter", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"input", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mscontentzoom", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"cuechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"msmanipulationstatechanged", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"online", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"message", listener:(ev:MessageEvent) => any, useCapture?:boolean): void;
addEventListener(type:"afterprint", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeprint", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"offline", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"unload", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"hashchange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeunload", listener:(ev:BeforeUnloadEvent) => any, useCapture?:boolean): void;
addEventListener(type:"storage", listener:(ev:StorageEvent) => any, useCapture?:boolean): void;
addEventListener(type:"popstate", listener:(ev:PopStateEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pageshow", listener:(ev:PageTransitionEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pagehide", listener:(ev:PageTransitionEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var HTMLBodyElement:{
prototype: HTMLBodyElement;
new(): HTMLBodyElement;
}
interface DocumentType extends Node {
name: string;
notations: NamedNodeMap;
systemId: string;
internalSubset: string;
entities: NamedNodeMap;
publicId: string;
}
declare var DocumentType:{
prototype: DocumentType;
new(): DocumentType;
}
interface SVGRadialGradientElement extends SVGGradientElement {
cx: SVGAnimatedLength;
r: SVGAnimatedLength;
cy: SVGAnimatedLength;
fx: SVGAnimatedLength;
fy: SVGAnimatedLength;
}
declare var SVGRadialGradientElement:{
prototype: SVGRadialGradientElement;
new(): SVGRadialGradientElement;
}
interface MutationEvent extends Event {
newValue: string;
attrChange: number;
attrName: string;
prevValue: string;
relatedNode: Node;
initMutationEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, relatedNodeArg:Node, prevValueArg:string, newValueArg:string, attrNameArg:string, attrChangeArg:number): void;
MODIFICATION: number;
REMOVAL: number;
ADDITION: number;
}
declare var MutationEvent:{
prototype: MutationEvent;
new(): MutationEvent;
MODIFICATION: number;
REMOVAL: number;
ADDITION: number;
}
interface DragEvent extends MouseEvent {
dataTransfer: DataTransfer;
initDragEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, viewArg:Window, detailArg:number, screenXArg:number, screenYArg:number, clientXArg:number, clientYArg:number, ctrlKeyArg:boolean, altKeyArg:boolean, shiftKeyArg:boolean, metaKeyArg:boolean, buttonArg:number, relatedTargetArg:EventTarget, dataTransferArg:DataTransfer): void;
msConvertURL(file:File, targetType:string, targetURL?:string): void;
}
declare var DragEvent:{
prototype: DragEvent;
new(): DragEvent;
}
interface HTMLTableSectionElement extends HTMLElement, HTMLTableAlignment, DOML2DeprecatedBackgroundColorStyle {
/**
* Sets or retrieves a value that indicates the table alignment.
*/
align: string;
/**
* Sets or retrieves the number of horizontal rows contained in the object.
*/
rows: HTMLCollection;
/**
* Removes the specified row (tr) from the element and from the rows collection.
* @param index Number that specifies the zero-based position in the rows collection of the row to remove.
*/
deleteRow(index?:number): void;
/**
* Moves a table row to a new position.
* @param indexFrom Number that specifies the index in the rows collection of the table row that is moved.
* @param indexTo Number that specifies where the row is moved within the rows collection.
*/
moveRow(indexFrom?:number, indexTo?:number): any;
/**
* Creates a new row (tr) in the table, and adds the row to the rows collection.
* @param index Number that specifies where to insert the row in the rows collection. The default value is -1, which appends the new row to the end of the rows collection.
*/
insertRow(index?:number): HTMLElement;
}
declare var HTMLTableSectionElement:{
prototype: HTMLTableSectionElement;
new(): HTMLTableSectionElement;
}
interface DOML2DeprecatedListNumberingAndBulletStyle {
type: string;
}
interface HTMLInputElement extends HTMLElement, MSDataBindingExtensions {
/**
* Sets or retrieves the width of the object.
*/
width: string;
status: boolean;
/**
* Retrieves a reference to the form that the object is embedded in.
*/
form: HTMLFormElement;
/**
* Gets or sets the starting position or offset of a text selection.
*/
selectionStart: number;
indeterminate: boolean;
readOnly: boolean;
size: number;
loop: number;
/**
* Gets or sets the end position or offset of a text selection.
*/
selectionEnd: number;
/**
* Sets or retrieves the URL of the virtual reality modeling language (VRML) world to be displayed in the window.
*/
vrml: string;
/**
* Sets or retrieves a lower resolution image to display.
*/
lowsrc: string;
/**
* Sets or retrieves the vertical margin for the object.
*/
vspace: number;
/**
* Sets or retrieves a comma-separated list of content types.
*/
accept: string;
/**
* Sets or retrieves a text alternative to the graphic.
*/
alt: string;
/**
* Sets or retrieves the state of the check box or radio button.
*/
defaultChecked: boolean;
/**
* Sets or retrieves how the object is aligned with adjacent text.
*/
align: string;
/**
* Returns the value of the data at the cursor's current position.
*/
value: string;
/**
* The address or URL of the a media resource that is to be considered.
*/
src: string;
/**
* Sets or retrieves the name of the object.
*/
name: string;
/**
* Sets or retrieves the URL, often with a bookmark extension (#name), to use as a client-side image map.
*/
useMap: string;
/**
* Sets or retrieves the height of the object.
*/
height: string;
/**
* Sets or retrieves the width of the border to draw around the object.
*/
border: string;
dynsrc: string;
/**
* Sets or retrieves the state of the check box or radio button.
*/
checked: boolean;
/**
* Sets or retrieves the width of the border to draw around the object.
*/
hspace: number;
/**
* Sets or retrieves the maximum number of characters that the user can enter in a text control.
*/
maxLength: number;
/**
* Returns the content type of the object.
*/
type: string;
/**
* Sets or retrieves the initial contents of the object.
*/
defaultValue: string;
/**
* Retrieves whether the object is fully loaded.
*/
complete: boolean;
start: string;
/**
* Returns the error message that would be displayed if the user submits the form, or an empty string if no error message. It also triggers the standard error message, such as "this is a required field". The result is that the user sees validation messages without actually submitting.
*/
validationMessage: string;
/**
* Returns a FileList object on a file type input object.
*/
files: FileList;
/**
* Defines the maximum acceptable value for an input element with type="number".When used with the min and step attributes, lets you control the range and increment (such as only even numbers) that the user can enter into an input field.
*/
max: string;
/**
* Overrides the target attribute on a form element.
*/
formTarget: string;
/**
* Returns whether an element will successfully validate based on forms validation rules and constraints.
*/
willValidate: boolean;
/**
* Defines an increment or jump between values that you want to allow the user to enter. When used with the max and min attributes, lets you control the range and increment (for example, allow only even numbers) that the user can enter into an input field.
*/
step: string;
/**
* Provides a way to direct a user to a specific field when a document loads. This can provide both direction and convenience for a user, reducing the need to click or tab to a field when a page opens. This attribute is true when present on an element, and false when missing.
*/
autofocus: boolean;
/**
* When present, marks an element that can't be submitted without a value.
*/
required: boolean;
/**
* Used to override the encoding (formEnctype attribute) specified on the form element.
*/
formEnctype: string;
/**
* Returns the input field value as a number.
*/
valueAsNumber: number;
/**
* Gets or sets a text string that is displayed in an input field as a hint or prompt to users as the format or type of information they need to enter.The text appears in an input field until the user puts focus on the field.
*/
placeholder: string;
/**
* Overrides the submit method attribute previously specified on a form element.
*/
formMethod: string;
/**
* Specifies the ID of a pre-defined datalist of options for an input element.
*/
list: HTMLElement;
/**
* Specifies whether autocomplete is applied to an editable text field.
*/
autocomplete: string;
/**
* Defines the minimum acceptable value for an input element with type="number". When used with the max and step attributes, lets you control the range and increment (such as even numbers only) that the user can enter into an input field.
*/
min: string;
/**
* Overrides the action attribute (where the data on a form is sent) on the parent form element.
*/
formAction: string;
/**
* Gets or sets a string containing a regular expression that the user's input must match.
*/
pattern: string;
/**
* Returns a ValidityState object that represents the validity states of an element.
*/
validity: ValidityState;
/**
* Overrides any validation or required attributes on a form or form elements to allow it to be submitted without validation. This can be used to create a "save draft"-type submit option.
*/
formNoValidate: string;
/**
* Sets or retrieves the Boolean value indicating whether multiple items can be selected from a list.
*/
multiple: boolean;
/**
* Creates a TextRange object for the element.
*/
createTextRange(): TextRange;
/**
* Sets the start and end positions of a selection in a text field.
* @param start The offset into the text field for the start of the selection.
* @param end The offset into the text field for the end of the selection.
*/
setSelectionRange(start:number, end:number): void;
/**
* Makes the selection equal to the current object.
*/
select(): void;
/**
* Returns whether a form will validate when it is submitted, without having to submit it.
*/
checkValidity(): boolean;
/**
* Decrements a range input control's value by the value given by the Step attribute. If the optional parameter is used, it will decrement the input control's step value multiplied by the parameter's value.
* @param n Value to decrement the value by.
*/
stepDown(n?:number): void;
/**
* Increments a range input control's value by the value given by the Step attribute. If the optional parameter is used, will increment the input control's value by that value.
* @param n Value to increment the value by.
*/
stepUp(n?:number): void;
/**
* Sets a custom error message that is displayed when a form is submitted.
* @param error Sets a custom error message that is displayed when a form is submitted.
*/
setCustomValidity(error:string): void;
}
declare var HTMLInputElement:{
prototype: HTMLInputElement;
new(): HTMLInputElement;
}
interface HTMLAnchorElement extends HTMLElement, MSDataBindingExtensions {
/**
* Sets or retrieves the relationship between the object and the destination of the link.
*/
rel: string;
/**
* Contains the protocol of the URL.
*/
protocol: string;
/**
* Sets or retrieves the substring of the href property that follows the question mark.
*/
search: string;
/**
* Sets or retrieves the coordinates of the object.
*/
coords: string;
/**
* Contains the hostname of a URL.
*/
hostname: string;
/**
* Contains the pathname of the URL.
*/
pathname: string;
Methods: string;
/**
* Sets or retrieves the window or frame at which to target content.
*/
target: string;
protocolLong: string;
/**
* Sets or retrieves a destination URL or an anchor point.
*/
href: string;
/**
* Sets or retrieves the shape of the object.
*/
name: string;
/**
* Sets or retrieves the character set used to encode the object.
*/
charset: string;
/**
* Sets or retrieves the language code of the object.
*/
hreflang: string;
/**
* Sets or retrieves the port number associated with a URL.
*/
port: string;
/**
* Contains the hostname and port values of the URL.
*/
host: string;
/**
* Contains the anchor portion of the URL including the hash sign (#).
*/
hash: string;
nameProp: string;
urn: string;
/**
* Sets or retrieves the relationship between the object and the destination of the link.
*/
rev: string;
/**
* Sets or retrieves the shape of the object.
*/
shape: string;
type: string;
mimeType: string;
/**
* Retrieves or sets the text of the object as a string.
*/
text: string;
/**
* Returns a string representation of an object.
*/
toString(): string;
}
declare var HTMLAnchorElement:{
prototype: HTMLAnchorElement;
new(): HTMLAnchorElement;
}
interface HTMLParamElement extends HTMLElement {
/**
* Sets or retrieves the value of an input parameter for an element.
*/
value: string;
/**
* Sets or retrieves the name of an input parameter for an element.
*/
name: string;
/**
* Sets or retrieves the content type of the resource designated by the value attribute.
*/
type: string;
/**
* Sets or retrieves the data type of the value attribute.
*/
valueType: string;
}
declare var HTMLParamElement:{
prototype: HTMLParamElement;
new(): HTMLParamElement;
}
interface SVGImageElement extends SVGElement, SVGStylable, SVGTransformable, SVGLangSpace, SVGTests, SVGExternalResourcesRequired, SVGURIReference {
y: SVGAnimatedLength;
width: SVGAnimatedLength;
preserveAspectRatio: SVGAnimatedPreserveAspectRatio;
x: SVGAnimatedLength;
height: SVGAnimatedLength;
}
declare var SVGImageElement:{
prototype: SVGImageElement;
new(): SVGImageElement;
}
interface SVGAnimatedNumber {
animVal: number;
baseVal: number;
}
declare var SVGAnimatedNumber:{
prototype: SVGAnimatedNumber;
new(): SVGAnimatedNumber;
}
interface PerformanceTiming {
redirectStart: number;
domainLookupEnd: number;
responseStart: number;
domComplete: number;
domainLookupStart: number;
loadEventStart: number;
msFirstPaint: number;
unloadEventEnd: number;
fetchStart: number;
requestStart: number;
domInteractive: number;
navigationStart: number;
connectEnd: number;
loadEventEnd: number;
connectStart: number;
responseEnd: number;
domLoading: number;
redirectEnd: number;
unloadEventStart: number;
domContentLoadedEventStart: number;
domContentLoadedEventEnd: number;
toJSON(): any;
}
declare var PerformanceTiming:{
prototype: PerformanceTiming;
new(): PerformanceTiming;
}
interface HTMLPreElement extends HTMLElement, DOML2DeprecatedTextFlowControl {
/**
* Sets or gets a value that you can use to implement your own width functionality for the object.
*/
width: number;
/**
* Indicates a citation by rendering text in italic type.
*/
cite: string;
}
declare var HTMLPreElement:{
prototype: HTMLPreElement;
new(): HTMLPreElement;
}
interface EventException {
code: number;
message: string;
name: string;
toString(): string;
DISPATCH_REQUEST_ERR: number;
UNSPECIFIED_EVENT_TYPE_ERR: number;
}
declare var EventException:{
prototype: EventException;
new(): EventException;
DISPATCH_REQUEST_ERR: number;
UNSPECIFIED_EVENT_TYPE_ERR: number;
}
interface MSNavigatorDoNotTrack {
msDoNotTrack: string;
removeSiteSpecificTrackingException(args:ExceptionInformation): void;
removeWebWideTrackingException(args:ExceptionInformation): void;
storeWebWideTrackingException(args:StoreExceptionsInformation): void;
storeSiteSpecificTrackingException(args:StoreSiteSpecificExceptionsInformation): void;
confirmSiteSpecificTrackingException(args:ConfirmSiteSpecificExceptionsInformation): boolean;
confirmWebWideTrackingException(args:ExceptionInformation): boolean;
}
interface NavigatorOnLine {
onLine: boolean;
}
interface WindowLocalStorage {
localStorage: Storage;
}
interface SVGMetadataElement extends SVGElement {
}
declare var SVGMetadataElement:{
prototype: SVGMetadataElement;
new(): SVGMetadataElement;
}
interface SVGPathSegArcRel extends SVGPathSeg {
y: number;
sweepFlag: boolean;
r2: number;
x: number;
angle: number;
r1: number;
largeArcFlag: boolean;
}
declare var SVGPathSegArcRel:{
prototype: SVGPathSegArcRel;
new(): SVGPathSegArcRel;
}
interface SVGPathSegMovetoAbs extends SVGPathSeg {
y: number;
x: number;
}
declare var SVGPathSegMovetoAbs:{
prototype: SVGPathSegMovetoAbs;
new(): SVGPathSegMovetoAbs;
}
interface SVGStringList {
numberOfItems: number;
replaceItem(newItem:string, index:number): string;
getItem(index:number): string;
clear(): void;
appendItem(newItem:string): string;
initialize(newItem:string): string;
removeItem(index:number): string;
insertItemBefore(newItem:string, index:number): string;
}
declare var SVGStringList:{
prototype: SVGStringList;
new(): SVGStringList;
}
interface XDomainRequest {
timeout: number;
onerror: (ev:ErrorEvent) => any;
onload: (ev:Event) => any;
onprogress: (ev:ProgressEvent) => any;
ontimeout: (ev:Event) => any;
responseText: string;
contentType: string;
open(method:string, url:string): void;
abort(): void;
send(data?:any): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"timeout", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var XDomainRequest:{
prototype: XDomainRequest;
new(): XDomainRequest;
create(): XDomainRequest;
}
interface DOML2DeprecatedBackgroundColorStyle {
bgColor: any;
}
interface ElementTraversal {
childElementCount: number;
previousElementSibling: Element;
lastElementChild: Element;
nextElementSibling: Element;
firstElementChild: Element;
}
interface SVGLength {
valueAsString: string;
valueInSpecifiedUnits: number;
value: number;
unitType: number;
newValueSpecifiedUnits(unitType:number, valueInSpecifiedUnits:number): void;
convertToSpecifiedUnits(unitType:number): void;
SVG_LENGTHTYPE_NUMBER: number;
SVG_LENGTHTYPE_CM: number;
SVG_LENGTHTYPE_PC: number;
SVG_LENGTHTYPE_PERCENTAGE: number;
SVG_LENGTHTYPE_MM: number;
SVG_LENGTHTYPE_PT: number;
SVG_LENGTHTYPE_IN: number;
SVG_LENGTHTYPE_EMS: number;
SVG_LENGTHTYPE_PX: number;
SVG_LENGTHTYPE_UNKNOWN: number;
SVG_LENGTHTYPE_EXS: number;
}
declare var SVGLength:{
prototype: SVGLength;
new(): SVGLength;
SVG_LENGTHTYPE_NUMBER: number;
SVG_LENGTHTYPE_CM: number;
SVG_LENGTHTYPE_PC: number;
SVG_LENGTHTYPE_PERCENTAGE: number;
SVG_LENGTHTYPE_MM: number;
SVG_LENGTHTYPE_PT: number;
SVG_LENGTHTYPE_IN: number;
SVG_LENGTHTYPE_EMS: number;
SVG_LENGTHTYPE_PX: number;
SVG_LENGTHTYPE_UNKNOWN: number;
SVG_LENGTHTYPE_EXS: number;
}
interface SVGPolygonElement extends SVGElement, SVGStylable, SVGTransformable, SVGLangSpace, SVGAnimatedPoints, SVGTests, SVGExternalResourcesRequired {
}
declare var SVGPolygonElement:{
prototype: SVGPolygonElement;
new(): SVGPolygonElement;
}
interface HTMLPhraseElement extends HTMLElement {
/**
* Sets or retrieves the date and time of a modification to the object.
*/
dateTime: string;
/**
* Sets or retrieves reference information about the object.
*/
cite: string;
}
declare var HTMLPhraseElement:{
prototype: HTMLPhraseElement;
new(): HTMLPhraseElement;
}
interface NavigatorStorageUtils {
}
interface SVGPathSegCurvetoCubicRel extends SVGPathSeg {
y: number;
y1: number;
x2: number;
x: number;
x1: number;
y2: number;
}
declare var SVGPathSegCurvetoCubicRel:{
prototype: SVGPathSegCurvetoCubicRel;
new(): SVGPathSegCurvetoCubicRel;
}
interface SVGTextContentElement extends SVGElement, SVGStylable, SVGLangSpace, SVGTests, SVGExternalResourcesRequired {
textLength: SVGAnimatedLength;
lengthAdjust: SVGAnimatedEnumeration;
getCharNumAtPosition(point:SVGPoint): number;
getStartPositionOfChar(charnum:number): SVGPoint;
getExtentOfChar(charnum:number): SVGRect;
getComputedTextLength(): number;
getSubStringLength(charnum:number, nchars:number): number;
selectSubString(charnum:number, nchars:number): void;
getNumberOfChars(): number;
getRotationOfChar(charnum:number): number;
getEndPositionOfChar(charnum:number): SVGPoint;
LENGTHADJUST_SPACING: number;
LENGTHADJUST_SPACINGANDGLYPHS: number;
LENGTHADJUST_UNKNOWN: number;
}
declare var SVGTextContentElement:{
prototype: SVGTextContentElement;
new(): SVGTextContentElement;
LENGTHADJUST_SPACING: number;
LENGTHADJUST_SPACINGANDGLYPHS: number;
LENGTHADJUST_UNKNOWN: number;
}
interface DOML2DeprecatedColorProperty {
color: string;
}
interface Location {
hash: string;
protocol: string;
search: string;
href: string;
hostname: string;
port: string;
pathname: string;
host: string;
reload(flag?:boolean): void;
replace(url:string): void;
assign(url:string): void;
toString(): string;
}
declare var Location:{
prototype: Location;
new(): Location;
}
interface HTMLTitleElement extends HTMLElement {
/**
* Retrieves or sets the text of the object as a string.
*/
text: string;
}
declare var HTMLTitleElement:{
prototype: HTMLTitleElement;
new(): HTMLTitleElement;
}
interface HTMLStyleElement extends HTMLElement, LinkStyle {
/**
* Sets or retrieves the media type.
*/
media: string;
/**
* Retrieves the CSS language in which the style sheet is written.
*/
type: string;
}
declare var HTMLStyleElement:{
prototype: HTMLStyleElement;
new(): HTMLStyleElement;
}
interface PerformanceEntry {
name: string;
startTime: number;
duration: number;
entryType: string;
}
declare var PerformanceEntry:{
prototype: PerformanceEntry;
new(): PerformanceEntry;
}
interface SVGTransform {
type: number;
angle: number;
matrix: SVGMatrix;
setTranslate(tx:number, ty:number): void;
setScale(sx:number, sy:number): void;
setMatrix(matrix:SVGMatrix): void;
setSkewY(angle:number): void;
setRotate(angle:number, cx:number, cy:number): void;
setSkewX(angle:number): void;
SVG_TRANSFORM_SKEWX: number;
SVG_TRANSFORM_UNKNOWN: number;
SVG_TRANSFORM_SCALE: number;
SVG_TRANSFORM_TRANSLATE: number;
SVG_TRANSFORM_MATRIX: number;
SVG_TRANSFORM_ROTATE: number;
SVG_TRANSFORM_SKEWY: number;
}
declare var SVGTransform:{
prototype: SVGTransform;
new(): SVGTransform;
SVG_TRANSFORM_SKEWX: number;
SVG_TRANSFORM_UNKNOWN: number;
SVG_TRANSFORM_SCALE: number;
SVG_TRANSFORM_TRANSLATE: number;
SVG_TRANSFORM_MATRIX: number;
SVG_TRANSFORM_ROTATE: number;
SVG_TRANSFORM_SKEWY: number;
}
interface UIEvent extends Event {
detail: number;
view: Window;
initUIEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, viewArg:Window, detailArg:number): void;
}
declare var UIEvent:{
prototype: UIEvent;
new(): UIEvent;
}
interface SVGURIReference {
href: SVGAnimatedString;
}
interface SVGPathSeg {
pathSegType: number;
pathSegTypeAsLetter: string;
PATHSEG_MOVETO_REL: number;
PATHSEG_LINETO_VERTICAL_REL: number;
PATHSEG_CURVETO_CUBIC_SMOOTH_ABS: number;
PATHSEG_CURVETO_QUADRATIC_REL: number;
PATHSEG_CURVETO_CUBIC_ABS: number;
PATHSEG_LINETO_HORIZONTAL_ABS: number;
PATHSEG_CURVETO_QUADRATIC_ABS: number;
PATHSEG_LINETO_ABS: number;
PATHSEG_CLOSEPATH: number;
PATHSEG_LINETO_HORIZONTAL_REL: number;
PATHSEG_CURVETO_CUBIC_SMOOTH_REL: number;
PATHSEG_LINETO_REL: number;
PATHSEG_CURVETO_QUADRATIC_SMOOTH_ABS: number;
PATHSEG_ARC_REL: number;
PATHSEG_CURVETO_CUBIC_REL: number;
PATHSEG_UNKNOWN: number;
PATHSEG_LINETO_VERTICAL_ABS: number;
PATHSEG_ARC_ABS: number;
PATHSEG_MOVETO_ABS: number;
PATHSEG_CURVETO_QUADRATIC_SMOOTH_REL: number;
}
declare var SVGPathSeg:{
prototype: SVGPathSeg;
new(): SVGPathSeg;
PATHSEG_MOVETO_REL: number;
PATHSEG_LINETO_VERTICAL_REL: number;
PATHSEG_CURVETO_CUBIC_SMOOTH_ABS: number;
PATHSEG_CURVETO_QUADRATIC_REL: number;
PATHSEG_CURVETO_CUBIC_ABS: number;
PATHSEG_LINETO_HORIZONTAL_ABS: number;
PATHSEG_CURVETO_QUADRATIC_ABS: number;
PATHSEG_LINETO_ABS: number;
PATHSEG_CLOSEPATH: number;
PATHSEG_LINETO_HORIZONTAL_REL: number;
PATHSEG_CURVETO_CUBIC_SMOOTH_REL: number;
PATHSEG_LINETO_REL: number;
PATHSEG_CURVETO_QUADRATIC_SMOOTH_ABS: number;
PATHSEG_ARC_REL: number;
PATHSEG_CURVETO_CUBIC_REL: number;
PATHSEG_UNKNOWN: number;
PATHSEG_LINETO_VERTICAL_ABS: number;
PATHSEG_ARC_ABS: number;
PATHSEG_MOVETO_ABS: number;
PATHSEG_CURVETO_QUADRATIC_SMOOTH_REL: number;
}
interface WheelEvent extends MouseEvent {
deltaZ: number;
deltaX: number;
deltaMode: number;
deltaY: number;
initWheelEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, viewArg:Window, detailArg:number, screenXArg:number, screenYArg:number, clientXArg:number, clientYArg:number, buttonArg:number, relatedTargetArg:EventTarget, modifiersListArg:string, deltaXArg:number, deltaYArg:number, deltaZArg:number, deltaMode:number): void;
getCurrentPoint(element:Element): void;
DOM_DELTA_PIXEL: number;
DOM_DELTA_LINE: number;
DOM_DELTA_PAGE: number;
}
declare var WheelEvent:{
prototype: WheelEvent;
new(): WheelEvent;
DOM_DELTA_PIXEL: number;
DOM_DELTA_LINE: number;
DOM_DELTA_PAGE: number;
}
interface MSEventAttachmentTarget {
attachEvent(event:string, listener:EventListener): boolean;
detachEvent(event:string, listener:EventListener): void;
}
interface SVGNumber {
value: number;
}
declare var SVGNumber:{
prototype: SVGNumber;
new(): SVGNumber;
}
interface SVGPathElement extends SVGElement, SVGStylable, SVGAnimatedPathData, SVGTransformable, SVGLangSpace, SVGTests, SVGExternalResourcesRequired {
getPathSegAtLength(distance:number): number;
getPointAtLength(distance:number): SVGPoint;
createSVGPathSegCurvetoQuadraticAbs(x:number, y:number, x1:number, y1:number): SVGPathSegCurvetoQuadraticAbs;
createSVGPathSegLinetoRel(x:number, y:number): SVGPathSegLinetoRel;
createSVGPathSegCurvetoQuadraticRel(x:number, y:number, x1:number, y1:number): SVGPathSegCurvetoQuadraticRel;
createSVGPathSegCurvetoCubicAbs(x:number, y:number, x1:number, y1:number, x2:number, y2:number): SVGPathSegCurvetoCubicAbs;
createSVGPathSegLinetoAbs(x:number, y:number): SVGPathSegLinetoAbs;
createSVGPathSegClosePath(): SVGPathSegClosePath;
createSVGPathSegCurvetoCubicRel(x:number, y:number, x1:number, y1:number, x2:number, y2:number): SVGPathSegCurvetoCubicRel;
createSVGPathSegCurvetoQuadraticSmoothRel(x:number, y:number): SVGPathSegCurvetoQuadraticSmoothRel;
createSVGPathSegMovetoRel(x:number, y:number): SVGPathSegMovetoRel;
createSVGPathSegCurvetoCubicSmoothAbs(x:number, y:number, x2:number, y2:number): SVGPathSegCurvetoCubicSmoothAbs;
createSVGPathSegMovetoAbs(x:number, y:number): SVGPathSegMovetoAbs;
createSVGPathSegLinetoVerticalRel(y:number): SVGPathSegLinetoVerticalRel;
createSVGPathSegArcRel(x:number, y:number, r1:number, r2:number, angle:number, largeArcFlag:boolean, sweepFlag:boolean): SVGPathSegArcRel;
createSVGPathSegCurvetoQuadraticSmoothAbs(x:number, y:number): SVGPathSegCurvetoQuadraticSmoothAbs;
createSVGPathSegLinetoHorizontalRel(x:number): SVGPathSegLinetoHorizontalRel;
getTotalLength(): number;
createSVGPathSegCurvetoCubicSmoothRel(x:number, y:number, x2:number, y2:number): SVGPathSegCurvetoCubicSmoothRel;
createSVGPathSegLinetoHorizontalAbs(x:number): SVGPathSegLinetoHorizontalAbs;
createSVGPathSegLinetoVerticalAbs(y:number): SVGPathSegLinetoVerticalAbs;
createSVGPathSegArcAbs(x:number, y:number, r1:number, r2:number, angle:number, largeArcFlag:boolean, sweepFlag:boolean): SVGPathSegArcAbs;
}
declare var SVGPathElement:{
prototype: SVGPathElement;
new(): SVGPathElement;
}
interface MSCompatibleInfo {
version: string;
userAgent: string;
}
declare var MSCompatibleInfo:{
prototype: MSCompatibleInfo;
new(): MSCompatibleInfo;
}
interface Text extends CharacterData, MSNodeExtensions {
wholeText: string;
splitText(offset:number): Text;
replaceWholeText(content:string): Text;
}
declare var Text:{
prototype: Text;
new(): Text;
}
interface SVGAnimatedRect {
animVal: SVGRect;
baseVal: SVGRect;
}
declare var SVGAnimatedRect:{
prototype: SVGAnimatedRect;
new(): SVGAnimatedRect;
}
interface CSSNamespaceRule extends CSSRule {
namespaceURI: string;
prefix: string;
}
declare var CSSNamespaceRule:{
prototype: CSSNamespaceRule;
new(): CSSNamespaceRule;
}
interface SVGPathSegList {
numberOfItems: number;
replaceItem(newItem:SVGPathSeg, index:number): SVGPathSeg;
getItem(index:number): SVGPathSeg;
clear(): void;
appendItem(newItem:SVGPathSeg): SVGPathSeg;
initialize(newItem:SVGPathSeg): SVGPathSeg;
removeItem(index:number): SVGPathSeg;
insertItemBefore(newItem:SVGPathSeg, index:number): SVGPathSeg;
}
declare var SVGPathSegList:{
prototype: SVGPathSegList;
new(): SVGPathSegList;
}
interface HTMLUnknownElement extends HTMLElement, MSDataBindingRecordSetReadonlyExtensions {
}
declare var HTMLUnknownElement:{
prototype: HTMLUnknownElement;
new(): HTMLUnknownElement;
}
interface HTMLAudioElement extends HTMLMediaElement {
}
declare var HTMLAudioElement:{
prototype: HTMLAudioElement;
new(): HTMLAudioElement;
}
interface MSImageResourceExtensions {
dynsrc: string;
vrml: string;
lowsrc: string;
start: string;
loop: number;
}
interface PositionError {
code: number;
message: string;
toString(): string;
POSITION_UNAVAILABLE: number;
PERMISSION_DENIED: number;
TIMEOUT: number;
}
declare var PositionError:{
prototype: PositionError;
new(): PositionError;
POSITION_UNAVAILABLE: number;
PERMISSION_DENIED: number;
TIMEOUT: number;
}
interface HTMLTableCellElement extends HTMLElement, HTMLTableAlignment, DOML2DeprecatedBackgroundStyle, DOML2DeprecatedBackgroundColorStyle {
/**
* Sets or retrieves the width of the object.
*/
width: number;
/**
* Sets or retrieves a list of header cells that provide information for the object.
*/
headers: string;
/**
* Retrieves the position of the object in the cells collection of a row.
*/
cellIndex: number;
/**
* Sets or retrieves how the object is aligned with adjacent text.
*/
align: string;
/**
* Sets or retrieves the color for one of the two colors used to draw the 3-D border of the object.
*/
borderColorLight: any;
/**
* Sets or retrieves the number columns in the table that the object should span.
*/
colSpan: number;
/**
* Sets or retrieves the border color of the object.
*/
borderColor: any;
/**
* Sets or retrieves a comma-delimited list of conceptual categories associated with the object.
*/
axis: string;
/**
* Sets or retrieves the height of the object.
*/
height: any;
/**
* Sets or retrieves whether the browser automatically performs wordwrap.
*/
noWrap: boolean;
/**
* Sets or retrieves abbreviated text for the object.
*/
abbr: string;
/**
* Sets or retrieves how many rows in a table the cell should span.
*/
rowSpan: number;
/**
* Sets or retrieves the group of cells in a table to which the object's information applies.
*/
scope: string;
/**
* Sets or retrieves the color for one of the two colors used to draw the 3-D border of the object.
*/
borderColorDark: any;
}
declare var HTMLTableCellElement:{
prototype: HTMLTableCellElement;
new(): HTMLTableCellElement;
}
interface SVGElementInstance extends EventTarget {
previousSibling: SVGElementInstance;
parentNode: SVGElementInstance;
lastChild: SVGElementInstance;
nextSibling: SVGElementInstance;
childNodes: SVGElementInstanceList;
correspondingUseElement: SVGUseElement;
correspondingElement: SVGElement;
firstChild: SVGElementInstance;
}
declare var SVGElementInstance:{
prototype: SVGElementInstance;
new(): SVGElementInstance;
}
interface MSNamespaceInfoCollection {
length: number;
add(namespace?:string, urn?:string, implementationUrl?:any): any;
item(index:any): any;
// [index: any]: any;
}
declare var MSNamespaceInfoCollection:{
prototype: MSNamespaceInfoCollection;
new(): MSNamespaceInfoCollection;
}
interface SVGCircleElement extends SVGElement, SVGStylable, SVGTransformable, SVGLangSpace, SVGTests, SVGExternalResourcesRequired {
cx: SVGAnimatedLength;
r: SVGAnimatedLength;
cy: SVGAnimatedLength;
}
declare var SVGCircleElement:{
prototype: SVGCircleElement;
new(): SVGCircleElement;
}
interface StyleSheetList {
length: number;
item(index?:number): StyleSheet;
[index: number]: StyleSheet;
}
declare var StyleSheetList:{
prototype: StyleSheetList;
new(): StyleSheetList;
}
interface CSSImportRule extends CSSRule {
styleSheet: CSSStyleSheet;
href: string;
media: MediaList;
}
declare var CSSImportRule:{
prototype: CSSImportRule;
new(): CSSImportRule;
}
interface CustomEvent extends Event {
detail: any;
initCustomEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, detailArg:any): void;
}
declare var CustomEvent:{
prototype: CustomEvent;
new(): CustomEvent;
}
interface HTMLBaseFontElement extends HTMLElement, DOML2DeprecatedColorProperty {
/**
* Sets or retrieves the current typeface family.
*/
face: string;
/**
* Sets or retrieves the font size of the object.
*/
size: number;
}
declare var HTMLBaseFontElement:{
prototype: HTMLBaseFontElement;
new(): HTMLBaseFontElement;
}
interface HTMLTextAreaElement extends HTMLElement, MSDataBindingExtensions {
/**
* Retrieves or sets the text in the entry field of the textArea element.
*/
value: string;
/**
* Sets or retrieves the value indicating whether the control is selected.
*/
status: any;
/**
* Retrieves a reference to the form that the object is embedded in.
*/
form: HTMLFormElement;
/**
* Sets or retrieves the name of the object.
*/
name: string;
/**
* Gets or sets the starting position or offset of a text selection.
*/
selectionStart: number;
/**
* Sets or retrieves the number of horizontal rows contained in the object.
*/
rows: number;
/**
* Sets or retrieves the width of the object.
*/
cols: number;
/**
* Sets or retrieves the value indicated whether the content of the object is read-only.
*/
readOnly: boolean;
/**
* Sets or retrieves how to handle wordwrapping in the object.
*/
wrap: string;
/**
* Gets or sets the end position or offset of a text selection.
*/
selectionEnd: number;
/**
* Retrieves the type of control.
*/
type: string;
/**
* Sets or retrieves the initial contents of the object.
*/
defaultValue: string;
/**
* Returns the error message that would be displayed if the user submits the form, or an empty string if no error message. It also triggers the standard error message, such as "this is a required field". The result is that the user sees validation messages without actually submitting.
*/
validationMessage: string;
/**
* Provides a way to direct a user to a specific field when a document loads. This can provide both direction and convenience for a user, reducing the need to click or tab to a field when a page opens. This attribute is true when present on an element, and false when missing.
*/
autofocus: boolean;
/**
* Returns a ValidityState object that represents the validity states of an element.
*/
validity: ValidityState;
/**
* When present, marks an element that can't be submitted without a value.
*/
required: boolean;
/**
* Sets or retrieves the maximum number of characters that the user can enter in a text control.
*/
maxLength: number;
/**
* Returns whether an element will successfully validate based on forms validation rules and constraints.
*/
willValidate: boolean;
/**
* Gets or sets a text string that is displayed in an input field as a hint or prompt to users as the format or type of information they need to enter.The text appears in an input field until the user puts focus on the field.
*/
placeholder: string;
/**
* Creates a TextRange object for the element.
*/
createTextRange(): TextRange;
/**
* Sets the start and end positions of a selection in a text field.
* @param start The offset into the text field for the start of the selection.
* @param end The offset into the text field for the end of the selection.
*/
setSelectionRange(start:number, end:number): void;
/**
* Highlights the input area of a form element.
*/
select(): void;
/**
* Returns whether a form will validate when it is submitted, without having to submit it.
*/
checkValidity(): boolean;
/**
* Sets a custom error message that is displayed when a form is submitted.
* @param error Sets a custom error message that is displayed when a form is submitted.
*/
setCustomValidity(error:string): void;
}
declare var HTMLTextAreaElement:{
prototype: HTMLTextAreaElement;
new(): HTMLTextAreaElement;
}
interface Geolocation {
clearWatch(watchId:number): void;
getCurrentPosition(successCallback:PositionCallback, errorCallback?:PositionErrorCallback, options?:PositionOptions): void;
watchPosition(successCallback:PositionCallback, errorCallback?:PositionErrorCallback, options?:PositionOptions): number;
}
declare var Geolocation:{
prototype: Geolocation;
new(): Geolocation;
}
interface DOML2DeprecatedMarginStyle {
vspace: number;
hspace: number;
}
interface MSWindowModeless {
dialogTop: any;
dialogLeft: any;
dialogWidth: any;
dialogHeight: any;
menuArguments: any;
}
interface DOML2DeprecatedAlignmentStyle {
align: string;
}
interface HTMLMarqueeElement extends HTMLElement, MSDataBindingExtensions, DOML2DeprecatedBackgroundColorStyle {
width: string;
onbounce: (ev:Event) => any;
vspace: number;
trueSpeed: boolean;
scrollAmount: number;
scrollDelay: number;
behavior: string;
height: string;
loop: number;
direction: string;
hspace: number;
onstart: (ev:Event) => any;
onfinish: (ev:Event) => any;
stop(): void;
start(): void;
addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerdown", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgotpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturedoubletap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerhover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturehold", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointermove", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturechange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturestart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointercancel", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgestureend", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturetap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerout", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msinertiastart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mslostpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerup", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"lostpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerenter", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"gotpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerleave", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mouseleave", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"keydown", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"move", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"keyup", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"reset", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"help", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"dragleave", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusin", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"seeked", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"durationchange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"blur", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"emptied", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"seeking", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"canplay", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"deactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"datasetchanged", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsdelete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"loadstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"losecapture", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragenter", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"controlselect", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"submit", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"change", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"layoutcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"beforeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"canplaythrough", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"filterchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"datasetcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"suspend", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mouseenter", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"errorupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"mouseout", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mousewheel", listener:(ev:MouseWheelEvent) => any, useCapture?:boolean): void;
addEventListener(type:"volumechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"cellchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowexit", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsinserted", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"propertychange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragend", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforepaste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragover", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseup", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragstart", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecopy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drag", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseover", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pause", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousedown", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"click", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"waiting", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"paste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"moveend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"stalled", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousemove", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforeeditfocus", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"ratechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dblclick", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"contextmenu", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadedmetadata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"afterupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"play", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizeend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"playing", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focusout", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dataavailable", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"readystatechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"keypress", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadeddata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforedeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"activate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"movestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"selectstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focus", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"timeupdate", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resize", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"cut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"select", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drop", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"copy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"ended", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"scroll", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"rowenter", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"input", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mscontentzoom", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"cuechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"msmanipulationstatechanged", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"bounce", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"start", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"finish", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var HTMLMarqueeElement:{
prototype: HTMLMarqueeElement;
new(): HTMLMarqueeElement;
}
interface SVGRect {
y: number;
width: number;
x: number;
height: number;
}
declare var SVGRect:{
prototype: SVGRect;
new(): SVGRect;
}
interface MSNodeExtensions {
swapNode(otherNode:Node): Node;
removeNode(deep?:boolean): Node;
replaceNode(replacement:Node): Node;
}
interface History {
length: number;
state: any;
back(distance?:any): void;
forward(distance?:any): void;
go(delta?:any): void;
replaceState(statedata:any, title:string, url?:string): void;
pushState(statedata:any, title:string, url?:string): void;
}
declare var History:{
prototype: History;
new(): History;
}
interface SVGPathSegCurvetoCubicAbs extends SVGPathSeg {
y: number;
y1: number;
x2: number;
x: number;
x1: number;
y2: number;
}
declare var SVGPathSegCurvetoCubicAbs:{
prototype: SVGPathSegCurvetoCubicAbs;
new(): SVGPathSegCurvetoCubicAbs;
}
interface SVGPathSegCurvetoQuadraticAbs extends SVGPathSeg {
y: number;
y1: number;
x: number;
x1: number;
}
declare var SVGPathSegCurvetoQuadraticAbs:{
prototype: SVGPathSegCurvetoQuadraticAbs;
new(): SVGPathSegCurvetoQuadraticAbs;
}
interface TimeRanges {
length: number;
start(index:number): number;
end(index:number): number;
}
declare var TimeRanges:{
prototype: TimeRanges;
new(): TimeRanges;
}
interface CSSRule {
cssText: string;
parentStyleSheet: CSSStyleSheet;
parentRule: CSSRule;
type: number;
IMPORT_RULE: number;
MEDIA_RULE: number;
STYLE_RULE: number;
NAMESPACE_RULE: number;
PAGE_RULE: number;
UNKNOWN_RULE: number;
FONT_FACE_RULE: number;
CHARSET_RULE: number;
KEYFRAMES_RULE: number;
KEYFRAME_RULE: number;
VIEWPORT_RULE: number;
}
declare var CSSRule:{
prototype: CSSRule;
new(): CSSRule;
IMPORT_RULE: number;
MEDIA_RULE: number;
STYLE_RULE: number;
NAMESPACE_RULE: number;
PAGE_RULE: number;
UNKNOWN_RULE: number;
FONT_FACE_RULE: number;
CHARSET_RULE: number;
KEYFRAMES_RULE: number;
KEYFRAME_RULE: number;
VIEWPORT_RULE: number;
}
interface SVGPathSegLinetoAbs extends SVGPathSeg {
y: number;
x: number;
}
declare var SVGPathSegLinetoAbs:{
prototype: SVGPathSegLinetoAbs;
new(): SVGPathSegLinetoAbs;
}
interface HTMLModElement extends HTMLElement {
/**
* Sets or retrieves the date and time of a modification to the object.
*/
dateTime: string;
/**
* Sets or retrieves reference information about the object.
*/
cite: string;
}
declare var HTMLModElement:{
prototype: HTMLModElement;
new(): HTMLModElement;
}
interface SVGMatrix {
e: number;
c: number;
a: number;
b: number;
d: number;
f: number;
multiply(secondMatrix:SVGMatrix): SVGMatrix;
flipY(): SVGMatrix;
skewY(angle:number): SVGMatrix;
inverse(): SVGMatrix;
scaleNonUniform(scaleFactorX:number, scaleFactorY:number): SVGMatrix;
rotate(angle:number): SVGMatrix;
flipX(): SVGMatrix;
translate(x:number, y:number): SVGMatrix;
scale(scaleFactor:number): SVGMatrix;
rotateFromVector(x:number, y:number): SVGMatrix;
skewX(angle:number): SVGMatrix;
}
declare var SVGMatrix:{
prototype: SVGMatrix;
new(): SVGMatrix;
}
interface MSPopupWindow {
document: Document;
isOpen: boolean;
show(x:number, y:number, w:number, h:number, element?:any): void;
hide(): void;
}
declare var MSPopupWindow:{
prototype: MSPopupWindow;
new(): MSPopupWindow;
}
interface BeforeUnloadEvent extends Event {
returnValue: string;
}
declare var BeforeUnloadEvent:{
prototype: BeforeUnloadEvent;
new(): BeforeUnloadEvent;
}
interface SVGUseElement extends SVGElement, SVGStylable, SVGTransformable, SVGLangSpace, SVGTests, SVGExternalResourcesRequired, SVGURIReference {
y: SVGAnimatedLength;
width: SVGAnimatedLength;
animatedInstanceRoot: SVGElementInstance;
instanceRoot: SVGElementInstance;
x: SVGAnimatedLength;
height: SVGAnimatedLength;
}
declare var SVGUseElement:{
prototype: SVGUseElement;
new(): SVGUseElement;
}
interface Event {
timeStamp: number;
defaultPrevented: boolean;
isTrusted: boolean;
currentTarget: EventTarget;
cancelBubble: boolean;
target: EventTarget;
eventPhase: number;
cancelable: boolean;
type: string;
srcElement: Element;
bubbles: boolean;
initEvent(eventTypeArg:string, canBubbleArg:boolean, cancelableArg:boolean): void;
stopPropagation(): void;
stopImmediatePropagation(): void;
preventDefault(): void;
CAPTURING_PHASE: number;
AT_TARGET: number;
BUBBLING_PHASE: number;
}
declare var Event:{
prototype: Event;
new(): Event;
CAPTURING_PHASE: number;
AT_TARGET: number;
BUBBLING_PHASE: number;
}
interface ImageData {
width: number;
data: number[];
height: number;
}
declare var ImageData:{
prototype: ImageData;
new(): ImageData;
}
interface HTMLTableColElement extends HTMLElement, HTMLTableAlignment {
/**
* Sets or retrieves the width of the object.
*/
width: any;
/**
* Sets or retrieves the alignment of the object relative to the display or table.
*/
align: string;
/**
* Sets or retrieves the number of columns in the group.
*/
span: number;
}
declare var HTMLTableColElement:{
prototype: HTMLTableColElement;
new(): HTMLTableColElement;
}
interface SVGException {
code: number;
message: string;
name: string;
toString(): string;
SVG_MATRIX_NOT_INVERTABLE: number;
SVG_WRONG_TYPE_ERR: number;
SVG_INVALID_VALUE_ERR: number;
}
declare var SVGException:{
prototype: SVGException;
new(): SVGException;
SVG_MATRIX_NOT_INVERTABLE: number;
SVG_WRONG_TYPE_ERR: number;
SVG_INVALID_VALUE_ERR: number;
}
interface SVGLinearGradientElement extends SVGGradientElement {
y1: SVGAnimatedLength;
x2: SVGAnimatedLength;
x1: SVGAnimatedLength;
y2: SVGAnimatedLength;
}
declare var SVGLinearGradientElement:{
prototype: SVGLinearGradientElement;
new(): SVGLinearGradientElement;
}
interface HTMLTableAlignment {
/**
* Sets or retrieves a value that you can use to implement your own ch functionality for the object.
*/
ch: string;
/**
* Sets or retrieves how text and other content are vertically aligned within the object that contains them.
*/
vAlign: string;
/**
* Sets or retrieves a value that you can use to implement your own chOff functionality for the object.
*/
chOff: string;
}
interface SVGAnimatedEnumeration {
animVal: number;
baseVal: number;
}
declare var SVGAnimatedEnumeration:{
prototype: SVGAnimatedEnumeration;
new(): SVGAnimatedEnumeration;
}
interface DOML2DeprecatedSizeProperty {
size: number;
}
interface HTMLUListElement extends HTMLElement, DOML2DeprecatedListSpaceReduction, DOML2DeprecatedListNumberingAndBulletStyle {
}
declare var HTMLUListElement:{
prototype: HTMLUListElement;
new(): HTMLUListElement;
}
interface SVGRectElement extends SVGElement, SVGStylable, SVGTransformable, SVGLangSpace, SVGTests, SVGExternalResourcesRequired {
y: SVGAnimatedLength;
width: SVGAnimatedLength;
ry: SVGAnimatedLength;
rx: SVGAnimatedLength;
x: SVGAnimatedLength;
height: SVGAnimatedLength;
}
declare var SVGRectElement:{
prototype: SVGRectElement;
new(): SVGRectElement;
}
interface ErrorEventHandler {
(event:Event, source:string, fileno:number, columnNumber:number): void;
}
interface HTMLDivElement extends HTMLElement, MSDataBindingExtensions {
/**
* Sets or retrieves how the object is aligned with adjacent text.
*/
align: string;
/**
* Sets or retrieves whether the browser automatically performs wordwrap.
*/
noWrap: boolean;
}
declare var HTMLDivElement:{
prototype: HTMLDivElement;
new(): HTMLDivElement;
}
interface DOML2DeprecatedBorderStyle {
border: string;
}
interface NamedNodeMap {
length: number;
removeNamedItemNS(namespaceURI:string, localName:string): Attr;
item(index:number): Attr;
[index: number]: Attr;
removeNamedItem(name:string): Attr;
getNamedItem(name:string): Attr;
// [name: string]: Attr;
setNamedItem(arg:Attr): Attr;
getNamedItemNS(namespaceURI:string, localName:string): Attr;
setNamedItemNS(arg:Attr): Attr;
}
declare var NamedNodeMap:{
prototype: NamedNodeMap;
new(): NamedNodeMap;
}
interface MediaList {
length: number;
mediaText: string;
deleteMedium(oldMedium:string): void;
appendMedium(newMedium:string): void;
item(index:number): string;
[index: number]: string;
toString(): string;
}
declare var MediaList:{
prototype: MediaList;
new(): MediaList;
}
interface SVGPathSegCurvetoQuadraticSmoothAbs extends SVGPathSeg {
y: number;
x: number;
}
declare var SVGPathSegCurvetoQuadraticSmoothAbs:{
prototype: SVGPathSegCurvetoQuadraticSmoothAbs;
new(): SVGPathSegCurvetoQuadraticSmoothAbs;
}
interface SVGPathSegCurvetoCubicSmoothRel extends SVGPathSeg {
y: number;
x2: number;
x: number;
y2: number;
}
declare var SVGPathSegCurvetoCubicSmoothRel:{
prototype: SVGPathSegCurvetoCubicSmoothRel;
new(): SVGPathSegCurvetoCubicSmoothRel;
}
interface SVGLengthList {
numberOfItems: number;
replaceItem(newItem:SVGLength, index:number): SVGLength;
getItem(index:number): SVGLength;
clear(): void;
appendItem(newItem:SVGLength): SVGLength;
initialize(newItem:SVGLength): SVGLength;
removeItem(index:number): SVGLength;
insertItemBefore(newItem:SVGLength, index:number): SVGLength;
}
declare var SVGLengthList:{
prototype: SVGLengthList;
new(): SVGLengthList;
}
interface ProcessingInstruction extends Node {
target: string;
data: string;
}
declare var ProcessingInstruction:{
prototype: ProcessingInstruction;
new(): ProcessingInstruction;
}
interface MSWindowExtensions {
status: string;
onmouseleave: (ev:MouseEvent) => any;
screenLeft: number;
offscreenBuffering: any;
maxConnectionsPerServer: number;
onmouseenter: (ev:MouseEvent) => any;
clipboardData: DataTransfer;
defaultStatus: string;
clientInformation: Navigator;
closed: boolean;
onhelp: (ev:Event) => any;
external: External;
event: MSEventObj;
onfocusout: (ev:FocusEvent) => any;
screenTop: number;
onfocusin: (ev:FocusEvent) => any;
showModelessDialog(url?:string, argument?:any, options?:any): Window;
navigate(url:string): void;
resizeBy(x?:number, y?:number): void;
item(index:any): any;
resizeTo(x?:number, y?:number): void;
createPopup(arguments?:any): MSPopupWindow;
toStaticHTML(html:string): string;
execScript(code:string, language?:string): any;
msWriteProfilerMark(profilerMarkName:string): void;
moveTo(x?:number, y?:number): void;
moveBy(x?:number, y?:number): void;
showHelp(url:string, helpArg?:any, features?:string): void;
captureEvents(): void;
releaseEvents(): void;
addEventListener(type:"mouseleave", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseenter", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"help", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focusout", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusin", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
interface MSBehaviorUrnsCollection {
length: number;
item(index:number): string;
}
declare var MSBehaviorUrnsCollection:{
prototype: MSBehaviorUrnsCollection;
new(): MSBehaviorUrnsCollection;
}
interface CSSFontFaceRule extends CSSRule {
style: CSSStyleDeclaration;
}
declare var CSSFontFaceRule:{
prototype: CSSFontFaceRule;
new(): CSSFontFaceRule;
}
interface DOML2DeprecatedBackgroundStyle {
background: string;
}
interface TextEvent extends UIEvent {
inputMethod: number;
data: string;
locale: string;
initTextEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, viewArg:Window, dataArg:string, inputMethod:number, locale:string): void;
DOM_INPUT_METHOD_KEYBOARD: number;
DOM_INPUT_METHOD_DROP: number;
DOM_INPUT_METHOD_IME: number;
DOM_INPUT_METHOD_SCRIPT: number;
DOM_INPUT_METHOD_VOICE: number;
DOM_INPUT_METHOD_UNKNOWN: number;
DOM_INPUT_METHOD_PASTE: number;
DOM_INPUT_METHOD_HANDWRITING: number;
DOM_INPUT_METHOD_OPTION: number;
DOM_INPUT_METHOD_MULTIMODAL: number;
}
declare var TextEvent:{
prototype: TextEvent;
new(): TextEvent;
DOM_INPUT_METHOD_KEYBOARD: number;
DOM_INPUT_METHOD_DROP: number;
DOM_INPUT_METHOD_IME: number;
DOM_INPUT_METHOD_SCRIPT: number;
DOM_INPUT_METHOD_VOICE: number;
DOM_INPUT_METHOD_UNKNOWN: number;
DOM_INPUT_METHOD_PASTE: number;
DOM_INPUT_METHOD_HANDWRITING: number;
DOM_INPUT_METHOD_OPTION: number;
DOM_INPUT_METHOD_MULTIMODAL: number;
}
interface DocumentFragment extends Node, NodeSelector, MSEventAttachmentTarget, MSNodeExtensions {
}
declare var DocumentFragment:{
prototype: DocumentFragment;
new(): DocumentFragment;
}
interface SVGPolylineElement extends SVGElement, SVGStylable, SVGTransformable, SVGLangSpace, SVGAnimatedPoints, SVGTests, SVGExternalResourcesRequired {
}
declare var SVGPolylineElement:{
prototype: SVGPolylineElement;
new(): SVGPolylineElement;
}
interface SVGAnimatedPathData {
pathSegList: SVGPathSegList;
}
interface Position {
timestamp: Date;
coords: Coordinates;
}
declare var Position:{
prototype: Position;
new(): Position;
}
interface BookmarkCollection {
length: number;
item(index:number): any;
[index: number]: any;
}
declare var BookmarkCollection:{
prototype: BookmarkCollection;
new(): BookmarkCollection;
}
interface PerformanceMark extends PerformanceEntry {
}
declare var PerformanceMark:{
prototype: PerformanceMark;
new(): PerformanceMark;
}
interface CSSPageRule extends CSSRule {
pseudoClass: string;
selectorText: string;
selector: string;
style: CSSStyleDeclaration;
}
declare var CSSPageRule:{
prototype: CSSPageRule;
new(): CSSPageRule;
}
interface HTMLBRElement extends HTMLElement {
/**
* Sets or retrieves the side on which floating objects are not to be positioned when any IHTMLBlockElement is inserted into the document.
*/
clear: string;
}
declare var HTMLBRElement:{
prototype: HTMLBRElement;
new(): HTMLBRElement;
}
interface MSNavigatorExtensions {
userLanguage: string;
plugins: MSPluginsCollection;
cookieEnabled: boolean;
appCodeName: string;
cpuClass: string;
appMinorVersion: string;
connectionSpeed: number;
browserLanguage: string;
mimeTypes: MSMimeTypesCollection;
systemLanguage: string;
language: string;
javaEnabled(): boolean;
taintEnabled(): boolean;
}
interface HTMLSpanElement extends HTMLElement, MSDataBindingExtensions {
}
declare var HTMLSpanElement:{
prototype: HTMLSpanElement;
new(): HTMLSpanElement;
}
interface HTMLHeadElement extends HTMLElement {
profile: string;
}
declare var HTMLHeadElement:{
prototype: HTMLHeadElement;
new(): HTMLHeadElement;
}
interface HTMLHeadingElement extends HTMLElement, DOML2DeprecatedTextFlowControl {
/**
* Sets or retrieves a value that indicates the table alignment.
*/
align: string;
}
declare var HTMLHeadingElement:{
prototype: HTMLHeadingElement;
new(): HTMLHeadingElement;
}
interface HTMLFormElement extends HTMLElement, MSHTMLCollectionExtensions {
/**
* Sets or retrieves the number of objects in a collection.
*/
length: number;
/**
* Sets or retrieves the window or frame at which to target content.
*/
target: string;
/**
* Sets or retrieves a list of character encodings for input data that must be accepted by the server processing the form.
*/
acceptCharset: string;
/**
* Sets or retrieves the encoding type for the form.
*/
enctype: string;
/**
* Retrieves a collection, in source order, of all controls in a given form.
*/
elements: HTMLCollection;
/**
* Sets or retrieves the URL to which the form content is sent for processing.
*/
action: string;
/**
* Sets or retrieves the name of the object.
*/
name: string;
/**
* Sets or retrieves how to send the form data to the server.
*/
method: string;
/**
* Sets or retrieves the MIME encoding for the form.
*/
encoding: string;
/**
* Specifies whether autocomplete is applied to an editable text field.
*/
autocomplete: string;
/**
* Designates a form that is not validated when submitted.
*/
noValidate: boolean;
/**
* Fires when the user resets a form.
*/
reset(): void;
/**
* Retrieves a form object or an object from an elements collection.
* @param name Variant of type Number or String that specifies the object or collection to retrieve. If this parameter is a Number, it is the zero-based index of the object. If this parameter is a string, all objects with matching name or id properties are retrieved, and a collection is returned if more than one match is made.
* @param index Variant of type Number that specifies the zero-based index of the object to retrieve when a collection is returned.
*/
item(name?:any, index?:any): any;
/**
* Fires when a FORM is about to be submitted.
*/
submit(): void;
/**
* Retrieves a form object or an object from an elements collection.
*/
namedItem(name:string): any;
[name: string]: any;
/**
* Returns whether a form will validate when it is submitted, without having to submit it.
*/
checkValidity(): boolean;
}
declare var HTMLFormElement:{
prototype: HTMLFormElement;
new(): HTMLFormElement;
}
interface SVGZoomAndPan {
zoomAndPan: number;
SVG_ZOOMANDPAN_MAGNIFY: number;
SVG_ZOOMANDPAN_UNKNOWN: number;
SVG_ZOOMANDPAN_DISABLE: number;
}
declare var SVGZoomAndPan:SVGZoomAndPan;
interface HTMLMediaElement extends HTMLElement {
/**
* Gets the earliest possible position, in seconds, that the playback can begin.
*/
initialTime: number;
/**
* Gets TimeRanges for the current media resource that has been played.
*/
played: TimeRanges;
/**
* Gets the address or URL of the current media resource that is selected by IHTMLMediaElement.
*/
currentSrc: string;
readyState: any;
/**
* The autobuffer element is not supported by Internet Explorer 9. Use the preload element instead.
*/
autobuffer: boolean;
/**
* Gets or sets a flag to specify whether playback should restart after it completes.
*/
loop: boolean;
/**
* Gets information about whether the playback has ended or not.
*/
ended: boolean;
/**
* Gets a collection of buffered time ranges.
*/
buffered: TimeRanges;
/**
* Returns an object representing the current error state of the audio or video element.
*/
error: MediaError;
/**
* Returns a TimeRanges object that represents the ranges of the current media resource that can be seeked.
*/
seekable: TimeRanges;
/**
* Gets or sets a value that indicates whether to start playing the media automatically.
*/
autoplay: boolean;
/**
* Gets or sets a flag that indicates whether the client provides a set of controls for the media (in case the developer does not include controls for the player).
*/
controls: boolean;
/**
* Gets or sets the volume level for audio portions of the media element.
*/
volume: number;
/**
* The address or URL of the a media resource that is to be considered.
*/
src: string;
/**
* Gets or sets the current rate of speed for the media resource to play. This speed is expressed as a multiple of the normal speed of the media resource.
*/
playbackRate: number;
/**
* Returns the duration in seconds of the current media resource. A NaN value is returned if duration is not available, or Infinity if the media resource is streaming.
*/
duration: number;
/**
* Gets or sets a flag that indicates whether the audio (either audio or the audio track on video media) is muted.
*/
muted: boolean;
/**
* Gets or sets the default playback rate when the user is not using fast forward or reverse for a video or audio resource.
*/
defaultPlaybackRate: number;
/**
* Gets a flag that specifies whether playback is paused.
*/
paused: boolean;
/**
* Gets a flag that indicates whether the the client is currently moving to a new playback position in the media resource.
*/
seeking: boolean;
/**
* Gets or sets the current playback position, in seconds.
*/
currentTime: number;
/**
* Gets or sets the current playback position, in seconds.
*/
preload: string;
/**
* Gets the current network activity for the element.
*/
networkState: number;
/**
* Specifies the purpose of the audio or video media, such as background audio or alerts.
*/
msAudioCategory: string;
/**
* Specifies whether or not to enable low-latency playback on the media element.
*/
msRealTime: boolean;
/**
* Gets or sets the primary DLNA PlayTo device.
*/
msPlayToPrimary: boolean;
textTracks: TextTrackList;
/**
* Gets or sets whether the DLNA PlayTo device is available.
*/
msPlayToDisabled: boolean;
/**
* Returns an AudioTrackList object with the audio tracks for a given video element.
*/
audioTracks: AudioTrackList;
/**
* Gets the source associated with the media element for use by the PlayToManager.
*/
msPlayToSource: any;
/**
* Specifies the output device id that the audio will be sent to.
*/
msAudioDeviceType: string;
/**
* Gets or sets the path to the preferred media source. This enables the Play To target device to stream the media content, which can be DRM protected, from a different location, such as a cloud media server.
*/
msPlayToPreferredSourceUri: string;
onmsneedkey: (ev:MSMediaKeyNeededEvent) => any;
/**
* Gets the MSMediaKeys object, which is used for decrypting media data, that is associated with this media element.
*/
msKeys: MSMediaKeys;
msGraphicsTrustStatus: MSGraphicsTrust;
/**
* Pauses the current playback and sets paused to TRUE. This can be used to test whether the media is playing or paused. You can also use the pause or play events to tell whether the media is playing or not.
*/
pause(): void;
/**
* Loads and starts playback of a media resource.
*/
play(): void;
/**
* Fires immediately after the client loads the object.
*/
load(): void;
/**
* Returns a string that specifies whether the client can play a given media resource type.
*/
canPlayType(type:string): string;
/**
* Clears all effects from the media pipeline.
*/
msClearEffects(): void;
/**
* Specifies the media protection manager for a given media pipeline.
*/
msSetMediaProtectionManager(mediaProtectionManager?:any): void;
/**
* Inserts the specified audio effect into media pipeline.
*/
msInsertAudioEffect(activatableClassId:string, effectRequired:boolean, config?:any): void;
msSetMediaKeys(mediaKeys:MSMediaKeys): void;
addTextTrack(kind:string, label?:string, language?:string): TextTrack;
HAVE_METADATA: number;
HAVE_CURRENT_DATA: number;
HAVE_NOTHING: number;
NETWORK_NO_SOURCE: number;
HAVE_ENOUGH_DATA: number;
NETWORK_EMPTY: number;
NETWORK_LOADING: number;
NETWORK_IDLE: number;
HAVE_FUTURE_DATA: number;
addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerdown", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgotpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturedoubletap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerhover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturehold", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointermove", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturechange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturestart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointercancel", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgestureend", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturetap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerout", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msinertiastart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mslostpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerup", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"lostpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerenter", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"gotpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerleave", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mouseleave", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"keydown", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"move", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"keyup", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"reset", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"help", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"dragleave", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusin", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"seeked", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"durationchange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"blur", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"emptied", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"seeking", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"canplay", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"deactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"datasetchanged", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsdelete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"loadstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"losecapture", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragenter", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"controlselect", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"submit", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"change", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"layoutcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"beforeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"canplaythrough", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"filterchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"datasetcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"suspend", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mouseenter", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"errorupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"mouseout", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mousewheel", listener:(ev:MouseWheelEvent) => any, useCapture?:boolean): void;
addEventListener(type:"volumechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"cellchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowexit", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsinserted", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"propertychange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragend", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforepaste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragover", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseup", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragstart", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecopy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drag", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseover", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pause", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousedown", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"click", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"waiting", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"paste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"moveend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"stalled", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousemove", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforeeditfocus", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"ratechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dblclick", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"contextmenu", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadedmetadata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"afterupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"play", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizeend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"playing", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focusout", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dataavailable", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"readystatechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"keypress", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadeddata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforedeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"activate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"movestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"selectstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focus", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"timeupdate", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resize", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"cut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"select", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drop", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"copy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"ended", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"scroll", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"rowenter", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"input", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mscontentzoom", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"cuechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"msmanipulationstatechanged", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msneedkey", listener:(ev:MSMediaKeyNeededEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var HTMLMediaElement:{
prototype: HTMLMediaElement;
new(): HTMLMediaElement;
HAVE_METADATA: number;
HAVE_CURRENT_DATA: number;
HAVE_NOTHING: number;
NETWORK_NO_SOURCE: number;
HAVE_ENOUGH_DATA: number;
NETWORK_EMPTY: number;
NETWORK_LOADING: number;
NETWORK_IDLE: number;
HAVE_FUTURE_DATA: number;
}
interface ElementCSSInlineStyle {
runtimeStyle: MSStyleCSSProperties;
currentStyle: MSCurrentStyleCSSProperties;
doScroll(component?:any): void;
componentFromPoint(x:number, y:number): string;
}
interface DOMParser {
parseFromString(source:string, mimeType:string): Document;
}
declare var DOMParser:{
prototype: DOMParser;
new(): DOMParser;
}
interface MSMimeTypesCollection {
length: number;
}
declare var MSMimeTypesCollection:{
prototype: MSMimeTypesCollection;
new(): MSMimeTypesCollection;
}
interface StyleSheet {
disabled: boolean;
ownerNode: Node;
parentStyleSheet: StyleSheet;
href: string;
media: MediaList;
type: string;
title: string;
}
declare var StyleSheet:{
prototype: StyleSheet;
new(): StyleSheet;
}
interface SVGTextPathElement extends SVGTextContentElement, SVGURIReference {
startOffset: SVGAnimatedLength;
method: SVGAnimatedEnumeration;
spacing: SVGAnimatedEnumeration;
TEXTPATH_SPACINGTYPE_EXACT: number;
TEXTPATH_METHODTYPE_STRETCH: number;
TEXTPATH_SPACINGTYPE_AUTO: number;
TEXTPATH_SPACINGTYPE_UNKNOWN: number;
TEXTPATH_METHODTYPE_UNKNOWN: number;
TEXTPATH_METHODTYPE_ALIGN: number;
}
declare var SVGTextPathElement:{
prototype: SVGTextPathElement;
new(): SVGTextPathElement;
TEXTPATH_SPACINGTYPE_EXACT: number;
TEXTPATH_METHODTYPE_STRETCH: number;
TEXTPATH_SPACINGTYPE_AUTO: number;
TEXTPATH_SPACINGTYPE_UNKNOWN: number;
TEXTPATH_METHODTYPE_UNKNOWN: number;
TEXTPATH_METHODTYPE_ALIGN: number;
}
interface HTMLDTElement extends HTMLElement {
/**
* Sets or retrieves whether the browser automatically performs wordwrap.
*/
noWrap: boolean;
}
declare var HTMLDTElement:{
prototype: HTMLDTElement;
new(): HTMLDTElement;
}
interface NodeList {
length: number;
item(index:number): Node;
[index: number]: Node;
}
declare var NodeList:{
prototype: NodeList;
new(): NodeList;
}
interface XMLSerializer {
serializeToString(target:Node): string;
}
declare var XMLSerializer:{
prototype: XMLSerializer;
new(): XMLSerializer;
}
interface PerformanceMeasure extends PerformanceEntry {
}
declare var PerformanceMeasure:{
prototype: PerformanceMeasure;
new(): PerformanceMeasure;
}
interface SVGGradientElement extends SVGElement, SVGUnitTypes, SVGStylable, SVGExternalResourcesRequired, SVGURIReference {
spreadMethod: SVGAnimatedEnumeration;
gradientTransform: SVGAnimatedTransformList;
gradientUnits: SVGAnimatedEnumeration;
SVG_SPREADMETHOD_REFLECT: number;
SVG_SPREADMETHOD_PAD: number;
SVG_SPREADMETHOD_UNKNOWN: number;
SVG_SPREADMETHOD_REPEAT: number;
}
declare var SVGGradientElement:{
prototype: SVGGradientElement;
new(): SVGGradientElement;
SVG_SPREADMETHOD_REFLECT: number;
SVG_SPREADMETHOD_PAD: number;
SVG_SPREADMETHOD_UNKNOWN: number;
SVG_SPREADMETHOD_REPEAT: number;
}
interface NodeFilter {
acceptNode(n:Node): number;
SHOW_ENTITY_REFERENCE: number;
SHOW_NOTATION: number;
SHOW_ENTITY: number;
SHOW_DOCUMENT: number;
SHOW_PROCESSING_INSTRUCTION: number;
FILTER_REJECT: number;
SHOW_CDATA_SECTION: number;
FILTER_ACCEPT: number;
SHOW_ALL: number;
SHOW_DOCUMENT_TYPE: number;
SHOW_TEXT: number;
SHOW_ELEMENT: number;
SHOW_COMMENT: number;
FILTER_SKIP: number;
SHOW_ATTRIBUTE: number;
SHOW_DOCUMENT_FRAGMENT: number;
}
declare var NodeFilter:NodeFilter;
interface SVGNumberList {
numberOfItems: number;
replaceItem(newItem:SVGNumber, index:number): SVGNumber;
getItem(index:number): SVGNumber;
clear(): void;
appendItem(newItem:SVGNumber): SVGNumber;
initialize(newItem:SVGNumber): SVGNumber;
removeItem(index:number): SVGNumber;
insertItemBefore(newItem:SVGNumber, index:number): SVGNumber;
}
declare var SVGNumberList:{
prototype: SVGNumberList;
new(): SVGNumberList;
}
interface MediaError {
code: number;
msExtendedCode: number;
MEDIA_ERR_ABORTED: number;
MEDIA_ERR_NETWORK: number;
MEDIA_ERR_SRC_NOT_SUPPORTED: number;
MEDIA_ERR_DECODE: number;
MS_MEDIA_ERR_ENCRYPTED: number;
}
declare var MediaError:{
prototype: MediaError;
new(): MediaError;
MEDIA_ERR_ABORTED: number;
MEDIA_ERR_NETWORK: number;
MEDIA_ERR_SRC_NOT_SUPPORTED: number;
MEDIA_ERR_DECODE: number;
MS_MEDIA_ERR_ENCRYPTED: number;
}
interface HTMLFieldSetElement extends HTMLElement {
/**
* Sets or retrieves how the object is aligned with adjacent text.
*/
align: string;
/**
* Retrieves a reference to the form that the object is embedded in.
*/
form: HTMLFormElement;
/**
* Returns the error message that would be displayed if the user submits the form, or an empty string if no error message. It also triggers the standard error message, such as "this is a required field". The result is that the user sees validation messages without actually submitting.
*/
validationMessage: string;
/**
* Returns a ValidityState object that represents the validity states of an element.
*/
validity: ValidityState;
/**
* Returns whether an element will successfully validate based on forms validation rules and constraints.
*/
willValidate: boolean;
/**
* Returns whether a form will validate when it is submitted, without having to submit it.
*/
checkValidity(): boolean;
/**
* Sets a custom error message that is displayed when a form is submitted.
* @param error Sets a custom error message that is displayed when a form is submitted.
*/
setCustomValidity(error:string): void;
}
declare var HTMLFieldSetElement:{
prototype: HTMLFieldSetElement;
new(): HTMLFieldSetElement;
}
interface HTMLBGSoundElement extends HTMLElement {
/**
* Sets or gets the value indicating how the volume of the background sound is divided between the left speaker and the right speaker.
*/
balance: any;
/**
* Sets or gets the volume setting for the sound.
*/
volume: any;
/**
* Sets or gets the URL of a sound to play.
*/
src: string;
/**
* Sets or retrieves the number of times a sound or video clip will loop when activated.
*/
loop: number;
}
declare var HTMLBGSoundElement:{
prototype: HTMLBGSoundElement;
new(): HTMLBGSoundElement;
}
interface Comment extends CharacterData {
text: string;
}
declare var Comment:{
prototype: Comment;
new(): Comment;
}
interface PerformanceResourceTiming extends PerformanceEntry {
redirectStart: number;
redirectEnd: number;
domainLookupEnd: number;
responseStart: number;
domainLookupStart: number;
fetchStart: number;
requestStart: number;
connectEnd: number;
connectStart: number;
initiatorType: string;
responseEnd: number;
}
declare var PerformanceResourceTiming:{
prototype: PerformanceResourceTiming;
new(): PerformanceResourceTiming;
}
interface CanvasPattern {
}
declare var CanvasPattern:{
prototype: CanvasPattern;
new(): CanvasPattern;
}
interface HTMLHRElement extends HTMLElement, DOML2DeprecatedColorProperty, DOML2DeprecatedSizeProperty {
/**
* Sets or retrieves the width of the object.
*/
width: number;
/**
* Sets or retrieves how the object is aligned with adjacent text.
*/
align: string;
/**
* Sets or retrieves whether the horizontal rule is drawn with 3-D shading.
*/
noShade: boolean;
}
declare var HTMLHRElement:{
prototype: HTMLHRElement;
new(): HTMLHRElement;
}
interface HTMLObjectElement extends HTMLElement, GetSVGDocument, DOML2DeprecatedMarginStyle, DOML2DeprecatedBorderStyle, DOML2DeprecatedAlignmentStyle, MSDataBindingExtensions, MSDataBindingRecordSetExtensions {
/**
* Sets or retrieves the width of the object.
*/
width: string;
/**
* Sets or retrieves the Internet media type for the code associated with the object.
*/
codeType: string;
/**
* Retrieves the contained object.
*/
object: any;
/**
* Retrieves a reference to the form that the object is embedded in.
*/
form: HTMLFormElement;
/**
* Sets or retrieves the URL of the file containing the compiled Java class.
*/
code: string;
/**
* Sets or retrieves a character string that can be used to implement your own archive functionality for the object.
*/
archive: string;
/**
* Sets or retrieves a message to be displayed while an object is loading.
*/
standby: string;
/**
* Sets or retrieves a text alternative to the graphic.
*/
alt: string;
/**
* Sets or retrieves the class identifier for the object.
*/
classid: string;
/**
* Sets or retrieves the name of the object.
*/
name: string;
/**
* Sets or retrieves the URL, often with a bookmark extension (#name), to use as a client-side image map.
*/
useMap: string;
/**
* Sets or retrieves the URL that references the data of the object.
*/
data: string;
/**
* Sets or retrieves the height of the object.
*/
height: string;
/**
* Retrieves the document object of the page or frame.
*/
contentDocument: Document;
/**
* Gets or sets the optional alternative HTML script to execute if the object fails to load.
*/
altHtml: string;
/**
* Sets or retrieves the URL of the component.
*/
codeBase: string;
declare: boolean;
/**
* Sets or retrieves the MIME type of the object.
*/
type: string;
/**
* Retrieves a string of the URL where the object tag can be found. This is often the href of the document that the object is in, or the value set by a base element.
*/
BaseHref: string;
/**
* Returns the error message that would be displayed if the user submits the form, or an empty string if no error message. It also triggers the standard error message, such as "this is a required field". The result is that the user sees validation messages without actually submitting.
*/
validationMessage: string;
/**
* Returns a ValidityState object that represents the validity states of an element.
*/
validity: ValidityState;
/**
* Returns whether an element will successfully validate based on forms validation rules and constraints.
*/
willValidate: boolean;
/**
* Gets or sets the path to the preferred media source. This enables the Play To target device to stream the media content, which can be DRM protected, from a different location, such as a cloud media server.
*/
msPlayToPreferredSourceUri: string;
/**
* Gets or sets the primary DLNA PlayTo device.
*/
msPlayToPrimary: boolean;
/**
* Gets or sets whether the DLNA PlayTo device is available.
*/
msPlayToDisabled: boolean;
readyState: number;
/**
* Gets the source associated with the media element for use by the PlayToManager.
*/
msPlayToSource: any;
/**
* Returns whether a form will validate when it is submitted, without having to submit it.
*/
checkValidity(): boolean;
/**
* Sets a custom error message that is displayed when a form is submitted.
* @param error Sets a custom error message that is displayed when a form is submitted.
*/
setCustomValidity(error:string): void;
}
declare var HTMLObjectElement:{
prototype: HTMLObjectElement;
new(): HTMLObjectElement;
}
interface HTMLEmbedElement extends HTMLElement, GetSVGDocument {
/**
* Sets or retrieves the width of the object.
*/
width: string;
/**
* Retrieves the palette used for the embedded document.
*/
palette: string;
/**
* Sets or retrieves a URL to be loaded by the object.
*/
src: string;
/**
* Sets or retrieves the name of the object.
*/
name: string;
hidden: string;
/**
* Retrieves the URL of the plug-in used to view an embedded document.
*/
pluginspage: string;
/**
* Sets or retrieves the height of the object.
*/
height: string;
/**
* Sets or retrieves the height and width units of the embed object.
*/
units: string;
/**
* Gets or sets the path to the preferred media source. This enables the Play To target device to stream the media content, which can be DRM protected, from a different location, such as a cloud media server.
*/
msPlayToPreferredSourceUri: string;
/**
* Gets or sets the primary DLNA PlayTo device.
*/
msPlayToPrimary: boolean;
/**
* Gets or sets whether the DLNA PlayTo device is available.
*/
msPlayToDisabled: boolean;
readyState: string;
/**
* Gets the source associated with the media element for use by the PlayToManager.
*/
msPlayToSource: any;
}
declare var HTMLEmbedElement:{
prototype: HTMLEmbedElement;
new(): HTMLEmbedElement;
}
interface StorageEvent extends Event {
oldValue: any;
newValue: any;
url: string;
storageArea: Storage;
key: string;
initStorageEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, keyArg:string, oldValueArg:any, newValueArg:any, urlArg:string, storageAreaArg:Storage): void;
}
declare var StorageEvent:{
prototype: StorageEvent;
new(): StorageEvent;
}
interface CharacterData extends Node {
length: number;
data: string;
deleteData(offset:number, count:number): void;
replaceData(offset:number, count:number, arg:string): void;
appendData(arg:string): void;
insertData(offset:number, arg:string): void;
substringData(offset:number, count:number): string;
}
declare var CharacterData:{
prototype: CharacterData;
new(): CharacterData;
}
interface HTMLOptGroupElement extends HTMLElement, MSDataBindingExtensions {
/**
* Sets or retrieves the ordinal position of an option in a list box.
*/
index: number;
/**
* Sets or retrieves the status of an option.
*/
defaultSelected: boolean;
/**
* Sets or retrieves the text string specified by the option tag.
*/
text: string;
/**
* Sets or retrieves the value which is returned to the server when the form control is submitted.
*/
value: string;
/**
* Retrieves a reference to the form that the object is embedded in.
*/
form: HTMLFormElement;
/**
* Sets or retrieves a value that you can use to implement your own label functionality for the object.
*/
label: string;
/**
* Sets or retrieves whether the option in the list box is the default item.
*/
selected: boolean;
}
declare var HTMLOptGroupElement:{
prototype: HTMLOptGroupElement;
new(): HTMLOptGroupElement;
}
interface HTMLIsIndexElement extends HTMLElement {
/**
* Retrieves a reference to the form that the object is embedded in.
*/
form: HTMLFormElement;
/**
* Sets or retrieves the URL to which the form content is sent for processing.
*/
action: string;
prompt: string;
}
declare var HTMLIsIndexElement:{
prototype: HTMLIsIndexElement;
new(): HTMLIsIndexElement;
}
interface SVGPathSegLinetoRel extends SVGPathSeg {
y: number;
x: number;
}
declare var SVGPathSegLinetoRel:{
prototype: SVGPathSegLinetoRel;
new(): SVGPathSegLinetoRel;
}
interface DOMException {
code: number;
message: string;
name: string;
toString(): string;
HIERARCHY_REQUEST_ERR: number;
NO_MODIFICATION_ALLOWED_ERR: number;
INVALID_MODIFICATION_ERR: number;
NAMESPACE_ERR: number;
INVALID_CHARACTER_ERR: number;
TYPE_MISMATCH_ERR: number;
ABORT_ERR: number;
INVALID_STATE_ERR: number;
SECURITY_ERR: number;
NETWORK_ERR: number;
WRONG_DOCUMENT_ERR: number;
QUOTA_EXCEEDED_ERR: number;
INDEX_SIZE_ERR: number;
DOMSTRING_SIZE_ERR: number;
SYNTAX_ERR: number;
SERIALIZE_ERR: number;
VALIDATION_ERR: number;
NOT_FOUND_ERR: number;
URL_MISMATCH_ERR: number;
PARSE_ERR: number;
NO_DATA_ALLOWED_ERR: number;
NOT_SUPPORTED_ERR: number;
INVALID_ACCESS_ERR: number;
INUSE_ATTRIBUTE_ERR: number;
INVALID_NODE_TYPE_ERR: number;
DATA_CLONE_ERR: number;
TIMEOUT_ERR: number;
}
declare var DOMException:{
prototype: DOMException;
new(): DOMException;
HIERARCHY_REQUEST_ERR: number;
NO_MODIFICATION_ALLOWED_ERR: number;
INVALID_MODIFICATION_ERR: number;
NAMESPACE_ERR: number;
INVALID_CHARACTER_ERR: number;
TYPE_MISMATCH_ERR: number;
ABORT_ERR: number;
INVALID_STATE_ERR: number;
SECURITY_ERR: number;
NETWORK_ERR: number;
WRONG_DOCUMENT_ERR: number;
QUOTA_EXCEEDED_ERR: number;
INDEX_SIZE_ERR: number;
DOMSTRING_SIZE_ERR: number;
SYNTAX_ERR: number;
SERIALIZE_ERR: number;
VALIDATION_ERR: number;
NOT_FOUND_ERR: number;
URL_MISMATCH_ERR: number;
PARSE_ERR: number;
NO_DATA_ALLOWED_ERR: number;
NOT_SUPPORTED_ERR: number;
INVALID_ACCESS_ERR: number;
INUSE_ATTRIBUTE_ERR: number;
INVALID_NODE_TYPE_ERR: number;
DATA_CLONE_ERR: number;
TIMEOUT_ERR: number;
}
interface SVGAnimatedBoolean {
animVal: boolean;
baseVal: boolean;
}
declare var SVGAnimatedBoolean:{
prototype: SVGAnimatedBoolean;
new(): SVGAnimatedBoolean;
}
interface MSCompatibleInfoCollection {
length: number;
item(index:number): MSCompatibleInfo;
}
declare var MSCompatibleInfoCollection:{
prototype: MSCompatibleInfoCollection;
new(): MSCompatibleInfoCollection;
}
interface SVGSwitchElement extends SVGElement, SVGStylable, SVGTransformable, SVGLangSpace, SVGTests, SVGExternalResourcesRequired {
}
declare var SVGSwitchElement:{
prototype: SVGSwitchElement;
new(): SVGSwitchElement;
}
interface SVGPreserveAspectRatio {
align: number;
meetOrSlice: number;
SVG_PRESERVEASPECTRATIO_NONE: number;
SVG_PRESERVEASPECTRATIO_XMINYMID: number;
SVG_PRESERVEASPECTRATIO_XMAXYMIN: number;
SVG_PRESERVEASPECTRATIO_XMINYMAX: number;
SVG_PRESERVEASPECTRATIO_XMAXYMAX: number;
SVG_MEETORSLICE_UNKNOWN: number;
SVG_PRESERVEASPECTRATIO_XMAXYMID: number;
SVG_PRESERVEASPECTRATIO_XMIDYMAX: number;
SVG_PRESERVEASPECTRATIO_XMINYMIN: number;
SVG_MEETORSLICE_MEET: number;
SVG_PRESERVEASPECTRATIO_XMIDYMID: number;
SVG_PRESERVEASPECTRATIO_XMIDYMIN: number;
SVG_MEETORSLICE_SLICE: number;
SVG_PRESERVEASPECTRATIO_UNKNOWN: number;
}
declare var SVGPreserveAspectRatio:{
prototype: SVGPreserveAspectRatio;
new(): SVGPreserveAspectRatio;
SVG_PRESERVEASPECTRATIO_NONE: number;
SVG_PRESERVEASPECTRATIO_XMINYMID: number;
SVG_PRESERVEASPECTRATIO_XMAXYMIN: number;
SVG_PRESERVEASPECTRATIO_XMINYMAX: number;
SVG_PRESERVEASPECTRATIO_XMAXYMAX: number;
SVG_MEETORSLICE_UNKNOWN: number;
SVG_PRESERVEASPECTRATIO_XMAXYMID: number;
SVG_PRESERVEASPECTRATIO_XMIDYMAX: number;
SVG_PRESERVEASPECTRATIO_XMINYMIN: number;
SVG_MEETORSLICE_MEET: number;
SVG_PRESERVEASPECTRATIO_XMIDYMID: number;
SVG_PRESERVEASPECTRATIO_XMIDYMIN: number;
SVG_MEETORSLICE_SLICE: number;
SVG_PRESERVEASPECTRATIO_UNKNOWN: number;
}
interface Attr extends Node {
expando: boolean;
specified: boolean;
ownerElement: Element;
value: string;
name: string;
}
declare var Attr:{
prototype: Attr;
new(): Attr;
}
interface PerformanceNavigation {
redirectCount: number;
type: number;
toJSON(): any;
TYPE_RELOAD: number;
TYPE_RESERVED: number;
TYPE_BACK_FORWARD: number;
TYPE_NAVIGATE: number;
}
declare var PerformanceNavigation:{
prototype: PerformanceNavigation;
new(): PerformanceNavigation;
TYPE_RELOAD: number;
TYPE_RESERVED: number;
TYPE_BACK_FORWARD: number;
TYPE_NAVIGATE: number;
}
interface SVGStopElement extends SVGElement, SVGStylable {
offset: SVGAnimatedNumber;
}
declare var SVGStopElement:{
prototype: SVGStopElement;
new(): SVGStopElement;
}
interface PositionCallback {
(position:Position): void;
}
interface SVGSymbolElement extends SVGElement, SVGStylable, SVGLangSpace, SVGFitToViewBox, SVGExternalResourcesRequired {
}
declare var SVGSymbolElement:{
prototype: SVGSymbolElement;
new(): SVGSymbolElement;
}
interface SVGElementInstanceList {
length: number;
item(index:number): SVGElementInstance;
}
declare var SVGElementInstanceList:{
prototype: SVGElementInstanceList;
new(): SVGElementInstanceList;
}
interface CSSRuleList {
length: number;
item(index:number): CSSRule;
[index: number]: CSSRule;
}
declare var CSSRuleList:{
prototype: CSSRuleList;
new(): CSSRuleList;
}
interface MSDataBindingRecordSetExtensions {
recordset: any;
namedRecordset(dataMember:string, hierarchy?:any): any;
}
interface LinkStyle {
styleSheet: StyleSheet;
sheet: StyleSheet;
}
interface HTMLVideoElement extends HTMLMediaElement {
/**
* Gets or sets the width of the video element.
*/
width: number;
/**
* Gets the intrinsic width of a video in CSS pixels, or zero if the dimensions are not known.
*/
videoWidth: number;
/**
* Gets the intrinsic height of a video in CSS pixels, or zero if the dimensions are not known.
*/
videoHeight: number;
/**
* Gets or sets the height of the video element.
*/
height: number;
/**
* Gets or sets a URL of an image to display, for example, like a movie poster. This can be a still frame from the video, or another image if no video data is available.
*/
poster: string;
msIsStereo3D: boolean;
msStereo3DPackingMode: string;
onMSVideoOptimalLayoutChanged: (ev:any) => any;
onMSVideoFrameStepCompleted: (ev:any) => any;
msStereo3DRenderMode: string;
msIsLayoutOptimalForPlayback: boolean;
msHorizontalMirror: boolean;
onMSVideoFormatChanged: (ev:any) => any;
msZoom: boolean;
msInsertVideoEffect(activatableClassId:string, effectRequired:boolean, config?:any): void;
msSetVideoRectangle(left:number, top:number, right:number, bottom:number): void;
msFrameStep(forward:boolean): void;
getVideoPlaybackQuality(): VideoPlaybackQuality;
addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerdown", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgotpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturedoubletap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerhover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturehold", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointermove", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturechange", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturestart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointercancel", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgestureend", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msgesturetap", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerout", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msinertiastart", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mslostpointercapture", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerover", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerup", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"lostpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerenter", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"gotpointercapture", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mspointerleave", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"mouseleave", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"keydown", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"move", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"keyup", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"reset", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"help", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"dragleave", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"focusin", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"seeked", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"durationchange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"blur", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"emptied", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"seeking", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"canplay", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"deactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"datasetchanged", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsdelete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"loadstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"losecapture", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragenter", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"controlselect", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"submit", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"change", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"layoutcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"beforeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"canplaythrough", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforeupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"filterchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"datasetcomplete", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"suspend", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mouseenter", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"errorupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"mouseout", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mousewheel", listener:(ev:MouseWheelEvent) => any, useCapture?:boolean): void;
addEventListener(type:"volumechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"cellchange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowexit", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"rowsinserted", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"propertychange", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"dragend", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforepaste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragover", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseup", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dragstart", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforecopy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drag", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"mouseover", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pause", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousedown", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"click", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"waiting", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"paste", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"moveend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"stalled", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mousemove", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"beforeeditfocus", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"ratechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dblclick", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"contextmenu", listener:(ev:MouseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadedmetadata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"afterupdate", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"play", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resizeend", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"playing", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focusout", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"dataavailable", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"readystatechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"keypress", listener:(ev:KeyboardEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadeddata", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"beforedeactivate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"activate", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"movestart", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"selectstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"focus", listener:(ev:FocusEvent) => any, useCapture?:boolean): void;
addEventListener(type:"timeupdate", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"resize", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"cut", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"select", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"drop", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"copy", listener:(ev:DragEvent) => any, useCapture?:boolean): void;
addEventListener(type:"ended", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"scroll", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"rowenter", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"input", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"mscontentzoom", listener:(ev:MSEventObj) => any, useCapture?:boolean): void;
addEventListener(type:"cuechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"msmanipulationstatechanged", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"msneedkey", listener:(ev:MSMediaKeyNeededEvent) => any, useCapture?:boolean): void;
addEventListener(type:"MSVideoOptimalLayoutChanged", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"MSVideoFrameStepCompleted", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"MSVideoFormatChanged", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var HTMLVideoElement:{
prototype: HTMLVideoElement;
new(): HTMLVideoElement;
}
interface ClientRectList {
length: number;
item(index:number): ClientRect;
[index: number]: ClientRect;
}
declare var ClientRectList:{
prototype: ClientRectList;
new(): ClientRectList;
}
interface SVGMaskElement extends SVGElement, SVGUnitTypes, SVGStylable, SVGLangSpace, SVGTests, SVGExternalResourcesRequired {
y: SVGAnimatedLength;
width: SVGAnimatedLength;
maskUnits: SVGAnimatedEnumeration;
maskContentUnits: SVGAnimatedEnumeration;
x: SVGAnimatedLength;
height: SVGAnimatedLength;
}
declare var SVGMaskElement:{
prototype: SVGMaskElement;
new(): SVGMaskElement;
}
interface External {
}
declare var External:{
prototype: External;
new(): External;
}
interface MSGestureEvent extends UIEvent {
offsetY: number;
translationY: number;
velocityExpansion: number;
velocityY: number;
velocityAngular: number;
translationX: number;
velocityX: number;
hwTimestamp: number;
offsetX: number;
screenX: number;
rotation: number;
expansion: number;
clientY: number;
screenY: number;
scale: number;
gestureObject: any;
clientX: number;
initGestureEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, viewArg:Window, detailArg:number, screenXArg:number, screenYArg:number, clientXArg:number, clientYArg:number, offsetXArg:number, offsetYArg:number, translationXArg:number, translationYArg:number, scaleArg:number, expansionArg:number, rotationArg:number, velocityXArg:number, velocityYArg:number, velocityExpansionArg:number, velocityAngularArg:number, hwTimestampArg:number): void;
MSGESTURE_FLAG_BEGIN: number;
MSGESTURE_FLAG_END: number;
MSGESTURE_FLAG_CANCEL: number;
MSGESTURE_FLAG_INERTIA: number;
MSGESTURE_FLAG_NONE: number;
}
declare var MSGestureEvent:{
prototype: MSGestureEvent;
new(): MSGestureEvent;
MSGESTURE_FLAG_BEGIN: number;
MSGESTURE_FLAG_END: number;
MSGESTURE_FLAG_CANCEL: number;
MSGESTURE_FLAG_INERTIA: number;
MSGESTURE_FLAG_NONE: number;
}
interface ErrorEvent extends Event {
colno: number;
filename: string;
error: any;
lineno: number;
message: string;
initErrorEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, messageArg:string, filenameArg:string, linenoArg:number): void;
}
declare var ErrorEvent:{
prototype: ErrorEvent;
new(): ErrorEvent;
}
interface SVGFilterElement extends SVGElement, SVGUnitTypes, SVGStylable, SVGLangSpace, SVGURIReference, SVGExternalResourcesRequired {
y: SVGAnimatedLength;
width: SVGAnimatedLength;
filterResX: SVGAnimatedInteger;
filterUnits: SVGAnimatedEnumeration;
primitiveUnits: SVGAnimatedEnumeration;
x: SVGAnimatedLength;
height: SVGAnimatedLength;
filterResY: SVGAnimatedInteger;
setFilterRes(filterResX:number, filterResY:number): void;
}
declare var SVGFilterElement:{
prototype: SVGFilterElement;
new(): SVGFilterElement;
}
interface TrackEvent extends Event {
track: any;
}
declare var TrackEvent:{
prototype: TrackEvent;
new(): TrackEvent;
}
interface SVGFEMergeNodeElement extends SVGElement {
in1: SVGAnimatedString;
}
declare var SVGFEMergeNodeElement:{
prototype: SVGFEMergeNodeElement;
new(): SVGFEMergeNodeElement;
}
interface SVGFEFloodElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
}
declare var SVGFEFloodElement:{
prototype: SVGFEFloodElement;
new(): SVGFEFloodElement;
}
interface MSGesture {
target: Element;
addPointer(pointerId:number): void;
stop(): void;
}
declare var MSGesture:{
prototype: MSGesture;
new(): MSGesture;
}
interface TextTrackCue extends EventTarget {
onenter: (ev:Event) => any;
track: TextTrack;
endTime: number;
text: string;
pauseOnExit: boolean;
id: string;
startTime: number;
onexit: (ev:Event) => any;
getCueAsHTML(): DocumentFragment;
addEventListener(type:"enter", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"exit", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var TextTrackCue:{
prototype: TextTrackCue;
new(startTime:number, endTime:number, text:string): TextTrackCue;
}
interface MSStreamReader extends MSBaseReader {
error: DOMError;
readAsArrayBuffer(stream:MSStream, size?:number): void;
readAsBlob(stream:MSStream, size?:number): void;
readAsDataURL(stream:MSStream, size?:number): void;
readAsText(stream:MSStream, encoding?:string, size?:number): void;
}
declare var MSStreamReader:{
prototype: MSStreamReader;
new(): MSStreamReader;
}
interface DOMTokenList {
length: number;
contains(token:string): boolean;
remove(token:string): void;
toggle(token:string): boolean;
add(token:string): void;
item(index:number): string;
[index: number]: string;
toString(): string;
}
declare var DOMTokenList:{
prototype: DOMTokenList;
new(): DOMTokenList;
}
interface SVGFEFuncAElement extends SVGComponentTransferFunctionElement {
}
declare var SVGFEFuncAElement:{
prototype: SVGFEFuncAElement;
new(): SVGFEFuncAElement;
}
interface SVGFETileElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
in1: SVGAnimatedString;
}
declare var SVGFETileElement:{
prototype: SVGFETileElement;
new(): SVGFETileElement;
}
interface SVGFEBlendElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
in2: SVGAnimatedString;
mode: SVGAnimatedEnumeration;
in1: SVGAnimatedString;
SVG_FEBLEND_MODE_DARKEN: number;
SVG_FEBLEND_MODE_UNKNOWN: number;
SVG_FEBLEND_MODE_MULTIPLY: number;
SVG_FEBLEND_MODE_NORMAL: number;
SVG_FEBLEND_MODE_SCREEN: number;
SVG_FEBLEND_MODE_LIGHTEN: number;
}
declare var SVGFEBlendElement:{
prototype: SVGFEBlendElement;
new(): SVGFEBlendElement;
SVG_FEBLEND_MODE_DARKEN: number;
SVG_FEBLEND_MODE_UNKNOWN: number;
SVG_FEBLEND_MODE_MULTIPLY: number;
SVG_FEBLEND_MODE_NORMAL: number;
SVG_FEBLEND_MODE_SCREEN: number;
SVG_FEBLEND_MODE_LIGHTEN: number;
}
interface MessageChannel {
port2: MessagePort;
port1: MessagePort;
}
declare var MessageChannel:{
prototype: MessageChannel;
new(): MessageChannel;
}
interface SVGFEMergeElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
}
declare var SVGFEMergeElement:{
prototype: SVGFEMergeElement;
new(): SVGFEMergeElement;
}
interface TransitionEvent extends Event {
propertyName: string;
elapsedTime: number;
initTransitionEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, propertyNameArg:string, elapsedTimeArg:number): void;
}
declare var TransitionEvent:{
prototype: TransitionEvent;
new(): TransitionEvent;
}
interface MediaQueryList {
matches: boolean;
media: string;
addListener(listener:MediaQueryListListener): void;
removeListener(listener:MediaQueryListListener): void;
}
declare var MediaQueryList:{
prototype: MediaQueryList;
new(): MediaQueryList;
}
interface DOMError {
name: string;
toString(): string;
}
declare var DOMError:{
prototype: DOMError;
new(): DOMError;
}
interface CloseEvent extends Event {
wasClean: boolean;
reason: string;
code: number;
initCloseEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, wasCleanArg:boolean, codeArg:number, reasonArg:string): void;
}
declare var CloseEvent:{
prototype: CloseEvent;
new(): CloseEvent;
}
interface WebSocket extends EventTarget {
protocol: string;
readyState: number;
bufferedAmount: number;
onopen: (ev:Event) => any;
extensions: string;
onmessage: (ev:MessageEvent) => any;
onclose: (ev:CloseEvent) => any;
onerror: (ev:ErrorEvent) => any;
binaryType: string;
url: string;
close(code?:number, reason?:string): void;
send(data:any): void;
OPEN: number;
CLOSING: number;
CONNECTING: number;
CLOSED: number;
addEventListener(type:"open", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"message", listener:(ev:MessageEvent) => any, useCapture?:boolean): void;
addEventListener(type:"close", listener:(ev:CloseEvent) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var WebSocket:{
prototype: WebSocket;
new(url:string, protocols?:string): WebSocket;
new(url:string, protocols?:string[]): WebSocket;
OPEN: number;
CLOSING: number;
CONNECTING: number;
CLOSED: number;
}
interface SVGFEPointLightElement extends SVGElement {
y: SVGAnimatedNumber;
x: SVGAnimatedNumber;
z: SVGAnimatedNumber;
}
declare var SVGFEPointLightElement:{
prototype: SVGFEPointLightElement;
new(): SVGFEPointLightElement;
}
interface ProgressEvent extends Event {
loaded: number;
lengthComputable: boolean;
total: number;
initProgressEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, lengthComputableArg:boolean, loadedArg:number, totalArg:number): void;
}
declare var ProgressEvent:{
prototype: ProgressEvent;
new(): ProgressEvent;
}
interface IDBObjectStore {
indexNames: DOMStringList;
name: string;
transaction: IDBTransaction;
keyPath: string;
count(key?:any): IDBRequest;
add(value:any, key?:any): IDBRequest;
clear(): IDBRequest;
createIndex(name:string, keyPath:string, optionalParameters?:any): IDBIndex;
put(value:any, key?:any): IDBRequest;
openCursor(range?:any, direction?:string): IDBRequest;
deleteIndex(indexName:string): void;
index(name:string): IDBIndex;
get(key:any): IDBRequest;
delete(key:any): IDBRequest;
}
declare var IDBObjectStore:{
prototype: IDBObjectStore;
new(): IDBObjectStore;
}
interface SVGFEGaussianBlurElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
stdDeviationX: SVGAnimatedNumber;
in1: SVGAnimatedString;
stdDeviationY: SVGAnimatedNumber;
setStdDeviation(stdDeviationX:number, stdDeviationY:number): void;
}
declare var SVGFEGaussianBlurElement:{
prototype: SVGFEGaussianBlurElement;
new(): SVGFEGaussianBlurElement;
}
interface SVGFilterPrimitiveStandardAttributes extends SVGStylable {
y: SVGAnimatedLength;
width: SVGAnimatedLength;
x: SVGAnimatedLength;
height: SVGAnimatedLength;
result: SVGAnimatedString;
}
interface IDBVersionChangeEvent extends Event {
newVersion: number;
oldVersion: number;
}
declare var IDBVersionChangeEvent:{
prototype: IDBVersionChangeEvent;
new(): IDBVersionChangeEvent;
}
interface IDBIndex {
unique: boolean;
name: string;
keyPath: string;
objectStore: IDBObjectStore;
count(key?:any): IDBRequest;
getKey(key:any): IDBRequest;
openKeyCursor(range?:IDBKeyRange, direction?:string): IDBRequest;
get(key:any): IDBRequest;
openCursor(range?:IDBKeyRange, direction?:string): IDBRequest;
}
declare var IDBIndex:{
prototype: IDBIndex;
new(): IDBIndex;
}
interface FileList {
length: number;
item(index:number): File;
[index: number]: File;
}
declare var FileList:{
prototype: FileList;
new(): FileList;
}
interface IDBCursor {
source: any;
direction: string;
key: any;
primaryKey: any;
advance(count:number): void;
delete(): IDBRequest;
continue(key?:any): void;
update(value:any): IDBRequest;
PREV: string;
PREV_NO_DUPLICATE: string;
NEXT: string;
NEXT_NO_DUPLICATE: string;
}
declare var IDBCursor:{
prototype: IDBCursor;
new(): IDBCursor;
PREV: string;
PREV_NO_DUPLICATE: string;
NEXT: string;
NEXT_NO_DUPLICATE: string;
}
interface SVGFESpecularLightingElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
kernelUnitLengthY: SVGAnimatedNumber;
surfaceScale: SVGAnimatedNumber;
specularExponent: SVGAnimatedNumber;
in1: SVGAnimatedString;
kernelUnitLengthX: SVGAnimatedNumber;
specularConstant: SVGAnimatedNumber;
}
declare var SVGFESpecularLightingElement:{
prototype: SVGFESpecularLightingElement;
new(): SVGFESpecularLightingElement;
}
interface File extends Blob {
lastModifiedDate: any;
name: string;
}
declare var File:{
prototype: File;
new(): File;
}
interface URL {
revokeObjectURL(url:string): void;
createObjectURL(object:any, options?:ObjectURLOptions): string;
}
declare var URL:URL;
interface IDBCursorWithValue extends IDBCursor {
value: any;
}
declare var IDBCursorWithValue:{
prototype: IDBCursorWithValue;
new(): IDBCursorWithValue;
}
interface XMLHttpRequestEventTarget extends EventTarget {
onprogress: (ev:ProgressEvent) => any;
onerror: (ev:ErrorEvent) => any;
onload: (ev:Event) => any;
ontimeout: (ev:Event) => any;
onabort: (ev:UIEvent) => any;
onloadstart: (ev:Event) => any;
onloadend: (ev:ProgressEvent) => any;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"timeout", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"loadend", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var XMLHttpRequestEventTarget:{
prototype: XMLHttpRequestEventTarget;
new(): XMLHttpRequestEventTarget;
}
interface IDBEnvironment {
msIndexedDB: IDBFactory;
indexedDB: IDBFactory;
}
interface AudioTrackList extends EventTarget {
length: number;
onchange: (ev:Event) => any;
onaddtrack: (ev:TrackEvent) => any;
onremovetrack: (ev:any /*PluginArray*/) => any;
getTrackById(id:string): AudioTrack;
item(index:number): AudioTrack;
[index: number]: AudioTrack;
addEventListener(type:"change", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"addtrack", listener:(ev:TrackEvent) => any, useCapture?:boolean): void;
addEventListener(type:"removetrack", listener:(ev:any /*PluginArray*/) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var AudioTrackList:{
prototype: AudioTrackList;
new(): AudioTrackList;
}
interface MSBaseReader extends EventTarget {
onprogress: (ev:ProgressEvent) => any;
readyState: number;
onabort: (ev:UIEvent) => any;
onloadend: (ev:ProgressEvent) => any;
onerror: (ev:ErrorEvent) => any;
onload: (ev:Event) => any;
onloadstart: (ev:Event) => any;
result: any;
abort(): void;
LOADING: number;
EMPTY: number;
DONE: number;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:"loadend", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"loadstart", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
interface SVGFEMorphologyElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
operator: SVGAnimatedEnumeration;
radiusX: SVGAnimatedNumber;
radiusY: SVGAnimatedNumber;
in1: SVGAnimatedString;
SVG_MORPHOLOGY_OPERATOR_UNKNOWN: number;
SVG_MORPHOLOGY_OPERATOR_ERODE: number;
SVG_MORPHOLOGY_OPERATOR_DILATE: number;
}
declare var SVGFEMorphologyElement:{
prototype: SVGFEMorphologyElement;
new(): SVGFEMorphologyElement;
SVG_MORPHOLOGY_OPERATOR_UNKNOWN: number;
SVG_MORPHOLOGY_OPERATOR_ERODE: number;
SVG_MORPHOLOGY_OPERATOR_DILATE: number;
}
interface SVGFEFuncRElement extends SVGComponentTransferFunctionElement {
}
declare var SVGFEFuncRElement:{
prototype: SVGFEFuncRElement;
new(): SVGFEFuncRElement;
}
interface WindowTimersExtension {
msSetImmediate(expression:any, ...args:any[]): number;
clearImmediate(handle:number): void;
msClearImmediate(handle:number): void;
setImmediate(expression:any, ...args:any[]): number;
}
interface SVGFEDisplacementMapElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
in2: SVGAnimatedString;
xChannelSelector: SVGAnimatedEnumeration;
yChannelSelector: SVGAnimatedEnumeration;
scale: SVGAnimatedNumber;
in1: SVGAnimatedString;
SVG_CHANNEL_B: number;
SVG_CHANNEL_R: number;
SVG_CHANNEL_G: number;
SVG_CHANNEL_UNKNOWN: number;
SVG_CHANNEL_A: number;
}
declare var SVGFEDisplacementMapElement:{
prototype: SVGFEDisplacementMapElement;
new(): SVGFEDisplacementMapElement;
SVG_CHANNEL_B: number;
SVG_CHANNEL_R: number;
SVG_CHANNEL_G: number;
SVG_CHANNEL_UNKNOWN: number;
SVG_CHANNEL_A: number;
}
interface AnimationEvent extends Event {
animationName: string;
elapsedTime: number;
initAnimationEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, animationNameArg:string, elapsedTimeArg:number): void;
}
declare var AnimationEvent:{
prototype: AnimationEvent;
new(): AnimationEvent;
}
interface SVGComponentTransferFunctionElement extends SVGElement {
tableValues: SVGAnimatedNumberList;
slope: SVGAnimatedNumber;
type: SVGAnimatedEnumeration;
exponent: SVGAnimatedNumber;
amplitude: SVGAnimatedNumber;
intercept: SVGAnimatedNumber;
offset: SVGAnimatedNumber;
SVG_FECOMPONENTTRANSFER_TYPE_UNKNOWN: number;
SVG_FECOMPONENTTRANSFER_TYPE_TABLE: number;
SVG_FECOMPONENTTRANSFER_TYPE_IDENTITY: number;
SVG_FECOMPONENTTRANSFER_TYPE_GAMMA: number;
SVG_FECOMPONENTTRANSFER_TYPE_DISCRETE: number;
SVG_FECOMPONENTTRANSFER_TYPE_LINEAR: number;
}
declare var SVGComponentTransferFunctionElement:{
prototype: SVGComponentTransferFunctionElement;
new(): SVGComponentTransferFunctionElement;
SVG_FECOMPONENTTRANSFER_TYPE_UNKNOWN: number;
SVG_FECOMPONENTTRANSFER_TYPE_TABLE: number;
SVG_FECOMPONENTTRANSFER_TYPE_IDENTITY: number;
SVG_FECOMPONENTTRANSFER_TYPE_GAMMA: number;
SVG_FECOMPONENTTRANSFER_TYPE_DISCRETE: number;
SVG_FECOMPONENTTRANSFER_TYPE_LINEAR: number;
}
interface MSRangeCollection {
length: number;
item(index:number): Range;
[index: number]: Range;
}
declare var MSRangeCollection:{
prototype: MSRangeCollection;
new(): MSRangeCollection;
}
interface SVGFEDistantLightElement extends SVGElement {
azimuth: SVGAnimatedNumber;
elevation: SVGAnimatedNumber;
}
declare var SVGFEDistantLightElement:{
prototype: SVGFEDistantLightElement;
new(): SVGFEDistantLightElement;
}
interface SVGFEFuncBElement extends SVGComponentTransferFunctionElement {
}
declare var SVGFEFuncBElement:{
prototype: SVGFEFuncBElement;
new(): SVGFEFuncBElement;
}
interface IDBKeyRange {
upper: any;
upperOpen: boolean;
lower: any;
lowerOpen: boolean;
}
declare var IDBKeyRange:{
prototype: IDBKeyRange;
new(): IDBKeyRange;
bound(lower:any, upper:any, lowerOpen?:boolean, upperOpen?:boolean): IDBKeyRange;
only(value:any): IDBKeyRange;
lowerBound(bound:any, open?:boolean): IDBKeyRange;
upperBound(bound:any, open?:boolean): IDBKeyRange;
}
interface WindowConsole {
console: Console;
}
interface IDBTransaction extends EventTarget {
oncomplete: (ev:Event) => any;
db: IDBDatabase;
mode: string;
error: DOMError;
onerror: (ev:ErrorEvent) => any;
onabort: (ev:UIEvent) => any;
abort(): void;
objectStore(name:string): IDBObjectStore;
READ_ONLY: string;
VERSION_CHANGE: string;
READ_WRITE: string;
addEventListener(type:"complete", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var IDBTransaction:{
prototype: IDBTransaction;
new(): IDBTransaction;
READ_ONLY: string;
VERSION_CHANGE: string;
READ_WRITE: string;
}
interface AudioTrack {
kind: string;
language: string;
id: string;
label: string;
enabled: boolean;
sourceBuffer: SourceBuffer;
}
declare var AudioTrack:{
prototype: AudioTrack;
new(): AudioTrack;
}
interface SVGFEConvolveMatrixElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
orderY: SVGAnimatedInteger;
kernelUnitLengthY: SVGAnimatedNumber;
orderX: SVGAnimatedInteger;
preserveAlpha: SVGAnimatedBoolean;
kernelMatrix: SVGAnimatedNumberList;
edgeMode: SVGAnimatedEnumeration;
kernelUnitLengthX: SVGAnimatedNumber;
bias: SVGAnimatedNumber;
targetX: SVGAnimatedInteger;
targetY: SVGAnimatedInteger;
divisor: SVGAnimatedNumber;
in1: SVGAnimatedString;
SVG_EDGEMODE_WRAP: number;
SVG_EDGEMODE_DUPLICATE: number;
SVG_EDGEMODE_UNKNOWN: number;
SVG_EDGEMODE_NONE: number;
}
declare var SVGFEConvolveMatrixElement:{
prototype: SVGFEConvolveMatrixElement;
new(): SVGFEConvolveMatrixElement;
SVG_EDGEMODE_WRAP: number;
SVG_EDGEMODE_DUPLICATE: number;
SVG_EDGEMODE_UNKNOWN: number;
SVG_EDGEMODE_NONE: number;
}
interface TextTrackCueList {
length: number;
item(index:number): TextTrackCue;
[index: number]: TextTrackCue;
getCueById(id:string): TextTrackCue;
}
declare var TextTrackCueList:{
prototype: TextTrackCueList;
new(): TextTrackCueList;
}
interface CSSKeyframesRule extends CSSRule {
name: string;
cssRules: CSSRuleList;
findRule(rule:string): CSSKeyframeRule;
deleteRule(rule:string): void;
appendRule(rule:string): void;
}
declare var CSSKeyframesRule:{
prototype: CSSKeyframesRule;
new(): CSSKeyframesRule;
}
interface SVGFETurbulenceElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
baseFrequencyX: SVGAnimatedNumber;
numOctaves: SVGAnimatedInteger;
type: SVGAnimatedEnumeration;
baseFrequencyY: SVGAnimatedNumber;
stitchTiles: SVGAnimatedEnumeration;
seed: SVGAnimatedNumber;
SVG_STITCHTYPE_UNKNOWN: number;
SVG_STITCHTYPE_NOSTITCH: number;
SVG_TURBULENCE_TYPE_UNKNOWN: number;
SVG_TURBULENCE_TYPE_TURBULENCE: number;
SVG_TURBULENCE_TYPE_FRACTALNOISE: number;
SVG_STITCHTYPE_STITCH: number;
}
declare var SVGFETurbulenceElement:{
prototype: SVGFETurbulenceElement;
new(): SVGFETurbulenceElement;
SVG_STITCHTYPE_UNKNOWN: number;
SVG_STITCHTYPE_NOSTITCH: number;
SVG_TURBULENCE_TYPE_UNKNOWN: number;
SVG_TURBULENCE_TYPE_TURBULENCE: number;
SVG_TURBULENCE_TYPE_FRACTALNOISE: number;
SVG_STITCHTYPE_STITCH: number;
}
interface TextTrackList extends EventTarget {
length: number;
onaddtrack: (ev:TrackEvent) => any;
item(index:number): TextTrack;
[index: number]: TextTrack;
addEventListener(type:"addtrack", listener:(ev:TrackEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var TextTrackList:{
prototype: TextTrackList;
new(): TextTrackList;
}
interface SVGFEFuncGElement extends SVGComponentTransferFunctionElement {
}
declare var SVGFEFuncGElement:{
prototype: SVGFEFuncGElement;
new(): SVGFEFuncGElement;
}
interface SVGFEColorMatrixElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
in1: SVGAnimatedString;
type: SVGAnimatedEnumeration;
values: SVGAnimatedNumberList;
SVG_FECOLORMATRIX_TYPE_SATURATE: number;
SVG_FECOLORMATRIX_TYPE_UNKNOWN: number;
SVG_FECOLORMATRIX_TYPE_MATRIX: number;
SVG_FECOLORMATRIX_TYPE_HUEROTATE: number;
SVG_FECOLORMATRIX_TYPE_LUMINANCETOALPHA: number;
}
declare var SVGFEColorMatrixElement:{
prototype: SVGFEColorMatrixElement;
new(): SVGFEColorMatrixElement;
SVG_FECOLORMATRIX_TYPE_SATURATE: number;
SVG_FECOLORMATRIX_TYPE_UNKNOWN: number;
SVG_FECOLORMATRIX_TYPE_MATRIX: number;
SVG_FECOLORMATRIX_TYPE_HUEROTATE: number;
SVG_FECOLORMATRIX_TYPE_LUMINANCETOALPHA: number;
}
interface SVGFESpotLightElement extends SVGElement {
pointsAtY: SVGAnimatedNumber;
y: SVGAnimatedNumber;
limitingConeAngle: SVGAnimatedNumber;
specularExponent: SVGAnimatedNumber;
x: SVGAnimatedNumber;
pointsAtZ: SVGAnimatedNumber;
z: SVGAnimatedNumber;
pointsAtX: SVGAnimatedNumber;
}
declare var SVGFESpotLightElement:{
prototype: SVGFESpotLightElement;
new(): SVGFESpotLightElement;
}
interface WindowBase64 {
btoa(rawString:string): string;
atob(encodedString:string): string;
}
interface IDBDatabase extends EventTarget {
version: string;
name: string;
objectStoreNames: DOMStringList;
onerror: (ev:ErrorEvent) => any;
onabort: (ev:UIEvent) => any;
createObjectStore(name:string, optionalParameters?:any): IDBObjectStore;
close(): void;
transaction(storeNames:any, mode?:string): IDBTransaction;
deleteObjectStore(name:string): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var IDBDatabase:{
prototype: IDBDatabase;
new(): IDBDatabase;
}
interface DOMStringList {
length: number;
contains(str:string): boolean;
item(index:number): string;
[index: number]: string;
}
declare var DOMStringList:{
prototype: DOMStringList;
new(): DOMStringList;
}
interface IDBOpenDBRequest extends IDBRequest {
onupgradeneeded: (ev:IDBVersionChangeEvent) => any;
onblocked: (ev:Event) => any;
addEventListener(type:"success", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"upgradeneeded", listener:(ev:IDBVersionChangeEvent) => any, useCapture?:boolean): void;
addEventListener(type:"blocked", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var IDBOpenDBRequest:{
prototype: IDBOpenDBRequest;
new(): IDBOpenDBRequest;
}
interface HTMLProgressElement extends HTMLElement {
/**
* Sets or gets the current value of a progress element. The value must be a non-negative number between 0 and the max value.
*/
value: number;
/**
* Defines the maximum, or "done" value for a progress element.
*/
max: number;
/**
* Returns the quotient of value/max when the value attribute is set (determinate progress bar), or -1 when the value attribute is missing (indeterminate progress bar).
*/
position: number;
/**
* Retrieves a reference to the form that the object is embedded in.
*/
form: HTMLFormElement;
}
declare var HTMLProgressElement:{
prototype: HTMLProgressElement;
new(): HTMLProgressElement;
}
interface MSLaunchUriCallback {
(): void;
}
interface SVGFEOffsetElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
dy: SVGAnimatedNumber;
in1: SVGAnimatedString;
dx: SVGAnimatedNumber;
}
declare var SVGFEOffsetElement:{
prototype: SVGFEOffsetElement;
new(): SVGFEOffsetElement;
}
interface MSUnsafeFunctionCallback {
(): any;
}
interface TextTrack extends EventTarget {
language: string;
mode: any;
readyState: number;
activeCues: TextTrackCueList;
cues: TextTrackCueList;
oncuechange: (ev:Event) => any;
kind: string;
onload: (ev:Event) => any;
onerror: (ev:ErrorEvent) => any;
label: string;
addCue(cue:TextTrackCue): void;
removeCue(cue:TextTrackCue): void;
ERROR: number;
SHOWING: number;
LOADING: number;
LOADED: number;
NONE: number;
HIDDEN: number;
DISABLED: number;
addEventListener(type:"cuechange", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var TextTrack:{
prototype: TextTrack;
new(): TextTrack;
ERROR: number;
SHOWING: number;
LOADING: number;
LOADED: number;
NONE: number;
HIDDEN: number;
DISABLED: number;
}
interface MediaQueryListListener {
(mql:MediaQueryList): void;
}
interface IDBRequest extends EventTarget {
source: any;
onsuccess: (ev:Event) => any;
error: DOMError;
transaction: IDBTransaction;
onerror: (ev:ErrorEvent) => any;
readyState: string;
result: any;
addEventListener(type:"success", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var IDBRequest:{
prototype: IDBRequest;
new(): IDBRequest;
}
interface MessagePort extends EventTarget {
onmessage: (ev:MessageEvent) => any;
close(): void;
postMessage(message?:any, ports?:any): void;
start(): void;
addEventListener(type:"message", listener:(ev:MessageEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var MessagePort:{
prototype: MessagePort;
new(): MessagePort;
}
interface FileReader extends MSBaseReader {
error: DOMError;
readAsArrayBuffer(blob:Blob): void;
readAsDataURL(blob:Blob): void;
readAsText(blob:Blob, encoding?:string): void;
}
declare var FileReader:{
prototype: FileReader;
new(): FileReader;
}
interface ApplicationCache extends EventTarget {
status: number;
ondownloading: (ev:Event) => any;
onprogress: (ev:ProgressEvent) => any;
onupdateready: (ev:Event) => any;
oncached: (ev:Event) => any;
onobsolete: (ev:Event) => any;
onerror: (ev:ErrorEvent) => any;
onchecking: (ev:Event) => any;
onnoupdate: (ev:Event) => any;
swapCache(): void;
abort(): void;
update(): void;
CHECKING: number;
UNCACHED: number;
UPDATEREADY: number;
DOWNLOADING: number;
IDLE: number;
OBSOLETE: number;
addEventListener(type:"downloading", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"updateready", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"cached", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"obsolete", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"checking", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"noupdate", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var ApplicationCache:{
prototype: ApplicationCache;
new(): ApplicationCache;
CHECKING: number;
UNCACHED: number;
UPDATEREADY: number;
DOWNLOADING: number;
IDLE: number;
OBSOLETE: number;
}
interface FrameRequestCallback {
(time:number): void;
}
interface PopStateEvent extends Event {
state: any;
initPopStateEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, stateArg:any): void;
}
declare var PopStateEvent:{
prototype: PopStateEvent;
new(): PopStateEvent;
}
interface CSSKeyframeRule extends CSSRule {
keyText: string;
style: CSSStyleDeclaration;
}
declare var CSSKeyframeRule:{
prototype: CSSKeyframeRule;
new(): CSSKeyframeRule;
}
interface MSFileSaver {
msSaveBlob(blob:any, defaultName?:string): boolean;
msSaveOrOpenBlob(blob:any, defaultName?:string): boolean;
}
interface MSStream {
type: string;
msDetachStream(): any;
msClose(): void;
}
declare var MSStream:{
prototype: MSStream;
new(): MSStream;
}
interface MSBlobBuilder {
append(data:any, endings?:string): void;
getBlob(contentType?:string): Blob;
}
declare var MSBlobBuilder:{
prototype: MSBlobBuilder;
new(): MSBlobBuilder;
}
interface DOMSettableTokenList extends DOMTokenList {
value: string;
}
declare var DOMSettableTokenList:{
prototype: DOMSettableTokenList;
new(): DOMSettableTokenList;
}
interface IDBFactory {
open(name:string, version?:number): IDBOpenDBRequest;
cmp(first:any, second:any): number;
deleteDatabase(name:string): IDBOpenDBRequest;
}
declare var IDBFactory:{
prototype: IDBFactory;
new(): IDBFactory;
}
interface MSPointerEvent extends MouseEvent {
width: number;
rotation: number;
pressure: number;
pointerType: any;
isPrimary: boolean;
tiltY: number;
height: number;
intermediatePoints: any;
currentPoint: any;
tiltX: number;
hwTimestamp: number;
pointerId: number;
initPointerEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, viewArg:Window, detailArg:number, screenXArg:number, screenYArg:number, clientXArg:number, clientYArg:number, ctrlKeyArg:boolean, altKeyArg:boolean, shiftKeyArg:boolean, metaKeyArg:boolean, buttonArg:number, relatedTargetArg:EventTarget, offsetXArg:number, offsetYArg:number, widthArg:number, heightArg:number, pressure:number, rotation:number, tiltX:number, tiltY:number, pointerIdArg:number, pointerType:any, hwTimestampArg:number, isPrimary:boolean): void;
getCurrentPoint(element:Element): void;
getIntermediatePoints(element:Element): void;
MSPOINTER_TYPE_PEN: number;
MSPOINTER_TYPE_MOUSE: number;
MSPOINTER_TYPE_TOUCH: number;
}
declare var MSPointerEvent:{
prototype: MSPointerEvent;
new(): MSPointerEvent;
MSPOINTER_TYPE_PEN: number;
MSPOINTER_TYPE_MOUSE: number;
MSPOINTER_TYPE_TOUCH: number;
}
interface MSManipulationEvent extends UIEvent {
lastState: number;
currentState: number;
initMSManipulationEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, viewArg:Window, detailArg:number, lastState:number, currentState:number): void;
MS_MANIPULATION_STATE_STOPPED: number;
MS_MANIPULATION_STATE_ACTIVE: number;
MS_MANIPULATION_STATE_INERTIA: number;
MS_MANIPULATION_STATE_SELECTING: number;
MS_MANIPULATION_STATE_COMMITTED: number;
MS_MANIPULATION_STATE_PRESELECT: number;
MS_MANIPULATION_STATE_DRAGGING: number;
MS_MANIPULATION_STATE_CANCELLED: number;
}
declare var MSManipulationEvent:{
prototype: MSManipulationEvent;
new(): MSManipulationEvent;
MS_MANIPULATION_STATE_STOPPED: number;
MS_MANIPULATION_STATE_ACTIVE: number;
MS_MANIPULATION_STATE_INERTIA: number;
MS_MANIPULATION_STATE_SELECTING: number;
MS_MANIPULATION_STATE_COMMITTED: number;
MS_MANIPULATION_STATE_PRESELECT: number;
MS_MANIPULATION_STATE_DRAGGING: number;
MS_MANIPULATION_STATE_CANCELLED: number;
}
interface FormData {
append(name:any, value:any, blobName?:string): void;
}
declare var FormData:{
prototype: FormData;
new(): FormData;
}
interface HTMLDataListElement extends HTMLElement {
options: HTMLCollection;
}
declare var HTMLDataListElement:{
prototype: HTMLDataListElement;
new(): HTMLDataListElement;
}
interface SVGFEImageElement extends SVGElement, SVGLangSpace, SVGFilterPrimitiveStandardAttributes, SVGURIReference, SVGExternalResourcesRequired {
preserveAspectRatio: SVGAnimatedPreserveAspectRatio;
}
declare var SVGFEImageElement:{
prototype: SVGFEImageElement;
new(): SVGFEImageElement;
}
interface AbstractWorker extends EventTarget {
onerror: (ev:ErrorEvent) => any;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
interface SVGFECompositeElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
operator: SVGAnimatedEnumeration;
in2: SVGAnimatedString;
k2: SVGAnimatedNumber;
k1: SVGAnimatedNumber;
k3: SVGAnimatedNumber;
in1: SVGAnimatedString;
k4: SVGAnimatedNumber;
SVG_FECOMPOSITE_OPERATOR_OUT: number;
SVG_FECOMPOSITE_OPERATOR_OVER: number;
SVG_FECOMPOSITE_OPERATOR_XOR: number;
SVG_FECOMPOSITE_OPERATOR_ARITHMETIC: number;
SVG_FECOMPOSITE_OPERATOR_UNKNOWN: number;
SVG_FECOMPOSITE_OPERATOR_IN: number;
SVG_FECOMPOSITE_OPERATOR_ATOP: number;
}
declare var SVGFECompositeElement:{
prototype: SVGFECompositeElement;
new(): SVGFECompositeElement;
SVG_FECOMPOSITE_OPERATOR_OUT: number;
SVG_FECOMPOSITE_OPERATOR_OVER: number;
SVG_FECOMPOSITE_OPERATOR_XOR: number;
SVG_FECOMPOSITE_OPERATOR_ARITHMETIC: number;
SVG_FECOMPOSITE_OPERATOR_UNKNOWN: number;
SVG_FECOMPOSITE_OPERATOR_IN: number;
SVG_FECOMPOSITE_OPERATOR_ATOP: number;
}
interface ValidityState {
customError: boolean;
valueMissing: boolean;
stepMismatch: boolean;
rangeUnderflow: boolean;
rangeOverflow: boolean;
typeMismatch: boolean;
patternMismatch: boolean;
tooLong: boolean;
valid: boolean;
}
declare var ValidityState:{
prototype: ValidityState;
new(): ValidityState;
}
interface HTMLTrackElement extends HTMLElement {
kind: string;
src: string;
srclang: string;
track: TextTrack;
label: string;
default: boolean;
readyState: number;
ERROR: number;
LOADING: number;
LOADED: number;
NONE: number;
}
declare var HTMLTrackElement:{
prototype: HTMLTrackElement;
new(): HTMLTrackElement;
ERROR: number;
LOADING: number;
LOADED: number;
NONE: number;
}
interface MSApp {
createFileFromStorageFile(storageFile:any): File;
createBlobFromRandomAccessStream(type:string, seeker:any): Blob;
createStreamFromInputStream(type:string, inputStream:any): MSStream;
terminateApp(exceptionObject:any): void;
createDataPackage(object:any): any;
execUnsafeLocalFunction(unsafeFunction:MSUnsafeFunctionCallback): any;
getHtmlPrintDocumentSource(htmlDoc:any): any;
addPublicLocalApplicationUri(uri:string): void;
createDataPackageFromSelection(): any;
getViewOpener(): MSAppView;
suppressSubdownloadCredentialPrompts(suppress:boolean): void;
execAsyncAtPriority(asynchronousCallback:MSExecAtPriorityFunctionCallback, priority:string, ...args:any[]): void;
isTaskScheduledAtPriorityOrHigher(priority:string): boolean;
execAtPriority(synchronousCallback:MSExecAtPriorityFunctionCallback, priority:string, ...args:any[]): any;
createNewView(uri:string): MSAppView;
getCurrentPriority(): string;
NORMAL: string;
HIGH: string;
IDLE: string;
CURRENT: string;
}
declare var MSApp:MSApp;
interface SVGFEComponentTransferElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
in1: SVGAnimatedString;
}
declare var SVGFEComponentTransferElement:{
prototype: SVGFEComponentTransferElement;
new(): SVGFEComponentTransferElement;
}
interface SVGFEDiffuseLightingElement extends SVGElement, SVGFilterPrimitiveStandardAttributes {
kernelUnitLengthY: SVGAnimatedNumber;
surfaceScale: SVGAnimatedNumber;
in1: SVGAnimatedString;
kernelUnitLengthX: SVGAnimatedNumber;
diffuseConstant: SVGAnimatedNumber;
}
declare var SVGFEDiffuseLightingElement:{
prototype: SVGFEDiffuseLightingElement;
new(): SVGFEDiffuseLightingElement;
}
interface MSCSSMatrix {
m24: number;
m34: number;
a: number;
d: number;
m32: number;
m41: number;
m11: number;
f: number;
e: number;
m23: number;
m14: number;
m33: number;
m22: number;
m21: number;
c: number;
m12: number;
b: number;
m42: number;
m31: number;
m43: number;
m13: number;
m44: number;
multiply(secondMatrix:MSCSSMatrix): MSCSSMatrix;
skewY(angle:number): MSCSSMatrix;
setMatrixValue(value:string): void;
inverse(): MSCSSMatrix;
rotateAxisAngle(x:number, y:number, z:number, angle:number): MSCSSMatrix;
toString(): string;
rotate(angleX:number, angleY?:number, angleZ?:number): MSCSSMatrix;
translate(x:number, y:number, z?:number): MSCSSMatrix;
scale(scaleX:number, scaleY?:number, scaleZ?:number): MSCSSMatrix;
skewX(angle:number): MSCSSMatrix;
}
declare var MSCSSMatrix:{
prototype: MSCSSMatrix;
new(text?:string): MSCSSMatrix;
}
interface Worker extends AbstractWorker {
onmessage: (ev:MessageEvent) => any;
postMessage(message:any, ports?:any): void;
terminate(): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"message", listener:(ev:MessageEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var Worker:{
prototype: Worker;
new(stringUrl:string): Worker;
}
interface MSExecAtPriorityFunctionCallback {
(...args:any[]): any;
}
interface MSGraphicsTrust {
status: string;
constrictionActive: boolean;
}
declare var MSGraphicsTrust:{
prototype: MSGraphicsTrust;
new(): MSGraphicsTrust;
}
interface SubtleCrypto {
unwrapKey(wrappedKey:ArrayBufferView, keyAlgorithm:any, keyEncryptionKey:Key, extractable?:boolean, keyUsages?:string[]): KeyOperation;
encrypt(algorithm:any, key:Key, buffer?:ArrayBufferView): CryptoOperation;
importKey(format:string, keyData:ArrayBufferView, algorithm:any, extractable?:boolean, keyUsages?:string[]): KeyOperation;
wrapKey(key:Key, keyEncryptionKey:Key, keyWrappingAlgorithm:any): KeyOperation;
verify(algorithm:any, key:Key, signature:ArrayBufferView, buffer?:ArrayBufferView): CryptoOperation;
deriveKey(algorithm:any, baseKey:Key, derivedKeyType:any, extractable?:boolean, keyUsages?:string[]): KeyOperation;
digest(algorithm:any, buffer?:ArrayBufferView): CryptoOperation;
exportKey(format:string, key:Key): KeyOperation;
generateKey(algorithm:any, extractable?:boolean, keyUsages?:string[]): KeyOperation;
sign(algorithm:any, key:Key, buffer?:ArrayBufferView): CryptoOperation;
decrypt(algorithm:any, key:Key, buffer?:ArrayBufferView): CryptoOperation;
}
declare var SubtleCrypto:{
prototype: SubtleCrypto;
new(): SubtleCrypto;
}
interface Crypto extends RandomSource {
subtle: SubtleCrypto;
}
declare var Crypto:{
prototype: Crypto;
new(): Crypto;
}
interface VideoPlaybackQuality {
totalFrameDelay: number;
creationTime: number;
totalVideoFrames: number;
droppedVideoFrames: number;
}
declare var VideoPlaybackQuality:{
prototype: VideoPlaybackQuality;
new(): VideoPlaybackQuality;
}
interface GlobalEventHandlers {
onpointerenter: (ev:PointerEvent) => any;
onpointerout: (ev:PointerEvent) => any;
onpointerdown: (ev:PointerEvent) => any;
onpointerup: (ev:PointerEvent) => any;
onpointercancel: (ev:PointerEvent) => any;
onpointerover: (ev:PointerEvent) => any;
onpointermove: (ev:PointerEvent) => any;
onpointerleave: (ev:PointerEvent) => any;
addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
interface Key {
algorithm: Algorithm;
type: string;
extractable: boolean;
keyUsage: string[];
}
declare var Key:{
prototype: Key;
new(): Key;
}
interface DeviceAcceleration {
y: number;
x: number;
z: number;
}
declare var DeviceAcceleration:{
prototype: DeviceAcceleration;
new(): DeviceAcceleration;
}
interface HTMLAllCollection extends HTMLCollection {
namedItem(name:string): Element;
// [name: string]: Element;
}
declare var HTMLAllCollection:{
prototype: HTMLAllCollection;
new(): HTMLAllCollection;
}
interface AesGcmEncryptResult {
ciphertext: ArrayBuffer;
tag: ArrayBuffer;
}
declare var AesGcmEncryptResult:{
prototype: AesGcmEncryptResult;
new(): AesGcmEncryptResult;
}
interface NavigationCompletedEvent extends NavigationEvent {
webErrorStatus: number;
isSuccess: boolean;
}
declare var NavigationCompletedEvent:{
prototype: NavigationCompletedEvent;
new(): NavigationCompletedEvent;
}
interface MutationRecord {
oldValue: string;
previousSibling: Node;
addedNodes: NodeList;
attributeName: string;
removedNodes: NodeList;
target: Node;
nextSibling: Node;
attributeNamespace: string;
type: string;
}
declare var MutationRecord:{
prototype: MutationRecord;
new(): MutationRecord;
}
interface MimeTypeArray {
length: number;
item(index:number): Plugin;
[index: number]: Plugin;
namedItem(type:string): Plugin;
// [type: string]: Plugin;
}
declare var MimeTypeArray:{
prototype: MimeTypeArray;
new(): MimeTypeArray;
}
interface KeyOperation extends EventTarget {
oncomplete: (ev:Event) => any;
onerror: (ev:ErrorEvent) => any;
result: any;
addEventListener(type:"complete", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var KeyOperation:{
prototype: KeyOperation;
new(): KeyOperation;
}
interface DOMStringMap {
}
declare var DOMStringMap:{
prototype: DOMStringMap;
new(): DOMStringMap;
}
interface DeviceOrientationEvent extends Event {
gamma: number;
alpha: number;
absolute: boolean;
beta: number;
initDeviceOrientationEvent(type:string, bubbles:boolean, cancelable:boolean, alpha:number, beta:number, gamma:number, absolute:boolean): void;
}
declare var DeviceOrientationEvent:{
prototype: DeviceOrientationEvent;
new(): DeviceOrientationEvent;
}
interface MSMediaKeys {
keySystem: string;
createSession(type:string, initData:Uint8Array, cdmData?:Uint8Array): MSMediaKeySession;
}
declare var MSMediaKeys:{
prototype: MSMediaKeys;
new(keySystem:string): MSMediaKeys;
isTypeSupported(keySystem:string, type?:string): boolean;
}
interface MSMediaKeyMessageEvent extends Event {
destinationURL: string;
message: Uint8Array;
}
declare var MSMediaKeyMessageEvent:{
prototype: MSMediaKeyMessageEvent;
new(): MSMediaKeyMessageEvent;
}
interface MSHTMLWebViewElement extends HTMLElement {
documentTitle: string;
width: number;
src: string;
canGoForward: boolean;
height: number;
canGoBack: boolean;
navigateWithHttpRequestMessage(requestMessage:any): void;
goBack(): void;
navigate(uri:string): void;
stop(): void;
navigateToString(contents:string): void;
captureSelectedContentToDataPackageAsync(): MSWebViewAsyncOperation;
capturePreviewToBlobAsync(): MSWebViewAsyncOperation;
refresh(): void;
goForward(): void;
navigateToLocalStreamUri(source:string, streamResolver:any): void;
invokeScriptAsync(scriptName:string, ...args:any[]): MSWebViewAsyncOperation;
buildLocalStreamUri(contentIdentifier:string, relativePath:string): string;
}
declare var MSHTMLWebViewElement:{
prototype: MSHTMLWebViewElement;
new(): MSHTMLWebViewElement;
}
interface NavigationEvent extends Event {
uri: string;
}
declare var NavigationEvent:{
prototype: NavigationEvent;
new(): NavigationEvent;
}
interface RandomSource {
getRandomValues(array:ArrayBufferView): ArrayBufferView;
}
interface SourceBuffer extends EventTarget {
updating: boolean;
appendWindowStart: number;
appendWindowEnd: number;
buffered: TimeRanges;
timestampOffset: number;
audioTracks: AudioTrackList;
appendBuffer(data:ArrayBuffer): void;
remove(start:number, end:number): void;
abort(): void;
appendStream(stream:MSStream, maxSize?:number): void;
}
declare var SourceBuffer:{
prototype: SourceBuffer;
new(): SourceBuffer;
}
interface MSInputMethodContext extends EventTarget {
oncandidatewindowshow: (ev:any) => any;
target: HTMLElement;
compositionStartOffset: number;
oncandidatewindowhide: (ev:any) => any;
oncandidatewindowupdate: (ev:any) => any;
compositionEndOffset: number;
getCompositionAlternatives(): string[];
getCandidateWindowClientRect(): ClientRect;
hasComposition(): boolean;
isCandidateWindowVisible(): boolean;
addEventListener(type:"candidatewindowshow", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"candidatewindowhide", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:"candidatewindowupdate", listener:(ev:any) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var MSInputMethodContext:{
prototype: MSInputMethodContext;
new(): MSInputMethodContext;
}
interface DeviceRotationRate {
gamma: number;
alpha: number;
beta: number;
}
declare var DeviceRotationRate:{
prototype: DeviceRotationRate;
new(): DeviceRotationRate;
}
interface PluginArray {
length: number;
refresh(reload?:boolean): void;
item(index:number): Plugin;
[index: number]: Plugin;
namedItem(name:string): Plugin;
// [name: string]: Plugin;
}
declare var PluginArray:{
prototype: PluginArray;
new(): PluginArray;
}
interface MSMediaKeyError {
systemCode: number;
code: number;
MS_MEDIA_KEYERR_SERVICE: number;
MS_MEDIA_KEYERR_HARDWARECHANGE: number;
MS_MEDIA_KEYERR_OUTPUT: number;
MS_MEDIA_KEYERR_DOMAIN: number;
MS_MEDIA_KEYERR_UNKNOWN: number;
MS_MEDIA_KEYERR_CLIENT: number;
}
declare var MSMediaKeyError:{
prototype: MSMediaKeyError;
new(): MSMediaKeyError;
MS_MEDIA_KEYERR_SERVICE: number;
MS_MEDIA_KEYERR_HARDWARECHANGE: number;
MS_MEDIA_KEYERR_OUTPUT: number;
MS_MEDIA_KEYERR_DOMAIN: number;
MS_MEDIA_KEYERR_UNKNOWN: number;
MS_MEDIA_KEYERR_CLIENT: number;
}
interface Plugin {
length: number;
filename: string;
version: string;
name: string;
description: string;
item(index:number): MimeType;
[index: number]: MimeType;
namedItem(type:string): MimeType;
// [type: string]: MimeType;
}
declare var Plugin:{
prototype: Plugin;
new(): Plugin;
}
interface MediaSource extends EventTarget {
sourceBuffers: SourceBufferList;
duration: number;
readyState: string;
activeSourceBuffers: SourceBufferList;
addSourceBuffer(type:string): SourceBuffer;
endOfStream(error?:string): void;
removeSourceBuffer(sourceBuffer:SourceBuffer): void;
}
declare var MediaSource:{
prototype: MediaSource;
new(): MediaSource;
isTypeSupported(type:string): boolean;
}
interface SourceBufferList extends EventTarget {
length: number;
item(index:number): SourceBuffer;
[index: number]: SourceBuffer;
}
declare var SourceBufferList:{
prototype: SourceBufferList;
new(): SourceBufferList;
}
interface XMLDocument extends Document {
}
declare var XMLDocument:{
prototype: XMLDocument;
new(): XMLDocument;
}
interface DeviceMotionEvent extends Event {
rotationRate: DeviceRotationRate;
acceleration: DeviceAcceleration;
interval: number;
accelerationIncludingGravity: DeviceAcceleration;
initDeviceMotionEvent(type:string, bubbles:boolean, cancelable:boolean, acceleration:DeviceAccelerationDict, accelerationIncludingGravity:DeviceAccelerationDict, rotationRate:DeviceRotationRateDict, interval:number): void;
}
declare var DeviceMotionEvent:{
prototype: DeviceMotionEvent;
new(): DeviceMotionEvent;
}
interface MimeType {
enabledPlugin: Plugin;
suffixes: string;
type: string;
description: string;
}
declare var MimeType:{
prototype: MimeType;
new(): MimeType;
}
interface PointerEvent extends MouseEvent {
width: number;
rotation: number;
pressure: number;
pointerType: any;
isPrimary: boolean;
tiltY: number;
height: number;
intermediatePoints: any;
currentPoint: any;
tiltX: number;
hwTimestamp: number;
pointerId: number;
initPointerEvent(typeArg:string, canBubbleArg:boolean, cancelableArg:boolean, viewArg:Window, detailArg:number, screenXArg:number, screenYArg:number, clientXArg:number, clientYArg:number, ctrlKeyArg:boolean, altKeyArg:boolean, shiftKeyArg:boolean, metaKeyArg:boolean, buttonArg:number, relatedTargetArg:EventTarget, offsetXArg:number, offsetYArg:number, widthArg:number, heightArg:number, pressure:number, rotation:number, tiltX:number, tiltY:number, pointerIdArg:number, pointerType:any, hwTimestampArg:number, isPrimary:boolean): void;
getCurrentPoint(element:Element): void;
getIntermediatePoints(element:Element): void;
}
declare var PointerEvent:{
prototype: PointerEvent;
new(): PointerEvent;
}
interface MSDocumentExtensions {
captureEvents(): void;
releaseEvents(): void;
}
interface MutationObserver {
observe(target:Node, options:MutationObserverInit): void;
takeRecords(): MutationRecord[];
disconnect(): void;
}
declare var MutationObserver:{
prototype: MutationObserver;
new (callback:(arr:MutationRecord[], observer:MutationObserver)=>any): MutationObserver;
}
interface MSWebViewAsyncOperation extends EventTarget {
target: MSHTMLWebViewElement;
oncomplete: (ev:Event) => any;
error: DOMError;
onerror: (ev:ErrorEvent) => any;
readyState: number;
type: number;
result: any;
start(): void;
ERROR: number;
TYPE_CREATE_DATA_PACKAGE_FROM_SELECTION: number;
TYPE_INVOKE_SCRIPT: number;
COMPLETED: number;
TYPE_CAPTURE_PREVIEW_TO_RANDOM_ACCESS_STREAM: number;
STARTED: number;
addEventListener(type:"complete", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var MSWebViewAsyncOperation:{
prototype: MSWebViewAsyncOperation;
new(): MSWebViewAsyncOperation;
ERROR: number;
TYPE_CREATE_DATA_PACKAGE_FROM_SELECTION: number;
TYPE_INVOKE_SCRIPT: number;
COMPLETED: number;
TYPE_CAPTURE_PREVIEW_TO_RANDOM_ACCESS_STREAM: number;
STARTED: number;
}
interface ScriptNotifyEvent extends Event {
value: string;
callingUri: string;
}
declare var ScriptNotifyEvent:{
prototype: ScriptNotifyEvent;
new(): ScriptNotifyEvent;
}
interface PerformanceNavigationTiming extends PerformanceEntry {
redirectStart: number;
domainLookupEnd: number;
responseStart: number;
domComplete: number;
domainLookupStart: number;
loadEventStart: number;
unloadEventEnd: number;
fetchStart: number;
requestStart: number;
domInteractive: number;
navigationStart: number;
connectEnd: number;
loadEventEnd: number;
connectStart: number;
responseEnd: number;
domLoading: number;
redirectEnd: number;
redirectCount: number;
unloadEventStart: number;
domContentLoadedEventStart: number;
domContentLoadedEventEnd: number;
type: string;
}
declare var PerformanceNavigationTiming:{
prototype: PerformanceNavigationTiming;
new(): PerformanceNavigationTiming;
}
interface MSMediaKeyNeededEvent extends Event {
initData: Uint8Array;
}
declare var MSMediaKeyNeededEvent:{
prototype: MSMediaKeyNeededEvent;
new(): MSMediaKeyNeededEvent;
}
interface LongRunningScriptDetectedEvent extends Event {
stopPageScriptExecution: boolean;
executionTime: number;
}
declare var LongRunningScriptDetectedEvent:{
prototype: LongRunningScriptDetectedEvent;
new(): LongRunningScriptDetectedEvent;
}
interface MSAppView {
viewId: number;
close(): void;
postMessage(message:any, targetOrigin:string, ports?:any): void;
}
declare var MSAppView:{
prototype: MSAppView;
new(): MSAppView;
}
interface PerfWidgetExternal {
maxCpuSpeed: number;
independentRenderingEnabled: boolean;
irDisablingContentString: string;
irStatusAvailable: boolean;
performanceCounter: number;
averagePaintTime: number;
activeNetworkRequestCount: number;
paintRequestsPerSecond: number;
extraInformationEnabled: boolean;
performanceCounterFrequency: number;
averageFrameTime: number;
repositionWindow(x:number, y:number): void;
getRecentMemoryUsage(last:number): any;
getMemoryUsage(): number;
resizeWindow(width:number, height:number): void;
getProcessCpuUsage(): number;
removeEventListener(eventType:string, callback:(ev:any) => any): void;
getRecentCpuUsage(last:number): any;
addEventListener(eventType:string, callback:(ev:any) => any): void;
getRecentFrames(last:number): any;
getRecentPaintRequests(last:number): any;
}
declare var PerfWidgetExternal:{
prototype: PerfWidgetExternal;
new(): PerfWidgetExternal;
}
interface PageTransitionEvent extends Event {
persisted: boolean;
}
declare var PageTransitionEvent:{
prototype: PageTransitionEvent;
new(): PageTransitionEvent;
}
interface MutationCallback {
(mutations:MutationRecord[], observer:MutationObserver): void;
}
interface HTMLDocument extends Document {
}
declare var HTMLDocument:{
prototype: HTMLDocument;
new(): HTMLDocument;
}
interface KeyPair {
privateKey: Key;
publicKey: Key;
}
declare var KeyPair:{
prototype: KeyPair;
new(): KeyPair;
}
interface MSMediaKeySession extends EventTarget {
sessionId: string;
error: MSMediaKeyError;
keySystem: string;
close(): void;
update(key:Uint8Array): void;
}
declare var MSMediaKeySession:{
prototype: MSMediaKeySession;
new(): MSMediaKeySession;
}
interface UnviewableContentIdentifiedEvent extends NavigationEvent {
referrer: string;
}
declare var UnviewableContentIdentifiedEvent:{
prototype: UnviewableContentIdentifiedEvent;
new(): UnviewableContentIdentifiedEvent;
}
interface CryptoOperation extends EventTarget {
algorithm: Algorithm;
oncomplete: (ev:Event) => any;
onerror: (ev:ErrorEvent) => any;
onprogress: (ev:ProgressEvent) => any;
onabort: (ev:UIEvent) => any;
key: Key;
result: any;
abort(): void;
finish(): void;
process(buffer:ArrayBufferView): void;
addEventListener(type:"complete", listener:(ev:Event) => any, useCapture?:boolean): void;
addEventListener(type:"error", listener:(ev:ErrorEvent) => any, useCapture?:boolean): void;
addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean): void;
addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean): void;
addEventListener(type:string, listener:EventListener, useCapture?:boolean): void;
}
declare var CryptoOperation:{
prototype: CryptoOperation;
new(): CryptoOperation;
}
interface WebGLTexture extends WebGLObject {
}
declare var WebGLTexture:{
prototype: WebGLTexture;
new(): WebGLTexture;
}
interface OES_texture_float {
}
declare var OES_texture_float:{
prototype: OES_texture_float;
new(): OES_texture_float;
}
interface WebGLContextEvent extends Event {
statusMessage: string;
}
declare var WebGLContextEvent:{
prototype: WebGLContextEvent;
new(): WebGLContextEvent;
}
interface WebGLRenderbuffer extends WebGLObject {
}
declare var WebGLRenderbuffer:{
prototype: WebGLRenderbuffer;
new(): WebGLRenderbuffer;
}
interface WebGLUniformLocation {
}
declare var WebGLUniformLocation:{
prototype: WebGLUniformLocation;
new(): WebGLUniformLocation;
}
interface WebGLActiveInfo {
name: string;
type: number;
size: number;
}
declare var WebGLActiveInfo:{
prototype: WebGLActiveInfo;
new(): WebGLActiveInfo;
}
interface WEBGL_compressed_texture_s3tc {
COMPRESSED_RGBA_S3TC_DXT1_EXT: number;
COMPRESSED_RGBA_S3TC_DXT5_EXT: number;
COMPRESSED_RGBA_S3TC_DXT3_EXT: number;
COMPRESSED_RGB_S3TC_DXT1_EXT: number;
}
declare var WEBGL_compressed_texture_s3tc:{
prototype: WEBGL_compressed_texture_s3tc;
new(): WEBGL_compressed_texture_s3tc;
COMPRESSED_RGBA_S3TC_DXT1_EXT: number;
COMPRESSED_RGBA_S3TC_DXT5_EXT: number;
COMPRESSED_RGBA_S3TC_DXT3_EXT: number;
COMPRESSED_RGB_S3TC_DXT1_EXT: number;
}
interface WebGLRenderingContext {
drawingBufferWidth: number;
drawingBufferHeight: number;
canvas: HTMLCanvasElement;
getUniformLocation(program:WebGLProgram, name:string): WebGLUniformLocation;
bindTexture(target:number, texture:WebGLTexture): void;
bufferData(target:number, data:ArrayBufferView, usage:number): void;
bufferData(target:number, data:ArrayBuffer, usage:number): void;
bufferData(target:number, size:number, usage:number): void;
depthMask(flag:boolean): void;
getUniform(program:WebGLProgram, location:WebGLUniformLocation): any;
vertexAttrib3fv(indx:number, values:number[]): void;
vertexAttrib3fv(indx:number, values:Float32Array): void;
linkProgram(program:WebGLProgram): void;
getSupportedExtensions(): string[];
bufferSubData(target:number, offset:number, data:ArrayBuffer): void;
bufferSubData(target:number, offset:number, data:ArrayBufferView): void;
vertexAttribPointer(indx:number, size:number, type:number, normalized:boolean, stride:number, offset:number): void;
polygonOffset(factor:number, units:number): void;
blendColor(red:number, green:number, blue:number, alpha:number): void;
createTexture(): WebGLTexture;
hint(target:number, mode:number): void;
getVertexAttrib(index:number, pname:number): any;
enableVertexAttribArray(index:number): void;
depthRange(zNear:number, zFar:number): void;
cullFace(mode:number): void;
createFramebuffer(): WebGLFramebuffer;
uniformMatrix4fv(location:WebGLUniformLocation, transpose:boolean, value:number[]): void;
uniformMatrix4fv(location:WebGLUniformLocation, transpose:boolean, value:Float32Array): void;
framebufferTexture2D(target:number, attachment:number, textarget:number, texture:WebGLTexture, level:number): void;
deleteFramebuffer(framebuffer:WebGLFramebuffer): void;
colorMask(red:boolean, green:boolean, blue:boolean, alpha:boolean): void;
compressedTexImage2D(target:number, level:number, internalformat:number, width:number, height:number, border:number, data:ArrayBufferView): void;
uniformMatrix2fv(location:WebGLUniformLocation, transpose:boolean, value:number[]): void;
uniformMatrix2fv(location:WebGLUniformLocation, transpose:boolean, value:Float32Array): void;
getExtension(name:string): any;
createProgram(): WebGLProgram;
deleteShader(shader:WebGLShader): void;
getAttachedShaders(program:WebGLProgram): WebGLShader[];
enable(cap:number): void;
blendEquation(mode:number): void;
texImage2D(target:number, level:number, internalformat:number, width:number, height:number, border:number, format:number, type:number, pixels:ArrayBufferView): void;
texImage2D(target:number, level:number, internalformat:number, format:number, type:number, image:HTMLImageElement): void;
texImage2D(target:number, level:number, internalformat:number, format:number, type:number, canvas:HTMLCanvasElement): void;
texImage2D(target:number, level:number, internalformat:number, format:number, type:number, video:HTMLVideoElement): void;
texImage2D(target:number, level:number, internalformat:number, format:number, type:number, pixels:ImageData): void;
createBuffer(): WebGLBuffer;
deleteTexture(texture:WebGLTexture): void;
useProgram(program:WebGLProgram): void;
vertexAttrib2fv(indx:number, values:number[]): void;
vertexAttrib2fv(indx:number, values:Float32Array): void;
checkFramebufferStatus(target:number): number;
frontFace(mode:number): void;
getBufferParameter(target:number, pname:number): any;
texSubImage2D(target:number, level:number, xoffset:number, yoffset:number, width:number, height:number, format:number, type:number, pixels:ArrayBufferView): void;
texSubImage2D(target:number, level:number, xoffset:number, yoffset:number, format:number, type:number, image:HTMLImageElement): void;
texSubImage2D(target:number, level:number, xoffset:number, yoffset:number, format:number, type:number, canvas:HTMLCanvasElement): void;
texSubImage2D(target:number, level:number, xoffset:number, yoffset:number, format:number, type:number, video:HTMLVideoElement): void;
texSubImage2D(target:number, level:number, xoffset:number, yoffset:number, format:number, type:number, pixels:ImageData): void;
copyTexImage2D(target:number, level:number, internalformat:number, x:number, y:number, width:number, height:number, border:number): void;
getVertexAttribOffset(index:number, pname:number): number;
disableVertexAttribArray(index:number): void;
blendFunc(sfactor:number, dfactor:number): void;
drawElements(mode:number, count:number, type:number, offset:number): void;
isFramebuffer(framebuffer:WebGLFramebuffer): boolean;
uniform3iv(location:WebGLUniformLocation, v:number[]): void;
uniform3iv(location:WebGLUniformLocation, v:Int32Array): void;
lineWidth(width:number): void;
getShaderInfoLog(shader:WebGLShader): string;
getTexParameter(target:number, pname:number): any;
getParameter(pname:number): any;
getShaderPrecisionFormat(shadertype:number, precisiontype:number): WebGLShaderPrecisionFormat;
getContextAttributes(): WebGLContextAttributes;
vertexAttrib1f(indx:number, x:number): void;
bindFramebuffer(target:number, framebuffer:WebGLFramebuffer): void;
compressedTexSubImage2D(target:number, level:number, xoffset:number, yoffset:number, width:number, height:number, format:number, data:ArrayBufferView): void;
isContextLost(): boolean;
uniform1iv(location:WebGLUniformLocation, v:number[]): void;
uniform1iv(location:WebGLUniformLocation, v:Int32Array): void;
getRenderbufferParameter(target:number, pname:number): any;
uniform2fv(location:WebGLUniformLocation, v:number[]): void;
uniform2fv(location:WebGLUniformLocation, v:Float32Array): void;
isTexture(texture:WebGLTexture): boolean;
getError(): number;
shaderSource(shader:WebGLShader, source:string): void;
deleteRenderbuffer(renderbuffer:WebGLRenderbuffer): void;
stencilMask(mask:number): void;
bindBuffer(target:number, buffer:WebGLBuffer): void;
getAttribLocation(program:WebGLProgram, name:string): number;
uniform3i(location:WebGLUniformLocation, x:number, y:number, z:number): void;
blendEquationSeparate(modeRGB:number, modeAlpha:number): void;
clear(mask:number): void;
blendFuncSeparate(srcRGB:number, dstRGB:number, srcAlpha:number, dstAlpha:number): void;
stencilFuncSeparate(face:number, func:number, ref:number, mask:number): void;
readPixels(x:number, y:number, width:number, height:number, format:number, type:number, pixels:ArrayBufferView): void;
scissor(x:number, y:number, width:number, height:number): void;
uniform2i(location:WebGLUniformLocation, x:number, y:number): void;
getActiveAttrib(program:WebGLProgram, index:number): WebGLActiveInfo;
getShaderSource(shader:WebGLShader): string;
generateMipmap(target:number): void;
bindAttribLocation(program:WebGLProgram, index:number, name:string): void;
uniform1fv(location:WebGLUniformLocation, v:number[]): void;
uniform1fv(location:WebGLUniformLocation, v:Float32Array): void;
uniform2iv(location:WebGLUniformLocation, v:number[]): void;
uniform2iv(location:WebGLUniformLocation, v:Int32Array): void;
stencilOp(fail:number, zfail:number, zpass:number): void;
uniform4fv(location:WebGLUniformLocation, v:number[]): void;
uniform4fv(location:WebGLUniformLocation, v:Float32Array): void;
vertexAttrib1fv(indx:number, values:number[]): void;
vertexAttrib1fv(indx:number, values:Float32Array): void;
flush(): void;
uniform4f(location:WebGLUniformLocation, x:number, y:number, z:number, w:number): void;
deleteProgram(program:WebGLProgram): void;
isRenderbuffer(renderbuffer:WebGLRenderbuffer): boolean;
uniform1i(location:WebGLUniformLocation, x:number): void;
getProgramParameter(program:WebGLProgram, pname:number): any;
getActiveUniform(program:WebGLProgram, index:number): WebGLActiveInfo;
stencilFunc(func:number, ref:number, mask:number): void;
pixelStorei(pname:number, param:number): void;
disable(cap:number): void;
vertexAttrib4fv(indx:number, values:number[]): void;
vertexAttrib4fv(indx:number, values:Float32Array): void;
createRenderbuffer(): WebGLRenderbuffer;
isBuffer(buffer:WebGLBuffer): boolean;
stencilOpSeparate(face:number, fail:number, zfail:number, zpass:number): void;
getFramebufferAttachmentParameter(target:number, attachment:number, pname:number): any;
uniform4i(location:WebGLUniformLocation, x:number, y:number, z:number, w:number): void;
sampleCoverage(value:number, invert:boolean): void;
depthFunc(func:number): void;
texParameterf(target:number, pname:number, param:number): void;
vertexAttrib3f(indx:number, x:number, y:number, z:number): void;
drawArrays(mode:number, first:number, count:number): void;
texParameteri(target:number, pname:number, param:number): void;
vertexAttrib4f(indx:number, x:number, y:number, z:number, w:number): void;
getShaderParameter(shader:WebGLShader, pname:number): any;
clearDepth(depth:number): void;
activeTexture(texture:number): void;
viewport(x:number, y:number, width:number, height:number): void;
detachShader(program:WebGLProgram, shader:WebGLShader): void;
uniform1f(location:WebGLUniformLocation, x:number): void;
uniformMatrix3fv(location:WebGLUniformLocation, transpose:boolean, value:number[]): void;
uniformMatrix3fv(location:WebGLUniformLocation, transpose:boolean, value:Float32Array): void;
deleteBuffer(buffer:WebGLBuffer): void;
copyTexSubImage2D(target:number, level:number, xoffset:number, yoffset:number, x:number, y:number, width:number, height:number): void;
uniform3fv(location:WebGLUniformLocation, v:number[]): void;
uniform3fv(location:WebGLUniformLocation, v:Float32Array): void;
stencilMaskSeparate(face:number, mask:number): void;
attachShader(program:WebGLProgram, shader:WebGLShader): void;
compileShader(shader:WebGLShader): void;
clearColor(red:number, green:number, blue:number, alpha:number): void;
isShader(shader:WebGLShader): boolean;
clearStencil(s:number): void;
framebufferRenderbuffer(target:number, attachment:number, renderbuffertarget:number, renderbuffer:WebGLRenderbuffer): void;
finish(): void;
uniform2f(location:WebGLUniformLocation, x:number, y:number): void;
renderbufferStorage(target:number, internalformat:number, width:number, height:number): void;
uniform3f(location:WebGLUniformLocation, x:number, y:number, z:number): void;
getProgramInfoLog(program:WebGLProgram): string;
validateProgram(program:WebGLProgram): void;
isEnabled(cap:number): boolean;
vertexAttrib2f(indx:number, x:number, y:number): void;
isProgram(program:WebGLProgram): boolean;
createShader(type:number): WebGLShader;
bindRenderbuffer(target:number, renderbuffer:WebGLRenderbuffer): void;
uniform4iv(location:WebGLUniformLocation, v:number[]): void;
uniform4iv(location:WebGLUniformLocation, v:Int32Array): void;
DEPTH_FUNC: number;
DEPTH_COMPONENT16: number;
REPLACE: number;
REPEAT: number;
VERTEX_ATTRIB_ARRAY_ENABLED: number;
FRAMEBUFFER_INCOMPLETE_DIMENSIONS: number;
STENCIL_BUFFER_BIT: number;
RENDERER: number;
STENCIL_BACK_REF: number;
TEXTURE26: number;
RGB565: number;
DITHER: number;
CONSTANT_COLOR: number;
GENERATE_MIPMAP_HINT: number;
POINTS: number;
DECR: number;
INT_VEC3: number;
TEXTURE28: number;
ONE_MINUS_CONSTANT_ALPHA: number;
BACK: number;
RENDERBUFFER_STENCIL_SIZE: number;
UNPACK_FLIP_Y_WEBGL: number;
BLEND: number;
TEXTURE9: number;
ARRAY_BUFFER_BINDING: number;
MAX_VIEWPORT_DIMS: number;
INVALID_FRAMEBUFFER_OPERATION: number;
TEXTURE: number;
TEXTURE0: number;
TEXTURE31: number;
TEXTURE24: number;
HIGH_INT: number;
RENDERBUFFER_BINDING: number;
BLEND_COLOR: number;
FASTEST: number;
STENCIL_WRITEMASK: number;
ALIASED_POINT_SIZE_RANGE: number;
TEXTURE12: number;
DST_ALPHA: number;
BLEND_EQUATION_RGB: number;
FRAMEBUFFER_COMPLETE: number;
NEAREST_MIPMAP_NEAREST: number;
VERTEX_ATTRIB_ARRAY_SIZE: number;
TEXTURE3: number;
DEPTH_WRITEMASK: number;
CONTEXT_LOST_WEBGL: number;
INVALID_VALUE: number;
TEXTURE_MAG_FILTER: number;
ONE_MINUS_CONSTANT_COLOR: number;
ONE_MINUS_SRC_ALPHA: number;
TEXTURE_CUBE_MAP_POSITIVE_Z: number;
NOTEQUAL: number;
ALPHA: number;
DEPTH_STENCIL: number;
MAX_VERTEX_UNIFORM_VECTORS: number;
DEPTH_COMPONENT: number;
RENDERBUFFER_RED_SIZE: number;
TEXTURE20: number;
RED_BITS: number;
RENDERBUFFER_BLUE_SIZE: number;
SCISSOR_BOX: number;
VENDOR: number;
FRONT_AND_BACK: number;
CONSTANT_ALPHA: number;
VERTEX_ATTRIB_ARRAY_BUFFER_BINDING: number;
NEAREST: number;
CULL_FACE: number;
ALIASED_LINE_WIDTH_RANGE: number;
TEXTURE19: number;
FRONT: number;
DEPTH_CLEAR_VALUE: number;
GREEN_BITS: number;
TEXTURE29: number;
TEXTURE23: number;
MAX_RENDERBUFFER_SIZE: number;
STENCIL_ATTACHMENT: number;
TEXTURE27: number;
BOOL_VEC2: number;
OUT_OF_MEMORY: number;
MIRRORED_REPEAT: number;
POLYGON_OFFSET_UNITS: number;
TEXTURE_MIN_FILTER: number;
STENCIL_BACK_PASS_DEPTH_PASS: number;
LINE_LOOP: number;
FLOAT_MAT3: number;
TEXTURE14: number;
LINEAR: number;
RGB5_A1: number;
ONE_MINUS_SRC_COLOR: number;
SAMPLE_COVERAGE_INVERT: number;
DONT_CARE: number;
FRAMEBUFFER_BINDING: number;
RENDERBUFFER_ALPHA_SIZE: number;
STENCIL_REF: number;
ZERO: number;
DECR_WRAP: number;
SAMPLE_COVERAGE: number;
STENCIL_BACK_FUNC: number;
TEXTURE30: number;
VIEWPORT: number;
STENCIL_BITS: number;
FLOAT: number;
COLOR_WRITEMASK: number;
SAMPLE_COVERAGE_VALUE: number;
TEXTURE_CUBE_MAP_NEGATIVE_Y: number;
STENCIL_BACK_FAIL: number;
FLOAT_MAT4: number;
UNSIGNED_SHORT_4_4_4_4: number;
TEXTURE6: number;
RENDERBUFFER_WIDTH: number;
RGBA4: number;
ALWAYS: number;
BLEND_EQUATION_ALPHA: number;
COLOR_BUFFER_BIT: number;
TEXTURE_CUBE_MAP: number;
DEPTH_BUFFER_BIT: number;
STENCIL_CLEAR_VALUE: number;
BLEND_EQUATION: number;
RENDERBUFFER_GREEN_SIZE: number;
NEAREST_MIPMAP_LINEAR: number;
VERTEX_ATTRIB_ARRAY_TYPE: number;
INCR_WRAP: number;
ONE_MINUS_DST_COLOR: number;
HIGH_FLOAT: number;
BYTE: number;
FRONT_FACE: number;
SAMPLE_ALPHA_TO_COVERAGE: number;
CCW: number;
TEXTURE13: number;
MAX_VERTEX_ATTRIBS: number;
MAX_VERTEX_TEXTURE_IMAGE_UNITS: number;
TEXTURE_WRAP_T: number;
UNPACK_PREMULTIPLY_ALPHA_WEBGL: number;
FLOAT_VEC2: number;
LUMINANCE: number;
GREATER: number;
INT_VEC2: number;
VALIDATE_STATUS: number;
FRAMEBUFFER: number;
FRAMEBUFFER_UNSUPPORTED: number;
TEXTURE5: number;
FUNC_SUBTRACT: number;
BLEND_DST_ALPHA: number;
SAMPLER_CUBE: number;
ONE_MINUS_DST_ALPHA: number;
LESS: number;
TEXTURE_CUBE_MAP_POSITIVE_X: number;
BLUE_BITS: number;
DEPTH_TEST: number;
VERTEX_ATTRIB_ARRAY_STRIDE: number;
DELETE_STATUS: number;
TEXTURE18: number;
POLYGON_OFFSET_FACTOR: number;
UNSIGNED_INT: number;
TEXTURE_2D: number;
DST_COLOR: number;
FLOAT_MAT2: number;
COMPRESSED_TEXTURE_FORMATS: number;
MAX_FRAGMENT_UNIFORM_VECTORS: number;
DEPTH_STENCIL_ATTACHMENT: number;
LUMINANCE_ALPHA: number;
CW: number;
VERTEX_ATTRIB_ARRAY_NORMALIZED: number;
TEXTURE_CUBE_MAP_NEGATIVE_Z: number;
LINEAR_MIPMAP_LINEAR: number;
BUFFER_SIZE: number;
SAMPLE_BUFFERS: number;
TEXTURE15: number;
ACTIVE_TEXTURE: number;
VERTEX_SHADER: number;
TEXTURE22: number;
VERTEX_ATTRIB_ARRAY_POINTER: number;
INCR: number;
COMPILE_STATUS: number;
MAX_COMBINED_TEXTURE_IMAGE_UNITS: number;
TEXTURE7: number;
UNSIGNED_SHORT_5_5_5_1: number;
DEPTH_BITS: number;
RGBA: number;
TRIANGLE_STRIP: number;
COLOR_CLEAR_VALUE: number;
BROWSER_DEFAULT_WEBGL: number;
INVALID_ENUM: number;
SCISSOR_TEST: number;
LINE_STRIP: number;
FRAMEBUFFER_INCOMPLETE_ATTACHMENT: number;
STENCIL_FUNC: number;
FRAMEBUFFER_ATTACHMENT_OBJECT_NAME: number;
RENDERBUFFER_HEIGHT: number;
TEXTURE8: number;
TRIANGLES: number;
FRAMEBUFFER_ATTACHMENT_OBJECT_TYPE: number;
STENCIL_BACK_VALUE_MASK: number;
TEXTURE25: number;
RENDERBUFFER: number;
LEQUAL: number;
TEXTURE1: number;
STENCIL_INDEX8: number;
FUNC_ADD: number;
STENCIL_FAIL: number;
BLEND_SRC_ALPHA: number;
BOOL: number;
ALPHA_BITS: number;
LOW_INT: number;
TEXTURE10: number;
SRC_COLOR: number;
MAX_VARYING_VECTORS: number;
BLEND_DST_RGB: number;
TEXTURE_BINDING_CUBE_MAP: number;
STENCIL_INDEX: number;
TEXTURE_BINDING_2D: number;
MEDIUM_INT: number;
SHADER_TYPE: number;
POLYGON_OFFSET_FILL: number;
DYNAMIC_DRAW: number;
TEXTURE4: number;
STENCIL_BACK_PASS_DEPTH_FAIL: number;
STREAM_DRAW: number;
MAX_CUBE_MAP_TEXTURE_SIZE: number;
TEXTURE17: number;
TRIANGLE_FAN: number;
UNPACK_ALIGNMENT: number;
CURRENT_PROGRAM: number;
LINES: number;
INVALID_OPERATION: number;
FRAMEBUFFER_INCOMPLETE_MISSING_ATTACHMENT: number;
LINEAR_MIPMAP_NEAREST: number;
CLAMP_TO_EDGE: number;
RENDERBUFFER_DEPTH_SIZE: number;
TEXTURE_WRAP_S: number;
ELEMENT_ARRAY_BUFFER: number;
UNSIGNED_SHORT_5_6_5: number;
ACTIVE_UNIFORMS: number;
FLOAT_VEC3: number;
NO_ERROR: number;
ATTACHED_SHADERS: number;
DEPTH_ATTACHMENT: number;
TEXTURE11: number;
STENCIL_TEST: number;
ONE: number;
FRAMEBUFFER_ATTACHMENT_TEXTURE_CUBE_MAP_FACE: number;
STATIC_DRAW: number;
GEQUAL: number;
BOOL_VEC4: number;
COLOR_ATTACHMENT0: number;
PACK_ALIGNMENT: number;
MAX_TEXTURE_SIZE: number;
STENCIL_PASS_DEPTH_FAIL: number;
CULL_FACE_MODE: number;
TEXTURE16: number;
STENCIL_BACK_WRITEMASK: number;
SRC_ALPHA: number;
UNSIGNED_SHORT: number;
TEXTURE21: number;
FUNC_REVERSE_SUBTRACT: number;
SHADING_LANGUAGE_VERSION: number;
EQUAL: number;
FRAMEBUFFER_ATTACHMENT_TEXTURE_LEVEL: number;
BOOL_VEC3: number;
SAMPLER_2D: number;
TEXTURE_CUBE_MAP_NEGATIVE_X: number;
MAX_TEXTURE_IMAGE_UNITS: number;
TEXTURE_CUBE_MAP_POSITIVE_Y: number;
RENDERBUFFER_INTERNAL_FORMAT: number;
STENCIL_VALUE_MASK: number;
ELEMENT_ARRAY_BUFFER_BINDING: number;
ARRAY_BUFFER: number;
DEPTH_RANGE: number;
NICEST: number;
ACTIVE_ATTRIBUTES: number;
NEVER: number;
FLOAT_VEC4: number;
CURRENT_VERTEX_ATTRIB: number;
STENCIL_PASS_DEPTH_PASS: number;
INVERT: number;
LINK_STATUS: number;
RGB: number;
INT_VEC4: number;
TEXTURE2: number;
UNPACK_COLORSPACE_CONVERSION_WEBGL: number;
MEDIUM_FLOAT: number;
SRC_ALPHA_SATURATE: number;
BUFFER_USAGE: number;
SHORT: number;
NONE: number;
UNSIGNED_BYTE: number;
INT: number;
SUBPIXEL_BITS: number;
KEEP: number;
SAMPLES: number;
FRAGMENT_SHADER: number;
LINE_WIDTH: number;
BLEND_SRC_RGB: number;
LOW_FLOAT: number;
VERSION: number;
}
declare var WebGLRenderingContext:{
prototype: WebGLRenderingContext;
new(): WebGLRenderingContext;
DEPTH_FUNC: number;
DEPTH_COMPONENT16: number;
REPLACE: number;
REPEAT: number;
VERTEX_ATTRIB_ARRAY_ENABLED: number;
FRAMEBUFFER_INCOMPLETE_DIMENSIONS: number;
STENCIL_BUFFER_BIT: number;
RENDERER: number;
STENCIL_BACK_REF: number;
TEXTURE26: number;
RGB565: number;
DITHER: number;
CONSTANT_COLOR: number;
GENERATE_MIPMAP_HINT: number;
POINTS: number;
DECR: number;
INT_VEC3: number;
TEXTURE28: number;
ONE_MINUS_CONSTANT_ALPHA: number;
BACK: number;
RENDERBUFFER_STENCIL_SIZE: number;
UNPACK_FLIP_Y_WEBGL: number;
BLEND: number;
TEXTURE9: number;
ARRAY_BUFFER_BINDING: number;
MAX_VIEWPORT_DIMS: number;
INVALID_FRAMEBUFFER_OPERATION: number;
TEXTURE: number;
TEXTURE0: number;
TEXTURE31: number;
TEXTURE24: number;
HIGH_INT: number;
RENDERBUFFER_BINDING: number;
BLEND_COLOR: number;
FASTEST: number;
STENCIL_WRITEMASK: number;
ALIASED_POINT_SIZE_RANGE: number;
TEXTURE12: number;
DST_ALPHA: number;
BLEND_EQUATION_RGB: number;
FRAMEBUFFER_COMPLETE: number;
NEAREST_MIPMAP_NEAREST: number;
VERTEX_ATTRIB_ARRAY_SIZE: number;
TEXTURE3: number;
DEPTH_WRITEMASK: number;
CONTEXT_LOST_WEBGL: number;
INVALID_VALUE: number;
TEXTURE_MAG_FILTER: number;
ONE_MINUS_CONSTANT_COLOR: number;
ONE_MINUS_SRC_ALPHA: number;
TEXTURE_CUBE_MAP_POSITIVE_Z: number;
NOTEQUAL: number;
ALPHA: number;
DEPTH_STENCIL: number;
MAX_VERTEX_UNIFORM_VECTORS: number;
DEPTH_COMPONENT: number;
RENDERBUFFER_RED_SIZE: number;
TEXTURE20: number;
RED_BITS: number;
RENDERBUFFER_BLUE_SIZE: number;
SCISSOR_BOX: number;
VENDOR: number;
FRONT_AND_BACK: number;
CONSTANT_ALPHA: number;
VERTEX_ATTRIB_ARRAY_BUFFER_BINDING: number;
NEAREST: number;
CULL_FACE: number;
ALIASED_LINE_WIDTH_RANGE: number;
TEXTURE19: number;
FRONT: number;
DEPTH_CLEAR_VALUE: number;
GREEN_BITS: number;
TEXTURE29: number;
TEXTURE23: number;
MAX_RENDERBUFFER_SIZE: number;
STENCIL_ATTACHMENT: number;
TEXTURE27: number;
BOOL_VEC2: number;
OUT_OF_MEMORY: number;
MIRRORED_REPEAT: number;
POLYGON_OFFSET_UNITS: number;
TEXTURE_MIN_FILTER: number;
STENCIL_BACK_PASS_DEPTH_PASS: number;
LINE_LOOP: number;
FLOAT_MAT3: number;
TEXTURE14: number;
LINEAR: number;
RGB5_A1: number;
ONE_MINUS_SRC_COLOR: number;
SAMPLE_COVERAGE_INVERT: number;
DONT_CARE: number;
FRAMEBUFFER_BINDING: number;
RENDERBUFFER_ALPHA_SIZE: number;
STENCIL_REF: number;
ZERO: number;
DECR_WRAP: number;
SAMPLE_COVERAGE: number;
STENCIL_BACK_FUNC: number;
TEXTURE30: number;
VIEWPORT: number;
STENCIL_BITS: number;
FLOAT: number;
COLOR_WRITEMASK: number;
SAMPLE_COVERAGE_VALUE: number;
TEXTURE_CUBE_MAP_NEGATIVE_Y: number;
STENCIL_BACK_FAIL: number;
FLOAT_MAT4: number;
UNSIGNED_SHORT_4_4_4_4: number;
TEXTURE6: number;
RENDERBUFFER_WIDTH: number;
RGBA4: number;
ALWAYS: number;
BLEND_EQUATION_ALPHA: number;
COLOR_BUFFER_BIT: number;
TEXTURE_CUBE_MAP: number;
DEPTH_BUFFER_BIT: number;
STENCIL_CLEAR_VALUE: number;
BLEND_EQUATION: number;
RENDERBUFFER_GREEN_SIZE: number;
NEAREST_MIPMAP_LINEAR: number;
VERTEX_ATTRIB_ARRAY_TYPE: number;
INCR_WRAP: number;
ONE_MINUS_DST_COLOR: number;
HIGH_FLOAT: number;
BYTE: number;
FRONT_FACE: number;
SAMPLE_ALPHA_TO_COVERAGE: number;
CCW: number;
TEXTURE13: number;
MAX_VERTEX_ATTRIBS: number;
MAX_VERTEX_TEXTURE_IMAGE_UNITS: number;
TEXTURE_WRAP_T: number;
UNPACK_PREMULTIPLY_ALPHA_WEBGL: number;
FLOAT_VEC2: number;
LUMINANCE: number;
GREATER: number;
INT_VEC2: number;
VALIDATE_STATUS: number;
FRAMEBUFFER: number;
FRAMEBUFFER_UNSUPPORTED: number;
TEXTURE5: number;
FUNC_SUBTRACT: number;
BLEND_DST_ALPHA: number;
SAMPLER_CUBE: number;
ONE_MINUS_DST_ALPHA: number;
LESS: number;
TEXTURE_CUBE_MAP_POSITIVE_X: number;
BLUE_BITS: number;
DEPTH_TEST: number;
VERTEX_ATTRIB_ARRAY_STRIDE: number;
DELETE_STATUS: number;
TEXTURE18: number;
POLYGON_OFFSET_FACTOR: number;
UNSIGNED_INT: number;
TEXTURE_2D: number;
DST_COLOR: number;
FLOAT_MAT2: number;
COMPRESSED_TEXTURE_FORMATS: number;
MAX_FRAGMENT_UNIFORM_VECTORS: number;
DEPTH_STENCIL_ATTACHMENT: number;
LUMINANCE_ALPHA: number;
CW: number;
VERTEX_ATTRIB_ARRAY_NORMALIZED: number;
TEXTURE_CUBE_MAP_NEGATIVE_Z: number;
LINEAR_MIPMAP_LINEAR: number;
BUFFER_SIZE: number;
SAMPLE_BUFFERS: number;
TEXTURE15: number;
ACTIVE_TEXTURE: number;
VERTEX_SHADER: number;
TEXTURE22: number;
VERTEX_ATTRIB_ARRAY_POINTER: number;
INCR: number;
COMPILE_STATUS: number;
MAX_COMBINED_TEXTURE_IMAGE_UNITS: number;
TEXTURE7: number;
UNSIGNED_SHORT_5_5_5_1: number;
DEPTH_BITS: number;
RGBA: number;
TRIANGLE_STRIP: number;
COLOR_CLEAR_VALUE: number;
BROWSER_DEFAULT_WEBGL: number;
INVALID_ENUM: number;
SCISSOR_TEST: number;
LINE_STRIP: number;
FRAMEBUFFER_INCOMPLETE_ATTACHMENT: number;
STENCIL_FUNC: number;
FRAMEBUFFER_ATTACHMENT_OBJECT_NAME: number;
RENDERBUFFER_HEIGHT: number;
TEXTURE8: number;
TRIANGLES: number;
FRAMEBUFFER_ATTACHMENT_OBJECT_TYPE: number;
STENCIL_BACK_VALUE_MASK: number;
TEXTURE25: number;
RENDERBUFFER: number;
LEQUAL: number;
TEXTURE1: number;
STENCIL_INDEX8: number;
FUNC_ADD: number;
STENCIL_FAIL: number;
BLEND_SRC_ALPHA: number;
BOOL: number;
ALPHA_BITS: number;
LOW_INT: number;
TEXTURE10: number;
SRC_COLOR: number;
MAX_VARYING_VECTORS: number;
BLEND_DST_RGB: number;
TEXTURE_BINDING_CUBE_MAP: number;
STENCIL_INDEX: number;
TEXTURE_BINDING_2D: number;
MEDIUM_INT: number;
SHADER_TYPE: number;
POLYGON_OFFSET_FILL: number;
DYNAMIC_DRAW: number;
TEXTURE4: number;
STENCIL_BACK_PASS_DEPTH_FAIL: number;
STREAM_DRAW: number;
MAX_CUBE_MAP_TEXTURE_SIZE: number;
TEXTURE17: number;
TRIANGLE_FAN: number;
UNPACK_ALIGNMENT: number;
CURRENT_PROGRAM: number;
LINES: number;
INVALID_OPERATION: number;
FRAMEBUFFER_INCOMPLETE_MISSING_ATTACHMENT: number;
LINEAR_MIPMAP_NEAREST: number;
CLAMP_TO_EDGE: number;
RENDERBUFFER_DEPTH_SIZE: number;
TEXTURE_WRAP_S: number;
ELEMENT_ARRAY_BUFFER: number;
UNSIGNED_SHORT_5_6_5: number;
ACTIVE_UNIFORMS: number;
FLOAT_VEC3: number;
NO_ERROR: number;
ATTACHED_SHADERS: number;
DEPTH_ATTACHMENT: number;
TEXTURE11: number;
STENCIL_TEST: number;
ONE: number;
FRAMEBUFFER_ATTACHMENT_TEXTURE_CUBE_MAP_FACE: number;
STATIC_DRAW: number;
GEQUAL: number;
BOOL_VEC4: number;
COLOR_ATTACHMENT0: number;
PACK_ALIGNMENT: number;
MAX_TEXTURE_SIZE: number;
STENCIL_PASS_DEPTH_FAIL: number;
CULL_FACE_MODE: number;
TEXTURE16: number;
STENCIL_BACK_WRITEMASK: number;
SRC_ALPHA: number;
UNSIGNED_SHORT: number;
TEXTURE21: number;
FUNC_REVERSE_SUBTRACT: number;
SHADING_LANGUAGE_VERSION: number;
EQUAL: number;
FRAMEBUFFER_ATTACHMENT_TEXTURE_LEVEL: number;
BOOL_VEC3: number;
SAMPLER_2D: number;
TEXTURE_CUBE_MAP_NEGATIVE_X: number;
MAX_TEXTURE_IMAGE_UNITS: number;
TEXTURE_CUBE_MAP_POSITIVE_Y: number;
RENDERBUFFER_INTERNAL_FORMAT: number;
STENCIL_VALUE_MASK: number;
ELEMENT_ARRAY_BUFFER_BINDING: number;
ARRAY_BUFFER: number;
DEPTH_RANGE: number;
NICEST: number;
ACTIVE_ATTRIBUTES: number;
NEVER: number;
FLOAT_VEC4: number;
CURRENT_VERTEX_ATTRIB: number;
STENCIL_PASS_DEPTH_PASS: number;
INVERT: number;
LINK_STATUS: number;
RGB: number;
INT_VEC4: number;
TEXTURE2: number;
UNPACK_COLORSPACE_CONVERSION_WEBGL: number;
MEDIUM_FLOAT: number;
SRC_ALPHA_SATURATE: number;
BUFFER_USAGE: number;
SHORT: number;
NONE: number;
UNSIGNED_BYTE: number;
INT: number;
SUBPIXEL_BITS: number;
KEEP: number;
SAMPLES: number;
FRAGMENT_SHADER: number;
LINE_WIDTH: number;
BLEND_SRC_RGB: number;
LOW_FLOAT: number;
VERSION: number;
}
interface WebGLProgram extends WebGLObject {
}
declare var WebGLProgram:{
prototype: WebGLProgram;
new(): WebGLProgram;
}
interface OES_standard_derivatives {
FRAGMENT_SHADER_DERIVATIVE_HINT_OES: number;
}
declare var OES_standard_derivatives:{
prototype: OES_standard_derivatives;
new(): OES_standard_derivatives;
FRAGMENT_SHADER_DERIVATIVE_HINT_OES: number;
}
interface WebGLFramebuffer extends WebGLObject {
}
declare var WebGLFramebuffer:{
prototype: WebGLFramebuffer;
new(): WebGLFramebuffer;
}
interface WebGLShader extends WebGLObject {
}
declare var WebGLShader:{
prototype: WebGLShader;
new(): WebGLShader;
}
interface OES_texture_float_linear {
}
declare var OES_texture_float_linear:{
prototype: OES_texture_float_linear;
new(): OES_texture_float_linear;
}
interface WebGLObject {
}
declare var WebGLObject:{
prototype: WebGLObject;
new(): WebGLObject;
}
interface WebGLBuffer extends WebGLObject {
}
declare var WebGLBuffer:{
prototype: WebGLBuffer;
new(): WebGLBuffer;
}
interface WebGLShaderPrecisionFormat {
rangeMin: number;
rangeMax: number;
precision: number;
}
declare var WebGLShaderPrecisionFormat:{
prototype: WebGLShaderPrecisionFormat;
new(): WebGLShaderPrecisionFormat;
}
interface EXT_texture_filter_anisotropic {
TEXTURE_MAX_ANISOTROPY_EXT: number;
MAX_TEXTURE_MAX_ANISOTROPY_EXT: number;
}
declare var EXT_texture_filter_anisotropic:{
prototype: EXT_texture_filter_anisotropic;
new(): EXT_texture_filter_anisotropic;
TEXTURE_MAX_ANISOTROPY_EXT: number;
MAX_TEXTURE_MAX_ANISOTROPY_EXT: number;
}
declare var Option:{ new(text?:string, value?:string, defaultSelected?:boolean, selected?:boolean): HTMLOptionElement; };
declare var Image:{ new(width?:number, height?:number): HTMLImageElement; };
declare var Audio:{ new(src?:string): HTMLAudioElement; };
declare var ondragend:(ev:DragEvent) => any;
declare var onkeydown:(ev:KeyboardEvent) => any;
declare var ondragover:(ev:DragEvent) => any;
declare var onkeyup:(ev:KeyboardEvent) => any;
declare var onreset:(ev:Event) => any;
declare var onmouseup:(ev:MouseEvent) => any;
declare var ondragstart:(ev:DragEvent) => any;
declare var ondrag:(ev:DragEvent) => any;
declare var screenX:number;
declare var onmouseover:(ev:MouseEvent) => any;
declare var ondragleave:(ev:DragEvent) => any;
declare var history:History;
declare var pageXOffset:number;
declare var name:string;
declare var onafterprint:(ev:Event) => any;
declare var onpause:(ev:Event) => any;
declare var onbeforeprint:(ev:Event) => any;
declare var top:Window;
declare var onmousedown:(ev:MouseEvent) => any;
declare var onseeked:(ev:Event) => any;
declare var opener:Window;
declare var onclick:(ev:MouseEvent) => any;
declare var innerHeight:number;
declare var onwaiting:(ev:Event) => any;
declare var ononline:(ev:Event) => any;
declare var ondurationchange:(ev:Event) => any;
declare var frames:Window;
declare var onblur:(ev:FocusEvent) => any;
declare var onemptied:(ev:Event) => any;
declare var onseeking:(ev:Event) => any;
declare var oncanplay:(ev:Event) => any;
declare var outerWidth:number;
declare var onstalled:(ev:Event) => any;
declare var onmousemove:(ev:MouseEvent) => any;
declare var innerWidth:number;
declare var onoffline:(ev:Event) => any;
declare var length:number;
declare var screen:Screen;
declare var onbeforeunload:(ev:BeforeUnloadEvent) => any;
declare var onratechange:(ev:Event) => any;
declare var onstorage:(ev:StorageEvent) => any;
declare var onloadstart:(ev:Event) => any;
declare var ondragenter:(ev:DragEvent) => any;
declare var onsubmit:(ev:Event) => any;
declare var self:Window;
declare var document:Document;
declare var onprogress:(ev:ProgressEvent) => any;
declare var ondblclick:(ev:MouseEvent) => any;
declare var pageYOffset:number;
declare var oncontextmenu:(ev:MouseEvent) => any;
declare var onchange:(ev:Event) => any;
declare var onloadedmetadata:(ev:Event) => any;
declare var onplay:(ev:Event) => any;
declare var onerror:ErrorEventHandler;
declare var onplaying:(ev:Event) => any;
declare var parent:Window;
declare var location:Location;
declare var oncanplaythrough:(ev:Event) => any;
declare var onabort:(ev:UIEvent) => any;
declare var onreadystatechange:(ev:Event) => any;
declare var outerHeight:number;
declare var onkeypress:(ev:KeyboardEvent) => any;
declare var frameElement:Element;
declare var onloadeddata:(ev:Event) => any;
declare var onsuspend:(ev:Event) => any;
declare var window:Window;
declare var onfocus:(ev:FocusEvent) => any;
declare var onmessage:(ev:MessageEvent) => any;
declare var ontimeupdate:(ev:Event) => any;
declare var onresize:(ev:UIEvent) => any;
declare var onselect:(ev:UIEvent) => any;
declare var navigator:Navigator;
declare var styleMedia:StyleMedia;
declare var ondrop:(ev:DragEvent) => any;
declare var onmouseout:(ev:MouseEvent) => any;
declare var onended:(ev:Event) => any;
declare var onhashchange:(ev:Event) => any;
declare var onunload:(ev:Event) => any;
declare var onscroll:(ev:UIEvent) => any;
declare var screenY:number;
declare var onmousewheel:(ev:MouseWheelEvent) => any;
declare var onload:(ev:Event) => any;
declare var onvolumechange:(ev:Event) => any;
declare var oninput:(ev:Event) => any;
declare var performance:Performance;
declare var onmspointerdown:(ev:any) => any;
declare var animationStartTime:number;
declare var onmsgesturedoubletap:(ev:any) => any;
declare var onmspointerhover:(ev:any) => any;
declare var onmsgesturehold:(ev:any) => any;
declare var onmspointermove:(ev:any) => any;
declare var onmsgesturechange:(ev:any) => any;
declare var onmsgesturestart:(ev:any) => any;
declare var onmspointercancel:(ev:any) => any;
declare var onmsgestureend:(ev:any) => any;
declare var onmsgesturetap:(ev:any) => any;
declare var onmspointerout:(ev:any) => any;
declare var msAnimationStartTime:number;
declare var applicationCache:ApplicationCache;
declare var onmsinertiastart:(ev:any) => any;
declare var onmspointerover:(ev:any) => any;
declare var onpopstate:(ev:PopStateEvent) => any;
declare var onmspointerup:(ev:any) => any;
declare var onpageshow:(ev:PageTransitionEvent) => any;
declare var ondevicemotion:(ev:DeviceMotionEvent) => any;
declare var devicePixelRatio:number;
declare var msCrypto:Crypto;
declare var ondeviceorientation:(ev:DeviceOrientationEvent) => any;
declare var doNotTrack:string;
declare var onmspointerenter:(ev:any) => any;
declare var onpagehide:(ev:PageTransitionEvent) => any;
declare var onmspointerleave:(ev:any) => any;
declare function alert(message?:any):void;
declare function scroll(x?:number, y?:number):void;
declare function focus():void;
declare function scrollTo(x?:number, y?:number):void;
declare function print():void;
declare function prompt(message?:string, _default?:string):string;
declare function toString():string;
declare function open(url?:string, target?:string, features?:string, replace?:boolean):Window;
declare function scrollBy(x?:number, y?:number):void;
declare function confirm(message?:string):boolean;
declare function close():void;
declare function postMessage(message:any, targetOrigin:string, ports?:any):void;
declare function showModalDialog(url?:string, argument?:any, options?:any):any;
declare function blur():void;
declare function getSelection():Selection;
declare function getComputedStyle(elt:Element, pseudoElt?:string):CSSStyleDeclaration;
declare function msCancelRequestAnimationFrame(handle:number):void;
declare function matchMedia(mediaQuery:string):MediaQueryList;
declare function cancelAnimationFrame(handle:number):void;
declare function msIsStaticHTML(html:string):boolean;
declare function msMatchMedia(mediaQuery:string):MediaQueryList;
declare function requestAnimationFrame(callback:FrameRequestCallback):number;
declare function msRequestAnimationFrame(callback:FrameRequestCallback):number;
declare function removeEventListener(type:string, listener:EventListener, useCapture?:boolean):void;
declare function dispatchEvent(evt:Event):boolean;
declare function attachEvent(event:string, listener:EventListener):boolean;
declare function detachEvent(event:string, listener:EventListener):void;
declare var localStorage:Storage;
declare var status:string;
declare var onmouseleave:(ev:MouseEvent) => any;
declare var screenLeft:number;
declare var offscreenBuffering:any;
declare var maxConnectionsPerServer:number;
declare var onmouseenter:(ev:MouseEvent) => any;
declare var clipboardData:DataTransfer;
declare var defaultStatus:string;
declare var clientInformation:Navigator;
declare var closed:boolean;
declare var onhelp:(ev:Event) => any;
declare var external:External;
declare var event:MSEventObj;
declare var onfocusout:(ev:FocusEvent) => any;
declare var screenTop:number;
declare var onfocusin:(ev:FocusEvent) => any;
declare function showModelessDialog(url?:string, argument?:any, options?:any):Window;
declare function navigate(url:string):void;
declare function resizeBy(x?:number, y?:number):void;
declare function item(index:any):any;
declare function resizeTo(x?:number, y?:number):void;
declare function createPopup(arguments?:any):MSPopupWindow;
declare function toStaticHTML(html:string):string;
declare function execScript(code:string, language?:string):any;
declare function msWriteProfilerMark(profilerMarkName:string):void;
declare function moveTo(x?:number, y?:number):void;
declare function moveBy(x?:number, y?:number):void;
declare function showHelp(url:string, helpArg?:any, features?:string):void;
declare function captureEvents():void;
declare function releaseEvents():void;
declare var sessionStorage:Storage;
declare function clearTimeout(handle:number):void;
declare function setTimeout(handler:any, timeout?:any, ...args:any[]):number;
declare function clearInterval(handle:number):void;
declare function setInterval(handler:any, timeout?:any, ...args:any[]):number;
declare function msSetImmediate(expression:any, ...args:any[]):number;
declare function clearImmediate(handle:number):void;
declare function msClearImmediate(handle:number):void;
declare function setImmediate(expression:any, ...args:any[]):number;
declare function btoa(rawString:string):string;
declare function atob(encodedString:string):string;
declare var msIndexedDB:IDBFactory;
declare var indexedDB:IDBFactory;
declare var console:Console;
declare var onpointerenter:(ev:PointerEvent) => any;
declare var onpointerout:(ev:PointerEvent) => any;
declare var onpointerdown:(ev:PointerEvent) => any;
declare var onpointerup:(ev:PointerEvent) => any;
declare var onpointercancel:(ev:PointerEvent) => any;
declare var onpointerover:(ev:PointerEvent) => any;
declare var onpointermove:(ev:PointerEvent) => any;
declare var onpointerleave:(ev:PointerEvent) => any;
declare function addEventListener(type:"mouseleave", listener:(ev:MouseEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mouseenter", listener:(ev:MouseEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"help", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"focusout", listener:(ev:FocusEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"focusin", listener:(ev:FocusEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"pointerenter", listener:(ev:PointerEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"pointerout", listener:(ev:PointerEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"pointerdown", listener:(ev:PointerEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"pointerup", listener:(ev:PointerEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"pointercancel", listener:(ev:PointerEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"pointerover", listener:(ev:PointerEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"pointermove", listener:(ev:PointerEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"pointerleave", listener:(ev:PointerEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"dragend", listener:(ev:DragEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"keydown", listener:(ev:KeyboardEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"dragover", listener:(ev:DragEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"keyup", listener:(ev:KeyboardEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"reset", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mouseup", listener:(ev:MouseEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"dragstart", listener:(ev:DragEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"drag", listener:(ev:DragEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mouseover", listener:(ev:MouseEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"dragleave", listener:(ev:DragEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"afterprint", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"pause", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"beforeprint", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mousedown", listener:(ev:MouseEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"seeked", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"click", listener:(ev:MouseEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"waiting", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"online", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"durationchange", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"blur", listener:(ev:FocusEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"emptied", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"seeking", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"canplay", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"stalled", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mousemove", listener:(ev:MouseEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"offline", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"beforeunload", listener:(ev:BeforeUnloadEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"ratechange", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"storage", listener:(ev:StorageEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"loadstart", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"dragenter", listener:(ev:DragEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"submit", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"progress", listener:(ev:ProgressEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"dblclick", listener:(ev:MouseEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"contextmenu", listener:(ev:MouseEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"change", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"loadedmetadata", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"play", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"playing", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"canplaythrough", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"abort", listener:(ev:UIEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"readystatechange", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"keypress", listener:(ev:KeyboardEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"loadeddata", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"suspend", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"focus", listener:(ev:FocusEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"message", listener:(ev:MessageEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"timeupdate", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"resize", listener:(ev:UIEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"select", listener:(ev:UIEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"drop", listener:(ev:DragEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mouseout", listener:(ev:MouseEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"ended", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"hashchange", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"unload", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"scroll", listener:(ev:UIEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mousewheel", listener:(ev:MouseWheelEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"load", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"volumechange", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"input", listener:(ev:Event) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mspointerdown", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"msgesturedoubletap", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mspointerhover", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"msgesturehold", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mspointermove", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"msgesturechange", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"msgesturestart", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mspointercancel", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"msgestureend", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"msgesturetap", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mspointerout", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"msinertiastart", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mspointerover", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"popstate", listener:(ev:PopStateEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mspointerup", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"pageshow", listener:(ev:PageTransitionEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"devicemotion", listener:(ev:DeviceMotionEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"deviceorientation", listener:(ev:DeviceOrientationEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mspointerenter", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:"pagehide", listener:(ev:PageTransitionEvent) => any, useCapture?:boolean):void;
declare function addEventListener(type:"mspointerleave", listener:(ev:any) => any, useCapture?:boolean):void;
declare function addEventListener(type:string, listener:EventListener, useCapture?:boolean):void;
/////////////////////////////
/// WorkerGlobalScope APIs
/////////////////////////////
// These are only available in a Web Worker
declare function importScripts(...urls:string[]):void;
/////////////////////////////
/// Windows Script Host APIS
/////////////////////////////
declare var ActiveXObject:{ new (s:string): any; };
interface ITextWriter {
Write(s:string): void;
WriteLine(s:string): void;
Close(): void;
}
declare var WScript:{
Echo(s:any): void;
StdErr: ITextWriter;
StdOut: ITextWriter;
Arguments: { length: number; Item(n:number): string; };
ScriptFullName: string;
Quit(exitCode?:number): number;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 795 |
^ doesn't matter, just get a 42x speed.
Sounds great Jason..If so take me off this individual vine. Also, I'll be away from July 10th until August 7th as I'm going to Israel..would I still be able to recieve the bundle?
^ leo, since you were originally going to receive all my old data discs you will be put towards the top of this list. sorry for the confusion!
^^ hopefully we can get the discs to the UK in time for you to copy them before your trip. if not i am sure they can take a detour on their way back to the states.
Okay then, keep me on the list until further notice.
Last edited by Anonymous on Sun May 28, 2006 8:21 am, edited 1 time in total.
we should all add our city and state after our name in parentheses so Guelah could possibly set up some sort of shipping deal where people who are in the same region can get in touch first. just an idea.
Yeah, help me, help you!!
well i just went thorugh and added as many locations as people had in thier profiles....AND i spelled my name correctly this time!
Now if we can get everyone else to put their cities, we can do this a lot easier than we could have.
Last edited by Anonymous on Mon May 29, 2006 11:21 pm, edited 1 time in total.
if anyone is going to GRAB in hartford they can send me a spindle of blank dvd's and i will bring them to you filled w/ the archive @ the show.
i just wanted to edit the list, cuz im back home for the summer and not in erie.
Last edited by Anonymous on Wed May 31, 2006 8:09 pm, edited 1 time in total.
Hey McNulty, considering I am like 16 people higher on the list, I could maybe meet up with you one day, get some blanks, and burn all these for you. Get them back sometime quickly maybe.
I don't know, or maybe I could just give you the originals, and you can send them off. Or is that like cutting in line? Anyways... carry on.
haha...um, i dunno. sounds like a good idea. we'll have to discuss this later.
if people are within the same city it just makes sense to meet up and pass on the discs. i dont think people will get bent out of shape if they're pushed 2-3 days down the line.
I think we're all greatful that at some point in time we will get these discs. Being bumped around is cool. For Chrissakes!!! 300+ shows and all for the cost of the blank discs adn S/H. IT'S A STEAL!!!!!!
after further research, I've concluded that DVD media is not currently capable of being written onto faster than 16x, however the burner can be up to 52x (the standard) for burning all other types of media: cd-r, cd+r, cd-rw, etc. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,886 |
module.exports = function(io){
io.users = {};
var connected = io.sockets.connected;
var defaultStatus = ["online", "busy", "away", "offline"];
io.emitToUser = function(userId, msg, data){
if(io.users[userId])
{
for(var s of io.users[userId].sockets){
if(connected[s]){
connected[s].emit(msg, data);
}
}
}
}
io.emitToUsers = function(users, msg, data){
for(var p of users){
io.emitToUser(p, msg, data);
}
}
io.on('connection', function(socket){
var userId = socket.request.user._id;
if(io.users[userId]){
// delete users[userId];
io.users[userId].sockets.push(socket.client.id);
}
else{
io.users[userId] = {
sockets: [socket.client.id],
info: {
_id: userId,
firstName: socket.request.user.firstName,
secondName: socket.request.user.secondName,
//koristiti status sa logouta
onlineStatus: "online"
}
}
io.emitToUsers(socket.request.user.partnerships, "online status", io.users[userId].info);
}
console.log("user connected", userId, "on socket", socket.client.id);
// console.log("number of clients:", io.engine.clientsCount);
socket.on('disconnect', function(reason){
// console.log(reason);
io.users[userId].sockets.splice(io.users[userId].sockets[socket.client.id], 1);
console.log('user disconnected', userId, "on socket", socket.client.id);
setTimeout(function(){
if(io.users[userId] && !io.users[userId].sockets.length){
io.emitToUsers(socket.request.user.partnerships, "online status", io.users[userId].info);
delete io.users[userId];
console.log("user disconnected", userId);
//spremit u bazu zadnji online status i datetime
}
}, 3000)
});
});
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\section{Introduction}
Galileons are four-dimensional higher-derivative field theories originally discovered as brane bending modes in decoupling limits of higher-dimensional
induced-gravity models such as the Dvali-Gabadadze-Porrati (DGP) model \cite{Dvali:2000hr,Luty:2003vm,Nicolis:2004qq}. Galileons have two key features: their equations of motion are second order (despite the appearance of higher derivatives in the action), and they possess novel non-linear global symmetries. Generalized and abstracted away from these origins \cite{Nicolis:2008in}, the galileons now describe a class of theories with interesting properties including radiative stability~\cite{Luty:2003vm,Hinterbichler:2010xn,Burrage:2010cu}, the successful implementation of the Vainshtein mechanism, and the presence of self-accelerating vacuum solutions (see Sec. 4.4 of \cite{Clifton:2011jh} for a review).
As ghost free, higher derivative field theories, the galileons have have been applied to inflation, late time acceleration, and a variety of other cosmological applications~\cite{Silva:2009km,Burrage:2010cu,Mizuno:2010ag,Creminelli:2010qf,DeFelice:2010as,Kobayashi:2011pc,Mota:2010bs,Wyman:2011mp,Agarwal:2011mg,RenauxPetel:2011dv,Gao:2011mz,Gao:2011qe,RenauxPetel:2011uk,Qiu:2011cy}. They have been used as alternative theories to inflation \cite{Creminelli:2010ba,Hinterbichler:2011qk,Levasseur:2011mw}, kinetic braiding theories \cite{Deffayet:2010qz,Deffayet:2011gz}, and appear naturally in ghost-free theories of massive gravity (\cite{deRham:2011by,deRham:2010gu,deRham:2010ik,deRham:2011ca}, see \cite{Hinterbichler:2011tt} for a review). The theory has been extended to the multi-galileon case \cite{Padilla:2010de,Hinterbichler:2010xn,Padilla:2010ir,Padilla:2010tj}, supersymmetrized \cite{Khoury:2011da}, and generalized to $p$-forms \cite{Deffayet:2009mn}.
Recent progress has been made towards covariantizing the galileons and putting them on curved backgrounds.
Naive covariantization of the galileons leads to third order equations of motion, a problem which is solvable by introducing appropriate non-minimal couplings between the galileons and the curvature tensor \cite{Deffayet:2009wt,Deffayet:2009mn,Deffayet:2011gz}. However, this construction destroys the the interesting global symmetries of the flat-space theory (see however \cite{Germani:2011bc}).
A method exists to put the galileons on a fixed curved background while preserving the global symmetries \cite{Goon:2011qf,Goon:2011uw,Burrage:2011bt}.
The method is based on a geometric interpretation in which the galileon field $\pi(x)$ is interpreted as an embedding function describing the position of a 3-brane living in a Minkowski bulk \cite{deRham:2010eu}. The galileon terms arise from the small field limit of 4D Lovelock invariants \cite{Lovelock:1971yv} and the 4D boundary terms associated with the 5D Lovelock invariants (the Myers terms \cite{Myers:1987yn,Miskovic:2007mg}). By extending this geometric construction to allow for an arbitrary 5D bulk geometry and and an arbitrary brane embedding \cite{Goon:2011qf,Goon:2011uw}, it is possible to put the galileons on any embeddable background. The non-linear shift symmetries for $\pi$ are then inherited from the isometries of the bulk geometry (for a short review on generalizing galileons, see \cite{Trodden:2011xh}).
The purpose of the present paper is to apply the brane construction to cosmological FRW spacetimes, and to identify the non-linear symmetries of the resulting theories (this possibility was commented on in \cite{Burrage:2011bt}).
In what follows, we construct galileons on an FRW background embedded in a flat 5D bulk, so that the symmetry group will be the 15-dimensional Poincare group of 5D flat space, of which the 6 symmetries of FRW (spatial translations and rotations) will be linearly realized.
After a short review of the general geometric construction of the galileons, we introduce bulk coordinates that define a foliation of 5D Minkowski space by spatially flat FRW slices, and we present expressions for the galileon Lagrangians and their symmetries. These are ghost-free higher derivative scalar theories that live on an FRW space with an arbitrary time dependence for the scale factor, and which possess 9 non-linearly realized shift-like symmetries. In the process, we provide expressions for a general Gaussian-normal embedding, of which FRW in flat space is just one example. For FRW, we display the shorter minisuperspace Lagrangians, which themselves may be useful in a number of cosmological settings. Additionally, we discuss the small $\pi$ limits and explore the existence and stability of simple solutions for $\pi$.
\subsection*{Conventions and notation}
We use the mostly plus metric signature convention. Tensors are symmetrized with unit weight, i.e $T_{(\mu\nu)}=\frac{1}{2} \left(T_{\mu\nu}+T_{\nu\mu}\right)$. Curvature tensors are defined by $\left [\nabla_{\mu},\nabla_{\nu}\right ]V^{\rho}=R^{\rho}{}_{\sigma\mu\nu}V^{\sigma}$ and $R_{\mu\nu}=R^{\rho}{}_{\mu\rho\nu}$, $R=R^\mu_{\ \mu}$.
\section{Review of the brane construction for DBI galileons}
The general geometric construction of galileons living on arbitrary curved backgrounds was derived in \cite{Goon:2011qf,Goon:2011uw} and will be briefly reviewed here. The procedure begins with a fixed 5D metric $G_{AB}(X)$ and a 3-brane defined by the embedding functions $X^{A}(x)$, $A\in\{0,1,2,3,5\}$ where $x^{\mu}$, $\mu\in\{0,1,2,3\}$ are the coordinates native to the hypersurface. The induced metric and extrinsic curvature on the brane are
\begin{align}
\bar{g}_{\mu\nu}&=e^{A}_{\mu}e^{B}_{\nu}G_{AB},\nonumber\\
K_{\mu\nu}&=e^{A}_{\mu}e^{B}_{\nu}\nabla_{A}n_{B} \ ,
\label{gandKinduced}
\end{align}
where $e^{A}_{\mu}=\frac{\partial X^{A}}{\partial x^{\mu}}$ are the tangent vectors to the brane, $n^{A}$ is the spacelike normal vector to the brane, and $\nabla_{A}$ is the covariant derivative with respect to the 5D metric $G_{AB}$.
The action on the brane is an action for the embedding variables $X^A(x)$, and is chosen to be a diffeomorphism scalar constructed from $\bar{g}_{\mu\nu}$, its covariant derivative and its curvature tensor, as well as the extrinsic curvature tensor,
\begin{align}
S&=\int d^{4}x\, \sqrt{-\bar g}\,\mathcal{L}(\bar{g}_{\mu\nu},\bar{\nabla}_{\mu},\bar{R}^{\alpha}{}_{\beta\mu\nu}, K_{\mu\nu})\ ,
\label{generallagrangian0}
\end{align}
so that it is invariant under under gauge symmetries which are reparameterizations of the brane coordinates,
\begin{align}
\delta X^{A}&=\xi^{\mu}(x)\partial_{\mu}X^{A} \ .
\label{branereparameter}
\end{align}
Given any bulk Killing vector $K^{A}(X)$ satisfying the bulk Killing equation
\begin{equation} \label{killingequation}
K^C\partial_C G_{AB}+\partial_AK^CG_{CB}+\partial_BK^CG_{AC}=0 \ ,
\end{equation}
both the induced metric and the extrinsic curvature tensor (\ref{gandKinduced}), and therefore the action (\ref{generallagrangian0}), are invariant under the action of the global symmetry
\begin{equation} \delta_{K} X^{A}=K^{A}(X).\label{gaugeglobalsym}\end{equation}
We fix the gauge symmetry by choosing
\begin{align}
X^{\mu}(x)=x^{\mu}, \ \ \ X^{5}(x)=\pi(x) \ ,
\label{preferredgauge}
\end{align}
thereby yielding an action solely for $\pi(x)$,
\begin{align}
S&=\int d^{4}x\, \sqrt{-\bar g}\,\mathcal{L}(\bar{g}_{\mu\nu},\bar{\nabla}_{\mu},\bar{R}^{\alpha}{}_{\beta\mu\nu}, K_{\mu\nu})\Big|_{X^{\mu}=x^{\mu},\,X^{5}=\pi(x)} \ ,
\label{generallagrangian}
\end{align}
which has no remaining gauge symmetry.
However, a global symmetry transformation (\ref{gaugeglobalsym}) will generally ruin the gauge choice (\ref{preferredgauge}) and to re-fix the gauge we must make a compensating coordinate transformation on the brane by using (\ref{branereparameter}) with $\xi^{\mu}=-K^{\mu}$. Thus the combined transformation
\begin{align}
\delta \pi&=-K^{\mu}(x,\pi)\partial_{\mu}\pi+K^{5}(x,\pi)
\label{gensymmetry}
\end{align}
is a global symmetry of the gauge fixed action (\ref{generallagrangian}).
Aside from their symmetries, the other defining characteristic of galileon field theories is the absence of derivatives higher than second order in the equations of motion. Generic choices for the Lagrangian in (\ref{generallagrangian}) will not meet this requirement, but the Lovelock terms and the Myers boundary terms will \cite{deRham:2010eu}. In 4D there are only four such terms:
\begin{align}
\mathcal{L}_{2}&= -\sqrt{-\bar g},\nonumber\\
\mathcal{L}_{3}&= \sqrt{-\bar g}K,\nonumber\\
\mathcal{L}_{4}&= -\sqrt{-\bar g}\bar{R},\nonumber\\
\mathcal{L}_{5}&= \frac{3}{2}\sqrt{-\bar g}\left [-\frac{1}{3}K^{3}+K_{\mu\nu}^{2}K-\frac{2}{3}K_{\mu\nu}^{3}-2\left (\bar{R}_{\mu\nu}-\frac{1}{2}\bar{R}\bar{g}_{\mu\nu}\right )K^{\mu\nu}\right ] \ ,
\label{deflagrangian2to5}
\end{align}
where all contractions of indices are performed using the induced metric $\bar{g}_{\mu\nu}$ and its inverse.
In addition, there exists a zero derivative ``tadpole" term which is not of the form (\ref{generallagrangian0}) but which obeys the same symmetries. This term can be interpreted as the proper volume between an $X^{5}=\rm{const.}$ surface and the brane position $\pi(x)$,
\begin{align}
S_{1}&=\int d^{4}x\, \int^{\pi(x)}d\pi'\, \sqrt{-\det{G_{AB}(\pi',x)}} \ .
\label{deflagrangiantadpole}
\end{align}
As we show in Appendix \ref{tadpoleappendix}, this term also respects the global symmetries (\ref{gensymmetry}).
\section{DBI Galileons on a Gaussian normal foliation}
In this section we calculate the Lagrangians (\ref{deflagrangian2to5}) and (\ref{deflagrangiantadpole}) in the general case of a background metric which is in Gaussian normal form. The FRW galileon will be a special case of this general form, and we will specialize to it in later sections.
The background metric in Gaussian normal form is
\begin{equation}
\label{metricform}
G_{AB}dX^AdX^B=f_{\mu\nu}(x,w)dx^\mu dx^\nu+dw^2 \ .
\end{equation}
Here $X^5=w$ denotes the Gaussian normal transverse coordinate, and $f_{\mu\nu}(x,w)$ is an arbitrary metric on the leaves of the foliation defined by the constant $w$ surfaces. Recall that in the physical gauge (\ref{preferredgauge}), the transverse coordinate of the brane is set equal to the scalar field, $w(x)=\pi(x)$. This extends our earlier analysis \cite{Goon:2011qf,Goon:2011uw}, by relaxing the condition that the extrinsic curvature of constant $\pi$ slices be proportional to the induced metric.
\subsection{Induced quantities and other ingredients}
The induced metric is
\begin{align}
\bar{g}_{\mu\nu}&=f_{\mu\nu}+\partial_{\mu}\pi\partial_{\nu}\pi,
\end{align}
and its inverse is
\begin{align}
\bar{g}^{\mu\nu}=f^{\mu\nu}-\gamma^{2}\partial^{\mu}\pi\partial^{\nu}\pi \ ,
\end{align}
where
\begin{equation}
\gamma\equiv 1/\sqrt{1+(\partial\pi)^{2}} \ ,
\end{equation}
and the indices on the derivatives are raised with $f^{\mu\nu}$, the inverse of $f_{\mu\nu}$.
To calculate the extrinsic curvature we need to find the normal vector $n^{A}$, which satisfies
\begin{align}
n^{A}e^{B}_{\nu}G_{AB}&=0 \ ,\nonumber\\
n^{A}n^{B}G_{AB}&=1 \ ,
\end{align}
where $e^{B}_{\nu}=\frac{\partial X^{B}}{\partial x^{\nu}}$ are the tangent vectors to the brane. Solving these equations in the gauge (\ref{preferredgauge}) yields
\begin{equation} n_{A}=\gamma(-\partial_{\mu}\pi,1).\end{equation}
The extrinsic curvature is given by
\begin{align}
K_{\mu\nu}&=e^{A}_{\mu}e^{B}_{\nu}\nabla_{A}n_{B} \ ,
\end{align}
which can be written as $K_{\mu\nu}=e^{B}_{\nu}\partial_\mu n_{B}-e^{A}_{\mu}e^{B}_{\nu}\Gamma^C_{AB} n_{C}.$
The $\nabla_{A}$ is a covariant derivative of the bulk metric and so the Christoffel $\Gamma^C_{AB}$ must be calculated with $X^{5}=w$. The replacement $w\to\pi(x)$ is then made at the end of the calculation. Using the bulk coordinates in the form (\ref{Xmetric}), the non-zero 5D Christoffels, $\Gamma^{A}_{BC}$, are
\begin{align}
\Gamma^{\lambda}_{\mu\nu}&=\Gamma^{\lambda}_{\mu\nu}(f),\nonumber\\
\Gamma^{5}_{\mu\nu}&=-\frac{1}{2}f'_{\mu\nu},\nonumber\\
\Gamma^{\mu}_{5\nu}&=\frac{1}{2}f^{\mu\lambda}f'_{\lambda\nu} \ ,
\end{align}
where primes denote derivatives with respect to $\pi$. Note that on the right-hand side of the first line, the Christoffels of $f_{\mu\nu}$ are to be calculated with the $\pi$ dependence held fixed. The extrinsic curvature then reads
\begin{align}
K_{\mu\nu}&=-\gamma{\nabla}_{\mu}{\nabla}_{\nu}\pi+\frac{1}{2}\gamma f'_{\mu\nu}+\gamma\partial^{\lambda}\pi\partial_{(\mu }\pi f'_{\nu) \lambda} \ ,
\end{align}
where ${\nabla}_{\mu}$ is the covariant derivative calculated from $f_{\mu\nu}$ at fixed $\pi$.
The only remaining components needed to calculate the Lagrangians (\ref{deflagrangian2to5}) are expressions for the induced curvature, $\bar{R}^{\rho}{}_{\sigma\mu\nu}$, which arise in $\mathcal{L}_{4}$ and $\mathcal{L}_{5}$. At this point, we will specialize to a flat bulk for which the 5D curvature tensor vanishes, so that the induced curvature tensor can be expressed solely in terms of the extrinsic curvature tensor and induced metric via the Gauss-Codazzi equations,
\begin{equation} R^{(5)}_{ABCD}e^A_{\ \mu}e^B_{\ \nu}e^C_{\ \rho}e^D_{\ \sigma}= 0=
\bar R_{\mu\nu\rho\sigma}-K_{\mu \rho}K_{\nu \sigma}+K_{\mu \sigma}K_{\nu \rho}\ . \end{equation}
The expressions for ${\cal L}_4$ and ${\cal L}_5$ in (\ref{deflagrangian2to5}) then reduce to
\begin{align}
\mathcal{L}_{4}&=-\sqrt{-\bar g}\left [K^{2}-K_{\mu\nu}^{2}\right ], \\
\mathcal{L}_{5}&=\sqrt{-\bar g}\left [K^{3}-3K_{\mu\nu}^{2}K
+2K_{\mu\nu}^{3}\right ].
\end{align}
These are all the elements necessary for the calculation of the Lagrangians.
\subsection{The Lagrangians}
We now present the explicit forms for the DBI galileon Lagrangians. In all cases, we use the definition
$\gamma=1/\sqrt{1+\left (\partial\pi\right )^{2}}$ to replace $(\partial\pi)^2$ in favor of $\gamma$ (recall that indices on the derivatives are raised with $f^{\mu\nu}$). In addition, we employ a shorthand notation. We define $\Pi_{\mu\nu}=\nabla_{\mu}\nabla_{\nu}\pi$, where the covariant derivative $\nabla_{\mu}$ is calculated from $f_{\mu\nu}$ at fixed $\pi$. $f'_{\mu\nu}$ denotes the derivative of $f_{\mu\nu}(x,\pi)$ with respect to $\pi$. We use angular brackets $\langle\ldots\rangle$ to denote traces of the enclosed product as matrices, with all contractions performed using $f^{\mu\nu}$. For example, we have
\begin{align}
\left [\partial f\right ]&=f^{\mu\nu}\partial_{\pi}f_{\mu\nu},\nonumber\\
\left [\Pi\partial f\right ]&=\Pi_{\mu\nu}f^{\nu\lambda}\(\partial_{\pi}f_{\lambda\sigma}\)f^{\sigma\mu},\nonumber\\
\langle \Pi^3\rangle&=\Pi_{\mu\nu}f^{\nu\lambda}\Pi_{\lambda\sigma}f^{\sigma\rho}\Pi_{\rho\kappa}f^{\kappa\mu}.
\end{align}
In addition, when $\pi$ appears within a angled bracket, it does so only at both ends, and denotes contraction with $\nabla_\mu\pi$, for example,
\begin{align}
\left [\pi^{2}\partial f\right ]&=\nabla_{\mu}\pi\, f^{\mu\nu}\( \partial_{\pi}f_{\nu\lambda}\)f^{\lambda\sigma} \nabla_{\sigma}\pi,\nonumber\\
\left [\pi^{2}\Pi\partial f\right ]&=\nabla_{\mu}\pi\, f^{\mu\nu}\Pi_{\nu\lambda} f^{\lambda\sigma}\( \partial_{\pi}f_{\sigma\rho}\)f^{\rho\kappa} \nabla_{\kappa}\pi \ .
\end{align}
Employing this notation, the Lagrangians (\ref{deflagrangiantadpole}) and (\ref{deflagrangian2to5}) are calculated to be (no integrations by parts have been made in obtaining these expressions)
\begin{align*}
\mathcal{L}_1&=\int^{\pi(x)} d\pi' \sqrt{-\det f_{\mu\nu}(x,\pi')}, \nonumber\\
\mathcal{L}_{2}&=-\sqrt{-f}\frac{1}{\gamma},\nonumber\\
\mathcal{L}_3&=\sqrt{- f}\Big[-\left [\Pi\right ]+\dfrac{1}{2} \left [\partial f\right ]+\gamma^{2}\left(\left [\pi^{3}\right ]+\dfrac{1}{2} \left [\pi^{2}\partial f\right ]\right)\Big],\nonumber\\
\mathcal{L}_{4}&=\sqrt{-f}\Big[-\frac{1}{2} \left [\pi^{2}\partial f\right ]^2 \gamma ^3-\left [\partial f\right ] \left [\pi^{3}\right ] \gamma ^3-2
\left [\pi^{4}\right ] \gamma ^3+2 \left [\pi^{3}\right ] \left [\Pi\right ] \gamma
^3\nonumber\\
&\quad -\frac{1}{2} \left [\partial f\right ] \left [\pi^{2}\partial f\right ] \gamma ^3+\left [\Pi\right ] \left [\pi^{2}\partial f\right ]
\gamma ^3-\frac{\left [\partial f\right ]^2 \gamma }{4}-\left [\Pi\right ]^2 \gamma
+\frac{\left [\partial f^{2}\right ] \gamma }{4}\nonumber\\
&\quad -\left [\Pi\partial f\right ] \gamma +\left [\partial f\right ] \left [\Pi\right ]
\gamma +\left [\Pi^{2}\right ] \gamma +\frac{\left [\pi^{2}\partial f^{2}\right ] \gamma }{2}\Big]
\end{align*}
\begin{align}
\mathcal{L}_{5}&=\sqrt{-f}\Big[3 \left [\pi^{3}\right ] \left [\Pi\right ]^2 \gamma ^4+\frac{3}{4} \left [\partial f\right ] \left [\pi^{2}\partial f\right ]^2
\gamma ^4-\frac{3}{2} \left [\Pi\right ] \left [\pi^{2}\partial f\right ]^2 \gamma ^4+\frac{3}{4}
\left [\partial f\right ]^2 \left [\pi^{3}\right ] \gamma ^4\nonumber\\
&\quad -\frac{3}{4} \left [\partial f^{2}\right ]
\left [\pi^{3}\right ] \gamma ^4+3 \left [\Pi\partial f\right ] \left [\pi^{3}\right ] \gamma ^4+6
\left [\pi^{5}\right ] \gamma ^4+3 \left [\partial f\right ] \left [\pi^{4}\right ] \gamma ^4\nonumber\\
&\quad -3
\left [\partial f\right ] \left [\pi^{3}\right ] \left [\Pi\right ] \gamma ^4-6 \left [\pi^{4}\right ]
\left [\Pi\right ] \gamma ^4-3 \left [\pi^{3}\right ] \left [\Pi^{2}\right ] \gamma
^4+\frac{3}{8} \left [\partial f\right ]^2 \left [\pi^{2}\partial f\right ] \gamma ^4\nonumber\\
&\quad +\frac{3}{2}
\left [\Pi\right ]^2 \left [\pi^{2}\partial f\right ] \gamma ^4-\frac{3}{8} \left [\partial f^{2}\right ] \left [\pi^{2}\partial f\right ]
\gamma ^4+\frac{3}{2} \left [\Pi\partial f\right ] \left [\pi^{2}\partial f\right ] \gamma ^4\nonumber\\
&\quad -\frac{3}{2}
\left [\partial f\right ] \left [\Pi\right ] \left [\pi^{2}\partial f\right ] \gamma ^4-\frac{3}{2} \left [\Pi^{2}\right ]
\left [\pi^{2}\partial f\right ] \gamma ^4-\frac{3}{2} \left [\pi^{3}\right ] \left [\pi^{2}\partial f^{2}\right ] \gamma
^4\nonumber\\
&\quad -\frac{3}{4} \left [\pi^{2}\partial f\right ] \left [\pi^{2}\partial f^{2}\right ] \gamma ^4-3 \left [\pi\Pi\partial f\Pi\pi\right ] \gamma
^4+3 \left [\pi^{2}\partial f\right ] \left [\pi^{2}\Pi\partial f\right ] \gamma ^4\nonumber\\
&\quad +\frac{\left [\partial f\right ]^3 \gamma
^2}{8}-\left [\Pi\right ]^3 \gamma ^2+\frac{3}{2} \left [\partial f\right ] \left [\Pi\right ]^2 \gamma
^2-\frac{3}{8} \left [\partial f\right ] \left [\partial f^{2}\right ] \gamma ^2+\frac{\left [\partial f^{3}\right ] \gamma
^2}{4}\nonumber\\
&\quad +\frac{3}{2} \left [\partial f\right ] \left [\Pi\partial f\right ] \gamma ^2-\frac{3 \left [\Pi\partial f^{2}\right ]
\gamma ^2}{2}-\frac{3 \left [\pi\partial f\Pi\partial f\pi\right ] \gamma ^2}{2}-\frac{3}{4}
\left [\partial f\right ]^2 \left [\Pi\right ] \gamma ^2\nonumber\\
&\quad +\frac{3}{4} \left [\partial f^{2}\right ] \left [\Pi\right ]
\gamma ^2-3 \left [\Pi\partial f\right ] \left [\Pi\right ] \gamma ^2-2 \left [\Pi^{3}\right ] \gamma
^2-\frac{3}{2} \left [\partial f\right ] \left [\Pi^{2}\right ] \gamma ^2\nonumber\\
&\quad +3 \left [\Pi\right ]
\left [\Pi^{2}\right ] \gamma ^2+3 \left [\pi^{2}\partial f\right ] \gamma ^2-\frac{3}{4} \left [\partial f\right ]
\left [\pi^{2}\partial f^{2}\right ] \gamma ^2+\frac{3}{2} \left [\Pi\right ] \left [\pi^{2}\partial f^{2}\right ] \gamma
^2+\frac{3 \left [\pi^{2}\partial f^{3}\right ] \gamma ^2}{4}\Big]\label{L5} \ .
\end{align}
The only dynamical field present is $\pi$, and it enters the Lagrangians both explicitly, and implicitly through the metric $f_{\mu\nu}(x,\pi)$ and its covariant derivatives. Despite the complicated higher derivative structure of these Lagrangians, the equations of motion will contain at most second order time derivatives, so that they describe only the $\pi$ degree of freedom.
\subsection{\label{globsym2}Global symmetries}
As mentioned in the introduction, if the bulk metric possesses Killing vectors $K^{A}(X)$, then the induced metric and extrinsic curvature, and hence actions of the form (\ref{generallagrangian}), are invariant under the transformations (\ref{gensymmetry}).
The algebra of Killing vectors of $G_{AB}$ contains a subalgebra consisting of those Killing vectors
for which $K^5=0$. This is the subalgebra of Killing vectors which are parallel to the foliation of constant $w$ surfaces, which generates the subgroup of isometries which preserve the foliation. For such a Killing vector, the $\mu5$ components of the Killing equations (\ref{killingequation}) tell us that $K^\mu$ is independent of $w$, and the $\mu\nu$ components of the Killing equations tell us that $K^\mu(x)$ is a Killing vector of $f_{\mu\nu}(x,w)$, for any $w$. We choose a basis of this subalgebra with elements indexed by ${\cal I}$,
\begin{equation}
K_{\cal I}^A(X)=\begin{cases} K_{\cal I}^\mu(x) & A=\mu \\ 0 & A=5\end{cases} \ .
\end{equation}
We now extend this basis to a basis for the algebra of all Killing vectors by adding a suitably chosen set of linearly independent Killing vectors with non-vanishing $K^5$. We index these with $I$, so that $(K_{\cal I},K_I)$ is a basis of the full algebra of Killing vectors. From the $55$ component of Killing's equation, we see that $K^5$ must be independent of $w$, so we may write $K^5(x)$.
A generic symmetry transformation takes the form
\begin{align}
\delta_{K}X^{A}&=a^{\cal I}K^{A}_{\cal I}(X)+a^{I}K_{I}(X) \ ,
\end{align}
where $a^{\cal I}$ and $a^I$ are constant parameters. It induces the gauge preserving shift symmetry (\ref{gensymmetry})
\begin{align}
(\delta_{K}+\delta_{g,\rm{comp}})\pi&=-a^{\cal I}K^{\mu}_{\cal I}(x)\partial_{\mu}\pi+a^{I}K_{I}^{5}(x)-a^{I}K_{I}^{\mu}(x,\pi)\partial_{\mu}\pi \ ,\label{gensymmetry2}
\end{align}
demonstrating that the $K_{\cal I}$ symmetries are linearly realized, whereas the $K_{I}$ symmetries are non-linearly realized, corresponding to the spontaneous breaking of the bulk symmetry algebra down to the subalgebra which preserves the leaves of the foliation. If the bulk metric (\ref{metricform}) has Killing vectors, the Lagrangians (\ref{L5}) will have the symmetries (\ref{gensymmetry2}).
\section{DBI Galileons on cosmological spaces}
We now specialize to the case where the brane metric is FRW. We thus need a Gaussian-normal foliation of 5D Minkowski space by FRW slices.
\subsection{Embedding 4D FRW in 5D Minkowski}
We consider the case of a spatially flat FRW 3-brane embedded in 5D Minkowski space. Starting from the bulk Minkowski metric with coordinates $Y^{A}$
\begin{align}
ds^{2}&=-\left (dY^{0}\right )^{2}+\left (dY^{1}\right )^{2}+\left (dY^{2}\right )^{2}+\left (dY^{3}\right )^{2}+\left (dY^{5}\right )^{2} \ ,
\label{Ycoords}
\end{align}
we make a change to coordinates to $t, x^{i},w$, where $i=1,2,3$ runs over the spatial indices on the brane\footnote{This is the transformation used in \cite{Deruelle:2000ge}, except that we have not imposed a $Z_{2}$ symmetry.},
\begin{align}
Y^{0}&=S(t,w)\left (\frac{x^{2}}{4}+1-\frac{1}{4H^{2}a^{2}}\right )-\frac{1}{2}\int dt\, \frac{\dot H}{H^{3}a},\nonumber\\
Y^{i}&=S(t,w)x^{i},\nonumber\\
Y^{5}&=S(t,w)\left (\frac{x^{2}}{4}-1-\frac{1}{4H^{2}a^{2}}\right )-\frac{1}{2}\int dt\, \frac{\dot H}{H^{3}a} \ .
\label{Xcoords}
\end{align}
Here, $a(t)$ is an arbitrary function of $t$ which will become the scale factor of the 4D space, and overdots denote derivatives with respect to $t$. We have defined $x^{2}\equiv x^{i}x^{j}\delta_{ij}$, $H\equiv{\dot a}/{a}$, and
\begin{equation}
S(t,w)\equiv a-\dot{a}w.
\end{equation}
The lower limits on the integrals in (\ref{Xcoords}) are arbitrary, and different choices merely shift the embedding. In the case of power law expansions $a(t)\sim t^{\alpha}$, $\alpha>0$, taking the lower limit to be zero puts the big bang at the origin of the embedding space.
In these new coordinates, the Minkowski metric reads
\begin{align}
ds^{2}&=-n^{2}(t,w)dt^{2}+S^{2}(t, w)\delta_{ij}dx^{i}dx^{j}+dw^{2} \ ,
\label{Xmetric}
\end{align}
where
\begin{align}
n(t,w)&\equiv1-\frac{\ddot a}{\dot a }w \ .
\end{align}
On any $w={\rm const.}$ slice, the induced metric is
\begin{align}
d\tilde{s}^{2}&=-n^{2}(t,w)dt^{2}+S^{2}(t, w)\delta_{ij}dx^{i}dx^{j},
\end{align}
and so after a slice by slice time redefinition $n(t,w)dt=dt'$ we verify that we have indeed foliated $M_{5}$ with spatially flat FRW slices. Furthermore, the coordinates are Gaussian normal with respect to this foliation. A plot of the embedding in the case $a\sim t^{1/2}$ is shown in Fig.(\ref{embeddingplot}).
\begin{figure}
\centering
\includegraphics[width=4.3in]{embedding.pdf}
\caption{The embedding of an FRW brane in 5D Minksowski space for the case $a(t)=t^{1/2}$.}
\label{embeddingplot}
\end{figure}
In the FRW case, the first two galileon Lagrangians (\ref{L5}) read (no integrations by parts have been made)
\begin{align}
\mathcal{L}_1&=a^3 \pi-\frac{a^2 \Big(3
\dot{a}^2+a \ddot{a}\Big) \pi ^2}{2 \dot{a}}+a
\Big(\dot{a}^2+a \ddot{a}\Big) \pi ^3-\frac{1}{4} \dot{a}
\Big(\dot{a}^2+3 a \ddot{a}\Big) \pi ^4+\frac{1}{5} \ddot{a} \dot{a}^2 \pi ^5,\nonumber\\
\mathcal{L}_{2}&=-(1-\frac{\ddot a}{\dot a}\pi)^{}(a-\dot a\pi)^{3}\sqrt{1-\left(1-\frac{\ddot a}{\dot a}\pi\right)^{-2}\dot\pi^{2}+(a-\dot a\pi)^{-2}(\vec\nabla\pi)^{2}}. \label{explicit12}
\end{align}
We relegate the expression for ${\cal L}_3$ to Appendix \ref{L3append}, due to its complexity, and opt not to write out explicit expressions for ${\cal L}_4$ and ${\cal L}_5$ due to their even more unmanageable length.
\subsection{Global symmetries for FRW}
As reviewed in Section \ref{globsym2}, identifying the relevant global symmetries reduces to the task of finding the Killing vectors of the bulk Minkowski metric in the brane-adapted coordinates (\ref{Xmetric}),
separating the Killing vectors into those with vanishing $K^{5}$ components, denoted by $K^{A}_{\cal I}$, and those which have non-vanishing $K^{5}$'s, denoted by $K^{A}_{I}$.
Let $Y^{A}$ be the cartesian coordinates used in (\ref{Ycoords}) with associated basis vectors $\bar{\partial}_{A}$. The Killing vectors in the $Y^{A}$ coordinates take the form of the ten rotations and boosts, $L_{AB}$, and the five translations $P_{A}$,
\begin{equation} L_{AB}=Y_{A}\bar{\partial}_{B}-Y_{B}\bar{\partial}_{A}, \ \ \ \ \ P_{A}=-\bar{\partial}_{A}.
\end{equation}
After rewriting these Killing vectors in terms of the brane-adapted coordinates $\{t,x^i,w\}$ and the associated basis vectors $\{\partial_{t},\partial_i,\partial_w\}$, we find the following combinations which contain no $K^5$ component,
\begin{equation}
L_{ij}=x^{i}\partial_{j}-x^{j}\partial_{i},\ \ \ \ -\frac{1}{2}\left [L_{i0}+L_{i5}\right] =-\partial_{i} \ .
\end{equation}
These generate the three rotations and three spatial translations of the FRW leaves. They are the $K^{A}_{\cal I}$.
The remaining vectors form the $K^{A}_{I}$, which we take to be the following combinations,
\begin{align}
v_{i}&=\frac{1}{2}\left [L_{i0}-L_{i5}\right ]= \frac{1}{2} x^{i}\dot{a} \left [\int dt \, \frac{\dot{H}}{H^3a}\right ] \partial_{w}+\frac{
x^{i} \big(a-\dot{a}\pi+ \dot{a}^{2}\int dt \, \frac{\dot{H}}{H^3a}
\big)}{2\dot{a}-2\pi \ddot{a}} \partial_{t}\nonumber\\
&\quad -\left [\frac{x^ix^i\dot{a}^{2}+1}{4\dot{a}^{2}} +\frac{
\int dt \, \frac{\dot{H}}{H^3a}
}{2a-2\pi
\dot{a}}\right ]\partial_{i}+\sum_{j\neq i}\left[-\frac{x^{i}x^{j}}{2}\partial_{j}+\frac{x^jx^j}{4}\partial_{i}\right ] ,\nonumber\\
k_{i}&=-P_{i}=\frac{1}{a-\pi \dot{a}}\partial_{i}+x^{i} \dot{a}
\big(\frac{\dot{a} }{\pi
\ddot{a}-\dot{a}}\partial_{t}-\partial_{w}\big),\nonumber\\
q&=-\frac{1}{2}\left [P_{0}+P_{5}\right ]=\dot{a} \big(\partial_{w}+\frac{ \dot{a}}{\dot{a}-\pi
\ddot{a}}\partial_{t}\big),\nonumber\\
u &=-\frac{1}{2}\left [P_{0}-P_{5}\right ]= \frac{x^{2}\dot{a}^{2}-1}{4\dot{a}}\partial_{w}+\frac{
x^{2} \dot{a}^2+1}{4\dot{a}-4\pi
\ddot{a}}\partial_{t}-\frac{1
}{2a-2\pi \dot{a}}\sum_i x^{i}\partial_{i},\nonumber\\
s&=L_{50} =\left [\frac{a-\pi \dot{a}+\dot{a}^{2}\int dt \, \frac{\dot{H}}{H^3a}}{\pi\ddot{a}-\dot{a}}\right ]\partial_{t}-\dot{a}\left[\int dt \, \frac{\dot{H}}{H^3a}\right]\partial_{w}+\sum_ix^{i}\partial_{i}\ ,
\end{align}
where $H=\dot a/a$, $x^{2}=\delta_{ij}x^{i}x^{j}$, and the summation convention has been suspended. The lower limits on the integrals should be the same as those in (\ref{Xcoords}).
The non-linear symmetries of the $\pi$ field are then obtained from (\ref{gensymmetry}),
\begin{align}
\delta_{v_{i}}\pi&=\frac{1}{2} x^{i}\dot{a} \int dt \, \frac{\dot{H}}{H^3a} -\frac{
x^{i} \big(a-\dot{a}\pi+ \dot{a}^{2}\int dt \, \frac{\dot{H}}{H^3a}
\big)}{2\dot{a}-2\pi \ddot{a}} \dot\pi\nonumber\\
&\quad +\left [\frac{x^ix^i\dot{a}^{2}+1}{4\dot{a}^{2}} +\frac{
\int dt \, \frac{\dot{H}}{H^3a}
}{2a-2\pi
\dot{a}}\right ]\partial_{i}\pi-\sum_{j\neq i}\left[-\frac{x^{i}x^{j}}{2}\partial_{j}\pi+\frac{ x^jx^j}{4}\partial_{i}\pi\right ],\nonumber\\
\delta_{k_{i}} \pi&=x^{i} \dot{a} \left(\frac{\dot{a} \dot{\pi}}{\dot{a}-\pi
\ddot{a}}-1\right)-\frac{\partial_{i}\pi }{a-\pi \dot{a}},\nonumber\\
\delta_q \pi&=\frac{\dot{\pi} \dot{a}^2}{\pi
\ddot{a}-\dot{a}}+\dot{a},\nonumber\\
\delta _u \pi &=\frac{x^{2}\dot{a}^{2}-1}{4\dot{a}}-\frac{
x^{2} \dot{a}^2+1}{4\dot{a}-4\pi
\ddot{a}}\dot\pi+\frac{1
}{2a-2\pi \dot{a}}\sum_i x^{i}\partial_{i}\pi,\nonumber\\
\delta_s \pi&= -\dot{a}\int dt \, \frac{\dot{H}}{H^3a} +\frac{\left(a-\dot{a}\pi+\dot{a}^{2}
\int dt \, \frac{\dot{H}}{H^3a}\right) \dot{\pi}}{\dot{a}-\pi \ddot{a}} -\sum x^{i}\partial_{i}\pi , \,\label{fullsyms}
\end{align}
where the replacement $w\to\pi(x^{\mu})$ was performed.
These non-linear transformations are the FRW analogues of the shift symmetries of the flat space galileon. They are symmetries of the Lagrangians (\ref{explicit12}) and (\ref{L3frw}) as well the ${\cal L}_4$, ${\cal L}_5$ which we did not write out. Together with the spatial rotation and translation symmetries of FRW, the commutation relations of these transformations are those of the 5D Poincare group. These are complicated and highly non-linear transformations, and without the brane formalism it would be nearly impossible to guess their form.
\subsection{Minisuperspace Lagrangians}
For cosmological applications where we are not considering fluctuations, we may be most interested in the limiting case in which spatial gradients are set to zero, so that $\pi=\pi(t)$. In this minisuperspace approximation, the Lagrangians simplify significantly, and we display their full forms here. In displaying these, the numerators are ordered by increasing powers of $\pi$, and then by patterns of derivatives on the $\pi$ fields. No integrations by parts have been made.
\begin{align*}
\mathcal{L}_{1}&=a^3 \pi-\frac{a^2 \Big(3
\dot{a}^2+a \ddot{a}\Big) \pi ^2}{2 \dot{a}}+a
\Big(\dot{a}^2+a \ddot{a}\Big) \pi ^3-\frac{1}{4} \dot{a}
\Big(\dot{a}^2+3 a \ddot{a}\Big) \pi ^4+\frac{1}{5} \ddot{a} \dot{a}^2 \pi ^5,\nonumber\\
\mathcal{L}_{2}&= -\Big(a-\pi \dot{a}\Big)^3\sqrt{ \Big(1-\frac{\pi
\ddot{a}}{\dot{a}}\Big) ^2-{\dot{\pi}^2}},\nonumber\\
\mathcal{L}_{3}&=\Big[3 a^2 \dot{a}^4+a^3 \ddot{a} \dot{a}^2
+\left (-6 a \dot{a}^5-12 a^2 \ddot{a}
\dot{a}^3-2 a^3 \ddot{a}^2 \dot{a}\right )\pi
-3 a^2 \dot{a}^4 \dot{\pi}
- a^3 \dot{a}^3 \ddot{\pi}\nonumber\\
&\quad+\left (3 \dot{a}^6+21 a \ddot{a}
\dot{a}^4+15 a^2 \ddot{a}^2 \dot{a}^2+
a^3 \ddot{a}^3\right )\pi^{2}
+(6 a \dot{a}^5+6
a^2 \ddot{a} \dot{a}^3\nonumber\\
&\quad -
a^3 \dddot a \dot{a}^2+
a^3 \ddot{a}^2 \dot{a})\pi\dot\pi+(
-3 a^2 \dot{a}^4-2 a^3 \ddot{a} \dot{a}^2)\dot\pi^{2}+ (3 a^2 \dot{a}^4+ a^3 \ddot{a} \dot{a}^2)\pi \ddot{\pi}\nonumber\\
&\quad+(-10 \ddot{a} \dot{a}^5-24 a
\ddot{a}^2 \dot{a}^3-6 a^2 \ddot{a}^3 \dot{a})\pi ^3
+ (-3 \dot{a}^6-12
a \ddot{a} \dot{a}^4+3
a^2 \dddot a \dot{a}^3\nonumber\\
&\quad-6
a^2 \ddot{a}^2 \dot{a}^2)\pi ^2 \dot{\pi}
+(6 a \dot{a}^5+9
a^2 \ddot{a} \dot{a}^3)\pi \dot{\pi}^2+(-3
a \dot{a}^5-3
a^2 \ddot{a} \dot{a}^3)\pi ^2 \ddot{\pi}\nonumber\\
&\quad
+3 a^2 \dot{a}^4 \dot{\pi}^3
+(9 a \dot{a}^2 \ddot{a}^3 +11 \dot{a}^4
\ddot{a}^2 )\pi^{4}
+(6 \ddot{a} \dot{a}^5-3
a \dddot a \dot{a}^4+9
a \ddot{a}^2 \dot{a}^3)\pi ^3 \dot{\pi}\nonumber\\
&\quad+(
-3 \dot{a}^6-12 a
\ddot{a} \dot{a}^4)\pi ^2 \dot{\pi}^2+
(\dot{a}^6+3 a \ddot{a} \dot{a}^4)\pi ^3 \ddot{\pi}
-6 a \dot{a}^5\pi \dot{\pi}^3
-4 \dot{a}^3 \ddot{a}^3\pi ^5\nonumber\\
&\quad
+ (\dot{a}^5 \dddot a -4
\dot{a}^4 \ddot{a}^2)\pi ^4 \dot{\pi}
+5 \dot{a}^5 \ddot{a}\pi ^3\dot{\pi}^2-
\dot{a}^5 \ddot{a} \ddot{\pi}\pi ^4 \nonumber\\
&\quad
+3 \dot{a}^6 \dot{\pi}^3\pi ^2
\Big]/\Big[\dot a \left(\left(\dot\pi^2-1\right) \dot a^2+2 \pi \ddot a
\dot a-\pi ^2 \ddot a^2\right)\Big],\nonumber\\
\end{align*}
\begin{align*}
\mathcal{L}_{4}&=\Big[-6 a \dot{a}^4-6 a^2 \ddot{a} \dot{a}^2
+(6 \dot{a}^5+30 a \ddot{a} \dot{a}^3+12 a^2 \ddot{a}^2 \dot{a})\pi
+6 a \dot{a}^4 \dot{\pi}
+6 a^2 \dot{a}^3 \ddot{\pi}\nonumber\\
&\quad+ (
-24 \ddot{a} \dot{a}^4-42 a
\ddot{a}^2 \dot{a}^2-6 a^2 \ddot{a}^3)\pi ^2
+(-6 \dot{a}^5-12
a \ddot{a} \dot{a}^3+6
a^2 \dddot a \dot{a}^2\nonumber\\
&\quad-6
a^2 \ddot{a}^2 \dot{a})\pi \dot{\pi}
+(6 a \dot{a}^4+12 a^2 \ddot{a} \dot{a}^2)\dot{\pi}^2+(-12 a \dot{a}^4-6 a^2 \ddot{a}
\dot{a}^2)\pi \ddot{\pi}\nonumber\\
&\quad
+(30 \ddot{a}^2 \dot{a}^3+18 a
\ddot{a}^3 \dot{a})\pi ^3
+(12 \ddot{a} \dot{a}^4-12
a \dddot a \dot{a}^3+18
a \ddot{a}^2 \dot{a}^2)\pi ^2 \dot{\pi}\nonumber\\
&\quad+ (
-6 \dot{a}^5-30 a
\ddot{a} \dot{a}^3)\pi \dot{\pi}^2+(6
\dot{a}^5+12 a \ddot{a} \dot{a}^3)\pi ^2 \ddot{\pi}
-6 a \dot{a}^4 \dot{\pi}^3-12 \dot{a}^2 \ddot{a}^3\pi ^4\nonumber\\
&\quad
+(6 \dot{a}^4 \dddot a -12 \dot{a}^3 \ddot{a}^2)\pi ^3 \dot{\pi}
+18 \dot{a}^4 \ddot{a}\pi ^2\dot{\pi}^2-6 \dot{a}^4 \ddot{a} \ddot{\pi}\pi ^3 \nonumber\\
&\quad
+6 \dot{a}^5 \pi \dot{\pi}^3\Big]/\Big[ \dot{a}\left(\dot{a} \left(\dot\pi +1\right)-\pi \ddot{a}\right) \sqrt{ \left(1-{\pi \ddot{a}\over \dot a}\right)^2 -{\dot\pi ^2}}\Big],
\end{align*}
\begin{align}
\mathcal{L}_{5}&=\Big[-6 \dot{a}^5-18 a \ddot{a} \dot{a}^3
+(36 \ddot{a} \dot{a}^4+36 a \ddot{a}^2
\dot{a}^2)\pi
+6 \dot{a}^5 \dot{\pi}\nonumber\\
&\quad +18 a \dot{a}^4 \ddot{\pi}+(
-54 \ddot{a}^2 \dot{a}^3-18 a
\ddot{a}^3 \dot{a})\pi ^2+ (
-12 \ddot{a} \dot{a}^4+18
a \dddot a \dot{a}^3\nonumber\\
&\quad -18
a \ddot{a}^2 \dot{a}^2)\pi \dot{\pi}
+(6 \dot{a}^5+36 a \ddot{a}
\dot{a}^3)\dot{\pi}^2+ (-18 \dot{a}^5-18 a \ddot{a} \dot{a}^3)\pi \ddot{\pi}
\nonumber\\
&\quad +24 \dot{a}^2 \ddot{a}^3\pi ^3
+(24 \dot{a}^3 \ddot{a}^2-18 \dot{a}^4 \dddot a )\pi ^2\dot{\pi}
+18 \dot{a}^4 \ddot{a} \pi ^2 \ddot{\pi}\nonumber\\
&\quad -42 \dot{a}^4 \ddot{a}\pi \dot{\pi}^2
-6 \dot{a}^5 \dot{\pi}^3\Big]/\Big[\dot{a} \left(\dot\pi+1\right)-\pi
\ddot a\Big]^{2}.\label{minisuperlag}
\end{align}
The $\pi$ equations of motion derived from these are second order in time derivatives. As before, the scale factor $a(t)$ describes the fixed background cosmological evolution, and does not represent a dynamical degree of freedom.
Of the symmetries (\ref{fullsyms}), only $\delta _q$ is free of explicit dependence on the spatial coordinates. It is a symmetry of the Lagrangians (\ref{minisuperlag}),
\begin{align}
\delta_q \pi&=\frac{\dot{\pi} \dot{a}^2}{\pi
\ddot{a}-\dot{a}}+\dot{a}.\nonumber\\
\end{align}
\section{Solutions, fluctuations, and small field limits}
In this section, we explore the existence and stability of simple solutions for $\pi$. In particular, we focus on the properties of the possible $\pi=0$ solutions.
\subsection{Simple solutions and stability}
Retaining all temporal and spatial derivatives, we expand the Lagrangians to second order in $\pi$, and find, after much integration by parts,
\begin{eqnarray}
\mathcal{L}_{1}&=& a^3 \pi-\frac{1}{2} \Big({ \ddot{a}a^3\over \dot a}+3
\dot{a}a^2\Big) \pi ^2+{\cal O}\(\pi^3\),\nonumber\\ \\
\mathcal{L}_{2}&=&\(3a^2 \dot a+{a^3\ddot a\over \dot a}\) \pi+\frac{1}{2} a^3 \dot\pi^2 -\frac{1}{2} a \(\vec\nabla\pi\)^2-3 \left (\ddot{a} a^2+ \dot{a}^2
a\right )\pi^2+{\cal O}\(\pi^3\), \nonumber\\\\
\mathcal{L}_{3}&=& 6 \(a\dot a^2+a^2 \ddot a\)\pi+3 \dot{a} a^2 \dot\pi^2-\left (
2\dot{a} +\frac{ a \ddot{a}}{\dot{a}} \right ) \(\vec\nabla\pi\)^2-3 \left (3 \dot{a} \ddot{a} a+
\dot{a}^3\right )\pi^2+{\cal O}\(\pi^3\), \nonumber\\\\
\mathcal{L}_{4}&=& 6 \(\dot a ^3+3a \dot a \ddot a\)\pi+9 \dot{a}^2 a\dot\pi^2 -3\left (\frac{\dot{a}^2}{a}+2\ddot a\right ) \(\vec\nabla\pi\)^2 -12 \dot{a}^2 \ddot{a}\pi^2 +{\cal O}\(\pi^3\),\nonumber\\\\
\mathcal{L}_{5}&=& 24 \dot a^2 \ddot a\,\pi+12 \dot{a}^3 \dot \pi^2 -12\frac{ \ddot{a}^2 \dot{a}}{a} \(\vec\nabla\pi\)^2+{\cal O}\(\pi^3\). \nonumber\\
\end{eqnarray}
Note that at quadratic order all the higher derivative terms have cancelled out up to total derivative, a consequence of the fact that the equations of motion are second order.
Consider a theory which is an arbitrary linear combination of the five Lagrangians,
\begin{equation} \mathcal{L}=\sum_{n=1}^5 c_n{\cal L}_n,\end{equation}
where the $c_n$ are (dimensionful) constants. If $\pi=0$ is to be a solution to the full equations of motion, the linear terms in $\mathcal{L}$ must vanish, which gives the condition
\begin{equation} c_1 a^3+c_2\(3a^2 \dot a+{a^3\ddot a\over \dot a}\) +6 c_3\(a\dot a^2+a^2 \ddot a\)+6 c_4\(\dot a ^3+3a \dot a \ddot a\)+24 c_5 \dot a^2 \ddot a=0.\label{tadpoleconstraint}\end{equation}
For generic values of the $c_n$, this is a non-linear second order equation for $a(t)$ which can be solved to yield a background for which $\pi=0$ is a solution. If we look for standard power-law solutions, $a(t)=\left (t/t_0\right )^{\alpha}$, the condition (\ref{tadpoleconstraint}) becomes
\begin{equation}
\big[24 c_5 (\alpha-1) \alpha^3+ 6 c_4 (4 \alpha-3) \alpha^2 t+6
c_3 \alpha (2 \alpha-1)t^2+c_2 (4 \alpha-1)t^3+c_1 t^4\big]
\left(\frac{t}{t_0}\right)^{3 \alpha}=0 \ .
\end{equation}
Each power of $ t$ must vanish independently, so we see that the only non-trivial power-law solutions are for $\alpha=1,3/4,1/2,1/4$. For these solutions, the corresponding $c_n$ must be non-zero and the others must be set to zero.
To test the stability around a given solution, we look at the quadratic part of the Lagrangian, which has the following form,
\begin{align}
\mathcal{L}&=\frac{1}{2}A(a(t),c_{n})\dot\pi^{2}-\frac{1}{2}B(a(t),c_{n})(\vec{\nabla}\pi)^{2}-\frac{1}{2}C(a(t),c_{n})\pi^{2} \ ,
\end{align}
where
\begin{align}
A(a(t),c_{n})&= c_2 a^3+6 c_3 \dot{a} a^2+18 c_4
\dot{a}^2 a+24 c_5 \dot{a}^3 , \nonumber\\
B(a(t),c_{n})&= c_2 a+2 c_3\left (
2\dot{a} +\frac{ a \ddot{a}}{\dot{a}} \right )+ 6 c_4 \left (\frac{\dot{a}^2}{a}+2\ddot a\right )+24 c_5\frac{ \ddot{a}
\dot{a}}{a}
\ddot{a}, \nonumber\\
C(a(t),c_n)&= c_1\left ( \frac{ \ddot{a} a^3}{\dot{a}}+3 \dot{a}
a^2\right )+6 c_2 \left (\ddot{a} a^2+ \dot{a}^2
a\right )+6c_3\left (3 \dot{a} \ddot{a} a+
\dot{a}^3\right )+24 c_4 \dot{a}^2 \ddot{a}\ .\label{coeffgen}
\end{align}
The stability of the theory against ghost and gradient instability, which is catastrophic at the shortest length scales, requires $A>0$ and $B\ge 0$. Freedom from tachyon-like instabilities requires $ C\ge 0$. However a tachyonic instability where $ C<0 $ only affects the large-scale stability of the field, and may be tolerable as long as the time scale associated with the tachyonic mass is of the same order or larger than the Hubble time.
The equations of motion take the form of a damped harmonic oscillator, $A\ddot{\pi}+\dot{A}\dot{\pi}-B\nabla^{2}\pi+C\pi=0.$
Thus, the time scale $\tau $ associated with a tachyonic mass term is given by $ \tau=\sqrt{A/|C|} $ and the tachyonic instability is tolerable if ${H\tau}\gtrsim 1 $.
\begin{table}\large
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\hline
$\alpha$ & $c_{1}$ & $c_{2}$ & $c_{3}$ & $c_{4}$ & $c_{5}$ & $A$ & $B$ & $C$ & ${H\tau}$\\ \hline \hline
$ 1 $ & $ 0 $ & $ 0 $ & $ 0 $ & $ 0 $ & $ c_5 $ & $ 24 \frac{c_5}{t_0^{3}} $ & $ 0 $& $ 0 $ & $ 0 $\\ \hline
$ \frac{3}{4} $ & $ 0 $ & $ 0 $ & $ 0 $ & $ c_4 $ & $ 0 $ & $ \frac{81}{8t^{2}}c_{4}\left (t/t_0\right )^{9/4}$& $ \frac{9}{8t^{2}}c_4\, \left (t/t_0\right )^{3/4} $& $ -\frac{81}{32t^{4}}c_4\, \left (t/t_0\right )^{9/4} $& $ 3/2$\\ \hline
$ \frac{1}{2} $ & $ 0 $ & $ 0 $ & $ c_3 $ & $ 0 $ & $ 0 $ & $ \frac{3}{t}c_{3}\left (t/t_0\right )^{3/2} $& $ \frac{1}{t}c_{3}\left (t/t_0\right )^{1/2} $& $ -\frac{3}{2t^{3}} c_3\, \left (t/t_0\right )^{3/2} $ & ${1\over \sqrt{2}} $\\ \hline
$ \frac{1}{4} $ & $ 0 $ & $ c_{2} $ & $ 0 $ & $ 0 $ & $ 0 $ & $ c_{2}\left (t/t_0\right )^{3/4} $& $ c_2\left (t/t_0\right )^{1/4} $& $ -\frac{3}{4t^{2}} c_{2} \left (t/t_0\right )^{3/4} $ & ${1\over 2\sqrt{3}} $\\ \hline
\end{tabular}\caption{Lagrangian coefficients, stability coefficients, and time scale comparisons for fluctuations about $\pi=0$ for all possible non-trivial power law solutions $a(t)=\left (t/t_0\right )^{n}$ .}\label{stabletable}
\end{table}
In Table \ref{stabletable}, we display the coefficients (\ref{coeffgen}) for the four possible power-law solutions. For the solution $a(t)\sim t$, the choice $c_5>0$ leads to a stable solution, albeit marginally so, since there is no mass or gradient energy. For each of the other three cases, choosing the relevant coefficient to be positive ensures that $A>0 $, $B>0$, at which point we necessarily have $C<0$ and hence a tachyonic instability. The tachyon time scale is however $ \tau H\sim 1$ (and happens to be independent of time). Therefore, each of the four power law solutions are stable to fluctuations over time scales shorter than the age of the universe.
Repeating the analysis in the case of a de-Sitter universe, the condition (\ref{tadpoleconstraint}) for a $\pi=0$ solution becomes
\begin{equation} c_1+4Hc_2+12 c_3 H^2+24
c_4 H^3+24 c_5 H^4=0,\end{equation}
and the coefficients (\ref{coeffgen}) of the quadratic part are
\begin{align}
A(a(t),c_{i})&= a_0^3 e^{3 H t}\( c_2+6 c_3 H+18
c_4 H^2+24 c_5 H^3\), \nonumber\\
B(a(t),c_{i})&= a_0 e^{H t}\( c_2+6 c_3 H+18
c_4 H^2+24 c_5 H^3\), \nonumber\\
C(a(t),c_{i})&= -4 a_0^3 e^{3 H t} H^2 \( c_2+6 c_3 H+18
c_4 H^2+24 c_5 H^3\).
\end{align}
All the coefficients share a common factor, so the field is either a ghost or a tachyon, in agreement with the findings in Section V.A of \cite{Goon:2011qf}. Comparing the tachyon time scale against $ 1/H $ gives $ {H\tau}=1/2 $, so the tachyon time scale is approximately the Hubble time. This would be disastrous for inflation, since the instability would manifest itself after one e-fold, but it may be tolerable for late-time cosmic acceleration.
\subsection{Small $\pi$ symmetries}
The small $\pi$ limits of the symmetries (\ref{fullsyms}) expanded to lowest order in $\pi$, are
\begin{align}
\delta_{v_i}\pi&= \frac{1}{2} x^{i}
\int dt\, \frac{\dot H}{H^{3}a}
\dot{a},\nonumber\\
\delta_{k_{i}}\pi&=-x^{i} \dot{a} ,\nonumber\\
\delta_{q}\pi&= \dot{a} ,\nonumber\\
\delta_{u}\pi&=\frac{
x^2 \dot{a}^2-1}{4 \dot{a}} ,\nonumber\\
\delta_{s}\pi&= -
\dot{a}\int dt\, \frac{\dot H}{H^{3}a} \ .
\end{align}
In the case where $\pi=0$ is a solution, these are symmetries of the quadratic action for $\pi$. Otherwise, they are symmetries of the action linear in $\pi$.
\subsection{Galileon-like limits}
When we generate galileon theories by foliating a maximally symmetric bulk by maximally symmetric branes, as in \cite{Goon:2011uw}, there exist small field limits which greatly simplify the Lagrangians (\ref{L5}). To take these limits, we form linear combinations $\bar{\mathcal{L}}_{n}=\sum_{m=1}^n c_{n,m}\mathcal{L}_{m}$ of the original Lagrangians, with constant coefficients $c_{n,m}$ chosen such that a perturbative expansion of ${\cal L}_n$ around a constant background $\pi\to\pi_0+\delta{\pi}$ begins at $\mathcal{O}(\delta\pi^{n})$. In particular, as first shown in \cite{deRham:2010eu}, when applied to the case of a flat brane in a flat bulk, this procedure reproduces the flat space galileons of \cite{Nicolis:2008in}.
The ability to carry out such an expansion appears to be an artifact of maximal symmetry. The small $\pi$ limit in the present case of a flat bulk and an FRW brane does not, for general $a(t)$, admit a choice of $c_{n,m}$ with the above mentioned properties.
One case which does work is $a(t)\sim e^{Ht}$, corresponding to a de Sitter brane, which has maximal symmetry. The induced metric on any $w={\rm const}$ hypersurface is
\begin{align}
ds^{2}&=(1-Hw)^{2}\left [-dt^{2}+e^{2Ht}d\vec{x}^{2}\right]\nonumber\\
&=(1-Hw)^{2}\, g_{\mu\nu}^{\,(dS)}dx^{\mu}dx^{\nu} \ ,
\end{align}
where $g_{\mu\nu}^{\,(dS)}$ is the 4D de Sitter metric in inflationary coordinates, and so we are simply foliating 5D minkowski by $dS_{4}$, returning to the setup of a maximally symmetric brane in a maximally symmetric bulk. In the gauge (\ref{preferredgauge}), the induced metric becomes
\begin{align}
\bar{g}_{\mu\nu}&=\left (-1+H\pi\right )^{2}g_{\mu\nu}^{\,(dS)}+\partial_{\mu}\pi\partial_{\nu}\pi \ .
\label{dsinduced}
\end{align}
If we then make the field redefinition $\tilde{\pi}=-1+H\pi$ and switch to coordinates $\hat{x}^{\mu}=Hx^{\mu}$, the Lagrangians calculated from the induced metric (\ref{dsinduced}) and associated extrinsic curvature take the forms of those in Sec. IV.C of \cite{Goon:2011qf}, from which small $\tilde \pi$ limits can be constructed.
\section{Conclusion}
The probe-brane construction has facilitated the development of entirely new four-dimensional scalar effective field theories with nontrivial symmetries stemming from the Killing symmetries of the higher-dimensional bulk. The simplest example of this construction~\cite{deRham:2010eu} yields flat space galileons~\cite{Nicolis:2008in}, of which the DGP cubic term represents the simplest nontrivial interaction term. In general, however, a much richer structure is possible, depending on the geometries of the bulk and the brane. In previous work~\cite{Goon:2011qf,Goon:2011uw} we have laid out the general framework for deriving new four-dimensional field theories in this way, and have applied the method to the examples in which bulk and brane are maximally symmetric spaces.
In this paper, we have extended the construction to background geometries with Gaussian normal foliations, of which the cosmological FRW spacetimes are a particularly useful example. We have derived the relevant operators allowed in the Lagrangians, and identified the highly nontrivial symmetry transformations under which they are invariant. These general expressions are much longer for FRW spacetimes than they are for maximally symmetric ones. By specializing to the minisuperspace approximation, in which the galileons depend only on cosmic time, we are able to provide somewhat more compact versions suitable for understanding the effects of galileons on the background cosmology. However, more complicated questions, such as those involving spatially dependent galileon perturbations, will require the full expressions. It is possible that integrations by parts would greatly simplify the expressions, but we have not attempted these here.
We have sought interesting small-field limits of the Lagrangians and their symmetry transformations, as was done for galileons propagating on maximally symmetric backgrounds. Due to the fewer isometries of FRW, the analogous expressions do not seem to exist, except in the special cases in which the FRW space coincides with de Sitter.
Finally, we have studied the stability of simple solutions, namely $\pi=0$ with $ a(t)=\left (t/t_0\right )^{n} $, and find that given a correct sign for coefficients in the Lagrangians, all four possible solutions are stable, at least on the time scales of the background. One of the four cases leads to a massless field without any gradient energy and the remaining three cases lead to scalar fields with tachyonic masses but the associated time scales are large enough to avoid the potential instability. For exponential scale factor growth, the $\pi=0$ solution also leads to a tachyon whose time scale is again large enough to stabilize the theory for one e-fold.
\bigskip
\goodbreak
\centerline{\bf Acknowledgements}
\noindent
\\
The authors are grateful to Clare Burrage, Claudia de Rham and Lavinia Heisenberg for helpful conversations. This work is supported in part by NASA ATP grant NNX08AH27G, NSF grant PHY-0930521, and by Department of Energy grant DE-FG05-95ER40893-A020. MT is also supported by the Fay R. and Eugene L. Langberg chair.
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{"url":"https:\/\/socratic.org\/questions\/how-do-you-solve-the-following-system-x-3y-20-6x-5y-4","text":"# How do you solve the following system: x-3y=20 , -6x + 5y = 4 ?\n\nJul 9, 2018\n\n$x = 48.615384 \\mathmr{and} y = 9.5384$\n\n#### Explanation:\n\n$x - 3 y = 20$----------------(1)\n\n$- 6 x + 5 y = 4$-----------------(2)\n\nWe can find the value off variable $x$ in terms of $y$ from equation (1) as the coefficient of $x$ is 1 .\n\n$\\implies x = 20 + 3 y$ ----------(3)\n\nNow, substitute this value of $x$ in equation (2),\n\n$\\implies - 6 \\left(20 + 3 y\\right) + 5 y = 4$\n\n$\\implies - 120 - 18 y + 5 y = 4$\n\n$\\implies - 18 y + 5 y = 4 + 120$\n\n$\\implies - 13 y = 124$\n\n$\\therefore y = \\frac{124}{-} 13 = 9.5384$------(4) --- (truncated value)\n\nNow substitute this value of $y$ in equation (3) to get the value of $x$,\n\n$\\implies x = 20 + 3 \\left(9.5384\\right) = 20 + 28.615384 = 48.615384$\n\n$\\therefore x = 48.615384 \\mathmr{and} y = 9.5384$\n\nNote: The values of $x$ and $y$ are truncated.","date":"2020-01-25 05:17:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 19, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.785841703414917, \"perplexity\": 393.91380788487885}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579251669967.70\/warc\/CC-MAIN-20200125041318-20200125070318-00257.warc.gz\"}"} | null | null |
Q: given a list of strings tell tensorflow to create a new one from learned of other strings? I have the following list thousands of strings like this:
gabaybagxppppapppx5qvxdncxcyPcxvNcPNxPPPdPxgaBQaBag
gcyvgxpppvNppxab5nxdpvbxvBaPvqxBQPvPvxP5PPxgN5y
gabcygxpppaBpapxab6xnvPdxvpcqaxvQvNvxPdPPPxgNvgaya
gvnagyaxappbvppxapapdxcPpqanxvBcPaxvPdPxPPaNaPayxgvQagNa
cqagayxvpdpxapapBgpaxpvPpcxvPnPcx5PPaxPPaQvyax5gag
6yaxpppvpppx8xvnvyaPxvPvPaPxvBpgcxPPdgaxdggv
gncgyaxp5ppxvp5xcPpbvxvq5xaQ6xPPPBvPPxgcyaNg
NabydxppapaQppx8xvb5xcncqx8xPPPvgPPxgNBagBya
8xvpcNax6pax5PBaxppvgnvx7yxPapvyaPxcgd
gabayangxpvpapppxnvBdxapaNPNaPx6PaxcPaQvxPaPaycxq5ba
How can tensorflow be trained to create a new one from learned?
Im using Jupyter Notebook with python 3
A: You should look into generative models.
1) Why are you intent on using tensorflow? There are non-DL generative models that MIGHT work here
2) What exactly are those strings? What do they mean? To me, they look like random noise.
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A battle the people of the United States should watch like hawks continues to unfold in Mexicali, the capital of Mexico's Baja California, over water claims, privatization, and an existential question particularly poignant and raw to Indigenous Peoples and Native American tribes across the continent — who has a right to clean, potable water?
Is water even a right?
Add the possibility a region sorely in need would see 750 permanent jobs, and this battle — whether or not Constellation Brands should be permitted to construct a government-endorsed, $1 billion state-of-the-art brewing facility for its various Modelo and Corona beer brands — has exploded into a hostile and bloody showdown between law enforcement and irate residents concerned for area farmers over the possibility of dwindling water supplies.
Proponents contend the brewery represents the grandest project for the region, possibly in decades — fueling massive and needed economic growth, employment, and likely even tourism.
Constellation Brands is the United States' third largest beermaker — its hotly contentious decision to construct the sleek facility considered a boon by many politicians, particularly in regard to expanding employment opportunities, with 4,000 to 5,000 positions enduring for the period of construction, and hundreds more permanently, after that.
Throngs of demonstrators, led since 2016 by a local group comprised of farmers and locals called Mexicali Resiste — (whose website as of publication had been suspended by its host, but which maintains active accounts with both Twitter and Facebook) — violently clashing with law enforcement sent to ensure construction comes to completion say the cost, and tandem trampling of their rights, hardly constitute sound justification for the plunder of precious, clean water.
Permits have been issued and the project appears poised to otherwise take off — but confrontations have accordingly intensified — and protester-residents, understandably worried over the diversion of water from agricultural and personal needs, aren't budging.
Five people were arrested by state and municipal law enforcement January 17, according to State Public Security Secretary Gerardo Sosa Olachea, after demonstrators effectively thwarted workers' efforts, halting construction of the final leg of an aqueduct set to supply the brewery. Stone-throwing demonstrators met police in the early morning hours — and the clash did not abate until at least five attempts eventually wore the protest down to a simmer.
"Following an act of vandalism that resulted in damage to a pipe that passes near the brewery site and supplies water to the state capital, personnel from the State Public Services Commission attended to shut off valves and repair a leak.
"However, they were also met by a group of people who attempted to obstruct their work by throwing things at them.
According to the Tribune noting reports from journalists on scene, one demonstrator landed a broken jaw in the prolonged encounter, during which police — unarmed, but heavily protected with riot gear — also took to throwing stones at the crowd in defense.
Neither of these skirmishes were the first, nor will they be the last, in the bitterly visceral fight to protect civilians' access to potable water — particularly as the local population claims to see past politicians' glowing jobs assessment. Skeptical, those who condemn Constellation Brands and its coziness with politicians cite rampant corruption and self-interested business dealings as the true impetus for the brewery and such vehemence in support of it.
No matter how vitriolic the ongoing dispute grows, it is the people of Mexicali and Baja California declaring an impassable line in the sand who continue without faltering to resist the designs of corporate-government corruption. Refusing to simply stand down or not to act in forceful defense of their most basic need was never a question for many taking a stand.
After all, the premise of the opposition highlights the specifically modern context of a coming war for what's left in the world, pitting profiteers shady and otherwise in partnership with governments ostensibly representational, against ordinary people who — devoid of the aggregate financial power corporations wield at whim — view the struggle very much as one of survival.
It is additionally a fight for the future — as much as it is also defense of the people's past.
Mexicali Resiste and opponents of the controversial product have enjoined the public to boycott Constellation Brands — and consider more carefully the choices casually made with money. | {
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Justia Patents Security Kernel Or UtilityUS Patent for Method, system and browser for executing active object of browser Patent (Patent # 10,218,767)
Method, system and browser for executing active object of browser
Aug 30, 2013 - Beijing Qihoo Technology Company Limited
The present disclosure discloses a method, a system and a browser for executing a browser active object. In the present invention, a proxy object is run in a page process and an active object is run in an independent process, so that a true plug-in is separated from the page process. The present invention further discloses an inter-process script execution method, system and browser. The present invention further discloses a browser active object executing method and system, and a browser.
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The present invention relates to the field of computer network technologies, and in particular, to a method, a system, and a browser for executing a browser active object.
When a browser is opened, a webpage progress of the browser will create an active object and then perform various operations of the webpage. The active object usually refers to a plug-in of the browser, such as an ActiveX plug-in.
The ActiveX plug-in is an extension interface manner which is supported by an IE kernel browser of Microsoft Inc. and enhances browsing experience. A typical plug-in comprises Flash, Baidu video, QVOD and the like. The ActiveX plug-in is a reusable software assembly. Through use of the ActiveX plug-in, a special function can be quickly added to a website, desktop application and a development tool. For example, a StockTicker plug-in can be used to add activity information to a webpage in real time, and an animation plug-in can be used to add an animation property to a webpage.
When the ActiveX plug-in runs in a webpage process of the IE browser, the average quality of the Active X plug-in is made poor due to the large number, complicated running environment and openness of ActiveX plug-ins. Once the ActiveX plug-in is confronted with issues such as a halt or failure, the whole webpage will be caused to exit, which affects the stability of the webpage operation. On the contrary, if the webpage process is confronted with issues such as a halt or failure for some reason, the ActiveX plug-in in the webpage will be caused to exit, which affects the stability of the ActiveX plug-in.
In addition, IPC (Inter-Process Communication) refers to some techniques or methods for transmitting data or signals between at least two processes or threads. A process is a minimum unit for a computer system to allocate resources. Each process has a portion of independent system resources of its own, and processes are isolated from one another. The inter-process communication is provided to enable different processes to access one another's resources and perform coordination. Different processes may run on the same computer or different computers connected via networks.
Generally speaking, the inter-process communication needs to be achieved in the following situations:
(1) Data transmission: a process needs to transmit its own data to another process, with the quantity of transmitted data being in a range between one byte and several mega bytes.
(2) Data sharing: a plurality of processes desire to operate shared data, when one process modifies the shared data, other processes are supposed to see the modification immediately.
(3) Notification of events: a process needs to transmit a message to another process or a group of processes to notify it (them) of the occurrence of a certain event (e.g., notify a parent process when the process terminates).
(4) Resource sharing: a plurality of processes share the same resource. To this end, a core is needed to provide a locking and synchronization mechanism.
(5) Process control: some process desires to completely control the execution of another process (e.g., a Debug process), at this point, the control process desires to intercept all traps and exceptions of another process and know status changes thereof in time.
However, the existing art has not yet provided a solution for executing inter-process script, and current methods cannot be used to implement interactive execution of inter-process scripts.
In view of the above problems, the present invention is proposed to provide a system for executing a browser active object, a browser, and a corresponding method for executing a browser active object, which can overcome the above problems or at least partially solve the above problems.
According to an aspect of the present invention, there is provided a method for executing a browser active object, the active object being an object corresponding to an ActiveX plug-in, the method comprising: before the active object is created, intercepting a webpage process to query for a safety interface of a pre-created active object, and returning information indicating the ActiveX plug-in is a safe plug-in; intercepting a procedure of the webpage process creating the active object, and creating a proxy object to replace an active object actually to be created, with the proxy object running in the webpage process; when the webpage process activates the proxy object, creating the active object actually to be created in an independent process independent from the webpage process, with the active object running in the independent process; a communication window is created respectively in the active object and the proxy object; the active object and the proxy object communicating via the communication windows, thus realizing that the proxy object invokes the active object and/or the active object invokes the proxy object.
According to another aspect of the present invention, there is provided a system for executing a browser active object, comprising: a webpage process module, configured to, before the active object is created, intercept a webpage process to query for a safety interface of a pre-created active object, and return information indicating the active object is a safe plug-in; and intercept a procedure of the webpage process creating the active object, and create a proxy object to replace an active object actually to be created, with the proxy object running in the webpage process; an independent process module, configured to, when the webpage process activates the proxy object, create the active object actually to be created in an independent process independent from the webpage process, with the active object running in the independent process; the proxy object is located in the webpage process module, the active object is located in the independent process module, and a communication window is created respectively in the active object and the proxy object; the active object is an active object corresponding to the ActiveX plug-in; the active object and the proxy object communicating via the communication windows, thus realizing that the proxy object invokes the active object and/or the active object invokes the proxy object.
According to a further aspect of the present invention, there is provided a browser comprising the above system for executing a browser active object.
According to a further aspect of the present invention, there is provided a computer program which comprises a computer readable code, wherein when the computer readable code is run on a server, the server executes the method for executing a browser active object according to any one of claims 1-9.
According to a further aspect of the present invention, there is provided a computer readable medium which stores the computer program.
In the present invention, with the proxy object running in the webpage process and the active object running in the independent process, a true ActiveX plug-in is made separate from the webpage process so that the ActiveX plug-in, when there is something wrong, will not affect the webpage process, and thereby improving stability of the browser; particularly when the ActiveX plug-in is confronted with a security issue such as occurrence of viruses, it will not exert an influence on the webpage process, thereby improving the safety of the browser; meanwhile, the webpage process, when being confronted with a problem, will not affect normal run of the ActiveX plug-in so that the stability of the ActiveX plug-in is improved.
According to a solution provided by one aspect of the present invention, after the active object transmits the script to the proxy object, the proxy object queries for an interface in the webpage process related to the script execution, obtains the script executing method according to the interface, and then executes the script transmitted by the active object according to the script executing method, thereby achieving the script execution between different processes and implementing control of the webpage running in the webpage process by the active object running in the independent process.
According to a solution provided by another aspect of the present invention, after the proxy object transmits the script to the active object, the proxy object invokes the scheduling interface of the active object to obtain the scheduling identification of the to-be-executed method in the script; after the proxy object feeds back the scheduling identification to the webpage, the proxy object intercepts an instruction in the webpage process executing the to-be-executed method in the script and transmits the instruction to the active object, the active object executes the instruction and returns an execution result resulting from the execution of the instruction to the proxy object, thereby achieving the script execution between different processes and implementing control of the active object running in the independent process by the webpage running in the webpage process.
In the present invention, with the proxy object running in the webpage process and the active object running in the independent process, a true ActiveX plug-in is made separate from the webpage process so that the ActiveX plug-in, when there is something wrong, will not affect the webpage process, and thereby improving stability of the browser; particularly when the ActiveX plug-in is confronted with a security issue such as occurrence of viruses, it will not exert an influence on the webpage process, thereby improving the safety of the browser; meanwhile, the webpage process, when being confronted with a problem, will not affect normal run of the ActiveX plug-in so that the stability of the ActiveX plug-in is improved. Furthermore, the method is compatible with all ActiveX plug-ins, including plug-ins whose script safety is declared in a registration table and plug-ins whose script safety is not declared in the registration table.
The above description only generalizes technical solutions of the present invention. The present invention may be implemented according to the content of the description in order to make technical means of the present invention more apparent. Specific embodiments of the present invention are exemplified to make the above and other objects, features and advantages of the present invention more apparent.
Various other advantages and merits will become apparent to those having ordinary skill in the art by reading through the following detailed description of preferred embodiments. Figures are only intended to illustrate preferred embodiments and not construed as limiting the present invention. In all figures, the same reference numbers denote the same part. In the figures:
FIG. 1 illustrates a flow chart of a method for executing a browser active object according to an embodiment of the present invention;
FIG. 2 illustrates a schematic diagram of a procedure of creating a proxy object and an active object in the present invention;
FIG. 3 illustrates a flowchart of a method of the proxy object executing a script of the active object in the present invention;
FIG. 4 illustrates a flowchart of a method of the active object executing a script of the proxy object in the present invention;
FIG. 5 illustrates a structural block diagram of a system for executing a browser active object according to an embodiment of the present invention;
FIG. 6 illustrates a flowchart of an inter-process script executing method according to an embodiment of the present invention;
FIG. 7 illustrates a flowchart of an inter-process script executing method according to another embodiment of the present invention;
FIG. 8 illustrates a structural block diagram of an inter-process script executing system according to an embodiment of the present invention;
FIG. 9 illustrates a structural block diagram of an inter-process script executing system according to another embodiment of the present invention;
FIG. 10 illustrates a flowchart of a method for executing a browser active object according to an embodiment of the present invention;
FIG. 11 illustrates a flowchart of a method of a proxy object executing a script of the active object in the present invention;
FIG. 12 illustrates a structural block diagram of a system for executing a browser active object according to an embodiment of the present invention;
FIG. 13 illustrates a structural block diagram of a system for executing a browser active object according to another embodiment of the present invention;
FIG. 14 schematically illustrates a block diagram of a server for executing the method according to the present invention; and
FIG. 15 schematically illustrates a memory unit for maintaining or carrying a program code for implementing the method according to the present invention.
The present invention will be further described below with reference to figures and specific embodiments.
FIG. 1 illustrates a flow chart of a method 100 for executing a browser active object according to an embodiment of the present invention. In the method, the active object is an object corresponding to the ActiveX plug-in, and the ActiveX plug-in may be a video play plug-in such as Baidu video or QVOD, but the present invention is not limited to this. As shown in FIG. 1, the method 100 begins with step S101, wherein before the active object is created, a webpage process is intercepted to query for a safety interface of a pre-created active object, and information indicating the ActiveX plug-in is a safe plug-in is returned. Specifically, in the case that there is a ActiveX plug-in in the webpage, before the webpage process of the browser creates the ActiveX plug-in, a registration table is firstly queried to look up whether the registration table related to the plug-in declares the safety of a script of the plug-in. Usually, regarding the type of plug-ins such as Baidu video or QVOD, the registration table generally does not declare safety of its script, so after the registration table is queried, an active object will be pre-created, the safety of the plug-in is determined by querying for a safety interface (IObjectSafety) of the pre-created active object, and the active object of the plug-in is truly created only when the plug-in is determined safe. Regarding the type of plug-ins such as Baidu video or QVOD, if the special processing is not performed here, information about warning or not creating an object will be displayed so that such type of plug-ins cannot operate normally. Therefore, in the method, the IObjectSafety interface for the webpage process to query for the pre-created active object is intercepted, and information indicating the plug-in is a safe plug-in is directly returned. For example, when the webpage process queries for the IObjectSafety interface of the pre-created active object, the interface is intercepted, the IObjectSafety interface indicative of plug-in safety is returned thereto, INTERFACESAFE_FOR_UNTRUSTED_CALLER| INTERFACESAFE_FOR_UNTRUSTED_DATA is returned via GetInterfaceSafetyOptions to indicate that the plug-in is safe in script and safe in initialization, and thereby safety verification can be smoothly passed.
FIG. 2 illustrates a schematic diagram of a procedure of creating a proxy object and an active object in the present invention. As shown in FIG. 2, a Web webpage and a proxy object run in the webpage process, and an empty webpage and an active object run in an independent process. Subsequent steps are further introduced below in detail with reference to FIG. 2.
After step S101, the method 100 proceeds to step S102, wherein a procedure of a webpage process creating an active object is intercepted, a proxy object is created to replace an active object actually to be created, and the proxy object runs in the webpage process. Specifically, when the webpage process of the browser creates an active object, CoGetClassObject is intercepted, and a proxy object is created to replace the active object actually to be created. The proxy object includes a series of interfaces, including many standard interfaces that will be invoked by an IUnknown pointer, such as IOleObject and IViewObject. Then, when CreateInstance of IClassFactory is invoked, the created proxy object is returned to the IUnknown pointer of the webpage process, whereby creation of the proxy object is completed.
Then, the method 100 proceeds to step S103, wherein when the webpage process activates the proxy object, the active object actually to be created is created in an independent process independent from the webpage process, and the active object is run in the independent process. Specifically, when the webpage process activates the proxy object, information related to the plug-in such as attribute, URL and size is extracted, and then an independent process is created. First, a document, namely, an empty webpage, is created, an object is inserted therein, the independent process invokes CoGetClassObject, intercepts the CoGetClassObject and creates an active object. The active object comprises a series of interfaces, including many standard interfaces that will be invoked by an IUnknown pointer, such as IOleObject and IViewObject. Then when CreateInstance of IClassFactory is invoked, the created active object is returned to the IUnknown pointer of the independent process, whereby creation of the active object is completed. The active object is an object which truly realizes the plug-in function.
During creation of the active object, the active object can still be normally created in the case that IWeBrowser2 interface may not be implemented for some plug-ins such as Flash plug-in. However, regarding the type of plug-ins such as Baidu video or QVOD, creation of the active object will fail if the IWeBrowser2 interface is not implemented. The IWeBrowser 2 is implemented mainly to perform functions such as skipping (Navigate method), obtainment of an URL (Get_LocationURL method) and obtainment of a webpage (IHtmlDocument interface under get_document method). In order to perform these functions, when the independent process queries for the IWeBrowser2 interface, it is intercepted, and then a self-created IWeBrowser2 interface is returned so as to create the active object of the above plug-in.
Then, the method 100 proceeds to step S104, wherein a communication window associated with a plug-in object is created respectively in the proxy object and the active object to enable the proxy object and the active object to communicate. Through the two communication windows, the two objects may interact with respect to size, focus, refresh, script and other information of the plug-in.
Then, the method 100 proceeds to step S105, wherein the active object and the proxy object communicate via the communication windows so that the proxy object invokes the active object and/or the active object invokes the proxy object so as to perform the function to be achieved by the plug-in.
Furthermore, if there is a nested structure in the Web webpage, i.e., the parent process runs a plurality of parent webpages and a plurality of subpages. Whenever the webpage process creates a subpage and its proxy object, a proxy object ID (may also include its URL) of the subpage and a proxy object ID (may also include its URL) of the subpage's parent webpage are sent to the independent process in which an active object is created according to a corresponding hierarchical structure. In this case, a plurality of proxy objects run in the webpage process, a plurality of active objects run in the dependent process, there is a one-to-one correspondence between the proxy objects and the active objects, the communication window of each proxy object has a corresponding attribute with the communication window of the active object corresponding to the proxy object, and communication between the active object and the corresponding proxy object is achieved through the two communication windows.
In this method, with the proxy object running in the webpage process and the active object running in the independent process, a true ActiveX plug-in is made separate from the webpage process so that the ActiveX plug-in, when there is something wrong, will not affect the webpage process, and thereby improving stability of the browser; particularly when the ActiveX plug-in is confronted with a security issue such as occurrence of viruses, it will not exert an influence on the webpage process, thereby improving the safety of the browser; meanwhile, the webpage process, when being confronted with a problem, will not affect normal run of the ActiveX plug-in so that the stability of the ActiveX plug-in is improved.
In the above method, since the true ActiveX plug-in is separate from the webpage process, when a plug-in window is expected to be displayed in the original Web webpage, a plug-in window corresponding to the active object may be arranged at a location of the plug-in window of the original Web webpage, the plug-in window corresponding to the active object may move and zoom without affecting the Web webpage.
According to the method provided by the above embodiment, when the Web webpage rolls, the plug-in window may not roll therewith. Specifically, according to the current operation, when the Web webpage rolls, a GetWindow method of an IOleWindow interface of the active object will be invoked to judge whether the active object has a corresponding window, a WM_MOVE message is sent to move the window if there is the window. However, according to the present invention, a null value can be returned when the GetWindow method of the IOleWindow interface of the active object is invoked so that the window will not roll therewith.
Furthermore, on the basis that the proxy object and the active object are created as stated above, the communication between the active object and the proxy object comprises an inter-process script executing procedure. The script executing procedure comprises a procedure of the proxy object executing a script of the active object, and a procedure of the active object executing a script of the proxy object, which will be introduced respectively below.
FIG. 3 illustrates a flowchart of a method 300 of the proxy object executing a script of the active object in the present invention. The method is adapted for the case that the proxy object running in the webpage process executes a script of the active object running in the independent process, i.e., the method is a method of the active object running in the independent process controlling the webpage in the webpage process. As shown in FIG. 3, the method 300 begins with step S301, wherein the active object transmits the script to the proxy object via the communication window. Taking a user clicking a button created on the plug-in window to trigger the webpage to become black as an example, when the user clicks the button on the plug-in window, the active object of the plug-in obtains a script corresponding to the clicking operation of the button and transmits the script to the proxy object via the communication window.
Subsequently, the method 300 proceeds to step S302, wherein by invoking a scheduling interface of the proxy object, the active object enables the proxy object to execute a method of obtaining a scheduling identification of a to-be-executed method in the script, and the proxy object, after obtaining the scheduling identification, returns the scheduling identification to the active object via the communication window. Specifically, first, the independent process parses the script corresponding to the clicking operation of the button, obtains a to-be-executed method in the script to allow the webpage to become black, then schedules a name of the to-be-executed method in the script to an IDispatch interface of the active object, the IDispatch interface is the scheduling interface used to invoke a function in a language program not supporting a virtual function table, the IDispatch interface has a GetIDsOfNames function and an Invoke function, wherein the GetIDsOfNames function provides a method of using the name of the method to return its scheduling ID, and the Invoke function provides an instruction of using the scheduling ID of the method to execute the method. As the webpage corresponding to the active object is an empty webpage, and it does not have a method of enabling the webpage to become black, the active object cannot obtain the scheduling identification (ID) of the method in the script, and the active object invokes the IDispatch interface of the proxy object via the communication window. Since the Web webpage corresponding to the proxy object is a complete webpage, the method provided by the GetIDsOfNames function of the proxy object is invoked to enable the proxy object to obtain the scheduling ID of the to-be-executed method in the script, and then the proxy object returns the scheduling ID to the active object via the communication window.
Then, the method 300 proceeds to step S303, wherein the active object intercepts an instruction in the independent process executing the to-be-executed method in the script and transmits the instruction to the proxy object. Specifically, after the active object obtains the scheduling ID of the to-be-executed method, the scheduling ID is fed back to the webpage, and the independent process will execute an instruction of the Invoke(ID) in next step according to a conventional flow. However, according to the method here, the instruction of Invoke(ID) of the independent process is intercepted, and the instruction of the Invoke(ID) is sent to the proxy object.
Then, the method 300 proceeds to step S304, wherein the proxy object executes the to-be-executed method in the script by executing the instruction, and then returns an execution result to the active object. Specifically, the proxy object executes the instruction of Invoke(ID) so as to execute the to-be-executed method in the script to make the webpage become black, and return a notification message of the effect that the webpage becomes black to the active object.
FIG. 4 illustrates a flowchart of a method 400 of the active object executing a script of the proxy object in the present invention. The method is adapted for the case that the active object running in the independent process executes a script of the proxy object running in the webpage process, i.e., the method is a method of the webpage running in the webpage process controlling the active object in the independent process. If the webpage desires to obtain information of the plug-in (such as attributes like version number, path and URL) or the webpage desires to operate the plug-in (e.g., the webpage desires to change a size of the plug-in window), it can be implemented by this method. As shown in FIG. 4, the method begins with step S401, wherein the proxy object transmits the script to the active object via the communication window. Take obtainment of the version number of the plug-in as an example. Since the proxy object in the webpage process is not a true plug-in object, it does not know the version number of the plug-in, so the proxy object cannot directly feed back the version number of the plug-in to the webpage. After the independent process completes creation of the active object, a variable of the active object is notified to the webpage, a webpage developer writes in the webpage a script for obtaining the version number of the plug-in according to the variable, and the to-be-executed method in the script is intended to obtain the version number of the plug-in. The proxy object transmits the script for obtaining the version number of the plug-in to the active object via the communication window.
Then, the method 400 proceeds to step S402, wherein by invoking a scheduling interface of the active object, the proxy object enables the active object to execute a method of obtaining a scheduling identification of a to-be-executed method in the script, and the active object, after obtaining the scheduling identification, returns the scheduling identification to the proxy object via the communication window. Specifically, first, the webpage parses the script, schedules name of the to-be-executed method in the script to an IDispatch interface of the proxy object, the IDispatch interface is the scheduling interface to invoke a function in a language program not supporting a virtual function table, the IDispatch interface has a GetIDsOfNames function and an Invoke function, wherein the GetIDsOfNames function provides a method of using the name of the method to return its scheduling ID, and the Invoke function provides an instruction of using the scheduling ID of the method to execute the method. Since the proxy object cannot obtain the scheduling identification (ID) of the to-be-executed method in the script, the proxy object invokes the IDispatch interface of the active object via the communication window. The active object is an object of the true plug-in, the method provided by the GetIDsOfNames function of the active object is invoked to enable the active object to obtain the scheduling ID of the to-be-executed method, and then the active object returns the scheduling ID to the proxy object via the communication window.
Then, the method 400 proceeds to step S403, wherein the proxy object intercepts an instruction in the webpage process executing the to-be-executed method in the script and transmits the instruction to the active object. Specifically, after the proxy object obtains the scheduling ID of the to-be-executed method, the scheduling ID is fed back to the webpage, and the webpage process will execute an instruction of the Invoke(ID) in next step according to a conventional flow. However, according to the method here, the instruction of Invoke(ID) of the webpage process is intercepted, and the instruction of the Invoke(ID) is sent to the active object.
Then, the method 400 proceeds to step S404, wherein the active object executes the to-be-executed method in the script by executing the instruction, and then returns an execution result to the proxy object. Specifically, the active object executes the instruction of Invoke(ID) so as to execute the to-be-executed method in the script, obtain the version number of the plug-in, and return the version number of the plug-in to the proxy object, and then the proxy object will feed back the version number of the plug-in to the webpage.
According to the above inter-process script executing method, the proxy object may execute the script of the active object, and the active object may execute the script of the proxy object, thereby achieving control of the webpage running in the webpage process by the active object running in the independent process and control of the active object running in the independent process by the webpage running in the webpage process.
FIG. 5 illustrates a structural diagram of a system for executing a browser active object according to an embodiment of the present invention. As shown in FIG. 5, the system comprises a webpage process module 510 and an independent process module 520, a proxy object 511 located in the webpage process module 510 and an active object 521 located in the independent process module 520, and a communication window 530 respectively created in the active object 521 and the proxy object 511. In the system, the active object is an active object corresponding to the ActiveX plug-in, and the ActiveX plug-in may be a video play plug-in such as Baidu video or QVOD, but the present invention is not limited to this.
The webpage process module 510 is configured to, before the active object is created, intercept a webpage process to query for a safety interface of a pre-created active object, and return information indicating the active object is a safe plug-in; and intercept a procedure of the webpage process creating an active object, and create a proxy object 511 to replace an active object actually to be created, with the proxy object 511 running in the webpage process. In the case that there is a ActiveX plug-in in the webpage, before the webpage process module 510 creates the ActiveX plug-in, a registration table is firstly queried to look up whether the registration table related to the plug-in declares the safety of a script of the plug-in. Usually, regarding the type of plug-ins such as Baidu video or QVOD, the registration table generally does not declare safety of its script, so after the registration table is queried, the webpage process module 510 pre-creates an active object, determines the safety of the plug-in by querying for a safety interface (IObjectSafety) of the pre-created active object, and truly creates the active object of the plug-in only when the plug-in is determined safe. Regarding the type of plug-ins such as Baidu video or QVOD, if the special processing is not performed here, information about warning or not creating an object will be displayed so that such type of plug-ins cannot operate normally. Therefore, the webpage process module 510 intercepts the IObjectSafety interface for the webpage process to query for the pre-created active object, and directly returns information indicating the plug-in is a safe plug-in. For example, when the webpage process queries for the IObjectSafety interface of the pre-created active object, the webpage process module 501 intercepts the interface, returns to it the IObjectSafety interface indicative of plug-in safety, returns INTERFACESAFE_FOR_UNTRUSTED_CALLER| INTERFACESAFE_FOR_UNTRUSTED_DATA via GetInterfaceSafetyOptions to indicate that the plug-in is safe in script and safe in initialization, and thereby safety verification can be smoothly passed. When the webpage process of the browser creates an active object, the webpage process module 510 intercepts CoGetClassObject, and creates a proxy object 511 to replace the active object actually to be created. The proxy object 511 includes a series of interfaces, including many standard interfaces that will be invoked by an IUnknown pointer, such as IOleObject and IViewObject. Then, when CreateInstance of IClassFactory is invoked, the created proxy object 511 is returned to the IUnknown pointer of the webpage process, whereby creation of the proxy object 511 is completed.
The independent process module 520 is configured to, when the webpage process activates the proxy object 511, create the active object actually to be created in the independent process independent from the webpage process, and run the active object 521 in the independent process. When the webpage process activates the proxy object 511, information related to the plug-in such as attribute, URL and size is extracted, and then an independent process is created. First, a document, namely, an empty webpage, is created, an object is inserted therein, the independent process invokes CoGetClassObject, the independent process module 520 intercepts the CoGetClassObject and creates the active object 521. The active object 521 comprises a series of interfaces, including many standard interfaces that will be invoked by an IUnknown pointer, such as IOleObject and IViewObject. Then, when CreateInstance of IClassFactory is invoked, the created active object 521 is returned to the IUnknown pointer of the independent process, whereby creation of the active object 521 is completed. The active object 521 is an object which truly realizes the plug-in function.
The independent process module 520 is further configured to intercept the independent process querying for the IWeBrowser2 interface, and return a self-created IWeBrowser2 interface so as to create the active object 521 actually to be created.
The active object 521 and the proxy object 511 communicate via the communication window 530 so that the proxy object 511 invokes the active object 521 and/or the active object 521 invokes the proxy object 511. The two objects may interact with respect to size, focus, refresh, script and other information of the plug-in through the two communication windows.
Furthermore, if there is a nested structure in the Web webpage, the webpage process module 510 comprises a plurality of proxy objects, the independent process module 520 comprises a plurality of active objects, there is a one-to-one correspondence between the proxy objects and the active objects, the communication window of the proxy object has a corresponding attribute to the communication window of the active object corresponding to the proxy object, and communication between the active object and the proxy object is achieved through the two communication windows.
In this system, with the proxy object running in the webpage process and the active object running in the independent process, a true ActiveX plug-in is made separate from the webpage process so that the ActiveX plug-in, when there is something wrong, will not affect the webpage process, and thereby improving stability of the browser; particularly when the ActiveX plug-in is confronted with a security issue such as occurrence of viruses, it will not exert an influence on the webpage process, thereby improving the safety of the browser; meanwhile, the webpage process, when being confronted with a problem, will not affect normal run of the ActiveX plug-in so that the stability of the ActiveX plug-in is improved.
Furthermore, in the system as shown in FIG. 5, the proxy object 511 and the active object 521 may interact scripts therebetween, i.e., the proxy object 511 may execute the script of the active object 521, and the active object 521 may execute the script of the proxy object 511.
The active object 521 comprises a first transmitting module 522, a first scheduling identification obtaining module 523 and a first intercepting module 524, wherein the first scheduling identification obtaining module 523 further comprises a first invoking module 525 and a first scheduling identification receiving module 526. The proxy object 511 comprises a first receiving module 512 and a first executing module 513.
The first transmitting module 522 is configured to transmit the script to the proxy object 511 via the communication window 530. Taking a user clicking a button created on the plug-in window to trigger the webpage to become black as an example, when the user clicks the button on the plug-in window, the active object 521 of the plug-in obtains a script corresponding to the clicking operation of the button, and the first transmitting module 522 transmits the script to the proxy object 511 via the communication window 530.
The first scheduling identification obtaining module 523 is configured to obtain a scheduling identification of the to-be-executed method in the script by invoking a scheduling interface of the proxy object 511, wherein the first invoking module 525 is configured to invoke the scheduling interface of the proxy object 511 and enable the proxy object 511 to execute a method of obtaining a scheduling identification of the to-be-executed in the script; the first scheduling identification receiving module 256 is configured to receive the scheduling identification returned by the proxy object 511 via the communication window 530. Specifically, first, the independent process module 520 parses the script corresponding to the clicking operation of the button, obtains the to-be-executed method in the script to allow the webpage to become black, and then schedules a name of the to-be-executed method in the script to an IDispatch interface of the active object 521. As the webpage corresponding to the active object 521 is an empty webpage, and it does not have a method of enabling the webpage to become black, the active object 521 cannot obtain the scheduling identification (ID) of the method in the script, and the first scheduling module 525 of the active object 521 invokes the IDispatch interface of the proxy object 511 via the communication window 530. Since the Web webpage corresponding to the proxy object 511 is a complete webpage, the method provided by the GetIDsOfNames function of the proxy object 511 is invoked to enable the proxy object 511 to obtain the scheduling ID of the to-be-executed method in the script, and then the proxy object 511 returns the scheduling ID to the first scheduling identification receiving module 526 of the active object 521 via the communication window 530.
The first intercepting module 524 is configured to intercept an instruction in the independent process executing the to-be-executed method in the script and transmit the instruction to the proxy object 511. After the active object 521 obtains the scheduling ID of the to-be-executed method, the scheduling ID is fed back to the webpage, and the independent process will execute an instruction of the Invoke(ID) in next step according to a conventional flow. However, the first intercepting module 524 intercepts the instruction of Invoke(ID) of the independent process and transmits the instruction of the Invoke(ID) to the proxy object 511.
The first receiving module 512 is configured to receive the script transmitted by the first transmitting module 522 of the active object 521 and the instruction transmitted by the first intercepting module 524; the first executing module 513 is configured to execute the to-be-executed method in the script by executing the instruction and then return an execution result to the active object 521. The first executing module 513 executes the instruction of Invoke(ID) so as to execute the to-be-executed method in the script to make the webpage become black, and return a notification message of the effect that the webpage becomes black to the active object 521.
The proxy object 511 further comprises a second transmitting module 514, a second scheduling identification obtaining module 515, a second intercepting module 516, wherein the second scheduling identification obtaining module 515 further comprises a second invoking module 517 and a second scheduling identification receiving module 518. The active object 521 comprises a second receiving module 527 and a second executing module 528.
The second transmitting module 514 is configured to transmit the script to the active object 521 via the communication window 530. Take obtainment of the version number of the plug-in as an example. Since the proxy object 511 in the webpage process is not a true plug-in object, it does not know the version number of the plug-in, so the proxy object 511 cannot directly feed back the version number of the plug-in to the webpage. After the independent process completes creation of the active object 521, a variable of the active object 521 is notified to the webpage, a webpage developer writes in the webpage a script for obtaining the version number of plug-in according to the variable, and the to-be-executed method in the script is intended to obtain the version number of the plug-in. The second transmitting module 514 of the proxy object 511 transmits the script for obtaining the version number of the plug-in to the active object 521 via the communication window 530.
The second scheduling identification obtaining module 515 is configured to obtain a scheduling identification of the to-be-executed method in the script by invoking the scheduling interface of the active object 521, wherein the second invoking module 517 is configured to invoke the scheduling interface of the active object 521 and enable the active object 521 to execute a method of obtaining a scheduling identification of the to-be-executed in the script; the second scheduling identification receiving module 518 is configured to receive the scheduling identification returned by the active object 521 via the communication window 530. First, the webpage process module 510 parses the script and schedules a name of the to-be-executed method in the script to an IDispatch interface of the proxy object 511. As the proxy object 511 cannot obtain the scheduling identification (ID) of the to-be-executed method in the script, the second scheduling module 517 of the proxy object 511 invokes the IDispatch interface of the active object 521 via the communication window 530. The active object 521 is an object of the true plug-in, the method provided by the GetIDsOfNames function of the active object 521 is invoked to enable the active object 521 to obtain the scheduling ID of the to-be-executed method, and then the active object 521 returns the scheduling ID to the second scheduling identification receiving module 518 of the proxy object 511 via the communication window 530.
The second intercepting module 516 is a configured to intercept an instruction in the webpage process executing the to-be-executed method in the script and transmit the instruction to the active object 521. After the proxy object 511 obtains the scheduling ID of the to-be-executed method, the scheduling ID is fed back to the webpage, and the webpage process will execute an instruction of the Invoke(ID) in next step according to a conventional flow. However, the second intercepting module 516 intercepts the instruction of Invoke(ID) of the webpage process and transmits the instruction of the Invoke(ID) to the active object 521.
The second receiving module 527 is configured to receive the script transmitted by the second transmitting module 514 of the proxy object 511 and the instruction transmitted by the second intercepting module 516; the second executing module 528 is configured to execute the to-be-executed method in the script by executing the instruction and then return an execution result to the proxy object 511.
According to the functions performed by the above function modules, the proxy object may execute the script of the active object, and the active object may execute the script of the proxy object, thereby achieving control of the webpage running in the webpage process by the active object running in the independent process and control of the active object running in the independent process by the webpage running in the webpage process.
The present invention further provides a browse comprising the system for executing a browser active object according to the above embodiment.
The inter-process script executing method provided by the present invention refers to a script executing method between the webpage process and the independent process, wherein the webpage process is a process for running the Web webpage, and the independent process is another process independent from the webpage process. In the case that the webpage has an ActiveX plug-in, the webpage process runs an ActiveX plug-in proxy object, the independent process runs an ActiveX plug-in active object, and the ActiveX plug-in active object is an object of a true ActiveX plug-in. In the following embodiment, illustration is presented by taking a Flash plug-in as the ActiveX plug-in, but the present invention is not limited to this.
Reference may be made to FIG. 2 and the corresponding depictions for a procedure for creating the Flash plug-in proxy object and Flash plug-in active object.
FIG. 6 illustrates a flowchart of an inter-process script executing method 600 according to an embodiment of the present invention. The method is adapted to the case that the Flash plug-in proxy object run in the webpage process executes a script of the Flash plug-in active object run in the independent process, namely, the method is a method of the Flash plug-in active object run in the independent process controlling the webpage in the webpage process. As shown in FIG. 6, the method 600 begins with step S601, wherein the Flash plug-in active object transmits a script to the Flash plug-in proxy object via the communication window. Taking a user clicking a button created on the Flash to trigger the webpage to become black as an example, when the user clicks the button on the Flash, the Flash plug-in active object obtains a script corresponding to the clicking operation of the button to transmit the script to the Flash proxy object via the communication window.
Then, the method 600 proceeds to step S602, wherein the Flash plug-in proxy object queries for an interface in the webpage process related to script execution, and obtains a script executing method according to the interface. After receiving the script corresponding to a button clicking operation, the Flash plug-in proxy object queries for and obtains an IHTML Window interface in the webpage process, the IHTML Window interface is an interface related to the script execution, and ExecScript in the IHTML Window interface is a function for executing the script, i.e., the ExecScript function provides the script executing method.
Then, the method 600 proceeds to step S603, wherein Flash plug-in proxy object executes the script according to the script executing method. According to the script executing method provided by the ExecScript function, the Flash plug-in proxy object execute the script corresponding to the above button clicking operation to make the webpage become black.
According to the inter-process script executing method provided by the present embodiment, after the active object transmits the script to the proxy object, the proxy object queries for an interface in the webpage process related to script execution, obtains the script executing method according to the interface, and thereby executes the script transmitted by the active object according to the script executing method, thereby achieving script execution between different processes and implementing control of the webpage running in the webpage process by the active object running in the independent process. Furthermore, in this method, with the proxy object running in the webpage process and the active object running in the independent process, a true ActiveX plug-in is made separate from the webpage process so that the ActiveX plug-in, when there is something wrong, will not affect the webpage process, and thereby improving stability of the browser; particularly when the ActiveX plug-in is confronted with a security issue such as occurrence of viruses, it will not exert an influence on the webpage process, thereby improving the safety of the browser; meanwhile, the webpage process, when being confronted with a problem, will not affect normal run of the ActiveX plug-in so that the stability of the plug-in is improved.
FIG. 7 illustrates a flowchart of an inter-process script executing method 700 according to another embodiment of the present invention. The method is adapted to the case that the Flash plug-in active object run in the independent process executes a script of the Flash plug-in proxy object run in the webpage process, namely, the method is a method of the webpage run in the webpage process controlling the Flash plug-in active object in the independent process. If the webpage desires to obtain information of the Flash (such as attributes like version number, path and URL) or the webpage desires to operate the Flash (e.g., the webpage desires to change a size of the Flash window), it can be implement by this method.
As shown in FIG. 7, the method 700 begins with step S701, wherein the Flash plug-in proxy object transmits a script to the Flash plug-in active object via the communication window. Take obtainment of the version number of the Flash as an example. Since the Flash plug-in in the webpage process is not a true Flash plug-in, it does not know the version number of the Flash, so the Flash plug-in proxy object cannot directly feed back the version number of the Flash to the webpage. After the independent process completes creation of the Flash plug-in active object, a variable of the Flash plug-in active object is notified to the webpage, a webpage developer writes in the webpage a script for obtaining the version number of Flash according to the variable, and the to-be-executed method in the script is intended to obtain the version number of the Flash. The Flash plug-in proxy object transmits the script for obtaining the version number of the Flash to the Flash plug-in active object via the communication window.
Then, the method 700 proceeds to step S702, wherein by invoking a scheduling interface of the Flash plug-in active object, the Flash plug-in proxy object enables the Flash plug-in active object to execute a method of obtaining a scheduling identification of a to-be-executed method in the script, and the Flash plug-in active object, after obtaining the scheduling identification, returns the scheduling identification to the Flash plug-in proxy object via the communication window. Specifically, first, the webpage parses the script, schedules a name of the to-be-executed method in the script to an IDispatch interface of the Flash plug-in proxy object, the IDispatch interface is the scheduling interface to invoke a function in a language program not supporting a virtual function table, the IDispatch interface has a GetIDsOfNames function and an Invoke function, wherein the GetIDsOfNames function provides a method of using the name of the method to return its scheduling ID, and the Invoke function provides an instruction of using the scheduling ID of the method to execute the method. Since the Flash plug-in proxy object cannot obtain the scheduling identification (ID) of the to-be-executed method in the script, the Flash plug-in proxy object invokes the IDispatch interface of the Flash plug-in active object via the communication window. The Flash plug-in active object is an object of the true Flash plug-in, the method provided by the GetIDsOfNames function of the Flash plug-in active object is invoked to enable the Flash plug-in active object to obtain the scheduling ID of the to-be-executed method, and then the Flash plug-in active object returns the scheduling ID to the Flash plug-in proxy object via the communication window.
Then, the method 700 proceeds to step S703, wherein Flash plug-in proxy object intercepts an instruction in the webpage process executing the to-be-executed method in the script and transmit the instruction to the Flash plug-in active object. Specifically, after the Flash plug-in proxy object obtains the scheduling ID of the to-be-executed method, the scheduling ID is fed back to the webpage, and the webpage process will execute an instruction of the Invoke(ID) in next step according to a conventional flow. However, according to the method here, the instruction of Invoke(ID) of the webpage process is intercepted, and the instruction of the Invoke(ID) is sent to the Flash plug-in active object.
Then, the method 700 proceeds to step S704, wherein the Flash plug-in active object executes the to-be-executed method in the script by executing the instruction, and then returns an execution result to the Flash plug-in proxy object. Specifically, the Flash plug-in active object executes the instruction of Invoke(ID) so as to execute the to-be-executed method in the script, obtain the version number of the Flash, and return the version number of the Flash to the Flash plug-in proxy object, and then the Flash plug-in proxy object will feed back the version number of the Flash to the webpage.
According to the inter-process script executing method provided by the present embodiment, after the proxy object transmits the script to the active object, the proxy object invokes the scheduling interface of the active object and obtain the scheduling ID of the to-be-executed method in the script; after the proxy object feeds back the scheduling ID to the webpage, the proxy object intercepts an instruction in the webpage process executing the to-be-executed method in the script and transmits the instruction to the active object, the active object executes the instruction and returns an execution result resulting from the execution of the instruction to the proxy object, thereby achieving script execution between different processes and implementing control of the active object running in the independent process by the webpage running in the webpage process. Furthermore, in this method, with the proxy object running in the webpage process and the active object running in the independent process, a true ActiveX plug-in is made separate from the webpage process so that the ActiveX plug-in, when there is something wrong, will not affect the webpage process, and thereby improving stability of the browser; particularly when the ActiveX plug-in is confronted with a security issue such as occurrence of viruses, it will not exert an influence on the webpage process, thereby improving the safety of the browser; meanwhile, the webpage process, when being confronted with a problem, will not affect normal run of the ActiveX plug-in so that the stability of the plug-in is improved.
FIG. 8 illustrates a structural block diagram of an inter-process script executing system according to an embodiment of the present invention. As shown in FIG. 8, the system comprises: a proxy object 810 running in a webpage process, an active object 820 running in an independent process and a pair of communication windows 830 respectively created in the proxy object 810 and the active object 820. The system is adapted to the case that the proxy object 810 executes a script of the active object 820, and the proxy object 810 and active object 820 communicate via the pair of communication windows 830.
The active object 820 comprises a transmitting module 821 which is configured to transmit the script to the proxy object 810 via the communication window 830. Taking a user clicking a button created on the Flash to trigger the webpage to become black as an example, when the user clicking the button on the Flash, the active object 820 obtains a script corresponding to the clicking operation of the button, and the transmitting module 821 transmits the script to the proxy object 810 via the communication window 830.
The proxy object 810 comprises: a query module 811 and an executing module 812. The query module 811 is configured to query for an interface related to script execution in the webpage process, and obtain a script executing method according to the interface; the executing module 812 is configured to execute the script according to the script executing method. After the proxy object 810 receives the script corresponding to a button clicking operation, the query module 811 queries for and obtains an IHTML Window interface in the webpage process, the IHTML Window interface is an interface related to the script execution, and ExecScript in the IHTML Window interface is a function for executing the script, i.e., the ExecScript function provides the script executing method. The executing module 812 executes the script corresponding to the above button clicking operation according to the script executing method provided by the ExecScript function to make the webpage become black.
The proxy object 810 running in the webpage process is created in place of the active object actually to be created by intercepting a procedure of the webpage process of the browser creating an active object; the active object 820 running in the independent process is created in the independent process independent from the webpage process when the webpage process activates the proxy object 810; the pair of communication windows 830 respectively created in the active object 820 and the proxy object 810 are used to enable the active object 820 and the proxy object 810 to communicate.
Furthermore, there are a plurality of proxy objects running in the webpage process, there are a plurality of active objects running in the independent process, there is a one-to-one correspondence between the proxy objects and the active objects, the communication window of the proxy object has a corresponding attribute with the communication window of the active object corresponding to the proxy object, and communication between the active object and the proxy object is achieved through the pair of communication windows.
According to the inter-process script executing system provided by the present embodiment, after the active object transmits the script to the proxy object, the proxy object queries for an interface related to script execution in the webpage process, and obtain a script executing method according to the interface, and thereby executes the script transmitted from the active object according to the script executing method, thereby achieving script execution between different processes and implementing control of the webpage running in the webpage process by the active object running in the independent process. Furthermore, in this system, with the proxy object running in the webpage process and the active object running in the independent process, a true ActiveX plug-in is made separate from the webpage process so that the ActiveX plug-in, when there is something wrong, will not affect the webpage process, and thereby improving stability of the browser; particularly when the ActiveX plug-in is confronted with a security issue such as occurrence of viruses, it will not exert an influence on the webpage process, thereby improving the safety of the browser; meanwhile, the webpage process, when being confronted with a problem, will not affect normal run of the ActiveX plug-in so that the stability of the plug-in is improved.
FIG. 9 illustrates a structural block diagram of an inter-process script executing system according to another embodiment of the present invention. As shown in FIG. 9, the system comprises: a proxy object 910 running in a webpage process, an active object 920 running in an independent process and a pair of communication windows 930 respectively created in the proxy object 910 and the active object 920. The system is adapted to the case that the active object 920 executes a script in the webpage process, and the proxy object 910 and active object 920 communicate via the pair of communication windows 930.
The proxy object 910 comprises a transmitting module 911, a scheduling identification obtaining module 912 and an intercepting module 913.
The transmitting module 911 is configured to transmit the script to the active object 920 via the communication window. After the independent process completes creation of the active object 920, a variable of the active object 920 is notified to the webpage, a webpage developer writes a script into the webpage according to the variable, and the transmitting module 911 transmits the script to the active object 920 via the communication window 930.
The scheduling identification obtaining module 912 is configured to obtain a scheduling identification of the to-be-executed method in the script by invoking a scheduling interface of the active object 920. Furthermore, the scheduling identification obtaining module 912 comprises an invoking module 914 and a scheduling identification receiving module 915, the invoking module 914 is configured to invoke the scheduling interface of the active object 920 and enable the active object 920 to execute a method of obtaining a scheduling identification of the to-be-executed in the script to obtain the scheduling identification; the scheduling identification receiving module 915 is configured to receive the scheduling identification returned by active object 920 via the communication window 930. Specifically, first, the webpage parses the script and schedules a name of the to-be-executed method in the script to an IDispatch interface of the proxy object 910, the IDispatch interface is the scheduling interface to invoke a function in a language program not supporting a virtual function table, the IDispatch interface has a GetIDsOfNames function and an Invoke function, wherein the GetIDsOfNames function provides a method of using the name of the method to return its scheduling ID, and the Invoke function provides an instruction of using the scheduling ID of the method to execute the method. As the proxy object 910 cannot obtain the scheduling identification (ID) of the to-be-executed method in the script, the scheduling module 914 invokes the IDispatch interface of the active object 920 via the communication window 930. The active object 920 is an object of the true plug-in, the method provided by the GetIDsOfNames function of the active object 920 is invoked to enable the active object 920 to obtain the scheduling ID of the to-be-executed method, and then the active object 920 returns the scheduling ID to the scheduling identification receiving module 915 via the communication window 930.
The intercepting module 913 is configured to intercept an instruction in the webpage process executing the to-be-executed method in the script and transmit the instruction to the active object 920. After the proxy object 910 obtains the scheduling ID of the to-be-executed method, the scheduling ID is fed back to the webpage, and the webpage process will execute an instruction of the Invoke(ID) in next step according to a conventional flow. However, the intercepting module 913 intercepts the instruction of Invoke(ID) of the webpage process and transmits the instruction of the Invoke(ID) to the active object 920.
The active object 920 comprises: a receiving module 921 and an executing module 922. The receiving module 921 is configured to receive the script transmitted by the transmitting module 911 of the proxy object 910 and the instruction transmitted by the intercepting module 913; the executing module 922 is configured to execute the to-be-executed method in the script by executing the instruction and then return an execution result to the proxy object 910. The executing module 922 executes the instruction of Invoke(ID) so as to execute the to-be-executed method in the script, and returns the execution result to the proxy object 910, and then the proxy object 910 will feed back the execution result to the webpage.
Furthermore, there are a plurality of proxy objects running in the webpage process, there are a plurality of active objects running in the independent process, there is a one-to-one correspondence between the proxy objects and the active objects, the communication window of the proxy object has a corresponding attribute to the communication window of the active object corresponding to the proxy object, and communication between the active object and the proxy object is achieved through the pair of communication windows.
According to the inter-process script executing system provided by the present embodiment, after the proxy object transmits the script to the active object, the proxy object invokes the scheduling interface of the active object and obtain the scheduling ID of the to-be-executed method in the script; after the proxy object feeds back the scheduling ID to the webpage, the proxy object intercepts an instruction in the webpage process executing the to-be-executed method in the script and transmits the instruction to the active object, the active object executes the instruction and returns an execution result resulting from the execution of the instruction to the proxy object, thereby achieving script execution between different processes and implementing control of the active object running in the independent process by the webpage running in the webpage process. Furthermore, in this system, with the proxy object running in the webpage process and the active object running in the independent process, a true ActiveX plug-in is made separate from the webpage process so that the ActiveX plug-in, when there is something wrong, will not affect the webpage process, and thereby improving stability of the browser; particularly when the ActiveX plug-in is confronted with a security issue such as occurrence of viruses, it will not exert an influence on the webpage process, thereby improving the safety of the browser; meanwhile, the webpage process, when being confronted with a problem, will not affect normal run of the ActiveX plug-in so that the stability of the plug-in is improved.
The present invention further provides a browse comprising any inter-process script executing system as described in the above embodiment.
FIG. 10 illustrates a flowchart of a method 1000 for executing a browser active object according to an embodiment of the present invention. In the method, the active object is an object corresponding to the ActiveX plug-in. As shown in FIG. 10, the method 1001 begins with step S1001, wherein before the active object is created, the type of the plug-in is obtained. Specifically, in the case that there is a ActiveX plug-in in the webpage, before the webpage process of the browser creates the ActiveX plug-in, a registration table is firstly queried to look up whether the registration table related to the plug-in declares the safety of a script of the plug-in. In the text here, plug-ins whose script safety has already been declared in the registration table are called the first type of plug-ins, e.g., a Flash plug-in belongs to the first type of plug-in; plug-ins whose script safety is not declared in the registration table are called the second type of plug-ins, e.g., a video play plug-in such as Baidu video or QVOD belongs to the second type of plug-in. Therefore, the type of the current plug-in can be known according to whether the registration table declares the safety of the script of the plug-in. If the current plug-in is the first type of plug-in, the flow will skip to step S1004; if the current plug-in is the second type of plug-in, step S1002 will be executed.
In step S1002, regarding the second type of plug-in, after the query is performed to the registration table, a pre-created active object is created.
After step S1002, the method 1000 proceeds to step S1003, a safety interface (IObjectSafety interface) of the pre-created active object the webpage process queried for is intercepted, and information indicating the plug-in is a safe plug-in is directly returned. Regarding the plug-in whose script safety is not declared in the registration table, the safety of the plug-in is determined by querying for the IObjectSafety of the pre-created active object of the plug-in, and the active object of the plug-in is truly created only when the plug-in is determined safe. Regarding the second type of plug-ins, if special processing is not performed here, information about warning or not creating an object will be displayed so that such type of plug-ins cannot operate normally. Therefore, when the webpage process queries for the IObjectSafety interface of the pre-created active object, the interface is intercepted, the IObjectSafety interface indicative of plug-in safety is returned thereto, INTERFACESAFE_FOR_UNTRUSTED_CALLER| INTERFACESAFE_FOR_UNTRUSTED_DATA is returned via GetInterfaceSafetyOptions to indicate that the plug-in is safe in script and safe in initialization, and thereby safety verification can be smoothly passed.
Referring to the above FIG. 2, a Web webpage and a proxy object run in the webpage process, and an empty webpage and an active object run in an independent process. Subsequent steps are further introduced below in detail with reference to FIG. 2.
After step S1003, the method 1000 proceeds to step S1004, wherein a procedure of a webpage process creating an active object is intercepted, a proxy object is created to replace an active object actually to be created, and the proxy object runs in the webpage process. Specifically, when the webpage process of the browser creates an active object, CoGetClassObject is intercepted, and a proxy object is created to replace the active object actually to be created. The proxy object includes a series of interfaces, including many standard interfaces that will be invoked by an IUnknown pointer, such as IOleObject and IViewObject. Then, when CreateInstance of IClassFactory is invoked, the created proxy object is returned to the IUnknown pointer of the webpage process, whereby creation of the proxy object is completed.
Then, the method 1000 proceeds to step S1005, wherein when the webpage process activates the proxy object, the active object actually to be created is created in an independent process independent from the webpage process, and the active object is run in the independent process. Specifically, when the webpage process activates the proxy object, information related to the plug-in such as attribute, URL and size is extracted, and then an independent process is created. First, a document, namely, an empty webpage, is created, an object is inserted therein, the independent process invokes CoGetClassObject, intercepts the CoGetClassObject and creates an active object. The active object comprises a series of interfaces, including many standard interfaces that will be invoked by an IUnknown pointer, such as IOleObject and IViewObject. Then, when CreateInstance of IClassFactory is invoked, the created active object is returned to the IUnknown pointer of the independent process, whereby creation of the active object is completed. The active object is an object which truly realizes the plug-in function.
During creation of the active object, the active object can still be normally created in the case that IWeBrowser2 interface may not be implemented for the first type off plug-ins. However, regarding the second type of plug-ins, creation of the active object will fail if the IWeBrowser2 interface is not implemented. The IWeBrowser 2 is implemented mainly to perform functions such as skipping (Navigate method), obtainment of an URL (Get_LocationURL method) and obtainment of a webpage (IHtmlDocument interface under get_document method). In order to perform these functions, when the plug-in is the second type of plug-in, the independent process queries for the IWeBrowser2 interface, it is intercepted, and then a self-created IWeBrowser2 interface is returned so as to create the active object of the above second type of plug-ins.
Then, the method 1000 proceeds to step S1006, wherein a communication window associated with a plug-in object is created respectively in the proxy object and the active object to enable the proxy object and the active object to communicate. Through the two communication windows, the two objects may interact with respect to size, focus, refresh, script and other information of the plug-in.
Then, the method 1000 proceeds to step S1007, wherein the active object and the proxy object communicate via the communication windows so that the proxy object invokes the active object and/or the active object invokes the proxy object so as to perform the function to be achieved by the plug-in.
Furthermore, if there is a nested structure in the Web webpage, i.e., the parent process runs a plurality of parent webpages and a plurality of subpages. Whenever the webpage process creates a subpage and its proxy object, a proxy object ID (may also include its URL) of the subpage and a proxy object ID (may also include its URL) of the subpage's parent webpage are sent to the independent process in which an active object is created according to a corresponding hierarchical structure. In this case, a plurality of proxy objects run in the webpage process, a plurality of active objects run in the dependent process, there is a one-to-one correspondence between the proxy and the active objects, the communication window of each proxy object has a corresponding attribute with the communication window of the active object corresponding to the proxy object, and communication between the active object and its corresponding proxy object is achieved through the two communication windows.
In this method, with the proxy object running in the webpage process and the active object running in the independent process, a true ActiveX plug-in is made separate from the webpage process so that the ActiveX plug-in, when there is something wrong, will not affect the webpage process, and thereby improving stability of the browser; particularly when the ActiveX plug-in is confronted with a security issue such as occurrence of viruses, it will not exert an influence on the webpage process, thereby improving the safety of the browser; meanwhile, the webpage process, when being confronted with a problem, will not affect normal run of the ActiveX plug-in so that the stability of the ActiveX plug-in is improved. Furthermore, the method is compatible with all ActiveX plug-ins, including plug-ins whose script safety is declared in the registration table and plug-ins whose script safety is not declared in the registration table.
In the above method, since the true ActiveX plug-in is separate from the webpage process, when a plug-in window is expected to be displayed in the original Web webpage, a plug-in window corresponding to the active object may be arranged at a location of the plug-in window of the original Web webpage, and the plug-in window corresponding to the active object may move and zoom without affecting the Web webpage.
Reference may be made to FIG. 6 which illustrates a flowchart of a method 600 of the proxy object executing the script of the active object according to the present invention.
FIG. 11 illustrates a flowchart of a method 1100 of a proxy object executing a script of the active object in the present invention. The method is adapted for the case that the proxy object of the second type of plug-in executes a script of the active object, i.e., the method is a method of the active object of the second type of plug-in running in the independent process controlling the webpage in the webpage process. As shown in FIG. 11, the method 1100 begins with step S1101, wherein the active object transmits the script to the proxy object via the communication window. Taking a user clicking a button created on the plug-in window to trigger the webpage to become black as an example, when the user clicks the button on the plug-in window, the active object of the plug-in obtains a script corresponding to the clicking operation of the button and transmits the script to the proxy object via the communication window.
Subsequently, the method 1100 proceeds to step S1102, wherein by invoking a scheduling interface of the proxy object, the active object enables the proxy object to execute a method of obtaining a scheduling identification of a to-be-executed method in the script, and the proxy object, after obtaining the scheduling identification, returns the scheduling identification to the active object via the communication window. Specifically, first, the independent process parses the script corresponding to the clicking operation of the button, obtains a to-be-executed method in the script to allow the webpage to become black, then schedules a name of the to-be-executed method in the script to an IDispatch interface of the active object, the IDispatch interface is the scheduling interface used to invoke a function in a language program not supporting a virtual function table, the IDispatch interface has a GetIDsOfNames function and an Invoke function, wherein the GetIDsOfNames function provides a method of using the name of the method to return its scheduling ID, and the Invoke function provides an instruction of using the scheduling ID of the method to execute the method. As the webpage corresponding to the active object is an empty webpage, and it does not have a method of enabling the webpage to become black, the active object cannot obtain the scheduling identification (ID) of the method in the script, and the active object invokes the IDispatch interface of the proxy object via the communication window. Since the Web webpage corresponding to the proxy object is a complete webpage, the method provided by the GetIDsOfNames function of the proxy object is invoked to enable the proxy object to obtain the scheduling ID of the to-be-executed method in the script, and then the proxy object returns the scheduling ID to the active object via the communication window.
Then, the method 1100 proceeds to step S1103, wherein the active object intercepts an instruction in the independent process executing the to-be-executed method in the script and transmits the instruction to the proxy object. Specifically, after the active object obtains the scheduling ID of the to-be-executed method, the scheduling ID is fed back to the webpage, and the independent process will execute an instruction of the Invoke(ID) in next step according to a conventional flow. However, according to the method here, the instruction of Invoke(ID) of the independent process is intercepted, and the instruction of the Invoke(ID) is sent to the proxy object.
Then, the method 1100 proceeds to step S1104, wherein the proxy object executes the to-be-executed method in the script by executing the instruction, and then returns an execution result to the active object. Specifically, the proxy object executes the instruction of Invoke(ID) so as to execute the to-be-executed method in the script to make the webpage become black, and return a notification message of the effect that the webpage becomes black to the active object.
Reference may be made to FIG. 7 which illustrates a flowchart of a method 700 of the active object executing the script of the proxy object according to the present invention.
FIG. 12 illustrates a structural block diagram of a system for executing a browser active object according to an embodiment of the present invention. As shown in FIG. 12, the system comprises a webpage process module 1210 and an independent process module 1220, a proxy object 1211 located in the webpage process module 1210 and an active object 1221 located in the independent process module 1220, and a communication window 1230 respectively created in the active object 1221 and the proxy object 1211. In the system, the active object is an active object corresponding to the ActiveX plug-in, and the ActiveX plug-in may be a video play plug-in such as Baidu video or QVOD, but the present invention is not limited to this.
The webpage process module 1210 is configured to, before the active object is created, obtain the type of the plug-in; and intercept a procedure of the webpage process creating an active object, and create a proxy object 1211 to replace an active object actually to be created according to the type of the plug-in, with the proxy object 1211 running in the webpage process. In the case that there is a ActiveX plug-in in the webpage, before the webpage process module 1210 creates the ActiveX plug-in, a registration table is firstly queried to look up whether the registration table related to the plug-in declares the safety of a script of the plug-in. If the registration table already declares the safety of the script of the plug-in, it is obtained that the plug-in is the first type of plug-in, e.g., a Flash plug-in belongs to the first type of plug-in; if the registration table does not declare the safety of the script of the plug-in, it is obtained that the plug-in is the second type of plug-in, e.g., a video play plug-in such as Baidu video or QVOD belongs to the second type of plug-in.
If the plug-in is determined as the second type of plug-in according to the type of the plug-in, the webpage process module 1210 is further configured to intercept the webpage process to query for the safety interface of the pre-created active object, and return information indicating the second type of plug-in is a safe plug-in. Regarding the second type of plug-in, after the query is performed to the registration table, a pre-created active object is created. When the webpage process module 1210 queries for the IObjectSafety interface of the pre-created active object, the webpage process module intercepts the interface, returns to it the IObjectSafety interface indicative of plug-in safety, returns INTERFACESAFE_FOR_UNTRUSTED_CALLER| INTERFACESAFE_FOR_UNTRUSTED_DATA via GetInterfaceSafetyOptions to indicate that the plug-in is safe in script and safe in initialization, and thereby safety verification can be smoothly passed.
When the webpage process of the browser creates an active object, the webpage process module 1210 intercepts CoGetClassObject, and creates a proxy object 1211 to replace the active object actually to be created. The proxy object 1211 includes a series of interfaces, including many standard interfaces that will be invoked by an IUnknown pointer, such as IOleObject and IViewObject. Then, when CreateInstance of IClassFactory is invoked, the created proxy object 1211 is returned to the IUnknown pointer of the webpage process, whereby creation of the proxy object 1211 is completed.
The independent process module 1220 is configured to, when the webpage process activates the proxy object 1211, create the active object 1211 actually to be created in the independent process independent from the webpage process according to the type of the plug-in, and run the active object 1221 in the independent process. When the webpage process activates the proxy object 1211, information related to the plug-in such as attribute, URL and size is extracted, and then an independent process is created. First, a document, namely, an empty webpage, is created, an object is inserted therein, the independent process invokes CoGetClassObject, the independent process module 1220 intercepts the CoGetClassObject and creates the active object 1221. The active object 1221 comprises a series of interfaces, including many standard interfaces that will be invoked by an IUnknown pointer, such as IOleObject and IViewObject. Then, when CreateInstance of IClassFactory is invoked, the created active object 1221 is returned to the IUnknown pointer of the independent process, whereby creation of the active object 1221 is completed. The active object 1221 is an object which truly realizes the plug-in function.
If the plug-in is determined as the second type of plug-in according to the type of the plug-in, the independent process module 1220 is further configured to intercept the independent process querying for the IWeBrowser2 interface, and return a self-created IWeBrowser2 interface so as to create the active object 1221 actually to be created.
The active object 1221 and the proxy object 1211 communicate via the communication window 1230 so that the proxy object 1211 invokes the active object 1221 and/or the active object 1221 invokes the proxy object 1211. The two objects may interact with respect to a size, focus, refresh, script and other information of the plug-in through the two communication windows.
Furthermore, if there is a nested structure in the Web webpage, the webpage process module 1210 comprises a plurality of proxy objects, the independent process module 1220 comprises a plurality of active objects, there is a one-to-one correspondence between the proxy objects and the active objects, the communication window of the proxy object has a corresponding attribute to the communication window of the active object corresponding to the proxy object, and communication between the active object and the proxy object is achieved through the two communication windows.
In this system, with the proxy object running in the webpage process and the active object running in the independent process, a true ActiveX plug-in is made separate from the webpage process so that the ActiveX plug-in, when there is something wrong, will not affect the webpage process, and thereby improving stability of the browser; particularly when the ActiveX plug-in is confronted with a security issue such as occurrence of viruses, it will not exert an influence on the webpage process, thereby improving the safety of the browser; meanwhile, the webpage process, when being confronted with a problem, will not affect normal run of the ActiveX plug-in so that the stability of the ActiveX plug-in is improved. Furthermore, the method is compatible with all ActiveX plug-ins, including plug-ins whose script safety is declared in the registration table and plug-ins whose script safety is not declared in the registration table.
FIG. 13 illustrates a structural block diagram of a system for executing a browser active object according to another embodiment of the present invention. On the basis of the system as shown in FIG. 12, the proxy object 1311 located in the webpage process module 1310 and the active object 1321 located in the independent process module 1320 in the system may interact scripts therebetween, i.e., the proxy object 1311 may execute the script of the active object 1321, and the active object 1321 may execute the script of the proxy object 1311. The proxy object 1311 and the active object 13211 in the system as shown in FIG. 13 are created when the webpage process module 1310 knows that the plug-in is the first type of plug-in.
The active object 1321 comprises a first transmitting module 1322. The proxy object 1311 comprises a query module 1312 and a first executing module 1313.
The first transmitting module 1322 is configured to transmit the script to the proxy object 1311 via the communication window 1330. Taking a user clicking a button created on the plug-in to trigger the webpage to become black as an example, when the user clicks the button on the plug-in, the active object 1321 obtains a script corresponding to a clicking operation of the button, and the first transmitting module 1322 transmits the script to the proxy object 1311 via the communication window 1330.
The first query module 1312 is configured to query for an interface related to script execution in the webpage process, and obtain a script executing method according to the interface; the first executing module 1313 is configured to execute the script according to the script executing method. After the proxy object 1311 receives the script corresponding to the above button clicking operation, the first query module 1312 queries for and obtains an IHTML Window interface in the webpage process, the IHTML Window interface is an interface related to the script execution, and ExecScript in the IHTML Window interface is a function for executing the script, i.e., the ExecScript function provides the script executing method. The first executing module 1313 executes the script corresponding to the above button clicking operation according to the script executing method provided by the ExecScript function to make the webpage become black.
The proxy object 1311 further comprises a third transmitting module 1314, a third scheduling identification obtaining module 1315, a third intercepting module 1316, wherein the third scheduling identification obtaining module 1315 further comprises a third invoking module 1317 and a third scheduling identification receiving module 1318. The active object 1321 further comprises a third receiving module 1323 and a third executing module 1324.
The third transmitting module 1314 is configured to transmit the script to the active object 1321 via the communication window 1330. Take obtainment of the version number of the plug-in as an example. Since the proxy object 1311 in the webpage process is not a true plug-in object, it does not know the version number of the plug-in, so the proxy object 1311 cannot directly feed back the version number of the plug-in to the webpage. After the independent process completes creation of the active object 1321, a variable of the active object 1321 is notified to the webpage, a webpage developer writes in the webpage a script for obtaining the version number of plug-in according to the variable, and the to-be-executed method in the script is intended to obtain the version number of the plug-in. The third transmitting module 1314 transmits the script for obtaining the version number of the plug-in to the active object 1321 via the communication window 1330.
The third scheduling identification obtaining module 1315 is configured to obtain a scheduling identification of the to-be-executed method in the script by invoking the scheduling interface of the active object 1321, wherein the third invoking module 1317 is configured to invoke the scheduling interface of the active object 1321 and enable the active object 1321 to execute a method of obtaining a scheduling identification of the to-be-executed in the script to obtain the scheduling identification; the third scheduling identification receiving module 1318 is configured to receive the scheduling identification returned by the active object 1321 via the communication window 1330. First, the webpage process module 1310 parses the script and schedules a name of the to-be-executed method in the script to an IDispatch interface of the proxy object 1311. As the proxy object 1311 cannot obtain the scheduling identification (ID) of the to-be-executed method in the script, the third scheduling module 1317 of the proxy object 1311 invokes the IDispatch interface of the active object 1321 via the communication window 1330. The active object 1321 is an object of the true plug-in, the method provided by the GetIDsOfNames function of the active object 1321 is invoked to enable the active object 1321 to obtain the scheduling ID of the to-be-executed method, and then the active object 1321 returns the scheduling ID to the third scheduling identification receiving module 1318 of the proxy object 1311 via the communication window 1330.
The third intercepting module 1316 is configured to intercept an instruction in the webpage process executing the to-be-executed method in the script and transmit the instruction to the active object 1321. After the proxy object 1311 obtains the scheduling ID of the to-be-executed method, the scheduling ID is fed back to the webpage, and the webpage process will execute an instruction of the Invoke(ID) in next step according to a conventional flow. However, the third intercepting module 1316 intercepts the instruction of Invoke(ID) of the webpage process and transmits the instruction of the Invoke(ID) to the active object 1321.
The third receiving module 1323 is configured to receive the script transmitted by the third transmitting module 1314 of the proxy object 1311 and the instruction transmitted by the third intercepting module 1316; the third executing module 1324 is configured to execute the to-be-executed method in the script by executing the instruction and then return an execution result to the proxy object 1311.
According to the functions performed by the above function modules, the proxy object of the first type of plug-in may execute the script of the active object, and the active object may execute the script of the proxy object, thereby achieving control of the webpage running in the webpage process by the active object running in the independent process and control of the active object running in the independent process by the webpage running in the webpage process.
Reference may be made to FIG. 5 which illustrates a structural block diagram of a system for executing a browser active object according to a further embodiment of the present invention.
The present invention further provides a browse comprising the system for executing any browser active object according to the above embodiment.
The embodiments of respective components of the present invention can be carried out in hardware, or in software modules run on one or more processors, or in the combination thereof. The skilled person in the art should understand that a microprocessor or a digital signal processor (DSP) can be used in practice to implement some or all functions of some or all components in the device for prompting information about an e-mail according to the embodiment of the present invention. The present invention can also be carried out as part or all of the device or device program (e.g., computer program and computer program product) for performing the method described here. Such a program for carrying out the present invention can be stored on a computer readable medium, or may have the form of one or more signals. Such signals can be downloaded from the internet website, or be provided on a carrier signal, or be provided in any other forms.
For example, FIG. 14 schematically illustrates a user terminal that may implement the present invention. The user terminal conventionally comprises a processor 1410 and a computer program product or computer-readable medium in the form of a memory 1420. The memory 1420 may be a flash memory, EEPROM (Electrically Erasable Programmable Read-Only Memory), EPROM, hard disk or ROM-like electronic memory. The memory 1420 has a storage space 1430 for a program code 1431 for executing any step of the above method. For example, the storage space 1430 for the program code may comprise program codes 1431 respectively for implementing steps of the above method. These program codes may be read from one or more computer program products or written into the one or more computer program products. These computer program products comprise program code carriers such as hard disk, compact disk (CD), memory card or floppy disk. Such computer program products are usually portable or fixed memory units as shown in FIG. 15. The memory unit may have a storage section, a storage space or the like arranged in a similar way to the memory 1420 in the user terminal of FIG. 14. The program code may for example be compressed in a suitable form. Usually, the memory unit includes a computer-readable code 1431′, namely, a code readable by a processor for example similar to 1410. When these codes are run by the server, the server is caused to execute steps of the method described above.
Reference herein to "one embodiment", "an embodiment", or to "one or more embodiments" means that a particular feature, structure, or characteristic described in connection with the embodiments is included in at least one embodiment of the invention. Further, it is noted that instances of the phrase "in one embodiment" herein are not necessarily all referring to the same embodiment.
The description as provided here describes a lot of specific details. However, it is appreciated that embodiments of the present invention may be implemented in the absence of these specific details. Some embodiments do not specify detail known methods, structures and technologies to make the description apparent.
It should be noted that the above embodiment illustrate the present invention but are not intended to limit the present invention, and those skilled in the art may design alternative embodiments without departing from the scope of the appended claims. In claims, any reference signs placed in parentheses should not be construed as limiting the claims. The word "comprising" does not exclude the presence of elements or steps not listed in a claim. The word "a" or "an" preceding an element does not exclude the presence of a plurality of such elements. The present invention may be implemented by virtue of hardware including several different elements and by virtue of a properly-programmed computer. In the apparatus claims enumerating several units, several of these units can be embodied by one and the same item of hardware. The usage of the words first, second and third, et cetera, does not indicate any ordering. These words are to be interpreted as names.
In addition, it should be noted that the language used in the specification has been principally selected for readability and instructional purposes, and may not have been selected to delineate or circumscribe the inventive subject matter. Therefore, those having ordinary skill in the art appreciate that many modifications and variations without departing from the scope and spirit of the appended claims are obvious. The disclosure of the present invention is intended to be illustrative, but not limiting, of the scope of the invention, which is set forth in the claims. Embodiments according to the present invention disclose A1: An inter-process script executing method, the method being adapted to a case that a proxy object running in a webpage process executes a script of an active object running in an independent process, the proxy object and the active object communication via pre-created communication windows, the method comprising: the active object transmits the script to the proxy object via the communication window; the proxy object queries for an interface in the webpage process related to script execution, and obtains a script executing method according to the interface; the proxy object executes the script according to the script executing method. A2. The method according to A1, the proxy object running in the webpage process is created in place of an active object actually to be created by intercepting a procedure of the webpage process of the browser creating an active object; the active object running in the independent process is created in the independent process independent from the webpage process when the webpage process activates the proxy object; the communication windows respectively created in the active object and the proxy object are used to enable the active object and the proxy object to communicate. A3. The method according to A2, there are a plurality of proxy objects running in the webpage process, there are a plurality of active objects running in the independent process, there is a one-to-one correspondence between the proxy objects and the active objects, the communication window of the proxy object has a corresponding attribute to the communication window of the active object corresponding to the proxy object, and communication between the active object and the proxy object is achieved through the two communication windows.
Embodiments according to the present invention further disclose B4: an inter-process script executing method, the method being adapted to a case that an active object running in an independent process executes a script of a proxy object running in a webpage process, the proxy object and the active object communication via pre-created communication windows, the method comprising: the proxy object transmits the script to the active object via the communication window; the proxy object obtains a scheduling identification of a to-be-executed method in the script by invoking a scheduling interface of the active object; the proxy object intercepts an instruction in the webpage process executing the to-be-executed method in the script and transmits the instruction to the active object; the active object executes the to-be-executed method in the script by executing the instruction, and then returns an execution result to the proxy object. B5. The method according to B4, the step of the proxy object obtaining a scheduling identification of a to-be-executed method in the script by invoking a scheduling interface of the active object comprises: by invoking the scheduling interface of the active object, the proxy object enables the active object to execute a method of obtaining a scheduling identification of a to-be-executed method in the script, and the active object returns the scheduling identification to the proxy object via the communication window. B6. The method according to B4 or B5, the proxy object running in the webpage process is created in place of an active object actually to be created by intercepting a procedure of the webpage process of the browser creating an active object; the active object running in the independent process is created in the independent process independent from the browser webpage process when the webpage process activates the proxy object; the communication windows respectively created in the active object and the proxy object are used to enable the active object and the proxy object to communicate. B7. The method according to B6, there are a plurality of proxy objects running in the webpage process, there are a plurality of active objects running in the independent process, there is a one-to-one correspondence between the proxy objects and the active objects, the communication window of the proxy object has a corresponding attribute with the communication window of the active object corresponding to the proxy object, and communication between the active object and the proxy object is achieved through the two communication windows.
Embodiments according to the present invention disclose C8: an inter-process script executing system comprising: a proxy object running in a webpage process, an active object running in an independent process and a pair of communication windows respectively created in the proxy object and the active object; the system is adapted to a case that the proxy object executes a script of the active object, and the proxy object and active object communicate via the pair of communication windows; the active object comprises a transmitting module which is configured to transmit a script to the proxy object via the communication window; the proxy object comprises: a query module configured to query for an interface related to script execution in the webpage process, and obtain a script executing method according to the interface; an executing module configured to execute the script according to the script executing method. C9. The system according to C8, the proxy object running in the webpage process is created in place of an active object actually to be created by intercepting a procedure of the webpage process of the browser creating an active object; the active object running in the independent process is created in the independent process independent from the webpage process when the webpage process activates the proxy object; the pair of communication windows respectively created in the active object and the proxy object are used to enable the active object and the proxy object to communicate. C10. The system according to C9, there are a plurality of proxy objects running in the webpage process, there are a plurality of active objects running in the independent process, there is a one-to-one correspondence between the proxy objects and the active objects, the communication window of the proxy object has a corresponding attribute to the communication window of the active object corresponding to the proxy object, and communication between the active object and the proxy object is achieved through the pair of communication windows.
Embodiments according to the present invention disclose D11: a browser comprising the inter-process script executing system according to any one of C8 to C10.
Embodiments according to the present invention disclose E12: an inter-process script executing system comprising: a proxy object running in a webpage process, an active object running in an independent process and a pair of communication windows respectively created on the proxy object and the active object; the system is adapted to a case that the active object executes a script in the webpage process, and the proxy object and active object communicate via the pair of communication windows; the proxy object comprises a transmitting module configured to transmit the script to the active object via the communication window; a scheduling identification obtaining module configured to obtain a scheduling identification of the to-be-executed method in the script by invoking the scheduling interface of the active object; an intercepting module configured to intercept an instruction in the webpage process executing the to-be-executed method in the script and transmit the instruction to the active object; the active object comprises: a receiving module configured to receive the script transmitted by a transmitting module in the proxy object and an instruction transmitted by an intercepting module; an executing module configured to execute the to-be-executed method in the script by executing the instruction and then return an execution result to the proxy object. E13. The system according to E12, the scheduling identification obtaining module comprises: an invoking module configured to invoke a scheduling interface of the active object and enable the active object to execute a method of obtaining a scheduling identification of the to-be-executed in the script to obtain the scheduling identification; a scheduling identification receiving module configured to receive the scheduling identification returned by active object via the communication window. E14. The system according to E12 or E13, the proxy object running in the webpage process is created in place of an active object actually to be created by intercepting a procedure of the webpage process of the browser creating an active object; the active object running in the independent process is created in the independent process independent from the browser webpage process when the webpage process activates the proxy object; the communication windows respectively created in the active object and the proxy object are used to enable the active object and the proxy object to communicate. E15. The system according to E14, there are a plurality of proxy objects running in the webpage process, there are a plurality of active objects running in the independent process, there is a one-to-one correspondence between the proxy objects and the active objects, the communication window of the proxy object has a corresponding attribute with the communication window of the active object corresponding to the proxy object, and communication between the active object and the proxy object is achieved through the pair of communication windows.
Embodiments according to the present invention disclose F16: a browser comprising the inter-process script executing system according to any one of E12 to E15.
Embodiments according to the present invention further disclose G1: a method of executing a browser active object, the active object being an object corresponding to ta plug-in, the method comprising: obtaining a type of the plug-in before the active object is created; and intercepting a procedure of a webpage process creating the active object, and creating a proxy object to replace an active object actually to be created according to the type of the plug-in, with the proxy object running in the webpage process; when the webpage process activates the proxy object, the active object actually to be created is created in an independent process independent from the webpage process, and the active object is run in the independent process; the active object and the proxy object communicating via the communication windows, thus realizing that the proxy object invokes the active object and/or the active object invokes the proxy object. E2. The method according to E1, if the plug-in is determined as a second type of plug-in according to the type of the plug-in, the step of intercepting a procedure of the webpage process creating the active object further comprises: intercepting the webpage process to query for a safety interface of a pre-created active object, and returning information indicating the second type of plug-in is a safe plug-in. E3. The method according to E2, the step of creating the active object actually to be created in an independent process independent from the webpage process comprises: intercepting the independent process querying for an IWeBrowser2 interface, and returning a self-created IWeBrowser2 interface so as to create the active object actually to be created. E4. The method according to E1 or E2 or E3, communication between the active object and the proxy object via the communication windows comprises a script executing procedure between the active object and the proxy object. E5. The method according to E4, the script executing procedure between the active object and the proxy object comprises execution of the script of the active object by the proxy object, if the plug-in is determined to be a first type of plug-in according to the type of the plug-in, the procedure of the proxy object executing the script of the active object comprises: the active object transmits the script to the proxy object via the communication window; the proxy object queries for an interface in the webpage process related to script execution, and obtains a script executing method according to the interface; the proxy object executes the script according to the script executing method. E6. The method according to E4, the script executing procedure between the active object and the proxy object comprises execution of the script of the active object by the proxy object, if the plug-in is determined to be the second type of plug-in according to the type of the plug-in, the procedure of proxy object executing the script of the active object comprises: the active object transmits the script to the proxy object via the communication window; the active object obtains a scheduling identification of the to-be-executed method in the script by invoking a scheduling interface of the proxy object; the active object intercepts an instruction in the independent process executing the to-be-executed method in the script and transmits the instruction to the proxy object; the proxy object executes the to-be-executed method in the script by executing the instruction, and then returns an execution result to the active object. E7. The method according to E6, the step of the active object obtaining a scheduling identification of the to-be-executed method in the script by invoking a scheduling interface of the proxy object comprises: by invoking a scheduling interface of the proxy object, the active object enables the proxy object to execute a method of obtaining a scheduling identification of a to-be-executed method in the script, and then the proxy object returns the scheduling identification to the active object via the communication window. E8. The method according to E4, the script executing procedure between the active object and the proxy object comprises execution of the script of the proxy object by the active object, and the procedure of the active object executing the script of the proxy object comprises: the proxy object transmits the script to the active object via the communication window; the proxy object obtains a scheduling identification of a to-be-executed method in the script by invoking a scheduling interface of the active object; the proxy object intercepts an instruction in the webpage process executing the to-be-executed method in the script and transmits the instruction to the active object; the active object executes the to-be-executed method in the script by executing the instruction, and then returns an execution result to the proxy object. E9. The method according to E8, the step of the proxy object obtaining a scheduling identification of a to-be-executed method in the script by invoking a scheduling interface of the active object comprises: by invoking the scheduling interface of the active object, the proxy object enables the active object to execute a method of obtaining a scheduling identification of a to-be-executed method in the script, and the active object returns the scheduling identification to the proxy object via the communication window. E10. The method according to any one of E1 to E9, there are a plurality of proxy objects running in the webpage process, there are a plurality of active objects running in the independent process, there is a one-to-one correspondence between the proxy objects and the active objects, the communication window of the proxy object has a corresponding attribute with the communication window of the active object corresponding to the proxy object, and communication between the active object and the proxy object is achieved through the two communication windows. E11. The method according to E5, the first type of plug-in is a Flash plug-in. E12. The method according to E6, the second type of plug-in is Baidu video or QVOD.
Embodiments according to the present invention further disclose F13: a system of executing a browser active object, comprising: a webpage process module configured to obtain a type of the plug-in before the active object is created; and intercept a procedure of a webpage process creating the active object, and create a proxy object to replace an active object actually to be created according to the type of the plug-in, with the proxy object running in the webpage process; an independent process module configured to, when the webpage process activates the proxy object, create the active object actually to be created in an independent process independent from the webpage process according to the type of the plug-in, with the active object running in the independent process; the proxy object is located in the webpage process module, the active object located in the independent process module, and a communication window is respectively created in the active object and the proxy object, the active object is an active object corresponding to the plug-in, the active object and the proxy object communicate via the communication windows, thus realizing that the proxy object invokes the active object and/or the active object invokes the proxy object. F14. The system according to F13, if the plug-in is determined as a second type of plug-in according to the type of the plug-in, the webpage process module is further configured to intercept the webpage process to query for a safety interface of a pre-created active object, and return information indicating the second type of plug-in is a safe plug-in. F15. The system according to F14, the independent process module is further configured to intercept the independent process querying for an IWeBrowser2 interface, and return a self-created IWeBrowser2 interface so as to create the active object actually to be created. F16. The system according to F13, if the plug-in is determined to be a first type of plug-in according to the type of the plug-in, the proxy object comprises: a first transmitting module configured to transmit the script to the proxy object via the communication window; the proxy object comprises: a first query module configured to query for an interface in the webpage process related to script execution, and obtain a script executing method according to the interface; a first executing module configured to execute the script according to the script executing method. F17. The system according to F14 or F15, if the plug-in is determined to be the second type of plug-in according to the type of the plug-in, the active object comprises: a second transmitting module configured to transmit the script to the proxy object via the communication window; a second scheduling identification obtaining module configured to obtain a scheduling identification of the to-be-executed method in the script by invoking a scheduling interface of the proxy object; a second intercepting module configured to intercept an instruction in the independent process executing the to-be-executed method in the script and transmit the instruction to the proxy object; the proxy object comprises: a second receiving module configured to receive the script transmitted by the second transmitting module of the active object and the instruction transmitted by the second intercepting module; a second executing module configured to execute the to-be-executed method in the script by executing the instruction and then return an execution result to the proxy object. F18. The system according to F17, the second scheduling identification obtaining module comprises: a second invoking module configured to invoke a scheduling interface of the proxy object and enable the proxy object to execute a method of obtaining a scheduling identification of a to-be-executed method in the script to obtain the scheduling identification; a second scheduling identification receiving module configured to receive the scheduling identification returned by the proxy object via the communication window. F19. The system according to F.13, or F15 or FIG. 15, the proxy object comprises: a third transmitting module configured to transmit the script to the active object via the communication window; a third scheduling identification obtaining module configured to obtain a scheduling identification of a to-be-executed method in the script by invoking a scheduling interface of the active object; a third intercepting module configured to intercept an instruction in the webpage process executing the to-be-executed method in the script and transmit the instruction to the active object; the active object comprises: a third receiving module configured to receive a script transmitted by the third transmitting module in the proxy object and an instruction transmitted by the third intercepting module; a third executing module configured to execute the to-be-executed method in the script by executing the instruction, and then return an execution result to the proxy object. F.20 The third scheduling identification obtaining module comprises: a third invoking module configured to, by invoking the scheduling interface of the active object, enable the active object to execute a method of obtaining a scheduling identification of a to-be-executed method in the script to obtain the scheduling identification; a third scheduling identification receiving module configured to receive the scheduling identification returned by the active object via the communication window. F21. The system according to any one of F13 to F20, the webpage process module comprises a plurality of proxy objects, the independent process module comprises a plurality of active objects, there is a one-to-one correspondence between the proxy objects and the active objects, the communication window of the proxy object has a corresponding attribute to the communication window of the active object corresponding to the proxy object, and communication between the active object and the proxy object is achieved through the two communication windows. F22. The system according to F16, the first type of plug-in is a Flash plug-in. F23. The system according to F17, the second type of plug-in is Baidu video or QVOD.
Embodiments according to the present invention further disclose G24: a browser comprising a system for executing a browser active object according to any one of F13 to F23.
1. A method for executing an active object of a browser, the active object being an object corresponding to an ActiveX plug-in, the method comprising:
before the active object is created, intercepting, by at least one processor, a webpage process to query for a safety interface of a pre-created active object corresponding to the active object, and returning information indicating the ActiveX plug-in is a safe plug-in;
intercepting, by the at least one processor, a procedure of the webpage process creating the active object, and creating a proxy object to replace the active object, with the proxy object running in the webpage process;
when the webpage process activates the proxy object, creating, by the at least one processor, the active object in an independent process independent from the webpage process, with the active object running in the independent process;
creating, by the at least one processor, a communication window for each of the active object and the proxy object; and
the active object and the proxy object communicating, by the at least one processor, via the communication windows, and the proxy object invoking the active object and/or the active object invoking the proxy object.
2. The method according to claim 1, wherein
creating the active object in the independent process independent from the webpage process comprises: intercepting the independent process to query for an IWeBrowser2 interface, and returning a self-created IWeBrowser2 interface to create the active object.
3. The method according to claim 1, wherein the communication between the active object and the proxy object via the communication windows comprises a script executing procedure between the active object and the proxy object.
4. The method according to claim 3, wherein the script executing procedure between the active object and the proxy object comprises an execution of a script of the active object by the proxy object, and the procedure of the proxy object executing the script of the active object comprises:
the active object transmitting the script to the proxy object via the communication window;
the active object obtaining a scheduling identification of a to-be-executed method in the script by invoking a scheduling interface of the proxy object;
the active object intercepting an instruction in the independent process executing the to-be-executed method in the script and transmitting the instruction to the proxy object; and
the proxy object executing the to-be-executed method in the script by executing the instruction and then returning an execution result to the active object.
5. The method according to claim 4, wherein the active object obtaining the scheduling identification of the to-be-executed method in the script by invoking the scheduling interface of the proxy object comprises:
by invoking the scheduling interface of the proxy object, the active object enabling the proxy object to execute a method of obtaining the scheduling identification of the to-be-executed method in the script, and then the proxy object returning the scheduling identification to the active object via the communication window.
6. The method according to claim 3, wherein the script executing procedure between the active object and the proxy object comprises an execution of a script of the proxy object by the active object, and the procedure of the active object executing the script of the proxy object comprises:
the proxy object transmitting the script to the active object via the communication window;
the proxy object obtaining a scheduling identification of a to-be-executed method in the script by invoking the scheduling interface of the active object;
the proxy object intercepting an instruction in the webpage process executing the to-be-executed method in the script and transmitting the instruction to the active object; and
the active object executing the to-be-executed method in the script by executing the instruction, and then returning an execution result to the proxy object.
7. The method according to claim 6, wherein the proxy object obtaining the scheduling identification of the to-be-executed method in the script by invoking the scheduling interface of the active object comprises:
by invoking the scheduling interface of the active object, the proxy object enabling the active object to execute a method of obtaining the scheduling identification of the to-be-executed method in the script, and then the active object returning the scheduling identification to the proxy object via the communication window.
there are a plurality of proxy objects running in the webpage process, there are a plurality of active objects running in the independent process, there is a one-to-one correspondence between the proxy objects and the active objects, the communication window of the proxy object has a corresponding attribute with the communication window of the active object corresponding to the proxy object, and communication between the active object and the proxy object is achieved through the two communication windows.
9. The method according to claim 1, wherein the ActiveX plug-in is Baidu video or QVOD.
10. A system for executing an active object of a browser, comprising at least one processor to execute:
before the active object is created, a webpage process to intercept a webpage process to query for a safety interface of a pre-created active object corresponding to the active object, and return information indicating the active object is a safe plug-in;
intercept a procedure of the webpage process creating the active object, and create a proxy object to replace the active object, with the proxy object running in the webpage process;
when the webpage process activates the proxy object, an independent process to create the active object independent from the webpage process, with the active object running in the independent process; and
the proxy object is located in the webpage process, the active object is located in the independent process, and a communication window is created for each of the active object and the proxy object, the active object being an active object corresponding to the ActiveX plug-in, the active object and the proxy object communicating via the communication windows, and the proxy object invoking the active object and/or the active object invoking the proxy object.
11. The system according to claim 10, wherein the independent process is further configured to intercept the independent process to query for an IWeBrowser2 interface, and return a self-created IWeBrowser2 interface to create the active object.
12. The system according to claim 10, wherein the active object comprises:
a first transmitting module to transmit a script to the proxy object via the communication window;
a first scheduling identification obtaining module to obtain a scheduling identification of a to-be-executed method in the script by invoking a scheduling interface of the proxy object; and
a first intercepting module to intercept an instruction in the independent process executing the to-be-executed method in the script and transmit the instruction to the proxy object;
the proxy object comprises:
a first receiving module to receive the script transmitted by the first transmitting module of the active object and the instruction transmitted by the first intercepting module; and
a first executing module to execute the to-be-executed method in the script by executing the instruction and then return an execution result to the active object.
13. The system according to claim 12, wherein the first scheduling identification obtaining module comprises:
a first invoking module to invoke the scheduling interface of the proxy object and enable the proxy object to execute a method of obtaining a scheduling identification of the to-be-executed method in the script to obtain the scheduling identification; and
a first scheduling identification receiving module to receive the scheduling identification returned by the proxy object via the communication window.
14. The system according to claim 10, wherein
a second transmitting module to transmit a script to the active object via the communication window;
a second scheduling identification obtaining module to obtain a scheduling identification of a to-be-executed method in the script by invoking a scheduling interface of the active object; and
a second intercepting module to intercept an instruction in the webpage process executing the to-be-executed method in the script and transmit the instruction to the active object;
the active object comprises:
a second receiving module to receive the script transmitted by the second transmitting module of the proxy object and the instruction transmitted by the second intercepting module; and
a second executing module to execute the to-be-executed method in the script by executing the instruction and then return an execution result to the proxy object.
15. The system according to claim 14, wherein the second scheduling identification obtaining module comprises:
a second invoking module to invoke the scheduling interface of the active object and enable the active object to execute a method of obtaining a scheduling identification of the to-be-executed method in the script to obtain the scheduling identification; and
a second scheduling identification receiving module to receive the scheduling identification returned by the active object via the communication window.
16. The system according to claim 10, wherein the webpage process comprises a plurality of proxy objects, the independent process comprises a plurality of active objects, there is a one-to-one correspondence between the proxy objects and the active objects, the communication window of the proxy object has a corresponding attribute with the communication window of the active object corresponding to the proxy object, and communication between the active object and the proxy object is achieved through the two communication windows.
17. The system according to claim 10, wherein the ActiveX plug-in is Baidu video or QVOD.
18. A non-transitory computer readable medium having instructions stored thereon that, when executed by at least one processor, cause the at least one processor to perform operations for executing an active object of a browser, the active object being an object corresponding to an ActiveX plug-in, the operations comprising:
before the active object is created, intercepting a webpage process to query for a safety interface of a pre-created active object corresponding to the active object, and returning information indicating the ActiveX plug-in is a safe plug-in;
intercepting a procedure of the webpage process creating the active object, and creating a proxy object to replace the active object, with the proxy object running in the webpage process;
when the webpage process activates the proxy object, creating the active object in an independent process independent from the webpage process, with the active object running in the independent process;
creating a communication window for each of the active object and the proxy object; and
the active object and the proxy object communicating via the communication windows, and the proxy object invoking the active object and/or the active object invoking the proxy object.
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Patent number: 10218767
Filed: Aug 30, 2013
Assignee: Beijing Qihoo Technology Company Limited (Beijing)
Inventors: Jinwei Li (Beijing), Yuesong He (Beijing), Zhi Chen (Beijing), Yu Fu (Beijing), Ming Li (Beijing), Huan Ren (Beijing)
Primary Examiner: Kostas J Katsikis
Current U.S. Class: Security Kernel Or Utility (713/164)
International Classification: H04L 29/08 (20060101); H04L 29/06 (20060101); G06F 9/54 (20060101); | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,731 |
Q: Reutilizar un objeto en otra funcion, sin pasarlo como argumento Hola a todos y perdón si no consigo explicarme bien.
Mi código es similar al que pego debajo.
La funcion getRestaurants(allData) crea una tabla dinamica, iterando sobre el objeto "allData".
Donde "allData" son distintos objetos que obtengo con llamadas a una API (es decir, que dependiendo de una función anterior, "allData" es un objeto u otro).
Y quiero conseguir , en la función goBack(), llamar de nuevo la funcion getRestaurants(allData) , para que se vuelva a generar la misma tabla.
En principio no es posible pasar el objeto como argumento de de goBack (no se podria hacer goBack(allData) ), porque crea otros problemas.
No se si queda claro, sino, decidme y veo si puedo mejorar el ejemplo
Ejemplo de codigo :
var allDataGlobal = {};
function getRestaurants(allData) {
for (i = 0, i < allData.lenght, i++) {
"genera una tabla dinamica con los datos del objeto"
allDataGlobal = allData;
console.log(allDataGlobal) //muestra el objeto correcto)
}}
function goBack() {
console.log(allDataGlobal) // muestra el objeto vacio
if ("caso A") {
myFunction1();
} if else {
getrestaurants(allData);
} else { otherFunction() }}
Añado la función fetch (el "cityId" varia segun el value de un Select anterior) :
function searchCity(cityId) {
var url = "https://developers.zomato.com/api/v2.1/search?entity_id=" + cityId + "&entity_type=city";
console.log(url);
fetch(url, {
method: "GET",
headers: {
"user-key": "myKey"
}
})
.then(function (res) {
return res.json();
})
.then(function (data) {
var allData = data.restaurants;
getRestaurans(allData);
console.log(allData[0].restaurant.location.address);
})
.catch(function (error) {
console.log(error)
});
};
A: Como te dijeron en los comentario, deberías almacenar los datos en una variable global una vez recibes la respuesta de la API, imaginando que haces la obtención de la información por medio de un AJAX el código sería algo así:
let allData;
let allDataGlobal;
$.ajax({
url: "xxx",
method: "GET",
success: function(response){
allData = response;
allDataGlobal = allData;
getRestaurants(allData); //Opcionalmente
}
})
function getRestaurants(allData) {
for (i = 0, i < allData.lenght, i++) {
"genera una tabla dinamica con los datos del objeto"
}
}
function goBack() { //El siguiente código correría mejor con un switch y no con if anidados
console.log(allDataGlobal);
if ("caso A") {
myFunction1();
} else if {
getrestaurants(allData);
} else {
otherFunction()
}
}
A: Simple como:
async function searchByCityId(cityId){
try{
let response = await (await fetch(url, { method: "GET", headers: { "user-key": "myKey" } })).json(),
restaurants = getRestaurants(response);
sessionStorage.setItem("currentRestaurantList", JSON.stringify(restaurants));
}catch(err){
console.log(err)
}
}
function getRestaurants(jsonResponse){
//Procesar respuesta JSON
}
//Para leerlo en otro ámbito seria
var restaurants = JSON.parse(sessionStorage.getItem('currentRestaurantList'))
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,892 |
Even if this is fake it's pretty funny. But I'm fairly certain it's real. When you click on the name her blogger profile comes up, with a picture that looks like her. And there's also a link to her blog in which she talks about being a Gamefly member. It seems like a pretty elaborate hoax to create all that just to leave a comment @LucidSportsFan as a joke. So Alex P, if you're reading this, I officially apologize. Sorry for not believing that you are a real gamer.
UPDATE 1/20: Read the comment below!
She's also a model (and actor? -- two profs. that seem to go hand-in-hand). | {
"redpajama_set_name": "RedPajamaC4"
} | 5,170 |
«Історія кам'яного століття» () — оповідання англійського письменника Герберта Веллса. Видане у 1897 році.
Сюжет
Печерний чоловік на ім'я Фу-Ломі вбиває свого суперника, фактичного ватажка племені Уя. Перебуваючи у вигнанні Фу-Ломі стає першою людиною, яка з'єднала камінь та дерево, та зробила нову сокиру. Він використовує цю зброю та свій розум щоб стати потрібним племені.
Оповідання Герберта Веллса | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,528 |
Antonio Jiménez Quiles (born 13 July 1934) is a Spanish former road cyclist. Professional from 1954 to 1963, he most notably finished 2nd overall at the 1955 Vuelta a España, while being the youngest competitor at 20 years old. He also won a stage of the 1958 Vuelta a España.
Major results
1955
1st Stage 4b Vuelta a Levante
1st Stage 2 GP Ayutamiento de Bilbao
2nd Overall Vuelta a España
2nd GP Goierri
6th Overall Volta a Catalunya
1956
1st Stage 3 Vuelta a Levante
4th Overall Vuelta a Andalucía
1957
1st National Hill-climb Championships
1st Stage 1 Gran Premio de la Bicicleta Eibarresa
3rd Road race, National Road Championships
4th Overall Vuelta a Andalucía
5th Overall Volta a Catalunya
1959
4th Trofeo Masferrer
1960
1st National Hill-climb Championships
1961
1st Stage 9 Volta a Portugal
References
External links
1934 births
Living people
Sportspeople from Granada
Spanish male cyclists
Spanish Vuelta a España stage winners
Cyclists from Andalusia | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,306 |
Luyendyk may refer to:
Arie Luyendyk (born 1953), Dutch racing driver
Arie Luyendyk Jr. (born 1981), son of the above and also a racing driver
Luyendyk Racing, racing team owned by Luyendyk Sr. for which Luyendyk Jr. raced in the 2006 Indianapolis 500
Bruce P. Luyendyk (born 1943), American geophysicist
Mount Luyendyk, mountain in Antarctica named for the above
See also | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,790 |
The planned meeting between President Maithripala Sirisena and former President Mahinda Rajapaksa is to be taken place at the Speaker's residence on Wednesday or Thursday, informed sources said yesterday.
Accordingly, SLFP MPs Dilan Perera, Mahinda Amaraweera and Lasantha Alagiyawanna had met President Sirisena on Saturday to discuss about the meeting. The MPs had reportedly told the President that the meeting between the two is essential in order to sort out the differences.
The MPs also highlighted the importance of sending a senior SLFPer to invite the former President for the meeting, such as the Opposition Leader, the General Secretary of the UPFA or the SLFP.
Be careful of that Talisman MR carrying everywhere. That might put a spell on Mr Sirisena.
Katussage kare raththran bandata uta raja wenda beha toiyane.
MS is looking for a scapegoat to win the next election. Without MR SLFP is zero.
I am saddened to note that after all those talks and attempts taken to remove MR from Power because he was not acceptable to the majority of this country, MS is planning to meet him? For what and Why?
All these are UNP fellows. Thay are scared. If SLFP is divided chance for the UNP fellows who wanted to give the country to LTTE.
Please sir, we are the 62 lakhs who voted for you and yaha palanaya and we beg you not to compromise on your promises. Let the course of justice prevail and let all the wrongdoers be sent behind the bars. Those of the old regime wanted that prevented - by threats, by cajoling and by any means. Please do not give in to their demands now.Let us look to the worst that may happen. Only possibility is that the SLFP may lose at the elections if MR and his catchers break away from it. What is the big deal! We were in opposition earlier and that is nothing to fear: we will have the President from our party. Some day we will come to office again.It is better to lose the elections after removing all the thieves, smugglers and murderers rather than win the elections with those rogues on our side.May you have the strength to stand for the principles of honesty and virtue.
Please remember the Old saying " Visagora Sarpaya Daka Naru Modaya"Will Maithree fall to the traps of MR and get himself destroy.
What is there to meet? Should not get caught to any trap. Some are trying to bring the king cobra to put into the sarong. The cobra will struggle inside for a while and the powerful venom will be injected and the outcome..... everybody knows.
Is My3 a president of this country or servant of MR?
The best solution to this problem is for Mr. Sirisena to hand over the Prersients position of the SLFP to Mahinda and for him to join either the UNP proper or its associate party JVP.
MS should not trust MR at all. Not even the Speaker. Ms must take care of his security.
Any discussion by a State leader/President with who ever it is should be held at Presidential Secretariat/ House.
MY3 has to be very careful when dealing with MR. MY3 is an innocent man and MR is a rogue. I mean my word. Sri Lankans still think MR is a nice man. I am sorry you all are wrong.
Dear President Maithripala Sirisena, we have seen articles, letters, news items condemning your meeting with Mahinda Rajapaksa. The Fascist who ruled the country with an iron fist, who insulted you wholesale when you announced your candidacy for presidency, plotted to assassinate you at least 3 times, your meeting with him will be the worst disastrous act that you would be doing.
If arrests are made illegally, the grieved party can take legal against those who did so as the Judicial system is free now unlike the time of MR, the King. Now, he wants to manipulate the judicial system by corporating with Mr. Maithreepala and tarnish the prestigious name of Mr. Maithree. Mr. MY3 must not fall to that trap as he received a mandate to eradicate all malpractices by punishing the culprits whoever they are. That is what matters.
Why is the Prez wasting his precious time meeting with this wanna be ? He is a sore and a bad loser. The Prez should have no time for these LOSER type of people.
If I were MS I would tell MR "Bring back your stolen billions being enjoyed by your cronies in Dubai, USA, etc, THEN we can talk".
It is frustrating the people who voted-out MR after suffering a prolonged years of tyranny and the only expectation was to see the dawn of a new era of peace and harmony The people who supported MS will not endorse any talks or ties or compromises with MR or his goons . If MS is cannot proceed without succumbed to pressures of parliamentarians of SLFP the political that opposed ,commended ,vilified ,ridiculed him at the presidential election but after he was elected the president he has chosen to lead it , for any strategic reasons of implementation of 100 day program or otherwise , MS should step down from being the head of the party that was not their founders expected and is now irreparable to reform and become apolitical without surrendering the peoples victory and insulting them.
Join hands and save the country from this rot.
May loka hour tika meet wala, President gay image aka nathikaranda apa.. Now there there is a big difference between M3 and MR. Moreover this meeting will be held where in speak residence. So something is fishy.
Dear President we humbly request you to think about Country first rather than any individual agenda .
What do you mean by unidentified?
He never gave up power, He was thrown out by the voters.
Who is prepare the agenda for the talks ? It looks like MR is speaking on behalf of those who prepare the topics.
Ranjith, did you just come from abroad??? Are you a foreigner??? Where were you over the past 10-15 years?
Why meet ? Mr.MR is still Power hungry. The Majority of the PEOPLE did not want him.We must Respect the People.
Former President should attend without his security or his security guards should be checked for loaded weapons .
MS has to be very cautious as lots of serpents around.
Invite former president CBK too.
Enjoy the retirement MR.Great people like Nelson Mandela when they gave up power.Did not interfere with politics once they gave up. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,142 |
Our Professional staff has over 40 years of combined experience.
Not sure of which style is best for you? Come in for a free consultation and we will be happy to help you decide the best look.
We have regular specials going on.
Welcome to The Barbershop on Little Greece where our team of experienced and professionally trained staff to give you best services. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,369 |
from pyiso import client_factory
from unittest import TestCase
import pytz
import logging
from io import StringIO
from datetime import datetime
class TestERCOT(TestCase):
def setUp(self):
self.c = client_factory('ERCOT')
handler = logging.StreamHandler()
self.c.logger.addHandler(handler)
self.c.logger.setLevel(logging.DEBUG)
self.load_html = StringIO(u'<html>\n\
<body class="bodyStyle">\n\
<table class="tableStyle" cellpadding="0" cellspacing="0" border="0" bgcolor="#ECECE2">\n\
<tr>\n\
<th colspan="5" valign="middle" class="hdr_port" >Real-Time System Conditions</th>\n\
</tr>\n\
<tr valign="top">\n\
<td colspan="5" class="labelClass" valign="middle"><span class="labelValueClass">Last Updated Sep 15 2014 13:50:20 CDT</span></td>\n\
</tr>\n\
<tr valign="top">\n\
<th colspan="5" valign="middle" class="headerClass"><span class="headerValueClass">Frequency</span></th>\n\
</tr>\n\
<tr valign="top">\n\
<td colspan = "4" valign="middle" class="labelClass"><span class="labelValueClass">Current Frequency</span></td>\n\
<td valign="middle" class="labelClassRight"><span class="labelValueClassBold">59.998</span></td>\n\
</tr>\n\
<tr valign="top">\n\
<td colspan = "4" valign="middle" class="labelClass"><span class="labelValueClass">Instantaneous Time Error</span></td>\n\
<td valign="middle" class="labelClassRight"><span class="labelValueClassBold">-29.222</span></td>\n\
</tr>\n\
<tr valign="top">\n\
<th colspan="5" valign="middle" class="headerClass"><span class="headerValueClass">Real-Time Data</span></th>\n\
</tr>\n\
<tr valign="top">\n\
<td colspan = "4" valign="middle" class="labelClass"><span class="labelValueClass">Actual System Demand</span></td>\n\
<td valign="middle" class="labelClassRight"><span class="labelValueClassBold">48681</span></td>\n\
</tr>\n\
<tr valign="top">\n\
<td colspan = "4" valign="middle" class="labelClass"><span class="labelValueClass">Total System Capacity (not including Ancillary Services)</span></td>\n\
<td valign="middle" class="labelClassRight"><span class="labelValueClassBold">54642</span></td>\n\
</tr>\n\
<tr valign="top">\n\
<td colspan = "4" valign="middle" class="labelClass"><span class="labelValueClass">Total Wind Output</span></td>\n\
<td valign="middle" class="labelClassRight"><span class="labelValueClassBold">885</span></td>\n\
</tr>\n\
<tr valign="top">\n\
<th colspan = "5" valign="middle" class="headerClass"><span class="headerValueClass">DC Tie Flows</span></th>\n\
</tr>\n\
<tr valign="top">\n\
<td colspan = "3" valign="middle" class="labelClass"><span class="labelValueClass">DC_E (East)</span></td>\n\
<td colspan = "2" valign="middle" class="labelClassRight"><span class="labelValueClassBold">-543</span></td>\n\
</tr>\n\
<tr valign="top">\n\
<td colspan = "3" valign="middle" class="labelClass"><span class="labelValueClass">DC_L (Laredo VFT)</span></td>\n\
<td colspan = "2" valign="middle" class="labelClassRight"><span class="labelValueClassBold">0</span></td>\n\
</tr>\n\
<tr valign="top">\n\
<td colspan = "3" valign="middle" class="labelClass"><span class="labelValueClass">DC_N (North)</span></td>\n\
<td colspan = "2" valign="middle" class="labelClassRight"><span class="labelValueClassBold">0</span></td>\n\
</tr>\n\
<tr valign="top">\n\
<td colspan = "3" valign="middle" class="labelClass"><span class="labelValueClass">DC_R (Railroad)</span></td>\n\
<td colspan = "2" valign="middle" class="labelClassRight"><span class="labelValueClassBold">0</span></td>\n\
</tr>\n\
<tr valign="top">\n\
<td colspan = "3" valign="middle" class="labelClass"><span class="labelValueClass">DC_S (Eagle Pass)</span></td>\n\
<td colspan = "2" valign="middle" class="labelClassRight"><span class="labelValueClassBold">5</span></td>\n\
</tr>\n\
</table>\n\
</body>\n\
</html>')
def test_utcify(self):
ts_str = '05/03/2014 02:00'
ts = self.c.utcify(ts_str)
self.assertEqual(ts.year, 2014)
self.assertEqual(ts.month, 5)
self.assertEqual(ts.day, 3)
self.assertEqual(ts.hour, 2+5-1)
self.assertEqual(ts.minute, 0)
self.assertEqual(ts.tzinfo, pytz.utc)
def test_parse_load(self):
data = self.c.parse_load(self.load_html)
self.assertEqual(len(data), 1)
expected_keys = ['timestamp', 'ba_name', 'load_MW', 'freq', 'market']
self.assertEqual(sorted(data[0].keys()), sorted(expected_keys))
self.assertEqual(data[0]['timestamp'], pytz.utc.localize(datetime(2014, 9, 15, 17, 50, 20)))
self.assertEqual(data[0]['load_MW'], 48681.0)
def test_request_report_gen_hrly(self):
# get data as list of dicts
data = self.c._request_report('gen_hrly')
# test for expected data
self.assertEqual(len(data), 1)
for key in ['SE_EXE_TIME_DST', 'SE_EXE_TIME', 'SE_MW']:
self.assertIn(key, data[0].keys())
def test_request_report_wind_hrly(self):
# get data as list of dicts
data = self.c._request_report('wind_hrly')
# test for expected data
self.assertEqual(len(data), 95)
for key in ['DSTFlag', 'ACTUAL_SYSTEM_WIDE', 'HOUR_BEGINNING']:
self.assertIn(key, data[0].keys())
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,860 |
We want to wish you luck today on the Object Detection and Avoidance Challenge! Everyone has put in to much work this week, you all should be so proud! As the day comes to an end someone will from the DPG staff will be around to help you get your materials in order for shipping and your mentor will help you get the car ready to go to Phoenix. The DPG staff will also be asking you for shipping details for your car when it is ready to be shipped back to your school in late May(after the retro fitting). Please let us know if you have any questions.
Good Luck and safe travels to Phoenix! | {
"redpajama_set_name": "RedPajamaC4"
} | 9,598 |
{"url":"https:\/\/search.r-project.org\/CRAN\/refmans\/DesignLibrary\/html\/mediation_analysis_designer.html","text":"mediation_analysis_designer {DesignLibrary} R Documentation\n\n## Create a design for mediation analysis\n\n### Description\n\nA mediation analysis design that examines the effect of treatment (Z) on mediator (M) and the effect of mediator (M) on outcome (Y) (given Z=0) as well as direct effect of treatment (Z) on outcome (Y) (given M=0). Analysis is implemented using an interacted regression model. Note this model is not guaranteed to be unbiased despite randomization of Z because of possible violations of sequential ignorability.\n\n### Usage\n\nmediation_analysis_designer(\nN = 200,\na = 1,\nb = 0.4,\nc = 0,\nd = 0.5,\nrho = 0,\nargs_to_fix = NULL\n)\n\n\n### Arguments\n\n N An integer. Size of sample. a A number. Parameter governing effect of treatment (Z) on mediator (M). b A number. Effect of mediator (M) on outcome (Y) when Z = 0. c A number. Interaction between mediator (M) and (Z) for outcome (Y). d A number. Direct effect of treatment (Z) on outcome (Y), when M = 0. rho A number in [-1,1]. Correlation between mediator (M) and outcome (Y) error terms. Non zero correlation implies a violation of sequential ignorability. args_to_fix A character vector. Names of arguments to be args_to_fix in design.\n\n### Details\n\nSee vignette online.\n\n### Value\n\nA mediation analysis design.\n\n### Examples\n\n# Generate a mediation analysis design using default arguments:\nmediation_1 <- mediation_analysis_designer()\ndraw_estimands(mediation_1)\n## Not run:\ndiagnose_design(mediation_1, sims = 1000)\n\n## End(Not run)\n\n# A design with a violation of sequential ignorability and heterogeneous effects:\nmediation_2 <- mediation_analysis_designer(a = 1, rho = .5, c = 1, d = .75)\ndraw_estimands(mediation_2)\n## Not run:\ndiagnose_design(mediation_2, sims = 1000)\n\n## End(Not run)\n\n\n\n[Package DesignLibrary version 0.1.10 Index]","date":"2022-05-25 19:53:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.36314839124679565, \"perplexity\": 11606.989606738336}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662593428.63\/warc\/CC-MAIN-20220525182604-20220525212604-00479.warc.gz\"}"} | null | null |
\section{Introduction}
\label{sec:introduction}
Segmentation has several medical applications, such as patient-specific surgical planning.
Due to limited resources of expert physicians, detailed manual annotations are often not possible, even when desired anatomy may be visible with sufficient contrast using non-invasive imaging modalities such as MRI and ultrasound.
Deep learning has shown encouraging performance for segmentation~\cite{baumgartner2017exploration,Chen2016DCAN}, but often only when sufficient amount of labeled data for a target anatomy is available.
Medical image data across different medical centers is often not uniform, for instance with respect to machine manufacturer, imaging settings, and cohort demographics. Thus, studies and corresponding annotations are only carried out in isolated datasets, with difficulties in merging information with data sharing, patient rights, and confidentiality concerns.
Hence, a sufficiently large dataset for a given task needs to be labeled.
\textit{Active learning} aims at maximizing the prediction performance through an intelligent sample querying system so that the limited expert annotation resources can be properly managed as opposed to training on a randomly selected next batch of samples which would contain a lot of redundancy.
In a clinical environment, one can imagine that expert(s) will allocate a fixed amount of annotation time per time interval (i.e.,\ week), hence the correct use of this time (i.e.,\ on most valuable samples) is essential.
Therefore, the segmentation framework would be initially provided a very limited labeled dataset, which will be extended with a certain batch size of samples intelligently selected at each iteration of the active learning.
Intuitively, the prediction confidence of a learned model can be used as a surrogate metric for its potential accuracy, in order to propose the most \emph{uncertain} predictions for future manual annotation.
In~\cite{gal2015dropout}, \textit{MC dropout} is proposed to sample from the approximate trained model posterior, which can be used to quantify an \textit{uncertainty} metric through variations in the model predictions for a given input.
Based on this, several approaches of querying the next batch of data are studied and compared with uniform random sampling in~\cite{gal2017deep}.
Unfortunately, it is intractable to assess conditional uncertainty of multiple samples; e.g.\, would $i^\mathrm{th}$ sample be still as uncertain as before once $j^\mathrm{th}$ sample is queried and trained for.
Thus, it is intuitive to select a \textit{representative} subset of these uncertain samples to reduce redundancy.
Using a simplified version of DCAN~\cite{Chen2016DCAN} architecture (which has won the first place in the 2015 MICCAI Gland Segmentation Challenge~\cite{Sirinukunwattana2017gland}) for the purpose of faster training, a state-of-the-art method was proposed in~\cite{yang2017suggestive} to select optimal sample images to annotate.
First, a batch of \emph{uncertain} samples is chosen based on the mean variance of multiple network predictions, followed by picking a subset of these using \emph{maximum set coverage}~\cite{Feige1998a} over the \textit{image descriptors} of these samples.
Recently in~\cite{ozdemir2018learn}, a \textit{content distance}~\cite{gatys2016image} concept was proposed to quantify the similarity between two images, for selecting representative samples in class-incremental learning.
Herein we propose two main novelties for querying samples at an active learning step:
(1)~we add an additional constraint on the \textit{abstraction layer}~\cite{ozdemir2018learn} activations during training to maximize information content at this level.
We show that this additional constraint improves sample suitability that boosts segmentation performance from active learning.
(2)~Instead of the two step sample querying procedure (i.e., first select based on \textit{uncertainty}, then cull using \textit{representativeness}), we propose a Borda-count based method.
This alone provides improvement over the state-of-the-art~\cite{yang2017suggestive}; and when used in conjunction with our novel constraint above, it yields even further segmentation improvement.
\section{Estimating Surrogate Metrics for Representativeness}
\label{sec:representativeness_metrics}
\textbf{Background.}
In~\cite{yang2017suggestive}, multiple FCNs were trained to estimate uncertainty for a given image through variation in their inferences.
To make the FCN predictions diverse, the annotated dataset was also bootstrapped when training each model.
However, training several models is a costly operation and with larger number of models, one should bootstrap a smaller portion of the already-minimal dataset available in the early stages of typical active learning scenarios.
\emph{In our work,} as a baseline, we implemented an improved version of the \textit{Suggestive Annotation} framework~\cite{yang2017suggestive}.
We added dropout layers (c.f.\ Fig.\ref{fig:mod_DCAN_architecture}) to allow for MC dropout~\cite{gal2015dropout}, through which one can compute the voxel-wise variance across $n_i$ inferences, and average it over all input voxels.
The first step in querying samples is to pick the most uncertain $n_\mathrm{unc}$ samples $S_\mathrm{unc}$ from the set of non-annotated data $D_\mathrm{pool}$.
For representativeness, ``image descriptor'' $I_i^c$ of every image $I_i \in D_\mathrm{pool}$ is computed as described in~\cite{yang2017suggestive} at the abstraction layer, $l_\mathrm{abst}$ (c.f. Fig.~\ref{fig:mod_DCAN_architecture}).
Using cosine similarity $d_\mathrm{sim}(I_i, I_j) = \cos(I_i^c, I_j^c)$ between the descriptors of images $I_i$ and $I_j$, the maximum set-cover~\cite{Feige1998a} over $D_\mathrm{pool}$ is computed using descriptors from $S_\mathrm{unc}$ for the top $n_\mathrm{rep}$ images.
We call this method of using uncertainty and the above image descriptor (ID) as UNC-ID hereafter.
\begin{figure}[t]
\centering
\includegraphics[width = 0.99\textwidth]{figs/schematics5.png}
\caption{DCAN network for \textit{Suggestive\,Annotation} with additional spatial dropout layers. $n_{ch}$ is the number of filters in respective block, BN is batch normalization, and $n_{cl}$ is the number of classes. In consecutive bottlenecks, the first uses convolution filter in shortcuts to match tensor size while the second does not.}
\label{fig:mod_DCAN_architecture}
\end{figure}
\noindent\textbf{Content Distance.}
The image descriptor $I_i^c$ averages the spatial information at the corresponding layer activations.
While this allows for a spatially invariant means of representing a given image at a very abstract level, higher order features extracted at this stage would be blurred by this process.
Alternatively, layer activation responses $R^l(I_i)$ of a pretrained classification network at a layer $l$ can be used to describe the content of an image $I_i$~\cite{gatys2016image}.
Then, content distance ($d_\mathrm{cont}$) between images $I_i$ and $I_j$ is defined as the mean squared error between their responses at layer $l$:
\begin{equation}
d_\mathrm{cont}(I_i,I_j) = \frac{1}{N}\sum^N (R^{l}(I_i) - R^{l}(I_j))^2
\label{eq:dcont}
\end{equation}
A similar notion can be applied to active learning problems, where input images are described by the activation response at the $l_\mathrm{abst}$ of the currently trained network (c.f. Fig.~\ref{fig:mod_DCAN_architecture}).
\noindent \textbf{Encoding Representativeness by Maximizing Entropy.}
Content distance defined in Eq.~(\ref{eq:dcont}) allows for finer content discrimination than image descriptors~\cite{yang2017suggestive}.
However, it has been suggested that activations at a single layer may not be sufficient for accurate content description~\cite{ozdemir2018learn}.
This is likely to particularly apply to segmentation networks, since network weights until $l_\mathrm{abst}$ are not optimized to describe the input image.
Therefore, it has been proposed to stack activations from multiple layers.
For a typical segmentation network, storing all layer activations of $D_\mathrm{pool}$ can quickly diverge to an unfeasible size.
Alternatively, one can try to increase information content at the $l_\mathrm{abst}$ through maximizing its activation entropy~\cite{shannon2001a} along the feature channels.
Entropy loss can then be defined as:
\begin{equation}
L_\mathrm{ent} = -\sum_x \mathrm{H}(R^{(l_\mathrm{abst}, x)})
\label{eq:entropy_loss}
\end{equation}
where $R^{(l_\mathrm{abst}, x)}$ are the input activations of all channels for spatial location $x$, and $x$ iterates over the width and height of the layer $l_\mathrm{abst}$.
Hence, total loss for the trained network becomes $L_\mathrm{total} = L_\mathrm{seg} + \lambda L_\mathrm{ent}$, where $L_\mathrm{seg}$ is the segmentation loss, and $\lambda$ is used to scale the entropy loss $L_\mathrm{ent}$.
Optimization of the network weights through entropy maximization is a novel regularization.
$L_\mathrm{ent}$ alone would have a tendency to alter network weights to only increase information, which may also encourage randomness.
With an appropriate $\lambda$, the network is forced to optimize parameters for the segmentation task while also increasing ``useful'' information content at the abstraction layer; as opposed to producing just noise at $l_\mathrm{abst}$.
Hence, additional content description for a given image can be retrieved from a single layer activation, making it a feasible alternative.
We refer to this method, where an entropy-based content distance (\textbf{E}CD) is used, as UNC-\textbf{E}CD.
\section{Sample Selection Strategy}
\label{sec:sample_selection}
For active learning, one should emphasize that the initial data size can be very small.
Until the model parameters are optimized for a sufficient coverage of the data distribution, the defined ``uncertainty'' metric might be misleading.
As a result, one can explore different ways to combine multiple metrics when querying samples instead of the conventional 2-step process.
An intuitive way to combine two metrics $m_k$ and $m_l$ would be to use $w_k m_k + w_l m_l$, where $w_k, w_l$ are weights.
However, \textit{uncertainty} and \textit{representativeness} metrics defined in Sec.~\ref{sec:representativeness_metrics} are not linearly combinable, even if normalized, due to non-linear unit increments.
Therefore, we propose to use Borda count, where samples are ranked for each metric, and the next query sample $I_{i^*}$ is picked based on the best combined rank:
\begin{equation}
i^* = \arg\min_i(\sum_{m_k \in S_m} f_\mathrm{rank}(m_k(I_i)))
\label{eq:rank}
\end{equation}
where $S_m$ is the set of metrics $m_k$ to combine, and the $f_\mathrm{rank}$ function sorts the images based on the metric $m_k$.
When we use the ranking in Eq.~(\ref{eq:rank}) for samples selection, we denote this in our results with ``+'', e.g.\ content distance with uncertainty is named UNC+\textbf{E}CD.
In an active learning framework, the methods mentioned until now can be denoted as UNC+ID, UNC+\textbf{E}CD for ranking based sample selection and UNC-ID, UNC-\textbf{E}CD for uncertainty selection followed by representativeness selection.
\section{Experiments and Results}
\label{sec:Experiments}
\begin{table}[t]
\centering
\caption{Dataset configuration}
\label{tbl:dataset}
\begin{tabular}{r|r|r|r|r|r|r|r}
\multicolumn{1}{c|}{Config} & \multicolumn{1}{c|}{\#volumes} & \multicolumn{1}{c|}{Left/Right} & \multicolumn{1}{c|}{vox res.\,{[}mm{]}} & \multicolumn{1}{c|}{image size\,{[}px{]}} & \multicolumn{1}{c|}{TR\,{[}s{]}} & \multicolumn{1}{c|}{TE\,{[}s{]}} & \multicolumn{1}{c}{FA\,[\si{\degree}]} \\ \hline
1 & 20 & 9/11 & 0.91 $\times$ 0.91 $\times$ 3.0 & 192 $\times$ 192 $\times$ 64 & 20 & 1.70 & 10 \\ \hline
2 & 16 & 8/8 & 0.83 $\times$ 0.83 $\times$ 3.0 & 144 $\times$ 144 $\times$ 56 & 20 & 2.39 & 10
\end{tabular}
\end{table}
\begin{figure}[]
\centering
\begin{subfigure}[b]{0.499\textwidth}
\centering
\includegraphics[width=0.999\textwidth, trim= 0cm 0.0cm 1.1cm 0cm, clip]{figs/q1dice4.png}
\caption{Dice score}
\label{fig:scores:dice}
\end{subfigure}%
\begin{subfigure}[b]{0.499\textwidth}
\centering
\includegraphics[width=0.999\textwidth, trim= 0cm 0.0cm 1.1cm 0cm, clip]{figs/q1assd4.png}
\caption{MSD [mm]}%
\label{fig:scores:assd}
\end{subfigure}%
\caption{Comparison between our implementation of the baseline method (UNC-ID) with random sampling (RAND) and only uncertainty-based (UNC) active learning methods.
Training on 100\% of the data ($D_\mathrm{pool}$) is shown as upperbound.
(a) Mean Dice score and (b) mean surface distance (MSD) with error bars covering the standard deviation of 5 hold-out experiments at every evaluation point.
}
\label{fig:scores}
\end{figure}
\begin{figure}[]
\centering
\begin{subfigure}[b]{0.499\textwidth}
\centering
\includegraphics[width=0.999\textwidth, trim= 0cm 0.0cm 1.1cm 0cm, clip]{figs/q2dice4.png}
\caption{Dice score}
\label{fig:scores2:dice}
\end{subfigure}%
\begin{subfigure}[b]{0.499\textwidth}
\centering
\includegraphics[width=0.999\textwidth, trim= 0cm 0.0cm 1.1cm 0cm, clip]{figs/q2assd4.png}
\caption{MSD [mm]}
\label{fig:scores2:assd}
\end{subfigure}%
\caption{Comparison of the baseline method (UNC-ID) with ranking based sample selection (UNC+ID) and the combination of our proposed extensions (UNC+\textbf{E}CD).
Training on 100\% of the data ($D_\mathrm{pool}$) is shown as upperbound.
(a) Mean Dice score and (b) mean surface distance (MSD) with error bars covering the standard deviation of 5 hold-out experiments at every evaluation point.
The mean Dice score of UNC+\textbf{E}CD was statistically significantly higher than the baseline in 4 of 5 experiments (one-sided paired t-test at the 0.05 level).}
\label{fig:scores2}
\end{figure}
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.35\textwidth}
\centering
\includegraphics[width=0.98\textwidth, trim= 0cm 0.0cm 0cm 0cm, clip]{figs/329332gs.png}
\caption{Gold Standard (GS)}
\label{fig:qualRes:GS}
\end{subfigure}%
\begin{subfigure}[b]{0.324\textwidth}
\centering
\includegraphics[width=0.98\textwidth, trim= 0cm 0.0cm 0cm 0cm, clip]{figs/329332m_uid_crop.png}
\caption{GS+Baseline}
\label{fig:qualRes:overlay_baseline}
\end{subfigure}%
\begin{subfigure}[b]{0.324\textwidth}
\centering
\includegraphics[width=0.98\textwidth, trim= 0cm 0.0cm 0cm 0cm, clip]{figs/329332erucd_crop.png}
\caption{GS+Proposed}
\label{fig:qualRes:overlay_proposed}
\end{subfigure}\\%
\begin{subfigure}[b]{0.495\textwidth}
\centering
\includegraphics[width=0.98\textwidth, trim= 0cm 0.0cm 0cm 0cm, clip]{figs/329332m_uid_arrow.png}
\caption{Baseline}
\label{fig:qualRes:baseline}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.495\textwidth}
\centering
\includegraphics[width=0.98\textwidth, trim= 0cm 0.0cm 0cm 0cm, clip]{figs/329332erucd.png}
\caption{Proposed}
\label{fig:qualRes:proposed}
\end{subfigure}%
\caption{Segmentation of a test volume comparing baseline (UNC-ID) with proposed method (UNC+\textbf{E}CD) after the first active learning step.
Segmentation of two muscles overlaid on GS annotation (red) for (b) baseline and (c) proposed method.
(d) Some of the substantial differences are pointed out by red arrows.
}
\label{fig:qualRes}
\end{figure}
We have conducted experiments on an MR dataset of 36 patients diagnosed with rotator cuff tear (shoulders) according to specifications shown on Table~\ref{tbl:dataset}.
In an effort to regularize the dataset, Config2 images have been resized to match the voxel resolution of Config1, and then zero padded to match the in-plane image size of Config1.
The data has expert annotations of two bones (humerus \& scapula) and two muscle groups (supraspinatus \& infraspinatus\,+\,teres minor).
Experiments have been conducted using NVIDIA Titan X GPU and Tensorflow library~\cite{tensorflow2015whitepaper}.
For all compared methods, we have used the modified DCAN architecture shown in Fig.\ref{fig:mod_DCAN_architecture}, training it on 2D in-plane slices with the parameters $n_{ch}$\,$=$\,$32$ and Adam optimizer.
When training the networks, learning rate of $5\times 10^{-4}$, dropout rate of 0.5, $n_i$$=$$17$, and minibatch size of 8 images were applied.
At each active learning stage, including the initial training, models were trained for 8000 steps, which took about 48\,mins.
Uncertainty metric is aggregated over the foreground classes to represent their mean uncertainty.
We used cross-entropy loss at the softmax layer (c.f. Fig.~\ref{fig:mod_DCAN_architecture}) for the $L_\mathrm{seg}$.
Weight $\lambda$ for scaling $L_\textrm{ent}$ in methods UNC-\textbf{E}CD and UNC+\textbf{E}CD is empirically set to $\lambda = 1 / (360 \times |R^{l_\mathrm{abst}}|)$.
To provide quantitative results, we have evaluated Dice score coefficient and mean surface distance (MSD).
In an effort to efficiently utilize the available dataset, we have generated 5 hold-out experiments
where the initial training set $D_\mathrm{an}$, the non-annotated set $D_\mathrm{pool}$, the validation set (all slices from 2 patients) and the test set (all slices from 9 patients) are randomly picked.
All experiments are initially trained on 64 slices.
For every active learning step, $n_\mathrm{rep}$\,$=$\,$32$ and $n_\mathrm{unc}$\,$=$\,$64$ is used.
In Figs.~\ref{fig:scores} \&~\ref{fig:scores2}, we show the Dice score and MSD of different methods evaluated for the test set at 11 stages of active learning ranging from $4\%$ up to $27\%$ of the $D_\mathrm{pool}$.
Conducted experiments are shown in two groups to increase clarity: (1) Comparison of our implementation of the baseline (UNC-ID) to uniform random sample querying (RAND) and sample querying based only on uncertainty (UNC) as seen in Fig.~\ref{fig:scores}; (2) Building on top of (1), improvements of ranking (UNC+ID) and the gain from $L_\mathrm{ent}$ during training and representativeness capabilities of $d_\mathrm{cont}$ for sample querying, UNC+\textbf{E}CD (c.f. Fig.~\ref{fig:scores2}).
In Fig.~\ref{fig:qualRes}, we show an example cross-section from a test volume, where segmentation superiority of our proposed method (UNC+\textbf{E}CD) when compared to baseline is already visible after a single active learning step.
We conducted one-sided paired-sample t-tests at the $5\%$ significance level on the mean Dice scores over all active learning steps for each hold-out experiment for UNC+\textbf{E}CD being superior to UNC-ID. Performance of UNC+\textbf{E}CD was statistically significantly better in 4 of 5 experiments.
\section{Discussions \& Conclusions}
\label{sec:discussions}
At early steps of active learning, one can see that the only uncertainty-based query sampling method (UNC) performs similar to random sample querying (RAND), with UNC only improving soon after $\approx$\,$12\%$ of $D_\mathrm{pool}$ is used in training (c.f. Fig.~\ref{fig:scores}).
While UNC-ID already yields better segmentation performance than just uncertainty-based sampling, by simply using ranking, one can see that the baseline method achieves a more substantial boost at early stages of active learning (see UNC+ID in Fig.~\ref{fig:scores2}).
This behavior suggests that the surrogate uncertainty metric can give a bad approximation when the trained data size is fairly low; i.e.,\ initial step(s).
However, the suboptimal segmentation performance gain can be compensated with representativeness, and even further improved when given a higher priority; i.e.,\ ranking instead of 2-step sample querying.
Upon combination of the proposed additional information maximization constraint during training and ranking combined with content distance at sample querying (UNC+\textbf{E}CD), we have observed the best Dice score on average at all active learning steps among the compared baseline and ranking extensions of the baseline methods.
Other possible combinations of our proposed extensions (UNC-CD, UNC+CD, UNC-\textbf{E}CD) yielded inferior performance to UNC+\textbf{E}CD, and hence are not included in the quantitative comparisons to reduce clutter.
In this paper, we have comparatively studied the impact of different sample selection methods in active learning for segmentation.
We have proposed 2 novel ways to query samples for active learning, which also can be combined to further boost performance during active learning steps.
Compared to a state-of-the-art method, we have shown our proposed method to yield statistically significant improvement of segmentation Dice scores.
\noindent\textbf{Acknowledgements.}
This work was funded by the Swiss National Science Foundation (SNSF), a Highly Specialized Medicine (HSM2) grant of the Canton of Zurich, and the EU's 7th Framework Program (Agreement No. 611889, TRANS-FUSIMO).
We acknowledge NVIDIA GPU Grant support.
\bibliographystyle{splncs03}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,521 |
\section*{Introduction}
The idea to build a device capable "to compute all that can be computed" had emerged a long time.
One can remember Blaise Pascal's Arithmetic Machine, Gottfried Wilhelm Leibniz's Stepped Reckoner, Charles Babbage's Difference
Engine and his Analytical Engine \cite{bib::white}.
But only in the thirties of last century, Alonzo Church \cite{bib::church}, Alan Mathison Turing \cite{bib::turing_machine}, and
Emil Leon Post \cite{bib::post} built mathematical models of the computational processes.
Although these models have different shapes each of them describes inherently the same class of processes.
The equivalence of the Turing's model and the Church's model, for example, was proved by A.M. Turing in 1937
\cite{bib::turing_thesis}.
In the late forties, hardware implementations of a universal computational system were developed and began to be used.
They are known now as computers.
The practice of using computers to solve real problems showed that besides answering the question "Can the problem be solved using computer?", an answer to the question "Do we have enough computational resources to solve the problem?" is important too.
Searches for methods to evaluate computational resources for computer-assisted problem solving led to the special scientific area
which is called theory of computational complexity (the brief historical overview one can see in \cite{bib::fortnow}).
Unfortunately, most important computational problems are complex ones.
In compliance with the generally accepted propositions of theoretical computing science the application field of classical
computers, i.e. hardware implementations of the universal Turing machine concept, is physically challenged by problems which have polynomial computational complexity.
However, modern science, technique, and technology are in need of me\-thods to solve problems whose complexity is higher than polynomial.
This situation stimulates research of non-classical approaches to computing, and quantum computing is one of these.
The idea to use quantum systems as computing devices appeared in the early eighties of the twentieth century.
The idea's authors considered it as a way to overcome computational complexity.
In the context Yuri Ivanovitch Manin's monograph
\footnote{
The monograph's introduction was translated into English \cite{bib::manin_essays}
}
\cite{bib::manin} and Richard Phillips Feynman's paper \cite{bib::feynman} should be noted.
Considering the possibility of using quantum machines for solving complex problems of simulation Yu.I. Manin wrote (cited by \cite{bib::manin_essays}): "\dots we need a mathematical theory of quantum automata. Such a theory would provide us with
mathematical models of deterministic processes with quite unusual properties. One reason for this is that the quantum state space
has far greater capacity then the classical one: for a classical system with $N$ states, its quantum version allowing
superposition (entanglement) accommodates $e^N$ states".
In \cite{bib::manin_essays}, Yu.I. Manin also sets requirements to the mathematical theory of quantum automata: "The first
difficulty we must overcome is the choice of the correct balance between the mathematical and the physical principles. The quantum automaton has to be an abstract one: its mathematical model must appeal only to the general principles of quantum physics, without prescribing a physical implementation. Then the model of evolution is the unitary rotation in a finite dimensional Hilbert space,
and the decomposition of the system into its virtual parts corresponds to the tensor product decomposition of the state space
("quantum entanglement"). Somewhere in this picture we must accommodate interaction, which is described by density matrices and probabilities".
R.P. Feynman had a similar opinion \cite{bib::feynman,bib::feynman_computer}.
This paper is an attempt to construct a mathematical model of quantum automata that fulfils requirements formulated by
Yu.I. Manin.
\section{Classical Computational Model}
In this section a mathematical model of a classical computational system is considered.
The approach was proposed by A.N.~Kolmogorov and V.A.~Uspensky \cite{bib::kolmogorov}.
Theory of abstract state machine (ASM) is the further development of the approach \cite{bib::gurevich_1,bib::gurevich_2}.
\subsection{Preliminary definitions}
\begin{definition}\label{dfn::algorithm}
Let $\mathfrak{A}$ denotes an {\bfseries algorithm}. It is determined by
\begin{itemize}
\item[$-$]
a set $\mathcal{C}(\mathfrak{A})$ of states{\rm;}
\item[$-$]
a subset $\mathcal{I}(\mathfrak{A})$ of $\mathcal{C}(\mathfrak{A})$ which elements are called {\bfseries initial} states;
\item[$-$]
a subset $\mathcal{T}(\mathfrak{A})$ of $\mathcal{C}(\mathfrak{A})$ which elements are called {\bfseries terminal} states;
\item[$-$]
a map $\tau_\mathfrak{A}\colon\mathcal{C}(\mathfrak{A})\rightarrow\mathcal{C}(\mathfrak{A})$ which defines one step of
the computational process;
\end{itemize}
and the next condition
\begin{equation}
\mathcal{I}(\mathfrak{A})\bigcap\mathcal{T}(\mathfrak{A})=\emptyset.
\end{equation}
\end{definition}
Note that elements of the set $\mathcal{C}(\mathfrak{A})$ correspond to complete state descriptions of the computational process
which is defined by the algorithm $\mathfrak{A}$.
\begin{definition}\label{dfn::run}
Let $\mathfrak{A}$ be an algorithm then a partial map $C\colon\mathbb{N}\dashrightarrow\mathcal{C}(\mathfrak{A})$
\footnote{
For two sets $A$ and $B$ by $f\colon A\dashrightarrow B$ a partial map from $A$ into $B$ is denoted.
For $a\in A$ by $f(a)\neq\emptyset$ the clause "$f(a)$ is defined" is denoted.
}
is called a {\bfseries run} of the algorithm if it satisfies the following conditions
\begin{itemize}
\item[{\rm(i)}]
$C(0)\in\mathcal{I}(\mathfrak{A});$
\item[{\rm(ii)}]
if $C(t)\neq\emptyset$ for some $t\in \mathbb{N}$ then $C(t')\neq\emptyset$ for all $t'\in \mathbb{N}$ such that $t'<t;$
\item[{\rm(iii)}]
if $C(t+1)\neq\emptyset$ for some $t\in \mathbb{N}$ then $C(t+1)=\tau_\mathfrak{A}(C(t))${\rm;}
\item[{\rm(iv)}]
if $\emptyset\neq C(t)\in\mathcal{T}(\mathfrak{A})$ then $C(t+1)=\emptyset$.
\end{itemize}
\end{definition}
From this definition it follows immediately that the domain of an arbitrary run is the set $\mathbb{N}$ or some set
$0~..~T=\{t\in\mathbb{N}\mid t\leq T\}$, where $T$ is a non-negative integer.
In the first case the algorithm {\bf diverges} on the initial state $C(0)$ (this is denoted by $\mathfrak{A}(C(0))\uparrow$).
In the second case the algorithm {\bf converges} on the initial state $C(0)$ to $C(T)$(this is denoted by
$\mathfrak{A}(C(0))\downarrow C(T)$).
\subsection{Abstract state machines with stochastic behaviour}
Let's refine the Definition \ref{dfn::algorithm} and Definition \ref{dfn::run} for such algorithms that have sets of states with some special structure.
Let's start refining with the following auxiliary definitions.
\begin{definition}
Let $\mathcal{N}$ and $\mathcal{A}$ be finite sets of nodes and arcs respectively, $\mathrm{dom}$ and $\mathrm{codom}$ be a maps that associate
with arcs their initial and terminal nodes respectively, then the tuple $(\mathcal{N},\mathcal{A},\mathrm{dom},\mathrm{codom})$ is called a
{\bfseries directed multigraph}.
\end{definition}
\begin{definition}
Let $\mathcal{G}=(\mathcal{N},\mathcal{A},\mathrm{dom},\mathrm{codom})$ be a directed multigraph, then an alternating sequence
$\alpha=n_1,a_1,\dots,a_k,n_{k+1}$ of nodes and arcs, beginning and ending with a node, is called a {\bfseries walk} if for all
$s=1,\dots,k$ the next condition holds: $\mathrm{dom}(a_s)=n_s$ and $\mathrm{codom}(a_s)=n_{s+1}$.
In this case we shall use the notation: $n_1\arc{a_1}\dots\arc{a_k}n_{k+1}$.
\end{definition}
\begin{definition}
Let $\alpha=n_1\arc{a_1}\dots\arc{a_k}n_{k+1}$ be a walk in the directed multigraph
$\mathcal{G}=(\mathcal{N},\mathcal{A},\mathrm{dom},\mathrm{codom})$ and $n$ be its node, then we shall say that
\begin{itemize}
\item[{\rm(i)}]
$n$ is the {\bfseries initial node} of $\alpha$ {\rm(}it is denoted by $\mathrm{dom}(\alpha)=n${\rm)} if $n=n_1${\rm;}
\item[{\rm(ii)}]
$n$ is the {\bfseries terminal node} of $\alpha$ {\rm(}it is denoted by $\mathrm{codom}(\alpha)=n${\rm)} if $n=n_{k+1}${\rm;}
\item[{\rm(iii)}]
$\alpha$ {\bfseries traverses} $n$ if for some $s\in\{1,\dots,k+1\}$ the equality $n=n_s$ holds.
\end{itemize}
\end{definition}
\begin{definition}
Let $(\mathcal{N},\mathcal{A},\mathrm{dom},\mathrm{codom},n_0,F)$ be a tuple such that the tuple\linebreak
$(\mathcal{N},\mathcal{A},\mathrm{dom},\mathrm{codom})$ is a directed multigraph, $n_0$ is a fixed node {\rm(}it is called the initial node{\rm)},
$F$ is a fixed subset of nodes{\rm(}its elements are called terminal nodes{\rm)}. The tuple is called a {\bfseries control graph}
if the next conditions hold:
\begin{itemize}
\item[{\rm(i)}]
$n_0\notin F${\rm;}
\item[{\rm(ii)}]
for each $n\in F$ there is no arc with initial node equals $n${\rm;}
\item[{\rm(iii)}]
for each node $n$ there is a walk such that its initial node equals $n_0$, its terminal node belongs to $F$, and it traverses $n$.
\end{itemize}
\end{definition}
Note that for the control graph $(\mathcal{N},\mathcal{A},\mathrm{dom},\mathrm{codom},n_0,F)$ and each $n\in\mathcal{N}\setminus F$ the set $\mathrm{Out}(n)=\{a\in\mathcal{A}\mid\mathrm{dom}(a)=n\}$ is not empty.
Let's assume that for an arbitrary algorithm $\mathfrak{A}$ the set of states has the next structure
$\mathcal{C}(\mathfrak{A})=\mathcal{N}(\mathfrak{A})\times\mathcal{S}(\mathfrak{A})$ where $\mathcal{N}(\mathfrak{A})$ is the
nodes set of some control graph $\mathcal{G}(\mathfrak{A})$ and $\mathcal{S}(\mathfrak{A})$ is some set of memory snapshots.
In this case suppose that the set of initial states is the next set
$\mathcal{I}(\mathfrak{A})=\{(n_0,S)\mid S\in\mathcal{S}(\mathfrak{A})\}$ and the set of terminal states is the following set
$\mathcal{T}(\mathfrak{A})=\{(n,S)\mid n\in F~\&~S\in\mathcal{S}(\mathfrak{A})\}$.
This supposition leads us to the next representation of the map $\tau_\mathfrak{A}$:
\begin{multline}\label{eqn::tau1}
\tau_\mathfrak{A}(n,S)=(\sigma_\mathfrak{A}(n,S),\gamma_\mathfrak{A}(n,S))\mbox{, where }\\
\sigma_\mathfrak{A}\colon\mathcal{N}(\mathfrak{A})\times\mathcal{S}(\mathfrak{A})\rightarrow\mathcal{N}(\mathfrak{A})\mbox{ and }
\gamma_\mathfrak{A}\colon\mathcal{N}(\mathfrak{A})\times\mathcal{S}(\mathfrak{A})\rightarrow\mathcal{S}(\mathfrak{A}).
\end{multline}
Suppose now that the map $\sigma_\mathfrak{A}$ has property of locality.
It means that for each $n\in\mathcal{N}(\mathfrak{A})\setminus F$ there exists a map
$h_n\colon\mathcal{S}(\mathfrak{A})\rightarrow\mathrm{Out}(n)$ and for each $a\in\mathcal{A}(\mathfrak{A})$
\footnote{By $\mathcal{A}(\mathfrak{A})$ is denoted the set of arcs for the control graph $\mathcal{G}(\mathfrak{A})$.}
there exists a map $g_a\colon\mathcal{S}(\mathfrak{A})\rightarrow\mathcal{S}(\mathfrak{A})$
such that the following equalities are true:
\begin{eqnarray}
\sigma_\mathfrak{A}(n,S)=&\mathrm{codom}(h_n(S));\label{eqn::sigma}\\
\gamma_\mathfrak{A}(n,S)=&g_{h_n(S)}(S)\label{eqn::gamma}.
\end{eqnarray}
From (\ref{eqn::tau1}), (\ref{eqn::sigma}), and (\ref{eqn::gamma}) it follows
\begin{equation}\label{eqn::tau2}
\tau_\mathfrak{A}(n,S)=(\mathrm{codom}(h_n(S)),g_{h_n(S)}(S)).
\end{equation}
Therefore, we can consider the computational process which is determined by the algorithm $\mathfrak{A}$ as a sequence of steps.
Each step begins when the current state is described by some control graph node $n$ and a memory snapshot $S$.
Then the map $h_n$ chooses the arc $a$ outgoing from the node $n$ depending on the snapshot $S$.
Finally, using the selected arc and the memory snapshot the new control graph node and the new memory snapshot
are determined in compliance with (\ref{eqn::tau2}).
Let's modify the computational model by rejecting the assumption about determinacy for the choosing process.
Definition \ref{dfn::asmr} describes this modification formally.
\begin{definition}\label{dfn::asmr}
Let $\mathcal{G}=(\mathcal{N},\mathcal{A},\mathrm{dom},\mathrm{codom},n_0,F)$ be a control graph, $\mathcal{S}$ be some set of memory snapshots, $\mathcal{P}=\{\prob{\cdot}{S,n}\mid n\in\mathcal{N}\setminus F,S\in\mathcal{S}\}$ be a family of probability distributions on $\mathcal{A}$, and $\mathcal{T}=\{g_a\mid a\in\mathcal{A}\}$ be a family of maps from $\mathcal{S}$ into itself then the tuple $(\mathcal{G},\mathcal{S},\mathcal{P},\mathcal{T})$ is called an {\bfseries abstract state machine with stochastic behaviour} if
the following condition holds
\begin{multline}
\mbox{for all }n\in\mathcal{N},S\in\mathcal{S},\mbox{ and }a\in\mathcal{A}\\
\prob{a}{S,n}=0\mbox{ follows from }a\notin\mathrm{Out}(n).
\end{multline}
\end{definition}
Dynamics of such machines is determined by the next definition.
\begin{definition}\label{dfn::rrun}
Let $(\mathcal{G},\mathcal{S},\mathcal{P},\mathcal{T})$ be an abstract state machine with stochastic behaviour,
$\mathcal{G}=(\mathcal{N},\mathcal{A},\mathrm{dom},\mathrm{codom},n_0,F)$ be its control graph then a partial map $C\colon\mathbb{N}\dashrightarrow\mathcal{N}\times\mathcal{A}$ is called a {\bfseries run} of the machine if it satisfies the
following conditions
\begin{itemize}
\item[{\rm(i)}]
$C(0)=(n_0,S)\mbox{, where }S\in\mathcal{S}${\rm;}
\item[{\rm(ii)}]
if $C(t)\neq\emptyset$ for some $t\in \mathbb{N}$ then $C(t')\neq\emptyset$ for all $t'\in \mathbb{N}$ such that $t'<t$
{\rm;}
\item[{\rm(iii)}]
if $\emptyset\neq C(t+1)=(n',S')$ for some $t\in \mathbb{N}$ and $C(t)=(n,S)$ then there exists $a\in\mathrm{Out}(n)$ such
that $\prob{a}{n,S}>0$, $n'=\mathrm{codom}(a)$, and $S'=g_a(S)${\rm;}
\item[{\rm(iv)}]
if $\emptyset\neq C(t)=(n,S)$ for $n\in F$ and $S\in\mathcal{S}$ then $C(t+1)=\emptyset$.
\end{itemize}
\end{definition}
Below such machines will be generalised for the quantum case.
\section{Mathematical Model of Finite-Level Quantum Systems}
In the section the model of quantum systems with finite quantity of levels (finite-level quantum systems) is described.
It is based on the approaches set forth in the works \cite{bib::holevo,bib::nielsen}.
\subsection{Postulates of finite-level quantum systems}
The postulates of finite-level quantum systems fix basic notions which are used to construct mathematical models for the systems.
{\bfseries Postulate 1:} an $n$-dimensional Hilbert space $\mathcal{H}_n$ is associated to any quantum physical system with $n$
levels.
This space is known as the state space of the system.
The system is completely described by its pure state, which is a one-dimensional subspace of the state space.
This subspace is uniquely represented by the ortho-projector $\ket{\psi}\bra{\psi}$ on a vector $\ket{\psi}$ which generates the subspace.
In contrast to pure states mixed states are used to describe quantum systems whose state is not completely known.
Rather more detailed suppose we know that a quantum system is in one of a number of states
$\{\ket{\psi_k}\bra{\psi_k}~\colon~k=1,\dots,m\}$ with respective probabilities $\{p_k\colon k=1,\dots,m\}$.
We shall call $\{p_k,\ket{\psi_k}\bra{\psi_k}~\colon~k=1,\dots,m\}$ an ensemble of pure states.
The density operator for the system is defined by the equation
\[
\rho=\sum\limits_{k=1}^m p_k\ket{\psi_k}\bra{\psi_k}.
\]
We identify mixed states with density operators
\footnote{The set of all density operators on the space $\mathcal{H}_n$ is denoted by $\mathfrak{S}_n$}.
Evidently, that each density operator is a non-negative defined operator which trace is equal to unit.
It is known that the inverse statement is true: a non-negative defined operator, which trace is equal to unit, is a density
operator \cite{bib::holevo}.
Of course, a one-dimensional ortho-projector is a density operator.
The set of all density operators is convex and its subset of one-dimensional ortho-projectors is the subset of its extreme points \cite{bib::holevo}.
This allows to consider pure states as indecomposable states.
{\bfseries Postulate 2:} the state space of a composite physical system is the tensor product of the state spaces of the component physical systems.
Moreover, if we have systems indexed by $k=1,\dots,m$, and the state of the system with number $k$ is described by the density operator $\rho_k$, then the joint state of the total system before any interactions is $\rho_1\otimes\cdots\otimes\rho_m$.
{\bfseries Postulate 3:} the evolution of a closed quantum system is described by a unitary transformation. That is, the state $\ket{\psi}\bra{\psi}$ of the system at time $t_1$ is related to the state $\ket{\psi '}\bra{\psi '}$ of the system at time $t_2$
by a unitary operator $U$ which depends only on the times $t_1$ and $t_2$,
\[
\ket{\psi '}\bra{\psi '}=U\ket{\psi}\bra{\psi}U^\ast.
\]
If we have an ensemble of pure states of the system which is described by the density operator $\rho$ at time $t_1$ then the density operator $\rho '$ of the system at time $t_2$ can be calculated by the formula
\[
\rho '=U\rho U^\ast.
\]
{\bfseries Postulate 4:} quantum measurements are described by an indexed finite family $\mathcal{K}=\{K(x)\colon x\in X\}$ of Kraus' operators, where $X$ is a finite set.
These are operators acting on the state space of the system being measured.
The index $x$ refers to the measurement outcome that may occur in the experiment.
If the state of the quantum system is described by the density operator $\rho$ immediately before the measurement then the
probability that result $x$ occurs is given by the following formula
\footnote{
By $\trace{\cdot}$ the usual operator trace is denoted
}
\begin{equation}\label{frm::prob}
\prob{x}{\rho}=\trace{\rho K(x)^\ast K(x)}
\end{equation}
and the state of the system immediately after the measurement is described by the density operator
\begin{equation}\label{frm::effect}
\effect{\rho}{x}=\frac{K(x)\rho K(x)^\ast}{\trace{\rho K(x)^\ast K(x)}}
\end{equation}
Any Kraus' family $\mathcal{K}=\{K(x)\colon x\in X\}$ satisfies the {\bfseries completeness condition}
\begin{equation}\label{frm::comp}
\sum\limits_{x\in X}K(x)^\ast K(x)=\mathbf{1},
\end{equation}
which ensures correctness of the definitions given by formulas (\ref{frm::prob}) and (\ref{frm::effect}).
\subsection{Measurements and isometric operators. Quantum operations}
Postulate 3 and Postulate 4 describe two different ways of changing a system state.
It looks non-naturally.
Hence, we can set the problem: find a unified description for evolutions and measurements of a finite-level quantum system.
To solve the problem let's introduce for a state space $\mathcal{H}_n$ of an $n$-level quantum system and a finite set $X$
operators $J(x)\colon\mathcal{H}_n\rightarrow\mathcal{H}_n\otimes l^2(X)$ by the formula
\begin{equation}\label{eqn::jop}
J(x)\ket{\psi}=\ket{\psi}\otimes\ket{x},
\end{equation}
where $\ket{x}\in l^2(X)$ such that $\ket{x}(\cdot)=\delta(x,\cdot)$.
Properties of operators from the family $\{J(x)\colon x\in X\}$ are established by the next proposition, which is proved by the
direct calculation.
\begin{proposition}
Let $\mathcal{H}_n$ be a state space of an $n$-level quantum system and $X$ be a finite set, then the operators family
$\{J(x)\colon x\in X\}$ defined by formula {\rm(\ref{eqn::jop})} satisfies the next identities
\begin{eqnarray}
&J(x)^\ast\sum\limits_{x'\in X}(\ket{\psi(x')}\otimes\ket{x'})=\ket{\psi(x)}\\
&J(x')^\ast J(x'')=\delta(x',x'')\cdot\mathbf{1}\\
&J(x')J(x'')^\ast =\mathbf{1}\otimes\ket{x'}\bra{x''}
\label{frm::prjdelta}
\end{eqnarray}
\end{proposition}
Now using a Kraus' family $\mathcal{K}=\{K(x)\colon x\in X\}$ for some measurement let's define an operator
$W_\mathcal{K}\colon\mathcal{H}_n\rightarrow\mathcal{H}_n\otimes l^2(X)$ by the formula
\begin{equation}\label{eqn::iso}
W_\mathcal{K}\ket{\psi}=\sum\limits_{x'\in X}(K(x')\ket{\psi}\otimes\ket{x'}).
\end{equation}
\begin{proposition}\label{prp::kraus2iso}
Let $\mathcal{K}=\{K(x)\colon x\in X\}$ be a Kraus' family, and $W_\mathcal{K}$ be the operator that is built by the formula {\rm(\ref{eqn::iso})}, then $W_\mathcal{K}$ is an isometric operator and the next identities hold for all $x\in X$:
\begin{equation}\label{eqn::krestore}
K(x)=J(x)^\ast W_\mathcal{K}
\end{equation}
\end{proposition}
\begin{proof}
Let $\ket{0},\dots,\ket{n-1}$ be some orthonormal basis in $\mathcal{H}_n$, then for any $k=0,\dots,n-1$ from (\ref{eqn::iso})
we have
\[
W_\mathcal{K}\ket{k}=\sum\limits_{x'\in X}(K(x')\ket{k}\otimes\ket{x'}).
\]
Hence,
\[
W_\mathcal{K}=\sum\limits_{k=0}^{n-1}\sum\limits_{x'\in X}(K(x')\ket{k}\otimes\ket{x'})\bra{k}
\]
and
\[
W_\mathcal{K}^\ast=\sum\limits_{k=0}^{n-1}\sum\limits_{x'\in X}\ket{k}(\bra{k}K(x')^\ast\otimes\bra{x'}).
\]
Therefore,
\begin{multline*}
W_\mathcal{K}^\ast W_\mathcal{K}=\\
\left(\sum\limits_{l=0}^{n-1}\sum\limits_{x''\in X}\ket{l}(\bra{l}K(x'')^\ast\otimes\bra{x''})\right)\cdot
\left(\sum\limits_{k=0}^{n-1}\sum\limits_{x'\in X}(K(x')\ket{k}\otimes\ket{x'})\bra{k}\right)=\\
\sum\limits_{k,l=0}^{n-1}\sum\limits_{x',x''\in X}\ket{l}\bra{l}K(x'')^\ast K(x')\ket{k}\braket{x''}{x'}\bra{k} =\\
\sum\limits_{k,l=0}^{n-1}\sum\limits_{x'\in X}\ket{l}\bra{l}K(x')^\ast K(x')\ket{k}\bra{k}.
\end{multline*}
Using the completeness condition one can obtain that
\[
W_\mathcal{K}^\ast W_\mathcal{K}=\sum\limits_{k,l=0}^{n-1}\ket{l}\braket{l}{k}\bra{k}=\sum\limits_{k=0}^{n-1}\ket{k}\bra{k}=
\mathbf{1}.
\]
The last equation ensures that $W_\mathcal{K}$ is an isometric operator.\\
Equation (\ref{eqn::krestore}) is proved by the direct calculation:
\[
J(x)^\ast W_\mathcal{K}\ket{\psi}=J(x)^\ast\sum\limits_{x'\in X}(K(x')\ket{\psi}\otimes\ket{x'})=K(x)\ket{\psi}.
\]
Proof is complete\qed
\end{proof}
Using Proposition \ref{prp::kraus2iso} one can rewrite formulae (\ref{frm::prob}) and (\ref{frm::effect}) in the following way:
\begin{align}
\prob{x}{\rho}=\trace{\rho W_\mathcal{K}^\ast(\mathbf{1}\otimes\ket{x}\bra{x})W_\mathcal{K}}
\tag{\ref{frm::prob}$'$}\label{frm::prob_bis}\\
\effect{\rho}{x}=\frac{J(x)^\ast W_\mathcal{K}\rho W_\mathcal{K}^\ast J(x)}
{\trace{\rho W_\mathcal{K}^\ast(\mathbf{1}\otimes\ket{x}\bra{x})W_\mathcal{K}}}
\tag{\ref{frm::effect}$'$}\label{frm::eff_bis}
\end{align}
Now we claim that this construction can be inverted.
Really, let $\mathcal{H}_n$ be a state space for an $n$-level quantum system, $X$ be a finite set of outcomes, and
$W\colon\mathcal{H}_n\rightarrow\mathcal{H}_n\otimes l^2(X)$ be an isometric operator.
Let's define a family $\mathcal{K}=\{K(x)\colon x\in X\}$ of operators on the space $\mathcal{H}_n$ by the formula
\begin{equation}\label{frm::kraus}
K(x)=J(x)^\ast W.
\end{equation}
\begin{proposition}\label{prp::iso2kraus}
Let $\mathcal{H}_n$ be a state space for an $n$-level quantum system, $X$ be a finite set of outcomes,
$W\colon\mathcal{H}_n\rightarrow\mathcal{H}_n\otimes l^2(X)$ be an isometric operator, and $\mathcal{K}=\{K(x)\colon x\in X\}$ be
the family of operators which is defined by formula {\rm(\ref{frm::kraus});} then
\begin{enumerate}
\item
$\mathcal{K}$ satisfies the completeness condition and, therefore,
it is a Kraus' family;
\item
$W_\mathcal{K}=W$.
\end{enumerate}
\end{proposition}
\begin{proof}
To prove the completeness condition let's calculate the left side of (\ref{frm::comp}) using (\ref{frm::prjdelta}) and the isometry property
\begin{multline*}
\sum\limits_{x\in X}K(x)^\ast K(x)=\\
\sum\limits_{x\in X}W^\ast J(x)J(x)^\ast W=\sum\limits_{x\in X}W^\ast(\mathbf{1}\otimes\ket{x}\bra{x})W=W^\ast W=\mathbf{1}.
\end{multline*}
To prove the second statement let's calculate using (\ref{frm::prjdelta})
\begin{multline*}
W_\mathcal{K}\ket{\psi} =\\
\sum\limits_{x\in X}(K(x)\ket{\psi}\otimes\ket{x})=\sum\limits_{x\in X}J(x)K(x)\ket{\psi}=
\sum\limits_{x\in X}J(x)(J^\ast(x)W)\ket{\psi}=\\
\sum\limits_{x\in X}(J(x)J^\ast(x))W\ket{\psi}=\sum\limits_{x\in X}(\mathbf{1}\otimes\ket{x}\bra{x})W\ket{\psi}=W\ket{\psi}.
\end{multline*}
Proof is complete\qed
\end{proof}
Proposition \ref{prp::kraus2iso} and \ref{prp::iso2kraus}, formulae (\ref{frm::prob_bis}) and (\ref{frm::eff_bis}) substantiate
replacing the Kraus' families by the corresponding isometric operators under studying the interaction of
quantum systems with classical systems.
This replacing leads us to unification of Postulate 3 and Postulate 4.
To stress such unification we will say that an isomeric operator describes the quantum operation by formulae (\ref{frm::prob_bis}) and (\ref{frm::eff_bis}).
\begin{definition}
Let $\mathcal{H}_n$ be a state space of an $n$-level quantum system, $X$ be a finite set of outcomes, then isometric operators $W_1,W_2\colon\mathcal{H}_n\rightarrow\mathcal{H}_n\otimes l^2(X)$ are called equivalent if for all $x\in X$ and for any density
operator $\rho$ the following equalities are true
\begin{align}
\trace{\rho W_1^\ast(\mathbf{1}\otimes\ket{x}\bra{x})W_1}&=\trace{\rho W_2^\ast(\mathbf{1}\otimes\ket{x}\bra{x})W_2},
\label{frm::probinv}\\
J(x)^\ast W_1\rho W_1^\ast J(x)&=J(x)^\ast W_2\rho W_2^\ast J(x).\label{frm::effinv}
\end{align}
Classes of this equivalence will be called {\bfseries quantum operations} with a set of outcomes $X$.
\end{definition}
Easy to see that isometric operators $W_1,W_2\colon\mathcal{H}_n\rightarrow\mathcal{H}_n\otimes l^2(X)$ describe the same quantum
operation if $J(x)^\ast W_2=\mathrm{e}^{i\theta(x)}J(x)^\ast W_1$ for any $\theta\colon X\rightarrow [0,2\pi)$.
We claim that the inverse statement is true too.
\begin{theorem}\label{thr::eqv}
Let $\mathcal{H}_n$ be a state space of an $n$-level quantum system, $X$ be a finite set of outcomes, $W_1,W_2\colon\mathcal{H}_n\rightarrow\mathcal{H}_n\otimes l^2(X)$ be equivalent isometric operators then
$J(x)^\ast W_2=\mathrm{e}^{i\theta(x)}J(x)^\ast W_1$ for some $\theta\colon X\rightarrow [0,2\pi)$.
\end{theorem}
\begin{proof}
It is evident, that each isometric operator $W_s\colon\mathcal{H}_n\rightarrow\mathcal{H}_n\otimes l^2(X)$, where $s=1,2$, can be represented by the formula
\begin{equation}\label{frm::isovrep}
W_s=\sum\limits_{x\in X}\sum\limits_{k=0}^{n-1}(\ket{\omega_k^{(s)}(x)}\otimes\ket{x})\bra{k},
\end{equation}
where $\{\ket{0},\dots,\ket{n-1}\}$ is an orthonormal basis in $\mathcal{H}_n$ and
$\ket{\omega_k^{(s)}(x)}=J(x)^\ast W_s\ket{k}$ for $k=0,\dots,n-1$ and $x\in X$.
Using representation (\ref{frm::isovrep}) we calculate $\mathrm{Pr}(x\mid \ket{k}\bra{k})$ for $W_s$ where $s=1,2$.
\begin{multline*}
\mathrm{Pr}(x\mid \ket{k}\bra{k})=\\
\trace{\ket{k}\bra{k}W_s^\ast(\mathbf{1}\otimes\ket{x}\bra{x})W_s}=\bra{k}W_s^\ast(\mathbf{1}\otimes\ket{x}\bra{x})W_s)\ket{k}=\\
\bra{k}W_s^\ast(\mathbf{1}\otimes\ket{x}\bra{x})\sum\limits_{x'\in X}\sum\limits_{l=0}^{n-1}
(\ket{\omega_l^{(s)}(x')}\otimes\ket{x'})\braket{l}{k}=
\bra{k}W_s^\ast(\ket{\omega_k^{(s)}(x)}\otimes\ket{x})=\\
\bra{k}\sum\limits_{x'\in X}\sum\limits_{l=0}^{n-1}\ket{l}(\bra{\omega_l^{(s)}(x')}\otimes\bra{x'})
(\ket{\omega_k^{(s)}(x)}\otimes\ket{x})=
\|\omega_k^{(s)}\|^2.
\end{multline*}
Using this and identity (\ref{frm::probinv}) one can derive that for all $k=0,\dots,n-1$ and $x\in X$ the next equality holds
\begin{equation}\label{frm::normeqv}
\|\omega_k^{(1)}(x)\|^2=\|\omega_k^{(2)}(x)\|^2.
\end{equation}
Let $I_s(x)=\{k\mid 0\leq k<n, \|\omega_k^{(s)}(x)\|\neq 0\}$ for each $x\in X$ and $s=0,1$.
From (\ref{frm::normeqv}) it follows that $I_1(x)=I_2(x)$, hence, we can denote this set by $I(x)$.
From (\ref{frm::effinv}) one can derive that for all $x\in X$ and $k\in I(x)$
\begin{equation}\label{eqv::temp}
\ket{\omega_k^{(2)}(x)}\bra{\omega_k^{(2)}(x)}=\ket{\omega_k^{(1)}(x)}\bra{\omega_k^{(1)}(x)}.
\end{equation}
The next equality is obtained by multiplying equality (\ref{eqv::temp}) from left by $\bra{\omega_k^{(1)}(x)}$ and from right by
$\ket{\omega_k^{(1)}(x)}$ and using equality (\ref{frm::normeqv}):
\begin{equation}\label{eqv::temp1}
|\braket{\omega_k^{(2)}(x)}{\omega_k^{(1)}(x)}|^2=\|\omega_k^{(2)}(x)\|^2\cdot\|\omega_k^{(1)}(x)\|^2.
\end{equation}
From the (\ref{eqv::temp1}) and (\ref{frm::normeqv}) it follows that for all $x\in X$ and $k\in I(x)$
\begin{equation}
\ket{\omega_k^{(2)}(x)}=\mathrm{e}^{i\theta(k,x)}\ket{\omega_k^{(1)}(x)},\mbox{ where }0\leq\theta(k,x)<2\pi.
\end{equation}
Further, from (\ref{frm::effinv}) it follows that for all $x\in X$ and $k,l\in I(x)$ the next equality is true:
\[
\ket{\omega_k^{(2)}(x)}\bra{\omega_l^{(2)}(x)}=\ket{\omega_k^{(1)}(x)}\bra{\omega_l^{(1)}(x)}.
\]
Therefore,
\[
\mathrm{e}^{i(\theta(k,x)-\theta(l,x))}\ket{\omega_k^{(1)}(x)}\bra{\omega_l^{(1)}(x)}=
\ket{\omega_k^{(1)}(x)}\bra{\omega_l^{(1)}(x)},
\]
and $\mathrm{e}^{i(\theta(k,x)-\theta(l,x))}=1$.
In summary, we obtain the next equality for all $x\in X$ and $k\in I(x)$
\begin{equation}\label{eqv::temp2}
\ket{\omega_k^{(2)}(x)}=\mathrm{e}^{i\theta(x)}\ket{\omega_k^{(1)}(x)}.
\end{equation}
Using (\ref{eqv::temp2}) for $x\in X$, $k\in I(x)$ and the equality $\ket{\omega_l^{(2)}(x)}=\ket{\omega_l^{(1)}(x)}=0$
for $l\in \{0,\dots, n-1\}\setminus I(x)$ one can get that equality (\ref{eqv::temp2}) is true for all $0\leq k<n$.
Therefore, $J(x)^\ast W_2=\mathrm{e}^{i\theta(x)}J(x)^\ast W_1$ for some $\theta\colon X\rightarrow [0,2\pi)$\qed
\end{proof}
\begin{corollary}
Two isometric operators $W_1,W_2\colon\mathcal{H}_n\rightarrow\mathcal{H}_n\otimes l^2(X)$ define the same quantum operation
if and only if for some $\theta\colon X\rightarrow [0,2\pi)$ the following equality holds
\[
W_2=\Theta W_1\mbox{, where }\Theta=\mathbf{1}\otimes\sum_{x\in X}e^{i\theta(x)}\ket{x}\bra{x}.
\]
\end{corollary}
\section{Abstract Quantum Automata}
Now we describe some class of mathematical models for quantum information processes.
This class we call the class of abstract quantum automata.
\subsection{The notion of an abstract quantum automaton}
\begin{definition}
Let $\mathcal{H}_m (m>1)$ be a state space of an $m$-level quantum system, and let
$\mathcal{G}=(\mathcal{N},\mathcal{A},\mathrm{dom},\mathrm{codom},n_0,F)$ be a control graph.
Suppose that each non-terminal node $n$ of the graph $\mathcal{G}$ is connected with a quantum operation for which $\mathcal{H}_m$
is the state space, $\mathrm{Out}(n)$ is the outcomes set, and
$W_n\colon\mathcal{H}_m\rightarrow\mathcal{H}_m\otimes l^2(\mathrm{Out}(n))$ is an isometric operator describing the operation.
Then the tuple
\[
(\mathcal{H}_m,\mathcal{G},\{W_n\mid n\in\mathcal{N}\setminus F\})
\]
is called an {\bfseries abstract quantum automaton}.
\end{definition}
The next definition describes the set of runs for an abstract quantum automaton similarly to Definition \ref{dfn::rrun}.
\begin{definition}
Let $(\mathcal{H}_m,\mathcal{G},\{W_n\mid n\in\mathcal{N}\setminus F\})$ be an abstract quantum automaton where the control graph
$\mathcal{G}$ is equal to $(\mathcal{N},\mathcal{A},\mathrm{dom},\mathrm{codom},n_0,F)$.
Then a partial map $C\colon\mathbb{N}\dashrightarrow\mathcal{N}\times\mathfrak{S}$ is called a {\bfseries run} of the automaton if
it satisfies the following conditions
\begin{itemize}
\item[{\rm(i)}]
$C(0)=(n_0,\rho)\mbox{ where }\rho\in\mathfrak{S}_m${\rm;}
\item[{\rm(ii)}]
if $C(t)\neq\emptyset$ for some $t\in \mathbb{N}$ then $C(t')\neq\emptyset$ for all $t'\in \mathbb{N}$ such that $t'<t$
{\rm;}
\item[{\rm(iii)}]
if $\emptyset\neq C(t+1)=(n',\rho')$ for some $t\in \mathbb{N}$ and $C(t)=(n,\rho)$ then there exists
$a\in\mathrm{Out}(n)$ such that $\prob{a}{n,\rho}=\trace{\rho W_n^\ast(\mathbf{1}\otimes\ket{a}\bra{a})W_n}>0$,
$n'=\mathrm{codom}(a)$, and $\rho'=\effect{\rho}{n,a}=\dfrac{J(a)^\ast W_n\rho W_n^\ast J(a)}{\prob{a}{n,\rho}}${\rm;}
\item[{\rm(iv)}]
if $\emptyset\neq C(t)=(n,\rho)$ for $n\in F$ and $\rho\in\mathfrak{S}_m$ then $C(t+1)=\emptyset$.
\end{itemize}
\end{definition}
\begin{example}
Let's consider a quantum information process which sets a qubit (2-level quantum system) into the state $\ket{0}\bra{0}$.
Evidently, that this problem can not be solved by any unitary transformation.
We shall specify an abstract quantum automaton that does it.
The control graph of the automaton is shown in Fig. \ref{fig::qucleaner}.
\begin{figure}[h]
\centering
\includegraphics[width=0.7\textwidth]{qucleaner.png}
\caption{Qubit cleaner}
\label{fig::qucleaner}
\end{figure}
As one can see $\mathrm{Out}(m)=\{0,1\}$.
Let's define $W_m\colon\mathcal{H}_2\rightarrow\mathcal{H}_2\otimes l^2(\{0,1\})$ by the formula
\[
W_m\ket{\psi}=\ket{0}\braket{0}{\psi}\otimes\ket{0}+\ket{1}\braket{1}{\psi}\otimes\ket{1}.
\]
Further, $\mathrm{Out}(f)$ is a singleton hence $W_f\colon\mathcal{H}_2\rightarrow\mathcal{H}_2$.
Let's define
\[
W_f\ket{\psi}=\ket{0}\braket{1}{\psi}+\ket{1}\braket{0}{\psi}.
\]
Easy to see that for an arbitrary initial state of a qubit its state after handling by the automaton is equal to $\ket{0}\bra{0}$.
Therefore, we have built the abstract quantum automaton that specifies the process of cleaning a qubit.
\end{example}
\begin{example}
This example deals with preparing an entangled pair of qubits.
We shall specify an abstract quantum automaton that does it.
The control graph of the automaton is shown in Fig. \ref{fig::entanglement}.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{entanglement.png}
\caption{Preparing an entangled pair of qubits}
\label{fig::entanglement}
\end{figure}
Let's define $W_h\colon\mathcal{H}_2\otimes\mathcal{H}_2\rightarrow\mathcal{H}_2\otimes\mathcal{H}_2$ by the formulae
\begin{eqnarray}
W_h(\ket{\psi}\otimes\ket{0})=\ket{\psi}\otimes\dfrac{1}{\sqrt{2}}(\ket{0}+\ket{1}),\notag\\
W_h(\ket{\psi}\otimes\ket{1})=\ket{\psi}\otimes\dfrac{1}{\sqrt{2}}(\ket{0}-\ket{1}).\notag
\end{eqnarray}
and define $W_c\colon\mathcal{H}_2\otimes\mathcal{H}_2\rightarrow\mathcal{H}_2\otimes\mathcal{H}_2$ by the formulae
\begin{eqnarray}
W_c(\ket{\psi}\otimes\ket{0})=&\ket{\psi}\otimes\ket{0},\notag\\
W_c(\ket{\psi}\otimes\ket{1})=&\ket{1}\braket{\psi}{0}\otimes\ket{1}+\ket{0}\braket{\psi}{1}\otimes\ket{1}.\notag
\end{eqnarray}
Easy to see that for an arbitrary initial state of a qubit pair its state after handling by the automaton is equal to
$\dfrac12(\ket{0}\otimes\ket{0}+\ket{1}\otimes\ket{1})(\bra{0}\otimes\bra{0}+\bra{1}\otimes\bra{1})$.
\end{example}
These examples demonstrate that modelling of quantum information processes by abstract quantum automata allows to describe
processes of initial preparation of a quantum memory for quantum computing devices.
\subsection{Quantum teleportation as an abstract quantum automaton}
To complete the paper let's consider the quantum teleportation process and let's show that it can be described by an abstract
quantum automaton.
Quantum teleportation is a process by which a qubit state can be transmitted exactly from one location to another, without the
qubit being transmitted through the intervening space.
This phenomenon has been confirmed experimentally \cite{bib::bou,bib::bos}.
The control graph of the automaton is shown in Fig. \ref{fig::teleportation}.
\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{teleportation.png}
\caption{Teleportation}
\label{fig::teleportation}
\end{figure}
Let's define
$W_c\colon\mathcal{H}_2\otimes\mathcal{H}_2\otimes\mathcal{H}_2\rightarrow
\mathcal{H}_2\otimes\mathcal{H}_2\otimes\mathcal{H}_2$, where the first qubit is Alice's qubit, the second and the third qubits
are the first and the second qubits of the entangled pair respectively, by the formulae
\begin{eqnarray}
W_c(\ket{0}\otimes\ket{0}\otimes\ket{\psi})=&\ket{0}\otimes\ket{0}\otimes\ket{\psi},\notag\\
W_c(\ket{0}\otimes\ket{1}\otimes\ket{\psi})=&\ket{0}\otimes\ket{1}\otimes\ket{\psi},\notag\\
W_c(\ket{1}\otimes\ket{0}\otimes\ket{\psi})=&\ket{1}\otimes\ket{1}\otimes\ket{\psi},\notag\\
W_c(\ket{1}\otimes\ket{1}\otimes\ket{\psi})=&\ket{1}\otimes\ket{0}\otimes\ket{\psi}.\notag
\end{eqnarray}
Further, $\mathrm{Out}(m)=\{00,01,10,11\}$ and the corresponding isometric operator is defined by formulae
\begin{eqnarray}
W_m(\ket{0}\otimes\ket{0}\otimes\ket{\psi})=&\ket{0}\otimes\ket{0}\otimes\ket{\psi}\otimes\ket{00},\notag\\
W_m(\ket{0}\otimes\ket{1}\otimes\ket{\psi})=&\ket{0}\otimes\ket{1}\otimes\ket{\psi}\otimes\ket{01},\notag\\
W_m(\ket{1}\otimes\ket{0}\otimes\ket{\psi})=&\ket{1}\otimes\ket{0}\otimes\ket{\psi}\otimes\ket{10},\notag\\
W_m(\ket{1}\otimes\ket{1}\otimes\ket{\psi})=&\ket{1}\otimes\ket{1}\otimes\ket{\psi}\otimes\ket{11}.\notag
\end{eqnarray}
By direct calculation one can prove that the initial state $\ket{\psi}\bra{\psi}\otimes\rho$ for an arbitrary
$\rho\in\mathfrak{S}_4$ is transformed by the automaton into the state
$\dfrac14(\ket{0}\bra{0}\otimes\ket{0}\bra{0}+\ket{0}\bra{0}\otimes\ket{1}\bra{1}+\ket{1}\bra{1}\otimes\ket{0}\bra{0}+
\ket{1}\bra{1}\otimes\ket{1}\bra{1})\otimes\ket{\psi}\bra{\psi}$.
\section*{Conclusion}
Summarising the above we can conclude:
\begin{itemize}
\item[$-$]
our attempt to solve the Yu. Manin's problem led us towards the notion of an abstract quantum automaton;
\item[$-$]
this notion is based on a computational model known as a machine of A. Kolmogorov and V. Uspensky;
\item[$-$]
abstract quantum automata can be used for formal specification of quantum information processes including non-invertible
processes like qubit cleaning, entangled pair preparing and quantum teleportation.
\end{itemize}
The authors know that quantum algorithms can be specified by using abstract quantum automata but corresponding results are not
given in the paper because they are cumbersome.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,081 |
\section{Introduction}
In these notes we begin the detailed development of the structures underlying quantum mechanics in linearly distributive categories (LDC) and,
thus, in $*$-autonomous categories. Motivating this development is the desire to handle infinite dimensional quantum processes in a categorically agreeable way.
Currently, categorical quantum mechanics is largely based on dagger compact closed categories. A well-known limitation of these settings is that,
while they support finite dimensional processes, they do not support infinite dimensional processes \cite{Heunen16}. A natural categorical generalization of compact closed categories,
which does not have this limitation, is $*$-autonomous categories and, more generally, LDCs. However, this means that one must show that the ideas of categorical
quantum mechanics can transfer to this setting.
At the very outset of such a project, there are some immediate -- and rather paradigm shattering -- problems to be faced. The most obvious question to be answered
is the question of what a dagger LDC ($\dagger$-LDC) might be. Of course, to answer this question is also to answer the question of what ``dagger linear logic'' is. Below we lay the
categorical groundwork for these ideas and, thereby, forge a tight connection between dagger linear logic and categorical quantum mechanics.
Recall that, following the lead of \cite{AC04,Sel07}, it is standard in categorical quantum mechanics to interpret the dagger functor as a stationary on objects involution.
However, in the setting of an LDC and linear logic one expects an involution to flip the tensor and par structure so that $A^\dagger \otimes B^\dagger = (A \oplus B)^\dagger$. This has
the -- perhaps uncomfortable -- effect of implying that the involution in this more general setting can no longer be viewed as being stationary on objects. Of course, having started
on this road it seems prudent also to replace the equality above by a coherent isomorphism $\lambda_\otimes: A^\dagger \otimes B^\dagger \@ifnextchar^ {\t@@}{\t@@^{}} (A \oplus B)^\dagger$ and the involution by an
isomorphism $\iota_A: A \@ifnextchar^ {\t@@}{\t@@^{}} (A^\dagger)^\dagger$.
At this juncture it is perhaps appropriate to acknowledge that these are not new ideas. Models for quantum mechanics in $*$-autonomous categories are often described
as ``toy models'' \cite{Abr12} and were, in particular discussed by Pavlovic \cite{Pav11} where some very similar directions were advocated. Indeed, Egger \cite{Egg11}, in initiating the
development of ``involutive'' categories , was also implicitly suggesting that dagger functors should not necessarily be regarded as being stationary on objects in these more
general settings. Section \ref{daggers-duals-conjugation} is essentially a realization of Egger's ideas.
There are, nonetheless, some significant problems with allowing a non-stationary dagger. The first is that one gets inundated by coherence issues. These notes do provide a path through
these coherence issues, and, hopefully show -- once one has assimilated all the structure -- that these issues are not nearly as terrible as might be expected. However, we are forced to concede that they are non-trivial. The next problem, once the coherences are under control, is that one would like to be able to say what a unitary isomorphism is with respect to such a non-stationary involution. How this may be accomplished seems, at first glance, less than obvious.
The fact that the dagger functor is an involution with a coherent isomorphism $\iota_A: A \@ifnextchar^ {\t@@}{\t@@^{}} (A^\dagger)^\dagger$ makes it natural to view a
{\bf unitary object} as an object with an isomorphism $\varphi_A: A \@ifnextchar^ {\t@@}{\t@@^{}} A^\dagger$, such that $\iota_A = \varphi_A (\varphi_A^{-1})^\dagger$: we refer to $\varphi_A$ as
the {\bf unitary structure} of $A$. Considering this, with the expected coherent behaviour of this unitary structure, leads one to realize that, for unitary objects, we
must have $A \otimes B \simeq A \oplus B$. This, in turn, leads one to ask how this can happen in an LDC. Fortunately, there is a theory which has been developed
for this situation: namely that of LDCs with mix \cite{CS97,Pav11}. An LDC with
mix has a coherent map ${\sf m}: \bot \@ifnextchar^ {\t@@}{\t@@^{}} \top$, called the {\bf mix map}, which, in turn, induces a map ${\sf mx}: A \otimes B \@ifnextchar^ {\t@@}{\t@@^{}} A \oplus B$, called the {\bf mixor}. In a mix LDC
we say an object $A$ is in the {\bf core} \cite{BCS00} in case the mixor for that object and with other object, ${\sf mx}_{A,X}: A \otimes X \@ifnextchar^ {\t@@}{\t@@^{}} A \oplus X$, is an isomorphism.
Clearly, if we are to interpret our earlier expectation as the requirement that unitary objects live in the core, then we need to be in a $\dagger$-LDC with mix, which
we shall refer to as a $\dagger$-mix category. If, further, we want our unitary objects to form a compact LDC (i.e one in which the mixor is always an isomorphism) then
we must also ask that the mix map, ${\sf m}$, be an isomorphism; that is that the category be an {\bf isomix} category. The first milestone of the paper is therefore to
collect all this structure into what we call a {\bf $\dagger$-isomix category}.
Amidst the introduction of all this structure, the astute reader may notice that we have completely failed to elucidate how the unitary maps arise. Let us hasten to correct
this oversight: a unitary isomorphism $f: A \@ifnextchar^ {\t@@}{\t@@^{}} B$ is an isomorphism between unitary objects which is (twisted) natural with respect to the unitary structures, $\varphi_A$ and $\varphi_B$,
\footnote{This formulation of unitary maps is not completely original as a lively discussion of whether $\dagger$-categories were ``evil'' led
Peter Lumsdaine to suggest in the math overflow forum \cite{Lums15} how they might be made a little less evil. These ideas never took off, perhaps because it was pointed
out by Peter Selinger that, when one regarded something as evil if it was not preserved by equivalence, then it was impossible for dagger not to be evil! Here we quite explicitly
have ``unitary structure'' which is of course is not only not necessarily unique but also will not necessarily be preserved by an equivalence. Thus, like all structure, it is thoroughly evil!}
in the sense that the following diagram is rendered commutative:
\[ \xymatrix{ A \ar[d]_{\varphi_A} \ar[rr]^f & & B \ar[d]^{\varphi_B} \\ A^\dagger && B^\dagger \ar[ll]^{f^\dagger} } \]
The coherences requirements on unitary structure then have the pleasing effect of forcing all the coherence isomorphisms, for the full subcategory of unitary objects,
to be unitary maps.
A {\bf unitary category} is a $\dagger$-isomix category in which all objects have a unitary structure. Unitary categories are necessarily compact LDCs and so are rather special.
In fact, we show that they are $\dagger$-linearly equivalent to the more standard categorical quantum mechanical notion of a dagger monoidal category -- and, furthermore, that a closed
unitary category is linearly equivalent to a dagger compact closed category. One may -- with some justification -- feel that one has come full circle at this stage as the standard
structures from categorical quantum mechanics seem to be emerging. However, once one has met this new notion of unitary it is hard to look back! Indeed, the $\dagger$-linear
equivalence is simply transferring the unitary structure into the functor (and its preservator) so that the fact that there is an equivalence -- although philosophically important -- is not practically so useful.
A mixed unitary category (MUC) is a strong $\dagger$-Frobenius functor $M: \mathbb{U} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$, which has domain a small unitary category and codomain the core of an ``large'' $\dagger$-isomix category.
One may think of the unitary category acting on the larger category, much as a field $K$ acts on a $K$-algebra as scalars. Expressing these categories in this manner allows an obvious notion of functor as a square of $\dagger$-Frobenius functors, whose component on the unitary categories preserves unitary structure, and which commutes up to a linear natural isomorphism. An important objective of these notes was to establish the functorial properties of these settings; the ability to move functorially between the different examples of these settings
allows one to see, on the one hand, important relationships between examples and, on the other, the naturality of constructions.
The fact that the unitary category sits in the core of a mix unitary category allows one to mimic, for example, the ${\sf CP}^\infty$ construction \cite{CH16} in a way which displays
some interesting features. To start with the ancillary objects must now necessarily be chosen from the unitary category and these can be supposed to be an essentially
small class (with respect to unitary maps) even if the overall category is large (which seems to happen in examples). Although all this is beyond the scope of these notes -- but
planned for future notes -- the resulting category is under certain reasonable assumptions a MUC which has environment structure \cite{Coecke10} in a suitable sense. Furthermore, In the presence of duals the whole construction can be made functorial.
These notes end by exploring some examples. These, in particular, show how to generate MUCs with both trivial and non-trivial unitary structure.
An important example is the $*$-autonomous category of finiteness modules (or matrices) over the complex numbers \cite{Ehrhard}.
Here the maps are infinite dimensional matrices whose support is carefully controlled by the finiteness structure. The dagger on the category is given, essentially,
by transposing and conjugating the matrices: on objects it is given by taking the dual finiteness space. Finiteness modules form an iso-mix $\dagger$-LDC which, furthermore,
is $*$-autonomous. An object is in the core if and only if it's underlying set is finite and these objects are also exactly the unitary objects. The unitary structure of a finite object
in this case is the identity map (so the unitary structure is ``trivial'') -- which does mean that the coherence requirements are immediately satisfied.
Another source of examples is from the Chu construction, \cite{Bar06}, where the dualizing object is set to the tensor unit. This is always produces an isomix $*$-autonomous category: to obtain a dagger, using the constructions in these notes, one needs in addition a conjugation. Considering the Chu construction over complex vector spaces there is an obvious notion of conjugation which means that this category forms a $\dagger$-isomix category. From there one can obtain a non-trivial MUC by pulling off the pre-unitary objects. More fully understanding, the MUCs generated, in this manner, from Chu categories is left for future work -- as is another interesting source of examples which use Joyal's bicompletion procedure.
\section{Linearly distributive categories}
This section recalls some background concepts from the theory of linearly distributive categories which shall be useful in this development. The definition of linearly distributive categories is available in \cite{CS97,BCST96}. Here we briefly recall the definition of linear functors and their transformations \cite{CS99}, the notion of a linear adjoint \cite{CKS00} -- which we shall refer to as a linear dual -- and the notion of a mix category and its core \cite{CS97a,BCS00}.
\subsection{Linearly distributive categories, functors, and transformations}
A {\bf linearly distributive category (LDC)} is a category with two monoidal structures $(\otimes, \top, a_\otimes, u_\otimes^L, u_\otimes^R)$ and $(\oplus, \bot, a_\oplus, u_\oplus^L, u_\oplus^R)$ linked by natural transformations called the linear distributors:
\begin{align*}
&\partial^L: A \otimes (B \oplus C) \rightarrow (A \otimes B) \oplus C \\
& \partial^R: (B \oplus C) \otimes A \rightarrow B \oplus (A \otimes C)
\end{align*}
such that the monoidal natural isomorphisms interact coherently with the linear distributors, see \cite{BCST96, CS97} for more details. A symmetric LDC is an LDC in which both monoidal structures are symmetric, with symmetry maps $c_\otimes$ and $c_\oplus$, such that
$\partial^R = c_\otimes (1 \otimes c_\oplus) \partial^L (c_\otimes \oplus 1) c_\oplus$.
\begin{definition} \cite[Definition 1]{CS97}
Given linearly distributive categories $\mathbb{X}$ and $\mathbb{Y}$, a linear functor $F: \mathbb{X} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{Y}$ consists of
\begin{enumerate}[(i)]
\item a pair of functors: $F_\otimes$ which is monoidal with respect to $\otimes$ and $F_\oplus$ which is comonoidal with respect to $\oplus$
\item natural transformations:
\begin{align*}
\nu_\otimes^R &: F_\otimes(A \oplus B) \@ifnextchar^ {\t@@}{\t@@^{}} F_\oplus(A) \oplus F_\otimes(B) \\
\nu_\otimes^L &: F_\otimes(A \oplus B) \@ifnextchar^ {\t@@}{\t@@^{}} F_\otimes(A) \oplus F_\oplus(B) \\
\nu_\oplus^R &: F_\otimes(A) \otimes F_\oplus(B) \@ifnextchar^ {\t@@}{\t@@^{}} F_\oplus(A \otimes B) \\
\nu_\oplus^L &: F_\oplus(A) \otimes F_\otimes(B) \@ifnextchar^ {\t@@}{\t@@^{}} F_\oplus( A \otimes B)
\end{align*}
\end{enumerate}
such that the following coherence conditions hold:
\begin{enumerate}[{\bf \small [LF.1]}]
\item
\begin{enumerate}[(a)]
\item $F_\otimes(u^L_\oplus) = \nu^R_\otimes (n_\bot \oplus 1) u^L_\oplus$
\[ \xymatrix{
F_\otimes( \bot \oplus A) \ar[r]^{F_\otimes(u_\oplus^L)} \ar[d]_{\nu_\otimes^R} & F_\otimes(A) \\
F_\oplus(\bot) \oplus F_\otimes(A) \ar[r]_{n_\bot \oplus 1} & \bot \oplus F_\otimes(A) \ar[u]_{u_\oplus^L}
} \]
\item $\nu_\otimes^L ( 1 \oplus n_\bot ) u_\oplus^R = F_\otimes( u_\oplus^R) $
\item $(u_\otimes^L)^{-1} (m_\top \otimes 1) \nu_\oplus^R = F_\oplus((u_\otimes^L)^{-1})$
\item $(u_\otimes^R)^{-1} (m_\top \otimes 1) \nu_\oplus^L = F_\oplus((u_\otimes^R)^{-1}) $
\end{enumerate}
\item
\begin{enumerate}[(a)]
\item $F_\otimes(a_\oplus) \nu^R_\otimes (1 \oplus \nu^R_\otimes) = \nu^R_\otimes (n_\oplus \oplus 1) a_\oplus$
\[ \xymatrix{
F_\otimes((A \oplus B) \oplus C) \ar[r]^{F_\otimes(a_\oplus)} \ar[r]^{F_\otimes(a_\oplus)} \ar[d]_{\nu_\otimes^R} &
F_\otimes(A \oplus (B \oplus C)) \ar[d]^{\nu_\otimes^R} \\
F_\oplus(A \oplus B) \oplus F_\otimes(C) \ar[d]_{n_\oplus \oplus 1} &
F_\oplus(A) \oplus F_\otimes (B \oplus C) \ar[d]^{1 \oplus \nu_\otimes^R} \\
(F_\oplus (A) \oplus F_\oplus(B)) \oplus F_\otimes(C) \ar[r]_{a_\oplus} &
F_\oplus(A) \oplus (F_\oplus(B) \oplus F_\oplus(C))
} \]
\item $F_\otimes(a_\oplus) \nu_\otimes^L (1 \oplus n_\oplus ) = \nu^L_\oplus (\nu^L \oplus 1 ) a_\oplus$
\item $(m_\otimes \otimes 1) \nu_\oplus^R F_\oplus(a_\otimes) = a_\otimes (1 \otimes \nu_\oplus^R) \nu_\oplus^R$
\item $(\nu^R_\oplus \otimes 1) \nu_\oplus^L F_\oplus(a_\otimes) = a_\otimes (1 \otimes m_\otimes) \nu_\oplus^L$
\end{enumerate}
\item
\begin{enumerate}[(a)]
\item $F_\otimes(a_\oplus)\nu^R_\otimes(1 \oplus \nu^L_\otimes) = \nu_\otimes^L (\nu^R_\otimes \oplus 1) a_\oplus$
\[ \xymatrix{
F_\otimes((A \oplus B) \oplus C) \ar[r]^{F_\otimes(a_\oplus)} \ar[d]_{\nu_\otimes^L} &
F_\otimes(A \oplus (B \oplus C)) \ar[d]^{\nu_\otimes^R} \\
F_\otimes(A \oplus B) \oplus F_\oplus(C) \ar[d]_{\nu_\otimes^R \oplus 1} &
F_\oplus(A) \oplus F_\otimes (B \oplus C) \ar[d]^{1 \oplus \nu_\otimes^L} \\
(F_\oplus (A) \oplus F_\otimes(B)) \oplus F_\oplus(C) \ar[r]_{a_\oplus}&
F_\oplus(A) \oplus (F_\otimes(B) \oplus F_\oplus(C))}
\]
\item $(\nu^R_\oplus \otimes 1) \nu^L_\oplus F_\oplus(a_\otimes) = a_\otimes (1 \otimes \nu_\oplus^L) \nu_\oplus^R$
\end{enumerate}
\item
\begin{enumerate}[(a)]
\item $(1 \otimes \nu^R_\otimes) \delta^L (\nu^R_\oplus \oplus 1) = m_\otimes F_\otimes(\delta^L) \nu^R_\otimes$
\[ \xymatrix{
F_\otimes(A) \otimes F_\otimes(B \oplus C) \ar[r]^{1 \otimes \nu_\otimes^R} \ar[d]_{m_\otimes} &
F_\otimes(A) \otimes (F_\oplus(B) \oplus F_\otimes(C)) \ar[d]^{\delta^L} \\
F_\otimes(A \otimes (B \oplus C)) \ar[d]_{F_\otimes(\delta^L)} &
(F_\otimes(A) \otimes F_\oplus(B)) \oplus F_\otimes(C) \ar[d]^{\nu_\oplus^R \oplus 1} \\
F_\otimes((A \otimes B) \oplus C) \ar[r]_{\nu_\otimes^R} &
F_\oplus(A \oplus B) \oplus F_\otimes(C)
} \]
\item $(\nu_\otimes^L \otimes 1) \delta^R (1 \oplus \nu_\oplus^L) = m_\otimes F_\otimes(\delta^R) \nu_\otimes^L$
\item $(1 \otimes \nu_\otimes^L) \delta^L (\nu_\oplus^L \oplus 1) = \nu_\oplus^L F_\oplus(\delta^L) n_\oplus$
\item $(\nu_\otimes^R \otimes 1) \delta^R (1 \oplus \nu_\oplus^R) = \nu_\oplus^R F_\oplus(\delta^R) n_\oplus $
\end{enumerate}
\item
\begin{enumerate}[(a)]
\item $(1 \otimes \nu^L_\otimes) \delta^L(m_\otimes \oplus 1) = m_\otimes F_\otimes(\delta^L) \nu^L_\otimes$
\[ \xymatrix{
F_\otimes(A) \otimes F_\otimes(B \oplus C) \ar[r]^{1 \otimes \nu_\otimes^L} \ar[d]_{m_\otimes} &
F_\otimes(A) \otimes (F_\otimes(B) \oplus F_\oplus(C)) \ar[d]^{\delta^L} \\
F_\otimes(A \otimes (B \oplus C)) \ar[d]_{F_\otimes(\delta^L)} &
(F_\otimes(A) \otimes F_\otimes(B)) \oplus F_\oplus(C) \ar[d]^{m_\otimes \oplus 1} \\
F_\otimes((A \otimes B) \oplus C) \ar[r]_{\nu_\otimes^L} &
F_\otimes(A \otimes B) \oplus F_\oplus(C)
}\]
\item $(\nu_\otimes^R \otimes 1) \delta^R (1 \oplus m_\otimes) = m_\otimes F_\otimes(\delta^R) \nu_\otimes^R$
\item $(1 \otimes n_\oplus) \delta^L (\nu_\oplus^R \oplus 1) = \nu_\oplus^R F_\oplus(\delta^L) n_\oplus $
\item $(n_\oplus \otimes 1) \delta^R (1 \oplus \nu_\oplus^L) = \nu_\oplus^L F_\oplus(\delta^R) n_ \oplus $
\end{enumerate}
\end{enumerate}
\end{definition}
The linear strengths are drawn in the graphical calclulus as follows:
\[
\nu_\oplus^L = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 2) {};
\node [style=none] (1) at (-1.5, 2) {};
\node [style=none] (2) at (-2.25, 2) {};
\node [style=none] (3) at (-2.25, 1) {};
\node [style=none] (4) at (-0.75, 1) {};
\node [style=none] (5) at (-0.75, 2) {};
\node [style=none] (6) at (-2, 2.75) {};
\node [style=none] (7) at (-1, 2.75) {};
\node [style=none] (61) at (-2.75, 2.75) {$F_\oplus(A)$};
\node [style=none] (71) at (-0.25, 2.75) {$F_\otimes(B)$};
\node [style=none] (8) at (-1.5, 0.25) {};
\node [style=none] (81) at (-2.25, 0) {$F_\oplus(A \otimes B) $};
\node [style=ox] (9) at (-1.5, 1.5) {};
\node [style=none] (10) at (-2, 1.25) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=-90, out=-90, looseness=1.25] (0.center) to (1.center);
\draw [bend right=15, looseness=1.00] (6.center) to (9);
\draw [bend right=15, looseness=0.75] (9) to (7.center);
\draw (2.center) to (3.center);
\draw (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (5.center) to (2.center);
\draw (9) to (8.center);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~~ \nu_\oplus^R = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-1.5, 2) {};
\node [style=none] (1) at (-1, 2) {};
\node [style=none] (2) at (-2.25, 2) {};
\node [style=none] (3) at (-2.25, 1) {};
\node [style=none] (4) at (-0.75, 1) {};
\node [style=none] (5) at (-0.75, 2) {};
\node [style=none] (6) at (-2, 2.75) {};
\node [style=none] (7) at (-1, 2.75) {};
\node [style=none] (61) at (-2.75, 2.75) {$F_\otimes(A)$};
\node [style=none] (71) at (-0.25, 2.75) {$F_\oplus(B)$};
\node [style=none] (8) at (-1.5, 0.25) {};
\node [style=none] (81) at (-2.25, 0) {$F_\oplus(A \otimes B) $};
\node [style=ox] (9) at (-1.5, 1.5) {};
\node [style=none] (10) at (-2, 1.25) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=-90, out=-90, looseness=1.25] (0.center) to (1.center);
\draw [bend right=15, looseness=1.00] (6.center) to (9);
\draw [bend right=15, looseness=0.75] (9) to (7.center);
\draw (2.center) to (3.center);
\draw (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (5.center) to (2.center);
\draw (9) to (8.center);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~~ \nu_\oplus^L = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.25, 1) {};
\node [style=none] (1) at (-2.25, 2) {};
\node [style=none] (2) at (-0.75, 2) {};
\node [style=none] (3) at (-0.75, 1) {};
\node [style=none] (4) at (-2, 0.25) {};
\node [style=none] (5) at (-1, 0.25) {};
\node [style=none] (6) at (-1.5, 2.75) {};
\node [style=none] (41) at (-2.75, 0.25) {$F_\otimes(A)$};
\node [style=none] (51) at (-0.25, 0.25) {$F_\oplus(B)$};
\node [style=none] (61) at (-2, 3) {$F_\otimes(A \oplus B)$};
\node [style=oa] (7) at (-1.5, 1.5) {};
\node [style=none] (8) at (-2, 1.75) {$F_\oplus$};
\node [style=none] (9) at (-2, 1) {};
\node [style=none] (10) at (-1.5, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=15, looseness=1.00] (4.center) to (7);
\draw [bend left=15, looseness=0.75] (7) to (5.center);
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (0.center);
\draw (7) to (6.center);
\draw [in=90, out=90, looseness=1.25] (9.center) to (10.center);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~~~ \nu_\otimes^R = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.25, 1) {};
\node [style=none] (1) at (-2.25, 2) {};
\node [style=none] (2) at (-0.75, 2) {};
\node [style=none] (3) at (-0.75, 1) {};
\node [style=none] (4) at (-2, 0.25) {};
\node [style=none] (5) at (-1, 0.25) {};
\node [style=none] (41) at (-2.75, 0.25) {$F_\oplus(A)$};
\node [style=none] (51) at (-0.25, 0.25) {$F_\otimes(B)$};
\node [style=none] (61) at (-2, 3) {$F_\otimes(A \oplus B)$};
\node [style=none] (6) at (-1.5, 2.75) {};
\node [style=oa] (7) at (-1.5, 1.5) {};
\node [style=none] (8) at (-2, 1.75) {$F$};
\node [style=none] (9) at (-1.5, 1) {};
\node [style=none] (10) at (-1, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=15, looseness=1.00] (4.center) to (7);
\draw [bend left=15, looseness=0.75] (7) to (5.center);
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (0.center);
\draw (7) to (6.center);
\draw [in=90, out=90, looseness=1.25] (9.center) to (10.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\[
m_\otimes = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.25, 2) {};
\node [style=none] (1) at (-2.25, 1) {};
\node [style=none] (2) at (-0.75, 1) {};
\node [style=none] (3) at (-0.75, 2) {};
\node [style=none] (4) at (-2, 2.75) {};
\node [style=none] (5) at (-1, 2.75) {};
\node [style=none] (41) at (-2.75, 2.75) {$F_\otimes(A)$};
\node [style=none] (51) at (-0.25, 2.75) {$F_\otimes(B)$};
\node [style=none] (6) at (-1.5, 0.25) {};
\node [style=none] (61) at (-2, 0) {$F_\otimes(A \otimes B)$};
\node [style=ox] (7) at (-1.5, 1.5) {};
\node [style=none] (8) at (-2, 1.25) {$F$};
\node [style=none] (9) at (-1.75, 1) {};
\node [style=none] (10) at (-1.25, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=15, looseness=1.00] (4.center) to (7);
\draw [bend right=15, looseness=0.75] (7) to (5.center);
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (0.center);
\draw (7) to (6.center);
\draw [in=90, out=90, looseness=1.25] (9.center) to (10.center);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~ m_\top = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (8) at (0, 1) {};
\node [style=circle] (0) at (0, -0) {$\top$};
\node [style=none] (1) at (0.75, -0.5) {};
\node [style=none] (2) at (2.75, -0.5) {};
\node [style=none] (3) at (0.75, -1.5) {};
\node [style=none] (4) at (2.75, -1.5) {};
\node [style=none] (5) at (1.75, -2.5) {};
\node [style=circle] (6) at (1.75, -1) {$\top$};
\node [style=circle, scale=0.5] (7) at (1.75, -2) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\draw (2.center) to (4.center);
\draw (4.center) to (3.center);
\draw (3.center) to (1.center);
\draw (6) to (5.center);
\draw [dotted, in=-165, out=-90, looseness=1.25] (0) to (7);
\draw (0) to (8.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (8) at (2.75, 1) {};
\node [style=circle] (0) at (2.75, -0) {$\top$};
\node [style=none] (1) at (2, -0.5) {};
\node [style=none] (2) at (0, -0.5) {};
\node [style=none] (3) at (2, -1.5) {};
\node [style=none] (4) at (0, -1.5) {};
\node [style=none] (5) at (1, -2.5) {};
\node [style=circle] (6) at (1, -1) {$\top$};
\node [style=circle, scale=0.5] (7) at (1, -2) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\draw (2.center) to (4.center);
\draw (4.center) to (3.center);
\draw (3.center) to (1.center);
\draw (6) to (5.center);
\draw (0) to (8.center);
\draw [dotted, in=-15, out=-90, looseness=1.25] (0) to (7);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~ n_\oplus = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.25, 1) {};
\node [style=none] (1) at (-2.25, 2) {};
\node [style=none] (2) at (-0.75, 2) {};
\node [style=none] (3) at (-0.75, 1) {};
\node [style=none] (4) at (-2, 0.25) {};
\node [style=none] (5) at (-1, 0.25) {};
\node [style=none] (41) at (-2.75, 0.25) {$F_\oplus(A)$};
\node [style=none] (51) at (-0.25, 0.25) {$F_\oplus(B)$};
\node [style=none] (61) at (-2, 3) {$F_\oplus(A \oplus B)$};
\node [style=none] (6) at (-1.5, 2.75) {};
\node [style=oa] (7) at (-1.5, 1.5) {};
\node [style=none] (8) at (-2, 1.75) {$F$};
\node [style=none] (9) at (-1.25, 2) {};
\node [style=none] (10) at (-1.75, 2) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=15, looseness=1.00] (4.center) to (7);
\draw [bend left=15, looseness=0.75] (7) to (5.center);
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (0.center);
\draw (7) to (6.center);
\draw [in=-90, out=-90, looseness=1.25] (9.center) to (10.center);
\end{pgfonlayer}
\end{tikzpicture}
~~~~~~~ n_\bot =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 2) {};
\node [style=none] (1) at (-3, 1) {};
\node [style=none] (2) at (-1, 2) {};
\node [style=none] (3) at (-1, 1) {};
\node [style=circle] (4) at (-2, 1.5) {$\bot$};
\node [style=circle] (5) at (0, 1) {$\bot$};
\node [style=none] (6) at (-2, 3) {};
\node [style=circle, scale=0.5] (7) at (-2, 2.5) {};
\node [style=none] (8) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (6.center) to (4);
\draw (8.center) to (5);
\draw [dotted, bend left=45, looseness=1.25] (7) to (5);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 2) {};
\node [style=none] (1) at (0, 1) {};
\node [style=none] (2) at (-2, 2) {};
\node [style=none] (3) at (-2, 1) {};
\node [style=circle] (4) at (-1, 1.5) {$\bot$};
\node [style=circle] (5) at (-3, 1) {$\bot$};
\node [style=none] (6) at (-1, 3) {};
\node [style=circle, scale=0.5] (7) at (-1, 2.5) {};
\node [style=none] (8) at (-3, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (6.center) to (4);
\draw (8.center) to (5);
\draw [dotted, bend right=45, looseness=1.25] (7) to (5);
\end{pgfonlayer}
\end{tikzpicture}
\]
These are subject to a very natural ``box eats box'' calculus described in \cite{CS99}. The ``ports'' which highlight one of the interfaces of these boxes is important as two boxes can only be amalgamated if (exactly) one box has a port on the interface.
When working in the categorical doctrine of {\em symmetric} LDCs we will expect the linear functors to preserve the symmetry. Thus, a {\bf symmetric linear functor} is a linear functor $F= (F_\otimes,F_\oplus)$
which satisfies in addition:
\[ \xymatrix{F_\otimes(A) \otimes F_\otimes(B) \ar[d]_{c_\otimes} \ar[rr]^{m_\otimes} & & F_\otimes(A \otimes B) \ar[d]^{F_\otimes(c_\otimes)} \\
F_\otimes(B) \otimes F_\otimes(A) \ar[rr]_{m_\otimes} & & F_\otimes(B \otimes A) }
~~~~~~
\xymatrix{F_\oplus(A \oplus B) \ar[d]_{F_\oplus(c_\oplus)} \ar[rr]^{n_\otimes} & & F_\oplus(A) \oplus F_\oplus(B)\ar[d]^{c_\oplus} \\
F_\oplus(B \otimes A) \ar[rr]_{n_\oplus} & & F_\oplus(B) \oplus F_\oplus(A) } \]
Natural transformations between linear functors also break into two components linking respectively the tensor functors and, in the opposite direction, the par functors:
\begin{definition} \cite[Definition 3]{CS99}
A {\bf linear transformation}, $\alpha: F \@ifnextchar^ {\t@@}{\t@@^{}} G$, between parallel linear functors $F,G: \mathbb{X} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{Y}$ consists of a pair of natural transformations $\alpha = (\alpha_\otimes,\alpha_\oplus)$ such that $\alpha_\otimes: F_\otimes \@ifnextchar^ {\t@@}{\t@@^{}} G_\otimes$ is a monoidal transformation and $\alpha_\oplus: G_\oplus \@ifnextchar^ {\t@@}{\t@@^{}} F_\oplus$ is a comonoidal transformation satisfying the following coherence conditions:
\begin{enumerate}[{\bf \small [LT.1]}]
\item $a_\otimes \nu^R_\otimes (a_\oplus \oplus 1) = \nu^R_\otimes (1 \oplus a_\otimes)$
\[
\xymatrix{
F_\otimes (A \oplus B) \ar[rr]^{\alpha_\otimes} \ar[d]_{\nu_\otimes^R} & & G_\otimes(A \oplus B) \ar[d]^{\nu_\otimes^R} \\
F_\oplus(A) \oplus F_\otimes(B) \ar[dr]_{1 \oplus \alpha_\otimes} & & G_\oplus(A) \oplus G_\otimes(B) \ar[ld]^{\alpha_\oplus \oplus 1} \\
& F_\oplus(A) \oplus G_\otimes(B)&
}
\]
\item $\alpha_\otimes \nu_\otimes^L (1 \oplus \alpha_\oplus) = \nu_\otimes^L (\alpha_\otimes \oplus 1)$
\item $(1 \otimes \alpha_\otimes) \nu_\oplus^L (\alpha_\oplus) = (\alpha_\oplus \otimes 1) \nu_\oplus^L$
\item $(\alpha_\otimes \otimes 1) \nu_\oplus^R \alpha_\oplus = (1 \otimes \alpha_\oplus) \nu_\oplus^R$
\end{enumerate}
\end{definition}
Conditions {\bf [LT.1]} - {\bf [LT.4]} are represented graphically as follows:
\[
\mbox{\small\bf [LT.1]}~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.5, -0.75) {};
\node [style=none] (1) at (-0.5, 0.25) {};
\node [style=none] (2) at (-2.5, 0.25) {};
\node [style=none] (3) at (-2.5, -0.75) {};
\node [style=oa] (4) at (-1.5, -0.25) {};
\node [style=none] (5) at (-1, -2) {};
\node [style=none] (6) at (-2, -2) {};
\node [style=none] (7) at (-1.5, 1.25) {};
\node [style=circle, scale=2] (8) at (-1, -1.25) {};
\node [style=none] (9) at (-1, -1.25) {$\alpha_\otimes$};
\node [style=none] (10) at (-0.75, -0) {$F$};
\node [style=none] (11) at (-1.25, -0.75) {};
\node [style=none] (12) at (-0.75, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw [in=90, out=-150, looseness=1.00] (4) to (6.center);
\draw (4) to (7.center);
\draw (5.center) to (8);
\draw [in=-30, out=90, looseness=1.00] (8) to (4);
\draw [bend left=90, looseness=1.25] (11.center) to (12.center);
\end{pgfonlayer}
\end{tikzpicture}
= \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.5, -0.75) {};
\node [style=none] (1) at (-0.5, 0.25) {};
\node [style=none] (2) at (-2.5, 0.25) {};
\node [style=none] (3) at (-2.5, -0.75) {};
\node [style=oa] (4) at (-1.5, -0.25) {};
\node [style=none] (5) at (-1, -2) {};
\node [style=none] (6) at (-2, -2) {};
\node [style=none] (7) at (-2, -1.25) {$\alpha_\oplus$};
\node [style=circle, scale=2] (8) at (-1.5, 1) {};
\node [style=none] (9) at (-1.5, 2) {};
\node [style=none] (10) at (-1.5, 1) {$\alpha_\otimes$};
\node [style=none] (11) at (-0.75, -0) {$G$};
\node [style=circle, scale=2] (12) at (-2, -1.25) {};
\node [style=none] (13) at (-1.5, -0.75) {};
\node [style=none] (14) at (-1, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw (9.center) to (8);
\draw (8) to (4);
\draw [in=90, out=-44, looseness=0.75] (4) to (5.center);
\draw [bend left=90, looseness=1.25] (13.center) to (14.center);
\draw (6.center) to (12);
\draw [bend left, looseness=1.00] (12) to (4);
\end{pgfonlayer}
\end{tikzpicture} ~~~\mbox{\small\bf [LT.2]}~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.5, -0.75) {};
\node [style=none] (1) at (-2.5, 0.25) {};
\node [style=none] (2) at (-0.5, 0.25) {};
\node [style=none] (3) at (-0.5, -0.75) {};
\node [style=oa] (4) at (-1.5, -0.25) {};
\node [style=none] (5) at (-2, -2) {};
\node [style=none] (6) at (-1, -2) {};
\node [style=none] (7) at (-1.5, 1.25) {};
\node [style=circle, scale=2] (8) at (-2, -1.25) {};
\node [style=none] (9) at (-2, -1.25) {$\alpha_\otimes$};
\node [style=none] (10) at (-0.75, 0) {$F$};
\node [style=none] (11) at (-1.75, -0.75) {};
\node [style=none] (12) at (-2.25, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw [in=90, out=-15, looseness=1.00] (4) to (6.center);
\draw (4) to (7.center);
\draw (5.center) to (8);
\draw [in=-165, out=90, looseness=1.25] (8) to (4);
\draw [bend right=90, looseness=1.25] (11.center) to (12.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.5, -0.75) {};
\node [style=none] (1) at (-2.5, 0.25) {};
\node [style=none] (2) at (-0.5, 0.25) {};
\node [style=none] (3) at (-0.5, -0.75) {};
\node [style=oa] (4) at (-1.5, -0.25) {};
\node [style=none] (5) at (-2, -2) {};
\node [style=none] (6) at (-1, -2) {};
\node [style=none] (7) at (-1, -1.25) {$\alpha_\oplus$};
\node [style=circle, scale=2] (8) at (-1.5, 1) {};
\node [style=none] (9) at (-1.5, 2) {};
\node [style=none] (10) at (-1.5, 1) {$\alpha_\otimes$};
\node [style=none] (11) at (-0.75, -0) {$G$};
\node [style=circle, scale=2] (12) at (-1, -1.25) {};
\node [style=none] (13) at (-1.5, -0.75) {};
\node [style=none] (14) at (-2, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw (9.center) to (8);
\draw (8) to (4);
\draw [in=90, out=-135, looseness=0.75] (4) to (5.center);
\draw [bend right=90, looseness=1.25] (13.center) to (14.center);
\draw (6.center) to (12);
\draw [bend right, looseness=1.00] (12) to (4);
\end{pgfonlayer}
\end{tikzpicture} ~~~ \mbox{\small\bf [LT.3]}~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.5, 0) {};
\node [style=none] (1) at (-0.5, -1) {};
\node [style=none] (2) at (-2.5, -1) {};
\node [style=none] (3) at (-2.5, -0) {};
\node [style=ox] (4) at (-1.5, -0.5) {};
\node [style=none] (5) at (-1, 1.25) {};
\node [style=none] (6) at (-2, 1.25) {};
\node [style=none] (7) at (-1.5, -2) {};
\node [style=circle, scale=2] (8) at (-1, 0.5) {};
\node [style=none] (9) at (-1, 0.5) {$\alpha_\oplus$};
\node [style=none] (10) at (-0.75, -0.5) {$F$};
\node [style=none] (11) at (-1.25, 0) {};
\node [style=none] (12) at (-0.75, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw [in=-90, out=165, looseness=1.00] (4) to (6.center);
\draw (4) to (7.center);
\draw (5.center) to (8);
\draw [in=15, out=-90, looseness=1.25] (8) to (4);
\draw [bend right=90, looseness=1.25] (11.center) to (12.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.5, 0.75) {};
\node [style=none] (1) at (-0.5, -0.25) {};
\node [style=none] (2) at (-2.5, -0.25) {};
\node [style=none] (3) at (-2.5, 0.75) {};
\node [style=ox] (4) at (-1.5, 0.25) {};
\node [style=none] (5) at (-1, 2) {};
\node [style=none] (6) at (-2, 2) {};
\node [style=none] (7) at (-2, 1.25) {$\alpha_\otimes$};
\node [style=circle, scale=2] (8) at (-1.5, -1) {};
\node [style=none] (9) at (-1.5, -2) {};
\node [style=none] (10) at (-1.5, -1) {$\alpha_\oplus$};
\node [style=none] (11) at (-0.75, 0.5) {$F$};
\node [style=circle, scale=2] (12) at (-2, 1.25) {};
\node [style=none] (13) at (-1.5, 0.75) {};
\node [style=none] (14) at (-1, 0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw (9.center) to (8);
\draw (8) to (4);
\draw [in=-90, out=44, looseness=0.75] (4) to (5.center);
\draw [bend right=90, looseness=1.25] (13.center) to (14.center);
\draw (6.center) to (12);
\draw [bend right, looseness=1.00] (12) to (4);
\end{pgfonlayer}
\end{tikzpicture}~~~ \mbox{\small\bf [LT.4]}~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.5, 0) {};
\node [style=none] (1) at (-2.5, -1) {};
\node [style=none] (2) at (-0.5, -1) {};
\node [style=none] (3) at (-0.5, 0) {};
\node [style=ox] (4) at (-1.5, -0.5) {};
\node [style=none] (5) at (-2, 1.25) {};
\node [style=none] (6) at (-1, 1.25) {};
\node [style=none] (7) at (-1.5, -2) {};
\node [style=circle, scale=2] (8) at (-2, 0.5) {};
\node [style=none] (9) at (-2, 0.5) {$\alpha_\oplus$};
\node [style=none] (10) at (-0.75, -0.25) {$G$};
\node [style=none] (11) at (-1.75, 0) {};
\node [style=none] (12) at (-2.25, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw [in=-90, out=15, looseness=1.00] (4) to (6.center);
\draw (4) to (7.center);
\draw (5.center) to (8);
\draw [in=165, out=-90, looseness=1.25] (8) to (4);
\draw [bend left=90, looseness=1.25] (11.center) to (12.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.5, 0.75) {};
\node [style=none] (1) at (-2.5, -0.25) {};
\node [style=none] (2) at (-0.5, -0.25) {};
\node [style=none] (3) at (-0.5, 0.75) {};
\node [style=ox] (4) at (-1.5, 0.25) {};
\node [style=none] (5) at (-2, 2) {};
\node [style=none] (6) at (-1, 2) {};
\node [style=none] (7) at (-1, 1.25) {$\alpha_\otimes$};
\node [style=circle, scale=2] (8) at (-1.5, -1) {};
\node [style=none] (9) at (-1.5, -2) {};
\node [style=none] (10) at (-1.5, -1) {$\alpha_\oplus$};
\node [style=none] (11) at (-0.75, 0.5) {$F$};
\node [style=circle, scale=2] (12) at (-1, 1.25) {};
\node [style=none] (13) at (-1.5, 0.75) {};
\node [style=none] (14) at (-2, 0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw (9.center) to (8);
\draw (8) to (4);
\draw [in=-90, out=136, looseness=0.75] (4) to (5.center);
\draw [bend left=90, looseness=1.00] (13.center) to (14.center);
\draw (6.center) to (12);
\draw [bend left, looseness=1.00] (12) to (4);
\end{pgfonlayer}
\end{tikzpicture}
\]
An adjunction of linear functors, $(\eta, \epsilon): F \dashv G$ is an adjunction in the usual sense (i.e. satisfying the triangle equalities) in the 2-category of LDCs with linear functors and
linear natural transformations. In particular, such an adjunction yields a pair of adjunctions: $(\eta_\otimes, \epsilon_\otimes): F_\otimes \dashv G_\otimes$ which is a monoidal adjunction, and $(\epsilon_\oplus, \eta_\oplus): G_\oplus \dashv F_\oplus$ which is a comonoidal adjunction. Using Kelly's results \cite{Kel97}, this allows us to observe:
\begin{lemma}
\label{Lemma: strong adjunction}
If $(\eta, \epsilon): F \dashvv G$ is an adjunction of linear functors, then $F_\otimes$ is iso-monoidal (or strong) with respect to $\otimes$ and $F_\oplus$ is iso-comonoidal making the
linear functor $F$ strong.
\end{lemma}
\begin{proof}
Since $(\eta_\otimes, \epsilon_\otimes): F_\otimes \dashv G_\otimes$ is a monoidal adjunction, the left adjoint $(F_\otimes, m_\otimes, m_\top)$ is a strong monoidal functor. Similarly, since $(\epsilon_\oplus, \eta_\oplus): G_\oplus \dashv F_\oplus$ is a comonoidal adjunction, the right adjoint $(F_\oplus, n_\oplus, n_\bot)$ is a strong comonoidal functor.
\end{proof}
A {\bf linear equivalence} is a linear adjunction in which the unit and counit are linear natural isomorphisms.
\subsection{Mix categories}
In this paper we shall be predominately concerned with LDCs which have a mix map:
\begin{definition} \cite{CS97a}
An LDC is a {\bf mix category} in case there is a {\bf mix map} ${\sf m}:\bot\@ifnextchar^ {\t@@}{\t@@^{}}\top$ in $\mathbb{X}$ such that:
\[
\xymatrixcolsep{4pc}
\xymatrix{
A \otimes B \ar[r]^{1 \otimes u_\oplus^L} \ar[d]_{u_\oplus^R \otimes 1} \ar@{.>}[ddrr]^{\mathsf{mx}_{A,B}} & A \otimes (\bot \oplus B) \ar[r]^{1 \otimes ({\sf m} \oplus 1)} & A \otimes ( \top \oplus B) \ar[d]^{\delta^L} \\
(A \oplus \bot) \otimes B \ar[d]_{\delta^R} & & ( A \otimes \top ) \oplus B \ar[d]^{u_\otimes^R} \\
A \oplus (\bot \otimes B) \ar[r]_{1 \oplus ({\sf m} \otimes 1)} & A \oplus (\top \otimes B) \ar[r]_{1 \oplus u_\otimes^L} & A \oplus B
}
\]
\end{definition}
The map $\mathsf{mx}_{A,B}$ is natural and called the {\bf mixor}. The coherence condition for the mix map has the following form in string diagrams (where the mix map is represented by an empty box):
\begin{align*}
\mathsf{mx}_{A,B}:=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=ox] (0) at (0, 0.2500001) {};
\node [style=circ] (1) at (0.5000001, -0.2500001) {};
\node [style=circ] (2) at (0, -1) {$\bot$};
\node [style=map] (3) at (0, -1.75) {~};
\node [style=circ] (4) at (0, -2.5) {$\top$};
\node [style=circ] (5) at (-0.5000001, -3.25) {};
\node [style=oa] (6) at (0, -3.75) {};
\node [style=nothing] (7) at (0, 0.7499999) {};
\node [style=nothing] (8) at (0, -4.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (7) to (0);
\draw (0) to (1);
\draw [in=45, out=-60, looseness=1.00] (1) to (6);
\draw [in=120, out=-135, looseness=1.00] (0) to (5);
\draw (5) to (6);
\draw (6) to (8);
\draw [densely dotted, in=-90, out=45, looseness=1.00] (5) to (4);
\draw (4) to (3);
\draw (3) to (2);
\draw [densely dotted, in=-135, out=90, looseness=1.00] (2) to (1);
\end{pgfonlayer}
\end{tikzpicture}
=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circ] (0) at (-0.5000001, -0.2500001) {};
\node [style=circ] (1) at (0, -1) {$\bot$};
\node [style=map] (2) at (0, -1.75) {~};
\node [style=circ] (3) at (0, -2.5) {$\top$};
\node [style=circ] (4) at (0.5000001, -3.25) {};
\node [style=nothing] (5) at (0, 0.7499999) {};
\node [style=nothing] (6) at (0, -4.25) {};
\node [style=oa] (7) at (0, -3.75) {};
\node [style=ox] (8) at (0, 0.2500001) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [densely dotted, in=-90, out=150, looseness=1.00] (4) to (3);
\draw (3) to (2);
\draw (2) to (1);
\draw [densely dotted, in=-45, out=90, looseness=1.00] (1) to (0);
\draw (8) to (5);
\draw (8) to (0);
\draw [in=135, out=-120, looseness=1.00] (0) to (7);
\draw (7) to (6);
\draw (7) to (4);
\draw [in=-45, out=60, looseness=1.00] (4) to (8);
\end{pgfonlayer}
\end{tikzpicture}
\end{align*}
When the map ${\sf m}$ is an isomorphism, then $\mathbb{X}$ is said to be an {\bf isomix category}. Recall that, when ${\sf m}$ is an isomorphism, the coherence requirement for the mix map is automatically satisfied (see \cite[Lemma 6.6]{CS97a}).
An isomix-LDC, $(\mathbb{X},\otimes, \oplus)$ always has two linear functors ${\sf Mx}_\downarrow: (\mathbb{X},\otimes, \otimes) \@ifnextchar^ {\t@@}{\t@@^{}} (\mathbb{X},\otimes,\oplus)$ and ${\sf Mx}_\uparrow: (\mathbb{X},\oplus, \oplus) \@ifnextchar^ {\t@@}{\t@@^{}} (\mathbb{X},\otimes,\oplus)$ given by the identity functor, that is $({\sf Mx}_\uparrow)_\otimes = ({\sf Mx}_\uparrow)_\oplus = {\sf Id} = ({\sf Mx}_\downarrow)_\otimes = ({\sf Mx}_\downarrow)_\oplus$. The linear strengths and monoidal maps are given by the inverse of the mix map and the mixor . These mix functors takes the degenerate linear structure on the tensor (respectively the par) and spread it out over both the tensor structures.
\begin{lemma} \label{mix-functor}
For any isomix-LDC $\mathbb{X}$ the functors ${\sf Mx}_\downarrow: (\mathbb{X},\oplus,\oplus) \@ifnextchar^ {\t@@}{\t@@^{}} (\mathbb{X},\otimes, \oplus)$ and ${\sf Mx}_\uparrow: (\mathbb{X},\otimes,\otimes) \@ifnextchar^ {\t@@}{\t@@^{}} (\mathbb{X},\otimes, \oplus)$ are linear functors.
\end{lemma}
\begin{proof}
We show that ${\sf Mx}_\downarrow: (\mathbb{X},\oplus, \oplus) \@ifnextchar^ {\t@@}{\t@@^{}} (\mathbb{X},\otimes,\oplus)$ is a linear functor: the monoidal and comonoidal components of the functor are given by $( 1, \mathsf{mx}, {\sf m}^{-1})$ and $(1, 1, 1)$ respectively. The linear strenghts are $\nu_\otimes^L = \nu_\otimes^R: A \otimes B \@ifnextchar^ {\t@@}{\t@@^{}} A \oplus B := \mathsf{mx}$ and $\nu_\oplus^L = \nu_\oplus^R: A \oplus B \@ifnextchar^ {\t@@}{\t@@^{}} A \oplus B := 1$.
First we show $( 1, \mathsf{mx}, {\sf m}^{-1}): (\mathbb{X},\oplus,\oplus) \@ifnextchar^ {\t@@}{\t@@^{}} (\mathbb{X},\otimes, \oplus)$ is a monoidal functor:
\begin{itemize}
\item The pentagon law for monoidal functors, $(\mathsf{mx} \otimes 1)~\mathsf{mx}~a_\oplus = a_\otimes~(1 \otimes \mathsf{mx})~\mathsf{mx}$, is satisfied:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle, scale=0.5] (0) at (-0.5, 2) {};
\node [style=circle, scale=0.5] (1) at (0.5, 1.25) {};
\node [style=map] (2) at (0, 1.75) {};
\node [style=ox] (3) at (0, 2.5) {};
\node [style=oa] (4) at (0, 0.75) {};
\node [style=ox] (5) at (1, 3) {};
\node [style=oa] (6) at (0, -1) {};
\node [style=oa] (7) at (1, -1.75) {};
\node [style=none] (8) at (1, -3) {};
\node [style=none] (9) at (1, 3.75) {};
\node [style=none] (10) at (-0.75, -3) {};
\node [style=circle, scale=0.5] (11) at (1.5, -1.25) {};
\node [style=map] (12) at (0.75, -0.5) {};
\node [style=circle, scale=0.5] (13) at (0, -0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted, in=90, out=-15, looseness=1.25] (0) to (2);
\draw [dotted, in=165, out=-90, looseness=1.25] (2) to (1);
\draw (9.center) to (5);
\draw [bend left=45, looseness=1.00] (5) to (7);
\draw [bend right=15, looseness=1.00] (5) to (3);
\draw [bend right=15, looseness=1.00] (6) to (7);
\draw (7) to (8.center);
\draw [in=90, out=-126, looseness=1.00] (6) to (10.center);
\draw [bend right=60, looseness=1.25] (3) to (4);
\draw (4) to (6);
\draw [bend left=60, looseness=1.25] (3) to (4);
\draw [dotted, in=90, out=0, looseness=1.50] (13) to (12);
\draw [dotted, in=165, out=-90, looseness=1.25] (12) to (11);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle, scale=0.5] (0) at (0.25, 1) {};
\node [style=circle, scale=0.5] (1) at (1.75, -0.5) {};
\node [style=map] (2) at (1, 0.5) {};
\node [style=ox] (3) at (-0.25, 1.75) {};
\node [style=ox] (4) at (1, 3) {};
\node [style=oa] (5) at (1, -1.75) {};
\node [style=none] (6) at (1, -3) {};
\node [style=none] (7) at (1, 3.75) {};
\node [style=none] (8) at (-1, -3) {};
\node [style=circle, scale=0.5] (9) at (0.5, -0.75) {};
\node [style=map] (10) at (-0.25, -0) {};
\node [style=circle, scale=0.5] (11) at (-0.75, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted, in=90, out=-15, looseness=1.25] (0) to (2);
\draw [dotted, in=165, out=-90, looseness=1.25] (2) to (1);
\draw (7.center) to (4);
\draw [bend left=45, looseness=1.00] (4) to (5);
\draw [bend right=15, looseness=1.00] (4) to (3);
\draw (5) to (6.center);
\draw [dotted, in=90, out=0, looseness=1.50] (11) to (10);
\draw [dotted, in=165, out=-90, looseness=1.25] (10) to (9);
\draw [in=90, out=-165, looseness=0.50] (3) to (8.center);
\draw [in=135, out=-45, looseness=1.00] (3) to (5);
\end{pgfonlayer}
\end{tikzpicture}
= \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle, scale=0.5] (0) at (0.25, 2.5) {};
\node [style=circle, scale=0.5] (1) at (1.5, -0.5) {};
\node [style=map] (2) at (1, 0.5) {};
\node [style=ox] (3) at (-0.25, 1.75) {};
\node [style=ox] (4) at (1, 3) {};
\node [style=oa] (5) at (1, -1.25) {};
\node [style=none] (6) at (1, -3) {};
\node [style=none] (7) at (1, 3.75) {};
\node [style=none] (8) at (-1, -3) {};
\node [style=circle, scale=0.5] (9) at (1, -2.25) {};
\node [style=map] (10) at (-0.25, -0) {};
\node [style=circle, scale=0.5] (11) at (-0.75, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted, in=90, out=-15, looseness=1.25] (0) to (2);
\draw [dotted, in=165, out=-90, looseness=1.25] (2) to (1);
\draw (7.center) to (4);
\draw [bend left=45, looseness=1.00] (4) to (5);
\draw [bend right=15, looseness=1.00] (4) to (3);
\draw (5) to (6.center);
\draw [dotted, in=90, out=0, looseness=1.50] (11) to (10);
\draw [dotted, in=165, out=-90, looseness=1.25] (10) to (9);
\draw [in=90, out=-165, looseness=0.50] (3) to (8.center);
\draw [in=135, out=-45, looseness=1.00] (3) to (5);
\end{pgfonlayer}
\end{tikzpicture}
\]
\item The unit laws for monoidal functors hold. Here is the pictorial proof of $(1 \otimes {\sf m}^{-1}) \mathsf{mx} = u_\otimes^L(u_\oplus^L)^{-1}$, where
the filled rectangles represent ${\sf m}^{-1}$:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=map, fill=black] (0) at (0, 2) {};
\node [style=circle, scale=0.5] (1) at (0, 1.5) {};
\node [style=circle, scale=0.5] (2) at (-1.75, -0) {};
\node [style=map] (3) at (-1, 0.75) {};
\node [style=none] (4) at (0, 3.5) {};
\node [style=none] (5) at (0, -1) {};
\node [style=none] (6) at (-1.75, -1) {};
\node [style=none] (7) at (-1.75, 3.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted, in=90, out=-165, looseness=1.00] (1) to (3);
\draw [dotted, in=30, out=-90, looseness=1.25] (3) to (2);
\draw (4.center) to (0);
\draw (7.center) to (6.center);
\draw (0) to (5.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=map, fill=black] (0) at (0, 2.25) {};
\node [style=circle, scale=0.5] (1) at (0, 1.5) {};
\node [style=circle, scale=0.5] (2) at (-1.75, -0.5) {};
\node [style=map] (3) at (-1, 0.25) {};
\node [style=none] (4) at (0, 3.75) {};
\node [style=none] (5) at (0.5, -1) {};
\node [style=none] (6) at (-1.75, -1) {};
\node [style=none] (7) at (-1.75, 3.75) {};
\node [style=circle] (8) at (0, 3) {$\top$};
\node [style=circle] (9) at (0, 0.25) {$\bot$};
\node [style=circle] (10) at (0.5, -0.5) {$\bot$};
\node [style=circle, scale=0.5] (11) at (0, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted, dotted, in=90, out=-165, looseness=1.00] (1) to (3);
\draw [dotted, dotted, in=30, out=-90, looseness=1.25] (3) to (2);
\draw (7.center) to (6.center);
\draw (8) to (0);
\draw (0) to (1);
\draw (1) to (9);
\draw (10) to (5.center);
\draw [bend left, looseness=1.00, dotted] (11) to (10);
\draw (4.center) to (8);
\end{pgfonlayer}
\end{tikzpicture}
= \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=map, fill=black] (0) at (0, 2.25) {};
\node [style=circle, scale=0.5] (1) at (0, 1.5) {};
\node [style=circle, scale=0.5] (2) at (-1.75, 0.5) {};
\node [style=map] (3) at (-1, 1) {};
\node [style=none] (4) at (0, 3.75) {};
\node [style=none] (5) at (0, -1) {};
\node [style=none] (6) at (-1.75, -1) {};
\node [style=none] (7) at (-1.75, 3.75) {};
\node [style=circle] (8) at (0, 3) {$\top$};
\node [style=circle] (9) at (0, 0.75) {$\bot$};
\node [style=circle] (10) at (0, -0.25) {$\bot$};
\node [style=circle, scale=0.5] (11) at (-1.75, -0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted, dotted, in=90, out=-165, looseness=1.00] (1) to (3);
\draw [dotted, dotted, in=30, out=-90, looseness=1.25] (3) to (2);
\draw (7.center) to (6.center);
\draw (8) to (0);
\draw (0) to (1);
\draw (1) to (9);
\draw (10) to (5.center);
\draw [bend left, looseness=1.00, dotted] (11) to (10);
\draw (4.center) to (8);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle, scale=0.5] (0) at (-1.75, 1.75) {};
\node [style=none] (1) at (0, 3.75) {};
\node [style=none] (2) at (0, -1) {};
\node [style=none] (3) at (-1.75, -1) {};
\node [style=none] (4) at (-1.75, 3.75) {};
\node [style=circle] (5) at (0, 3) {$\top$};
\node [style=circle] (6) at (0, -0.25) {$\bot$};
\node [style=circle, scale=0.5] (7) at (-1.75, 0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4.center) to (3.center);
\draw (6) to (2.center);
\draw [dotted, bend left, looseness=1.00] (7) to (6);
\draw (1.center) to (5);
\draw [dotted, bend left=45, looseness=1.00] (5) to (0);
\end{pgfonlayer}
\end{tikzpicture}
\]
The other unit law holds similarly.
\end{itemize}
${\sf Mx}_\downarrow: (\mathbb{X},\otimes, \oplus) \@ifnextchar^ {\t@@}{\t@@^{}} (\mathbb{X},\oplus,\oplus)$ satisfies all the coherence requirements of a linear functor:
{\bf [LF.1]}, {\bf [LF.2]}, and {\bf [LF.3]} hold because $({\sf Mx}_\downarrow)_\otimes$ and $({\sf Mx}_\downarrow)_\oplus$ are monoidal and comonoidal respectively, {\bf [LF.4]}(a) becomes
$\mathsf{mx} a_\oplus^{-1} = \delta^L (\mathsf{mx} \oplus 1)$ and holds because:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=oa] (0) at (0, 2.25) {};
\node [style=oa] (1) at (0, 1) {};
\node [style=oa] (2) at (0, -1) {};
\node [style=oa] (3) at (-1, -2) {};
\node [style=none] (4) at (-1, -3) {};
\node [style=none] (5) at (0.5, -3) {};
\node [style=none] (6) at (-2, 3) {};
\node [style=none] (7) at (0, 3) {};
\node [style=map] (8) at (-1.25, 0.5) {};
\node [style=circle, scale=0.5] (9) at (0, -0.25) {};
\node [style=circle, scale=0.5] (10) at (-2, 1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (6.center) to (10);
\draw (7.center) to (0);
\draw [bend left=60, looseness=1.25] (0) to (1);
\draw [bend right=60, looseness=1.25] (0) to (1);
\draw (1) to (2);
\draw [in=90, out=-45, looseness=1.00] (2) to (5.center);
\draw [in=30, out=-150, looseness=1.50] (2) to (3);
\draw [in=-89, out=135, looseness=1.00] (3) to (10);
\draw (3) to (4.center);
\draw [in=90, out=-15, looseness=1.25,dotted] (10) to (8);
\draw [in=180, out=-90, looseness=1.25, dotted] (8) to (9);
\end{pgfonlayer}
\end{tikzpicture}
= \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=oa] (0) at (0, 2.25) {};
\node [style=oa] (1) at (0, -0) {};
\node [style=oa] (2) at (0, -1) {};
\node [style=oa] (3) at (-1, -2) {};
\node [style=none] (4) at (-1, -3) {};
\node [style=none] (5) at (0.5, -3) {};
\node [style=none] (6) at (-2, 3) {};
\node [style=none] (7) at (0, 3) {};
\node [style=map] (8) at (-1.25, 1.5) {};
\node [style=circle,scale=0.5] (9) at (-0.75, 0.75) {};
\node [style=circle, scale=0.5] (10) at (-2, 2.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (6.center) to (10);
\draw (7.center) to (0);
\draw [bend left=60, looseness=1.25] (0) to (1);
\draw [bend right=60, looseness=1.25] (0) to (1);
\draw (1) to (2);
\draw [in=90, out=-45, looseness=1.00] (2) to (5.center);
\draw [in=30, out=-150, looseness=1.50] (2) to (3);
\draw [in=-89, out=135, looseness=1.00] (3) to (10);
\draw (3) to (4.center);
\draw [dotted, in=90, out=-15, looseness=1.25] (10) to (8);
\draw [dotted, in=180, out=-90, looseness=1.25] (8) to (9);
\end{pgfonlayer}
\end{tikzpicture}
= \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=oa] (0) at (0, 2.25) {};
\node [style=oa] (1) at (-1, -2) {};
\node [style=none] (2) at (-1, -3) {};
\node [style=none] (3) at (0.5, -3) {};
\node [style=none] (4) at (-2, 3) {};
\node [style=none] (5) at (0, 3) {};
\node [style=map] (6) at (-1.25, 1) {};
\node [style=circle, scale=0.5] (7) at (-0.75, -0.5) {};
\node [style=circle, scale=0.5] (8) at (-2, 2.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4.center) to (8);
\draw (5.center) to (0);
\draw [in=-89, out=135, looseness=1.00] (1) to (8);
\draw (1) to (2.center);
\draw [dotted, in=90, out=-15, looseness=1.25] (8) to (6);
\draw [dotted, in=150, out=-90, looseness=1.00] (6) to (7);
\draw [in=90, out=-45, looseness=0.50] (0) to (3.center);
\draw [in=60, out=-150, looseness=0.75] (0) to (1);
\end{pgfonlayer}
\end{tikzpicture}
\]
{\bf [LF.4]} (b) - (d) and {\bf [LF.5]} (a) - (d) are satisfied similarly.
Thus, ${\sf Mx}_\downarrow$ is a linear functor.
The proof that ${\sf Mx}_\uparrow$ is a linear functor is (linearly) dual
\end{proof}
In fact, these linear functors and are examples of {\em isomix Frobenius functors\/}, which we shall introduce formally in the next section.
\begin{definition} \cite{BCS00}
The {\bf core} of a mix-LDC, $\mathsf{Core}(\mathbb{X}) \subseteq \mathbb{X}$, is the full subcategory with objects $U$ such that the mixors
\[ U \otimes (\_) \@ifnextchar^ {\t@@}{\t@@^{}}^{\mathsf{mx}_{U,(\_)}} U \oplus (\_) ~~~~\mbox{and}~~~~ (\_) \otimes \@ifnextchar^ {\t@@}{\t@@^{}}^{\mathsf{mx}_{(\_),U}} (\_) \oplus U \]
are isomorphisms.
\end{definition}
\begin{proposition} \cite[Proposition 3]{BCS00}
If $\mathbb{X}$ is a mix-LDC and $A,B \in \mathsf{Core}(\mathbb{X})$ then $A \oplus B$ and $A \otimes B \in \mathsf{Core}(\mathbb{X})$ (and $A \oplus B \simeq A \otimes B$). If $\mathbb{X}$ is an isomix-LDC, then $\top, \bot \in \mathsf{Core}(\mathbb{X})$.
\end{proposition}
A {\bf compact LDC} is an isomix category in which each mixor $\mathsf{mx}_{A,B}$ is an isomorphism. In any isomix category, $\mathbb{X}$, the core, $\mathsf{Core}(\mathbb{X})$, forms a compact LDC as
here both $\mathsf{mx}_{A,B}: A \otimes B \@ifnextchar^ {\t@@}{\t@@^{}} A \oplus B$ and ${\sf m}: \top \@ifnextchar^ {\t@@}{\t@@^{}} \bot$ are isomorphisms.
\begin{corollary} \label{compact-mix-functor}
When $\mathbb{X}$ is a compact LDC, the mix functors, ${\sf Mx}_\downarrow$ and ${\sf Mx}_\uparrow$, are linear isomorphisms.
\end{corollary}
We shall denote the inverse of ${\sf Mx}_\downarrow$ by ${\sf Mx}^{*}_\downarrow: (\mathbb{X},\oplus,\oplus) \@ifnextchar^ {\t@@}{\t@@^{}} (\mathbb{X},\otimes,\oplus)$: this is the identity functor as a mere functor, strict on the par structure, and on the tensor structure having as the unit laxor ${\sf m}$ and as the tensor laxor ${\sf mx}^{-1}$. Similarly we shall denote the inverse of ${\sf Mx}_\uparrow$ by ${\sf Mx}^{*}_\uparrow$.
\subsection{Linear duals}
A key notion in the theory of LDCs is the notion of a linear adjoint \cite{CKS00}. Here we shall refer to linear adjoints as ``linear duals'' in order to avoid any confusion with an adjunction of linear functors.
\begin{definition} Suppose $\mathbb{X}$ is a LDC and $A,B \in\mathbb{X}$, then $B$ is {\bf left linear dual} (or left linear adjoint) to $A$ -- or $A$ is {\bf right linear dual} (right linear adjoint) to $B$ -- written $(\eta, \epsilon): B \dashv \!\!\!\!\! \dashv A$, if there exists $\eta: \top \rightarrow B \oplus A$ and $\epsilon: A \otimes B \rightarrow \bot$ such that the following diagrams commute:
\[
\xymatrix{
B \ar[r]^{(u_\otimes^L)^{-1}} \ar@{=}[d]
& \top \otimes B \ar[r]^{\eta \otimes 1}
& (B \oplus A) \otimes B \ar[d]^{\partial_R} \\
B
& B \oplus \bot \ar[l]^{u_\oplus^R}
& B \oplus (A \otimes B) \ar[l]^{1 \oplus \epsilon}
}
~~~~~
\xymatrix{
A \ar[r]^{(u_\otimes^R)^{-1}} \ar@{=}[d]
& A \otimes \top \ar[r]^{1 \otimes \eta}
& A \otimes (B \oplus A)\ar[d]^{\partial_L} \\
A
& \bot \oplus A \ar[l]^{u_\oplus^L}
& (A \otimes B) \oplus A \ar[l]^{ \epsilon \oplus 1} }
\]
\end{definition}
The commuting diagrams are called often referred to as ``snake diagrams'' because of their shape when drawn in string calculus:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-3, 1.25) {$\eta$};
\node [style=circle] (1) at (-2, 0) {$\epsilon$};
\node [style=none] (2) at (-3.5, -0.5) {};
\node [style=none] (3) at (-1.5, 2) {};
\node [style=none] (4) at (-1.75, 2) {$A$};
\node [style=none] (5) at (-3.25, -0.5) {$A$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=none, in=-90, out=30, looseness=1.00] (1) to (3.center);
\draw [style=none, in=150, out=-15, looseness=1.25] (0) to (1);
\draw [style=none, in=90, out=-165, looseness=1.00] (0) to (2.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\draw (0,2.5) -- (0,0);
\end{tikzpicture} ~~~~~~~~~~
\begin{tikzpicture
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-3, 0.25) {$\epsilon$};
\node [style=circle] (1) at (-2, 1.5) {$\eta$};
\node [style=none] (2) at (-3.5, 2) {};
\node [style=none] (3) at (-1.5, -0.5) {};
\node [style=none] (4) at (-3.25, 2) {$B$};
\node [style=none] (5) at (-1.75, -0.5) {$B$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=none, in=90, out=-30, looseness=1.00] (1) to (3.center);
\draw [style=none, in=-150, out=15, looseness=1.25] (0) to (1);
\draw [style=none, in=-90, out=165, looseness=1.00] (0) to (2.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\draw (0,2.5) -- (0,0);
\end{tikzpicture}
\]
\begin{lemma} \cite{BCS00}
\begin{enumerate}[(i)]
\item In an LDC if $(\eta,\epsilon): B \dashvv A$ and $(\eta',\epsilon'): C \dashvv A$, then $B$ and $C$ are isomorphic;
\item In a symmetric LDC $(\eta, \epsilon): B \dashvv A$ if and only if $(\eta c_\oplus, c_\otimes \epsilon): A \dashvv B$;
\item In a mix-LDC if $B \in \mathsf{Core}(\mathbb{X})$ and $B \dashvv A$, then $A \in \mathsf{Core}(\mathbb{X})$.
\end{enumerate}
\end{lemma}
\begin{lemma} \cite{CKS00}
\label{Lemma: linear adjoints}
Linear functors preserve linear duals: when $F: \mathbb{X} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{Y}$ is a linear functor and $(\eta, \epsilon): A \dashvv B \in \mathbb{X}$, then $F_\otimes(A) \dashvv F_\oplus(B)$ and $F_\oplus(A) \dashvv F_\otimes(B)$.
\end{lemma}
\begin{proof}
The unit and counit of the adjunction $(\eta', \epsilon'): F_\otimes(A) \dashvv F_\oplus(B)$ is given as follows:
\[ \eta' := \top \xrightarrow{m_\top} F_\otimes(\top) \xrightarrow{F_\otimes(\eta)} F_\otimes( A \oplus B) \xrightarrow{\nu_\otimes^L} F_\otimes(A) \oplus F_\oplus(B) \]
\[ \epsilon' := F_\oplus(B) \otimes F_\otimes(A) \xrightarrow{\nu_\oplus^L} F_\oplus(B \otimes A) \xrightarrow{F_\oplus(\epsilon)} F_\oplus(\bot) \xrightarrow{n_\bot} \bot \]
\end{proof}
An LDC in which every object has a chosen left and right linear dual, respectively $(\eta{*},\epsilon{*}): A^{*} \dashvv A$ and $({*}\eta,{*}\epsilon): A \dashvv \!~^{*}A$, is a {\bf $*$-autonomous category}. In the symmetric case a left linear dual gives a right linear dual using the symmetry: thus, it is standard to assume the existence of just the left dual with the right being the same object with the unit and counit given by symmetry (as above).
Just as compact LDCs are linearly equivalent to monoidal categories so compact $*$-autonomous categories are linearly equivalent to a compact closed categories. The equivalence is given by ${\sf Mx}_\uparrow$ which spreads the par onto two tensor structures (or, indeed, by ${\sf Mx}_\downarrow$ which shows how to spread out a compact closed
structure on the tensor).
In a symmetric $*$-autonomous category the left dual is always canonically isomorphic to the right dual. Moreover, even in non-symmetric $*$-autonomous categories, it is often the case that the two duals are coherently isomorphic:
\begin{definition}\cite{EggMcCurd12}
A {\bf cyclor} in a $*$-autonomous category $(\mathbb{X}, \otimes, \top, \oplus, \bot, ~^{*}(\_), (\_)^*)$ is a natural isomorphism $A^* \@ifnextchar^ {\t@@}{\t@@^{}}^{\psi} \!~^{*}A$ satisfying the following coherence conditions:
\[
\xymatrix{
\bot^* \ar[rr]^{\psi} \ar[dr] & \ar@{}[d]|{\mbox{\tiny \bf [C.1]}} & ~^{*}\bot \ar[ld]^{} \\
& \top &
} ~~~~~~~~ \xymatrix{
& A \ar[ld] \ar[dr] \ar@{}[d]|{\mbox{\tiny \bf [C.2]}} & \\
( ~^{*}A)^* \ar[r]_{\psi^*} & (A^*)^* \ar[r]_{\psi_{A^*}} & ~^{*}(A^*)
} ~~~~~~~~ \xymatrix{
\top^* \ar[rr]^{\psi} \ar[dr] & \ar@{}[d]|{\mbox{\tiny \bf [C.3]}} & ~^{*}\top \ar[ld]^{} \\
& \bot &
}
\]
\[
\xymatrixcolsep{4pc}
\xymatrix{
(A \otimes B)^* \ar[r]^{\psi} \ar[d]_{} \ar@{}[dr]|{\mbox{\tiny \bf [C.4]}} & ~^{*}(A \otimes B) \ar[d]^{} \\
(B^* \oplus A^*) \ar[r]_{\psi \oplus \psi} & ~^{*}B \oplus ~^{*}A
} ~~~~~~~~~ \xymatrix{
(A \oplus B)^* \ar[r]^{\psi} \ar[d]_{} \ar@{}[dr]|{\mbox{\tiny \bf [C.5]}} & ~^{*}(A \oplus B) \ar[d]^{} \\
(B^* \otimes A^*) \ar[r]_{\psi \otimes \psi} & ~^{*}B \otimes ~^{*}A
}
\]
A $*$-autonomous category with a cyclor is said to be {\bf cyclic}.
\end{definition}
The coherence conditions are not independent of each other: being cyclic is equivalent to any of the following four pairs of conditions: ({\bf [C.1]}, {\bf [C.5]}), ({\bf [C.2]}, {\bf [C.5]}), ({\bf [C.4]}, {\bf [C.2]}) and ({\bf [C.4]}, {\bf [C.3]}) (see \cite{EggMcCurd12}).
Condition {\bf [C.2]} which is used extensively in Section \ref{daggers-duals-conjugation} is pictorially represented as follows:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-2, 1) {$\psi_{A^*}$};
\node [style=circle] (1) at (-0.75, 1) {$\psi_A$};
\node [style=none] (2) at (-2, 1.75) {};
\node [style=none] (3) at (-0.75, 1.75) {};
\node [style=none] (4) at (-2, -1) {};
\node [style=none] (5) at (-0.75, -0) {};
\node [style=none] (6) at (0.5, -0) {};
\node [style=none] (7) at (0.5, 3) {};
\node [style=none] (8) at (-1.5, 2.75) {$\eta*$};
\node [style=none] (9) at (0, -0.75) {$*\epsilon$};
\node [style=none] (10) at (-2.5, 1.75) {$A^{**}$};
\node [style=none] (11) at (-2.7, -0.7) {$~^*(A^*)$};
\node [style=none] (12) at (-0.5, 2) {$A^*$};
\node [style=none] (13) at (-0.45, 0.25) {$~^*A$};
\node [style=none] (14) at (0.75, 2.5) {$A$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (4.center);
\draw (2.center) to (0);
\draw [bend left=90, looseness=2.00] (2.center) to (3.center);
\draw (3.center) to (1);
\draw (1) to (5.center);
\draw [bend right=90, looseness=1.50] (5.center) to (6.center);
\draw (6.center) to (7.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 0.25) {};
\node [style=none] (1) at (-0.75, 0.25) {};
\node [style=none] (2) at (-2, 3) {};
\node [style=none] (3) at (-0.75, 2) {};
\node [style=none] (4) at (0.5, 2) {};
\node [style=none] (5) at (0.5, -1) {};
\node [style=none] (6) at (-0.25, 2.75) {$*\eta$};
\node [style=none] (7) at (-1.5, -0.75) {$\epsilon*$};
\node [style=none] (8) at (1, -0.7) {$~^*(A^*)$};
\node [style=none] (9) at (-0.5, 1.15) {$A^*$};
\node [style=none] (10) at (-2.25, 2.75) {$A$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=2.00] (0.center) to (1.center);
\draw [bend left=90, looseness=1.50] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (3.center) to (1.center);
\draw (2.center) to (0.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
The requirement {\bf [C.2]} implies:
\[
\text{\bf [C.2]$^{-1}$}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (0, 1) {$\psi^{-1}_A$};
\node [style=circle] (1) at (2, 1) {$\psi^{-1}_{A^*}$};
\node [style=none] (2) at (0, -0) {};
\node [style=none] (3) at (2, -0) {};
\node [style=none] (4) at (2, 3) {};
\node [style=none] (5) at (0, 2) {};
\node [style=none] (6) at (-1, 2) {};
\node [style=none] (7) at (-1, -1.25) {};
\node [style=none] (8) at (1, -1.1) {$\epsilon*$};
\node [style=none] (9) at (-0.5, 2.75) {$*\eta$};
\node [style=none] (10) at (2.5, 0.25) {$A^{**}$};
\node [style=none] (11) at (-0.5, 0.25) {$A^*$};
\node [style=none] (12) at (2.5, 2.75) {$~^*(A^*)$};
\node [style=none] (13) at (-0.5, 1.75) {$~^*A$};
\node [style=none] (14) at (-1.25, -0.75) {$A$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2.center);
\draw [bend right=90, looseness=1.50] (2.center) to (3.center);
\draw (3.center) to (1);
\draw (4.center) to (1);
\draw (5.center) to (0);
\draw [bend right=90, looseness=2.00] (5.center) to (6.center);
\draw (6.center) to (7.center);
\end{pgfonlayer}
\end{tikzpicture} = \left(
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-2, 1) {$\psi_{A^*}$};
\node [style=circle] (1) at (-0.75, 1) {$\psi_A$};
\node [style=none] (2) at (-2, 1.75) {};
\node [style=none] (3) at (-0.75, 1.75) {};
\node [style=none] (4) at (-2, -1) {};
\node [style=none] (5) at (-0.75, -0) {};
\node [style=none] (6) at (0.5, -0) {};
\node [style=none] (7) at (0.5, 3) {};
\node [style=none] (8) at (-1.5, 2.75) {$\eta*$};
\node [style=none] (9) at (0, -0.75) {$*\epsilon$};
\node [style=none] (10) at (-2.5, 1.75) {$A^{**}$};
\node [style=none] (11) at (-2.7, -0.7) {$~^*(A^*)$};
\node [style=none] (12) at (-0.5, 2) {$A^*$};
\node [style=none] (13) at (-0.45, 0.25) {$~^*A$};
\node [style=none] (14) at (0.75, 2.5) {$A$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (4.center);
\draw (2.center) to (0);
\draw [bend left=90, looseness=2.00] (2.center) to (3.center);
\draw (3.center) to (1);
\draw (1) to (5.center);
\draw [bend right=90, looseness=1.50] (5.center) to (6.center);
\draw (6.center) to (7.center);
\end{pgfonlayer}
\end{tikzpicture} \right)^{-1} = \left( \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 0.25) {};
\node [style=none] (1) at (-0.75, 0.25) {};
\node [style=none] (2) at (-2, 3) {};
\node [style=none] (3) at (-0.75, 2) {};
\node [style=none] (4) at (0.5, 2) {};
\node [style=none] (5) at (0.5, -1) {};
\node [style=none] (6) at (-0.25, 2.75) {$*\eta$};
\node [style=none] (7) at (-1.5, -0.75) {$\epsilon*$};
\node [style=none] (8) at (1, -0.7) {$~^*(A^*)$};
\node [style=none] (9) at (-0.5, 1.15) {$A^*$};
\node [style=none] (10) at (-2.25, 2.75) {$A$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=2.00] (0.center) to (1.center);
\draw [bend left=90, looseness=1.50] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (3.center) to (1.center);
\draw (2.center) to (0.center);
\end{pgfonlayer}
\end{tikzpicture} \right)^{-1}
= \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 0.25) {};
\node [style=none] (1) at (-0.75, 0.25) {};
\node [style=none] (2) at (-2, 3) {};
\node [style=none] (3) at (-0.75, 2) {};
\node [style=none] (4) at (0.5, 2) {};
\node [style=none] (5) at (0.5, -1) {};
\node [style=none] (6) at (-0.25, 2.75) {$\eta*$};
\node [style=none] (7) at (-1.5, -0.75) {$*\epsilon$};
\node [style=none] (8) at (-2.75, 2.75) {$~^*(A^*)$};
\node [style=none] (9) at (-0.5, 1) {$A^*$};
\node [style=none] (10) at (0.75, -0.75) {$A$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=2.00] (0.center) to (1.center);
\draw [bend left=90, looseness=1.50] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (3.center) to (1.center);
\draw (2.center) to (0.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
Symmetric $*$-autonomous categories always have a canonical cyclor. It is given pictorially by:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.25, 2.25) {};
\node [style=none] (1) at (-0.25, 0.5) {};
\node [style=none] (12) at (-0.85, 1.7) {$*\eta$};
\node [style=none] (2) at (-1.5, 0.5) {};
\node [style=none] (3) at (-1.5, 1) {};
\node [style=none] (4) at (-0.5, 1) {};
\node [style=none] (34) at (-0.85, -0.4) {$\epsilon*$};
\node [style=none] (5) at (-0.5, -1) {};
\node [style=none] (6) at (0, 2) {$A^{*}$};
\node [style=none] (7) at (-0.15, -0.75) {${~^*A}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend left=90, looseness=1.75] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend left=90, looseness=1.75] (3.center) to (4.center);
\draw (4.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\section{Frobenius functors and daggers}
We shall be interested here in linear functors between LDCs called Frobenius functors which come in various flavours, including mix functors and isomix functors, as illustrated in Table \ref{linear-functor-family}. These functors are directly related to the Frobenius monoidal functors of \cite{DP08}. Furthermore, we have already seen two rather basic examples, namely, ${\sf Mx}_\uparrow$ and ${\sf Mx}_\downarrow$.
\begin{table}
\begin{center}
\iffalse
\begin{tikzpicture}[scale=1.5]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 2) {};
\node [style=none] (1) at (-3, -0) {};
\node [style=none] (2) at (1, -0) {};
\node [style=none] (3) at (1, 2) {};
\node [style=none] (4) at (-4, 3) {};
\node [style=none] (5) at (-4, -1) {};
\node [style=none] (6) at (2, -1) {};
\node [style=none] (7) at (2, 3) {};
\node [style=none] (8) at (-5, -2) {};
\node [style=none] (9) at (3, -2) {};
\node [style=none] (10) at (3, 4) {};
\node [style=none] (11) at (-5, 4) {};
\node [style=none] (12) at (-6, 5) {};
\node [style=none] (13) at (4, 5) {};
\node [style=none] (14) at (4, -3) {};
\node [style=none] (15) at (-6, -3) {};
\node [style=none] (16) at (-1, 4.5) {{\bf Linear functors}};
\node [style=none] (17) at (-1, 3.5) {{\bf Frobenius functors}};
\node [style=none] (18) at (-1, 2.5) {{\bf Mix functors}};
\node [style=none] (19) at (-1, 1.5) {{\bf Isomix functors}};
\node [style=none] (20) at (-1, 0.5) {$m_\top F({\sf m}^{-1})n_\bot = {\sf m}^{-1}$};
\node [style=none] (21) at (-1, -0.5) {$n_\bot{\sf m} m_\top = F({\sf m})$};
\node [style=none] (22) at (-1, -1.5) {$F_\otimes = F_\oplus,~ m_\otimes = \nu_\oplus^L = \nu_\oplus^R, ~n_\oplus = \nu_\otimes^L = \nu_\otimes^R$};
\node [style=none] (23) at (-1, -2.5) {$((F_\otimes, m_\otimes, m_\top), (F_\oplus, n_\oplus, n_\bot))$ };
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (0.center);
\draw (4.center) to (5.center);
\draw (5.center) to (6.center);
\draw (6.center) to (7.center);
\draw (7.center) to (4.center);
\draw (11.center) to (10.center);
\draw (10.center) to (9.center);
\draw (9.center) to (8.center);
\draw (8.center) to (11.center);
\draw (12.center) to (13.center);
\draw (13.center) to (14.center);
\draw (12.center) to (15.center);
\draw (15.center) to (14.center);
\end{pgfonlayer}
\end{tikzpicture}
\fi
\begin{tikzpicture} [scale=1.5]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 2) {};
\node [style=none] (1) at (-3, -0) {};
\node [style=none] (2) at (1, -0) {};
\node [style=none] (3) at (1, 2) {};
\node [style=none] (4) at (-4, 3) {};
\node [style=none] (5) at (-4, -1) {};
\node [style=none] (6) at (2, -1) {};
\node [style=none] (7) at (2, 3) {};
\node [style=none] (8) at (-5, -2) {};
\node [style=none] (9) at (3, -2) {};
\node [style=none] (10) at (3, 4) {};
\node [style=none] (11) at (-5, 4) {};
\node [style=none] (12) at (-6, 5) {};
\node [style=none] (13) at (4, 5) {};
\node [style=none] (14) at (4, -3) {};
\node [style=none] (15) at (-6, -3) {};
\node [style=none] (16) at (-1, 4.5) {Linear functors};
\node [style=none] (17) at (-1, 3.5) {Frobenius functors};
\node [style=none] (18) at (-1, 2.5) {Mix functors};
\node [style=none] (19) at (-1, 1.5) {Isomix functors};
\node [style=none] (20) at (-1, 0.5) {$m_\top F({\sf m}^{-1}n_\bot = {\sf m}^{-1}$};
\node [style=none] (21) at (-1, -0.5) {$n_\bot{\sf m} m_\top = F({\sf m})$};
\node [style=none] (22) at (-1, -1.5) {$F_\otimes = F_\oplus; m_\otimes = \nu_\oplus^L = \nu_\oplus^R; n_\oplus = \nu_\otimes^L = \nu_\otimes^R$};
\node [style=none] (23) at (-1, -2.75) {};
\node [style=none] (24) at (0, 6) {};
\node [style=none] (25) at (0, -4) {};
\node [style=none] (26) at (5.75, 6) {};
\node [style=none] (27) at (5.75, -4) {};
\node [style=none] (28) at (-0.9999999, 7) {};
\node [style=none] (29) at (-0.9999999, -5) {};
\node [style=none] (30) at (6.75, -5) {};
\node [style=none] (31) at (6.75, 7) {};
\node [style=none] (32) at (3, 6.75) {cyclic functors};
\node [style=none] (33) at (3, 5.5) {symmetric functors};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (0.center);
\draw (4.center) to (5.center);
\draw (5.center) to (6.center);
\draw (6.center) to (7.center);
\draw (7.center) to (4.center);
\draw (11.center) to (10.center);
\draw (10.center) to (9.center);
\draw (9.center) to (8.center);
\draw (8.center) to (11.center);
\draw (12.center) to (13.center);
\draw (13.center) to (14.center);
\draw (12.center) to (15.center);
\draw (15.center) to (14.center);
\draw[dotted] (28.center) to (29.center);
\draw[dotted] (29.center) to (30.center);
\draw[dotted] (30.center) to (31.center);
\draw[dotted] (31.center) to (28.center);
\draw[dotted] (24.center) to (25.center);
\draw[dotted] (25.center) to (27.center);
\draw[dotted] (27.center) to (26.center);
\draw[dotted] (26.center) to (24.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{center}
\caption{Linear functor family}
\label{linear-functor-family}
\end{table}
Frobenius functors preserve linear duals and they allow a straightforward requirement which ensures that mix map is preserved. The notion of a dagger, which is central to this development, is based on a Frobenius functor. An objective of this section is to show how the coherence requirements for a dagger on an LDC are implied by requiring that the dagger functor be a Frobenius involutive equivalence. Once the dagger is understood we can consider $\dagger$-mix categories and their functors: these we shall take to be mix Frobenius functors with a
further requirement concerning the preservation of the dagger.
\subsection{Frobenius functors}
\begin{definition}
A {\bf Frobenius functor} is a linear functor $F$ such that:
\begin{enumerate}[{\bf \small [FLF.1]}]
\item $F_\otimes = F_\oplus $
\item $m_\otimes = \nu_\oplus^R = \nu_\oplus^L $
\item $n_\oplus = \nu_\otimes^L = \nu_\otimes^R$
\end{enumerate}
\end{definition}
A Frobenius functor is {\bf symmetric} if as a linear functor it preserves the symmetries of the tensor and par.
The left and right linear strengths of $\otimes$ and $\oplus$ coinciding with the $m_\otimes$ and $n_\oplus$ respectively means that in the diagrammatic calculus, ports can be moved around freely:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 2) {};
\node [style=none] (1) at (-1.5, 2) {};
\node [style=none] (2) at (-2.25, 2) {};
\node [style=none] (3) at (-2.25, 1) {};
\node [style=none] (4) at (-0.75, 1) {};
\node [style=none] (5) at (-0.75, 2) {};
\node [style=none] (6) at (-2, 2.75) {};
\node [style=none] (7) at (-1, 2.75) {};
\node [style=none] (61) at (-2.75, 2.75) {$F_\oplus(A)$};
\node [style=none] (71) at (-0.25, 2.75) {$F_\otimes(B)$};
\node [style=none] (8) at (-1.5, 0.25) {};
\node [style=none] (81) at (-2.25, 0) {$F_\oplus(A \otimes B) $};
\node [style=ox] (9) at (-1.5, 1.5) {};
\node [style=none] (10) at (-2, 1.25) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=-90, out=-90, looseness=1.25] (0.center) to (1.center);
\draw [bend right=15, looseness=1.00] (6.center) to (9);
\draw [bend right=15, looseness=0.75] (9) to (7.center);
\draw (2.center) to (3.center);
\draw (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (5.center) to (2.center);
\draw (9) to (8.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-1.5, 2) {};
\node [style=none] (1) at (-1, 2) {};
\node [style=none] (2) at (-2.25, 2) {};
\node [style=none] (3) at (-2.25, 1) {};
\node [style=none] (4) at (-0.75, 1) {};
\node [style=none] (5) at (-0.75, 2) {};
\node [style=none] (6) at (-2, 2.75) {};
\node [style=none] (7) at (-1, 2.75) {};
\node [style=none] (61) at (-2.75, 2.75) {$F_\otimes(A)$};
\node [style=none] (71) at (-0.25, 2.75) {$F_\oplus(B)$};
\node [style=none] (8) at (-1.5, 0.25) {};
\node [style=none] (81) at (-2.25, 0) {$F_\oplus(A \otimes B) $};
\node [style=ox] (9) at (-1.5, 1.5) {};
\node [style=none] (10) at (-2, 1.25) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=-90, out=-90, looseness=1.25] (0.center) to (1.center);
\draw [bend right=15, looseness=1.00] (6.center) to (9);
\draw [bend right=15, looseness=0.75] (9) to (7.center);
\draw (2.center) to (3.center);
\draw (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (5.center) to (2.center);
\draw (9) to (8.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.25, 2) {};
\node [style=none] (1) at (-2.25, 1) {};
\node [style=none] (2) at (-0.75, 1) {};
\node [style=none] (3) at (-0.75, 2) {};
\node [style=none] (4) at (-2, 2.75) {};
\node [style=none] (5) at (-1, 2.75) {};
\node [style=none] (41) at (-2.75, 2.75) {$F_\otimes(A)$};
\node [style=none] (51) at (-0.25, 2.75) {$F_\otimes(B)$};
\node [style=none] (6) at (-1.5, 0.25) {};
\node [style=none] (61) at (-2, 0) {$F_\otimes(A \otimes B)$};
\node [style=ox] (7) at (-1.5, 1.5) {};
\node [style=none] (8) at (-2, 1.25) {$F$};
\node [style=none] (9) at (-1.75, 1) {};
\node [style=none] (10) at (-1.25, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=15, looseness=1.00] (4.center) to (7);
\draw [bend right=15, looseness=0.75] (7) to (5.center);
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (0.center);
\draw (7) to (6.center);
\draw [in=90, out=90, looseness=1.25] (9.center) to (10.center);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.25, 1) {};
\node [style=none] (1) at (-2.25, 2) {};
\node [style=none] (2) at (-0.75, 2) {};
\node [style=none] (3) at (-0.75, 1) {};
\node [style=none] (4) at (-2, 0.25) {};
\node [style=none] (5) at (-1, 0.25) {};
\node [style=none] (6) at (-1.5, 2.75) {};
\node [style=none] (41) at (-2.75, 0.25) {$F_\otimes(A)$};
\node [style=none] (51) at (-0.25, 0.25) {$F_\oplus(B)$};
\node [style=none] (61) at (-2, 3) {$F_\otimes(A \oplus B)$};
\node [style=oa] (7) at (-1.5, 1.5) {};
\node [style=none] (8) at (-2, 1.75) {$F$};
\node [style=none] (9) at (-2, 1) {};
\node [style=none] (10) at (-1.5, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=15, looseness=1.00] (4.center) to (7);
\draw [bend left=15, looseness=0.75] (7) to (5.center);
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (0.center);
\draw (7) to (6.center);
\draw [in=90, out=90, looseness=1.25] (9.center) to (10.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.25, 1) {};
\node [style=none] (1) at (-2.25, 2) {};
\node [style=none] (2) at (-0.75, 2) {};
\node [style=none] (3) at (-0.75, 1) {};
\node [style=none] (4) at (-2, 0.25) {};
\node [style=none] (5) at (-1, 0.25) {};
\node [style=none] (41) at (-2.75, 0.25) {$F_\oplus(A)$};
\node [style=none] (51) at (-0.25, 0.25) {$F_\otimes(B)$};
\node [style=none] (61) at (-2, 3) {$F_\otimes(A \oplus B)$};
\node [style=none] (6) at (-1.5, 2.75) {};
\node [style=oa] (7) at (-1.5, 1.5) {};
\node [style=none] (8) at (-2, 1.75) {$F$};
\node [style=none] (9) at (-1.5, 1) {};
\node [style=none] (10) at (-1, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=15, looseness=1.00] (4.center) to (7);
\draw [bend left=15, looseness=0.75] (7) to (5.center);
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (0.center);
\draw (7) to (6.center);
\draw [in=90, out=90, looseness=1.25] (9.center) to (10.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.25, 1) {};
\node [style=none] (1) at (-2.25, 2) {};
\node [style=none] (2) at (-0.75, 2) {};
\node [style=none] (3) at (-0.75, 1) {};
\node [style=none] (4) at (-2, 0.25) {};
\node [style=none] (5) at (-1, 0.25) {};
\node [style=none] (41) at (-2.75, 0.25) {$F_\oplus(A)$};
\node [style=none] (51) at (-0.25, 0.25) {$F_\oplus(B)$};
\node [style=none] (61) at (-2, 3) {$F_\oplus(A \oplus B)$};
\node [style=none] (6) at (-1.5, 2.75) {};
\node [style=oa] (7) at (-1.5, 1.5) {};
\node [style=none] (8) at (-2, 1.75) {$F$};
\node [style=none] (9) at (-1.25, 2) {};
\node [style=none] (10) at (-1.75, 2) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=15, looseness=1.00] (4.center) to (7);
\draw [bend left=15, looseness=0.75] (7) to (5.center);
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (0.center);
\draw (7) to (6.center);
\draw [in=-90, out=-90, looseness=1.25] (9.center) to (10.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\[ \nu_\oplus^L = \nu_\otimes^R = m_\otimes ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \nu_\oplus^L = \nu_\oplus^R = n_\oplus \]
This, of course, means the ports can be omitted.
\begin{lemma}
\label{Lemma: Frobenius}
Suppose $\mathbb{X}$ and $\mathbb{Y}$ are LDCs. The following are equivalent:
\begin{enumerate}[(a)]
\item $F: \mathbb{X} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{Y}$ is a Frobenius linear functor.
\item $F$ is $\otimes$-monoidal and $\oplus$-comonoidal such that
\[
\hspace{-1cm}
\xymatrixcolsep{2pc}
\xymatrix{
F(A) \otimes F(B \oplus C) \ar[r]^{1 \otimes n_\oplus} \ar[d]_{m_\otimes} \ar@{}[dr]|{\tiny{\bf [F.1]}} & F(A) \otimes (F(B) \oplus F(C)) \ar[d]^{\delta^L} \\
F(A \otimes (B \oplus C)) \ar[d]_{F(\delta^L)} & (F(A) \otimes F(B)) \oplus F(C) \ar[d]^{m_\otimes \oplus 1} \\
F((A \oplus B) \oplus C) \ar[r]_{n_\oplus} & F(A \oplus B) \oplus F(C)
}~~~~\xymatrix{
F( A \oplus B) \otimes F(C) \ar[r]^{n_\oplus \otimes 1} \ar[d]_{m_\otimes} \ar@{}[dr]|{\tiny{\bf [F.2]}} & (F(A) \oplus F(B)) \otimes F(C) \ar[d]^{\delta^R} \\
F( (A \oplus B) \otimes C) \ar[d]_{F(\delta^R)} & F(A) \oplus (F(B) \otimes F(C)) \ar[d]^{1 \oplus m_\otimes} \\
F(A \oplus (B \otimes C)) \ar[r]_{n_\oplus} & F(A) \oplus F(B \otimes C)
}
\]
\end{enumerate}
\end{lemma}
\begin{proof}
For (a)$\Rightarrow$(b), fix $F := F_\otimes = F_\oplus$, then $F$ is $\otimes$-monoidal and $\oplus$-comonoidal. Conditions {\bf \small [F.1]} and {\bf \small [F.2]} are given by {\bf \small [LF.7]-(a)} and {\bf \small [LF.7]-(b)}. For the other direction, define $F_\otimes = F_\oplus := F$. Then it is straightforward to check that all the axioms of Frobenius linear functors are satisfied by $(F_\otimes, F_\oplus)$.
\end{proof}
Conditions {\bf \small [F.1]} and {\bf \small [F.2]} in Lemma \ref{Lemma: Frobenius} are diagrammatically represented as follows:
\[
{\bf [F.1]}~~~~~~~~
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=oa] (0) at (-3, 2) {};
\node [style=ox] (1) at (-4, 1) {};
\node [style=none] (2) at (-4.5, 3) {};
\node [style=none] (3) at (-3, 3) {};
\node [style=none] (4) at (-2.5, -0) {};
\node [style=none] (5) at (-4, -0) {};
\node [style=none] (6) at (-4.75, 2.5) {};
\node [style=none] (7) at (-4.75, 0.5) {};
\node [style=none] (8) at (-2.25, 0.5) {};
\node [style=none] (9) at (-2.25, 2.5) {};
\node [style=none] (10) at (-2.5, 2.25) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=15, out=-150, looseness=1.00] (0) to (1);
\draw [in=-90, out=135, looseness=1.00] (1) to (2.center);
\draw (0) to (3.center);
\draw [in=90, out=-45, looseness=1.00] (0) to (4.center);
\draw (1) to (5.center);
\draw (6.center) to (9.center);
\draw (9.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=oa] (0) at (-3, 2.5) {};
\node [style=ox] (1) at (-4, 1) {};
\node [style=none] (2) at (-4.5, 3.5) {};
\node [style=none] (3) at (-3, 3.5) {};
\node [style=none] (4) at (-2.5, -0) {};
\node [style=none] (5) at (-4, -0) {};
\node [style=none] (6) at (-4.75, 1.5) {};
\node [style=none] (7) at (-4.75, 0.5) {};
\node [style=none] (8) at (-3.25, 0.5) {};
\node [style=none] (9) at (-3.25, 1.5) {};
\node [style=none] (10) at (-3.75, 2) {};
\node [style=none] (11) at (-2.25, 2) {};
\node [style=none] (12) at (-2.25, 3) {};
\node [style=none] (13) at (-3.75, 3) {};
\node [style=none] (14) at (-2.5, 2.75) {$F$};
\node [style=none] (15) at (-3.5, 1.25) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=60, out=-150, looseness=1.00] (0) to (1);
\draw [in=-90, out=135, looseness=1.00] (1) to (2.center);
\draw (0) to (3.center);
\draw [in=90, out=-45, looseness=1.00] (0) to (4.center);
\draw (1) to (5.center);
\draw (6.center) to (9.center);
\draw (9.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\draw (13.center) to (10.center);
\draw (10.center) to (11.center);
\draw (11.center) to (12.center);
\draw (12.center) to (13.center);
\end{pgfonlayer}
\end{tikzpicture}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~ {\bf [F.2]}~~~~~~~~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=oa] (0) at (-3.25, 1) {};
\node [style=ox] (1) at (-4, 1.75) {};
\node [style=none] (2) at (-4, 3) {};
\node [style=none] (3) at (-2.75, 3) {};
\node [style=none] (4) at (-3.25, -0) {};
\node [style=none] (5) at (-4.5, -0) {};
\node [style=none] (6) at (-4.75, 2.5) {};
\node [style=none] (7) at (-4.75, 0.5) {};
\node [style=none] (8) at (-2.25, 0.5) {};
\node [style=none] (9) at (-2.25, 2.5) {};
\node [style=none] (10) at (-2.5, 2.25) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=-45, out=165, looseness=1.25] (0) to (1);
\draw [in=-90, out=90, looseness=1.00] (1) to (2.center);
\draw [in=-90, out=45, looseness=1.00] (0) to (3.center);
\draw [in=90, out=-90, looseness=1.00] (0) to (4.center);
\draw [in=90, out=-150, looseness=0.75] (1) to (5.center);
\draw (6.center) to (9.center);
\draw (9.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=oa] (0) at (-3.25, 0.5) {};
\node [style=ox] (1) at (-4, 1.75) {};
\node [style=none] (2) at (-4, 3) {};
\node [style=none] (3) at (-2.75, 3) {};
\node [style=none] (4) at (-3.25, -0.5) {};
\node [style=none] (5) at (-4.5, -0.5) {};
\node [style=none] (6) at (-2.5, 1) {};
\node [style=none] (7) at (-2.5, -0) {};
\node [style=none] (8) at (-3.75, -0) {};
\node [style=none] (9) at (-3.75, 1) {};
\node [style=none] (10) at (-2.5, 2.25) {};
\node [style=none] (11) at (-3.25, 2.25) {};
\node [style=none] (12) at (-3.25, 1.25) {};
\node [style=none] (13) at (-4.5, 2.25) {};
\node [style=none] (14) at (-4.5, 1.25) {};
\node [style=none] (15) at (-3.4, 1.5) {$F$};
\node [style=none] (16) at (-2.75, 0.25) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=-45, out=135, looseness=1.25] (0) to (1);
\draw [in=-90, out=90, looseness=1.00] (1) to (2.center);
\draw [in=-90, out=45, looseness=1.00] (0) to (3.center);
\draw [in=90, out=-90, looseness=1.00] (0) to (4.center);
\draw [in=90, out=-150, looseness=0.75] (1) to (5.center);
\draw (6.center) to (9.center);
\draw (9.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\draw (11.center) to (13.center);
\draw (13.center) to (14.center);
\draw (14.center) to (12.center);
\draw (12.center) to (11.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\begin{lemma}
The composite of Frobenius functors is a Frobenius functor.
\end{lemma}
\begin{proof}
Composition is defined as the usual linear functor composition. It is straightforward to check the necessary axioms for a Frobenius linear functor hold.
\end{proof}
It is immediate from Lemma \ref{Lemma: linear adjoints} that Frobenius functors preserve linear duals. In fact if $F: \mathbb{X} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{Y}$ is a Frobenius functor
and $A \dashvv B$ is a linear dual, as the duals $F_\otimes(A) \dashvv F_\oplus(B)$ and $F_\oplus(A) \dashvv F_\otimes(B)$ now coincide, we just obtain the
one dual $F(A) \dashvv F(B)$. In the case when the Frobenius functor is between $*$-autonomous categories which are cyclic we expect the
functor to be {\bf cyclor preserving} in the following sense:
\[ \mbox{\bf [CFF]} ~~~~~\xymatrix{ F(X^{*}) \ar[d]_{\cong} \ar[rr]^{F(\psi)} & & F(\!\!~^{*}X) \ar[d]^{\cong} \\
F(X)^{*} \ar[rr]_{\psi} & & \!\!~^{*}F(X) } \]
where the vertical arrows are respectively the maps:
\[ (u^R_\otimes)^{-1} (\eta* \otimes 1) \delta^R (1 \oplus (m^F_\otimes F(\epsilon*) n^F_\bot) u^R_\oplus
~~~\mbox{and}~~~(u^R)^{-1} (1 \otimes *\eta) \delta^L (m^F_\otimes \oplus 1)((F(*\epsilon)n_\bot^F) \oplus 1) u^L_\oplus \]
The cyclor preserving condition maybe pictorially represented as follows:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-1, 4) {};
\node [style=none] (1) at (3, 4) {};
\node [style=none] (2) at (-1, 2) {};
\node [style=none] (3) at (3, 2) {};
\node [style=none] (4) at (0, 3.25) {};
\node [style=none] (5) at (2, 3.25) {};
\node [style=circle, scale=2] (6) at (-2, 1) {};
\node [style=none] (7) at (-2, -0.5) {};
\node [style=none] (8) at (2, 6) {};
\node [style=none] (9) at (0, 5) {};
\node [style=none] (10) at (-2, 5) {};
\node [style=none] (11) at (-2, 1) {$\psi$};
\node [style=none] (12) at (2.7, 2.3) {$F$};
\node [style=none] (13) at (2.7, 5.75) {$F(X^*)$};
\node [style=none] (14) at (2.35, 3.5) {$X^*$};
\node [style=none] (15) at (1, 2.25) {$\epsilon*$};
\node [style=none] (16) at (-0.25, 3.5) {$X$};
\node [style=none] (17) at (-1, 6.25) {$\eta*$};
\node [style=none] (18) at (-2.7, 2) {$F(X)^*$};
\node [style=none] (19) at (-2.7, -0.3) {$~^*F(X)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (1.center);
\draw (1.center) to (0.center);
\draw (8.center) to (5.center);
\draw [bend left=90, looseness=1.25] (5.center) to (4.center);
\draw (4.center) to (9.center);
\draw [bend left=90, looseness=1.50] (10.center) to (9.center);
\draw (10.center) to (6);
\draw (6) to (7.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (1.5, 5) {};
\node [style=none] (1) at (-2.5, 5) {};
\node [style=none] (2) at (1.5, 2) {};
\node [style=none] (3) at (-2.5, 2) {};
\node [style=none] (4) at (0.5, 3.25) {};
\node [style=none] (5) at (-1.5, 3.25) {};
\node [style=none] (6) at (-1.5, 6) {};
\node [style=none] (7) at (0.5, 5) {};
\node [style=none] (8) at (2.5, 5) {};
\node [style=none] (9) at (1.2, 2.3) {$F$};
\node [style=none] (10) at (-2.2, 5.75) {$F(X^*)$};
\node [style=none] (11) at (-1.85, 3.5) {$~^*X$};
\node [style=none] (12) at (-0.5, 2.25) {$*\epsilon$};
\node [style=none] (13) at (0.85, 3.5) {$X$};
\node [style=none] (14) at (1.5, 6.15) {$*\eta$};
\node [style=none] (15) at (3.2, -0.25) {$~^*F(X)$};
\node [style=circle, scale=2] (16) at (-1.5, 4.25) {};
\node [style=none] (17) at (-1.5, 4.25) {$\psi$};
\node [style=none] (18) at (2.5, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (1.center);
\draw (1.center) to (0.center);
\draw [bend right=90, looseness=1.25] (5.center) to (4.center);
\draw (4.center) to (7.center);
\draw [bend right=90, looseness=1.50] (8.center) to (7.center);
\draw (6.center) to (16);
\draw (16) to (5.center);
\draw (8.center) to (18.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\begin{lemma}
\label{Lemma: cyclic Frob}
Suppose $F$ is a cyclor preserving Frobenius linear functor, then
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle, scale=2.5] (0) at (2, -4.75) {};
\node [style=circle, scale=2.5] (1) at (4, -3.25) {};
\node [style=none] (2) at (2, -7) {};
\node [style=none] (3) at (4, -4.25) {};
\node [style=none] (4) at (5.999999, -4) {};
\node [style=none] (5) at (5.999999, 0.9999999) {};
\node [style=none] (6) at (2, -2) {};
\node [style=none] (7) at (4, -2) {};
\node [style=none] (8) at (3.25, -2) {};
\node [style=none] (9) at (3.25, -5.5) {};
\node [style=none] (10) at (6.75, -5.5) {};
\node [style=none] (11) at (6.75, -2) {};
\node [style=none] (12) at (6.499999, -5) {$F$};
\node [style=none] (13) at (6.499999, -0.25) {$F(X)$};
\node [style=none] (14) at (5, -5) {$*\epsilon$};
\node [style=none] (15) at (4, -3.25) {$\psi$};
\node [style=none] (16) at (2, -4.75) {$\psi$};
\node [style=none] (17) at (3, -0.7499999) {$\eta*$};
\node [style=none] (18) at (6.25, -2.75) {$X$};
\node [style=none] (19) at (4.5, -4) {$~^*X$};
\node [style=none] (20) at (4.5, -2.5) {$X^*$};
\node [style=none] (21) at (4.25, -1.25) {$F(X^*)$};
\node [style=none] (22) at (1.25, -2.25) {$F(X^*)^*$};
\node [style=none] (23) at (1.25, -6.25) {$~^*F(X^*)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (6.center) to (0);
\draw (0) to (2.center);
\draw [bend left=90, looseness=1.50] (6.center) to (7.center);
\draw (7.center) to (1);
\draw (1) to (3.center);
\draw [bend right=90, looseness=1.25] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (8.center) to (9.center);
\draw (9.center) to (10.center);
\draw (10.center) to (11.center);
\draw (11.center) to (8.center);
\end{pgfonlayer}
\end{tikzpicture}
= \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (7.25, -2) {};
\node [style=none] (1) at (5.25, -1.75) {};
\node [style=none] (2) at (5.25, 3.5) {};
\node [style=none] (3) at (4.75, 3.25) {$F(X)$};
\node [style=none] (4) at (5, -0.4999999) {$X$};
\node [style=none] (5) at (7.25, 1.5) {};
\node [style=none] (6) at (9.500001, 1.25) {};
\node [style=none] (7) at (4.5, 0.7499999) {};
\node [style=none] (8) at (4.5, -3.5) {};
\node [style=none] (9) at (8.25, -3.5) {};
\node [style=none] (10) at (8.25, 0.7499999) {};
\node [style=none] (11) at (7.999999, -3.25) {$F$};
\node [style=none] (12) at (9.500001, -4.75) {};
\node [style=none] (13) at (10.5, -4.25) {$~^*F(X^*)$};
\node [style=none] (14) at (9.500001, 1.25) {};
\node [style=none] (15) at (8.25, 2.75) {$*\eta$};
\node [style=none] (16) at (6.25, -3) {$\epsilon*$};
\node [style=none] (17) at (4.25, -4) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=90, looseness=1.25] (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw [bend right=90, looseness=1.50] (6.center) to (5.center);
\draw (7.center) to (10.center);
\draw (10.center) to (9.center);
\draw (9.center) to (8.center);
\draw (8.center) to (7.center);
\draw (14.center) to (12.center);
\draw (5.center) to (0.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\end{lemma}
\begin{proof}
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle, scale=2.5] (0) at (2, -4.75) {};
\node [style=circle, scale=2.5] (1) at (4, -3.25) {};
\node [style=none] (2) at (2, -7) {};
\node [style=none] (3) at (4, -4.25) {};
\node [style=none] (4) at (5.999999, -4) {};
\node [style=none] (5) at (5.999999, 0.9999999) {};
\node [style=none] (6) at (2, -2) {};
\node [style=none] (7) at (4, -2) {};
\node [style=none] (8) at (3.25, -2) {};
\node [style=none] (9) at (3.25, -5.5) {};
\node [style=none] (10) at (6.75, -5.5) {};
\node [style=none] (11) at (6.75, -2) {};
\node [style=none] (12) at (6.499999, -5) {$F$};
\node [style=none] (13) at (6.499999, -0.25) {$F(X)$};
\node [style=none] (14) at (5, -5) {$*\epsilon$};
\node [style=none] (15) at (4, -3.25) {$\psi$};
\node [style=none] (16) at (2, -4.75) {$\psi$};
\node [style=none] (17) at (3, -0.7499999) {$\eta*$};
\node [style=none] (18) at (6.25, -2.75) {$X$};
\node [style=none] (19) at (4.5, -4) {$~^*X$};
\node [style=none] (20) at (4.5, -2.5) {$X^*$};
\node [style=none] (21) at (4.25, -1.25) {$F(X^*)$};
\node [style=none] (22) at (1.25, -2.25) {$F(X^*)^*$};
\node [style=none] (23) at (1.25, -6.25) {$~^*F(X^*)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (6.center) to (0);
\draw (0) to (2.center);
\draw [bend left=90, looseness=1.50] (6.center) to (7.center);
\draw (7.center) to (1);
\draw (1) to (3.center);
\draw [bend right=90, looseness=1.25] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (8.center) to (9.center);
\draw (9.center) to (10.center);
\draw (10.center) to (11.center);
\draw (11.center) to (8.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle, scale=2.5] (0) at (2, -4.75) {};
\node [style=circle, scale=2.5] (1) at (6.75, -4.25) {};
\node [style=none] (2) at (2, -7.5) {};
\node [style=none] (3) at (6.75, -5.25) {};
\node [style=none] (4) at (8.75, -5) {};
\node [style=none] (5) at (8.75, 1) {};
\node [style=none] (6) at (2, -2) {};
\node [style=none] (7) at (4, -2) {};
\node [style=none] (8) at (3.25, -2) {};
\node [style=none] (9) at (3.25, -6.5) {};
\node [style=none] (10) at (9.500001, -6.5) {};
\node [style=none] (11) at (9.500001, -2) {};
\node [style=none] (12) at (9.25, -6) {$F$};
\node [style=none] (13) at (9.25, -0.25) {$F(X)$};
\node [style=none] (14) at (7.75, -6) {$*\epsilon$};
\node [style=none] (15) at (6.75, -4.25) {$\psi$};
\node [style=none] (16) at (2, -4.75) {$\psi$};
\node [style=none] (17) at (3, -0.7499999) {$\eta*$};
\node [style=none] (18) at (9.000001, -3.75) {$X$};
\node [style=none] (19) at (7.25, -5) {$~^*X$};
\node [style=none] (20) at (7.25, -3.5) {$X^*$};
\node [style=none] (21) at (4.25, -1.25) {$F(X^*)$};
\node [style=none] (22) at (1.25, -2.25) {$F(X^*)^*$};
\node [style=none] (23) at (1.25, -6.25) {$~^*F(X^*)$};
\node [style=none] (24) at (4, -5) {};
\node [style=none] (25) at (5.499999, -5) {};
\node [style=none] (26) at (5.499999, -3.5) {};
\node [style=none] (27) at (6.75, -3.5) {};
\node [style=none] (28) at (5.999999, -2.75) {$\eta*$};
\node [style=none] (29) at (4.75, -6) {$\epsilon*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (6.center) to (0);
\draw (0) to (2.center);
\draw [bend left=90, looseness=1.50] (6.center) to (7.center);
\draw (1) to (3.center);
\draw [bend right=90, looseness=1.25] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (8.center) to (9.center);
\draw (9.center) to (10.center);
\draw (10.center) to (11.center);
\draw (11.center) to (8.center);
\draw (7.center) to (24.center);
\draw [bend right=90, looseness=1.75] (24.center) to (25.center);
\draw (26.center) to (25.center);
\draw [bend right=90, looseness=1.50] (27.center) to (26.center);
\draw (27.center) to (1);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle, scale=2.5] (0) at (2, -4.75) {};
\node [style=circle, scale=2.5] (1) at (6.75, -1.25) {};
\node [style=none] (2) at (2, -7.5) {};
\node [style=none] (3) at (6.75, -2.25) {};
\node [style=none] (4) at (8.75, -2) {};
\node [style=none] (5) at (8.75, 1.75) {};
\node [style=none] (6) at (2, -2) {};
\node [style=none] (7) at (4, -2) {};
\node [style=none] (8) at (3.25, -3.75) {};
\node [style=none] (9) at (3.25, -6.5) {};
\node [style=none] (10) at (5.999999, -6.5) {};
\node [style=none] (11) at (5.75, -6) {$F$};
\node [style=none] (12) at (9.25, 1.5) {$F(X)$};
\node [style=none] (13) at (7.75, -3.25) {$*\epsilon$};
\node [style=none] (14) at (6.75, -1.25) {$\psi$};
\node [style=none] (15) at (2, -4.75) {$\psi$};
\node [style=none] (16) at (3, -0.7499999) {$\eta*$};
\node [style=none] (17) at (9.000001, -0.7499999) {$X$};
\node [style=none] (18) at (7.25, -2) {$~^*X$};
\node [style=none] (19) at (7.25, -0.4999999) {$X^*$};
\node [style=none] (20) at (4.25, -1.25) {$F(X^*)$};
\node [style=none] (21) at (1.25, -2.25) {$F(X^*)^*$};
\node [style=none] (22) at (1.25, -6.25) {$~^*F(X^*)$};
\node [style=none] (23) at (4, -5) {};
\node [style=none] (24) at (5.499999, -5) {};
\node [style=none] (25) at (5.499999, -0.7499999) {};
\node [style=none] (26) at (6.75, -0.4999999) {};
\node [style=none] (27) at (6.25, 0.25) {$\eta*$};
\node [style=none] (28) at (4.75, -6) {$\epsilon*$};
\node [style=none] (29) at (5.999999, -3.75) {};
\node [style=none] (30) at (5.25, 0.7499999) {};
\node [style=none] (31) at (5.25, -3.5) {};
\node [style=none] (32) at (9.500001, -3.5) {};
\node [style=none] (33) at (9.500001, 0.7499999) {};
\node [style=none] (34) at (9.25, -3.25) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (6.center) to (0);
\draw (0) to (2.center);
\draw [bend left=90, looseness=1.50] (6.center) to (7.center);
\draw (1) to (3.center);
\draw [bend right=90, looseness=1.25] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (8.center) to (9.center);
\draw (9.center) to (10.center);
\draw (7.center) to (23.center);
\draw [bend right=90, looseness=1.75] (23.center) to (24.center);
\draw (25.center) to (24.center);
\draw [bend right=90, looseness=1.50] (26.center) to (25.center);
\draw (26.center) to (1);
\draw (10.center) to (29.center);
\draw (29.center) to (8.center);
\draw (30.center) to (33.center);
\draw (33.center) to (32.center);
\draw (32.center) to (31.center);
\draw (31.center) to (30.center);
\end{pgfonlayer}
\end{tikzpicture} \stackrel{\tiny {\bf [CFF]}}{=}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle, scale=2.5] (0) at (6.75, -1.25) {};
\node [style=none] (1) at (6.75, -2.25) {};
\node [style=none] (2) at (8.75, -2) {};
\node [style=none] (3) at (8.75, 1.75) {};
\node [style=none] (4) at (9.25, 1.5) {$F(X)$};
\node [style=none] (5) at (7.75, -3.25) {$*\epsilon$};
\node [style=none] (6) at (6.75, -1.25) {$\psi$};
\node [style=none] (7) at (9.000001, -0.7499999) {$X$};
\node [style=none] (8) at (7.25, -2) {$~^*X$};
\node [style=none] (9) at (7.25, -0.4999999) {$X^*$};
\node [style=none] (10) at (5.499999, -0.7499999) {};
\node [style=none] (11) at (6.75, -0.4999999) {};
\node [style=none] (12) at (6.25, 0.25) {$\eta*$};
\node [style=none] (13) at (5.25, 0.7499999) {};
\node [style=none] (14) at (5.25, -3.5) {};
\node [style=none] (15) at (9.500001, -3.5) {};
\node [style=none] (16) at (9.500001, 0.7499999) {};
\node [style=none] (17) at (9.25, -3.25) {$F$};
\node [style=none] (18) at (4.5, -5.25) {};
\node [style=none] (19) at (6.999999, -5.5) {};
\node [style=none] (20) at (6.25, -7.5) {$\epsilon*$};
\node [style=none] (21) at (7.75, -8) {};
\node [style=none] (22) at (8.75, -9) {};
\node [style=none] (23) at (9.500001, -7.75) {$~^*F(X^*)$};
\node [style=none] (24) at (4.5, -8) {};
\node [style=none] (25) at (8.75, -5.5) {};
\node [style=none] (26) at (6.999999, -6.75) {};
\node [style=none] (27) at (7.75, -5.25) {};
\node [style=none] (28) at (7.75, -4.5) {$*\eta$};
\node [style=circle, scale=2.5] (29) at (5.499999, -6.25) {};
\node [style=none] (30) at (5.499999, -6.25) {$\psi$};
\node [style=none] (31) at (7.499999, -7.75) {$F$};
\node [style=none] (32) at (5.499999, -6.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1.center);
\draw [bend right=90, looseness=1.25] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend right=90, looseness=1.50] (11.center) to (10.center);
\draw (11.center) to (0);
\draw (13.center) to (16.center);
\draw (16.center) to (15.center);
\draw (15.center) to (14.center);
\draw (14.center) to (13.center);
\draw (25.center) to (22.center);
\draw (18.center) to (27.center);
\draw [bend right=90, looseness=1.50] (25.center) to (19.center);
\draw (21.center) to (24.center);
\draw (24.center) to (18.center);
\draw (19.center) to (26.center);
\draw (27.center) to (21.center);
\draw (29) to (10.center);
\draw [bend right=75, looseness=1.25] (32.center) to (26.center);
\draw (29) to (32.center);
\end{pgfonlayer}
\end{tikzpicture} =
\]
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle, scale=2.5] (0) at (6.75, -1.25) {};
\node [style=none] (1) at (6.75, -2.25) {};
\node [style=none] (2) at (8.75, -2) {};
\node [style=none] (3) at (8.75, 1.75) {};
\node [style=none] (4) at (9.25, 1.5) {$F(X)$};
\node [style=none] (5) at (7.75, -3.25) {$*\epsilon$};
\node [style=none] (6) at (6.75, -1.25) {$\psi$};
\node [style=none] (7) at (9.000001, -0.7499999) {$X$};
\node [style=none] (8) at (7.25, -2) {$~^*X$};
\node [style=none] (9) at (7.25, -0.4999999) {$X^*$};
\node [style=none] (10) at (5.499999, -0.7499999) {};
\node [style=none] (11) at (6.75, -0.4999999) {};
\node [style=none] (12) at (6.25, 0.25) {$\eta*$};
\node [style=none] (13) at (4.5, 0.7499999) {};
\node [style=none] (14) at (4.5, -3.5) {};
\node [style=none] (15) at (9.500001, -3.5) {};
\node [style=none] (16) at (9.500001, 0.7499999) {};
\node [style=none] (17) at (9.25, -3.25) {$F$};
\node [style=none] (18) at (4.5, -5.25) {};
\node [style=none] (19) at (6.999999, -5.5) {};
\node [style=none] (20) at (6.25, -7.5) {$*\epsilon$};
\node [style=none] (21) at (7.75, -8) {};
\node [style=none] (22) at (8.75, -9) {};
\node [style=none] (23) at (9.500001, -7.75) {$~^*F(X^*)$};
\node [style=none] (24) at (4.5, -8) {};
\node [style=none] (25) at (8.75, -5.5) {};
\node [style=none] (26) at (6.999999, -6.75) {};
\node [style=none] (27) at (7.75, -5.25) {};
\node [style=none] (28) at (7.75, -4.5) {$*\eta$};
\node [style=circle, scale=2.5] (29) at (5.499999, -1.25) {};
\node [style=none] (30) at (5.499999, -1.25) {$\psi$};
\node [style=none] (31) at (7.499999, -7.75) {$F$};
\node [style=none] (32) at (5.499999, -6.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1.center);
\draw [bend right=90, looseness=1.25] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend right=90, looseness=1.50] (11.center) to (10.center);
\draw (11.center) to (0);
\draw (13.center) to (16.center);
\draw (16.center) to (15.center);
\draw (15.center) to (14.center);
\draw (14.center) to (13.center);
\draw (25.center) to (22.center);
\draw (18.center) to (27.center);
\draw [bend right=90, looseness=1.50] (25.center) to (19.center);
\draw (21.center) to (24.center);
\draw (24.center) to (18.center);
\draw (19.center) to (26.center);
\draw (27.center) to (21.center);
\draw (29) to (10.center);
\draw [bend right=75, looseness=1.25] (32.center) to (26.center);
\draw (29) to (32.center);
\end{pgfonlayer}
\end{tikzpicture} \stackrel{F({\bf [C.2]})}{=}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (7.25, -2) {};
\node [style=none] (1) at (5.25, -1.75) {};
\node [style=none] (2) at (5.25, 2) {};
\node [style=none] (3) at (4.75, 1.75) {$F(X)$};
\node [style=none] (4) at (5, -0.4999999) {$X$};
\node [style=none] (5) at (7.25, -0.7499999) {};
\node [style=none] (6) at (8.75, -0.7499999) {};
\node [style=none] (7) at (4.5, 0.7499999) {};
\node [style=none] (8) at (4.5, -3.5) {};
\node [style=none] (9) at (9.500001, -3.5) {};
\node [style=none] (10) at (9.500001, 0.7499999) {};
\node [style=none] (11) at (9.25, -3.25) {$F$};
\node [style=none] (12) at (7.75, -5) {};
\node [style=none] (13) at (10.25, -5.25) {};
\node [style=none] (14) at (9.500001, -7.25) {$*\epsilon$};
\node [style=none] (15) at (11, -7.75) {};
\node [style=none] (16) at (12, -8.75) {};
\node [style=none] (17) at (12.75, -7.5) {$~^*F(X^*)$};
\node [style=none] (18) at (7.75, -7.75) {};
\node [style=none] (19) at (12, -5.25) {};
\node [style=none] (20) at (10.25, -6.5) {};
\node [style=none] (21) at (11, -5) {};
\node [style=none] (22) at (11, -4.25) {$*\eta$};
\node [style=none] (23) at (10.75, -7.5) {$F$};
\node [style=none] (24) at (8.75, -6.5) {};
\node [style=none] (25) at (6.25, -3) {$\epsilon*$};
\node [style=none] (26) at (7.999999, 0.25) {$*\eta$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=90, looseness=1.25] (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw [bend right=90, looseness=1.50] (6.center) to (5.center);
\draw (7.center) to (10.center);
\draw (10.center) to (9.center);
\draw (9.center) to (8.center);
\draw (8.center) to (7.center);
\draw (19.center) to (16.center);
\draw (12.center) to (21.center);
\draw [bend right=90, looseness=1.50] (19.center) to (13.center);
\draw (15.center) to (18.center);
\draw (18.center) to (12.center);
\draw (13.center) to (20.center);
\draw (21.center) to (15.center);
\draw [bend right=75, looseness=1.25] (24.center) to (20.center);
\draw (6.center) to (24.center);
\draw (5.center) to (0.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (7.25, -2) {};
\node [style=none] (1) at (5.25, -1.75) {};
\node [style=none] (2) at (5.25, 3.5) {};
\node [style=none] (3) at (4.75, 3.25) {$F(X)$};
\node [style=none] (4) at (5, -0.4999999) {$X$};
\node [style=none] (5) at (7.25, 1.5) {};
\node [style=none] (6) at (9.500001, 1.25) {};
\node [style=none] (7) at (4.5, 0.7499999) {};
\node [style=none] (8) at (4.5, -3.5) {};
\node [style=none] (9) at (8.25, -3.5) {};
\node [style=none] (10) at (8.25, 0.7499999) {};
\node [style=none] (11) at (7.999999, -3.25) {$F$};
\node [style=none] (12) at (9.500001, -4.75) {};
\node [style=none] (13) at (10.5, -4.25) {$~^*F(X^*)$};
\node [style=none] (14) at (9.500001, 1.25) {};
\node [style=none] (15) at (8.25, 2.75) {$*\eta$};
\node [style=none] (16) at (6.25, -3) {$\epsilon*$};
\node [style=none] (17) at (4.25, -4) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=90, looseness=1.25] (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw [bend right=90, looseness=1.50] (6.center) to (5.center);
\draw (7.center) to (10.center);
\draw (10.center) to (9.center);
\draw (9.center) to (8.center);
\draw (8.center) to (7.center);
\draw (14.center) to (12.center);
\draw (5.center) to (0.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\end{proof}
\begin{definition}
Suppose $\mathbb{X}$ and $\mathbb{Y}$ are mix categories. $F: \mathbb{X} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{Y}$ is a {\bf mix functor} if it is a Frobenius functor such that
\[
\mbox{\bf{[mix-FF]}}~~~~~
\xymatrix{
F(\bot) \ar@/_2pc/[rrr]_{F({\sf m})} \ar[r]^{n_\bot} & \bot \ar[r]^{{\sf m}} & \top \ar[r]^{m_\top} & F(\top) \\
}
\]
\end{definition}
This is diagrammatically represented as follows:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 2) {};
\node [style=none] (1) at (-3, 1) {};
\node [style=none] (2) at (-1, 2) {};
\node [style=none] (3) at (-1, 1) {};
\node [style=circle] (4) at (-2, 1.5) {$\bot$};
\node [style=circle] (5) at (0, 1.3) {$\bot$};
\node [style=none] (6) at (0, 0.5) {};
\node [style=circle] (7) at (0, -0.3) {$\top$};
\node [style=none] (8) at (-2, 3) {};
\node [style=none] (9) at (0.75, -0.5) {};
\node [style=none] (10) at (2.75, -0.5) {};
\node [style=none] (11) at (0.75, -1.5) {};
\node [style=none] (12) at (2.75, -1.5) {};
\node [style=none] (13) at (1.75, -2.5) {};
\node [style=circle] (14) at (1.75, -1) {$\top$};
\node [style=map] (15) at (0, 0.5) {};
\node [style=circle, scale=0.5] (16) at (1.75, -2) {};
\node [style=circle, scale=0.5] (17) at (-2, 2.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (9.center) to (10.center);
\draw (10.center) to (12.center);
\draw (12.center) to (11.center);
\draw (11.center) to (9.center);
\draw (8.center) to (4);
\draw (14) to (13.center);
\draw [dotted, bend left=45, looseness=1.25] (17) to (5);
\draw [dotted, in=-165, out=-90, looseness=1.25] (7) to (16);
\draw (5) to (6.center);
\draw (6.center) to (7);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, 0.5) {};
\node [style=none] (2) at (0, -0.5) {};
\node [style=map] (3) at (0, 0.5) {};
\node [style=none] (4) at (0, -1.75) {};
\node [style=none] (5) at (0, 2.5) {};
\node [style=none] (6) at (-1.25, 2.2) {};
\node [style=none] (7) at (-1.25, -1.2) {};
\node [style=none] (8) at (1.25, 2.2) {};
\node [style=none] (9) at (1.25, -1.2) {};
\node [style=none] (10) at (-1, 1.8) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1.center);
\draw (1.center) to (2);
\draw (5.center) to (0);
\draw (2) to (4.center);
\draw (6.center) to (8.center);
\draw (8.center) to (9.center);
\draw (9.center) to (7.center);
\draw (7.center) to (6.center);
\end{pgfonlayer}
\end{tikzpicture} \]
\begin{lemma}
\label{Lemma: Mix Frobenius linear functor}
Mix functors preserve the mix map:
\[
\xymatrix{
F(A) \otimes F(B) \ar[r]^{\mathsf{mx}} \ar[d]_{m_\otimes} \ar[r]^{\mathsf{mx}} & F(A) \oplus F(B) \\
F(A \otimes B) \ar[r]_{F(\mathsf{mx})} \ar[r]_{F(\mathsf{mx})} & F(A \oplus B) \ar[u]_{n_\oplus}
}
\]
\end{lemma}
\begin{proof}
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.25, 2.5) {};
\node [style=none] (1) at (-2.25, -2.25) {};
\node [style=none] (2) at (1.5, 2.5) {};
\node [style=none] (3) at (1.5, -2.25) {};
\node [style=none] (4) at (-1, -4.5) {};
\node [style=circle] (5) at (0, 1.5) {$\bot$};
\node [style=none] (6) at (0, 0.5) {};
\node [style=circle] (7) at (0, -0.5) {$\top$};
\node [style=none] (8) at (-1, 4.5) {};
\node [style=none] (9) at (1, -4.5) {};
\node [style=none] (10) at (1, 4.5) {};
\node [style=map] (11) at (0, 0.5) {};
\node [style=circle, scale=0.5] (12) at (1, -2) {};
\node [style=circle, scale=0.5] (13) at (-1, 2.25) {};
\node [style=none] (14) at (-1.75, 2) {$F$};
\node [style=none] (15) at (-1.5, 4.5) {$F(A)$};
\node [style=none] (16) at (1.5, 4.5) {$F(B)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (8.center) to (4.center);
\draw (10.center) to (9.center);
\draw [dotted, in=90, out=3, looseness=1.00] (13) to (5);
\draw [dotted, in=180, out=-90, looseness=1.00] (7) to (12);
\draw (5) to (6.center);
\draw (6.center) to (7);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.25, 3) {};
\node [style=none] (1) at (-2.25, 1.5) {};
\node [style=none] (2) at (0.5000002, 3) {};
\node [style=none] (3) at (0.5000002, 1.5) {};
\node [style=none] (4) at (-0.9999997, -4) {};
\node [style=circle] (5) at (0, 2) {$\bot$};
\node [style=circle] (6) at (0, -1.5) {$\top$};
\node [style=none] (7) at (-0.9999997, 4.5) {};
\node [style=none] (8) at (0.9999997, -4) {};
\node [style=none] (9) at (0.9999997, 4.5) {};
\node [style=map] (10) at (0, 0.2500001) {};
\node [style=circle, scale=0.5] (11) at (0.9999997, -2.25) {};
\node [style=circle, scale=0.5] (12) at (-0.9999997, 2.75) {};
\node [style=none] (13) at (-1.75, 2.5) {};
\node [style=none] (14) at (-0.5000002, -1) {};
\node [style=none] (15) at (2, -1) {};
\node [style=none] (16) at (-0.5000002, -2.75) {};
\node [style=none] (17) at (2, -2.75) {};
\node [style=none] (18) at (1.5, -2.5) {};
\node [style=none] (19) at (-0.5000002, 0.75) {};
\node [style=none] (20) at (-0.5000002, -0.2500001) {};
\node [style=none] (21) at (0.5000002, -0.2500001) {};
\node [style=none] (22) at (0.5000002, 0.75) {};
\node [style=none] (23) at (-1.75, 1.75) {$F$};
\node [style=none] (24) at (0, -2.5) {$F$};
\node [style=none] (25) at (-0.3, -0.1) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (7.center) to (4.center);
\draw (9.center) to (8.center);
\draw [dotted, in=90, out=3, looseness=1.00] (12) to (5);
\draw [dotted, in=180, out=-90, looseness=1.00] (6) to (11);
\draw (14.center) to (16.center);
\draw (16.center) to (17.center);
\draw (17.center) to (15.center);
\draw (15.center) to (14.center);
\draw (5) to (10);
\draw (10) to (6);
\draw (19.center) to (20.center);
\draw (20.center) to (21.center);
\draw (21.center) to (22.center);
\draw (22.center) to (19.center);
\end{pgfonlayer}
\end{tikzpicture} \stackrel{\text{ {\bf[mix-FF]}}}{=}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.25, 2.25) {};
\node [style=none] (1) at (-4, 2.25) {};
\node [style=none] (2) at (-2.25, 1) {};
\node [style=none] (3) at (-4, 1) {};
\node [style=none] (4) at (-3.5, 3) {};
\node [style=none] (5) at (-3.5, -6.25) {};
\node [style=circle, scale=0.5] (6) at (-3.5, 2) {};
\node [style=circle] (7) at (-2.75, 1.5) {$\bot$};
\node [style=circle] (8) at (-1.5, -2) {$\top$};
\node [style=circle] (9) at (-1.5, -0.25) {$\bot$};
\node [style=none] (10) at (-0.75, -2) {};
\node [style=none] (11) at (0.25, -2) {};
\node [style=none] (12) at (0.25, -3) {};
\node [style=none] (13) at (-3.25, -0) {};
\node [style=none] (14) at (-2.25, -1) {};
\node [style=circle] (15) at (-0.25, -2.5) {$\top$};
\node [style=none] (16) at (-0.75, -3) {};
\node [style=none] (17) at (-2.25, -0) {};
\node [style=circle] (18) at (-2.75, -0.5) {$\bot$};
\node [style=circle, scale=0.5] (19) at (-2.75, 0.5) {};
\node [style=none] (20) at (-3.25, -1) {};
\node [style=circle, scale=0.5] (21) at (-0.25, -3.5) {};
\node [style=map] (22) at (-1.5, -1.25) {};
\node [style=none] (23) at (1.25, -4) {};
\node [style=none] (24) at (0.75, -6.25) {};
\node [style=none] (25) at (-0.75, -4) {};
\node [style=circle] (26) at (-0.25, -4.5) {$\top$};
\node [style=none] (27) at (1.25, -5.25) {};
\node [style=none] (28) at (0.75, 3) {};
\node [style=none] (29) at (-0.75, -5.25) {};
\node [style=circle, scale=0.5] (30) at (0.75, -5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (4.center) to (5.center);
\draw [bend left, looseness=1.25, dotted] (6) to (7);
\draw (13.center) to (20.center);
\draw (20.center) to (14.center);
\draw (14.center) to (17.center);
\draw (17.center) to (13.center);
\draw (10.center) to (11.center);
\draw (11.center) to (12.center);
\draw (12.center) to (16.center);
\draw (16.center) to (10.center);
\draw [dotted, bend left=45, looseness=1.25] (19) to (9);
\draw [dotted, in=-165, out=-90, looseness=1.25] (8) to (21);
\draw (9) to (22);
\draw (22) to (8);
\draw (29.center) to (27.center);
\draw (27.center) to (23.center);
\draw (23.center) to (25.center);
\draw (25.center) to (29.center);
\draw (24.center) to (28.center);
\draw [dotted, bend left=45, looseness=1.25] (30) to (26);
\draw (15) to (26);
\draw (7) to (18);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.25, -0.25) {};
\node [style=none] (1) at (-4, -0.25) {};
\node [style=none] (2) at (-2.25, -2.75) {};
\node [style=none] (3) at (-4, -2.75) {};
\node [style=none] (4) at (-3.5, 2.5) {};
\node [style=none] (5) at (-3.5, -6.5) {};
\node [style=circle, scale=0.5] (6) at (-3.5, -0.5) {};
\node [style=circle] (7) at (-2.75, -1.25) {$\bot$};
\node [style=circle] (8) at (-1.5, -2) {$\top$};
\node [style=circle] (9) at (-1.5, -0.25) {$\bot$};
\node [style=circle] (10) at (-0.25, -1.25) {$\top$};
\node [style=circle] (11) at (-2.75, -2.25) {$\bot$};
\node [style=circle, scale=0.5] (12) at (-3.5, 0.5) {};
\node [style=circle, scale=0.5] (13) at (0.75, -4.25) {};
\node [style=map] (14) at (-1.5, -1.25) {};
\node [style=none] (15) at (1.25, -0.75) {};
\node [style=none] (16) at (0.75, -6.5) {};
\node [style=none] (17) at (-0.75, -0.75) {};
\node [style=circle] (18) at (-0.25, -2.25) {$\top$};
\node [style=none] (19) at (1.25, -3.5) {};
\node [style=none] (20) at (0.75, 2.5) {};
\node [style=none] (21) at (-0.75, -3.5) {};
\node [style=circle, scale=0.5] (22) at (0.75, -3.25) {};
\node [style=none] (23) at (-0.5, -3.25) {$F$};
\node [style=none] (24) at (-3.75, -2.5) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (4.center) to (5.center);
\draw [bend left, looseness=1.25, dotted] (6) to (7);
\draw [dotted, bend left, looseness=1.25] (12) to (9);
\draw [dotted, in=-165, out=-90, looseness=1.25] (8) to (13);
\draw (9) to (14);
\draw (14) to (8);
\draw (21.center) to (19.center);
\draw (19.center) to (15.center);
\draw (15.center) to (17.center);
\draw (17.center) to (21.center);
\draw (16.center) to (20.center);
\draw [dotted, bend left=45, looseness=1.25] (22) to (18);
\draw (10) to (18);
\draw (7) to (11);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.9999997, -4) {};
\node [style=circle] (1) at (0, 2) {$\bot$};
\node [style=circle] (2) at (0, -1.5) {$\top$};
\node [style=none] (3) at (-0.9999997, 4.5) {};
\node [style=none] (4) at (0.9999997, -4) {};
\node [style=none] (5) at (0.9999997, 4.5) {};
\node [style=map] (6) at (0, 0.2500001) {};
\node [style=circle, scale=0.5] (7) at (0.9999997, -2.25) {};
\node [style=circle, scale=0.5] (8) at (-0.9999997, 2.75) {};
\node [style=none] (9) at (-1.5, 4.25) {$F(A)$};
\node [style=none] (10) at (1.5, 4.25) {$F(B)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (3.center) to (0.center);
\draw (5.center) to (4.center);
\draw [dotted, in=90, out=3, looseness=1.00] (8) to (1);
\draw [dotted, in=180, out=-90, looseness=1.00] (2) to (7);
\draw (1) to (6);
\draw (6) to (2);
\end{pgfonlayer}
\end{tikzpicture}
\]
\end{proof}
Linear natural isomorphisms between Frobenius functors $\alpha: F \@ifnextchar^ {\t@@}{\t@@^{}} G$ often take a special form with $\alpha_\otimes = \alpha_\oplus^{-1}$: this allows the coherence
requirements to be simplified. The next results describe some basic circumstances in which this happens:
\begin{lemma}
\label{Lemma: Frobenius linear transformation}
Suppose $F: \mathbb{X} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{Y}$ are Frobenius linear functors and $\alpha := (\alpha_\otimes, \alpha_\oplus): F \Rightarrow G$ is a linear natural transformation. Then, the following are equivalent:
\begin{enumerate}[(i)]
\item One of \[
\mbox{\bf \small [nat.1]} ~~~
\xymatrixcolsep{5pc}
\xymatrix{
\top \ar[r]^{m_\top} \ar[dr]_{m_\top} & G(\top) \ar[d]^{\alpha_\oplus} \ar@{}[dl]|(.35){\tiny{(a)}} \\
& F(\top)
} ~~~~~ \text{ or }~~~~~ \xymatrix{
F(\bot) \ar[r]^{\alpha_\otimes} \ar[dr]_{n_\bot} & G(\bot) \ar[d]^{n_\bot} \ar@{}[dl]|(.35){\tiny{(b)}} \\
& \bot
}
\] commutes and either $\alpha_\otimes$ or $\alpha_\oplus$ is an isomorphism.
\item One of {\bf [nat.1](a)} or {\bf [nat.1](b)} holds and one of
\[ \mbox{\bf \small [nat.2]}~~~~
\xymatrix{
G(A) \otimes F(B) \ar[r]^{1 \otimes \alpha_\otimes} \ar[d]_{\alpha_\oplus \otimes 1} \ar@{}[ddr]|{\tiny{(a)}} & G(A) \otimes G(B) \ar[dd]^{m_\otimes^G} \\
F(A) \otimes F(B) \ar[d]_{m_\otimes^F} & \\
F(A \otimes B) \ar[r]_{\alpha_\otimes} & G(A \otimes B)
} ~~~ \text{ or }~~~ \xymatrix{
F(A) \otimes G(B) \ar[r]^{\alpha_\otimes \otimes 1} \ar[d]_{1 \otimes \alpha_\oplus} \ar@{}[ddr]|{\tiny{(b)}} & G(A) \otimes G(B) \ar[dd]^{m_\otimes^G} \\
F(A) \otimes F(B) \ar[d]_{m_\otimes^F} & \\
F(A \otimes B) \ar[r]_{\alpha_\otimes} & G(A \otimes B)
}
\]
\[
\text{or}~~~~~
\xymatrix{
G(A \oplus B) \ar[r]^{n_\oplus^G} \ar[d]_{\alpha_\oplus} \ar@{}[ddr]|{\tiny{(c)}} & G(A) \oplus G(B) \ar[dd]^{1 \oplus \alpha_\oplus} \\
F(A \oplus B) \ar[d]_{n_\oplus^F} & \\
F(A) \oplus F(B) \ar[r]_{\alpha_\otimes \oplus 1} & G(A) \oplus F(B)
} ~~~\text{or}~~~
\xymatrix{
G(A \oplus B) \ar[r]^{n_\oplus^G} \ar[d]_{\alpha_\oplus} \ar@{}[ddr]|{\tiny{(d)}} & G(A) \oplus G(B) \ar[dd]^{\alpha_\oplus \oplus 1} \\
F(A \oplus B) \ar[d]_{n_\oplus} & \\
F(A) \oplus F(B) \ar[r]_{1 \otimes \alpha_\otimes} & F(A) \oplus G(B)
}
\] holds.
\item $\alpha_\otimes^{-1} = \alpha_\oplus$
\item $\alpha' := (\alpha_\oplus, \alpha_\otimes): G \Rightarrow F$ is a linear transformation.
\end{enumerate}
\end{lemma}
Conditions {\bf [nat.2]} are as follows in the graphical calculus:
\[ \mbox{\small (a)}~~
\begin{tikzpicture
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.5, -0) {};
\node [style=none] (1) at (-2.5, -1) {};
\node [style=none] (2) at (-0.5, -1) {};
\node [style=none] (3) at (-0.5, -0) {};
\node [style=ox] (4) at (-1.5, -0.5) {};
\node [style=none] (5) at (-2, 1.25) {};
\node [style=none] (6) at (-1, 1.25) {};
\node [style=none] (7) at (-1.5, -2) {};
\node [style=circle, scale=2] (8) at (-2, 0.5) {};
\node [style=none] (9) at (-2, 0.5) {$\alpha_\otimes$};
\node [style=none] (10) at (-0.75, -0.25) {$G$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw [in=-90, out=30, looseness=1.00] (4) to (6.center);
\draw (4) to (7.center);
\draw (5.center) to (8);
\draw [in=150, out=-90, looseness=1.00] (8) to (4);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.5, 0.75) {};
\node [style=none] (1) at (-0.5, -0.25) {};
\node [style=none] (2) at (-2.5, -0.25) {};
\node [style=none] (3) at (-2.5, 0.75) {};
\node [style=ox] (4) at (-1.5, 0.25) {};
\node [style=none] (5) at (-1, 2) {};
\node [style=none] (6) at (-2, 2) {};
\node [style=circle, scale=2] (7) at (-1, 1.25) {};
\node [style=none] (8) at (-1, 1.25) {$\alpha_\oplus$};
\node [style=circle, scale=2] (9) at (-1.5, -1) {};
\node [style=none] (10) at (-1.5, -2) {};
\node [style=none] (11) at (-1.5, -1) {$\alpha_\otimes$};
\node [style=none] (12) at (-0.75, 0.5) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw [in=-90, out=150, looseness=1.00] (4) to (6.center);
\draw (5.center) to (7);
\draw [in=30, out=-90, looseness=1.00] (7) to (4);
\draw (10.center) to (9);
\draw (9) to (4);
\end{pgfonlayer}
\end{tikzpicture}~~~~~~ \mbox{\small (b)} ~~\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.5, -0) {};
\node [style=none] (1) at (-0.5, -1) {};
\node [style=none] (2) at (-2.5, -1) {};
\node [style=none] (3) at (-2.5, -0) {};
\node [style=ox] (4) at (-1.5, -0.5) {};
\node [style=none] (5) at (-1, 1.25) {};
\node [style=none] (6) at (-2, 1.25) {};
\node [style=none] (7) at (-1.5, -2) {};
\node [style=circle, scale=2] (8) at (-1, 0.5) {};
\node [style=none] (9) at (-1, 0.5) {$\alpha_\otimes$};
\node [style=none] (10) at (-0.75, -0.25) {$G$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw [in=-90, out=150, looseness=1.00] (4) to (6.center);
\draw (4) to (7.center);
\draw (5.center) to (8);
\draw [in=30, out=-90, looseness=1.00] (8) to (4);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.5, 0.75) {};
\node [style=none] (1) at (-2.5, -0.25) {};
\node [style=none] (2) at (-0.5, -0.25) {};
\node [style=none] (3) at (-0.5, 0.75) {};
\node [style=ox] (4) at (-1.5, 0.25) {};
\node [style=none] (5) at (-2, 2) {};
\node [style=none] (6) at (-1, 2) {};
\node [style=circle, scale=2] (7) at (-2, 1.25) {};
\node [style=none] (8) at (-2, 1.25) {$\alpha_\oplus$};
\node [style=circle, scale=2] (9) at (-1.5, -1) {};
\node [style=none] (10) at (-1.5, -2) {};
\node [style=none] (11) at (-1.5, -1) {$\alpha_\otimes$};
\node [style=none] (12) at (-2.25, 0.5) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw [in=-90, out=30, looseness=1.00] (4) to (6.center);
\draw (5.center) to (7);
\draw [in=150, out=-90, looseness=1.00] (7) to (4);
\draw (10.center) to (9);
\draw (9) to (4);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~~~ \mbox{\small (c)}~~\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.5, -0.75) {};
\node [style=none] (1) at (-0.5, 0.25) {};
\node [style=none] (2) at (-2.5, 0.25) {};
\node [style=none] (3) at (-2.5, -0.75) {};
\node [style=oa] (4) at (-1.5, -0.25) {};
\node [style=none] (5) at (-1, -2) {};
\node [style=none] (6) at (-2, -2) {};
\node [style=none] (7) at (-1.5, 1.25) {};
\node [style=circle, scale=2] (8) at (-1, -1.25) {};
\node [style=none] (9) at (-1, -1.25) {$\alpha_\oplus$};
\node [style=none] (10) at (-0.75, -0) {$G$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw [in=90, out=-150, looseness=1.00] (4) to (6.center);
\draw (4) to (7.center);
\draw (5.center) to (8);
\draw [in=-30, out=90, looseness=1.00] (8) to (4);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.5, -0.75) {};
\node [style=none] (1) at (-2.5, 0.25) {};
\node [style=none] (2) at (-0.5, 0.25) {};
\node [style=none] (3) at (-0.5, -0.75) {};
\node [style=oa] (4) at (-1.5, -0.25) {};
\node [style=none] (5) at (-2, -2) {};
\node [style=none] (6) at (-1, -2) {};
\node [style=circle, scale=2] (7) at (-2, -1.25) {};
\node [style=none] (8) at (-2, -1.25) {$\alpha_\otimes$};
\node [style=circle, scale=2] (9) at (-1.5, 1) {};
\node [style=none] (10) at (-1.5, 2) {};
\node [style=none] (11) at (-1.5, 1) {$\alpha_\oplus$};
\node [style=none] (12) at (-0.75, -0) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw [in=90, out=-30, looseness=1.00] (4) to (6.center);
\draw (5.center) to (7);
\draw [in=-150, out=90, looseness=1.00] (7) to (4);
\draw (10.center) to (9);
\draw (9) to (4);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~~ \mbox{\small (d)}~~\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.5, -0.75) {};
\node [style=none] (1) at (-2.5, 0.25) {};
\node [style=none] (2) at (-0.5, 0.25) {};
\node [style=none] (3) at (-0.5, -0.75) {};
\node [style=oa] (4) at (-1.5, -0.25) {};
\node [style=none] (5) at (-2, -2) {};
\node [style=none] (6) at (-1, -2) {};
\node [style=none] (7) at (-1.5, 1.25) {};
\node [style=circle, scale=2] (8) at (-2, -1.25) {};
\node [style=none] (9) at (-2, -1.25) {$\alpha_\oplus$};
\node [style=none] (10) at (-0.75, -0) {$G$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw [in=90, out=-30, looseness=1.00] (4) to (6.center);
\draw (4) to (7.center);
\draw (5.center) to (8);
\draw [in=-150, out=90, looseness=1.00] (8) to (4);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.5, -0.75) {};
\node [style=none] (1) at (-0.5, 0.25) {};
\node [style=none] (2) at (-2.5, 0.25) {};
\node [style=none] (3) at (-2.5, -0.75) {};
\node [style=oa] (4) at (-1.5, -0.25) {};
\node [style=none] (5) at (-1, -2) {};
\node [style=none] (6) at (-2, -2) {};
\node [style=circle,scale=2] (7) at (-1, -1.25) {};
\node [style=none] (8) at (-1, -1.25) {$\alpha_\otimes$};
\node [style=circle, scale=2] (9) at (-1.5, 1) {};
\node [style=none] (10) at (-1.5, 2) {};
\node [style=none] (11) at (-1.5, 1) {$\alpha_\oplus$};
\node [style=none] (12) at (-0.75, -0) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (0.center) to (3.center);
\draw [in=90, out=-150, looseness=1.00] (4) to (6.center);
\draw (5.center) to (7);
\draw [in=-30, out=90, looseness=1.00] (7) to (4);
\draw (10.center) to (9);
\draw (9) to (4);
\end{pgfonlayer}
\end{tikzpicture}
\]
\begin{proof}
\begin{description}
\item{$(i) \Rightarrow (iii)$:} Here is the proof assuming {\bf \small [nat.1](a)} that $\alpha_\otimes \alpha_\oplus = 1$:
\[ \hspace{-0.75cm}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5, 3.5) {};
\node [style=none] (1) at (0.5, -2) {};
\node [style=none] (2) at (0, 3.25) {$F(X)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5, 3.5) {};
\node [style=none] (1) at (0.5, -2) {};
\node [style=none] (2) at (0, 2.5) {};
\node [style=none] (3) at (0, -1) {};
\node [style=none] (4) at (1.5, -1) {};
\node [style=none] (5) at (1.5, 2.5) {};
\node [style=none] (6) at (1.25, -0.5) {$F$};
\node [style=none] (7) at (0.25, -1) {};
\node [style=none] (8) at (0.75, -1) {};
\node [style=none] (9) at (0, 3.25) {$F(X)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (2.center) to (5.center);
\draw (5.center) to (4.center);
\draw (4.center) to (3.center);
\draw (3.center) to (2.center);
\draw [bend left=90, looseness=1.25] (7.center) to (8.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (1.75, 1.25) {$\top$};
\node [style=circle] (1) at (1.75, -0) {$\top$};
\node [style=none] (2) at (0.5, 3.5) {};
\node [style=none] (3) at (0.5, -2) {};
\node [style=circle, scale=0.5] (4) at (0.5, -0.75) {};
\node [style=none] (5) at (0, 2) {};
\node [style=none] (6) at (0, -1.5) {};
\node [style=none] (7) at (2.5, -1.5) {};
\node [style=none] (8) at (2.5, 2) {};
\node [style=none] (9) at (2.25, -1) {$F$};
\node [style=none] (10) at (0.25, -1.5) {};
\node [style=none] (11) at (0.75, -1.5) {};
\node [style=none] (12) at (0, 3.25) {$F(X)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (3.center);
\draw [dotted, in=0, out=-90, looseness=1.25] (1) to (4);
\draw (5.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\draw (6.center) to (5.center);
\draw [bend left=90, looseness=1.25] (10.center) to (11.center);
\draw (0) to (1);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (2, 2) {$\top$};
\node [style=circle] (1) at (2, -0) {$\top$};
\node [style=none] (2) at (0.5, 3.5) {};
\node [style=none] (3) at (0.5, -2) {};
\node [style=circle, scale=0.5] (4) at (0.5, -0.75) {};
\node [style=none] (5) at (0, 0.5) {};
\node [style=none] (6) at (0, -1.5) {};
\node [style=none] (7) at (2.5, -1.5) {};
\node [style=none] (8) at (2.5, 0.5) {};
\node [style=none] (9) at (1.5, 2.5) {};
\node [style=none] (10) at (3, 2.5) {};
\node [style=none] (11) at (1.5, 1.25) {};
\node [style=none] (12) at (3, 1.25) {};
\node [style=none] (13) at (2.25, -1.25) {$F$};
\node [style=none] (14) at (0.25, -1.5) {};
\node [style=none] (15) at (0.75, -1.5) {};
\node [style=none] (16) at (1.75, 1.25) {};
\node [style=none] (17) at (2.25, 1.25) {};
\node [style=none] (18) at (0, 3.25) {$F(X)$};
\node [style=none] (19) at (2.75, 1.5) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (3.center);
\draw [dotted, in=0, out=-90, looseness=1.25] (1) to (4);
\draw (5.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\draw (6.center) to (5.center);
\draw (9.center) to (11.center);
\draw (11.center) to (12.center);
\draw (12.center) to (10.center);
\draw (10.center) to (9.center);
\draw [bend left=90, looseness=1.25] (14.center) to (15.center);
\draw [bend left=90, looseness=1.25] (16.center) to (17.center);
\draw (0) to (1);
\end{pgfonlayer}
\end{tikzpicture}
=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (2, 1.25) {$\alpha_\oplus$};
\node [style=circle] (1) at (2, 2.75) {$\top$};
\node [style=circle] (2) at (2, -0.25) {$\top$};
\node [style=none] (3) at (0.5, 3.5) {};
\node [style=none] (4) at (0.5, -2) {};
\node [style=circle, scale=0.5] (5) at (0.5, -1) {};
\node [style=none] (6) at (0, 0.25) {};
\node [style=none] (7) at (0, -1.5) {};
\node [style=none] (8) at (2.5, -1.5) {};
\node [style=none] (9) at (2.5, 0.25) {};
\node [style=none] (10) at (1.5, 3.25) {};
\node [style=none] (11) at (3, 3.25) {};
\node [style=none] (12) at (1.5, 2) {};
\node [style=none] (13) at (3, 2) {};
\node [style=none] (14) at (2.25, -1.25) {$F$};
\node [style=none] (15) at (0.25, -1.5) {};
\node [style=none] (16) at (0.75, -1.5) {};
\node [style=none] (17) at (1.75, 2) {};
\node [style=none] (18) at (2.25, 2) {};
\node [style=none] (19) at (2.75, 2.25) {$G$};
\node [style=none] (20) at (0, 3.25) {$F(X)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1) to (0);
\draw (0) to (2);
\draw (3.center) to (4.center);
\draw [dotted, in=0, out=-90, looseness=1.25] (2) to (5);
\draw (6.center) to (9.center);
\draw (9.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\draw (10.center) to (12.center);
\draw (12.center) to (13.center);
\draw (13.center) to (11.center);
\draw (11.center) to (10.center);
\draw [bend left=105, looseness=1.00] (15.center) to (16.center);
\draw [bend left=90, looseness=1.25] (17.center) to (18.center);
\end{pgfonlayer}
\end{tikzpicture}=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (2, 1.25) {$\alpha_\oplus$};
\node [style=circle] (1) at (2, 2.75) {$\top$};
\node [style=circle] (2) at (2, -0.25) {$\top$};
\node [style=none] (3) at (0.5, 3.5) {};
\node [style=none] (4) at (0.5, -2) {};
\node [style=circle, scale=0.5] (5) at (0.5, -1) {};
\node [style=none] (6) at (0, 0.5) {};
\node [style=none] (7) at (0, -1.5) {};
\node [style=none] (8) at (2.5, -1.5) {};
\node [style=none] (9) at (2.5, 0.5) {};
\node [style=none] (10) at (1.5, 3.25) {};
\node [style=none] (11) at (3, 3.25) {};
\node [style=none] (12) at (1.5, 2) {};
\node [style=none] (13) at (3, 2) {};
\node [style=none] (14) at (2.25, -1.25) {$F$};
\node [style=none] (15) at (1.75, 0.5) {};
\node [style=none] (16) at (2.25, 0.5) {};
\node [style=none] (17) at (1.75, 2) {};
\node [style=none] (18) at (2.25, 2) {};
\node [style=none] (19) at (2.75, 2.25) {$G$};
\node [style=none] (20) at (0, 3.25) {$F(X)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1) to (0);
\draw (0) to (2);
\draw (3.center) to (4.center);
\draw [dotted, in=0, out=-90, looseness=1.25] (2) to (5);
\draw (6.center) to (9.center);
\draw (9.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\draw (10.center) to (12.center);
\draw (12.center) to (13.center);
\draw (13.center) to (11.center);
\draw (11.center) to (10.center);
\draw [bend right=105, looseness=1.00] (15.center) to (16.center);
\draw [bend left=90, looseness=1.25] (17.center) to (18.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (1.75, 3) {$\top$};
\node [style=circle] (1) at (1.75, 0.75) {$\top$};
\node [style=none] (2) at (0.5, 3.5) {};
\node [style=none] (3) at (0.5, -2.5) {};
\node [style=circle, scale=0.5] (4) at (0.5, -0) {};
\node [style=none] (5) at (0, 1.5) {};
\node [style=none] (6) at (0, -0.75) {};
\node [style=none] (7) at (2.25, -0.75) {};
\node [style=none] (8) at (2.25, 1.5) {};
\node [style=none] (9) at (1.25, 3.5) {};
\node [style=none] (10) at (2.75, 3.5) {};
\node [style=none] (11) at (1.25, 2) {};
\node [style=none] (12) at (2.75, 2) {};
\node [style=none] (13) at (2, -0.5) {$G$};
\node [style=circle, scale=2] (14) at (0.5, 2.5) {};
\node [style=circle, scale=2] (15) at (0.5, -1.5) {};
\node [style=none] (16) at (0.5, 2.5) {$\alpha_\otimes$};
\node [style=none] (17) at (0.5, -1.5) {$\alpha_\oplus$};
\node [style=none] (18) at (2.5, 2.25) {$F$};
\node [style=none] (19) at (1.5, 2) {};
\node [style=none] (20) at (2, 2) {};
\node [style=none] (21) at (-0.3, 3.25) {$F(X)$};
\node [style=none] (24) at (-0.3, -2) {$F(X)$};
\node [style=none] (25) at (2, 1.5) {};
\node [style=none] (26) at (1.5, 1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted, in=0, out=-105, looseness=1.25] (1) to (4);
\draw (5.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\draw (6.center) to (5.center);
\draw (9.center) to (11.center);
\draw (11.center) to (12.center);
\draw (12.center) to (10.center);
\draw (10.center) to (9.center);
\draw (0) to (1);
\draw (2.center) to (14);
\draw (14) to (4);
\draw (4) to (15);
\draw (15) to (3.center);
\draw [bend left=75, looseness=1.50] (19.center) to (20.center);
\draw [bend right=75, looseness=1.50] (26.center) to (25.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (1.75, 3) {$\top$};
\node [style=circle] (1) at (1.75, 0.75) {$\top$};
\node [style=none] (2) at (0.5, 3.5) {};
\node [style=none] (3) at (0.5, -2.5) {};
\node [style=circle, scale=0.5] (4) at (0.5, -0) {};
\node [style=none] (5) at (0, 1.5) {};
\node [style=none] (6) at (0, -0.75) {};
\node [style=none] (7) at (2.25, -0.75) {};
\node [style=none] (8) at (2.25, 1.5) {};
\node [style=none] (9) at (1.25, 3.5) {};
\node [style=none] (10) at (2.75, 3.5) {};
\node [style=none] (11) at (1.25, 2) {};
\node [style=none] (12) at (2.75, 2) {};
\node [style=none] (13) at (2, -0.5) {$G$};
\node [style=circle, scale=2] (14) at (0.5, 2.5) {};
\node [style=circle, scale=2] (15) at (0.5, -1.5) {};
\node [style=none] (16) at (0.5, 2.5) {$\alpha_\otimes$};
\node [style=none] (17) at (0.5, -1.5) {$\alpha_\oplus$};
\node [style=none] (18) at (2.5, 2.25) {$G$};
\node [style=none] (19) at (1.5, 2) {};
\node [style=none] (20) at (2, 2) {};
\node [style=none] (21) at (-0.3, 3.25) {$F(X)$};
\node [style=none] (24) at (-0.3, -2) {$F(X)$};
\node [style=none] (25) at (0.75, -0.75) {};
\node [style=none] (26) at (0.25, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted, in=0, out=-105, looseness=1.25] (1) to (4);
\draw (5.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\draw (6.center) to (5.center);
\draw (9.center) to (11.center);
\draw (11.center) to (12.center);
\draw (12.center) to (10.center);
\draw (10.center) to (9.center);
\draw (0) to (1);
\draw (2.center) to (14);
\draw (14) to (4);
\draw (4) to (15);
\draw (15) to (3.center);
\draw [bend left=75, looseness=1.50] (19.center) to (20.center);
\draw [bend left=75, looseness=1.50] (26.center) to (25.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5, 3.5) {};
\node [style=none] (1) at (0.5, -2) {};
\node [style=none] (2) at (0, 3.25) {$F(X)$};
\node [style=circle] (3) at (0.5, 2) {$\alpha_\otimes$};
\node [style=circle] (4) at (0.5, -0) {$\alpha_\oplus$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (3);
\draw (3) to (4);
\draw (4) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
if either $\alpha_\otimes$ or $\alpha_\oplus$ are isomorphisms this implies $\alpha_\oplus \alpha_\otimes =1$.
\item{$(ii) \Rightarrow (iii)$:}
The assumption of {\bf [nat.1](a)} or {\bf (b)} yields, as above, that $\alpha_\otimes \alpha_\oplus =1$:
using {\bf [nat.2](c)} for example gives $\alpha_\oplus \alpha_\otimes =1$:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5, 3.5) {};
\node [style=none] (1) at (0.5, -2) {};
\node [style=none] (2) at (0, 3.25) {$G(X)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5, 3.5) {};
\node [style=none] (1) at (0.5, -2) {};
\node [style=none] (2) at (0, 2.5) {};
\node [style=none] (3) at (0, -1) {};
\node [style=none] (4) at (1.5, -1) {};
\node [style=none] (5) at (1.5, 2.5) {};
\node [style=none] (6) at (1.25, -0.5) {$G$};
\node [style=none] (7) at (0.25, 2.5) {};
\node [style=none] (8) at (0.75, 2.5) {};
\node [style=none] (9) at (0, 3.25) {$G(X)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (2.center) to (5.center);
\draw (5.center) to (4.center);
\draw (4.center) to (3.center);
\draw (3.center) to (2.center);
\draw [bend right=90, looseness=1.25] (7.center) to (8.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (1.75, -1) {$\bot$};
\node [style=circle] (1) at (1.75, 1) {$\bot$};
\node [style=none] (2) at (0.5, -2.5) {};
\node [style=none] (3) at (0.5, 3.5) {};
\node [style=circle, scale=0.5] (4) at (0.5, 1.75) {};
\node [style=none] (5) at (0, 0.5) {};
\node [style=none] (6) at (0, 2.5) {};
\node [style=none] (7) at (2.5, 2.5) {};
\node [style=none] (8) at (2.5, 0.5) {};
\node [style=none] (9) at (1.25, -1.5) {};
\node [style=none] (10) at (2.75, -1.5) {};
\node [style=none] (11) at (1.25, -0.25) {};
\node [style=none] (12) at (2.75, -0.25) {};
\node [style=none] (13) at (2.25, 0.75) {$G$};
\node [style=none] (14) at (0.25, 2.5) {};
\node [style=none] (15) at (0.75, 2.5) {};
\node [style=none] (16) at (1.5, -0.25) {};
\node [style=none] (17) at (2, -0.25) {};
\node [style=none] (18) at (2.5, -1.25) {$G$};
\node [style=none] (19) at (0, 3.25) {$G(X)$};
\node [style=none] (20) at (0, -2) {$G(X)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted, in=0, out=90, looseness=1.25] (1) to (4);
\draw (5.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\draw (6.center) to (5.center);
\draw (9.center) to (11.center);
\draw (11.center) to (12.center);
\draw (12.center) to (10.center);
\draw (10.center) to (9.center);
\draw [bend right=90, looseness=1.50] (14.center) to (15.center);
\draw [bend right=90, looseness=1.25] (16.center) to (17.center);
\draw (3.center) to (4);
\draw (4) to (2.center);
\draw (1) to (0);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (1.75, -1.75) {$\bot$};
\node [style=circle] (1) at (1.75, 1) {$\bot$};
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\node [style=none] (3) at (0.5, 3.5) {};
\node [style=circle, scale=0.5] (4) at (0.5, 1.75) {};
\node [style=none] (5) at (0, 0.5) {};
\node [style=none] (6) at (0, 2.5) {};
\node [style=none] (7) at (2.5, 2.5) {};
\node [style=none] (8) at (2.5, 0.5) {};
\node [style=none] (9) at (1.25, -2.25) {};
\node [style=none] (10) at (2.75, -2.25) {};
\node [style=none] (11) at (1.25, -1) {};
\node [style=none] (12) at (2.75, -1) {};
\node [style=none] (13) at (2.25, 0.75) {$G$};
\node [style=none] (14) at (0.25, 2.5) {};
\node [style=none] (15) at (0.75, 2.5) {};
\node [style=none] (16) at (1.5, -1) {};
\node [style=none] (17) at (2, -1) {};
\node [style=none] (18) at (2.5, -2) {$F$};
\node [style=none] (19) at (0, 3.25) {$G(X)$};
\node [style=none] (20) at (0, -2) {$G(X)$};
\node [style=circle] (21) at (1.75, -0.25) {$\alpha_\oplus$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted, in=0, out=90, looseness=1.25] (1) to (4);
\draw (5.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\draw (6.center) to (5.center);
\draw (9.center) to (11.center);
\draw (11.center) to (12.center);
\draw (12.center) to (10.center);
\draw (10.center) to (9.center);
\draw [bend right=90, looseness=1.50] (14.center) to (15.center);
\draw [bend right=90, looseness=1.25] (16.center) to (17.center);
\draw (3.center) to (4);
\draw (4) to (2.center);
\draw (1) to (21);
\draw (21) to (0);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (1.75, -1.75) {$\bot$};
\node [style=circle] (1) at (1.75, 0.5) {$\bot$};
\node [style=none] (2) at (0.5, -2.5) {};
\node [style=none] (3) at (0.5, 3.5) {};
\node [style=circle, scale=0.5] (4) at (0.5, 1.25) {};
\node [style=none] (5) at (0, -0) {};
\node [style=none] (6) at (0, 2) {};
\node [style=none] (7) at (2.5, 2) {};
\node [style=none] (8) at (2.5, -0) {};
\node [style=none] (9) at (1.25, -2.25) {};
\node [style=none] (10) at (2.75, -2.25) {};
\node [style=none] (11) at (1.25, -1) {};
\node [style=none] (12) at (2.75, -1) {};
\node [style=none] (13) at (2.25, 0.25) {$F$};
\node [style=none] (14) at (0.25, 2) {};
\node [style=none] (15) at (0.75, 2) {};
\node [style=none] (16) at (1.5, -1) {};
\node [style=none] (17) at (2, -1) {};
\node [style=circle] (18) at (0.5, 2.75) {$\alpha_\oplus$};
\node [style=circle] (19) at (0.5, -1.25) {$\alpha_\otimes$};
\node [style=none] (20) at (2.5, -2) {$F$};
\node [style=none] (21) at (0, 3.5) {$G(X)$};
\node [style=none] (22) at (0, -2) {$G(X)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted, in=0, out=90, looseness=1.25] (1) to (4);
\draw (5.center) to (8.center);
\draw (8.center) to (7.center);
\draw (7.center) to (6.center);
\draw (6.center) to (5.center);
\draw (9.center) to (11.center);
\draw (11.center) to (12.center);
\draw (12.center) to (10.center);
\draw (10.center) to (9.center);
\draw [bend right=90, looseness=1.50] (14.center) to (15.center);
\draw [bend right=90, looseness=1.25] (16.center) to (17.center);
\draw (1) to (0);
\draw (3.center) to (18);
\draw (18) to (4);
\draw (4) to (19);
\draw (19) to (2.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5, 3.5) {};
\node [style=none] (1) at (0.5, -2) {};
\node [style=none] (2) at (0, 3.25) {$G(X)$};
\node [style=circle] (3) at (0.5, 2) {$\alpha_\oplus$};
\node [style=circle] (4) at (0.5, -0) {$\alpha_\otimes$};
\node [style=none] (5) at (0, -1.5) {$G(X)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (3);
\draw (3) to (4);
\draw (4) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
Since, $\alpha_\otimes \alpha_\oplus = 1$ and $\alpha_\otimes \alpha_\oplus = 1$ we have $\alpha_\otimes = \alpha_\oplus^{-1}$. The other combinations are used in similar fashion.
\item{$(iii) \Rightarrow (iv)$:}
If $\alpha_\otimes = \alpha_\oplus^{-1}$, then
$(\alpha_\oplus \otimes \alpha_\oplus) m_\otimes^F = m_\otimes^G \alpha_\otimes: G(A) \otimes G(B) \@ifnextchar^ {\t@@}{\t@@^{}} F(A \otimes B)$
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 1.25) {};
\node [style=none] (1) at (-2, 0.25) {};
\node [style=none] (2) at (0.25, 0.25) {};
\node [style=none] (3) at (0.25, 1.25) {};
\node [style=ox] (4) at (-1, 0.75) {};
\node [style=circle] (5) at (-1.75, 2) {$\alpha_\oplus$};
\node [style=circle] (6) at (-0.25, 2) {$\alpha_\oplus$};
\node [style=none] (7) at (-1.75, 2.75) {};
\node [style=none] (8) at (-0.25, 2.75) {};
\node [style=none] (9) at (-1, -1.25) {};
\node [style=none] (10) at (0, 0.5) {};
\node [style=none] (11) at (-1.25, 0.25) {};
\node [style=none] (12) at (-0.75, 0.25) {};
\node [style=none] (15) at (0, 0.5) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (7.center) to (5);
\draw (8.center) to (6);
\draw (0.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (1.center);
\draw (1.center) to (0.center);
\draw [bend right=15, looseness=1.00] (5) to (4);
\draw [bend right=15, looseness=1.00] (4) to (6);
\draw (4) to (9.center);
\draw [bend left=75, looseness=1.00] (11.center) to (12.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 1.25) {};
\node [style=none] (1) at (-2, 0.25) {};
\node [style=none] (2) at (0.25, 0.25) {};
\node [style=none] (3) at (0.25, 1.25) {};
\node [style=ox] (4) at (-1, 0.75) {};
\node [style=circle] (5) at (-1.75, 2) {$\alpha_\oplus$};
\node [style=circle] (6) at (-0.25, 2) {$\alpha_\oplus$};
\node [style=none] (7) at (-1.75, 2.75) {};
\node [style=none] (8) at (-0.25, 2.75) {};
\node [style=none] (9) at (-1, -2.5) {};
\node [style=none] (10) at (0, 0.5) {};
\node [style=none] (11) at (-1.25, 0.25) {};
\node [style=none] (12) at (-0.75, 0.25) {};
\node [style=circle] (13) at (-1, -0.5) {$\alpha_\otimes$};
\node [style=circle] (14) at (-1, -1.5) {$\alpha_\oplus$};
\node [style=none] (15) at (0, 0.5) {$F$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (7.center) to (5);
\draw (8.center) to (6);
\draw (0.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (1.center);
\draw (1.center) to (0.center);
\draw [bend right=15, looseness=1.00] (5) to (4);
\draw [bend right=15, looseness=1.00] (4) to (6);
\draw [bend left=75, looseness=1.00] (11.center) to (12.center);
\draw (4) to (13);
\draw (13) to (14);
\draw (14) to (9.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 0.25) {};
\node [style=none] (1) at (-2, -0.75) {};
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\node [style=none] (3) at (0.25, 0.25) {};
\node [style=ox] (4) at (-1, -0.25) {};
\node [style=circle] (5) at (-1.75, 2) {$\alpha_\oplus$};
\node [style=circle] (6) at (-0.25, 2) {$\alpha_\oplus$};
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\node [style=none] (12) at (-0.75, -0.75) {};
\node [style=circle] (13) at (-1, -1.5) {$\alpha_\oplus$};
\node [style=circle] (14) at (-1.75, 1) {$\alpha_\otimes$};
\node [style=circle] (15) at (-0.25, 1) {$\alpha_\otimes$};
\node [style=none] (16) at (0, -0.5) {$G$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (7.center) to (5);
\draw (8.center) to (6);
\draw (0.center) to (3.center);
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\draw (2.center) to (1.center);
\draw (1.center) to (0.center);
\draw [bend left=75, looseness=1.00] (11.center) to (12.center);
\draw (13) to (9.center);
\draw (5) to (14);
\draw (6) to (15);
\draw [bend right, looseness=1.00] (14) to (4);
\draw [bend right, looseness=1.00] (4) to (15);
\draw (4) to (13);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 1.25) {};
\node [style=none] (1) at (-2, 0.25) {};
\node [style=none] (2) at (0.25, 0.25) {};
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\node [style=none] (8) at (-1.25, 0.25) {};
\node [style=none] (9) at (-0.75, 0.25) {};
\node [style=circle] (10) at (-1, -1.5) {$\alpha_\oplus$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (1.center);
\draw (1.center) to (0.center);
\draw [bend left=75, looseness=1.00] (8.center) to (9.center);
\draw (10) to (7.center);
\draw [bend right=15, looseness=1.00] (5.center) to (4);
\draw [bend right=15, looseness=1.00] (4) to (6.center);
\draw (4) to (10);
\end{pgfonlayer}
\end{tikzpicture}
\]
$m_\top^F = m_\top^G \alpha_\oplus: \top \@ifnextchar^ {\t@@}{\t@@^{}} F(\top)$
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (1.25, 2.25) {$\top$};
\node [style=none] (1) at (2, 1.5) {};
\node [style=none] (2) at (2, 3) {};
\node [style=none] (3) at (0.5, 3) {};
\node [style=none] (4) at (0.5, 1.5) {};
\node [style=circle, scale=0.5] (5) at (1.25, 1) {};
\node [style=circle] (6) at (-0.75, 3.25) {$\top$};
\node [style=none] (7) at (1.5, 1.5) {};
\node [style=none] (8) at (1, 1.5) {};
\node [style=none] (9) at (1.25, -2) {};
\node [style=none] (10) at (1.75, 1.75) {};
\node [style=none] (11) at (1.75, 1.75) {$F$};
\node [style=none] (12) at (-0.75, 4.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (4.center);
\draw (4.center) to (1.center);
\draw (5) to (0);
\draw [in=-90, out=180, looseness=1.25, dotted] (5) to (6);
\draw [bend right=75, looseness=1.25] (7.center) to (8.center);
\draw (5) to (9.center);
\draw (6) to (12.center);
\end{pgfonlayer}
\end{tikzpicture}=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (1.25, 2.25) {$\top$};
\node [style=none] (1) at (2, 1.5) {};
\node [style=none] (2) at (2, 3) {};
\node [style=none] (3) at (0.5, 3) {};
\node [style=none] (4) at (0.5, 1.5) {};
\node [style=circle, scale=0.5] (5) at (1.25, 1) {};
\node [style=circle] (6) at (-0.75, 3.25) {$\top$};
\node [style=circle] (7) at (1.25, 0.25) {$\alpha_\otimes$};
\node [style=none] (8) at (1.5, 1.5) {};
\node [style=none] (9) at (1, 1.5) {};
\node [style=circle] (10) at (1.25, -1) {$\alpha_\oplus$};
\node [style=none] (11) at (1.75, 1.75) {$F$};
\node [style=none] (12) at (1.25, -2) {};
\node [style=none] (13) at (-0.75, 4.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (4.center);
\draw (4.center) to (1.center);
\draw (5) to (0);
\draw [in=-90, out=180, looseness=1.25, dotted] (5) to (6);
\draw [bend right=75, looseness=1.25] (8.center) to (9.center);
\draw (10) to (7);
\draw (7) to (5);
\draw (10) to (12.center);
\draw (6) to (13.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (1.25, 3) {$\top$};
\node [style=none] (1) at (2, 2.25) {};
\node [style=none] (2) at (2, 3.75) {};
\node [style=none] (3) at (0.5, 3.75) {};
\node [style=none] (4) at (0.5, 2.25) {};
\node [style=circle, scale=0.5] (5) at (1.25, 1.75) {};
\node [style=circle] (6) at (-0.75, 4) {$\top$};
\node [style=none] (7) at (1.25, -1) {};
\node [style=none] (8) at (1.5, 2.25) {};
\node [style=none] (9) at (1, 2.25) {};
\node [style=circle] (10) at (1.25, -0.25) {$\alpha_\oplus$};
\node [style=none] (11) at (1.75, 2.5) {$G$};
\node [style=none] (12) at (-0.75, 5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (4.center);
\draw (4.center) to (1.center);
\draw (5) to (0);
\draw [dotted, in=-90, out=180, looseness=1.25] (5) to (6);
\draw [bend right=75, looseness=1.25] (8.center) to (9.center);
\draw (7.center) to (10);
\draw (10) to (5);
\draw (6) to (12.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
Thus, $\alpha_\otimes$ is comonoidal. Similarly, it can be proven that $\alpha_\oplus$ is monoidal. The axioms {\bf \small [LT.4] (a)}-{\bf \small (d)} for a linear transformation are satisfied for $(\alpha_\oplus, \alpha_\otimes)$ because $\alpha_\oplus = \alpha_\otimes^{-1}$.
\item{$(iv) \Rightarrow (i)$ and $(ii)$:} The axioms {\bf \small [nat.1]} and {\bf \small [nat.2]} are given by the fact that $(\alpha_\oplus, \alpha_\otimes)$ is a linear transformation.
\end{description}
\end{proof}
Frobenius functors between isomix categories are especially important in this development and they satisfy an additional property:
\begin{definition}
A Frobenius functor between isomix categories is an {\bf isomix functor} in case it is a mix functor which satisfies, in addition, the following diagram:
\[ \mbox{\bf{[isomix-FF]}}~~~~~\xymatrix{ \top \ar@/^/[rrr]^{{\sf m}^{-1}} \ar[dr]_{m_\top} & & & \bot \\ & F(\top) \ar[r]_{F({\sf m}^{-1})} & F(\bot) \ar[ur]_{n_\top} } \]
\end{definition}
Recall that a linear functor is {\bf normal} in case bot $m_\top$ and $n_\bot$ are isomorphisms. We observe:
\begin{lemma}
For a mix Frobenius functor, $F: \mathbb{X} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{Y}$, between isomix categories the following are equivalent:
\begin{enumerate}[(i)]
\item $n_\bot: F(\bot) \@ifnextchar^ {\t@@}{\t@@^{}} \bot$ or $m_\top: \top \@ifnextchar^ {\t@@}{\t@@^{}} F(\top)$ is an isomorphism;
\item $F$ is a normal functor;
\item $F$ is an isomix functor.
\end{enumerate}
\end{lemma}
\begin{proof}~
\begin{description}
\item{$(i) \Rightarrow (ii)$:} Note that, as $F$ is a mix functor $F({\sf m}) = n_\bot {\sf m}~m_\top$. As the mix map ${\sf m}$ is an isomorphism so is $F({\sf m})$. Then if either $n_\bot$ or $m_\top$ is an isomorphism the other must be an isomorphism. Thus, $F$ will be a normal functor.
\item{$(ii) \Rightarrow (iii)$:} If $F$ is normal then $n_\bot$ and $m_\top$ are isomorphisms and so
\[ \infer={m_\top F({\sf m}^{-1}) n_\bot = {\sf m}^{-1}}{\infer={F({\sf m}^{-1}) = m_\top^{-1} {\sf m}^{-1} n_\bot^{-1}}{F({\sf m}) = n_\bot {\sf m}~m_\top}} \]
\item{$(iii) \Rightarrow (i)$:} The mix preservation for $F$ makes $n_\bot$ a section ($m_\top$ a retraction) while isomix preservation makes $m_\bot$ a retraction (and $m_\top$ a section.
This means $n_\bot$ is an isomorphism ($m_\top$ is an isomorphism).
\end{description}
\end{proof}
\begin{corollary} \label{normal-nat-iso}
If $\alpha$ is a linear natural isomorphism between isomix Frobenius linear functors if and only if $\alpha_\otimes = \alpha_\oplus^{-1}$.
\end{corollary}
\begin{proof}
Note that if we can establish {\bf [nat.1](a)} or {\bf (b)} then we can prove that $\alpha_\otimes\alpha_\oplus =1$ and, as $\alpha_\otimes$ is an isomorphism it follows that $\alpha_\oplus\alpha_\otimes =1$.
Thus, it suffices to show that {\bf [nat.1](a)} holds for this we have:
\[ m_\top \alpha_\oplus G({\sf m}^{-1}) n_\bot = m_\top F({\sf m}^{-1})\alpha_\oplus n_\bot = m_\top F({\sf m}^{-1}) n_\bot = {\sf m}^{-1} = m_\top G({\sf m}^{-1}) n_\bot \]
As $G({\sf m}^{-1}) n_\bot$ is an isomorphism it follows that $m_\top \alpha_\oplus = m_\top$.
\end{proof}
Lemma \ref{Lemma: Frobenius linear transformation} and Corollary \ref{normal-nat-iso} are generalizations of \cite[Proposition 7]{DP08} which states that given a natural transformation $\alpha: F \Rightarrow G$ which is both monoidal and comonoidal between Frobenius monoidal functors, and an object $A$ which has a dual $(\eta, \epsilon): A \dashv B$, then $\alpha_A$ is invertible. In Lemma \ref{Lemma: Frobenius linear transformation}, when $A \dashvv B \in \mathbb{X}$, then $\alpha_\oplus$ is defined as follows:
\[
\alpha_\oplus: G(A) \@ifnextchar^ {\t@@}{\t@@^{}} F(A) = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-0.25, -0) {$\alpha_\otimes$};
\node [style=none] (1) at (1.25, -2.75) {};
\node [style=none] (2) at (-0.25, -1) {};
\node [style=none] (3) at (-1.75, -1) {};
\node [style=none] (4) at (-1.75, 2.75) {};
\node [style=circle] (5) at (-1, -1.75) {$\epsilon$};
\node [style=none] (6) at (-0.25, 1) {};
\node [style=none] (7) at (1.25, 1) {};
\node [style=circle] (8) at (0.5, 1.75) {$\eta$};
\node [style=none] (9) at (1.5, 2.5) {};
\node [style=none] (10) at (1.5, 1) {};
\node [style=none] (11) at (-0.5, 1) {};
\node [style=none] (12) at (-0.5, 2.5) {};
\node [style=none] (13) at (-2, -0.75) {};
\node [style=none] (14) at (0, -0.75) {};
\node [style=none] (15) at (0, -2.5) {};
\node [style=none] (16) at (-2, -2.5) {};
\node [style=none] (17) at (-2.25, 2.5) {$G(A)$};
\node [style=none] (18) at (1.75, -2.5) {$F(A)$};
\node [style=none] (19) at (0.25, 0.65) {$F(B)$};
\node [style=none] (20) at (0.5, -0.5) {$G(B)$};
\node [style=none] (21) at (1.25, 2.25) {$F$};
\node [style=none] (22) at (-0.25, -2.25) {$G$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2.center);
\draw (3.center) to (4.center);
\draw [in=0, out=-90, looseness=1.25] (2.center) to (5);
\draw [in=180, out=-90, looseness=1.25] (3.center) to (5);
\draw (11.center) to (10.center);
\draw (12.center) to (11.center);
\draw (10.center) to (9.center);
\draw (9.center) to (12.center);
\draw (7.center) to (1.center);
\draw [in=90, out=0, looseness=1.25] (8) to (7.center);
\draw (6.center) to (0);
\draw [in=180, out=90, looseness=1.25] (6.center) to (8);
\draw (13.center) to (16.center);
\draw (16.center) to (15.center);
\draw (15.center) to (14.center);
\draw (14.center) to (13.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
For these special linear isomorphisms with $\alpha_\otimes = \alpha_\oplus^{-1}$ we can simplify the coherence requirements:
\begin{lemma} \label{simplifying-coherences}
Suppose $F$ and $G$ are Frobenius functors and $\alpha$ is a natural isomorphism then:
\begin{enumerate}[(i)]
\item If $\alpha: F \@ifnextchar^ {\t@@}{\t@@^{}} G$ is $\otimes$-monoidal and $\oplus$-comonoidal then $(\alpha,\alpha^{-1})$ is a linear transformation;
\item If $F$ and $G$ are strong Frobenius functors and $\alpha$ is $\otimes$-monoidal and $\oplus$-monoidal then $(\alpha,\alpha^{-1})$ is a linear transformation.
\end{enumerate}
\end{lemma}
\begin{proof}~
\begin{enumerate}[{\em (i)}]
\item If $\alpha$ is $\otimes$-monoidal and $\oplus$-comonoidal then so is $\alpha^{-1}$ supporting the possibility that it is a component of a linear transformation.
Considering {\bf [LT.1]} we show that $(\alpha,\alpha^{-1})$ satisfies this requirement as:
\[ \xymatrix{ F(A \oplus B) \ar[rr]^{\alpha_\otimes} \ar[d]_{n_\oplus = \nu^R_\otimes} & & G(A \oplus B) \ar[d]^{n_\oplus = \nu^R_\otimes} \\
F(A) \oplus F(B) \ar[dr]_{1 \oplus \alpha} \ar[rr]^{\alpha \oplus \alpha} & & G(A) \oplus G(B) \ar[dl]^{\alpha^{-1} \oplus 1} \\
& F(A) \oplus G(B) } \]
The remaining requirements follow in a similar manner.
\item When the laxors for the functors are isomorphisms then being monoidal implies being comonoidal.
\end{enumerate}
\end{proof}
\subsection{Dagger mix categories}
\label{Section: dagger LDC}
Conventionally, in categorical quantum mechanics a dagger is defined as a contravariant functor which is an involution that is stationary on objects, . Before proceeding to define dagger for LDCs, the notion of the opposite of an LDC and whence the notion of a contravariant linear functors have to be developed. For LDCs we cannot expect the dagger to be stationary on objects, however, it is still possible that it can act like an involution.
If $(\mathbb{X}, \otimes, \top, \oplus, \bot)$ is a linear distributive category, the {\bf opposite linear distributive category} is $(\mathbb{X}, \otimes, \top, \oplus, \bot)^{\mathsf{op}} := (\mathbb{X}^{\mathsf{op}}, \oplus, \bot, \otimes, \top)$ where $\mathbb{X}^{\mathsf{op}}$ is the usual opposite category with the monoidal structures are flipped as follows:
\[\otimes^{\mathsf{op}} := \oplus ~~~~~~~ \top^{\mathsf{op}} : \bot ~~~~~~~ \oplus^{\mathsf{op}} := \otimes ~~~~~~~ \bot^{\mathsf{op}} := \top\]
Note that $(\mathbb{X}, \otimes, \top, \oplus, \bot)^{{\mathsf{op}}~{\mathsf{op}}} = (\mathbb{X}, \otimes, \top, \oplus, \bot) $. A contravariant linear functor $F$ is then a linear functor $F: \mathbb{Y}^{\mathsf{op}} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$. If $(F_\otimes, F_\oplus): (\mathbb{X}, \otimes, \top, \oplus, \bot)^{\mathsf{op}} \@ifnextchar^ {\t@@}{\t@@^{}} (\mathbb{X}, \otimes, \top, \oplus, \bot)$ is a contravariant linear functor, then the opposite linear functor $(F_\otimes, F_\oplus)^{\mathsf{op}}: (\mathbb{X}, \otimes, \top, \oplus, \bot) \@ifnextchar^ {\t@@}{\t@@^{}} (\mathbb{X}, \otimes, \top, \oplus, \bot)^{\mathsf{op}}$ given by the pair of opposite functors $(F_\oplus^{\mathsf{op}}, F_\otimes^{\mathsf{op}})$. Observe that $F^{\mathsf{op}}$ is a mix Frobenius linear functor if and only if $F$ is.
\begin{definition}
\label{Definition: daggerLDC succinct}
A {\bf dagger linearly distributive category}, $\dagger$-LDC, is an LDC, $\mathbb{X}$, with a contravariant Frobenius linear functor $(\_)^\dagger: \mathbb{X}^{\mathsf{op}} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$ which is a linear involutive equivalence $(\_)^\dagger ~\dashvv ~ (\_)^{\dagger^{\mathsf{op}}}: \mathbb{X}^{\mathsf{op}} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$.
\end{definition}
First note that saying this is an {\bf involutive} equivalence asserts that the unit and counit of the equivalence are the same map (although one is in the opposite category). Thus, the adjunction expands to take the form $(\imath,\imath): (\_)^\dagger ~\dashvv ~ (\_)^{\dagger^{\mathsf{op}}}: \mathbb{X}^{\mathsf{op}} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$. However, the unit and counit are linear natural transformations so $\imath$ expands to $\imath = (\imath_\otimes,\imath_\oplus)$. As the dagger functor is
a left adjoint, it is strong and, thus, is normal. Furthermore, as the unit of an equivalence, $\imath$ is a linear natural isomorphism. This means $\imath = (\imath_\otimes,\imath_\oplus)$ satisfies the requirements of Lemma \ref{Lemma: Frobenius linear transformation}, implying that $\imath_\otimes^{-1} = \imath_\oplus$. Simplifying notation we shall set $\iota:= \imath_\oplus$ so the unit linear transformation $\imath := (\iota^{-1},\iota)$ we then can simplify the requirements to those of this $\iota: A \@ifnextchar^ {\t@@}{\t@@^{}} (A^\dagger)^\dagger$ which we refer to as the {\bf involutor}.
A {\bf symmetric} $\dagger$-LDC is a $\dagger$-LDC which is a symmetric LDC for which the dagger is a symmetric linear functor. A {\bf cyclic} $\dagger$-$*$-autonomous category is a
$\dagger$-LDC with chosen left are right duals and a cyclor which is preserved by the dagger. A $\dagger$-{\bf mix} category is a $\dagger$-LDC for which $(\_)^\dagger: \mathbb{X}^{\mathsf{op}} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$
is a mix functor. As the dagger functor is strong (and so normal) if the category is an isomix category then being {\bf $\dagger$-mix} already implies that
the dagger is an isomix functor. Thus, a {\bf $\dagger$-\bf isomix} category is a $\dagger$-mix category which happens to be an isomix category.
In the remainder of the section, we unfold the definition of a $\dagger$-isomix categories and give the coherences requirements explicitly.
\begin{proposition}
\label{Definition: daggerLDC elaborate}
A dagger linearly distributive category is an LDC with a functor $(\_)^\dag:\mathbb{X}^\mathsf{op}\@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$ and
natural isomorphisms
\begin{align*}
\text{ \bf laxors: } &A^\dag \otimes B^\dag \xrightarrow{ \lambda_\otimes} (A\oplus B)^\dag ~~~~~ A^\dag \oplus B^\dag \xrightarrow{ \lambda_\oplus} (A\otimes B)^\dag \\
&\top \xrightarrow{\lambda_\top} \bot^\dag ~~~~~~~~~~~~~~~~~~~~~ \bot \xrightarrow{\lambda_\bot} \top^\dag \\
\text{ \bf and involutor: } & A \xrightarrow{\iota} (A^\dag)^\dag
\end{align*}
such that the following coherences hold:
\begin{enumerate}[{\bf [$\dagger$-ldc.1]}]
\item Interaction of $\lambda_\otimes, \lambda_\oplus$ with associators:
\[
\begin{tabular}{cc}
\xymatrix{
A^\dag \otimes (B^\dag \otimes C^\dag) \ar@{->}[r]^{a_\otimes} \ar@{->}[d]_{1 \otimes \lambda_\otimes}
& (A^\dag \otimes B^\dag) \otimes C^\dag \ar@{->}[d]^{\lambda_\otimes \otimes 1} \\
A^\dag \otimes ( B \oplus C)^\dag \ar@{->}[d]_{\lambda_\otimes}
& (A \oplus B)^\dag \oplus C^\dag \ar@{->}[d]^{\lambda_\otimes} \\
(A \oplus (B \oplus C))^\dag \ar@{->}[r]_{(a_\oplus^{-1})^\dag}
& ( (A\oplus B) \oplus C)^\dag
} &\xymatrix{
A^\dag \oplus (B^\dag \oplus C^\dag) \ar@{->}[r]^{a_\oplus} \ar@{->}[d]_{1 \oplus \lambda_\oplus }
& (A^\dag \oplus B^\dag) \oplus C^\dag \ar@{->}[d]^{\lambda_\oplus \oplus 1} \\
A^\dag \oplus (B \otimes C)^\dag \ar@{->}[d]_{\lambda_\oplus}
& (A \otimes B)^\dag \oplus C^\dag \ar@{->}[d]^{\lambda_\oplus} \\
(A \otimes (B\otimes C))^\dag \ar@{->}[r]_{(a_\otimes^{-1})^\dag}
& ((A\otimes B) \otimes C)^\dag
}
\end{tabular}
\]
\item Interaction of $\lambda_\top, \lambda_\bot$ with unitors:
\begin{center}
\begin{tabular}{cc}
\xymatrix{
\top \otimes A^\dag \ar@{->}[rr]^{\lambda_\top\otimes 1} \ar@{->}[d]_{u_\otimes^R}
& & \bot^\dag\otimes A^\dag \ar@{->}[d]^{\lambda_\otimes}\\
A^\dag
& & (\bot \oplus A)^\dag \ar@{<-}[ll]^{(u_\oplus^R)^\dag}\\
} &
\xymatrix{
\bot \oplus A^\dag \ar@{->}[rr]^{\lambda_\bot\oplus 1} \ar@{->}[d]_{u_\oplus^R}
& & \top^\dag\oplus A^\dag \ar@{->}[d]^{\lambda_\oplus}\\
A^\dag
& & (\top \otimes A)^\dag \ar@{<-}[ll]^{(u_\otimes^R)^\dag}\\
}
\end{tabular}
\end{center}
and two symmetric diagrams for $u_\otimes^R$ and $u_\oplus^R$ must also be satisfied.
\item Interaction of $\lambda_\otimes, \lambda_\oplus$ with linear distributors:
\begin{center}
\begin{tabular}{cc}
\xymatrix{
A^\dag \otimes(B^\dag\oplus C^\dag) \ar@{->}[r]^{\delta^L} \ar@{->}[d]_{1\otimes\lambda_\oplus}
& (A^\dag \otimes B^\dag)\oplus C^\dag \ar@{->}[d]_{\lambda_\otimes\oplus 1}\\
A^\dag \otimes (B\otimes C)^\dag \ar@{->}[d]_{\lambda_\otimes}
& (A\oplus B)^\dag \oplus C^\dag \ar@{->}[d]^{\lambda_\oplus}\\
(A\oplus (B\otimes C))^\dag \ar@{->}[r]_{(\delta^R)^\dag}
& ((A\oplus B)\otimes C)^\dag
} &
\xymatrix{
(A^\dag \oplus B^\dag) \otimes C^\dag \ar@{->}[r]^{\delta^R} \ar@{->}[d]_{\lambda_\oplus\otimes 1}
& A^\dag \oplus (B^\dag \otimes C^\dag) \ar@{->}[d]_{1\oplus \lambda_\otimes}\\
(A\otimes B)^\dag \otimes C^\dag \ar@{->}[d]_{\lambda_\otimes}
& A^\dag \oplus (B\oplus C)^\dag \ar@{->}[d]^{\lambda_\oplus}\\
((A\otimes B)\oplus C)^\dag \ar@{->}[r]_{(\delta^L)^\dag}
& (A\otimes (B\oplus C))^\dag
}
\end{tabular}
\end{center}
\item Interaction of $\iota: A \rightarrow A^{\dagger\dagger}$ with $\lambda_\otimes$, $\lambda_\oplus$:
\begin{center}
\begin{tabular}{cc}
\xymatrix{
A\oplus B \ar@{->}[r]^{\iota} \ar@{->}[d]_{\iota \oplus \iota}
& ((A\oplus B)^\dag)^\dag \ar@{->}[d]^{\lambda_\otimes^\dag}\\
(A^\dag)^\dag \oplus (B^\dag)^\dag \ar@{->}[r]_{\lambda_\oplus}
& (A^\dag\otimes B^\dag)^\dag
} & \xymatrix{
A\otimes B \ar@{->}[r]^{\iota} \ar@{->}[d]_{\iota \otimes \iota}
& ((A\otimes B)^\dag)^\dag \ar@{->}[d]^{\lambda_\oplus^\dag}\\
(A^\dag)^\dag \otimes (B^\dag)^\dag \ar@{->}[r]_{\lambda_\otimes}
& (A^\dag\oplus B^\dag)^\dag
}
\end{tabular}
\end{center}
\item Interaction of $\iota: A \rightarrow A^{\dagger\dagger}$ with $\lambda_\top$, $\lambda_\bot$:
\begin{center}
\begin{tabular}{cc}
$\begin{matrix} \xymatrix{
&\bot \ar@{->}[r]^{\iota} \ar@{->}[dr]_{\lambda_\bot}
& (\bot^\dag)^\dag \ar@{->}[d]^{\lambda_\top^\dag}\\
&{}
& \top^\dag
} \end{matrix}$ &
$\begin{matrix} \xymatrix{
&\top \ar@{->}[r]^{\iota} \ar@{->}[dr]_{\lambda_\top}
& (\top^\dag)^\dag \ar@{->}[d]^{\lambda_\bot^\dag} \\
&{}
& \bot^\dag
} \end{matrix}$
\end{tabular}
\end{center}
\item $\iota_{A^\dagger} = (\iota_A^{-1})^\dagger: A^\dagger \@ifnextchar^ {\t@@}{\t@@^{}} A^{\dagger\dagger\dagger}$
\end{enumerate}
\end{proposition}
\begin{proof}
The structure is presented using strong monoidal laxors: to form a linear functor the laxor $\lambda_\oplus$ needs to be reversed by taking its inverse. Once this
adjustment is made all the required coherences for a linear functor are present.
Note that {\bf [$\dagger$-ldc.6]} equivalently expresses the triangle identities of the adjunction.
The coherences for the involutor are given as it is a monoidal transformation for both the tensor and par: by Lemma \ref{simplifying-coherences} (ii) this suffices.
\end{proof}
A {\bf symmetric $\dagger$-LDC} is a $\dagger$-LDC which is a symmetric LDC and for which the following additional diagrams commute:
\begin{enumerate}[{\bf [$\dagger$-ldc.7]}]
\item Interaction of $\lambda_\otimes , \lambda_\oplus$ with symmetry maps:
\[
\begin{tabular}{cc}
\xymatrix{
A^\dag \otimes B^\dag \ar@{->}[r]^{\lambda_\otimes} \ar@{->}[d]_{c_\otimes}
& (A\oplus B)^\dag \ar@{->}[d]^{c_\oplus^\dag}\\
B^\dag \otimes A^\dag \ar@{->}[r]_{\lambda_\otimes}
& (B\oplus A)^\dag\\
} & \xymatrix{
A^\dag \oplus B^\dag \ar@{->}[r]^{\lambda_\oplus} \ar@{->}[d]_{c_\oplus}
& (A\otimes B)^\dag \ar@{->}[d]^{c_\otimes^\dag}\\
B^\dag \oplus A^\dag \ar@{->}[r]_{\lambda_\oplus}
& (B\otimes A)^\dag\\
}
\end{tabular}
\]
\end{enumerate}
A {\bf $\dagger$-mix category} is a $\dagger$-LDC which has a mix map and satisfies the following additional coherence:
\[ \mbox{\bf [$\dagger$-\text{mix}]} ~~~~\begin{array}[c]{c}
\xymatrix{
\bot \ar@{->}[r]^{{\sf m}} \ar@{->}[d]_{\lambda_\bot}
& \top \ar@{->}[d]^{\lambda_\top}\\
\top^\dag \ar@{->}[r]_{{\sf m}^\dag}
& \bot^\dag
}
\end{array} \]
If ${\sf m}$ is an isomorphism, then $\mathbb{X}$ is an {\bf $\dagger$-isomix category} and, since $(\_)^\dagger$ is normal, $(\_)^\dagger$ is an isomix Frobenius functor.
\begin{lemma}
\label{lemma:mixdagger}
Suppose $\mathbb{X}$ is a $\dagger$-mix category then the following diagram commutes:
\[
\xymatrix{ A^\dag \oplus B^\dag \ar[r]^{\mathsf{mx}} \ar[d]_{\lambda_\oplus}& A^\dag \otimes B^\dag \ar[d]^{\lambda_\otimes} \\
(A \otimes B)^\dag \ar[r]_{\mathsf{mx}^\dag} & (A \oplus B)^\dag }
\]
\end{lemma}
\begin{proof}
The proof follows directly from Lemma \ref{Lemma: Mix Frobenius linear functor}.
\end{proof}
\begin{lemma}
\label{Lemma: mixdagger}
Suppose $\mathbb{X}$ is a $\dagger$-mix category and $A \in \mathsf{Core}(\mathbb{X})$ then $A^\dagger \in$ $\mathsf{Core}(\mathbb{X})$.
\end{lemma}
\begin{proof}
The natural transformation $A^\dagger \otimes X \xrightarrow{\mathsf{mx}} A^\dagger \oplus X$ is an isomorphism as:
\[
\xymatrix{
A^\dagger \otimes X \ar[r]^{1 \otimes \iota} \ar[d]_{\mathsf{mx}} \ar@{}[dr]|{\scalebox{0.95}{\tiny\bf (nat. {\sf mx})}}
& A^\dagger \otimes X^{\dagger\dagger} \ar[r]^{\lambda_\otimes} \ar[d]_{\mathsf{mx}} \ar@{}[dr]|{\scalebox{.95}{\tiny\bf {lem. \ref{Lemma: mixdagger}}}}
& (A \oplus X^\dagger)^\dagger \ar[d]^{\mathsf{mx}^\dagger} \\
A^\dagger \oplus X \ar[r]_{1 \oplus \iota}
& A^\dagger \oplus A^{\dagger \dagger} \ar[r]_{\lambda_\oplus}
&(A \otimes X^\dagger)^\dagger
}
\]
\end{proof}
\begin{lemma}
Let $\mathbb{X}$ be $\dagger$-LDC. If $A \dashvv B$ then $B^\dagger \dashvv A^\dagger$.
\end{lemma}
\begin{proof}
The statement follows from Lemma \ref{Lemma: linear adjoints}: Frobenius functors preserve linear adjoints. Explicitly, if $(\eta,\epsilon): A \dashvv B$ then $(\lambda_\top\epsilon^\dag\lambda_\oplus^{-1},\lambda_\otimes\eta^\dagger \lambda_\bot^{-1}): B^\dagger \dashvv A^\dagger$.
\end{proof}
Suppose $\mathbb{X}$ is a $\dagger$-$*$-autonomous category and $(\eta*, \epsilon*): A^* \dashvv A$, then $((\epsilon*)^\dagger, (\eta*)^\dagger): A^\dagger \dashvv (A^*)^\dagger$, where $((\epsilon*)^\dagger, (\eta*)^\dagger) := (\lambda_\top\epsilon*^\dag\lambda_\oplus^{-1},\lambda_\otimes\eta*^\dagger \lambda_\bot^{-1})$. We draw $(\epsilon*)^\dagger$ and $(*\epsilon)^\dagger$ as dagger cups, and $(\eta*)^\dagger$ and $(*\eta)^\dagger$ as dagger caps which are pictorially represented as follows:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-1, 2) {};
\node [style=none] (1) at (1, 2) {};
\node [style=none] (2) at (-1, 0.5) {};
\node [style=none] (3) at (1, 0.5) {};
\node [style=none] (4) at (-1.5, 1.5) {$X^\dagger$};
\node [style=none] (5) at (1.5, 1.5) {$(^*X)^\dagger$};
\node [style=none] (6) at (0, -1) {$(*\eta)^\dagger$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (0.center);
\draw [bend right=90, looseness=2.00] (2.center) to (3.center);
\draw (3.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~~~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-1, 2) {};
\node [style=none] (1) at (1, 2) {};
\node [style=none] (2) at (-1, 0.5) {};
\node [style=none] (3) at (1, 0.5) {};
\node [style=none] (4) at (-1.5, 1.5) {$X^{*\dagger}$};
\node [style=none] (5) at (1.5, 1.5) {$X^\dagger$};
\node [style=none] (6) at (0, -1) {$(\eta*)^\dagger$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (0.center);
\draw [bend right=90, looseness=2.00] (2.center) to (3.center);
\draw (3.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~~~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-1, -1) {};
\node [style=none] (1) at (1, -1) {};
\node [style=none] (2) at (-1, 0.5) {};
\node [style=none] (3) at (1, 0.5) {};
\node [style=none] (4) at (-1.5, -0.5) {$X^\dagger$};
\node [style=none] (5) at (1.5, -0.5) {$(^*X)^\dagger$};
\node [style=none] (6) at (0, 2) {$(*\epsilon)^\dagger$ };
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (0.center);
\draw [bend left=90, looseness=2.00] (2.center) to (3.center);
\draw (3.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~~~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-1, -1) {};
\node [style=none] (1) at (1, -1) {};
\node [style=none] (2) at (-1, 0.5) {};
\node [style=none] (3) at (1, 0.5) {};
\node [style=none] (4) at (-1.5, -0.5) {$X^{*^\dagger}$};
\node [style=none] (5) at (1.5, -0.5) {$X^\dagger$};
\node [style=none] (6) at (0, 2) {$(\epsilon*)^\dagger$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (0.center);
\draw [bend left=90, looseness=2.00] (2.center) to (3.center);
\draw (3.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
A $\dagger$-$*$-autonomous category is a {\bf cyclic} $\dagger$-$*$-autonomous category when the dagger preserves the cyclor:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (5.5, 2.75) {};
\node [style=none] (1) at (5.5, -0.9999999) {};
\node [style=none] (2) at (4, -0.9999999) {};
\node [style=none] (3) at (4, 0.9999999) {};
\node [style=none] (4) at (2.75, 0.9999999) {};
\node [style=circle, scale=1.2] (5) at (2.75, -0.9999999) {$\phi^\dagger$};
\node [style=circle] (6) at (2.75, -2.5) {$\phi^{{-1}^\dagger}$};
\node [style=none] (7) at (2.75, -4.25) {};
\node [style=none] (8) at (6, 2.25) {$(A^\dagger)^*$};
\node [style=none] (9) at (4.5, -2) {$\epsilon*$};
\node [style=none] (10) at (3, 2) {$(*\epsilon)^\dagger$};
\node [style=none] (11) at (2.25, -3.75) {$(^*A)^\dagger$};
\node [style=none] (12) at (4.5, -0) {$A^\dagger$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend left=90, looseness=1.25] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend right=90, looseness=1.25] (3.center) to (4.center);
\draw (4.center) to (5);
\draw (5) to (6);
\draw (6) to (7.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (5.5, -0.9999999) {};
\node [style=none] (1) at (4, -1) {};
\node [style=none] (2) at (4, -3) {};
\node [style=none] (3) at (2.5, -3) {};
\node [style=circle, scale=1.3] (4) at (2.5, -0.9999999) {$\phi$};
\node [style=circle] (5) at (5.5, -4) {$\phi^{{-1}^\dagger}$};
\node [style=none] (6) at (2.5, 0.9999999) {};
\node [style=none] (7) at (1.25, -0.25) {$(A^\dagger)^*$};
\node [style=none] (8) at (4.75, -0.25) {$(\epsilon*)^\dagger$};
\node [style=none] (9) at (3.25, -3.75) {$*\epsilon$};
\node [style=none] (10) at (6.999999, -5) {$(^*A(+)^\dagger$};
\node [style=none] (11) at (4.5, -2) {$A^\dagger$};
\node [style=none] (12) at (5.5, -5.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=1.25] (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw [bend left=90, looseness=1.25] (2.center) to (3.center);
\draw (3.center) to (4);
\draw (0.center) to (5);
\draw (5) to (12.center);
\draw (4) to (6.center);
\end{pgfonlayer}
\end{tikzpicture} \]
\begin{lemma}
\label{Lemma: cyclic dagger}
In a cyclic, $\dagger$-$*$-autonomous category,
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (5.999999, 0.9999999) {};
\node [style=none] (1) at (3.999998, 0.9999999) {};
\node [style=none] (2) at (4, -1.75) {};
\node [style=none] (3) at (2, -1.75) {};
\node [style=circle, scale=1.25] (4) at (2, -1) {$\phi$};
\node [style=none] (5) at (2, 2.25) {};
\node [style=none] (6) at (4.999999, 2) {$(*\epsilon)^\dagger$};
\node [style=none] (7) at (3, -2.75) {$*\epsilon$};
\node [style=none] (8) at (6, -3) {};
\node [style=circle] (9) at (3.999998, -0.7499999) {$\phi^\dagger$};
\node [style=none] (10) at (1.5, 2) {$A^{*\dagger*}$};
\node [style=none] (11) at (6.499999, -2.75) {$A^\dagger$};
\node [style=none] (12) at (1.25, -1.5) {$~^*(A^{\dagger*})$};
\node [style=none] (13) at (4.5, -1.25) {$A^{\dagger*}$};
\node [style=none] (14) at (4.499998, 0.4999999) {$~^*(A^\dagger)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=1.25] (0.center) to (1.center);
\draw [bend left=90, looseness=1.25] (2.center) to (3.center);
\draw (3.center) to (4);
\draw (4) to (5.center);
\draw (0.center) to (8.center);
\draw (1.center) to (9);
\draw (9) to (2.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (6, -2.75) {};
\node [style=none] (1) at (4, -2.75) {};
\node [style=none] (2) at (4, -0.5) {};
\node [style=none] (3) at (2, -0.5) {};
\node [style=none] (4) at (3, 0.4999999) {$(\epsilon*)^\dagger$};
\node [style=none] (5) at (5, -3.75) {$\epsilon*$};
\node [style=none] (6) at (6, 1.25) {};
\node [style=none] (7) at (6.75, 0.7499999) {$A^{*\dagger*}$};
\node [style=none] (8) at (1.25, -3) {$A^\dagger$};
\node [style=none] (9) at (2, -4.25) {};
\node [style=none] (10) at (4.5, -1.75) {$A^{*\dagger}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=90, looseness=1.25] (0.center) to (1.center);
\draw [bend right=90, looseness=1.25] (2.center) to (3.center);
\draw (0.center) to (6.center);
\draw (2.center) to (1.center);
\draw (3.center) to (9.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\end{lemma}
\begin{proof}
Proved by direct application of Lemma \ref{Lemma: cyclic Frob}.
\end{proof}
\subsection{Functors for $\dagger$-LDCs}
Clearly the functors and transformations between $\dagger$-LDCs must ``preserve'' the dagger in some sense. More precisely we have:
\begin{definition}
$F: \mathbb{X} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{Y}$ is a {\bf $\dagger$-linear functor} between $\dagger$-LDCs when $F$ is a linear functor equipped with a linear natural isomorphism
$\rho^F= (\rho_\otimes^F: F_\otimes(A^\dagger) \@ifnextchar^ {\t@@}{\t@@^{}} F_\oplus(A)^\dagger ,\rho_\oplus^F: F_\otimes(A)^\dagger \@ifnextchar^ {\t@@}{\t@@^{}} F_\oplus(A^\dagger))$ called the {\bf preservator},
such that the following diagrams commute:
\[
\xymatrix{
F_\otimes(X) \ar[r]^{\iota} \ar[d]_{F_\otimes(\iota)} \ar@{}[dr]|{\mbox{\tiny {\bf [$\dagger$-LF.1]}}} &
F_\otimes(X)^{\dagger \dagger} \ar@{<-}[d]^{(\rho^F_\oplus)^\dagger} \\
F_\otimes(X^{\dagger \dagger}) \ar[r]_{\rho^F_\otimes} & F_\oplus(X^\dagger)^\dagger
} ~~~~~~~~~ \xymatrix{
F_\oplus(X) \ar[r]^{\iota} \ar[d]_{F_\oplus(\iota)} \ar@{}[dr]|{\mbox{\tiny {\bf [$\dagger$-LF.2]}}} & F_\oplus(X)^{\dagger \dagger} \ar[d]^{(\rho^F_\otimes)^\dagger} \\
F_\oplus(X^{\dagger \dagger}) \ar@{<-}[r]_{\rho^F_\oplus} & F_\otimes(X^\dagger)^\dagger
}
\]
\end{definition}
Observe that in the case $F$ is a normal mix functor between $\dagger$-isomix categories, then $F_\otimes= F_\oplus$ and the preservators become inverses, $\rho^F_\otimes = (\rho^F_\oplus)^{-1}$.
This means the squares {\bf [$\dagger$-LF.1]} and {\bf [$\dagger$-LF.2]} coincide to give a single condition for the tensor preservator:
\[ \xymatrix{
F(X) \ar[r]^{\iota} \ar[d]_{F(\iota)} \ar@{}[dr]|{\mbox{\tiny {\bf [$\dagger$-isomix]}}} &
F(X)^{\dagger \dagger} \ar@{->}[d]^{(\rho^F_\otimes)^\dagger} \\
F(X^{\dagger \dagger}) \ar[r]_{\rho^F_\otimes} & F(X^\dagger)^\dagger
} \]
For linear natural transformations $\beta: F \@ifnextchar^ {\t@@}{\t@@^{}} G$ between $\dagger$-linear functors we demand that $\beta_\otimes$ and $\beta_\oplus$ are related by:
\[ \xymatrix{F_\otimes(A^\dagger) \ar[d]_{\rho^F_\otimes} \ar[rr]^{\beta_\otimes} && G_\otimes(A^\dagger) \ar[d]^{\rho^G_\otimes} \\
(F_\oplus(X))^\dagger \ar[rr]_{\beta_\oplus^\dagger} & & (G_\oplus(X))^\dagger}
~~~~~~
\xymatrix{(G_\otimes(X))^\dagger \ar[d]_{\rho^G_\oplus} \ar[rr]^{\beta_\otimes^\dagger} && (F_\otimes(X))^\dagger \ar[d]^{\rho^F_\oplus} \\
G_\oplus(A^\dagger) \ar[rr]_{\beta_\oplus} & & F_\oplus(A^\dagger)} \]
Notice that this means that $\beta_\otimes$ is completely determined by $\beta_\oplus$ in the following sense:
\[ \xymatrix{ F_\otimes(A) \ar[d]_{F_\otimes(\iota)} \ar[rr]^{\beta_\otimes} && G_\otimes(A) \ar[d]^{G_\otimes(\iota)} \\
F_\otimes(A^{\dagger\dagger}) \ar[d]_{\rho^F_\otimes} \ar[rr]^{\beta_\otimes} & & G_\otimes(A^{\dagger\dagger}) \ar[d]^{\rho^G_\otimes} \\
F_\oplus(A^\dagger)^\dagger \ar[rr]_{\beta_\oplus^\dagger} && G_\oplus(A^\dagger)^\dagger } \]
Because the vertical maps are isomorphisms, this diagram can be used to express $\beta_\otimes$ in terms of $\beta_\oplus$. Similarly $\beta_\oplus$ can be expressed in terms
of $\beta_\otimes$. Thus, it is possible to express the coherences in terms of just one of these transformations.
\section{Daggers, duals, and conjugation}
\label{daggers-duals-conjugation}
The goal of this section is to review the interaction of the dualizing, conjugation and dagger functors. In dagger compact closed categories, $(\_)^\dagger$ and the dualizing functor $(\_)^*$ commute with each other and their composite gives the conjugate functor $(\_)_*$. Similary, $(\_)_*$ and $(\_)^*$ when composed gives the dagger functor. Our aim is to generalize these interactions to $\dagger$-LDCs and to achieve this at a reasonable level of abstraction. To achieve this we shall need the notion which we here refer to as ``conjugation'' but was investigated by Egger in \cite{Egg11} under the moniker of ``involution'' (which clashes with our usage).
\subsection{Duals}
The reverse of an LDC, $\mathbb{X}$, written $\mathbb{X}^\mathsf{rev} := (\mathbb{X}, \otimes, \top, \oplus, \bot)^{\sf rev} = (\mathbb{X}, \otimes^{\sf rev}, \top, \oplus^{\sf rev}, \bot)$ where,
\[ A \otimes^{\sf rev} B := B \otimes A ~~~~~~~~~~~ A \oplus^{\sf rev} B := B \oplus A \]
and the associators and distributors are adjusted accordingly. Similar to the opposite of an LDC, we have $(\mathbb{X}^{\sf rev})^{\sf rev} = \mathbb{X}$.
In a $*$-autonomous category, taking the left (or right) linear dual of an object extends to a Frobenius linear functor as follows:
\begin{align*}
(\_)^*: (\mathbb{X}^{\sf op})^{\sf rev} &\@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X} ; ~~ A \mapsto A^* ; ~~
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 2) {};
\node [style=none] (1) at (-0.5, 1) {};
\node [style=none] (2) at (0.5, 1) {};
\node [style=none] (3) at (-0.5, -0) {};
\node [style=none] (4) at (0.5, -0) {};
\node [style=none] (5) at (0, 1) {};
\node [style=none] (6) at (0, -1) {};
\node [style=none] (7) at (0, -0) {};
\node [style=none] (8) at (0, 0.5) {$f$};
\node [style=none] (9) at (0.25, 1.75) {$A$};
\node [style=none] (10) at (0.25, -0.75) {$B$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (3.center);
\draw (3.center) to (4.center);
\draw (4.center) to (2.center);
\draw (2.center) to (1.center);
\draw (0.center) to (5.center);
\draw (7.center) to (6.center);
\end{pgfonlayer}
\end{tikzpicture} \mapsto \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (-0.5, 1) {};
\node [style=none] (2) at (0.5, 1) {};
\node [style=none] (3) at (-0.5, -0) {};
\node [style=none] (4) at (0.5, -0) {};
\node [style=none] (5) at (0, 1) {};
\node [style=none] (6) at (0, -0.5) {};
\node [style=none] (7) at (0, -0) {};
\node [style=none] (8) at (-1, 1.5) {};
\node [style=none] (9) at (-1, -1) {};
\node [style=none] (10) at (1, -0.5) {};
\node [style=none] (11) at (1, 2) {};
\node [style=none] (12) at (0, 0.5) {$f$};
\node [style=none] (13) at (1.25, 1.75) {$B^*$};
\node [style=none] (14) at (-1.25, -0.75) {$A^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (3.center);
\draw (3.center) to (4.center);
\draw (4.center) to (2.center);
\draw (2.center) to (1.center);
\draw (0.center) to (5.center);
\draw (7.center) to (6.center);
\draw [bend right=90, looseness=1.25] (0.center) to (8.center);
\draw (8.center) to (9.center);
\draw [bend right=90, looseness=1.25] (6.center) to (10.center);
\draw (10.center) to (11.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{align*}
The $(\_)^*$ functor is both contravariant and reverses the tensors with the monoidal and comonoidal components defined as follows:
\[ m_{\otimes}: A^* \otimes B^* \@ifnextchar^ {\t@@}{\t@@^{}} (B \oplus A)^* := \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.25, 2) {};
\node [style=ox] (1) at (-0.25, 1) {};
\node [style=none] (2) at (-0.5, 0.25) {};
\node [style=none] (3) at (0.25, -0) {};
\node [style=none] (4) at (-1.25, 0.25) {};
\node [style=none] (5) at (-2, -0) {};
\node [style=oa] (6) at (-1.5, 1) {};
\node [style=none] (7) at (-1.5, 1.5) {};
\node [style=none] (8) at (-2.25, 1.5) {};
\node [style=none] (9) at (-2.25, -1) {};
\node [style=none] (10) at (0.5, 1.75) {$B^* \otimes A^*$};
\node [style=none] (11) at (-0.8, 0.5) {$B^*$};
\node [style=none] (12) at (0.5, 0.5) {$A^*$};
\node [style=none] (13) at (-3.2, -0.75) {$(A \oplus B)^* $};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=90, out=-45, looseness=1.00] (1) to (3.center);
\draw [in=90, out=-135, looseness=1.00] (1) to (2.center);
\draw [bend left=90, looseness=1.25] (2.center) to (4.center);
\draw [in=-60, out=90, looseness=1.00] (4.center) to (6);
\draw (6) to (7.center);
\draw [bend right=90, looseness=1.25] (7.center) to (8.center);
\draw (8.center) to (9.center);
\draw [in=90, out=-135, looseness=0.75] (6) to (5.center);
\draw [in=-90, out=-90, looseness=1.00] (5.center) to (3.center);
\draw (1) to (0.center);
\end{pgfonlayer}
\end{tikzpicture}
~~~~~~~~~~~~~~~ m_\top: \top \@ifnextchar^ {\t@@}{\t@@^{}} \bot^* := \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (0, 1.75) {$\top$};
\node [style=none] (1) at (0, 3) {};
\node [style=circle, scale=0.5] (2) at (-1, 1) {};
\node [style=circle] (3) at (-1, -0.25) {$\bot$};
\node [style=none] (4) at (-2, -1) {};
\node [style=none] (5) at (-2, 1.5) {};
\node [style=none] (6) at (-1, 1.5) {};
\node [style=none] (7) at (0.25, 2.75) {$\top$};
\node [style=none] (8) at (-2.5, -0.75) {$\bot^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (0);
\draw (3) to (6.center);
\draw [bend left=90, looseness=2.00] (5.center) to (6.center);
\draw (5.center) to (4.center);
\draw [in=0, out=-90, looseness=1.50, dotted] (0) to (2);
\end{pgfonlayer}
\end{tikzpicture} \]
\[ n_{\oplus}: (A \otimes B)^* \@ifnextchar^ {\t@@}{\t@@^{}} B^* \oplus A^* := \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (2, 2) {};
\node [style=ox] (1) at (1, 0.25) {};
\node [style=none] (2) at (2, -0.5) {};
\node [style=none] (3) at (1, -0.5) {};
\node [style=none] (4) at (0.5, 0.75) {};
\node [style=none] (5) at (1.5, 0.75) {};
\node [style=oa] (6) at (-0.75, 0.25) {};
\node [style=none] (7) at (-0.75, -1.75) {};
\node [style=none] (8) at (-0.25, 0.75) {};
\node [style=none] (9) at (-1.25, 0.75) {};
\node [style=none] (10) at (2, 2.25) {$(A \otimes B)^*$};
\node [style=none] (11) at (-0.75, -2) {$B^* \oplus A^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (2.center);
\draw [bend left=90, looseness=1.50] (2.center) to (3.center);
\draw (3.center) to (1);
\draw (6) to (7.center);
\draw [in=150, out=-90, looseness=1.00] (9.center) to (6);
\draw [in=-90, out=11, looseness=1.00] (6) to (8.center);
\draw [in=165, out=-90, looseness=1.25] (4.center) to (1);
\draw [in=-90, out=15, looseness=1.25] (1) to (5.center);
\draw [bend left=90, looseness=2.00] (8.center) to (4.center);
\draw [bend left=90, looseness=1.25] (9.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~~~~~~~~~ n_\bot: \top^* \@ifnextchar^ {\t@@}{\t@@^{}} \bot := \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-2, 1) {$\top$};
\node [style=none] (1) at (-1, 2) {};
\node [style=none] (2) at (-2, -0) {};
\node [style=none] (3) at (-1, -0) {};
\node [style=circle] (4) at (0, -1) {$\bot$};
\node [style=circle, scale=0.5] (5) at (-1, 1) {};
\node [style=none] (6) at (0, -2) {};
\node [style=none] (7) at (-0.6, 1.75) {$\top^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2.center);
\draw [bend right=90, looseness=1.75] (2.center) to (3.center);
\draw (3.center) to (1.center);
\draw [in=90, out=-15, looseness=1.25, dotted] (5) to (4);
\draw (6.center) to (4);
\end{pgfonlayer}
\end{tikzpicture} \]
These maps are isomorphisms and tensor reversing laxors which satisfy the obvious coherences. Thus, $(\_)^*$ is a strong Frobenius linear functor.
\begin{lemma}
$(\_)^*: \mathbb{X}^{\mathsf{op}\mathsf{rev}} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$ is a strong Frobenius functor which, when $\mathbb{X}$ is an isomix category, is a strong isomix Frobenius functor .
\end{lemma}
\begin{proof}
To prove that $(\_)^*$ preserves mix we need to show that $n_\bot \!~{\sf m}~ m_\top = {\sf m}^* : \top^* \@ifnextchar^ {\t@@}{\t@@^{}} \bot^* $:
\[
n_\bot {\sf m} m_\top = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-3, 2.75) {$\top$};
\node [style=none] (1) at (-2, 2) {};
\node [style=none] (2) at (-2, 3) {};
\node [style=circle] (3) at (-1, 1.25) {$\bot$};
\node [style=map] (4) at (-1, 0.5) {};
\node [style=circle] (5) at (-1, -0.25) {$\top$};
\node [style=circle] (6) at (-2, -2) {$\bot$};
\node [style=none] (7) at (-3, -1.25) {};
\node [style=none] (8) at (-3, -2.25) {};
\node [style=circle, scale=0.5] (9) at (-2, 1.75) {};
\node [style=circle, scale=0.5] (10) at (-2, -0.75) {};
\node [style=none] (11) at (-3, 2) {};
\node [style=none] (12) at (-2, -1.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1.center) to (2.center);
\draw (7.center) to (8.center);
\draw [bend left, looseness=1.00, dotted] (9) to (3);
\draw [bend left=45, looseness=1.00, dotted] (5) to (10);
\draw (3) to (4);
\draw (4) to (5);
\draw [bend right=90, looseness=3.25] (11.center) to (1.center);
\draw (0) to (11.center);
\draw [bend left=90, looseness=3.75] (7.center) to (12.center);
\draw (12.center) to (6);
\end{pgfonlayer}
\end{tikzpicture} =\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (1.25, 3) {$\top$};
\node [style=none] (1) at (0.25, 3) {};
\node [style=circle] (2) at (-1, 1.25) {$\bot$};
\node [style=map] (3) at (-1, 0.5) {};
\node [style=circle] (4) at (-1, -0.25) {$\top$};
\node [style=circle] (5) at (-2, -2.25) {$\bot$};
\node [style=none] (6) at (-3, -2.25) {};
\node [style=circle, scale=0.5] (7) at (0.25, 2) {};
\node [style=circle, scale=0.5] (8) at (-2, -0.75) {};
\node [style=none] (9) at (0.25, 1.5) {};
\node [style=none] (10) at (1.25, 1.5) {};
\node [style=none] (11) at (-2, -0.25) {};
\node [style=none] (12) at (-3, -0.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted, bend right, looseness=1.25] (7) to (2);
\draw [dotted, bend left=45, looseness=1.00] (4) to (8);
\draw (2) to (3);
\draw (3) to (4);
\draw (1.center) to (9.center);
\draw [bend right=90, looseness=2.00] (9.center) to (10.center);
\draw (10.center) to (0);
\draw (5) to (11.center);
\draw (12.center) to (6.center);
\draw [bend left=90, looseness=1.75] (12.center) to (11.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (1.25, 3) {$\top$};
\node [style=none] (1) at (0.25, 2.75) {};
\node [style=circle] (2) at (-1, 1.25) {$\bot$};
\node [style=map] (3) at (-1, 0.5) {};
\node [style=circle] (4) at (-1, -0.25) {$\top$};
\node [style=circle] (5) at (-2, -2.25) {$\bot$};
\node [style=none] (6) at (-3, -2) {};
\node [style=circle, scale=0.5] (7) at (0.25, 2) {};
\node [style=circle, scale=0.5] (8) at (-2, -0.75) {};
\node [style=none] (9) at (0.25, -1.75) {};
\node [style=none] (10) at (1.25, -1.75) {};
\node [style=none] (11) at (-2, 2.5) {};
\node [style=none] (12) at (-3, 2.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted, bend right, looseness=1.25] (7) to (2);
\draw [dotted, bend left=45, looseness=1.00] (4) to (8);
\draw (2) to (3);
\draw (3) to (4);
\draw (1.center) to (9.center);
\draw [bend right=90, looseness=2.00] (9.center) to (10.center);
\draw (10.center) to (0);
\draw (5) to (11.center);
\draw (12.center) to (6.center);
\draw [bend left=90, looseness=1.75] (12.center) to (11.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (1.25, 3) {$\top$};
\node [style=none] (1) at (0.25, 2.75) {};
\node [style=circle] (2) at (-0.75, 1.25) {$\bot$};
\node [style=map] (3) at (-0.75, 0.5) {};
\node [style=circle] (4) at (-0.75, -0.25) {$\top$};
\node [style=circle] (5) at (-2, -2.25) {$\bot$};
\node [style=none] (6) at (-3, -2) {};
\node [style=circle, scale=0.5] (7) at (-2, 2) {};
\node [style=circle, scale=0.5] (8) at (0.25, -0.75) {};
\node [style=none] (9) at (0.25, -1.75) {};
\node [style=none] (10) at (1.25, -1.75) {};
\node [style=none] (11) at (-2, 2.5) {};
\node [style=none] (12) at (-3, 2.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted, bend left, looseness=1.25] (7) to (2);
\draw [dotted, bend right=45, looseness=1.00] (4) to (8);
\draw (2) to (3);
\draw (3) to (4);
\draw (1.center) to (9.center);
\draw [bend right=90, looseness=2.00] (9.center) to (10.center);
\draw (10.center) to (0);
\draw (5) to (11.center);
\draw (12.center) to (6.center);
\draw [bend left=90, looseness=1.75] (12.center) to (11.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=map] (0) at (-0.75, 1) {};
\node [style=none] (1) at (-1.75, -1.75) {};
\node [style=none] (2) at (-0.75, -0.25) {};
\node [style=none] (3) at (0.25, -0.25) {};
\node [style=none] (4) at (-0.75, 2.25) {};
\node [style=none] (5) at (-1.75, 2.25) {};
\node [style=none] (6) at (0.25, 3.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=2.00] (2.center) to (3.center);
\draw (5.center) to (1.center);
\draw [bend left=90, looseness=1.75] (5.center) to (4.center);
\draw (4.center) to (0);
\draw (0) to (2.center);
\draw (6.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture} = {\sf m}^*
\]
\end{proof}
\begin{lemma}
$(\eta, \epsilon): (\_)^* \dashvv ~ {^*(\_)^{\mathsf{op}\mathsf{rev}}}$
\[
\eta_\otimes: X \@ifnextchar^ {\t@@}{\t@@^{}} ~^*(X^*) := \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (1, -2) {};
\node [style=none] (1) at (1, 0.75) {};
\node [style=none] (2) at (0, 0.75) {};
\node [style=none] (12) at (0.5, 1.75) {$*\eta$};
\node [style=none] (3) at (0, -0.25) {};
\node [style=none] (4) at (-1, -0.25) {};
\node [style=none] (12) at (-0.5, -1.5) {$\epsilon*$};
\node [style=none] (5) at (-1, 2.5) {};
\node [style=none] (6) at (-1.25, 2) {$X$};
\node [style=none] (7) at (-0.35, 0.5) {$X^*$};
\node [style=none] (8) at (1.6, -1.75) {$^*(X^*)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4.center) to (5.center);
\draw [bend right=90, looseness=3.25] (4.center) to (3.center);
\draw (3.center) to (2.center);
\draw [bend left=90, looseness=2.50] (2.center) to (1.center);
\draw (1.center) to (0.center);
\end{pgfonlayer}
\end{tikzpicture} \in \mathbb{X}
~~~~~~~~~~~~ \eta_\oplus := \eta_\otimes^{-1} \]
\[
\epsilon_\otimes: (^*X)^* \@ifnextchar^ {\t@@}{\t@@^{}} X := \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.75, -2) {};
\node [style=none] (1) at (-0.75, 0.75) {};
\node [style=none] (2) at (0.25, 0.75) {};
\node [style=none] (12) at (-0.5, 1.75) {$\eta*$};
\node [style=none] (3) at (0.25, -0.25) {};
\node [style=none] (4) at (1.25, -0.25) {};
\node [style=none] (34) at (0.65, -1.5) {$*\epsilon$};
\node [style=none] (5) at (1.25, 2.5) {};
\node [style=none] (6) at (1.65, 2) {$X$};
\node [style=none] (7) at (0.6, 0.5) {$^*X$};
\node [style=none] (8) at (-1.25, -1.75) {$(^*X)^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4.center) to (5.center);
\draw [bend left=90, looseness=3.25] (4.center) to (3.center);
\draw (3.center) to (2.center);
\draw [bend right=90, looseness=2.50] (2.center) to (1.center);
\draw (1.center) to (0.center);
\end{pgfonlayer}
\end{tikzpicture} \in \mathbb{X} ~~~~~~~~~~~ \epsilon_\oplus := \epsilon_\otimes^{-1}
\]
is a linear equivalence of Frobenius linear functors.
\end{lemma}
\begin{proof}
The proof is straightforward in the graphical calculus.
\end{proof}
When the $*$-autonomous category is cyclic, so there is a cyclor $\psi: A^* \@ifnextchar^ {\t@@}{\t@@^{}}^{\varphi} \!~^{*}\!A$, then we may straighten out this equivalence to be a dualizing involutive equivalence
(i.e. so that the unit and counit are equal):
\begin{lemma}
$(\eta', \epsilon'): (\_)^* \dashvv ((\_)^{*})^{\mathsf{op}\mathsf{rev}}$ where $\eta'_\otimes = {\eta'_\oplus}^{-1} := \eta_\otimes \varphi^{-1}$, $\epsilon'_\otimes = \epsilon'_\oplus:= \epsilon \varphi^*$ and $\eta' = \epsilon'$.
\end{lemma}
\begin{proof}
The unit and counit are drawn as follows:
\[
\eta_\otimes' = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 3) {};
\node [style=none] (1) at (-3, 1) {};
\node [style=none] (2) at (-2, 1) {};
\node [style=none] (3) at (-2, 2) {};
\node [style=none] (4) at (-1, 2) {};
\node [style=circle] (5) at (-1, -0) {$\phi^{-1}$};
\node [style=none] (6) at (-1, -1) {};
\node [style=none] (7) at (-2.5, 0.25) {$\epsilon*$};
\node [style=none] (8) at (-1.5, 2.75) {$~^*\eta$};
\node [style=none] (9) at (-3.25, 2.75) {$X$};
\node [style=none] (10) at (-0.5, 1.75) {$~^*(X^*)$};
\node [style=none] (11) at (-0.5, -0.7) {$X^{**}$};
\node [style=none] (12) at (-2.35, 1.5) {$X^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend right=90, looseness=1.75] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend left=90, looseness=2.00] (3.center) to (4.center);
\draw (4.center) to (5);
\draw (5) to (6.center);
\end{pgfonlayer}
\end{tikzpicture} \in \mathbb{X} ~~~~~~~~~~ \epsilon_\otimes' = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.75, -0.5) {};
\node [style=none] (1) at (-0.75, 2.25) {};
\node [style=none] (2) at (0.25, 2.25) {};
\node [style=none] (3) at (0.25, 1) {};
\node [style=none] (4) at (1.25, 1) {};
\node [style=none] (5) at (1.25, 3) {};
\node [style=none] (6) at (0.75, 0.25) {$*\epsilon$};
\node [style=none] (7) at (-0.25, 3) {$\eta*$};
\node [style=none] (8) at (1.5, 2.5) {$X^*$};
\node [style=none] (9) at (-1.75, -0.5) {};
\node [style=circle, scale=2] (10) at (-1.75, 0.5) {};
\node [style=none] (11) at (-1.75, 1.25) {};
\node [style=none] (12) at (-2.75, 1.25) {};
\node [style=none] (13) at (-2.75, -1.25) {};
\node [style=none] (14) at (-1.25, -1.25) {$\epsilon*$};
\node [style=none] (15) at (-2.25, 2) {$\eta*$};
\node [style=none] (16) at (-1.75, 0.5) {$\phi$};
\node [style=none] (17) at (-3.25, -0.75) {$X^{**}$};
\node [style=none] (18) at (-0.25, -0) {$~^*(X^*)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend right=90, looseness=2.00] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (1.center) to (0.center);
\draw [bend right=90, looseness=1.75] (9.center) to (0.center);
\draw (11.center) to (10);
\draw (10) to (9.center);
\draw [bend right=90, looseness=1.75] (11.center) to (12.center);
\draw (12.center) to (13.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.75, 3) {};
\node [style=none] (1) at (-0.75, -0) {};
\node [style=none] (2) at (-1.75, -0) {};
\node [style=none] (3) at (-1.75, 2) {};
\node [style=none] (4) at (-2.75, 2) {};
\node [style=none] (5) at (-2.75, -1) {};
\node [style=none] (6) at (-1.25, -0.75) {$*\epsilon$};
\node [style=none] (7) at (-2.25, 2.75) {$\eta^*$};
\node [style=none] (8) at (-0.5, 2.75) {$X$};
\node [style=none] (9) at (-3.25, -0.5) {$X^{**}$};
\node [style=circle] (10) at (-1.75, 1) {$\phi$};
\node [style=none] (11) at (-2.25, 0.25) {$~^*X$};
\node [style=none] (12) at (-2.1, 1.7) {$X^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend left=90, looseness=1.75] (1.center) to (2.center);
\draw [bend right=90, looseness=2.00] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (3.center) to (10);
\draw (10) to (2.center);
\end{pgfonlayer}
\end{tikzpicture} \stackrel{{\bf \tiny [C.2]}}{=}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 3) {};
\node [style=none] (1) at (-3, 1) {};
\node [style=none] (2) at (-2, 1) {};
\node [style=none] (3) at (-2, 2) {};
\node [style=none] (4) at (-1, 2) {};
\node [style=circle] (5) at (-1, -0) {$\phi^{-1}$};
\node [style=none] (6) at (-1, -1) {};
\node [style=none] (7) at (-2.5, 0.25) {$\epsilon*$};
\node [style=none] (8) at (-1.5, 2.75) {$~^*\eta$};
\node [style=none] (9) at (-3.25, 2.75) {$X$};
\node [style=none] (10) at (-0.5, 1.75) {$~^*(X^*)$};
\node [style=none] (11) at (-0.5, -0.7) {$X^{**}$};
\node [style=none] (12) at (-2.35, 1.5) {$X^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend right=90, looseness=1.75] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend left=90, looseness=2.00] (3.center) to (4.center);
\draw (4.center) to (5);
\draw (5) to (6.center);
\end{pgfonlayer}
\end{tikzpicture} \in \mathbb{X}
\]
The cyclor is a linear transformation which is an isomorphism as it monoidal with respect to both tensor and par ( see ) and adjoints are determined only upto isomorphism. It remains to check that the triangle identities hold:
\[
\eta_{\otimes'_{X^*}}(\epsilon_\otimes')^* =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (0.75, 0.25) {$\phi^{-1}$};
\node [style=none] (1) at (0.75, -2.5) {};
\node [style=none] (2) at (0.75, 2.25) {};
\node [style=none] (3) at (-0.25, 2.25) {};
\node [style=none] (4) at (-0.25, 1) {};
\node [style=none] (5) at (-1.25, 1) {};
\node [style=none] (6) at (-1.25, 3) {};
\node [style=none] (7) at (1.5, -0.5) {$X^{***}$};
\node [style=none] (8) at (-0.75, 0.25) {$\epsilon*$};
\node [style=none] (9) at (0.25, 3) {$*\eta$};
\node [style=none] (10) at (-1.5, 2.5) {$X^*$};
\node [style=none] (11) at (-1, -2.5) {};
\node [style=circle] (12) at (-1, -1.5) {$\phi^{-1}$};
\node [style=none] (13) at (-1, -0.75) {};
\node [style=none] (14) at (-2, -0.75) {};
\node [style=none] (15) at (-2, -3.25) {};
\node [style=none] (16) at (-1.5, -0.1) {$*\eta$};
\node [style=none] (17) at (0, -3.35) {$\epsilon*$};
\node [style=none] (18) at (-2.5, -3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (0);
\draw (0) to (1.center);
\draw [bend right=90, looseness=2.00] (2.center) to (3.center);
\draw (3.center) to (4.center);
\draw [bend left=90, looseness=2.00] (4.center) to (5.center);
\draw (5.center) to (6.center);
\draw [bend right=75, looseness=1.25] (11.center) to (1.center);
\draw (11.center) to (12);
\draw (12) to (13.center);
\draw [bend right=90, looseness=1.50] (13.center) to (14.center);
\draw (14.center) to (15.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, -2.5) {};
\node [style=none] (1) at (0, 2.5) {};
\node [style=none] (2) at (1, 2.5) {};
\node [style=none] (3) at (1, 1.25) {};
\node [style=none] (4) at (2, 1.25) {};
\node [style=none] (5) at (2, 3.25) {};
\node [style=none] (6) at (-0.5, 1.25) {$X^{***}$};
\node [style=none] (7) at (1.5, 0.5) {$*\epsilon$};
\node [style=none] (8) at (0.5, 3.25) {$\eta*$};
\node [style=none] (9) at (2.25, 2.75) {$X^*$};
\node [style=none] (10) at (-1, -2.5) {};
\node [style=circle] (11) at (-1, -1.5) {$\phi^{-1}$};
\node [style=none] (12) at (-1, -0.75) {};
\node [style=none] (13) at (-2, -0.75) {};
\node [style=none] (14) at (-2, -3.25) {};
\node [style=none] (15) at (-1.5, -0) {$*\eta$};
\node [style=none] (16) at (-0.5, -3) {$\epsilon*$};
\node [style=circle] (17) at (1, 2) {$\phi$};
\node [style=none] (18) at (-2.25, -2.75) {$X^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw [bend right=90, looseness=2.00] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw [bend right=75, looseness=1.25] (10.center) to (0.center);
\draw (10.center) to (11);
\draw (11) to (12.center);
\draw [bend right=90, looseness=1.50] (12.center) to (13.center);
\draw (13.center) to (14.center);
\draw (1.center) to (0.center);
\draw (2.center) to (17);
\draw (17) to (3.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-1, -1) {};
\node [style=none] (1) at (0, -1) {};
\node [style=none] (2) at (0, 3.25) {};
\node [style=none] (3) at (-0.5, -1.75) {$*\epsilon$};
\node [style=none] (4) at (0.25, 2.75) {$X^*$};
\node [style=circle] (5) at (-1, 1.5) {$\phi^{-1}$};
\node [style=none] (6) at (-1, 2.25) {};
\node [style=none] (7) at (-2, 2.25) {};
\node [style=none] (8) at (-2, -3.25) {};
\node [style=none] (9) at (-1.5, 3) {$*\eta$};
\node [style=circle] (10) at (-1, -0.25) {$\phi$};
\node [style=none] (11) at (-2.25, -2.75) {$X^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=2.00] (0.center) to (1.center);
\draw (1.center) to (2.center);
\draw (5) to (6.center);
\draw [bend right=90, looseness=1.50] (6.center) to (7.center);
\draw (7.center) to (8.center);
\draw (10) to (0.center);
\draw (5) to (10);
\end{pgfonlayer}
\end{tikzpicture} = ~ \begin{tikzpicture} \draw (2, 3.25) -- (2,-3.5); \end{tikzpicture} ~ = ~ 1
\] The other triangle identity holds similarly.
\end{proof}
The equality of $\eta'$ and $\epsilon'$ is immediate from {\bf [C.2]} for cyclors, the map $\eta'=\epsilon'$ is the {\bf dualizor}. In the symmetric case, the dualizor of this
equivalence may be drawn as:
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 3) {};
\node [style=none] (1) at (-3, 1) {};
\node [style=none] (2) at (-2, 1) {};
\node [style=none] (3) at (-2, 2) {};
\node [style=none] (4) at (-1, 2) {};
\node [style=circle] (5) at (-1, -0) {$\phi^{-1}$};
\node [style=none] (6) at (-1, -1) {};
\node [style=none] (7) at (-2.5, 0.25) {$\epsilon*$};
\node [style=none] (8) at (-1.5, 2.75) {$*\eta$};
\node [style=none] (9) at (-3.25, 2.75) {$A$};
\node [style=none] (10) at (-0.5, 1.75) {$~^*(A^*)$};
\node [style=none] (11) at (-0.5, -0.7) {$A^{**}$};
\node [style=none] (12) at (-2.25, 1.5) {$A^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend right=90, looseness=1.75] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend left=90, looseness=2.00] (3.center) to (4.center);
\draw (4.center) to (5);
\draw (5) to (6.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-1.75, 2) {};
\node [style=none] (1) at (-1.75, -0) {};
\node [style=none] (2) at (0, -0) {};
\node [style=none] (12) at (-0.75, -1) {$\epsilon*$};
\node [style=none] (3) at (0, 1) {};
\node [style=none] (4) at (-1, 1) {};
\node [style=none] (34) at (-0.5, 1.75) {$\eta*$};
\node [style=none] (5) at (-1, -2.75) {};
\node [style=none] (6) at (-2, 1.75) {$A$};
\node [style=none] (8) at (-1.5, -2.25) {$A^{**}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend right=90, looseness=1.50] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend right=90, looseness=1.75] (3.center) to (4.center);
\draw (4.center) to (5.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.75, 3) {};
\node [style=none] (1) at (-0.75, -0) {};
\node [style=none] (2) at (-1.75, -0) {};
\node [style=none] (3) at (-1.75, 2) {};
\node [style=none] (4) at (-2.75, 2) {};
\node [style=none] (5) at (-2.75, -1) {};
\node [style=none] (6) at (-1.25, -0.75) {$*\epsilon$};
\node [style=none] (7) at (-2.25, 2.75) {$\eta*$};
\node [style=none] (8) at (-0.5, 2.75) {$A$};
\node [style=none] (9) at (-3.25, -0.5) {$A^{**}$};
\node [style=circle] (10) at (-1.75, 1) {$\phi$};
\node [style=none] (11) at (-2.25, 0.25) {$~^*A$};
\node [style=none] (12) at (-2, 1.7) {$A^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend left=90, looseness=1.75] (1.center) to (2.center);
\draw [bend right=90, looseness=2.00] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (3.center) to (10);
\draw (10) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\subsection{Conjugation}
Recall the following structure from Egger \cite{Egg11}:
\begin{definition}
A {\bf conjugation} for a monoidal category $(X, \otimes, I)$ consists of a functor $\overline{(\_)}: \mathbb{X}^\mathsf{rev} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$ with natural isomorphisms:
\[ \bar{A} \otimes \bar{B} \@ifnextchar^ {\t@@}{\t@@^{}}^{\chi} \bar{B \otimes A} ~~~~~~~~~~~~~ \bar{\bar{A}} \@ifnextchar^ {\t@@}{\t@@^{}}^{\varepsilon} A \]
called respectively the (tensor reversing) {\bf conjugating laxor} and the {\bf conjugator}
such that
\[\bar{\bar{\bar{A}}} \@ifnextchar^ {\t@@}{\t@@^{}}^{\bar{\varepsilon_A} = \varepsilon_{\bar{A}} } \bar{A} \]
and
\[
\xymatrix{
(\bar{A} \otimes \bar{B}) \otimes \bar{C} \ar[rr]^{a_\otimes} \ar[d]_{\chi \otimes 1} \ar@{}[ddrr]|{\bf [CF.1]_\otimes} & & \bar{A} \otimes (\bar{B} \otimes \bar{C}) \ar[d]^{1 \otimes \chi} \\
\bar{(B \otimes A)} \otimes \bar{C} \ar[d]_{\chi} & & \bar{A} \otimes \bar{(C \otimes B)} \ar[d]^{\chi} \\
\bar{C \otimes (B \otimes A)} \ar[rr]_{\bar{a_\otimes^{-1}}} & & \bar{(C \otimes B) \otimes A}
} ~~~~~~~~ \xymatrix{
\bar{\bar{A}} \otimes \bar{\bar{B}} \ar[rr]^{\chi} \ar[dd]_{\varepsilon \otimes \varepsilon} \ar@{}[ddrr]|{\bf [CF.2]_\otimes} & & \bar{\bar{B} \otimes \bar{A}} \ar[dd]^{\bar{\chi}} \\
& & \\
A \otimes B & & \bar{\bar{A \otimes B}} \ar[ll]^{\varepsilon}
}
\]
\end{definition}
A monoidal category is {\bf conjugative} when it has a conjugation.
A symmetric monoidal category, which is conjugative, is {\bf symmetric conjugative} in case it satisfies the additional coherence:
\[
\xymatrix{
\bar{A} \otimes \bar{B} \ar[d]_{c_\otimes} \ar[rr]^{\chi} \ar@{}[drr]|{\bf [CF.3]_\otimes} & & \bar{B \otimes A} \ar[d]^{\bar{c_\otimes}} \\
\bar{B} \otimes \bar{A} \ar[rr]_{\chi} & & \bar{A \otimes B}
}
\]
Notice that we have not specified any coherence for the unit $I$ this is because the expected coherences are automatic:
\begin{lemma}
\label{Lemma: unit conjugate} \cite[Lemma 2.3]{Egg11}
For every conjugative monoidal category, there exists a unique isomorphism $I \xrightarrow{\chi^{\!\!\!\circ}} \bar{I}$ such that
\[
\xymatrix{
I \otimes \bar{A} \ar[rr]^{\chi^{\!\!\!\circ} \otimes 1} \ar[d]_{u_\otimes} \ar@{}[drr]|{\bf [CF.4]_\top} & & \bar{I} \otimes \bar{A} \ar[d]^{\chi} \\
\bar{A} \ar[rr]_{\bar{u_\otimes^{-1}}} & & \bar{A \otimes I } } ~~~~~
\xymatrix{ \bar{A} \otimes I \ar[rr]^{1 \otimes \chi^{\!\!\!\circ}} \ar[d]_{u_\otimes} \ar@{}[drr]|{\bf [CF.5]_\top} & & \bar{A} \otimes \bar{I} \ar[d]^{\chi} \\
\bar{A} \ar[rr]_{\bar{u_\otimes^{-1}}} & & \bar{I \otimes A} } ~~~~~
\xymatrix{
I \ar[rr]^{\chi^{\!\!\!\circ}} \ar@{=}[d] \ar@{}[drr]|{\bf [CF.6]_\top} & & \bar{I} \ar[d]^{\bar{\chi^{\!\!\!\circ}}} \\
I \ar[rr]_{\varepsilon^{-1}} & & \bar{\bar{I}}
}
\]
\end{lemma}
\begin{definition} \cite{Egg11}
A {\bf conjugative LDC} is a linearly distributive category $(\mathbb{X}, \otimes, \top, \oplus, \bot)$ together with a conjugating functor $\bar{(\_)}: \mathbb{X} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$ and natural isomorphisms:
\[ \bar{A} \otimes \bar{B} \xrightarrow{\chi_\otimes} \bar{B \otimes A} ~~~~~~~~~~~~ \bar{A \oplus B} \xrightarrow{\chi_\oplus} \bar{B} \oplus \bar{A} ~~~~~~~~~~~~ \bar{\bar{A}} \xrightarrow{\varepsilon} A \]
\end{definition}
such that $(\mathbb{X}, \otimes, \top, \chi_\otimes, \varepsilon)$ and $(\mathbb{X}, \oplus, \bot, \chi_\oplus^{-1}, \varepsilon)$ are conjugative (symmetric) monoidal categories with respect to the conjugating functor and the following diagrams commute:
\[
\xymatrix{
\bar{B \oplus C} \otimes \bar{A} \ar[rr]^{\chi_\oplus \otimes 1} \ar[d]_{\chi_\otimes} \ar@{}[ddrr]|{\bf [CF.7]} & & (\bar{C} \oplus \bar{B}) \otimes \bar{A} \ar[d]^{\delta} \\
\bar{(A \otimes (B \oplus C))} \ar[d]_{\bar{\delta}} & & \bar{C} \oplus ( \bar{B} \otimes \bar{A}) \ar[d]^{1 \oplus \chi_\otimes} \\
\bar{((A \otimes B) \oplus C)} \ar[rr]_{\chi_\oplus} & & \bar{C} \oplus \bar{A \otimes B}
} ~~~~~~~~~~
\xymatrix{
\bar{A} \otimes \bar{C \oplus B} \ar[rr]^{\chi_\otimes} \ar[d]_{1 \otimes \chi_\oplus} \ar@{}[ddrr]|{\bf [CF.8]} & & \bar{(C \oplus B) \otimes A} \ar[d]^{\bar{\delta}} \\
\bar{A} \otimes (\bar{B} \oplus \bar{C}) \ar[d]_{\delta} & & \bar{C \oplus (B \otimes A)} \ar[d]^{\chi_\oplus} \\
(\bar{A} \otimes \bar{B}) \oplus \bar{C} \ar[rr]_{\chi_\otimes \oplus 1} & & \bar{(B \otimes A)} \oplus \bar{C}
}
\]
Note, by Lemma \ref{Lemma: unit conjugate}, there exists canonical isomorphisms $\top \xrightarrow{\chi^{\!\!\!\circ}_\top} \bar{\top}$ and $\bot \xrightarrow{\chi^{\!\!\!\circ}_\bot} \bar{\bot}$ thes ensure conjugation is a normal functor. However, if the category is a mix category there is no guarantee that conjugation is a mix functor. To ensure this we must demand that
conjugation is a mix functor. Explicitly this means that the following extra condition must be satisfied:
\[ {\bf \small [CF.9]}~~~~~ \xymatrix{\overline{\bot} \ar[rd]_{(\chi^{\!\!\!\circ}_\bot)^{-1}} \ar@/^/[rrr]^{\overline{{\sf m}}} & & & \overline{\top} \\ & \bot \ar[r]_{{\sf m}} & \top \ar[ur]_{\chi^{\!\!\!\circ}_\top} } \]
\begin{proposition}
A conjugative LDC is precisely a LDC $\mathbb{X}$, with a Frobenius adjoint $(\epsilon^{-1}, \epsilon): \overline{(\_)} \dashv \overline{(\_)}^\mathsf{rev}: \mathbb{X}^\mathsf{rev} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$ where $\epsilon := (\varepsilon,\varepsilon^{-1})$. Furthermore, if the category is an isomix category and conjugation is a mix functor then conjugation is an isomix equivalence.
\end{proposition}
\begin{proof}
It is clear that $\overline{(\_)}$ is a strong Frobenius functor so being mix implies being isomix. Also $\varepsilon$ is clearly monoidal for tensor and par. The triangle equalities give $\varepsilon \overline{\varepsilon^{-1}} = 1: \overline{A} \@ifnextchar^ {\t@@}{\t@@^{}} \overline{A}$ which shows $\varepsilon = \overline{\varepsilon}$.
\end{proof}
Clearly conjugation should flip left duals into right duals:
\begin{lemma}
\label{Lemma: involutive linear adjoint}
If $B \dashvv A$ is a linear dual then $\bar{A} \dashvv \bar{B}$ is a linear dual.
\end{lemma}
\begin{proof}
Suppose $(\eta, \varepsilon): B \dashvv A$. Then, $(\chi^{\!\!\!\circ}_\top \bar{\eta} \chi_\oplus,\chi_\otimes \bar{\varepsilon} \chi^{\!\!\!\circ}_\bot): \bar{A} \dashvv \bar{B}$.
\end{proof}
When a $*$-autonomous category is cyclic one expects that conjugation should interact with the cyclor in a coherent fashion:
\begin{definition}\cite{EggMcCurd12}
\label{Defn: conjugative cyclic}
A {\bf conjugative cyclic $*$-autonomous category} is a conjugative $*$-autonomous category together with a cyclor $A^* \@ifnextchar^ {\t@@}{\t@@^{}}^{\psi} \!\!~^{*}\!A$ such that
\[
\xymatrix{
(\bar{A})^* \ar[rr]^{\psi} \ar[d]_{\simeq} & & ^{*}(\bar{A}) \ar[d]^{\simeq} \\
\bar{(^{*}A)} \ar[rr]_{\bar{\psi^{-1}}} & & \bar{(A^*)}
}
\]
which gives a map $\sigma: (\overline{A})^{*} \@ifnextchar^ {\t@@}{\t@@^{}} \overline{(A^{*})}$.
\end{definition}
The above condition is drawn as follows:
\[ \sigma =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-1, 0.75) {$\overline{\psi^{-1}}$};
\node [style=circle, scale=2.5] (0) at (-1, 0.75) {};
\node [style=none] (1) at (-1, -0.5) {};
\node [style=none] (2) at (-1, 2.25) {};
\node [style=none] (3) at (0, 2.25) {};
\node [style=none] (4) at (0, 1) {};
\node [style=none] (5) at (1, 1) {};
\node [style=none] (6) at (1, 3) {};
\node [style=none] (7) at (1.5, 2.75) {$(\overline{A})^*$};
\node [style=none] (8) at (0.5, 0.2) {$\epsilon^*$};
\node [style=none] (9) at (-0.5, 3.1) {$\overline{~^*\eta}$};
\node [style=none] (10) at (-1.5, -0.25) {$\overline{A^*}$};
\node [style=none] (12) at (-0.25, 1.5) {$\overline{A}$};
\node [style=none] (11) at (-1.5, 1.75) {$\overline{~^*A}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (0);
\draw (0) to (1.center);
\draw [bend left=90, looseness=2.00] (2.center) to (3.center);
\draw (3.center) to (4.center);
\draw [bend right=90, looseness=2.00] (4.center) to (5.center);
\draw (5.center) to (6.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-1, 2.5) {$\psi$};
\node [style=none] (1) at (-1, 3.25) {};
\node [style=none] (2) at (-1, 0.5) {};
\node [style=none] (3) at (0, 0.5) {};
\node [style=none] (4) at (0, 1.75) {};
\node [style=none] (5) at (1, 1.75) {};
\node [style=none] (6) at (1, -0.25) {};
\node [style=none] (7) at (-1.75, 3.25) {$(\overline{A})^*$};
\node [style=none] (8) at (0.5, 2.6) {$\overline{\eta^*}$};
\node [style=none] (9) at (-0.5, -0.35) {$~^*\epsilon$};
\node [style=none] (10) at (1.5, 0.25) {$\overline{A^*}$};
\node [style=none] (11) at (-1.75, 1.5) {$~^*(\overline{A})$};
\node [style=none] (12) at (-0.25, 1.25) {$\overline{A}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (0);
\draw (0) to (1.center);
\draw [bend right=90, looseness=2.00] (2.center) to (3.center);
\draw (3.center) to (4.center);
\draw [bend left=90, looseness=2.00] (4.center) to (5.center);
\draw (5.center) to (6.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
When the $*$-autonomous category is symmetric, (symmetric) conjugation automatically preserves the canonical cyclor.
\begin{lemma}
\label{Lemma: varepsi monoidal}
In a conjugative $*$-autonomous category,
\[
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (0, 1) {$\varepsilon$};
\node [style=none] (1) at (0, -0) {};
\node [style=none] (2) at (-1.75, -0) {};
\node [style=none] (3) at (-1.75, 2) {};
\node [style=none] (4) at (0, 2) {};
\node [style=none] (5) at (-1, 3) {$\overline{\overline{\eta*}}$};
\node [style=none] (6) at (-2.25, 0.3) {$\overline{\overline{X^*}}$};
\node [style=none] (7) at (0.5, 1.7) {$\overline{\overline{X}}$};
\node [style=none] (8) at (0.5, 0.25) {$X$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4.center) to (0);
\draw (0) to (1.center);
\draw [bend left=90, looseness=1.50] (3.center) to (4.center);
\draw (3.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-1.75, 1) {$\varepsilon^{-1}$};
\node [style=none] (1) at (-1.75, -0) {};
\node [style=none] (2) at (0, -0) {};
\node [style=none] (3) at (0, 2) {};
\node [style=none] (4) at (-1.75, 2) {};
\node [style=none] (5) at (-1, 3) {$\eta*$};
\node [style=none] (6) at (-2.5, 0.3) {$\overline{\overline{X^*}}$};
\node [style=none] (7) at (0.5, 0.5) {$X$};
\node [style=none] (8) at (-2.25, 1.7) {$X^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4.center) to (0);
\draw (0) to (1.center);
\draw [bend right=90, looseness=1.50] (3.center) to (4.center);
\draw (3.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\[
\chi_\top^{\!\!\!\circ}~ \overline{\chi_\top^{\!\!\!\circ}} ~ \overline{\overline{\eta*}} ~ \overline{\chi_\oplus}^{-1} \chi_\oplus^{-1} (1 \oplus \varepsilon) = \eta* (\varepsilon^{-1} \oplus 1) : \top \@ifnextchar^ {\t@@}{\t@@^{}} \overline{\overline{X^*}} \oplus X
\]
\end{lemma}
\begin{proof}
\begin{align*}
\chi_\top^{\!\!\!\circ} ~ \overline{\chi_\top^{\!\!\!\circ}} ~ \overline{\overline{\eta*}} ~ \overline{\chi}^{-1} \chi^{-1} (1 \oplus \varepsilon) &= \chi_\top^{\!\!\!\circ} ~ \overline{\chi_\top^{\!\!\!\circ}} ~ \overline{\overline{\eta*}} ~ \overline{\chi_\top}^{-1} \chi_\top^{-1} (\varepsilon \varepsilon^{-1} \oplus \varepsilon) \\
&=\chi_\top^{\!\!\!\circ} ~ \overline{\chi_\top^{\!\!\!\circ}} ~ \overline{\overline{\eta*}} ~ \overline{\chi}^{-1} \chi^{-1} (\varepsilon \oplus \varepsilon) (\varepsilon^{-1} \oplus 1) \\
&\stackrel{{\bf \small [CF.2]_\oplus}}{=} \chi^{\!\!\!\circ} ~ \overline{\chi_\top^{\!\!\!\circ}} ~ \overline{\overline{\eta*}} \varepsilon (\varepsilon^{-1} \oplus 1) \\
&\stackrel{{\bf \small nat.}}{=} \chi_\top^{\!\!\!\circ} ~ \overline{\chi_\top^{\!\!\!\circ}} ~ \varepsilon \eta\!* (\varepsilon^{-1} \oplus 1) \\
&\stackrel{{\bf \small [CF.6]_\top}}{=} \eta\!* (\varepsilon^{-1} \oplus 1)
\end{align*}
\end{proof}
\subsection{Dagger and conjugation}
The interaction of the dagger and conjugation, in the presence of the dualizing functor for cyclic $*$-autonomous categories is illustrated by the following diagram:
\[
\xymatrixcolsep{5pc}
\xymatrixrowsep{5pc}
\xymatrix{
\mathbb{X}^{\mathsf{op}} \ar@/^1pc/[rr]^{(\_)^\dagger} \ar@/^1pc/[dr]|{((\_)^*)^\mathsf{rev}} \ar@{}[rr]|{\bot} &~ \ar@{}[d]|{\cong} &
\mathbb{X} \ar@/^1pc/[ll]|{((\_)^{\dagger})^{\mathsf{op}}} \ar@/_1pc/[dl]|{\overline{(\_)}^\mathsf{rev}} \ar@{}[dl]|{\dashv} \\
& \mathbb{X}^\mathsf{rev} \ar@/^1pc/[ul]^{(\_)^{*^{\mathsf{op}}}} \ar@/_1pc/[ur]_{\overline{(\_)}} \ar@{}[ul]|{\dashv} &
}
\]
Specifically we have:
\begin{theorem}
Every cyclic, conjugative $*$-autonomous category is also a $\dagger$-$*$-autonomous category.
\end{theorem}
\begin{proof}
Let $\mathbb{X}$ be a cyclic, conjugative $*$-autonomous category then $\bar{(\_)^*} \dashv \bar{(\_)}^*$ is an equivalence. To build a dagger we need an equivalence on the same
functor: we obtain this by using the natural equivalence $\sigma: \bar{(\_)^*} \@ifnextchar^ {\t@@}{\t@@^{}} \bar{(\_)}^*$ from Definition \ref{Defn: conjugative cyclic}. An involutive equivalence, in addition,
requires the unit and counit of the (contravariant) equivalence to be the same map (which we called the involutor, $\iota$). We show that this is the case:
The unit and counit of the equivalence is given by {\em (a)} and {\em (b)} respectively;
\[ \mbox{\em (a)}~~~~~~~~~ \xymatrix{
\mathbb{X}^{\mathsf{op}} \ar@{=}[rrrr] \ar[dr]_{(\_)^{*^{\sf rev}}} &
&
\ar@{}[d]|{\Downarrow ~ \eta_\otimes'}&
&
\mathbb{X}^{\mathsf{op}} \ar@{<-}[ld]^{(\_)^{*^{\sf op}}}
& \\
&
\mathbb{X}^{\sf rev} \ar@{=}[rr] \ar[dr]_{\bar{(\_)}} &
\ar@{}[d]|{\Downarrow ~ \varepsilon^{-1}} &
\mathbb{X}^{\sf rev} \ar@{<-}[ld]^{\bar{(\_)}^{\sf rev}} \ar@{}[rr]_{\sigma} \ar@{}[rr]^{\Longrightarrow} &
&
\mathbb{X}^{\sf oprev} \ar@{<-}@/^1pc/[llld]^{(\_)^{*^{\sf oprev}}} \ar@/_1pc/[ul]_{(\_)^\mathsf{op}} \\
& & \mathbb{X} & &
}
\]
\[
\mbox{\em (b)}~~~~~~~~~~~~~~ \xymatrix{
& & &
\mathbb{X}^{\mathsf{op}} \ar@{<-}@/_1pc/[llld]_{\bar{(\_)}^\mathsf{op}} \ar@{<-}[dl]_{(\_)^{*^{\mathsf{op}}}} \ar[dr]^{(\_)^{*^{\sf rev}}} \ar@{}[d]|{\Downarrow ~ \epsilon'_\otimes} & &\\
\mathbb{X}^{\sf op rev} \ar@{<-}[rd]_{(\_)^{*^{\sf oprev}}} \ar@{}[rr]_{\sigma^{-1}} \ar@{}[rr]^{\Longrightarrow} & &
\mathbb{X}^{\sf rev} \ar@{=}[rr] \ar@{<-}[dl]^{\bar{(\_)}^{\sf rev}} & \ar@{}[d]|{\Downarrow ~ \varepsilon} &
\mathbb{X}^{\sf rev} \ar[dr]^{\bar{(\_)}} \\
& \mathbb{X} \ar@{=}[rrrr]& & & & \mathbb{X} }
\] where $\sigma: \bar{A}^* \@ifnextchar^ {\t@@}{\t@@^{}} \bar{A^*}$ is given in Definition \ref{Defn: conjugative cyclic}. Below we show that the unit and counit coincide as maps in $\mathbb{X}$.
\[
(a)~~~~~
\begin{tikzpicture}
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\node [style=none] (0) at (-2, 3.5) {};
\node [style=circle, scale=2.5] (1) at (-1, 2.75) {};
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\node [style=none] (3) at (-1, 2) {};
\node [style=none] (4) at (-1, 3.5) {};
\node [style=none] (5) at (-1, -1.25) {};
\node [style=none] (6) at (-2, -1.25) {};
\node [style=circle, scale=2.5] (7) at (-2, -0.5) {};
\node [style=none] (8) at (-2, 0.25) {};
\node [style=none] (9) at (-3, 0.25) {};
\node [style=circle, scale=2.5] (10) at (-3, -2.5) {};
\node [style=none] (11) at (-3, -3.5) {};
\node [style=none] (12) at (-2, -3.5) {};
\node [style=none] (13) at (-2, -3) {};
\node [style=none] (14) at (-1, -3) {};
\node [style=none] (15) at (-1, -5) {};
\node [style=none] (16) at (-2.75, 2) {};
\node [style=none] (17) at (-2.75, 4.5) {};
\node [style=none] (18) at (-1, 2.75) {$\phi^{-1}$};
\node [style=none] (19) at (-2, -0.5) {$\varepsilon^{-1}$};
\node [style=none] (20) at (-3, -2.5) {$\phi$};
\node [style=none] (21) at (-3.25, 4) {$\bar{\left(\bar{X^*}\right)^*}$};
\node [style=none] (22) at (-2.5, 1.5) {$\bar{\epsilon*}$};
\node [style=none] (23) at (-2.25, 2.75) {$\bar{\bar{X^*}}$};
\node [style=none] (24) at (-1.5, 4.25) {$*\eta$};
\node [style=none] (25) at (-0.4, 3.5) {$~^*\!\!\left(\bar{\bar{X^*}}\right)$};
\node [style=none] (26) at (-0.4, 2) {$\left(\bar{\bar{X^*}}\right)^*$};
\node [style=none] (27) at (-1.75, -2) {$\epsilon*$};
\node [style=none] (28) at (-1, -2.5) {$\eta*$};
\node [style=none] (29) at (-2.5, 1) {$\eta*$};
\node [style=none] (30) at (-3.5, -1) {$X^{**}$};
\node [style=none] (31) at (-1.5, -0) {$X^*$};
\node [style=none] (32) at (-1.5, -1.25) {$\bar{\bar{X^*}}$};
\node [style=none] (33) at (-2.5, -4.25) {$*\epsilon$};
\node [style=none] (34) at (-3.75, -3.25) {$~^*(X^*)$};
\node [style=none] (35) at (-2.5, -3.25) {$X^*$};
\node [style=none] (36) at (-0.5, -4.75) {$X$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (14.center) to (15.center);
\draw [bend right=90, looseness=1.50] (14.center) to (13.center);
\draw (13.center) to (12.center);
\draw [bend left=90, looseness=1.50] (12.center) to (11.center);
\draw (10) to (11.center);
\draw (10) to (9.center);
\draw [bend left=90, looseness=1.75] (9.center) to (8.center);
\draw (8.center) to (7);
\draw (7) to (6.center);
\draw [bend right=90, looseness=1.75] (6.center) to (5.center);
\draw (3.center) to (5.center);
\draw (3.center) to (1);
\draw (1) to (4.center);
\draw [bend right=90, looseness=1.75] (4.center) to (0.center);
\draw (0.center) to (2.center);
\draw [bend left=75, looseness=1.50] (2.center) to (16.center);
\draw (16.center) to (17.center);
\end{pgfonlayer}
\end{tikzpicture}\stackrel{{\bf [C.2]}}{=}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 3.5) {};
\node [style=circle, scale=2.5] (1) at (-1, 2.75) {};
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\node [style=none] (3) at (-1, 2) {};
\node [style=none] (4) at (-1, 3.5) {};
\node [style=none] (5) at (-1, -1.25) {};
\node [style=none] (6) at (-2, -1.25) {};
\node [style=circle, scale=2.5] (7) at (-2, -0.5) {};
\node [style=none] (8) at (-2, 0.25) {};
\node [style=none] (9) at (-3, 0.25) {};
\node [style=none] (10) at (-3, -4.5) {};
\node [style=none] (11) at (-4, -4.5) {};
\node [style=none] (12) at (-4, -3) {};
\node [style=none] (13) at (-5, -3) {};
\node [style=none] (14) at (-5, -5.25) {};
\node [style=none] (15) at (-2.75, 2) {};
\node [style=none] (16) at (-2.75, 4.5) {};
\node [style=none] (17) at (-1, 2.75) {$\phi^{-1}$};
\node [style=none] (18) at (-2, -0.5) {$\varepsilon^{-1}$};
\node [style=none] (19) at (-3.25, 4) {$\bar{\left(\bar{X^*}\right)^*}$};
\node [style=none] (20) at (-2.5, 1.5) {$\bar{\epsilon*}$};
\node [style=none] (21) at (-2.25, 2.75) {$\bar{\bar{X^*}}$};
\node [style=none] (22) at (-1.5, 4.25) {$*\eta$};
\node [style=none] (23) at (-0.4, 3.5) {$~^*\!\!\left(\bar{\bar{X^*}}\right)$};
\node [style=none] (24) at (-0.4, 2) {$\left(\bar{\bar{X^*}} \right)^*$};
\node [style=none] (25) at (-1.75, -2) {$\epsilon*$};
\node [style=none] (26) at (-2.5, 1) {$\eta*$};
\node [style=none] (27) at (-2.5, -3.5) {$X^{**}$};
\node [style=none] (28) at (-1.5, -0) {$X^*$};
\node [style=none] (29) at (-1.5, -1.25) {$\bar{\bar{X^*}}$};
\node [style=none] (30) at (-3.5, -5.25) {$\epsilon*$};
\node [style=none] (31) at (-4.5, -4.35) {$X^*$};
\node [style=none] (32) at (-5.25, -4.75) {$X$};
\node [style=circle, scale=2.5] (33) at (-4, -3.75) {};
\node [style=none] (34) at (-4, -3.75) {$\varphi^{-1}$};
\node [style=none] (35) at (-4.5, -2.3) {$*\eta$};
\node [style=none] (36) at (-3.75, -3) {$~^*X$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (13.center) to (14.center);
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\draw [bend right=90, looseness=1.50] (11.center) to (10.center);
\draw [bend left=90, looseness=1.75] (9.center) to (8.center);
\draw (8.center) to (7);
\draw (7) to (6.center);
\draw [bend right=90, looseness=1.75] (6.center) to (5.center);
\draw (3.center) to (5.center);
\draw (3.center) to (1);
\draw (1) to (4.center);
\draw [bend right=90, looseness=1.75] (4.center) to (0.center);
\draw (0.center) to (2.center);
\draw [bend left=75, looseness=1.50] (2.center) to (15.center);
\draw (15.center) to (16.center);
\draw (9.center) to (10.center);
\draw (12.center) to (33);
\draw (33) to (11.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 3.5) {};
\node [style=circle, scale=2.5] (1) at (-1, 2.75) {};
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\node [style=none] (3) at (-1, 2) {};
\node [style=none] (4) at (-1, 3.5) {};
\node [style=none] (5) at (-1, -3.5) {};
\node [style=none] (6) at (-2, -3.5) {};
\node [style=circle, scale=2.5] (7) at (-2, -2.75) {};
\node [style=none] (8) at (-2, -2) {};
\node [style=none] (9) at (-2, -2) {};
\node [style=none] (10) at (-2, -0.5) {};
\node [style=none] (11) at (-3, -0.5) {};
\node [style=none] (12) at (-3, -5.25) {};
\node [style=none] (13) at (-2.75, 2) {};
\node [style=none] (14) at (-2.75, 4.5) {};
\node [style=none] (15) at (-1, 2.75) {$\psi^{-1}$};
\node [style=none] (16) at (-2, -2.75) {$\varepsilon^{-1}$};
\node [style=none] (17) at (-3.25, 4) {$\bar{\left(\bar{X^*}\right)^*}$};
\node [style=none] (18) at (-2.5, 1.5) {$\bar{\epsilon*}$};
\node [style=none] (19) at (-2.25, 2.75) {$\bar{\bar{X^*}}$};
\node [style=none] (20) at (-1.5, 4.25) {$*\eta$};
\node [style=none] (21) at (-0.5, 3.5) {$~^*(\bar{\bar{X^*}})$};
\node [style=none] (22) at (-0.5, 2) {$(\bar{\bar{X^*}})^*$};
\node [style=none] (23) at (-1.75, -4.25) {$\epsilon*$};
\node [style=none] (24) at (-1.5, -2.25) {$X^*$};
\node [style=none] (25) at (-1.5, -3.5) {$\bar{\bar{X^*}}$};
\node [style=none] (26) at (-2.5, -1.75) {$X^*$};
\node [style=none] (27) at (-3.25, -4.75) {$X$};
\node [style=circle, scale=2.5] (28) at (-2, -1.25) {};
\node [style=none] (29) at (-2, -1.25) {$\varphi^{-1}$};
\node [style=none] (30) at (-2.5, 0.25) {$*\eta$};
\node [style=none] (31) at (-1.75, -0.5) {$~^*X$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (11.center) to (12.center);
\draw [bend left=90, looseness=1.50] (11.center) to (10.center);
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\draw (7) to (6.center);
\draw [bend right=90, looseness=1.75] (6.center) to (5.center);
\draw (3.center) to (5.center);
\draw (3.center) to (1);
\draw (1) to (4.center);
\draw [bend right=90, looseness=1.75] (4.center) to (0.center);
\draw (0.center) to (2.center);
\draw [bend left=75, looseness=1.50] (2.center) to (13.center);
\draw (13.center) to (14.center);
\draw (10.center) to (28);
\draw (28) to (9.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 3.5) {};
\node [style=circle, scale=2.5] (1) at (-1, 2.75) {};
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\node [style=none] (4) at (-1, 3.5) {};
\node [style=none] (5) at (-1, -3.5) {};
\node [style=none] (6) at (-3, -3.5) {};
\node [style=none] (7) at (-3, -2) {};
\node [style=none] (8) at (-3, -2) {};
\node [style=none] (9) at (-3, -0.5) {};
\node [style=none] (10) at (-4, -0.5) {};
\node [style=none] (11) at (-4, -5.25) {};
\node [style=none] (12) at (-2.75, 2) {};
\node [style=none] (13) at (-2.75, 4.5) {};
\node [style=none] (14) at (-1, 2.75) {$\psi^{-1}$};
\node [style=none] (15) at (-1, -1) {$\varepsilon^{-1^*}$};
\node [style=none] (16) at (-3.25, 4) {$\bar{\left(\bar{X^*}\right)^*}$};
\node [style=none] (17) at (-2.5, 1.5) {$\bar{\epsilon*}$};
\node [style=none] (18) at (-2.25, 2.75) {$\bar{\bar{X^*}}$};
\node [style=none] (19) at (-1.5, 4.25) {$*\eta$};
\node [style=none] (20) at (-0.5, 3.5) {$~^*(\bar{\bar{X^*}})$};
\node [style=none] (21) at (-0.5, -0) {$(\bar{\bar{X^*}})^*$};
\node [style=none] (22) at (-2, -4.75) {$\epsilon*$};
\node [style=none] (23) at (-2.5, -2) {$X^*$};
\node [style=none] (24) at (-4.25, -4.75) {$X$};
\node [style=circle, scale=2.5] (25) at (-3, -1.25) {};
\node [style=none] (26) at (-3, -1.25) {$\varphi^{-1}$};
\node [style=none] (27) at (-3.5, 0.25) {$*\eta$};
\node [style=none] (28) at (-2.75, -0.5) {$~^*X$};
\node [style=circle, scale=2.5] (29) at (-1, -1) {};
\node [style=none] (30) at (-0.5, -2) {$X^{**}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
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\draw (3.center) to (1);
\draw (1) to (4.center);
\draw [bend right=90, looseness=1.75] (4.center) to (0.center);
\draw (0.center) to (2.center);
\draw [bend left=75, looseness=1.50] (2.center) to (12.center);
\draw (12.center) to (13.center);
\draw (9.center) to (25);
\draw (25) to (8.center);
\draw (7.center) to (6.center);
\draw (3.center) to (29);
\draw (29) to (5.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
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\node [style=none] (0) at (-2, 3.5) {};
\node [style=circle, scale=2.5] (1) at (-1, 2.5) {};
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\node [style=none] (12) at (-2.75, 2) {};
\node [style=none] (13) at (-2.75, 4.5) {};
\node [style=none] (14) at (-3.25, 4) {$\bar{\left(\bar{X^*}\right)^*}$};
\node [style=none] (15) at (-2.5, 1.5) {$\bar{\epsilon*}$};
\node [style=none] (16) at (-2.25, 2.75) {$\bar{\bar{X^*}}$};
\node [style=none] (17) at (-1.5, 4.25) {$*\eta$};
\node [style=none] (18) at (-0.5, 3.5) {$~^*(\bar{\bar{X^*}})$};
\node [style=none] (19) at (-2, -4.75) {$\epsilon*$};
\node [style=none] (20) at (-2.5, -2) {$X^*$};
\node [style=none] (21) at (-4.25, -4.75) {$X$};
\node [style=circle, scale=2.5] (22) at (-3, -1.25) {};
\node [style=none] (23) at (-3, -1.25) {$\psi^{-1}$};
\node [style=none] (24) at (-3.5, 0.25) {$*\eta$};
\node [style=none] (25) at (-2.75, -0.5) {$~^*X$};
\node [style=circle, scale=2.5] (26) at (-1, -1) {};
\node [style=none] (27) at (-1, -1) {$\psi^{-1}$};
\node [style=none] (28) at (-1, 2.5) {$~^*\varepsilon^{-1}$};
\node [style=none] (29) at (-0.5, 0.75) {$~^*(X^*)$};
\node [style=none] (30) at (-0.5, -2.5) {$X^{**}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (10.center) to (11.center);
\draw [bend left=90, looseness=1.50] (10.center) to (9.center);
\draw [bend right=90, looseness=1.75] (6.center) to (5.center);
\draw (3.center) to (1);
\draw (1) to (4.center);
\draw [bend right=90, looseness=1.75] (4.center) to (0.center);
\draw (0.center) to (2.center);
\draw [bend left=75, looseness=1.50] (2.center) to (12.center);
\draw (12.center) to (13.center);
\draw (9.center) to (22);
\draw (22) to (8.center);
\draw (7.center) to (6.center);
\draw (3.center) to (26);
\draw (26) to (5.center);
\end{pgfonlayer}
\end{tikzpicture} =
\]
\[
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\node [style=none] (0) at (-3, 3.5) {};
\node [style=none] (1) at (-3, 2) {};
\node [style=none] (2) at (0.5, 0.25) {};
\node [style=none] (3) at (-1, 3.5) {};
\node [style=none] (4) at (0.5, -1.75) {};
\node [style=none] (5) at (-3, -1.75) {};
\node [style=none] (6) at (-3, -2) {};
\node [style=none] (7) at (-3, -2) {};
\node [style=none] (8) at (-3, -0.5) {};
\node [style=none] (9) at (-4, -0.5) {};
\node [style=none] (10) at (-4, -5.25) {};
\node [style=none] (11) at (-3.75, 2) {};
\node [style=none] (12) at (-3.75, 4.5) {};
\node [style=none] (13) at (-4.25, 4) {$\bar{\left(\bar{X^*}\right)^*}$};
\node [style=none] (14) at (-3.5, 1.5) {$\bar{\epsilon*}$};
\node [style=none] (15) at (-3.25, 2.75) {$\bar{\bar{X^*}}$};
\node [style=none] (16) at (-2, 4.75) {$*\eta$};
\node [style=none] (17) at (-0.5, 3.5) {$~^*(\bar{\bar{X^*}})$};
\node [style=none] (18) at (-1, -4) {$\epsilon*$};
\node [style=none] (19) at (-2.5, -2) {$X^*$};
\node [style=none] (20) at (-4.25, -4.75) {$X$};
\node [style=circle, scale=2.5] (21) at (-3, -1.25) {};
\node [style=none] (22) at (-3, -1.25) {$\psi^{-1}$};
\node [style=none] (23) at (-3.5, 0.25) {$*\eta$};
\node [style=none] (24) at (-2.75, -0.5) {$~^*X$};
\node [style=circle, scale=2.5] (25) at (0.5, -1) {};
\node [style=none] (26) at (0.5, -1) {$\psi^{-1}$};
\node [style=none] (27) at (0.75, -0.25) {$~^*(X^*)$};
\node [style=none] (28) at (1, -2) {$X^{**}$};
\node [style=circle, scale=2.5] (29) at (-0.25, 1.75) {};
\node [style=none] (30) at (-0.25, 2.5) {};
\node [style=none] (31) at (-0.25, 1) {};
\node [style=none] (32) at (-1, 1) {};
\node [style=none] (33) at (0.5, 2.5) {};
\node [style=none] (34) at (0.5, 0.25) {};
\node [style=none] (35) at (-1, 3.5) {};
\node [style=none] (36) at (-0.25, 1.75) {$\varepsilon^{-1}$};
\node [style=none] (37) at (0.25, 3) {$~^*\eta$};
\node [style=none] (38) at (-0.75, 0.5) {$~^*\epsilon$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (9.center) to (10.center);
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\draw [bend right=90, looseness=1.75] (3.center) to (0.center);
\draw (0.center) to (1.center);
\draw [bend left=75, looseness=1.50] (1.center) to (11.center);
\draw (11.center) to (12.center);
\draw (8.center) to (21);
\draw (21) to (7.center);
\draw (6.center) to (5.center);
\draw (2.center) to (25);
\draw (25) to (4.center);
\draw (29) to (31.center);
\draw [bend left=90, looseness=1.25] (31.center) to (32.center);
\draw (32.center) to (35.center);
\draw (30.center) to (29);
\draw [bend left=90, looseness=1.50] (30.center) to (33.center);
\draw (33.center) to (34.center);
\end{pgfonlayer}
\end{tikzpicture}
=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5, 0.25) {};
\node [style=none] (1) at (0.5, -1.75) {};
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\node [style=none] (3) at (-3, -2) {};
\node [style=none] (4) at (-3, -2) {};
\node [style=none] (5) at (-3, -0.5) {};
\node [style=none] (6) at (-4, -0.5) {};
\node [style=none] (7) at (-4, -5.25) {};
\node [style=none] (8) at (-1.5, 2.75) {$\bar{\left(\bar{X^*}\right)^*}$};
\node [style=none] (9) at (-1, -4) {$\epsilon*$};
\node [style=none] (10) at (-2.5, -2) {$X^*$};
\node [style=none] (11) at (-4.25, -4.75) {$X$};
\node [style=circle, scale=2.5] (12) at (-3, -1.25) {};
\node [style=none] (13) at (-3, -1.25) {$\psi^{-1}$};
\node [style=none] (14) at (-3.5, 0.25) {$*\eta$};
\node [style=none] (15) at (-2.75, -0.5) {$~^*X$};
\node [style=circle, scale=2.5] (16) at (0.5, -1) {};
\node [style=none] (17) at (0.5, -1) {$\psi^{-1}$};
\node [style=none] (18) at (1, -0.25) {$~^*(X^*)$};
\node [style=none] (19) at (1, -2) {$X^{**}$};
\node [style=circle, scale=2.5] (20) at (-0.25, 1.75) {};
\node [style=none] (21) at (-0.25, 2.5) {};
\node [style=none] (22) at (-0.25, 1) {};
\node [style=none] (23) at (-1, 1) {};
\node [style=none] (24) at (0.5, 2.5) {};
\node [style=none] (25) at (0.5, 0.25) {};
\node [style=none] (26) at (-1, 3.5) {};
\node [style=none] (27) at (-0.25, 1.75) {$\varepsilon^{-1}$};
\node [style=none] (28) at (0.25, 3) {$~^*\eta$};
\node [style=none] (29) at (-0.75, 0.5) {$\bar{\epsilon*}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (6.center) to (7.center);
\draw [bend left=90, looseness=1.50] (6.center) to (5.center);
\draw [bend right=90, looseness=1.75] (2.center) to (1.center);
\draw (5.center) to (12);
\draw (12) to (4.center);
\draw (3.center) to (2.center);
\draw (0.center) to (16);
\draw (16) to (1.center);
\draw (20) to (22.center);
\draw [bend left=90, looseness=1.25] (22.center) to (23.center);
\draw (23.center) to (26.center);
\draw (21.center) to (20);
\draw [bend left=90, looseness=1.50] (21.center) to (24.center);
\draw (24.center) to (25.center);
\end{pgfonlayer}
\end{tikzpicture} \stackrel{{\bf \small [C.2](inv)}}{=}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5, -0) {};
\node [style=none] (1) at (0.5, -2) {};
\node [style=none] (2) at (1.5, -2) {};
\node [style=none] (3) at (1.5, -0.5) {};
\node [style=none] (4) at (2.5, -0.5) {};
\node [style=none] (5) at (2.5, -5.25) {};
\node [style=none] (6) at (-1.5, 2.75) {$\bar{\left(\bar{X^*}\right)^*}$};
\node [style=none] (7) at (2.75, -4.75) {$X$};
\node [style=circle, scale=2.5] (8) at (-0.25, 1.75) {};
\node [style=none] (9) at (-0.25, 2.5) {};
\node [style=none] (10) at (-0.25, 1) {};
\node [style=none] (11) at (-1, 1) {};
\node [style=none] (12) at (0.5, 2.5) {};
\node [style=none] (13) at (0.5, -0) {};
\node [style=none] (14) at (-1, 3.5) {};
\node [style=none] (15) at (-0.25, 1.75) {$\varepsilon$};
\node [style=none] (16) at (0.25, 3) {$*\eta$};
\node [style=none] (17) at (-0.75, 0.5) {$\bar{\epsilon*}$};
\node [style=none] (18) at (-0.2, -1) {$~^*(X^*)$};
\node [style=none] (19) at (-0.5, 2.25) {};
\node [style=none] (20) at (1, -2.75) {$*\epsilon$};
\node [style=none] (21) at (1.85, -1) {$X^*$};
\node [style=none] (22) at (2, 0.25) {$\eta*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4.center) to (5.center);
\draw [bend right=90, looseness=1.50] (4.center) to (3.center);
\draw [bend left=90, looseness=1.75] (2.center) to (1.center);
\draw (8) to (10.center);
\draw [bend left=90, looseness=1.25] (10.center) to (11.center);
\draw (11.center) to (14.center);
\draw (9.center) to (8);
\draw [bend left=90, looseness=1.50] (9.center) to (12.center);
\draw (12.center) to (13.center);
\draw (0.center) to (1.center);
\draw (2.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (1.25, 0.5) {};
\node [style=none] (1) at (1.25, 2) {};
\node [style=none] (2) at (2.5, 2) {};
\node [style=none] (3) at (2.5, -5.25) {};
\node [style=none] (4) at (-0.75, 2.75) {$\bar{\left(\bar{X^*}\right)^*}$};
\node [style=none] (5) at (2.75, -4.75) {$X$};
\node [style=circle, scale=2.5] (6) at (1.25, -0.25) {};
\node [style=none] (7) at (1.25, 0.5) {};
\node [style=none] (8) at (1.25, -1) {};
\node [style=none] (9) at (0, -1) {};
\node [style=none] (10) at (0, 3.5) {};
\node [style=none] (11) at (1.25, -0.25) {$\varepsilon^{-1}$};
\node [style=none] (12) at (0.75, -1.75) {$\bar{\epsilon*}$};
\node [style=none] (13) at (1, 0.25) {};
\node [style=none] (14) at (1.5, 1.5) {$X^*$};
\node [style=none] (15) at (1.75, 2.75) {$\eta*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (3.center);
\draw [bend right=90, looseness=1.50] (2.center) to (1.center);
\draw (6) to (8.center);
\draw [bend left=90, looseness=1.25] (8.center) to (9.center);
\draw (9.center) to (10.center);
\draw (7.center) to (6);
\draw (0.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture} =: \iota^{-1}
\]
\[
(b)~~~~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 4.75) {};
\node [style=none] (1) at (0, -0.5) {};
\node [style=none] (2) at (-0.75, -0.5) {};
\node [style=none] (3) at (-0.75, 2) {};
\node [style=none] (4) at (-1.5, 2) {};
\node [style=none] (5) at (-1.5, -0) {};
\node [style=none] (6) at (-2.5, -0) {};
\node [style=circle, scale=2.5] (7) at (-2.5, 1) {};
\node [style=none] (8) at (-2.5, 2.5) {};
\node [style=none] (9) at (-3.75, 2.5) {};
\node [style=none] (10) at (-3.75, -2.75) {};
\node [style=none] (11) at (-2.5, 1) {$\psi$};
\node [style=none] (12) at (-0.5, -1.25) {$\epsilon*$};
\node [style=none] (13) at (-1.25, 2.75) {$\bar{\eta*}$};
\node [style=none] (14) at (-2, -0.75) {$*\epsilon$};
\node [style=none] (15) at (-3, 3.5) {$\eta*$};
\node [style=circle, scale=2.5] (16) at (-3.75, -3.75) {};
\node [style=none] (17) at (-3.75, -4.75) {};
\node [style=none] (18) at (-2.5, -4.75) {};
\node [style=none] (19) at (-2.5, -3.5) {};
\node [style=none] (20) at (-1.25, -3.5) {};
\node [style=circle, scale=2.5] (21) at (-1.25, -6.75) {};
\node [style=none] (22) at (-1.25, -8) {};
\node [style=none] (23) at (-3.75, -3.75) {$\psi$};
\node [style=none] (24) at (-1.25, -6.75) {$\varepsilon$};
\node [style=none] (25) at (-2, -2.75) {$\eta*$};
\node [style=none] (26) at (-3, -5.5) {$*\epsilon$};
\node [style=none] (27) at (-4.75, 3.75) {};
\node [style=none] (28) at (1.25, 3.75) {};
\node [style=none] (29) at (-4.75, -1.5) {};
\node [style=none] (30) at (1.25, -1.5) {};
\node [style=none] (31) at (-5, -2.25) {};
\node [style=none] (32) at (-0.25, -2.25) {};
\node [style=none] (33) at (-0.25, -6) {};
\node [style=none] (34) at (-5, -6) {};
\node [style=none] (35) at (0.5, 4.5) {$\bar{\left( \bar{X^*} \right)^*}$};
\node [style=none] (36) at (-2.75, -2) {$\bar{\left(\bar{X} \right)^{**}}$};
\node [style=none] (37) at (-0.75, -7.5) {$X$};
\node [style=none] (38) at (-0.75, -5.25) {$\bar{X}$};
\node [style=none] (39) at (-2, -4) {$\bar{X}^*$};
\node [style=none] (40) at (-4.25, -4.5) {$~^*(\bar{X}^*)$};
\node [style=none] (41) at (-4.25, -3) {$\bar{X}^{**}$};
\node [style=none] (42) at (-4, 2) {$\bar{X}^{**}$};
\node [style=none] (43) at (0.5, 2.5) {$\bar{X^*}^*$};
\node [style=none] (44) at (-1.25, 1) {$\bar{X^*}$};
\node [style=none] (45) at (-3, 0.25) {$~^*\left(\bar{X^*} \right)$};
\node [style=none] (46) at (-3, 2) {$\left(\bar{X^*} \right)^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend right=90, looseness=1.50] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw [bend left=90, looseness=1.25] (5.center) to (6.center);
\draw (6.center) to (7);
\draw (7) to (8.center);
\draw [bend right=90, looseness=1.75] (8.center) to (9.center);
\draw (9.center) to (10.center);
\draw (10.center) to (16);
\draw (16) to (17.center);
\draw [bend right=90, looseness=1.50] (17.center) to (18.center);
\draw (18.center) to (19.center);
\draw [bend left=90, looseness=1.50] (19.center) to (20.center);
\draw (20.center) to (21);
\draw (21) to (22.center);
\draw (27.center) to (28.center);
\draw (28.center) to (30.center);
\draw (30.center) to (29.center);
\draw (29.center) to (27.center);
\draw (31.center) to (34.center);
\draw (34.center) to (33.center);
\draw (32.center) to (33.center);
\draw (32.center) to (31.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 4.75) {};
\node [style=none] (1) at (0, -0.5) {};
\node [style=none] (2) at (-0.75, -0.5) {};
\node [style=none] (3) at (-0.75, 2) {};
\node [style=none] (4) at (-1.5, 2) {};
\node [style=none] (5) at (-1.5, -0) {};
\node [style=none] (6) at (-2.5, -0) {};
\node [style=circle, scale=2.5] (7) at (-2.5, 1) {};
\node [style=none] (8) at (-2.5, 2.5) {};
\node [style=none] (9) at (-3.75, 2.5) {};
\node [style=none] (10) at (-3.75, -2.75) {};
\node [style=none] (11) at (-2.5, 1) {$\psi$};
\node [style=none] (12) at (-0.5, -1.25) {$\epsilon*$};
\node [style=none] (13) at (-1.25, 2.75) {$\bar{\eta*}$};
\node [style=none] (14) at (-2, -0.75) {$*\epsilon$};
\node [style=none] (15) at (-3, 3.5) {$\eta*$};
\node [style=circle, scale=2.5] (16) at (-3.75, -3.75) {};
\node [style=none] (17) at (-3.75, -4.75) {};
\node [style=none] (18) at (-2.5, -4.75) {};
\node [style=none] (19) at (-2.5, -3.5) {};
\node [style=none] (20) at (-1.25, -3.5) {};
\node [style=circle, scale=2.5] (21) at (-1.25, -6.75) {};
\node [style=none] (22) at (-1.25, -8) {};
\node [style=none] (23) at (-3.75, -3.75) {$\psi$};
\node [style=none] (24) at (-1.25, -6.75) {$\varepsilon$};
\node [style=none] (25) at (-2, -2.75) {$\eta*$};
\node [style=none] (26) at (-3, -5.5) {$*\epsilon$};
\node [style=none] (27) at (-5.5, 3.75) {};
\node [style=none] (28) at (1.25, 3.75) {};
\node [style=none] (29) at (1.25, 3.75) {};
\node [style=none] (30) at (1.25, -6) {};
\node [style=none] (31) at (-5.5, -6) {};
\node [style=none] (32) at (0.5, 4.5) {$\bar{\left( \bar{X^*} \right)^*}$};
\node [style=none] (33) at (-2.75, -2) {$\bar{\left(\bar{X} \right)^{**}}$};
\node [style=none] (34) at (-0.75, -7.5) {$X$};
\node [style=none] (35) at (-0.75, -5.25) {$\bar{X}$};
\node [style=none] (36) at (-2, -4) {$\left(\bar{X}\right)^*$};
\node [style=none] (37) at (-4.5, -4.5) {$~^*(\left(\bar{X}\right)^*)$};
\node [style=none] (38) at (-4.5, -3) {$(\bar{X}^{**}$};
\node [style=none] (39) at (-4, 2) {$\bar{X}^{**}$};
\node [style=none] (40) at (0.5, 2.5) {$\bar{X^*}^*$};
\node [style=none] (41) at (-1.25, 1) {$\bar{X^*}$};
\node [style=none] (42) at (-3, 0.25) {$~^*\left(\bar{X^*} \right)$};
\node [style=none] (43) at (-3, 2) {$\left(\bar{X^*} \right)^*$};
\node [style=none] (44) at (0.75, -5.5) {$\bar{(\_)}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend right=90, looseness=1.50] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw [bend left=90, looseness=1.25] (5.center) to (6.center);
\draw (6.center) to (7);
\draw (7) to (8.center);
\draw [bend right=90, looseness=1.75] (8.center) to (9.center);
\draw (9.center) to (10.center);
\draw (10.center) to (16);
\draw (16) to (17.center);
\draw [bend right=90, looseness=1.50] (17.center) to (18.center);
\draw (18.center) to (19.center);
\draw [bend left=90, looseness=1.50] (19.center) to (20.center);
\draw (20.center) to (21);
\draw (21) to (22.center);
\draw (27.center) to (28.center);
\draw (31.center) to (30.center);
\draw (29.center) to (30.center);
\draw (27.center) to (31.center);
\end{pgfonlayer}
\end{tikzpicture} \stackrel{{\bf [C.2]}}{=}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 4.75) {};
\node [style=none] (1) at (0, -0.5) {};
\node [style=none] (2) at (-0.75, -0.5) {};
\node [style=none] (3) at (-0.75, 1.25) {};
\node [style=none] (4) at (-4.5, 1.25) {};
\node [style=none] (5) at (-4.5, -0.25) {};
\node [style=none] (6) at (-1.75, -4.75) {};
\node [style=none] (7) at (-1.75, -3.5) {};
\node [style=none] (8) at (-0.5, -3.5) {};
\node [style=circle, scale=2.5] (9) at (-0.5, -6.75) {};
\node [style=none] (10) at (-0.5, -8) {};
\node [style=none] (11) at (-0.5, -6.75) {$\varepsilon$};
\node [style=none] (12) at (-1.25, -2.75) {$\eta*$};
\node [style=none] (13) at (-5, 3.75) {};
\node [style=none] (14) at (1.25, 3.75) {};
\node [style=none] (15) at (1.25, 3.75) {};
\node [style=none] (16) at (1.25, -6) {};
\node [style=none] (17) at (-5, -6) {};
\node [style=none] (18) at (0.5, 4.5) {$\bar{\left( \bar{X^*} \right)^*}$};
\node [style=none] (19) at (0, -7.5) {$X$};
\node [style=none] (20) at (0, -5.25) {$\bar{X}$};
\node [style=none] (21) at (-1.25, -4) {$\left(\bar{X}\right)^*$};
\node [style=none] (22) at (0.5, 2.5) {$\bar{X^*}^*$};
\node [style=none] (23) at (0.75, -5.5) {$\bar{(\_)}$};
\node [style=none] (24) at (-2.25, -4.75) {};
\node [style=none] (25) at (-2.25, -1.25) {};
\node [style=none] (26) at (-3.25, -1.25) {};
\node [style=none] (27) at (-3.25, -2.5) {};
\node [style=none] (28) at (-4.5, -2.5) {};
\node [style=none] (29) at (-0.5, -1.25) {$\epsilon*$};
\node [style=none] (30) at (-2.5, 3.25) {$\bar{\eta*}$};
\node [style=none] (31) at (-4, -3.75) {$\epsilon*$};
\node [style=none] (32) at (-2.75, -0.75) {$*\eta$};
\node [style=none] (33) at (-2, -5.25) {$*\epsilon$};
\node [style=none] (34) at (-1.25, 0.75) {$\bar{X^*}$};
\node [style=none] (35) at (-4, 1) {$\bar{X}$};
\node [style=none] (36) at (-3.5, -2) {$\bar{X}^*$};
\node [style=none] (37) at (-1.75, -2.25) {$~^*(\bar{X}^*)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend right=90, looseness=1.50] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (6.center) to (7.center);
\draw [bend left=90, looseness=1.50] (7.center) to (8.center);
\draw (8.center) to (9);
\draw (9) to (10.center);
\draw (13.center) to (14.center);
\draw (17.center) to (16.center);
\draw (15.center) to (16.center);
\draw (13.center) to (17.center);
\draw [bend left=90, looseness=1.75] (6.center) to (24.center);
\draw (24.center) to (25.center);
\draw (27.center) to (26.center);
\draw [bend left=90, looseness=1.25] (26.center) to (25.center);
\draw [bend right=90, looseness=2.50] (28.center) to (27.center);
\draw (5.center) to (28.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 4.75) {};
\node [style=none] (1) at (0, -0.5) {};
\node [style=none] (2) at (-0.75, -0.5) {};
\node [style=none] (3) at (-0.75, 1.25) {};
\node [style=none] (4) at (-4.5, 1.25) {};
\node [style=none] (5) at (-4.5, -0.25) {};
\node [style=circle, scale=2.5] (6) at (-2.25, -6.75) {};
\node [style=none] (7) at (-2.25, -8) {};
\node [style=none] (8) at (-2.25, -6.75) {$\varepsilon$};
\node [style=none] (9) at (-5, 3.75) {};
\node [style=none] (10) at (1.25, 3.75) {};
\node [style=none] (11) at (1.25, 3.75) {};
\node [style=none] (12) at (1.25, -6) {};
\node [style=none] (13) at (-5, -6) {};
\node [style=none] (14) at (0.5, 4.5) {$\bar{\left( \bar{X^*} \right)^*}$};
\node [style=none] (15) at (-1.75, -7.5) {$X$};
\node [style=none] (16) at (0.5, 2.5) {$\bar{X^*}^*$};
\node [style=none] (17) at (0.75, -5.5) {$\bar{(\_)}$};
\node [style=none] (18) at (-2.25, -1.25) {};
\node [style=none] (19) at (-3.25, -1.25) {};
\node [style=none] (20) at (-3.25, -2.5) {};
\node [style=none] (21) at (-4.5, -2.5) {};
\node [style=none] (22) at (-0.5, -1.25) {$\epsilon*$};
\node [style=none] (23) at (-2.5, 3.25) {$\bar{\eta*}$};
\node [style=none] (24) at (-4, -3.75) {$\epsilon*$};
\node [style=none] (25) at (-2.75, -0.75) {$\eta*$};
\node [style=none] (26) at (-1.25, 0.75) {$\bar{X^*}$};
\node [style=none] (27) at (-4, 1) {$\bar{X}$};
\node [style=none] (28) at (-3.5, -2) {$\bar{X}^*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend right=90, looseness=1.50] (3.center) to (4.center);
\draw (4.center) to (5.center);
\draw (6) to (7.center);
\draw (9.center) to (10.center);
\draw (13.center) to (12.center);
\draw (11.center) to (12.center);
\draw (9.center) to (13.center);
\draw (20.center) to (19.center);
\draw [bend left=90, looseness=1.25] (19.center) to (18.center);
\draw [bend right=90, looseness=2.50] (21.center) to (20.center);
\draw (5.center) to (21.center);
\draw (18.center) to (6);
\end{pgfonlayer}
\end{tikzpicture}= \]
\
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 4.75) {};
\node [style=none] (1) at (0, -0.5) {};
\node [style=none] (2) at (-0.75, -0.5) {};
\node [style=none] (3) at (-0.75, 2) {};
\node [style=none] (4) at (-2.25, 2) {};
\node [style=circle, scale=2.5] (5) at (-2.25, -2.75) {};
\node [style=none] (6) at (-2.25, -4) {};
\node [style=none] (7) at (-2.25, -2.75) {$\varepsilon$};
\node [style=none] (8) at (-5, 3.75) {};
\node [style=none] (9) at (1.25, 3.75) {};
\node [style=none] (10) at (1.25, 3.75) {};
\node [style=none] (11) at (1.25, -2) {};
\node [style=none] (12) at (-5, -2) {};
\node [style=none] (13) at (0.75, 4.5) {$\bar{\left( \bar{X^*} \right)^*}$};
\node [style=none] (14) at (-1.75, -3.5) {$X$};
\node [style=none] (15) at (0.5, 2.5) {$\bar{X^*}^*$};
\node [style=none] (16) at (0.7499999, -1.5) {$\bar{(\_)}$};
\node [style=none] (17) at (-2.25, -1.25) {};
\node [style=none] (18) at (-0.5, -1.25) {$\epsilon*$};
\node [style=none] (19) at (-1.5, 3) {$\bar{\eta*}$};
\node [style=none] (20) at (-1.25, 0.75) {$\bar{X^*}$};
\node [style=none] (21) at (-1.75, 1.75) {$\bar{X}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend right=90, looseness=1.50] (3.center) to (4.center);
\draw (5) to (6.center);
\draw (8.center) to (9.center);
\draw (12.center) to (11.center);
\draw (10.center) to (11.center);
\draw (8.center) to (12.center);
\draw (17.center) to (5);
\draw (4.center) to (17.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.5, 4.75) {};
\node [style=none] (1) at (-2.5, -0.5) {};
\node [style=none] (2) at (-1, -0.5) {};
\node [style=none] (3) at (-1, 2) {};
\node [style=none] (4) at (0.5, 2) {};
\node [style=circle, scale=2.5] (5) at (0.4999999, -1.25) {};
\node [style=none] (6) at (0.4999999, -2.5) {};
\node [style=none] (7) at (0.4999999, -1.25) {$\varepsilon$};
\node [style=none] (8) at (-3, 4.5) {$\bar{\left( \bar{X^*} \right)^*}$};
\node [style=none] (9) at (0.9999999, -2) {$X$};
\node [style=none] (10) at (0.5, -1.25) {};
\node [style=none] (11) at (-1.75, -1.75) {$\bar{\epsilon*}$};
\node [style=none] (12) at (-0.25, 3) {$\bar{\bar{\eta*}}$};
\node [style=none] (13) at (-1.5, 1) {$\bar{\bar{X^*}}$};
\node [style=none] (14) at (1, 1) {$\bar{\bar{X}}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend right=90, looseness=2.00] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend left=90, looseness=1.50] (3.center) to (4.center);
\draw (5) to (6.center);
\draw (10.center) to (5);
\draw (4.center) to (10.center);
\end{pgfonlayer}
\end{tikzpicture} \stackrel{\ref{Lemma: varepsi monoidal}}{=}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (1.25, 0.5) {};
\node [style=none] (1) at (1.25, 2) {};
\node [style=none] (2) at (2.5, 2) {};
\node [style=none] (3) at (2.5, -2.5) {};
\node [style=none] (4) at (-0.75, 2.75) {$\bar{\bar{X^*}^*}$};
\node [style=none] (5) at (2.75, -2) {$X$};
\node [style=circle, scale=2.5] (6) at (1.25, -0.25) {};
\node [style=none] (7) at (1.25, 0.5) {};
\node [style=none] (8) at (1.25, -1) {};
\node [style=none] (9) at (0, -1) {};
\node [style=none] (10) at (0, 3.5) {};
\node [style=none] (11) at (1.25, -0.25) {$\varepsilon^{-1}$};
\node [style=none] (12) at (0.75, -1.75) {$\bar{\epsilon*}$};
\node [style=none] (13) at (1, 0.25) {};
\node [style=none] (14) at (1.5, 1.5) {$X^*$};
\node [style=none] (15) at (1.75, 2.75) {$\eta*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (3.center);
\draw [bend right=90, looseness=1.50] (2.center) to (1.center);
\draw (6) to (8.center);
\draw [bend left=90, looseness=1.25] (8.center) to (9.center);
\draw (9.center) to (10.center);
\draw (7.center) to (6);
\draw (0.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture} =: \iota^{-1}
\]
\end{proof}
\begin{theorem}
Every cyclic, $\dagger$-$*$-autonomous category is a conjugative $*$-autonomous category.
\end{theorem}
\begin{proof}
Let $\mathbb{X}$ be a cyclic, $\dagger$-$*$-autonomous category then composing adjoints gives the equivalence $(\_)^{\dagger^*} \dashv (\_)^{*^\dagger}$. To build a conjugation, however,
we need an equivalence between the same functors: to obtain such an equivalence we use the natural equivalence $\omega: (\_)^{*\dagger} \@ifnextchar^ {\t@@}{\t@@^{}} (\_)^{\dagger*}$ from the cyclor preserving condition for Frobenius linear functors. A conjugative equivalence, in addition, requires that the unit and counit of the equivalence be inverses of each of other.
The unit and counit of the equivalence are given by {\em (a)} and {\em (b)} respectively;
\[ \mbox{\em (a)}~~~~~~~~~ \xymatrix{
\mathbb{X}^{\sf rev} \ar@{=}[rrrr] \ar[dr]_{(\_)^{*^\mathsf{op}}} &
&
\ar@{}[d]|{\Downarrow ~ \eta_\otimes'}&
&
\mathbb{X}^{\sf rev} \ar@{<-}[ld]^{(\_)^{*^\mathsf{op}}}
& \\
&
\mathbb{X}^{\mathsf{op}} \ar@{=}[rr] \ar[dr]_{\dagger} &
\ar@{}[d]|{\Downarrow ~ \iota^{-1}} &
\mathbb{X}^{\sf op} \ar@{<-}[ld]^{\dagger^\mathsf{op}} \ar@{}[rr]_{\omega} \ar@{}[rr]^{\Longrightarrow} &
&
\mathbb{X}^{\sf oprev} \ar@{<-}@/^1pc/[llld]^{(\_)^{*^{\sf oprev}}} \ar@/_1pc/[ul]_{\dagger^\mathsf{rev}} \\
& & \mathbb{X} & &
}
\]
\[
\mbox{\em (b)}~~~~~~~~~~~~~~ \xymatrix{
& & &
\mathbb{X}^{\mathsf{rev}} \ar@{<-}@/_1pc/[llld]_{\dagger^\mathsf{rev}} \ar@{<-}[dl]_{(\_)^{*^{\mathsf{rev}}}} \ar[dr]^{(\_)^{*^{\sf op}}} \ar@{}[d]|{\Downarrow ~ \epsilon'_\otimes} & &\\
\mathbb{X}^{\sf op rev} \ar@{<-}[rd]_{(\_)^{*^{\sf oprev}}} \ar@{}[rr]_{\omega^{-1}} \ar@{}[rr]^{\Longrightarrow} & &
\mathbb{X}^{\sf op} \ar@{=}[rr] \ar@{<-}[dl]^{\dagger^\mathsf{op}} & \ar@{}[d]|{\Downarrow ~ \iota^{-1}} &
\mathbb{X}^{\sf op} \ar[dr]^{\dagger} \\
& \mathbb{X} \ar@{=}[rrrr]& & & & \mathbb{X} }
\] where the isomorphism $\omega: (\_)^{\dagger*} \@ifnextchar^ {\t@@}{\t@@^{}} (\_)^{*\dagger}$ is from the cyclor preserving condition for Frobenius linear functors.
\[
\omega := \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 5) {};
\node [style=none] (1) at (0, 3) {};
\node [style=none] (2) at (-1, 3) {};
\node [style=none] (3) at (-1, 4) {};
\node [style=none] (4) at (-2, 4) {};
\node [style=circle, scale=2.5] (5) at (-2, 2.25) {};
\node [style=none] (6) at (-2, 1) {};
\node [style=none] (7) at (-2, 2.25) {$\psi^\dagger$};
\node [style=none] (8) at (0.5, 4.5) {$X^{\dagger*}$};
\node [style=none] (9) at (-2.5, 1.5) {$X^{*\dagger}$};
\node [style=none] (10) at (-0.5, 2.25) {$\epsilon*$};
\node [style=none] (11) at (-1.5, 4.75) {$(*\epsilon)^\dagger$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend right=90, looseness=1.75] (3.center) to (4.center);
\draw (4.center) to (5);
\draw (5) to (6.center);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~~~~~~~ \omega^{-1} := \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 1) {};
\node [style=none] (1) at (-2, 3) {};
\node [style=none] (2) at (-1, 3) {};
\node [style=none] (3) at (-1, 2) {};
\node [style=none] (4) at (0, 2) {};
\node [style=circle, scale=2.5] (5) at (0, 3.75) {};
\node [style=none] (6) at (0, 5) {};
\node [style=none] (7) at (0, 3.75) {$\psi^\dagger$};
\node [style=none] (8) at (-2.5, 1.5) {$X^{\dagger*}$};
\node [style=none] (9) at (0.5, 4.5) {$X^{*\dagger}$};
\node [style=none] (10) at (-1.5, 3.75) {$\eta*$};
\node [style=none] (11) at (-0.5, 1.25) {$(*\eta^{-1})^\dagger$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend right=90, looseness=1.75] (3.center) to (4.center);
\draw (4.center) to (5);
\draw (5) to (6.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
It remains to show that the unit and the counit maps are inverses of each other in $\mathbb{X}$:
\[ (a) ~~~~~
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 5) {};
\node [style=none] (1) at (-2, 3) {};
\node [style=none] (2) at (-1, 3) {};
\node [style=none] (3) at (-1, 4) {};
\node [style=none] (4) at (0, 4) {};
\node [style=circle, scale=2.5] (5) at (0, 3) {};
\node [style=none] (6) at (0, 2) {};
\node [style=none] (7) at (0, 3) {$\psi^{-1}$};
\node [style=none] (8) at (-1.5, 2.25) {$\epsilon*$};
\node [style=none] (9) at (-0.5, 4.75) {$*\eta$};
\node [style=none] (10) at (0, 2) {};
\node [style=none] (11) at (-1.75, 1.75) {$\eta*$};
\node [style=none] (12) at (-1, -1) {};
\node [style=none] (13) at (-2.5, 0.75) {};
\node [style=none] (14) at (-2.5, -1.25) {};
\node [style=none] (15) at (-1, 0.75) {};
\node [style=none] (16) at (-0.5, -1.75) {$\epsilon*$};
\node [style=none] (17) at (0, -1) {};
\node [style=none] (18) at (-3.5, -1.25) {};
\node [style=none] (19) at (-3.5, -0.25) {};
\node [style=none] (20) at (-4.5, -0.25) {};
\node [style=circle, scale=2.5] (21) at (-4.5, -1.25) {};
\node [style=none] (22) at (-4.5, -2.25) {};
\node [style=none] (23) at (-4.5, -1.25) {$\psi^\dagger$};
\node [style=circle, scale=2.5] (24) at (-1, -0) {};
\node [style=none] (25) at (-1, -0) {$\iota^{-1}$};
\node [style=none] (26) at (-5.5, -2) {$X^{*\dagger*\dagger}$};
\node [style=none] (27) at (-4, 0.5) {$(*\epsilon)^\dagger$};
\node [style=none] (28) at (-3, -2) {$\epsilon*$};
\node [style=none] (29) at (-1.75, 0.5) {$(X^*)^{\dagger \dagger}$};
\node [style=none] (30) at (-1.5, -0.75) {$X^*$};
\node [style=none] (31) at (0.5, 0.5) {$X^{**}$};
\node [style=none] (32) at (-2.5, 4.5) {$X$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend right=90, looseness=2.00] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend left=90, looseness=1.75] (3.center) to (4.center);
\draw (4.center) to (5);
\draw (5) to (6.center);
\draw (10.center) to (17.center);
\draw [bend left=90, looseness=2.00] (17.center) to (12.center);
\draw [bend right=90, looseness=1.75] (15.center) to (13.center);
\draw (14.center) to (13.center);
\draw [bend right=90, looseness=1.50] (18.center) to (14.center);
\draw (18.center) to (19.center);
\draw [bend right=90, looseness=1.50] (19.center) to (20.center);
\draw (20.center) to (21);
\draw (21) to (22.center);
\draw (15.center) to (24);
\draw (24) to (12.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 5) {};
\node [style=none] (1) at (-2, 3) {};
\node [style=none] (2) at (-1, 3) {};
\node [style=none] (3) at (-1, 4) {};
\node [style=none] (4) at (0, 4) {};
\node [style=circle, scale=2.5] (5) at (0, 3) {};
\node [style=none] (6) at (0, 2) {};
\node [style=none] (7) at (0, 3) {$\psi^{-1}$};
\node [style=none] (8) at (-1.5, 2.25) {$\epsilon*$};
\node [style=none] (9) at (-0.5, 4.75) {$*\eta$};
\node [style=none] (10) at (0, 2) {};
\node [style=none] (11) at (-1, -1) {};
\node [style=none] (12) at (-2.5, 0.75) {};
\node [style=none] (13) at (-1, 0.75) {};
\node [style=none] (14) at (-0.5, -1.75) {$\epsilon*$};
\node [style=none] (15) at (0, -1) {};
\node [style=circle, scale=2.5] (16) at (-2.5, -1) {};
\node [style=none] (17) at (-2.5, -2) {};
\node [style=none] (18) at (-2.5, -1) {$\psi^\dagger$};
\node [style=circle, scale=2.5] (19) at (-1, -0) {};
\node [style=none] (20) at (-1, -0) {$\iota^{-1}$};
\node [style=none] (21) at (-3.5, -1.75) {$X^{*\dagger*\dagger}$};
\node [style=none] (22) at (-2, 1.75) {$(*\epsilon)^\dagger$};
\node [style=none] (23) at (-1.75, 0.5) {$(X^*)^{\dagger \dagger}$};
\node [style=none] (24) at (-1.5, -0.75) {$X^*$};
\node [style=none] (25) at (0.5, 0.5) {$X^{**}$};
\node [style=none] (26) at (-2.5, 4.5) {$X$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend right=90, looseness=2.00] (1.center) to (2.center);
\draw (2.center) to (3.center);
\draw [bend left=90, looseness=1.75] (3.center) to (4.center);
\draw (4.center) to (5);
\draw (5) to (6.center);
\draw (10.center) to (15.center);
\draw [bend left=90, looseness=2.00] (15.center) to (11.center);
\draw [bend right=90, looseness=1.75] (13.center) to (12.center);
\draw (16) to (17.center);
\draw (13.center) to (19);
\draw (19) to (11.center);
\draw (12.center) to (16);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (2, 4.75) {};
\node [style=none] (1) at (2, 2.75) {};
\node [style=none] (2) at (1, 2.75) {};
\node [style=none] (3) at (1, 3.75) {};
\node [style=none] (4) at (0, 3.75) {};
\node [style=none] (5) at (0, 2) {};
\node [style=none] (6) at (1.5, 2) {$*\epsilon$};
\node [style=none] (7) at (0.5, 4.5) {$\eta*$};
\node [style=none] (8) at (0, 2) {};
\node [style=none] (9) at (-1, -1) {};
\node [style=none] (10) at (-2.5, 0.75) {};
\node [style=none] (11) at (-1, 0.75) {};
\node [style=none] (12) at (-0.5, -1.75) {$\epsilon*$};
\node [style=none] (13) at (0, -1) {};
\node [style=circle, scale=2.5] (14) at (-2.5, -1) {};
\node [style=none] (15) at (-2.5, -2) {};
\node [style=none] (16) at (-2.5, -1) {$\psi^{\dagger}$};
\node [style=circle, scale=2.5] (17) at (-1, -0) {};
\node [style=none] (18) at (-1, -0) {$\iota^{-1}$};
\node [style=none] (19) at (-3.5, -1.75) {$X^{*\dagger*\dagger}$};
\node [style=none] (20) at (-2, 1.75) {$(*\epsilon)^\dagger$};
\node [style=none] (21) at (-1.75, 0.5) {$(X^*)^{\dagger \dagger}$};
\node [style=none] (22) at (-1.5, -0.75) {$X^*$};
\node [style=none] (23) at (0.5, 0.5) {$X^{**}$};
\node [style=none] (24) at (2.5, 4.25) {$X$};
\node [style=circle, scale=2.5] (25) at (1, 3.25) {};
\node [style=none] (26) at (1, 3.25) {$\psi$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw [bend right=90, looseness=1.75] (3.center) to (4.center);
\draw (8.center) to (13.center);
\draw [bend left=90, looseness=2.00] (13.center) to (9.center);
\draw [bend right=90, looseness=1.75] (11.center) to (10.center);
\draw (14) to (15.center);
\draw (11.center) to (17);
\draw (17) to (9.center);
\draw (10.center) to (14);
\draw (4.center) to (5.center);
\draw (3.center) to (25);
\draw (25) to (2.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 4.5) {};
\node [style=none] (1) at (0, -0.5) {};
\node [style=none] (2) at (-1, -0.5) {};
\node [style=none] (3) at (-1, 1.5) {};
\node [style=none] (4) at (-0.5, -1.25) {$*\epsilon$};
\node [style=none] (5) at (-1, 1.5) {};
\node [style=none] (6) at (-2.5, 3.25) {};
\node [style=none] (7) at (-1, 3.25) {};
\node [style=circle, scale=2.5] (8) at (-2.5, 0.75) {};
\node [style=none] (9) at (-2.5, -2) {};
\node [style=none] (10) at (-2.5, 0.75) {$\psi^{\dagger}$};
\node [style=circle, scale=2.5] (11) at (-1, 2.5) {};
\node [style=none] (12) at (-1, 2.5) {$\iota^{-1}$};
\node [style=none] (13) at (-3.25, -1.75) {$X^{*\dagger*\dagger}$};
\node [style=none] (14) at (-2, 4.25) {$(*\epsilon)^\dagger$};
\node [style=none] (15) at (-1.75, 3) {$X^{*^{\dagger \dagger}}$};
\node [style=none] (16) at (-1.5, 1.75) {$X^*$};
\node [style=none] (17) at (0.5, 4) {$X$};
\node [style=circle, scale=2.5] (18) at (-1, 0.75) {};
\node [style=none] (19) at (-1, 0.75) {$\psi$};
\node [style=none] (20) at (-1.5, -0) {$~^*X$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw [bend right=90, looseness=1.75] (7.center) to (6.center);
\draw (8) to (9.center);
\draw (7.center) to (11);
\draw (11) to (5.center);
\draw (6.center) to (8);
\draw (3.center) to (18);
\draw (18) to (2.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 4.5) {};
\node [style=none] (1) at (0, -0.5) {};
\node [style=none] (2) at (-1, -0.5) {};
\node [style=none] (3) at (-1, 1.5) {};
\node [style=none] (4) at (-0.5, -1.25) {$*\epsilon$};
\node [style=none] (5) at (-1, 1.5) {};
\node [style=none] (6) at (-2.5, 3.25) {};
\node [style=none] (7) at (-1, 3.25) {};
\node [style=circle, scale=2.5] (8) at (-2.5, 0.75) {};
\node [style=none] (9) at (-2.5, -2) {};
\node [style=none] (10) at (-2.5, 0.75) {$\psi^{\dagger}$};
\node [style=circle, scale=2.5] (11) at (-1, 2.5) {};
\node [style=none] (12) at (-1, 2.5) {$\psi^{\dagger \dagger}$};
\node [style=none] (13) at (-3, -1.75) {$X^{*\dagger*\dagger}$};
\node [style=none] (14) at (-2, 4.25) {$(*\epsilon)^\dagger$};
\node [style=none] (15) at (-1.75, 3) {$(X^*)^{\dagger \dagger}$};
\node [style=none] (16) at (-1.75, 1.75) {$(~^*X)^{\dagger \dagger}$};
\node [style=none] (17) at (0.25, 4) {$X$};
\node [style=circle, scale=2.5] (18) at (-1, 0.75) {};
\node [style=none] (19) at (-1, 0.75) {$\iota^{-1}$};
\node [style=none] (20) at (-1.5, -0) {$~^*X$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw [bend right=90, looseness=1.75] (7.center) to (6.center);
\draw (8) to (9.center);
\draw (7.center) to (11);
\draw (11) to (5.center);
\draw (6.center) to (8);
\draw (3.center) to (18);
\draw (18) to (2.center);
\end{pgfonlayer}
\end{tikzpicture} =
\]
\[
=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5, 4.5) {};
\node [style=none] (1) at (0.5, -0.5) {};
\node [style=none] (2) at (-1, -0.5) {};
\node [style=none] (3) at (-2.5, 3.25) {};
\node [style=none] (4) at (-1, 3.25) {};
\node [style=circle, scale=2.5] (5) at (-2.5, 0.75) {};
\node [style=none] (6) at (-2.5, -2) {};
\node [style=none] (7) at (-2.5, 0.75) {$\psi^{\dagger}$};
\node [style=circle, scale=2.5] (8) at (-1, 1.75) {};
\node [style=none] (9) at (-1, 1.75) {$\psi^{\dagger \dagger}$};
\node [style=none] (10) at (-3, -1.75) {$X^{*\dagger*\dagger}$};
\node [style=none] (11) at (-2, 4.25) {$(*\epsilon)^\dagger$};
\node [style=none] (12) at (-1.75, 3) {$(X^*)^{\dagger \dagger}$};
\node [style=none] (13) at (-1.5, -0) {$(~^*X)^{\dagger \dagger}$};
\node [style=none] (14) at (0.75, 4) {$X$};
\node [style=circle, scale=2.5] (15) at (0.5, 1.5) {};
\node [style=none] (16) at (0.5, 1.5) {$\iota^{-1}$};
\node [style=none] (17) at (-0.5, -1.75) {$*\epsilon^{\dagger \dagger}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw [bend right=90, looseness=1.75] (4.center) to (3.center);
\draw (5) to (6.center);
\draw (4.center) to (8);
\draw (3.center) to (5);
\draw (0.center) to (15);
\draw (15) to (1.center);
\draw (8) to (2.center);
\end{pgfonlayer}
\end{tikzpicture} \stackrel{*}{=} \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5, 6.25) {};
\node [style=none] (1) at (0.5, 2) {};
\node [style=none] (2) at (-1, 2) {};
\node [style=none] (3) at (-2.5, 0.25) {};
\node [style=none] (4) at (-1, 0.25) {};
\node [style=circle, scale=2.5] (5) at (-2.5, 1) {};
\node [style=none] (6) at (-2.5, -2.75) {};
\node [style=none] (7) at (-2.5, 1) {$\psi$};
\node [style=circle, scale=2.5] (8) at (-1, 1) {};
\node [style=none] (9) at (-1, 1) {$\psi^\dagger$};
\node [style=none] (10) at (-3, -2.5) {$X^{*\dagger*\dagger}$};
\node [style=none] (11) at (-2, -0.75) {$(*\epsilon)^\dagger$};
\node [style=none] (12) at (-1.75, 0.25) {$(X^*)^{\dagger}$};
\node [style=none] (13) at (-1.75, 1.75) {$(~^*X)^{\dagger}$};
\node [style=none] (14) at (0.75, 5.75) {$X$};
\node [style=circle, scale=2.5] (15) at (0.5, 5) {};
\node [style=none] (16) at (0.5, 5) {$\iota$};
\node [style=none] (17) at (-0.5, 3.25) {$*\epsilon^\dagger$};
\node [style=none] (18) at (-3.75, 4) {};
\node [style=none] (19) at (1.75, 4) {};
\node [style=none] (20) at (1.75, -1.5) {};
\node [style=none] (21) at (-3.75, -1.5) {};
\node [style=none] (22) at (-2.5, 4) {};
\node [style=none] (23) at (0.5, -1.5) {};
\node [style=none] (24) at (1.5, -1.25) {$\dagger$};
\node [style=none] (25) at (0.5, 4) {};
\node [style=none] (26) at (-2.5, -1.5) {};
\node [style=none] (27) at (-3.25, 0.25) {$~^*(X^{*\dagger})$};
\node [style=none] (28) at (-3, 2.5) {$X^{*\dagger*}$};
\node [style=none] (29) at (0, -1) {$X^{\dagger}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=2.00] (1.center) to (2.center);
\draw [bend left=90, looseness=1.75] (4.center) to (3.center);
\draw (4.center) to (8);
\draw (3.center) to (5);
\draw (0.center) to (15);
\draw (8) to (2.center);
\draw (1.center) to (23.center);
\draw (5) to (22.center);
\draw (18.center) to (19.center);
\draw (19.center) to (20.center);
\draw (20.center) to (21.center);
\draw (21.center) to (18.center);
\draw (15) to (25.center);
\draw (26.center) to (6.center);
\end{pgfonlayer}
\end{tikzpicture} \stackrel{\ref{Lemma: cyclic dagger}}{=}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.75, 6.25) {};
\node [style=none] (1) at (-2.5, 2) {};
\node [style=none] (2) at (-1, 2) {};
\node [style=none] (3) at (0.5, 0.25) {};
\node [style=none] (4) at (-1, 0.25) {};
\node [style=none] (5) at (-1, -3.25) {};
\node [style=none] (6) at (-1.75, -2.5) {$X^{*\dagger*\dagger}$};
\node [style=none] (7) at (-0.5, 5.75) {$X$};
\node [style=circle, scale=2.5] (8) at (-0.75, 5) {};
\node [style=none] (9) at (-0.75, 5) {$\iota$};
\node [style=none] (10) at (-1.5, 3.25) {$*\epsilon^\dagger$};
\node [style=none] (11) at (-3.75, 4) {};
\node [style=none] (12) at (1.75, 4) {};
\node [style=none] (13) at (1.75, -1.5) {};
\node [style=none] (14) at (-3.75, -1.5) {};
\node [style=none] (15) at (0.5, 4) {};
\node [style=none] (16) at (-2.5, -1.5) {};
\node [style=none] (17) at (1.5, -1.25) {$\dagger$};
\node [style=none] (18) at (-0.75, 4) {};
\node [style=none] (19) at (-1, -1.5) {};
\node [style=none] (20) at (-0.25, 4.5) {$X^{\dagger}$};
\node [style=none] (21) at (-3, -1) {$X^\dagger$};
\node [style=none] (22) at (1, 3.5) {$X^{*\dagger*}$};
\node [style=none] (23) at (-1.5, 1) {$(X^*)^\dagger$};
\node [style=none] (24) at (-0.25, -0.75) {$\epsilon*$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=90, looseness=2.00] (1.center) to (2.center);
\draw [bend right=90, looseness=1.75] (4.center) to (3.center);
\draw (0.center) to (8);
\draw (11.center) to (12.center);
\draw (12.center) to (13.center);
\draw (13.center) to (14.center);
\draw (14.center) to (11.center);
\draw (8) to (18.center);
\draw (19.center) to (5.center);
\draw (15.center) to (3.center);
\draw (2.center) to (4.center);
\draw (1.center) to (16.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\[
(b) ~~~~~~
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 3.5) {};
\node [style=none] (1) at (-3, -2.5) {};
\node [style=none] (2) at (4.5, 3.5) {};
\node [style=none] (3) at (4.5, -2.5) {};
\node [style=none] (4) at (3, 3.5) {};
\node [style=none] (5) at (3, -1) {};
\node [style=none] (6) at (-0.25, -1) {};
\node [style=none] (7) at (-0.25, 1) {};
\node [style=none] (8) at (0.75, 1) {};
\node [style=none] (9) at (0.75, -0.25) {};
\node [style=none] (10) at (2, -0.25) {};
\node [style=circle, scale=2.5] (11) at (2, 0.75) {};
\node [style=none] (12) at (2, 1.75) {};
\node [style=none] (13) at (-2, 1.75) {};
\node [style=none] (14) at (-2, -2.5) {};
\node [style=none] (15) at (0, 3.25) {$\eta*$};
\node [style=none] (16) at (0.25, 1.75) {$\eta*$};
\node [style=none] (17) at (1.5, -1.25) {$(*\eta)^\dagger$};
\node [style=none] (18) at (1.5, -2.25) {$\epsilon*$};
\node [style=none] (19) at (3.5, 3) {$X^{\dagger**}$};
\node [style=none] (20) at (2.5, 1.5) {$X^{*\dagger}$};
\node [style=none] (21) at (2.5, -0) {$(^*X)^\dagger$};
\node [style=none] (22) at (0.25, 0.5) {$X^*$};
\node [style=none] (23) at (-0.75, -0.5) {$X^{\dagger*}$};
\node [style=none] (24) at (-2.5, -2) {$X^{*\dagger*}$};
\node [style=none] (25) at (1, 3.5) {};
\node [style=none] (26) at (1, 5.5) {};
\node [style=none] (27) at (1.5, 5) {$X^{*\dagger*\dagger}$};
\node [style=none] (28) at (2, 0.75) {$\psi^{-1^\dagger}$};
\node [style=none] (29) at (-2, -4) {};
\node [style=none] (30) at (-2, -6) {};
\node [style=none] (31) at (-0.75, -6) {};
\node [style=none] (32) at (-0.75, -5.5) {};
\node [style=none] (33) at (1, -5.5) {};
\node [style=none] (34) at (1, -8.5) {};
\node [style=circle, scale=2.5] (35) at (1, -7) {};
\node [style=none] (99) at (1, -7) {$\psi^{-1}$};
\node [style=none] (36) at (0, -4.5) {$*\eta$};
\node [style=none] (37) at (-1.5, -6.75) {$\epsilon*$};
\node [style=none] (38) at (1.5, -6.25) {$~^*(X^{\dagger*})$};
\node [style=none] (39) at (1.5, -8) {$X^{\dagger * *}$};
\node [style=none] (40) at (-0.25, -5.75) {$X^{\dagger*}$};
\node [style=none] (41) at (-2.5, -4.5) {$X^\dagger$};
\node [style=none] (42) at (-3, -4) {};
\node [style=none] (43) at (2.5, -4) {};
\node [style=none] (44) at (2.5, -8.5) {};
\node [style=none] (45) at (-3, -8.5) {};
\node [style=none] (46) at (0, -2.5) {};
\node [style=none] (47) at (0, -4) {};
\node [style=none] (48) at (0.5, -3.25) {$X^{\dagger**\dagger}$};
\node [style=circle, scale=2.5] (49) at (0, -9.5) {};
\node [style=none] (50) at (0, -10.5) {};
\node [style=none] (51) at (0, -8.5) {};
\node [style=none] (52) at (0, -9.5) {$\iota^{-1}$};
\node [style=none] (53) at (-0.5, -9) {$X^{\dagger \dagger}$};
\node [style=none] (54) at (-0.5, -10.25) {$X$};
\node [style=none] (55) at (4, -2) {$\dagger$};
\node [style=none] (56) at (2, -8.25) {$\dagger$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (4.center) to (5.center);
\draw [bend left=90, looseness=1.00] (5.center) to (6.center);
\draw (6.center) to (7.center);
\draw [bend left=90, looseness=1.50] (7.center) to (8.center);
\draw (8.center) to (9.center);
\draw [bend right=90, looseness=1.50] (9.center) to (10.center);
\draw (10.center) to (11);
\draw (11) to (12.center);
\draw [bend right=90, looseness=1.00] (12.center) to (13.center);
\draw (13.center) to (14.center);
\draw (26.center) to (25.center);
\draw (29.center) to (30.center);
\draw [bend right=90, looseness=1.50] (30.center) to (31.center);
\draw (31.center) to (32.center);
\draw [bend left=90, looseness=1.25] (32.center) to (33.center);
\draw (33.center) to (35);
\draw (35) to (34.center);
\draw (43.center) to (44.center);
\draw (44.center) to (45.center);
\draw (45.center) to (42.center);
\draw (42.center) to (43.center);
\draw (46.center) to (47.center);
\draw (51.center) to (49);
\draw (49) to (50.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 2.5) {};
\node [style=none] (1) at (-3, -8.5) {};
\node [style=none] (2) at (4.5, 2.5) {};
\node [style=none] (3) at (4.5, -8.5) {};
\node [style=none] (4) at (3, -2) {};
\node [style=none] (5) at (3, -7) {};
\node [style=none] (6) at (-0.25, -7) {};
\node [style=none] (7) at (-0.25, -5) {};
\node [style=none] (8) at (0.75, -5) {};
\node [style=none] (9) at (0.75, -6.25) {};
\node [style=none] (10) at (2, -6.25) {};
\node [style=circle, scale=2.5] (11) at (2, -5.25) {};
\node [style=none] (12) at (2, -4.25) {};
\node [style=none] (13) at (-2, -4.25) {};
\node [style=none] (14) at (-2, -8.5) {};
\node [style=none] (15) at (0, -2.75) {$\eta*$};
\node [style=none] (16) at (0.25, -4.25) {$\eta*$};
\node [style=none] (17) at (1.5, -7.25) {$(*\eta)^\dagger$};
\node [style=none] (18) at (1.5, -8.25) {$\epsilon*$};
\node [style=none] (19) at (2.5, -4.5) {$X^{*\dagger}$};
\node [style=none] (20) at (2.5, -6) {$(^*X)^\dagger$};
\node [style=none] (21) at (0.25, -5.5) {$X^*$};
\node [style=none] (22) at (-0.75, -6.5) {$X^{\dagger*}$};
\node [style=none] (23) at (-2.5, -8) {$X^{*\dagger*}$};
\node [style=none] (24) at (0.9999999, 2.5) {};
\node [style=none] (25) at (0.9999999, 4.5) {};
\node [style=none] (26) at (1.5, 4) {$X^{*\dagger*\dagger}$};
\node [style=none] (27) at (2, -5.25) {$\psi^{-1^\dagger}$};
\node [style=none] (28) at (0.5, 1.75) {$X^\dagger$};
\node [style=none] (29) at (0, -8.5) {};
\node [style=circle, scale=2.5] (30) at (0, -11) {};
\node [style=none] (31) at (0, -12.25) {};
\node [style=none] (32) at (0, -11) {$\iota^{-1}$};
\node [style=none] (33) at (-0.9999999, -9.75) {$X^{\dagger \dagger}$};
\node [style=none] (34) at (-0.4999999, -12) {$X$};
\node [style=none] (35) at (4, -8) {$\dagger$};
\node [style=none] (36) at (1.25, 1) {};
\node [style=none] (37) at (1.25, 0.4999999) {};
\node [style=none] (38) at (0, 2.5) {};
\node [style=circle, scale=2.5] (39) at (3, -0.7499999) {};
\node [style=none] (39) at (3, -0.7499999) {$\psi^{-1}$};
\node [style=none] (40) at (0, 0.4999999) {};
\node [style=none] (41) at (1.75, 0.7500001) {$X^{\dagger*}$};
\node [style=none] (42) at (2, 2) {$*\eta$};
\node [style=none] (43) at (3, 1) {};
\node [style=none] (44) at (3.5, 0.25) {$~^*(X^{\dagger*})$};
\node [style=none] (45) at (3, -2) {};
\node [style=none] (46) at (0.4999999, -0.25) {$\epsilon*$};
\node [style=none] (47) at (3.5, -2.75) {$X^{\dagger * *}$};
\node [style=none] (48) at (2, 2.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (4.center) to (5.center);
\draw [bend left=90, looseness=1.00] (5.center) to (6.center);
\draw (6.center) to (7.center);
\draw [bend left=90, looseness=1.50] (7.center) to (8.center);
\draw (8.center) to (9.center);
\draw [bend right=90, looseness=1.50] (9.center) to (10.center);
\draw (10.center) to (11);
\draw (11) to (12.center);
\draw [bend right=90, looseness=1.00] (12.center) to (13.center);
\draw (13.center) to (14.center);
\draw (25.center) to (24.center);
\draw (30) to (31.center);
\draw (38.center) to (40.center);
\draw [bend right=90, looseness=1.50] (40.center) to (37.center);
\draw (37.center) to (36.center);
\draw [bend left=90, looseness=1.25] (36.center) to (43.center);
\draw (43.center) to (39);
\draw (39) to (45.center);
\draw (29.center) to (30);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 2.5) {};
\node [style=none] (1) at (-3, -8.5) {};
\node [style=none] (2) at (6.75, 2.5) {};
\node [style=none] (3) at (6.75, -8.5) {};
\node [style=none] (4) at (-0.25, -7) {};
\node [style=none] (5) at (-0.25, -5) {};
\node [style=none] (6) at (0.75, -5) {};
\node [style=none] (7) at (0.75, -6.25) {};
\node [style=none] (8) at (2, -6.25) {};
\node [style=circle, scale=2.5] (9) at (2, -5.25) {};
\node [style=none] (10) at (2, -4.25) {};
\node [style=none] (11) at (-2, -4.25) {};
\node [style=none] (12) at (-2, -8.5) {};
\node [style=none] (13) at (0, -2.75) {$\eta*$};
\node [style=none] (14) at (0.25, -4.25) {$\eta*$};
\node [style=none] (15) at (1.5, -7.25) {$(*\eta)^\dagger$};
\node [style=none] (16) at (1.5, -8.25) {$\epsilon*$};
\node [style=none] (17) at (0.25, -5.5) {$X^*$};
\node [style=none] (18) at (-0.75, -6.5) {$X^{\dagger*}$};
\node [style=none] (19) at (-2.5, -8) {$X^{*\dagger*}$};
\node [style=none] (20) at (0.9999999, 2.5) {};
\node [style=none] (21) at (0.9999999, 4.5) {};
\node [style=none] (22) at (1.5, 4) {$X^{*\dagger*\dagger}$};
\node [style=none] (23) at (2, -5.25) {$\psi^{-1^\dagger}$};
\node [style=none] (24) at (0, -8.5) {};
\node [style=circle, scale=2.5] (25) at (0, -11) {};
\node [style=none] (26) at (0, -12.25) {};
\node [style=none] (27) at (0, -11) {$\iota^{-1}$};
\node [style=none] (28) at (-0.9999999, -9.75) {$X^{\dagger \dagger}$};
\node [style=none] (29) at (-0.4999999, -12) {$X$};
\node [style=none] (30) at (6.25, -8) {$\dagger$};
\node [style=none] (31) at (2, 2.5) {};
\node [style=none] (32) at (4.5, 1.25) {};
\node [style=circle, scale=2.5] (33) at (4.5, -0) {};
\node [style=none] (34) at (4.5, -0.9999999) {};
\node [style=none] (35) at (5.75, -0.9999999) {};
\node [style=none] (36) at (5.75, 2.5) {};
\node [style=none] (37) at (4.5, -0) {$\psi$};
\node [style=none] (38) at (5, -2) {$*\epsilon$};
\node [style=none] (39) at (5.25, 2) {$X^\dagger$};
\node [style=none] (40) at (4, 0.7499999) {$(X^{\dagger*}$};
\node [style=none] (41) at (4, -0.9999999) {$~^*(X^\dagger)$};
\node [style=none] (42) at (3, 1.25) {};
\node [style=none] (43) at (2.5, -6) {$(^*X)^\dagger$};
\node [style=none] (44) at (3, -2) {};
\node [style=none] (45) at (2.5, -4.5) {$X^{*\dagger}$};
\node [style=none] (46) at (3.5, -2.75) {$X^{\dagger * *}$};
\node [style=none] (47) at (3.75, 2) {$\eta*$};
\node [style=none] (48) at (3, -7) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\draw (1.center) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (0.center);
\draw (4.center) to (5.center);
\draw [bend left=90, looseness=1.50] (5.center) to (6.center);
\draw (6.center) to (7.center);
\draw [bend right=90, looseness=1.50] (7.center) to (8.center);
\draw (8.center) to (9);
\draw (9) to (10.center);
\draw [bend right=90, looseness=1.00] (10.center) to (11.center);
\draw (11.center) to (12.center);
\draw (21.center) to (20.center);
\draw (25) to (26.center);
\draw (24.center) to (25);
\draw (32.center) to (33);
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\draw [bend right=90, looseness=2.00] (34.center) to (35.center);
\draw (35.center) to (36.center);
\draw [bend left=90, looseness=1.00] (48.center) to (4.center);
\draw (42.center) to (48.center);
\draw [bend left=90, looseness=1.25] (42.center) to (32.center);
\end{pgfonlayer}
\end{tikzpicture} = \]
\[ \begin{tikzpicture}
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\node [style=none] (0) at (-3, 2.5) {};
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\node [style=none] (2) at (4.25, 2.5) {};
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\node [style=none] (4) at (-0.25, -3.25) {};
\node [style=none] (5) at (-0.25, -0.7499999) {};
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\node [style=none] (10) at (2, -0) {};
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\node [style=none] (12) at (-2, -8.5) {};
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\node [style=none] (15) at (1.5, -2.75) {$(*\eta)^\dagger$};
\node [style=none] (16) at (0.25, -1.25) {$X^*$};
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\node [style=none] (20) at (0.9999999, 4.5) {};
\node [style=none] (21) at (1.5, 4) {$X^{*\dagger*\dagger}$};
\node [style=none] (22) at (2, -0.9999999) {$\psi^{-1^\dagger}$};
\node [style=none] (23) at (0, -8.5) {};
\node [style=circle, scale=2.5] (24) at (0, -11) {};
\node [style=none] (25) at (0, -12.25) {};
\node [style=none] (26) at (0, -11) {$\iota^{01}$};
\node [style=none] (27) at (-0.9999999, -9.75) {$X^{\dagger \dagger}$};
\node [style=none] (28) at (-0.4999999, -12) {$X$};
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\node [style=none] (30) at (2, 2.5) {};
\node [style=circle, scale=2.5] (31) at (2, -5.75) {};
\node [style=none] (32) at (2, -6.25) {};
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\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
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\draw (9) to (10.center);
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\draw (20.center) to (19.center);
\draw (24) to (25.center);
\draw (23.center) to (24);
\draw (31) to (32.center);
\draw [bend right=90, looseness=2.00] (32.center) to (33.center);
\draw (33.center) to (34.center);
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\end{pgfonlayer}
\end{tikzpicture} \stackrel{*}{=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3, 2.5) {};
\node [style=none] (1) at (-3, -8.5) {};
\node [style=none] (2) at (4.5, 2.5) {};
\node [style=none] (3) at (4.5, -8.5) {};
\node [style=none] (4) at (2, -1.5) {};
\node [style=none] (5) at (2, -0.7499999) {};
\node [style=none] (6) at (1, -0.7499999) {};
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\node [style=none] (9) at (-0.25, 0.7500001) {};
\node [style=none] (10) at (-2, 0.7500001) {};
\node [style=none] (11) at (-2, -8.5) {};
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\node [style=none] (13) at (-2.5, -8) {$X^{*\dagger*}$};
\node [style=none] (14) at (0.9999999, 2.5) {};
\node [style=none] (15) at (0.9999999, 4.5) {};
\node [style=none] (16) at (1.5, 4) {$X^{*\dagger*\dagger}$};
\node [style=none] (17) at (0, -8.5) {};
\node [style=circle, scale=2.5] (18) at (0, -11) {};
\node [style=none] (19) at (0, -12.25) {};
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\node [style=none] (22) at (-0.4999999, -12) {$X$};
\node [style=none] (23) at (4, -8) {$\dagger$};
\node [style=none] (24) at (2, 2.5) {};
\node [style=circle, scale=2.5] (25) at (2, -5.75) {};
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\node [style=none] (31) at (3, 2) {$X^\dagger$};
\node [style=circle, scale=2.5] (32) at (2, -3.25) {};
\node [style=none] (33) at (2, -5) {};
\node [style=none] (34) at (2, -3.25) {$\psi^{-1}$};
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\node [style=none] (36) at (1.5, -0) {$\eta*$};
\end{pgfonlayer}
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\draw (17.center) to (18);
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\draw (33.center) to (25);
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\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2.5, 2.5) {};
\node [style=none] (1) at (-2.5, -8.5) {};
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\node [style=none] (13) at (0.9999999, 4.5) {};
\node [style=none] (14) at (1.5, 4) {$X^{*\dagger*\dagger}$};
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\node [style=none] (22) at (2.5, -3.75) {};
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\]
$(*)$ holds because $\dagger$ preserves the cyclor. Thus, $(a)$ and $(b)$ are inverses of each other.
\end{proof}
\section{Unitary structure and mixed unitary categories}
The objective of this section is to introduce mixed unitary categories (MUCs) and their morphisms. A mixed unitary category consist of
a unitary category, $\mathbb{U}$, with a $\dagger$-isomix Frobenius functor $V$ into the core of a ``large'' isomix category, so that $V: \mathbb{U} \@ifnextchar^ {\t@@}{\t@@^{}} {\sf Core}(\mathbb{X}) \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$.
The unitary category is to be regarded as providing the analogue of scalars for the larger category much as a field provides scalars
for an algebra over that field.
The section starts by describing the general notion of unitary structure in a $\dagger$-isomix category. This allows the definition of a unitary category as
a compact $\dagger$-isomix category in which all objects have unitary structure. We then show how to extract a unitary category from any $\dagger$-isomix
category using pre-unitary objects (introduced below). This is a useful construction, however, it should be born in mind that it can deliver a trivial unitary
category -- trivial in the sense that all objects are isomorphic to the units. This means that, in any application of the construction, it is important to identify
non-trivial pre-unitary objects to ensure that one is getting something worthwhile out.
Next we show, using the isomix functors ${\sf Mx}_\uparrow$ (or ${\sf Mx}_\downarrow$) that unitary categories are $\dagger$-linearly equivalent to
$\dagger$-monoidal categories and, furthermore, that closed unitary categories are equivalent to $\dagger$-compact closed categories. This provides an
explicit connection to the standard notions from categorical quantum mechanics. One contribution of this more general perspective is that through the
constructions in this section one can obtain examples not only of mix unitary categories but also of $\dagger$-monoidal and $\dagger$-compact closed
categories which might otherwise have been difficult to realize.
The final subsection introduces the notion of a mixed unitary category which forms the basis for our approach to infinite dimensional categorical quantum
mechanics.
\subsection{Unitary structure}
We are now ready to introduce the key notion of ``unitary'' which is natural for the setting of $\dagger$-isomix categories. It will transpire that one can only have
unitary isomorphisms between certain object, which we call ``unitary objects'', and that these must be in the core of the isomix category:
\begin{definition}
A $\dagger$-isomix category, $\mathbb{X}$, has {\bf unitary structure} in case there is an essentially small class of objects, ${\cal U}$, called the {\bf unitary objects} of $\mathbb{X}$ such that:
\begin{enumerate}[{\bf [U.1]}]
\item Every unitary object is in the core
\item Each unitary object $A$ comes equipped with an isomorphism, called the {\bf unitary structure map} of $A$, $\varphi_A: A \@ifnextchar^ {\t@@}{\t@@^{}} A^\dag$
\item Unitary objects are closed to $(\_)^\dag$ so that $\varphi_{A^\dag} = ((\varphi_A)^{-1})^\dag$
\item The following diagram commutes:
\[ \xymatrix{ A \ar[d]_{\varphi_A} \ar[drrr]^{\iota} & \\ A^\dag \ar[rrr]_{\varphi_{A^\dag}} & & & (A^\dag)^\dag } \]
\item The monoidal units $\top$ and $\bot$ are unitary objects and the following diagrams commute:
\[ \begin{matrix} \xymatrix{ \bot \ar[d]_{{\sf m}} \ar[dr]^{\varphi_\bot} \ar[r]^{\lambda_\bot} & \top ^\dagger \ar[d]^{({\sf m})^{\dagger}} \\ \top \ar[r]_{\lambda_\top} & \bot^\dag } \end{matrix}~~~~~
\begin{matrix} \xymatrix{ \top \ar[d]_{{\sf m}^{-1}} \ar[dr]^{\varphi_\top} \ar[r]^{\lambda_\top} & \bot ^\dagger \ar[d]^{({\sf m}^{-1})^{\dagger}} \\ \bot \ar[r]_{\lambda_\bot} & \top^\dag } \end{matrix}
\]
\item If $A$ and $B$ are unitary objects then $A \otimes B$ and $A \oplus B$ are unitary objects and the following diagrams commute:
\[ \begin{matrix} \xymatrix{A\otimes B \ar[r]^{\varphi_A\otimes \varphi_B} \ar[d]_{\mathsf{mx}} & A^\dag \otimes B^\dag \ar[d]^{\lambda_\otimes}\\
A\oplus B \ar[r]_{\varphi_{A\oplus B}} & (A\oplus B)^\dag } \end{matrix}~~~~~
\begin{matrix} \xymatrix{A\otimes B \ar[r]^{\varphi_{A\otimes B}} \ar[d]_{\mathsf{mx}} & (A \otimes B)^\dag \\
A\oplus B \ar[r]_{\varphi_{A}\oplus \varphi_B} & A^\dag\oplus B^\dag \ar@{->}[u]_{\lambda_\oplus} } \end{matrix}
\]
\end{enumerate}
A $\dagger$-isomix category with unitary structure in which all objects are unitary is a {\bf unitary category}.
\end{definition}
Unitary structure is {\em structure\/}, thus, a given category can have many different unitary structures. The requirements, however, do mean that there is always a smallest unitary
structure which generated from the requirement that the tensor units be present: this we refer to this as the ``trivial'' unitary structure. Our next observations show how this is generated.
If $A = \bot$, then by {\bf [U.5](a)} $\varphi_\bot = {\sf m} \lambda_\top$, and by {\bf [U.3]}, $\varphi_{\bot^\dagger} = (({\sf m} \lambda_\top)^{-1})^\dagger$, and this means {\bf [U.4]} hold
automatically as:
\begin{eqnarray*}
{\sf m} \lambda_\top (({\sf m} \lambda_\top)^{-1})^\dagger
&=& {\sf m} \lambda_\top (\lambda_\top^{-1} {\sf m}^{-1})^\dagger \\
&=& {\sf m} \lambda_\top ({\sf m}^{-1})^\dag (\lambda_\top^{-1})^\dag \\
&\stackrel{\text{ \tiny{ \bf{[$\dagger$-mix]}}}}{=}& {\sf m} {\sf m}^{-1} \lambda_\bot (\lambda_\top^{-1})^\dag \\ &\stackrel{\text{ \tiny{ \bf{[$\dagger$-ldc.6(a)]}}}}{=}& \iota
\end{eqnarray*}
Similarly, if $A = A \oplus B$, then by {\bf [U.6](a)} $\varphi_{A \oplus B} = {\mathsf{mx}^{-1} }\varphi_A \otimes \varphi_B \lambda_\otimes$, and by {\bf [U.3]}, $\varphi_{\bot^\dagger} = (({\mathsf{mx}^{-1} }\varphi+A \otimes \varphi_B \lambda_\otimes)^{-1})^\dagger$, and again {\bf [U.4]} holds as:
\begin{lemma}
\label{Lemma: square root tensor unitary}
When $A$ and $B$ are unitary objects in a $\dagger$-isomix category then
\begin{enumerate}[(i)]
\item ${\mathsf{mx}^{-1} }(\varphi_A \otimes \varphi_B) \lambda_\otimes (({\mathsf{mx}^{-1} }(\varphi_A \otimes \varphi_B) \lambda_\otimes)^{-1})^\dagger = \iota$;
\item $\varphi_{A^{\dagger\dagger}} = (\varphi_A)^{\dagger \dagger}: A^{\dagger\dagger} \@ifnextchar^ {\t@@}{\t@@^{}} A^{\dagger \dagger \dagger}$.
\end{enumerate}
\end{lemma}
\begin{proof}~
\begin{enumerate}[{\em (i)}]
\item
\begin{eqnarray*}
\lefteqn{ {\mathsf{mx}^{-1} }(\varphi_A \otimes \varphi_B) \lambda_\otimes (({\mathsf{mx}^{-1} }(\varphi_A \otimes \varphi_B) \lambda_\otimes)^{-1})^\dagger} \\
&=& {\mathsf{mx}^{-1} }(\varphi_A \otimes \varphi_B) \lambda_\otimes \mathsf{mx}^\dagger (\varphi_A^{-1} \otimes \varphi_B^{-1})^\dagger (\lambda_\otimes^{-1})^\dagger \\
&\stackrel{\text{nat. } \mathsf{mx}}{=}& {\mathsf{mx}^{-1} }(\varphi_A \otimes \varphi_B) \mathsf{mx} ~ \lambda_\oplus (\varphi_A^{-1} \otimes \varphi_B^{-1})^\dagger (\lambda_\otimes^{-1})^\dagger \\
&\stackrel{\text{nat. } \mathsf{mx}}{=}& (\varphi_A \oplus \varphi_B) \lambda_\oplus (\varphi_A^{-1} \otimes \varphi_B^{-1})^\dagger (\lambda_\otimes^{-1})^\dagger \\
& \stackrel{\text{nat} ~\lambda_\oplus}{=}& (\varphi_A)((\varphi_A)^{-1})^\dag (\varphi_B)((\varphi_B)^{-1})^\dag \lambda_\oplus (\lambda_\otimes^{-1})^\dagger \\
& \stackrel{\text{\tiny {\bf [U.3] \& [U.4]}}}{=}& (\iota \oplus \iota) \lambda_\oplus (\lambda_\otimes^{-1})^\dagger \\
& \stackrel{\text{\tiny {\bf [$\dagger$-ldc.6](a)}}}{=}& \iota
\end{eqnarray*}
\item We have:
\[ \varphi_{(A^\dagger)^{\dagger}} = ((\varphi_A^\dagger)^{-1})^{\dagger} = ((((\varphi_A)^{-1})^\dagger)^{-1})^\dagger = ((((\varphi_A)^{-1})^{-1})^\dagger)^\dagger = ((\varphi_A)^\dagger)^\dagger \]
\end{enumerate}
\end{proof}
These observations show that the definition of the unitary structure for objects can be inductively pushed down onto its tensor and par components and, thus, if such exists, onto atomic objects.
In particular, objects generated from the units will therefore have their unitary structure completely determined. This shows that these objects with their unitary structure give the smallest unitary structure possible on the category (which we have called the trivial unitary structure).
Often we shall want the unitary objects to have linear adjoints (or duals) but we shall need the analogue of $\dagger$-duals from categorical quantum mechanics:
\begin{definition} \label{unitary-duals}
A {\bf unitary linear dual} $(\eta, \epsilon): A \dashvv_{~u} B$ is a linear dual, $A \dashvv B$ with $A$ and $B$ being unitary objects satisfying in addition:
\[
\xymatrix{
\top \ar@{}[ddrr]|{(a)} \ar[rr]^{\eta_A} \ar[d]_{\lambda_\top} & & A \oplus B \ar[d]^{\varphi_A \oplus \varphi_B} \\
\bot^\dagger \ar[d]_{\epsilon^\dag} & & A^\dagger \oplus B^\dagger \ar[d]^{c_\oplus} \\
(B \otimes A)^\dag \ar[rr]_{\lambda_\oplus^{-1}} & & B^\dagger \oplus A^\dagger}
~~~~~~~~~~~~~~~~~
\xymatrix{
A \otimes B \ar@{}[ddrr]|{(b)} \ar[rr]^{\varphi_A \otimes \varphi_B} \ar[d]_{c_\otimes} & & A^\dag \otimes B^\dag \ar[d]^{\lambda_\otimes} \\
B \otimes A \ar[d]_{\epsilon_A} & & (A \oplus B)^\dagger \ar[d]^{\eta_A^\dagger} \\
\bot \ar[rr]_{\lambda_\bot} & & \top^\dagger }
\]
\end{definition}
A unitary category in which every unitary object has a unitary linear dual is a called a {\bf closed unitary category}. We shall see a more precise correspondence to categorical
quantum mechanical $\dagger$-duals in Proposition \ref{unitary-2-dagger}.
\begin{lemma}
$\top \dashvv_{~u} \bot$ and whenever $(\eta_1, \epsilon_1): V_1 \dashvv_{~u} U_1$ and $(\eta_2, \epsilon_2): V_2 \dashvv_{~u} U_2$ then $(V_1 \otimes V_2) \dashvv_{~u} (U_1 \oplus U_2)$ .
\end{lemma}
\begin{proof}
Define $(\eta', \epsilon'): (V_1 \otimes V_2) \dashvv_{~u} (U_1 \oplus U_2)$ where
$\eta' = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-4, 3) {$\eta_1$};
\node [style=circle] (1) at (-2, 3) {$\eta_2$};
\node [style=ox] (2) at (-4, 1.75) {};
\node [style=oa] (3) at (-2, 1.75) {};
\node [style=none] (4) at (-4, 1) {};
\node [style=none] (5) at (-2, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=none, in=15, out=-165, looseness=1.00] (1) to (2);
\draw [style=none, bend left, looseness=1.25] (1) to (3);
\draw [style=none, in=180, out=-15, looseness=1.00] (0) to (3);
\draw [style=none, bend left=45, looseness=1.25] (2) to (0);
\draw [style=none] (2) to (4.center);
\draw [style=none] (3) to (5.center);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~
\epsilon' = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-4, 1) {$\epsilon_1$};
\node [style=circle] (1) at (-2, 1) {$\epsilon_2$};
\node [style=oa] (2) at (-4, 2.25) {};
\node [style=ox] (3) at (-2, 2.25) {};
\node [style=none] (4) at (-4, 3) {};
\node [style=none] (5) at (-2, 3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [style=none, in=-15, out=165, looseness=1.00] (1) to (2);
\draw [style=none, bend right, looseness=1.25] (1) to (3);
\draw [style=none, in=180, out=15, looseness=1.00] (0) to (3);
\draw [style=none, bend right=45, looseness=1.25] (2) to (0);
\draw [style=none] (2) to (4.center);
\draw [style=none] (3) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}
$
then this is easily checked to be a unitary linear adjoint.
\end{proof}
This means a unitary category alway has a unitary nucleus which is a unitary category consisting of those objects with unitary duals.
Finally we can define what it means for an isomorphism to be unitary:
\begin{definition}
Suppose $A$ and $B$ are unitary objects in a $\dagger$-isomix category with unitary structure. An isomorphism $A \@ifnextchar^ {\t@@}{\t@@^{}}^f B$ between unitary objects is said to be {\bf unitary} if the following diagram commutes:
\[ \xymatrix{A \ar[r]^{\varphi_A} \ar[d]_{f} \ar[r]^{\varphi_A} & A^\dag \ar@{<-}[d]^{f^\dag} \\ B \ar[r]_{\varphi_B} & B^\dag } \]
\end{definition}
Observe that $\varphi$ is ``twisted'' natural for all unitary isomorphisms, thus, it is easily seen that unitary isomorphisms compose and identity maps are unitary. Our next objective is to show that all the coherence isomorphisms between unitary objects are unitary maps when one has unitary structure. In particular the linear distributors on unitary objects are isomorphisms as
everything is in the core and they are unitary isomorphisms.
First a warm up:
\begin{lemma}
\label{lemma:MUCProperties}
In a $\dagger$-isomix category with unitary structure:
\begin{enumerate}[(i)]
\item If $f$ is unitary, then so is $f^\dagger$;
\item If $f$ and $g$ are unitary, then so are $f \otimes g$ and $f \oplus g$;
\item Unitary isomorphisms are closed to composition.
\end{enumerate}
\end{lemma}
\begin{proof}~
\begin{enumerate}[{\em (i)}]
\item Recall that $\varphi_{A^\dag} = (\varphi_A^{-1})^\dag$ then $f^\dagger$ is unitary because
\[ \xymatrix{B^\dag \ar[d]_{(\varphi_B^{-1})^\dag = \varphi_{B^\dag}} \ar[rr]^{f^\dag} & & A^\dag \ar[d]^{(\varphi_A^{-1})^\dag = \varphi_{A^\dag}} \\
B^{\dag\dag} & & A^{\dag\dag} \ar[ll]^{f^{\dag\dag}}} \]
is just the dagger functor applied to the unitary diagram of $f$.
\item Suppose $f$ and $g$ are unitary morphisms, then:
\[
\xymatrix{
A \otimes B \ar@{->}[rrr]^{\varphi_{A \otimes B}} \ar[ddd]_{f \otimes g} \ar[dr]_{\mathsf{mx}} \ar@{}[dddr]|{\mbox{\tiny \bf (nat. $\mathsf{mx}$)}~~~}
& \ar@{}[dr]|{\mbox{\tiny {\bf [U.6(b)]}}} & & (A \otimes B)^\dagger \ar@{}[lddd]|{~~~~~\mbox{\tiny \bf (nat. $\lambda_\oplus)$}} \\
& A \oplus B \ar[r]^{\varphi_A \oplus \varphi_B} \ar[d]_{f \oplus g}
& A^\dagger \oplus B^\dagger \ar[ur]_{\lambda_\oplus} &
\\ & A' \oplus B' \ar[r]_{\varphi_{A'} \otimes \varphi_{B'}} \ar@{}[dr]|{\mbox{\tiny { \bf [U.6(b)]}}}
& A'^\dagger \oplus B'^\dagger \ar[dr]^{\lambda_\oplus} \ar[u]_{f^\dagger \oplus g^\dagger}
& \\ A' \otimes B' \ar[rrr]_{\varphi_{A' \otimes B'}} \ar[ur]_{\mathsf{mx}}
& & & (A' \otimes B')^\dagger \ar[uuu]_{(f \otimes g)^\dagger}
}
\]
The inner square commutes because $f$ and $g$ are unitary maps.
Similarly, using coherence {\bf [U.6(b)]}, one can show that if $f$ and $g$ are unitary, then $f \oplus g$ is unitary.
\item
The proof is trivial.
\end{enumerate}
\end{proof}
\begin{lemma}
\label{lemma:cohUnitary}
Suppose $\mathbb{X}$ is a $\dagger$-isomix category with unitary structure and $A$, $B$, and $C$ are unitary objects then the following are unitary maps:
\begin{multicols}{2}
\begin{enumerate}[(i)]
\item $\lambda_\otimes: A^\dagger \otimes B^\dagger \rightarrow (A \oplus B)^\dagger$
\item $\lambda_\oplus: A^\dagger \oplus B^\dagger \rightarrow (A \otimes B)^\dagger$
\item $\lambda_\top: \top \rightarrow \bot^\dagger$
\item $\lambda_\bot: \bot \@ifnextchar^ {\t@@}{\t@@^{}} \top^\dagger$
\item $\varphi_A: A \rightarrow A^\dagger$
\item $m: \top \rightarrow \bot$
\item $\mathsf{mx}_{A,B}: A \otimes B \rightarrow A \oplus B$
\item $\iota : A \rightarrow (A^{\dagger})^\dagger$
\item $a_\otimes: A \otimes (B \otimes C) \rightarrow (A \otimes B) \otimes C$
\item $a_\oplus: A \oplus ( B \oplus C) \rightarrow (A \oplus B) \oplus C$
\item $c_\otimes: A \otimes B \rightarrow B \otimes A$
\item $c_\oplus: A \oplus B \rightarrow B \oplus A$
\item $\partial_L: A \otimes (B \oplus C) \rightarrow (A \otimes B) \oplus C$
\item $\partial_R: (A \oplus R) \otimes C \rightarrow A \oplus (B \otimes C)$
\end{enumerate}
\end{multicols}
\end{lemma}
\begin{proof}~
\begin{enumerate}[(i)]
\item $\lambda_\otimes: A^\dagger \otimes B^\dagger \rightarrow (A \oplus B)^\dagger$ is a unitary map because:
\[\xymatrixcolsep{4pc}\xymatrix{
{}&&&&\\
A^\dag\otimes B^\dag \ar[r]^{\phi_A^{-1}\otimes\phi_B^{-1}} \ar[d]^{\lambda_\otimes} \ar@{=}@/^3pc/[rr] \ar@{}[dr]|{\mbox{\tiny {\bf [U.6(a)] }}}
& A\otimes B \ar[r]^{\phi_A\otimes \phi_B} \ar[d]^{\mathsf{mx}} \ar@{}[dr]|{\mbox{\tiny {\bf nat.}}}
& A^\dag \otimes B^\dag \ar[r]^{\phi_{A^\dag \otimes B^\dag}} \ar[d]^{\mathsf{mx}} \ar@{}[dr]|{\mbox{\tiny {\bf [U.6(a)]}}}
& (A^\dag \otimes B^\dag)^\dag \ar[d]_{\lambda_\oplus^{-1}} \ar@{=}@/^4pc/[ddd]\\
(A\oplus B)^\dag \ar[r]^{\phi_{A\oplus B}^{-1}} \ar[ddr]_{\phi_{(A\oplus B)^\dag}} \ar@{}[dr]|{\mbox{\tiny { \bf [U.4]}}}
& A\oplus B \ar[r]^{\phi_A\oplus\phi_B} \ar@{=}[d]
& A^\dag \oplus B^\dag \ar[r]^{\phi_{A^\dag}\oplus\phi_{B^\dag}} \ar@{}[d]|{\mbox{\tiny { \bf [U.4]}}\ \oplus\mbox{\tiny { \bf [U.4]}}}
& (A^\dag)^\dag \oplus (B^\dag)^\dag \ar@{=}[d]\\
{}
& A \oplus B \ar[rr]^{\iota \oplus \iota} \ar[d]^{\iota}
& {} \ar@{}[d]|{\mbox{\tiny {\bf [$\dagger$-ldc.5(a)]}}}
& (A^\dag)^\dag \oplus (B^\dag)^\dag \ar[d]_{\lambda_\oplus}\\
{}
& ((A \oplus B)^\dag)^\dag \ar[rr]_{\lambda_\oplus^\dag}
& {}
& (A^\dag \otimes B^\dag)^\dag
}\]
\item $\lambda_\oplus$ is unitary because:
\[
\xymatrix{
A^\dag\oplus B^\dag \ar[rrr]^{\phi_{A^\dag\oplus B^\dag}} \ar[dr]^{\mathsf{mx}^{-1}} \ar[ddd]_{\lambda_\oplus} \ar@{}[dddr]|{\mbox{\tiny {\bf Lem. \ref{lemma:mixdagger}}}}
&
&
&
(A^\dag\oplus B^\dag)^\dag \ar@{}[dddl]|{\mbox{\tiny {\bf (Lem. \ref{lemma:mixdagger})}}^\dag} \\
{}
& A^\dag\otimes B^\dag \ar[r]^{\phi_{A^\dag\otimes B^\dag}} \ar[d]_{\lambda_\otimes} \ar@{}[ur]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (vi)}}} \ar@{}[dr]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (i)}}}
& (A^\dag\otimes B^\dag)^\dag \ar[ur]^{(\mathsf{mx}^{-1})^\dag}
&\\
{}
& (A\oplus B)^\dag \ar[r]^{\phi_{(A\oplus B)^\dag}} \ar@{}[dr]|{\mbox{\tiny {\bf Lems. \ref{lemma:cohUnitary} (vi), \ref{lemma:MUCProperties} (i)}}}
& ((A\oplus B)^\dag)^\dag \ar[u]_{\lambda_\otimes^\dag} \ar[dr]^{((\mathsf{mx}^{-1})^\dag)^\dag}
&\\
(A\otimes B)^\dag \ar[rrr]^{\phi_{(A\oplus B)^\dag}} \ar[ur]^{(\mathsf{mx}^{-1})^\dag}
&
&
&
((A\otimes B)^\dag)^\dag \ar[uuu]_{\lambda_\oplus^\dag}
}
\]
\item $\lambda_\bot: \bot \rightarrow \top^\dagger$ is unitary because:
\[
\xymatrix{
\bot \ar[d]_{\lambda_\bot} \ar[rr]^{\varphi_\bot} & & \bot^\dagger \ar[d]^{(\lambda_\bot^{-1})^{\dagger}} \\
\top^\dagger \ar[urr]_{m^\dagger} \ar[rr]_{\varphi_{\top^\dagger} = (\varphi_{\top}^{-1})^\dagger} & &\top^{\dagger \dagger}
}
\]
The left triangle commutes by {\bf [U.5(a)]}. The right triangle commutes by {\bf [U.5(b)]} and the functoriality of $\dag$.
\item $\lambda_\top: \top \rightarrow \bot^\dagger$ is unitary because:
\[
\xymatrix{
\top \ar[d]_{\lambda_\top} \ar[rr]^{\varphi_\top} & & \top^\dagger \ar[d]^{(\lambda_\top^{-1})^{\dagger}} \\
\bot^\dagger \ar[urr]_{(m^{-1})^\dagger} \ar[rr]_{\varphi_{\bot^\dagger} = (\varphi_{\bot}^{-1})^\dagger} & &\bot^{\dagger \dagger}
}
\]
The left triangle commutes by {\bf [U.5(a)]}. The right triangle commutes by {\bf [U.5(b)]} and the functoriality of $\dag$.
\item $\varphi_A$ is unitary because the following square commutes by U.5(a).
\[
\xymatrix{
A \ar[r]^{\varphi_A} \ar[d]_{\varphi_A} & A^\dagger \ar[d]^{(\varphi^{-1})^\dagger} \\
A^\dagger \ar[r]^{\varphi_{A^\dagger}} & A^{\dagger \dagger}
}
\]
\item $m: \bot \rightarrow \top$ is unitary because:
\[
\xymatrix{
\bot \ar[r]^{\varphi_\bot} \ar[d]_{{\sf m}} & \bot^\dagger \ar[d]^{({\sf m}^{-1})^\dagger} \\
\top \ar[r]_{\varphi_\top} \ar[ur]^{\lambda_\top} & \top^\dagger
}
\]
The left and right triangles commute by {\bf [U.5(a)]} and {\bf [U.5(b)]} respectively. Hence, the outer squares commutes.
\item $\mathsf{mx}_{A,B}: A \otimes B \rightarrow A \oplus B$ is unitary as:
\[ \xymatrix{
{}
&
&
&\\
A \otimes B \ar[dd]_\mathsf{mx}\ar@/^2.5pc/[rrr]^{\varphi_{A \otimes B}} \ar[r]^\mathsf{mx} \ar@/_1.5pc/[drr]_{\varphi_A \otimes \varphi_B} \ar@{}[drr]|{\mbox{\tiny {\bf nat.}}} \ar@{}[urrr]|{\mbox{\tiny {\bf [U.3]}}}
& A \oplus B \ar[r]^{\varphi_A \oplus \varphi_B}
& A^\dag \oplus B^\dag \ar[r]^{\lambda_\oplus}
& (A \otimes B)^\dag \\
{}
&
& A^\dag \otimes B^\dag \ar[u]^\mathsf{mx} \ar[dr]^{\lambda_\otimes} \ar@{}[ur]|{\mbox{\tiny {\bf Lem. \ref{lemma:mixdagger}}}} \\
A \oplus B \ar[rrr] _{\varphi_{A\oplus B}} \ar@{}[urr]|{\mbox{\tiny {\bf [U.3]}}}
&
&
& (A \oplus B)^\dag \ar[uu]_{\mathsf{mx}^\dag}
}
\]
\item $\iota: A \rightarrow A^{\dagger \dagger}$ is unitary as in
\[
\xymatrix{
A \ar[d]_{\iota} \ar[r]^{\varphi_A} & A^\dagger \ar[d]^{(\iota^{-1})^\dagger} \ar[ld]_{\varphi_{A^\dagger}} \\
A^{\dagger \dagger} \ar[r]_{\varphi_{A^{\dagger \dagger}}} & A^{\dagger \dagger \dagger}
}
\]
the left triangle commutes by {\bf [U.4]} and the right triangle commutes by:
\begin{eqnarray*}
(\iota^{-1})^\dagger &= & ((\varphi_{A^\dagger})^{-1} \varphi_A^{-1})^\dagger = (((\varphi_{A}^{-1})^\dagger)^{-1} \varphi_A^{-1})^\dagger \\
& = & ((\varphi_A^\dagger)(\varphi_A^{-1}) )^\dagger = ( \varphi_A^{-1} )^{\dagger} ( \varphi_A )^{\dagger \dagger} \\
& = & \varphi_{A^\dagger} (\varphi_A)^{\dagger \dagger} = \varphi_{A^\dagger} (\varphi_{A^{\dagger \dagger}})
\end{eqnarray*}
\item $a_\otimes$ is unitary as:
\[
\xymatrix{
(A \otimes B) \otimes C \ar[rrr]^{\varphi_{(A \otimes B) \otimes C}} \ar[ddddd]_{a_\otimes} \ar[dr]_{\mathsf{mx}} \ar@{}[drrr]|{\mbox{\tiny {\bf [U.3]}}}
& {}
& {}
& ( (A \otimes B) \otimes C )^\dagger \\
& {}
(A \otimes B) \oplus C \ar[r]^{\varphi_{A\otimes B} \oplus \varphi_{C}} \ar[d]_{\mathsf{mx}\oplus 1} \ar@{}[dr]|{\mbox{\tiny {\bf [U.6]$\oplus$(id)}}}
& (A \otimes B)^\dag \oplus C^\dag \ar[d]^{ \lambda_\oplus^{-1}\oplus 1} \ar[ur]_{\lambda_\oplus} \ar@{}[dddr]|{\mbox{\tiny {\bf [\dag-ldc.1]}}}
& {} \\
{}
& (A \oplus B) \oplus C \ar[r]_{(\varphi_A \oplus \varphi_B) \oplus \varphi_C} \ar[d]_{a_\oplus} \ar@{}[dr]|{\mbox{\tiny {\bf nat.}}}
& (A^\dag \oplus B^\dag) \oplus C^\dag \ar[d]^{a_\oplus}
& {} \\
{}
& A \oplus (B \oplus C) \ar[r]_{\phi_A \oplus (\phi_B \oplus \phi_C)} \ar@{}[dr]|{\mbox{\tiny {\bf (id)$\oplus$[U.6] }}}
& A^\dag \oplus (B^\dag \oplus C^\dag) \ar[d]^{1\oplus\lambda_\oplus}
& {} \\
{}
& A \oplus (B \otimes C) \ar[r]_{\varphi_{A} \oplus \varphi_{B\otimes C}} \ar[u]^{1\oplus \mathsf{mx}}
& A^\dag \oplus (B \otimes C)^\dag \ar[dr]^{\lambda_\oplus}
& {} \\
A \otimes (B \otimes C) \ar[rrr]^{\varphi_{A \otimes (B \otimes C)}} \ar[ur]^{\mathsf{mx}} \ar@{}[urrr]|{\mbox{\tiny {\bf [U.3]}}}
& {}
& {}
& ( A \otimes (B \otimes C) )^\dagger \ar[uuuuu]_{a_\otimes^\dag}
}
\]
\item $a_\oplus$ is unitary because:
\[
\xymatrix{
(A \oplus B) \oplus C \ar[rrr]^{\varphi_{(A \oplus B) \oplus C}} \ar[ddddd]_{a_\oplus} \ar[dr]_{\mathsf{mx}^{-1}} \ar@{}[drrr]|{\mbox{\tiny {\bf [U.3]}}}
& {}
& {}
& ( (A \oplus B) \oplus C )^\dagger \\
& {}
(A \oplus B) \otimes C \ar[r]^{\varphi_{A\oplus B} \otimes \varphi_{C}} \ar[d]_{\mathsf{mx}^{-1}\otimes 1} \ar@{}[dr]|{\mbox{\tiny {\bf [U.6]$\otimes$(id)}}}
& (A \oplus B)^\dag \otimes C^\dag \ar[d]^{ \lambda_\otimes^{-1}\otimes 1} \ar[ur]_{\lambda_\oplus} \ar@{}[dddr]|{\mbox{\tiny {\bf [\dag-ldc.1]}}}
& {} \\
{}
& (A \otimes B) \otimes C \ar[r]_{(\varphi_A \oplus \varphi_B) \oplus \varphi_C} \ar[d]_{a_\otimes} \ar@{}[dr]|{\mbox{\tiny {\bf nat.}}}
& (A^\dag \otimes B^\dag) \otimes C^\dag \ar[d]^{a_\otimes}
& {} \\
{}
& A \otimes (B \otimes C) \ar[r]_{\phi_A \otimes (\phi_B \otimes \phi_C)} \ar@{}[dr]|{\mbox{\tiny {\bf (id)$\otimes$[U.6] }}}
& A^\dag \otimes (B^\dag \otimes C^\dag) \ar[d]^{1\otimes\lambda_\otimes}
& {} \\
{}
& A \otimes (B \oplus C) \ar[r]_{\varphi_{A} \otimes \varphi_{B\oplus C}} \ar[u]^{1\otimes \mathsf{mx}^{-1}}
& A^\dag \otimes (B \oplus C)^\dag \ar[dr]^{\lambda_\otimes}
& {} \\
A \oplus (B \oplus C) \ar[rrr]^{\varphi_{A \oplus (B \oplus C)}} \ar[ur]^{\mathsf{mx}^{-1}} \ar@{}[urrr]|{\mbox{\tiny {\bf [U.3]}}}
& {}
& {}
& ( A \oplus (B \oplus C) )^\dagger \ar[uuuuu]_{a_\oplus^\dag}
}
\]
\item $c_\otimes$ is unitary because:
\[
\xymatrix{
A\otimes B \ar[rrr]^{\phi_{A\otimes B}} \ar[dr]^{\mathsf{mx}} \ar[ddd]_{c_\otimes}
& {} \ar@{}[dr]|{\mbox{\tiny {\bf [U.6(b)]}}}
& {}
& (A\otimes B)^\dag \\
{}
& A \oplus B \ar[r]^{\phi_A \oplus \phi_B} \ar[d]_{c_\oplus} \ar@{}[dr]|{\mbox{\tiny {\bf nat.}}}
& A^\dag \oplus B^\dag \ar[d]^{c_\oplus} \ar[ur]^{\lambda_\oplus} \ar@{}[dr]|{\mbox{\tiny {\bf [$\dagger$-ldc.2(b)]}}}
& {} \\
{}
& B \oplus A \ar[r]^{\phi_B \oplus \phi_A}
& B^\dag \oplus A^\dag \ar[dr]^{\lambda_\oplus}
& {} \\
B\otimes A \ar[ur]^{\mathsf{mx}} \ar[rrr]^{\phi_{B\otimes A}}
& {} \ar@{}[ur]|{\mbox{\tiny {\bf [U.6(b)]}}}
& {}
& (B\otimes A)^\dag \ar[uuu]_{(c_\otimes^{-1})^\dag = c_\otimes^\dag}\\
}
\]
where the left square commutes commutes because
$$
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circ] (0) at (0.5, -0.25) {};
\node [style=circ] (1) at (0, -1) {$\top$};
\node [style=map] (2) at (0, -1.75) {};
\node [style=circ] (3) at (0, -2.5) {$\bot$};
\node [style=circ] (4) at (-0.5, -3.25) {};
\node [style=nothing] (5) at (0.5, -3.25) {};
\node [style=nothing] (6) at (-0.5, -4.25) {};
\node [style=nothing] (7) at (0.5, -4.25) {};
\node [style=nothing] (8) at (-0.5, 0.5) {};
\node [style=nothing] (9) at (0.5, 0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [densely dotted, in=-90, out=45, looseness=1.00] (4) to (3);
\draw (3) to (2);
\draw (2) to (1);
\draw [densely dotted, in=-135, out=90, looseness=1.00] (1) to (0);
\draw [style=simple] (0) to (9);
\draw [style=simple] (8) to (4);
\draw [style=simple] (0) to (5);
\draw [style=simple, in=90, out=-90, looseness=1.00] (5) to (6);
\draw [style=simple, in=90, out=-90, looseness=1.00] (4) to (7);
\end{pgfonlayer}
\end{tikzpicture}
=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circ] (0) at (0, -1.25) {$\top$};
\node [style=map] (1) at (0, -2) {};
\node [style=circ] (2) at (0.5, -3.5) {};
\node [style=circ] (3) at (0, -2.75) {$\bot$};
\node [style=circ] (4) at (-0.5, -0.5) {};
\node [style=nothing] (5) at (0.5, -0.5) {};
\node [style=nothing] (6) at (-0.5, 0.5) {};
\node [style=nothing] (7) at (0.5, 0.5) {};
\node [style=nothing] (8) at (0.5, -4.25) {};
\node [style=nothing] (9) at (-0.5, -4.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [densely dotted, in=-90, out=150, looseness=1.25] (2) to (3);
\draw (3) to (1);
\draw (1) to (0);
\draw [densely dotted, in=-45, out=90, looseness=1.00] (0) to (4);
\draw [style=simple] (8) to (2);
\draw [style=simple] (9) to (4);
\draw [style=simple, in=-90, out=90, looseness=1.00] (4) to (7);
\draw [style=simple, in=-90, out=90, looseness=1.00] (5) to (6);
\draw [style=simple] (5) to (2);
\end{pgfonlayer}
\end{tikzpicture}
=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circ] (0) at (0.5, -0.25) {};
\node [style=circ] (1) at (0, -1) {$\top$};
\node [style=map] (2) at (0, -1.75) {};
\node [style=circ] (3) at (0, -2.5) {$\bot$};
\node [style=circ] (4) at (-0.5, -3.25) {};
\node [style=nothing] (5) at (-0.5, -4) {};
\node [style=nothing] (6) at (0.5, -4) {};
\node [style=nothing] (7) at (-0.5, 0.75) {};
\node [style=nothing] (8) at (0.5, 0.75) {};
\node [style=nothing] (9) at (-0.5, -0.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [densely dotted, in=-90, out=45, looseness=1.00] (4) to (3);
\draw (3) to (2);
\draw (2) to (1);
\draw [densely dotted, in=-135, out=90, looseness=1.00] (1) to (0);
\draw [style=simple] (6) to (0);
\draw [style=simple] (5) to (4);
\draw [style=simple] (4) to (9);
\draw [style=simple, in=-90, out=90, looseness=1.00] (9) to (8);
\draw [style=simple, in=-90, out=90, looseness=1.00] (0) to (7);
\end{pgfonlayer}
\end{tikzpicture}
$$
\item $c_\oplus$ is unitary because:
\[
\xymatrix{
A\oplus B \ar[rrr]^{\phi_{A\oplus B}} \ar[dr]^{\mathsf{mx}^{-1}} \ar[ddd]_{c_\oplus} \ar@{}[drrr]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (vii)}}}
&
&
& (A\oplus B)^\dag \\
{}
& A \otimes B \ar[r]^{\phi_{A\otimes B}} \ar[d]_{c_\otimes} \ar@{}[dr]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (xi)}}}
& (A\otimes B)^\dag \ar[ur]^{(\mathsf{mx}^{-1})^\dag} &\\
{}
& B \otimes A \ar[r]^{\phi_{B\otimes A}}
& (B \otimes A)^\dag \ar[dr]^{(\mathsf{mx}^{-1})^\dag} \ar[u]_{c_\otimes^\dag} &\\
B\oplus A \ar[rrr]^{\phi_{B\oplus A}} \ar[ur]^{\mathsf{mx}^{-1}} \ar@{}[urrr]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (vii)}}}
&
&
& (B\oplus A)^\dag \ar[uuu]_{c_\oplus^\dag}
}
\]
where the left square commutes for the same reason and the right square is the dagger of the left square.
\item $\partial_L$ is unitary because:
\newpage
\begin{sideways}
\scalebox{.89}{
$
\xymatrix{
{}
& {}
& {}
& {}
& {}
& {}
& {}
& {}
& {}\\
{}
& {}
& {}
& {}
& {}
& {}
& {}
& {}
& {}\\
{}
& {}
& A\otimes (B \oplus C) \ar[rr]^{\phi_{A}\otimes \phi_{B\oplus C}} \ar[d]_{\mathsf{mx}} \ar@/^4pc/[rrrrrr]^{\phi_{A\otimes (B\oplus C)}} \ar@/_6pc/[dddddd]_{\partial_L}
& {} \ar@{}[d]|{\mbox{\tiny {\bf [U.6(b)]}}}
& A^\dag \otimes (B \oplus C)^\dag \ar@=[r] \ar@=[d] \ar@{}[dr]|{\mbox{\tiny {\bf id}}}
& A^\dag \otimes (B \oplus C)^\dag \ar[r]^{\mathsf{mx}} \ar[d]^{1 \otimes \lambda_\otimes^{-1} } \ar@{}[dr]|{\mbox{\tiny {\bf nat.}}} \ar@{}[u]|{\mbox{\tiny {\bf [U6.(b)]}}}
& A^\dag \oplus (B \oplus C)^\dag \ar[rr]^{\lambda_\oplus} \ar[d]^{1 \oplus \lambda_\otimes^{-1}} \ar@{}[ddrr]|{\mbox{\tiny {\bf [$\dagger$-ldc.4(b)]}}}
& {}
& (A\otimes (B\oplus C))^\dag \\
{}
& {}
& A\oplus (B \oplus C) \ar[r]^{\phi_{A\oplus (B \oplus C)}} \ar[d]_{a_\oplus} \ar@{}[dr]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (ix)}}}
& (A\oplus (B\oplus C))^\dag \ar[r]^{\lambda_\otimes^{-1}} \ar[d]^{(a_\oplus^{-1})^\dag}
& A^\dag \otimes (B\oplus C)^\dag \ar[r]^{1\otimes \lambda_\otimes^{-1}} \ar@{}[d]|{\mbox{\tiny {\bf [$\dagger$-ldc.1(b)]}}}
& A^\dag \otimes (B^\dag \otimes C^\dag) \ar[r]^{\mathsf{mx} } \ar[d]^{a_\otimes^{-1}} \ar@{}[dr]|{\mbox{\tiny {\bf mix}}}
& A^\dag \oplus (B^\dag \otimes C^\dag)
& {}
& {}\\
{}
& {}
& (A\oplus B)\oplus C \ar[r]^{\phi_{(A\oplus B)\oplus C}} \ar@=[d]
& ((A\oplus B)\oplus C)^\dag \ar[r]^{\lambda_\otimes^{-1}} \ar@{}[d]|{\mbox{\tiny {\bf [U.6(a)]}}}
& (A\oplus B)^\dag \otimes C^\dag \ar[r]^{\lambda_\otimes^{-1}\otimes 1} \ar[d]_{\mathsf{mx}} \ar@{}[dr]|{\mbox{\tiny {\bf nat.}}}
& (A^\dag \otimes B^\dag) \otimes C^\dag \ar[r]^{ \mathsf{mx}\otimes 1} \ar[d]^{\mathsf{mx}} \ar@{}[dr]|{\mbox{\tiny {\bf nat.}}}
& (A^\dag \oplus B^\dag) \otimes C^\dag \ar[d]^{\mathsf{mx}}\ar@=[r] \ar[u]_{\partial_R} \ar@{}[ddddr]|{\mbox{\tiny {\bf nat.}}}
& (A^\dag \oplus B^\dag) \otimes C^\dag \ar[dddd]^{\lambda_\oplus\otimes 1}
& {}\\
{}
& {}
& (A\oplus B)\oplus C \ar[rr]^{\phi_{A\oplus B}\oplus \phi_{C}} \ar[d]_{\mathsf{mx}^{-1}\oplus 1} \ar@{}[ll]|{\mbox{\tiny {\bf mix}}} \ar@{}[drr]|{\mbox{\tiny{\bf (id)$\otimes$(Lem. \ref{lemma:cohUnitary} (vii))} } }
& {}
& (A\oplus B)^\dag \oplus C^\dag \ar[r]^{\lambda_\otimes^{-1}\oplus 1} \ar[d]_{\mathsf{mx}^\dag\oplus 1} \ar@{}[dr]|{\mbox{\tiny {\bf 1$\otimes$(Lem. \ref{lemma:mixdagger}) }}}
& (A^\dag \otimes B^\dag) \oplus C^\dag \ar[d]^{\mathsf{mx}\oplus 1} \ar[r]^{\mathsf{mx}\oplus 1} \ar@{}[ddr]|{\mbox{\tiny {\bf id}}}
& (A^\dag \oplus B^\dag) \oplus C^\dag \ar@=[dd]
& {}
& {}\\
{}
& {}
& (A\oplus B)\otimes C \ar[rr]^{\phi_{A\oplus B}\otimes \phi_{C}} \ar@=[d]
& {} \ar@{}[d]|{\mbox{\tiny {\bf [U.6(a)] }}}
& (A\oplus B)^\dag \otimes C^\dag \ar[r]^{\lambda_\otimes^{-1}\oplus 1} \ar[d]_{\lambda_\otimes} \ar@{}[dr]|{\mbox{\tiny {\bf id}}}
& (A^\dag\otimes B^\dag) \otimes C^\dag \ar[d]^{\lambda_\otimes\ox 1}
& {}
& {}
& {}\\
{}
& {}
& (A\oplus B)\otimes C \ar[r]^{\mathsf{mx}} \ar@=[d]
& (A\oplus B)\oplus C \ar[r]^{\phi_{(A\oplus B)\oplus C}} \ar@{}[dr]|{\mbox{\tiny {\bf [U.6(a)]}}}
& ((A\oplus B)\oplus C)^\dag \ar[r]^{\lambda_\otimes^{-1}}
& (A\oplus B)^\dag \otimes C^\dag \ar[r]^{ \lambda_\otimes^{-1} \otimes 1} \ar@=[d] \ar@{}[dr]|{\mbox{\tiny {\bf id}}}
& (A^\dag \otimes B^\dag) \otimes C^\dag \ar[d]^{\lambda_\otimes \otimes 1}
& {}
& {}\\
{}
& {}
& (A\oplus B) \otimes C \ar[rrr]^{\phi_{A\oplus B}\otimes\phi_{C}} \ar@/_4pc/[rrrrrr]_{\phi_{(A\oplus B) \otimes C}}
& {}
& {}
& (A \oplus B)^\dag \otimes C^\dag \ar@=[r] \ar@{}[d]|{\mbox{\tiny {\bf [U.6(a)]}}}
& (A \oplus B)^\dag \otimes C^\dag \ar[r]^{\mathsf{mx}}
& (A \oplus B)^\dag \oplus C^\dag \ar[r]^{\lambda_\oplus}
& ((A \oplus B) \otimes C)^\dag \ar[uuuuuu]^{\partial_L^\dag}\\
{}
& {}
& {}
& {}
& {}
& {}
& {}
& {}
& {}\\
}
$
}
\end{sideways}
\newpage
\item $\partial_R$ is unitary because:
\[ \hspace{-1.25cm}
\xymatrix{
{}
& {}
& {}
& {}
& {}
& {}\\
(\!A\!\oplus\! B\!) \!\otimes\! C \ar[r]^{\mathsf{mx}} \ar[d]_{\partial_R} \ar@/^4pc/[rrrrr]^{\phi_{(A\oplus B) \otimes C }}
& (\!A\!\oplus\! B\!) \!\oplus\! C \ar[r]^{\mathsf{mx}^{-1}\oplus 1} \ar@{}[d]|{\mbox{\tiny {\bf [\dag-ldc.4]}}}
& (\!A\!\otimes\! B\!) \!\oplus\! C \ar[r]^{\phi_{(A\otimes B) \oplus C}} \ar@{}[dr]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (xiii)}}} \ar@{}[ur]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (vii), \ref{lemma:MUCProperties}}}}
& (\!(\!A\!\otimes\! B\!) \!\oplus\! C \!)^\dag \ar[r]^{(\mathsf{mx}^{-1}\oplus 1 )^\dag} \ar[d]_{\partial_L^\dag}
& (\!(\!A\!\oplus\! B\!) \!\oplus\! C \!)^\dag \ar[r]^{\mathsf{mx}^\dag} \ar@{}[d]|{\mbox{\tiny {\bf [\dag-ldc.4]}}^\dag}
& (\!(\!A\!\oplus\! B\!) \!\otimes\! C \!)^\dag \\
A\!\oplus\! (\!B \!\otimes\! C\!) \ar[r]^{\mathsf{mx}^{-1}} \ar@/_4pc/[rrrrr]_{\phi_{A\oplus (B \otimes C) }}
& A\!\oplus\! (\!B \!\oplus\! C\!) \ar[r]^{1 \oplus \mathsf{mx}}
& A\!\otimes\! (\!B \!\oplus\! C\!) \ar[r]^{\phi_{A\otimes (B \oplus C)}} \ar[u]_{\partial_L} \ar@{}[dr]|{\mbox{\tiny {\bf Lem. \ref{lemma:cohUnitary} (vii), \ref{lemma:MUCProperties}}}}
& (\!A\!\otimes\! (\!B\!\oplus\! C\!) \!)^\dag \ar[r]^{(\mathsf{mx} \oplus 1 )^\dag}
& (\!A\!\oplus\! (\!B \!\oplus\! C\!) \!)^\dag \ar[r]^{(\mathsf{mx}^{-1})^\dag}
& (\!A\!\oplus\! (\!B \!\otimes\! C\!) \!)^\dag \ar[u]_{\partial_R^\dag}\\
{}
& {}
& {}
& {}
& {}
& {}
}
\]
\end{enumerate}
\end{proof}
\subsection{Unitary categories}
In this section, we provide a general construction of a unitary category from a $\dagger$-isomix category. This provides a general and important technique of constructing unitary categories. The construction is based on identifying objects with pre-unitary structure: the tensor units always have a canonical pre-unitary structure so the construction always produces a non-empty category. However, to ensure that an application of the construction yields a unitary category in which there are object which are not isomorphic to the units, one must exhibit concretely such objects. Fortunately this is often not difficult to do, making the construction quite useful.
\begin{definition}
In a $\dagger$-mix category a {\bf pre-unitary object} is an object $U$, which in the core, together with an isomorphism $\alpha: U \@ifnextchar^ {\t@@}{\t@@^{}} U^\dagger$ such that $\alpha (\alpha^{-1})^\dagger = \iota$.
\end{definition}
\begin{definition}
\label{Def: Unitary construction}
Suppose $\mathbb{X}$ is a $\dagger$-mix category, then define ${\sf Unitary}(\mathbb{X})$, the {\bf unitary core} of $\mathbb{X}$, as follows:
\begin{description}
\item[Objects:] Pre-unitary objects $(U, \alpha)$,
\item[Maps:] $(U, \alpha) \@ifnextchar^ {\t@@}{\t@@^{}}^f (V, \beta)$ where $U \@ifnextchar^ {\t@@}{\t@@^{}}^f V $ is in $\mathbb{X}$.
\end{description}
\end{definition}
\begin{lemma}
For and $\dagger$-isomix category the unitary core, $\mathsf{Unitary}(\mathbb{X})$, is a compact $\dagger$-LDC with tensor and par defined by
\[ (\top,{\sf m}^{-1}\lambda_\bot: \top \@ifnextchar^ {\t@@}{\t@@^{}} \top^\dagger) ~~~~~(A, \alpha) \otimes (B, \beta) := (A \otimes B, \mathsf{mx}^{-1}(\alpha \oplus \beta) \lambda_\oplus: A \otimes B \@ifnextchar^ {\t@@}{\t@@^{}} (A \otimes B)^\dagger)\]
\[ (\bot, {\sf m} ~\lambda_\top: \bot \@ifnextchar^ {\t@@}{\t@@^{}} \bot^\dagger) ~~~~~(A, \alpha) \oplus (B, \beta) := (A \oplus B, \mathsf{mx}(\alpha \otimes \beta) \lambda_\otimes: A \oplus B \@ifnextchar^ {\t@@}{\t@@^{}} (A \oplus B)^\dagger) \]
and $(U,\alpha)^\dagger := (U^\dagger, (\alpha^{-1})^\dagger)$.
\end{lemma}
\begin{proof}
The proof uses the techniques of Lemma \ref{Lemma: square root tensor unitary}.
Note that, as the map and tensor structure is inherited, it suffices to show that these objects are all pre-unitary objects. Starting with $(U \alpha)^\dagger$
we have:
\[ (\alpha^{-1})^\dagger (((\alpha^{-1})^\dagger)^{-1})^\dagger = (\alpha^{-1})^\dagger (\alpha^\dagger)^\dagger = (\alpha^\dagger \alpha^{-1})^\dagger = (\iota^{-1})^\dagger = \iota \]
For the tensor and par we have:
\begin{eqnarray*}
{\sf m}^{-1}\lambda_\bot (({\sf m}^{-1}\lambda_\bot)^{-1})^\dagger & = & {\sf m}^{-1}\lambda_\bot {\sf m}^\dagger \lambda_\bot^\dagger \\
& = & {\sf m}^{-1} {\sf m} \lambda_\top \lambda_\bot^\dagger = \iota \\
\mathsf{mx}^{-1}(\alpha \oplus \beta) \lambda_\oplus ((\mathsf{mx}^{-1}(\alpha \oplus \beta) \lambda_\oplus)^{-1})^\dagger
& = & \mathsf{mx}^{-1}(\alpha \oplus \beta) \lambda_\oplus (\mathsf{mx}^\dagger) (\alpha^{-1} \oplus \beta^{-1})^\dagger (\lambda_\oplus^{-1})^\dagger \\
& = & \mathsf{mx}^{-1}(\alpha \oplus \beta) \mathsf{mx} \lambda_\otimes (\alpha^{-1} \oplus \beta^{-1})^\dagger (\lambda_\oplus^{-1})^\dagger \\
& = & (\alpha \otimes \beta) \lambda_\otimes (\alpha^{-1} \oplus \beta^{-1})^\dagger (\lambda_\oplus^{-1})^\dagger \\
& = & (\alpha \oplus \beta) ((\alpha^{-1})^\dagger \oplus (\alpha^{-1})^\dagger) \lambda_\otimes (\lambda_\oplus^{-1})^\dagger \\
& = & (\iota \oplus \iota) \lambda_\otimes (\lambda_\oplus^{-1})^\dagger = \iota \\
{\sf m} \lambda_\top (({\sf m} \lambda_\top)^{-1})^\dagger & = & {\sf m} \lambda_\top ({\sf m}^{-1})^\dagger (\lambda_\top^{-1})^\dagger \\
& = & {\sf m} ~{\sf m}^{-1} \lambda_\bot (\lambda_\top^{-1})^\dagger = \iota \\
\mathsf{mx}(\alpha \otimes \beta) \lambda_\otimes ((\mathsf{mx}(\alpha \otimes \beta) \lambda_\otimes)^{-1})^\dagger
& = & \mathsf{mx}(\alpha \otimes \beta) \lambda_\otimes (\mathsf{mx}^{-1})^\dagger (\alpha^{-1} \otimes \beta^{-1})^\dagger (\lambda_\otimes^{-1})^\dagger \\
& = & (\alpha \oplus \beta) \mathsf{mx}~ \mathsf{mx}^{-1} \lambda_\oplus (\alpha^{-1} \otimes \beta^{-1})^\dagger (\lambda_\otimes^{-1})^\dagger \\
&= & (\alpha \oplus \beta) ((\alpha^{-1})^\dagger \oplus (\beta^{-1})^\dagger) \lambda_\oplus (\lambda_\otimes^{-1})^\dagger \\
& = & (\iota \oplus \iota) \lambda_\oplus (\lambda_\otimes^{-1})^\dagger = \iota.
\end{eqnarray*}
\end{proof}
This makes $\mathsf{Unitary}(\mathbb{X})$ into a compact $\dagger$-isomix category with all the structure inherited directly from $\mathbb{X}$.
However, more is true: each object now has an obvious unitary structure. This gives:
\begin{proposition}
If $\mathbb{X}$ is any $\dagger$-isomix category, then $\mathsf{Unitary}(\mathbb{X})$ is a unitary category with a full and faithful underlying $\dagger$-isomix functor to $U: {\sf Unitary}(\mathbb{X}) \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$.
\end{proposition}
\begin{proof}
The laxors are all identity maps so that the underlying functors is immediately a $\dagger$-mix functor.
It remains to show that every object is unitary: we set the unitary structure of an object to be $\alpha: (X,\alpha) \@ifnextchar^ {\t@@}{\t@@^{}} (X,\alpha)^\dagger$. However, {\bf [U.1]} -- {\bf [U.6]} are immediately satisfied by construction implying this provides unitary structure for every object.
\end{proof}
A unitary category being a compact LDC is linearly equivalent, using ${\sf Mx}^*_\uparrow: (\mathbb{X}, \oplus,\oplus) \@ifnextchar^ {\t@@}{\t@@^{}} (\mathbb{X},\otimes,\oplus)$ (see Corollary \ref{compact-mix-functor}) to the underlying monoidal category based on the par using the mix functor. We now show that for a unitary category one can induce a stationary on objects dagger on $(\mathbb{X},\oplus,\oplus)$. We denote this dagger by
$(\_)^\ddagger$ and define it by $f^\ddagger := \varphi_Bf^\dagger\varphi_A^{-1}$ as illustrated by the left diagram below:
\[ \xymatrix{ B \ar[d]_{\varphi_B}\ar[rr]^{f^\ddagger} \ar@{}[rrd]|{:=}& & A \ar[d]^{\varphi_A} \\
B^\dagger \ar[rr]_{f^\dagger} && A^\dagger} ~~~~~~~~~~~~~
\xymatrix{ A \ar@/_1pc/[dd]_{\iota} \ar[d]^{\varphi_A}\ar[rr]^{f^{\ddagger\ddagger}} & & B \ar[d]_{\varphi_B} \ar@/^1pc/[dd]^{\iota} \\
A^\dagger \ar[d]^{(\varphi_A^{-1})^\dagger} \ar[rr]_{(f^\ddagger)^\dagger} && B^\dagger \ar[d]_{(\varphi_B^{-1})^\dagger} \\
A^{\dagger\dagger} \ar[rr]_{f^{\dagger\dagger}} & & B^{\dagger\dagger} } \]
This new dagger clearly preserves composition and is also a stationary on objects involution as proven by the second diagram:
the lower square of this diagram is the dagger of the inverted definition and the resulting outer square is the naturality of $\iota$ forcing $f^{\ddagger\ddagger} = f$.
We next observe that if $u: X \@ifnextchar^ {\t@@}{\t@@^{}} Y$ is unitary in $\mathbb{X}$ if and only if $u^{-1}= u^\ddagger$. This makes unitary in the tradition sense of categorical quantum mechanics coincide
with the notion introduced here. Thus, $u$ unitary in the sense here if and only if the diagram below commutes
\[ \xymatrix{ B \ar[d]_{\varphi_B} \ar[rr]^{u^{-1}} && A \ar[d]^{\varphi_A} \\ B^\dagger \ar[rr]_{u^\dagger} && A} \]
but this diagram commutes if and only if $u^{-1} = u^\ddagger$.
\begin{definition} \label{preserving-unitary-structure}
A $\dagger$-Frobenius functor, $F: \mathbb{X} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{Y}$, between isomix categories with unitary structure {\bf preserves unitary structure} in case whenever $\varphi_A$ is unitary in $\mathbb{X}$ then
$F(\varphi_A)\rho^F$ is unitary structure in $\mathbb{Y}$.
\end{definition}
We now show that ${\sf Mx}_\downarrow: (\mathbb{X},\oplus,\oplus) \@ifnextchar^ {\t@@}{\t@@^{}} (\mathbb{X},\otimes,\oplus)$ provides a unitary structure preserving equivalence of a dagger monoidal category into a unitary category:
\begin{proposition} \label{unitary-2-dagger}
Unitary categories are $\dagger$-linearly equivalent via the mix functor ${\sf Mx}_\downarrow: (\mathbb{X},\oplus,\oplus) \@ifnextchar^ {\t@@}{\t@@^{}} (\mathbb{X},\otimes,\oplus)$ to the underlying dagger monoidal category on the par.
Furthermore, closed unitary categories under this equivalence become dagger compact closed categories.
\end{proposition}
\begin{proof}
We must exhibit a preservator, that is a natural transformation showing that the involution is preserved:
\[ \infer={A \@ifnextchar^ {\t@@}{\t@@^{}}_{\varphi_A} A^\dagger}{{\sf Mx}_\downarrow(A^\ddagger) \@ifnextchar^ {\t@@}{\t@@^{}}^{\varphi_A} {\sf Mx}_\downarrow(A)^\dagger} \]
Note that $\varphi$ is a natural transformation by the definition of $(\_)^\ddagger$ and its coherence requirements
make it a linear natural equivalence. Making this the preservator immediately means that unitary structure is preserved.
Finally we must show that unitary linear duals under ${\sf Mx}^{*}_\downarrow$ become $\ddagger$-duals. Given $(\eta,\epsilon): A \dashvv_u B$ we must
show that under ${\sf Mx}^{*}_\downarrow$ this produces a dagger dual. ${\sf Mx}^{*}_\downarrow(\eta) = {\sf m} ~\eta: \bot \@ifnextchar^ {\t@@}{\t@@^{}} A \oplus B$ and
${\sf Mx}^{*}_\downarrow(\epsilon) = {\sf mx}^{-1} \epsilon: B \oplus A \@ifnextchar^ {\t@@}{\t@@^{}} \bot$
We then require that $c_\oplus {\sf Mx}^{*}_\downarrow(\epsilon) = {\sf Mx}^{*}_\downarrow(\eta)^\ddagger$. This is provided by:
\[ \xymatrix{A \oplus B \ar[d]^{{\sf mx}^{-1}} \ar@/_2pc/[ddd]_{\varphi_{A \oplus B}} \ar[r]^{c_\oplus} & B \oplus A \ar@/^1pc/[rr]^{{\sf Mx}^{*}_\downarrow(\epsilon)} \ar[r]_{{\sf mx}^{-1}}
& A \otimes B \ar[r]_{\epsilon} & \bot \ar[dd]_{{\sf m}} \ar@/^2pc/[ddd]^{\varphi_\bot} \ar@/_/[dddl]_{\lambda_\bot}\\
A \otimes B \ar[d]^{\varphi_A \otimes \varphi_B} \ar@/_/[rru]^{c_\otimes} \\
A^\dagger \otimes B^\dagger \ar[d]^{\lambda_\otimes} & & & \top \ar[d]_{\lambda_\top} \\
(A \oplus )^\dagger \ar@{}[rrruuu]|{{\rm Defn.} ~\ref{unitary-duals}~(b)} \ar@/_2pc/[rrr]_{{\sf Mx}^{*}_\downarrow(\eta)^\dagger} \ar[rr]_{\eta^\dagger} & & \top^\dagger \ar[r]_{{\sf m}^\dagger} & \bot^\dagger } \]
\end{proof}
\subsection{Mixed unitary categories}
A {\bf mixed unitary category} (MUC) is a strong isomix Frobenius functor $M: \mathbb{U} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$ from a unitary category to a $\dagger$-isomix category, $\mathbb{X}$, such that $M$ factors through the core of $\mathbb{X}$.
We already have various sources of examples of MUCs. The inclusions of the full subcategory of unitary objects of any $\dagger$-isomix category with unitary structure is a MUC. More interestingly, for any $\dagger$-isomix category, $\mathbb{X}$, $U: {\sf Unitary}(\mathbb{X}) \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$ is a MUC.
Our main objective in this section is to explain the couniversal property of the unitary construction.
Mix unitary categories organize themselves into a 2-category ${\sf MUC}$ (although we shall not discuss the 2-cell structure):
\begin{description}
\item[0-cells:] Are mix unitary categories $M: \mathbb{U} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$;
\item[1-cells:] Are MUC morphisms: these are squares of $\dagger$-mix Frobenius functors $(F',F,\gamma): M \@ifnextchar^ {\t@@}{\t@@^{}} N$ commuting up to a $\dagger$-linear natural isomorphism $\gamma$:
\[ \xymatrix{ \mathbb{U} \ar[d]_{F'} \ar@{}[drr]|{\Downarrow~\gamma} \ar[rr]^M & & \mathbb{X} \ar[d]^F \\ \mathbb{V} \ar[rr]_{N} & & \mathbb{Y}} \]
The functor $F': \mathbb{U} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{V}$ is between unitary categories and we demand of it that preserves unitary structure in the
sense of Definition \ref{preserving-unitary-structure}, thus, whenever ${\varphi_A}$ is the unitary structure then $F(\varphi_A) \rho^F$ is unitary structure.
\item[2-cells:] These are ``pillows'' of natural transformations.
\end{description}
We remark that we have observed that any MUC can be ``simplified'' to a dagger monoidal category with a strong $\dagger$-mix Frobenius functor into a $\dagger$--isomix category: this is achieved by precomposing with ${\sf Mx}_\downarrow$. This may seem a worthwhile simplification but it should be recognized that it simply transfers complexity from the unitary category itself onto the preservator which must now ``create'' unitary structure:
\[ \xymatrix{\mathbb{U} \ar[d]_{{\sf Mx}^{*}_\downarrow} \ar[rr]^M & & \mathbb{C} \ar@{=}[d] \\
\mathbb{U}_\downarrow \ar[rr]_{{\sf Mx}_\downarrow;M} & & \mathbb{C}} \]
Here $\mathbb{U}_\downarrow = (\mathbb{U},\oplus,\oplus)$ is viewed as a dagger monoidal category and ${\sf Mx}_\downarrow^{*}$ is the inverse of ${\sf Mx}_\downarrow$.
The point is that the preservator of the lower arrow ${\sf Mx}_\downarrow;M$ is non-trivial as it must encode the unitary structure of $\mathbb{U}$.
Our objective is now to show that the unitary construction of the previous section gives rise to a right adjoint to the underlying 2-functor $U: {\sf MUC} \@ifnextchar^ {\t@@}{\t@@^{}} {\sf MCC}$ where
the 2-category ${\sf MCC}$ is defined as:
\begin{description}
\item[0-cells:] Its objects are {\bf mix core categories}, that is strong $\dagger$-Frobenius functors $V: \mathbb{C} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{Y}$ where $\mathbb{C}$ is a compact $\dagger$-LDC, $\mathbb{Y}$ is a $\dagger$-isomix category,
and $V$ factors through the core of $\mathbb{Y}$. An example of a mix core category is, of course, the inclusion of the core into a $\dagger$-isomix category $C:{\sf Core}(\mathbb{X}) \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{X}$;
\item[1-cells:] The 1-cells are squares of mix Frobenius functors which commute up to a linear natural isomorphism;
\item[2-cells:] Are pillows of natural transformations (which we shall ignore).
\end{description}
Preliminary to our next result we prove that Frobenius functors preserve pre-unitary objects:
\begin{lemma}
Frobenius functors between compact LDCs preserve preunitary objects. Thus, suppose $F: \mathbb{X} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{Y}$ is a Frobenius $\dagger$-functor between compact $\dagger$-LDCs, then if $(A,\varphi)$ is a preunitary object and then $(F(A),F(\varphi)\rho)$ is a preunitary object in $\mathbb{Y}$.
\end{lemma}
\begin{proof}
To prove that $(F_\otimes(A)F(\varphi)\rho)$ is a preunitary objects, one has the following computation:
\begin{eqnarray*}
F(\varphi)\rho ((F(\varphi) \rho)^{-1})^\dagger
& = & F(\varphi)\rho (F(\varphi^{-1})^\dagger (\rho^{-1})^\dagger \\
& = & F(\varphi (\varphi^{-1})^\dagger) \rho (\rho^{-1})^\dagger \\
& = & F(\iota) \rho (\rho^{-1})^\dagger = \iota.
\end{eqnarray*}
\end{proof}
\begin{proposition}
$U: {\sf MUC} \@ifnextchar^ {\t@@}{\t@@^{}} {\sf MCC}$ has a right adjoint ${\sf Unitary}: {\sf MCC} \@ifnextchar^ {\t@@}{\t@@^{}} {\sf MUC}; [\mathbb{C} \@ifnextchar^ {\t@@}{\t@@^{}}^C \mathbb{X}] \mapsto [{\sf Unitary}(\mathbb{C}) \@ifnextchar^ {\t@@}{\t@@^{}}^{U;C} \mathbb{X}]$.
\end{proposition}
\begin{proof}
The couniversal diagram is as follows:
\[ \xymatrix{ {\mathbb{U} \@ifnextchar^ {\t@@}{\t@@^{}}^M \mathbb{X}} \ar[rr]^{(F,G,\gamma)} \ar[d]_{(F^\flat,G,\gamma^\flat)} && {\mathbb{C} \@ifnextchar^ {\t@@}{\t@@^{}}^V \mathbb{Y}} \\
{{\sf Unitary}(\mathbb{C}) \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{Y}} \ar[urr]_{\epsilon} } \]
where $\epsilon$ is the square on the left and $(F^\flat,G,\gamma^\flat)$ is the square on the right:
\[ \xymatrix{{\sf Unitary}(\mathbb{C}) \ar[d]_U \ar[r]^{~~~~U} & \mathbb{C} \ar[r]^V & \mathbb{Y} \ar@{=}[d] \\
\mathbb{C} \ar[rr]_V && \mathbb{Y}}
~~~~~~~~
\xymatrix{\mathbb{U} \ar[dr]_F \ar[d]^{F^\flat} \ar@{}[drr]|{~~~~~~~~~\uparrow~\gamma} \ar[rr]^M & & \mathbb{X} \ar[d]^{G} \\
{\sf Unitary}(\mathbb{C}) \ar[r]_{~~~~U} & \mathbb{C} \ar[r]_V & \mathbb{Y}} \]
It remains only to explain the triangle in the second square $F$ is a Fobenius functor and so preserves pre-unitary structure.
This means that each $(U,\varphi_U)$ in $\mathbb{U}$ is carried by $F$ onto a pre-unitary object in $\mathbb{C}$, $(F(U),F(\varphi)\rho^F)$. But a pre-unitary object in
$\mathbb{C}$ is an object of ${\sf Unitary}(\mathbb{C})$ and this determines $F^\flat$. The functor is uniquely determined as it must preserve the unitary structure.
\end{proof}
This proposition means that in building a MUC from a $\dagger$-LDC it suffices to show that the core contains non-trivial pre-unitary objects. We shall use this fact below.
\section{Examples of mixed unitary categories}
In this section we present a number of examples of MUCs. We have already noted that dagger monoidal categories are automatically unitary categories in which the unitary
structure is given by identity maps. The identity functors then gives a rather trivial MUC. More excitingly -- and beyond the scope of these notes (although we hope to return to
describe this example in more detail later) -- one can take the bicompletion of the $\dagger$-monoidal category: this is a non-trivial $\dagger$-isomix $*$-autonomous category
extension of the original $\dagger$-monoidal category and provides, thus, an interesting example of how MUCs arise.
Our purpose in this section is to exhibit some non-trivial manifestations of the various structural components of a MUC. To this end we discuss in some detail three basic examples.
\subsection{Finite dimensional framed vector spaces}
\label{subsection:fdfv}
Our first example is significant as it provides an example of a unitary category which is non-trivial (in the sense that the
unitary structure is not given by identity maps). It is the category of ``framed'' finite dimensional vector spaces, where a frame in this context is just a
choice of basis. Thus the objects in this category are vector spaces with a chosen basis while the maps, ignoring the basis, are simply homomorphisms of
the vector spaces.
The category of finite dimensional framed vectors spaces, ${\sf FFVec}_K$, is a monoidal category defined as follows:
\begin{description}
\item[Objects:] The objects are pairs $(V,{\cal V})$ where $V$ is a finite dimensional $K$-vector space and ${\cal V} = \{ v_1,...,v_n \}$ is a basis.
\item[Maps:] A map $(V,{\cal V}) \xrightarrow{f} (W,{\cal W})$ is a linear map $V \xrightarrow{f} W$ in ${\sf FdVec}_K$.
\item[Tensor:] $(V,{\cal V}) \otimes (W,{\cal W}) = (V \otimes W,\{ v \otimes w | v \in {\cal V}, w \in {\cal W} \})$ where $V \otimes W$ is the usual tensor product. The unit is
$(K,\{ e \})$ where $e$ is the unit of the field $K$.
\end{description}
To define the ``dagger'' we must first choose an involution $\overline{(\_)}: K \@ifnextchar^ {\t@@}{\t@@^{}} K$, that is a field homomorphism with $k = \overline{(\overline{k})}$. The
canonical example being conjugation of the complex numbers, however, the involution can be arbitrarily chosen -- so could also, for example, be the identity.
This involution then can be extended to a (covariant) functor:
\[ \overline{(\_)}: {\sf FFVec}_K \@ifnextchar^ {\t@@}{\t@@^{}} {\sf FFVec}_K; \begin{array}[c]{c} \xymatrix{(V,{\cal V}) \ar[d]^{f} \\ (W,{\cal W})} \end{array}
\mapsto \begin{array}[c]{c} \xymatrix{\overline{(V,{\cal V}) } \ar[d]^{\overline{f}} \\ \overline{(W,{\cal W}) }} \end{array} \]
where $\overline{(V,{\cal V})}$ is the vector space with the same basis but with the conjugate action $c~\overline{\cdot}~v = \overline{c} \cdot v$. The conjugate
homomorphism, $\overline{f}$, is then the same underlying map which is homomorphism between the conjugate spaces.
${\sf FFVec}_K$ is also a compact closed category with $(V,{\cal B})^{*} = (V^{*}, \{ \widetilde{b_i} | b_i \in {\cal B} \})$ where
\[ V^{*} = V \ensuremath{\!-\!\!\circ} K~~~~\mbox{and}~~~~\widetilde{b_i}: V \@ifnextchar^ {\t@@}{\t@@^{}} K; \left(\sum_j \beta_j \cdot b_j \right) \mapsto \beta_i \]
This makes $(\_)^{*}: {\sf FFVec}_K^{\rm op} \@ifnextchar^ {\t@@}{\t@@^{}} {\sf FFVec}_K$ a contravariant functor whose action is determined by precomposition. Finally we
define the ``dagger'' to be the composite $(V,{\cal B})^\dagger = \overline{(V,{\cal B})^{*}}$.
This is a compact LDC with tensor and par being identified (so the linear distribution is the associator) and is isomix. We must show that it is a $\dagger$-LDC.
Towards this aim we define the required natural transformations on the basis:
\[ \lambda_\otimes = \lambda_\oplus: (V,{\cal V})^\dag \otimes (W,{\cal W})^\dag \@ifnextchar^ {\t@@}{\t@@^{}} ((V,{\cal V}) \otimes (W,{\cal W}))^\dag; \widetilde{v_i} \otimes \widetilde{w_j} \mapsto \widetilde{v_i \otimes w_j} \]
\[ \lambda_\top = \lambda_\bot: (K,\{ e\}) \@ifnextchar^ {\t@@}{\t@@^{}} (K,\{ e\})^\dag; k \mapsto \overline{k} \]
\[ \iota: (V,{\cal V}) \@ifnextchar^ {\t@@}{\t@@^{}} ((V,{\cal V})^\dag)^\dag; v \mapsto \lambda f. f(v) \]
Note that the last transformation is given in a basis independent manner. Importantly, it may also be given in a basis dependent manner as
$\iota(v_i) = \widetilde{\widetilde{v_i}}$ as the behaviour of these two maps is the same when applied to the basis of $(V,{\cal V})^\dag$ namely
the elements $\widetilde{v_j}$:
\[ \iota(v_i)(\widetilde{v_j}) = (\lambda f. f(v_i)) \widetilde{v_j} = \widetilde{v_j} v_i = \delta_{i,j} = \widetilde{\widetilde{v_i}}(\widetilde{v_j}) \]
Also note that $\widetilde{v_i \otimes w_j} = (\widetilde{v_i} \otimes \widetilde{w_j}) u_\otimes$, where $u_\otimes: K \otimes K \@ifnextchar^ {\t@@}{\t@@^{}} K$ is the multiplication of the field.
With these definitions in hand it is straightforward to check that this gives a $\dagger$-LDC by checking the required coherences on basis elements.
To demonstrate the technique consider the coherence {\bf [$\dagger$-ldc.4]}:
\[ \xymatrix{A \oplus B \ar[d]_{\iota \oplus \iota} \ar[rr]^\iota & & ((A \oplus B)^\dag)^\dag \ar[d]^{\lambda_\otimes^\dag} \\
(A^\dag)^\dag \oplus (B^\dag)^\dag \ar[rr]_{\lambda_\oplus} & & (A^\dag \otimes B^\dag)^\dag} \]
We must show (identifying tensor and par) that $\lambda_\otimes^\dag (\iota(a_i \otimes b_j)) = \lambda_\otimes(\iota \otimes \iota(a_i \otimes b_j))$. Now the result is a
higher-order term so it suffices to show the evaluations on basis elements are the same. This means we need to show:
$\lambda_\otimes^\dag (\iota(a_i \otimes b_j))(\widetilde{a_p} \otimes \widetilde{b_q}) = \lambda_\otimes(\iota \otimes \iota(a_i \otimes b_j))(\widetilde{a_p} \otimes \widetilde{b_q})$
\begin{eqnarray*}
(\lambda_\otimes(\iota \otimes \iota(a_i \otimes b_j)))(\widetilde{a_p} \otimes \widetilde{b_q}) & = & (\lambda_\otimes(\widetilde{\widetilde{a_i}} \otimes \widetilde{\widetilde{b_j}}))(\widetilde{a_p} \otimes \widetilde{b_q}) \\
& = & (\widetilde{\widetilde{a_i} \otimes \widetilde{b_j}}) (\widetilde{a_p} \otimes \widetilde{b_q}) \\
& = & (\widetilde{a_p} \otimes \widetilde{b_q})(\widetilde{\widetilde{a_i}} \otimes \widetilde{\widetilde{b_j}}) u_\otimes ~~~\mbox{(diagrammatic order)}\\
& = & \delta_{p,i} \delta_{q,j} \\
(\lambda_\otimes^\dag (\iota(a_i \otimes b_j)))(\widetilde{a_p} \otimes \widetilde{b_q})
& = & (\lambda_\otimes^\dag (\widetilde{\widetilde{a_i \otimes b_j}}))(\widetilde{a_p} \otimes \widetilde{b_q}) \\
& = & (\widetilde{a_p} \otimes \widetilde{b_q}) \lambda_\otimes \widetilde{\widetilde{a_i \otimes b_j}} ~~~\mbox{(diagrammatic order)}\\
& = & \widetilde{a_p \otimes b_q} \widetilde{\widetilde{a_i \otimes b_j}} \\
& = & \delta_{p,i} \delta_{q,j}
\end{eqnarray*}
The unitary structure for this example is defined by:
\[ \varphi_{(V,{\cal V})}: (V,{\cal V}) \@ifnextchar^ {\t@@}{\t@@^{}} (V,{\cal V})^\dag; v_i \mapsto \widetilde{v_i} \]
and it remains to check the coherences {\bf [U.3]}--{\bf [U.6]}. First note that {\bf [U.4]} holds immediately by the observation above that
$\iota(v_i) = \widetilde{\widetilde{v_i}}$. For {\bf [U.3]} we require that $\varphi_{A^\dag}(\widetilde{a_i}) = (\varphi_A^{-1})^\dag)(\widetilde{a_i})$
the result is a higher-order term so we may check that the evaluations are the same on basis elements:
\begin{eqnarray*}
(\varphi_{A^\dag}(\widetilde{a_i}) ) (\widetilde{a_j}) & = & \widetilde{\widetilde{a_i}}(\widetilde{a_j}) = \delta_{i,j} \\
((\varphi_{A}^{-1})^\dag(\widetilde{a_i}))(\widetilde{a_j}) & = & \widetilde{a_i} (\varphi_{A}^{-1}(\widetilde{a_j})) = \widetilde{a_i}(a_j) = \delta_{i,j}
\end{eqnarray*}
Note that {\bf [U.5]}(a) and {\bf [U.5]}(b), in this example, requires $\lambda_\top = \varphi_\top$ which can easily be verified as each reduces to conjugation.
{\bf [U.6]}(a) and {\bf [U.6]}(b), in this example, are the same requirement which is verified by:
\[ \lambda_\otimes(\varphi_A \otimes \varphi_B(a_i \otimes b_j) ) = \lambda_\otimes (\widetilde{a_i} \otimes \widetilde{b_j}) = \widetilde{a_i \otimes b_j} = \varphi_{A \otimes B} (a_i \otimes b_j) \]
This gives:
\begin{proposition}
${\sf FFVec}_K$ with the unitary structure above is a MUC.
\end{proposition}
This raises the question of what precisely the unitary maps of this example are. To elucidate this we note that a functor can easily be constructed
$U:{\sf FFVec}_K \@ifnextchar^ {\t@@}{\t@@^{}} {\sf Mat}(K)$ where, for each object in ${\sf FFVec}_K$ we choose a total order on the elements of the basis and note that
any map is then given by a matrix acting on the bases: thus a matrix in ${\sf Mat}(K)$ with the appropriate dimensions. We now observe:
\begin{lemma}
An isomorphism $u: (A,{\cal A}) \@ifnextchar^ {\t@@}{\t@@^{}} (B,{\cal B})$ in ${\sf FFVec}_K$ is unitary if and only if $U(f)$ is unitary in ${\sf Mat}(\mathbb{X})$.
\end{lemma}
\proof While $U$ does not preserve $(\_)^\dag$ on the nose it does so up to the natural equivalence determined by $U(\varphi_A)$ which being a basis
permutation is a unitary equivalence. Thus, it is not hard to see that the following diagram commutes:
\[ \xymatrix{ U(B,{\cal B}) \ar[d]_{U(f)^\dag} \ar[rr]^{U(\varphi_B)} & &U((B,{\cal B})^\dag) \ar[d]^{U(f^\dag)} \\
U(A,{\cal A}) \ar[rr]_{U(\varphi_A)} & & U((A,{\cal A})^\dag) } \]
where recall in the category of matrices dagger is stationary on objects so $U(B,{\cal B}) = U(B,{\cal B})^\dag$.
Now suppose $u$ is unitary in ${\sf FFVec}_K$ then $u^{-1} = \varphi_B u^\dagger \varphi_A^{-1}$ so that
\[ U(u)^{-1} = U(u^{-1}) = U(\varphi_B u^\dagger \varphi_A^{-1}) = U(\varphi_B) U(u^\dagger) U(\varphi_A^{-1}) = U(u)^\dagger \]
so that its underlying map is unitary. Conversely, if $U(u)$ is unitary then
\[ U(u^{-1}) = U(u)^{-1} = U(u)^\dag = U(\varphi_B u^\dagger \varphi_A^{-1}) \]
which immediately implies, as $U$ is faithful, that $u$ is unitary in ${\sf FFVec}_K$.
\endproof
One might reasonably regard this as a rather round about way to describe the standard notion of a unitary map. However, two things of importance have been
achieved. First an example of a unitary category with a non-stationary dagger and, thus, a non-identity unitary structure, has been exhibited. Second we have
shown how the standard unitary structure may be re-expressed in this formalism using straightforward non-stationary constructs.
\subsection{Finiteness spaces}
Finiteness spaces were introduced by \cite{Ehrhard} as a model of linear logic. The type system can then be used to produce a typed system for infinite dimensional matrix multiplication in which no sums become infinite. The system of infinite dimensional matrices forms an isomix $*$-autonomous category. If one take matrices over the complex numbers then there is a natural notion of conjugation and this, in turn, gives a $\dagger$-isomix category: by taking the unitary core one can then obtain a MUC.
\begin{definition}
Let $X$ be a set and $\mathcal{F}$ be a subset of $\mathcal{P} (X)$. Define the set,
\[\mathcal{F}^\perp:=\{ x' \subseteq X |\forall x \in \mathcal{F}, x\cap x'\text{ is finite} \}\]
A {\bf finiteness space} is a pair $\mathbb{X}:=(X,\mathcal{F})$ so that $\mathcal{F}^{\perp\perp} = \mathcal{F}$. $X$ is called the {\bf web}, and the elements of $\mathcal{F}$ are called the finitary sets of the finiteness space.
\end{definition}
Observe that if the web, $X$, is finite, then $\mathcal{F}$ is forced to be the whole powerset of $X$. Finiteness spaces organize themselves into a (symmetric) $*$-autonomous category:
\begin{description}
\item[Objects: ] Finiteness spaces $\mathbb{X}:=(X, \mathcal{F})$.
\item[Maps: ] A map $R:(X, \mathcal{F}) \@ifnextchar^ {\t@@}{\t@@^{}} (Y, \mathcal{G})$ is a relation $R:X\@ifnextchar^ {\t@@}{\t@@^{}} Y$ so that:
\begin{itemize}
\item For all $x \in \mathcal{F}$, $xR \in \mathcal{G}$.
\item For all $y \in \mathcal{G}^\perp$, $Ry \in \mathcal{F}^\perp$
\end{itemize}
\item[Composition and identities: ] same as in sets and relations.
\item[Monoidal tensor: ] Given maps $R:\mathbb{X}_1\@ifnextchar^ {\t@@}{\t@@^{}}\mathbb{Y}_1$ and $S:\mathbb{X}_2\@ifnextchar^ {\t@@}{\t@@^{}}\mathbb{Y}_2$:
\[R\otimes S := \{((x_1, x_2), (y_1,y_2))| (x_1,y_1)\in R, (x_2,y_2) \in S \}\]
\item[Monoidal tensor unit: ] $\top := (\{*\}, \mathcal{P}(\{*\}))$
\item[Dualizing functor: ] $(X, \mathcal{F})^* := (X, \mathcal{F}^\perp)$
\end{description}
Finiteness spaces form an isomix $*$-autonomous category, where the core is the full subcategory determined by objects whose webs are finite sets.
From the category of finiteness spaces we can build a category of finiteness matrices over the complex numbers, ${\sf FMat}(\mathbb{C})$, its objects are
finiteness spaces and its maps are matrices whose support is a finiteness relation between the finiteness spaces. This category is an isomix
$*$-auonomous category. However, it also has a natural conjugation -- essentially by taking conjugates of the matrices -- thus, it has a natural dagger
structure.
The core of ${\sf FMat}(\mathbb{C})$ is the subcategory of finite sets and this is a well-known $\dagger$-compact closed category and the inclusion
${\cal I}: {\sf Mat}(\mathbb{C}) \@ifnextchar^ {\t@@}{\t@@^{}} {\sf FMat}(\mathbb{C})$ provides an important example of a MUC. Note, however, that the unitary structure is ``trivial'' in the
sense that it is given by identity maps.
\subsection{Chu Spaces give MUCs}
The Chu construction over a closed involutive monoidal category, which has pullbacks, produces a $\dagger$-isomix LDC, ${\sf Chu}_\mathbb{X}(I)$, whose pre-unitary objects can be used, as above, to form a mix unitary category. To get the $*$-autonomous category and $\dagger$-structure on ${\sf Chu}_\mathbb{X}(I)$ we shall start by explaining how one can produce involutive structure on the ${\sf Chu}$ category. To achieve this we iteratively develop the structure of this category, starting with an involutive closed monoidal category, $\mathbb{X}$, which is not necessarily symmetric. Note that the fact that it is involutive means that it is both left and right closed which allows us to consider the non-commutative ${\sf Chu}$ construction: in this regard we shall follow J\"urgen Koslowski's construction \cite{Jurgen} using simplified ``Chu-cells'' on the same dualizing object to obtain not a ${*}$-linear bicategory but a cyclic $*$-autonomous category. Furthermore, we shall choose a dualizing object which is involutive in order to obtain an involutive cyclic $*$-autonomous category.
Recall that an involutive object is an object $D$ of $\mathbb{X}$ with an isomorphism $d: \overline{D} \@ifnextchar^ {\t@@}{\t@@^{}} D$ such that $\overline{d}d = \varepsilon$.
We can then define ${\sf Chu}_\mathbb{X}(D)$ as follows:
\begin{description}
\item[Objects:] $(A, B, \psi_0, \psi_1)$ where $\psi_0: A \otimes B \@ifnextchar^ {\t@@}{\t@@^{}} D$ and $\psi_1: B \otimes A \@ifnextchar^ {\t@@}{\t@@^{}} D$ in $\mathbb{X}$ (these are the simplified Chu cells).
\item[Arrows:] $(f,g): (A, B, \psi_0,\psi_1) \@ifnextchar^ {\t@@}{\t@@^{}} (A', B', \psi_0',\psi_1')$ where $f: A \@ifnextchar^ {\t@@}{\t@@^{}} A'$ and $g: B' \@ifnextchar^ {\t@@}{\t@@^{}} B$ and the following diagrams commutes:
\[ \xymatrix{
& A \otimes B' \ar[dl]_{1 \otimes g} \ar[dr]^{f \otimes 1} & \\
A \otimes B \ar[dr]_{\psi_0} & & A' \otimes B' \ar[ld]^{\psi'_0} \\
& D &}
~~~~~\xymatrix{
& B' \otimes A \ar[dl]_{g \otimes 1} \ar[dr]^{1 \otimes f} & \\
B \otimes A \ar[dr]_{\psi_1} & & B' \otimes A' \ar[ld]^{\psi'_1} \\
& D &}
\]
\item[Compositon:] $(f,g)(f',g') := (ff', g'g)$. Composition is well-defined as:
\[
\xymatrix{
&& A \otimes B'' \ar[dl]_{1 \otimes g'} \ar[dr]^{f \otimes 1} && \\
& A \otimes B' \ar[dr]_{f' \otimes 1} \ar[dl]_{1 \otimes g} & & A' \otimes B'' \ar[dl]^{1 \otimes g'} \ar[dr]^{f' \otimes 1} \\
A \otimes B \ar[drr]^{\psi_0} & & A' \otimes B' \ar[d]^{\psi'_0} && A'' \otimes B'' \ar[lld]_{\psi_0''} \\
& & D & &
}
\]
and similarly for the reverse Chu-maps $\psi_1$, $\psi'_1$ and $\psi_1''$.
The identity maps are $(1_A,1_B): (A, B, \psi_0,\psi_1) \@ifnextchar^ {\t@@}{\t@@^{}} (A, B, \psi_0,\psi_1)$ as expected.
\end{description}
It is standard that ${\sf Chu}_\mathbb{X}(D)$ is a (non-commutative) $*$-autonomous category. Furthermore, it is cyclic because
$$~^{*}(A,B,\psi_0,\psi_1) = (A,B,\psi_0,\psi_1)^{*} = (B,A,\psi_1,\psi_0).$$
In addition, ${\sf Chu}_\mathbb{X}(D)$ is involutive with
$$\overline{(A,B,\psi_0,\psi_1)} := (\overline{A},\overline{B},\chi \overline{\psi_1}d,\chi \overline{\psi_0}d)$$
and $\overline{(f,g)} = (\overline{f},\overline{g})$.
Finally being involutive cyclic $*$-autonomous implies that one has a dagger!
In the case that $\mathbb{X}$ is a symmetric monoidal closed category we recapture the usual Chu construction \cite{Bar06}, which we denote ${\sf Chus}_\mathbb{X}(D)$. Consider the full subcategory
of Chu-objects with special Chu-cells of the form $(A,B, \psi,c_\otimes \psi)$ in which the symmetry map is used to obtain the second cell, this gives an inclusion
${\sf Chus}_\mathbb{X}(D) \@ifnextchar^ {\t@@}{\t@@^{}} {\sf Chu}_\mathbb{X}(D)$. We observe that when $\mathbb{X}$ is symmetric involutive that this subcategory is closed to the involution:
\begin{lemma}
If $\mathbb{X}$ is an involutive symmetric monoidal closed category and $d: \overline{D} \@ifnextchar^ {\t@@}{\t@@^{}} D$ is an involutive object, then ${\sf Chus}_\mathbb{X}(D)$ is an involutive monoidal category.
\end{lemma}
\begin{proof}
It suffices to observe that the Chu-cells of $\overline{(A,B,\psi,c_\otimes\psi)}$ have the right form. Using the coherence of the
involution with symmetry, the first Chu-cell of this object has $\chi \overline{c_\otimes \psi}d = c_\otimes \chi \overline{\psi}d$
which is exactly the symmetry map applied to the second Chu-cell of the object as desired.
\end{proof}
Of course, to obtain a MUC we must first have an isomix category, in this regard we shall use the well-known fact that ${\sf Chus}_\mathbb{X}(I)$ is an isomix category simply because
the unit for tensor and par are the same (namely $\top = \bot = (I,I)$). Of course, the tensor unit is always an involutive object since
$(\chi^{\!\!\!\circ})^{-1}: \overline{I} \@ifnextchar^ {\t@@}{\t@@^{}} I$; therefore, this is immediately an involutive symmetric $*$-autonomous category. Composing the involution with the dualizing functor gives us the dagger. What remains is
to identify the objects which are in the core of ${\sf Chus}_\mathbb{X}(I)$ and to identify specific non-trivial preunitary objects.
A closed symmetric monoidal category is (degenerately) a compact linearly distributive category and thus there may be objects which have linear adjoints:
called {\bf nuclear} objects. They form a compact closed subcategory, thus in ${\sf Vec}_\mathbb{C}$ all the finite dimensional vector spaces are in the nucleus. It is not hard then to
show that if $(\eta, \epsilon): A \dashv\!\!\!\dashv B$ that the object $(A,B,\epsilon,c_\otimes\epsilon)$ is in the core of ${\sf Chus}_\mathbb{X}(I)$.
We want to show that there are non-trivial examples of pre-unitary objects. To achieve this we consider an object $H$ for which $(e, n): H \dashv\!\!\!\dashv \overline{H}$
and such that $e$ satisfies:
\[ \xymatrix{\overline{\overline{H}} \otimes \overline{H} \ar[d]_{\varepsilon \otimes 1} \ar[r]^{\chi} & \overline{\overline{H} \otimes \overline{H}} \ar[dd]^{\overline{e}}\\
H \otimes \overline{H} \ar[d]_{e} \\
I \ar[r]_{\chi^{\!\!\!\circ}} & \overline{I} } \]
In ${\sf Vec}_{\mathbb{C}}$ $e: H \otimes \overline{H} \@ifnextchar^ {\t@@}{\t@@^{}} \mathbb{C}$ is a ``sesquilinear form'' and the diagram above asserts that it is in addition a symmetric form. Any finite dimensional
Hilbert space with its inner product satisfies the above conditions.
For such an object we note:
\begin{align*}
{ \overline{(H,\overline{H},e,c_\otimes e)}}^* & = (\overline{H},\overline{\overline{H}},\chi\overline{c_\otimes e} (\chi^{\!\!\!\circ})^{-1},\chi\overline{e} (\chi^{\!\!\!\circ})^{-1})^* \\
& = (\overline{\overline{H}},\overline{H},\chi\overline{e} (\chi^{\!\!\!\circ})^{-1},\chi\overline{c_\otimes e} (\chi^{\!\!\!\circ})^{-1})
\end{align*}
This makes
$$(\varepsilon^{-1},1) : (H,\overline{H},e,c_\otimes e) \@ifnextchar^ {\t@@}{\t@@^{}} (\overline{\overline{H}},\overline{H},\chi\overline{e} (\chi^{\!\!\!\circ})^{-1},\chi\overline{c_\otimes e} (\chi^{\!\!\!\circ})^{-1})$$
a preunitay map. Note that it is a Chu map by the commuting diagram above and as $\overline{\varepsilon} =\varepsilon$ we have
$$(\varepsilon^{-1},1) (1,\overline{\varepsilon}) = (\varepsilon^{-1},1) (1,\varepsilon) = (\varepsilon^{-1},\varepsilon)$$
where the latter is the involutor.
Of course, given such an object $H$ one can take an object isomorphic to $\overline{H}$ to obtain a core object with a non-identity pre-unitary structure (which looks less familiar). Thus, this rather long story ends up giving, by the forming the unitary core, examples which are neither compact nor having unitary structure which is
given (uniformly) by identity maps.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,723 |
Just as I thought the consistently superior jumper, Savigne, won the gold medal. She quickly established her dominance and no other athlete could match her distances. If you look at the most recent history; winning the gold is exactly the place where Savigne should be. The fifth place finish in Beijing is an aberration compared to her frequent high profile victories. Savigne, however, needs to work more on her arm swing to get within the world record.
Apparently, the Cubans have found the secret to triple jumping for women. Gay, the silver medalist, rounded into form just in time. I think, however, that the distances were really weak in this competition, probably due to the swirling winds of the Berlin Olympic Stadium. I don't see anything special about the jumping technique that shows something new so I am going to assume it is their physical and mental preparedness that got them on the podium. I look forward to seeing if the Cuban men can duplicate the women's success. I predict it will not happen.
It looks like it is time to close the doors on the "old guard" like Lebedeva. Lebedeva's form is so difficult to maintain and puts unnecessary strain on her hips and lower back, it is surprising to see her make it this long at the top of the world lists.
The Russian, Pyatykh, pulled a nice jump out but was not even a personal best. It was really a fairly bland competition for Pyatykh. She is capable of jumping over 15 meters but unfortunately, does not seem to have the consistency lately. Anyway, the jumping surface does not seem conducive to jumping far, at least not with the swirling winds.
I hope that the US can put together a new training regimen for the women to get them over the 14m65 mark. It seems silly that we have speedy jumpers without the ability to transfer the speed into distance. US women need to work on power with arm and leg swing if they are ever going to catch the best in the world.
This entry was posted on August 17, 2009 at 3:31 pm and is filed under Jump. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,966 |
Q: why does the pseudo selector on p not work? I have a div tag containing ul in which there are 3 li tags in which there are p and h2.
I can't set CSS properties on all but the first li:
For some reason, in the first li, p is also shifted.
.features li:not(:first-child) h2, p { margin-left: 94px; }
enter image description here
A: The comma in your selector splits it like this:
✅ .features li:not(:first-child) h2 or p.
You seem to expect it to split it like this:
❌ .features li:not(:first-child) h2 or .features li:not(:first-child) p.
CSS doesn't have syntax which allows that kind of shorthand. you have to be explicit:
.features li:not(:first-child) h2,
.features li:not(:first-child) p { ... }
There are preprocessors which provide more convenient syntax. For example, if you were writing SASS using the SCSS syntax:
.features li:not(:first-child) {
h2, p { ... }
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,030 |
{"url":"http:\/\/clay6.com\/qa\/34692\/pyramid-of-numbers-deals-with-the-number-of-","text":"Pyramid of numbers deals with the number of :-\n\n$\\begin{array}{1 1}(a)\\;\\text{Species in an area}\\\\(b)\\;\\text{Subspecies in a community}\\\\(c)\\;\\text{Individuals in a community}\\\\(d)\\;\\text{Individuals of a trophic level}\\%\\end{array}$\n\n1 Answer\n\nPyramid of numbers deals with the number of Individuals of a trophic level\nHence (d) is the correct answer.\nanswered Mar 28, 2014\n\n1 answer\n\n1 answer\n\n1 answer\n\n1 answer\n\n1 answer\n\n1 answer\n\n1 answer","date":"2017-12-12 23:48:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.33234959840774536, \"perplexity\": 9910.072726097706}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-51\/segments\/1512948520042.35\/warc\/CC-MAIN-20171212231544-20171213011544-00187.warc.gz\"}"} | null | null |
{"url":"https:\/\/plainmath.net\/discrete-math\/82830-for-example-say-children-together-ga","text":"beatricalwu\n\n2022-07-17\n\nI have a prove by contradiction question. For example, say 15 children together gathered 100 marbles.\nHow do I prove by contradiction that some pair of children gathered the same number?\n\nphravincegrln2\n\nExpert\n\nStep 1\nYou begin by assuming that the desired result is false: no two children gathered the same number of marbles. In that case what is the smallest possible number of marbles that they could have gathered altogether? The smallest set of 15 different non-negative integers is\n$M=\\left\\{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14\\right\\};$\nif each child gathered a different number of marbles, they must have gathered at least as are in the set M. How many is that? Is that possible if they gathered 100 marbles altogether?\nStep 2\nIf the answer to that last question is no, you\u2019ve reached the desired contradiction: you\u2019ve shown that it\u2019s impossible for each of them to have gathered a different number of marbles. This of course implies that two of them must have gathered the same number.\n\nNoelanijd\n\nExpert\n\nStep 1\nJust to expand on Brian's answer, to outline the general approach when writing a proof by contradiction:\n- We have the \"givens\", which is one or more \"knowns\" or premises that we accept as true (as given). Call them premises P.\n- We have a conclusion we need to prove: Call this C.\nSo our objective is to show that $P\\to C$. That is, if the premise(s) is\/are true, then the conclusion must be true.\nA proof by contradiction then explores what would happen if C were to be false, with the aim of finding a contradiction:\nStep 2\nWe start by assuming $\\mathrm{\u00ac}C.\\phantom{\\rule{thickmathspace}{0ex}}$.\n- \"Suppose P is true, and assume it is not the case that C holds.\"\nthen \u22ee\nthen \u22ee\n(Having assumed $\\mathrm{\u00ac}C$, we are led to conclude something that simply CANNOT BE TRUE, given the premises and what we know to be true.)\n- Therefore, if P is true, it cannot be the case that that our assumption $\\mathrm{\u00ac}C$ is correct; that is, we reject the assumption \"NOT C\" , in order to avoid a contradiction, and are left, by default, to conclude that it is C which must follow from P\n- Therefore, $P\\to C$\n\nDo you have a similar question?","date":"2023-01-30 14:25:51","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 32, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7040091156959534, \"perplexity\": 360.65091078715716}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499819.32\/warc\/CC-MAIN-20230130133622-20230130163622-00283.warc.gz\"}"} | null | null |
<awesome-slider/>
=================
The awesome-slider Vanilla JS custom element web component is a cool option to quickly add a slider/banner/gallery to your web projects.
## Live demo
[Access the live demo here.](https://caferati.me/demo/awesome-slider)
## Browser support
### Quick usage
1. Load the WebComponents polyfill:
```html
<script src="https://cdnjs.cloudflare.com/ajax/libs/webcomponentsjs/0.7.5/webcomponents.min.js"></script>
```
2. Import the custom element:
```html
<link rel="import" href="/awesome-slider.html">
```
3. Use it:
```html
<awesome-slider pre-image="/images/logo.svg" autostart="true" bullets="true">
<item source="/images/image-1.jpg"></item>
<item source="/images/image-2.jpg"></item>
<item source="/images/image-3.jpg"></item>
</awesome-slider>
```
## Install
Install the component using [Bower](http://bower.io/):
```sh
$ bower install awesome-slider --save
```
Or [download the zip file](https://github.com/rcaferati/awesome-slider/archive/master.zip).
## Troubleshooting
Join the discussion at the [article page](https://caferati.me/labs/awesome-slider)
License
-------
MIT: http://mit-license.org/
Copyright 2015 [Rafael Caferati](https://caferati.me)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,745 |
Q: Not getting Keyboard Hide notification I am noticing that after I register for the UIKeyboardWillHideNotification notification, I do not get the callback keyboardDisappeared:(NSNotification*)note
I am trying to hit the "down keyboard button" on my keyboard but nothing is firing.
This is how I register:
[[NSNotificationCenter defaultCenter]addObserver:self selector:@selector(keyboardDisappeared:) name:UIKeyboardWillHideNotification object:self];
And this is my callback:
- (void)keyboardDisappeared:(NSNotification*)note
{
NSLog@("called");
}
Also this method :
- (BOOL)textFieldShouldEndEditing:(UITextField *)textField
is not getting called. But this one:
- (void)textViewDidBeginEditing:(UITextView *)textView
IS getting called...
Any thoughts or suggestions?
Thanks,
A: You need to implement the UITextField delegate method, textFieldShouldReturn:, and have your text field resign first responder status in that method.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,045 |
\section{Introduction}
We discuss a data market technique based on intrinsic (relevance and uniqueness) as well as extrinsic value (influenced by supply and demand) of data. For intrinsic value, we explain how to perform valuation of data in absolute terms (i.e just by itself), or relatively (i.e in comparison to multiple datasets) or in conditional terms (i.e valuating new data given currently existing data).
\section{Motivation for creating Data Markets}
\textbf{AI will benefit from Liquid Markets:}
Data is increasingly concentrated in large firms. For startups, and small organizations it is increasingly difficult to compete as the lack of availability of data can stymie any and all efforts to build better machine learning algorithms. Algorithmic capability indeed increases with the availability and quality of data. One way to tackle this is the marketplace approach. By creating conditions such that data, the raw material for AI, can be bought and sold with security, privacy and consent safeguarded, specialized niches will be created and firms will be able to tackle a smaller subset of the problem. A precondition for this large scale collaboration to occur is the existence of liquid markets at various steps of the value chain.
\textbf{Example:} Consider a diagnostic healthcare company aiming to do an acquisition of labeled X-ray images from various hospitals for developing state of the art diagnostics.
The key problem in such a setting is: ``How can value of datasets from each hospital be estimated to decide their price?" The data for some hospitals can belong to unique health traits and demographics and can be very valuable for the diagnostic use-case of the company while data from some other hospitals may be of a relatively much lower value.\\
A key problem therefore is that obtaining large amounts of diverse, yet useful data costs a lot of resources. There are also diminishing returns at some point when additional data does not improve the capabilities of the algorithm, if additional data is not acquired intelligently in a cost-effective manner.
\section{Data Valuation for AI}
\textbf{Absolute, relative or conditional data purchase:}
Another required facet to setting up a data market is to build a capability to perform valuation of data in absolute terms (i.e just by itself), or relatively (i.e in comparison to multiple datasets) or in conditional terms (i.e valuating new data given currently existing data). \par \textbf{Intrinsic or extrinsic data valuation:} Any of these data valuation use-cases can be performed via intrinsic factors of evaluation such as based on quality of information within the dataset or via extrinsic factors of evaluation such as based on demand-supply, market economics, game theoretic mechanisms and speculative market forces or via a combination of both as in \citep{koutris2015query,koutris2013toward,balazinska2013discussion,deep2017qirana,li2014theory,zheng2019arete,zheng2017trading,zheng2017online}. \par \textbf{Goal dependent or independent data trading:} An additional slicing to this problem includes goal specific or goal independent data valuation depending on whether there is a specific well-defined goal for the data purchase or if it is exploratory by design for a goal that is currently undefined; but would be drafted later on. \par \textbf{Horizontal or vertical data acquisition:} In addition to all these situations of data valuation, yet another categorization is based on whether the data acuiqition is being done vertically (in terms of acquiring attributes/columns) or horizontally (acquiring records/rows) as in\citep{ghorbani2019data,jia2019towards}. This terminology of 'vertical partitioning' and 'horizontal partitioning' extends from the databases as well as distributed systems research communities.
\begin{figure}[H]
\centering
\includegraphics[width=70mm,height=62mm]{dm.png}
\caption{Data market showing data providers, data customers, notions of market basket and data pricing}
\label{fig:boat0}
\end{figure}
\textbf{Privacy aware data valuation:} Ideally such a data valuation needs to be acheived by looking at as few records per data source as possible or via privacy aware AI \citep{vepakomma2018split,vepakomma2018leakage,vepakomma2018no,gupta2018distributed}. Pooling of all data at a centralized location defeats the central purpose and the data sharing constraints of privacy, security, safety, fairness and resource efficiency need to be kept in consideration with regards to a data valuation solution for data markets. \par
\textbf{Relevance and diversity of data acquisition:} An optimal data purchase under these constraints needs to cater to high utility and low redundancy (high diversity) of data in terms of incremental benefit obtained. There is often a tradeoff of utility vs. diversity of data that needs to be considered in realistic settings. This concept has been the guding principle for techniques like sure independence screening (SIS) and conditional sure independence screening \citep{fan2008sure,zhong2015iterative,barut2016conditional} currently actively being studied in the field of statistics and in min redundancy max relevance (mRMR) \cite{peng2005feature} in the field of data mining, during the precursory periods of current day AI and machine learning.
As shown in Figure 1, a robust data valuation acts as a good input for data pricing as well as for building an optimal market basket of data for every data consumer. \par
\par The intent of sharing these possibilities is to motivate further discussion and research. We summarize some of these points with regards to data valuation in the context of data markets as shown below:
\begin{figure}[H]
\centering
\includegraphics[width=120mm,height=55mm]{dm2.png}
\caption{Landscape of data valuation problems for data markets}
\label{fig:boat0}
\end{figure}
\section{Data is very complex to price}
The value of an incremental unit of data is also conditionally dependent on data already possessed by the prospective data buyer entity that is valuating it. This is because one would like to obtain relevant yet diverse data from what is already available in-house. In addition, data can be acquired for performing either a similar or a more diverse task in comparison to the current use-cases being applied on data that is already available in-house. Also, there are so many archetypes that it is difficult to find a proxy variable (like weight or number in the case of other goods) that can be used to define the data. Since seamless discovery and a small spread in price is essential for a marketplace to function well, it has been challenging thus far to create a functioning data marketplace. A thorough data pricing startegy needs to adhere to the following guiding principles.\\\\
\textbf{Data Pricing Guidelines}
\begin{enumerate}
\item Liquidity: models freshness of dat in terms of value vs diminished/increased value over time
\item Traceability: can be only 'sold' once, or sold non-exclusively
\item Consent: maintains privacy of owner, tracks consent over time, and reduces friction with smart contracts or data concierges.
\item Neutrality: accessible to all buyers to prevent unfair trading practices. Otherwise, it would encourage some players (be it large or small) to unfairly price out the rest of prospective buyers during the trading.
\item Recourse: Allows for calling back, provides right to be forgotten, allows for some course correction, broadly remains self-sustaining.
\end{enumerate}
\section{Data sharing challenges that data markets need to address}
Although acquiring the right amount of quality data is ideal, data sharing is heavily impeded by friction caused by lack of trust, data sharing regulations such as HIPAA/GDPR, lack of ease and lack of incentive. We further expand on these factors that cause data friction.
\begin{enumerate}
\item \textbf{Lack of incentives:}
\begin{enumerate}
\item Large organizations need incentive mechanisms to share data with small players. For example, an incentive for data sharing between large centralized hospitals and local clinics, testing centers could be to foster better provision of health.
\item Big tech players have taken a lead and are rapidly collecting and hoarding data while monopolizing the data resources and are preventing small players from entering into data acquisition. This stifles innovation.
\item Individuals need incentives to share their data as they happen to generate and own tremendous amount of data on a daily basis. But this leads to the burden of consent management which is too complex to manage granularly across different modalities, time horizons, and trust-levels in data buyers.
\item Governments and non-profit are often not allowed to sell data for monetary gains .
\end{enumerate}
\item \textbf{Lack of ease of sharing data:}
Due to lack of automated processes, digitization, access to data pre-processing pipelines, compatible data schemas, lack of standardization across data sources and other forms of siloing of socially beneficial data; seamless data sharing is restricted. To summarize, these factors include:
\begin{enumerate}
\item Lack of digitization and lack of use cases
\item Lack of data standardization across multiple sources
\item Collection of data currently will likely cost more than market price of data
\item Socially beneficial good data is locked away (e.g. with government, non-profits, hospitals, remote sensing data)
\end{enumerate}
\item \textbf{Lack of trust:} Data sharing can also be impeded by the factors of market forces, need for maintaining trade secrets, competitive economy that impedes trust, fear of losing control and accountability over future usage of data for adversarial purposes. To summarize, these factors include cases when:
\begin{enumerate}
\item Data owner does not trust what the buyer will do with
data in a competing environment
\item Data indirectly contains trade secrets of the data owner
\item Fear of adversarial future usage of shared data
\end{enumerate}
\item \textbf{Regulations:} Data sharing is regulated for privacy, security, fairness and safety and therefore any data transactions for performing basic data analysis or for any advanced AI/ML usecases has to be aware of these constraints and be able to safely circumvent these friction points while also maintaining compliance with the law. To summarize:
\begin{enumerate}
\item In sectors such as health, finance and cybersecurity that are tightly governed by local, federal and international data sharing regulations such as HIPAA, GDPR, COX, PCI, SHIELD, we need a new strategy for safe data sharing.
\item Policies for inter as well as intra organizational data sharing have to be adhered to.
\item The origination of data may have country specific regulations on usage. Therefore international regulations need to be adhered to with respect to both the data provider as well as the data consumer.
\item There are policies where data cannot physically leave the premises of the data owners.
\end{enumerate}
\section{Governance and encouragement for a data market ecosystem} In addition from the perspective of governance, the following would be key to support the setup as well as to sustain a good ecosystem for data markets \cite{aalekhTalk}.
\begin{enumerate}
\item Need to support technological solution vs market solution vs policy-driven solutions.
\item Data governance policies by undertaking a study of changes needed in existing legal/regulatory frameworks.
\item Standardization of data sharing
\item Setting up national 'nodes' of servers for data exchange (like stock exchanges)
\item 'Clean Data' credits like 'clear air' carbon credits
\item Treat data as labor (it's from activity, that creates value)
\item Ethics and bias: self certification as well as audits
\end{enumerate}
\end{enumerate}
\printbibliography
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,264 |
Farkaždin (cyr. Фаркаждин) – wieś w Serbii, w Wojwodinie, w okręgu środkowobanackim, w mieście Zrenjanin. W 2011 roku liczyła 1179 mieszkańców.
Przypisy
Miejscowości w okręgu środkowobanackim | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,337 |
Q: Custom Settings with ApplicationSettingsBase saving List of custom objects in C# I am trying to use ApplicationSettingsBase with saving list of custom objects. Here is my application settings class:
internal class ApplicationSettings: ApplicationSettingsBase
{
[DefaultSettingValue(""), UserScopedSetting]
public List<MyAwesomeClass> MyObjects
{
get
{
try
{
return this["MyObjects"] as List<MyAwesomeClass>;
}
catch (SettingsPropertyNotFoundException)
{
return null;
}
}
}
public void Add(MyAwesomeClass val)
{
MyObjects.Add(val);
this["MyObjects"] = MyObjects;
}
}
Here is my custom class:
[Serializable]
public class MyAwesomeClass
{
public Guid Id { get; set; }
public string SomeStringProperty{ get; set; }
public bool SomeBooleanProperty{ get; set; }
}
And here is the usage:
_applicationSettings = new ApplicationSettings();
_applicationSettings.Add(new MyAwesomeClass {/*Some initial values*/});
_applicationSettings.Save();
But when application restarts _applicationSettings.MyObjects is always have zero count. I don't know why my list doesn't want saving. Can anybody help me with this? Thanks!
A: You need to decorate the property with [SettingsSerializeAs(SettingsSerializeAs.Binary)]:
[DefaultSettingValue(""), UserScopedSetting]
[SettingsSerializeAs(SettingsSerializeAs.Binary)]
public List<MyAwesomeClass> MyObjects
Application Settings Architecture » Setting Persistence » Settings Serialization
If you implement your own settings class, you can use the
SettingsSerializeAsAttribute to mark a setting for either binary or
custom serialization using the SettingsSerializeAs enumeration.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,055 |
\section{Introduction}
The mapping class group ${\rm Mod}(\Sigma_g)$ of a compact connected orientable
surface $\Sigma_g$ without boundary is the group of
orientation--preserving diffeomorphisms of $\Sigma_g\to \Sigma_g$ up to isotopy. This study aims to explore
generation of ${\rm Mod}(\Sigma_g)$ by two torsion elements of small orders.
Korkmaz is the first person to find a generating set consisting of two torsions, elements of finite order, for the mapping class group ${\rm Mod}(\Sigma_g)$. He~\cite{Korkmaz} proved that the group ${\rm Mod}(\Sigma_g)$
is generated by two elements of order $4g+2$. Korkmaz asked in~\cite{Korkmaz2012} the following problem:
What are the other numbers $k$ (less than $4g+2$) such that ${\rm Mod}(\Sigma_g)$ is generated by two elements of order $k$. What is the smallest such $k$?
Dan Margalit~\cite{Margalit} asked a quite similar question, too:
For which $k$ can ${\rm Mod}(\Sigma_g)$ be generated by two elements of order $k$?
We proved on Theorem ~\ref{thm:2} that the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by
two elements of order $g$ if $g \geq 6$. So, we showed that $k=g$ satisfies the desired result for $g \geq 6$.
Moreover, we found generating sets of two elements of smaller orders as shown in Theorem ~\ref{thm:1} and Theorem ~\ref{thm:3}.
\begin{theorem} \label{thm:2}
The mapping class group ${\rm Mod}(\Sigma_g)$ is generated by
two elements of order g for $g \geq 6$.
\end{theorem}
\begin{theorem} \label{thm:1}
For $g\geq 7$ the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by
two elements of order $g$ and order $g'$ where $g'$ is the least divisor of $g$ such that $g'>2$.
\end{theorem}
\begin{theorem} \label{thm:3}
For $g\geq 3k^2+4k+1$ and any positive integer $k$, the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by
two elements of order $g/$$\gcd(g,k)$.
\end{theorem}
Since there is a surjective
homomorphism from ${\rm Mod}(\Sigma_g)$ onto the symplectic group
${\rm Sp} (2g,{\mathbb{Z}})$,
we have the following immediate result:
\begin{corollary}
The symplectic group ${\rm Sp}(2g,{\mathbb{Z}})$ is generated
by two elements of order g for $g\geq 6$.
\end{corollary}
Firstly, Dehn~\cite{Dehn}
showed that ${\rm Mod}(\Sigma_g)$ is generated by $2g(g-1)$ many Dehn twists. Lickorish~\cite{Lickorish}
decreased this number to $3g-1$.
Humphries~\cite{Humphries} introduced a generating set of $2g+1$ many Dehn twists and proved that this is the least such number.
Lu~\cite{Lu} found a generating set of three elements and two of the generators are of finite order.
Maclachlan~\cite{Maclachlan} proved that ${\rm Mod}(\Sigma_g)$ can be generated by only using torsions. McCarthy and Papadopoulos~\cite{McCarthyPapadopoulos} showed
${\rm Mod}(\Sigma_g)$ can be generated by involutions. Stukow~\cite{Stukow2}
proved index five subgroup of ${\rm Mod}(\Sigma_2)$ is generated by using involutions.
Luo~\cite{Luo} found an upper bound that is necessary to generate ${\rm Mod}(\Sigma_g)$ by involutions.
After that, Brendle and Farb~\cite{BrendleFarb} showed that six involutions are enough.
Kassabov~\cite{Kassabov} decreased this to $4$ for $g\geq 7$ .
Recently, Korkmaz~\cite{Korkmaz3inv} introduced a generating set consisting of three involutions for $g\geq 8$ and of four involutions for $g\geq 3$.
Finally, we ~\cite{Yildiz} showed that ${\rm Mod}(\Sigma_g)$ is generated by three involutions if $g\geq 6$.
Since the mapping class group cannot be generated by two involutions for homological reasons, three was the least possible such number. Now, we are interested in
generating ${\rm Mod}(\Sigma_g)$ by two torsion elements of small orders.
Wajnryb~\cite{Wajnryb1983} found a presentation for the mapping class group of an orientable surface.
Wajnryb~\cite{Wajnryb1996} also proved that
${\rm Mod}(\Sigma_g)$ can be generated by two elements; one is of order $4g+2$ and the other
is a product of opposite Dehn twists.
After that, Korkmaz~\cite{Korkmaz} showed that ${\rm Mod}(\Sigma_g)$
is generated by an element of order $4g+2$ and a Dehn twist, improving Wajnryb's result.
He also proved that ${\rm Mod}(\Sigma_g)$ can be generated by two torsion elements of order $4g+2$.
Recently, Baykur and Korkmaz~\cite{Baykur} proved that the mapping class group can be generated by two commutators.
Moreover, Monden~\cite{Monden} proved that ${\rm Mod}(\Sigma_g)$ can be generated by three elements of order $3$ and by four elements of order $4$.
Lanier~\cite{Lanier} showed ${\rm Mod}(\Sigma_g)$ is generated by three elements of any order $k\geq 6$ if $g\geq (k-1)^2+1$.
See ~\cite{Du2} for generating set of torsion elements for the extended mapping class group.
\bigskip
\section{An overview of mapping class groups}
Throughout the paper only closed connected orientable surfaces, $\Sigma_g$,
where all genera are distributed as in Figure ~\ref{fig1} are considered. Note that the rotation $R$ by $2\pi$$/g$ degrees about $z$-axis is a well-defined
self-diffeomorphism of $\Sigma_g$. The mapping class group ${\rm Mod}(\Sigma_g)$ of a closed connected orientable
surface $\Sigma_g$ is the group of orientation--preserving diffeomorphisms of $\Sigma_g\to \Sigma_g$ up to isotopy.
Diffeomorphisms and curves are classified up to isotopy.
We refer to~\cite{FarbMargalit} for all other information on the mapping class groups.
We only deal with simple types of simple closed curves
$a_i$'s, $b_i$'s and $c_i$'s as shown on Figure ~\ref{fig1} where $1 \leq i \leq g$.
In order to align with the notation in ~\cite{Yildiz},
we show simple closed curves by lowercase letters $a_i$, $b_i$, $c_i$
and corresponding positive Dehn twists by uppercase letters $A_i$, $B_i$, $C_i$ or the usual notation $t_{a_i}$,$t_{b_i}$,$t_{c_i}$, respectively.
All indices are to be considered as modulo $g$.
For the composition of diffeomorphisms,
$f_1f_2$ means that $f_2$ is first and then $f_1$ comes second as usual.
Commutativity, Braid Relation and the following basic facts on the mapping class group are used along the paper for many times:
For any simple closed curves $c_1$ and $c_2$ on $\Sigma_g$ and diffeomorphism $f:\Sigma_g \to \Sigma_g$,
$ft_{c_1}f^{-1}=t_{f(c_1)}$; $c_1$ is isotopic to $c_2$ if and only if $t_{c_1}=t_{c_2}$ in ${\rm Mod}(\Sigma_g)$; and if $c_1$ and $c_2$ are disjoint, then $t_{c_1}(c_2)=c_2$.
After Dehn and Lickorish proved the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by $3g-1$ Dehn twists about nonseparating
simple closed curves, Humphries~\cite{Humphries} stated the following theorem.
\begin{theorem} {\rm\bf(Dehn-Lickorish-Humphries)}\label{thm:Humphries}
The mapping class group ${\rm Mod}(\Sigma_g)$ is generated by the set
$\{ A_1,A_2,B_1,B_2,\ldots ,B_g,C_1,C_2,\ldots ,C_{g-1}\}.$
\end{theorem}
Let $R$ denote the rotation by $2\pi/g$ about the $x$--axis represented in Figure~\ref{fig1}.
Then, \mbox{$R(a_k)=a_{k+1}$,} $R(b_k)=b_{k+1}$ and
$R(c_k)=c_{k+1}$.
Korkmaz~\cite{Korkmaz3inv} showed on the following handy theorem deducing from Theorem~\ref{thm:Humphries} that the mapping class group is generated the four elements. First element is the rotation element $R$
and others are products of one positive and one negative Dehn twists.
\begin{theorem} \label{thm:thmKorkmaz}
If $g\geq 3$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by the four elements
\(
R, A_1A_2^{-1},B_1B_2^{-1}, C_1C_2^{-1}. \)
\end{theorem}
The next result is easily deduced from Theorem ~\ref{thm:thmKorkmaz}.
\begin{corollary} \label{cor:thmKorkmaz}
If $g\geq 3$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by the four elements
\(
R, A_1B_1^{-1}, B_1C_1^{-1}, C_1B_2^{-1}. \)
\end{corollary}
\begin{proof}
Let $H$ be the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set\\
$\{ R, A_1B_1^{-1}, B_1C_1^{-1}, C_1B_2^{-1}.\}$.
Then, $B_2A_2^{-1} = R(B_1A_1^{-1})R^{-1}$$\in$$H$ and
$B_2C_2^{-1} = R(B_1C_1^{-1})R^{-1}\in$$H$.
$B_1B_2^{-1}=(B_1C_1^{-1})(C_1B_2^{-1})$$\in$$H$,
$C_1C_2^{-1}=(C_1B_2^{-1})(B_2C_2^{-1})$$\in$$H$ and
$A_1A_2^{-1}=(A_1B_1^{-1})(B_1B_2^{-1})(B_2A_2^{-1})$$\in$$H$.
It follows from Theorem~\ref{thm:thmKorkmaz} that
$H={\rm Mod}(\Sigma_g)$, completing the proof of the corollary.
\end{proof}
\begin{figure}
\begin{tikzpicture}[scale=0.45]
\begin{scope} [xshift=7cm]
\draw[very thick, violet] (0,0) circle [radius=5.5cm];
\draw[very thick, violet] (0,2.7) circle [radius=0.8cm];
\draw[very thick, violet, rotate=80] (0,2.7) circle [radius=0.8cm];
\draw[very thick, violet, rotate=-80] (0,2.7) circle [radius=0.8cm];
\draw[very thick, violet, fill ] (0,-2.7) circle [radius=0.03cm];
\draw[very thick, violet, fill, rotate=15] (0,-2.7) circle [radius=0.03cm];
\draw[very thick, violet, fill, rotate=-15] (0,-2.7) circle [radius=0.03cm];
\draw[thick, green, rounded corners=10pt] (-0.05, 3.5) ..controls (-0.6,3.8) and (-0.6,5.2).. (-0.05,5.5) ;
\draw[thick, green, dashed, rounded corners=10pt] (0.05,3.5)..controls (0.6,3.8) and (0.6,5.2).. (0.05,5.5) ;
\node at (-1,4.7) {$a_1$};
\draw[thick, green, rotate=80, rounded corners=10pt]
(-0.05, 3.5) ..controls (-0.6,3.8) and (-0.6,5.2).. (-0.05,5.5) ;
\draw[thick, green, dashed, rotate=80, rounded corners=10pt]
(0.05,3.5)..controls (0.6,3.8) and (0.6,5.2).. (0.05,5.5) ;
\node at (-4.6,-0.1) {$a_g$};
\draw[thick, green, rotate=-80, rounded corners=10pt]
(-0.05, 3.5) ..controls (-0.6,3.8) and (-0.6,5.2).. (-0.05,5.5) ;
\draw[thick, green, dashed, rotate=-80, rounded corners=10pt]
(0.05,3.5)..controls (0.6,3.8) and (0.6,5.2).. (0.05,5.5) ;
\node at (4.7,-0.1) {$a_2$};
\draw[thick, blue] (0,2.7) circle [radius=1.1cm];
\draw[thick, rotate=80, blue] (0,2.7) circle [radius=1.1cm];
\draw[thick, blue, rotate=-80] (0,2.7) circle [radius=1.1cm];
\node at (-3.5,2) {$b_g$};
\node at (3.2,2) {$b_2$};
\node at (1.4,3.7) {$b_1$};
\draw[thick, red, rounded corners=15pt] (-2.15, 1.05)-- (-1.856,2.233) -- (-0.7,2.28) ;
\draw[thick, red, dashed, rounded corners=15pt] (-2.03, 1)-- (-0.96,1.16) -- (-0.6,2.18) ;
\node at (-1.9,2.4) {$c_g$};
\draw[thick, red, rounded corners=15pt] (2.15, 1.05)-- (1.856,2.233) -- (0.7,2.28) ;
\draw[thick, red, dashed, rounded corners=15pt] (2.03, 1)-- (0.96,1.16) -- (0.6,2.18) ;
\node at (1.9,2.4) {$c_1$};
\draw[thick, red, rotate=80, rounded corners=10pt] (-2.1, 2)-- (-1.5,2.433) -- (-0.7,2.28) ;
\draw[thick, red, dashed, rotate=80, rounded corners=8pt] (-1.3, 1)-- (-0.9,1.36) -- (-0.6,2.18) ;
\node at (-3.2,-1.4) {$c_{g-1}$};
\draw[thick, red, rotate=-80, rounded corners=10pt] (2.1, 2)-- (1.5,2.433) -- (0.7,2.28) ;
\draw[thick, red, dashed, rotate=-80, rounded corners=8pt] (1.3, 1)-- (0.9,1.36) -- (0.6,2.18) ;
\node at (3,-1.4) {$c_2$};
\draw[->, rounded corners=20pt, rotate=-19.5] (0, 6.2)..controls (1.395, 6.039) and (2.716, 5.574).. (3.906,4.817);
\node[rotate=-40] at (4.28,5.284) {$R$};
\end{scope}
\node at (0.7+7, 6.7+0.5) {$y$};
\node at (6.7+7+0.5, 0.7) {$x$};
\draw[->, thick] (0+7,5.8)--(0+7, 8.3);
\draw[->, thick] (5.8+7,0)--(8.3+7,0);
\end{tikzpicture}
\caption{The curves $a_i,b_i,c_i$ and the rotation $R$
on the surface $\Sigma_g$.}
\label{fig1}
\end{figure}
\bigskip
\section{Twelve new generating sets for ${\rm Mod}(\Sigma_g)$.}
In this section, we obtain twelve new generating sets to
generate the mapping class group by two elements of small orders.
We follow the idea of Korkmaz in~\cite{Korkmaz3inv} to create our generating sets in the next corollaries to Theorem~\ref{thm:thmKorkmaz}.
Our idea is to use the rotation element $R$, a torsion element of order $g$ in the group ${\rm Mod}(\Sigma_g)$, as the first element and products of Dehn twists as the second element.
We use the first four corollaries to create generating sets of elements of order $g$.
In order to follow Theorem~\ref{thm:thmKorkmaz}, we need the rotation element $R$, which is of order $g$, in our generating set. So, we cannot decrease order $g$ by following this method.
But indeed, we can reduce the order of the second element. Applying Lemma~\ref{lem:order}, we see that most suitable candidates are the divisors of $g$. Unfortunately, it is not possible
to find an order $2$ element, an involution, as the second element in our generating sets by following this method. Instead, we found the least divisor of $g$ greater than $2$ as the minimal order for the second element.
We use the next six corollaries after the first four ones to create generating sets of elements of order $g$ and $g'$ where $g'$ is the least divisor of $g$ such that $g'>2$.
\begin{corollary} \label{cor:gen1}
If $g=6$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by
the two elements
$R$ and $C_1B_4A_6A_1^{-1}B_5^{-1}C_2^{-1}$.
\end{corollary}
\begin{proof}
Let $F_1=C_1B_4A_6A_1^{-1}B_5^{-1}C_2^{-1}$. Let us denote by
$H$ the subgroup of ${\rm Mod}(\Sigma_6)$ generated by the set
$\{ R, F_1\}$.
If $H$ contains the elements $A_1A_2^{-1}$, $B_1B_2^{-1}$ and $C_1C_2^{-1}$ then we are done by Theorem~\ref{thm:thmKorkmaz}.
Let
\begin{eqnarray*}
F_2
&=& RF_1R^{-1} \\
&=& R(C_1B_4A_6A_1^{-1}B_5^{-1}C_2^{-1})R^{-1}\\
&=& RC_1R^{-1}RB_4R^{-1}RA_6R^{-1}RA_1^{-1}R^{-1}RB_5^{-1}R^{-1}RC_2^{-1}R^{-1}\\
&=& Rt_{c_1}R^{-1}Rt_{b_4}R^{-1}Rt_{a_6}R^{-1}Rt_{a_1}^{-1}R^{-1}Rt_{b_5}^{-1}R^{-1}Rt_{c_2}^{-1}R^{-1}\\
&=& t_{R(c_1)}t_{R(b_4)}t_{R(a_6)}t_{R(a_1)}^{-1}t_{R(b_5)}^{-1}t_{R(c_2)}^{-1}\\
&=& t_{c_2}t_{b_5}t_{a_1}t_{a_2}^{-1}t_{b_6}^{-1}t_{c_3}^{-1}\\
&=& C_2B_5A_1A_2^{-1}B_6^{-1}C_3^{-1}
\end{eqnarray*}
and \[
F_3 = F_2^{-1} = C_3B_6A_2A_1^{-1}B_5^{-1}C_2^{-1}.
\]
\begin{figure}
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\node[scale=0.8] at (-1.5,2.5) {$F_1$};
\node[scale=0.8] at (4.3,2.5) {$F_2$};
\node[scale=0.8] at (10.1,2.5) {$F_3$};
\node[scale=0.8] at (7.2,-3.1) {$F_4$};
\node[scale=0.8] at (1.4,-3.1) {$F_5$};
\node[scale=0.8] at (1.4,-8.7) {$F_6$};
\node[scale=0.8] at (7.2,-8.7) {$F_7$};
\draw[ ->, rounded corners=5pt] (2.4,1.5)--(2.9,1.6)-- (3.4,1.5);
\node[scale=0.8] at (2.9,2) {$R$};
\draw[ ->, rounded corners=5pt] (11.1,-2.7)--(11, -3.3)--(10.7,-3.7);
\node[scale=0.8] at (11.6,-3.2) {$F_3F_1$};
\draw[ ->, rounded corners=5pt] (5.7+0.97,-3.92)--(5.7+0.07,-3.84)--(5.7-0.83,-3.92) ;
\node[scale=0.8] at (5.7,-3.53) {$ \cdot (C_2A_3^{-1})$};
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\caption{Proof of Corollary~\ref{cor:gen1}.}
\end{figure}
Then, $F_3F_1(c_3,b_6,a_2,a_1,b_5,c_2)=(b_4,a_6,a_2,a_1,b_5,c_2)$ so that\\
$F_4=B_4A_6A_2A_1^{-1}B_5^{-1}C_2^{-1}\in H$. Note that $F_3F_1(c_3) = b_4$
since \begin{eqnarray*}
t_{F_3F_1(c_3)}
&=& (F_3F_1)t_{c_3}(F_3F_1)^{-1}\\
&=& F_3F_1C_3F_1^{-1}F_3^{-1}\\
&=& C_3B_4C_3B_4^{-1}C_3^{-1}\\
&=& (t_{c_3}t_{b_4})t_{c_3}(t_{c_3}t_{b_4})^{-1}\\
&=& t_{t_{c_3}t_{b_4}(c_3)}\\
&=& t_{b_4}.
\end{eqnarray*}
$F_1F_4^{-1}=C_1A_2^{-1}$$\in$$H$ and then by conjugating $C_1A_2^{-1}$ with $R$ iteratively, we get $C_iA_{i+1}^{-1}$$\in$$H$ $\forall i$.
Let \[
F_5 = F_4(C_2A_3^{-1}) = B_4A_6A_2A_1^{-1}B_5^{-1}A_3^{-1}
\]
\[
F_6 = RF_5R^{-1} = B_5A_1A_3A_2^{-1}B_6^{-1}A_4^{-1}.
\]
and
\[
F_7 = F_5F_6 = B_4A_6B_6^{-1}A_4^{-1}.
\]
$(C_4A_5^{-1})F_7(c_4,a_5)=(b_4,a_5)$ so that $B_4A_5^{-1}$$\in$$H$ and then \mbox{$B_iA_{i+1}^{-1}$$\in$$H$ $\forall i$}.\\
$B_iC_i^{-1}=(B_iA_{i+1}^{-1})(A_{i+1}C_i^{-1})$$\in$$H$ $\forall i$.
$(A_4B_3^{-1})F_7(a_4,b_3)=(b_4,b_3)$ so that $B_4B_3^{-1}$$\in$$H$ and then \mbox{$B_{i+1}B_i^{-1}$$\in$$H$ $\forall i$}.
In particular, $B_1B_2^{-1}$$\in$$H$.
$C_1C_2^{-1}=(C_1B_1^{-1})(B_1B_2^{-1})(B_2C_2^{-1})$$\in$$H$.
$A_1A_2^{-1}=(A_1B_6^{-1})(B_6B_1^{-1})(B_1A_2^{-1})$$\in$$H$.
It follows from Theorem~\ref{thm:thmKorkmaz} that
$H={\rm Mod}(\Sigma_6)$, completing the proof of the corollary.
\end{proof}
\begin{corollary} \label{cor:gen2}
If $g=7$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by two elements
$R$ and $C_1B_4A_6A_7^{-1}B_5^{-1}C_2^{-1}$.
\end{corollary}
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\draw[thick, blue, rotate=360/7*6, rounded corners=8pt] (-1.09,1.17)--(-0.58, 1.27)--(-0.28,1.5);
\draw[thick, blue, dashed, rotate=360/7*6, rounded corners=8pt] (-1.09,1.21)--(-0.7, 1.44)--(-0.3,1.54);
\draw[thick, red, rotate=360/7*5, rounded corners=8pt] (-1.09,1.17)--(-0.58, 1.27)--(-0.28,1.5);
\draw[thick, red, dashed, rotate=360/7*5, rounded corners=8pt] (-1.09,1.21)--(-0.7, 1.44)--(-0.3,1.54);
\draw[thick, blue, rotate=360/7*4] (0,1.6) circle [radius=0.3cm];
\draw[thick, red, rotate=360/7*3] (0,1.6) circle [radius=0.3cm];
\draw[thick, blue, rotate=360/7*2, rounded corners=6pt] (0.04,1.9)--(0.14, 2.15)--(0.04,2.4);
\draw[thick, blue, dashed, rotate=360/7*2, rounded corners=6pt] (-0.04,1.9)--(-0.14, 2.15)--(-0.04,2.4);
\draw[thick, dashed, red, rotate=360/7, rounded corners=6pt] (0.04,1.9)--(0.14, 2.15)--(0.04,2.4);
\draw[thick, red, rotate=360/7, rounded corners=6pt] (-0.04,1.9)--(-0.14, 2.15)--(-0.04,2.4);
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\node[scale=0.6, red] at (1.2,0.3) {$-$};
\node[scale=0.6, blue] at (-2.1,-0.2) {$+$};
\node[scale=0.6, blue] at (0.5,-1.0) {$+$};
\node[scale=0.6, red] at (-0.5,-1.0) {$-$};
\end{scope}
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\draw[very thick, violet] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*2] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*3] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*4] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*5] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*6] (0,1.6) circle [radius=0.2cm];
\draw[thick, blue, rotate=360/7*6, rounded corners=8pt] (-1.09,1.17)--(-0.58, 1.27)--(-0.28,1.5);
\draw[thick, blue, dashed, rotate=360/7*6, rounded corners=8pt] (-1.09,1.21)--(-0.7, 1.44)--(-0.3,1.54);
\draw[thick, red, rotate=360/7*5, rounded corners=8pt] (-1.09,1.17)--(-0.58, 1.27)--(-0.28,1.5);
\draw[thick, red, dashed, rotate=360/7*5, rounded corners=8pt] (-1.09,1.21)--(-0.7, 1.44)--(-0.3,1.54);
\draw[thick, blue, rotate=360/7*4] (0,1.6) circle [radius=0.3cm];
\draw[thick, red, rotate=360/7*3] (0,1.6) circle [radius=0.3cm];
\draw[thick, blue, rotate=360/7*2, rounded corners=6pt] (0.04,1.9)--(0.14, 2.15)--(0.04,2.4);
\draw[thick, blue, dashed, rotate=360/7*2, rounded corners=6pt] (-0.04,1.9)--(-0.14, 2.15)--(-0.04,2.4);
\draw[thick, dashed, red, rotate=360/7, rounded corners=6pt] (0.04,1.9)--(0.14, 2.15)--(0.04,2.4);
\draw[thick, red, rotate=360/7, rounded corners=6pt] (-0.04,1.9)--(-0.14, 2.15)--(-0.04,2.4);
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\node[scale=0.6, blue] at (-2.1,-0.2) {$+$};
\node[scale=0.6, blue] at (0.5,-1.0) {$+$};
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\end{scope}
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\draw[very thick, violet] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*2] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*3] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*4] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*5] (0,1.6) circle [radius=0.2cm];
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\draw[thick, red, dashed, rotate=360/7*6, rounded corners=8pt] (-1.09,1.21)--(-0.7, 1.44)--(-0.3,1.54);
\draw[thick, blue, rotate=360/7*5, rounded corners=8pt] (-1.09,1.17)--(-0.58, 1.27)--(-0.28,1.5);
\draw[thick, blue, dashed, rotate=360/7*5, rounded corners=8pt] (-1.09,1.21)--(-0.7, 1.44)--(-0.3,1.54);
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\draw[thick, red, dashed, rotate=360/7*2, rounded corners=6pt] (-0.04,1.9)--(-0.14, 2.15)--(-0.04,2.4);
\draw[thick, dashed, blue, rotate=360/7, rounded corners=6pt] (0.04,1.9)--(0.14, 2.15)--(0.04,2.4);
\draw[thick, blue, rotate=360/7, rounded corners=6pt] (-0.04,1.9)--(-0.14, 2.15)--(-0.04,2.4);
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\node[scale=0.6, blue] at (-0.5,-1.0) {$+$};
\end{scope}
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\draw[very thick, violet] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*2] (0,1.6) circle [radius=0.2cm];
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\draw[thick, red, rotate=360/7*6, rounded corners=8pt] (-1.09,1.17)--(-0.58, 1.27)--(-0.28,1.5);
\draw[thick, red, dashed, rotate=360/7*6, rounded corners=8pt] (-1.09,1.21)--(-0.7, 1.44)--(-0.3,1.54);
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\draw[thick, red, dashed, rotate=360/7*2, rounded corners=6pt] (-0.04,1.9)--(-0.14, 2.15)--(-0.04,2.4);
\draw[thick, dashed, blue, rotate=360/7, rounded corners=6pt] (0.04,1.9)--(0.14, 2.15)--(0.04,2.4);
\draw[thick, blue, rotate=360/7, rounded corners=6pt] (-0.04,1.9)--(-0.14, 2.15)--(-0.04,2.4);
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\end{scope}
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\draw[very thick, violet] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*2] (0,1.6) circle [radius=0.2cm];
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\draw[very thick, violet, rotate=360/7*4] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*5] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*6] (0,1.6) circle [radius=0.2cm];
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\draw[thick, red, dashed, rotate=360/7*6, rounded corners=8pt] (-1.09,1.21)--(-0.7, 1.44)--(-0.3,1.54);
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\draw[thick, red, dashed, rotate=360/7*2, rounded corners=6pt] (-0.04,1.9)--(-0.14, 2.15)--(-0.04,2.4);
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\node[scale=0.6, blue] at (-1.45,1.5) {$+$};
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\node[scale=0.6, blue] at (1.1,-0.1) {$+$};
\node[scale=0.6, red] at (-2.1,-0.2) {$-$};
\node[scale=0.6, red] at (0.5,-1.0) {$-$};
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\end{scope}
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\draw[very thick, violet] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*2] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*3] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*4] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*5] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*6] (0,1.6) circle [radius=0.2cm];
\draw[thick, blue, rotate=360/7*6, rounded corners=8pt] (-1.09,1.17)--(-0.58, 1.27)--(-0.28,1.5);
\draw[thick, blue, dashed, rotate=360/7*6, rounded corners=8pt] (-1.09,1.21)--(-0.7, 1.44)--(-0.3,1.54);
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\draw[thick, blue, dashed, rotate=360/7*2, rounded corners=6pt] (-0.04,1.9)--(-0.14, 2.15)--(-0.04,2.4);
\draw[thick, dashed, red, rotate=360/7, rounded corners=6pt] (0.04,1.9)--(0.14, 2.15)--(0.04,2.4);
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\node[scale=0.6, blue] at (-2.1,-0.2) {$+$};
\node[scale=0.6, blue] at (0.5,-1.0) {$+$};
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\end{scope}
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\draw[very thick, violet] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*2] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*3] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*4] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*5] (0,1.6) circle [radius=0.2cm];
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\draw[thick, blue, rotate=360/7*4] (0,1.6) circle [radius=0.3cm];
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\draw[thick, blue, rotate=360/7*2, rounded corners=6pt] (0.04,1.9)--(0.14, 2.15)--(0.04,2.4);
\draw[thick, blue, dashed, rotate=360/7*2, rounded corners=6pt] (-0.04,1.9)--(-0.14, 2.15)--(-0.04,2.4);
\draw[thick, dashed, red, rotate=360/7, rounded corners=6pt] (0.04,1.9)--(0.14, 2.15)--(0.04,2.4);
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\node[scale=0.6, blue] at (0.5,-1.0) {$+$};
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\end{scope}
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\draw[very thick, violet] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7] (0,1.6) circle [radius=0.2cm];
\draw[very thick, violet, rotate=360/7*2] (0,1.6) circle [radius=0.2cm];
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\draw[very thick, violet, rotate=360/7*4] (0,1.6) circle [radius=0.2cm];
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\draw[thick, blue, rotate=360/7*4] (0,1.6) circle [radius=0.3cm];
\draw[thick, red, rotate=360/7*3] (0,1.6) circle [radius=0.3cm];
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\draw[thick, blue, dashed, rotate=360/7*2, rounded corners=6pt] (-0.04,1.9)--(-0.14, 2.15)--(-0.04,2.4);
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\end{scope}
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\end{scope}
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\node[scale=0.8] at (-1.5,2.5) {$F_1$};
\node[scale=0.8] at (4.3,2.5) {$F_2$};
\node[scale=0.8] at (10.1,2.5) {$F_3$};
\node[scale=0.8] at (7.2,-3.1) {$F_4$};
\node[scale=0.8] at (1.4,-3.1) {$F_5$};
\node[scale=0.8] at (-1.5,-8.7) {$F_6$};
\node[scale=0.8] at (4.3,-8.7) {$F_7$};
\node[scale=0.8] at (10.1,-8.7) {$F_8$};
\node[scale=0.8] at (-1.5,-14.3) {$F_9$};
\node[scale=0.8] at (4.3,-14.3) {$F_{10}$};
\node[scale=0.8] at (10.1,-14.3) {$F_{11}$};
\node[scale=0.8] at (-1.5,-14.3-5.6) {$F_{14}$};
\node[scale=0.8] at (4.3,-14.3-5.6) {$F_{13}$};
\node[scale=0.8] at (10.1,-14.3-5.6) {$F_{12}$};
\draw[ ->, rounded corners=5pt] (2.4,1.5)--(2.9,1.6)-- (3.4,1.5);
\node[scale=0.8] at (2.9,2) {$R$};
\draw[ ->, rounded corners=5pt] (11.1,-2.7)--(11, -3.3)--(10.7,-3.7);
\node[scale=0.8] at (11.6,-3.2) {$F_3F_1$};
\draw[ ->, rounded corners=5pt] (3.3+3,1.5-5.8)--(2.8+3,1.6-5.8)-- (2.3+3,1.5-5.8);
\node[scale=0.8] at (5.8,2-5.8) {$R$};
\draw[ ->, rounded corners=5pt] (2.4,-9.8)--(2.9,-9.7)-- (3.4,-9.8);
\node[scale=0.8] at (2.83,-9.3) {$F_6F_4$};
\draw[ ->, rounded corners=5pt] (8.2,-9.8)--(8.7,-9.7)-- (9.2,-9.8);
\node[scale=0.8] at (8.7,-9.4) {$R$};
\draw[ ->, rounded corners=5pt] (8.2-5.8,-15.4)--(8.7-5.8,-15.3)-- (9.2-5.8,-15.4);
\node[scale=0.8] at (8.7-5.8,-15.0) {$F_9F_7$};
\draw[ ->, rounded corners=5pt] (11.1+2.2,-2.7-16.3)--(11+2.2, -3.3-16.3)--(10.7+2.2,-3.7-16.3);
\node[scale=0.8] at (11.6+2.2,-3.2-16.3) {$R^{-1}$};
\draw[ ->, rounded corners=5pt] (9.2-5.8,-15.4-5.8)--(8.7-5.8,-15.3-5.8)--(8.2-5.8,-15.4-5.8);
\node[scale=0.8] at (8.7-5.8,-15.0-5.8) {$R$};
\end{tikzpicture}
\caption{Proof of Corollary~\ref{cor:gen2}.}
\end{figure}
\begin{proof}
Let $F_1=C_1B_4A_6A_7^{-1}B_5^{-1}C_2^{-1}$. Let us denote by
$H$ the subgroup of ${\rm Mod}(\Sigma_7)$ generated by the set
$\{ R, F_1\}$.
Let
\[
F_2 = RF_1R^{-1} = C_2B_5A_7A_1^{-1}B_6^{-1}C_3^{-1}
\]
and \[
F_3 = F_2^{-1} = C_3B_6A_1A_7^{-1}B_5^{-1}C_2^{-1}.
\]
Then, $F_3F_1(c_3,b_6,a_1,a_7,b_5,c_2)=(b_4,a_6,a_1,a_7,b_5,c_2)$ so that\\
$F_4=B_4A_6A_1A_7^{-1}B_5^{-1}C_2^{-1}\in H$.
Let \[
F_5 = RF_4R^{-1} = B_5A_7A_2A_1^{-1}B_6^{-1}C_3^{-1}
\]
and \[
F_6 = F_5^{-1} = C_3B_6A_1A_2^{-1}A_7^{-1}B_5^{-1}.
\]
Then, $F_6F_4(c_3,b_6,a_1,a_2,a_7,b_5)=(b_4,a_6,a_1,a_2,a_7,b_5)$ so that\\
$F_7=B_4A_6A_1A_2^{-1}A_7^{-1}B_5^{-1}$$\in$$H$.
Let \[
F_8 = RF_7R^{-1} = B_5A_7A_2A_3^{-1}A_1^{-1}B_6^{-1}
\]
and \[
F_9 = F_8^{-1} = B_6A_1A_3A_2^{-1}A_7^{-1}B_5^{-1}.
\]
$F_9F_7(b_6,a_1,a_3,a_2,a_7,b_5)=(a_6,a_1,a_3,a_2,a_7,b_5)$ so that\\
$F_{10}=A_6A_1A_3A_2^{-1}A_7^{-1}B_5^{-1}$$\in$$H$.
$F_{10}F_8=A_6B_6^{-1}$$\in$$H$ and then by conjugating $A_6B_6^{-1}$ with $R$ iteratively, we get $A_iB_i^{-1}$$\in$$H$ $\forall i$.
Let \[
F_{11} = (B_6A_6^{-1})F_4 = B_4B_6A_1A_7^{-1}B_5^{-1}C_2^{-1}
\]
and \[
F_{12} = R^{-1}F_{11}R = B_3B_5A_7A_6^{-1}B_4^{-1}C_1^{-1}.
\]
$F_{12}F_1=B_3C_2^{-1}$$\in$$H$ and then $B_{i+1}C_i^{-1}$$\in$$H$ $\forall i$.
Let \[
F_{13} = (B_6A_6^{-1})F_1(A_7B_7^{-1}) = C_1B_4B_6B_7^{-1}B_5^{-1}C_2^{-1}
\]
and \[
F_{14} = RF_{13}R^{-1} = C_2B_5B_7B_1^{-1}B_6^{-1}C_3^{-1}.
\]
Then $F_{13}F_{14}(C_3B_4^{-1})=C_1B_1^{-1}$$\in$$H$ and then $C_iB_i^{-1}$$\in$$H$ $\forall i$.
It follows from Corollary~\ref{cor:thmKorkmaz} that
$H={\rm Mod}(\Sigma_7)$, completing the proof of the corollary.
\end{proof}
\begin{corollary} \label{cor:gen3}
If $g=8$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by two elements
$R$ and $B_1C_4A_7A_8^{-1}C_5^{-1}B_2^{-1}$.
\end{corollary}
\begin{figure}
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\end{scope}
\node[scale=0.8] at (-1.5,2.5) {$F_1$};
\node[scale=0.8] at (4.3,2.5) {$F_2$};
\node[scale=0.8] at (10.1,2.5) {$F_3$};
\node[scale=0.8] at (7.2,-3.1) {$F_4$};
\node[scale=0.8] at (1.4,-3.1) {$F_5$};
\node[scale=0.8] at (-1.5,-8.7) {$F_6$};
\node[scale=0.8] at (4.3,-8.7) {$F_7$};
\node[scale=0.8] at (10.1,-8.7) {$F_8$};
\node[scale=0.8] at (-1.5,-14.3) {$F_9$};
\node[scale=0.8] at (4.3,-14.3) {$F_{10}$};
\node[scale=0.8] at (10.1,-14.3) {$F_{11}$};
\draw[ ->, rounded corners=5pt] (2.4,1.5)--(2.9,1.6)-- (3.4,1.5);
\node[scale=0.8] at (2.9,2) {$R$};
\draw[ ->, rounded corners=5pt] (11.1,-2.7)--(11, -3.3)--(10.7,-3.7);
\node[scale=0.8] at (11.6,-3.2) {$F_3F_1$};
\draw[ ->, rounded corners=5pt] (0.3,-2.7)--(0.4,-3.3)-- (0.7,-3.7);
\node[scale=0.8] at (0.0,-3.2) {$R^2$};
\draw[ ->, rounded corners=5pt] (2.0,-9.22)--(2.9,-9.14)-- (3.8,-9.22);
\node[scale=0.8] at (2.83,-8.83) {$(B_2A_2^{-1}) \cdot $};
\node[scale=0.8] at (2.88,-9.5) {$ \cdot (A_1B_1^{-1})$};
\draw[ ->, rounded corners=5pt] (8.2,-9.8)--(8.7,-9.7)-- (9.2,-9.8);
\node[scale=0.8] at (8.7,-9.4) {$F_7F_1$};
\draw[ ->, rounded corners=5pt] (3.4,-13.3)--(2.5,-13.7)-- (1.6,-14.4);
\node[scale=0.8] at (2.4,-13.5) {$R$};
\draw[ ->, rounded corners=5pt] (8.2,-15.4)--(8.7,-15.3)-- (9.2,-15.4);
\node[scale=0.8] at (8.7,-15.0) {$F_{10}F_8$};
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\caption{Proof of Corollary~\ref{cor:gen3}.}
\end{figure}
\begin{proof}
Let $F_1=B_1C_4A_7A_8^{-1}C_5^{-1}B_2^{-1}$. Let us denote by
$H$ the subgroup of ${\rm Mod}(\Sigma_8)$ generated by the set
$\{ R, F_1\}$.
Let
\[
F_2 = RF_1R^{-1} = B_2C_5A_8A_1^{-1}C_6^{-1}B_3^{-1}
\]
and \[
F_3 = F_2^{-1} = B_3C_6A_1A_8^{-1}C_5^{-1}B_2^{-1}.\]
Then, $F_3F_1(b_3,c_6,a_1,a_8,c_5,b_2)=(b_3,c_6,b_1,a_8,c_5,b_2)$ so that\\
$F_4=B_3C_6B_1A_8^{-1}C_5^{-1}B_2^{-1}\in H$.
$F_4F_3^{-1}=B_1A_1^{-1}$$\in$$H$ and then by conjugating $B_1A_1^{-1}$ with $R$ iteratively, we get $B_iA_i^{-1}$$\in$$H$ $\forall i$.
Let \[
F_5 = R^2F_1R^{-2} = B_3C_6A_1A_2^{-1}C_7^{-1}B_4^{-1}
\]
\[
F_6 = F_5^{-1} = B_4C_7A_2A_1^{-1}C_6^{-1}B_3^{-1}\]
and \[
F_7 = (B_2A_2^{-1})F_6(A_1B_1^{-1}) = B_4C_7B_2B_1^{-1}C_6^{-1}B_3^{-1}.\]
Then, $F_7F_1(b_4,c_7,b_2,b_1,c_6,b_3)=(c_4,c_7,b_2,b_1,c_6,b_3)$ so that\\
$F_8=C_4C_7B_2B_1^{-1}C_6^{-1}B_3^{-1}$$\in$$H$.
$F_8F_7^{-1}=C_4B_4^{-1}$$\in$$H$ and then we get $C_iB_i^{-1}$$\in$$H$ $\forall i$.
Let \[
F_9 = RF_7R^{-1} = B_5C_8B_3B_2^{-1}C_7^{-1}B_4^{-1} \]
and \[
F_{10} = (C_4B_4^{-1})F_9^{-1}(B_5C_5^{-1}) = C_4C_7B_2B_3^{-1}C_8^{-1}C_5^{-1}.
\]
Then, $F_{10}F_8(c_4,c_7,b_2,b_3,c_8,c_5)=(c_4,c_7,b_2,b_3,b_1,c_5)$ so that\\
$F_{11}=C_4C_7B_2B_3^{-1}B_1^{-1}C_5^{-1}$$\in$$H$.
$F_{10}^{-1}F_{11}=C_8B_1^{-1}$$\in$$H$ and then we get $C_iB_{i+1}^{-1}$$\in$$H$ $\forall i$.
It follows from Corollary~\ref{cor:thmKorkmaz} that
$H={\rm Mod}(\Sigma_8)$, completing the proof of the corollary.
\end{proof}
\begin{corollary} \label{cor:gen4}
If $g\geq 9$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by two elements
$R$ and $C_1B_4A_7A_8^{-1}B_5^{-1}C_2^{-1}$.
\end{corollary}
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\node[scale=0.8] at (-1.5,2.5) {$F_1$};
\node[scale=1.3, violet] at (-1.5+0.56,2.5-1.31) {*};
\node[scale=0.8] at (4.3,2.5) {$F_2$};
\node[scale=1.3, violet] at (4.3+0.56,2.5-1.31) {*};
\node[scale=0.8] at (10.1,2.5) {$F_3$};
\node[scale=1.3, violet] at (10.1+0.56,2.5-1.31) {*};
\node[scale=0.8] at (10.1,-3.1) {$F_4$};
\node[scale=1.3, violet] at (10.1+0.56,-3.1-1.31) {*};
\node[scale=0.8] at (4.3,-3.1) {$F_5$};
\node[scale=1.3, violet] at (4.3+0.56,-3.1-1.31) {*};
\node[scale=0.8] at (-1.5,-3.1) {$F_6$};
\node[scale=1.3, violet] at (-1.5+0.56,-3.1-1.31) {*};
\node[scale=0.8] at (-1.5,-8.7) {$F_7$};
\node[scale=1.3, violet] at (-1.5+0.56,-8.7-1.31) {*};
\node[scale=0.8] at (4.3,-8.7) {$F_8$};
\node[scale=1.3, violet] at (4.3+0.56,-8.7-1.31) {*};
\node[scale=0.8] at (10.1,-8.7) {$F_9$};
\node[scale=1.3, violet] at (10.1+0.56,-8.7-1.31) {*};
\node[scale=0.8] at (-1.5,-14.3) {$F_{12}$};
\node[scale=1.3, violet] at (-1.5+0.56,-14.3-1.31) {*};
\node[scale=0.8] at (4.3,-14.3) {$F_{11}$};
\node[scale=1.3, violet] at (4.3+0.56,-14.3-1.31) {*};
\node[scale=0.8] at (10.1,-14.3) {$F_{10}$};
\node[scale=1.3, violet] at (10.1+0.56,-14.3-1.31) {*};
\draw[ ->, rounded corners=5pt] (2.4,1.5)--(2.9,1.6)-- (3.4,1.5);
\node[scale=0.8] at (2.9,2) {$R$};
\draw[ ->, rounded corners=5pt] (11.1+2.2,-2.7+0.4)--(11+2.2, -3.3+0.4)--(10.7+2.2,-3.7+0.4);
\node[scale=0.8] at (11.6+2.2,-3.2+0.4) {$F_3F_1$};
\draw[ <-, rounded corners=5pt] (2.0+5.8,-9.32+5.5)--(2.9+5.8,-9.24+5.5)-- (3.8+5.8,-9.32+5.5);
\node[scale=0.8] at (2.88+5.8,-8.93+5.5) {$ \cdot (C_2B_3^{-1})$};
\draw[ <-, rounded corners=5pt] (8.2-5.8,-9.8+5.6)--(8.7-5.8,-9.7+5.6)-- (9.2-5.8,-9.8+5.6);
\node[scale=0.8] at (8.7-5.8,-9.4+5.6) {$R^{-2}$};
\draw[ ->, rounded corners=5pt] (8.2-5.8,-9.8)--(8.7-5.8,-9.7)-- (9.2-5.8,-9.8);
\node[scale=0.8] at (8.7-5.8,-9.4) {$F_7F_5$};
\draw[ ->, rounded corners=5pt] (1.6+6.0,-13.3+5.4)--(2.5+6,-13.7+5.4)-- (3.4+6,-14.4+5.4);
\node[scale=0.8] at (2.4+5.5,-13.7+5) {$ \cdot (A_8B_8^{-1})$};
\node[scale=0.8] at (2.4+6.6,-13.7+5+0.7) {$ \cdot (B_8C_7^{-1})$};
\draw[ ->, rounded corners=5pt] (11.1+2.2,-2.7+0.4-11.2)--(11+2.2, -3.3+0.4-11.2)--(10.7+2.2,-3.7+0.4-11.2);
\node[scale=0.8] at (11.6+2.2,-3.2+0.4-11.2) {$R^{-1}$};
\draw[ <-, rounded corners=5pt] (8.2-5.8,-15.4)--(8.7-5.8,-15.3)-- (9.2-5.8,-15.4);
\node[scale=0.8] at (8.7-5.8,-15.0) {$F_{11}F_9$};
\node[scale=1.0] at (6.0-5.8,-15.5-4.85) {Note that};
\node[scale=1.2, violet] at (0+1.8,-15.5-4.95) {*};
\node[scale=1.0] at (9.7-5.8+1.1,-15.5-4.9) {represents $g-9$ genera.};
\end{tikzpicture}
\caption{Proof of Corollary~\ref{cor:gen4}.}
\end{figure}
\begin{proof}
Let $F_1=C_1B_4A_7A_8^{-1}B_5^{-1}C_2^{-1}$. Let us denote by
$H$ the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set
$\{ R, F_1\}$.
Let
\[
F_2 = RF_1R^{-1} = C_2B_5A_8A_9^{-1}B_6^{-1}C_3^{-1}
\]
and \[
F_3 = F_2^{-1} = C_3B_6A_9A_8^{-1}B_5^{-1}C_2^{-1}.
\]
Then, $F_3F_1(c_3,b_6,a_9,a_8,b_5,c_2)=(b_4,b_6,a_9,a_8,b_5,c_2)$ so that\\
$F_4=B_4B_6A_9A_8^{-1}B_5^{-1}C_2^{-1}\in H$.
$F_4F_3^{-1}=B_4C_3^{-1}$$\in$$H$ and then by conjugating $B_4C_3^{-1}$ with $R$ iteratively, we get $B_{i+1}C_i^{-1}$$\in$$H$ $\forall i$.
Let \[
F_5 = F_4(C_2B_3^{-1}) = B_4B_6A_9A_8^{-1}B_5^{-1}B_3^{-1}
\]
\[
F_6 = R^{-2}F_5R^2 = B_2B_4A_7A_6^{-1}B_3^{-1}B_1^{-1}
\]
and \[
F_7 = F_6^{-1} = B_1B_3A_6A_7^{-1}B_4^{-1}B_2^{-1}.
\]
Then, $F_7F_5(b_1,b_3,a_6,a_7,b_4,b_2)=(b_1,b_3,b_6,a_7,b_4,b_2)$ so that\\
$F_8=B_1B_3B_6A_7^{-1}B_4^{-1}B_2^{-1}$$\in$$H$.
Then, $F_8F_7^{-1}= B_6A_6^{-1}$$\in$$H$ and then \mbox{$B_iA_i^{-1}$$\in$$H$ $\forall i$}.
Let \[
F_9 = F_5(A_8B_8^{-1})(B_8C_7^{-1}) = B_4B_6A_9C_7^{-1}B_5^{-1}B_3^{-1}
\]
\[
F_{10} = R^{-1}F_9R = B_3B_5A_8C_6^{-1}B_4^{-1}B_2^{-1}
\]
and \[
F_{11} = F_{10}^{-1} = B_2B_4C_6A_8^{-1}B_5^{-1}B_3^{-1}.
\]
Then, $F_{11}F_9(b_2,b_4,c_6,a_8,b_5,b_3)=(b_2,b_4,b_6,a_8,b_5,b_3)$ so that\\
$F_{12}=B_2B_4B_6A_8^{-1}B_5^{-1}B_3^{-1}$$\in$$H$.
Then, $F_{12}F_{11}^{-1}= B_6C_6^{-1}$$\in$$H$ and then \mbox{$B_iC_i^{-1}$$\in$$H$ $\forall i$}.
It follows from Corollary~\ref{cor:thmKorkmaz} that
$H={\rm Mod}(\Sigma_g)$, completing the proof of the corollary.
\end{proof}
\begin{corollary} \label{cor:gen12}
If $g=8$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by two elements
$R$ and $B_1A_5C_5C_7^{-1}A_7^{-1}B_3^{-1}$.
\end{corollary}
\begin{proof}
Let $F_1=B_1A_5C_5C_7^{-1}A_7^{-1}B_3^{-1}$. Let us denote by
$H$ the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set
$\{ R, F_1\}$.
Let
\[
F_2 = RF_1R^{-1} = B_2A_6C_6C_8^{-1}A_8^{-1}B_4^{-1}
\]
and \[
F_3 = F_2^{-1} = B_4A_8C_8C_6^{-1}A_6^{-1}B_2^{-1}.
\]
Then, $F_3F_1(b_4,a_8,c_8,c_6,a_6,b_2)=(b_4,a_8,b_1,c_6,a_6,b_2)$ so that\\
$F_4=B_4A_8B_1C_6^{-1}A_6^{-1}B_2^{-1}\in H$.
$F_4F_3^{-1}=B_1C_8^{-1}$$\in$$H$ and then by conjugating $B_1C_8^{-1}$ with $R$ iteratively, we get $B_{i+1}C_i^{-1}$$\in$$H$ $\forall i$.
Let \[
F_5 = RF_4R^{-1} = B_5A_1B_2C_7^{-1}A_7^{-1}B_3^{-1}.
\]
Then, $F_5F_4(b_5,a_1,b_2,c_7,a_7,b_3)=(b_5,b_1,b_2,c_7,a_7,b_3)$ so that\\
$F_6=B_5B_1B_2C_7^{-1}A_7^{-1}B_3^{-1}$$\in$$H$.
Then, $F_6F_5^{-1}= B_1A_1^{-1}$$\in$$H$ and then \mbox{$B_iA_i^{-1}$$\in$$H$ $\forall i$}.
Let \[
F_7 = (C_4B_5^{-1})F_6(C_7B_8^{-1})(A_7B_7^{-1}) = C_4B_1B_2B_3^{-1}B_8^{-1}B_7^{-1}
\]
\[
F_8 = RF_9R^{-1} = C_5B_2B_3B_4^{-1}B_1^{-1}B_8^{-1}
\]
and \[
F_9 = F_8^{-1} = B_8B_1B_4B_3^{-1}B_2^{-1}C_5^{-1}.
\]
Then, $F_9F_7(b_8,b_1,b_4,b_3,b_2,c_5)=(b_8,b_1,c_4,b_3,b_2,c_5)$ so that\\
$F_{10}=B_8B_1C_4B_3^{-1}B_2^{-1}C_5^{-1}$$\in$$H$.
Then, $F_{10}F_9^{-1}= C_4B_4^{-1}$$\in$$H$ and then \mbox{$C_iB_i^{-1}$$\in$$H$ $\forall i$}.
It follows from Corollary~\ref{cor:thmKorkmaz} that
$H={\rm Mod}(\Sigma_8)$, completing the proof of the corollary.
\end{proof}
\begin{corollary} \label{cor:gen13}
If $g=9$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by two elements
$R$ and $B_1A_3C_5C_8^{-1}A_6^{-1}B_4^{-1}$.
\end{corollary}
\begin{proof}
Let $F_1=B_1A_3C_5C_8^{-1}A_6^{-1}B_4^{-1}$. Let us denote by
$H$ the subgroup of ${\rm Mod}(\Sigma_9)$ generated by the set
$\{ R, F_1\}$.
Let
\[
F_2 = RF_1R^{-1} = B_2A_4C_6C_9^{-1}A_7^{-1}B_5^{-1}
\]
and \[
F_3 = F_2^{-1} = B_5A_7C_9C_6^{-1}A_4^{-1}B_2^{-1}.
\]
Then, $F_3F_1(b_5,a_7,c_9,c_6,a_4,b_2)=(c_5,a_7,b_1,c_6,b_4,b_2)$ so that\\
$F_4=C_5A_7B_1C_6^{-1}B_4^{-1}B_2^{-1}\in H$.
Let \[
F_5 = RF_4R^{-1} = C_6A_8B_2C_7^{-1}B_5^{-1}B_3^{-1}
\]
and \[
F_6 = F_5^{-1} = B_3B_5C_7B_2^{-1}A_8^{-1}C_6^{-1}.
\]
Then, $F_6F_4(b_3,b_5,c_7,b_2,a_8,c_6)=(b_3,c_5,c_7,b_2,a_8,c_6)$ so that\\
$F_7=B_3C_5C_7B_2^{-1}A_8^{-1}C_6^{-1}\in H$.
$F_7F_6^{-1}=C_5B_5^{-1}$$\in$$H$ and then by conjugating $C_5B_5^{-1}$ with $R$ iteratively, we get $C_iB_i^{-1}$$\in$$H$ $\forall i$.
Let \[
F_8 = (B_7C_7^{-1})F_6(C_6B_6^{-1}) = B_3B_5B_7B_2^{-1}A_8^{-1}B_6^{-1}
\]
\[
F_9 = RF_8R^{-1} = B_4B_6B_8B_3^{-1}A_9^{-1}B_7^{-1}
\]
and \[
F_{10} = F_9^{-1} = B_7A_9B_3B_8^{-1}B_6^{-1}B_4^{-1}.
\]
Then, $F_{10}F_8(b_7,a_9,b_3,b_8,b_6,b_4)=(b_7,a_9,b_3,a_8,b_6,b_4)$ so that\\
$F_{11}=B_7A_9B_3A_8^{-1}B_6^{-1}B_4^{-1}$$\in$$H$.
Then, $F_{11}^{-1}F_{10}= A_8B_8^{-1}$$\in$$H$ and then \mbox{$A_iB_i^{-1}$$\in$$H$ $\forall i$}.
Then, $F_4(B_4A_4^{-1})F_2(B_5C_5^{-1})= B_1C_9^{-1} $$\in$$H$ and then \mbox{$B_{i+1}C_i^{-1}$$\in$$H$ $\forall i$}.
It follows from Corollary~\ref{cor:thmKorkmaz} that
$H={\rm Mod}(\Sigma_9)$, completing the proof of the corollary.
\end{proof}
\begin{corollary} \label{cor:gen5}
If $g=10$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by two elements
$R$ and $A_1C_1B_3B_7^{-1}C_5^{-1}A_5^{-1}$.
\end{corollary}
\begin{proof}
Let $F_1=A_1C_1B_3B_7^{-1}C_5^{-1}A_5^{-1}$. Let us denote by
$H$ the subgroup of ${\rm Mod}(\Sigma_{10})$ generated by the set
$\{ R, F_1\}$.
Let
\[
F_2 = RF_1R^{-1} = A_2C_2B_4B_8^{-1}C_6^{-1}A_6^{-1}.
\]
Then, $F_2F_1(a_2,c_2,b_4,b_8,c_6,a_6)=(a_2,b_3,b_4,b_8,b_7,a_6)$ so that\\
$F_3=A_2B_3B_4B_8^{-1}B_7^{-1}A_6^{-1}\in H$.
Let \[
F_4 = R^4F_3R^{-4} = A_6B_7B_8B_2^{-1}B_1^{-1}A_{10}^{-1}
\]
and \[
F_5 = F_4^{-1} = A_{10}B_1B_2B_8^{-1}B_7^{-1}A_6^{-1}.
\]
Then, $F_5F_3(a_{10},b_1,b_2,b_8,b_7,a_6)=(a_{10},b_1,a_2,b_8,b_7,a_6)$ so that\\
$F_6=A_{10}B_1A_2B_8^{-1}B_7^{-1}A_6^{-1}$$\in$$H$.
$F_6F_5^{-1}=A_2B_2^{-1}$$\in$$H$ and then by conjugating $A_2B_2^{-1}$ with $R$ iteratively, we get $A_iB_i^{-1}$$\in$$H$ $\forall i$.
Let \[
F_7 = (B_2A_2^{-1})(A_3B_3^{-1})F_3(B_7A_7^{-1})(A_6B_6^{-1}) = B_2A_3B_4B_8^{-1}A_7^{-1}B_6^{-1}
\]
\[
F_8 = RF_2F_3^{-1}R^{-1}F_7 = B_2A_3C_3C_7^{-1}A_7^{-1}B_6^{-1}
\]
\[
F_9 = F_8^{-1} = B_6A_7C_7C_3^{-1}A_3^{-1}B_2^{-1}
\]
and \[
F_{10} = R^4F_9R^{-4} = B_{10}A_1C_1C_7^{-1}A_7^{-1}B_6^{-1}.
\]
Then, $F_{10}F_8(b_{10},a_1,c_1,c_7,a_7,b_6)=(b_{10},a_1,b_2,c_7,a_7,b_6)$ so that\\
$F_{11}=B_{10}A_1B_2C_7^{-1}A_7^{-1}B_6^{-1}$$\in$$H$.
Then, $F_{11}F_{10}^{-1}= B_2C_1^{-1}$$\in$$H$ and then \mbox{$B_{i+1}C_i^{-1}$$\in$$H$ $\forall i$}.
Let \[
F_{12} =(B_2A_2^{-1})F_3(A_6B_6^{-1})(B_6C_5^{-1})(B_7A_7^{-1})(B_8A_8^{-1})= B_2B_3B_4A_8^{-1}A_7^{-1}C_5^{-1}
\]
\[
F_{13} = F_{12}^{-1} = C_5A_7A_8B_4^{-1}B_3^{-1}B_2^{-1}
\]
and \[
F_{14} = RF_{13}R^{-1} = C_6A_8A_9B_5^{-1}B_4^{-1}B_3^{-1}.
\]
Then, $F_{14}F_{12}(c_6,a_8,a_9,b_5,b_4,b_3)=(c_6,a_8,a_9,c_5,b_4,b_3)$ so that\\
$F_{15}=C_6A_8A_9C_5^{-1}B_4^{-1}B_3^{-1}$$\in$$H$.
Then, $F_{15}^{-1}F_{14}= C_5B_5^{-1}$$\in$$H$ and then \mbox{$C_iB_i^{-1}$$\in$$H$ $\forall i$}.
It follows from Corollary~\ref{cor:thmKorkmaz} that
$H={\rm Mod}(\Sigma_{10})$, completing the proof of the corollary.
\end{proof}
\begin{corollary} \label{cor:gen6}
If $g \geq 13$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by two elements
$R$ and $A_1B_4C_8C_{10}^{-1}B_6^{-1}A_3^{-1}$.
\end{corollary}
\begin{proof}
Let $F_1=A_1B_4C_8C_{10}^{-1}B_6^{-1}A_3^{-1}$. Let us denote by
$H$ the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set
$\{ R, F_1\}$.
Let
\[
F_2 = RF_1R^{-1} = A_2B_5C_9C_{11}^{-1}B_7^{-1}A_4^{-1}
\]
and \[
F_3 = F_2^{-1} = A_4B_7C_{11}C_9^{-1}B_5^{-1}A_2^{-1}.
\]
Then, $F_3F_1(a_4,b_7,c_{11},c_9,b_5,a_2)=(b_4,b_7,c_{11},c_9,b_5,a_2)$ so that\\
$F_4=B_4B_7C_{11}C_9^{-1}B_5^{-1}A_2^{-1}\in H$.
$F_4F_3^{-1}=B_4A_4^{-1}$$\in$$H$ and then by conjugating $B_4A_4^{-1}$ with $R$
iteratively, we get $B_iA_i^{-1}$$\in$$H$ $\forall i$.
Let \[
F_5 = R^2F_1R^{-2} = A_3B_6C_{10}C_{12}^{-1}B_8^{-1}A_5^{-1}
\]
and \[
F_6 = F_5^{-1} = A_5B_8C_{12}C_{10}^{-1}B_6^{-1}A_3^{-1}.
\]
Then, $F_6F_1(a_5,b_8,c_{12},c_{10},b_6,a_3)=(a_5,c_8,c_{12},c_{10},b_6,a_3)$ so that\\
$F_7=A_5C_8C_{12}C_{10}^{-1}B_6^{-1}A_3^{-1}$$\in$$H$.
Then, $F_7F_6^{-1}= C_8B_8^{-1}$$\in$$H$ and then
\mbox{$C_iB_i^{-1}$$\in$$H$ $\forall i$}.
Let \[
F_8 = (A_4B_4^{-1})F_1(A_3B_3^{-1}) = A_1A_4C_8C_{10}^{-1}B_6^{-1}B_3^{-1}
\]
\[
F_9 = R^3F_8R^{-3} = A_4A_7C_{11}C_{13}^{-1}B_9^{-1}B_6^{-1}
\]
and \[
F_{10} = F_9^{-1} = B_6B_9C_{13}C_{11}^{-1}A_7^{-1}A_4^{-1}.
\]
Then, $F_{10}F_8(b_6,b_9,c_{13},c_{11},a_7,a_4)=(b_6,c_8,c_{13},c_{11},a_7,a_4)$ so
that\\
$F_{11}=B_6C_8C_{13}C_{11}^{-1}A_7^{-1}A_4^{-1}$$\in$$H$.
Then, $F_{11}F_{10}^{-1}= C_8B_9^{-1}$$\in$$H$ and then
\mbox{$C_iB_{i+1}^{-1}$$\in$$H$ $\forall i$}.
It follows from Corollary~\ref{cor:thmKorkmaz} that
$H={\rm Mod}(\Sigma_g)$, completing the proof of the corollary.
\end{proof}
\begin{corollary} \label{cor:gen7}
If $g \geq 12$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by two elements
$R$ and $B_1A_3C_6C_{10}^{-1}A_7^{-1}B_5^{-1}$.
\end{corollary}
\begin{proof}
Let $F_1=B_1A_3C_6C_{10}^{-1}A_7^{-1}B_5^{-1}$. Let us denote by
$H$ the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set
$\{ R, F_1\}$.
Let
\[
F_2 = RF_1R^{-1} = B_2A_4C_7C_{11}^{-1}A_8^{-1}B_6^{-1}
\]
and \[
F_3 = F_2^{-1} = B_6A_8C_{11}C_7^{-1}A_4^{-1}B_2^{-1}.
\]
Then, $F_3F_1(b_6,a_8,c_{11},c_7,a_4,b_2)=(c_6,a_8,c_{11},c_7,a_4,b_2)$ so that\\
$F_4=C_6A_8C_{11}C_7^{-1}A_4^{-1}B_2^{-1}\in H$.
$F_4F_3^{-1}=C_6B_6^{-1}$$\in$$H$ and then by conjugating $C_6B_6^{-1}$ with $R$
iteratively, we get $C_iB_i^{-1}$$\in$$H$ $\forall i$.
Let \[
F_5 = F_1(C_{10}B_{10}^{-1})(B_5C_5^{-1}) = B_1A_3C_6B_{10}^{-1}A_7^{-1}C_5^{-1}
\]
and \[
F_6 = R^2F_5R^{-2} = B_3A_5C_8B_{12}^{-1}A_9^{-1}C_7^{-1}.
\]
Then, $F_6F_5(b_3,a_5,c_8,b_{12},a_9,c_7)=(a_3,a_5,c_8,b_{12},a_9,c_7)$ so that\\
$F_7=A_3A_5C_8B_{12}^{-1}A_9^{-1}C_7^{-1}$$\in$$H$.
Then, $F_7F_6^{-1}= A_3B_3^{-1}$$\in$$H$ and then
\mbox{$A_iB_i^{-1}$$\in$$H$ $\forall i$}.
Let \[
F_8 = (C_1B_1^{-1})(B_3A_3^{-1})F_1(B_5C_5^{-1}) = C_1B_3C_6C_{10}^{-1}A_7^{-1}C_5^{-1}
\]
and \[
F_9 = RF_8R^{-1} = C_2B_4C_7C_{11}^{-1}A_8^{-1}C_6^{-1}.
\]
Then, $F_9F_8(c_2,b_4,c_7,c_{11},a_8,c_6)=(b_3,b_4,c_7,c_{11},a_8,c_6)$ so that\\
$F_{10}=B_3B_4C_7C_{11}^{-1}A_8^{-1}C_6^{-1}$$\in$$H$.
Then, $F_{10}F_9^{-1}= B_3C_2^{-1}$$\in$$H$ and then
\mbox{$B_{i+1}C_i^{-1}$$\in$$H$ $\forall i$}.
It follows from Corollary~\ref{cor:thmKorkmaz} that
$H={\rm Mod}(\Sigma_g)$, completing the proof of the corollary.
\end{proof}
\begin{lemma} \label{lem:gen9}
If $g \geq 11$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated
by two elements $R$ and $A_1B_2C_4C_{g-1}^{-1}B_{g-3}^{-1}A_{g-4}^{-1}$.
\end{lemma}
\begin{proof}
Let $F_1=A_1B_2C_4C_{g-1}^{-1}B_{g-3}^{-1}A_{g-4}^{-1}$. Let us denote by
$H$ the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set
$\{ R, F_1\}$.
Let
\[
F_2 = RF_1R^{-1} = A_2B_3C_5C_g^{-1}B_{g-2}^{-1}A_{g-3}^{-1}.
\]
Then, $F_2F_1(a_2,b_3,c_5,c_g,b_{g-2},a_{g-3})=(b_2,b_3,c_5,c_g,b_{g-2},b_{g-3})$ so
that\\
$F_3=B_2B_3C_5C_g^{-1}B_{g-2}^{-1}B_{g-3}^{-1}\in H$.
Let \[
F_4 = R^{-1}F_3R = B_1B_2C_4C_{g-1}^{-1}B_{g-3}^{-1}B_{g-4}^{-1}
\]
and \[
F_5 = F_3^{-1} = B_{g-3}B_{g-2}C_gC_5^{-1}B_3^{-1}B_2^{-1}.
\]
Then, $F_5F_4(b_{g-3},b_{g-2},c_g,c_5,b_3,b_2)=(b_{g-3},b_{g-2},b_1,c_5,b_3,b_2)$ so
that\\
$F_6=B_{g-3}B_{g-2}B_1C_5^{-1}B_3^{-1}B_2^{-1}$$\in$$H$.
$F_6F_5^{-1}=B_1C_g^{-1}$$\in$$H$ and then by conjugating $B_1C_g^{-1}$ with $R$
iteratively, we get $B_{i+1}C_i^{-1}$$\in$$H$ $\forall i$.
Let \[
F_7 = (C_{g-3}B_{g-2}^{-1})(C_{g-4}B_{g-3}^{-1})F_6 = C_{g-3}C_{g-4}B_1C_5^{-1}B_3^{-1}B_2^{-1}
\]
and \[
F_8 = R^2F_7R^{-2} = C_{g-1}C_{g-2}B_3C_7^{-1}B_5^{-1}B_4^{-1}.
\]
Then, $F_8F_7(c_{g-1},c_{g-2},b_3,c_7,b_5,b_4)=(c_{g-1},c_{g-2},b_3,c_7,c_5,b_4)$ so
that\\
$F_9=C_{g-1}C_{g-2}B_3C_7^{-1}C_5^{-1}B_4^{-1}$$\in$$H$.
Then, $F_9F_8^{-1}= C_5B_5^{-1}$$\in$$H$ and then
\mbox{$C_iB_i^{-1}$$\in$$H$ $\forall i$}.
Let \[
F_{10} = F_1(B_{g-3}C_{g-3}^{-1}) = A_1B_2C_4C_{g-1}^{-1}C_{g-3}^{-1}A_{g-4}^{-1}
\]
and \[
F_{11} = RF_{10}R^{-1} = A_2B_3C_5C_g^{-1}C_{g-2}^{-1}A_{g-3}^{-1}.
\]
Then,
$F_{11}F_{10}(a_2,b_3,c_5,c_g,c_{g-2},a_{g-3})=(b_2,b_3,c_5,c_g,c_{g-2},a_{g-3})$\\
so that $F_{12}=B_2B_3C_5C_g^{-1}C_{g-2}^{-1}A_{g-3}^{-1}$$\in$$H$.
Then, $F_{12}F_{11}^{-1}= B_2A_2^{-1}$$\in$$H$ and then
\mbox{$B_iA_i^{-1}$$\in$$H$ $\forall i$}.
It follows from Corollary~\ref{cor:thmKorkmaz} that
$H={\rm Mod}(\Sigma_g)$, completing the proof of the lemma.
\end{proof}
\begin{lemma} \label{lem:gen10}
If $g \geq 13$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated
by two elements $R$ and $A_1B_2C_4C_{g-2}^{-1}B_{g-4}^{-1}A_{g-5}^{-1}$.
\end{lemma}
\begin{proof}
Let $F_1=A_1B_2C_4C_{g-2}^{-1}B_{g-4}^{-1}A_{g-5}^{-1}$. Let us denote by
$H$ the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set
$\{ R, F_1\}$.
Let
\[
F_2 = RF_1R^{-1} = A_2B_3C_5C_{g-1}^{-1}B_{g-3}^{-1}A_{g-4}^{-1}.
\]
Then,
$F_2F_1(a_2,b_3,c_5,c_{g-1},b_{g-3},a_{g-4})=(b_2,b_3,c_5,c_{g-1},b_{g-3},b_{g-4})$
so that $F_3=B_2B_3C_5C_{g-1}^{-1}B_{g-3}^{-1}B_{g-4}^{-1}\in H$.
Let \[
F_4 = F_2F_3^{-1} = A_2B_2^{-1}A_{g-4}^{-1}B_{g-4}
\]
\[
F_5 = RF_4R^{-1} = A_3B_3^{-1}A_{g-3}^{-1}B_{g-3}
\]
\[
F_6 = F_5F_3 = B_2A_3C_5C_{g-1}^{-1}A_{g-3}^{-1}B_{g-4}^{-1}
\]
\[
F_7 = R^{-2}F_6R^2 = B_gA_1C_3C_{g-3}^{-1}A_{g-5}^{-1}B_{g-6}^{-1}
\]
and \[
F_8 = F_7^{-1} = B_{g-6}A_{g-5}C_{g-3}C_3^{-1}A_1^{-1}B_g^{-1}.
\]
Then,
$F_8F_6(b_{g-6},a_{g-5},c_{g-3},c_3,a_1,b_g)=(b_{g-6},a_{g-5},c_{g-3},c_3,a_1,c_{g-1})$
so that $F_9=B_{g-6}A_{g-5}C_{g-3}C_3^{-1}A_1^{-1}C_{g-1}^{-1}$$\in$$H$.
$F_9F_8^{-1}=C_{g-1}B_g^{-1}$$\in$$H$ and then by conjugating $C_{g-1}B_g^{-1}$ with
$R$
iteratively, we get $C_iB_{i+1}^{-1}$$\in$$H$ $\forall i$.
Let \[
F_{10} = F_3(C_{g-1}B_g^{-1}) = B_2B_3C_5B_g^{-1}B_{g-3}^{-1}B_{g-4}^{-1}
\]
and \[
F_{11} = R^2F_{10}R^{-2} = B_4B_5C_7B_2^{-1}B_{g-1}^{-1}B_{g-2}^{-1}.
\]
Then,
$F_{11}F_{10}(b_4,b_5,c_7,b_2,b_{g-1},b_{g-2})=(b_4,c_5,c_7,b_2,b_{g-1},b_{g-2})$\\
so that $F_{12}=B_4C_5C_7B_2^{-1}B_{g-1}^{-1}B_{g-2}^{-1}$$\in$$H$.
Then, $F_{12}F_{11}^{-1}= C_5B_5^{-1}$$\in$$H$ and then
\mbox{$C_iB_i^{-1}$$\in$$H$ $\forall i$}.
Let \[
F_{13} = F_1(B_{g-4}C_{g-4}^{-1}) = A_1B_2C_4C_{g-2}^{-1}C_{g-4}^{-1}A_{g-5}^{-1}
\]
and \[
F_{14} = RF_{13}R^{-1} = A_2B_3C_5C_{g-1}^{-1}C_{g-3}^{-1}A_{g-4}^{-1}.
\]
Then,
$F_{14}F_{13}(a_2,b_3,c_5,c_{g-1},c_{g-3},a_{g-4})=(b_2,b_3,c_5,c_{g-1},c_{g-3},a_{g-4})$
so that $F_{15}=B_2B_3C_5C_{g-1}^{-1}C_{g-3}^{-1}A_{g-4}^{-1}$$\in$$H$.
Then, $F_{15}F_{14}^{-1}= B_2A_2^{-1}$$\in$$H$ and then
\mbox{$B_iA_i^{-1}$$\in$$H$ $\forall i$}.
It follows from Corollary~\ref{cor:thmKorkmaz} that
$H={\rm Mod}(\Sigma_g)$, completing the proof of the corollary.
\end{proof}
\begin{lemma} \label{lem:gen11}
If $k \geq 7$ and $g \geq 2k+1$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated
by elements $R$ and $A_1B_2C_4C_{g-k+4}^{-1}B_{g-k+2}^{-1}A_{g-k+1}^{-1}$.
\end{lemma}
\begin{proof}
Let $F_1=A_1B_2C_4C_{g-k+4}^{-1}B_{g-k+2}^{-1}A_{g-k+1}^{-1}$. Let us denote by
$H$ the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set
$\{ R, F_1\}$.
Let
\[
F_2 = R^{k-3}F_1R^{3-k} = A_{k-2}B_{k-1}C_{k+1}C_1^{-1}B_{g-1}^{-1}A_{g-2}^{-1}
\]
and \[
F_3 = F_2^{-1} = A_{g-2}B_{g-1}C_1C_{k+1}^{-1}B_{k-1}^{-1}A_{k-2}^{-1}.
\]
$F_3F_1(a_{g-2},b_{g-1},c_1,c_{k+1},b_{k-1},a_{k-2})=(a_{g-2},b_{g-1},b_2,c_{k+1},b_{k-1},a_{k-2})$ so
that $F_4=A_{g-2}B_{g-1}B_2C_{k+1}^{-1}B_{k-1}^{-1}A_{k-2}^{-1}\in H$.
$F_4F_3^{-1}=B_2C_1^{-1}$$\in$$H$ and then by conjugating $B_2C_1^{-1}$ with $R$
iteratively, we get $B_{i+1}C_i^{-1}$$\in$$H$ $\forall i$.
Let \[
F_5 = F_1(B_{g-k+2}C_{g-k+1}^{-1}) = A_1B_2C_4C_{g-k+4}^{-1}C_{g-k+1}^{-1}A_{g-k+1}^{-1}
\]
and \[
F_6 = RF_5R^{-1} = A_2B_3C_5C_{g-k+5}^{-1}C_{g-k+2}^{-1}A_{g-k+2}^{-1}.
\]
$F_6F_5(a_2,b_3,c_5,c_{g-k+5},c_{g-k+2},a_{g-k+2})=(b_2,b_3,c_5,c_{g-k+5},c_{g-k+2},a_{g-k+2})$\\
so that $F_7=B_2B_3C_5C_{g-k+5}^{-1}C_{g-k+2}^{-1}A_{g-k+2}^{-1}$$\in$$H$.
Then, $F_7F_6^{-1}= B_2A_2^{-1}$$\in$$H$ and then
\mbox{$B_iA_i^{-1}$$\in$$H$ $\forall i$}.
Let \[
F_8 = R^{k-2}F_6R^{2-k} = A_kB_{k+1}C_{k+3}C_3^{-1}Cg^{-1}A_g^{-1}
\]
and \[
F_9 = F_8^{-1} = A_gC_gC_3C_{k-3}^{-1}B_{k+1}^{-1}A_k^{-1}.
\]
Then,
$F_9F_6(a_g,c_g,c_3,c_{k+3},b_{k+1},a_k)=(a_g,c_g,b_3,c_{k+3},b_{k+1},a_k)$\\
so that $F_{10}=A_gC_gB_3C_{k-3}^{-1}B_{k+1}^{-1}A_k^{-1}$$\in$$H$.
Then, $F_{10}F_9^{-1}= B_3C_3^{-1}$$\in$$H$ and then
\mbox{$B_iC_i^{-1}$$\in$$H$ $\forall i$}.
It follows from Corollary~\ref{cor:thmKorkmaz} that
$H={\rm Mod}(\Sigma_g)$, completing the proof of the lemma.
\end{proof}
\begin{corollary} \label{cor:gen8}
If $k \geq 5$ and $g \geq 2k+1$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated
by elements $R$ and $A_1B_2C_4C_{g-k+4}^{-1}B_{g-k+2}^{-1}A_{g-k+1}^{-1}$.
\end{corollary}
\begin{proof}
It directly follows from Lemma~\ref{lem:gen9}, Lemma~\ref{lem:gen10} and Lemma~\ref{lem:gen11}.
\end{proof}
\bigskip
\section{Main Results}
\begin{lemma} \label{lem:order}
If $R$ is an element of order k in a group $G$ and if $x$ and $y$ are
elements in $G$ satisfying $RxR^{-1}=y$, then the order of $Rxy^{-1}$ is also k.
\end{lemma}
\begin{proof}
$(Rxy^{-1})^{k}=(yRy^{-1})^{k}=yR^ky^{-1}=1$.\\
On the other hand, if $(Rxy^{-1})^{l}=1$ then $(Rxy^{-1})^{l}=(yRy^{-1})^{l}=yR^ly^{-1}=1$ i.e. $R^l=1$ and hence $k \mid l$.
\end{proof}
Now, we can prove Theorem~\ref{thm:1}.
\begin{proof}
If $g=10$, let $H_{10}$ be the subgroup of ${\rm Mod}(\Sigma_{10})$ generated by the set $\{ R, R^4A_1C_1B_3B_7^{-1}C_5^{-1}A_5^{-1}\}$.
Then, $H_{10} =$ ${\rm Mod}(\Sigma_{10})$ by Corollary ~\ref{cor:gen5}.
Then, we are done by Lemma ~\ref{lem:order} since $R^4(A_1C_1B_3)R^{-4}=A_5C_5B_7$.
Note that, order of $R^4$ is clearly $5$ and hence order of the element $R^4(A_1C_1B_3)(A_5C_5B_7)^{-1}$ is also $5$ by Lemma ~\ref{lem:order}
since $R^4(a_1)=a_5$, $R^4(c_1)=c_5$ and $R^4(b_3)=b_7$ implies $R^4(A_1C_1B_3)R^{-4} = A_5C_5B_7$.
If $g=9$, let $H_9$ be the subgroup of ${\rm Mod}(\Sigma_9)$ generated by the set $\{ R, R^3B_1A_3C_5C_8^{-1}A_6^{-1}B_4^{-1}\}$.
Then, $H_9 =$ ${\rm Mod}(\Sigma_9)$ by Corollary ~\ref{cor:gen13}.
Then, we are done by Lemma ~\ref{lem:order} since $R^3(B_1A_3C_5)R^{-3}=B_4A_6C_8$.
If $g=8$, let $H_8$ be the subgroup of ${\rm Mod}(\Sigma_8)$ generated by the set $\{ R, R^2B_1A_5C_5C_7^{-1}A_7^{-1}B_3^{-1}\}$.
Then, $H_8 =$ ${\rm Mod}(\Sigma_8)$ by Corollary ~\ref{cor:gen12}.
Then, we are done by Lemma ~\ref{lem:order} since $R^2(B_1A_5C_5)R^{-2}=B_3A_7C_7$.
If $g=7$, let $H_7$ be the subgroup of ${\rm Mod}(\Sigma_7)$ generated by the set $\{ R, RC_1B_4A_6A_7^{-1}B_5^{-1}C_2^{-1}\}$.
Then, $H_7 =$ ${\rm Mod}(\Sigma_7)$ by Corollary ~\ref{cor:gen2}.
Then, we are done by Lemma ~\ref{lem:order} since $R(C_1B_4A_6)R^{-1}=C_2B_5A_7$.
The remaining part of the proof is the case of $g\geq 11$.
Let $k=g/g'$ so that $k$ is the greatest divisor of $g$ such that $k$ is strictly less than $g/2$.
Clearly, k can be any positive integer but three.
If $k=2$, let $K_2$ be the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set $\{ R, R^2A_1B_4C_8C_{10}^{-1}B_6^{-1}A_3^{-1}\}$.
Then, $K_2 =$ ${\rm Mod}(\Sigma_g)$ by Corollary ~\ref{cor:gen6}.
Then, we are done by Lemma ~\ref{lem:order} since $R^2(A_1B_4C_8)R^{-2}=A_3B_6C_{10}$.
If $k=4$, let $K_4$ be the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set $\{ R, R^4B_1A_3C_6C_{10}^{-1}A_7^{-1}B_5^{-1}\}$.
Then, $K_4 =$ ${\rm Mod}(\Sigma_g)$ by Corollary ~\ref{cor:gen7}.
Then, we are done by Lemma ~\ref{lem:order} since $R^4(B_1A_3C_6)R^{-4}=B_5A_7C_{10}$.
If $k=1$ or $k=5$, let $K_5$ be the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set $\{R, R^{-5}A_1B_2C_4C_{g-1}^{-1}B_{g-3}^{-1}A_{g-4}^{-1}\}$.
Then, $K_5 =$ ${\rm Mod}(\Sigma_g)$ by Corollary ~\ref{cor:gen8}.
Then, we are done by Lemma ~\ref{lem:order} since $R^{-5}(A_1B_2C_4)R^5=A_{g-4}B_{g-3}C_{g-1}$.
If $k=6$, let $K_6$ be the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set $\{ R, R^{-6}A_1B_2C_4C_{g-2}^{-1}B_{g-4}^{-1}A_{g-5}^{-1}\}$.
Then, $K_6 =$ ${\rm Mod}(\Sigma_g)$ by Corollary ~\ref{cor:gen8}.
Then, we are done by Lemma~\ref {lem:order} since $R^{-6}(A_1B_2C_4)R^6=A_{g-5}B_{g-4}C_{g-2}$.
If $k\geq 7$, let $K$ be the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set $\{ R, R^{-k}A_1B_2C_4C_{g-k+4}^{-1}B_{g-k+2}^{-1}A_{g-k+1}^{-1}\}$.
Then, $K =$ ${\rm Mod}(\Sigma_g)$ by Corollary ~\ref{cor:gen8}.
Then, we are done by Lemma~\ref {lem:order} since $R^{-k}(A_1B_2C_4)R^k = A_{g-k+1}B_{g-k+2}C_{g-k+4}$.
\end{proof}
Finally, we prove Theorem~\ref{thm:2}.
\begin{proof}
If $g=6$, let $H_6$ be the subgroup of ${\rm Mod}(\Sigma_6)$ generated by the set $\{ R, RC_1B_4A_6A_1^{-1}B_5^{-1}C_2^{-1}\}$.
Then, $H_6 =$ ${\rm Mod}(\Sigma_6)$ by Corollary ~\ref{cor:gen1}.
Then, we are done by Lemma ~\ref{lem:order} since $R(C_1B_4A_6)R^{-1}=C_2B_5A_1$.
Note that, since $R(c_1)=c_2$, $R(b_4)=b_5$ and $R(a_6)=a_1$, we have
$R(C_1B_4A_6)R^{-1} = C_2B_5A_1$
which implies order of the element $R(C_1B_4A_6)(C_2B_5A_1)^{-1}$ is $g$.
If $g=7$, let $H_7$ be the subgroup of ${\rm Mod}(\Sigma_7)$ generated by the set $\{ R, RC_1B_4A_6A_7^{-1}B_5^{-1}C_2^{-1}\}$.
Then, $H_7 =$ ${\rm Mod}(\Sigma_7)$ by Corollary ~\ref{cor:gen2}.
Then, we are done by Lemma ~\ref{lem:order} since $R(C_1B_4A_6)R^{-1}=C_2B_5A_7$.
If $g=8$, let $H_8$ be the subgroup of ${\rm Mod}(\Sigma_8)$ generated by the set $\{ R, RB_1C_4A_7A_8^{-1}C_5^{-1}B_2^{-1}\}$.
Then, $H_8 =$ ${\rm Mod}(\Sigma_8)$ by Corollary ~\ref{cor:gen3}.
Then, we are done by Lemma ~\ref{lem:order} since $R(B_1C_4A_7)R^{-1}=B_2C_5A_8$.
If $g\geq 9$, let $H_9$ be the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set $\{ R, RC_1B_4A_7A_8^{-1}B_5^{-1}C_2^{-1}\}$.
Then, $H_9 =$ ${\rm Mod}(\Sigma_g)$ by Corollary ~\ref{cor:gen4}.
Then, we are done by Lemma ~\ref{lem:order} since $R(C_1B_4A_7)R^{-1}=C_2B_5A_8$.
\end{proof}
\section{Further Results}
In this section, we prove Theorem ~\ref{thm:3} which states as: for $g\geq 3k^2+4k+1$ and any positive integer $k$, the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by
two elements of order $g/\gcd (g,k)$.
Korkmaz showed the following in the proof of Theorem ~\ref{thm:thmKorkmaz}.
\begin{theorem} \label{thm:thmKorkmaz2}
If $g\geq 3$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by the elements
\(
A_iA_j^{-1},B_iB_j^{-1}, C_iC_j^{-1} \forall i, j. \)
\end{theorem}
Sketch of the proof is as follows:
$A_1A_2^{-1}B_1B_2^{-1}(a_1,a_3) = (b_1,a_3)$.
$B_1A_3^{-1}C_1C_2^{-1}(b_1,a_3) = (c_1,a_3)$.
Then, Korkmaz showed that $A_3$ can be generated by these elements by using lantern relation.
Hence, $A_i = (A_iA_3^{-1})A_3$,
$B_i = (B_iB_1^{-1})(B_1A_3^{-1})A_3$ and
$C_i = (C_iC_1^{-1})(C_1A_3^{-1})A_3$ are generated by given elements.
This finishes the proof.
Now, we prove the next statement as a corollary to Theorem ~\ref{thm:thmKorkmaz2}.
\begin{corollary} \label{generating}
If $g\geq 3$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by the elements
\(
A_iB_i^{-1},C_iB_i^{-1}, C_iB_{i+1}^{-1} \forall i. \)
\end{corollary}
\begin{proof}
Let us denote by $H$ the subgroup generated by the elements
\(
A_iB_i^{-1},C_iB_i^{-1}, C_iB_{i+1}^{-1} \forall i. \)
\\
$B_iB_j^{-1}=(B_iC_i^{-1})(C_iB_{i+1}^{-1})\cdots (B_{j-1}C_{j-1}^{-1})(C_{j-1}B_j^{-1})$$\in$$H$$\forall i, j$\\
$C_iC_j^{-1}=(C_iB_i^{-1})(B_iB_j^{-1})(B_jC_j^{-1})$$\in$$H$ $ \forall i, j$\\
$A_iA_j^{-1}=(A_iB_i^{-1})(B_iB_j^{-1})(B_jA_j^{-1})$$\in$$H$ $ \forall i, j$
It follows from Theorem~\ref{thm:thmKorkmaz2} that
$H={\rm Mod}(\Sigma_g)$, completing the proof of the lemma.
\end{proof}
\begin{theorem} \label{thm:r2}
If $g\geq 21$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by the elements
\(
R^2,B_1B_2A_5A_8C_{11}C_{14}C_{16}^{-1}C_{13}^{-1}A_{10}^{-1}A_7^{-1}B_4^{-1}B_3^{-1}. \)
\end{theorem}
\begin{proof}
Let $F_1=B_1B_2A_5A_8C_{11}C_{14}C_{16}^{-1}C_{13}^{-1}A_{10}^{-1}A_7^{-1}B_4^{-1}B_3^{-1}$. Let us denote by
$H$ the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set
$\{ R^2, F_1\}$.
Let
\[
F_2 = R^2F_1R^{-2} = B_3B_4A_7A_{10}C_{13}C_{16}C_{18}^{-1}C_{15}^{-1}A_{12}^{-1}A_9^{-1}B_6^{-1}B_5^{-1}.
\]
and \[
F_3 = F_2^{-1} = B_5B_6A_9A_{12}C_{15}C_{18}C_{16}^{-1}C_{13}^{-1}A_{10}^{-1}A_7^{-1}B_4^{-1}B_3^{-1}.
\]
Then, $F_3F_1(b_5,b_6, \cdots ,b_3)=(a_5,b_6, \cdots ,b_3)$ so that\\
$F_4=A_5B_6A_9A_{12}C_{15}C_{18}C_{16}^{-1}C_{13}^{-1}A_{10}^{-1}A_7^{-1}B_4^{-1}B_3^{-1}\in H$.
Note that $\cdots$ refers to the elements remaining fixed under the given maps.
$F_4F_3^{-1}=A_5B_5^{-1}$$\in$$H$ and then by conjugating $A_5B_5^{-1}$ with $R^2$ iteratively, we get $A_{2i+1}B_{2i+1}^{-1}$$\in$$H$ $\forall i$.
Let
\[
F_5 = R^4F_1R^{-4} = B_5B_6A_9A_{12}C_{15}C_{18}C_{20}^{-1}C_{17}^{-1}A_{14}^{-1}A_{11}^{-1}B_8^{-1}B_7^{-1}.
\]
and \begin{eqnarray*}
F_6
&=& (A_7B_7^{-1})F_5^{-1}(B_5A_5^{-1})\\
&=& A_7B_8A_{11}A_{14}C_{17}C_{20}C_{18}^{-1}C_{15}^{-1}A_{12}^{-1}A_9^{-1}B_6^{-1}A_5^{-1}.
\end{eqnarray*}
Then, $F_6F_1(a_7,b_8,a_{11}, \cdots ,b_6,a_5)=(a_7,a_8,a_{11}, \cdots ,b_6,a_5)$ so that\\
$F_7=A_7A_8A_{11}A_{14}C_{17}C_{20}C_{18}^{-1}C_{15}^{-1}A_{12}^{-1}A_9^{-1}B_6^{-1}A_5^{-1}\in H$.
$F_7F_6^{-1}=A_8B_8^{-1}$$\in$$H$ and then by conjugating $A_8B_8^{-1}$ with $R^2$ iteratively, we get $A_{2i}B_{2i}^{-1}$$\in$$H$ $\forall i$.\\
Hence, $A_iB_i^{-1}$$\in$$H$ $\forall i$.
Let
\[
F_8 = (B_{12}A_{12}^{-1})F_4 = A_5B_6A_9B_{12}C_{15}C_{18}C_{16}^{-1}C_{13}^{-1}A_{10}^{-1}A_7^{-1}B_4^{-1}B_3^{-1}.
\]
Then, $F_8F_1( \cdots ,b_{12}, \cdots )=( \cdots ,c_{11}, \cdots )$ so that\\
$F_9=A_5B_6A_9C_{11}C_{15}C_{18}C_{16}^{-1}C_{13}^{-1}A_{10}^{-1}A_7^{-1}B_4^{-1}B_3^{-1}\in H$.
$F_9F_8^{-1}=C_{11}B_{12}^{-1}$$\in$$H$ and then by conjugating $C_{11}B_{12}^{-1}$ with $R^2$ iteratively, we get $C_{2i+1}B_{2i+2}^{-1}$$\in$$H$ $\forall i$.
Let \[
F_{10} = (B_{11}A_{11}^{-1})F_7 = A_7A_8B_{11}A_{14}C_{17}C_{20}C_{18}^{-1}C_{15}^{-1}A_{12}^{-1}A_9^{-1}B_6^{-1}A_5^{-1}.
\]
Then, $F_{10}F_1( \cdots ,b_{11}, \cdots )=( \cdots ,c_{11}, \cdots )$ so that\\
$F_{11}=A_7A_8C_{11}A_{14}C_{17}C_{20}C_{18}^{-1}C_{15}^{-1}A_{12}^{-1}A_9^{-1}B_6^{-1}A_5^{-1}$$\in$$H$.
Then, $F_{11}F_{10}^{-1}= C_{11}B_{11}^{-1}$$\in$$H$ and then, we get $C_{2i+1}B_{2i+1}^{-1}$$\in$$H$ $\forall i$.
Let
\[
F_{12} = (B_{15}C_{15}^{-1})F_4 = A_5B_6A_9A_{12}B_{15}C_{18}C_{16}^{-1}C_{13}^{-1}A_{10}^{-1}A_7^{-1}B_4^{-1}B_3^{-1}.
\]
Then, $F_{12}F_1( \cdots ,b_{15}, \cdots )=( \cdots ,c_{14}, \cdots )$ so that\\
$F_{13}=A_5B_6A_9A_{12}C_{14}C_{18}C_{16}^{-1}C_{13}^{-1}A_{10}^{-1}A_7^{-1}B_4^{-1}B_3^{-1}$$\in$$H$.
Then, $F_{13}F_{12}^{-1}= C_{14}B_{15}^{-1}$$\in$$H$ and then, we get $C_{2i}B_{2i+1}^{-1}$$\in$$H$ $\forall i$.\\
Hence, $C_iB_{i+1}^{-1}$$\in$$H$ $\forall i$.
Let
\[
F_{14} = F_7(C_{15}B_{16}^{-1}) = A_7A_8A_{11}A_{14}C_{17}C_{20}C_{18}^{-1}B_{16}^{-1}A_{12}^{-1}A_9^{-1}B_6^{-1}A_5^{-1}.
\]
Then, $F_{14}F_1(\cdots ,b_{16}, \cdots )=( \cdots ,c_{16}, \cdots )$ so that\\
$F_{15}=A_7A_8A_{11}A_{14}C_{17}C_{20}C_{18}^{-1}C_{16}^{-1}A_{12}^{-1}A_9^{-1}B_6^{-1}A_5^{-1}$$\in$$H$.
Then, $F_{15}^{-1}F_{14}= C_{16}B_{16}^{-1}$$\in$$H$ and then, we get $C_{2i}B_{2i}^{-1}$$\in$$H$ $\forall i$.\\
Hence, $C_iB_i^{-1}$$\in$$H$ $\forall i$.
It follows from Corollary~\ref{generating} that
$H={\rm Mod}(\Sigma_g)$, completing the proof of the theorem.
\end{proof}
\begin{corollary} \label{cor:r2}
If $g$ is even and $g\geq 22$, then the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by two elements
of order $g/2$.
\end{corollary}
\begin{proof}
Let $H$ be the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set\\
$\{
R^2, R^2B_1B_2A_5A_8C_{11}C_{14}C_{16}^{-1}C_{13}^{-1}A_{10}^{-1}A_7^{-1}B_4^{-1}B_3^{-1}\}$. \\
Then, $H=$ ${\rm Mod}(\Sigma_g)$ by Theorem ~\ref{thm:r2}.
Then, we are done by Lemma ~\ref{lem:order} since
$R^2(B_1B_2A_5A_8C_{11}C_{14})R^{-2}=B_3B_4A_7A_{10}C_{13}C_{16}$.
\end{proof}
Generalization of Theorem ~\ref{thm:r2} and Corollary ~\ref{cor:r2} is as follows:
\begin{theorem} \label{generalization}
For $k\geq 2$ and $g\geq 3k^2+4k+1$, the mapping class group ${\rm Mod}(\Sigma_g)$ is generated by the elements
\(
R^k,R^kF(R^kF^{-1}R^{-k})\) where $F=B_1B_2\cdots B_kA_{2k+1}A_{3k+2}\cdots A_{k^2+2k}C_{k^2+3k+1}C_{k^2+4k+2}\cdots C_{2k^2+3k}$.
\end{theorem}
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\node[scale=0.6, rotate=-360/44*10] at (4.85,0.6) {$3k$};
\node[scale=0.6, rotate=-360/44*11] at (4.4,0.1) {$3k+1$};
\node[scale=0.6, rotate=-360/44*12] at (4.3,-0.7) {$3k+2$};
\node[scale=0.6, rotate=-360/44*13] at (4.65,-1.4) {$3k+3$};
\node[scale=0.6, rotate=-360/44*16] at (3.75,-3.15) {$4k+1$};
\node[scale=0.6, rotate=-360/44*17] at (2.95,-3.25) {$4k+2$};
\node[scale=0.6, rotate=-360/44*18] at (2.3,-3.7) {$4k+3$};
\node[scale=0.6, rotate=-360/44*19] at (2.0,-4.5) {$4k+4$};
\node[scale=0.6, rotate=-360/44*22] at (0.1,-4.9) {$k^2+3k-1$};
\node[scale=0.6, rotate=-360/44*23] at (-0.7,-4.3) {$k^2+3k$};
\node[scale=0.6, rotate=-360/44*24] at (-1.5,-4.65) {$k^2+3k+1$};
\node[scale=0.6, rotate=-360/44*30] at (-3.9,-1.9) {$k^2+4k+1$};
\node[scale=0.6, rotate=-360/44*31] at (-4.7,-1.4) {$k^2+4k+2$};
\node[scale=0.6, rotate=-360/44*37] at (-3.7,2.2) {$2k^2+3k-1$};
\node[scale=0.6, rotate=-360/44*38] at (-3.85,3.15) {$2k^2+3k$};
\end{tikzpicture}
\caption{Generator for Theorem~\ref{thm:3}.}
\end{figure}
\begin{proof}
We define an algorithm to prove the desired result.\\
Let $F=B_1B_2\cdots B_iA_{2k+1}A_{3k+2}\cdots A_{k^2+2k}C_{k^2+3k+1}C_{k^2+4k+2}\cdots C_{2k^2+3k}$
and $F_1=F(R^kF^{-1}R^{-k})$. Let us denote by
$H$ the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set
$\{ R^k, F_1\}$.
A) Use conjugation of $F_1$ with $R^k, R^{2k} , \cdots , R^{k^2}$ with proper multiplications to get $A_{k+1}B_{k+1}^{-1}$$\in$$H$, $A_{k+2}B_{k+2}^{-1}$$\in$$H$, $\cdots$ , $A_{2k-1}B_{2k-1}^{-1}$$\in$$H$, $A_{2k}B_{2k}^{-1}$$\in$$H$, respectively. Hence, $A_iB_i^{-1}$$\in$$H$ $\forall i$.
B) Follow the next $k$ steps.
1) Use conjugation of $F_1$ with $R^{kl}$ for some positive integers $l$'s with proper multiplications to get $C_{ik+1}B_{ik+1}^{-1}$$\in$$H$ and $C_{ik+1}B_{ik+2}^{-1}$$\in$$H$ $\forall i$.
2) Use conjugation of $F_1$ with $R^{kl}$ for some positive integers $l$'s with proper multiplications to get $C_{ik+2}B_{ik+2}^{-1}$$\in$$H$ and $C_{ik+2}B_{ik+3}^{-1}$$\in$$H$ $\forall i$.\\
$\cdots $
k) Use conjugation of $F_1$ with $R^{kl}$ for some positive integers $l$'s with proper multiplications to get $C_{ik}B_{ik}^{-1}$$\in$$H$ and $C_{ik}B_{ik+1}^{-1}$$\in$$H$ $\forall i$.\\
Hence, $C_iB_i^{-1}$$\in$$H$ and $C_iB_{i+1}^{-1}$$\in$$H$ $\forall i$.
It follows from Corollary~\ref{generating} that
$H={\rm Mod}(\Sigma_g)$, completing the proof of the theorem.
See Theorem ~\ref{thm:r2} for an example usage of the algorithm.
\end{proof}
Now, we prove Theorem~\ref{thm:3}.
\begin{proof}
For $k\geq 2$ and $g\geq 3k^2+4k+1$, let $H$ be the subgroup of ${\rm Mod}(\Sigma_g)$ generated by the set
$\{R^k, R^kF(R^kF^{-1}R^{-k})\}$.
Then, $H =$ ${\rm Mod}(\Sigma_g)$ by Theorem ~\ref{generalization}.
Hence, we are done by Lemma ~\ref{lem:order} since the orders of $R^k$ and $R^kF(R^kF^{-1}R^{-k})$ are $g/d$ where $d$ is the greatest common divisor of $g$ and $k$.
If $k=1$, we are done by Theorem ~\ref{thm:2}.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,032 |
Q: UIScrollView objectAtIndex In my UIScrollView,I have 30 images in my array. I tried getting the objectAtIndex in my array.
I want to be able to do is display an animation in a specific image when UIScrollView stop scrolling.
So I tried this code below:
- (void) scrollViewDidEndScrollingAnimation:(UIScrollView *)scrollView{
if([images objectAtIndex:0]){
//Animation 1
} else if([images objectAtIndex:1]){
//Animation 2
} else if{
...
}
But I cant make it do my animation. Is my condition possible or is there other way?
A: Your code:
if([images objectAtIndex:0])
will evaluate if the object stored in the array images at index position 0 is nil or not. And presumably if the array is in bounds then it will return true. And if the array is out of bounds then that statement will cause a crash. So the result of your if statements will most likely be true or crash (unless you are storing nil pointers in the array for some reason).
What are you trying to evaluate? Are you trying to determine what image is showing when your scroll view stops scrolling?
A: Im applying NSNumber or NSString to my array. Like this:
for (int i=0; i<30;i++)
{
//for NSString
[arrayName addObject:[NSString stringWithFormat:@"%d", i]];
//for NSNumber
[arrayName addObject:[NSNumber numberWithInt:i]];
}
Then in your scrollViewDidEndScrollingAnimation,implement either of this condition:
if ([[arrayName objectAtIndex:anIndex] intValue] == 1)
if ([arrayName objectAtIndex:anIndex] isEqualToString:@"1"])
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,738 |
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"redpajama_set_name": "RedPajamaC4"
} | 8,426 |
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"redpajama_set_name": "RedPajamaC4"
} | 7,630 |
Q: Smooth complete intersections Let $X_{2,3}\subset\mathbb{P}^n$, with $n\geq 5$, be a complete intersection of a quadric $X_2$ and a cubic $X_3$ containing a $2$-plane $H$. Assume $X_2$ and $X_3$ to be general among the hypersurface of the same degree containing $H$. In particular $X_2$ and $X_3$ are smooth.
Question: If $n = 5$ is $X_{2,3}$ necessarily singular? Is $X_{2,3}$ smooth for $n\geq 6$?
A: If $X \subset \mathbb{P}^n$ is a non-degenerate, smooth complete intersection variety of dimension at least $3$, then the restriction map $$\operatorname{Pic}(\mathbb{P}^n) \to \operatorname{Pic}(X)$$ is an isomorphism by Grothendieck-Lefschetz theorem. So, we get $\operatorname{Pic}(X) = \mathbb{Z}$, generated by the hyperplane section.
This implies that $X$ contains no linear spaces of codimension $1$; in particular, the threefold $X_{2,3} \subset \mathbb{P}^5$ is necessarily singular.
A: If $n=5$ then let $\mathbb P^5$ have coordinates $x_0,\ldots,x_5$ and suppose the plane is $H=\mathbb P^2_{(x_0:x_1:x_2)}$. The two equations of $X$ are necessarily of the form
$$ \begin{pmatrix}
A_1 & B_1 & C_1 \\
D_2 & E_2 & F_2 \end{pmatrix}\begin{pmatrix}
x_3 \\ x_4 \\ x_5 \end{pmatrix} = 0 $$
for some polynomials $A_1,\ldots,F_2\in\mathbb C[x_i]$ of the indicated degree. The scheme $Z\subset \mathbb P^5$ defined by the $2\times 2$ minors of the $2\times 3$ matrix has codimension 2, and thus $Z\cap H$ is nonempty (generically given by some points). Since both equations have order of vanishing $\geq2$ along $(Z\cap H)\subset X$, the 3-fold $X$ must be singular there.
In dimension $n\geq6$ the scheme $Z$ (obtained by the analogous argument) has codimension $n-3\geq3$ so in the general case $Z\cap H=\emptyset$ and the $n$-fold $X$ is smooth (along $H$ at least). Essentially, since the matrix never drops rank we can always use the equations to eliminate two of the variables $x_3,\ldots,x_n$, showing that $X$ is smooth of dimension $n$ at every point along $H$.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,259 |
Військо́во-тра́нспортна авіа́ція (ВТА) або транспортна авіація — один з видів військової авіації (ВПС), призначений для десантування повітряних десантів, перевезення військ, здійснення маневру військ повітрям, доставки озброєння, боєприпасів, ракет, пального, продовольства і інших матеріальних засобів, евакуації поранених і хворих.
Транспортна авіація
У 1930-х роках у складі ВПС найбільш розвинених держав були створені окремі авіаційні підрозділи для перевезення військ і військових вантажів; вони були оснащені переобладнаними літаками бомбардувальної авіації і транспортними літаками цивільної авіації. У роки, що передували Другій світовій війні, в низці країн (СРСР, США, Велика Британія, Німеччина, Франція тощо) було створено спеціальні військово-транспортні літаки, а також вантажні десантні планери, а в ході війни сформовано авіаційні транспортні частини та з'єднання.
Під час німецько-радянської війни радянські транспортні авіаційні частини здійснювали викидання (висадку) повітряних десантів, доправляли військам боєприпаси і пальне, евакуювали поранених, надавали допомогу партизанам. 1946 року авіаційно-транспортні частини ВПС були об'єднані в десантно-транспортну авіацію, пізніше вона отримала назву транспортно-десантна. У 1955 транспортно-десантна авіація була перетворена на військово-транспортну, що стала самостійним видом авіації радянських ВПС.
В інших країнах найбільш розвинену ВТА мають збройні сили США. Вона поділяється на стратегічну, тактичну і армійську. Стратегічна і тактична транспортна авіація об'єднані у військово-транспортне авіаційне командування (ВТАК) ВПС США, яке здійснює перевезення військ і вантажів (міжконтинентальні і в межах ТВД), а також висадку (викидання) повітряних десантів у межах ТВД. Армійська транспортна авіація входить до складу з'єднань і частин сухопутних військ. Вона призначена для виконання завдань безпосередньо в районах бойових дій з перекидання військ, висадки тактичних повітряних десантів і диверсійних груп, доставці матеріальних засобів, евакуації поранених і хворих. Для висадки і забезпечення морських десантів у складі ВМС США і ряду інших країн є частини і підрозділи військово-транспортних літаків і транспортно-десантних вертольотів.
Сучасна ВТА оснащується спеціальними військово-транспортними літаками, що відрізняються за конструкцією та устаткуванням від інших типів військових літаків. За своїм призначенням і вантажопідйомності літаки ВТА поділяються на важкі, середні й легкі. До важких відносяться Ан-22 (СРСР, Україна, Росія), C-17 «Глоубмайстер» III, С-141 «Старліфтер», С-5А «Гелаксі» (США), до середніх — Ан-12 (СРСР, Україна), С-130 «Геркулес» (США), С-160 «Трансалл» (Франція, ФРН); до легких — Ан-24, Ан-26 (СРСР, Україна), С-123 «Провайдер», С-7А «Карібу» (Канада, США) та інші. Для перевезення військ і вантажів широко використовуються також транспортно-десантні вертольоти.
Під час здійснення десантування військ і доставки вантажів, ВТА залежно від умов обстановки, може застосовувати парашутний або посадковий спосіб. В окремих випадках використовується безпарашутне скидання вантажів.
У складі Українських Повітряних сил (ПС) перебувають формування транспортної авіації, яка призначена для перекидання та десантування аеромобільних й повітряно-десантних підрозділів, повітряних перевезень військ і бойової техніки, доставки вантажів, евакуації поранених тощо. На її озброєнні знаходяться два основні типи літаків.
Ан-26 — легкий транспортний літак.
Іл-76МД — важкий транспортний літак з чотирма турбореактивними двигунами.
У найближчому майбутньому планується замінити три вищезгадані типи одним середнім транспортним літаком нового покоління Ан-70, створеним в Україні.
Див. також
Повітряні сили
Бомбардувальна авіація
Ан-12
Ан-22
Ан-225 «Мрія»
Повітряно-десантна операція
Протиповітряна оборона
Протиракетна оборона
Авіаційне пальне
Джерела
Посилання
Военно-транспортная авиация
Література
Тактика подразделений воздушно-десантных войск. — Москва, Воениздат, 1985.
Військово-транспортна авіація
Військова авіація | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,119 |
Spotlights can obscure even as they illuminate.
Robert Wells Rhapsody in Rock with Smoke and Fire (2009). Photograph by Bengt Nyman.
Recently on Twitter a buddy posed a question: What's the greatest guitar solo of the 20th century?
But never mind what I think — why not let a master do the talking?
Years ago I picked up this interesting tidbit on the radio (since confirmed by Wikipedia, so this is an air-tight fact): Jimmy Page's favorite guitar solo ever is the one from Steely Dan's "Reelin' In The Years." This knowledge just might affect your opinion of the song; I know it changed the way I hear it. There isn't even a single definitive solo section; the brilliant solo work is woven throughout the entire composition.
You don't have to be a Steely Dan fan to recognize that Elliott Randall's work on that song — their biggest hit — was inspired. Here's the thing: Elliott Randall was never in Steely Dan. He was in the studio that day because he's a session guitarist. A highly accomplished one at that.
Who was that guitarist? Well, it was a session musician. Unlike most session musicians though, you might've heard of this one. It was Duane Allman, and he was a regular part of the renowned Muscle Shoals Rhythm Section (a group of regular studio players also known as the Swampers) until his younger brother Gregg offered him a spot in this new band he was forming.
Duane Allman died at age 24, but his tragically foreshortened career was enough to land him at the very top of some "Best Guitarists of All Time Lists." NME had him at number 1 on one of theirs, and Rolling Stone more recently placed him at number 2, adding that he'd be high on their list even if judged solely by his session work prior to the Allman Brothers Band.
Many people are aware that session musicians exist, but it's safe to say that most underestimate just how much popular music they're responsible for. The songs heard on the radio often weren't recorded by the bands they're attributed to. Studio time and resources are too precious to allow front acts to play imperfectly over and over till at last they get it right, or close to right. This was especially true in the analog days of tape. But even in these days of digital recording, studio musicians are called in to lay down tracks precisely, exquisitely. They are hired because they rarely make mistakes and they often make songs better than their original conceptions in surprising ways. They are hired because they are the best.
It's not an understatement to say that to be a session musician requires nothing less than virtuosity. They are as close to perfect as their profession can produce. But unless they pull a Duane Allman and leave the studio handlers behind to create their own music, not many people outside the music industry know who they are.
Occasionally, though, they get the spotlight they had long deserved but had enjoyed only vicariously.
In the 1960s and '70s in L.A., a select group of session musicians were responsible for an extraordinary quantity and quality of output. Though rarely if ever credited at the time, they were the players behind the recordings of artists as varied as Elvis Presley, Frank Sinatra, the Beach Boys, Simon and Garfunkel, the Mamas and the Papas, Nat King Cole, the Partridge Family, the Monkees, the Byrds, the Righteous Brothers, and Leonard Cohen.
How good was the Wrecking Crew?
So good that in between proper sessions they often would jam, and end up liking something so much that they'd develop it into a song, record it, and release it under a fake band name. Quite a few "one-hit wonders" are actually one-off songs written and recorded on the fly by different combinations of Wrecking Crew players.
So good that sometimes they would embellish their material to such a degree that the original artists who wrote the song would struggle to frantically try to learn some of the Wrecking Crew's intricate parts before going out on tour.
So good that for six years in a row (1966–71) the Grammy for Record of the Year was played by Wrecking Crew musicians.
The good news is that these remarkable musicians now receive more widespread recognition. Kent Hartman recently published The Wrecking Crew: The Inside Story of Rock and Roll's Best-Kept Secret, and they're also the subject of a documentary directed by Danny Tedesco (that's the trailer above). Other famous clusters of session players, like Motown's Funk Brothers, James Brown's J.B.'s, and the aforementioned Swampers of Muscle Shoals are also more well known now than they were when they were recording, though they'll never achieve the fame of the artists they recorded for.
The music industry of course shouldn't be singled out here.
For any large-scale endeavor, it would be unreasonable and impossible to equally distribute credit among everyone involved. Furthermore, individual leaders often do play outsize roles in these accomplishments. It's often appropriate that they receive the lion's share of accolades.
But you can't help but feel that too often there are important people behind the scenes who get short shrift. Think of Andy Warhol's Factory. An assembly line of his Superstars would work to produce his famous silkscreens. Chances are that any physical piece of art you see attributed to Andy Warhol was actually produced by one these many apprentices. He was the idea guy (no small feat). But this sort of thing is common in the art world.
Put on Saturday Night Live or The Tonight Show. The players and the host get the laughs and the press, but the material is crafted by a dozen or more writers locked in some room somewhere. Like the session players whose music we often hear without knowing who's playing it, these writers are often known to fellow comedians and other industry people, but not so much to the public.
Casual fans will know the quarterback but not each member of the offensive line that enables his every success. Likewise, we'll know who directed a film but not who held the boom. Only rarely are we able to name a single other person on the crew — though at least they're always credited.
You also see this in the tech world. It's hard to think of a more understated group of people than the coders who work behind the scenes to make products work. But of course founders get all the love, even if they aren't developers themselves. Even on the grandest scale, brilliant businesspeople and marketers (Steve Jobs) get the spotlight while ingenious builders (Steve Wozniak) hang back behind the curtain (which is often right where they want to be).
Wozniak isn't exactly a secret. In fact he's idolized by scores of people he inspired. But he always was and always will be in Steve Jobs's shadow.
Rock stars, quarterbacks, film directors, founders, generals. All are essential, all are rightfully celebrated. Face it — it's only human to generalize in this way. And yet.
This isn't to suggest that Wrecking Crews are prevalent only in glamorous industries. Unsung heroes are all around. To take one random example, think about how society would be affected if our garbage wasn't ferried away every week. Things would rapidly get out of control. We all take out our garbage, but how often do we think about what happens next, or how minor our role in the matter really is?
We're all the stars of our own plays, the centers of our own universes. But surely every person has a Wrecking Crew to be grateful for, hanging back in the shadows. This isn't to suggest that the results of your labor are not your own, or that you're taking credit for someone else's work. The Wrecking Crew did not technically compose the songs they recorded for others — but they were quintessential collaborators, elevating other artists' material, honing their creations, adding depth, and ultimately helping those artists reach a wider audience. We all can name people who made us look better along the way, who maybe provided a little bit of polish to our efforts, who gave us valuable inspiration. Parents, family, friends, partners, coworkers, teachers, mentors, coaches, doctors, authors, artists, pets. And countless people we'll never know, whose actions affected us down the line in unknowable ways.
And you've likely been on a Wrecking Crew for somebody else, at a job or in your personal life. There are worse things in the world than making someone else look good, helping them to succeed.
Sometimes, Wrecking Crews are impossible to identify. Which brings us back to legendary guitarist Jimmy Page. In the documentary It Might Get Loud, he tells a fascinating story of serendipity. When he was about nine years old, his family moved to a new house. He recalls the moment they first walked through the front door. As you can imagine, there was no furniture anywhere. The entire place was empty — except for one item. For some inexplicable reason, the prior owners had left behind an old guitar, which stood all alone in a corner. Who were these people? Who left that instrument behind, and why? Needless to say, nine-year-old Jimmy Page was drawn to it.
Everyone knows what happened after. Led Zeppelin happened after. But before all that, even before he joined the Yardbirds, Jimmy Page had a different gig.
He was a session guitarist. | {
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} | 4,099 |
Tishman Speyer and BlackRock committed to orderly transition.
We have spent the last few weeks negotiating in good faith to restructure the debt and ownership of Stuyvesant Town/Peter Cooper Village. It was our hope in these discussions that our partnership would remain as part of the long-term ownership.
We have no intention of putting ST/PCV into bankruptcy. We make this decision as we feel a battle over the property or a contested bankruptcy proceeding is not in the long-term interest of the property, its residents, our partnership or the City.
We are fully committed to an efficient transition of the property's operation and are offering to continue managing the property during this period to make the transition smooth and seamless for the residents. | {
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{"url":"https:\/\/math.stackexchange.com\/questions\/1001499\/sign-of-the-real-part-of-two-complex-numbers","text":"# Sign of the real part of two complex numbers\n\nLet $z_1$ and $z_2$ be two complex numbers of modulus $1$. Denote by $Re$, the real part of a complex number.\n\nUsing maple, I believe that the sign of $Re((1-z_1)(1-z_2))$ is equal to the sign of $Re((z_1-1)(z_2-1)(1-z_1z_2))$. Is there a nice way to prove this?\n\n\u2022 I think you can use $\\Re(z)=\\frac{z+\\overline{z}}{2}$ with $z$ being all those expressions and conclude something from there. Nov 1, 2014 at 19:27\n\nExpress the complex numbers as\n\n$$z_1=a_1+ib_1$$\n\n$$z_2=a_2+ib_2$$\n\nExpanding the products should tell you if the result is true.\n\n\u2022 Unfortunately, the signs of the resulting expressions seem quite hard to determine?! Nov 1, 2014 at 20:13\n\nFor $z_{1,2} \\approx 1$ and $\\Im z_{1,2}>0$, we have that $1-z_1$, $1-z_2$, $1-z_1z_2$ are approximately multiples of $i$ with slightly negative real part, i.e. their arguments are slightly above $\\frac\\pi2$. We conclude that $(1-z_1)(1-z_2)$ has argument $\\approx \\pi$ and negative real part, whereas $(1-z_1)(1-z_2)(1-z_1z_2)$ has argument slightly above $\\frac32\\pi$, i.e. positive real part.\n\n\u2022 $1-z_1$ etc. are approximately multiples of $i$ with slightly positive real part, I think. Which unfortunately spoils your nice argument. Nov 1, 2014 at 19:53","date":"2022-07-04 14:54:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.979297399520874, \"perplexity\": 192.8977894239431}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104432674.76\/warc\/CC-MAIN-20220704141714-20220704171714-00327.warc.gz\"}"} | null | null |
Q: Parse a string into characters using ruby I have a requirement in my ruby code.I'm using ruby 1.9.2 and rails 3.0.
I have a string like "SR2G1M1D2".Now i want to split it and extract values like [S,R2,G1,M1,D2] .It's like whenever next value is character it should split.Is there any ruby function or code available.
Thanx
A: "SR2G1M1D2".scan(/\D\d*/)
=> ["S", "R2", "G1", "M1", "D2"]
Hope this helps.
A: Just use #split with a RegExp:
ruby-1.9.2-p180 :002 > "SR2G1M1D2".split(/(?=[a-zA-Z])/)
=> ["S", "R2", "G1", "M1", "D2"]
ruby-1.9.2-p180 :005 >
A: split with a regex should do this. Basically a regex that does .\D with a positive lookahead on the \D. sadly i haven't used lookaheads enough to know the format
| {
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July 26, 2017 10:35am Comment Doug Lynch
Nova Launcher Partners with Sesame Shortcuts for a New Search Feature
Nova Launcher is one of the most popular custom launchers available on Android right now. The application has a ton of customization features so you can tweak the look and feel of your launcher to how you like it. It may take some time to go through the plethora of options included, but it's generally worth it to get your launcher looking perfect. Now, thanks to a new partnership with Sesame, the launcher is now getting an application and shortcut search feature.
Sesame Shortcuts is an application in the Play Store right now from a developer team calling themselves the Sesame Crew. With their application, they've been working on gathering deep linking shortcuts to some of the most popular applications. Meaning, you can use one of these shortcuts to go directly to a specific part of an application instead of having to open it and manually navigate to it. This might sound a lot like Android 7.1 Nougat's App Shortcuts feature, and that's because it is.
However, a lot of those have been limited to just people using Android 7.1 Nougat. As we saw earlier this month, only 0.9% of the Android community are currently running Android 7.1 Nougat. Some of these shortcuts are able to be backported as far back as Android 5.1 Lollipop, but this selection is limited and many are left wanting more. This is where Sesame Shortcuts comes into play and its new integration into Nova Launcher makes it even more accessible.
Those running Android 5.0 Lollipop and higher will be able to take advantage of this new feature. How it will work is you tap the Home button when you're already at the Home Screen and it will launch a search prompt. From here, you can start typing for anything you want to go directly to. This can be someone's name that you're talking to via SMS or Hangouts, it can be a song from a Spotify playlist, anything that Sesame Shortcuts supports can be searched from here.
You then just need to tap it and it will launch the application and take you straight there. The new Sesame Shortcuts feature is already integrated in 5.4-beta1 of Nova Launcher so you can check it out right now. You will need to manually download the Sesame Shortcuts application yourself. It comes with a 14-day free trial, but then has a $3 in-app purchase to continue using it. Check out this GIF to see the new feature in action.
Source: TeslaCoil Apps
Tags nova launcherSesame Shortcutsteslacoil software
XDA » News Brief » Nova Launcher Partners with Sesame Shortcuts for a New Search Feature
Doug Lynch
When I am passionate about something, I go all in and thrive on having my finger on the pulse of what is happening in that industry. This has transitioned over the years from PCs and video games, but for close to a decade now all of my attention has gone toward smartphones and Android. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,578 |
Q: Angularjs - Show True/False as Yes/No Is there a simple way to show a true/false value as yes/no?
I am retrieving from the Database a JSON that contains :
[Object, Object, "WithCertification": true]
This is the html:
With Certification {{elem.WithCertification}}
Is displaying this:
With Certification true
I want it to display this:
With Certification yes
Is there a way without having to go to the controller to change the true/false to yes/no?
Anyhow I need to change it, any suggestions?
Thank you.
A: {{elem.WithCertification == true ? "Yes" : "No"}}
A: You can use either a filter or a ternary operator. The filter would probably be better from what i've read ternary operators are frowned upon.
app.filter('YesNo', function(){
return function(text){
return text ? "Yes" : "No";
}
})
The code is fairly simple set your app.filter, give it a name, inside pass an anon function, returning a function that takes your true,false as a param, then return the text and determine if true or false, if true return yes, otherwise no.
Alternatively do it inside the tag.
<td>{{item.value ? "Yes":"No"}}</td>
You can see this working here http://plnkr.co/edit/altgrjKhXHACmczOvGFw
A: You can have ternary operators inside Angular Expressions, so just do this:
With Certification {{elem.WithCertification?'yes':'no'}}
A: The best way is to create a custom filter, as it is best to put as less logic in the view as possible. Below is a sample code showing you how to achieve this.
angular.module('myApp', [])
.filter('yesOrNo', function() {
return function(input) {
return input === 'true' ? 'yes' : 'no' ;
};
})
In html, you would just do:-
With Certification {{elem.WithCertification | yesOrNo}}
A: If this is a one time case, I agree with Joseps answer. Use:
With Certification {{elem.WithCertification?'yes':'no'}}
However, if you intend to use this more often, I would recommend using a filter. A very similar case is explained in the AngularJS tutorial step 9
| {
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Home > Operators > Compliance & enforcement > Investigations
The Office of the National Rail Safety Regulator (ONRSR) may conduct compliance investigations for the purpose of establishing whether a breach of the Rail Safety National Law has occurred.
A compliance investigation may be conducted in a range of circumstances which include but are not limited in response to:
a notifiable occurrence;
an adverse finding from an audit;
outcomes identified from a inspection;
outcome identified from a investigation;
matters identified trend analysis;
findings from a rail transport operator's investigation report;
confidential or other intelligence reports; or
a written direction from a responsible Minister for a participating jurisdiction on a rail safety matter relating to that jurisdiction.
ONRSR will investigate in order to determine:
causes;
whether there has been a breach of legislation;
whether action has been taken or needs to be taken to prevent a recurrence of an incident and / or to secure compliance with the law;
lessons to be learnt and whether there is a requirement to influence the law and industry guidance; and
what response is appropriate to a breach of the law.
To maintain a proportionate response, ONRSR will devote most resources available for investigation to the more serious incidents.
ONRSR will always carry out a site investigation of a reportable work-related death, unless there are other specific reasons for not doing so, in which case those reasons will be recorded.
In selecting which complaints or reports of incidents or injury to investigate, and in deciding the level of resources to be used, ONRSR will take account of the following factors:
the severity and scale of potential or actual harm;
the seriousness of any potential or actual breach of the law;
knowledge of the Rail Transport Operator's (RTO's) past performance in terms of compliance with the law;
ONRSR's enforcement priorities;
the likelihood of the investigation leading to successful enforcement action against a RTO or a meaningful improvement in their behaviour; and
the wider relevance of the event, including serious public concern.
The Australian Transport Safety Bureau (ATSB) is responsible for identifying the causes of accidents it investigates in order to improve safety; it does not allocate blame.
Working arrangements agreed between ONRSR and the ATSB reflect both their respective statutory duties and the need to ensure efficient and effective liaison. ONRSR and the ATSB have a Memorandum of Understanding in place.
Compliance and Enforcement policy
Rail Safety Audit policy
Last updated: 9 September 2013 | {
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} | 2,510 |
{"url":"https:\/\/www.maths.usyd.edu.au\/s\/scnitm\/dhauer-PDESeminar-PetrGurka-Asym","text":"SMS scnews item created by Daniel Hauer at Wed 7 Aug 2019 0618\nType: Seminar\nDistribution: World\nExpiry: 12 Aug 2019\nCalendar1: 12 Aug 2019 1200-1300\nCalLoc1: AGR Carslaw 829\nCalTitle1: Asymptotic estimates of s-numbers of Sobolev-type embeddings\nAuth: dhauer@203.54.34.178 (dhauer) in SMS-WASM\n\n# Asymptotic estimates of s-numbers of Sobolev-type embeddings\n\n### Petr Gurka\n\nPetr Gurka\nCzech University of Life Sciences Prague, Czech Republic\nMon 12th Aug 2019, 12-1pm, Carslaw Room 829 (AGR)\n\n## Abstract\n\nIn a quite recent paper of D. E. Edmunds and J. Lang, Asymptotic formulae for $s$-numbers of a Sobolev embedding and a Volterra type operator (published in [Rev. Mat. Complut., 29(1), 2016]) the authors obtained sharp upper and lower estimates of the approximation numbers of a Sobolev embedding involving second derivatives and of a corresponding integral operator of Volterra type. We discuss possible extensions of these results for higher order derivatives. Namely, we obtain estimates for the embedding of Sobolev type involving derivatives of order four.\n\nActions:","date":"2022-12-05 06:59:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3939674496650696, \"perplexity\": 5973.8320159299}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446711013.11\/warc\/CC-MAIN-20221205064509-20221205094509-00552.warc.gz\"}"} | null | null |
{"url":"http:\/\/www.math.cmu.edu\/CCF\/ccfseminar.php?SeminarSelect=606","text":"Faculty in \u00a0Mathematical\u00a0\u00a0Finance \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Math Finance Home Conferences Seminars People Open Positions Contact Probability and Computational Finance Seminar Robert Aguirre Carnegie Mellon University Title: An Arbitrage-Free Proportional Volatility Term Structure Model Abstract: We construct a continuous-time term structure model in the risk-neutral measure via a family of instantaneous forward rates $f(t, T)$ such that: (1) $df\/f$ has deterministic volatility, (2) $f(t,T)$ explodes with positive probability for positive $t$ and $T$, and (3) the family of $B(t,T)$ constructed in the expected way create an arbitrage-free collection of assets with an appropriate choice of numeraire.Date: Monday, November 12, 2012Time: 5:00 pmLocation: Wean Hall 6423Submitted by:\u00a0\u00a0Steve Shreve","date":"2018-05-24 03:59:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8844647407531738, \"perplexity\": 3258.3112315308094}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-22\/segments\/1526794865913.52\/warc\/CC-MAIN-20180524033910-20180524053910-00229.warc.gz\"}"} | null | null |
if these specs r true then samsung wont compete with others in this year.
Qualcomm?! Blah, hope it's not true.
Snapdragon, with Adreno 205? ADRENO TWO O FIVE? That's much, much weaker than Hummingbird in the original Galaxy.
Are they serious? I hope this info is false. | {
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Sattenapalle is a town in Palnadu district of the Indian state of Andhra Pradesh. It is a municipality and the headquarters of Sattenapalle mandal under Sattenapalle revenue division.
History
The region is an important region of the ancient Kammanadu. This region is also part of the Palnadu region.
Demographics
Census of India, Sattenapalle had a population of 56,721. The total population constitute, 28,350 males and 28,371 females —a sex ratio of 1001 females per 1000 males. 5,827 children are in the age group of 0–6 years, of which 3,046 are boys and 2,781 are girls —a ratio of 913 per 1000. The average literacy rate stands at 73.58% with 37,449 literates, significantly higher than the state average of 67.41%.
Civic administration
On 1 April 1984, the municipality was formed as 3rd grade which is now a second grade municipality. The municipality constitutes 30 election and 11 revenue wards. The civic services and infrastructure of the municipality include, supply of 135 litres of per capita water supply, 49 public taps, 415 bore–wells etc. Maintaining 148 km of roads, 136 km of drains etc.
Economy
Sattenapalle is a part of Guntur–Sattenapalle growth corridor.
Transport
The town has a total road length of . APSRTC operates buses from Sattenapalle bus station to most of the cities and towns in the district like Guntur, Macherla, Amaravati, Narasaraopet, Piduguralla etc.. NH 167AG passes through the town. Sattenapalle railway station is located on Pagidipalli–Nallapadu section of Guntur railway division.
See also
Villages in Sattenapalle mandal
References
External links
Towns in Guntur district
Mandal headquarters in Guntur district
Towns in Andhra Pradesh Capital Region | {
"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"https:\/\/socratic.org\/questions\/is-7-6-a-irrational-number","text":"# Is 7\/6 a irrational number?\n\nMar 30, 2016\n\n$\\frac{7}{6}$ is not irrational\n\n#### Explanation:\n\nAs $\\frac{7}{6}$ is a fraction, it is a rational number and not irrational number. Irrational numbers cannot be written as a fraction or ratio of two integers.\n\nRational numbers, when expressed in decimal form have either terminating decimals (when denominator of the fraction in its smallest form has only $2$ and $5$ as prime factors) on non-terminating but repeating decimals.\n\nIf a number written in decimal form has non-terminating as well as non-repeating decimals, then it cannot be written as a ratio of two integers and is hence irrational.\n\nAs $\\frac{7}{6} = 1.16161616 \\ldots \\ldots .$, in which $16$ get repeated (till infinity), it can be expressed as a fraction and is not irrational.","date":"2021-10-22 12:54:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 6, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9062842726707458, \"perplexity\": 534.234683330472}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585507.26\/warc\/CC-MAIN-20211022114748-20211022144748-00134.warc.gz\"}"} | null | null |
Fruits of Viet Nam: Waterapple (Quả Roi, Quả mận)
A bell-shaped fruit, mostly 5cm in diameter, the Waterapple is abundant in Vietnam, its neighbours in Southeast Asia as well as in the Pacific Islands. It may be named waterapple, but it does not, in any way, taste like an apple. The fruits' color could vary from pale green to ruby red, though sweetest when at its red-most. With a few brown seeds found in its hollow core that sometimes contain a loose weave of cotton candy-like mesh, its white wooly fibrous flesh holds a thirst-relieving juice that is slightly acidic but refreshing to the taste. Its flavour is likened to that of a snow pear, but with slight hint of bitter aftertaste. When young, the waterapple could be very sour. Because of its vibrant color, the subtle sheen of its outer texture and its unique shape, it is popularly used in the country to adorn altars.
The waterapple could be eaten in a variety of ways. Most would prefer to cut the fruit in parts, but those who greatly enjoy the appeal of the bell-shape would rather eat is as whole, taking bitefuls from it. Chilled or fresh, having a taste of the waterapple is always a delightful experience. Its seeds are not edible, though, so be careful not to swallow them. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,645 |
package ru.maizy.ambient7.core.config.reader
import ru.maizy.ambient7.core.config.options.DbOptions
import ru.maizy.ambient7.core.config.{ Ambient7Options, Defaults }
trait MainDbConfigReader extends UniversalConfigReader {
import UniversalConfigReader._
private def mainDbOpts(opts: Ambient7Options)(fill: DbOptions => DbOptions): Ambient7Options = {
opts.copy(mainDb = Some(fill(opts.mainDb.getOrElse(DbOptions()))))
}
private def appendDbOptsCheck(check: DbOptions => CheckResult): Unit =
appendCheck { appOpts =>
appOpts.mainDb match {
case Some(dbOpts) => check(dbOpts)
case _ => failure("DB opts not defined")
}
}
def fillDbOptions(): Unit = {
// TODO: uni config rules
cliParser.opt[String]("db-url")
.abbr("d")
.valueName(s"<${Defaults.DB_URL}>")
.action { (value, opts) => mainDbOpts(opts)(_.copy(url = Some(value))) }
.text(s"URL for connecting to h2 database")
appendSimpleOptionalConfigRule[String]("db.url") { (value, opts) =>
mainDbOpts(opts)(_.copy(url = Some(value)))
}
appendDbOptsCheck{ dbOpts =>
if (dbOpts.url.isDefined) {
success
} else {
failure("db-url is required")
}
}
cliParser.opt[String]("db-user")
.action { (value, opts) => mainDbOpts(opts)(_.copy(user = value)) }
.text("database user")
appendSimpleOptionalConfigRule[String]("db.user") {
(value, opts) => mainDbOpts(opts)(_.copy(user = value))
}
cliParser.opt[String]("db-password")
.action { (value, opts) => mainDbOpts(opts)(_.copy(password = value)) }
.text("database password")
appendSimpleOptionalConfigRule[String]("db.password") { (value, opts) =>
mainDbOpts(opts)(_.copy(password = value))
}
()
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,546 |
CR tests for heavy metal levels in spices and herbs
Posted on December 30, 2021 by KOAM TV 7
Most of us shake spices on our food and add them to our recipes without much thought, but Consumer Reports found there could be problems with some of them.
"Of all the spices that we investigated, one-third of them had concerning levels of heavy metals: lead, cadmium or arsenic," says James Dickerson, chief scientific officer with Consumer Reports.
Dickerson says Consumer Reports tested 126 herbs and spices from 38 different companies. Concerning levels of heavy metals were detected in every brand of oregano and thyme tested. The metals were also found in almost all brands of ginger and basil tested. And in around half of the paprika and turmeric brands that were tested.
Studies have shown that frequent exposure to lead, cadmium, and arsenic can be damaging over time. "For children, during their early stages of development, these heavy metals can adversely impact their neurological development as well as the respiratory development," Dickerson says.
Spices are often grown overseas where contaminated water can lead to heavy metals in the soil. The American Spice Trade Association says an analysis shows spices make up less than .1% of dietary lead exposure in children ages one to six, and the risk is low in adults.
Dickerson says, "don't panic." He says the good news is every brand tested of curry, garlic powder, black pepper, coriander, sesame seed, and saffron did not have concerning levels of metals.
"Think about what herbs and spices that you and your family use and diversify the amount that you use, so don't overload on a particular spice," Dickerson says.
In the Consumer Reports tests, organic products did not perform any better than non-organic spices.
Correspondent: Naomi Ruchim
Producer: Chris Stein
This entry was posted in Local News and tagged Food, Health, Top Stories by KOAM TV 7. Bookmark the permalink. | {
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{"url":"https:\/\/web2.0calc.com\/questions\/what-is-the-remainder-when-x-1-2010-is-divided-by-x-2-x-1","text":"We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.\n\n+0\n\n# What is the remainder when (x+1)^2010 is divided by x^2+x+1?\n\n0\n174\n1\n\nWhat is the remainder when (x+1)^2010\u00a0is divided by\u00a0x^2+x+1?\n\nMar 27, 2019\n\n### 1+0 Answers\n\n#1\n+23516\n+3\n\nWhat is the remainder when $$(x+1)^{2010}$$\u00a0is divided by\u00a0$$x^2+x+1$$?\n\nI assume\n\nLook at the low powers of x.\n\nEventually they will cycle.\n\n$$\\begin{array}{|r|c|c|} \\hline n & & \\text{remainder} \\\\ \\hline 0 & \\dfrac{(x+1)^0}{x^2+x+1} & \\color{red}1 \\\\ \\hline 1 & \\dfrac{(x+1)^1}{x^2+x+1} & x+1 \\\\ \\hline 2 & \\dfrac{(x+1)^2}{x^2+x+1} & x \\\\ \\hline 3 & \\dfrac{(x+1)^3}{x^2+x+1} & -1 \\\\ \\hline 4 & \\dfrac{(x+1)^4}{x^2+x+1} & -x-1 \\\\ \\hline 5 & \\dfrac{(x+1)^5}{x^2+x+1} & -x \\\\ \\hline 6 & \\dfrac{(x+1)^6}{x^2+x+1} & \\color{red}1 \\\\ \\hline 7 & \\dfrac{(x+1)^7}{x^2+x+1} & x+1 \\\\ \\hline 8 & \\dfrac{(x+1)^8}{x^2+x+1} & x \\\\ \\hline \\ldots &\\ldots & \\ldots \\\\ \\hline 2010 & \\dfrac{(x+1)^{2010}}{x^2+x+1} & \\color{red}1 \\\\ \\hline \\end{array}$$\n\n$$(x+1)^n \/ (x^2+x+1)$$ is cyclic of order 6\nSo we get $$(x+1)^n \\equiv (x+1)^{n\\pmod{6}}\\pmod{x^2+x+1}$$\n\n$$(x+1)^{2010} \\equiv (x+1)^{6\u00b7335+0} \\equiv(x+1)^{0} \\equiv 1 \\pmod{x^2+x+1}$$\n\nThe remainder when $$(x+1)^{2010}$$ is divided by $$x^2+x+1$$ is 1\n\nMar 28, 2019","date":"2019-12-07 06:39:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.17627085745334625, \"perplexity\": 3024.901945064622}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575540496492.8\/warc\/CC-MAIN-20191207055244-20191207083244-00488.warc.gz\"}"} | null | null |
{"url":"https:\/\/socratic.org\/questions\/the-volume-of-a-gas-is-10-2-l-when-the-temperature-is-9-5-o-c-if-the-volume-is-i","text":"# The volume of a gas is 10.2 L when the temperature is 9.5\"^oC. If the volume is increased to 64.9 L without changing the pressure what is the new temperature?\n\nNov 20, 2016\n\nThe new temperature is $= 1524.5$\u00baC\n\n#### Explanation:\n\nwe need to use Charles' law\n\n${V}_{1} \/ {T}_{1} = {V}_{2} \/ {T}_{2}$\n\nThe temperatures are in Kelvin (K)\n\nK=\u00baC+273\n\n${V}_{1} = 10.2 l$\n\nT_1=9.5\u00baC=9.5+273=282.5K\n\n${V}_{2} = 64.9 l$\n\nso, ${T}_{2} = {V}_{2} \\cdot {T}_{1} \/ {V}_{1} = 64.9 \\cdot \\frac{282.5}{10.2} = 1797.5 K$\n\nT_2=1797.5-273=1524.5\u00baC","date":"2019-09-23 05:18:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 8, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6263192892074585, \"perplexity\": 2742.6287677681385}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514576047.85\/warc\/CC-MAIN-20190923043830-20190923065830-00101.warc.gz\"}"} | null | null |
{"url":"https:\/\/xianblog.wordpress.com\/tag\/probability-basics\/","text":"## Why do we draw parameters to draw from a marginal distribution that does not contain the\u00a0parameters?\n\nPosted in Statistics with tags , , , , , , , on November 3, 2019 by xi'an\n\nA revealing question on X validated of a simulation concept students (and others) have trouble gripping with. Namely using auxiliary variates to simulate from a marginal distribution, since these auxiliary variables are later dismissed and hence appear to them (students) of no use at all. Even after being exposed to the accept-reject algorithm. Or to multiple importance sampling. In the sense that a realisation of a random variable can be associated with a whole series of densities in an importance weight, all of them being valid (but some more equal than others!).\n\n## a probabilistic proof to a quasi-Monte Carlo\u00a0lemma\n\nPosted in Books, Statistics, Travel, University life with tags , , , , , on November 17, 2014 by xi'an\n\nAs I was reading in the Paris m\u00e9tro a new textbook on Quasi-Monte Carlo methods, Introduction to Quasi-Monte Carlo Integration and Applications, written by Gunther Leobacher and Friedrich Pillichshammer, I came upon the lemma that, given two sequences on (0,1) such that, for all i\u2019s,\n\n$|u_i-v_i|\\le\\delta\\quad\\text{then}\\quad\\left|\\prod_{i=1}^s u_i-\\prod_{i=1}^s v_i\\right|\\le 1-(1-\\delta)^s$\n\nand the geometric bound made me wonder if there was an easy probabilistic proof to this inequality. Rather than the algebraic proof contained in the book. Unsurprisingly, there is one based on associating with each pair (u,v) a pair of independent events (A,B) such that, for all i\u2019s,\n\n$A_i\\subset B_i\\,,\\ u_i=\\mathbb{P}(A_i)\\,,\\ v_i=\\mathbb{P}(B_i)$\n\nand representing\n\n$\\left|\\prod_{i=1}^s u_i-\\prod_{i=1}^s v_i\\right| = \\mathbb{P}(\\cap_{i=1}^s A_i) - \\mathbb{P}(\\cap_{i=1}^s B_i)\\,.$\n\nObviously, there is no visible consequence to this remark, but it was a good way to switch off the m\u00e9tro hassle for a while! (The book is under review and the review will hopefully be posted on the \u2018Og as soon as it is completed.)","date":"2020-05-25 04:57:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 3, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7788216471672058, \"perplexity\": 707.7689899093921}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347387219.0\/warc\/CC-MAIN-20200525032636-20200525062636-00036.warc.gz\"}"} | null | null |
\section{Introduction}
A hypergraph $H$ is called \emph{$k$-uniform} if each edge of $H$ contains exactly $k$ vertices.
The eigenvalues of the adjacency tensor of $H$ are called the eigenvalues of $H$ \cite{cooper2012spectra}.
The $k$-power hypergraph $G^{(k)}$ is the $k$-uniform hypergraph that is obtained by adding $k-2$ new vertices to each edge of a graph $G$ (where each edge of $G$ gets different new vertices).
Zhou, Sun, Wang, and Bu \cite[Thm.~16]{Zhou2014Some} showed that the complex solutions $\lambda$ of $\lambda^k=\beta^{2}$ are eigenvalues of $G^{(k)}$ if $\beta$ is an eigenvalue of $G$, and that also the spectral radius of $G^{(k)}$ can be obtained this way. Moreover, they showed that also the eigenvalues of subgraphs of $G$ give rise to eigenvalues of $G^{(k)}$, for $k \geq 4$.
In \cite[Thm.~3]{cardoso2020spectrum}, it was stated that all distinct eigenvalues of a (so-called) generalized power hypergraph $H_s^{(k)}$ can be generated from eigenvalues of subgraphs of the $r$-uniform hypergraph $H$.
When we restrict that result to the case that $H$ is a graph $G$ (i.e., $r=2$), then we obtain the following statement.
\begin{state}(incorrect)\label{chenshu1}
The complex number $\lambda$ is an eigenvalue of $G^{(k)}$ if and only if \\
\noindent(a) some induced subgraph of $G$ has an eigenvalue $\beta$ such that $\beta^2=\lambda^k$, when $k=3$;\\
\noindent(b) some subgraph of $G$ has an eigenvalue $\beta$ such that $\beta^2=\lambda^k$, when $k\geq4$.
\end{state}
However, this result is incorrect, as we shall see in Section \ref{sec:3.1}.
In this paper, we shall extend the result of Zhou et al.~\cite{Zhou2014Some}, and fix the above incorrect statement by using the spectra of signed subgraphs of $G$.
A signed graph $G_\pi$ is a pair $(G, \pi)$, where $G=(V,E)$ is a graph and $\pi:E \rightarrow \{+1,-1\}$ is the edge sign function.
We use $i \sim j$ to denote that the vertices $i$ and $j$ are adjacent in the graph $G$.
The adjacency matrix $A(G_\pi)=(A_{ij})$ of the signed graph $G_\pi$ is the symmetric $\{0,+1,-1\}$-matrix, where
\begin{align*}
A_{ij}=\left\{ \begin{array}{l}
\pi(i,j),{\kern 35pt} i \sim j, \\
0, {\kern 57pt}\mathrm{ otherwise}. \\
\end{array} \right.
\end{align*}
The eigenvalues of $A(G_{\pi})$ are called the eigenvalues of $G_{\pi}$.
An (induced) subgraph of the signed graph $G_{\pi}$ is called a signed (induced) subgraphs of $G$.
Using the eigenvalues of signed (induced) subgraphs of $G$, we can obtain all distinct eigenvalues of $G^{(k)}$ as follows.
\begin{thm}\label{dingli2}
The complex number $\lambda$ is an eigenvalue of $G^{(k)}$ if and only if \\
\noindent(a) some\textbf{ signed induced subgraph} of $G$ has an eigenvalue $\beta$ such that $\beta^2=\lambda^k$, when $k=3$;\\
\noindent(b) some \textbf{signed subgraph} of $G$ has an eigenvalue $\beta$ such that $\beta^2=\lambda^k$, when $k\geq4$.
\end{thm}
The rest of this paper is organized as follows. In Section \ref{zhunbeizhangjie}, some notation and basic definitions are introduced.
In Section \ref{zhuyaojieguo}, we will first extend the result of Zhou et al.~\cite{Zhou2014Some}, then give a counterexample to Statement \ref{chenshu1}, and finish with the proof of Theorem \ref{dingli2}.
\section{Preliminaries}\label{zhunbeizhangjie}
For a positive integer $n$, let $\left[ n \right] = \left\{ {1, \ldots ,n} \right\}$.
A $k$-order $n$-dimensional complex tensor $T= \left( {{t_{{i_1} \cdots {i_k}}}} \right) $ is a multidimensional array with $n^k$ entries over the complex number field $\mathbb{C}$, where ${i_j} \in \left[ n \right]$, for $j = 1, \ldots ,k$.
For $\mathbf{x} = {\left( {{x_1}, \ldots ,{x_n}} \right)^{\top}} \in {\mathbb{C}^n}$, we define ${\mathbf{x}^{\left[ {k - 1} \right]}} = {\left( {x_1^{k - 1}, \ldots ,x_n^{k - 1}} \right)^{\top}}$.
Moreover, $T{\mathbf{x}^{k - 1}}$ denotes a vector in $\mathbb{C}^{n}$ whose $i$-th component is
\[{\left( {T{\mathbf{x}^{k - 1}}} \right)_i} = \sum\limits_{{i_2}, \ldots ,{i_k}=1}^n {{t_{i{i_2} \cdots {i_k}}}{x_{{i_2}}} \cdots {x_{{i_k}}}} .\]
If there exist a nonzero vector $\mathbf{x} \in {\mathbb{C}^n}$ such that $T{\mathbf{x}^{k - 1}} = \lambda {\mathbf{x}^{\left[ {k - 1} \right]}}$, then $\lambda \in \mathbb{C}$ is called an \emph{eigenvalue} of $T$ and $\mathbf{x}$ is an \emph{eigenvector} of $T$ corresponding to $\lambda$.
The pair $(\lambda,\mathbf{x})$ is called an eigenpair of $T$ \cite{lim2005singular,qi2005eigenvalues}.
A hypergraph $H=(V,E)$ is called \emph{$k$-uniform} if each edge of $H$ contains exactly $k$ vertices.
Similar to the relation between graphs and matrices, there is a natural correspondence between uniform hypergraphs and (symmetric) tensors.
Indeed, for a $k$-uniform hypergraph $H$ with $n$ vertices, its \emph{adjacency tensor} ${A}_H=(a_{i_1i_2\ldots i_k})$ is a $k$-order $n$-dimensional tensor, where
\[{a_{{i_1}{i_2} \ldots {i_k}}} = \left\{ \begin{array}{l}
\frac{1}{{\left( {k - 1} \right)!}},{\kern 37pt}{ \left\{ {{i_1},{i_2},\ldots ,{i_k}} \right\} \in {E}}, \\
0, {\kern 57pt}\mathrm{ otherwise}. \\
\end{array} \right.\]
For a graph $G=(V(G),E(G))$ and $e \in E(G)$, we use $N_{e}$ to denote the set of added vertices of $G^{(k)}$ on the edge $e$.
Thus, the set $e\cup N_e$ is a hyperedge of $G^{(k)}$.
By $E_i(G^{(k)})$, we denote the set of hyperedges containing $i$
Let $x^{S}=\prod_{s \in S}x_s$ for $S\subseteq V(G^{(k)})$.
Then it follows easily that $(\lambda,\mathbf{x})$ is an eigenpair of $G^{(k)}$,
if and only if
\begin{align}\label{shizi1}
\lambda x^{k-1}_i&=\sum_{h \in E_i(G^{(k)})}{x^{h\setminus \{i\}}}\notag\\
&=\sum_{j \sim i}{x_jx^{N_{\{i,j\}}}}
\end{align}
for every $i \in V(G)$ and
\begin{align}\label{shizi2}
\lambda x^{k-1}_v = x_ix_jx^{N_{\{i,j\}}\setminus \{v\}}
\end{align}
for every $v \in N_{\{i,j\}}$ and $\{i,j\} \in E(G)$.
\section{All eigenvalues of the power hypergraph}\label{zhuyaojieguo}
In this section, we will prove our main result and give a counterexample to Statement \ref{chenshu1}.
\subsection{More eigenvalues from signed subgraphs and a counterexample to Statement \ref{chenshu1}}\label{sec:3.1}
First, we will extend the result of Zhou et al.~\cite{Zhou2014Some} by showing how to obtain more eigenvalues of the power hypergraph by using signed subgraphs.
\begin{lem}\label{xinyinli}
Let $(\beta, \mathbf{y})$ be an eigenpair of some signed (induced, if $k=3$) subgraph $\widehat{G}_{\pi}$ of $G$, with $\beta \neq 0$.
Let $\lambda \in \mathbb{C}$ be such that $\lambda^k=\beta^2$.
For each $\{i,j\}\in E({\widehat{G}})$, we fix $v_{ij}$ as one of the vertices in $N_{\{i,j\}}$.
Let $\mathbf{x}$ be the vector with entries
\begin{align*}
x_v=\left\{ \begin{array}{ll}
y^{\frac{2}{k}}_i, & \mathrm{for}~v=i \in V({\widehat{G}}),\\
\pi(i,j)^{\frac{3}{k}}(y_iy_j/\beta)^{\frac{1}{k}}, & \mathrm{for}~ v=v_{ij} \in N_{\{i,j\}} ~ \mathrm{and} ~\{i,j\} \in E({\widehat{G}}),\\
\pi(i,j)^{\frac{1}{k}}(y_iy_j/\beta)^{\frac{1}{k}}, &\mathrm{for}~ v\in N_{\{i,j\}}\setminus \{v_{ij}\} ~ \mathrm{and} ~\{i,j\} \in E({\widehat{G}}),\\
0, &\mathrm{otherwise}.
\end{array} \right.
\end{align*}
Then $(\lambda, \mathbf{x})$ is an eigenpair of $G^{(k)}$.
\end{lem}
\begin{proof}
What is essential in the definition is that for each $\{i,j\}\in E({\widehat{G}})$ and each $v \in N_{\{i,j\}}$, we have that
$$x_v^k=\frac{x_ix_j}{\lambda}x^{N_{\{i,j\}}}= \pi(i,j) \frac{y_iy_j}{\beta},$$
as is easily checked. The left equality then shows \eqref{shizi2}, whereas the right implies \eqref{shizi1}.
\end{proof}
Note that for odd $k$, one can replace the factors $\pi(i,j)^{\frac{3}{k}}$ and $\pi(i,j)^{\frac{1}{k}}$ in the definition of $\mathbf{x}$ by $\pi(i,j)$, to obtain a somewhat simpler expression.
From Lemma \ref{xinyinli}, we can easily get a counterexample for Statement \ref{chenshu1}. Indeed, let $K_4$ be the complete graph with four vertices, and consider the signed subgraph $K_4^{-}$ by signing one of its edges, say $\{1,2\}$, negative. This signed subgraph has eigenvalue $\sqrt{5}$ with eigenvector $(\sqrt{5}-1,\sqrt{5}-1,2,2)^{\top}$, as one can easily check. Thus, by Lemma \ref{xinyinli}, the power hypergraph $K^{(3)}_4$ has an eigenvalue $5^{\frac{1}{3}}$.
According to Statement \ref{chenshu1}, some induced subgraph of $K_4$ should therefore have an eigenvalue $\pm \sqrt{5}$.
But this is clearly not the case, because the induced subgraphs are complete graphs, which only have integer eigenvalues.
\subsection{Characterizing all eigenvalues of the power hypergraph}
To finish, we will show that each eigenvalue of a power hypergraph must be obtained from a signed subgraph, thus proving Theorem \ref{dingli2}.
\begin{proofb}
First of all, note that $0$ occurs as an eigenvalue of the induced subgraphs $K_1$. On the other hand, it is known that a $k$-uniform hypergraph always has an eigenvalue $0$ for $k\geq 3$ \cite{Qi2014Heigenvalue} (and indeed, any vector having only one non-zero entry is an eigenvector of the hypergraph for eigenvalue $0$). Thus, for the remainder of the proof, we only need to consider the case of $\lambda \neq 0$.
Clearly, Lemma \ref{xinyinli} shows one implication of Theorem \ref{dingli2}. Thus, what remains to show is that every nonzero eigenvalue $\lambda$ of the power hypergraph gives rise to an (appropriate) eigenvalue $\beta$ of a signed (induced) subgraph.
Let $(\lambda,\mathbf{x})$ be an eigenpair of $G^{(k)}$ with $\lambda\neq 0$.
We will indeed prove that there exists a signed (induced) subgraph of $G$ with an eigenvalue $\beta$ such that $\beta^2=\lambda^k$.
Using \eqref{shizi2}, we have that
\begin{align*}
\lambda^{k-2}(x^{N_{\{i,j\}}})^{k-1}&=\prod_{v\in {N_{\{i,j\}}}}\lambda x_v^{k-1}\\
&=\prod_{v\in {N_{\{i,j\}}}}x_ix_jx^{N_{\{i,j\}}\setminus \{v\}}\\
&=(x_ix_j)^{k-2}(x^{N_{\{i,j\}}})^{k-3},
\end{align*}
that is
\begin{align*}
(x^{N_{\{i,j\}}})^{k-3}\left(\lambda^{k-2}(x^{N_{\{i,j\}}})^{2}-(x_ix_j)^{k-2}\right)=0.
\end{align*}
Therefore, we have that $x^{N_{\{i,j\}}}= \mathrm{sgn}(i,j)(\frac{x_ix_j}{\lambda})^{\frac{k-2}{2}}$, where
\begin{align}\label{signing}
\mathrm{sgn}(i,j) \in \left\{ \begin{array}{l}
\{\pm1\},{\kern 27pt} k=3,\\
\{\pm1,0\} ,{\kern 17pt} k\geq 4. \\
\end{array} \right.
\end{align}
Next, we substitute $x^{N_{\{i,j\}}}= \mathrm{sgn}(i,j)(\frac{x_ix_j}{\lambda})^{\frac{k-2}{2}}$ into \eqref{shizi1} to obtain
\begin{align}\label{shizi3}
\lambda x^{k-1}_i =\sum_{j \sim i}{\mathrm{sgn}(i,j)}\left(\frac{x_ix_j}{\lambda}\right)^{\frac{k-2}{2}}x_j,
\end{align}
for every $i \in V(G)$.
Consider now the induced subgraph $\widehat{G}$ on the vertices $ i \in V(G)$ with $x_i \neq 0$. Note that $x_i=0$ for all $i \in V(G)$ is impossible, because of \eqref{shizi2} and $\lambda \neq 0$.
From \eqref{shizi3}, we have that
\begin{align}\label{eigenequation}
\lambda^{\frac{k}{2}}x_i^{\frac{k}{2}}=\sum_{\{i,j\}\in E(\widehat{G})}\mathrm{sgn}(i,j)x_j^{\frac{k}{2}}
\end{align}
for every $i \in V(\widehat{G})$.
Now note that \eqref{signing} defines a sign function on a subgraph of $\widehat{G}$ (or on $\widehat{G}$ itself for $k=3$), and \eqref{eigenequation} shows that this signed subgraph has eigenvalue $\lambda^{\frac{k}{2}}$.
\end{proofb}
\section*{References}
\bibliographystyle{plain}
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Q: lib clang.dylib: change installation path I have a problem using libclang:
I built libclang locally. It resides somewhere like clang-llvm/…/libclang.3.4.dylib.
Then I developed a foundation tool using that dylib. (exactly: I copied a version to my project folder and linked against this.) The foundation tool works fine. But, of course, at load time it uses the dylib in my local build folder. This is unacceptable, because the user of the tool has to install clang to use my tool.
So I copied libclang.3.4.dylib to a location inside /usr/…/libclang.3.4.dylib and changed the installation path to that location using install_name_path -id /usr/…/libclang.3.4.dylib /usr/…/libclang.3.4.dylib.
After that my tool finds the dylib there but does not work since the parser cannot find stdarg.h any more in the file, that is parsed by my tool.
/Applications/Xcode.app/Contents/Developer/Platforms/MacOSX.platform/Developer/SDKs/MacOSX10.9.sdk/System/Library/Frameworks/CoreFoundation.framework/Headers/CoreFoundation.h:12:10: fatal error: 'stdarg.h' file not found
How can I set the installation path of libclang.3.4.dylib to something public?
A: Amin, my good friend.
<sarcasm>
From what you wrote it should be OBVIOUS to EVERYONE that you have to create a release build of your tool and NOT a debug build. Xcode should have told you that in the form of CLEAR and EASY to understand error messages.
</sarcasm>
Solution: Use a release build of your tool instead of a debug build.
:)
| {
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There are now at least 125 domain names that have been recently registered relating to the explosions at the Boston Marathon today and most troubling many that look like charitable domains that can be be used to raise money for the victims.
Over 20 of the domain .com/.net domains registered today sound like they could be used for fundraising efforts for the victims so we need to watch those to make sure they are only used by licensed and regulated charities.
We noted earlier today, just minutes after the reports of the explosions hit the news, domain names related to the bombings were already registered and some parked by people looking to make money off the tragedy. While we don't know every registrants intention, we do know historically that many of the domain names registered immediately after were done to get traffic and make money parking domains or worse.
While many of the domains were registered under privacy some of the registrants were brave enough (orstupid enough) to put their name as the registrant of these domains.
Even several "relief" domain was registered, which is especially troubling unless it was registered by a recognized charity.
As the editor of TheDomains and as a domainer personally I hate to see these type of registrations, so here the registrant information for some of the domain names that don't have privacy on the registrations.
Bostonexplosion.com and bostonexplosions.com are registered to FusionWorks LLC of Saratoga, the domain name bostonmarathonbomb.com who is registered to a Mr. Myers of Queen Anne, MD, bostonmarathonbombs.com is registered to Web Service Resource Associates inc of merritt island, Florida.
The domain bostonmarathonrelief.com was registered to Thomas Shaffer of Fort Worth, Texas.
were registered by Jason Ischia of Melrose, Massachusetts.
Bostonrelief2013.com was registered to Earl Sranton, of Scranton PA.
How did this guy know??
and guess who reg the singular one??
Boston Marathon is a TM'ed term, so some of them may be lost at UDRP. Crap like this gives domainers a bad name.
You need to look for a recent registration before today.
I will get a big REWARD!!!
I suppose there's always a chance that some of these registrants bought the domains to keep them out of the hands of ne'er-do-wells, but I'm not optimistic.
Team needs a new name….
Lets hope at least a few of these are such registrations.
Now, will the owner renew the domain??
Many more are being regged like BOSTONMARATHONMASSACRE.COM.
Not to mention that many of the name listed above have taken in .net and .org as well!
Sometimes I wonder about the intelligence of the domain community. If you do about 15 seconds worth of research you can see that the "Boston Bombers" are a women's semi pro basketball team. Incredible!!
Good work, Mike. I love it when a plan comes together.
This is non-news. This will happen every time there is a tragedy and no amount of reporting will stop it. If anything it will give more people the idea to be prepared to waste their money on the next tragedy. | {
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} | 3,118 |
Source files for the documentation and website of [Rails Event Store](https://railseventstore.org).
## Serving the docs locally
Provided you have Ruby installed, you can serve the docs locally at `http://localhost:4567`
```
bundle install
npm install
bundle exec middleman server
```
## Publishing the docs
Documentation and website are deployed continuously and automatically from the `master` branch.
## About
<img src="http://arkency.com/images/arkency.png" alt="Arkency" width="20%" align="left" />
Rails Event Store is funded and maintained by Arkency. Check out our other [open-source projects](https://github.com/arkency).
You can also [hire us](http://arkency.com) or [read our blog](http://blog.arkency.com).
| {
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} | 2,963 |
The Most Important Challenge For Colleges Isn't Price—It's Attention
By Kara Miller
As universities look for ways to teach more students at a reasonable price, they will move online -- into the clicky, crowded, distractable world of the Web. Get ready for Infotainment U.
yusunkwon/Flickr
This is one of my favorite anecdotes: Last year, the University of Phoenix enlisted renowned Harvard Business School Professor Clayton Christensen to record a lecture. The university reserved a harbor-view room for Christensen and populated it with young people, so that the camera operators could record their reactions.
Before he began to speak, Christensen noticed that the audience appeared unusually engaged and attractive.
"What school do you guys go to?" he asked.
"We're not students," a young man told him. "We're models."
When Christensen told me this story, I laughed. (Hear the whole interview here.) But the University of Phoenix is serious -- and smart. Putting a Harvard professor in front of a lecture hall filled with models is an acknowledgment that, in a Web-recorded lecture, appearance counts -- even the few seconds of cutaways to reactions from gorgeous, engaged "students."
Education now competes in a world shaped by the Kardashians, "X Factor," and "Call of Duty." And in response to this assault on students' attention, Phoenix has embraced the power of editing, graphics, and cut-aways. In our media-filled Internet landscape, Phoenix understands that it's not enough to give a good lecture. You have to put on a show that wins the war of attention.
For as long as secret notes and daydreamers have existed, colleges have had to vie for students' focus. But in the next few years, they'll have to raise their game. As they try to deliver more education at the same price, schools will move into the crowded and distractable world of the Web.
WHY ARE UNIVERSITIES GOING ONLINE, ANYWAY?
Last month, Harvard announced that it will begin offering free online courses this fall in collaboration with MIT -- a move that will make this year's high school graduates the first to truly inhabit the educational landscape of the future. Indeed, Harvard and MIT's move capped a year of increasingly troublesome news for more traditional forms of higher education.
Billionaire Peter Thiel famously told 60 Minutes that college just doesn't seem worth it anymore. "We have a society where successful people are encouraged to go to college," he said. "But it's a mistake to think that that's what makes people successful."
Of course, the central issue in the "is college worth it?" debate is a number: $1,000,000,000,000.
That's how much Americans owe in student loans. More than four times what we owed in 2000. And more -- to Suze Orman's chagrin, I'm sure -- than we owe in credit card debt (a mere $800 billion).
So, how did we get here? In part, because of another number: $40,000.
That's the approximate cost of a year at private college. And hundreds of schools -- including Boston University ($56,184), Amherst College ($56,260), Emory ($54, 980), and Stanford ($54,508) -- will be sending out far heftier bills this fall.
Though many students will get some form of financial assistance, even a healthy dose of aid could leave middle-class and upper middle-class families struggling to pay. Much like health care, higher education has been unable to keep costs under control, routinely hiking tuition far above the rate of inflation. Since 1980, according to the Census, personal income has quadrupled. But the cost of private, four-year colleges has increased six-fold -- an increase that's clearly untenable.
At state institutions -- like the University of Massachusetts, where I teach -- the pop in prices particularly hurts low-income families, who view public schools as a way to give their children an inexpensive, high-quality education. Now that UMass charges over $20,000 for those who live on-campus, lots of students take out loan after loan to scrape by, racking up tens of thousands of dollars in debt.
Though professors at four-year colleges don't want to think about it, an innovative solution is coming. The only real question is what College 2.0 will look like.
Thiel -- a co-founder of PayPal and an early funder of Facebook -- bestows $100,000 on each of 20 high school graduates each year, provided that they skip college and, instead, launch cutting-edge projects. It's a costly, time-intensive venture, which seems destined to stay small-scale.
THE UNIVERSITY OF INFOTAINMENT
But it brings up a useful question: What's college for?
If it's to learn -- rather than drink or meet your future bridesmaids -- the next frontier may be affordable, at-home infotainment, rather than ivy-covered walls.
Already, infotainment has made some notable forays into top-tier institutions. This year, Yale, Harvard, and Bard offered freshmen a class called "Great Big Ideas," which brings together lectures from luminaries like Steven Pinker and Larry Summers from Harvard, Michio Kaku from CUNY, and Joel Cohen from Columbia. "Great Big Ideas" is -- like Christensen's recording for the University of Phoenix -- edited and enhanced with whatever special effects the editors find desirable. When Michio Kaku talks about computers, a video of a monkey typing on a computer appears briefly; when he talks about lasers, a laser-projecting robot walks towards the audience.
I suspect Harvard and MIT already know that online education -- and infotainment, in particular -- is where we're headed. Which is why they will not only offer free courses online this fall, they'll also gather data about students -- an explicit goal of the project. Quite likely, that data will show that students like being entertained. And that -- with a few graphics and some editing -- we may be able to find a high-gloss, low-cost way of delivering education.
The seminal question is whether anything will be lost when professors start to seem as polished as Diane Sawyer and lecture halls become populated with Discovery-Channel-like graphics.
Get ready to find out. | {
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Home » Local » Gate of Heaven celebrates double anniversary
Gate of Heaven celebrates double anniversary
On: 11/19/2004, By , In: Local
On the occasion of the 140th anniversary of the founding of Gate of Heaven Parish in South Boston and the 125th anniversary of the parish school, Archbishop Seán P. O'Malley came to the church on the evening of Nov. 6 to offer a Mass of Thanksgiving. Prior to the entrance procession of the Mass, a group of children in their school uniforms lined up in front of the sanctuary to sing "Standing On The Shoulders." Music for the Mass was led by a guest choir from Holy Family Parish in Amesbury.
Welcoming all to the anniversary Mass and Archbishop O'Malley was Father Robert Casey, current pastor of Gate of Heaven Parish. Among the many priests in the sanctuary was Father Gilbert Phinn, pastor from 1983 to 1994 and to the delight of many, Father A. Paul White who now resides at nearby Marian Manor. For his homily, Archbishop O'Malley climbed the steps of the pulpit and expressed his best wishes to all on the occasion of the anniversary of the parish and school. Following the Mass, parishioners boarded buses in front of the church for a ride to the new Boston Convention and Exhibition Center for an evening of dinner and dancing. Awards were presented to Sister Joan Duffy and Sister Prudence McCarthy, representing the Sisters of St. Joseph, and Father Paul White for his many years of CYO work in South Boston and with the present day students of Gate of Heaven Grammar School. Boston College professor Thomas O'Connor, a "Gatey" alumnus, delivered a historical review of the parish.
Planning for the anniversary celebration got underway over the summer after Gate of Heaven Parish received word last May that the parish would not be suppressed as part of the overall reconfiguration of parishes throughout the archdiocese. The theme of the anniversary was a "Faithful Past and a Faith-Filled Future." Many volunteers met over the summer to plan the celebration chaired by honorary chairman, state Sen. Jack Hart.
Gate of Heaven Parish began as an offshoot of Sts. Peter and Paul Parish during the Civil War years, under the pastorate of Father Patrick F. Lyndon. The cornerstone of the first church was laid on May 1, 1862. The church was dedicated on Mach 19, 1863. The first pastor, Father James F. Sullivan took charge on Jan. 11, 1865. In 1879, Father Michael F. Higgins, pastor from 1873 to 1886, erected a convent at the corner of I and East Fifth Streets for a group of Sisters of St. Joseph. That same year, the Sisters opened St. Agnes School in September with an enrollment of 375 girls, establishing a school that exists to this day and has provided a fine education to generations of Catholic youth in South Boston. The present school building was built in 1922 during the pastorate of Father George A. Lyons (1916-1932). The present Gate of Heaven Church was built during the pastorate of Father Robert J. Johnson (1890-1916). The lower church opened in June 1900 and was greatly needed to meet the needs of a growing Catholic population. The upper church was dedicated on May 12, 1912 by Cardinal O'Connell. Gate of Heaven Church is a magnificent building of French Gothic architecture.
After the departure of Father Alexander Keenan, pastor from 1994 to 2002, Father Robert Casey was appointed administrator of Gate of Heaven, while remaining pastor of St. Brigid Parish. The two parishes have begun collaborating in sharing resources and programs. As announced at the dinner dance, Gate of Heaven Parish has begun a $2.5 million capital campaign to refurbish the interior of the church. | {
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Q: Ember: How to use ember i18n translation in view In our project we have built some components that are actually default views, is it possible to translate view properties, for example we pass title for each page.
E.g
This is my view that has a dynamic title to be shown for each page {{view.titleToShow}}
..templates/view/simple-navbar.hbs
<div class="navbar-header pull-left">
<div class="navbar-brand">
{{view.titleToShow}}
</div>
</div>
..templates/cars/cars.hbs
{{view 'simple-navbar'
titleToShow='Projects'
...
}}
..translations/eng.js
import Ember from 'ember';
export default Ember.Object.create({
eng: {
General: {
SequenceAnalyze: "Sequence Analysis",
UNITESH: "Unite Species Hypotheses",
},
Specimen : {
},
})
And an example of regular usage in template {{i18n-t 'General.Save'}}
<button {{action 'createStudy'}} class="btn btn-default"><span class="glyphicon glyphicon-ok"></span> {{i18n-t 'General.Save'}}</button>
A: Should be
titleToShow: Ember.I18n.t('General.Save')
| {
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} | 6,832 |
Scunthorpe United FC (celým názvem: Scunthorpe United Football Club) je anglický fotbalový klub, který sídlí ve městě Scunthorpe v nemetropolitním hrabství Lincolnshire. Založen byl v roce 1899. Od sezóny 2014/15 hraje ve třetí nejvyšší anglické soutěži EFL League One. Klubové barvy jsou rudá a světle modrá.
Své domácí zápasy odehrává na stadionu Glanford Park s kapacitou 9 088 diváků.
Historické názvy
Zdroj:
1899 – Scunthorpe United FC (Scunthorpe United Football Club)
1910 – fúze s Lindsey United FC ⇒ Scunthorpe & Lindsey United FC (Scunthorpe & Lindsey United Football Club)
1958 – Scunthorpe United FC (Scunthorpe United Football Club)
Úspěchy v domácích pohárech
Zdroj:
FA Cup
5. kolo: 1957/58, 1969/70
EFL Cup
4. kolo: 2009/10
EFL Trophy
Finále: 2008/09
Umístění v jednotlivých sezonách
Stručný přehled
Zdroj:
1912–1950: Midland Football League
1950–1958: Football League Third Division North
1958–1964: Football League Second Division
1964–1968: Football League Third Division
1968–1972: Football League Fourth Division
1972–1973: Football League Third Division
1973–1983: Football League Fourth Division
1983–1984: Football League Third Division
1984–1992: Football League Fourth Division
1992–1999: Football League Third Division
1999–2000: Football League Second Division
2000–2004: Football League Third Division
2004–2005: Football League Two
2005–2007: Football League One
2007–2008: Football League Championship
2008–2009: Football League One
2009–2011: Football League Championship
2011–2013: Football League One
2013–2014: Football League Two
2014–2016: Football League One
2016– : English Football League One
Jednotlivé ročníky
Zdroj:
Legenda: Z - zápasy, V - výhry, R - remízy, P - porážky, VG - vstřelené góly, OG - obdržené góly, +/- - rozdíl skóre, B - body, červené podbarvení - sestup, zelené podbarvení - postup, fialové podbarvení - reorganizace, změna skupiny či soutěže
Odkazy
Reference
Externí odkazy
Anglické fotbalové kluby
Fotbalové kluby založené v roce 1899 | {
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Respect is one of the firm's fundamental values: respect for the client, respect for respondents and respect for staff members. At BIP, clients are welcomed with empathy and treated as true partners. With respondents, interviewers create an atmosphere of trust and of openness, which constitute the key to the success of any good interview.
BIP is renowned for its methodical rigor and its operational efficiency. The firm manages its computerized center, which has 44 work stations. Operations are supervised on an ongoing basis and each step is verified by Quality Control. By adopting this rigorous approach, BIP can ensure that projects proceed smoothly and that timelines are observed.
BIP is renowned for its ability to deliver high quality research data. The BIP team indeed built its credibility on successfully fulfilling particularly complex and demanding mandates. The know-how gained from working on such major projects is carried over to all other projects with which the firm is involved, thus assuring clients reliable results. | {
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She made the decision to protest the parade two weeks ago, when the New England Journal of Medicine published a Harvard study estimating that the U.S. government had dramatically undercounted the number of people who died on the island as a result of Hurricane Maria. The study estimated that 4,645 Puerto Ricans lost their lives because of the hurricane – 4,581 more than the official government estimate of 64 deaths. The number is only slightly less than the total deaths of 9/11 and Hurricane Katrina combined.
The revised tally was published the same week ABC canceled Roseanne after its namesake star tweeted that if "muslim brotherhood & planet of the apes had a baby" the result would be former Obama advisor Valerie Jarrett. The major television networks – NBC, MSNBC, CNN and Fox News – devoted eight-and-a-half-hours to the fallout from Barr's tweet and just 32 minutes to the Harvard study, according to a count by Media Matters. The reaction, galling as Clemente found it, didn't exactly shock her.
If there's any good to come out of the tragedy, though, Clemente, a long-time proponent for Puerto Rican independence, says it's the fact that the apparent indifference of both the media and the government has stirred more interest in independence than she's seen in the past. | {
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Sulphur (franska: Soufre) är en stad i Calcasieu Parish i den amerikanska delstaten Louisiana med en yta av 26,3 km² och en folkmängd, som uppgår till 20 272 invånare (2016).
Referenser
Externa länkar
Sulphur på Louisiana Travel
Orter i Louisiana
Calcasieu Parish | {
"redpajama_set_name": "RedPajamaWikipedia"
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ACCEPTED
#### According to
Index Fungorum
#### Published in
null
#### Original name
Selenophoma eremuri Koshk.
### Remarks
null | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,406 |
{"url":"https:\/\/www.eng-tips.com\/viewthread.cfm?qid=433536","text":"\u00d7\nINTELLIGENT WORK FORUMS\nFOR ENGINEERING PROFESSIONALS\n\nAre you an\nEngineering professional?\nJoin Eng-Tips Forums!\n\u2022 Talk With Other Members\n\u2022 Be Notified Of Responses\n\u2022 Keyword Search\nFavorite Forums\n\u2022 Automated Signatures\n\u2022 Best Of All, It's Free!\n\n*Eng-Tips's functionality depends on members receiving e-mail. By joining you are opting in to receive e-mail.\n\n#### Posting Guidelines\n\nPromoting, selling, recruiting, coursework and thesis posting is forbidden.\n\n# Train Derailment3\n\n## Train Derailment\n\n(OP)\nIn Washington state, derailment killed 3 people and some still seriously injured. Part of the problem it seems is the design of the rail. From the BBC.\n\n\"A US passenger train that derailed, killing three people, was travelling at 80mph (130km\/h) on a curve with a speed limit of 30mph, data from the train's rear engine indicates.\"\n\nThe rail was supposed to be a high speed rail and it seems really silly to have a 30mph curve on it.\n\nDik\n\n### RE: Train Derailment\n\nLatest word is that it was travelling at 128 km\/h in a zone where the max is 48 km\/h.\n\n### RE: Train Derailment\n\n(OP)\nYup... but it seems silly that you would design a high speed rail with corners that were only suitable for 30mph. I wouldn't have expected that, and it was the 'maiden' run. The rail line was billed as a 'high speed line'.\n\nDik\n\n### RE: Train Derailment\n\n30 mph was high speed in 1829.\n\n----------------------------------------\n\nThe Help for this program was created in Windows Help format, which depends on a feature that isn't included in this version of Windows.\n\n### RE: Train Derailment\n\nDik,\n\nA few years, I sat through a presentation at a conference about a \"high speed rail\" line in the planning phase between Des Moines, IA and Chicago (with future expansion to Omaha, NE). It was to go at 80 mph where it could, but wherever it went through a community (town, village, city), it still had to slow down to 30mph (or slower) since the plan was to use the existing freight track. So, calling it \"high speed\" is more of a marketing\/branding angle, and not completely representative of operation. Without knowing ANYTHING about this event, I would guess the high speed train is utilizing existing track, and the operator (engineer?) of the train missed that they had to slow down.\n\nAt any rate, \"high speed rail line\" doesn't necessarily mean 80mph all the time, at least, not if it's using existing track.\n\n### RE: Train Derailment\n\nIt's not a high-speed rail. It's a shorter path than the other route. It's only the train that is designed for moderate speeds. \"The train was running on track previously used for occasional freight and military transport,\" http:\/\/www.cnn.com\/2017\/12\/19\/us\/amtrak-derailment..., so it's not a purpose built route for unrestricted high-speed rail.\n(ETA - I was responding to dik)\n\n### RE: Train Derailment\n\nFor some reason yet unknown, the automated speed control safety ststem was not activated.\n\nAlso, the run may have been inaugural, but previous runs and testing had been made.\nwith no issues as far as has been reported.\n\nThe route has long been a controversial topic, being in the courts, too, over the years.\n\nMike McCann, PE, SE (WA)\n\n### RE: Train Derailment\n\nAnd there are issues with actions taken that prevented the use of technology that could have mitigated some of the problems:\n\nhttps:\/\/www.huffingtonpost.com\/entry\/washington-tr...\n\nJohn R. Baker, P.E. (ret)\nEX-Product 'Evangelist'\nIrvine, CA\nSiemens PLM:\nUG\/NX Museum:\n\nThe secret of life is not finding someone to live with\nIt's finding someone you can't live without\n\n### RE: Train Derailment\n\n(OP)\nwinelandv:\n\n80 mph is not high speed for rails... Europe has several lines that are several times faster than that. If marketing is the reason for calling it 'high speed' then marketing is partly to blame. This was the initial run, and, the engineer may not have been sufficiently trained.\n\nDik\n\n### RE: Train Derailment\n\nIt was the initial run with passengers. It was not the first time an engineer ran the train down this route.\n\nAs for the design that included the need to drop to 30 MPH, the alignment had to cross from running parallel to one side of the highway to running on the other side. Unless it crosses at a shallow skew elevated above the highway for extended length of track, it has to zig and zag to get across.\n\nI like to blame marketing for a lot of things. But this is a stretch.\n\n### RE: Train Derailment\n\nTake a look at Google Earth for Dupont, WA and this area and you will see the railroad, an old alignment, is winding through some hilly area with several curves. No way for an 80 mph speed un0less the rails are banked to allow for it. That's very unlikely since the rail line also is used for freight. Reminds me of the passenger trains in Sweden and their twisty alignments. Instead of banking the curves, can you imagine they bank the position of the passenger cars on their trucks. Riding in the cars you get tilted back and forth as if the rails are banked. Aside from wearing the wheel flanges and rail edges it seems to work, but unlikely would work for an 80 mph speed on a 35 mph curve.\n\n### RE: Train Derailment\n\n@dik, many railways (also in Europe) are very old. Some sections gets upgraded and then some dont. Its a matter of money.\n\n### RE: Train Derailment\n\n@JohnRBaker, but why did they delay the implementation of a safety system? Because the industry asked for it to be delayed and threatened to stop service!\n\n### RE: Train Derailment\n\nPerhaps, but then the auto industry objected to seat belts, air bags and emission controls, but they're all standard equipment now on every vehicle sold in America.\n\nJohn R. Baker, P.E. (ret)\nEX-Product 'Evangelist'\nIrvine, CA\nSiemens PLM:\nUG\/NX Museum:\n\nThe secret of life is not finding someone to live with\nIt's finding someone you can't live without\n\n### RE: Train Derailment\n\nTrain was going 81.1MPH at a 30MPH turn.\n\nTwo 30MPH speed limit signs, one 2 miles out, one close to the turn.\n\nTrain has been running that track for 6 weeks.\n\nI believe an engineer is required to pass that track 8 times before passengers are allowed to travel with that engineer on that stretch of track.\n\nThose engines have inward facing cameras that can be called up for realtime high res color video viewable at dispatch.\n\nAll new engines have cellphone detectors that alert dispatch of ANY cellphones that are ON in the cab and result in immediate response from dispatch.\n\nThe cars involved in the WA wreck are made in Spain and are very lightly built with two cars sharing three sets of axles. (That's why the wreck seemed to have lots of paired cars). They are built like motor coaches (buses) unlike normal heavy duty rail cars. They are not allowed in most states due to their not meeting federal guidelines. They are allowed in WA and 5 other states under special federal wavers.\n\nThey spent $180.7M to put that shortcut into service to save, (I believe), 15 minutes. Worth it? The lead locomotive ended up more than 120 feet from the tracks. All engineers are handed between one and about 8 sheets of paper showing all speed restrictions on the pending trip. There are often more than a dozen special speed restrictions on typical trips. Speed restrictions can be caused by things happening near tracks and weather. The engines are extremely new and made by Siemens. They have had so many problems it's taken a year to actually start putting them into service. In some units the throttles work in reverse to the historic normal. (Can you imagine an airliner where the throttles work in reverse?) I have not been able to find out if these particular engine are the wacky reverse throttle types. I believe there are seven complete trains for that particular run. Two are owned by the state of WA, two by Amtrak, and the last three I don't recall. The one that just went on the ground was owned by WA. PTC Positive train control implementation has so far directly cost the rail companies approximately one billion dollars. It's been very hard to implement. (Note the changing pages of speed limits noted above.) Keith Cress kcress - http:\/\/www.flaminsystems.com ### RE: Train Derailment \"The engines are extremely new and made by Siemens\" The P42 that was pushing was most definitely not built by Siemens, it's a GE product. I'm assuming you're referring to the cab car. ### RE: Train Derailment Interesting. https:\/\/www.talgo.com\/en\/projects\/usa\/serie_6_usa\/ Looks like the series 6 tilting carriages used on this line, but the series 8 for the rest of the US system. it will be interesting to see how many times the driver ACTUALLY ran the line in that direction. One warning sign 2 miles out doesn't sound enough to me, but I'm not a train driver or designer. Maybe a little more graduation in speed limits required. Remember - More details = better answers Also: If you get a response it's polite to respond to it. ### RE: Train Derailment I was most impressed with the way the embankment seemed to have kept the engine upright after it left the track. The side of the engine looked severely smashed and there's a new notch in the embankment at the curve that seems to match. I can see the hesitation about PTC. One looks at the fleet cost of implementation along with the track indicators for short-range communications and then factor in that it's a system that overrides the engineer and compare the current rates of failure of engineers while also considering that there will be new failure modes. I'm not certain that PTC is an overall best solution to the problem of engineer's operating problems. One solution that I would have pushed for is a GPS\/Cellular comms based location system that alerted the engineer, much like terrain and collision avoidance systems in aircraft typically do. This could have been deployed cheaply and without the headache of adding additional failure modes to train control. However, doing this as an interim solution would certainly have ended the push to full autonomous taking of control. The big advantage to an alerting system is that multiple systems could be deployed on a single train, allowing the conductors to intervene if required. The disadvantage is that there is no good autonomous way to detect which of any parallel tracks a train is on, so mis-tracked trains could still be a problem. ### RE: Train Derailment I can very nearly imagine the recommendations from the report now - some rehash of: \"Having regard, therefore, to all the circumstances of this serious accident and to the criticism, to which my attention was particularly drawn, that both drivers concerned, though running approximately on time, may have been exceeding a speed reasonably justified by visibility conditions, I recommend that the Company should take early steps to reach decisions, with a view to applying Warning Control to high speed services on their Trunk Routes.\" (Extract from Report by Lt.-Colonel A. H. L. Mount, C.B., C.B.E. on the Collision between two Passenger Trains which occurred on 10th December, 1937, at CASTLECARY on the London and North Eastern Railway, HMSO, 1938). So why does this issue keep coming up without ever really getting implemented very enthusiastically or thoroughly? On the one hand, the systems are expensive and the proportion of rail accidents they might have any influence on is genuinely low. On the other hand, the accidents they do prevent tend to be the catastrophic ones that grab world headlines. A. ### RE: Train Derailment \"They spent$180.7M to put that shortcut into service to save, (I believe), 15 minutes. Worth it?\"\n\nThe re-route also put the train across a lot more level crossings, increasing pedestrian\/auto risks. And, for passengers, takes you down the \"scenic\" I-5 corridor, rather than along the shoreline of Puget Sound. Pretty much a crappy decision all around. I think there was also a feeling that the high clay banks along the shoreline put the trains at risk due to mudslides (as happens fairly regularly a bit farther north between Seattle and Everett). Lots of politics leading up to the change.\n\n### RE: Train Derailment\n\nIs 30+ years of successful operation long enough to declare a system mature and dependable?\nSkyTrain\nMass transportation system\nSkyTrain is the metropolitan rail system of Greater Vancouver, British Columbia, Canada. SkyTrain has 79.6 km of track and uses fully automated trains on grade-separated tracks running on underground ... Wikipedia\nAverage speed: 45 km\/h\nBegan operation: December 11, 1985\nDaily ridership: 454,600 (December 2016)\nAnnual ridership: 137.4 million (2016)\nTop speed: 80 km\/h (50 mph) (Expo and Millennium Lines); 80 km\/h (50 mph) (Canada Line)\nDid you know: SkyTrain is the ninth-busiest North American rapid transit system by annual ridership (137,380,000)\n\nMuch of the system uses Linear Induction Motors with regeneration.\nAs the trains pull into the stations, it is easy to hear when the LIM regeneration cuts off and the mechanical brakes apply.\nThe distance to stop from the point that regeneration ends: About 3 to 5 feet.\nFull automation is available for anyone who wants to use it.\n\nBill\n--------------------\n\"Why not the best?\"\nJimmy Carter\n\n### RE: Train Derailment\n\nSo a train running on it's dedicated tracks without possibility of traffic conflict is safer? You don't say. I bet the Disney World Monorail also has a pretty good safety record as well.\n\nIt looks like Sky Train also pulls about $0.5 Billion from outside taxes to support it on top of fares and advertising sales. This means they cost Canada about$5 per rider. That might be a good deal, but it's not a universal solution. And it's not entirely without fatalities.\n\nhttps:\/\/www.straight.com\/life\/458271\/skytrain-deat...\n\nAnd yes, many, but not all Skytrain deaths are probably suicide, but they still look at the money before deciding on measures to mitigate it.\n\n### RE: Train Derailment\n\nFROM OUT OF LEFT ,ER I MEAN RIGHT FIELD: It appears that some on the far-Right would like you to believe that it was members of a left-leaning anti-fascism group who sabotaged the AMTRAK train in Washington:\n\nJohn R. Baker, P.E. (ret)\nEX-Product 'Evangelist'\nIrvine, CA\nSiemens PLM:\nUG\/NX Museum:\n\nThe secret of life is not finding someone to live with\nIt's finding someone you can't live without\n\n### RE: Train Derailment\n\nJohn, remember that the very act of linking your posts to lunatic news actually encourages it - even if your goal is to laugh at it.\nThe number click-throughs that Newsweek will get from the link on a high traffic website like Eng-Tips will only serve to validate that muck-raking. Your ridicule is not a factor in their web stats or their advertising revenue.\n\nSTF\n\n### RE: Train Derailment\n\n\"They spent $180.7M to put that shortcut into service to save, (I believe), 15 minutes. Worth it?\" A bit more context: On the old path Amtrak shared the rails with freight trains, so there were often highly unpredictable delays. Being on time 7 out of 8 times and two hours late the 8th averages out to 15 minutes of delay. When I was regularly commuting along this route, the train's reputation of highly variable arrival times discouraged me from going by train. ### RE: Train Derailment I'm a big believer in the concept that sunlight is an effective disinfectant. Besides, the article from 'Newsweek' is critical of and goes to great length to undermine the premise that it was some sort of plot by a Left=leaning faction of society, as was being promoted by these far-Right wackos. John R. Baker, P.E. (ret) EX-Product 'Evangelist' Irvine, CA Siemens PLM: UG\/NX Museum: The secret of life is not finding someone to live with It's finding someone you can't live without ### RE: Train Derailment #### Quote (SparWeb) John, remember that the very act of linking your posts to lunatic news actually encourages it - even if your goal is to laugh at it. The number click-throughs that Newsweek will get from the link on a high traffic website like Eng-Tips will only serve to validate that muck-raking. Your ridicule is not a factor in their web stats or their advertising revenue. What is it I'm missing about Newsweek's reporting that was supposed to make it so awful? It's not as if John linked to the actual conspiracy drivel. ### RE: Train Derailment quote: On the old path Amtrak shared the rails with freight trains, so there were often highly unpredictable delays. ditto... I recently took the Via-Rail Canadian from Toronto across Canada to Vancouver. From what I saw and experienced, the amount of east bound rail freight originating from Vancouver and I would imagine the northwestern ports in general, being the closest to the Asian Pacific, is staggering, with significant delays to passenger rail. Passenger rail traffic is last in priority on shared freight rails, due to track ownership, ecomonics, and that the freight trains are far too long in length to fit on just about all sidings. After getting into Vancouver proper, it was a 2 1\/2 hour delay to get past the freight assembly yards (and east bound freight) to the passenger station just a few miles ahead. ### RE: Train Derailment DanEE, \"...the amount of east bound rail freight....is staggering...\" On average, the number of east-bound and west-bound freight rail cars should be very nearly identical. Otherwise the rail cars would tend to pile up at one end. You would have passed more east-bound traffic since you were heading west. !! ### RE: Train Derailment I saw a report many years ago, before Vietnam, that The Port of Vancouver handled over half of the tonnage on the entire West Coast, from Mexico to Alaska. The percentage may have slipped somewhat, but it still carries a lot of tonnage and a lot of it is bound for eastern markets. And as far as west bound traffic, there is a lot of grain heading west. Bill -------------------- \"Why not the best?\" Jimmy Carter ### RE: Train Derailment Pretty good <> Perfect. Didn't derail. ### RE: Train Derailment (OP) ### RE: Train Derailment First point, the locomotive engineer failed to slow to the 40 mph speed limit at Mounts Road. The train probably would not have derailed had he done so. The second error was the failure to slow to 30 mph speed limit for the curve approximately 1\/4 mile after Mounts Road. The speed limits are shown in the brochure and with signage on the railroad route. http:\/\/www.wsdot.wa.gov\/NR\/rdonlyres\/20790BB4-7A4E... \"Upgrades tracks and improves existing connection to BNSF Railway main line so trains can travel up to 40 mph from Nisqually to Mounts Road and 79 mph from Mounts Road to Bridgeport Way.\" The locomotive engineer announced an over-speed condition approximately 6 seconds (or 650 feet) before the crash. The locomotive engineer should engage legal counsel as he will probably face manslaughter charges. ### RE: Train Derailment In the many dozens of mainstream news reports on this, and the many interviews on radio programs about the overspeed, it is noteworthy to see that the local \"politicians\" and \"officials\" are really straining to NOT \"blame the operator\" - although I see nothing but \"operator error\" in running too fast. ALL these \"officials\" are so very willing to mention \"not installing the speed regulators\" .... Odd attitude. ### RE: Train Derailment It used to be a challenge commuting by car into Vancouver on the Lougheed Highway. There were a couple of level crossings in Burnaby. Inbound freight trains would have to wait for clearance to enter the freight yards. There was a limit to the amount of time a train could sit stationery blocking a level crossing. If the time was exceeded, the train crew was supposed to break the train and clear the crossing. Rather than break the train, the train would proceed at about 2 MPH and block the crossings for a very long time. Now the track through Burnaby alongside the Highway is built on very soft ground. You could see the rails subside as each loaded truck passed by. Came the day that the grain cars of a slow moving freight train started to rock and a harmonic frequency must have been found. The rocking progressed until a large number of grain cars were laying on the ground beside the tracks. After that, the scheduling was revised so that trains could waste time further away from the city without inconveniencing commuters. Too slow may also be a problem. Bill -------------------- \"Why not the best?\" Jimmy Carter ### RE: Train Derailment #### Quote: ...to see that the local \"politicians\" and \"officials\" are really straining to NOT \"blame the operator\" ... I see nothing but \"operator error\" in running too fast... I'm watching this trend too. Remind yourself who was at fault during the many Toyota \"stuck accelerator\" accidents a decade ago. The company paid the fine. All the people who explained why the car probably wasn't at fault were ignored. Anybody demonstrating how the subject vehicle engines couldn't overpower the brakes were called company shils, and ignored. I'm wondering if this is going to be a similar case. My local media has already imagined a link between this derailment and a local city transit train accident in Calgary. Nobody was hurt, but a locomotive had to be scrapped. It left the tracks, clearly due to operator error, any yet still the city transit system had to install magnetic brakes to stop trains overrunning the end of the tracks. STF ### RE: Train Derailment Does this mean they can stop putting up guard rails on mountain roads and high bridges? It would be nice to have an unobstructed view. The fines were mostly for not reporting incidents to the feds. It's tougher to manage safety if the maker is hiding safety related information. The crash and burn Toyota had another driver report the same problem days before to the dealership. He also was unable to stop the car with the brakes while in gear. I have no idea how a cop would not know to use neutral, but that should not be required by having a car configured to create the problem. http:\/\/www.sandiegouniontribune.com\/sdut-report-lo... It's also the case that drivers will tend not to initially put full force effort into the brakes at the outset, so the brakes rapidly heat, glaze, and fade, unlike the behavior of those 'proving' some contention about how the brakes, under different usage, could work. Flight-sim pilots were often able to successfully pilot a plane under the circumstances that brought a DC-10 down in Chicago, but only after they were fully informed as to the exact defect and given a chance to plan a response, time and information the original crew did not have. ### RE: Train Derailment I have wondered why the Toyota driver could not just turn the ignition off. In every car that I have ever driven it was possible to turn off the ignition at any time even if it was not possible to turn the key further to lock the steering or to remove the key. Were we seeing a Darwin Award competition? Talking about Flight-sim attempts. I understand that Boeing set up a simulation of the Gimli Glider in their flight simulator and the first three attempts by Boeing test pilots ended in simulated crashes. Bill -------------------- \"Why not the best?\" Jimmy Carter ### RE: Train Derailment Gillian Glider - last I\u2019d heard it had never been successful repeated in a simulator. ### RE: Train Derailment The Toyota had push-button start. When the car is in motion it requires holding the button for several seconds to shut the engine off, obviously to prevent the result of a kid playing with the buttons while driving. There is no key in the dash. It was a loaner replacement, so the driver had never used the car before. (edit to clarify) ### RE: Train Derailment Thanks for the explanation Dave. Bill -------------------- \"Why not the best?\" Jimmy Carter ### RE: Train Derailment IMO the way cars have adopted keyless engine start should not have been allowed. Keyless - No problem. But it should have been done using a rotary selector switch with the same positions as a normal key switch and located in the same place in the vehicle. Or, done the way motorcycles and race cars do it - a pushbutton for starting, but also in conjunction with the big red button for shutting it down in a hurry. But, we digress ... ### RE: Train Derailment I think the biggest problem for Toyota, aside from a heart-wrenching 911 call recording playing over and over and the photographed aftermath of an incinerated and pulverized family with no instant explanation, was Toyota had been 'quietly' dealing with a couple of driveability issues related to the throttle. One problem was the formation of tin whiskers in one pedal sensor which caused the throttle response to be non-linear - from the idle position to part throttle the ECU didn't see a resistance change and when the whisker lost contact it looked to the ECU like a sudden throttle input; not WOT, just dead-band and then a bit of voom, which startled drivers. Depending on how the whisker was positioned the symptoms would be irregular and testing with a typical ohm-meter could be enough to damage the whisker such that the pedal tests OK only for the symptoms to return. There was also a pedal design issue. In an 'old-fashioned' car there is some sticktion due to the throttle linkage and cable so that a driver's foot could vary pressure slightly without moving the pedal. In drive-by-wire, there is just a pedal return spring and slight variations in pressure result in variations in throttle which results in slight surge\/sag of power. So they added a friction source to produce sticktion and, in some cases, this meant that the return spring didn't have enough force to ensure the pedal returned all the way to idle when released. When the accident happened it resulted in every leaf being turned over to explain why the family was incinerated so it came to light that some of this had not been divulged. The 'trapped' pedal concept was advanced by Toyota both because that's what really caused the crash, the wrong floor mat was identified early on as a most-probable cause and, I think, to provide a simple to implement fix. It was also a dodge as there would have been hundreds of videos of pedals trapped by floor mats on YouTube. As far as I can tell, there was only one video, where a guy wadded a floor mat and shoved it between the foot well wall and the pedal. The other source of trouble for Toyota was the lack of an obvious fault in the ECU that would explain the non-existent ECU related problems, leading to investigations into the software development practices at Toyota. These investigations lacked any demonstrations of realistic failure modes. I suspect it's true the ECU software wasn't made with significant fault tolerance in mind, but no one demonstrated any actual faults to be tolerant of. This led to the grand-standing of an expert and further unsubstantiated guesses increasing the speculation that there was something to hide. And let's not forget the driver who falsely claimed an out-of-control condition that seemed to be an extortion attempt that also implicated every Toyota, even those with entirely different ECUs. In contrast is the VW ECU\/Diesel lie, where independent software and hardware investigators were able to identify the place in the software and verify by bench test and testing in the vehicle that they had been programmed to cheat the federal emissions testing. Anyone could duplicate the observations\/reproduce the results - they could look at the inputs and the state of the outputs. In spite of the obvious value in confirming such a flaw in the Toyota ECUs I don't recall seeing anyone demonstrate a clear runaway causing condition. Out of it all, one feature that eventually did make it into the software was a check to give priority to the brake input such that some amount of brake application would cause the ECU to ignore the throttle input and set the engine back to idle. This is a handy change to make, but I doubt that it makes much difference except in the case that the pedal is physically restrained, which doesn't seem to happen often, and maybe only ever happened on the one car. (Though articles claim there was a prior problem with all-weather mats, it seems so unlikely to be true; all the cell phones and no one put up a video showing their runaway death-traps) The majority cause of unexpected acceleration is the same as always - pressing on the accelerator when intending to press the brake and then being startled by the sudden motion and pressing harder on the 'brake,' which just makes the control loop worse. Some (most?) of this has been dealt with by interlocking the shift out of Park with application of the brake, so that the car can't move from Park without the driver pressing the brake. One thing that seems ignored is that the pedal problem is a side effect of cost reduction. Originally most accelerator pedals were hinged at the floor, which was advantageous to the placement of the pull-cable housing mount in the firewall. With the hinge at the bottom the worst a floor mat could do is run up the pedal and provide slight pressure with little moment arm. It required some time to install the pedal in that location. The 'electronic' pedal meant that it could be integrated into the dash assembly and fit before installing the dash into the car as one unit. This exposes the end of the pedal to bypass the edge of the floor mat. If the user is able to push the pedal into the carpeting, an oversized mat edge can ride up and prevent its return, applying its load at the point of maximum leverage. An all-weather mat makes this worse by being significantly stiffer than the carpet mat and might as well be a wedge. I expect one reason few people noticed this, aside from not having the wrong mats, is that it requires a very high level of pedal force. In the accident vehicle there was a report that the car seemed to have trouble keeping speed and then suddenly shot forward in traffic. If the mat was blocking the pedal travel, preventing ordinary application, and the driver got frustrated and stamped as hard as he could to overcome the obstruction, it would fit the observation. Why the driver just didn't put the transmission in neutral is a question - maybe he did and the sound frightened him, believing the engine would explode. (Hint everyone - Let the engine manage itself, especially if it's a loaner.) ### RE: Train Derailment 3D dave, Whilst this is off the subject of the train de-railment I would point out that the guy killed in Santee was not \"Just a cop\", he was a California Highway patrol officer , these guys\/gals get extensive car handling training including skidpan work. I also used to commute to work on that road. At the time of the accident , the road was unfinished with a Tee junction at the end of a high speed downgrade into a road work area The area across the other side of the tee was a river valley with large boulders in it. I am sure given his training, if there was a way of stopping the car that he knew about ,he would have done so. Anyway this is off the subject of train de-railments. B.E. You are judged not by what you know, but by what you can do. ### RE: Train Derailment Two things before we get back on the rails. I believe that particular model of Toyota would not go into neutral above a certain RPM threshold. And regarding the testing that showed that brakes could overcome an engine, it was only if they were firmly and constantly applied without releasing them. In the case of sudden acceleration on the highway, cockpit resource management becomes a lot more challenging. Your first instinct is not to stand on the brakes. It\u2019s to try to get the car below 80 MPH while you troubleshoot. So you ride the brake a bit maybe. Try not to hit the cars in front of you. Release the brake to clear the accelerator with your foot. Reapply. Ride it some more. Try to turn it off. Etc. At that point the brakes have soaked up so much heat it\u2019s game over. Especially in \u201cfamily cars\u201d pushing 300 HP but without the brakes to match (have to keep that weight down for MPG). That was what happened to that patrol officer. No doubt. ### RE: Train Derailment Spartan5 - if that was true it would have been documented. The exact same car was brought to a stop a few days earlier due to the same problem by a driver who shifted it into neutral and pulled over. He dislodged the floor mat and reported it to the dealership when he dropped the car off. I'm not sure there's an advantage to forcing the transmission to remain engaged; the ECU can look after the motor to keep it from detonating while unloaded. There's at least one comment that the US DOT requires that vehicles always be able to shift to neutral, though the I didn't find a rule to that effect. The fact that the Officer Saylor did't succeed using that let the start that some huge programming problem prevented it from happening and therefore starting rumors that there had to be a coverup. Bershire - A different driver of the exact same car had the exact same problem and brought the car to safe halt. Training cannot make up for panic and I doubt that any skidpad training included WOT latching on. I expect the additional burden of having his immediate family in the car also added to the cognitive load, causing him to exclude more survivable alternatives to heading off an embankment, such as grinding along a guard rail or sliding into a ditch. The question for the train derailment is that certain systems can offset operator error and without looking at how operators get into good or terrible situations, allows for future problems. This train was run with a single engineer so any mistake made had no co-pilot to alert him or take action. This driver wasn't able to observe what made previous runs on the refreshed line successful, such as noting the positions and indications on speed control signs. Had the conductor been given a device that plotted the location, speed limit, and current speed and sounded an alarm for over-speed, the conductor could have accessed the emergency brake or radioed the engineer and stopped the train on the way to the curve. Frankly, I'm a bit surprised that the railroad enthusiasts weren't aware of the impending situation; it's a new route and they would certainly be interested as to exactly where they were and could know what the track speeds should be. Perhaps they had too much confidence in the system to recognize the danger. I expect the immediate cause is the engineer was explaining something to the conductor-trainee and they just failed to notice what was happening outside the cab due to the distraction. I would not be surprised if it was the engineer's first run on the route. ### RE: Train Derailment Sometimes the problem is just that driving a train is so boring that people lose track of where they are. There's a tendency to shout at drivers after they've had accidents like this in the hope of encouraging their colleagues to be more careful in future, but it never really works very well. Human beings are vulnerable to lapses of concentration under conditions of low arousal - engineering in an independent layer of protection may feel like an expensive way of solving the problem but it really is \"low hanging fruit\" compared to the futility of trying to bully a driver (who is already quite motivated to not kill him\/herself and a trainload of passengers) into hours of unbroken concentration. Some interesting parallels with a recent tram crash in the UK. A. ### RE: Train Derailment Do locomotives still have so-called 'dead-man switches' to assure that the engineer is actually 'driving' the train at all times? I would think that something like that would be mandatory with a single-man cab. John R. Baker, P.E. (ret) EX-Product 'Evangelist' Irvine, CA Siemens PLM: UG\/NX Museum: The secret of life is not finding someone to live with It's finding someone you can't live without ### RE: Train Derailment 3DDave - Not sure where I saw that, but it must have seemed reputable or I might not have logged it. Perhaps it's wrong. More than likely, as I said, it boiled down to cockpit resource management (CRM) in an unfamiliar vehicle. Take a look at this shift gate and tell me where neutral is. Now imagine trying to figure that out at 110 MPH in traffic with everyone in the car freaking out. Another thing to consider with regards to braking, is that at wide open throttle you only get a few pumps on the pedal before the boost is gone and you're left with manual braking. Though it was clearly documented in this lexus that the brakes were thoroughly cooked. Sad story all around. Especially considering that the officer was made out by some to be at fault due to his perceived incompetence. I guess the moral of the story, perhaps as we may even find in this train wreck, is that all of this complexity we are building into things (push button start, trick automatic transmissions, POWER!) is causing CRM issues. ### RE: Train Derailment The problem with the car \"OFF\" \"rotary\" switch as implemented in the US was exaggerated by the design of the OFF rotary switch was right between three OTHER rotary control switches, all of sear-identical size, height, diameter, and \"feel\". The AC fan Off-speed selection rotary knob, the radio On-Volume control rotary knob, the transmission Reverse-Neutral-Drive-Low selection control rotary knob, and the engine Start-Off rotary selection knob. But. If you \"turn off\" the wrong knob, the engine does NOT turn off but the transmission IS locked into its last (drive or reverse) position. If you turn \"off\" the transmission selection, it changed to the Reverse position, and - again - the engine does NOT turn off. The Key fob is a remote control sensor - The Key does NOT have to be pulled from the key slot at any time. So, getting out of the car seat (with the key now in your pocket) means nothing: The engine is still running, and the transmissio is still in \"Drive\". The radio is Off though. ### RE: Train Derailment ^ I'm missing some context. Which specific vehicle are you talking about here? Very few vehicles use a rotary switch for a keyless-ignition system and the few that I know of that do, have that switch in the same place as where a normal rotary key switch would be (which IMO is the right way to do it). Very few vehicles also use a rotary switch for transmission selection (certain late model Chryslers and Jaguars are the only ones I can think of) and the ones that do, don't also use a rotary switch for keyless-ignition, nevermind having such a switch similarly arranged as the ignition switch ... and they're not shaped similarly to the HVAC controls. So, you must be referring to a specific make, model, year that I haven't seen. What is it? ### RE: Train Derailment The locomotive I road in a few years back still had a so-called 'dead-man switch' alertness monitor. It monitored control inputs, then started a flashing light and a buzzer if too much time elapsed without operator input. At 80 MPH, a train can travel quite a distance before a monitor would take any action. I don't recall the exact timing, but it seemed more like minutes than seconds. ### RE: Train Derailment The problem with such a switch is precisely how much input is really required? A 10-mile stretch of straight track should require no inputs for 7 minutes. Such as system cannot adequately capture the variation of track length and turns. The positive train control that was supposed to have been completed in 2015 would be far superior to any sort of dead man switch. Even an Arduino coupled with a GPS and a detailed track program could have prevented such an accident by warning the operator that the speed was excessive for that portion of the track, as well as the previous portion, since the operator probably needed to have slowed down well in advance of the slow section. TTFN (ta ta for now) I can do absolutely anything. I'm an expert! https:\/\/www.youtube.com\/watch?v=BKorP55Aqvg FAQ731-376: Eng-Tips.com Forum Policies forum1529: Translation Assistance for Engineers Entire Forum list http:\/\/www.eng-tips.com\/forumlist.cfm ### RE: Train Derailment I can't help thinking that a tech school class could probably design a good reliable safety system for a couple of thousand dollars in hardware. Is this reasonable or an I blowing smoke? Bill -------------------- \"Why not the best?\" Jimmy Carter ### RE: Train Derailment Given that your normal average GPS knows what speed limits are, it really shouldn't be all that hard. GPS is not infallible, but it knows when it's not getting a signal, and a good many places where it won't have a signal (e.g. tunnels) are predictable in advance, which means you can do something about it. Even without a GPS signal, the path of a train is governed by the tracks that it's running on, which is known, and the distance that it has covered along those tracks can be established by wheel sensors on non-driving wheels to eliminate the possibility of wheelspin. That ought to provide enough coverage for the periods where it doesn't have a GPS signal or where the signal is ambiguous. For that matter, the distance-since-trip-start (or since a known \"reset\" location - a station, a track switch) could be the primary control with GPS only used to refine the position accuracy. \"31.7 km into this trip, reset maximum speed to 70 km\/h, then 33.2 km into this trip, reset maximum speed to 120 km\/h, then 55.4 km into this trip, download next instruction set depending on which position the track switch sends the train down\", that sort of thing. I'm sure someone can toss enough FMEA darts at this to find theoretical holes that this strategy doesn't cover, but compare it to what the current system provides ... nothing. ### RE: Train Derailment #### Quote (3DDave) Flight-sim pilots were often able to successfully pilot a plane under the circumstances that brought a DC-10 down in Chicago, but only after they were fully informed as to the exact defect and given a chance to plan a response, time and information the original crew did not have. For those who haven't seen Sully yet... same situation. The board swore up and down multiple sims showed he could have landed at one of several airports. Once they reset the time limit to more appropriately reflect what would likely happen in the cockpit, none of the sim pilots could make it. Dan - Owner http:\/\/www.Hi-TecDesigns.com ### RE: Train Derailment To continue the digression just a bit, I had not heard of the Gimli Glider incident (accident?). What an amazing feat of piloting. Other than running out of fuel in the first place :) The problem with sloppy work is that the supply FAR EXCEEDS the demand ### RE: Train Derailment OK, so a GPS\/INS + Arduino could be had for around$150 from Sparkfun, and triple redundancy would be slightly more than triple to account for the voting hardware. I would think that the existing train routing software already has the speed limits database, and the programmed route information could easily include the limits along with the GPS coordinates of the track segments.\n\nExisting route planners from other industries can already autonomously program flight paths and speeds for UAVs well enough to avoid enemy radars; adapting them to plan a train route shouldn't be that complicated.\n\nThe biggest issue, of course, is a fundamental lack of desire. The rail companies neither want to spend the money or even to do the job in the first place. That's the only rational explanation for an already 3-yr slip in implementation of positive train control. Any time safety equipment is demanded by the public or the government, companies resist, until they're back up against a wall. Then, the implementation is PDQ, and the companies laud all their safety features, after the fact.\n\nTTFN (ta ta for now)\nI can do absolutely anything. I'm an expert! https:\/\/www.youtube.com\/watch?v=BKorP55Aqvg\nFAQ731-376: Eng-Tips.com Forum Policies forum1529: Translation Assistance for Engineers Entire Forum list http:\/\/www.eng-tips.com\/forumlist.cfm\n\n### RE: Train Derailment\n\nGreat piloting and an amazing coincidence.\nIn the cockpit were two pilots who had the combined experience and skill set to land successfully.\nOne pilot had glider experience and the other had first hand experience flying out of the Gimli airport.\nWhat are the odds that those two pilots would be in the cockpit on that flight?\n\nBill\n--------------------\n\"Why not the best?\"\nJimmy Carter\n\n### RE: Train Derailment\n\nWaross,\nSome years ago the two pilots who did this were guest speakers at a Soaring Society of America convention , It was a very interesting story they told among other things they mentioned , The aircraft's fuel gauges were inoperative because of an electronic fault indicated on the instrument panel and airplane logs. They relied only on the quantity put on board which of course was done in pounds instead of kilos so they thought they were getting more fuel than they really did.\nBut again there was a comedy of errors prior to their taking off which later resulted in the two pilots and three maintenance workers getting suspended.\nB.E.\n\nYou are judged not by what you know, but by what you can do.\n\n### RE: Train Derailment\n\nSo often the case. Several seeming unrelated minor problems add up to one big problem. Turned out OK that time, too often does not.\n\nThe problem with sloppy work is that the supply FAR EXCEEDS the demand\n\n### RE: Train Derailment\n\nThe last chance to avoid failure was they did not floatstick after fueling to confirm what they thought was on board was actually on board. Failing to repeat the measurement they relied on to determine fuel need was a critical step that would have exposed the calculation flaw. Additionally, they probably failed to close the loop with the fuelers about the range they expected out of the amount put on board. There is no way the plane would be twice as efficient as anything else in the air.\n\n### RE: Train Derailment\n\nRegarding: The biggest issue, of course, is a fundamental lack of desire. The rail companies neither want to spend the money or even to do the job in the first place. That's the only rational explanation for an already 3-yr slip in implementation of positive train control. Any time safety equipment is demanded by the public or the government, companies resist, until they're back up against a wall. Then, the implementation is PDQ, and the companies laud all their safety features, after the fact.\n\nRonald Batory \u2014 President Trump\u2019s nominee to lead the Federal Railroad Administration (FRA) \u2014 will be pushing for the controversial self-regulatory approach to safety known as \u201cperformance-based regulations,\u201d according to his July 26 statement for the U.S. Senate Committee on Commerce, Science and Transportation.\n\nhttps:\/\/www.desmogblog.com\/2017\/09\/27\/federal-rail...\n\nThe engineers at VW took advantage of the performance-based regulations when they installed modifications to get around the diesel emissions regulations.\n\n### RE: Train Derailment\n\n#### Quote (IRStuff)\n\nThe biggest issue, of course, is a fundamental lack of desire\n\nAre you sure it's not something more human, such as the threat to job security, or the reduction of the driver's responsibility to the point of uselessness?\nWhen comparing rail safety records, one should ask: What do they do in Japan?\nNorth America's rail system is pretty sad compared to the Shinkansen.\n\"In 2011, 27 shinkansen trains were skimming the country the afternoon of March 11 when a 9.0 megaquake struck... There were no fatalities or injuries.\"\n\nThere is no way a human could make the split-second decisions needed to minutely control a 300 KPH train all day every day. Automation of rail transport has already been solved. A 130kph train is trivial in comparison.\n\nSTF\n\n### RE: Train Derailment\n\nSparWeb has it exactly. Follow the money from the unions to the politicians. Should be an easy connect-the-dots exercise.\n\n### RE: Train Derailment\n\n\"Are you sure it's not something more human, such as the threat to job security, or the reduction of the driver's responsibility to the point of uselessness?\"\n\nSince when has that really stopped any company from executing a corporate desire? It certainly didn't stop the fireman and conductor from disappearing. It certainly hasn't stopped the airlines from reducing cockpit crews from 4 to 2. And when have the unions successfully won anything in the last 20 years? Does anyone really believe that the unions can buy more politicians than the railroads?\n\nTTFN (ta ta for now)\nI can do absolutely anything. I'm an expert! https:\/\/www.youtube.com\/watch?v=BKorP55Aqvg\nFAQ731-376: Eng-Tips.com Forum Policies forum1529: Translation Assistance for Engineers Entire Forum list http:\/\/www.eng-tips.com\/forumlist.cfm\n\n### RE: Train Derailment\n\nJapan spends a lot for national pride on a fairly narrow scope project. It didn't translate well into their nuclear program though and they don't ship enough tonnage by rail to be more than a round off error to US rail. Not having to contend with freight trains makes passenger trains a lot easier. I guess if a country wants to be really good at something they are going to spend a ton of money to do so.\n\nAnyway - the 2015 article has some 2017 followup https:\/\/www.dmagazine.com\/frontburner\/2017\/12\/has-... It's not dead, but it's not a source of pride to a lot of Texans.\n\n### RE: Train Derailment\n\n#### Quote (waross)\n\nI can't help thinking that a tech school class could probably design a good reliable safety system for a couple of thousand dollars in hardware.\nIs this reasonable or an I blowing smoke?\n\nAs others point out, this is all about cost. The technology has been around a long time. There is nothing to \"invent\", only the implementation. The standards have been all hashed out and every manufacturer of equipment has solutions available.\n\nThe New York underground system had positive train control when built in 1904. It was centrally dispatched, with remote controlled switches and automatic signals. A mechanical trip rising from besides the track indicated clear, restricted or stop. Passing restricted too fast or passing stop would apply the air brake to emergency.\n\nThe London Underground started automated train control in 1964. It uses wayside coils to indicate target speed by a number of different frequencies. The operator only controls the doors and issues a \"go\" command and after that the central control system sets speed and onboard controls regulate the speed including the final stop in place.\n\n#### Quote (SparWeb)\n\nAre you sure it's not something more human, such as the threat to job security, or the reduction of the driver's responsibility to the point of uselessness?\n\nAs you point out, there are plenty of examples of automated systems. They still have an operator though. Automated systems don't do well with unexpected disruptions like objects on the tracks, people blocking doors, etc..\n\nOK, so a GPS\/INS + Arduino could be had for around $150 from Sparkfun, and triple redundancy would be slightly more than triple to account for the voting hardware. I would think that the existing train routing software already has the speed limits database, and the programmed route information could easily include the limits along with the GPS coordinates of the track segments. Rail equipment isn't a since fair project. I know for certain you've never designed anything that goes onboard rail equipment if you think COTS will work without a bunch of modifications. Rail is BRUTAL for vibration and impact. That said, the real cost isn't in the moving equipment, it is in the wayside equipment, communications and software. Far more often the problem isn't a train going too fast for the location, it is unauthorized movement - going against switches and \/ or running into another train. For that you've got to communicate who's got authority, where everyone is, etc.. All of the technology is developed and agreed upon in the US. It is a matter of spending the money to put it in. ### RE: Train Derailment A tidbit. Last year I put an LTE Ethernet router into a railcar to provide me with a link into the system controller. Just for the heck of it I \"checked the box\" to add GPS to the router. I installed the complex antenna on a flat deck already existing on the car's left side roof, about over the rail on that side. I can VPN into the router and ask it for it's immediate GPS location. If I copy that location and paste it into Google Maps then switch to satellite mode and zoom in, with out exception, I can always tell which way the car is pointing just by that two foot offset of the antenna. I was quite amazed when I realized that. Keith Cress kcress - http:\/\/www.flaminsystems.com ### RE: Train Derailment Hi MatthewDB, Please be careful when quoting others. It makes for confusing reading for everyone else, when you don't cut-and-paste correctly. Now that that's been taken care of, I want to add that I understand IRStuff when IRStuff wrote the following: #### Quote (IRStuff) OK, so a GPS\/INS + Arduino could be had for around$150 from Sparkfun, and triple redundancy would be slightly more than triple to account for the voting hardware. I would think that the existing train routing software already has the speed limits database, and the programmed route information could easily include the limits along with the GPS coordinates of the track segments.\n\nI believe IRStuff was NOT making an engineering design recommendation about how to automate a rail system. My reading of IRStuff's comment was more to place some ridicule on an industry that has fallen woefully behind in providing its operators with electronic assistance, when every other transport system has done similar things for their operators. You clearly understand the mechanism of doing this in the rail system better than I do, and perhaps better than IRStuff (I won't speak for them) but didn't notice the implied scorn, for not an industry that has not widely implemented it decades ago, when it became possible.\n\nSTF\n\n### RE: Train Derailment\n\nIt's not a matter of cost, it's a matter of cost NOW. One would think that millions of dollars of freight would be offset by a $60k (finished\/hardened\/qualified) system, but the aggregate cost is that multiplied by thousands of engines, so no one wants to spend the money after the fact (BTDT). Nevertheless, it seems that the freight companies won't even broach the subject with inexpensive COTS demo programs, because that'll just make it harder for them to refuse to implement the full Monte. When such a system is incorporated into new engines, the cost would be less than half, and no one would even blink an eye if the cost of a new engine were$20k higher, since that would simply get amortized over the freight costs over the lifetime of the engine. Engine additions are much easier to justify, compared to trading between bullets and safety equipment in military. That's been an ongoing losing proposition for at least 20 years. Military helos are routinely lost due to self-induced brownouts caused by the downwash; the technology exists to deal with that, but the aggregate cost constantly makes such systems fall below the budget line.\n\nWhile rail transport environment is harsh, it's nowhere close to impossible, and nowhere close to military truck transport or naval 901D shock. Anything that a human bottom can handle for 8 hrs is benign, by definition. And note that I was describing a simple warning system, which isn't even close to positive control, and does not require interaction with any other part of the engine, other than power and external antenna.\n\nTTFN (ta ta for now)\nI can do absolutely anything. I'm an expert! https:\/\/www.youtube.com\/watch?v=BKorP55Aqvg\nFAQ731-376: Eng-Tips.com Forum Policies forum1529: Translation Assistance for Engineers Entire Forum list http:\/\/www.eng-tips.com\/forumlist.cfm\n\n### RE: Train Derailment\n\nSo, rail guys, here's a question for you. There are a number of lines thru my city that are posted that the engines may be unmanned, remote controlled or some such wording. How they doing that?\n\nThe problem with sloppy work is that the supply FAR EXCEEDS the demand\n\n### RE: Train Derailment\n\nUnmanned is easy enough, however controlled, local or remote, I'm not sure.\n\n### RE: Train Derailment\n\nI believe they are radio controlled with an observer. https:\/\/en.wikipedia.org\/wiki\/Remote_control_locom... The caution is that people would think that the lack of a person in the cab means the locomotive won't move and therefore increase the odds for a collision by dodging around crossing-arms, for example.\n\n### RE: Train Derailment\n\n#### Quote (SnTMan)\n\nSo, rail guys, here's a question for you. There are a number of lines thru my city that are posted that the engines may be unmanned, remote controlled or some such wording. How they doing that?\n\nThey are locally controlled via radio from a console worn in a harness. They are mainly used to allow for one man operation in switching service. Most of the time the operator will be in the cab, but when it comes time to back up, make a couple, throw a switch and set it back, etc... the operator will dismount and run the train from the ground. They are always operating with the end of the train in the direction of movement visible.\n\n### RE: Train Derailment\n\nMatthewDB, yeah that makes sense, where I am most used to seeing them is near a yard. Thx :)\n\nThe problem with sloppy work is that the supply FAR EXCEEDS the demand\n\n### RE: Train Derailment\n\nI expect a preliminary report in under 6 months.\n\n### RE: Train Derailment\n\nSome info and tidbits:\nThe train is a Talgo manufactured train which has been is service for many years by Amtrak. They are rather odd looking, with an engine on both ends, higher than the cars in between. The passenger cars share wheel trucks, single axle between cars.\n\nA train on the alternate, older route alongside Puget Sound suffered a derailment last July. There is a lift bridge on that route and the train failed to stop for a bridge opening. There is an old (1920's or older) vertical lift bridge over Steilacoom creek on the Western track which I suspect was involved (news website didn't say).\n\nThe old Western route follows the shore of Puget Sound and enters a long tunnel with a quite tight turn on the West side of Tacoma, under Point Defiance Park. It is a picturesque route, but subject to mud slides from the high bluffs along the Sound.\n\nA lot of construction has been going on over the last couple of years on the new route. The tracks were regraded and reinstalled with new concrete ties. I believe the commuter \"Sounder\" trains will be sharing this track, but only as far as South Tacoma in the near future - maybe about 5 miles North of the accident site.\n\nThe bridge where it happened is the span over Interstate 5 South bound lanes. The East pier used to be a favorite site for the WA State Patrol to hide out and catch speeders on the downgrade. Haven't seen them there lately?\n\n### RE: Train Derailment\n\nI think the Talgo sets use a locomotive on one end, the other end is a control car \/ HEP car, but not a locomotive. In this case, it seems to have been a GE unit pushing the train, but I understand the plan was to use EMD locomotives painted to match.\n\n### RE: Train Derailment\n\n#### Quote (bimr)\n\nThe locomotive engineer should engage legal counsel as he will probably face manslaughter charges.\n\nThe engineer is in a union, and will be represented at no cost of his own.\n\nTrain derailments are under the jurisdiction of the NTSB, and get treated in a very similar way as commercial plane accidents... So I'd bet the NTSB\/Union reps\/Lawyers were on board within about 10 minutes of the accident.\n\n### RE: Train Derailment\n\nIn a recent 'Railway Age' magazine, I saw an article about how bad Amtrak's safety culture is; it was based on the inquiry into the accident in NJ (I think) where a train hit a work crew. It will be interesting to see how this plays out.\n\n### RE: Train Derailment\n\n#### Quote (bimr)\n\n\u201cperformance-based regulations\u201d\nIs that the same logic that decides a stoplight will only be installed after the second child is killed by a speeding motorist, after years of pleading by neighbourhood parents?\n\n\"Everyone is entitled to their own opinions, but they are not entitled to their own facts.\"\n\n### RE: Train Derailment\n\n#### Quote:\n\nIs that the same logic that decides a stoplight will only be installed after the second child is killed by a speeding motorist, after years of pleading by neighbourhood parents?\n\nNo.\n\nAround here, it takes three fatalities to get a stoplight.\n\nLocal rules will vary, and will not be recorded in a place where just any citizen has access.\n\nMike Halloran\nPembroke Pines, FL, USA\n\n### RE: Train Derailment\n\n#### Quote (jgKRI's (Mechanical))\n\nThe engineer is in a union, and will be represented at no cost of his own.\n\nTrain derailments are under the jurisdiction of the NTSB, and get treated in a very similar way as commercial plane accidents... So I'd bet the NTSB\/Union reps\/Lawyers were on board within about 10 minutes of the accident.\n\nNot necessarily. The locomotive engineer would have to be in the union and the union contract would have to include liability coverage, both of which are unknown\n\nHere is an example:\n\n### RE: Train Derailment\n\n#### Quote (ironic metallurgist (Materials))\n\nIs that the same logic that decides a stoplight will only be installed after the second child is killed by a speeding motorist, after years of pleading by neighbourhood parents?\n\nPerhaps a better terminology would be \"Tombstone Engineering\".\n\n### RE: Train Derailment\n\n#### Quote (bimr)\n\nPerhaps a better terminology would be \"Tombstone Engineering\".\n\nThe downside to that is far to often, the wrong thing is done after a death. It's a natural reaction to \"we must do something\" after someone is killed. Particularly when the dead person is politically connected, or related to someone politically corrected.\n\nA good example is when a pedestrian is killed by someone turning right on a red. We're not going to ban right turns on red, so they will ban it at only that one intersection, even if there is nothing there that makes that one intersection particularly dangerous.\n\n### RE: Train Derailment\n\n#### Quote (bimr)\n\nAmtrak's engineers are fully unionized. I expect them to circle the wagons for this guy, whether he deserves their protection or not.\n\n### RE: Train Derailment\n\nALL Americans have a right to legal consul, PERIOD! It's NOT an issue of whether he deserves it or not, it's a Constitutional right. As for who provides it, that's totally irrelevant. In fact, if he was not a member of a union and he was being charged with a criminal offense and he was not able to afford his own legal representation, the state would be obligated to provide him consul at no cost to himself.\n\nJohn R. Baker, P.E. (ret)\nEX-Product 'Evangelist'\nIrvine, CA\nSiemens PLM:\nUG\/NX Museum:\n\nThe secret of life is not finding someone to live with\nIt's finding someone you can't live without\n\n### RE: Train Derailment\n\nI would hope the union helps ensure he gets treated to the rights he has.\n\nThat's kinda the point of a union.\n\n### RE: Train Derailment\n\n#### Quote (jgKRI (Mechanical))\n\nAmtrak's engineers are fully unionized. I expect them to circle the wagons for this guy, whether he deserves their protection or not.\n\nIs the following from a similar incident an example of \"circle the wagons\"?\n\n\"The executive director of the Rail Employees Union now says 46-year-old engineer William Rockefeller, who was injured in crash, has said he caught himself \"nodding off\" at the controls.\"\n\nhttp:\/\/www.texomashomepage.com\/news\/national-news\/...\n\nhttp:\/\/www.cnn.com\/2013\/12\/03\/us\/new-york-train-cr...\n\nTo surmise that a union will obstruct an investigation seems to be somewhat cynical.\n\n### RE: Train Derailment\n\nIf y'all read my post as implying that the engineer in this accident will unduly avoid any consequences because of his union membership... that's not what I meant.\n\n### RE: Train Derailment\n\n#### Quote (CNN article)\n\nLate Tuesday, the NTSB said the rail union has been kicked out of its investigation of the derailment for violating confidentiality rules.\nOops...\n\nDan - Owner\nhttp:\/\/www.Hi-TecDesigns.com\n\nOuch.\n\n### RE: Train Derailment\n\nHappens all the time:\n\n\"UPS, union ousted from inquiry into crash of one of carrier service\u2019s cargo planes\"\n\nhttps:\/\/www.washingtonpost.com\/local\/trafficandcom...\n\nAll take a vow of silence until the NTSB delivers its final report on the accident. So secret is the process that for some portions of the inquest the partners gather in a secure section of the NTSB building that is equipped with a unique computer system that allows no communication outside the room. Partners at those sessions take notes on color-coded paper that is collected before they leave the room.\n\n#### Red Flag This Post\n\nPlease let us know here why this post is inappropriate. Reasons such as off-topic, duplicates, flames, illegal, vulgar, or students posting their homework.\n\n#### Red Flag Submitted\n\nThank you for helping keep Eng-Tips Forums free from inappropriate posts.\nThe Eng-Tips staff will check this out and take appropriate action.\n\nClose Box\n\n# Join Eng-Tips\u00ae Today!\n\nJoin your peers on the Internet's largest technical engineering professional community.\nIt's easy to join and it's free.\n\nHere's Why Members Love Eng-Tips Forums:\n\n\u2022 Talk To Other Members\n\u2022 Notification Of Responses To Questions\n\u2022 Favorite Forums One Click Access\n\u2022 Keyword Search Of All Posts, And More...\n\nRegister now while it's still free!","date":"2018-05-22 17:36:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2083301991224289, \"perplexity\": 4010.2805982093523}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-22\/segments\/1526794864837.40\/warc\/CC-MAIN-20180522170703-20180522190703-00187.warc.gz\"}"} | null | null |
package it.unibz.inf.ontop.dbschema.impl.json;
import com.fasterxml.jackson.annotation.*;
import com.google.common.collect.ImmutableList;
import it.unibz.inf.ontop.dbschema.*;
import it.unibz.inf.ontop.dbschema.impl.AbstractRelationDefinition;
import it.unibz.inf.ontop.dbschema.impl.AttributeImpl;
import it.unibz.inf.ontop.dbschema.impl.DatabaseTableDefinition;
import it.unibz.inf.ontop.exception.MetadataExtractionException;
import it.unibz.inf.ontop.model.type.DBTypeFactory;
import it.unibz.inf.ontop.utils.ImmutableCollectors;
import javax.annotation.Nullable;
import java.util.List;
import java.util.Optional;
import java.util.stream.Stream;
@JsonInclude(JsonInclude.Include.NON_NULL)
@JsonPropertyOrder({ // Why is this "reversed order"?
"uniqueConstraints",
"otherFunctionalDependencies",
"foreignKeys",
"columns",
"name"
})
public class JsonDatabaseTable extends JsonOpenObject {
@JsonInclude(value= JsonInclude.Include.NON_EMPTY)
public final List<JsonUniqueConstraint> uniqueConstraints;
@JsonInclude(value= JsonInclude.Include.NON_EMPTY)
public final List<JsonFunctionalDependency> otherFunctionalDependencies;
@JsonInclude(value= JsonInclude.Include.NON_EMPTY)
public final List<JsonForeignKey> foreignKeys;
public final List<Column> columns;
public final List<String> name;
@JsonInclude(value= JsonInclude.Include.NON_EMPTY)
public final List<List<String>> otherNames;
@JsonCreator
public JsonDatabaseTable(@JsonProperty("uniqueConstraints") List<JsonUniqueConstraint> uniqueConstraints,
@JsonProperty("otherFunctionalDependencies") List<JsonFunctionalDependency> otherFunctionalDependencies,
@JsonProperty("foreignKeys") List<JsonForeignKey> foreignKeys,
@JsonProperty("columns") List<Column> columns,
@JsonProperty("name") List<String> name,
@JsonProperty("otherNames") List<List<String>> otherNames) {
this.uniqueConstraints = Optional.ofNullable(uniqueConstraints).orElse(ImmutableList.of());
this.otherFunctionalDependencies = Optional.ofNullable(otherFunctionalDependencies).orElse(ImmutableList.of());
this.foreignKeys = Optional.ofNullable(foreignKeys).orElse(ImmutableList.of());
this.columns = columns;
this.name = name;
this.otherNames = Optional.ofNullable(otherNames).orElse(ImmutableList.of());
}
public JsonDatabaseTable(NamedRelationDefinition relation) {
this.name = JsonMetadata.serializeRelationID(relation.getID());
this.otherNames = relation.getAllIDs().stream()
.filter(id -> !id.equals(relation.getID()))
.map(JsonMetadata::serializeRelationID)
.collect(ImmutableCollectors.toList());
this.columns = relation.getAttributes().stream()
.map(Column::new)
.collect(ImmutableCollectors.toList());
this.foreignKeys = relation.getForeignKeys().stream()
.map(JsonForeignKey::new)
.collect(ImmutableCollectors.toList());
this.uniqueConstraints = relation.getUniqueConstraints().stream()
.map(JsonUniqueConstraint::new)
.collect(ImmutableCollectors.toList());
this.otherFunctionalDependencies = relation.getOtherFunctionalDependencies().stream()
.map(JsonFunctionalDependency::new)
.collect(ImmutableCollectors.toList());
}
public DatabaseTableDefinition createDatabaseTableDefinition(DBParameters dbParameters) {
DBTypeFactory dbTypeFactory = dbParameters.getDBTypeFactory();
QuotedIDFactory idFactory = dbParameters.getQuotedIDFactory();
RelationDefinition.AttributeListBuilder attributeListBuilder = AbstractRelationDefinition.attributeListBuilder();
for (Column attribute: columns)
attributeListBuilder.addAttribute(
idFactory.createAttributeID(attribute.name),
attribute.datatype != null
? dbTypeFactory.getDBTermType(attribute.datatype)
:dbTypeFactory.getAbstractRootDBType(),
attribute.isNullable);
ImmutableList<RelationID> allIDs = Stream.concat(Stream.of(name), otherNames.stream())
.map(s -> JsonMetadata.deserializeRelationID(dbParameters.getQuotedIDFactory(), s))
.collect(ImmutableCollectors.toList());
return new DatabaseTableDefinition(allIDs, attributeListBuilder);
}
public void insertIntegrityConstraints(NamedRelationDefinition relation, MetadataLookup lookupForFk) throws MetadataExtractionException {
for (JsonUniqueConstraint uc: uniqueConstraints)
uc.insert(relation, lookupForFk.getQuotedIDFactory());
for (JsonFunctionalDependency fd: otherFunctionalDependencies)
fd.insert(relation, lookupForFk.getQuotedIDFactory());
for (JsonForeignKey fk : foreignKeys) {
if (!fk.from.relation.equals(this.name))
throw new MetadataExtractionException("Table names mismatch: " + name + " != " + fk.from.relation);
fk.insert(relation, lookupForFk);
}
}
@JsonInclude(JsonInclude.Include.NON_NULL)
@JsonPropertyOrder({
"name",
"isNullable",
"datatype"
})
public static class Column extends JsonOpenObject {
public final String name;
public final Boolean isNullable;
@Nullable
public final String datatype;
@JsonCreator
public Column(@JsonProperty("name") String name,
@JsonProperty("isNullable") Boolean isNullable,
@Nullable @JsonProperty("datatype") String datatype) {
this.name = name;
this.isNullable = isNullable;
this.datatype = datatype;
}
public Column(Attribute attribute) {
this.name = attribute.getID().getSQLRendering();
this.isNullable = attribute.isNullable();
this.datatype = attribute.getTermType().isAbstract()
? null
: ((AttributeImpl)attribute).getSQLTypeName();
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,875 |
If you were in a tournament where you can have your own coach, who would you choose?
Similar Threads for: If you were in a tournament where you can have your own coach, who would you choose?
An excellent professional player and a personal teacher!
Eric Seidel or Darryl Fish.
Stu Ungar and Daniel Negreanu.
I am not a fan of Helmuth's antics, but I would choose him. You can't argue with his continued results.
i like sammy farah game,and i would like to have him in my corner during a tourny!
google alpha go or google AI - the worlds unbeatable 1st top player.
Phil Helmuth because he had won many bracelete.
Shame on all of you! Not recognizing that one of the sh!ttiest players at CC, is a helluva coach, where's you insight and vision in my abilities in that regard. Forget all of you; and please now get back to your generic, name-branded, fantasy coach tournament mentor picks. You could of got me for a rock bottom price, nope, another missed value deal for the know it alls.
Doug Polk or Daniel as I know they can coach. Sometimes great players aren't good teachers so with that said I would take Ivey if he can teach.
re: Poker & If you were in a tournament where you can have your own coach, who would you choose?
Doyle Brunson(R.I.P)or Phil Helmuth would be my choice.
I would chose Jonathan Little or Feder Holtz.
I would only choose Nagreanu. He is balanced, calm and at the same time a strong player.
there are people I would like to meet... Brunson, etc.
but if we are talking about learning from someone, it would be Negreanu or Ivey.
though I would love to shove 7-6s on the opening hand and then ask Helmuth if I made a mistake?
Parker Talbot or Daniel Negreanu - with both I would spend best beker hours of my life. Doug Polk is also an option here hmm... but nr 1 Parker cause he is neverlucky as I am.
By far i would have to say daniel negranyu.
Do you have information about that player that you share with us?
I'd go with Danielle negreana and Phil ivy.
Brian Paris or Doug Polk they are both great tourney pros and teachers those would be my top two choices and I have watched both of them live stream on the net while them bink hugh prize tourneys I also like Jason Sommerville he can play too.
If it's a tournament, Daniel Negreanu or Phil Laak. I tend to be talkative and a goofball myself at the table so Those two would help me harness it haha.
What? no one picks a female poker pro????
It depends.... what is the buy-in? what is the structure? where in the world is the tournament? who is expected to be in the field?
Some of these posts are so creative... it's awesome. That being said - I think having Barry as my couch would be more hurt than help. I only say that because I would literally fall asleep listening to that dude talk. Super monotone. Not to mention im pretty sure hes 70 now.
I'm gonna have to go with Kidpoker. Daniel just seems so full of life and his reads are next level. That or Stu Unger.... but i'm not sure he would have been able to stay off cocaine long enough to coach someone. Seems more of the type of guy that when you ask him a question he just shrugs like... " yeah everyone is a wizard and can soul read people, right?"
Nano is the best multi-tabler ever. Used to be a mid stakes crusher online and holds the Guinness record for most hands played in eight hours with profit. Has recently moved to bigger live games and just recorded a 6th place finish in the Aussie Millions.
As far as my answer....I haven't seen anyone mention tonkaaaa so I'll choose him to be different (and we can share a bong on break).
« Previous Should I have made this play? | {
"redpajama_set_name": "RedPajamaC4"
} | 3,933 |
Both surgeries are closed for training from 1pm to 4pm on every third Thursday of the month.please telephone 111 if you require a doctor urgently during this time. Only the evening surgery at the Windhill surgery is extended until 7.45pm on Thursdays.
The Saturday morning surgery at Windhill is for booked appointments only. No extra emergency or walk-in appointments will be available. Once all the appointments are booked patients will be directed to telephone 111 to access the Out of Hours Service.The telephones will be transferred to the 111 service as soon as the appointments have been filled. This may mean you are unable to telephone the surgery at all on a Saturday morning. However we are open for you to pick up prescriptions or letters.
A limited number of appointments are also available between 8.00 - 8:30am on selected days.
The service provided out of normal practice hours is for emergencies, which cannot wait until your surgery re-opens.
If a doctor is required in an emergency telephone 01274 584223.
The practice is a member of West Yorkshire Urgent Care. If you are unwell and the situation cannot reasonably wait until the surgery opens, telephone the NHS service on 111. If you telephone the surgery your call will be automatically redirected to the NHS 111 service. Some emergency calls can be dealt with by telephone advice. Otherwise, you may be offered an emergency appointment at Eccleshill Community Hospital. In some circumstances, the doctor may decide to visit the patient at home, if the person was too ill or frail to be moved. In a life-threatening emergency you should still dial 999 and ask for an ambulance.
The practice is a member of Local Care Direct. If you are unwell and the situation cannot reasonably wait until the surgery opens, telephone 01274 584223, your call will re-directed. In case of difficulty in contacting the out-of-hours service please ring NHS Direct, a 24-hour nurse led advice line, on 0845 46 47.
Some emergency calls can be dealt with by telephone advice. Otherwise, you may be offered an emergency appointment at the nearest urgent care centre. In some circumstances, the doctor may decide to visit the patient at home, if the person was too ill or frail to be moved. In a life-threatening emergency you should still dial 999 and ask for an ambulance. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,564 |
During this time, lending and mortgages policies were much more lenient! There was 100% financing available, 40-year amortizations, cash back mortgages, 95% refinancing, 5% down payment required for rental properties, and qualifications for FIXED terms under 5 years and VARIABLE mortgages at discounted contract rate. There was also NO LIMIT for your GROSS DEBT SERVICING (GDS) if your credit was strong enough. Relaxed lending guidelines when debt servicing secured and unsecured lines of credits and heating costs for non-subject and subject properties.
We saw the elimination of 100% financing, the decrease of amortizations from 40-35 years and the introduction of minimum required credit scores, which all took place during this time period. It was also the time in which the Total Debt Servicing (TDS) could only be maxed to 45%.
This time period saw Variable Rate Mortgages having to be qualified at the 5-year Bank of Canada's posted rate along with 1-4 year Fixed Term Mortgages qualified at the same. There was also the introduction of a minimum of 20% down vs. 5% on investment properties and an introduction of new guidelines on looking at rental income, property taxes and heat.
The 35-year Amortization dropped to 30 years for conventional mortgages, refinancing dropped to 85% from 90% and the elimination of mortgage insurance on secured lines of credit.
Increase in default insurance premiums.
Exemption for BC Property Transfer Tax on NEW BUILDS regardless if one was a 1st time home buyer with a purchase price of $750,000 or less.
Still fresh in our minds, the introduction of the foreign tax stating that an ADDITIONAL 15% Property Transfer Tax is applied for all non residents or corporations that are not incorporated in Canada purchasing property in British Columbia.
INSURED mortgages with less than 20% down Have to qualify at Bank of Canada 5 year posted rate.
Portfolio Insured mortgages (monoline lenders) greater than 20% have new conditions with regulations requiring qualification at the Bank of Canada 5 year posted rate, maximum amortization of 25 years, max purchase price of $1 million and must be owner-occupied.
•Lenders will be required to enhance their LTV (loan to value) limits so that they will be responsive to risk. This means LTV's will need to change as the housing market and economic environment change.
•Restrictions will be placed on lending arrangements that are designed to circumvent LTV limits. This means bundled mortgages will no longer be permitted.
*A bundled mortgage is when you have a primary mortgage and pair it with a second loan from an alternative lender. It is typically done when the borrower is unable to have the required down payment to meet a specific LTV.
BOTTOM LINE: WHERE DO WE GO FROM HERE?
As you can see, the industry has always been one that has changed, shifted and altered based on the economy and what is currently going on in Canada. However, with the new changes that have come into effect this year, we recognize that many are concerned about the financial implications the 2018 changes may have.
The one piece of advice that we promised you at the start of this blog, and one that has helped all our clients get through these changes is this: work with a Dominion Lending Centres mortgage broker!
We cannot emphasis the importance of this enough. We have up to date, industry knowledge, access to all of the top lenders and we are free to use! We guarantee to not only get you the sharpest rate, but also the right product for your mortgage.
Geoff is part of DLC GLM Mortgage Group based in Vancouver, BC. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,485 |
\section{Introduction}
Inclusive electron scattering, particularly around the quasielastic
peak is probably one of the problems that has attracted more attention in
nuclear physics. One of the reasons for the proliferation of theoretical
work was the persistent difficulty of simple pictures based on the shell
model of the nucleus, or the Fermi gas approach, to reproduce simultaneously
the longitudinal and the transverse response functions around the quasielastic
peak. Particularly, the longitudinal response appeared systematically
to be much larger than experiment \cite{MEZ,ALT,BAR}.
Even more worrisome was the fact that the integrated strength of the
experimental longitudinal response function showed large discrepancies with
the expected result according to the Coulomb sum rule \cite{GIU}. At large
values of the momentum transfer $q$ this integral should be the
charge of the nucleus and there was some apparent missing strength.
Those experimental results have been generally accepted, even when some
other experimental results seemed to challenge them. Indeed the results of
the experiment of \cite{BLA} in $^{238}$U did not show the expected
suppression of the longitudinal response. More recently an experiment at Bates
on $^{40}$Ca \cite{KAR} showed a longitudinal response larger than in
ref. \cite{MEZ} with only 20$\%$ reduction over the simple shell
model expectations.
The growing experimental discrepancies stimulated the thorough and
thoughtful work of \cite{JOU}.
In this work the author analysed the world set of data and made
further improvements in the analysis, coming with new results for the
response functions which show a smaller reduction of the longitudinal
response than previously assumed. At the heart of the issue was the fact
that, in a modified Rosenbluth plot in terms of the variable $\epsilon$
(which runs
from 0 to 1), the Saclay data~\cite{MEZ,ALT,BAR} concentrated their points
in the region of $\epsilon < 0.5$
which induces large errors in the slope of the straight
line which correlates the points of the
plot. Those results, complemented by others at SLAC and Bates, which
fill up the region of $\epsilon \sim 1$
, lead to a more accurate determination of
the slope and thus the longitudinal response. The problems with the
Coulomb sum rule are then automatically solved \cite{JOU} to the relief
of all \cite{DEN}.
As usually happens, some theoretical calculations were ``successful'' in
explaining the abnormal reductions of the longitudinal response, through
the use of small effective masses, swelling of nucleons, abnormal nucleon
form factors, correlations, etc. But in the benefit of the theoretical
community it should be said that whenever this happened, the agreement
with the transverse response was spoiled, although excuses and good wishes
beyond the ideas used for the longitudinal response were invoked as
possible solutions. Hence, a convincing simultaneous explanation of both the
longitudinal and transverse response functions was never obtained.
The interesting thing is that the existence of a ``problem'' induced so much
work that practically all resources of nuclear physics and many body theory
have been used in this topic and a lot of things have been learned.
In the present work we shall benefit from all this previous work and by
using a selfcontained many body formalism we will incorporate all these
effects which convincingly proved to be relevant in previous works. We also
incorporate some new ingredients which come out naturally within our many
body expansion and furthermore we include Delta--hole ($\Delta h$)
excitations, additional
to the particle--hole ($ph$) excitations, which allows us to simultaneously study the
quasielastic peak, the $\Delta$ peak and the ``dip'' region between the
two peaks in the inclusive $(e,e^\prime)$ cross section.
\section{Brief review of different approaches and introduction to our
approach.}
The amount of ideas which have been studied in connection with the
inclusive $(e, e^\prime)$ scattering is quite large. We shall discuss them
briefly:
\subsection{Modification of the nucleon form factor (swelling of the nucleon)}
These ideas were soon invoked and explored within different frameworks
~\cite{NOB,SHA}. They appear naturally in some microscopic models of the
nucleon form factor when the intermediate baryon propagators are replaced
by those in the nuclear medium. This is the case, for instance in ref.~\cite{GOE}, where the underlying elementary model is the
Nambu-Jona-Lasinio model
of the nucleon. Obviously there are many other medium effects not
taken into account in these schemes, as we shall see. Furthermore,
although in a different language and in terms of a few relevant physical
magnitudes, such medium modifications appear in a systematic many body
expansion, where they can be classified as vertex corrections.
\subsection{Relativistic effects}
Theories using relativistic scalar and vector potentials like in the Walecka
model \cite{WAL} have been popular. The scalar and vector potentials are
about one
order of magnitude larger than the ordinary non relativistic potential
which is roughly the sum of the two. This cancellation is missed in many
applications of these relativistic potentials leading to unrealistic
predictions. The appealing thing of the relativistic approach is the small
effective mass, $M_N^* \simeq M_N/2$
of the nucleon and the fact that the nucleon response
for $ph$ excitations is roughly proportional to $M_N^*$. This reduces
the longitudinal response but also the transverse one.
A clarifying view of these problems is exposed in \cite{HOR} where the
necessity to go beyond the relativistic mean field approach is shown in
order to avoid the pathological predictions tied to the small effective
masses, like in the
computation of the nuclear response functions or the large relativistic
enhancements that drive magnetic moments outside their Schmidt lines.
In \cite{HOR} a relativistic RPA calculation is used. Similar conclusions are
found in \cite{PRI}, stating that ``selfconsistent calculations''
show cancellations
between large relativistic effects on the single nucleon current and on
the many nucleon wave functions. In \cite{GIM} it is also shown, by
solving numerically the Dirac equation with the relativistic potential, that
the genuine relativistic effects in the enhancement of the axial charge amount
to 20-30$\%$, while perturbative calculations give as much as 70-80$\%$
enhancement \cite{RIS,GRA}.
Even when improved with some selfconsistent steps, relativistic calculations
still rely on the concept of the effective mass~\cite{HOR,HOD,SUZ},
which is only an
approximation to the richer content of the nucleon self-energy.
The nucleon self-energy is a function of the energy and momentum, as
independent variables, and
leads to important dynamical properties of the nucleus
~\cite{MAH} not contained
in static pictures like the mean field theories. One of the consequences is
that the nucleon effective mass is a strongly dependent function of the
energy, with a peak around the Fermi surface \cite{MAH}. Furthermore, as shown
in \cite{PAL}, RPA correlations tied to the pionic degrees of freedom are
essential
in order to provide the energy dependence of the nucleon self-energy and hence
the dynamical properties of the nucleus. Actually it is quite interesting to
see that recent sophisticated calculations in light nuclei using path integral
Monte Carlo methods \cite{CARL} find that ``pion degrees of freedom in both
nuclear interaction and currents play a crucial role in reproducing the
experimental data''.
From this discussion it looks clear that improvements along the relativistic
line for the present problem should include ``selfconsistency'' in the
sense of ref.~\cite{PRI}, in order to
exploit the large cancellations between the scalar and vector potentials.
\subsection{Pionic effects}
As mentioned above \cite{CARL}, the pionic degrees of freedom play an important
role in quasielastic electron scattering. This has also been emphasized
in \cite{WAN} where meson exchange currents driven by pion exchange are
evaluated, putting special emphasis in fulfilling the continuity equation
and preserving gauge invariance in the many body system. This imposition
has as a consequence some changes in the results with respect to former
works along similar lines \cite{MAG,ORD,MIS}
In ref. \cite{RIN} similar ideas, but using the formalism of path integrals is
followed. RPA correlations are automatically generated in that scheme leading
to some quenching of the longitudinal response from a reduction in the
isoscalar channel.
Pions are also explicitly used in approaches which include meson exchange
currents, as we shall see below. They are usually taken static, as in
~\cite{WAN,RIN}, meaning that the energy carried by the pion is neglected.
While this is a fair approximation for the exchange currents at energies
below pion production threshold, at higher energies the need to work will the
full pion propagator becomes apparent. This is particularly true if one wishes
to account for real pion production in the same many body scheme, as we
shall do.
In the resonance region primary pion production accounts for the largest
part of the response function (although some of the pions are absorbed in
their way out of the nucleus and show up in $2N$ or $3N$ emission channels).
Hence, the explicit treatment of pionic degrees of freedom allowing pions
to be produced, both as virtual as well as real states, becomes a necessity
in this region. Our scheme puts a special emphasis on pions. In fact it
follows a different path to other schemes, beginning with real pion
production and ensuring that a proper hand on the $(e, e' \pi)$ reaction is
held. Then exchange currents and further corrections in the many body
system are generated from the model for the $e N \rightarrow
e' N \pi$ reaction.
\subsection{Meson exchange currents}
A large fraction of work has been devoted to the role played by meson
exchange currents (MEC) in this reaction~\cite{WAN,MAG,ORD,MIS,GCO,AMA,LAN}.
The standard seagull, pion in flight and $\Delta$ terms are included mediated
by pions. The indirect
effect of short range correlations in these terms is, however, neglected in those
works. The work of \cite{TAI} incorporates terms in the scheme which account
for virtual photon absorption on correlated pairs, hence accounting for
ground state correlations. Although the same concepts are shared in the
previous
approaches, differences in the input and the way to implement them lead
to different results.
Our approach differs from the quoted works although conceptually it is quite
similar. First we realize that the two body currents appear in $(e, e')$ as
corrections to the main one body contribution. This is because virtual
photons can be absorbed by one nucleon. This is opposite to the case of
real photons which require at least two nucleons to be absorbed (we are
thinking in terms of infinite nuclear matter). Thus it is clear that the
laboratory to test the effect of two nucleon currents is real photons not
virtual ones. This is the reason why prior to the present work we devoted
energies to the problem of real photon absorption \cite{CAR}. Second,
in order to
minimize sources of uncertainties, the MEC were generated from the model
$\gamma N \rightarrow \pi N$
by allowing the pion to be produced in a virtual state and be
absorbed by a second nucleon. After this is done, long range correlations
from polarization phenomena, as well as short range correlations, are taken
into account. The model for the $\gamma N \rightarrow \pi N$
reactions was tested against experimental
data and was found to be good. This gives one some confidence in
the strength that one generates for the MEC. The reliability of the
method gets extra support from recent measurements of two body photon
absorption \cite{CRO,GRO,helh} where the agreement with the
$(\gamma, np)$ emission channels
is rather
good. Some discrepancies remain in the $(\gamma, pp)$
channel, but this channel has
an experimental cross section nearly one order of magnitude smaller than
the $(\gamma, np)$ one, thus for the purpose of the total
$(\gamma, NN)$ emission or the two
body MEC in $(e,e')$ such discrepancies will not play an important role.
In view of the success in the real photon case, we adopt here the same
scheme and follow the same steps simply substituting the real photon by
the virtual one. This means that we begin with a model for the
$e N \rightarrow e' N \pi$ reaction
and construct the MEC from it following identical steps to those of ref.
~\cite{CAR}. For this reason we begin in next section by showing our model for
the $e N \rightarrow e' N\pi$
reaction and contrasting it with the experimental results.
\subsection{RPA correlations}
Several works have emphasized the role played by RPA correlations allowing
for a $ph$ excitation which propagates in the nuclear medium mediated by
some residual $ph$ interaction \cite{HOR,RIN,DRE,WAB}. These works share the
feature that
a reduction is produced in the longitudinal channel. We shall also incorporate
these long range correlations or polarization effects. In addition
we shall also include $\Delta h$
excitations, as a source for polarization which
will be relevant in the transverse channel as shown in \cite{WAG}.
\subsection{Final state interaction (FSI)}
This is another topic which has received some attention and a thorough work
devoted to this issue can be seen in \cite{CHI}. It is clear that once a
$ph$ excitation is produced by the virtual photon, the outgoing nucleon can
collide many times, thus inducing the emission of other nucleons. A distorted
wave approximation with an optical (complex) nucleon nucleus potential
would remove all these events. However, if we want to evaluate the inclusive
$(e, e')$ cross section these events should be kept and one must sum over all
open final state channels \cite{NIMAI,CHI}. This is done explicitly in
~\cite{CHI}
and the result of it is a certain quenching of the quasielastic peak of the
simple $ph$ excitation calculation and a spreading of the strength to the
sides of it, or widening of the peak. The integrated strength over energies
is not much affected though.
The use of correlated wave functions, evaluated from realistic $NN$ forces and
incorporating the effects of the nucleon force in the nucleon pairs has
also been advocated in connection with the effects from final state
interaction \cite{FAN}. If one incorporates
two particle--two hole ($2p 2h$) components in the final excited
states one gets the spreading of the peaks as found in~\cite{CHI}. For the
purpose of the response function it is like exciting $ph$ components which
have a decay width into the $2p 2h$ channel. This gives rise to the
quenching of the peak and spreading of the strength. Another source of these
effects is the momentum dependence of the nucleon self-energy which is also
accounted for in the scheme and which sometimes is taken into account
approximately in terms of an effective mass (although our position on
evaluations which use this variable has been already
made clear). The approach of \cite{FAN} using an orthogonal correlated basis with
functions obtained with variational methods also incorporates RPA
correlations discussed in point 5) but does not account for the MEC discussed
in point 4), which rely upon the coupling of the photon to the pion or the
$\Delta$ excitations.
In our many body scheme we will account for this FSI by using nucleon
propagators properly dressed with a realistic self-energy in the medium,
which depends explicitly on the energy and the momentum \cite{PED}. This
self-energy leads to nucleon spectral functions in good agreement with other
accurate more microscopic approaches like the ones
in \cite{ANG,ART}. The self-energy
of \cite{PED}
has the proper energy--momentum dependence plus an imaginary part from
the coupling to the
$2p 2h$ components, hence it has the ingredients to account for the FSI
effects discussed in \cite{FAN} although using a different calculational
scheme.
Nuclear spectral functions with a language closer to the one we shall
follow
have also been used in \cite{CIO}. They include the interesting result,
which we will employ here, that keeping the width of the particle
states is important but one can disregard the width of the hole states.
\subsection{$\Delta$ excitation}
While many efforts have been devoted to the quasielastic peak very little
attention has been given to the $\Delta$ region and the dip region between
the quasielastic and $\Delta$ peaks. One exception is the work of
\cite{TJON}
which looks at the effects of MEC in the dip region.
A recent work \cite{ANGH} presents some experimental results and a theoretical
analysis of the $\Delta$ region based on the
$e N \rightarrow e' N \pi $ model of \cite{NOZ}. One interesting
conclusion of the work is that the data have a broader
energy spectrum than the theoretical one based on the properties of a
free $\Delta$ width. This suggest a larger $\Delta$ width
from $\Delta$ coupling to many body channels,
additional to the natural decay width with effects of Fermi motion included.
This is actually well known from pion physics \cite{WOL,THI}.
Our aim in this section is also to evaluate as accurately as possible the
response function in this region, for which we shall use results for the
$\Delta$ in a nuclear medium \cite{LOR} which have been tested thoroughly in a
variety of pionic reactions: elastic~\cite{CARMEN}, quasielastic,
charge exchange, absorption, etc... \cite{MAN}.
The discussion in this sections has served to expose relevant works done
in the literature and to present our approach in connection to the ideas
exposed in these works. We shall follow a microscopic many body description
of the inclusive $(e, e')$ process and will incorporate in our approach the
important effects discussed in this section. We are thus aiming
at an evaluation as
accurate as possible at the present moment. A small sacrifice is made
in order to allow one to treat microscopically in a tractable way all
the effects
discussed above: we use an infinite nuclear medium and obtain results for
finite nuclei using the local density approximation (LDA). This was also
used for the evaluation of the total cross section of real photons with
nuclei with good results \cite{CAR} and a mathematical derivation was made there
to justify the accuracy of the approximation. A direct comparison of results
with the LDA with finite nucleus results, using the same input was done
in \cite{EUG} for deep inelastic lepton scattering, showing that indeed the LDA
is an excellent approximation to deal with volume processes (i.e no
screening or absorption effects). In \cite{ANT}
a comparison of the results for finite nuclei with those of the Fermi gas
around the quasielastic
$ (e,e')$ peak is done,
showing that for some average Fermi momentum, the
results of the Fermi gas and those of finite nuclei are nearly identical,
proposing that choice as an even better prescription than the LDA. We
shall follow the LDA, already tested for real photons, and sufficiently
good for our purposes.
Another feature which is novel in our approach and rather important
in practical terms is the following: we shall use a method which
relates the $(e,e')$
cross section to the imaginary part of the virtual photon self-energy. This
is actually not new and has been used before~\cite{HOR,ALB}. The
novelty is that
by using properly Cutkosky rules one can relate the different sources of
imaginary part to different channels which contribute to the inclusive
$(e,e')$ cross section. In this way, we lay the grounds to evaluate from
the present input the cross sections for exclusive processes $(e, e'N)$,
$(e, e'NN)$, $(e, e' \pi) \, (e, e' N \pi)$
etc., which will be the subject of a forthcoming
paper \cite{JUA}. The results of the present paper and those of \cite{JUA}
are part
of a PhD thesis \cite{AMP} where additional details to those given here
can be found if desired.
\section{The $e N \rightarrow e' N \pi$ reaction}
\subsection{Formalism}
We shall follow the model used in ref.~\cite{CAR} for
$\gamma N \rightarrow \pi N$ at intermediate
energies generalizing it to virtual photons. In fact this model is
essentially the same as used for the
$e N \rightarrow e N \pi$ reaction in \cite{NOZ}. However, having
in mind the application to nuclei we make a reduction of the relativistic
amplitudes keeping terms up to
$O (\frac{P}{M_N})$, with,
$P$,$M_N$ the nucleon momentum and mass.
The neglect of the $O (\frac{P}{M_N})^2$ terms is justified
numerically as we shall see in the results, so we construct this non
relativistic amplitude for the $e N \rightarrow
e N \pi$ process ready to be used with ordinary
non relativistic nucleon wave functions.
\vspace*{0.2cm}
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=vertices.ps,width=6.9cm}}}
\vskip 0.1cm
\noindent
{\small {\bf Fig.3.1} Basic couplings of the virtual photon and the pion
to the nucleon and to the $N \Delta$ transition.}
\vspace*{0.3cm}
The basic couplings which we need are those depicted in fig.3.1 which
account for the coupling of the photon and the pion to the nucleon and
to the $N \Delta$ transition, plus the Kroll Ruderman term (KR), and the
coupling of the photon to the pion. The Kroll Ruderman term appears as a
gauge invariant term through minimal substitution when a pseudovector
$\pi N N$
coupling is used, as we do. The analytical expressions for these vertices
are given in appendix A. For convenience, from the KR term of the appendix
coming from gauge invariance, and which we call there seagull term, we
construct the KR term displayed in this chapter such that it contains all the
non vanishing pieces of the amplitude when $p_\pi \rightarrow
0$. We will come back to this
point later on.
\vspace*{1cm}
\subsection{The amplitudes for the $e N \rightarrow e N \pi$ model}
The Feynman diagrams which are considered in the model for
$\gamma N \rightarrow \pi N$ of \cite{CAR}
or in $\gamma^*
N \rightarrow \pi N$ of \cite{NOZ}
($\gamma^*$ will stand from now on for the virtual photon) are
depicted in fig. 3.2.
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=amplitudes.ps,width=11.5cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.3.2} Feynman diagrams considered for the
$\gamma^* N \rightarrow \pi N$ reaction. }
\vspace*{0.1cm}
They are the nucleon pole direct (NP) term (a), the nucleon
pole crossed (NPC) term (b), the pion pole (PP) term (c),
the delta pole direct (DP) term (d),
delta pole crossed (DPC) term (e) and Kroll Ruderman (KR) term (f).
The expressions for
these amplitudes are obtained by doing the nonrelativistic reduction of
the relativistic amplitudes of \cite{NOZ}. There is some small
contribution from
the negative energy intermediate nucleon states which is kept in
our expressions,
which as we mentioned neglect only terms of order
$O (\frac{P}{M_{N}})^2$. The corresponding
expressions are:
\begin{equation}
\begin{array}{rcl}
{\cal{M}}^{\mu}_{NP} & = &-e\frac{\displaystyle{f_{\pi N N}}}
{\displaystyle{m_{\pi}}}
B(N,N^{\prime} \pi)\frac{\displaystyle{1}}{\displaystyle{\sqrt{s}-M_{N}}}
F_{\pi}((q-k)^2)\times\\
& & \\
& &\!\!\!\!\!\!\!\!\!\!\times\left(
\begin{array}{c}
F_{1}^{N}(q^{2})\vec{\sigma}\vec{k}\\
\\
F_{1}^{N}(q^{2})\vec{\sigma}\vec{k}\left[ \frac{\displaystyle{
2\vec{p}+\vec{q}}}{\displaystyle{2M_{N}}}\right]
+i\frac{\displaystyle{\vec{\sigma}\vec{k}}}{\displaystyle{2M_{N}}}(
\vec{\sigma}\times\vec{q}\,)G^{N}_{M}(q^{2})
\end{array}
\right)
\end{array}
\end{equation}
\begin{equation}\label{eq:KR}
{\cal{M}}^{\mu}_{KR} =e\frac{\displaystyle{f_{\pi N N}}}
{\displaystyle{m_{\pi}}}
B(N,N^{\prime} \pi)F_{A}(q^{2})C^{\mu}(\pi,N^{\prime})F_{\pi}((q-k)^2)
\end{equation}
\noindent
where
$$
\begin{array}{lll}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!C^{\mu}(\pi^{-}p)=
\left(
\begin{array}{c}
\frac{\displaystyle{\vec{\sigma}\vec{q}}}
{\displaystyle{2M_{N}}}\\
\\
\left(
1+\frac{\displaystyle{q^{0}}}{\displaystyle{2M_{N}}}\right)\vec{\sigma}
\end{array}
\right)
&\,\,\,\, ; &
C^{\mu}(\pi^{0} n)=0
\end{array}
$$
$$
\begin{array}{lll}
C^{\mu}(\pi^{+}n)=
\left(
\begin{array}{c}
\frac{\displaystyle{\vec{\sigma}\vec{q}}}
{\displaystyle{2M_{N}}}\\
\\
-\left(
1-\frac{\displaystyle{q^{0}}}{\displaystyle{2M_{N}}}\right)\vec{\sigma}
\end{array}
\right)
& ; &
C^{\mu}(\pi^{0}p)=
\left(
\begin{array}{c}
\frac{\displaystyle{\vec{\sigma}\vec{q}}}
{\displaystyle{M_{N}}}\\
\frac{\displaystyle{q^{0}}}{\displaystyle{M_{N}}}\vec{\sigma}
\end{array}
\right)
\end{array}
$$
\begin{equation}
{\cal{M}}^{\mu}_{PP} =-e_{\pi}\frac{\displaystyle{f_{\pi N N}}}
{\displaystyle{m_{\pi}}}
B(N,N^{\prime} \pi)\vec{\sigma}(\vec{k}-\vec{q}\,)F_{\gamma \pi \pi}(q^{2})
\frac{\displaystyle{(2k-q)^{\mu}}}{\displaystyle{(k-q)^{2}-m_{\pi}^{2}}}
F_{\pi}((q-k)^2)
\end{equation}
\begin{equation}
\begin{array}{lll}
{\cal{M}}^{\mu}_{NPC} = -e\frac{\displaystyle{f_{\pi N N}}}
{\displaystyle{m_{\pi}}}
B(N,N^{\prime} \pi)\frac{\displaystyle{1}}{\displaystyle{p^{0}-k^{0}-
E(\vec{p}-\vec{k})}}F_{\pi}((q-k)^2)
\times & &\\
& &
\\
\times
\left(
\begin{array}{c}
F_{1}^{N^{\prime}}(q^{2})\vec{\sigma}\vec{k}\\
F_{1}^{N^{\prime}}(q^{2})\left\{
\frac{\displaystyle{2\vec{p}+\vec{q}-2\vec{k}}}{\displaystyle{
2M_{N}}}\right\}\vec{\sigma}\vec{k}+G_{M}^{N^{\prime}}(q^{2})
i\frac{\displaystyle{
(\vec{\sigma}\times\vec{q})\vec{\sigma}\vec{k}}}{\displaystyle{2M_{N}}}
\end{array}\right)& &
\end{array}
\end{equation}
\noindent
with
$$
\begin{array}{l}
B(n,n \pi^0)=-1\\
B(n,p \pi^-)=\sqrt{2}\\
B(p,p \pi^0)=1\\
B(p,n \pi^+)=\sqrt{2}
\end{array}
$$
\noindent
If we consider,
$$
\begin{array}{l}
I(\pi^{0})=I_{c}(\pi^{0})=2/3\\
I(\pi^{+})=-I_{c}(\pi^{+})=-\sqrt{2}/3\\
I(\pi^{-})=-I_{c}(\pi^{-})=\sqrt{2}/3
\end{array}
$$
\noindent
we get
\begin{equation}
\begin{array}{rl}
{\cal{M}}^{\mu}_{DP}=&-if^{*}\frac{\displaystyle{f_{\gamma}(q^{2})}}
{\displaystyle{m_{\pi}^{2}}}\frac{\displaystyle{\vec{S}\left[
\vec{k}-\frac{k^{0}}{\sqrt{s}}\vec{p}_{\Delta}\right]}}
{\displaystyle{\sqrt{s}-M_{\Delta}+i\frac{\Gamma (s)}{2}}}
\Frac{\sqrt{s}}{M_{\Delta}}I(\pi)\times
\\
& \\
& \times
\left(
\begin{array}{c}
\Frac{\vec{p}_{\Delta}}{\sqrt{s}}\left(\vec{S}^{\dagger}
\times\vec{q}\right)\\
\\
\Frac{p^{0}_{\Delta}}{\sqrt{s}}\left[
\vec{S}^{\dagger}\times\left(\vec{q}-\Frac{q^{0}}{p^{0}_{\Delta}}\vec{p}_{\Delta}
\right)\right]
\end{array}\right)
\end{array}
\end{equation}
\begin{equation}
\begin{array}{rl}
{\cal{M}}^{\mu}_{DPC}=&-if^{*}\frac{\displaystyle{f_{\gamma}(q^{2})}}
{\displaystyle{m_{\pi}^{2}}}\left(
\frac{\displaystyle{M_{N}}}{\displaystyle{M_{\Delta}}}\right)
\frac{\displaystyle{(E_{\Delta}+M_{\Delta})}}{\displaystyle{
(p^{2}_{\Delta}-M^{2}_{\Delta})}}I_{c}(\pi)\times
\\
& \\
& \times
\left(
\begin{array}{c}
0\\
(\vec{S}\times\vec{q}\,)(\vec{S}^{\dagger}\vec{k}\,)-\vec{B}
\end{array}
\right)
\end{array}
\end{equation}
\noindent
with
$$
\vec{B}=\frac{1}{3}\left\{ i \frac{\displaystyle{(k^{0}+a)}}{\displaystyle{
E_{\Delta}+M_{\Delta}}}(\vec{q}^{\,2}\vec{\sigma}-({\vec{q}}\,\vec{\sigma})
\vec{q}\,)+\frac{\displaystyle{(k^{0}-a)}}{\displaystyle{2M_N}}
[({\vec{q}\,}\times\vec{p}\,)-i(\vec{q}\,\vec{p}\,)\vec{\sigma}
+i\vec{p}\,(\vec{\sigma}\vec{q}\,)]\right\}
$$
$$
a=(p_{\Delta}k)\frac{1}{M_{\Delta}}
$$
\newpage
\noindent
where $\sqrt{s}$ is the invariant energy of the system virtual photon-initial
nucleon, $e (e > 0)$ is the electron charge, $m_\pi=139.5$ MeV is the
pion mass,
$M_\Delta = 1238$ MeV, the $\Delta$ mass \cite{CAR}
and the free decay width of the $\Delta$ is given by
\begin{equation}
\Gamma ({s})=
\frac{\displaystyle{1}}{\displaystyle{6\pi}}
\left(\frac{\displaystyle{f^*}}{\displaystyle{m_{\pi}}}\right)^2
\frac{\displaystyle{M_N}}{\displaystyle{\sqrt{s}}}
|\vec{k}_{cm}|^3\Theta(\sqrt{s}-M_{N}-m_{\pi})
\end{equation}
On the other hand, $f_{\pi NN},
f_\gamma, f^*, F_1, G_M, F_A, F_{\gamma \pi \pi}, F_{\pi}$,
are coupling constants and form factors which
we show in the appendix, as well as $\vec{S}$, the spin transition operator from
spin 3/2 to 1/2.
The expressions given keep the Lorentz covariance up to terms $O (\frac{
P}{m_{N}})^2$. The
small $\Delta$ crossed term of eq.(6) holds strictly in the frame where
the outgoing nucleon is at rest.
As mentioned before, in the KR term of eq.(~\ref{eq:KR}) we have
included corrections
in the zero component which come from the zero component of the NP
and NCP terms when using the vertex of eq.(~\ref{eq:piNN})
(this means from positive
energy intermediate states). On the other hand the $q^0/2M_N$
terms in the spatial
components of the KR term come from the
intermediate
negative energy
components of the NP and NCP relativistic amplitudes. This trick serves
us to concentrate on the KR term all the contributions which do not vanish
when the pion momentum goes to zero.
\subsection{Gauge invariance and form factors}
Gauge invariance implies that $q_\mu
{\cal{M}}^\mu = 0 $ where ${\cal{M}}^\mu = \sum_{i} {\cal{M}}^{\mu}_{i}$.
From the expressions given above we can
see that the delta terms are gauge invariant by themselves. As for the rest of
the amplitudes they form a block of gauge invariance terms in the absence of
form factors. However, we must impose some restrictions in order to keep
gauge invariance when the form factors are included. If we consider the
$\gamma^* n \rightarrow p \pi^-$
amplitude we see
\begin{equation}
\begin{array}{c}
q_{\mu}{\cal{M}}^{\mu}\Frac{1}{e\frac{f_{\pi NN}}{m_{\pi}}B(N,N^{\prime}\pi)}= \
\\
=-\Frac{1}{p^{0}-k^{0}-E(\vec{p}-\vec{k})}
\left\{\left(
q_{0}F_{1}^{p}-\frac{\displaystyle{(2\vec{p}\,\vec{q}-
2\vec{k}\vec{q}+{\vec{q}\,}^{2})}}
{\displaystyle{2M_{N}}}F_{1}^{p}\right)\vec{\sigma}\vec{k}\right\}\\
\\
-\vec{\sigma}\vec{q}F_{A}-F_{\gamma\pi\pi}\vec{\sigma}(\vec{k}-\vec{q})=0
\end{array}
\end{equation}
\noindent
which together imply
\begin{equation}
\Rightarrow \left[F_{1}^{p}(q^{2})=F_{\gamma\pi\pi}(q^{2})=
F_{A}(q^{2})\right]
\end{equation}
Gauge invariance in other isospin channels does not require extra
relationships. In the Appendix we can see that $\Lambda
\simeq M_A \simeq \sqrt{2} p_\pi$ and hence eq. (9)
is fulfilled to a good degree of approximation. However, in order to keep
strict gauge invariance we take only one form factor for all, which we
choose to be $F_1^p$ of eq.(~\ref{eq:f1p}).
We have checked that by taking any of the other form
factors the changes induced in the cross sections are much smaller than
the experimental errors (see fig.3.6),
so we are rather safe with any of these choices.
On the other hand, since in the pion pole term we have a form factor
corresponding to the coupling $\pi NN$ with a virtual pion,
$F_{\pi} ((q - k)^{2})$, we have included
this form factor in the amplitudes of this block (NP,NCP,PP,KR) in order to
preserve gauge invariance.
\subsection{Unitarity}
Another refinement introduced in the model is unitarity. Watson's theorem
implies that the phase of the $\pi N \to \pi N$
and $ \pi N \to \gamma N $ amplitudes in each term of the
partial wave decomposition must be the same. Such as our model stands, we
have a $\Delta$ term which by itself satisfies the theorem if we assume
the $\pi N \rightarrow \pi N$
amplitude dominated by the $\Delta$, as it is the case. However, in
the $\gamma N \rightarrow \pi N$
amplitude we have a sizeable background that in our model is real
and the sum of the terms does not satisfy Watson's theorem. Although this
violation of unitary does not result in important numerical changes in the
cross section, we nevertheless unitarize the model as was done for real
photons \cite{CAR}. We follow the procedure of \cite{OLS} introducing
a small phase
$\phi (\sqrt{s}, q^2)$ which corrects the $\Delta$ term, where
$\sqrt{s}$ is the invariant energy
of the virtual photon-initial nucleon system and $q^2$ the four--momentum
squared of the virtual photon. By means of an iterative method we find
$\phi (\sqrt{s}, q^2)$ such that
\begin{equation}
Im\left[ (T_{\Delta}(q^2)e^{i\phi (\sqrt{s}\,,\,q^2)}
+ T_{B}(q^2))^{(3/2,3/2)}e^{-i\delta_{(3/2,3/2)}(q^2) }\right]=0
\end{equation}
\noindent
where $T_\Delta(q^2)$ represents the $\Delta$ pole direct term amplitude,
$T_B(q^2)$ is the
contribution of the rest of the terms to the (3/2, 3/2)
channel (see details in
~\cite{CAR} for the projection of these terms in the (3/2, 3/2) channel)
and $\delta_{(3/2,3/2)}(q^2)$ are the $\pi N$ phase shifts in the 3/2,3/2
spin-isospin channel.
\vspace*{0.4cm}
\subsection{Cross sections for the $e N \rightarrow e N \pi$ process}
We follow here the steps and nomenclature of ref.~\cite{AML}. In fig.3.3 we
show diagrammatically the process with the different variables which we use,
$k_e, k'_e, q, p, p'$
and $k$ representing the momenta of the
incoming electron, outgoing electron, virtual photon,
initial nucleon, final nucleon and
pion respectively.
The unpolarized cross section is given by \cite{AML}:
\begin{equation}
\frac{\displaystyle{d^{2}\sigma}}{\displaystyle{d{\Omega}^{\prime}_{e}
d{E}^{\prime}_{e}}}=\frac{\displaystyle{\alpha^{2}}}{\displaystyle{q^{4}}}
\frac{\displaystyle{|\vec{k}_{e}^{\,\prime}|}}{\displaystyle{|\vec{k}_{e}}|}
L_{\mu\nu}(e,e^{\prime}) \bar{W}^{\mu\nu}_{e.m.}
\end{equation}
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=elementa.eps,width=8cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.3.3} Feynman diagram for the
$e N \rightarrow e ^{\prime} N \pi$ process. }
\vspace*{0.8cm}
\noindent
where $\alpha = 1/137.036$ is the fine structure constant $(e^2/4 \pi)$
and $L_{\mu \nu}$ the leptonic
tensor defined as
\begin{equation}
L_{\mu\nu}(e,e^{\prime})=
2({k_{e}}_{\mu}^{\prime}{k_{e}}_{\nu}+{k_{e}}_{\nu}^{\prime}{k_{e}}_{\mu}+
\frac{\displaystyle{q^2}}{\displaystyle{2}}g_{\mu\nu})
\end{equation}
The hadronic tensor is given by
\begin{equation}
\begin{array}{ll}
\bar{W}^{\mu \nu}_{em}=& \overline{\displaystyle \sum_{{}_{spin}}}
\displaystyle{\int}
\frac{\displaystyle{d^{3}p_{N}^{\prime}}}{\displaystyle{(2\pi)^{3}}}
\frac{\displaystyle{M_N}}{\displaystyle{E^{\prime}}}\int
\frac{\displaystyle{d^{3}k }}{\displaystyle{(2\pi)^{3}}}
\frac{\displaystyle{1}}{\displaystyle{2E_{\pi}}}(2\pi)^{3}\times
\\
&\\
&\!\!\!\!\!\!\!\!\!\!\!\!
\times \delta^{4}(p_{N}^{\prime}\!+\!k\!-\!p_{N}\!-\!q)
<N^{\prime} \pi | j^{\mu}_{em}|N>^{*} <N^{\prime}
\pi | j_{em}^{\nu}
| N>
\end{array}
\end{equation}
\noindent
with $j^{\mu}_{em}$ the $\gamma^* N$ to $N^{\prime} \pi $ amplitude defined
in sect. 3.2.
We can separate from there the angular dependence of the pion and get
\begin{equation}
\frac{\displaystyle{d^{3}\sigma}}{\displaystyle{d{\Omega}^{\prime}_{e}
dE^{\prime}_{e}d\Omega_{\pi}}}=
\frac{\displaystyle{\alpha^{2}}}{\displaystyle{q^{4}}}
\frac{\displaystyle{|\vec{k}_{e}^{\,\prime}|}}{\displaystyle{|\vec{k}_{e}|}}
L_{\mu\nu}(e,e^{\prime}) {W}^{\mu\nu}_{e.m.}(N)
\end{equation}
\noindent
where now
\begin{equation}
\begin{array}{ll}
{W}^{\mu \nu}_{em}=&\overline{\displaystyle \sum_{{}_{spin}}}
\displaystyle{\int}
\frac{\displaystyle{d^{3}p_{N}^{\prime}}}{\displaystyle{(2\pi)^{3}}}
\frac{\displaystyle{M_N}}{\displaystyle{E^{\prime}}}\int
\frac{\displaystyle{d k\,{\vec{k}^{2}}}}
{\displaystyle{(2\pi)^{3}2E_{\pi}}}
(2\pi)^{3}\times\\
&\\
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \times \delta^{4}
(p_{N}^{\prime}\!+\!k\!-\!p_{N}\!-q)
<N^{\prime} \pi | j^{\mu}_{em}|N>^{*}
<N^{\prime} \pi | j_{em}^{\nu}
| N>
\end{array}
\end{equation}
Now by taking $\vec{q}$ along the $z$ direction, using gauge invariance and the
explicit expressions for $L_{\mu \nu}$
we can write, following exactly the same steps
as in \cite{AML,NOZ}
\newpage
\begin{equation}
\begin{array}{ll}
\frac{\displaystyle{d^{5}\sigma}}{\displaystyle{d\Omega_{e}^{\prime}
dE_{e}^{\prime} d\Omega^{*}_{\pi}}}= & \Gamma \left\{
\frac{\displaystyle{d\sigma_{T}}}{\displaystyle{d\Omega^{*}_{\pi}}}
+{\cal \epsilon}
\frac{\displaystyle{d\sigma_{L}}}{\displaystyle{d\Omega^{*}_{\pi}}}
+{\cal \epsilon}
\frac{\displaystyle{d\sigma_{p}}}{\displaystyle{d\Omega^{*}_{\pi}}}
cos2\Phi^{*}_{\pi}+ \right.\\
&\\
& \left. +\sqrt{\displaystyle{2{\cal \epsilon}(1+{\cal \epsilon})}}
\frac{\displaystyle{d\sigma_{I}}}{\displaystyle{d\Omega^{*}_{\pi}}}
cos\Phi^{*}_{\pi}\right\}
\end{array}
\end{equation}
\noindent
where
$$
\epsilon=\left[ 1- 2\frac{\displaystyle{|\vec{q}\,|^{2}}}{\displaystyle{
q^{2}}}tg^{2}\left(\frac{\displaystyle{\theta_e}}{\displaystyle{2}}
\right)\right]^{-1}
$$
\begin{equation}
\begin{array}{l}
\Gamma=\frac{\displaystyle{\alpha}}{\displaystyle{2\pi^{2}}}
\frac{\displaystyle{|\vec{k}_{e}^{\,\prime} |}}
{\displaystyle{|\vec{k}_{e} |}}
\left[-\frac{\displaystyle{1}}{\displaystyle{q^{2}}}\right]
\frac{\displaystyle{k_{\gamma}}}{\displaystyle{1-{\cal \epsilon}}}\\
\\
k_{\gamma}=\frac{\displaystyle{s-M_{N}^{2}}}{\displaystyle{2M_{N}}}
\end{array}
\end{equation}
\noindent
and
\begin{equation}
\begin{array}{l}
\frac{\displaystyle{d\sigma_{T}}}{\displaystyle{d\Omega^{*}_{\pi}}}
=\frac{\displaystyle{e^{2}}}{\displaystyle{64\pi^{2}k_{\gamma}}}
\frac{\displaystyle{M_{N}}}{\displaystyle{\sqrt{s}}}
|\vec{k}_{CM}|\left[(J^{xx}+J^{yy})\right]^{CM}_{\Phi^{*}_{\pi}=0}\\
\\
\frac{\displaystyle{d\sigma_{L}}}{\displaystyle{d\Omega^{*}_{\pi}}}
=\frac{\displaystyle{-q^{2}}}{\displaystyle{(q^{0}_{cm})^{2}}}
\frac{\displaystyle{e^{2}}}{\displaystyle{32\pi^{2}k_{\gamma}}}
\frac{\displaystyle{M_{N}}}{\displaystyle{\sqrt{s}}}|\vec{k}_{CM}|
\left[ J^{zz}_{CM}\right]_{\Phi^{*}_{\pi}=0}
\\
\\
\frac{\displaystyle{d\sigma_{p}}}{\displaystyle{d\Omega^{*}_{\pi}}}
=\frac{\displaystyle{e^{2}}}{\displaystyle{64\pi^{2}k_{\gamma}}}
\frac{\displaystyle{M_{N}}}{\displaystyle{\sqrt{s}}}|\vec{k}_{CM}|
\left[(J^{xx}-J^{yy})\right]^{CM}_{\Phi^{*}_{\pi}=0}
\\
\\
\frac{\displaystyle{d\sigma_{I}}}{\displaystyle{d\Omega^{*}_{\pi}}}=
-\sqrt{\displaystyle{\frac{\displaystyle{-q^{2}}}
{\displaystyle{(q^{0}_{cm})^{2}}}}}
\frac{\displaystyle{e^{2}}}{\displaystyle{64\pi^{2}k_{\gamma}}}
\frac{\displaystyle{M_{N}}}{\displaystyle{\sqrt{s}}}
|\vec{k}_{CM}|\left[(J^{zx}+J^{xz})\right]^{CM}_{\Phi^{*}_{\pi}=0}
\end{array}
\end{equation}
\noindent
where the variables of the electron are in the lab frame while those of
the pion are in the $\gamma^* N$ CM frame. $J^{\mu \nu}$
in the former expressions is given
by
\begin{equation}
J^{\mu\nu}=Tr(j^{\dagger\mu}_
{em}j^{\nu}_{em})
\end{equation}
\noindent
The angular variables $\theta_e, \theta^{*}_{\pi}, \Phi^{*}_\pi$
are depicted in fig. 3.4. The variables $k_e, k'_e$ determine the
$(e, e')$ reaction plane and $\vec{q}$, the virtual photon
momentum, determines the $z$ direction. The $\vec{q}$ direction and the pion
momentum $\vec{k}$ determine the $\pi N$ reaction plane, and the angle
between this plane and the ($e, e')$ plane is the
angle $\Phi^{*}_\pi $.
The normalization of $d \sigma_i/ d \Omega^*_\pi$
is chosen in such a way that in the limit
of real photons $d \sigma_i/ d \Omega^*_\pi$ coincides with the unpolarized
cross section of $\gamma N \rightarrow \pi N$
with real photons. In this case the variable $k_\gamma$
becomes the lab momentum
of the real photon. The variables $\sigma_T, \sigma_L,
\sigma_p, \sigma_I$ are the so called transverse,
longitudinal, polarization and interference cross sections, respectively.
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=planoes.ps,width=11cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.3.4} Angular variables
$\theta_e, \theta^{*}_{\pi}, \Phi^{*}_\pi$ in the
$e N \rightarrow e^{\prime} N \pi$ process.}
\vspace*{0.1cm}
\vspace*{1cm}
\subsection{Results for $e N \rightarrow e N \pi$}
In fig. 3.5 we show the $W=\sqrt{s}$ dependence of the longitudinal (lower line)
and
transverse (upper line) cross
sections summing over the final charge of the pion and integrating over
$\Omega_\pi^*$.
The peak position, the strength and the shape of our results are in good
agreement with the data of \cite{BAE}. In general terms our results are very
close to those obtained in \cite{NOZ} with a similar good agreement with
experiment as found there.
In fig. 3.6, and as an illustration of our discussion about gauge invariance
in sect. 3.3,
we show the differences between the results
for the inclusive cross sections
$\sigma_T$ and $\sigma_L$, obtained:
by using $F_A$ and taking
(as gauge invariance forces)
$F_{1}^{p}=F_{\gamma \pi \pi}=F_A$ (dotted lines);
by using $F_{\gamma \pi \pi}$ and taking
$F_{1}^{p}=F_A=F_{\gamma \pi \pi}$ (dash lines);
by using $F_{1}^{p}$ and taking
$F_A=F_{\gamma \pi \pi}=F_{1}^{p}$ (full lines) and,
finally, by using $F_A$, $F_{\gamma \pi \pi}$ and $F_{1}^{p}$
(dash-dotted lines). As one can see, the differences are negligible,
relative to present experimental errors.
In fig. 3.7 we show the results for
$d \sigma_I / d \Omega^{*}_{\pi} / (\sin \theta_\pi^{*} \sqrt{2})$
in the channel $e p \rightarrow e n \pi^+$ and compare
them with the experimental data of \cite{BRE}. We can see that the agreement
is reasonably good and so is the case for
$ d \sigma_P / d \Omega_{\pi}^{*} / \sin^2\theta_\pi^{*} $
shown in fig. 3.8, where the
data are again from \cite{BRE}.
In fig. 3.9 we show the $\Phi_{\pi}^{*}$ dependence of the cross section
in the channel
$e p \rightarrow e n \pi^+$
for three
different kinematics. The data are from~\cite{BRE} and we see again a
reasonable agreement with experiment.
Finally in fig. 3.10 we show the $\Phi_\pi^{*}$ dependence for another channel, the
$e p \rightarrow e p \pi^0$,
in order to show a case where the agreement with the data, in this
case from ref.~\cite{MIST}, is not as good as in general terms. Given the fact
that the angle $\Phi_\pi^{*}$ will be integrated in the $(e,e')$
reactions in nuclei,
such punctual discrepancies will not matter in our study of the
nuclear processes.
\newpage
\vspace*{2.cm}
\centerline{\protect\hbox{\psfig{file=a1.ps,width=11.cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.3.5}
$W=\sqrt{s}$ dependence of the longitudinal (lower line)
and
transverse (upper line) cross
sections summing over the final charge of the pion and integrating over
$\Omega_\pi^*$. Experimental data from \cite{BAE}.
}
\newpage
\vspace*{2.cm}
\centerline{\protect\hbox{\psfig{file=nogauge2.ps,width=11.cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.3.6}
Inclusive cross sections
$\sigma_T$ and $\sigma_L$ obtained:
by using $F_A$ and taking
$F_{1}^{p}=F_{\gamma \pi \pi}=F_A$ (dotted lines);
by using $F_{\gamma \pi \pi}$ and taking
$F_{1}^{p}=F_A=F_{\gamma \pi \pi}$ (dash lines);
by using $F_{1}^{p}$ and taking
$F_A=F_{\gamma \pi \pi}=F_{1}^{p}$ (full lines) and,
finally, by using $F_A$, $F_{\gamma \pi \pi}$ and $F_{1}^{p}$
(dash-dotted lines). Experimental data from \cite{BAE}.}
\newpage
\vspace*{0.8cm}
\centerline{\protect\hbox{\psfig{file=sigmai.ps,width=12.cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.3.7}
Calculation of
$d \sigma_I / d \Omega^{*}_{\pi} / (\sin \theta_\pi^{*} \sqrt{2})$
in the $e p \rightarrow e n \pi^+$ channel . Experimental data from
\cite{BRE}.}
\vspace*{0.8cm}
\centerline{\protect\hbox{\psfig{file=sigmap.ps,width=10.5cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.3.8}
Calculation of $ d \sigma_P / d \Omega_{\pi}^{*} / \sin^2\theta_\pi^{*} $
in the $e p \rightarrow e n \pi^+$ channel. Experimental data from
\cite{BRE}.}
\begin{minipage}[c]{14cm}
\begin{minipage}[c]{6.8cm}
\centerline{\protect\hbox{\psfig{file=a.ps,width=6.2cm}}}
\end{minipage}
\begin{minipage}[c]{6.8cm}
\centerline{\protect\hbox{\psfig{file=b.ps,width=6.2cm}}}
\end{minipage}
\end{minipage}
\vspace*{0.6cm}
\centerline{\protect\hbox{\psfig{file=c.ps,width=6.5cm}}}
\vspace*{0.3cm}
\noindent
{\small {\bf Fig.3.9}
Calculation of the $\Phi^{*}_{\pi}$ dependence of the
cross section in the $e p \rightarrow e n \pi^+$ channel.
Experimental data from \cite{BRE}.}
\newpage
\vskip 0.5cm
\centerline{\protect\hbox{\psfig{file=pi0.ps,width=12.5cm}}}
\noindent
{\small {\bf Fig.3.10} Calculation of the $\Phi^{*}_{\pi}$ dependence of the
cross section in the
$e p \rightarrow e n \pi^0$ channel.
Experimental data from \cite{MIST}. }
\vspace*{0.1cm}
In the next sections we shall use this model to evaluate the pion
production contribution to the $(e,e')$ cross section, as well as the exchange
currents which contribute to the $2 N$ emission channel.
\section{The $(e, e')$ reaction in nuclei.}
\subsection{Formalism}
We want to use a covariant many body formalism to evaluate
the $(e, e')$ cross section. For this purpose we evaluate the electron
self-energy for an electron moving in infinite nuclear matter.
Diagrammatically this is depicted in fig. 4.1
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=autoincl.eps,width=6cm}}}
\vskip 0.2cm
{\small {\bf Fig.4.1} Diagrammatic representation of the electron self-energy
in nuclear matter.}
\vspace*{0.1cm}
The electron disappears
from the elastic flux, by inducing $1p 1h, 2p 2h...$ excitations or creating
pions, etc., at a rate given by
\begin{equation}
\Gamma(k)=-2\Frac{m_e}{E_e}Im\Sigma.
\end{equation}
\noindent
where $Im \Sigma$
is the imaginary part of the electron self-energy. This latter
magnitude can be readily evaluated from the diagram of fig. 4.1 and we
find:
\begin{equation}
\Sigma_{r}(k)=ie^{2}{\displaystyle \int \frac{\displaystyle{d^{4}q}}
{\displaystyle{(2\pi)^{4}}}\bar{u}_{r}(k)\gamma_{\mu}\frac{
\displaystyle{(\slash \!\!\! k^{\prime}+{m}_{e})}}
{\displaystyle{k^{\prime\, 2}
-{m}_{e}^{2}+
i\epsilon}}\gamma_{\nu}u_{r}(k)\frac{\displaystyle{\Pi^{\mu\nu}_{\gamma}(q)}}
{\displaystyle{(q^{2}+i\epsilon)^2}}}
\end{equation}
\noindent
where $\Pi^{\mu \nu}_\gamma$
is the virtual photon self-energy. Eq.(21)
displays explicitly the electron propagator (fraction after ${\gamma}_{\mu}$)
and the photon propagator $(q^2+i \epsilon)^{-1}$ which appears twice.
By averaging over the spin
of the electron, $r$, we find
\begin{equation}
\Sigma(k)=\frac{\displaystyle{ie^{2}}}{\displaystyle{2m_{e}}}\displaystyle{
\int\frac{\displaystyle{d^{4}q}}
{\displaystyle{(2\pi)^{4}}} \frac{\displaystyle{L_{\mu\nu}
\Pi_{\gamma}^{\mu\nu}(q)}}{\displaystyle{q^{4}}}\frac{\displaystyle{
1}}{\displaystyle{(k^{\prime\, 2}-{m}_{e}^{2}+i\epsilon)}}
}
\end{equation}
\noindent
and since we are interested in the imaginary part of $\Sigma$ we can obtain
it by following the prescription of the Cutkosky's rules. In this case we
cut with a straight horizontal line the intermediate $e'$ state and those
implied by the photon polarization (shaded region). Those states are then
placed on shell by taking the imaginary part of the propagator,
self-energy, etc. Technically the rules to obtain $Im \Sigma$ reduce to making
the substitutions:
\begin{equation}
\begin{array}{l}
\Sigma(k)\rightarrow 2iIm\Sigma(k)\Theta(k^{0})\\
\\
\Xi (k^{\prime})\rightarrow 2i Im\Xi(k^{\prime})\Theta({k^{\prime}}^{0})\\
\\
\Pi^{\mu\nu}(q)\rightarrow 2i Im\Pi^{\mu\nu}(q)\Theta(q^{0})
\end{array}
\end{equation}
\noindent
where
\begin{equation}
\Xi(k^{\prime})=\frac{\displaystyle{1}}
{\displaystyle{k^{\prime\, 2}-{m}_{e}^{2}+i\epsilon}}
\end{equation}
\noindent
and $\Theta$ is the Heaviside, or step, function.
By proceeding according to these rules we obtain
\begin{equation}
Im\Sigma(k)=\frac{\displaystyle{2\pi\alpha}}{\displaystyle{m_{e}}}
{\displaystyle \int\frac{\displaystyle{d^{3}q}}{\displaystyle{
(2\pi)^{3}}}\left(Im\Pi_{\gamma}^{\mu\nu}L_{\mu\nu}(k,k^{\prime})\right)
\frac{\displaystyle{1}}{\displaystyle{q^{4}}}\frac{\displaystyle{1}}{
\displaystyle{2E_{e}(\vec{k}^{\prime})}}
}\Theta(q^{0})
\end{equation}
The relationship of $Im \Sigma$ to the $(e, e')$ cross section is easy:
$\Gamma dt dS$ provides a probability times a differential of area, which is a
contribution to a cross section. Hence we find
\begin{equation}
d\sigma=\Gamma (k)dt dS=
-\frac{\displaystyle{2m}}{\displaystyle{E_e}}
Im\Sigma dl\,dS=
-\frac{\displaystyle{2m}}{\displaystyle{|\vec{k}\,|}}
Im\Sigma d^{3}r
\end{equation}
\noindent
and hence the nuclear cross section is given by
\begin{equation}
\sigma= -{\displaystyle \int d^{3}r \frac{\displaystyle{2m}}{\displaystyle{
|\vec{k}\,|}}Im\Sigma(k, \rho(\vec{r}\,))
}
\end{equation}
\noindent
where we have substituted $\Sigma$ as a function of the nuclear
density at each
point of the nucleus and integrate over the whole nuclear
volume. Eq. (27)
assumes the local density approximation, which, as shown in \cite{CAR},
is an excellent approximation for volume processes like here, hence we are
neglecting the electron screening and using implicitly plane waves for the
electrons (corrections to account for the small distortion are usually
done in the experimental analysis of the data, see \cite{GUE} ).
Coming back to eq. (25) we find then
\begin{equation}
\frac{\displaystyle{d^{2}\sigma}}{\displaystyle{d\Omega^{\prime}_e
dE^{\prime}_e}} =
-\frac{\displaystyle{\alpha}}{\displaystyle{q^{4}}}\frac{
\displaystyle{|\vec{k}^{\,\prime}|}}{\displaystyle{|\vec{k}|}}\frac{
\displaystyle{1}}{\displaystyle{(2\pi)^{2}}}{\displaystyle\int d^{3}r
\left(Im\Pi_{\gamma}^{\mu\nu}L_{\mu\nu}\right)
}
\end{equation}
\noindent
which gives us the $(e,e')$ differential cross section in terms of the
imaginary part of the photon self-energy.
If one compares eq. (28) with the general expression for the inclusive
$(e, e')$ cross section \cite{MUL,GIO} (see also eq. (11))
\begin{equation}
\frac{\displaystyle{d^{2}\sigma}}{\displaystyle{d\Omega^{\prime}_e
dE^{\prime}_e}}
=\frac{\displaystyle{\alpha^{2}}}{\displaystyle{q^{4}}}\frac{\displaystyle{
|\vec{k}^{\,\prime}|}}{\displaystyle{|\vec{k}|}}L^{\mu\nu}W_{\mu\nu}
\end{equation}
\noindent
we find
\begin{equation}
W^{\mu\nu}=-\frac{\displaystyle{1}}{\displaystyle{\pi e^{2}}}
{\displaystyle \int d^{3}r
\frac{\displaystyle{1}}{\displaystyle{2}}
\left(Im\Pi^{\mu\nu}+Im\Pi^{\nu\mu}\right)
}
\end{equation}
Once again, by choosing $\vec{q}$ in the $z$ direction and using gauge
invariance one
can write the cross section in terms of the longitudinal and transverse
structure functions $W_L, W_T$ as
\begin{equation}
\frac{\displaystyle{d^{2}\sigma}}{\displaystyle{d\Omega^{\prime}_e
dE^{\prime}_e}}=
\left(\frac{\displaystyle{d\sigma}}{\displaystyle{d\Omega}}\right)_{Mott}
\left(-\frac{\displaystyle{q^{2}}}{\displaystyle{|\vec{q}\,|^{2}}}\right)
\left\{W_{L}(\omega ,|\vec{q}\,|)+
\frac{\displaystyle{W_{T}(\omega,|\vec{q}\,|)}}
{\displaystyle{\epsilon}}\right\}
\end{equation}
\noindent
where
\begin{equation}
\begin{array}{c}
q^{2}=\omega^{2}-|\vec{q}\,|^{2}\\
\\
\left.\frac{\displaystyle{d\sigma}}{\displaystyle{d\Omega}}\right|_{Mott}=
\frac{\displaystyle{\alpha^{2}\cos_{e}^{2}(\theta /2)}}{\displaystyle{
4E_{e}^{2}\sin_{e}^{4}(\theta /2)}}\\
\\
\end{array}
\end{equation}
\noindent
and
\begin{equation}
\begin{array}{l}
W_{L}=-\frac{\displaystyle{q^{2}}}{\displaystyle{\omega^{2}}}W^{zz}
=-\frac{\displaystyle{q^{2}}}{\displaystyle{|\vec{q}\,|^{2}}}W^{00}\\
\\
W_{T}=W^{xx}
\end{array}
\end{equation}
Hence using eq. (30) we can write $W_L$ and
$W_T$ in terms of the photon self-energy as
\begin{equation}
\begin{array}{l}
W_{L}=\frac{\displaystyle{q^{2}}}{\displaystyle{\pi e^{2} |\vec{q}\,|^{2}}}
{\displaystyle \int d^{3}r Im \Pi^{00}(q,\rho(\vec{r}\,))
}\\
\\
W_{T}=-\frac{\displaystyle{1}}{\displaystyle{\pi e^{2}}}{\displaystyle
\int d^{3}r Im \Pi^{xx}(q,\rho(\vec{r}\,))}
\end{array}
\end{equation}
\noindent
where we see that we only need the components $\Pi^{00}$ and $\Pi^{xx}$.
\subsection{The virtual photon self-energy in pion production}
We must construct a self-energy diagram for the photon which contains pion
production in the intermediate states. This is readily accomplished by
taking any generic diagram of the
$\gamma^* N \rightarrow \pi N$ amplitude of fig. 3.2 and folding
it with itself. One gets then the diagram of fig. 4.2 where the circle
stands for any of the 6 terms of the elementary model for
$\gamma^* N \rightarrow \pi N$.
The
lines going up and down in fig. 4.2. follow the standard many body
nomenclature and stand for particle and hole states respectively.
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=burbujas8.ps,width=3.cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.4.2} Photon self-energy obtained by folding the
$\gamma^* N \rightarrow \pi N$ amplitude.}
\vspace*{0.1cm}
The photon
self-energy corresponding to this diagram (actually 36 diagrams) is
readily evaluated and gives
\begin{equation}
\begin{array}{ll}
\Pi_{NN^{\prime}}^{\mu\nu}(q)=i{\displaystyle \int \frac{\displaystyle{
d^{4}k}}{\displaystyle{(2\pi)^{4}}}} & 2 {\displaystyle \int\frac
{\displaystyle{d^{3}p}}{
\displaystyle{(2\pi)^{3}}}\frac{\displaystyle{n_{N}(p)[1-n_{N^{\prime}} (
p+q-k)]}}{\displaystyle{q^{0}-k^{0}+E(p)-E(p+q-k)+i\epsilon}}
}\times\\
&\\
&\times D_{\pi}(k)\frac{1}{2}Tr^{Spin}(T^{\mu}{{T}}
^{\dagger \nu})_{NN^{\prime}}
\end{array}
\end{equation}
\noindent
where $T^\mu$ is the amplitude for
$\gamma^* N \rightarrow \pi N$. The indices $N, N'$ in eq. (35) stand for
the hole and particle nucleon states respectively and
$n_N (\vec{p})$ is the occupation number
in the Fermi local sea. $E (\vec{p})$ is the energy of the nucleon
$\sqrt{\vec{p}\,^2 + M_N^2}$ and $D_\pi$ is the
pion propagator
\begin{equation}
D_{\pi}(k)=\frac{\displaystyle{1}}
{\displaystyle{k^{2}-m_{\pi}^{2}+i\epsilon}}
\end{equation}
A further simplification can be done by evaluating the $T^\mu$
amplitudes at
an average Fermi momentum. Explicit integration over $\vec{p}$ and also this
approximation were done is \cite{CAR} and the approximation was found to be
rather good. We take $< \vec{p} >= \sqrt{\frac{3}{5}} k_F$
with $k_F$ the local Fermi momentum ($3\pi^2 \rho (r)/2)^{1/3}$
and a direction
orthogonal to that of the virtual photon. The errors induced by
this approximation are
smaller than 5$\%$. Then we can use the Lindhard function $\bar{U}_{N,N'}$
defined as
\begin{equation}
\bar{U}_{r,s}(q-k)=2{\displaystyle \int \frac{\displaystyle{d^{3}k}}
{\displaystyle{(2\pi)^{3}}}\frac{\displaystyle{n_{r}(\vec{p}\,)[1-n_{s} (
\vec{p}+\vec{q}-\vec{k}\,)]}}{\displaystyle{q^{0}-k^{0}+{E}(\vec{p}\,)
-{E}(\vec{p}+\vec{q}-\vec{k}\,)+i\epsilon}}}
\end{equation}
\noindent
where the indices, $r,s$ correspond to protons or neutrons.
For the evaluation of the imaginary part we need an extra Cutkosky rule
$$
U (p) \rightarrow 2 i Im U (p) \Theta (p^0)
$$
\noindent
which is the general rule considering that the Lindhard
function plays the role of a $ph$ propagator.
Hence we apply the Cutkosky
rules of eq. (23) and the former one and we find
\begin{equation}
\begin{array}{ll}
Im\Pi^{\mu\nu}=& {\displaystyle \int\frac{\displaystyle{d^{3}k}}
{\displaystyle{(2\pi)^{3}}}
Im
\bar{U}_{NN^{\prime}}(q-k)\frac{\displaystyle{1}}{\displaystyle{2
\omega(\vec{k}\,)}}
\theta(q^{0}-\omega(\vec{k}\,))
}\times\\
&\\
&\times \left.\frac{1}{2}Tr^{Spin}(T^{\mu}T^{\dagger\nu}_{NN^{\prime}})
\right|_{k^{0}=\omega(\vec{k}\,)}
\end{array}
\end{equation}
Since there are analitycal expressions for $Im\bar{U}_{NN^{\prime}}$
(see Appendix B of \cite{CAR}), the
approximation done saves us three integrals and a considerable amount of
computational time.
There is an interesting test to eq.(38). Indeed, in
the limit of small densities
the Pauli blocking factor $1-n$ becomes 1 and
$Im\bar{U}_{NN^\prime} (q)
\simeq - \pi \rho_N \delta (q^0 - \frac{\vec{q}\,^2}{2 M_N})$. Substituting
this into eq. (38) and (29),(30) one easily obtains that
$\sigma_{eA} = \sigma_{ep} Z + \sigma_{en} N$, the
strict impulse approximation. By performing the integral in eq. (38) one
accounts for Pauli blocking and in an approximate way for Fermi motion. Later
on we shall introduce other corrections due to medium polarization.
\subsection{The $\Delta$ excitation term}
One of the terms implicit in eq. (38) is the one where one picks up the
$\Delta$ excitation term both in $T^\mu$ and is $T^{\dagger \nu}$.
This term is depicted diagrammatically in fig. 4.3(a) and, like in
pion-nuclear and photo-nuclear reactions at intermediate energies,
plays a major role in this reaction.
In order to evaluate this piece one can
go back to eq. (35) and perform the $d^4 k$
integration to factorize the $\Delta$
width and on the other hand one will also have the modulus
squared of the $\Delta$ propagator
present in the $\Delta$ term of eq. (5). This, however, can be obtained
more economically by reinterpreting the diagram 4.3 (a) as a $\Delta h$
excitation with a $\Delta$ width.
We can also divert a little from the
general formulation, and in order to gain some extra accuracy we can
implement Lorentz covariance exactly simply boosting the tensor
$ \Pi^{\mu \nu}$ from
a frame where the $\Delta$ is at rest
$(\vec{q} + \vec{p} = \vec{p}_\Delta = 0)$,
where the amplitude of eq.
(5) would be (by construction) equivalent to the relativistic amplitude.
\vskip 0.15cm
\centerline{\protect\hbox{\psfig{file=deltacontri.ps,width=8cm}}}
\vskip 0.1cm
\noindent
{\small {\bf Fig.4.3} Diagrammatic representation of the $\Delta h$
photonuclear excitation piece.}
Hence we get
\vspace*{-0.2cm}
\begin{equation}
\begin{array}{ll}
Im\Pi_{\Delta}^{\mu\nu}=&\sum_{ij}|c_{ij}|^{2}\frac{\displaystyle{f_{\gamma}^{2}
(q^{2})}}{\displaystyle{m_{\pi}^{2}}}{\displaystyle \int \frac{\displaystyle{
d^{3}p}}{\displaystyle{(2\pi)^{3}}}n_{i}(p)\Lambda^{\mu}_{m}(p,q)
\Lambda^{\nu}_{l}(p,q)
}\\
&\\
&\times Tr\left((\vec{S}^{\dagger}\times\vec{q}_{cm})^{m}
(\vec{S}\times\vec{q}_{cm})^{l}\right)\frac{\displaystyle{s}}{
\displaystyle{M_{\Delta}^{2}}}\times
\\
&\\
&\times \frac{\displaystyle{Im\Sigma^{j}_{\Delta}(p+q)-
\frac{\displaystyle{\bar{\Gamma}}}{\displaystyle{2}}(p+q)}}
{\displaystyle{\left|\sqrt{s}-M_{\Delta}+i\frac
{\displaystyle{\bar{\Gamma}(s)}}
{\displaystyle{2}}-\Sigma_{\Delta}(s)\right|^{2}}}
\end{array}
\end{equation}
The Lorentz matrix $\Lambda$ is such that
${\Lambda_\gamma}^\mu_\nu q^\nu_{cm} = q^\mu$ and $s=(p+q)^{2}$.
The coefficients $C_{ij}$, $i,j= 1,2$ stands
for proton or neutron, account for isospin and are given by
\begin{equation}
|C_{ij}|=\left(\frac{2}{3}\right)^{2}\delta_{ij}+\frac{2}{9}
\delta_{i,j+1}+\frac{2}{9}\delta_{i,j-1}
\end{equation}
Eq. (39), however, seems to
neglect the Pauli blocking factor $1-n$ of eq. (35).
This factor, however, is taken into account implicitly and leads to the Pauli
blocked width $\bar{\Gamma}$, which is
evaluated in \cite{RAFAPI}. Furthermore,
in a nuclear
medium the $\Delta$ is renormalized and acquires a self-energy
$\Sigma_\Delta$, which
is also accounted for in eq.(39). The results of \cite{LOR}
for $\Sigma_\Delta$ are used
in the calculation.
This self-energy accounts for the diagrams
depicted in fig. 4.3, where the double dashed line stands for the effective
spin -isospin interaction, while the serrated line accounts for the induced
interaction. The effective spin-isospin interaction is originated by a pion
exchange in the presence of short range correlations and includes
$\rho$-exchange as well. It is obtained by substituting
\begin{equation}
\hat{q}_{i}\hat{q}_{j} D_{\pi}(q)
\rightarrow \hat{q}_{i}\hat{q}_{j}V_{l}(q) +({\delta}_{ij}-
\hat{q}_{i}\hat{q}_{j})V_{t}(q)
\end{equation}
\noindent
and expressions for $V_l, V_t$ are found in \cite{LOR}
( $(f_{\pi NN}/m_{\pi})^{2}V_{l,t}$ here is equivalent to $V_{l,t}$
of \cite{LOR}). The induced interaction
accounts for the series of diagrams depicted in fig. 4.4.
\vspace*{0.2cm}
\centerline{\protect\hbox{\psfig{file=interpaper.ps,width=8cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.4.4} Feynman diagrams
included in the evaluation of the $\Delta$
self-energy.}
\vspace*{0.1cm}
There is an RPA sum
through $ph$ and $\Delta h$ excitation and is readily obtained as
\begin{equation}
\begin{array}{lll}
V_{ind}&=&
\hat{q}_i\hat{q}_j\,
\Frac{V_l(q)}{1-U(q)V_l(q)\left(\Frac{f_{\pi NN}}{m_{\pi}}\right)^2}
+\\
&&\\
&&
(\delta_{ij}-\hat{q}_i\hat{q}_j)
\Frac{V_t(q)}{1-U(q)V_t(q)\left(\Frac{f_{\pi NN}}{m_{\pi}}\right)^2}
\end{array}
\end{equation}
\noindent
where now $U (q) = U_N (q) + U_\Delta (q)$
is the Lindhard function for $ph\,+\,\Delta h$
excitations including forward
going and backward going bubbles \cite{LOR} in contrast to $\bar{U}$
which only contains
the forward going bubble of a $ph$ excitation (the only one which
contributes to
$Im U_N $ for
$q^0 > 0$). $U_N$ in addition incorporates a factor two of isospin
with respect to
$\bar{U}$, such that $Im U_N =
2 Im \bar{U}$ for symmetric nuclear matter.
However, all the work which goes into the evaluation of
$\Sigma_\Delta$
is done in
ref. \cite{LOR}, where a useful analytical parameterization of the numerical
results is given that we use here. The imaginary part is parametrized
as
\begin{equation}
Im\Sigma_{\Delta}=-\left\{C_{Q}(\rho /\rho_{0})^{\alpha}+C_{A_{2}}
(\rho /\rho_{0})^{\beta}+C_{A_{3}}(\rho /\rho_{0})^{\gamma}\right\}
\end{equation}
\noindent
where the different coefficients are given in \cite{LOR} as a function of the
energy.
The separation of terms in eq. (43) is useful because the term $C_Q$ comes
from the diagrams (c) and (d)
of fig. 4.3
when the lines cut by the dotted line are
placed on shell, and hence the term is related to the $(
\gamma^*, \pi)$ channel, while
$C_{A2}, C_{A3}$
come from the diagrams (b) and (e) and are related to two and three
body absorption. Hence, the separation in this formula allows us to separate
the final cross section into different channels.
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=burbujas9.ps,width=8.cm}}}
\vskip 0.2cm
\vspace*{-1.cm}
\noindent
{\small {\bf Fig.4.5} Irreducible pieces in the $\Delta h$ channel from the
$\Delta h$ interaction.}
\vspace*{0.1cm}
It is also easy to realize that the RPA sum of $\Delta$h excitations, shown
in fig. 4.5, can be taken into account by substituting $Re \Sigma_\Delta$
by \cite{CAR}
\begin{equation}
Re\Sigma_{\Delta}\rightarrow Re\Sigma_{\Delta}+\frac{4}{9}
\left(\frac{\displaystyle{f^{*}}}{\displaystyle{m_{\pi}}}\right)^{2}
\rho V_{t}
\end{equation}
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=burbuja10.ps,width=4.3cm}}}
\vskip 0.2cm
\vspace*{-0.5cm}
\noindent
{\small {\bf Fig.4.6} Diagrammatic representation of the inclusion
of a $ph$ excitation between $\Delta h$ excitations.}
\vspace*{0.1cm}
\noindent
and furthermore, if we wish to include some $ph$ excitation in between,
see fig. 4.6, (which is actually not relevant numerically), this is done
easily by substituting $Re \Sigma_\Delta$ by
\begin{equation}
Re \Sigma_{\Delta}+\frac{4}{9}
\left(\frac{f^{*}}{m_{\pi}}\right)^{2}\frac{\displaystyle{V_{t}}}{
\displaystyle{\left(1-\frac{{f}_{\pi NN}^{2}}{{m}_{\pi}^{2}}{U_N}
V_{t}\right)}}
\rho
\end{equation}
\subsection{Results for the $\Delta$ contribution}
In fig. 4.7 we show the results coming from the $\Delta$ term discussed
in the former section. The experimental data are coming from \cite{BAR}.
We
have separated the contribution from the different channels. Besides
the upper solid line which stands for the total contribution, looking from
up to down at about $\omega = 350$ MeV
the next line corresponds to pion production,
the following one is two nucleon absorption and the lowest one three body absorption.
We can see that most of the experimental strength in the $\Delta$ region
is provided by this $\Delta$
excitation term, but there is still some strength missing.
In fig. 4.8 we show the contribution of the delta piece for the $^{208}$ Pb
nucleus. The data are now
from \cite{ZGH}. The results are similar to those found in
$^{12}$C and there is
still some strength missing. The study of this missing strength will
occupy the next sections.
\centerline{\protect\hbox{\psfig{file=deltacarbon.ps,width=10.cm}}}
\vskip 0.2cm
\noindent
{\small{\bf Fig.4.7} Contribution of the
$\Delta$ piece to the $(e,e^{\prime})$
cross section in $^{12}$C. Experimental data from \cite{BAR}.
See text for different contributions.}
\vskip 0.1cm
\centerline{\protect\hbox{\psfig{file=plomodelta.ps,width=10.cm}}}
\vskip 0.1cm
\noindent
{\small{\bf Fig.4.8}
Contribution of the
$\Delta$ piece to the $(e,e^{\prime})$
cross section in $^{208}$Pb. Experimental data from \cite{ZGH}.
}
\subsection{Two body photoabsorption}
Let us go back to the generic diagram of pion electroproduction of
fig. 4.2. Let us take the pion line and allow the pion to excite a
$ph$. This leads us to the diagram of fig. 4.9.
\vskip 0.1cm
\centerline{\protect\hbox{\psfig{file=burbuja7.ps,width=4.7cm}}}
\vskip 0.1cm
\noindent
{\small {\bf Fig.4.9} Photon self-energy obtained from the one in fig. 4.2
when the pion is allowed to excite a $ph$.}
\vspace*{0.1cm}
This is still a generic diagram which actually
contains 36 diagrams when in the shaded circle we put each one of the
terms of the $\gamma^* N \rightarrow \pi N$ amplitude of fig. 3.2. One
must avoid the temptation of factorizing these amplitudes in order
to evaluate these diagrams since some of them might be symmetric
and then have a symmetry factor 1/2. This is the case here with one diagram
implicit in fig. 4.9, which is the one corresponding to the pion pole
term in each one of the $\gamma^* N \rightarrow \pi N$ amplitudes.
This diagram is shown explicitly in fig. 4.10.
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=polopion.ps,width=4.4cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.4.10} Pion pole term included in fig. 4.9.}
\vspace*{0.1cm}
One can see that the diagrams in fig. 4.9 contribute to $Im \Pi$
according to Cutkosky rules when we cut by
a horizontal line and the $2 p 2h$ are placed on shell.
The contribution of the diagram of fig. 4.9 is readily done. We obtain
\begin{equation}
\begin{array}{ll}
\Pi^{(2) \mu\nu}(q)=
& {\displaystyle {\sum_{N,N^{\prime}}}}i
{\displaystyle{\int}}\Frac{d^{4}k}{(2\pi)^{4}}
\Frac{d^{3}p}{(2\pi)^{3}}\Frac{n_{N}(p)[1-n_{N^{\prime}}(p+q-k)]}
{q^{0}-k^{0}+E(p)-E(p+q-k)+i\epsilon} \times
\\
&
\\
& D_{\pi}^{2}(k)\Frac{{f}_{\pi NN}^{2}}{m_{\pi}^{2}}\vec{k}^{2}
{U}_{\lambda}(k)Tr^{Spin}(T^{\mu}{T}^{\dagger \nu})_{NN^{\prime}}
S_{\alpha}F_{\pi}^{4}(k)
\end{array}
\end{equation}
\noindent
where $U_\lambda$ is the Lindhard function for $ph$ by an object of charge
$\lambda$: this is, twice $\bar{U}_{p,n}$ or $\bar{U}_{p,n}$ for the
excitation by a charged pion or $\bar{U}_{p,p}+\bar{U}_{n,n}$ for
the excitation by a neutral pion and $\vec{k}$ is the pion momentum.
The factor $F^4_\pi (k)$, where $F_\pi$ is the pion form factor
appears because now the pions are off shell. Recall that we also
take all form factors equal in order to preserve gauge invariance
(eq. (9)). The factor $S_\alpha$ is the symmetry factor, unity for all
diagrams and 1/2 for the symmetric one of fig. (4.10).
We can again simplify the expression by taking an average nucleon
momentum of the Fermi sea to evaluate the matrix elements of
$T^\mu \, T^{\dagger\nu}$. This allows us to factorize the Lindhard
function and we get
\begin{equation}
\begin{array}{ll}
\Pi^{(2)\,\,\mu\nu}(q)=
& {\displaystyle{\sum_{NN^{\prime}}}} i {\displaystyle{\int}}
\Frac{d^{4}k}{(2\pi)^{4}}\bar{{U}}_{NN^{\prime}}(q-k)
D_{\pi}^{2}(k)\Frac{{f}_{\pi NN}^{2}}{m_{\pi}^{2}}
\\
&\\
& {\vec{k}}^{2}
{U}_{\lambda}(k)\frac{1}{2}Tr^{Spin}(T^{\mu}T^{\dagger \nu})_{NN^{\prime}}
S_{\alpha}F_{\pi}^{4}(k)
\end{array}
\end{equation}
By applying Cutkosky
rules we find
\begin{equation}
\begin{array}{ll}
Im\Pi^{(2)\,\,\mu\nu}=&-{\displaystyle{\sum_{NN^{\prime}}}}{\displaystyle{
\int}}\Frac{d^{4}k}{(2\pi)^{4}}\Theta(q^{0}-k^{0})Im\bar{{U}}_{N
N^{\prime}}(q-k)\Theta (k^{0})\times\\
&\\
& Im {U}_{\lambda}(k)D_{\pi}^{2}(k)\Frac{{f}_{\pi NN}^{2}}{m_{\pi}^{2}}
{\vec{k}}^{2} F_{\pi}^{4}(k)S_{\alpha}\times\\
&\\
& Tr^{Spin}(T^{\mu}T^{\dagger \nu})_{NN^{\prime}}
\end{array}
\end{equation}
The cut which places the two $ph$ on shell in the diagrams of
fig. 4.9 is not the only possible one. In fig. 4.11 we show a different
cut (dotted line) which places one $ph$ and
the pion on shell.
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=burbuja12.ps,width=5.5cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.4.11} Same as fig. 4.9 and showing the cut which places
one $ph$ and the pion on shell.}
\vspace*{0.2cm}
This contribution is taken into account in the
$\Delta$ excitation term by means of the term $C_Q$. As done for
real photons in \cite{CAR}, we neglect this contribution in the
other terms, because at low energies where the background pieces are
important, the $(\gamma^*, \pi)$ channel is small and at high
energies where the $(\gamma^*, \pi)$ contribution is important, this channel
is dominated by the $\Delta$ excitation and there this correction is
taken into account.
We have also considered two body diagrams where each photon couples
to different bubbles: As found is \cite{CAR} only one of them is
relevant, the one in fig. (4.12), which involves the KR term alone
and which we take into account.
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=burbuja13.ps,width=4.9cm}}}
\vskip 0.1cm
\noindent
{\small {\bf Fig.4.12} Feynman diagram related to the KR term of fig. 4.9
with outgoing photon from the second nucleon.}
\vspace*{0.3cm}
Following the same rules as above,
this term is readily evaluated and gives
\begin{equation}
\begin{array}{ll}
Im\Pi^{00}=& -2e^{2}\left(\Frac{f^{2}}{m_{\pi}^{2}}\right)^{2}{2}
F_{A}^{2}(q^{2})\times\\
&\\
& {\displaystyle{\int}}\Frac{d^{4}k}{(2\pi)^{4}}D_{0}(k)D_{0}(k-q)
\Frac{(\vec{k}\vec{q})(\vec{k}-\vec{q})\vec{q}}{M_{N}^{2}}\times\\
&\\
& F_{\pi NN}^{2}(k)F_{\pi NN}^{2}(k-q)\times\\
&\\
& \left[Im\bar{{U}}_{pp}(q)Im\bar{{U}}_{pp}(k-q)+
Im\bar{{U}}_{pn}
(q)Im\bar{{U}}_{np}(k-q)+\right.\\
& \\
& +\left.Im\bar{{U}}_{np}(q)Im\bar{{U}}_{pn}(k-q)\right]
\Theta(k^{0})\Theta (k^{0}-q^{0})
\end{array}
\end{equation}
$$
\begin{array}{ll}
Im(\Pi^{xx}+\Pi^{yy})= & -2e^{2}\left(\Frac{f^{2}}{m_{\pi}^{2}}\right)^{2}{2}
F_{A}^{2}(q^{2})\times\\
&\\
& {\displaystyle{\int}}\Frac{d^{4}k}{(2\pi)^{4}}D_{0}(k)D_{0}(k-q)
|\vec{k}|^{2}sin^{2}\theta F_{\pi NN}^{2}(k)F_{\pi NN}^{2}(k-q)\times\\
&\\
& \left\{\Frac{1}{2}\left(\Frac{q^{0}}{M_{N}}\right)^{2}
Im\bar{{U}}_{pp}(q)Im\bar{{U}}_{pp}(k-q)\right.+
\end{array}
$$
\newpage
\begin{equation}
\begin{array}{ll}
&\left[-\left(1-\Frac{q^{0}}{2M}\right)\sqrt{2}\right]^{2}
Im
\bar{{U}}_{pn}(q)Im\bar{{U}}_{np}(k-q)\\
&\\
& +\left.\left[\left(1+\Frac{q^{0}}{2M}\right)\sqrt{2}\right]^{2}
Im\bar{{U}}_{np}(q)Im\bar{{U}}_{pn}(k-q)\right\}\times\\
&\\
& \Theta(k^{0})\Theta(k^{0}-q^{0})
\end{array}
\end{equation}
The contribution of this term is roughly 1/2 of the $KR \times KR$
term in the generic diagram of fig. 4.9.
\subsection{Contributions tied to the ($\gamma^*, 2 \pi)$ channel}
The $\gamma N \rightarrow \pi \pi N$ reaction has been the subject of
recent detailed experimental analyses \cite{BRA,STR} and also of
recent theoretical analyses, some of
them spanning a large energy range \cite{GOM1,GOM2} and
others concentrating only very close to threshold in order
to test predictions of chiral perturbation theory \cite{ULF,BEN}.
The model in ref. \cite{GOM1} for the $\gamma p \rightarrow
\pi^+ \pi^- p$ uses 67 Feynman diagrams, while ref. \cite{GOM2},
where the model is extended to the other isospin channels uses
only 20 diagrams which are necessary below $E_\gamma = 800 \, MeV$,
where the new data have been measured.
Although the model is rather elaborate and contains many terms,
one can see that the gross features of the reaction
can be obtained with the two
terms of fig. 4.13,
accounting
for about 80$\%$ of the cross section.
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=2pionespaper.ps,width=8.cm}}}
\noindent
{\small {\bf Fig.4.13} Relevant Feynman diagrams that enter in the evaluation
of the $\gamma^* N \rightarrow N 2 \pi$ cross section. }
\vspace*{0.1cm}
Since here we are only concerned about corrections to the more
important terms which we have discussed above, it is sensible to just
take these two diagrams. The diagram in fig. (a) is the $\Delta
N \pi \gamma$ Kroll Ruderman (KR) term, while the one in fig. (b) is the
pion pole term. In both terms the $\Delta$ is excited. The KR term,
which appears from the $\Delta N \pi$ vertex by minimal coupling,
is given by,
\begin{equation}
\begin{array}{ll}
\cal{M}^{\mu}= &
\left\{
\begin{array}{ccc}
0 & , & \pi^{0}\\
1 & , & \pi^{\pm}
\end{array}\right\}
\left(e\Frac{f^{*}}{m_{\pi}}\right)\times\\
& \\
& \left(1 \frac{1}{2} \frac{3}{2} \left| m_{\pi}
M_{N} M_{\Delta}\right.\right)
\left(
\begin{array}{c}
\Frac{\vec{S}^{\dagger}\vec{p}_{\Delta}}{\sqrt{s}}\\
\\
\vec{S}^{\dagger}
\end{array}\right)
\end{array}
\end{equation}
\noindent
corresponding to $\pi^\pm \Delta$ production in the $\gamma^* N
\rightarrow \Delta \pi$ vertex.
By following the same steps as before we obtain the many
body diagrams of fig. (4.14), where the dashed circle indicates any of
the two terms of fig. 4.13.
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=burbujas6.ps,width=6.cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.4.14} Photon self-energy diagrams obtained by folding
the terms of fig. 4.13. Diagram (b) is obtained when the pion is allowed
to produce a $ph$ excitation.}
\vspace*{0.8cm}
Furthermore, as discussed in ref.
~\cite{CAR}, in the diagrams of fig. 4.14 (b) we keep only
the term with the $KR \times KR$ in the vertices. This is done since
the pion in flight term, can be considered as a two step process of a
$\gamma^* N \rightarrow \pi N$ with $\pi$, a real pion, followed by the
$\pi N \rightarrow \Delta$ excitation. The two step processes
redistribute strength but do not change the cross section and
hence are not included in our approach.
Since the $\Delta$ in the $\Delta h$ excitation in fig. 4.14 is also
renormalized, we are accounting for the physical channels depicted
in fig. 4.15 when placing on shell the states cut by the dotted
line.
\vskip 0.4cm
\centerline{\protect\hbox{\psfig{file=burbujas3.ps,width=13.5cm}}}
\vspace*{-1.3cm}
\noindent
{\small {\bf Fig.4.15} Detail of fig. 4.14 indicating the physical channels
associated to the cuts.}
\vspace*{0.8cm}
As one can see there, (1) accounts for $1p 1h \, 2 \pi$ excitation,
(2) and (3) for $2 p 2 h \,1 \pi$ excitation and (4) for $3p 3h$
excitation.
The evaluation of these pieces follows exactly the same steps as
for figs. 4.2 and 4.9, simply replacing the $\gamma^* N
\rightarrow \pi N$ by the $\gamma^* N \rightarrow \pi \Delta$
amplitudes and one nucleon propagator by the $\Delta$ propagator.
The contribution of these terms below $\omega = 350 \, MeV$ is
very small. Their importance increases with the energy and at
$\omega = 450 \; MeV$ they account for about 1/5 of the
cross section, as found for real photons.
\subsection{Polarization (RPA) effects}
In the diagrams of fig. 4.9 we can consider the $ph$ as just
the first order of a series of the RPA excitations through
$ph$ and $\Delta h$ excitations. If one replaces the $ph$ by the RPA series,
one is led to the terms implicit in fig. 4.16. A similar
series would appear for the case of the $(\gamma, \pi)$
process depicted in fig. 4.2.
In practical
terms this is done in a simple way by having a bookkeeping of both the
spin longitudinal and spin transverse parts and replacing
\begin{equation}\label{eq:pol}
Im U_N \rightarrow a \frac{Im U_N}{|1 - U_\lambda (q) V_l|^2}
+ b \frac{Im U_N}{|1 - U_\lambda V_t|^2}
\end{equation}
\noindent
where $a,b$ measure the strength of the longitudinal and
transverse parts.
For the transverse part of the photon self-energy $\Pi^{xx},
\Pi^{yy}$ the procedure to follow is identical to the one explained
in section 9 of \cite{CAR} and we refer the reader to this
paper (see also \cite{AMP}). The only novelty here is $\Pi^{00}$,
but this component is of spin longitudinal character and is
renormalized by means of eq.(~\ref{eq:pol}) with $a = 1,
b= 0$. In the $\Delta$ term the polarization effects are already
included in the self-energy of ref. \cite{LOR}, hence, no
further corrections are needed.
\section{Short range correlations}
So far the calculations have been done using implicitly
plane waves for the nucleon states. Short range nuclear correlations
modify the two nucleon relative wave function and this has a repercusion
in some of the matrix elements which we have calculated. This is
particularly true in those matrix elements which involve
a $p$-wave coupling in the vertex for each of the two nucleons,
because the pion exchange generates a $\delta (\vec{r})$ function,
which is rended inoperative in the presence of short range
correlations. This is not exactly the case if finite size effects
by means of form factors are taken into account, but the need
to implement the effects of the short range correlations
remains. The correlations can also introduce spin transverse components
in the p-wave-p-wave terms which were originally of the spin
longitudinal nature \cite{WES}. Hence, at the same time that one
introduces the effect of correlations,
one takes advantage of this and introduces the $\rho$ meson exchange
in this case. In this way, we generated $V_l$ and $V_t$ of eq.(41).
The method to introduce the effects of correlations is to substitute
a two nucleon amplitude $V (q)$ by
$$
V (\vec{q}) \rightarrow \frac{1}{(2 \pi)^3} \int d^3k V (\vec{k})
\Omega (\vec{q} - \vec{k})
$$
\noindent
where $\Omega (\vec{p})$ is the Fourier transform of a nuclear
correlation function.
Once again the techniques to make these corrections can be seen
in \cite{CAR} (appendix D, see also \cite{AMP}), and we do not
repeat them here.
\newpage
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=burbujas.ps,width=14cm}}}
\vskip 0.4cm
\vspace*{0.3cm}
\noindent
{\small {\bf Fig.4.16} Terms of the KR and pion pole
block implicit in fig. 4.9 showing the medium
polarization through RPA $ph$ and $\Delta h$ excitations induced by the pion.}
\newpage
\section{Detailed study of the quasielastic peak}
\subsection{Formalism}
So far we have studied the $(\gamma^*, \pi)$ process and the
$\gamma^*$ absorption by a pair or trio of particles. This was
done keeping a parallelism to the real photon case. However, unlike
the case with real photons, a virtual photon can be absorbed by
one nucleon leading to the quasielastic peak of the response
function. We have left this problem till the end in order to
introduce the concepts of many body which proved relevant in the
scattering of real photons with nuclei and in the equivalent
channels of virtual photons studied before.
Then we can use the same concepts and ideas here in order to
introduce the appropriate many body corrections to the quasielastic
peak.
Thus we begin by evaluating $\Pi^{\mu \nu}$ for the $1ph$
excitation driven by the virtual photon, as depicted in fig. 6.1.
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=burbuja16.ps,width=4cm}}}
\vskip 0.2cm
{\small {\bf Fig.6.1} Photon self-energy diagram for the 1$ph$ excitation
driven by the virtual photon.}
\vspace*{0.5cm}
The
photon self-energy associated to this diagram is given by
\begin{equation}
\begin{array}{ll}
-i\Pi^{\mu\nu}={\displaystyle\int}\Frac{d^{4}p}{(2\pi)^{4}} & \Frac{i
n_{i}(p)}{p^{0}-E(\vec{p}\,)+i\epsilon}\times\\
&\\
& \times\Frac{i(1-n_{j}(p+q))}
{p^{0}+q^{0}-{E}(\vec{p}+\vec{q}\,)-i\epsilon}
Tr(V^{\mu}V^{\dagger\nu})
\end{array}
\end{equation}
\noindent
where $V^\mu$ represents the $\gamma NN$ vertex and $E (\vec{p})$
is the nucleon kinetic energy. The vertex
$V^\mu$ is given by
\begin{equation}
V^{\mu}=\bar{u}_r(\vec{p}\,)\left\{eF_1\gamma^{\mu}-ie\Frac{G_M}{2M_N}
\mu_n\sigma^{\mu \rho}q_{\rho}\right\}u_{{r}^{\prime}}(\vec{p}\,+\vec{q}\,)
\end{equation}
Once again the application of Cutkosky rules leads to
\begin{equation}
\begin{array}{ll}
Im\Pi^{00}=-{\displaystyle\int}\Frac{d^{3}p}{(2\pi)^{2}} &
n_{i}(p)(1-n_{j}(p+q))\times\\
&\\
& \times\delta (p^{0}+q^{0}-{E}(\vec{p}+\vec{q}\,)\,)
Tr(V^{0}V^{\dagger 0})
\end{array}
\end{equation}
\begin{equation}
\begin{array}{ll}
Im\Pi^{xx}=-{\displaystyle\int}\Frac{d^{3}p}{(2\pi)^{2}} &
n_{i}(p)(1-n_{j}(p+q))\times\\
&\\
& \times\delta (p^{0}+q^{0}-{E}(\vec{p}+\vec{q}\,)\,)
Tr(V^{x}V^{\dagger x})
\end{array}
\end{equation}
\noindent
and if we average $Tr (V^\mu V^{\dagger\nu})$ over the nucleon momentum
in the Fermi sea we can write
\begin{equation}
Im\Pi^{00}=\Frac{1}{2}Im\bar{U}(q,\rho)\langle Tr(V^{0}V^{\dagger 0})\rangle
\end{equation}
\begin{equation}
Im\Pi^{xx}=\Frac{1}{2}Im\bar{U}(q,\rho)\langle
Tr(V^{x}V^{\dagger x})\rangle
\end{equation}
The average over the Fermi momentum can be done keeping terms
up to $q^2 /M_N^2$ and we find in terms of
$$
A^{\mu \nu} = \frac{1}{e^2} < Tr (V^\mu V^{\dagger \nu}) >
$$
\begin{equation}
A^{00}=\Frac{1}{{M}_{N}^{2}}\left\{
\Frac{1}{1-\Frac{q^{2}}{4{M}_{N}^{2}}}
\left[{G}_{E}^{2}(q)-\Frac{q^{2}}{4{M}_{N}^{2}}{G}_{M}^{2}(q)\right]
\Frac{1}{2}(2p^{0}+q^{0})^{2}-\Frac{1}{2}{\vec{q}\,}^{2}{G}_{M}^{2}(q)
\right\}
\end{equation}
\begin{equation}
A^{xx}=\Frac{1}{{M}_{N}^{2}}\left\{
\Frac{1}{1-\Frac{q^{2}}{4{M}_{N}^{2}}}
\left[{G}_{E}^{2}(q)-\Frac{q^{2}}{4{M}_{N}^{2}}{G}_{M}^{2}(q)\right]
\Frac{2}{5}{k}_{F}^{2}-\Frac{1}{2}{q}^{2}{G}_{M}^{2}(q)
\right\}
\end{equation}
\noindent
where $p^0 = M_N + \frac{3}{5} \frac{k_F^2}{2M_N}$ and $G_E, G_M$
are the Sachs form factors \cite{AML,MUL}.
\vspace*{1.3cm}
\subsection{Spectral function description and final state interaction}
One of the corrections to the bare $ph$ excitation studied above is
the one induced by final state interaction, as we indicated in
section 2, which in our approach can be taken into account by
dressing up the nucleon propagator of the particle state in the $ph$
excitation, as depicted in fig. 6.2 (there the dashed line would
account for the whole $NN$ interaction not just pion exchange). However,
some caution must be exerted when talking about this diagram.
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=burbuja15.ps,width=5.cm}}}
\vskip 0.2cm
\vspace*{-0.2cm}
\noindent
{\small {\bf Fig.6.2} Photon self-energy diagram obtained from fig. 6.1
by dressing up the nucleon pro\-pa\-ga\-tor of the particle state in the
$ph$ excitation.
}
\vspace*{0.3cm}
In the first place, this is one of the terms implicit in the generic
diagram of fig. 4.9 when the nucleon pole term is
taken in each of the $\gamma^* N \rightarrow N N$ amplitudes. This term poses
no problem for real photons and leads to a small fraction of the two
nucleon absorption. However, for virtual photons this diagram is
divergent. The reason is that when placing the $2p 2h$ excitation on shell
through Cutkosky rules, we still have the square of the nucleon propagator
with
momentum $p + q$ in the figure. This propagator can be placed
on shell for virtual photons (not for real photons) and we get a
divergence.
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=burbuja17.ps,width=3.5cm}}}
\vskip 0.2cm
\vspace*{-0.2cm}
\noindent
{\small {\bf Fig.6.3} Insertion of the nucleon
self-energy on the nucleon line of the particle state.}
\vspace*{0.3cm}
The divergence is physical in the sense that its meaning is the
probability per unit time of absorbing a virtual photon by one
nucleon times the probability of collision of the final nucleon with
other nucleons in the infinite Fermi sea in the lifetime of this
nucleon \cite{STROT}. Since this nucleon is real, its lifetime is infinite
and thus the probability infinite. The problem is physically
solved \cite{STROT} by recalling that the nucleon in the Fermi sea has
a self-energy with an imaginary part which gives it a finite lifetime
(for collisions). This is also immediately taken into account technically
by iterating the nucleon self-energy insertion of fig. 6.3 on the
nucleon line, following the Dyson equation, hence substituting
the particle nucleon propagator by a renormalized nucleon propagator
including the nucleon self-energy in the medium,
\begin{equation}
G(p^{0},\vec{p}\,)=\Frac{1}{p^{0}-\Frac{{\vec{p}\,}^{2}}{2M_N}-\Sigma(p^{0},
\vec{p}\,)}
\end{equation}
\noindent
where $\sum (p^0 , \vec{p})$ is the nucleon self-energy. Alternatively
one can use the spectral function representation \cite{WAK}
\begin{equation}\label{eq:spsh}
G (p^0 \vec{p}) = \int_{- \infty}^\mu d \omega \frac{S_h (\omega,
\vec{p})}{p^0 - \omega - i \epsilon} +
\int_{\mu}^\infty \frac{S_p (\omega, \vec{p})}{p^0 - \omega + i
\epsilon} d \omega
\end{equation}
\noindent
where $S_h, S_p$ are the hole and particle spectral functions related to
$\Sigma$ by means of \cite{ANG}
\begin{equation}
\begin{array}{lcl}
\mbox{* If}\,\,\, \omega \geq\mu & , & S_{p}(\omega,p)=-\Frac{1}{\pi}
Im G(\omega, p)=-\Frac{1}{\pi}\Frac{Im\Sigma(\omega, p)}{A+B}\\
\\
\mbox{* If}\,\,\, \omega \leq\mu & , & S_{h}(\omega, p)=\Frac{1}{\pi}
Im G(\omega, p)=\Frac{1}{\pi}\Frac{Im\Sigma(\omega, p)}{A+B}
\end{array}
\end{equation}
\noindent
and $\mu$ is the chemical potential and
$$
\begin{array}{l}
A=\left( \omega-\Frac{{\vec{p}\,}^{2}}{2M_N}-Re\Sigma(\omega,\vec{p}\,)\right)^{2}\\
\\
B=\left( Im\Sigma(\omega,{\vec{p}\,})\right)^{2}
\end{array}
$$
By means of eq.(62)
we can write the $ph$ propagator or new Lindhard
function incorporating the effects of the nucleon self-energy in
the medium and we have for $Im \bar{U}$
\begin{equation}
\begin{array}{ll}
Im\,\bar{{U}}(q)=-\Frac{1}{2\pi}& \!\!\!
{\displaystyle \int_{0}^{\infty}}
dp\, p^{2}
{\displaystyle \int_{-1}^{1} dx}{\displaystyle \int_{\mu-q^{0}}
^{\mu}}d\omega S_{h}(\omega, p)\times\\
& \\
& \times S_{p}(q^{0}+\omega ,{\displaystyle
\sqrt{{\vec{p}\,}^{2}+{\vec{q}\,}^{2}+2pqx}})
\end{array}
\end{equation}
We use the spectral functions calculated in \cite{PED}, but since the
imaginary part of the nucleon self-energy for the hole states is much
smaller than that of the particle states under consideration we make
the approximation of setting to zero $Im \Sigma$ for the hole
states. This was found to be a good approximation in \cite{CIO}. Thus,
we take
\begin{equation}
S_{h}(\omega,p)=\delta(\omega-\tilde{E}(\vec{p}\,))\Theta(\mu-\tilde{E}(p))
\end{equation}
\noindent
where $\tilde{E} (p)$ is the energy associated to a momentum $\vec{p}$
obtained selfconsistently by means of the equation
\vspace{0.2cm}
\begin{equation}
\tilde{E}(\vec{p}\,)=\Frac{{\vec{p}\,}^{2}}{2M_N}+Re\Sigma (\tilde{E}(\vec{p}\,),
\vec{p}\,)
\end{equation}
The chemical potential was then taken as
$$
\mu = \frac{k_F^2}{2M_N} + Re \Sigma (\mu, k_F)
$$
\noindent
where $k_F$ is the Fermi momentum.
It must be stressed that it is important to keep the real part of
$\Sigma$ in the hole states when renormalizing the particle states because
there are pieces in the nucleon self-energy largely independent of the
momentum and which cancel in the $ph$ propagator, where the two
selfenergies subtract.
The effect of the use of the spectral function, accounting for FSI is
a quenching of the quasielastic peak and a spreading of the
strength at higher energy as can be seen in fig. 6.4.
\vskip 0.3cm
\centerline{\protect\hbox{\psfig{file=spectralnew.ps,width=10cm}}}
\vskip 0.1cm
\noindent
{\small {\bf Fig.6.4} Effect of the use of the spectral function in the
evaluation of the Lindhard function.
The uncorrelated Fermi sea results are obtained from eqs.(57),(58).
Those with the medium spectral function, with the same equations
substituting the bare Lindhard function $\bar{{U}}$ by the medium
modified one of eq.(64).
}
\subsection{Polarization (RPA) effects in the quasielastic contribution}
We take now into account polarization effects in the $1p 1h$ excitation,
substituting it by an RPA response as shown diagrammatically in fig.
6.5.
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=burbujas14.ps,width=4cm}}}
\vskip 0.2cm
\vspace*{-0.5cm}
{\small {\bf Fig.6.5} Diagrammatic representation of the polarization
effects in the
1$ph$ excitation.}
\vspace*{0.25cm}
For that purpose we use an effective interaction of the Landau-Migdal
type
\begin{equation}
\begin{array}{ll}
V(\vec{r}_{1},\vec{r}_{2})= & c_{0}\delta(\vec{r}_{1}-\vec{r}_{2})\left\{
f_{0}(\rho)+f_{0}^{\prime}(\rho)\vec{\tau}_{1}\vec{\tau}_{2}\right.+\\
& \\
& +\left.g_{0}(\rho)\vec{\sigma}_{1}\vec{\sigma}_{2}+g_{0}^{\prime}(\rho)
(\vec{\sigma}_{1}\vec{\sigma}_{2})
\vec{\tau}_{1}\vec{\tau}_{2}
\right\}
\end{array}
\end{equation}
and we take the parameterization for the coefficients from ref.
~\cite{SPH}
\begin{equation}
f_{i}(\rho (r))=\Frac{\rho (r)}{\rho (0)} f_{i}^{(in)}+
\left[ 1-\Frac{\rho (r)}{\rho (0)}\right] f_{i}^{(ex)}
\end{equation}
\begin{equation}
\begin{array}{ll}
f_{0}^{(in)}=0.07 & f_{0}^{\prime (ex)}=0.45\\
f_{0}^{(ex)}=-2.15 & c_{0}=380\, MeV fm^{3}\\
f_{0}^{\prime (in)}= 0.33 & \\
g_{0}^{(in)}=g_{0}^{(ex)}=g_{0}=0.575\\
g_{0}^{\prime (in)}=g_{0}^{\prime (ex)}=g_{0}^{\prime}=0.725
\end{array}
\end{equation}
For consistency, in the $S = 1 = T$ channel ($\vec{\sigma} \vec{\sigma}
\vec{\tau} \vec{\tau}$ operator) we have continued to use the interaction used in
~\cite{CAR} which has been used for the renormalization of the pionic
and pion related channels studied in the former sections. There is only
a minor difference of about 4$\%$ in $g^\prime_0$ between the two
parametrizations.
Recalling that we had
\begin{equation}
\Pi^{\mu\nu}_{(1)}=\frac{1}{2} \bar{U}_{N}(q)A_{N}^{\nu\mu}(q)e^{2}
\end{equation}
\noindent
let us take the nonrelativistic reduction of $A^{\nu \mu}$ in
order to see the effects of the RPA renormalizations
\begin{equation}\label{eq:amunu}
\begin{array}{ll}
A^{\mu\nu}\equiv & {\displaystyle \sum_{r,r^{\prime}}}
\chi_{r}\left[F_{1}^{p}(q)\delta^{\mu 0}-i\Frac{\mu_{p}G_{M}(q)}
{2M_{N}}(\vec{\sigma}\times\vec{q}\,)_{i}\delta^{\mu i}
+F_{1}^{p}\Frac{(2\vec{p}+\vec{q}\,)_{i}}{2M_{N}}\delta^{\mu i}
\right]\chi_{r^{\prime}}\times \\
& \\
&\times\chi_{r^{\prime}}\left[ F_{1}^{p}(q)\delta^{\nu 0}+
F_{1}^{p}\Frac{(2\vec{p}+\vec{q}\,)_{i}}{2M_{N}}\delta^{\nu i}
+i\Frac{\mu_{p}G_{M}(q)}
{2M_{N}}(\vec{\sigma}\times\vec{q}\,)_{i}\delta^{\nu i}
\right]\chi_{r}+\\
& \\
& +(p\leftrightarrow n)
\end{array}
\end{equation}
Given the spin-isospin structure,
the electric and magnetic components will be renormalized in the following
way:
{\bf a)} Interaction $\vec{\sigma} \vec{\sigma} \vec{\tau} \vec{\tau}$: is the
one we used to renormalize the pionic related channels in former
sections. It affects only the magnetic components. If we write
\begin{equation}
\begin{array}{ll}
A_{mag.}^{\mu\nu}= & {\displaystyle \sum_{r, r^{\prime}}}
\chi_{r}\left[-i\Frac{\mu_{p}G_{M}(q)}
{2M_{N}}(\vec{\sigma}\times\vec{q})_{i}\delta^{\mu i}
\right]\chi_{r^{\prime}}\times\\
& \\
& \times \chi_{r^{\prime}}\left[i\Frac{\mu_{p}G_{M}(q)}
{2M_{N}}(\vec{\sigma}\times\vec{q})_{i}\delta^{\nu i}
\right]\chi_{r}
\Frac{(1+\tau_{3})}{2}+\\
& \\
& + \mbox{[neutrons]}\Frac{(1-\tau_{3})}{2}
\end{array}
\end{equation}
\noindent
it is easy to see that the magnetic part of $\Pi^{ij}$ becomes
\begin{equation}\label{eq:a}
\begin{array}{ll}
\Pi^{ij}_{mag.}= & \frac{1}{2}\bar{{U}}_{N}A^{ij}_{mag.}(q)e^{2}+
\Frac{e^{2}}{4{M}_{N}^{2}}\Frac{{f}_{\pi NN}^{2}}{{m}_{\pi}^{2}}\Frac{
V_{t}(q)}
{1-\Frac{{f}_{\pi NN}^{2}}{{m}_{\pi}^{2}}V_{t}(q){U}(q)}\times
\\
&
\\
& \times ({\vec{q}\,}^{2}\delta^{ij}-q^{i}q^{j})G_{M}^{2}(q)
\left( \mu_{p}\bar{{U}}_{p}-\mu_{n}\bar{{U}}_{n}\right)^{2}
\end{array}
\end{equation}
\noindent
where $U = U_N + U_\Delta$.
{\bf b)} Interaction $\vec{\tau} \vec{\tau}$:
This interaction selects the non magnetic components of $V^\mu$.
Thus $A^{00}$ and the convective terms of $A^{ij}$ (term with $2 \vec{p} +
\vec{q}$ in eq.(~\ref{eq:amunu})) are renormalized.
However, given the smallness of the convective terms (about 10$\%$
contribution to the transverse response) we shall not consider their
renormalization.
Thus we consider only the modification to $A^{00}$ from this source.
Since $A^{00}$ is given by
\begin{equation}
\begin{array}{ll}
A^{00} & = \left[ {\displaystyle \sum_{r,r^{\prime}}}\chi_{r}F_{1}^{p}(q)\chi_{
^{\prime}}\chi_{r^{\prime}}F_{1}^{p}(q)\chi_{r}\right]
\Frac{(1+\tau_{3})}{2}+\\
& \\
& + \left[ {\displaystyle \sum_{r,r^{\prime}}}\chi_{r}F_{1}^{n}(q)\chi_{r
^{\prime}}\chi_{r^{\prime}}F_{1}^{n}(q)\chi_{r}\right]
\Frac{(1-\tau_{3})}{2}
\end{array}
\end{equation}
\noindent
the renormalized expression for $\Pi^{00}$ will be
\begin{equation}
\Pi^{00}=e^{2}\left\{(F_{1}^{p})^{2}\bar{{U}}_{p}+
(F_{1}^{n})^{2}\bar{{U}}_{n}+\Frac{c_{0}f_{0}^{\prime}(
F_{1}^{p}\bar{{U}}_{p}-F_{1}^{n}\bar{{U}}_{n})^{2}}{
1-c_{0}f_{0}^{\prime}{U}_{N}(q)}\right\}
\end{equation}
\noindent
where in the denominator we do not have now $U_\Delta$ since the operator
$\vec{\tau} \vec{\tau}$ cannot excite $\Delta$ components.
{\bf c)} Interaction $\vec{\sigma} \vec{\sigma}$:
Here again, like in case a), only the magnetic components are modified.
We find
\begin{equation}\label{eq:c}
\begin{array}{ll}
\Pi^{ij}_{mag.}= & \frac{1}{2}\bar{{U}}_{N}A^{ij}_{mag.}(q)e^{2}+
\Frac{e^{2}}{4M^{2}}c_{0}g_{0}\Frac{
1}{1-c_{0}g_{0}{U}_{N}(q)}\times \\
&
\\
& \times ({\vec{q}\,}^{2}\delta^{ij}-q^{i}q^{j})G_{M}^{2}(q)
\left( \mu_{p}\bar{{U}}_{p}+\mu_{n}\bar{{U}}_{n}\right)^{2} \\
&
\\
&+\,\,\, effect\,\,\,of\,\,\,(a)
\end{array}
\end{equation}
The correction from the RPA sum is taken into account by means of the
second term of the right hand side of eq.(~\ref{eq:c}) to which we
should add the same term in eq.(~\ref{eq:a}) coming from the
renormalization with the $\vec{\sigma} \vec{\sigma} \vec{\tau} \vec{\tau}$
operator.
{\bf d)} Scalar interaction:
This one affects $A^{00}$ and we find
\begin{equation}
\Pi^{00}=e^{2}\left\{(F_{1}^{p})^{2}\bar{{U}}_{p}+
(F_{1}^{n})^{2}\bar{{U}}_{n}+\Frac{c_{0}f_{0}(\rho)(
F_{1}^{p}\bar{{U}}_{p}+F_{1}^{n}\bar{{U}}_{n})^{2}}{
1-c_{0}f_{0}{U}_{N}(q)}\right\}\,\,+\,\,\, effect\,\, of\,\,(b)
\end{equation}
We show in figs. 6.6, 6.7, 6.8 and 6.9 the effects
on $R_L$ and $R_T$ ($R_L=-\Frac{ |\vec{q}\,|^{2} }{q^2}W_L$ and $R_T=2W_T$)
of the different polarization terms. The solid line corresponds to the
calculation including these effects and the dashed line, to the calculation
without polarization effects (we are using spectral functions in the
calculation of the Lindhard function):
\vspace*{0.2cm}
\centerline{\protect\hbox{\psfig{file=rlf0esc.ps,width=11.3cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.6.6} Polarization (RPA)
effect (solid line) in the evaluation of
$R_L$: scalar interaction.}
\vspace*{0.1cm}
\centerline{\protect\hbox{\psfig{file=rlfprima0.ps,width=11.3cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.6.7}
Polarization (RPA) effect (solid line) in the evaluation of
$R_L$: $\vec{\tau}\vec{\tau}$ interaction.
}
\newpage
\vspace*{0.4cm}
\centerline{\protect\hbox{\psfig{file=rtss.ps,width=11.5cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.6.8}
Polarization (RPA) effect (solid line) in the evaluation of
$R_T$: $\vec{\sigma}\vec{\sigma}$ interaction.}
\vspace*{0.7cm}
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=rtsstt.ps,width=11.5cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.6.9} Polarization (RPA) effect (solid line) in the evaluation of
$R_T$:
$\vec{\sigma}\, \vec{\sigma}\,\vec{\tau}\,\vec{\tau}$
interaction.}
\vspace*{0.1cm}
\newpage
One possible source of renormalization not yet considered is the one
shown in fig. 6.10, where a $ph$ is attached to one photon and a
$\Delta h$ to the other one, plus any other possible $ph$
or $\Delta h$ excitations in between.
\vspace*{0.3cm}
\centerline{\protect\hbox{\psfig{file=burbuja18.ps,width=5.9cm}}}
\vskip 0.2cm
\vspace*{-1.5cm}
\noindent
{\small {\bf Fig.6.10} Diagrammatic representation of a possible
source of renormalization: a $ph$ is attached to one photon and a
$\Delta h$ to the other one, plus any other possible $ph$
or $\Delta h$ excitations in between.}
\vspace*{0.5cm}
This contribution affects the transverse part and both the quasielastic
as well as the $\Delta$ peak. However, given the small interference
between $ph$ and $\Delta h$ excitations, the contribution of these terms is
not significant. We give, however, the expression here for completeness
\begin{equation}
\begin{array}{ll}
\Pi^{ij}= e\left(\Frac{4}{3}\right)^{2}&\!\! \Frac{f_{\gamma}}{m_{\pi}}
\Frac{\rho}{\left(\sqrt{s}-M_{\Delta}+i\frac{\bar{\Gamma}}{2}-\Sigma_{\Delta}
\right)}i\times\\
&\\
& \times\Frac{V_{t}}{\left(1-{U}
\Frac{{f}_{\pi NN}^{2}}{{m}_{\pi}^{2}}
V_{t}\right)}\Frac{f_{\pi NN}}{m_{\pi}}\Frac{
f^{*}}{m_{\pi}}[{\vec{q}\,}^{2}\delta^{ij}-q^{i}q^{j}]\times\\
& \\
& \times\Frac{G_{M}}{4M_N}[\mu_{p}\bar{{U}}_{p}-\mu_{n}\bar{{U}}_{n}]
\end{array}
\end{equation}
The effect of the polarization is moderate, but relevant when aiming
at a precise description of the process.
\vspace*{0.8cm}
We show in figs. 6.11 and 6.12 the effects of the polarization
in the longitudinal and transverse response functions. The net effect
in the cross section is a quenching in the quasielastic peak and
a spreading of the strength at higher energies.
\vskip 0.1cm
\centerline{\protect\hbox{\psfig{file=sinpolrl.ps,width=11.cm}}}
\vskip 0.1cm
\noindent
{\small {\bf Fig.6.11}
Polarization (RPA) effect in the evaluation of
$R_L$. }
\vspace*{0.1cm}
\vskip 0.6cm
\centerline{\protect\hbox{\psfig{file=sinpolrt.ps,width=11.cm}}}
\vskip 0.1cm
\noindent
{\small {\bf Fig.6.12}
Polarization (RPA) effect in the evaluation of
$R_T$.}
\vspace*{0.1cm}
\subsection{Further considerations}
Some other terms appearing in the generic diagram of fig. 4.11
require some special thought. These are the terms in which one of the
vertices contains the nucleon pole term of the $\gamma^* N \rightarrow
\pi N$ amplitude, while the other one contains all terms but
that one. This is depicted in fig. 6.13.
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=burbujas19.ps,width=9.5cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.6.13} Photon self-energy diagrams in which one of
the vertices contains the
nucleon pole term of the $\gamma^* N \rightarrow
\pi N$ amplitude, while the other one contains all terms but that one.}
\vspace*{0.5cm}
Unlike the case of real photons where only the cut (a) exciting
$2p 2h$ gives rise to an imaginary part, now the cut (b)
placing the $ph$ on shell is a source of imaginary part which produces
strength is the quasielastic peak. The diagrams in fig. (6.13)
for the cut (b) could
be then considered an ordinary $ph$ excitation with a renormalized
vertex. We have evaluated the two sources of
imaginary part in $\Pi^{\mu \nu}$ and find for the cut (a)
\begin{equation}
\begin{array}{ll}
Im \Pi^{\mu\nu}=&\Frac{{f}_{\pi NN}^{2}}{m_{\pi}^{2}}
{\displaystyle \sum_{ij}}{\displaystyle \int}\Frac{d^{4}p}{(2\pi)^{4}}
n_{i}(\vec{p})(2\pi)\Theta (p^{0})
\delta\left(p^{0}-{E}(\vec{p}\,)-Re\Sigma
\left(\Frac{{\vec{p}\,}^{2}}{2M_{N}},\vec{p}\,\right)\right)\times\\
& \\
& \times {\displaystyle \int} \Frac{d^{4}k}{(2\pi)^{4}}F_{\pi}^{4}(k)
\Frac{{\vec{k}}^{2}_{CM}\Theta (k^{0})}{(k^{2}-m_{\pi}^{2})^{2}}2\pi
\Theta(p^{0}+q^{0}-k^{0})\times \\
& \\
&\delta\left(p^{0}+q^{0}-k^{0}-{E}(\vec{p}+\vec{q}-\vec{k}\,)
-Re\Sigma\left( \Frac{(\vec{p}+\vec{q}-\vec{k}\,)^{2}}{2M_N} ,
\vec{p}+\vec{q}-\vec{k}\,\right) \right)\times\\
& \\& \\
&\times (1-n_{j}(\vec{p}+\vec{q}-\vec{k}\,))
\Frac{Im {U}_{\lambda}(k)}{1-\frac{{f}_{\pi NN}^{2}}{m_{\pi}^{2}}V_{l}(k)
{U}(k)}(1-n_{i}(\vec{p}+\vec{q}\,))\times\\
& \\
& \times 2 Re\left\{\Frac{1}{p^{0}+q^{0}-{E}(\vec{p}+\vec{q}\,)
-\Sigma_{N}\left(\frac{(\vec{p}+\vec{q}\,)^{2}}{2M_N}, \vec{p}+\vec{q}\,\right)}
\right.\times \\
& \\
& \left.Tr\left(\bar{\cal{M}}^{\mu}_{NP}(i\rightarrow j)
\bar{\cal{M}}^{\dagger \nu}_{{}_
{KR,PP,NPC,\Delta,\Delta C}}
(i\rightarrow j)\right)\right\}
\end{array}
\end{equation}
\noindent
where $\bar{\cal{M}}_{NP}$ is the nucleon pole amplitude of $\gamma^* N
\rightarrow \pi N$ omitting the nucleon propagator. On the other
hand the contribution of the cut (b) is given by
\begin{equation}
\begin{array}{ll}
Im\Pi^{\mu\nu}=& -{\displaystyle \sum_{ij}}{\displaystyle \int}
\Frac{d^{4}p}{(2\pi)^{4}}n_{i}(p) \delta\left(p^{0}
-E(\vec{p}\,)-Re\Sigma\left(\Frac{{\vec{p}\,}^{2}}{2M_N}, \vec{p}\,\right)
\right)
\times\\
& \\
&\times\Theta (p^{0}) (2\pi)^{2}(1-n_{i}(\vec{p}+\vec{q}\,))
\Theta(p^{0}+q^{0})\times\\
&\\
&\times\delta\left(p^{0}+q^{0}-{E}(\vec{p}+\vec{q}\,)
-Re\Sigma\left( \Frac{(\vec{p}+\vec{q}\,)^{2}}{2M_N} ,
\vec{p}+\vec{q}\,\right)\right)\times\\
&\\
&{\displaystyle \int }
\Frac{d^{4}k}{(2\pi)^{4}}F_{\pi}^{2}(k) Im\left\{\Frac{(1-n_{j}(p+q-k))}{
p^{0}+q^{0}-k^{0}-{E}(\vec{p}+\vec{q}-\vec{k}\,)+i\epsilon}\right.\times\\
& \\
& \times\left(\Frac{1}{k^{2}-m_{\pi}^{2}-\Pi}-\Frac{1}{
{k}^{2}-m_{\pi}^{2}+i\epsilon}\right)\times\\
& \\
& \times Tr\left.\left(\bar{\cal{M}}^{\nu}_{{}_
{KR,PP,NPC,\Delta,\Delta C}}
(i\rightarrow j)\bar{\cal{M}}^{\dagger\mu}_{NP}(i\rightarrow j)
\right)\right\}
\end{array}
\end{equation}
\noindent
where $\Pi$ is the pion self-energy in the nuclear medium
\begin{equation}
\Pi={\vec{k}}^2_{CM}\left(\Frac{f_{\pi NN}}{m_{\pi}}\right)^2
{F}^{2}_{\pi}(k^2)\Frac{U_{\lambda}(k)}{1-g^{\prime}
\left(\Frac{f_{\pi NN}}{m_{\pi}}\right)^2
U(k)}
\end{equation}
\noindent
the subtraction of the free pion propagator
appearing in eq.(80) guarantees that in the limit
$\rho \rightarrow 0$ the correction to the $\gamma NN$ vertex
vanishes as it should be.
\subsection{Considerations on gauge invariance}
Most of the theoretical models found in the literature
on inclusive $(e,e^\prime)$ scattering from nuclei do not preserve
gauge invariance. The requirement of invariance under gauge transformations
leads to relations between the components (charge and
spatial current) of the hadronic current which determines the nuclear
response. Actually, the longitudinal (charge) multipoles are related
to the two transverse (spatial current) multipoles of the electric
type~\cite{Hei}. Several prescriptions have been used to restore
gauge invariance~\cite{Hei},~\cite{Friar}. However,
in a recent work~\cite{AAL} the arbitrariness
of the most common prescriptions
is discussed in detail
with the conclusion that
the standard procedures to impose gauge invariance in
calculations, based on models which do not verify it, are misleading
and for very low nuclear excitation energies
do not ensure at all that a better, or a more reasonable,
description of the data will be obtained.
Our model, as we shall see, is gauge invariant at the lowest order
(impulse approximation) of
the density expansion and this symmetry is
only partially broken when some non-leading density corrections are
included. In what follows, we will study the consequences of the
partial breaking
of the gauge symmetry for the kinematics studied in this paper (from the
quasielastic peak to the $\Delta$ excitation region), which involves
larger nuclear excitation energies than those studied
in~\cite{AAL}. We will also study the feasibility of non-gauge
invariant models to disentangle between longitudinal and transverse channels.
The unpolarized cross section for inclusive $(e,e^\prime)$ scattering from
nuclei is given by (eq.(29)):
\begin{equation}\label{eq:j1}
\frac{\displaystyle{d^{2}\sigma}}{\displaystyle{d\Omega^{\prime}_e
dE^{\prime}_e}}
=\frac{\displaystyle{\alpha^{2}}}{\displaystyle{q^{4}}}\frac{\displaystyle{
|\vec{k}^{\,\prime}|}}{\displaystyle{|\vec{k}|}}L^{\mu\nu}W_{\mu\nu}
\end{equation}
Lorentz, space-inversion, time-reversal and gauge invariance
constraint the form of the hadronic
tensor $W_{\mu\nu}$, which determines the nuclear
response. Indeed, the most general expression for this tensor assuming
the latter symmetries is given by~\cite{Muta}:
\begin{eqnarray}\label{eq:j2}
W^{\mu\nu} & = & \{\frac{q^\mu q^\nu}{q^2}-g^{\mu\nu}\}W_1 \nonumber\\
& + & \left \{ \left ( P^\mu -\frac{P.q}{q^2}q^\mu\right )
\left ( P^\nu -\frac{P.q}{q^2}q^\nu\right ) \frac{W_2}{M^2_A} \right \}
\end{eqnarray}
with $q$ and $P$ the virtual photon and initial hadronic system four-momenta
respectively and $M_A^2 = P^2$.
The structure functions $W_{1,2}$ are unknown scalar functions
of the virtual photon variables which determine the nuclear response.
Using the expression for the hadronic tensor of eq.~(\ref{eq:j2})
and taking $\vec{q}$ in the $z$ direction the cross
section of eq.~(\ref{eq:j1}) in the lab system becomes (eq. (31)):
\begin{equation}\label{eq:j3}
\frac{\displaystyle{d^{2}\sigma}}{\displaystyle{d\Omega^{\prime}_e
dE^{\prime}_e}}=
\left(\frac{\displaystyle{d\sigma}}{\displaystyle{d\Omega}}\right)_{Mott}
\left(-\frac{\displaystyle{q^{2}}}{\displaystyle{|\vec{q}\,|^{2}}}\right)
\left\{W_{L}(\omega ,|\vec{q}\,|)+
\frac{\displaystyle{W_{T}(\omega,|\vec{q}\,|)}}
{\displaystyle{\epsilon}}\right\}
\end{equation}
\noindent
where the structure functions $W_L$ and $W_T$ are given in terms of
$W_1$ and $W_2$ by:
\begin{eqnarray}\label{eq:j4}
W_L &\equiv& -W_1 -\frac{|\vec{q}\,|^2}{q^2}W_2 =
-\frac{q^2}{\omega^2}W^{zz}= -\frac{q^2}{|\vec{q}\,|^2}W^{00}=
-\frac{q^2}{\omega|\vec{q}\,|}W^{0z}\\
W_T &\equiv& W_1 = W^{xx}= W^{yy}
\end{eqnarray}
Let us now suppose that the hadronic model does not preserve gauge
invariance. In these circumstances the hadronic tensor (we will call
it ${\cal W}^{\mu\nu}$ to differentiate it from the one defined in
eq.~(\ref{eq:j2})) is not conserved (ie, $q_\mu {\cal W}^{\mu\nu} \ne
0$, $q_\nu {\cal W}^{\mu\nu} \ne
0$ ), and is now given in terms of four independent
functions:
\begin{eqnarray}\label{eq:j5}
{\cal W}^{\mu\nu} & = & \{\frac{q^\mu q^\nu}{q^2}-g^{\mu\nu}\}W_1 \nonumber\\
& + & \left \{ \left ( P^\mu -\frac{P.q}{q^2}q^\mu\right )
\left ( P^\nu -\frac{P.q}{q^2}q^\nu\right ) \frac{W_2}{M^2_A} \right \}
\nonumber\\
& + & W_3 \frac{q^\mu q^\nu}{q^2} + W_4 \frac{P^\mu P^\nu}{M^2_A}
\end{eqnarray}
Because of the loss of gauge invariance now ${\cal W}^{00}$,
${\cal W}^{0z}$ and ${\cal W}^{zz}$ are no longer related and become
independent. Thus, we have now:
\begin{eqnarray}\label{eq:j6}
W_L &\equiv& -W_1 -\frac{|\vec{q}\,|^2}{q^2}W_2 =
\frac{|\vec{q}\,|}{\omega}{\cal W}^{0z}-{\cal W}^{zz}\\
W_T &\equiv& W_1 = {\cal W}^{xx}= {\cal W}^{yy}\\
W_3 & = & \frac{\omega}{|\vec{q}\,|}{\cal W}^{0z}-{\cal W}^{zz}\\
W_4 & = & {\cal W}^{00} + {\cal W}^{zz} - \left
(\frac{|\vec{q}\,|}{\omega} + \frac{\omega}{|\vec{q}\,|}\right ){\cal W}^{0z}
\end{eqnarray}
If gauge invariance is restored and therefore the hadronic tensor is
conserved, the response functions $W_3$ and $W_4$ vanish and $W_L$
reduces to any of the expressions of eq.~(\ref{eq:j4}). With this new
hadronic tensor the differential cross section is now given by:
\begin{equation}\label{eq:j7}
\frac{\displaystyle{d^{2}\sigma}}{\displaystyle{d\Omega^{\prime}_e
dE^{\prime}_e}}=
\left(\frac{\displaystyle{d\sigma}}{\displaystyle{d\Omega}}\right)_{Mott}
\left(-\frac{\displaystyle{q^{2}}}{\displaystyle{|\vec{q}\,|^{2}}}\right)
\left\{ {\cal W}_{L}(\omega ,|\vec{q}\,|) +
\frac{\displaystyle{W_{T}(\omega,|\vec{q}\,|)}}
{\displaystyle{\epsilon}}\right\}
\end{equation}
with
\begin{eqnarray}\label{eq:j8}
{\cal W}_L (\omega ,|\vec{q}\,|) &=& \left ( W_{L}(\omega ,|\vec{q}\,|) -
\frac{|\vec{q}\,|^2 }{q^2}W_{4}(\omega ,|\vec{q}\,|)\right
)\nonumber\\
&=& -\frac{\omega^2}{q^2}{\cal W}^{zz}-\frac{|\vec{q}\,|^2}{q^2}{\cal
W}^{00} + 2 \frac{|\vec{q}\,|\omega}{q^2} {\cal
W}^{0z}
\end{eqnarray}
The function $W_3$ does not appear in the expression for the cross
section because the leptonic tensor is conserved. Note that, one can
still factor out the differential cross section in the form $A+B/\epsilon$
and therefore, despite the breaking of gauge invariance, one can still
compare the results to the experimental response functions obtained via the
Rosenbluth plot. Then, the breaking of gauge invariance in the theoretical
model for the nuclear response leads to a redefinition of the response
function which has to be compared to the experimental one, in the
longitudinal channel, and thus one should compute ${\cal W}_L$ given
in eq.~(\ref{eq:j8}) instead of $W_L$ of eq.~(\ref{eq:j4}). In fact,
the latter one is not well defined and there is an arbitrariness in
its definition because the $00$, $0z$ and $zz$ components of
${\cal W}^{\mu\nu}$ are no longer related.
Traditionally, the longitudinal response function is calculated from
the ``charge-charge'' component of the hadronic tensor
($ -q^2/|\vec{q}\,|^2{\cal W}^{00}$). In general, the response
function calculated in this way will differ from that calculated by
means of eq.~(\ref{eq:j8}). Now we will examine
the difference between both approaches
as a function of the energy and momentum
transferred to the nucleus. In order to do that, we define
\begin{eqnarray}\label{eq:j9}
{\cal W}^{zz} & = & \frac{\omega^2}{|\vec{q}\,|^2}{\cal W}^{00} +
\Delta {\cal W}^{zz}\\
{\cal W}^{0z} & = & \frac{\omega}{|\vec{q}\,|}{\cal W}^{00} +
\Delta {\cal W}^{0z}
\end{eqnarray}
where $\Delta {\cal W}^{zz}$ and $\Delta {\cal W}^{0z}$ account for
the breaking of the gauge symmetry. By construction one expects:
\begin{eqnarray}\label{eq:j10}
\frac{\Delta {\cal W}^{zz}}{{\cal W}^{00}} &\approx &
\delta_1\frac{\omega^2}{|\vec{q}\,|^2} \\
\frac{\Delta {\cal W}^{0z}}{{\cal W}^{00}} &\approx &
\delta_2\frac{\omega}{|\vec{q}\,|}
\end{eqnarray}
\noindent
where $\delta_1$ and $\delta_2$ would be 1 in the case that
$\Delta {\cal W}^{zz}={\cal W}^{zz}$ and $\Delta {\cal W}^{0z}={\cal W}^{0z}$.
We certainly expect that in our case $\delta_1$ and $\delta_2$ are smaller
than 1 because at order $\rho$ in the density expansion
we exactly fulfill gauge invariance (see comments below).
Taking a conservative point of view one sees that
the ratios
in eqs. (96), (97) are at most of order 1. Using the definitions of
eq.~(\ref{eq:j9}), we can now write
\begin{equation}\label{eq:j11}
{\cal W}_L = -\frac{q^2}{|\vec{q}\,|^2}{\cal W}^{00} \left\{
1 + \frac{\omega^2|\vec{q}\,|^2}{q^4}\frac{\Delta
{\cal W}^{zz}}{{\cal W}^{00}} - 2\frac{\omega|\vec{q}\,|^3}{q^4}\frac{\Delta
{\cal W}^{0z}}{{\cal W}^{00}} \right\}
\end{equation}
The size of the corrections to the ``charge-charge'' prescription (the
term proportional to 1 in the formula)
traditionally used in the literature depends on the kinematics
under study. Here we will pay a special attention to three different
regions (keeping always the momentum transferred
to the nucleus smaller than about 500 MeV) : $\Delta$-resonance and quasielastic peaks and
the region
({\it dip}) between both peaks.
\begin{itemize}
\item {\underbar{Quasielastic peak}}: In this region we
have $\omega \approx |\vec{q}\,|^2/2M_N$ and then the coefficients
of the ratios $\frac{\Delta {\cal W}^{zz}}{{\cal W}^{00}}$ and
$\frac{\Delta {\cal W}^{0z}}{{\cal W}^{00}}$ in eq.~(\ref{eq:j10})
turn out to be
\begin{eqnarray}
\frac{\omega^2|\vec{q}\,|^2}{q^4}&\approx&
\frac{|\vec{q}\,|^2}{4M_N^2} \\
2\frac{\omega|\vec{q}\,|^3}{q^4}&\approx&\frac{|\vec{q}\,|}{M_N}
\end{eqnarray}
Thus, taking also into account the estimates of eq.~(\ref{eq:j10}),
one finds corrections to the ``charge-charge'' prescription of the
order of $\delta_2|\vec{q}\,|^2/2M_N^2$ which at most could be of the order of
$5-10\%$ for the momenta and energies transfers studied in this
paper,
assuming $\delta_2\approx 1$,
which is certainly an overestimate for the
reasons pointed above. From this discussion, in this region
we have decided to use the traditional
prescription ``charge-charge'' to compute the longitudinal response
function.
\item {\underbar{{\it Dip} area}}: In this region the conclusions are
similar to those drawn in the previous
point. However, we would like to point out that gauge symmetry
breaking corrections to the longitudinal response function are not now
as small as before.
\item {\underbar{$\Delta$-resonance peak}}: In this region
$\omega/|\vec{q}\,| \approx 1 $ and the situation is
radically different. The corrections due to gauge
symmetry breaking are significantly more important than in the quasi-free
scattering region. For instance, taking the incoming electron energy
equal to 620 MeV and the outgoing electron scattering angle equal to
$60^0$, one finds values for the coefficients
of the ratios $\frac{\Delta {\cal W}^{zz}}{{\cal W}^{00}}$ and
$\frac{\Delta {\cal W}^{0z}}{{\cal W}^{00}}$ in eq.~(\ref{eq:j11}) of the
order of two.
Thus the corrections to unity
in the bracket of eq.~(\ref{eq:j11}), are of the order
of $2(\delta_1-\delta_2)$, much larger than in the quasielastic peak.
We have evaluated $\delta_1$ and $\delta_2$ in this region
and we find $\delta_1 \approx 0.01-0.06$, $\delta_2 \approx 0.01\,-\,0.02$
and $2(\delta_1-\delta_2)\approx 0.0\,-\,0.08$. Hence a 10$\%$ error in
the longitudinal response due to the breaking of gauge invariance
of our results seems realistic in this region.
In any case, we should mention that the contribution of the
longitudinal response to the cross section is very small
here and hence theoretical cross sections are largely free
of uncertainties due to the small breaking of gauge invariance.
Since there is no experimental separation of $R_L$ and $R_T$
in this region, we do not give these results either.
\end{itemize}
We finish this section discussing the origin of the breaking of the
gauge invariance within our model. Let us consider the
diagram of fig.~6.1 whose contribution to the virtual photon
self-energy in the medium is
given in eqs.(53) and (54). One can easily check that the imaginary
part (when the intermediate nucleons are put on shell)
of this self-energy is gauge invariant
($q_\mu\Pi^{\mu\nu}\propto q_\mu V^\mu=0$). The
first medium correction is given by the diagram
depicted in fig.6.2. When the two $ph$
excitations are put on shell, this diagram contributes to the
imaginary part of the photon self-energy. This new
contribution is not gauge invariant because the intermediate nucleon
with four momentum $p+q$ is not on shell and then the contraction
$q_\mu V^\mu$ does not vanish now.
However, to this level we restore gauge invariance
because we consider not only the term of fig. 6.2, but also
all terms implicit in fig. 4.9. Though the $NP$ amplitude in not gauge
invariance by itself, the thick dots of fig. 4.9 account for the six
amplitudes (NP+NPC+KR+PP+DP+DPC) of our model for the $eN \to e^\prime
N\pi$ reaction. In section 3, we fixed the different form-factors
entering in the amplitudes to end up with
a gauge invariant model (Eqs.(8-9)).
Thus, the leading terms in our density expansion (fig.6.1, fig. 4.2 and
fig. 4.9) lead to a gauge invariant photon self-energy.
However, we break again the gauge invariance in section 4.7
when we include the polarization corrections to the 36 diagrams of
fig. 4.9: we do not
renormalize in the same way for instance the $KR \times KR $ term
(which is purely longitudinal and therefore gets renormalized only
with $V_l$) than the $NP \times NP $ term (which
contributes to both longitudinal and transverse channels and thus
gets renormalized not only
with $V_l$ but also with $V_t$). As a consequence, the cancellations
in eq.(8) which ensured the gauge invariance of the model are altered.
\section{Results}
We have already shown results on the different effects in previous
sections. Here we will show results with emphasis in comparison with
experiment.
Let us first show results in the quasielastic peak in figs. 7.1, 7.2
we show results for $R_L$ and $R_T$
for $^{12}$C and compare them to the
data of \cite{DLA}. The lower line shows the results obtained
with the medium spectral function, while the upper one includes also the rest of the
effects discussed in the former section.
\vskip 0.3cm
\centerline{\protect\hbox{\psfig{file=rlc12300.ps,width=11.5cm}}}
\vskip 0.1cm
\noindent
{\small {\bf Fig.7.1} Calculation of $R_L$ for $^{12}$C. The lower line
in the high energy region
corresponds to the result obtained with
the contribution of the $1p1h$ excitation (fig. 6.3) using the
medium spectral function of eq.(64). The upper line is the result
when one adds the rest of contributions: vertex corrections
(fig. 6.13), two body absorption diagrams (fig. 4.9), $(\gamma ^*, \pi)$
terms (fig. 4.2), $(\gamma ^*, 2 \pi)$ related terms (fig. 4.15), etc.
Experimental data from \cite{DLA}.}
\vspace*{0.1cm}
\centerline{\protect\hbox{\psfig{file=rtcarbon300.ps,width=11.5cm}}}
\vskip 0.1cm
\noindent
{\small {\bf Fig.7.2}
Calculation of $R_T$ for $^{12}$C. Same meaning of the lines
as in fig. 7.1. Experimental data from \cite{DLA}.
}
\vspace*{0.6cm}
In fig. 7.3, 7.4, we show results for $^{40}$Ca compared to the data
of~\cite{JOU} (fig. (7.3) and lower points in fig. (7.4))
and those of~\cite{MEZ} (upper points in fig. (7.4)).
\centerline{\protect\hbox{\psfig{file=rlca410.ps,width=11.5cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.7.3}
Calculation of $R_L$ for $^{40}$Ca.
Same meaning of the lines
as in fig. 7.1.
Experimental data from \cite{JOU}.
}
\vspace*{1cm}
\centerline{\protect\hbox{\psfig{file=rtcalcio410.ps,width=11.5cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.7.4} Calculation of $R_T$ for $^{40}$Ca.
Same meaning of the lines
as in fig. 7.1.
Experimental data from \cite{JOU} (lower points) and \cite{MEZ} (upper points). }
\vspace*{0.1cm}
As one can see, we find a good agreement with the recent
reanalysis of \cite{JOU}.
On the other hand much of the work done here has gone into the
evaluation of two body mechanisms. In fig. 7.5 we show the results
for two body photon absorption (solid line) and compare them to pion
production (dotted line).
Similarly, in fig. 7.6 we show the
contribution of three body photon absorption (solid line) versus the
two body one (dotted line). We can see that at low energies,
the contribution of three body absorption is negligible while at
energies around 450 MeV the three body contribution becomes sizeable.
These results agree qualitatively with the findings of \cite{AO}
for real photons. Let us recall that this classification corresponds
to the primary step in the collision. The particles produced still
undergo secondary collisions in their way out of the nucleus. This
does not change the inclusive cross section but redistributes the
strength. The treatment of this FSI and the evaluation of the exclusive
channels will be treated in a forthcoming paper \cite{JUA}.
\centerline{\protect\hbox{\psfig{file=1pion2body.ps,width=11.3cm}}}
\vskip 0.2cm
\noindent
{\small {\bf Fig.7.5}
Two body photon absorption (solid line) versus pion
production (dotted line) for $^{12}$C.
}
\vspace*{0.1cm}
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=3bodysec.ps,width=11.3cm}}}
\vskip 0.2cm
\noindent
\centerline{\small {\bf Fig.7.6}
Three body photon absorption (solid line) versus the
two body one (dotted line) for $^{12}$C.
}
\vspace*{0.1cm}
Finally let us see the global results including the quasielastic
peak, the dip region and the delta region. They can be
seen in figs. 7.7, 7.8 and 7.9 for the nuclei of $^{12}C$ and $^{208}Pb$.
The global agreement is good and the three regions are well reproduced (a bit
overestimated for $^{208}Pb$).
\centerline{\protect\hbox{\psfig{file=newcarbonsec.ps,width=11.5cm}}}
\vskip 0.2cm
\noindent
{\small{\bf Fig.7.7} Inclusive $(e,e^{\prime})$ cross section for $^{12}C$.
$E_e=620$ MeV and $\theta_e=60^0$. The dotted line
corresponds to the pion production contribution.
Experimental data from \cite{BAR}.
}
\centerline{\protect\hbox{\psfig{file=carbon36inclusivo.ps,width=11.5cm}}}
\vskip 0.2cm
\noindent
{\small{\bf Fig.7.8}
Inclusive $(e,e^{\prime})$ cross section for $^{12}C$.
$E_e=680$ MeV and $\theta_e=36^0$. Experimental data from \cite{BAR}.
}
\vspace*{0.1cm}
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=plomoinclusivo.ps,width=12cm}}}
\vskip 0.2cm
\noindent
{\small{\bf Fig.7.9}
Inclusive $(e,e^{\prime})$ cross section for $^{208}Pb$.
$E_e=645$ MeV and $\theta_e=60^0$. The dotted line
corresponds to the $1p 1h$ excitation
contribution. Experimental data from \cite{ZGH}.
}
\vspace*{0.1cm}
In fig. 7.7 we also show with a dotted line the results for pion
production.
In fig. 7.9 instead we show with a dotted line the results for
the $1p 1h$ excitation alone.
\section{Conclusions}
We have undertaken the task of constructing a microscopic many body
model of the $(e,e^{\prime})$ reaction including all the reaction channels
which appear below $\omega=500-600$ MeV, and which is suited to study
the inclusive $(e,e^{\prime})$ reaction from the quasielastic peak
up to the $\Delta$ peak, passing through the dip region. Although
many studies have been devoted to particular
energy regions of the spectrum, this is the first work, to our knowledge,
which ranges this wide energy spectrum.
Our model has no free parameters. All the input consists of basic couplings
of photons to nucleons and isobars, and some phenomenological inputs,
as correlations, which has been tested in former pionic reactions.
We include explicitly the $1N$ knockout channel, the virtual photon
absorption by pairs or trios of particles, the pion production
plus exchange currents mechanisms tied to the $(\gamma^*,2\pi)$
channel and which contribute to $(\gamma^*,NN\pi)$ or
$(\gamma^*,NNN)$ channels.
We include effects which have been found important in earlier works,
like polarization, renormalization of $\Delta$ properties
in a nuclear medium, FSI effects through the use of spectral
functions and meson exchange currents.
The meson exchange currents are generated in
a systematic way from a model
for the elementary pion electroproduction on the nucleon, which
reproduces accurately the experimental data.
We have payed some attention to the question of gauge invariance, showing
that it is preserved in our approach in leading order
of the density expansion. We also show that the appropriate
prescription to evaluate the longitudinal response
is from the $W^{00}$ component of the hadronic tensor, which minimizes the breaking
of gauge invariance at higher orders in $\rho$.
We evaluate cross sections in the energy range from the quasielastic peak
to the $\Delta$ peak and find good agreement with experimental
data. The three traditional regions: quasielastic peak, dip
region and delta peak, are well reproduced in our scheme.
We also separate the longitudinal and transverse response functions
in the quasielastic peak and find good agreement with the latest
results of the analysis of Jourdan from the world set of data.
We have used the technique of the local density
approximation, which has been shown before to be particularly suited
to deal with inclusive cross sections and which makes
unnecessary the use of sophisticated finite nuclei wave functions.
Finally, the method used here allows the separation of the
contribution of different channels to the inclusive
cross section. This information is the seed to produce
exclusive cross sections like $(e,e^{\prime}N)$,
$(e,e^{\prime}NN)$, $(e,e^{\prime}\pi)$, $(e,e^{\prime}\pi N)$
etc. However, this still requires to follow the fate of all the particles
produced from their production point in the nucleus,
which is usually done using Monte Carlo simulation techniques,
and this will be the subject of some future work.
\vspace*{0.5cm}
We would like to acknowledge useful discussions with
R.C. Carrasco, C. Garc\'{\i}a-Recio and
A. Lallena. This paper is partially
supported by CICYT contract no. AEN 96-1719.
One of us (J. Nieves) thanks to DGES contract PB95-1204.
\newpage
{\Large {\bf Appendix}}
\vspace*{0.6cm}
The Galilean invariant vertices
which appear in the model for
$e N \rightarrow e \pi N$, are:
\begin{description}
\item{{\bf (a)}} $\gamma NN$ vertex (fig. 3.1(a)):
\begin{equation}
\begin{array}{rcl}
{V}^{\mu}_{\gamma N N} & = &-ie
\left\{
\begin{array}{c}
F_{1}^{N}(q^{2})\\
\\
F_{1}^{N}(q^{2})\left[ \frac{\displaystyle{
\vec{p}+{\vec{p}\,}^{\prime}}}{\displaystyle{2M_{N}}}\right]
+i\frac{\displaystyle{\vec{\sigma}\times\vec{q}}}{\displaystyle{2M_{N}}}
G^{N}_{M}(q^{2})
\end{array}
\right\}
\end{array}
\end{equation}
\item{{\bf (b)}} $\gamma N\Delta$ vertex
(fig. 3.1(c)):
\begin{equation}
\begin{array}{rcl}
{V}^{\mu}_{\gamma N \Delta} & = &
\displaystyle{\sqrt{\Frac{2}{3} }}
\frac{\displaystyle{f_{\gamma}(q^{2})}}{\displaystyle{m_{\pi}}}
\frac{\displaystyle{\sqrt{s}}}{\displaystyle{M_{\Delta}}}
\left\{
\begin{array}{c}
\frac{\displaystyle{\vec{p}_{\Delta}}}{\displaystyle{\sqrt{s}}}
(\vec{S}^{\dagger}\times \vec{q}\,) \\
\\
\frac{\displaystyle{{p}^{0}_{\Delta}}}{\displaystyle{\sqrt{s}}}
\left\{
\vec{S}^{\dagger}\times \left(\vec{q}\,-\,
\frac{\displaystyle{{q}^{0}}}{\displaystyle{{p}^{0}_{\Delta}}}
\vec{p}_{\Delta}
\right) \right\}
\end{array}
\right\}
\end{array}
\end{equation}
\item{{\bf (c)}} $\pi N\Delta$ vertex
(fig. 3.1(d)):
\begin{equation}\label{eq:pind}
V_{\pi N \Delta}
=I
\frac{\displaystyle{{f}^{*}}}{\displaystyle{{m}_{\pi}}}
\vec{S}^{\dagger}. \left(\vec{k}\,-\,
\frac{\displaystyle{{k}^{0}}}{\displaystyle{\sqrt{s}}}
\vec{p}_{\Delta}
\right)
\end{equation}
\item{{\bf (d)}} $\pi N N$ vertex
(fig. 3.1(b)):
\begin{equation}\label{eq:piNN}
V_{\pi N N}
=\frac{\displaystyle{{f}_{\pi NN} }}{\displaystyle{{m}_{\pi}}}
B(N,N^{\prime}\pi)
\left\{\vec{\sigma}\,\vec{k}\,-\,
\frac{\displaystyle{{k}^{0}}}{\displaystyle{2M_{N}}}
\vec{\sigma}\,(\vec{p}\,+\,\vec{p}\,^{\prime})
\right\}
\end{equation}
\end{description}
\noindent
where $\vec{q}$, $\vec{p}$, ${\vec{p}\,}^{\prime}$,
$\vec{p}_{\Delta}$ y $\vec{k}$ are
the photon, incoming nucleon, outgoing nucleon and pion
momenta, respectively;
$\sqrt{s}$, the invariant energy in the
$\gamma^{*}\,N$ system
and $M_N$, $m_{\pi}$ and $M_{\Delta}$
are the nucleon, pion and delta resonance masses.
In equations (\ref{eq:pind})
and (\ref{eq:piNN}) we are including
the corresponding isospin factors
$I$ y $B(N,N^{\prime}\pi)$, respectively.
Besides the vertices shown in fig. 3.1,
in our elementary model for electroproduction two more vertices appear:
\vskip 0.2cm
\centerline{\protect\hbox{\psfig{file=verticesapendice.ps,width=6.7cm}}}
In fig. A.1(a) one can see the seagull vertex. It appears from the
${\cal{L}}_{\pi NN}$ lagrangian via minimal coupling.
This vertex is exactly zero for the
${\pi}^{0} n$ and
${\pi}^{0} p$ channels and has the following expression:
\begin{equation}
{V}^{\mu}_{seagull}=e
\frac{\displaystyle{{f}_{\pi NN} }}{\displaystyle{{m}_{\pi}}}
B(N,N^{\prime}\pi)F_AC^{\mu}
\end{equation}
\noindent
where
$$
\begin{array}{lll}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\,\,\,\,\,\,\,\,\,
\,\,\,C^{\mu}(\pi^{-}p)=
\left(
\begin{array}{c}
\frac{\displaystyle{\vec{\sigma}\,(2\vec{p}\,+\,\vec{q}\,-\,\vec{k})}}
{\displaystyle{2M_{N}}}\\
\\
\vec{\sigma}
\end{array}
\right)
&\,\,\,\, ; &
C^{\mu}(\pi^{0} n)=0
\end{array}
$$
$$
\begin{array}{lll}
C^{\mu}(\pi^{+}n)=
\left(
\begin{array}{c}
\frac{\displaystyle{\vec{\sigma}(\vec{k}\,-\,\vec{q}\,-\,2\vec{p})}}
{\displaystyle{2M_{N}}}\\
\\
\vec{\sigma}
\end{array}
\right)
& ; &
C^{\mu}(\pi^{0}p)=0
\end{array}
$$
In figure A.1, the vertex (b) corresponds to the
$\pi \pi \gamma^{*}$ coupling and it is defined as:
\begin{equation}
{V}^{\mu}_{\pi \pi \gamma^{*}}=ie(k^{\mu}\,+\,{k^{\prime}}^{\mu})
\end{equation}
With respect to form factors and coupling constants,
their expressions are the following:
\vspace*{0.3cm}
We use Sachs Form Factors:
\begin{equation}
\begin{array}{ll}
{G}^{N}_{M}(q^2) =
\frac{\displaystyle{{\mu}_{N}}}{\displaystyle{
\left(1-
\frac{\displaystyle{{q}^{2}}}{\displaystyle{{\Lambda}^{2}}}
\right)^2
}} & ; \,\,\,
{G}^{N}_{E}(q^2) =
\frac{\displaystyle{1}}{\displaystyle{
\left(1-
\frac{\displaystyle{{q}^{2}}}{\displaystyle{{\Lambda}^{2}}}
\right)^2
}}
\end{array}
\end{equation}
\noindent
with $\Lambda^{2}=0.71\,\,GeV^2$; $\mu_{p}=2.793$; $\mu_n=-1.913$.
The relationship between ${F}_{1}^{p}(q^2)$ (Dirac form factor)
and ${G}_{E}^{p}$ is:
\begin{equation}\label{eq:f1p}
{F}^{p}_{1}(q^2) ={G}_{E}^{p}
\frac{\displaystyle{
\left(1-
\frac{\displaystyle{{q}^{2}}}{\displaystyle{4{M_N}^{2}}}
\mu_p
\right)
}}{\displaystyle{
\left(1-
\frac{\displaystyle{{q}^{2}}}{\displaystyle{4{M_N}^{2}}}
\right)
}}
\end{equation}
\noindent
and ${F}^{n}_{1}=0$.
\vspace*{0.2cm}
For the rest of form factors and coupling constants we take:
\begin{equation}
\begin{array}{lll}
\frac{\displaystyle{{f}^{2}_{\pi NN}}}{\displaystyle{4\pi}}\, =\, 0.08\,\,;&
\frac{\displaystyle{{{f}^{*}}^{2}}}{\displaystyle{4\pi}}\, =\, 0.36\,\,;&
{F}_{A}(q^2) =
\frac{\displaystyle{1}}{\displaystyle{
\left(1-
\frac{\displaystyle{{q}^{2}}}{\displaystyle{{M_{A}}^{2}}}
\right)^2
}}
\end{array}
\end{equation}
\noindent
where ${M}_{A}=1.08\,GeV$.
\begin{equation}
{F}_{\gamma \pi \pi}(q^2) =
\frac{\displaystyle{1}}{\displaystyle{
\left(1-
\frac{\displaystyle{{q}^{2}}}{\displaystyle{p^{2}_{\pi}}}
\right)
}}
\end{equation}
\noindent
with ${p}^{2}_{\pi}=0.47\,GeV^2$.
\begin{equation}
{F}_{\pi}(q^2) =
\Frac{{\Lambda}_{\pi}^2-{m}_{\pi}^2}{{\Lambda}_{\pi}^2-q^2}
\,\,\,;\,\,\,\,{\Lambda}_{\pi}\sim 1250 \,MeV
\end{equation}
\begin{equation}
{f}_{\gamma}(q^2) =
f_{\gamma}(0)
\frac{\displaystyle{
\left(1-
\frac{\displaystyle{{q}^{2}}}{\displaystyle{
\left(M_{\Delta}+M_{N}\right)^{2}}}
\right)
}}
{\displaystyle{
\left(1-
\frac{\displaystyle{{q}^{2}}}{\displaystyle{4{M_{N}}^{2}}}
\right)
}}\,
\frac{\displaystyle{{G}_{M}^{p}(q^2)}}{\displaystyle{\mu_{p}}}
\frac{\displaystyle{ \left(M_{\Delta}+M_{N}\right)^{2} }}
{\displaystyle{({M_{\Delta}+M_{N}})^{2}-q^2}}
\end{equation}
\noindent
where $f_{\gamma}(0)=0.122$. It is
the $\gamma N \Delta$ coupling constant
for real photons.
With respect to the $\vec{S}$ and $\vec{T}$ operators
(transition operator between $\Frac{3}{2}$ spin states
to
$\Frac{1}{2}$ spin states and respectively
between
$\Frac{3}{2}$ isospin states to
$\Frac{1}{2}$ isospin states),
their normalization is:
\begin{equation}
<\frac{3}{2},M|S^{\dagger}_{\lambda}|\frac{1}{2},m>=
(1\frac{1}{2}\frac{3}{2}|\lambda m M)
\end{equation}
\begin{equation}
<\frac{3}{2},M|T^{\dagger}_{\lambda}|\frac{1}{2},m>=
(1\frac{1}{2}\frac{3}{2}|\lambda m M)
\end{equation}
\noindent
where $\lambda$ is a spherical basis index.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,991 |
Q: _reactNative.useState is not a function hi everyone i'm just learning hooks in react native and i get an error when i try to execute the function written below. The code is:
aa(){
const [booblenavalue,setboolenavalue]=useState(false)
useEffect(() => {
setboolenavalue(true)
}, [])
}
My code don't work.My error is:" TypeError: (0 , _reactNative.useState) is not a function". How can I fix this error?
A: import React,{useState} from 'react';
const aa = () => {
const [booblenavalue,setboolenavalue]=useState(false)
useEffect(() => {
setboolenavalue(true)
}, []);
}
export default aa;
Maybe this must be the format of your code
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,108 |
{"url":"https:\/\/www.hpmuseum.org\/forum\/thread-11891-post-108945.html","text":"Some HP-50G Questions\n12-09-2018, 07:42 PM\nPost: #21\n John Keith Senior Member Posts: 435 Joined: Dec 2013\nRE: Some HP-50G Questions\nOlder MicroSD cards are slow just like old SD cards. Newer ones claim to be fast enough to record 4K video.\n12-09-2018, 11:31 PM\nPost: #22\n cdmackay Senior Member Posts: 343 Joined: Sep 2018\nRE: Some HP-50G Questions\n(12-09-2018 06:41 PM)HP67 Wrote: \u00a0It seems to me there should be no technical difference of cards based on physical format alone. If you compare specs on both you should find the difference.\n\nyup, sorry, my comment wasn't clear: micro SD and SD should be of equivalent performance, and using an adaptor shouldn't affect that at all.\n\nBut I thought I'd noticed that a similarly branded card in micro SD format often had a lower performance rating than the same branded SD. But perhaps I was imagining that.\n\nAnd counterfeit, as you say\u2026\n\nCambridge, UK\n41CL, 12C\/15C, DM15\/16, 71B, 17B, 28S, DM42, 48GX, 17bII+, 50g (& newRPL), 35s, 30b (WP 34S), Prime G2\n& Casios, Rockwell 18R :)\n12-10-2018, 04:42 PM (This post was last modified: 12-10-2018 04:45 PM by Claudio L..)\nPost: #23\n Claudio L. Senior Member Posts: 1,589 Joined: Dec 2013\nRE: Some HP-50G Questions\n(12-09-2018 11:31 PM)cdmackay Wrote:\n(12-09-2018 06:41 PM)HP67 Wrote: \u00a0It seems to me there should be no technical difference of cards based on physical format alone. If you compare specs on both you should find the difference.\n\nyup, sorry, my comment wasn't clear: micro SD and SD should be of equivalent performance, and using an adaptor shouldn't affect that at all.\n\nBut I thought I'd noticed that a similarly branded card in micro SD format often had a lower performance rating than the same branded SD. But perhaps I was imagining that.\n\nAnd counterfeit, as you say\u2026\n\nHere are some quick facts, that may or may not clarify anything at all\n\n* microSD and SD are electrically identical, the adapter is just tracks to put each pin to the right place.\n* SD standard has a negotiation in voltage and there's a maximum current too. The faster the card, the more current it draws, and the hotter it gets.\n* Faster cards use newer protocols, they aren't any faster if an older host doesn't support the high speed protocol (the 50g being SD 1.1 compatible falls into this category, quite slow).\n* microSD cards in older generations were meant for low-powered devices, therefore they were usually slower to limit power consumption. Back then, big SD cards were thought for PCs and camcorders, so performance to record video without skipping was more important.\n* The previous statement is 100% untrue nowadays. microSD became the standard for phones, which have a lot of power AND demand a lot of performance. There's virtually no difference between new microSD and big SD cards.\n* I had microSD cards that were very fast, but they would shut down due to overheating after less than a minute working at full blast (copying large files). I don't know if to blame the card or the host's socket poor heat dissipation. In other devices the same card would work without any issues.\n\nRegarding boot delays:\n* SD cards have a power-up delay that varies from manufacturer to manufacturer. Some cards are ready to receive commands in a few milliseconds, I had others that take 300 ms before they accept the first command. The 50g cuts down power to the card after every single operation, making it unbearably slow on those cards. I had a case of a fast card that tested on a PC was more than twice the speed of another one, but on the 50g it was 4x slower. Investigating the issue I discovered it had 190 ms startup time, ruining the performance on the 50g.\n* The above statement is not true when using SDFiler, hpgcc 2.0, hpgcc3 or newRPL, all of which don't power off the card (merely stop its clock, meaning instant wake-up at the expense of a tiny current draw).\n* The boot delay on the 50g is due to the ROM computing the free space, for which it needs to read the entire FAT table. The amount of data to read is therefore proportional to the number of clusters, just lower the number of clusters and your 50g boots faster.\n* The recommendation to use FAT16 is only because FAT16 is limited to 65536 clusters.\n* Since most objects in the 50g are small, using a large cluster size is also a waste, so it's best to use a smaller partition if possible. Now that Windows finally mounts all partitions, I think it's wise to create a first partition with 128 MB (FAT16) and the rest in a FAT32 volume for bigger files.\n\n12-10-2018, 08:20 PM (This post was last modified: 12-10-2018 08:23 PM by edryer.)\nPost: #24\n edryer Member Posts: 102 Joined: Dec 2013\nRE: Some HP-50G Questions\nWow, thank you Claudio!\n\nWould you happen to know if we used a 4GB SD (the non HC rare ones that follow the SD spec) would it work? These are difficult to find I know. I did have one a long time ago in a camera that could only take 2GB max and it worked fine.\n\nI can see them on eBay for around 20 Euros for a non name brand (may not even be real). I expect a decent named brand 4GB SD non HC to reach quite a bit more... probably more than a 128GB or even 256GB card. Crazy.\n\nHP-50G (2013 model), HP-48G (1996 model)\n12-10-2018, 10:30 PM\nPost: #25\n rprosperi Senior Member Posts: 3,541 Joined: Dec 2013\nRE: Some HP-50G Questions\n(12-10-2018 08:20 PM)edryer Wrote: \u00a0Wow, thank you Claudio!\n\nWould you happen to know if we used a 4GB SD (the non HC rare ones that follow the SD spec) would it work? These are difficult to find I know. I did have one a long time ago in a camera that could only take 2GB max and it worked fine.\n\nI can see them on eBay for around 20 Euros for a non name brand (may not even be real). I expect a decent named brand 4GB SD non HC to reach quite a bit more... probably more than a 128GB or even 256GB card. Crazy.\n\nHere's a 2GB Transcend card for < $9.00 from Amazon. Known to work fine with the 50g, I've used many of these. If you need more than 2GB of storage in your 50g, then you're quite probably doing something very wrong... --Bob Prosperi 12-10-2018, 10:40 PM Post: #26 Claudio L. Senior Member Posts: 1,589 Joined: Dec 2013 RE: Some HP-50G Questions (12-10-2018 08:20 PM)edryer Wrote: Wow, thank you Claudio! Would you happen to know if we used a 4GB SD (the non HC rare ones that follow the SD spec) would it work? These are difficult to find I know. I did have one a long time ago in a camera that could only take 2GB max and it worked fine. I can see them on eBay for around 20 Euros for a non name brand (may not even be real). I expect a decent named brand 4GB SD non HC to reach quite a bit more... probably more than a 128GB or even 256GB card. Crazy. Unfortunately, while the SD specification allowed a lot of freedom to card manufacturers, software developers didn't care to implement all possible cases. Large SD cards use larger sectors (this Wikipedia article might interest you), and developers simply reused their old code from DOS era with 512-byte hard-coded sector size. The thing is, the SD card specification allows partial sector read\/write, so you could have a card with 2048 byte sectors and still allow the host to read or write 512-byte chunks and so far so good. But the specification didn't make that part mandatory! The manufacturers went completely wild and they supported whatever bits and pieces were convenient to them. Developers on the other hand, went completely conservative and never touched their 512-byte sector size in their code. After all, why bother supporting a feature if it will be available in only certain cards from certain manufacturers? You want your device to read all of them. As a result, the more \"special\" the card was, the more chances for trouble. By the time SD 2.0 came out, the specification for SDHC made 512-byte sector support mandatory and discarded all the partial reads\/writes support to keep it simple and universal. Back to your 4GB cards: it will be hit and miss. 4 GB cards use larger sectors, if you get it from a responsible manufacturer that supported partial read\/writes at 512-bytes, your poorly programmed device will be able to read it. Keep in mind that the SD card specification says 2GB is the maximum capacity for SD cards, so 4GB cards are non-standard (but unless there's a hard-coded limit, a proper driver should be able to read them). The bad part is that poorly programmed devices also had trouble doing math so close to the limit of the 32-bit integers, hence some 4GB cards are seen as 2GB cards by some devices. I'd say stay away from the 4 GB cards, they are most likely trouble. 2 GB and lower are fine for the 50g. Unfortunately HP never updated the firmware of the 50g to support SDHC. 12-10-2018, 10:54 PM Post: #27 DavidM Senior Member Posts: 748 Joined: Dec 2013 RE: Some HP-50G Questions Claudio, any plans to release an updated SDLIB that could be used in an updated SDFiler? I seem to recall you mentioning in another thread that you had updated the library after it was included with SDFiler 1.3. I know, I know... oldRPL isn't exactly on your radar screen these days. But it never hurts to ask! 12-10-2018, 11:13 PM Post: #28 cdmackay Senior Member Posts: 343 Joined: Sep 2018 RE: Some HP-50G Questions thanks very much Claudio, very helpful! (12-10-2018 04:42 PM)Claudio L. Wrote: * Since most objects in the 50g are small, using a large cluster size is also a waste, so it's best to use a smaller partition if possible. Now that Windows finally mounts all partitions, I think it's wise to create a first partition with 128 MB (FAT16) and the rest in a FAT32 volume for bigger files. I don't quite get the above; the 50g would only see the first partition, wouldn't it? So why would you do that? thanks. Cambridge, UK 41CL, 12C\/15C, DM15\/16, 71B, 17B, 28S, DM42, 48GX, 17bII+, 50g (& newRPL), 35s, 30b (WP 34S), Prime G2 & Casios, Rockwell 18R :) 12-11-2018, 02:57 PM (This post was last modified: 12-11-2018 02:57 PM by edryer.) Post: #29 edryer Member Posts: 102 Joined: Dec 2013 RE: Some HP-50G Questions Quote:Here's a 2GB Transcend card for <$9.00 from Amazon. Known to work fine with the 50g, I've used many of these. If you need more than 2GB of storage in your 50g, then you're quite probably doing something very wrong...\n\nI guess I'd never use the 4GB capacity in the 50G !! However I was thinking of getting a few of these cards for some old electronics I have (2004-2006 era) and was curious about the 50G.\n\nClaudio's note on how the spec is implemented was interesting, so I would prefer to try and find some old 4GB SD non HC cards from a reputable manufacturer.\n\nActually looking I can see they are quite cheap (Transcend on eBay) I very much doubt though they are genuine cards - my experience of eBay and SD Cards is that perhaps over 50% are fakes, whether manufacturer or even reported capacity in some cases.\n\nHP-50G (2013 model), HP-48G (1996 model)\n12-12-2018, 07:40 PM\nPost: #30\n Claudio L. Senior Member Posts: 1,589 Joined: Dec 2013\nRE: Some HP-50G Questions\n(12-10-2018 11:13 PM)cdmackay Wrote: \u00a0thanks very much Claudio, very helpful!\n\n(12-10-2018 04:42 PM)Claudio L. Wrote: \u00a0* Since most objects in the 50g are small, using a large cluster size is also a waste, so it's best to use a smaller partition if possible. Now that Windows finally mounts all partitions, I think it's wise to create a first partition with 128 MB (FAT16) and the rest in a FAT32 volume for bigger files.\n\nI don't quite get the above; the 50g would only see the first partition, wouldn't it? So why would you do that?\n\nthanks.\n\nAssume you want to limit the number of clusters to 65535. If you do that with a 2G partition , you end up with huge cluster size to store thy files. As a result you'll waste 99% of the space. If you create a small partition, you can store the same amount of small files, and you still have the rest of the card for other uses in the second partition, just not for the calculator or I should say only when using SDFiler.\n12-12-2018, 10:01 PM\nPost: #31\n cdmackay Senior Member Posts: 343 Joined: Sep 2018\nRE: Some HP-50G Questions\n(12-12-2018 07:40 PM)Claudio L. Wrote: \u00a0Assume you want to limit the number of clusters to 65535. If you do that with a 2G partition , you end up with huge cluster size to store thy files. As a result you'll waste 99% of the space. If you create a small partition, you can store the same amount of small files, and you still have the rest of the card for other uses in the second partition, just not for the calculator or I should say only when using SDFiler.\n\nthanks!\n\nCambridge, UK\n41CL, 12C\/15C, DM15\/16, 71B, 17B, 28S, DM42, 48GX, 17bII+, 50g (& newRPL), 35s, 30b (WP 34S), Prime G2\n& Casios, Rockwell 18R :)\n12-13-2018, 08:11 PM (This post was last modified: 12-13-2018 08:12 PM by pier4r.)\nPost: #32\n pier4r Senior Member Posts: 2,008 Joined: Nov 2014\nRE: Some HP-50G Questions\n(12-10-2018 10:54 PM)DavidM Wrote: \u00a0Claudio, any plans to release an updated SDLIB that could be used in an updated SDFiler? I seem to recall you mentioning in another thread that you had updated the library after it was included with SDFiler 1.3.\n\nI know, I know... oldRPL isn't exactly on your radar screen these days. But it never hurts to ask!\n\nI second the question and I thank Claudio L. for the info about the SD cards. Finding such detailed recaps nowadays is quite tricky.\n\nWikis are great, Contribute :)\n12-14-2018, 10:29 PM\nPost: #33\n Claudio L. Senior Member Posts: 1,589 Joined: Dec 2013\nRE: Some HP-50G Questions\n(12-10-2018 10:54 PM)DavidM Wrote: \u00a0Claudio, any plans to release an updated SDLIB that could be used in an updated SDFiler? I seem to recall you mentioning in another thread that you had updated the library after it was included with SDFiler 1.3.\n\nI know, I know... oldRPL isn't exactly on your radar screen these days. But it never hurts to ask!\n\nThat was a loooong time ago. I don't know if I can build SDLIB anymore. I'm not setup to build hpgcc 2.0 programs, and to be honest I don't even know if hpgcc 2.0 would build using a new gcc version.\nThe answer is: if you like SDLIB, it's actually included in newRPL! (you saw that one coming from a mile away, but at least pretend to be surprised ), do you have any plans to write a Filer for newRPL? (you didn't see THAT one coming, didn't you? )\n12-14-2018, 11:46 PM\nPost: #34\n DavidM Senior Member Posts: 748 Joined: Dec 2013\nRE: Some HP-50G Questions\nTouch\u00e9, Claudio\n\nWell, I wasn't sure if you still had a compiled SDLIB somewhere that was newer than the one included with SDFiler but still compatible with the ancient O\/S.\n\nI imagine (some)one could write a Filer-like app using standard newRPL instead of having to resort to C, and its performance would still be quite good! Maybe after I finish a few other non-calculator projects I'll give newRPL some more brain cycles.\n \u00ab Next Oldest | Next Newest \u00bb\n\nUser(s) browsing this thread: 1 Guest(s)","date":"2019-09-17 02:21:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.27292606234550476, \"perplexity\": 4156.171868916179}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514573011.59\/warc\/CC-MAIN-20190917020816-20190917042816-00245.warc.gz\"}"} | null | null |
\section{Introduction}
\setcounter{equation}{0}
It is well-known that a $m\times m$ matrix Schr\"{o}dinger equation on
$-\infty<x<\infty$ is defined by the following expression \cite{1} :
$$L\psi(x,k)=\lambda\psi(x,k),\quad\lambda=k^2,$$
where
\begin{eqnarray}
&&L=-(\partial^2/\partial x^2)I+U(x),\nonumber\\
&&I=(\delta_{ij}),\quad U(x)=\left(u_{ij}(x)\right);\;\;i,j=1,\ldots,m,
\nonumber\\
&&\psi(x,k)=\left[\psi_1(x,k),\psi_2(x,k),\ldots,\psi_m(x,k)\right].\nonumber
\end{eqnarray}
Further, let $\eta^i$ be the components of deviation vector between two
infinitesimally nearby geodesic lines. Then the components $\eta^i$ satisfy
to the Jacoby equation \cite{2}
\begin{equation}\label{i1}
v^i\nabla_i(v^j\nabla_j\eta^l)=-v^iR^l_{ikm}v^m\eta^k,
\end{equation}
where $v^i$ are the components of the tangent vector along a geodesic line
$\gamma$, $R^i_{jkl}$ is the curvature tensor of the metrics
$$ds^2=g_{ij}dx^idx^j.$$
In a special system of coordinates, where axis $x^j$ is a geodesic line,
equation (\ref{i1}) has the following form \cite{2}-\cite{4}
\begin{equation}\label{i2}
\frac{d^2\eta^j}{{dx^i}^2}+R^j_{ili}\eta^l=0.
\end{equation}
In the paper \cite{5} it has been shown that in the case of
three-dimensional space
with the metrics
\begin{equation}\label{i3}
ds^2=dx^2+A(x,y,z)dy^2+2B(x,y,z)dydz+C(x,y,z)dz^2
\end{equation}
the equations of geodesic deviations
\begin{eqnarray}
\frac{d^2\eta^2}{dx^2}+R^2_{121}\eta^2+R^2_{131}\eta^3&=&0,\nonumber\\
\frac{d^2\eta^3}{dx^2}+R^3_{121}\eta^2+R^3_{131}\eta^3&=&0 \label{i4}
\end{eqnarray}
may be represented in the form of the $2\times 2$ matrix Schr\"{o}dinger
equation
\begin{eqnarray}
-\frac{d^2\eta^2}{dx^2}+(-R^2_{121}+\lambda^2)\eta^2+(-R^2_{131})\eta^3
=\lambda^2\eta^2,\nonumber\\
-\frac{d^2\eta^3}{dx^2}+(-R^3_{121})\eta^2+(-R^3_{131}+\lambda^2)\eta^3
=\lambda^2\eta^3.\label{i5}
\end{eqnarray}
On the other hand, it is known that AKNS-system \cite{6}
\begin{eqnarray}
\frac{\partial\psi_1}{\partial x}+i\lambda\psi_1&=&q(x,y,z)\psi_2,\nonumber\\
\frac{\partial\psi_2}{\partial x}-i\lambda\psi_2&=&r(x,y,z)\psi_1\label{i6}
\end{eqnarray}
can be rewritten in the form of a Schr\"{o}dinger-like equation \cite{1}
\begin{equation}\label{i7}
\left[-\left(\begin{array}{cc}1 & 0 \\ 0 & 1 \end{array}\right)
\frac{\partial^2}{\partial x^2}+\left(\begin{array}{cc} rq & q_x \\
r_x & rq\end{array}\right)\right]\begin{pmatrix} \psi_1 \\ \psi_2\end{pmatrix}
=\lambda^2\begin{pmatrix}\psi_1 \\ \psi_2\end{pmatrix}.
\end{equation}
The comparison of the systems (\ref{i7}) and (\ref{i5}) gives the following
conditions on the curvature tensor
\begin{equation}\label{i8}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{rl}
\lambda^2-R^2_{121}=rq,& \lambda^2-R^3_{131}=rq,\\
R^2_{131}=-q_x,& R^3_{121}=-r_x.
\end{array}}
\end{equation}
Analogously, in the case of 4-dimensional space with a geodesic coordinate
system
\begin{equation}\label{i9}
ds^2=dt^2+g_{ab}dx^adx^b
\end{equation}
the geodesic deviations equation has the form \cite{5}
\begin{eqnarray}
\frac{d^2\eta^1}{dt^2}+R^1_{010}\eta^1+R^1_{020}\eta^2+R^1_{030}\eta^3
&=&0,\nonumber\\
\frac{d^2\eta^2}{dt^2}+R^2_{010}\eta^1+R^2_{020}\eta^2+R^2_{030}\eta^3
&=&0,\label{i10}\\
\frac{d^2\eta^3}{dt^2}+R^3_{010}\eta^1+R^3_{020}\eta^2+R^3_{030}\eta^3
&=&0.\nonumber
\end{eqnarray}
In the present paper we consider solutions of the equations (\ref{i4}) and
(\ref{i10}) obtained by the inverse scattering transform. Our consideration
is realized on the basis of a Chandrasekhar metrics \cite{7,8} (the so-called
space-time of a sufficiently general structure), which includes as
particular cases the static and spherically symmetric solutions
(Schwarzschild and Reissner-Nordstr\"{o}m metrics), and also stationary
and axially symmetric solutions (Kerr and Kerr-Newman metrics) and so on.
In section 2 we introduce a three-dimensional analog of the
Chandrasekhar metrics,
the particular case of which is coincide with the metrics (\ref{i3}). It is
shown that in the orthonormal basis, related with this metrics, solutions of
the system (\ref{i4}) are reduced to the solutions of the Zakharov-Shabat
problem \cite{9}. Thus, a dependence of the potential $u$ on parameters
$y$ and $z$ is described by the modified Korteweg-de Vries (mKdV) equations.
Different particular cases, in which the vector of geodesic deviation
$\boldsymbol{\eta}$ is explicitly expressed via the fundamental solutions
(Jost functions) of the Zakharov-Shabat problem, are considered at the end of
section 2. In section 3 we introduce a $3\times 3$ matrix Schr\"{o}dinger
equation which then is associated with the system of type (\ref{i10}).
Further, a dependence on parameters is reduced to evolution equations of
the well-known problem of three-wave interaction, the explicit solutions
of which was obtained by Zakharov and Manakov in 1973 \cite{10,11,12}.
It is shown that in the case of decay instability and reality of potential
matrix, the system of equations of geodesic deviation (\ref{i10}) has a
wide class of particular solutions.
\section{Three-dimensional space}
\setcounter{equation}{0}
\subsection{The three-dimensional Chandrasekhar metrics}
Let us consider in the three-dimensional space with a signature $(-,-,-)$ a
metrics of the following form
\begin{equation}\label{e1}
{\rm d} s^{2}=-\sum_{A}e^{2\mu_{A}}({\rm d} x^{A})^{2}-e^{2\psi}({\rm d} x^{3}-
\sum_{A}q_{A}{\rm d} x^{A})^{2},
\end{equation}
where $A=1,2$. $\psi,\,\mu_{A}$ and $q_{A}$ are the functions on variables
$x^{1},\, x^{2},\,x^{3}$.
The orthonormal basis, related with this metrics, is defined by the
following covariant and contravariant vectors
$$e_{(1)i}=(0,\,0,\,-e^{\mu_{1}}),\quad e_{(2)i}=(0,\,-e^{\mu_{2}},\,0),$$
\begin{equation}\label{e2}
e_{(3)i}=(-e^{\psi},\,q_{1}e^{\psi},\,q_{2}e^{\psi}).
\end{equation}
$$e^{i}_{(1)}=(q_{2}e^{-\mu_{1}},\,0,\,e^{-\mu_{1}}),\quad
e^{i}_{(2)}=(q_{1}e^{-\mu_{2}},\,e^{-\mu_{2}},\,0),$$
\begin{equation}\label{e3}
e^{i}_{(3)}=(e^{-\psi},\,0,\,0).
\end{equation}
From (\ref{e2}) and (\ref{e3}) it follows that
$$e^{i}_{(a)}e_{(b)i}=\eta_{(a)(b)}=\left|
\begin{array}{ccc}
-1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & -1
\end{array}\right|.$$
Let
\begin{equation}\label{e4}
\boldsymbol{\omega}^{A}=e^{\mu_{A}}{\rm d} x^{A},\quad \boldsymbol{\omega}^{3}=e^{\psi}({\rm d} x^{3}-
\sum_{A}q_{A}{\rm d} x^{A})
\end{equation}
be the basis 1-forms.
It is easy to see that inverse relations for (\ref{e4}) have the form
\begin{equation}\label{e5}
{\rm d} x^{A}=e^{-\mu_{A}}\boldsymbol{\omega}^{A},\quad {\rm d} x^{3}=e^{-\psi}\boldsymbol{\omega}^{3}+
\sum_{A}e^{-\mu_{A}}q_{A}\boldsymbol{\omega}^{A}.
\end{equation}
\begin{sloppypar}
Expressing the exterior derivatives of the forms $\boldsymbol{\omega}^{i}$ via the basis
2-forms $\boldsymbol{\omega}^{i}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{j}\;(i\neq j,\,i,j=1,2,3)$, we have\end{sloppypar}
\begin{multline}
{\rm d}\boldsymbol{\omega}^{A}=\sum_{B}e^{\mu_{A}}\mu_{A,B}{\rm d} x^{B}{\scriptstyle\bigwedge}{\rm d} x^{A}+
e^{\mu_{A}}\mu_{A,3}{\rm d} x^{3}{\scriptstyle\bigwedge}{\rm d} x^{A}= \\
=\sum_{B}e^{-\mu_{B}}\mu_{A,B}\boldsymbol{\omega}^{B}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{A}+\mu_{A,3}\left[e^{-\psi}
\boldsymbol{\omega}^{3}+\sum_{B}e^{-\mu_{B}}q_{B}\boldsymbol{\omega}^{B}\right]{\scriptstyle\bigwedge}\boldsymbol{\omega}^{A}= \\
=\sum_{B}e^{-\mu_{B}}(\mu_{A,B}+q_{B}\mu_{A,3})\boldsymbol{\omega}^{B}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{A}+
e^{-\psi}\mu_{A,3}\boldsymbol{\omega}^{3}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{A}.\label{e6}
\end{multline}
For the brevity of exposition let us introduce a
derivative of the function $f(x^{1},x^{2},x^{3})$ on a coordinate
$x^{A}\;(A=1,2)$ which we will denote as $f_{:A}$,
\begin{equation}\label{e7}
f_{:A}=f_{,A}+q_{A}f_{,3}.
\end{equation}
This operation is the differentiation, since it satisfies to a Leibnitz rule
$$(fg)_{:A}=fg_{:A}+gf_{:A}.$$
Using (\ref{e7}), we can rewrite
the equation (\ref{e6}) in the form
\begin{equation}\label{e8}
{\rm d}\boldsymbol{\omega}^{A}=-\sum_{B}e^{-\mu_{B}}\mu_{A:B}\boldsymbol{\omega}^{A}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{B}-
e^{-\psi}\mu_{A,3}\boldsymbol{\omega}^{A}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}.
\end{equation}
In like manner we have
\begin{equation}\label{e9}
{\rm d}\boldsymbol{\omega}^{3}=\sum_{A}e^{-\mu_{A}}(\psi_{:A}+q_{A,3})\boldsymbol{\omega}^{A}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}-
\sum_{A,B}e^{\psi-\mu_{A}-\mu_{B}}q_{A:B}\boldsymbol{\omega}^{A}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{B}.
\end{equation}
Further, the equations
\begin{eqnarray}
&&\frac{1}{2}T^j={\rm d}\boldsymbol{\omega}^j+\boldsymbol{\omega}^j_l{\scriptstyle\bigwedge}\boldsymbol{\omega}^l=\Omega^j,\\
&&\frac{1}{2}R^j_{lkm}\boldsymbol{\omega}^k{\scriptstyle\bigwedge}\boldsymbol{\omega}^m=\Omega^j_l
\end{eqnarray}
are called respectively the first and second Cartan structure equations,
where the Cartan 2-form $\Omega^j_l$ is
$$\Omega^j_l={\rm d}\boldsymbol{\omega}^j_l+\boldsymbol{\omega}^j_k{\scriptstyle\bigwedge}\boldsymbol{\omega}^k_l.$$
Owing to absence of torsion ($T^j=0$) the first Cartan structure equation gives
\begin{equation}\label{e10}
{\rm d}\boldsymbol{\omega}^{3}=-\sum_{A}\boldsymbol{\omega}^{3}_{A}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{A},
\end{equation}
\begin{equation}\label{e11}
{\rm d}\boldsymbol{\omega}^{A}=-\sum_{B}\boldsymbol{\omega}^{A}_{B}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{B}-\boldsymbol{\omega}^{A}_{3}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}.
\end{equation}
These equations allow us to define the connection 1-forms $\boldsymbol{\omega}^{3}_{A}$ and
$\boldsymbol{\omega}^{A}_{B}$ if the forms ${\rm d}\boldsymbol{\omega}^{3}$ and ${\rm d}\boldsymbol{\omega}^{A}$ are known.
Since the 1-forms $\boldsymbol{\omega}^{3}$ and $\boldsymbol{\omega}^{A}$ are the basis forms, then
\begin{equation}\label{e12}
\boldsymbol{\omega}^{i}_{j}=-\boldsymbol{\omega}^{j}_{i}\quad (i,j=1,2,3).
\end{equation}
Comparing the equations (\ref{e8}) and (\ref{e9}) with the equations
(\ref{e10}) and (\ref{e11}), we obtain
\begin{eqnarray}
\boldsymbol{\omega}^{3}_{A}&\!\!=\!\!&-\boldsymbol{\omega}^{A}_{3}=e^{-\mu_{A}}\Psi_{A}\boldsymbol{\omega}^{3}-
e^{-\psi}\mu_{A,3}\boldsymbol{\omega}^{A}+
\frac{1}{2}\sum_{B}e^{\psi-\mu_{A}-\mu_{B}}Q_{AB}\boldsymbol{\omega}^{B},\label{e13}\\
\boldsymbol{\omega}^{A}_{B}&\!\!=\!\!&-\boldsymbol{\omega}^{B}_{A}=-\frac{1}{2}e^{\psi-\mu_{A}-\mu_{B}}
Q_{AB}\boldsymbol{\omega}^{3}
+e^{-\mu_{B}}\mu_{A:B}\boldsymbol{\omega}^{A}-e^{-\mu_{A}}\mu_{B:A}\boldsymbol{\omega}^{B},\label{e14}
\end{eqnarray}
where
\begin{eqnarray}
Q_{AB}&=&q_{A:B}-q_{B:A}, \\
\Psi_{A}&=&\psi_{:A}+q_{A,3}.
\end{eqnarray}
From (\ref{e13}) and (\ref{e14}) for the different connection forms we have
\begin{eqnarray}
\boldsymbol{\omega}^{1}_{3}&=&e^{-\psi}\mu_{1,3}\boldsymbol{\omega}^{1}-\frac{1}{2}e^{\psi-\mu_{1}-\mu_{2}}
Q_{12}\boldsymbol{\omega}^{2}-e^{-\mu_{1}}\Psi_{1}\boldsymbol{\omega}^{3},\nonumber \\
\boldsymbol{\omega}^{2}_{3}&=&-\frac{1}{2}e^{\psi-\mu_{1}-\mu_{2}}Q_{21}\boldsymbol{\omega}^{1}+
e^{-\psi}\mu_{2,3}\boldsymbol{\omega}^{2}-e^{-\mu_{2}}\Psi_{2}\boldsymbol{\omega}^{3},\label{e17} \\
\boldsymbol{\omega}^{1}_{2}&=&e^{-\mu_{2}}\mu_{1:2}\boldsymbol{\omega}^{1}-e^{-\mu_{1}}\mu_{2:1}\boldsymbol{\omega}^{2}-
\frac{1}{2}e^{\psi-\mu_{1}-\mu_{2}}Q_{12}\boldsymbol{\omega}^{3}.\nonumber
\end{eqnarray}
Further, in order to culculate the components of the Riemann tensor from
the second Cartan structure equation
\begin{equation}\label{e18}
\frac{1}{2}R^{i}_{jkl}\boldsymbol{\omega}^{k}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{l}=\Omega^{i}_{j}={\rm d}\boldsymbol{\omega}^{i}_{j}+
\boldsymbol{\omega}^{i}_{k}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{k}_{j},
\end{equation}
it is necessary at first to calculate the exterior derivatives of the connection
forms (\ref{e17}).
\begin{lem}[{\rm Chandrasekhar \cite{8}}] If $F$ is an arbitrary functions of
the arguments
$x^{1},x^{2}$ and $x^{3}$, then
\begin{equation}\label{e19}
{\rm d}(F\boldsymbol{\omega}^{3})=\sum_{A}e^{-\psi-\mu_{A}}\mathcal{D}_{A}(Fe^{\psi})\boldsymbol{\omega}^{A}
{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}
+\frac{1}{2}\sum_{A,B}Fe^{\psi-\mu_{A}-\mu_{B}}Q_{AB}\boldsymbol{\omega}^{A}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{B},
\end{equation}
\begin{equation}\label{e20}
{\rm d}(F\boldsymbol{\omega}^{A})=\sum_{B}e^{-\mu_{A}-\mu_{B}}(e^{\mu_{A}}F)_{:B}\boldsymbol{\omega}^{B}{\scriptstyle\bigwedge}
\boldsymbol{\omega}^{A}+e^{-\psi-\mu_{A}}(e^{\mu_{A}}F)_{,3}\boldsymbol{\omega}^{3}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{A},
\end{equation}
where $\mathcal{D}_{A}$ is an operator, the action of which on an arbitrary
function $f(x^1,x^2,x^3)$ is defined by the following expression
\begin{equation}\label{e21}
\mathcal{D}_{A}f=f_{:A}+q_{A,3}f=f_{,A}+(q_{A}f)_{,3}.
\end{equation}
\end{lem}
Using this lemma, we obtain
\begin{multline}\label{e22}
{\rm d}\boldsymbol{\omega}^{1}_{2}=-\sum_{A}e^{-\psi-\mu_{A}}\mathcal{D}_{A}(\frac{1}{2}
e^{2\psi-\mu_{1}-\mu_{2}}Q_{12})\boldsymbol{\omega}^{A}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}-\\
-e^{-\psi-\mu_{1}}(e^{\mu_{1}-\mu_{2}}\mu_{1:2})_{,3}\boldsymbol{\omega}^{1}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}+
e^{-\psi-\mu_{2}}(e^{\mu_{2}-\mu_{1}}\mu_{2:1})_{,3}\boldsymbol{\omega}^{2}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}+\\
+\boldsymbol{\omega}^{1}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{2}\left\{-\frac{1}{4}e^{2\psi-2\mu_{1}-2\mu_{2}}Q^{2}_{12}
\right.-\\
-\biggl.e^{-\mu_{1}-\mu_{2}}\left[(e^{\mu_{1}-\mu_{2}}\mu_{1:2})_{:2}+
(e^{\mu_{2}-\mu_{1}}\mu_{2:1})_{:1}\right]\biggr\},
\end{multline}
\begin{multline}\label{e23}
\boldsymbol{\omega}^{1}_{3}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{2}_{3}=\left[e^{-2\psi}\mu_{1,3}\mu_{2,3}-\frac{1}{4}
e^{2\psi-2\mu_{1}-2\mu_{2}}Q_{12}Q_{21}\right]\boldsymbol{\omega}^{1}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{2}-\\
-\left[e^{-\psi-\mu_{2}}\Psi_{2}\mu_{1,3}+\frac{1}{2}e^{\psi-2\mu_{1}-\mu_{2}}
\Psi_{1}Q_{21}\right]\boldsymbol{\omega}^{1}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}+\\
+\left[\frac{1}{2}e^{\psi-\mu_{1}-2\mu_{2}}Q_{12}\Psi_{2}+e^{-\psi-\mu_{1}}
\Psi_{1}\mu_{2,3}\right]\boldsymbol{\omega}^{2}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3},
\end{multline}
\begin{multline}\label{e24}
{\rm d}\boldsymbol{\omega}^{2}_{3}=-\sum_{A}e^{-\psi-\mu_{A}}\mathcal{D}_{A}(e^{\psi-\mu_{2}}
\Psi_{2})\boldsymbol{\omega}^{A}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}+\\
+e^{-\psi-\mu_{1}}(\frac{1}{2}e^{\psi-\mu_{2}}Q_{21})_{,3}\boldsymbol{\omega}^{1}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}
-e^{-\psi-\mu_{2}}(e^{\mu_{2}-\psi}\mu_{2,3})_{,3}\boldsymbol{\omega}^{2}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}+\\
+\boldsymbol{\omega}^{1}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{2}\left\{-\frac{1}{2}e^{\psi-\mu_{1}-2\mu_{2}}\Psi_{2}
Q_{12}+\right.\\
+\biggl.e^{-\mu_{1}-\mu_{2}}\left[(e^{\mu_{2}-\psi}\mu_{2,3})_{:1}+(\frac{1}{2}
e^{\psi-\mu_{2}}Q_{21})_{:2}\right]\biggr\},
\end{multline}
\begin{multline}\label{e25}
\boldsymbol{\omega}^{1}_{2}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{1}_{3}=\left[e^{-\psi-\mu_{1}}\mu_{2:1}\mu_{1,3}-
\frac{1}{2}e^{\psi-\mu_{1}-2\mu_{2}}Q_{12}\mu_{1:2}\right]\boldsymbol{\omega}^{1}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{2}+\\
+\left[\frac{1}{2}e^{-\mu_{1}-\mu_{2}}Q_{12}\mu_{1,3}-e^{-\mu_{1}-\mu_{2}}
\Psi_{1}\mu_{1:2}\right]\boldsymbol{\omega}^{1}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}+\\
+\left[e^{-2\mu_{1}}\Psi_{1}\mu_{2:1}-\frac{1}{4}e^{-2\psi-2\mu_{1}-2\mu_{2}}
Q^{2}_{12}\right]\boldsymbol{\omega}^{2}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3},
\end{multline}
\begin{multline}\label{e26}
{\rm d}\boldsymbol{\omega}^{1}_{3}=-\sum_{A}e^{-\psi-\mu_{A}}\mathcal{D}_{A}
(e^{\psi-\mu_{1}}\Psi_{1})\boldsymbol{\omega}^{A}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}-\\
-e^{-\psi-\mu_{1}}(e^{\mu_{1}-\psi}\mu_{1,3})_{,3}\boldsymbol{\omega}^{1}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}+
e^{-\psi-\mu_{2}}(\frac{1}{2}e^{\psi-\mu_{1}}Q_{12})_{,3}\boldsymbol{\omega}^{2}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}+\\
+\boldsymbol{\omega}^{1}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{2}\left\{-\frac{1}{2}e^{\psi-2\mu_{1}-\mu_{2}}\Psi_{1}
Q_{12}-\right.\\
-\biggl.e^{-\mu_{1}-\mu_{2}}\left[(e^{\mu_{1}-\psi}\mu_{1,3})_{:2}+
(\frac{1}{2}e^{\psi-\mu_{1}}Q_{12})_{:1}\right]\biggr\},
\end{multline}
\begin{multline}\label{e27}
\boldsymbol{\omega}^{1}_{2}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{2}_{3}=\left[e^{-\psi-\mu_{2}}\mu_{1:2}\mu_{2,3}-
\frac{1}{2}e^{\psi-2\mu_{1}-\mu_{2}}\mu_{2:1}Q_{21}\right]\boldsymbol{\omega}^{1}{\scriptstyle\bigwedge}\boldsymbol{\omega}^2- \\
-\left[e^{-2\mu_{2}}\mu_{1:2}\Psi_{2}-\frac{1}{4}e^{2\psi-2\mu_{1}-2\mu_{2}}
Q_{12}Q_{21}\right]\boldsymbol{\omega}^{1}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}+\\
+\left[e^{-\mu_{1}-\mu_{2}}\mu_{2:1}\Psi_{2}+\frac{1}{2}e^{-\mu_{1}-\mu_{2}}
\mu_{2,3}Q_{12}\right]\boldsymbol{\omega}^{2}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}.
\end{multline}
Further, from the equations (\ref{e12}) and (\ref{e18}) we obtain
\begin{equation}\label{e28}
\frac{1}{2}R^{1}_{2kl}\boldsymbol{\omega}^{k}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{l}=\Omega^{1}_{2}={\rm d}\boldsymbol{\omega}^{1}_{2}-
\boldsymbol{\omega}^{1}_{3}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{2}_{3}.
\end{equation}
Substituting (\ref{e22})-(\ref{e23}) into this equation and collecting
the coefficients at $\boldsymbol{\omega}^k{\scriptstyle\bigwedge}\boldsymbol{\omega}^l$, we
obtain the components $R^{1}_{2kl}$ of the curvature tensor.
For example, with the object to calculate the component
$R^{1}_{212}$ we must collect the coefficients at $\boldsymbol{\omega}^{1}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{2}$
in the expression for $\Omega^{1}_{2}$. Analogously, the components
$R^{1}_{213}$ and $R^{1}_{223}$ are obtained from $\Omega^{1}_{2}$ via the
comparison of the coefficients at $\boldsymbol{\omega}^{1}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}$ and
$\boldsymbol{\omega}^{2}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{3}$. In like manner from the equation
\begin{equation}\label{e29}
\frac{1}{2}R^{2}_{3kl}\boldsymbol{\omega}^{k}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{l}=\Omega^{2}_{3}={\rm d}\boldsymbol{\omega}^{2}_{3}-
\boldsymbol{\omega}^{1}_{2}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{1}_{3}
\end{equation}
and equations (\ref{e24})-(\ref{e25}) we obtain the components
$R^{2}_{323}$ and $R^{2}_{313}$. Analogously, from equation
\begin{equation}\label{e30}
\frac{1}{2}R^{1}_{3kl}\boldsymbol{\omega}^{k}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{l}=\Omega^{1}_{3}={\rm d}\boldsymbol{\omega}^{1}_{3}+
\boldsymbol{\omega}^{1}_{2}{\scriptstyle\bigwedge}\boldsymbol{\omega}^{2}_{3}
\end{equation}
and equations (\ref{e26})-(\ref{e27}) we have the component $R^{1}_{313}$.
Finally, we have the following six essential components of the curvature tensor:
\begin{multline}\label{e31}
R^{1}_{212}=-\frac{1}{4}e^{2\psi-1\mu_{1}-2\mu_{2}}Q^2_{12}-
e^{-\mu_{1}-\mu_{2}}\left[(e^{\mu_{1}-\mu_{2}}\mu_{1:2})_{:2}+
(e^{\mu_{2}-\mu_{1}}\mu_{2:1})_{:1}\right]-\\
-e^{-2\psi}\mu_{1,3}\mu_{2,3}+\frac{1}{4}e^{2\psi-2\mu_{1}-2\mu_{2}}Q_{12}
Q_{21},
\end{multline}
\begin{multline}\label{e32}
R^{1}_{213}=-e^{-\psi-\mu_{1}}\mathcal{D}_{1}(1/2e^{2\psi-\mu_{1}-
\mu_{2}}Q_{12})-e^{-\psi-\mu_{1}}(e^{\mu_{1}-\mu_{2}}\mu_{1:2})_{,3}+\\
+e^{-\psi-\mu_{2}}\Psi_{2}\mu_{1,3}+\frac{1}{2}e^{\psi-2\mu_{1}-\mu_{2}}
\Psi_{1}Q_{21},
\end{multline}
\begin{multline}\label{e33}
R^{1}_{223}=-e^{-\psi-\mu_2}\mathcal{D}_2(1/2e^{2\psi-\mu_1-\mu_2}Q_{12})
-e^{-\psi-\mu_2}(e^{\mu_2-\mu_1}\mu_{2:1})_{,3}-\\
-\frac{1}{2}e^{\psi-\mu_1-2\mu_2}Q_{12}\Psi_2-e^{-\psi-\mu_1}\Psi_1\mu_{2,3},
\end{multline}
\begin{multline}\label{e34}
R^{2}_{323}=-e^{-\psi-\mu_2}\mathcal{D}_2(e^{\psi-\mu_2}\Psi_2)-
e^{-\psi-\mu_2}(e^{\mu_2-\psi}\mu_{2,3})_{,3}-\\
-e^{-2\mu_1}\Psi_1\mu_{2:1}+\frac{1}{4}e^{2\psi-2\mu_1-2\mu_2}Q^2_{12},
\end{multline}
\begin{multline}\label{e35}
R^2_{313}=-e^{-\psi-\mu_1}\mathcal{D}_1(e^{\psi-\mu_2}\Psi_2)+
e^{-\psi-\mu_1}(1/2e^{\psi-\mu_2}Q_{21})_{,3}-\\
-e^{-\mu_1-\mu_2}\left[1/2Q_{12}\mu_{1,3}-\Psi_1\mu_{1:2}\right],
\end{multline}
\begin{multline}\label{e36}
R^1_{313}=-e^{-\psi-\mu_1}\mathcal{D}_1(e^{\psi-\mu_1}\Psi_1)-
e^{-\psi-\mu_1}(e^{\mu_1-\psi}\mu_{1,3})_{,3}-\\
-e^{-2\mu_2}\mu_{1:2}\Psi_2-\frac{1}{4}e^{2\psi-2\mu_1-2\mu_2}Q_{12}Q_{21}.
\end{multline}
\subsection{Solutions of equations of geodesic deviation
in\protect\newline the three-dimensional space}
Let us consider a particular case $(\mu_1=q_1=0)$ of the metrics (\ref{e1}).
In this case the metrics (\ref{e1}) is coincide with the
three-dimensional metrics
considered in \cite{5} if suppose
$$A(x,y,z)=-\left(e^{2\mu_2}+q^2_2e^{2\psi}\right),
\quad B(x,y,z)=q_2e^{2\psi},$$
\begin{equation}\label{e37}
C(x,y,z)=-e^{2\psi}.
\end{equation}
At the condition $\mu_1=q_1=0$ the covariant and contravariant vectors
(\ref{e2})-(\ref{e3}) take the form
$$e_{(1)i}=(0,\,0,\,-1),\quad e_{(2)i}=(0,\,-e^{\mu_2},\,0),$$
\begin{equation}\label{e38}
e_{(3)i}=(-e^{\psi},\,0,\,q_2e^{\psi});
\end{equation}
$$e^i_{(1)}=(q_2,\,0,\,1),\quad e^i_{(2)}=(0,\,e^{-\mu_2},\,0),$$
\begin{equation}\label{e39}
e^i_{(3)}=(e^{-\psi},\,0,\,0).
\end{equation}
It is easy to see that in this orthonormal basis for the components of
the curvature tensor we have
\begin{equation}\label{e40}
R^n_{jkl}=-R_{ijkl}.
\end{equation}
It is well-known that the Riemann tensor $R_{ijkl}$ has the following
symmetry properties:
\begin{eqnarray}
R_{ijkl}&=&R_{klij},\nonumber\\
R_{ijkl}&=&-R_{jikl},\label{e41}\\
R_{ijkl}&=&-R_{ijlk}.\nonumber
\end{eqnarray}
\begin{sloppypar}
It is easy to show that the symmetry properties (\ref{e41}) decrease the
number of independent (essential) components of the Riemann tensor from $n^4$
to $n^2(n^2-1)/12$, where $n$ is a dimensionality of the space. In the case of
three-dimensional space we have six independent components of
the curvature tensor:
$R_{1212},\,R_{1213},\,R_{1223},\,R_{1313},\,R_{2313},\,R_{2323}$.
Further, using (\ref{e40})-(\ref{e41}), we see that in the
system (\ref{i4}) among the four components of the curvature tensor only
three are independent, namely, $R^2_{121},\,R^3_{131}$ and $R^2_{131}$
(or $R^3_{121}$). The latter two components are coincide with each other
in virtue of (\ref{e40})-(\ref{e41}).
Therefore,\end{sloppypar}
$$R^3_{121}=-R^2_{113}.$$
Hence it immediately follows that the conditions (\ref{i8}) and the system
(\ref{i7}) are reduced to the form
\begin{equation}\label{e42}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{rl}
\lambda^2-R^2_{121}=-u^2,& \lambda^2-R^3_{131}=-u^2,\\
R^3_{121}=&-R^2_{113}=u_x;
\end{array}}
\end{equation}
\begin{equation}\label{e43}
\left[-\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\frac{\partial^2}
{\partial x^2}+\begin{pmatrix} -u^2 & u_x \\ -u_x & -u^2 \end{pmatrix}
\right]\begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix}=\lambda^2
\begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix}.
\end{equation}
It is easy to see that the matrix equation (\ref{e43}) corresponds to the
Zakharov-Shabat system \cite{9}
\begin{eqnarray}
\frac{\partial\psi_1}{\partial x}+i\lambda\psi_1&=&u\psi_2,\nonumber\\
\frac{\partial\psi_2}{\partial x}-i\lambda\psi_2&=&-u\psi_1.\label{e44}
\end{eqnarray}
Thus, {\it in the orthonormal basis (\ref{e38})-(\ref{e39}), related with
the metrics (\ref{e1}), at the condition $\mu_1=q_1=0$ the AKNS-system
for the equations of geodesic deviation is
reduced to the Zakharov-Shabat system.}
Moreover, instead the two potentials in AKNS-system we have
now only one potential
in ZS-system.
Let us calculate the independent components of the
curvature tensor in the system
(\ref{i4}) for the metrics (\ref{e1}) at the condition $\mu_1=q_1=0$.
From (\ref{e31}), (\ref{e32}) and (\ref{e36}) we have
\begin{eqnarray}
R^1_{212}&=&-\mu_{2,11}-\mu^2_{2,1}-\frac{1}{4}e^{-2\mu_2}q^2_{2,1}
(e^{2\psi}+1),\label{e45}\\
R^1_{213}&=&\frac{1}{2}e^{\psi-\mu_2}q_{2,11}+\frac{3}{2}e^{\psi-\mu_2}
\psi_{,1}q_{2,1}-\frac{1}{2}e^{\psi-\mu_2}\mu_{2,1}q_{2,1},\label{e46}\\
R^1_{313}&=&-\psi_{,11}-\psi^2_{,1}+\frac{1}{4}e^{2\psi-2\mu_2}q^2_{2,1}.
\label{e47}
\end{eqnarray}
So, in the case of the metrics (\ref{e1}) our problem
of solving of the equations of geodesic deviation
is reduced to the Zakharov-Shabat problem
(\ref{e44}). It is known that fundamental solutions (Jost functions)
of ZS-problem are defined by the following expressions \cite{6,13,14}
\begin{equation}\label{e48}
{\renewcommand{\arraystretch}{2.0}
\begin{array}{ccl}
\varphi^-_1(x,\lambda)&=&e^{-i\lambda x}+{\displaystyle\int\limits^x_{-\infty}}
dx^\prime A_1(x,x^\prime)e^{-i\lambda x^\prime},\\
\varphi^-_2(x,\lambda)&=&{\displaystyle\int\limits^x_{-\infty}}
dx^\prime A_2(x,x^\prime)e^{-i\lambda x^\prime};
\end{array}}
\end{equation}
\begin{equation}\label{e49}
{\renewcommand{\arraystretch}{2.0}
\begin{array}{ccl}
\varphi^+_1(x,\lambda)&=&{\displaystyle\int\limits^\infty_x}
dx^\prime B_1(x,x^\prime)e^{i\lambda x^\prime},\\
\varphi^+_2(x,\lambda)&=&e^{i\lambda x}+{\displaystyle\int\limits^\infty_x}
dx^\prime
B_2(x,x^\prime)e^{i\lambda x^\prime}.
\end{array}}
\end{equation}
These solutions are linearly dependent:
\begin{eqnarray}
\varphi^-(x,\lambda)&=&c_{11}(\lambda)\varphi^+(x,\lambda)+
c_{12}(\lambda)\bar{\varphi}^+(x,\lambda),\label{e50}\\
\varphi^+(x,\lambda)&=&c_{21}(\lambda)\bar{\varphi}^-(x,\lambda)+c_{22}(\lambda)
\varphi^-(x,\lambda),\label{e51}
\end{eqnarray}
where
\begin{equation}\label{e52}
\varphi^\mp(x,\lambda)=\begin{pmatrix} \varphi^\mp_1(x,\lambda)\\
\varphi^\mp_2(x,\lambda)
\end{pmatrix},\quad \bar{\varphi}^\mp(x,\lambda)=\begin{pmatrix}
\varphi^\mp_2(x,-\lambda)\\
-\varphi^\mp_1(x,-\lambda)\end{pmatrix}.
\end{equation}
Further, the pair of Gel'fand-Levitan-Marchenko integral equations
can be derived
from (\ref{e50}) by means of the Fourier transform:
$$-A_2(x,y)+\Omega_L(x+y)+\int\limits^x_{-\infty}dx^\prime A_1(x,x^\prime)
\Omega_L(x^\prime+y)=0,$$
\begin{equation}\label{e54}
A_1(x,y)+\int\limits^x_{-\infty}dx^\prime A_2(x,x^\prime)\Omega_L(x^\prime
+y)=0,
\end{equation}
$$x>y.$$
Analogously, from (\ref{e51}) we have the following pair
$$B_2(x,y)+\int\limits^\infty_xdx^\prime B_1(x,x^\prime)\Omega_R(x^\prime
+y)=0,$$
\begin{equation}\label{e55}
-B_1(x,y)+\Omega_R(x+y)+\int\limits^\infty_xdx^\prime B_2(x,x^\prime)
\Omega_R(x^\prime+y)=0,
\end{equation}
$$x<y,$$
where
\begin{eqnarray}
\Omega_L&=&r_L(z)-i\sum^N_{l=1}\frac{c_{22}(\lambda_l)}{\dot{c}_{12}(\lambda_l)}
e^{-i\lambda_l z},\label{e56}\\
\Omega_R&=&r_R(z)+i\sum^N_{l=1}\frac{c_{11}(\lambda_l)}{\dot{c}_{21}(\lambda_l)}
e^{i\lambda_l z}.\label{e57}
\end{eqnarray}
Thus, the potential $u(x)$ is expressed via the kernels $A_1,\,A_2$ and
$B_1,\,B_2$ as follows
\begin{equation}\label{e58}
{\renewcommand{\arraystretch}{2.0}
\begin{array}{lcl}
u=-2A_2(x,x),&&u=-2B_1(x,x),\\
u^2=2\frac{\displaystyle dA_1(x,x)}{\displaystyle dx},&&
u^2=-2\frac{\displaystyle dB_2(x,x)}{\displaystyle dx}.
\end{array}}
\end{equation}
In the case of a reflectionless potential $(r_L(z)=0)$ the system of
Gel'fand-Levitan-Marchenko integral equations may be solved explicitly.
In this case we obtain
\begin{equation}\label{e59}
\Omega_L=-i\sum^N_{l=1}\frac{c_{22}(\lambda_l)}{\dot{c}_{12}(\lambda_l)}
e^{-i\lambda_lz}.
\end{equation}
The system (\ref{e54}) is reduced to algebraic equations and the potential
$u(x)$ is defined by a following expression
\begin{equation}\label{e60}
u=2\frac{d}{dx}\arctan\left[\frac{\mbox{Im}\det(I-iM)}
{\mbox{Re}\det(I-iM)}\right],
\end{equation}
where
\begin{eqnarray}
M_{ij}&=&\frac{im_{Lj}}{\kappa_i+\kappa_j}e^{-i(\kappa_i+\kappa_j)x},
\label{e61}\\
m_{Lj}&=&-i\frac{c_{22}(\lambda_j)}{\dot{c}_{12}(\lambda_j)}.\nonumber
\end{eqnarray}
Here $\kappa_i$ are the poles of the transmission coefficient $T_L(\lambda)=
\frac{\displaystyle 1}{\displaystyle c_{12}(\lambda)}$.
Analogous relations take place in the case
of system (\ref{e55}).
Let us return to the equations of geodesic deviation. From the conditions on
the curvature tensor (\ref{e42}) it follows that
\begin{eqnarray}
R^2_{121}&=&R^3_{131},\nonumber\\
R^3_{121}&=&u_x.\nonumber
\end{eqnarray}
Substituting the expressions (\ref{e45})-(\ref{e47}) into the latter
equations, we obtain
\begin{eqnarray}
\psi_{,11}+\psi^2_{,1}-\mu_{2,11}-\mu^2_{2,1}-\frac{1}{2}e^{-2\mu_2}
q^2_{2,1}(e^{2\psi}+\frac{1}{2})&=&0,\label{e62}\\
\frac{1}{2}e^{\psi-\mu_2}q_{2,11}+\frac{3}{2}e^{\psi-\mu_2}\psi_{,1}
q_{2,1}-\frac{1}{2}e^{\psi-\mu_2}\mu_{2,1}q_{2,1}&=&u_{,1}.\label{e63}
\end{eqnarray}
Thus, we have the system of differential equations (\ref{e62})-(\ref{e63})
as the conditions on the potential $u(x)$. The explicit form of $u(x)$
we will find by means of the inverse scattering problem.
Moreover, the potential
$u(x)$ depends parametrically
on variables $y$ and $z$. According to
widely accepted methods \cite{6,13,14}, the dependence on variables $y$ and
$z$ may be represented by a nonlinear integrable equation. Indeed,
the dependence on $y$ for $\psi_1$ and $\psi_2$ from (\ref{e44}) may be
expressed in general form
\begin{eqnarray}
\psi_{1y}&=&A\psi_1+B\psi_2,\nonumber\\
\psi_{2y}&=&C\psi_1+D\psi_2.\label{e64}
\end{eqnarray}
The compatibility conditions of (\ref{e64}) with (\ref{e44}) give
(at this point, $\lambda_y=0$):
\begin{eqnarray}
A_x&=&u(B+C),\\
B_x+2i\lambda B&=&-2uA+u_y,\label{e66}\\
C_x-2i\lambda C&=&-2uA-u_y,\label{e67}
\end{eqnarray}
here $D_x=-A_x$. Further, let us suppose that
$A=\sum^3_0a_n\lambda^n,\;B=\sum^3_0b_n
\lambda^n$ and $C=\sum^3_0c_n\lambda^n$. Substituting these series into
(\ref{e66})-(\ref{e67}), we obtain for the coefficients $a_n,\,b_n$ and
$c_n$ the following expressions
\begin{eqnarray}
a_3=a_3(y),&&b_3=c_3=0,\nonumber\\
a_2=a_2(y),&&b_2=-c_2=ia_3u,\nonumber\\
a_1=-\frac{1}{2}a_3u^2,&&b_1=-\frac{1}{2}a_3u_x+ia_2u,\quad
c=-\frac{1}{2}a_3u_x-ia_2u,\nonumber\\
a_0=-\frac{1}{2}a_2u^2,&&b_0=c_0=\frac{i}{4}a_3(u_{xx}+2u^3)-\frac{1}{2}
a_2u_x.\label{e68}
\end{eqnarray}
In the equations (\ref{e66})-(\ref{e67}) the components independing on
$\lambda$ give the evolution equation $u_y=b_{0x}+2a_0u$. Using the
obtained above expressions (\ref{e68}), we obtain
for the coefficients $a_0$ and $b_0$:
$$u_y=-\frac{1}{4}ia_3(6u^2u_x+u_{xxx})-a_2(u^3+\frac{1}{2}u_{xx}).$$
Suppose $a_2=0$ and $a_3=4i$ we have the modified Korteweg-de Vries equation
\begin{equation}\label{e69}
u_y+6u^2u_x+u_{xxx}=0.
\end{equation}
Thus, the dependence on parameter $y$ for the potential $u$ is defined by the
mKdV equation. Thus, the system (\ref{e64}) has a form
\begin{eqnarray}
\psi_{1y}&=&2i\lambda(u^2-2\lambda^2)\psi_1+(4\lambda^2u+2i\lambda u_x
-2u^3-u_{xx})\psi_2,\nonumber\\
\psi_{2y}&=&(-4\lambda^2u+2i\lambda u_x+2u^3+u_{xx})\psi_1-2i\lambda
(u^2-2\lambda^2)\psi_2.\label{e70}
\end{eqnarray}
Solutions of the modified Korteweg-de Vries equation can be found
by the standard
procedure \cite{6,13,14}. When $u\rightarrow 0$ we see that the
dependence on $y$ is described
by a limiting form of the equations (\ref{e70})
\begin{eqnarray}
\psi_{1y}&=&-4i\lambda^3\psi_1,\nonumber\\
\psi_{2y}&=&4i\lambda^3\psi_2.\label{e71}
\end{eqnarray}
Let us assume that $\psi=\begin{pmatrix}\psi_1\\ \psi_2\end{pmatrix}$ is
proportional to the fundamental solution $\varphi^-$ at $x\rightarrow -\infty$.
Then, at $x\rightarrow-\infty$ we have $\psi(x,y)=f(y)\varphi^-\rightarrow
f(y)e^{-i\lambda x}\begin{pmatrix}0\\ 1\end{pmatrix}$. Substituting
$\psi_1=f(y)e^{-i\lambda x}$ into the first equation from (\ref{e71}), we
obtain $f(y)=f(0)\exp(-4i\lambda^3y)$. From (\ref{e50}) at $x\rightarrow
+\infty$ it follows that
\begin{equation}\label{e72}
\psi=f(y)\varphi^-\longrightarrow f(0)e^{-4i\lambda^3y}\left[c_{11}(\lambda,
y)e^{i\lambda x}\begin{pmatrix} 0 \\ 1\end{pmatrix}+c_{12}(\lambda,y)
e^{-i\lambda x}\begin{pmatrix} 0 \\ 1\end{pmatrix}\right].
\end{equation}
Substituting it again into (\ref{e71}),
we obtain that $\dot{c}_{12}=0,\;\dot{c}_{11}
=8i\lambda^3$, whence
\begin{eqnarray}
c_{12}(\lambda,y)&=&c_{12}(\lambda,0),\nonumber\\
c_{11}(\lambda,y)&=&c_{11}(\lambda,0)e^{8i\lambda^3y}.\label{e73}
\end{eqnarray}
The analogous calculations for $\psi\sim\varphi^+$ give
\begin{equation}\label{e73'}
c_{22}(\lambda,y)=c_{22}(\lambda,0)e^{-8i\lambda^3y}.
\end{equation}
Using the dependence on parameter $y$ given by the relations (\ref{e73})-
(\ref{e73'}), we have for (\ref{e61}) and (\ref{e56}) the following
expressions
\begin{equation}\label{e74}
m_{Ll}(\kappa_l,y)=-i\frac{c_{22}(\kappa_l,y)}{\dot{c}_{12}(\kappa_l,0)}=
m_{Ll}(\kappa_l,0)e^{-8i\lambda^3_ly},
\end{equation}
\begin{equation}\label{e75}
\Omega_L(z,y)=\int\limits^\infty_{-\infty}\frac{d\lambda}{2\pi}R_L(\lambda,0)
e^{-8i\lambda^3y-i\lambda z}+\sum^N_{l=1}m_{Ll}(\kappa_l,0)
e^{-8i\kappa_l^3y-i\kappa_lz},
\end{equation}
where $R_L(\lambda,0)=-c_{22}(\lambda)/c_{21}(\lambda)$. Further, from
(\ref{e58}) it follows that the potential $u(x,y)$ is expressed by the kernel
of Gel'fand-Levitan-Marchenko equations (\ref{e54}) as
$$u(x,y)=-2A_2(x,x,y).$$
In case of the reflectionless potential $(R_L(\lambda,0)=0)$ the integral
equations (\ref{e54}) are solved explicitly. In this case,
the potential $u(x,y)$ is
defined by the formula (\ref{e60}) with the matrix $M$ of the following
form
\begin{equation}\label{e76}
M_{ij}=i\frac{m_{Lj}(\kappa_j,y)}{\kappa_i+\kappa_j)}e^{-i(\kappa_i+\kappa_j)x}.
\end{equation}
In the simplest case of the one-soliton solution $(N=1)$, the matrix $M$
is reduced
to a scalar $M=i(m_1/2\kappa_1)\exp(-2i\kappa_1x)$. Taking into account
the relation (\ref{e74}), we see that the matrix $M$ at
$\kappa=i\lambda$ can be written as
$$M=\frac{m_1(0)}{2\lambda}e^{2\lambda x-8\lambda^3y}.$$
Thus, in case of the one-soliton solution, the potential, defined by the
expression (\ref{e60}) with the matrix (\ref{e76}), is reduced to the form
\begin{equation}\label{e77}
u(x,y)=-2\frac{\partial}{\partial x}\arctan\left[\frac{m_1(0)}{2\lambda}
e^{2\lambda x -8\lambda^3y}\right]
\end{equation}
or
\begin{equation}\label{e78}
u(x,y)=\pm 2\lambda\,\mbox{sech}\,(2\lambda x -8\lambda^3+\delta),
\end{equation}
where $\delta=\ln\left[m_1(0)/2\lambda\right]$. For $m_1(0)<0$ we take the
upper sign, for $m_1(0)>0$ the lower sign.
Supposing the analogous dependence on the parameter $z$,
that is, defining it by the
modified Korteweg-de Vries equation of the form
\begin{equation}\label{e79}
u_z+6u^2u_x+u_{xxx}=0,
\end{equation}
we came in the case of the one-soliton solution to the following dependence
\begin{equation}\label{e80}
u(x,y,z)=\pm 2\lambda\mbox{sech}(2\lambda x -8\lambda^3y-8\lambda^3z+\delta).
\end{equation}
Thus, we see that dependence of the potential $u$ on
the parameters $y$ and $z$
is given by the mKdV equations (\ref{e69}) and (\ref{e79}).
Let us consider now how
the vector of geodesic deviation $\boldsymbol{\eta}$ may be expressed via
the fundamental solutions $\varphi^\mp$ of
the Zakharov-Shabat problem (\ref{e44}).
We will consider here two particular cases of the system (\ref{e62})-
(\ref{e63}).
\subsubsection{$\psi=0$}
In this case, the coefficients (\ref{e37}) of the metrics (\ref{i3}) are
\begin{equation}\label{e81}
A=-\left(e^{2\mu_2}+q^2_2\right),\quad B=q_2,\quad C=-1.
\end{equation}
And the system (\ref{e62})-(\ref{e63}) is reduced to the form
\begin{eqnarray}
\mu_{2,11}+\mu^2_{2,1}+\frac{3}{4}e^{-2\mu_2}q^2_{2,1}&=&0,\label{e82}\\
\frac{1}{2}e^{-\mu_2}q_{2,11}-\frac{1}{2}e^{-\mu_2}\mu_{2,1}q_{2,1}&=&u_{,1}.
\label{e83}
\end{eqnarray}
The latter equation, obviously, may be written as
$$\frac{1}{2}\left(e^{-\mu_2}q_{2,1}\right)_{,1}=u_{,1},$$
whence
\begin{equation}\label{e84}
u=\frac{1}{2}e^{-\mu_2}q_{2,1}.
\end{equation}
On the other hand, in virtue of (\ref{e80}) we have
\begin{equation}\label{e85}
\frac{1}{2}e^{-\mu_2}q_{2,1}=\pm 2\lambda\mbox{sech}(2\lambda x-8\lambda^3y
-8\lambda^3z+\delta).
\end{equation}
Using the well-known relation $\sinh 2A=2\cosh A\sinh A$, we can write
(choosing the upper sign) (\ref{e85}) in the form
\begin{equation}\label{e86}
e^{-\mu_2}q_{2,1}=\frac{8\lambda\sinh(2\lambda x -8\lambda^3y-8\lambda^3z
+\delta)}{\sinh(4\lambda x-16\lambda^3y-16\lambda^3z+2\delta)}.
\end{equation}
Whence, supposing
\begin{eqnarray}
e^{-\mu_2}&=&\frac{1}{\sinh(4\lambda x-16\lambda^3y-16\lambda^3z+2\delta)},
\nonumber\\
q_{2,1}&=&8\lambda\sinh(2\lambda x-8\lambda^3y-8\lambda^3z+\delta),\nonumber
\end{eqnarray}
we obtain
\begin{eqnarray}
\mu_2&=&\ln\sinh(4\lambda x-16\lambda^3y-16\lambda^3z+2\delta),\label{e87}\\
q_2&=&4\lambda\cosh(2\lambda x-8\lambda^3y-8\lambda^3z+\delta),\label{e88}
\end{eqnarray}
or
\begin{eqnarray}
\mu_2&=&\ln\sinh(4\lambda x-16\lambda^3y-16\lambda^3z+2\delta),\label{e87'}\\
q_2&=&-4\lambda\cosh(2\lambda x-8\lambda^3y-8\lambda^z+\delta)\label{e88'}
\end{eqnarray}
in the case $m_1(0)>0$.
Thus, using (\ref{e48})-(\ref{e49}) and (\ref{e58}), we obtain that solutions
of the equations of geodesic deviation (\ref{i4}) in the case of the metrics
(\ref{e81}) are expressed via the fundamental solutions of the Zakharov-Shabat
problem as follows
\begin{eqnarray}
\eta^2&\sim&\varphi^-_1(x,\lambda)=e^{-i\lambda x}+\frac{1}{2}\int
\limits^x_{-\infty}dx^\prime\int u^2dxe^{-i\lambda x^\prime},\label{e89}\\
\eta^3&\sim&\varphi^-_2(x,\lambda)=-\frac{1}{2}\int\limits^x_{-\infty}
dx^\prime ue^{-i\lambda x^\prime}\label{e90}
\end{eqnarray}
or
\begin{eqnarray}
\eta^2&\sim&\varphi^+_1(x,\lambda)=-\frac{1}{2}\int\limits^\infty_x
dx^\prime ue^{i\lambda x^\prime},\label{e91}\\
\eta^3&\sim&\varphi^+_2(x,\lambda)=e^{i\lambda x}-\frac{1}{2}\int
\limits^\infty_xdx^\prime\int u^2dxe^{i\lambda x},\label{e92}
\end{eqnarray}
where
\begin{equation}\label{e93}
u=\frac{1}{2}e^{-\mu_2}q_{2,1},
\end{equation}
and the functions $\mu_2$ and $q_2$ are related by the equation
\begin{equation}\label{e94}
\mu_{2,11}+\mu^2_{2,1}+\frac{1}{4}e^{-2\mu_2}q^2_{2,1}=0.
\end{equation}
For example, in the case of the parameter dependence on $y$ and $z$ described
by the mKdV equations (\ref{e69}) and (\ref{e79}) at $N=1$ (one-soliton
solution) we have the following integral representations
\begin{eqnarray}
\eta^2&\sim&e^{-i\lambda x}+\lambda\int\limits^x_{-\infty}dx\tanh(2\lambda
x-8\lambda^3y-8\lambda^3z+\delta)e^{-i\lambda x},\nonumber\\
\eta^3&\sim&\mp\lambda\int\limits^x_{-\infty}dx\,\mbox{sech}(2\lambda
x-8\lambda^3y-8\lambda^3z+\delta)e^{-i\lambda x}
\end{eqnarray}
or
\begin{eqnarray}
\eta^2&\sim&\mp\lambda\int\limits^\infty_xdx\,\mbox{sech}(2\lambda x-8\lambda^3y
-8\lambda^3z+\delta)e^{i\lambda x},\\
\eta^3&\sim&e^{i\lambda x}-\lambda\int\limits^\infty_xdx\tanh(2\lambda x
-8\lambda^3y-8\lambda^3z+\delta)e^{i\lambda x},
\end{eqnarray}
where for $m_1(0)<0$ we take the upper sign and the lower sign for $m_1(0)>0$.
The constraint (\ref{e94}) gives (here the functions $\mu_2$ and $q_2$ are
defined by (\ref{e87})-(\ref{e88}) or (\ref{e87'})-(\ref{e88'})):
$$\cosh^2(4\lambda x-16\lambda^3y-16\lambda^3z+2\delta)+3\sinh^2(2\lambda x
-8\lambda^3y-8\lambda^3z+\delta)=1.$$
\subsubsection{$\mu_2=0$}
In this case, the coefficients (\ref{e37}) of the metrics (\ref{i3}) are
\begin{equation}\label{e95}
A(x,y,z)=-\left(1+q^2_2e^{2\psi}\right),\quad B(x,y,z)=q_2e^{2\psi},
\quad C(x,y,z)=-e^{2\psi}.
\end{equation}
The system (\ref{e62})-(\ref{e63}) is reduced to the form
\begin{eqnarray}
&&\psi_{,11}+\psi^2_{,1}-\frac{1}{2}q^2_{2,1}(e^{2\psi}+\frac{1}{2})=0,
\label{e96}\\
&&\frac{1}{2}e^{\psi}q_{2,11}+\frac{3}{2}e^{\psi}\psi_{,1}q_{2,1}=u_{,1}.
\label{e97}
\end{eqnarray}
Using the substitution $\theta=\frac{\displaystyle\psi}{\displaystyle 3}$,
we obtain from the latter equation
\begin{equation}\label{e98}
\frac{1}{2}\left(e^{\theta/3}q_{2,1}\right)_{,1}=u_{,1}.
\end{equation}
Therefore, in this case the vector of geodesic deviation $\boldsymbol{\eta}$
is also expressed via the fundamental solutions $\varphi^\mp(x,\lambda)$
of the form (\ref{e89})-(\ref{e90}) or (\ref{e91})-(\ref{e92}).
At this point,
\begin{equation}\label{e99}
u=\frac{1}{2}e^{\theta/3}q_{2,1}
\end{equation}
and the functions $\theta$ and $q_2$ are related by the equation
\begin{equation}\label{e100}
\theta_{,11}+\frac{1}{3}\theta^2_{,1}-\frac{3}{2}q_{2,1}(e^{2/3\theta}+
\frac{1}{2})=0.
\end{equation}
In the case of the one-soliton solution, a potential $u$
is defined by the expression
(\ref{e80}), and the functions $\theta$ and $q_2$ are respectively equal to
\begin{eqnarray}
\theta&=&3\ln\mbox{csch}(4\lambda x-16\lambda^3y-16\lambda^3z+2\delta),\\
q_2&=&\pm\cosh(2\lambda x-8\lambda^3y-8\lambda^3z+\delta).
\end{eqnarray}
More complicated case $\mu_2\neq 0,\;\psi\neq 0$ and also multi-soliton
solutions will be considered in a separate paper.
\section{Four-dimensional space}
\setcounter{equation}{0}
\subsection{$3\times 3$ matrix Schr\"{o}dinger equation}
Let us consider the following linear problem
\begin{equation}\label{f1}
\psi_{,4}=i\zeta D\psi+N\psi,
\end{equation}
where $\psi_{,4}=\frac{\displaystyle\partial}{\displaystyle\partial x^4}
\psi,\;x^4=it;\;\zeta$ is a spectral parameter and $\psi$ is a
$3\times 1$ matrix (vector) of the form
$$\psi=\begin{pmatrix}
\psi_1\\
\psi_2\\
\psi_3
\end{pmatrix}.$$
The $3\times 3$ matrices $D$ and $N$ (a potential matrix) are
$$D=\begin{pmatrix}
\mp d_1 & 0 & 0\\
0 & \pm d_2 & 0\\
0 & 0 & \mp d_3
\end{pmatrix},\quad N=\begin{pmatrix}
0 & N_{12} & N_{13}\\
N_{21} & 0 & N_{23}\\
N_{31} & N_{32} & 0
\end{pmatrix}.$$
The system (\ref{f1}) may be rewritten (see Appendix) in the following form
($3\times 3$ matrix Schr\"{o}dinger equation)
\begin{equation}\label{f2}
-I\psi_{,44}+\mathfrak{N}\psi=d^2\zeta^2\psi,
\end{equation}
where $I$ is a $3\times 3$ unit matrix, $d=(d_1,\,d_2,\,d_3)$ and
$$\mathfrak{N}=
\begin{pmatrix}
N_{12}N_{21}+N_{13}N_{31}& N_{12,4}+N_{13}N_{32}& N_{13,4}+N_{12}N_{23}\\
N_{21,4}+N_{23}N_{31} & N_{21}N_{12}+N_{23}N_{32} & N_{23,4}+N_{21}N_{13}\\
N_{31,4}+N_{32}N_{21} & N_{32,4}+N_{31}N_{12} & N_{31}N_{13}+N_{32}N_{23}
\end{pmatrix}.$$
Further, it is easy to see that for the metrics (\ref{i9}) the geodesic
deviation equation (\ref{i10}),
\begin{eqnarray}
\eta^1_{,44}+R^1_{414}\eta^1+R^1_{424}\eta^2+R^1_{434}\eta^3&=&0,\nonumber\\
\eta^2_{,44}+R^2_{414}\eta^1+R^2_{424}\eta^2+R^2_{434}\eta^3&=&0,\label{f3}\\
\eta^3_{,44}+R^3_{414}\eta^1+R^3_{424}\eta^2+R^3_{434}\eta^3&=&0,\nonumber
\end{eqnarray}
can be rewritten in the form of the $3\times 3$ matrix Schr\"{o}dinger operator
\begin{eqnarray}
-\eta^1_{,44}+(-R^1_{414}+d^2_1\zeta^2)\eta^1+(-R^1_{424})\eta^2+(-R^1_{434})
\eta^3&=&d^2_1\zeta^2\eta^1,\nonumber\\
-\eta^2_{,44}+(-R^2_{414})\eta^1+(-R^2_{242}+d^2_2\zeta^2)\eta^2+(-R^2_{434})
\eta^3&=&d^2_2\zeta^2\eta^2,\label{f4}\\
-\eta^3_{,44}+(-R^3_{414})\eta^1+(-R^3_{424})\eta^2+(-R^3_{434}+d^2_3\zeta^2)
\eta^3&=&d^2_3\zeta^2\eta^3.\nonumber
\end{eqnarray}
Comparing these equations with equations (\ref{f2}), we obtain the following
conditions on the curvature tensor:
\begin{eqnarray}
&&\phantom{-}d^2_1\zeta^2-R^1_{414}=N_{12}N_{21}+N_{13}N_{31},\nonumber\\
&&\phantom{-}d^2_2\zeta^2-R^2_{424}=N_{21}N_{12}+N_{23}N_{32},\nonumber\\
&&\phantom{-}d^2_3\zeta^2-R^3_{434}=N_{31}N_{13}+N_{32}N_{23},\nonumber\\
&&-R^1_{424}=N_{12,4}+N_{13}N_{32},\nonumber\\
&&-R^1_{434}=N_{13,4}+N_{12}N_{23},\nonumber\\
&&-R^2_{414}=N_{21,4}+N_{23}N_{31},\nonumber\\
&&-R^2_{434}=N_{23,4}+N_{21}N_{13},\nonumber\\
&&-R^3_{414}=N_{31,4}+N_{32}N_{21},\nonumber\\
&&-R^3_{414}=N_{31,4}+N_{32}N_{21}.\label{f5}
\end{eqnarray}
\subsection{Chandrasekhar metrics}
In the 4-dimensional space with a signature $(-,-,-,-)$ the Chandrasekhar
metrics is defined by the following expression \cite{8}
\begin{equation}\label{f7}
{\rm d} s^{2}=-\sum_{A}e^{2\mu_{A}}({\rm d} x^{A})^{2}-e^{2\psi}({\rm d} x^{1}-
\sum_{A}q_{A}{\rm d} x^{A})^{2},
\end{equation}
where $A=2,3,4$. $\psi,\,\mu_{A}$ and $q_{A}$ are the functions on variables
$x^{1},\, x^{2},\,x^{3},\,x^{4}$.
The orthonormal tetrad, related with the metrics
(\ref{f7}), is defined by the following covariant
basis vectors:
\begin{eqnarray}
e_{(4)i}=(-e^{\mu_{4}},\,0,\,0,\,0), && e_{(1)i}=(q_{4}e^{\psi},\,-e^{\psi},\,
q_{2}e^{\psi},\,q_{3}e^{\psi}),\nonumber \\
e_{(2)i}=(0,\,0,\,-e^{\mu_{2}},\,0), && e_{(3)i}=(0,\,0,\,0,\,-e^{\mu_{3}}).
\label{f8}
\end{eqnarray}
And also the contravariant basis vectors are
\begin{eqnarray}
e^{i}_{(4)}=(e^{-\mu_{4}},\,q_{4}e^{-\mu_{4}},\,0,\,0), && e^{i}_{(1)}=(0,
\,e^{-\psi},\,
0,\,0),\nonumber \\
e^{i}_{(2)}=(0,\,q_{2}e^{-\mu_{2}},\,e^{-\mu_{2}},\,0), && e^{i}_{(3)}=(0,\,
q_{3}e^{-\mu_{3}},\,0,\,e^{-\mu_{3}}).\label{f9}
\end{eqnarray}
From (\ref{f8}) and (\ref{f9}) it is easy to see that
$$e^{i}_{(a)}e_{(b)i}=\eta_{(a)(b)}=\left|
\begin{array}{cccc}
-1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{array}\right|.$$
Therefore, in this orthonormal basis for the components of the curvature tensor
we have always
\begin{equation}\label{f10}
R^{n}_{klm}=-R_{iklm}.
\end{equation}
Moreover, at ${\rm d} t=-i{\rm d} x^4,\;\nu=\mu_4$ and $\omega=iq_4$ there exists
an analytic continuation of the basis (\ref{f8})-(\ref{f9}) with the signature
$(-,\,-,\,-,\,-)$ onto a basis with a signature $(-,\,-,\,-,\,+)$, the
covariant and contravariant vectors of which have the form
\begin{equation}
{\renewcommand{\arraystretch}{1.3}
\begin{array}{lcl}
e_{(1)i}=(\omega e^\psi,\,-e^\psi,\,q_2e^\psi,\,q_3e^\psi), &&
e_{(2)i}=(0,\,0,\,-e^{\mu_2},\,0),\\
e_{(3)i}=(0,\,0,\,0,\,-e^{\mu_3}), &&
e_{(4)i}=(e^\nu,\,0,\,0,\,0).\\
e^i_{(1)}=(0,\,e^{-\psi},\,0,\,0), &&
e^i_{(2)}=(0,\,q_2e^{-\mu_2},\,e^{-\mu_2},\,0),\\
e^i_{(3)}=(0,\,q_3e^{-\mu_3},\,0,\,e^{-\mu_3}), &&
e^i_{(4)}=(e^{-\nu},\,\omega e^{-\nu},\,0,\,0).
\end{array}}
\end{equation}
It is obvious that in the orthonormal basis (\ref{f8})-(\ref{f9}) among
the nine components of the curvature tensor of the system (\ref{f3})
only six are independent, namely,
$$R^1_{414},\;R^2_{424},\;R^3_{434},\;R^1_{424},\;R^1_{434},\;R^2_{434}.$$
In the orthonormal basis (\ref{f8})-(\ref{f9}) for the metrics (\ref{f7})
these components have the form \cite{8}
\begin{multline}\label{f13}
-R_{1414}=-e^{-\psi-\mu_4}\mathcal{D}_4\left(e^{\psi-\mu_4}\Psi_4\right)-
e^{-2\mu_2}\Psi_2\mu_{4:2}-e^{-2\mu_3}\Psi_3\mu_{4:3}-\\
-e^{-\psi-\mu_4}\left(e^{\mu_4-\psi}\mu_{4,1}\right)_{,1}+\frac{1}{4}
e^{2\psi-2\mu_4}\left[e^{-2\mu_2}Q^2_{24}+e^{-2\mu_3}Q^2_{34}\right],
\end{multline}
\begin{multline}\label{f14}
-R_{2424}=-e^{-\mu_2-\mu_4}\left[\left(e^{\mu_2-\mu_4}\mu_{2:4}\right)_{:4}
+\left(e^{\mu_4-\mu_2}\mu_{4:2}\right)_{:2}\right]-\\
-e^{-2\mu_3}\mu_{4:3}\mu_{2:3}-\frac{3}{4}e^{2\psi-2\mu_2-2\mu_4}Q^2_{24}
-e^{-2\psi}\mu_{2,1}\mu_{4,1},
\end{multline}
\begin{multline}\label{f15}
-R_{3434}=-e^{-\mu_3-\mu_4}\left[\left(e^{\mu_3-\mu_4}\mu_{3:4}\right)_{:4}
+\left(e^{\mu_4-\mu_3}\mu_{4:3}\right)_{:3}\right]-\\
-e^{-2\mu_2}\mu_{4:2}\mu_{3:2}-\frac{3}{4}e^{2\psi-2\mu_3-2\mu_4}Q^2_{34}
-e^{-2\psi}\mu_{3,1}\mu_{4,1},
\end{multline}
\begin{multline}\label{f16}
R_{1424}=e^{\psi-2\mu_4-\mu_2}Q_{42}\left(\Psi_4-\frac{1}{2}\mu_{2:4}\right)
+\frac{1}{2}e^{-\mu_4-\mu_2}\left(e^{\psi-\mu_4}Q_{42}\right)_{:4}+\\
+\frac{1}{2}e^{\psi-2\mu_3-\mu_2}Q_{32}\mu_{4:3}+e^{-\mu_4-\mu_2}\left(
e^{-\psi+\mu_4}\mu_{4,1}\right)_{:2}-e^{-\psi-\mu_2}\mu_{2,1}\mu_{4:2},
\end{multline}
\begin{multline}\label{f17}
R_{1434}=e^{\psi-2\mu_4-\mu_3}Q_{43}\left(\Psi_4-\frac{1}{2}\mu_{3:4}\right)
+\frac{1}{2}e^{-\mu_4-\mu_3}\left(e^{\psi-\mu_4}Q_{43}\right)_{:4}+\\
+\frac{1}{2}e^{\psi-2\mu_2-\mu_3}Q_{23}\mu_{4:2}+e^{-\mu_4-\mu_3}\left(
e^{-\psi+\mu_4}\mu_{4,1}\right)_{:3}-e^{-\psi-\mu_3}\mu_{3,1}\mu_{4:3},
\end{multline}
\begin{multline}\label{f18}
R_{2434}=e^{-\mu_2-\mu_3}\left[\mu_{4:32}+\mu_{4:3}\left(\mu_4-\mu_3\right)
_{:2}-\mu_{4:2}\mu_{2:3}\right]-\\
-\frac{3}{4}e^{2\psi-\mu_3-2\mu_4-\mu_2}Q_{34}Q_{42}-\frac{1}{2}e^{-\mu_3-
\mu_2}Q_{32}\mu_{4,1},
\end{multline}
where
\begin{eqnarray}
\mathcal{D}_Af&=&f_{,A}+\left(q_Af\right)_{,1},\nonumber\\
\Psi_A&=&\psi_{:A}+q_{A,1},\nonumber\\
Q_{AB}&=&q_{A:B}-q_{B:A},\nonumber\\
f_{:A}&=&f_{,A}+q_{A}f_{,1}.\nonumber
\end{eqnarray}
\subsection{Solutions of equations of geodesic deviation
in\protect\newline the four-dimensional space}
It is easy to see that the Chandrasekhar metrics (\ref{f7}) coincides with the
metrics (\ref{i9}) at $\mu_4=q_4=0$. In this case, the orthonormal basis
is reduced to the form
$$
{\renewcommand{\arraystretch}{1.3}
\begin{array}{lcl}
e_{(1)i}=(0,\,-e^\psi,\,q_2e^\psi,\,q_3e^\psi),&&
e_{(2)i}=(0,\,0,\,-e^{\mu_2},\,0),\\
e_{(3)i}=(0,\,0,\,0,\,-e^{\mu_3}),&&
e_{(4)i}=(-1,\,0,\,0,\,0);\\
e^i_{(1)}=(0,\,e^{-\psi},\,0,\,e^{-\mu_3}),&&
e^i_{(2)}=(0,\,q_2e^{-\mu_2},\,e^{-\mu_2},\,0),\\
e^i_{(3)}=(0,\,q_3e^{-\mu_3},\,0,\,e^{-\mu_3}),&&
e^i_{(4)}=(1,\,0,\,0,\,0).
\end{array}}$$
It is obvious that in this basis we have $R^n_{jkl}=-R_{ijkl}$, and the
components of the curvature tensor (\ref{f13})-(\ref{f18}) are
\begin{eqnarray}
-R_{1414}&=&-\psi_{,44}-\psi^2_{,4}+\frac{1}{4}e^{2\psi}\left[e^{-2\mu_2}
q^2_{2,4}+e^{-2\mu_3}q^2_{3,4}\right],\label{f18'}\\
-R_{2424}&=&-\mu_{2,44}-\mu^2_{2,4}-\frac{3}{4}e^{2\psi-2\mu_2}q^2_{2,4},\\
-R_{3434}&=&-\mu_{3,44}-\mu^2_{3,4}-\frac{3}{4}e^{2\psi-2\mu_3}q^2_{3,4},\\
R_{1424}&=&-\frac{1}{2}e^{\psi-\mu_2}\left[q_{2,44}-\mu_{2,4}q_{2,4}+
3\psi_{,4}q_{2,4}\right],\\
R_{1434}&=&-\frac{1}{2}e^{\psi-\mu_3}\left[q_{3,44}-\mu_{3,4}q_{3,4}+
3\psi_{,4}q_{3,4}\right],\\
R_{2434}&=&\frac{3}{4}e^{2\psi-\mu_2-\mu_3}q_{2,4}q_{3,4}.\label{f19}
\end{eqnarray}
Further, let us define now the evolution equations related with the problem
(\ref{f1}). We consider the following system
\begin{eqnarray}
\psi_{,4}&=&i\zeta D\psi+N\psi,\nonumber\\
\psi_{,1}&=&Q\psi,\nonumber
\end{eqnarray}
where $\psi_{,1}=\frac{\displaystyle\partial}{\displaystyle\partial x^1},\;
x^1$ is a parameter of the considered problem. $D,\,N,\,Q$ be the $3\times 3$
matrices. At this point, $D$ is diagonal: $D=d_i\delta_{ij},\;d_i=const$;
$N$ is such a matrix that $N_{ii}=0$. From the compatibility condition
$\psi_{,14}=\psi_{,41}$ and the requirement $\zeta_{,1}=0$ we obtain
$$Q_{,4}=N_{,1}+i\zeta[D,Q]+[N,Q].$$
Decomposing $Q$ in the form
$$Q=Q^{(1)}\zeta+Q^{(0)},$$
we have $Q^{(0)}_{,4}=N_{,1}+[N,Q^{(0)}]$,
whence we obtain the
system of $n(n-1)$ equations (see \cite{6}):
\begin{equation}\label{f20}
N_{lj,1}-a_{lj}N_{lj,4}=\sum_k(a_{lk}-a_{kj})N_{lk}N_{kj},
\end{equation}
where
$$a_{lj}=\frac{1}{i}\frac{q_l-q_j}{d_l-d_j}=a_{jl}.$$
Equations (\ref{f20}) may be reduced to the standard system of nonlinear
equations of three-wave interaction. Namely, we obtain
\begin{eqnarray}
Q_{1,1}+C_1Q_{1,4}&=&i\gamma_1Q^\ast_2Q^\ast_3,\nonumber\\
Q_{2,1}+C_2Q_{2,4}&=&i\gamma_2Q^\ast_1Q^\ast_3,\label{f21}\\
Q_{3,1}+C_3Q_{3,4}&=&i\gamma_3Q^\ast_1Q^\ast_2,\nonumber
\end{eqnarray}
where $\gamma_1\gamma_2\gamma_3=-1$, $\gamma_i=\pm 1$ and
\begin{equation}\label{f22}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{lcl}
N_{12}=-iQ_3/\sqrt{\beta_{13}\beta_{23}},&&
N_{31}=-iQ_2/\sqrt{\beta_{12}\beta_{23}},\\
N_{23}=+iQ_1/\sqrt{\beta_{12}\beta_{13}},&&
N_{13}=-\gamma_1\gamma_3N^\ast_{31},\\
N_{32}=\gamma_2\gamma_3N^\ast_{23},&&
N_{21}=\gamma_1\gamma_2N^\ast_{12},
\end{array}}
\end{equation}
here
$$q_j=-i\frac{C_1C_2C_3}{C_j},\quad\beta_{lj}=d_l-d_j=C_j-C_l,$$
$$C_3>C_2>C_1.$$
In the system (\ref{f21}) there is a decay instability (for the waves with
positively defined energy) if the sign of one $\gamma_n$ is different from
the other,
and also there is an explosive instability when $\gamma_1=\gamma_2=\gamma_3=-1$.
Solutions of the system (\ref{f21}) was obtained by Zakharov and Manakov in
1973 \cite{10,11,12}. They have the form
\begin{multline}\label{f23}
Q_1=\sqrt{\beta_{12}\beta_{13}}\frac{2\chi_3}{\mathfrak{D}}e^{i\xi_3(x^4-C_1x^1-
\bar{\varphi}_1)}\biggl[e^{\chi_1(x^4-C_3x^1-\varphi_3}-\biggr.\\
-\biggl.\gamma_1\gamma_2\frac{\bar{\zeta}_1-\bar{\zeta}_3}{\zeta^\ast_1-
\zeta_3}e^{-\chi_1(x^4-C_3x^1-\varphi_3)}\biggr],
\end{multline}
\begin{equation}\label{f24}\!\!\!\!\!\!
Q_2=\frac{-4\chi_1\chi_3\beta_{13}\gamma_2\gamma_3}{\sqrt{\beta_{12}\beta_{23}}
(\bar{\zeta}_1-\zeta^\ast_3)\mathfrak{D}}e^{-i\xi_1(x^4-C_3x^1-\bar{\varphi}_3)}
e^{-i\xi_3(x^4-C_1x^1-\bar{\varphi}_1)},
\end{equation}
\begin{multline}\label{f25}
Q_3=\sqrt{\beta_{13}\beta_{23}}\gamma_1\gamma_2\frac{2\chi_1}{\mathfrak{D}}
e^{i\xi_1(x^4-C_3x^1-\bar{\varphi}_3)}\biggl[e^{\chi_3(x^4-C_1x^1-\varphi_1)}
-\biggr.\\
-\biggl.\gamma_2\gamma_3\frac{\bar{\zeta}^\ast_1-\zeta^\ast_3}
{\bar{\zeta}^\ast_1-\zeta_3}e^{-\chi_3(x^4-C_1x^1-\varphi_1)}\biggr],
\end{multline}
where
\begin{multline}
\mathfrak{D}=\left[e^{\chi_1(x^4-C_3x^1-\varphi_3)}-\gamma_1
\gamma_2e^{-\chi_1(x^4-
C_3x^1-\varphi_3)}\right]\times\\
\times\left[e^{\chi_3(x^4-C_1x^1-\varphi_1)}-\gamma_2\gamma_3e^{-\chi_3(x^4-
C_1x^1-\varphi_1)}\right]+\\
+\gamma_1\gamma_3\frac{\left(\bar{\zeta}_1-\bar{\zeta}^\ast_1\right)\left(
\zeta_3-\zeta^\ast_3\right)}{\left(\bar{\zeta}_1-\zeta^\ast_3\right)\left(
\bar{\zeta}^\ast_1-\zeta_3\right)}e^{-\chi_1(x^4-C_3x^1-\varphi_3)}
e^{-\chi_3(x^4-C_1x^1-\varphi_1)},
\end{multline}
$$\bar{\zeta}_1=\frac{\xi_1-i\chi_1}{\beta_{12}},\quad\zeta_3=\frac{\xi_3-
i\chi_3}{\beta_{23}}.$$
Supposing now that the matrix $N$ is real ($N^\ast=N$) and choosing $\gamma_1=
\gamma_3=1,\;\gamma_2=-1$, we obtain from (\ref{f22})
\begin{equation}\label{f26}
{\renewcommand{\arraystretch}{1.5}
\begin{array}{lcl}
N_{12}=-\mbox{Re}(iQ_3/\sqrt{\beta_{13}\beta_{23}}),&&
N_{31}=-\mbox{Re}(iQ_2/\sqrt{\beta_{12}\beta_{23}}),\\
N_{23}=+\mbox{Re}(iQ_1/\sqrt{\beta_{12}\beta_{13}}),&&
N_{13}=-N_{31},\\
N_{32}=-N_{23},&&
N_{21}=-N_{12}.
\end{array}}
\end{equation}
It is obvious that the latter three conditions in (\ref{f26}) are equivalent
to antisymmetry of the matrix $N$.
Thus, we assume that the potential matrix $N$ is real and antisymmetric.
Taking it into account and also the expressions (\ref{f18'})-(\ref{f19}),
we obtain from the conditions on the curvature tensor (\ref{f5}) the following
system of differential equations:
\begin{eqnarray}
&&-\psi_{,44}-\psi^2_{,4}+\frac{1}{4}e^{2\psi}\left[e^{-2\mu_2}q^2_{2,4}+
e^{-2\mu_3}q^2_{3,4}\right]=N^2_{12}+N^2_{13}+d^2_1\zeta^2,\label{f27}\\
&&-\mu_{2,44}-\mu^2_{2,4}-\frac{3}{4}e^{2\psi-2\mu_2}q^2_{2,4}=N^2_{12}+
N^2_{23}+d^2_2\zeta^2,\\
&&-\mu_{3,44}-\mu^2_{3,4}-\frac{3}{4}e^{2\psi-2\mu_3}q^2_{3,4}=N^2_{13}+
N^2_{23}+d^2_3\zeta^2,\\
&&-\frac{1}{2}e^{\psi-\mu_2}\left[q_{2,44}-\mu_{2,4}q_{2,4}+3\psi_{,4}q_{2,4}
\right]=N_{23}N_{31},\\
&&-\frac{1}{2}e^{\psi-\mu_3}\left[q_{3,44}-\mu_{3,4}q_{3,4}+3\psi_{,4}q_{3,4}
\right]=N_{12}N_{23},\\
&&\phantom{-}\frac{3}{4}e^{2\psi-\mu_2-\mu_3}q_{2,4}q_{3,4}=N_{31}N_{12}.
\label{f32}
\end{eqnarray}
Obviously, this system has a great number of particular cases. For example,
let us consider one simplest case.
\subsubsection{$q_3=\text{const},\;\psi=\mu_2=0$}
In this case for the metrics (\ref{i9}) we have
$$
{\renewcommand{\arraystretch}{1.5}
\begin{array}{lll}
g_{11}=-1,&\quad g_{22}=-(1+q^2_2),&\quad g_{33}=-(e^{2\mu_3}+\text{const}^2),
\\
g_{12}=2q_2,&\quad g_{13}=2\text{const},&\quad g_{23}=-2\text{const}q_2.
\end{array}}
$$
The system (\ref{f27})-(\ref{f32}) is reduced to the form
\begin{eqnarray}
&&\phantom{-}q^2_{2,4}=N^2_{12}+N^2_{13}+d^2_1\zeta^2,\label{f33}\\
&&-\frac{3}{4}q^2_{2,4}=N^2_{12}+N^2_{23}+d^1_2\zeta^2,\\
&&-\mu_{3,44}-\mu^2_{3,4}=N^2_{13}+N^2_{23}+d^2_3\zeta^2,\\
&&-\frac{1}{2}q_{2,44}=N_{23}N_{31},\\
&&\phantom{-}N_{12}N_{23}=0,\\
&&\phantom{-}N_{31}N_{12}=0.\label{f38}
\end{eqnarray}
From the latter two equations it follows that $N_{12}=0$.
Therefore, in this case, the
potential matrix $N$ has a form
$$\begin{pmatrix}
0 & 0 & -N_{31}\\
0 & 0 & N_{23}\\
N_{31} & -N_{23} & 0
\end{pmatrix}.$$
Further, at $d^2_1=-4/3d^2_2$ from (\ref{f33})-(\ref{f38}) it
follows that
\begin{eqnarray}
&&-\frac{1}{2}q_{2,44}=N_{23}N_{31},\label{f39}\\
&&-\mu_{3,44}-\mu^2_{3,4}=-\frac{1}{3}N^2_{23}+d^2_3\zeta^2,\\
&&\phantom{-}N^2_{13}+\frac{4}{3}N^2_{23}=0.\label{f41}
\end{eqnarray}
In accordance with (\ref{f26}), the components $N_{23}$ and
$N_{31}$ are defined as
\begin{multline}
N_{23}=\frac{2\chi_3}{\mathfrak{D}}\Biggl[\frac{2(\beta_{23}\chi_1-\beta_{12}
\chi_3)}{(\beta_{23}\xi_1-\beta_{12}\xi_3)^2+(\beta_{23}\chi_1-\beta_{12}
\chi_3)^2}\Biggr.\times\\
\times\cos\xi_3\left(x^4-C_1x^1-\bar{\varphi}_1\right)e^{-\chi_1\left(x^4-
C_3x^1-\varphi_3\right)}-\\
-\Biggl.2\sin\xi_3\left(x^4-C_1x^1-\bar{\varphi}_1\right)\cosh\chi_1\left(
x^4-C_3x^1-\varphi_3\right)\Biggr].
\end{multline}
\begin{multline}
N_{31}=-\frac{4\chi_1\chi_3\beta_{13}(\beta_{23}\xi_1-\beta_{12}\xi_3)}
{(\beta_{23}\xi_1-\beta_{12}\xi_3)^2+(\beta_{23}\chi_1-\beta_{12}\chi_3)^2
\mathfrak{D}}\times\\
\times\Biggl[\sin\xi_1\left(x^4-C_3x^1-\bar{\varphi}_3\right)\cos\xi_3\left(
x^4-C_1x^1-\bar{\varphi}_1\right)+\Biggr.\\
+\Biggl.\cos\xi_1\left(x^4-C_3x^1-\bar{\varphi}_3\right)\sin\xi_3\left(
x^4-C_1x^1-\bar{\varphi}_1\right)\Biggr],
\end{multline}
where
\begin{multline}
\mathfrak{D}=4\cosh\chi_1\left(x^4-C_3x^1-\varphi_3\right)\cosh\chi_3\left(
x^4-C_1x^1-\varphi_1\right)+\\
+\frac{4\chi_1\chi_3}{\beta^2_{12}(\xi^2_3+\chi^2_3)+2\beta_{12}\beta_{23}
(\xi_1\xi_3+\chi_1\chi_3)+\beta^2_{23}(\xi^2_1+\chi^2_1)}\times\\
\times e^{-\chi_1\left(x^4-C_3x^1-\varphi_3\right)}e^{-\chi_3\left(x^4-C_1x^1
-\varphi_1\right)}.
\end{multline}
After very cumbersome but elementary calculations it is easy to verify
that solutions of the system (\ref{f39})-(\ref{f41}) exist.
\section*{Appendix}
\setcounter{equation}{0}
\renewcommand{\theequation}{A.\arabic{equation}}
Let us consider the system (\ref{f1}) with the matrix
$$D=\begin{pmatrix}
-d_1 & 0 & 0\\
0 & d_2 & 0\\
0 & 0 & -d_3\\
\end{pmatrix}.$$
Differentiating (\ref{f1}) and excluding the first derivatives $\psi_{,4}$,
we obtain the following system
\begin{multline}\label{a1}
-\psi_{1,44}+(N_{12}N_{21}+N_{13}N_{31})\psi_1+(N_{12,4}+N_{13}N_{32}
-i\zeta d_1N_{12}+i\zeta d_2N_{12})\psi_2+\\
+(N_{13,4}+N_{12}N_{23}-i\zeta d_1N_{13}-i\zeta d_3N_{13})\psi_3=
\zeta^2d^2_1\psi_1,
\end{multline}
\begin{multline}
-\psi_{2,44}+(N_{21,4}+N_{23}N_{31}+i\zeta d_2N_{21}-i\zeta d_1N_{21})\psi_1
+(N_{21}N_{12}+N_{23}N_{32})\psi_2+\\
+(N_{23,4}+N_{21}N_{13}+i\zeta d_2N_{23}-i\zeta d_3N_{23})\psi_3=
\zeta^2d^2_2\psi_2,
\end{multline}
\begin{multline}\label{a3}
-\psi_{3,44}+(N_{31,4}+N_{32}N_{21}-i\zeta d_3N_{31}-i\zeta d_1N_{31})\psi_1
+(N_{32,4}+N_{31}N_{12}-i\zeta d_3N_{32}+\\
+i\zeta d_2N_{32})\psi_2+(N_{31}N_{13}+N_{32}N_{23})\psi_3=\zeta^2d^2_3\psi_3.
\end{multline}
Analogously, for the matrix
$$D=\begin{pmatrix}
d_1 & 0 & 0\\
0 & -d_2 & 0\\
0 & 0 & d_3
\end{pmatrix}$$
we have the system
\begin{multline}\label{a4}
-\psi_{1,44}+(N_{12}N_{21}+N_{13}N_{31})\psi_1+
(N_{12,4}+N_{13}N_{32}+i\zeta d_1N_{12}-i\zeta d_2N_{12})\psi_2+\\
+(N_{13,4}+N_{12}N_{23}+i\zeta d_1N_{13}+i\zeta d_3N_{13})\psi_3=
\zeta^2d^2_1\psi_1,
\end{multline}
\begin{multline}
-\psi_{2,44}+(N_{21,4}+N_{23}N_{31}-i\zeta d_2N_{21}+i\zeta d_1N_{21})\psi_1+
(N_{21}N_{12}+N_{23}N_{32})\psi_2+\\
+(N_{23,4}+N_{21}N_{13}-i\zeta d_2N_{23}+i\zeta d_3N_{23})\psi_3=
\zeta^2d^2_2\psi_2,
\end{multline}
\begin{multline}\label{a6}
-\psi_{3,44}+(N_{31,4}+N_{32}N_{21}+i\zeta d_3N_{31}+i\zeta d_1N_{31})\psi_1
+(N_{32,4}+N_{31}N_{12}+i\zeta d_3N_{32}-\\
-i\zeta d_2N_{32})\psi_2++(N_{31}N_{13}+N_{32}N_{23})\psi_3=\zeta^2d^2_3\psi_3.
\end{multline}
Adding the systems (\ref{a1})-(\ref{a3}) and (\ref{a4})-(\ref{a6}),
we obtain in the result
\begin{multline}
-\psi_{1,44}+(N_{12}N_{21}+N_{13}N_{31})\psi_1+(N_{12,4}+N_{13}N_{32})\psi_2+\\
+(N_{13,4}+N_{12}N_{23})\psi_3=\zeta^2d^2_1\psi_1,
\end{multline}
\begin{multline}
-\psi_{2,44}+(N_{21,4}+N_{23}N_{31})\psi_1+(N_{21}N_{12}+N_{23}N_{32})\psi_2+\\
+(N_{23,4}+N_{21}N_{13})\psi_3=\zeta^2d^2_2\psi_2,
\end{multline}
\begin{multline}
-\psi_{3,44}+(N_{31,4}+N_{32}N_{21})\psi_1+(N_{32,4}+N_{31}N_{12}
)\psi_2+\\
+(N_{31}N_{13}+N_{32}N_{23})\psi_3=\zeta^2d^2_3\psi_3.
\end{multline}
It is easy to see that the latter system can be rewritten in the form
of $3\times 3$ matrix Schr\"{o}dinger equation (\ref{f2}).
\section*{Acknowledgement}
I am deeply grateful to Prof. B. G. Konopelchenko for useful discussions.
| {
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Q: SpringCloud's /refresh not effective on properties used in camel routes Camel version : 2.22.0
SpringBoot Version : 2.0.2.RELEASE
Observation: the config properties, which ideally have to be updated as they are changed in the configuration service and when a refresh is done at the Client service, does not take effect in case of properties that are used inside a Camel Route.
Note: there is a ticket in Jira whihc says fixed. but which release has this? https://issues.apache.org/jira/browse/CAMEL-8482
Adding the code segment:
The class which contains below code segment is annotated with @Component and @RefreshScope
@Value("${prop}")
String prop;
@Override
public void configure() throws Exception {
from("direct:route1").routeId("Child-1")
.setHeader( Exchange.CONTENT_ENCODING, simple("gzip"))
.setBody(simple("RESPONSE - [ { \"id\" : \"bf383eotal length is 16250]]"))
.log(prop + "${body}")
;
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Олекса Марченко (, с. Яловенкове, Харківський район, Харківська область, УРСР — , Харків, Україна) — український поет. Лауреат премії ім. Олександра Олеся. Член Національної спілки письменників України (2002).
Біографія
Марченко Олекса Андрійович народився у селі Яловенкове на Харківщині (нині у межах міста Люботина Харківської області.
По закінченні школи навчався на філологічному факультеті Харківського педінституту ім. Г. С. Сковороди. Закінчив інститут 1960 року.
Спочатку працював учителем в середній школі №124 на Салтівці. З 1966 до 1970 року викладав українську мову та літературу в середній школі №36 міста Харкова. Пізніше перейшов на роботу до редакції газети «Вечірній Харків».
Був одним із провідних співробітників і літературним консультантом. Працював разом з поетами Олександром Черевченком, Аркадієм Філатовим, Робертом Третяковим. Як журналіст, писав свої поетичні нариси про трудові будні краю прозою.
Обраний членом Національної спілки письменників України з 9.12.2002
9 березня 2003 року брав участь у акції «Повстань, Україно!» у Харкові.
У грудні 2003 року став лауреатом творчої премії виконкому Харківської міської ради у галузі літератури.
Творчість
Автор збірок
«Друге народження»,
«Кровообіг»,
«Забрость»,
«Полюддя».
Є автором слів до відомої пісні В'ячеслава Корепанова «Харків моя любов».
Писав вірші українською ї російською мовами.
Поет Казаков Анатолій Сергійович присвятив йому коротенького вірша<ref>Олексе Марченко </ref
Лауреат премії ім. Олександра Олеся.
Вийшли друком:
Полюддя / Марченко Олекса Андрійович. — Харків : Поліком, 2000. — 160 с.
Забрость : поезії / Олекса Марченко. — Харків : Слобожанщина, 2005. — 343 с. ISBN 966-7814-07-6
Примітки
Джерела
Національна спілка письменників України. Письменницький довідник. Марченко Олекса
Посилання
Харківська обласна організація НСПУ
Національна спілка письменників України. Письменницький довідник. Марченко Олекса
Українські пісні
Contemporary Ukrainian Literary Authors List.
Випускники Харківського педагогічного інституту
Письменники Харкова
Українські поети
Уродженці Харківського району
Члени НСПУ
Журналісти Харкова | {
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