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\begin{document}
\title{Finding the Dominant Roots of a Time Delay System without Using the Principal Branch of the Lambert W Function}
\author{Rudy Cepeda-Gomez\\
\textit{a+i Engineering, Berlin, Germany}\\
\texttt{rudy.cepeda-gomez@a--i.com}}
\maketitle
\begin{abstract}
This brief note complements some results regarding a recently developed technique for the stability analysis of linear time-invariant, time delay systems using the matrix Lambert W function. By means of a numeric example, it is shown that there are cases for which the dominant roots of the system can be found without using the principal branch of this multi-valued function, contradicting the main proposition of the methodology.
\end{abstract}
\section{Introduction}
Consider a $n$-th order Linear Time Invariant-Time Delay System (\textsc{lti-tds}), represented by a Delay-Differential Equation (\textsc{dde}) of the form
\begin{equation}
\dot{x}\left(t\right)=Ax\left(t\right)+Bx\left(t-\tau\right),
\label{eq:lti-tds}
\end{equation}
with $x\in\mathbb{R}^n$, $A,B\in\mathbb{R}^{n\times n}$ and $\tau>0$. A framework for analyzing \textsc{ddes} like \eqref{eq:lti-tds} based on the Lambert W function has been developed in the past decade \cite{Asl2003,Yi2009,Yi2010d}, expanding earlier works like \cite{Wright1959}. The main idea of this methodology is to express the solution of a \textsc{dde} as the sum of a series of infinitely many exponential functions. The characteristic roots of the system are found analytically in terms of the Lambert W function. While the problem remains infinite dimensional, a one-to-one correspondence between the characteristic roots of the system and the branches of this multi-valued function is assumed. The stability question is then solved by earmarking the dominant characteristic roots of the system with the branches of the Lambert W function, such that only a few branches have to be considered to determine whether a solution is stable or not.
Although the basic foundation of this methodology, i.e., the assumption that the principal branch of the Lambert W function defines the stability of the system, was proven for first order systems (when $n=1$ in \eqref{eq:lti-tds}) in \cite{Shinozaki2006} and \cite{Asl2003} independently, for higher order systems this is not the case. It was shown in \cite{AUT} that under mild conditions it is possible to find the complete spectrum of a second order system using only two branches of the matrix Lambert W function, namely $k=0$ and $k=-1$.
Nevertheless, the example presented in \cite{AUT} shows that the 22 dominant roots are found using the principal branch. This may prompt the reader to believe that there is still some of the correspondence observed for first order systems. In this technical note, we present an example in which \emph{all roots of the system can be found without using the principal branch of the matrix Lambert W function}. All the roots in this case can be found using the branch corresponding to $k=-1$.
To avoid unnecessary repetition, this note does not describe the steps of the stability analysis methodology based on the matrix Lambert W function, which can be found in references such as \cite{Yi2009,Yi2006a,Yi2006b}, nor the foundation and the steps used to create the counterexamples, which are presented in \cite{AUT}. For a discussion on the definition and properties of the Lambert W function, the reader is referred to \cite{Corless1996}.
\section{Finding the Spectrum Without using The Principal Branch}
For a system described by \eqref{eq:lti-tds} with
\begin{equation}
A=\left[\begin{array}{rr}0&1\\-5&10\end{array}\right]\quad B=\left[\begin{array}{rr}0&0\\-3&-3\end{array}\right],
\end{equation}
and $\tau=1$, the QPmR algorithm \cite{Vhylidal2009} finds the 10 dominant roots observed in Fig.~\ref{fig:rts}. Following the reverse engineering approach described in \cite{AUT}, the two real roots of the system, $\lambda_1=0.8070$ and $\lambda_2=-2.1854$, are combined to create the matrix
\begin{equation}
S=\left[\begin{array}{cc}0&1\\1.7636&-1.3784\end{array}\right],
\end{equation}
which in turns generates
\begin{equation}
W_k(M_k)=\tau\left(S-A\right)=\left[\begin{array}{cc}0&0\\6.7636&-11.3784\end{array}\right].
\label{eq:wkm}
\end{equation}
As discussed in \cite{AUT}, since $W(m_{22})\in\left(-\infty,\,-1\right)$, which is the range of the branch $k=-1$ of the Lambert W function, there is an $M$ matrix such that \eqref{eq:wkm} is satisfied for $k=-1$. That matrix is
\begin{equation}
M_{-1}=\left[\begin{array}{cc}0&0\\0.0774&-0.1302\end{array}\right].
\end{equation}
and from it, a $Q_{-1}$ matrix like
\begin{equation}
Q_{-1}=\left[\begin{array}{rr}2&1\\-2&-1\end{array}\right].
\end{equation}
can be used as a starting value in the \emph{LambertDDE} Matlab toolbox \cite{Yi2012a} to obtain these roots as a solution with $k=-1$. Notice that this pair of roots includes the dominant root of the system, which according to the methodology under study should have been found using $k=0$.
\begin{figure}[tb]
\centering
\includegraphics[scale=1]{roots.eps}
\caption{Dominant roots of the system under study.}
\label{fig:rts}
\end{figure}
If the second dominant pair of roots of the system, i.e. $s=-1.4928\pm j6.6027$, is used to create a matrix like
\begin{equation}
S=\left[\begin{array}{cc}0&1\\-45.8241&-2.9855\end{array}\right],
\end{equation}
we obtain
\begin{equation}
W_k(M_k)=\tau\left(S-A\right)=\left[\begin{array}{cc}0&0\\-40.8241&-12.9855\end{array}\right],
\label{eq:wkm2}
\end{equation}
which also belongs to the range of the $k=-1$ branch.
As we move further to the left, the pairs of complex conjugate roots will always create matrices for which $W\left(m_{22}\right)\in\left(-\infty,\,-1\right)$. From the way in which $W_k(M)$ is created, we have that:
\begin{equation}
W_k\left(m_{22}\right)=\tau\left(2\Re\left(\lambda\right)-a_{22}\right).
\end{equation}
Since $a_{22}>0$ for this system and all the roots further to the left have negative real part with increasing absolute value, $W_k\left(m_{22}\right)$ will be always decreasing, thus remaining within the range of the $k=-1$ branch.
\section{Closing Remarks}
This short note presented a numerical example in which the full spectrum of a second order system can be found using only the $k=-1$ branch of the matrix Lambert w function, provided proper initial conditions. These results complement the remarks made by previous works on the applicability of the methodology under scrutiny.
The code used to create the example is available at \texttt{http://bit.ly/2uN8NZG} or can be requested via email to the author.
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The bridge on the mighty river Missouri. Perhaps it's the wrong season, or the wrong place, but it didn't seem that mighty. I've always been fascinated by bridges, though, so this picture is really my favourite from the whole trip.
Blue skies, bay and mud-brown earth. Coffee and music as we speed by. Heaven in a package. Not much of a picture, but go with it, go with it.
The St. Louis gateway arch, designed by Eero Saarinen and completed in 1965, towers 630 feet above the Mississippi river, commemorating America's westward exploration in the 19th century. Taller than the Washington Monument and more than twice as tall as the Statue of Liberty, even the giants from Mount Rushmore, had they bodies proportionate to the sie of their heads would be able to stroll through the arch.
The romance of the arch, however, lies in the idea of pioneering expeditions to the west, in intrepid explorers in boats and on horseback, crossing the river to go where no man they knew had gone before.
The Arch refused to fit into a single frame, no matter how we tried. Here's how it looks from below.
I had thought the view from the top would be something spectacular. It wasn't, in the normal sense of things. But for the first time, I could see the curvature of the earth. I checked this photograph with a ruler, just to be sure I wasn't imagining it. Yes, you can really see the curve.
Photographs courtesey R.
2 comments:
..and i can even spot your shadow on the water :p
Not my shadow, I'm shaped just a teensy bit differently from a horse-shoe!
| 296,576
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7 November - A dry day, apart for a
bit of drizzle at the moment when I decided to go outside with my camera
tonight. After a relatively mild night (min 8C) today it got up to 12C.
I was looking for something on a shelf
under my computer desk when I pulled out a UV lamp intended for use with
security pens. As it was dark outside I decided to take it down the garden
to see if I could find anything that is fluorescent.
The
only places that I found positive results were around the bee hotels. First
of all, a couple of areas on one of the pieces of timber showed up well.
This first one is a knot, The left-hand
image was taken using flash, and the right-hand image was a long exposure
under UV light.
Towards the end of the piece of timber
there is a wood grain pattern which shows the position where a branch would
have been growing out from the tree.
In both examples, as well as a bright
yellow/green fluorescence, there are also patches of almost magenta
colouring.
This last pair of images shows the effect
of pointing the UV light at the sealed nest holes of the solitary bees (Heriades
truncorum).
Like the grain patterns in the picture
above, the material used to seal the holes fluoresces yellow-green, which
perhaps makes sense as the bees use resin to seal the holes. You can also
see how some of the grains of sand (that were added to the surface of the
resin) are glowing red - fascinating.
At some point I may spend more time
investigating this.
I pointed the UV lamp at many places,
including the leaves of the Birch to see if the mildew would fluoresce - it
did show up a little bit with that lamp, but nothing like the images above.
The Orange Ladybirds certainly didn't glow!
They continue to be active on the tree, and
I counted twelve in the short time I was by the tree. They are usually
solitary, and just after I took this picture, this pair had gone their own
ways.
9 November - Another sunny day,
although some four degrees cooler than yesterday's high of 15C. I see
that water levels in the south-east are still very low, so while these sunny
days are nice, we do need to get some prolonged rain.
While
they are still staying securely attached, the leaves on the Birch are
looking really sorry for themselves now. In addition to the white mildew
with the black 'pin-head' spore bodies, There are now areas underneath many
leaves that are covered with clumps of yellow filaments.
This pair of images were taken at the same
scale, and the lines in the middle represent 0.25mm divisions.
This second image shows more clearly how
the yellow fungus(?) seems to appear out of isolated areoles - these are the
small areas between the network of fine veins that spread out across the
leaf.
While the surrounding areoles still contain
green chlorophyl, the affected ones lose their green colour.
The right-hand image of this pair show how
the upper leaf surface also shows a change when areoles are infected with
the fungus. Although the yellow colour shows through, the fungus remains
below the surface, only breaking out of the leaf on the underside.
Yesterday
I decided to dismantle the scaffold tower for the moment as there are now
very few Ivy flowers. I may well put it back up with a hide on top once the
berries start to ripen in the hope of getting photographs of visiting birds
coming to feed.
The picture shows some of the furthest
developed berries. To the right of the image you can see some of the
'failed' flowers that will drop off the plant soon.
I was out for most of the day so I have
little else to report from the garden. However, first thing this morning we
had a visit from a Song Thrush, which tucked into the fast disappearing crop
of Pyracantha berries. Also, over the last few days the local Sparrowhawk
has started visiting more frequently again. Yesterday I saw it here twice.
11 November - It looks like being a
second dull day in a row. Yesterday was a
mild, dry day after a cold start, and when I wandered down the garden at 11pm it
was still 14C! This morning it is very grey, and it is drizzling at 10.30am.
The balmy conditions last night certainly had an
effect on the frogs, and I counted at least thirty in sight in and around
the two ponds.
In contrast, in the gloomy light of this
morning I can only see one.
Despite what you read about frogs being in
hibernation by now, they are still very active here after dark, both in the pond and
wandering around the garden in search of worms etc.
I've finally got started on next year's
calendar. Good progress yesterday means that the basic layout is nearly complete and over the next few days I
shall start to sort out the pictures. It will be available for downloading
towards the end of December.
13 November - A dull, overcast day
with just a few minutes of sunshine, and it's getting colder. The outside
temperature reached only 9C today.
I've done very little this weekend, and
today's pictures were all taken through our patio window (I must clean the
glass!).
The feeders are in constant use now with
Goldfinches here more or less all the time.
There were at least three Blue Tits back and forth constantly, and a pair of
Great Tits hardly left all day. This one has the less bold stripe down the
front and is possible the female.
It will be encouraging if the pair continue
to visit regularly through the Winter. I must get on with jobs I need to do
in and around the nestbox.
Down under the Hawthorn the Robin and
Dunnock(s) continue to squabble over feeding rights. I was just in time to
get this photograph before the Robin chased it off once again.
The Blackbird male made several visits this
morning and now there are very few berries left on the Pyracantha now.
The female hasn't been to them at all, preferring to feed on the ground.
Another visitor was somewhat less welcome
this morning - a Brown Rat! We haven't seen one in the garden for a very
long time, since I had to deal with one that tried to set up home here
before I started the diaries.
It only stayed for a minute or two, so
hopefully was just passing through.
14
November - For the first time this season the temperature dipped down to
0C first thing this morning and brought a touch of frost to the garden.
It was most obvious on the roof of the
caravan shelter, and the neighbours would have been treated to me up a
ladder in my dressing gown to grab a couple of shots.
In this shot you can see how the shelter
provided by these birch leaves was enough to prevent the frost extending
underneath them.
The day was bright, with hazy sunshine not
lifting the temperature much above 8C, and to complement the frosty start it ended with a fiery, but rather
untidy sunset (with hardly a natural cloud to be seen amongst the vapour
trails in our very busy airspace).
Despite the good weather I did very little in the course of the day, besides
searching for some pictures that I thought I had stored on a CD and then
lost (but found, eventually, safely stored in an external hard drive).
It
gave me plenty of time to watch the garden. It was very noticeable how
slowly bird activity picked up in the morning after the cold start.
I didn't see any Goldfinches until nearly
lunchtime, and although there is a picture of one in yesterday's entry I'm
adding this one because it is one of the few I've managed to get away from
the feeders.
The male Blackbird didn't appear at all,
and I saw his partner only once. I'm pleased to note that here was no sign
of the rat today.
Lastly, a couple of House Sparrow pictures, both were taken at the same
perch. I have reversed the left-hand image so that the male and female (on
the right, with her bold eyebrow) can be compared.
The male is in autumn plumage - its black
bib with become more pronounced during the winter.
15 November - A dull and damp
morning, but that cleared up later, and tonight the sky is starry with a
full moon high above us.
Following my entry about the Thistle seed
feeder on the 1st today I bought a new feeder with eight perches and with
narrow openings for access to the seeds inside. It will be interesting to
watch the amount of seed that falls into the tray from this one. I now have
my fingers crossed that the Goldfinch family doesn't desert us!
Just
one photograph today, or rather, tonight.
The cloudy start to the day meant that the
temperature got up to over 10C in the daytime, and at 8pm it has just dipped
below 8C. In the ponds there are only a few frogs in sight, and this pair
presented the best pose, although I couldn't get really close to them.
No sign of the rat today.
16 November - A dry, bright day,
although high, broken cloud meant that the sun couldn't warm us up, and it
stayed at around 6-7C all day.
I have spent a bit of time in the big
birdbox today, starting to sort things out in the hope that we have birds
move in for next Spring. There is no sign that it has been used for roosting
(but this is normal).
The Sparrowhawk came at least twice during
the day, but didn't try very hard, leaving each time before I could grab my
camera.
However,
I had more luck with a welcome visitor that is becoming a winter regular.
I walked out of our back door and disturbed
this male Great Spotted Woodpecker. Fortunately, when I retreated back into
the house it returned and stayed long enough for me to get the camera and
dash upstairs to our bedroom.
In the right-hand image, the bird is a bit
out of focus, but it does show the banding on the underside of the wings.
Once the wings are fully opened these appear as rows of white patches
against a black backround.
While I was photographing the lichen at the
beginning of the month I noticed, but didn't mention that the moss was
sending up dark-tipped spikes, seen here in front of that strange looking
lichen growth.
Today I took another look at them and, as
the picture shows, the tops of the spikes swell as they develop into the
Sporogonia. These grow from the female moss plants after sexual reproduction
took place back in May. In the Sporogonia, spores are now developing, but it
will take until next April for them to be ready for dispersal.
On a couple of the heads you can see the
cap, or operculum which protects the special, toothed structure (peristome)
which will allow the wind to shake spores out of the sporongium during
dry weather only.
I
continue to check the Birch tree for Orange Ladybirds that are still active.
Today I only saw a couple during the day, but after dark (5pm) I have just
found this small group under a leaf about 10ft up the tree.
I know that ladybirds group together to
hibernate during the winter, but as the leaf could fall at any time, I guess
that this is just an overnight roost. I find it amazing how they end up in a
group like this when the tree still has so many leaves to choose from and
there is quite a small population this year.
I should point out that I also saw a couple
of individuals elsewhere on the tree, so they didn't all come together!
Click on the images
to see larger versions -
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Come
Next thread: Wimborne 5 - 3 Boro by Earlsmead part-timer18/8/2018 19:00Sat Aug 18 19:00:52 2018view thread
Our new League
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Just been on the league website. Sadly only shows half the leagues results. Football web pages doing likewise. Any idea why anyone?
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Re: Our new League
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All the scores are here -
Footballwebpages doesn't seem to be used that much by Southern League clubs. I don't know why that is. I just say this because over the past few years when using footballwebpages to check non league results, results from the Southern League are rarely posted.
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\begin{document}
\maketitle
\begin{abstract}
We define a new logic-induced notion of bisimulation (called $\rho$-bisimulation)
for coalgebraic modal logics given by a logical connection, and investigate its
properties. We show that it is structural in the sense that it is defined only
in terms of the coalgebra structure and the one-step modal semantics and, moreover,
can be characterised by a form of relation lifting.
Furthermore we compare $\rho$-bisimulations to several well-known equivalence notions,
and we prove that the collection of bisimulations between two models often forms a
complete lattice.
The main technical result is a Hennessy-Milner type theorem which states that,
under certain conditions, logical equivalence implies $\rho$-bisimilarity.
In particular, the latter does \emph{not} rely on a duality between functors
$\fun{T}$ (the type of the coalgebras) and $\fun{L}$ (which gives the logic),
nor on properties of the logical connection $\rho$.
\end{abstract}
\section{Introduction}\label{sec:intro}
In this paper, we investigate when logical equivalence for a given modal
language can be captured by a structural semantic equivalence notion,
understood as a form of bisimulation.
Our investigation is carried out in the setting of coalgebraic modal logic
\cite{KupPat11}, where semantic structures are given by coalgebras
for a functor $\fun{T} \colon \cat{C} \to \cat{C}$ \cite{Rut00}.
This allows for a uniform treatment of a wide variety of modal
logics~\cite{KupPat11,Pat03b,Sch08}.
Coalgebras come with general notions of \emph{behavioural equivalence}
and \emph{bisimilarity}, and a logic is said to be \emph{expressive} if
logical equivalence implies behavioural equivalence,
in which case we have a generalisation of the classic Hennessy-Milner
theorem \cite{HenMil85}.
For $\cat{Set}$-coalgebras, i.e., when $\cat{C} = \cat{Set}$,
it has been shown that a coalgebraic modal logic is expressive if
the language has sufficiently large conjunctions and
the set $\Lambda$ of modalities is \emph{separating},
meaning that they separate points in $\fun T X$ \cite{Mos99,Pat04,Sch08}.
In the more abstract setting of coalgebraic modal logic,
where a logic is given by a functor and its semantics by a natural
transformation $\rho$ \cite{BonKur05,Kli07},
a sufficient condition for a logic being expressive is that the so-called
mate of $\rho$ is pointwise monic \cite[Theorem 4.2]{Kli07}.
In this line of research,
modal logics are often viewed as specification languages for coalgebras.
Therefore behavioural equivalence is a given, and the aim is to find expressive logics.
However, sometimes the modal language is of primary interest \cite{BalCin18}
and the relevant modalities need not be separating, see e.g.~\cite{FanWanDit14,BakDitHan17}.
This leads us to consider the following question:
\begin{center}
Given a possibly non-expressive coalgebraic modal logic, can we \\
characterise logical equivalence by a notion of bisimulation?
\end{center}
Such investigations have been carried out earlier in \cite{BakHan17}
where the notion of $\Lambda$-bisimulation was proposed for $\cat{Set}$-coalgebras
and coalgebraic modal logics with a classical propositional base.
Here we generalise and extend the work of \cite{BakHan17} beyond $\cat{Set}$
using the formulation of coalgebraic modal logic via dual adjunctions
\cite{BonKur05,Kli07,KurRos12}.
Examples include coalgebras over ordered and topological spaces and
modal logics on different propositional bases.
After recalling basic definitions of coalgebraic modal logic in Section~\ref{sec:cml-rel},
we define the concept of a \emph{$\rho$-bisimulation} in Section~\ref{sec:rho-bis}.
For $\cat{Set}$-coalgebras, this is a relation $B$ between coalgebras for which the
so-called $B$-coherent pairs \cite{HanKupPac09,BalCin18} give rise to a congruence
between complex algebras.
The definition of $\rho$-bisimulation is structural in the sense that it is
defined in terms of the coalgebra structure and the one-step modal semantics $\rho$.
Moreover, it can often be characterised as a greatest fixpoint via relation lifting.
For coalgebras on finite sets, this means that $\rho$-bisimilarity can be
computed by a partition refinement algorithm.
We also prove results concerning truth-preservation,
composition and lattice structure.
The main technical results are found in Section~\ref{sec:disting}
and concern the distinguishing power of {$\rho$-bisimulations}.
We first compare $\rho$-bisimulations with other coalgebraic equivalence notions.
Subsequently, we prove a Hennessy-Milner style theorem (Theorem~\ref{thm:hm})
in which we give conditions that guarantee that logical equivalence {is a} $\rho$-bisimulation.
We emphasise that the logic is \emph{not} assumed to be expressive and
$\rho$-bisimilarity will generally differ from bisimilarity for $\fun{T}$-coalgebras.
Finally, we define a notion of translation between logics and show that if
the language of $\rho'$ is a propositional extension of the language of $\rho$,
then $\rho$-bisimulations are also $\rho'$-bisimulations (Proposition~\ref{prop:tau-flat}).
By instantiating Proposition~\ref{prop:tau-flat},
we obtain that for labelled transition systems the $\rho$-bisimilarity notions
for Hennessy-Milner logic \cite{HenMil85} and trace logic \cite{Kli07} coincide
and are equal to the standard notion of bisimilarity even without assuming image-finiteness.
These two logics have the same modalities, which are separating,
but trace logic has $\top$ as the only propositional connective.
\paragraph{Earlier version}
This is the extended version of an AIML paper \cite{GroHanKur20}
with the same name.
The current paper includes proofs that were left out in \cite{GroHanKur20}.
Besides, it includes an additional example of a logic for linear weighted
automata that matches precisely the logic from \cite[Section 3.2]{BonEA12}
(Example \ref{exm:trace-vec}),
and a Hennessy-Milner result for it (Example \ref{exm:trans-vec}).
\section{Coalgebraic modal logic}\label{sec:cml-rel}
We review some background on coalgebraic logic, categorical algebra, and Stone duality.
For more details, e.g.~\cite{Rut00,KupPat11,AdaHerStr90,ARV,Joh82}.
We write $\cat{Set}$ for the category of sets and functions.
Coalgebraic modal logic generalises modal logic from Kripke frames to
coalgebras for a functor $\fun{T}$.
\medskip\noindent\textbf{Coalgebras} can be understood as generalised,
state-based systems defined parametrically in the system type $\fun{T}$.
Formally, we require $\fun{T}$ to be an endofunctor on a category $\cat{C}$.
A $\fun{T}$-\emph{coalgebra} is then a pair $(X, \gamma)$
such that $\gamma : X \to \fun{T}X$ is a morphism in $\cat{C}$.
The object $X$ is the state space, and the arrow $\gamma$
is the coalgebra structure map.
A \emph{$\fun{T}$-coalgebra morphism} from $(X, \gamma)$ to
$(X', \gamma')$ is a $\cat{C}$-morphism $f : X \to X'$ satisfying
$\gamma' \circ f = \fun{T}f \circ \gamma$.
Together, $\fun{T}$-coalgebras and $\fun{T}$-coalgebra morphisms form a category
which we write as $\cat{Coalg}(\fun{T})$.
\medskip\noindent An \textbf{algebra} for a functor is the dual notion of a coalgebra.
Given an endofunctor $\fun{L} \colon \cat{A} \to \cat{A}$,
an \emph{$\fun{L}$-algebra} is a pair $(A, \alpha)$ such that $\alpha : \fun{L}A \to A$
is a morphism in $\cat{A}$. An \emph{$\fun{L}$-algebra morphism} from $(A,\alpha)$ to
$(A',\alpha')$ is an $\cat{A}$-morphism $h: A \to A'$ such that
$h \circ \alpha = \alpha' \circ \fun{L}h$. We write $\cat{Alg}(\fun{L})$
for the category of $\fun{L}$-algebras and $\fun{L}$-algebra morphisms.
\begin{example}\label{ex:Kripke-coalg}
A Kripke frame $(X, R\subseteq X \times X)$ is a coalgebra for the
covariant powerset functor $\Pow\colon \cat{Set}\to\cat{Set}$ which maps
a set to its set of subsets, and a function $f\colon X \to Y$ to the
direct image map $f[-] \colon \Pow X \to \Pow Y$,
by defining $\gamma \colon X \to \Pow X$ as $\gamma(x) = R[x] = \{ y \in X \mid xRy \}$.
Similarly, a Kripke model $(X, R, V)$, where $V$ is a valuation of a set $P_0$
of atomic propositions, is a coalgebra for the $\cat{Set}$-functor
$\Pow(-) \times \Pow(P_0)$ (which is constant in its second component)
by taking $\gamma(x) = (R[x],V'(x))$, with $V'(x) = \{ p \in P_0 \mid x \in V(p)\}$.
It can be verified that the ensuing notion of coalgebra morphism
coincides with the usual notion of bounded morphism for Kripke frames and Kripke models,
respectively.
\end{example}
\begin{example}\label{ex:LTS-coalg}
Labelled transition systems (LTSs) are coalgebras for
the $\cat{Set}$-functor $\fun{T} = \Pow(-)^A$
where $\Pow$ is the covariant powerset functor and $A$ is the set of labels.
A coalgebra $\gamma \colon X \to \Pow(X)^A$
specifies for each state $x \in X$ and label $a \in A$,
the set $\gamma(x)(a)$ of $a$-successors of $x$.
In other words, an LTS is an $A$-indexed multi-relational Kripke frame.
One readily verifies that coalgebra morphisms are $A$-indexed bounded morphisms.
\end{example}
\paragraph{Logical connections}
To investigate logics for $\fun{T}$-coalgebras in this generality,
we use the Stone duality approach to modal logic \cite{Gol76,Abr91},
but rather than a full duality, here one requires only a dual adjunction
$
\begin{tikzcd}[sep=1.5em, cramped]
\fun{P} : \cat{C}
\arrow[r, shift left=1.7pt]
& \cat{A} : \fun{S}
\arrow[l, shift left=1.7pt]
\end{tikzcd}
$
(sometimes called a \emph{logical connection})
between a category $\cat{C}$ of state spaces
and a category $\cat{A}$ of algebras that encode a propositional base logic.
We emphasise that the functors $\fun{P}$ and $\fun{S}$ are contravariant.
The classic example is then the instance
$
\begin{tikzcd}[sep=1.5em, cramped]
\Pba : \cat{Set}
\arrow[r, shift left=1.7pt]
& \cat{BA} : \Uf
\arrow[l, shift left=1.7pt]
\end{tikzcd}
$
where $\Pba$ maps a set to its Boolean algebra of predicates (i.e., subsets),
and $\Uf$ maps a Boolean algebra to its set of ultrafilters.
We denote the units of a dual adjunction
$
\begin{tikzcd}[sep=1.5em, cramped]
\fun{P} : \cat{C}
\arrow[r, shift left=1.7pt]
& \cat{A} : \fun{S}
\arrow[l, shift left=1.7pt]
\end{tikzcd}
$
by $\unitc\colon \Id_{\cat{C}} \to \fun{SP}$ and
$\unita \colon \Id_{\cat{A}} \to \fun{PS}$,
and the bijection of Hom-sets $\cat{C}(C,\fun{S}A) \cong \cat{A}(A,\fun{P}C)$
in both directions by $\trans{(-)}$.
Recall that for $f \colon C \to \fun{S}A$, the adjoint transpose of $f$
is $\trans{f} = \fun{P}f \circ \unita_A$, and for $g\colon A \to \fun{P}C$,
the adjoint is $\trans{g} = \fun{S}g \circ \unitc_C$.
\paragraph{Coalgebraic Modal Logic}
Given a dual adjunction
$
\begin{tikzcd}[sep=1.5em, cramped]
\fun{P} : \cat{C}
\arrow[r, shift left=1.7pt]
& \cat{A} : \fun{S}
\arrow[l, shift left=1.7pt]
\end{tikzcd}
$
and an endofunctor $\fun{T}$ on $\cat{C}$,
a \emph{modal logic for $\fun{T}$-coalgebras}
is a pair $(\fun{L}, \rho)$ consisting of an endofunctor
$\fun{L}\colon \cat{A}\to \cat{A}$ (defining modalities)
and a natural transformation $\rho : \fun{LP} \to \fun{PT}$,
(defining the \emph{one-step modal semantics}).
This data gives rise to a functor
$\cat{Coalg}(\fun{T}) \to \cat{Alg}(\fun{L})$ which sends a coalgebra
$(X, \gamma)$ to its \emph{complex algebra} $(\fun{P}X, \gamma^*)$,
where $\gamma^* = \fun{P}\gamma \circ \rho_X$.
Assuming that $\cat{Alg}(\fun{L})$ has an initial algebra
$\alpha : \fun{L}\Phi \to \Phi$, which generalises the Lindenbaum-Tarski algebra,
the semantics of (equivalence classes of) formulae is obtained
as the unique $\cat{Alg}(\fun{L})$-morphism
$\sem{-}_\gamma \colon (\Phi,\alpha) \to (\fun{P}X, \gamma^*)$.
Viewing the semantics as an $\cat{A}$-morphism $\sem{-}_\gamma \colon \Phi \to \fun{P}X$,
its adjoint
$\th_{\gamma} = \trans{\sem{-}_{\gamma}} \colon X \to \fun{S}\Phi$,
is called the \emph{theory map}, since in the classic case
it maps a state in $X$ to the ultrafilter of $\fun{L}$-formulae it satisfies.
By their definitions, the semantics and the theory map make the following diagrams commute:
\[
\begin{tikzcd}[row sep=1.4em, cramped]
\fun{L}\Phi
\arrow[rr, "\alpha"]
\arrow[d, "\fun{L}\llb \cdot \rrb_{\gamma}" left]
&
& \Phi
\arrow[d, "\llb \cdot \rrb_{\gamma}"]
& X \arrow[rr, "\th_{\gamma}"]
\arrow[d, "\gamma"]
&
& \fun{S}\Phi
\arrow[d, "\fun{S}\alpha"] \\
\fun{LP}X
\arrow[r, "\rho_X"]
& \fun{PT}X
\arrow[r, "\fun{P}\gamma"]
& \fun{P}X
& \fun{T}X
\arrow[r, "\fun{T}\th_{\gamma}"]
& \fun{TS}\Phi
\arrow[r, "\rho_{\Phi}^{\flat}"]
& \fun{SL}\Phi
\end{tikzcd}
\]
Here $\mate{\rho}\colon \fun{TS} \to \fun{SL}$ is the so-called mate of $\rho$.
This is the natural transformation obtained (component-wise) as the adjoint of $\rho_{\fun{S}} \circ \fun{L} \unita$.
\begin{example}\label{ex:LTS-trace-logic}
Consider the self-dual adjunction
$
\begin{tikzcd}[sep=1.5em, cramped]
\cPow : \cat{Set}
\arrow[r, shift left=1.7pt]
& \cat{Set} : \cPow
\arrow[l, shift left=1.7pt]
\end{tikzcd}
$
given in both directions by the contravariant powerset functor $\cPow$,
which maps a set to its powerset $2^X$, and a function $f\colon X \to Y$
to its inverse image map $f^{-1}\colon 2^Y \to 2^X$.
In this case, the adjoints are given by transposing.
That is, for $f \colon X \to 2^Y$, $\trans{f}\colon Y \to 2^X$ is defined by
$\trans{f}(y)(x) = f(x)(y)$.
Considering LTSs as $\Pow(-)^A$-coalgebras over $\cat{Set}$ (cf.~Example~\ref{ex:LTS-coalg}),
we obtain \emph{trace logic for LTSs} \cite[Example~3.2]{Kli07}
by taking $\Ltr \colon \cat{Set}\to\cat{Set}$ to be the functor
$\Ltr = 1+A\times(-)$ (where $1 = \{*\}$ is a set with one element).
This encodes a modal signature with a constant modality $\top$
and a unary modality for each $a \in A$.
Since $\cat{A} = \cat{Set}$, trace logic has no other connectives.
The initial $\Ltr$-algebra consists of finite sequences over $A$
with the empty word as constant, and prefixing with elements from $A$
as the unary operations. That is, $\Ltr$-formulae are of the form
$\diam{a_1} \cdots \diam{a_k}\top$, where $k \geq 0$.
We obtain the usual semantics of $\top$ and $A$-labelled diamonds
by defining the modal semantics $\rhotr\colon 1+A\times\cPow(-) \to \cPow(\Pow(-)^A)$ as
$\rhotr_X(*) = \Pow(X)^A$ and
$\rhotr_X(a,U) = \{ t \in \Pow(X)^A \mid t(a) \cap U \neq \emptyset\}$.
Hence for an LTS $(X,\gamma)$,
$\sem{\diam{a_1} \cdots \diam{a_k}\top}_\gamma$ is the subset of $X$ consisting of
states $x$ that can execute the trace $a_1 \cdots a_k$.
\end{example}
\begin{example}\label{ex:LTS-HM-logic}
Again consider LTSs as $\Pow(-)^A$-coalgebras over $\cat{Set}$
(cf.~Example~\ref{ex:LTS-coalg}), but now take the classic dual adjunction
$
\begin{tikzcd}[sep=1.5em, cramped]
\Pba : \cat{Set}
\arrow[r, shift left=1.7pt]
& \cat{BA} : \Uf
\arrow[l, shift left=1.7pt]
\end{tikzcd}
$.
Hennessy-Milner logic \cite{HenMil85} (or equivalently, normal multi-modal logic)
is here defined as classical propositional logic extended with
join-preserving diamonds.
This is achieved by defining $\Lhm\colon \cat{BA} \to\cat{BA}$ as follows:
For a Boolean algebra $B$, $\Lhm B$ is the free Boolean algebra generated
by the set $\{ \diam{a}b \mid b \in B, a \in A \}$ modulo the congruence
generated by the usual diamond equations, i.e.,
$$
\diam{a}\bot = \bot
\quad\text{and}\quad
\diam{a}(\phi_1 \lor \phi_2) = \diam{a}\phi_1 \lor \diam{a}\phi_2
$$
for all $a \in A$.
The modal semantics $\rhohm \colon \Lhm\Pba \to \Pba(\Pow(-)^A)$
is essentially the Boolean extension of $\rhotr$. In particular,
$\rhohm_X(\diam{a}U) = \{ t \in \Pow(X)^A \mid t(a) \cap U \neq \emptyset\}$.
\end{example}
The above description of Hennessy-Milner logic is a special case of a
more general approach described in the next example.
\begin{example}\label{exm:pl}
If $\cat{A}$ in the dual adjunction is a variety of algebras,
we can define a logic $(\fun{L}, \rho)$ for $\fun{T} : \cat{C} \to \cat{C}$ by
\emph{predicate liftings and axioms} as in \cite[Definition~4.2]{KupKurPat04}
and \cite[Theorems 4.7 and 8.8]{KurRos12}.
An \emph{$n$-ary predicate lifting} is a natural transformation
$$
\lambda : \fun{UP}^n \to \fun{UPT},
$$
where $\fun{P}^nX$ is the $n$-fold product of $\fun{P}X$ in $\cat{A}$ and
$\fun{U} : \cat{A} \to \cat{Set}$ is the forgetful functor.
Together with a suitable notion of \emph{axioms},
a collection $\Lambda$ of such predicate liftings yields a functor
$
\fun{L} : \cat{A} \to \cat{A}
$
sending $A \in \cat{A}$ to the free algebra generated by
$\{ \und{\lambda}(a_1, \ldots, a_n) \mid \lambda \in \Lambda, a_i \in A \}$
modulo (instantiations of) the axioms.
Define $\rho : \fun{LP} \to \fun{PT}$ on generators by
$\rho_X(\und{\lambda}(a_1, \ldots, a_n)) = \lambda_X(a_1, \ldots, a_n) \in \fun{PT}X$.
If $\rho$ is well-defined then it is natural transformation and
$(\fun{L}, \rho)$ is a logic for $\cat{Coalg}(\fun{T})$.
All logics in e.g. \cite{BakHan17,BezGroVen19-report,KupPat11} are instances hereof.
\end{example}
Next, we interpret positive modal logic \cite{Dun95,CelJan99},
whose coalgebraic semantics over posets can be found in \cite[Example 2.4]{KapKurVel14},
in topological spaces:
\begin{example}\label{exm:pml-top}
Consider the dual adjunction
$
\begin{tikzcd}[cramped, sep=1.5em]
\fun{\Omega} : \cat{Top}
\arrow[r, shift left=1.7pt]
& \cat{DL} : \fun{pf}
\arrow[l, shift left=1.7pt]
\end{tikzcd}
$,
where $\fun{\Omega}$ takes open subsets of a topological space, viewed as a
distributive lattice, and $\fun{pf}$ takes prime filters of a distributive
lattice topologised by the subbase $\{ \tilde{a} \mid a \in A \}$,
where $\tilde{a} = \{ p \in \fun{pf}A \mid a \in p \}$.
The Vietoris functor $\fun{V} : \cat{Top} \to \cat{Top}$ takes
$X \in \cat{Top}$ to its collection of compact subsets topologised by the subbase consisting of
$\dbox a = \{ c \in \fun{V}X \mid c \subseteq a \}$ and
$\ddiamond a = \{ c \in \fun{V}X \mid c \cap a \neq \emptyset \}$,
where $a$ ranges over the opens of $X$.
For a continuous map $f : X \to X'$ the map $\fun{V}f$ takes
direct images.
Positive modal logic is given by the functor $\fun{N} : \cat{DL} \to \cat{DL}$
that sends a distributive lattice $A$ to the free distributive lattice
generated by the set $\{ \Box a, \Diamond a \mid a \in A \}$ modulo the axioms
\begin{align*}
\Box\top &= \top
& \Diamond\bot &= \bot \\
\Box a \wedge \Box b &= \Box(a \wedge b)
& \Diamond a \vee \Diamond b &= \Diamond(a \vee b) \\
\Diamond a \wedge \Box b &\leq \Diamond(a \wedge b)
& \Box(a \vee b) &\leq \Box a \vee \Diamond b
\end{align*}
The interpretation of this logic in $\fun{V}$-coalgebras is given by
the natural transformation $\rho : \fun{N\Omega} \to \fun{\Omega V}$,
defined on generators by $\Box a \mapsto \dbox a$ and $\Diamond a \mapsto \ddiamond a$.
\end{example}
We now recall \emph{linear weighted automata}, see e.g.~\cite[Section 3.2]{BonEA12}.
This is particularly interesting because it is an example of a
\emph{many-valued} logic, with truth values in some field $\Bbbk$.
\begin{example}\label{exm:trace-vec}
Let $\Bbbk$ be a field and
let $\cat{Vec}_{\Bbbk}$ be the category of vector spaces over $\Bbbk$.
For a set $A$ of labels, define the endofunctor $\fun{W}$ on $\cat{Vec}_{\Bbbk}$
by $\fun{W} = \Bbbk \times (-)^A$, where $(-)^A$ is the collection of
maps $A \to (-)$ with a pointwise vector space structure.
Then \emph{linear weighted automata} are $\fun{W}$-coalgebras.
We wish to interpret \emph{linear trace logic} in such coalgebras,
that is, formulae in the language given by the grammar
$$
\phi ::= p \mid \langle a \rangle \phi
$$
In order to do this in the abstract coalgebraic framework,
we use the dual adjunction between $\cat{Vec}_{\Bbbk}$ and $\cat{Set}$.
In one direction this is given by the hom-functor
$(-)^{\circ} = \Hom(-, \Bbbk) : \cat{Vec}_{\Bbbk} \to \cat{Set}$.
Conversely, for a set $X$ define $X^{\wedge}$ to be the collection
$\Hom(X, \Bbbk)$ with pointwise vector space structure.
It is easy to see that this yields a functor $\cat{Set} \to \cat{Vec}_{\Bbbk}$.
The interpretation of $p$ is given by the nullary predicate lifting
$\lambda^p \in (\fun{W}-)^{\circ}$ given by
$\lambda^p_X : \fun{W}X \to \Bbbk : (r, t) \mapsto r$.
Then $\llb p \rrb = \lambda^p_X \circ \gamma : X \to \Bbbk$.
The interpretation of the diamonds is given by the unary predicate lifting
$\lambda^{\langle a \rangle} : \fun{U}(-)^{\circ} \to \fun{U}(\fun{W}-)^{\circ}$
defined by
$$
\lambda^{\langle a \rangle}_X(m) : \fun{W}X \to \Bbbk : (r, t) \mapsto m(t(a)).
$$
(Note that in this case $\fun{U}$ is the identity functor on $\cat{Set}$.)
Concretely, this means that if $\llb p \rrb_{\gamma}(y) = r \in \Bbbk$
and there is an $a$-transition $x \overset{a}{\too} y$, then
$\llb \langle a \rangle p \rrb(x) = r$.
\end{example}
Since for $V \in \cat{Vec}_{\Bbbk}$ the set $\Hom(V, \Bbbk)$
forms a vector space, rather than just a set,
we can also interpret vector space operations in $\fun{W}$-coalgebras.
We make this modification in the following example.
\begin{example} \label{exm:mod-vec}
Let $\Bbbk$ be a field and
$
\begin{tikzcd}[cramped, sep=1.5em]
\cat{Vec}_{\Bbbk}
\arrow[r, shift left=1.7pt]
& \cat{Vec}_{\Bbbk}
\arrow[l, shift left=1.7pt]
\end{tikzcd}
$
the dual adjunction between vector spaces over $\Bbbk$
given in both directions by taking dual vector space via the contravariant functor
$(-)^{\vee} = \Hom(-, \Bbbk) : \cat{Vec}_{\Bbbk} \to \cat{Vec}_{\Bbbk}$.
(Note that the functors $(-)^{\circ}$ and $(-)^{\vee}$ are related via
$(-)^{\circ} = \fun{U}_{\cat{Vec}} \circ (-)^{\vee}$,
where $\fun{U}_{\cat{Vec}} : \cat{Vec} \to \cat{Set}$ is the forgetful functor.)
We extend linear trace logic with vector space operations,
and work with the language given by the grammar
$$
\phi ::= 0 \mid p \mid r \cdot \phi \mid \phi + \phi \mid \langle a \rangle\phi,
$$
where $a \in A$, $r \in \Bbbk$, and $p$ is a single proposition letter
(the termination predicate).
We refer to this \emph{linear Hennessy-Milner logic}.
The interpretation of a formula $\phi$ in this (many-valued) setting
is a linear map $\llb \phi \rrb : X \to \Bbbk$.
The connectives $0$, $+$ and $r$ are interpreted via the corresponding operations
in vector spaces, and for $p$ and $\langle a \rangle$ we use
the predicate liftings from Example \ref{exm:trace-vec}.
Together with the axioms
$\langle a \rangle(\phi + \psi) = \langle a \rangle \phi + \langle a \rangle \psi$
and $r \cdot \langle a \rangle \phi = \langle a \rangle (r \cdot \phi)$
this gives rise to an endofunctor $\fun{L} : \cat{Vec}_{\Bbbk} \to \cat{Vec}_{\Bbbk}$,
and a logic $(\fun{L}, \rho)$ for linear weighted automata.
One can show that logical equivalence coincides with language semantics
if the state-space is finite-dimensional.
\end{example}
\paragraph{Relations as jointly mono spans}
We are interested in giving certain relations a special status.
In $\cat{Set}$, a binary relation $B \subseteq X \times X$ corresponds
to an injective map $B \injr X \times X$.
This generalises to an arbitrary category (possibly lacking products)
via the notion of a \emph{jointly mono span}:
A span
$
\begin{tikzcd}[cramped, sep=1.5em]
X_1
& B \arrow[l, swap, "\pi_1"] \arrow[r, "\pi_2"]
& X_2
\end{tikzcd}
$
in a category $\cat{C}$ is called \emph{jointly mono} if for all
$\cat{C}$-arrows $h, h'$ with codomain $B$ it satisfies:
$
\text{if } \pi_1\circ h = \pi_1\circ h' \text{ and } \pi_2\circ h = \pi_2\circ h' \text{ then } h = h'.
$
We sometimes write the above span as $(B,\pi_1,\pi_2)$, leaving codomains implicit.
If $\cat{C}$ has products, then $(B, \pi_1, \pi_2)$ is a jointly mono span if and
only if the pairing $\langle \pi_1, \pi_2 \rangle : B \to X_1 \times X_2$ is
monic.
The collection of jointly mono spans between two objects $X_1, X_2 \in \cat{C}$
can be ordered as follows: $(B, \pi_1, \pi_2) \leq (B', \pi_1', \pi_2')$
if there exists a (necessarily monic) map $k : B \to B'$ such that $\pi_i = \pi_i' \circ k$.
If
$(B, \pi_1, \pi_2) \leq (B', \pi_1', \pi_2')$ \emph{and}
$(B', \pi_1', \pi_2') \leq (B, \pi_1, \pi_2)$, then the two spans must be
isomorphic. We write $\cat{Rel}(X_1, X_2)$ for the poset of jointly mono spans
between $X_1$ and $X_2$ up to isomorphism.
\paragraph{Image factorisations and regular epis}
We will also need a generalisation of image factorisation.
A category $\cat{C}$ is said to have \emph{$(\ms{E}, \ms{M})$-factorisations}
for some classes $\ms{E}$ and $\ms{M}$ of $\cat{C}$-morphisms,
if every morphism $f \in \cat{C}$ factorises as $f = m \circ e$ with
$e \in \ms{E}$ and $m \in \ms{M}$. We say that $\cat{C}$ has an
\emph{$(\ms{E}, \ms{M})$-factorisation system}
\cite[Definition~14.1]{AdaHerStr90} if moreover both $\ms{E}$ and $\ms{M}$
are closed under composition, and whenever
$g \circ e = m \circ f$, with $e \in \ms{E}$ and $m \in \ms{M}$,
there exists a unique diagonal fill-in $d$ such that $f = d\circ e$ and $g = m\circ d$.
In a diagram:
$$
\begin{tikzcd}
{} \arrow[r, two heads, "e"]
\arrow[d, "f" swap]
& [.5em] {}
\arrow[d, "g"]
\arrow[dl, dashed, "d", shorten <=4pt, shorten >=4pt] \\ [.5em]
{} \arrow[r, >->, "m" swap]
& {}
\end{tikzcd}
$$
An epi $e$ is \emph{regular} if it is a coequalizer.
In a variety, the regular epis are precisely the surjective morphisms.
The categories $\cat{Set}, \cat{Pos}, \cat{Top}, \cat{Vec}, \cat{SL}, \cat{Stone}$
all have a $(\ms{R}eg\ms{E}pi, \ms{M}ono)$-factorisation system.
\section{Logic-induced bisimulations}
\label{sec:rho-bis}
We are now ready to define our logic-induced notion of bisimulation.
Throughout this section, we fix a dual adjunction
$
\begin{tikzcd}[sep=1.5em, cramped]
\fun{P} : \cat{C}
\arrow[r, shift left=1.7pt]
& \cat{A} : \fun{S}
\arrow[l, shift left=1.7pt]
\end{tikzcd}
$,
an endofunctor $\fun{T}$ on $\cat{C}$, and a logic $(\fun{L}, \rho)$
for $T$-coalgebras.
Moreover, we assume that $\cat{C}$ has pull\-backs and, in addition,
that $\cat{A}$ has pullbacks or $\cat{C}$ has pushouts.
Both conditions hold in all examples given in Section \ref{sec:cml-rel}.
In particular, if $\cat{A}$ is variety of algebras then pullbacks exist and
are computed as in $\cat{Set}$.
\subsection{Definition and first examples}\label{subsec:def}
The basic ingredient for the definition of $\rho$-bisimulation is
the notion of a \emph{dual span}: A jointly mono span
$
\begin{tikzcd}[cramped, sep=1.5em]
X_1
& B \arrow[l, swap, "\pi_1"] \arrow[r, "\pi_2"]
& X_2
\end{tikzcd}
$
in $\cat{C}$
is mapped by $\fun{P}$ to a cospan
$
\begin{tikzcd}[cramped, sep=1.5em]
\fun{P}X_1
& \fun{P}B
\arrow[l, latex-, swap, "\fun{P}\pi_1"]
\arrow[r, latex-, "\fun{P}\pi_2"]
& \fun{P}X_2
\end{tikzcd}
$
in $\cat{A}$.
Taking its pullback we obtain a jointly mono span in $\cat{A}$,
which we denote by $(\dpo{B}, \dpo{\pi}_1, \dpo{\pi}_2)$ and refer to as
the \emph{dual span} of $(B, \pi_1, \pi_2)$. In a diagram:
\begin{equation*}
\begin{tikzcd}[row sep=0em, column sep=1.5em]
& \dpo{B}
\arrow[dl, bend right=5, "\ov{\pi}_1" {swap,pos=.4}]
\arrow[dr, bend left=5, "\ov{\pi}_2" {pos=.4}]
& \\
\fun{P}X_1
\arrow[dr, bend right=5, "\fun{P}\pi_1" swap]
&
& \fun{P}X_2
\arrow[dl, bend left=5, "\fun{P}\pi_2"] \\ [-.3em]
& \fun{P}B
&
\end{tikzcd}
\end{equation*}
If $\cat{C}$ has pushouts, dual spans exist because dual adjoints send
pushouts to pullbacks.
In the classic case where $\fun{P} = \Pba\colon \cat{Set} \to\cat{BA}$
maps a set to its Boolean algebra of subsets,
the dual span $(\dpo{B}, \ov{\pi}_1, \ov{\pi}_2)$ consists of
\emph{$B$-coherent pairs (of subsets of $X_1$ and $X_2$)},
that is, pairs $(a_1, a_2) \in \fun{P}X_1 \times \fun{P}X_2$ of subsets
satisfying $B[a_1] \subseteq a_2$ and $B^{-1}[a_2] \subseteq a_1$.
This notion of $B$-coherent pairs has been used in the definitions of
$\Lambda$-bisimulation~\cite{BakHan17},
neighbourhood bisimulation \cite{HanKupPac09},
and conditional bisimulation \cite{BalCin18}.
We proceed to the definition of a $\rho$-bisimulation.
\begin{definition}\label{def:bisim}
Let $\gamma_1 \colon X_1 \to \fun{T}X_1$ and $\gamma_2 \colon X_2 \to \fun{T}X_2$ be $\fun{T}$-coalgebras.
A jointly mono span
$
\begin{tikzcd}[cramped, sep=1.5em]
X_1
& B \arrow[l, swap, "\pi_1"] \arrow[r, "\pi_2"]
& X_2
\end{tikzcd}
$
is a \emph{$\rho$-bisimulation between $\gamma_1$ and $\gamma_2$} if
\begin{equation}\label{eq:rho-bis}
\fun{P}\pi_1 \circ \gamma_1^* \circ \fun{L}\ov{\pi}_1
= \fun{P}\pi_2 \circ \gamma_2^* \circ \fun{L}\ov{\pi}_2.
\end{equation}
\end{definition}
Definition~\ref{def:bisim}
is structural in the sense that it is defined in terms of the coalgebra structure and the one-step modal semantics $\rho$ (via the complex algebras $\gamma_i^*$). In particular, it does not refer to the collection of all formulae nor to the initial $\fun{L}$-algebra.
Equation~\eqref{eq:rho-bis} provides a coherence condition that can be checked in concrete settings. We provide examples below.
First, we give a more conceptual characterisation in terms of dual spans.
\begin{proposition}\label{prop:pb-alg }
A jointly mono span $(B,\pi_1,\pi_2)$
is a $\rho$-bisimulation between $(X_1, \gamma_1)$ and $(X_2, \gamma_2)$
if and only if its dual span $(\dpo{B},\ov{\pi}_1,\ov{\pi}_2)$
is a congruence between $\gamma_1^*$ and $\gamma_2^*$.
\end{proposition}
\begin{proof}
Suppose $(B, \pi_1, \pi_2)$ is a $\rho$-bisimulation,
i.e., equation \eqref{eq:rho-bis} holds true. Then the
outer shell of the diagram in \eqref{eq:def-bisim} commutes and
the universal property of the pullback $\dpo{B}$ yields a morphism
$\beta : \fun{L}\dpo{B} \to \dpo{B}$ such that all squares in
\eqref{eq:def-bisim} commute.
\begin{equation}\label{eq:def-bisim}
\begin{tikzcd}[row sep=0em, column sep=1.5em]
& \fun{L}\dpo{B}
\arrow[dl, bend right=5, "\fun{L}\ov{\pi}_1" swap]
\arrow[dr, bend left=5, "\fun{L}\ov{\pi}_2"]
\arrow[dd, dashed, "\beta"]
& \\
\fun{LP}X_1
\arrow[dd, "\gamma_1^*" left]
&
& \fun{LP}X_2
\arrow[dd, "\gamma_2^*" right] \\
& \dpo{B}
\arrow[dl, bend right=5, "\ov{\pi}_1" {swap,pos=.4}]
\arrow[dr, bend left=5, "\ov{\pi}_2" {pos=.4}]
& \\
\fun{P}X_1
\arrow[dr, bend right=5, "\fun{P}\pi_1" swap]
&
& \fun{P}X_2
\arrow[dl, bend left=5, "\fun{P}\pi_2"] \\ [-.3em]
& \fun{P}B
&
\end{tikzcd}
\end{equation}
Conversely, the existence of such a $\beta$ making the inner squares
commute implies commutativity of the outer shell of the diagram.
\end{proof}
We instantiate the definition
for some of the examples of Section~\ref{sec:cml-rel}.
\begin{example}\label{exm:rho-bis-lambda}
Recall the setting of Example~\ref{exm:pl} where $\cat{A}$ is a variety and
$(\fun{L}, \rho)$ is given by predicate liftings and axioms,
and let $(X_1, \gamma_1)$ and $(X_2, \gamma_2)$ be two $\fun{T}$-coalgebras.
If $\cat{C}$ is concrete, then a jointly mono span $\Bspan$
is a $\rho$-bisimulation between $(X_1,\gamma_1)$ and $(X_2,\gamma_2)$
if for all $(x_1, x_2) \in B$, $\lambda \in \Lambda$
and all $B$-coherent $(a_1, a_2) \in \fun{P}X_1 \times \fun{P}X_2$ we have:
\[
\gamma_1(x_1) \in \lambda_{X_1}(a_1)
\quad\text{iff}\quad
\gamma_2(x_2) \in \lambda_{X_2}(a_2).
\]
The notion of a $\rho$-bisimulation thus generalises that of a
$\Lambda$-bisimulation from \cite{BakHan17,BezGroVen19-report},
where $\Lambda$ denotes a collection of (open) predicate liftings.
Examples~\ref{exm:pml-top-bis}, \ref{exm:mod-vec-bis} and \ref{exm:bis-join-logic}
below are instances hereof.
\end{example}
\begin{example}\label{exm:pml-top-bis}
In the setting of positive modal logic from Example~\ref{exm:pml-top},
a $\rho$-bisimulation between $(X_1, \gamma_1)$
and $(X_2, \gamma_2)$ is a subspace $B \subseteq X_1 \times X_2$
with projections $\pi_i : B \to X_i$ satisfying for all $(x_1, x_2) \in B$
and all $B$-coherent pairs of opens $(a_1, a_2) \in \fun{\Omega}X_1 \times \fun{\Omega}X_2$:
\begin{center}
$\gamma_1(x_1) \subseteq a_1$ iff $\gamma_2(x_2) \subseteq a_2$
\quad and \quad
$\gamma_1(x_1) \cap a_1 \neq \emptyset$ iff $\gamma_2(x_2) \cap a_2 \neq \emptyset$.
\end{center}
\end{example}
\begin{example}\label{exm:mod-vec-bis}
In the setting of linear Hennessy-Milner logic from Example~\ref{exm:mod-vec},
a jointly mono span between the state-spaces of $\fun{W}$-coalgebras
$(X_1, \gamma_1)$ and $(X_2, \gamma_2)$ is a linear subspace of $X_1 \times X_2$.
The dual span of $\Bspan$ is the linear subspace of $X_1^{\vee} \times X_2^{\vee}$
consisting of those pairs of $\Bbbk$-valued, linear predicates
$(h_1, h_2) \in X_1^{\vee} \times X_2^{\vee}$
such that $(x_1, x_2) \in B$ implies $h_1(x_1) = h_2(x_2)$.
Unravelling the definitions shows that $(B, \pi_1, \pi_2)$ is
a $\rho$-bisimulation between $(X_1, \gamma_1)$ and $(X_2, \gamma_2)$,
if for all $(x_1, x_2) \in B$, we have
$\llb p \rrb_{\gamma}(x_1) = \llb p \rrb_{\gamma}(x_2)$, and:
\begin{center}
if $x_1 \overset{a}{\too} y_1$ and $x_2 \overset{a}{\too} y_2$,
then $h_1(y_1) = h_2(y_2)$ for all $(h_1, h_2) \in \dpo{B}$.
\end{center}
\end{example}
\begin{example}\label{exm:id-rho-bis}
More abstractly, suppose given any logical connection
$
\begin{tikzcd}[sep=1.5em, cramped]
\fun{P} : \cat{C}
\arrow[r, shift left=1.7pt]
& \cat{A} : \fun{S}
\arrow[l, shift left=1.7pt]
\end{tikzcd}
$,
functor $\fun{T} : \cat{C} \to \cat{C}$ and logic $(\fun{L}, \rho)$ for
$\fun{T}$-coalgebras.
Then for every $\fun{T}$-coalgebra $(X, \gamma)$ the
jointly mono span $(X, \id_X, \id_X)$ is a $\rho$-bisimulation on $(X, \gamma)$.
\end{example}
We complete this subsection by showing that the notion of a $\rho$-bisimulation
is adequate, that is, $\rho$-bisimulations preserve truth.
We say that a span $(B, \pi_1, \pi_2)$ between $(X_1, \gamma_1)$ and
$(X_2, \gamma_2)$ is \emph{truth preserving}
if $\th_{\gamma_1} \circ\; \pi_1 = \th_{\gamma_2} \circ\; \pi_2$.
If $\cat{C}$ is concrete, this means that if $(x_1,x_2) \in B$
then $x_1$ and $x_2$ have the same theory, i.e., satisfy the same formulae.
\begin{proposition}\label{prop:adequacy}
If
$
\begin{tikzcd}[cramped, sep=1.5em]
X_1
& B \arrow[l, swap, "\pi_1"] \arrow[r, "\pi_2"]
& X_2
\end{tikzcd}
$
is a $\rho$-bisimulation between $\fun{T}$-coalgebras
$(X_1, \gamma_1)$ and $(X_2, \gamma_2)$,
then $\th_{\gamma_1} \circ\, \pi_1 = \th_{\gamma_2} \circ\, \pi_2$.
\end{proposition}
\begin{proof}
Let $\beta : \fun{L}\dpo{B} \to \dpo{B}$ be given as in \eqref{eq:def-bisim},
and let $h_\beta\colon \Phi \to \dpo{B}$ be the unique morphism from the initial
$\fun{L}$-algebra to $(\dpo{B}, \beta)$.
By construction of $\beta$,
$\dpo{\pi}_i\colon (\dpo{B},\beta) \to (\fun{P}X_i,\gamma_i^*)$ are $\fun{L}$-algebra morphisms.
By uniqueness of initial morphisms,
$\sem{-}_{\gamma_i} = \dpo{\pi}_i \circ h_{\beta}$,
and hence
$\fun{S}\sem{-}_{\gamma_i} = \fun{S}h_\beta \circ \fun{S}\dpo{\pi}_i$, for $i = 1, 2$.
Combining this with $\fun{S}\dpo{\pi}_1 \circ \fun{SP}\pi_1 = \fun{S}\dpo{\pi}_2 \circ \fun{SP}\pi_2$ (obtained by applying $\fun{S}$ to the pullback square of $(\dpo{B},\dpo{\pi}_1,\dpo{\pi}_2)$),
it follows that
$\fun{S}\llb \cdot \rrb_{\gamma_1} \circ \fun{SP}\pi_1
= \fun{S}\llb \cdot \rrb_{\gamma_2} \circ \fun{SP}\pi_2$.
Recall that the theory map is the adjoint of the semantic map, i.e.,
$\th_{\gamma_i} = \fun{S}\sem{-}_{\gamma_i} \circ \unitc_{X_i}$
where $\unitc:\Id_{\cat{C}}\to\fun{SP}$ is a unit of the logical connection
$
\begin{tikzcd}[sep=1.5em, cramped]
\fun{P} : \cat{C}
\arrow[r, shift left=1.7pt]
& \cat{A} : \fun{S}
\arrow[l, shift left=1.7pt]
\end{tikzcd}
$.
It then follows from naturality of $\unitc$ that:
\begin{alignat*}{3}
\th_{\gamma_1} \circ\; \pi_1
&= \fun{S}\llb \cdot \rrb_{\gamma_1} \circ \unitc_{X_1} \circ \pi_1
&&= \fun{S}\llb \cdot \rrb_{\gamma_1} \circ \fun{SP}\pi_1 \circ \unitc_B
&& \\
&= \fun{S}\llb \cdot \rrb_{\gamma_2} \circ \fun{SP}\pi_2 \circ \unitc_B
&&= \fun{S}\llb \cdot \rrb_{\gamma_2} \circ \unitc_{X_2} \circ \pi_2
&&= \th_{\gamma_2} \circ\; \pi_2
\end{alignat*}
as desired.
\end{proof}
\subsection{Lattice structure and composition of $\rho$-bisimulations}\label{subsec:lat}
In the remainder of Section~\ref{sec:rho-bis}
we assume that $\cat{C}$ is finitely complete and well-powered,
hence $\cat{Rel}(X_1, X_2)$ is simply the poset of subobjects of $X_1\times X_2$.
Besides, assume that $\cat{C}$ has an $(\ms{E}, \ms{M})$-factorisation system
with $\ms{M} = \ms{M}ono$.
Again, all examples in Section \ref{sec:cml-rel} satisfy these assumptions.
It is well known that bisimulations for $\cat{Set}$-based coalgebras
are closed under composition if and only if
the coalgebra functor preserves weak pullbacks \cite{Rut00}.
We know from \cite[Example~3.3]{BakHan17} that $\Lambda$-bisimulations
do not always compose, even for weak pullback-preserving functors,
so as a consequence of Example~\ref{exm:rho-bis-lambda} the same failure
occurs for $\rho$-bisimulations.
However, in special cases we \emph{can} compose.
Let us first define what we mean by the composition of two relations.
\begin{definition}\label{def:comp-rel}
The composition of two jointly mono spans $(B, \pi_1, \pi_2)$ in $\cat{Rel}(X_1, X_2)$
and $(B', \pi_2', \pi_3)$ in $\cat{Rel}(X_2, X_3)$ is given as follows:
The pullback $(C, c_1, c_3)$ of $\pi_2$ and $\pi_2'$ yields projections
$\pi_i\circ c_i : C \to X_i$, and we
define $B \circ B'$ via the $(\ms{E}, \ms{M}ono)$-factorisation of
$\langle \pi_1\circ c_1, \pi_3\circ c_3 \rangle$:
$$
\begin{tikzcd}[]
C \arrow[dr, ->>, bend right=5]
\arrow[rr, "{\langle \pi_1\circ c_1, \pi_3\circ c_3 \rangle}"]
& [1em]
& [1em]
X_1 \times X_3. \\ [-2em]
& B \circ B'
\arrow[ur, hook, bend right=5]
&
\end{tikzcd}
$$
\end{definition}
Call a $\rho$-bisimulation \emph{full} if both projections are split epi, that is, they
have a section.
For $\cat{Set}$-based coalgebras this means that
the projections are surjective, i.e.,
each state in $(X_1,\gamma_1)$ is $\rho$-bisimilar to some state in $(X_2,\gamma_2)$,
and vice versa.
\begin{lemma}\label{lem:rho-bis-span}
Let
$
\begin{tikzcd}[cramped, sep=1.5em]
X_1
& S \arrow[l, swap, "\zeta_1"] \arrow[r, "\zeta_2"]
& X_2
\end{tikzcd}
$
and
$
\begin{tikzcd}[cramped, sep=1.5em]
X_1
& B \arrow[l, swap, "\pi_1"] \arrow[r, "\pi_2"]
& X_2
\end{tikzcd}
$
be spans between $\fun{T}$-coalgebra $(X_1, \gamma_1)$ and $(X_2, \gamma_2)$
and suppose $e : S \to B$
is an epimorphism such that $\zeta_i = \pi_i \circ e$.
Then $(S, \zeta_1, \zeta_2)$ satisfies \eqref{eq:rho-bis} if and only if
$(B, \pi_1, \pi_2)$ does.
\end{lemma}
\begin{proof}
Since $S \to B$ is epic $\fun{P}B \to \fun{P}S$ is monic.
It follows that the pullback of $(S, \zeta_1, \zeta_2)$ coincides with
the pullback to $(B, \pi_1, \pi_2)$. With this observation the
proof of the proposition follows from a straightforward computation.
\end{proof}
Now we can show that full bisimulations compose.
\begin{proposition}\label{prop:comp}
Full bisimulations are closed under composition.
\end{proposition}
\begin{proof}
Let $(B, \pi_1, \pi_2)$ be a $\rho$-bisimulation between $(X_1, \gamma_1)$ and
$(X_2, \gamma_2)$, and $(B', \pi_2', \pi_3)$ a $\rho$-bisimulation between
$(X_2, \gamma_2)$ and $(X_3, \gamma_3)$.
Write $(C, c_1, c_3)$ for the pullback of $\pi_2$ and $\pi_2'$
and define $(\dpo{C}, \dpo{c}_1, \dpo{c}_3)$ to be the pullback
of $\dpo{\pi}_2$ and $\dpo{\pi}_2'$.
By Lemma \ref{lem:rho-bis-span} it suffices to show that
$(C, \pi_1c_1, \pi_3c_3)$ satisfies the $\rho$-bisimulation condition.
Since all the $\pi_i$ are split epic, so are $c_1$ and $c_3$
(cf.~Lemma \ref{lem-pb-split}).
According to Lemma \ref{lem-pb-po}
this implies that the pullback square
\begin{equation*}
\begin{tikzcd}[row sep=.2em]
& C
\arrow[dl, "c_1" swap]
\arrow[dr, "c_2"]
& \\
B_1 \arrow[dr, "\pi_2" swap]
&
& B_2 \arrow[dl, "\pi_2'"] \\
& X_2
&
\end{tikzcd}
\end{equation*}
is also a pushout.
Therefore square \circled{4} below is a pullback, while \circled{1}, \circled{2}
and \circled{3} are pullbacks by definition.
It follows from repeated application of the pullback lemma that the
outer square is a pullback as well.
\begin{equation}\label{eq:pbs}
\begin{tikzcd}[row sep=.2em]
&
& \dpo{C}
\arrow[dl, "\ov{c}_1" swap] \arrow[dr, "\ov{c}_3"]
\arrow[dd, phantom, "\circled{1}" pos=.5]
&
& \\
& \dpo{B}_1
\arrow[dl, "\ov{\pi}_1" swap]
\arrow[dr, "\ov{\pi}_2"]
\arrow[dd, phantom, "\circled{2}"]
&
& \dpo{B}_2
\arrow[dl, "\ov{\pi}_2'" swap]
\arrow[dr, "\ov{\pi}_3"]
\arrow[dd, phantom, "\circled{3}"]
& \\
\fun{P}X_1
\arrow[dr, "\fun{P}\pi_1" swap]
&
& \fun{P}X_2
\arrow[dl, "\fun{P}\pi_2" swap]
\arrow[dr, "\fun{P}\pi_2'"]
\arrow[dd, phantom, "\circled{4}"]
&
& \fun{P}X_3 \arrow[dl, "\fun{P}\pi_3"] \\
& \fun{P}B_1 \arrow[dr, "\fun{P}c_1" swap]
&
& \fun{P}B_2 \arrow[dl, "\fun{P}c_3"]
& \\
&
& \fun{P}C
&
&
\end{tikzcd}
\end{equation}
As a consequence
$(\ov{\pi}_1\ov{c}_1, \ov{\pi}_3\ov{c}_3)$ is jointly monic.
In order to show that $(C, \pi_1c_1, \pi_3c_3)$ is a $\rho$-bisimulation
we need to prove
\begin{equation}\label{eq:composition}
\fun{P}c_1 \circ \fun{P}\pi_1 \circ \gamma_1^*
\circ \fun{L}\ov{\pi}_1 \circ \fun{L}\ov{c}_1
= \fun{P}c_3 \circ \fun{P}\pi_3 \circ \gamma_3^*
\circ \fun{L}\ov{\pi}_3 \circ \fun{L}\ov{c}_3
\end{equation}
This is the outer shell of the following diagram:
$$
\begin{tikzcd}[row sep=tiny]
&
& \fun{L}\dpo{C}
\arrow[dl, "\fun{L}\ov{c}_1" swap]
\arrow[dr, "\fun{L}\ov{c}_3"]
&
& \\
& \fun{L}\dpo{B}_1
\arrow[dl, "\fun{L}\ov{\pi}_1" swap]
\arrow[dr, "\fun{L}\ov{\pi}_2"]
\arrow[dd, "\beta_1"]
&
& \fun{L}\dpo{B}_2
\arrow[dl, "\fun{L}\ov{\pi}_2'" swap]
\arrow[dr, "\fun{L}\ov{\pi}_3"]
\arrow[dd, "\beta_2"]
& \\
\fun{LP}X_1 \arrow[dd, "\gamma_1^*" swap]
&
& \fun{LP}X_2 \arrow[dd, "\gamma_2^*"]
&
& \fun{LP}X_3 \arrow[dd, "\gamma_3^*"] \\
& \dpo{B}_1 \arrow[dl, "\ov{\pi}_1" swap] \arrow[dr, "\ov{\pi}_2"]
&
& \dpo{B}_2 \arrow[dl, "\ov{\pi}_2'" swap] \arrow[dr, "\ov{\pi}_3"]
& \\
\fun{P}X_1 \arrow[dr, "\fun{P}\pi_1" swap]
&
& \fun{P}X_2 \arrow[dl, "\fun{P}\pi_2" swap] \arrow[dr, "\fun{P}\pi_2'"]
&
& \fun{P}X_3 \arrow[dl, "\fun{P}\pi_3"] \\
& \fun{P}B_1 \arrow[dr, "\fun{P}c_1" swap]
&
& \fun{P}B_2 \arrow[dl, "\fun{P}c_2"]
& \\
&
& \fun{P}C
&
&
\end{tikzcd}
$$
Commutativity of the top square follows from applying $\fun{L}$ to cell
\circled{1} in \eqref{eq:pbs}.
The bottom square commutes by definition of $(C, c_1, c_3)$.
All other squares commute because $(B_1, \pi_1, \pi_2)$ and
$(B_2, \pi_2', \pi_3)$ are assumed to be $\rho$-bisimulations.
Thus \eqref{eq:composition} holds, and this ultimately proves that
full $\rho$-bisimulations are closed under composition.
\end{proof}
Another well-known result for bisimulations on $\cat{Set}$-coalgebras
is that they form a complete lattice \cite{Rut00}.
We now show that, provided $\cat{C}$
has all coproducts,
this also holds for $\rho$-bisimulations.
Recall that the empty coproduct $\coprod\emptyset =: \initobj$ is an initial object,
i.e., for all $C \in \cat{C}$ there is a unique morphism
$!_C \colon \initobj \to C$.
\begin{definition}\label{def:union}
The \emph{join} of a family of relations $(B_i,\pi_{i,1},\pi_{i,2})$, $i \in I$,
in $\cat{Rel}(X_1, X_2)$, is the jointly mono span
$\bigcup_{i\in I} B_i$ that arises from the factorisation
\[
\begin{tikzcd}[column sep=3.5em]
\coprod_i B_i \arrow[r, two heads]
\arrow[rr, bend left=15, shift left=2pt, "{\coprod_i \langle \pi_{i,1}, \pi_{i,2} \rangle }"]
& \bigcup_i B_i \arrow[r, hook]
& X_1 \times X_2
\end{tikzcd}
\]
The \emph{bottom element} $(I, \iota_1, \iota_2)$ in $\cat{Rel}(X_1, X_2)$
is defined by the factorisation of the initial morphism:
$
\begin{tikzcd}[cramped, column sep=1.5em]
\initobj \arrow[r, two heads]
& I \arrow[r, hook,
"{\langle \iota_1, \iota_2 \rangle}"]
&[2em] X_1 \times X_2
\end{tikzcd}
$.
\end{definition}
Indeed, $\bigcup_i B_i$ is an upper bound in $\cat{Rel}(X_1, X_2)$.
Suppose $(B_i, \pi_{i,1}, \pi_{i,2}) \leq (S, s_1, s_2)$ for all $i$,
then there are $t_i : B_i \to S$ such that $\pi_{i,j} = s_j \circ t_i$.
From the coproduct we get $t : \coprod_{i \in I} B_i \to S$ and this makes
the outer shell of the diagram below commute.
$$
\begin{tikzcd}[row sep=0em, column sep=1.7em]
& \bigcup_i B_i
\arrow[dr, hook]
\arrow[dd, dashed, "d"]
& \\
\coprod_i B_i
\arrow[ru, ->>]
\arrow[rd, "t"]
&
& X_1 \times X_2 \\
& S \arrow[ru, hook]
&
\end{tikzcd}
$$
The factorisation system now gives a diagonal $d : \bigcup_{i \in I} B_i \to S$
witnessing that $S$ is bigger than $\bigcup_{i \in I}B_i$ in $\cat{Rel}(X_1, X_2)$.
\begin{example}
Let $X_1$ and $X_2$ be objects in $\cat{C}$ and $(B_i, \pi_{i,1}, \pi_{i,2})$
relations in $\cat{Rel}(X_1, X_2)$, where $i$ ranges over some index set $I$.
View these as subobjects of $X_1 \times X_2$.
We describe the join of all $B_i$.
\begin{enumerate}
\item If $\cat{C} = \cat{Set}$, $\cat{C} = \cat{Pos}$ or $\cat{C} = \cat{Top}$
the join is given by the union in $X_1 \times X_2$.
\item If $\cat{C} = \cat{Stone}$ then the join of the $B_i$ is
the closure of $\bigcup B_i$ viewed as a subspace of $X_1 \times X_2$.
\item If $\cat{C} = \cat{Vec}$ then the join of a family of relations
$B_i \subseteq X_1 \times X_2$ is the smallest subspace of $X_1 \times X_2$
containing $\bigcup B_i$. That is, $\bigvee B_i$ contains all
vectors $v \in X_1 \times X_2$ of the form
$v = v_{i_1} + \cdots + v_{i_n}$, with $v_{i_j} \in B_{i_j}$.
\end{enumerate}
\end{example}
\begin{proposition}\label{prop:rel-jsl}
If $\cat{C}$ has an $(\ms{E}, \ms{M}ono)$-factorisation system,
binary products and all coproducts, then
the poset $\cat{Rel}(X_1,X_2)$ is a complete join-semilattice with
join $\bigcup$ and bottom element $(I, \iota_1, \iota_2)$.
\end{proposition}
\begin{proof}
Commutativity and associativity of the join follow from the
fact that coproducts are commutative and associative.
For idempotency note that for every $\Bspan$ in $\cat{Rel}(X_1,X_2)$
we have an $(\ms{E}, \ms{M}ono)$-factorisation
$
\begin{tikzcd}[cramped, sep=1.5em]
B + B \arrow[r, two heads, "\nabla"]
&[.5em] B \arrow[r, hook]
& X_1 \times X_2,
\end{tikzcd}
$
where $\nabla$ is the codiagonal, so $B \cup B = B$.
Next, we show that $(I, \iota_1, \iota_2)$ is the bottom element in $(\cat{Rel}(X_1,X_2), \cup)$.
That is, for all $\Bspan$ in $\cat{Rel}(X_1,X_2)$,
$B \cup I$ is isomorphic to $B$. By the definition of a coproduct,
$$
\begin{tikzcd}[]
B \arrow[rr, bend left=15, shift left=2pt, "i"]
\arrow[r, "\cong" below]
& \initobj + B
\arrow[->>, r, "!_I + \id_B" below]
&[1em] I + B
\end{tikzcd}
$$
commutes, where $i$ is the inclusion that arises from the coproduct.
Since $\ms{E}$ is closed under composition, the map
$i : B \to I + B$ is in $\ms{E}$.
By definition of the join, the following commutes:
\[
\begin{tikzcd}[column sep=3.5em]
B \arrow[r, bend right=0, two heads]
\arrow[rrr, bend left=12, shift left=1pt, hook, "{\langle \pi_1, \pi_2 \rangle}"]
& I + B
\arrow[r, two heads, shift right=2pt]
\arrow[rr, bend right=15, shift right=2pt, "{[ \langle \iota_1, \iota_2 \rangle,
\langle \pi_1 , \pi_2 \rangle ]}" below]
& I \cup B \arrow[r, hook]
& X_1 \times X_2
\end{tikzcd}
\]
Since factorisation systems are unique up to isomorphism, we get an isomorphism
$B \cong B \cup I$.
\end{proof}
We define \emph{$\rho$-bisimilarity} as the join of all $\rho$-bisimulations
in $\cat{Rel}(X_1, X_2)$.
The following proposition tells us that $\rho$-bisimilarity
is itself a $\rho$-bisimulation.
Given two $\fun{T}$-coalgebras
$(X_1, \gamma_1)$ and $(X_2, \gamma_2)$, we denote by
$\rho\hyphen\cat{Bis}(\gamma_1, \gamma_2)$ the sub-poset of
$\cat{Rel}(X_1, X_2)$ of $\rho$-bisimulations
between $(X_1, \gamma_1)$ and $(X_2, \gamma_2)$.
\begin{proposition}\label{prop:bis-jsl}
Under the assumptions of Proposition~\ref{prop:rel-jsl},
$\rho$-$\cat{Bis}(\gamma_1, \gamma_2)$
is closed under joins and bottom element in $\cat{Rel}(X_1, X_2)$.
Consequently, $\rho$-$\cat{Bis}(\gamma_1, \gamma_2)$
is a complete join-semilattice, and hence also a complete lattice.
\end{proposition}
\begin{proof}
We give the proof for the binary case. This is easily adapted to arbitrary joins.
As a consequence of Lemma \ref{lem:rho-bis-span} it suffices to
prove that the bisimulation condition holds for the span
$
\begin{tikzcd}[column sep=3em]
X_1
& B + S
\arrow[l, "{[\pi_1, \sigma_1]}" {swap,pos=.4}]
\arrow[r, "{[\pi_2, \sigma_2]}" {pos=.4}]
& X_2.
\end{tikzcd}
$
We write $(\dpo{B + S}, \dpo{[\pi_1,\sigma_1]}, \dpo{[\pi_2, \sigma_2]})$
for its dual span.
Since $\fun{P}$ is part of a dual adjunction it turns colimits into limits,
hence $\fun{P}(B + S) = \fun{P}B \times \fun{P}S$ and
$\fun{P}([\pi_i, \sigma_i]) = \langle \fun{P}\pi_i, \fun{P}\sigma_i \rangle$ for $i = 1, 2$.
Denote by $\theta_B$ and $\theta_S$ the projections of
$\fun{P}B \times \fun{P}S$ to $\fun{P}B$ and $\fun{P}S$.
By the property of a product, in order to show that
$\fun{P}([\pi_1, \sigma_1]) \circ \gamma_1^* \circ \fun{L}\dpo{[\pi_1, \sigma_1]}
= \fun{P}([\pi_2, \sigma_2]) \circ \gamma_2^* \circ \fun{L}\dpo{[\pi_2, \sigma_2]}$
holds, it suffices to show that
\begin{equation}\label{eq-rb-coprod1}
\theta_j \circ \fun{P}([\pi_1, \sigma_1]) \circ \gamma_1^* \circ \fun{L}\dpo{[\pi_1, \sigma_1]}
= \theta_j \circ \fun{P}([\pi_2, \sigma_2]) \circ \gamma_2^*\circ \fun{L}\dpo{[\pi_2, \sigma_2]}
\end{equation}
for $j = B, S$.
Since $\fun{P}([\pi_i, \sigma_i]) = \langle \fun{P}\pi_i, \fun{P}\sigma_i \rangle$ this
reduces to proving
\begin{align}
\fun{P}\pi_1 \circ \gamma_1^* \circ \fun{L}\dpo{[\pi_1, \sigma_1]}
&= \fun{P}\pi_2 \circ \gamma_2^* \circ \fun{L}\dpo{[\pi_2, \sigma_2]}
\label{eq-rb-coprod2} \\
\fun{P}\sigma_1 \circ \gamma_1^* \circ \fun{L}\dpo{[\pi_1, \sigma_1]}
&= \fun{P}\sigma_2 \circ \gamma_2^* \circ \fun{L}\dpo{[\pi_2, \sigma_2]}.
\label{eq-rb-coprod3}
\end{align}
We focus on the first equation, the second being similar.
Let $(\dpo{B}, \dpo{\pi}_1, \dpo{\pi}_2)$ be the pullback of $(\fun{P}\pi_1, \fun{P}\pi_2)$.
Since
$
\fun{P}\pi_1 \circ q_1
= \theta_1 \circ \fun{P}(\pi_1 + \sigma_1) \circ q_1
= \theta_1 \circ \fun{P}(\pi_2 + \sigma_2) \circ q_2
= \fun{P}\pi_2 \circ q_2
$,
the triple $(\dpo{B + S}, \dpo{[\pi_1, \sigma_1]}, \dpo{[\pi_2, \sigma_2]})$
forms
\begin{equation}
\begin{tikzcd}[column sep=3.5em, row sep=1.4em, cramped]
& \dpo{B + S} \arrow[dl, bend right=20, "\dpo{[\pi_1, \sigma_1]}" swap]
\arrow[dr, bend left=10, "\dpo{[\pi_2, \sigma_2]}"]
\arrow[d, dashed, "h"]
& \\
\fun{P}X_1
\arrow[dr, bend right=10, "\fun{P}(\pi_1 + \sigma_1)"]
\arrow[ddr, bend right=20, "\fun{P}\pi_1" swap]
& \dpo{B}
\arrow[l, "\dpo{\pi}_1" swap]
\arrow[r, "\dpo{\pi}_2"]
& \fun{P}X_2
\arrow[dl, bend left=10, "\fun{P}(\pi_2 + \sigma_2)" swap]
\arrow[ddl, bend left=20, "\fun{P}\pi_2"] \\
& \fun{P}B \times \fun{P}S
\arrow[d, "\theta_1"]
& \\
& \fun{P}B
&
\end{tikzcd}
\end{equation}
a cone of the pullback diagram of $\dpo{B}$, and
hence we get a mediating map $h : \dpo{B + S} \to \dpo{B}$ making the diagram
commute.
The equality in \eqref{eq-rb-coprod2} now follows from applying $\fun{L}$ to
this diagram and using that $B$ is a $\rho$-bisimulation.
To see that $I$ is a $\rho$-bisimulation,
it suffices to show that $\initobj$ with the unique maps to $X_1$ and $X_2$ satisfies
\eqref{eq:rho-bis}. This follows immediately from the fact that
$\fun{P}(\initobj)$ is final in $\cat{D}$.
\end{proof}
While $\rho$-$\cat{Bis}(\gamma_1, \gamma_2)$ is
a complete sub-semilattice of $\cat{Rel}(X_1, X_2)$,
it need not inherit the meets.
This resembles the situation for Kripke bisimulations,
which are generally not closed under intersections.
\begin{example}
The categories $\cat{Set}$, $\cat{Top}$ and $\cat{Vec}_{\Bbbk}$ from Examples
\ref{exm:pml-top-bis}, \ref{exm:rho-bis-lambda} and \ref{exm:mod-vec-bis}
are well-powered, complete and cocomplete, and as mentioned in Section~\ref{sec:cml-rel}
have a $(\ms{R}eg\ms{E}pi, \ms{M}ono)$-factorisation system.
Hence $\rho$-bisimulations for positive modal logic,
linear Hennessy-Milner logic and
coalgebraic geometric logic form complete lattices,
and we recover the similar result for $\Lambda$-bisimulations in
\cite[Proposition 3.7]{BakHan17} and \cite[Proposition 8.6]{BezGroVen19-report}.
\end{example}
\subsection{Characterisation via relation lifting}\label{subsec:rel-lift}
Another property of
bisimulations for $\cat{Set}$-coalgebras is that they can
be characterised via relation lifting (see e.g.~\cite[Section~2.2]{Sta11}),
and that bisimilarity on a coalgebra $(X,\gamma)$ is a greatest fixpoint of a monotone operator
on the lattice of relations $\Pow(X \times X)$.
In this subsection and the following, we show that these results
generalise to realm of $\rho$-bisimulations.
Given $X_1, X_2$ in $\cat{C}$, we shall define a monotone map
$$
\fun{T}^{\rho} : \cat{Rel}(X_1, X_2) \to \cat{Rel}(\fun{T}X_1, \fun{T}X_2)
$$
which lifts $(B, \pi_1, \pi_2)$ in $\cat{Rel}(X_1, X_2)$ to
$(\fun{T}^{\rho}B,\fun{T}^{\rho}\pi_1,\fun{T}^{\rho}\pi_2)$
in $\cat{Rel}(\fun{T}X_1, \fun{T}X_2)$.
In order to do so, consider the
composition,
\begin{equation}\label{eq:sigma}
\begin{tikzcd}[column sep=2em]
\sigma_i : \fun{T}X_i \arrow[r, "\unitc_{\fun{T}X_i}"]
& \fun{SPT}X_i \arrow[r, "\fun{S}\rho_{X_i}"]
& [.5em] \fun{SLP}X_i \arrow[r, "\fun{SL}\ov{\pi}_i"]
& [.5em] \fun{SL}\dpo{B},
\end{tikzcd}
\end{equation}
for $i = 1, 2$.
For a concrete example of $\sigma_i$, see Example~\ref{exm:Trho-set} below.
\begin{definition}
Given $\Bspan$
in $\posRel$,
define
$\fun{T}^{\rho}(B, \pi_1, \pi_2) = (\fun{T}^{\rho}B, \fun{T}^{\rho}\pi_1, \fun{T}^{\rho}\pi_2)$
in $\cat{Rel}(\fun{T}X_1, \fun{T}X_2)$ as the pullback of
\begin{tikzcd}[cramped, sep=1.5em]
\fun{T}X_1 \arrow[r, "\sigma_1"]
& \fun{SL}\dpo{B}
& \fun{T}X_2 \arrow[l, swap, "\sigma_2"]
\end{tikzcd}.
\end{definition}
Observe that $(\fun{T}^{\rho}B, \fun{T}^{\rho}\pi_1, \fun{T}^{\rho}\pi_2)$
is a jointly mono span because it is a pullback.
Monotonicity of $\fun{T}^{\rho}$ follows from unravelling the definitions.
We can now characterise $\rho$-bisimulations as in \cite{HerJac98}
using the relation lifting $\fun{T}^\rho$.
\begin{theorem}\label{thm:beta-bis}
A jointly mono span $(B, \pi_1, \pi_2)$ between two $\fun{T}$-coalgebras
$(X_1, \gamma_1)$ and $(X_2, \gamma_2)$ is a $\rho$-bi\-sim\-ula\-tion
if and only if there exists a morphism $\delta : B \to \fun{T}^{\rho}B$
in $\cat{C}$ making diagram \eqref{eq:rho-bis-fill-in} commute.
\begin{equation}\label{eq:rho-bis-fill-in}
\begin{tikzcd}[row sep=1.5em]
X_1 \arrow[d, "\gamma_1" swap]
& B \arrow[l, "\pi_1" swap]
\arrow[r, "\pi_2"]
\arrow[d, dashed, "\delta"]
& X_2
\arrow[d, "\gamma_2"] \\
\fun{T}X_1
& \fun{T}^{\rho}B
\arrow[l, swap, "\fun{T}^{\rho}\pi_1"]
\arrow[r, "\fun{T}^{\rho}\pi_2"]
& \fun{T}X_2
\end{tikzcd}
\end{equation}
\end{theorem}
\begin{proof}
{\sl If $\delta$ exists, then $B$ is a $\rho$-bisimulation.} \;
Suppose such a $\delta$ exists. In order to show that $B$ is
a $\rho$-bisimulation, we need to show that the outer shell of the left diagram
below commutes.
Recall that $\unitc$ and $\unita$
are the units of the dual adjunction
$
\begin{tikzcd}[cramped, sep=1.5em]
\fun{P} : \cat{C} \arrow[r, shift left=1.7pt]
& \cat{A} : \fun{S} \arrow[l, shift left=1.7pt]
\end{tikzcd}
$.
\begin{equation}\label{prop-eq-beta-bis}
\begin{tikzcd}[column sep=2.5em, row sep=1.7em]
& \fun{L}\dpo{B}
\arrow[dl, bend right=10, "\fun{L}\ov{\pi}_1" swap]
\arrow[dr, bend left=10, "\fun{L}\ov{\pi}_2"]
\arrow[d, "\unita_{\fun{L}\dpo{B}}"]
&
& [0em] \fun{L}\dpo{B}
\arrow[r, "\unita_{\fun{L}\dpo{B}}"]
\arrow[d, "\fun{L}\dpo{\pi}_i"]
& \fun{PSL}\dpo{B}
\arrow[d, "\fun{PSL}\dpo{\pi}_i" left]
\arrow[ddd, bend left=30, shift left=10pt, "\fun{P}\sigma_i"] \\
\fun{LP}X_1
\arrow[d, "\rho_{X_1}" swap]
& \fun{PSL}\dpo{B}
\arrow[dl, bend right=5, "\fun{P}\sigma_1" {swap, pos=.2}]
\arrow[dr, bend left=5, "\fun{P}\sigma_2" {pos=.3}]
& \fun{LP}X_2
\arrow[d, "\rho_{X_2}"]
& \fun{LP}X_i
\arrow[r, "\unita_{\fun{LP}X_i}"]
\arrow[d, "\rho_{X_i}"]
& \fun{PSLP}X_i
\arrow[d, "\fun{PS}\rho_{X_i}" left] \\
\fun{PT}X_1
\arrow[d, "\fun{P}\gamma_1" left]
& \fun{PT}^{\rho}B
\arrow[l, <-, "\fun{PT}^{\rho}\pi_1"]
\arrow[r, <-, "\fun{PT}^{\rho}\pi_2" swap]
\arrow[d, "\fun{P}\delta"]
& \fun{PT}X_2
\arrow[d, "\fun{P}\gamma_2" right]
& \fun{PT}X_i
\arrow[r, "\unita_{\fun{PT}X_i}"]
\arrow[dr, equal, bend right=20]
& \fun{PSPT}X_i
\arrow[d, "\fun{P}\unitc_{\fun{T}X_i}" left] \\
\fun{P}X_1
\arrow[r, "\fun{P}\pi_1" swap]
& \fun{P}B
& \fun{P}X_2
\arrow[l, "\fun{P}\pi_2"]
&
& \fun{PT}X_i
\end{tikzcd}
\end{equation}
Commutativity of the bottom two squares follows from applying $\fun{P}$ to
the diagram in \eqref{eq:rho-bis-fill-in}.
The middle square commutes because of the definition of $\fun{T}^{\rho}B$.
The top two squares commute because they are the outer shell of the
right diagram in \eqref{prop-eq-beta-bis}.
In \eqref{prop-eq-beta-bis}, the right square commutes by definition of $\sigma_i$ (Equation~\ref{eq:sigma}).
The other two squares commute by naturality of $\unita$
and the lower triangle
is a triangle identity of the dual adjunction.
Therefore the outer shell commutes.
\bigskip\noindent
{\sl If $B$ is a $\rho$-bi\-sim\-ulation, then we can find $\delta$.} \;
Suppose $(B, \pi_1, \pi_2)$ is a $\rho$-bi\-sim\-ulation. If we can prove that
$
\sigma_1 \circ \gamma_1 \circ \pi_1 = \sigma_2 \circ \gamma_2 \circ \pi_2
$
then we obtain $\delta$ as the
mediating map induced by the pullback which defines $\fun{T}^{\rho}B$,
as shown below:
\begin{equation}\label{prop-bis-beta-2}
\begin{tikzcd}[row sep=0em]
& B \arrow[dl, bend right=5, "\pi_1" {swap,pos=.4}]
\arrow[dr, bend left=5, "\pi_2" {pos=.4}]
\arrow[dd, dashed, "\delta"]
& \\
X_1 \arrow[dd, "\gamma_1" swap]
&
& X_2
\arrow[dd, "\gamma_2"] \\
& \fun{T}^{\rho}B
\arrow[dl, bend right=3, "\fun{T}^{\rho}\pi_1" {swap, pos=.2}]
\arrow[dr, bend left=3, "\fun{T}^{\rho}\pi_2" {pos=.3}]
& \\
\fun{T}X_1
\arrow[dr, bend right=5, "\sigma_1" swap]
&
& \fun{T}X_2
\arrow[dl, bend left=5, "\sigma_2"] \\
& \fun{SL}Q
&
\end{tikzcd}
\end{equation}
We claim that the following diagram commutes. Since its outer shell is
the same as the outer shell of \eqref{prop-bis-beta-2}, this proves the proposition.
So consider:
$$
\begin{tikzcd}[row sep=1.7em]
& X_1
\arrow[d, "\unitc_{X_1}"]
\arrow[ddl, bend right=30, "\gamma_1" swap]
& B \arrow[l, "\pi_1" swap]
\arrow[r, "\pi_2"]
\arrow[d, "\unitc_B"]
& X_2
\arrow[d, "\unitc_{X_2}"]
\arrow[ddr, bend left=30, "\gamma_2"]
& \\
& \fun{SP}X_1
\arrow[d, "\fun{SP}\gamma_1"]
& \fun{SP}B
\arrow[l, "\fun{SP}\pi_1"]
\arrow[r, "\fun{SP}\pi_2" swap]
& \fun{SP}X_2
\arrow[d, "\fun{SP}\gamma_1"]
& \\
\fun{T}X_1
\arrow[r, "\unitc_{\fun{T}X_1}"]
\arrow[ddrr, bend right=35, "\sigma_1" swap]
& \fun{SPT}X_1
\arrow[d, "\fun{S}\rho_{X_1}"]
&
& \fun{SPT}X_2
\arrow[d, "\fun{S}\rho_{X_2}"]
& \fun{T}X_2
\arrow[l, "\unitc_{\fun{T}X_2}" swap]
\arrow[ddll, bend left=35, "\sigma_2"] \\
& \fun{SLP}X_1
\arrow[dr, "\fun{SL}\ov{\pi}_1"]
&
& \fun{SLP}X_2
\arrow[dl, "\fun{SL}\ov{\pi}_2" swap]
& \\ [-1em]
&
& \fun{SL}Q
&
&
\end{tikzcd}
$$
Commutativity of the middle part follows from the fact that
$B$ is a $\rho$-bisimulation. The four top squares commute because $\unitc$ is
a natural transformation. The two remaining squares commute by definition of $\sigma_i$.
\end{proof}
We work out the explicit description of $\fun{T}^{\rho}$ in a special case:
\begin{example}\label{exm:Trho-set}
Suppose we work with the classic dual adjunction
$
\begin{tikzcd}[cramped, sep=1.5em]
\Pba : \cat{Set} \arrow[r, shift left=1.7pt]
& \cat{BA} : \Uf \arrow[l, shift left=1.7pt]
\end{tikzcd}
$,
$\fun{T}$ is an endofunctor on $\cat{Set}$, and the logic $(\fun{L}, \rho)$
is given by predicate liftings and axioms (cf.~Example \ref{exm:pl}).
Then the type of $\sigma_i$ is $\fun{T}X_i \to \Uf\fun{L}\dpo{B}$ and
the ultrafilter $\sigma_i(t_i)$
is determined by the elements of the form $\und{\lambda}(a_1, a_2)$ it contains,
where $\lambda \in \Lambda$ and $(a_1, a_2) \in \dpo{B}$.
Therefore the action of $\fun{T}^{\rho}$ on $(B, \pi_1, \pi_2)$
is given by
\begin{equation*}
\begin{split}
\fun{T}^{\rho}B = \{ (t_1, t_2) \in \fun{T}X_1 \times \fun{T}X_2 \mid
\forall &\lambda \in \Lambda \text{ and $B$-coherent } (a_1, a_2) \\
&\text{ we have } t_1 \in \lambda_{X_1}(a_1)
\Leftrightarrow t_2 \in \lambda_{X_2}(a_2) \}.
\end{split}
\end{equation*}
Informally, these are the pairs in $\fun{T}X_1 \times \fun{T}X_2$
that cannot be distinguished by lifted $B$-coherent predicates.
\end{example}
\subsection{Characterisation as a (post)fixpoint}\label{subsec:fix}
As for $\cat{Set}$-coalgebras, given a relation lifting of $\fun{T}$
and $\fun{T}$-coalgebras $(X_1,\gamma_1)$, $(X_2,\gamma_2)$,
we can define a map $\fun{T}^{\rho}_{\gamma_1, \gamma_2} \colon\cat{Rel}(X_1, X_2) \to \cat{Rel}(X_1, X_2)$ by,
essentially, taking inverse images under the $\gamma_i$.
This is a relational version of a predicate transformer on a coalgebra.
\begin{definition}\label{def:T-rho-gamma}
Given $\fun{T}$-coalgebras $(X_1, \gamma_1)$ and $(X_2, \gamma_2)$
and a jointly mono span $(B, \pi_1, \pi_2)$ between $X_1$ and $X_2$,
define $\fun{T}^{\rho}_{\gamma_1, \gamma_2}(B, \pi_1, \pi_1) =
(\fun{T}^{\rho}_{\gamma_1, \gamma_2}B, \fun{T}^{\rho}_{\gamma_1, \gamma_2}\pi_1,
\fun{T}^{\rho}_{\gamma_1, \gamma_2}\pi_2) \in \cat{Rel}(X_1, X_2)$ via the pullback
$$
\begin{tikzcd}[row sep=0em]
&
& \fun{T}_{\gamma_1,\gamma_2}^{\rho}B
\arrow[dll, "\fun{T}_{\gamma_1,\gamma_2}^{\rho}\pi_1" swap]
\arrow[drr, "\fun{T}_{\gamma_1,\gamma_2}^{\rho}\pi_1"]
&
& \\ [.5em]
X_1 \arrow[dr, "\gamma_1" swap]
&
&
&
& X_2
\arrow[dl, "\gamma_2"] \\ [-.5em]
& \fun{T}X_1
\arrow[dr, "\sigma_1" swap]
&
& \fun{T}X_2
\arrow[dl, "\sigma_2"]
& \\ [-.5em]
&
& \fun{SL}\dpo{B}
&
&
\end{tikzcd}
$$
This is well defined because pullbacks are jointly mono spans.
\end{definition}
\begin{lemma}\label{lem:Trhogam-mon}
The map $\fun{T}^{\rho}_{\gamma_1, \gamma_2}\colon \cat{Rel}(X_1, X_2) \to \cat{Rel}(X_1, X_2)$ is monotone.
\end{lemma}
\begin{proof}
If $(B, \pi_1, \pi_2) \leq (B', \pi_1', \pi_2')$ then there exists an
$m : B \to B'$ such that $\pi_i = \pi_i' \circ m$.
As a consequence the pullback $\dpo{B}'$ is a cone for $\dpo{B}$ and we have
a mediating map $k : \dpo{B}' \to \dpo{B}$ satisfying $\dpo{\pi}_i' = \dpo{\pi}_i \circ k$.
Unravelling the definitions reveals that $\fun{T}^{\rho}_{\gamma_1, \gamma_2}B$
with its projections is a cone for $\fun{T}^{\rho}_{\gamma_1, \gamma_2}B'$,
hence there is a (unique) map $t : \fun{T}^{\rho}_{\gamma_1, \gamma_2}B \to
\fun{T}^{\rho}_{\gamma_1, \gamma_2}B'$ such that
$\fun{T}^{\rho}_{\gamma_1, \gamma_2}\pi_i = \fun{T}^{\rho}_{\gamma_1, \gamma_2}\pi_i' \circ t$
which witnesses that
$\fun{T}^{\rho}_{\gamma_1, \gamma_2}(B, \pi_1, \pi_2) \leq
\fun{T}^{\rho}_{\gamma_1, \gamma_2}(B', \pi_1', \pi_2')$.
\end{proof}
As announced,
$\rho$-bisimulations are precisely the post-fixpoints of
$\fun{T}^{\rho}_{\gamma_1, \gamma_2}$.
\begin{theorem}\label{thm:rho-post-fix}
A relation
$
\begin{tikzcd}[cramped, sep=1.5em]
X_1
& B \arrow[l, swap, "\pi_1"] \arrow[r, "\pi_2"]
& X_2
\end{tikzcd}
$
is a $\rho$-bisimulation between $(X_1, \gamma_1)$
and $(X_2, \gamma_2)$ if and only if
$(B, \pi_1, \pi_2) \leq \fun{T}^{\rho}_{\gamma_1, \gamma_2}(B, \pi_1, \pi_2)$.
\end{theorem}
\begin{proof}
If $\Bspan$ is a $\rho$-bisimulation, then by Theorem \ref{thm:beta-bis}
there is a map $\beta\colon B \to \fun{T}^\rho B$ such that
diagram \eqref{eq:rho-bis-fill-in} commutes. We then get a
map $\beta'\colon B \to \fun{T}^\rho_{\gamma_1, \gamma_2}B$
from the pullback property of $\fun{T}^\rho_{\gamma_1, \gamma_2}B$.
Conversely, given $\beta'\colon B \to \fun{T}^\rho_{\gamma_1, \gamma_2}B$,
we obtain $\beta\colon B \to \fun{T}^\rho B$ from the pullback property of $\fun{T}^\rho B$.
\end{proof}
Monotonicity of $\fun{T}^{\rho}_{\gamma_1, \gamma_2}$ and
the Knaster-Tarski fixpoint theorem imply:
\begin{corollary}
Under the assumptions of Proposition~\ref{prop:rel-jsl},
$\fun{T}^{\rho}_{\gamma_1, \gamma_2}$ has a greatest fixpoint,
and this greatest fixpoint is $\rho$-bisimilarity.
\end{corollary}
This result encompasses a similar result in \cite[Proposition 6]{BakDitHan17}
which states that $\Lambda$-bisimilarity for contingency logic
is a bisimulation.
\begin{example}\label{exm:Trhogamma-set}
We return to the classic setting of Example~\ref{exm:Trho-set}.
Let $(B, \pi_1, \pi_2)$ be a relation
between $\fun{T}$-coalgebras $(X_1, \gamma)$ and $(X_2, \gamma_2)$.
Then
\begin{align*}
\fun{T}^{\rho}_{\gamma_1, \gamma_2}B
&= \{ (x_1, x_2) \in X_1 \times X_2 \mid
(\gamma_1(x_1), \gamma_2(x_2)) \in \fun{T}^{\rho}B \}.
\end{align*}
Informally, $\fun{T}^{\rho}_{\gamma_1, \gamma_2}B$
consists of all pairs of worlds whose
one-step behaviours are indistiguishable by lifted $B$-coherent predicates.
\end{example}
\section{Distinguishing power}
\label{sec:disting}
In this section
we compare the distinguishing power of $\rho$-bisimulations
with that of other semantic equivalence notions, and with logical equivalence.
We make the same assumptions here as at the start of Section~\ref{sec:rho-bis}.
Given a cospan $(X_1,\gamma_1) \rightarrow (Y,\delta) \leftarrow (X_2,\gamma_2)$ in $\CoalgT$,
we call $(Y,\delta)$ a \emph{congruence} (of $\fun{T}$-coalgebras).
\subsection{Comparison with known equivalence notions}\label{subsec:other}
We briefly recall three coalgebraic equivalence notions, in descending order of
distinguishing power.
For more details, see e.g.~\cite[Definition~3.9]{BakHan17}.
\begin{definition}\label{def:equiv}
Let $\Bspan$ be a jointly mono span between $(X_1,\gamma_1)$ and $(X_2,\gamma_2)$.
Then $\Bspan$ is called a:
\begin{enumerate}
\item \emph{$\fun{T}$-bisimulation} if there is $t\colon B \to \fun{T}B$
such that the $\pi_i$ become coalgebra morphisms;
\item \emph{precocongruence}
if its pushout $\po{\pi}_1: X_1 \to \po{B} \la X_2 : \po{\pi}_1$
can be turned into a congruence between $(X_1,\gamma_1)$ and
$(X_2,\gamma_2)$, more precisely, if there is
$t\colon \po{B} \to \fun{T}\po{B}$ such that
$$
\begin{tikzcd}[row sep=0em]
& B \arrow[dl, bend right=5, "\pi_1" {swap, pos=.35}]
\arrow[dr, bend left=5, "\pi_2"]
& \\
X_1 \arrow[dr, bend right=3, "\po{\pi}_1" {swap, pos=.65}]
\arrow[dd, "\gamma_1" swap]
&
& X_2
\arrow[dl, bend left=3, "\po{\pi}_2"]
\arrow[dd, "\gamma_2"] \\ [-.3em]
& \po{B}
\arrow[dd, dashed, "t"]
& \\ [.3em]
\fun{T}X_1
\arrow[dr, bend right=5, "\fun{T}\po{\pi}_1" {swap,pos=.6}]
&
& \fun{T}X_2
\arrow[dl, bend left=5, "\fun{T}\po{\pi}_2"] \\ [-.2em]
& \fun{T}\po{B}
&
\end{tikzcd}
$$
commutes.
$\po{\pi}_1$ and $\po{\pi}_2$ become coalgebra morphisms;
\item \emph{behavioural equivalence} if it is a pullback in $\cat{C}$ of
some cospan $(X_1,\gamma_1) \rightarrow (Y,\delta) \leftarrow (X_2,\gamma_2)$
in $\CoalgT$.
\end{enumerate}
\end{definition}
When $\fun{T}$ preserves weak pullbacks, all three notions coincide
(when considering associated ``bisimilarity'' notions),
but in general, they may differ.
In particular, expressive logics can generally only
capture behavioural equivalence \cite{HanKupPac09}.
The next proposition can be proved in the same way as \cite[Proposition~3.10]{BakHan17}.
\begin{proposition}
\label{it:T-bis}
(i) Every $\fun{T}$-bisimulation is a $\rho$-bisimulation.
\label{it:preco}
(ii) Every precocongruence is a $\rho$-bisimulation.
\end{proposition}
The converse direction requires additional assumptions.
\begin{proposition}\label{prop:bis-preco}
Suppose $\cat{C}$ has pushouts, $\fun{P}$ is faithful, and either
\begin{itemize}
\item[(i)] $\rho$ is pointwise epic; or
\item[(ii)] $\mate{\rho}$ is pointwise monic and $\fun{T}$ preserves monos.
\end{itemize}
Then every $\rho$-bisimulation is a precocongruence.
If, in addition, $\fun{T}$ preserves weak pullbacks, then
$\rho$-bisimilarity coincides with
all three notions in Definition~\ref{def:equiv}.
\end{proposition}
\begin{proof}
Suppose
$
\begin{tikzcd}[sep=1.5em, cramped]
X_1
& B \arrow[l, "\pi_1" above]
\arrow[r, "\pi_2" above]
& X_2
\end{tikzcd}
$
is a $\rho$-bisimulation with pushout $(\po{B}, \po{\pi}_1, \po{\pi}_2)$ be the pushout.
We need to find a coalgebra structure $\zeta : \po{B} \to \fun{T}\po{B}$ which turns
$\po{\pi}_1$ and $\po{\pi}_2$ into coalgebra morphisms. It suffices to show that
$
\fun{T}\po{\pi}_1 \circ \gamma_1 \circ \pi_1 = \fun{T}\po{\pi}_2 \circ \gamma_2 \circ \pi_2,
$
because then the universal property of the pushout yields the desired $\zeta$.
If $\fun{P}$ is faithful and $\rho$ is pointwise epic, then it suffices to
prove that
$\fun{P}\pi_1 \circ \fun{P}\gamma_1 \circ \fun{PT}\po{\pi}_1 \circ \rho_{\po{B}}
= \fun{P}\pi_2 \circ \fun{P}\gamma_2 \circ \fun{PT}\po{\pi}_2 \circ \rho_{\po{B}}$.
This follows from the left diagram below, where the outer shell commutes
because $(B, \pi_1, \pi_2)$ is a $\rho$-bisimulation and the top two squares
commute by naturality of $\rho$.
$$
\begin{tikzcd}[row sep=0em]
& \fun{LP}\po{B}
\arrow[dl, bend right=5, "\fun{LP}\po{\pi}_1" {swap, pos=.25}]
\arrow[dr, bend left=5, "\fun{LP}\po{\pi}_2" {pos=.25}]
\arrow[dd, "\rho_{\po{B}}"]
&
& [1em]
& \fun{SLP}\po{B}
& \\
\fun{LP}X_1
\arrow[dd, "\rho_{X_1}" left]
&
& \fun{LP}X_2
\arrow[dd, "\rho_{X_2}" right]
& \fun{SLP}X_1
\arrow[ru, bend left=5, "\fun{SLP}\po{\pi}_1" {pos=.8}]
&
& \fun{SLP}X_2
\arrow[lu, bend right=5, "\fun{SLP}\po{\pi}_2" {swap,pos=.7}] \\
& \fun{PT}\po{B}
\arrow[dl, bend right=5, "\fun{PT}\po{\pi}_1" {swap, pos=.25}]
\arrow[dr, bend left=5, "\fun{PT}\po{\pi}_2" {pos=.25}]
&
&
& \fun{T}\po{B}
\arrow[uu, "\trans{\rho}_{\po{B}}"]
& \\
\fun{PT}X_1
\arrow[dd, "\fun{P}\gamma_1" left]
&
& \fun{PT}X_2
\arrow[dd, "\fun{P}\gamma_2" right]
& \fun{T}X_1
\arrow[ru, bend left=5, "\fun{T}\po{\pi}_1" {pos=.7}]
\arrow[uu, "\trans{\rho}_{X_1}" left]
&
& \fun{T}X_2
\arrow[lu, bend right=5, "\fun{T}\po{\pi}_2" {swap,pos=.6}]
\arrow[uu, "\trans{\rho}_{X_2}" right] \\
& \fun{P}\po{B}
\arrow[dl, bend right=5, "\fun{P}\po{\pi}_1" {swap, pos=.25}]
\arrow[dr, bend left=5, "\fun{P}\po{\pi}_2" {pos=.25}]
&
&
& \po{B}
& \\
\fun{P}X_1
\arrow[dr, bend right=5, "\fun{P}\pi_1" swap]
&
& \fun{P}X_2
\arrow[dl, bend left=5, "\fun{P}\pi_2"]
& X_1
\arrow[ru, bend left=5, "\po{\pi}_1" {pos=.7}]
\arrow[uu, "\gamma_1" left]
&
& X_2
\arrow[lu, bend right=5, "\po{\pi}_2" {swap,pos=.6}]
\arrow[uu, "\gamma_2" right]
\\
& \fun{P}B
&
&
& B
\arrow[ul, bend left=5, "\pi_1"]
\arrow[ur, bend right=5, "\pi_2" swap] & \\
\end{tikzcd}
$$
Alternatively, suppose $\fun{P}$ is faithful
(hence $\unitc\colon \Id_{\cat{C}} \to \fun{SP}$ is pointwise monic),
$\rho^{\flat}$ is pointwise monic and $\fun{T}$ preserves monos.
Then the transpose $\trans{\rho}_{\po{B}} : \fun{T}\po{B} \to \fun{SLP}\po{B}$
of $\rho_{\po{B}}$ is monic, because
$$
\trans{\rho}_{\po{B}}
= \fun{S}\rho_{\po{B}} \circ \unitc_{\fun{T}\po{B}}
= \rho^{\flat}_{\fun{P}\po{B}} \circ \fun{T}\unitc_{\po{B}},
$$
so it suffices to show that
$ \trans{\rho}_{\po{B}}
\circ \fun{T}\po{\pi}_1
\circ \gamma_1
\circ \pi_1
= \trans{\rho}_{\po{B}}
\circ \fun{T}\po{\pi}_2
\circ \gamma_2
\circ \pi_2.
$
But this follows from transposing the left diagram above, which yields
the diagram on the right.
When $\fun{T}$ preserves weak pullbacks, $\fun{T}$-bisimilarity coincides
with behavioural equivalence \cite{Rut00}, and hence also with the largest
precocongruence and $\rho$-bisimilarity.
\end{proof}
We note that condition (ii) in Proposition~\ref{prop:bis-preco}
entails that $(\fun{L},\rho)$ is expressive \cite[Thm.~4.2]{Kli07},
i.e., that logical equivalence implies behavioural equivalence.
In our abstract setting, \emph{logical equivalence} with respect to $(\fun{L},\rho)$ is
the kernel pair $(B,\pi,\pi')$ of the theory map $\th:X\to\fun{S}\Phi$.
Hence, $(\fun{L},\rho)$ is \emph{expressive} if $(B,\pi,\pi')$ is below
a behavioural equivalence in $\cat{Rel}(X,X)$.
\subsection{Hennessy-Milner type theorem}\label{subsec:hm}
We now prove a partial converse to Proposition~\ref{prop:adequacy} (truth-preservation).
We show that under certain conditions logical equivalence implies $\rho$-bisimilarity.
\begin{theorem}\label{thm:ex}\label{thm:hm}
Let
$
\begin{tikzcd}[cramped, sep=1.2em]
\cat{C'} \arrow[r, shift left=1.7pt]
& \cat{A'} \arrow[l, shift left=1.7pt]
\end{tikzcd}
$
be the dual equivalence induced by the dual adjunction
$
\begin{tikzcd}[cramped, sep=1.2em]
\cat{C} \arrow[r, shift left=1.7pt]
& \cat{A} \arrow[l, shift left=1.7pt]
\end{tikzcd}
$.
Suppose that
\begin{itemize}
\item $\cat{C}$ has $\RegEpiMono$-factorisations for morphisms with
domain $\in \cat{C'}$;
\item $\cat{C'}$ is closed under regular epimorphic images;
\item $\fun{S}$ is faithful and $\fun{L}$ preserves epis.
\end{itemize}
Then for all $\fun{T}$-coalgebras $(X, \gamma)$ with $X \in \cat{C'}$,
logical equivalence, i.e., the
kernel pair $(B, \pi, \pi')$ of $\th_{\gamma} : X \to \fun{S}\Phi$, is a $\rho$-bisimulation.
\end{theorem}
\begin{proof}
In order to prove that $(B, \pi, \pi')$ is a $\rho$-bisimulation, we need to show
that the outer shell of
\begin{equation}
\begin{tikzcd}[row sep=1.3em]
& \fun{L}\dpo{B}
\arrow[dl, bend right=20, "\fun{L}\ov{\pi}" swap]
\arrow[dr, bend left=20, "\fun{L}\ov{\pi}'"]
& \\
\fun{LP}X
\arrow[dd, "\gamma^*" swap]
& \fun{L}\Phi
\arrow[l, "\fun{L}\llb \cdot \rrb_{\gamma}"]
\arrow[r, "\fun{L}\llb \cdot \rrb_{\gamma}" swap]
\arrow[d, "\alpha"]
\arrow[u, dashed, "\fun{L}h"]
& \fun{LP}X \arrow[dd, "\gamma^*"] \\
& \Phi
\arrow[dl, "\llb \cdot \rrb_{\gamma}" swap]
\arrow[dr, "\llb \cdot \rrb_{\gamma}"]
\arrow[d, dashed, "h"]
& \\
\fun{P}X
\arrow[dr, bend right=15, "\fun{P}\pi" swap]
& \dpo{B}
\arrow[l, "\ov{\pi}"]
\arrow[r, "\ov{\pi}'" swap]
& \fun{P}X
\arrow[dl, bend left=15, "{\fun{P}\pi'}"]\\
& \fun{P}B
&
\end{tikzcd}
\end{equation}
commutes.
From $B$ being the kernel pair of $\th_{\gamma}$ we have that
$(\Phi, \llb \cdot \rrb_{\gamma}, \llb \cdot \rrb_{\gamma})$ is a cone
for the pullback $\dpo{B}$. Hence we get a morphism
$h : \Phi \to \dpo{B}$ such that the triangles left and right of $h$ commute,
and it is easy to see that all the inner squares and triangles in the diagram
on the right commute.
Thus, in order to show that the outer shell commutes, it suffices to show that
$\fun{L}h$ is epic.
By the assumption that $\fun{L}$ preserves epis,
it suffices to show that
$h : \Phi \to \dpo{B}$ is epic.
Let $m \circ e$ be the $\RegEpiMono$-factorisation of $\th_{\gamma}$. Then
the left diagram in \eqref{eq-t1-fact} commutes.
Since $m$ is monic the upper square is a pullback,
and by Lemma \ref{lem-pb-po}
it is also a pushout.
As a consequence, the lower square in the
right diagram of \eqref{eq-t1-fact}, obtained from dualising the left one, is a pullback.
\begin{equation}\label{eq-t1-fact}
\begin{tikzcd}[row sep=0em, cramped]
& B \arrow[dl, bend right=5, "\pi" swap]
\arrow[dr, bend left=5, "\pi'"]
& \\ [-0.3em]
X \arrow[dr, ->>, bend right=5, "e" {pos=.4}]
\arrow[ddr, bend right=20, "\th_{\gamma}" swap]
&
& X \arrow[dl, ->>, bend left=5, "e" {swap,pos=.4}]
\arrow[ddl, bend left=20, "\th_{\gamma}"] \\ [-.3em]
& A \arrow[d, hook, "m"]
& \\ [1.5em]
& \fun{S}\Phi
&
\end{tikzcd}
\qquad\quad
\begin{tikzcd}[row sep=0, cramped]
& \Phi
\arrow[d, ->, "h"]
\arrow[ddl, bend right=20, "\llb \cdot \rrb_{\gamma}" swap]
\arrow[ddr, bend left=20, "\llb \cdot \rrb_{\gamma}"]
& \\ [1.5em]
& \fun{P}A
\arrow[dl, hook, bend right=5, "\fun{P}e" {swap, pos=.4}]
\arrow[dr, hook, bend left=5, "\fun{P}e" {pos=.4}]
& \\ [-.3em]
\fun{P}X
\arrow[dr, hook, bend right=5, "\fun{P}\pi" swap]
&
& \fun{P}X
\arrow[dl, hook, bend left=5, "\fun{P}\pi'"] \\ [-.3em]
& \fun{P}B
&
\end{tikzcd}
\end{equation}
Here $h$ denotes the adjoint transpose of $m$.
Applying $\fun{S}$ to $h$ gives the morphism
$\fun{S}h : \fun{SP}A \to \fun{S}\Phi$ which by assumption is isomorphic to
$m$ (because $A \cong \fun{SP}A$). Since $\fun{S}$
is faithful and $m$ is monic, $h$ and therefore $\fun{L}h$ are epic.
\end{proof}
\begin{example}\label{exm:classic-hm}
In the classic case,
$
\begin{tikzcd}[cramped, sep=1.5em]
\cat{Set}
\arrow[r, shift left=1.7pt, ""]
& \cat{BA}
\arrow[l, shift left=1.7pt, ""]
\end{tikzcd}
$
restricts to the full duality between finite sets and finite Boolean algebras.
$\cat{Set}$ has $\RegEpiMono$-factorisations \cite[Example~14.2(2)]{AdaHerStr90}.
In $\cat{Set}$ and $\cat{BA}$, all epis are regular and coincide with surjections \cite{AdaHerStr90,Ban10},
and finite sets are closed under surjective images.
The ultrafilter functor $\fun{S}$ is faithful.
If the logic functor $\fun{L}$ is given by predicate liftings and relations,
then by \cite[Remark 4.10]{KurRos12} it preserves regular epis,
and since all epis are regular, $\fun{L}$ preserves epis.
Applying Theorem~\ref{thm:hm}, we recover \cite[Theorem 4.5]{BakHan17}, and thereby all examples given there.
In particular,
taking $(\fun{L},\rho)$ to be Hennessy-Milner logic
(Example~\ref{ex:LTS-HM-logic}), then
we recover from Theorem~\ref{thm:hm} that over finite labelled transition systems,
logical equivalence implies $\rho$-bisimilarity for Hennessy-Milner logic.
\end{example}
\begin{remark}\label{exm:posml--not-hm}
For positive modal logic from Examples~\ref{exm:pml-top} and \ref{exm:pml-top-bis},
we have not been able to show that the logic functor
$\fun{N} : \cat{DL} \to \cat{DL}$ preserves epis.
\end{remark}
\begin{example}\label{exm:hm-vec}
We return to linear Hennessy-Milner logic from
Examples~\ref{exm:mod-vec} and \ref{exm:mod-vec-bis}.
The dual adjunction
$
\begin{tikzcd}[cramped, sep=1.2em]
\cat{Vec}_{\Bbbk} \arrow[r, shift left=1.7pt]
& \cat{Vec}_{\Bbbk} \arrow[l, shift left=1.7pt]
\end{tikzcd}
$
restricts to the well-known self-duality of finite-dimensional vector spaces
$\cat{FinVec}_{\Bbbk}$.
The category $\cat{Vec}_{\Bbbk}$ has
$\RegEpiMono$-factorisations \cite[Example~14.2]{AdaHerStr90}
and the regular epis in both $\cat{Vec}_{\Bbbk}$ and $\cat{FinVec}_{\Bbbk}$
are the surjections \cite[Example~7.72]{AdaHerStr90}.
Moreover, the surjective image of a finite-dimensional vector space is
again fi\-nite-dimensional, and the functor $(-)^{\vee}$ is faithful.
Finally, since $\fun{L}$ is generated by
predicate liftings and axioms it preserves surjections,
so we can apply Theorem \ref{thm:hm} to conclude that
logical equivalence and $\rho$-bisimilarity coincide
on $\fun{W}$-coalgebras state-spaces in $\cat{FinVec}_{\Bbbk}$.
\end{example}
\begin{example}\label{exm:trace-not-hm}
An example where logical equivalence does not imply $\rho$-bisimilarity
is given by trace logic for labelled transitions systems (Example~\ref{ex:LTS-trace-logic}).
The conditions for Theorem~\ref{thm:hm} hold for trace logic, but
the induced dual equivalence is in this case trivial, i.e., $\cat{C'}$ and $\cat{A}'$
are the empty category, hence Theorem~\ref{thm:hm} does not tell us anything.
\end{example}
\subsection{Invariance under translations}\label{subsec:trans}
In this section we assume that $\cat{C}$ has pushouts.
The example of Hennessy-Milner logic (Example~\ref{ex:LTS-HM-logic}) and
trace logic (Examples~\ref{ex:LTS-trace-logic}~and~\ref{exm:trace-not-hm})
is a situation where one logic is a reduct of the other.
This can be considered a special case of translating a logic into another.
We will show under which conditions $\rho$-bisimilarity is preserved under translations.
To make this formal, we first generalise \cite[Definition 4.1]{KurLea12}.
\begin{definition}
Assume we are given a ``triangle situation'' as in diagram (\ref{eq:triangle}(a))
such that $\fun{P} = \fun{UP'}$,
and we have modal semantics $\rho' : \fun{L'P'} \to \fun{P'T}$ and $\rho : \fun{LP} \to \fun{PT}$.
A \emph{translation} from $(\fun{L}',\rho')$ to $(\fun{L},\rho)$ is a natural transformation
$\tau : \fun{LP} \to \fun{UL'P'}$ such that $\rho = \fun{U}\rho' \circ \tau$,
see diagram (\ref{eq:triangle}(b)).
\begin{equation}\label{eq:triangle}
\begin{tikzcd}[row sep=.3em]
& [1em]
\cat{A'}
\arrow[dd, shift left=3pt, bend left=9, "\fun{U}"]
\arrow[dd, phantom, "\dashv"]
\arrow[dd, <-, shift right=3pt, bend right=9, "\fun{F}" swap]
\arrow[loop right, "\fun{L'}"]
& [2em]
\fun{LP}
\arrow[dd, "\rho"]
\arrow[r, "\tau"]
& \fun{UL'P'}
\arrow[dd, "\fun{U}\rho'"]
& [.5em]
\fun{FLP}
\arrow[r, "\trans{\tau}"]
\arrow[dd, "\fun{F}\rho" swap]
& \fun{L'P'}
\arrow[dd, "\rho'"]
\\
\cat{C}
\arrow[ru, "\fun{P'}"]
\arrow[rd, "\fun{P}" swap]
\arrow[loop left, "\fun{T}"]
&
&
&
&
& \\
& \cat{A}
\arrow[loop right, "\fun{L}"]
& \fun{PT}
\arrow[r, "="]
& \fun{UP'T}
& \fun{FPT}
\arrow[r, "\epsilon_{\fun{P}'\fun{T}}"]
& \fun{P'T} \\ [.8em]
{} \arrow[r, phantom, "\text{(a)}"]
& {}
& {} \arrow[r, phantom, "\text{(b)}"]
& {}
& {} \arrow[r, phantom, "\text{(c)}"]
& {}
\end{tikzcd}
\end{equation}
\end{definition}
\vspace{-.2em}\noindent
In (c), $\epsilon$ is the counit of $\fun{F} \dashv \fun{U}$
(which is adjoint to the identity) because $\fun{P} = \fun{UP'}$,
and $\trans{\tau}$ is the ($\fun{F} \dashv \fun{U}$)-adjoint of $\tau$.
\begin{proposition}\label{prop:trans-easy}
Suppose $\tau$ is a translation from $\rho'$ to $\rho$
and $(B, \pi_1, \pi_2)$ is a $\rho'$-bisimulation.
Then it is also a $\rho$-bisimulation.
\end{proposition}
\begin{proof}
Let $(\hat{B}, \hat{\pi}_1, \hat{\pi}_2)$ be the pushout of $(B, \pi_1, \pi_2)$.
Since $B$ is assumed to be a $\rho'$-bisimulation the diagram on the left
commutes.
$$
\begin{tikzcd}[row sep=0em, column sep=2.2em]
&
&
& [2em]
& \fun{LP}\hat{B}
\arrow[dl, "\fun{LP}\hat{\pi}_1" swap]
\arrow[dr, "\fun{LP}\hat{\pi}_2"]
\arrow[dd, "\tau_{\hat{B}}"]
& \\
& \fun{L'P'}\hat{B}
\arrow[dl, "\fun{L'P'}\hat{\pi}_1" swap]
\arrow[dr, "\fun{L'P'}\hat{\pi}_2"]
&
& \fun{LP}X_1
\arrow[dd, "\tau_{X_1}" swap]
\arrow[dddd, bend right=60, shift right=5pt, "\rho_{X_1}" swap]
&
& \fun{LP}X_2
\arrow[dd, "\tau_{X_2}"]
\arrow[dddd, bend left=60, shift left=5pt, "\rho_{X_2}"] \\
\fun{L'P'}X_1
\arrow[dd, "\rho'_{X_1}" swap]
&
& \fun{L'P'}X_2
\arrow[dd, "\rho'_{X_2}"]
&
& \fun{UL'P'}\hat{B}
\arrow[dl, "\fun{UL'P'}\hat{\pi}_1" {swap, pos=.3}]
\arrow[dr, "\fun{UL'P'}\hat{\pi}_2" pos=.3]
& \\
&
&
& \fun{UL'P'}X_1
\arrow[dd, "\fun{U}\rho'_{X_1}"]
&
& \fun{UL'P'}X_2
\arrow[dd, "\fun{U}\rho'_{X_2}"] \\
\fun{P'T}X_1
\arrow[dd, "\fun{P'}\gamma_1" swap]
&
& \fun{P'T}X_2
\arrow[dd, "\fun{P'}\gamma_2"]
&
&
& \\
& \fun{P'}\hat{B}
\arrow[dl, "\fun{P'}\hat{\pi}_1" swap]
\arrow[dr, "\fun{P'}\hat{\pi}_2"]
&
& \fun{PT}X_1
\arrow[dd, "\fun{P}\gamma_1" swap]
&
& \fun{PT}X_2
\arrow[dd, "\fun{P}\gamma_2"] \\
\fun{P'}X_1
\arrow[dr, "\fun{P'}\pi_1" swap]
&
& \fun{P'}X_2
\arrow[dl, "\fun{P'}\pi_2"]
&
& \fun{P}\hat{B}
\arrow[dl, "\fun{P}\hat{\pi}_1" swap]
\arrow[dr, "\fun{P}\hat{\pi}_2"]
& \\
& \fun{P'}B
&
& \fun{P}X_1
\arrow[dr, "\fun{P}\pi_1" swap]
&
& \fun{P}X_2
\arrow[dl, "\fun{P}\pi_2"] \\
&
&
&
& \fun{P}B
&
\end{tikzcd}
$$
Applying $\fun{U}$ to this diagram and putting the translation
$\tau$ on top then yields the right diagram (using that $\fun{P} = \fun{UP'}$),
and this proves that $(B, \pi_1, \pi_2)$ is a $\rho$-bisimulation.
\end{proof}
A sufficient condition for the converse is that the transpose $\trans{\tau}$ of $\tau$ is epic,
see diagram (\ref{eq:triangle}(c)).
Note that due to the adjunction $\fun{F} \dashv \fun{U}$, diagram (b) commutes if and only if (c) does.
Intuitively, $\trans{\tau} : \fun{FLP} \to \fun{L'P'}$ being epic formalises that
every modality in $\fun{L}'$ is a propositional combination of a modal formula of $\fun{L}$.
\begin{proposition}\label{prop:tau-flat}
Suppose that $\trans{\tau}$ is pointwise epic. Then every $\rho$-bisimulation
is a $\rho'$-bisimulation.
\end{proposition}
\begin{proof}
Commutativity of the outer shell of the following diagram will prove that
$
\begin{tikzcd}[cramped, sep=1.5em]
X_1
& B \arrow[l, swap, "\pi_1"] \arrow[r, "\pi_2"]
& X_2
\end{tikzcd}
$
is a $\rho'$-bisimulation:
$$
\begin{tikzcd}[row sep=.1em, column sep=large]
&
& \fun{L'P'}\po{B}
\arrow[dl, "\fun{L'U}\po{\pi}_1" swap]
\arrow[dr, "\fun{L'U}\po{\pi}_2"]
&
& \\ [-.6em]
& \fun{L'P'}X_1
\arrow[ddddl, bend right=30, "\rho'_{X_1}" swap]
\arrow[ddddl, phantom, bend left=5, "\circled{1}"]
\arrow[dr, phantom, "\circled{2}"]
&
& \fun{L'P'}X_2
\arrow[ddddr, bend left=30, "\rho'_{X_2}"]
\arrow[ddddr, phantom, bend right=5, "\circled{4}"]
\arrow[dl, phantom, "\circled{3}"]
& \\ [.6em]
&
& \fun{FLP}\po{B}
\arrow[uu, "\trans{\tau}_{\po{B}}"]
\arrow[dl, "\fun{FLP}\po{\pi}_1"]
\arrow[dr, "\fun{FLP}\po{\pi}_2" swap]
\arrow[dddd, phantom, "\circled{5}" {pos=.65}]
&
& \\ [-.6em]
& \fun{FLP}X_1
\arrow[uu, "\trans{\tau}_{X_1}"]
\arrow[dd, "\fun{F}\rho_{X_1}" swap]
&
& \fun{FLP}X_2
\arrow[uu, "\trans{\tau}_{X_2}" swap]
\arrow[dd, "\fun{F}\rho_{X_2}"]
& \\
&
&
&
& \\ [15pt]
\fun{P'T}X_1
\arrow[ddddr, bend right=30, "\fun{P'}\gamma_1" swap]
\arrow[ddr, phantom, bend right=20, "{\circled{7}}" {pos=.65}]
& \fun{FPT}X_1
\arrow[dd, "\fun{FP}\gamma_1" swap]
\arrow[l, "\epsilon_{\fun{P'T}X_1}"]
&
& \fun{FPT}X_2
\arrow[dd, "\fun{FP}\gamma_2"]
\arrow[r, "\epsilon_{\fun{P'T}X_2}" swap]
& \fun{P'T}X_2
\arrow[ddddl, bend left=30, "\fun{P'}\gamma_2"]
\arrow[ddl, phantom, bend left=20, "{\circled{10}}" {pos=.65}] \\ [-.8em]
&
& \fun{FP}\po{B}
\arrow[dl, bend right=5, "\fun{FP}\po{\pi}_1" {swap,pos=.25}]
\arrow[dr, bend left=5, "\fun{FP}\po{\pi}_2" {pos=.3}]
\arrow[dd, phantom, "\circled{6}"]
&
& \\ [.8em]
& \fun{FP}X_1
\arrow[dd, "\epsilon_{\fun{P'}X_1}" swap]
\arrow[dr, "\fun{FP}\pi_1" swap]
&
& \fun{FP}X_2
\arrow[dd, "\epsilon_{\fun{P'}X_1}"]
\arrow[dl, "\fun{FP}\pi_2"]
& \\ [-.6em]
&
& \fun{FP}B
\arrow[dd, "\epsilon_{\fun{P'B}}"]
&
& \\ [.6em]
& \fun{P'}X_1
\arrow[dr, "\fun{P'}\pi_1" swap]
\arrow[ur, phantom, bend right=6, "\circled{8}"]
&
& \fun{P'}X_2
\arrow[dl, "\fun{P'}\pi_2"]
\arrow[ul, phantom, bend left=6, "\circled{9}"]
& \\ [-.6em]
&
& \fun{P'}B
&
&
\end{tikzcd}
$$
Cells 1 and 4 commute by diagram (c) in \eqref{eq:triangle},
and cells 2 and 3 by naturality of $\trans{\tau}$.
Commutativity of 5 and 6 together follows from applying $\fun{F}$ to the diagram
witnessing the fact that
$
\begin{tikzcd}[cramped, sep=1.5em]
X_1
& B \arrow[l, swap, ""] \arrow[r, ""]
& X_2
\end{tikzcd}
$
is a $\rho$-bisimulation.
Commutativity of the remaining cells follows from the naturality
of the counit $\epsilon$.
\end{proof}
As a first example, we give a translation between Hennessy-Milner logic
and trace logic.
\begin{example}\label{exle:triangle-hm-trace}
Recall trace logic (Example~\ref{ex:LTS-trace-logic}) and Henessy-Milner
logic (Example~\ref{ex:LTS-HM-logic}) for LTSs.
Filling in the categories and functors in diagram (\ref{eq:triangle}(a)) we get:
\begin{equation}\label{eq:triangle-hm-pos}
\begin{tikzcd}[row sep=1em]
& [1em]
\cat{BA}
\arrow[dd, shift left=3pt, bend left=9, "\fun{U}"]
\arrow[loop right, "\Lhm"]
\arrow[dd, <-, shift right=3pt, bend right=9, "\fun{F}" swap]
\arrow[dd, phantom, "\dashv"]
\arrow[dl, bend right=11, shift right=0pt, "\Uf" {pos=.7}] \\
\cat{Set}
\arrow[ru, bend left=14, shift left=4pt, "\Pba"]
\arrow[rd, bend right=11, shift right=0pt, "\cPow" {pos=.3}]
\arrow[loop left, "\fun{T}"]
& \\
& \cat{Set}
\arrow[loop right, "\Ltr"]
\arrow[lu, bend left=14, shift left=4pt, "\cPow"]
\end{tikzcd}
\end{equation}
We can define a translation $\tau : \Ltr \cPow \to \fun{U}\Lhm\Pow$
by
$$
\tau_X
: \Ltr \cPow X \to \fun{U}\Lhm\Pow X
: \left\{ \begin{array}{rl}
1 &\!\mapsto\, \top_{\Lhm\Pow X} \\
(a, b) &\!\mapsto\, \langle a \rangle b
\end{array}\right.
$$
Here $a \in A$, the set of labels, and $b \in \cPow X$.
Then $\trans{\tau} : \fun{F}\Ltr\cPow X \to \Lhm\Pba X$ is surjective
because each generator $\langle a \rangle b$ of $\Lhm\Pba X$
is seen by some element in $\fun{F}\Ltr\cPow X$.
Concretely, this is the case
because formulae of Hennessy-Milner logic are precisely the Boolean
combinations of trace logic formulae.
Hence, in particular, $\trans{\tau}$ has surjective components,
and in $\cat{BA}$ epis are the surjective Boolean homomorphisms.
It now follows from Proposition~\ref{prop:tau-flat} that a
$\rhotr$-bisimulation is a $\rhohm$-bisimulation (and the converse also holds).
\end{example}
In the setting of Examples~\ref{ex:LTS-HM-logic}, \ref{exm:pl} and \ref{exm:rho-bis-lambda},
where $\cat{A'}$ is a variety of algebras
and the logic $(\fun{L}, \rho)$ is given by predicate liftings and axioms,
we can consider the special case of \eqref{eq:triangle}
where $(\fun{L},\rho)$ is the ``modal reduct'' of $(\fun{L}', \rho')$.
\begin{example}\label{exm:translation-Var}
Suppose $\cat{A}$ is a variety of algebras with
free-forgetful adjunction $\fun{F} \dashv \fun{U}$. Let
$(\fun{L}, \rho)$ be a logic for $\fun{T}$-coalgebras given by a collection $\Lambda$
of predicate liftings and axioms (Example~\ref{exm:pl}).
Then we can define $\fun{P_0} = \fun{U} \circ \fun{P}$,
which has dual adjoint $\fun{S_0} = \fun{SF}$,
where $\fun{S}$ is the dual adjoint of $\fun{P}$.
Define the logic functor $\fun{L_0} : \cat{Set} \to \cat{Set}$ by
$\fun{L_0}X = \{ \und{\lambda}_0(a_1, \ldots, a_n) \mid \lambda \in \Lambda, a_i \in X \}$
and $\fun{L_0}f(\und{\lambda}_0(a_1, \ldots, a_n)) = \und{\lambda}_0(fa_1, \ldots, fa_n)$.
\begin{equation}
\begin{tikzcd}[row sep=.2em]
& \cat{A}
\arrow[dd, shift left=3pt, bend left=9, "\fun{U}"]
\arrow[dd, phantom, "\dashv"]
\arrow[dd, <-, shift right=3pt, bend right=9, "\fun{F}" swap]
\arrow[loop right, "\fun{L}"] \\
\cat{C}
\arrow[ru, "\fun{P}"]
\arrow[rd, "\fun{P_0}" {swap,pos=.6}]
\arrow[loop left, "\fun{T}"]
& \\
& \cat{Set}
\arrow[loop right, "\fun{L_0}"]
\end{tikzcd}
\end{equation}
Define $\tau : \fun{L_0P_0} \to \fun{ULP}$ by
$\tau_X(\und{\lambda}_0(a_1, \ldots, a_n)) = \und{\lambda}(a_1, \ldots, a_n) \in \fun{ULP}X$.
The logic $(\fun{L}, \rho)$ gives rise to the logic $(\fun{L_0}, \rho_0)$, where
$\rho_0 = \fun{U}\rho \circ \tau : \fun{L_0P_0} \to \fun{P_0T}$.
Then $\tau$ is a translation.
One can verify that, in this situation, $\trans{\tau}$ is
pointwise epic. Therefore a jointly mono span
$
\begin{tikzcd}[cramped, sep=1.5em]
X_1
& B \arrow[l, swap, ""] \arrow[r, ""]
& X_2
\end{tikzcd}
$
in $\cat{C}$ is a $\rho$-bisimulation if and only if it is a $\rho_0$-bisimulation.
Hence it suffices to look at the underlying sets when verifying
whether a jointly mono span is a $\rho$-bisimulation.
\end{example}
\begin{example}\label{exm:trans-vec}
If we apply the procedure from Example \ref{exm:translation-Var}
to linear Hennessy-Milner logic (Example~\ref{exm:mod-vec}) we precisely get
linear trace logic (Example~\ref{exm:trace-vec}).
$$
\begin{tikzcd}[row sep=.2em]
& \cat{Vec}_{\Bbbk}
\arrow[dd, shift left=3pt, bend left=9, "\fun{U}"]
\arrow[dd, phantom, "\dashv"]
\arrow[dd, <-, shift right=3pt, bend right=9, "\fun{F}" swap]
\arrow[loop right, "\fun{L}"] \\
\cat{Vec}_{\Bbbk}
\arrow[ru, "(-)^{\vee}"]
\arrow[rd, "(-)^{\circ}" {swap,pos=.6}]
\arrow[loop left, "\fun{W}"]
& \\
& \cat{Set}
\arrow[loop right, "\fun{L_0}"]
\end{tikzcd}
$$
Thus a span relation between two vector spaces
(i.e., a linear subspace of $X_1 \times X_2$)
is a linear trace logic bisimulation if and only if it is a
linear Hennessy-Milner logic bisimulation.
We can use this to transfer the Hennessy-Milner result from Example
\ref{exm:hm-vec} for vector space logic with all vector space operators
interpreted in finite linear weighted automata to the setting
of linear trace logic.
This relies on the fact that logical equivalence with respect to
linear trace logic implies logical equivalence with respect to linear
Hennessy-Milner logic. To see this, note that, using the axioms,
we can rewrite any formula in linear HM logic to an equivalent formula
of the form
$$
\phi ::= p \mid r \cdot \phi \mid \phi + \phi \mid \llangle a \rrangle p
$$
where $r \in \Bbbk$ and $\llangle a \rrangle$ is a finite sequence of the form
$\langle a_1 \rangle\langle a_2 \rangle \cdots \langle a_n \rangle p$,
with $a_i \in A$.
Intuitively, this is the case
because modalities are linear,
and because we can view $0$ as shorthand for $0_{\Bbbk} \cdot p$.
Now suppose two states $x, x'$ satisfy the same linear trace logic formulae,
then we have $x \Vdash p$ iff $x' \Vdash p$ and
$x \Vdash \llangle a \rrangle p$ iff $x' \Vdash \llangle a \rrangle p$.
Since the interpretation of $r \cdot \phi$ and $\phi + \phi$ is computed
pointwise, this implies that $x$ and $x'$ satisfy the same linear Hennessy-Milner
logic formulae.
As a consequence logical equivalence with respect to trace logic
implies the existence of a bisimulation for linear Hennessy-Milner logic
linking $x$ and $x'$, which in turn is also a linear trace logic bisimulation.
\end{example}
Finally, we compare several logics that can be interpreted
in topological spaces.
\begin{example}
We squeeze the topological semantics for positive modal logic
from Example \ref{exm:pml-top} between two
other logics with varying base logics as in the following diagram:
\begin{equation}
\begin{tikzcd}[row sep=1.5em]
& \cat{Frm}
\arrow[d, "\fun{U'}"]
\arrow[loop right, "\fun{N'}"] \\
\cat{Top}
\arrow[ru, "\fun{\Omega'}", bend left=22]
\arrow[r, "\fun{\Omega}"]
\arrow[rd, "\fun{\Omega_0}" swap, bend right=22]
\arrow[loop left, "\fun{V}"]
& \cat{DL}
\arrow[d, "\fun{U}"]
\arrow[loop right, "\fun{N}"] \\
& \cat{Set}
\arrow[loop right, "\fun{N_0}"]
\end{tikzcd}
\end{equation}
Here $\cat{Frm}$ is the category of frames and
$\fun{\Omega}' : \cat{Top} \to \cat{Frm}$ is the functor that
sends a topological space to its frame of opens. Let
$\fun{N'} : \cat{Frm} \to \cat{Frm}$ be the functor given as in
\cite[Section III4.3]{Joh82} (known also as the \emph{Vietoris locale})
and define $\rho' : \fun{N'\Omega'} \to \fun{\Omega'V}$
on generators by $\Box a \mapsto \dbox a$ and $\Diamond a \mapsto \ddiamond a$.
The translation $\tau : \fun{N\Omega} \to \fun{U'N'\Omega'}$ given by
$\Box a \mapsto \Box a$ and $\Diamond a \mapsto \Diamond a$
is such that $\trans{\tau}$ is epic,
thus satisfies the assumptions of Proposition \ref{prop:tau-flat}.
The bottom triangle is an instance of
Example~\ref{exm:translation-Var}.
We conclude that a jointly mono span
$
\begin{tikzcd}[cramped, sep=1.5em]
X_1
& B \arrow[l, swap, ""] \arrow[r, ""]
& X_2
\end{tikzcd}
$
between $\fun{V}$-coalgebras $(X_1, \gamma_1)$ and $(X_2, \gamma_2)$ is a $\rho$-bisimulation
if and only if it is a $\rho'$-bisimulation if and only if it
is a $\rho_0$-bisimulation.
\end{example}
\section{Conclusion}
\label{sec:conc}
Our main question was whether we can characterise logical equivalence
for (possibly non-expressive) coalgebraic logics by a notion of bisimulation.
Towards this goal, we
generalised the logic-induced bisimulations in \cite{BakHan17}
for coalgebraic logics for $\cat{Set}$-coalgebras
to coalgebraic logics parameterised by a dual adjunction.
We identified sufficient conditions for when logical equivalence
coincides with logic-induced bisimilarity (Thm.~\ref{thm:ex}).
These are conditions on the categories in the dual adjunction,
and \emph{not} on the natural transformation $\rho$ defining (the semantics of)
the logic. In particular, we do not require the logic to be expressive.
We found that the distinguishing power of $\rho$-bisimulations depends on the modalities of the language but not on the propositional connectives. More generally, we showed that certain translations between logics preserve $\rho$-bisimilarity (Prop.~\ref{prop:tau-flat}).
Furthermore, as in the expressivity result of \cite{Kli07}, $\rho$-bisimilarity agrees with behavioural equivalence if the mate of $\rho$ is pointwise monic (Prop.~\ref{prop:bis-preco}). However, Example~\ref{exle:triangle-hm-trace} shows that this is not a necessary condition which raises the question whether one can characterise, purely in terms of $\rho$, when $\rho$-bisimilarity coincides with behavioural equivalence.
There are many other avenues for further research.
When is a congruence on complex algebras induced by a $\rho$-bisimulation?
Can we drop
in Theorem~\ref{thm:hm} the restriction to the subcategory if $\fun T$ is finitary?
Can we take quotients with respect to (the largest) $\rho$-bisimulation on a $\fun{T}$-coalgebra?
Moreover, the definition of $\rho$-bisimulation has a natural generalisation
to the order-enriched setting. This gives rise to \emph{$\rho$-simulations}.
Can one prove an ordered Hennessy-Milner theorem where ``logical inequality'' is recognised by $\rho$-simulations?
Since this question naturally falls into the realm of order-enriched category theory, we will also seek a generalisation to the quantale-enriched setting, accounting for metric versions of simulation.
\bibliographystyle{plain}
\bibliography{bib-modal-bisim.bib}
\appendix
\section{Appendix}
\subsection{Some lemmas}
\begin{lemma}\label{lem-pb-split}
Pullbacks preserve split epimorphisms.
\end{lemma}
\begin{proof}
Suppose
$$
\begin{tikzcd}[row sep=1.4em, cramped]
B' \arrow[rr, bend left=20, "\bar{g}"]
\arrow[dr, bend right=10, "\id" swap]
\arrow[r, dashed, "g^*" swap]
& A' \arrow[r, "v" swap] \arrow[d, "g"] & A \arrow[d, "f"] \\
& B' \arrow[r, "w"] & B
\end{tikzcd}
$$
is a pullback square and $f$ is split epic.
Define $\bar{g} : B' \to A$ by $\bar{g} = f^*w$.
Then we have $w = ff^*w = f\bar{g}$ hence a cone of the pullback.
The fill-in $g^*$ satisfies $gg^* = \id_{B'}$, hence $g$ is split epic.
\end{proof}
\begin{lemma}\label{lem-pb-po}
Let
\begin{equation}\label{eq-sq-pb-po}
\begin{tikzcd}[row sep=1.4em]
A' \arrow[r, "v"]
\arrow[d, "f'"]
& A \arrow[d, "f"] \\
B' \arrow[r, "w"]
& B
\end{tikzcd}
\end{equation}
be a pullback square such that $w, f$ are regular epic and
$v, f'$ are split epic. Then the square is also a pushout.
\end{lemma}
\begin{proof}
Note that every split epi is regular.
Denote by $w_1, w_2$ the kernel pair of $w$
(note that $w$ is the coequalizer of this pair) and similar for the other maps.
Then we get the following diagram:
$$
\begin{tikzcd}[row sep=1.4em, cramped]
& C'
\arrow[r, dashed, "u"]
\arrow[d, shift right=2pt, "f_1'" left]
\arrow[d, shift left=2pt, "f_2'" right]
& C \arrow[d, shift right=2pt, "f_1" left]
\arrow[d, shift left=2pt, "f_2" right] \\
A'' \arrow[d, dashed, "f''" left]
\arrow[r, shift left=2pt, "v_1" above]
\arrow[r, shift right=2pt, "v_2" below]
& A'
\arrow[d, "f'"]
\arrow[r, "v"]
& A \arrow[d, "f"] \\
B''
\arrow[r, shift left=2pt, "w_1" above]
\arrow[r, shift right=2pt, "w_2" below]
& B'
\arrow[r, "w"]
& B
\end{tikzcd}
$$
The dashed arrow $f''$ can be obtained as the pullback of
$ww_1 (= ww_2)$ and $vv_1 (= vv_2)$,
and makes the horizontal coequalizers commute. (Similarly for $u$.)
Moreover, $f''$ and $u$ are (split) epis by Lemma \ref{lem-pb-split}.
Now let $h : B' \to D$ and $k : A \to D$ be such that $hf' = kv$.
Then we have
$$
hw_1f'' = hf'v_1 = kvv_1 = kvv_2 = hf'v_2 = hw_2f''
$$
and since $f''$ is (split) epic it follows that $hw_1 = hw_2$.
Since $w, w_1, w_2$ form a coequalizer, there exists $t : B \to D$
such that $tw = h$.
In a similar way we obtain $t' : B \to D$ such that $t'f = k$.
Since $wf'$ is epic, it follows that $t = t'$.
In general, unicity of $t$ follows from $wf'$ being epic.
This proves that the square in \eqref{eq-sq-pb-po} is indeed a pushout square.
\end{proof}
\end{document}
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TITLE: Non-isomorphic graphs with isomorphic edge vectors
QUESTION [8 upvotes]: Let $G$ and $H$ be graphs on the vertex set $\{1, \ldots, n\}$ and let $(e_i)$ be the standard basis of $\mathbb{R}^n$. For each edge $\{i,j\}$ define edge vectors $e_i - e_j$ and $e_j - e_i$ in $\mathbb{R}^n$.
Question 1: If there is a linear isomorphism of $\mathbb{R}^n$ with itself that takes the edge vectors of $G$ bijectively onto the edge vectors of $H$, must $G$ and $H$ be isomorphic?
The answer to this question is no: if $G$ and $H$ are both trees then there is such a linear isomorphism. Aside from $e_i - e_j$ being the negation of $e_j - e_i$, there are no linear dependences among the edge vectors, and the edge vectors of $G$ can be mapped to the edge vectors of $H$ in any manner.
Question 2: Same as Question 1, but now assuming that $G$ and $H$ both have central vertices, i.e., each of them has a vertex which is adjacent to every other vertex.
I assumed a counterexample to Question 1 would easily yield a counterexample to Question 2, but I don't see this. A counterexample to Question 2 is what I need.
REPLY [2 votes]: Attempt to show that there is no example for Question 2.
Let $G$ be a graph with central vertex $v_0$, $H$ be a graph with central vertex $u_0$ and cycle structures (cyclic matroids) of $G$ and $H$ are isomorphic. I claim that $H$ and $G$ themselves are isomorphic as graphs. Let $T$ be a spanning tree in $G$ formed by edges incident to $v_0$. It corresponds to some spanning tree $f(T)$ in $H$, here $f$ is an isomorphism of matroids (so, $f$ is defined on edges of $G$). Note that in $G$ any edge $e\notin T$ belongs to a triangle with two edges from $T$. Thus the same holds in $H$. Apply this to edges in $H$ incident to $u_0$ but not coming from $u_0$. We see that maximal path in $f(T)$ going from $u_0$ consists at most two edges. Let $u_0u_1,\dots,u_0u_k$ be edges incident to $u_0$ and belonging to $f(T)$, $v_0v_i$ be their $f$-preimages. Next, if $u_i$, $1\leqslant i\leqslant k$, is incident to some edge $u_iu_m\in f(T)$, $m>k$, then denote by $v_0v_m$ $f$-preimage of the edge $u_0u_m\in H$. Then $v_i\rightarrow u_i,i=0,1,\dots$ is isomorphism of $G$ and $H$.
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\begin{document}
\begin{abstract} Let $m$ be a positive integer and $A$ an elementary abelian group of order $q^r$ with $r\geq 2$ acting on a finite $q'$-group $G$. We show that if for some integer $d$ such that $2^{d}\leq r-1$ the $d$th derived group of $C_{G}(a)$ has exponent dividing $m$ for any $a \in A^{\#}$, then $G^{(d)}$ has $\{m,q,r\}$-bounded exponent and if $\gamma_{r-1}(C_G(a))$ has exponent dividing $m$ for any $a\in A^\#$, then $\gamma_{r-1}(G)$ has $\{m,q,r\}$-bounded exponent.
\end{abstract}
\maketitle
\section{Introduction}
\label{introduction}
Let $A$ be a finite group acting coprimely on a finite
group $G$. It is well known that the structure of
the centralizer $C_G(A)$ (the fixed-point subgroup) of
$A$ has strong influence over the structure of $G$.
To exemplify this we mention the following results.
The celebrated theorem of Thompson \cite{T} says that
if $A$ is of prime order and $C_G(A)=1$, then $G$ is
nilpotent. On the other hand, any nilpotent
group admitting a fixed-point-free automorphism of prime
order $q$ has nilpotency class bounded by some
function $h(q)$ depending on $q$ alone. This result
is due to Higman \cite{Higman}. The reader can find in
\cite{Khu1} and \cite{Khu2} an account on the more recent
developments related to these results.
The next result is a consequence of the classification
of finite simple groups \cite{Wang}: If $A$ is a group
of automorphisms of $G$ whose order is coprime to that
of $G$ and $C_G(A)$ is nilpotent or has odd order,
then $G$ is soluble. Once the group $G$ is known to be soluble, there is
a wealth of results bounding the Fitting height of $G$ in terms of the order of $A$ and the Fitting height of $C_G(A)$. This direction of research was started by Thompson in \cite{thompson2}. The proofs mostly use representation theory in the spirit of the Hall-Higman work \cite{HH}. A general discussion of these methods and their use in numerous fixed-point theorems can be found in Turull \cite{Tu}.
Following the solution of the restricted Burnside problem it was discovered that the exponent of $C_G(A)$ may have strong impact over the exponent of $G$. Remind that a group $G$ is said to have exponent $m$ if $x^m=1$ for every $x\in G$ and $m$ is the minimal positive integer with this property. The next theorem was obtained in \cite{KS}.
\begin{theorem}
\label{q2}
Let $q$ be a prime, m a positive integer and $A$ an elementary abelian group of order $q^{2}$. Suppose that $A$ acts as a coprime group of automorphisms on a finite group $G$ and assume that $C_{G}(a)$ has exponent dividing $m$ for each $a\in A^{\#}$. Then the exponent of $G$ is $\{m,q\}$-bounded.
\end{theorem}
Here and throughout the paper $A^{\#}$ denotes the set of nontrivial elements of $A$. The proof of the above result involves a number of deep ideas. In
particular, Zelmanov's techniques that led to the solution of the restricted Burnside problem \cite{Z1} are combined with the Lubotzky--Mann theory of powerful $p$-groups \cite{LM}, Lazard's criterion for a pro-$p$~group to be $p$-adic analytic \cite{L}, and a theorem of Bakhturin and Zaicev on Lie algebras admitting a group of automorphisms whose fixed-point subalgebra is PI \cite{BZ}.
Another quantitative result of similar nature was proved in the paper of Guralnick and the second author \cite{GS}.
\begin{theorem}
\label{q3}
Let $q$ be a prime, $m$ a positive integer. Let $G$ be a finite $q'$-group acted on by an elementary abelian group $A$ of order $q^{3}$. Assume that $C_{G}(a)$ has derived group of exponent dividing $m$ for each $a\in A^{\#}$. Then the exponent of $G'$ is $\{m,q\}$-bounded.
\end{theorem}
Note that the assumption that $|A|=q^{3}$ is essential here and the theorem fails if $|A|=q^{2}$. The proof of Theorem \ref{q3} depends on the classification of finite simple groups.
It was natural to expect that Theorems \ref{q2} and \ref{q3} admit a common generalization that would show that both theorems are part of a more general phenomenon. Let us denote by $\gamma_i(H)$ the $i$th term of the lower central series of a group $H$ and by $H^{(i)}$ the $i$th term of the derived series of $H$. The following conjecture was made in \cite{drei}.
\begin{conjecture}
\label{255}
Let $q$ be a prime, $m$ a positive integer and $A$ an elementary
abelian group of order $q^r$ with $r\ge 2$ acting
on a finite $q'$-group $G$.\begin{enumerate}
\item If $\gamma_{r-1}(C_G(a))$ has exponent dividing
$m$ for any $a\in A^\#$, then $\gamma_{r-1}(G)$
has $\{m,q,r\}$-bounded exponent;
\item If, for some integer $d$ such that $2^d\le r-1$,
the $d$th derived group of $C_G(a)$ has exponent dividing
$m$ for any $a\in A^\#$, then the $d$th derived
group $G^{(d)}$ has $\{m,q,r\}$-bounded exponent.
\end{enumerate}
\end{conjecture}
The main purpose of the present paper is to confirm Conjecture \ref{255}. Theorem \ref{PR} and Theorem \ref{gamma_PR} show that both parts of the conjecture are correct. The main novelty of the paper is the introduction of the concept of $A$-special subgroups of $G$ (see Section 3). Using the classification of finite simple groups it is shown in Section 4 that the $A$-invariant Sylow $p$-subgroups of $G^{(d)}$ are generated by their intersections with $A$-special subgroups of degree $d$. This enables us to reduce the proof of Conjecture \ref{255} to the case where $G$ is a $p$-group, which can be treated via Lie methods. The idea of this kind of reduction has been anticipated already in \cite{GS}. In Section 6 we give a detailed proof of part (2) of Conjecture \ref{255}. In Section 7 we briefly describe how the developed techniques can be used to prove part (1) of Conjecture \ref{255}.
Throughout the article we use the term ``$\{a,b,c,\dots\}$-bounded" to mean ``bounded from above
by some function depending only on the parameters $a,b,c,\dots$".
\section{Preliminary Results}
\label{preliminary result}
We start with the following elementary lemma.
\begin{lemma}
\label{generation_and_product}
Suppose that a nilpotent group $G$ is generated by subgroups $G_{1},\ldots,G_{t}$ such that
$\gamma_{i}(G)=\langle \gamma_{i}(G)\cap G_{j} \mid 1 \leq j \leq t \rangle$ for all $i\geq 1$. Then $G=G_{1}G_{2}\cdots G_{t}$.
\end{lemma}
\begin{proof}
We argue by induction on the nilpotency class $c$ of $G$. If $c=1$, then $G$ is abelian and the result is clear. Assume that $c\geq 2$. Let $K=\gamma_{c}(G)$. Since $K$ is central, it is abelian and we have
$K=K_{1}K_{2}\cdots K_{t}$,
where
$K_{j}=K\cap G_{j}$ for $j=1,\ldots,t$.
By induction we have
$$
G=G_{1}G_{2}\cdots G_{t}K=G_{1}G_{2}\cdots G_{t}K_{1}K_{2}\cdots K_{t}.
$$
Since each subgroup $K_{j}$ is central in $G$ and $K_j\leq G_{j}$, it follows that $G=G_{1}G_{2}\cdots G_{t}$, as required.
\end{proof}
We now collect some facts about coprime automorphisms of finite groups.
The two following lemmas are well known (see \cite[5.3.16, 6.2.2, 6.2.4]{GO}).
\begin{lemma}
\label{FG1}
Let $A$ be a group of automorphisms of the finite group $G$ with $(|A|,|G|)=1$.
\begin{enumerate}
\item If $N$ is an $A$-invariant normal subgroup of $G$, then \\$C_{G/N}(A)=C_G(A)N/N$;
\item If $H$ is any $A$-invariant $p$-subgroup of $G$, then $H$ is contained in an $A$-invariant Sylow $p$-subgroup of $G$.
\end{enumerate}
\end{lemma}
\begin{lemma}
\label{FG2}
Let $q$ be a prime, $G$ a finite $q'$-group acted on by an elementary abelian $q$-group $A$ of rank at least $2$.\ Let $A_1, \dots,A_s$ be the maximal subgroups of $A$.\ If $H$ is an $A$-invariant subgroup of $G$ we have
$H=\langle C_H(A_1),\dots,C_H(A_s)\rangle$. Furthermore if $H$ is nilpotent then $H=\prod_{i} C_{H}(A_{i})$.
\end{lemma}
We also need the following result, which is a well-known corollary of the classification of finite simple groups.
\begin{lemma}
\label{simple}
Let $G$ be a finite simple group and $A$ a group of automorphisms of $G$ with $(|A|,|G|)=1.$ Then $A$ is cyclic.
\end{lemma}
We conclude this section by citing an important theorem due to Gasch\"{u}tz. The proof can be found in \cite[p.\ 121]{Huppert} or in \cite[p.\ 191]{Ro}.
\begin{theorem}
\label{gaschutz thm}
Let $N$ be a normal abelian $p$-subgroup of a finite group $G$ and let $P$ be a Sylow $p$-subgroup of $G$. Then $N$ has a complement in $G$ if and only if $N$ has a complement in $P$.
\end{theorem}
\section{$A$-special subgroups}
\label{Aspecial}
In this section we introduce the concept of \emph{$A$-special subgroups} of $G$. For every integer $k\geq 0$ we define $A$-special subgroups of $G$ of degree $k$ in the following way.
\begin{definition}
\label{DAspecial}
Let $q$ be a prime and $A$ an elementary abelian $q$-group acting on a finite $q'$-group $G$.
Let $A_{1},\ldots,A_{s}$ be the subgroups of index $q$ in $A$ and $H$ a subgroup of $G$.
\begin{itemize}
\item We say that $H$ is an $A$-special subgroup of $G$ of degree $0$ if and only if $H=C_{G}(A_{i})$ for suitable $i\leq s$.
\item Suppose that $k\geq 1$ and the $A$-special subgroups of $G$ of degree $k-1$ are defined. Then $H$ is an $A$-special subgroup of $G$ of degree $k$ if and only if there exist $A$-special subgroups $J_{1},J_{2}$ of $G$ of degree $k-1$ such that $H=[J_{1},J_{2}]\cap C_{G}(A_{j})$ for suitable $j\leq s$.
\end{itemize}
\end{definition}
Here as usual $[J_{1},J_{2}]$ denotes the subgroup generated by all commutators $[x,y]$ where $x\in J_1$ and $y\in J_2$.
We note that all $A$-special subgroups of $G$ of any degree are $A$-invariant. Assume that $A$ has order $q^r$. It is clear that for a given integer $k$ the number of $A$-special subgroups of $G$ of degree $k$ is $\{q,r,k\}$-bounded. Let us denote this number by $s_{k}$.
The $A$-special subgroups have certain properties that will be crucial for the proof of the main result of this paper.
\begin{proposition}
\label{PAspecial}
Let $A$ be an elementary abelian $q$-group of order $q^{r}$ with $r\geq 2$ acting on a finite $q'$-group $G$ and let $A_{1},\ldots,A_{s}$ be the maximal subgroups of $A$. Let $k\geq 0$ be an integer.
\begin{enumerate}
\item If $k\geq 1$, then every $A$-special subgroup of $G$ of degree $k$ is contained in some $A$-special subgroup of $G$ of degree $k-1$.
\item Let $K$ be an $A$-invariant subgroup of $G$ and let $K_{1},\ldots,K_{t}$ be all the subgroups of the form $K\cap H$, where $H$ is some $A$-special subgroup of $G$ of degree $k$. Let $L_{1},\ldots,L_{u}$ be all the subgroups of the form $K'\cap J$ where $J$ is some $A$-special subgroup of $G$ of degree $k+1$. If $K=\langle K_{1},\ldots,K_{t} \rangle$, then $K'=\langle L_{1},\ldots,L_{u} \rangle$.
\item Let $R_{k}$ be the subgroup generated by all $A$-special subgroups of $G$ of degree $k$. Then $R_{k}=G^{(k)}$.
\item If $2^{k}\leq r-1$ and $H$ is an $A$-special subgroup of $G$ of degree $k$, then $H$ is contained in the $k$th derived group of $C_{G}(B)$ for some subgroup $B\leq A$ such that $|A/B|\leq q^{2^{k}}$.
\item Suppose that $G=G'$ and let $N$ be an $A$-invariant subgroup such that $N=[N,G]$. Then for every $k\geq 0$ the subgroup $N$ is generated by subgroups of the form $N\cap H$, where $H$ is some $A$-special subgroup of $G$ of degree $k$.
\item Let $H$ be an $A$-special subgroup of $G$. If $N$ is an $A$-invariant normal subgroup of $G$, then the image of $H$ in $G/N$ is an $A$-special subgroup of $G/N$.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) If $k=1$ and $H$ is an $A$-special subgroup of $G$ of degree $1$, then $H=[J_{1},J_{2}]\cap C_{G}(A_{j})$ for a suitable $j\leq s$. Observe that $H\leq C_{G}(A_{j})$ and the centralizer $C_{G}(A_{j})$ is an $A$-special subgroup of $G$ of degree $0$. Assume that $k\geq 2$ and use induction on $k$. Let $H$ be an $A$-special subgroup of degree $k$. We know that there exist $A$-special subgroups $J_{1}, J_{2}$ of $G$ of degree $k-1$ such that $H=[J_{1},J_{2}]\cap C_{G}(A_{j})$ for suitable $j\leq s$. By induction $J_{i}$ is contained in some $A$-special subgroup $L_{i}$ of $G$ of degree $k-2$. Observe that $[L_{1},L_{2}]\cap C_{G}(A_{j})$ is an $A$-special subgroup of $G$ of degree $k-1$ and $H\leq [L_{1},L_{2}]\cap C_{G}(A_{j})$, so the result follows.
(2) Set $M=\langle [K_{i},K_{j}] \mid 1\leq i,j \leq t\rangle$. It is clear that each of the subgroups $[K_{i},K_{j}]$ is $A$-invariant. Thus, by Lemma \ref{FG2} each subgroup $[K_{i},K_{j}]$ is generated by subgroups of the form $[K_{i},K_{j}]\cap C_{G}(A_{l})$, where $l=1,\ldots,s$. Note that each subgroup $[K_{i},K_{j}]\cap C_{G}(A_{l})$ is contained in an $A$-special subgroup of $G$ of degree $k+1$. Hence $M$ is generated by subgroups of the form $M\cap D$, where $D$ ranges through the set of all $A$-special subgroups of $G$ of degree $k+1$. If $M^*=M\cap D^*$ is such a subgroup we claim that $[M^*,K_{j}]\leq M$ for every $1\leq j\leq t$. Indeed, by (1) we know that there exists some $A$-special subgroup $H$ of $G$ of degree $k$ such that $D^*\leq H$. This implies that $M^*$ is contained in some $K_{l}$ and so we have $[M^*,K_{j}]\leq[K_{l},K_{j}]\leq M$, as desired. Therefore $M$ is normal in $K$ and we conclude that $M=K'$. The result now follows.
(3) If $k=0$ the result is immediate from Lemma \ref{FG2}. Therefore we assume that $k\geq 1$ and set $N=R_{k-1}$. By induction on $k$ we assume that $N=G^{(k-1)}$. Let $D_{1},D_{2},\ldots,D_{s_{k-1}}$ be the $A$-special subgroups of $G$ of degree $k-1$ and $H_{1},H_{2},\ldots,H_{s_{k}}$ be the $A$-special subgroups of $G$ of degree $k$. It follows from (2) that $G^{(k)}=\langle [D_{i},D_{j}] \mid 1\leq i,j \leq s_{k-1}\rangle$. Since each subgroup $[D_{i},D_{j}]$ is $A$-invariant, it follows from Lemma \ref{FG2} that it is generated by subgroups of the form $[D_{i},D_{j}]\cap C_{G}(A_{l})$, where $l=1,\ldots,s$. These are precisely $A$-special subgroups of $G$ of degree $k$ so the result follows.
(4) If $k=0$ this is clear because $H=C_{G}(A_{i})$ for a suitable $i\leq s$ and $|A/A_{i}|=q$.
Assume that $k\geq 1$ and use induction on $k$. We have $H=[J_{1},J_{2}]\cap C_{G}(A_{j})$ for a suitable $j\leq s $ and $A$-special subgroups $J_{1},J_{2}$ of $G$ of degree $k-1$. By induction there exist subgroups $B_{1},B_{2}\leq A$ such that $|A/B_{i}|\leq q^{2^{k-1}}$ and $J_{i}\leq C_{G}(B_{i})^{(k-1)}$ where $i=1,2$. Set $B=B_{1}\cap B_{2}$. Observe that
$H\leq [J_{1},J_{2}]\leq [C_{G}(B_{1})^{(k-1)},C_{G}(B_{2})^{(k-1)}]\leq [C_{G}(B)^{(k-1)}, C_{G}(B)^{(k-1)}]$. Thus $H\leq C_{G}(B)^{(k)}$ and $|A/B|\leq q^{2^{k}}$, as required.
(5) For $k=0$ this follows from Lemma \ref{FG2}. Assume that $k\geq 1$ and use induction on $k$.
Let $N_{1},\ldots, N_{t}$ be all the subgroups of the form $N\cap H$ where $H$ is some $A$-special subgroup of degree $k$ and set $M=\langle N_{1},\ldots, N_{t}\rangle$. We want to show that $N=M$.
Since $G=G'$ by (3) $G$ can be generated by all $A$-special subgroups of degree $k$, for any $k\geq1$. Thus $G=\langle H_{1},\ldots, H_{s_{k}}\rangle$, where $H_{j}$ is $A$-special subgroup of degree $k$. Lemma \ref{FG2} shows that for all $i$ and $j$ the commutator $[N_{i},H_{j}]$ is generated by subgroups of the form $[N_{i},H_{j}]\cap C_{G}(A_{l})$, where $l=1,\dots,s$. Note that each subgroup $[N_{i},H_{j}]\cap C_{G}(A_{l})$ is contained in $N$ since $N=[N,G]$ and on the other hand it is also contained in some $A$-special subgroup of degree $k$, so $[N_{i},H_{j}]\cap C_{G}(A_{l})\leq N_{m}$ for a suitable $m\leq t$. This implies that $M$ is normal in $G$.
Let now $L_{1},\ldots, L_{u}$ be all the subgroups of the form $N\cap K$ where $K$ is some $A$-special subgroup of degree $k-1$, so by induction we can assume that $N=\langle L_{1},\ldots,L_{u} \rangle$. For all $i$ and $j$, using the argument as above, it is easy to show that $[L_{i},H_{j}]$ is generated by subgroups of the form $[L_{i},H_{j}]\cap C_{G}(A_{l})$, where $l=1,\dots,s$ and that each such a subgroup is contained in some $N_{m}$, for a suitable $m\leq t$.
If $M=1$, then, for all $m\leq t$, $N_{m}=1$ and so $[L_{i},H_{j}]=1$ for all $i$ and $j$. Hence $N$ is central in $G$ but this is a contradiction because $N=[N,G]$. Assume now that $M$ is a nontrivial subgroup strictly contained in $N$. Since we have shown that $M$ is normal we can pass to the quotient $G/M$. In the quotient $N/M$ is central so $[N,G]\leq M$ but this contradicts the assumption that $M<N$. Thus we conclude that $M$ must be equal to $N$.
(6) This is immediate from Lemma \ref{FG1}(1) and the definitions.
\end{proof}
\section{Some generation results}
\label{generation_results}
Throughout this section let $q$ be a prime, $G$ a finite $q'$-group and $A$ an elementary abelian group of order $q^{r}$ acting on $G$.
We will show that if $P$ is an $A$-invariant Sylow $p$-subgroup of $G^{(d)}$, then it can be generated by its intersections with $A$-special subgroups of $G$ of degree $d$.
\begin{theorem}
\label{generation1}
Assume $r\geq 2$. Let $P$ be an $A$-invariant Sylow $p$-subgroup of $G^{(d)}$ for some fixed integer $d\geq 0$. Let $P_{1},\ldots,P_{t}$ be the subgroups of the form $P\cap H$ where $H$ is some $A$-special subgroup of $G$ of degree $d$. Then $P=\langle P_{1},\dots,P_{t}\rangle$.
\end{theorem}
We first handle the case where $G$ is a direct product of simple groups.
\begin{lemma}
\label{product_of_simple}
Assume that $r\geq 2$ and $G$ is a direct product of nonabelian simple groups. Let $P$ be an $A$-invariant Sylow $p$-subgroup of $G$, and for some fixed integer $d\geq 0$ let $P_{1},\ldots,P_{t}$ be all the subgroups of the form $P\cap H$, where $H$ is some $A$-special subgroup of $G$ of degree $d$. Then $P=\langle P_{1},\dots,P_{t}\rangle$.
\end{lemma}
\begin{proof} Let $G=S_{1}\times \cdots \times S_{m}$. By induction on the order of $G$ we may assume that $A$ permutes transitively the simple factors $S_{1},\ldots,S_{m}$.
We will now use induction on $r$ to show that without loss of generality it can be assumed that $A$ acts on $G$ faithfully. Suppose that some element $a\in A^{\#}$ acts on $G$ trivially. Thus, $C_{G}(a)=G$.
Since $G$ is a product of nonabelian simple groups it follows that $[C_{G}(a),C_{G}(a)]=G$ and $[C_{G}(a),C_{G}(a)]\cap C_{G}(a)=G$. Thus if $r=2$, then $G$ itself is an $A$-special subgroup of degree $1$. From this it is easy to see that $G$ is an $A$-special subgroup of degree $d$ and the lemma follows immediately.
Suppose now that $r\geq 3$ and let $A_{1},\ldots,A_{s}$ be the maximal subgroups of $A$. Put $\overline{A}=A/\langle a\rangle$. If $A_{1},\ldots A_{t}$ are the maximal subgroups of $A$ containing $a$, then $\overline{A_{1}},\ldots, \overline{A_{t}}$ are maximal subgroups of $\overline{A}$. Then $C_{G}(\overline{A_{i}})=C_{G}(A_{i})$ for all $i \leq s$, and so we can consider $\overline{A}$ instead of $A$ and use induction on $r$.
Thus, from now on we assume that $A$ is faithful on $G$. Let $B$ be the stabilizer of $S_{1}$ in $A$. Then by Lemma \ref{simple} $B$ is cyclic. Remark that if $b\notin B $, then $C_{G}(b)$ is a product of simple groups. Indeed if we consider all the $b$-orbits, then it is not difficult to see that $C_{G}(b)$ is the product of the diagonal subgroups of these $b$-orbits, i.e., $C_{G}(b)$ is a product of simple groups, one for each $b$-orbit.
Suppose that $r=2$ and $B\neq 1$. Let $a$ be a nontrivial element of $B$ and choose $b\in A$ that permutes $S_{1},\ldots,S_{q}$. Observe that the case where $A=B$ does not happen because of Lemma \ref{simple}. Since $b\notin B$ from the above remark we know that $C_{G}(b)=diag(S_{1}\times \cdots \times S_{q})$ is a diagonal subgroup of $G$. On the other hand it follows from the Thompson Theorem \cite{T} that $C_{S_{1}}(a) \neq 1$ and this holds also for the other factors $S_2,\dots,S_q$ because $a$ normalizes each of the simple factors. Thus $C_{G}(a)=C_{S_{1}}(a)\times \cdots \times C_{S_{q}}(a)$ and we have
\begin{equation}
\label{eq1}
\begin{split}
[C_{G}(a),C_{G}(b)]= &\\
[C_{S_{1}}(a),diag(S_{1}\times &\cdots \times S_{q})]\times\cdots \times [C_{S_{q}}(a),diag(S_{1}\times \cdots \times S_{q})].
\end{split}\end{equation}
Furthermore observe that for any $j=1,\ldots,q$
\begin{equation}
\label{eq2}
[C_{S_{j}}(a),diag(S_{1}\times \cdots \times S_{q})]=[C_{S_{j}}(a),S_{j}]=S_{j},
\end{equation}
where the first equality follows from the fact that the simple factors commute each other and the second one holds since $[C_{S_{j}}(a),S_{j}]$ is a nontrivial normal subgroup of $S_{j}$. By (\ref{eq1}) and (\ref{eq2}) we see that $[C_{G}(a),C_{G}(b)]=S_{1}\times \cdots \times S_{q}=G.$
Thus, for any $c\in A^{\#}$, $C_{G}(c)=[C_{G}(a),C_{G}(b)]\cap C_{G}(c)$ and so the centralizer $C_{G}(c)$ is also an $A$-special subgroup of degree $1$. We deduce that, for any $a\in A^{\#}$, the centralizer $C_{G}(a)$ is an $A$-special subgroup of $G$ of any degree. Since Lemma \ref{FG2} tells us that $P$ can be generated by subgroups of the form $P\cap C_{G}(A_{i})$ where $A_{i}$ are the maximal subgroups of $A$, the result follows.
Next, assume that $r=2$ and $B=1$. Note that $A$ permutes the factors $S_{1},\ldots,S_{q^{2}}$ and, for any $a\in A^{\#}$, the centralizer $C_{G}(a)$ is a product of $q$ simple groups, one for each $a$-orbit. Thus $C_{G}(a)$ is perfect and, in particular, it is an $A$-special subgroup of any degree for all $a\in A^{\#}$. The lemma follows.
Finally assume that $r\geq3$. Since $B$ is cyclic we have $|A:B|\geq q^{2}$. Hence we can choose a subgroup $E$ of type $(q,q)$ that intersects $B$ trivially. Note that for all $a\in E^{\#}$, the centralizer $C_{G}(a)$ is a product of simple groups. Moreover Lemma \ref{FG2} shows that $P=\prod_{a\in E^{\#}}C_{P}(a)$. Therefore it is sufficient to prove that for each $a\in E^{\#}$ the subgroup $C_{G}(a)\cap P$ is generated by its intersections with all the $A$-special subgroups of $G$ of degree $d$.
Fix $a$ in $E^{\#}$. Let $D$ be the group of automorphisms induced on $C_{G}(a)$ by $A$. By induction $C_{G}(a)\cap P$ is generated by subgroups of the form $(C_{G}(a)\cap P)\cap H$, where $H$ ranges through the set of $D$-special subgroups of $C_{G}(a)$ of degree $d$. We now remark that any $D$-special subgroup of $C_{G}(a)$ of any degree is in fact an $A$-special subgroup of $G$ of the same degree. This follows form Definition \ref{DAspecial} and from the fact that if $A_{i}$ is a maximal subgroup of $A$ containing $a$, then there exists a maximal subgroup $D_{j}$ of $D$ such that $C_{C_{G}(a)}(D_{j})=C_{G}(A_{i})$. Thus we can conclude that $C_{G}(a)\cap P$ is generated by subgroups of the form $(C_{G}(a)\cap P)\cap H$, where now $H$ can be regarded as an $A$-special subgroup of $G$ of degree $d$. The proof is now complete.
\end{proof}
We now are ready to complete the proof of Theorem \ref{generation1}.
\begin{proof}[Proof of Theorem \ref{generation1}]
Let $G$ be a counterexample of minimal order and let $N$ be a minimal normal $A$-invariant subgroup of $G$. Set $X=\langle P_{1},\ldots,P_{t}\rangle$. By minimality and Proposition \ref{PAspecial}(6) $PN=XN$. To prove that $P=X$ it is sufficient to show that $P\cap N \leq X$.
First suppose that $N$ is a $p'$-group. In this case the intersection $P\cap N$ is trivial and there is nothing to prove.
Next suppose that $N$ is perfect. Since $N$ is characteristically simple, $N$ is a product of nonabelian simple groups. It follows from Lemma \ref{product_of_simple} that $P\cap N$ is contained in $X$ and we are done.
Thus, it remains to consider the case where $N$ is a $p$-group. Suppose that $G \neq G'$. By induction we know that every $A$-invariant Sylow $p$-subgroup of $G^{(d+1)}$ is generated by its intersections with all the $A$-special subgroups of $G'$ of degree $d$ . Therefore we can pass to the quotient $G/G^{(d+1)}$ and assume that $G^{(d+1)}=1$. This implies that $G^{(d)}$ is abelian and so we may assume that $G^{(d)}$ is a $p$-group. Then $G^{(d)}=P$. It follows from Proposition \ref{PAspecial}(3) that $P$ is generated by $A$-special subgroups of $G$ of degree $d$ and the result holds.
We are reduced to the case that $G=G'$. Since $N$ is minimal, either $N=[N,G]$ or $N\leq Z(G)$.
If $N=[N,G]$ we note that $P\cap N=N$ because $N$ is contained in $P$. Since $N=[N,G]$, Proposition \ref{PAspecial}(5) shows that $N$ is generated by its intersections with all the $A$-special subgroups of $G$ of degree $d$ and so $N\leq X$, as desired.
Now suppose $N$ central in $G$. Then $N$ is of order $p$ and either $N$ is contained in every maximal subgroup of $P$ or there exists a maximal subgroup $S$ in $P$ such that $P=NS$.
In the former case $N\leqslant \Phi(P)$. Since we know that $P=XN$, it follows that $P=X$, as required.
In the latter case, by Theorem \ref{gaschutz thm}, $N$ is also complemented in $G$ and so $G=NH$ for some subgroup $H\leq G$. Since $N$ is central, we have $G=N\times H$. This yields a contradiction because we have assumed that $G=G'$.
\end{proof}
We note some consequences of Theorem \ref{generation1}. These facts will be useful later on.
\begin{lemma}
\label{generation2}
Under the hypothesis of Theorem \ref{generation1} let $P^{(l)}$ be the $l$th derived group of $P$. Then $P^{(l)}= \langle P^{(l)} \cap P_{j} \mid 1 \leq j \leq t \rangle$.
\end{lemma}
\begin{proof}
First we want to establish the following fact:
\begin{equation}
\label{generation}
\begin{aligned}
&\text{The group}\, P^{(l)} \, \text{is generated by all the subgroups of the form}\\
&P^{(l)}\cap D,\, \text{where}\, D \, \text{ranges through the set of all}\, A\text{-special sub-}\\
&\text{groups of} \, G \, \text{of degree}\, d+l.\\
\end{aligned}
\end{equation}
Indeed if $l=0$ then (\ref{generation}) is exactly Theorem \ref{generation1}. Assume that $l\geq 1$ and use induction on $l$. Let $L_{1},\ldots, L_{u}$ be all the subgroups of the form $P^{(l)}\cap J$, where $J$ is some $A$-special subgroup of $G$ of degree $d+l$. By induction $P^{(l-1)}$ is generated by subgroups of the form $P^{(l-1)}\cap D$, where $D$ is some $A$-special subgroup of $G$ of degree $d+(l-1)$. It now follows from Proposition \ref{PAspecial}(2) that $P^{(l)}=\langle L_{1},\ldots, L_{u}\rangle$ and this concludes the proof of (\ref{generation}).
Now for $l=0$ the lemma is obvious since by Theorem \ref{generation1} $P=\langle P_{1},\ldots, P_{t}\rangle$, where each subgroup $P_{j}$ is of the form $P\cap H$ for some $A$-special subgroup $H$ of $G$ of degree $d$.
Assume that $l\geq 1$. Proposition \ref{PAspecial}(1) tells us that every $A$-special subgroup $D$ of degree $d+l$ is contained in some $A$-special subgroup $H$ of degree $d$. Combined with (\ref{generation}) this implies that each subgroup of the form $P^{(l)}\cap D$ is contained in $P_{j}$, for a suitable $j\leq t$. Thus $P^{(l)}=\langle P^{(l)}\cap P_{j} \mid 1 \leq j \leq t \rangle$, as required.
\end{proof}
Combining Theorem \ref{generation1} with Lemma \ref{generation_and_product} we obtain a further refinement of Theorem \ref{generation1}.
\begin{corollary}
\label{generation3}
$P=P_{1}P_{2}\cdots P_{t}$.
\end{corollary}
\begin{proof}
By Theorem \ref{generation1} we have $P=\langle P_{1},\dots,P_{t}\rangle$. Since $P$ is nilpotent, in view of Lemma \ref{generation_and_product} it is sufficient to show that
\begin{equation}
\label{eq}
\gamma_{i}(P)=\langle \gamma_{i}(P)\cap P_{j} \mid 1\leq j \leq t\rangle,
\end{equation}
for all $i\geq 1$.
For $i=1$ the equality (\ref{eq}) is Theorem \ref{generation1}.
Assume that $i\geq 2$. Set $N_{j}=\gamma_{i}(P)\cap P_{j}$ for $j=1,\ldots,t$ and $N=\langle N_j \mid 1\leq j \leq t \rangle$.
By Lemma \ref{FG2} $[N_{j},P_{k}]$ can be generated by subgroups of the form $[N_{j},P_{k}]\cap C_{G}(A_{l})$, where $l=1,\ldots,s$ and each of them is contained in some $N_{u}$ for suitable $u\leq t$. Indeed, on the one hand $[N_{j},P_{k}]\cap C_{G}(A_{l})$ is obviously contained in $\gamma_{i}(P)$. On the other hand it follows from Proposition \ref{PAspecial}(1) that $[N_{j},P_{k}]\cap C_{G}(A_{l})$ is contained in some $A$-special subgroup $H$ of degree $d$. Hence
$$
[N_{j},P_{k}]\cap C_{G}(A_{l})\leq\gamma_{i}(P)\cap (P\cap H)= \gamma_{i}(P)\cap P_{u}
$$
for some $u\leq t$. So $[N_{j},P_{k}]\cap C_{G}(A_{l})$ is contained in some $N_{u}$ as desired. This implies that $[N_{j},P_{k}]\leq N$ for all $j$ and $k$. Therefore $N$ is normal in $P$.
We can now consider the quotient $P/N$ and observe that for $j=1,\ldots,t$ the image of the subgroup $\gamma_{i}(P)\cap P_{j}$ is trivial. Therefore $\gamma_i(P)\leq N$. Since the subgroup $N$ is obviously contained in $\gamma_{i}(P)$ we conclude that $N=\gamma_{i}(P)$ and we have (\ref{eq}).
\end{proof}
We will also require the following result that is a little stronger than Corollary \ref{generation3}.
\begin{corollary}
\label{generation4}
For all $l\geq 1$ the $l$th derived group $P^{(l)}$ is the product of the subgroups of the form $P^{(l)}\cap P_{j}$, where $j=1,\ldots,t$.
\end{corollary}
\begin{proof}
Recall that by Lemma \ref{generation2} we have
$$
P^{(l)}= \langle P^{(l)} \cap P_{j} \mid 1 \leq j \leq t \rangle
$$ for all $l\geq 1$. By using the same argument as in the proof of Corollary \ref{generation3} the result follows.
\end{proof}
\section{Useful Lie-theoretic machinery}
\label{Lie_machinery}
Let $L$ be a Lie algebra over a field ${\mathfrak k}$. Let $k$ be a positive integer and let $x_1,x_2,\dots,x_k$ be elements of $L$. We define inductively
$$[x_1]=x_1;\ [x_1,x_2,\dots,x_k]=[[x_1,x_2,\dots,x_{k-1}],x_k].$$
An element $a\in L$ is called ad-nilpotent if there exists a positive integer $n$ such that
$$[x,\underset{n}{\underbrace{a,\ldots,a}}]=0 \quad \text{for all}\, x\in L.$$
If $n$ is the least integer with the above property then we say that $a$ is ad-nilpotent of index $n$. Let $X\subseteq L$ be any subset of $L$. By a commutator in elements of $X$ we mean any element of $L$ that can be obtained as a Lie product of elements of $X$ with some system of brackets.
Denote by $F$ the free Lie algebra over
${\mathfrak k}$ on countably many free generators $x_1,x_2,\dots$. Let $f=f(x_1,x_2,\dots,x_n)$ be a non-zero element of $F$. The algebra $L$ is said to satisfy the identity $f\equiv 0$ if $f(a_1,a_2,\dots,a_n)=0$ for any $a_1,a_2,\dots,a_n\in L$. In this case we say that $L$ satisfies a polynomial identity, in short, is PI. A deep result of Zelmanov \cite{Z0}, which has numerous important applications to group theory (in particular see \cite{OS} for examples where the theorem is used), says that if a Lie algebra $L$ is PI and is generated by finitely many elements all commutators in which are ad-nilpotent, then $L$ is nilpotent. From Zelmanov's result the following theorem can be deduced \cite{KS}.
\begin{theorem}\label{liealgbnilp}
Let $L$ be a Lie algebra over a field ${\mathfrak k}$ generated by $a_1,a_2,\dots,a_m$.\ Assume that $L$ satisfies an identity $f\equiv 0$ and that each commutator in the generators $a_1,a_2,\dots,a_m$ is ad-nilpotent of index at most $n$.\ Then $L$ is nilpotent of $\{f,n,m,{\mathfrak k}\}$-bounded class.
\end{theorem}
The next theorem provides an important criterion for a Lie algebra to be PI. It was proved by Bakhturin and Zaicev for soluble groups $A$ \cite{BZ} and later extended by Linchenko to the general case \cite{LI}.
\begin{theorem}
\label{LichBak}
Assume that a finite group $A$ acts on a Lie algebra $L$ by automorphisms in such a manner that $C_L(A)$, the subalgebra formed by fixed elements, is PI. Assume further that the characteristic of the ground field of $L$ is either 0 or prime to the order of $A$. Then $L$ is PI.
\end{theorem}
We will need a corollary of the previous result.
\begin{corollary}[\cite{Shu}]
\label{polynomialidentity}
Let $F$ be the free Lie algebra of countable rank over $\mathfrak k$. Denote by $F^{*}$ the set of non-zero elements of $F$. For any finite group $A$ there exists a mapping
$$\phi: F^{*}\rightarrow F^{*}$$
such that if $L$ and $A$ are as in Theorem \ref{LichBak}, and if $C_{L}(A)$ satisfies an identity $f\equiv 0$, then $L$ satisfies the identity $\phi(f)\equiv 0$.
\end{corollary}
Now we turn to groups and for the rest of this section $p$ will denote a fixed prime number.
Let $G$ be any group. A series of subgroups
\begin{equation*}
(*)\quad \quad G=G_{1}\geq G_{2}\geq \cdots
\end{equation*}
is called an $N_{p}$-series if $[G_{i},G_{j}]\leq G_{i+j}$ and $G_{i}^{p}\leq G_{pi}$ for all $i,j$. With any $N_{p}$-series $(*)$ of $G$ one can associate a Lie algebra $L^{*}(G)=\oplus L^{*}_{i}$ over the field with $p$ elements $\F_{p}$, where we view each $L^{*}_{i}=G_{i}/G_{i+1}$ as a linear space over $\F_{p}$. If $x \in G$, let $i=i(x)$ be the largest integer such that $x \in G_i$. We denote by $x^*$ the element $xG_{i+1}$ of $L^{*}(G)$. The following lemma tells us something about the relationship between the group $G$ and the associated Lie algebra $L^{*}(G)$.
\begin{lemma}[Lazard, \cite{L}]
\label{Laz}
For any $x\in G$ we have $(ad\, x^*)^p=ad\, (x^p)^*$. Consequently, if $x$ is of finite order $p^{t}$, then $x^*$ is ad-nilpotent of index at most $p^{t}$.
\end{lemma}
Let $w=w(x_1,x_2,\dots,x_n)$ be nontrivial group-word, i.e., a nontrivial element of the free group on free generators $x_1,x_2,\dots,x_n$. We say that $G$ satisfies the identity $w\equiv 1$ if $w(g_1,\dots,g_n)=1$ for any $g_1,g_2,\dots,g_n\in G$. The next proposition follows from the proof of Theorem 1 in the paper of Wilson and Zelmanov \cite{WZ}.
\begin{proposition}
\label{WilZel}
Let $G$ be a group satisfying an identity $w\equiv 1$. Then there exists a non-zero multilinear Lie polynomial $f$ over $\F_{p}$, depending only on $p$ and $w$, such that for any $N_{p}$-series $(*)$ of $G$ the corresponding algebra $L^{*}(G)$ satisfies the identity $f\equiv 0$.
\end{proposition}
In general a group $G$ has many $N_{p}$-series; one of the most important is the so-called Jennings-Lazard-Zassenhaus series that can be defined as follows.
Let $\gamma_j(G)$ denote the $j$th term of the lower central series of $G$.
Set $D_i=D_i(G)= \prod_{jp^{k}\geq i}\gamma_{j}(G)^{p^{k}}$. The subgroup $D_{i}$ is also known as the $i$th-dimension subgroup of $G$ in characteristic $p$. These subgroups form an $N_{p}$-series of $G$ known as the Jennings-Lazard-Zassenhaus series. Let $L_{i}=D_{i}/D_{i+1}$ and $L(G)=\oplus L_{i}$. Then $L(G)$ is a Lie algebra over the field $\F_p$ (see \cite[Chapter 11]{GA} for more detail). The subalgebra of $L(G)$ generated by $L_{1}=D_1/D_2$ will be denoted by $L_p(G)$. The next lemma is a ``finite'' version of Lazard's criterion for a pro-$p$ group to be $p$-adic analytic. The proof can be found in \cite{KS}.
\begin{lemma}
\label{PowerfulR}
Suppose that $P$ is a $d$-generator finite $p$-group such that the Lie algebra $L_{p}(P)$ is nilpotent of class $c$. Then $P$ has a powerful characteristic subgroup of $\{p,c,d\}$-bounded index.
\end{lemma}
Remind that powerful $p$-groups were introduced by Lubotzky and Mann in \cite{LM}. A finite $p$-group $G$ is said to be powerful if and only if $[G,G]\leq G^{p}$ for $p \neq 2$ (or $[G,G]\leq G^{4}$ for $p=2$). These groups have some nice properties. In particular we will use the following property: if $G$ is a powerful $p$-group generated by elements of order $e=p^{k}$, then the exponent of $G$ is $e$.
Every subspace (or just an element) of $L(G)$ that is contained in $D_i/D_{i+1}$ for some $i$ will be called homogeneous. Given a subgroup $H$ of the group $G$, we denote by $L(G,H)$ the linear span in $L(G)$ of all homogeneous elements of the form $hD_{i+1}$, where $h\in D_{i}\cap H$. Clearly, $L(G,H)$ is always a subalgebra of $L(G)$. Moreover, it is isomorphic with the Lie algebra associated with $H$ using the $N_{p}$-series of $H$ formed by $H_{i}=D_{i}\cap H$.
We also set $L_{p}(G,H)=L_{p}(G)\cap L(G,H)$. The proof of the following lemma can be found in \cite{GS}.
\begin{lemma}
\label{L(GH)}
Suppose that any Lie commutator in homogeneous elements $x_{1},\ldots,x_{r}$ of $L(G)$ is ad-nilpotent of index at most $t$.\ Let $K=\langle x_{1},\ldots,x_{r} \rangle$ and assume that $K\leq L(G,H)$ for some subgroup $H$ of $G$ satisfying a group identity $w\equiv 1$. Then there exists some $\{r,t,w,p\}$-bounded number $u$ such that:
$$[L(G),\underset{u}{\underbrace{K,\ldots,K}}]=0.$$
\end{lemma}
Lemma \ref{FG1}(1) has important implications in the context of associated Lie algebras and their automorphisms. Let $G$ be a group with a coprime automorphism $a$. Obviously $a$ induces an automorphism of every quotient $D_i/D_{i+1}$. This action extends to the direct sum $\oplus D_i/D_{i+ 1}$. Thus, $a$ can be viewed as an automorphism of $L(G)$ (or of $L_p(G)$). Set $C_i=D_i \cap C_G(a)$.\ Then Lemma \ref{FG1}(1) shows that
\begin{equation}
\label{LandCand}
C_{L(G)}(a)=\oplus C_iD_{i+1}/D_{i + 1},
\end{equation}
and that
\begin{equation}
\label{CLp}
C_{L_{p}(G)}(a)=L_{p}(G,C_{G}(a)).
\end{equation}
This implies that the properties of $C_{L(G)}(a)$ are very much related to those of $C_G(a)$. In particular, Proposition \ref{WilZel} shows that if $C_G(a)$ satisfies a certain identity, then $C_{L(G)}(a)$ is PI.
\section{Proof of the main result}
\label{derived subgroups case}
Our goal in this section is to prove that part (2) of Conjecture \ref{255} is correct. More precisely we have the following result.
\begin{theorem}\label{PR}
Let m be a positive integer, $q$ a prime, and $A$ an elementary abelian group of order $q^r$, with $r\geq2$. Suppose that $A$ acts as a coprime group of automorphisms on a finite group $G$. If, for some integer $d$ such that $2^{d}\leq r-1$, the $d$th derived group of $C_G(a)$ has exponent dividing $m$ for any $a \in A^{\#}$, then the $d$th derived group $G^{(d)}$ has $\{m,q,r\}$-bounded exponent.
\end{theorem}
First we will consider the particular case where $G$ is a powerful $p$-group.
\begin{lemma}
\label{powerfulth}
Theorem \ref{PR} is valid if $G$ is powerful.
\end{lemma}
\begin{proof}
It follows from \cite[Exercise 2.1]{GA} that $G^{(d)}$ is also powerful. Furthermore, by Proposition \ref{PAspecial}(3), $G^{(d)}$ is generated by $A$-special subgroups of $G$ of degree $d$. Since $2^{d}\leq r-1$, Proposition \ref{PAspecial}(4) shows that any $A$-special subgroup $H$ of $G$ of degree $d$ is contained in $C_{G}(B)^{(d)}$ for some nontrivial subgroup $B\leq A$ and so $H$ is also contained in $C_{G}(a)^{(d)}$ for some $a\in A^{\#}$. This implies that $G^{(d)}$ is generated by elements of order dividing $m$, and so it follows from \cite[Lemma 2.5]{GA} that the exponent of $G^{(d)}$ divides $m$.
\end{proof}
We will now handle the case of an arbitrary $p$-group. The Lie-theoretic techniques that we have described in Section \ref{Lie_machinery} will play a fundamental role in the subsequent arguments.
\begin{lemma}
\label{pgroupth}
Theorem \ref{PR} is valid if $G$ is a $p$-group.
\end{lemma}
\begin{proof} Assume that $G$ is a $p$-group. By Corollary \ref{generation3} we have
\begin{equation}
\label{producto}
G^{(d)}=G_{1}G_{2}\cdots G_{t},
\end{equation}
where each $G_{j}$ is an $A$-special subgroup of $G$ of degree $d$. It is clear that the number $t$ is $\{q,r\}$-bounded.
Let $x$ be any element of $G^{(d)}$. In view of (\ref{producto}) we can write $x=x_{1}x_{2}\cdots x_{t}$, where each $x_{j}$ belongs to $G_{j}$. Since $2^{d}\leq r-1$, by Proposition \ref{PAspecial}(4) each $G_{j}$ is contained in $C_{G}(B)^{(d)}$ for some subgroup $B\leq A$ such that $|A/B|\leq q^{2^{d}}$. Thus each $x_{j}$ is contained in some $C_{G}(a)^{(d)}$ for a suitable $a\in A^{\#}$.
Let $Y$ be the subgroup of $G$ generated by the orbits $x_{j}^{A}$ for $j=1,\ldots,t$. Each orbit contains at most $q^{r-1}$ elements so it follows that $Y$ has at most $q^{r-1}t$ generators, each of order dividing $m$. Since $x\in Y$ and we wish to bound the order of $x$, it is enough to show that the exponent of $Y$ is $\{m,q,r\}$-bounded.
Set $Y_{j}= G_{j}\cap Y$ for $j=1,\dots,t$ and note that every $Y_{j}\leq C_{G}(a)^{(d)}$ for a suitable $a\in A^{\#}$.
Since $Y=\langle x_{1}^{A},\ldots, x_{t}^{A}\rangle$ and every $G_{j}$ is an $A$-invariant subgroup we have $Y=\langle Y_{1},\ldots,Y_{t}\rangle$. By applying Lemma \ref{generation_and_product} we see that $Y=Y_{1}Y_{2}\cdots Y_{t}$.
Let $L=L_{p}(Y)$ and let $V_{1},\ldots,V_{t}$ be the images of $Y_{1},\ldots,Y_{t}$ in $Y/\Phi(Y)$. It follows that the Lie algebra $L$ is generated by $V_{1},\ldots,V_{t}$.
Let $W$ be a subspace of $L$. We say that $W$ is a \emph{special subspace} of weight $1$ of $L$ if and only if $W=V_{j}$ for some $j\leq t$ and say that $W$ is a special subspace of weight $\varphi\geq 2$ if $W=[W_{1},W_{2}]\cap C_{L}(A_{k})$, where $W_{1},W_{2}$ are some special subspaces of $L$ of weight $\varphi_{1}$ and $\varphi_{2}$ such that $\varphi_{1}+\varphi_{2}=\varphi$ and $A_{k}$ is some maximal subgroup of $A$ for a suitable $k$.
We wish to show that every special subspace $W$ of $L$ corresponds to a subgroup of an $A$-special subgroup of $G$ of degree $d$. We argue by induction on the weight $\varphi$. If $\varphi=1$, then $W=V_{j}$ and so $W$ corresponds to $Y_{j}$ for some $j\leq t$. Assume that $\varphi\geq 2$ and write $W=[W_{1},W_{2}]\cap C_{L}(A_{k})$. By induction we know that $W_{1}, W_{2}$ correspond respectively to some $J_{1},J_{2}$ which are subgroups of some $A$-special subgroups of $G$ degree $d$. Note that $[W_{1},W_{2}]$ is contained in the image of $[J_{1},J_{2}]$. This implies that the special subspace $W$ corresponds to a subgroup of $[J_{1},J_{2}]\cap C_{G}(A_{k})$ which, by Proposition \ref{PAspecial}(1), is contained in some $A$-special subgroup of $G$ of degree $d$, as desired. Moreover it follows from Proposition \ref{PAspecial}(4) that every element of $W$ corresponds to some element of $C_{G}(a)^{(d)}$ for some $a\in A^{\#}$ and so, by Lemma \ref{Laz}, it is ad-nilpotent of index at most $m$.
From the previous argument we deduce that $L=\langle V_{1}, \ldots, V_{t}\rangle$ is generated by ad-nilpotent elements of index at most $m$ but we cannot claim that every Lie commutator in these generators is again in some special subspace of $L$ and hence it is ad-nilpotent of bounded index. To overcome this difficulty we extend the ground field $\F_{p}$ by a primitive $q$th root of unity $\omega$ and put $\overline{L}=L\otimes\F_{p}[\omega]$. We view $\overline{L}$ as a Lie algebra over $\F_{p}[\omega]$ and it is natural to identify $L$ with the $\F_{p}$-subalgebra $L\otimes 1$ of $\overline{L}$.
In what follows we write $\overline{X}$ to denote $X\otimes \F_{p}[\omega]$ for some subspace $X$ of $L$. Note that if an element $x \in L$ is ad-nilpotent, then the ``same" element $x \otimes 1$ is also ad-nilpotent in $\overline{L}$. We will say that an element of $\overline{L}$ is homogeneous if it belongs to $\overline{S}$ for some homogeneous subspace $S$ of $L$.
Let $W$ be a special subspace of $L$. We claim that
\begin{equation}
\label{eigenvectorsadnilp}
\begin{aligned}
& \text{there exists an}\, \{m,q\}\text{-bounded number}\, u\, \text {such that every}\\
& \text{element}\, w\, \text{of}\, \overline{W}\, \text{is ad-nilpotent of index at most}\, u. \\
\end{aligned}
\end{equation}
Since $w$ is a homogeneous element of $\overline{L}$ it can be written as
$$
w=l_{0}\otimes1+l_{1}\otimes \omega+\cdots+l_{q-2}\otimes \omega^{q-2},
$$
for suitable homogeneous elements $l_{0},\ldots,l_{q-2}$ of $W$. The elements $l_{0},\ldots,l_{q-2}$ correspond to some $x_{0},\ldots,x_{q-2}$ of $Y$ that belong to some $A$-special subgroup of degree $d$ and so in particular $x_{0},\ldots,x_{q-2}$ are elements of $C_{G}(a)^{(d)}$ for some $a\in A^{\#}$. Set $H=\langle x_{0},\ldots,x_{q-2}\rangle$ and $K=\langle l_{0},\ldots, l_{q-2}\rangle$. Since $H$ has exponent $m$ and $K\leq L_{p}(Y, H)$, Lemma \ref{L(GH)} shows that there exists an $\{m,q\}$-bounded number $u$ such that
\begin{equation}
\label{Kequ}
[L, \underset{u}{\underbrace{K,\ldots, K}}]=0
\end{equation}
Obviously (\ref{Kequ}) implies that
\begin{equation}
\label{Kbar}
[\overline{L}, \underset{u}{\underbrace{\overline{K},\ldots, \overline{K}}}]=0.
\end{equation}
Since $w$ lies in $\overline{K}$, (\ref{eigenvectorsadnilp}) follows.
The group $A$ acts naturally on $\overline{L}$ and now the ground field is a splitting field for $A$. Since $Y$ can be generated by at most $q^{r-1}t$ elements, we can choose elements $v_{1},\ldots, v_{s}$ in $\overline{V_{1}}\cup \cdots \cup \overline{V_{t}}$ with $s\leq q^{r-1}t$ that generate the Lie algebra $\overline{L}$ and each of them is a common eigenvector for all transformations from $A$.
Now let $v$ be any Lie commutator in $v_{1},\ldots,v_{s}$. We wish to show that $v$ belongs to some $\overline{W}$, where $W$ is a special subspace of $L$. If $v$ has weight $1$ there is nothing to prove. Assume $v$ has weight at least 2. Write $v=[w_{1},w_{2}]$ for some $w_{1}\in \overline{W_{1}}$ and $w_{2}\in \overline{W_{2}}$, where $W_{1}, W_{2}$ are two special subspaces of $L$ of smaller weights. It is clear that $v$ belongs to $[\overline{W_{1}},\overline{W_{2}}]$. Note that any commutator in common eigenvectors is again a common eigenvector. Therefore $v$ is a common eigenvector and it follows that there exists some maximal subgroup $A_{l}$ of $A$ such that $v\in C_{\overline{L}}(A_{l})$. Thus $v\in [\overline{W_{1}},\overline{W_{2}}]\cap C_{\overline{L}}(A_{l})$. Hence $v$ lies in $\overline{W}$, where $W$ is the special subspace of $L$ of the form $[W_{1},W_{2}]\cap C_{L}(A_{l})$ and so by (\ref{eigenvectorsadnilp}) $v$ is ad-nilpotent of bounded index.
This proves that
\begin{multline}
\label{commutatorsvi}
\text{any commutator in}\, v_{1},\ldots,v_{s}\, \text{is ad-nilpotent of index at most}\, u.
\end{multline}
Remind that $C_{L}(a)=L_{p}(Y,C_{Y}(a))$. Proposition \ref{WilZel} shows that $C_{L}(a)$ satisfies a multilinear polynomial identity of $\{m,q\}$-bounded degree. This also holds in $C_{\overline{L}}(a)=\overline{C_{L}(a)}$. Therefore Corollary \ref{polynomialidentity} implies that $\overline{L}$ satisfies a polynomial identity of $\{m,q\}$-bounded degree. Combining this with (\ref{eigenvectorsadnilp}) and (\ref{commutatorsvi}) we are now able to apply Theorem \ref{liealgbnilp}. Thus $\overline{L}$ is nilpotent of $\{m,q,r\}$-bounded class and the same holds for $L$.
Since $Y$ is a $p$-group and $L=L_{p}(Y)$ is nilpotent of bounded class, it follows from Lemma \ref{PowerfulR} that $Y$ has a characteristic powerful subgroup $K$ of $\{m,q,r\}$-bounded index. By Lemma \ref{powerfulth} $K^{(d)}$ has bounded exponent and so we can pass to the quotient $Y/K^{(d)}$ and assume that $Y$ is of $\{m,q,r\}$-bounded derived length. We now recall that $Y=Y_{1}Y_{2}\ldots Y_{t}$ and each $Y_{j}$ is contained is some $G_j$. From the results obtained in Section 4 also each derived group $Y^{(i)}$ is a product of subgroups of the form $Y^{(i)}\cap Y_j$. Thus every $Y^{(i)}$ can be generated by elements whose orders divide $m$. Since the derived length of $Y$ is bounded, we conclude that $Y$ has $\{m,q,r\}$-bounded exponent, as required.
\end{proof}
Finally we are ready to complete the proof of Theorem \ref{PR}.
\begin{proof}[Proof of Theorem \ref{PR}]
Note that it suffices to prove that there is a bound, depending only on $m,q$ and $r$, on the exponent of a Sylow $p$-subgroup of $G^{(d)}$ for each prime $p$.
Indeed, let $\pi(G^{(d)})$ be the set of prime divisors of $|G^{(d)}|$. Choose $p\in \pi(G^{(d)})$. It follows from Lemma \ref{FG1}(2) that $G^{(d)}$ possesses an $A$-invariant Sylow $p$-subgroup, say $P$. By Corollary \ref{generation3}, $P=P_{1}P_{2}\cdots P_{t}$, where each $P_{j}$ is of the form $P\cap H$ for some $A$-special subgroup $H$ of $G$ of degree $d$. Combining this fact with Proposition \ref{PAspecial}(4) we see that each $P_{j}$ is contained in $C_{G}(B)^{(d)}$ for a suitable subgroup $B$ of $A$ and thus $P_{j}\leq C_{G}(a)^{(d)}$, for some $a\in A^{\#}$. Since the exponent of $C_{G}(a)^{(d)}$ divides $m$, so does $p$.
From Lemma \ref{pgroupth} we know that $P^{(d)}$ has $\{m,q,r\}$-bounded exponent. Moreover by Lemma \ref{generation2} the subgroup $P^{(d-1)}$ is generated by subgroups of the form $P^{(d-1)}\cap P_{j}$, for $j=1,\ldots,t$, so in particular $P^{(d-1)}$ is generated by elements of order dividing $m$. Since $P^{(d)}=(P^{(d-1)})'$ has bounded exponent it is clear that also the exponent of $P^{(d-1)}$ is $\{m,q,r\}$-bounded. Repeating the same argument several times we see that all subgroups $P^{(d-2)},\ldots,P'$ and $P$ are generated by elements whose orders divide $m$ and so we conclude that $P$ has $\{m,q,r\}$-bounded exponent, as desired. This completes the proof.
\end{proof}
\section{The other part of the conjecture}
\label{gamma case}
In this last section we will deal with part (1) of Conjecture \ref{255}. The proof of that part is similar to that of part (2) but in fact it is easier. Therefore we will not give a detailed proof here but rather describe only steps where the proof of part (1) is somewhat different from that of part (2).
The definition of $A$-special subgroups of $G$ needs to be modified in the following way.
\begin{definition}
\label{gammaAspecial}
Let $A$ be an elementary abelian $q$-group acting on a finite $q'$-group $G$.
Let $A_{1},\ldots,A_{s}$ be the subgroups of index $q$ in $A$ and $H$ a subgroup of $G$.
\begin{itemize}
\item
We say that $H$ is a $\gamma$-$A$-special subgroup of $G$ of degree $1$ if and only if $H=C_{G}(A_{i})$ for suitable $i\leq s$.
\item Suppose that $k\geq 2$ and the $\gamma$-$A$-special subgroups of $G$ of degree $k-1$ are defined. Then $H$ is a $\gamma$-$A$-special subgroup of $G$ of degree $k$ if and only if there exists a $\gamma$-$A$-special subgroup $J$ of $G$ of degree $k-1$ such that $H=[J,C_{G}(A_{i})]\cap C_{G}(A_{j})$ for suitable $i,j\leq s$.
\end{itemize}
\end{definition}
The next proposition is similar to Proposition \ref{PAspecial}.
\begin{proposition}
\label{gammaPAspecial}
Let $A$ be an elementary abelian $q$-group of order $q^{r}$ with $r\geq 2$ acting on a finite $q'$-group $G$
and $A_{1},\ldots,A_{s}$ the maximal subgroups of $A$. Let $k\geq 1$ be an integer.
\begin{enumerate}
\item If $k\geq 2$, then every $\gamma$-$A$-special subgroup of $G$ of degree $k$ is contained in some $\gamma$-$A$-special subgroup of $G$ of degree $k-1$.
\item Let $R_{k}$ be the subgroup generated by all $\gamma$-$A$-special subgroups of $G$ of degree $k$. Then $R_{k}=\gamma_{k}{(G)}$.
\item If $k\leq r-1$ and $H$ is a $\gamma$-$A$-special subgroup of $G$ of degree $k$, then $H\leq \gamma_{k}(C_{G}(B))$ for some subgroup $B\leq A$ such that $|A/B|\leq q^{k}$.
\item Suppose that $G=G'$ and let $N$ be an $A$-invariant subgroup such that $N=[N,G]$. Then for every $k\geq 1$ the subgroup $N$ is generated by subgroups of the form $N\cap H$, where $H$ is some $\gamma$-$A$-special subgroup of $G$ of degree $k$.
\item Let $H$ be a $\gamma$-$A$-special subgroup of $G$. If $N$ is an $A$-invariant normal subgroup of $G$, then the image of $H$ in $G/N$ is a $\gamma$-$A$-special subgroup of $G/N$.
\end{enumerate}
\end{proposition}
The above properties of $\gamma$-$A$-special subgroups are essential in the proof
of the following generation result, which is analogous to Theorem \ref{generation1}.
\begin{theorem}
\label{gamma_generation1}
Assume $r\geq 2$. Let $P$ be an $A$-invariant Sylow $p$-subgroup of $\gamma_{r-1}(G)$. Let $P_{1},\ldots,P_{t}$ be all the subgroups of the form $P\cap H$ where $H$ is some $\gamma$-$A$-special subgroup of $G$ of degree $r-1$. Then $P=\langle P_{1},\dots,P_{t}\rangle$.
\end{theorem}
From this one can deduce
\begin{theorem}
\label{gamma_PR}
Let $m$ be a positive integer, $q$ a prime and $A$ an elementary
abelian group of order $q^r$ with $r\ge 2$ acting
on a finite $q'$-group $G$. If $\gamma_{r-1}(C_G(a))$ has exponent dividing
$m$ for any $a\in A^\#$, then $\gamma_{r-1}(G)$
has $\{m,q,r\}$-bounded exponent.
\end{theorem}
The above theorem shows that part (1) of Conjecture \ref{255} is correct. The proof of Theorem
\ref{gamma_PR} can be obtained in the same way as that of Theorem \ref{PR} with only obvious changes required. Thus, we omit the further details.
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Korea Electric Power Corp. (KEPCO), the largest power provider in South Korea will introduce blockchain technology as part of its innovations towards its new microgrid (MG).
In a statement on Monday, Nov. 19, the company announced plans to replace its current microgrid with a more eco-friendly one and that would be done with the help of blockchain.
KEPCO explained the role of blockchain in the process. The statement read:
“By using P2G technology, the remaining electricity can be converted into hydrogen and saved, and it can be converted to electric energy through the fuel cell when necessary. It can increase the energy independence rate and efficiency more than the existing microgrid, and it is eco-friendly because it does not emit greenhouse gas.”
It earlier noted that the current MG consists of small photovoltaics, wind turbines, and energy storage devices, which made it difficult to supply stable power. It, therefore, plans to replace the same with a new MG known as the ‘KEPCO Open MG’ which is a localized energy source connected to the main grid that can operate separately on its own. KEPCO explains that the new open MG will use power-to-gas (P2G) technology that converts electricity into gas fuel and it has plans to make it the first mega-wattage (MW) energy-independent microgrid.
According to President and CEO of the company, Kim Jong-gap, decentralization is one of the three major trends in the future of the energy industry. Others being decarbonization and digitalization.
Blockchain For efficient Power and Energy Systems
As Smartereum reported, blockchain technology—Ethereum blockchain, in particular, is being used in Germany to improve the efficiency in the energy sector. Early this month, the project Lition which helps residents find cheaper energy sources was already serving 700 hundred households in 10 cities having gotten licensed in 12 German cities including Hamburg, Berlin and Munich.
Yet, Lition is only a small player in a larger decentralized energy market. Singapore Power Group, the energy utility provider in Singapore had in October launched a decentralized marketplace for renewable energy trading. Using blockchain technology, the system will efficiently issue Energy tradable certificates (RECs) which can be generated from renewable energy generation like solar and traded with a buyer who will be matched automatically on the blockchain platform.
Similarly, KEPCO noted that through its new MG project it will show the speed of new and renewable energy generation and energy efficiency projects. “We will take a step forward as an energy platform provider to drive energy conversion and digital conversion,” President Kim remarked.
Last month, KEPCO has partnered with Mitsubishi UFJ Bank, IT service management company Nihon Unisys, and the University of Tokyo on a joint research project on how to utilize blockchain for distributed electricity supply.
| 338,526
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Report Abuse
Graph Betweenness is used to compute either the Betweenness metric of an edge or node in a graph, which are metrics related to centrality.
This is a part of a series of custom modules based on the CRAN igraph package. Graph Betweenness is used to compute either Betweenness metric of an edge or node in a graph, which are metrics related to the edge or node centrality, respectively. sorted. ![enter image description here][4] And here is an example of the output when computing Node Betweenness without sorting the results. ![enter image description here][5] [1]: [2]: [3]: [4]: [5]:
| 12,319
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\begin{document}
\begin{abstract}
We extend Deligne's weight filtration to the integer cohomology of complex analytic spaces
(endowed with an equivalence class of compactifications). In general, the weight filtration that we obtain is
not part of a mixed Hodge structure.
Our purely geometric proof is based on cubical descent
for resolution of singularities and Poincar\'{e}-Verdier duality.
Using similar techniques, we introduce the singularity filtration on the cohomology of compactificable analytic spaces.
This is a new and natural analytic invariant which does not depend on the equivalence class of compactifications and is related to the weight filtration.
\end{abstract}
\maketitle
\section*{Introduction}
The weight filtration was introduced by Deligne \cite{DeHII}, \cite{DeHIII} following Grothendieck's yoga of weights,
and as the key ingredient of mixed Hodge theory.
This is an increasing filtration defined functorially on the rational cohomology of every complex algebraic variety,
expressing the way in which its cohomology is related to
cohomologies of smooth projective varieties. In Deligne's approach, the weight filtration on the cohomology
of a singular complex algebraic variety $X$, supposed compact to simplify, is the
filtration induced by a smooth hypercovering $X_\bullet\to X$ of $X$. Indeed,
the induced spectral
sequence
$$E_1^{p,q}(X;A)=H^q(X_p;A)\Rightarrow H^{p+q}(X;A)$$
defines a filtration on $H^n(X;A)$ for any given coefficient ring $A$.
Using Hodge theory, Deligne proved that when $A=\QQ$, the above spectral sequence
degenerates at the second stage and that the
filtration on the rational cohomology is well-defined and functorial.
In \cite{GiSo}, Gillet and Soul\'{e} gave
an alternative proof of the well-definedness of the weight filtration
using smooth hypercoverings and algebraic K-theory.
Their more geometric approach allowed them to
obtain the result with integer coefficients, at least for compact support cohomology.
For a general coefficient ring $A$, the above spectral sequence
does not necessarily degenerate at the second stage. However, they proved that, from the second stage onwards,
the corresponding spectral sequence for cohomology with compact supports is a well-defined algebraic invariant of the variety.
An analogous construction yielded a weight filtration on algebraic K-theory with compact supports (see also the work of Pascual-Rubi\'{o} \cite{PR}).
Based on cubical hyperresolutions (see \cite{GNPP}),
Guill\'{e}n and Navarro-Aznar \cite{GN} developed a general descent theory which
allows one to extend contravariant
functors compatible with elementary acyclic squares (smooth blow-ups) on the category of smooth
schemes, to functors on the category of all schemes in such a way that the extended
functor is compatible with acyclic squares (abstract blow-ups).
Totaro observed in his ICM lecture \cite{To} that using the results on cohomological descent
of \cite{GN}, the weight filtration is well-defined on the cohomology with compact supports of any
complex or real analytic space with a given equivalence class of compactifications, for which
Hironaka's resolution of singularities holds.
Following this idea, and using Poincar\'{e} duality for real manifolds with
$\ZZ_2$-coefficients, McCrory and Parusinski obtained the weight filtration for
real algebraic varieties on Borel-Moore $\ZZ_2$-homology in \cite{MCPI}, and on compactly supported
$\ZZ_2$-homology in \cite{MCPII}.
Let $X$ be a compactificable complex analytic space.
Every cubical hyperresolution $X_\bullet \to X$ induces a spectral sequence
$$E_1^{p,q}(X;A)=\bigoplus_{|\alpha|=p}H^q(X_\alpha;A)\Rightarrow H^{p+q}(X;A)$$
converging to a filtration of $H^n(X;A)$, which we call \textit{singularity filtration}. In this note we prove
that from the $E_2$-term onwards, this spectral sequence does not depend
on the choice of the hyperresolution (Theorem $\ref{singan}$).
In the same vein, in Theorem $\ref{weightan}$ we obtain a generalization of the weight filtration
on the cohomology $H^n(X;A)$, when $X$ is endowed with an
equivalence class of compactifications (see Definition $\ref{equivclass}$).
In particular, if $X$ is a complex algebraic variety, the weight filtration is well-defined on $H^n(X;A)$.
For smooth manifolds, the singularity filtration is trivial, while for compact analytic spaces, it
coincides with the weight filtration.
To obtain these results we give
an analytic version of the extension criterion of functors
of \cite{GN} (see Theorem $\ref{descens1an}$).
The analytic setting differs from the algebraic setting appearing in loc.cit., mainly
due to the weaker formulation of Chow-Hironaka's Lemma and certain finiteness issues,
and it may find applications to study other topological invariants.
For instance, it should allow one to define a Hodge and a weight
filtration on the rational homotopy type of complex analytic spaces,
extending the filtrations obtained by Morgan \cite{Mo}, to the analytic setting.
We will present this multiplicative theory elsewhere.
Our study of the singularity and the weight filtrations is also valid for the cohomology
with $\ZZ_2$ coefficients of real analytic spaces.
Other cohomology theories such as Borel-Moore homology or cohomology
with compact supports could also be studied using parallel techniques,
allowing a comparison with the quoted results of Gillet-Soul\'{e} and McCrory-Parusinski.
\\
Section $\ref{prelims}$ contains a brief exposition of the main results of \cite{GN}
on cohomological descent categories and the extension criterion of functors.
In Section $\ref{fcx}$ we show that the category of filtered complexes over an abelian category
admits a cohomological descent structure with respect to the class of weak equivalences given by $E_r$-quasi-isomorphisms:
morphisms of filtered complexes inducing a quasi-isomorphism
at the $r$-stage of the associated spectral sequence (see Theorem $\ref{cohdescentfiltcomplexes}$).
In Section $\ref{ext_anal}$ we establish an extension criterion of functors from smooth to singular
complex analytic spaces (Theorem $\ref{descens1an}$) as well as a relative version (Theorem $\ref{descens2an}$).
In Section $\ref{acyclicity}$ we study the behavior of the cohomology functor with respect to acyclic squares of analytic spaces.
We then study the Gysin complex of a smooth compactification $U\hookrightarrow X$ with $D=X-U$ a normal crossings divisor.
In Sections $\ref{sec_sing}$ and $\ref{sec_weightan}$ we use the results of the previous sections to
define the singularity and weight filtrations respectively, on the cohomology with coefficients in an arbitrary ring $A$,
of compactificable complex analytic spaces (Theorem $\ref{singan}$ and Theorem $\ref{weightan}$).
\section{Preliminaries}\label{prelims}
The extension criterion of functors of \cite{GN} is based on the assumption that the target category
is a \textit{cohomological descent category}, a variant of the triangulated categories of Verdier.
This is essentially a category $\Dd$ endowed with a saturated class of \textit{weak equivalences} $\Ee$,
and a \textit{simple functor} $\mathbf{s}$ sending every cubical codiagram of $\Dd$ to an object of $\Dd$ and
satisfying certain axioms analogous to those of the total complex of a double complex.
The simple functor can be viewed as the homotopy limit, and allows to define
realizable homotopy limits for diagrams indexed by finite categories (see \cite{Bea}).
The basic idea of the extension criterion of functors is that, given a functor compatible with smooth
blow-ups from the category of smooth schemes to a cohomological descent category, there
exists an extension to all schemes. Furthermore, the extension is essentially unique,
and is compatible with general blow-ups.
Let us first recall some features of cubical codiagrams and cohomological descent categories.
We refer to \cite{GN} for the precise definitions.
\begin{nada}
Given a set $\{0,\cdots,n\}$, with $n\geq 0$, the set of its non-empty parts,
ordered by the inclusion, defines the category $\square_n$.
Likewise, any non-empty
finite set $S$ defines the category $\square_S$.
Every injective map $u:S\to T$ between non-empty finite sets
induces a functor $\square_u:\square_S\to\square_T$
defined by $\square_u(\alpha)=u(\alpha)$.
Denote by $\Pi$
the category whose objects are finite products of categories $\square_S$ and whose morphisms
are the functors associated with injective maps in each component.
\end{nada}
\begin{defi}
Let $\Dd$ be an arbitrary category. A \textit{cubical codiagram} of $\Dd$ is
a pair $(X,\square)$, where $\square$ is an object of $\Pi$ and $X$ is a functor $X:\square\to \Dd$.
A \textit{morphism} $(X,\square)\to (Y,\square')$ between cubical codiagrams is given by a pair $(a,\delta)$ where
$\delta:\square'\to \square$ is a morphism of $\Pi$ and $a:\delta^*X:=X\circ\delta \to Y$ is a natural transformation.
\end{defi}
Denote by $CoDiag_{\Pi}\Dd$ the category of cubical codiagrams of $\Dd$.
\begin{defi}
A \textit{cohomological descent category} is given by
a cartesian category $\Dd$ provided with an initial object $1$, together with
a saturated class of morphisms $\Ee$ of $\Dd$ which is stable by products,
called \textit{weak equivalences}, and a contravariant functor
$\mathbf{s}:CoDiag_{\,\Pi}\Dd\to \Dd$, called the \textit{simple functor}.
The data $(\Dd,\Ee,\mathbf{s})$ must satisfy the axioms of Definition 1.5.3 of \cite{GN}.
Objects weakly equivalent to the initial object $1$ are called \textit{acyclic}.
We shall denote by $\Ho(\Dd):=\Dd[\Ee^{-1}]$ the localized category of $\Dd$ with respect to the class of weak equivalences.
\end{defi}
The primary example of a cohomological descent category is given by the category of complexes $\Cx{\Aa}$ of an abelian category $\Aa$,
with the weak equivalences being quasi-isomorphisms and the simple
functor $\mathbf{s}$ defined via the total complex.
Let $\Dd$ be a category with initial object and a simple functor $\mathbf{s}:CoDiag_{\,\Pi}\Dd\to \Dd$.
In a large class of examples, a cohomological descent structure on $\Dd$ is given after lifting the class of
quasi-isomorphisms on $\Cx{\Aa}$ via a functor $\Psi:\Dd\lra \Cx{\Aa}$, provided that $\Psi$ is compatible
with the simple functor (see Proposition 1.5.12 of \cite{GN}).
\begin{examples} The following categories inherit a cohomological descent structure via functors with values in categories of complexes:
\begin{enumerate}[(1)]
\item The category $\mathrm{Top}$ of topological spaces, with the simple functor $\mathbf{s}$ given by the geometric realization of cubical diagrams,
via the functor $S_*:\mathrm{Top}\lra C_+(\ZZ{\mathrm{-mod}})$ of singular chains.
\item The category $\dga{}{\kk}$ of differential graded algebras over a field $\kk$ of characteristic 0,
with the Thom-Whitney simple functor $\mathbf{s}_{TW}$ (see \cite{Na}), via the functor
$\dga{}{\kk}\lra \Cx{\mathrm{Vect}(\kk)}$ defined by forgetting the algebra structures.
\item The category $\mathbf{MHC}$ of mixed Hodge complexes with the cubical analog of Deligne's
simple functor $\mathbf{s}_D$ (see \cite{DeHIII}),
via the forgetful functor $\mathbf{MHC}\lra \Cx{\mathrm{Vect}(\QQ)}$.
\end{enumerate}
\end{examples}
In the Section $\ref{fcx}$ we will lift the cohomological descent structure on $\Cx{\Aa}$
to a family of structures for filtered complexes, via the functor defined by the $r$-stage of
the associated spectral sequences.
\begin{nada}[$\Phi$-rectified functors]
Let $\Dd$ be a cohomological descent category and let
$\square\in \Pi$. Denote by $\Dd^\square:=Fun(\square,\Dd)$ the category of diagrams of type $\square$ in $\Dd$.
The simple functor induces a functor $\Ho(\Dd^\square)\to \Ho(\Dd)$.
In general, we are interested in cubical diagrams in $\Ho(\Dd)$
and we do not have a simple functor $\Ho(\Dd)^\square\to \Ho(\Dd)$.
The notion of \textit{$\Phi$-rectified functor} corresponds, roughly speaking, to functors
$F:\Cc\to \Ho(\Dd)$ which are defined on all cubical diagrams in the form
$F^\square:\Cc^\square\to \Ho(\Dd^\square)$,
so that we can take the composition $\Cc^\square\to \Ho(\Dd^\square)\to \Ho(\Dd)$
(see 1.6 of \cite{GN}).
\end{nada}
Denote
by $\Sch{\kk}$ the category of reduced schemes, that are separated and of finite type over a field $\kk$
of characteristic 0. Denote by $\Sm{\kk}$ the full subcategory of smooth schemes.
\begin{defi}\label{defelement1_alg}
A cartesian diagram of $\Sch{\kk}$
$$
\xymatrix{
\ar[d]_g\wt Y\ar[r]^j&\wt X\ar[d]^f\\
Y\ar[r]^i&X
}
$$
is said to be an \textit{acyclic square} if $i$ is a closed immersion, $f$ is proper and it induces an isomorphism
$\wt X-\wt Y\to X-Y$.
It is an \textit{elementary acyclic square} if, in addition,
all the objects in the diagram are in $\Sm{\kk}$, and $f$ is the blow-up of $X$ along $Y$.
\end{defi}
\begin{teo}[\cite{GN}, Theorem. 2.1.5]\label{descens1alg}Let $\Dd$ be a cohomological descent category
and
$F:\Sm{\kk}\to \Ho(\Dd)$
a contravariant $\Phi$-rectified functor satisfying:
\begin{enumerate}
\item [{(F1)}] $F(\emptyset)$ is the final object of $\Dd$ and $F(X\sqcup Y)\to F(X)\times F(Y)$ is an isomorphism.
\item [{(F2)}] If $X_\bullet$ is an elementary acyclic square of $\Sm{\kk}$, then
$\mathbf{s}F(X_\bullet)$ is acyclic.
\end{enumerate}
Then there exists a contravariant $\Phi$-rectified
functor
$F':\Sch{\kk}\lra \Ho(\Dd)$
such that:
\begin{enumerate}
\item [(1)]If $X$ is an object of $\Sm{\kk}$, then $F'(X)\cong F(X)$.
\item [(2)]If $X_{\bullet}$ is an acyclic square of $\Sch{\kk}$, then $\mathbf{s}F'(X_{\bullet})$ is acyclic.
\end{enumerate}
In addition, the functor $F'$ is essentially unique.
\end{teo}
The main applications of the extension criterion appearing in \cite{GN}
concern the study of
the algebraic de Rham homotopy theory,
the Hodge filtration for complex analytic spaces and
the theory of Grothendieck motives (see Sections 3,4 and 5 of loc.cit.).
The extension criterion has been successfully applied on other algebraic situations, such as
the weight filtration in algebraic K-theory \cite{PR},
the mixed Hodge structures in rational homotopy \cite{Na}, \cite{CG1}, \cite{cirici}
or the weight filtration for real algebraic varieties \cite{MCPII}.
\section{Cohomological descent structures on the category of filtered complexes}\label{fcx}
In this section we show that the category of filtered complexes over an abelian category admits a cohomological descent structure,
where the weak equivalences are given by $E_r$-quasi-isomorphisms.
Let $\Aa$ be an abelian category. Denote by
$\mathbf{F}\Aa$ the category of filtered objects of $\Aa$,
and by $\FCx{\Aa}$ the category of complexes over objects of $\mathbf{F}\Aa$.
\begin{defi}
Let $r\geq 0$ be an integer.
A morphism of filtered complexes $f:K\to L$ is called \textit{$E_r$-quasi-isomorphism} if
the induced morphism $E_r(f):E_r(K)\to E_r(L)$ of the associated spectral sequences is a quasi-isomorphism
(the map $E_{r+1}(f)$ is an isomorphism).
\end{defi}
Denote by $\Ee_r$ the class of $E_r$-quasi-isomorphisms.
\begin{defi}\label{rderived}
The \textit{$r$-derived category} of $\mathbf{F}\Aa$ is the localized category
$$\mathbf{D}^+_r(\mathbf{F}\Aa):=\FCx{\Aa}[\Ee_r^{-1}]$$
of complexes of $\mathbf{F}\Aa$
with respect to the class of $E_r$-quasi-isomorphisms.
\end{defi}
Let $s>r$ and consider the functor $E_{s}:\FCx{\Aa}\to \Cx{\Aa}$ defined by sending a filtered complex to
the $E_s$-stage of its associated spectral sequence.
Since it sends morphisms of $\Ee_r$ to isomorphisms, there is a functor
$E_{s}:\mathbf{D}^+_r(\mathbf{F}\Aa)\to \Cx{\Aa}$.
Deligne introduced the d\'{e}calage of a filtered complex and
proved that its associated spectral sequences are related by a shift of indexing.
This proves to be a key tool in the study of filtered complexes and their cohomological descent properties.
\begin{defi}\label{decalage_defi_algs}
The \textit{d\'{e}calage} $\Dec K$ of a filtered complex $K$ is the filtered complex defined by
$$(\Dec W)_pK^n=W_{p-n}K^n\cap d^{-1}(W_{p-n-1}K^{n-1}).$$
\end{defi}
\begin{prop}[\cite{DeHII}, Prop. 1.3.4] \label{deligne_decalage}
The canonical morphism
$E_0^{p,n-p}(\Dec K)\to E_1^{p+n,-p}(K)$ is a quasi-isomorphism.
For all $r>0$, the induced morphism
$E_r^{p,n-p}(\Dec K)\to E_{r+1}^{p+n,-p}(K)$
is an isomorphism. In particular $\Ee_{r}=\Dec^{-1}(\Ee_{r-1})$ for all $r>0$.
\end{prop}
\begin{defi}
The \textit{$r$-simple} of a codiagram of filtered
complexes $K=(K,W)^\bullet$ is the filtered complex
$\mathbf{s}^r(K):=(\mathbf{s}(K),W(r))$
defined by
$$W(r)_p(\mathbf{s}(K))=\int_\alpha C^*(\Delta^{|\alpha|})\otimes W_{p+r|\alpha|}K^\alpha=
\bigoplus_{|\alpha|=0} W_pK^\alpha\oplus \bigoplus_{|\alpha|=1} W_{p+r}K^\alpha[-1]\oplus\cdots
$$
\end{defi}
Note that $\mathbf{s}^0$ and $\mathbf{s}^1$ correspond to the
filtered total complexes defined via the convolution with the trivial and the b\^{e}te filtrations respectively,
introduced by Deligne in \cite{DeHIII}. By forgetting the filtrations on $\mathbf{s}^r$
we recover the simple functor
$\mathbf{s}$ of complexes.
\begin{prop}\label{decala_simple_cxos}Let $K$ be a codiagram of filtered complexes.
Then for $r\geq 0$,
$$\Dec\left(\mathbf{s}^{r+1}(K)\right)\cong \mathbf{s}^r(\Dec K).$$
\end{prop}
\begin{proof}
The category $\FCx{\Aa}$ is complete. Furthermore, since
the d\'{e}calage has a left adjoint defined by the shift of a filtration (see \cite{CG2}), it commutes with pull-backs.
It also commutes with $r$-translations $(K,W)\mapsto (K[r], W(-r))$. We have:
$$\Dec\int_\alpha (C^*(\Delta^{|\alpha|})\otimes W_{p+r|\alpha|}K^\alpha)\cong\int_\alpha \Dec(C^*(\Delta^{|\alpha|})\otimes W_{p+r|\alpha|}K^\alpha)\cong
\int_\alpha C^*(\Delta^{|\alpha|})\otimes \Dec W_pK^\alpha.$$
This gives an isomorphism of the filtrations $\Dec(W(r+1))$ and $(\Dec W)(r)$ on $\mathbf{s}(K)$.
\end{proof}
\begin{prop}\label{Ercommutasimple}Let $K$ be a codiagram of filtered complexes.
For $r\geq 0$, there is a chain of quasi-isomorphisms $E_r^{*,q}(\mathbf{s}^r(K))\stackrel{\sim}{\longleftrightarrow}\mathbf{s}E_r^{*,q}(K)$.
\end{prop}
\begin{proof}
For $r=0$ we have an isomorphism $E_0^{*,q}(\mathbf{s}^0(K))\cong\mathbf{s}E_0^{*,q}(K)$.
Assume inductively that the proposition is true for $r-1$. We then have a chain of quasi-isomorphisms
$$E_r^{*,q}(\mathbf{s}^r(K))\stackrel{\sim}{\leftarrow}E_{r-1}^{-q,*+2q}(\Dec(\mathbf{s}^r(K)))\cong
E_{r-1}^{-q,*+2q}(\mathbf{s}^{r-1}(\Dec K))\cong \mathbf{s} E_{r-1}^{-q,*+2q}(\Dec K)\stackrel{\sim}{\to} \mathbf{s}E_r^{*,q}(K),$$
where the first and last quasi-isomorphisms follow from Proposition $\ref{deligne_decalage}$ and the isomorphisms follow from
Proposition $\ref{decala_simple_cxos}$ and the induction hypothesis
respectively.
\end{proof}
\begin{teo}\label{cohdescentfiltcomplexes}
Let $\Aa$ be an abelian category. The triple $(\FCx{\Aa}{},\Ee_r,\mathbf{s}^r)$ is a cohomological descent category for all $r\geq 0$.
\end{teo}
\begin{proof}
Consider the functor $E_r:\FCx{\Aa}\to \Cx{\Aa}$ defined by sending every filtered complex to the $r$-stage of its associated spectral sequence.
Then $\Ee_r=E_r^{-1}(\Ee)$, where $\Ee$ denotes the class of quasi-isomorphisms of $\Cx{\Aa}$.
Furthermore, by Proposition
$\ref{Ercommutasimple}$, the complexes
$E_r(\mathbf{s}^r(K))$ and $\mathbf{s}E_r(K)$ are isomorphic in the derived category $\mathbf{D}^+(\Aa)=\Cx{\Aa}[\Ee^{-1}]$, for every codiagram $K$ in $\FCx{\Aa}$.
This isomorphism is compatible with the morphisms $\mu$ and $\lambda$ of Definition 1.5.3 of \cite{GN}.
By Proposition 1.7.2 of \cite{GN} the triple $(\Cx{\Aa},\Ee,\mathbf{s})$ is a cohomological descent category.
Hence by Proposition 1.5.12 of loc.cit., this lifts to a
cohomological descent structure for the triple $(\FCx{\Aa},\Ee_r,\mathbf{s}^r)$.
\end{proof}
\begin{rmk}
For all $r\geq 0$, Deligne's d\'{e}calage is compatible with the cohomological descent structures
$\Dec:(\FCx{\Aa},\Ee_{r+1},\mathbf{s}^{r+1})\to (\FCx{\Aa},\Ee_{r},\mathbf{s}^{r})$. Furthermore,
it induces an equivalence of categories
$\Dec:\mathbf{D}^+_{r+1}(\mathbf{F}\Aa){\lra}\mathbf{D}^+_r(\mathbf{F}\Aa)$
(see Theorem 2.19 of \cite{CG2}).
\end{rmk}
\section{Extension criterion of functors for analytic spaces}\label{ext_anal}
Let $\An{\CC}$ denote the category of complex analytic spaces that are reduced, separated and of finite dimension.
Denote by $\Man{\CC}$ the full subcategory of smooth manifolds.
\begin{defi}\label{defelement1}
A cartesian diagram of $\An{\CC}$
$$
\xymatrix{
\ar[d]_g\wt Y\ar[r]^j&\wt X\ar[d]^f\\
Y\ar[r]^i&X
}
$$
is said to be an \textit{acyclic square} if $i$ is a closed immersion, $f$ is proper and it induces an isomorphism
$\wt X-\wt Y\to X-Y$.
It is an \textit{elementary acyclic square} if, in addition,
all the objects in the diagram are in $\Man{\CC}$, and $f$ is the blow-up of $X$ along $Y$.
In the latter case, the map $f$ is said to be an
\textit{elementary proper modification}.
\end{defi}
\begin{rmk}\label{finiteness}
In the analytic setting, we still have Hironaka's resolution of singularities.
However, in order to provide an extension criterion valid for analytic spaces,
we need to address certain issues concerning finiteness.
The first of this issues is Chow-Hironaka's Lemma (\cite{Hi}, 0.5),
stating that every proper birational map of irreducible schemes
factors as a composition of a finite sequence of blow-ups with smooth centers.
This result allows the passage
from acyclic squares to elementary acyclic squares in the hypotheses of the extension criterion.
In the analytic setting, the
factorization is made through the composition of a possibly infinite sequence of blow-ups, which is locally finite.
This is a consequence of
Hironaka's Flattening Theorem \cite{Hi2}.
The second issue concerns the finiteness of $\nu(X)=(n,c_n(X),\cdots,c_0(X))$,
where $c_i(X)$ is the number of irreducible components of dimension $i$, of a variety $X$ of dimension $n$,
which contain the singular points of $X$.
If $X$ is an algebraic variety, then $c_i(X)$ is finite for all $i$.
However, for an analytic space this may not be the case.
For compactificable analytic spaces, these two drawbacks disappear.
\end{rmk}
Denote by $\Manc{\CC}$ (resp. $\Anc{\CC}$) the full subcategory of $\Man{\CC}$ (resp. $\An{\CC}$) of compactificable analytic spaces.
The following is an analytic version of the extension criterion of functors defined over smooth schemes (see Theorem 2.1.10 of \cite{GN}).
\begin{teo}\label{descens1an}Let $\Dd$ be a cohomological descent category,
and let
$$F:\Manc{\CC}\lra \Ho(\Dd)$$
be a contravariant $\Phi$-rectified functor satisfying:
\begin{enumerate}
\item [{(F1)}] $F(\emptyset)$ is the final object of $\Dd$ and $F(X\sqcup Y)\to F(X)\times F(Y)$ is an isomorphism.
\item [{(F2)}] If $X_\bullet$ is an elementary acyclic square of $\Manc{\CC}$, then
$\mathbf{s}F(X_\bullet)$ is acyclic.
\end{enumerate}
Then there exists a contravariant $\Phi$-rectified
functor
$$F':\Anc{\CC}\lra \Ho(\Dd)$$
such that:
\begin{enumerate}
\item [(1)]If $X$ is an object of $\Manc{\CC}$, then $F'(X)\cong F(X)$.
\item [(2)]If $X_{\bullet}$ is an acyclic square of $\Anc{\CC}$, then $\mathbf{s}F'(X_{\bullet})$ is acyclic.
\end{enumerate}
In addition, the functor $F'$ is essentially unique.
\end{teo}
\begin{proof}
With the notations of 2.1.10 of \cite{GN}, this is equivalent to prove that the inclusion functor
$\Manc{\CC}\to \Anc{\CC}$ verifies the extension property.
It suffices to replace $\Mm'=\Sm{\kk}$ by $\Manc{\CC}$ and $\Mm=\Sch{\kk}$ by $\Anc{\CC}$ in the proof of Theorem 2.1.5 of loc.cit.,
which by Remark $\ref{finiteness}$ is valid for compactificable analytic spaces.
\end{proof}
To prove the invariance of the weight filtration we will use a relative version of the above result.
Let $\An{\CC}^2$ denote the category of pairs $(X,U)$ where $X$ is an analytic space and
$U$ is an open subset of $X$ such that $D=X-U$ is a closed analytic subspace of $X$.
Likewise, let $\Man{\CC}^2$ be the full subcategory of $\An{\CC}^2$ of those pairs
$(X,U)$ with $X$ smooth and $D=X-U$ a normal crossings divisor in $X$ which is a union of smooth divisors.
\begin{defi}\label{defelement2}
A commutative diagram of $\An{\CC}^2$
$$
\xymatrix{
\ar[d]_g(\wt Y,\wt U\cap \wt Y)\ar[r]^j&(\wt X,\wt U)\ar[d]^{f}\\
(Y,U\cap Y)\ar[r]^i&(X,U)
}
$$
is said to be an \textit{acyclic square} if $f:\wt X\to X$ is proper, $i:Y\to X$ is a closed immersion,
the diagram of the first components is cartesian,
$f^{-1}(U)=\wt U$ and the diagram of the second components is an acyclic square of $\An{\CC}$.
\end{defi}
\begin{defi}
A morphism $f:(\wt X,\wt U)\to (X,U)$ in $\Man{\CC}^2$ is called \textit{proper elementary modification} if
$f:\wt X\to X$ is the blow-up of $X$ along a smooth center $Y$ which has normal crossings with the
complementary of $U$ in $X$, and if $\wt U=f^{-1}(U)$.
\end{defi}
\begin{defi}
An acyclic square of objects of $\Man{\CC}^2$ is said to be an \textit{elementary acyclic square}
if the map $f:(\wt X,\wt U)\to (X,U)$ is a proper elementary modification,
and the diagram of the second components is an elementary acyclic square of $\Man{\CC}$.
\end{defi}
Let $\An{\CC}^2_{comp}$ denote the full subcategory of $\An{\CC}^2$
given by those pairs $(X,U)$ such that $X$ is compact. Define $\Man{\CC}^2_{comp}$ similarly.
In particular, if $(X,U)\in \An{\CC}^2_{comp}$ we have that both $X$ and $U$ are objects of $\Anc{\CC}$.
\begin{nada}
Denote by $\gamma:\An{\CC}^2_{comp}\to \An{\CC}$ the forgetful functor $(X,U)\mapsto U$, and let $\Sigma$ be the class of morphisms
$s$ of $\An{\CC}^2_{comp}$ such that $\gamma(s)$ is an isomorphism. Then $\gamma$ induces a functor
$$\eta:\An{\CC}^2_{comp}[\Sigma^{-1}]\lra \An{\CC}.$$
In the algebraic situation, Nagata's Compactification Theorem implies that the functor $\eta^{alg}$
induces an equivalence of categories
$$\eta^{alg}:\Sch{\CC}^2_{comp}[\Sigma^{-1}]\stackrel{\sim}{\lra} \Sch{\CC}.$$
This does not hold in the analytic case. However,
the localized category $\An{\CC}^2_{comp}[\Sigma^{-1}]$
is equivalent to the category $\An{\CC}_\infty$ defined as follows.
\end{nada}
\begin{defi}[\cite{GN}, 4.7]\label{equivclass}
An \textit{object} of $\An{\CC}_\infty$ is given by an equivalence class of objects $(X,U)$ of $\An{\CC}^2_{comp}$,
where $(X,U)$ and $(X',U')$ are said to be \textit{equivalent} if
$U=U'$ and there exists a third compactification of $U$ that dominates $X$ and $X'$.
Two compactifications
$f_1:X_1\to X_1'$ and $f_2:X_2\to X_2'$ of a morphism of
analytic spaces $f:U\to U'$ are said to be \textit{equivalent}
if there exists a third compactification $f_3:X_3\to X_3'$ which dominates them.
A \textit{morphism} of $\An{\CC}_\infty$ is given by an equivalence class of morphisms of $\An{\CC}^2_{comp}$.
\end{defi}
An \textit{acyclic square} in $\An{\CC}_\infty$ is a square induced by an acyclic square of $\An{\CC}^2_{comp}$.
The following is an analytic version of Theorem 2.3.6 of \cite{GN}.
\begin{teo}\label{descens2an}Let $\Dd$ be a cohomological descent category,
and let
$$F:\Man{\CC}^2_{comp}\lra \Ho(\Dd)$$
be a contravariant $\Phi$-rectified functor satisfying:
\begin{enumerate}
\item [{(F1)}] $F(\emptyset,\emptyset)$ is the final object of $\Dd$ and $F((X,U)\sqcup (Y,V))\to F(X,U)\times F(Y,V)$ is an isomorphism.
\item [{(F2)}] If $(X_\bullet,U_\bullet)$ is an elementary acyclic square of $\Man{\CC}^2_{comp}$, then
$\mathbf{s}F(X_\bullet,U_\bullet)$ is acyclic.
\end{enumerate}
Then there exists a contravariant $\Phi$-rectified
functor
$$F':\An{\CC}_\infty\lra \Ho(\Dd)$$
such that:
\begin{enumerate}
\item [(1)]If $(X,U)$ is an object of $\Man{\CC}^2_{comp}$, then $F'(U_\infty)\cong F(X,U)$.
\item [(2)]If $U_{\infty\bullet}$ is an acyclic square of $\An{\CC}_{\infty}$, then $\mathbf{s}F'(U_{\infty\bullet})$ is acyclic.
\end{enumerate}
In addition, the functor $F'$ is essentially unique.
\end{teo}
\begin{proof}If $(X,U)$ is an object of $\An{\CC}^2_{comp}$, then both $U$ and $X$ are objects of $\Anc{\CC}$.
Hence by Remark $\ref{finiteness}$, the proof of Theorem 2.3.3 of \cite{GN}
applies to show that the inclusion functor
$\Man{\CC}^2_{comp}\to \An{\CC}^2_{comp}$ verifies the extension property 2.1.10 of loc. cit..
Therefore there exists a $\Phi$-rectified functor $F':\An{\CC}^2_{comp}\to \Ho(\Dd)$ satisfying:
\begin{enumerate}
\item [(1')]If $(X,U)$ is an object of $\Man{\CC}^2_{comp}$, then $F'(X,U)\cong F(X,U)$.
\item [(2')]If $(X_\bullet,U_\bullet)$ is an acyclic square of $\An{\CC}^2_{comp}$, then $\mathbf{s}F'(X_\bullet,U_\bullet)$ is acyclic.
\end{enumerate}
Furthermore, the functor $F'$ is essentially unique.
The remaining of the proof follows analogously to that of Theorem 2.3.6 of loc. cit., via
the equivalence of categories $$\eta:\An{\CC}^2_{comp}[\Sigma^{-1}]\stackrel{\sim}{\lra} \An{\CC}_\infty$$
given in 4.7 of loc.cit..
\end{proof}
\begin{rmk}\label{real}
The results of this section are also valid in the real setting, replacing $\CC$ by $\RR$, since Hironaka's resolution of singularities and
the analytic version of
Chow-Hironaka's Lemma
hold for real analytic spaces (see \cite{Hi}).
\end{rmk}
\section{Acyclic squares and Gysin complex}\label{acyclicity}
In this section we study the behavior of the cohomology functor with respect to certain acyclic squares of smooth analytic spaces.
We then introduce the Gysin complex of a pair $(X,U)$, where $U\hookrightarrow X$ is a smooth compactification with $D=X-U$ a normal crossings divisor
and describe its behavior with respect to elementary acyclic squares.
Let $X$ be a complex analytic space.
Given a commutative ring $A$ we will denote by $\underline{A}_X$ the constant sheaf over $X$ associated to $A$
and by $H^*(X;A)$ the singular cohomology of $X$ with coefficients in $A$.
For a continuous map $f:X\to Y$ we will denote $\Rr f_*:=f_*\Cc^{\bullet}_{Gdm}$, where $\Cc^{\bullet}_{Gdm}$ is the Godement resolution.
\begin{prop}\label{element1}
For every acyclic square of $\Man{\CC}$ as in Definition $\ref{defelement1}$, the sequence
$$0\to H^q(X;A)\xra{(f^*,-i^*)} H^q(\wt X;A)\oplus H^q(Y;A)\xra{j^*+g^*} H^q(\wt Y;A)\to 0$$
is exact for all $q$.
\end{prop}
\begin{proof}
We have a Mayer-Vietoris long exact sequence
$$\cdots \to H^q(X;A)\to H^q(\wt X;A)\oplus H^q(Y;A)\to H^q(\wt Y;A)\to H^{q+1}(X;A)\to\cdots$$
Therefore it suffices to see that the map $f^*:H^q(X;A)\to H^q(\wt X;A)$ is injective.
This is a well known consequence of Poincar\'{e}-Verdier duality, which gives the existence of a trace morphism
$f_{\sharp}$ such that $f_{\sharp}f^*=1$.
We recall the proof.
The map $f^*$ is induced by a morphism of sheaves $\underline{A}_X\to \Rr f_*\underline{A}_{\wt X}$. Since $f$ is proper,
$\Rr f_*=\Rr f_{!}$, and we have an adjunction $\Rr f_*\vdash f^!$. Since $\wt X$ and $X$ are smooth and of the same pure dimension,
there is a quasi-isomorphism $f^!\underline{A}_X\stackrel{\sim}{\to}\underline{A}_{\wt X}$.
The trace map $f_\sharp:\Rr f_*\underline{A}_{\wt X}\to \underline{A}_X$ is deduced by adjunction from the identity morphism
$1_X\in Hom(\underline{A}_X,\underline{A}_X)$
of
$$\Hom(\Rr f_*\underline{A}_{\wt X},\underline{A}_X)\to \Hom(\underline{A}_{\wt X},f^!\underline{A}_X)\cong
\Hom(\underline{A}_{\wt X},\underline{A}_{\wt X}).$$
Lastly, since $f$ is birational, the composition $f_{\sharp}\circ f^*:\underline{A}_X\to\Rr f_*\underline{A}_{\wt X}\to \underline{A}_X$
is the identity, since it coincides with the identity
over an open dense subset of $X$. Hence $f_\sharp$ induces a left inverse
of $f^*$, and $f^*$ is injective.
\end{proof}
We shall also use the following blow-up formula for cohomology.
A proof can be found in Theorem VI.4.5 of \cite{FL}, which is an axiomatization
of Theorem VII.3.7 of \cite{SGA6}.
\begin{prop}\label{blowup}
Consider an elementary acyclic square of $\Man{\CC}$ as in Definition $\ref{defelement1}$.
Let $m={\normalfont codim}_XY$, and let $\wt g^*=c_{m-1}(E)\cdot g^*:H^{*-2m}(Y;A)\to H^{*-2}(\wt Y;A) $, where
$c_{m-1}(E)\in H^{2m-2}(Y;A)$ denotes the $(m-1)$th-Chern class of the normal bundle $E=N_{Y/X}$ of $Y$ in $X$.
For all $q\geq 0$, there is a commutative square
$$
\xymatrix{
H^{q-2}(\wt Y;A)\ar[r]^{j_*}&H^q(\wt X;A)\\
\ar[u]^{\wt g^*}H^{q-2m}(Y;A)\ar[r]^{i_*}&H^q(X;A)\ar[u]_{f^*}
}
$$
such that the following sequence is exact
$$0\to H^{q-2m}(Y;A)\xra{(\wt g^*,-i_*)} H^{q-2}(\wt Y;A)\oplus H^q(X;A)\xra{j_*+f^*} H^q(\wt X;A)\to 0.$$
\end{prop}
\begin{nada}[Gysin complex]
Let $(X,U)\in \Man{\CC}^2$.
We may write $D:=X-U=D_1\cup\cdots D_N$ as the union of irreducible smooth divisors meeting transversally.
Let $D^{(0)}=X$ and for $0<p\leq N$ let
$D^{(p)}$ be the disjoint union of all $p$-fold intersections $D_I=D_{i_1}\cap\cdots\cap D_{i_p}$
with $I=\{i_1,\cdots,i_p\}\subset\{1,\cdots,N\}$.
Since $D$ is a normal crossings divisor, $D^{(p)}$ is smooth. For all $q\geq 0$, the
\textit{Gysin complex} $G^q(X,U)$ is the cochain complex defined by
$$G^q(X,U)^p:=H^{q+2p}(D^{(-p)};A),$$
with $d^p:G^q(X,U)^p\to G^q(X,U)^{p+1}$ defined by the alternated sum of Gysin morphisms
$$i_{*}(I,J):H^{q+2p}(D_{J};A)\lra H^{q+2(p+1)}(D_{I};A),$$ where $I\subset J\subset\{1,\cdots,N\}$ and $|J|=|I|+1=-p$.
\end{nada}
\begin{lem}
For all $q\geq 0$, the Gysin complex defines a contravariant functor
$$G^q:\Man{\CC}^2\to \Cx{\Rmod}.$$
\end{lem}
\begin{proof}
Let
$f:(X',U')\to (X,U)$ be a morphism
in $\Man{\CC}^2$.
Let $D=X-U=D_1\cup\cdots\cup D_N$ and $D'=X'-U'=D'_1\cup\cdots\cup D'_{M}$. For every irreducible component $D_i$ of $D$, its inverse image divisor is a sum
$$f^{-1}(D_i)=\sum_{j=1}^{M} m_{ij} D_j'$$ of irreducible components of $D'$. Let $M_f=(m_{ij})$ denote the matrix of multiplicities of $f$.
We next define $G^q(f)^*:G^q(X,U)^*\to G^q(X',U')^*$.
Let $G^q(f)^0=f^*$. Let $I\subset\{1,\cdots,N\}$ and $J\subset\{1,\cdots,M\}$ be two sets with $|I|=|J|=p>0$.
Let $m_{IJ}$ denote the determinant of the minor of $M_f$ of indices $(I,J)$.
If $I$ and $J$ are such that $f(D_J')\subset D_I$, we define a morphism
$G^q(f)_{IJ}:H^q(D_I)\to H^q(D'_J)$
by letting $G^q(f)_{IJ}:=m_{IJ}f_{IJ}^*$, where
$f_{IJ}:D'_{J}\to D_I$ denotes the restriction of $f$.
If $I$ and $J$ are such that $f(D_J')\nsubseteq D_I$, we let $G^q(f)_{IJ}=0$.
Then the morphisms $G^*_{IJ}(f)$ are the components of $G^q(f)^p:G^q(X,U)^p\to G^q(X',U')^p$.
It follows from the decomposition property of determinants, that this is a map of complexes (see \cite{GN}, pag. 84).
If $g:(X'',U'')\to (X',U')$ is a morphism of $\Man{\CC}^2$,
then the matrix of multiplicities $M_{f\circ g}$ of $f\circ g$ is the product of the multiplicity matrices $M_f$ and $M_g$ of $f$ and $g$.
The functoriality of $G^q$ then follows from the functoriality of the determinants.
\end{proof}
\begin{prop}\label{element2}
Consider an elementary acyclic square of $\Man{\CC}^2$ as in Definition $\ref{defelement2}$.
\begin{enumerate}
\item If $Y\nsubseteq D$ then the simple of the double complex
\begin{equation}0\to G^q(X,U)\to G^q(\wt X,\wt U)\oplus G^q(Y,U\cap Y)\to G^q(\wt Y,\wt U\cap \wt Y)\to 0
\tag{$\ast$}
\end{equation}
is acyclic for all $q$.
\item If $Y\subset D$ then the map $G^q(X,U)\to G^q(\wt X,\wt U)$ is a quasi-isomorphism for all $q$.
\end{enumerate}
\end{prop}
\begin{proof}
We adapt the proof of Proposition 5.9 of \cite{GN} in the motivic setting
(see also \cite{MCPII}, Sections 5 and 6).
Assume that $Y\nsubseteq D$. We proceed by induction on the number $N$ of smooth irreducible components of $D$.
If $N=0$ then $G^q(X,U)=H^q(X;A)$ is concentrated in degree 0 and the sequence ($\ast$) becomes
that of Proposition $\ref{element1}$.
Assume that $N>0$. Let $D=D''\cup X'$ and $D'=D''\cap X'$, where $X'$ is a component of $D$.
From the definition of the Gysin complex we obtain an exact sequence
$$0\to G(X,X-D'')\to G(X,X-D)\to G(X',X'-D')[1]\to 0.$$
Denote by $(X_\bullet,X_\bullet-D''_\bullet)$ the commutative square
$$
\xymatrix{
\ar[d](\wt Y,\wt Y- \wt E'')\ar[r]&(\wt X,\wt X-\wt D'')\ar[d]\\
(Y,Y- E'')\ar[r]^i&(X,X- D'')
}
$$
where $E''=D''\cap Y$, $\wt D''=f^{-1}(D'')$ and $\wt E''=\wt D''\cap \wt Y$.
Consider the blow-up $\wt X'$ of $X'$ along $Y'=Y\cap X'$, and denote by
$(X_\bullet',X_\bullet'- D'_\bullet)$
the commutative square
$$
\xymatrix{
\ar[d](\wt Y',\wt Y'- \wt E')\ar[r]&(\wt X',\wt X'-\wt D')\ar[d]\\
(Y',Y'- E')\ar[r]^i&(X',X'- D')
}
$$
where
$E'=D'\cap Y'$, $\wt Y'=\wt X'\cap \wt Y$, $\wt D'=\wt X'\cap f^{-1}(D')$ and $\wt E'=\wt D'\cap \wt Y'$.
We then have a short exact sequence
$$0\to \mathbf{s}G(X_\bullet,X_\bullet- D''_\bullet)\to \mathbf{s}G(X_\bullet,X_\bullet- D_\bullet)\to
\mathbf{s}G(X'_\bullet,X'_\bullet- D'_\bullet)[1]\to 0.$$
By induction hypothesis, both $\mathbf{s}G(X_\bullet,X_\bullet- D''_\bullet)$ and
$\mathbf{s}G(X'_\bullet,X'_\bullet- D'_\bullet)$ are acyclic complexes.
Therefore the middle complex is acyclic, as desired. This proves (1).
Assume that $Y\subset D$. We proceed by induction over the number of components
$r$ of $D$ which contain $Y$, and the number $s$ of components
which do not contain $Y$.
Assume that $(r,s)=(1,0)$, so that $D$ is smooth irreducible and $Y\subset D$.
Then $G^q(X,X- D)$ is the simple of the morphism $H^{q-2}(D;A)\to H^q(X;A).$
Denote by $\widehat D$ the proper transform of $D$, and let $\widehat E=\wt Y\cap \widehat D$.
Denote by
$$\xymatrix{
\widehat E\ar[d]_{\widehat g}\ar[r]^{i_{\widehat E,\widehat D}}&\widehat{D}\ar[d]^{\widehat f}\\
Y\ar[r]^{i_{Y,D}}&D
}$$
the induced diagram.
Then
$\wt D=\wt Y\cup \widehat D$, and
$G^q(\wt X,\wt X- \wt D)$ is the simple of the square
$$
\xymatrix{
H^{q-4}(\widehat E;A)\ar[d]_{i_{\widehat E,\widetilde Y*}}\ar[r]^{i_{\widehat E,\widehat D*}}&H^{q-2}(\widehat D;A)\ar[d]^{i_{\widehat D,\widetilde X*}}\\
H^{q-2}(\wt Y;A)\ar[r]^{i_{\widetilde Y,\widetilde X*}}&H^{q}(\wt X;A)&.
}
$$
Therefore it suffices to show that the following complex is acyclic:
$$H^{q-2}(D;A)\oplus H^{q-4}(\widehat E;A)\xra{\alpha} H^{q-2}(\widehat D;A)\oplus H^{q-2}(\wt Y;A)\oplus H^q(X;A)\xra{\beta} H^{q}(\wt X;A),$$
where $\beta=i_{\widehat D,\widetilde X*}+i_{\widetilde Y,\widetilde X*}+f^*$ and $\alpha$ is given by the matrix
$$\left(
\begin{smallmatrix}
-\widehat f^*&-i_{\widehat E,\widehat D*}\\
-(i_{Y,D}\circ g)^*&i_{\widehat E,\widetilde Y*}\\
i_{DX*}&0
\end{smallmatrix}\right).
$$
After adding the acyclic complex $H^{q-2m}(Y;A)\lra H^{q-2m}(Y;A)$ and rearranging factors, we obtain the following complex
$$
\xymatrix@C=0pt@R=10pt{
& H^{q-2}(\wt Y;A) \ar[rrrrrrr]
& &&&&&& H^{q}(\wt X;A) \ar@{-}[dd]
\\
H^{q-2m}(Y;A) \ar[ur]\ar[rrrrrrr]
& &&&&&& H^{q}(X;A) \ar[ur]
\\
& H^{q-4}(\widehat E;A) \ar'[rrrrr]'[rrrrrr][rrrrrrr] \ar'[u][uu]
& &&&&&& H^{q-2}(\widehat D;A) \ar[uu]
\\
H^{q-2m}(Y;A) \ar[rrrrrrr]\ar[ur]\ar[uu]
&&&&&&& H^{q-2}(D;A) \ar[ur]\ar[uu] \ar'[uulllll][uuullllll]
}
$$
where the top and bottom faces of the cube are squares of blow-up type which are acyclic by Proposition $\ref{blowup}$.
Therefore the total complex of this complex is acyclic.
This proves (2) for the case $(r,s)=(1,0)$.
Assume that $r=1$ and $s>0$. Let $D=D''\cup X'$, where $Y\subsetneq X'$, and $D'=D''\cap X'$.
We have a commutative diagram with acyclic rows
$$
\xymatrix{
0\ar[r]&\ar[d]^{f^{''*}}G(X,X- D'')\ar[r]&\ar[d]^{f^{*}}G(X,X- D)\ar[r]&\ar[d]^{f^{'*}}G(X',X'- D')[1]\ar[r]&0\\
0\ar[r]&G(\wt X,\wt X- \wt D'')\ar[r]&G(\wt X,\wt X- \wt D)\ar[r]&G(\wt X',\wt X'- \wt D')[1]\ar[r]&0\\
}
$$
By induction hypothesis, the maps $f^{''*}$ and $f^{'*}$ are quasi-isomorphisms. Therefore $f^{*}$ is a quasi-isomorphism.
Assume that $r>1$ and consider a decomposition $D=D''\cup X'$ such that $Y\subset X'$.
An argument parallel to the previous case, by induction over $r$, shows that (2) is satisfied in the general case.
\end{proof}
\begin{rmk}\label{real_acyclic}
The results of this section have their analogues in the real setting,
by taking cohomology with $\ZZ_2$-coefficients, for which Poincar\'{e}-Verdier duality holds.
Indeed, the same proof of Proposition $\ref{element1}$ can be carried out in this case.
Proposition $\ref{blowup}$ appears in Section 4 of \cite{MCPII} for $\ZZ_2$-homology.
Note that one needs to adjust the degree of the cohomology groups appearing in Proposition $\ref{blowup}$ and in the definition of the Gysin complex
as done in loc.cit., since
a closed immersion $f:X\hookrightarrow Y$ of real algebraic varieties
induces a Gysin map $f_*:H^k(X;A)\to H^{k+m}(Y;A)$, where $m=\normalfont{codim}_YX$.
The proof of Proposition $\ref{element2}$ follows analogously, with the Gysin complex defined by
$G^q(X,U)^p:=H^{q+p}(D^{(-p)};\ZZ_2)$.
\end{rmk}
\section{Singularity filtration}\label{sec_sing}
The singularity filtration is an analytic invariant that appears naturally when we extend the
functor of singular chains with the trivial filtration, from smooth to singular analytic spaces,
using the $\Ee_1$-cohomological descent structure on filtered complexes.
Let $X$ be a complex manifold.
Given a commutative ring $A$ denote by $S^*(X;A)$ the complex of singular cochains of $X$ with
values in $A$, so that $H^n(S^*(X;A))=H^n(X;A)$.
Together with the trivial filtration defined on $S^*(X;A)$ this defines a functor
$\Ss:\Man{\CC}\lra \FCx{\Rmod}$.
Any extended functor $\Anc{\CC}\to \Ho(\Dd)$ defined via Theorem $\ref{descens1an}$
depends strongly on the cohomological descent structure that we consider on the category $\Dd$.
In our case of interest, we may extend the functor $\Ss$ to a functor
$\Anc{\CC}\lra \mathbf{D}_0^+({\mathbf{F}\Rmod})$,
using the cohomological descent structure on $\FCx{\Rmod}$ associated with the class of $E_0$-quasi-isomorphisms.
It is easy to see that this is an empty exercise: the extended filtration of the trivial filtration is also trivial.
However, if we consider the cohomological descent structure associated with $E_1$-quasi-isomorphisms,
we obtain a non-trivial filtration which for compact spaces coincides with the weight filtration.
\begin{teo}\label{singan}
There exists a $\Phi$-rectified functor
$\Ss':\Anc{\CC}\lra \mathbf{D}^+_1(\mathbf{F}\Rmod)$
such that:
\begin{enumerate}[(1)]
\item If $X\in \Anc{\CC}$ then $H^n(\Ss'(X))\cong H^n(X;A)$.
\item If $X$ is a smooth manifold then $\Ss'(X)=(S^*(X;A), L)$, where $L$ is the trivial filtration.
\item For every $p,q\in\ZZ$ and every acyclic square of $\Anc{\CC}$ as in Definition $\ref{defelement1}$
there is a long exact sequence
$$\cdots\to E_2^{p,q}(\Ss'(X))\to
E_2^{p,q}(\Ss'(\wt X))\oplus E_2^{p,q}(\Ss'(Y)) \to
E_2^{p,q}(\Ss'(\wt Y))\to E_2^{p+1,q}(\Ss'(X))\to \cdots$$
\item If $X$ is a compact complex algebraic variety and $A=\QQ$ then the filtration induced in cohomology
coincides with Deligne's weight filtration after d\'{e}calage.
\end{enumerate}
\end{teo}
\begin{proof}
By Theorem $\ref{cohdescentfiltcomplexes}$ the triple $(\FCx{\Rmod}{},\Ee_1,\mathbf{s}^1)$
is a cohomological descent category. Therefore it suffices to show
that the functor
$$\Manc{\CC}\stackrel{\Ss}{\lra}\FCx{\Rmod}\stackrel{\gamma}{\lra}\mathbf{D}^+_1(\mathbf{F}\Rmod)$$
given by $\Ss(X)=(S^*(X;A),t)$, where $t$ denotes the trivial filtration,
satisfies properties (F1) and (F2) of Theorem $\ref{descens1an}$.
Property (F1) is trivial.
Let us prove (F2).
This is equivalent to the condition that
for every elementary acyclic square
$X_\bullet\to X$ of $\Manc{\CC}$,
the map
$\Ss(X)\to \mathbf{s}^1(\Ss(X_\bullet))$ is an $E_1$-quasi-isomorphism.
By Proposition $\ref{Ercommutasimple}$, given a codiagram of filtered complexes $K^\bullet$, we have a chain of quasi-isomorphisms
$E_1^{*,q}(\mathbf{s}^1(K^\bullet))\stackrel{\sim}{\longleftrightarrow}\mathbf{s}E_1^{*,q}(K^\bullet).$
Hence
it suffices to check that for all $q\in\ZZ$, the sequence
$$\cdots\to E_2^{*,q}(\Ss(X))\to
E_2^{*,q}(\Ss(\wt X))\oplus E_2^{*,q}(\Ss(Y)) \to E_2^{*,q}(\Ss(\wt Y))\to E_2^{*+1,q}(\Ss(X))\to \cdots
$$
is exact.
Since the filtrations are trivial,
$E_1^{0,q}(\Ss(-))=H^q(S^*(-;A))=H^q(-;A)$ and
$E_1^{p,q}(\Ss(-))=0$ for $p\neq 0$.
Therefore it suffices to see that the sequence
$$0\to H^q(X;A)\lra H^q(\wt X;A)\oplus H^q(Y;A)\lra H^q(\wt Y;A)\to 0$$
is exact. This follows from Proposition $\ref{element1}$.
\end{proof}
\begin{defi}
Let $X$ be a compactificable complex analytic space.
The \textit{singularity spectral sequence} is the spectral sequence associated with the
filtered complex $\Ss'(X)$ of Theorem $\ref{singan}$.
Let $L'$ denote the increasing filtration induced on $H^{*}(X;A)$. The
\textit{singularity filtration} $L_p$ on $H^{*}(X;A)$ is defined by $L_pH^{n}(X;A):=L_{p-n}'H^{n}(X;A)$.
\end{defi}
\begin{cor}
Let $X$ be a compactificable complex analytic space. Then for every $n\geq 0$, its cohomology
$H^n(X;A)$ with values in any commutative ring $A$ carries a singularity filtration
$$0=L_{-1}\subset L_{0} \cdots\subset L_{n}=H^n(X;A)$$
which is functorial for morphisms in $\Anc{\CC}$ and satisfies:
\begin{enumerate}[1)]
\item If $X$ is smooth then $L$ is the trivial filtration.
\item If $X$ is a complex projective variety and $A=\QQ$ then $L$ coincides with Deligne's weight filtration.
\end{enumerate}
\end{cor}
Note that by Theorem $\ref{singan}$, the $_LE_2$-term of the singularity spectral sequence is well-defined.
The first term $_LE_1$, which is well-defined up to quasi-isomorphism, admits a description in terms of resolutions as follows:
let $X_\bullet\to X$ be a resolution of a compactificable complex analytic space $X$.
Then:
$$_LE_1^{p,q}(X;A)=\bigoplus_{|\alpha|=p} H^q(X_{\alpha};A) \Longrightarrow H^{p+q}(X;A).$$
If $X$ is a projective complex variety and $A=\QQ$ this corresponds to the analogous formula for the weight filtration
appearing in Theorem 8.1.15 of \cite{DeHIII} (see also IV.3 of \cite{GNPP}).
\begin{rmk}
The same arguments give a filtration $L$ on the homology with compact supports and on the Borel-Moore homology of a compactificable complex analytic space $X$.
In \cite{GZeeman}, Deligne's weight filtration $W$ and Zeeman's filtration $S$ are compared in the homology of a compact variety, giving the relation
$S^{2N-i-q}\subset W^{i-q}$ on $H_i(X;\QQ)$, where $N=\mathrm{dim}X$. The same proof would give the relation
$S^{2N-i-q}\subset L^{i-q}$ for the singularity filtration on the Borel-Moore homology $H_i^{BM}(X;A)$.
\end{rmk}
\begin{example}
Let $X$ be the open singular variety obtained by taking a Riemann surface of genus one,
identifying any two points and removing any two other points, as shown in the figure below.
The weight filtration on $H^1(X)\cong \QQ^4$ has length 3, and it is given by
$$Gr_0^WH^1(X)\cong \QQ[d],\, Gr_1^WH^1(X)\cong \QQ[a]\oplus\QQ[b]\text{ and }Gr_2^WH^1(X)\cong \QQ[c].$$
The singularity filtration has length 2, and it is given by
$$Gr_0^LH^1(X)\cong \QQ[d]\text{ and } Gr_1^LH^1(X)\cong \QQ[a]\oplus\QQ[b]\oplus\QQ[c].$$
\begin{center}
\includegraphics[width=6cm]{figura.png}
\end{center}
\end{example}
Note that in the above example, the singularity filtration is simpler than the weight filtration, since it only captures the singularity of the variety.
In certain situations, the singularity filtration results in a finer invariant than the weight filtration,
since the contributions from the singular part and the part at the infinity may cancel out, as in the example below.
\begin{example}
Let $N$ denote the rational node, i.e. $\CC\PP^1$ with two points identified, and let $Y=N\times\CC^*$.
The weight and singularity filtrations on $H^1(Y)\cong \QQ^2$ have both length 2, and are given by $Gr_0^WH^1(Y)=Gr_0^LH^1(Y)\cong \QQ$ and
$Gr_2^WH^1(Y)=Gr_1^LH^1(Y)\cong \QQ$.
The weight filtration on $H^2(Y)\cong \QQ^2$ is pure of weight $2$, so
$Gr_2^WH^2(Y)\cong \QQ^2$. In contrast, the singularity filtration
has two non-trivial graded pieces $Gr_1^LH^2(Y)\cong \QQ$ and $Gr_2^LH^2(Y)\cong \QQ$.
\end{example}
\section{Weight filtration}\label{sec_weightan}
Recall that the \textit{canonical filtration} $\tau$ is defined on any given complex $K$ by truncation:
$$\tau_{\leq p}K=\{\cdots\to K^{p-1}\to \Ker\,d\to 0\to 0\to\cdots\}.$$
Given $(X,U)\in \Man{\CC}^2_{comp}$, let $j:U\hookrightarrow X$ denote the inclusion,
and $(\Rr j_*\underline{A}_U,\tau)$ the filtered complex of sheaves given by the direct image of the constant sheaf $\underline{A}_U$,
together with the canonical filtration.
Taking the right derived functor of global sections we obtain a $\Phi$-rectified functor
$$\Ww:\Man{\CC}^2_{comp}\lra \mathbf{D}^+_1(\mathbf{F}\Rmod)$$
with values in the 1-derived category of filtered complexes of $A$-modules (see Definition $\ref{rderived}$),
given by
$$\Ww(X,U)=\Rr\Gamma(X,(\Rr j_*\underline{A}_U,\tau)).$$
By the properties of the global sections functor and the derived direct image functor, we have an isomorphism $H^n(\Ww(X,U))\cong H^n(U;A)$.
\begin{teo}\label{weightan}
There exists a $\Phi$-rectified functor
${\Ww}':\An{\CC}_\infty\lra \mathbf{D}^+_1(\mathbf{F}\Rmod)$
such that:
\begin{enumerate}[(1)]
\item If $U_\infty\in \An{\CC}_\infty$ then $H^n({\Ww}'(U_\infty))\cong H^n(U;A)$.
\item If $(X,U)$ is an object of $\Man{\CC}^2_{comp}$ then ${\Ww}'(U_\infty)\cong \Ww(X,U)$.
\item For every $p,q\in\ZZ$ and every acyclic square of $\An{\CC}_\infty$
$$
\xymatrix{
\ar[d]\wt Y_\infty\ar[r]&\wt X_\infty\ar[d]\\
Y_\infty\ar[r]&X_\infty
}
$$
there is a long exact sequence
$$\cdots\to E_2^{p,q}(\Ww'(X_\infty))\to
E_2^{p,q}(\Ww'(\wt X_\infty))\oplus E_2^{p,q}(\Ww'(Y_\infty)) \to
E_2^{p,q}(\Ww'(\wt Y_\infty))\to E_2^{p+1,q}(\Ww'(X_\infty))\to \cdots$$
\item If $X$ is a complex algebraic variety and $A=\QQ$ then the filtration induced in cohomology
coincides with Deligne's weight filtration after d\'{e}calage.
\end{enumerate}
\end{teo}
\begin{proof}
By Theorem $\ref{cohdescentfiltcomplexes}$ the triple $(\FCx{\Rmod}{},\Ee_1,\mathbf{s}^1)$
is a cohomological descent category. Therefore by Theorem $\ref{descens2an}$ it suffices to show
that the functor
$$\Ww:\Man{\CC}^2_{comp}\lra \mathbf{D}^+_1(\mathbf{F}\Rmod)$$
given by $\Ww(X,U)=\Rr\Gamma(X,(\Rr j_*\underline{A}_U,\tau))$
satisfies properties (F1) and (F2).
Property (F1) is trivial.
Condition (F2)
is equivalent to the condition that
the map
${\Ww}(X,U)\to \mathbf{s}^1(\Ww(X_\alpha,U_\alpha))$ is an $E_1$-quasi-isomorphism
for every elementary acyclic square
$(X_\alpha,U_\alpha)\to (X,U)$ of $\Man{\CC}^2_{comp}$.
As in the proof of Theorem $\ref{singan}$, it suffices to check that
for all $q\in\ZZ$, the sequence
$$\cdots\to E_2^{*,q}({\Ww}(X,U))\to
E_2^{*,q}({\Ww}(\wt X,\wt U))\oplus E_2^{*,q}({\Ww}(Y,U\cap Y)) \to E_2^{*,q}({\Ww}(\wt Y,\wt U\cap\wt Y))\to \cdots$$
is exact.
Since $E_1^{*,q}({\Ww}(X,U))$ is the shifted Leray spectral sequence of the inclusion $j:U\hookrightarrow X$,
it is isomorphic to the Gysin complex
$G^q(X,U)^*$ for all $q$ (see \cite{DeHII}, 3.1.9 and 3.2.4, see also Section 4.3 of \cite{PS}).
Hence the exactness of this sequence follows from Proposition $\ref{element2}$.
\end{proof}
\begin{defi}
Let $X_\infty$ be a complex analytic space with an equivalence class of compactifications.
The \textit{weight spectral sequence} is the spectral sequence associated with the
filtered complex $\Ww'(X_\infty)$.
If $W'$ denotes the induced filtration on $H^{*}(X;A)$, the \textit{weight filtration} $W_p$
on $H^{*}(X;A)$ is defined by $W_pH^{n}(X;A):=W_{p-n}'H^{n}(X;A)$.
\end{defi}
\begin{cor}
Let $X_\infty$ be a complex analytic space with an equivalence class of compactifications. For every $n\geq 0$, its cohomology
$H^n(X;A)$ with values in any commutative ring $A$ carries a weight filtration
$$0=W_{-1}\subset W_{0} \cdots\subset W_{2 n}=H^n(X;A)$$
which is functorial for morphisms in $\An{\CC}_\infty$ and satisfies:
\begin{enumerate}[1)]
\item If $X$ is smooth then $0=W_{n-1}\subset H^n(X;A)$.
\item If $X$ is compact then $W_{n}=H^n(X;A)$.
\item If $X$ is a complex algebraic variety and $A=\QQ$ then $W$ is Deligne's weight filtration.
\end{enumerate}
\end{cor}
Note that by Theorem $\ref{weightan}$ the weight spectral sequence of $X_\infty$ is well-defined
from the $E_2$-term onwards. The first term $_WE_1$ of the weight spectral sequence admits a description in terms of compactifications and resolutions as follows:
1) Assume that $X_\infty$ is smooth. Choose a representative $X\hookrightarrow \overline{X}$ of the compactification
class $X_\infty$ with $D=\overline{X}-X$
a normal crossings divisor. Denote by $D^{(p)}$ the disjoint union of all $p$-fold intersections of the smooth irreducible components of $D$.
Then:
$$_WE_1^{-p,q}(X_\infty;A)=H^{q-2p}(D^{(p)};A)\Longrightarrow H^{q-p}(X;A).$$
If $X$ is algebraic and $A=\QQ$ we recover Deligne's formula 3.2.4 of \cite{DeHII}.
2) Assume that $X$ is compact. Let $X_\bullet\to X$ be a cubical hyperresolution of $X$.
Then:
$$_WE_1^{p,q}(X;A)=\bigoplus_{|\alpha|=p}H^q(X_{\alpha};A)\Longrightarrow H^{p+q}(X;A).$$
3) For the general case, let $X\hookrightarrow \overline{X}$ be a representative of $X_\infty$,
and let $(\overline{X}_\bullet,X_\bullet)\to (\overline{X},X)$ be a resolution of $(\overline{X},X)$.
These are resolutions $X_\bullet\to X$ and $\overline{X}_\bullet\to \overline{X}$ such that
the complement $D_\alpha=\overline{X}_\alpha- X_\alpha$ is a normal crossings divisor for each $\alpha$.
Then:
$$_WE_1^{-p,q}(X_\infty;A)=\bigoplus_{\alpha}{\color{white}.}_WE_1^{-p-|\alpha|,q}(X_{\alpha};A)=\bigoplus_{\alpha}H^{q-2p-2|\alpha|}(D^{(p+|\alpha|)}_\alpha;A) \Longrightarrow H^{q-p}(X;A).$$
If $X$ is algebraic and $A=\QQ$ this corresponds to the analogous formula appearing in Theorem 8.1.15 of \cite{DeHIII} (see also IV.3 of \cite{GNPP}).
\begin{rmk}In general, the weight filtration $W$ on the cohomology of a compactificable complex analytic space depends on the class of its compactification,
and it is functorial for morphisms in $\An{\CC}_\infty$ (see Example $\ref{compadep}$ below).
If $X$ is a complex algebraic variety, by Nagata's Theorem, it admits a unique equivalence class of compactifications.
Therefore we obtain a weight filtration on $H^*(X;A)$ for an arbitrary coefficient ring $A$,
which is independent of the compactification and is functorial for morphisms of algebraic varieties.
\end{rmk}
\begin{example}\label{compadep}
Serre constructed two non-equivalent analytic compactifications on $U=\CC^*\times\CC^*$
(see \cite{Har}, Appendix B, 2.0.1).
The first is $\CC\PP^1\times\CC\PP^1$, and gives a weight filtration of pure weight 2 on $H^1(U;\QQ)$.
The second compactification is defined as the total space of a certain $\PP^1$-bundle, and gives a weight filtration of pure weight 1 on $H^1(U;\QQ)$
(see Example 4.19 of \cite{PS} for details).
Hence $U^{an}$ is an example of a complex analytic space having two different classes of compactifications, which give different weight filtrations.
\end{example}
\begin{rmk}The techniques used previously are also applicable to real analytic spaces, leading to the existence of a functorial
weight filtration on the $\ZZ_2$-cohomology of every real analytic space with an equivalence class of compactifications. Indeed,
by Remarks $\ref{real}$ and $\ref{real_acyclic}$, the extension criterion and the results of Section $\ref{acyclicity}$ are valid
in the real setting. A similar proof of Theorem $\ref{weightan}$ can be carried out in this case, extending the functor
$\Ww:\Man{\RR}^2_{comp}\lra \mathbf{D}^+_1(\mathbf{F}\ZZ_2-\text{mod})$
given by $\Ww(X,U)=\mathbf{s}G^*(X,U)^*$, with the filtration by the second degree.
\end{rmk}
\begin{example}[cf. \cite{MCPII}]
Let $U=\RR^2-\{*\}$ be the punctured plane and
$V=\RR\PP^1\times\RR$ the infinite cylinder.
The weight filtration on $H^1(U;\ZZ_2)$ is pure of weight 2, while on $H^1(V;\ZZ_2)$ it is pure of weight 1.
Note that while $U$ and $V$ are not isomorphic algebraic varieties, they are isomorphic as real analytic spaces.
Hence $U^{an}$ is an example of a real analytic space having two different classes of compactifications
which give different weight filtrations,
corresponding to $\overline{U}=\widetilde {\RR\PP}^2$ and $\overline{V}=\RR\PP^1\times \RR\PP^1$,
where $\widetilde {\RR\PP}^2$ is the blow-up of $\RR\PP^2$ at a point.
\end{example}
\section*{Acknowledgments}
The results of this paper are based on a non-published manuscript by the second named author together with V. Navarro-Aznar.
We deeply thank him for his generosity and useful comments.
\bibliographystyle{amsplain}
\bibliography{bibliografia}
\end{document}
| 105,159
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Sun and ESI Group have been working together for over 15 years and ESI Group uses Sun Studio compilers and Sun HPC Cluster Tools to develop applications. The companies have recently re-enforced the relationship with ESI's Group's commitment to port PAM-CRASH 2G and PAM-STAMP 2G to the Solaris 10 OS..
We need to hear what's on your mind! Please share your thoughts and ideas about Sun and ESI Group. Contact us today.
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How Disney Changes the Theme Park Business
By: Nickie Harber-Frankart
A new year brings new opportunities. For our clients, that means a new level of understanding of how consumers’ desires for convenience and personalization continually drive business strategy. A recent New York Times article about Disney’s new MyMagic+ technology is a perfect example of this.
MyMagic+ utilizes radio-frequency identification (RFID) technology embedded in rubber bracelets (called MagicBands), which when worn by park visitors simplifies the daunting task of keeping track of paper tickets, fumbling with credit cards, and tracking room keys, while watching small children in a theme park. Customers (at their own discretion) can also encode personal data on the bands which park employees can access. This makes it possible for me or my sister to opt in and personalize my niece’s first visit to Walt Disney World where Minnie Mouse greets her saying, “Nice to meet you, Hailey!”
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\section{Effect of the Cyclic Learning Rate Schedule}
In this Appendix, we provide the effect of different cyclic learning rate strategies during exploitation phases in our three sequential ensemble methods. We explore the stepwise (our approach), cosine \cite{huang2017snapshot}, linear-fge \cite{FGE}, and linear-1 cyclic learning rate schedules.
{\bf Cosine.} The cyclic cosine learning rate schedule reduces the higher learning rate of 0.01 to a lower learning rate of 0.001 using the shifted cosine function \citep{huang2017snapshot} in each exploitation phase.
{\bf Linear-fge.} In the cyclic linear-fge learning rate schedule, we first drop the high learning rate of 0.1 used in the exploration phase to 0.01 linearly in ${\rm t_{ex}}/2$ epochs and then further drop the learning rate to 0.001 linearly for the remaining ${\rm t_{ex}}/2$ epochs during the first exploitation phase. Afterwards, in each exploitation phase, we linearly increase the learning rate from 0.001 to 0.01 for ${\rm t_{ex}}/2$ and then linearly decrease it back to 0.001 for the next ${\rm t_{ex}}/2$ similar to \cite{FGE}.
{\bf Linear-1.} In the linear-1 cyclic learning rate schedule, we linearly decrease the learning rate from 0.01 to 0.001 for ${\rm t_{ex}}$ epochs in each exploitation phase and then suddenly increase the learning to 0.01 after each sequential perturbation step.
In Figure~\ref{fig:lr_schedules}, we present the plots of the cyclic learning rate schedules considered in this Appendix.
\begin{figure}[H]
\centering
\includegraphics[width=0.75\textwidth]{Figures/LR_schedule_plot.png}
\caption{Cyclic Learning Rate Schedules. The red dots represent the converged models after each exploitation phase used in our final sequential ensemble.}
\label{fig:lr_schedules}
\end{figure}
In Table~\ref{eff_lr_Cifar10_expts}, we present the results for our three sequential ensemble methods under the four cyclic learning rate schedules mentioned above. We observe that, in all three sequential ensembles, the cyclic stepwise learning rate schedule yields the best performance in almost all criteria compared to the rest of the learning rate schedules in each sequential ensemble method. In Table~\ref{Table:eff_lr_diversity_analysis}, we present the prediction disagreement and KL divergence metrics for the experiments described in this Appendix. We observe that, in \texttt{SeBayS}-No Freeze ensemble, cyclic stepwise schedule generates highly diverse subnetworks, which also leads to high predictive performance. Whereas, in the BNN sequential and \texttt{SeBayS}-Freeze ensemble, we observe lower diversity metrics for the cyclic stepwise learning rate schedule compared to the rest of the learning rate schedules.
\begin{table}[H]
\small
\caption{ResNet-32/CIFAR10 experiment results: we mark the best results out of different learning rate (LR) schedules under a given method in bold. Ensemble size is fixed at $M=3$.}
\label{eff_lr_Cifar10_expts}
\centering
\begin{tabular}{l*{3}{c}*{1}{H}*{2}{c}*{1}{H}}
\toprule
Methods & LR Schedule & Acc ($\uparrow$) & NLL ($\downarrow$) & ECE ($\downarrow$) & cAcc ($\uparrow$) & cNLL ($\downarrow$) & cECE ($\downarrow$) \\
\midrule
SeBayS-Freeze Ensemble & stepwise & \textbf{92.5} & 0.273 & 0.039 & \textbf{70.4} & \textbf{1.344} & 0.183 \\
SeBayS-Freeze Ensemble & cosine & 92.3 & 0.301 & 0.044 & 69.8 & 1.462 & 0.198 \\
SeBayS-Freeze Ensemble & linear-fge & \textbf{92.5} & \textbf{0.270} & 0.037 & 70.1 & 1.363 & 0.185 \\
SeBayS-Freeze Ensemble & linear-1 & 92.1 & 0.310 & 0.045 & 69.8 & 1.454 & 0.197 \\
\midrule
SeBayS-No Freeze Ensemble & stepwise & \textbf{92.4} & \textbf{0.274} & 0.039 & 69.8 & \textbf{1.356} & 0.185 \\
SeBayS-No Freeze Ensemble & cosine & 92.2 & 0.294 & 0.044 & 69.9 & 1.403 & 0.191 \\
SeBayS-No Freeze Ensemble & linear-fge & \textbf{92.4} & 0.276 & 0.039 & \textbf{70.0} & 1.379 & 0.187 \\
SeBayS-No Freeze Ensemble & linear-1 & 92.2 & 0.296 & 0.043 & 69.7 & 1.412 & 0.193 \\
\midrule
BNN Sequential Ensemble & stepwise & \textbf{93.8} & \textbf{0.265} & 0.037 & \textbf{73.3} & \textbf{1.341} & 0.184 \\
BNN Sequential Ensemble & cosine & 93.7 & 0.279 & 0.042 & 72.7 & 1.440 & 0.195 \\
BNN Sequential Ensemble & linear-fge & 93.5 & 0.270 & 0.039 & 73.1 & 1.342 & 0.182 \\
BNN Sequential Ensemble & linear-1 & 93.4 & 0.287 & 0.040 & 72.2 & 1.430 & 0.196 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[H]
\small
\caption{Diversity metrics: prediction disagreement $(d_{\rm dis})$ and KL divergence $(d_{\rm KL})$. We mark the best results out of different learning rate (LR) schedules under a given method in bold. Ensemble size is fixed at $M=3$.}
\label{Table:eff_lr_diversity_analysis}
\centering
\begin{tabular}{lccccc}
\toprule
\multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{ResNet-32/CIFAR10} \\
\cmidrule(lr){3-5}
Methods & LR Schedule & $(d_{\rm dis})$ ($\uparrow$) & $d_{\rm KL}$ ($\uparrow$) & Acc ($\uparrow$) \\
\midrule
BNN Sequential Ensemble & stepwise & 0.061 & 0.201 & \textbf{93.8} \\
BNN Sequential Ensemble & cosine & 0.068 & 0.256 & 93.7 \\
BNN Sequential Ensemble & linear-fge & 0.070 & 0.249 & 93.5 \\
BNN Sequential Ensemble & linear-1 & \textbf{0.071} & \textbf{0.275} & 93.4 \\
\midrule
SeBayS-Freeze Ensemble & stepwise & 0.060 & 0.138 & \textbf{92.5} \\
SeBayS-Freeze Ensemble & cosine & 0.072 & 0.204 & 92.3 \\
SeBayS-Freeze Ensemble & linear-fge & \textbf{0.076} & \textbf{0.215} & \textbf{92.5} \\
SeBayS-Freeze Ensemble & linear-1 & 0.074 & 0.209 & 92.1 \\
\midrule
SeBayS-No Freeze Ensemble & stepwise & \textbf{0.106} & \textbf{0.346} & \textbf{92.4} \\
SeBayS-No Freeze Ensemble & cosine & 0.078 & 0.222 & 92.2 \\
SeBayS-No Freeze Ensemble & linear-fge & 0.074 & 0.199 & \textbf{92.4} \\
SeBayS-No Freeze Ensemble & linear-1 & 0.077 & 0.217 & 92.2 \\
\bottomrule
\end{tabular}
\end{table}
| 35,346
|
TITLE: maximal spectrum of $\mathbb{F}_2[x,y]$
QUESTION [3 upvotes]: What are the maximal ideals in $\mathbb{F}_2[x,y]$? In particular, I'm trying to answer the following question from my commutative algebra class: What is the number of maximal ideals in $\mathbb{F}_2[x,y]$ with quotient ring of order 8?
So far, the only maximal ideals I have found are $(x,y)$ and $(x+1,y+1)$.Since $\mathbb{F}_2$ is not algebraically closed (as $x^2+x+1$ has no root in $\mathbb{F}_2$) , I don't think Hilbert's Nullstellensatz applies.
For the quotient ring to have order $8$, it must be generated by some elements $1,a,b$ where $a^2,b^2,ab$ are in the maximal ideal $m$, or $1,a,a^2$ where $a^3 \in m$. But I'm not sure how this helps me find $m$.
REPLY [7 votes]: Warning
Despite appearances this is a quite subtle and non trivial question!
Solution
Clearly to any maximal ideal $\mathfrak m \subset \mathbb F_2[x,y])$ with residue field $\kappa(\mathfrak m)$ of order $8$ we can associate an $\mathbb F_2$-algebra morphism $\mathbb F_2[x,y]\to \mathbb F_8$.
Since there are $64=8^2$ such morphisms (send $x,y$ arbitrarily to $F_8$) the answer to our problem is $64$, right?
Wrong! We have to dismiss the points with residue field of order $2$ (corresponding to morphisms $\mathbb F_2[x,y]\to \mathbb F_2$) of which we have $4$.
OK, so the required number is $64-4=60$, right?
Wrong again! The subtle point is that for each $\mathfrak m$ with residue field of order $8$ we have three morphisms $\kappa(\mathfrak m)\to \mathbb F_8$ : they are obtained by composing one of them with the $3$ elements of $\operatorname {Gal} (\mathbb F_8/\mathbb F_2)$. Dividing by $3$ we obtain the required result:
There are 20 maximal ideals in $\mathbb F_2[x,y]$ with residue field of order 8.
| 206,270
|
Calendar
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Come by the Harbourfront Library to see the Canadian Museum of Immigration's travelling exhibit, "Canada: Day 1." Originally created to …Find out more >
Movie Matinee: On the Basis of Sex
The film tells an inspiring and spirited true story that follows young lawyer Ruth Bader Ginsburg as she teams with …Find out more >
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| 237,812
|
\begin{document}
\begin{abstract} For $p>0$, let ${\mathcal B}^p(\BB_n)$ and ${\mathcal L}_p(\BB_n)$ respectively
denote the $p$-Bloch and holomorphic $p$-Lipschitz spaces of the
open unit ball $\BB_n$ in $\CC^n$. It is known that ${\mathcal
B}^p(\BB_n)$ and $\li_{1-p}(\BB_n)$ are equal as sets when
$p\in(0,1)$. We prove that these spaces are additionally
norm-equivalent, thus extending known results for $n=1$ and the
polydisk. As an application, we generalize work by Madigan on the
disk by investigating boundedness of the composition operator
${\mathfrak C}_\phi$ from ${\mathcal L}_p(\BB_n)$ to ${\mathcal
L}_q(\BB_n)$.
\end{abstract}
\maketitle
\section{Background and Terminology}
Let $n\in\NN$, and suppose that ${\DD}$ is a domain in $\CC^n$.
Denote the linear space of complex-valued, holomorphic functions on
${\DD}$ by ${\mathcal H}({\DD})$. If ${\mathcal X}$ is a linear
subspace of ${\mathcal H}({\DD})$ and $\phi:{\DD}\to{\DD}$ is
holomorphic, then one can define the linear operator ${\mathfrak
C}_\phi:{\mathcal X}\rightarrow{\mathcal H}({\DD})$ by ${\mathfrak
C}_\phi(f)=f\circ\phi$ for all $f\in {\mathcal X}$. ${\mathfrak
C}_\phi$ is called the {\em composition operator}$\,$ induced by
$\phi$.
The problem of relating properties of symbols $\phi$ and operators
such as ${\mathfrak C}_\phi$ that are induced by these symbols is of
fundamental importance in concrete operator theory. However,
efforts to obtain characterizations of self-maps that induce bounded
composition operators on many function spaces have not yielded
completely satisfactory results in the several-variable case,
leaving a wealth of basic, open problems.
In this paper, we try to make progress toward the goal of
characterizing the holomorphic self-maps of the open unit ball
$\BB_n$ in $\CC^n$ that induce bounded composition operators between
holomorphic $p$-Lipschitz spaces ${\mathcal L}_p(\BB_n)$ for $0<p<1$
by translating the problem to $(1-p)$-Bloch spaces ${\mathcal
B}^{1-p}(\BB_n)$ via an auxiliary Hardy/Littlewood-type
norm-equivalence result of potential independent interest. This
method was also used in \cite{m} for $\BB_1$ and in \cite{csz} for
the unit polydisk $\Delta^n$.
The function-theoretic characterization of analytic self-maps of
$\BB_1$ that induce bounded composition operators on $\li_p(\BB_1)$
for $0<p<1$ is due to K. Madigan \cite{m}, and the case of
$\Delta^n$ was handled in a joint paper by the present authors with
Z. Zhou \cite{csz}, in which a full characterization of the
holomorphic self-maps $\phi$ of $\Delta^n$ that induce bounded
composition operators between ${\mathcal L}_p(\Delta^n)$ and
${\mathcal L}_q(\Delta^n)$, and, more generally, between Bloch
spaces ${\mathcal B}^p(\Delta^n)$ and ${\mathcal B}^q(\Delta^n)$, is
obtained for $p,q\in(0,1)$, along with analogous characterizations
of compact composition operators between these spaces.
Although our main results concerning composition operators, Theorem
\ref{main} and Corollary \ref{lip}, are not full characterizations,
they do generalize Madigan's result for the disk to $\BB_n$; on the
other hand, we obtain a complete Hardy-Littlewood norm-equivalence
result for $p$-Bloch and $(1-p)$-Lipschitz spaces of $\BB_n$ for all
$n\in\NN$. This norm-equivalence result should lead to an eventual
extension to $\BB_n$ of the characterizations of bounded composition
operators established on $\BB_1$ in \cite{m} and on $\Delta^n$ in
\cite{csz}.
Most of our several complex variables notation is adopted from
\cite{r}. If $z=(z_1,...,z_n)$ and $w=(w_1,...,w_n)$ are points in
$\CC^n$, then we define a complex inner product by $\langle
z,\om\rangle=\sum_{k=1}^n z_k\bar{w_k}$ and put $|z|:=\sqrt{\langle
z, z\rangle}$. We call $\BB_n:=\{z\in\CC^n:|z|<1\}$ the {\em (open)
unit ball} of $\CC^n$.
Let $p\in (0,\infty).$ The {\em $p$-Bloch space ${\mathcal
B}^p(\BB_n)$} consists of the set of all $f\in {\mathcal H}(\BB_n)$
with the property that there is an $M\geq 0$ such that
$$b(f,z,p):=(1-|z|^2)^p|\nabla f(z)|\leq M\,\,{\text{ for all }}z\in \BB_n$$
${\mathcal B}^p(\BB_n)$ is a Banach space with norm
$||f||_{{\mathcal B}^p}$ given by
\[
||f||_{{\mathcal B}^p}=|f(0)|+\sup_{z\in \BB_n}b(f,z,p).
\]
The {\em little $p$-Bloch space} ${\mathcal B}^p_0(\BB_n)$ is
defined as the closed subspace of ${\mathcal B}^p(\BB_n)$ consisting of the
functions that satisfy
$$\lim_{z\to\pt \BB_n}(1-|z|^2)^p|\nabla f(z)|=0.$$
For $p\in(0,1)$, $\li_p(\BB_n)$ denotes the {\em holomorphic $p$-Lipschitz space}, which is the set of all $f\in
{\mathcal H}(\BB_n)$ such that for some $C>0$,
\begin{equation}\label{heaven}
|f(z)-f(w)|\leq C|z-w|^{p}\,\,\,\,\mbox{for every}\; z,w\in \BB_n.
\end{equation}
These functions extend continuously to ${\overline \BB_n}$ (cf.
\cite[~Lemma~4.4]{csz}). Therefore, if $A(\BB_n)$ is the ball
algebra \cite[~Ch.~6]{r}, then $${\mathcal L}_p(\BB_n)={\text
{Lip}}_p(\BB_n)\cap A(\BB_n),$$ where Lip$_p(\BB_n)$ is the set of
all $f:\BB_n\rightarrow \CC$ satisfying Equation (\ref{heaven}) for
some $C>0$ and all $z\in{\overline \BB_n}$. ${\mathcal L}_p(\BB_n)$
is endowed with a complete norm $||\cdot||_{{\mathcal L}_p}$ that is
given by
\begin{equation}\label{haven} ||f||_{{\mathcal L}_p}=|f(0)|+\sup_{z\not= w:
z,w\in{\overline \BB_n}}\left\{\frac{|f(z)-f(w)|}{|z-w|^p}\right\}.
\end{equation}
In Equations (\ref{heaven}) and (\ref{haven}), $\BB_n$ and
${\overline \BB_n}$ are interchangeable, since functions in
${\mathcal L}_p(\BB_n)$ extend continuously to ${\overline
\BB_n}$. The supremum above is called the {\em Lipschitz
constant} for $f$. As in \cite[~p.~13]{r}, $\sigma$ represents the
unique rotation-invariant positive Borel measure on $\partial
\BB_n$ for which $\sigma(\partial \BB_n)=1$, and for $f\in
L^1(\sigma)$, $C[f]$ denotes Cauchy integral of $f$ on $\BB_n$
(see \cite[~p.~38]{r}).
Let $u\in\partial \BB_n$ and $f\in {\mathcal H}(\BB_n)$. The {\em
directional derivative} of $f$ at $z\in\BB_n$ in the direction of
$u\in\pt\BB_n$ is given by
\[D_uf(z)=\lim_{\lambda\rightarrow 0,\lambda\in\CC}\frac{f(z+\lambda
u)-f(z)}{\lambda}.
\]
Observe that
\begin{equation}\label{gradi}
D_uf(z)=\langle \nabla f(z),{\overline u}\rangle.
\end{equation}We define the partial differential operators
$D_j$ as in \cite[~Ch.~1]{r}. The radial derivative operator
\cite[~p.~103]{r} in $\CC^n$ will be denoted by ${\mathfrak R}$ and
is linear. Let ${\UU}=\left\{u_1,u_2,\ldots u_n\right\}$ be an
orthonormal basis for the Hilbert space $\CC^n$ with its usual
Euclidean structure. We define a gradient operator $\nabla^{\UU}$
on ${\mathcal H}({\DD})$ with respect to ${\UU}$ by
\[\nabla^{\UU}f(z)=(D_{u_1}f(z),D_{u_2}f(z),\ldots,
D_{u_n}f(z)),
\]
and we can denote $\nabla^{\UU}$ by $\nabla$ when ${\UU}$ is the
typically ordered standard basis for $\CC^n$.
Let $x$ and $y$ be two positive variable quantities. We write
$x\asymp y$ (and say that $x$ and $y$ are {\em comparable}) if and
only if $x/y$ is bounded above and below.
\section{Main Results on Composition Operators}
Our norm-equivalence result (Theorem \ref{bigone}) ties our results
concerning ${\mathfrak C}_\phi$ between $p$-Lipschitz spaces of
$\BB_n$ to the following result for general Bloch spaces:
\begin{theorem}\label{main}
Let $p,q\in (0,\infty),$ and suppose that $\phi:\BB_n\rightarrow
\BB_n$ is holomorphic. Then the following statements hold:
(A) If there is an $M\geq 0$ such that for all $z\in\BB_n$ and
$j\in\{1,\ldots,n\}$,
\[
\frac{(1-|z|^2)^q}{(1-|\phi(z)|^2)^p} |\nabla \phi_j(z)|\leq M,
\]
then ${\mathfrak C}_\phi$ is bounded from ${\mathcal B}^p(\BB_n)$
(respectively, ${\mathcal B}^p_0(\BB_n)$) to ${\mathcal
B}^q(\BB_n)$.
(B) If ${\mathfrak C}_\phi$ is bounded from ${\mathcal B}^p(\BB_n)$
(respectively, ${\mathcal B}^p_0(\BB_n)$) to ${\mathcal
B}^q(\BB_n)$, then there is an $M'\geq 0$ such that for all $z\in
\BB_n$ and $u\in\pt\BB_n$,
\[
\frac{(1-|z|^2)^q}{(1-|\langle\phi(z),u\rangle|^2)^p} |\nabla
\langle\phi(z),u\rangle|\leq M'.
\]
\end{theorem}
Theorem \ref{main} above and Corollary \ref{lip} below for $0<p=q<1$
appear in \cite[~Ch.~4]{cl}. It should be pointed out that Theorem
\ref{main}, Part (A) is similar to a statement that is proved in
\cite{zh100}; furthermore, \cite{zh100} contains a result that is in
the same direction as Part (B) of Theorem \ref{main} and that is
proven using different testing functions. Unlike \cite{zh100},
however, the present paper addresses composition operators between
$\li_p(\BB_n)$ and $\li_q(\BB_n)$ and the coincidence and
norm-equivalence of ${\mathcal B}^{1-p}(\BB_n)$ and ${\mathcal
L}_p(\BB_n)$, respectively.
It is natural to consider the application of corresponding
``little-oh" arguments to obtain a compactness result analogous to
Theorem \ref{main}, in which ``bounded" is replaced by ``compact"
and the limit of the left hand side of each inequality in the
statement is taken as $|\phi(z)|\rightarrow 1^-$, with inequality
replaced by equality to $0$. However, in the case that $p\in(0,1)$,
${\mathcal B}^p(\BB_n)$ is the same as and norm-equivalent to
${\mathcal L}_{1-p}(\BB_n)$, whose compact composition operators are
known (by a result due to J. H. Shapiro) to be generated precisely
by holomorphic self-maps $\phi$ of $\BB_n$ with supremum norm
strictly less than $1$ (see \cite[~Ch.~4]{cm}).
The following corollary follows from Theorems \ref{main} and
\ref{bigone} and extends the main result of \cite{m}:
\begin{corollary}\label{lip}
Let $\,\,p,q\in(0,1)$, and suppose that $\phi:\BB_n\rightarrow
\BB_n$ is holomorphic. Then the following statements hold:
(A) If there is an $M\geq 0$ such that
\[
\frac{(1-|z|^2)^{1-q}}{(1-|\phi(z)|^2)^{1-p}} | \nabla
\phi_j(z)|\leq M,
\]
for all $j\in\{1,2\ldots,n\}$ and $z\in\BB_n$, then ${\mathfrak
C}_\phi$ is a bounded operator from ${\mathcal L}_p(\BB_n)$ to
${\mathcal L}_q(\BB_n)$.
(B) If ${\mathfrak C}_\phi$ is a bounded operator from ${\mathcal
L}_p(\BB_n)$ to ${\mathcal L}_q(\BB_n)$, then there is an $M'\geq 0$
such that for all $z\in \BB_n$ and $u\in\pt\BB_n$,
\[
\frac{(1-|z|^2)^{1-q}}{(1-|\langle\phi(z),u\rangle|^2)^{1-p}}
|\nabla \langle\phi(z),u\rangle|\leq M'.
\]
\end{corollary}
Choosing $n=1$, $p=q\in(0,1)$, and $u=1$ in Corollary \ref{lip}
leads to the following result, which is due to K. Madigan \cite{m}:
\begin{theorem}\label{Madig}
Let $\,\,0<p<1$, and suppose that $\phi$ is an analytic self-map of
$\BB_1$. Then ${\mathfrak C}_\phi$ is bounded on ${\mathcal
L}_p(\BB_1)$ if and only if
\[
\sup_{z\in \BB_1}\left\{\left(\frac{1-|z|^2}{1-|\phi(z)|}\right)^
{1-p}|\phi'(z)|\right\}<\infty.
\]
\end{theorem}
\section{Norm Equivalence of ${\mathcal L}_p(\BB_n)$ and ${\mathcal B}^{1-p}(\BB_n)$ for $0<p<1$.}
To generalize Theorem \ref{Madig} to $\BB_n$, we need Theorem
\ref{bigone}, which is the ball analogue of the following result for
the disk (Lemma 2 in \cite{m}). The first statement in Theorem
\ref{H-L} can be derived from a classical theorem of
Hardy/Littlewood for $n=1$ (see \cite{hl}, \cite[~p.~74]{d}, and
\cite[~p.~176]{cm}).
\begin{theorem}\label{H-L}
Let $0<p<1$. If $f:\BB_1\rightarrow \CC$ is analytic, then $f\in $
${\mathcal L}_p(\BB_1)$ if and only if
\[
|f'(z)|=O\left(\frac{1}{1-|z|^2}\right)^{1-p}.
\]
Furthermore, the Lipschitz constant of $f$ and the quantity
\[
\sup_{z\in \BB_1}\{(1-|z|^2)^{1-p}|f'(z)|\}
\]
are comparable as $f$ varies through ${\mathcal L}_p(\BB_1)$.
\end{theorem}
We remark that the polydisk version of Theorem \ref{H-L} is stated
and proved in \cite{csz}. However, the argument used there cannot
be applied to $\BB_n$, so we need a different approach for that
domain. We will proceed by listing some lemmas, which together
eseentially form the norm equivalence Theorem \ref{bigone}.
For $0<p<1$, we can define a norm $||f||_{{\mathcal
B}^{1-p}}^{\mathfrak R}$ on $\li_p(\BB_n)$ by
\[
||f||_{{\mathcal
B}^{1-p}}^{\mathfrak R}=|f(0)|+\sup_{z\in \BB_n}
\{(1-|z|^2)^{1-p}|({\mathfrak R}f)(z)|\}.\] The following lemma is
part of our norm equivalence result, Theorem \ref{bigone}:
\begin{lemma}\label{dont2}
Suppose that $0<p<1$. Furthermore, there is a $C_p\geq 0$ such
that for all $f\in \li_p(\BB_n)$,
\[
||f||_{{\mathcal B}^{1-p}}^{\mathfrak R}\leq C_p||f||_{{\mathcal
L}_p}.
\]
\end{lemma}
\begin{proof}
The proof of the first statement is standard and left to the reader.
Since functions in ${\mathcal L}_p(\BB_n)$ extend continuously to
${\overline \BB_n}$, they are automatically in $L^1(\sigma)$
\cite[~Remark,~p.~107]{r} and since the quotients of these functions
and their $\li_p$-norms satisfy \cite[~Equation~(1),~p.~107]{r}, the
second statement is obtained from \cite[~Theorem~6.4.9]{r}.
\end{proof}
The following lemma is also a portion of Theorem \ref{bigone}:
\begin{lemma}\label{grr}
If $p\in(0,1)$, then ${\mathcal B}^{1-p}(\BB_n)\subset\li_p(\BB_n)$,
and
\[||f||_{{\mathcal L}_p}\leq (2+2p^{-1})||f||_{{\mathcal
B}^{1-p}}\,\,\mbox{ for all }\,f\in {\mathcal B}^{1-p}(\BB_n).
\]
\end{lemma}
\begin{proof} Suppose that $f\in{\mathcal B}^{1-p}(\BB_n)$. If $f=0$ then $f\in{\mathcal L}_p(\BB_n)$
trivially, so assume henceforward that $f\not=0$.
A well-known result \cite[~Ch.~6]{r} applied to $f/||f||_{{\mathcal B}^{1-p}}$ implies that for all $z,w\in B_n$,
\[
\frac{1}{||f||_{{\mathcal B}^{1-p}}}|f(z)-f(w)|\leq
(1+2p^{-1})|z-w|^p,
\]
from which the first statement of the lemma follows. Moreover,
\begin{eqnarray*}
||f||_{{\mathcal L}_p}&=&
|f(0)|+\sup_{z,w\in\BB_n:z\not=w}\frac{|f(z)-f(w)|}{|z-w|^{p}}\\
&\leq &|f(0)|+(1+2p^{-1})||f||_{{\mathcal
B}^{1-p}}\\
&\leq &(2+2p^{-1})||f||_{{\mathcal B}^{1-p}}.
\end{eqnarray*}
\end{proof}
The following fact also constitutes part of Theorem \ref{bigone}:
\begin{lemma}\label{st}
If $p>0$, then $f\in{\mathcal B}^p(\BB_n)$ if and only if there
exists $M\geq 0$ such that $|({\mathfrak R}f)(z)|(1-|z|^2)^p\leq M$
for all $z\in\BB_n$. Also, $||\cdot||_{{\mathcal B}^p}^{\mathfrak
R}$ given by $||f||_{{\mathcal B}^p}^{\mathfrak
R}:=|f(0)|+\sup_{z\in\BB_n}|({\mathfrak R}f)(z)|(1-|z|^2)^p$ is a
norm on ${\mathcal B}^p(\BB_n).$ If $p\in (0,1]$, then there is a
$C_p\geq 0$ such that $||f||_{{\mathcal B}^p}\leq
C_p||f||_{{\mathcal B}^p}^{\mathfrak R}$ for all $f\in{\mathcal
B}^p(\BB_n).$
\end{lemma}
\begin{proof}
For a proof of the first statement, see \cite[Proposition 1]{yo}.
The second statement follows from subsequent applications of the
first statement in Lemma \ref{grr} and Lemma \ref{dont2}. To prove
the final statement, we use the weighted Bergman projection $P_s$
with kernel $K_s$ and the map $L_s$ defined on $P_s[L^\infty(B_n)]$
by
\[
(L_sg)(z)=(s+1)^{-1}(1-|z|^2)\left[(n+s+1)g(z)+({\mathfrak
R}g)(z)\right]\mbox{ for all }z\in\BB_n,
\]
where $s\in\CC$ satisfies Re\,\,$s>-1$ (see \cite{c}). By
\cite[Corollary 13]{c}, we have that $P_s\circ L_s$ is the identity
on ${\mathcal B}^1(\BB_n)$ for all such values of $s$. In
particular, $P_0\circ L_0$ is the identity on ${\mathcal
B}^p(\BB_n)$, since this set is contained in ${\mathcal
B}^1(\BB_n)$. Note that the assumption $p\in(0,1]$ is used here.
We then obtain that there is a $C\geq 0$ such that for all $z\in
B_n$ and $f\in{\mathcal B}^1(\BB_n)$,
\begin{eqnarray*}
f(z)&=& (P_0\circ L_0)(f)(z)\\
&=&C \int_{\BB_n} (1-|w|^2) K_0(z,w)\biggl[(n+1) f(w)+{\mathfrak R}
f(w)\biggr] dV(w). \end{eqnarray*} Hence, there is a $C'\geq 0$
such that for all $f\in{\mathcal B}^p(\BB_n)$ and $z\in \BB_n$,
\begr |\nabla f(z)| &\le& C' \int_{\BB_n} (1-|w|^2) |\nabla K_0(z,w)|\, |f(w)| dV(w)\nonumber\\
&&+C' \int_{\BB_n} (1-|w|^2) |\nabla K_0(z,w)|\, |{\mathfrak R}
f(w)| dV(w).\nonumber\endr
\noindent Let $\ve\in (1-p,1)$. Subsequent applications of the
above inequality, \cite[~Lemma~2]{s}, and \cite[Theorem 1.4.10]{r}
imply that there are non-negative constants $C''$ and $C'''$ such
that for all $z\in B_n$ and $f\in {\mathcal B}^p(\BB_n)$, the
following chain of inequalities holds: \begr|\nabla f(z)|
&\le& C' \int_{\BB_n} \frac{(1-|w|^2) |w|}{|1-\langle z, w\rangle|^{n+2}} |f(w)| dV(w)\nonumber\\
&&+C' \int_{\BB_n} \frac{(1-|w|^2) |w|}{|1-\langle z, w\rangle|^{n+2}}|{\mathfrak R} f(w)| dV(w)\nonumber\\
&\le&C'' ||f||_{{\mathcal
B}^p}^{\mathfrak R}\int_{\BB_n} \frac{(1-|w|^2)^{\ve}}{|1-\langle z, w \rangle|^{n+2}} dV(w)\nonumber\\
&&+C'' ||f||_{{\mathcal
B}^p}^{\mathfrak R}\int_{\BB_n} \frac{(1-|w|^2)^{1-p}}{|1-\langle z, w \rangle|^{n+2}} dV(w)\le\nonumber\\
&\le& C'''||f||_{{\mathcal B}^p}^{\mathfrak
R}\frac1{(1-|z|)^{1-\ve}}+ C'''||f||_{{\mathcal B}^p}^{\mathfrak R}\frac1{(1-|z|)^{p}}.\nonumber\endr\\
It follows that for all $f\in{\mathcal B}^p(\BB_n)$ and $z\in\BB_n$,
$$(1-|z|^2)^p|\nabla f(z)|\leq 2^{p+1}C'''||f||_{{\mathcal B}^p}^{\mathfrak R}.$$
The final statement in the lemma now follows from the above
statement and an application of \cite[~Lemma~2]{s} at $z=0$.
\end{proof}
Next, we state and prove this section's main result, the analogue of
Theorem \ref{H-L} for $\BB_n$. We emphasize that while the
statement of equality in the theorem is known and can be obtained,
for example, from \cite{zhu}, the norm equivalence portion requires
additional work that includes the previous lemmas and the proof
below. Furthermore, neither this result nor its proof has appeared
previously in any literature that is known to the authors, though it
seems to be part of the folklore. The proof of this rather
fundamental theorem seems to be non-trivial and worthy of recording.
\begin{theorem}\label{bigone}
If $0<p<1$, then ${\mathcal B}^{1-p}(\BB_n)=\li_p(\BB_n)$;
furthermore,
\[||f||_{{\mathcal B}^{1-p}}\asymp ||f||_{{\mathcal
B}^{1-p}}^{\mathfrak R}\asymp ||f||_{{\mathcal L}_p}
\]
as $f$ varies through ${\mathcal L}_p(\BB_n)$.
\end{theorem}
\begin{proof} The first statement is known, since ${\mathcal L}_p(\BB_n)=A(\BB_n)\cap $Lip$_\alpha(\BB_n)$
(see \cite[~Ch.~6]{r}), which is set-theoretically equal to
${\mathcal B}^{1-p}(\BB_n)$ (see \cite{yo}). By Lemma \ref{st}, it
follows that there is a $C_p\geq 0$ such that for all
$f\in\li_p(\BB_n)$, $||f||_{{\mathcal B}^{1-p}}\leq
C_p||f||_{{\mathcal B}^{1-p}}^{\mathfrak R}$. It follows from Lemma
\ref{dont2} that there is a $C_p'\geq 0$ such that for all
$f\in\li_p(\BB_n)$, $||f||_{{\mathcal B}^{1-p}}\leq
C_p||f||_{{\mathcal B}^{1-p}}^{\mathfrak R}\leq
C_pC_p'||f||_{{\mathcal L}_p}$, which is less than or equal to
$C_pC_p'(2+2p^{-1})||f||_{{\mathcal B}^{1-p}}$ by Lemma \ref{grr}.
The second statement in Theorem \ref{bigone} follows.
\end{proof}
\section{Proof of Theorem \ref{main}}\label{simp}
In the proof of Theorem \ref{main}, part (B), we will use part of
the following lemma, which is obtained by straightforward estimates
involving Equation (\ref{gradi}) (see \cite[~Ch.~4]{cl}):
\begin{lemma}\label{ugrad} Let $f\in {\mathcal H}({\DD}),$ where ${\DD}$ is an open subset of $\CC^n$, and suppose that ${\UU}$ is an
orthonormal basis for $\CC^n.$ Then for all $z\in{\DD}$,
\[
|\nabla^{\UU}f(z)|\asymp|\nabla f(z)|.
\]
\end{lemma}
We are now ready to prove Theorem \ref{main}.\bsk
{\it Proof of Theorem \ref{main}. (A)} Suppose that for some $M\geq
0$,
\begin{eqnarray}
\quad\,\,\,\frac{(1-|z|^2)^q}{(1-|\phi(z)|^2)^p} |\nabla
\phi_j(z)|\leq M\label{condi}\mbox{ for all } \,z\in
\BB_n,\,j\in\{1,2,\ldots ,n\}.
\end{eqnarray}
If $z\in\BB_n$ and $F(z)=(1-|z|^2)^q|\nabla({\mathfrak C}_\phi
f)(z)|$. Then we have that
\begin{eqnarray}
F(z)&=&(1-|z|^2)^q\sqrt{\sum_{i=1}^n\left|D_i(f\circ\phi)(z)\right|^2}\nonumber\\
&\leq&
(1-|z|^2)^q\sum_{i=1}^n \left|D_i(f\circ\phi)(z)\right|\nonumber\\
&\leq &(1-|z|^2)^qn\sum_{j=1}^n \left|\nabla f(\phi(z))\right|\left|\nabla \phi_j(z)\right|\nonumber\\
&=&n \left|\nabla f(\phi(z))\right|(1-|\phi(z)|^2)^p \frac{(1-|z|^2)^q}{(1-|\phi(z)|^2)^p}\sum_{j=1}^n \left|\nabla \phi_j(z)\right|\nonumber\\
&\leq&n \sup_{w\in \BB_n}\left\{\left|\nabla
f(w)\right|(1-|w|^2)^p\right\}
\sum_{j=1}^n \frac{(1-|z|^2)^q}{(1-|\phi(z)|^2)^p}\left|\nabla \phi_j(z)\right|\nonumber\\
&\leq& n||f||_{{\mathcal B}^p}nM\label{hy},
\end{eqnarray}
by Inequality (\ref{condi}). It follows that $||{\mathfrak C}_\phi
f||_{{\mathcal B}^q}\leq (1+n^2M)||f||_{{\mathcal B}^p}$ for every
$f\in {\mathcal B}^p(\BB_n)$, thus completing the proof of Theorem
\ref{main}, Part (A).
{\it (B).} We proceed by modifying the argument given in
\cite[~p.~187-188]{cm} for $n=1$. For $a\in \BB_n$, define
$f_a:\BB_n\rightarrow \CC$ to be function that vanishes at $0$ and
is the antiderivative of $\psi_a:\BB_n\rightarrow\CC$ given by
$\psi_a(t)=(1-\bar a t)^{-p}$. Let $w\in \BB_n$ and $u\in\partial
\BB_n$. Define $F_{w,u}:\BB_n\rightarrow\CC$ by
\[
F_{w,u}(z)=f_{\langle w,u\rangle}(\langle z,u\rangle).
\]
Define $\phi_u:\BB_n\rightarrow \BB_1$ by $\phi_u(z)=\langle
\phi(z),u\rangle$. Let $u^{(1)}:=u$, and choose
$u^{(2)},u^{(3)},\ldots ,u^{(n)}$ so that
${\UU}=\{u^{(1)},u^{(2)},u^{(3)},\ldots ,u^{(n)}\}$ is an
orthonormal basis for $\CC^n$. For all $z\in \BB_n$ and
\,$j\in\{2,3,\ldots,n\}$, we have that
\begin{eqnarray}
D_{u^{(j)}}F_{w,u}(z)&=&\lim_{\lambda\rightarrow 0}\frac{F_{w,u}(z+\lambda u^{(j)})-F_{w,u}(z)}{\lambda}\nonumber\\
&=&\lim_{\lambda\rightarrow 0}\frac{f_{\langle w,u^{(1)}\rangle}(\langle z+\lambda u^{(j)},u^{(1)}\rangle)- f_{\langle w,u^{(1)}
\rangle}(\langle z,u^{(1)}\rangle)}{\lambda}\nonumber\\
&=&0\label{tes}.
\end{eqnarray}
On the other hand, for every $z\in \BB_n$,
\begin{eqnarray}
D_{u^{(1)}}F_{w,u}(z)
&=&\lim_{\lambda\rightarrow 0}\frac{F_{w,u}(z+\lambda u)-F_{w,u}(z)}{\lambda}\nonumber\\
&=&\lim_{\lambda\rightarrow 0}\frac{f_{\langle w,u\rangle}(\langle
z,u\rangle+\lambda) -
f_{\langle w,u\rangle}(\langle z,u\rangle)}{\lambda}\nonumber\\
&=& \psi_{\langle w,u\rangle}(\langle z,u\rangle)\label{tes1}.
\end{eqnarray}
From Equations (\ref{tes}) and (\ref{tes1}), it follows that
\begin{equation}\label{tes3}
|\nabla^{\UU}F_{w,u}(z)|=|\psi_{\langle w,u\rangle}(\langle
z,u\rangle)|= \left|1-\overline{\langle w,u\rangle}\langle
z,u\rangle\right|^{-p}.
\end{equation}
We observe that the quantity above is bounded when $u$ is fixed.
This fact and Lemma \ref{ugrad} together imply that $F_{w,u}\in
{\mathcal B}_0^p(\BB_n)$. Also, we have
\[F_{w,u}(0)=f_{\langle w,u\rangle}(\langle 0,u\rangle)=f_{\langle w,u\rangle}(0)=0.\]
Furthermore, by Lemma \ref{ugrad}, we have that
\begin{eqnarray}
\sup_{z\in \BB_n}(1-|z|^2)^{p}|\nabla F_{w,u}(z)|
&=&\sup_{z\in \BB_n}(1-|z|^2)^{p}|\nabla^{\UU} F_{w,u}(z)|\nonumber\\
&=&\sup_{z\in \BB_n}(1-|z|^2)^{p}\left|1-{\overline{\langle
w,u\rangle}}\langle z,u\rangle\right|^{-p}\label{van1}.
\end{eqnarray}
Note that
\[
\left|1-{\overline{\langle w,u\rangle}}\langle
z,u\rangle|\right|^{-p}\leq (1-|z|)^{-p}\leq
\frac{2^{p}}{(1-|z|^2)^{p}}.
\]
It follows that Quantity (\ref{van1}) is less than or equal to
$2^{p}$. Hence, $F_{w,u}\in {\mathcal B}^p(\BB_n)$ for every $w\in
\BB_n$ and $u\in\pt\BB_n$; moreover, the set
\[\{||F_{w,u}||_{{\mathcal
B}^p}: u\in\pt\BB_n,\,\,w\in \BB_n\}
\]
is bounded. This fact and the hypothesis together imply that there
exist $C$ and $M\geq 0$ such that for every $w\in \BB_n$ and
$u\in\pt\BB_n$,
\[
||F_{w,u}\circ\phi||_{{\mathcal B}^q}\leq C||F_{w,u}||_{{\mathcal
B}^p}\leq CM.
\]
Therefore, we obtain that
\begin{equation}\label{tric}
\sup_{u\in\pt\BB_n,\,\,z,w\in \BB_n}\left\{|\nabla (f_{\langle
w,u\rangle}\circ\phi_{u})(z)|(1-|z|^2)^{q}\right\}\leq CM.
\end{equation}
Now for each $j\in\{1,2,\ldots,n\}$, we have that
\begin{eqnarray*}
D_{j} (f_{\langle w,u\rangle}\circ\phi_{u})(z)
&=& f'_{\langle w,u\rangle}(\langle \phi(z),u\rangle)D_j\langle \phi(z),u\rangle\\
&=& \left(1-{\overline{\langle w,u\rangle}}\langle
\phi(z),u\rangle\right)^{-p} D_j\langle \phi(z),u\rangle.
\end{eqnarray*}
It follows that
\begin{equation}\label{glass}
\nabla(f_{\langle w,u\rangle}\circ\phi_{u})(z)
=\left(1-{\overline{\langle w,u\rangle}}\langle
\phi(z),u\rangle\right)^{-p} \nabla\langle \phi(z),u\rangle
\end{equation}
Using Equation (\ref{glass}), we can rewrite Equation (\ref{tric}) as
\[
\sup_{u\in\pt\BB_n,\,\,z,w\in \BB_n}
\frac{(1-|z|^2)^{q}}{\left|1-{\overline{\langle w,u\rangle}}\langle
\phi(z),u\rangle\right|^{p}}|\nabla\langle \phi(z),u\rangle|\leq CM.
\]
In particular, we have that
\begin{equation}\label{endy}
\sup_{u\in \pt \BB_n,\,\, z\in
\BB_n}\frac{(1-|z|^2)^{q}}{(1-|\langle
\phi(z),u\rangle|)^{p}}|\nabla\langle \phi(z),u\rangle|\leq CM,
\end{equation}
from which the statement of Theorem \ref{main}, Part (B) follows.\bsk
By restricting the values of $u$, one obtains various necessary
conditions for compactness of $C_\phi$ from Part (B) of Theorem
\ref{main}. Two such conditions are listed in Corollary
\ref{062005} below. We point out that the boundedness of Quantity
(\ref{pet}) below when ${\mathfrak C}_\phi$ is bounded from
${\mathcal B}^p(\BB_n)$ to ${\mathcal B}^q(\BB_n)$ is a result given
by Zhou in \cite{zh100}.
\begin{corollary}\label{062005} Let $p,q>0$. If ${\mathfrak C}_\phi$ is a bounded operator from ${\mathcal B}^p(\BB_n)$ (respectively,
${\mathcal B}^p_0(\BB_n))$ to ${\mathcal B}^q(\BB_n)$, then there is
an $M\geq 0$ such that the following statements hold:
(i) For all $z\in \BB_n$ with $\phi(z)\not=0$, we have that
\begin{eqnarray}
\frac{(1-|z|^2)^q}{(1-|\phi(z)|^2)^p}\frac{|J_\phi(z)^T\cdot\phi(z)|}{|\phi(z)|}\leq
M.\label{pet}
\end{eqnarray}
(ii) For all $z\in\BB_n$ and $j\in\{1,2,\ldots,n\}$,
\begin{equation}\label{ebob}
\frac{(1-|z|^2)^q}{(1-|\phi_j(z)|^2)^p}|\nabla \phi_j(z)|\leq M.
\end{equation}
\end{corollary}
\begin{proof} Putting $u:={\overline{\phi(z)}}/|\phi(z)|$ in Theorem \ref{main}, Part (B),
one obtains that Quantity (\ref{pet}) is no larger than some $M'\geq
0$ for all $z\in B_n$ such that $\phi(z)\not=0$. Successively
replacing $u\in\pt\BB_n$ in Theorem \ref{main}, Part (B) by the
typically ordered standard basis elements $e_j$ of $\CC^n$ for
$j=1,2,\ldots,n$, we see that the left side of Inequality
(\ref{ebob}) is no larger than some $M''\geq 0$, so that we can
choose $M:=\max(M',M'')$.
\end{proof}
\section{Acknowledgements}\nonumber
The authors would like to thank W. Wogen for kindly pointing out
that an earlier, coordinate-dependent version of Theorem \ref{main},
Part (B) could be improved to its current coordinate-free form.
| 69,240
|
\begin{document}
\title[Gravitational wave of the Bianchi VII universe: particle trajectories, geodesic deviation and tidal accelerations]{Gravitational wave of the Bianchi VII universe: particle trajectories, geodesic deviation and tidal accelerations}
\author*[1,2]{\fnm{Konstantin} \sur{Osetrin}}\email{osetrin@tspu.edu.ru}
\author[1]{\fnm{Evgeny} \sur{Osetrin}}\email{evgeny.osetrin@tspu.edu.ru}
\equalcont{These authors contributed equally to this work.}
\author[1]{\fnm{Elena} \sur{Osetrina}}\email{elena.osetrina@tspu.edu.ru}
\equalcont{These authors contributed equally to this work.}
\affil*[1]{\orgdiv{Center for Mathematical and Computer Physics}, \orgname{Tomsk State Pedagogical University}, \orgaddress{\street{Kievskaya str. 60}, \city{Tomsk}, \postcode{634061},
\country{Russia}}}
\affil[2]{
\orgname{National Research Tomsk State University}, \orgaddress{\street{Lenina pr. 36}, \city{Tomsk}, \postcode{634050},
\country{Russia}}}
\abstract{
For the gravitational wave model based on the type III Shapovalov wave space-time, test particle trajectories and the exact solution of geodesic deviation equations for the Bianchi type VII universe are obtained.
Based on the found 4-vector of deviation, tidal accelerations in a gravitational wave are calculated.
For the obtained solution in a privileged coordinate system, an explicit form of transformations into a synchronous reference system is found, which allows time synchronization at any points of space-time with separation of time and spatial coordinates.
The synchronous reference system used is associated with a freely falling observer on the base geodesic.
In a synchronous coordinate system, an explicit form of the gravitational wave metric, a 4-vector of geodesic deviation, and a 4-vector of tidal accelerations in a gravitational wave are obtained.
The exact solution describes a variant of the primordial gravitational wave.
The results of the work can be used to study the plasma radiation generated by tidal accelerations of a gravitational wave.
}
\keywords{
gravitation wave, cosmology, deviation of geodesics, tidal acceleration, Hamilton-Jacobi equation, Bianchi universes, Stäckel spaces, Shapovalov spaces}
\maketitle
\section{Introduction}\label{sec1}
Gravitational waves are currently both an object of academic interest and an object of experimental observation, which provides new data about the Universe
\cite{PhysRevLett.116.061102,PhysRevX.9.031040,PhysRevX.11.021053}.
In addition, from the point of view of theoretical cosmology, the discovery of primordial gravitational waves can provide a weighty argument in support of the inflationary model of the initial stage of the Universe.
Obtaining exactly integrable models in this direction, as the basic elements of the theory, makes a significant contribution to the development of gravitational-wave research.
This paper presents new results on primordial gravitational waves in Bianchi type VII spaces, which are spatially homogeneous but not isotropic like the standard Friedmann–Robertson–Walker model, inspired by observational data on the anisotropy of the electromagnetic microwave background of the Universe \cite{Bennett2013} and is of interest for studying the initial stages of the dynamics of the Universe.
The main manifestation of the gravitational field from the point of view of the motion of test particles is the deviation of the geodesics along which the particles move. The deviation of geodesics leads to the appearance of tidal accelerations in the gravitational wave, which can cause various physical effects. For example, the primordial gravitational wave, acting on the primary plasma, excites plasma density waves, which leads to the appearance of additional electromagnetic fields that contributed to the characteristics of the background microwave radiation, which is currently detected using satellite telescopes \cite{Bennett2013}.
The calculation in such problems is quite complicated and is carried out mainly by approximate and numerical methods. In this regard, finding exact solutions to the equations of deviation of geodesics in a gravitational wave seems to be an important basic element of the theory, which makes it possible to exact calculate tidal accelerations and the resulting physical effects. Exact solutions in any physical theory are the cornerstones that make it possible to understand the qualitative behavior of physical objects in more complex cases. In addition, exact solutions make it possible to effectively debug approximate and numerical methods for more complex physical models, as well as to train computer ''artificial intelligence'' in recognizing gravitational waves.
As an initial model for a gravitational wave, we use Shapovalov type III wave spaces \cite{Osetrin2020Symmetry} in this paper, which allow the existence of privileged coordinate systems, where the metric depends on only one wave variable, along which the space-time interval vanishes. Shapovalov's wave spaces make it possible to construct a complete integral of the equation of motion of test particles in the Hamilton-Jacobi formalism and, consequently, to find the trajectories of motion of test particles in a gravitational wave in quadratures. The presence of the complete integral of the Hamilton-Jacobi equation also allows one to obtain the exact solution of the geodesic deviation equations, and then calculate the tidal accelerations for particles in a gravitational wave.
The privileged coordinate systems used to integrate the Hamilton-Jacobi equation are based on the three commuting Killing vectors allowed by the Shapovalov type III wave space, which determine the so-called "complete set" of integrals of the motion of test particles and make it possible to carry out analytical integration of the Hamilton-Jacobi equation by the method of complete separation of variables \cite{ Shapovalov1978I, Shapovalov1978II, Shapovalov1979}.
An additional advantage of Shapovalov wave spaces is the ability to construct an explicit coordinate transformation from a privileged coordinate system, where the exact integration of the equations of motion of test particles in the Hamilton-Jacobi formalism is allowed, into synchronous reference systems, where time and spatial coordinates are separated, and an observer freely falling in a gravitational field is at rest . Synchronous reference systems make it possible to synchronize time at different points of space-time and clarify the physical content of the obtained exact models.
Spaces that allow the existence of privileged coordinate systems, where the Hamilton-Jacobi equation of test particles can be integrated by the method of complete separation of variables, are widely used in the theory of gravity, since they allow integrating field equations and obtaining exactly integrable models. These are models with an electromagnetic field \cite{Obukhov1988,Obukhov2022632,Obukhov2021695},
models with dust and pure radiation \cite{OsetrinDust2016,OsetrinVaidya1996,OsetrinVaidya2009,OsetrinRadiation2017},
models with a scalar field \cite{OsetrinScalar2018,Obukhov2022142,Obukhov2021134,Obukhov2021183}.
Obtaining realistic models of primordial gravitational waves of the Universe on the basis of Shapovalov's wave spaces is also possible for gravity models taking into account quantum and other corrections (for example, f(R) - gravity, etc. \cite{Odintsov2007,Odintsov2011,Capozziello2011,Odintsov2017}).
Shapovalov's wave spaces make it possible to obtain exact wave solutions in the theory of gravity not only in Einstein's general theory of relativity, but also in modified theories of gravity \cite{OsetrinScalar312020,OsetrinScalar212020,OsetrinSymmetry2021}, which makes it possible to compare analogous solutions and select realistic theories and models.
Shapovalov's wave spaces allow selection of classes of spatially homogeneous models, which are interpreted as cosmological models of the Universe and, accordingly, describe primordial gravitational waves. So, for example, type III Shapovalov spaces allow IV, VI and VII types of Bianchi Universes (see for example \cite{OsetrinHomog312002,OsetrinHomog2006}), type II Shapovalov spaces allow Bianchi universes of type III \cite{OsetrinHomog212020}.
In this paper, we will consider the Shapovalov wave spaces related to Bianchi Universes of type VII.
Trajectories of test particles will be found, exact solutions of the geodesic deviation equations in a gravitational wave will be found, and tidal accelerations of particles in a gravitational wave will be calculated. The results are presented both in a privileged coordinate system, where the equations for test particle trajectories are integrated, and in a laboratory synchronous coordinate system associated with a freely falling observer.
\section{
Shapovalov wave spacetimes
}
The Hamilton-Jacobi equation for a test particle of mass $m$ in a gravitational field with the metric tensor $g_{{\alpha}{\beta}}(x^{\gamma})$
has the form (see \cite{LandauEng1}):
\begin{equation}
g^{{\alpha}{\beta}}\frac{\partial S}{\partial x^{\alpha}}\frac{\partial S}{\partial x^{\beta}}=m^2c ^2
,
\label{HJE}
\end{equation}
$$
{\alpha},{\beta},{\gamma} ...
=0,...,(n-1),
$$
where the capital letter $S$ denotes the test particle action function, $n$ is the space dimension. We will set the speed of light $c$ to be equal to unity.
In what follows, we will also use the lowercase letter $s$ to denote a space-time interval.
\begin{definition}
\label{stack_space}
If the space allows the existence of a privileged coordinate system $\{x^{\alpha}\}$, where the Hamilton-Jacobi equation (\ref{HJE}) can be integrated by the complete separation of variables method, when the complete integral $S$ for the test particle action function can be written in ''separated'' form:
\begin{equation}
S=\sum_{\alpha=0}^{n-1}\phi_\alpha(x^\alpha,\lambda_0,...,\lambda_{n-1})
,
\end{equation}
$$
\lambda_0,...,\lambda_{n-1}-\mbox{const},
\qquad
\det \Bigl\lvert \frac{\partial{}^2 S}{\partial x^{\alpha}\partial \lambda_{\beta}}\Bigr\rvert \ne 0
,
$$
moreover, one of the non-ignored variables (on which the metric depends) is a wave (null), i.e. the space-time interval along this variable vanishes, then such a space is called the Shapovalov wave space.
\end{definition}
Shapovalov spaces represent gravitational-wave models of space-time, which allow integrating the equations of motion of test particles and radiation in the Hamilton-Jacobi formalism in quadratures, i.e. make it possible to find an explicit form of geodesic lines along which test particles move.
Shapovalov wave spaces are a subclass of wave models for the broader class of {St{\"{a}}ckel} spaces (Paul {St{\"{a}}ckel}, see \cite{Stackel1891thesis,Stackel1897145}), which allow separation of variables in the Hamilton-Jacobi equation. The theory of {St{\"{a}}ckel} spaces was built by the efforts of a large number of researchers and was basically completed by the construction of a classification and a detailed description of these spaces in the works of Vladimir Shapovalov \cite{Shapovalov1978I,Shapovalov1978II,Shapovalov1979}.
Shapovalov spaces admit a ''complete set'' of Killing vectors and Killing tensors of the second rank, which correspond to sets of integrals of motion of test particles. Shapovalov wave spaces are classified according to the Abelian group of motions they admit, which specifies the number of ignored variables (on which the metric in the privileged coordinate system does not depend) and, in the case of space-time, can admit from one to three commuting Killing vectors in a ''complete set'' and, accordingly, for the case of space-time there are types I, II and III of Shapovalov wave spaces.
Shapovalov's wave spaces allow explicit construction of the complete integral of the Hamilton-Jacobi equation for test particles and make it possible to obtain an analytical form of their motion trajectories in the Hamilton-Jacobi formalism.
The test particle coordinates $x^{\alpha}$ are functions of proper time $\tau$ on the base geodesic line along which the test particle moves, and is given in the Hamilton-Jacobi formalism by the particle trajectory equations in the form:
\begin{equation}
\label{MovEqu}
\frac{\partial S (x^{\alpha},\lambda_{k})}{\partial \lambda_{j}}=\sigma_{j},
\quad
\tau=S (x^{\alpha},\lambda_{k})\rvert_{m=1},
\end{equation}
$$
{i},{j},{k}=1,2,3;
$$
where $\lambda_{k}$, $\sigma_{k}$ are independent constant parameters determined by the initial or boundary data for the motion of a test particle along the base geodesic line, the variable $\tau$ is the proper time of the particle.
Shapovalov type III wave space-time metric in a privileged coordinate system depends on one wave variable and can be reduced to the following form
(see f.e. \cite{OsetrinHomog312002,OsetrinHomog2006,OsetrinHomog312020}):
$$
{ds}^2=2dx^0dx^1
+g_{ab}(x^0)\Bigl(dx^a+g^a(x^0) \, dx^1\Bigr)
$$
\begin{equation}
\Bigl(dx^b +g^b(x^0) \, dx^1\Bigr)
\label{MetricShapovalovIII}
,\end{equation}
where indices a, b run through the values 2, 3. Thus, in the general case, the metric includes five arbitrary functions of the wave (null) variable $x^0$.
Einstein's equations with cosmological constant $\Lambda$ in vacuum
\begin{equation}
R_{\alpha\beta}=\Lambda g_{\alpha\beta}
\label{EinsteinEqs}
\end{equation}
for the metric (\ref{MetricShapovalovIII}) lead to the following necessary conditions:
\begin{equation}
\Lambda=g^a=0
.\end{equation}
The space-time metric for a gravitational wave (\ref{MetricShapovalovIII}) has type N according to Petrov's classification.
\section{
Deviation of geodesics in Shapovalov wave spaces
}
The deviation of geodesics in the general theory of relativity and in modified theories of gravity is the main manifestation of the gravitational field and gravitational waves in particular. Geodesic deviation is used to detect the gravitational field and provides information about the space-time curvature tensor, and also allows you to calculate various effects of gravity on physical objects, incl. tidal effects of the gravitational field.
The geodesic deviation equation has the following form:
\begin{equation}
\label{deviation}
\frac{D^2 \eta^{\alpha}}{{d\tau}^2}=
R^{\alpha}{}_{{\beta}{\gamma}{\delta}}u^{\beta} u^{\gamma}\eta^{\delta}.
\end{equation}
Here
$u^{\alpha}(\tau)$ is the four-velocity of the test particle on the base geodesic line,
$D$ is the covariant derivative,
$\eta^{\alpha}(\tau)$ is the geodesic deviation vector, $R^{\alpha}{}_{{\beta}{\gamma}{\delta}}$ is the Riemann curvature tensor.
This equation is considered along the base geodesic line of a test particle with four-velocity $u^{\alpha}$ and, therefore, the coordinates $x^{\alpha}$ are parametrized by one parameter (the proper time of the test particle $\tau$).
For the four-velocity test particle, the normalization condition is satisfied
\begin{equation}
\label{Norm}
g^{{\alpha}{\beta}}u_{\alpha} u_{\beta} =1.
\end{equation}
For four-velocity, assuming the mass of the test particle to be equal to unity, we obtain the expression:
\begin{equation}
\label{General4U}
u_{\alpha}=\frac{\partial S}{\partial x^{\alpha}}
\,\biggl\rvert_{m=1}
,\end{equation}
where $S$ is the test particle action function.
Thus, first finding the form of the metric from the field equations for the corresponding gravitational model, then obtaining the trajectories of the test particles from the Hamilton-Jacobi equations (\ref{HJE}) and calculating the four-velocities of the test particles from the relation (\ref{General4U}), one can write the deviation equations (\ref{deviation}) explicitly and integrate the resulting system of differential equations in a direct way.
On the other hand, if we are able to find the complete integral of the Hamilton-Jacobi equation for a test particle
$S (x^{\alpha} ,\lambda_{a})$,
then the deviation vector of geodesics
$\eta^{\alpha}$
can be found also
according to the method proposed in the work
\cite{Bazanski19891018}. Using the variational method, a generalization of the Hamilton-Jacobi equations for the complete integral of geodesics was found, which made it possible to obtain equations for determining the deviation of geodesics through the complete integral of the Hamilton-Jacobi equation.
According to this method, the deviation vector $\eta^{\alpha}$ can be found as a solution of a linear algebraic system of equations on the trajectory of the ''base'' test particle
\begin{equation}
\label{Deviation1}
\eta^{\alpha}\,
\frac{\partial u_{\alpha}(x^{\beta},\lambda_{j})}{\partial\lambda_{i}}
+\rho_{k}\frac{\partial^2 S(x^{\beta},\lambda_{j})}{\partial\lambda_{i}\partial\lambda_{k}}=\vartheta_{i}
,
\end{equation}
\begin{equation}
\label{Deviation2}
u_{\alpha}(x^{\beta},\lambda_{i})\,\eta^{\alpha}=0
,
\end{equation}
$$
{i},{j},{k} = 1\ldots 3;
\qquad
{\alpha}, {\beta}, {\gamma}=0\ldots 3,
$$
where $\lambda_{i}$, $\rho_{i}$, $\vartheta_{i}$ are independent constant parameters.
Here the constants $\lambda_{i}$ are determined by the initial data or boundary conditions for the velocity (momentum) of the test particle on the base geodesic line.
The constants $\rho_{i}$ are determined by the initial data for the deviation vector (that is, they are related to the initial data adjacent geodetic line).
The values ot the constants $\vartheta_{i}$ are related to the definition of the origin of the variables $x^{\alpha}$.
The equation (\ref{Deviation2}) expresses the condition that the deviation vector $\eta^{\alpha}$ is orthogonal to the base geodesic line.The dependence of the variables $x^{\alpha}$ on the proper time $\tau$ on the base geodesic line along which the test particle moves is established in the Hamilton-Jacobi formalism by the equations (\ref{MovEqu}) for the particle trajectory.
Thus, the Shapovalov wave spaces make it possible to obtain exact solutions for gravitational waves both of the equations of motion of test particles and to find exact solutions of the geodesic deviation equations in a gravitational wave.
\section{Spatially homogeneous models of Shapovalov spaces}
Earlier, in the study of spatially homogeneous symmetries of Shapovalov spaces of type III
it was shown that three types of spatially homogeneous models can be built on the basis of these spaces: types IV, VI and VII according to Bianchi (see f.e. \cite{OsetrinHomog312002,OsetrinHomog2006,OsetrinHomog312020}). Moreover, gravitational waves in Shapovalov spaces of type IV and VI types of Bianchi are aperiodic in the wave variable.
Bianchi type VII Shapovalov wave space metric
in a privileged coordinate system, where the metric depends only on the wave (null) variable $x^0$ (along the variable $x^0$ the space-time interval vanishes), can be represented as \cite{OsetrinHomog2006}:
$$
{ds}^2=2dx^0dx^1
$$
$$
\mbox{}
-
\frac{{x^0}^{2 \omega } }{\gamma \left({\delta}^2-1\right)}
\,\biggl[
\left(1-{\delta} \cos \left(\theta -2\log {x^0}\right)\right) \,\left({dx^2}\right)^2
$$
$$
\mbox{}
-
2 \sin \left(\theta -2\log {x^0}\right)\,{dx^2}{dx^3}
$$
\begin{equation}
\mbox{}
+
\left(1+{\delta} \cos \left(\theta -2\log {x^0}\right)\right)
\,\left({dx^3}\right)^2
\,\biggr]
\label{Metric}
,\end{equation}
\begin{equation}
{g=\det g_{ij}=}\frac{-{x^0}^{4 \omega }}{\gamma ^2 \left({1-\delta}^2\right)}
,\qquad
{\delta}^2<1
,\end{equation}
where $x^0$ is the wave variable,
the constants ${\gamma}$, ${\delta}$, $\theta$ and ${\omega}$
are independent parameters of the gravitational wave model. The parameter ${\omega}$ is the parameter of the Bianchi type VII spatial homogeneity group.
The space-time model under consideration can be interpreted as a model of a propagating primordial gravitational wave against the background of an expanding Universe.
The space with the metric (\ref{Metric}) admits a covariantly constant vector $K$ and, therefore, is a plane-wave space:
\begin{equation}
\nabla_{\beta} K_{\alpha}=0
\quad
\to
\quad
K_{\alpha}=\bigl( K_0,0,0,0 \bigl)
,\end{equation}
where $K_0$ is a {constant}.
A space with a metric (\ref{Metric}) admits a spatial homogeneity group with Killing vectors $X_{(1)}$, $X_{(2)}$, and $X_{(3)}$, which can be chosen in the privileged coordinate system in the form
\begin{equation}
X^{\alpha}_{(1)}=\bigl(0,0,1,0\bigr),
\qquad
X^{\alpha}_{(2)}=\bigl(0,0,0,1\bigr),
\end{equation}
\begin{equation}
X^{\alpha}_{(3)}=\bigl(-x^0, \,x^1, \,\omega\, x^2-x^3, \,x^2+\omega\,x^3 \bigr)
.\end{equation}
The additional fourth Killing vector associated with the choice of the privileged coordinate system commutes with the vectors $X_{(1)}$ and $X_{(2)}$ and has the form in this coordinate system
\begin{equation}
X^{\alpha}_{(0)}=\bigl(0,1,0,0\bigr)
.
\end{equation}
The Killing vector $X_{(0)}$ is a null vector, because $g_{{\alpha}{\beta}}X^{\alpha}_{(0)}X^{\beta}_{(0)}=0$.
The commutation relations for the Killing vectors have the form:
\begin{equation}
\left[X_{(0)},X_{(1)}\right]=0
,\qquad
\left[X_{(0)},X_{(2)}\right]=0
,\end{equation}
\begin{equation}
\left[X_{(0)},X_{(3)}\right]=X_{(0)}
,\end{equation}
\begin{equation}
\left[X_{(1)},X_{(2)}\right]=0
,\end{equation}
\begin{equation}
\left[X_{(1)},X_{(3)}\right]= {\omega}X_{(1)}+X_{(2)}
,\end{equation}
\begin{equation}
\left[X_{(2)},X_{(3)}\right]=-X_{(1)}+ {\omega}X_{(2)}
.
\end{equation}
The Killing vectors $X_{(0)}$, $X_{(1)}$, $X_{(2)}$ and $X_{(3)}$ generate a 4-dimensional motion group, the $X_{(0 )}$, $X_{(1)}$, $X_{(2)}$ generate a 3-dimensional abelian subgroup, vectors $X_{(1)}$, $X_{(2)}$ and $X_{ (3)}$ generate a 3-dimensional subgroup of the spatial homogeneity of the type VII Bianchi model.
\section{Gravitational Wave for Type VII Bianchi Models in Einstein's Theory of Gravity}
Consider the solution of the Einstein equations in vacuum (\ref{EinsteinEqs}) for the gravitational wave metric (\ref{Metric}).
As a result, we obtain the following exhaustive restrictions on the parameters of the gravitational wave:
\begin{equation}
\label{deltaParameterRange}
{\delta}^2 = \frac{\omega (\omega -1) }{\omega ^2-\omega -1}
,\end{equation}
\begin{equation}
\label{omegaParameterRange}
0\le\omega\le 1
,\qquad
0\le\delta^2\le 1/5
.\end{equation}
Thus, three independent constant parameters of the gravitational wave remain in the model under consideration: the Bianchi type VII homogeneity subgroup parameter $\omega$, the $\gamma$ parameter related to the wave amplitude, and the angular parameter $\theta$ related to the wave phase.
The $(+,-,-,-)$ metric signature imposes additional restrictions on the choice of $\gamma$ sign:
\begin{equation}
\gamma<0.
\end{equation}
To shorten the formulas, we will further use the notation ${\delta}$, understanding it as an expression of the following form:
\begin{equation}
\label{deltaOmega}
{\delta}(\omega) =\pm \sqrt{ \frac{\omega (\omega -1) }{\omega ^2-\omega -1} }
.\end{equation}
The nonzero components of the Riemann curvature tensor $R_{\alpha\beta\gamma\delta}$ for the metric (\ref{Metric}), subject to the constraints (\ref{deltaParameterRange}) and (\ref{omegaParameterRange}) in the privileged coordinate system, take the following form:
$$
{R}_{0202} =
-\frac{\left(\omega ^2-\omega -1\right) {x^0}^{2 \omega -2}
}{\gamma }
\Bigl[
-2 (\omega -1) \omega
$$
$$
\mbox{}
+
2 \left(\omega ^2-\omega -1\right) {\delta} \cos \left( \theta -2\log {x^0} \right)
$$
\begin{equation}
\mbox{}
+(2 \omega -1) {\delta} \sin \left( \theta -2\log {x^0} \right)
\Bigr]
,\end{equation}
$$
{R}_{0302} =
\frac{\left(\omega ^2-\omega -1\right) {\delta} {x^0}^{2 \omega -2}
}{\gamma }
\Bigl[
$$
$$
(2 \omega -1) \cos \left( \theta -2\log {x^0} \right)
$$
\begin{equation}
\mbox{}
+
2 \left(-\omega ^2+\omega +1\right) \sin \left( \theta -2\log {x^0} \right)
\Bigr]
,\end{equation}
$$
{R}_{0303} = \frac{\left(\omega ^2-\omega -1\right) {x^0}^{2 \omega -2}
}{\gamma }
\Bigl[
2 (\omega -1) \omega
$$
$$
\mbox{}
+
2 \left(\omega ^2-\omega -1\right) {\delta} \cos \left( \theta -2\log {x^0} \right)
$$
\begin{equation}
\mbox{}
+(2 \omega -1) {\delta} \sin \left( \theta -2\log {x^0} \right)
\Bigr]
.\end{equation}
Thus, for the value of the parameter $\omega=0,1$ the parameter $\delta$ and the Riemann curvature tensor $R_{\alpha\beta\gamma\delta}$
vanish, the model degenerates, and space-time becomes flat.
\section{Integration of the Hamilton-Jacobi equation for test particles (case of $\omega\ne 1/2$)}
To obtain the trajectories of motion of test particles in a gravitational wave (\ref{Metric}), we consider the Hamilton-Jacobi equation (\ref{HJE}) in the given space. In accordance with the general properties of Shapovalov spaces, we will look for the complete integral of the Hamilton-Jacobi equation in a separated form:
\begin{equation}
S(x^\alpha,\lambda_k)=\phi_0(x^0)+\sum_{k=1}^3 \lambda_k x^k
,\end{equation}
where the independent constant parameters $\lambda_k$ are determined by the initial or boundary conditions for the motion of a test particle.
Then the Hamilton-Jacobi equation (\ref{HJE}) gives for the function $\phi_0(x^0)$ (${\lambda_1}\ne0$):
$$
\phi_0{}' =
\frac{1}{2 {\lambda_1}}
+
\frac{\gamma {x^0}^{-2 \omega }
}{2 {\lambda_1}}
\biggl[
-{\lambda_2}^2 -{\lambda_3}^2
$$
$$
\mbox{}
-2{\delta} {\lambda_2} {\lambda_3} \sin \left(\theta -2\log {x^0}\right)
$$
\begin{equation}
\label{PhiEq}
\mbox{}
- {\delta} \left({\lambda_2}^2-{\lambda_3}^2\right) \cos \left(\theta -2\log {x^0}\right)
\biggr]
.\end{equation}
The integration of the equation (\ref{PhiEq}) has a singularity at $\omega=1/2$. Therefore, we will further consider the case $\omega=1/2$ separately.
In this section, we will assume that $\omega\ne 1/2$. Then the equation (\ref{PhiEq}) gives
for the function $\phi_0(x^0)$:
$$
\phi_0 (x^0) =
\frac{x^0}{2 {\lambda_1} }
+
\frac{
{x^0}^{1-2 \omega } \gamma({\lambda_2}^2 +{\lambda_3}^2 )
}{2 {\lambda_1} (2 \omega -1) }
$$
$$
\mbox{}
+
\frac{
{\gamma} {\delta}
{x^0}^{1-2 \omega }
}{2 {\lambda_1}
\left(4 \omega ^2-4 \omega +5\right)}
\Bigl[
$$
$$
2
\left({\lambda_2}^2
+{\lambda_2} {\lambda_3}
(2 \omega -1)
-{\lambda_3}^2\right) \sin \left(\theta -2\log {x^0}\right)
$$
\begin{equation}
\mbox{}
+
\Bigl(
(2 \omega -1)
\left( {\lambda_2}^2+{\lambda_3}^2 \right)
-4 {\lambda_2} {\lambda_3}
\Bigr)
\cos \left(\theta -2\log {x^0}\right)
\Bigr]
. \end{equation}
Thus, we have found the complete integral of the Hamilton-Jacobi equation of test particles $S(x^\alpha,\lambda_k)$.
Now we can write and solve particle trajectory equations (\ref{MovEqu}) in the Hamilton-Jacobi formalism.
\onecolumn
Omitting obvious calculations, we present the result of solving the equations (\ref{MovEqu}) in the form of a test particle trajectory in a privileged coordinate system:
\begin{equation}
x^0 (\tau) = {\lambda_1} {\tau}
, \end{equation}
$$
x^1 (\tau) =
\frac{
({\lambda_1} {\tau})^{1-2 \omega }
}{2 {\lambda_1}^2
(2 \omega -1 )
}
\Bigl(
(2 \omega -1) ({\lambda_1} {\tau})^{2 \omega }
+\gamma({\lambda_2}^2 +{\lambda_3}^2 )
\Bigr)
$$
$$
\mbox{}
+
\frac{
\gamma {\delta}
({\lambda_1} {\tau})^{1-2 \omega}
}{2 {\lambda_1}^2
(4 \omega ^2 - 4 \omega + 5)}
\Bigl[
2
\Bigl(
{\lambda_2}^2+{\lambda_2} {\lambda_3} (2 \omega -1)-{\lambda_3}^2
\Bigr)
\sin \left(\theta -2\log \left({\lambda_1} {\tau}\right)\right)
$$
\begin{equation}
\mbox{}
+
\Bigl(
{\lambda_2}^2 (2 \omega -1)-4 {\lambda_2} {\lambda_3}+{\lambda_3}^2 (1-2 \omega )
\Bigr)
\cos \left(\theta -2\log \left({\lambda_1} {\tau}\right)\right)
\Bigr]
, \end{equation}
$$
x^2 (\tau) =
-\frac{
\gamma
({\lambda_1} {\tau})^{1-2 \omega }
}{{\lambda_1} (2 \omega -1 ) (4 \omega ^2 - 4 \omega + 5)}
\Bigl[
{\lambda_2} \left(4 \omega ^2-4 \omega +5\right)
$$
$$
\mbox{}
+
(2 \omega -1) {\delta} (2 {\lambda_2}+{\lambda_3} (2 \omega -1)) \sin \left(\theta -2\log \left({\lambda_1} {\tau}\right)\right)
$$
\begin{equation}
+(2 \omega -1) {\delta} ({\lambda_2} (2 \omega -1)-2 {\lambda_3}) \cos \left(\theta -2\log \left({\lambda_1} {\tau}\right)\right)
\Bigr]
, \end{equation}
$$
x^3 (\tau) =
\frac{\gamma
({\lambda_1} {\tau})^{1-2 \omega }
}{{\lambda_1}(2 \omega -1 ) (4 \omega ^2 - 4 \omega + 5)}
\Bigl[
{\lambda_3} \left(-4 \omega ^2+4 \omega -5\right)
$$
$$
\mbox{}
-(2 \omega -1) {\delta} ({\lambda_2} (2 \omega -1)-2 {\lambda_3}) \sin \left(\theta -2\log \left({\lambda_1} {\tau}\right)\right)
$$
\begin{equation}
\mbox{}
+(2 \omega -1) {\delta} (2 {\lambda_2}+{\lambda_3} (2 \omega -1)) \cos \left(\theta -2\log \left({\lambda_1} {\tau}\right)\right)
\Bigr]
, \end{equation}
where $\tau$ is the proper time of the test particle. In the process of integrating the equations (\ref{MovEqu}), we set the constants $\sigma_k$ equal to zero by choosing the origin of the variables $x^\alpha$ and the proper time $\tau$.
Obtaining an explicit form of test particle trajectories in a gravitational wave here demonstrates the possibilities of exact integration of wave models in privileged coordinate systems in Shapovalov spaces.
For the solutions obtained, we find the 4-velocity of the test particle in the privileged coordinate system:
\begin{equation}
u^{\alpha}(\tau) =\frac{D x^{\alpha}}{d\tau}
, \end{equation}
\begin{equation}
u^{0} = {\lambda_1}
,\qquad
x^0= {\lambda_1} \tau
\label{uPrivilUp0}
, \end{equation}
\begin{equation}
u^{1} =
\frac{
{x^0}^{-2 \omega }
}{2 {\lambda_1}}
\Bigl[
{x^0}^{2 \omega }
-\gamma {\delta} \left({\lambda_2}^2-{\lambda_3}^2\right) \cos \left(\theta -2\log {x^0}\right)
-2 {\lambda_2} {\lambda_3} \gamma {\delta} \sin \left(\theta -2\log {x^0}\right)
-\gamma ({\lambda_2}^2 +{\lambda_3}^2)
\Bigr]
\label{uPrivilUp1}
, \end{equation}
\begin{equation}
\label{uPrivilUp2}
u^{2} = \gamma {x^0}^{-2 \omega }
\left[
{\lambda_2} {\delta} \cos \left(\theta -2\log {x^0}\right)
+{\lambda_3} {\delta} \sin \left(\theta -2\log {x^0}\right)
+{\lambda_2}
\right]
, \end{equation}
\begin{equation}
\label{uPrivilUp3}
u^{3} = \gamma {x^0}^{-2 \omega }
\left[
{\lambda_2} {\delta} \sin \left(\theta -2\log {x^0}\right)
-{\lambda_3}{\delta} \cos \left(\theta -2\log {x^0}\right)+{\lambda_3}
\right]
, \end{equation}
Having obtained complete the integral of the Hamilton-Jacobi equation $S(x^\alpha,\lambda_k)$ and the 4-velocity of test particles, we can now write and solve equations for the components of the geodesic deviation vector
(\ref{Deviation1})-(\ref{Deviation2}).
Omitting the obvious calculations for solving the system of algebraic equations (\ref{Deviation1})-(\ref{Deviation2}), we present the exact solution for the geodesic deviation vector $\eta^\alpha (\tau)$ in the privileged coordinate system ($\omega\ne 1/2$):
\begin{equation}
\label{DeviationSolutionPriv0}
\eta^0 (\tau) = {\rho_1} \tau -{\lambda_1} {\Omega}
,\qquad
x^0(\tau)=\lambda_1\tau
, \end{equation}
$$
\eta^1 (\tau) =
{\vartheta_1}
-\frac{
\gamma {R_2} {x^0}^{1-2 \omega }
}{{\lambda_1}^3 (2 \omega -1) \left(4 \omega ^2-4 \omega +5\right)}
\Bigl[
{\lambda_2} \left(4 \omega ^2-4 \omega +5\right)
$$
$$
+
(2 \omega -1) {\delta} (2 {\lambda_2}+{\lambda_3} (2 \omega -1)) \sin \left(\theta -2\log {x^0}\right)
$$
$$
+(2 \omega -1) {\delta} ({\lambda_2} (2 \omega -1)-2 {\lambda_3}) \cos \left(\theta -2\log {x^0}\right)
\Bigr]
$$
$$
+
\frac{
\gamma {R_3}{x^0}^{1-2 \omega }
}{{\lambda_1}^3 (2 \omega -1) \left(4 \omega ^2-4 \omega +5\right)}
\Bigl[
{\lambda_3} \left(-4 \omega ^2+4 \omega -5\right)
$$
$$
-(2 \omega -1) {\delta} ({\lambda_2} (2 \omega -1)-2 {\lambda_3}) \sin \left(\theta -2\log {x^0}\right)
$$
$$
+(2 \omega -1) {\delta} (2 {\lambda_2}+{\lambda_3} (2 \omega -1)) \cos \left(\theta -2\log {x^0}\right)
\Bigr]
$$
$$
+\frac{
{x^0}^{-2 \omega }
}{2 {\lambda_1}^3}
\Bigl[
\gamma {\delta} \left({\lambda_2}^2-{\lambda_3}^2\right) \left({\lambda_1}^2 {\Omega}-{\rho_1} {x^0}\right) \cos \left(\theta -2\log {x^0}\right)
$$
$$
+2 {\lambda_2} {\lambda_3} \gamma {\delta} \left({\lambda_1}^2 {\Omega}-{\rho_1} {x^0}\right) \sin \left(\theta -2\log {x^0}\right)
-{\rho_1} {x^0}^{2 \omega +1}
$$
\begin{equation}
\mbox{}
- {\lambda_1}^2{\Omega} {x^0}^{2 \omega }
+({\lambda_2}^2+{\lambda_3}^2 )\gamma ({\lambda_1}^2{\Omega}- {\rho_1} {x^0})
\Bigr]
\label{DeviationSolutionPriv1}
, \end{equation}
$$
\eta^2 (\tau) =
{\vartheta_2}
-\frac{
\gamma {x^0}^{-2 \omega }
\left(
{\lambda_1}^2 {\Omega}-{\rho_1} {x^0}
\right)
}{{\lambda_1}^2}
\Bigl[
{\lambda_2}
+{\lambda_2} {\delta} \cos \left(\theta -2\log {x^0}\right)
$$
$$
+{\lambda_3} {\delta} \sin \left(\theta -2\log {x^0}\right)
\Bigr]
$$
$$
+\frac{
\gamma {R_2} {x^0}^{1-2 \omega }
}{{\lambda_1}^2 (2 \omega -1) \left(4 \omega ^2-4 \omega +5\right)}
\Bigl[
4 \omega ^2-4 \omega +5
$$
$$
+2 (2 \omega -1) {\delta} \sin \left(\theta -2\log {x^0}\right)
+(1-2 \omega )^2 {\delta} \cos \left(\theta -2\log {x^0}\right)
\Bigr]
$$
\begin{equation}
-\frac{\gamma {R_3} {\delta} {x^0}^{1-2 \omega }
\left[
(1-2 \omega ) \sin \left(\theta -2\log {x^0}\right)+2 \cos \left(\theta -2\log {x^0}\right)
\right]
}{{\lambda_1}^2 \left(4 \omega ^2-4 \omega +5\right)}
\label{DeviationSolutionPriv2}
, \end{equation}
$$
\eta^3 (\tau) =
{\vartheta_3}
+
\frac{\gamma {x^0}^{-2 \omega } \left({\lambda_1}^2 {\Omega}-{\rho_1} {x^0}\right)
}{{\lambda_1}^2}
\Bigl[
-{\lambda_2} {\delta} \sin \left(\theta -2\log {x^0}\right)
$$
$$
+{\lambda_3} {\delta} \cos \left(\theta -2\log {x^0}\right)-{\lambda_3}
\Bigr]
$$
$$
-\frac{\gamma {R_2} {\delta} {x^0}^{1-2 \omega }
\left[
(1-2 \omega ) \sin \left(\theta -2\log {x^0}\right)+2 \cos \left(\theta -2\log {x^0}\right)
\right]
}{{\lambda_1}^2 \left(4 \omega ^2-4 \omega +5\right)}
$$
$$
-\frac{
\gamma {R_3} {x^0}^{1-2 \omega }
}{{\lambda_1}^2 (2 \omega -1) \left(4 \omega ^2-4 \omega +5\right)}
\Bigl[
-4 \omega ^2+4 \omega -5
$$
\begin{equation}
+2 (2 \omega -1) {\delta} \sin \left(\theta -2\log {x^0}\right)
+(1-2 \omega )^2 {\delta} \cos \left(\theta -2\log {x^0}\right)
\Bigr]
\label{DeviationSolutionPriv3}
, \end{equation}
where, for brevity, the auxiliary notation $R_2$, $R_3$ and $\Omega$ is introduced:
\begin{equation}
\label{R23}
R_2 = {\lambda_2} {\rho_1}-{\lambda_1} {\rho_2}
,\qquad
R_3 = {\lambda_3} {\rho_1}-{\lambda_1} {\rho_3}
,\end{equation}
\begin{equation}
\label{constOmega}
\Omega = {\lambda_1} {\vartheta_1}+{\lambda_2} {\vartheta_2}+{\lambda_3} {\vartheta_3}
.\end{equation}
The exact solution obtained above for the deviation vector $\eta^\alpha (\tau)$ includes a number of independent parameters.
The parameters $\omega$, $\gamma$ and $\theta$ determine the characteristics of the gravitational wave.
The parameters $\lambda_k$ define the motion of the test particle on the base geodesic and are determined by the initial or boundary conditions for the particle velocity on the base geodesic.
The parameters ${\rho_k}$ and ${\vartheta_k}$ are determined by the initial or boundary conditions for the velocity and relative position of the particle on the adjacent geodesic.
One of the important manifestations of geodesic deviation in a gravitational field is tidal accelerations $A^\alpha = D^2\eta^\alpha/d^2\tau$, which affect physical objects in a gravitational wave.
In a privileged coordinate system, the form of tidal accelerations in a gravitational wave is quite cumbersome, so we will not present it here, we only note that
component $A^0=0$. Further, passing to the synchronous frame of reference, we will find a compact form for tidal accelerations.
\section{Gravitational wave in synchronous frame of reference (case of~$\omega\ne 1/2$)}
Synchronous frames of reference are physically distinguished frames of reference, since they allow one to synchronize time at different points in space-time \cite{LandauEng1}.
Also, in contrast to the privileged coordinate system, which we used to integrate the Hamilton-Jacobi equations, in the synchronous reference frame, time and spatial coordinates are separated, which is important for the observer. Therefore, the possibility of passing to a synchronous frame of reference is an advantage in the physical analysis of solutions. Shapovalov wave spaces provide such an opportunity, since the presence of the complete integral of the Hamilton-Jacobi equation and the knowledge of trajectories for test particles allows one to find an explicit analytical form of such a transition. Moreover, the proper time of an observer freely falling along the base geodesic $\tau$ in a synchronous frame of reference becomes a common time variable. According to the algorithm described in \cite{LandauEng1}, the transformation from the privileged coordinate system $x^\alpha$ to the synchronous frame $\tilde x{}^\alpha$ has the form:
\begin{equation}
x^{\alpha} \to \tilde x{}^{\alpha}=\left(\tau,\lambda_1,\lambda_2,\lambda_3 \right)
, \end{equation}
\begin{equation}
x^0 = {\tilde x{}^1} {\tau}
\label{ToSynchr0}
, \end{equation}
$$
x^1 =
\frac{
\tau^2
({\tilde x{}^1} {\tau})^{-2 \omega-1 }
}{2
(2 \omega -1 ) (4 \omega ^2 - 4 \omega + 5)}
\Bigl[
\left(4 \omega ^2-4 \omega +5\right)
\Bigl(
(2 \omega -1) ({\tilde x{}^1} {\tau})^{2 \omega }
+\gamma({\tilde x{}^2}^2 +{\tilde x{}^3}^2 )
\Bigr)
$$
$$
+
2 \gamma (2 \omega -1) {\delta}
\Bigl(
{\tilde x{}^2}^2+{\tilde x{}^2} {\tilde x{}^3} (2 \omega -1)-{\tilde x{}^3}^2
\Bigr)
\sin \left(\theta -2\log \left({\tilde x{}^1} {\tau}\right)\right)
$$
\begin{equation}
+\gamma (2 \omega -1) {\delta}
\Bigl(
{\tilde x{}^2}^2 (2 \omega -1)-4 {\tilde x{}^2} {\tilde x{}^3}+{\tilde x{}^3}^2 (1-2 \omega )
\Bigr)
\cos \left(\theta -2\log \left({\tilde x{}^1} {\tau}\right)\right)
\Bigr]
\label{ToSynchr1}
, \end{equation}
$$
x^2 =
-\frac{
\gamma {\tau} ({\tilde x{}^1} {\tau})^{-2 \omega }
}{(2 \omega -1 ) (4 \omega ^2 - 4 \omega + 5)}
\Bigl[
{\tilde x{}^2} \left(4 \omega ^2-4 \omega +5\right)
$$
$$
+
(2 \omega -1) {\delta} (2 {\tilde x{}^2}+{\tilde x{}^3} (2 \omega -1)) \sin \left(\theta -2\log \left({\tilde x{}^1} {\tau}\right)\right)
$$
\begin{equation}
+(2 \omega -1) {\delta} ({\tilde x{}^2} (2 \omega -1)-2 {\tilde x{}^3}) \cos \left(\theta -2\log \left({\tilde x{}^1} {\tau}\right)\right)
\Bigr]
\label{ToSynchr2}
, \end{equation}
$$
x^3 =
\frac{\gamma {\tau} ({\tilde x{}^1} {\tau})^{-2 \omega }
}{(2 \omega -1 ) (4 \omega ^2 - 4 \omega + 5)}
\Bigl[
{\tilde x{}^3} \left(-4 \omega ^2+4 \omega -5\right)
$$
$$
-(2 \omega -1) {\delta} ({\tilde x{}^2} (2 \omega -1)-2 {\tilde x{}^3}) \sin \left(\theta -2\log \left({\tilde x{}^1} {\tau}\right)\right)
$$
\begin{equation}
+(2 \omega -1) {\delta} (2 {\tilde x{}^2}+{\tilde x{}^3} (2 \omega -1)) \cos \left(\theta -2\log \left({\tilde x{}^1} {\tau}\right)\right)
\Bigr]
\label{ToSynchr3}
. \end{equation}
The chosen synchronous reference system is connected with the observer on the base geodesic, which in the synchronous reference system will have constant spatial coordinates. Four-velocity components of a freely falling observer on a base geodesic
(\ref{uPrivilUp0})-(\ref{uPrivilUp3}) are converted to the following form:
\begin{equation}
\tilde u^\alpha=\Bigl\{1,0,0,0\Bigr\}
. \end{equation}
Thus, the observer is at rest on the base geodesic relative to the chosen synchronous frame of reference.
Now, by applying the transformations (\ref{ToSynchr0})-(\ref{ToSynchr3}), we can write down the form of the gravitational wave metric in the synchronous frame of reference:
\begin{equation}
ds^2=d \tau^2-dl^2=d \tau^2+\tilde g{}_{{i}{j}}(\tau, {\tilde x{}^{k}})
\,d{\tilde x{}^{i}}d{\tilde x{}^{j}}
,\qquad
i,j,k=1,2,3;
\end{equation}
where $\tau$ is the time variable (the observer's proper time on the base geodesic), $dl$ is the spatial distance element, ${\tilde x{}^{k}}$ are the spatial coordinates.
The components of the gravitational wave metric (\ref{Metric}) in the synchronous frame take the following form ($\omega\ne 1/2$):
\begin{equation}
\label{metricSynchr1k}
\tilde g{}^{00} = 1
,\qquad
\tilde g{}^{01} =
\tilde g{}^{02} =
\tilde g{}^{03} = 0
, \end{equation}
\begin{equation}
\tilde g{}^{1k} = -\frac{{\tilde x{}^1} {\tilde x{}^k}}{{\tau}^2}
, \end{equation}
$$
\tilde g{}^{22} =
\frac{
(1-2 \omega )^2 \left(\omega ^2-\omega -1\right) ({\tau} {\tilde x{}^1})^{2 \omega }
}{5 \gamma {\tau}^2}
\biggl[
-4 \omega ^2+4 \omega -5
$$
$$
\mbox{}
+
(2\omega+1)(2\omega-3)
{\delta}
\cos \Bigl(\theta -2 \log ({\tau} {\tilde x{}^1})\Bigr)
$$
\begin{equation}
\mbox{}
+4 (2 \omega -1) {\delta} \sin \Bigl(\theta -2 \log ({\tau} {\tilde x{}^1})\Bigr)
\biggr]
-\frac{{\tilde x{}^2}^2}{{\tau}^2}
\label{metricSynchr22}
, \end{equation}
$$
\tilde g{}^{23} =
\frac{
(1-2 \omega )^2 \left(\omega ^2-\omega -1\right) {\delta} ({\tau} {\tilde x{}^1})^{2 \omega }
}{5 \gamma {\tau}^2}
\biggl[
4(1-2 \omega ) \cos \Bigl(\theta -2 \log ({\tau} {\tilde x{}^1})\Bigr)
$$
\begin{equation}
\mbox{}
+
(2\omega+1)(2\omega-3)
\sin \Bigl(\theta -2 \log ({\tau} {\tilde x{}^1})\Bigr)
\biggr]
-\frac{{\tilde x{}^2} {\tilde x{}^3}}{{\tau}^2}
\label{metricSynchr23}
, \end{equation}
$$
\tilde g{}^{33} =
-\frac{
(1-2 \omega )^2 \left(\omega ^2-\omega -1\right) ({\tau} {\tilde x{}^1})^{2 \omega }
}{5 \gamma {\tau}^2}
\biggl[
4 \omega ^2-4 \omega +5
$$
$$
\mbox{}
+
(2\omega+1)(2\omega-3)
{\delta} \cos \Bigl(\theta -2 \log ({\tau} {\tilde x{}^1})\Bigr)
$$
\begin{equation}
\mbox{}
+4 (2 \omega -1) {\delta} \sin \Bigl(\theta -2 \log ({\tau} {\tilde x{}^1})\Bigr)
\biggr]
-\frac{{\tilde x{}^3}^2}{{\tau}^2}
\label{metricSynchr33}
, \end{equation}
where $\omega$, $\gamma$ and $\theta$ are independent parameters of the gravitational wave model, and the constant ${\delta}$ is given by the relation (\ref{deltaOmega}).
Converting the components of the deviation vector $\eta{}^\alpha (\tau)$ from the preferred coordinate system (\ref{DeviationSolutionPriv0})-(\ref{DeviationSolutionPriv3}) to the synchronous reference system gives the following result:
\begin{equation}
\tilde \eta{}^0= 0
\label{DeviationSolutionSynchr0}
,\end{equation}
\begin{equation}
\tilde \eta{}^1 (\tau) = {\rho_1}-\frac{{\lambda_1} {\Omega}}{{\tau}}
\label{DeviationSolutionSynchr1}
,\end{equation}
$$
\tilde \eta{}^2 (\tau) =
\frac{(1-2 \omega ) \left(1+\omega -\omega ^2\right) {\lambda_1}{\vartheta_2} ({\lambda_1} {\tau})^{2 \omega-1 }
}{5 \gamma }
\Bigl[ 2 (1-2 \omega) {\delta} \sin \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
$$
$$
\mbox{}
-(1-2 \omega )^2 {\delta} \cos \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)+4 \omega ^2-4 \omega +5\Bigr]
$$
$$
\mbox{}
-\frac{(1-2 \omega )^2 \left(1+\omega -\omega ^2\right) {\delta} {\lambda_1}{\vartheta_3} ({\lambda_1} {\tau})^{2 \omega-1 }
}{5 \gamma }
\Bigl[
2 \cos \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
$$
\begin{equation}
\mbox{}
+(1-2 \omega ) \sin \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
\Bigr]
-\frac{{\lambda_2} {\Omega}}{{\tau}}+{\rho_2}
\label{DeviationSolutionSynchr2}
,\end{equation}
$$
\tilde \eta{}^3 (\tau) =
-\frac{(1-2 \omega )^2 \left(1+\omega -\omega ^2\right)
{\delta}{\lambda_1} {\vartheta_2} ({\lambda_1} {\tau})^{2 \omega -1}
}{5 \gamma }
\Bigl[
2 \cos \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
$$
$$
\mbox{}
+ (1-2 \omega ) \sin \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
\Bigr]
$$
$$
\mbox{}
+\frac{(1-2 \omega ) \left(1+\omega -\omega ^2\right) {\lambda_1}{\vartheta_3} ({\lambda_1} {\tau})^{2 \omega -1}
}{5 \gamma }
\Bigl[
2 (2 \omega -1) {\delta} \sin \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
$$
\begin{equation}
\mbox{}
+(1-2 \omega )^2 {\delta} \cos \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)+4 \omega ^2-4 \omega +5
\Bigr]
-\frac{{\lambda_3} {\Omega}}{{\tau}}+{\rho_3}
\label{DeviationSolutionSynchr3}
.\end{equation}
Here
the parameters $\omega$, $\gamma$ and $\theta$ determine the gravitational wave model,
the constants $\lambda_k$, $\rho_k$ and $\vartheta_k$ are given by the initial or boundary conditions for the velocities and mutual positions of particles on neighboring geodesics,
the auxiliary constant $\Omega$ is determined by the relation (\ref{constOmega}), the constant $\delta$ is determined by the relation (\ref{deltaOmega}).
Let us now find the form of tidal accelerations $\tilde A{}^\alpha$ of a gravitational wave (\ref{Metric})
in the synchronous reference system:
\begin{equation}
\tilde A{}^\alpha =\frac{ D^2\tilde\eta{}^\alpha}{d^2\tau}
, \end{equation}
\begin{equation}
\tilde A{}^0= \tilde A{}^1= 0
, \end{equation}
$$
\tilde A{}^2=
\frac{(2\omega -1)(\omega ^2-\omega -1) {\delta} {\vartheta_2} ({\lambda_1} {\tau})^{2 \omega }
}{\gamma {\tau}^3}
\Bigl[
2 \cos \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
$$
$$
\mbox{}
+(1-2 \omega ) \sin \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
\Bigr]
$$
$$
\mbox{}
+\frac{(2\omega -1)(\omega ^2-\omega -1){\delta} {\vartheta_3} ({\lambda_1} {\tau})^{2 \omega }
}{\gamma {\tau}^3}
\Bigl[
2 \sin \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
$$
$$
\mbox{}
+(2 \omega -1) \cos \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
\Bigr]
$$
$$
\mbox{}
+\frac{{R_2} \left(\omega ^2-\omega -1\right) {\delta}
}{{\lambda_1} {\tau}^2}
\Bigl[
(1-2 \omega ) \sin \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)+2 \cos \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
\Bigr]
$$
\begin{equation}
\mbox{}
+\frac{{R_3}
}{{\lambda_1} {\tau}^2}
\Bigl[
2 \left(\omega ^2-\omega -1\right) {\delta} \sin \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
$$
$$
\mbox{}
+(2\omega -1)(\omega ^2-\omega -1) {\delta} \cos \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)+\omega (2\omega-1)(1-\omega)
\Bigr]
, \end{equation}
$$
\tilde A{}^3= \frac{(2\omega -1)(\omega ^2-\omega -1){\delta} {\vartheta_2} ({\lambda_1} {\tau})^{2 \omega }
}{\gamma {\tau}^3}
\Bigl[
2 \sin \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
$$
$$
\mbox{}
+(2 \omega -1) \cos \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
\Bigr]
$$
$$
\mbox{}
-\frac{(2\omega -1)(\omega ^2-\omega -1) {\delta} {\vartheta_3} ({\lambda_1} {\tau})^{2 \omega }
}{\gamma {\tau}^3}
\Bigl[
2 \cos \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
$$
$$
\mbox{}
+
(1-2 \omega ) \sin \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
\Bigr]
$$
$$
\mbox{}
+\frac{{R_2}
}{{\lambda_1} {\tau}^2}
\Bigl[
\omega (2\omega-1)(\omega-1)
+2 \left(\omega ^2-\omega -1\right) {\delta} \sin \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
$$
$$
\mbox{}
+(2\omega -1)(\omega ^2-\omega -1) {\delta} \cos \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
\Bigr]
$$
$$
\mbox{}
-\frac{{R_3} \left(\omega ^2-\omega -1\right) {\delta}
}{{\lambda_1} {\tau}^2}
\Bigl[
2 \cos \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
$$
\begin{equation}
\mbox{}
+(1-2 \omega ) \sin \Bigl(\theta -2 \log ({\lambda_1} {\tau})\Bigr)
\Bigr]
, \end{equation}
where the parameters $\omega$, $\gamma$ and $\theta$ determine the characteristics of the gravitational wave, the parameters $\lambda_k$ are determined
by the initial or boundary values of the particle velosity on the base geodesic,
the parameters ${\rho_k}$ and ${\vartheta_k}$ are determined
by the initial or boundary values of the velosity
and relative positions of the particle on the adjacent geodesic, auxiliary constants $R_2$ and $R_3$ are defined by the relations (\ref{R23}).
In the synchronous reference frame used, the components of the gravitational wave tidal acceleration are not equal to zero only in the plane of variables ${\tilde x{}^{2}}$, ${\tilde x{}^{3}}$. The gravitational wave propagates along the space variable ${\tilde x{}^{1}}$.
\twocolumn
\section{A special case of a wave at $\omega=1/2$ in a privileged coordinate system}
When integrating the Hamilton-Jacobi equation of test particles for the Shapovalov type III space in the Bianchi type VII cosmological model for the metric (\ref{Metric}), a special case arises when
$\omega=1/2$. In this case, the wave metric acquires the following form in the privileged coordinate system
$$
{ds}^2=
2dx^0dx^1
-
\frac{
{x^0}
}{\gamma ^2-\alpha ^2-\beta ^2}
\Bigl[
$$
$$
\left(
\alpha \cos{\bigl(2\log{x^0}\bigr)}+\beta \sin{\bigl(2\log{x^0}\bigr)}-\gamma
\right)
{dx^2}^2
$$
$$
+
\left(
\beta \cos{\bigl(2\log{x^0}\bigr)}-\alpha \sin{\bigl(2\log{x^0}\bigr)}
\right)
{dx^2}{dx^3}
$$
\begin{equation}
+
\left(
-\alpha \cos{\bigl(2\log{x^0}\bigr)}-\beta \sin{\bigl(2\log{x^0}\bigr)}-\gamma
\right)
{dx^3}^2
\Bigr]
,\end{equation}
\begin{equation}
{g=\det g_{ij}=}-\frac{{x^0}^2}{\gamma ^2-\alpha ^2-\beta ^2}
.\end{equation}
The Einstein vacuum equations (\ref{EinsteinEqs}) in this case lead to a condition on the constants $\alpha$, $\beta$ and $\gamma$ of the form
\begin{equation}
\gamma^2 = 5 \left(\alpha ^2+\beta^2\right)
\label{constgamma}
,\end{equation}
which allows us to introduce a two-parameter representation ($\gamma$, $\theta$) using the constant angular parameter $\theta$ instead of the parameters $\alpha$ and $\beta$:
\begin{equation}
\label{GammaTheta}
\alpha=\frac{\gamma\sin{\theta}}{\sqrt{5}}
,\qquad
\beta=\frac{\gamma\cos{\theta}}{\sqrt{5}}
.\end{equation}
The metric signature $(+,-,-,-)$ leads to a negative value of the constant $\gamma$:
\begin{equation}
\gamma = -\sqrt{5 \left(\alpha ^2+\beta^2\right)} <0
,\end{equation}
Integrating the Hamilton-Jacobi equation of a test particle for the special case $\omega=1/2$, we obtain the complete integral in the following form
$$
S=
\sum_{k=1}^{3}{\lambda_k}{x^k}
+
\frac{
1
}{4 {\lambda_1}}
\biggl[
2 {x^0} -2 \gamma \left({\lambda_2}^2+{\lambda_3}^2\right) \log{x^0}
$$
$$
\mbox{}
+
\Bigl(
\beta \left({\lambda_2}^2-{\lambda_3}^2\right)-2 \alpha {\lambda_2} {\lambda_3}
\Bigr)
\cos{\left( 2\log{{x^0}}\right)}
$$
\begin{equation}
\mbox{}
+
\Bigl(
\alpha \left({\lambda_3}^2-{\lambda_2}^2\right)-2 \beta {\lambda_2} {\lambda_3}
\Bigr)
\sin{\left(2\log{x^0}\right)}
\biggr]
,\end{equation}
where $\lambda_k$ are constant parameters,
determined by the initial or boundary values of the velocity of the test particle.
From the equations of motion in the Hamilton-Jacobi formalism, we find the general form of the trajectories of motion of test particles in a gravitational wave in a privileged coordinate system:
\begin{equation}
x^0 (\tau) = {\lambda_1}{\tau}
,\end{equation}
$$
x^1 (\tau) =
-\frac{
1
}{4 {\lambda_1}^2}
\Bigl[
$$
$$
\left(\alpha \left({\lambda_2}^2-{\lambda_3}^2\right)+2 \beta {\lambda_2} {\lambda_3}\right) \sin{\bigl(2\log{({\lambda_1}{\tau})}\bigr)}
$$
$$
+\left(2 \alpha {\lambda_2} {\lambda_3}+\beta \left({\lambda_3}^2-{\lambda_2}^2\right)\right) \cos{\bigl(2\log{({\lambda_1}{\tau})}\bigr)}
$$
\begin{equation}
+2 \gamma \left({\lambda_2}^2+{\lambda_3}^2\right) \log{({\lambda_1}{\tau})}-2 {\lambda_1}{\tau}
\Bigr]
,\end{equation}
$$
x^2 (\tau) =
\frac{
1}{2 {\lambda_1}}
\Bigl[
(\alpha {\lambda_2}+\beta {\lambda_3}) \sin{\bigl(2\log{({\lambda_1}{\tau})}\bigr)}
$$
\begin{equation}
+(\alpha {\lambda_3}-\beta {\lambda_2}) \cos{\bigl(2\log{({\lambda_1}{\tau})}\bigr)}+2 {\lambda_2} \gamma \log{({\lambda_1}{\tau})}
\Bigr]
,\end{equation}
$$
x^3 (\tau) =
\frac{
1}{2 {\lambda_1}}
\Bigl[
(\beta {\lambda_2}-\alpha {\lambda_3}) \sin{\bigl(2\log{({\lambda_1}{\tau})}\bigr)}
$$
\begin{equation}
+(\alpha {\lambda_2}+\beta {\lambda_3}) \cos{\bigl(2\log{({\lambda_1}{\tau})}\bigr)}+2 {\lambda_3} \gamma \log{({\lambda_1}{\tau})}
\Bigr]
,\end{equation}
where $\tau$ is the proper time of the test particle.
The components of the 4-velocity vector $u^\alpha$ of test particles in a gravitational wave take the form:
\begin{equation}
u^{0} = {\lambda_1}
, \end{equation}
$$
u^{1} (\tau) =
\frac{
1}{2 {\lambda_1} {x^0}}
\Bigl[
{x^0}
-\gamma ({\lambda_2}^2 +{\lambda_3}^2 )
$$
$$
\mbox{}
+
\sin{\bigl(2\log{x^0}\bigr)} \left(2 \alpha {\lambda_2} {\lambda_3}-\beta {\lambda_2}^2+\beta {\lambda_3}^2\right)
$$
\begin{equation}
\mbox{}
+\cos{\bigl(2\log{x^0}\bigr)} \left(\alpha \left({\lambda_3}^2-{\lambda_2}^2\right)-2 \beta {\lambda_2} {\lambda_3}\right)
\Bigr]
, \end{equation}
$$
u^{2} (\tau) =
\frac{
1}{{x^0}}
\Bigl[
\sin{\bigl(2\log{x^0}\bigr)} (\beta {\lambda_2}-\alpha {\lambda_3})
$$
\begin{equation}
\mbox{}
+\cos{\bigl(2\log{x^0}\bigr)} (\alpha {\lambda_2}+\beta {\lambda_3})
+{\lambda_2} \gamma
\Bigr]
, \end{equation}
$$
u^{3} (\tau) =
\frac{
1
}{{x^0}}
\Bigl[
\cos{\bigl(2\log{x^0}\bigr)} (\beta {\lambda_2}-\alpha {\lambda_3})
$$
\begin{equation}
\mbox{}
-\sin{\bigl(2\log{x^0}\bigr)} (\alpha {\lambda_2}+\beta {\lambda_3})
+{\lambda_3} \gamma
\Bigr]
, \end{equation}
where $x^0 = {\lambda_1}{\tau}$.
\onecolumn
Solving the equations (\ref{Deviation1})-(\ref{Deviation2}) for the deviation vector $\eta^\alpha (\tau)$ , we obtain its following form in the privileged coordinate system:
\begin{equation}
\eta^0 (\tau) = {\rho_1}{\tau} - {\lambda_1} {\Omega}
, \qquad
x^0 = {\lambda_1}{\tau}
, \end{equation}
$$
\eta^1 (\tau) =
\frac{{R_2}
}{2 {\lambda_1}^3}
\Bigl[
\sin{\bigl(2\log{x^0}\bigr)}
(\alpha {\lambda_2}+\beta {\lambda_3})
$$
$$
+
\cos \left(2\log{{x^0}} \right)
(\alpha {\lambda_3}-\beta {\lambda_2})
+2 \gamma {\lambda_2} \log ({x^0})
\Bigr]
$$
$$
\mbox{}
+
\frac{{R_3}
}{2 {\lambda_1}^3}
\Bigl[
\sin{\bigl(2\log{x^0}\bigr)} (\beta {\lambda_2}-\alpha {\lambda_3})
$$
$$
+\cos{\bigl(2\log{x^0}\bigr)} (\alpha {\lambda_2}+\beta {\lambda_3})+2 \gamma {\lambda_3} \log ({x^0})
\Bigr]
$$
$$
+
\frac{
1 }{2 {\lambda_1}^3 {x^0}}
\Bigl[
\sin{\bigl(2\log{x^0}\bigr)} \left(-2 \alpha {\lambda_2} {\lambda_3}+\beta {\lambda_2}^2-\beta {\lambda_3}^2\right) \left({\lambda_1}^2 {\Omega}-{\rho_1} {x^0}\right)
$$
$$
+\cos{\bigl(2\log{x^0}\bigr)} \left(\alpha \left({\lambda_2}^2-{\lambda_3}^2\right)+2 \beta {\lambda_2} {\lambda_3}\right) \left({\lambda_1}^2 {\Omega}-{\rho_1} {x^0}\right)
$$
\begin{equation}
\mbox{}
-{\rho_1} {x^0}^2
-
{x^0}
\left(
{\lambda_1}^2 {\Omega}
+\gamma {\rho_1}
\left(
{\lambda_2}^2
+{\lambda_3}^2
\right)
\right)
+
{\lambda_1}^2 \gamma {\Omega}
\left(
{\lambda_2}^2 + {\lambda_3}^2
\right)
\Bigr]
+{\vartheta_1}
, \end{equation}
$$
\eta^2 (\tau) =
-\frac{
\left({\lambda_1}^2 {\Omega}-{\rho_1} {x^0}\right)
}{{\lambda_1}^2 {x^0}}
\Bigl[
\sin{\bigl(2\log{x^0}\bigr)} (\beta {\lambda_2}-\alpha {\lambda_3})
$$
$$
+\cos{\bigl(2\log{x^0}\bigr)} (\alpha {\lambda_2}+\beta {\lambda_3})
+{\lambda_2} \gamma
\Bigr]
$$
$$
+\frac{
{R_2}
}{2 {\lambda_1}^2}
\Bigl[
-\alpha \sin{\bigl(2\log{x^0}\bigr)}+\beta \cos{\bigl(2\log{x^0}\bigr)}-2 \gamma \log ({x^0})
\Bigr]
$$
\begin{equation}
\mbox{}
-\frac{{R_3}
}{2 {\lambda_1}^2}
\Bigl[
\alpha \cos{\bigl(2\log{x^0}\bigr)}+\beta \sin{\bigl(2\log{x^0}\bigr)}
\Bigr]
+{\vartheta_2}
, \end{equation}
$$
\eta^3 (\tau) =
-\frac{
\left({\lambda_1}^2 {\Omega}-{\rho_1} {x^0}\right)
}{{\lambda_1}^2 {x^0}}
\Bigl[
-\sin{\bigl(2\log{x^0}\bigr)} (\alpha {\lambda_2}+\beta {\lambda_3})
$$
$$
+\cos{\bigl(2\log{x^0}\bigr)} (\beta {\lambda_2}-\alpha {\lambda_3})+{\lambda_3} \gamma
\Bigr]
$$
$$
-\frac{
{R_3}
}{2 {\lambda_1}^2}
\Bigl[
-\alpha \sin{\bigl(2\log{x^0}\bigr)}+\beta \cos{\bigl(2\log{x^0}\bigr)}+2 \gamma \log ({x^0})
\Bigr]
$$
\begin{equation}
\mbox{}
-\frac{
{R_2}
}{2 {\lambda_1}^2}
\Bigl[
\alpha \cos{\bigl(2\log{x^0}\bigr)}+\beta \sin{\bigl(2\log{x^0}\bigr)}
\Bigr]
+{\vartheta_3}
, \end{equation}
where the constant parameters ${\rho_k}$ and ${\vartheta_k}$ are determined by the initial or boundary values of the relative positions and velocities of neighboring geodesics. Auxiliary constants ${R_2}$, ${R_3}$ and $\Omega$ are introduced for brevity and are determined by the relations (\ref{R23})-(\ref{constOmega}).
Having obtained the solution for the deviation vector, we can now write down the form of tidal accelerations $A^\alpha (\tau)$ of neighboring geodesics. Calculating the covariant derivative $A^\alpha =D^2\eta^\alpha/{d\tau}^2$, we obtain the tidal acceleration components in the preferred coordinate system in the form:
\begin{equation}
A^0 = 0
, \qquad
x^0 = {\lambda_1}{\tau}
,\end{equation}
$$
A^1 (\tau) =
\frac{
{R_2}
}{4 {\lambda_1} {x^0}^2}
\biggl[
10 {\lambda_2} \log ({x^0})
\left(\alpha \cos{\bigl(2\log{x^0}\bigr)}+\beta \sin{\bigl(2\log{x^0}\bigr)}\right)
$$
$$
-{\lambda_3}
\left(10 \alpha \log ({x^0}) \sin{\bigl(2\log{x^0}\bigr)}-10 \beta \log ({x^0}) \cos{\bigl(2\log{x^0}\bigr)}+\gamma \right)
\biggr]
$$
$$
+
\frac{
{R_3}
}{4 {\lambda_1} {x^0}^2}
\biggl[
{\lambda_2}
\left(-10 \alpha \log ({x^0}) \sin{\bigl(2\log{x^0}\bigr)}+10 \beta \log ({x^0}) \cos{\bigl(2\log{x^0}\bigr)}+\gamma \right)
$$
$$
- 10 {\lambda_3} \log ({x^0})
\left(\alpha \cos{\bigl(2\log{x^0}\bigr)}+\beta \sin{\bigl(2\log{x^0}\bigr)} \right)
\biggr]
$$
$$
-
\frac{
{5 {\lambda_1}\vartheta_2}
}{2 \gamma {x^0}^2}
\biggl[
{\lambda_2}
\left(\alpha \cos{\bigl(2\log{x^0}\bigr)}+\beta \sin{\bigl(2\log{x^0}\bigr)}\right)
$$
$$
+ {\lambda_3}
\left(\beta \cos{\bigl(2\log{x^0}\bigr)} -\alpha \sin{\bigl(2\log{x^0}\bigr)}\right)
\biggr]
$$
$$
+
\frac{
{5 {\lambda_1}\vartheta_3}
}{2 \gamma {x^0}^2}
\biggl[
{\lambda_3}
\left(\alpha \cos{\bigl(2\log{x^0}\bigr)} +\beta \sin{\bigl(2\log{x^0}\bigr)}\right)
$$
\begin{equation}
- {\lambda_2}
\left(\beta \cos{\bigl(2\log{x^0}\bigr)}-\alpha \sin{\bigl(2\log{x^0}\bigr)} \right)
\biggr]
,\end{equation}
$$
A^2 (\tau) =
\frac{5 {\lambda_1}^2 {\vartheta_2}}{2 \gamma {x^0}^2}
\left(\alpha \cos{\bigl(2\log{x^0}\bigr)}+\beta \sin{\bigl(2\log{x^0}\bigr)}\right)
$$
$$
+\frac{5 {\lambda_1}^2 {\vartheta_3} }{2 \gamma {x^0}^2}
\left(\beta \cos{\bigl(2\log{x^0}\bigr)}-\alpha \sin{\bigl(2\log{x^0}\bigr)}\right)
$$
$$
-\frac{{R_3}}{4 {x^0}^2}
\left(-10 \alpha \log ({x^0}) \sin{\bigl(2\log{x^0}\bigr)}
+10 \beta \log ({x^0}) \cos{\bigl(2\log{x^0}\bigr)}+\gamma \right)
$$
\begin{equation}
-\frac{5 {R_2} \log ({x^0}) }{2 {x^0}^2}
\left(\alpha \cos{\bigl(2\log{x^0}\bigr)}+\beta \sin{\bigl(2\log{x^0}\bigr)}\right)
,\end{equation}
$$
A^3 (\tau) =
\frac{5 {\lambda_1}^2 {\vartheta_2} }{2 \gamma {x^0}^2}
\left(\beta \cos{\bigl(2\log{x^0}\bigr)}-\alpha \sin{\bigl(2\log{x^0}\bigr)}\right)
$$
$$
-\frac{5 {\lambda_1}^2 {\vartheta_3} }{2 \gamma {x^0}^2}
\left(\alpha \cos{\bigl(2\log{x^0}\bigr)}+\beta \sin{\bigl(2\log{x^0}\bigr)}\right)
$$
$$
+\frac{{R_2} }{4 {x^0}^2}
\left(10 \alpha \log ({x^0}) \sin{\bigl(2\log{x^0}\bigr)}
-10 \beta \log ({x^0}) \cos{\bigl(2\log{x^0}\bigr)}+\gamma \right)
$$
\begin{equation}
+\frac{5 {R_3} \log ({x^0})
}{2 {x^0}^2}
\left(\alpha \cos{\bigl(2\log{x^0}\bigr)}+\beta \sin{\bigl(2\log{x^0}\bigr)}\right)
,\end{equation}
where the constant parameters of the gravitational wave $\alpha$, $\beta$, $\gamma$ are related by the relation (\ref{GammaTheta}), the constants $\lambda_k$, $\rho_k$, ${\vartheta_k}$ are given by the initial conditions for velocities and relative positions of particles on neighboring geodesics, the constants ${R_2}$ and ${R_3}$ are determined by the relations
(\ref{R23}).
The obtained exact solution for tidal acceleration in a gravitational wave can be used to calculate various physical effects when a wave passes through a material medium.
\twocolumn
\section{Special case of a wave for $\omega=1/2$
in synchronous reference system}
To study the physical effects of a gravitational wave and the action of tidal accelerations, it is convenient to switch to a synchronous reference frame, where the observer is at rest on the base geodesic, time and spatial coordinates are separated, and the observer’s proper time on the base geodesic is a temporal variable, and time can be synchronized between different points in space. time.
The transition to a similar synchronous reference frame $\tilde x{}^\alpha$ with a known complete integral of the Hamilton-Jacobi equation of test particles is carried out according to the well-known \cite{LandauEng1} algorithm by transformation to spatial variables, which are kept constant on the trajectory of the basic test particle:
\begin{equation}
x^\alpha \to \tilde x{}^\alpha=\left(\tau,\lambda_1,\lambda_2,\lambda_3 \right)
,\end{equation}
In our case $(\omega=1/2)$ the coordinate transformation takes the form:
\begin{equation}
x^0 = {\tilde x{}^1}{\tau}
,\end{equation}
$$
x^1 =
-\frac{
1
}{4 {\tilde x{}^1}^2}
\Bigl[
$$
$$
\left(\alpha \left({\tilde x{}^2}^2-{\tilde x{}^3}^2\right)+2 \beta {\tilde x{}^2} {\tilde x{}^3}\right)
\sin{\bigl(2\log{({\tilde x{}^1}{\tau})}\bigr)}
$$
$$
+\left(2 \alpha {\tilde x{}^2} {\tilde x{}^3}+\beta \left({\tilde x{}^3}^2-{\tilde x{}^2}^2\right)\right)
\cos{\bigl(2\log{({\tilde x{}^1}{\tau})}\bigr)}
$$
\begin{equation}
+2 \gamma \left({\tilde x{}^2}^2+{\tilde x{}^3}^2\right) \log{({\tilde x{}^1}{\tau})}-2 {\tilde x{}^1}{\tau}
\Bigr]
,\end{equation}
$$
x^2 =
\frac{
1}{2 {\tilde x{}^1}}
\Bigl[
(\alpha {\tilde x{}^2}+\beta {\tilde x{}^3}) \sin{\bigl(2\log{({\tilde x{}^1}{\tau})}\bigr)}
$$
$$
+(\alpha {\tilde x{}^3}-\beta {\tilde x{}^2})
\cos{\bigl(2\log{({\tilde x{}^1}{\tau})}\bigr)}
$$
\begin{equation}
+2 {\tilde x{}^2} \gamma \log{({\tilde x{}^1}{\tau})}
\Bigr]
,\end{equation}
$$
x^3 =
\frac{
1}{2 {\tilde x{}^1}}
\Bigl[
(\beta {\tilde x{}^2}-\alpha {\tilde x{}^3})
\sin{\bigl(2\log{({\tilde x{}^1}{\tau})}\bigr)}
$$
$$
+(\alpha {\tilde x{}^2}+\beta {\tilde x{}^3})
\cos{\bigl(2\log{({\tilde x{}^1}{\tau})}\bigr)}
$$
\begin{equation}
+2 {\tilde x{}^3} \gamma \log{({\tilde x{}^1}{\tau})}
\Bigr]
,\end{equation}
Upon transition to the considered synchronous frame of reference, the components of the gravitational wave metric will take the following form:
\begin{equation}
{ds}^2={d\tau}^2-{dl}^2={d\tau}^2+\tilde g{}_{ij}(\tau,\tilde x{}^k)\,d\tilde x{}^id\tilde x{}^j
,\end{equation}
where ${dl}$ is the spatial distance element, which in the synchronous reference frame is a function of time $\tau$ and spatial variables $\tilde x{}^k$.
The four-velocity of the particle on the base geodesic in the chosen synchronous frame of reference, as expected, takes the following form:
\begin{equation}
\tilde u{}^\alpha=\Bigl\{ 1,0,0,0 \Bigr\}
,\end{equation}
those, in the selected synchronous frame of reference, the observer rests on the base geodesic.
The metric of a gravitational wave in the Bianchi type VII space in the considered synchronous reference frame in the case of $\omega=1/2$ will take the following form:
\begin{equation}
\tilde g{}^{00} = 1
,\qquad
\tilde g{}^{01} =
\tilde g{}^{02} =
\tilde g{}^{03} = 0
,\end{equation}
\begin{equation}
\tilde g{}^{1k} = -\frac{{\tilde x{}^1} {\tilde x{}^k}}{{\tau}^2}
, \end{equation}
$$
\tilde g{}^{22} =
-\frac{\left({\tilde x{}^2}\right)^2}{{\tau}^2}
$$
$$
\mbox{}
+
\frac{
20\, {\tilde x{}^1}
}{\gamma^2{\tau} \Bigl( 1 -20 \log ^2({\tilde x{}^1}{\tau}) \Bigr)^2}
\biggl[
\gamma
+
20 \gamma \log ^2({\tilde x{}^1}{\tau})
$$
$$
+
\sin{\bigl(2\log{({\tilde x{}^1}{\tau})}\bigr)}
\left(
20 \beta \log ^2({\tilde x{}^1}{\tau})
-20 \alpha \log{({\tilde x{}^1}{\tau})}-\beta
\right)
$$
\begin{equation}
+\cos{\bigl(2\log{({\tilde x{}^1}{\tau})}\bigr)}
\left(
20 \alpha \log ^2({\tilde x{}^1}{\tau}) +20 \beta \log{({\tilde x{}^1}{\tau})}
-\alpha
\right)
\biggr]
, \end{equation}
$$
\tilde g{}^{23} =
-\frac{{\tilde x{}^2} {\tilde x{}^3}}{{\tau}^2}
+
\frac{
20\, {\tilde x{}^1}
}{\gamma^2{\tau} \Bigl( 1 -20 \log ^2({\tilde x{}^1}{\tau}) \Bigr)^2}
\biggl[
$$
$$
\Bigl(\alpha
- 20
\log{({\tilde x{}^1}{\tau})}
\left(
\alpha \log{({\tilde x{}^1}{\tau})} + \beta
\right)
\Bigr)
\sin{\bigl(2\log{({\tilde x{}^1}{\tau})}\bigr)}
$$
\begin{equation}
+\left(20 \beta \log ^2({\tilde x{}^1}{\tau}) - 20 \alpha \log{({\tilde x{}^1}{\tau})}-\beta \right)
\cos{\bigl(2\log{({\tilde x{}^1}{\tau})}\bigr)}
\biggr]
, \end{equation}
$$
\tilde g{}^{33} =
-\frac{\left({\tilde x{}^3}\right)^2}{{\tau}^2}
$$
$$
\mbox{}
+
\frac{
20\, {\tilde x{}^1}
}{\gamma^2{\tau} \Bigl( 1 -20 \log ^2({\tilde x{}^1}{\tau})\Bigr)^2}
\biggl[
\gamma
\left(
1
+
20 \log ^2({\tilde x{}^1}{\tau})
\right)
$$
$$
\mbox{}
+
\cos{\bigl(2\log{({\tilde x{}^1}{\tau})}\bigr)}
\left(
\alpha
-20 \beta \log{({\tilde x{}^1}{\tau})}
-20 \alpha \log ^2({\tilde x{}^1}{\tau})
\right)
$$
\begin{equation}
\mbox{}
+
\sin{\bigl(2\log{({\tilde x{}^1}{\tau})}\bigr)}
\left(
\beta
+
20 \alpha \log{({\tilde x{}^1}{\tau})}-20 \beta \log ^2({\tilde x{}^1}{\tau})
\right)
\biggr]
, \end{equation}
where the observer's proper time on the base geodesic $\tau$ is the unified time of the synchronous frame of reference, and the variables ${\tilde x{}^k}$ are the spatial coordinates.
The resulting coordinate transformation also allows calculating the components of the geodesic deviation vector, and since the base geodesic in the used synchronous reference system has become a time line on which the spatial coordinates do not change, the deviation vector, up to a shift in the reference point, becomes simply the spatial position (radius vector) of the test particles on an adjacent geodesic.
The components of the 4-vector deviation $\tilde\eta{}^\alpha(\tau)$ in the synchronous frame of reference in the case of $\omega=1/2$ take the form:
\begin{equation}
\tilde\eta{}^0= 0
, \end{equation}
\begin{equation}
\tilde\eta{}^1 (\tau) = -\frac{{\lambda_1} {\Omega}}{{\tau}}
+{\rho_1}
, \end{equation}
$$
\tilde\eta{}^2 (\tau) =
\frac{
10 {\lambda_1} {\vartheta_2}
}{\gamma ^2 \left(20 \log ^2({\lambda_1}{\tau})-1\right)}
\Bigl[
-\alpha \sin{\bigl(2\log{({\lambda_1}{\tau})}\bigr)}
$$
$$
+\beta \cos{\bigl(2\log{({\lambda_1}{\tau})}\bigr)}+2 \gamma \log ({\lambda_1}{\tau})
\Bigr]
$$
$$
\mbox{}
-\frac{
10 {\lambda_1} {\vartheta_3}
}{\gamma ^2 \left(20 \log ^2({\lambda_1}{\tau})-1\right)}
\Bigl[
\alpha \cos{\bigl(2\log{({\lambda_1}{\tau})}\bigr)}
$$
\begin{equation}
\mbox{}
+\beta \sin{\bigl(2\log{({\lambda_1}{\tau})}\bigr)}
\Bigr]
-\frac{{\lambda_2} {\Omega}}{{\tau}}
+{\rho_2}
, \end{equation}
$$
\tilde\eta{}^3 (\tau) =
\frac{
10 {\lambda_1} {\vartheta_3}
}{\gamma ^2 \left(20 \log ^2({\lambda_1}{\tau})-1\right)}
\Bigl[
\alpha \sin{\bigl(2\log{({\lambda_1}{\tau})}\bigr)}
$$
$$
-\beta \cos{\bigl(2\log{({\lambda_1}{\tau})}\bigr)}
+2 \gamma \log{({\lambda_1}{\tau})}
\Bigr]
$$
$$
\mbox{}
-\frac{
10 {\lambda_1} {\vartheta_2}
}{\gamma ^2 \left(20 \log ^2({\lambda_1}{\tau})-1\right)}
\Bigl[
\alpha \cos{\bigl(2\log{({\lambda_1}{\tau})}\bigr)}
$$
\begin{equation}
\mbox{}
+\beta \sin{\bigl(2\log{({\lambda_1}{\tau})}\bigr)}
\Bigr]
-\frac{{\lambda_3} {\Omega}}{{\tau}}
+{\rho_3}
, \end{equation}
where the quantities $\lambda_k$ now determine the constant spatial coordinates of the observer on the base geodesic, $\tau$ is the time, the constant parameters ${\rho_k}$ determine the asymptotic values of the spatial components of the deviation vector
for large values of $\tau$.
\onecolumn
Let us find in the case of $\omega=1/2$ in the synchronous frame of reference
4-vector of tidal acceleration $\tilde A{}^\alpha(\tau) = D^2\tilde\eta{}^\alpha/{d\tau}^2$
in a gravitational wave:
\begin{equation}
\tilde A^0= 0
,\qquad
\tilde A^1= 0
, \end{equation}
$$
\tilde A^2 (\tau) =
\frac{50 {\lambda_1} {\vartheta_2} \log{({\lambda_1}{\tau})}
\left(\alpha \cos{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}
+\beta \sin{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}\right)}{\gamma ^2{\tau}^2 \left(20 \log ^2({\lambda_1}{\tau})-1\right)}
$$
$$
\mbox{}
-\frac{5 {\lambda_1} {\vartheta_3}
\left(10 \alpha \log ({\lambda_1} {\tau}) \sin{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}
-10 \beta \log{({\lambda_1}{\tau})} \cos{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}-\gamma \right)
}{\gamma ^2 {\tau}^2 \left(20 \log ^2({\lambda_1}{\tau})-1\right)}
$$
$$
\mbox{}
-\frac{5 {R_3}
}{2 {\lambda_1} \gamma {\tau}^2 \left(20 \log ^2({\lambda_1}{\tau})-1\right)}
\Bigl[
-\alpha \left(20 \log ^2({\lambda_1}{\tau})+1\right)
\sin{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}
$$
$$
\mbox{}
+\beta \left(20 \log ^2({\lambda_1}{\tau})+1\right) \cos{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}+4 \gamma \log{({\lambda_1}{\tau})}
\Bigr]
$$
\begin{equation}
\mbox{}
-\frac{5 {R_2}
\left(20 \log ^2({\lambda_1}{\tau})+1\right)
\left[\alpha \cos{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}
+\beta \sin{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}\right]
}{2 {\lambda_1} \gamma {\tau}^2
\left(20 \log ^2({\lambda_1}{\tau})-1\right)
}
, \end{equation}
$$
\tilde A^3 (\tau) =
-\frac{50 {\lambda_1} {\vartheta_3} \log{({\lambda_1}{\tau})}
\left[
\alpha \cos{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}
+\beta \sin{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}
\right]
}{\gamma ^2 {\tau}^2 \left(20 \log ^2({\lambda_1}{\tau})-1\right)}
$$
$$
\mbox{}
-\frac{5 {\lambda_1} {\vartheta_2}
\left[
10 \alpha \log{({\lambda_1}{\tau})}
\sin{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}-10 \beta \log ({\lambda_1}{\tau}) \cos{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}+\gamma
\right]
}{\gamma ^2{\tau}^2 \left(20 \log ^2({\lambda_1}{\tau})-1\right)}
$$
$$
\mbox{}
+\frac{5 {R_2}
}{2 {\lambda_1} \gamma {\tau}^2 \left(20 \log ^2({\lambda_1}{\tau})-1\right)}
\Bigl[
\alpha \left(20 \log ^2({\lambda_1}{\tau})+1\right)
\sin{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}
$$
$$
\mbox{}
-\beta \left(20 \log ^2({\lambda_1}{\tau})+1\right) \cos{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}+4 \gamma \log{({\lambda_1}{\tau})}
\Bigr]
$$
\begin{equation}
\mbox{}
+\frac{5 {R_3}
\left(20 \log ^2({\lambda_1}{\tau})+1\right)
\left[
\alpha \cos{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}
+\beta \sin{\bigl( 2\log{({\lambda_1}{\tau})}\bigr)}
\right]
}{2 {\lambda_1} \gamma {\tau}^2
\left(20 \log ^2({\lambda_1}{\tau})-1\right)
}
, \end{equation}
where $\tau$ is the time in the synchronous frame of reference, the constants $\lambda_k$
are the values of the spatial coordinates
$\tilde x{}^k$ of the observer on the base geodesic, the constants $\alpha$, $\beta$ and $\gamma$ are the parameters of a gravitational wave in type VII Bianchi spaces, related by an additional relation (\ref{constgamma}).
In the synchronous reference frame used, tidal accelerations appear only in the plane of variables $\tilde x{}^2$ and $\tilde x{}^3$, while the wave propagates along the coordinate $\tilde x{}^1$.
\section{On retarded potentials generated by a charge in a gravitational wave}
As an example of the application of the obtained exact solutions, consider the problem of retarded potentials.
In the synchronous reference frame we use, the observer is at rest on the base geodesic, and the motion of the test particle on the neighboring geodesic is carried out with tidal acceleration, so if this particle has a charge, we can formulate the problem of finding retarded potentials for a charge moving along a given trajectory in a gravitational wave with respect to resting observer in a synchronous frame of reference.
The exact solution of this problem requires the integration of Maxwell's equations in space-time with the metric (\ref{metricSynchr1k})-(\ref{metricSynchr33}), when the charge motion is given by the deviation vector (\ref{DeviationSolutionSynchr0})-(\ref{DeviationSolutionSynchr3}).
\twocolumn
The solution of such a problem in flat space-time is given, as is known, by the Lienard-Wiechert potentials $A^\alpha$ (in this section $A^\alpha$ is used to denote electromagnetic potentials), which can be written as \cite{LandauEng1}:
\begin{equation}
A^\alpha(\tau, \vec{r})=e\frac{u^\alpha}{R_{\beta} u^{\beta}}
\label{PotentialEq}
,\end{equation}
where $e$ is the charge of the particle, $u^\alpha$ is the 4-velocity of the charge at time $\tau'$,
$R^{\beta}=\{c(\tau-\tau'), \vec{r}-\vec{r}\,{}'\}$ is the difference of the observer's 4-vector at time
$\tau$ and the 4-vector of the charge at the time $\tau'$, $c$ is the speed of light, $\vec{r}$ is the spatial radius vector of the charge, and the time $\tau'$ and the components of the radius vector charges $\vec{r}\,{}'$ are related to $\tau$ and $\vec{r}$ by the following relation:
\begin{equation}
R_\alpha R^\alpha=0
.\end{equation}
In the weak field approximation, when the space-time differs little from the Minkowski space, we can apply the Lienard-Wiechert potential formula for an approximate, qualitative study of the problem, but taking into account the specifics of the exact solution we obtained for the deviation vector $\eta^\alpha$.
In the weak field approximation, for the charge vector $x^\alpha(\tau)$ and the vector $R^\alpha$ in the synchronous reference frame, one can take the expressions
\begin{equation}
\tilde x{}^\alpha (\tau)=
\left\{c\tau, \, \vec{\eta}(\tau)+\vec{\lambda}\,\right\}
,\end{equation}
\begin{equation}
R^\alpha=\left\{
c(\tau-\tau'),-\vec{\eta}(\tau')
\right\}
,\end{equation}
\begin{equation}
\vec{\eta}=\left\{
\tilde\eta{}^1, \tilde\eta{}^2, \tilde\eta{}^3
\,\right\}
,\quad
\vec{\lambda}=\left\{
{\lambda_1}, {\lambda_2}, {\lambda_3}
\,\right\}
.\end{equation}
Then the retarded potentials of the charge in the gravitational wave at the observer's point on the base geodesic can be represented as (\ref{PotentialEq}), where $\tilde\eta{}^\alpha$ is the deviation vector (\ref{DeviationSolutionSynchr0})-(\ref{DeviationSolutionSynchr3}), and in the expression for the potentials $A^\alpha(\tau)$
the values on the right
are taken at the moment time ${\tau'}$, which is related to the time $\tau$ in the synchronous reference frame by the relation
\begin{equation}
c^2 (\tau-\tau')^2 +\tilde g{}^{kl}
\,\tilde \eta{}_k \tilde \eta{}_l
\,
\Bigr\rvert_{x^\alpha=\{c\tau',\,\vec{\lambda} \} }=0
,\end{equation}
where $k,l=1,2,3$ and the gravitational wave metric $\tilde g{}^{\alpha\beta}$ in the synchronous reference frame is taken from the relations (\ref{metricSynchr1k})-(\ref{metricSynchr33}).
\section{Discussion}
The exact solutions of the geodesic deviation equation obtained in this work are the basis for calculating various physical effects in a gravitational wave, including
to calculate the radiation of charges moving with tidal acceleration in a gravitational wave.
These solutions make it possible to estimate the influence of primordial gravitational waves on the parameters of the microwave background of the Universe. Even a “naive” direct substitution into the Lienard–Wiechert formulas of the solutions obtained in this work for the deviation vector gives, in the weak field approximation, a qualitative description of the radiation of charged plasma particles in the primordial gravitational wave.
Obtaining exact mathematical models of geodesic deviation in primordial gravitational waves with different types of space-time symmetries also makes it possible to evaluate the influence of primordial gravitational waves on the formation of the modern stage of homogeneous and isotropic space-time in order to understand the process of "isotropization" in the early stages of the Universe (including for describing violations of the homogeneity and isotropy of the observed microwave electromagnetic background of the Universe), which is important for constructing theoretical models of the early Universe.
Exact models of primordial gravitational waves for different types of space-time symmetries will make it possible to reveal those parameters of the electromagnetic microwave background that can characterize the presence of certain types of spatial homogeneity symmetries in the initial stages of the Universe, which is significant for constructing theoretical models of the early Universe.
The approach proposed in this paper for obtaining solutions to geodesic deviation equations in models of primordial gravitational waves for Universes with Bianchi symmetries is applicable both in Einstein's theory of gravity and in modified theories of gravity. This allows, using exact solutions for the deviation vector in various models and modified theories of gravitation, to obtain different options for the influence of primary plasma radiation and to find out the differences that may arise in the parameters of the microwave electromagnetic background of the Universe at the present stage under the influence of primordial gravitational waves in various models and modified theories of gravity.
\section{Conclusion}
An exact solution of the geodesic deviation equations in a gravitational wave is obtained for Bianchi type VII cosmological models in Einstein's theory of gravitation. The models of gravitational waves under consideration refer to Shapovalov wave spaces of type III, which made it possible to obtain a complete integral for the Hamilton-Jacobi equation of test particles in a privileged coordinate system. The presence of the complete integral of the Hamilton-Jacobi equation made it possible, in turn, to obtain both the trajectories of motion of test particles and the solution of the geodesic deviation equations in the models under consideration.
The obtained exact solutions of the geodesic deviation equation are presented both in the privileged coordinate system and in the synchronous (laboratory) coordinate system, relative to which the freely falling observer is at rest. The transition to a synchronous coordinate system is based on the use of the trajectories of motion of test particles obtained in the work.
An explicit form of metrics, geodesic deviation vectors, and an explicit form of tidal force acceleration in a gravitational wave are obtained in this work both for a privileged coordinate system and for a synchronous coordinate system.
The approach presented in the paper can be used both in the general theory of relativity and in modified theories of gravity. The exact models obtained describe the primordial gravitational waves of the Universe and can be used to calculate the secondary physical effects that arise during the passage of a wave, as well as to debug approximate and numerical methods of detection and comparative analysis of the characteristics of gravitational waves in terms of their effect on test particles and charges.
\section*{Acknowledgments}
The authors thank the administration of the Tomsk State Pedagogical University for the technical support of the scientific project.
The study was supported by the Russian Science Foundation, grant \mbox{No. 22-21-00265},
\url{https://rscf.ru/project/22-21-00265/}
\section*{Statements and Declarations}
\subsection*{Data availability statement}
All necessary data and references to external sources are contained in the text of the manuscript.
All information sources used in the work are publicly available and refer to open publications in scientific journals and textbooks.
\subsection*{Compliance with Ethical Standards}
The authors declare no conflict of interest.
\subsection*{Competing Interests and Funding}
The study was supported by the Russian Science Foundation,
grant \mbox{No. 22-21-00265},
\url{https://rscf.ru/project/22-21-00265/}
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One of the downsides to having such a packed lineup is the earlier start. Considering this was a midweek show, the fans started to trickle in as Savage After Midnight took to the stage. Unfortunately due to some unforeseen circumstances, we weren't able to see them perform, but we were able to hear them. One highlight was their cover of Canada's own Alanis Morissette's hit, You Oughta Know.
While it took a few songs for the crowd to start jumping after a long work day, Asking Alexandria eventually got them moving -- especially in the mosh pit that opened up in the middle of the floor during When The Lights Come On.
PAPA ROACH
After having released their 10th studio album earlier this year, Papa Roach kicked off their set with the title track off of that album, Who Do You Trust.
Frontman Jacoby Shaddix took the time to get up close and personal with the crowd jumping up to the barricade during their performance of Elevate -- also off of their latest release. Performing with such joy and enthusiasm is refreshing to see especially with a band that's been around for twenty plus years.
Fans seeing the band for the first time were welcomed to the Papa Roach family before hearing one of their old school hits, Getting Away With Murder, which also lead to the mosh pit opening up even more -- this time with bonus crowd surfers.
Before playing Blood Brothers off of their hit album Infest, Jacoby pumped up the crowd letting them know that they were the best crowd of the tour so far.
Come Around, and Forever, were up next, and as expected, the crowd in Place Bell let it all out when they played their hits Scars and Last Resort. To close out their incredible set were Born For Greatness, and To Be Loved.
SHINEDOWN
Papa Roach is a tough act to follow, but headliners Shinedown, were up for the challenge. As the curtain dropped, Devil, Diamond Eyes, and Enemies got their set started with a bang -- and we mean that literally... flames shot up from the stage (a photographers dream!)
Monsters was up next and when the first few notes of I'll Follow You rang out on the piano, Place Bell was ready to have their voices heard singing along with every word. Guitarist Zach Myers, and bassist Eric Bass were constantly on the moving making use of the huge stage while drummer Barry Kerch was all smiles perched in the middle of it.
Attention Attention, Unity and the old school fan favourite 45 were up next followed by Bully, and The Crow & the Butterfly. Since the last Shinedown show in Montreal was close to 7 years ago, they decided to play a song they haven't been playing lately, State Of My Head.
Similarly to I'll Follow You, as soon as the first chords of Second Chance were strummed, the crowd was willing, and ready, to provide some powerful backing vocals for the band. Their cover of Lynyrd Skynyrd's Simple Man was next followed by Cut The Cord.
Brent Smith offered up some words of advice before their next song telling the crowd to not be afraid of failing, and that failure will only lead to a new path. He added, that we shouldn't be worried about using a plan B; if something knocks you down, GET UP! <- which was also the name of the next song played.
Their massive hit, Sound Of Madness was up next followed by the last song of the night, Brilliant. The only thing I would've change about their entire set, was that I felt they should've ended with Sound Of Madness instead, but hey, it was still an incredible night.
As the band prepared to leave the stage, Brent said, it's never goodbye, it's just till next time; let's just hope it's not another 7 years until we see them again!
Stay tuned for our interview with drummer Barry Kerch in the coming days, and for more information on Shinedown, visit their official site. As usual, for all of your Montreal concert needs, visit our friends at Evenko.
Enjoy some more of our photos below
Asking Alexandria
Papa Roach
Shinedown
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I've been pinning, reading, and researching like a mad woman! I want to educate myself about what I am putting into and onto my body.
It's scary, folks.
Educate yourself.
Did you know that there are banned substances that the FDA allows in our foods? If they are banned, why are they allowed?
Did you know that most major food companies pay scientists to find ways to make the food they seel more addictive?
A friend of my recently directed me to a pin from that you must see as well.
Here's challenge: take this list and go to your pantry.
Pull out the items you use most in your cooking. See any of these ingredients?
Pull out your go-to snacks for you and your children. See any?
Surprised to see sugar in EVERYTHING?
It's there!
Get rid of it. I'm not kidding.
We've been eating mostly sugar free and fairly clean for a few months now. Look how much more I had to get rid of. I'll admit, some of it we haven't been eating, it was just sitting in there and I knew it had to go. But, a lot of it was pretty shocking.
The amount of HYDROGENATED and PARTIALLY HYDROGENATED OILS was the most shocking for me.
That's TRANS FATS, folks. The stuff that isn't allowed anymore - it's still there! It is allowed as long as it can be rounded down to zero on the nutrition label.
I call that a lie.
Don't tell me zero trans fats when you know it's in there. They may think, "hey, it's .5 grams per serving" but they know most folks have more than a serving size at a time.
And think on this: If I eat 4 items that have .5 trans fats in them in one day, I'm getting 2 grams of trans fats and my risk of heart disease, cancer, alzheimer's, etc.
Have I ranted enough yet?
I think you get the idea.
Read the list. Check your labels.
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Heritage
From prehistory to industrial heritage we’re interested. Are you?
It’s so easy to miss what is on our doorstep when we jet off to foreign climes but our willing heritage team arrange trips to local places as diverse as the last surviving Methodist Chapel built above a barn and a behind the scenes visit to the Hippodrome Theatre in Colne.
One day trips have included visits to a pottery museum in Stoke on Trent and a boat journey on the Ship Canal from Manchester to Liverpool and a tour of Saltaire.
We have an annual trip which may simply involve an overnight stop in North Wales as we learn about the bridges across the Menai Straits and history on Anglesey or even a 6 day journey to immerse ourselves in the prehistory of Orkney!
Unfortunately all these trips have been and gone so why not take a look at our forthcoming events?
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History Month with a fourth and final post. And what better thing to talk about than Mardi Gras!
While the oldest and most prestigious parade in the city is put on by Rex, whose king is legally in charge of the city today (yes, really), another long-running krewe is the historically black Zulu Social Aid and Pleasure Club. Zulu’s parade rolls right after Rex, though their route takes them through predominantly black neighborhoods, while Rex follows the uptown route along St. Charles Ave. Trying to explain Zulu is a bit hard, but here goes. The imagery the krewe uses in their parade and court is essentially a satire of white caricatures of blacks from the early 20th century: the riders present themselves as “Zulu Warriors,” all the members ride in black-face (even putting white makeup around their eyes and lips!) in imitation of the minstrel shows popular during that period, and they distribute what is now one of the most coveted Mardi Gras throws: hand-decorated coconuts. But the krewe isn’t just for blacks. Back in the 90s all krewes were forced to integrate if they wanted to use public roads for their parades, which means Zulu also has a few white members. Reread this paragraph and sit on that for a second. This definitely wouldn’t fly anywhere else, but like so many things in New Orleans it works because it’s traditional and because it’s tongue-in-cheek, and everybody is happy to have a good time. Zulu is also unique for electing its King democratically and allowing him to pick his own queen–those roles being doled out by elite subgroups within most other krewes. It’s hard to put your head around, but it all makes for a great parade. While Mallory and I head out early to see it in person, you can check out a more detailed history on this page worked up by Zulu’s historians.
Another delightful part of Carnival culture is Mardi Gras Indians. While the origin of these groups is disputed, the long and short of it is this: black men don fantastically elaborate, hand-beaded “Indian” costumes and parade through the city looking for other “tribes” to challenge. While these meetings sometimes led to violent conflict in the past, the current tradition is for tribes to engage in a musical call-and-response contest to determine which sounds better and also to decide which “Big Chief” is prettier. The loser has to bow down before the winner and let him pass through his neighborhood. The beaded designs at the center of the costumes are closely guarded–so much that only a lucky few get to see a Big Chief’s main embroidery–and even the techniques they use to make them are kept secret from all but the tribe’s inner circle. And these costumes are only used for a single season! After that, they’re broken down and an entirely new design is made for the next year. This may be familiar to some of you from TV: if you saw the first season of HBO’s Treme you’ll remember the strange, awesome scene where Albert Lambreaux puts on his headdress for the first time after The Storm. Even though Indians don’t get much attention in popular representations outside the city, they really add a magical layer to Mardi Gras celebrations, and it’s always a thrill to hear the drumbeat and rush outside to catch a glimpse when they pass by. To learn more, you can also check out this page that was worked up by the city’s tourism board.
Okay, that’s all I’ve got for today. Have a happy Mardi Gras, and make sure you get in some good food and drink before Lenten fasting and repentance begin on Ash Wednesday!
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\begin{document}
\title[The space of full-ranked one-forms]{A diffeomorphism-invariant metric on the space of vector-valued one-forms}
\author{Martin Bauer}
\address{Department of Mathematics, Florida State University, USA}
\email{bauer@math.fsu.edu}
\author{Eric Klassen}
\address{Department of Mathematics, Florida State University, USA}
\email{klassen@math.fsu.edu}
\author[S.C. Preston]{Stephen C. Preston}
\address{Department of Mathematics, Brooklyn College and the Graduate Center, City University of New York, NY 11106, USA}
\email{stephen.preston@brooklyn.cuny.edu}
\author{Zhe Su}
\address{Department of Mathematics, Florida State University, USA}
\email{zsu@math.fsu.edu}
\keywords{Space of Riemannian metrics, Ebin metric, Sectional Curvature, Shape analysis}
\date{\today}
\begin{abstract}
In this article we introduce a diffeomorphism-invariant Riemannian metric on the space of vector valued one-forms.
The particular choice of metric is motivated by potential future applications in the field of functional data and shape analysis and
by connections to the Ebin metric on the space of all Riemannian metrics.
In the present work we calculate the geodesic equations and obtain
an explicit formula for the solutions to the corresponding initial value problem.
Using this we show that it is a geodesically and metrically incomplete space and study the existence of totally geodesic subspaces.
Furthermore, we calculate the sectional curvature and observe that, depending on the dimension of the base manifold and the target space, it either has a semidefinite sign or admits both signs.
\end{abstract}
\maketitle
\section{Introduction}
Motivated by applications in the field of mathematical shape analysis we introduce a diffeomorphism-invariant Riemannian metric on the space of full-ranked
$\mathbb R^n$-valued one-forms $\Omega^1_+(M,\mathbb R^n)$, where $M$ is a smooth, orientable, compact manifold (possibly with boundary) of dimension $m$ with $m\leq n$.
The definition of our metric will not include any derivatives of the tangent vectors. For this reason we call the metric an $L^2$-type metric, which however differs, due to the appearance of the foot point $\alpha$, from the standard $L^2$-metric. The main reason for introducing this particular dependence on the foot point is the invariance of the resulting metric under the action of the diffeomorphism group $\Diff(M)$, see Lemma~\ref{lem:invariances}.
\\
\subsection*{Contributions of the article}
In this article we will initiate a detailed study of the induced geometry of the proposed Riemannian metric.
The point-wise nature of the metric will allow us to reduce many of
the investigations of the metric to the study of a finite dimensional space of matrices. Using this we are able to obtain
explicit formulas for geodesics and curvature.
Our main results of the article are as follows:
\begin{enumerate}
\item The induced geodesic distance on the space of full ranked, vector valued one-forms $\Omega^1_+(M,\mathbb R^n)$ is non-degenerate; see Theorem~\ref{thm:distpointwise} where a
lower bound for the geodesic distance is obtained.
\item The geodesic equation on the space of full ranked, vector valued one-forms $\Omega^1_+(M,\mathbb R^n)$ has explicit solutions for any initial conditions as presented in
Theorem~\ref{thm.geodesicformula.a}.
\item Depending on the values of $m$ and $n$ the sectional curvature is either sign-semidefinite or admits both signs.
\item The metric is linked via a Riemannian submersion to the Ebin metric on the space of all Riemannian metrics.
\end{enumerate}
As a consequence of the explicit formula for geodesics we will obtain the metric and geodesic incompleteness of the space $\Omega^1_+(M,\mathbb R^n)$.
For the finite-dimensional space of matrices we will characterize its metric completion, which consists of a quotient space of
matrices, where two matrices are identified if they have less than full rank. In future work, we plan to use this characterization to determine
the metric completion of the space of full ranked one-forms, using a similar strategy as in \cite{clarke2013completion}. Finally, in Section~\ref{shape:analysis},
we will discuss potential applications in the field of shape analysis, that have been further developed in the application-oriented article~\cite{su2019shape}.
\subsection*{Background and motivation}
In the following we will further motivate the study of this metric from two different angles.\\
\noindent\emph{Connections to shape analysis.}
The field of functional data analysis is concerned with describing and comparing data, where each data point can be a function \cite{srivastava2016functional,younes2010shapes,dryden2016statistical,bauer2014overview}. In this context the difficulties lie both in the infinite dimensionality as well as in the non-linearity of the involved spaces. Infinite dimensional Riemannian geometry has proven to provide the necessary tools to tackle some of the problems and applications in this field. A space that is of particular interest in this area of research is the space of (unparametrized) curves or surfaces, which appears e.g., in the study of human organs, trajectory detection, body motions, or in general computer graphics applications. In order to obtain a Riemannian framework on the space of unparametrized surfaces (curves resp.),
one needs to consider metrics on the space of parametrized surfaces (curves resp.) that are invariant with respect to the reparametrization group \cite{michor2007overview,klassen2004analysis}.
Given a parametrized surface (curve resp.) $f\colon M\to \mathbb R^n$, we can view $df$ as a full-ranked one-form.
Hence, one can construct invariant Riemannian metrics on the space of parametrized surfaces (curves resp.)
as the pullback of invariant Riemannian metrics on the space of full-ranked one-forms, which puts us directly in the setup of this article.
A similar strategy has proven extremely efficient for shape analysis of unparametrized curves and has yielded the so-called SRV-framework \cite{klassen2004analysis,bauer2014constructing}. For surfaces the situation is more intricate. A generalization of the SRV-framework has been proposed in \cite{laga2017numerical}.
This framework, called the square root normal field (SRNF), has proved successful in applications but has some mathematical limitations, see e.g., the discussions in \cite{su2019shape}. The representation proposed in the current article will allow us to obtain a better mathematical understanding of the properties of the induced metric on the space of surfaces. The main reason is the simpler characterization of the image of the map $f\mapsto df$, as compared to the SRNF. In fact we obtain the isometric immersion:
\begin{align}
\operatorname{Imm}(M,\mathbb R^n) \longrightarrow\Omega_{+,\operatorname{ex}}^1(M,\mathbb R^n)\subset \Omega_+^1(M,\mathbb R^n)\;,
\end{align}
where $\Omega_{+,\operatorname{ex}}^1(M,\mathbb R^n)$ denotes the subset of exact one-forms (assuming that the topology of $M$ is sufficiently simple).
The present article will focus mainly on the geometry on the larger space of all full-ranked one-forms; we plan to study the submanifold geometry of
the space of exact one-forms in future work.
This strategy is similar to that of Ebin-Marsden~\cite{ebin1970groups}, who considered the $L^2$-geometry of $\Diff(M)$ where all the geometry may be done point-wise, then considered the submanifold of volume-preserving diffeomorphisms under the induced metric (where geodesics describe ideal fluid motion).
In Figures~\ref{CurvesExamples}, ~\ref{SurfaceExample1}, and~\ref{SurfaceExample2} one can see examples of geodesics in the space of immersions, equipped with the pull-back of the generalized Ebin metric studied in this article. These examples have been calculated using the numerical framework for the Riemannian metric studied in this paper as developed in~\cite{su2019shape}\footnote{An open source implementation of the corresponding numerical framework can be found at~\url{https://github.com/zhesu1/elasticMetrics}.}, where the spherical parametrizations of the boundary surfaces have been obtained using the code of Laga et al.~\cite{kurtek2013landmark}.
\begin{figure*}
\begin{center}
\includegraphics[width=0.65\linewidth]{curves3d_1}
\end{center}
\caption{A geodesic in the space of regular curves modulo translations with respect to the Younes-metric~\eqref{eq.metric.curve}, a special case of our metric.}
\label{CurvesExamples}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.9\linewidth]{cat17_geo_winter}\\
\vspace{.25cm}
\includegraphics[width=0.89\linewidth]{horse03_geo_winter}
\end{center}
\caption{Examples of geodesics in the space of surfaces modulo translations with respect to the generalized Ebin metric~\eqref{metric}. These examples have been calculated using the numerical framework for the Riemannian metric studied in this paper as developed in~\cite{su2019shape}.}
\label{SurfaceExample1}
\end{figure*}
\noindent
\emph{Connections to the Ebin-metric on the space of all Riemannian metrics.}
Another motivation for the present article can be found in the connection of the proposed metric to the Ebin metric on the space of all Riemannian metrics, which has been introduced by Ebin~\cite{ebin1970manifold}; see also the article of DeWitt \cite{DeWitt67}. Motivated by applications in Teichm\"uller theory, K\"ahler geometry and mathematical statistics, the geometry of this metric has been studied in detail by Clarke, Freed, Groisser, Michor, and others~\cite{gil1991riemannian,freed1989basic,clarke2010metric,clarke2013completion,clarke2013geodesics,clarke2011riemannian,bauer2013sobolev}. The proposed metric is closely related to the Ebin metric as they are connected via the Riemannian submersion:
\begin{equation}
\Omega_+^1(M,\mathbb R^n) \to \operatorname{Met}(M),\quad \alpha\mapsto \alpha^{T}\alpha;
\end{equation}
see Section~\ref{Sect:submersiontoEbin} for more details. Furthermore, the proposed metric shares many of the geometric features of the original Ebin metric, such as non-degenerate geodesic distance, existence of explicit solutions to the geodesic equation, and geodesic and metric incompleteness. On the other hand, we will see that the sectional curvature can admit both signs, which is in stark contrast to the Ebin metric on the space of Riemannian metrics, which always has non-positive curvature.
\subsection{Acknowledgements}
The authors want to thank the anonymous referees for their careful remarks that greatly improved the quality of the article. We are grateful to Hamid Laga for providing the parametrization of the boundary surfaces in Figure~\ref{SurfaceExample1}.
M. Bauer and Z. Su were partially supported by NSF-grant 1912037 (collaborative research in connection with NSF-grant 1912030). E. Klassen was partially supported by Simons Foundation, Collaboration Grant for Mathematicians, no. 317865 and
S. C. Preston was partially supported by Simons Foundation, Collaboration Grant for Mathematicians, no. 318969.
\section{Notation}\label{notation}
\subsection{Spaces of matrices}\label{notation:matrices}
In large parts of the article the pointwise nature of the metric will allow us to reduce the analysis to the study of a corresponding Riemannian metric on a finite dimensional space of matrices. Therefore we introduce, for
$m\leq n \in \mathbb N$, the space of all full rank $n\times m$ matrices:
$$M_{+}(n,m):=\left\{a\in \mathbb R^{n\times m}|\;\operatorname{rank}(a)=m\right\}\;.$$
The space $M_{+}(n,m)$ is an open subset of the vector space of all $n\times m$-matrices $M(n,m)$ and is thus a manifold of dimension $n\times m$.
The full-rank condition on the elements of $M_{+}(n,m)$ allows us to consider the Moore-Penrose pseudo inverse $a^+$ of a matrix $a\in M_{+}(n,m)$, which is defined by
$a^+=(a^Ta)^{-1}a^T$. The most important property of $a^+$ is $a^+a=I_{m\times m}$, i.e., $a^+$ is a left-inverse. Here $I_{m\times m}$ denotes the $m\times m$ identity matrix. In general we will use lower-case letters $u$ to denote $n\times m$ matrices in the tangent space, $u\in T_aM(n,m)\cong M(n,m)$, and we will use upper-case letters $U$ to denote the $n\times n$ square matrix $U = ua^+$. Note that the map $u\mapsto U = ua^+$, from $T_aM(n,m)$ to $M(n,n)$, is injective.
Related to the space of full rank $n\times m$ matrices is the space of positive definite symmetric $m\times m$-matrices:
\begin{equation}
\operatorname{Sym}_+(m):=\left\{a\in M(m,m): a^T=a \text{ and } a \text{ is positive definite} \right\}\;.
\end{equation}
Similarly to the space of all full rank $n\times m$ matrices, the space $\operatorname{Sym}_+(m)$ is a manifold as it is an open subset of a vector space, namely of the
space of all symmetric $m\times m$-matrices $\operatorname{Sym}(m)$.
In the remainder of the article we will also use the group of all invertible $m$-dimensional matrices $\on{GL}(m)$, the
groups of special orthogonal matrices $\on{SO}(n)$ and $\on{SO}(m)$, and the groups of orthogonal matrices $\on{O}(n)$ and $\on{O}(m)$.
\subsection{Spaces of one forms, diffeomorphisms and Riemannian metrics}
Suppose $M$ is a compact $m$-dimensional manifold $M$ and recall that $m \leq n$. Let $\Omega^1(M,\mathbb R^n)$ denote the space of smooth $\mathbb R^n$-valued one-forms on $M$. Recall that an $\mathbb R^n$-valued one-form $\alpha$ on $M$ is a choice, for each $x\in M$, of a linear transformation $\alpha(x):T_xM\to \mathbb R^n$ that varies smoothly with $x\in M$. Note that $\Omega^1(M,\mathbb R^n)$ is -- with the usual addition and scalar multiplication on $\mathbb R^n$ -- an infinite dimensional vector space.
If $\alpha(x)$ is injective for all $x\in M$, we say that $\alpha$ is a {\it full-ranked} one-form and we denote by $\Omega_+^1(M,\mathbb R^n)$ the space of full-ranked one-forms. We immediately obtain the following result concerning the manifold structure of $\Omega^1_+(M,\mathbb R^n)$ (see e.g., \cite{hamilton1982inverse} for an introduction to Fr\'echet manifolds):
\begin{lem}
The space of all full-ranked one-forms $\Omega^1_+(M,\mathbb R^n)$ is a smooth Fr\'echet manifold with tangent space the space of all one-forms
$\Omega^1(M,\mathbb R^n)$.
\end{lem}
\begin{proof}
By definition we have $\Omega^1_+(M,\mathbb R^n)\subset \Omega^1(M,\mathbb R^n)$. The full-rank condition is an open condition and thus $\Omega^1_+(M,\mathbb R^n)$ is an open subset of an infinite dimensional Fr\'echet space, which implies
the result.
\end{proof}
Related to this space is the infinite dimensional manifold of all smooth Riemannian metrics $\operatorname{Met}(M)$. For an overview on different Riemannian structures on this space and in particular to the Ebin metric, we refer to the vast literature; see e.g., \cite{gil1991riemannian,freed1989basic,clarke2010metric,clarke2013completion,clarke2013geodesics,clarke2011riemannian,bauer2013sobolev}.
On both of the spaces we consider the action of the diffeomorphism group
\begin{equation}
\operatorname{Diff}(M):=\left\{\varphi \in C^{\infty}(M,M)|\; \varphi \text{ is bijective and }\varphi^{-1}\in C^{\infty}(M,M) \right\}
\end{equation}
via pullback:
\begin{align*}
&\Omega^1_+(M,\mathbb R^n)\times \operatorname{Diff}(M)\mapsto \Omega^1(M,\mathbb R^n),\quad (\alpha,\varphi)\rightarrow \varphi^*\alpha(x)=\alpha(\varphi(x))\circ d\varphi(x)\,,\\
&\operatorname{Met}(M)\times \operatorname{Diff}(M)\mapsto \operatorname{Met}(M),\quad (g,\varphi)\rightarrow \varphi^* g(x)=d\varphi^T(x) g(\varphi(x))d\varphi(x)\,.
\end{align*}
\section{A Riemannian metric on the space of full rank $n\times m$-matrices}
The main results of this article will be concerned with a diffeomorphism-invariant Riemannian metric on an infinite dimensional manifold of mappings, as introduced in the introduction \eqref{metric}.
The pointwise nature of the metric will allow us to reduce many aspects of the study of the corresponding geometry to the study of a corresponding metric on a (finite dimensional) manifold of matrices, which will be the object of interest in the following section.
Therefore we consider the space of full rank $n\times m$ matrices $M_+(n,m)$ with $m \leq n$ as introduced in Section~\ref{notation:matrices}.
For $a\in M_+(n, m)$ and $u, v\in T_aM_+(n, m)$ we define the Riemannian metric:
\begin{align}\label{eq.metric.a}
\langle u, v \rangle_a = \tr(u(a^Ta)^{-1}v^T)\sqrt{\det(a^Ta)}.
\end{align}
Using the Moore-Penrose inverse $a^+ = (a^Ta)^{-1}a^T$ of $a\in M_+(n, m)$,
we obtain an alternative formula for the metric that will turn out to be useful later:
\begin{align}\label{eq.metric.a.simplified}
\langle u, v \rangle_a = \tr(UV^T)\sqrt{\det(a^Ta)}, \qquad U = ua^+, \quad V = va^+.
\end{align}
As a first result we will describe a series of invariance properties of the
Riemannian metric that will be of importance in the remainder of the article:
\begin{lem}\label{matmetinvar}
Let $a\in M_+(n, m)$ and $u, v\in T_aM_+(n, m)$.
\begin{enumerate}
\item The metric \eqref{eq.metric.a} is invariant under the left action of the orthogonal group:
$$\langle zu, zv\rangle_{za} = \langle u, v\rangle_a \text{ for } z\in \operatorname{O}(n);$$
\item The metric \eqref{eq.metric.a} satisfies the following transformation rule under the right action of the group of invertible matrices:
$$\langle uc, vc\rangle_{ac} = \langle u, v\rangle_a|\det(c)| \text{ for } c\in\operatorname{GL}(m);$$
\item The metric \eqref{eq.metric.a} is invariant under the right action of the group of determinant one or minus one matrices:
$$\langle uc, vc\rangle_{ac} = \langle u, v\rangle_a \text{ for } c\in\operatorname{GL}(m), \det(c)= \pm1;$$
\end{enumerate}
\end{lem}
\begin{proof}
The proof consists of elementary matrix operations. For $z\in \operatorname{O}(n)$ we have
\begin{align}
\langle zu, zv\rangle_{za} &= \tr(zu(a^Tz^Tza)^{-1}v^Tz^T)\sqrt{\det(a^Tz^Tza)}\\
&= \tr(u(a^Ta)^{-1}v^T)\sqrt{\det(a^Ta)} = \langle u, v\rangle_a,
\end{align}
which proves the invariance under the action of $\operatorname{O}(n)$.
To see the second property we calculate for
$c\in\operatorname{GL}(m)$:
\begin{align}
\langle uc, vc\rangle_{ac} &= \tr(uc(c^Ta^Tac)^{-1}c^Tv)\sqrt{\det(c^Ta^Tac)}\\
&= \tr(ucc^{-1}(a^Ta)^{-1}(c^T)^{-1}c^Tv)\sqrt{\det(a^Ta)}|\det(c)|\\
&= \langle u, v\rangle_a|\det(c)|.
\end{align}
The third statement follows immediately from the second one, which concludes the proof.
\end{proof}
\subsection{The space of symmetric $m\times m$-matrices}\label{sec:symmetric}
In this section we will describe the relation of our metric to a well-studied Riemannian metric on the space of symmetric matrices.
Therefore we recall the definition of the finite dimensional version of the Ebin-metric, as studied by \cite{freed1989basic,clarke2010metric}:
\begin{align}\label{eq.metric.sym}
\langle h, k \rangle^{\operatorname{Sym}}_g = \frac14\tr(hg^{-1}kg^{-1})\sqrt{\det(g)},
\end{align}
where $g\in \operatorname{Sym}_+(m)$ and $h,k\in T_g \operatorname{Sym}_+(m)=\operatorname{Sym}(m)$.
Our main result in this section will show that the projection
\begin{equation}\label{projection}
\pi: M_+(n,m) \to \operatorname{Sym}_+(m), \qquad a \mapsto a^T a
\end{equation}
is a Riemannian submersion, where the spaces are equipped with their respective Riemannian metrics.
Note that $\operatorname{O}(n)$ acts by left multiplication on $M_+(n,m)$. The following proposition tells us that the orbits under this action are precisely the fibers of the map $\pi:M_+(n,m) \to \operatorname{Sym}_+(m)$ defined earlier.
\begin{prop}\label{prop.quotientidentification}
Let $a, b\in M_+(n, m)$. Then $a^Ta = b^Tb$ if and only if there is $z\in \operatorname{O}(n)$ such that $a = zb$.
\end{prop}
\begin{proof}
It is easy to see that if $a = zb$ for some $z\in \operatorname{O}(n)$, then
\begin{equation}
a^Ta = (zb)^Tzb = b^Tz^Tzb = b^Tb.
\end{equation}
Conversely, denote by $p\in\operatorname{Sym}_+(m)$ the positive definite symmetric square root of $a^Ta$. Then we have
$$a^Ta = b^Tb = p^2 = \begin{pmatrix}
p & 0
\end{pmatrix}\begin{pmatrix}
p\\0
\end{pmatrix}, \text{ where }
\tilde p= \begin{pmatrix}
p\\
0
\end{pmatrix}\in M_+(n,m).$$ It is enough to show that there is $z\in \operatorname{O}(n)$ such that $a = z\tilde p$. Let $z_1 = ap^{-1}$. We have
\begin{equation}
z_1^Tz_1 = p^{-1}a^Tap^{-1} = I_{m\times m},
\end{equation}
which means that the columns in $z_1$ form a set of orthonormal vectors in $\mathbb R^n$. Let $z_2$ be an $n\times (n-m)$ matrix whose columns form an orthonormal basis of the orthogonal complement of the span of the columns of $z_1$. Let $z = \begin{pmatrix}
z_1 & z_2
\end{pmatrix}$. Then $a = z_1 p = \begin{pmatrix}
z_1 & z_2
\end{pmatrix}\tilde p = z\tilde p$. Now the conclusion follows by using
\begin{equation}
z^Tz = \begin{pmatrix}
z_1^T\\z_2^T
\end{pmatrix}\begin{pmatrix}
z_1 & z_2
\end{pmatrix} = \begin{pmatrix}
z_1^Tz_1 & z_1^Tz_2\\
z_2^Tz_1 & z_2^Tz_2
\end{pmatrix} = I_{n\times n}.
\end{equation}
\end{proof}
Proposition~\ref{prop.quotientidentification} implies that
$\pi$ induces a diffeomorphism
\begin{equation}\label{eq.identification}
\operatorname{O}(n)\backslash M_+(n,m)\cong\operatorname{Sym}_+(m),
\end{equation}
where $\operatorname{O}(n)\backslash M_+(n,m)$ denotes the space of orbits under the $O(n)$ action.
Furthermore, for any $a \in M_+(n,m)$ we obtain a (non-unique) decomposition
\begin{equation}\label{decomposition}
a= z \begin{pmatrix}
s\\
0_{(n-m)\times m}
\end{pmatrix},\qquad \text{with } z\in \operatorname{O}(n), \text{ and } s\in \operatorname{Sym}_+(m)\;.
\end{equation}
In the following theorem we describe the corresponding Riemannian submersion picture:
\begin{thm}\label{thm:matrices:submersion}
The mapping $\pi: M_+(n,m) \to \operatorname{Sym}_+(m)$ is a Riemannian submersion, where $M_+(n,m)$ is equipped with the metric~\eqref{eq.metric.a}
and where $\operatorname{Sym}_+(m)$ carries the metric~\eqref{eq.metric.sym}. The corresponding vertical and horizontal bundles are given by:
\begin{align}
\mathcal V_a &= \{u\in M_+(n,m)\,|\, u = Xa, X \in \mathfrak{so}(n)\}\\
\mathcal H_a &= \{v\in M_+(n,m)\,|\, va^+\in \operatorname{Sym}(n)\}.
\end{align}
\end{thm}
\begin{proof}
In the following we identify the space of all symmetric matrices $\operatorname{Sym}_+(m)$ with the quotient space $\operatorname{O}(n)\backslash M_+(n,m)$. The Riemannian metric on $M_+(n,m)$ descends to a Riemannian metric on the quotient space due to the invariance under the left action of $\operatorname{O}(n)$.
To determine the induced metric on the quotient space we need to calculate the vertical and horizontal bundle.
It is immediate that the vertical bundle of $\pi$ at $a\in M_+(n,m)$ consists of all matrices $u$ such that $ u = Xa$ with $X\in\mathfrak{so}(n)$. A matrix $v$ is in the horizontal bundle if and only if it is orthogonal to all elements in the vertical bundle.
Letting $V=va^+$, we obtain
\begin{align}
0 = \langle Xa, v\rangle_a &= \tr(Xa(a^Ta)^{-1}v^T)\sqrt{\det(a^Ta)}\\
&= \tr(XV^T)\sqrt{\det(a^Ta)}.
\end{align}
for all $X\in\mathfrak{so}(n)$.
It follows that $V$ has to be a symmetric matrix, proving the expressions for the vertical and horizontal bundles given in the statement of the Theorem.
To show that the differential $d\pi_a$ induces an isometry $\mathcal H_a\to T_{\pi(a)}\operatorname{Sym}_+(m)$ we calculate
\begin{align}
d\pi_a(v) = a^Tv + v^Ta.
\end{align}
For a horizontal tangent vector $v$ we have
\begin{align}
&\langle d\pi_a(v), d\pi_a(v)\rangle_{\pi(a)}^{\operatorname{Sym}}\\
&\qquad=\frac14\tr((a^Ta)^{-1}(v^Ta + a^Tv)(a^Ta)^{-1}(v^Ta + a^Tv))\sqrt{\det(a^Ta)}\\
&\qquad= \frac12\tr((a^Ta)^{-1}v^Ta(a^Ta)^{-1}v^Ta)\sqrt{\det(a^Ta)}\\
&\qquad\qquad\quad + \frac12\tr((a^Ta)^{-1}v^Ta(a^Ta)^{-1}a^Tv)\sqrt{\det(a^Ta)}
\end{align}
Using the cyclic permutation property of the trace and the fact that $V = va^+$ is symmetric we obtain
\begin{align}
&\tr((a^Ta)^{-1}v^Ta(a^Ta)^{-1}v^Ta)\sqrt{\det(a^Ta)}\\
&\qquad= \tr(a(a^Ta)^{-1}v^Ta(a^Ta)^{-1}v^T)\sqrt{\det(a^Ta)}\\
&\qquad= \tr(V^TV^T)\sqrt{\det(a^Ta)}= \tr(VV^T)\sqrt{\det(a^Ta)} = \langle v,v\rangle_a.
\end{align}
A similar calculation for the second term shows the statement.
\end{proof}
\subsection{The Geodesic Equation}
In this section we will present the geodesic equation of the Riemannian metric \ref{eq.metric.a} and derive an explicit solution.
\begin{thm}\label{thm.geodesic.a}
The geodesic equation on $M_+(n, m)$ with respect to the metric \eqref{eq.metric.a} is given by
\begin{equation}
\label{eq.geodesic.a}
\begin{aligned}
a_{tt} = &a_t(a^Ta)^{-1}a_t^Ta + a_t(a^Ta)^{-1}a^Ta_t
-a(a^Ta)^{-1}a_t^Ta_t\\
&\qquad\qquad\qquad+\frac12\tr\left(a_t(a^Ta)^{-1}a_t^T\right)a-\tr\left(a_t(a^Ta)^{-1}a^T\right)a_t.
\end{aligned}\end{equation}
\end{thm}
\begin{proof}
Let $a(t)$ be a smooth curve in $M_+(n, m)$ defined on the unit interval $I = [0,1]$ and $\delta a$ be a smooth variation of $a$ that vanishes at the endpoints $t = 0$ and $t = 1$. The energy of $a$ in $M_+(n, m)$ is
given by
\begin{align}
E(a) &= \int_I\langle a_t, a_t\rangle_a dt\\
&=\int_I\tr(a_t(a^Ta)^{-1}a_t^T)\sqrt{\det(a^Ta)}dt.
\end{align}
The directional derivative of the energy function $E$ at $a$ in the direction of $\delta a$ can be calculated as:
\begin{align}
\delta E(a)(\delta a)
&=\delta\left(\int_I \tr\left(a_t(a^Ta)^{-1}a_t^T\right)\sqrt{\det(a^Ta)} dt\right)(\delta a)\\
&=2\int_I \tr\left((\delta a)_t(a^Ta)^{-1}a_t^T\right)\sqrt{\det(a^Ta)} dt\\
&\qquad-2\int_I \tr\left(a_t(a^Ta)^{-1}(\delta a)^Ta(a^Ta)^{-1}a_t^T\right)\sqrt{\det(a^Ta)} dt\\
&\qquad+\int_I \tr\left(a_t(a^Ta)^{-1}a_t^T\right)\delta\left(\sqrt{\det(a^Ta)}\right) dt.
\end{align}
Note that for any smooth matrix function $B:\mathbb R\to \operatorname{GL}(m)$ we have
$$\dfrac{d}{dt} \det B = \tr\left(B_tB^{-1}\right)\det B; \quad \dfrac{d}{dt}B^{-1} = -B^{-1}B_tB^{-1}.$$
Using integration by parts and the above formulas we obtain
\begin{align}\label{eq.directional_derivative}
\delta E(a)(\delta a) = \int_I\langle\mathcal{T}(a),\delta a\rangle_{a}dt,
\end{align}
where
\begin{equation}
\label{eq.T}
\begin{aligned}
\mathcal{T} (a) &=-2\tr\left(a^Ta_t(a^Ta)^{-1}\right)a_t-2a_{tt}+2a_t(a^Ta)^{-1}(a^Ta)_t\\
&\qquad\qquad\qquad-2a(a^Ta)^{-1}(a_t)^Ta_t +\tr\left(a_t(a^Ta)^{-1}a_t^T\right)a.
\end{aligned}\end{equation}
Now the result follows, since $a$ is a geodesic if and only if $\mathcal{T}(a) = 0$.
\end{proof}
Using the Moore-Penrose inverse $a^+=(a^T a)^{-1}a^T$ a simpler form of the geodesic equation can be obtained:
\begin{lem}\label{lem.geodesic.L}
Let $L = a_ta^+$. Then $a$ is a geodesic if and only if $L$ satisfies the equation:
\begin{align}\label{eq.geodesic.L}
L_t + \tr(L)L + (L^TL-LL^T) - \frac{1}{2} \tr(L^TL) aa^+ = 0
\end{align}
\end{lem}
\begin{proof}
We have
\begin{align}
L_t &= (a_ta^+)_t = a_{tt}a^+ + a_t \big((a^Ta)^{-1}a^T\big)_t \\
&= a_{tt} a^+ - a_t (a^Ta)^{-1} (a^T_ta+a^Ta_t) (a^Ta)^{-1}a^T + a_t (a^Ta)^{-1}a^T_t \\
&= a_{tt}a^+ - a_t (a^Ta)^{-1}a^T_t aa^+ - L^2 + a_t (a^Ta)^{-1}a^T_t.
\end{align}
Now equation \eqref{eq.geodesic.L} is obtained by inserting the expression of $a_{tt}$ in \eqref{eq.geodesic.a}.
\end{proof}
This form of the geodesic equation allows us to obtain an analytic formula for the solution of the geodesic initial value problem, which constitutes the first of the main results of this article:
\begin{thm}\label{thm.geodesicformula.a}
Let $\delta = \tr(L^TL)$ and $\tau = \tr(L)$. The solution of \eqref{eq.geodesic.a} with initial values $a(0)$ and $L(0) = a_t(0)a(0)^+$ is given by
\begin{equation}\label{eq.geodesicFormula}
a(t) = f(t)^{1/m}e^{-s(t)\omega_0}a(0)e^{s(t)P_0},
\end{equation}
where
\begin{align}
f(t) &= \frac{m\delta(0)}{4}t^2+\tau(0)t+1,\qquad
s(t) = \int_0^t\frac{d\sigma}{f(\sigma)},\\
\omega_0 &= L^T(0) - L(0),\qquad P_0 = (a(0)^Ta(0))^{-1}(a_t(0)^Ta(0)) - \frac{\tau(0)}{m}I_{m\times m},
\end{align}
and $I_{m\times m}$ is the $m\times m$ identity matrix .
\end{thm}
\begin{proof}
This result can be shown by a direct calculation, substituting our solution into the geodesic equation. We can easily compute for example that
$$ L(t) = \frac{1}{f(t)} e^{-s(t)\omega_0} \left( \frac{f'(t)}{m} a_0 - \omega_0 a_0 + a_0 P_0 \right) a_0^+ e^{s(t)\omega_0},$$
and from here verify the formula \eqref{eq.geodesic.L}.
A more instructive proof of this result, along the lines of Freed-Groisser~\cite{freed1989basic} is presented in the Appendix~\ref{appendix.A}.
\end{proof}
In Figure~\ref{geo:rectangles} one can see a visualization of a geodesic in the space $M_+(3,2)$, where we visualize the matrices via their action on the unit rectangle.
\begin{figure}
\begin{center}
\includegraphics[width=0.40\linewidth]{rectangles.png}
\includegraphics[width=0.40\linewidth]{rectangles2.png}
\end{center}
\caption{Geodesics in the space $M_+(3,2)$. The matrices are visualized via their action on the unit rectangle. Note that the geodesic in the right figure leaves the space of full-ranked matrices in the middle of the geodesic.}
\label{geo:rectangles}
\end{figure}
As a direct consequence we obtain the following result concerning the incompleteness of $M_+(n,m)$:
\begin{cor}\label{cor.aGeodesic}
For any initial conditions $a(0) = a_0$ and $a_t(0)$ with $L_0 = a_t(0)a_0^+$, the geodesic $a(t)$ in $M_+(n,m)$ exists for all time $t\geq 0$ if and only if $a_t(0)$ is not a constant multiple $c$ of $a_0$ for some $c<0$. If $a_t(0)$ is a negative multiple of $a_0$, then the geodesic reaches the zero matrix at time $T = \frac{2}{|c|m}$.
\end{cor}
\begin{proof}
Note that $L_0 = a_t(0)a_0^+ = a_t(0)a_0^+a_0a_0^+ = L_0a_0a_0^+$. Using the Cauchy-Schwarz inequality, we have
\begin{align*}
(\tr(L_0))^2 &=(\tr(L_0a_0a^+_0))^2 \leq\tr(L_0L_0^T)\tr(a_0a_0^+(a_0a_0^+)^T)\\
&=\tr(L_0L_0^T)\tr(a_0a_0^+a_0a_0^+) =\tr(L_0L_0^T)\tr(a_0a_0^+)\\
&=m\tr(L_0L_0^T).
\end{align*}
Then we conclude that $\tau_0^2\leq m\delta_0$ with $\tau_0 = \tau(0)$ and $\delta_0 = \delta(0)$ in the notation of Theorem~\ref{thm.geodesicformula.a}, and the only way the equality holds is if there is a number $c$ such that $L_0 =a_t(0)a_0^+= a_0a_0^+$, i.e., $a_t(0) = ca_0$. Thus if $a_t(0)$ is not a multiple of $a_0$, we must have $\tau_0^2< m\delta_0$, and therefore
$$ f(t) = \epsilon^2 t^2 + (1+\tfrac{1}{2} \tau_0 t)^2, \qquad s(t) = \frac{1}{\epsilon} \, \arctan{\left( \frac{2\epsilon t}{2+\tau_0 t}\right)}, \qquad \epsilon = \sqrt{m\delta_0 - \tau_0^2}.$$
Thus $f(t)$ is never zero and $s(t)$ is well-defined for all $t>0$.
On the other hand, if $a_t(0)= c a_0$, then $m\delta_0 =\tau_0^2$ and $\tau_0 = cm$, and we have
$$ f(t) = (1 + \tfrac{cmt}{2})^2, \qquad s(t) = \frac{2t}{2+cm t}.$$
Hence $f(t)$ approaches zero in finite time, and as it does, $s(t)$ approaches positive infinity. Note however that in this case $\omega_0=0$, and
$$P_0 = c (a_0^T a_0)^{-1} (a_0^T a_0) - \frac{\tau_0}{m} I_m = c I_m- cI_m = 0.$$
Thus the solution \eqref{eq.geodesicFormula} becomes
$$ a(t) = (1+\tfrac{cmt}{2})^{2/m} a_0, $$
and the result follows.
\end{proof}
\subsection{Totally Geodesic Subspaces}
In this section we will study two families of totally geodesic subspaces of the space $M_+(n,m)$:
\begin{thm}\label{thm.totallyGeodesicSubspace.M}
The following spaces are totally geodesic subspaces of $M_+(n,m)$ with respect to the metric \eqref{eq.metric.a}:
\begin{enumerate}
\item the space $\operatorname{Scal}(b) :=\left \{ \lambda b|\lambda \in\mathbb R_{>0}\right\}$, where $b$ is any fixed element of $M_+(n,m)$;
\item the space $\operatorname{GL}(m)$, where elements in $\operatorname{GL}(m)$ are extended to $n\times m$ matrices by zeros.
\end{enumerate}
\end{thm}
\begin{proof}
The first result follows directly from the last sentence of the proof of Corollary~\ref{cor.aGeodesic}.
To prove that each component of $\operatorname{GL}(m)$ is a totally geodesic submanifold, consider the map $M_+(n,m)\to M_+(n,m)$ defined by $a\mapsto Ja$, where
$J$ is the matrix given in block diagonal form by
\begin{equation}
J=\begin{pmatrix} I_{m\times m}&0_{m\times (n-m)} \cr 0_{(n-m)\times m}&-I_{(n-m)\times (n-m)}\end{pmatrix}\;.
\end{equation} We know that this map is an isometry by the first invariance proved in Lemma~\ref{matmetinvar}, since $J\in \operatorname{O}(n)$. Clearly, its fixed point set is $\operatorname{GL}(m)$. It is well known that each component of the fixed point set of any set of isometries is a totally geodesic submanifold -- see, for example \cite[Proposition 24]{petersen2006riemannian}. This proves that each component of $\operatorname{GL}(m)$ is a totally geodesic submanifold.
\end{proof}
\subsection{The Riemannian Curvature}
In this part we will calculate the Riemannian curvatures of the metric~\eqref{eq.metric.a}. We will then show that the sectional curvature admits in general both signs.
There exists, however, an interesting subspace where the curvature is negative. In addition we will see that for the special case $m=1$, all sectional curvatures are non-negative.
Since $M_+(n,m)$ is an open subset of the vector space of all matrices $M(n,m)$,
we have a global chart. Using this chart, we will always identify tangent vectors
of $M_+(n, m)$ with elements of $M(n,m)$.
To calculate the Riemannian curvature tensor, we will use the following curvature formula, which is true in local coordinates:
\begin{multline}\label{curvatureformulachris}
R_a(u,v)w = -d\Gamma_a(u)(v,w) + d\Gamma_a(v)(u, w) \\+ \Gamma_a(u, \Gamma_a(v,w)) - \Gamma_a(v,\Gamma_a(u,w)),
\end{multline}
where $\Gamma: M_+(n, m)\times M(n, m)\times M(n, m)\to M(n, m)$ denotes the Christoffel symbols of the metric.
We can obtain the formula for the Christoffel symbol
by polarization of the right side of the geodesic equation $a_{tt} = \Gamma_{a}(a_t, a_t)$.
Using formula~\eqref{eq.geodesic.a} we thus get:
\begin{align}\label{eq.christoffel}
\Gamma_{a}(u, v) &= \dfrac12\big(u(a^Ta)^{-1}v^Ta + v(a^Ta)^{-1}u^Ta + ua^+v + va^+u - (ua^+)^Tv \\
&\qquad - (va^+)^Tu + \tr(u(a^Ta)^{-1}v^T)a - \tr(ua^+)v - \tr(va^+)u\big).
\end{align}
From here it is a straightforward calculation to obtain the formula for the Riemannian curvature:
\begin{lem}\label{lem.RieCurvature}
Using the notation $U = ua^+, V = va^+, W = wa^+$ the Riemannian curvature of $M_+(n, m)$ is given by
\begin{equation} \label{eq.RieCurvature}
\begin{aligned}
4(R_a&(u, v)w)a^+ \\&=[V, U^T]W^Taa^+ + W[U^T,V^T]aa^+ + WUV^Taa^++ W^TUV^Taa^+ \\
&\qquad + UWV^Taa^+ - [U, V^T]W^Taa^+ - WVU^Taa^+ - W^TVU^Taa^+ \\
&\qquad - VWU^Taa^+ + 2VU^Taa^+W + WU^Taa^+V + VW^Taa^+U\\
&\qquad - 2UV^Taa^+W - WV^Taa^+U - UW^Taa^+V + 2aa^+VU^TW\\
&\qquad + aa^+VW^TU + aa^+WU^TV - 2aa^+UV^TW - aa^+UW^TV\\
&\qquad - aa^+WV^TU + [[V, U], W]+ [V^T, U^T]W + 2UW^TV\\
&\qquad + 2UV^TW + V^TUW + W^TU^TV + V^TWU - 2VW^TU\\
&\qquad - 2VU^TW - U^TVW - W^TV^TU - U^TWV\\
&\qquad + \tr(VW^T)\tr(U)aa^+ - \tr(V)\tr(WU^T)aa^+ + m\tr(UW^T)V \\
&\qquad - m\tr(VW^T)U + \tr(W)\tr(V)U -\tr(W)\tr(U)V\
\end{aligned}
\end{equation}
Furthermore, if any of the tangent vectors
of $u, v, w, s$ is of the form $\lambda a$ for $\lambda \in \mathbb R$, then
\begin{equation}
\langle R_a(u,v)w,s\rangle_a = 0.
\end{equation}
\end{lem}
\begin{proof}
The proof is a very long, but basic computation using
the curvature formula \eqref{curvatureformulachris}
and the following formula for the differential of the Christoffel symbol;
\begin{align}
&2d\Gamma(u)(v, w)a^+ \\
=& -VU^TW^Taa^+ - VUW^Taa^+ + VW^TU - WU^TV^Taa^+\\
&\quad - WUV^Taa^+ + WV^TU - VU^Taa^+W - VUW + VU^TW \\
&\quad - WU^Taa^+V - WUV + WU^TV + aa^+UV^TW + U^TV^TW\\
&\quad - UV^TW +aa^+UW^TV + U^TW^TV - UW^TV -\tr(VU^TW^T)aa^+\\
&\quad - \tr(VUW^T)aa^+ + \tr(VW^T)U + \tr(VU^Taa^+)W + \tr(VU)W\\
&\quad - \tr(VU^T)W + \tr(WU^Taa^+)V + \tr(WU)V - \tr(WU^T)V.
\end{align}
\end{proof}
In the following we will decompose the tangent space of the space $M_+(n, m)$ in a scaling part -- i.e., changing only the determinant of the linear mapping -- and the complement.
Therefore we recall that any square matrix $U$ can be decomposed into a traceless part and a remainder as follows:
\begin{align}
U = U - \frac{\tr(U)}{m}aa^+ + \frac{\tr(U)}{m}aa^+:=U_0+ \frac{\tr(U)}{m}aa^+.
\end{align}
Analogously we define for a non-square matrix $u\in T_aM_+(n,m)$ the decomposition
\begin{align}
u = u - \frac{\tr(ua^+)}{m}a + \frac{\tr(ua^+)}{m}a:=u_0+\frac{\tr(ua^+)}{m}a.
\end{align}
Note that these two terms in the formula above are orthogonal with respect to the metric \eqref{eq.metric.a}. We will call $u_0$ the \emph{traceless part} and $\tfrac{\tr(ua^+)}{m}a$ the \emph{pure trace part} of $u$. It is easy to see that $U_0 = u_0a^+$.
We have seen in Lemma~\ref{lem.RieCurvature} that the curvature tensor vanishes if pure trace directions are involved. As a consequence, the
sectional curvature will only depend on the traceless part of the tangent vectors $u$ and $v$:
\begin{thm}\label{thm.SectionalCurvature}
The sectional curvature of $M_+(n, m)$ at $a$ is given by
\begin{align*}
&4\mathcal K_a(u, v)/\sqrt{\det(a^Ta)} = 4\langle R(u,v)v,u\rangle_a/\sqrt{\det(a^Ta)}\\
= &2\tr([V_0,U_0][V_0^T,U_0])+ 2\tr([V_0,U_0^T][V_0,U_0]) + 2\tr(V_0U_0V_0^TU_0^T)\\
&\quad + \tr(V_0V_0^TU_0^TU_0) - 4\tr(V_0V_0U_0^TU_0^T) + 4\tr(V_0U_0^TU_0V_0^T) \\
&\quad + \tr(V_0^TV_0U_0U_0^T) - 2\tr(V_0V_0^TU_0U_0^T) - 2\tr(V_0U_0^TV_0U_0^T)\\
&\quad + 6\tr(V_0U_0^TV_0U_0^Taa^+) - 3\tr(V_0U_0^TU_0V_0^Taa^+) - 3\tr(U_0V_0^TV_0U_0^Taa^+) \\
&\quad - m\tr(V_0V_0^T)\tr(U_0U_0^T) + m(\tr(U_0V_0^T))^2,
\end{align*}
where $u,v\in T_aM_+(n, m)$ are orthonormal with respect to the metric \eqref{eq.metric.a}, and $U_0, V_0$ are the traceless parts of $U =ua^+$ and $V = va^+$, respectively.
Furthermore, we have:
\begin{enumerate}
\item If one of the tangent vectors $u, v$ is a pure trace direction, then the sectional curvature is zero.
\item If $m\geq 2$ and $u, v\in T_aM_+(n, m)$ such that $U= ua^+$ and $V = va^+$ are symmetric -- i.e., for horizontal tangent vectors with respect to the projection~\eqref{projection} --
then the sectional curvature is negative.
\item If $m=1$, then all sectional curvatures are non-negative, and they vanish identically for $n=m+1=2$.\label{sec:curvature:m1}
\item If $ m\in \left\{2,3\right\}$ and $n\geq m+2$, then the sectional curvature always admits both signs.
\end{enumerate}
\end{thm}
\begin{rem}[Open cases and conjecture]
Using extensive testing with random matrices in MATLAB, we did not find any positive sectional curvatures for any of the open cases, i.e., for $m>3$, or for $m=\{2,3\}$ and $n=m+1$.
This leads us to the conjecture that the sectional curvature is non-positive in these cases.
In Figure~\ref{randomexperiments} we show histogram plots of our random-matrix experiments, that also demonstrate the scarcity of positive sectional curvature in the case
$m=\{2,3\}$ and $n\geq m+2$.
\end{rem}
\begin{figure}
\begin{center}
\includegraphics[width=0.31\linewidth]{curvaturesm2n3.pdf}
\includegraphics[width=0.31\linewidth]{curvaturesm2n4.pdf}
\includegraphics[width=0.31\linewidth]{curvaturesm3n5.pdf}
\end{center}
\caption{Histogram plots demonstrating the scarcity of positive sectional curvature: $x$-axis: value of the sectional curvature; $y$-axis: number of 2-planes that attained this value. Left figure: $m=2$, $n=3$. Percentage of positive sectional curvature: zero.
Middle figure: $m=2$ , $n=4$. Percentage of positive sectional curvature: 3.041\%. Right figure: $m=3$ , $n=5$. Percentage of positive sectional curvature: 0.007\%.
The figures have been created in MATLAB using $10^7$ runs with random matrices for each choice of $m$ and $n$.}
\label{randomexperiments}
\end{figure}
\begin{proof}[Proof of Theorem~\ref{thm.SectionalCurvature}]
The formula for $\mathcal K$ at $a\in M_+(n,m)$ can be obtained by direct computation.
W.l.o.g. we assume that $m$ and $n$ are not both one, as for this case the space is one-dimensional and the curvature is trivial.
For orthonormal $u$ and $v$ we have
\begin{align}
\mathcal K_a(u, v) = \langle R_a(u,v)v, u\rangle_a = \langle R_a(u_0,v_0)v_0, u_0\rangle_a,
\end{align}
where the second equality is obtained by Lemma~\ref{lem.RieCurvature}.
Statement~(1) follows directly from the curvature formula. To see~(2) we calculate
\begin{align}
&4\langle R(u,v)v, u\rangle_a/\sqrt{\det(a^Ta)}\\
&\qquad= 14(\tr(U_0V_0U_0V_0) - \tr(U_0U_0V_0V_0))\\
&\qquad\qquad + m\tr(U_0V_0)\tr(V_0U_0) - m\tr(U_0U_0)\tr(V_0V_0)\\
&\qquad= 7\tr\left([U_0, V_0]^2\right) + m\left(\left(\tr(U_0V_0)\right)^2 - \tr(U_0^2)\tr(V_0^2)\right).
\end{align}
Note that $U, V$ being symmetric implies that $U_0, V_0$ are symmetric. Thus their commutator is antisymmetric and then $\tr\left([U_0, V_0]^2\right)\leq 0$. In addition, by the Cauchy-Schwarz inequality we have
\begin{align}
\left(\tr(U_0V_0)\right)^2 = \left(\tr(U_0V_0^T)\right)^2 \leq \tr(U_0U_0^T)\tr(V_0V_0^T) = \tr(U_0^2)\tr(V_0^2).
\end{align}
Therefore, $\mathcal K_a(u, v)\leq 0$. Note that we needed $m \geq 2$ to construct two linear independent tangent vectors $u$ and $v$ with $U$ and $V$ being symmetric.
Furthermore the inequality is strict if $U=U_0$ and $V=V_0$, i.e., if the linearly independent vectors $u,v$ are traceless. Note that such pairs always exist for $m \geq 2$.
For point~(3), we first observe that in the situation $m=1$, $u_0^Tv_0 = \norm{a}_a^2\langle u_0, v_0 \rangle_a$ with $\norm{a}_a^2 = \sqrt{a^Ta}$, and
$a^+ = (a^Ta)^{-1}a^T = \norm{a}^{-4}a^T$,
where the norm $\norm{\cdot}$ is with respect to the metric \eqref{eq.metric.a}. By direct calculation we obtain
\begin{align}
&U_0U_0=V_0V_0=U_0V_0=V_0U_0=0_{n\times n},\quad U_0^Ta = V_0^Ta=0_{n\times 1},
\end{align}
and
\begin{align}
U_0^TU_0 &= (a^+)^Tu_0^Tu_0a^+ = \norm{a}_a^{-6}\norm{u_0}_a^2aa^T,\\
V_0^TV_0 &= (a^+)^Tv_0^Tv_0a^+ = \norm{a}_a^{-6}\norm{v_0}_a^2aa^T,\\
U_0^TV_0 &= (a^+)^Tu_0^Tv_0a^+ = \norm{a}_a^{-6}\langle u_0, v_0\rangle_aaa^T.
\end{align}
Substituting these formulas into the formula of the sectional curvature for the general case and simplifying it, we have
\begin{align}\label{eq.sectional.curvature.d=1}
\mathcal{K}_a(u, v) = \frac34\norm{a}_a^{-2}\left(\norm{u_0}_a^2\norm{v_0}_a^2 - \langle u_0, v_0\rangle_a^2\right).
\end{align}
By the Cauchy-Schwarz inequality, the sectional curvature is therefore non-negative. If $n =2$ in addition, we have at each $a\in M_+(2,1)$ only one $2$-dim tangent plane. Let $u, v\in M_+(2,1)$ be a pair of orthonormal tangent vectors respect to the metric \eqref{eq.metric.a} such that $u$ is in the direction of $a$. Then $u_0=0$, and thus by formula~\ref{eq.sectional.curvature.d=1} the sectional curvature vanishes.
Finally for statement (4), i.e., $m\in \{2,3\}$ and $n\geq m+2$, we let
\begin{align}a = \begin{pmatrix}
\operatorname{Id}_{m\times m}\\
0_{(n-m)\times m} \\
\end{pmatrix},
u = \begin{pmatrix}
0 & \cdots & 0 & 0\\
\vdots & & \vdots & \vdots\\
0 & \cdots & 0 & 0\\
0 & \cdots & 0 & 1\\
0 & \cdots & 0 & 0\\
\end{pmatrix} , v = \begin{pmatrix}
0 & \cdots & 0 & 0\\
\vdots & & \vdots & \vdots\\
0 & \cdots & 0 & 0\\
0 & \cdots & 0 & 0\\
0 & \cdots & 0 & 1\\
\end{pmatrix},
\end{align}
where $\operatorname{Id}_{m\times m}$ denotes the $m\times m$ identity matrix and $0_{(n-m)\times m}$ the $(n-m)\times m$ zero matrix.
It is easy to check that $u$ and $v$ are orthonormal tangent vectors at $a$ with respect to the metric \eqref{eq.metric.a}. Plugging $a$ and $u, v$ into the formula of the sectional curvature we obtain
\begin{align}
\mathcal K_a(u,v) = 4 - m,
\end{align}
which proves the last statement.
\end{proof}
\subsection{The metric completion}
In Corollary~\ref{cor.aGeodesic} we have seen that $M_+(m,n)$ with the metric \eqref{eq.metric.a} is geodesically incomplete. By the theorem of Hopf-Rinow that implies that the corresponding metric space is also metrically incomplete.
In this section we will study its metric completion. For technical reasons we will restrict ourself to the case $n>m$, as the space $M_+(m,m)=\operatorname{Gl}(m)$ is not connected and thus one would have to study the completion of each of the two connected components separately. To keep the presentation simple we will not treat this special case.
We first recall the formula for the geodesic distance function on $M_+(n,m)$ with respect to the metric \eqref{eq.metric.a}:
\begin{multline}\label{eq.distance.nbym}
\on{dist}^{n\times m}(a_0, a_1) = \inf_{a}\Big\{L(a) = \int_I\|a_t(t)\|_{a(t)}dt\, \Big\vert \, a\colon[0,1]\to M_+(n,m)\\ \text{ is piecewise differentiable with } a(0) = a_0, a(1) = a_1\Big\},
\end{multline}
where the norm $\|\cdot\|$ is induced by the metric \eqref{eq.metric.a} on $M_+(n,m)$.
We first calculate an upper bound for the geodesic distance:
\begin{lem}\label{lem.inequality.distance}
Let $a, b\in M_+(n,m)$ with $n>m$. Then
\begin{align*}
\on{dist}^{n\times m}(a, b)\leq \dfrac{2}{\sqrt{m}}\left(\sqrt[4]{\det(a^Ta)}+ \sqrt[4]{\det(b^Tb)}\right).
\end{align*}
\end{lem}
\begin{proof}
Let $a,b \in M_+(n,m)$. Using the invariance properties of the metric -- c.f. item (2) in Lemma \ref{matmetinvar} -- we observe that the geodesic distance between scaled versions of the matrices $a$ and $b$ can be made arbitrary small, i.e., given $\epsilon>0$ there exists $\delta>0$ such that $\on{dist}^{n\times m}(\delta a, \delta b)\leq \epsilon$.
We will now calculate an upper bound for the geodesic distance between a matrix to a scaled version of the same matrix. Assume $\epsilon, \delta$ are as above and let $a_1\in M_+(n,m)$. We consider the path $a(t) =(1-t)a_1$ for $t\in (0,1-\delta)$. Using $a_t(t) = -a_1$ we calculate
\begin{align}\label{eq.ine1}
\on{dist}^{n\times m}(a_1,\delta a_1 ) \leq& \int_0^{1-\delta}\|a_t(t)\|_{a(t)}dt
\leq \int_0^{1}\|a_t(t)\|_{a(t)}dt\notag\\
=&\int_0^1\left(\tr\left(a_t\left(a^T(t)a(t)\right)^{-1}a_t^T\right)\sqrt{\det(a^T(t)a(t))}\right)^{1/2}dt\notag\\
=&\int_0^1\left(mt^{m-2}\sqrt{\det(a_1^Ta_1)}\right)^{1/2}dt = \dfrac{2}{\sqrt{m}}\sqrt[4]{\det(a_1^Ta_1)}.
\end{align}
Now the statement follows from the triangle inequality:
\begin{align*}
\on{dist}^{n\times m}(a, b) \leq& \on{dist}^{n\times m}(a, \delta a) + \on{dist}^{n\times m}(\delta a, \delta b) +\on{dist}^{n\times m}(\delta b, b)\\
=&\dfrac{2}{\sqrt{m}}\left(\sqrt[4]{\det(a^Ta)}+ \sqrt[4]{\det(b^Tb)}\right)+\epsilon\;,
\end{align*}
which proves the result.
\end{proof}
Using this result we are able to characterize the metric completion of $M_+(n,m)$:
\begin{thm}
Let $n>m$. The metric completion of the space $M_+(n,m)$ with respect to the geodesic distance \eqref{eq.distance.nbym} is given by
$M(n,m)/\sim$ where $a\sim b$ if $\on{rank}(a)<m$ and $\on{rank}(b) <m$.
\end{thm}
\begin{proof}
In the following let $\{a_k\}$ and $\{b_k\}$ be Cauchy sequences with respect to the geodesic distance function
$\on{dist}^{n\times m}$.
First we consider the case that $\operatorname{det}(a_k^Ta_k) \to 0$ and $\operatorname{det}(b_k^Tb_k) \to 0$ as
$k$ goes to infinity.
By Lemma~\ref{lem.inequality.distance} we have $\on{dist}^{n\times m}(a_k, b_k)\to 0$ and thus any two such sequences are identified with each other in the metric completion. This new point corresponds to the identification of all matrices with non-maximal rank.
It remains to consider the case in which $\operatorname{det}(a_k^Ta_k)\not\to 0$ as $k\to\infty$. In this case, there exists a subsequence of $\tilde a_k$, an $\eta>0$ and $K_0\in \mathbb N$ such that $\operatorname{det}(\tilde a_k^T \tilde a_k)>\eta$ for all $k>K_0$.
By the identification~\eqref{eq.identification} we write $\tilde a_k=z_ks_k$ with $z_k\in \on{O}(n)$ and $s_k\in \on{Sym}^+(m)$ (extended to a $n\times m$ matrix with zeros). We will view $s_k$ both as an $n \times m$ and as an $m\times m$ matrix, depending on which form is more convenient for our purposes. Since $\on{O}(n)$ is compact we can always pass to a convergent subsequence and using the left invariance of the Riemannian metric (and thus of the induced geodesic distance function) we may assume that this limit is the identity matrix, i.e., $\lim_{k\to \infty}z_k =I_{n\times n}$. It remains to show that $s_k$ converges. Let $\epsilon>0$. Since $\tilde a_k$ is a Cauchy sequence, for all $k, l$ sufficiently large we have
\begin{align*}
\epsilon &> \on{dist}^{n\times m}(z_ks_k,z_ls_l)=
\on{dist}^{n\times m}(s_k,z_k^T z_l s_l)
\geq \inf_{z\in \on{O}(n)} \on{dist}^{n\times m}(s_k,zs_l).
\end{align*}
The mapping $\pi: a\mapsto a^Ta$ is a Riemannian submersion onto the space of symmetric matrices with the metric \eqref{eq.metric.sym} and thus the last expression is equal to the geodesic distance induced by \eqref{eq.metric.sym} of $s_k^Ts_k$ and $s_l^Ts_l$.
Thus we have shown that $s_k^Ts_k \in \on{Sym}^+(m)$
is a Cauchy sequence with respect to the geodesic distance of the metric~\eqref{eq.metric.sym}. By a result of Clarke~\cite[Proposition 4.11]{clarke2013completion} and the assumption on the determinant, there exists a constant $C$ such that $(s_k^Ts_k)_{ij}\leq C$ for all $k >K_0$. It follows that $$(s_k^Ts_k)_{jj} = \sum_i(s_k)_i^j(s_k)_i^j\leq C$$ and thus $|(s_k)_i^j|\leq \sqrt{C}$. Therefore $s_k$ is in a bounded and closed subset of $\mathbb R^{m\times m}$ and thus, by taking a further subsequence, we can conclude that $s_k$ converges to a unique element $s \in \operatorname{Sym}^+(m)$.
\end{proof}
\begin{rem}[The space of symmetric matrices (revisited)]
Using the Riemannian submersion structure as described in Section~\ref{sec:symmetric} to study the geometry
of the space of symmetric matrices~\ref{eq.metric.sym}, one can regain several classical results of \cite{freed1989basic,gil1991riemannian,clarke2013completion,ebin1970manifold}, including the solution
for the geodesic equation and the non-positivity of the sectional curvature. We will present the alternative derivation
of these results in Appendix~\ref{appendix:submersion}.
\end{rem}
\section{The generalized Ebin metric}
In this section we will introduce the generalized Ebin metric on the space of one-forms $\Omega_+^1(M,\RR^n)$.
Therefore let $\alpha\in\Omega_+^1(M,\RR^n)$. Then the tensor product $\alpha^T\otimes\alpha$, which is defined for each $x\in M$ as the pull back of the Euclidean scalar product under $\alpha$, defines a Riemannian metric on $M$. For simplicity, we will just denote this tensor product (Riemannian metric resp.) by $\alpha^T\alpha$. Consequently, the inner product $(\alpha^T\alpha)_x = (\alpha^T\alpha)(x)$ induces for each $x\in M$ an inner product on the cotangent space $T_x^*M$, which is given by $(\alpha^T\alpha)^{-1}_x = ((\alpha^T\alpha)(x))^{-1}$.
Given $\zeta,\eta\in T_{\alpha}\Omega_+^1(M,\RR^n)$, we can now define a Riemannian metric on $\Omega_+^1(M,\RR^n)$ as the integral over $M$ of the point-wise inner product of $\zeta_x = \zeta(x)$ and $\eta_x = \eta(x)$ with respect to the volume form $\on{vol}(\alpha)$ associated with the metric $\alpha^T\alpha$ on $M$:
\begin{align}\label{metric}
G_{\alpha}(\zeta, \eta) = \int_M (\alpha^T\alpha)^{-1}_x(\zeta_x, \eta_x)\on{vol}(\alpha).
\end{align}
In the following we will derive an expression of this metric in local coordinates
$\{x^i, i=1,\cdots,m\}$. With respect to the corresponding basis $\{\frac{\partial}{\partial x^i}\}$ on $T_xM$ and the standard basis on $\RR^n$,
the one-forms $\alpha_x, \zeta_x$ and $\eta_x$ can be represented by $n\times m$ matrices, which we will still denote by $\alpha_x, \zeta_x$ and $\eta_x$. Furthermore, the metric $(\alpha^T\alpha)_x$ can be identified with the $m\times m$ matrix $\alpha_x^T\alpha_x$. Thus we obtain the local formula of the Riemannian metric~\eqref{metric} as:
\begin{align}\label{eq.metricLocal}
G_{\alpha}(\zeta, \eta) = \int_M \tr(\zeta_x(\alpha^T\alpha)^{-1}_x \eta_x^T)\sqrt{\det(\alpha^T\alpha)_x}dx.
\end{align}
It is easy to see that by definition our metric $G$ \eqref{metric} is independent of the original metric on $M$. In addition, it follows from the local formula and the second invariance of Lemma~\ref{matmetinvar} that the metric $G$ does not depend on the choice of coordinates near $x\in M$. The following lemma gives two important invariances of our metric $G$ on $\Omega^1_+(M,\mathbb R^n)$.
\begin{lem}\label{lem:invariances}
Let $\alpha\in\Omega^1_+(M,\mathbb R^n)$ and $\zeta, \eta\in T_\alpha\Omega^1_+(M,\mathbb R^n)$.
\begin{enumerate}
\item The metric \eqref{metric} is invariant under pointwise left multiplication with $\on{O}(n)$, i.e., for any smooth function $z: M\to\on{O}(n)$, we have
\begin{equation*}
G_{\alpha}(\zeta, \eta)=G_{z\alpha}(z\zeta,z\eta)
\end{equation*}
\item The metric \eqref{metric} is invariant under the right action of the diffeomorphism group, i.e., for any $\varphi \in \Diff(M)$ we have
\begin{equation*}
G_{\alpha}(\zeta, \eta)=G_{\varphi^*\alpha}(\varphi^*\zeta,\varphi^*\eta)
\end{equation*}
\end{enumerate}
\end{lem}
\begin{proof}
The proof of the first invariance property is the same as for the finite dimensional metric on $M_+(n,m)$ from Lemma \ref{matmetinvar}. For the second invariance property we calculate
\begin{align*}
&G_{\varphi^*\alpha}(\varphi^*\zeta,\varphi^*\eta)= \int_M \tr \left( \varphi^*\zeta\;((\varphi^*\alpha)^T\varphi^*\alpha)^{-1}(\varphi^*\eta)^T \right) \sqrt{\operatorname{det}\left((\varphi^*\alpha)^T\varphi^*\alpha\right)}\; \mu\\
&=\int_M \tr \left(\zeta\circ\varphi\;((\alpha\circ\varphi)^T\alpha\circ\varphi)^{-1}(\eta\circ\varphi)^T \right) \sqrt{\operatorname{det}\left((\alpha\circ\varphi)^T\alpha\circ\varphi \right)}|\det(d\varphi)|\; \mu\\
&=G_{\alpha}(\zeta, \eta)
\qedhere
\end{align*}
\end{proof}
\subsection{Connection to the Ebin metric}\label{Sect:submersiontoEbin}
In this section we will show that the metric defined in \eqref{metric} on the space $\Omega_+^1(M, \mathbb R^n)$ is connected to the Ebin metric on the space of Riemannian metrics $\operatorname{Met}(M)$ on $M$. This will be a consequence of the previous result for the finite dimensional spaces of matrices and the point-wise nature of the metric.
The main difficulty in the infinite-dimensional situation is proving the surjectivity of the projection map.
Following \cite{ebin1970manifold} we will first recall the definition of the Ebin metric.
The space of Riemannian metrics $\operatorname{Met}(M)$ is a open subset of the space of all smooth symmetric $(0,2)$ tensor fields on $M$, denoted by $\Gamma(S^2T^*M)$, and thus the tangent space at each element $g$ is $\Gamma(S^2T^*M)$ itself. Let $g\in \operatorname{Met}(M)$ and $h, k\in T_g{\operatorname{Met}(M)}=\Gamma(S^2T^*M)$.
We can then introduce the metric via
\begin{align}\label{eq.metric.metricsOnM}
(h, k)_g = \dfrac14\int_M\tr_g(hk)\mu_g,
\end{align}
where $\mu_g$ is the volume form induced by $g$ and
where at any $x\in M$, we define the integrand by replacing $g, h, k$ by the associated symmetric $m\times m$ matrices $g(x), h(x), k(x)$ with
respect to an arbitrary basis of $T_xM$ and where we define $$\tr_g(hk)(x)=\tr(h(x)g(x)^{-1}k(x)g(x)^{-1}).$$
Recall that in Section~\ref{sec:symmetric} we have shown that the mapping
$\pi: M_+(n,m)\to\operatorname{Sym}_+(m)$, $
\pi(a)=a^Ta$
is a Riemannian submersion, where the metric on $M_+(n,m)$ is given by \eqref{eq.metric.a} and the metric on $\operatorname{Sym}_+(m)$ is given by \eqref{eq.metric.sym}. Similarly, we can define a mapping
\begin{align}\label{induced_metric_oneform}
\tilde{\pi}: \Omega_+^1(M, \mathbb R^n) \to \operatorname{Met}(M),\quad \alpha \mapsto \alpha^T \alpha,
\end{align}
where $(\alpha^T\alpha)$ is a section of $S^2_*TM$ which for $x\in M$ (the pullback of the Euclidean scalar product under $\alpha$.
We have the following result:
\begin{thm}
Let $M$ and $n$ be such that there exists at least one full-ranked $\mathbb R^n$ valued one-form on $M$, i.e., $\Omega^1_+(M,\mathbb R^n)\neq \emptyset$.
Then the mapping $\tilde{\pi}: \Omega_+^1(M, \mathbb R^n) \to \operatorname{Met}(M)$ is a Riemannian submersion, where $\Omega_+^1(M, \mathbb R^n)$ is equipped with the metric \eqref{metric} and $\operatorname{Met}(M)$ carries the multiple of the Ebin metric, as defined in \eqref{eq.metric.metricsOnM}.
\end{thm}
\begin{proof}
We first need to show that $\tilde \pi$ is a surjective map, i.e., given $g\in \operatorname{Met}(M)$ we need to construct $\beta(x)\in \Omega_+^1(M, \mathbb R^n)$
with $\tilde \pi(\beta)=g$. Therefore let $\al_0 \in \Omega_+^1(M, \mathbb R^n)$ be any fixed full-ranked one-form and let $g_0$ be the Riemannian metric induced by $\al_0$
via pulling back the Euclidean scalar product, see \eqref{induced_metric_oneform}.
Then for each $x\in M$, the operator $Y_x = g_x(g_0)_x^{-1}$ from $T_xM$ to itself,
defined by $g_0(Y(u),v) = g(u,v)$ for $u,v\in T_xM$, is positive-definite and
symmetric with respect to the Riemannian metric $(g_0)_x$. Since $g$ and $\alpha_0$ are smooth tensor fields,
$Y_x$ depends smoothly on $x$. The pointwise positive-definite square root $\sqrt{Y_x}$ is uniquely determined,
and it is a smooth function of $x$ as well (see Kato, Perturbation Theory, II.6~\cite{Kato}). We then define
$\beta_x = (\alpha_0)_x\circ \sqrt{Y_x}$, which is again smooth in $x$ and maps each $T_xM$ to $\mathbb{R}^n$.
We verify that
$$ \langle \beta(u), \beta(v)\rangle_{\mathbb{R}^n} = g_0\big(\sqrt{Y}(u),\sqrt{Y}(v)\big) = g_0(\sqrt{Y}{}^T \sqrt{Y}(u), v) = g(u,v)$$
for all $u,v\in T_xM$, so that $\beta^T \beta = g$.
It follows that $\pi(\beta)=g$.
Since the metric on $\Omega_+^1(M, \mathbb R^n)$ and the metric on $ \operatorname{Met}(M)$ are both point-wise, the remainder of the result is now an immediate consequence of Theorem~\ref{thm:matrices:submersion}.
\end{proof}
\subsection{A product structure for the space of one-forms}
We begin this section by fixing a volume form $\mu$ on $M$. Whenever we refer to a matrix operation on a 1-form (e.g., trace or transpose), it is assumed that we have expressed that form locally as a matrix field, using a basis of the tangent space that has unit volume with respect to $\mu$.
Following the work of \cite{freed1989basic} we will decompose the space of 1-forms as the product
of the space of volume forms on $M$ with the space of 1-forms that induce the fixed volume form $\mu$, i.e.,
$\Omega_+^1(M, \mathbb R^n)\equiv \operatorname{Vol}(M)\times \Omega^1_{\mu}(M, \mathbb R^n)$, where $\Omega^1_{\mu}(M, \mathbb R^n)$ denotes the set of all 1-forms such that $\operatorname{det}\left(\alpha^T\alpha \right)=1$. A straight-forward calculation shows that the tangent space of $\Omega^1_{\mu}(M, \mathbb R^n)$ consists of all tangent vectors $h\in T_{\alpha}\Omega^1_{\mu}(M, \mathbb R^n)$ such that
$\on{tr} (\alpha^+ h)=0$ with $\alpha^+=(\alpha^T \alpha)^{-1}\alpha^T$ being the Moore-Penrose pseudo-inverse. In the following lemma we calculate the formula of the metric $G$
in this product decomposition:
\begin{lem}
In the identification $\Omega_+^1(M, \mathbb R^n)\equiv \operatorname{Vol}(M)\times \Omega_{\mu}^1(M, \mathbb R^n)$ the metric \eqref{metric} takes the form
\begin{equation}
\bar G_{(\rho,\beta)}\left((\nu_1,h_1),(\nu_2,h_2)\right)=
\int_M \tr \left(h_1\;(\beta^T\beta)^{-1}h_2^T \right) \rho\mu+\frac1{m}
\int_M \frac{\nu_1}{\rho}\frac{\nu_2}{\rho} \rho\mu\\
\end{equation}
The metric $\bar G$ is not a product metric, since the foot-point volume density $\rho$ appears in both terms above. Note, however, that the decomposition of the tangent space into directions tangent to $\on{Vol}(M)$ and directions tangent to $\Omega_{\mu}(M, \mathbb R^n)$ are orthogonal to each other with respect to the metric $\bar G$. Such a metric is also called an almost product metric, see~\cite{gil1992pseudoriemannian}.
\end{lem}
\begin{proof}
We first construct a bijection from $\operatorname{Vol}(M)\times \Omega^1_{\mu}(M, \mathbb R^n)$ to the space of full-ranked one-forms. Therefore we let
\begin{equation}
\Phi(\alpha):= (\rho,\beta)=\left(\sqrt{\operatorname{det}(\alpha^T\alpha)}, \rho^{-1/m}\alpha \right)\qquad \Phi^{-1}(\rho,\beta) := \rho^{1/m} \beta\;.
\end{equation}
To see that this mapping has the required properties, we calculate
\begin{align*}
\sqrt{\operatorname{det}(\beta^T\beta)}= \rho^{-1} \sqrt{\operatorname{det}(\alpha^T\alpha)}=1\;.
\end{align*}
To calculate the induced metric on the product we have to calculate the variation of the inverse mapping.
We have
\begin{equation}
d\Phi^{-1}(\rho, \beta)(\nu,h)= \rho^{1/m} h+\frac1m\rho^{1/m-1}\nu\beta
\end{equation}
Thus we obtain the formula of the metric on the product space:
\begin{align*}
&\bar G_{(\rho,\beta)}\left((\nu_1,h_1),(\nu_2,h_2)\right)\\&\qquad=
G_{\Phi^{-1}(\rho,\beta)}\left( d\Phi^{-1}(\rho, \beta)(\nu_1,h_1),d\Phi^{-1}(\rho, \beta)(\nu_2,h_2)\right)\\
&\qquad=
G_{\rho^{1/m} \beta}\left(\rho^{1/m} h_1+\frac1m\rho^{1/m-1}\nu_1\beta,\rho^{1/m} h_2+\frac1m\rho^{1/m-1}\nu_2\beta\right)\\
&\qquad=
\int_M \tr \left(h_1\;(\beta^T\beta)^{-1}h_2^T \right) \rho\mu+\frac1{m}
\int_M \frac{\nu_1}{\rho}\frac{\nu_2}{\rho} \rho\mu\\&\qquad\qquad\qquad+\frac{v_2}{m}
\int_M \tr \left(h_1\;(\beta^T\beta)^{-1}\beta^T \right) \mu
+
\frac{v_1}{m}\int_M \tr \left(\beta\;(\beta^T\beta)^{-1}h_2^T \right) \mu
\end{align*}
Now the result follows since any tangent vector $h$ to $\Omega^1_{\mu}$ satisfies
$$\tr \left(h\;(\beta^T\beta)^{-1}\beta^T \right)=0.$$
Note that, by standard properties of the trace, this also shows that last term vanishes.
\end{proof}
\begin{rem}
If one restricts the metric to the space of volume forms $\on{Vol}(M)$ one obtains the Fisher-Rao metric.
For this metric the geometry is well-studied and completely understood, see e.g., \cite{friedrich1991fisher,khesin2013geometry}. Furthermore, it has been
shown that the Fisher-Rao metric is up to a constant the unique Riemannian metric on the space of volume densities that is invariant under the action of the diffeomorphism group \cite{ay2015information,bauer2016uniqueness,cencov2000statistical}.
\end{rem}
\subsection{The geodesic distance}
Any Riemannian metric (on a finite or infinite dimensional manifold) gives rise to a (pseudo) distance on the manifold, the geodesic distance. In finite dimensions this distance function is always a true metric, i.e.,
symmetric, satisfies the triangle inequality and non-degenerate. In infinite dimensions it has been shown that the third property might fail, see \cite{eliashberg1993biinvariant,michor2005vanishing,bauer2013geodesic,bauer2018vanishing}. In this section we will observe that the geodesic distance function of the metric \eqref{metric} can be written as an integral over the geodesic distance function of a finite dimensional space of matrices and thus we will obtain the non-degeneracy of the geodesic distance on the infinite dimensional space of all full ranked one-forms. This is essentially the same proof as for the Ebin-metric on the space of all Riemannian metrics; see the work of Clarke \cite{clarke2013geodesics}.
To formulate this result we recall the finite dimensional Riemannian metric on the space $M_+(n,m)$:
\begin{equation}
\langle u,v\rangle_a= \tr \left(u\;(a^Ta)^{-1}v^T \right) \sqrt{\operatorname{det}\left(a^Ta \right)}\;.
\end{equation}
Furthermore we denote the corresponding geodesic distance by $\operatorname{dist}^{n\times m}(\cdot,\cdot)$. Note that $\operatorname{dist}^{n\times m}$ is non-degenerate as
the space of $n\times m$ matrices is finite dimensional.
With this notation we immediately obtain the following result concerning the geodesic distance on the infinite dimensional manifold
of all full-ranked one-forms:
\begin{thm}\label{thm:distpointwise}
The geodesic distance on the manifold $\Omega^1_+(M,\mathbb R^n)$ is non-degenerate and satisfies
\begin{equation}\label{distance:pointwise}
\operatorname{dist}^{\Omega^1_+}(\alpha,\beta)^2 \geq \int_M \operatorname{dist}^{n\times m}(\alpha(x),\beta(x))^2\; \mu\;.
\end{equation}
\end{thm}
\begin{proof}
To prove this result we only need to show the inequality \eqref{distance:pointwise}.
The non-degeneracy of the geodesic distance follows then directly from the non-degeneracy of the geodesic distance on finite dimensional manifolds
and the face that two distinct elements of $\Omega^1_+(M,\mathbb R^n)$ have to differ on a set of positive measure.
The proof of the above inequality is exactly the same as in \cite[Thm. 2.1]{clarke2013geodesics}
\end{proof}
\begin{rem}
For the Ebin metric on the space of all Riemannian metrics it has been shown
that the analogue of the inequality~\eqref{distance:pointwise} is actually an equality, i.e.,
that
\begin{equation}\label{distance:pointwise:eq}
\operatorname{dist}^{\on{Met}}(\alpha,\beta)^2 = \int_M \operatorname{dist}^{m\times m}(\alpha(x),\beta(x))^2\; \mu\;.
\end{equation}
It is easy to generalize this result to the situation studied here by allowing paths of one-forms that
are only of class $L^2$ in $x\in M$. Therefore one simply chooses for each $x\in M$
a short path in the finite dimensional manifold $\mathbb R^{n\times m}$, which immediately yields the equality. Here a short path means a path of matrices $a(t)$ such that
$\operatorname{len}(a(t))\leq \operatorname{dist}^{n\times m}(a(0),a(1))+\epsilon$ for some $\epsilon >0$. To prove the result in the smooth category is much harder.
We believe, however, that a similar analysis as in \cite{clarke2013geodesics} might be used to obtain this result. We leave this question open for future research.
\end{rem}
\subsection{Geodesics and curvature}\label{sec:geodesicequation_forms}
The point-wise nature of the metric will allow us to directly use our results for the space of matrices to obtain the following result concerning geodesics and curvature, c.f. \cite{Misiolek1993Stability,Bao1993nonlinear}.
\begin{thm}\label{thm.SectionalCurvatureandGeodesics_forms}
The geodesic equation of the generalized Ebin metric on the space of full-ranked one-forms decouples in space and time. Thus for each $x\in M$ it is given by the ODE
\eqref{eq.geodesic.a} with explicit solution as presented in Theorem~~\ref{thm.geodesicformula.a}. Similarly, the sectional curvature is simply the integral over the pointwise sectional
curvatures and thus the statements on sign-definiteness of Theorem~\ref{thm.SectionalCurvature} hold also in this infinite dimensional situation.
\end{thm}
\subsection{On totally geodesic subspaces}
In this section we will show that the space $\Omega^1_+(M,\mathbb R^n)$ contains two remarkable totally geodesic subspaces. To understand one of these subspaces, we need some preliminaries. Let $\hbox{Gr}(m,n)$ denote the Grassmannian manifold of all $m$-dimensional linear subspaces of $\mathbb R^n$. Define a map
$$W:\Omega^1_+(M,\mathbb R^n)\to C^\infty(M,\hbox{Gr}(m,n))$$ by
$$W(\alpha)(x)=\alpha(T_xM).$$
Let $\xi$ denote the canonical $m$-plane bundle over $\hbox{Gr}(m,n)$.
Given $f\in C^\infty(M,\hbox{Gr}(m,n))$, it is easy to see that $f\in W(\Omega^1_+(M,\mathbb R^n))$ if and only if $TM\cong f^*(\xi)$. This is because $W(\alpha)=f$ if and only if $\alpha$ is a bundle isomorphism $TM\to f^*(\xi)$.
\begin{thm}\label{thm.totallyGeodesicSubspace.2}
The following spaces are totally geodesic subspaces of the space $\Omega^1_+(M,\mathbb R^n)$ equipped with the generalized Ebin metric:
\begin{enumerate}
\item any one-dimensional space of scalings $\mathcal A :=\left \{ t\alpha_0 \,|\,t\in \mathbb R_{>0}\right\}$, where $\alpha_0$ is a fixed element of $\Omega^1_+(M,\mathbb R^n)$,
\item the space $\mathcal B :=\left \{\alpha \in \Omega^1_+(M,\mathbb R^n) | W(\alpha)=f_0\right\},$ where $f_0$ is any fixed element of $C^\infty(M,\hbox{Gr}(m,n))$. (Note that this space is empty unless $TM\cong f_0^*(\xi)$, by the remark just above this Lemma).
\end{enumerate}
\end{thm}
\begin{proof}
Here we use the point-wise nature of the metric \eqref{metric} on the space $\Omega_+^1(M, \mathbb R^n)$. Let $x\in M$ and $\{e_i, 1\leq i\leq m\}$ be an orthonormal basis of $T_xM$. Choosing the standard basis for $\mathbb R^n$, (1) follows immediately from the first statement of Theorem~\ref{thm.totallyGeodesicSubspace.M}.
Now we prove (2), i.e., the space $\mathcal B$ is a totally geodesic subspace. Since $\alpha$ is a bundle isomorphism $TM\to f_0^*(\xi)$, for each $x\in M$ the image of the orthonormal basis under $\alpha_x$, denoted by $\{\tilde{e}_i = \alpha_x(e_i)\}$, forms an orthonormal basis of $\xi_{f_0(x)}$. Note that $\xi_{f_0(x)} = f_0(x)$ is a $m$-plane. So we can extend this orthonormal basis to get an orthonormal basis $\{\tilde{e}_i, 1\leq i\leq n\}$ of $\mathbb R^n$. With respect to this basis $\{e_i\}$ of $T_xM$ and the basis $\{\tilde{e_i}\}$ of $\mathbb R^n$,
it is easy to see that each linear transformation in $\left\{\alpha_x: T_xM\to \mathbb R^n\,|\, W(\alpha)(x)=f_0(x)\right\}$ corresponds to a matrix in $\operatorname{GL}(m)$ (extended to a $n\times m$ matrix with zeros). Thus the result follows from the second statement of Theorem~\ref{thm.totallyGeodesicSubspace.M}.
\end{proof}
\subsection{Metric and geodesic incompleteness}
As a consequence of the fact that scaling of a full ranked one-form is totally geodesic, we immediately obtain the geodesic and metric
incompleteness of the metric:
\begin{thm}
The space $\Omega^1_+(M,\mathbb R^n)$ is metrically and geodesically incomplete.
\end{thm}
\begin{proof}
This follows directly from the fact that scaling of a metric yields geodesic curves that leave the space in finite time, c.f. Theorem~\ref{thm.totallyGeodesicSubspace.2}.
\end{proof}
To obtain the metric completion we believe that a similar strategy as in \cite{clarke2013completion} will lead to the following result:
\begin{conj}
The metric completion of the space $\Omega^1_+(M,\mathbb R^n)$ equipped with the geodesic distance function of the generalized Ebin metric is the space of $L^2$-sections
of the vector bundle $\left(T^*M\otimes \mathbb R^n\right) \to M$ modulo the equivalence relation $\sim$,
where $\alpha \sim \beta$ if the statement
\begin{equation*}
\alpha(x)\neq \beta(x) \Longleftrightarrow \operatorname{rank}(\alpha(x)) < m \text{ and } \operatorname{rank}(\beta(x)) < m
\end{equation*}
holds almost everywhere.
\end{conj}
The proof of \cite{clarke2013completion} used rather heavy machinery from geometric measure theory. To develop this theory in the current context is out of the scope of the present article.
Thus we leave this question open for future research.
\section{An application: Reparametrization invariant metrics on the space of open curves}\label{shape:analysis}
In this section we will describe the relation of our proposed metric to the square root framework as developed for shape analysis
of curves \cite{younes1998computable,younes2008metric,Srivastava2011Shape}. In contrast to the aforementioned framework, our construction is not limited to one-dimensional objects, but has a direct
generalization to higher dimensional objects, notably to the space of surfaces. We plan to develop this line of research in a future
application-oriented article and will focus mainly on the simpler space of curves in this section.
In the following we denote the space of immersed curves in $\mathbb R^n$ by
\begin{equation}
\operatorname{Imm}([0,1],\mathbb R^n):=\left\{c\in C^{\infty}([0,1],\mathbb R^n): |c'|\neq 0 \right\}\;.
\end{equation}
Here $c'$ denotes the derivative of $c$ with respect to $\theta \in [0,1]$.
We can map each curve to a $\mathbb R^n$-valued one-form on $[0,1]$ via
$c \mapsto c' d\theta$. The immersion condition ensures that
the resulting one-form actually has full rank and thus we obtain
a bijection
\begin{equation}\label{eq.differential.Phi}
\Phi: \operatorname{Imm}([0,1],\mathbb R^n)/\operatorname{trans} \to \Omega^1_+([0,1],\mathbb R^n)\;.
\end{equation}
To see that the map $\Phi$ is surjective, note that all one-forms are closed since we are in dimension one, and all closed one-forms are exact since the first cohomology of $[0,1]$ vanishes.
Furthermore, we had to identify curves that differ only by a translation as they all get mapped to the same one-form.
Pulling back the metric~\eqref{metric} on $\Omega^1_+([0,1],\mathbb R^n)$, one obtains a reparametrization invariant
metric on the space of curves modulo translations. It turns out that this metric is exactly the Younes-metric as studied in
\cite{younes1998computable}:
\begin{equation}\label{eq.metric.curve}
(\Phi^*G)_c(h,k)= G_{\Phi(c)}\left(d\Phi(c)(h),d\Phi(c)(k)\right)=\int_0^1 \frac{h_{\th}\cdot k_\th}{|c'|}d\theta = \int_0^1 D_sh\cdot D_sk ds,
\end{equation}
where $c\in \operatorname{Imm}([0,1],\mathbb R^n)$ and $h,k \in T_c \operatorname{Imm}([0,1],\mathbb R^n)$. Here
$D_s=\frac{1}{|c'|} \frac{d}{d\theta}$ denotes arc-length differentiation, and $ds=|c'|d\theta$ denotes integration with respect to arc length.
In the article \cite{younes2008metric} the authors introduced a transformation for this metric that yields explicit formulas for geodesics between open and closed curves in the plane.
Implicitly this has been extended to open curves in arbitrary dimension in the article \cite{needham2018shape}.
By considering the formulas of Section~\ref{sec:geodesicequation_forms} in the special case studied in this section, we obtain an explicit formula for geodesics for curves in arbitrary dimension, and in addition we obtain the non-negativity of the sectional curvature:
\begin{thm}
Let $c_0\in \operatorname{Imm}([0,1], \mathbb R^n)$ and $h\in T_{c_0}\operatorname{Imm}([0,1], \mathbb R^n)$.
The geodesic on the space of open curves modulo translations $\operatorname{Imm}([0,1], \mathbb R^n)/\operatorname{trans}$ starting at $c_0$ in the direction $h$ with respect to the metric \eqref{eq.metric.curve} is given by
\begin{align}\label{eq.geodesic.curve}
c(t, \theta) = \int_0^{\theta}f(t,\lambda)e^{-s(t,\lambda)\left(V^T(\lambda)-V(\lambda)\right)}c'_0(\lambda)d\lambda,
\end{align}
where
\begin{align*}
&V(\theta) = h(\theta)\,c_0'^+(\theta),\qquad \delta_0(\theta) = \operatorname{tr}(V^T(\theta)V(\theta)),\qquad \tau_0(\theta) = \operatorname{tr}(V(\theta)),\\
&f(t,\theta) = \dfrac{\delta_0(\theta)}{4} t^2 + \tau_0(\theta) t + 1,\qquad s(t,\theta) = \int_0^t 1/f(\sigma,\theta) {d\sigma}\,.
\end{align*}
Furthermore, the sectional curvature of $\operatorname{Imm}([0,1], \mathbb R^n)/\operatorname{trans}$ with respect to the metric \eqref{eq.metric.curve} is always non-negative for $n\geq2$ and vanishes for $n = 2$.
\end{thm}
\begin{proof}
To prove the statement on the explicit solution we consider the formula given in Theorem~\ref{thm.geodesicformula.a} for $m = 1$.
Let $c(t,\theta)$ be the geodesic starting at $c_0 = c(0, \theta)$ in direction $h$. We will use $c'$ to denote the derivative with respect to $\theta$ and $c_t$ to denote the derivative in time $t$.
Using the notation $V(\theta) = c'_t(0,\theta)c'^+(0,\theta) = hc_0'^+$
we have:
\begin{align*}
c_{\theta}(t,\theta) = f(t,\theta)e^{-s(t,\theta)\omega_0}c'_0(\theta)e^{s(t,\theta)P_0},
\end{align*}
where $f(t,\theta)$ and $s(t,\theta)$ are as in Theorem~\ref{thm.geodesicformula.a}, $\omega_0 = V^T(\theta) - V(\theta)$ and
$$P_0 = \left(c_{\theta}^Tc_{\theta}\right)^{-1}v^Tc_{\theta}-\tau_0 =0\;,$$
Taking the integral with respect to $\theta$ formula \eqref{eq.geodesic.curve} follows.
The result on the sectional curvature follows directly from statement \eqref{sec:curvature:m1}
of Theorem~\ref{thm.SectionalCurvature} and Theorem~\ref{thm.SectionalCurvatureandGeodesics_forms}.
\end{proof}
In Figure~\ref{CurvesExamples}, we present one example of a geodesic that was computed using the explicit formula derived above.
\subsection{The space of surfaces}
In this section we will briefly comment on the difficulties that arise for using the same method to obtain a framework for shape analysis of surfaces. As mentioned above, in the case of curves, the mapping $\Phi$ in \eqref{eq.differential.Phi} gives us a bijection between the space of curves modulo translations and the space of full rank $\RR^n$-valued one-forms on $[0,1]$. Thus the preimage of a geodesic in the space $\Omega_+^1([0,1], \mathbb R^n)$ gives a geodesic in the space of immersed curves in $\mathbb R^n$. However, in the case of (two-dimensional) surfaces in $\mathbb R^3$ (here typically $n$ will be $3$), the operator $d: \operatorname{Imm}(S^2,\mathbb R^3)/\operatorname{trans}\to \Omega_+^1(S^2, \mathbb R^3)$ only induces a bijection between $\operatorname{Imm}(S^2,\mathbb R^3)/\operatorname{trans}$ and the space of full rank and \emph{exact} one forms, denoted by ${\Omega}_{+,\on{ex}}^1(S^2, \mathbb R^3)$, which is a proper subspace of $\Omega_+^1(S^2, \mathbb R^3)$.
Furthermore ${\Omega}_{+,\on{ex}}^1(S^2, \mathbb R^3)$ is not a totally geodesic submanifold of $\Omega_+^1(S^2, \mathbb R^3)$ and so geodesics in $\Omega_+^1(S^2, \mathbb R^3)$ do not give rise to geodesics in $\operatorname{Imm}(S^2,\mathbb R^3)/\operatorname{trans}$. Note that the same would be true for $S^2$ replaced with the sheet $[0,1]\times [0,1]$.
Thus using this representation for shape analysis of surfaces will require some extra work. A potential approach is to study the submanifold geometry of ${\Omega}_{+,\operatorname{ex}}^1(S^2, \mathbb R^3)$ in more detail to obtain an explicit solution in this space. Alternatively one could work in the space of all full rank one-forms $\Omega_+^1(S^2, \mathbb R^3)$ and project the geodesic onto the submanifold ${\Omega}_{+,\operatorname{ex}}^1(S^2, \mathbb R^3)$. In Figure~\ref{SurfaceExample1} and~\ref{SurfaceExample2}, we present examples of geodesics between two parametrized surfaces with respect to the pull-back metric, that have been calculated using a discretization of the space of full ranked exact one-forms. These examples have been calculated using the numerical framework for the Riemannian metric studied in this paper as developed in~\cite{su2019shape}, where the spherical parametrizations of the boundary surfaces have been obtained using the code of Laga et al.~\cite{kurtek2013landmark}.
\begin{figure*}
\begin{center}
\includegraphics[width=0.9\linewidth]{surface1}
\end{center}
\caption{A geodesic in the space of surfaces modulo translations with respect to the generalized Ebin metric~\eqref{metric}.}
\label{SurfaceExample2}
\end{figure*}
\appendix
\section{The computation of the geodesic formula in the space $M_+(n,m)$}\label{appendix.A}
In this appendix we give the computation of the geodesic formula in the space $M_+(n,m)$ with respect to the metric \eqref{eq.metric.a}. Recall that the geodesic equation on $M_+(n,m)$ is given by
\begin{equation}
\label{eq.appendix.geodesic.a}
\begin{aligned}
a_{tt} = &a_t(a^Ta)^{-1}a_t^Ta + a_t(a^Ta)^{-1}a^Ta_t
-a(a^Ta)^{-1}a_t^Ta_t\\
&\qquad\qquad\qquad+\frac12\tr\left(a_t(a^Ta)^{-1}a_t^T\right)a-\tr\left(a_t(a^Ta)^{-1}a^T\right)a_t,
\end{aligned}\end{equation}
and a simpler form of the geodesic equation for $L = a_ta^+$ is given by
\begin{align}\label{eq.appendix.geodesic.L}
L_t + \tr(L)L + (L^TL-LL^T) - \frac{1}{2} \tr(L^TL) aa^+ = 0.
\end{align}
To solve the equation \eqref{eq.appendix.geodesic.a}, we start with the equation \eqref{eq.appendix.geodesic.L} for $L(t)$ and we have the following proposition.
\begin{prop}\label{prop.appendix.taudelta}
Suppose $a$ and $L$ are as in \eqref{eq.appendix.geodesic.a} and \eqref{eq.appendix.geodesic.L}. Define $\delta = \tr(L^TL)$ and $\tau = \tr(L)$. Then
$\tau$ and $\delta$ satisfy the differential equations
\begin{align}
\begin{cases}\label{eq.appendix.taudelta}
\tau_t + \tau^2 - \tfrac{m}{2} \delta = 0, \qquad &\tau(0)=\tau_0 = \tr(L(0))\\
\delta_t + \tau \delta = 0, \qquad &\delta(0)=\delta_0 = \tr(L(0)^TL(0)).
\end{cases}
\end{align}
The solution of these equations is
\begin{equation}\label{eq.appendix.taudelta.solutions}
\tau(t) = \frac{f_t(t)}{f(t)}, \qquad \delta(t) = \frac{\delta_0}{f(t)},
\end{equation}
where
\begin{equation}\label{eq.appendix.f}
f(t) = \frac{m\delta_0}{4} t^2 + \tau_0 t + 1.
\end{equation}
\end{prop}
\begin{proof}
The trace of \eqref{eq.appendix.geodesic.L} yields the first equation in \eqref{eq.appendix.taudelta} since $\tr{(aa^+)} = \tr(a^+a) = \tr(I_{m\times m}) = m$. Notice that $Laa^+ = L$. We have
\begin{align*}
\tr(L^Taa^+) &= \tr(aa^+L^T) = \tr((Laa^+)^T) = \tr(L^T) = \tr(L).
\end{align*}
Multiplying \eqref{eq.appendix.geodesic.L} on the left by $L^T$
yields the second equation in \eqref{eq.appendix.taudelta}.
The system \eqref{eq.appendix.taudelta} is exactly the same system as in the work of Freed-Groisser~\cite{freed1989basic}. Thus we can use the same trick to solve it. Write $\tau(t) = f_t(t)/f(t)$ where $f(0) = 1$, and the first equation in \eqref{eq.appendix.taudelta} becomes
\begin{align}
f_{tt}(t) = \dfrac{m}{2}\delta(t)f(t),\quad f(0) = 1,\, f_t(0) = \tau_0.
\end{align}
Meanwhile the second equation in \eqref{eq.appendix.taudelta} becomes $\dfrac{d}{dt}(\delta f) = 0$, which can immediately be solved to give $\delta(t) = \delta_0/f(t)$.
So the second derivative $f_{tt}(t) = \dfrac{m\delta_0}{2}$ is constant, and with $f_t(0)=\tau_0$ and $f(0) = 1$, we get the solution $f(t) = \dfrac{m\delta_0}{4} t^2 + \tau_0 t + 1.$ Formula \eqref{eq.appendix.taudelta.solutions} follows.
\end{proof}
With explicit solutions for $\tau(t)$ and $\delta(t)$ in hand, we can now solve the rest of the geodesic equation \eqref{eq.appendix.geodesic.L} with initial $L(0)$, given by
\begin{equation} \label{eq.appendix.geo.L}
L_t + \frac{f_t}{f} L + (L^TL-LL^T) - \frac{\delta_0}{2f}aa^+ = 0.
\end{equation}
\begin{lem}
Let $M(t) = L(t) - \dfrac{\tau(t)}{m}a(t)a^+(t)$. Then $L$ satisfies \eqref{eq.appendix.geo.L} if and only if $M$ satisfies
\begin{align}\label{eq.appendix.geo.M}
M_t + \frac{f_t}{f}M + (M^TM-MM^T) = 0.
\end{align}
\end{lem}
\begin{proof}
We first compute
\begin{align*}
(aa^+)_t =& \left(a(a^Ta)^{-1}a^T\right)_t\\
=& a_t(a^Ta)^{-1}a^T -a(a^Ta)^{-1}\left(a^T_ta+a^Ta_t\right)(a^Ta)^{-1}a^T + a(a^Ta)^{-1}a^T_t\\
=& L - L^Taa^+ - aa^+L + L^T\\
=& M - M^Taa^+ - aa^+M +M^T\;.
\end{align*}
Here we used that $L = a_ta^+ = a_t(a^Ta)^{-1}a^T$, that $\tau_t = \dfrac{m}{2}\delta - \tau^2$ and that $Maa^+ = M$. Thus we obtain
\begin{align*}
L_t =& M_t + \dfrac{\delta_0}{2f}aa^+ - \dfrac{\tau^2}{m}aa^+ + \dfrac{\tau}{m}\left(M - M^Taa^+ -aa^+M+ M^T\right),\\\
\dfrac{f_t}{f}L =& \dfrac{f_t}{f}M + \dfrac{\tau^2}{m}aa^+,\\
L^TL - LL^T =& M^TM -MM^T + \frac{\tau}{m}aa^+M + \frac{\tau}{m}M^Taa^+ - \frac{\tau}{m}M^T - \frac{\tau}{m}M.
\end{align*}
Replacing the terms in \eqref{eq.appendix.geo.L} with the formulas above we obtain equation \eqref{eq.appendix.geo.M} and thus the statement follows.
\end{proof}
\begin{prop}
The solution of \eqref{eq.appendix.geo.L} satisfies
\begin{equation}\label{eq.appendix.L}
L(t) = \frac{1}{f(t)}e^{-s(t)\omega_0}M_0e^{s(t)\omega_0} + \frac{f_t(t)}{mf(t)}a(t)a(t)^+,
\end{equation}
where $\omega_0 = L(0)^T - L(0)$, $s(t) = \int_0^t \dfrac{d\sigma}{f(\sigma)}$ and
$
M_0 = L(0) - \dfrac{\tau_0}{m}a(0)a^+(0).
$
\end{prop}
\begin{proof}
Use equation \eqref{eq.appendix.geo.M} and set $M(t) = N(t)/f(t)$. Then $N$ satisfies
\begin{equation*}
N_t + \frac{1}{f}(N^TN- NN^T) = 0.
\end{equation*}
Changing variables to $s(t) = \int_0^t d\sigma/f(\sigma)$ we obtain
\begin{equation}\label{eq.Nequation}
N_s + N^TN - NN^T = 0.
\end{equation}
Note that the transpose of \eqref{eq.Nequation} is
$ N^T_s + N^TN - NN^T = 0.$
It follows that $\omega = N^T-N$ is constant in time, and thus $ \omega = \omega_0 = N^T(0) - N(0) = M^T(0) - M(0) = L^T(0)-L(0)$.
We can rewrite \eqref{eq.Nequation} as
\begin{align*}
N_s = NN^T - N^TN = -\omega_0 N + N \omega_0 = [-\omega_0, N].
\end{align*}
Then we obtain the solution
\begin{align}\label{eq.appendix.N}
N(s) = e^{-s \omega_0} N(0) e^{s\omega_0}.
\end{align}
Translate \eqref{eq.appendix.N} back into
\begin{equation}
L(t) = M(t) + \frac{\tau}{m}a(t)a^+(t) = \frac{1}{f(t)} N(t) + \frac{\tau}{m}a(t)a^+(t),
\end{equation}
we obtain \eqref{eq.appendix.L}.
\end{proof}
Using formula \eqref{eq.appendix.L} of $L(t)$ we are now able to obtain a solution for the flow equation $a_t(t) = L(t)a(t)$.
\begin{thm}
Let $f(t)$ be of the same form as in \eqref{eq.appendix.f}. Then the solution of the flow $a_t(t) = L(t)a(t)$ with initial data $a(0)$ is given by
\begin{align}\label{eq.appendix.a}
a(t) = f(t)^{1/m}e^{-s(t)\omega_0}a(0)e^{s(t)P_0},
\end{align}
where $\omega_0 = L^T(0)-L(0)$ and
\begin{align*}
P_0 = \left(a^T(0)a(0)\right)^{-1}a_t(0)^Ta(0) - \dfrac{\tau_0}{m}I_{m\times m}.
\end{align*}
\end{thm}
\begin{proof}
Using \eqref{eq.appendix.L}, the equation for $a(t)$ becomes
\begin{align*}
a_t = L a = \dfrac{1}{f }e^{-s\omega_0}M_0e^{s\omega_0}a + \dfrac{f_t }{mf}a .
\end{align*}
Write $a(t) = f(t)^{1/m}Q(t)$ to eliminate the second term. Then we have
\begin{align*}
Q_t = \dfrac{1}{f}e^{-s\omega_0}M_0e^{s\omega_0}Q.
\end{align*}
Changing variables to $s(t) = \int_0^t\dfrac{d\sigma}{f(\sigma)}$ we obtain
\begin{align*}
Q_s = e^{-s\omega_0}M_0e^{s\omega_0}Q.
\end{align*}
Now let $Q(s) = e^{-s\omega_0}R(s)$. Then $R(s)$ satisfies the differential equation
\begin{align*}
R_s =& \omega_0R + M_0R = M_0^TR\\
=& (L^T(0) - \dfrac{\tau_0}{m}a(0)a^+(0))R\\
=& a(0)\left((a^T(0)a(0))^{-1}a_t^T(0) - \dfrac{\tau_0}{m}a^+(0)\right)R.
\end{align*}
Notice that the initial $R(0) = a(0)$ and $R_s$ is always of the form $a(0)$ times a $m\times m$ matrix. Therefore we must have $R(s) = a(0)B(s)$ for some $m\times m$ matrix $B$, which satisfies $B(0) = I_{m\times m}$ and
\begin{align}\label{eq.appendix.B}
B_s = \left(\left(a^T(0)a(0)\right)^{-1}a_t(0)^Ta(0) - \dfrac{\tau_0}{m}I_{m\times m}\right)B(s).
\end{align}
Let $P_0 = \left(a^T(0)a(0)\right)^{-1}a_t(0)^Ta(0) - \dfrac{\tau_0}{m}I_{m\times m}$. The solution of the equation \eqref{eq.appendix.B} with initial $B(0) = I_{m\times m}$ is
\begin{align*}
B(s) = e^{sP_0}.
\end{align*}
Changing back to $t$ variables, formula \eqref{eq.appendix.a} follows immediately.
\end{proof}
\section{The space of symmetric matrices (revisited)}\label{appendix:submersion}
In this appendix we re-derive some classical results by \cite{freed1989basic,gil1991riemannian,clarke2013completion,ebin1970manifold} concerning the (finite-dimensional version of the) Ebin-metric on the space of symmetric matrices using our Riemannian submersion picture.
We first present the geodesic equation on $\on{Sym}_+(m)$, which corresponds to the horizontal geodesic equation on $M_+(n,m)$:
\begin{cor}
The geodesic equation on $\operatorname{Sym}_+(m)$ with respect to the metric \ref{eq.metric.sym} is given by
\begin{align}
g_{tt} = g_tg^{-1}g_t + \dfrac14\tr(g^{-1}g_tg^{-1}g_t)g - \dfrac12\tr(g^{-1}g_t)g_t.
\end{align}
\end{cor}
\begin{proof}
We identify the space of symmetric matrices $\operatorname{Sym}_+(m)$ with the quotient space $\operatorname{SO}(n)\backslash M_+(n,m)$ and consider the horizontal geodesic equation on $M_+(n, m)$, which is given by
\begin{align}\label{eq.geodesicHorizontal}
a_{tt} = a_ta^+a_t + \dfrac12\tr(a_t(a^Ta)^{-1}a_t^T)a - \tr(a_ta^+)a_t.
\end{align}
This is a straight-forward calculation using that $a_ta^+$ is symmetric.
Now consider a smooth curve $g(t)$ in the space of symmetric matrices $\operatorname{Sym}_+(m)$. Then $g(t) = \pi(a(t)) = a(t)^Ta(t)$ for some horizontal lift $a(t)\in M_+(n,m)$ and
\begin{align}
g_t = a_t^Ta + a^Ta_t;\qquad g_{tt} = a_{tt}^Ta + 2a_t^Ta_t + a^Ta_{tt}.
\end{align}
Inserting the expression of $a_{tt}$ in \eqref{eq.geodesicHorizontal} we obtain
\begin{align}
g_{tt} =& a_{tt}^Ta + 2a_t^Ta_t + a^Ta_{tt}\\
=& a_t^Ta(a^Ta)^{-1}a_t^Ta + 2a_t^Ta_t + a^Ta_t(a^Ta)^{-1}a^Ta_t\\
&\quad + \tr(a_t(a^Ta)^{-1}a_t^T)a^Ta - \tr(a_ta^+)(a_t^Ta+a^Ta_t)
\end{align}
Notice that $a^+a = I$ and $a_ta^+$ is symmetric. It is easy to check that
\begin{align}
g_tg^{-1}g = a_t^Ta(a^Ta)^{-1}a_t^Ta + 2a_t^Ta_t + a^Ta_t(a^Ta)^{-1}a^Ta_t.
\end{align}
Similar to the calculation in Theorem~\ref{thm:matrices:submersion} we obtain
\begin{align}
\dfrac14\tr(g^{-1}g_tg^{-1}g_t)=& \dfrac14\tr((a^Ta)^{-1}(a_t^Ta + a^Ta_t)(a^Ta)^{-1}(a_t^Ta + a^Ta_t))\\
=& \tr(a_t(a^Ta)^{-1}a_t^T),
\end{align}
and
\begin{align}
\dfrac12\tr(g^{-1}g_t) &= \dfrac14\tr((a^Ta)^{-1}(a_t^Ta + a^Ta_t)(a^Ta)^{-1}(a^Ta + a^Ta))\\
=& \tr(a_t(a^Ta)^{-1}a^T) = \tr(a_ta^+).
\end{align}
The conclusion follows.
\end{proof}
Using Theorem~\ref{thm.SectionalCurvature} and O'Neill's curvature formula we obtain the curvature of the space of symmetric matrices, which agrees with the formula of \cite{freed1989basic}:
\begin{cor}
The space $\left(\operatorname{Sym}_+(m),\langle \cdot, \cdot \rangle^{\operatorname{Sym}}\right)$ has non-positive sectional curvature given by:
\begin{align*}
\mathcal K_g^{\operatorname{Sym}}(h,k) =& \dfrac{1}{16}\big[\tr([g^{-1}h,g^{-1}k]^2) + \frac{m}{4}\left(\tr(g^{-1}hg^{-1}k)\right)^2 \\
&\qquad\qquad-\frac{m}{4}\tr\left((g^{-1}h)^2\right)\tr\left((g^{-1}k)^2\right)\big]\sqrt{\det(g)}
\end{align*}
\end{cor}
\begin{proof}
Similarly as in Section~\ref{sec:symmetric}, we identify the space of symmetric matrices $\operatorname{Sym_+(m)}$ with the quotient space $\operatorname{SO}(n)\backslash M_+(n,m)$.
Using the fact that the metrics on $M_+(m,n)$ and $\operatorname{Sym}(m)$ are connected via a Riemannian submersion, we can calculate the curvature of the quotient space using O'Neill's curvature formula.
Let $g\in\operatorname{Sym}_+(m)$ and $h,k\in T_{g}\operatorname{Sym}_+(m)$ be two orthonormal tangent vectors with respect to the metric \eqref{eq.metric.sym}. Then we have a lift $a\in M_+(n,m)$ and the horizontal lifts $\tilde{h},\tilde{k}\in T_a\left(M_+(n,m)\right)$ of $h,k$ such that
\begin{align*}
\pi(a) = g,\quad d\pi_a(\tilde{h}) = h,\quad d\pi_a(\tilde{k}) = k.
\end{align*}
Since $d\pi_a$ is an isometry, $\tilde{h},\tilde{k}$ are orthonormal with respect to the metric \eqref{eq.metric.a}. Recall from Theorem~\ref{thm:matrices:submersion} that any horizontal tangent vector $u$ at $a\in M_+(n,m)$ has the property that $U = ua^+$ is symmetric.
Thus by Theorem~\ref{thm.SectionalCurvature} the sectional curvature $\mathcal K$ at $a$ for $\tilde{h},\tilde{k}\in T_a(M_+(n,m))$ is given by:
\begin{multline}
\mathcal K_a(\tilde{h},\tilde{k})\\
= \left(\frac74\tr\left([\tilde{H}_0,\tilde{K}_0]^2\right) + \frac{m}{4}\left(\tr(\tilde{H}_0\tilde{K}_0)\right)^2 - \frac{m}{4}\tr(\tilde{H}_0^2)\tr(\tilde{K}_0^2)\right)\sqrt{\det(a^Ta)}.
\end{multline}
It remains to calculate O'Neill's curvature term. We have
\begin{align}
[\tilde{h},\tilde{k}]a^+ = (\tilde{h}a^+\tilde{k} - \tilde{k}a^+\tilde{h})a^+ = \tilde{H}\tilde{K} - \tilde{K}\tilde{H} = [\tilde{H},\tilde{K}],
\end{align}
where the commutator on the right side is the usual matrix commutator, which is defined for any two square matrices. Notice that for symmetric $\tilde{H}$ and $\tilde{K}$, the commutator $[\tilde{H}, \tilde{K}]$ is skew-symmetric and thus $[\tilde{h}, \tilde{k}] = \tilde{h}a^+\tilde{k} - \tilde{k}a^+\tilde{h}$ is in the vertical bundle. Therefore the O'Neill term is given by
\begin{align}
\frac34\langle [\tilde{H}, \tilde{K}],[\tilde{H},\tilde{K}]\rangle_a = -\frac34\tr([\tilde{H}, \tilde{K}]^2)\sqrt{\det(a^Ta)}.
\end{align}
Notice that $\tr([\tilde{H}, \tilde{K}]^2) = \tr([\tilde{H}_0, \tilde{K}_0]^2)$. Using O'Neill's curvature formula we then obtain the sectional curvature on the quotient space:
\begin{align*}
&\mathcal K_g^{\operatorname{Sym}}(h,k)\\
=& \left(\tr\left([\tilde{H}_0,\tilde{K}_0]^2\right) + \frac{m}{4}\left(\tr(\tilde{H}_0\tilde{K}_0)\right)^2 - \frac{m}{4}\tr(\tilde{H}_0^2)\tr(\tilde{K}_0^2)\right)\sqrt{\det(a^Ta)}.
\end{align*}
It is straightforward calculation to show that
\begin{align*}
\tr\left([\tilde{H}_0,\tilde{K}_0]^2\right) =& \dfrac{1}{16}\tr([g^{-1}h,g^{-1}k]^2);\\
\tr(\tilde{H}_0\tilde{K}_0) =& \dfrac{1}{4} \tr(g^{-1}hg^{-1}k);\\
\tr(\tilde{H}_0^2)\tr(\tilde{K}_0^2) =& \dfrac{1}{16}\tr\left((g^{-1}h)^2\right)\tr\left((g^{-1}k)^2\right).
\end{align*}
Therefore, the result follows.
\end{proof}
\bibliographystyle{abbrv}
\bibliography{refs}
\end{document}
| 90,824
|
TITLE: Compact MU or BP Modules
QUESTION [2 upvotes]: Is there a classification of the compact MU or BP modules in any category of spectra? Can the periodicity theorem be finagled to give a MU-module structure on finite spectra?
REPLY [11 votes]: No nonzero finite spectrum admits an $MU$-module structure. Indeed, suppose $F$ is a finite spectrum with an $MU$-module structure. Then for all $n$, $F$ has a map $v_n:\Sigma^{2p^n-2}F\to F$, which induces an isomorphism on $K(n)_*F$ (there's a subtlety here in that it's not obvious that the $v_n$ map on $F$ and the $v_n$ map on $K(n)$ give rise to the same map on $K(n)\wedge F$; see eg the end of the proof of Lemma 7 of http://math.harvard.edu/~lurie/252xnotes/Lecture33.pdf). Thus for each $n$, $F/v_n$ is $K(n)$-acyclic. But $F/v_n$ is finite, so this implies it is also $K(m)$-acyclic for all $m<n$, and so $v_n$ is also an isomorphism on $K(m)_*F$. But by finiteness of $F$, for any $m$ sufficiently large we can find $n>m$ for which $v_n$ must be $0$ on $K(m)_*F$ just for reasons of degree (by the AHSS for $K(m)_*F$). Thus $K(m)_*F=0$ for all sufficiently large $m$, which implies $F=0$.
I would also add that even if you did have a finite spectrum with an $MU$-module structure, it could not possibly be compact as an $MU$-module. Indeed, if it were, after smashing with $H\mathbb{Z}$ it would be a compact $H\mathbb{Z}\wedge MU$-module. But $\pi_*(H\mathbb{Z}\wedge MU)$ is a polynomial ring on infinitely many generators, and so all but finitely many of those generators have to act non-nilpotently on any compact module (basically, any "finite presentation" of a compact module can only involve finitely many of the polynomial generators). Since $\pi_*(H\mathbb{Z}\wedge F)=H_*(F)$ vanishes in all but finitely many degrees for $F$ finite, this is impossible.
| 126,526
|
TITLE: multivariable limit problem
QUESTION [1 upvotes]: I have a confusion regarding this problem.
Problem: $\displaystyle f(x,y)=\frac{\sin^2|x+2y|}{x^2+y^2}$ is continuous for all $(x,y)\neq (0,0)$. True or false?
I think that the limit does not exist so the function is not continuous.
How to prove that limit does not exist?
Any help will be appreciated. Thanks in advance
REPLY [0 votes]: Try approaching the limit by the line $y=k\cdot x$.
Applying the standard limit $$\lim_{t\rightarrow 0} \frac{\sin(t)}{t}$$
You can prove that the limit depends on the constant K, hence does not exist
| 68,917
|
Music Gear News
At Gear4Music.com we update our news channel daily to ensure you always get the latest and most important music news.
Shakira kicks-off World Cup
Posted on 11 Jun 2010 10:00 to category : Music News
Black Eyed Peas, Shakira and Alicia Keys took to the stage in South Africa to open the World Cup last night (10.06.10).
The music stars performed for 40,000 people at the FIFA World Cup Kick-Off Celebration Concert in Johannesburg’s Orlando Stadium, in an evening inspired by African culture, including drumming, flag waving and dancing.
Archbishop Desmond Tutu thanked all the fans in attendance and welcomed them to the nation.
The Associated Press reports him telling the audience: "We welcome you all. For Africa is the cradle of humanity, so we welcome you home."
The performances by the three global acts were interspersed with songs by local musicians, including trumpeter Hugh Masekela, who opened the evening with 1968 chart-topper ‘Grazin’ in the Grass’.
The Black Eyed Peas – consisting of will.i.am, apl.de.ap, Taboo and Fergie - were the first global stars to take to the stage, where they performed a medley of their biggest hits.
They were followed by blind duo Amadou and Mariam from Mali, before ‘Hips Don’t Lie’ hitmaker Shakira – who was wearing an animal print dress - belted out the soccer tournament’s official track ‘Waka Waka (This Time for Africa)’.
‘Fallin’ hitmaker Alicia performed several tracks, while other artists who performed included South African stars Lira and BLK JKS and Colombian rocker Juanes, who performed his latest single ‘Yerbatero’.
Concertgoer Nana Masithela praised the “showcase” evening for being a positive event for black people.
She said: "This is a showcase. We are showcasing ourselves, to say, 'Blacks can do it!' "
The soccer tournament kicks off today (11.06.10) with a match between the host nation and Mex:
| 305,652
|
\begin{document}
\maketitle
\begin{abstract}
The theory of mean field games studies the limiting behaviors of large systems where the agents interact with each other in a certain symmetric way. The running and terminal costs are critical for the agents to decide the strategies. However, in practice they are often partially known or totally unknown for the agents, while the total cost is known at the end of the game. To address this challenging issue, we propose and study several inverse problems for mean field games. When the Lagrangian is a kinetic energy, we first establish unique identifiability results, showing that one can recover either the running cost or the terminal cost from knowledge of the total cost. If the running cost is limited to the time-independent class, we can further prove that one can simultaneously recover both the running and the terminal costs. Finally, we extend the results to the setup with general Lagrangians.
\end{abstract}
\tableofcontents
\section{Introduction}
The theory of mean field games (MFGs) was introduced and studied by Caines-Huang-Malham\'e \cite{HCM06,HCM071,HCM072,HCM073} and Lasry-Lions \cite{LL06a, LL06b, LL07a, Lions} independently in 2006. It has received considerable attentions in the literature in recent years. We refer to Lions \cite{Lions}, Cardaliaguet \cite{Cardaliaguet} and Bensoussan-Frehse-Yam \cite{BFY} for introductions of the subject in its early stage and Carmona-Delarue \cite{CarDel-I, CarDel-II} and Cardaliaguet-Porretta \cite{CarPor} for comprehensive accounts on the state-of-the-art developments in the literature.
\smallskip
We first briefly introduce the mathematical setup of our study and shall supplement more details in Section 2. In its typical formulation, an MFG can be described as follows. Let $n\in\mathbb{N}$ and the quotient space $\mathbb{T}^n:=\mathbb{R}^n\backslash\ \mathbb{Z}^n$ be the $n$-dimensional torus, which signifies a state space. Given $x\in\mathbb T^n$ and the flow of probability measures $\{\rho_t\}_{t\in [0,T]}$ on $\mathbb T^n$ with $\rho_0=m_0$, one aims at minimizing the cost functional over all the admissible closed-loop controls:
\begin{equation}\label{eq:mfgproblem}
J(x;\{\rho_t\}_{t\in[0,T]},\alpha)=\inf_{\alpha}\mathbb E\left\{\int_0^T L(X_t^{x,\alpha},\alpha(t, X_t^{x,\alpha}))+F(X_t^{x,\alpha},\rho_t)dt+G(X_T,\rho_T)\right\},
\end{equation}
such that
\begin{equation}\label{eq:mfgconstrain}
X_t^{x,\alpha}=x+\int_0^t\alpha(s,X_s^{x,\alpha})ds+\sqrt{2} B_s+\mathbb Z^n\quad\text{on $[0,T]$,}
\end{equation}
where $L:\mathbb T^n\times\mathbb R^n\to\mathbb R$ is a Lagrangian, $F:\mathbb T^n\times\mathcal{P}(\mathbb T^n)\to\mathbb R$ is a running cost and $G:\mathbb T^n\times\mathcal{P}(\mathbb T^n)\to\mathbb R$ is a terminal cost. We call $(\alpha^*,\{\rho_{t}^*\}_{t\in[0,T]})$ a mean field equilibrium if
\[
\rho_0^*=m_0\quad\text{and}\quad\alpha^*:=\arg\min_{\alpha}J(x;\{\rho_t^*\}_{t\in[0,T]},\alpha),
\]
and the law of $X_t^{\xi,\alpha^*}$ on $\mathbb T^n$ is $\rho_t^*$ where
\begin{equation}\label{eq:mfgpopulation}
X_t^{\xi,\alpha^*}=\xi_0+\int_0^{t}\alpha^*(s,X_s^{x,\alpha^*})ds+\sqrt{2}B_t+\mathbb Z^n\quad \text{on [0,T]},
\end{equation}
and its initial status $\xi_0$ is a random variable with the law $m_0$ on $\mathbb T^n$. The mean field equilibrium can be characterized by the following MFG system:
\begin{equation}\label{eq:mfg}
\left\{
\begin{array}{ll}
-\partial_t u(x,t) -\Delta u(t,x)+ H\big(x,\nabla u(x,t)\big)-F(x,t,m(x,t))=0,& {\rm{in}}\ \mathbb T^n\times (0,T),\medskip\\
\partial_tm(x,t)-\Delta m(x,t)-{\rm div} \big(m(x,t) \nabla_pH(x, \nabla u(x,t)\big)=0, & {\rm{in}}\ \mathbb T^n\times(0,T),\medskip\\
u(x,T)=G(x,m(x,T)),\ m(x,0)=m_0(x), & {\rm{in}}\ \mathbb T^n.
\end{array}
\right.
\end{equation}
In \eqref{eq:mfg}, $\Delta$ and ${\rm}$ are the Laplacian and divergent operators with respect to the $x$-variable, respectively. The Hamiltonian $H$ is the Legendre-Fenchel transform of the Lagrangian $L$ in \eqref{eq:mfgproblem}. Here, $H(x, \nabla u)=H(x, p)$ with $(x, p):=(x, \nabla u)\in \mathbb{T}^n\times\mathbb{R}^{n}$ being the canonical coordinates. In the physical setup, $u$ is the value function of each player; $m$ signifies the population distribution; $F$ is the running cost function which signifies the interaction between the agents and the population; $m_0$ represents the initial population distribution and $G$ signifies the terminal cost. All the functions involved are real valued and periodically extended from $\mathbb{T}^n$ to $\mathbb{R}^n$, which means that we are mainly concerned with periodic boundary conditions for the MFG system \eqref{eq:mfg}. The mean field equilibrium can be formally represented by $\alpha^*=-\nabla_pH(x,\nabla u(x,t))$. In Section~2 in what follows, we shall supplement more background introduction on the MFG system.
\smallskip
The well-posedness of the MFG system \eqref{eq:mfg} is well-understood in various settings. The first results date back to the original works of Lasry and Lions and have been presented in Lions \cite{Lions} and see also Caines-Huang-Malhame \cite{HCM06}. Many progresses have been made afterwards. Regarding $F$ and $G$, one can consider both nonlocal and local dependences on the measure $m$. The well-posedness of the MFG system \eqref{eq:mfg} is known in Cardaliaguet \cite{Cardaliaguet}, Cardaliaguet-Porretta \cite{CarPor}, Carmona-Delarue \cite{CarDel-I}, Meszaros-Mou \cite{MM} in the case of nonlocal data $F$ and $G$; and Ambrose\cite{Amb:18, Amb:21}, Cardaliaguet\cite{Car}, Cardaliaguet-Graber\cite{CarGra}, Cardaliaguet-Graber-Porretta-Tonon\cite{CarGraPorTon}, Cardaliaguet-Porretta \cite{CarPor}, Cirant-Gianni-Mannucci\cite{CirGiaMan}, Cirant-Goffi\cite{CirGof}, Ferreira-Gomez\cite{FerGom}, Ferreira-Gomez-Tada\cite{FerGomTad}, Gomez-Pimentel-Sanchez Morgado\cite{GomPimSan:15,GomPimSan:16}, Porretta \cite{Por} in the case that $F,G$ are locally dependent on the measure variable $m$.
\smallskip
We term the above well-posed MFG system \eqref{eq:mfg} to be the forward problem. In this paper, we are mainly concerned with the inverse problem of determining the running cost $F$ or the terminal cost $G$ by knowledge of the total cost associated with the above MFG system. To that end, we introduce a measurement map $\mathcal{M}_{F,G}$ as follows:
\begin{equation}\label{eq:M}
\mathcal{M}_{F, G}(m_0(x))=u(x,t)\big|_{t=0},\quad x\in\mathbb{T}^n,
\end{equation}
where $m_0(x)$ and $u(x, t)$ are given in the MFG system \eqref{eq:mfg}. That is, for a given pair of $F$ and $G$, $\mathcal{M}_{F, G}$ sends a prescribed initial population distribution $m_0$ to $u(x, 0)$, which signifies the total cost of the MFG \eqref{eq:mfg}. In Section 3, we shall show that $\mathcal{M}_{F,G}$ is well-defined in proper function spaces. The inverse problem mentioned above can be formulated as:
\begin{equation}\label{eq:ip1}
\mathcal{M}_{F,G}\longrightarrow F\ \mbox{or/and}\ G.
\end{equation}
In the mean field game theory, the running cost $F$ and the terminal cost $G$ are critical for the agents to decide the strategies. However, in practice they are often partially known or totally unknown for the agents, while the total cost $u(\cdot, 0)$ can be measured at the end of the game. This is a major motivation for us to propose and study the inverse problem \eqref{eq:ip1}. In this paper, we are mainly concerned with the unique identifiability issue, which is of primary importance for a generic inverse problem. In its general formulation, the unique identifiability asks whether one can establish the following one-to-one correspondence:
\begin{equation}\label{eq:ip2}
\mathcal{M}_{F_1, G_1}=\mathcal{M}_{F_2, G_2}\quad\mbox{if and only if}\quad (F_1, G_1)=(F_2, G_2),
\end{equation}
where $(F_j, G_j)$, $j=1,2$, are two configurations.
\smallskip
Unlike the forward problem of MFGs, the theory of the inverse problem has not yet been well-established. To the best of our knowledge, only some numerical studies have been conducted to the inverse problem of MFGs. It starts from the recent work Ding-Li-Osher-Yin \cite{DingLiOsherYin}. The authors reconstructed the running cost from the observation of the distribution of the population and the agents' strategy. The running cost consists of a kinetic energy (with an unknown underlining metric) and a convolution-type running cost. The main goal there is to numerically recover the underlining metric and the convolution kernel. Another numerical work Chow-Fung-Liu-Nurbekyan-Osher \cite{ChowFungLiuNurbekyanOsher} considered a different inverse problem of MFGs. The work focused on the recovery of the running cost from a finite number of the boundary measurements of the population profile and boundary movement. All the above MFG models consider the data which are nonlocally dependent on the measure variable $m$.
\smallskip
In our study of the inverse problem \eqref{eq:ip1}, we are mainly concerned with the data locally depending on the measure variable, i.e. $F(x, t, m(\cdot, t)):=F(x, t, m(x, t))$ and $G(x,m(\cdot, T)):=G(x,m(x, T))$. The model is motivated from the traffic flow and the crowd motion problems. For the problems, the cost depends only on the distribution of the population locally. We assume all the agents are rational and the observer only knows the total cost of agents at the end. The main goal is to recover the running or/and terminal costs. Let us briefly introduce the main results we prove in the paper. When the Lagrangian is a kinetic energy, we first show that the running cost $F$ is uniquely determined by the measurement map $\mathcal{M}_{F,G}$ by assuming the terminal cost $G$ is a-priori known. We then show that the terminal cost $G$ is uniquely identifiable by the measurement map $\mathcal{M}_{F,G}$ by assuming the running cost $F$ is a-priori known. We emphasize that, for both inverse problems, we assume that the running and the terminal costs satisfy $F(x, t, 0)=G(x,0)=0$ and we justify that the assumption is necessary for both unique identifiability problems. Moreover, the running cost is allowed to be time-dependent for both inverse problems. If the running cost is limited to the time-independent class, we further show that we can recover both the running and the terminal costs with the given measurement map $\mathcal{M}_{F,G}$. Finally, we extend a large extent of the above unique identifiability results to general Lagrangians. To establish those theoretical unique identifiability results, we develop novel mathematical strategies that make full use of the intrinsic structure of the MFG system. Our study opens up a new field of research on inverse problems for mean field games with many potential developments.
\medskip
The rest of the paper is organized as follows. We introduce the admissibility assumptions on $F$ and $G$ and state the main results of this paper in Section 2. In Section 3, we establish certain well-posedness results of the forward MFG system, which shall be needed for the inverse problems. We discuss the admissibility assumptions in Section 4. By counter examples, we show that those assumptions are unobjectionable for the inverse problems. Finally, we show various unique identifiability results in Section 5 and some generalizations in Section 6.
\section{Preliminaries and Statement of Main Results}
\subsection{Notations and Basic Setting}
As introduced earlier, we let $n\in\mathbb{N}$ and $\mathbb{T}^n:=\mathbb{R}^n\backslash\ \mathbb{Z}^n$ be the $n$-dimensional torus. Set $x=(x_1,x_2,\ldots, x_n)\in\mathbb{R}^n$.
If $f(x): x\in\mathbb{T}^n\to \mathbb{R}$ is smooth and $l=(l_1,l_2,...,l_n)\in\mathbb{N}_0^n$ is a multi-index with $\mathbb{N}_0:=\mathbb{N}\cup\{0\}$, then $D^lf$ stands for the
derivative $\frac{\p ^{l_1}}{\p x_1^{l_1}}...\frac{\p ^{l_n}}{\p x_n^{l_n}}f$. Given $\nu\in\mathbb{S}^{n-1}:=\{x\in\mathbb{R}^n; |x|=1\}$,
we also denote by $\p_{\nu}f$ the directional derivative of $f$ in the direction $\nu$. For $k\in\mathbb{N}_0$ and $\alpha\in [0,1)$, we say $f\in C^{k+\alpha} (k\in\mathbb{N}_0)$ if $D^lf$ exists and $\alpha$- H\"older continuous for any $l\in\mathbb{N}_0^n$ with $|l|\leq k$.
For functions $f:\mathbb T^n\times (0,T)\to\mathbb R$, we say $f$ belongs to $C^{k+\alpha,\frac{k+\alpha}{2}}$ if $D^l D_t^jf$ exists for any $l\in\mathbb{N}_0^n$ and $j\in\mathbb{N}_0$ with $ |l|+2j\leq k$ and
\[
\sup_{(x_1,t_1),(x_2,t_2)\in \mathbb T^n\times (0,T)}\frac{|D^lD_t^jf(x_1,t_1)-D^lD_t^jf(x_2,t_2)|}{|x_1-x_2|^\alpha+|t_1-t_2|^{\frac{\alpha}{2}}}<\infty,
\]
for any $l\in\mathbb N_0^n$ and $j\in\mathbb N_0$ with $|l|+2j=k$.
Throughout the paper, for a function $f$ define on $\mathbb{T}^n$ or $\mathbb{T}^n\times(0,T)$, it means that it is a periodic-$1$ function with respect to the space variable $x_j$, $1\leq j\leq N$. That is, it is a periodic-$(1,1,\ldots,1)$ function with respect to $x\in \mathbb{R}^n$.
\subsection{Mean Field Game}
Let $\mathcal{P}(\mathbb T^n)$ and $\mathcal{P}(\mathbb R^n)$ denote the set of probability measures on $\mathbb T^n$ and $\mathbb R^n$ respectively. Let $(\Omega,\mathscr{F},\mathbb F,\mathbb P)$ be a filtered probability space; $B$ be an $\mathbb F$-adapted Brownian motion on $\mathbb R^n$; and we assume $\mathscr{F}_0$ is rich enough to support $\mathcal{P}(\mathbb T^n)$. For any $\mathscr{F}$-measurable random variable $\xi$, we denote the law of $\xi$ on $\mathbb R^n$ by $\mathcal{L}_\xi\in\mathcal{P}(\mathbb R^n)$ and the law of $\xi$ on $\mathbb T^n$ by $\mathcal{L}_{\xi+\mathbb Z^n}\in\mathcal{P}(\mathbb T^n)$. Moreover, for any sub-$\sigma$-algebra $\mathcal{G}\subset \mathscr{F}$ and any $m\in\mathcal P(\mathbb T^n)$, $\mathbb M(\mathcal{G};m)$ denotes the set of $\mathcal{G}$-measurable random variables $\xi$ on $\mathbb R^n$ such that $\mathcal{L}_{\xi+\mathbb Z^n}=m$.
Our mean field game depends on the following data:
\[
L:\mathbb T^n\times\mathbb R^n\to\mathbb R,\quad F:\mathbb T^n\times\mathcal{P}(\mathbb T^n)\to\mathbb R\quad\text{and}\quad G:\mathbb T^n\times\mathcal{P}(\mathbb T^n)\to\mathbb R.
\]
Let $T>0$. For any $t_0\in [0,T]$, we let $\mathscr{A}_{t_0}$ denote the set of admissible controls $\alpha:[t_0,T]\times\mathbb T^n\to\mathbb R^d$ which are Borel measurable, and uniformly Lipschitz continuous in $x$. We also denote $B_t^{t_0}:=B_t-B_{t_0}$, $B_t^{0,t_0}:=B_t^0-B_{t_0}^0$, $t\in[t_0,T]$.
Given $x\in\mathbb T^n$, $\alpha\in\mathscr{A}_{t_0}$ and the flow of probability measures $\{\rho_t\}_{t\in[0,T]}\subset \mathcal{P}(\mathbb T^n)$ with $\rho_0=m_0$,
the state of an agent satisfies the following controlled SDE (stochastic differential equation) on $[t_0,T]$:
\begin{equation}\label{eq:ind}
X_t^{t_0,x,\alpha}=x+\int_{t_0}^t\alpha(s,X_s^{t_0,x,\alpha})ds+\sqrt{2}B_t^{t_0}+\mathbb Z^n.
\end{equation}
Consider the conditionally expected cost for the mean field game:
\begin{eqnarray}\label{eq:cost}
&&J(t_0,x;\{\rho_{t}\}_{t\in [0,T]},\alpha):=\inf_{\alpha\in\mathscr{A}_{t_0}}\mathbb E\Big[\int_{t_0}^TL(X_t^{t_0,x,\alpha},\alpha(t,X_t^{t_0,x,\alpha}))+F(X_t^{t_0,x,\alpha},\rho_t)dt\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad+G(X_T^{t_0,x,\alpha},\rho_T)\Big].
\end{eqnarray}
\begin{defi}
We say that $(\alpha^*,\{\rho_t^{*}\}_{t\in[0,T]})$ is a mean field equilibrium (MFE) if it satisfies the following properties:\\
(i) $\rho_0^*=m_0$;\\
(ii) for any $\xi_0\in \mathbb{M}(\mathcal{F}_0,m_0)$, we have $\mathcal{L}_{X_t^{0,\xi_0,\alpha^*}}=\rho_t^{*}$ where
\[
X_t^{0,\xi_0,\alpha^*}=\xi_0+\int_{0}^t\alpha^*(s,X_s^{0,\xi_0,\alpha^*})ds +\sqrt{2}B_t+\mathbb Z^n;
\]
(iii) for any $(t_0,x)\in[0,T]\times\mathbb T^n$, we have
\begin{equation*}
J(t_0,x;\{\rho_t^*\}_{t\in [0,T]},\alpha^*)=\inf_{\alpha\in\mathscr{A}_{t_0}}J(t_0,x;\{\rho_t^*\}_{t\in [0,T]},\alpha),\quad \text{for $\rho^*_{t_0}$-a.e. $x\in\mathbb T^n$.}
\end{equation*}
\end{defi}
When there is a unique MFE $(\alpha^*,\{\rho_t^{*}\}_{t\in[0,T]})$, then the mean field game leads to the following value function of the agent:
\[
u(t_0,x):=J(t_0,x;\{\rho_t^*\}_{t\in [0,T]},\alpha^*).
\]
Let $m(\cdot,t_0)=\rho_{t_0}^*$. Then $(u,m)$ solves the following mean field game system (cf. \cite{CarPor,Lions}):
\begin{equation}\label{main_equation}
\begin{cases}
\displaystyle{-\p_tu(x,t)-\Delta u(x,t)+\frac 1 2 |\nabla u(x, t)|^2= F(x,t,m(x,t)),}& \text{ in } \mathbb{T}^n\times(0,T),\smallskip\\
\p_t m(x,t)-\Delta m(x,t)-{\rm div}\big(m(x,t)\nabla u(x,t)\big)=0,&\text{ in }\mathbb{T}^n\times(0,T),\smallskip \\
u(x,T)=G(x,m(T,x)), \quad m(x,0)=m_0(x) & \text{ in } \mathbb{T}^n,
\end{cases}
\end{equation}
where as mentioned earlier, periodic boundary conditions are imposed on $\partial \mathbb{T}^n$ for $u$ and $m$.
\subsection{Inverse Problems}
Define the set
\[
\begin{split}
\mathcal{E}_{F,G}:=& \{m_0\in C^{2+\alpha}(\mathbb{T}^n) : \text{the system }\eqref{main_equation}\\
& \text{ has a unique solution in the sense described in Section 3 in what follows } \}.
\end{split}
\]
We introduce the following measurement map $\mathcal{M}_{F,G}$:
\begin{align}\label{eq:G}
\begin{split}
\mathcal{M}_{F,G}: \mathcal{E}_{F,G} & \rightarrow L^2(\mathbb{T}^n) , \\
m_0&\mapsto u(x,t) \Big|_{t=0},
\end{split}
\end{align}
where $u(x,t)$ is the solution of $\eqref{main_equation}$ with intinal data $m(x,0)=m_0(x).$
In the first setup of our study, we consider the case that $F$ and $G$ belong
to an analytic class. Henceforth, we set
\begin{equation}\label{eq:Q}
Q=\overline{\mathbb{T}^n\times(0,T) }.
\end{equation}
\begin{defi}\label{Admissible class1}
We say $U(x,t,z):\mathbb{T}^n\times \mathbb{R}\times\mathbb{C}\to\mathbb{C}$ is admissible, denoted by $U\in \mathcal{A}$, if it satisfies the following conditions:
\begin{enumerate}
\item[(i)]~The map $z\mapsto U(\cdot,\cdot,z)$ is holomorphic with value in $C^{2+\alpha,1+\frac{\alpha}{2}}(Q)$ for some $\alpha\in(0,1)$;
\item[(ii)] $U(x,t,0)=0$ for all $(x,t)\in\mathbb{T}^n\times (0,T).$
\end{enumerate}
Clearly, if (1) and (2) are fulfilled, then $U$ can be expanded into a power series as follows:
\begin{equation}\label{eq:F}
U(x,t,z)=\sum_{k=1}^{\infty} U^{(k)}(x,t)\frac{z^k}{k!},
\end{equation}
where $ U^{(k)}(x,t)=\frac{\p^k U}{\p z^k}(x,t,0)\in C^{2+\alpha,1+\frac{\alpha}{2}}(Q).$
\end{defi}
\begin{defi}\label{Admissible class2}
We say $U(x,z):\mathbb{T}^n\times\mathbb{C}\to\mathbb{C}$ is admissible, denoted by $U\in\mathcal{B}$, if it satisfies the following conditions:
\begin{enumerate}
\item[(i)] The map $z\mapsto U(\cdot,z)$ is holomorphic with value in $C^{2+\alpha}(\mathbb{T}^n)$ for some $\alpha\in(0,1)$;
\item[(ii)] $U(x,0)=0$ for all $x\in\mathbb{T}^n.$
\end{enumerate}
Clearly, if (1) and (2) are fulfilled, then $U$ can be expanded into a power series as follows:
\begin{equation}\label{eq:G}
U(x,z)=\sum_{k=1}^{\infty} U^{(k)}(x)\frac{z^k}{k!},
\end{equation}
where $ U^{(k)}(x)=\frac{\p^kU}{\p z^k}(x,0)\in C^{2+\alpha}(\mathbb{T}^n).$
\end{defi}
\begin{rmk}\label{rem:1}
The admissibility conditions in Definitions~\ref{Admissible class1} and \ref{Admissible class2} shall be imposed as a-priori conditions on the unknowns $F$ and $G$ in what follows for our inverse problem study. It is remarked that as noted earlier that both $F$ and $G$ are functions of real variables. However, for technical reasons, we extend the functions to the complex plane with respect to the $z$-variable, namely $U(\cdot,z)$ and $ U(\cdot,\cdot,z)$, and assume that they are holomorphic as functions of the complex variable $z$. This also means that we shall assume $F$ and $G$ are restrictions of those holomorphic functions to the real line. This technical assumption shall be used to show the well-posedness of the MFG system in section $\ref{section wp}.$ Throughout the paper, we also assume that in the series expansions \eqref{eq:F} and \eqref{eq:G}, the coefficient functions $U^{(k)}$ are real-valued.
\end{rmk}
\begin{rmk}
We would like to emphasise that the zero conditions, namely the admissibility conditions (ii) in Definitions~\ref{Admissible class1} and \ref{Admissible class2}, are unobjectionable to our inverse problem study. In fact, in Section~4 in what follows, we shall construct several MFG examples where the zero admissibility conditions are violated and the associated inverse problems have no unique identifiability results.
\end{rmk}
We are in a position to state the first unique recovery result for the inverse problem \eqref{eq:ip1}, which shows that one can recover either the running cost $F$ or the terminal cost $G$ from the measurement map $\mathcal{M}$. Here and also in what follows, we sometimes drop the dependence on $F, G$ of $\mathcal{M}$, and in particular in the case that one quantity is a-priori known, say e.g. $\mathcal{M}_F$ or $\mathcal{M}_G$, which should be clear from the context.
\begin{thm}\label{der F}
Assume $F_j \in\mathcal{A}$ ($j=1,2$), $G\in\mathcal{B}$. Let $\mathcal{M}_{F_j}$ be the measurement map associated to
the following system:
\begin{equation}\label{eq:mfg1}
\begin{cases}
-\p_tu(x,t)-\Delta u(x,t)+\frac 1 2 {|\nabla u(x,t)|^2}= F_j(x,t,m(x,t)),& \text{ in } \mathbb{T}^n\times(0,T),\medskip\\
\p_t m(x,t)-\Delta m(x,t)-{\rm div} (m(x,t) \nabla u(x,t)),&\text{ in } \mathbb{T}^n\times (0,T),\medskip\\
u(x,T)=G(x,m(x,T)), & \text{ in } \mathbb{T}^n,\medskip\\
m(x,0)=m_0(x), & \text{ in } \mathbb{T}^n.\\
\end{cases}
\end{equation}
If for any $m_0\in C^{2+\alpha}(\mathbb{T}^n)$, one has
$$\mathcal{M}_{F_1}(m_0)=\mathcal{M}_{F_2}(m_0),$$ then it holds that
$$F_1(x,t,z)=F_2(x,t,z)\ \text{ in } \mathbb{T}^n\times \mathbb{R}.$$
\end{thm}
\begin{thm}\label{der g}
Assume $F \in\mathcal{A}$, $G_j\in\mathcal{B}$ ($j=1,2$). Let $\mathcal{M}_{G_j}$ be the measurement map associated to
the following system:
\begin{equation}\label{eq:mfg2}
\begin{cases}
-\p_tu(x,t)-\Delta u(x,t)+\frac 1 2 {|\nabla u(x,t)|^2}= F(x,t,m(x,t)),& \text{ in } \mathbb{T}^n\times (0,T),\medskip\\
\p_t m(x,t)-\Delta m(x,t)-{\rm div}(m(x,t)\nabla u(x,t))=0,&\text{ in }\mathbb{T}^n\times(0,T),\medskip\\
u(x,T)=G_j(x,m(x,T)), & \text{ in } \mathbb{T}^n,\medskip\\
m(x,0)=m_0(x), & \text{ in } \mathbb{T}^n.\\
\end{cases}
\end{equation}
If for any $m_0\in C^{2+\alpha}(\mathbb{T}^n)$, one has
$$\mathcal{M}_{G_1}(m_0)=\mathcal{M}_{G_2}(m_0),$$ then it holds that
$$G_1(x,z)=G_2(x,z)\ \text{ in } \mathbb{T}^n\times \mathbb{R}.$$
\end{thm}
Notice that in Theorems~\ref{der F} and \ref{der g} we allow $F$ to depend on time. If we assume $n\geq 3$ and $F$ depends only on $x$ and $m(x,t)$, we can determine $F$ and $G$ simultaneously.
\begin{thm}\label{der F,g}
Assume $F_j,G_j \in\mathcal{B}$ ($j=1,2$). Let $\mathcal{M}_{F_j,G_j}$ be the measurement map associated to
the following system:
\begin{equation}\label{eq:mfg3}
\begin{cases}
-\p_tu(x,t)-\Delta u(x,t)+\frac 1 2 {|\nabla u(x,t)|^2}= F_j(x,m(x,t)),& \text{ in }\mathbb{T}^n\times(0,T),\medskip\\
\p_t m(x,t)-\Delta m(x,t)-{\rm div}(m(x,t)\nabla u(x,t))=0,&\text{ in }\mathbb{T}^n\times (0,T),\medskip\\
u(x,T)=G_j(x,m(x,T)), & \text{ in } \mathbb{T}^n,\medskip\\
m(x,0)=m_0(x), & \text{ in } \mathbb{T}^n.\\
\end{cases}
\end{equation}
If for any $m_0\in C^{2+\alpha}(\mathbb{T}^n)$, one has
$$\mathcal{M}_{F_1,G_1}(m_0)=\mathcal{M}_{F_2,G_2}(m_0),$$ then it holds that
$$(G_1(x,z),F_1(x,z))=(G_2(x,z),F_2(x,z)) \ \text{ in } \mathbb{T}^n\times \mathbb{R}.$$
\end{thm}
In Theorems~\ref{der F}, \ref{der g} and \ref{der F,g}, the Lagrangian is of a quadratic form, namely $H(x,\nabla u)$ in \eqref{eq:mfg} is of the form $\frac 1 2 |\nabla u|^2$ (see \eqref{eq:mfg1}--\eqref{eq:mfg3}). In fact, we can extend a large extent of the results in those theorems to the case with a general Lagrangian. We choose to postpone the statement of those results in Section~6 along with their proofs.
\section{Well-posedness of the forward problems}\label{section wp}
In this section, we show the well-posedness of the MFG systems in our study. The key point is the infinite differentiability of the equation with respect to a given (small) input $m_0(x).$ As a preliminary, we recall the well-posedness result
for linear parabolic equations \cite{Lady}\cite[Lemma 3.3]{ CarDelLasLio} .
\begin{lem}\label{linear app unique}
Consider the parabolic equation
\begin{equation}\label{linearapp wellpose}
\begin{cases}
-\p_tv(x,t)-\Delta v(x,t)+{\rm div} ( a(x,t)\cdot\nabla v(x,t))= f(x,t),& \text{ in }\mathbb{T}^n\times(0,T),\medskip\\
v(x,0)=v_0(x), & \text{ in } \mathbb{T}^n,
\end{cases}
\end{equation}
where the periodic boundary condition is imposed on $v$. Suppose $a,f\in C^{\alpha,\frac{\alpha}{2}}(Q) $ and $v_0\in C^{2+\alpha}(\mathbb{T}^n)$, then $\eqref{linearapp wellpose}$ has a unique classical solution $v\in C^{2+\alpha,1+\frac{\alpha}{2}}(Q).$
\end{lem}
The following result is somewhat standard (especially Theorem \ref{local_wellpose}-(a)), while our technical conditions could be different from those in the literature. For completeness we provide a proof here. The idea is to differentiate the equation infinitely many times with respect to the (small) input $m_0(x)$. We recall that $Q$ is defined in \eqref{eq:Q} and periodic boundary conditions are imposed to the MFG systems.
\begin{thm}\label{local_wellpose}
Suppose that $F\in\mathcal{A}$ and $G\in\mathcal{B}$. The following results holds:
\begin{enumerate}
\item[(a)]
There exist constants $\delta>0$ and $C>0$ such that for any
\[
m_0\in B_{\delta}(C^{2+\alpha}(\mathbb{T}^n)) :=\{m_0\in C^{2+\alpha}(\mathbb{T}^n): \|m_0\|_{C^{2+\alpha}(\mathbb{T}^n)}\leq\delta \},
\]
the MFG system $\eqref{main_equation}$ has a solution $u \in
C^{2+\alpha,1+\frac{\alpha}{2}}(Q)$ which satisfies
\begin{equation}\label{eq:nn1}
\|(u,m)\|_{ C^{2+\alpha,1+\frac{\alpha}{2}}(Q)}:= \|u\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(Q)}+ \|m\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(Q)}\leq C\|m_0\|_{ C^{2+\alpha}(\mathbb{T}^n)}.
\end{equation}
Furthermore, the solution $(u,m)$ is unique within the class
\begin{equation}\label{eq:nn2}
\{ (u,m)\in C^{2+\alpha,1+\frac{\alpha}{2}}(Q)\times C^{2+\alpha,1+\frac{\alpha}{2}}(Q): \|(u,m)\|_{ C^{2+\alpha,1+\frac{\alpha}{2}}(Q)}\leq C\delta \}.
\end{equation}
\item[(b)] Define a function
\[
S: B_{\delta}(C^{2+\alpha}(\mathbb{T}^n))\to C^{2+\alpha,1+\frac{\alpha}{2}}(Q)\times C^{2+\alpha,1+\frac{\alpha}{2}}(Q)\ \mbox{by $S(m_0):=(u,v)$}.
\]
where $(u,v)$ is the unique solution to the MFG system \eqref{main_equation}.
Then for any $m_0\in B_{\delta}(C^{2+\alpha}(\mathbb T^n))$, $S$ is holomorphic.
\end{enumerate}
\end{thm}
\begin{proof}
Let
\begin{align*}
&X_1:= C^{2+\alpha}(\mathbb{T}^n ), \\
&X_2:=C^{2+\alpha,1+\frac{\alpha}{2}}(Q)\times C^{2+\alpha,1+\frac{\alpha}{2}}(Q),\\
&X_3:=C^{2+\alpha}(\mathbb{T}^n)\times C^{2+\alpha}(\mathbb{T}^n)\times C^{\alpha,\frac{\alpha}{2}}(Q )\times C^{\alpha,\frac{\alpha}{2}}(Q ),
\end{align*} and we define a map $\mathscr{K}:X_1\times X_2 \to X_3$ by that for any $(m_0,\tilde u,\tilde m)\in X_1\times X_2$,
\begin{align*}
&
\mathscr{K}( m_0,\tilde u,\tilde m)(x,t)\\
:=&\big( \tilde u(x,T)-G(x,\tilde m(x,T)), \tilde m(x,0)-m_0(x) ,
-\p_t\tilde u(x,t)-\Delta \tilde u(x,t)\\ &+\frac{|\nabla \tilde u(x,t)|^2}{2}- F(x,t,\tilde m(x,t)),
\p_t \tilde m(x,t)-\Delta \tilde m(x,t)-{\rm div}(\tilde m(x,t)\nabla \tilde u(x,t)) \big) .
\end{align*}
First, we show that $\mathscr{K} $ is well-defined. Since the
H\"older space is an algebra under the point-wise multiplication, we have $|\nabla u|^2, {\rm div}(m(x,t)\nabla u(x,t)) \in C^{\alpha,\frac{\alpha}{2}}(Q ).$
By the Cauchy integral formula,
\begin{equation}\label{eq:F1}
F^{(k)}\leq \frac{k!}{R^k}\sup_{|z|=R}\|F(\cdot,\cdot,z)\|_{C^{\alpha,\frac{\alpha}{2}}(Q ) },\ \ R>0.
\end{equation}
Then there is $L>0$ such that for all $k\in\mathbb{N}$,
\begin{equation}\label{eq:F2}
\left\|\frac{F^{(k)}}{k!}m^k\right\|_{C^{\alpha,\frac{\alpha}{2}}(Q )}\leq \frac{L^k}{R^k}\|m\|^k_{C^{\alpha,\frac{\alpha}{2}}(Q )}\sup_{|z|=R}\|F(\cdot,\cdot,z)\|_{C^{\alpha,\frac{\alpha}{2}}(Q ) }.
\end{equation}
By choosing $R\in\mathbb{R}_+$ large enough and by virtue of \eqref{eq:F1} and \eqref{eq:F2}, it can be seen that the series \eqref{eq:F} converges in $C^{\alpha,\frac{\alpha}{2}}(Q )$ and therefore $F(x,m(x,t))\in C^{\alpha,\frac{\alpha}{2}}(Q ).$ Similarly, we have $G(x,m(x,T))\in C^{2+\alpha}(\mathbb{T}^n).$
This implies that $\mathscr{K} $ is well-defined.
Let us show that $\mathscr{K}$ is holomorphic. Since $\mathscr{K}$ is clearly locally bounded, it suffices to verify that it is weakly holomorphic. That is we aim to show the map
$$\lambda\in\mathbb C \mapsto \mathscr{K}((m_0,\tilde u,\tilde m)+\lambda (\bar m_0,\bar u,\bar m))\in X_3,\quad\text{for any $(\bar m_0,\bar u,\bar m)\in X_1\times X_2$}$$
is holomorphic. In fact, this follows from the condition that $F\in\mathcal{A}$ and $G\in\mathcal{B}$.
Note that $ \mathscr{K}(0,0,0)=0$. Let us compute $\nabla_{(\tilde u,\tilde m)} \mathscr{K} (0,0,0)$:
\begin{equation}\label{Fer diff}
\begin{aligned}
\nabla_{(\tilde u,\tilde m)} \mathscr{K}(0,0,0) (u,m) =(& u|_{t=T}-G^{(1)}m(x,T), m|_{t=0}, \\
&-\p_tu(x,t)-\Delta u(x,t)-F^{(1)}m, \p_t m(x,t)-\Delta m(x,t)).
\end{aligned}
\end{equation}
By Lemma $\ref{linear app unique}$,
If $\nabla_{(\tilde u,\tilde m)} \mathscr{K} (0,0,0)=0$, we have
$ \tilde m=0$ and then $\tilde u=0$. Therefore, the map is injective.
On the other hand, letting $ (r(x),s(x,t))\in C^{2+\alpha}(\mathbb{T}^n)\times C^{\alpha,\frac{\alpha}{2}}(Q ) $,
and by Lemma $\ref{linear app unique}$, there exists $a(x,t)\in C^{2+\alpha,1+\frac{\alpha}{2}}(Q)$ such that
\begin{equation*}
\begin{cases}
\p_t a(x,t)-\Delta a(x,t)=s(x,t) &\text{ in } \mathbb{T}^n,\medskip\\
a(x,0)=r(x) & \text{ in } \mathbb{T}^n .
\end{cases}
\end{equation*}
Then letting $ (r'(x),s'(x,t))\in C^{2+\alpha}(\mathbb{T}^n)\times C^{\alpha,\frac{\alpha}{2}}(Q ) $, one can show that there exists $ b(x,t)\in C^{2+\alpha,1+\frac{\alpha}{2}}(Q)$ such that
\begin{equation*}
\begin{cases}
-\p_t b(x,t)-\Delta b(x,t)-F^{(1)}a=s'(x,t) &\text{ in } \mathbb{T}^n,\medskip\\
b(x,T)=G^{(1)}a(x,T)+r'(x) & \text{ in } \mathbb{T}^n.
\end{cases}
\end{equation*}
Therefore, $\nabla_{(\tilde u,\tilde m)} \mathscr{K} (0,0,0)$ is a linear isomorphism between $X_2$ and $X_3$. Hence, by the implicit function theorem, there exist $\delta>0$ and a unique holomorphic function $S: B_{\delta}(\mathbb{T}^n)\to X_2$ such that $\mathscr{K}(m_0,S(m_0))=0$ for all $m_0\in B_{\delta}(\mathbb{T}^n) $.
By letting $(u,m)=S(m_0)$, we obtain the unique solution of the MFG system \eqref{main_equation}. Let $ (u_0,v_0)=S(0)$. Since $S$ is Lipschitz, we know that there exist constants $C,C'>0$ such that
\begin{equation*}
\begin{aligned}
&\|(u,m)\|_{ C^{2+\alpha,1+\frac{\alpha}{2}}(Q)^2}\\
\leq& C'\|m_0\|_{B_{\delta}(\mathbb{T}^n)} +\|u_0\|_ { C^{2+\alpha,1+\frac{\alpha}{2}}(Q)}+\|v_0\|_{ C^{2+\alpha,1+\frac{\alpha}{2}}(Q)}\\
\leq& C \|m_0\|_{B_{\delta}(\mathbb{T}^n)}.
\end{aligned}
\end{equation*}
The proof is complete.
\end{proof}
\begin{rmk}
Regarding the local well-posedness, several remarks are in order.
\begin{enumerate}
\item[(a)] The conditions on $F$ and $G$ (Definition \ref{Admissible class1}-(i) and $G$ satisfies Definition \ref{Admissible class2}-(i) ) are not essential and it is for convenience to
apply implicit function theorem . Also, the analytic conditions on $F$ and $G$ can be replayed by weaker regularity conditions in the proof of the local well-posedness \cite{Lions} , but these conditions will be utilized in our
inverse problem study.
\item[(b)] In order to apply the higher order linearization method that shall be developed in Section 5 for the inverse problems, we need the infinite differentiability of the equation with respect to the given input $m_0(x)$, it is shown by the fact that the solution map $S$ is holomorphic.
\item[(c)] In the proof of Theorem $\ref{local_wellpose}$, we show the solution map $S$ is holomorphic. As a corollary, the measurement map $\mathcal{M}=(\pi_1\circ S)\Big|_{t=0}$ is also holomorphic, where $\pi_1$ is the projection map with respect to the first variable.
\end{enumerate}
\end{rmk}
\section{Non-uniqueness and discussion on the zero admissibility conditions}
In this section, we show that the zero admissibility conditions, namely $F(x,t, 0)=0$ and $G(x,0)=0$ in Definitions~\ref{Admissible class1} and \ref{Admissible class2} are unobjectionably necessary if one intends to uniquely recover $F$ or $G$ by knowledge of the measurement operator $\mathcal{M}_{F, G}$ for the inverse problem \eqref{eq:ip1}. For simplicity, we only consider the case that the space dimension $n=1$ without the periodic boundary conditions. That is, we consider the following MFG system:
\begin{equation}\label{dim1}
\begin{cases}
-\p_tu_j(x,t)-\p_{xx} u_j(x,t)+\frac 1 2 {|\p_x u_j(x)|^2}= F_j(x,t,v_j(x,t)),& \text{ in } \mathbb{R}\times (0,T),\medskip\\
\p_t v_j(x,t)-\p_{xx} v_j(x,t)-\p_x(v_j(x,t)\p_x u_j(x,t))=0,&\text{ in } \mathbb{R}\times(0,T),\medskip\\
u_j(x,T)= G_j(x,v_j(x,T)), & \text{ in } \mathbb{R},\medskip\\
v_j(x,0)=m_0(x), & \text{ in } \mathbb{R}.\\
\end{cases}
\end{equation}
Furthermore, we assume $T$ is small enough such that the solution of the MFG system \eqref{dim1} is unique \cite{Amb:18,Amb:21,Amb22,Cira,Lions}. In what follows, we construct examples to show that if the zero admissibility conditions are violated then the corresponding inverse problems do not have uniqueness.
\begin{prop}
Consider the system $\eqref{dim1}$. There exist $F_1=F_2\in C^{\infty}(\mathbb{R}\times\mathbb{R}\times\mathbb{R})$ and $G_1\neq G_2\in C^{\infty}(\mathbb{R}\times\mathbb{R})$ (but we do not have $G_1(x,0)=G_2(x,0)=0$) such that the corresponding two systems admit the same measurement map, i.e. $\mathcal{M}_{G_1}=\mathcal{M}_{G_2}$.
\end{prop}
\begin{proof}
Set
\[
F_1=F_2=-\sin(x)+\frac{1}{4}(e^t-1)^2\cos^2(x),
\]
and
\[
G_1=(e^T-1)\sin(x),\quad G_2=(1-e^T)\sin(x).
\]
It can be directly verified that
\[
u_1(x,t)=(e^t-1)\sin(x)\quad\mbox{and}\quad u_2(x,t)=(1-e^t)\sin(x),
\]
satisfy the corresponding system. In this case, we have $\mathcal{M}_{G_1}(m_0)=\mathcal{M}_{G_2}(m_0)=0$ for any admissible $m_0$.
\end{proof}
\begin{prop}\label{2}
Consider the syetem $\eqref{dim1}$. There exist $G_1=G_2\in C^{\infty}(\mathbb{R}\times\mathbb{R})$ and $F_1\neq F_2\in C^{\infty}(\mathbb{R}\times\mathbb{R}\times\mathbb{R})$ (but we do not have $F_j(x,t,0)=0$, $j=1,2$) such that the corresponding two systems admit the same measurement map, i.e. $\mathcal{M}_{F_1}=\mathcal{M}_{F_2}$.
\end{prop}
\begin{proof}
Set
\[
F_1=-x(2t-T)+\frac{t^2(t-T)^2}{2},\quad F_2=-2x(2t-T)+2t^2(t-T)^2,
\]
and
\[
G_1=G_2=0.
\]
Here, it is noted that $F_1$ and $F_2$ are independent of $v$. In such a case, it is straightforward to verify that $u_j(x,t)=jxt(t-T)$ is the solution of the corresponding system \eqref{dim1}. Clearly, one has $\mathcal{M}_{F_1}(m_0)=\mathcal{M}_{F_2}(m_0)=0$ for any admissible $m_0$.
\end{proof}
Moreover, we can find $F_1, F_2\in C^{\infty}(\mathbb{R}\times\mathbb{R})$ which are independent of $t$ such that Proposition $\ref{2}$ holds.
\begin{proof}
Define
$$Lu_j:=-\p_tu(x,t)-\p_{xx} u(x,t)+\frac{|\p_x u|^2}{2}.$$
It is sufficient for us to show that there exist $u_1(x,t),u_2(x,t)$ such that
\begin{enumerate}
\item[(1)] $L u_1\neq L u_2$ and $\p_t (Lu_j)=0 $ for $ j=1,2$;
\item[(2)] $u_1(x,0)=u_2(x,0)$ and $u_1(x,T)=u_2(x,T)$.
\end{enumerate}
In fact, if this is true, we can set $F_j= Lu_j$ and $G(x)=u_1(x,T)$. Then one has $G_1=G_2.$
Without loss of generality, we assume $T=1.$
Let $p(t)$ be a non-zero solution of the following ordinary differential equation (ODE):
\begin{equation*}
( \ln( p'(t)))'=\frac{\sqrt{1+4t}}{2},
\end{equation*}
and $q(t)$ be a solution of the ODE:
\begin{equation*}
\begin{cases}
&2q'(t)+\sqrt{1+4t}\, q''(t)=p(t)p'(t)\sqrt{1+4t},\medskip\\
&q(0)=0.
\end{cases}
\end{equation*}
With $p(t)$ and $q(t)$ given above, we can set
\[
u_1(x,t)=p(t(t-1))x+q(t(t-1))\quad\mbox{and}\quad u_2(x,t)=q(t(t-1))x+2q(t(t-1)).
\]
It can be directly verified that $u_1$ and $u_2$ fulfil the requirements (1) and (2) stated above.
\end{proof}
Finally, we would like to remark that by following a similar spirit, one may construct similar examples as those in Proposition 4.1 and 4.2 to the MFG system \eqref{dim1} associated with a periodic boundary condition. However, this shall involve a bit more tedious calculations and is not the focus of the current study. We choose not to explore further. As also stated earlier, it is unobjectionable to see that the zero admissibility conditions are necessary for the inverse problem study.
\section{Proofs of Theorems~\ref{der F}, \ref{der g} and \ref{der F,g}}
In this section, we present the proofs of the three main theorems, namely Theorems~\ref{der F}, \ref{der g} and \ref{der F,g}.
To that end, we first introduce a higher order linearization procedure associated with the MFG system \eqref{main_equation} which shall be repeatedly used in the proofs. We also refer to \cite{LLLZ} where a higher order linearization procedure was considered for a semilinear parabolic equation.
Throughout the current section, if $f$ is a function defined on $\mathbb{T}^n$, we still use $f$ to denote the function obtained by extending $f$ to $\mathbb{R}^n$ periodically.
\subsection{Higher-order linearization}\label{HLM}
This method depends on the infinite differentiability of the solution with respect to a given input $m_0(x)$, which was derived in Theorem~$\ref{local_wellpose}$.
First, we introduce the basic setting of this higher order
linearization method. Consider the system $\eqref{main_equation}$. Let
$$m_0(x;\varepsilon)=\sum_{l=1}^{N}\varepsilon_lf_l,$$
where $f_l\in C^{2+\alpha}(\mathbb{T}^n)$ and $\varepsilon=(\varepsilon_1,\varepsilon_2,...,\varepsilon_N)\in\mathbb{R}^N$ with
$|\varepsilon|=\sum_{l=1}^{N}|\varepsilon_l|$ small enough. Then by Theorem $\ref{local_wellpose}$, there exists a unique solution $(u(x,t;\varepsilon),m(x,t;\varepsilon) )$ of $\eqref{main_equation}$. Let $(u(x,t;0),m(x,t;0) ) $ be the solution of $\eqref{main_equation}$ when $\varepsilon=0.$
Let
$$u^{(1)}:=\p_{\varepsilon_1}u|_{\varepsilon=0}=\lim\limits_{\varepsilon\to 0}\frac{u(x,t;\varepsilon)-u(x,t;0) }{\varepsilon_1},$$
$$m^{(1)}:=\p_{\varepsilon_1}m|_{\varepsilon=0}=\lim\limits_{\varepsilon\to 0}\frac{m(x,t;\varepsilon)-m(x,t;0) }{\varepsilon_1}.$$
The idea is that we consider a new system of $(u^{(1)},m^{(1)}).$ If $F\in\mathcal{A}$, $g\in\mathcal{B}$, we have
\[
(u(x,t;0),m(x,t;0) )=(0,0)
\]
and hence
\begin{align*}
&-\p_tu^{(1)}(x,t)-\Delta u^{(1)}(x,t)\\
=& \lim\limits_{\varepsilon\to 0}\frac{1}{\varepsilon_1}[\frac{|\nabla u(x,t;\varepsilon)|^2-|\nabla u(x,t;0)|^2}{2}+ F(x,t,m(x,t;\varepsilon))-F(x,t;m(x,t;0)) ]\\
=&\nabla u^{(1)}\cdot (\lim\limits_{\varepsilon\to 0}\frac{\nabla u(x,t;\varepsilon)+\nabla u(x,t;0)}{2})+ F^{(1)}(x,t)( m(x,t;\varepsilon)-m(x,t;0))\\
=&F^{(1)}(x,t)( m(x,t;\varepsilon)-m(x,t;0)).
\end{align*}
Similary, we can compute
\begin{align*}
&\p_t m^{(1)}(x,t)-\Delta m^{(1)}(x,t)\\
=&\lim\limits_{\varepsilon\to 0}\frac{1}{\varepsilon_1}[{\rm div} ( m(x,t;\varepsilon)\nabla u(x,t;\varepsilon)-m(x,t;0)\nabla u(x,t;0) )]\\
=&\lim\limits_{\varepsilon\to 0}\frac{1}{\varepsilon_1} [ \nabla m(x,t;\varepsilon)\cdot\nabla u(x,t;\varepsilon)+m(x,t;\varepsilon)\Delta u(x,t;\varepsilon) -\\
& \nabla m(x,t;0)\cdot\nabla u(x,t;0)-m(x,t;0)\Delta u(x,t;0) ]\\
=&0.
\end{align*}
Now, we have that $(u_{j}^{(1)},m_{j}^{(1)} )$ satisfies the following system:
\begin{equation}\label{linear l=1,eg}
\begin{cases}
-\p_tu^{(1)}(x,t)-\Delta u^{(1)}(x,t)= F^{(1)}(x,t)m^{(1)}(x,t),& \text{ in } \mathbb{T}^n\times (0,T),\medskip\\
\p_t m^{(1)}(x,t)-\Delta m^{(1)}(x,t)=0,&\text{ in }\mathbb{T}^n\times (0,T),\medskip\\
u^{(1)}_j(x,T)=G^{(1)}(x)m^{(1)}(x,T), & \text{ in } \mathbb{T}^n,\medskip\\
m^{(1)}_j(x,0)=f_1(x). & \text{ in } \mathbb{T}^n.\\
\end{cases}
\end{equation}
Then we can define $$u^{(l)}:=\p_{\varepsilon_l}u|_{\varepsilon=0}=\lim\limits_{\varepsilon\to 0}\frac{u(x,t;\varepsilon)-u(x,t;0) }{\varepsilon_l},$$
$$m^{(l)}:=\p_{\varepsilon_l}m|_{\varepsilon=0}=\lim\limits_{\varepsilon\to 0}\frac{m(x,t;\varepsilon)-m(x,t;0) }{\varepsilon_l},$$
for all $l\in\mathbb{N}$ and obtain a sequence of similar systems.
In the proof of Theorem $\ref{der F}$ in what follows, we recover the first Taylor coefficient of $F$ or $G$ by considering this new system $\eqref{linear l=1,eg}$. In order to recover the higher order Taylor coefficients,
we consider
\begin{equation}\label{eq:ht1}
u^{(1,2)}:=\p_{\varepsilon_1}\p_{\varepsilon_2}u|_{\varepsilon=0},
m^{(1,2)}:=\p_{\varepsilon_1}\p_{\varepsilon_2}m|_{\varepsilon=0}.
\end{equation}
By direct calculations, we have from \eqref{eq:ht1} that
\begin{equation}\label{eq:ht2}
\begin{split}
&-\p_tu^{(1,2)}(x,t)-\Delta u^{(1,2)}(x,t)\\
=& -\nabla u^{(1)}\cdot \nabla u^{(2)}-\nabla u^{(1,2)}\cdot \nabla u(x,t;0) \\
& +F_j^{(1)}m^{(1,2)}+F^{(2)}_j(x,t)m^{(1)}m^{(2)},
\end{split}
\end{equation}
and
\begin{equation}\label{eq:ht3}
\begin{split}
&\p_t m^{(1,2)}(x,t)-\Delta m^{(1,2)}(x,t)\\
=& \p_{\varepsilon_1}\p_{\varepsilon_2}{\rm div} (m\nabla u)|_{\varepsilon=0}\medskip\\
=&\nabla m^{(1,2)}\nabla u(x,t;0)+\nabla m(x,t;0)\nabla u^{(1,2)}+m^{(1,2)}\Delta u(x,t;0)+m(x,t;0)\Delta u^{(1,2)}\medskip\\
&+ {\rm div} (m^{(1)}\nabla u^{(2)})+{\rm div}(m^{(2)}\nabla u^{(1)})\medskip\\
=& {\rm div} (m^{(1)}\nabla u^{(2)})+{\rm div}(m^{(2)}\nabla u^{(1)}).
\end{split}
\end{equation}
Combining \eqref{eq:ht2} and \eqref{eq:ht3}, we have the second order linearization as follows:
\begin{equation}\label{linear l=1,2 eg}
\begin{cases}
-\p_tu^{(1,2)}-\Delta u^{(1,2)}(x,t)+\nabla u^{(1)}\cdot \nabla u^{(2)}\medskip\\
\hspace*{3cm}= F^{(1)}(x,t)m^{(1,2)}+F^{(2)}(x,t)m^{(1)}m^{(2)},& \text{ in } \mathbb{T}^n\times(0,T),\medskip\\
\p_t m^{(1,2)}-\Delta m^{(1,2)}= {\rm div} (m^{(1)}\nabla u^{(2)})+{\rm div}(m^{(2)}\nabla u^{(1)}) ,&\text{ in } \mathbb{T}^n\times (0,T),\medskip\\
u^{(1,2)}(x,T)=G^{(1)}(x)m^{(1,2)}(x,T)+G^{(2)}(x)m^{(1)}m^{(2)}(x,T), & \text{ in } \mathbb{T}^n,\medskip\\
m^{(1,2)}(x,0)=0, & \text{ in } \mathbb{T}^n.\\
\end{cases}
\end{equation}
Notice that the non-linear terms of the system $\eqref{linear l=1,2 eg}$ depend on the first order linearised system $\eqref{linear l=1,eg}$. This shall be an important ingredient in the proofs of Theorem $\ref{der F}$ and $\ref{der g}$ in what follows.
Inductively, for $N\in\mathbb{N}$, we consider
\begin{equation*}
u^{(1,2...,N)}=\p_{\varepsilon_1}\p_{\varepsilon_2}...\p_{\varepsilon_N}u|_{\varepsilon=0},
\end{equation*}
\begin{equation*}
m^{(1,2...,N)}=\p_{\varepsilon_1}\p_{\varepsilon_2}...\p_{\varepsilon_N}m|_{\varepsilon=0}.
\end{equation*}
we can obtain a sequence of parabolic systems, which shall be employed again in determining the higher order Taylor coefficients of the unknowns $F$ and $G$.
\subsection{Unique determination of single unknown function}
We first present the proofs of Theorems $\ref{der F}$ and $\ref{der g}$.
\begin{proof}[Proof of Theorem $\ref{der F}$]
Consider the following systems for $j=1, 2$:
\begin{equation}\label{j=1,2for F}
\begin{cases}
-\p_tu_j(x,t)-\Delta u_j(x,t)+\frac 1 2{|\nabla u_j(x,t)|^2}= F_j(x,t,m_j(x,t)),& \text{ in }\mathbb{T}^n\times(0,T),\medskip\\
\p_t m_j(x,t)-\Delta m_j(x,t)-{\rm div}(m_j(x,t)\nabla u_j(x,t))=0,&\text{ in } \mathbb{T}^n\times(0,T),\medskip\\
u_j(x,T)=G(x,m_j(x,T)), & \text{ in } \mathbb{T}^n,\medskip\\
m_j(x,0)=m_0(x), & \text{ in } \mathbb{T}^n.
\end{cases}
\end{equation}
Following the discussion in Section $\ref{HLM}$, we consider the case $N=1.$ Let
$$u_{j}^{(1)}:=\p_{\varepsilon_1}u_{j}|_{\varepsilon=0}=\lim\limits_{\varepsilon\to 0}\frac{u_j(x,t;\varepsilon)-u_j(x,t;0) }{\varepsilon_1},$$
$$m_{j}^{(1)}:=\p_{\varepsilon_1}m_{j}|_{\varepsilon=0}=\lim\limits_{\varepsilon\to 0}\frac{m_j(x,t;\varepsilon)-m_j(x,t;0) }{\varepsilon_1}.$$
Direct computations imply that $(u_{j}^{(1)},m_{j}^{(1)} )$ satisfies the following system:
\begin{equation}\label{linear l=1}
\begin{cases}
-\p_tu_j^{(1)}(x,t)-\Delta u^{(1)}_j(x,t)= F^{(1)}_j(x,t)m_j^{(1)}(x,t),& \text{ in } \mathbb{T}^n\times (0,T),\medskip\\
\p_t m^{(1)}_j(x,t)-\Delta m^{(1)}_j(x,t)=0,&\text{ in }\mathbb{T}^n\times (0,T),\medskip\\
u^{(1)}_j(x,T)=G^{(1)}(x)m^{(1)}_j(x,T), & \text{ in } \mathbb{T}^n,\medskip\\
m^{(1)}_j(x,0)=f_1(x), & \text{ in } \mathbb{T}^n.\\
\end{cases}
\end{equation}
We extend $f_l$ to $\mathbb{R}^n$ periodically, and still denote it by $f_l$. Then we can solve $\eqref{linear l=1}$ by solving $m^{(1)}_j$ first and then obtain $u^{(1)}_j.$ The solution is
\begin{equation}\label{solution m F}
m_j^{(1)}(x,t)= \int_{\mathbb{R}^n}\Phi(x-y,t)f_{1}(y)\, dy,
\end{equation}
\begin{equation}\label{solution u F}
\begin{aligned}
u_j^{(1)}(x,t)&= \int_{\mathbb{R}^n}\Phi(x-y,T-t)G^{(1)}(y)m^{(1)}_j(y,T) )\, dy\\
&+\int_{0}^{T-t}\int_{\mathbb{R}^n}\Phi(x-y,T-t-s)F^{(1)}_j(y,T-s)\overline{m}_j^{(1)}(y,s)\, dyds,
\end{aligned}
\end{equation}
where $\overline{m}_j^{(1)}(x,t)= m_j^{(1)}(x,T-t)$ and $\Phi$ is the fundamental solution of heat equation:
\begin{equation}\label{eq:fund1}
\Phi(x,t)= \frac{1}{(4\pi t)^{n/2}}e^{-\frac{|x|^2}{4t}}.
\end{equation}
Since $\mathcal{M}_{F_1}=\mathcal{M}_{F_2}$, we have $$ u_1^{(1)}(x,0)=u_2^{(1)}(x,0),$$ for all $f_1\in C^{2+\alpha}(\mathbb{T}^n).$ Then $\eqref{solution m F}$ and $\eqref{solution u F}$ readily yield that
\begin{equation}\label{eq:dd1}
\int_{0}^{T}\int_{\mathbb{R}^n}\Phi(x-y,T-s)(F^{(1)}_1(y,T-s)-F^{(1)}_2(y,T-s))\overline{m}_1^{(1)}(y,s)\, dyds= 0.
\end{equation}
We can rewrite \eqref{eq:dd1} as
\begin{equation}\label{eq:dd2}
\int_{0}^{T}[\Phi*((F^{(1)}_1-F^{(1 )}_2)m_1^{(1)})](x,T-s) ds=0,
\end{equation}
which holds for all $f_1\in C^{2+\alpha}(\mathbb{T}^n),$ and hence for all solution $ m_1^{(1)}$ of $\eqref{linear l=1}.$
Recall the periodic Fourier transform :
\begin{equation}\label{eq:fourier}
\hat{\varphi}(\boldsymbol {\xi})=\mathcal{F}(\varphi)(\boldsymbol {\xi}):=\int_{\mathbb{T}^n} \varphi(x) e^{-\mathrm{i}2\pi\boldsymbol {\xi}\cdot x}\, dx;\quad
\end{equation}
where $\mathrm{i}:=\sqrt{-1}$, $\boldsymbol {\xi}\in\mathbb{Z}^n$. Applying the above Fourier transform $\mathcal{F}$ to the both sides of \eqref{eq:dd2} (with respect to the space variable), we can obtain
\begin{equation}\label{eq:dd3}
\int_{0}^{T}[\mathcal{F}(\Phi)\mathcal{F}((F^{(1)}_1-F^{(1 )}_2)m_1^{(1)})](\boldsymbol {\xi},T-s) ds=0,
\end{equation}
for all $\boldsymbol {\xi}\in\mathbb{Z}^n$.
Since $F^{(1)}_1-F^{(1 )}_2\in C^{2+\alpha,1+\frac{\alpha}{2}}(Q)$, there exist $\hat{F}_{\boldsymbol{\eta}}(t)$ such that
$$F^{(1)}_1-F^{(1 )}_2=\sum_{\eta\in\mathbb{Z}^n} \hat{F}_{\boldsymbol{\eta}}(t)e^{2\pi \mathrm{i}\boldsymbol{\eta}\cdot x}. $$
We may choose $ m_1^{(1)}(x,t)=\exp(-4\pi^2|\boldsymbol{\zeta}|^2t-2\pi \mathrm{i} \boldsymbol{\zeta}\cdot x),$ $\boldsymbol{\zeta}\in\mathbb{Z}^n.$ Then $\eqref{eq:dd3}$ implies that
\begin{equation}\label{eq:dd4}
\int_{0}^{T}[\mathcal{F}(\Phi)(\boldsymbol{\xi},T-s) \hat{F}_{\boldsymbol{\xi}+\boldsymbol{\zeta}}(t)\exp(-4\pi^2|\boldsymbol{\zeta}|^2(T-s) ) ds=0.
\end{equation}
By the Weierstrass' approximation theorem, we have $\{\exp(-4\pi^2 kt )\}_{k=0}^{\infty}$ is dense in $C^1(0,T)$. It follows that $ \hat{F}_{\boldsymbol{\eta}}(t)=0$ for all $\boldsymbol{\eta}\in\mathbb{Z}^n$.
Therefore, we have $$ F^{(1)}_1(x,t)=F^{(1 )}_2(x,t).$$
We proceed to consider the case $N=2.$ Let
$$u_{j}^{(1,2)}:=\p_{\varepsilon_1}\p_{\varepsilon_2}u_{j}|_{\varepsilon=0},
m_{j}^{(1,2)}:=\p_{\varepsilon_1}\p_{\varepsilon_2}m_{j}|_{\varepsilon=0},$$
and
$$u_{j}^{(2)}:=\p_{\varepsilon_2}u_{j}|_{\varepsilon=0},m_{j}^{(2)}:=\p_{\varepsilon_2}m_{j}|_{\varepsilon=0}.$$
Then we can deal with the seceond-order linearization:
\begin{equation}\label{linear l=1,2}
\begin{cases}
-\p_tu_j^{(1,2)}(x,t)-\Delta u^{(1,2)}_j(x,t)+\nabla u_{j}^{(1)}\cdot \nabla u_{j}^{(2)}\\
\hspace*{3cm} = F_j^{(1)}m_j^{(1,2)}+F^{(2)}_j(x,t)m_j^{(1)}m_j^{(2)},& \text{ in }\mathbb{T}^n\times (0,T),\\
\p_t m^{(1,2)}_j(x,t)-\Delta m^{(1,2)}_j(x,t)\medskip \\
\hspace*{3cm} = {\rm div} (m_{j}^{(1)}\nabla u_j^{(2)})+{\rm div}(m_j^{(2)}\nabla u_j^{(1)}) ,&\text{ in }\mathbb{T}^n\times(0,T),\medskip\\
u^{(1,2)}_j(x,T)=G^{(1)}(x)m_j^{(1,2)}(x,T)+G^{(2)}(x)m_j^{(1)}m_j^{(2)}(x,T), & \text{ in } \mathbb{T}^n,\medskip\\
m^{(1,2)}_j(x,0)=0, & \text{ in } \mathbb{T}^n.\\
\end{cases}
\end{equation}
Noticing that by the same argument in the case $N=1$ (considering $m_0=\varepsilon_2f_2$ ), we have
$$ u^{(1)}_1(x,t)= u^{(1)}_2(x,t), u^{(2)}_1(x,t)=u^{(2)}_2(x,t), $$
and
$$m^{(1)}_1(x,t)= m^{(1)}_2(x,t) , m^{(2)}_1(x,t)= m^{(2)}_2(x,t).$$
Denote
\[
p(x,t)={\rm div} (m_{j}^{(1)}\nabla u_j^{(2)})+{\rm div}(m_j^{(2)}\nabla u_j^{(1)}),\ \ q(x,t)= -\nabla u_{j}^{(1)}\cdot \nabla u_{j}^{(2)}.
\]
Then we can also solve system \eqref{linear l=1,2} as follows:
\begin{equation*}
m^{(1,2)}_j(x,t)=\int_{0}^{t} \int_{\mathbb{R}^n} \Phi(x-y,t-s)p(y,s)\, dyds,
\end{equation*}
\begin{equation*}
\begin{aligned}
u_j^{(1,2)}(x,t)= &\int_{\mathbb{R}^n}\Phi(x-y,T-t) [G^{(1)}(x)m_j^{(1,2)}(x,T)+G^{(2)}(x)m_j^{(1)}m_j^{(2)}(x,T) ]\, dy\\
+&\int_{0}^{T-t}\int_{\mathbb{R}^n}\Phi(x-y,T-t-s)(F^{(2)}_j(y,T-s)m_j^{(1)}m_j^{(2)}(y,T-s) -\overline{q}(y,s))\, dyds,
\end{aligned}
\end{equation*}
where $\overline{q}(y,s)=q(y,T-s).$
Since $$u_1^{(1,2)}(x,0)= u_2^{(1,2)}(x,0),$$ we have
$$\int_{0}^{T}\int_{\mathbb{R}^n}\Phi(x-y,T-s)(F^{(2)}_1(y,T-s)-F^{(2)}_2(y,T-s))m_j^{(1)}(y,T-s)m_j^{(2)}(y,T-s)\,
dyds= 0.$$
Next, by a similar argument in the case $N=1$, we can prove that $F^{(2)}_1(x,t)=F^{(2)}_2(x,t). $
Finally, by the mathematical induction, we can show the same result for $N\geq 3$. That is, for any $k\in\mathbb{N},$ we have $F^{(k)}_1(x,t)=F^{(k)}_2(x,t).$ Therefore, we have $F_1(x,t,z)=F_2(x,t,z).$
The proof is complete.
\end{proof}
Next, we give the proof of Theorem $\ref{der g}$. Let us first remark that the proof of Theorem \ref{der g} is similar to that for Theorem \ref{der F}. Hence, we only provide the necessary modifications.
\begin{proof}[Proof of Theorem $\ref{der g}$]
Consider the following systems for $j=1,2$:
\begin{equation}\label{j=1,2for g}
\begin{cases}
-\p_tu_j(x,t)-\Delta u_j(x,t)+\frac 1 2 {|\nabla u_j|^2}= F(x,t,m_j(x,t)),& \text{ in }\mathbb{T}^n\times (0,T),\medskip\\
\p_t m_j(x,t)-\Delta m_j(x,t)-{\rm div}(m_j(x,t)\nabla u_j(x,t))=0,&\text{ in }\mathbb{T}^n\times(0,T),\medskip\\
u_j(x,T)=G_j(x,m_j(x,T)), & \text{ in } \mathbb{T}^n,\medskip\\
m_j(x,0)=m_0(x), & \text{ in } \mathbb{T}^n.\\
\end{cases}
\end{equation}
By the successive linearization procedure, we first consider the case $N=1.$ Let
$$u_{j}^{(1)}:=\p_{\varepsilon_1}u_{j}|_{\varepsilon=0},\quad m_{j}^{(1)}:=\p_{\varepsilon_1}m_{j}|_{\varepsilon=0}.$$
Direct computations show that $(u_{j}^{(1)},v_{j}^{(1)} )$ satisfies the following system
\begin{equation}\label{linear l=1for g}
\begin{cases}
-\p_tu_j^{(1)}(x,t)-\Delta u^{(1)}_j(x,t)= F^{(1)}(x)m_j^{(1)}(x,t),& \text{ in }\mathbb{T}^n\times(0,T),\medskip\\
\p_t m^{(1)}_j(x,t)-\Delta m^{(1)}_j(x,t)=0,&\text{ in }\mathbb{T}^n\times(0,T),\medskip\\
u^{(1)}_j(x,T)=G_j^{(1)}(x)m^{(1)}_j(x,T), & \text{ in } \mathbb{T}^n,\medskip\\
m^{(1)}_j(x,0)=f_1(x), & \text{ in } \mathbb{T}^n.
\end{cases}
\end{equation}
We can solve the system \eqref{linear l=1for g} by first deriving $m^{(1)}_j$ and then obtaining $u^{(1)}_j.$ In doing so, we can obtain that the solution is
$$ m_j^{(1)}(x,t)= \int_{\mathbb{R}^n}\Phi(x-y,t)f_{1}(y)\, dy,$$
\begin{equation*}
\begin{aligned}
u_j^{(1)}(x,t)&= \int_{\mathbb{R}^n}\Phi(x-y,T-t)G_j^{(1)}(y)m^{(1)}_j(y,T) )\, dy\\
&+\int_{0}^{T-t}\int_{\mathbb{R}^n}\Phi(x-y,T-t-s)F^{(1)}(y,T-s)\overline{m}_j^{(1)}(y,s)\, dyds,
\end{aligned}
\end{equation*}
where $\overline{m}_j^{(1)}(x,t)= m_j^{(1)}(x,T-t)$ and $\Phi$ is the fundamental solution of the heat equation given in $\eqref{eq:fund1}$.
Since $\mathcal{M}_{G_1}=\mathcal{M}_{G_2}$, we have $$ u_1^{(1)}(x,0)=u_2^{(1)}(x,0),$$ for all $f_1\in C^{2+\alpha}(\mathbb{T}^n).$ This implies that
$$ \int_{\mathbb{R}^n}\Phi(x-y,T)[G_1^{(1)}(y)m_1^{(1)}(y,T))-G_2^{(1)}(y)m_2^{(1)}(y,T)) ]\, dy=0.$$
Noticing that $m_1^{(1)}(y,T)=m_2^{(1)}(y,T)$ and it is true for all $m_1^{(1)}(y,T)=m_2^{(1)}(y,T)\in C^{\alpha,\frac{\alpha}{2}}(\mathbb{T}^n)$,
we have $G_1^{(1)}(x)=G_2^{(1)}(x)$
Finally, by following a similar argument to that in the proof of Theorem~\ref{der F}, we can conduct the successive linearization procedure to show that $G_1^{(k)}(x)=G_2^{(k)}(x)$ for all $k\in\mathbb{N}$. Hence, $G_1(x,z)=G_2(x,z).$
The proof is complete.
\end{proof}
\subsection{Simultaneous recovery results for inverse problems}
In this section, we aim to determinate $F$ and $G$ simultaneously. To that end, we first derive an auxiliary lemma as follows.
\begin{lem}\label{dense}
Let $u$ be a solution of the heat equation
\begin{equation}\label{per heat}
\p_t w(x,t)-\Delta w(x,t)=0 \text{ in } \mathbb{T}^n.
\end{equation}
Let $f(x)\in C^{2+\alpha}(\mathbb{T}^n)$ for some $\alpha\in(0,1)$. Suppose
\begin{equation}\label{fuv=0}
\int_{\mathbb{T}^n\times(0,T)} f(x)u(x,t)dxdt=0,
\end{equation}
for all $u\in C^{\infty}(\mathbb{T}^n\times(0,T))$. Then one has $f=0.$
\end{lem}
\begin{proof}
Let $\boldsymbol {\xi}\in\mathbb{Z}^n$. It is directly verified that
$$
u(x,t)=\exp(- 2\pi\mathrm{i}\boldsymbol {\xi}\cdot x-4\pi^2|\boldsymbol {\xi}|^2t ), \quad \mathrm{i}:=\sqrt{-1},
$$
is a solution of $\eqref{per heat}$. Then $\eqref{fuv=0}$ implies that
$$\int_{\mathbb{T}^n} \frac{1-\exp(-4\pi^2|\boldsymbol {\xi}|^2T)}{4\pi^2|\boldsymbol {\xi}|^2}f(x)e^{-2\pi \mathrm{i}\boldsymbol {\xi}\cdot x } dx=0.$$
Hence, the Fourier series of $f(x)$ is $0$. Since $f(x)\in C^{2+\alpha}(\mathbb{T}^n)$, its Fourier series converges to $f(x)$ uniformly.
Therefore, $f(x)=0.$
\end{proof}
We are now in a position to present the proof of Theorem $\ref{der F,g}$.
\begin{proof}[Proof of Thoerem $\ref{der F,g}$]
Consider the following systems
\begin{equation}\label{j=1,2for Fg}
\begin{cases}
-\p_tu_j(x,t)-\Delta u_j(x,t)+\frac 1 2 {|\nabla u_j|^2}= F_j(x,m_j(x,t)),& \text{ in }\mathbb{T}^n\times(0,T),\medskip\\
\p_t m_j(x,t)-\Delta m_j(x,t)-{\rm div}(m_j(x,t)\nabla u_j(x,t))=0,&\text{ in }\mathbb{T}^n\times(0,T),\medskip\\
u_j(x,T)=G_j(x,m_j(x,T)), & \text{ in } \mathbb{T}^n,\medskip\\
m_j(x,0)=m_0(x), & \text{ in } \mathbb{T}^n.
\end{cases}
\end{equation}
Following a similar method we used in the proof of Theorem $\ref{der F}$, we let
$$m_0(x;\varepsilon)=\sum_{l=1}^{N}\varepsilon_lf_l,$$
where $f_l\in C^{2+\alpha,1+\frac{\alpha}{2}}(Q)$ and $\varepsilon=(\varepsilon_1,\varepsilon_2,...,\varepsilon_N)\in\mathbb{R}^N$ with
$|\varepsilon|=\sum_{l=1}^{N}|\varepsilon_l|$ small enough.
Consider the case $N=1.$ Let
$$u_{j}^{(1)}:=\p_{\varepsilon_1}u_{j}|_{\varepsilon=0},$$
$$m_{j}^{(1)}:=\p_{\varepsilon_1}m_{j}|_{\varepsilon=0}.$$
Then direct computations imply that $(u_{j}^{(1)},v_{j}^{(1)} )$ satisfies the following system:
\begin{equation}\label{linear l=1for F g}
\begin{cases}
-\p_tu_j^{(1)}(x,t)-\Delta u^{(1)}_j(x,t)= F_j^{(1)}(x)m_j^{(1)}(x,t),& \text{ in }\mathbb{T}^n\times(0,T),\medskip \\
\p_t m^{(1)}_j(x,t)-\Delta m^{(1)}_j(x,t)=0,&\text{ in }\mathbb{T}^n\times (0,T),\medskip \\
u^{(1)}_j(x,T)=G_j^{(1)}(x)m^{(1)}_j(x,T), & \text{ in } \mathbb{T}^n,\medskip\\
m^{(1)}_j(x,0)=f_1(x), & \text{ in } \mathbb{T}^n.
\end{cases}
\end{equation}
Then we have $ m_1^{(1)}=m_2^{(1)}:=m^{(1)}(x,t)$ .
Let $ \overline{u}=u_1^{(1)}-u_2^{(1)}$, $\eqref{linear l=1for F g}$ implies that
\begin{equation}\label{u1-u2 }
\begin{cases}
&-\p_t\overline{u}-\Delta\overline{u}= (F_1^{(1)}-F_2^{(1)})m^{(1)}(x,t),\medskip\\
&\overline{u}(x,T)=(G_1^{(1)}-G_2^{(1)})m^{(1)}(x,T).
\end{cases}
\end{equation}
Now let $w$ be a solution of the heat equation $\p_t w(x,t)-\Delta w(x,t)=0$ in $\mathbb{T}^n$. Then
\begin{equation}
\begin{aligned}
&\int_Q (F_1^{(1)}-F_2^{(1)})m^{(1)}(x,t)w\, dxdt\medskip\\
=&\int_Q (-\p_t\overline{u}-\Delta\overline{u})w\, dxdt\medskip\\
=&\int_{\mathbb{T}^n} (\overline{u}w)\big|_0^T\, dx +\int_Q \overline{u}\p_tw- \overline{u}\Delta w\medskip\\
=& \int_{\mathbb{T}^n} (\overline{u}w)\big|_0^T\, dx.
\end{aligned}
\end{equation}
Since $\mathcal{M}_{F_1,G_1}=\mathcal{M}_{F_2,G_2}$, we have $$\overline{u}(x,0)=0.$$ It follows that
\begin{equation}\label{integral by part}
\int_Q (F_1^{(1)}-F_2^{(1)})m^{(1)}(x,t)w(x,t)\, dxdt= \int_{\mathbb{T}^n} w(x,T)(G_1^{(1)}-G_2^{(1)})m^{(1)}(x,T)\, dx,
\end{equation}
for all solutions $w(x,t),m^{(1)}(x,t)$ of the heat equation in $\mathbb{T}^n$.
Here, we cannot apply Lemma $\ref{dense}$ directly. Nevertheless, we use the same construction.
Let $\boldsymbol {\xi_1},\boldsymbol {\xi_2}\in\mathbb{Z}^n\backslash\{0\}$ and $\boldsymbol {\xi}=\boldsymbol {\xi_1}+\boldsymbol {\xi_2}$ .
Let
$$w(x,t)=\exp(- 2\pi\mathrm{i}\boldsymbol {\xi_1}\cdot x-4\pi^2|\boldsymbol {\xi_1}|^2t ),$$
and
$$m(x,t)=\exp(- 2\pi\mathrm{i}\boldsymbol {\xi_2}\cdot x-4\pi^2|\boldsymbol {\xi_2}|^2t ).$$
Then $\eqref{integral by part}$ implies that
\begin{equation}\label{Fourier}
\begin{aligned}
&\int_{\mathbb{T}^n} \frac{1-\exp(-4\pi^2T(|\boldsymbol {\xi_1}|^2+|\boldsymbol {\xi_2}|^2))}{4\pi^2( |\boldsymbol {\xi_1}|^2+|\boldsymbol {\xi_2}|^2)}(F_1^{(1)}-F_2^{(1)} )e^{-2\pi \mathrm{i}\boldsymbol {\xi}\cdot x }\, dx\\
=&-\int_{\mathbb{T}^n}\exp(-4\pi^2T(|\boldsymbol {\xi_1}|^2+|\boldsymbol {\xi_2}|^2)) (G_1^{(1)}-G_2^{(1)})e^{-2\pi \mathrm{i}\boldsymbol {\xi}\cdot x }\, dx.
\end{aligned}
\end{equation}
Let
$$(F_1^{(1)}-F_2^{(1)} )=\sum_{\boldsymbol {\eta}\in\mathbb{Z}^n}a_{\boldsymbol {\eta}} e^{2\pi \mathrm{i}\boldsymbol {\eta}\cdot x},$$
$$(G_1^{(1)}-G_2^{(1)} )=\sum_{\boldsymbol {\eta}\in\mathbb{Z}^n}b_{\boldsymbol {\eta}} e^{2\pi \mathrm{i}\boldsymbol {\eta}\cdot x},$$
be the Fourier series of $ (F_1^{(1)}-F_2^{(1)} )$ and $(G_1^{(1)}-G_2^{(1)} )$.
Then $\eqref{Fourier}$ readily yields that
$$\frac{1-\exp(-4\pi^2T(|\boldsymbol {\xi_1}|^2+|\boldsymbol {\xi_2}|^2))}{4\pi^2( |\boldsymbol {\xi_1}|^2+|\boldsymbol {\xi_2}|^2)}a_{\boldsymbol {\xi} }+\exp(-4\pi^2T(|\boldsymbol {\xi_1}|^2+|\boldsymbol {\xi_2}|^2))b_{\boldsymbol {\xi}}=0.$$
For a given $\boldsymbol {\xi}\in\mathbb{Z}^n $, there exist $\boldsymbol {\xi_1},\boldsymbol {\xi_2},\boldsymbol {\xi_1}',\boldsymbol {\xi_2}' \in\mathbb{Z}^n\backslash\{0\}$ such that $\boldsymbol {\xi}=\boldsymbol {\xi_1}+\boldsymbol {\xi_2}=\boldsymbol {\xi_1}'+\boldsymbol {\xi_2}'$ and $|\boldsymbol {\xi_1}|^2+|\boldsymbol {\xi_2}|^2\neq |\boldsymbol {\xi_1}'|^2+|\boldsymbol {\xi_2}'|^2 .$ Therefore, $a_{\boldsymbol {\xi}}=b_{\boldsymbol {\xi}}=0$ for all $\boldsymbol {\xi}\in\mathbb{Z}^n$. It follows that $F_1^{(1)}-F_2^{(1)}=G_1^{(1)}-G_2^{(1)}=0$.
Next, we consider the case $N=2.$ Let
\begin{equation}\label{eq:ss1}
u_{j}^{(1,2)}:=\p_{\varepsilon_1}\p_{\varepsilon_2}u_{j}|_{\varepsilon=0},\quad
m_{j}^{(1,2)}:=\p_{\varepsilon_1}\p_{\varepsilon_2}m_{j}|_{\varepsilon=0},
\end{equation}
and
\begin{equation}\label{eq:ss2}
u_{j}^{(2)}:=\p_{\varepsilon_2}u_{j}|_{\varepsilon=0},\quad m_{j}^{(2)}:=\p_{\varepsilon_2}m_{j}|_{\varepsilon=0}.
\end{equation}
By the second-order linearization in \eqref{eq:ss1} and \eqref{eq:ss2}, we can obtain
\begin{equation}
\begin{cases}
-\p_tu_j^{(1,2)}(x,t)-\Delta u^{(1,2)}_j(x,t)+\nabla u_{j}^{(1)}\cdot \nabla u_{j}^{(2)}\\
\hspace*{3cm} = F_j^{(1)}m_j^{(1,2)}+F^{(2)}_j(x)m_j^{(1)}m_j^{(2)},& \text{ in }\mathbb{T}^n\times(0,T),\medskip\\
\p_t m^{(1,2)}_j(x,t)-\Delta m^{(1,2)}_j(x,t)\\
\hspace*{3cm}= {\rm div} (m_{j}^{(1)}\nabla u_j^{(2)})+{\rm div}(m_j^{(2)}\nabla u_j^{(1)}) ,&\text{ in }\mathbb{T}^n\times (0,T),\medskip\\
u^{(1,2)}_j(x,T)=G^{(1)}(x)m_j^{(1,2)}(x,T)+G^{(2)}(x)m_j^{(1)}m_j^{(2)}(x,T), & \text{ in } \mathbb{T}^n,\medskip\\
m^{(1,2)}_j(x,0)=0, & \text{ in } \mathbb{T}^n.
\end{cases}
\end{equation}
By following a similar argument in the case $N=1$ , we have
$$ u^{(1)}_1(x,t)= u^{(1)}_2(x,t), u^{(2)}_1(x,t)=u^{(2)}_2(x,t),$$
and
$$ m^{(1)}_1(x,t)= m^{(1)}_2(x,t) , m^{(2)}_1(x,t)= m^{(2)}_2(x,t).$$
Let $\overline{u}^2=u_1^{(1,2)}(x,t)-u_2^{(1,2)}(x,t) $. We have
\begin{equation}\label{u1-u2,2 }
\begin{cases}
&-\p_t\overline{u}^2-\Delta\overline{u}^2= (F_1^{(1)}-F_2^{(1)})m^{(1)}(x,t)m_1^{(2)}(x,t),\medskip\\
&\overline{u}(x,T)=(G_1^{(1)}-G_2^{(1)})m^{(1)}(x,T)m_1^{(2)}(x,t).
\end{cases}
\end{equation}
Let $w$ be a solution of the heat equation $\p_t w(x,t)-\Delta w(x,t)=0$ in $\mathbb{T}^n$. Then by following a similar argument in the case $N=1$, we can show that
\begin{equation}\label{integral by part2}
\begin{split}
& \int_Q (F_1^{(1)}-F_2^{(1)})m^{(1)}m_1^{(2)}w(x,t)\, dxdt\\
=& \int_{\mathbb{T}^n} w(x,T)(G_1^{(1)}-G_2^{(1)})m^{(1)}(x,T)m_1^{(2)}(x,T)\, dx.
\end{split}
\end{equation}
To proceed further, by using the construction in Lemma $\ref{dense}$ again, we have from \eqref{integral by part2} that
\[
F_1^{(1)}-F_2^{(1)}=G_1^{(1)}-G_2^{(1)}.
\]
Finally, via a mathematical induction, we can derive the same result for $N\geq 3$. That is, for any $k\in\mathbb{N},$ we have $F^{(k)}_1(x)=F^{(k)}_2(x).$ Hence,
$$(F_1(x,z),F_2(x,z))=(G_1(x,z),G_2(x,z)),\text{ in } \mathbb{R}^n\times \mathbb{R}.$$
The proof is complete.
\end{proof}
\begin{rmk}
Theorem $\ref{der F,g}$ is not strictly stronger than Theorem $\ref{der F}$ or Theorem $\ref{der g}$. We need $F(x,z)$ is independ of $t$ in the proof of Theorem $\ref{der F,g}$ but we do not need this condition in the proof of Theorem $\ref{der F}$ or Theorem $\ref{der g}$.
\end{rmk}
\section{Inverse Problems for MFGs with General Lagrangians}
In the previous sections, we established the unique identifiability results for the inverse problems by assuming that the Hamiltonian involved is of a quadratic form, which represents a kinetic energy. In this section, we show that one can extend a large part of the previous results to the case with general Lagrangians if $F$ is independ of $t$.
In what follows, we let $T>0$ and $n\in\mathbb{N}$ and consider the following system of nonlinear PDEs :
\begin{equation}\label{general H}
\begin{cases}
-\p_tu(x,t)-\Delta u(x,t)+ H(x,\nabla u)= F(x,m(x,t)),& \text{ in }\mathbb{T}^n\times (0,T),\medskip\\
\p_t m(x,t)-\Delta m(x,t)-{\rm div}(m(x,t) H_p (x,\nabla u))=0,&\text{ in }\mathbb{T}^n\times (0,T),\medskip \\
u(x,T)=G(x,m_T), & \text{ in } \mathbb{T}^n,\medskip \\
m(x,0)=m_0(x), & \text{ in } \mathbb{T}^n.
\end{cases}
\end{equation}
We study the inverse problem \eqref{eq:ip1}-\eqref{eq:ip2} associated with \eqref{general H}. In order to apply the method developed in the previous sections to this general case, we first introduce a new analytic class.
\begin{defi}
Let $H(x,z_1,z_2,...,z_n)$ be a function mapping from $\mathbb{R}\times\mathbb{C}^n $ to $\mathbb{C}$. We say that $H$ is admissible and write $H \in \mathcal{I}$ if it fulfils the following conditions:
\begin{enumerate}
\item[(1)]~The map $(z_1,z_2,...,z_n)\to H(\cdot,z_1,z_2,...,z_n)$ is holomorphic with value in $C^{2+\alpha}(\mathbb{T}^n)$, $\alpha\in(0,1)$;
\item[(2)] $H(x,0)=0, $ for all $x\in\mathbb{T}^n.$
\end{enumerate}
It is clear that $H$ can be expanded into a power series:
\begin{equation}\label{eq:sss1}
H(x,z)=\sum_{|\beta|=1}^{\infty} H^{(\beta)}(x)\frac{z^{\beta}}{k!},
\end{equation}
where $ H^{(\beta)}(x)\in C^{2+\alpha}(\mathbb{T}^n)$ and $\beta$ is a muti-index.
\end{defi}
Similar to our discussion in Remark~\ref{rem:1}, we always assume that the coefficient functions $H^{(\beta)}$ in \eqref{eq:sss1} are real-valued. We first state the main theorems of the results for the inverse problems associated with \eqref{general H}. The corresponding proofs are given in Section $\ref{proof H}$.
\begin{thm}\label{der F 2}
Assume $F_j\in\mathcal{B}$ ($j=1,2$), $G\in\mathcal{B}$ and $H\in\mathcal{I}$. Let $\mathcal{M}_{F_j}$ be the measurement map associated to
the following system ($j=1,2$):
\begin{equation}
\begin{cases}
-\p_tu(x,t)-\Delta u(x,t)+ H(x,\nabla u)= F_j(x,m(x,t)),& \text{ in }\mathbb{T}^n\times (0,T),\medskip\\
\p_t m(x,t)-\Delta m(x,t)-{\rm div}(m(x,t) H_p (x,\nabla u))=0,&\text{ in }\mathbb{T}^n\times (0,T),\medskip\\
u(x,T)=G(x,m_T), & \text{ in } \mathbb{T}^n,\medskip\\
m(x,0)=m_0(x), & \text{ in } \mathbb{T}^n.
\end{cases}
\end{equation}
If for any $m_0\in C^{2+\alpha}(\mathbb{T}^n)$, one has
$$\mathcal{M}_{F_1}(m_0)=\mathcal{M}_{F_2}(m_0),$$
then it holds that
$$F_1(x,z)=F_2(x,z) \text{ in } \mathbb{T}^n\times \mathbb{R}.$$
\end{thm}
\begin{thm}\label{der g2}
Assume $F \in\mathcal{B}$, $G_j\in\mathcal{B}$ ($j=1,2$) and $H\in\mathcal{I}$. Let $\mathcal{M}_{G_j}$ be the measurement map associated to the following system ($j=1,2$):
\begin{equation}
\begin{cases}
-\p_tu(x,t)-\Delta u(x,t)+ H(x,\nabla u)= F(x,t,m(x,t)),& \text{ in }\mathbb{T}^n\times (0,T),\medskip\\
\p_t m(x,t)-\Delta m(x,t)-{\rm div}(m(x,t) H_p (x,\nabla u))=0,&\text{ in } \mathbb{T}^n\times (0,T),\medskip\\
u(x,T)=G_j(x,m_T), & \text{ in } \mathbb{T}^n,\medskip\\
m(x,0)=m_0(x), & \text{ in } \mathbb{T}^n.
\end{cases}
\end{equation}
If for any $m_0\in C^{2+\alpha}(\mathbb{T}^n)$, one has
$$\mathcal{M}_{G_1}(m_0)=\mathcal{M}_{G_2}(m_0)m,$$
then it holds that
$$G_1(x,z)=G_2(x,z) \text{ in } \mathbb{T}^n\times \mathbb{R}.$$
\end{thm}
\subsection{Well-posedness of the general system}
\begin{lem}\label{localwellpose2}
Suppose $F,G\in\mathcal{B}$ ,$H\in\mathcal{I}$. Then
there exist $\delta>0$, $C>0$ such that for any $m_0\in B_{\delta}(\mathbb{T}^n) :=\{m_0\in C^{\alpha}(\mathbb{T}^n): \|m_0\|_{C^{2+\alpha}(\mathbb{T}^n)}\leq\delta \}$, the MFG system $\eqref{general H}$ has a solution $u = u_{m_0} \in
C^{2+\alpha,1+\frac{\alpha}{2}}(Q)$ which satisfies
\begin{equation}\label{eq:nn4}
\|u\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(Q}+ \|m\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(Q)}\leq C\|m_0\|_{ C^{2+\alpha}(\mathbb{T}^n)}.
\end{equation}
Furthermore, the solution $(u,m)$ is unique within the class
\begin{equation}\label{eq:nn5}
\{ (u,m)\in C^{2+\alpha,1+\frac{\alpha}{2}}(Q)^2 : \|(u,m)\|_{ C^{2+\alpha,1+\frac{\alpha}{2}}(Q)^2}\leq C\delta \},
\end{equation}
where
\begin{equation}\label{eq:nn6}
\|(u,m)\|_{ C^{2+\alpha,1+\frac{\alpha}{2}}(Q)^2}:= \|u\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(Q)}+ \|m\|_{C^{2+\alpha,1+\frac{\alpha}{2}}(Q)},
\end{equation}
and it depends holomorphically on $m_0\in C^{2+\alpha}(\mathbb{T}^n)$.
\end{lem}
The proof of Lemma $\ref{localwellpose2}$ follows from a similar argument to that of Lemma $\ref{local_wellpose}$. We choose to skip it.
\subsection{Proofs of Theorem $\ref{der F 2}$ and $\ref{der g2}$}\label{proof H}
We first introduce the general heat kernl to recover the unknown functions in a parabolic system. The construction and basic properties of the general heat kernel can be found in \cite{ito11}.
\begin{lem}\label{general heat ker}
Let $F_1,F_2,f\in C^{2+\alpha,1+\frac{\alpha}{2}}(Q)$, $g\in C^{2+\alpha}(\mathbb{T}^n)$ and $A(x)\in C^{2+\alpha,1+\frac{\alpha}{2}}(\mathbb{T}^n)^n$. Consider the following system
\begin{equation}
\begin{cases}
\p_tu_i(x,t)-\Delta u_i(x,t)+ A(x)\cdot \nabla u_i= F_i(x)v(x,t)+f(x,t),& \text{ in }\mathbb{T}^n\times (0,T),\medskip\\
u_i(x,0)= g(x) , & \text{ in } \mathbb{T}^n .\\
\end{cases}
\end{equation}
Suppose for any $v\in C^{2+\alpha,1+\frac{\alpha}{2}}(Q)$, we have $u_1(x,T;v)=u_2(x,T;v)$. Then it holds that $F_1=F_2.$
\begin{proof}
Let $L=\partial_t-\Delta+A\cdot\nabla(\cdot)$ and $K(x,y,t)$ be the solution of the following Cauchy problem
\begin{equation*}
\begin{cases}
&L (K(x,t))=0,\ \ t>0,\ \ x\in\mathbb{R}^n,\medskip\\
&K(x,0)=\delta(0).
\end{cases}
\end{equation*}
Then one has that
\begin{equation*}
\begin{aligned}
u_i(x,t)=&\int_{\mathbb{T}^n}K(x-y,t)g(y)dy\\
+&\int_{0}^t\int_{\mathbb{T}^n}K(x-y,t-s)(F_i(y)v(y,s)+f(y,s))\, dyds.
\end{aligned}
\end{equation*}
Since we have $u_1(x,T;v)=u_2(x,T;v)$, it follows that
\begin{equation}\label{implies F1=F2}
\int_{0}^T\int_{\mathbb{T}^n}K(x-y,T-s)(F_1(y)-F_2(y))v(y,s)\, dyds=0.
\end{equation}
By absurdity, we assume that there is $y_0\in \mathbb{T}^n$ such that $ F_1(y_0)\neq F_2(y_0)$. Then there is a neighborhood $U$ of $ y_0 $ such that $F_1-F_2>0$ or $F_1-F_2<0$ in $U$. Since $K(x-y,T-s)>0$ and $\eqref{implies F1=F2}$ holds for all $v\in C^{2+\alpha,1+\frac{\alpha}{2}}(Q)$. We may choose $v$ such that
$K(x-y,T-s)(F_1(y)-F_2(y))v(y,s)>0$ in $U$ and $K(x-y,T-s)(F_1(y)-F_2(y))v(y,s)=0$ in $\mathbb{T}^n\backslash U$. It is a contradiction. Therefore, we have $F_1=F_2.$
The proof is complete.
\end{proof}
\end{lem}
Before we present the proofs for Theorems $\ref{der F 2}$ and $\ref{der g2}$, we first perform the higher order
linearization for the MFG system $\eqref{general H}$, which follows a similar strategy to that developed in Section $\ref{HLM}$.
Let
$$m_0(x;\varepsilon)=\sum_{l=1}^{N}\varepsilon_lf_l,$$
where $f_l\in C^{2+\alpha}(\mathbb{T}^n)$ and $\varepsilon=(\varepsilon_1,\varepsilon_2,...,\varepsilon_N)\in\mathbb{R}^N$ with
$|\varepsilon|=\sum_{l=1}^{N}|\varepsilon_l|$ small enough. Then by Lemma $\ref{localwellpose2}$, there exists a unique solution $(u(x,t;\varepsilon),m(x,t;\varepsilon) )$ of $\eqref{general H}$. Let $(u(x,t;0),m(x,t;0) ) $ be the solution of $\eqref{general H}$ when $\varepsilon=0.$
Notice that if $H\in\mathcal{I}, $ then $(u(x,t;0),m(x,t;0) ) =(0,0).$
Let
$$u^{(1)}:=\p_{\varepsilon_1}u|_{\varepsilon=0},$$
$$m^{(1)}:=\p_{\varepsilon_1}m|_{\varepsilon=0}.$$
Suppose $H\in\mathcal{I}$, $F\in\mathcal{A}$ and $G\in\mathcal{B}, $ we have
\begin{equation}\label{compute for H1}
\begin{aligned}
&\p_t m^{(1)}_j(x,t)-\Delta m^{(1)}_j(x,t)\\
=&\lim\limits_{\varepsilon\to 0}\frac{1}{\varepsilon_l} [ -H(x,\nabla u(x,t;\varepsilon)) +H(x;u(x,t;0))+ F(x,u(x,t;\varepsilon))-F(x;u(x,t:0)) ]\\
=&\lim\limits_{\varepsilon\to 0}\frac{1}{\varepsilon_l} [ \sum_{|\beta|=1}^{\infty} H^{(\beta)}(x)\frac{z^{\beta}}{k!}]+F^{(1)}(x)m_j^{(1)}(x,t)\\
=& -A^{(1)}(x)\cdot \nabla u+F^{(1)}(x)m_j^{(1)}(x,t),
\end{aligned}
\end{equation}
where $A^{(1)}(x)=(H^{(1,0,0,...,0)}(x),H^{(0,1,0,...,0)}(x),...,H^{(0,0,...,1)}(x) ).$
Moreover, we have
\begin{equation}\label{compute for H2}
\begin{aligned}
&\p_{\varepsilon_1} {\rm div}(m(x,t) H_p (x,\nabla u)) |_{\varepsilon=0}\\
=&\p_{\varepsilon_1} {\rm div} ( m(x,t) A^{(1)}(x) )+ \p_{\varepsilon_1} {\rm div}(m(x,t) B^{(1)}(x) \cdot \nabla u)|_{\varepsilon=0}\\
=&\p_{\varepsilon_1} {\rm div} ( m(x,t) A^{(1)}(x) ),
\end{aligned}
\end{equation}
where
\[
\begin{split}
B^{(1)}(x)=&(\sum_{|\beta|=1}H^{(1,\beta)}(x),\sum_{|\alpha|+|\beta|=1,\alpha\in\mathbb{R}}H^{(\alpha,1,\beta)}(x),\\
&\sum_{|\alpha|+|\beta|=1,\alpha\in\mathbb{R}^2}H^{(\alpha,1,\beta)}(x),.....,\sum_{|\alpha|=1,\alpha\in\mathbb{R}^{n-1}}H^{(\alpha,1)}(x) ).
\end{split}
\]
Hence, we can see that $(u^{(1)},m^{(1)} )$ satisfies the following system:
\begin{equation}\label{H linear l=1 eg}
\begin{cases}
-\p_tu^{(1)}(x,t)-\Delta u^{(1)}(x,t)+ A^{(1)}(x)\cdot \nabla u= F^{(1)}(x)m^{(1)}(x,t),& \text{ in }\mathbb{T}^n\times (0,T),\medskip\\
\p_t m^{(1)}(x,t)-\Delta m^{(1)}(x,t)-{\rm div} ( m^{(1)}(x,t) A^{(1)}(x))=0,&\text{ in }\mathbb{T}^n\times (0,T),\medskip\\
u^{(1)}(x,T)=G^{(1)}(x)m^{(1)}(x,T), & \text{ in } \mathbb{T}^n,\medskip\\
m^{(1)}(x,0)=f_1(x), & \text{ in } \mathbb{T}^n,
\end{cases}
\end{equation}
Here, we make a key observation that the non-linear terms and source terms in higher-order
linearization system only depend on the solutions of the lower-order
linearization system. Hence, as an illustrative case for our argument, we only compute the second order linearization system.
Let
$$u^{(1,2)}:=\p_{\varepsilon_1}\p_{\varepsilon_2}u|_{\varepsilon=0},
m^{(1,2)}:=\p_{\varepsilon_1}\p_{\varepsilon_2}m|_{\varepsilon=0},$$
and
$$u^{(2)}:=\p_{\varepsilon_2}u|_{\varepsilon=0},m^{(2)}:=\p_{\varepsilon_2}m|_{\varepsilon=0}.$$
Recall the derivation of the system $\eqref{linear l=1,2 eg}$ in Section $\ref{HLM}$. By direct calculations, we have
\begin{equation}\label{compute H 12 eg}
\begin{aligned}
&-\p_tu^{(1,2)}-\Delta u^{(1,2)}\\
=&-\p_{\varepsilon_1}\p_{\varepsilon_2}H(x,\nabla u)|_{\varepsilon=0}+F^{(1)}(x)m^{(1,2)}+F^{(2)}(x)m^{(1)}m^{(2)}\\
=&-\p_{\varepsilon_1}\p_{\varepsilon_2}(\sum_{|\beta|=1}^{2} H^{(\beta)}(x)\frac{z^{\beta}}{k!})|_{\varepsilon=0}+F^{(1)}(x)m^{(1,2)}+F^{(2)}(x)m^{(1)}m^{(2)}\\
=&-A^{(1)}\cdot\nabla u_j^{(1,2)}-\sum_{|\beta|=2}H^{(\beta)}(x)u_j^{(1)}u_j^{(2)}++F^{(1)}(x)m^{(1,2)}+F^{(2)}(x)m^{(1)}m^{(2)}.
\end{aligned}
\end{equation}
Now, with the discussion above at hand and Lemma $\ref{general heat ker}$, we are now in a position to present the proofs of Theorems $\ref{der F 2}$ and $\ref{der g2}.$
\begin{proof}[Proof of Theorem $\ref{der F 2}$]
Consider the following MFG systems for $j=1,2$:
\begin{equation}\label{general H for F}
\begin{cases}
-\p_tu_j(x,t)-\Delta u_j(x,t)+ H(x,\nabla u_j)= F_j(x,m(x,t)),& \text{ in }\mathbb{T}^n\times (0,T),\medskip\\
\p_t m_j(x,t)-\Delta m_j(x,t)-{\rm div} (m_j(x,t) H_p (x,\nabla u_j))=0,&\text{ in }\mathbb{T}^n\times (0,T),\medskip\\
u_j(x,T)=G(x,m_T), & \text{ in } \mathbb{T}^n,\medskip\\
m_j(x,0)=m_0(x), & \text{ in } \mathbb{T}^n.
\end{cases}
\end{equation}
Recall the higher order linearization method in Section $\ref{HLM}$.
Let
$$u_{j}^{(1)}:=\p_{\varepsilon_1}u_{j}|_{\varepsilon=0},$$
$$m_{j}^{(1)}:=\p_{\varepsilon_1}m_{j}|_{\varepsilon=0}.$$
By combining $\eqref{compute for H1}$, $\eqref{compute for H2}$ and $\eqref{H linear l=1 eg}$, we can deduce that
\begin{equation}\label{H linear l=1}
\begin{cases}
-\p_tu_j^{(1)}(x,t)-\Delta u^{(1)}_j(x,t)+ A^{(1)}(x)\cdot \nabla u_j= F^{(1)}_j(x)m_j^{(1)}(x,t),& \text{ in }\mathbb{T}^n\times (0,T),\medskip\\
\p_t m^{(1)}_j(x,t)-\Delta m^{(1)}_j(x,t)-{\rm div} ( m_j^{(1)}(x,t) A^{(1)}(x))=0,&\text{ in }\mathbb{T}^n\times (0,T),\medskip\\
u^{(1)}_j(x,T)=G^{(1)}(x)m^{(1)}_j(x,T), & \text{ in } \mathbb{T}^n,\medskip\\
m^{(1)}_j(x,0)=f_1(x), & \text{ in } \mathbb{T}^n,\medskip\\
\end{cases}
\end{equation}
where
\[
A^{(1)}(x)=(H^{(1,0,0,...,0)}(x),H^{(0,1,0,...,0)}(x),...,H^{(0,0,...,1)}(x) ).
\]
We extend $f_l$ from $\mathbb{T}^n$ to $\mathbb{R}^n$ periodically, and still denote it by $f_l$. By Lemma $\ref{linearapp wellpose}$, $m_j^{(1)}$ is unique determined by $f_1(x)$. We use change of variables as well as a similar strategy in the proof of Lemma $\ref{general heat ker}$. Here, we can choose $f_1(x)$ such that $m_j^{(1)}$ satisfies the conditions in the proof of Lemma $\ref{general heat ker}$ ( in fact, one can see a simple example by considering $f_1(x)$ depends only on one space variable). In doing so, we can show that
$F_1^{(1)}(x)=F_2^{(1)}(x).$
Next, we can consider the case $N=2.$ Let
$$u_{j}^{(1,2)}:=\p_{\varepsilon_1}\p_{\varepsilon_2}u_{j}|_{\varepsilon=0},\quad
m_{j}^{(1,2)}:=\p_{\varepsilon_1}\p_{\varepsilon_2}m_{j}|_{\varepsilon=0},$$
and
$$u_{j}^{(2)}:=\p_{\varepsilon_2}u_{j}|_{\varepsilon=0},\quad m_{j}^{(2)}:=\p_{\varepsilon_2}m_{j}|_{\varepsilon=0}.$$
We can conduct the second-order linearization. Following a similar process as that in $\eqref{compute H 12 eg}$, we can deduce that
\begin{equation}
\begin{cases}
-\p_tu_j^{(1,2)}-\Delta u^{(1,2)}_j+A^{(1)}\cdot\nabla u_j^{(1,2)}+R_1(x,t)\\
\hspace*{3cm}= F_j^{(1)}(x)m_j^{(1,2)}+F^{(2)}_j(x)m_j^{(1)}m_j^{(2)},& \text{ in }\mathbb{T}^n\times (0,T),\medskip\\
\p_t m^{(1,2)}_j(x,t)-\Delta m^{(1,2)}_j(x,t)-{\rm div} ( m_j^{(1)}(x,t) A^{(1)}(x))\\
\hspace*{3cm}= R_2(x,t) ,&\text{ in }\mathbb{T}^n\times (0,T),\medskip\\
u^{(1,2)}_j(x,T)=G^{(2)}(x)m^{(1,2)}_j(x,T), & \text{ in } \mathbb{T}^n,\medskip\\
m^{(1,2)}_j(x,0)=0, & \text{ in } \mathbb{T}^n.
\end{cases}
\end{equation}
where $$R_1(x,t)= \sum_{|\beta|=2}H^{(\beta)}(x)u_j^{(1)}u_j^{(2)}, $$ and
$$R_2(x,t)={\rm div}(m_j^{(1)} U^{(2)})+{\rm div}(m_j^{(2)} U^{(1)}).$$
Here, the $l$-th component of $U^{(1)}$ is
$$U_l^{1}=\sum_{i=1}^{n}\frac{\p^2H}{\p z_l\p z_i}(x,0)\frac{\p u_j^{(2)}}{\p x_l},$$
and the $l$-th component of $U^{(2)}$ is
$$U_l^{1}=\sum_{i=1}^{n}\frac{\p^2H}{\p z_l\p z_i}(x,0)\frac{\p u_j^{(1)}}{\p x_l}.$$
Following a similar argument to the case $N=1$ (considering $m_0=\varepsilon_2f_2$ ), we have
$$u^{(1)}_1(x,t)= u^{(1)}_2(x,t),\quad u^{(2)}_1(x,t)=u^{(2)}_2(x,t),$$
and
$$ m^{(1)}_1(x,t)= m^{(1)}_2(x,t),\quad m^{(2)}_1(x,t)= m^{(2)}_2(x,t).$$
By Lemma $\ref{linearapp wellpose}$, $m_j^{(1,2)}$ is unique determined by $f_1(x),f_2(x)$ and $G^{(1)}(x)$. Applying Lemma $\ref{general heat ker}$ again, we readily have $F_1^{(2)}(x)=F_2^{(2)}(x).$
Finally, by a mathematical induction, we can show the same result holds for $N\geq 3$. That is, for any $k\in\mathbb{N},$ we have $F^{(k)}_1(x)=F^{(k)}_2(x).$ Therefore, we have $F_1(x,z)=F_2(x,z).$
The proof is complete.
\end{proof}
We proceed with the proof of Theorem $\ref{der g2}$. To that end, we first state an auxiliary lemma, which is an analogue to Lemma $\ref{general heat ker}$, and omit its proof.
\begin{lem}\label{general heat ker2}
Let $g_1,g_2\in C^{2+\alpha}(\mathbb{T}^n)$ and $A(x)\in C^{2+\alpha}(\mathbb{T}^n)^n$. Consider the following systems with $f\in C^{2+\alpha,1+\frac{\alpha}{2}}(Q)$ and $j=1,2$:
\begin{equation}
\begin{cases}
\p_tu_j(x,t)-\Delta u_j(x,t)+ A(x)\cdot \nabla u_j= f(x,t),& \text{ in } \mathbb{T}^n\times(0,T),\medskip\\
u_j(x,0)= g_j(x)v(x,T) , & \text{ in } \mathbb{T}^n .
\end{cases}
\end{equation}
Suppose for any $v\in C^{2+\alpha,1+\frac{\alpha}{2}}(Q)$, we have $u_1(x,T;v)=u_2(x,T;v)$. Then it holds that $g_1(x)=g_2(x).$
\end{lem}
Next, we give the proof Theorem $\ref{der g2}.$
\begin{proof}[Proof of Theorem $\ref{der g2} $]
We shall follow a similar strategy that was developed for the proof of Theorem $\ref{der F 2}$.
Consider the following systems for $j=1,2$:
\begin{equation}\label{general H for g}
\begin{cases}
-\p_tu_j(x,t)-\Delta u_j(x,t)+ H(x,\nabla u_j)= F(x,m(x,t)),& \text{ in }\mathbb{T}^n\times (0,T),\medskip\\
\p_t m_j(x,t)-\Delta m_j(x,t)-{\rm div} (m_j(x,t) H_p (x,\nabla u_j))=0,&\text{ in }\mathbb{T}^n\times (0,T),\medskip\\
u_j(x,T)=G_j(x,m_T), & \text{ in } \mathbb{T}^n,\medskip\\
m_j(x,0)=m_0(x), & \text{ in } \mathbb{T}^n.\\
\end{cases}
\end{equation}
We next perform the successive linearization process. Consider the case $N=1.$ Let
$$u_{j}^{(1)}:=\p_{\varepsilon_1}u_{j}|_{\varepsilon=0},$$
$$m_{j}^{(1)}:=\p_{\varepsilon_1}m_{j}|_{\varepsilon=0}.$$
By direct computations, one can show that $(u_{j}^{(1)},v_{j}^{(1)} )$ satisfies the following system:
\begin{equation}
\begin{cases}
-\p_tu_j^{(1)}(x,t)-\Delta u^{(1)}_j(x,t)+ A^{(1)}(x)\cdot \nabla u_j= F^{(1)}(x)m_j^{(1)}(x,t),& \text{ in }\mathbb{T}^n\times (0,T),\medskip\\
\p_t m^{(1)}_j(x,t)-\Delta m^{(1)}_j(x,t)-{\rm div} ( m_j^{(1)}(x,t) A^{(1)}(x))=0,&\text{ in }\mathbb{T}^n\times (0,T),\medskip\\
u^{(1)}_j(x,T)=G_j^{(1)}(x)m^{(1)}_j(x,T), & \text{ in } \mathbb{T}^n,\medskip\\
m^{(1)}_j(x,0)=f_1(x), & \text{ in } \mathbb{T}^n.
\end{cases}
\end{equation}
We can solve this system by first deriving $m^{(1)}_j$ and then obtaining $u^{(1)}_j.$
Since $\mathcal{M}_{G_1}=\mathcal{M}_{G_2}$, we have $$ u_1^{(1)}(x,0)=u_2^{(1)}(x,0),$$ for all $f_1\in C^{2+\alpha,1+\frac{\alpha}{2}}(Q).$ By Lemma $\ref{general heat ker2}$, we readily see that $ G_1^{(1)}(x)=G_2^{(2)}(x).$
Finally, by following a similar argument in the proof of Theorem~\ref{der F 2}, we can conduct the higher-order linearization process to show that $G_1^{(k)}(x)=G_2^{(k)}(x)$ for all $k\in\mathbb{N}$. Hence, $G_1(x,z)=G_2(x,z).$
The proof is complete.
\end{proof}
\section*{Acknowledgment}
The work of H Liu was supported by Hong Kong RGC General Research Funds (project numbers, 11300821, 12301420 and 12302919) and the NSFC/RGC Joint Research Grant (project number, N\_CityU101/21).
\vskip0.5cm
| 58,868
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\begin{document}
\title{A mathematical formalism of non-Hermitian quantum mechanics and observable-geometric phases}
\author{Zeqian Chen}
\address{State Key Laboratory of Resonances and Atomic and Molecular Physics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, 30 West District, Xiao-Hong-Shan, Wuhan 430071, China.}
\thanks{Key words: Spectral operator; para-Hermitian operator; non-Hermitian quantum mechanics; non-Hermitian Born formula; observable-geometric phase.}
\date{}
\maketitle
\markboth{Zeqian Chen}
{Non-Hermitian quantum mechanics}
\begin{abstract}
This paper presents a mathematical formalism of non-Hermitian quantum mechanics, following the Dirac-von Neumann formalism of quantum mechanics. In this formalism, the state postulate is the same as in the Dirac-von Neumann formalism, but the observable postulate should be changed to include para-Hermitian operators (spectral operators of scalar type with real spectrum) representing observable, as such both the measurement postulate and the evolution postulate must be modified accordingly. This is based on a Stone type theorem as proved here that the dynamics of non-Hermitian quantum systems is governed by para-unitary time evolution. The novelty of this formalism is the Born formula on the expectation of an observable at a certain state, which is proved to be equal to the usual Born rule for every Hermitian observable, but for a non-Hermitian one it may depend on measurement via the choice of a metric operator associated with the non-Hermitian observable under measurement. This non-Hermitian Born formula satisfies probability conservation for both Hermitian and non-Hermitian observables. Our formalism is nether Hamiltonian-dependent nor basis-dependent, but can recover both PT-symmetric and biorthogonal quantum mechanics, and it reduces to the Dirac-von Neumann formalism of quantum mechanics in the Hermitian setting. As application, we study geometric phases for non-Hermitian quantum systems, focusing on the observable-geometric phase of a time-dependent non-Hermitian quantum system.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}\label{Intro}
Non-Hermitian quantum theory regards as observable some non-Hermitian (not necessarily self-adjoint) operators, such as $PT$-symmetric, pseudo-Hermitian or biorthogonal quantum mechanics. In $PT$-symmetric quantum mechanics developed by Bender {\it et al.} \cite{BB1998, BBM1999}, the Hamiltonian $H$ is not necessarily Hermitian, but has the unbroken $PT$-symmetry so that all its spectra are real. Bender {\it et al.} \cite{BBJ2002} have shown that the Hamiltonian $H$ with the unbroken $PT$-symmetry is always Hermitian (or {\it self-adjoint} in mathematical texts) in a new inner product defined by a symmetric operator $C$ associated with $H.$ However, this physical Hilbert-space inner product is dependent on the Hamiltonian $H$ itself. This idea was further developed by Mostafazadeh \cite{Mosta2010} who presented pseudo-Hermitian quantum mechanics by employing the concept of pseudo Hermiticity first introduced by Dirac and Pauli \cite{Pauli1943}. On the other hand, in biorthogonal quantum mechanics as developed by Brody \cite{Brody2014}, the observables are determined by a previously chosen (unconditional) basis in the associated Hilbert space. In such a theory, since an unconditional basis is not necessarily orthogonal in a Hilbert space \cite{LT1977}, some observables are not represented by Hermitian operators, and meanwhile, some Hermitian operators are excluded from the observable when given a basis.
Namely, $PT$-symmetric or pseudo-Hermitian quantum mechanics is Hamiltonian-dependent in the sense that the physical Hilbert-space inner product is determined by the non-Hermitian Hamiltonian of a system, while biorthogonal quantum mechanics is basis-dependent in the sense that observables are determined by a chosen basis in the Hilbert space of a system. Note that the conventional quantum mechanics, that is the Dirac-von Neumann formalism \cite{Dirac1958, vN1955}, is nether a Hamiltonian-dependent nor basis-dependent theory. To this end, we need to present a mathematical formalism of non-Hermitian quantum mechanics, which is nether Hamiltonian-dependent nor basis-dependent, but can recover both $PT$-symmetric and biorthogonal quantum mechanics, and it reduces to the Dirac-von Neumann formalism in the Hermitian setting.
In our formalism, the (pure) states are represented by ray lines or nonzero vectors as the same as the usual, but the observable is represented by a spectral operator of scalar type with real spectrum ({\it para-Hermitian operators} in our notion, including all Hermitian operators), as such both the measurement postulate and the evolution postulate must be modified accordingly. This is based on a theorem of Antoine and Trapani \cite{AT2014}, stating that a densely defined closed operator $T$ is para-Hermitian if and only if there is a metric operator $G$ such that $G^\frac{1}{2}TG^{-\frac{1}{2}}$ is Hermitian (see Section \ref{Pre:para-Hermi}). In particular, a Stone type theorem associated with the para-unitary operators is proved that the dynamics of non-Hermitian quantum systems is governed by para-unitary time evolution (see Section \ref{Pre:Stone} for the details). There are more general concepts than para-Hermitian operators commonly found in the literature, such as {\it quasi-Hermitian} and {\it pseudo-Hermitian} operators (cf. \cite{AT2014, Mosta2010}). However, there seems not to exist a general theory of functional calculus for them (cf. \cite{DS1971}), yet a Stone type theorem cannot hold in general for such classes of operator. Mathematically, this explains the reason why we chose para-Hermitian operators representing observables in the formalism.
The novelty of this formalism is the Born formula expressing the expectation of an observable at a certain state, which we give in terms of a metric operator associated with the observable under measurement. This formula is proved to be equal to the usual Born rule for any Hermitian observable, but for a non-Hermitian observable, it is usually dependent on the choice of a metric operator involved for the measurement of the associated non-Hermitian observable. This non-Hermitian Born formula satisfies probability conservation for both Hermitian and non-Hermitian observables. Thus, in our formalism, a metric operator associated with a para-Hermitian operator plays a role of measurement only, as pointed out in \cite{SGH1992}, but it needs not to introduce a new inner product.
\begin{comment}
Roughly speaking, the measurement of a non-Hermitian observable has to be contextual in general, contrary to the basic fact that the measurement of a Hermitian observable is always non-contextual in the conventional quantum mechanics. Accordingly, our formalism is mathematically different from the existed formulation of non-Hermitian quantum mechanics, and explicitly utilizes the measurement contextuality in place of the Hamiltonian-dependence of $PT$-symmetric quantum mechanics and the basis-dependence of biorthogonal quantum mechanics mentioned above.\end{comment}
The paper is organized as follows. In Section \ref{Pre}, we include some definitions and preliminary results on para-Hermitian and para-unitary operators and evolution systems. In particular, we prove a Stone type theorem associated with the para-unitary operator. In Section \ref{Axiom}, we present a mathematical formalism of non-Hermitian quantum mechanics and give some examples for illustration. In Section \ref{PTBIqm}, we explain how to recover PT-symmetric and biorthogonal quantum mechanics from our formalism. As application, in Section \ref{GeoPhase}, we study geometric phases for non-Hermitian quantum systems, focusing on the observable-geometric phase of a time-dependent non-Hermitian quantum system. We give a summary in Section \ref{Sum}. Finally, we include an appendix, namely Section \ref{App}, on the geometry of non-Hermitian observable space, which is needed for a geometrical description of the observable-geometric phase associated with a non-Hermitian quantum system.
\section{Preliminaries}\label{Pre}
In what follows, we utilize the standard notions and notations from functional analysis (cf. \cite{Rudin1991}). $\mathbb{C}$ denotes the complex field and $\mathbb{C}_* = \mathbb{C} \setminus \{0\}.$ We denote by $\mathbb{H}$ a complex separable Hilbert space with an inner product $\langle \cdot, \cdot \rangle,$ linear in the second entry, and $\mathbb{H}_* = \mathbb{H} \setminus \{0\}.$ By an {\it operator} $T$ in $\mathbb{H}$ we shall mean a linear mapping whose domain $\mathcal{D} (T)$ is a (not necessarily closed) linear subset of $\mathbb{H}$ and whose range $\mathcal{R} (T)$ lies in $\mathbb{H}.$ We always use $I$ denote the identity operator; $T^*$ the adjoint operator for any densely defined operator $T$ in $\mathbb{H};$ $\mathcal{B} (\mathbb{H})$ the algebra of all bounded operators; $\mathcal{O} (\mathbb{H})$ the set of all Hermitian (self-adjoint) operators; $\mathcal{P} (\mathbb{H})$ the set of all orthogonal projections; $\mathcal{U} (\mathbb{H})$ the group of all unitary operators on $\mathbb{H},$ and $\mathcal{T} (\mathbb{H})$ the group of all bounded operators with bounded inverse on $\mathbb{H}.$ Note that $\mathcal{P} (\mathbb{H}),$ $\mathcal{U} (\mathbb{H})$ and $\mathcal{T} (\mathbb{H})$ are all subsets of $\mathcal{B} (\mathbb{H}).$
\subsection{Spectral operators}\label{Pre:SpecOper}
If an operator $P \in \mathcal{B} (\mathbb{H})$ satisfies $P^2 = P,$ it is called a {\it projection} as in \cite{DS1971,RS1980I,Rudin1991} (or a {\it skew projection} in some literatures, cf.\cite{Ovch1993}), and is an {\it orthogonal projection} if in addition $P^* =P.$ For a projection $P,$ its adjoint operator $P^*$ and complementary operator $P^\bot = I -P$ are both projections. We denote by $\tilde{\mathcal{P}} (\mathbb{H})$ the set of all projections in $\mathbb{H},$ and thus $\tilde{\mathcal{P}} (\mathbb{H}) \supset \mathcal{P} (\mathbb{H}).$ For two commuting projections $P, Q \in \tilde{\mathcal{P}} (\mathbb{H}),$ the intersection $P\wedge Q$ is defined by
\be
P\wedge Q = P Q
\ee
with the range $P\wedge Q (\mathbb{H}) = P (\mathbb{H}) \cap Q (\mathbb{H}),$ and the union $P \vee Q$ by
\be
P \vee Q = P+Q - P Q,
\ee
with the range $P \vee Q (\mathbb{H}) = P (\mathbb{H}) + Q (\mathbb{H}) = \overline{\mathrm{span}} [P (\mathbb{H}) \cup Q (\mathbb{H})],$ the closed subspace of $\mathbb{H}$ spanned by the sets $P (\mathbb{H})$ and $Q (\mathbb{H}).$ The order $P \le Q$ between two commuting projections $P, Q \in \tilde{\mathcal{P}} (\mathbb{H})$ is defined to be $P (\mathbb{H}) \subset Q (\mathbb{H}).$ A Boolean algebra of projections in $\mathbb{H}$ is a subset of $\tilde{\mathcal{P}} (\mathbb{H})$ which is a Boolean algebra under operations $\wedge, \vee$ and $\le$ together with its zero and unit elements being the operators $0$ and $I$ in $\mathcal{B} (\mathbb{H})$ respectively.
\begin{definition}\label{df:SpecMeasure}{\rm (cf. \cite{Dunf1958, DS1971})}\;
Let $\Sigma$ be a $\sigma$-field of subsets of a non-empty set $\Omega.$ A spectral measure on $\Sigma$ is a map $\mathbf{E}$ from $\Sigma$ into a Boolean algebra of projections in $\mathbb{H}$ satisfying the following conditions:
\begin{enumerate}[$1)$]
\item $\mathbf{E} (\emptyset) =0$ and $\mathbf{E} (\Omega) =I.$
\item For any $A, B \in \Sigma,$ $\mathbf{E} (\Omega \setminus A) = \mathbf{E} (A)^\bot,$
\be
\mathbf{E} (A \cap B) = \mathbf{E} (A) \wedge \mathbf{E} (B), \quad \mathbf{E} (A \cup B) = \mathbf{E} (A) \vee \mathbf{E} (B).
\ee
\item $\mathbf{E} (A)$ is countably additive in $A$ in the strong operator topology, i.e., for every sequence $\{A_n\}$ of mutually disjoint sets in $\Sigma,$
\be
\mathbf{E} (\cup_n A_n) x = \sum_n \mathbf{E} (A_n)x
\ee
holds for any $x \in \mathbb{H},$ where the series of the right hand side converges in $\mathbb{H}$ in the norm topology.
\end{enumerate}
\end{definition}
\begin{remark}\label{rk:DualSpecMeasure}\rm
For a spectral measure $\mathbf{E}$ on $\Sigma,$ define $\mathbf{E}^* (A) = [\mathbf{E} (A)]^*$ for every $A \in \Sigma.$ Then $\mathbf{E}^*$ is also a spectral measure $\mathbf{E}$ on $\Sigma,$ called the dual of $\mathbf{E}.$
\end{remark}
Note that every spectral measure $\mathbf{E}$ on $\Sigma$ is bounded, i.e., $\sup_{A \in \Sigma} \| \mathbf{E} (A) \| < \8.$ In this case, the integral $\int_\Omega f(\omega) \mathbf{E} (d \omega)$ may be defined for every bounded $\Sigma$-measurable (complex-valued) function defined $\mathbf{E}$-almost everywhere on $\Omega.$ Recall that a function $f$ is defined $\mathbf{E}$-almost everywhere on $\Omega,$ if there exists $\Omega_0 \in \Sigma$ such that $\mathbf{E} (\Omega_0) = I$ and $f$ is well defined for every $\omega \in \Omega_0.$ It was shown (cf. \cite[X.1]{DS1963}) that this integral is a bounded homomorphism of the $C^*$-algebra of $\mathcal{B}(\Omega, \Sigma)$ of bounded $\Sigma$-measurable functions in $\Omega$ with the norm $\|f\| = \sup_{\omega \in \Omega} |f(\omega)|$ into the $C^*$-algebra $\mathcal{B} (\mathbb{H}),$ that is, for any $\alpha, \beta \in \mathbb{C}$ and for $f, g \in \mathcal{B}(\Omega, \Sigma),$
\be\begin{split}
\int_\Omega [\alpha f (\omega) + \beta g (\omega)] \mathbf{E} (d \omega) & = \alpha \int_\Omega f (\omega) \mathbf{E} (d \omega) + \beta \int_\Omega g (\omega) \mathbf{E} (d \omega),\\
\int_\Omega f (\omega) g (\omega) \mathbf{E} (d \omega) & = \int_\Omega f (\omega) \mathbf{E} (d \omega) \int_\Omega g (\omega) \mathbf{E} (d \omega),\\
\Big \| \int_\Omega f (\omega) \mathbf{E} (d \omega) \Big \| & \le C_\mathbf{E} \sup_{\omega \in \Omega} |f(\omega)|,
\end{split}\ee
where $C_\mathbf{E}$ is a positive constant depending only upon the spectral measure $\mathbf{E}.$
In the sequel, we will focus on the spectral measures on the $\sigma$-field of Borel sets in the complex plane $\mathbb{C},$ denoted by $\mathcal{B}_\mathbb{C}.$
\begin{definition}\label{df:SpecOp}{\rm (cf. \cite[Definition XVIII.2.1]{DS1971})}\;
A densely defined closed operator $T$ in $\mathbb{H}$ with the domain $\mathcal{D} (T)$ is called a spectral operator, if there is a spectral measure $\mathbf{E}$ on $\mathcal{B}_\mathbb{C}$ such that
\begin{enumerate}[$1)$]
\item $\mathbf{E}$ is regular, i.e., for any $x,y \in \mathbb{H},$ the complex-valued measure $A \mapsto \langle x, \mathbf{E} (A) y \rangle$ is regular on $\mathcal{B}_\mathbb{C},$
\item for any bounded set $A \in \mathcal{B}_\mathbb{C},$ $\mathbf{E} (A) \mathbb{H} \subset \mathcal{D} (T),$
\item for any $B \in \mathcal{B}_\mathbb{C},$ $\mathbf{E} (B) \mathcal{D} (T) \subset \mathcal{D} (T)$ and
\be
T \mathbf{E} (B) x = \mathbf{E} (B) T x
\ee
for all $x \in \mathcal{D} (T),$
\item for any $B \in \mathcal{B}_\mathbb{C},$ the spectral set $\sigma (T|\mathbf{E} (B) (\mathbb{H}))$ of the restriction $T|\mathbf{E} (B) (\mathbb{H})$ of $T$ to $\mathbf{E} (B) (\mathbb{H})$ is contained in the closure $\bar{B}$ of $B,$ i.e.,
\be
\sigma (T|\mathbf{E} (B) (\mathbb{H})) \subset \bar{B},
\ee
where the domain $\mathcal{D} (T|\mathbf{E} (B) (\mathbb{H})) = \mathcal{D} (T) \cap \mathbf{E} (B) (\mathbb{H}).$
\end{enumerate}
The spectral measure $\mathbf{E}$ is called the {\it spectral resolution} or {\it resolution of the identity} for $T.$
\end{definition}
Note that the spectral measure $\mathbf{E}$ is uniquely determined by $T,$ i.e., the spectral resolution of a densely defined closed operator in $\mathbb{H}$ is unique whenever it exists (cf. \cite[Theorem XVIII.2.5]{DS1971}).
\begin{definition}\label{df:SpecOpScalar}{\rm (cf. \cite[Definition XVIII.2.12]{DS1971})}\;
A densely defined closed operator $T$ in $\mathbb{H}$ with the domain $\mathcal{D} (T)$ is of scalar type, if there is a spectral measure $\mathbf{E}$ on $\mathcal{B}_\mathbb{C}$ such that
\be
\mathcal{D} (T) = \{ x \in \mathbb{H}: \lim_n T_n x\;\text{exists}\}
\ee
and
\be
T x = \lim_n T_n x,\quad \forall x \in \mathcal{D} (T),
\ee
where
\be
T_n = \int_{\{z \in\mathbb{C}: |z| \le n\} } z \mathbf{E} (d z).
\ee
The spectral measure $\mathbf{E}$ is said to be the spectral resolution for $T.$
\end{definition}
\begin{remark}\label{rk:ScalarTypeOp}\rm
It is shown in \cite[Lemma XVIII.2.13]{DS1971} that a scalar type operator $T$ in the sense of Definition \ref{df:SpecOpScalar} is a spectral operator in the sense of Definition \ref{df:SpecOp} and the spectral resolution of $T$ is unique. Thus, a scalar type operator is also called a {\it spectral operator of scalar type}. A theorem of Wermer (cf. \cite[Theorem XV.6.2.4]{DS1971}) states that a bounded spectral operator $T$ of scalar type is equivalent to a normal operator, that is, there exists a bounded self-adjoint operator $K$ with bounded inverse $K^{-1}$ such that the operator $K T K^{-1}$ is a normal operator.
\end{remark}
\subsection{Para-Hermitian operators}\label{Pre:para-Hermi}
Now we are ready to introduce the notion of {\it para-Hermitian} operators, which plays an essential role in the mathematical formulation of non-Hermitian quantum mechanics.
\begin{definition}\label{df:paraHermiOp}
A densely defined closed operator $T$ in $\mathbb{H}$ is called a para-Hermitian operator, if it is a spectral operator of scalar type with real spectrum, namely $\sigma (T) \subset \mathbb{R}.$
\end{definition}
We denote by $\tilde{\mathcal{O}} (\mathbb{H})$ the set of all para-Hermitian operators. Thus, $\mathcal{O} (\mathbb{H}) \subset \tilde{\mathcal{O}} (\mathbb{H}).$
Recall that an operator $G$ in $\mathbb{H}$ is called a {\it metric operator} (cf. \cite{AT2014, Mosta2010}), if $G$ is a bounded and strictly positive self-adjoint operator having bounded inverse $G^{-1}.$ Given a metric operator $G$ in $\mathbb{H},$ we can define a new inner product $\langle \cdot, \cdot \rangle_G$ in $\mathbb{H}$ by $\langle u, v \rangle_G = \langle u, Gv \rangle$ for any $u,v \in \mathbb{H}.$ Then the induced norm $\|u\|_G = \|G^\frac{1}{2} u \|$ is really equivalent to the original norm of $\mathbb{H}.$
\begin{proposition}\label{prop:paraHop}{\rm (cf. \cite[Proposition 3.12]{AT2014})}
Let $T$ be a densely defined closed operator in $\mathbb{H}.$ Then the following statements are
equivalent:
\begin{enumerate}[\rm 1)]
\item $T$ is a para-Hermitian operator.
\item There exists a metric operator $G$ such that $T$ is self-adjoint with respect to the inner $\langle \cdot, \cdot \rangle_G.$
\item There exists a metric operator $G$ such that $G^\frac{1}{2} T G^{-\frac{1}{2}}$ is self-adjoint.
\end{enumerate}
\end{proposition}
\begin{remark}\label{rk:MetricOp}\rm
The metric operator $G$ associated with a para-Hermitian operator $T$ in the above proposition is dependent on $T$ itself, and needs not to be unique in general (see Example \ref{ex:NHObM} below). We denote by $\mathcal{M} (T)$ the set of all metric operator $G$ associated with a para-Hermitian operator $T.$ Evidently, for any Hermitian operator $T$ the identity operator $I \in \mathcal{M} (T)$ but $I$ is not necessarily a unique metric operator associated with a Hermitian operator.
\end{remark}
There are more general concepts than para-Hermitian operators commonly found in the literature, such as {\it quasi-Hermitian} and {\it pseudo-Hermitian} operators (cf. \cite{AT2014, Mosta2010}), we include their definitions here for the sake of convenience.
\begin{definition}\label{df:quasipseudoHermiOp}
Let $T$ be a densely defined closed operator in $\mathbb{H}.$
\begin{enumerate}[\rm 1)]
\item $T$ is called a quasi-Hermitian operator, if there exists a bounded and strictly positive self-adjoint operator $G$ such that
\be
G T = T^* G.
\ee
\item $T$ is called a pseudo-Hermitian operator, if there exists a bounded self-adjoint operator $\eta$ with bounded inverse $\eta^{-1},$ such that
\be
T^* = \eta T \eta^{-1}.
\ee
\end{enumerate}
\end{definition}
\begin{remark}\label{rk:QuasiPseudoOp}\rm
By definition, a quasi-Hermitian operator is para-Hermitian if the operator $G$ has bounded inverse $G^{-1},$ while a pseudo-Hermitian operator is para-Hermitian if the operator $\eta$ is a positive operator. Note that, the definitions of quasi-Hermitian and pseudo-Hermitian operators have been respectively adapted to the cases of a unbounded metric operator $G$ (cf. \cite{AT2014}) and a unbounded self-adjoint operator $\eta$ (cf. \cite{Mosta2013}).
\end{remark}
\begin{definition}\label{df:FunctCalculusparaHop}{\rm (cf. \cite[Definition XVIII.2.10]{DS1971})}\;
Let $T$ be a spectral operator of scalar type with the spectral resolution $\mathbf{E}$ on $\mathcal{B}_\mathbb{C}.$ For any $\mathcal{B}_\mathbb{C}$-measurable function $f,$ we define $f (T)$ by
\be
f(T) x = \lim_n T(f_n) x,\quad \forall x \in \mathcal{D} (f(T)),
\ee
where
\be\begin{split}
\mathcal{D} (f(T)) =& \{x \in \mathbb{H}: \lim_n T(f_n) x\;\text{exists}\},\\
T(f_n) = & \int_\mathbb{C} f_n (z) \mathbb{E} (d z),
\end{split}\ee
and
\be
f_n (z) = \left \{\begin{split} & f(z),\quad |f(z)| \le n,\\
& 0, \quad |f(z)| >0.
\end{split}\right.
\ee
\end{definition}
\begin{remark}\label{rk:FunctCalculusparaHop}\rm
It is shown in \cite[Theorem XVIII.2.17]{DS1971} that $f(T)$ in the above definition is a spectral operator of scalar type with the spectral resolution $\mathbf{E}_f (E) = \mathbf{E} (f^{-1} (E))$ for any $E \in \mathcal{B}_\mathbb{C}.$
\end{remark}
Thus, we have the well-defined functional calculus for para-Hermitian operators, which plays a role in the dynamics of non-Hermitian quantum mechanics as called the Stone-type theorem in the sequel. However, there seems no such functional calculus for either {\it quasi-Hermitian} or {\it pseudo-Hermitian} operators. Mathematically, this is the reason why we use para-Hermitian operators representing the observable beyond quasi-Hermitian and pseudo-Hermitian operators.
\subsection{A Stone-type theorem}\label{Pre:Stone}
At first, we need to introduce the notion of a {\it para-unitary operator}, corresponding to the one of a para-Hermitian operator.
\begin{definition}\label{df:paraUOp}
A bounded spectral operator $U$ of scalar type is said to be para-unitary if $\sigma (U) \subset \mathbb{T},$ namely $|\lambda| =1$ for all $\lambda \in \sigma (U).$
\end{definition}
We denote by $\tilde{\mathcal{U}} (\mathbb{H})$ the set of all para-unitary operators in $\mathbb{H}.$ Thus, $\mathcal{U} (\mathbb{H}) \subset \tilde{\mathcal{U}} (\mathbb{H}).$
\begin{proposition}\label{prop:paraUop}
A bounded spectral operator $U$ of scalar type in $\mathbb{H}$ is para-unitary if and only if there exists a metric operator $G$ such that $G^\frac{1}{2} U G^{-\frac{1}{2}}$ is unitary.
\end{proposition}
\begin{proof}
Suppose that $U$ is a para-unitary operator. By a theorem of Wermer (cf. \cite[Theorem XV.6.4]{DS1971}), there exists a bounded self-adjoint operator $K$ with bounded inverse $K^{-1}$ such that $K U K^{-1}$ is normal. Since $\sigma (U) \subset \mathbb{T},$ then $\sigma (K U K^{-1}) \subset \mathbb{T}$ and so $K U K^{-1}$ is unitary with the spectral decomposition
\be
K U K^{-1} = \int_\mathbb{T} \lambda \mathbf{E} (d \lambda),
\ee
where $\mathbf{E}$ is a self-adjoint spectral resolution. Putting $G = |K|^2,$ by the polar decomposition we have $K = V G^\frac{1}{2}$ with $V$ unitary such that
\be
G^\frac{1}{2} U G^{-\frac{1}{2}} = \int_\mathbb{T} \lambda \mathbf{F} (d \lambda),
\ee
where $\mathbf{F} (\cdot) = V^{-1}\mathbf{E} (\cdot) V$ is a self-adjoint spectral resolution. Thus, $G^\frac{1}{2} U G^{-\frac{1}{2}}$ is unitary.
Conversely, if there exists a bounded and strictly positive self-adjoint operator $G$ with bounded inverse $G^{-1},$ such that $G^\frac{1}{2} U G^{-\frac{1}{2}}$ is unitary, then
\be
U = \int_\mathbb{T} \lambda G^{-\frac{1}{2}} \mathbf{E} G^\frac{1}{2}(d \lambda)
\ee
where $\mathbf{E}$ is a self-adjoint spectral resolution. Clearly, $\mathbf{F} (\cdot) = G^{-\frac{1}{2}}\mathbf{E} (\cdot) G^\frac{1}{2}$ is a spectral resolution for $U$ and $\sigma (U) \subset \mathbb{T}.$ Hence, $U$ is a para-unitary operator. This completes the proof.
\end{proof}
The following proposition shows the relationship between para-Hermitian and para-unitary operators through function calculus about spectral operators of scalar type.
\begin{proposition}\label{prop:paraHUop}
Let $H$ be a para-Hermitian operator. If $f(z) = e^{\mathrm{i}z},$ then $f(H)$ is a para-unitary operator, denoted by $e^{\mathrm{i} H}.$
\end{proposition}
\begin{proof}
Let $H$ be a para-Hermitian operator and $f(z) = e^{\mathrm{i}z}.$ By \cite[Theorem XVIII.2.21]{DS1971}, $\sigma (f(H)) = \overline{f(\sigma(H))} \subset \mathbb{T}.$ Therefore, by \cite[Theorem XVIII.2.11(c) and Theorem XVIII.2.17]{DS1971}, $e^{\mathrm{i}H}$ is a para-unitary operator.
\begin{comment}
Conversely, let $U$ be a para-unitary operator. We peak $\theta \in (0, 2 \pi]$ so that $e^{\mathrm{i} \theta} \notin \sigma (U),$ and define
\be
H = \mathrm{i} \mathrm{Ln}_\theta U,
\ee
where $\mathrm{Ln}_\theta$ is the complex logarithm with the branch cut along the half-line $z = r e^{\mathrm{i} \theta}$ with $r \ge 0,$ i.e.,
$$
\mathrm{Ln}_\theta (r e^{\mathrm{i} \phi} ) = \ln r + \mathrm{i} \phi
$$
for $r >0$ and for $\theta - 2 \pi < \phi < \theta.$ By definition, $\mathrm{Ln}_\theta$ is analytic in a neighborhood of $\sigma (U),$ and by \cite[Theorem XVIII.2.21]{DS1971} again, $\sigma (H) = \overline{\mathrm{i}\mathrm{Ln}_\theta \sigma (U)} \subset \mathbb{R}.$ Therefore, by \cite[Theorem XVIII.2.17]{DS1971} again, $H$ is a para-Hermitian operator such that $U = e^{\mathrm{i}H}.$ This completes the proof.\end{comment}
\end{proof}
In what follows, we will prove Stone's theorem for the one-parameter group of para-unitary operators, which is fundamental for the dynamics of non-Hermitian quantum mechanics.
\begin{proposition}\label{prop:paraUgro}
Let $H$ be a para-Hermitian operator and define $U(t) = e^{\mathrm{i} t H}$ for every $t \in \mathbb{R}.$ Then
\begin{enumerate}[\rm 1)]
\item For each $t \in \mathbb{R},$ $U(t)$ is a para-unitary operator and
\be
U(t+s) = U(t) U(s),\quad \forall t,s \in \mathbb{R}.
\ee
\item For each $x \in \mathbb{H},$ $\lim_{t \to t_0} U(t) x = U(t_0) x$ in $\mathbb{H},$ that is, $t \mapsto U(t)$ is strongly continuous.
\item $U(0) = I$ and $\{U(t): t \in \mathbb{R} \}$ is a bounded Abelian group of para-unitary operators such that $U(t)^{-1} = U(-t)$ for every $t \in \mathbb{R}.$
\item For every $x \in \mathcal{D} (H),$
\be
\lim_{t \to 0} \frac{U(t) x - x}{t} = \mathrm{i} H x.
\ee
\item If $x \in \mathbb{H}$ such that $\lim_{t \to 0} (U(t) x - x)/t$ exists, then $x \in \mathcal{D} (H).$
\end{enumerate}
\end{proposition}
\begin{proof}
1) follows immediately from Proposition \ref{prop:paraHUop} and the functional calculus for the complex-valued function $e^{\mathrm{i} t z}.$ To prove 2) note that
\be
\|e^{\mathrm{i} t H} x - x \|^2 = \int_\mathbb{R} \int_\mathbb{R} (e^{\mathrm{i} t \lambda} -1) (e^{\mathrm{i} t \gamma} -1) \langle \mathbf{E}^* (d \gamma) \mathbf{E} (d \lambda)x, x \rangle,
\ee
where $\mathbf{E}$ is the spectral resolution of $H$ and $\mathbf{E}^*$ is the dual of $\mathbf{E}$ (cf. Remark \ref{rk:DualSpecMeasure}). Define $\mathbf{F} (A \times B) = \mathbf{E}^* (A) \mathbf{E} (B)$ for any $A,B \in \mathcal{B}_\mathbb{R}$ (the $\sigma$-algebra of Borel sets in $\mathbb{R}$). Then $\mathbf{F}$ extends to a bounded operator-valued measure on $\mathcal{B}_{\mathbb{R}^2}$ (the $\sigma$-algebra of Borel sets in $\mathbb{R}^2$) such that
\be
\|e^{\mathrm{i} t H} x - x \|^2 = \int_{\mathbb{R}^2} (e^{\mathrm{i} t \lambda} -1) (e^{\mathrm{i} t \gamma} -1) \langle \mathbf{F} (d \gamma \times d \lambda)x, x \rangle.
\ee
Note that $\mu_x (K) = \langle \mathbf{F} (K)x, x \rangle$ is a complex measure on $\mathcal{B}_{\mathbb{R}^2}.$ Since $|(e^{\mathrm{i} t \lambda} -1) (e^{\mathrm{i} t \gamma} -1)| \le 4$ and
\be
\|e^{\mathrm{i} t H} x - x \|^2 \le \int_{\mathbb{R}^2} |(e^{\mathrm{i} t \lambda} -1) (e^{\mathrm{i} t \gamma} -1)| |\mu_x| (d \gamma \times d \lambda),
\ee
we conclude that $\lim_{t \to 0} \|e^{\mathrm{i} t H} x - x \|^2 =0$ by the Lebesgue dominated convergence theorem. Thus $t \mapsto U(t)$ is strongly continuous at $t=0,$ which implies by the group property that $t \mapsto U(t)$ is strongly continuous at any $t \in \mathbb{R}.$
For 3), by Proposition \ref{prop:paraHop} there is a metric operator $G$ such that $G^\frac{1}{2} H G^{-\frac{1}{2}}$ is self-adjoint. By the uniqueness of spectral resolution and functional calculus, we conclude that $G^\frac{1}{2} U(t) G^{-\frac{1}{2}} = e^{\mathrm{i} t G^\frac{1}{2} H G^{-\frac{1}{2}}}$ is unitary for every $t \in \mathbb{R}.$ Thus,
\be
\| U(t) \| = \| G^{-\frac{1}{2}} e^{\mathrm{i} t G^\frac{1}{2} H G^{-\frac{1}{2}}} G^\frac{1}{2} \| \le \| G^{-\frac{1}{2}} \| \| G^\frac{1}{2} \|
\ee
for all $t \in \mathbb{R},$ i.e., $\{U(t): t \in \mathbb{R} \}$ is a bounded set of operators.
For 4) and 5), as above there is a metric operator $G$ such that $G^\frac{1}{2} U(t) G^{-\frac{1}{2}} = e^{\mathrm{i} t G^\frac{1}{2} H G^{-\frac{1}{2}}}$ is unitary for every $t \in \mathbb{R}.$ Thus, by \cite[Theorem VIII.7 (c) and (d)]{RS1980I} we obtain 4) and 5), since $x \in \mathcal{D} (H)$ if and only if $G^\frac{1}{2} x \in \mathcal{D} (G^\frac{1}{2} H G^{-\frac{1}{2}}).$
\end{proof}
\begin{remark}\label{rk:SkewUnitaryGroup}\rm
An operator-valued function $t \mapsto U(t)$ from $\mathbb{R}$ into $\tilde{\mathcal{U}} (\mathbb{H})$ satisfying $1)$ and $2)$ is called {\it a strongly continuous one-parameter para-unitary group}.
\end{remark}
The following theorem says that every strongly continuous bounded one-parameter para-unitary group arises as the exponential of a para-Hermitian operator, that is, Stone's theorem holds true in the para-Hermitian case.
\begin{theorem}\label{th:StoneTh}
Let $(U(t):\; t \in \mathbb{R})$ be a strongly continuous bounded one-parameter para-unitary group on a Hilbert space $\mathbb{H}.$ Then there is a para-Hermitian operator $H$ on $\mathbb{H}$ so that $U(t) = e^{\mathrm{i}t H}$ for every $t \in \mathbb{R}.$
\end{theorem}
\begin{proof}
By \cite[Lemma XV.6.1]{DS1971}, there exists a bounded self-adjoint operator $K$ with bounded inverse $K^{-1}$ such that $K U(t) K^{-1}$ is unitary for every $t \in \mathbb{R}.$ By the polar decomposition of $K,$ $G^\frac{1}{2} U(t) G^{-\frac{1}{2}}$ is unitary for every $t \in \mathbb{R},$ where $G = |K|^2$ is a bounded and strictly positive self-adjoint operator with bounded inverse $G^{-1}.$ Since $(G^\frac{1}{2} U(t) G^{-\frac{1}{2}}: t \in \mathbb{R})$ is a strongly continuous one-parameter unitary group, by Stone's theorem (cf.\cite[Theorem VIII.8]{RS1980I}) there exists a self-adjoint operator $A$ such that $e^{\mathrm{i} t A} = G^\frac{1}{2} U(t) G^{-\frac{1}{2}}$ for every $t \in \mathbb{R}.$ By the uniqueness of spectral resolution and functional calculus, we conclude that
\be
U(t) = G^{-\frac{1}{2}} e^{\mathrm{i} t A} G^\frac{1}{2} = e^{\mathrm{i} t H},\quad \forall t \in \mathbb{R},
\ee
where $H = G^{-\frac{1}{2}} A G^\frac{1}{2}$ is a para-Hermitian operator by Proposition \ref{prop:paraHop}.
\end{proof}
\begin{remark}\label{rk:paraUnitaryGroupInfGenerator}\rm
If $(U(t):\; t \in \mathbb{R})$ is a strongly continuous bounded one-parameter para-unitary group, then the para-Hermitian operator $H$ with $U(t)= e^{\mathrm{i}t H}$ ($\forall t \in \mathbb{R}$) is called the {\it infinitesimal generator} of $(U(t):\; t \in \mathbb{R}).$
\end{remark}
\begin{remark}\label{rk:paraUnitaryGroupPl}\rm
Concerning Theorem \ref{th:StoneTh}, a question arises: Whether does there exist a strongly continuous one-parameter para-unitary group on an infinite-dimensional Hilbert space $\mathbb{H}$ which is unbounded as a subset of $\mathcal{B} (\mathbb{H})$? At the time of this writing, we have no such example.
\end{remark}
\subsection{Evolution systems}\label{Pre:EvoSys}
Any two-parameter family of bounded operators $\{U(t,s) \in \mathcal{B} (\mathbb{H}): s,t \in [0, T]\}$ is said to be an {\it evolution system} (cf. \cite{SG2017}), if it satisfies the following conditions:
\begin{enumerate}[(i)]
\item $U(t,t) = I$ and $U(t, r) U (r, s) = U(t,s)$ for all $s,r,t \in [0, T];$
\item $(t,s) \mapsto U(t,s)$ is strongly continuous on $[0,T] \times [0, T].$
\end{enumerate}
Note that by (i), $U(t,s)$ are all bounded operators with bounded inverse and $U(t,s)^{-1} = U(s,t),$ namely $U(t,s) \in \mathcal{T} (\mathbb{H}).$ In some literatures, an evolution system is only assumed to satisfy (i) and (ii) on the triangle region $0\le s \le t \le T$ (cf. \cite{Pazy1983}). In this case, $U(t,s)$ need not to be invertible.
Let $\{A(t): t \in [0, T]\}$ be a family of densely defined closed operators in $\mathbb{H}$ with a property that there exists a dense subset $\mathbb{D}$ of $\mathbb{H}$ such that $\mathbb{D} \subset \mathcal{D} (A(t))$ for all $t \in [0,T].$ If a evolution system $\{U(t,s) \in \mathcal{T} (\mathbb{H}): s,t \in [0, T]\}$ satisfies the condition that $U(t,s) \mathbb{D} \subset \mathbb{D}$ for all $s,t \in[0,T],$ and for any $v \in \mathbb{D}$ and $s \in [0,T],$ the map $t \mapsto U(t,s) v$ is continuously differentiable in $[0, T]$ such that
\begin{equation}\label{eq:EvoSysEqu}
\frac{d}{d t} U(t,s)v = A(t) U(t,s)v, \quad \forall t \in [0,T],
\end{equation}
then it is called an evolution system for $A(t)$ on $\mathbb{D}$ (cf. \cite{SG2017}).
It is well known that if $A(t)$'s are all bounded operators such that $[0,T] \ni t \mapsto A(t) \in \mathcal{B}(\mathbb{H})$ is strongly continuous, then there exists a unique evolution system $\{U(t,s):t,s \in [0,T]\}$ such that the evolution equations
\begin{equation}\label{eq:EvoSysEquI}
\frac{d}{d t} U(t,s) = A(t) U(t,s),
\end{equation}
and
\begin{equation}\label{eq:EvoSysEquII}
\frac{d}{d t} U(s,t) = -U(s,t) A(t)
\end{equation}
hold in the strong topology of $\mathcal{B} (\mathbb{H})$ for any $s \in [0, T]$ (see \cite{Pazy1983} for the details).
We refer to \cite{NZ2009, Schmid2016, SG2017} for the details on the existence of the evolution systems for a family of densely defined closed operators in a Hilbert space. In fact, by \cite[Theorem 2.1]{SG2017}, we have the following result:
\begin{proposition}\label{prop:paraEvoSys}
Let $\{h(t): t \in [0, T]\}$ be a family of para-Hermitian operators in $\mathbb{H},$ having the same domain $\mathbb{D},$ namely $\mathbb{D} = \mathcal{D} (h(t))$ for all $t \in [0,T].$ If there exists $\omega >0$ such that
\be
\| e^{-\mathrm{i} s h(t)} \| \le e^{\omega |s|},\quad \forall s \in \mathbb{R},\forall t \in [0,T],
\ee
and if the map $[0,T] \ni t \mapsto h(t) \in \mathcal{B} (\mathbb{D}, \mathbb{H})$ is continuous and of bounded variation, where $\mathbb{D}$ is endowed with the graph norm of $h(0),$ then there exists a unique evolution system $\{U(t,s):t,s \in [0,T]\}$ for $A(t)= - \mathrm{i} h(t)$ on $\mathbb{D}.$
\end{proposition}
\begin{proof}
This is so, because every $A(t) = - \mathrm{i} h(t)$ generates a strongly continuous group $\{e^{-\mathrm{i} s h(t)}: s \in \mathbb{R}\}$ by Proposition \ref{prop:paraUgro}.
\end{proof}
\section{Mathematical axiom}\label{Axiom}
Following the Dirac-von Neumann formalism of quantum mechanics \cite{Dirac1958, vN1955}, we present a mathematical formalism of non-Hermitian quantum mechanics in what follows. Precisely, this formalism includes the following five postulates:
\begin{definition}\label{df:MathFNHQM}
The mathematical formalism of non-Hermitian quantum mechanics is defined by a set of postulates as follows:
\begin{enumerate}[$(P_1)$]
\item {\bf The state postulate}\; Associated with a non-Hermitian quantum system is a complex separable Hilbert space $\mathbb{H},$ the system at any given time is described by a state, which is determined by a nonzero vector in $\mathbb{H}.$
\item {\bf The observable postulate}\; Each observable for a non-Hermitian quantum system associated with a complex separable Hilbert space $\mathbb{H}$ is represented by a para-Hermitian operator in $\mathbb{H}.$
\item {\bf The measurement postulate}\; For an observable represented by a para-Hermitian operator $A$ in $\mathbb{H},$ if $G$ is a metric operator associated with $A,$ then $G$ introduces a measurement context for the observable $A$ such that the expectation of $A$ at a certain state determined by a nonzero vector $\psi$ with $ G^{-\frac{1}{2}} \psi \in \mathcal{D} (A)$ is given by
\beq\label{eq:BornRuleNH}
\langle A \rangle_{\psi, G} = \frac{\langle \psi, G^\frac{1}{2} A G^{-\frac{1}{2}} \psi \rangle}{\| \psi \|^2}.
\eeq
In particular, if $A$ has a discrete spetrum $\{\lambda_n\}_{n \ge 1},$ whose eigenstates $\{e_n\}_{n \ge 1}$ is a unconditional basis in $\mathbb{H},$ then the expectation of $A$ at $\psi$ with $G^{-\frac{1}{2}} \psi \in \mathcal{D} (A),$ where $G = \sum_{n \ge 1} |e^*_n\rangle \langle e^*_n|$ being a metric operator for $A,$ is given by
\beq\label{eq:BornRuleNHV}
\langle A \rangle_{\psi, \Pi} = \sum^\8_{n = 1} \lambda_n \frac{|\langle e^*_n, G^{-\frac{1}{2}} \psi\rangle |^2}{\| \psi \|^2},
\eeq
under the measurement $\Pi = \{|e_n\rangle \langle e^*_n|: n \ge 1 \}$ or equivalently in the measurement context of $G.$ In this case, $\psi$ will be changed to the state $e_n$ with probability
\beq\label{eq:TransProba}
p(\psi | e_n) = \frac{|\langle e^*_n, G^{-\frac{1}{2}} \psi\rangle |^2}{\| \psi \|^2}
\eeq
for each $n \ge 1.$
\item {\bf The evolution postulate}\; The system described by vectors is changed with time according to the Schr\"{o}dinger equation
\beq\label{eq:BiSchrEqu}
\mathrm{i} \frac{d \psi (t) }{d t} = H \psi (t),
\eeq
where $H$ is a para-Hermitian operator, which is called the energy operator of the system.
\item {\bf The composite-systems postulate}\; The Hilbert space associated with a composite non-Hermitian quantum system is the Hilbert space tensor product of the Hilbert spaces of its components. If systems numbered $1$ through $n$ are prepared in states $\psi_k,$ $k=1,\ldots, n,$ then the joint state of the composite total system is the tensor product $\psi_1 \otimes \cdots \otimes \psi_n.$
\end{enumerate}
\end{definition}
\begin{remark}\label{rk:NHQM}\rm
\begin{enumerate}[$1)$]
\item Both postulates $(P_1)$ and $(P_5)$ are the same as in the Dirac-von Neumann formalism of quantum mechanics. For the sake of completeness, we include them here.
\item Let $A$ be a para-Hermitian such that $A = \sum_{n \ge 1} \lambda_n |e_n\rangle \langle e^*_n|$ with a discrete spectrum whose eigenstates $(e_n)_{n \ge 1}$ constitute a unconditional basis in $\mathbb{H}.$ Define $G = \sum_{n \ge 1} |e^*_n\rangle \langle e^*_n|.$ Then $e^*_n = G e_n$ for every $n \ge 1,$ and $(e_n)_{n \ge 1}$ is orthogonal in the inner product $\langle \phi, \psi\rangle_G = \langle \phi, G \psi\rangle$ defined by $G.$ Moreover, we have
\be
\| \psi \|^2 = \langle G^{-\frac{1}{2}}\psi, G^{-\frac{1}{2}}\psi\rangle_G = \sum_{n \ge 1} |\langle e^*_n, G^{-\frac{1}{2}} \psi\rangle|^2,
\ee
and $\sum_n p(\psi | e_n) = 1.$ Thus, the formula \eqref{eq:BornRuleNHV} of the discrete case coincides with \eqref{eq:BornRuleNH}.
\item By the non-Hermitian Born formula \eqref{eq:BornRuleNH}, each state is scalar free and uniquely determined by a complex line through the origin of $\mathbb{H},$ i.e.,
\beq\label{eq:BornRuleScalarfree}
\langle A \rangle_{\alpha \psi, G} = \langle A \rangle_{\psi, G}
\eeq
for any nonzero scalar $\alpha \in \mathbb{C}.$ In what follows, without specified otherwise, we always use a unit vector to represent a state, simply called a {\it vector state}.
\end{enumerate}
\end{remark}
\begin{proposition}\label{prop:BornRuleHermit}\rm
Let $A$ be a Hermitian operator in $\mathbb{H}.$ If $G$ is a metric operator associated with $A,$ then
\beq\label{eq:BornRuleHermit}
\langle A \rangle_{\psi, G} = \langle A \rangle_\psi : = \frac{\langle \psi, A \psi \rangle}{\| \psi \|^2}
\eeq
for any nonzero $\psi \in \mathbb{H}.$
\end{proposition}
\begin{proof}
Since $G$ and $G^\frac{1}{2} A G^{-\frac{1}{2}}$ are both self-adjoint, it follows that
\be
G^\frac{1}{2} A G^{-\frac{1}{2}} = G^{-\frac{1}{2}} A G^\frac{1}{2}
\ee
and so $GA = AG,$ i.e., $G$ commutes with $A.$ Note that $\sigma (G)$ is a bounded closed set in $(0, \8)$ by the assumption. Thus $f(x) = x^\frac{1}{2}$ is a Borel function in $\sigma (G).$ By the spectral theorem (cf. \cite[Theorem 13.33]{Rudin1991}) and functional calculus, we conclude that $G^\frac{1}{2} A = A G^\frac{1}{2}.$ This implies the required \eqref{eq:BornRuleHermit}.
\end{proof}
\begin{remark}\label{rk:NHQMtoDvN}\rm
By Proposition \ref{prop:BornRuleHermit}, the non-Hermitian Born formula \eqref{eq:BornRuleNH} is independent of the choice of a metric operator $G$ associated with a Hermitian operator $A.$ Therefore, the mathematical formalism of non-Hermitian quantum mechanics as in Definition \ref{df:MathFNHQM} is an extension of the Dirac-von Neumann formalism of quantum mechanics to the non-Hermitian setting.
\end{remark}
The following example shows that the non-Hermitian Born formula \eqref{eq:BornRuleNH} is not equal to the usual Born rule for a non-Hermitian observable in general.
\begin{example}\rm
Consider the operator $A: \mathbb{C}^2 \mapsto \mathbb{C}^2$ defined by the matrix
\be
A = \left (
\begin{matrix}
0 & 1 \\
4 & 0
\end{matrix} \right )
\ee
in the standard basis of $\mathbb{C}^2$ and is not Hermitian. Define $G: \mathbb{C}^2 \mapsto \mathbb{C}^2$ by the matrix
\be
G = \left (
\begin{matrix}
1 & 0 \\
0 & \frac{1}{4}
\end{matrix} \right ).
\ee
Then $G^\frac{1}{2} A G^{- \frac{1}{2}} = 2 \sigma_x $ is Hermitian and so $A$ is para-Hermitian in $\mathbb{C}^2,$ that is, $A$ is a non-Hermitian observable. For $\psi = \frac{1}{\sqrt{2}} \left ( \begin{matrix} 1 \\ - \mathrm{i} \end{matrix} \right ),$ we have
\be
\langle A \rangle_\psi = \frac{3}{2} \mathrm{i}, \quad \langle A \rangle_{\psi, G} = 0,
\ee
and thus $\langle A \rangle_{\psi, G} \not= \langle A \rangle_\psi.$
\end{example}
The following example shows that the measurement of a non-Hermitian observable represented by a para-Hermitian operator may depend on the choice of a metric operator associated with it.
\begin{example}\label{ex:NHObM}\rm (cf. \cite[Section 3.5]{Mosta2010})\;
Consider the operator $A: \mathbb{C}^2 \mapsto \mathbb{C}^2$ defined by the matrix
\be
A = \frac{1}{2}\left (
\begin{matrix}
1+\delta & -1+\delta \\
1-\delta & -1 - \delta
\end{matrix} \right )
\ee
in the standard basis of $\mathbb{C}^2,$ where $\delta >0.$ It has two eigenvalues $\lambda_\pm = \pm \delta^\frac{1}{2}$ with the corresponding eigenstates
\be
e_+ (A) = c_+ \left (
\begin{matrix}
1 + \delta^\frac{1}{2}\\
1 - \delta^\frac{1}{2}
\end{matrix} \right ),\quad
e_- (A) = c_- \left (
\begin{matrix}
1 - \delta^\frac{1}{2}\\
1 + \delta^\frac{1}{2}
\end{matrix} \right ),
\ee
where $c_1, c_2 \in \mathbb{C}_*,$ and with the dual eigenstaes
\be
e^*_+ (A) = \frac{1}{4 \bar{c}_+} \left (
\begin{matrix}
1 + \delta^{-\frac{1}{2}}\\
1 - \delta^{-\frac{1}{2}}
\end{matrix} \right ),\quad
e^*_- (A) & = \frac{1}{4 \bar{c}_-} \left (
\begin{matrix}
1 - \delta^{-\frac{1}{2}}\\
1 + \delta^{-\frac{1}{2}}
\end{matrix} \right ),
\ee
such that
\be
A = \delta^\frac{1}{2} |e_+ (A) \rangle \langle e^*_+ (A)| - \delta^\frac{1}{2} |e_- (A)\rangle \langle e^*_- (A)|.
\ee
Hence, $A$ is para-Hermitian for all $\delta>0,$ and is Hermitian only when $\delta =1.$
Define $G: \mathbb{C}^2 \mapsto \mathbb{C}^2$ by the matrix
\be
G = r_+ \left (
\begin{matrix}
(1 + \delta^{-\frac{1}{2}})^2 & 1 - \delta^{-1} \\
1 - \delta^{-1} & (1 - \delta^{-\frac{1}{2}})^2
\end{matrix} \right ) + r_- \left (
\begin{matrix}
(1 - \delta^{-\frac{1}{2}})^2 & 1 - \delta^{-1} \\
1 - \delta^{-1} & (1 + \delta^{-\frac{1}{2}})^2
\end{matrix} \right ),
\ee
in the standard basis, where $r_+ = |4 c_+|^{-2}$ and $r_- = |4 c_-|^{-2}.$ By computation, one finds that $A$ is Hermitian in the inner product $(\cdot, \cdot)_G,$ and thus $G$ is a metric operator associated with $A.$ Note that $G$ has two positive eigenvalues
\be
\lambda_\pm = (1+ \delta^{-1}) (r_+ + r_-) \pm [(1+ \delta^{-1})^2 (r_+ + r_-)^2 - 4^2 \delta^{-1} r_+ r_-]^\frac{1}{2}.
\ee
Taking $\delta = \frac{1}{4},$ we have $\lambda_\pm = 5(r_+ + r_-) \pm \gamma$ with the corresponding eigenstates
\be\begin{split}
e_+ (G) = & \left (
\begin{matrix}
-3(r_+ + r_-)\\
\gamma -4(r_+ - r_-)
\end{matrix} \right ),\\
e_-(G) = & \left (
\begin{matrix}
-3(r_+ + r_-)\\
-\gamma -4(r_+ - r_-)
\end{matrix} \right ),
\end{split}\ee
where $\gamma = [25( r_+^2 + r_-^2) - 14 r_+ r_-]^\frac{1}{2}.$ It is then easy to see that the expectation $\langle A \rangle_{|0\rangle, G}$ of $A$ at $|0\rangle = \left (
\begin{matrix}
1\\
0
\end{matrix} \right )$ is dependent on the values of $r_+$ and $r_-,$ and thus $\langle A \rangle_{|0\rangle, G}$ depends on the choice of the metric operator $G$ associated with it.
\end{example}
\begin{comment}
\begin{proposition}\label{prop:BornRuleEqua}\rm
Let $A$ be a para-Hermitian operator having discrete spectrum, whose eigenstates $\{e_n\}_{n \ge 1}$ constitute an unconditional basis of $\mathbb{H}.$ Define $G = \sum_{n \ge 1} |e^*_n\rangle \langle e^*_n|.$ For a nonzero vector $\psi$ with $G^{-\frac{1}{2}} \psi \in \mathcal{D} (A),$ if the series $\sum_{n \ge 1} \langle e^*_n, G^{-\frac{1}{2}} \psi\rangle e^*_n$ converges in $\mathbb{H},$ denoted by $\psi^*$ the limit, then the expectation of $A$ at the state $\psi$ is equal to
\beq\label{eq:BornRuleEqua}
\langle A \rangle_{\psi,G} = \langle A \rangle_{\psi,\Pi} = \frac{\langle \psi^*, A G^{-\frac{1}{2}} \psi \rangle}{\|\psi\|^2},
\eeq
under the measurement $\Pi = \{|e_n\rangle \langle e^*_n|: n \ge 1 \}$ or equivalently in the measurement context of $G.$
\end{proposition}
\begin{proof}
Let $A$ be a para-Hermitian operator having discrete spectrum $\{\lambda_n\}_{n \ge 1}$ with eigenstates $\{e_n\}_{n \ge 1}.$ By the continuity of inner product, one has
\be\begin{split}
\langle \psi^*, A G^{-\frac{1}{2}} \psi \rangle = &\lim_{N \to \8} \sum^N_{n=1} \overline{\langle e^*_n, G^{-\frac{1}{2}} \psi\rangle} \lim_{M \to \8} \sum^M_{m=1} \langle e^*_n, A e_m \rangle \langle e^*_m, G^{-\frac{1}{2}} \psi\rangle\\
=& \lim_{N \to \8} \sum^N_{n=1} \lambda_n | \langle e^*_n, G^{-\frac{1}{2}} \psi\rangle|^2.
\end{split}\ee
This follows \eqref{eq:BornRuleEqua} from \eqref{eq:BornRuleNHV}.
\end{proof}
\end{comment}
\begin{example}\label{ex:para-PaulMat}\rm
Consider the non-Hermitian qubit system associated with the Hilbert space $\mathbb{C}^2.$ Given a real number $-\frac{\pi}{2} < \omega < \frac{\pi}{2},$ the non-Hermitian (deformed) Pauli matrices are defined by (cf. \cite{Brody2014})
\beq\label{eq:biPauliM}
\left \{
\begin{split}
\sigma^\omega_x & = \frac{1}{\cos \omega} \left (
\begin{matrix}
- \mathrm{i} \sin \omega & 1 \\
1 & \mathrm{i} \sin \omega
\end{matrix} \right ),\\
\sigma^\omega_y & = \left (
\begin{matrix}
0 & - \mathrm{i} \\
\mathrm{i} & 0
\end{matrix} \right ),\\
\sigma^\omega_z & = \frac{1}{\cos \omega} \left (
\begin{matrix}
1 & \mathrm{i}\sin \omega \\
\mathrm{i}\sin \omega & -1
\end{matrix} \right ).
\end{split}\right.\eeq
All $\sigma^\omega_x, \sigma^\omega_y,$ and $\sigma^\omega_z$ have eigenvalues $1$ and $-1,$ and satisfy the canonical commutation relations
\beq\label{eq:PauliCommuLation}
\sigma^\omega_x \sigma^\omega_y = \mathrm{i} \sigma^\omega_z,\; \sigma^\omega_y \sigma^\omega_z = \mathrm{i} \sigma^\omega_x,\;\sigma^\omega_z \sigma^\omega_x = \mathrm{i} \sigma^\omega_y.
\eeq
The eigenstates of $\sigma^\omega_x$ are
\beq\label{eq:Xeigenstate}
\left \{ \begin{split}
e_+ (\sigma^\omega_x) & = \frac{1}{\sqrt{2}} \left (
\begin{matrix}
1 \\
e^{\mathrm{i}\omega}
\end{matrix} \right ),\\
e_- (\sigma^\omega_x) & = \frac{1}{\sqrt{2}} \left (
\begin{matrix}
1 \\
- e^{-\mathrm{i}\omega}
\end{matrix} \right ),
\end{split}
\right.\eeq
and
\beq\label{eq:XstarEigenstate}
\left \{\begin{split}
e^*_+ (\sigma^\omega_x) & = \frac{1}{\sqrt{2} \cos \omega} \left (
\begin{matrix}
e^{\mathrm{i}\omega}\\
1
\end{matrix} \right ),\\
e^*_- (\sigma^\omega_x) & = -\frac{1}{\sqrt{2} \cos \omega} \left (
\begin{matrix}
- e^{-\mathrm{i}\omega}\\
1
\end{matrix} \right ),
\end{split}
\right.\eeq
where we have written $e_+$ for $e_1$ and $e_-$ for $e_2.$ Note that
\be
\langle e_- (\sigma^\omega_x), e_+ (\sigma^\omega_x) \rangle = \frac{1}{2} ( 1 - e^{2\mathrm{i}\omega}) \not=0
\ee
for $\omega \not=0,$ namely if $\omega \not=0,$ $e_+$ and $e_-$ are not orthogonal, due to the fact that $\sigma^\omega_x$ is not a self-adjoint operator.
Define
\be
G= |e^*_+ (\sigma^\omega_x) \rangle \langle e^*_+ (\sigma^\omega_x)| + |e^*_- (\sigma^\omega_x)\rangle \langle e^*_- (\sigma^\omega_x)| = \frac{1}{ \cos^2 \omega} \left (
\begin{matrix}
1 & \mathrm{i} \sin \omega\\
-\mathrm{i}\sin \omega & 1
\end{matrix} \right )
\ee
which has eigenvalues $\lambda_\pm = \frac{1}{\cos^2 \omega} (1 \pm \sin \omega),$ whose eigenstates are respectively
\be
e_+ (G) = \frac{1}{\sqrt{2}} \left (
\begin{matrix}
1\\
- \mathrm{i}
\end{matrix} \right ), \quad
e_- (G) = \frac{1}{\sqrt{2}} \left (
\begin{matrix}
- \mathrm{i}\\
1
\end{matrix} \right ).
\ee
Then
\be\begin{split}
G^\frac{1}{2} = & \frac{(1 + \sin \omega)^\frac{1}{2}}{\cos \omega} |e_+ (G) \rangle \langle e_+ (G)| + \frac{(1 - \sin \omega)^\frac{1}{2}}{\cos \omega} |e_- (G)\rangle \langle e_- (G)|\\
= & \frac{(1 + \sin \omega)^\frac{1}{2}}{2 \cos \omega} \left (
\begin{matrix}
1 & \mathrm{i} \\
-\mathrm{i} & 1
\end{matrix} \right )
+
\frac{(1 - \sin \omega)^\frac{1}{2}}{2\cos \omega} \left (
\begin{matrix}
1 & - \mathrm{i} \\
\mathrm{i} & 1
\end{matrix} \right )
\end{split}\ee
and
\be
G^{-\frac{1}{2}}= \frac{\cos \omega}{2(1 + \sin \omega)^\frac{1}{2}} \left (
\begin{matrix}
1 & \mathrm{i} \\
-\mathrm{i} & 1
\end{matrix} \right ) + \frac{\cos \omega}{2(1 - \sin \omega)^\frac{1}{2}} \left (
\begin{matrix}
1 & - \mathrm{i} \\
\mathrm{i} & 1
\end{matrix} \right ).
\ee
Thus,
\be
G^\frac{1}{2} \sigma^\omega_x G^{-\frac{1}{2}}= \frac{(1 - \sin \omega)^\frac{1}{2}}{\cos \omega} \left (
\begin{matrix}
0 & 1 \\
1 & 0
\end{matrix} \right ),
\ee
and so $\sigma^\omega_x$ is a para-Hermitian operator in $\mathbb{C}^2.$ Then, for a state determined by
\beq\label{eq:psi}
\psi = \cos \frac{\theta}{2} \left (
\begin{matrix}
1 \\
0
\end{matrix} \right ) + e^{\mathrm{i} \phi}\sin \frac{\theta}{2} \left (
\begin{matrix}
0 \\
1
\end{matrix} \right )
= \left (
\begin{matrix}
\cos \frac{\theta}{2} \\
e^{\mathrm{i} \phi}\sin \frac{\theta}{2}
\end{matrix} \right ), \quad \theta,\phi \in [0, 2 \pi),
\eeq
the expectation of $\sigma^\omega_x$ at $\psi$ is
\be
\langle \sigma^\omega_x \rangle_{\psi, G} = \frac{(1 - \sin \omega)^\frac{1}{2}}{\cos \omega} \sin \theta \cos \phi
\ee
under the measurement $\{ |e_+ (\sigma^\omega_x) \rangle \langle e^*_+ (\sigma^\omega_x)|, |e_- (\sigma^\omega_x)\rangle \langle e^*_- (\sigma^\omega_x)|\}$ or equivalently in the measurement context of $G.$
Since $\sigma^\omega_y = \sigma_y,$ two eigenstates of it are
\beq\label{eq:Yeigenstate}
\left \{\begin{split}
e_+ (\sigma^\omega_y)& =\frac{1}{\sqrt{2}} \left (
\begin{matrix}
1 \\
\mathrm{i}
\end{matrix} \right )= e^*_+ (\sigma^\omega_y),\\
e_- (\sigma^\omega_y)&= \frac{1}{\sqrt{2}} \left (
\begin{matrix}
\mathrm{i} \\
1
\end{matrix} \right )= e^*_- (\sigma^\omega_y).
\end{split}
\right.\eeq
Then the expectation of $\sigma^\omega_x$ at $\psi$ is
\be
\langle \sigma^\omega_y \rangle_\psi = \langle \sigma^\omega_y \rangle_{\psi, G} = \sin \theta \sin \phi
\ee
in the measurement context of $G,$ where $G$ is any strictly positive self-adjoint operator such that $G^\frac{1}{2} \sigma_y G^{-\frac{1}{2}}$ is self-adjoint in $\mathbb{C}^2$ or equivalently $G$ commutes with $\sigma_y.$
Finally, the eigenstates of $\sigma^\omega_z$ are
\beq\label{eq:Zeigenstate}
\left \{\begin{split}
e_+ (\sigma^\omega_z) & = \frac{(1+ \cos \omega)^\frac{1}{2}}{\sqrt{2}}\left (
\begin{matrix}
1 \\
\frac{\mathrm{i}\sin \omega}{1+ \cos \omega}
\end{matrix} \right ),\\
e_- (\sigma^\omega_z) & = \frac{(1+ \cos \omega)^\frac{1}{2}}{\sqrt{2}} \left (
\begin{matrix}
-\frac{\mathrm{i}\sin \omega}{1+ \cos \omega} \\
1
\end{matrix} \right ),
\end{split}
\right.\eeq
and so,
\beq\label{eq:ZstarEigenstate}
\left \{\begin{split}
e^*_+ (\sigma^\omega_z) & =\frac{(1+ \cos \omega)^\frac{1}{2}}{\sqrt{2} \cos \omega} \left (
\begin{matrix}
1 \\
-\frac{\mathrm{i}\sin \omega}{1+ \cos \omega}
\end{matrix} \right ),\\
e^*_- (\sigma^\omega_z) & =\frac{(1+ \cos \omega)\frac{1}{2}}{\sqrt{2} \cos \omega} \left (
\begin{matrix}
\frac{\mathrm{i}\sin \omega}{1+ \cos \omega}\\
1
\end{matrix} \right ).
\end{split}
\right.\eeq
Note that
\be
\langle e_- (\sigma^\omega_z), e_+ (\sigma^\omega_z) \rangle =\mathrm{i} \sin \omega \not=0
\ee
if $\omega \not=0,$ in which case $\sigma^\omega_z$ is not a self-adjoint operator. Define
\be
G = |e^*_+ (\sigma^\omega_z) \rangle \langle e^*_+ (\sigma^\omega_z)| + |e^*_- (\sigma^\omega_z)\rangle \langle e^*_- (\sigma^\omega_z)| = \frac{1}{\cos^2 \omega} \left (
\begin{matrix}
1 & \mathrm{i} \sin \omega\\
-\mathrm{i}\sin \omega & 1
\end{matrix} \right )
\ee
which is the same $G$ as appearing in the case of $\sigma^\omega_x.$ Then
\be
G^\frac{1}{2} \sigma^\omega_z G^{-\frac{1}{2}}= \left (
\begin{matrix}
1 & 0 \\
0 & -1
\end{matrix} \right ),
\ee
and so $\sigma^\omega_z$ is a para-Hermitian operator in $\mathbb{C}^2.$ Therefore, the expectation of $\sigma^\omega_z$ at $\psi$ is
\be
\langle \sigma^\omega_z \rangle_{\psi, G} = \langle \sigma_z \rangle_\psi = \cos \theta
\ee
under the measurement $\{ |e_+ (\sigma^\omega_z) \rangle \langle e^*_+ (\sigma^\omega_z)|, |e_- (\sigma^\omega_z)\rangle \langle e^*_- (\sigma^\omega_z)|\}$ or equivalently in the measurement context of $G.$
\end{example}
\begin{example}\label{ex:qubitParaHermitian}\rm
Consider a two-dimensional Hamiltonian of the form
\beq\label{eq:2-dParaHamiltonian}
H = \left (
\begin{matrix}
r e^{\mathrm{i}\theta} & \gamma\\
\gamma & r e^{-\mathrm{i}\theta}
\end{matrix} \right )
\eeq
where the three parameters $r, \theta, $ and $\gamma$ are real numbers. Then $H$ has two real eigenvalues $\lambda_\pm = r \cos \theta \pm \sqrt{\gamma^2 - r^2 \sin^2 \theta}$ provided that $\gamma^2 > r^2 \sin^2 \theta,$ and the associated eigenstates of $H$ are
\be
e_+ (H) = \frac{1}{\sqrt{2}}\left (
\begin{matrix}
e^{\mathrm{i} \phi /2} \\
e^{-\mathrm{i} \phi /2}
\end{matrix} \right ),\quad
e_- (H) = \frac{1}{\sqrt{2}}\left (
\begin{matrix}
\mathrm{i} e^{-\mathrm{i} \phi /2} \\
-\mathrm{i} e^{\mathrm{i} \phi /2}
\end{matrix} \right ),
\ee
where the real number $\phi$ is defined by $\sin \phi = \frac{r}{\gamma} \sin \theta,$ and so
\be
e^*_+ (H)= \frac{1}{\sqrt{2}\cos \phi}\left (
\begin{matrix}
e^{-\mathrm{i} \phi /2} \\
e^{\mathrm{i} \phi /2}
\end{matrix} \right ),\quad
e^*_- (H)= \frac{1}{\sqrt{2}\cos \phi}\left (
\begin{matrix}
\mathrm{i} e^{\mathrm{i} \phi /2} \\
-\mathrm{i} e^{-\mathrm{i} \phi /2}
\end{matrix} \right ).
\ee
Note that
\be
\langle e_- (H), e_+ (H) \rangle = 2 \sin \phi \not=0,
\ee
when $r, \theta \not=0,$ in this case $H$ is not a self-adjoint operator.
Define
\be
G = |e^*_+ (H) \rangle \langle e^*_+ (H)| + |e^*_- (H)\rangle \langle e^*_- (H)| = \frac{1}{\cos^2 \phi} \left (
\begin{matrix}
1 & -\mathrm{i}\sin \phi\\
\mathrm{i}\sin \phi & 1
\end{matrix} \right )
\ee
which is the same $G$ ($\phi = -\omega$) as appearing in the case of $\sigma^\omega_x$ in Example \ref{ex:para-PaulMat}. Then
\be
G^\frac{1}{2} H G^{-\frac{1}{2}}= \cos \theta \left (
\begin{matrix}
r & 0 \\
0 & r
\end{matrix} \right ) + |\cos \phi| \gamma \left (
\begin{matrix}
0 & 1 \\
1 & 0
\end{matrix} \right ),
\ee
and so $H$ is a para-Hermitian operator in $\mathbb{C}^2$ provided $r, \theta \not=0$ such that $\gamma^2 > r^2 \sin^2 \theta.$ Note that, for $\gamma =1$ and $\theta = \frac{\pi}{2},$ the non-Hermitian operator $H =\sigma_r = \sigma_x - \mathrm{i} r \sigma_z$ is para-Hermitian for $0 < r < 1.$
Moreover, the expectation of $H$ at $\psi = \left (
\begin{matrix}
a \\
b
\end{matrix} \right )$ ($|a|^2 + |b|^2 =1$) is
\be
\langle H \rangle_{\psi, G} = r \cos \theta + \gamma |\cos \phi| (a \bar{b} + \bar{a} b)
\ee
under the measurement $\{ |e_+ (H) \rangle \langle e^*_+ (H)|, |e_- (H)\rangle \langle e^*_- (H)|\}$ or equivalently in the measurement context of $G.$
\end{example}
\section{PT-symmetric and biorthogonal quantum mechanics}\label{PTBIqm}
This section shows how the above formalism can recover PT-symmetric and biorthogonal quantum mechanics.
\subsection{PT-symmetric quantum mechanics}\label{PT-qm}
Recall that the linear parity operator $\mathcal{P}$ is defined by $\mathcal{P} f (x) = f(-x),$ whereas the anti-linear time-reversal operator $\mathcal{T}$ is defined by $\mathcal{T} f (x) = \overline{f(x)},$ so that $\mathcal{P}\mathcal{T} f (x) = \overline{f(-x)}.$ In \cite{BB1998}, Bender and Boettcher found that non-Hermitian Hamiltonian $H = p^2 + x^2 (\mathrm{i} x)^\nu$ ($\nu \ge 0$) may have real and positive spectrum, by noticing that while $H$ is not symmetric under $\mathcal{P}$ or $\mathcal{T}$ separately, it is invariant under their combined operation $\mathcal{P}\mathcal{T},$ that is, $H$ is $\mathcal{P}\mathcal{T}$-symmetric. The reality of the spectrum of $H$ is a consequence of its unbroken $\mathcal{P}\mathcal{T}$ symmetry. The unbroken $\mathcal{P}\mathcal{T}$ symmetry of $H$ means that eigenfunctions of $H$ are also eigenfunctions of $\mathcal{P}\mathcal{T}.$ In this case, there appears to be a natural choice for an inner produce given by
\be
(f, g)_{\mathcal{P}\mathcal{T}} = \int [\mathcal{P}\mathcal{T} f (x)] g (x) d x
\ee
such that the eigenfunctions $\{\psi_n\}$ of $H$ are orthonormal, i.e., $(\psi_m, \psi_n)_{\mathcal{P}\mathcal{T}} = (-1)^n \delta_{mn}.$ However, the $\mathcal{P}\mathcal{T}$ inner product $ (f, g)_{\mathcal{P}\mathcal{T}}$ is indefinite, because the norms of the eigenfunctions via this inner product may be negative.
In \cite{BBJ2002}, Bender {\it et al.} observed that for any $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian $H$ having an unbroken $\mathcal{P}\mathcal{T}$ symmetry, there exists a symmetry of $H$ described by a linear operator $\mathcal{C}$ such that the eigenfunctions $\{\psi_n\}$ of $H$ are orthogonal in the $\mathcal{C}\mathcal{P}\mathcal{T}$ inner product defined by
\be
(f, g)_{\mathcal{C}\mathcal{P}\mathcal{T}} = \int [\mathcal{C}\mathcal{P}\mathcal{T} f (x)] g (x) d x,
\ee
that is, $(\psi_m, \psi_n)_{\mathcal{C}\mathcal{P}\mathcal{T}} = \delta_{mn}$ and so the inner product $(f, g)_{\mathcal{C}\mathcal{P}\mathcal{T}}$ is positively definite. This means that any $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian $H$ having an unbroken $\mathcal{P}\mathcal{T}$ symmetry has a metric operator $G =\mathcal{C}\mathcal{P}\mathcal{T}$ so that $H$ is a Hermitian operator in the $\mathcal{C}\mathcal{P}\mathcal{T}$ inner product $(f, g)_{\mathcal{C}\mathcal{P}\mathcal{T}}.$ Also, they construct the operator $\mathcal{C}$ explicitly in terms of eigenfunctions $\{\psi_n\}$ of $H$ as follows
\be
C(x,y)= \sum_n \psi_n(x) \psi_n(y).
\ee
Note that $C$ commutes with Hamiltonian $H$ and the operator $\mathcal{P}\mathcal{T}.$
Mathematically, the $\mathcal{P}\mathcal{T}$ symmetry of a Hamiltonian provides a way to construct a metric operator for the Hamiltonian. Once having the metric operator $G$ associated with a $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian $H,$ the measurement and dynamics relative to $H$ can be resolved in the mathematical formwork of Definition \ref{df:MathFNHQM}. In this sense, we say that the mathematical formalism of non-Hermitian quantum mechanics given by Definition \ref{df:MathFNHQM} recovers $PT$-symmetric quantum mechanics.
For illustration, in what follows, we show the fact, which was previously found by Bender {\it et al.} \cite{BBJM2007} in the formwork of $PT$-symmetric quantum mechanics, that the transformation between a pair of orthogonal states according to non-Hermitian quantum mechanics in the formwork of Definition \ref{df:MathFNHQM} can be arbitrarily faster than Hermitian quantum mechanics under the same energy constraint.
To this end, for the sake of clarity, let us consider the transformation from the initial state $|0\rangle = \left (
\begin{matrix}
1 \\
0
\end{matrix} \right )$ to the final state $|1\rangle = \left (
\begin{matrix}
0 \\
1
\end{matrix} \right )$ in a qubit system associated with the two-dimensional Hilbert space $\mathbb{C}^2.$ As shown in \cite{BBJM2007}, for any two-dimensional Hermitian Hamiltonian $H,$ the smallest time required to transform $|0\rangle$ to $|1\rangle$ is $\tau = \frac{\pi \hbar}{\omega},$ i.e., $|1\rangle = \alpha e^{- \mathrm{i}H\tau /\hbar} |0\rangle$ with some phase factor $\alpha \not= 0,$ where $\omega$ is the difference between the energy eigenvalues of the Hamiltonian.
However, for a para-Hermitian Hamiltonian of the form
\be\label{eq:2-dParaHamiltonian}
H = \left (
\begin{matrix}
r e^{\mathrm{i}\theta} & \gamma\\
\gamma & r e^{-\mathrm{i}\theta}
\end{matrix} \right )
\ee
as shown in Example \ref{ex:qubitParaHermitian}, one has
\be
e^{- \mathrm{i}H t /\hbar} |0\rangle = \frac{e^{- \mathrm{i} r t \cos \theta /\hbar}}{\cos \phi} \left (
\begin{matrix}
\cos (\frac{\omega}{2 \hbar} t - \phi) \\
-\mathrm{i} \sin(\frac{\omega t}{2 \hbar})
\end{matrix} \right ),
\ee
where $\omega = 2 (\gamma^2 - r^2 \sin^2 \theta )^\frac{1}{2}$ is the difference between the eigenvalues of $H,$ and $\phi$ is defined by $\sin \phi = \frac{r}{\gamma} \sin \theta.$ Then we see that the evolution time to reach $|1\rangle$ from $|0\rangle$ is $t = (2 \phi + \pi)\hbar /\omega.$ Taking allowable values for $r, \gamma,$ and $\theta,$ we can make $\phi$ approaching $- \pi/2$ such that the optimal time $\tau$ tends to $0.$ Thus, the transformation from the initial state $|0\rangle$ to the final state $|0\rangle$ according to non-Hermitian quantum mechanics in the formwork of Definition \ref{df:MathFNHQM} can be arbitrarily faster than Hermitian quantum mechanics under the same energy constraint.
\subsection{Biorthogonal quantum mechanics}\label{Bi-qm}
Given an unconditional basis $\{e_n\}$ in a Hilbert space $\mathbb{H}$ with the unique dual basis $\{ e^*_n\}$ ($\{e_n, e^*_n\}$ is called a biorthogonal basis, cf. \cite{LT1977}), a densely defined closed operator $T$ in $\mathbb{H}$ can be expressed by
\beq\label{eq:BiOpRep}
T = \sum_{n,m} f_{n m} |e_n \rangle \langle e^*_m |
\eeq
with $f_{n m} = \langle e^*_n, T e_m \rangle,$ provided $\{e_n\} \subset \mathcal{D} (T).$ In biorthogonal quantum mechanics \cite{Brody2014}, such a operator $T$ is said to be a `biorthogonally Hermitian' operator with respect to the biorthogonal basis $\mathcal{F} =\{e_n, e^*_n\},$ if $\bar{f}_{n m} = f_{m n}$ for any $n,m.$
Given a biorthogonal basis $\mathcal{F}=\{e_n, e^*_n\},$ as assumed in \cite{Brody2014}, each observable is represented by a `biorthogonally Hermitian' operator $T$ relative to $\mathcal{F},$ and the expectation of $T$ at a state $\psi$ is defined by
\beq\label{eq:BiOpExp}
\langle T \rangle_{\psi, \mathcal{F}} = \frac{\langle \tilde{\psi}, T \psi \rangle}{\langle \tilde{\psi}, \psi \rangle},
\eeq
where $\tilde{\psi} = \sum_n a_n e^*_n$ with $a_n = \langle e^*_n, \psi \rangle$ (noticing that $\psi = \sum_n a_n e_n$). Then $\langle T \rangle_{\psi, \mathcal{F}}$ defined by \eqref{eq:BiOpExp} is real for any `biorthogonally Hermitian' operator $T$ relative to a biorthogonal basis $\mathcal{F}$ and for all states $\psi,$ since
\be
\langle T \rangle_{\psi, \mathcal{F}} = \frac{\sum_{n,m} \bar{a}_n a_m f_{n m}}{\sum_n |a_n|^2}.
\ee
Note that a `biorthogonally Hermitian' operator $T$ relative to a biorthogonal basis $\mathcal{F} =\{e_n, e^*_n\}$ is not necessarily Hermitian if $\{e_n\}$ is not an orthogonal basis, and so $\langle \psi, T \psi \rangle/ \langle \psi, T \psi \rangle$ is not real for most states $\psi,$ as noted in \cite{Brody2014}.
\begin{proposition}\label{prop:BiHop}\rm
Let $\mathcal{F} =\{e_n, e^*_n\}$ be a biorthogonal basis in $\mathbb{H}.$ A densely defined closed operator $T$ with $\{e_n\} \subset \mathcal{D} (T)$ is a `biorthogonally Hermitian' operator $T$ relative to $\mathcal{F}$ if and only if $T$ is self-adjoint with respect to the inner product $(\cdot, \cdot)_G,$ where $G = \sum_n |e^* \rangle \langle e^*_n|$ is a metric operator associated with $T.$ Consequently, if a densely defined closed operator is `biorthogonally Hermitian' then it is a para-Hermitian operator. Moreover,
\be
\langle T \rangle_{\psi, \mathcal{F}} = \langle T \rangle_{G^\frac{1}{2} \psi, G}
\ee
for any nonzero $\psi \in \mathbb{H}.$
\end{proposition}
\begin{proof}
Note that
\be
\langle u, T v \rangle_G = \langle u, \sum_{n,m} f_{n m} |e^*_n \rangle \langle e^*_m | v \rangle = \langle \sum_{n,m} \bar{f}_{n m} |e_m \rangle \langle e^*_n |u, G v \rangle = \langle T u, v \rangle_G
\ee
for any $u,v \in \mathbb{H},$ whenever $\bar{f}_{n m} = f_{m n}$ for any $n,m.$ This concludes the first assertion.
For the second assertion, since
\be
\langle \psi, T \psi \rangle_G = \langle \psi, G T \psi \rangle = \sum_{n,m} \bar{a}_n a_m f_{n m}
\ee
for $\psi = \sum_n a_n e_n,$ and $\|G^\frac{1}{2} \psi\| = \sum_n |a_n|^2$ (see Remark \ref{rk:NHQM}), this follows the required equality.
\end{proof}
By this proposition, we see that a densely defined closed `biorthogonally Hermitian' operator is a para-Hermitian operator, and the expectation value formula \eqref{eq:BiOpExp} reduces to the non-Hermitian Born formula \eqref{eq:BornRuleNH}. In this sense, we say that the mathematical formalism of non-Hermitian quantum mechanics given by Definition \ref{df:MathFNHQM} recovers biorthogonal quantum mechanics.
\section{Geometric phase}\label{GeoPhase}
The notion of the geometric phase for a quantum system was introduced by Berry (cf. \cite{Berry1984, Simon1983}), on which there exist extensive works (cf. \cite{SMC2016} and references therein). The geometric phases for non-Hermitian quantum systems have been studied in \cite{CZ2012, GW1988, MM2008, SB1988, ZW2019}, etc. As usual, these geometric phases are associated with the quantum state. Recently, the notion of the geometric phase for the observable (the so-called observable-geometric phase) was introduced in \cite{Chen2020}, which is defined as a sequence of phases associated with a complete set of eigenstates of the observable. In this section, we will study the observable-geometric phase in the non-Hermitian setting. In order to compare, we first give a description of the geometric phase for the state in the non-Hermitian case via the mathematical framework of non-Hermitian quantum mechanics given by Definition \ref{df:MathFNHQM}.
\subsection{Geometric phases for the state}\label{SGP}
Consider a non-Hermitian quantum system with a time-dependent Hamiltonian $\{h(t): t \in [0,T]\},$ where $h(t)$'s are all para-Hermitian operators in $\mathbb{H}.$ Suppose that the state $\psi (t) \in \mathbb{H}_* = \mathbb{H} \setminus \{0\}$ evolves according to the time-dependent Schr\"{o}dinger equation
\beq\label{eq:SchrEquNHtime}
\mathrm{i} \frac{d \psi (t) }{d t} = h(t) \psi (t),
\eeq
such that $\psi (T) = e^{\mathrm{i} \theta} \psi (0),$ where $\theta \in \mathbb{C}.$ Then $\psi (t)$ defines a curve $C: [0, T] \mapsto \mathbb{H}_*$ with $\hat{C} = \mathcal{P} (C)$ being a closed curve in the projective Hilbert space $\mathcal{P} (\mathbb{H})$ of rays of $\mathbb{H},$ where $\mathcal{P}: \mathbb{H}_* \mapsto \mathcal{P} (\mathbb{H})$ is the projection from $\mathbb{H}_*$ into $\mathcal{P}(\mathbb{H}).$ Following \cite{SB1988}, we define
\beq\label{eq:GP-NH}
\beta = \theta + \int^T_0 \frac{\langle \psi (t) | h (t) | \psi (t) \rangle}{\| \psi (t) \|^2} d t,
\eeq
and
\begin{equation}\label{eq:Parallelstate}
\tilde{\psi} (t) = e^{ \mathrm{i} \int^t_0 \frac{\langle \psi (s) | h (s) | \psi (s) \rangle}{\| \psi (s) \|^2} d s } \psi (t).
\end{equation}
Then $\tilde{\psi} (T) = e^{\mathrm{i} \beta} \psi (0)$ and from \eqref{eq:SchrEquNHtime}, we conclude
\begin{equation}\label{eq:ParallelCondState}
\langle \tilde{\psi} (t) | \frac{d}{d t} |\tilde{\psi} (t) \rangle =0.
\end{equation}
For any closed curve in $\mathbb{H}_*$
\begin{equation}\label{eq:ClosedCurvestate}
\bar{\psi} (t) = e^{- \mathrm{i} \alpha (t)} \psi (t)
\end{equation}
where $\alpha: [0, T] \mapsto \mathbb{C}$ is a continuously differential function satisfying $\alpha (T) - \alpha (0) = \theta,$ i.e., $\bar{\psi} (T)= \bar{\psi} (0),$ differentiating both sides of \eqref{eq:ClosedCurvestate} we obtain
\be
\frac{d}{d t} \bar{\psi} (t) = - \mathrm{i} \frac{d \alpha (t)}{d t} \bar{\psi} (t) + e^{- \mathrm{i} \alpha (t)} \frac{d}{d t} \psi (t).
\ee
Taking the scalar product of this expression with $\bar{\psi} (t)$ and from \eqref{eq:SchrEquNHtime} again, we have
\begin{equation}\label{eq:GP-NHClosed}
\beta = \int^T_0 \mathrm{i} \frac{\langle \bar{\psi} (t) | \frac{d}{d t} |\bar{\psi} (t) \rangle}{\| \bar{\psi} (t) \|^2} d t.
\end{equation}
Clearly, the same closed curve $\bar{\psi} (t)$ can be chosen for every $C$ for which $\mathcal{P}(C) = \hat{C},$ by appropriate choice of $\alpha (t).$ Thus, $\beta$ defined by \eqref{eq:GP-NH}, is independent of $\theta$ and $h(t)$ for a given closed curve $\hat{C}$ in $\mathcal{P} (\mathbb{H}).$ Also, from \eqref{eq:GP-NHClosed}, $\beta$ is independent of the parameter $t$ of $C,$ and is uniquely defined up to $2 \pi k$ ($k$ being integer).
According to \cite{AA1987, SB1988}, we get the notion of the geometric phase in the non-Hermitian setting as follows.
\begin{definition}\label{df:GP-NP}
Using the above notations, $\beta$ is defined to be the geometric phase of the period evolution of the state $\psi (t).$
\end{definition}
Indeed, we can rewrite \eqref{eq:GP-NHClosed} as
\beq\label{eq:GP-NHGauge}
\beta = \oint_{\hat{C}} \mathrm{i} \frac{\langle \bar{\psi} | d | \bar{\psi} \rangle}{\| \bar{\psi} \|^2},
\eeq
where $\bar{\psi} (t)$ corresponds to any of the closed lifts of $\hat{C}.$ And, $\mathcal{A} = \mathrm{i} \frac{\langle \psi | d | \psi \rangle}{\|\psi\|^2}$ is a connection one-form. It satisfies the transformation rule as follows
\be
\mathcal{A} = \mathrm{i} \frac{\langle \psi | d | \psi \rangle}{\|\psi\|^2} \mapsto \mathcal{A}' = \mathrm{i} \frac{\langle \psi | d | \psi \rangle}{\|\psi\|^2} - d \xi,
\ee
under the gauge transformation $\psi (t) \mapsto \psi'(t) = e^{\mathrm{i} \xi (t)} \psi (t).$ Note that $e^{\mathrm{i} \beta}$ is a geometric property of the unparametrized image of $\hat{C}$ in $\mathcal{P} (\mathbb{H})$ only, which can be described by a principal fiber bundle $(\mathbb{H}_*, \mathcal{P} (\mathbb{H}), \mathcal{P}, \mathbb{C}_*)$ over $\mathcal{P} (\mathbb{H})$ with the structure group $\mathbb{C}_* = GL (1, \mathbb{C})$ (the group of nonzero complex numbers). We omit the details (cf. \cite[5.4]{BMKNZ2003}).
\begin{remark}\label{rk:GP-NH}\rm
The non-Hermitian geometric phase $\beta$ as defined in \eqref{eq:GP-NH} is different from the one found in \cite{CZ2012} and references therein. They used the adjoint wave function to define the geometric phase in the non-Hermitian case. Precisely, let $\psi^\dag (t)$ be the solution to the adjoint Schr\"{o}dinger equation
\beq\label{eq:SchrEquNHtimeAdj}
\mathrm{i} \frac{d \psi^\dag (t) }{d t} = h^\dag(t) \psi^\dag (t),
\eeq
with $\psi^\dag (0)$ satisfying $\langle \psi^\dag (0), \psi (0) \rangle =1,$ where $h^\dag (t)$ is the adjoint operator of $h(t),$ such that
\beq\label{eq:Normpreserve}
\langle \psi^\dag (t), \psi (t) \rangle =1,
\eeq
for all $t,$ since $\frac{d }{d t} \langle \psi^\dag (t), \psi (t) \rangle =0.$ The geometric phase of the cyclic evolution $\psi (t)$ with period $T$ is then defined to be equal to
\begin{equation}\label{eq:GP-nonHermitian}
\check{\beta} = \theta + \int^T_0 \langle \psi^\dag (t)|h(t) |\psi (t) \rangle d t = \int^T_0 \mathrm{i} \langle \bar{\psi}^\dag (t) | \frac{d}{d t} |\bar{\psi} (t) \rangle d t,
\end{equation}
where $\bar{\psi}^\dag (t) = e^{- \mathrm{i} \alpha (t)} \psi^\dag (t).$
Although the same closed curve $\bar{\psi} (t)$ can be chosen for every $C$ for which $\mathcal{P}(C) = \hat{C}$ by appropriate choice of $\alpha (t),$ the curve $\bar{\psi}^\dag (t)$ can be different so that $\check{\beta}$ may change. This means that $\check{\beta}$ may depend on $h(t),$ since $h^\dag(t)$ depends on $h(t).$ Hence, in general, the geometric phase defined in terms of \eqref{eq:GP-nonHermitian} seems not to be a geometric quantity associated with $\mathcal{P}(C)$ in $\mathcal{P}(\mathbb{H}).$
\end{remark}
If the system evolves adiabatically (cf. \cite{BMKNZ2003}), $h(t)$ varies slowly with $h(t) |\bar{n}(t)\rangle = E_n (t) |\bar{n}(t) \rangle$ for a unconditional basis $\{\bar{n}(t)\}$ in $\mathbb{H},$ i.e., $h(t) = \sum_n E_n (t) |\bar{n}(t) \rangle \langle \bar{n}^* (t)|,$ where $\{\bar{n}^* (t)\}$ is the dual basis of $\{\bar{n}(t)\}.$ Assume that $E_n(t)$'s are non-degenerate and $\bar{n}(t)$'s are continuously differential in $t.$ If we write
\beq\label{eq:AdiabaticEvoluState}
\psi (t) = \sum_n a_n (t) e^{-\mathrm{i} \int^t_0 E_n(s) d s} \bar{n}(t),
\eeq
differentiating both sides of this expression and using \eqref{eq:SchrEquNHtime}, we have
\be
\sum_n \dot{a}_n (t) e^{-\mathrm{i} \int^t_0 E_n(s) d s} \bar{n}(t) + \sum_n a_n (t) e^{-\mathrm{i} \int^t_0 E_n(s) d s} \dot{\bar{n}}(t) =0,
\ee
where the dot denotes derivative in time $t.$ Taking the scalar product of this expression with $\bar{m}^* (t),$ we have
\be
\dot{a}_m (t) = - a_m (t) \langle \bar{m}^*(t) | \dot{\bar{m}} (t) \rangle - \sum_{n \not= m} a_n (t) e^{-\mathrm{i} \int^t_0 [E_m(s) - E_n(s)] d s} \langle \bar{m}^*(t) | \dot{\bar{n}}(t)\rangle.
\ee
In adiabatic approximation, the second term on the right-hand side of the above equation can be neglected so that
\beq\label{eq:AdiabaticEvoluCoefficient}
a_m (t) = e^{- \int^t_0 \langle \bar{m}^*(s) | \dot{\bar{m}} (s) \rangle d s} a_m (0).
\eeq
Then if $a_n (0) = \delta_{n m},$ the system would continues as an eigenstate of $h(t)$ to a good approximation. In this case, for a cyclic adiabatic evolution with $h(T) = h(0)$ so that $\bar{n}(T) = \bar{n}(0),$ i.e., $\bar{n}(t)$ is a closed curve in $\mathbb{H}_*,$ we have
\beq\label{eq:AdiabaticEvoluEigenstate}
n(T) = e^{\mathrm{i} (\int^T_0 \mathrm{i}\langle \bar{n}^*(s) | \dot{\bar{n}} (s) \rangle d s - \int^T_0 E_n(s) d s)} \bar{n}(0),
\eeq
and so the total phase $\theta = \int^T_0 \mathrm{i}\langle \bar{n}^*(s) | \dot{\bar{n}} (s) \rangle d s - \int^T_0 E_n(s) d s.$ Hence,
\be\begin{split}
\check{\beta} & = \int^T_0 \mathrm{i}\langle \bar{n}^*(s) | \dot{\bar{n}} (s) \rangle d s\\
& = \theta + \int^T_0 \frac{\langle n(t) | h (t) | n(t) \rangle}{\| n(t) \|^2} d t\\
& = \int^T_0 \mathrm{i}\frac{\langle \bar{n}(t) | \dot{\bar{n}} (t) \rangle}{\| \bar{n}(t) \|^2} d t = \beta,
\end{split}\ee
because $\frac{\langle n(t) | h (t) | n(t) \rangle}{\| n(t) \|^2} = E_n (t)$ for all $t.$ This shows that the geometric phase defined in \cite{GW1988, MM2008, ZW2019} for a cyclic adiabatic evolution in the non-Hermitian setting is indeed a geometric quantity with respect to the unparametrized image of $\hat{C} = \mathcal{P} [n(t)]$ in $\mathcal{P} (\mathbb{H}).$
\begin{remark}\label{rk:GP-NHadiabatic}\rm
For a cyclic non-adiabatic evolution, $\check{\beta} \not= \beta$ in general (see Example \ref{ex:GP-qubit} below), and $\check{\beta}$ seems not to be a geometric quantity in the non-adiabatic case as remarked above. However, $\beta$ defined in \eqref{eq:GP-NH} is always a geometric term in both adiabatic and non-adiabatic cases.
\end{remark}
For illustrating the geometric phase in a non-Hermitian quantum system, we consider a qubit case, namely the Hilbert space $\mathbb{H} = \mathbb{C}^2.$
\begin{example}\label{ex:GP-qubit}\rm
Consider a non-Hermitian qubit system, whose Hamiltonian is $H = -\sigma^\omega_z$ (see Example \ref{ex:para-PaulMat}). Recall that $\sigma^\omega_z$ has eigenvalues $1$ and $-1,$ and the corresponding eigenstates are
\be\label{eq:Zeigenstate}
\left \{ \begin{split}
e_+ (\sigma^\omega_z) & = \left (
\begin{matrix}
1 \\
\frac{\mathrm{i}\sin \omega}{1+ \cos \omega}
\end{matrix} \right ),\\
e_- (\sigma^\omega_z) & = \left (
\begin{matrix}
-\frac{\mathrm{i}\sin \omega}{1+ \cos \omega} \\
1
\end{matrix} \right ),
\end{split}\right.\ee
and
\be\label{eq:ZstarEigenstate}
\left \{ \begin{split}
e^*_+ (\sigma^\omega_z) & =\frac{1+ \cos \omega}{2 \cos \omega} \left (
\begin{matrix}
1 \\
-\frac{\mathrm{i}\sin \omega}{1+ \cos \omega}
\end{matrix} \right ),\\
e^*_- (\sigma^\omega_z) & =\frac{1+ \cos \omega}{2\cos \omega} \left (
\begin{matrix}
\frac{\mathrm{i}\sin \omega}{1+ \cos \omega}\\
1
\end{matrix} \right ),
\end{split}
\right.\ee
(see Example \ref{ex:para-PaulMat}). Then
\be\begin{split}
e^{\mathrm{i}t \sigma^\omega_z}& =e^{\mathrm{i}t} |e_+ (\sigma^\omega_z) \rangle \langle e^*_+ (\sigma^\omega_z)| + e^{-\mathrm{i} t} |e_- (\sigma^\omega_z) \rangle \langle e^*_- (\sigma^\omega_z)|,\\
e^{\mathrm{i}t (\sigma^\omega_z)^\dag}& =e^{\mathrm{i}t} |e^*_+ (\sigma^\omega_z)\rangle \langle e_+ (\sigma^\omega_z)| + e^{-\mathrm{i} t} |e^*_- (\sigma^\omega_z)\rangle \langle e_- (\sigma^\omega_z)|.
\end{split}\ee
For $\psi (0) = \left ( \begin{matrix} \cos \frac{\phi}{2} \\
\sin \frac{\phi}{2}
\end{matrix}\right )$
in $\mathbb{C}^2,$ we have
\be
\begin{split}
\psi (t) = & e^{\mathrm{i}t \sigma^\omega_z} \psi (0) = \frac{e^{\mathrm{i}t}}{2 \cos \omega} \left ( \begin{matrix} (1 + \cos \omega) \cos \frac{\phi}{2} + \mathrm{i} \sin \omega \sin \frac{\phi}{2} \\
- \frac{\sin^2 \omega \sin \frac{\phi}{2}}{1+\cos \omega} + \mathrm{i} \sin \omega \cos \frac{\phi}{2}
\end{matrix}\right ) + \frac{e^{-\mathrm{i}t}}{2 \cos \omega} \left ( \begin{matrix} - \frac{\sin^2 \omega \cos \frac{\phi}{2}}{1+\cos \omega} - \mathrm{i} \sin \omega \sin \frac{\phi}{2} \\
(1 + \cos \omega) \sin \frac{\phi}{2} - \mathrm{i} \sin \omega \cos \frac{\phi}{2}
\end{matrix}\right ),\\
\psi^\dag (t) =& e^{\mathrm{i}t (\sigma^\omega_z)^\dag} \psi (0) = \frac{e^{\mathrm{i}t}}{2 \cos \omega} \left ( \begin{matrix} (1 + \cos \omega) \cos \frac{\phi}{2} - \mathrm{i} \sin \omega \sin \frac{\phi}{2} \\
- \frac{\sin^2 \omega \sin \frac{\phi}{2}}{1+\cos \omega} - \mathrm{i} \sin \omega \cos \frac{\phi}{2}
\end{matrix}\right ) + \frac{e^{-\mathrm{i}t}}{2 \cos \omega} \left ( \begin{matrix} -\frac{\sin^2 \omega \cos \frac{\phi}{2}}{1+\cos \omega} + \mathrm{i} \sin \omega \sin \frac{\phi}{2} \\
(1 + \cos \omega) \sin \frac{\phi}{2} + \mathrm{i} \sin \omega \cos \frac{\phi}{2}
\end{matrix}\right ).
\end{split}\ee
Then, the evolution $\psi(t)$ is periodic with period $T = \pi,$ namely $\psi (\pi) = e^{\mathrm{i} \pi} \psi (0).$
By \eqref{eq:GP-NH}, we have
\be
\beta = \pi - \int^\pi_0 \frac{\langle \psi (t) | \sigma^\omega_z | \psi (t) \rangle}{\|\psi (t) \|^2} d t = \pi - \int^\pi_0 \frac{\cos \phi + \frac{a}{2 \cos \omega}e^{-2\mathrm{i}t} - \frac{\bar{a}}{2 \cos \omega}e^{2\mathrm{i}t}}{\frac{1}{\cos^2 \omega} - \frac{a}{2 \cos^2 \omega}e^{-2\mathrm{i}t} - \frac{\bar{a}}{2 \cos^2 \omega}e^{2\mathrm{i}t}}d t,
\ee
where $a = \sin^2 \omega + \mathrm{i} \sin \omega \cos \omega \sin \phi.$ Denoting by $I$ the last integral term of the above equality, we have
\be
I = \frac{\cos \omega}{2 \mathrm{i}} \int_{\{z \in \mathbb{C}: |z| =1\}} \frac{\bar{a} z^2- 2( \cos \omega\cos \phi) z - a}{z (\bar{a} z^2 - 2 z + a)} d z = \frac{\pi \cos^2 \omega \cos \phi}{(1-|a|^2)^\frac{1}{2}} = \frac{\pi |\cos \omega| \cos \phi}{(1+ \sin^2 \omega \cos^2 \phi)^\frac{1}{2}}.
\ee
Hence,
\be
\beta = \pi \Big ( 1 - \frac{|\cos \omega| \cos \phi}{(1+ \sin^2 \omega \cos^2 \phi)^\frac{1}{2}}\Big ).
\ee
On the other hand, by \eqref{eq:GP-nonHermitian} we have
\be
\check{\beta} = \pi - \int^\pi_0 \langle \psi^\dag (t) | \sigma^\omega_z | \psi (t) \rangle d t = \frac{ \pi(\cos \omega - \cos \phi)}{\cos \omega} - \frac{\mathrm{i}\pi \sin \omega \sin \phi}{\cos \omega}.
\ee
Therefore, $\check{\beta} \not=\beta$ in general.
\end{example}
\subsection{Observable-geometric phases}\label{OGP}
This subsection is devoted to the study of observable-geometric phases for non-Hermitian quantum systems, which was introduced in \cite{Chen2020} for a quantum system in the Hermitian case. We first define this notion in the non-Hermitian case using the evolution system mentioned in Section \ref{Pre:EvoSys}. Then we give the geometric interpretation of it using the geometry of the observable space based on the group $\mathcal{T} (\mathbb{H})$ of invertible bounded operators (see Section \ref{App} for the details).
Consider a non-Hermitian quantum system with a time-dependent Hamiltonian $\{h(t): t \in [0,T]\},$ where $h(t)$'s are all para-Hermitian operators. Assume that $h(t)$'s have the same domain $\mathbb{D}$ and there exists an evolution system $\{U(t,s) \in \mathcal{T} (\mathbb{H}): t, s \in [0,T]\}$ for $A(t) = - \mathrm{i}h(t)$ on $\mathbb{D}.$ By \eqref{eq:EvoSysEqu} and $U(t,s)^{-1} = U(s,t),$ we then have
the Schr\"{o}dinger equation
\begin{equation}\label{eq:SchrodingerEquPropagatorI}
\mathrm{i} \frac{d}{d t} U(t,s)\phi = h(t) U(t,s)\phi, \quad \forall \phi \in \mathbb{D},
\end{equation}
and the skew Schr\"{o}dinger equation
\begin{equation}\label{eq:SchrodingerEquPropagatorII}
\mathrm{i} \frac{d}{d t} U(s,t) \phi = - \tilde{h}_s(t)U(s,t) \phi, \quad \forall \phi \in \mathbb{D},
\end{equation}
where $\tilde{h}_s(t) = U(s,t) h(t) U(t, s).$ In this case, the evolution system $\{U(t,s): t, s \in [0,T]\}$ is also called the {\it time evolution operator} or {\it propagator} generated by $h(t)$ (cf. \cite[X.69]{RS1980II}).
Note that, by \eqref{eq:SchrodingerEquPropagatorI}, for any $s \in [0,T)$ and $\phi \in \mathbb{H},$ $\phi_s (t) = U (t,s) \phi$ is the unique solution of the time-dependent Schr\"{o}dinger equation
\begin{equation}\label{eq:SchrodingerEquTime}
\mathrm{i} \frac{d}{d t} \phi_s(t) = h(t) \phi_s (t),\quad \phi_s (s) = \phi.
\end{equation}
Given any observable $X_0,$ namely a para-Hermitian operator on $\mathbb{H},$ by \eqref{eq:SchrodingerEquPropagatorI} and \eqref{eq:SchrodingerEquPropagatorII} we conclude that $X(t) = U(0,t) X_0 U(t,0)$ is the unique solution of the time-dependent Heisenberg equation
\begin{equation}\label{eq:HeisenbergEquTime}
\mathrm{i} \frac{d X (t)}{d t} = [X (t), \tilde{h}(t)],\quad X(0) = X_0,
\end{equation}
where $\tilde{h}(t) = U(0,t) h(t) U(t, 0).$ If there exists $\tau \in (0, T)$ such that $X(\tau) = X(0),$ the time evolution of observable $X(t)$ is then called {\it cyclic} with period $\tau,$ and $X_0 = X(0)$ is said to be a cyclic observable.
Suppose that the observable $X_0$ has a non-degenerate eigenvalue associated with every eigenstate $\psi_n \in \mathbb{D}$ for $n \ge 1,$ and $X(t) = U(0,t) X_0 U(t,0)$ is cyclic with period $\tau \in (0,T),$ namely $X(\tau) = X_0.$ Then $U(0, \tau)\psi_n = e^{\mathrm{i} \theta_n}\psi_n$ with some $\theta_n \in \mathbb{C}$ for $n \ge 1.$ Denoting $\psi_n (t) = U(0,t) \psi_n$ for $n \ge 1,$ which are the eigenstates of $X(t),$ by \eqref{eq:SchrodingerEquPropagatorII} we conclude that $\psi_n (t)$ satisfies the skew (time-dependent) Schr\"{o}dinger equation
\begin{equation}\label{eq:SchrodingerEquTimeEigenstate}
\mathrm{i} \frac{d}{d t} \psi_n (t) = - \tilde{h}(t) \psi_n (t),\quad \psi_n (0) = \psi_n.
\end{equation}
Note that $\{\psi_n (t): n \ge 1\}$ is an unconditional basis in $\mathbb{H}$ for every $t.$ Let $\psi^*_n (t) = U^*(t,0) \psi^*_n$ for $n \ge 1,$ then $\{\psi^*_n (t): n \ge 1\}$ is the dual basis of $\{\psi_n (t): n \ge 1\}$ such that $\langle \psi^*_n (t),\psi_m (t) \rangle = \delta_{n m}.$
For each $n \ge 1,$ we define
\begin{equation}\label{eq:Parallelvect}
|\tilde{\psi}_n (t) \rangle = e^{ - \mathrm{i} \int^t_0 \langle \psi^*_n (s) | \tilde{h} (s) | \psi_n (s) \rangle d s } |\psi_n (t) \rangle.
\end{equation}
Since
\be
|\tilde{\psi}_n (t) \rangle = e^{ - \mathrm{i} \int^t_0 \langle \psi^*_n (0) | h (s) | \psi_n (0) \rangle d s } |\psi_n (t) \rangle,
\ee
then $|\tilde{\psi}_n (\tau) \rangle = e^{\mathrm{i} \beta_n}|\psi_n (0) \rangle,$ where
\beq\label{eq:ObGP-NH}
\beta_n = \theta_n - \int^\tau_0 \langle \psi^*_n (0) | h(t) | \psi_n (0)\rangle d t.
\eeq
Moreover, from \eqref{eq:SchrodingerEquTimeEigenstate} we conclude
\begin{equation}\label{eq:ParallelCondVect}
\langle \tilde{\psi}_n (t) | \frac{d}{d t} |\tilde{\psi}_n (t) \rangle =0.
\end{equation}
Also, for any closed path
\begin{equation}\label{eq:ClosedCurve}
|\bar{\psi}_n (t) \rangle = e^{- \mathrm{i} \alpha_n (t)} |\psi_n (t) \rangle,
\end{equation}
where $\alpha_n: [0, \tau) \mapsto \mathbb{C}$ is continuously differential and $\alpha_n (\tau) - \alpha_n (0) = \theta_n$ for every $n \ge 1,$ i.e., $|\bar{\psi}_n (\tau) \rangle = |\bar{\psi}_n (0) \rangle,$ we have
\begin{equation}\label{eq:ObGP-ClosedCurve}
\beta_n = \int^\tau_0\mathrm{i} \langle \bar{\psi}^*_n (t) | \frac{d}{d t} |\bar{\psi}_n (t) \rangle d t,
\end{equation}
where $\bar{\psi}^*_n (t) = e^{-\mathrm{i} \alpha_n (t)} |\psi^*_n (t) \rangle$ for every $n \ge 1.$
Following \cite{Chen2020}, this leads to the notion of the observable-geometric phase in the non-Hermitian setting as follows.
\begin{definition}\label{df:ObGeoPhase}
Using the above notations, the observable-geometric phases of the periodic evolution of observable $X(t)$ in a non-Hermitian quantum system are defined by
\begin{equation}\label{eq:q-GeoPhase}
\beta_n = \theta_n - \int^\tau_0 \langle \psi^*_n (0) | h(t) | \psi_n (0)\rangle d t
\end{equation}
which is uniquely defined up to $2 \pi k$ ($k$ is integer) for every $n \ge 1.$
\end{definition}
\begin{remark}\rm
\begin{enumerate}[{\rm 1)}]
\item Note that for every $n \ge 1,$ $\beta_n$ may be a complex number (see Example \ref{ex:qubit} below). This is different from the ones of a quantum system in the Hermitian case as defined in \cite{Chen2020}.
\item If $h(t)$'s are all Hermitian, then $\psi^*_n (t) = \psi_n (t)$ and the observable-geometric phases $\beta_n$'s are all real and coincide with the ones defined in \cite{Chen2020}.
\item When some eigenvalues of the initial observable $X_0$ are degenerate as eigenstates, this would lead to the notion of non-Abelian observable-geometric phase as similar to the usual non-Abelian geometric phase (cf. \cite{Anandan1988,STAHJS2012}). We will discuss it elsewhere.
\item We can also discuss the adiabatic case of the observable-geometric phase in the non-Hermitian setting, as done in \cite{Chen2020} in the Hermitian case. We omit the details.
\end{enumerate}
\end{remark}
For illustrating the observable-geometric phase in a non-Hermitian quantum system, we consider a qubit case, namely the Hilbert space $\mathbb{H} = \mathbb{C}^2.$
\begin{example}\label{ex:qubit}\rm
Consider a non-Hermitian qubit system, whose Hamiltonian is $H = -\sigma^\omega_z$ (see Example \ref{ex:para-PaulMat}). Given a spin observable $X_0$ with two non-degenerate eigenstates
\be\label{eq:InitalObsQubit}
\psi_1 = \left ( \begin{matrix} \cos \frac{\phi}{2} \\
\sin \frac{\phi}{2}
\end{matrix}\right ),\; \psi_2 = \left ( \begin{matrix} -\sin \frac{\phi}{2} \\
\cos \frac{\phi}{2}
\end{matrix}\right )
\ee
in $\mathbb{C}^2,$ $X(t) = U(0,t) X_0 U(t,0)$ satisfies Eq.\eqref{eq:HeisenbergEquTime} with
\be
\tilde{h} (t) = h (t)= -\sigma^\omega_z
\ee
and $U(t,0) = e^{\mathrm{i}t \sigma^\omega_z}.$ Note that $\sigma^\omega_z$ has eigenvalues $1$ and $-1,$ and the corresponding eigenstates are
\be\label{eq:Zeigenstate}
\left \{ \begin{split}
e_+ (\sigma^\omega_z) & = \left (
\begin{matrix}
1 \\
\frac{\mathrm{i}\sin \omega}{1+ \cos \omega}
\end{matrix} \right ),\\
e_- (\sigma^\omega_z) & = \left (
\begin{matrix}
-\frac{\mathrm{i}\sin \omega}{1+ \cos \omega} \\
1
\end{matrix} \right ),
\end{split}\right.\ee
and so,
\be\label{eq:ZstarEigenstate}
\left \{ \begin{split}
e^*_+ (\sigma^\omega_z) & =\frac{1+ \cos \omega}{2 \cos \omega} \left (
\begin{matrix}
1 \\
-\frac{\mathrm{i}\sin \omega}{1+ \cos \omega}
\end{matrix} \right ),\\
e^*_- (\sigma^\omega_z) & =\frac{1+ \cos \omega}{2\cos \omega} \left (
\begin{matrix}
\frac{\mathrm{i}\sin \omega}{1+ \cos \omega}\\
1
\end{matrix} \right ).
\end{split}
\right.\ee
Then, by Definition \ref{df:FunctCalculusparaHop} we have
\be\begin{split}
U(t,0)& =e^{\mathrm{i}t} |e_+ (\sigma^\omega_z) \rangle \langle e^*_+ (\sigma^\omega_z)| + e^{-\mathrm{i} t} |e_- (\sigma^\omega_z) \rangle \langle e^*_- (\sigma^\omega_z)|,\\
U(0,t)& =e^{-\mathrm{i}t} |e_+ (\sigma^\omega_z) \rangle \langle e^*_+ (\sigma^\omega_z)| + e^{\mathrm{i} t} |e_- (\sigma^\omega_z) \rangle \langle e^*_- (\sigma^\omega_z)|.
\end{split}\ee
Define $\psi_n (t) = U(0,t) \psi_n$ for $n=1,2,$ we have
\be
\begin{split}
\psi_1 (t) = & \frac{e^{-\mathrm{i}t}}{2 \cos \omega} \left ( \begin{matrix} (1 + \cos \omega) \cos \frac{\phi}{2} + \mathrm{i} \sin \omega \sin \frac{\phi}{2} \\
- \frac{\sin^2 \omega \sin \frac{\phi}{2}}{1+\cos \omega} + \mathrm{i} \sin \omega \cos \frac{\phi}{2}
\end{matrix}\right ) + \frac{e^{\mathrm{i}t}}{2 \cos \omega} \left ( \begin{matrix} - \frac{\sin^2 \omega \cos \frac{\phi}{2}}{1+\cos \omega} - \mathrm{i} \sin \omega \sin \frac{\phi}{2} \\
(1 + \cos \omega) \sin \frac{\phi}{2} - \mathrm{i} \sin \omega \cos \frac{\phi}{2}
\end{matrix}\right ),\\
\psi_2 (t) = & \frac{e^{-\mathrm{i}t}}{2 \cos \omega} \left ( \begin{matrix} - (1 + \cos \omega) \sin \frac{\phi}{2} + \mathrm{i} \sin \omega \cos \frac{\phi}{2} \\
- \frac{\sin^2 \omega \cos \frac{\phi}{2}}{1+\cos \omega} - \mathrm{i} \sin \omega \sin \frac{\phi}{2}
\end{matrix}\right ) + \frac{e^{\mathrm{i}t}}{2 \cos \omega} \left ( \begin{matrix} \frac{\sin^2 \omega \sin \frac{\phi}{2}}{1+\cos \omega} - \mathrm{i} \sin \omega \cos \frac{\phi}{2} \\
(1 + \cos \omega) \cos \frac{\phi}{2} + \mathrm{i} \sin \omega \sin \frac{\phi}{2}
\end{matrix}\right ),
\end{split}\ee
which satisfies the skew Schr\"{o}dinger equation \eqref{eq:SchrodingerEquTimeEigenstate}, namely
$$
\mathrm{i} \frac{d \psi_n (t) }{d t} =\sigma^\omega_z \psi_n (t),\quad n=1,2.
$$
The evolution $X(t)$ is periodic with period $\tau = \pi,$ precisely $\psi_n (\pi) = e^{\mathrm{i} \pi} \psi_n (0)$ for $n=1,2.$
Since $\psi^*_n = \psi_n$ for $n=1,2,$ by \eqref{eq:q-GeoPhase} we have
\be
\beta_1 = \pi + \int^\pi_0 \langle \psi_1 | \sigma^\omega_z | \psi_1\rangle d t = \pi + \frac{\cos \phi + \mathrm{i} \sin \omega \sin \phi}{\cos \omega} \pi = \pi \big (1 + \frac{\cos \phi}{\cos \omega} \big ) +\frac{\mathrm{i} \pi \sin \omega \sin \phi}{\cos \omega},
\ee
and
\be
\beta_2 = \pi + \int^\pi_0 \langle \psi_2 | \sigma^\omega_z | \psi_2 \rangle d t = \pi \big (1 - \frac{\cos \phi}{\cos \omega} \big ) - \frac{\mathrm{i} \pi\sin \omega \sin \phi}{\cos \omega}.
\ee
Both are complex numbers if $\omega, \phi \not= 0.$
\end{example}
Finally, we give a geometric interpretation of $\beta_n$'s defined as in \eqref{eq:q-GeoPhase}, involving the geometry of the non-Hermitian observable space developed in Section \ref{App}.
Given a point $O_0 = \{|e_n \rangle \langle e_n^*|\}_{n \ge 1}$ in $\tilde{\mathcal{W}} (\mathbb{H}),$ using the above notations, we define $\tilde{V} (t) \in \mathcal{T} (\mathbb{H})$ for $0 \le t \le \tau$ by
$$
\tilde{V} (t) = \sum_{n \ge 1} |\tilde{\psi}_n (t) \rangle \langle e_n^* |,
$$
where $|\tilde{\psi}_n (t) \rangle$'s are defined in \eqref{eq:Parallelvect}. Then,
$$
\tilde{C}_P: \; [0, \tau] \ni t \longmapsto \tilde{V} (t) \in \mathcal{T} (\mathbb{H})
$$
is a smooth $O_0$-lift of $C_W: [0, \tau] \ni t \mapsto O(t) = \{ |\psi_n (t) \rangle \langle \psi_n^* (t)|\}_{n \ge 1}.$ Since $\tilde{V}^{-1} (t) = \sum_{n \ge 1} |e_n \rangle \langle \tilde{\psi}^*_n (t)|,$ by \eqref{eq:ParallelCondVect}, we have
\begin{equation}\label{eq:CanonicalParallelCond}
\check{\Omega}_{\tilde{V} (t)} \Big [ \frac{d \tilde{V} (t)}{d t} \Big ] = 0
\end{equation}
for all $t \in [0,\tau],$ where $\check{\Omega}$ is the canonical quantum connection (cf. Example \ref{Ex:CanonicalConnection}). This means that $[0, \tau] \ni t \mapsto \tilde{V} (t)$ is the parallel transportation along $C_W$ with respect to the {\it canonical connection} $\check{\Omega}$ on $\xi_{O_0}.$ Therefore, $\tilde{C}_P$ is the {\it horizontal} $O_0$-lift of $C_W$ with respect to $\check{\Omega}$ in the principal bundle $\xi_{O_0}$ such that
\be
\tilde{V} (\tau) |e_n \rangle = |\tilde{\psi}_n (\tau) \rangle = e^{\mathrm{i} \beta_n} |\psi_n \rangle,\quad \forall n \ge 1,
\ee
and so
\begin{equation}\label{eq:HolonomyUnitaryOper}
\tilde{V} (\tau) = \sum^d_{n = 1} e^{\mathrm{i} \beta_n} | \psi_n \rangle \langle e_n^* |
\end{equation}
is the holonomy element associated with the connection $\check{\Omega}, C_W,$ and $V_0 = \sum_{n \ge 1}| \psi_n \rangle \langle e_n^* |$ in $\xi_{O_0}.$
In conclusion, we have the following theorem.
\begin{theorem}\label{thm:q-GeoPhase}\rm
\begin{enumerate}[\rm (1)]
\item For every $n \ge 1,$ the geometric phase $\beta_n$ defined in \eqref{eq:q-GeoPhase} is given by
\begin{equation}\label{eq:q-GeoPhaseExpression}
\beta_n =\langle e^*_n | \mathrm{i} \int^\tau_0 \check{\Omega}_{\bar{V} (t)} \Big [ \frac{d \bar{V} (t)}{d t} \Big ] d t |e_n \rangle = \langle e^*_n | \mathrm{i} \oint_{C_W} \bar{V}^{-1} \star d \bar{V} |e_n \rangle,
\end{equation}
where $\bar{C}_P: [0, \tau] \ni t \mapsto \bar{V} (t) \in \mathcal{T} (\mathbb{H})$ corresponds to any of the closed smooth $O_0$-lifts of $C_W$ with $\bar{V} (0) = \bar{V}_0,$ and $\check{\Omega}_V = V^{-1} \star d V$ is the canonical connection on $\xi_{O_0} (\mathbb{H}).$ Thus, $\beta_n$'s are independent of the choice of the time parameterization of $V (t),$ namely the speed with which $V (t)$ traverses its closed path. It is also independent of the choice of the Hamiltonian as long as the Heisenberg equations \eqref{eq:HeisenbergEquTime} involving these Hamiltonians describe the same closed path $C_W$ in $\tilde{\mathcal{W}} (\mathbb{H}).$
\item The set $\{ \beta_n: n \ge 1\}$ is independent of the choice of the starting point $V_0.$
\item The set $\{ \beta_n: n \ge 1\}$ is independent of the choice of the measurement point $O_0.$ Therefore, this number set is considered to be a set of geometric invariants for $C_W.$
\end{enumerate}
\end{theorem}
\begin{remark}\label{rk:q-ObsGeoPhase}\rm
In the definition \eqref{eq:q-GeoPhase}, the $\beta_n$'s are in fact independent of the choice of measurement points $O_0$ and background geometry over the fiber bundle $\xi_{O_0}.$
\end{remark}
\begin{proof}
(1).\; Let $\bar{C}_P: [0, \tau] \ni t \longmapsto \bar{V} (t) \in \mathcal{F}^{O(t)}_{O_0}$ be a smooth $O_0$-lift of $C_W$ such that $\bar{V} (\tau) = \bar{V} (0)=\bar{V}_0.$ By definition, $\bar{V} (t) = \sum_{n \ge 1} |\bar{\psi}_n (t) \rangle \langle e_n^* |$ and $\bar{V}^{-1} (t) = \sum_{n \ge 1} |e_n \rangle \langle \bar{\psi}^*_n (t) |,$ where $\bar{\psi}_n (\tau) = \bar{\psi}_n (0)$ for all $n \ge 1.$ By \eqref{eq:ObGP-ClosedCurve}, we
conclude \eqref{eq:q-GeoPhaseExpression}.
(2).\; For any $\check{V}_0 \in \mathcal{F}^{O(0)}_{O_0}$ there exists some $G = \sum_{n\ge 1} c_n| e_{\sigma (n)} \rangle \langle e^*_n | \in \mathcal{G}_{O_0}$ with $\sigma \in \Pi (d)$ and $(c_n)_{n \ge 1} \in \mathcal{G}_\8 (\mathbb{C}_*)$ such that $\check{V}_0 = V_0 G.$ Then $\check{C}_P: [0, T] \ni t \longmapsto \check{V}(t) = \tilde{V} (t) G$ is the horizontal $O_0$-lift of $C_W$ with the starting point $\check{V}(0) = V_0 G$ such that $\check{V}(T) |e_n\rangle = e^{\mathrm{i} \beta_{\sigma (n)}} \check{V} (0) |e_n\rangle$ for all $n \ge 1.$ Thus, the set $\{\beta_n: n \ge 1 \}$ is invariant for any starting point $V_0 \in \mathcal{F}^{O(0)}_{O_0}.$ Combining this fact with \eqref{eq:q-GeoPhaseExpression} yields
$$
\{\beta_n: n \ge 1 \} = \Big \{ \langle e^*_n | \mathrm{i} \oint_{C_W} \bar{V} \star d \bar{V} | e_n \rangle:\; n \ge 1 \Big \}
$$
for any closed smooth $O_0$-lift $\bar{C}_P$ of $C_W.$ Therefore, the observable-geometric phases are independent of the choice of the starting point and only depends on the geometry of the curve $C_W$ with respect to the $O_0$-connection $\check{\Omega}.$
(3).\; Let $\tilde{C}_P: [0, \tau] \ni t \longmapsto \tilde{V} (t)$ be the horizontal $O_0$-lift of $C_W$ with respect to $\Omega$ with the starting point $\tilde{V} (0) = V_0.$ For any $O'_0 = \{ | e_n' \rangle \langle (e'_n)^*|: n \ge 1\} \in \tilde{\mathcal{W}} (\mathbb{H})$ there exists some $T \in \mathcal{T} (\mathbb{H})$ such that $O'_0= T O_0 T^{-1}$ with $| e_n' \rangle = T | e_n \rangle$ for $n\ge 1.$ Then $\Omega' = \{\Omega'_P: P \in \mathcal{T} (\mathbb{H}) \}$ is a $O'_0$-connection on $\xi_{O'_0},$ where $\Omega'_P (Q) = T \Omega_{P T} (Q T) T^{-1}$ for any $P \in \mathcal{T} (\mathbb{H})$ and for all $Q \in T_P \xi_{O'_0} (\mathbb{H}).$ By computation, we conclude that $\tilde{C}'_P: [0, T] \ni t \longmapsto \tilde{V}' (t) = \tilde{V} (t) T^{-1}$ is the horizontal $O'_0$-lift of $C_W$ with respect to $\Omega'$ with the starting point $\tilde{V}' (0) = V_0 T^{-1}.$ Therefore,
$$
\tilde{V}' (\tau) | e_n' \rangle = \tilde{V} (\tau) T^{-1} | e_n' \rangle = \tilde{V} (\tau) | e_n \rangle = e^{\mathrm{i} \beta_n} \tilde{V} (0) | e_n \rangle = e^{\mathrm{i} \beta_n} \tilde{V}' (0) | e_n' \rangle,
$$
and hence the set of the geometric phases of $C_W$ with respect to $\Omega'$ is the same as that of $\Omega.$
\end{proof}
\section{Summary}\label{Sum}
Based on a theorem of Antoine and Trapani \cite{AT2014}, we introduce the notions of para-Hermitian and para-unitary operators, and prove a Stone type theorem for the one-parameter group of the para-unitary operators. Using the para-Hermitian and para-unitary operators, we present a mathematical formalism of non-Hermitian quantum mechanics, including the five postulates: the state postulate, the observable postulate, the measurement postulate, the evolution postulate, and the composite-systems postulate. These postulates are non-Hermitian analogies of those found in the Dirac-von Neumann formalism of quantum mechanics. Thus, our formalism is an extension of the Dirac-von Neumann formalism of quantum mechanics to the non-Hermitian setting. In the framework of this formalism, we study geometric phases in the non-Hermitian setting, and generalize the notion of the observable-geometric phase \cite{Chen2020} to the non-Hermitian setting. We hope this formalism could play a role of a mathematical foundation for non-Hermitian quantum mechanics and its application to quantum computation and quantum information theory.
\section{Appendix: Geometry of non-Hermitian observable space}\label{App}
\subsection{Non-Hermitian observable space}\label{ObSpaceNH}
A complete decomposition in $\mathbb{H}$ is defined as a set $\{ | n \rangle \langle n^*|: n \ge 1 \}$ of projections of rank one satisfying
\begin{equation}\label{eq:OrthDecomp}
\sum_{n \ge 1} | n \rangle \langle n^*| = I,\quad \langle n^* |m \rangle = \delta_{n m}.
\end{equation}
We denote by $\tilde{\mathcal{W}} (\mathbb{H})$ the set of all complete decompositions in $\mathbb{H}.$ Note that a complete decomposition $O= \{ | n \rangle \langle n^*|: n \ge 1 \}$ determines uniquely a unconditional basis $\{| n \rangle \}_{n \ge 1}$ up to phases for basic vectors. Conversely, a unconditional basis uniquely defines a complete decomposition in $\mathbb{H}.$ Since a non-Hermitian observable $X$ represented by a para-Hermitian operator with discrete spectrum has a complete decomposition, the evolution of a non-Hermitian quantum system by the Heisenberg equation
\begin{equation}\label{eq:HeisenbergEqu}
\mathrm{i} \frac{d X}{d t} = [X, H]
\end{equation}
for the observable $X,$ gives rise to a curve in $\tilde{\mathcal{W}} (\mathbb{H}).$ This is the reason why $\tilde{\mathcal{W}} (\mathbb{H})$ can be regarded as the observable space, whose geometry induces a geometric structure for a non-Hermitian quantum system.
We equip $\tilde{\mathcal{W}} (\mathbb{H})$ with the Hausdorff distance $D_{\tilde{\mathcal{W}}}$ defined by
\begin{equation}\label{eq:HausdDist}
D_{\tilde{\mathcal{W}}} ( O, O') = \max_{a \in O} \inf_{b \in O'} \| a - b \| + \max_{a \in O'} \inf_{b \in O} \| a - b \|,\quad \forall O, O' \in \tilde{\mathcal{W}}(\mathbb{H}).
\end{equation}
Then $\tilde{\mathcal{W}} (\mathbb{H})$ is a complete metric space under the distance $D_{\tilde{\mathcal{W}}}.$ Also, we define $\tilde{\mathcal{X}} (\mathbb{H})$ to be the set of all ordered sequences $(|n\rangle \langle n^* |)_{n \ge 1},$ where $\{|n\rangle \langle n^* |:\; n \ge 1\}$'s are all complete decompositions in $\mathbb{H}.$ We equip $\tilde{\mathcal{X}} (\mathbb{H})$ with the distance $D_{\tilde{\mathcal{X}}}$ defined as follows: For $(|n\rangle \langle n^* |)_{n \ge 1}, (|\bar{n}\rangle \langle \bar{n}^* |)_{n \ge 1} \in \tilde{\mathcal{X}} (\mathbb{H}),$
$$
D_{\tilde{\mathcal{X}}} ((|n\rangle \langle n^* |)_{n \ge 1}, (|\bar{n}\rangle \langle \bar{n}^* |)_{n \ge 1}) = \max_{n \ge 1} \| |n\rangle \langle n^* | - |\bar{n}\rangle \langle \bar{n}^* | \|.
$$
Then $\tilde{\mathcal{X}} (\mathbb{H})$ is a complete metric space under $D_{\tilde{\mathcal{X}}}$ such that
$$
\tilde{\mathcal{W}} (\mathbb{H}) \cong \frac{\tilde{\mathcal{X}} (\mathbb{H})}{\Pi (d)},
$$
where $\Pi (d)$ denotes the permutation group of $d$ objects ($d$ denotes the dimension of $\mathbb{H}$), which has a representation in $\mathbb{H}$ as follows: For a given unconditional basis $\{|e_n\rangle \}_{n \ge 1}$ of $\mathbb{H},$
\begin{equation}\label{eq:PermutationGroupRepresentation}
\Pi (d) = \bigg \{ V_\sigma = \sum_{n \ge 1} |e_{\sigma (n)} \rangle \langle e_n^* | \in \mathcal{T} (\mathbb{H}):\; \forall \sigma \in \Pi (d) \bigg \}.
\end{equation}
We denote by
\be
\mathcal{G}_\8 (\mathbb{C}_*) = \big \{ (c_n)_{n \ge 1} \in \mathbb{C}_*^d:\; 0< \inf_{n \ge 1} |c_n| \le \sup_{n \ge 1} |c_n|< \8 \big \}.
\ee
Then $\mathcal{G}_\8 (\mathbb{C}_*)$ is an abelian topological group under pointwise multiplication and has a representation in $\mathbb{H}$ as follows: For a given unconditional basis $\{|e_n\rangle \}_{n \ge 1}$ of $\mathbb{H},$
\begin{equation}\label{eq:ScalarGroupRepresentation}
\mathcal{G}_\8 (\mathbb{C}_*) = \bigg \{ V_{(c_n)} = \sum_{n \ge 1} c_n |e_n \rangle \langle e_n^* | \in \mathcal{T} (\mathbb{H}):\; \forall (c_n)_{n \ge 1} \in \mathcal{G}_\8 (\mathbb{C}_*) \bigg \}.
\end{equation}
\begin{proposition}\label{prop:TopoSpaceQ-system}\rm
For a given unconditional basis $\{|e_n\rangle \}_{n \ge 1}$ of $\mathbb{H},$
$$
\tilde{\mathcal{W}} (\mathbb{H}) \cong \{ \mathcal{G} (V):\; V \in \mathcal{T} (\mathbb{H})\}
$$
with
\begin{equation}\label{eq:FiberForm}
\mathcal{G} (V) = \Big \{ \sum_{n \ge 1} c_n |\sigma (n) \rangle \langle e_n^* |:\; \forall \sigma \in \Pi (d), \forall (c_n)_{n \ge 1} \in \mathcal{G}_\8 (\mathbb{C}_*) \Big \},
\end{equation}
where $|n\rangle = V |e_n\rangle$ for any $n \ge 1,$ and the distance between two elements is defined by
$$
d (\mathcal{G} (V), \mathcal{G} (V')) = \inf \{ \| K - G \|: K \in \mathcal{G} (V), G \in \mathcal{G} (V') \}.
$$
\end{proposition}
\begin{proof}
We need to prove that
$$
\mathcal{X} (\mathbb{H}) \cong \frac{\mathcal{T} (\mathbb{H})}{\mathcal{G}_\8 (\mathbb{C}_*)},
$$
from which we conclude the result.
Indeed, for a fixed unconditional basis $\{|e_n\rangle \}_{n \ge 1}$ of $\mathbb{H},$ we have that $\frac{\mathcal{T} (\mathbb{H})}{\mathcal{G}_\8 (\mathbb{C}_*)} = \{ [V]:\; V \in \mathcal{T} (\mathbb{H}) \}$ with
$$
[V] = V \cdot \mathbb{C}_*^d = \Big \{ \sum_{n \ge 1} c_n |n \rangle \langle e_n^* |:\; \forall (c_n)_{n \ge 1} \in \mathcal{G}_\8 (\mathbb{C}_*) \Big \},
$$
where $|n\rangle = V |e_n\rangle$ for $n \ge 1.$ Define $T: \tilde{\mathcal{X}} (\mathbb{H}) \mapsto \frac{\mathcal{T} (\mathbb{H})}{\mathcal{G}_\8 (\mathbb{C}_*)}$ by
$$
T [(|n\rangle \langle n^* |)_{n \ge 1}] \longmapsto [V]
$$
for any $(|n\rangle \langle n^* |)_{n \ge 1} \in \tilde{\mathcal{X}} (\mathbb{H}),$ where $V$ is the invertible operator so that $|n\rangle = V |e_n\rangle$ for $n \ge 1.$ Then, $T$ is surjective and isometric, and so the required assertion follows. This completes the proof.
\end{proof}
\begin{comment}
For illustration, we consider the topology of $\tilde{\mathcal{W}} (\mathbb{H})$ in the qubit case of $\mathbb{H} = \mathbb{C}^2.$ Indeed, we have
$$
\mathcal{X} (\mathbb{C}^2) \cong \frac{\mathcal{T} (\mathbb{C}^2)}{\mathbb{C}_* \times \mathbb{C}_*} \cong ?,
$$
and so
$$
\tilde{\mathcal{W}} (\mathbb{C}^2) \cong \frac{\mathbb{S}^2}{\mathbb{Z}_2},
$$
where we have used the fact $\Pi (2) = \mathbb{Z}_2.$ This has a simple geometrical interpretation, since every element in $\mathcal{X} (\mathbb{C}^2)$ has the form $X_{\vec{n}} = ( |-\vec{n} \rangle \langle -\vec{n}|, | \vec{n}\rangle \langle \vec{n}|)$ with $\vec{n} = (n_x, n_y, n_z) \in \mathbb{S}^2,$ where
$$
| \pm \vec{n}\rangle \langle \pm \vec{n}| = \frac{1}{2} (I \pm \vec{n} \cdot \vec{\sigma}).
$$
Although $X_{\vec{n}} \not= X_{-\vec{n}}$ in $\mathcal{X} (\mathbb{C}^2),$ they both correspond to the same element in $\tilde{\mathcal{W}} (\mathbb{C}^2).$ This implies that $\tilde{\mathcal{W}} (\mathbb{C}^2)$ may have non-trivial topology: There are exactly two topologically distinct classes of loops in $\tilde{\mathcal{W}} (\mathbb{C}^2),$ one corresponds to the trivial class $X_{\vec{n}} \longmapsto X_{\vec{n}}$ while the other to the nontrivial class $X_{\vec{n}} \longmapsto X_{-\vec{n}}.$ Then the first fundamental group $\pi_1 ( \tilde{\mathcal{W}} (\mathbb{C}^2) ) \cong \mathbb{Z}_2,$ and thus the topology of the observable space $\tilde{\mathcal{W}} (\mathbb{C}^2)$ for the qubit system is nontrivial.
\end{comment}
\subsection{Fibre bundles over the non-Hermitian observable space}\label{FibleBundleObNH}
According to \cite{Isham1999}, a bundle is a triple $(E, \pi, B),$ where $E$ and $B$ are two Hausdorff topological spaces, and $\pi: E \mapsto B$ is a continuous map which is always assumed to be surjective. The space $E$ is called the total space, the space $B$ is called the base space, and the map $\pi$ is called the projection of the bundle. For each $b \in B,$ the set $\pi^{-1} (b)$ is called the fiber of the bundle over $b.$ Given a topological space $F,$ a bundle $(E, \pi, B)$ is called a fiber bundle with the fiber $F$ provided every fiber $\pi^{-1} (b)$ for $b \in B$ is homeomorphic to $F.$ For a topological group $G,$ a bundle $(E, \pi, B)$ is called a $G$-bundle, denoted by $(E, \pi, B, G),$ provided $G$ acts on $E$ from the right preserving the fibers of $E$ such that the map $f$ from the quotient space $E/G$ onto $B$ defined by $f (x G) = \pi (x)$ for $x G \in E/G$ is a homeomorphism, namely
\[
\xymatrix{
E \ar[d]_{P_G} \ar[rr]^{id} & & E \ar[d]^\pi \\
E/G \ar[rr]^{f:\cong} & & B }
\]
where $P_G$ is the usual projection. A $G$-bundle $(E, \pi, B, G)$ is principal if the action of $G$ on $E$ is free in the sense that $x g = x$ for some $x \in E$ and $g \in G$ implies $g=1,$ and the group $G$ is then called the structure group of the bundle $(E, \pi, B, G)$ (in physical literatures $G$ is also called the gauge group, cf. \cite{BMKNZ2003}). Note that, in a principal $G$-bundle $(E, \pi, B, G),$ every fiber $\pi^{-1} (b)$ for $b \in B$ is homeomorphic to $G$ by the freedom of the $G$-action, hence it is a fiber bundle $(E, \pi, B, G)$ with the fiber $G$ and is simply called a principal fiber bundle with the structure group $G.$
Next, we construct principal fiber bundles over the observable space $\tilde{\mathcal{W}} (\mathbb{H}).$ To this end, fix a point $O_0 = \{ | e_n \rangle \langle e_n^*|: n \ge 1 \}$ in $\tilde{\mathcal{W}} (\mathbb{H}).$ For any $O \in \tilde{\mathcal{W}} (\mathbb{H}),$ we write
$$
\mathcal{F}^O_{O_0} = \{ V \in \mathcal{T} (\mathbb{H}):\; V^{-1} O V = O_0 \},
$$
that is, $V \in \mathcal{F}^O_{O_0}$ if and only if $\{ V| e_n \rangle: n \ge 1 \}$ is an unconditional basis such that $O = \{ V| e_n \rangle \langle e_n^*| V^{-1}: n \ge 1 \}.$ Indeed, if $O = \{ |n \rangle \langle n^*|: n \ge 1 \},$ then
$$
\mathcal{F}^O_{O_0} = \mathcal{G} (V) = \bigg \{ \sum_{n \ge 1} c_n | \sigma (n) \rangle \langle e_n^* |:\; \forall \sigma \in \Pi (d), \forall (c_n)_{n \ge 1} \in \mathcal{G}_\8 (\mathbb{C}_*) \bigg \},
$$
where $V$ is an invertible operator so that $|n\rangle = V |e_n\rangle$ for $n \ge 1.$ Also, define
\begin{equation}\label{eq:GaugeGroup}
\mathcal{G}_{O_0} = \bigg \{ \sum_{n \ge 1} c_n |e_{\sigma (n)} \rangle \langle e_n^* |:\; \forall \sigma \in \Pi (d), \forall (c_n)_{n \ge 1} \in \mathcal{G}_\8 (\mathbb{C}_*) \bigg \}.
\end{equation}
By \eqref{eq:PermutationGroupRepresentation} and \eqref{eq:ScalarGroupRepresentation}, $\mathcal{G}_{O_0}$ is a (non-abelian) subgroup of $\mathcal{T} (\mathbb{H})$ generated by $\mathcal{G}_\8 (\mathbb{C}_*)$ and $\Pi (d).$
The (right) action of $\mathcal{G}_{O_0}$ on $\mathcal{F}^O_{O_0}$ is defined as: For any $G \in \mathcal{G}_{O_0},$
$$
(G, V) \mapsto V G
$$
for all $V \in \mathcal{F}^O_{O_0}.$ Evidently, this action is free and invariant, namely $\mathcal{F}^O_{O_0}\cdot G = \mathcal{F}^O_{O_0}$ for any $G \in \mathcal{G}_{O_0}$ and every $O \in \tilde{\mathcal{W}} (\mathbb{H}).$ Note that
$$
\mathcal{T} (\mathbb{H}) = \bigcup_{O \in \tilde{\mathcal{W}} (\mathbb{H})} \mathcal{F}^O_{O_0},
$$
and $\mathcal{F}^O_{O_0}$ is homeomorphic to $\mathcal{G}_{O_0}$ as topological spaces since $\mathcal{F}^O_{O_0} = \mathcal{G} [V]$ for some $V \in \mathcal{T} (\mathbb{H})$ such that $O = \{ V| e_n \rangle \langle e_n^*| V^{-1}: n \ge 1 \}.$
The following is then principal fiber bundles over the observable space.
\begin{definition}\label{df:PrincipalFiber}
Given $O_0 \in \tilde{\mathcal{W}} (\mathbb{H}),$ a principal fiber bundle over $\tilde{\mathcal{W}} (\mathbb{H})$ associated with $O_0$ is defined to be
$$
\xi_{O_0} (\mathbb{H}) = (\mathcal{T} (\mathbb{H}), \Pi_{O_0}, \tilde{\mathcal{W}} (\mathbb{H}), \mathcal{G}_{O_0}),
$$
where $\mathcal{T} (\mathbb{H})$ is the total space, and the bundle projection $\Pi_{O_0}: \mathcal{T} (\mathbb{H}) \mapsto \tilde{\mathcal{W}} (\mathbb{H})$ is defined by
$$
\Pi_{O_0} (V) = O
$$
provided $V \in \mathcal{F}^O_{O_0}$ for (unique) $O \in \tilde{\mathcal{W}} (\mathbb{H}),$ namely $\Pi^{-1} (O) = \mathcal{F}^O_{O_0}$ for every $O \in \tilde{\mathcal{W}} (\mathbb{H}).$
We simply denote this bundle by $\xi_{O_0} = \xi_{O_0} (\mathbb{H}).$
\end{definition}
\begin{remark}\rm
In the sequel, we will see that the fixed point $O_0 \in \tilde{\mathcal{W}} (\mathbb{H})$ physically plays a role of measurement. On the other hand, the point $O_0$ induces a differential structure over the base space $\tilde{\mathcal{W}} (\mathbb{H})$ and determines the geometric structure of $\xi_{O_0},$ namely quantum connection and parallel transportation.
\end{remark}
For any two points $O_0, \bar{O}_0 \in \tilde{\mathcal{W}} (\mathbb{H})$ with $O_0 = \{ | e_n \rangle \langle e_n^*|: n \ge 1 \}$ and $\bar{O}_0 = \{ | \bar{e}_n \rangle \langle \bar{e}_n^* |: n \ge 1 \},$ we define an invertible operator $V_0$ by $V_0 |e_n \rangle = | \bar{e}_n \rangle$ for $n \ge 1.$ Then the map $T: \xi_{O_0} \mapsto \xi_{O'_0}$ defined by $T V = V V^{-1}_0$ for all $V \in \mathcal{T} (\mathbb{H})$ is an isometric isomorphism on $\mathcal{T} (\mathbb{H})$ such that $T$ maps the fibers of $\xi_{O_0}$ onto the fibers of $\xi_{O'_0}$ over the same points in the base space $\tilde{\mathcal{W}} (\mathbb{H}),$ namely the following diagram is commutative:
\[
\xymatrix{
\mathcal{T} (\mathbb{H}) \ar[dr]_{\Pi_{O_0}} \ar[rr]^{T} & & \mathcal{T} (\mathbb{H}) \ar[dl]^{\Pi_{O'_0}} \\
& \tilde{\mathcal{W}} (\mathbb{H}) & }
\]
that is, $\Pi_{O_0} = \Pi_{O'_0} \circ T.$ Thus, $\xi_{O_0}$ and $\xi_{O'_0}$ are isomorphic as principal fiber bundles (cf. \cite{Isham1999}).
\subsection{Quantum connection}\label{q-connection}
In order to define the suitable concepts of quantum connection and parallel transportation over the principal fiber bundle $\xi_{O_0},$ we need to introduce a differential structure over $\tilde{\mathcal{W}} (\mathbb{H})$ associated with each fixed $O_0 \in \tilde{\mathcal{W}} (\mathbb{H}).$ Indeed, we will introduce a geometric structure over $\xi_{O_0}$ in a certain operator-theoretic sense (cf. \cite{Chen2020}).
Let us begin with the definition of tangent vectors for $\mathcal{G}_{O_0}$ in the operator-theoretic sense. We denote $\mathcal{Q} (\mathbb{H})$ to be the set of all densely defined closed operators in $\mathbb{H}.$
\begin{definition}\label{df:q-tangvectorUgroup}
Fix $O_0 \in \tilde{\mathcal{W}} (\mathbb{H}).$ For a given $V \in \mathcal{G}_{O_0},$ an operator $Q \in \mathcal{Q} (\mathbb{H})$ is called a tangent vector at $V$ for $\mathcal{G}_{O_0},$ if there is a curve $\chi: (-\varepsilon, \varepsilon) \ni t \mapsto V(t) \in \mathcal{G}_{O_0}$ with $\chi (0) = V$ such that for every $h \in \mathcal{D} (Q),$ the limit
$$
\lim_{t \to 0} \frac{V(t) (h) - V (h)}{t} = Q (h)
$$
in $\mathbb{H},$ denoted by $Q = \frac{d \chi (t)}{d t} \big |_{t=0}.$ The set of all tangent vectors at $V$ is denoted by $T_V \mathcal{G}_{O_0},$ and $T \mathcal{G}_{O_0}= \bigcup_{V \in \mathcal{G}_{O_0}} T_V \mathcal{G}_{O_0}.$ In particular, we denote $\mathrm{g}_{O_0} = T_V \mathcal{G}_{O_0}$ if $V =I.$
\end{definition}
Note that given $V \in \mathcal{G}_{O_0}$ with the form $V =\sum_{n \ge 1} c_n |e_{\sigma (n)} \rangle \langle e_n^* |$ for some $\sigma \in \Pi (d)$ and $(c_n)_{n \ge 1} \in \mathcal{G}_\8 (\mathbb{C}_*),$ for every $Q \in T_V \mathcal{G}_{O_0}$ there exists a unique sequence of complex number $(\alpha_n)_{n \ge 1}$ such that
\begin{equation}\label{eq:VertVectStrucGroupExpress}
Q = \sum_{n \ge 1} \alpha_n |e_{\sigma (n)} \rangle \langle e_n^* |.
\end{equation}
In particular, each element $Q \in \mathrm{g}_{O_0}$ is of form
\begin{equation}\label{eq:VertVectLieAlg}
Q = \sum_{n \ge 1} \alpha_n |e_n \rangle \langle e_n^* |,
\end{equation}
where $(\alpha_n)_{n \ge 1}$ is a sequence of complex number. Thus, $T_V \mathcal{G}_{O_0}$ is a linear subspace of $\mathcal{Q} (\mathbb{H}).$
The following is the tangent space for the base space $\tilde{\mathcal{W}} (\mathbb{H})$ in the operator-theoretic sense.
\begin{definition}\label{df:q-tangvectorBaseSpace}
\begin{enumerate}[{\rm 1)}]
\item Fix $O_0 \in \tilde{\mathcal{W}} (\mathbb{H}).$ A continuous curve $\chi: [a, b] \ni t \mapsto O(t) \in \tilde{\mathcal{W}} (\mathbb{H})$ is said to be differential at a fixed $t_0 \in (a, b)$ relative to $O_0,$ if there is a nonempty subset $\mathcal{A}$ of $\mathcal{Q} (\mathbb{H})$ satisfying that for any $Q \in \mathcal{A}$ there exist $\varepsilon>0$ such that $(t_0 -\varepsilon, t_0 + \varepsilon) \subset [a,b]$ and a strongly continuous curve $\gamma: (t_0 -\varepsilon, t_0 + \varepsilon) \ni t \mapsto V_t \in \mathcal{F}^{O(t)}_{O_0}$ such that the limit
$$
\lim_{t \to t_0} \frac{V_t (h) - V_{t_0} (h)}{t - t_0} = Q (h)
$$
for any $h \in \mathcal{D} (Q).$ In this case, $\mathcal{A}$ is called a tangent vector of $\chi$ at $t=t_0$ and denoted by
$$
\mathcal{A} = \frac{d O(t)}{d t}\big |_{t = t_0} = \frac{d \chi (t)}{d t} \big |_{t = t_0}.
$$
We can define the left (or, right) tangent vector of $\chi$ at $t = a$ (or, $t =b$) in the usual way.
\item Fix $O_0 \in \tilde{\mathcal{W}} (\mathbb{H}).$ Given $O \in \tilde{\mathcal{W}} (\mathbb{H}),$ a tangent vector of $\tilde{\mathcal{W}} (\mathbb{H})$ at $O$ relative to $O_0$ is define to be a nonempty subset $\mathcal{A}$ of $\mathcal{Q} (\mathbb{H}),$ provided $\mathcal{A}$ is a tangent vector of some continuous curve $\chi$ at $t=0,$ where $\chi: (-\varepsilon, \varepsilon) \ni t \mapsto O(t) \in \tilde{\mathcal{W}} (\mathbb{H})$ with $\chi (0) = O,$ i.e., $\mathcal{A} = \frac{d O (t)}{d t} \big |_{t=0}.$ We denote by $T_O \tilde{\mathcal{W}} (\mathbb{H})$ the set of all tangent vectors at $O,$ and write $T \tilde{\mathcal{W}} (\mathbb{H}) = \bigcup_{O \in \tilde{\mathcal{W}} (\mathbb{H})} T_O \tilde{\mathcal{W}} (\mathbb{H}).$
\end{enumerate}
\end{definition}
Note that, the tangent vectors for the base space $\tilde{\mathcal{W}} (\mathbb{H})$ is dependent on the choice of a measurement point $O_0.$ This is the same for the total space $\mathcal{T} (\mathbb{H})$ as follows.
\begin{definition}\label{df:q-tangvectorFiberSpace}
\begin{enumerate}[{\rm 1)}]
\item Fix $O_0 = \{ | e_n \rangle \langle e_n^*|: n \ge 1 \}$ in $\tilde{\mathcal{W}} (\mathbb{H}).$ A strongly continuous curve $\gamma: [a, b] \ni t \mapsto T(t) \in \mathcal{T} (\mathbb{H})$ is said to be differential at a fixed $t_0 \in (a, b)$ relative to $O_0,$ if there is an operator $Q \in \mathcal{Q} (\mathbb{H})$ such that $\{e_n\}_{n \ge 1} \subset \mathcal{D} (Q)$ and the limit
$$
\lim_{t \to t_0} \frac{T(t) (h) - T (t_0) (h)}{t - t_0} = Q (h)
$$
for all $h \in \mathcal{D} (Q).$ In this case, $Q$ is called the tangent vector of $\gamma$ at $t=t_0$ and denoted by
$$
Q = \frac{d \gamma (t)}{d t} \Big |_{t = t_0} = \frac{d T (t)}{d t} \Big |_{t = t_0}.
$$
We can define the left (or, right) tangent vector of $\gamma$ at $t = a$ (or, $t =b$) in the usual way.
Moreover, $\gamma$ is called a smooth curve relative to $O_0,$ if $\gamma$ is differential at each point $t \in [a, b]$ relative to $O_0,$ and for any $n \ge 1,$ the $\mathbb{H}$-valued function $t \mapsto \frac{d \gamma (t)}{d t} (e_n)$ is continuous in $[a, b].$
\item Fix $O_0 \in \tilde{\mathcal{W}} (\mathbb{H}).$ For a given $P \in \mathcal{T} (\mathbb{H}),$ an operator $Q \in \mathcal{Q} (\mathbb{H})$ is called a tangent vector of $\xi_{O_0}$ at $P,$ if there exists a strongly continuous curve $\gamma: (-\varepsilon, \varepsilon) \ni t \mapsto P_t \in \mathcal{T} (\mathbb{H})$ with $\gamma (0) = P,$ such that $\gamma$ is differential at $t=0$ relative to $O_0,$ and $Q = \frac{d \gamma (t)}{d t} \big |_{t =0}.$ Denote $T_P \xi_{O_0} (\mathbb{H})$ to be the set of all tangent vectors of $\xi_{O_0}$ at $P$ relative to $O_0,$ and write
$$
T \xi_{O_0} (\mathbb{H}) = \bigcup_{P \in \mathcal{T} (\mathbb{H})} T_P \xi_{O_0} (\mathbb{H}).
$$
\item Fix $O_0 \in \tilde{\mathcal{W}} (\mathbb{H}).$ Given $P \in \mathcal{T} (\mathbb{H}),$ a tangent vector $Q \in T_P \xi_{O_0} (\mathbb{H})$ is said to be vertical, if there is a strongly continuous curve $\gamma: (-\varepsilon, \varepsilon)\ni t \mapsto P_t \in \mathcal{F}^{\Pi (P)}_{O_0}$ with $\gamma (0) = P$ such that $\gamma$ is differential at $t=0$ relative to $O_0,$ and $Q = \frac{d \gamma (t)}{d t} \big |_{t =0}.$ We denote $V_P \xi_{O_0} (\mathbb{H})$ to be the set of all vertically tangent vectors at $P.$
\end{enumerate}
\end{definition}
\begin{remark}\rm
Note that for a given $P \in \mathcal{T} (\mathbb{H}),$ every $Q \in V_P \xi_{O_0} (\mathbb{H})$ with $O_0 = \{|e_n \rangle \langle e_n^* |\}_{n \ge 1}$ has the form
\begin{equation}\label{eq:VertTangentVect}
Q = \sum_{n \ge 1} \alpha_n P |e_n \rangle \langle e_n^* |,
\end{equation}
where $(\alpha_n)_{n \ge 1}$ is a sequence of complex number.
\end{remark}
Given $O_0 \in \tilde{\mathcal{W}} (\mathbb{H}),$ for each $G \in \mathcal{G}_{O_0},$ the right action $R_G$ of $\mathcal{G}_{O_0}$ on $\xi_{O_0}$ is defined by
$$
R_G (V) = V G,\quad \forall V \in \mathcal{T} (\mathbb{H}).
$$
This induces a map $(R_G)_*: T_P \xi_{O_0} (\mathbb{H}) \mapsto T_{R_G(P)} \xi_{O_0} (\mathbb{H})$ for each $P \in \mathcal{T} (\mathbb{H})$ such that
$$
(R_G)_* (Q) = Q G,\quad \forall Q \in T_P \xi_{O_0} (\mathbb{H}).
$$
Since $R_G$ preserves the fibers of $\xi_{O_0},$ then $(R_G)_*$ maps $V_P \xi_{O_0} (\mathbb{H})$ into $V_{R_G(P)} \xi_{O_0} (\mathbb{H}).$
Now, we are ready to define the concept of quantum connection over the observable space.
\begin{definition}\label{df:q-connetion}
Fix $O_0 = \{|e_n \rangle \langle e_n^* |\}_{n \ge 1}$ in $\tilde{\mathcal{W}} (\mathbb{H}).$ A connection on the principal fiber bundle $\xi_{O_0}= (\mathcal{T} (\mathbb{H}), \Pi_{O_0}, \tilde{\mathcal{W}} (\mathbb{H}), \mathcal{G}_{O_0})$ is a family of linear operators $\Omega = \{\Omega_P:\; P \in \mathcal{T} (\mathbb{H}) \},$ where $\Omega_P$ is a linear mapping from $T_P \xi_{O_0} (\mathbb{H})$ into $\mathrm{g}_{O_0}$ for $P \in \mathcal{T} (\mathbb{H}),$ satisfying the following conditions:
\begin{enumerate}[{\rm (1)}]
\item For any $P \in \mathcal{T} (\mathbb{H}),$
\begin{equation}\label{eq:q-ConnectionVertTangVect}
\Omega_P (Q) = P^{-1} Q, \quad \forall Q \in V_P \xi_{O_0} (\mathbb{H}).
\end{equation}
\item $\Omega_P$ depends continuously on $P$ in the sense that if $P_k$ converges to $P_0$ in $\mathcal{T} (\mathbb{H})$ in the uniform operator topology, and if $Q_k \in T_{P_k} \xi_{O_0} (\mathbb{H}), Q_0 \in T_{P_0} \xi_{O_0} (\mathbb{H})$ such that $\lim_k Q_k (e_n) = Q_0 (e_n)$ for all $n \ge 1,$ then
\be
\lim_k \Omega_{P_k} (Q_k) (e_n) = \Omega_{P_0} (Q_0) (e_n), \quad \forall n \ge 1.
\ee
\item For any $G \in \mathcal{G}_{O_0}$ and $P \in \mathcal{T} (\mathbb{H}),$
\begin{equation}\label{eq:GaugeTransConnection}
\Omega_{R_G(P)} [(R_G)_* (Q )] = G^{-1} \Omega_P (Q) G,\quad \forall Q \in T_P \xi_{O_0} (\mathbb{H}),
\end{equation}
namely, $\Omega$ transforms according to \eqref{eq:GaugeTransConnection} under the right action of $\mathcal{G}_{O_0}$ on $\xi_{O_0} (\mathbb{H}).$
\end{enumerate}
Such a connection is simply called an $O_0$-connection.
\end{definition}
Next, we present a canonical example of such quantum connections, which plays a crucial role in the expression of non-Hermitian observable-geometric phases.
\begin{example}\label{Ex:CanonicalConnection}\rm
Fix $O_0 = \{|e_n \rangle\langle e_n^*| \}_{n \ge 1}\in \tilde{\mathcal{W}} (\mathbb{H}),$ we define $\check{\Omega} = \{\check{\Omega}_P: P \in \mathcal{T} (\mathbb{H}) \}$ as follows: For each $P \in \mathcal{T} (\mathbb{H}),$ $\check{\Omega}_P : T_P \xi_{O_0} (\mathbb{H}) \mapsto \mathrm{g}_{O_0}$ is given by
\begin{equation}\label{eq:CanonConnction}
\check{\Omega}_P (Q) = P^{-1} \star Q,\quad \forall Q \in T_P \xi_{O_0} (\mathbb{H}),
\end{equation}
where
$$
P^{-1} \star Q = \sum_{n \ge 1} \langle e_n^* | P^{-1} Q | e_n \rangle |e_n \rangle \langle e_n^*|.
$$
By \eqref{eq:VertTangentVect}, one has $P^{-1} \star Q = P^{-1} Q \in \mathrm{g}_{O_0}$ for any $Q \in V_P \xi_{O_0} (\mathbb{H}),$ namely $\check{\Omega}_P$ satisfies \eqref{eq:q-ConnectionVertTangVect}. The conditions (2) and (3) of Definition \ref{df:q-connetion} are clearly satisfied by $\check{\Omega}.$ Hence, $\check{\Omega}$ is an $O_0$-connection on $\xi_{O_0}.$ In this case, we write $\check{\Omega}_P = P^{-1} \star d P$ for any $P \in \mathcal{T} (\mathbb{H}).$
\end{example}
\subsection{Quantum parallel transportation}\label{q-ParallelTransport}
This section is devoted to the study of quantum parallel transport over the observable space.
\begin{definition}\label{df:q-lift}
Fix a point $O_0 \in \tilde{\mathcal{W}} (\mathbb{H}).$ For a continuous curve $C_W: [a, b] \ni t \longmapsto O (t) \in \tilde{\mathcal{W}} (\mathbb{H}),$ a lift of $C_W$ with respect to $O_0$ is defined to be a continuous curve
$$
C_P: [a, b] \ni t \longmapsto V(t) \in \mathcal{T} (\mathbb{H})
$$
satisfying the condition that $V(t) \in \mathcal{F}^{O(t)}_{O_0}$ for any $t\in [a, b].$
\end{definition}
\begin{remark}\label{rk:q-lift}\rm
Note that, a lift of $C_W$ depends on the choice of the point $O_0;$ for the same curve $C_W,$ lifts are distinct for different points $O_0.$ For this reason, such a lift $C_P$ is called a $O_0$-lift of $C_W.$
\end{remark}
\begin{definition}\label{df:SmoothCurve}
Fix a point $O_0 \in \tilde{\mathcal{W}} (\mathbb{H}).$ A continuous curve $C_W: [a, b] \ni t \longmapsto O (t) \in \tilde{\mathcal{W}} (\mathbb{H})$ is said to be smooth relative to $O_0,$ if it has a $O_0$-lift $C_P: [a, b] \ni t \longmapsto V(t) \in \mathcal{T} (\mathbb{H})$ which is a smooth curve relative to $O_0.$ In this case, $C_P$ is called a smooth $O_0$-lift of $C_W.$
\end{definition}
Note that, if a continuous curve $C_W: [a, b] \ni t \longmapsto O (t) \in \tilde{\mathcal{W}} (\mathbb{H})$ is smooth relative to $O_0,$ then it is differential at every point $t \in [a, b]$ relative to $O_0.$ Indeed, suppose that $C_P: [a, b] \ni t \longmapsto V(t) \in \mathcal{T} (\mathbb{H})$ is a smooth $O_0$-lift of $C_W.$ For each $t \in [a, b],$ we have $\frac{d C_P (t)}{d t} \in \frac{d O(t)}{d t},$ namely $\frac{d O(t)}{d t}$ is a nonempty subset of $\mathcal{Q} (\mathbb{H}),$ and hence $C_W$ is differential at $t$ relative to $O_0.$
\begin{definition}\label{df:q-HorizontalLift}
Fix $O_0 \in \tilde{\mathcal{W}} (\mathbb{H})$ and suppose $\Omega$ be an $O_0$-connection on $\xi_{O_0} (\mathbb{H}).$ Let $C_W: [0, T] \ni t \longmapsto O (t) \in \tilde{\mathcal{W}} (\mathbb{H})$ be a smooth curve. If $C_P: [0, T] \ni t \longmapsto \tilde{V} (t) \in \mathcal{T} (\mathbb{H})$ is a smooth $O_0$-lift of $C_W$ such that
\begin{equation}\label{eq:ParallelTransfConnectionCond}
\Omega_{\tilde{V} (t)} \Big [ \frac{d \tilde{V} (t)}{d t}\Big ] =0
\end{equation}
for every $t \in [0, T],$ then $C_P$ is called a horizontal $O_0$-lift of $C_W$ with respect to $\Omega.$
In this case, the curve $C_P: t \mapsto \tilde{V} (t)$ is also called the parallel transportation along $C_W$ with the starting point $C_P (0) = \tilde{V}(0)$ with respect to the connection $\Omega$ on $\xi_{O_0} (\mathbb{H}).$
\end{definition}
The following proposition shows the existence of the horizontal lifts in the case of finite dimension.
\begin{proposition}\label{prop:ParallelTransf}\rm
Let $\mathbb{H}$ be a finite-dimensional Hilbert space. Fix $O_0 \in \tilde{\mathcal{W}} (\mathbb{H})$ and let $\Omega$ be an $O_0$-connection on $\xi_{O_0} (\mathbb{H}).$ If $C_W: [0, T] \ni t \longmapsto O (t) \in \tilde{\mathcal{W}} (\mathbb{H})$ is a smooth curve, then for any $V_0 \in \mathcal{F}^{O(0)}_{O_0},$ there exists a unique horizontal $O_0$-lift $\tilde{C}_P$ of $C_W$ with respect to $\Omega$ such that $\tilde{C}_P (0) = V_0.$
\end{proposition}
\begin{proof}
Let $\Gamma: [0, T] \ni t \longmapsto V(t) \in \mathcal{T} (\mathbb{H})$ be a smooth $O_0$-lift of $C_W$ with respect to $\Omega$ with $\Gamma (0) = V_0.$ Note that if $\mathbb{H}$ is a Hilbert space of finite dimension, the condition (2) of Definition \ref{df:q-connetion} implies that the function $t \mapsto \Omega_{\Gamma (t)} \big [ \frac{d \Gamma (t)}{d t} \big ]$ is continuous in $[0, T].$ Then,
\begin{equation}\label{eq:GeodesicEquaGaugeTransf}
\frac{d G (t)}{d t} = - \Omega_{\Gamma (t)} \Big [ \frac{d \Gamma (t)}{d t} \Big ] \cdot G (t)
\end{equation}
with $G (0) = I$ has the unique solution in $[0, T].$ Therefore, $\tilde{C}_P (t) = \Gamma (t) \cdot G (t)$ is the required horizontal $O_0$-lift of $C_W$ for the initial point $V_0 \in \mathcal{F}^{O(0)}_{O_0}.$
To prove the uniqueness, suppose $\check{C}_P: [0, T] \ni t \longmapsto \check{V} (t) \in\mathcal{T} (\mathbb{H})$ be another horizontal $O_0$-lift of $C_W$ for the initial point $V \in \mathcal{F}^{O(0)}_{O_0}.$ Then, for every $t \in [0, T]$ there exists a unique $\check{G} (t) \in \mathcal{G}_0$ such that $\check{C}_P (t) = \tilde{C}_P (t) \cdot \check{G} (t)$ and $\check{G}(0) = I.$ Since
$$
0 = \Omega_{\check{V}(t)} \Big [ \frac{d \check{V}(t)}{d t} \Big ] = \check{G}(t)^{-1} \frac{d \check{G}(t)}{d t},
$$
this follows that $\check{G}(t) = I$ for all $t \in [0, T].$ Hence, the horizontal $O_0$-lift of $C_W$ is unique for the initial point $U \in \mathcal{F}^{O(0)}_{O_0}.$
\end{proof}
\begin{example}\label{Ex:QuantumParallelTransport}\rm
Let $C_P: [0,T] \ni t \mapsto V(t) \in \mathcal{T} (\mathbb{H})$ be a time evolution satisfying the Schr\"{o}dinger equation
\begin{equation}\label{eq:SchrodingerEquTimeUnitaryEvolution}
\mathrm{i} \frac{d V(t)}{d t} = h(t) V (t)
\end{equation}
where $h(t)$ are time-dependent para-Hermitian operators in $\mathbb{H}.$ Given a fixed point $O_0 = \{|e_n \rangle\langle e_n^*|\}_{n \ge 1}$ in $\tilde{\mathcal{W}} (\mathbb{H}),$ define $C_W: [0, T] \ni t \longmapsto O (t) \in \tilde{\mathcal{W}} (\mathbb{H})$ by $O(t) = V(t) O_0 V^{-1}(t)$ for all $t \in [0, T].$ We define $\tilde{C}_P: [0,T] \ni t \mapsto \tilde{V}(t) \in \tilde{\mathcal{U}} (\mathbb{H})$ by
$$
\tilde{V} (t) = \sum_{n \ge 1} \exp \Big ( - \int^t_0\langle e_n^* | \Big [ V^{-1}(s) \frac{V(s)}{d s} \Big ] | e_n \rangle d s \Big )V(t) | e_n \rangle \langle e_n^*|
$$
for every $t \in [0, T],$ along with the initial point $\tilde{V}(0) = V(0) \in \mathcal{F}^{O(0)}_{O_0}.$ Then $\tilde{C}_P$ is a smooth $O_0$-lift of $C_W$ such that
$$
\check{\Omega}_{\tilde{V} (t)} \Big [ \frac{d \tilde{V}(t)}{d t} \Big ] =0
$$
for all $t \in [0, T],$ where $\check{\Omega}$ is the canonical $O_0$-connection introduced in Example \ref{Ex:CanonicalConnection}. Thus, $\tilde{C}_P$ is the horizontal $O_0$-lift of $C_W$ with respect to $\check{\Omega},$ namely $\tilde{C}_p$ is the parallel transportation along $C_W$ with the starting point $C_P (0) = U(0)$ with respect to the connection $\check{\Omega}$ on $\xi_{O_0} (\mathbb{H}).$
\end{example}
\
{\it Acknowledgments}\; This work is partially supported by the Natural Science Foundation of China under Grant No.11871468, and also in part supported by MOST under Grant No. 2017YFA0304500.
\bibliography{apssamp}
| 76,108
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Desperate times, goes the old saying, call for desperate measures. But before members of Congress push ahead with proposals that would legalize, and tax, online poker and other nonsports betting games, they must ask themselves whether the country's financial problems justify this degree of desperation. Even with huge federal budget deficits and a massive national debt, the answer should be no.
A bill approved by the House Financial Services Committee had bipartisan support, and the "pros" cited by a number of backers really boiled down to a single benefit: money. Estimates of federal revenues should such activities be taxed are put at $42 billion over the next 10 years. And supporters try to salve their argument by noting that many such gambling activities - Web-based casinos is one term used - are simply driven offshore anyway, where they operate outside the reach of the law. Why not grab that tax money by making them legal?
That was not the view in 2006, when lawmakers instituted a federal ban on such activities.
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\begin{document}
\title{\textbf{The Effects of El Ni\~no on the Global Weather and Climate}}
\author[1]{\textbf{Marat Akhmet}\thanks{Corresponding Author Tel.: +90 312 210 5355, Fax: +90 312 210 2972, E-mail: marat@metu.edu.tr}$^{,}$}
\author[2]{\textbf{Mehmet Onur Fen}}
\author[1]{\textbf{Ejaily Milad Alejaily}}
\affil[1]{\textbf{Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey}}
\affil[2]{\textbf{Department of Mathematics, TED University, 06420 Ankara, Turkey}}
\date{}
\maketitle
\begin{abstract}
\noindent This paper studies the chaotic behavior of hydrosphere and its influence on global weather and climate. We give mathematical arguments for the sea surface temperature (SST) to be unpredictable over the global ocean. The impact of SST variability on global climate is clear during global climate patterns, which involve large-scale ocean-atmosphere fluctuations similar to the El Ni\~no-Southern Oscillation (ENSO). Sensitivity (unpredictability) is the core ingredient of chaos. Several researches suggested that the ENSO might be chaotic. It was Vallis [\citen{Vallis}] who revealed unpredictability of ENSO by reducing his model to the Lorenz equations. Interactions of ENSO and other global climate patterns may transmit chaos. We discuss the unpredictability as a global phenomenon through extension of chaos ``horizontally'' and ``vertically'' in coupled Vallis ENSO models, Lorenz systems, and advection equations by using theoretical as well as numerical analyses. To perform theoretical research, we apply our recent results on replication of chaos and unpredictable solutions of differential equations, while for numerical analysis, we combine results on unpredictable solutions with numerical analysis of chaos in the advection equation.
\vspace{.2cm}
\noindent \textbf{Keywords:} El Ni\~no-Southern Oscillation; Vallis model; Advection equation; Lorenz system; Weather unpredictability; Climate catastrophes
\end{abstract}
\vspace{.5cm}
\noindent \textbf{The famous Lorenz equations give birth to the weather related observations. One of them is the unpredictability of weather in long period of time, which is a meteorological concept, and another one is that small changes of the climate and even weather at present may cause catastrophes for the human life in future. Issuing from this, we have taken into account the following three features of the Lorenz system, to emphasize the actuality of the present study. Firstly, it is a regional model. Secondly, for some values of its parameters the equations are non-chaotic. Finally, the model is of the atmosphere, but not of the hydrosphere. Therefore, one has to make additional investigations to reveal that the unpredictability of weather is a {\it global} phenomenon, and climatic catastrophes can be caused by physical processes at {\it any point} on the surface of the globe. The present paper is concerned with all of the three factors issuing from the ocean surface dynamics of ENSO type, and results of our former research.}
\section{Introduction and Preliminaries}
\subsection{Unpredictability of Weather and Deterministic Chaos}
Global climate change has gained the attention of scientists and policymakers. The reason for that lies in its remarkable impact on human life on the Earth \cite{Roy}. Climate change affects and controls many social, economic and political human activities. It was an essential motive of human migration throughout history.
Weather is defined by the condition of the atmosphere at a specific place and time measured in terms of temperature, humidity, air pressure, wind, and precipitation, whereas climate can be viewed as the average of weather of a large area over a long period of time \cite{Petersen}. Some definitions of climate expand to include the conditions of not only the atmosphere, but also the rest components of the climate system: hydrosphere, cryosphere, lithosphere, biosphere and according to Vernadsky no{\"o}sphere \cite{Eppelbaum}.
During the last few decades, big efforts have been made to develop weather and climate change forecasting models. Due to the chaotic nature of weather, the forecasting range of weather prediction models is limited to only a few days. Climate models are more complicated than ordinary weather forecasting models, since they need to include additional factors of climate system that are not important in the weather forecast \cite{Schmandt}. Understanding the concepts of chaos is an important step toward better comprehension of the natural variability of the climate system on different time scales. This involves determining what the reasons and sources that stand behind of presence of chaos in weather and climate models. Any progress made in this path will be helpful to adjust the conception of climate change and find solutions for climate control.
Chaos can be defined as aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions \cite{Strogatz}. Predictability consists of constructing a relationship between cause and effect by which we can predict and estimate the future behavior of a physical property. Unpredictability means the failure of such empirical or theoretical relationships to predict due to consisting of noise term(s), mathematical nature of the relationships or intrinsic irregularity of the physical property itself. Mathematically unpredictability is considered as a result of the sensitive dependence on initial conditions, which is an essential feature of Devaney chaos \cite{Devaney}. Recently, it is theoretically proved that a special kind of Poisson stable trajectory, called an unpredictable trajectory, gives rise to the existence of Poincar\'{e} chaos \cite{AkhmetExistence,AkhmetPoincare,AkhmetUnpredictable}.
Unpredictability in the dynamics of weather forecast models was firstly observed by E. N. Lorenz. He developed a heat convection model consisting of twelve equations describing the relationship between weather variables such as temperature and pressure. Lorenz surprisingly found that his system was extremely sensitive to initial conditions. Later, in his famous paper [\citen{Lorenz}], he simplified another heat convection model to a three-equation model that has the same sensitivity property \cite{Robinson}. This model is defined by the following nonlinear system of ordinary differential equations:
\begin{equation} \label{LorenzSystem}
\begin{split}
& \frac{dx}{dt} =-\sigma\,x + \sigma y,\\
& \frac{dy}{dt} = r\,x - xz-y, \\
& \frac{dz}{dt} = xy - b \,z,
\end{split}
\end{equation}
where the variable $ x $ represents the velocity of the convection motion, the variable $ y $ is proportional to the temperature difference between the ascending and descending currents, and the variable $ z $ is proportional to the deviation of the vertical temperature profile from linearity, whereas the constants $ \sigma $, $ r $, and $ b $ are positive physical parameters. Model (\ref{LorenzSystem}) describes the thermal convection of a fluid heated from below between two layers. With certain values of these parameters, Lorenz system possesses intrinsic chaos and produces the so-called Lorenz butterfly attractor.
The paper [\citen{AkhmetLorenz}] was concerned with the extension of chaos through Lorenz systems. It was demonstrated in [\citen{AkhmetLorenz}] that Lorenz systems can be unidirectionally coupled such that the drive system influences the response system, which is non-chaotic in the absence of driving, in such a way that the latter also possesses chaos. A possible connection of these results to the global weather dynamics was also provided in that study.
\subsection{Ocean-Atmosphere Interaction}
Coupled ocean-atmosphere models are the most fundamental tool for understanding the natural processes that affect climate. These models have been widely applied to interpret and predict global climate phenomena such as ENSO \cite{Siedler}. In meteorology and climate science, SST is considered as a very important factor in ocean-atmosphere interaction, where it plays a basic role in determining the magnitude and direction of the current velocity, as well as the ocean surface wind speed. It is difficult to give a precise definition of SST due to the complexity of the heat transfer operations in the mixed layer of upper ocean. In general, however, it can be defined as the bulk temperature of the oceanic mixed layer with a depth varies from $ 1 \, m $ to $ 20 \, m $ depending on the measurement method used \cite{Barale}. The importance of SST stems from the fact that the world's oceans cover over $ 70 \, \% $ of the whole surface of the globe. This large contact area gives way to an active ocean-atmosphere interaction and sometimes becomes a fertile place for complex feedbacks between the ocean and atmosphere that drive an irregular climate change.
The most important example of the interactions and feedbacks between the ocean and the atmosphere is El Ni\~no and Southern Oscillation (ENSO) which is defined as a global coupled ocean-atmosphere phenomenon occurs irregularly in the Pacific Ocean about every $2$ to $7$ years \cite{Stuecker}. This phenomenon is accompanied by undesirable changes in weather across the tropical Pacific and losses in agricultural and fishing industries especially in South America. The El Ni\~no mechanism can be briefly summarized as follows: During normal conditions in the equatorial Pacific, trade winds blow from east to west driving the warm surface current in the same direction. As a consequence of this, warm water accumulates in the western Pacific around southeast Asia and northern Australia. On the opposite side of the ocean around central and south America, the warm water, pushed to the west, is replaced by upwelling cold deep water. During El Ni\~no conditions, the trade winds are much weaker than normal. Because of this and due to SST difference, warm water flows back towards the western Pacific. This situation involves large changes in air pressure and rainfall patterns in the tropical Pacific. The cool phase of this phenomenon is called La Nina, which is an intensification of the normal situation. The term ``Southern Oscillation'' is usually used to refer to the difference of the sea-level pressure (SLP) between Tahiti and Darwin, Australia. Bjerknes \cite{Bjerknes} conclude that El Ni\~no and the Southern Oscillation are merely two different results of the same phenomenon. These phases of the phenomenon are scientifically called El Ni\~no Southern Oscillation or shortly ENSO. From the above mechanism we can note that the ENSO dynamics is a perfect example of self-excited oscillating systems.
The ramifications of El Ni\~no are not restricted to the Pacific basin alone, but have widespread effects which severely disrupt global weather patterns. In the last few decades scientists developed theories about the climatic engine which produced El Ni\~no, and they are trying to explain how that engine interact with the great machine of global climate. Although remarkable progress has been made in monitoring and forecasting the onset of El Ni\~no, it is still challenging to predict its intensity and the impact of the event on global weather. Study of ENSO is considered as a key to understanding climate change, it is a significant stride toward the meteorology's ultimate goal, ``accurate prediction and control of world weather''.
Besides the ENSO, there are several other atmospheric patterns that occur in different regions of the Earth. These phenomena are interacting in very complicated ways. Many researchers paid attention to the mutual influence of these phenomena and investigated if there is any co-occurrence relationship or interaction between them.
The most similar atmosphere-ocean coupled phenomenon to ENSO is the Indian Ocean Dipole (IOD), which occur in the tropical Indian Ocean, and it is sometimes called the Indian Ni\~no. IOD has normal (neutral), negative and positive phases. During neutral phase, Pacific warm water, driven by the Pacific trade winds, cross between south Asia and Australia and flow toward the Indian Ocean. Because of the westerly wind, the warm water accumulates in the eastern basin of Indian Ocean. In the negative IOD phase with the coincidence of strength of the westerly wind, warmer water concentrate near Indonesia and Australia, and cause a heavy rainfall weather in these regions and cooler SST and droughts in the opposite side of the Indian Ocean basin around the eastern coast of Africa. The positive phase is the reversal mode of the negative phase, i.e., what happened in the east side will happen in west side and vice versa.
From the above we can see that there is a symmetry between the IOD and ENSO mechanisms. Indeed, SST data shows that the Indian Ocean warming appears as a near mirror image of ENSO in the Pacific \cite{Chambers}. In addition, the IOD is likely to have a link with ENSO events, where a positive (negative) IOD often occurs during El Ni\~no (La Nina) \cite{Eamus, Yamagata}. Luo et al. \cite{Luo} investigated the ENSO-IOD interactions, and they suggest that IOD may significantly enhance ENSO and its onset forecast, and vice versa. Several other researchers like \cite{Behera, Roxy} studied the relationship and interaction between ENSO and IOD. It should be noted here that (as in all these studies) the SST considered as the major variable, indicator and index for these events.
Other important atmosphere-ocean coupled phenomena are briefly described in Table \ref{T1} (Appendix) and Figure \ref{Wmap} (Appendix) shows the places of occurrence of them. They have significant influences on weather and climate variability throughout the world. Similar to the relationship between ENSO and IOD, various studies show expected relationships between these phenomena and mutual effects on their predictability. These pattern modes have different degrees of effect on SST. In Table \ref{T1} (Appendix) we see that the patterns that remarkably influence the ocean temperature are indexed by SST, whereas those that are most correlated with air pressure, the main indexes of them are based on SLP.
\subsection{El Ni\~no Chaos} \label{ElNinoChaos}
The SST behavior associated with ENSO indicates irregular fluctuations. The ENSO indicator NINO3.4 index, for example, is one of the most commonly used indices, where the SST anomaly averaged over the region bounded by $ 5^{\circ}$N--$5^{\circ}$S, $170^{\circ}$--$120^{\circ}$W \cite{Bunge}. Figure \ref{SST_Curve} shows the oscillatory behavior of SST in the NINO3.4 region. Data from the Hadley Centre Sea-Ice and SST dataset HadISST1 \cite{Rayner} is used to generate the figure. This behavior encourages many scientists to answer the question: Is ENSO a self-sustained chaotic oscillation or a damped one, requiring external stochastic forcing to be excited? \cite{Sheinbaum}.
\begin{figure}[H]
\centering
\includegraphics[width=1.0\linewidth]{SSTHadleyData}
\caption{Sea surface temperature anomalies of NINO3.4 region. The data utilized in the figure is from the Hadley Centre Sea-Ice and SST dataset HadISST1 \cite{Rayner}.}
\label{SST_Curve}
\end{figure}
There are different hypotheses for the source of chaos in ENSO. According to Neelin and Latif \cite{Neelin}, deterministic chaos within the nonlinear dynamics of coupled system, uncoupled atmospheric weather noise and secular variation in the climatic state are the possible source of ENSO irregularity. Tiperman et al. \cite{Tziperman} concluded that the chaotic behavior of ENSO is caused by the irregular jumping of the ocean-atmosphere system among different nonlinear resonances. Several studies like \cite{Battisti, Penland} support this assumption and attributed the irregularity and the unpredictability of ENSO to influence of stochastic forcing generated by weather noise. Other studies like \cite{Zebiak, Munnich} infer that ENSO is intrinsically chaotic, which means that the irregularity and the loss of predictability are independent of the chaotic nature of weather.
Practically, investigating chaos in ENSO needs long time-series of data, which make the task quite difficult experimentally. Vallis \cite{Vallis}, developed a conceptual model of ENSO and suggested that the ENSO oscillation exhibits a chaotic behavior. Vallis used finite difference formulation to reduce two dimensional versions of advection and continuity equations to a set of ordinary differential equations. In addition, he assumed that the zonal current is driven by the surface wind, which is in turn proportional to the temperature difference across the ocean. The model is described by the set of equations
\begin{equation} \label{VallisModel}
\begin{split}
\frac{du}{dt} & =\beta \,(T_e-T_w)-\lambda \,(u-u^*),\\
\frac{dT_w}{dt} & = \frac{u}{2\Delta x} \,(\bar{T}-T_e)-\alpha \,(T_w-T^*) , \\
\frac{dT_e}{dt} & = \frac{u}{2\Delta x} \,(T_w-\bar{T})-\alpha \,(T_e-T^*),
\end{split}
\end{equation}
where $ u $ represents the zonal velocity, $ T_w $ and $ T_e $ are the SST in the eastern and western ocean respectively, $ \bar{T} $ is the deep ocean temperature, $ T^* $ is the steady state temperature of ocean, $ u^* $ represents the effect of the mean trade winds, $ \Delta x $ is the width of the ocean basin, and $\alpha,$ $\beta$ and $\lambda$ are constants.
By nondimensionalizing Equations (\ref{VallisModel}) and forming the sum and difference of the two temperature equations, one can see that these equations have the same structure as the Lorenz system (\ref{LorenzSystem}). Vallis utilized the fact that the Lorenz system, with specific parameters, is intrinsically chaotic, and showed that a chaotic behavior of the sum and difference of the west and east SST can be obtained.
ENSO, as mentioned above, occurs as a result of the interaction of the ocean and atmosphere. Therefore, modeling of ENSO would be a good instrument to research unpredictability not only in the atmosphere but also in the hydrosphere. Nevertheless, ENSO provides the arguments that unpredictability is also proper for sea water parameters which possibly can be reduced to a single one, the SST, if one excludes flow characteristics.
Vallis saved in the model only hydrosphere variables ignoring the variation of atmosphere parameters when he considers chaos problem. In our opinion, however, the model is appreciated as a pioneer one, and furthermore, it implies chaos presence in the Pacific ocean water. Hopefully, in the future, ENSO with both atmosphere and hydrosphere characteristics being variable will be modeled, but this time we focus on chaotic effects of ENSO by utilizing the Vallis model.
\subsection{SST Advection Equation}
The temporal and spatial evolution of the SST is governed by a first order quasi-linear partial differential equation, the advection equation. If we denote the SST by $ T $, the temperature advection equation of mixed layer of fixed depth can be written in the form \cite{Willebrand,Lucas}
\begin{equation} \label{AdvectionEq}
\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z}= f(t,x,y,z,T),
\end{equation}
where $ u, v, w $ are the zonal, meridional and vertical components of current velocity, respectively. These velocities theoretically must satisfy the continuity equation
\begin{equation} \label{ContinuityEq}
\frac{\partial}{\partial x}(\rho u) + \frac{\partial}{\partial y}(\rho v) + \frac{\partial}{\partial z}(\rho w) = - \frac{\partial \rho}{\partial t},
\end{equation}
where $ \rho $ is the seawater density.
The inhomogeneous (forcing) term $ f $ on the right-hand side of Equation (\ref{AdvectionEq}) consists of the shortwave flux, the evaporative heat flux, the combined long-wave back-radiation and sensible heat flux and heat flux due to vertical mixing \cite{Kessler}. This term can be described by \cite{Gent, Stevenson, Jochum}
\begin{equation} \label{ForceTermEq}
f \approx \frac{1}{h \rho \, C_p} \, \frac{\partial q}{\partial z} + D,
\end{equation}
where $ h $ is the mixed layer depth, $ C_p $ is the heat capacity of seawater, $ q $ is radiative and diffusive heat flux, and $ D $ is the thermal damping (the numerical diffusion operator).
The spatial and temporal domain of Equation (\ref{AdvectionEq}) depend on the region and the nature of the phenomenon under study. For studying ENSO or IOD, for instance, there are various regions for monitoring SST. NINO3.4 is one of the most commonly used indices for ENSO. Dipole Mode Index (DMI) is usually used for IOD, and it depends on the difference in average SST anomalies between the western $50^{\circ}$E--$70^{\circ}$E, $10^{\circ}$N--$10^{\circ}$S and the eastern $90^{\circ}$E--$110^{\circ}$E, $0^{\circ}$--$10^{\circ}$S boxes \cite{Saji}. The mixed layer depth $ h $ varies with season and depends on the vertical heat flux through the upper layers of the ocean. The average of mixed layer depth is about $ 30 \, \text{m} $ \cite{Lukas}. Different studies of ocean-atmosphere coupled models considered different regions of various sizes. Zebiak and Cane \cite{Zebiak}, for example, developed a model of ENSO. They considered a rectangular model extending from $ 124^{\circ} $E to $ 80^{\circ} $W and $ 29^{\circ} $N to $ 29^{\circ} $S, with constant mixed layer depth of $ 50 \, \text{m} $ and 90 years simulation.
From the above we find that the domain of Equation (\ref{AdvectionEq}) depends on the purpose of the study. To study ENSO, for instance, we would cover a big region of pacific ocean basin, and if we choose the origin of coordinates to be at $160^{\circ}$E on the Equator, we can write the domain of (\ref{AdvectionEq}) as follows
\begin{equation*} \label{DomainMainEq}
t \geq 0, \quad 0\leq x \leq 9000 \, \text{km} , \quad -3000 \, \text{km} \leq y \leq 3000 \, \text{km}, \quad -100 \, \text{m} \leq z \leq 0.
\end{equation*}
The inhomogeneous term in Equation (\ref{AdvectionEq}), which includes mixing processes of heat transfer, plays the main role for chaotic dynamics. In addition to this term, a chaotic behavior in ocean current velocity terms may also produce an unpredictable behavior in SST. These causes of unpredictability are proved analytically and numerically by perturbing these terms by unpredictable functions. In this study we treat Equation (\ref{AdvectionEq}) mathematically without paying attention to the dimensions of the physical quantities. The important thing to us is the possibility of presence of chaos in this advection equation endogenously or be acquired from other equation or system. The advection equation, in addition to the Vallis model and the Lorenz system, will be used to demonstrate the extension of unpredictability ``horizontally'' through the global ocean and ``vertically'' between ocean and atmosphere.
\subsection{Unpredictable Functions and Chaos} \label{SubSec:UnpFuncChaos}
There are different types and definitions of chaos. Devaney \cite{Devaney} and Li-Yorke \cite{Yorke} chaos are the most frequently used types, which are characterized by transitivity, sensitivity, frequent separation and proximality. Another common type is the period-doubling cascade, which is a sort of route to chaos through local bifurcation \cite{Feigenbaum80,Scholl,SanderYorke11}. In the papers [\citen{AkhmetUnpredictable,AkhmetPoincare}], Poincar\'{e} chaos was developed by introducing the theory of unpredictable point and unpredictable function, which are built on the concepts of Poisson stable point and function. We define unpredictable point as follows. Let $(X, d)$ be a metric space and $ \pi : \mathbb{T} \times X \to X $ be a flow on X, where $ \mathbb{T} $ refer to either the set of real numbers or the set of integers. We assume that the triple $ ( \pi, X, d ) $ defines a dynamical system.
\begin{defn} ([\citen{AkhmetUnpredictable}]) \label{UnpPointDiff}
A point $ p \in X $ and the trajectory through it are unpredictable if there exist a positive number $ \epsilon $ (the unpredictability constant) and sequences $\left\{t_n\right\}$, $\left\{\tau_n\right\}$ both of which diverge to infinity such that $ \displaystyle{\lim_{n \to \infty}} \pi(t_n , p) = p $ and $ d[ \pi(t_n + \tau_n , p) , \pi(\tau_n , p)] \geq \epsilon $ for each $ n \in \mathbb{N} $.
\end{defn}
Definition \ref{UnpPointDiff} describes unpredictability as individual sensitivity for a motion, i.e., it is formulated for a single trajectory. The Poincar\'{e} chaos is distinguished by Poisson stable motions instead of periodic ones. Existence of infinitely many unpredictable Poisson stable trajectories that lie in a compact set meet all requirements of chaos. Based on this, chaos can be appeared in the dynamics on the quasi-minimal set which is the closure of a Poisson stable trajectory. Therefore, the Poincar\'{e} chaos is referred to as the dynamics on the quasi-minimal set of trajectory initiated from unpredictable point.
The definition of an unpredictable function is as follows.
\begin{defn} ([\citen{AkhmetUnpredSolnDE}]) \label{UnpFuncDiff}
A uniformly continuous and bounded function $ \varphi: \mathbb{R} \rightarrow \mathbb{R}^m $ is unpredictable if there exist positive numbers $\epsilon$, $\delta$ and sequences $\left\{t_n\right\}$, $\left\{\tau_n\right\}$ both of which diverge to infinity such that $ \| \varphi(t+t_n) - \varphi(t) \| \rightarrow 0 $ as $ n \rightarrow \infty $ uniformly on compact subsets of $ \mathbb{R} $, and $ \| \varphi(t+t_n) - \varphi(t) \| \geq \epsilon $ for each $ t \in [ \tau_n - \delta , \tau_n +\delta ] $ and $ n \in \mathbb{N} $.
\end{defn}
To determine unpredictable functions, we apply the uniform convergence topology on compact subsets of the real axis. This topology allows us to construct Bebutov dynamical system on the set of the bounded functions \cite{AkhmetExistence,Sell}. Consequently, the unpredictable functions imply presence of the Poincar\'{e} chaos.
\subsection{Global Weather and Climate}
The topic of weather and climate is one of the most profoundly important issues concerning the international community. It becomes very actual since the catastrophic phenomena such as global warming, hurricanes, droughts, and floods. This is why weather and climate are agenda of researches in physics, geography, meteorology, oceanography, hydrodynamics, aerodynamics and other fields. The problem is global, that is a comprehensive model would include the interactions of all major climate system components, howsoever, for a specific aspect of the problem, a appropriate model combination can be considered \cite{Stocker}. In the second half of the last century, it was learned \cite{Lorenz} that the weather dynamics is irregular and sensitive to initial conditions. Thus the chaos was considered as a characteristic of weather which can not be ignored. Moreover, chaos can be controlled \cite{Ott,Pyragas}. These all make us optimistic that the researches of weather and climate considering chaos effect may be useful not only for the deep comprehension of their processes but also for control of them. In our research \cite{AkhmetReplication}, we have shown how a local control of chaos can be expanded globally.
It is not wrong to say that in meteorological studies, chaos is considered as a severe limiting factor in the ability to predict weather events accurately \cite{Saravanan}. Beside this one can say that chaos is also a responsible factor for climate change if it is considered as a weather consequence. This is true, firstly, because of the weather unpredictability, since predictability can be considered as a useful feature of climate with respect to living conditions, and secondly, as the small weather change may cause a global climate change in time. Accordingly, it is possible to say that the control of weather, even a limited artificial one, bring us to a change of climate.
The chaotic behavior has also been observed in several models of ENSO \cite{Neelin}. Presence of chaos in the dynamic of this climate event provides other evidence of the unpredictable nature of the global weather. Besides the Lorenz chaos of atmosphere, ``Vallis chaos'' takes place in the hydrosphere. Without exaggerating, we can say that chaos seems to be inherent at the essence of any deterministic climate model. Therefore, unpredictability can be globally widespread phenomenon through constructive interactions between the components of the climate system.
To give a sketch how chaos is related globally to weather and climate, we will use, in the present research, information on dynamics of ENSO which will mainly utilize the Vallis model as will as the SST advection equation and the Lorenz equations. They will be properly coupled to have the global effect. It is apparent that, in the next research, the models will possibly be replaced by more developed ones, but our main idea is to demonstrate a feasible approach to the problem by constructing a special net of differential equations system. Obviously, one can consider the net as an instrument which can be subdued to an improvement by arranging new perturbation connections.
Proceeding from aforementioned remarks and as a part of the scientific work, we focus on one possible aspect of global weather and climate dynamics based on El Ni\~no phenomenon. To address this aim, we first review the Vallis model research for El Ni\~no in Subsection \ref{ElNinoChaos}, then, in Section \ref{UnpAdvEq} we analyze the presence of chaos in isolated models for the SST advection equation. In Section \ref{HorizontalExtension}, the extension of chaos in hydrosphere discussed through coupling of advection equation, the Vallis model and also mixing advection equation with the Vallis model. In paper \cite{AkhmetLorenz} we considered chaos as a global phenomenon in atmosphere, but it is clear that, to say about the globe weather one should take into account hydrosphere as well as the interaction processes between atmosphere and seas. For this reason Section \ref{VerticalExtension} is written where chaos extension from ocean to air and vice versa is discussed on the base of the Lorenz and Vallis models. So, finalizing the introduction we can conclude that the present paper is considered as an attempt to give a sketch of the global effects of chaos on weather and climate. This results are supposed to be useful for geographers, oceanographers, climate researchers and those mathematician who are looking for chaotic models and theoretical aspects of chaos researching.
\section{Unpredictable Solution of the Advection Equation} \label{UnpAdvEq}
In this section we study the presence of Poincar\'{e} chaos in the dynamics of Equation (\ref{AdvectionEq}). We expect that the behavior of the solutions of (\ref{AdvectionEq}) depends on the function $ f $ and the current velocity components $ u, v, w $, which are used in the equation. From Equation (\ref{ForceTermEq}), we see that the forcing term $ f $ depends mainly on the heat fluxes between the sea surface and atmosphere which is governed by SST, air temperature and wind speed, as well as between layers of sea which is caused by sea temperature gradient and vertical (entrainment) velocity. Therefore, this forcing term can be a natural source of noise and irregularity. Ocean currents are mainly driven by wind forces, as well as temperature and salinity differences \cite{Coley}. Thence again we can deduce that the irregular fluctuations of wind may be reflected in the behavior of SST.
To demonstrate the role of the function $ f $ in the dynamics of Equation (\ref{AdvectionEq}), let us take into account the equation
\begin{eqnarray} \label{Assump2Eq}
\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z}= -0.7 \, T+0.3 \, w_1 \, T+5 \, \sin(xt),
\end{eqnarray}
where the current velocity components are defined by $u= \sin(\frac{_x}{^{2}})+\sin(t)+3,$ $v= -0.02,$ and $w= -\frac{_1}{^{2}} \cos(\frac{_x}{^{2}}) z.$
Figure \ref{SSTforAssump2Eq} represents the solution of (\ref{Assump2Eq}) corresponding to the initial data $ T(0, 0, 0, 0)= 0.5 $. It is seen in Figure \ref{SSTforAssump2Eq} that the solution of Equation (\ref{Assump2Eq}) has an irregular oscillating behavior, whereas in the absence of the term $ 5 \sin(xt) $ in the function $ f $, the solution approaches the steady state. Even though the behavior of this numerical solution depends on the step size of the numerical scheme used, this situation leads us to consider that the forcing term $ f $ has a dominant role in the behavior of SST. \\
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{FMainSinxt}
\caption{The solution of Equation (\ref{Assump2Eq}) with the initial condition $T(0, 0, 0, 0)= 0.5$. The figure shows that the forcing term $ f $ has a significant role in the dynamics of (\ref{AdvectionEq}).}
\label{SSTforAssump2Eq}
\end{figure}
To investigate the existence of an unpredictable solution in the dynamics of Equation (\ref{AdvectionEq}) theoretically, let us apply the method of characteristics. If we parametrize the characteristics by the variable $ t $ and suppose that the initial condition is given by $ T(t_0,x,y,z)=\Phi (x,y,z) $, where $ t_0 $ is the initial time, then we obtain the system
\begin{equation} \label{ParamSysEq}
\begin{split}
& \frac{dx}{dt} = u(t,x,y,z,T) , \\
& \frac{dy}{dt} = v(t,x,y,z,T) , \\
& \frac{dz}{dt} = w(t,x,y,z,T) , \\
& \frac{dT}{dt} = f(t,x,y,z,T) ,
\end{split}
\end{equation}
with the initial conditions
\begin{equation*} \label{InitialCondParamSysEq}
x(t_0)=x_0, \quad y(t_0)=y_0, \quad z(t_0)=z_0, \quad T(t_0,x_0,y_0,z_0)=\Phi (x_0,y_0,z_0).
\end{equation*}
In system (\ref{ParamSysEq}), we assume that $ u, v, w $, and $ f $ are functions of $ x,y,z,t $, and $ T $, and they have the forms
\begin{equation} \label{Velo&TfuncEq}
\begin{split}
& u=a_1 \, x+a_2 \, y+a_3 \, z+a_4 \, T+U(x, y, z, T), \\
& v=b_1 \, x+b_2 \, y+b_3 \, z+b_4 \, T+V(x, y, z, T), \\
& w=c_1 \, x+c_2 \, y+c_3 \, z+c_4 \, T+W(x, y, z, T), \\
& f=d_1 \, x+d_2 \, y+d_3 \, z+d_4 \, T+F(x, y, z, T),
\end{split}
\end{equation}
where $ a_i $, $ b_i $, $ c_i $, $ d_i $, $i=1,2,3,4$, are real constants and the functions $ U, V, W, F $ are continuous in all their arguments. System (\ref{ParamSysEq}) can be expressed in the form
\begin{equation} \label{SysODEEq}
X'(t) = A X(t) + Q(t),
\end{equation}
in which
\begin{equation} \label{Eq1}
X(t)=
\begin{bmatrix}
x \\
y \\
z \\
T
\end{bmatrix}, \qquad
A=
\begin{bmatrix}
a_1 & a_2 & a_3 & a_4 \\
b_1 & b_2 & b_3 & b_4 \\
c_1 & c_2 & c_3 & c_4 \\
d_1 & d_2 & d_3 & d_4
\end{bmatrix}, \qquad
Q=
\begin{bmatrix}
U \\
V \\
W \\
F
\end{bmatrix}.
\end{equation}
The following theorem is needed to verify the existence of Poincar\'{e} chaos in the dynamics of Equation (\ref{AdvectionEq}).
\begin{theorem} \label{Thm1} ([\citen{AkhmetPoincare}])
Consider the system of ordinary differential equations
\begin{equation} \label{SysODEThmEq}
X'(t)= A \, X(t) + G(X(t))+H(t),
\end{equation}
where the $ n \times n $ constant matrix $ A $ has eigenvalues all with negative real parts, the function $ G: \mathbb{R}^n \rightarrow \mathbb{R}^n $ is Lipschitzian with a sufficiently small Lipschitz constant, and $ H: \mathbb{R} \rightarrow \mathbb{R}^n $ is a uniformly continuous and bounded function. If the function $ H(t) $ is unpredictable, then system (\ref{SysODEThmEq}) possesses a unique uniformly exponentially stable unpredictable solution, which is uniformly continuous and bounded on the real axis. \\
\end{theorem}
In the remaining parts of the present section, we will discuss the unpredictability when SST is chaotified by external irregularity. For that purpose let us consider the logistic map
\begin{equation} \label{logistic}
\eta_{j+1}= 3.91 \, \eta_j \, (1- \eta_j), \ j\in \mathbb Z.
\end{equation}
According to Theorem $4.1$ [\citen{AkhmetPoincare}], the map (\ref{logistic}) is Poincar\'{e} chaotic such that it possesses an unpredictable trajectory.
Next, we define a function $ \phi(t) $ by
\begin{equation} \label{UnpFunInEq}
\phi(t)=\int_{-\infty}^t e^{-2(t-s)} \gamma^*(s)\,ds,
\end{equation}
where
\begin{equation}\label{relayfunc}
\gamma^*(t) =
\begin{cases}
1.5, & \zeta^*_{2j} < t \leq \zeta^*_{2j+1}, \quad j \in \mathbb{Z}, \\
0.3, & \zeta^*_{2j-1} < t \leq \zeta^*_{2j}, \quad j \in \mathbb{Z}, \\
\end{cases}
\end{equation}
is a relay function. In (\ref{relayfunc}), the sequence $ \{\zeta^*_j \} $ of switching moments is generated through the equation $ \zeta^*_{j}= j+\eta^*_{j}, \; j \in \mathbb{Z} $, where $ \{\eta^*_j \} $ is an unpredictable trajectory of the logistic map (\ref{logistic}).
One can confirm that $ \phi(t) $ is bounded such that $\displaystyle \sup_{t\in \mathbb R} |\phi(t)|\leq 3/4$. It was shown in paper [\citen{AkhmetPoincare}] that the function $ \phi(t) $ is the unique uniformly exponentially stable unpredictable solution of the differential equation
\begin{equation} \label{UnpFunODEq}
\vartheta' (t)= -2\vartheta(t)+\gamma^*(t).
\end{equation}
It is not an easy task to visualize the unpredictable function $ \phi(t) $. Therefore, in order to illustrate the chaotic dynamics, we take into account the differential equation
\begin{equation} \label{UnpFunODEq2}
\vartheta' (t)= -2\vartheta(t)+\gamma(t),
\end{equation}
where
\begin{equation}\label{relayfunc2}
\gamma(t) =
\begin{cases}
1.5, & \zeta_{2j} < t \leq \zeta_{2j+1}, \quad j \in \mathbb{Z}, \\
0.3, & \zeta_{2j-1} < t \leq \zeta_{2j}, \quad j \in \mathbb{Z}, \\
\end{cases}
\end{equation}
and the sequence $ \{\zeta_j \} $ satisfies the equation $ \zeta_{j}= j+\eta_{j}, \; j \in \mathbb{Z} $, in which $ \{\eta_j \} $ is a solution of (\ref{logistic}) with $ \eta_0=0.4 $. The coefficient $ 3.91 $ used in the logistic map (\ref{logistic}) and the initial data $\eta_0=0.4$ were considered for shadowing analysis in the paper [\citen{Hammel}].
We depict in Figure \ref{UnpFunc} the solution of Equation (\ref{UnpFunODEq2}) with $ \vartheta(0)=0.3 $. It is seen in Figure \ref{UnpFunc} that the behavior of the solution is irregular, and this support that the function $ \phi(t) $ is unpredictable.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{UnprFunc}
\caption{The solution of Equation (\ref{UnpFunODEq2}) with $ \vartheta(0)=0.3 $. The figure support that the function $ \phi(t) $ is unpredictable.}
\label{UnpFunc}
\end{figure}
\subsection{Unpredictability Due to the Forcing Source Term}
Let us perturb Equation (\ref{AdvectionEq}) with the unpredictable function $ \phi(t) $ defined by (\ref{UnpFunInEq}) and set up the equation
\begin{equation} \label{PrturMainEq}
\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z}= f(t,x,y,z,T)+\psi(\phi(t)),
\end{equation}
where $ u, v, w $, and $ f $ are in the form of (\ref{Velo&TfuncEq}) and $ \psi: [-3/4,3/4] \to \mathbb R $ is a continuous function.
Using the method of characteristics, one can reduce Equation (\ref{PrturMainEq}) to system (\ref{ParamSysEq}) that can be expressed in the form of (\ref{SysODEThmEq}) with
\begin{equation*} \label{Eq2}
G(X(t))=
\begin{bmatrix}
U \\
V \\
W \\
F
\end{bmatrix}, \qquad
H(t)=
\begin{bmatrix}
0 \\
0 \\
0 \\
\psi(\phi(t))
\end{bmatrix}.
\end{equation*}
According to the result of Theorem 5.2 [\citen{AkhmetPoincare}], if there exist positive constants $ L_1 $ and $ L_2 $ such that
\begin{eqnarray} \label{bilipschitz}
L_1\left| s_1-s_2\right| \leq \left|\psi(s_1)-\psi(s_2) \right| \leq L_2\left| s_1-s_2\right|
\end{eqnarray}
for all $ s_1,s_2 \in [-3/4,3/4]$, then the function $H(t)$ is also unpredictable.
Now, in Equation (\ref{PrturMainEq}), let us take $u= -0.03 x+0.1 \sin(\frac{x}{80})+0.4,$ $v= -0.01 y-0.05 \sin(y),$ $w= -0.02 z+(0.05 \cos(y) - 0.00125 \cos(\frac{x}{80})) z$, $\psi(s)=6s$, and $f(t,x,y,z,T)= -1.5 T+0.1 w_2 T$. Since the conditions of Theorem \ref{Thm1} are valid and inequality (\ref{bilipschitz}) holds for these choices of $\psi$, $f$, $u$, $v$, and $w$, Equation (\ref{PrturMainEq}) exhibits Poincar\'{e} chaos.
In order to simulate the chaotic behavior, we consider the equation
\begin{equation} \label{PrturMainEq2}
\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z}= f(t,x,y,z,T)+\psi(\vartheta(t)),
\end{equation}
where $\vartheta(t)$ is the function depicted in Figure \ref{UnpFunc}, and $ u, v, w, f, \psi $ are the same as above. Figure \ref{ChoBehSST} shows the solution $ T(t, x, y, z) $ of (\ref{PrturMainEq2}) corresponding to the initial condition $ T(0, 0, 0, 0)= 0.5 $. It is seen in Figure \ref{ChoBehSST} that the behavior of the solution is chaotic, and this supports the result of Theorem \ref{Thm1} such that Equation (\ref{PrturMainEq}) admits an unpredictable solution.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{FMainPhi}
\caption{The solution of Equation (\ref{PrturMainEq2}) with the initial condition $T(0, 0, 0, 0)= 0.5$. The figure reveals the presence of an unpredictable solution in the dynamics of (\ref{PrturMainEq}).}
\label{ChoBehSST}
\end{figure}
Next, we will visualize the chaotic dynamics in the integral surface of SST. For that purpose, we omit the term of the meridional advection $ v \frac{\partial T}{\partial y} $ in (\ref{PrturMainEq}), which has less effect on SST compared with the zonal and vertical advections \cite{Bonjean}, and set up the equation
\begin{equation} \label{2DimApproxAdv-PhiEq}
\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + w \frac{\partial T}{\partial z}= -1.5 \, T+w \, T+6 \, \vartheta(t),
\end{equation}
where $u= 1.2+0.1 \sin(\frac{x}{80})+0.05 \sin(3t)$ and $w= 0.1 - 0.00125 \cos(\frac{x}{80}) z$. In (\ref{2DimApproxAdv-PhiEq}), $\vartheta(t)$ is again the function shown in Figure \ref{UnpFunc}.
We apply a finite difference scheme to solve Equation (\ref{2DimApproxAdv-PhiEq}) directly. In such a scheme, we need to specify boundary conditions along with an initial condition. Using the initial condition $ T(0, x, z)= 5 $ and the boundary conditions $ T(t, 0, z)=T(t, x, 0)= 0.5$, we represent in Figure \ref{SSTSerfacePhi} the integral surface of (\ref{2DimApproxAdv-PhiEq}) with respect to $t,$ $x$, and the fixed value $z=0$ for $5\leq x\leq 20$ and $0\leq t \leq 100$. Figure \ref{SSTSerfacePhi} supports the result of Theorem \ref{Thm1} one more time such that Poincar\'{e} chaos is present in the dynamics.
\begin{figure}[H]
\centering
\includegraphics[width=0.6\linewidth]{SSTSurfPhi}
\caption{The integral surface of (\ref{2DimApproxAdv-PhiEq}). The chaotic behavior in the SST is observable in the figure.}
\label{SSTSerfacePhi}
\end{figure}
\subsection{Unpredictability Due to the Current Velocity}
This subsection is devoted to the investigation of SST when the current velocity behaves chaotically. Here, we will make use of the unpredictable function $\phi(t)$ defined by (\ref{UnpFunInEq}) to apply perturbations to the zonal and vertical components of current velocity in Equation (\ref{AdvectionEq}).
We begin with considering the equation
\begin{equation} \label{E1ZonalPerturbEq}
\frac{\partial T}{\partial t} + [u +\psi(\phi(t))]\frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z}= f(t, x, y, z, T),
\end{equation}
where, in a similar way to (\ref{PrturMainEq}), $u, v, w$, and $f$ are in the form of (\ref{Velo&TfuncEq}), and $ \psi: [-3/4,3/4] \to \mathbb R $ is a continuous function.
One can confirm that Theorem \ref{Thm1} can be used to verify the existence of Poincar\'{e} chaos in the dynamics of (\ref{E1ZonalPerturbEq}) since it can be reduced by means of the method of characteristics to a system of the form (\ref{SysODEThmEq}) with
\begin{equation*}
H(t)=
\begin{bmatrix}
\psi(\phi(t)) \\
0 \\
0 \\
0
\end{bmatrix},
\end{equation*}
which is an unpredictable function provided that $\psi$ satisfies the condition (\ref{bilipschitz}).
In order to demonstrate the chaotic dynamics of (\ref{E1ZonalPerturbEq}), we take $u= -0.003 \,x+0.2 \sin(\frac{{x}}{3})+0.4, $ $v= -0.001\,y$, $w= -0.002\,z- \frac{{0.2}}{3} \cos(\frac{{x}}{3})\,z$, $\psi(s)=3s$, $f=-1.5 \, T -3 \sin(3x)+0.2$, and consider the equation
\begin{equation} \label{E1ZonalPerturbEq2}
\frac{\partial T}{\partial t} + [u +\psi(\vartheta(t))]\frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + w \frac{\partial T}{\partial z}= f(t, x, y, z, T),
\end{equation}
where $\vartheta(t)$ is the function shown in Figure \ref{UnpFunc}.
The time series of the solution of (\ref{E1ZonalPerturbEq2}) with $ T(0, 0, 0, 0)= 0.5 $ is depicted in Figure \ref{SSTWithUnpZonal-AlongCharac}. One can observe in the figure that the time series is chaotic, and this confirms the result of Theorem \ref{Thm1} such that Equation (\ref{E1ZonalPerturbEq}) possesses an unpredictable solution. More precisely, the perturbation of the zonal velocity component in Equation (\ref{AdvectionEq}) by the unpredictable function $ \psi(\phi(t)) $ affects the dynamics in such a way that the perturbed equation (\ref{E1ZonalPerturbEq}) is Poincar\'{e} chaotic.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{ZonalPhi}
\caption{The solution of (\ref{E1ZonalPerturbEq}) with $ T(0, 0, 0, 0)= 0.5 $. The chaotic behavior of the solution is apparent in the figure.}
\label{SSTWithUnpZonal-AlongCharac}
\end{figure}
Next, we will examine the case when the vertical velocity component in Equation (\ref{AdvectionEq}) is perturbed by the unpredictable function $\phi(t)$.
For this aim we set up the equation
\begin{equation} \label{E1meridionalPerturbEq3}
\frac{\partial T}{\partial t} + u\frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + [w +\psi(\phi(t))] \frac{\partial T}{\partial z}= f(t, x, y, z, T),
\end{equation}
where the function $\psi: [-3/4,3/4] \to \mathbb R$ is continuous. If we take $ u= -0.001 \,x+0.2 \sin(\frac{{x}}{3})+0.4,$ $v=-0.001\,y,$ $w= -0.03 z- \frac{{0.2}}{3} \cos(\frac{{x}}{3}) z+3 \vartheta(t) $, $\psi(s)=3s$, and $ f= -1.7 \, T + 0.5 \, z+1.6 $, then Equation (\ref{E1meridionalPerturbEq3}) admits an unpredictable solution in accordance with Theorem \ref{Thm1}.
We represent in Figure \ref{SSTWithUnpVertical-AlongCharac} the solution of the equation
\begin{equation} \label{E1meridionalPerturbEq4}
\frac{\partial T}{\partial t} + u\frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} + [w +\psi(\vartheta(t))] \frac{\partial T}{\partial z}= f(t, x, y, z, T),
\end{equation}
corresponding to the initial data $ T(0, 0, 0, 0)= 0.5 $. Here, we use the same $u,$ $v$, $w$, $\psi$, and $f$ as in (\ref{E1meridionalPerturbEq3}), and $\vartheta(t)$ is again the function whose time series is depicted in Figure \ref{UnpFunc}. The irregular fluctuations seen in the figure uphold the result of Theorem \ref{Thm1}.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{VerticalPhi}
\caption{Chaotic behavior of SST due to the perturbation of the vertical component of current velocity. The figure shows the solution of (\ref{E1meridionalPerturbEq4}) with $ T(0, 0, 0, 0)= 0.5 $.}
\label{SSTWithUnpVertical-AlongCharac}
\end{figure}
We end up this subsection by illustrating the influence of the chaotic current velocity on the integral surface of SST. Figure \ref{CurrentVeloPerturb-SurfFg} (a) shows the integral surface of (\ref{E1ZonalPerturbEq}) with
$u= 1.5+0.5 \sin x, $ $v=0$, $w= 1-0.5 \cos x$, $\psi(s)=2s$, and $f=-1.2 \, T -3 \sin(3x)$ at $z=0$. The initial condition $ T(0, x, y, z)= \sin (xz) +1$ and the boundary conditions $ T(t, 0, y, z)=T(t, x, y, 0)= 0.5 $ are utilized in the simulation. One can see in Figure \ref{CurrentVeloPerturb-SurfFg} (a) that the SST has chaotic behavior in keeping with the result of Theorem \ref{Thm1}. On the other hand, using the same initial and boundary conditions, we represent in Figure \ref{CurrentVeloPerturb-SurfFg} (b) the integral surface of (\ref{E1meridionalPerturbEq3}) with $u= 1, $ $v=0$, $w= 1$, $\psi(s)=2s$, and $f=-1.2 \, T +3 \sin(3z)$ at $z=1.5$. Figure \ref{CurrentVeloPerturb-SurfFg} (b) also manifests that the applied perturbation on the vertical component of current velocity make the Equation (\ref{E1meridionalPerturbEq3}) behave chaotically even if it is initially non-chaotic in the absence of the perturbations.
\begin{figure}[H]
\subfigure[The integral surface of (\ref{E1ZonalPerturbEq}) at $ z= 0 $]{\includegraphics[width = 3.27in]{ZonalPhiSurf}\label{SSTWithUnpZonal-Surf}}
\subfigure[The integral surface of (\ref{E1meridionalPerturbEq3}) at $ z= 1.5 $]{\includegraphics[width = 3.27in]{VerticalPhiSurf}\label{SSTWithUnpVertical-Surf}}
\caption{Chaotic behavior of SST due to the current velocity with initial condition $ T(0, x, y, z)= \sin (xz) +1$, and boundary conditions $T(t, 0, y, z)=T(t, x, y, 0)= 0.5$. Both pictures in (a) and (b) reveal that chaotic behavior in the current velocity leads to the presence of chaos in SST.}
\label{CurrentVeloPerturb-SurfFg}
\end{figure}
\section{Extension of Chaos through the Globe Ocean } \label{HorizontalExtension}
Chaotic behavior may transmit from one model to another \cite{AkhmetReplication}. This transmission interprets, for instance, why the unpredictability in one stock market or in the weather of one area is affected by another. Chaos in SST may be gained from another endogenous chaotic system like air temperature or wind speed. We can deal with the global ocean as a finite union of subregions. Each of these subregions may be controlled by different models depending on the position and circumstances. An assumption of the existence of chaotic and non-chaotic subregions for SST behavior is very probable. However, it seems quite unreasonable to imagine a predictable SST for one region whereas its neighbor region is characterized by an unpredictable SST. The mutual effect in SST between neighbor regions can be seen by coupling their controlling models.
\subsection{Coupling of Advection Equations}
In this part of the paper we deal with the extension of chaos in coupled advection equations. For that purpose, we consider a Poincar\'{e} chaotic advection equation of the form (\ref{PrturMainEq}) as the drive, and we take into account the equation
\begin{equation} \label{response_advec_eq}
\frac{\partial \tilde{T}}{\partial t} + \tilde{u} \frac{\partial \tilde{T}}{\partial x} + \tilde{v} \frac{\partial \tilde{T}}{\partial y} + \tilde{w} \frac{\partial \tilde{T}}{\partial z}= \tilde{f}(t,x,y,z,\tilde{T}) + g(T)
\end{equation}
as the response, in which $g$ is a continuous function and $T$ is a solution of the drive equation (\ref{PrturMainEq}).
We assume that the response does not possess chaos in the absence of the perturbation, i.e., we suppose that the advection equation
\begin{equation} \label{response_advec_eq2}
\frac{\partial \tilde{T}}{\partial t} + \tilde{u} \frac{\partial \tilde{T}}{\partial x} + \tilde{v} \frac{\partial \tilde{T}}{\partial y} + \tilde{w} \frac{\partial \tilde{T}}{\partial z}= \tilde{f}(t,x,y,z,\tilde{T})
\end{equation}
is non-chaotic.
To demonstrate the extension of chaos numerically, let us consider the response equation (\ref{response_advec_eq}) with $u=1.2$, $v=0$, $w=0.3$, $f=-1.5 \tilde{T} + 0.2$, and $g(T)=T$. Using the solution $T$ of Equation (\ref{2DimApproxAdv-PhiEq}) satisfying $T(0, 0, 0, 0)= 0.5$ as the perturbation in Equation (\ref{response_advec_eq}), we depict in Figure \ref{ChaosExten} the solution $\tilde{T}$ of (\ref{response_advec_eq}) corresponding to the initial data $\tilde{T}(0, 0, 0, 0)= 0.5$. Figure \ref{ChaosExten} reveals the extension of chaos in the coupled system (\ref{PrturMainEq})-(\ref{response_advec_eq}).
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{ChaosExten}
\caption{The solution of the response equation (\ref{response_advec_eq}) with initial condition $T(0, 0, 0, 0)= 0.5$. The figure manifests the extension of chaos in the coupled system (\ref{PrturMainEq})-(\ref{response_advec_eq}).}
\label{ChaosExten}
\end{figure}
\subsection{Coupling of the Advection Equation with Vallis Model}
The Lorenz-like form of the Vallis model is given by \cite{Vallis}
\begin{equation} \label{LorenzFormVallis}
\begin{split}
& \frac{du}{dt} =B\,T_d-C\,u,\\
& \frac{dT_d}{dt} = u\,T_s-T_d , \\
& \frac{dT_s}{dt} = -u\,T_d-T_s+1,
\end{split}
\end{equation}
where $ u $ represents the zonal velocity, $ T_d=(T_e-T_w)/2 $, $ T_s=(T_e+T_w)/2 $, $ T_e $ and $ T_w $ are the SST in the eastern and western ocean respectively, and $B$ and $C$ are constants. It was shown in paper [\citen{Vallis}] that system (\ref{LorenzFormVallis}) with the parameters $B = 102 $ and $C = 3$ is chaotic. The existence of chaos in the dynamics of Vallis systems was also investigated in the studies [\citen{Garay,Borghezan}].
Next, we take into account the equations
\begin{equation} \label{CoupledAdvecLorenz-Td.Eq1}
\frac{\partial T_1}{\partial t} + 1.2 \frac{\partial T_1}{\partial x} + 0.3 \frac{\partial T_1}{\partial z}= -1.2\, T_1 - 1 + 2 \sin x,
\end{equation}
\begin{equation} \label{CoupledAdvecLorenz-u.Eq1}
\frac{\partial T_2}{\partial t} + 1.2 \frac{\partial T_2}{\partial x} + 0.3 \frac{\partial T_2}{\partial z}= -2 \, T_2 + 4 \sin x,
\end{equation}
\begin{equation} \label{CoupledAdvecLorenz-Td-u.Eq1}
\frac{\partial T_3}{\partial t} + 0.6 \frac{\partial T_3}{\partial x} + 0.5 \frac{\partial T_3}{\partial z}= -2\, T_3 -1 + 3 \sin x,
\end{equation}
and
\begin{equation} \label{2CoupledAdvecLorenz.Eq1}
\frac{\partial T_4}{\partial t} + 1.2 \frac{\partial T_4}{\partial x} + 0.3 \frac{\partial T_4}{\partial z}= -1.5 \, T_4.
\end{equation}
One can verify that the equations (\ref{CoupledAdvecLorenz-Td.Eq1}), (\ref{CoupledAdvecLorenz-u.Eq1}), (\ref{CoupledAdvecLorenz-Td-u.Eq1}), and (\ref{2CoupledAdvecLorenz.Eq1}) are all non-chaotic such that they admit asymptotically stable regular solutions. By applying perturbations to these equations, we set up the following ones:
\begin{equation} \label{CoupledAdvecLorenz-Td.Eq}
\frac{\partial T_1}{\partial t} + 1.2 \frac{\partial T_1}{\partial x} + 0.3 \frac{\partial T_1}{\partial z}= -1.2\, T_1 - 1 + 2 \sin x +4.6 \, T_s,
\end{equation}
\begin{equation} \label{CoupledAdvecLorenz-u.Eq}
\frac{\partial T_2}{\partial t} + (1.2+0.8 \, u) \frac{\partial T_2}{\partial x} + 0.3 \frac{\partial T_2}{\partial z}= -2 \, T_2 + 4 \sin x,
\end{equation}
\begin{equation} \label{CoupledAdvecLorenz-Td-u.Eq}
\frac{\partial T_3}{\partial t} + (0.6+ u) \frac{\partial T_3}{\partial x} + 0.5 \frac{\partial T_3}{\partial z}= -2\, T_3 -1 + 3 \sin x +4 \, T_s,
\end{equation}
\begin{equation} \label{2CoupledAdvecLorenz.Eq}
\frac{\partial T_4}{\partial t} + 1.2 \frac{\partial T_4}{\partial x} + 0.3 \frac{\partial T_4}{\partial z}= -1.5 \, T_4 + 2.7 \, T_2,
\end{equation}
where $(u, T_d, T_s)$ is the solution of the chaotic Vallis model (\ref{LorenzFormVallis}) with $B = 102 $, $C = 3$ and the initial conditions $u(0)=2,$ $T_d(0)=0.2$, and $T_s(0)=0.4$.
In Equation (\ref{CoupledAdvecLorenz-Td.Eq}) the forcing term is perturbed by the SST average, $T_s$, whereas in Equation (\ref{CoupledAdvecLorenz-u.Eq}) the zonal velocity of Vallis model, $u$, is used as perturbation. On the other hand, in Equation (\ref{CoupledAdvecLorenz-Td-u.Eq}) both the forcing term and the zonal velocity components are perturbed with the solution of (\ref{LorenzFormVallis}). Moreover, the solution $T_2$ of (\ref{CoupledAdvecLorenz-u.Eq}) is used as a perturbation in the forcing term of Equation (\ref{2CoupledAdvecLorenz.Eq}).
Figure \ref{CoupledAdvecLorenzAlongCharacFg} (a) and (b) respectively show the solutions $T_2$, $T_3$ of Equations (\ref{CoupledAdvecLorenz-u.Eq}) and (\ref{CoupledAdvecLorenz-Td-u.Eq}), respectively. The initial data $ T_2(0, 0, 0, 0)= 0.5 $ and $ T_3(0, 0, 0, 0)= 0.5 $ are used in the simulation. Figure \ref{CoupledAdvecLorenzAlongCharacFg} reveals that the chaos of the model (\ref{LorenzFormVallis}) is extended by Equations (\ref{CoupledAdvecLorenz-u.Eq}) and (\ref{CoupledAdvecLorenz-Td-u.Eq}).
\begin{figure}[H]
\subfigure[]{\includegraphics[width = 6.4in]{SSTExtVallAdv-u}\label{SSTWithVallispZonal-AlongCharac}}
\subfigure[]{\includegraphics[width = 6.4in]{SSTExtVallAdv-Ts-u}\label{SSTWithVallispZonal-T-AlongCharac}}
\caption{The extension of the chaotic behavior by Equations (\ref{CoupledAdvecLorenz-u.Eq}) and (\ref{CoupledAdvecLorenz-Td-u.Eq}). (a) The time series of the solution of Equation (\ref{CoupledAdvecLorenz-u.Eq}), (b) The time series of the solution of Equation (\ref{CoupledAdvecLorenz-Td-u.Eq}). The initial data $T_2(0, 0, 0, 0)= 0.5$ and $T_3(0, 0, 0, 0)= 0.5$ are used.}
\label{CoupledAdvecLorenzAlongCharacFg}
\end{figure}
On the other hand, we depict in Figure \ref{CoupledAdvecLorenzSurfFg} (a) and (b) the $3$ dimensional integral surfaces corresponding to Equations (\ref{CoupledAdvecLorenz-Td.Eq}) and (\ref{2CoupledAdvecLorenz.Eq}), respectively. Here, we make use of the boundary conditions $T_1(0, x, z)=T_1(t, 0, z)=T_1(t, x, 0)= 0.5$ and $T_4(0, x, z)=T_4(t, 0, z)=T_4(t, x, 0)= 0.5$. The figure confirms one more time that the chaos of system (\ref{LorenzFormVallis}) is extended.
\begin{figure}[H]
\subfigure[]{\includegraphics[width = 3.27in]{SSTVallisSurf-Ts}\label{SSTWithVallisTs-Surf}}
\subfigure[]{\includegraphics[width = 3.27in]{SSTVallisSurf-zonal}\label{SSTWithVallispZonal-Surf}}
\caption{Extension of chaos by Equations (\ref{CoupledAdvecLorenz-Td.Eq}) and (\ref{2CoupledAdvecLorenz.Eq}). (a) The integral surface of Equation (\ref{CoupledAdvecLorenz-Td.Eq}), (b) The integral surface of Equation (\ref{2CoupledAdvecLorenz.Eq}).}
\label{CoupledAdvecLorenzSurfFg}
\end{figure}
\subsection{Coupling of Vallis Models}
Our purpose in this subsection is to demonstrate numerically our suggestion that chaos can be extended between the regions of some global climate variabilities. We assume that there are intermediate subregions located between these main regions and chaos can transmit from one region to another in a sequential way.
We also suggest that the IOD can be described by a Vallis model in the form of (\ref{LorenzFormVallis}) with parameters appropriate to the Indian Ocean. Evaluation of these parameters is rather difficult. However, for simplicity we can choose these values such that system (\ref{LorenzFormVallis}) does not exhibit chaotic behavior. Similar arguments can also be supposed for the AMO and SAM.
To demonstrate the extension of chaos, let us consider the perturbed Vallis system
\begin{equation} \label{LorenzVallisIOD}
\begin{split}
& \frac{d \tilde u}{dt} = \tilde B\, \tilde T_d- \tilde C\, \tilde u +1.5 \, u,\\
& \frac{d \tilde T_d}{dt} = \tilde u\, \tilde T_s- \tilde T_d +0.3 \, T_d, \\
& \frac{d \tilde T_s}{dt} = - \tilde u\, \tilde T_d- \tilde T_s+1 +0.2 \, T_s,
\end{split}
\end{equation}
where $(u, T_d, T_s)$ is the solution of the chaotic Vallis system (\ref{LorenzFormVallis}) with $B = 102$ and $C = 3$ corresponding to the initial conditions $u(0)=2,$ $T_d(0)=0.2$ and $T_s(0)=0.4$.
We use the parameters $\tilde B = 20$ and $\tilde C = 7$ in (\ref{LorenzVallisIOD}) and assume that the unperturbed Vallis model
\begin{equation} \label{VallisIOD2}
\begin{split}
& \frac{d \tilde u}{dt} = \tilde B\, \tilde T_d- \tilde C\, \tilde u, \\
& \frac{d \tilde T_d}{dt} = \tilde u\, \tilde T_s- \tilde T_d, \\
& \frac{d \tilde T_s}{dt} = - \tilde u\, \tilde T_d- \tilde T_s+1
\end{split}
\end{equation}
represents the IOD with these parameter values. Utilizing the initial conditions $\tilde u(0)=2$, $\tilde T_d(0)=0.2$, and $ \tilde T_s(0)=0.4$, we represent in Figure \ref{CoupledTwoVallis} the time series of $\tilde u,$ $\tilde T_d$, and $\tilde T_s$ coordinates of the solution of system (\ref{LorenzVallisIOD}). One can see in Figure \ref{CoupledTwoVallis} that system (\ref{LorenzVallisIOD}) possesses chaotic behavior.
\begin{figure}[H]
\subfigure[]{\includegraphics[width = 6.4in]{CoupTwoVallis-u}\label{ENSO-IOD-u}} \\
\subfigure[]{\includegraphics[width = 6.4in]{CoupTwoVallis-Td}\label{ENSO-IOD-Td}}
\subfigure[]{\includegraphics[width = 6.4in]{CoupTwoVallis-Ts}\label{ENSO-IOD-Ts}}
\caption{The solution of system (\ref{LorenzVallisIOD}) which reveals chaos extension between a pair of Vallis systems}
\label{CoupledTwoVallis}
\end{figure}
\section{Ocean-Atmosphere Unpredictability Interaction} \label{VerticalExtension}
In this section, we discuss the possibility of the ``vertical'' extension of unpredictability, i.e. the transmission of chaotic dynamics from ocean to atmosphere and vice versa. To demonstrate this interaction we apply the Lorenz system (\ref{LorenzSystem}) for the atmosphere and the Vallis model (\ref{LorenzFormVallis}) for the ocean. Vallis model is constructed for the domain length of $ 7500 \, \text{km} $, however, depending on the method of construction, the model can be applied for more localized region to be compatible with the Lorenz model.
Heat and momentum exchanges are two important ways of interaction between ocean and atmosphere. The heat exchange is mainly controlled by the air-sea temperature gradient, and, on the other hand, the momentum transfer is determined by the sea-surface stress caused by wind and currents \cite{Gallego}. These characteristics are represented in both Lorenz system (\ref{LorenzSystem}) and Vallis model (\ref{LorenzFormVallis}). Two coordinates in the Lorenz system represent temperature, whereas the third one is related to velocity, and the same could be said for the Vallis system. Therefore, the interaction between ocean and atmosphere can be modeled by coupling the Lorenz and Vallis models.
Let us consider the coupled Lorenz-Vallis systems
\begin{equation} \label{GenLorenzVallisCouple}
\begin{split}
& \frac{dx}{dt} =\sigma(y-x) + f_1(u,T_d,T_s),\\
& \frac{dy}{dt} = x(r-z)-y+f_2(u,T_d,T_s), \\
& \frac{dz}{dt} = xy - b \,z+f_3(u,T_d,T_s),
\end{split}
\end{equation}
and
\begin{equation} \label{GenVallisLorenzCouple}
\begin{split}
& \frac{du}{dt} =B\,T_d-C\,u+g_1(x,y,z),\\
& \frac{dT_d}{dt} = u\,T_s-T_d+g_2(x,y,z), \\
& \frac{dT_s}{dt} = -u\,T_d-T_s+1+g_3(x,y,z),
\end{split}
\end{equation}
where $f_i$, $g_i$, $i=1,2,3$, are continuous functions. The coupled model (\ref{GenLorenzVallisCouple})--(\ref{GenVallisLorenzCouple}) is in a sufficiently general form of interaction between the atmosphere and the ocean, where the functions $ f_i, g_i, \, i=1, 2, 3 $ are given in most general form.
To demonstrate the transmission of chaos between the atmosphere and ocean, we consider specific forms of the coupled model (\ref{GenLorenzVallisCouple})--(\ref{GenVallisLorenzCouple}). This technique relies on the theoretical investigations of replication of chaos introduced in [\citen{AkhmetReplication}].
In the case of upward transmission of chaos from the ocean to the atmosphere, we consider (\ref{GenLorenzVallisCouple}) with specific choices of the perturbation functions $ f_1, f_2 $ and $ f_3 $ to set up the following system,
\begin{equation} \label{LorenzVallisCouple}
\begin{split}
& \frac{dx}{dt} =\sigma(y-x) + 3 \sin u,\\
& \frac{dy}{dt} = x(r-z)-y+6\,T_d, \\
& \frac{dz}{dt} = xy - b \,z+0.5\,T_s^2,
\end{split}
\end{equation}
where $(u, T_d, T_s)$ is the solution of the chaotic Vallis system (\ref{LorenzFormVallis}) with $ B = 102 $, $ C = 3 $ and the initial data $u(0)=2,$ $T_d(0)=0.2$, $T_s(0)=0.4$. We use the parameter values $ \sigma = 10 $, $ r = 0.35 $ and $ b = 8/3 $ in (\ref{LorenzVallisCouple}) such that the corresponding unperturbed Lorenz system (\ref{LorenzSystem}) does not possess chaos \cite{Sparrow}.
Figure \ref{CoupledLorenzVallis} shows the time series of the $x$, $y$, and $z$ components of the solution of system (\ref{LorenzVallisCouple}). The initial data $ x(0)=0,$ $y(0)=0.5$, $z(0)=0.3$ are used in the figure. The irregular behavior in each component reveals that the chaotic behavior of the atmosphere can be gained from the chaoticity of the hydrosphere characteristics.
\begin{figure}[H]
\subfigure[]{\includegraphics[width = 6.4in]{CoupLorVall-x}\label{LorVall-x}} \\
\subfigure[]{\includegraphics[width = 6.4in]{CoupLorVall-y}\label{LorVall-y}}
\subfigure[]{\includegraphics[width = 6.4in]{CoupLorVall-z}\label{LorVall-z}}
\caption{The chaotic solution of the perturbed Lorenz system (\ref{LorenzVallisCouple})}
\label{CoupledLorenzVallis}
\end{figure}
For the downward chaos transmission from the atmosphere to the ocean, we consider the perturbed Vallis system
\begin{equation} \label{VallisLorenzCouple}
\begin{split}
& \frac{du}{dt} =B\,T_d-C\,u+0.7 x,\\
& \frac{dT_d}{dt} = u\,T_s-T_d+0.3 \cos y +0.4 y , \\
& \frac{dT_s}{dt} = -u\,T_d-T_s+1+0.5 z,
\end{split}
\end{equation}
where $(x,y,z)$ is the solution of the Lorenz system (\ref{LorenzSystem}) with the parameters $ \sigma = 10 $, $ r = 28 $ and $ b = 8/3 $ and the initial data $x(0)=0$, $y(0)=1$, $ z(0)=0 $. System (\ref{LorenzSystem}) possesses a chaotic attractor with these choices of the parameter values \cite{Lorenz,Sparrow}.
Let us take $ B = 20 $ and $ C = 7 $ in system (\ref{VallisLorenzCouple}). One can verify in this case that the corresponding unperturbed system (\ref{LorenzFormVallis}) is non-chaotic such that it possesses an asymptotically stable equilibrium. Figure \ref{CoupledVallisLorenz} depicts the solution of (\ref{VallisLorenzCouple}) with $ u(0)=2, $ $ T_d(0)=0.2 $, and $ T_s(0)=0.4 $. It is seen in Figure \ref{CoupledVallisLorenz} that the chaotic behavior of the Lorenz system (\ref{LorenzSystem}) is transmitted to (\ref{VallisLorenzCouple}). In other words, system (\ref{VallisLorenzCouple}) admits chaos even if it is initially non-chaotic in the absence of the perturbation.
\begin{figure}[H]
\subfigure[]{\includegraphics[width = 6.4in]{CoupVallLor-u}\label{VallLor-u}} \\
\subfigure[]{\includegraphics[width = 6.4in]{CoupVallLor-Td}\label{VallLor-Td}}
\subfigure[]{\includegraphics[width = 6.4in]{CoupVallLor-Ts}\label{VallLor-Ts}}
\caption{Chaotic behavior of system (\ref{VallisLorenzCouple})}
\label{CoupledVallisLorenz}
\end{figure}
\section{Conclusion}
In this paper we discuss the possible unpredictable behavior of climate variables on a global scale. Some ENSO-like climate variabilities have a significant influence on global weather and climate. ENSO variability is suggested to be chaotic by many studies. The well-known Vallis ENSO chaotic model is one among several ENSO models that exhibit irregular behavior. The presence of chaos in ENSO can be indicated by the behavior of SST as well as ocean current velocity. We describe the dynamics of SST by the advection equation. The forcing term, based on ocean-atmosphere interaction, and the current velocity in this equation can be a source of unpredictability in SST. We prove the presence of chaos in SST dynamics by utilizing the concept of unpredictable function. The relationship and interaction between the climate variabilities, like the ones between ENSO and IOD, have attracted attention in recent literature. Constructing and understanding the dynamic models driving these phenomena are the main steps to investigate the mutual influences between these global events. The SST anomalies are closely linked to some climate variabilities teleconnections in different parts of the global ocean. We suggest that the hydrosphere characteristics can behave chaotically through the possibility of transmission of chaos between ocean neighbor subregions. We verified this transmission by different ``toy'' couples of advection equations and Vallis models. The simulations of these couples show that unpredictability can be transmitted from a local region controlled by a chaotic model into its neighbor which is described by a non-chaotic model.
We proposed to apply the same technique for the ``vertical'' unpredictability exchange between atmosphere and hydrosphere. In this case, the Lorenz system and the Vallis model are assigned for the atmosphere and ocean, respectively. Physically, this exchange may be done in the midst of interaction between ocean and atmosphere associated with, for example, heat exchange. By this procedure, the global unpredictability of oceanic oscillation can be viewed as accompaniment to weather unpredictability.
Our approach provides a basic framework for mathematical interpretation to the irregular behavior of some global climate characteristics. It gives a way to link the local unpredictability in a component of climate system to more global scope. Further investigation can done by including different models for more climate components. Another important and interesting problem is \textit{controlling weather}. Even though the weather is too complicated to modify, a vital step can be taken toward this goal by modify the ENSO oscillation through control of chaos in its models and study the ``extension of the control'' between ENSO-like models and weather models. Chaos control in Lorenz system is still not effectively developed in the literature, where the most proposed methods are mainly depend on forcing the system into a single stable periodic behavior \cite{Chen,Yau}, and this is not adequate for real life applications. It is known that the chaos control can be achieved by using small perturbation to some parameters or variables of the system. This idea may be practically applied by making a small local artificial effect in atmosphere or hydrosphere. If we consider the positive tenor of the Lorenz's famous question, ``Does the flap of a butterfly's wing in Brazil set off a tornado in Texas?'', we can say that the small artificial climate change may prevent the occurrence or at least decrease the intensity of some extreme weather events such as cyclones, hurricanes, droughts, and floods.
\newpage
\appendix
\setcounter{secnumdepth}{0}
\section{Appendix}
\label{Appendix}
In addition to ENSO and IOD, in this appendix we give a short review of the major atmospheric patterns, namely Pacific Decadal Oscillation (PDO), Atlantic Multidecadal Oscillation (AMO), Southern Annular Mode (SAM), Tropical Atlantic Variability (TAV), North Atlantic Oscillation (NAO), Arctic Oscillation/Northern Annular Mode (AO/NAM), Madden-Julian Oscillation (MJO), Pacific/North American pattern (PNA), Quasi-Biennial Oscillation (QBO) and Western Pacific pattern (WP). Figure \ref{Wmap} shows the places of occurrence of these patterns \cite{Rosenzweig,Lehr}, and Table \ref{T1} gives brief descriptions of them \cite{Rosenzweig,Vuille,Lehr}.
\vspace{-0.2cm}
\begin{figure}[H]
\centering
\includegraphics[width=0.8\linewidth]{WorldMap}
\vspace{-0.2cm}
\caption{The major global climate patterns}
\label{Wmap}
\end{figure}
\vspace{-0.3cm}
\begin{table}[H]
\begin{center}
\begin{tabular}{|M{1.5cm}|c|M{1.8cm}|M{2.2cm}|} \hline
Term & Descriptions & Main Index & Timescale \\ \hline
ENSO & \multicolumn{1}{m{10cm}|}{An irregularly periodical variation in sea surface temperatures over the tropical eastern Pacific Ocean} & SST & 3--7 years \\\hline
QBO & \multicolumn{1}{m{10cm}|}{An oscillation of the equatorial zonal wind in the tropical stratosphere} & SLP & 26--30 months \\\hline
PDO & \multicolumn{1}{m{10cm}|}{A low-frequency pattern similar to ENSO occurs primarily in the Northeast Pacific near North America} & SST & 20--30 years \\\hline
PNA & \multicolumn{1}{m{10cm}|}{An atmospheric pressure pattern driven by the relationship between the warm ocean water near Hawaii and the cool one near the Aleutian Islands of Alaska} & SLP & 7--8 days \\\hline
AO/NAM & \multicolumn{1}{m{10cm}|}{Defined by westerly winds changes driven by temperature contrasts between the tropics and northern polar areas} & SLP & 1--9 months \\\hline
NAO & \multicolumn{1}{m{10cm}|}{Large scale of pressure varies in opposite directions in the North Atlantic near Iceland in the north and the Azores in the south} & SLP & 9--10 days \\\hline
TAV & \multicolumn{1}{m{10cm}|}{Like ENSO, but it exhibits a north-south low frequency oscillation of the SST gradient across the equatorial Atlantic Ocean} & SST & 10--15 years \\\hline
AMO & \multicolumn{1}{m{10cm}|}{A mode of natural variability occurring in the North Atlantic Ocean and affects the SST on different modes on multidecadal timescales} & SST & 55--80 years \\\hline
SAM & \multicolumn{1}{m{10cm}|}{Defined by westerly winds changes driven by temperature contrasts between the tropics and southern polar areas} & SLP & 30--70 days \\\hline
IOD & \multicolumn{1}{m{10cm}|}{An irregular oscillation of sea-surface temperatures in equatorial areas of the Indian Ocean} & SST & 2--5 years \\\hline
WP & \multicolumn{1}{m{10cm}|}{A low-frequency variability characterized by north-south dipolar anomalies in pressure over the Far East and western North Pacific} & SLP & 7--8 days \\\hline
MJO & \multicolumn{1}{m{10cm}|}{An equatorial traveling pattern of anomalous rainfall Located in the tropical Pacific and Indian oceans} & SLP & 40--50 days \\ \hline
\end{tabular}
\end{center}
\vspace{-0.2cm}
\caption{The major climate variability systems}
\label{T1}
\end{table}
\newpage
| 1,620
|
CJLC began by asking Professor Arsic about non-capitalist ontologies of things, and then expanded the definition of consumption to discuss the porosity of the self, which led us to think about reading as a tool of perception in the art of ordinary living.
Devika: In your essay “Our Things”, you discuss the modes in which material culture is accessed and how Thoreau’s material ontologies set him apart from our capitalist way of thinking about material culture. To begin, can you give us an understanding of what you think an ontology of things or objects is within this framework, in which accessing, buying, owning are able to give meanings to things in a certain way?
Branka: That’s a very big topic and a field these days — “thing theory.” Beyond the traditional understanding of Marx’s critique of possession and reification, I find in some more recent thinking about things that actually builds on 17th, 18th, and 19th century theories of things. In the works of Thoreau, Melville, and some Romantic poets, it’s possible to differentiate between what philosophy at that point called an “object” and what might be called a “thing.” An object is something that exists necessarily and always only in relation to the subject, and is available for appropriation; a “thing” is something that exists regardless, resists appropriation; it has no inherent value, market value, trade value, nor even aesthetic value – just idiosyncratic value, not referencing anything outside of itself. There are actually not a lot of verbs in our language that can articulate the difference between thing and object.
Thoreau visits estate sales and tries to salvage certain things from the property of people he never met and did not know. Those things typically absolutely no value – they could not be sold, traded, or even exchanged. Thoreau approached things thinking he would incorporate them in a circulation of his own thinking – not in the manner in which Benjamin says commodities speak to their future owners, but in a manner in which they would give him thoughts, and themselves become indistinguishable from thinking.
I am also interested in this relatively recent theory proposed by Jonathan Lamb. He wrote a really interesting book called The Things Things Say about various modalities philosophers and authors of 18th and 19th century England thought about things as opposed to objects. He differentiates objects as entities that depend on human recognition and appropriation, and then in return enact reification, as opposed to things, which can exist with or without humans, in landscapes that are not anthropomorphic.
D: As you were talking about the subject-object relation, you seem to suggest that the process through which things become objects is through language grafting that meaning onto them. I was wondering what ways of knowing things, through and in language, we can have that don’t have that violence outside of renaming them.
B: I think there are many ways of knowing things that actually do not – are we talking about things or objects here?
Melanie: Perhaps things becoming objects.
B: To clarify, objects are things involved in a circulation of trade, exchange, and implication. For that reason, they cannot exist but in relation to a subject or an owner. Objects are known through their market value or exchange value, which makes them, in Marxist terms, little fetishes. But one can know a thing in ways that are not necessarily linguistic — through an affect, or through a memory.
Different things exist in different ontologies. These days I’m very preoccupied by cosmologies of South Pacific peoples, and doing a lot of research on that because of my book on Melville. They found a really amazing way of integrating the non-human forms of life and things. A canoe is a transformation of a tree, and things at large exist as an archive of the living. This is not animism; it’s an integrated way of understanding life, where the living matter out of which a thing is made of continues to live in the thing and then gives the thing a special status. The canoe either becomes sacred, or is integrated into the floor of churches to be part of the life of the living.
There’s a certain kind of respect that’s incurred, that has been afforded to that kind of keeps that thing in a circulation of life but in a way that cannot be either reified or economically valued because the thing is removed into the realm of the sacred, because it’s made out of a living being. The tree sacrificed for a canoe is just one of many examples – these cosmologies can teach us a lot about possible way of lives of things that would not be capitalist objects.
Thoreau critiques people who spend their lives paying off the houses not because he was claiming that people don’t need houses but because these houses are valued according to their market values, or geography (which is again their market values). Instead of developing a relation to one’s own space, these people relate to their houses as object of exchange and profit potential. These days, people will say “my apartment or my house appreciated so much,” as though this makes it better — this is the object of his critique.
M: That brings to mind the contemporary art market, in which people collect art objects to store wealth and accrue value. Even a creative production can be commodified and made into an object. What is the opposite of commodifying art? Your example of the canoe, an extension of a tree, is imbued with meaning from natural life. But could a thing contain human to human relations that are non-economic?
B: There is a way of relating to things from the point of view of a collector, in Benjamin’s sense of the word – there are collectors who collect with the idea of reselling, like what you said in terms of contemporary art. There are people who collect, like I do with books, with absolutely no idea of exchanging or doing anything. It is fueled by an intense obsession with not just the world of things but with what that collection would make to the world of the collector. Some people collect worthless things.
While one can say as long as you collect, you’re piling up and manifesting a capitalist drive to have more, possess more, I would say there’s a deeper way, a way in which the thing enters the bodily emotional and intellectual space of the collector. I am a collector of Japanese stationery, not because I think I can sell it, but because it does something to my writing. I have a tactile relationship to the surface of the paper and so I almost have this little ritual of choosing which paper is right for which sentence, which chapter – it breathes good energy into my thinking, and I write better. I’m sure a lot of people would be successful in doing tracing my attitude to the logic of capitalism, but I think that that would be reductive. There’s so much more to our way of relating to not only things, but things as representatives of the embodied world.
M: That answers to a lot of concerns I have about the way I relate to objects and sort of store myself in objects in my room. I don’t have any intention to exchange those objects, but they quickly accumulate as I curate more things into my life.
I am also thinking about how it’s impossible not to always be consuming. We eat first thing in the morning, which is a kind of consumptive act. From food to clothing, almost all the things we bring into our lives are bought or acquired in some kind of way, especially in this city. In the context of perpetual consumption, does the thing-object distinction present us with a more ethical way to relate to the things we consume?
B: It’s important to note that what we consume is not necessarily always things, and that leads us into a slightly different zone: the question of consuming and consummation, because we consume all kinds of things, from living beings to very sophisticated artifacts, say movies at the theater. So when we start talking about consuming, we are talking about a realm that is kind of broader than the realm of things.
Here, there is a set of questions that is absolutely related to capitalism, most obviously the way we consume energy, enacting geological transformation, climate change, all kinds of stuff to the Earth. There is also a question of ethics that was with us even before capitalism, and only escalated with capitalism’s power, globality, and tremendous technological force. But the question of consuming, for instance life, did not necessarily appear with capitalism. We have to eat — that’s kind of obvious, because if we don’t we put in jeopardy our own life, which is also unethical. But then there’s always the question of how and what we eat. Today, we have to think about that at very many levels, right. Most obviously, we have to think about fair trade, we have to think – a lot of people think about – health issues, organic food, responsible growing. But then there’s also this question of what kind of food, ontologically speaking, people should eat. Should we eat animals? That’s not a question invented by capitalism; it goes very deeply into the question of how we treat life and other living beings, and has a very long tradition, from Vedas and the Upanishads through Greek philosophy. And then we encountered it again in a very powerful way in the 19th century, with vegetarian movements, across the globe, really, in India, in the United States, in Thoreau’s Higher Laws and his call to vegetarianism. If we move the question of consuming in this direction, which I also think we should, then it becomes far more complicated, and it’s not something that’s necessarily related to capitalism, it’s related to how we treat other life forms, and what do we do to decrease the suffering of other living beings.
I would say that the ethics and politics consumption includes a responsibility toward resources, towards our environment, fair trade and labor. But I also want to think that when people think about consuming they think about the lives of beings other than humans, and do their best to respect them, maybe not even consume. I am proposing vegetarianism, but I do not want to sound moral or preachy here. Obviously people eat meat, but if they have to then there are more and less responsible ways of eating meat.
D: I wonder if this distinction between morals and ethics, especially with relation to consumption, comes back to what Melanie said about storing herself in objects. Sometimes, ethical acts can be ways to mark yourself or to be in a certain way, in a more concrete and unified way. But I know you think a lot about moving away from the self or reducing its boundaries.
B: I think about self as something that’s very porous, and in that sense ecological. Self is not something that’s not in a strict opposition, therefore, to the external world, whatever that world is at the moment, whether looking at a thing or a human being or a piece of clothing or artwork. Self is something that is not something that exists in some kind of formed, stable, fixed interiority into which all of this exteriority comes and I kind of process it, but keep it under control. Rather, self is something that’s always in the process of becoming thanks to its interaction with the external world.
Emerson would always say that we find ourselves in a certain mood. A mood is not something we generate, it’s rather that we keep asking ourselves: why might I today be feeling like this, a little bit low? Is it the rainy day? And often when we come to the answer to the question the mood evaporates, which only tells you that our moods have us rather than the opposite. That tells you it’s not that we do not have a self, it’s not either/or. It’s not kind of some hard division, and I anyway don’t believe in strict limits and borderlines of any sort, frontiers even.
Self is constantly being re-negotiated through not just some interiority that, say, psychoanalysis posits with some unconsciousness then that kind of presses on us and wants to get out, but through the external encounters and external world. Therefore, when I fall for an object or for a work of art, there are some works of art where I can acknowledge their aesthetic value, but I pass by that, and there are some that obsess me and I keep going back to them. And the reason for that I suspect is that they act on who I am and remake me.
What I’m saying is that even objects are a part of our self — not because they appropriate it, but because they acted on us, sometimes even generating our desire.
I think the same about clothing — it is not something that is not outside of who I am. So you can say, you live in a capitalist society and there are only so many place people can shop for clothes, which is true. But on the other hand, we do not look the same. You go down the streets and you see people are differently dressed, so that tells you something about their psyches. I refer to Proust and I refer to Deleuze, who thought that our desires are not invested, necessarily, in some sublime minds that are somewhere behind, but that when we see a person we see a person integrated in a landscape, dressed in a certain way — all of that is “person.” When Proust sees Albertine in a beach, she’s a part of the beach and the sea and the atmosphere and all of that, for him, is she.
M: If self is produced by collecting external bits of clothing, curating all that’s supposedly exterior, and becoming at each instant, I wonder if we have a baseline natural desire to possess or have things that come from outside of us, that we then assign the name of a capitalist urge to acquire things in the process of history.
B: I don’t think that one thing would exclude the other. What I am resisting is a reductionist approach that would interpret everything as just the simple outcome of the circulation of capital. Edgar Allen Poe, for instance, wrote a beautiful letter in which he wonders whether people who live thousands of years ago were somehow substantially different from us, on the basis of technology. He said something they had very complicated network relays of desire and encounters that made them feel happy or sad in the way in which we are.
The economic structure changes us, but also doesn’t — not because there is an essential self in us, but because of exactly what you were saying, and that is that porousness of ourselves, the fact that we are made of so many other beings. Think about ancestral religions — the basic reasoning behind them is that we are made of so many psyches that already were, that we are, in a sense, multiplicitous. I think many people realized, a long time ago, what some theorists do not realize today: that we are many. That there are many memories in us that are not ours but are given to us, that become us, that there is a circulation of minds and even gestures or words that make us and remake us in a way that we can’t control, so that we are, actually, made of relations including not just humans, but the environment and animals and plants and the elements. I’m a very different person where I’m close to the ocean. And much better.
D: Is this to say we can’t trace the precise source of our desires, or the precise ways in which things act on us? Is it simply a ‘magical interaction’ produced by this dense conglomeration of relations that you talk about?
B: With this I go in the direction again of Emerson and pragmatism. Emerson thought — and Proust also has a similar understanding, a similar phrase — that there are ‘involuntary perceptions. When I talk about porousness that’s what I mean. We perceive so many things without even knowing when and how and why they work upon us. Those are, in fact, affects that work in us and re-work us before we can even figure out the kinds of changes that have been initiated in us.
This is not to say that we are absolutely doomed or delivered to so many contingent affects, perceptions and sensations that we can’t control. It is rather to say that things start happening to us much before we actually became aware of their happening to us, and then we can then think about whether we can act against them to react to them. But whatever we do to them, we are also, by the moment we start acting on them on them — on our affects or our sensations or our perceptions — touched by them. And that’s what I mean when I say we are acted on or upon — not magically but in a way in which our affects precede us.
But I’m not saying that we are in their control. We can actually look and say, “Here’s what I feel. What should I do about that?” It’s the feeling you find in yourself, and the fact that it’s there changes you, even if of no other reason than for the amount of force that it will require for you to act on it, to act back on it. And that interaction, again, will change you. So I’m not saying that there is not something we can call “rational agency,” but that rational agency is a very intricate and complicated thing made up of a lot of irrational things.
D: The way in which we come to abstractions, especially in the context and the conditions of the university environment, neglects this system of irrationality or seeks to deny it. I was wondering what you thought about that and how you navigate your place here within the university and within the expectations of being in the English department?
B: So for somebody who is so convinced in the porousness of the self and the weakness of rational agency, I don’t want to say that I act as a rational agent all the time. It’s not that, you know, I wake up and have a clear plan of my day hour to hour, as if this is what’s going to happen. I also have a child and that means I’m not in possession of my time.
Research is going on for me 24/7. Sometimes, for instance, I can’t fall asleep and then I start thinking and figure out a paragraph that I couldn’t figure out that morning when I was writing and then I immediately get up and I write it. When I talk now to you I’m thinking “Oh this thing I said about affect I should follow up on.” In a sense, I always write.
The department has a pretty organized set of customs and expectations and behaviors regarding what a good citizen should be or do. And I’m trying to be a good citizen in every realm of my political life, not just in the department but in general.
M: I am thinking about what your conception of rational agency means for us as students of literature — if we define consumption more broadly like you were speaking of earlier, we consume ideas at school, consume what we read and consume in our dialogues with others so as to become, right now, as we learn. I struggle not to be the product of what I’m reading, and am wondering if that’s a problem.
B: Of course! But if it’s a good reading then you’re going to be a good product!
M: But where do we exercise control? When we choose what to read?
B: See that’s the thing! Well, there’s several things, but it’s the thing I’m constantly working through with my graduate students who are in position to write their own dissertation, which is expected to be a piece of of original research and thinking. I’m always advising them that they should not be too original, but just a little bit original. I try to say to them that it’s kind of all happening at the same time. It’s kind of a “mesh,” as it were, but that’s how life is in general.
You never take everything from what you read, that’s impossible. I don’t even think that religious people take absolutely everything from religious texts. I think there’s always some way of filtering. Some things stay with you, and some things you don’t know have stayed with you. And then years later — this happens to me very often — I realize, “Oh, didn’t Kant say that? Where did I read that?” Then I spend days searching through my books and typically even what I’m looking for is not even what I underlined three years ago.
It somehow stays with you and in different periods of time of your life and your thinking, you get back to the books you read and take very different things from them. All that is to say that you filter. You take something that speaks to you, that takes the objects we were talking about “out of your mind,” as Emerson would say.
What does that mean? It means it takes you out of what you already know. That’s when thinking really starts, if you experience this wonderful moment of a thought or a philosopher or an author or an artist or whoever taking you out of what you know to a different place, making you learn new things, seeing things differently then things start happening in a very interesting way. And then it’s all a mesh, because then you read a little bit and then you stop and then you start writing a little bit but then the more you write about it the clearer things that you read become, then you go back and you read which then in turn illuminates some difficult spots in your writing, and so on and so forth.
Good writers really are taken by what they read, really think about it. One thing produces another. My theory is that one can’t write well if one is not a good reader and one cannot think well if one is not a good reader. It all depends on great reading that generates both attention to language and good writing and thought.
B: I just wanted to add that there’s a whole aspect of contemporary world — this world of the web and digital reality where the question of programming what people are going to be and how they’re going to feel is prevalent — that I don’t know about. I have sense that it’s almost a new world – a total world – and so much that we still don’t know about the ways in which that can actually be more in tune with the logic of capitalism and generating and possessing and manipulating minds, generating persons even in a certain way. I say this not as an informed claim, as I am really a very old fashioned person. I write my books in handwriting. I don’t have Facebook. But it is some sense that I’m getting from that whole general reality.
D: Part of the contemporary situation is that the information we start taking in abstraction happens at a disembodied level and the more that happens the harder sometimes it is to interact with materials and allow them to work on you. I was wondering how we might think of more embodied forms of reading of taking in information?
B: Well, that again can be given a narrow and broad answer. For instance Benjamin’s thesis that perception is reading, or Emerson’s, also, is that we involuntary perceptions and voluntary ones. Voluntary is a kind of reading where we select and and choose — in which sense everything is reading. There are things we decide not to read even though they are right in front of our nose in our everyday lives, which is precisely why to teach people the skill of reading and then like really attentive reading is in fact not just a skill from which they reading of literature would benefit but a political skill.
Persons who are skilled in very patient slow attentive reading will miss less disturbing signs, say even in the realm of the political. How do you decide which article you read in New York Times? All of that requires a very developed — but even conversing with other people, trying to figure out what is it their trying to tell me. All of that, you know, the skill of reading well… Attention to reading is not something only students of English and philosophy should be skilled in. I think everybody should be educated in the art of reading because it’s the most essential art for ordinary living. It is, in the end, the art of paying attention.
| 245,412
|
\begin{document}
\title {
On the skein polynomial for links
}
\author {
Boju Jiang
}
\author {
Jiajun Wang
}
\author {
Hao Zheng
}
\address {
Department of Mathematics\\
Peking University\\
Beijing 100871\\
China
}
\email {
bjjiang@math.pku.edu.cn
}
\email {
wjiajun@math.pku.edu.cn
}
\email {
hzheng@math.pku.edu.cn
}
\thanks {Partially supported by NSFC grant \#11131008}
\subjclass [2010]{Primary 57M25; Secondary 20F36}
\keywords {the skein polynomial, HOMFLY polynomial, Jones polynomial,
Alexander-Conway polynomial, skein relations}
\date{}
\begin{abstract}
We give characterizations of the skein polynomial for links
(as well as Jones and Alexander-Conway polynomials derivable from it),
avoiding the usual ``smoothing of a crossing'' move.
As by-products we have characterizations of these polynomials for knots,
and for links with any given number of components.
\end{abstract}
\maketitle
\section {Introduction}
\label{sec:Intro}
The skein polynomial (as called in \cite[Chapter 8]{K1},
also known as HOMFLY or HOMFLY-PT polynomial),
$P_L(a,z) \in \mathbb Z[a^{\pm1},z^{\pm1}]$, is an invariant for oriented links.
Here $\mathbb Z[a^{\pm1},z^{\pm1}]$ is the ring of Laurent polynomials in two variables $a$ and $z$, with integer coefficients.
It is defined to be the invariant of oriented links satisfying the axioms
\begin{gather*}
a^{-1}\cdot P
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1a_dd-0}
\end{pmatrix}
-a\cdot P
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1b_dd-0}
\end{pmatrix}
=z\cdot P
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1o_dd-0}
\end{pmatrix} ;
\tag*{\rm(I)}
\\
P
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/O-0}
\end{pmatrix}
=1 .
\tag*{\rm(O)}
\end{gather*}
The Alexander-Conway polynomial $\Delta_L \in \mathbb Z[t^{\pm\frac12}]$ and
the Jones polynomial $V_L \in \mathbb Z[t^{\pm\frac12}]$ are related to the skein polynomial:
\[
\Delta_L(t)=P_L(1,t^{\frac12}-t^{-\frac12}),
\qquad
V_L(t)=P_L(t,t^{\frac12}-t^{-\frac12}).
\]
Our main result is
\begin{thm}
\label{thm:HOMFLY}
The skein polynomial $P_L \in \mathbb Z[a^{\pm1},z^{\pm1}]$ is
the invariant of oriented links
determined uniquely by the following four axioms.
{\allowdisplaybreaks
\begin{gather*}
a^{-2}\cdot P
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1a1a_dd-0}
\end{pmatrix}
+a^2\cdot P
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1b1b_dd-0}
\end{pmatrix}
=(2+z^2)
\cdot P
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1oo_dd-0}
\end{pmatrix} ;
\tag*{\rm(II)}
\\
a^{-1}\cdot P
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1a2a1b_ddd-0}
\end{pmatrix}
-a\cdot P
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1a2b1b_ddd-0}
\end{pmatrix}
=
a^{-1}\cdot P
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1b2a1a_ddd-0}
\end{pmatrix}
-a\cdot P
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1b2b1a_ddd-0}
\end{pmatrix} ;
\tag*{\rm(III)}
\\
P
\begin{pmatrix}
\includegraphics[height=1.2cm]{fig_012/IO-0}
\end{pmatrix}
=z^{-1}(a^{-1}-a)\cdot
P
\begin{pmatrix}
\;\;
\includegraphics[height=1.2cm]{fig_012/I-0}
\;\;
\end{pmatrix} ;
\tag*{\rm(IO)}
\\
P
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/O-0}
\end{pmatrix}
=1 .
\tag*{\rm(O)}
\end{gather*}
}
\end{thm}
A parallel result is for the Jones polynomial.
It is not a direct corollary of the above theorem,
because the substitutions $a\mapsto t$ and $z\mapsto (t^{\frac12}-t^{-\frac12})$
do not send $\mathbb Z[a^{\pm1},z^{\pm1}]$ into $Z[t^{\pm\frac12}]$.
\begin{thm}
\label{thm:Jones}
The Jones polynomial $V_L \in \mathbb Z[t^{\pm\frac12}]$ is
the invariant of oriented links
determined uniquely by the following four axioms.
{\allowdisplaybreaks
\begin{gather*}
t^{-2}\cdot V
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1a1a_dd-0}
\end{pmatrix}
+t^2\cdot V
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1b1b_dd-0}
\end{pmatrix}
=(t+t^{-1})
\cdot V
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1oo_dd-0}
\end{pmatrix} ;
\tag*{\rm(II$_V$)}
\\
t^{-1}\cdot V
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1a2a1b_ddd-0}
\end{pmatrix}
-t\cdot V
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1a2b1b_ddd-0}
\end{pmatrix}
=
t^{-1}\cdot V
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1b2a1a_ddd-0}
\end{pmatrix}
-t\cdot V
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1b2b1a_ddd-0}
\end{pmatrix} ;
\tag*{\rm(III$_V$)}
\\
V
\begin{pmatrix}
\includegraphics[height=1.2cm]{fig_012/IO-0}
\end{pmatrix}
=-(t^{\frac12}+t^{-\frac12})\cdot
V
\begin{pmatrix}
\;\;
\includegraphics[height=1.2cm]{fig_012/I-0}
\;\;
\end{pmatrix} ;
\tag*{\rm(IO$_V$)}
\\
V
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/O-0}
\end{pmatrix}
=1 .
\tag*{\rm(O$_V$)}
\end{gather*}
}
\end{thm}
For the Alexander-Conway polynomial, the result takes a slightly different form.
We switch to a ($\Phi$)-type axiom because the (IO)-type one degenerates into a
consequence of (II) and (III) (see Corollary~\ref{cor:stabilization_Alexander-Conway}).
\begin{thm}
\label{thm:Alexander-Conway}
The Alexander-Conway polynomial $\Delta_L \in \mathbb Z[t^{\pm\frac12}]$ is
the invariant of oriented links
determined uniquely by the following four axioms.
{\allowdisplaybreaks
\begin{gather*}
\Delta
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1a1a_dd-0}
\end{pmatrix}
+ \Delta
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1b1b_dd-0}
\end{pmatrix}
=(t+t^{-1})
\cdot \Delta
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1oo_dd-0}
\end{pmatrix} ;
\tag*{\rm(II$_\Delta$)}
\\
\Delta
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1a2a1b_ddd-0}
\end{pmatrix}
- \Delta
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1a2b1b_ddd-0}
\end{pmatrix}
=
\Delta
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1b2a1a_ddd-0}
\end{pmatrix}
- \Delta
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1b2b1a_ddd-0}
\end{pmatrix} ;
\tag*{\rm(III$_\Delta$)}
\\
\Delta
\begin{pmatrix}
\includegraphics[height=1.2cm]{fig_012/Phi+-0}
\end{pmatrix}
=(t^{\frac12}-t^{-\frac12})\cdot
\Delta
\begin{pmatrix}
\;\;
\includegraphics[height=1.2cm]{fig_012/I-0}
\;\;
\end{pmatrix} ;
\tag*{($\Phi_\Delta$)}
\\
\Delta
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/O-0}
\end{pmatrix}
=1 .
\tag*{\rm(O$_\Delta$)}
\end{gather*}
}
\end{thm}
If we restrict our attention to oriented links with a fixed number $\mu>0$ of components,
the axiom (IO) becomes irrelevant but we must pick a suitable normalization.
Let $U_\mu$ denote the $\mu$-component unlink, and
let $C_\mu$ denote the $\mu$-component oriented chain
where adjacent rings have linking number $+1$.
(In terms of closed braids, $U_\mu$ is the closure of the trivial braid $e\in B_\mu$,
and $C_\mu$ is the closure of the braid $\sigma_1^2\sigma_2^2\dots\sigma_{\mu-1}^2\in B_\mu$.)
We can use either $U_\mu$ or $C_\mu$ (but $U_\mu$ is preferred) to normalize the skein or Jones polynomial,
but for Alexander-Conway polynomial we can only use $C_\mu$.
\begin{thm}
\label{thm:HOMFLY_knot}
The skein polynomial $P_L$ is
the invariant of oriented $\mu$-compo\-nent links
determined uniquely by the following three axioms.
{\allowdisplaybreaks
\begin{gather*}
a^{-2}\cdot P
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1a1a_dd-0}
\end{pmatrix}
+a^2\cdot P
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1b1b_dd-0}
\end{pmatrix}
=(2+z^2)
\cdot P
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1oo_dd-0}
\end{pmatrix} ;
\tag*{\rm(II)}
\\
a^{-1}\cdot P
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1a2a1b_ddd-0}
\end{pmatrix}
-a\cdot P
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1a2b1b_ddd-0}
\end{pmatrix}
=
a^{-1}\cdot P
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1b2a1a_ddd-0}
\end{pmatrix}
-a\cdot P
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1b2b1a_ddd-0}
\end{pmatrix} ;
\tag*{\rm(III)}
\\[2ex]
P(U_\mu) = (z^{-1}(a^{-1}-a))^{\mu-1} .
\tag*{\rm(U)}
\end{gather*}
}
\end{thm}
\begin{thm}
\label{thm:Jones_knot}
The Jones polynomial $V_K$ is
the invariant of oriented $\mu$-component links
determined uniquely by the following three axioms.
{\allowdisplaybreaks
\begin{gather*}
t^{-2}\cdot V
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1a1a_dd-0}
\end{pmatrix}
+t^2\cdot V
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1b1b_dd-0}
\end{pmatrix}
=(t+t^{-1})
\cdot V
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1oo_dd-0}
\end{pmatrix} ;
\tag*{\rm(II$_V$)}
\\
t^{-1}\cdot V
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1a2a1b_ddd-0}
\end{pmatrix}
-t\cdot V
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1a2b1b_ddd-0}
\end{pmatrix}
=
t^{-1}\cdot V
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1b2a1a_ddd-0}
\end{pmatrix}
-t\cdot V
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1b2b1a_ddd-0}
\end{pmatrix} ;
\tag*{\rm(III$_V$)}
\\[2ex]
V(U_\mu) = (-(t^{\frac12}+t^{-\frac12}))^{\mu-1} .
\tag*{\rm(U$_V$)}
\end{gather*}
}
\end{thm}
\begin{thm}
\label{thm:Alexander-Conway_knot}
The Alexander-Conway polynomial $\Delta_K$ is
the invariant of oriented $\mu$-component links
determined uniquely by the following three axioms.
{\allowdisplaybreaks
\begin{gather*}
\Delta
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1a1a_dd-0}
\end{pmatrix}
+ \Delta
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1b1b_dd-0}
\end{pmatrix}
=(t+t^{-1})
\cdot \Delta
\begin{pmatrix}
\includegraphics[width=.8cm]{fig_012/1oo_dd-0}
\end{pmatrix} ;
\tag*{\rm(II$_\Delta$)}
\\
\Delta
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1a2a1b_ddd-0}
\end{pmatrix}
- \Delta
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1a2b1b_ddd-0}
\end{pmatrix}
=
\Delta
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1b2a1a_ddd-0}
\end{pmatrix}
- \Delta
\begin{pmatrix}
\includegraphics[width=1.2cm]{fig_3icons/1b2b1a_ddd-0}
\end{pmatrix} ;
\tag*{\rm(III$_\Delta$)}
\\
\Delta(C_\mu) = (t^{\frac12}-t^{-\frac12})^{\mu-1} .
\tag*{\rm(C$_\Delta$)}
\end{gather*}
}
\end{thm}
Note that the foundational relation (I) cannot appear in
Theorems~\ref{thm:HOMFLY_knot}--\ref{thm:Alexander-Conway_knot}
because it involves links with different number of components.
Our approach is via closed braids.
We explain the language of relators in Section~\ref{sec:relators} and
give an algebraic reduction lemma in Section~\ref{sec:key_lem}.
This approach is adapted from the corresponding sections of \cite{J1}
on Conway's potential function for colored links.
The current context of uncolored links makes the reduction argument more transparent.
Section~\ref{sec:stabilizations} discusses closed braids with different number of strands.
The theorems are proved in the last two sections.
\section{Braids and skein relators}
\label{sec:relators}
For braids, we use the following conventions:
Braids are drawn from top to bottom.
The strands of a braid are numbered at the top of the braid, from left to right.
The product $\beta_1\cdot\beta_2$ of two $n$-braids is obtained by drawing $\beta_2$ below $\beta_1$.
The set $B_n$ of all $n$-braids forms a group under this multiplication,
with standard generators $\sigma_1,\sigma_2,\dots,\sigma_{n-1}$.
It is well known that links can be presented as closed braids.
The closure of a braid $\beta\in B_n$ will be denoted $\wh\beta$.
Two braids (possibly with different number of strands) have isotopic closures
if and only if they can be related by a finite sequence of two types of moves:
\begin{enumerate}
\item Conjugacy move: $\beta$ $\leftrightsquigarrow$ $\beta'$ where $\beta,\beta'$ are conjugate in a braid group $B_n$;
\item Markov move: $\beta\in B_n$ $\leftrightsquigarrow$ $\beta\sigma_n^{\pm1}\in B_{n+1}$.
\end{enumerate}
Let $\Lambda$ be the Laurent polynomial ring $\mathbb Z[a^{\pm1},z^{\pm1}]$.
Let $\Lambda B_n$ be the group-algebra on $B_n$ with coefficients in $\Lambda$.
\begin{defn}
\label{defn:skein_relation}
We say that an element
\[
\lambda_1\cdot\beta_1+\dots+\lambda_k\cdot\beta_k
\]
of $\Lambda B_n$ is a \emph{skein relator}, or equivalently, say that the corresponding formal equation
(in which $P_{L_{\beta_h}}$ stands for the $P$ of the link $L_{\beta_h}$)
\[
\lambda_1\cdot P_{L_{\beta_1}}+\dots+\lambda_k\cdot P_{L_{\beta_k}}=0
\]
is a \emph{skein relation}, if the following condition is satisfied:
For any links $L_{\beta_1},\dots,L_{\beta_k}$ that are identical except in a cylinder
where they are represented by the braids $\beta_1,\dots,\beta_k$ respectively,
the formal equation becomes an equality in $\Lambda$.
\end{defn}
\begin{exam}
\label{exam:relator_vs_relation}
To every element $\lambda_1\cdot\beta_1+\dots+\lambda_k\cdot\beta_k\in \Lambda B_n$,
by taking braid closures we have a corresponding element
\[
\lambda_1\cdot P_{\wh\beta_1}+\dots +\lambda_k\cdot P_{\wh\beta_k} \in \Lambda.
\]
The latter vanishes if the former is a skein relator.
\end{exam}
\begin{exam}
The skein relations (I), (II) and (III) in Section~\ref{sec:Intro}
correspond to the following relators, respectively:
(The symbol $e$ stands for the trivial braid.)
{\allowdisplaybreaks
\begin{gather*}
\text{\rm(I$_\text{B}$)}:=
a^{-1}\cdot\sigma_1^2 -a\cdot\sigma_1^{-2} -z\cdot{e} ;
\\
\text{\rm(II$_\text{B}$)}:=
a^{-2}\cdot\sigma_1^2 +a^2\cdot\sigma_1^{-2} -(2+z^2)\cdot{e} ;
\\
\text{\rm(III$_\text{B}$)}:=
\begin{aligned}[t]
&a^{-1}\cdot{\sigma_1\sigma_2\sigma_1^{-1}}
+ a\cdot{\sigma_1^{-1}\sigma_2^{-1}\sigma_1}
\\
&- a^{-1}\cdot{\sigma_1^{-1}\sigma_2\sigma_1}
- a\cdot{\sigma_1\sigma_2^{-1}\sigma_1^{-1}} .
\end{aligned}
\end{gather*}
}
\end{exam}
\begin{prop}
\label{prop:relator_ideal}
Assume that
\[
\lambda_1\cdot P_{L_{\beta_1}}+\dots+\lambda_k\cdot P_{L_{\beta_k}}=0
\]
is a skein relation. Then for any given braid $\alpha\in B_n$,
the following equations are also skein relations:
\begin{gather*}
\lambda_1\cdot P_{L_{(\beta_1\alpha)}}+\dots+\lambda_k\cdot P_{L_{(\beta_k\alpha)}}=0;
\\
\lambda_1\cdot P_{L_{(\alpha\beta_1)}}+\dots+\lambda_k\cdot P_{L_{(\alpha\beta_k)}}=0.
\end{gather*}
Hence skein relators form a two-sided ideal\/ $\mathfrak R_n$
(called the \emph{relator ideal\/})
in $\Lambda B_n$.
\end{prop}
\begin{proof}
Look at the cylinder where the links $L_{(\beta_1\alpha)},\dots,L_{(\beta_k\alpha)}$
are represented differently by braids $\beta_1\alpha,\dots,\beta_k\alpha$, respectively.
In the upper half cylinder they are represented by braids $\beta_1,\dots,\beta_k$.
So the assumption implies the first equality.
Similarly for the second equality.
By Definition~\ref{defn:skein_relation}, this means skein relators form a two-sided ideal.
\end{proof}
\section{An algebraic reduction lemma}
\label{sec:key_lem}
\begin{defn}
\label{defn:equivalence}
Let $\mathfrak I_n$ be the two-sided ideal in $\Lambda B_n$
generated by $\text{\rm(II$_\text{B}$)}$ and $\text{\rm(III$_{\text{B}}$)}$.
(When $n=2$ we ignore $\text{\rm(III$_{\text{B}}$)}$.)
Two elements of the algebra $\Lambda B_n$ are \emph {equivalent modulo $\mathfrak I_n$}
(denoted by $\sim$\,) if
their difference is in $\mathfrak I_n$.
\end{defn}
For example, by conjugation in $B_n$ we have
$a^{-2}\cdot \sigma_i^2 +a^2\cdot \sigma_i^{-2} -(2+z^2)\cdot e \sim 0$ and
$a^{-1}\cdot\sigma_i\sigma_{i+1}\sigma_i^{-1}
+a\cdot \sigma_i^{-1}\sigma_{i+1}^{-1}\sigma_i
-a^{-1}\cdot \sigma_i^{-1}\sigma_{i+1}\sigma_i
-a\cdot \sigma_i\sigma_{i+1}^{-1}\sigma_i^{-1}
\sim 0$,
for any $i$.
\begin{lem}
\label{lem:reduction}
Modulo $\mathfrak I_n$, every braid $\beta\in B_n$ is
equivalent to a $\Lambda$-linear combination of braids of the form
$\alpha\sigma_{n-1}^k \gamma$ with $\alpha,\gamma\in B_{n-1}$ and $k\in\{0,\pm1,2\}$.
\end{lem}
A braid $\beta\in B_n$ can be written as
\[
\beta=\beta_0\sigma_{n-1}^{k_1} \beta_1\sigma_{n-1}^{k_2} \dots \sigma_{n-1}^{k_r}\beta_r
\]
where $\beta_j\in B_{n-1}$ and $k_j\neq0$.
We allow that $\beta_0$ and $\beta_r$ be trivial,
but assume other $\beta_j$'s are nontrivial.
The number $r$ will be denoted as $r(\beta)$.
The lemma will be proved by an induction on the double index $(n,r)$.
Note that the lemma is trivial when $n=2$, or $r(\beta)\leq1$.
It is enough to consider the case $r=2$, because induction on $r$ works beyond $2$.
Indeed, if $r(\beta)>2$, let $\beta'=\beta_1\sigma_{n-1}^{k_2} \dots \sigma_{n-1}^{k_r}\beta_r$, then
$r(\beta')<r(\beta)$. By inductive hypothesis $\beta'$ is equivalent to a
linear combination of elements of the form $\alpha'\sigma_{n-1}^{k'} \gamma'$,
hence $\beta$ is equivalent to a linear combination of elements of the form
$\beta_0\sigma_{n-1}^{k_1} \alpha'\sigma_{n-1}^{k'} \gamma'$.
This brings the problem back to the $r=2$ case.
Henceforth we assume $r=2$.
Since the initial and terminal part of $\beta$, namely $\beta_0$ and $\beta_r$,
do not affect the conclusion of the lemma, we can drop them.
So we assume $\beta=\sigma_{n-1}^{k_1} \beta_1\sigma_{n-1}^{k_2}$, where $\beta_1\in B_{n-1}$.
By the induction hypothesis on $n$, $\beta_1\in B_{n-1}$ is a linear combination of
elements of the form $\alpha_1\sigma_{n-2}^\ell \gamma_1$.
Note that $\alpha_1,\gamma_1\in B_{n-2}$ commute with $\sigma_{n-1}$.
So it suffices to focus on braids of the form
$\beta=\sigma_{n-1}^{k}\sigma_{n-2}^{\ell}\sigma_{n-1}^{m}$.
For the sole purpose of controlling the length of displayed formulas,
we assume $n=3$ below.
The proof for a general $n$ can be obtained by a simple change of subscripts,
replacing $\sigma_1,\sigma_2$ with $\sigma_{n-2},\sigma_{n-1}$ and
replacing $t_1,t_2,t_3$ with $t_{n-2},t_{n-1},t_{n}$, respectively.
Thus, Lemma~\ref{lem:reduction} has been reduced to the following
\begin{lem}
\label{lem:key_reduction}
Every $\sigma_2^{k}\sigma_1^{\ell}\sigma_2^{m}$ is equivalent \textup{(modulo $\mathfrak I_n$)} to a linear combination
of braids of the form $\sigma_1^{k'}\sigma_2^{\ell'}\sigma_1^{m'}$ where $\ell'$ is $0$, $\pm1$ or $2$.
\end{lem}
\begin{proof}
Modulo $\text{(II$_\text{B}$)}$, we may restrict the exponent
$k$ to take values $1$, $2$ and $3$ (we are done if $k$ is $0$).
If $k>1$ we can decrease $k$ by looking at
$\sigma_2^{k-1}(\sigma_2\sigma_1^{\ell}\sigma_2^m)$, so it suffices to prove the case $k=1$.
Again modulo $\text{(II$_\text{B}$)}$, we can restrict the exponents $\ell, m$ to the values $\pm1$ and $2$.
There are altogether 9 cases to verify.
{\it 5 trivial cases (braid identities) }:
\begin{alignat*}{3}
&\sigma_2\sigma_1\sigma_2=\sigma_1\sigma_2\sigma_1, \quad
&&\sigma_2\sigma_1\sigma_2^{-1}=\sigma_1^{-1}\sigma_2\sigma_1, \quad
&&\sigma_2\sigma_1^{-1}\sigma_2^{-1}=\sigma_1^{-1}\sigma_2^{-1}\sigma_1, \quad
\\
&\sigma_2\sigma_1\sigma_2^2=\sigma_1^2\sigma_2\sigma_1, \quad
&&\sigma_2\sigma_1^2\sigma_2^{-1}=\sigma_1^{-1}\sigma_2^2\sigma_1. \quad
\end{alignat*}
{\it The case $\sigma_2\sigma_1^{-1}\sigma_2$ }:
Multiplying $\text{\rm(III$_{\text{B}}$)}$ by $\sigma_2$ on the right and $\sigma_1^{-1}$ on the left,
and taking braid identities into account, we get the relation
\[
a^{-1}\cdot \sigma_2\sigma_1^{-1}\sigma_2
+ a\cdot \sigma_1^{-1}\sigma_2\sigma_1^{-1}
- a^{-1}\cdot \sigma_1^{-1}\sigma_2\sigma_1
- a\cdot \sigma_1\sigma_2^{-1}\sigma_1^{-1}
\sim 0.
\]
Then $\sigma_2\sigma_1^{-1}\sigma_2$ is equivalent to a linear combination of
braids of the form $\sigma_1^{\pm1}\sigma_2^{\pm1}\sigma_1^{\pm1}$.
So the case $\sigma_2\sigma_1^{-1}\sigma_2$ is verified.
{\it The case $\sigma_2\sigma_1^{-1}\sigma_2^2$ }:
Multiplying the previous relation by $\sigma_2$ on the right,
and taking braid identities into account, we see that
\[
a^{-1}\cdot \sigma_2\sigma_1^{-1}\sigma_2^2
+ a\cdot \sigma_1^{-1}(\sigma_2\sigma_1^{-1}\sigma_2)
- a^{-1}\cdot \sigma_2\sigma_1
- a\cdot \sigma_1^2\sigma_2^{-1}\sigma_1^{-1}
\sim 0.
\]
Similar to the above case, this reduces $\sigma_2\sigma_1^{-1}\sigma_2^2$
to the verified case $\sigma_2\sigma_1^{-1}\sigma_2$.
{\it The case $\sigma_2\sigma_1^2\sigma_2$ }:
Multiplying $\text{\rm(III$_{\text{B}}$)}$ on the right by $\sigma_1\sigma_2\sigma_1$, we get
\[
a^{-1}\cdot \sigma_1\sigma_2^2\sigma_1
+ a\cdot \sigma_2^2
- a^{-1}\cdot \sigma_2\sigma_1^2\sigma_2
- a\cdot \sigma_1^2
\sim 0.
\tag*{$\text{\rm(III$'_{\text{B}}$)}$}
\]
This verifies the case $\sigma_2\sigma_1^2\sigma_2$.
{\it The case $\sigma_2\sigma_1^2\sigma_2^2$ }:
Multiplying $\text{\rm(III$'_{\text{B}}$)}$ by $\sigma_2$ on the right,
we get
\[
a^{-1}\cdot \sigma_1^2\sigma_2\sigma_1^2
+ a\cdot \sigma_2^3
- a^{-1}\cdot \sigma_2\sigma_1^2\sigma_2^2
- a\cdot \sigma_1^2\sigma_2
\sim 0.
\]
The case $\sigma_2\sigma_1^2\sigma_2^2$ is also verified.
We have verified all 9 cases.
Modulo $\text{\rm(II$_\text{B}$)}$ we can assume $\ell'\in\{0,\pm1,2\}$.
Thus Lemma~\ref{lem:key_reduction} is proved.
The inductive proof of Lemma~\ref{lem:reduction} is now complete.
\end{proof}
The resulting $\Lambda$-linear combination of braids of the form
$\alpha\sigma_{n-1}^k \gamma$ with $\alpha,\gamma\in B_{n-1}$ in the Lemma is not unique,
but the inductive proof gives us a recursive algorithm to find one.
To compare the ideal $\mathfrak I_n$ with the relator ideal $\mathfrak R_n$ of Section~\ref{sec:relators}, we have
\begin{prop}
$\mathfrak I_n\subset \mathfrak R_n$ but
$\mathfrak I_n\neq \mathfrak R_n$.
\end{prop}
\begin{proof}
The inclusion is easy.
Indeed, $(\text{I}_\text{B})$ is in the relator ideal $\mathfrak R_n$, and
\begin{gather*}
\text{\rm(II$_\text{B}$)}=
\text{\rm(I$_\text{B}$)}^2 +2z\cdot \text{\rm(I$_\text{B}$)} ,
\\
\text{\rm(III$_\text{B}$)}=
\sigma_2^{-1}\cdot \text{\rm(I$_\text{B}$)} \cdot\sigma_2 -\sigma_2\cdot\text{\rm(I$_\text{B}$)} \cdot\sigma_2^{-1} .
\end{gather*}
So both $\text{\rm(II$_\text{B}$)}$ and $\text{\rm(III$_\text{B}$)}$ are in $\mathfrak R_n$.
Therefore $\mathfrak I_n\subset \mathfrak R_n$.
To show they are not equal, we need the notion of homogeneity.
Each $n$-braid $\beta$ has an \emph{underlying permutation} of $\{1,\dots,n\}$, denoted $i\mapsto i^{\beta}$,
where $i^{\beta}$ is the position of the $i$-th strand at the bottom of $\beta$.
In this way the braid group $B_n$ projects onto the symmetric group $\mathfrak S_n$.
An element of $\Lambda B_n$ is called \emph{homogeneous} if all its terms
(with nonzero coefficients) have the same underlying permutation.
As a $\Lambda$-module, $\Lambda B_n$ splits into a direct sum
according to underlying permutations of braids.
Under this splitting, every element of $\Lambda B_n$ decomposes into a sum of its
\emph{homogeneous components}.
Since $\text{\rm(II$_\text{B}$)}$ and $\text{\rm(III$_\text{B}$)}$ are homogeneous,
the ideal $\mathfrak I_n\subset\Lambda B_n$ is generated by homogeneous elements.
Then every homogeneous component of any element of $\mathfrak I_n$ is also in $\mathfrak I_n$.
Now the relator $\text{\rm(I$_\text{B}$)}\in \mathfrak R_n$ has a homogeneous component $-z\cdot e$
which is not a relator.
Hence $\text{\rm(I$_\text{B}$)}$ is not in $\mathfrak I_n$.
Thus $\mathfrak I_n$ is strictly smaller than $\mathfrak R_n$.
\end{proof}
\section{Stabilizations}
\label{sec:stabilizations}
Suppose a braid $\beta\in B_n$ is written as a word in the standard generators
$\sigma_1$, $\sigma_2$, \dots, $\sigma_{n-1}$.
The same word $\beta$ gives a braid in $B_{n+k}$ for any $k\ge0$.
Thus $B_n$ is standardly embedded in $B_{n+k}$.
However, when talking about a closed braid $\wh\beta$, the number of strands in $\beta$ does matter.
We shall use the notation $[\beta]_n$ to emphasize that $\beta$ is regarded as an $n$-braid,
and use $[\beta]_n^{\;\wh{}}$ for its closure.
For example, $[\beta]_{n+1}^{\;\wh{}}$ adds a free circle to $[\beta]_n^{\;\wh{}}$.
The Markov move says $[\beta\sigma_n^{\pm1}]_{n+1}^{\;\wh{}}$ is isotopic to $[\beta]_n^{\;\wh{}}$.
For a braid $\beta\in B_n$ and an integer $k\ge0$, we shall use
$\beta^{\triangleright k} \in B_{n+k}$ to denote the $k$-th shifted version of $\beta$, i.e.,
the braid obtained from the word $\beta$ by replacing each generator $\sigma_i$ with $\sigma_{i+k}$.
Its closure $[\beta^{\triangleright k}]_{n+k}^{\;\wh{}}$ is isotopic to
$[\beta]_{n+k}^{\;\wh{}}$.
Suppose $\beta,\beta'\in B_n$ and $\gamma\in B_p$.
Observe from the diagram defining braid closure that
the closed braid $[\beta\sigma_n^{\pm1} \gamma^{\triangleright n}\beta']_{n+p}^{\;\wh{}}$
is isotopic to $[\beta\gamma^{\triangleright (n-1)}\beta']_{n+p-1}^{\;\wh{}}$
(which is in fact a connected sum of oriented links $[\beta\beta']_n^{\;\wh{}}$ and $[\gamma]_p^{\;\wh{}}$).
By an abuse of language, we will call this a \emph{Markov move}.
If $\beta'$ brings the $n$-th position at its top to the same position at its bottom,
then $[\beta\gamma^{\triangleright (n-1)}\beta']_{n+p-1}^{\;\wh{}}$
is isotopic to $[\beta\beta'\gamma^{\triangleright (n-1)}]_{n+p-1}^{\;\wh{}}$.
We will refer to it as a \emph{slide} move
(in the connected sum,
sliding $[\gamma]_p^{\;\wh{}}$ down the last strand of $\beta'$).
\begin{lem}
\label{lem:stabilization3}
Assume that $P_L \in \mathbb Z[a^{\pm1},z^{\pm1}]$ is
an invariant of oriented links that satisfies skein relations \textup{(II)} and \textup{(III)}.
Then for $\beta\in B_n$ and $\gamma\in B_p$ we have
\[
(1+z^2-a^2)\cdot P\left( [\beta\gamma^{\triangleright n}]_{n+p}^{\;\wh{}} \right) =
(a^{-2}-1)\cdot P\left( [\beta\sigma_{n}^2\gamma^{\triangleright n}]_{n+p}^{\;\wh{}} \right) .
\]
\end{lem}
\begin{proof}
The braid form of axioms (II) and (III) are the relators
(II$_{\text{B}}$) and (III$_{\text{B}}$), respectively.
Multiplying (III$_{\text{B}}$) by $\sigma_2\sigma_1^{-1}$ on the right we get another relator
\[
a^{-1}\cdot \sigma_2^{-2}\sigma_1^2\sigma_2
+ a\cdot \sigma_2\sigma_1^{-2}
- a^{-1}\cdot \sigma_2
- a\cdot \sigma_1^2\sigma_2^{-1}\sigma_1^{-2} .
\]
It gives us an equality between the $P$'s of closed $(n+p+1)$-braids:
\begin{align*}
&a^{-1}\cdot P\left( [\beta(\sigma_{n+1}^{-2}\sigma_{n}^2\sigma_{n+1})\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\wh{}} \right)
+ a\cdot P\left( [\beta(\sigma_{n+1}\sigma_{n}^{-2})\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\wh{}} \right)
\\
& - a^{-1}\cdot P\left( [\beta\sigma_{n+1}\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\wh{}} \right)
- a\cdot P\left( [\beta(\sigma_{n}^2\sigma_{n+1}^{-1}\sigma_{n}^{-2})\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\wh{}} \right)
=0 .
\end{align*}
These closed braids can be simplified via isotopy moves
(c=conjugacy, M=Markov and s=slide):
\begin{align*}
[\beta\sigma_{n+1}^{-2}\sigma_{n}^2\sigma_{n+1}\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\wh{}}
& \overset{\text{c}}\rightsquigarrow
[\beta\sigma_{n}^2\sigma_{n+1}\gamma^{\triangleright (n+1)}\sigma_{n+1}^{-2}]_{n+p+1}^{\;\wh{}}
\\
& \overset{\text{s}}{\rightsquigarrow}
[\beta\sigma_{n}^2\sigma_{n+1}^{-1}\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\wh{}}
\overset{\text{M}}{\rightsquigarrow}
[\beta\sigma_{n}^2\gamma^{\triangleright n}]_{n+p}^{\;\wh{}} ;
\\
[\beta\sigma_{n+1}\sigma_{n}^{-2}\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\wh{}}
& \overset{\text{M}}{\rightsquigarrow}
[\beta\gamma^{\triangleright n}\sigma_{n}^{-2}]_{n+p}^{\;\wh{}}
\overset{\text{s}}\rightsquigarrow
[\beta\sigma_{n}^{-2}\gamma^{\triangleright n}]_{n+p}^{\;\wh{}} ;
\\
[\beta\sigma_{n+1}\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\wh{}}
& \overset{\text{M}}{\rightsquigarrow}
[\beta\gamma^{\triangleright n}]_{n+p}^{\;\wh{}} ;
\\
[\beta\sigma_{n}^2\sigma_{n+1}^{-1}\sigma_{n}^{-2}\gamma^{\triangleright (n+1)}]_{n+p+1}^{\;\wh{}}
& \overset{\text{M}}{\rightsquigarrow}
[\beta\sigma_{n}^2\gamma^{\triangleright n}\sigma_{n}^{-2}]_{n+p}^{\;\wh{}}
\overset{\text{s}}{\rightsquigarrow}
[\beta\gamma^{\triangleright n}]_{n+p}^{\;\wh{}} .
\end{align*}
Since $P_L$ is isotopy invariant, the above equality becomes
\begin{gather*}
a^{-1}\cdot P\left( [\beta\sigma_{n}^2\gamma^{\triangleright n}]_{n+p}^{\;\wh{}} \right)
+ a\cdot P\left( [\beta\sigma_{n}^{-2}\gamma^{\triangleright n}]_{n+p}^{\;\wh{}} \right)
= (a^{-1}+a)\cdot P\left( [\beta\gamma^{\triangleright n}]_{n+p}]_{n+1}^{\;\wh{}} \right) .
\\
\intertext{Comparing it with the equality (from (II$_{\text{B}}$))}
a^{-2}\cdot P\left( [\beta\sigma_{n}^2\gamma^{\triangleright n}]_{n+p}^{\;\wh{}} \right)
+ a^2\cdot P\left( [\beta\sigma_{n}^{-2}\gamma^{\triangleright n}]_{n+p}^{\;\wh{}} \right)
= (2+z^2)\cdot P\left( [\beta\gamma^{\triangleright n}]_{n+p}^{\;\wh{}} \right) ,
\end{gather*}
we get the desired conclusion.
\end{proof}
\begin{cor}
\label{cor:stabilization}
Under the assumption of the above lemma, the following two relations are equivalent to each other:
\begin{gather*}
P
\begin{pmatrix}
\includegraphics[height=1.2cm]{fig_012/IO-0}
\end{pmatrix}
=z^{-1}(a^{-1}-a)\cdot
P
\begin{pmatrix}
\;\;
\includegraphics[height=1.2cm]{fig_012/I-0}
\;\;
\end{pmatrix} ;
\tag*{\rm(IO)}
\\
P
\begin{pmatrix}
\includegraphics[height=1.2cm]{fig_012/Phi+-0}
\end{pmatrix}
=az^{-1}(1+z^2-a^2)\cdot
P
\begin{pmatrix}
\;\;
\includegraphics[height=1.2cm]{fig_012/I-0}
\;\;
\end{pmatrix} .
\tag*{($\Phi$)}
\end{gather*}
\end{cor}
\begin{proof}
The braid form of these two relations are, respectively,
\begin{alignat}{2}
P\left( [\beta]_{n+1}^{\;\wh{}} \right) &= z^{-1}(a^{-1}-a)\cdot P\left( [\beta]_{n}^{\;\wh{}} \right)
&&\quad \text{for any braid } \beta\in B_n;
\tag*{(IO$_{\text{B}}$)}
\\
P\left( [\beta\sigma_{n}^2]_{n+1}^{\;\wh{}} \right) &= az^{-1}(1+z^2-a^2)\cdot P\left( [\beta]_{n}^{\;\wh{}} \right)
&&\quad \text{for any braid } \beta\in B_n.
\tag*{($\Phi_{\text{B}}$)}
\end{alignat}
They are equivalent to each other by the above lemma with $[\gamma]_p:=[e]_1$.
\end{proof}
There is a parallel statement for Jones polynomial:
\begin{cor}
\label{cor:stabilization_Jones}
Assume that $V_L \in \mathbb Z[t^{\pm\frac12}]$ is
an invariant of oriented links that satisfies skein relations \textup{(II$_V$)} and \textup{(III$_V$)}.
Then the following two relations are equivalent to each other:
\begin{gather*}
V
\begin{pmatrix}
\includegraphics[height=1.2cm]{fig_012/IO-0}
\end{pmatrix}
=-(t^{\frac12}+t^{-\frac12})\cdot
V
\begin{pmatrix}
\;\;
\includegraphics[height=1.2cm]{fig_012/I-0}
\;\;
\end{pmatrix} ;
\tag*{\rm(IO$_V$)}
\\
V
\begin{pmatrix}
\includegraphics[height=1.2cm]{fig_012/Phi+-0}
\end{pmatrix}
=-t^{\frac32}(t+t^{-1})\cdot
V
\begin{pmatrix}
\;\;
\includegraphics[height=1.2cm]{fig_012/I-0}
\;\;
\end{pmatrix} .
\tag*{($\Phi_V$)}
\end{gather*}
\end{cor}
For the Alexander-Conway polynomial, we have:
\begin{cor}
\label{cor:stabilization_Alexander-Conway}
Assume that $\Delta_L \in \mathbb Z[t^{\pm\frac12}]$ is
an invariant of oriented links that satisfies skein relations \textup{(II$_\Delta$)} and \textup{(III$_\Delta$)}.
Then $\Delta(L)=0$ for any split link $L$.
In particular, the following relation holds true:
\[
\Delta
\begin{pmatrix}
\includegraphics[height=1.2cm]{fig_012/IO-0}
\end{pmatrix}
=0 .
\tag*{\rm(IO$_\Delta$)}
\]
\end{cor}
\begin{proof}
For links $L_1=[\beta]_n^{\;\wh{}}$ and $L_2=[\gamma]_p^{\;\wh{}}$,
the split link $L=L_1\sqcup L_2=[\beta\gamma^{\triangleright n}]_{n+p}^{\;\wh{}}$.
Then apply Lemma~\ref{lem:stabilization3}
with substitutions $a\mapsto 1$ and $z\mapsto (t^{\frac12}-t^{-\frac12})$.
\end{proof}
\section{Proof of Theorems \ref{thm:HOMFLY}--\ref{thm:Alexander-Conway}}
\label{sec:proof_Theorems}
We shall focus on Theorem~\ref{thm:HOMFLY}, then remark on the other two.
\begin{proof}[Proof of Theorem~\ref{thm:HOMFLY}]
Let us forget about the original definition of the skein polynomial, and
regard the symbol $P_L$ as a well-defined invariant of oriented links
which satisfies the axioms (II), (III), (IO) and (O).
By Corollary~\ref{cor:stabilization}, $P_L$ also satisfies axiom ($\Phi$).
We shall show that such an invariant $P_L$ is computable,
hence uniquely determined.
It suffices to prove the following claim by induction on $n$.
\subsection* {Inductive Claim($n$)}
For every $n$-braid $\beta\in B_n$,
$P \left( [\beta]_n^{\;\wh{}} \right)$ is computable.
\vspace{1ex}
When $n=1$, Claim($1$) is true because there is only one $1$-braid $[e]_1$.
Its closure is the trivial knot, whose $P$ must be $1$ by axiom (O).
Now assume inductively that Claim($n-1$) is true, we shall prove that Claim($n$) is also true.
Suppose $\beta$ is an $n$-braid.
By Lemma~\ref{lem:reduction}, the braid $\beta\in B_n$ is equivalent to (in a computable way)
a $\Lambda$-linear combination of
braids of the form $\alpha\sigma_{n-1}^k \gamma$ with $\alpha,\gamma\in B_{n-1}$ and $k\in\{0,\pm1,2\}$.
By Example~\ref{exam:relator_vs_relation} the (mod $\mathfrak I_n$) equivalence preserves
the $P$ of closure of braids.
So $P \left( [\beta]_n^{\;\wh{}} \right)$ is a $\Lambda$-linear combination (with computable coefficients) of
$P \left( [\alpha\sigma_{n-1}^k \gamma]_n^{\;\wh{}} \right)$'s.
For $k\in\{\pm1,0,2\}$, respectively, we have
\begin{alignat*}{2}
P \left( [\alpha\sigma_{n-1}^{\pm1} \gamma]_n^{\;\wh{}} \right)
&= P \left( [\alpha \gamma]_{n-1}^{\;\wh{}} \right)
&&\qquad\text{by isotopy},
\\
P \left( [\alpha\sigma_{n-1}^0 \gamma]_n^{\;\wh{}} \right)
&= z^{-1}(a^{-1}-a) \cdot P \left( [\alpha \gamma]_{n-1}^{\;\wh{}} \right)
&&\qquad\text{by (IO)},
\\
P \left( [\alpha\sigma_{n-1}^2 \gamma]_n^{\;\wh{}} \right)
&= az^{-1}(1+z^2-a^2) \cdot P \left( [\alpha \gamma]_{n-1}^{\;\wh{}} \right)
&&\qquad\text{by $(\Phi)$}.
\end{alignat*}
Since $P \left( [\alpha \gamma]_{n-1}^{\;\wh{}} \right)$
is computable by the inductive hypothesis Claim($n-1$),
we see $P \left( [\alpha\sigma_{n-1}^k \gamma]_n^{\;\wh{}} \right)$ is also computable.
Thus Claim($n$) is proved.
The induction on $n$ is now complete.
Hence $P$ is computable for every closed braid.
\end{proof}
\begin{rem}
The induction above, together with the reduction argument of Section \ref{sec:key_lem},
provides a recursive algorithm for computing $P \left( [\beta]_n^{\;\wh{}} \right)$.
\end{rem}
\begin{rem}
A remarkable feature of this algorithm is that
it never increases the number of components of links.
In fact, all the reductions in Section~\ref{sec:key_lem} are by
axioms (II) and (III) which respect the components,
while in this Section, components could get removed
but never added, by axioms (IO) and ($\Phi$).
So if we start off with a knot, we shall always get knots along the way,
the axioms (IO) and ($\Phi$) becoming irrelevant.
This observation works even for links with any given number of components,
once we set up a suitable normalization.
Hence the Theorem~\ref{thm:HOMFLY_knot}.
\end{rem}
\begin{rem}
For the Jones polynomial, the proof above works well with the substitutions
$a\mapsto t$ and $z\mapsto (t^{\frac12}-t^{-\frac12})$.
\end{rem}
\begin{rem}
The case of Alexander-Conway polynomial is only slightly different.
Corollary~\ref{cor:stabilization_Alexander-Conway} says (IO$_\Delta$) is a
consequence of axioms (II$_\Delta$) and (III$_\Delta$),
and ($\Phi_\Delta$) is taken as an axiom.
So the proof above also works through with the substitutions
$a\mapsto 1$ and $z\mapsto (t^{\frac12}-t^{-\frac12})$.
Actually, the argument in Section~\ref{sec:stabilizations} can be adapted to
work for Conway potential function of colored links,
to the effect that in \cite[Main Theorem]{J1},
the relation (IO) is a consequence of axioms (II) and (III)
hence can be removed from the list of axioms.
\end{rem}
\section{Proof of Theorems \ref{thm:HOMFLY_knot}--\ref{thm:Alexander-Conway_knot}}
\label{sec:proof_Theorems2}
Suppose $\mu$ is a given positive integer.
Regard $P_L$ as a well-defined invariant of oriented $\mu$-compo\-nent links
which satisfies the axioms (II), (III) and (U).
We shall temporarily expand the ring $\Lambda:=\mathbb Z[a^{\pm1},z^{\pm1}]$,
where the invariant $P_L$ takes value,
to $\wt\Lambda:=\mathbb Z[a^{\pm1},z^{\pm1},(a^{-2}-1)^{-1}]$,
to allow fractions with denominator a power of $(a^{-2}-1)$.
We shall show that such an invariant $P_L$ is computable,
hence uniquely determined.
It is the normalization (U) that brings the value $P_L$ back into the original $\Lambda$.
\begin{lem}
\label{lem:reduction_HOMFLY}
Suppose $\beta\in B_n$, $p\ge0$, and $[\beta]_{n+p}^{\;\wh{}}$ has $\mu$ components.
If $n>1$, then $P \left( [\beta]_{n+p}^{\;\wh{}} \right)$ is
computable as a $\wt\Lambda$-linear combination of terms
of the form $P \left( [\beta']_{n'+p'}^{\;\wh{}} \right)$, each with $\mu$ components,
$\beta'\in B_{n'}$, $n'<n$, $p'\ge p$, and
the $(a^{-2}-1)^{-1}$-exponent of the corresponding coefficient is at most $p'-p$.
\end{lem}
\begin{proof}
By Lemma~\ref{lem:reduction}, the braid $\beta\in B_n$ is equivalent to (in a computable way)
a $\Lambda$-linear combination of
braids of the form $\alpha\sigma_{n-1}^k \gamma$ with $\alpha,\gamma\in B_{n-1}$ and $k\in\{0,\pm1,2\}$.
So $P \left( [\beta]_{n+p}^{\;\wh{}} \right)$ is a $\wt\Lambda$-linear combination (with computable coefficients) of
$P \left( [\alpha\sigma_{n-1}^k \gamma]_{n+p}^{\;\wh{}} \right)$'s.
For $k\in\{\pm1,0,2\}$, respectively, we have
\begin{alignat*}{2}
P \left( [\alpha\sigma_{n-1}^{\pm1} \gamma]_{n+p}^{\;\wh{}} \right)
&= P \left( [\gamma\alpha \sigma_{n-1}^{\pm1}]_{n+p}^{\;\wh{}} \right)
&&\quad\text{by braid conjugation},
\\
&= P \left( [\gamma\alpha ]_{(n-1)+p}^{\;\wh{}} \right)
&&\quad\text{by Markov move},
\\
P \left( [\alpha\sigma_{n-1}^0 \gamma]_{n+p}^{\;\wh{}} \right)
&= P \left( [\alpha \gamma]_{(n-1)+(p+1)}^{\;\wh{}} \right)
&&\quad\text{obvious},
\\
P \left( [\alpha\sigma_{n-1}^2 \gamma]_{n+p}^{\;\wh{}} \right)
&= P \left( [\gamma\alpha\sigma_{n-1}^2]_{n+p}^{\;\wh{}} \right)
&&\quad\text{by braid conjugation}
\\
&= \frac{1+z^2-a^2}{a^{-2}-1} \cdot P \left( [\gamma \alpha]_{(n-1)+(p+1)}^{\;\wh{}} \right)
&&\quad\text{by Lemma~\ref{lem:stabilization3}}.
\end{alignat*}
The $P$'s on the right hand sides satisfy the required conditions.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:HOMFLY_knot}]
Suppose a link $L$ with $\mu$ components is presented as $[\beta]_{n}^{\;\wh{}}=[\beta]_{n+0}^{\;\wh{}}$.
Apply Lemma~\ref{lem:reduction_HOMFLY} repeatedly until no such reduction is possible.
Then $P \left( [\beta]_{n+0}^{\;\wh{}} \right)$ is computed as a $\wt\Lambda$-linear combination of
terms $P \left( [\beta']_{n'+p'}^{\;\wh{}} \right)$, each with $n'=1$ hence $\beta'=[e]_1$, such that
\begin{itemize}
\item[(1)] every $[\beta']_{n'+p'}^{\;\wh{}}=[e]_{1+p'}^{\;\wh{}}$ has $\mu$ components, hence $p'=\mu-1$,
and $[\beta']_{n'+p'}^{\;\wh{}} =[e]_\mu^{\;\wh{}} =U_\mu$;
and
\item[(2)] the $(a^{-2}-1)^{-1}$-exponent of every coefficient is at most $p'-0=\mu-1$.
\end{itemize}
Therefore, $P (L)$ is computable and, by axiom (U),
every term $P \left( [\beta']_{n'+p'}^{\;\wh{}} \right) $ has a factor $(a^{-2}-1)^{\mu-1}$
that can cancel the $(a^{-2}-1)^{-1}$-exponent in its coefficient,
so $P (L) \in \Lambda$.
\end{proof}
Theorem~\ref{thm:Jones_knot} can be proved similarly, but
Theorem~\ref{thm:Alexander-Conway_knot} needs modifications.
Define $\delta_p:=\sigma_1^2\sigma_2^2\dots\sigma_{p-1}^2 \in B_p$ whose closure
$[\delta_p]_p^{\;\wh{}}$ is the oriented $p$-component chain $C_p$.
\begin{lem}
\label{lem:reduction_Alexander-Conway}
Suppose $\beta\in B_{n+1}$, $p\ge1$, and $[\beta\delta_p^{\triangleright n}]_{n+p}^{\;\wh{}}$
has $\mu$ components.
If $n>0$, then $\Delta \left( [\beta\delta_p^{\triangleright n}]_{n+p}^{\;\wh{}} \right)$ is
computable as a $\mathbb Z[t^{\pm\frac12}]$-linear combination of terms
of the form $\Delta \left( [\beta'\delta_{p'}^{\triangleright n'}]_{n'+p'}^{\;\wh{}} \right)$,
each with $\mu$ components, $\beta'\in B_{n'+1}$, $n'<n$ and $p'\ge p$.
\end{lem}
\begin{proof}
By Lemma~\ref{lem:reduction} (with substitutions $a\mapsto 1$ and $z\mapsto (t^{\frac12}-t^{-\frac12})$),
the braid $\beta\in B_{n+1}$ is equivalent to (in a computable way)
a $\mathbb Z[t^{\pm\frac12}]$-linear combination of
braids of the form $\alpha\sigma_{n}^k \gamma$ with $\alpha,\gamma\in B_{n}$ and $k\in\{0,\pm1,2\}$.
Multiplication by $\delta_p^{\triangleright n}$ makes the braid
$\beta\delta_p^{\triangleright n}$ equivalent to a linear combination of braids
of the form $\alpha\sigma_{n}^k \gamma\delta_p^{\triangleright n}$,
so $\Delta \left( [\beta\delta_p^{\triangleright n}]_{n+p}^{\;\wh{}} \right)$ is
computed as a linear combination of the
$\Delta \left( [\alpha\sigma_{n}^k \gamma\delta_p^{\triangleright n}]_{n+p}^{\;\wh{}} \right)$'s.
For $k\in\{\pm1,0,2\}$, respectively, we have
\begin{alignat*}{2}
\Delta \left( [\alpha\sigma_{n}^{\pm1} \gamma\delta_p^{\triangleright n}]_{n+p}^{\;\wh{}} \right)
&= \Delta \left( [\gamma\alpha\sigma_{n}^{\pm1} \delta_p^{\triangleright n}]_{n+p}^{\;\wh{}} \right)
&&\quad\text{by braid conjugacy},
\\
&= \Delta \left( [\gamma\alpha \delta_p^{\triangleright (n-1)}]_{n+p-1}^{\;\wh{}} \right)
&&\quad\text{by a Markov move},
\\
\Delta \left( [\alpha\sigma_{n}^0 \gamma\delta_p^{\triangleright n}]_{n+p}^{\;\wh{}} \right)
&= 0
&&\quad\text{by Corollary~\ref{cor:stabilization_Alexander-Conway}},
\\
\Delta \left( [\alpha\sigma_{n}^2 \gamma\delta_p^{\triangleright n}]_{n+p}^{\;\wh{}} \right)
&= \Delta \left( [\gamma\alpha\sigma_{n}^2 \delta_p^{\triangleright n}]_{n+p}^{\;\wh{}} \right)
&&\quad\text{by braid conjugacy}
\\
&= \Delta \left( [\gamma\alpha \delta_{p+1}^{\triangleright (n-1)}]_{n+p}^{\;\wh{}} \right)
&&\quad\text{by definition}.
\end{alignat*}
The $\Delta$'s on the right hand sides are in the desired form with $n'=n-1$
and with $\mu$ components.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:Alexander-Conway_knot}]
Suppose a link $L$ with $\mu$ components is presented as
$[\beta]_{n+1}^{\;\wh{}}=[\beta\delta_1^{\triangleright n}]_{n+1}^{\;\wh{}}$.
Apply Lemma~\ref{lem:reduction_Alexander-Conway} repeatedly until no such reduction is possible.
Then $\Delta \left( [\beta\delta_1^{\triangleright n}]_{n+1}^{\;\wh{}} \right)$
is computed as a $\mathbb Z[t^{\pm\frac12}]$-linear combination of
terms $\Delta \left( [\beta'\delta_{p'}^{\triangleright n'}]_{n'+p'}^{\;\wh{}} \right)$,
with $n'=0$.
Hence each $\beta'=[e]_1$, $p'=\mu$ the number of components,
and each $[\beta'\delta_{p'}^{\triangleright n'}]_{n'+p'}^{\;\wh{}}
= [\delta_\mu]_\mu^{\;\wh{}} = C_\mu$.
Therefore $\Delta (L)$ is computable and moreover, by axiom (C$_\Delta$),
divisible by $\Delta(C_\mu) = (t^{\frac12}-t^{-\frac12})^{\mu-1}$.
\end{proof}
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TITLE: Question about $\mathbb R^n$ and mathematical space(s)
QUESTION [1 upvotes]: In the context of Euclidean and real coordinate spaces ($\mathbb R^n$), does n (or $\mathbb N$) include 0?
$\mathbb R^1$ is the 1-dimensional real number line
$\mathbb R^2$ is the 2-dimensional coordinate plane
$\mathbb R^3$ is the 3-dimensional coordinate space
And so on…
There are of course higher dimensional spaces, for example it's mentioned in this video from Khan Academy:
https://www.youtube.com/watch?v=lCsjJbZHhHU&t=5m40s
But is it possible to have lower dimensional spaces? As in $\mathbb R^0$. I am unsure because not all authors include 0 in the set of natural numbers.
Professor Norman J. Wildberger briefly mentions 0-dimensional spaces here, but this is in the context of a “theory of mathematical space which doesn’t involve the infinities that are usually associated with a real number treatment” as he puts it:
http://www.youtube.com/watch?v=2WH6NTciV2Q&t=3m0s
And to mention a literary source, there’s “Pointland” in the novella Flatland: A Romance of Many Dimensions by Edwin A. Abbott.
Any recommendations as to further reading would also be greatly appreciated. Thank you
REPLY [1 votes]: Sure! The notation $\mathbb R^n$ really just means "the set of $n$-tuples of real numbers" - that is, ordered lists $(a_1,a_2,\ldots,a_n)$ where each $a_i$ is a real number. By this reasoning $\mathbb R^0$ is just the set of ordered lists of $0$ real numbers - and there is exactly one such list of zero real numbers: $()$. So, $\mathbb R^0$ is just a single point and it happens to be a vector space of dimension zero.
More generally, if you want to write $\mathbb R^n$, all that $n$ needs to do is specify the size of a set* - and zero is a perfectly acceptable value here. The exponent doesn't even need to be a natural number - you can happily talk about $\mathbb R^{\mathbb N}$ as the set of sequences $(a_1,a_2,a_3,\ldots)$ with countably many terms (or, more formally, of functions $\mathbb N\rightarrow\mathbb R$) or even do this with larger sets in the exponent (then meaning "an sequence of real numbers indexed by that set").
(*This idea is known as cardinality in general; the purpose of the natural numbers in this context is that they count how big finite collections are, which is a good hint that zero is a valid value, since collections can be empty)
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Six boxes of common cold relief, United States,.
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\begin{document}
\begin{frontmatter}
\title{Analytic solution for grand confluent hypergeometric function}
\author{Yoon Seok Choun
}
\ead{yoon.choun@baruch.cuny.edu; ychoun@gc.cuny.edu; ychoun@gmail.com}
\address{Baruch College, The City University of New York, Natural Science Department, A506, 17 Lexington Avenue, New York, NY 10010}
\begin{abstract}
In Ref.\cite{Chou2012a} I construct an approximative solution of the power series expansion in closed forms of Grand Confluent Hypergeometric (GCH) function only up to one term of $A_n$'s. And I obtain normalized constants and orthogonal relations of GCH function.
In this paper I will apply three term recurrence formula (3TRF) \cite{Chou2012b} to the power series expansion in closed forms of GCH function (for infinite series and polynomial which makes $B_n$ term terminated) including all higher terms of $A_n$'s.
In general most of well-known special function with two recursive coefficients only has one eigenvalue for the polynomial case. However this new function with three recursive coefficients has infinite eigenvalues that make $B_n$'s term terminated at specific value of index $n$ because of 3TRF \cite{Chou2012b}.
This paper is 9th out of 10 in series ``Special functions and three term recurrence formula (3TRF)''. See section 6 for all the papers in the series. The previous paper in series deals with generating functions of Lame polynomial in the Weierstrasss form\cite{Chou2012h}. The next paper in the series describes the integral formalism and the generating function of GCH function\cite{Chou2012j}.
\end{abstract}
\begin{keyword}
Biconfluent Heun Equation, Three term recurrence formula, Asymptotic expansion
\PACS{02.30.Hq, 02.30.Ik, 02.30.Gp, 03.65.Ge, 03.65.-w}
\MSC{33E30, 34A30, 34B30, 34E05}
\end{keyword}
\end{frontmatter}
\section{\label{sec:level1}Introduction}
Biconfluent Heun (BCH) function, a confluent form of Heun function\cite{Heun1889,Ronv1995}, is the special case of Grand Confluent Hypergeometric (GCH)\cite{Chou2012a}\footnote{For the canonical form of BCH equation \cite{Ronv1995}, replace $\mu $, $\varepsilon $, $\nu $, $\Omega $ and $\omega $ by $-2$, $-\beta $, $ 1+\alpha $, $\gamma -\alpha -2 $ and $ 1/2 (\delta /\beta +1+\alpha )$ in (\ref{eq:1}). For DLFM version \cite{NIST} or in ref.\cite{Slavy2000}, replace $\mu $ and $\omega $ by 1 and $-q/\varepsilon $ in (\ref{eq:1}).}: this has a regular singularity at $x=0$, and an irregular singularity at $\infty$ of rank 2. For example, BCH function is included in the radial Schr$\ddot{\mbox{o}}$dinger equation with rotating harmonic oscillator and a class of confinement potentials: recently it's started to appear in theoretical modern physics \cite{Slav1996,Ralk2002,Kand2005,Hortacsu:2011rr,Arri1991}.
In \cite{Chou2012c,Chou2012d}, I construct the power series expansion in closed form and its integral representation of Heun function by applying 3TRF. Heun function is applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, addition of three quantum spins, solutions of the Schr$\ddot{\mbox{o}}$dinger equation of quantum mechanics. \cite{Hortacsu:2011rr,Take2008,Suzu1999,Suzu1998}
In Ref.\cite{Chou2012a} I show an analytic solution of GCH function only up to one term of $A_n$'s. In this paper I construct the power series expansion of GCH equation in closed forms and asymptotic behaviors including all higher terms of $A_n$'s by applying 3TRF \cite{Chou2012b}.
\begin{equation}
x \frac{d^2{y}}{d{x}^2} + \left( \mu x^2 + \varepsilon x + \nu \right) \frac{d{y}}{d{x}} + \left( \Omega x + \varepsilon \omega \right) y = 0
\label{eq:1}
\end{equation}
(\ref{eq:1}) is a Grand Confluent Hypergeometric (GCH) differential equation where $\mu$, $\varepsilon$, $\nu $, $\Omega$ and $\omega$ are real or imaginary parameters.\cite{Chou2012a} It has a regular singularity at the origin and an irregular singularity at the in¯infinity. Biconfluent Heun Equation is derived, the special case of GCH equation, by putting coefficients $\mu =1$ and $\omega =-q/\varepsilon $.\cite{NIST}
$y(x)$ has a series expansion of the form
\begin{equation}
y(x)= \sum_{n=0}^{\infty } c_n x^{n+\lambda }
\label{eq:2}
\end{equation}
where $\lambda$ is an indicial root. Plug (\ref{eq:2}) into (\ref{eq:1}).
\begin{equation}
c_{n+1}=A_n \;c_n +B_n \;c_{n-1} \hspace{1cm};n\geq 1
\label{eq:3}
\end{equation}
where,
\begin{subequations}
\begin{equation}
A_n = -\frac{\varepsilon (n+\omega +\lambda )}{(n+1+\lambda )(n+\nu +\lambda )}
\label{eq:4a}
\end{equation}
\begin{equation}
B_n = -\frac{\Omega +\mu (n-1+\lambda )}{(n+1+\lambda )(n+\nu +\lambda )}
\label{eq:4b}
\end{equation}
\begin{equation}
c_1= A_0 \;c_0
\label{eq:4c}
\end{equation}
\end{subequations}
We have two indicial roots which are $\lambda = 0$ and $ 1-\nu $
\section{\label{sec:level2}Power series}
\subsection{Polynomial which makes $B_n$ term terminated}
\begin{thm}
In Ref.\cite{Chou2012b}, the general expression of power series of $y(x)$ for polynomial which makes $B_n$ term terminated is defined by
\begin{eqnarray}
y(x)&=& \sum_{n=0}^{\infty } c_n x^{n+\lambda }= y_0(x)+ y_1(x)+ y_2(x)+y_3(x)+\cdots \nonumber\\
&=& c_0 \Bigg\{ \sum_{i_0=0}^{\beta _0} \left( \prod _{i_1=0}^{i_0-1}B_{2i_1+1} \right) x^{2i_0+\lambda } \nonumber\\
&&+ \sum_{i_0=0}^{\beta _0}\left\{ A_{2i_0} \prod _{i_1=0}^{i_0-1}B_{2i_1+1} \sum_{i_2=i_0}^{\beta _1} \left( \prod _{i_3=i_0}^{i_2-1}B_{2i_3+2} \right)\right\} x^{2i_2+1+\lambda }\nonumber\\
&& + \sum_{N=2}^{\infty } \Bigg\{ \sum_{i_0=0}^{\beta _0} \Bigg\{A_{2i_0}\prod _{i_1=0}^{i_0-1} B_{2i_1+1} \prod _{k=1}^{N-1} \Bigg( \sum_{i_{2k}= i_{2(k-1)}}^{\beta _k} A_{2i_{2k}+k}\prod _{i_{2k+1}=i_{2(k-1)}}^{i_{2k}-1}B_{2i_{2k+1}+(k+1)}\Bigg)\nonumber\\
&&\times \sum_{i_{2N} = i_{2(N-1)}}^{\beta _N} \Bigg( \prod _{i_{2N+1}=i_{2(N-1)}}^{i_{2N}-1} B_{2i_{2N+1}+(N+1)} \Bigg) \Bigg\} \Bigg\} x^{2i_{2N}+N+\lambda }\Bigg\}
\label{eq:5}
\end{eqnarray}
For a polynomial, we need a condition:
\begin{equation}
B_{2\beta _i + (i+1)}=0 \hspace{1cm} \mathrm{where}\; i,\beta _i= 0,1,2,\cdots
\label{eq:6}
\end{equation}
\end{thm}
In this paper Pochhammer symbol $(x)_n$ is used to represent the rising factorial: $(x)_n = \frac{\Gamma (x+n)}{\Gamma (x)}$.
On the above $ \beta _i$ is an eigenvalue that makes $B_n$ term terminated at certain value of index $n$. (\ref{eq:6}) makes each $y_i(x)$ where $i=0,1,2,\cdots$ as the polynomial in (\ref{eq:5}). Substitute (\ref{eq:4a})-(\ref{eq:4c}) into (\ref{eq:5}) by using (\ref{eq:6}).
The general expression of power series of GCH equation for polynomial which makes $B_n$ term terminated is given by
\begin{eqnarray}
y(x)&=& c_0 x^{\lambda } \Bigg\{\sum_{i_0=0}^{\beta _0} \frac{(-\beta _0)_{i_0}}{(1+\frac{\lambda }{2})_{i_0}(\gamma +\frac{\lambda }{2})_{i_0}}z^{i_0}
+ \Bigg\{\sum_{i_0=0}^{\beta _0} \frac{(i_0+\frac{\lambda }{2}+\frac{\omega }{2})}{(i_0+\frac{1}{2}+\frac{\lambda }{2})(i_0-\frac{1}{2}+\gamma +\frac{\lambda }{2})} \frac{(-\beta _0)_{i_0}}{(1+\frac{\lambda }{2})_{i_0}(\gamma +\frac{\lambda }{2})_{i_0}}\nonumber\\
&&\times \sum_{i_1=i_0}^{\beta _1} \frac{(-\beta _1)_{i_1}(\frac{3}{2}+\frac{\lambda }{2})_{i_0}(\gamma +\frac{1}{2}+ \frac{\lambda }{2})_{i_0}}{(-\beta _1)_{i_0}(\frac{3}{2}+\frac{\lambda }{2})_{i_1}(\gamma +\frac{1}{2}+\frac{\lambda }{2})_{i_1}} z^{i_1} \Bigg\} \tilde{\varepsilon }\nonumber\\
&&+ \sum_{n=2}^{\infty } \Bigg\{ \sum_{i_0=0}^{\beta _0} \frac{(i_0+\frac{\lambda }{2}+\frac{\omega }{2})}{(i_0+\frac{1}{2}+\frac{\lambda }{2})(i_0-\frac{1}{2}+\gamma +\frac{\lambda }{2})} \frac{(-\beta _0)_{i_0}}{(1+\frac{\lambda }{2})_{i_0}(\gamma +\frac{\lambda }{2})_{i_0}} \nonumber\\
&&\times \prod _{k=1}^{n-1} \Bigg\{ \sum_{i_k=i_{k-1}}^{\beta _k} \frac{(i_k+\frac{\lambda }{2}+\frac{\omega }{2}+\frac{k}{2})}{(i_k+\frac{1}{2}+\frac{\lambda }{2}+\frac{k}{2})(i_k-\frac{1}{2}+\gamma + \frac{k}{2}+\frac{\lambda }{2})}
\frac{(-\beta _k)_{i_k}(1+\frac{k}{2}+\frac{\lambda }{2})_{i_{k-1}}(\frac{k}{2}+\gamma +\frac{\lambda }{2})_{i_{k-1}}}{(-\beta _k)_{i_{k-1}}(1+\frac{k}{2}+\frac{\lambda }{2})_{i_k}(\frac{k}{2}+\gamma +\frac{\lambda }{2})_{i_k}}\Bigg\} \nonumber\\
&&\times \sum_{i_n= i_{n-1}}^{\beta _n} \frac{(-\beta _n)_{i_n}(1+\frac{n}{2}+\frac{\lambda }{2})_{i_{n-1}}(\frac{n}{2}+\gamma +\frac{\lambda }{2})_{i_{n-1}}}{(-\beta _n)_{i_{n-1}}(1+\frac{n}{2}+\frac{\lambda }{2})_{i_n}(\frac{n}{2}+\gamma +\frac{\lambda }{2})_{i_n}} z^{i_n}\Bigg\} \tilde{\varepsilon }^n \Bigg\}
\label{eq:7}
\end{eqnarray}
where
\begin{equation}
\begin{cases} z = -\frac{1}{2}\mu x^2 \cr
\tilde{\varepsilon } = -\frac{1}{2}\varepsilon x\cr
\gamma = \frac{1}{2}(1+\nu ) \cr
\Omega = -\mu (2\beta _i+i+\lambda )\;\;\mbox{as}\;i=0,1,2,\cdots \;\;\mbox{and}\;\; \beta_i= 0,1,2,\cdots \cr
\mbox{As}\; \beta _i\leq \beta _j\;\;\mbox{only}\;\;\mbox{if}\;\;i\leq j
\end{cases}
\label{eq:8}
\end{equation}
Put $c_0$= $\frac{\Gamma (\gamma +\beta _0)}{\Gamma (\gamma )}$ as $\lambda $=0 in (\ref{eq:7}).
\begin{rmk}
The power series expansion of GCH equation of the first kind for polynomial which makes $B_n$ term terminated about $x=0 $ as $\Omega = -2\mu (\beta _i+\frac{i}{2})$ where $i, \beta _i = 0,1,2,\cdots$ is
\begin{eqnarray}
y(x)&=& QW_{\beta _i}\left( \beta _i=-\frac{\Omega }{2\mu }-\frac{i}{2} , \omega, \gamma =\frac{1}{2}(1+\nu );\; \tilde{\varepsilon }= -\frac{1}{2}\varepsilon x;\; z=-\frac{1}{2}\mu x^2 \right) \nonumber\\
&&= \frac{\Gamma (\gamma +\beta _0)}{\Gamma (\gamma )} \Bigg\{\sum_{i_0=0}^{\beta _0 } \frac{(-\beta _0)_{i_0}}{(1)_{i_0}(\gamma)_{i_0}}z^{i_0}+ \Bigg\{ \sum_{i_0=0}^{\beta _0 }\frac{(i_0+\frac{\omega }{2})}{(i_0+\frac{1}{2})(i_0-\frac{1}{2}+\gamma )} \frac{(-\beta _0)_{i_0}}{(1)_{i_0}(\gamma)_{i_0}}\nonumber\\
&&\times \sum_{i_1=i_0}^{\beta _1} \frac{(-\beta _1)_{i_1}(\frac{3}{2})_{i_0}(\gamma +\frac{1}{2})_{i_0}}{(-\beta _1)_{i_0}(\frac{3}{2})_{i_1}(\gamma +\frac{1}{2})_{i_1}} z^{i_1} \Bigg\} \tilde{\varepsilon }
+ \sum_{n=2}^{\infty } \Bigg\{ \sum_{i_0=0}^{\beta _0} \frac{(i_0+\frac{\omega }{2})}{(i_0+\frac{1}{2})(i_0-\frac{1}{2}+\gamma)} \frac{(-\beta _0)_{i_0}}{(1)_{i_0}(\gamma )_{i_0}} \nonumber\\
&&\times \prod _{k=1}^{n-1} \Bigg\{ \sum_{i_k=i_{k-1}}^{\beta _k} \frac{(i_k+\frac{\omega }{2}+\frac{k}{2})}{(i_k+\frac{1}{2}+\frac{k}{2})(i_k-\frac{1}{2}+\gamma + \frac{k}{2})} \frac{(-\beta _k)_{i_k}(1+\frac{k}{2})_{i_{k-1}}(\frac{k}{2}+\gamma )_{i_{k-1}}}{(-\beta _k)_{i_{k-1}}(1+\frac{k}{2})_{i_k}(\frac{k}{2}+\gamma )_{i_k}}\Bigg\} \nonumber\\
&&\times \sum_{i_n= i_{n-1}}^{\beta _n} \frac{(-\beta _n)_{i_n}(1+\frac{n}{2})_{i_{n-1}}(\frac{n}{2}+\gamma )_{i_{n-1}}}{(-\beta _n)_{i_{n-1}}(1+\frac{n}{2})_{i_n}(\frac{n}{2}+\gamma )_{i_n}} z^{i_n}\Bigg\} \tilde{\varepsilon }^n \Bigg\}\nonumber
\end{eqnarray}
\end{rmk}
put $c_0= \left( -\frac{1}{2}\mu \right)^{1-\gamma } \frac{\Gamma (\psi _0+2-\gamma )}{\Gamma (2-\gamma )}$ as $\lambda = 1-\nu = 2(1-\gamma )$ in (\ref{eq:7}) with replacing $\beta _i$ by $\psi _i$.
\begin{rmk}
The power series expansion of GCH equation of the second kind for polynomial which makes $B_n$ term terminated about $x=0 $ as $\Omega = -2\mu (\psi _i +1-\gamma +\frac{i}{2})$ where $i, \psi _i= 0,1,2,\cdots$ is
\begin{eqnarray}
y(x)&=& RW_{\psi _i}\left( \psi _i=-\frac{\Omega }{2\mu }+\gamma -1-\frac{i}{2}, \omega, \gamma =\frac{1}{2}(1+\nu );\; \tilde{\varepsilon }= -\frac{1}{2}\varepsilon x;\; z=-\frac{1}{2}\mu x^2 \right) \nonumber\\
&&= z^{1-\gamma }\frac{\Gamma (\psi _0+2-\gamma )}{\Gamma (2-\gamma )} \Bigg\{\sum_{i_0=0}^{\psi _0} \frac{(-\psi _0)_{i_0}}{(1)_{i_0}(2-\gamma)_{i_0}}z^{i_0}\nonumber\\
&&+ \Bigg\{ \sum_{i_0=0}^{\psi _0}\frac{(i_0+1-\gamma +\frac{\omega }{2})}{(i_0+\frac{1}{2})(i_0+\frac{3}{2}-\gamma )} \frac{(-\psi _0)_{i_0}}{(1)_{i_0}(2-\gamma)_{i_0}}
\sum_{i_1=i_0}^{\psi _1} \frac{(-\psi _1)_{i_1}(\frac{3}{2})_{i_0}(\frac{5}{2}-\gamma )_{i_0}}{(-\psi _1)_{i_0}(\frac{3}{2})_{i_1}(\frac{5}{2}-\gamma )_{i_1}} z^{i_1} \Bigg\} \tilde{\varepsilon }\nonumber\\
&&+ \sum_{n=2}^{\infty } \Bigg\{ \sum_{i_0=0}^{\psi _0} \frac{(i_0+1-\gamma +\frac{\omega }{2})}{(i_0+\frac{1}{2})(i_0+\frac{3}{2}-\gamma)} \frac{(-\psi _0)_{i_0}}{(1)_{i_0}(2-\gamma )_{i_0}} \nonumber\\
&&\times \prod _{k=1}^{n-1} \Bigg\{ \sum_{i_k=i_{k-1}}^{\psi _k} \frac{(i_k+1-\gamma +\frac{\omega }{2}+\frac{k}{2})}{(i_k+\frac{1}{2}+\frac{k}{2})(i_k+\frac{3}{2}-\gamma + \frac{k}{2})} \frac{(-\psi _k)_{i_k}(1+\frac{k}{2})_{i_{k-1}}(2-\gamma +\frac{k}{2})_{i_{k-1}}}{(-\psi _k)_{i_{k-1}}(1+\frac{k}{2})_{i_k}(2-\gamma +\frac{k}{2})_{i_k}}\Bigg\} \nonumber\\
&&\times \sum_{i_n= i_{n-1}}^{\psi _n} \frac{(-\psi _n)_{i_N}(1+\frac{n}{2})_{i_{n-1}}(2-\gamma +\frac{n}{2})_{i_{n-1}}}{(-\psi _n)_{i_{n-1}}(1+\frac{n}{2})_{i_n}(2-\gamma +\frac{n}{2})_{i_n}} z^{i_n}\Bigg\}\tilde{\varepsilon }^n \Bigg\}\nonumber
\end{eqnarray}
\end{rmk}
\subsection{Infinite series}
\begin{thm}
In Ref.\cite{Chou2012b}, the general expression of power series of $y(x)$ for infinite series is defined by
\begin{eqnarray}
y(x) &=& \sum_{n=0}^{\infty } y_{n}(x)= y_0(x)+ y_1(x)+ y_2(x)+ y_3(x)+\cdots \nonumber\\
&=& c_0 \Bigg\{ \sum_{i_0=0}^{\infty } \left( \prod _{i_1=0}^{i_0-1}B_{2i_1+1} \right) x^{2i_0+\lambda }
+ \sum_{i_0=0}^{\infty }\left\{ A_{2i_0} \prod _{i_1=0}^{i_0-1}B_{2i_1+1} \sum_{i_2=i_0}^{\infty } \left( \prod _{i_3=i_0}^{i_2-1}B_{2i_3+2} \right)\right\} x^{2i_2+1+\lambda } \nonumber\\
&& + \sum_{N=2}^{\infty } \Bigg\{ \sum_{i_0=0}^{\infty } \Bigg\{A_{2i_0}\prod _{i_1=0}^{i_0-1} B_{2i_1+1}
\prod _{k=1}^{N-1} \Bigg( \sum_{i_{2k}= i_{2(k-1)}}^{\infty } A_{2i_{2k}+k}\prod _{i_{2k+1}=i_{2(k-1)}}^{i_{2k}-1}B_{2i_{2k+1}+(k+1)}\Bigg)\nonumber\\
&& \times \sum_{i_{2N} = i_{2(N-1)}}^{\infty } \Bigg( \prod _{i_{2N+1}=i_{2(N-1)}}^{i_{2N}-1} B_{2i_{2N+1}+(N+1)} \Bigg) \Bigg\} \Bigg\} x^{2i_{2N}+N+\lambda }\Bigg\}
\label{eq:11}
\end{eqnarray}
\end{thm}
Substitute (\ref{eq:4a})-(\ref{eq:4c}) into (\ref{eq:11}).
The general expression of power series of GCH equation for infinite series is given by
\begin{eqnarray}
y(x) &=& \sum_{m=0}^{\infty } y_{m}(x)= y_0(x)+ y_1(x)+ y_2(x)+ y_3(x)+\cdots \nonumber\\
&=& c_0 x^{\lambda } \Bigg\{\sum_{i_0=0}^{\infty } \frac{(\frac{\Omega }{2\mu }+ \frac{\lambda }{2})_{i_0}}{(1+\frac{\lambda }{2})_{i_0}(\gamma +\frac{\lambda }{2})_{i_0}}z^{i_0}
+ \Bigg\{ \sum_{i_0=0}^{\infty }\frac{(i_0+\frac{\lambda }{2}+\frac{\omega }{2})}{(i_0+\frac{1}{2}+\frac{\lambda }{2})(i_0-\frac{1}{2}+\gamma +\frac{\lambda }{2})} \frac{(\frac{\Omega }{2\mu }+\frac{\lambda }{2})_{i_0}}{(1+\frac{\lambda }{2})_{i_0}(\gamma +\frac{\lambda }{2})_{i_0}}\nonumber\\
&&\times \sum_{i_1=i_0}^{\infty }\frac{(\frac{\Omega }{2\mu }+\frac{1}{2}+\frac{\lambda }{2})_{i_1}(\frac{3}{2}+\frac{\lambda }{2})_{i_0}(\gamma +\frac{1}{2}+ \frac{\lambda }{2})_{i_0}}{(\frac{\Omega }{2\mu }+\frac{1}{2}+\frac{\lambda }{2})_{i_0}(\frac{3}{2}+\frac{\lambda }{2})_{i_1}(\gamma +\frac{1}{2}+\frac{\lambda }{2})_{i_1}} z^{i_1} \Bigg\} \tilde{\varepsilon }\nonumber\\
&&+ \sum_{n=2}^{\infty } \Bigg\{ \sum_{i_0=0}^{\infty } \frac{(i_0+\frac{\lambda }{2}+\frac{\omega }{2})}{(i_0+\frac{1}{2}+\frac{\lambda }{2})(i_0-\frac{1}{2}+\gamma +\frac{\lambda }{2})} \frac{(\frac{\Omega }{2\mu }+\frac{\lambda }{2})_{i_0}}{(1+\frac{\lambda }{2})_{i_0}(\gamma +\frac{\lambda }{2})_{i_0}} \nonumber\\
&&\times \prod _{k=1}^{n-1} \Bigg\{ \sum_{i_k=i_{k-1}}^{\infty } \frac{(i_k+\frac{\lambda }{2}+\frac{\omega }{2}+\frac{k}{2})}{(i_k+\frac{1}{2}+\frac{\lambda }{2}+\frac{k}{2})(i_k-\frac{1}{2}+\gamma + \frac{k}{2}+\frac{\lambda }{2})} \frac{(\frac{\Omega }{2\mu }+\frac{k}{2}+\frac{\lambda }{2})_{i_k}(1+\frac{k}{2}+\frac{\lambda }{2})_{i_{k-1}}(\frac{k}{2}+\gamma +\frac{\lambda }{2})_{i_{k-1}}}{(\frac{\Omega }{2\mu }+\frac{k}{2}+\frac{\lambda }{2})_{i_{k-1}}(1+\frac{k}{2}+\frac{\lambda }{2})_{i_k}(\frac{k}{2}+\gamma +\frac{\lambda }{2})_{i_k}}\Bigg\} \nonumber\\
&&\times \sum_{i_n= i_{n-1}}^{\infty } \frac{(\frac{\Omega }{2\mu }+\frac{n}{2}+\frac{\lambda }{2})_{i_n}(1+\frac{n}{2}+\frac{\lambda }{2})_{i_{n-1}}(\frac{n}{2}+\gamma +\frac{\lambda }{2})_{i_{n-1}}}{(\frac{\Omega }{2\mu }+\frac{n}{2}+\frac{\lambda }{2})_{i_{n-1}}(1+\frac{n}{2}+\frac{\lambda }{2})_{i_n}(\frac{n}{2}+\gamma +\frac{\lambda }{2})_{i_n}} z^{i_n}\Bigg\} \tilde{\varepsilon }^n \Bigg\}
\label{eq:12}
\end{eqnarray}
Put $c_0$= $\frac{\Gamma (\gamma -\frac{\Omega }{2\mu })}{\Gamma (\gamma )}$ as $\lambda =0$ for the first independent solution of GCH equation and $c_0= \left( -\frac{1}{2}\mu \right)^{1-\gamma } \frac{\Gamma (1-\frac{\Omega }{2\mu })}{\Gamma (2-\gamma )}$ as $\lambda = 1-\nu = 2(1-\gamma )$ for the second one in (\ref{eq:12})
\begin{rmk}
The power series expansion of GCH equation of the first kind for infinite series about $x=0 $ is
\begin{eqnarray}
y(x)&=& QW\left(\omega, \gamma =\frac{1}{2}(1+\nu );\; \tilde{\varepsilon }= -\frac{1}{2}\varepsilon x;\; z=-\frac{1}{2}\mu x^2 \right) \nonumber\\
&&= \frac{\Gamma (\gamma -\frac{\Omega }{2\mu })}{\Gamma (\gamma )} \Bigg\{\sum_{i_0=0}^{\infty } \frac{(\frac{\Omega }{2\mu })_{i_0}}{(1)_{i_0}(\gamma)_{i_0}}z^{i_0}+ \Bigg\{\sum_{i_0=0}^{\infty } \frac{(i_0+\frac{\omega }{2})}{(i_0+\frac{1}{2})(i_0-\frac{1}{2}+\gamma )} \frac{(\frac{\Omega }{2\mu })_{i_0}}{(1)_{i_0}(\gamma)_{i_0}}\nonumber\\
&&\times \sum_{i_1=i_0}^{\infty } \frac{(\frac{\Omega }{2\mu }+\frac{1}{2})_{i_1}(\frac{3}{2})_{i_0}(\gamma +\frac{1}{2})_{i_0}}{(\frac{\Omega }{2\mu }+\frac{1}{2})_{i_0}(\frac{3}{2})_{i_1}(\gamma +\frac{1}{2})_{i_1}} z^{i_1} \Bigg\} \tilde{\varepsilon }
+ \sum_{n=2}^{\infty } \Bigg\{ \sum_{i_0=0}^{\infty } \frac{(i_0+\frac{\omega }{2})}{(i_0+\frac{1}{2})(i_0-\frac{1}{2}+\gamma)} \frac{(\frac{\Omega }{2\mu })_{i_0}}{(1)_{i_0}(\gamma )_{i_0}} \nonumber\\
&&\times \prod _{k=1}^{n-1} \Bigg\{ \sum_{i_k=i_{k-1}}^{\infty } \frac{(i_k+\frac{\omega }{2}+\frac{k}{2})}{(i_k+\frac{1}{2}+\frac{k}{2})(i_k-\frac{1}{2}+\gamma + \frac{k}{2})} \frac{(\frac{\Omega }{2\mu }+\frac{k}{2})_{i_k}(1+\frac{k}{2})_{i_{k-1}}(\frac{k}{2}+\gamma )_{i_{k-1}}}{(\frac{\Omega }{2\mu }+\frac{k}{2})_{i_{k-1}}(1+\frac{k}{2})_{i_k}(\frac{k}{2}+\gamma )_{i_k}}\Bigg\} \nonumber\\
&&\times \sum_{i_n= i_{n-1}}^{\infty } \frac{(\frac{\Omega }{2\mu }+\frac{n}{2})_{i_n}(1+\frac{n}{2})_{i_{n-1}}(\frac{n}{2}+\gamma )_{i_{n-1}}}{(\frac{\Omega }{2\mu }+\frac{n}{2})_{i_{n-1}}(1+\frac{n}{2})_{i_n}(\frac{n}{2}+\gamma )_{i_n}} z^{i_n}\Bigg\} \tilde{\varepsilon }^n \Bigg\}\nonumber
\end{eqnarray}
\end{rmk}
\begin{rmk}
The power series expansion of GCH equation of the second kind for infinite series about $x=0 $ is
\begin{eqnarray}
y(x)&=& RW\left(\omega, \gamma =\frac{1}{2}(1+\nu );\; \tilde{\varepsilon }= -\frac{1}{2}\varepsilon x;\; z=-\frac{1}{2}\mu x^2 \right) \nonumber\\
&&= z^{1-\gamma }\frac{\Gamma (1-\frac{\Omega }{2\mu })}{\Gamma (2-\gamma )} \Bigg\{\sum_{i_0=0}^{\infty } \frac{(\frac{\Omega }{2\mu }+1-\gamma )_{i_0}}{(1)_{i_0}(2-\gamma)_{i_0}}z^{i_0}
+ \Bigg\{ \sum_{i_0=0}^{\infty }\frac{(i_0+1-\gamma +\frac{\omega }{2})}{(i_0+\frac{1}{2})(i_0+\frac{3}{2}-\gamma )} \frac{(\frac{\Omega }{2\mu }+1-\gamma )_{i_0}}{(1)_{i_0}(2-\gamma)_{i_0}}\nonumber\\
&&\times \sum_{i_1=i_0}^{\infty } \frac{(\frac{\Omega }{2\mu }+\frac{3}{2}-\gamma )_{i_1}(\frac{3}{2})_{i_0}(\frac{5}{2}-\gamma )_{i_0}}{(\frac{\Omega }{2\mu }+\frac{3}{2}-\gamma )_{i_0}(\frac{3}{2})_{i_1}(\frac{5}{2}-\gamma )_{i_1}} z^{i_1} \Bigg\} \tilde{\varepsilon } \nonumber\\
&&+ \sum_{n=2}^{\infty } \Bigg\{ \sum_{i_0=0}^{\infty }\frac{(i_0+1-\gamma +\frac{\omega }{2})}{(i_0+\frac{1}{2})(i_0+\frac{3}{2}-\gamma)} \frac{(\frac{\Omega }{2\mu }+1-\gamma )_{i_0}}{(1)_{i_0}(2-\gamma )_{i_0}}\nonumber\\
&&\times \prod _{k=1}^{n-1} \Bigg\{ \sum_{i_k=i_{k-1}}^{\infty } \frac{(i_k+1-\gamma +\frac{\omega }{2}+\frac{k}{2})}{(i_k+\frac{1}{2}+\frac{k}{2})(i_k+\frac{3}{2}-\gamma + \frac{k}{2})}\frac{(\frac{\Omega }{2\mu }+1-\gamma + \frac{k}{2})_{i_k}(1+\frac{k}{2})_{i_{k-1}}(2-\gamma +\frac{k}{2})_{i_{k-1}}}{(\frac{\Omega }{2\mu }+1-\gamma + \frac{k}{2})_{i_{k-1}}(1+\frac{k}{2})_{i_k}(2-\gamma +\frac{k}{2})_{i_k}}\Bigg\} \nonumber\\
&&\times \sum_{i_n= i_{n-1}}^{\infty } \frac{(\frac{\Omega }{2\mu }+1-\gamma + \frac{n}{2})_{i_n}(1+\frac{n}{2})_{i_{n-1}}(2-\gamma +\frac{n}{2})_{i_{n-1}}}{(\frac{\Omega }{2\mu }+1-\gamma + \frac{n}{2})_{i_{n-1}}(1+\frac{n}{2})_{i_n}(2-\gamma +\frac{n}{2})_{i_n}} z^{i_n}\Bigg\} \tilde{\varepsilon }^n \Bigg\}\nonumber
\end{eqnarray}
\end{rmk}
When $\nu $ is integer, one of two solution of the GCH equation does not have any meaning, because $RW_{\psi _i}\left( \psi _i=-\frac{\Omega }{2\mu }+\gamma -1-\frac{i}{2}, \gamma =\frac{1}{2}(1+\nu );\; \tilde{\varepsilon };\; z \right)$ can be described as $QW_{\beta _i}\left( \beta _i=-\frac{\Omega }{2\mu }-\frac{i}{2} , \gamma =\frac{1}{2}(1+\nu );\; \tilde{\varepsilon };\; z\right)$ as long as $|\lambda _1-\lambda _2|=|\nu -1|=$ integer. As we see remarks 1--4, it is required that $\nu \ne 0,-1,-2,\cdots$ for the first kind of independent solution of GCH equation for polynomial and infinite series. By similar reason, $\nu \ne 2,3,4,\cdots$ is required for the second kind of independent solution of GCH equation.
\section{Asymptotic behavior of the function $\bm{y(x)}$ and the boundary condition for $\bm{x}$}
\subsection{The case of $|\mu| \ll 1$ or $ |\mu| \ll |\varepsilon | $}
As $n\gg 1$, (\ref{eq:4a}) and (\ref{eq:4b}) are
\begin{subequations}
\begin{equation}
\lim_{n\gg 1} A_n = A= -\frac{\varepsilon }{n}
\label{eq:37a}
\end{equation}
And,
\begin{equation}
\lim_{n\gg 1} B_n = B= -\frac{\mu }{n}
\label{eq:37b}
\end{equation}
\end{subequations}
Since $|\mu| \ll 1$ or $ |\mu| \ll |\varepsilon | $, (\ref{eq:37b}) is negligible.
Its recurrence relation is
\begin{equation}
c_{n+1} \approx -\frac{\varepsilon }{n} c_n
\label{eq:39}
\end{equation}
Plug (\ref{eq:39}) into the power series expansion where $\displaystyle{\sum_{n=0}^{\infty } c_n x^n}$, putting $c_0= 0 $ and $c_1= 1 $ for simplicity.
\begin{equation}
\lim_{n\gg 1}y(x) \approx x \left( e^{-\varepsilon x}-1\right)\hspace{1cm}\mbox{where}\;-\infty <x< \infty
\label{eq:40}
\end{equation}
\subsection{The case of $|\varepsilon | \ll 1$ or $ |\varepsilon | \ll |\mu | $}
Let assume that $|\varepsilon | \ll 1$ or $ |\varepsilon | \ll |\mu | $. Then (\ref{eq:37a}) is negligible.
Its recurrence relation is
\begin{equation}
c_{n+1} \approx -\frac{\mu }{n} c_{n-1}
\label{eq:42}
\end{equation}
We can classify $c_n$ as to even and odd terms in (\ref{eq:42}).
\begin{equation}
c_{2n} = \frac{\big(-\frac{1}{2}\big)!}{\big(n-\frac{1}{2}\big)!}\Big( -\frac{1}{2}\mu \Big)^n c_0 \hspace{1cm}c_{2n+1} = \frac{1 }{n!}\Big( -\frac{1}{2}\mu \Big)^n c_1\hspace{1cm}\mbox{where}\;n\geq 1
\label{eq:43}
\end{equation}
$c_1 = A c_0 =0$ in (\ref{eq:43}). Because $A$ is negligible since $|\varepsilon | \ll 1$ or $ |\varepsilon | \ll |\mu | $.
Put $c_{2n}$ in (\ref{eq:43}) into the power series expansion where $\displaystyle{\sum_{n=0}^{\infty } c_n x^n}$, putting $c_0 =1$ for simplicity.
\begin{eqnarray}
\lim_{n\gg 1}y(x) &=& 1+ \sqrt{ -\frac{\pi }{2}\mu x^2} \mbox{Erf}\left(\sqrt{ -\frac{1}{2}\mu x^2}\right) e^{ -\frac{1}{2}\mu x^2} \label{eq:44}\\
&&\mbox{where}\;-\infty <x< \infty \nonumber
\end{eqnarray}
On the above $\mbox{Erf(y)} $ is an error function which is
\begin{equation}
\mbox{Erf(y)} = \frac{2}{\sqrt{\pi }} \int_{0}^{y} dt\; e^{-t^2}\nonumber
\end{equation}
\section{Application}
I show the power series expansion in closed forms and asymptotic behaviors of the GCH function in this paper.
We can apply this new special function into many physics areas. I show three examples of GCH equation as follows:
\subsection{the rotating harmonic oscillator}
For example, there are quantum-mechanical systems whose radial Schr$\ddot{\mbox{o}}$dinger equation may be reduced to a Biconfluent Heun function\cite{Leau1986,Mass1983}, namely the rotating harmonic oscillator and a class of confinement potentials. Its radial Schr$\ddot{\mbox{o}}$dinger equation is given by
\begin{equation}
\Psi^{''}(r)+ \bigg\{ \frac{2\lambda _m+1}{2\omega } -\frac{(r-1)^2}{4\omega ^2}- \frac{l_m (l_m+1)}{r^2}\bigg\} \Psi(r) =0
\label{eq:47}
\end{equation}
where $0\leq r < \infty $, ¸$\lambda _m$ is the eigenvalue, $l_m$ is the rotational quantum number and $\omega$ is a
coupling parameter.
By means of the changes of variable,
\begin{equation}
\Psi(r) = r^{l_m+1} \exp\left( -\frac{(r-1)^2}{2\omega } \right) U(r) \hspace{.5cm}\mbox{and}\hspace{.5cm} r=\sqrt{2\omega }x
\label{eq:48}
\end{equation}
the above becomes the following Biconfluent Heun equation:
\begin{equation}
x U^{''}(x)+ (1+\alpha -\beta x -2 x^2)U^{'}(x)+ \left\{ (\gamma -\alpha -2)x - \frac{1}{2}[\delta +\beta (1+\alpha )] \right\} U(x) =0
\label{eq:49}
\end{equation}
where the four Heun parameters are
\begin{equation}
\alpha =2l_m+1 \hspace{.5cm}\beta = -\sqrt{\frac{2}{\omega }}\hspace{.5cm} \delta =0 \hspace{.5cm}\gamma = 1+2\lambda_m
\label{eq:50}
\end{equation}
If we compare (\ref{eq:49}) with (\ref{eq:1}), all coefficients on the above are correspondent to the following way.
\begin{equation}
\begin{split}
& \mu \longleftrightarrow -2 \\ & \varepsilon \longleftrightarrow -\beta \\ & \nu \longleftrightarrow 1+\alpha \\
& \Omega \longleftrightarrow \gamma -\alpha -2 \\ & \omega \longleftrightarrow \frac{1}{2\beta }[\delta +\beta (1+\alpha )]
\end{split}\label{eq:56}
\end{equation}
Put (\ref{eq:50}) in (\ref{eq:56}).
\begin{equation}
\begin{split}
& \mu \longleftrightarrow -2 \\ & \varepsilon \longleftrightarrow \sqrt{\frac{2}{\omega }} \\ & \nu \longleftrightarrow 2(l_m+1) \\
& \Omega \longleftrightarrow 2(\lambda _m-l_m-1) \\ & \omega \longleftrightarrow l_m+1
\end{split}\label{eq:57}
\end{equation}
Let's investigate function $\Psi(r)$ as $n$ and $r$ go to infinity. I assume that $U(x)$ is infinite series in (\ref{eq:49}). Since $\varepsilon \ll 1$ in (\ref{eq:57}), put (\ref{eq:44}) in (\ref{eq:48}) with replacing $\mu $ by $-2$.
\begin{equation}
\lim_{n\gg 1}\Psi(r) \approx r^{l_m+1} \exp\left(-\frac{(r-1)^2}{2\omega }\right) \left( 1+ \sqrt{\frac{\pi}{2\omega}} \mbox{Erf}\left(\frac{r}{\sqrt{ 2\omega }} \right) r e^{\frac{r^2}{2\omega}}\right)
\label{eq:58}
\end{equation}
In (\ref{eq:58}) if $r\rightarrow \infty $, then $\displaystyle {\lim_{n\gg 1}\Psi(r)\rightarrow \infty }$. It is unacceptable that wave function $\Psi(r)$ is divergent as $r$ goes to infinity in the quantum mechanical point of view. Therefore the function $U(x)$ must to be polynomial in (\ref{eq:49}) in order to make the wave function $\Psi(r)$ being convergent even if $r$ goes to infinity. $RW_{\psi _i}\left( \psi _i, \omega, \gamma;\; \tilde{\varepsilon }= -\frac{r}{2\omega};\; z=\frac{r^2}{2\omega} \right)\rightarrow \infty $ as $r\rightarrow 0$ because of $ \gamma =l_m+ \frac{3}{2}$ in Remark 2. But $QW_{\beta _i}\left( \beta _i,\omega, \gamma ;\; \tilde{\varepsilon }= -\frac{r}{2\omega};\; z=\frac{r^2}{2\omega} \right)\rightarrow 0$ as $r\rightarrow 0$ in Remark 1. So I choose Remark 1 as eigenfunction for (\ref{eq:48}). Put (\ref{eq:57}) in Remark 1 replacing $x$ and $y(x)$ by $\frac{r}{\sqrt{2\omega}}$ and $U(r)$.
\begin{eqnarray}
U(r)&=& QW_{\beta _i}\left( \beta _i=\frac{\lambda _m-l_m-1-i}{2} ,\omega =l_m+1, \gamma =l_m+ \frac{3}{2};\; \tilde{\varepsilon }= -\frac{r}{2\omega};\; z=\frac{r^2}{2\omega} \right)\nonumber\\
&&= \frac{\Gamma (\gamma +\beta _0)}{\Gamma (\gamma )} \Bigg\{\sum_{i_0=0}^{\beta _0 } \frac{(-\beta _0)_{i_0}}{(1)_{i_0}(\gamma)_{i_0}}z^{i_0}+ \Bigg\{\sum_{i_0=0}^{\beta _0 } \frac{(i_0+\frac{\omega }{2})}{(i_0+\frac{1}{2})(i_0-\frac{1}{2}+\gamma )} \frac{(-\beta _0)_{i_0}}{(1)_{i_0}(\gamma)_{i_0}}\nonumber\\
&&\times \sum_{i_1=i_0}^{\beta _1}\frac{(-\beta _1)_{i_1}(\frac{3}{2})_{i_0}(\gamma +\frac{1}{2})_{i_0}}{(-\beta _1)_{i_0}(\frac{3}{2})_{i_1}(\gamma +\frac{1}{2})_{i_1}} z^{i_1} \Bigg\} \tilde{\varepsilon }
+ \sum_{n=2}^{\infty } \Bigg\{ \sum_{i_0=0}^{\beta _0} \frac{(i_0+\frac{\omega }{2})}{(i_0+\frac{1}{2})(i_0-\frac{1}{2}+\gamma)} \frac{(-\beta _0)_{i_0}}{(1)_{i_0}(\gamma )_{i_0}} \nonumber\\
&&\times \prod _{k=1}^{n-1} \Bigg\{ \sum_{i_k=i_{k-1}}^{\beta _k} \frac{(i_k+\frac{\omega }{2}+\frac{k}{2})}{(i_k+\frac{1}{2}+\frac{k}{2})(i_k-\frac{1}{2}+\gamma + \frac{k}{2})} \frac{(-\beta _k)_{i_k}(1+\frac{k}{2})_{i_{k-1}}(\frac{k}{2}+\gamma )_{i_{k-1}}}{(-\beta _k)_{i_{k-1}}(1+\frac{k}{2})_{i_k}(\frac{k}{2}+\gamma )_{i_k}}\Bigg\} \nonumber\\
&&\times \sum_{i_n= i_{n-1}}^{\beta _n} \frac{(-\beta _n)_{i_n}(1+\frac{n}{2})_{i_{n-1}}(\frac{n}{2}+\gamma )_{i_{n-1}}}{(-\beta _n)_{i_{n-1}}(1+\frac{n}{2})_{i_n}(\frac{n}{2}+\gamma )_{i_n}} z^{i_n}\Bigg\} \tilde{\varepsilon }^n \Bigg\}
\label{eq:59}
\end{eqnarray}
Put (\ref{eq:59}) in (\ref{eq:48}). The wave function for the rotating harmonic oscillator is given by
\begin{eqnarray}
\Psi(r) &=& N r^{l_m+1} \exp\left(-\frac{(r-1)^2}{2\omega }\right) QW_{\beta _i}\left( \beta _i=\frac{\lambda _m-l_m-1-i}{2} ,\omega =l_m+1, \gamma =l_m+ \frac{3}{2}\right.\nonumber\\
&&;\left. \tilde{\varepsilon }= -\frac{r}{2\omega};\; z=\frac{r^2}{2\omega} \right)
\label{eq:60}
\end{eqnarray}
N is normalized constant. Eigenvalue $\lambda _m$ is
\begin{equation}
\lambda _m =2\beta _i+l_m+1+i \hspace{1cm} \mathrm{where}\; i,\beta _i= 0,1,2,\cdots \nonumber
\end{equation}
In general most of well-known special function with two recursive coefficients (Bessel, Legendre, Kummer, Laguerre and hypergeometric functions, etc) only has one eigenvalue for the polynomial case. However the GCH function with three recursive coefficients has infinite eigenvalues as we see (\ref{eq:59}).
\subsection{Confinement potentials}
Following Chaudhuri and Mukherjee, there is the radial Schr$\ddot{\mbox{o}}$dinger equation.\cite{Leau1986,Chau1983,Chau1984}:
\begin{equation}
\Psi^{''}(r)+ \bigg\{ \bigg( \frac{2\mu }{\hbar ^2} \bigg) \bigg( E+ \frac{a}{r} -br- cr^2\bigg) -\frac{l(l+1)}{r^2} \bigg\} \Psi(r) =0
\label{eq:51}
\end{equation}
with $E$ being the energy. By means of the consecutive changes of variable
\begin{equation}
\Psi(r) = r^{l+1} \exp\left( -\frac{1}{2} \alpha _F r^2 -\beta _F r \right) U(r) \hspace{.5cm}\mbox{and}\hspace{.5cm} x=\sqrt{\alpha _F}r
\label{eq:52}
\end{equation}
the above becomes also the following Biconfluent Heun equation:
\begin{equation}
x U^{''}(x)+ (1+\alpha -\beta x -2 x^2)U^{'}(x)+ \left\{ (\gamma -\alpha -2)x - \frac{1}{2}[\delta +\beta (1+\alpha )] \right\} U(x) =0
\label{eq:53}
\end{equation}
where the four Heun parameters are
\begin{eqnarray}
&&\alpha = 2l+1,\hspace{.5cm} \gamma = \frac{\epsilon _F}{\alpha _F},\nonumber\\
&& \beta = 2\frac{\beta _F}{\sqrt{\alpha _F}},\hspace{.5cm} \delta = -\frac{4\mu }{\hbar ^2}\frac{a}{\sqrt{\alpha _F}}
\label{eq:54}
\end{eqnarray}
where,
\begin{equation}
\alpha_F = \bigg( \frac{2\mu c}{\hbar ^2} \bigg)^{1/2} , \hspace{.5cm} \beta _F = b\bigg( \frac{ \mu }{2\hbar ^2 c}\bigg)^{1/2} , \hspace{.5cm}\epsilon _F = \beta _F^2 +\frac{2\mu }{\hbar ^2} E
\label{eq:55}
\end{equation}
Put (\ref{eq:54}) and (\ref{eq:55}) in (\ref{eq:56}).
\begin{equation}
\begin{split}
& \mu \longleftrightarrow -2 \\ & \varepsilon \longleftrightarrow -2\frac{\beta _F}{\sqrt{\alpha _F}} \\ & \nu \longleftrightarrow 2(l+1) \\
& \Omega \longleftrightarrow \frac{\epsilon _F}{ \alpha _F } -2\left( l+\frac{3}{2}\right) = \frac{1}{ \alpha _F }\left( \beta _F^2 +\frac{2\mu }{\hbar ^2} E\right) -2\left( l+\frac{3}{2}\right) \\ & \omega \longleftrightarrow -\frac{\mu a }{\hbar ^2 \beta _F}+ l+1
\end{split}\label{eq:61}
\end{equation}
Let's investigate function $\Psi(r)$ as $n$ and $r$ go to infinity. I assume that $U(x)$ is infinite series in (\ref{eq:53}). Since $ |\varepsilon |\ll 1 $ in (\ref{eq:61}), put (\ref{eq:44}) in (\ref{eq:52}) with replacing $\mu $ and $x$ by $-2$ and $\sqrt{\alpha _F}r$.
\begin{equation}
\lim_{n\gg 1}\Psi(r) \approx r^{l+1} \exp\left(-\frac{\alpha _F}{2}r^2-\beta _F r\right) \left( 1+ \sqrt{\pi \alpha _F} \mbox{Erf}\left(\sqrt{ \alpha _F }r \right) r e^{\alpha _F r^2}\right)
\label{eq:62}
\end{equation}
In (\ref{eq:62}) if $r\rightarrow \infty $, then $\displaystyle {\lim_{n\gg 1}\Psi(r)\rightarrow \infty }$. It is unacceptable that wave function $\Psi(r)$ is divergent as $r$ goes to infinity in the quantum mechanical point of view. Therefore the function $U(x)$ must to be polynomial in (\ref{eq:53}) in order to make the wave function $\Psi(r)$ being convergent even if $r$ goes to infinity. $RW_{\psi _i}\left( \psi _i, \omega, \gamma;\; \tilde{\varepsilon }= -\beta _F r;\; z= \alpha _F r^2 \right)\rightarrow \infty $ as $r\rightarrow 0$ because of $ \gamma = l+\frac{3}{2}$ in Remark 2. But $QW_{\beta _i}\left( \beta _i,\omega, \gamma ;\; \tilde{\varepsilon }= -\beta _F r;\; z=\alpha _F r^2 \right)\rightarrow 0$ as $r\rightarrow 0$ in Remark 1. So I choose Remark 1 as eigenfunction for (\ref{eq:52}). Put (\ref{eq:61}) in Remark 1 replacing $x$ and $y(x)$ by $\sqrt{\alpha _F}r$ and $U(r)$.
\begin{eqnarray}
U(r)&=& QW_{\beta _i}\left( \beta _i= \frac{1}{4 \alpha _F }\left( \beta _F^2 +\frac{2\mu }{\hbar ^2}E \right)-\frac{1}{2}\left(i+l+\frac{3}{2} \right),\omega =-\frac{\mu a }{\hbar ^2 \beta _F}+ l+1 \right.\nonumber\\
&&,\left. \gamma = l+ \frac{3}{2};\; \tilde{\varepsilon }= -\beta _F r;\; z=\alpha _F r^2\right) \nonumber\\
&&= \frac{\Gamma (\gamma +\beta _0)}{\Gamma (\gamma )} \Bigg\{\sum_{i_0=0}^{\beta _0 } \frac{(-\beta _0)_{i_0}}{(1)_{i_0}(\gamma)_{i_0}}z^{i_0}+ \Bigg\{ \sum_{i_0=0}^{\beta _0 }\frac{(i_0+\frac{\omega }{2})}{(i_0+\frac{1}{2})(i_0-\frac{1}{2}+\gamma )} \frac{(-\beta _0)_{i_0}}{(1)_{i_0}(\gamma)_{i_0}}\nonumber\\
&&\times \sum_{i_1=i_0}^{\beta _1} \frac{(-\beta _1)_{i_1}(\frac{3}{2})_{i_0}(\gamma +\frac{1}{2})_{i_0}}{(-\beta _1)_{i_0}(\frac{3}{2})_{i_1}(\gamma +\frac{1}{2})_{i_1}} z^{i_1} \Bigg\}\tilde{\varepsilon }
+ \sum_{n=2}^{\infty } \Bigg\{ \sum_{i_0=0}^{\beta _0} \frac{(i_0+\frac{\omega }{2})}{(i_0+\frac{1}{2})(i_0-\frac{1}{2}+\gamma)} \frac{(-\beta _0)_{i_0}}{(1)_{i_0}(\gamma )_{i_0}} \nonumber\\
&&\times \prod _{k=1}^{n-1} \Bigg\{ \sum_{i_k=i_{k-1}}^{\beta _k} \frac{(i_k+\frac{\omega }{2}+\frac{k}{2})}{(i_k+\frac{1}{2}+\frac{k}{2})(i_k-\frac{1}{2}+\gamma + \frac{k}{2})} \frac{(-\beta _k)_{i_k}(1+\frac{k}{2})_{i_{k-1}}(\frac{k}{2}+\gamma )_{i_{k-1}}}{(-\beta _k)_{i_{k-1}}(1+\frac{k}{2})_{i_k}(\frac{k}{2}+\gamma )_{i_k}}\Bigg\} \nonumber\\
&&\times \sum_{i_n= i_{n-1}}^{\beta _n} \frac{(-\beta _n)_{i_n}(1+\frac{n}{2})_{i_{n-1}}(\frac{n}{2}+\gamma )_{i_{n-1}}}{(-\beta _n)_{i_{n-1}}(1+\frac{n}{2})_{i_n}(\frac{n}{2}+\gamma )_{i_n}} z^{i_n}\Bigg\} \tilde{\varepsilon }^n \Bigg\}
\label{eq:63}
\end{eqnarray}
Put (\ref{eq:63}) in (\ref{eq:52}). The wave function for confinement potentials is given by
\begin{eqnarray}
\Psi(r) &=& N r^{l+1} \exp\left( -\frac{1}{2}r^2\alpha _F -\beta _F r \right) QW_{\beta _i}\left( \beta _i= \frac{1}{4 \alpha _F }\left( \beta _F^2 +\frac{2\mu }{\hbar ^2}E \right)-\frac{1}{2}\left(i+l+\frac{3}{2} \right) \right.\nonumber\\
&&,\left. \omega =-\frac{\mu a }{\hbar ^2 \beta _F}+ l+1, \gamma = l+ \frac{3}{2};\; \tilde{\varepsilon }= -\beta _F r;\; z=\alpha _F r^2\right) \label{eq:65}
\end{eqnarray}
N is normalized constant. Energy $E$ is
\begin{equation}
E= \frac{\hbar ^2}{2\mu } \left( 4 \alpha _F \left( \beta _i+\frac{i+l+\frac{3}{2}}{2} \right) -\beta _F^2\right)\hspace{1cm} \mathrm{where}\; i,\beta _i= 0,1,2,\cdots \nonumber
\end{equation}
Again the GCH function with three recursive coefficients has infinite eigenvalues.
\subsection{ The spin free Hamiltonian involving only scalar potential for the $q-\bar{q}$ system }
Following G\"{u}rsey and his colleagues, there is the spin free Hamiltonian involving only scalar potential for the $q-\bar{q}$ system:\cite{2011,1985,1988,1991}
\begin{equation}
H^2 = 4\left[(m+ \frac{1}{2}br)^2 + P_r^2 + \frac{l(l+1)}{r^2}\right]
\label{eq:67}
\end{equation}
where $P_r^2 = - \frac{\partial ^2}{\partial r^2} - \frac{2}{r} \frac{\partial}{\partial r} $, $m$= mass, $b$= real positive, and $l$= angular momentum quantum number. When wave function $\Psi (r) = \exp\left(-\frac{b}{4}\left(r+\frac{2m}{b}\right)^2\right) r^l y(r)Y_l^{m^{\star}}(\theta ,\phi)$ acts on both sides of (\ref{eq:67}), it becomes
\begin{equation}
r\frac{\partial^2{y}}{\partial{r}^2} + \left( - b r^2 -2m r +2(l+1)\right) \frac{\partial{y}}{\partial{r}} + \left( \left(\frac{E^2}{4}- b\left(l+\frac{3}{2}\right)\right) r -2m(l+1)\right) y = 0
\label{eq:68}
\end{equation}
If we compare (\ref{eq:68}) with (\ref{eq:1}), all coefficients on the above are correspondent to the following way.
\begin{equation}
\begin{split}
& \mu \longleftrightarrow -b \\ & \varepsilon \longleftrightarrow -2m \\ & \nu \longleftrightarrow 2(l+1) \\
& \Omega \longleftrightarrow \frac{E^2}{4}-b\left( l+\frac{3}{2}\right) \\ & \omega \longleftrightarrow l+1
\end{split}\label{eq:69}
\end{equation}
Let's investigate function $\Psi(r)$ as $n$ and $r$ go to infinity. I assume that $y(r)$ is infinite series in (\ref{eq:68}). Since $\varepsilon \ll 1$ in (\ref{eq:69}), put (\ref{eq:44}) in $\Psi (r) = \exp\left(-\frac{b}{4}\left(r+\frac{2m}{b}\right)^2\right) r^l y(r)Y_l^{m^{\star}}(\theta ,\phi)$ with replacing $x$ and $\mu $ by $r$ and $-b$ .
\begin{equation}
\lim_{n\gg 1}\Psi(r) \approx r^l \exp\left(-\frac{b}{4}\left(r+\frac{2m}{b}\right)^2\right) \left\{ 1+ \sqrt{ \frac{1 }{2}b\pi} \mbox{Erf}\left(\sqrt{\frac{1}{2}b r^2}\right) r e^{ \frac{1}{2}b r^2}\right\}Y_l^{m^{\star}}(\theta ,\phi)
\label{eq:70}
\end{equation}
In (\ref{eq:70}) if $r\rightarrow \infty $, then $\displaystyle {\lim_{n\gg 1}\Psi(r)\rightarrow \infty }$. It is unacceptable that wave function $\Psi(r)$ is divergent as $r$ goes to infinity in the quantum mechanical point of view. Therefore the function $y(r)$ must to be polynomial in (\ref{eq:68}) in order to make the wave function $\Psi(r)$ being convergent even if $r$ goes to infinity. $RW_{\psi _i}\left( \psi _i, \omega, \gamma;\; \tilde{\varepsilon }= mr;\; z=\frac{b}{2}\mu r^2\right)\rightarrow \infty $ as $r\rightarrow 0$ because of $\gamma = l+\frac{3}{2}$ in Remark 2. But $QW_{\beta _i}\left( \beta _i,\omega, \gamma ;\; \tilde{\varepsilon }= mr;\; z=\frac{b}{2}\mu r^2\right)\rightarrow 0$ as $r\rightarrow 0$ in Remark 1. So I choose Remark 1 as eigenfunction for (\ref{eq:68}). Put (\ref{eq:69}) in Remark 1 with replacing $x$ by $r$.
\begin{eqnarray}
y(r)&=& QW_{\beta _i}\left( \beta _i= \frac{1}{2}\left( \frac{E^2}{4b}-\bigg( i+l+\frac{3}{2}\bigg)\right), \omega =l+1; \gamma = l+\frac{3}{2};\; \tilde{\varepsilon }= mr;\; z=\frac{b}{2}\mu r^2 \right) \nonumber\\
&&= \frac{\Gamma (\gamma +\beta _0)}{\Gamma (\gamma )} \Bigg\{\sum_{i_0=0}^{\beta _0 } \frac{(-\beta _0)_{i_0}}{(1)_{i_0}(\gamma)_{i_0}}z^{i_0}+ \Bigg\{\sum_{i_0=0}^{\beta _0 } \frac{(i_0+\frac{\omega }{2})}{(i_0+\frac{1}{2})(i_0-\frac{1}{2}+\gamma )} \frac{(-\beta _0)_{i_0}}{(1)_{i_0}(\gamma)_{i_0}}\nonumber\\
&&\times \sum_{i_1=i_0}^{\beta _1} \frac{(-\beta _1)_{i_1}(\frac{3}{2})_{i_0}(\gamma +\frac{1}{2})_{i_0}}{(-\beta _1)_{i_0}(\frac{3}{2})_{i_1}(\gamma +\frac{1}{2})_{i_1}} z^{i_1} \Bigg\} \tilde{\varepsilon }
+ \sum_{n=2}^{\infty } \Bigg\{ \sum_{i_0=0}^{\beta _0} \frac{(i_0+\frac{\omega }{2})}{(i_0+\frac{1}{2})(i_0-\frac{1}{2}+\gamma)} \frac{(-\beta _0)_{i_0}}{(1)_{i_0}(\gamma )_{i_0}} \nonumber\\
&&\times \prod _{k=1}^{n-1} \Bigg\{ \sum_{i_k=i_{k-1}}^{\beta _k} \frac{(i_k+\frac{\omega }{2}+\frac{k}{2})}{(i_k+\frac{1}{2}+\frac{k}{2})(i_k-\frac{1}{2}+\gamma + \frac{k}{2})} \frac{(-\beta _k)_{i_k}(1+\frac{k}{2})_{i_{k-1}}(\frac{k}{2}+\gamma )_{i_{k-1}}}{(-\beta _k)_{i_{k-1}}(1+\frac{k}{2})_{i_k}(\frac{k}{2}+\gamma )_{i_k}}\Bigg\} \nonumber\\
&&\times \sum_{i_n= i_{n-1}}^{\beta _n} \frac{(-\beta _n)_{i_n}(1+\frac{n}{2})_{i_{n-1}}(\frac{n}{2}+\gamma )_{i_{n-1}}}{(-\beta _n)_{i_{n-1}}(1+\frac{n}{2})_{i_n}(\frac{n}{2}+\gamma )_{i_n}} z^{i_n}\Bigg\} \tilde{\varepsilon }^n \Bigg\}
\label{eq:71}
\end{eqnarray}
Put (\ref{eq:71}) in $\Psi (r) = \exp\left(-\frac{b}{4}\left(r+\frac{2m}{b}\right)^2\right) r^l y(r)Y_l^{m^{\star}}(\theta ,\phi)$. The wave function for the spin free Hamiltonian involving only scalar potential for the $q-\bar{q}$ system is given by
\begin{eqnarray}
\Psi(r) &=& N r^l \exp\left(-\frac{b}{4}\left(r+\frac{2m}{b}\right)^2\right) QW_{\beta _i}\left( \beta _i= \frac{1}{2}\left( \frac{E^2}{4b}-\bigg( i+l+\frac{3}{2}\bigg)\right), \omega =l+1; \gamma = l+\frac{3}{2}\right.\nonumber\\
&&;\left. \tilde{\varepsilon }= mr;\; z=\frac{b}{2}\mu r^2 \right) Y_l^{m^{\star}}(\theta ,\phi)
\label{eq:73}
\end{eqnarray}
N is normalized constant. Energy $E^2$ is
\begin{equation}
E^2= 4b\left( 2\beta _i + i+l+\frac{3}{2} \right) \hspace{1cm} \mathrm{where}\; i,\beta _i= 0,1,2,\cdots \nonumber
\end{equation}
The GCH function with three recursive coefficients has infinite eigenvalues.
\section{Conclusion}
Any special functions with two recursive coefficients (such as Bessel, Legendre, Kummer, Laguerre, hypergeometric, Coulomb wave function, etc) only have one eigenvalue for the polynomial case. However the GCH function with three recursive coefficients has infinite eigenvalues that make $B_n$'s term terminated as we see (\ref{eq:60}), (\ref{eq:65}) and (\ref{eq:73}).
I show the power series expansion in closed forms of the GCH function in this paper. As we see analytic power series expansion of the GCH function by applying 3TRF \cite{Chou2012b}, denominators and numerators in all $B_n$ terms arise with Pochhammer symbol: the meaning of this is that the analytic solutions of GCH equation with three recursive coefficients can be described as hypergoemetric function in a strict mathematical way. Since this function is described as hypergeometric function, we can transform this function to other well-known special functions having two term recurrence relation: understanding the connection between other special functions is important in the mathematical and physical points of views as we all know.
In my next paper I derive the integral representation of GCH equation including all higher
terms of $A_n$s by applying 3TRF \cite{Chou2012b}. From integral forms of the GCH function, we can investigate how this function is associated with other well known special functions such as Bessel, Laguerre, Kummer, hypergeometric functions, etc. And I show the generating function for the GCH polynomial. The generating function is really useful in order to derive orthogonal relations, recursion relations and expectation values of any physical quantities as we all recognize; i.e. the normalized wave function of hydrogen-like atoms and expectation values of its physical quantities such as position and momentum.
\vspace{5mm}
\section{Series ``Special functions and three term recurrence formula (3TRF)''}
This paper is 9th out of 10.
\vspace{3mm}
1. ``Approximative solution of the spin free Hamiltonian involving only scalar potential for the $q-\bar{q}$ system'' \cite{Chou2012a} - In order to solve the spin-free Hamiltonian with light quark masses we are led to develop a totally new kind of special function theory in mathematics that generalize all existing theories of confluent hypergeometric types. We call it the Grand Confluent Hypergeometric Function. Our new solution produces previously unknown extra hidden quantum numbers relevant for description of supersymmetry and for generating new mass formulas.
\vspace{3mm}
2. ``Generalization of the three-term recurrence formula and its applications'' \cite{Chou2012b} - Generalize three term recurrence formula in linear differential equation. Obtain the exact solution of the three term recurrence for polynomials and infinite series.
\vspace{3mm}
3. ``The analytic solution for the power series expansion of Heun function'' \cite{Chou2012c} - Apply three term recurrence formula to the power series expansion in closed forms of Heun function (infinite series and polynomials) including all higher terms of $A_n$s.
\vspace{3mm}
4. ``Asymptotic behavior of Heun function and its integral formalism'', \cite{Chou2012d} - Apply three term recurrence formula, derive the integral formalism, and analyze the asymptotic behavior of Heun function (including all higher terms of $A_n$s).
\vspace{3mm}
5. ``The power series expansion of Mathieu function and its integral formalism'', \cite{Chou2012e} - Apply three term recurrence formula, analyze the power series expansion of Mathieu function and its integral forms.
\vspace{3mm}
6. ``Lame equation in the algebraic form'' \cite{Chou2012f} - Applying three term recurrence formula, analyze the power series expansion of Lame function in the algebraic form and its integral forms.
\vspace{3mm}
7. ``Power series and integral forms of Lame equation in Weierstrass's form and its asymptotic behaviors'' \cite{Chou2012g} - Applying three term recurrence formula, derive the power series expansion of Lame function in Weierstrass's form and its integral forms.
\vspace{3mm}
8. ``The generating functions of Lame equation in Weierstrass's form'' \cite{Chou2012h} - Derive the generating functions of Lame function in Weierstrass's form (including all higher terms of $A_n$'s). Apply integral forms of Lame functions in Weierstrass's form.
\vspace{3mm}
9. ``Analytic solution for grand confluent hypergeometric function'' \cite{Chou2012i} - Apply three term recurrence formula, and formulate the exact analytic solution of grand confluent hypergeometric function (including all higher terms of $A_n$'s). Replacing $\mu $ and $\varepsilon \omega $ by 1 and $-q$, transforms the grand confluent hypergeometric function into Biconfluent Heun function.
\vspace{3mm}
10. ``The integral formalism and the generating function of grand confluent hypergeometric function'' \cite{Chou2012j} - Apply three term recurrence formula, and construct an integral formalism and a generating function of grand confluent hypergeometric function (including all higher terms of $A_n$'s).
\section*{Acknowledgment}
I thank Bogdan Nicolescu. The discussions I had with him on number theory was of great joy.
\vspace{3mm}
\bibliographystyle{model1a-num-names}
\bibliography{<your-bib-database>}
| 60,282
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Samsung has its own plans for not only the coolest high-end phones, but also entry-level phones. Among other things, the Galaxy A23 is one of those plans, the synthetic test of which was recently carried out by the manufacturer’s crew. At the same time as the standard results appear also appearsWhich Galaxy A23 5G is listed.
The Samsung Galaxy A23 appeared as a model number in the Geébench database under the model number SM-A235F. The listing for the synthetic test app reveals that the Qualcomm Snapdragon 680 chipset will do the computations inside the phone, and that the tested prototype has 4GB of RAM. The device runs Android 12, and it is clear that the final version will be on the shelves.
The Samsung Galaxy A23 5G, which is in the renders shown here, may be released with the latest MediaTek Dimensity 700 chipset. The processor in question may be located behind a completely flat 6.55-inch screen, at the top of which a drop-shaped sensor island may act as the home of the front camera.
For rear photo shooters, the primary camera module can be 50MP, but the kit also includes an 8MP wide-angle macro, 2MP macro sensor, and a 2MP depth sensor module.
The Samsung Galaxy A23 5G will also have a side-mounted fingerprint reader and headphone output, dimensions of 165.4 x 77 x 8.5 mm, and the thickness of the camera bump will be 10.3 mm.
The source who posted the presented images says that the Galaxy A23 variant, which is equipped with an LTE modem, will be equipped with a 6.4-inch IPS LCD screen, also with a drop-shaped hole for the front camera. It is said that the Samsung Galaxy A23 models will be announced before the end of March. Distribution details and therefore expected purchase prices are still unknown.
follow you too NapiDroid.huFor the latest Android
| 327,619
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). Review and approval of agreements for acquisitions for Florida Greenways and Trails Program properties pursuant to chapter 260 may be waived by the department in any contract with nonprofit corporations that have agreed to assist the department with this program. If the contribution of the acquiring agency exceeds $100 million in any one fiscal year, the agreement shall be submitted to and approved by the Legislative Budget Commission. for final purchase price approval shall explicitly state that payment of the final purchase price is subject to an appropriation from the Legislature. The consideration for such an option may not exceed $1,000 or 0.01 percent of the estimate by the department of the value of the parcel, whichever amount is greater.
Therefore, it is the intent of the Legislature that public land acquisition agencies develop programs to pursue alternatives to fee simple acquisition and to educate private landowners about such alternatives and the benefits of such alternatives. It is also the intent of the Legislature that a portion of the shares of.
| 164,684
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\begin{document}
\maketitle
\begin{abstract}
In topological data science, categories with a flow have become ubiquitous, including as special cases examples like persistence modules and sheaves. With the flow comes an interleaving distance, which has proven useful for applications. We give simple, categorical conditions which guarantee metric completeness of a category with a flow, meaning that every Cauchy sequence has a limit. We also describe how to find a metric completion of a category with a flow by using its Yoneda embedding. The overarching goal of this work is to prepare the way for a theory of convergence of probability measures on these categories with a flow.
\end{abstract}
\section{Introduction}\label{sec:Introduction}
It is common in applied topology to take a data set and assign to it an object from some category. Examples include persistence modules \cite{Chazal09}, multiparameter persistence modules \cite{Lesnick}, and derived sheaves \cite{KS}; see \cite{MunchCatsFlow} for more examples of categories used in this way. If this category can be given an interleaving distance, we can often treat it as a metric space (up to some technical complications).\\
A general way to get an interleaving distance is to have a flow as defined in \cite{MunchCatsFlow}. On a category $\Cat$, a flow is a functor $T_\e : \Cat\to \Cat$ for every $\e \in [0,\infty)$ satisfying certain properties. It is natural to ask whether $(\Cat, T_\e)$ is metrically complete\footnote{It is inconvenient that many terms we would like to use (such as \emph{limit}, \emph{complete}, \emph{dense}, and even \emph{Cauchy complete category}) have already be defined by category theorists in other contexts; see \cite{Lawvere,CauchyCompleteCatTheory}. We will try to avoid confusion by using the adjective \emph{metric} or adverb \emph{metrically} to denote terms related to limits coming from the interleaving distance and by using the adjective \emph{categorical} to refer to the category theory versions.}; i.e. if every Cauchy sequence has a metric limit. In particular, one might hope that there are categorical conditions we can place on $\Cat$ to ensure it is. \\
In this paper, we study the issue of metric completeness of categories with a flow. We are particularly interested in the connection between categorical properties of $\Cat$ and $T_\e$ and their metric properties. The following is an example of the kind of theorem that is proved:
\begin{theorem*} Let $(\Cat, T_\e)$ be a category with a flow. Then $(\Cat, T_\e)$ is metrically complete if $\Cat$ is categorically complete and $T_\e$ preserves limits.
\end{theorem*}
These conditions are in fact stronger than what we actually need; see Theorem \ref{thm:CategorialConditionsForCompleteness}. The proof involves constructing a diagram out of a given Cauchy sequence; then (under conditions on $T_\e$), the categorical limit of the diagram is a metric limit of the Cauchy sequence. We also describe the closure of a subcategory with a strict flow in a larger, complete category. This gives an analog of Cauchy completion using the (co-)Yoneda Embedding. \\
We can use this general framework to study completeness for many specific examples of categories with interleavings. Generalized persistence modules and derived sheaves are two more new-to-the-literature contexts in which we address completeness. \\
Let's give some indication of the motivation for this work. One might try to study convergence of probability measures directly on these categories with interleavings. Lots of previous work has studied stochastic processes at the level of simplicial complexes (e.g. Erd\"os-Renyi simplicial complexes) and at the level of persistence diagrams, persistence landscapes, betti numbers, and other invariants. However, working at the intermediate level of categories with interleaving distances would be preferable in many contexts, especially when a sufficiently descriptive and well-behaved invariant has not been found. Multi-parameter persistence modules come to mind. Very little work on the convergence of probability measures has been done at this level, in part because there does not yet exist strong foundations for this study. What is needed is an understanding at least of metric completion and separability of the category, and hopefully also some idea of the precompact subsets\footnote{Perhaps we should say precompact subcategories.}. Prokhorov's Theorem is a prime example of these three concepts in play.\\
While the majority of this paper is focused around studying the metric completeness of a category with a flow, the last section situates this paper in the broader context of Polish spaces and their applications to the convergence of probability measures.
\subsection*{Overview}
Section \ref{sec:Background} gives background and terminology for categories with a flow, heavily influenced by \cite{MunchCatsFlow}. It also introduces the dual notion of categories with a coflow. Section \ref{sec:Main} defines Cauchy sequences in categories with flows and gives categorical conditions on when they have metric limits. Section \ref{sec:Completions} explains how to use the Yoneda embedding to densely embed any category with a strict flow into a metrically complete category. Section \ref{sec:Examples} studies some example categories and shows whether they are complete or not. Section \ref{sec:Polish} gives some indication of the context and broader interest of these results by giving an application towards finding categories with a flow which are Polish spaces.
\subsection*{Acknowledgements}
I'd like to acknowledge and thank the IMA and the CMO for hosting conferences this year which gave me access to several very useful conversations. In particular, the idea for this paper was conceived at the IMA's conference ``Bridging Sheaves and Statistics''. At both these conferences, many people were very helpful. In particular, Justin Curry, Peter Bubenik, and Nikola Milicevic all gave important observations and pointers that made this paper much better.
\section{Background on Categories with Flows}\label{sec:Background}
The interleaving distance was first defined in the context of persistence modules \cite{Chazal09}. It has since been generalized in many different ways. Lesnick \cite{Lesnick} defines an interleaving distance on multiparameter persistence modules. Bubenik, de Silva, and Scott \cite{Bub15} define an interleaving distance for generalized persistence modules, which are functor categories $[\mathcal{P},\mathcal{D}]$, where $\mathcal{D}$ is any category and $\mathcal{P}$ is a preordered set. Kashiwara and Schapira \cite{KS} define an interleaving distance for constructible derived sheaves on $\R^n$.\\
De Silva, Munch, and Stefanou \cite{MunchCatsFlow,StefanouThesis} make a definition which generalizes all other current examples, at least to the author's knowledge. They define a \emph{category with a flow}, and show that this induces an interleaving distance with the required properties, and that this distance agrees with the proposed distance in many other circumstances. Further, they were able to provide a very general stability theorem using a notion of flow-equivariant functors.
\subsection{Definitions and Main Results}
All the definitions and results from this subsection can be found in \cite{MunchCatsFlow}.
\begin{definition} A flow on a category $\Cat$ is an assignment of a endofunctor $T_\e : \Cat \to \Cat$ for every non-negative real number $\e\geq 0$ along with natural transformations $T_{\de \leq \e}: T_\de \Rightarrow T_\e$ for $\de \leq \e$ so that for all $\de\leq \e\leq \zeta$, the following commutes
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
T_\de & & T_{\zeta}\\
& T_{\e}&\\
};
\path[-stealth]
(m-1-1) edge node [above] {$T_{\de \leq \zeta}$} (m-1-3)
edge node[xshift=2pt,yshift=-5pt] [left] {$T_{\de \leq \e}$ } (m-2-2)
(m-2-2) edge node[xshift=1pt,yshift=-5pt] [right] {$T_{\e \leq \zeta}$ } (m-1-3);
\end{tikzpicture}
\end{center}
Said another way, a flow is a functor $T: [0,\infty) \to \mathbf{End}(\Cat)$ from the poset category of non-negative reals to the category of endofunctors of $\Cat$. Further, we require any flow to come with natural transformations $\mu_{\e, \de}: T_\e T_\de \Rightarrow T_{\e + \de}$ and $u: Id_{\Cat} \Rightarrow T_0$ satifying the following relations:
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
& T_\e & \\
T_0 T_\e & & T_\e\\
};
\path[-stealth]
(m-1-2) edge[double, double distance=2pt, -] node [right] { } (m-2-3)
edge node[xshift=5pt,yshift=10pt] [left] {$u I_{T_\e}$ } (m-2-1)
(m-2-1) edge node [below] {$\mu_{0, \e}$} (m-2-3);
\end{tikzpicture}
\hspace{0.5in}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
& T_\e & \\
T_\e T_0 & & T_\e\\
};
\path[-stealth]
(m-1-2) edge[double, double distance=2pt, -] node [right] { } (m-2-3)
edge node[xshift=5pt,yshift=10pt] [left] {$I_{T_\e} u$ } (m-2-1)
(m-2-1) edge node [below] {$\mu_{0, \e}$} (m-2-3);
\end{tikzpicture}
\end{center}
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
T_\e T_\zeta T_\de & & T_\e T_{\zeta + \de} \\
T_{\e + \zeta} T_\de & & T_{\e + \zeta + \de}\\
};
\path[-stealth]
(m-1-1) edge node [above] {$ Id_{T_\e}\mu_{\zeta,\de }$ } (m-1-3)
edge node [left] {$\mu_{\e, \zeta} Id_{T_\de}$ } (m-2-1)
(m-1-3) edge node [right] {$\mu_{\e, \zeta + \de}$} (m-2-3)
(m-2-1) edge node [below] {$\mu_{\e + \zeta, \de}$ } (m-2-3);
\end{tikzpicture}
\hspace{0.5in}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
T_\e T_\zeta & & T_{\e + \zeta} \\
T_\de T_\kappa & & T_{\de + \kappa}\\
};
\path[-stealth]
(m-1-1) edge node [above] {$ \mu_{\e, \zeta}$ } (m-1-3)
edge node [left] {$T_{\e \leq \de}T_{\zeta \leq \kappa}$ } (m-2-1)
(m-1-3) edge node [right] {$T_{\e + \zeta \leq \de + \kappa}$} (m-2-3)
(m-2-1) edge node [below] {$\mu_{\de,\kappa}$ } (m-2-3);
\end{tikzpicture}
\end{center}
A flow is \emph{strict} if the natural transformations $Id_\Cat \Rightarrow T_0$ and $T_\e T_\de \Rightarrow T_{\e + \de}$ are identities for all $\e, \de\in [0,\infty)$. This implies that $T_0 = Id_\Cat$ and $T_\e T_\de = T_{\e + \de}$.\footnote{The implication does not go the other way. There are non-strict flows such that $T_0 = Id_\Cat$ and $T_\e T_\de = T_{\e + \de}$.} A flow is \emph{essentially strict} if the natural transformations $T_\e T_\de \Rightarrow T_{\e + \de}$ and $Id_\Cat \Rightarrow T_0$ are natural isomorphisms.
\end{definition}
\begin{definition}
Two objects $A,B\in \mathcal{C}$ are weakly $\e$-interleaved if there are maps $\alpha: A \to T_\e B$ and $\beta: B \to T_\e A$ and a commutative diagram
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
T_0 A & A & B & T_0 B \\
& T_\e A & T_\e B & \\
T_{2\e} A & T_\e T_\e A & T_\e T_\e B & T_{2\e} B \\
};
\path[-stealth]
(m-1-1) edge node [left] { } (m-3-1)
(m-1-2) edge node [left] { } (m-1-1)
edge node[xshift=-15pt, yshift=5pt] [left] {$\alpha$ } (m-2-3)
(m-1-3) edge node [left] { } (m-1-4)
edge[shorten >=1.5cm,-] node[xshift=15pt, yshift=5pt] [right] {$\beta$ } (m-2-2)
edge[shorten <=1.5cm] node[xshift=15pt, yshift=5pt] [right] {$\beta$ } (m-2-2)
(m-1-4) edge node [left] { } (m-3-4)
(m-2-2) edge node[xshift=-15pt, yshift=5pt] [left] {$T_\e \alpha$ }(m-3-3)
(m-2-3) edge[shorten >=1.5cm,-] node[xshift=15pt, yshift=5pt] [right] {$T_\e \beta$ } (m-3-2)
edge[shorten <=1.5cm] node[xshift=15pt, yshift=5pt] [right] { } (m-3-2)
(m-3-2) edge node [above] { } (m-3-1)
(m-3-3) edge node [right] { } (m-3-4);
\end{tikzpicture}
\end{center}
and we call such a diagram a weak $\e$-interleaving of $A$ and $B$.
\end{definition}
\begin{definition}
The interleaving distance on a category with a flow $(\Cat, T_\e)$ is
$$d_{(\Cat, T_\e)}(A,B) = \inf \{\infty\} \cup \{\e\, \, | A\text{ and } B\text{ are weakly }\e\text{-interleaved}\}$$
When the context is clear, we will drop the subscript and simply write $d(A,B)$.
\end{definition}
\begin{remark}
In the past, most contexts with an interleaving distance used standard $\e$-interleavings (not weak ones). In fact, many contexts rewritten in this framework involve a category with a \emph{strict} flow. For categories with strict flows, weak $\e$-interleavings and the usual $\e$-interleavings are the same, so the interleaving distance is the usual one.
\end{remark}
\begin{theorem} The interleaving distance has several desirable properties:
\begin{itemize}
\item $d(A,B) = 0$ if $A$ and $B$ are isomorphic.
\item $d(A,B) = d(B,A)$
\item $d(A,C) \leq d(A,B) + d(B,C)$
\end{itemize}
\end{theorem}
\begin{remark}
The interleaving distance is not quite a metric. Distances can be infinite, and distances can be zero even if the two objects are not isomorphic.
\end{remark}
\begin{definition}
A strict flow-equivariant functor $H:(\Cat, T_\e) \to (\mathcal{D}, S_\e)$ between categories with a flow is an ordinary functor $H:\Cat \to \mathcal{D}$ so that $HT_\e = S_\e H$ and the following diagrams commute
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
H & H\\
HT_0 & S_0 H\\
};
\path[-stealth]
(m-1-1) edge[double, double distance=2pt, -] node [right] { } (m-1-2)
edge node [left] { } (m-2-1)
(m-1-2) edge node [right] { } (m-2-2)
(m-2-1) edge[double, double distance=2pt, -] node [right] { } (m-2-2);
\end{tikzpicture}
\hspace{0.5in}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
HT_\de & S_\de H \\
HT_\e & S_\e H\\
};
\path[-stealth]
(m-1-1) edge[double, double distance=2pt, -] node [right] { } (m-1-2)
edge node [left] { } (m-2-1)
(m-1-2) edge node [right] { } (m-2-2)
(m-2-1) edge[double, double distance=2pt, -] node [right] { } (m-2-2);
\end{tikzpicture}
\hspace{0.5in}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
HT_\e T_\de & S_\e S_\de H\\
HT_{\e + \de} & S_{\e + \de} H\\
};
\path[-stealth]
(m-1-1) edge[double, double distance=2pt, -] node [right] { } (m-1-2)
edge node [left] { } (m-2-1)
(m-1-2) edge node [right] { } (m-2-2)
(m-2-1) edge[double, double distance=2pt, -] node [right] { } (m-2-2);
\end{tikzpicture}
\end{center}
\end{definition}
Note that our terminology differs slightly from \cite{MunchCatsFlow}.\footnote{We do this because ``flow-equivariant'' can be dualized to ``coflow-equivariant'' when we talk about categories with coflows. The alternative was to call the analogous functors for coflows ``$[0,\infty)^{op}$-equivariant'', which seemed worse.} Also, while this definition is strong enough for the purposes of this paper, \cite{MunchCatsFlow} defines a more general kind of functor and proves a Stability theorem at this level of generality. We will content ourselves with using the more specific version stated here.
\begin{theorem} (Soft Stability Theorem) Let $H$ be a strict flow-equivariant functor between $(\Cat, T_\e)$ and $(\mathcal{D}, S_\e)$. Then $H$ is 1-Lipschitz with respect to the categories' interleaving distances.
\end{theorem}
\subsection{Coflows}
It is not surprising that a categorical structure like a category with a flow would have a dual structure. We call this dual structure a category with a coflow. \\
There is a duality between categories with a flow and categories with a coflow. The following can be thought of as a definition, lemma, or remark, depending on your point of view:
\begin{definition} $(\Cat, T_\e)$ is a category with a coflow if and only if $(\Cat^{op}, T_\e^{op})$ is a category with a flow.
\end{definition}
Some might prefer the language of actegories. Then if a category with a flow is a $[0,\infty)$-actegory, a category with a coflow is a $[0,\infty)^{op}$-actegory.\\
All of the previous section can be dualized: there is a notion of weak $\e$-interleavings, of an interleaving distance, and of strict coflow-equivariant functors for categories with coflows. Likewise, the Triangle Inequality for the interleaving distance and the Soft Stability Theorem are still true for categories with a coflow.\\
In general, categories with coflows come up in cohomological contexts, while categories with flows come up in homological contexts. For example, persistence modules model persistent homology and form a category with a flow. Persistent cohomology lives in the opposite category, which is a category with a coflow. The category of sheaves and the category of derived sheaves are two more examples of categories with interleaving distances which come from a coflow.
\section{Cauchy Sequences in Categories with Interleavings}\label{sec:Main}
\subsection{Definition and Basic Results}
We will use the following lemma repeatedly and implicitly in this section.
\begin{lemma} The natural transformation $T_\de \Rightarrow T_{\e + \de}$ factors through $T_\de \Rightarrow T_0T_\de \Rightarrow T_\e T_\de \Rightarrow T_{\e + \de}$.
\end{lemma}
\begin{proof}
Combining two diagrams from section 2, we get the commutative diagram
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
& T_\de & & T_{\e + \de} \\
T_\de & T_0 T_\de & & T_\e T_\de \\
};
\path[-stealth]
(m-1-2) edge node [above] {$ T_{\de \leq \e+\de}$ } (m-1-4)
(m-2-1) edge node [below] {$u\, I_{T_\de}$} (m-2-2)
edge[double, double distance=2pt, -] node [above] { } (m-1-2)
(m-2-2) edge node [right] {$\mu_{0,\de}$ } (m-1-2)
edge node [below] {$T_{0\leq \e} T_{\de\leq \de }$} (m-2-4)
(m-2-4) edge node [right] {$ \mu_{\e, \de}$ } (m-1-4);
\end{tikzpicture}
\end{center}
\end{proof}
\begin{definition} Let $\Cat$ be a category with a flow. Define a sequence of objects $\{A_n\}$ to be \emph{Cauchy} if for every $\e > 0$, there is an $N_\e$ so that for all $n,m \geq N_\e$, $d(A_n, A_m) < \e$.\\
$A$ is a \emph{metric limit} of the sequence $\{A_n\}$ if for every $\e > 0$, there is an $N_\e$ so that for all $n > N_\e$, $d(A_n, A) < \e$. If every Cauchy sequence $\{A_n\}$ in $\Cat$ has a metric limit, we say $\Cat$ is \emph{metrically complete}.
\end{definition}
\begin{remark}This definition is probably not surprising, but there is a small reason to include it. Cauchy sequences are usually defined for sets with metrics (and quasi-metrics, and extended quasi-metrics); that is to say, for sets. But in this context, we are not guaranteed that the collection of objects of $\Cat$ forms a proper set. Nonetheless, this definition still makes sense.
\end{remark}
\begin{remark} Metric limits of a Cauchy sequence of objects of $\Cat$ are not necessarily unique, even up to isomorphism. However, we have the following lemma
\end{remark}
\begin{lemma} Let $A$ be a metric limit of a Cauchy sequence $\{A_n\}$. Then $B$ is also a metric limit of the same Cauchy sequence if and only if $d(A,B) = 0$.
\end{lemma}
\begin{proof}
Say $d(A,B) = 0$. Then for all $\e$, there is an $N_\e$ so that for all $n > N_\e$, $d(A_n, A) < \e$. Then $d(A_n, B) \leq d(A_n, A) + d(A,B) = d(A_n, A) < \e$, so $B$ is a metric limit of $\{A_n\}$. \\
Say $B$ is another metric limit of $\{A_n\}$. There is some $N_\e$ so that $d(A_n, A) < \e$ and $d(A_n, B) < \e$ for all $n \geq N_\e$. Therefore, $d(A, B) \leq d(A_n, A) + d(A_n, B) \leq 2\e$. Since $\e$ was arbitrary, $d(A,B) = 0$.
\end{proof}
Let $\{\tilde{A}_n\}$ be a Cauchy sequence of elements in $\Cat$. Let $\e_k = 2^{-k}$. Let $N_k$ be a sequence of natural numbers so that $N_k < N_{k+1}$ and for all $n,m \geq N_k$, $d(\tilde{A}_n, \tilde{A}_m) < \e_k/2 = \e_{k+1}$. Set $A_k = \tilde{A}_{N_k}$.
\begin{lemma}\label{lemma:subsequence} $\{A_k\}$ is a Cauchy sequence, and if $A$ is a metric limit of $\{\tilde{A}_k\}$ if and only if $A$ is a metric limit of $\{A_k\}$.
\end{lemma}
The proof of this lemma is the same as the proof that a subsequence of a Cauchy sequence in a metric space is Cauchy with the same limit, which can be found in any undergraduate analysis text.
\subsection{Categorical conditions implying metric completeness}
Take a Cauchy sequence $\{A_k\}$ so that $d(A_k, A_n) < \e_{k+1} = 2^{-(k+1)}$ for all $n > k$. For each $k \geq 1$, choose an $\e_{k+1}$-interleaving between $A_k$ and $A_{k+1}$. This gives us a map $$T_{\e_{k+1}}A_{k+1} \to T_{\e_{k+1}}T_{\e_{k+1}}A_{k+1} \to T_{2\e_{k+1}}A_k = T_{\e_k}A_k$$
Patching these together, we get the diagram
\begin{equation*}
\cdots \to T_{\e_{k+1}}A_{k+1} \to T_{\e_k}A_k \to \cdots \to T_{\e_2}A_2 \to T_{\e_1}A_1\label{eq:MainSequence}
\end{equation*}
If the categorical limit exists, set $A = \lim (\cdots \to T_{\e_{k+1}}A_{k+1} \to T_{\e_k}A_k \to \cdots \to T_{\e_2}A_2 \to T_{\e_1}A_1)$. Denote the limit maps $\phi_k: A \to A_k$.
\begin{theorem}\label{thm:CatLimitEqualsMetricLimit} Let $(\Cat, T_\e)$ be a category with a flow where $T_\e$ preserves categorical limits. Let $\{A_k\}$ be a sequence of objects so that $d(A_k, A_n) < \e_{k+1}$ for all $k$ and $n > k$. For all $k$, pick a weak $\e_{k+1}$-interleaving between $A_k$ and $A_{k+1}$, so you get a map $T_{\e_{k+1}}A_{k+1} \to T_{\e_k} A_k$. Let $A$ be a categorical limit of the diagram
$$\cdots \to T_{\e_{k+1}}A_{k+1} \to T_{\e_k} A_k \to \cdots \to T_{\e_2}A_2 \to T_{\e_1} A_1$$
Then $A$ is a metric limit of $\{A_k\}$.
\end{theorem}
\begin{proof}We will show that $d(A_k, A) < \e_k$ by exhibiting a weak $\e_k$-interleaving between $A_k$ and $A$. This will give us that $A$ is a metric limit of $\{A_k\}$. We will actually give a strict $\e_k$-interleaving.\\
First, consider the following commutative diagram
\begin{center}
\begin{tikzpicture}[scale = 0.94, every node/.style={scale=0.94}]
\matrix (m) [matrix of math nodes,row sep=3em,column sep=2.5em,minimum width=2em]
{
A_k & T_{\e_k}A_k & T_0T_{\e_k}A_k & T_{\e_k}T_{\e_k} A_k\\
T_{\e_{k+1}}A_{k+1} & T_{\e_{k+1}}T_{\e_{k+1}} A_{k+1}& T_{\e_{k+1}}T_{\e_{k+1}}A_{k+1}&T_{\e_k}T_{\e_{k+1}} A_{k+1}\\
T_{\e_{k+1}}T_{\e_{k+2}}A_{k+2} & T_{\e_{k+1}}T_{\e_{k+2}}T_{\e_{k+2}} A_{k+2}& T_{\e_{k+1}+\e_{k+2}}T_{\e_{k+2}}A_{k+2} &T_{\e_k}T_{\e_{k+2}} A_{k+2}\\
T_{\e_{k+1}}T_{\e_{k+2}}T_{\e_{k+3}}A_{k+3} & T_{\e_{k+1}}T_{\e_{k+2}}T_{\e_{k+3}}T_{\e_{k+3}} A_{k+3}& T_{\e_{k+1}+\e_{k+2}+\e_{k+3}} T_{\e_{k+3}} A_{k+3}&T_{\e_k}T_{\e_{k+3}} A_{k+3}\\
};
\path[-stealth]
(m-1-1) edge node [above] { } (m-1-2)
edge node [above] { } (m-2-1)
(m-1-2) edge node [above] { } (m-1-3)
(m-1-3) edge node [above] { } (m-1-4)
(m-2-1) edge node [above] { } (m-2-2)
edge node [above] { } (m-3-1)
edge node [above] { } (m-1-2)
(m-2-2) edge node [above] { } (m-2-3)
(m-2-3) edge node [above] { } (m-2-4)
(m-2-4) edge node [above] { } (m-1-4)
(m-3-1) edge node [above] { } (m-3-2)
edge node [above] { } (m-2-2)
edge node [above] { } (m-4-1)
(m-3-2) edge node [above] { } (m-3-3)
(m-3-3) edge node [above] { } (m-3-4)
(m-3-4) edge node [above] { } (m-2-4)
(m-4-1) edge node [above] { } (m-4-2)
edge node [above] { } (m-3-2)
(m-4-2) edge node [above] { } (m-4-3)
(m-4-3) edge node [above] { } (m-4-4)
(m-4-4) edge node [above] { } (m-3-4);
\end{tikzpicture}
\end{center}
The first triangle we get from the weak $\e_{k+1}$-interleaving between $A_k$ and $A_{k+1}$, the second triangle we get from applying $T_{\e_{k+1}}$ to the weak $\e_{k+2}$-interleaving between $A_{k+1}$ and $A_{k+2}$, and so on. The trapezoids we get from the natural transformation $T_{\e_{k+1}}T_{\e_{k+2}} ... T_{\e_{k+m}} \Rightarrow T_{\e_k}$.\\
For $m \geq k$, denote by $\psi_m: A_k \to T_{\e_k}T_{\e_m}A_m$ the map which we get from this diagram. For $m < k$, denote by $\psi_m: A_k \to T_{\e_k}T_{\e_m}A_m$ the map which makes the rest of the following diagram commute:
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
& & A_k & \\
\cdots & T_{\e_k}T_{\e_{m+1}} A_{m+1} & T_{\e_k}T_{\e_m}A_m & \cdots \\
};
\path[-stealth]
(m-1-3) edge node[xshift=-10pt, yshift=0pt] [above] {$\psi_{m+1}$ } (m-2-2)
edge node [left] {$\psi_m$ } (m-2-3)
(m-2-1) edge node [below] { }(m-2-2)
(m-2-2) edge node [right] { } (m-2-3)
(m-2-3) edge node [right] { } (m-2-4);
\end{tikzpicture}
\end{center}
The collection of maps $\psi_m$ induces a map $\psi: A_k \rightarrow T_{\e_k}A$, because $T_{\e_k}A$ is the limit of that bottom diagram.\\
Now we have the following maps:
\begin{itemize}
\item $\phi_k: A \to T_{\e_k}A_{\e_k}$. This comes from the definition of $A$ as a limit.
\item $T_{\e_k}\phi_k: T_{\e_k}A \to T_{\e_k}T_{\e_k} A_k$. This comes from applying $T_{\e_k}$ to $\phi_i$. By hypothesis, it is also the limit map of $T_{\e_k}A = \lim\left(\cdots \to T_{\e_k}T_{\e_2}A_2 \to T_{\e_k}T_{\e_1}A_1\right)$.
\item $\psi: A_k \to T_{\e_k}A$, which we just defined.
\item $T_{\e_k}\psi: T_{\e_k}A_k \to T_{\e_k}T_{\e_k}A $. This comes from applying $T_{\e_k}$ to $\psi$.
\end{itemize}
We must check that these pentagons commute:
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
T_0 A_k &A_k & \\
& &T_{\e_k} A \\
T_{2\e_k}A_k & T_{\e_k}T_{\e_k} A_k & \\
};
\path[-stealth]
(m-1-1) edge node [left] { } (m-3-1)
(m-1-2) edge node[xshift=0pt, yshift=5pt] [right] {$\psi$ } (m-2-3)
edge node [left] { } (m-1-1)
(m-2-3) edge node[xshift=0pt, yshift=-7pt] [right] {$T_{\e_k}\phi_k$ } (m-3-2)
(m-3-2) edge node [right] { } (m-3-1);
\end{tikzpicture}
\hspace{0.5in}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
& A & T_0 A\\
T_{\e_k}A_k & & \\
& T_{\e_k}T_{\e_k} A_k & T_{2\e_k} A_k\\
};
\path[-stealth]
(m-1-2) edge node[xshift=0pt, yshift=7pt] [left] {$T_{\e_k}\psi$ } (m-2-1)
edge node [right] { } (m-1-3)
(m-2-1) edge node[xshift=0pt, yshift=-5pt] [left] {$\phi_k$} (m-3-2)
(m-3-2) edge node [left] { } (m-3-3)
(m-1-3) edge node [left] { } (m-3-3);
\end{tikzpicture}
\end{center}
We will do this by showing the stronger statement that these triangles commute:
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
A_k & & T_{\e_k}T_{\e_k}A_k \\
& T_{\e_k}A&\\
};
\path[-stealth]
(m-1-1) edge node [left] { } (m-1-3)
edge node[xshift=-2pt, yshift=-3pt] [left] {$\psi$ } (m-2-2)
(m-2-2) edge node[xshift=5pt, yshift=-3pt] [right] {$T_{\e_k}\phi_k$ } (m-1-3);
\end{tikzpicture}
\hspace{0.5in}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
& T_{\e_k}A_k & \\
A & & T_{\e_k}T_{\e_k}A\\
};
\path[-stealth]
(m-1-2) edge node[xshift=-3pt, yshift=6pt] [right] {$T_{\e_k}\psi$ } (m-2-3)
(m-2-1) edge node[xshift=3pt, yshift=6pt] [left] {$\phi_k$} (m-1-2)
edge node [left] { } (m-2-3);
\end{tikzpicture}
\end{center}
The first commutes by the definition of $\psi$. The second triangle is harder. Notice we have the commutative diagram
\begin{center}
\begin{tikzpicture}[scale = 0.94, every node/.style={scale=0.94}]
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
T_{\e_k}A_k & & & T_{\e_k}A_k \\
T_{\e_{k+1}}T_{\e_{k+1}}A_k & T_{\e_{k+1}}T_{\e_{k+1}}T_{\e_{k+1}}A_{k+1} & T_{\e_{k}}T_{\e_{k+1}}A_{k+1} &\\
T_{\e_{k+1}}A_{k+1} & & &T_{\e_k}T_{\e_{k+1}}A_{k+1}\\
T_{\e_{k+2}}T_{\e_{k+2}}A_{k+1} & T_{\e_{k+2}}T_{\e_{k+2}}T_{\e_{k+2}}A_{k+2}& T_{\e_{k+1}}T_{\e_{k+2}}A_{k+2} &\\
T_{\e_{k+2}}A_{k+2} & & &T_{\e_k}T_{\e_{k+1}}T_{\e_{k+2}}A_{k+2}\\
};
\path[-stealth]
(m-1-1) edge node [right] { } (m-1-4)
edge node [left] { } (m-2-3)
(m-1-4) edge node[xshift=-50pt, yshift=20pt] [left] {$\textbf{D}$ } (m-3-4)
(m-2-1) edge node[xshift=50pt, yshift=-5pt] [right] {$\textbf{A}$ } (m-1-1)
edge node [left] { } (m-2-2)
(m-2-2) edge node [left] { } (m-2-3)
(m-3-1) edge node[xshift=50pt, yshift=5pt] [right] {$\textbf{B}$ } (m-2-1)
edge node [left] { } (m-2-3)
edge node[xshift=75pt, yshift=15pt] [above] { $\textbf{C}$} (m-3-4)
edge node [left] { } (m-4-3)
(m-2-3) edge node [left] { } (m-3-4)
(m-3-4) edge node [left] { } (m-5-4)
(m-4-1) edge node [left] { } (m-3-1)
edge node [left] { } (m-4-2)
(m-5-1) edge node [left] { } (m-4-1)
edge node [left] { } (m-4-3)
edge node [left] { } (m-5-4)
(m-4-2) edge node [left] { } (m-4-3)
(m-4-3) edge node [left] { } (m-5-4) ;
;
\end{tikzpicture}
\end{center}
The $\textbf{A}$ triangles come from applying the natural transformations $T_{\e_{k+m+1}}T_{\e_{k+m+1}} \Rightarrow T_{\e_{k+m}}$ to the interleaving maps $A_{k+m} \to T_{\e_{k+m+1}}A_{k+m+1}$. The $\textbf{B}$ triangles come from the weak $\e_{k+m+1}$-interleaving between $A_{k+m}$ and $A_{k+m+1}$. The horizontal maps are defined so that the $\textbf{C}$ triangles commute. Lastly, we know the $\mathbf{D}$ diagrams commute by applying the natural transformations $Id \Rightarrow T_{\e_k}T_{\e_{k+1}}...T_{\e_{k+m}}$ to the maps $A_{k+m} \to T_{\e_{k+m+1}}A_{k+m+1}$.\\
We can concatenate this diagram with $T_{\e_k}$ applied to an earlier diagram to get
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=3em,minimum width=2em]
{
A & T_{\e_k}A_k & T_{\e_k}A_k & T_{\e_k}T_{\e_k}T_{\e_k} A_k\\
& T_{\e_{k+1}} A_{k+1} & T_{\e_k}T_{\e_{k+1}}A_{k+1} & T_{\e_k}T_{\e_k}T_{\e_{k+1}} A_{k+1}\\
& T_{\e_{k+2}} A_{k+2}& T_{\e_k}T_{\e_{k+1}}T_{\e_{k+2}}A_{k+2} A_{k+2}& T_{\e_k}T_{\e_k}T_{\e_{k+2}} A_{k+2}\\
};
\path[-stealth]
(m-1-1) edge node [above] {$\phi_k$} (m-1-2)
edge node [above] { } (m-2-2)
edge node [above] { } (m-3-2)
(m-1-2) edge node [above] { } (m-1-3)
(m-2-2) edge node [above] { } (m-1-2)
edge node [above] { } (m-2-3)
(m-3-2) edge node [above] { } (m-2-2)
edge node [above] { } (m-3-3)
(m-1-3) edge node [above] { } (m-1-4)
edge node [above] { } (m-2-3)
(m-2-3) edge node [above] { } (m-2-4)
edge node [above] { } (m-3-3)
(m-2-4) edge node [above] { } (m-1-4)
(m-3-3) edge node [above] { } (m-3-4)
(m-3-4) edge node [above] { } (m-2-4);
\end{tikzpicture}
\end{center}
This diagram tells us $T_{\e_k}T_{\e_k}T_{\e_m}\phi_m = T_{\e_k}\psi_m\circ \phi_k$. These maps induce a map from $A$ to $T_{\e_k}T_{\e_k}A$. By continuity of $T_{\e_k}$, $T_{\e_k}T_{\e_k}T_{\e_m}\phi_m$ induces the map $T_{\e_k}T_{\e_k}T_{\e_m}\phi$, and $T_{\e_k}\psi_m\circ \phi_k$ induces the map $T_{\e_k}\psi\circ \phi_k$. Therefore, our second triangle commutes.\\
This shows there is an $\e_k$-interleaving between $A_k$ and $A$, and therefore $d(A_k, A) < \e_k$. Thus, $A$ is a metric limit of $\{A_k\}$.
\end{proof}
\begin{theorem}\label{thm:CategorialConditionsForCompleteness}A category with a flow $(\Cat, T_\e)$ is metrically complete if
\begin{itemize}
\item $\Cat$ contains all limits of the form $\cdots\to\bullet \to \bullet \to \bullet$, and
\item for all $\e > 0$, $T_\e$ preserves categorical limits.
\end{itemize}
Dually, a category with a coflow $(\Cat, T_\e)$ is metrically complete if
\begin{itemize}
\item $\Cat$ contains all colimits of the form $\bullet \to \bullet \to \bullet\to \cdots $, and
\item for all $\e > 0$, $T_\e$ preserves (categorical) colimits.
\end{itemize}
\end{theorem}
\begin{proof}
We show the proof for the category with a flow; dualize this to get a proof for a category with a coflow. \\
Let $\{\tilde{A}_k\}$ be a Cauchy sequence in $(\Cat, T_\e)$. As described before, there is a subsequence $\{A_k\}$ so that $d(A_k, A_n) < \e_{k+1}$ for all $n > k$. Then we can create a diagram
$$\cdots \to T_{\e_{k+1}}A_{k+1} \to T_{\e_k} A_k \to \cdots \to T_{\e_2}A_2 \to T_{\e_1} A_1$$
$\Cat$ has categorical limits of all diagrams of this form; denote this limit $A$. Then by Theorem \ref{thm:CatLimitEqualsMetricLimit}, $A$ is a metric limit of $\{A_k\}$, and by Lemma \ref{lemma:subsequence}, $A$ is also a metric limit of $\{\tilde{A}_k\}$. \\
Thus, every Cauchy sequence has a limit, and $(\Cat, T_\e)$ is metrically complete.
\end{proof}
\section{Metric Completions}\label{sec:Completions}
In this section, we will work with categories with a coflow for convenience's sake. The results we reference from outside sources are more familiar in the coflow case. The metric completion of a category with a flow can be found by dualizing and completing the resulting category with a coflow.\\
\begin{lemma}\label{lemma:definitionOfCompletion}Let $(\Cat, T_\e)$ and $(\mathcal{D}, S_\e)$ be categories with coflows, and $(\Cat, T_\e)$ be a strict coflow-equivariant, full subcategory of $(\mathcal{D}, S_\e)$. Let $T_\e$ be a strict flow, $\mathcal{D}$ be cocomplete, and $S_\e$ preserve colimits. Then there is a full subcategory $\mathcal{Z}$ of $\mathcal{D}$ so that
\begin{enumerate}
\item $\Cat$ is a full subcategory of $\mathcal{Z}$.
\item $\Cat$ is metrically dense in $\mathcal{Z}$; i.e. for every object $Z\in \mathcal{Z}$ and real number $\e > 0$, there is an object $A \in \Cat$ so that $d(A,Z) < \e$.
\item $S_\e$ preserves $\mathcal{Z}$.
\item $(\mathcal{Z}, S_\e)$ is metrically complete.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\{\tilde{A}_k\}$ be a Cauchy sequence in $\mathcal{C}$, and let $\{A_k\}$ be a subsequence so that $d(A_n,A_m) < \e_{k+1}$ for all $n,m \geq k$, as in section \ref{sec:Main}. Then we get as before a diagram
\begin{equation*}
T_{\e_1}A_1 \to T_{\e_2}A_2 \to \cdots \to T_{\e_k}A_k \to T_{\e_{k+1}}A_{k+1}\to\cdots
\end{equation*}
We can consider this a diagram in $\mathcal{D}$. It has a colimit $A$ in $\mathcal{D}$, because that category has all small colimits.\\
Let the objects of $\mathcal{Z}$ be the objects of $\mathcal{C}$ along with any object of $\mathcal{D}$ which is a colimit $A$ of a diagram of the form given above. Then $\mathcal{Z}$ is defined to be the full subcategory of $\mathcal{D}$ with that collection of objects.\\
We will now show that $\mathcal{Z}$ satisfies the properties of this corollary. (1) and (2) follow directly from the definition. To show (3), first note that if $Z$ is an object of $\mathcal{Z}$, $Z$ is the colimit of some diagram
\begin{equation*}
T_{\e_1}A_1 \to T_{\e_2}A_2 \to \cdots \to T_{\e_k}A_k \to T_{\e_{k+1}}A_{k+1}\to\cdots
\end{equation*}
of the type above. Then $S_\e Z$ is the colimit of the diagram
\begin{equation*}
S_\e T_{\e_1}A_1 \to S_\e T_{\e_2}A_2 \to \cdots \to S_\e T_{\e_k}A_k \to S_\e T_{\e_{k+1}}A_{k+1}\to\cdots
\end{equation*}
Because the embedding preserves the flow, this diagram is equivalent to
\begin{equation*}
T_\e T_{\e_1}A_1 \to T_\e T_{\e_2}A_2 \to \cdots \to T_\e T_{\e_k}A_k \to T_\e T_{\e_{k+1}}A_{k+1}\to\cdots
\end{equation*}
which by strictness of $T_\e$ is the same as
\begin{equation*}
T_{\e_1}\left(T_\e A_1\right) \to T_{\e_2}\left(T_\e A_2\right) \to \cdots \to T_{\e_k}\left(T_\e A_k\right) \to T_{\e_{k+1}}\left( T_\e A_{k+1}\right)\to\cdots
\end{equation*}
Therefore, $S_\e Z$ is also an object of $\mathcal{Z}$, and we've shown (3). \\
Lastly, we'll show (4). Take a Cauchy sequence $\{Z_k\}$ of objects of $\mathcal{Z}$. Then by (3), for each $Z_k$ there is an object $A_k$ in $\mathcal{C}$ so that $\ds d(A_k, Z_k) < 2^{-k}$. Then $\{A_k\}$ is also a Cauchy sequence with the same set of metric limits as $\{Z_k\}$. Because $\{A_k\}$ consists of objects of $\mathcal{C}$, it has a metric limit in $\mathcal{Z}$. Therefore, $\{Z_k\}$ also has a metric limit in $\mathcal{Z}$.
\end{proof}
\begin{definition}\label{def:Closure} We denote the largest subcategory satisfying the properties of Lemma \ref{lemma:definitionOfCompletion} as the \emph{closure of $(\Cat, T_\e)$ in $(\mathcal{D},S_\e)$}.
\end{definition}
\begin{proof}This is one of those definitions that requires proof. Let $A$ be an object of $\mathcal{X}$ if and only if $A$ is an object of a subcategory $\mathcal{Z}$ of $\mathcal{D}$ satisfying the conditions of Lemma \ref{lemma:definitionOfCompletion}; Let $\mathcal{X}$ be the full subcategory with this class of objects. It is clear that any category that satisfies the conditions of Lemma \ref{lemma:definitionOfCompletion} must be a subcategory of $\mathcal{X}$. We will show $\mathcal{X}$ also satisfies those conditions. \\
Conditions 1, 2, and 3 are easy to verify. Condition 4 can be shown as follows. Take a Cauchy sequence $\{A_k\} \subset \mathcal{X}$. By condition 2, for each $k$ there is an object $B_k \in \Cat$ so that $d(A_k, B_k) \leq \frac{1}{k}$. Now $\{B_k\}$ is a Cauchy sequence of objects in $\Cat$, and an object $A\in \mathcal{D}$ is a metric limit of $\{A_k\}$ if and only if it is a metric limit of $\{B_k\}$. Since Lemma \ref{lemma:definitionOfCompletion} says there is a subcategory $\mathcal{Z}$ which is metrically complete and contains each $B_k$, $\{B_k\}$ has a metric limit $A \in \mathcal{Z} \subseteq \mathcal{X}$. Thus, $\{A_k\}$ has a metric limit in $\mathcal{X}$, and $\mathcal{X}$ is metrically complete.
\end{proof}
Let $\Cat$ be a locally small category (i.e. for every pair of objects $A,B \in \Cat$, $Hom(A,B)$ is a proper set). Let $h_{-}$ be the Yoneda functor which takes an object $A$ in $\Cat$ and sends it to a contravariant functor from $\Cat$ to $\mathbf{Set}$ like so:
$$h_- : A \mapsto h_A, \text{ where } h_A(B) = Hom(B,A)$$
The Yoneda Lemma tells us that the natural transformations between the functors $h_A$ and $h_B$ are in one-to-one correspondence with the morphisms between $A$ and $B$. In other words, $Nat(h_A, h_B) = Hom(A,B)$. \\
One interpretation of this is that $\mathcal{C}$ can be embedded as a full subcategory of $\left[\Cat^{op},\mathbf{Set}\right]$, the category of contravariant functors from $\Cat$ to $\mathbf{Set}$.\footnote{This is sometimes called the category of set-valued presheaves on $\Cat$.} This is called the Yoneda embedding.\footnote{ Information on the Yoneda embedding and on category theory more generally can be found in \cite{Riehl} and \cite{MacLane}.}
\begin{lemma}\label{lemma:YonedaFacts} Assume $(\Cat, T_\e)$ is a small category. Then
\begin{enumerate}
\item $\left[\Cat^{op},\mathbf{Set}\right]$ is locally small.
\item $\left[\Cat^{op},\mathbf{Set}\right]$ is complete and cocomplete, meaning it contains all small limits and small colimits. In particular, it contains all colimits of diagrams of the form $\bullet \to \bullet \to \bullet \to \cdots$.
\item For every $\e > 0$, there is a pointwise left Kan extension $L_\e = \text{\emph{Lan}}_{h_-} h_-\circ T_\e$
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
\Cat & \Cat & \left[\Cat^{op},\mathbf{Set}\right]\\
\left[\Cat^{op},\mathbf{Set}\right]\\
};
\path[-stealth]
(m-1-1) edge node [above] {$T_\e$ } (m-1-2)
edge node [left] {$h_-$ } (m-2-1)
(m-1-2) edge node [above] {$h_-$ } (m-1-3)
(m-2-1) edge[dashed] node [below] {$L_\e$ } (m-1-3);
\end{tikzpicture}
\end{center}
and we can choose $L_\e$ so that $L_\e\circ h_- = h_-\circ T_\e$.
\item $L_\e$ preserves all small colimits.
\item $L_\e$ has a right adjoint.
\item $L_0 = Id_{[\Cat^{op}, \mathbf{Set}]}$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) follows directly from the Yoneda Lemma. Limits and colimits in functor categories are computed ``pointwise'', so (2) follows from the completion and cocompletion of $\mathbf{Set}$. The diagram above satisfies the conditions of Corollary 6.2.6 in \cite{Riehl}, which says the pointwise left Kan extension exists; that we can choose it to be a true extension is indicated in Corollary 4 in Section X.3 of \cite{MacLane}. (4) is a property of any pointwise left Kan extension along Yoneda; see Section 2.7 of \cite{KSCategoriesAndSheaves}. Further, because $[\Cat^{op}, \mathbf{Set}]$ is locally presentable, (5) follows from (4) by the Adjoint Functor Theorem. (6) is a famous result; another way this is said is that the Yoneda embedding is (categorically) codense (see Section X.6 in \cite{MacLane}).
\end{proof}
\begin{theorem} Assume $(\Cat, T_\e)$ is a category with a strict flow, and define the functors $L_\e$ as in Lemma \ref{lemma:YonedaFacts}. Then
\begin{enumerate}
\item $L_\e$ defines a coflow on $\left[\Cat^{op},\mathbf{Set}\right]$.
\item $\left(\Cat, T_\e\right) \xrightarrow{h_-} \left(\left[\Cat^{op},\mathbf{Set}\right], L_\e\right)$ is a full, strict coflow-equivariant embedding of categories.
\item $\left(\left[\Cat^{op},\mathbf{Set}\right], L_\e\right)$ is a metrically complete category.
\end{enumerate}
\end{theorem}
\begin{proof}
To show $L_\e$ defines a coflow, we must give natural transformations $L_\e \Rightarrow L_\de$ for $\de \leq \e$, $L_0 \Rightarrow Id$, and $L_{\e+\de} \Rightarrow L_\e L_\de$ so that all the diagrams from Section \ref{sec:Background} commute.\\
First, we can define the natural transformations $L_\e \Rightarrow L_\de$ by applying the universal property of a Left Kan extension to the diagram
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
\Cat & \Cat & \left[\Cat^{op},\mathbf{Set}\right]\\
\Cat & \Cat & \left[\Cat^{op},\mathbf{Set}\right]\\
\left[\Cat^{op},\mathbf{Set}\right]\\
};
\path[-stealth]
(m-1-1) edge node [above] {$T_\e$ } (m-1-2)
edge[double, double distance=2pt, -] node [left] { } (m-2-1)
(m-1-2) edge node [above] {$h_-$ } (m-1-3)
edge[double, double distance=2pt, -] node [above] { } (m-2-2)
(m-1-3) edge[double, double distance=2pt, -] node [above] { } (m-2-3)
(m-2-1) edge node [above] {$T_\de$ } (m-2-2)
edge node [left] {$h_-$ } (m-3-1)
(m-2-2) edge node [above] {$h_-$ } (m-2-3)
(m-3-1) edge node [below] {$L_\de$ } (m-2-3);
\end{tikzpicture}
\end{center}
This also shows why $L_\e$ is functorial in the $\e$. Lemma \ref{lemma:YonedaFacts}.6 says $L_0 = Id$, so that natural transformation is the identity.\\
Notice also that $L_\e$, being a left adjoint, preserves left Kan extensions; see Theorem 1 of Section X.5 in \cite{MacLane}. In particular,
\begin{align*}
L_\e \circ L_\de &\Leftarrow L_\e \circ \text{Lan}_{h_-} (h_- \circ T_\de)\\
&\Leftarrow \text{Lan}_{h_-} \left(L_\e \circ h_- \circ T_\de\right) \\
&= \text{Lan}_{h_-} \left(h_- \circ T_\e \circ T_\de \right)\\
&= \text{Lan}_{h_-} \left(h_- \circ T_{\e + \de}\right)\\
&\Leftarrow L_{\e + \de}
\end{align*}
where the arrows are the unique natural isomorphisms making the appropriate diagram commute. This is the natural transformation $L_{\e + \de} \Rightarrow L_\e L_\de$ we choose.\\
The only thing left to proving (1) is verifying that the four coflow relation diagrams commute. The ones involving $L_0$ are trivial because $L_0 = Id$. The other two follow from the properties of a Kan extension.\\
Yoneda says $h_-$ is a full embedding of categories. Lemma \ref{lemma:YonedaFacts}.3 says this is a strict co-equivariant embedding of categories. This gives (2).\\
Lastly, Lemmas \ref{lemma:YonedaFacts}.2 and \ref{lemma:YonedaFacts}.4 tell us that $[\Cat^{op}, \mathbf{Set}]$ satisfies the conditions of Theorem \ref{thm:CategorialConditionsForCompleteness}. This gives (3).
\end{proof}
\begin{corollary}\label{cor:ExistenceOfCompletion} For any category $(\Cat, T_\e)$ with a strict flow, there is a full subcategory $\overline{\Cat}$ of $\left[\Cat^{op},\mathbf{Set}\right]$ so that
\begin{enumerate}
\item $\Cat$ is a full subcategory of $\overline{\Cat}$.
\item $\Cat$ is metrically dense in $\overline{\Cat}$; i.e. for every object $Z\in \overline{\Cat}$ and real number $\e > 0$, there is an object $A \in \Cat$ so that $d(A,Z) < \e$.
\item $L_\e$ preserves $\overline{\Cat}$.
\item $(\overline{\Cat}, L_\e)$ is metrically complete.
\item $\overline{\Cat}$ is the largest subcategory of $\left[\Cat^{op},\mathbf{Set}\right]$ to satisfy properties 1-4.
\end{enumerate}
\end{corollary}
\begin{proof}
The theorem is essentially checking the conditions of Lemma \ref{lemma:definitionOfCompletion}, of which this corollary is a direct application. We then just apply Definition \ref{def:Closure}.
\end{proof}
\begin{definition} We call $(\overline{\Cat}, L_\e)$ the \emph{metric completion of $(\Cat, T_\e)$}.
\end{definition}
\section{Examples}\label{sec:Examples}
\subsection{$(\Q, \geq)$}
Consider the category $(\Q, \geq)$, with one object for every rational number, and $\ds \Hom(q,p) = \left\{\begin{array}{cl} \ast & \text{ if }q \geq p\\ \emptyset & \text{ otherwise}\end{array}\right.$, where $\ast$ is the set with one element and $\emptyset$ is the empty set. Define a coflow $T_\e q = q + \e$ (this is technically only defined for rational $\e$, but all our previous work goes through anyway). Then $q$ and $p$ are $\e$-interleaved if and only if $|q-p| \leq \e$. \\
A sequence $\{\tilde{q}_k\}$ is Cauchy in the context of this paper if and only if it is Cauchy in the usual sense. We can find a subsequence $\{q_k\}$ so that $q_k$ and $q_{k+1}$ are $\ds \e_{k+1}$-interleaved. Then our diagram looks like $ q_1 + \e_1 \geq q_2 + \e_2 \geq \cdots$, which has a colimit in $(\Q, \geq)$ if and only if $\{q_k\}$ converges to a rational number in the usual sense. \\
The Yoneda embedding is a functor $h_-: (\Q, \geq) \to [(\Q, \geq)^{op}, \mathbf{Set}]$. In particular, $$h_q(p) = \Hom(p,q) = \left\{\begin{array}{cl} \ast & \text{ if }p \geq q\\ \emptyset & \text{ otherwise}\end{array}\right. ,$$ and $h_q$ is a contravariant functor where $h_-$ sends the inequality $q_1 \geq q_2$ to the inclusion $h_{q_2}(p) \supseteq h_{q_1}(p)$. Thus, $h_q$ is essentially a Dedekind cut, and if $r$ is the traditional limit of $\{q_k\}$ in $\R$, then the categorical colimit of $h_{q_1 + \e_1}\to h_{q_2 + \e_2} \to \cdots $ in $[(\Q, \geq)^{op}, \mathbf{Set}]$ is ``$h_r$'', where $$h_r(p) = \left\{\begin{array}{cl} \ast & \text{ if }p \geq r\\ \emptyset & \text{ otherwise}\end{array}\right.$$
One can show the completion $\overline{(\Q, \geq)}$ is equivalent to the category $(\R, \geq)$. However, it is not true that the two are isomorphic as categories. In particular, $(\R, \geq)$ is a skeletal category, while the completion $\overline{(\Q, \geq)}$ is not.
\subsection{Persistence Modules}
Persistence modules form the most popular category with a flow, though there are really many different categories depending on which of many different types of conditions we impose. A recent paper by Bubenik and Vergili \cite{Bub18} studies metric completeness for several of the most used categories of persistence modules (along with a host of other properties from general topology).\\
A persistence module is a functor from the order category\footnote{Notice this category is the opposite category of the one considered in the last section, i.e. $(\R, \leq) = (\R, \geq)^{op}$.} $(\R, \leq)$ to the category of $\mathbf{k}$-vector spaces $\mathbf{Vect_k}$. The category of persistence modules can be denoted $[(\R,\leq), \mathbf{Vect_k}]$. The interleaving functor $T_\e$ on a persistence module is given by
$$(T_\e M)(a) = M(a + \e)\hspace{1.5in} (T_\e M)(a\leq b) = M(a+\e \leq b+\e)$$
\begin{example}$[(\R,\leq), \mathbf{Vect_k}]$ with the flow $T_\e$ is metrically complete.\end{example}
\begin{proof} We must show $[(\R,\leq), \mathbf{Vect_k}]$ with $T_\e$ satisfies the conditions of Corollary \ref{thm:CategorialConditionsForCompleteness}. Taking limits of functors is done pointwise; since $\mathbf{Vect_k}$ is complete, so is $[(\R,\leq), \mathbf{Vect_k}]$. Further, $T_\e$ is an autoequivalence of categories, so it preserves basically all categorical constructions, including limits.
\end{proof}
Let $\mathbf{Vect_k^{fin}}$ be the category of finite dimensional $\mathbf{k}$-vector spaces. Then $[(\R,\leq), \mathbf{Vect_k^{fin}}]$ is the category of persistence modules $M$ where $M(a)$ is finite dimensional for each $a\in \R$. We can use the flow $T_\e$ as before because $T_\e$ preserves $[(\R,\leq), \mathbf{Vect_k^{fin}}]$.
\begin{example}$[(\R,\leq), \mathbf{Vect_k^{fin}}]$ with $T_\e$ is \emph{not} metrically complete.
\end{example}
\begin{proof}Let $[a,b)$ be the interval module where
$$M(s) = \left\{\begin{array}{cl}\mathbf{k} & \text{ for }s\in [a,b)\\ 0 & \text{ otherwise}\end{array}\right. \hspace{1in} M(s\leq t) = \left\{\begin{array}{cl}Id_\mathbf{k} & \text{ for }s,t\in [a,b)\\ 0 & \text{ otherwise}\end{array}\right.$$
Define the sequence of persistence modules $A_n = \bigoplus_{k=1}^n \left[\frac{-1}{k}, \frac{1}{k}\right)$. Then $\{A_n\}$ is a Cauchy sequence but has no metric limit. $\{A_n\}$ is Cauchy because for every $\e > 0$, there is an $N$ so that $T_\e A_n = T_\e A_m$ for all $n,m > N$, so the $\e$-interleaving is easy to find. It has no metric limit, because any such limit $M$ would need an infinite dimensional vector space $M(0)$.
\end{proof}
In \cite{Bub18}, Bubenik and Vergili make similar calculations for categories of persistence modules with many different types of conditions. Note that their proof for completeness is different from ours and uses slightly different hypotheses (in general neither weaker nor stronger).
\subsection{Generalized Persistence Modules}\label{ex:GPM}
Bubenik, de Silva, and Scott \cite{Bub15} introduced the notion of a generalized persistence module (GPM). A category of generalized persistence modules is a functor category $[\mathcal{P},\mathcal{D}]$, where $\mathcal{P}$ is a poset category and $\mathcal{D}$ is some arbitrary category. To calculate interleaving distances, they also introduced the notion of a superlinear family of translations. Then Munch, de Silva, and Stefanou \cite{MunchCatsFlow} showed how generalized persistence modules with a superlinear family of translations can be thought of a flow on the category $[\mathcal{P},\mathcal{D}]$, where the flow functors $T_\e$ are pullbacks along translations.\\
The important observation in checking metric completeness for this case is that limits in functor categories are computed pointwise. This has two corollaries:
\begin{enumerate}
\item Pulling back a functor along a translation preserves limits.
\item If $\mathcal{D}$ is a complete category, so is $[\mathcal{P},\mathcal{D}]$.
\end{enumerate}
Therefore, we get the following:
\begin{corollary} Assume $\mathcal{P}$ is a poset category, $\mathcal{D}$ is a (categorically) complete category, and $[\mathcal{P}, \mathcal{D}]$ is the category of functors from $\mathcal{P}$ to $\mathcal{D}$. Choose a superlinear family of translations, and use it to define an interleaving distance as in \cite{Bub15} or \cite{MunchCatsFlow}. Then $[\mathcal{P}, \mathcal{D}]$ is metrically complete.
\end{corollary}
Interestingly, it is complete no matter which superlinear family of translations is chosen.
\subsection{The Derived Category of Sheaves}\label{ex:sheaves}
Recently, Kashiwara and Schapira \cite{KS} defined a flow on the derived category of sheaves of $\mathbf{k}$-vector spaces on $\R^m$, denoted $D(\mathbf{k}_{\R^m})$. We will not rehash all the details of that paper, but the broad strokes are these:\\
Let $s: \R^m \times \R^m \to \R^m$ be the addition map $s(x,y) = x+y$. Let $\mathbf{k}$ be a vector space. Let $K_\e = \mathbf{k}_{\{\|x\| \leq \e\}}$ for $\e\geq 0$. Then for an object $A\in D(\mathbf{k}_{\R^m})$, define\footnote{It deserves to be pointed out that while $T_\e$ makes sense as a functor on $D(\mathbf{k}_{\R^m})$ (without the bounded or constructible conditions) as well as on $D_c^b(\mathbf{k}_{\R^m})$ (with both the bounded and constructible conditions), it doesn't make sense on $D_c(\mathbf{k}_{\R^n})$; specifically, if $A\in D_c(\mathbf{k}_{\R^n})$ is not bounded, it is very possible that $T_\e A$ is not constructible.} $T_\e A = K_\e \star A = Rs_!(K_\e \boxtimes A)$. \\
It is shown in \cite{KS} that $T_\e$ forms a strict coflow on $D(\mathbf{k}_{\R^m})$. \\
\begin{corollary}
$D(\mathbf{k}_{\R^m})$ under the interleaving distance is metrically complete.
\end{corollary}
\begin{proof}We must show that convolving with $K_\e$ preserves colimits and that $D(\mathbf{k}_{\R^m})$ has enough colimits. Convolution with $K_\e$ is an autoequivalence of categories, and so it preserves basically every categorical construction, including colimits.\\
Thus, it suffices to show that a diagram $\triangle$ of the form
$$A_1 \to A_2 \to ... $$
has colimits in $D(\mathbf{k}_{\R^m})$. Certainly $\triangle$ has colimits in $Sh(\mathbf{k}_{\R^m})$, because $Sh(\mathbf{k}_{\R^m})$ is cocomplete. The functor $\colim_\triangle: Sh(\mathbf{k}_{\R^m})^\triangle \to Sh(\mathbf{k}_{\R^m})$ is right exact, because all colimit functors are right exact. If we can show it is left exact too, then $\colim_\triangle$ would be exact, and it would extend to a functor $\colim_\triangle : D(\mathbf{k}_{\R^m})^\triangle \to D(\mathbf{k}_{\R^m})$. This would mean $\triangle$ has colimits in $D(\mathbf{k}_{\R^m})$.\\
Therefore, it suffices to show that $\colim_\triangle$ is left exact. Say we have the exact sequence of diagrams
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]
{
0 & 0 & 0 & \cdots & \\
A_1 & A_2 & A_3 & \cdots & A\\
B_1 & B_2 & B_3 & \cdots & B\\
};
\path[-stealth]
(m-1-1) edge node [left] { } (m-2-1)
(m-1-2) edge node [left] { } (m-2-2)
(m-1-3) edge node [left] { } (m-2-3)
(m-2-1) edge node [left] {$p_1$} (m-3-1)
(m-2-2) edge node [left] {$p_2$} (m-3-2)
(m-2-3) edge node [left] {$p_3$} (m-3-3)
(m-2-5) edge node [left] {$p$} (m-3-5)
(m-2-1) edge node [above] { }(m-2-2)
(m-2-2) edge node [above] { }(m-2-3)
(m-2-3) edge node [above] {}(m-2-4)
(m-2-4) edge node [above] { }(m-2-5)
(m-3-1) edge node [above] { }(m-3-2)
(m-3-2) edge node [above] { }(m-3-3)
(m-3-3) edge node [above] { }(m-3-4)
(m-3-4) edge node [above] { }(m-3-5);
\end{tikzpicture}
\end{center}
where $A$ and $B$ are colimits of their respective diagrams, and all the $A_k, B_k$ are sheaves. We wish to show that $p$ is injective. This amounts to showing that $p$ is injective on stalks. But because taking colimits commutes with taking stalks, we can assume the $A_k$, $B_k$, etc. in the above diagram are vector spaces over $k$. \\
Now we must show the direct limit of direct system of vector spaces is an exact functor. This is a standard result, but a proof is included here because I could not find one in the literature. We can explicitly define $A$ as
$$A = \bigoplus_{n=1}^\infty A_n \Big/\left(a \sim \phi(a)\right)\hspace{0.5in} B = \bigoplus_{n=1}^\infty B_n \Big/\left(b \sim \psi(b)\right)$$
where $\phi$ and $\psi$ are the ``shift'' maps on $\bigoplus_{n=1}^\infty A_n$ and $\bigoplus_{n=1}^\infty B_n$, respectively. Say that $p([a]) = 0$ for some $[a]\in A$. We can represent $[a]$ which some $a\in A_n$ for some $n$, and $p([a]) = [p_n(a)] \in B$. Therefore, $[p_n(a)] = 0$, so $\psi^k(p_n(a)) = 0$ for some $k$. By commutativity of the diagram, $p_{n+k}(\phi^k(a)) = 0$. By injectivity of $p_{n+k}$, $\phi^k(a) = 0$. Therefore, $[a] = [\phi^k(a)] = 0$. This shows that $p$ is injective, which completes our proof.
\end{proof}
\begin{example} As an important example, $D^b_c(\mathbf{k}_{\R^m})$ is not metrically complete. Let $F_n = \bigoplus_{k=1}^N \mathbf{1}_{[0, 2^{-k})}[-k]$. Then $\{F_n\}$ is a Cauchy sequence, but the limit does not exist in $D^b_c(\mathbf{k}_{\R^m})$. \\
Nor is it only unboundedness that is the problem. Define $C_n = \bigcup_{k=1}^n \mathbf{1}_{(2^{-k},2^{-(k-1)}]}$ and $C = \bigcup_{n=1}^\infty C_n$. Set $F_n = \mathbf{1}_{C_n}$ and $F =\mathbf{1}_C$ . Each of the $F_n \in D^b_c(\mathbf{k}_{\R^n})$ and $\lim_{n\to \infty} F_n$ exists in $D(X)$ and is bounded, but $\lim_{n\to \infty} F_n$ is not constructible.
\end{example}
\section{Categories with a flow and Polish Spaces}\label{sec:Polish}
\footnote{This section will be more informal, but it will hopefully be useful for giving context for this paper. In particular, we will not rigorously define what we mean by separability. For the purposes of this paper, separability can mean that the induced extended quasimetric on the coskeleton is separable.}A pressing motivation for the research in this paper was in providing conditions for categories with a flow to be Polish spaces, i.e. complete and separable. Polish spaces are important because they are the foundation for many results from statistics and probability. Particularly when considering convergence of probability measures, it is important for those probability measures to be on a complete, separable space; see, for example, Prokhorov's Theorem or the result that every probability measure on a Polish space is tight \cite{Billingsley}.\\
This paper thus far has only addressed one side of that issue, that of completeness. Separability of categories with a flow seems difficult to characterize in general. Certainly specific cases are tractable, and it does seem to be related to the notion of $\kappa$-accessibility and similar concepts. However, an all-purpose, easy-to-check test remains elusive. \\
Often what happens in practice is that we have a ``little'' category which is separable sitting inside a ``big'' category which is complete, but neither is Polish. In this situation, we can find a Polish space between the two. In particular, the closure of the ``little category'' in the ``big category'' (see Definition \ref{def:Closure}) is a Polish space. \\
As an example, take the space of bounded, constructible sheaves $D^b_c(\mathbf{k}_{\R^m})$. This space is not complete, as shown in Section \ref{ex:sheaves}, but it is separable as long as $\mathbf{k}$ is countable, which can be shown with the help of Theorem 2.11 of \cite{KS}. On the other hand, $D(\mathbf{k}_{\R^m})$ is not separable for $m\geq 2$, but it is complete. Neither space is an appropriate space in which to do statistics, but we can use Lemma \ref{lemma:definitionOfCompletion} to create a Polish space of sheaves. \\
A similar question is ``What category of persistence modules should be used from applications?'' On the one hand, the whole category of persistence modules is metrically complete. However, in \cite{Bub18} it is shown that the collection of isomorphism classes of persistence modules is not only non-separable, but not even a set! Thus the whole category of persistence modules is too big. \\
On the other hand, there is a useful subcategory which is separable, namely the constructible persistence modules; see Remark 3.1 of \cite{Bub18}. However, this space is not complete. To see why this is a problem, consider the probability measure on persistence modules we get from looking at the persistence modules obtained from Brownian motion as was done, more or less, in \cite{Adler}. This measure is supported on persistence modules which are not constructible (nor even pointwise finite-dimensional). Thus, the category of constructible persistence modules is too little.\footnote{This was noticed in the ``decategorified'' setting of persistence diagrams by Mileyko, Mukherjee, and Harer \cite{MMH}.}\\
In fact, \cite{Bub18} considers over ten different categories of persistence modules, and none of them were both complete and separable except the ones which were trivially so, like the category of only the zero module or the category of ephemeral modules. One way to solve this problem is to take a separable subcategory and find its closure in all persistence modules using Definition \ref{def:Closure}.\\
The closure $\Cat$ of the constructible persistence modules in all persistence modules seems like a useful candidate category for using persistence modules in applications for two reasons. First, it is Polish, so we can use the relevant theorems from probability and statistics. Second, the acclaimed 1-Lipschitz map from the Stability Theorem \cite{Harer} sending bounded, continuous functions to persistence modules has its image in $\Cat$. In particular, it seems large enough to handle the expected examples of random persistence modules.
\bibliography{sources}
\bibliographystyle{plain}
\large
Department of Mathematics, Duke University\\
\indent \url{joshua.cruz@duke.edu}
\end{document}
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10.31.06
Chicago 10th Anniversary Cast Recording - John Kander
Legendary composer John Kander shares the story of the musical Chicago.
Featuring The Music of Masterworks and Masterworks Broadway Podcast Theatre
Legendary composer John Kander shares the story of the musical Chicago.
Legendary music orchestrator Jonathan Tunick discusses the new production and cast recording of A Chorus Line.
Tony-Award winner Ann Reinking talks dance, Fosse and about the revival of the musical Chicago.
Original cast member and revival choreographer Baayork Lee shares the ACL story from the start to the revival as well as discussing the Michael Bennett dance style.
Director Walter Bobbie describes the magic of Chicago’s stage revival.
Legendary choreographer, producer and director Bob Avian discusses the revival of the Pulitzer Prize winning musical “A Chorus Line.”
Bebe Neuwirth celebrates the 10th Anniversary of the stage production of the musical Chicago.
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TITLE: Does this two equation system have a solution for l(1)?
QUESTION [2 upvotes]: I'm having a debate with a friend over this problem:
$$\eqalign{
k(x) &= l(x)+8 \cr
l(x) &= 14-k(x)
}$$
What is the value of $l(1)$?
I think this isn't solvable because of the circular reference/dependency, but my friend thinks that it can be done with substitution, resulting in constant answers:
$$\eqalign{
l(x)&=14-[l(x)+8] \cr
2l(x)&=6 \cr
l(x)&=3
}$$
which leads to
$$\eqalign{
k(x)&=3+8 \cr
k(x)&=11
}$$
Thoughts?
REPLY [1 votes]: I think this isn't solvable because of the circular reference/dependency
It's just a linear system of 2 equations in 2 variables, where it does not matter that $k$ and $\ell$ are functions. One of the following cases will occur:
The system has no solutions because it's over-determined. Example would be if the equations are, say $\ell=1$ and $2\ell=-1$ etc.
There is a unique solution
There is a space of solutions, i.e. the solution is not unique. The equations just limit the degree of freedom of the 2-dimensional space in which $k$ and $\ell$ live.
In your case, the system of equations is
$$\begin{align}
k - \ell &= 8 \\
k + \ell &= 14
\end{align}$$
so that adding these equations gives an equation for $k$, and subtracting the equations gives an equation for $\ell$. It is case 1. from above, i.e. the solution is unique.
If $k$ and $\ell$ are functions (of $x$), then they only satisfy the system of equations if they are constant functions with the values you computed.
| 14,698
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TITLE: Projective representations of extensions of $PSL_2(q)$
QUESTION [3 upvotes]: Let $K=PSL_2(q)$ where $q=p^a$ for some odd prime $p$, and let $G$ be a group such that $G/O(G)\cong K$. (Here $O(G)$ is the largest odd-order normal subgroup of $G$.)
I have a homomorphism $\phi: G\to PGL_n(\overline{\mathbb{F}_r})$ whose image is non-solvable. (Here $r$ is a prime distinct from $p$.) I am interested in giving a lower bound for $n$.
In the case where $O(G)$ is trivial, a classical result originating with work of Frobenius gives a sharp lower bound for $n$, namely $\frac12(q-1)$. (Provided $q\neq9$, but let's ignore this exception.)
I'd like to prove that this bound is best possible, i.e.
Q1. For arbitrary $G$ of the given form, prove that $n\geq \frac12(q-1)$.
I reckon I can do this by brute force using Aschbacher's classification of subgroups of $GL_n(q)$ but I'd prefer a more elegant solution. A couple of easy reductions allow me to assume that $G$ acts absolutely irreducibly, that $G$ is center-less, and that $G$ is a minimal non-split extension of $K$, i.e. there does not exist a proper subgroup $H$ such that $H$ is an extension of $K$.
These reductions led me to wonder about a related (but probably much harder) question:
Q2. What are the center-less minimal non-split extensions of $K$?
I know that examples of these things exist where $G$ is not isomorphic to $K$ (see, for instance this MO question, in particular an example mentioned in the answer of @nkrempel). However I cannot find a systematic treatment of these things in the literature.
Final note: the word `extension' has two meanings in group theory. In the definition I'm using the group $G$ is an example of an extension of $K$.
REPLY [2 votes]: I would guess that the most efficient approach to this is to use the arguments used in the proof of the Aschbacher Theorem, which you can do without just citing the theorem and ploughing through the cases. There is a nice proof of a basic version of Aschbacher's Theorem in Theorem 3.5 (page 85) of "The Finite Simple Groups" by Robert A. Wilson.
It goes roughly as follows. Let $N$ be the socle of the image of $G$ in ${\rm PGL}_n(r^k)$ (for suitable $k$). So $N$ will have odd order in your case. If $N$ is not homogeneous, then $G$ acts imprimitively, so you get a reduction in degree or to a permutation group. If $N$ is homogeneous but reducible, then you get a tensor product decomposition for $G$, which again gives you a degree reduction. The same applies if $N$ is not the unique minimal normal subgroup, and if it is, then you are in the symplectic normalizer case, which again gives you an easy degree reduction.
| 3,715
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TITLE: The hairy ball theorem, from Brouwer's fixed point.
QUESTION [2 upvotes]: EDIT : The question is now the following. I know this statement of the hairy ball theorem :
Theorem : Let $n \geq 3$ be an odd number, and $f:\mathbb{S}^{n-1} \rightarrow \mathbb{R}^n$ be a continuous map such that $\langle f(x),x \rangle = 0$ for every $x \in \mathbb{S}^{n-1}$. Then there exists $x_0 \in \mathbb{S}^{n-1}$ such that $f(x_0)=0$.
In the linked paper provided by C.F.G., there is this version of hairy ball theorem :
Theorem : For any continuous map $v : B \rightarrow R^n$ such that $\langle v(x), x \rangle \leq 0$ for all $x \in \mathbb{S}^{n-1}$, there
exists some $z \in B$ such that $v(z)=0$.
To be honest, I don't really see how to go from one of these versions to the other : how the even dimension is replaced, in the second version, by the hypothesis $\langle v(x), x \rangle \leq 0$ ? Is someone could explain how these two statements are linked, that would be great !
$$-------------------------------------$$
Original question :
My question is rather simple today : is there an easy proof of hairy ball theorem that uses Brouwer fixed point theorem ?
I know that these two results are similar in some ways, and I know that they have proofs that rely on the same kind of arguments (I saw Milnor proofs for these two results), but my concern is : let's suppose that you know that Brouwer's fixed point theorem is true ; is there a way to deduce the hairy ball theorem with only elementary steps from there ?
I guess there may be an obstruction, due to the fact that Brouwer theorem is true in every dimension, whereas the hairy ball theorem is not.
If someone knows a short proof, or has a reference, it would be really nice.
Thanks !
REPLY [2 votes]: You can find a proof of this fact in the appendix of
Penot, Jean-Paul, Analysis. From concepts to applications, Universitext. Cham: Springer (ISBN 978-3-319-32409-8/pbk; 978-3-319-32411-1/ebook). xxiii, 669 p. (2016). ZBL1366.26002.
that is accessible freely in publisher website.
| 152,048
|
TITLE: Inverting a system of equations
QUESTION [0 upvotes]: $$a_{out} = a\ \mathrm{\cos}(\theta) + i b\ \mathrm{\sin}(\theta)$$
$$b_{out} = b\ \mathrm{\cos}(\theta) + i a\ \mathrm{\sin}(\theta)$$
What procedure would one use to invert this to get:
$$a = a_{out}\ \mathrm{\cos}(\theta) - i b_{out}\ \mathrm{\sin}(\theta)$$
$$b = b_{out}\ \mathrm{\cos}(\theta) - i a_{out}\ \mathrm{\sin}(\theta)$$
I can't seem to reproduce this result by solving the first system. Am I missing something obvious?
REPLY [2 votes]: This is a system of linear equations. If we add and subtract them we get $$a_{out}+b_{out}=(a+b)(\cos \theta + i \sin \theta)\\a_{out}-b_{out}=(a-b)(\cos \theta - i \sin \theta)\\a+b=\frac {a_{out}+b_{out}}{\cos \theta + i \sin \theta}\\a-b=\frac {a_{out}-b_{out}}{\cos \theta - i \sin \theta}\\a=\frac 12\left(\frac {a_{out}+b_{out}}{\cos \theta + i \sin \theta}+\frac {a_{out}-b_{out}}{\cos \theta - i \sin \theta}\right)\\a=\frac 12(a_{out}+b_{out})(\cos \theta -i\sin \theta)+\frac 12(a_{out}-b_{out})(\cos \theta +i\sin \theta)\\a=a_{out}\cos \theta -b_{out}i\sin \theta\\b=-a_{out}i\sin \theta +b_{out}\cos \theta$$
| 145,641
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TITLE: Prove that any vector can be written as the sum of any three non-coplanar vectors
QUESTION [2 upvotes]: I'm trying to prove the above in the following form:
$ \boldsymbol{V} = (\boldsymbol{V} \centerdot \boldsymbol{a}^1)\boldsymbol{a}_1 + (\boldsymbol{V} \centerdot \boldsymbol{a}^2)\boldsymbol{a}_2 + (\boldsymbol{V} \centerdot \boldsymbol{a}^3)\boldsymbol{a}_3 $,
where $ \boldsymbol{a}_1 $, $ \boldsymbol{a}_2 $ and $ \boldsymbol{a}_3 $ are any three non-coplanar vectors, and $ \boldsymbol{a}^1 $, $ \boldsymbol{a}^2 $ and $ \boldsymbol{a}^3 $ are corresponding "reciprocal" vectors, such that:
$ \boldsymbol{a}^1 = \dfrac{\boldsymbol{a}_2 \times \boldsymbol{a}_3}{\boldsymbol{a}_1 \centerdot \boldsymbol{a}_2 \times \boldsymbol{a}_3} $,
$ \boldsymbol{a}^2 = \dfrac{\boldsymbol{a}_3 \times \boldsymbol{a}_1}{\boldsymbol{a}_1 \centerdot \boldsymbol{a}_2 \times \boldsymbol{a}_3} $,
$ \boldsymbol{a}^3 = \dfrac{\boldsymbol{a}_1 \times \boldsymbol{a}_2}{\boldsymbol{a}_1 \centerdot \boldsymbol{a}_2 \times \boldsymbol{a}_3} $.
Is there a way to complete the proof without doing a lengthy expansion? What is an intuitive reason for why the scaling factors $ \boldsymbol{V} \centerdot \boldsymbol{a}^i $ work?
For example, taking $ \boldsymbol{V} \centerdot \boldsymbol{a}^1 $ I've got as far as supposing that if $ \boldsymbol{a}_2 \times \boldsymbol{a}_3 $ is a vector representing the base of parallelepiped $ \boldsymbol{a}_1 \centerdot \boldsymbol{a}_2 \times \boldsymbol{a}_3 $, then dividing top and bottom, $ \boldsymbol{V} \centerdot \boldsymbol{a}_2 \times \boldsymbol{a}_3 $ and $ \boldsymbol{a}_1 \centerdot \boldsymbol{a}_2 \times \boldsymbol{a}_3 $, by $ |\boldsymbol{a}_2 \times \boldsymbol{a}_3| $, leaves the length of $ \boldsymbol{V} $ relative to the length of $ \boldsymbol{a}_1 $ as measured along the vector $ \boldsymbol{a}_2 \times \boldsymbol{a}_3 $. I'm not sure though how to relate this to the rectangular coordinate system common to $ \boldsymbol{V} $ and $ \boldsymbol{a}_1 $.
REPLY [0 votes]: If you rewrite the starting identity as
$$\boldsymbol{V} = b_1\boldsymbol{a}_1 + b_2\boldsymbol{a}_2 + b_3\boldsymbol{a}_3 $$
and put it into matricial form
$$\boldsymbol{V} = \boldsymbol{A}\, \boldsymbol b $$
where $\boldsymbol{A}$ is the matrix composed by the vertical vectors $\boldsymbol{a}_k$, and $\boldsymbol b $ is the vertical vector with components $b_k$, then
$$det(\boldsymbol {A})={\boldsymbol{a}_1 \centerdot \boldsymbol{a}_2 \times \boldsymbol{a}_3}$$
and your starting identity is just an alternative way to write the Cramer's rule, in solving $\boldsymbol{V} = \boldsymbol{A}\, \boldsymbol b $ for $\boldsymbol b $.
| 159,939
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Kingstonian Win A Home Game
In a 46-game league season , we can all be guilty of too eagerly labelling certain matches “must-win”. But, without doubt, after three straight 1-0 home defeats, and without a victory at Kingsmeadow since mid-August, this Monday night fixture really did matter.
Manager Tommy Williams had complained of too many “similar” displays in recent weeks, with players going through the motions in an orthodox 4-4-2. A change in formation was a bold move – and it paid off.
K’s lined up in a 4-3-3 with Pelayo Pico Gomez the central, elusive striker, while Chris Henry and new loanee striker/winger Loick Pires played as inside-forwards –an Isthmian version of peak Spain/Barca.
Much of the summer was spent talking about keeping the 2013/14 team together but circumstances have dictated major changes to the frontline.
When Met Police took the lead through an Aaron Goode own goal in the 12th minute, it looked like 2014/15 was on the line. Then something happened that had nothing to do with tactics and everything to do with hard work – Pico Gomez pressed a lost cause, took advantage of a goalkeeping error and poked the ball into the net, reminiscent of a similar strike at Witham Town earlier this month.
At 1-1, K’s were transformed. After the ball broke from a corner in the 30th minute, Steve Laidler showed superb vision to knock the ball through a crowd of players to Matt Drage, recalled in Sam Page’s absence, who was lurking on the far side of the 6-yard box. Drage, a ball-playing centre-half if ever there was one, shaped to shoot first-time and curled the ball into the corner.
Two minutes later K’s made it 3-1 with another fine strike. Luke Pigden, signed from Wealdstone to add balance and mobility to midfield, arcked the ball over the keeper’s head with a fluid swing of his left boot.
In truth, aside from those mesmeric moments, K’s created little thereafter. But the need for a win was so great, and the threat from Met Police (albeit minus the injured Matt Pattison) serious enough, that few would begrudge a quiet second half.
Match report by Taimour Lay.
| 349,328
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Blue Turtle announces Parasoft’s Application Security Compliance solution
Parasoft’s unique automated infrastructure unobtrusively drives the development process to ensure that secure software is delivered on time and on budget.
Blue Turtle, the Parasoft reseller in southern Africa, adds Parasoft’s unique Application Security Compliance solution to its arsenal. Parasoft, a leading provider of solutions and services that deliver quality as a continuous process, announced the availability of enhanced data flow analysis capabilities that help organizations rapidly identify high-risk runtime security vulnerabilities as well as monitor security policy compliance.
With its newly integrated features, Parasoft’s Application Security Compliance now offers the following additional capabilities:
Security as a Continuous Process
Parasoft’s Application Security Solution establishes a continuous process that ensures security verification and remediation tasks are deployed across every stage of the SDLC and ingrained into the team’s workflow.
Establish, Apply, and Monitor Adherence to Policies
Parasoft’s policy-driven approach defines the organization’s expectations around quality while ensuring consistent, unobtrusive policy application. The automated infrastructure automatically monitors policy compliance for visibility and auditability.
Out-of-the-box Support for Critical Security Standards and Initiatives
Achieve compliance with industry best-practices, standards, and guidelines, including PCI DSS, OWASP, CWE/SANS, NIST SAMATE, and more
Easily Configure Custom Rules for Enforcing Coding Best Practices
The extensive, continually-expanding knowledge base of rules can be easily customized (graphically, without coding) to enable automated monitoring of custom best practices. The result is more realistic and accurate validation that is aligned with the team’s security priorities.
Robust Security Validation and Verification Practices
Parasoft automates a broad spectrum of application security activities for C/C++, Java, .NET, SOA, Web, and RIA– the most comprehensive in the industry, including:
- Static code analysis – Coding standards, data flow, metrics–including preconfigured PCI DSS 6 and OWASP configurations.
- Peer review – Workflow management and automation.
- Penetration testing – Message layer and web interface.
- Message Layer Policy Validation – Authentication, encryption, and access control.
- Runtime analysis – Buffer overflows and other memory errors.
- Unit testing – Verification of input validation methods.
Facilitates Remediation—Not Just Detection
To promote rapid remediation, each vulnerability detected is prioritized, automatically correlated to the developer who introduced it, then distributed to his or her IDE with direct links to the problematic code. Eventually, developers start writing more secure code as a matter of habit.
Extensive Centralized Reports
Parasoft’s centralized reporting system provides real-time visibility into overall security status and processes, documents improvements, and helps you determine what additional actions are needed to safeguard security.
| 214,796
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Mather spoke about the foreign-born players’ ability to speak English, the team’s player personnel decisions and financial situation, and about the tactics the team used to limit the serving time of young prospects, a maneuver that allows the team to retain control over the player for longer. .
Stanton said he was “extremely disappointed” when he learned of Mather’s comments to the Rotary Club.
“I want to apologize to all members of the Seattle Mariners organization, especially our players and our fans,” Mather said, according to MLB.com. “There is no excuse for my behavior, and I take full responsibility for my terrible misjudgment (sic).”
The outrage over Mather’s comments spread throughout the league, including from some of the people mentioned in Mather’s comments.
In the team’s statement announcing Mather’s resignation, Stanton said “[Mather’s] the comments were inappropriate and do not represent our organization’s feelings about our players, staff and fans. “
“There is no excuse for what was said, and I will not try to make one,” Stanton added. “I offer my sincere apologies on behalf of the club and my partners to our players and fans. We must be, and do, better.”
Stanton will assume the role of interim president and CEO of the team until a successor is announced.
The MLB Players Association was also highly critical of Mather’s comments. In a statement, the MLBPA called Mather’s video “an extremely disturbing but critically important window on how players are actually viewed by management” and said the comments provide “an unfiltered look at the Club’s thinking.”
“It is offensive, and it is not surprising that fans and others around the game are also offended,” the MLBPA statement read.
Mather also faced controversy in 2018
This is not the first time Mather has had to apologize for his actions while working as an executive for the Mariners. In 2018, she apologized after it came to light that two team employees complained about inappropriate language and Mather’s actions between 2009 and 2010, while Mather served as the team’s executive vice president.
“Throughout my career, I have tried to treat people with respect and professionalism,” Mather said in 2018. “As I rose through the ranks, I thought I had to be a demanding manager, but I realized that that I found it intimidating or even cruel at times. I also participated in jokes and at times it was too familiar to me, in ways that I found were inappropriate in the workplace.
“At the time, I did not recognize how my actions were affecting the people around me. I really feel sorry for the people I hurt and how I found myself. It was a humbling experience, and I have tried to learn from my mistakes. I take full responsibility. for my actions, and I appreciated the opportunity to change my behavior and the management training I received. I have worked to become a better co-worker, a better leader, and a better person. “
The Mariners said at the time that after an investigation into the allegations against Mather, the organization “imposed appropriate discipline, handling and sensitivity training, and other corrective actions.”
The Mariners also revealed that the team had “made amends” for the employees involved.
“Kevin learned from the experience and has been an outstanding manager and executive ever since,” Stanton said in a statement in 2018. “The Mariners’ owners took this into consideration, as well as nearly 20 years of work history and performance. from Kevin, considering him for promotion to president in 2014 and CEO last year. We would not have promoted Kevin if we had any doubts about his ability to lead and meet our high standards. “
| 212,814
|
TITLE: convergence in law in $\mathbb{R}^d$
QUESTION [1 upvotes]: In my probability professor's notes there's this lemma given as obvious, but I am not able to prove its truth.
Let $(X_n)_{n \in \mathbb{N}}$ and $X$ be random variables with values in $\mathbb{R}^d$.
Prove that the following to statements are equivalent:
the sequence $(X_n)_{n \in \mathbb{N}}$ converges in law to $X$.
for any $u \in \mathbb{R}^d$ we have that $(\langle u,X_n \rangle)_{n \in \mathbb{N}}$ converges in law to $\langle u,X \rangle$.
I am able to prove the easy direction (i.e. 1. $\implies$ 2.).
Can somebody provide me a proof of 2. $\implies$ 1. ?
REPLY [2 votes]: This is the Cramér-Wold theorem. A proof is given in Billingsley's Probability and Measure (p.383 Theorem 29.4). This is a very standard reference.
The way in Billingsley to prove "2 implies 1" is using the characteristic functions (and the continuity theorem). A key observation is that for any $t\in{\bf R}^d$,
$$
\varphi_X(t)= \varphi_{\langle t,X\rangle}(1)
$$
where $\varphi_X$ denotes the characteristic function of the random vector $X$ and $\varphi_{\langle t,X\rangle}$ is the characteristic function of the random variable $\langle t,X\rangle$.
| 190,127
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\begin{document}
\title[$p$-adic Invariant Quantum Fields\ via \ White Noise \ Analysis]{Construction of $p$-adic Covariant Quantum Fields in the Framework of White
Noise Analysis }
\author{Edilberto Arroyo-Ortiz}
\author{W. A. Z\'{u}\~{n}iga-Galindo}
\address{Centro de Investigaci\'{o}n y de Estudios Avanzados del Instituto
Polit\'{e}cnico Nacional\\
Departamento de Matem\'{a}ticas, Unidad Quer\'{e}taro\\
Libramiento Norponiente \#2000, Fracc. Real de Juriquilla. Santiago de
Quer\'{e}taro, Qro. 76230\\
M\'{e}xico.}
\email{earroyo@math.cinvestav.mx, \ wazuniga@math.cinvestav.edu.mx}
\thanks{The second author was partially supported by Conacyt Grant No. 250845.}
\subjclass[2000]{Primary 60H40, 81E05; Secondary 11Q25, 46S10}
\keywords{White noise, Kondratiev spaces, Hida spaces, Euclidean quantum fields,
non-Archimedean analysis.}
\begin{abstract}
In this article we construct a large class of interacting Euclidean quantum
field theories, over a $p$-adic space time, by using white noise calculus. We
introduce $p$-adic versions of the Kondratiev and Hida spaces in order to use
the Wick calculus on the Kondratiev spaces. The quantum fields introduced here
fulfill all the Osterwalder-Schrader axioms, except the reflection positivity.
\end{abstract}
\maketitle
\section{Introduction}
In this article, we construct interacting Euclidean quantum field theories,
over a $p$-adic spacetime, in arbitrary dimension, which satisfy all the
Osterwal\-der-Schrader axioms \cite{Osterwalder-Schrader} except for
reflection positivity. More precisely, we present a $p$-adic analogue of the
interacting field theories constructed by Grothaus and Streit in
\cite{GS1999}. The basic objects of an Euclidean quantum field theory are
probability measures on distributions spaces, in the classical case, on the
space of tempered distributions $\mathcal{S}^{\prime}(\mathbb{R}^{n})$. In
conventional quantum field theory (QFT) there have been some studies devoted
to the optimal choice of the space of test functions. In \cite{Jaffe}, Jaffe
discussed this topic (see also \cite{Lopu} and \cite{Strocchi}); his
conclusion was that, rather than an optimal choice, there exists a set of
conditions that must be satisfied by the candidate space, and any class of
test functions with these properties should be considered as valid. The main
condition is that the space of test functions must be a nuclear countable
Hilbert one. This fact constitutes the main mathematical motivation the study
of QFT on general nuclear spaces.
A physical motivation for studying QFT in the $p-$adic setting comes from the
conjecture of Volovich stating that spacetime has a non-Archimedean nature at
the Planck scale, \cite{Vol}, see also \cite{Var1}. The existence of the
Planck scale implies that below it the very notion of measurement as well as
the idea of `infinitesimal length' become meaningless, and this fact
translates into the mathematical statement that the Archimedean axiom is no
longer valid, which in turn drives to consider models based on $p$-adic
numbers. In the \ $p$-adic framework, the relevance of constructing quantum
field theories was stressed in \cite{V-V-Z} and \cite{Var2}. In the last 35
years \ $p$-adic QFT has attracted a lot of attention of physicists and
mathematicians, see e.g. \cite{Abd et al}, \cite{Dra01}-\cite{EU66},
\cite{Gubser}, \cite{Koch-Sait}-\cite{Koch}, \cite{Kh1}-\cite{LM89},
\cite{Mendoza-Zuniga}-\cite{Mis2}, \cite{Smirnov}-\cite{Smirnov-2},
\cite{Var1}-\cite{Zuniga-LNM-2016}, and the references therein.
A $p$-adic number is a sequence of the form
\begin{equation}
x=x_{-k}p^{-k}+x_{-k+1}p^{-k+1}+\ldots+x_{0}+x_{1}p+\ldots,\text{ with }
x_{-k}\neq0\text{,} \label{p-adic-number}
\end{equation}
where $p$ denotes a fixed prime number, and the $x_{j}$s \ are $p$-adic
digits, i.e. numbers in the set $\left\{ 0,1,\ldots,p-1\right\} $. There are
natural field operations, sum and multiplication, on series of form
(\ref{p-adic-number}). The set of all possible $p$-adic sequences constitutes
the field of $p$-adic numbers $\mathbb{Q}_{p}$. The field $\mathbb{Q}_{p}$ can
not be ordered. There is also a natural norm in $\mathbb{Q}_{p}$ defined as
$\left\vert x\right\vert _{p}=p^{k}$, for a nonzero $p$-adic number $x$ of the
form (\ref{p-adic-number}). The field of $p$-adic numbers with the distance
induced by $\left\vert \cdot\right\vert _{p}$ is a complete ultrametric space.
The ultrametric property refers to the fact that $\left\vert x-y\right\vert
_{p}\leq\max\left\{ \left\vert x-z\right\vert _{p},\left\vert z-y\right\vert
_{p}\right\} $ for any $x$, $y$, $z$ in $\mathbb{Q}_{p}$. As a topological
space, $\left( \mathbb{Q}_{p},\left\vert \cdot\right\vert _{p}\right) $ is
completely disconnected, i.e. the connected components are points. The field
of $p$-adic numbers has a fractal structure, see e.g. \cite{A-K-S},
\cite{V-V-Z}. All these results can be extended easily to $\mathbb{Q}_{p}^{N}
$, see Section \ref{Sction_2}.
In \cite{Zuniga-FAA-2017}, see also \cite[Chapter 11]{KKZuniga}, the second
author introduced a class of non-Archimedean massive Euclidean fields, in
arbitrary dimension, which are constructed as solutions of certain covariant
$p$-adic stochastic pseudodifferential equations, by using techniques of white
noise calculus. In particular a new non-Archimedean Gel'fand triple was
introduced. By using this new triple, here we introduce non-Archimedean
versions of the Kondratiev and Hida spaces, see Section \ref{Section_4}. The
non-Archimedean Kondratiev spaces, denoted as $\left( \mathcal{H}_{\infty
}\right) ^{1}$, $\left( \mathcal{H}_{\infty}\right) ^{-1}$, play a central
role in this article.
Formally an interacting field theory with interaction $V$ \ has associated a
measure of the form
\begin{equation}
d\mu_{V}=\frac{\exp\left( -\int_{\mathbb{Q}_{p}^{N}}V\left( \boldsymbol{\Phi
}\left( x\right) \right) d^{N}x\right) d\mu}{\int\exp\left(
-\int_{\mathbb{Q}_{p}^{N}}V\left( \boldsymbol{\Phi}\left( x\right) \right)
d^{N}x\right) d\mu}, \label{Eq_A1}
\end{equation}
where $\mu$ is the Gaussian white noise measure, $\boldsymbol{\Phi}\left(
x\right) $ is a random process at the point $x\in\mathbb{Q}_{p}^{N}$. \ In
general $\boldsymbol{\Phi}\left( x\right) $ is not an integrable function
rather a distribution, thus a natural problem is how to define $V\left(
\boldsymbol{\Phi}\left( x\right) \right) $. For a review about the
techniques for regularizing $V\left( \boldsymbol{\Phi}\left( x\right)
\right) $ and the construction of the associated measures, the reader may
consult \cite{Glimm-Jaffe}, \cite{GS1999}, \cite{Simon}, \cite{Strocchi} and
the references therein.
Following \cite{GS1999}, we consider the following generalized white
functional:
\begin{equation}
\boldsymbol{\Phi}_{H}=\exp^{\lozenge}\left( -\int_{\mathbb{Q}_{p}^{N}
}H^{\lozenge}\left( \boldsymbol{\Phi}\left( x\right) \right)
d^{N}x\right) , \label{Eq_A2}
\end{equation}
where $H$ is analytic function at the origin satisfying $H(0)=0$. The Wick
ana\-lytic function $H^{\lozenge}\left( \boldsymbol{\Phi}\left( x\right)
\right) $ of process $\boldsymbol{\Phi}\left( x\right) $ coincides with the
usual Wick ordered function $:H\left( \boldsymbol{\Phi}\left( x\right)
\right) :$ when $H$ is a polynomial function. It turns out that $H^{\lozenge
}\left( \boldsymbol{\Phi}\left( x\right) \right) $ is a distribution from
the Kondratiev space $\left( \mathcal{H}_{\infty}\right) ^{-1}$, \ and
consequently, its integral belong to $\left( \mathcal{H}_{\infty}\right)
^{-1}$, if it exists. In general we cannot take the exponential of \ $-\int
H^{\lozenge}\left( \boldsymbol{\Phi}\left( x\right) \right) d^{N}x$,
however, by using the Wick calculus in $\left( \mathcal{H}_{\infty}\right)
^{-1}$, see Section \ref{Section Wick_analytic_fun}, we can take the Wick
exponential $\exp^{\lozenge}\left( \cdot\right) $.
In certain cases, for instance when $H$ is linear or is a polynomial of even
degree, see \cite{Koch-Sait}, and if we integrate only over a compact subset
$K$ of $\mathbb{Q}_{p}^{N}$ (the space cutoff), the function $\boldsymbol{\Phi
}_{H}$ is integrable, and we have a direct correspondence between
(\ref{Eq_A1}) and (\ref{Eq_A2}), i.e.
\[
\boldsymbol{\Phi}_{H}d\mu=\left\{ \frac{\exp\left( -\int_{K}H^{\lozenge
}\left( \boldsymbol{\Phi}\left( x\right) \right) d^{N}x\right) }{\int
\exp\left( -\int_{K}H^{\lozenge}\left( \boldsymbol{\Phi}\left( x\right)
\right) d^{N}x\right) d\mu}\right\} d\mu.
\]
In general the distribution $\boldsymbol{\Phi}_{H}$\ is not necessarily
positive, and for a large class of functions $H$, there are no measures
representing $\boldsymbol{\Phi}_{H}$. It turns out that $\boldsymbol{\Phi}
_{H}$ can be represented by a measure if and only if $-H(it)+\frac{1}{2}t^{2}
$, $t\in\mathbb{R}$, is a L\'{e}vy characteristic, see Theorem \ref{Theorem2A}
. These measures are called generalized white noise measures.
Generalized white measures were considered in \cite{Zuniga-FAA-2017}, in the
$p$-adic framework, and in the Archimedean case in \cite{Albeverio-et-al-3}
-\cite{Albeverio-et-al-4}. Euclidean random fields over $\mathbb{Q}_{p}^{N}$
were constructed by convolving generalized white noise with the fundamental
solutions of certain $p$-adic pseudodifferential equations. These fundamental
solutions are invariant under the action of a $p$-adic version of the
Euclidean group, see Section \ref{Sect_symmetries}.
For all convoluted generalized white noise measures such that their L\'{e}vy
characteristics have an analytic extension at the origin, we can give an
explicit formula for the generalized density with respect to the white noise
measure, see Theorem \ref{Theorem2}. In addition, there exists a large class
of distributions $\boldsymbol{\Phi}_{H}$ of type (\ref{Eq_A2}) that do not
have an associated measure, see Remark \ref{Nota_Theorem_3}. We also prove
that the Schwinger functions corresponding to convoluted generalized functions
\ satisfy Osterwalder-Schrader axioms (axioms OS1, OS2, OS4, OS5 in the
notation used in \cite{GS1999}) except for reflect positivity, see Lemma
\ref{Lemma1}, Theorems \ref{Theorem2}, \ref{Theorem3}, just like in the
Archimedean case presented in \cite{GS1999}.
The $p$-adic spacetime $\left( \mathbb{Q}_{p}^{N},\mathfrak{q}\left(
\xi\right) \right) $ is a $\mathbb{Q}_{p}$-vector space of dimension $N$
with an elliptic quadratic form $\mathfrak{q}\left( \xi\right) $, i.e.
$\mathfrak{q}\left( \xi\right) =0\Leftrightarrow\xi=0$. This spacetime
differs from the classical spacetime $\left( \mathbb{R}^{N},\xi_{1}
^{2}+\cdots+\xi_{N}^{2}\right) $ in several aspects. The $p$-adic spacetime
is not an `infinitely divisible continuum', because $\mathbb{Q}_{p}^{N}$ is a
completely disconnected topological space, the connected components (the
points) play the role of `spacetime quanta'. Since $\mathbb{Q}_{p}$ is not an
ordered field, the notions of past and future do not exist, then any $p$-adic
QFT is an acausal theory. The reader may consult the introduction of
\cite{Mendoza-Zuniga} for an in-depth discussion of this matter. Consequently,
the reflection positivity, if it exists in the $p$-adic framework, requires a
particular formulation, that we do not know at the moment. The study of the
$p$-adic Wightman functions via the reconstruction theorem is an open problem.
Another important difference between the classical case and the $p$-adic one
comes from the fact that in the $p$-adic setting there are no elliptic
quadratic forms in dimension $N\geq5$. We replace $\mathfrak{q}\left(
\xi\right) $ by an elliptic polynomial $\mathfrak{l}\left( \xi\right) $,
which is a homogeneous polynomial satisfying $\mathfrak{l}\left( \xi\right)
=0\Leftrightarrow\xi=0$. For any dimension $N$ there are elliptic polynomials
of degree $d\geq2$. We use $\left\vert \mathfrak{l}\left( \xi\right)
\right\vert _{p}^{\frac{2}{d}}$ as a replacement of $\left\vert \mathfrak{q}
\left( \xi\right) \right\vert _{p}$. This approach is particularly useful to
define the $p$-adic Laplace equation that the (free) covariance function
$C_{p}\left( x-y\right) $ satisfies, this equation has the following form:
\[
\left( \boldsymbol{L}_{\alpha}+m^{2}\right) C_{p}\left( x-y\right)
=\delta\left( x-y\right) \text{, \ }x\text{, }y\in\mathbb{Q}_{p}^{N},
\]
where $\alpha>0$, $m>0$ and $\boldsymbol{L}_{\alpha}$, is the
pseudodifferential operator
\[
\boldsymbol{L}_{\alpha}\varphi\left( x\right) =\mathcal{F}_{\xi\rightarrow
x}^{-1}(\left\vert \mathfrak{l}\left( \xi\right) \right\vert _{p}^{\alpha
}\mathcal{F}_{x\rightarrow\xi}\varphi),
\]
here $\mathcal{F}$ denotes the Fourier transform. The QFTs presented here are
families depending on several parameters, among them, $p$, $\alpha$, $m$,
$\mathfrak{l}\left( \xi\right) $.
The $p$-adic free covariance $C_{p}\left( x-y\right) $ may have
singularities at the origin depending on the parameters $\alpha$, $d$, $N$,
and has a `polynomial' decay at infinity, see Section \ref{Section_C_p}. The
$p$-adic cluster property holds under the condition $\alpha d>N$. Under this
hypothesis the covariance function \ does not have singularities at the
origin. Since $\alpha$ is a `free' parameter, this condition can be satisfied
in any dimension. We think that the condition $\alpha d>N$ is completely
necessary to have the cluster property due to the fact that our test functions
do not decay exponentially at infinity, see Remark \ref{Nota_Cluster}.
\section{\label{Sction_2}$p$\textbf{-}Adic Analysis: Essential Ideas}
In this section we collect some basic results about $p$-adic analysis that
will be used in the article. For an in-depth review of the $p$-adic analysis
the reader may consult \cite{A-K-S}, \cite{Taibleson}, \cite{V-V-Z}.
\subsection{The field of $p$-adic numbers}
Along this article $p$ will denote a prime number. The field of $p-$adic
numbers $
\mathbb{Q}
_{p}$ is defined as the completion of the field of rational numbers
$\mathbb{Q}$ with respect to the $p-$adic norm $|\cdot|_{p}$, which is defined
as
\[
\left\vert x\right\vert _{p}=\left\{
\begin{array}
[c]{lll}
0 & \text{if} & x=0\\
& & \\
p^{-\gamma} & \text{if} & x=p^{\gamma}\frac{a}{b}\text{,}
\end{array}
\right.
\]
where $a$ and $b$ are integers coprime with $p$. The integer $\gamma:=ord(x)
$, with $ord(0):=+\infty$, is called the\textit{\ }$p-$\textit{adic order of}
$x$.
Any $p-$adic number $x\neq0$ has a unique expansion of the form
\[
x=p^{ord(x)}\sum_{j=0}^{\infty}x_{j}p^{j},
\]
where $x_{j}\in\{0,\dots,p-1\}$ and $x_{0}\neq0$. By using this expansion, we
define \textit{the fractional part of }$x\in\mathbb{Q}_{p}$, denoted
$\{x\}_{p}$, as the rational number
\[
\left\{ x\right\} _{p}=\left\{
\begin{array}
[c]{lll}
0 & \text{if} & x=0\text{ or }ord(x)\geq0\\
& & \\
p^{ord(x)}\sum_{j=0}^{-ord_{p}(x)-1}x_{j}p^{j} & \text{if} & ord(x)<0.
\end{array}
\right.
\]
In addition, any non-zero $p-$adic number can be represented uniquely as
$x=p^{ord(x)}ac\left( x\right) $ where $ac\left( x\right) =\sum
_{j=0}^{\infty}x_{j}p^{j}$, $x_{0}\neq0$, is called the \textit{angular
component} of $x$. Notice that $\left\vert ac\left( x\right) \right\vert
_{p}=1$.
We extend the $p-$adic norm to $
\mathbb{Q}
_{p}^{N}$ by taking
\[
||x||_{p}:=\max_{1\leq i\leq N}|x_{i}|_{p},\text{ for }x=(x_{1},\dots
,x_{N})\in
\mathbb{Q}
_{p}^{N}.
\]
We define $ord(x)=\min_{1\leq i\leq N}\{ord(x_{i})\}$, then $||x||_{p}
=p^{-ord(x)}$.\ The metric space $\left(
\mathbb{Q}
_{p}^{N},||\cdot||_{p}\right) $ is a complete ultrametric space. For
$r\in\mathbb{Z}$, denote by $B_{r}^{N}(a)=\{x\in
\mathbb{Q}
_{p}^{N};||x-a||_{p}\leq p^{r}\}$ \textit{the ball of radius }$p^{r}$
\textit{with center at} $a=(a_{1},\dots,a_{N})\in
\mathbb{Q}
_{p}^{N}$, and take $B_{r}^{N}(0):=B_{r}^{N}$. Note that $B_{r}^{N}
(a)=B_{r}(a_{1})\times\cdots\times B_{r}(a_{N})$, where $B_{r}(a_{i}):=\{x\in
\mathbb{Q}
_{p};|x_{i}-a_{i}|_{p}\leq p^{r}\}$ is the one-dimensional ball of radius
$p^{r}$ with center at $a_{i}\in
\mathbb{Q}
_{p}$. The ball $B_{0}^{N}$ equals the product of $N$ copies of $B_{0}
=\mathbb{Z}_{p}$, \textit{the ring of }$p-$\textit{adic integers of }$
\mathbb{Q}
_{p}$. We also denote by $S_{r}^{N}(a)=\{x\in\mathbb{Q}_{p}^{N};||x-a||_{p}
=p^{r}\}$ \textit{the sphere of radius }$p^{r}$ \textit{with center at}
$a=(a_{1},\dots,a_{N})\in
\mathbb{Q}
_{p}^{N}$, and take $S_{r}^{N}(0):=S_{r}^{N}$. We notice that $S_{0}
^{1}=\mathbb{Z}_{p}^{\times}$ (the group of units of $\mathbb{Z}_{p}$), but
$\left( \mathbb{Z}_{p}^{\times}\right) ^{N}\subsetneq S_{0}^{N}$. The balls
and spheres are both open and closed subsets in $
\mathbb{Q}
_{p}^{N}$. In addition, two balls in $
\mathbb{Q}
_{p}^{N}$ are either disjoint or one is contained in the other.
As a topological space $\left(
\mathbb{Q}
_{p}^{N},||\cdot||_{p}\right) $ is totally disconnected, i.e. the only
connected \ subsets of $
\mathbb{Q}
_{p}^{N}$ are the empty set and the points. A subset of $
\mathbb{Q}
_{p}^{N}$ is compact if and only if it is closed and bounded in $
\mathbb{Q}
_{p}^{N}$, see e.g. \cite[Section 1.3]{V-V-Z}, or \cite[Section 1.8]{A-K-S}.
The balls and spheres are compact subsets. Thus $\left(
\mathbb{Q}
_{p}^{N},||\cdot||_{p}\right) $ is a locally compact topological space.
We will use $\Omega\left( p^{-r}||x-a||_{p}\right) $ to denote the
characteristic function of the ball $B_{r}^{N}(a)$. We will use the notation
$1_{A}$ for the characteristic function of a set $A$. Along the article
$d^{N}x$ will denote a Haar measure on $\left(
\mathbb{Q}
_{p}^{N},+\right) $ normalized so that $\int_{
\mathbb{Z}
_{p}^{N}}d^{N}x=1.$
\subsection{Some function spaces}
A complex-valued function $\varphi$ defined on $
\mathbb{Q}
_{p}^{N}$ is \textit{called locally constant} if for any $x\in
\mathbb{Q}
_{p}^{N}$ there exist an integer $l(x)\in\mathbb{Z}$ such that
\[
\varphi(x+x^{\prime})=\varphi(x)\text{ for }x^{\prime}\in B_{l(x)}^{N}.
\]
A function $\varphi:
\mathbb{Q}
_{p}^{N}\rightarrow\mathbb{C}$ is called a \textit{Bruhat-Schwartz function
(or a test function)} if it is locally constant with compact support. The
$\mathbb{C}$-vector space of Bruhat-Schwartz functions is denoted by
$\mathcal{D}:=\mathcal{D}(
\mathbb{Q}
_{p}^{N})$. Let $\mathcal{D}^{\prime}:=\mathcal{D}^{\prime}(
\mathbb{Q}
_{p}^{N})$ denote the set of all continuous functional (distributions) on
$\mathcal{D}$.
We will denote by $\mathcal{D}_{\mathbb{R}}:=\mathcal{D}_{\mathbb{R}}(
\mathbb{Q}
_{p}^{N})$, the $\mathbb{R}$-vector space of test functions, and by
$\mathcal{D}_{\mathbb{R}}^{\prime}:=\mathcal{D}_{\mathbb{R}}^{\prime}(
\mathbb{Q}
_{p}^{N})$, the $\mathbb{R}$-vector space of distributions.
Given $\rho\in\lbrack0,\infty)$, we denote by $L^{\rho}:=L^{\rho}\left(
\mathbb{Q}
_{p}^{N}\right) :=L^{\rho}\left(
\mathbb{Q}
_{p}^{N},d^{N}x\right) ,$ the $
\mathbb{C}
-$vector space of all the complex valued functions $g$ satisfying $\int_{
\mathbb{Q}
_{p}^{N}}\left\vert g\left( x\right) \right\vert ^{\rho}d^{N}x<\infty$, and
$L^{\infty}\allowbreak:=L^{\infty}\left(
\mathbb{Q}
_{p}^{N}\right) =L^{\infty}\left(
\mathbb{Q}
_{p}^{N},d^{N}x\right) $ denotes the $
\mathbb{C}
-$vector space of all the complex valued functions $g$ such that the essential
supremum of $|g|$ is bounded. The corresponding $\mathbb{R}$-vector spaces are
denoted as $L_{\mathbb{R}}^{\rho}\allowbreak:=L_{\mathbb{R}}^{\rho}\left(
\mathbb{Q}
_{p}^{N}\right) =L_{\mathbb{R}}^{\rho}\left(
\mathbb{Q}
_{p}^{N},d^{N}x\right) $, $1\leq\rho\leq\infty$.
Set
\[
\mathcal{C}_{0}(
\mathbb{Q}
_{p}^{N},\mathbb{C}):=\left\{ f:
\mathbb{Q}
_{p}^{N}\rightarrow
\mathbb{C}
;\text{ }f\text{ is continuous and }\lim_{||x||_{p}\rightarrow\infty
}f(x)=0\right\} ,
\]
where $\lim_{||x||_{p}\rightarrow\infty}f(x)=0$ means that for every
$\epsilon>0$ there exists a compact subset $B(\epsilon)$ such that
$|f(x)|<\epsilon$ for $x\in
\mathbb{Q}
_{p}^{N}\backslash B(\epsilon).$ We recall that $(\mathcal{C}_{0}(
\mathbb{Q}
_{p}^{N},\mathbb{C}),||\cdot||_{L^{\infty}})$ is a Banach space. The
corresponding $\mathbb{R}$-vector space will be denoted as $\mathcal{C}_{0}(
\mathbb{Q}
_{p}^{N},\mathbb{R})$.
\subsection{Fourier transform}
Set $\chi_{p}(y):=\exp(2\pi i\{y\}_{p})$ for $y\in
\mathbb{Q}
_{p}$. The map $\chi_{p}(\cdot)$ is an additive character on $
\mathbb{Q}
_{p}$, i.e. a continuous map from $\left(
\mathbb{Q}
_{p},+\right) $ into $S$ (the unit circle considered as multiplicative group)
satisfying $\chi_{p}(x_{0}+x_{1})=\chi_{p}(x_{0})\chi_{p}(x_{1})$,
$x_{0},x_{1}\in
\mathbb{Q}
_{p}$. The additive characters of $
\mathbb{Q}
_{p}$ form an Abelian group which is isomorphic to $\left(
\mathbb{Q}
_{p},+\right) $, the isomorphism is given by $\xi\rightarrow\chi_{p}(\xi x)$,
see e.g. \cite[Section 2.3]{A-K-S}.
Given $x=(x_{1},\dots,x_{N}),$ $\xi=(\xi_{1},\dots,\xi_{N})\in
\mathbb{Q}
_{p}^{N}$, we set $x\cdot\xi:=\sum_{j=1}^{N}x_{j}\xi_{j}$. If $f\in L^{1}$ its
Fourier transform is defined by
\[
(\mathcal{F}f)(\xi)=\int_{
\mathbb{Q}
_{p}^{N}}\chi_{p}(\xi\cdot x)f(x)d^{N}x,\quad\text{for }\xi\in
\mathbb{Q}
_{p}^{N}.
\]
We will also use the notation $\mathcal{F}_{x\rightarrow\xi}f$ and
$\widehat{f}$\ for the Fourier transform of $f$. The Fourier transform is a
linear isomorphism from $\mathcal{D}(
\mathbb{Q}
_{p}^{N})$ onto itself satisfying
\begin{equation}
(\mathcal{F}(\mathcal{F}f))(\xi)=f(-\xi), \label{FF(f)}
\end{equation}
for every $f\in\mathcal{D}(
\mathbb{Q}
_{p}^{N}),$ see e.g. \cite[Section 4.8]{A-K-S}. If $f\in L^{2},$ its Fourier
transform is defined as
\[
(\mathcal{F}f)(\xi)=\lim_{k\rightarrow\infty}\int_{||x||_{p}\leq p^{k}}
\chi_{p}(\xi\cdot x)f(x)d^{N}x,\quad\text{for }\xi\in
\mathbb{Q}
_{p}^{N},
\]
where the limit is taken in $L^{2}.$ We recall that the Fourier transform is
unitary on $L^{2},$ i.e. $||f||_{L^{2}}=||\mathcal{F}f||_{L^{2}}$ for $f\in
L^{2}$ and that (\ref{FF(f)}) is also valid in $L^{2}$, see e.g. \cite[Chapter
III, Section 2]{Taibleson}.
The Fourier transform $\mathcal{F}\left[ W\right] $ of a distribution
$W\in\mathcal{D}^{\prime}\left(
\mathbb{Q}
_{p}^{N}\right) $ is defined by
\[
\left( \mathcal{F}\left[ W\right] ,\varphi\right) =\left( W,\mathcal{F}
\left[ \varphi\right] \right) \text{ for all }\varphi\in\mathcal{D}(
\mathbb{Q}
_{p}^{N})\text{.}
\]
The Fourier transform $W\rightarrow\mathcal{F}\left[ W\right] $ is a linear
isomorphism from $\mathcal{D}^{\prime}\left(
\mathbb{Q}
_{p}^{N}\right) $\ onto itself. Furthermore, $W=\mathcal{F}\left[
\mathcal{F}\left[ W\right] \left( -\xi\right) \right] $. We also use the
notation $\mathcal{F}_{x\rightarrow\xi}W$ and $\widehat{W}$ for the Fourier
transform of $W.$
\section{$p$-Adic White Noise}
In this section we review some basic aspects of the white noise calculus in
the $p$-adic setting. For a in-depth exposition on the white noise calculus on
arbitrary nuclear spaces the reader may consult \cite{Ber-Kon},
\cite{Gelfan-Vilenkin}, \cite{Hida et al}, \cite{Huang-Yang}, \cite{Obata}. We
will use white noise calculus on the nuclear spaces $\mathcal{H}_{\infty}$
introduced by Z\'{u}\~{n}iga-Galindo in \cite{Zuniga-FAA-2017}, see also
\cite[Chapters 10, 11]{KKZuniga}.
\subsection{A class of non-Archimedean nuclear spaces}
\subsubsection{\label{Section1}$\mathcal{H}_{\infty}$, a non-Archimedean
analog of the Schwartz space}
We denote the set on non-negative integers by $\mathbb{N}$, and set $\left[
\xi\right] _{p}:=[\max(1,\Vert\xi\Vert_{p})]$ for $\xi\in\mathbb{Q}_{p}^{N}$.
We define for $\varphi$, $\theta\in\mathcal{D}(\mathbb{Q}_{p}^{N})$, and
$l\in\mathbb{N}$, the following scalar product:
\[
\left\langle \varphi,\theta\right\rangle _{l}=\int_{\mathbb{Q}_{p}^{N}}\left[
\xi\right] _{p}^{l}\overline{\widehat{\varphi}\left( \xi\right) }
\widehat{\theta}\left( \xi\right) d^{N}\xi\text{,}
\]
where the overbar denotes the complex conjugate. We also set $\left\Vert
\varphi\right\Vert _{l}:=\left\langle \varphi,\varphi\right\rangle _{l}$.
Notice that $\left\Vert \cdot\right\Vert _{l}\leq\left\Vert \cdot\right\Vert
_{m}$ for $l\leq m$. We denote by $\mathcal{H}_{l}(\mathbb{C}):=$
$\mathcal{H}_{l}(\mathbb{Q}_{p}^{N},\mathbb{C})$ the complex Hilbert space
obtained by completing $\mathcal{D}(\mathbb{Q}_{p}^{N})$ with respect to
$\left\langle \cdot,\cdot\right\rangle _{l}$. Then $\mathcal{H}_{m}
(\mathbb{C})\hookrightarrow\mathcal{H}_{l}(\mathbb{C})$ for $l\leq m$. Now we
set
\[
\mathcal{H}_{\infty}(\mathbb{C}):=\mathcal{H}_{\infty}(\mathbb{Q}_{p}
^{N},\mathbb{C})=
{\displaystyle\bigcap\nolimits_{l\in\mathbb{N}}}
\mathcal{H}_{l}(\mathbb{C}).
\]
Notice that $\mathcal{H}_{\infty}(\mathbb{C})\subset L^{2}$. With the topology
induced by the family of seminorms $\left\{ \left\Vert \cdot\right\Vert
_{l}\right\} _{l\in\mathbb{N}}$, $\mathcal{H}_{\infty}(\mathbb{C})$ becomes a
locally convex space, which is metrizable. Indeed,
\[
d(f,g):=\max_{l\in\mathbb{N}}\left\{ 2^{-l}\frac{\left\Vert f-g\right\Vert
_{l}}{1+\left\Vert f-g\right\Vert _{l}}\right\} \text{, with }f\text{, }
g\in\mathcal{H}_{\infty}(\mathbb{C})\text{, }
\]
is a metric for the topology of $\mathcal{H}_{\infty}(\mathbb{C})$. The
projective topology $\tau_{P}$ of $\mathcal{H}_{\infty}(\mathbb{C})$ coincides
with the topology induced by the family of seminorms $\left\{ \left\Vert
\cdot\right\Vert _{l}\right\} _{l\in\mathbb{N}}$. The space $\mathcal{H}
_{\infty}(\mathbb{C})$ endowed with the topology $\tau_{P}$ is a countably
Hilbert space in the sense of Gel'fand-Vilenkin. Furthermore, $\left(
\mathcal{H}_{\infty}(\mathbb{C}),\tau_{P}\right) $ is metrizable and complete
and hence a Fr\'{e}chet space, cf. \cite[Lemma 10.3]{KKZuniga}, see also
\cite{Zuniga-FAA-2017}.
The space $(\mathcal{H}_{\infty}(\mathbb{C}),d)$ is the completion of
$(\mathcal{D}(\mathbb{Q}_{p}^{N}),d)$ with respect to $d$, and since
$\mathcal{D}(\mathbb{Q}_{p}^{N})$ is nuclear, then $\mathcal{H}_{\infty
}(\mathbb{C})$ is a nuclear space, which is continuously embedded in
$C_{0}(\mathbb{Q}_{p}^{N},\mathbb{C})$, the space of complex-valued bounded
functions vanishing at infinity. In addition, $\mathcal{H}_{\infty}
(\mathbb{C})\subset L^{1}\cap L^{2}$, cf. \cite[Theorem 10.15]{KKZuniga}.
\begin{remark}
(i) We denote by $\mathcal{H}_{l}(\mathbb{R}):=\mathcal{H}_{l}(\mathbb{Q}
_{p}^{N},\mathbb{R})$ the real Hilbert space obtained by completing
$\mathcal{D}_{\mathbb{R}}(\mathbb{Q}_{p}^{N})$ with respect to $\left\langle
\cdot,\cdot\right\rangle _{l}$. We also set $\mathcal{H}_{\infty}
(\mathbb{Q}_{p}^{N},\mathbb{R}):=\mathcal{H}_{\infty}(\mathbb{R})=\cap
_{l\in\mathbb{N}}\mathcal{H}_{l}(\mathbb{R})$. In the case in which the ground
field ($\mathbb{R}$ or $\mathbb{C)}$ is clear, we shall use the simplified
notation $\mathcal{H}_{l}$, $\mathcal{H}_{\infty}$. All the above announced
results \ for the spaces $\mathcal{H}_{l}(\mathbb{C})$, $\mathcal{H}_{\infty
}(\mathbb{C})$ are valid for the spaces $\mathcal{H}_{l}(\mathbb{R})$,
$\mathcal{H}_{\infty}(\mathbb{R})$. In particular, $\mathcal{H}_{\infty
}(\mathbb{R})$ is a nuclear countably Hilbert space.
(ii) The following characterization of the space $\mathcal{H}_{\infty
}(\mathbb{C})$ is very useful:
\begin{align*}
\mathcal{H}_{\infty}(\mathbb{C}) & =\left\{ f\in L^{2}\left(
\mathbb{Q}_{p}^{N}\right) ;\left\Vert f\right\Vert _{l}<\infty\text{ for any
}l\in\mathbb{N}\right\} \\
& =\left\{ W\in\mathcal{D}^{\prime}\left( \mathbb{Q}_{p}^{N}\right)
;\left\Vert W\right\Vert _{l}<\infty\text{ for any }l\in\mathbb{N}\right\} ,
\end{align*}
cf. \cite[Lemma 10.8]{KKZuniga}. An analog result is valid for $\mathcal{H}
_{\infty}(\mathbb{R})$.
(iii) The spaces $\mathcal{H}_{l}(\mathbb{R})$, $\mathcal{H}_{l}(\mathbb{C})$,
for any $l\in\mathbb{N}$, are nuclear and consequently they are separable, cf.
\cite[Chapter I, Section 3.4]{Gelfan-Vilenkin}.
\end{remark}
The spaces $\mathcal{H}_{\infty}(\mathbb{Q}_{p}^{N},\mathbb{C})\mathcal{\ }
$and $\mathcal{H}_{\infty}(\mathbb{Q}_{p}^{N},\mathbb{R})$ were introduced in
\cite{Zuniga-FAA-2017}, see also \cite{KKZuniga}. These spaces are invariant
under the action of a large class of pseudodifferential operators.
\subsubsection{The dual space of $\mathcal{H}_{\infty}$}
For $m\in\mathbb{N}$, and $W\in\mathcal{D}^{\prime}\left( \mathbb{Q}_{p}
^{N}\right) $ such that $\widehat{W}$ is a measurable function, we set
\[
\left\Vert W\right\Vert _{-m}^{2}:=\int_{\mathbb{Q}_{p}^{N}}\left[
\xi\right] _{p}^{-m}\left\vert \widehat{W}\left( \xi\right) \right\vert
^{2}d^{N}\xi\text{.}
\]
Then
\begin{equation}
\mathcal{H}_{-m}(\mathbb{C}):=\mathcal{H}_{-m}(\mathbb{Q}_{p}^{N}
,\mathbb{C})=\left\{ W\in\mathcal{D}^{\prime}\left( \mathbb{Q}_{p}
^{N}\right) ;\left\Vert W\right\Vert _{-m}<\infty\right\} \label{Eq_A}
\end{equation}
is a complex Hilbert space. If $\mathcal{X}$ is a locally convex, we denote by
$\mathcal{X}^{\ast}$ the dual space endowed with the strong dual topology or
the topology of the bounded convergence. We denote by $\mathcal{H}_{m}^{\ast
}(\mathbb{C})$ the dual of $\mathcal{H}_{m}(\mathbb{C})$ for $m\in\mathbb{N}$,
we identify $\mathcal{H}_{m}^{\ast}(\mathbb{C})$ with $\mathcal{H}
_{-m}(\mathbb{C})$, by using the bilinear form:
\begin{equation}
\left\langle W,g\right\rangle =\int_{\mathbb{Q}_{p}^{N}}\overline{\widehat
{W}\left( \xi\right) }\widehat{g}\left( \xi\right) d^{N}\xi\text{ for
}W\in\mathcal{H}_{-m}(\mathbb{C})\text{ and }g\in\mathcal{H}_{m}
(\mathbb{C})\text{.} \label{pairing}
\end{equation}
Then
\begin{align*}
\mathcal{H}_{\infty}^{\ast}(\mathbb{Q}_{p}^{N},\mathbb{C}) & :=\mathcal{H}
_{\infty}^{\ast}(\mathbb{C})=\bigcup\limits_{m\in\mathbb{N}}\mathcal{H}
_{-m}(\mathbb{C})\\
& =\left\{ W\in\mathcal{D}^{\prime}\left( \mathbb{Q}_{p}^{N}\right)
;\left\Vert W\right\Vert _{-m}<\infty\text{ for some }m\in\mathbb{N}\right\}
.
\end{align*}
We consider $\mathcal{H}_{\infty}^{\ast}(\mathbb{C})$ endowed with the strong
topology. We use (\ref{pairing}) as pairing between $\mathcal{H}_{\infty
}^{\ast}(\mathbb{C})$ and $\mathcal{H}_{\infty}(\mathbb{C})$. By a similar
construction one obtains the space $\mathcal{H}_{\infty}^{\ast}(\mathbb{R}
):=\mathcal{H}_{\infty}^{\ast}(\mathbb{Q}_{p}^{N},\mathbb{R})$. The above
announced results are also valid for $\mathcal{H}_{\infty}^{\ast}(\mathbb{R}
)$. If there is no danger of confusion we use $\mathcal{H}_{\infty}^{\ast}$
instead of $\mathcal{H}_{\infty}^{\ast}(\mathbb{C})$ or $\mathcal{H}_{\infty
}^{\ast}(\mathbb{R})$.
\begin{remark}
\label{Nota1}(i) For complex and real spaces, $\left\Vert \cdot\right\Vert
_{\pm l}$ denotes the norm on $\mathcal{H}_{l}$ and $\mathcal{H}_{-l}$. We
denote by $\left\langle \cdot,\cdot\right\rangle $ the dual pairings between
$\mathcal{H}_{-l}$ and $\mathcal{H}_{l}$ and between $\mathcal{H}_{\infty}$
and $\mathcal{H}_{\infty}^{\ast}$. We preserve this notation for the norm and
pairing on tensor powers of these spaces.
(ii) If $\left\{ \mathcal{X}_{l}\right\} _{l\in A}$ is a family of locally
convex spaces, we denote by $\underleftarrow{\lim}_{l\in\mathbb{N}}
\mathcal{X}_{l}$ the projective limit of the family, and by $\underrightarrow
{\lim}_{l\in\mathbb{N}}\mathcal{X}_{l}$ the inductive limit of the family.
(iii) If $\mathcal{N}$ is a nuclear space, which is the projective limit of
the Hilbert spaces $H_{l}$, $l\in\mathbb{N}$,the $n$-th symmetric tensor
product of $\mathcal{N}$ is defined as $\mathcal{N}^{\widehat{\otimes}
n}=\underleftarrow{\lim}_{l\in\mathbb{N}}H_{l}^{\widehat{\otimes}n}$. This is
a nuclear space. The dual space is $\mathcal{N}^{\ast\widehat{\otimes}
n}=\underrightarrow{\lim}_{l\in\mathbb{N}}H_{-l}^{\widehat{\otimes}n}$.
\end{remark}
\subsection{Non-Archimedean Gaussian measures}
The spaces
\[
\mathcal{H}_{\infty}(\mathbb{R})\hookrightarrow L_{\mathbb{R}}^{2}\left(
\mathbb{Q}_{p}^{N}\right) \hookrightarrow\mathcal{H}_{\infty}^{\ast
}(\mathbb{R})
\]
form a Gel'fand triple, that is, $\mathcal{H}_{\infty}(\mathbb{R})$ is a
nuclear countably Hilbert space which is densely and continuously embedded in
$L_{\mathbb{R}}^{2}$ and $\left\Vert g\right\Vert _{0}^{2}=\left\langle
g,g\right\rangle _{0}$ for $g\in\mathcal{H}_{\infty}(\mathbb{R})$. This triple
was introduced in \cite{Zuniga-FAA-2017}, see also \cite[Chapter 10]
{KKZuniga}. The inner product and the norm of $\left( L_{\mathbb{R}}
^{2}\left( \mathbb{Q}_{p}^{N}\right) \right) ^{\otimes m}\simeq
L_{\mathbb{R}}^{2}\left( \mathbb{Q}_{p}^{Nm}\right) $ are denoted by
$\left\langle \cdot,\cdot\right\rangle _{0}$ and $\left\Vert \cdot\right\Vert
_{0}$. From now on, we consider $\mathcal{H}_{\infty}^{^{\widehat{\otimes}n}
}\left( \mathbb{R}\right) $ as subspace of $\mathcal{H}_{\infty}^{^{\otimes
n}}\left( \mathbb{R}\right) $, then $\left\langle \cdot,\cdot\right\rangle
_{\mathcal{H}_{\infty}^{^{\widehat{\otimes}n}}\left( \mathbb{R}\right)
}=n!\left\langle \cdot,\cdot\right\rangle _{0}$.
We denote by $\mathcal{B}:=\mathcal{B}(\mathcal{H}_{\infty}^{\ast}
(\mathbb{R}))$ the $\sigma$-algebra generated by the cylinder subsets of
$\mathcal{H}_{\infty}^{\ast}(\mathbb{R})$. The mapping
\[
\begin{array}
[c]{cccc}
\mathcal{C}: & \mathcal{H}_{\infty}(\mathbb{R}) & \rightarrow & \mathbb{C}\\
& f & \rightarrow & e^{-\frac{1}{2}\left\Vert f\right\Vert _{0}^{2}}
\end{array}
\]
defines a characteristic functional, i.e. $\mathcal{C}$ is continuous,
positive definite and $\mathcal{C}\left( 0\right) =1$. By the Bochner-Minlos
theorem, see e.g. \cite{Ber-Kon}, \cite{Hida et al}, there exists a
probability measure $\mu$, called \textit{the canonical Gaussian measure} on
$\left( \mathcal{H}_{\infty}^{\ast}(\mathbb{R}),\mathcal{B}\right) $, given
by its characteristic functional as
\[
\int_{\mathcal{H}_{\infty}^{\ast}(\mathbb{R})}e^{i\langle W,f\rangle}
d\mu(W)=e^{-\frac{1}{2}\left\Vert f\right\Vert _{^{0}}^{2}}\text{,}
\ \ f\in\mathcal{H}_{\infty}(\mathbb{R})\text{.}
\]
We set $\left( L_{\mathbb{C}}^{2}\right) :=L^{2}\left( \mathcal{H}_{\infty
}^{\ast}(\mathbb{R}),\mu;\mathbb{C}\right) $ to denote the complex vector
space of measu\-rable functions $\Psi:\mathcal{H}_{\infty}^{\ast}
(\mathbb{R})\rightarrow\mathbb{C}$ satisfying
\[
\left\Vert \Psi\right\Vert _{\left( L_{\mathbb{C}}^{2}\right) }^{2}
=\int_{\mathcal{H}_{\infty}^{\ast}(\mathbb{R})}\left\vert \Psi\left(
W\right) \right\vert ^{2}d\mu(W)<\infty\text{.}
\]
The space $\left( L_{\mathbb{R}}^{2}\right) :=L^{2}\left( \mathcal{H}
_{\infty}^{\ast}(\mathbb{R}),\mu;\mathbb{R}\right) $ is defined in a similar
way. The pairing $\mathcal{H}_{\infty}^{\ast}(\mathbb{R})\times\mathcal{H}
_{\infty}(\mathbb{R})$ can be extended to $\mathcal{H}_{\infty}^{\ast
}(\mathbb{R})\times L^{2}(\mathbb{Q}_{p}^{N})$ as an $\left( L_{\mathbb{C}
}^{2}\right) $-function on $\mathcal{H}_{\infty}^{\ast}(\mathbb{R})$, this
fact follows from
\begin{equation}
\int_{\mathcal{H}_{\infty}^{\ast}(\mathbb{R})}\left\vert \left\langle
W,g\right\rangle \right\vert ^{2}d\mu(W)=\left\Vert g\right\Vert _{0}^{2},
\label{Eq_3}
\end{equation}
see e.g. \cite[Lemma 2.1.5]{Obata}. If $g\in L_{\mathbb{R}}^{2}$, then
$W\rightarrow\left\langle W,g\right\rangle $ belongs to $\left(
L_{\mathbb{R}}^{2}\right) $.
Let $f\in\mathcal{H}_{\infty}(\mathbb{R})$ and $W_{f}(J):=\left\langle
J,f\right\rangle $, $J\in$ $\mathcal{H}_{\infty}^{\ast}(\mathbb{R})$. Then
$W_{f}$ is a Gaussian random variable on $\left( \mathcal{H}_{\infty}^{\ast
}(\mathbb{R}),\mu\right) $ satisfying
\[
\mathbb{E}_{\mu}(W_{f})=0\text{, \ }\mathbb{E}_{\mu}(W_{f}^{2})=\left\Vert
f\right\Vert _{0}^{2}.
\]
Then the linear map
\[
\begin{array}
[c]{ccc}
\mathcal{H}_{\infty}(\mathbb{R}) & \rightarrow & \left( L_{\mathbb{R}}
^{2}\right) \\
& & \\
f & \rightarrow & W_{f}
\end{array}
\]
can be extended to a linear isometry from $L^{2}(\mathbb{Q}_{p}^{N})$\ to
$\left( L_{\mathbb{C}}^{2}\right) $.
\subsection{Wick-ordered polynomials}
Let $\mathcal{P}_{n}(\mathbb{R})$, respectively $\mathcal{P}_{n}(\mathbb{C})$,
be the vector space of finite linear combinations of functions of the form
\[
W\rightarrow\left\langle W,f\right\rangle ^{n}=\left\langle W^{\otimes
n},f^{\otimes n}\right\rangle \text{, with }W\in\mathcal{H}_{\infty}^{\ast
}(\mathbb{R})\text{,}
\]
where $f$ runs over $\mathcal{H}_{\infty}(\mathbb{R})$, respectively
$\mathcal{H}_{\infty}(\mathbb{C})$. Notice that $\mathcal{P}_{n}
(\mathbb{C})=\mathcal{P}_{n}(\mathbb{R})+i\mathcal{P}_{n}(\mathbb{R})$. An
element of the direct algebraic sums
\[
\mathcal{P}(\mathbb{R}):=\bigoplus\nolimits_{n=0}^{\infty}\mathcal{P}
_{n}(\mathbb{R})\text{, \ }\mathcal{P}(\mathbb{C}):=\bigoplus\nolimits_{n=0}
^{\infty}\mathcal{P}_{n}(\mathbb{C})\text{\ }
\]
is called a \textit{polynomial }on the Gaussian space $\mathcal{H}_{\infty
}^{\ast}(\mathbb{R})$. These functions are not very useful because they do not
satisfy orthogonality relations. This is the main motivation to introduce and
utilize the Wick-ordered polynomials.
For $W\in\mathcal{H}_{\infty}^{\ast}\left( \mathbb{R}\right) $ and
$f\in\mathcal{H}_{\infty}$, we define \textit{the Wick-ordered monomial} as
\begin{align*}
\left\langle :W^{\otimes n}:,f^{\otimes n}\right\rangle & =\sum
\limits_{k=0}^{\left[ \frac{n}{2}\right] }\frac{n!}{k!\left( n-2k\right)
!}\left( \frac{-1}{2}\left\langle f,f\right\rangle _{0}\right)
^{k}\left\langle W,f\right\rangle ^{n-2k}\\
& =\left\Vert f\right\Vert _{0}^{n}\boldsymbol{H}_{n}\left( \left\Vert
f\right\Vert _{0}^{-1}\left\langle W,f\right\rangle \right) ,
\end{align*}
where $\boldsymbol{H}_{n}$ denotes the $n$-th Hermite polynomial. Then
$:W^{\otimes n}:\in\mathcal{H}_{\infty}^{\ast\widehat{\otimes}n}$, in
addition, any polynomial $\Phi\in\mathcal{P}(\mathbb{R})$, respectively
$\mathcal{P}(\mathbb{C})$, is expressed as
\begin{equation}
\Phi\left( W\right) =\sum\limits_{n=0}^{\infty}\left\langle :W^{\otimes
n}:,\phi_{n}\right\rangle , \label{Eq_4}
\end{equation}
where $\phi_{n}$ belong to the symmetric $n$-fold algebraic tensor product
$\left( \mathcal{H}_{\infty}(\mathbb{R})\right) ^{\widehat{\otimes}n}$\ of
$\mathcal{H}_{\infty}(\mathbb{R})$, respectively of $\mathcal{H}_{\infty
}(\mathbb{C})$, and the sum symbol involves only a finite number of non-zero
terms. \ A function of type (\ref{Eq_4}) \ is called a \textit{Wick-ordered
polynomial}. For two polynomials $\Phi$, $\Psi\in\mathcal{P}(\mathbb{C})$
given respectively by (\ref{Eq_4}) with $\phi_{n}\in\left( \mathcal{H}
_{\infty}(\mathbb{C})\right) ^{\widehat{\otimes}n}$, and by
\begin{equation}
\Psi\left( W\right) =\sum\limits_{n=0}^{\infty}\left\langle :W^{\otimes
n}:,\psi_{n}\right\rangle ,\text{ with }\psi_{n}\in\left( \mathcal{H}
_{\infty}(\mathbb{C})\right) ^{\widehat{\otimes}n}\text{,} \label{Eq_5}
\end{equation}
it holds that
\[
\int_{\mathcal{H}_{\infty}^{\ast}(\mathbb{R})}\Phi\left( W\right)
\Psi\left( W\right) d\mu(W)=\sum\nolimits_{n=0}^{\infty}n!\left\langle
\varphi_{n},\psi_{n}\right\rangle _{0},
\]
where $\left\langle \cdot,\cdot\right\rangle _{0}$ denotes the scalar product
in $\left( L^{2}\left( \mathbb{Q}_{p}^{N}\right) \right) ^{\otimes n}$. In
parti\-cular,
\[
\left\Vert \Phi\right\Vert _{\left( L_{\mathbb{C}}^{2}\right) }^{2}
=\sum\nolimits_{n=0}^{\infty}n!\left\Vert \phi_{n}\right\Vert _{0}^{2},
\]
where $\left\Vert \cdot\right\Vert _{0}$ denotes the norms of $\left(
L^{2}\left( \mathbb{Q}_{p}^{N}\right) \right) ^{\otimes n}$, see e.g.
\cite[Proposition 2.2.10]{Obata}. Consequently, each $\Psi\in\mathcal{P}
(\mathbb{C})$ is uniquely expressed as a Wick-ordered polynomial.
\begin{remark}
\label{Nota2}We denote by $I_{n}(f_{n})$ the linear extension to $\left(
L^{2}\left( \mathbb{Q}_{p}^{N}\right) \right) ^{\widehat{\otimes}n}$ of the
map $f_{n}\rightarrow\left\langle :W^{\otimes n}:,f_{n}\right\rangle $,
$W\in\mathcal{H}_{\infty}^{\ast}\left( \mathbb{R}\right) $, then
\[
I_{n}(f^{\otimes n})=\left\Vert f\right\Vert _{0}^{n}\boldsymbol{H}
_{n}(\left\Vert f\right\Vert _{0}^{-1}W_{f})\text{, \ }f\in L^{2},
\]
and
\[
\int_{\mathcal{H}_{\infty}^{\ast}(\mathbb{R})}I_{n}(f_{n})I_{m}(g_{m}
)d\mu=\delta_{nm}n!\left\langle f_{n},g_{m}\right\rangle _{0}\text{, \ }
f_{n}\in L^{2\widehat{\otimes}n}\text{, }g_{m}\in L^{2\widehat{\otimes}m}.
\]
We shall also use $\left\langle :W^{\otimes n}:,f_{n}\right\rangle $ to denote
$I_{n}(f_{n})$ formally. In this case the symbol $\left\langle \cdot
,\cdot\right\rangle $ should not be confused with the bilinear form on
$\mathcal{H}_{\infty}^{\ast}\times\mathcal{H}_{\infty}$.
\end{remark}
\subsection{Wiener-It\^{o}-Segal isomorphism}
Let $\Gamma\left( L^{2}\left( \mathbb{Q}_{p}^{N}\right) \right) $ be the
space of sequences $\boldsymbol{f}=\left\{ f_{n}\right\} _{n\in\mathbb{N}}$,
$f_{n}\in\left( L^{2}\left( \mathbb{Q}_{p}^{N}\right) \right)
^{\widehat{\otimes}n}$, such that
\[
\left\Vert \boldsymbol{f}\right\Vert _{\Gamma\left( L^{2}\left(
\mathbb{Q}_{p}^{N}\right) \right) }^{2}:=\sum\nolimits_{n=0}^{\infty
}n!\left\Vert f_{n}\right\Vert _{0}^{2}<\infty.
\]
The Hilbert space $\Gamma\left( L^{2}\left( \mathbb{Q}_{p}^{N}\right)
\right) $ is called \textit{the Boson Fock Space on} $L^{2}\left(
\mathbb{Q}_{p}^{N}\right) $. The Wiener-It\^{o}-Segal theorem asserts that
for each $\Phi\in\left( L_{\mathbb{C}}^{2}\right) $ there exists a sequence
$\boldsymbol{\phi}=\left\{ \phi_{n}\right\} _{n\in\mathbb{N}}$ in
$\Gamma\left( L^{2}\left( \mathbb{Q}_{p}^{N}\right) \right) $ such that
(\ref{Eq_4}) holds in the $\left( L_{\mathbb{C}}^{2}\right) $-sense, but
with $\phi_{n}\in\left( L^{2}\left( \mathbb{Q}_{p}^{N}\right) \right)
^{\widehat{\otimes}n}$, see Remark \ref{Nota2}. Conversely, for any
$\boldsymbol{\phi}=\left\{ \phi_{n}\right\} _{n\in\mathbb{N}}\in
\Gamma\left( L_{\mathbb{C}}^{2}\left( \mathbb{Q}_{p}^{N}\right) \right) $,
(\ref{Eq_4}) defines a function in $\left( L_{\mathbb{C}}^{2}\right) $. In
this case
\[
\left\Vert \Phi\right\Vert _{\left( L_{\mathbb{C}}^{2}\right) }^{2}
=\sum\limits_{n=0}^{\infty}n!\left\Vert \phi_{n}\right\Vert _{0}
^{2}=\left\Vert \boldsymbol{\phi}\right\Vert _{\Gamma\left( L^{2}\left(
\mathbb{Q}_{p}^{N}\right) \right) }^{2},
\]
see e.g. \cite[Theorem 2.3.5]{Obata}, \cite{Segal}
\section{\label{Section_4}Non-Archimedean Kondratiev Spaces of Test Functions
and Distributions}
In this section we introduce non-Archimedean versions of Kondratiev-type
spaces of test functions and distributions.
\subsection{Kondratiev-type spaces of test functions}
We define for $l$, $k\in\mathbb{N}$, and $\beta\in\left[ 0,1\right] $ fixed,
the following norm on $\left( L_{\mathbb{C}}^{2}\right) $:
\[
\left\Vert \Phi\right\Vert _{l,k,\beta}^{2}=\sum\limits_{n=0}^{\infty}\left(
n!\right) ^{1+\beta}2^{nk}\left\Vert \phi_{n}\right\Vert _{l}^{2},
\]
where $\Phi$ is given in (\ref{Eq_4}), and $\left\Vert \cdot\right\Vert _{l}$
denotes the norm on $\mathcal{H}_{l}^{\widehat{\otimes}n}$.
We now define
\[
\mathcal{H}_{l,k,\beta}=\left\{ \Phi\left( W\right) =\sum\limits_{n=0}
^{\infty}\left\langle :W^{\otimes n}:,\phi_{n}\right\rangle \in\left(
L_{\mathbb{C}}^{2}\right) ;\left\Vert \Phi\right\Vert _{l,k,\beta}^{2}
<\infty\right\} .
\]
The space $\mathcal{H}_{l,k,\beta}$ is a Hilbert space with inner product
\[
\left\langle \Phi,\Psi\right\rangle _{l,k,\beta}=\sum\limits_{n=0}^{\infty
}\left( n!\right) ^{1+\beta}2^{nk}\left\langle \phi_{n},\psi_{n}
\right\rangle _{l},
\]
where $\Phi$, $\Psi\in\left( L_{\mathbb{C}}^{2}\right) $ are as in
(\ref{Eq_4})-(\ref{Eq_5}), and $\left\langle \cdot,\cdot\right\rangle _{l}$
denotes the inner product on $\mathcal{H}_{l}^{\widehat{\otimes}n}$.
The Kondratiev space of test functions $\left( \mathcal{H}_{\infty}\right)
^{\beta}$ is defined to be the projective limit of the spaces $\mathcal{H}
_{l,k,\beta}$:
\[
\left( \mathcal{H}_{\infty}\right) ^{\beta}=\underleftarrow{\lim}
_{l,k\in\mathbb{N}}\mathcal{H}_{l,k,\beta}.
\]
As a vector space $\left( \mathcal{H}_{\infty}\right) ^{\beta}=\cap
_{l,k\in\mathbb{N}}\mathcal{H}_{l,k,\beta}$. The space of test functions
$\left( \mathcal{H}_{\infty}\right) ^{\beta}$ is a nuclear countable Hilbert
space, which is continuously and densely embedded in $\left( L_{\mathbb{C}
}^{2}\right) $. Moreover, $\left( \mathcal{H}_{\infty}\right) ^{\beta}$ and
its topology do not depend on the family of Hilbertian norms $\left\{
\left\Vert \cdot\right\Vert _{l}\right\} _{l\in\mathbb{N}}$, see e.g.
\cite[Theorem 1]{KLS96}, \cite[Chapter IV, Theorem 1.4]{Huang-Yang}.
The construction used to obtain the spaces $\left( \mathcal{H}_{\infty
}\right) ^{\beta}$ can be carried out starting with an arbitrary nuclear
space $\mathcal{N}$. For $0\leq\beta\leq1$, the spaces $\left( \mathcal{N}
\right) ^{\beta}$ were studied by Kondratiev, Leukert and Streit in
\cite{KphD}, \cite{KS93}, \cite{KLS96}, see also \cite[Chapter IV]
{Huang-Yang}. In the case $\beta=0$ and $\mathcal{N=S}$, the Schwartz space in
$\mathbb{R}^{n}$, the space $\left( \mathcal{N}\right) ^{0}$ is the Hida
space of test functions, see e.g. \cite{Hida et al}.
\subsection{Kondratiev-type spaces of distributions}
Let $\mathcal{H}_{-l,-k,-\beta}$ be the dual with respect to $(L_{\mathbb{C}
}^{2})$ of $\mathcal{H}_{l,k,\beta}$ and let \ $(\mathcal{H}_{\infty}
)^{-\beta}$ be the dual with respect to\ $(L_{\mathbb{C}}^{2})$ of
$(\mathcal{H}_{\infty})^{\beta}$. We denote by $\left\langle \left\langle
\cdot,\cdot\right\rangle \right\rangle $\ the corresponding dual pairing which
is given by the extension of the scalar product on $(L_{\mathbb{C}}^{2})$. We
define the expectation of a distribution $\boldsymbol{\Phi}\in(\mathcal{H}
_{\infty})^{-\beta}$ as $\mathbb{E}_{\mu}(\boldsymbol{\Phi})=\left\langle
\left\langle \boldsymbol{\Phi},1\right\rangle \right\rangle $.
The dual space of \ $(\mathcal{H}_{\infty})^{-\beta}$ is given by
\[
(\mathcal{H}_{\infty})^{-\beta}=\underset{l,k\in\mathbb{N}}{{\LARGE \cup}
}\mathcal{H}_{-l,-k,-\beta},
\]
see \cite[Chapter IV, Theorem 1.5]{Huang-Yang}. We will consider
$(\mathcal{H}_{\infty})^{-\beta}$ with the inductive limit topology. In
particular, we know that every distribution is of finite order, i.e. for any
$\boldsymbol{\Phi}\in(\mathcal{H}_{\infty})^{-\beta}$ there exist
$l,k\in\mathbb{N}$ such that $\boldsymbol{\Phi}\in\mathcal{H}_{-l,-k,-\beta}$.
The chaos decomposition introduces a natural decomposition of
$\boldsymbol{\Phi}\in(\mathcal{H}_{\infty})^{-\beta}$ into generalized kernels
$\Phi_{n}\in(\mathcal{H}_{\infty}^{\ast}\mathbb{(C}))^{\widehat{\otimes}n}$.
Let $\Phi_{n}\in(\mathcal{H}_{\infty}^{\ast}\mathbb{(C}))^{\widehat{\otimes}
n}$ be given. Then there is a distribution, denoted as $\left\langle \Phi
_{n},:W^{\otimes n}:\right\rangle $, in $(\mathcal{H}_{\infty})^{-\beta}$
acting on $\Psi\in$\ $(\mathcal{H}_{\infty})^{\beta}$ ($\Psi=\sum
\limits_{n=0}^{\infty}\left\langle :\cdot^{\otimes n}:,\psi_{n}\right\rangle
,$ with $\psi_{n}\in\left( \mathcal{H}_{\infty}(\mathbb{C})\right)
^{\widehat{\otimes}n}$)\ as
\[
\left\langle \left\langle \left\langle \Phi_{n},:W^{\otimes n}:\right\rangle
,\Psi\right\rangle \right\rangle =n!\left\langle \Phi_{n},\psi_{n}
\right\rangle .
\]
Any $\boldsymbol{\Phi}\in(\mathcal{H}_{\infty})^{-\beta}$ has a unique
decomposition of the form
\[
\boldsymbol{\Phi}=\overset{\infty}{\underset{n=0}{\sum}}\left\langle \Phi
_{n},:W^{\otimes n}:\right\rangle \text{, }\Phi_{n}\in(\mathcal{H}_{\infty
}^{\ast}\mathbb{(C}))^{\widehat{\otimes}n}\text{,}
\]
where the series converges in $(\mathcal{H}_{\infty})^{-\beta}$, in addition,
we have
\[
\left\langle \left\langle \boldsymbol{\Phi},\Psi\right\rangle \right\rangle
=\overset{\infty}{\underset{n=0}{\sum}}n!\left\langle \Phi_{n},\psi
_{n}\right\rangle \text{,\ }\Psi\in(\mathcal{H}_{\infty})^{\beta}.
\]
Now, $\mathcal{H}_{-l,-k,-\beta}$\ is a Hilbert space, that can be described
as follows:
\[
\mathcal{H}_{-l,-k,-\beta}=\left\{ \boldsymbol{\Phi}\in(\mathcal{H}_{\infty
})^{-\beta};\text{ }\left\Vert \boldsymbol{\Phi}\right\Vert _{-l,-k,-\beta
}<\infty\right\} ,
\]
where
\begin{equation}
\left\Vert \boldsymbol{\Phi}\right\Vert _{-l,-k,-\beta}^{2}=\underset
{n=0}{\overset{\infty}{\sum}}\left( n!\right) ^{1-\beta}2^{-nk}\left\Vert
\Phi_{n}\right\Vert _{-l}^{2}, \label{Eq_6}
\end{equation}
see \ \cite[Chapter IV, Theorem 1.5]{Huang-Yang}.
\begin{remark}
Notice that
\begin{align*}
(\mathcal{H}_{\infty})^{1} & \subset\cdots\subset(\mathcal{H}_{\infty
})^{\beta}\subset\cdots\subset(\mathcal{H}_{\infty})^{0}\subset(L_{\mathbb{C}
}^{2})\\
& \subset(\mathcal{H}_{\infty})^{-0}\subset\cdots\subset(\mathcal{H}_{\infty
})^{-\beta}\subset\cdots\subset(\mathcal{H}_{\infty})^{-1}.
\end{align*}
Following Kondratiev, Leukert and Streit, in this article we work with the
Gel'fand triple $(\mathcal{H}_{\infty})^{1}\subset(L_{\mathbb{C}}^{2}
)\subset(\mathcal{H}_{\infty})^{-1}$.
\end{remark}
\subsection{The $S$-transform and the characterization of $(\mathcal{H}
_{\infty})^{-1}$}
\subsubsection{The $S$-transform}
We first consider the Wick exponential:
\[
:\exp\left\langle W,g\right\rangle :=\exp\left( \left\langle W,g\right\rangle
-\frac{1}{2}\left\Vert g\right\Vert _{0}^{2}\right) =\overset{\infty
}{\underset{n=0}{\sum}}\frac{1}{n!}\left\langle :W^{\otimes n}:,g^{\otimes
n}\right\rangle \text{, }
\]
\ for $W\in\mathcal{H}_{\infty}^{\ast}\left( \mathbb{R}\right) $,
$g\in\mathcal{H}_{\infty}\left( \mathbb{C}\right) $. Then $:\exp\left\langle
W,g\right\rangle :\in(L_{\mathbb{C}}^{2})$ and its $l$, $k$, $1$-norm is given
by
\[
\left\Vert :\exp\left\langle \cdot,g\right\rangle :\right\Vert _{l,k,1}
^{2}=\underset{n=0}{\overset{\infty}{\sum}}(n!)^{2}2^{nk}\left\Vert \frac
{1}{n!}g^{\otimes n}\right\Vert _{l}^{2}=\underset{n=0}{\overset{\infty}{\sum
}}\left( 2^{k}\left\Vert g\right\Vert _{l}^{2}\right) ^{n}.
\]
This norm is finite if and only if $2^{k}\left\Vert g\right\Vert _{l}^{2}<1$,
i.e. $:\exp\left\langle W,g\right\rangle :\in\mathcal{H}_{l,k,\beta}$ if and
only if $g$ belongs to the following neighborhood of zero:
\[
\mathcal{U}_{l,k}=\left\{ f\in\mathcal{H}_{\infty}\left( \mathbb{C}\right)
;\left\Vert f\right\Vert _{l}<\frac{1}{2^{\frac{k}{2}}}\right\} .
\]
Therefore the Wick exponential does not belong to $(\mathcal{H}_{\infty})^{1}
$, i.e. it is not a test function, in contrast to usual white noise analysis.
Let $\boldsymbol{\Phi}\in(\mathcal{H}_{\infty})^{-1},$ then there \ exist
$l,k$ such that $\boldsymbol{\Phi}\in$ $\mathcal{H}_{-l,-k,-1}$. For all
$f\in\mathcal{U}_{l,k}$, we define the (local) $S$-transform of
$\boldsymbol{\Phi}$ as
\begin{equation}
{\LARGE S}\boldsymbol{\Phi}\left( f\right) =\left\langle \left\langle
\boldsymbol{\Phi},:\exp\left\langle \cdot,f\right\rangle :\right\rangle
\right\rangle =\underset{n=0}{\overset{\infty}{\sum}}\left\langle \Phi
_{n},f^{\otimes n}\right\rangle . \label{Eq_7}
\end{equation}
Hence, for $\boldsymbol{\Phi}\in$ $\mathcal{H}_{-l,-k,-1}$, (\ref{Eq_7})
defines the $S$-transform for all $f$ $\in\mathcal{U}_{l,k}$.
\subsubsection{Holomorphic functions on $\mathcal{H}_{\infty}(\mathbb{C})$}
Let $\mathcal{V}_{l,\epsilon}=\left\{ f\in\mathcal{H}_{\infty}\left(
\mathbb{C}\right) ;\left\Vert f\right\Vert _{l}<\epsilon\right\} $ be a
neighborhood of zero in $\mathcal{H}_{\infty}\left( \mathbb{C}\right) $. A
map $F:\mathcal{V}_{l,\epsilon}\rightarrow\mathbb{C}$ is called
\textit{holomorphic} in $\mathcal{V}_{l,\epsilon}$, if it satisfies the
following two conditions: (i) for each $g_{0}\in\mathcal{V}_{l,\epsilon}$,
$g\in\mathcal{H}_{\infty}\left( \mathbb{C}\right) $ there exists a
neighborhood $V_{g_{0},g}$ in $\mathbb{C}$ around the origin such that the map
$z\rightarrow F\left( g_{0}+zg\right) $ is holomorphic in $V_{g_{0},g}$.
(ii) For each $g\in\mathcal{V}_{l,\epsilon}$ there exists an open set
$\mathcal{U}\subset\mathcal{V}_{l,\epsilon}$ containing $g$ such that
$F\left( \mathcal{U}\right) $\ is bounded.
By identifying two maps $F_{1}$ and $F_{2}$ coinciding in a neighborhood of
zero, we define $Hol_{0}(\mathcal{H}_{\infty}(\mathbb{C}))$ as the space of
germs of holomorphic maps around the origin.
\subsubsection{\label{Section_Characterization}Characterization of
$(\mathcal{H}_{\infty})^{-1}$}
A key result is the following: the mapping
\[
\begin{array}
[c]{cccc}
{\LARGE S}: & (\mathcal{H}_{\infty})^{-1} & \rightarrow & Hol_{0}
(\mathcal{H}_{\infty}(\mathbb{C}))\\
& \boldsymbol{\Phi} & \rightarrow & S\boldsymbol{\Phi}
\end{array}
\]
is a well-defined bijection, see \cite[Theorem 3]{KLS96}, \ \cite[Chapter IV,
Theorem 2.13]{Huang-Yang}.
\subsubsection{Integration of distributions}
Let $\left( \mathfrak{L},\mathcal{A},\nu\right) $ be a measure space, and
\[
\begin{array}
[c]{ccc}
\mathfrak{L} & \rightarrow & (\mathcal{H}_{\infty})^{-1}\\
\mathfrak{l} & \rightarrow & \boldsymbol{\Phi}_{\mathfrak{l}}
\end{array}
.
\]
Assume that there exists an open neighborhood $\mathcal{V}\subset
\mathcal{H}_{\infty}(\mathbb{C})$ of zero such that (i) $S\boldsymbol{\Phi
}_{\mathfrak{l}}$, $\mathfrak{l}\in\mathfrak{L}$, is holomorphic in
$\mathcal{V}$; (ii) the mapping $\mathfrak{l}\rightarrow{\LARGE S}
\boldsymbol{\Phi}_{\mathfrak{l}}\left( g\right) $ is measurable for every
$g\in\mathcal{V}$; and (iii) there exists a function $C(\mathfrak{l})\in
L^{1}\left( \mathfrak{L},\mathcal{A},\nu\right) $ such that $\left\vert
{\LARGE S}\boldsymbol{\Phi}_{\mathfrak{l}}\left( g\right) \right\vert \leq
C(\mathfrak{l})$ for all $g\in\mathcal{V}$ and for $\nu$-almost $\mathfrak{l}
\in\mathfrak{L}$. Then there exist $l_{0}$, $k_{0}$ $\in\mathbb{N}$ such that
$\int_{\mathfrak{L}}\boldsymbol{\Phi}_{\mathfrak{l}}d\nu\left( \mathfrak{l}
\right) $ exists as a Bochner integral in $\mathcal{H}_{-l_{0},-k_{0},-1}$,
in particular,
\begin{equation}
{\LARGE S}\left( \int\nolimits_{\mathfrak{L}}\boldsymbol{\Phi}_{\mathfrak{l}
}d\nu\left( \mathfrak{l}\right) \right) \left( g\right) =\int
\nolimits_{\mathfrak{L}}{\LARGE S}\boldsymbol{\Phi}_{\mathfrak{l}}\left(
g\right) d\nu\left( \mathfrak{l}\right) \text{, for any }g\in
\mathcal{V}\text{,} \label{Eq_C}
\end{equation}
cf. \cite[Theorem 6]{KLS96}, \cite[Chapter IV, Theorem 2.15]{Huang-Yang}.
\subsubsection{The Wick product}
Given $\boldsymbol{\Phi}$, $\boldsymbol{\Psi}\in(\mathcal{H}_{\infty})^{-1}$,
we define the \textit{Wick product} of them as
\[
\boldsymbol{\Phi}\Diamond\boldsymbol{\Psi}=S^{-1}\left( {\LARGE S}
\boldsymbol{\Phi}{\LARGE S}\boldsymbol{\Psi}\right) .
\]
This product is well-defined because $Hol_{0}(\mathcal{H}_{\infty}
(\mathbb{C}))$\ is an algebra. The map
\[
\begin{array}
[c]{ccc}
(\mathcal{H}_{\infty})^{-1}\times(\mathcal{H}_{\infty})^{-1} & \rightarrow &
(\mathcal{H}_{\infty})^{-1}\\
\left( \boldsymbol{\Phi},\boldsymbol{\Psi}\right) & \rightarrow &
\boldsymbol{\Phi}\Diamond\boldsymbol{\Psi}
\end{array}
\]
is well-defined and continuous. Furthermore, if $\boldsymbol{\Phi}
\in\mathcal{H}_{-l_{1},-k_{1},-1}$, $\boldsymbol{\Psi}\in\mathcal{H}
_{-l_{2},-k_{2},-1}$, and $l:=\max\left\{ l_{1},l_{2}\right\} $,
$k:=k_{1}+k_{2}+1$, then
\[
\left\Vert \boldsymbol{\Phi}\Diamond\boldsymbol{\Psi}\right\Vert
_{-l,-k,-1}\leq\left\Vert \boldsymbol{\Phi}\right\Vert _{-l_{1},-k_{1}
,-1}\left\Vert \boldsymbol{\Psi}\right\Vert _{-l_{2},-k_{2},-1},
\]
cf. \cite[Proposition 11]{KLS96}. The Wick product leaves $(\mathcal{H}
_{\infty})$ invariant. By induction on $n$, we can define the Wick powers:
\[
\boldsymbol{\Phi}^{\Diamond n}={\LARGE S}^{-1}(\left( {\LARGE S}
\boldsymbol{\Phi}\right) ^{n})\in(\mathcal{H}_{\infty})^{-1}.
\]
Consequently $\sum_{n=0}^{m}a_{n}\boldsymbol{\Phi}^{\Diamond n}\in
(\mathcal{H}_{\infty})^{-1}$.
\subsubsection{\label{Section Wick_analytic_fun}Wick analytic functions in
$(\mathcal{H}_{\infty})^{-1}$}
Assume that $F$ is an analytic function in a neighborhood of \ the point
$z_{0}=\mathbb{E}_{\mu}\left( \boldsymbol{\Phi}\right) $ in $\mathbb{C}$,
with $\boldsymbol{\Phi}\in(\mathcal{H}_{\infty})^{-1}$. Then $F^{\Diamond
}(\boldsymbol{\Phi})=S^{-1}(F(S\boldsymbol{\Phi}))$ exists in $(\mathcal{H}
_{\infty})^{-1}$, cf. \cite[Theorem 12]{KLS96}. \ In addition, if $F$ is
analytic in $z_{0}=\mathbb{E}_{\mu}\left( \boldsymbol{\Phi}\right) $, with
power series $F(z)=\sum_{n=0}^{\infty}c_{n}\left( z-z_{0}\right) ^{n}$, then
the Wick series $\sum_{n=0}^{\infty}c_{n}\left( \boldsymbol{\Phi}
-z_{0}\right) ^{\Diamond n}$ converges in $(\mathcal{H}_{\infty})^{-1}$ and
$F^{\Diamond}(\boldsymbol{\Phi})=\sum_{n=0}^{\infty}c_{n}\left(
\boldsymbol{\Phi}-z_{0}\right) ^{\Diamond n}$.
\section{Schwinger Functions and Euclidean Quantum Field Theory}
\subsection{Schwinger functions}
\begin{definition}
Let $f_{1},\ldots,f_{n}\in\mathcal{H}_{\infty}\left( \mathbb{R}\right) $,
$n\in\mathbb{N}$. The $n$-th Schwinger function corresponding to
$\boldsymbol{\Phi}\in(\mathcal{H}_{\infty})^{-1}$, with $\mathbb{E}_{\mu
}\left( \boldsymbol{\Phi}\right) =1$, is defined as
\begin{equation}
\mathcal{S}_{n}^{\boldsymbol{\Phi}}\left( f_{1}\otimes\cdots\otimes
f_{n}\right) \left( W\right) =\left\{
\begin{array}
[c]{lll}
1 & \text{if} & n=0\\
& & \\
\left\langle \left\langle \boldsymbol{\Phi},\left\langle W,f_{1}\right\rangle
\cdots\left\langle W,f_{n}\right\rangle \right\rangle \right\rangle &
\text{if} & n\geq1,
\end{array}
\right. \label{Eq_8}
\end{equation}
for $W\in\mathcal{H}_{\infty}^{\ast}\left( \mathbb{R}\right) $.
\end{definition}
The pairing in (\ref{Eq_8}) is well-defined because the Wick polynomials
$\mathcal{P}\left( \mathcal{H}_{\infty}^{\ast}\left( \mathbb{R}\right)
\right) $ are dense in $\left( \mathcal{H}_{\infty}\right) ^{1}$.
The ${\LARGE T}$-transform of a distribution is defined as
\begin{equation}
{\LARGE T}\boldsymbol{\Phi}\left( g\right) =\exp\left( \frac{-1}
{2}\left\Vert g\right\Vert _{0}^{2}\right) S\boldsymbol{\Phi}\left(
ig\right) \label{Eq_T}
\end{equation}
for $\boldsymbol{\Phi}\in(\mathcal{H}_{\infty})^{-1}$ and $g\in\mathcal{U}$,
where $\mathcal{U}$ is neighborhood of zero in $\mathcal{H}_{\infty}\left(
\mathbb{C}\right) $. The Schwinger functions can be computed by using the $T$-transform:
\begin{lemma}
[{\cite[Proposition III.3]{GS1999}}]\label{Lemma0}Let $f_{1},\ldots,f_{n}
\in\mathcal{H}_{\infty}\left( \mathbb{R}\right) $, $n\in\mathbb{N}$. The
$n$-th Schwinger function corresponding to $\boldsymbol{\Phi}\in
(\mathcal{H}_{\infty})^{-1}$ is given by
\[
\mathcal{S}_{n}^{\boldsymbol{\Phi}}\left( f_{1}\otimes\cdots\otimes
f_{n}\right) =(-i)^{n}\frac{\partial^{n}}{\partial t_{1}\cdots\partial t_{n}
}{\LARGE T}\boldsymbol{\Phi}\left( t_{1}f_{1}+\cdots+t_{n}f_{n}\right)
{\LARGE \mid}_{t_{1}=\cdots=t_{n}=0}.
\]
\end{lemma}
\begin{lemma}
\label{Lemma1}For each distribution $\boldsymbol{\Phi}\in(\mathcal{H}_{\infty
})^{-1}$, with $\mathbb{E}_{\mu}\left( \boldsymbol{\Phi}\right) =1 $, the
Schwinger functions $\left\{ \mathcal{S}_{n}^{\boldsymbol{\Phi}}\right\}
_{n\in\mathbb{N}}$\ satisfy the following conditions:
\begin{enumerate}
\item[(OS1)] the sequence $\left\{ \mathcal{S}_{n}^{\boldsymbol{\Phi}
}\right\} _{n\in\mathbb{N}}$, with $\mathcal{S}_{n}^{\boldsymbol{\Phi}}
\in\left( \mathcal{H}_{\infty}^{\ast}\left( \mathbb{C}\right) \right)
^{\otimes n}$, satisfies
\[
\left\vert \mathcal{S}_{n}^{\boldsymbol{\Phi}}\left( f_{1}\otimes
\cdots\otimes f_{n}\right) \right\vert \leq KC^{n}n!
{\textstyle\prod\nolimits_{i=1}^{n}}
\left\Vert f_{i}\right\Vert _{l},
\]
for some $l$, $k$ $\in\mathbb{N}$, where $K=\sqrt{I_{0}(2^{-k})}\left\Vert
\Phi\right\Vert _{-l.-k-1}$, here $I_{0}$ is the modified Bessel function of
order zero, which satisfies $I_{0}(2^{-k})<1.3$, $C=e2^{\frac{k}{2}}$, and for
any $f_{1},\cdots,f_{n}\in\mathcal{H}_{\infty}\left( \mathbb{R}\right) $;
\item[(OS4)] for $n\geq2$ and all $\sigma\in\mathfrak{S}_{n}$, the permutation
group of order $n$, it holds that
\[
\mathcal{S}_{n}^{\boldsymbol{\Phi}}\left( f_{1}\otimes\cdots\otimes
f_{n}\right) =\mathcal{S}_{n}^{\boldsymbol{\Phi}}\left( f_{\sigma\left(
1\right) }\otimes\cdots\otimes f_{\sigma\left( n\right) }\right) ,
\]
for any $f_{1},\cdots,f_{n}\in\mathcal{H}_{\infty}\left( \mathbb{R}\right) $.
\end{enumerate}
\end{lemma}
\begin{proof}
Estimation (OS1) is given in the proof of Theorem 2 in \cite{KSW95}. The
Schwinger functions $\left( \mathcal{S}_{n}^{\boldsymbol{\Phi}}\right) $ are
symmetric by definition.
\end{proof}
\subsection{\label{Section_white_noise_process}A white-noise process}
For $t\in\mathbb{Q}_{p}$, $\overrightarrow{x}\in\mathbb{Q}_{p}^{N-1}$, we set
$x=\left( t,\overrightarrow{x}\right) $. We denote by $\delta_{x}
:=\delta_{\left( t,\overrightarrow{x}\right) }$, the Dirac distribution at
$\left( t,\overrightarrow{x}\right) $.
\begin{lemma}
$\delta_{\left( t,\overrightarrow{x}\right) }\in\left( \mathcal{H}_{\infty
}\right) ^{-1}$.
\end{lemma}
\begin{proof}
We first notice that
\[
\left\Vert \delta_{\left( t,\overrightarrow{x}\right) }\right\Vert _{-l}
^{2}=\int\nolimits_{\mathbb{Q}_{p}^{N}}\frac{d^{N}\xi}{\left[ \xi\right]
_{p}^{l}}<\infty\text{ for }l>N\text{,}
\]
which implies that $\delta_{\left( t,\overrightarrow{x}\right) }
\in\mathcal{H}_{-l}(\mathbb{C})$ for all $l>N$, see (\ref{Eq_A}). Now, we
define $\left\{ \Phi_{n}\right\} _{n\in\mathbb{N}}$, with $\Phi_{n}
\in\left( \mathcal{H}_{\infty}^{\ast}\left( \mathbb{C}\right) \right)
^{\widehat{\otimes}n}$, as $\Phi_{n}=0$ if $n\neq1$ and $\Phi_{1}
=\delta_{\left( t,\overrightarrow{x}\right) }$. Then
\[
\sum_{n}\left\langle \Phi_{n},:W^{\otimes n}:\right\rangle =\left\langle
\delta_{\left( t,\overrightarrow{x}\right) },:W:\right\rangle \in\left(
\mathcal{H}_{\infty}\right) ^{-1}.
\]
In addition, for $\psi\in\mathcal{H}_{\infty}\left( \mathbb{C}\right) $, we
have
\begin{align*}
\left\langle \left\langle \left\langle \delta_{\left( t,\overrightarrow
{x}\right) },:W:\right\rangle ,\psi\right\rangle \right\rangle &
=\left\langle \delta_{\left( t,\overrightarrow{x}\right) },\psi\right\rangle
=\int\nolimits_{\mathbb{Q}_{p}^{N}}\chi_{p}\left( -\xi\cdot x\right)
\widehat{\psi}\left( x\right) d^{N}\xi\\
& =\psi\left( t,\overrightarrow{x}\right) ,
\end{align*}
where we used that $\psi$ is a continuous function in $L^{1}\cap L^{2}$, see
Section \ref{Section1}\ and \cite[Theorem 10.15]{KKZuniga}.
\end{proof}
We now set
\[
\boldsymbol{\Phi}\left( t,\overrightarrow{x}\right) :=\left\langle
\delta_{\left( t,\overrightarrow{x}\right) },:W:\right\rangle \in\left(
\mathcal{H}_{\infty}\right) ^{-1}.
\]
Then $\boldsymbol{\Phi}\left( t,\overrightarrow{x}\right) $\ is a
white-noise process with $\mathbb{E}_{\mu}\left( \boldsymbol{\Phi}\left(
t,\overrightarrow{x}\right) \right) =0$.
Assume that
\[
H(z)=\sum\nolimits_{k=0}^{\infty}\frac{1}{k!}H_{k}z^{k}\text{, }z\in
U\subset\mathbb{C}\text{,}
\]
is a holomorphic function \ in $U$, an open neighborhood of $0=\mathbb{E}
_{\mu}\left( \boldsymbol{\Phi}\left( t,\overrightarrow{x}\right) \right)
$. By \cite[Theorem 12]{KLS96}, see also Section
\ref{Section Wick_analytic_fun}, we can define
\begin{align*}
H^{\lozenge}(\boldsymbol{\Phi}\left( t,\overrightarrow{x}\right) ) &
=\sum\nolimits_{k=0}^{\infty}\frac{1}{k!}H_{k}\boldsymbol{\Phi}\left(
t,\overrightarrow{x}\right) ^{\lozenge k}\\
& =\sum\nolimits_{k=0}^{\infty}\frac{1}{k!}H_{k}\left\langle \delta_{\left(
t,\overrightarrow{x}\right) }^{\otimes k},:W^{\otimes k}:\right\rangle
\in\left( \mathcal{H}_{\infty}\right) ^{-1}.
\end{align*}
Our next goal is the construction of the potential
\begin{equation}
\int\nolimits_{\mathbb{Q}_{p}^{N}}H^{\lozenge}(\boldsymbol{\Phi}\left(
x\right) )d^{N}x \label{Eq_12}
\end{equation}
as a white-noise distribution. This goal is accomplished through the following result:
\begin{theorem}
\label{Theorem1}(i) Let $H$ be a holomorphic function at zero such that
$H(0)=0$. Then (\ref{Eq_12}) exists as a Bochner integral in a suitable
subspace of $\left( \mathcal{H}_{\infty}\right) ^{-1}$.
\noindent(ii) The distribution
\[
\boldsymbol{\Phi}_{H}:=\exp^{\lozenge}\left( -\int\nolimits_{\mathbb{Q}
_{p}^{N}}H^{\lozenge}(\boldsymbol{\Phi}\left( x\right) )d^{N}x\right)
\]
is an element of $\left( \mathcal{H}_{\infty}\right) ^{-1}$.
\noindent(iii) The ${\LARGE T}$-transform of $\boldsymbol{\Phi}_{H}$\ is given
by
\[
{\LARGE T}\boldsymbol{\Phi}_{H}\left( g\right) =\exp\left( -\int
\nolimits_{\mathbb{Q}_{p}^{N}}H(ig\left( x\right) )+\frac{1}{2}\left(
g\left( x\right) \right) ^{2}\text{ }d^{N}x\right)
\]
for all $g$ in a neighborhood $\mathcal{U\subset H}_{\infty}\mathcal{(}
\mathbb{C}\mathcal{)}$ of the zero. In particular, $\mathbb{E}_{\mathbb{\mu}
}(\boldsymbol{\Phi}_{H})=1$.
\end{theorem}
\begin{proof}
(i) The result follows from the discussion presented in Section
\ref{Section_Characterization}, see also \cite[Theorem 6]{KLS96}, as follows.
Let $r>0$ be the radius of convergence of the Taylor series of $H$ at the
origin. We set $C(N):=\sqrt{\int_{\mathbb{Q}_{p}^{N}}\frac{d^{N}\xi}{\left[
\xi\right] _{p}^{l}}}$, for a fixed $l>N$, and
\[
\mathcal{U}_{0}:=\left\{ g\in\mathcal{H}_{\infty}\left( \mathbb{C}\right)
;\left\Vert g\right\Vert _{l}<\frac{r}{C(N)}\right\} .
\]
Then, for $g\in\mathcal{U}_{0}$ we have
\begin{align}
{\LARGE S}H^{\lozenge}(\boldsymbol{\Phi}\left( x\right) )\left( g\right)
& =\sum_{k=1}^{\infty}\frac{1}{k!}H_{k}\left\langle \delta_{x}^{\otimes
k},g^{\otimes k}\right\rangle =\sum_{k=1}^{\infty}\frac{1}{k!}H_{k}
g(x)^{k}\label{Eq_13}\\
& =\sum_{k=1}^{\infty}\frac{1}{k!}H_{k}\left\{ \frac{g(x)}{r}\right\}
^{k}r^{k}.\nonumber
\end{align}
By Claim A, $\left\vert \frac{g(x)}{r}\right\vert <1$, and from (\ref{Eq_13})
we obtain that
\begin{equation}
\left\vert {\LARGE S}H^{\lozenge}(\boldsymbol{\Phi}\left( x\right) )\left(
g\right) \right\vert \leq\left\vert g(x)\right\vert \sum_{k=1}^{\infty}
\frac{1}{k!}\left\vert H_{k}\right\vert r^{k-1}\in L^{1}\left( \mathbb{Q}
_{p}^{N}\right) , \label{Eq_14}
\end{equation}
because $\mathcal{H}_{\infty}\left( \mathbb{C}\right) \subset L^{1}\left(
\mathbb{Q}_{p}^{N}\right) $, cf. \cite[Theorem 10.15]{KKZuniga}. Estimation
(\ref{Eq_14}) implies the holomorphy of ${\LARGE S}H^{\lozenge}
(\boldsymbol{\Phi}\left( x\right) )\left( g\right) $ for any
$g\in\mathcal{U}_{0}$. Since ${\LARGE S}H^{\lozenge}(\boldsymbol{\Phi}\left(
x\right) )\left( g\right) $ is measurable by \cite[Theorem 6]{KLS96}, we
conclude that (\ref{Eq_12}) is an element of $\left( \mathcal{H}_{\infty
}\right) ^{-1}$.
\textbf{Claim A.} $\ \mathcal{U}_{0}\subset\mathcal{U}:=\left\{
g\in\mathcal{H}_{\infty}\left( \mathbb{C}\right) ;\left\Vert g\right\Vert
_{L^{\infty}}<r\right\} .$
The Claim follows from the fact that
\[
\left\Vert g\right\Vert _{L^{\infty}}\leq C(N)\left\Vert g\right\Vert
_{l}\text{, for\ }g\in\mathcal{H}_{\infty}\left( \mathbb{C}\right) .
\]
\ This last fact is verified as follows: by using that $g\in L^{1}\left(
\mathbb{Q}_{p}^{N}\right) \cap L^{2}\left( \mathbb{Q}_{p}^{N}\right) $, and
the Cauchy-Schwarz inequality, we have
\begin{align*}
\left\vert g\left( x\right) \right\vert & =\left\vert \int
\nolimits_{\mathbb{Q}_{p}^{N}}\chi_{p}\left( -\xi\cdot x\right) \widehat
{g}\left( \xi\right) d^{N}\xi\right\vert \leq\int\nolimits_{\mathbb{Q}
_{p}^{N}}\left\vert \widehat{g}\left( \xi\right) \right\vert d^{N}\xi\\
& =\int\nolimits_{\mathbb{Q}_{p}^{N}}\frac{1}{\left[ \xi\right] _{p}
^{\frac{l}{2}}}\text{ }\left\{ \left[ \xi\right] _{p}^{\frac{l}{2}
}\left\vert \widehat{g}\left( \xi\right) \right\vert \right\} d^{N}\xi\leq
C(N)\left\Vert g\right\Vert _{l}.
\end{align*}
(ii) Since $\exp$ is analytic in a neighborhood of $0=\mathbb{E}_{\mu}\left(
\boldsymbol{\Phi}\left( t,\overrightarrow{x}\right) \right) $, then
\[
\exp^{\lozenge}\left( -\int\nolimits_{\mathbb{Q}_{p}^{N}}H^{\lozenge
}(\boldsymbol{\Phi}\left( x\right) )d^{N}x\right) ={\LARGE S}^{-1}\left(
\exp\left( {\LARGE S}\left( -\int\nolimits_{\mathbb{Q}_{p}^{N}}H^{\lozenge
}(\boldsymbol{\Phi}\left( x\right) )d^{N}x\right) \right) \right) ,
\]
and by (i), $-\int\nolimits_{\mathbb{Q}_{p}^{N}}H^{\lozenge}(\boldsymbol{\Phi
}\left( x\right) )d^{N}x\in\left( \mathcal{H}_{\infty}\right) ^{-1}$, and
then its $S$-transform is analytic at the origin, and its composition with
$\exp$ gives again an analytic function at the origin, whose inverse
$S$-transform gives an element of $\left( \mathcal{H}_{\infty}\right) ^{-1}
$, cf. \cite[Theorem 12]{KLS96}.
(iii) The calculation of the ${\LARGE T}$-transform uses (\ref{Eq_T}),
$\exp^{\lozenge}\left( \cdot\right) ={\LARGE S}^{-1}\left( \exp\left(
{\LARGE S}\left( \cdot\right) \right) \right) $, and (\ref{Eq_13}) as
follows:
\begin{align*}
\left( {\LARGE T}\boldsymbol{\Phi}_{H}\right) \left( g\right) &
=\exp\left( -\frac{1}{2}\left\Vert g\right\Vert _{0}^{2}\right) \exp\left(
{\LARGE S}\left( -\int_{\mathbb{Q}_{p}^{N}}H^{\lozenge}\left(
\boldsymbol{\Phi}\left( x\right) \right) d^{N}x\right) \left( ig\right)
\right) \\
& =\exp\left( -\frac{1}{2}\left\Vert g\right\Vert _{0}^{2}\right)
\exp\left( -\int_{\mathbb{Q}_{p}^{N}}\left\langle \left\langle H^{\lozenge
}\left( \boldsymbol{\Phi}\left( x\right) \right) ,:\exp\left\langle
\cdot,ig\right\rangle :\right\rangle \right\rangle d^{N}x\right) \\
& =\exp\left( -\frac{1}{2}\left\Vert g\right\Vert _{0}^{2}\right)
\exp\left( -\int_{\mathbb{Q}_{p}^{N}}{\LARGE S}H^{\lozenge}\left(
\boldsymbol{\Phi}\left( x\right) \right) \left( ig\right) d^{N}x\right)
\\
& =\exp\left( -\int_{\mathbb{Q}_{p}^{N}}H\left( ig\left( x\right)
\right) +\frac{1}{2}g\left( x\right) ^{2}\text{ }d^{N}x\right) .
\end{align*}
In particular $\mathbb{E}_{\mu}(\boldsymbol{\Phi}_{H})={\LARGE T}
\boldsymbol{\Phi}_{H}\left( 0\right) =1$.
\end{proof}
\subsection{\label{Sect4}Pseudodifferential Operators and Green Functions}
A non-constant homogeneous polynomial $\mathfrak{l}\left( \xi\right)
\in\mathbb{Z}_{p}\left[ \xi_{1},\cdots,\xi_{N}\right] $ of degree $d$ is
called \textit{elliptic\ }if it\textit{ }satisfies $\mathfrak{l}\left(
\xi\right) =0\Leftrightarrow\xi=0$. There are infi\-nitely many elliptic
polynomials, cf. \cite[Lemma 24]{Zuniga-LNM-2016}. A such polynomial
satisfies
\begin{equation}
C_{0}\left( \alpha\right) \left\Vert \xi\right\Vert _{p}^{\alpha d}
\leq\left\vert \mathfrak{l}\left( \xi\right) \right\vert _{p}^{\alpha}\leq
C_{1}\left( \alpha\right) \left\Vert \xi\right\Vert _{p}^{\alpha d},
\label{basic_estimate}
\end{equation}
for some positive constants $C_{0}\left( \alpha\right) $, $C_{1}\left(
\alpha\right) $, cf. \cite[Lemma 25]{Zuniga-LNM-2016}. We define an
\textit{elliptic pseudodifferential operator with symbol }$\left\vert
\mathfrak{l}\left( \xi\right) \right\vert _{p}^{\alpha}$, with $\alpha>0$,
as
\begin{equation}
\left( \mathbf{L}_{\alpha}h\right) \left( x\right) =\mathcal{F}
_{\xi\rightarrow x}^{-1}\left( \left\vert \mathfrak{l}\left( \xi\right)
\right\vert _{p}^{\alpha}\mathcal{F}_{x\rightarrow\xi}h\right) ,
\label{ellipticoperaator}
\end{equation}
for $h\in\mathcal{D}(\mathbb{Q}_{p}^{N})$. We define $G:=G\left(
x;m,\alpha\right) \in\mathcal{D}^{\prime}(\mathbb{Q}_{p}^{N})$, with
$\alpha>0$, $m>0$, to be the solution of
\[
\left( \boldsymbol{L}_{\alpha}+m^{2}\right) G=\delta\text{ in \ }
\mathcal{D}^{\prime}(\mathbb{Q}_{p}^{N}).
\]
We will say that the \textit{Green function} $G\left( x;m,\alpha\right) $ is
a \textit{fundamental solution} of the equation
\begin{equation}
\left( \boldsymbol{L}_{\alpha}+m^{2}\right) u=h,\;\text{with }
h\in\mathcal{D}(\mathbb{Q}_{p}^{N}),\text{ }m>0. \label{equation1}
\end{equation}
As a distribution \ from $\mathcal{D}^{\prime}(\mathbb{Q}_{p}^{N})$, the Green
function $G\left( x;m,\alpha\right) $\ is given by
\begin{equation}
G\left( x;\alpha,m\right) =\mathcal{F}_{\xi\rightarrow x}^{-1}\left(
\frac{1}{\left\vert \mathfrak{l}\left( \xi\right) \right\vert _{p}^{\alpha
}+m^{2}}\right) . \label{GreenFunc}
\end{equation}
Notice that by (\ref{basic_estimate}), we have
\[
\frac{1}{\left\vert \mathfrak{l}\left( \xi\right) \right\vert _{p}^{\alpha
}+m^{2}}\in L^{1}\left( \mathbb{Q}_{p}^{N},d^{N}\xi\right) \text{\ \ for
\ }\alpha d>N,
\]
and in this case, $G\left( x;\alpha,m\right) $ is an $L^{\infty}$-function.
There exists a Green function $G\left( x;\alpha,m\right) $ for the operator
$\boldsymbol{L}_{\alpha}+m^{2}$, which is continuous and non-negative on
$\mathbb{Q}_{p}^{n}\smallsetminus\left\{ 0\right\} $, and tends to zero at
infinity. The equation
\begin{equation}
\left( \boldsymbol{L}_{\alpha}+m^{2}\right) u=g\text{, } \label{equation3A}
\end{equation}
\ with $g\in\mathcal{H}_{\mathbb{\infty}}\left( \mathbb{R}\right) $, has a
unique solution $u(x)=G\left( x;\alpha,m\right) \ast g(x)\in\mathcal{H}
_{\mathbb{\infty}}\left( \mathbb{R}\right) $, cf. \cite[Theorem
11.2]{KKZuniga}.
As a consequence one obtains that \ the mapping
\begin{equation}
\begin{array}
[c]{cccc}
\mathcal{G}_{\alpha,m}: & \mathcal{H}_{\mathbb{\infty}}\left( \mathbb{R}
\right) & \rightarrow & \mathcal{H}_{\mathbb{\infty}}\left( \mathbb{R}
\right) \\
& g\left( x\right) & \rightarrow & G\left( x;\alpha,m\right) \ast g(x),
\end{array}
\label{Mapping_G}
\end{equation}
is continuous, cf. \cite[Corollary 11.3]{KKZuniga}.
\begin{remark}
\label{Nota3}For $\alpha>0$, $\beta>0$, $m>0$, we set
\[
\left( \mathbf{L}_{\alpha,\beta,m}h\right) \left( x\right) =\mathcal{F}
_{\xi\rightarrow x}^{-1}\left( \left( \left\vert \mathfrak{l}\left(
\xi\right) \right\vert _{p}^{\alpha}+m^{2}\right) ^{\beta}\mathcal{F}
_{x\rightarrow\xi}h\right) ,
\]
for $h\in\mathcal{D}(\mathbb{Q}_{p}^{N})$. We denote by $G\left(
x;\alpha,\beta,m\right) $ the associated Green function. By using the fact
that
\[
C_{0}\left( \alpha,\beta,m\right) \left[ \xi\right] _{p}^{\alpha\beta
d}\leq\left( \left\vert \mathfrak{l}\left( \xi\right) \right\vert
_{p}^{\alpha}+m^{2}\right) ^{\beta}\leq C_{1}\left( \alpha,\beta,m\right)
\left[ \xi\right] _{p}^{\alpha\beta d},
\]
all the results presented in this section for operators $\boldsymbol{L}
_{\alpha}+m^{2}$ can be extended to operators $\mathbf{L}_{\alpha,\beta,m}$.
In particular,
\begin{equation}
\begin{array}
[c]{cccc}
\mathcal{G}_{\alpha,\beta,m}: & \mathcal{H}_{\mathbb{\infty}}\left(
\mathbb{R}\right) & \rightarrow & \mathcal{H}_{\mathbb{\infty}}\left(
\mathbb{R}\right) \\
& g\left( x\right) & \rightarrow & G\left( x;\alpha,\beta,m\right) \ast
g(x),
\end{array}
\label{Mapping_G_2}
\end{equation}
gives rise to a continuous mapping. \ As operators on $\mathcal{H}
_{\mathbb{\infty}}\left( \mathbb{R}\right) $, we can identify $\mathcal{G}
_{\alpha,\beta,m}$ with the operator $\left( \boldsymbol{L}_{\alpha}
+m^{2}\right) ^{-\beta}$, which is a pseudodifferential operator with symbol
$\left( \left\vert \mathfrak{l}\left( \xi\right) \right\vert _{p}^{\alpha
}+m^{2}\right) ^{-\beta}$.
\end{remark}
\begin{remark}
\label{Nota_tensor_2}The mapping
\[
\begin{array}
[c]{cccc}
\mathcal{G}_{\alpha,m}^{\otimes2}-1: & \mathcal{H}_{\mathbb{\infty}}
^{\otimes2} & \rightarrow & \mathcal{H}_{\mathbb{\infty}}^{\otimes2}\\
& f\otimes g & \rightarrow & \mathcal{G}_{\alpha,m}\left( f\right)
\otimes\mathcal{G}_{\alpha,m}\left( g\right) -f\otimes g
\end{array}
\]
is well-defined and continuous. By using \cite[Proposition 1.3.6]{Obata}, any
element $h$ of $\mathcal{H}_{\mathbb{\infty}}^{\otimes2}$ can be represented
as an absolutely convergent series of the form $h=\sum_{i}f_{i}\otimes g_{i}$,
consequently, $\sum_{i}\mathcal{G}_{\alpha,m}\left( f_{i}\right)
\otimes\mathcal{G}_{\alpha,m}\left( g_{i}\right) $ is an element of
$\mathcal{H}_{\mathbb{\infty}}^{\otimes2}$, which implies that $\mathcal{G}
_{\alpha,m}^{\otimes2}-1$ is a well-defined mapping. On the other hand, the
space $\mathcal{H}_{\mathbb{\infty}}^{\otimes2}$ is locally convex, the
topology is defined by the seminorms
\[
\left\Vert h\right\Vert _{l,k}=\inf\sum_{i}\left\Vert f_{i}\right\Vert
_{l}\otimes\left\Vert g_{i}\right\Vert _{k}\text{, \ }h\in\mathcal{H}
_{\mathbb{\infty}}\otimes_{\text{alg}}\mathcal{H}_{\mathbb{\infty}},
\]
where the infimum is taken over all the pairs $\left( f_{i},g_{j}\right) $
satisfying $h=\sum_{j}f_{j}\otimes g_{j}$. The continuity of $\mathcal{G}
_{\alpha,m}^{\otimes2}-1$ is equivalent to
\[
\left\Vert \left( \mathcal{G}_{\alpha,m}^{\otimes2}-1\right) h\right\Vert
_{l,k}\leq C\left\Vert h\right\Vert _{l^{\prime},k^{\prime}},
\]
where the indices $l^{\prime}$, $k^{\prime}$ depend on $l$, $k$. This
condition can be verified easily using the continuity of \ $\mathcal{G}
_{\alpha,m}$.
\end{remark}
\begin{remark}
\label{Nota_Trace}We denote by $Tr$ (the trace), which is\ the unique element
of $\ \mathcal{H}_{\mathbb{\infty}}^{\ast\widehat{\otimes}2}$ determined by
the formula
\[
\left\langle Tr,f\otimes g\right\rangle =\left\langle f,g\right\rangle
_{0}\text{, for }f,g\in\mathcal{H}_{\mathbb{\infty}}\text{.}
\]
We define $\left( \mathcal{G}_{\alpha,m}^{\otimes2}-1\right) Tr\in
\mathcal{H}_{\mathbb{\infty}}^{\ast\widehat{\otimes}2}$ as
\[
\left\langle \left( \mathcal{G}_{\alpha,m}^{\otimes2}-1\right) Tr,f\otimes
g\right\rangle =\left\langle Tr,\left( \mathcal{G}_{\alpha,m}^{\otimes
2}-1\right) \left( f\otimes g\right) \right\rangle ,
\]
where $\left\langle \cdot,\cdot\right\rangle $ is the pairing between
$\mathcal{H}_{\mathbb{\infty}}^{\ast\widehat{\otimes}2}$ and $\mathcal{H}
_{\mathbb{\infty}}^{\widehat{\otimes}2}$. For a general construction of this
type of operators the reader may consult \cite[Theorem 9.11]{Kuo}.
\end{remark}
\subsection{L\'{e}vy characteristics}
We recall that an infinitely divisible probability distribution $P$ is a
probability distribution having the property that for each $n\in
\mathbb{N\smallsetminus}\left\{ 0\right\} $ there exists a probability
distribution $P_{n}$ such that $P=P_{n}\ast\cdots\ast P_{n}$ ($n$-times). By
the L\'{e}vy-Khinchine Theorem, see e.g. \cite{Lukacs}, the characteristic
function $C_{P}$ of $P$ satisfies
\begin{equation}
C_{P}(t)=\int\nolimits_{\mathbb{R}}e^{ist}dP(s)=e^{\digamma\left( t\right)
}\text{, }t\in\mathbb{R}\text{,} \label{char_function}
\end{equation}
where $\digamma:\mathbb{R}\rightarrow\mathbb{C}$ is a continuous function,
called the \textit{L\'{e}vy characteristic of} $P$, which is uniquely
represented as follows:
\[
\digamma\left( t\right) =iat-\frac{\sigma^{2}t^{2}}{2}+\int
\nolimits_{\mathbb{R\smallsetminus}\left\{ 0\right\} }\left( e^{ist}
-1-\frac{ist}{1+s^{2}}\right) dM(s)\text{, }t\in\mathbb{R}\text{,}
\]
where $a$, $\sigma\in\mathbb{R}$, with $\sigma\geq0$, and the measure $dM(s)$
satisfies
\begin{equation}
\int\nolimits_{\mathbb{R\smallsetminus}\left\{ 0\right\} }\min\left(
1,s^{2}\right) dM(s)<\infty. \label{dM(s)}
\end{equation}
On the other hand, given a triple $\left( a,\sigma,dM\right) $\ with
$a\in\mathbb{R}$, $\sigma\geq0$, and $dM$ a measure on
$\mathbb{R\smallsetminus}\left\{ 0\right\} $ satisfying (\ref{dM(s)}), there
exists a unique infinitely divisible probability distribution $P$ such that
its L\'{e}vy characteristic is given by (\ref{char_function}).
Let $\digamma$ be a L\'{e}vy characteristic defined by (\ref{char_function}).
Then there exists a unique probability measure \textrm{P}$_{\digamma}$ on
$\left( \mathcal{H}_{\infty}^{^{\ast}}\left( \mathbb{R}\right)
,\mathcal{B}\right) $ such that the `Fourier transform' of \textrm{P}
$_{\digamma}$ satisfies
\begin{equation}
\int\nolimits_{\mathcal{H}_{\infty}^{^{\ast}}\left( \mathbb{R}\right)
}e^{i\left\langle W,f\right\rangle }\mathrm{dP}_{\digamma}\left( W\right)
=\exp\left\{ \int\nolimits_{\mathbb{Q}_{p}^{N}}\digamma\left( f\left(
x\right) \right) d^{N}x\right\} \text{, }f\in\mathcal{H}_{\infty}\left(
\mathbb{R}\right) , \label{Eq_the_zuniga}
\end{equation}
cf. \cite[Theorem 5.2]{Zuniga-FAA-2017}, alternatively \cite[Theorem
11.6]{KKZuniga}.
We will say that a distribution $\boldsymbol{\Theta}\in\left( \mathcal{H}
_{\infty}\right) ^{-1}$ is \textit{represented by a probability measure}
$\mathrm{P}$ on $\left( \mathcal{H}_{\mathbb{R}}^{^{\ast}}\left(
\infty\right) ,\mathcal{B}\right) $ if
\begin{equation}
\left\langle \left\langle \boldsymbol{\Theta},\Psi\right\rangle \right\rangle
=
{\textstyle\int\nolimits_{\mathcal{H}_{\mathbb{\infty}}^{^{\ast}}\left(
\mathbb{R}\right) }}
\Psi\left( W\right) d\mathrm{P}\left( W\right) \text{ for any }\Psi
\in\left( \mathcal{H}_{\infty}\right) ^{1}. \label{Eq_definition}
\end{equation}
We will denote this fact as $d\mathrm{P}=\boldsymbol{\Theta}d\mu$. In this
case $\boldsymbol{\Theta}$ may be regarded as the generalized Radon-Nikodym
derivative $\frac{d\mathrm{P}}{d\mu}$ of $\mathrm{P}$ with respect to $\mu$.
By using this result, Theorem \ref{Theorem1}-(iii), and assuming that
\begin{equation}
\digamma\left( t\right) =-H(it)-\frac{1}{2}t^{2}\text{, }t\in\mathbb{R}
\label{Levy_char}
\end{equation}
is a L\'{e}vy characteristic, there exists a probability measure
$\mathrm{P}_{H}$ on $\left( \mathcal{H}_{\mathbb{R}}^{^{\ast}}\left(
\infty\right) ,\mathcal{B}\right) $ such that
\begin{equation}
{\LARGE T}\boldsymbol{\Phi}_{H}\left( f\right) =
{\textstyle\int\nolimits_{\mathcal{H}_{\mathbb{\infty}}^{^{\ast}}\left(
\mathbb{R}\right) }}
\exp\left( i\left\langle W,f\right\rangle \right) d\mathrm{P}_{H}\left(
W\right) \text{, \ }f\in\mathcal{H}_{\infty}\left( \mathbb{R}\right)
\text{.} \label{Eq_the_zuniga_2}
\end{equation}
\begin{theorem}
\label{Theorem2A}Assume that $H$ is a holomorphic function at the origin
satisfying $H(0)=0$. Then $d\mathrm{P}_{H}=\boldsymbol{\Phi}_{H}d\mu$ if and
only if $\digamma\left( t\right) $ is a L\'{e}vy characteristic.
\end{theorem}
\begin{proof}
Assume that $\digamma\left( t\right) $ is a L\'{e}vy characteristic. By
(\ref{Eq_the_zuniga_2}), we have
\begin{equation}
{\LARGE T}\boldsymbol{\Phi}_{H}\left( \lambda f\right) =
{\textstyle\int\nolimits_{\mathcal{H}_{\mathbb{\infty}}^{^{\ast}}\left(
\mathbb{R}\right) }}
\exp\left( \lambda i\left\langle W,f\right\rangle \right) d\mathrm{P}
_{H}\left( W\right) =\left\langle \left\langle \boldsymbol{\Phi}_{H}
,\exp\left( \lambda i\left\langle W,f\right\rangle \right) \right\rangle
\right\rangle , \label{Eq_16AA}
\end{equation}
for any $\lambda\in\mathbb{R}$.
In order to establish (\ref{Eq_definition}), it is sufficient to show that
(\ref{Eq_definition}) holds for $\Psi$ in a dense subspace of \ $\left(
L_{\mathbb{C}}^{2}\right) $, we can choose the linear span of the exponential
functions of the form $\exp\alpha\left\langle W,f\right\rangle $ for
$\alpha\in\mathbb{C}$, $f\in\mathcal{H}_{\infty}\left( \mathbb{R}\right) $,
cf. \cite[Proposition 1.9]{Hida et al}. On the other hand, since
$\boldsymbol{\Phi}_{H}\in\mathcal{H}_{-l,-k.-1}(\mathbb{C})$ for some $l$,
$k\in\mathbb{N}$, and $\left( L_{\mathbb{C}}^{2}\right) $ is dense in
$\mathcal{H}_{-l,-k.-1}(\mathbb{C})$, it is sufficient to establish
(\ref{Eq_definition}) when $\boldsymbol{\Phi}_{H}\in\left( L_{\mathbb{C}}
^{2}\right) $. \ Now the result follows from (\ref{Eq_16AA}) by using the
fact that
\[
\lambda\rightarrow{\LARGE T}\boldsymbol{\Phi}_{H}\left( \lambda f\right) =
{\textstyle\int\nolimits_{\mathcal{H}_{\mathbb{\infty}}^{^{\ast}}\left(
\mathbb{R}\right) }}
\exp\left( \lambda i\left\langle W,f\right\rangle \right) d\mathrm{\mu
}\left( W\right) \text{, }\lambda\in\mathbb{R}\text{,}
\]
has an entire analytic extension, cf. \cite[Proposition 2.2]{Hida et al}.
Conversely, assume that $d\mathrm{P}_{H}=\boldsymbol{\Phi}_{H}d\mu$, then by
Theorem \ref{Theorem1}-(iii), we have
\begin{gather}
{\textstyle\int\nolimits_{\mathcal{H}_{\mathbb{\infty}}^{^{\ast}}\left(
\mathbb{R}\right) }}
e^{i\left\langle W,f\right\rangle }d\mathrm{P}_{H}\left( W\right) =
{\textstyle\int\nolimits_{\mathcal{H}_{\mathbb{\infty}}^{^{\ast}}\left(
\mathbb{R}\right) }}
e^{i\left\langle W,f\right\rangle }\boldsymbol{\Phi}_{H}\left( W\right)
d\mu\left( W\right) \label{Eq_16BB}\\
=\left\langle \left\langle \boldsymbol{\Phi}_{H},e^{i\left\langle
\cdot,f\right\rangle }\right\rangle \right\rangle ={\LARGE T}\boldsymbol{\Phi
}_{H}\left( f\right) =\exp\left\{ \int\nolimits_{\mathbb{Q}_{p}^{N}
}\digamma\left( f\left( x\right) \right) d^{N}x\right\} \text{,
}\nonumber
\end{gather}
for $f\in\mathcal{H}_{\infty}\left( \mathbb{R}\right) $. We now take
$f\left( x\right) =t1_{\mathbb{Z}_{p}^{N}}\left( x\right) $, where
$t\in\mathbb{R}$ and $1_{\mathbb{Z}_{p}^{N}}$\ is the characteristic function
of $\mathbb{Z}_{p}^{N}$. By using that $H(0)=0$, we have
\begin{equation}
\exp\left\{ \int\nolimits_{\mathbb{Q}_{p}^{N}}\digamma\left( f\left(
x\right) \right) d^{N}x\right\} =\exp\digamma(t). \label{Eq_16CC}
\end{equation}
Now, we consider the random variable:
\[
\begin{array}
[c]{cccc}
\left\langle \cdot,1_{\mathbb{Z}_{p}^{N}}\right\rangle : & \left(
\mathcal{H}_{\infty}^{\ast}\left( \mathbb{R}\right) ,\mathcal{B}
,\mathrm{P}_{H}\right) & \rightarrow & \left( \mathbb{R},\mathcal{B}
(\mathbb{R})\right) \\
& & & \\
& W & \rightarrow & \left\langle W,1_{\mathbb{Z}_{p}^{N}}\right\rangle ,
\end{array}
\]
with probability distribution $\nu_{\left\langle \cdot,1_{\mathbb{Z}_{p}^{N}
}\right\rangle }\left( A\right) =\mathrm{P}_{H}\left\{ W\in\mathcal{H}
_{\infty}^{\ast}\left( \mathbb{R}\right) ;\left\langle W,1_{\mathbb{Z}
_{p}^{N}}\right\rangle \in A\right\} $, where $A$ is a Borel subset of
$\mathbb{R}$. Then, by (\ref{Eq_16BB})-(\ref{Eq_16CC}),
\begin{equation}
{\textstyle\int\nolimits_{\mathcal{H}_{\mathbb{\infty}}^{^{\ast}}\left(
\mathbb{R}\right) }}
e^{it\left\langle W,f\right\rangle }d\mathrm{P}_{H}\left( W\right)
=\int\nolimits_{\mathbb{R}}e^{itz}d\nu_{\left\langle \cdot,1_{\mathbb{Z}
_{p}^{N}}\right\rangle }\left( z\right) =\exp\digamma(t). \label{Eq_16DD}
\end{equation}
\end{proof}
We call these measures \textit{generalized white noise measures}. The moments
of the measure $\mathrm{P}_{H}$ are the Schwinger functions $\left\{
\mathcal{S}_{n}^{\boldsymbol{\Phi}_{H}}\right\} _{n\in\mathbb{N}}$.
Since $\left( \mathcal{G}_{\alpha,m}f\right) \left( x\right) :=G\left(
x;\alpha,m\right) \ast f\left( x\right) $ gives rise to a continuous
mapping from $\mathcal{H}_{\mathbb{R}}(\infty)$ into itself, then, the
conjugate operator $\widetilde{\mathcal{G}}_{\alpha,m}:\mathcal{H}
_{\mathbb{R}}^{^{\ast}}(\infty)\rightarrow\mathcal{H}_{\mathbb{R}}^{^{\ast}
}(\infty)$ is a measurable mapping from $\left( \mathcal{H}_{\mathbb{R}
}^{^{\ast}}\left( \infty\right) ,\mathcal{B}\right) $ into itself. For the
sake of simplicity, we use $\mathcal{G}$ instead of $\mathcal{G}_{\alpha,m}$
and $G$ instead of $G\left( x;\alpha,m\right) $. We set $\mathrm{P}_{H}^{G}$
to be the image probability measure of $\mathrm{P}_{H}$ under $\widetilde
{\mathcal{G}}$, i.e. $\mathrm{P}_{H}^{G}$ is the measure on $\left(
\mathcal{H}_{\mathbb{R}}^{^{\ast}}\left( \infty\right) ,\mathcal{B}\right)
$ defined by
\begin{equation}
\mathrm{P}_{H}^{G}\left( A\right) =\mathrm{P}_{H}\left( \widetilde
{\mathcal{G}}^{-1}\left( A\right) \right) \text{, for }A\in\mathcal{B}
\text{.} \label{probability}
\end{equation}
The Fourier transform of $\mathrm{P}_{H}^{G}$ is given by
\begin{equation}
{\textstyle\int\nolimits_{\mathcal{H}_{\mathbb{\infty}}^{^{\ast}}\left(
\mathbb{R}\right) }}
e^{i\left\langle W,f\right\rangle }d\mathrm{P}_{H}^{G}\left( W\right)
=\exp\left\{ \int\nolimits_{\mathbb{Q}_{p}^{N}}\digamma\left\{
\int\nolimits_{\mathbb{Q}_{p}^{N}}G\left( x-y;\alpha,m\right) f\left(
y\right) d^{N}y\right\} d^{N}x\right\} \text{,} \label{Eq_16A}
\end{equation}
for $f\in\mathcal{H}_{\mathbb{R}}\left( \infty\right) $, where $\digamma$ is
given as in (\ref{Levy_char}), cf. \cite[Proposition 6.2]{Zuniga-FAA-2017},
alternatively \cite[Proposition 11.12]{KKZuniga}. Finally, (\ref{Eq_16A}) is
also valid if we replace $G=$ $G\left( x;\alpha,m\right) $ by $G\left(
x;\alpha,\beta,m\right) $.
\subsection{The free Euclidean Bose field}
An important difference between the real and $p$-adic Euclidean quantum field
theories comes from the `ellipticity' of the quadratic form $\mathfrak{q}
_{N}\left( \xi\right) =\xi_{1}^{2}+\cdots+\xi_{N}^{2}$. In the real case
$\mathfrak{q}_{N}\left( \xi\right) $\ is elliptic for any $N\geq1$. In the
$p$-adic case, $\mathfrak{q}_{N}\left( \xi\right) $ is not elliptic for
$N\geq5$. In the case $N=4$, there is a unique elliptic quadratic form, up to
linear equivalence, which is $\xi_{1}^{2}-s\xi_{2}^{2}-p\xi_{3}^{2}+s\xi
_{4}^{2}$, where $s\in\mathbb{Z\smallsetminus}\left\{ 0\right\} $ is a
quadratic non-residue, i.e. $\left( \frac{s}{p}\right) =-1$.
\subsubsection{The Archimedean free covariance function}
The free covariance function $C(x-y;m):=C(x-y)$ is the solution of the Laplace
equation
\[
\left( -\Delta+m^{2}\right) C(x-y)=\delta\left( x-y\right) ,
\]
where $\Delta=\sum_{i=1}^{N}\frac{\partial^{2}}{\partial x_{i}^{2}}$. As a
distribution from $\mathcal{S}^{\prime}(\mathbb{R}^{N})$, the free covariance
is given by
\[
C(x-y)=\frac{1}{\left( 2\pi\right) ^{N}}\int\limits_{\mathbb{R}^{N}}
\frac{\exp\left( -ik\cdot\left( x-y\right) \right) }{k^{2}+m^{2}}d^{N}k,
\]
where $k$, $x$, $y\in\mathbb{R}^{N}$, $d^{N}k$ is the Lebesgue measure of
$\mathbb{R}^{N}$, $k^{2}=k\cdot k$, and $k\cdot x=\sum_{i=1}^{N}k_{i}x_{i}$.
Notice that the quadratic form used in the definition of the Fourier transform
$k^{2}$ is the same as the one used in the propagator $\frac{1}{k^{2}+m^{2}}$,
this situation does not occur in the $p$-adic case. In particular the group of
symmetries of $C(x-y)$ is the $SO(N,\mathbb{R})$. The function $C(x-y)$ has
the following properties (see \cite[Proposition 7.2.1]{Glimm-Jaffe}.):
\noindent(i) $C(x-y)$ is positive and analytic for $x-y\neq0$;
\noindent(ii) $C(x-y)\leq\exp\left( -m\left\Vert x-y\right\Vert \right) $ as
$\left\Vert x-y\right\Vert \rightarrow\infty$;
\noindent(iii) for $N\geq3$ and $m\left\Vert x-y\right\Vert $ in a
neighborhood of zero,
\[
C(x-y)\sim\left\Vert x-y\right\Vert ^{-N+2},
\]
\noindent(iv) for $N=2$ and $m\left\Vert x-y\right\Vert $ in a neighborhood of
zero,
\[
C(x-y)\sim-\ln\left( m\left\Vert x-y\right\Vert \right) .
\]
\subsubsection{The Archimedean free Euclidean Bose field}
Take $H_{m}$ to be the Hilbert space defined as the closure of $\mathcal{S}
(\mathbb{R}^{N})$ with respect to the norm $\left\Vert \cdot\right\Vert _{m}$
induced by the scalar product
\[
\left( f,g\right) _{m}:=\int\nolimits_{\mathbb{R}^{N}}f\left( x\right)
\left( -\Delta+m^{2}\right) ^{-1}g\left( x\right) d^{N}x=\left( f,\left(
-\Delta+m^{2}\right) ^{-1}g\right) _{L^{2}(\mathbb{R}^{N})}.
\]
Then $\mathcal{S}(\mathbb{R}^{N})\hookrightarrow H_{m}\hookrightarrow
\mathcal{S}^{\prime}(\mathbb{R}^{N})$ form a Gel'fand triple. The probability
space $\left( \mathcal{S}^{\prime}(\mathbb{R}^{N}),\mathcal{B},\nu\right) $,
where $\nu$\ is the centered Gaussian measure on $\mathcal{B}$ (the $\sigma
$-algebra of cylinder sets) with covariance
\[
\int\nolimits_{\mathcal{S}^{\prime}(\mathbb{R}^{N})}\left\langle
W,f\right\rangle \left\langle W,g\right\rangle d\nu\left( W\right) =\left(
f,\left( -\Delta+m^{2}\right) ^{-1}g\right) _{L^{2}(\mathbb{R}^{N})},
\]
for $f$, $g$ $\in\mathcal{S}(\mathbb{R}^{N})$, jointly with the coordinate
process $W\rightarrow\left\langle W,f\right\rangle $, with \ fixed
$f\in\mathcal{S}(\mathbb{R}^{N})$, is called the free Euclidean Bose field of
mass $m$ in $N$ dimensions.
\subsubsection{\label{Section_C_p}The non-Archimedean free covariance
function}
The $p$-adic free covariance $C_{p}(x-y;m):=C_{p}(x-y)$ is the solution of the
pseudodifferential equation
\[
\left( \boldsymbol{L}_{\alpha}+m^{2}\right) C(x-y)=\delta\left( x-y\right)
,
\]
where $\boldsymbol{L}_{\alpha}$ is the pseudodifferential operator defined in
(\ref{ellipticoperaator}). As a distribution from $\mathcal{D}^{\prime
}(\mathbb{Q}_{p}^{N})$, the free covariance is given by
\[
C_{p}(x-y)=\int\limits_{\mathbb{Q}_{p}^{N}}\frac{\chi_{p}\left( -\xi
\cdot\left( x-y\right) \right) }{\left\vert \mathfrak{l}(\xi)\right\vert
_{p}^{\alpha}+m^{2}}d^{N}\xi,
\]
where $k$, $x$, $y\in\mathbb{Q}_{p}^{N}$, $d^{N}\xi$ is the Haar measure of
$\mathbb{Q}_{p}^{N}$, $\mathfrak{l}(k)$ is an elliptic polynomial of degree
$d$, and $k\cdot x=\sum_{i=1}^{N}k_{i}x_{i}$.\ In this case $\mathfrak{l}
(k)\neq k\cdot k$, and then the symmetries of $C_{p}(x-y)$ form a subgroup of
\ the $p$-adic orthogonal group attached to the quadratic form $k\cdot k$.
There are other possible propagators, for instance
\[
\frac{1}{\left( \left\vert \mathfrak{l}(k)\right\vert _{p}+m^{2}\right)
^{\alpha}}\text{, }\alpha>0.
\]
For a discussion on the possible scalar propagators, in the $p$-adic setting,
the reader may consult \cite{Smirnov}.
The function $C_{p}(x-y)$ satisfies (see \cite[Proposition 4.1]
{Zuniga-FAA-2017}, or \cite[Proposition 11.1]{KKZuniga}):
\noindent(i) $C_{p}(x-y)$ is positive and locally constant for $x-y\neq0$;
\noindent(ii) $C_{p}(x-y)\leq C\left\Vert x-y\right\Vert _{p}^{-\alpha d-N}$
as $\left\Vert x-y\right\Vert _{p}\rightarrow\infty$;
\noindent(iii) for $0<\alpha d<N$ and $\left\Vert x-y\right\Vert _{p}\leq1$,
\[
C_{p}(x-y)\leq C\left\Vert x-y\right\Vert _{p}^{\alpha d-N};
\]
\noindent(iv) for $N=\alpha d$ and $\left\Vert x-y\right\Vert _{p}\leq1$,
\[
C_{p}(x-y)\leq C_{0}-C_{1}\ln\left\Vert x-y\right\Vert _{p}.
\]
\subsubsection{The non-Archimedean free Euclidean Bose field}
Take $H_{m}$ to be the Hilbert space defined as the closure of $\mathcal{D}
_{\mathbb{R}}(\mathbb{Q}_{p}^{N})$ with respect to the norm $\left\Vert
\cdot\right\Vert _{m}$ induced by the scalar product
\[
\left( f,g\right) _{m}:=\int\nolimits_{\mathbb{Q}_{p}^{N}}\overline
{\widehat{f}\left( \xi\right) }\widehat{g}\left( \xi\right) \frac{d^{N}
\xi}{\left\vert \mathfrak{l}(\xi)\right\vert _{p}^{\alpha}+m^{2}}=\left(
f,\left( \boldsymbol{L}_{\alpha}+m^{2}\right) ^{-1}g\right) _{L_{\mathbb{R}
}^{2}(\mathbb{Q}_{p}^{N})}.
\]
By using that
\[
C_{0}\left[ \xi\right] _{p}^{\left\lfloor d\alpha\right\rfloor }
\leq\left\vert \mathfrak{l}(\xi)\right\vert _{p}^{\alpha}+m^{2}\leq
C_{1}\left[ \xi\right] _{p}^{\left\lceil d\alpha\right\rceil },
\]
where $\left\lceil t\right\rceil =\min\left\{ m\in\mathbb{Z};m\geq x\right\}
$ and $\left\lfloor t\right\rfloor =\max\left\{ m\in\mathbb{Z};m\leq
x\right\} $, we have
\[
\mathcal{H}_{-\left\lfloor d\alpha\right\rfloor }(\mathbb{R})\hookrightarrow
H_{m}\hookrightarrow\mathcal{H}_{-\left\lceil d\alpha\right\rceil }
(\mathbb{R}).
\]
Then $\mathcal{H}_{\infty}(\mathbb{R})\hookrightarrow H_{m}\hookrightarrow
\mathcal{H}_{\infty}^{\ast}(\mathbb{R})$ from a Gel'fand triple. The
probability space $\left( \mathcal{H}_{\infty}^{\ast}(\mathbb{R}
),\mathcal{B},\nu_{d,\alpha}\right) $, where $\nu_{d,\alpha}$\ is the
centered Gaussian measure on $\mathcal{B}$ (the $\sigma$-algebra of cylinder
sets) with covariance
\[
\int\nolimits_{\mathcal{H}_{\infty}^{\ast}(\mathbb{R})}\left\langle
W,f\right\rangle \left\langle W,g\right\rangle d\nu_{d,\alpha}(W)=\left(
f,\left( \boldsymbol{L}_{\alpha}+m^{2}\right) ^{-1}g\right) _{L_{\mathbb{R}
}^{2}(\mathbb{Q}_{p}^{N})},
\]
for $f$, $g$ $\in\mathcal{H}_{\infty}(\mathbb{R})$, jointly with the
coordinate process $W\rightarrow\left\langle W,f\right\rangle $, with \ fixed
$f\in\mathcal{H}_{\infty}(\mathbb{R}))$, is called the non-Archimedean free
Euclidean Bose field of mass $m$ in $N$ dimensions.
If $N=4$ and $d=2$, then there is a unique elliptic quadratic form up to
linear equivalence. If $N\geq5$ and $\mathfrak{l}(\xi)$ is an elliptic
polynomial of degree $d$, then $\left\vert \mathfrak{l}(\xi)\right\vert
_{p}^{\frac{2}{d}}$ is a homogeneous function of degree $2$ that vanishes only
at the origin. We can use this function as the symbol for a pseudodifferential
operator, such operator is a $p$-adic analogue of $-\Delta$ in dimension $N$.
If we use the propagator $\frac{1}{\left( \left\vert \mathfrak{l}
(k)\right\vert _{p}+m^{2}\right) ^{\alpha}}$ instead of $\frac{1}{\left\vert
\mathfrak{l}(k)\right\vert _{p}^{\alpha}+m^{2}}$, similar results are obtained
due to the fact that
\[
\text{and }C_{0}^{\prime}\left[ k\right] _{p}^{\left\lfloor d\alpha
\right\rfloor }\leq\left( \left\vert \mathfrak{l}(k)\right\vert _{p}
+m^{2}\right) ^{\alpha}\leq C_{1}^{\prime}\left[ k\right] _{p}^{\left\lceil
d\alpha\right\rceil }.
\]
We prefer using propagator $\frac{1}{\left\vert \mathfrak{l}(k)\right\vert
_{p}^{\alpha}+m^{2}}$ because the corresponding `Laplace equation' has been
studied extensively in the literature. On the other hand, $\frac{\partial
u\left( x,t\right) }{\partial t}+\boldsymbol{L}_{\alpha}u\left( x,t\right)
=0$, with $x\in\mathbb{Q}_{p}^{N}$, $t>0$, behaves like a `heat equation',
i.e. the semigroup associated to\ this equation is a Markov semigroup, see
\cite[Chapter 2]{Zuniga-LNM-2016}, which means that $-\boldsymbol{L}_{\alpha}$
can be considered \ as $p$-adic version of \ the Laplacian.
\subsection{\label{Sect_symmetries}Symmetries}
Given a polynomial $\mathfrak{a}\left( \xi\right) \in\mathbb{Q}_{p}\left[
\xi_{1},\cdots,\xi_{n}\right] $ and $\Lambda\in GL_{N}\left( \mathbb{Q}
_{p}\right) $, we say that $\Lambda$ \textit{preserves} $\mathfrak{a}$ if
$\mathfrak{a}\left( \xi\right) =\mathfrak{a}\left( \Lambda\xi\right) $,
for all $\xi\in\mathbb{Q}_{p}^{N}$. By simplicity, we use $\Lambda x$ to mean
$\left[ \Lambda_{ij}\right] x^{T}$, $x=\left( x_{1},\cdots,x_{N}\right)
\in\mathbb{Q}_{p}^{N}$, where we identify $\Lambda$ with the matrix $\left[
\Lambda_{ij}\right] $.
Let $\mathfrak{q}_{N}\left( \xi\right) =\xi_{1}^{2}+\cdots+\xi_{N}^{2}$ be
the elliptic quadratic form used in the definition of the Fourier transform,
and let $\mathfrak{l}\left( \xi\right) $ be the elliptic polynomial that
appears in the symbol of the operator $\boldsymbol{L}_{\alpha}$. We define the
homogeneous Euclidean group of $\mathbb{Q}_{p}^{N}$ relative to $\mathfrak{q}
\left( \xi\right) $\ and $\mathfrak{l}\left( \xi\right) $, denoted as
$E_{0}\left( \mathbb{Q}_{p}^{N}\right) :=E_{0}\left( \mathbb{Q}_{p}
^{N};\mathfrak{q},\mathfrak{l}\right) $, as the subgroup of $GL_{N}\left(
\mathbb{Q}_{p}\right) $ whose elements preserve $\mathfrak{q}\left(
\xi\right) $\ and $\mathfrak{l}\left( \xi\right) $ simultaneously. Notice
that if $\mathbb{O}(\mathfrak{q}_{N})$ is the orthogonal group of
$\mathfrak{q}_{N}$, then $E_{0}\left( \mathbb{Q}_{p}^{N}\right) $ is a
subgroup of $\mathbb{O}(\mathfrak{q}_{N})$. We define the inhomogeneous
Euclidean group, denoted as $E\left( \mathbb{Q}_{p}^{N}\right)
\allowbreak:=E\left( \mathbb{Q}_{p}^{N};\mathfrak{q},\mathfrak{l}\right) $,
to be the group of transformations of the form $\left( a,\Lambda\right)
x=a+\Lambda x$, for $a,x\in\mathbb{Q}_{p}^{N}$, $\Lambda\in E_{0}\left(
\mathbb{Q}_{p}^{N}\right) $.
In the real case $\mathfrak{q}_{N}=\mathfrak{l}\left( \xi\right) $ and thus
the homogeneous Euclidean group is $SO(N,\mathbb{R})$. In the $p$-adic case,
$E_{0}\left( \mathbb{Q}_{p}^{N};\mathfrak{q},\mathfrak{l}\right) $ \ is a
subgroup of $\mathbb{O}(\mathfrak{q}_{N})$, in addition, it is not a
straightforward matter to decide whether or not $E_{0}\left( \mathbb{Q}
_{p}^{N};\mathfrak{q},\mathfrak{l}\right) $ is non trivial. For this reason,
we approach the Green kernels in a different way than do in \cite{GS1999},
which is based on \cite{Streater and Wightman}.
Notice that $\left( a,\Lambda\right) ^{-1}x=\Lambda^{-1}\left( x-a\right)
$. Let $\left( a,\Lambda\right) $ be a transformation in $E\left(
\mathbb{Q}_{p}^{N}\right) $, the action of $\left( a,\Lambda\right) $ on a
function $f\in\mathcal{H}_{\mathbb{\infty}}$ is defined by
\[
\left( \left( a,\Lambda\right) f\right) \left( x\right) =f\left(
\left( a,\Lambda\right) ^{-1}x\right) \text{, for\ }x\in\mathbb{Q}_{p}
^{N},
\]
and on a functional $W\in\mathcal{H}_{\mathbb{\infty}}^{^{\ast}}$, by
\[
\left\langle \left( a,\Lambda\right) W,f\right\rangle :=\left\langle
W,\left( a,\Lambda\right) ^{-1}f\right\rangle \text{, for }f\in
\mathcal{H}_{\mathbb{\infty}}\left( \mathbb{R}\right) .
\]
These definitions can be extended to elements of the spaces $\mathcal{H}
_{\mathbb{\infty}}^{\otimes n}$ and $\mathcal{H}_{\mathbb{\infty}}
^{\ast\otimes n}$, by taking
\[
\left( a,\Lambda\right) \left( f_{1}\otimes\cdots\otimes f_{n}\right)
:=\left( a,\Lambda\right) ^{-1}f_{1}\otimes\cdots\otimes\left(
a,\Lambda\right) ^{-1}f_{n}.
\]
In general, if $F:\mathcal{H}_{\mathbb{\infty}}^{\otimes n}\rightarrow
\mathcal{X}$ is linear $\mathcal{X}$-valued functional, where $\mathcal{X}$ is
a vector space, we define
\[
\left( \left( a,\Lambda\right) F\right) \left( f_{1}\otimes\cdots\otimes
f_{n}\right) =F\left( \left( a,\Lambda\right) \left( f_{1}\otimes
\cdots\otimes f_{n}\right) \right) ,
\]
and we say that $F$ is \textit{Euclidean invariant} if and only if $\left(
a,\Lambda\right) F=F$ for any $\left( a,\Lambda\right) \in E\left(
\mathbb{Q}_{p}^{N}\right) $.
\begin{definition}
We call a distribution $\boldsymbol{\Phi}=\sum_{n=0}^{\infty}\left\langle
\Phi_{n},:\cdot^{\otimes n}:\right\rangle \in\left( \mathcal{H}
_{\mathbb{\infty}}\right) ^{-1}$, with $\Phi_{n}\in\mathcal{H}
_{\mathbb{\infty}}^{^{\ast\widehat{\otimes}n}}$, Euclidean invariant if and
only if the functional $\left\langle \Phi_{n},\cdot\right\rangle $ is
Euclidean invariant for any $n\in\mathbb{N}$.
\end{definition}
It follows from this definition that $\boldsymbol{\Phi}\in\left(
\mathcal{H}_{\mathbb{\infty}}\right) ^{-1}$ is Euclidean invariant if and
only if ${\LARGE S}\boldsymbol{\Phi}$\ and ${\LARGE T}\boldsymbol{\Phi}$ are
Euclidean invariant.
\section{Schwinger functions and convoluted white noise}
We set $G:=$ $G\left( x;m,\alpha\right) $ for the Green function
(\ref{GreenFunc}). For $\boldsymbol{\Phi}\in\left( \mathcal{H}
_{\mathbb{\infty}}\right) ^{-1}$, we define $\boldsymbol{\Phi}^{G}$\ as
\begin{equation}
\left( {\LARGE T}\boldsymbol{\Phi}^{G}\right) \left( g\right) =\left(
{\LARGE T}\boldsymbol{\Phi}\right) \left( G\ast g\right) \text{, \ }
g\in\mathcal{U}\text{,}\label{Eq_15}
\end{equation}
where $\mathcal{U}$ is an open neighborhood of zero. Since $\mathcal{G}
:\mathcal{H}_{\mathbb{\infty}}\left( \mathbb{R}\right) \rightarrow
\mathcal{H}_{\mathbb{\infty}}\left( \mathbb{R}\right) $, see
(\ref{Mapping_G}), is linear and continuous, cf. \cite[Corollary
11.3]{KKZuniga}, by the characterization theorem, cf. \cite[Theorem 3]{KLS96},
or Section \ref{Section_Characterization}, $\boldsymbol{\Phi}^{G}$ is a
well-defined and unique element of $\left( \mathcal{H}_{\mathbb{\infty}
}\right) ^{-1}$.
\begin{remark}
\label{Nota5}By using that $\left\langle \delta_{x},G\ast f\right\rangle
=\left\langle G\ast\delta_{x},f\right\rangle $ for any $f\in\mathcal{H}
_{\mathbb{\infty}}$, we have that the white-noise process introduced in
Section \ref{Section_white_noise_process} satisfies
\[
\left\langle \left\langle G\ast\boldsymbol{\Phi}\left( x\right)
,\Psi\right\rangle \right\rangle =\left\langle \left\langle \boldsymbol{\Phi
}\left( x\right) ,G\ast\Psi\right\rangle \right\rangle ,
\]
because $\left\langle \left\langle \boldsymbol{\Phi}\left( x\right)
,G\ast\Psi\right\rangle \right\rangle =\left\langle \delta_{x},G\ast\psi
_{1}\right\rangle $, where $\Psi=\sum_{n=0}^{\infty}\left\langle
:\cdot^{\otimes n}:,\psi_{n}\right\rangle $, $\psi_{n}\in\mathcal{H}_{\infty
}^{\widehat{\otimes}n}$.
\end{remark}
We denote by $\left\{ S_{n}^{H,G}\right\} _{n\in N}$ the Schwinger functions
attached to $\boldsymbol{\Phi}_{H}^{G}$.
\begin{theorem}
\label{Theorem2}With $H$ and $\boldsymbol{\Phi}_{H}$ as in Theorem
\ref{Theorem1}, then distribution $\boldsymbol{\Phi}_{H}^{G}\in\left(
\mathcal{H}_{\infty}\right) ^{-1}$ is Euclidean invariant and is given by
\begin{equation}
\boldsymbol{\Phi}_{H}^{G}=\exp^{\lozenge}\left( -\int\nolimits_{\mathbb{Q}
_{p}^{N}}H^{\lozenge}\left( G\ast\boldsymbol{\Phi}\left( x\right) \right)
d^{N}x+\frac{1}{2}\left\langle \left( \mathcal{G}^{\otimes2}-1\right)
Tr,:\cdot^{\otimes2}:\right\rangle \right) , \label{Eq_16}
\end{equation}
where $Tr\in\left( \mathcal{H}_{\mathbb{\infty}}^{^{\ast}}\left(
\mathbb{Q}_{p}^{N},\mathbb{C}\right) \right) ^{\widehat{\otimes}2}$ denotes
the trace kernel defined by $\left\langle Tr,f\otimes g\right\rangle
=\left\langle f,g\right\rangle _{0}$, $f$, $g\in\mathcal{H}_{\mathbb{\infty}
}\left( \mathbb{Q}_{p}^{N},\mathbb{R}\right) $. The Schwinger functions
$\left\{ S_{n}^{H,G}\right\} _{n\in N}$ satisfy the conditions (OS1) and
(OS4) given in Lemma \ref{Lemma1}, and
\begin{equation}
\text{(Euclidean invariance) \ \ \ }S_{n}^{H,G}\left( \left( a,\Lambda
\right) f\right) =S_{n}^{H,G}\left( f\right) \text{, }f\in\left(
\mathcal{H}_{\mathbb{\infty}}\left( \mathbb{C}\right) \right) ^{\otimes n},
\tag{OS2}
\end{equation}
for any $\left( a,\Lambda\right) \in E\left( \mathbb{Q}_{p}^{N}\right) $.
\end{theorem}
\begin{proof}
By definition (\ref{Eq_15}) and Theorem \ref{Theorem1}-(iii), we have
\begin{equation}
\left( {\LARGE T}\boldsymbol{\Phi}_{H}^{G}\right) \left( g\right)
=\exp\left( -\int\nolimits_{\mathbb{Q}_{p}^{N}}H(iG\ast g\left( x\right)
)+\frac{1}{2}\left( G\ast g\left( x\right) \right) ^{2}\text{ }
d^{N}x\right) . \label{Eq_17}
\end{equation}
On the other hand, by taking the ${\LARGE T}$-transform in (\ref{Eq_16}) and
using (\ref{Eq_13}) and Remarks \ref{Nota_tensor_2}-\ref{Nota5}, we obtain
\begin{multline*}
\left( {\LARGE T}\boldsymbol{\Phi}_{H}^{G}\right) \left( g\right)
=\exp\left( \frac{-1}{2}\left\Vert g\right\Vert _{0}^{2}\right) \times\\
\exp\left\{ -{\LARGE S}\left( \int\nolimits_{\mathbb{Q}_{p}^{N}}H^{\lozenge
}\left( G\ast\boldsymbol{\Phi}\left( x\right) \right) d^{N}x\right)
\left( ig\right) -\frac{1}{2}{\LARGE S}\left( \left\langle \left(
\mathcal{G}^{\otimes2}\mathcal{-}1\right) Tr,:\cdot^{\otimes2}:\right\rangle
\right) \left( ig\right) \right\}
\end{multline*}
\begin{multline*}
=\exp\left( \frac{-1}{2}\left\Vert g\right\Vert _{0}^{2}\right) \times\\
\exp\left\{ -\int\nolimits_{\mathbb{Q}_{p}^{N}}H\left( iG\ast g\left(
x\right) \right) d^{N}x\right\} \exp\left\{ \frac{1}{2}{\LARGE S}\left(
\left\langle \left( \mathcal{G}^{\otimes2}\mathcal{-}1\right) Tr,:\cdot
^{\otimes2}:\right\rangle \right) \left( ig\right) \right\} \\
=\exp\left( \frac{-1}{2}\left\Vert g\right\Vert _{0}^{2}\right) \exp\left\{
-\int\nolimits_{\mathbb{Q}_{p}^{N}}H\left( iG\ast g\left( x\right) \right)
d^{N}x-\frac{1}{2}\left\langle \left( \mathcal{G}^{\otimes2}\mathcal{-}
1\right) Tr,g\otimes g\right\rangle \right\} \\
=\exp\left( \frac{-1}{2}\left\Vert g\right\Vert _{0}^{2}\right) \exp\left\{
-\int\nolimits_{\mathbb{Q}_{p}^{N}}H\left( iG\ast g\left( x\right) \right)
d^{N}x-\frac{1}{2}\left\langle Tr,G\ast g\otimes G\ast g-g\otimes
g\right\rangle \right\}
\end{multline*}
\begin{equation}
=\exp\left\{ -\int\nolimits_{\mathbb{Q}_{p}^{N}}H\left( iG\ast g\left(
x\right) \right) d^{N}x-\frac{1}{2}\left\langle G\ast g\otimes G\ast
g\right\rangle _{0}\right\} \label{Eq_18AA}
\end{equation}
Formula (\ref{Eq_16}) follows from (\ref{Eq_17})-(\ref{Eq_18AA}). Since
$\mathbb{E}_{\mu}(\boldsymbol{\Phi}_{H}^{G})=1$, conditions (OS1) and (OS4)
follow from Lemma \ref{Lemma1}, and condition (OS2) follows from Lemma
\ref{Lemma0} by using the Euclidean invariance of $\boldsymbol{\Phi}_{H}^{G}$.
\end{proof}
\begin{remark}
(i) Set $\mathcal{G}_{_{\frac{1}{2}}}:=\mathcal{G}_{\alpha,\frac{1}{2}
,m}=\left( \boldsymbol{L}_{\alpha}+m^{2}\right) ^{-\frac{1}{2}}$, and
$\mathcal{G}_{_{\frac{1}{2}}}\left( f\right) :=G_{\frac{1}{2}}\ast f$ for
$f\in\mathcal{H}_{\infty}(\mathbb{R})$. By taking $H\equiv0$, we obtain the
free Euclidean field. Indeed, $f\rightarrow\exp\left\{ -\frac{1}
{2}\left\langle G_{\frac{1}{2}}\ast f,G_{\frac{1}{2}}\ast f\right\rangle
_{0}\right\} $ defines a characteristic functional. Let denote by
$\nu_{G_{\frac{1}{2}}}$ the probability measure on $\left( \mathcal{H}
_{\infty}^{\ast}(\mathbb{R}),\mathcal{B}\right) $ provided by the
Bochner-Minlos theorem. Then
\begin{align*}
\left( {\LARGE T}\boldsymbol{\Phi}_{0}^{G_{\frac{1}{2}}}\right) \left(
g\right) & =\exp\left\{ -\frac{1}{2}\left\langle G_{\frac{1}{2}}\ast
g,G_{\frac{1}{2}}\ast g\right\rangle _{0}\right\} \\
& =\left\langle \left\langle \boldsymbol{\Phi}_{0}^{G_{\frac{1}{2}}},\exp
i\left\langle \cdot,g\right\rangle \right\rangle \right\rangle =\int
\nolimits_{\mathcal{H}_{\infty}^{\ast}(\mathbb{R})}\exp i\left\langle
W,g\right\rangle d\nu_{G_{\frac{1}{2}}}(W).
\end{align*}
(ii )Assuming that $\digamma\left( t\right) $, see (\ref{Levy_char}), is a
L\'{e}vy characteristic, Theorem \ref{Theorem2} implies that the probability
measure $\mathrm{P}_{H}^{G}$, see (\ref{probability}), admits
$\boldsymbol{\Phi}_{H}^{G}$ as a generalized density with respect to to white
noise measure $\mu$, i.e. $\mathrm{P}_{H}^{G}=\boldsymbol{\Phi}_{H}^{G}\mu$.
Indeed, by (\ref{Eq_16A}) and (\ref{Eq_18AA}), we have
\begin{multline*}
\int\nolimits_{\mathcal{H}_{\mathbb{\infty}}^{^{\ast}}\left( \mathbb{R}
\right) }e^{i\left\langle W,f\right\rangle }d\mathrm{P}_{H}^{G}\left(
W\right) =\exp\left\{ \int\nolimits_{\mathbb{Q}_{p}^{N}}\digamma\left(
G\left( x;\alpha,m\right) \ast f\left( x\right) \right) yd^{N}x\right\}
\\
=\exp\left\{ -\int\nolimits_{\mathbb{Q}_{p}^{N}}H\left( iG\ast f\left(
x\right) \right) d^{N}x-\frac{1}{2}\left\langle G\ast f,G\ast f\right\rangle
_{0}\right\} =\left( {\LARGE T}\boldsymbol{\Phi}_{H}^{G}\right) \left(
f\right) \\
=\left\langle \left\langle \boldsymbol{\Phi}_{H}^{G},\exp i\left\langle
\cdot,f\right\rangle \right\rangle \right\rangle .
\end{multline*}
\end{remark}
\subsection{Truncated Schwinger functions and the cluster property}
We denote by $P^{(n)}$ the collection of all partitions $I$ of $\left\{
1,\ldots,n\right\} $ into disjoint subsets.
\begin{definition}
Let $\left\{ S_{n}^{H,G}\right\} _{n\in\mathbb{N}}$ be a sequence of
Schwinger functions, with $S_{0}^{H,G}=1$, and $S_{n}^{H,G}\in\mathcal{H}
_{\infty}^{\ast}\left( \mathbb{Q}_{p}^{Nn},\mathbb{C}\right) $ for $n\geq1$.
The truncated Schwinger functions $\left\{ S_{n,T}^{H,G}\right\}
_{n\in\mathbb{N}}$ are defined recursively by the formula
\[
S_{n}^{H,G}\left( f_{1}\otimes\cdots\otimes f_{n}\right) =\sum\limits_{I\in
P^{(n)}}\prod\limits_{\left\{ j_{1},\ldots,j_{l}\right\} }S_{l,T}
^{H,G}\left( f_{j_{1}}\otimes\cdots\otimes f_{j_{l}}\right) ,
\]
for $n\geq1$. Here for $\left\{ j_{1},\ldots,j_{l}\right\} \in I$ we assume
that $j_{1}<\ldots<j_{l}$.
\end{definition}
\begin{remark}
By the kernel theorem, the sequence $\left\{ S_{n}^{H,G}\right\}
_{n\in\mathbb{N}}$ uniquely determines the sequence $\left\{ S_{n,T}
^{H,G}\right\} _{n\in\mathbb{N}}$ and vice versa. \ All the $S_{n}^{H,G}$ are
Euclidean (translation) invariant if and only if all the $S_{n,T}^{H,G}$ are
Euclidean (translation) invariant. The same equivalence holds for
`temperedness' ( i.e. membership to $\left( \mathcal{H}_{\infty}\right)
^{-1}$).
\end{remark}
\begin{definition}
Let $a\in\mathbb{Q}_{p}^{N}$, $a\neq0$, and $\lambda\in\mathbb{Q}_{p}$. Let
$T_{a\lambda}$ denote the representation of the translation by $a\lambda$ on
$\mathcal{H}_{\infty}\left( \mathbb{Q}_{p}^{Nn},\mathbb{R}\right) $. Take
$n$, $m\geq1$, $f_{1},\cdots,f_{n}\in\mathcal{H}_{\infty}\left(
\mathbb{Q}_{p}^{N},\mathbb{R}\right) $.
(OS5)(\textbf{Cluster property}) A sequence of Schwinger functions $\left\{
S_{n}^{H,G}\right\} _{n\in\mathbb{N}}$ has the cluster property if for all
$n$, $m\geq1$, it verifies that
\begin{align}
& \lim_{\left\vert \lambda\right\vert _{p}\rightarrow\infty}\left\{
S_{m+n}^{H,G}\left( f_{1}\otimes\cdots\otimes f_{m}\otimes T_{a\lambda
}\left( f_{m+1}\otimes\cdots\otimes f_{m+n}\right) \right) \right\}
\label{Cluster_1}\\
& =S_{m}^{H,G}\left( f_{1}\otimes\cdots\otimes f_{m}\right) S_{n}
^{H,G}\left( f_{m+1}\otimes\cdots\otimes f_{m+n}\right) .\nonumber
\end{align}
(\textbf{Cluster property of truncated Schwinger functions}) A sequence of
truncated Schwinger functions $\left\{ S_{n,T}^{H,G}\right\} _{n\in
\mathbb{N}}$ has the cluster property, if for all $n$, $m\geq1$, it verifies
that
\begin{equation}
\lim_{\left\vert \lambda\right\vert _{p}\rightarrow\infty}S_{m+n,T}
^{H,G}\left( f_{1}\otimes\cdots\otimes f_{m}\otimes T_{a\lambda}\left(
f_{m+1}\otimes\cdots\otimes f_{m+n}\right) \right) =0. \label{Cluster_2}
\end{equation}
\end{definition}
\begin{remark}
\label{Nota_Cluster}In the Archimedean case, it is possible to replace
$\lim_{\lambda\rightarrow\infty}\left( \cdot\right) $ in (\ref{Cluster_1})
and (\ref{Cluster_2}) by $\lim_{\lambda\rightarrow\infty}\left\vert
\lambda\right\vert ^{m}\left( \cdot\right) $ for arbitrary $m$, cf.
\cite[Remark 4.4]{Albeverio-et-al-3}. This is possible because Schwartz
functions decay at infinity faster than any polynomial function. This is not
possible in the $p$-adic case, because the elements of our `$p$-adic Schwartz
space $\mathcal{H}_{\infty}\left( \mathbb{Q}_{p}^{N},\mathbb{R}\right) $'
only have a polynomial decay at infinity. For instance, consider \ the
one-dimensional $p$-adic heat kernel \ $Z(x;t)=\mathcal{F}_{\xi\rightarrow
x}^{-1}\left( e^{-t\left\vert \xi\right\vert _{p}^{\alpha}}\right) $, \ for
$t>0$, and $\alpha>0$, which is an element of $\mathcal{H}_{\infty}\left(
\mathbb{Q}_{p},\mathbb{R}\right) $. The Fourier transform $e^{-t\left\vert
\xi\right\vert _{p}^{\alpha}}$\ of \ $Z(x;t)$ decays faster that any
polynomial function in $\left\vert \xi\right\vert _{p}$. However, $Z(x;t)$ has
only a polynomial decay at infinity, \ more precisely,
\[
Z(x;t)\leq C\frac{t}{\left( t^{\frac{1}{\alpha}}+\left\vert x\right\vert
_{p}\right) ^{\alpha+1}}\text{, }t>0\text{, }x\in\mathbb{Q}_{p},
\]
cf. \cite[Lemma 4.1]{Koch}.
\end{remark}
\begin{lemma}
\label{Lemma_4}Let $H(z)=
{\textstyle\sum\nolimits_{n=0}^{\infty}}
H_{n}z^{n}$, $z\in U\subset\mathbb{C}$, and $G$ as in Theorem \ref{Theorem2},
and $f_{1},\cdots,f_{n}\in\mathcal{H}_{\infty}\left( \mathbb{Q}_{p}
^{N},\mathbb{R}\right) $. Assume that $\digamma\left( t\right)
=-H(it)-\frac{1}{2}t^{2}$, $t\in\mathbb{R}$ is a L\'{e}vy characteristic, then
the truncated Schwinger functions are given by
\begin{equation}
S_{n,T}^{H,G}\left( f_{1}\otimes\cdots\otimes f_{n}\right) =\left\{
\begin{array}
[c]{lll}
-H_{n}\int\nolimits_{\mathbb{Q}_{p}^{N}}\prod\limits_{i=1}^{n}G\ast
f_{i}\left( x\right) d^{N}x & \text{for } & n\geq2\\
& & \\
(-H_{2}+1)\int\nolimits_{\mathbb{Q}_{p}^{N}}G\ast f_{1}\left( x\right) G\ast
f_{2}\left( x\right) d^{N}x & \text{for} & n=2.
\end{array}
\right. \label{Eq_20}
\end{equation}
\end{lemma}
\begin{proof}
The result follows from the formula for the Schwinger functions given in
Theorem 7.7 in \cite{Zuniga-FAA-2017}, and the uniqueness of the truncated
Schwinger functions. The coefficients in front of the integrals in
(\ref{Eq_20}) are the $n$-th derivatives of the L\'{e}vy characteristic
divided by $i^{n}$. For the general $H$ as in Theorem \ref{Theorem2} these
coefficients are the $n$-th derivatives of \ $-\left( H(iz)+\frac{1}{2}
z^{2}\right) $, $z\in U$.
\end{proof}
\begin{lemma}
\label{Lemma_5}Assume that $\alpha d>N$. Let $\boldsymbol{\Phi}$, $H$, $G$ as
in Theorem \ref{Theorem2}. Then the sequence of truncated Schwinger functions
$\left\{ S_{n,T}^{H,G}\right\} _{n\in\mathbb{N}}$ has the cluster property.
\end{lemma}
\begin{proof}
Fix $a\in\mathbb{Q}_{p}^{N}$ and take $\lambda\in\mathbb{Q}_{p}$, $m$,
$n\geq1$, $f_{1},\cdots,f_{m+n}\in\mathcal{H}_{\infty}\left( \mathbb{Q}
_{p}^{N},\mathbb{R}\right) $. By Lemma \ref{Lemma_4}, we have
\begin{align*}
& \left\vert S_{n,T}^{H,G}\left( f_{1}\otimes\cdots\otimes f_{n}\right)
\otimes T_{a\lambda}\left( f_{m+1}\otimes\cdots\otimes f_{m+n}\right)
\right\vert \\
& =\left\vert -H_{m+n}\right\vert \left\vert \int\nolimits_{\mathbb{Q}
_{p}^{N}}\prod\limits_{i=1}^{m}\left( G\ast f_{i}\right) \left( x\right)
\text{ }\prod\limits_{i=m+1}^{m+n}T_{a\lambda}\left( G\ast f_{i}\right)
\left( x\right) \text{ }\right\vert
\end{align*}
We now use that $G\ast f_{i}\in\mathcal{H}_{\infty}\left( \mathbb{Q}_{p}
^{N},\mathbb{R}\right) $ and that $\mathcal{H}_{\infty}\left( \mathbb{Q}
_{p}^{N},\mathbb{R}\right) \subset\mathcal{C}_{0}\left( \mathbb{Q}_{p}
^{N},\mathbb{R}\right) $ to get
\begin{align*}
& \left\vert S_{n,T}^{H,G}\left( f_{1}\otimes\cdots\otimes f_{n}\right)
\otimes T_{a\lambda}\left( f_{m+1}\otimes\cdots\otimes f_{m+n}\right)
\right\vert \\
& \leq\left\vert H_{m+n}\right\vert \prod\limits_{i=1}^{m}\left\Vert G\ast
f_{i}\right\Vert _{L^{\infty}}\text{ }\prod\limits_{i=m+1}^{m+n-1}\left\Vert
T_{a\lambda}\left( G\ast f_{i}\right) \right\Vert _{L^{\infty}}
\int\nolimits_{\mathbb{Q}_{p}^{N}}\left\vert T_{a\lambda}\left( G\ast
f_{m+n}\right) \left( x\right) \right\vert d^{N}x.
\end{align*}
Now, the announced result follows from the following fact:
\textbf{Claim.} If $\alpha d>N$, for any $f\in\mathcal{H}_{\infty}\left(
\mathbb{Q}_{p}^{N},\mathbb{R}\right) $, it verifies that
\[
\lim_{\left\vert \lambda\right\vert _{p}\rightarrow\infty}\int
\nolimits_{\mathbb{Q}_{p}^{N}}G\left( x-\lambda a-y\right) \left\vert
f_{m+n}\left( y\right) \right\vert \text{ }d^{N}y=0.
\]
Since $\alpha d>N$, by the Riemann-Lebesgue theorem, $G\in\mathcal{C}
_{0}\left( \mathbb{Q}_{p}^{N},\mathbb{R}\right) $, and consequently
$G\left( x-\lambda a-y\right) \left\vert f_{m+n}\left( y\right)
\right\vert \leq\left\Vert G\right\Vert _{L^{\infty}}\left\vert f_{m+n}\left(
y\right) \right\vert \in L_{\mathbb{R}}^{1}\left( \mathbb{Q}_{p}^{N}\right)
$. Now the Claim follows by applying the dominated convergence theorem.
\end{proof}
\begin{theorem}
\label{Theorem3}With $H$, $G$ and $\boldsymbol{\Phi}_{H}^{G}\in\left(
\mathcal{H}_{\infty}\right) ^{-1}$as in Theorem \ref{Theorem2}. If $\alpha
d>N$, then the sequence of Schwinger functions $\left\{ S_{n}^{H,G}\right\}
_{n\in\mathbb{N}}$ has the cluster property (OS5).
\end{theorem}
\begin{proof}
In \cite[Theorem 4.5]{Albeverio-et-al-3} was established that the cluster
property and the truncated cluster property are equivalent. By using this
result, the announced result follows from Lemma \ref{Lemma_5}.
\end{proof}
\begin{remark}
\label{Nota_Theorem_3}The class of Schwinger functions $\left\{ S_{n}
^{H,G}\right\} _{n\in\mathbb{N}}$ corresponding to a distribution
$\boldsymbol{\Phi}_{H}^{G}\in\left( \mathcal{H}_{\infty}\right) ^{-1}$as in
Theorem \ref{Theorem2} differs of the class of Schwinger functions
corresponding to the convoluted \ generalized white noise introduced in
\cite{Zuniga-FAA-2017}. In \ order to explain the differences, let us compare
the properties of the Levy characteristic used in \ \cite{Zuniga-FAA-2017}
with the properties of the function $H$ used in this article, where
$\digamma\left( t\right) =-H(it)-\frac{1}{2}t^{2}$, $t\in U\subset
\mathbb{R}$. We require only that function $H$ be holomorphic at zero and
$H(0)=0$, as in \cite{GS1999}. This only impose a restriction in choosing the
coefficients in front of the integrals corresponding to the $n$-th truncated
\ Schwinger function, see (\ref{Eq_20}). On the other hand in
\cite{Zuniga-FAA-2017}, the author requires the condition that the measure $M$
\ has finite moments of all orders. This implies that $\digamma$ belongs to
$C^{\infty}(\mathbb{R})$, but $\digamma$ does not have to have a holomorphic
extension. Furthermore, since $\exp s\digamma\left( t\right) $ is positive
definite for any $s>0$, cf. \cite[Proposition 5.5]{Zuniga-FAA-2017}, and by
using $\digamma\left( 0\right) =0$ and a result due Schoenberg, cf.
\cite[Theorem 7.8]{Berg-Gunnar}, we have $-\digamma\left( t\right)
:\mathbb{R}\rightarrow\mathbb{C}$ is a negative definite analytic function.
Since $\left\vert -\digamma\left( t\right) \right\vert \leq C\left\vert
t\right\vert ^{2}$ for any $\left\vert t\right\vert \geq1$, \cite[Corollary
7.16]{Berg-Gunnar}, we conclude that $-\digamma\left( t\right) $ is a
polynomial of the degree at most $2$, \ and then $H_{n}=0$ for $n\geq3$.
\end{remark}
\bigskip
| 129,429
|
Colorado Mass Shooting Suspect Described As 'Recluse'."
There are still many questions surrounding the shooting and the alleged shooter. But we're beginning to fill in a sketch of Holmes.
Here's what we know so far:
-- The Denver Post has perhaps the best profile of Holmes. It reports that in an apartment rental application last year, Holmes said he was a student who was "quiet and easy going." Other tenants told the paper that Holmes was a "recluse."
-- The Associated Press reports that Holmes dropped out of the University of Colorado School of Medicine in Denver last month.
Spokeswoman Jaque Montgomery said "she did not know when he started school or why he withdrew."
-- The Post says the building in which Holmes lived was reserved for medical students. The paper adds:
."
-- After a first look at public records, NPR's Margot Williams reports that there doesn't appear to be any prior cases against Holmes.
-- CBS News, quoting a federal official, says that Holmes "appears to have been "under the radar."
We'll add to this post as more information becomes available.
Update at 6:39 p.m. ET. 'Really Smart':
Holmes spent most of his life in California. The Los Angeles Times reports that he went to Westview High School in San Diego and as we reported, he went to college at the University of California, Riverside.
The Times spoke to people who knew him. They report:
.'"
KTLA went to the Riverside campus and the chancellor of the school described him an intelligent student who graduated with honors. The KTLA reporter said the chancellor called him the "top of the top."
Update at 6:23 p.m. ET. 'A Normal Guy':
KUNC's Kirk Siegler filed a profile of the suspected shooter for All Things Considered. He spoke to one of Holmes' family's next door neighbors, 16-year-old Anthony Mai. He said Holmes was a nice guy, but reserved.
"He didn't seem like a person who would do that, you know," he told Siegler. "Like, I've known him my whole life. I didn't know him that well, but, I mean he just felt like a normal guy, and I don't think he would going around with guns shooting people." 2:28 p.m. ET. Attended University Of California, Riverside:
The University of California, Riverside confirmed that the suspect graduated from UCR with "a BS in neuroscience in the Spring of 2010."
"His last known address was in San Diego," the university said in a statement.
Update at 1:56 p.m. ET. One Traffic Summons:
Dan Oates, the Aurora Police chief, said police only found that Holmes had been given one trafic summons for speeding.
Oates said the suspect was arrested outside the theater near his car. Oates said that police were not prepared to speculate as to a motive.
9(MDAxNDQ2NDAxMDEyNzU2NzM2ODA3ZGI1ZA001))
| 104,525
|
\begin{document}
\title[High order flow of curves]{High order curvature flows of plane curves with generalised Neumann boundary conditions}
\author[J. McCoy]{James McCoy*} \thanks{* Corresponding author}
\address{Priority Research Centre Computer Assisted Research Mathematics and Applications, School of Mathematical and Physical Sciences, University of Newcastle, Australia and Okinawa Institute for Science and Technology Graduate University, Japan}
\email{James.McCoy@newcastle.edu.au}
\author[G. Wheeler]{Glen Wheeler}
\address{Institute for Mathematics and its Applications, University of Wollongong}
\email{glenw@uow.edu.au}
\author[Y. Wu]{Yuhan Wu}
\address{Institute for Mathematics and its Applications, University of Wollongong}
\email{yw120@uowmail.edu.au}
\thanks{The research of the first author was supported by Discovery Project DP180100431 of the Australian Research Council. Part of this work was completed while the first author was a Visiting Professor at the Okinawa Institute for Science and Technology. The research of the third author was supported by a University of Wollongong Faculty of Engineering and Information Sciences Postgraduate research scholarship. The authors are grateful for the support provided by these facilities.}
\begin{abstract}
We consider the parabolic polyharmonic diffusion and $L^2$-gradient flows of the $m$-th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in $L^2$, then the evolving curve converges exponentially in the $C^\infty$ topology to a straight horizontal line segment. The same behaviour is shown for the $L^2$-gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on $m$.
\end{abstract}
\keywords{curvature flow, high order parabolic equation, Neumann boundary condition}
\subjclass[2010]{53C44}
\maketitle
\section{Introduction} \label{S:intro}
\newtheorem{main1}{Theorem}[section]
\newtheorem{main2}[main1]{Theorem}
Higher order geometric evolution problems have interesting practical applications that have motivated increasing attention in recent years to their theoretical behavior.
As fourth order examples for evolving curves we have the curve diffusion flow and the $L^2$-gradient flow of the
elastic energy, and for surfaces the corresponding surface diffusion and Willmore flows.
Flows of higher even order than four have been less thoroughly investigated,
but they and their elliptic counterparts are well-motivated given, for example, applications in
computer design, where higher order equations allow more
flexibility in terms of prescribing boundary conditions \cite{LX}. Such
equations have also found applications in medical imaging \cite{UW}. In \cite{MPW}, the first and second author together with Parkins considered the sixth order geometric triharmonic flow for closed surfaces while Parkins and the third order considered in \cite{PW16} even order flows of closed, planar curves. There the flows were of general even order of polyharmonic form as we will consider here, but we will also in this article consider the $L^2$-gradient flows of the $m$-th arclength derivative of curvature for general $m\in \mathbb{N}\cup \left\{ 0\right\}$. Our work here generalises \cite{MWW} where we considered the $L^2$-gradient flow for the energy
$$\int_\gamma k_s^2 ds \mbox{;}$$
$k_s$ denotes the first derivative of curvature with respect to the arc
length parameter $s$. Our work is also the arbitrary even order generalisation of \cite{WW} by the third author and V-M Wheeler, where the fourth order curve diffusion and elastic flow of curves between parallel lines were investigated. Other relevant works on fourth order flow of curves with boundary
conditions are \cites{DLP14, DP14, L12}. By way of comparison, closed curves without boundary evolving by higher order equations have been more thoroughly studied; see for example \cites{DKS02, EGBMWW14,
GI99, PW16, W13}.
Let $\gamma_0:\left[ -1, 1\right] \rightarrow \mathbb{R}^2$ be a (suitably) smooth embedded or immersed regular curve whose ends $\gamma_0\left( \pm 1\right)$ meet orthogonally two parallel lines $\eta_{\pm}$ separated by distance $d_0$. In this article we are interested in one-parameter families of curves $\gamma\left( \cdot, t\right)$ satisfying either the polyharmonic curve diffusion flow
\begin{equation} \label{E:PHCD}
\frac{\partial \gamma}{\partial t} = \left( -1\right)^{m+1} k_{s^{2m+2}} \nu
\end{equation}
or the flow
\begin{equation} \label{E:theflow}
\frac{\partial \gamma}{\partial t} = \left[ \left( -1\right)^{m+1} k_{s^{2m+2}} - \sum_{j=1}^m \left( -1\right)^j k\, k_{s^{m+j}} k_{s^{m-j}} - \frac{1}{2} k\, k_{s^m}^2\right] \nu \mbox{,}
\end{equation}
with $\gamma\left( \cdot, 0 \right) := \gamma_0$ and generalised Neumann boundary conditions. The equation \eqref{E:theflow} corresponds to the $L^2$-gradient flow for the energy
\begin{equation} \label{E:E}
\int_\gamma k_{s^m}^2 ds \mbox{.}
\end{equation}
Above $k_{s^m}$ denotes the $m$th iterated derivative of curvature with respect to the arc
length parameter $s$; $\nu$ is the smooth choice of unit normal such that the above flows are parabolic in the generalised sense. As discussed in \cite{Pthesis} for example, \eqref{E:PHCD} can also be considered as a gradient flow in an appropriate corresponding Sobolev space. The `generalised Neumann boundary conditions' we assume are not the most general possible for either flow, but they are mathematically a natural choice: we take classical Neumann boundary conditions, as shown in Figure 1, together with no curvature flux on the boundary ($k_s\left( \pm 1, t\right)=0$) and we additionally specify that all odd derivatives of curvature up to order $2m+1$ are equal to zero on the boundary. For each of our flows, induction arguments analogous to those in \cite[Lemma 2.6]{WW} then show that all higher odd curvature derivatives are also equal to zero on the boundary under the flow, so we have for every $\ell \in \mathbb{N}$, as long a solution to the flow equation exists,
\begin{equation} \label{E:bdy}
k_{s^{2\ell-1}}\left( \pm1, t\right) = 0 \mbox{.}
\end{equation}
\noindent \emph{Remark:} If we think of the corresponding higher order elliptic ordinary differential equation, an order $2m+4$ equation should normally have $2m+4$ boundary conditions for a unique solution. These correspond to the classical Neumann condition and all odd curvature derivatives up to order $2m+1$ equal to zero at $x=\pm 1$. On the other hand, from the point of view of partial differential equations, it is more natural to think of each pair of boundary requirements at $\pm 1$ as one condition, giving a total of $m+2$ conditions. We will use the latter description of the number of boundary conditions throughout the article.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw (0,0) node[below] {$\eta_-$} --(0,5) ;
\draw (4,0) node[below] {$\eta_+$}--(4,5);
\draw [ cyan] (0,2.2) to (0.2, 2.2) to [out=0,in=180] (1.5,2.8) to [out=0,in=180] (2.3,2.2) node[below] {$\gamma$} to [out=0,in=-180] (3.8, 3) to (4,3);
\draw (0,2.4)-- (0.2,2.4) -- (0.2,2.2);
\draw (4,2.8)-- (3.8,2.8) -- (3.8,3);
\draw[line width=1pt, -latex, blue] (4,3) --node[auto]{$e$} (5,3);
\draw[line width=1pt, -latex, blue] (4,3) -- node[auto]{$\nu$}(4,4);
\draw [blue] (0,1.4) to [out=0,in=180] (1.8,1.4) node[] {$ \ \ \ d_0$};
\draw [blue] (0,1.4) -- (0.2, 1.5);
\draw [blue] (0,1.4) -- (0.2, 1.3);
\draw [blue] (2.2,1.4) to [out=0,in=-180] (4,1.4);
\draw [blue] (4, 1.4)--(3.8, 1.3);
\draw [blue] (4, 1.4)--(3.8, 1.5);
\end{tikzpicture}
\ \
Figure 1
\end{center}
\end{figure}
Throughout this article we use $\omega$ to denote the \emph{winding number}, defined here by
\[
\omega := \frac{1}{2\pi} \int_{\gamma} k \, ds \mbox{.}
\]
A simple calculation shows that under quite general flows with Neumann boundary conditions on parallel lines, the winding number is constant \cite[Lemma 2.5]{WW}.\\[8pt]
Our main results are as follows:
\begin{main1} \label{T:main1}
Let $\gamma_0: \left[ -1, 1\right] \rightarrow \mathbb{R}^2$ be a smooth embedded or immersed curve with $\omega=0$, whose ends meet the parallel lines $\eta_{\pm}$ with $m+2$ generalised Neumann boundary conditions as described above. If the curvature $\kappa$ of $\gamma_0$ is sufficiently small in $L^2$, that is
$$\int \kappa^2 ds \leq \varepsilon$$
for some $\varepsilon>0$ depending only on $m$, then there exists a smooth solution $\gamma :\left[ 0, \infty\right) \rightarrow \mathbb{R}^2$ to \eqref{E:PHCD} with $\gamma\left( \cdot, 0 \right) = \gamma_0$. The solution $\gamma$ is unique up to parametrisation, smooth and converges exponentially to a horizontal line segment whose distance from $\gamma_0$ is finite.\\
\end{main1}
\begin{main2} \label{T:main2}
Let $\gamma_0: \left[ -1, 1\right] \rightarrow \mathbb{R}^2$ be a smooth embedded or immersed curve with $\omega=0$, whose ends meet the parallel lines $\eta_{\pm}$ with $m+2$ generalised Neumann boundary conditions as described above. If $\gamma_0$ has sufficiently small energy, that is
$$\int \kappa_{s^m}^2 ds \leq \varepsilon$$
for some $\varepsilon>0$ depending only on $m$, then there exists a smooth solution $\gamma :\left[ 0, \infty\right) \rightarrow \mathbb{R}^2$ to the $L^2$-gradient flow for \eqref{E:E} with $\gamma\left( \cdot, 0 \right) = \gamma_0$. The solution $\gamma$ is unique up to parametrisation, smooth and converges exponentially to a horizontal line segment whose distance from $\gamma_0$ is finite.\\
\end{main2}
We remark that local existence of a smooth regular solution $\gamma:\left[ -1, 1\right] \times \left( 0, T\right)\rightarrow \mathbb{R}^2$ to each of the above problems for some $T>0$ is standard. Such solutions are unique up to parametrisation. If $\gamma_0$ satisfies appropriate compatibility conditions, then the solution is smooth on $\left[ 0, T\right)$. It is possible to consider such flows with less smooth initial data but we will not do so in this article. An overview of the procedure for proving short-time existence in this setting is given in \cite[Theorem 2.1]{MWW}. In particular, the initial curve $\gamma_0$ need not be a graph over the horizontal line segment although of course, the later results show that the solution does indeed eventually become so.\\
The structure of the rest of this article is as follows. In Section \ref{S:prelim} we will state fundamental analytical tools that will be used in the analysis of each of our flow problems. We also give the general structure of some evolution equations that is useful in both cases. In Section \ref{S:PHCD} we prove Theorem \ref{T:main1}, the polyharmonic curve diffusion case. The proof has the same structure as the proof of Theorem \ref{T:main2}, thus illustrating the key ideas, however the estimates are much simpler to establish. In Section \ref{S:setup} we take the normal variation of the energy \eqref{E:E} to obtain the corresponding $L^2$-gradient
flow \eqref{E:theflow}, then we prove Theorem \ref{T:main2}.
\section{Preliminaries} \label{S:prelim}
\newtheorem{PSW}{Lemma}[section]
\newtheorem{interp}[PSW]{Proposition}
\newtheorem{evlneqns}[PSW]{Lemma}
We begin with the following standard result for functions of one variable.
\begin{PSW}[Poincar\'{e}-Sobolev-Wirtinger (PSW) inequalities] \label{T:PSW}
Suppose $f:\left[ 0, L \right] \rightarrow \mathbb{R}$, $L>0$ is absolutely continuous.
\begin{itemize}
\item If $\int_0^L f \,ds =0$ then
$$\int_0^L f^2 ds \leq \frac{L^2}{\pi^2} \int_0^L f_s^2 ds \mbox{ and } \left\| f\right\|_\infty^2 \leq \frac{2L}{\pi} \int_0^L f_s^2 ds \mbox{.}$$
\item Alternatively, if $f\left( 0\right) = f\left( L \right) =0$ then
$$\int_0^L f^2 ds \leq \frac{L^2}{\pi^2} \int_0^L f_s^2 ds \mbox{ and } \left\| f\right\|_\infty^2 \leq \frac{L}{\pi} \int_0^L f_s^2 ds \mbox{.}$$
\end{itemize}
\end{PSW}
To state the next interpolation inequality we will use, we first need to set up
some notation. For normal tensor fields $S$ and $T$ we denote by $S \star T$
any linear combination of $S$ and $T$. In our setting, $S$ and $T$ will be
simply curvature $k$ or its arclength derivatives. Denote by $P_n^m\left(
k\right)$ any linear combination of terms of type $\partial_s^{i_1} k \star
\partial_s^{i_2}k \star \ldots \star \partial_s^{i_n}k$ where $m=i_1 + \ldots+
i_n$ is the total number of derivatives.
It is convenient to use the following scale-invariant norms: we define
$$\left\| k \right\|_{\ell, p} := \sum_{i=0}^\ell \left\| \partial_s^i k \right\|_p$$
where
$$\left\| \partial_s^i k \right\|_p = L^{i+1-\frac{1}{p}} \left( \int \left| \partial_s^i k \right|^p ds\right)^{\frac{1}{p}} \mbox{.}$$
The following interpolation inequality for closed curves appears in
\cite{DKS02}; for our setting with boundary we refer to \cite{DP14}.
\begin{interp} \label{T:int}
Let $\gamma: I \rightarrow \mathbb{R}^2$ be a smooth closed curve. Then for any term $P_\nu^\mu\left( k\right)$ with $\nu \geq 2$ that contains derivatives of $k$ of order at most $\ell-1$,
$$\int_I \left| P_\nu^\mu\left( k \right)\right| ds \leq c \, L^{1-\mu-\nu} \left\| k \right\|_2^{\nu-p} \left\| k \right\|_{\ell, 2}^{p}$$
where $p = \frac{1}{\ell} \left( \mu + \frac{1}{2} \nu - 1\right)$ and $c=c\left( \ell, \mu, \nu \right)$. Moreover, if $\mu+ \frac{1}{2} \nu < 2\ell+1$ then $p<2$ and for any $\varepsilon>0$,
\begin{equation*}
\int_I \left| P_\nu^\mu\left( k \right)\right| ds \leq \varepsilon \int_I \left| \partial_s^\ell k \right|^2 ds
+ c\, \varepsilon^{\frac{-p}{2-p}} \left( \int_I \left| k\right|^2 ds\right)^{\frac{\nu-p}{2-p}} + c\left( \int_I \left| k\right|^2 ds \right)^{\mu+\nu-1}\mbox{.}
\end{equation*}
\end{interp}
We conclude this section with the evolution equations for some geometric quantities under the normal curvature flow
\begin{equation} \label{E:generalflow}
\frac{\partial \gamma}{\partial t} = - F\, \nu \mbox{.}
\end{equation}
Here $\nu$ is a smooth choice of unit normal vector and the sign is chosen to ensure that \eqref{E:generalflow} is parabolic in the generalised sense. Throughout this article $L=L\left[ \gamma\right]$ will denote the length of the curve $\gamma$.
The following evolution equations are straightforward to derive using techniques as in \cite{WW}, for example.
\begin{evlneqns} \label{T:evlneqns}
Under the flow \eqref{E:theflow} we have the following evolution equations:
\begin{enumerate}
\item[\textnormal{(i)}] $\frac{d}{d t} L = -\int_\gamma k \, F\, ds$;\\
\noindent For each $\ell = 0, 1, 2, \ldots$,
\item[\textnormal{(ii)}] $\frac{\partial}{\partial t} k_{s^\ell} = F_{s^{\ell+2}} + \sum_{j=0}^\ell \partial_{s^j} \left( k\, k_{s^{\ell-j}}F\right)$.
\end{enumerate}
\end{evlneqns}
\section{The polyharmonic curve diffusion flow} \label{S:PHCD}
In this section we establish our result for the polyharmonic curve diffusion flow \eqref{E:PHCD} for each fixed $m\in \mathbb{N}\cup \left\{ 0 \right\}$. When $m=0$ we have the classical curve diffusion flow. The case $m=1$ can be considered the geometric triharmonic heat flow of curves in view of the relationship between curvature and derivatives of $\gamma$. Since curvature depends upon second spatial derivatives of $\gamma$, the flow \eqref{E:PHCD} has order $2m+4$.
\newtheorem{PHCDL}{Lemma}[section]
\newtheorem{PHCDexp}[PHCDL]{Proposition}
\newtheorem{PHCDL2}[PHCDL]{Corollary}
\newtheorem{PHCDL3}[PHCDL]{Corollary}
\begin{PHCDL} \label{T:PHCDL}
While a solution to the flow \eqref{E:PHCD} with generalised Neumann boundary conditions exists, we have
$$\frac{d}{dt} L\left( t\right) = - \int_\gamma k_{s^{m+1}}^2 ds$$
\end{PHCDL}
\noindent \textbf{Proof:} The result follows directly using Lemma \ref{T:evlneqns} (i) and $m+1$ integrations by parts, noting in each case the boundary term will contain an odd derivative of $k$ that is equal to zero in view of the boundary conditions.\hspace*{\fill}$\Box$\\[8pt]
\noindent \textbf{Remark:} In fact, under the flow \eqref{E:PHCD}, the length $L\left( t\right)$ is strictly decreasing unless $\gamma$ is a straight line segment, because the only smooth solutions that satisfy $k_{s^{m+1}} \equiv 0$ and the boundary conditions are horizontal line segments.\\[8pt]
In view of Lemma \ref{T:PHCDL} and the separation $d_0$ of the supporting parallel lines $\eta_{\pm}$, the length $L\left( t\right)$ of the evolving curve $\gamma\left( \cdot, t\right)$ remains bounded above and below under the flow \eqref{E:PHCD}.\\
Next we show directly that, provided initially small, $\int k^2 ds$ decays exponentially under the flow. As in the statement of the main theorem, $\kappa$ denotes the curvature of the initial curve $\gamma_0$.
\begin{PHCDexp} \label{T:PHCDexp}
There exists a constant $\varepsilon>0$, depending only on $m$, such that, if $\gamma_0$ satisfies
\begin{equation} \label{E:PHCDsmallness}
\int \kappa^2 ds \leq \varepsilon \mbox{,}
\end{equation}
then, while a solution to \eqref{E:PHCD} exists,
$$ \int k^2 ds \leq \int \kappa^2 ds \cdot \exp\left( -\delta t \right) \mbox{.}$$
Here $\delta>0$ depends on $\varepsilon$ and $L_0$, the length of $\gamma_0$.
\end{PHCDexp}
\noindent \textbf{Proof:} Under the flow \eqref{E:PHCD}, a straightforward computation using Lemma \ref{T:evlneqns}, integration by parts and the boundary conditions shows that
\begin{equation} \label{E:k2}
\frac{d}{dt} \int k^2 ds =-2 \int k_{s^{m+2}}^2 ds + \int \left( k^3 \right)_{s^{m+1}} k_{s^{m+1}} ds
= -2 \int k_{s^{m+2}}^2 ds + \int P_4^{2m + 2}\left( k\right) ds \mbox{.}
\end{equation}
Since the highest order derivative in $P_4^{2m+2}\left( k\right)$ is $k_{s^{m+1}}$, we have using Proposition \ref{T:int}
$$\int P_4^{2m+2}\left( k \right) ds \leq c\, L^{-\left( 2m+5\right)} \left\| k \right\|_2^{\frac{2m+5}{m+2}} \left\| k \right\|_{m+2, 2}^{\frac{2m+3}{m+2}} \mbox{.}$$
In view of Lemma \ref{T:PSW},
$$\left\| k \right\|_{m+2, 2} \leq c\left( m\right) L^{m + \frac{5}{2}} \left( \int k_{s^{m+2}}^2 ds \right)^{\frac{1}{2}} \mbox{.}$$
We now estimate, again using Lemma \ref{T:PSW},
$$\left\| k \right\|_2^{\frac{2m+5}{m+2}} = L^{\frac{2m+5}{2m+4}} \int k^2 ds \left( \int k^2 ds \right)^{\frac{1}{2m+4}} \leq \frac{1}{\pi} L^{2+ \frac{1}{2m+4}} \int k^2 ds \left( \int k_{s^{m+2}}^2 ds \right)^{\frac{1}{2m+4}} \mbox{.}$$
Combining these, we have
\begin{equation} \label{E:P4}
\int P_4^{2m+2}\left( k \right) ds \leq c\, L \int k^2 ds \int k_{s^{m+2}}^2 ds
\end{equation}
and so from \eqref{E:k2},
$$\frac{d}{dt} \int k^2 ds \leq \left( -2 + c\, L \int k^2 ds \right) \int k_{s^{m+2}}^2 ds \mbox{.}$$
Suppose initially $c\, L \int k^2 ds \leq 2- 2 \tilde \delta$, for some $\tilde \delta >0$. Then, at least for a short time, $c\, L \int k^2 ds \leq 2- \tilde \delta$. While this is the case,
$$\frac{d}{dt} \int k^2 ds \leq - \tilde \delta \int k_{s^{m+2}}^2 ds \leq - \tilde \delta \left( \frac{\pi^2}{L^2} \right)^{m+2} \int k^2 ds \leq - \tilde \delta \left( \frac{\pi^2}{L_0^2} \right)^{m+2} \int k^2 ds$$
where we have used again Lemma \ref{T:PSW} and also Lemma \ref{T:PHCDL}, with $L_0$ denoting the length of $\gamma_0$. The result follows.\hspace*{\fill}$\Box$\\[8pt]
\noindent \textbf{Remarks:}
\begin{enumerate}
\item Without the smallness requirement, we can show similarly as in \cite[Theorem 3.1]{DKS02} that if the maximal existence time $T$ of a solution to \eqref{E:PHCD} is finite, then the curvature must blow up in $L^2$. Specifically, using the second statement of Proposition \ref{T:int} we have from \eqref{E:k2}
$$ \frac{d}{dt} \int k^2 ds \leq c\, \left( \int k^2 ds\right)^{2m+5}$$
from which it follows that
$$\int k^2 ds \geq c\left( T-t\right)^{-\frac{1}{2m+4}} \mbox{.}$$
\item The smallness requirement here may be compared with the requirement for exponential convergence in the case of closed curves evolving by the polyharmonic curvature flow \cite{PW16}. Denoting by $\overline{k}$ the average of the curvature over a closed curve, the scale-invariant $\left\| k \right\|_2$ is replaced by
$$K_{osc} = L \int \left( k - \overline{k} \right)^2 ds$$
and a smallness condition on this quantity (together with a condition on the isoperimetric ratio) facilitates exponential convergence of $K_{osc}$. The two quantities again appear in parallel in the Poincar\'{e}-Sobolev-Wirtinger inequalities for curves with boundary and closed curves.\\[8pt]
\end{enumerate}
The next step is to show that under the flow, $L^2$ norms of all curvature derivatives remain bounded. This proof here is considerably more direct than in the subsequent section for the flow \eqref{E:theflow}.
\begin{PHCDL2} \label{T:PHCDL2}
Suppose $\gamma_0$ satisfies the conditions of Theorem \ref{T:main1} including the smallness condition \eqref{E:PHCDsmallness}. Then, while a solution to the flow \eqref{E:PHCD} with generalised Neumann boundary conditions exists, we have for all $\ell \in \mathbb{N}\cup \left\{ 0\right\}$,
$$\int k_{s^\ell}^2 ds \leq C_{m, \ell} \mbox{,}$$
for constants $C_{m, \ell}$.
\end{PHCDL2}
\noindent \textbf{Proof:} Under the flow \eqref{E:PHCD}, a straightforward computation using Lemma \ref{T:evlneqns} and integration by parts with repeated application of the boundary conditions and the consequence \eqref{E:bdy}, gives that for each $\ell$,
\begin{equation} \label{E:evlnL2kl}
\frac{d}{dt} \int k_{s^\ell}^2 ds = -2 \int k_{s^{m+\ell+2}}^2 ds + \int P_4^{2m + 2\ell + 2}\left( k\right) ds \mbox{,}
\end{equation}
where the highest order of derivatives of $k$ in the second above term is $m+\ell+1$. Using Proposition \ref{T:int} we have for any $\varepsilon >0$,
\begin{equation} \label{E:P4}
\int P_4^{2m + 2\ell + 2}\left( k\right) ds \leq \varepsilon \int k_{s^{m+\ell+2}}^2 ds + c\left( m, \ell, \varepsilon \right) \left( \int k^2 ds\right)^{2m + 2\ell +5} \mbox{,}
\end{equation}
so from \eqref{E:evlnL2kl} we obtain by taking $\varepsilon=1$
$$ \frac{d}{dt} \int k_{s^\ell}^2 ds \leq - \int k_{s^{m+\ell+2}}^2 ds + c \left( \int k^2 ds\right)^{2m + 2\ell +5} \mbox{.}$$
Using now Lemma \ref{T:PSW} and Lemma \ref{T:PHCDL} we obtain
$$ \frac{d}{dt} \int k_{s^\ell}^2 ds \leq - \left( \frac{\pi^2}{L_0^2}\right)^{m+2} \int k_{s^\ell}^2 ds + c \left( \int k^2 ds\right)^{2m + 2\ell +5}$$
from which the result follows since $\int k^2 ds$ is bounded in view of Proposition \ref{T:PHCDexp}.\hspace*{\fill}$\Box$\\[8pt]
\noindent \textbf{Remark:} In view of Corollary \ref{T:PHCDL2}, a standard contradiction argument using short-time existence implies that in fact the solution to \eqref{E:PHCD} exists for all time, that is, $T=\infty$.\\[8pt]
Proposition \ref{T:PHCDexp} and Corollary \ref{T:PHCDL2} imply via interpolation that all curvature derivatives decay exponentially in $L^2$ and, via Lemma \ref{T:PSW}, in $L^\infty$.
\begin{PHCDL3} \label{T:PHCDL3}
Suppose $\gamma_0$ satisfies the conditions of Theorem \ref{T:main1} including the smallness condition \eqref{E:PHCDsmallness}. Under the flow \eqref{E:PHCD}, there exist $\delta_{\ell, m}>0$, depending only on $\varepsilon$ and $L_0$ such that, for all $\ell \in \mathbb{N} \cup \left\{ 0 \right\}$,
$$\int k_{s^\ell}^2 ds \leq \tilde C_{m, \ell} \exp \left( - \delta_{\ell, m} t\right) \mbox{.}$$
The quantities $\left\| k_{s^\ell} \right\|_\infty$ also decay exponentially for all $\ell$.
\end{PHCDL3}
\noindent \textbf{Proof:} The curvature derivative decay in $L^2$ follows by standard integration by parts; we give the first two calculations:
$$\int k_s^2 ds = \left[ k_s k \right]_{\partial \gamma} - \int k_{ss} k \, ds \leq \left( \int k_{ss}^2 ds\right)^{\frac{1}{2}} \left( \int k^2 ds\right)^{\frac{1}{2}} \mbox{.}$$
Here we have used that the boundary term contains an odd derivative of $k$ so is equal to zero. The $\int k_{ss}^2 ds$ factor is bounded by Corollary \ref{T:PHCDL2} and Lemma \ref{T:PHCDL} so then the exponential convergence follows from Proposition \ref{T:PHCDexp}.
We next compute
$$\int k_{ss}^2 ds = \left[ k_{ss} k_s \right]_{\partial \gamma} - \int k_{sss} k_s \, ds \leq \left( \int k_{sss}^2 ds\right)^{\frac{1}{2}} \left( \int k_s^2 ds\right)^{\frac{1}{2}} \mbox{.}$$
Again we have used that the boundary term contains an odd derivative of $k$. The $\int k_{sss}^2 ds $ factor is bounded by Corollary \ref{T:PHCDL2} and Lemma \ref{T:PHCDL} so then the exponential convergence follows from the previous step.
We can continue this way to obtain all curvature derivatives in $L^2$ decay exponentially. The $L^\infty$ decay then follows from Lemma \ref{T:PSW} and Lemma \ref{T:PHCDL}.\hspace*{\fill}$\Box$
\mbox{}\\[8pt]
Using now the evolution equation \eqref{E:PHCD} we obtain uniform bounds on all derivatives of the immersion $\gamma: \left[ -1, 1\right] \times \left[ 0, \infty\right) \rightarrow \mathbb{R}^2$. This implies there exists an immersion $\gamma_\infty : \left[ -1, 1\right] \rightarrow \mathbb{R}^2$ satisfying the boundary conditions and a subsequence $t_j \rightarrow \infty$ such that $\gamma\left( \cdot, t_j \right) \rightarrow \gamma_\infty$ in $C^\infty\left( \left[ -1, 1\right], \mathbb{R}^2 \right)$. Since $\left\| k \right\|_\infty \rightarrow 0$, the curve $\gamma_\infty$ is a straight line segment which is horizontal in view of the Neumann boundary condition. Exponential convergence in $C^\infty$ of $\gamma$ to $\gamma_\infty$ now follows by the same argument as in \cite{MWW} using exponential convergence of the curvature and its derivatives. This completes the proof of Theorem \ref{T:main1}. \hspace*{\fill}$\Box$\\[8pt]
\noindent \textbf{Remark:} While we don't know the precise height of the limiting straight horizontal line segment, that $\left\| k_{s^{2m+2}} \right\|_\infty$ decays exponentially shows that the solution curve remains within a bounded distance of the initial curve: for any $x$,
$$\left| \gamma\left( x, \tilde t\right) - \gamma\left( x, 0 \right) \right| \leq \int_0^{\tilde t} \left| \frac{\partial \gamma}{\partial t} \left( x, t\right) \right| dt \leq c \int_0^{\tilde t} e^{-\delta \, t} dt = \frac{c}{\delta} \left( 1- e^{-\delta \tilde t} \right) \mbox{.}$$
\section{The gradient flow for $\int k_{s^m}^2 ds$} \label{S:setup}
\newtheorem{BCs}{Lemma}[section]
\newtheorem{BD2}[BCs]{Lemma}
\newtheorem{Length}[BCs]{Lemma}
\newtheorem{L2bounds}[BCs]{Proposition}
\newtheorem{kexp}[BCs]{Proposition}
\newtheorem{dexp}[BCs]{Corollary}
For a suitably smooth curve $\gamma$ as described in Section \ref{S:intro}, we are interested in the associated curvature-dependent energies
$$E\left[ \gamma\right] = \frac{1}{2} \int_\gamma k_{s^m}^2\, ds$$
and the corresponding $L^2$-gradient flows with suitable associated generalised Neumann boundary conditions. As the energy involves $m$ derivative of curvature, so $m+2$ derivatives of $\gamma$, the gradient flow will be of order $2m+4$.
Under a normal variation $\tilde \gamma = \gamma + \varepsilon F \nu$
straightforward calculations yield
\begin{multline} \label{E:1}
\left. \frac{d}{d\varepsilon} E\left[ \tilde \gamma \right] \right|_{\varepsilon=0}
= - 2 \int_\gamma \left[ \left( -1\right)^{m+1} k_{s^{2m+2}} - \sum_{j=1}^m \left( -1\right)^j k\, k_{s^{m+j}} k_{s^{m-j}} - \frac{1}{2} k\, k_{s^m}^2 \right] F \, ds\\
+ 2 \left[ \sum_{j=0}^{m+1} \left( -1\right)^j k_{s^{m+j}} \partial_{s^{m+1-j}} F + \sum_{j=1}^{m} \sum_{\ell=0}^{j-1} \left( -1\right)^\ell k_{s^{m+\ell}} \partial_{s^{j-1-\ell}} \left( k\, k_{s^{m-j}} F \right) \right]_{\partial \gamma} \mbox{.}
\end{multline}
In particular, the above follows from the variations
$$\left. \frac{\partial}{\partial \varepsilon} k_{s^m} \right|_{\varepsilon=0}= \partial_{s^{m+2}} F + \sum_{j=0}^m \partial_{s^j} \left( k\, k_{s^{m-j}} F \right)$$
and
$$\left. \frac{\partial}{\partial \varepsilon} ds \right|_{\varepsilon=0} = -k\, F \, ds \mbox{,}$$
where the boundary terms in \eqref{E:1} appear via repeated integration by parts. Details behind these calculations may be found for example in \cite{WW}.
`Natural boundary conditions' for the corresponding $L^2$-gradient flow would
ensure that the boundary term in \eqref{E:1} is equal to zero. Assuming for now it is equal to zero we would take the normal flow speed
\begin{equation} \label{E:speed}
F= \left( -1\right)^{m+1} k_{s^{2m+2}} - \sum_{j=1}^m \left( -1\right)^j k\, k_{s^{m+j}} k_{s^{m-j}} - \frac{1}{2} k\, k_{s^m}^2
\end{equation}
and the corresponding $L^2$-gradient flow is then \eqref{E:theflow}.\\
Let us now establish a mathematically-reasonably choice of boundary conditions. Beginning with the classical Neumann boundary condition and differentiating in time (see also \cite[Lemma 2.5]{WW} for example) we have
\begin{equation} \label{E:Fs}
F_s\left( \pm 1, t\right) = 0 \mbox{.}
\end{equation}
If we assume as in previous work the `no curvature flux condition' at the boundary,
\begin{equation} \label{E:noflux}
k_s\left( \pm 1, t\right) =0 \mbox{,}
\end{equation}
then from the evolution equation for $k_s$,
$$\frac{\partial}{\partial t} k_s = F_{s^3} + \sum_{j=0}^1 \partial_{s^j} \left( k\, k_{s^{1-j}} F \right)$$
we see that, on the boundary, we must also have $F_{s^3}\left( \pm 1, t\right) \equiv 0$. More generally, by similar arguments in turn, for each odd derivative of $k$ equal to zero on the boundary, we see that the next odd derivative of $F$ is also equal to zero on the boundary. Assuming then that all odd derivatives of $k$ up to order $2m+1$ are equal to zero on the boundary and taking into account the corresponding behaviour of the odd derivatives of $F$ on the boundary, we see that this choice of boundary conditions does render the boundary term in \eqref{E:1} equal to zero. Moreover, we then have by an inductive argument similar to that in \cite{WW}:
\begin{BCs} \label{T:BCs}
With classical Neumann conditions and all odd derivatives of $k$ up to order $2m+1$ equal to zero on the boundary, a solution to the flow \eqref{E:theflow} satisfies $k_{s^{2\ell-1}}=0$ and $F_{s^{2\ell -1}} =0$ on the boundary for all $\ell \in \mathbb{N}$.
\end{BCs}
Throughout our work it will be necessary to check that various boundary terms arising by parts are equal to zero. These boundary terms typically are sums of products. In many cases, each product has three factors that are each either an \emph{even} or an \emph{odd} number of iterated spatial derivatives of $k$ or of $F$. For certain products, we need to know that an odd number of derivatives always produces an odd iterated derivative factor of $k$ or of $F$, while for others we need to know that an even number of derivatives always produces an odd derivative factor. The presence of such a factor in each term then ensures by Lemma \ref{T:BCs} that the boundary term is equal to zero.
We introduce the notation $e$ to denote an even (or zeroth order) derivative factor of $k$ or of $F$ and $o$ to denote an odd derivative factor of $k$ or of $F$. We allow sums of such factors in the notation $e\, o$ etc. We also use subscripts $e$ and $o$ to denote respectively an even (or zero) number or an odd number of spatial derivatives. Some of the results that we will need later can now be stated as follows:
\begin{BD2} \label{T:BD2}
Terms of the form $\left( e\, e\, e\right)_o$, $\left( e\, o\, o\right)_o$ and $\left( e\, e\, o \right)_e$ always contain an $o$ factor.
\end{BD2}
\noindent \textbf{Proof:} Using the product rule we begin with
$$\left( e\, e\, e\right)_s = e\, e\, o \mbox{,}$$
which has the required form. Differentiating a second time,
$$\left( e\, e\, e\right)_{ss} = \left( e\, e\, o \right)_s = e\, o\, o + e\, e\, e \mbox{,}$$
and a third time
$$\left( e\, e\, e\right)_{s^3}= o\, o\, o + e\, e\, o \mbox{.}$$
We see that the third derivative also consists of terms with an $o$ factor. Continuing
$$\left( e\, e\, e\right)_{s^4}= o\, o\, e + e\, o\, o + e\, e\, e $$
and
$$\left( e\, e\, e\right)_{s^5}= o\, o\, o + o\, e\, e \mbox{,}$$
which is the same form as the third derivative. We conclude the result for all odd derivatives $\left( e\, e\, e\right)_o$.\\
For $\left( e\, o\, o\right)_o$, we begin with
$$\left( e\, o\, o\right)_s = e\, e\, o + o\, o\, o\mbox{,}$$
which has the required form. Differentiating a second time,
$$\left( e\, o\, o\right)_{ss} = e\, o\, o + e\, e\, e$$
and a third time
$$\left( e\, o\, o\right)_{s^3} = e\, e\, o + o\, o\, o \mbox{,}$$
which is the same form as the first derivative, so we conclude the result for all odd derivatives $\left( e\, o\, o\right)_o$.\\
For $\left( e\, e\, o\right)$, the zeroth order derivative has an $o$ factor as required. We compute
$$\left( e\, e\, o\right)_s = e\, o\, o + e\, e\, e $$
and so
$$\left( e\, e\, o\right)_{ss} = e\, e\, o + o\, o\, o \mbox{;}$$
so each term has an $o$ factor as required. Continuing
$$\left( e\, e\, o\right)_{s^3} = e\, o\, o + e\, e\, e \mbox{;}$$
this is the same form as $\left( e\, e\, o\right)_s$, so therefore even derivatives, like $\left( e\, e\, o\right)_{ss}$, will always consist of sums of terms containing $o$ factors, as required.\hspace*{\fill}$\Box$
\mbox{}\\[8pt]
\noindent \textbf{Remark:} We will occasionally need results related to the above, in particular when `square' factors appear in the products to be considered. We will develop those results directly where they are needed to follow.\\
Notwithstanding our earlier comments on short-time existence, our first result for the flow \eqref{E:theflow} shows that if the initial energy is small, then the length of the evolving curve does not increase.
\begin{Length} \label{T:Length}
If the initial curve $\gamma_0$ has sufficiently small energy \eqref{E:E} depending only on $m$, then, under the flow \eqref{E:theflow} with normal speed \eqref{E:speed}, the length of $\gamma$ does not increase.
\end{Length}
\noindent \textbf{Proof:} We have using Lemma \ref{T:evlneqns}, (i),
\begin{equation} \label{E:Lksm}
\frac{d}{dt} L = - \int k \left[ \left( -1\right)^{m+1} k_{s^{2m+2}} + \sum_{j=1}^m \left( -1\right)^{j+1} k\, k_{s^{m+j}} k_{s^{m-j}} - \frac{1}{2} k\, k_{s^m}^2 \right] ds \mbox{.}
\end{equation}
By integrating by parts the first term $m+1$ times and using the boundary conditions, we have
$$-\left( -1\right)^{m+1}\int k \, k_{s^{2m+2}} ds = - \int k_{s^{m+1}}^2 ds \mbox{,}$$
while integrating by parts the $j$th term in the sum $j$ times we can see that the rest of the terms have the form $\int P_4^{2m}\left( k\right) ds$ with the highest order derivative of $k$ being $k_{s^m}$. Thus we estimate using Proposition \ref{T:int}
$$\int P_4^{2m}\left( k\right) ds \leq c\left( m\right) L^{-\left( 2m+3\right)} \left\| k \right\|_2^{\frac{2m+3}{m+1}} \left\| k\right\|_{m+1, 2}^{\frac{2m+1}{m+1}} \mbox{.}$$
Using now Lemma \ref{T:PSW} we have
$$ \left\| k\right\|_{m+1, 2} \leq c\left( m\right) L^{m+ \frac{3}{2}} \left( \int k_{s^{m+1}}^2 ds \right)^{\frac{1}{2}}$$
and
$$\int k^2 ds \leq \left( \frac{L^2}{\pi^2}\right)^{m+1} \left( \int k_{s^{m+1}}^2 ds \right) \mbox{,}$$
so
$$\int P_4^{2m}\left( k\right) ds \leq c\left( m\right) \left\| k\right\|_2^2 \int k_{s^{m+1}}^2 ds$$
and from \eqref{E:Lksm} we obtain
$$ \frac{d}{dt} L \leq \left( -1 + c\left( m\right) L^{2m+1} \int k_{s^{m}}^2 ds \right) \int k_{s^{m+1}}^2 ds \mbox{.}$$
Since $\int k_{s^{m}}^2 ds$ is nonincreasing under the flow by construction, it follows that if $\gamma_0$ has\\ $L^{2m+3} \int k_{s^{m}}^2 ds$ sufficiently small, depending only on $m$, then $L$ does not increase under the flow \eqref{E:theflow}. \hspace*{\fill}$\Box$
\begin{L2bounds} \label{T:L2bounds}
If the initial curve $\gamma_0$ has sufficiently small energy \eqref{E:E} depending only on $m$, then, under the flow \eqref{E:theflow} with normal speed \eqref{E:speed}, there are constants $C_{m, \ell}$ depending only on $L_0$ and the initial energy such that
$$\int k_{s^\ell}^2 ds \leq C_{m, \ell}$$
for all $\ell \in \mathbb{N}\cup \left\{ 0\right\}$.
\end{L2bounds}
\noindent \textbf{Proof:} For $\ell \leq m$, the result is immediate via Lemma \ref{T:PSW} and Lemma \ref{T:Length}. For any $\ell$, we have via Lemma \ref{T:evlneqns}
\begin{equation} \label{E:ksl2}
\frac{d}{dt} \int k_{s^\ell}^2 ds = 2 \int k_{s^\ell} \left[ F_{s^{\ell+2}} + \sum_{j=0}^\ell \partial_{s^j} \left( k\, k_{s^{\ell-j}} F\right) \right] ds - \int k_{s^\ell}^2 k\, F \, ds \mbox{.}
\end{equation}
For each $\ell >m$ we examine each of the terms on the right hand side of \eqref{E:ksl2} in turn. Since the leading term of $F_{s^{\ell+2}}$ is $k_{s^{2m+\ell+4}}$, we will integrate by parts the first term in \eqref{E:ksl2} $\left( m+2\right)$ times:
$$\int k_{s^\ell} F_{s^{\ell+2}} ds = \left[ k_{s^\ell} F_{s^{\ell+1}} \right]_{\partial \gamma} - \int k_{s^{\ell+1}} F_{s^{\ell+1}} ds \mbox{.}$$
Regardless of $\ell$, the boundary term above will have an odd derivative, so is equal to zero by Lemma \ref{T:BCs}. Integrating by parts again,
$$\int k_{s^\ell} F_{s^{\ell+2}} ds = \left[ k_{s^{\ell+1}} F_{s^{\ell}} \right]_{\partial \gamma} - \int k_{s^{\ell+2}} F_{s^{\ell}} ds \mbox{.}$$
Again, the boundary term will have an odd derivative so is equal to zero. With a further $m$ integrations by parts, observing that the boundary terms are always equal to zero, we obtain
\begin{align} \label{E:ksl2aux} \nonumber
& \int k_{s^\ell} F_{s^{\ell+2}} ds\\ \nonumber
&= \left( -1\right)^m \int k_{s^{\ell+m+2}} F_{s^{\ell-m}} ds\\ \nonumber
&= \left( -1\right)^m \int k_{s^{\ell+m+2}} \left[ \left( -1\right)^{m+1} k_{s^{2m+2}} - \sum_{j=1}^m k\, k_{s^{m+j}} k_{s^{m-j}} - \frac{1}{2} k\, k_{s^m}^2 \right]_{s^{\ell - m}}ds\\ \nonumber
&= -\int k_{s^{\ell+m+2}}^2 ds - \left( -1\right)^m \sum_{j=1}^m \int k_{s^{\ell+m+2}}\left( k\, k_{s^{m+j}} k_{s^{m-j}} \right)_{s^{\ell-m}} ds\\
& \quad- \frac{ \left( -1\right)^m }{2}\int k_{s^{\ell+m+2}} \left( k\, k_{s^m}^2 \right)_{s^{\ell - m}} ds \mbox{.}
\end{align}
We want to confirm that the summation and last terms here have the form $\int P_4^{2m+2\ell +2}\left( k\right) ds$ with highest order derivative $k_{s^{\ell+m+1}}$. Integrating by parts the last term,
\begin{equation} \label{E:extra1}
\int k_{s^{\ell + m + 2}} \left( k\, k_{s^m}^2 \right)_{s^{\ell-m}} ds = \left[ \left( k\, k_{s^m}^2\right)_{s^{\ell-m}} k_{s^{\ell+m+1}}\right]_{\partial \gamma} - \int k_{s^{\ell+m+1}} \left( k\, k_{s^m}^2 \right)_{s^{\ell -m+1}} ds \mbox{.}
\end{equation}
For the above boundary term, if $\ell+m+1$ is odd then we have an odd derivative of $k$ factor which is zero by Lemma \ref{T:BCs}. If, on the other hand, $\ell+m+1$ is even, then $\ell+m$ is odd and so is $\ell-m$. Using the notation of Lemma \ref{T:BD2}, we have multiplying $k_{s^{\ell-m}}$ a term either of the form $\left( e\, e\, e\right)_o$ or $\left( e\, o\, o\right)_o$. Lemma \ref{T:BD2} gives that either of these consist of terms all with odd derivatives of $k$, thus the boundary term is equal to zero in this case also. The remaining integral term in \eqref{E:extra1} has the form $\int P_4^{2m+2\ell+2}\left( k\right) ds$ with highest order derivative $k_{s^{m+\ell+1}}$.
For the summation term in \eqref{E:ksl2aux} we again need one integration by parts: for each $j$,
\begin{multline*}
\int k_{s^{\ell+m+2}}\left( k\, k_{s^{m+j}} k_{s^{m-j}} \right)_{s^{\ell-m}} ds\\
= \left[ k_{s^{\ell+m+1}}\left( k\, k_{s^{m+j}} k_{s^{m-j}} \right)_{s^{\ell-m}} \right]_{\partial \gamma} - \int k_{s^{\ell+m+1}}\left( k\, k_{s^{m+j}} k_{s^{m-j}} \right)_{s^{\ell-m+1}} ds \mbox{.}
\end{multline*}
If $\ell+m+1$ is odd then again clearly the boundary term is equal to zero. Otherwise, $\ell - m$ is odd and the other factor in the boundary term has the form $\left( e\, e\, e\right)_o$ or $\left( e\, o\, o\right)_o$. As before, the boundary term is again equal to zero for each $i$ and the remaining integral terms have the correct form.\\
Returning now to \eqref{E:ksl2}, using \eqref{E:speed} we have
\begin{equation} \label{E:ksl2aux2}
\int k_{s^\ell}^2 k\, F\, ds = \int k\, k_{s^\ell}^2 \left[ \left( -1\right)^{m+1} k_{s^{2m+2}} - \sum_{j=1}^m k\, k_{s^{m+j}} k_{s^{m-j}} - \frac{1}{2} k\, k_{s^m}^2 \right] ds \mbox{.}
\end{equation}
We want to show that all these terms have either the form $\int P_{4}^{2\ell + 2m +2}\left( k \right) ds$ with highest derivative $k_{s^{\ell+m+1}}$ or $\int P_6^{2m+2\ell}\left( k\right) ds$ with no higher derivative than $k_{s^{\ell+m}}$. The last term above already fits this latter form. On the first term we will need to integrate by parts $m+1$ times: first
$$\int k\, k_{s^\ell}^2 k_{s^{2m+2}} ds = \left[ k\, k_{s^\ell}^2 k_{s^{2m+1}} \right]_{\partial \gamma} - \int \left( k\, k_{s^\ell}^2 \right)_s k_{s^{2m+1}} ds \mbox{.}$$
The boundary term here has an odd derivative of $k$ so is equal to zero. Integrating by parts a second time,
$$\int k\, k_{s^\ell}^2 k_{s^{2m+2}} ds = -\left[ \left( k\, k_{s^\ell}^2 \right)_s k_{s^{2m}} \right]_{\partial \gamma} + \int \left( k\, k_{s^\ell}^2 \right)_{ss} k_{s^{2m}} ds \mbox{.}$$
This time $k_{s^{2m}}$ in the boundary term is an even derivative so we look at the other factor. Depending on $\ell$, the other factor has the form $\left( e\, e\, e\right)_o$ or $\left( e\, o\, o\right)_o$; so in both cases these will be equal to zero by Lemma \ref{T:BD2} and Lemma \ref{T:BCs}. With each of the further integrations by parts, the boundary terms will be one of the two types above, so we are left with
$$\int k\, k_{s^\ell}^2 k_{s^{2m+2}} ds = \left( -1\right)^{m+1} \int \left( k\, k_{s^\ell}^2 \right)_{s^{m+1}} k_{s^{m+1}} ds = \int P_4^{2m+2\ell+2}\left( k \right) ds\mbox{.}$$
with no higher derivative of $k$ than $k_{s^{m+\ell+1}}$ appearing.
The terms in the summation in \eqref{E:ksl2aux2} require some integration by parts if $j$ is large relative to $\ell$. Specifically, for the $j$th term we should do $j$ integrations by parts to ensure no derivative of order higher than $m+\ell$ appears. We have
$$\int k^2 k_{s^\ell}^2 k_{s^{m-j}} k_{s^{m+j}} ds = \left[ k^2 k_{s^\ell}^2 k_{s^{m-j}} k_{s^{m+j-1}} \right]_{\partial \gamma} -\int \left( k^2 k_{s^\ell}^2 k_{s^{m-j}} \right)_s k_{s^{m+j-1}} ds \mbox{.}$$
The boundary term above will always have an odd derivative of $k$ so is equal to zero. Continuing in the case $j>1$ we have
$$\int k^2 k_{s^\ell}^2 k_{s^{m-j}} k_{s^{m+j}} ds =- \left[ \left( k^2 k_{s^\ell}^2 k_{s^{m-j}}\right)_s k_{s^{m+j-2}} \right]_{\partial \gamma} +\int \left( k^2 k_{s^\ell}^2 k_{s^{m-j}} \right)_{ss} k_{s^{m+j-2}} ds \mbox{.}$$
If $m+ j$ is odd, then the boundary term is equal to zero. In the case $m+j$ is even, then $m-j$ is also even and applying the product rule to expand the first derivative, we see that every term will have an odd derivative so is equal to zero. Clearly, similar terms will arise with further integrations by parts. We need to see that an odd derivative of a product of the form $\mbox{square } \times \mbox{square } \times e$ is always equal to zero. Expanding out such a derivative using the binomial theorem, the terms with an odd derivative of $e$ are already equal to zero via Lemma \ref{T:BCs} so we only need to check those terms with an odd derivative of a square. We have
$$\left( e\, e\right)_s = e\, o$$
$$\left( e\, e\right)_{ss} = e\, e + o\, o$$
$$\left( e\, e\right)_{s^3} = e\, o$$
and so generally $\left( e\, e\right)_o = e\, o$. Similarly
$$\left( o\, o\right)_s = e\, o$$
so the same pattern will occur and we see that all odd derivatives of squares contain an $o$ factor, thus the boundary terms generated through repeated integration by parts will always be equal to zero. Therefore we have
$$\int k^2 k_{s^\ell}^2 k_{s^{m-j}} k_{s^{m+j}} ds =\left( -1\right)^j \int \left( k^2 k_{s^\ell}^2 k_{s^{m-j}} \right)_{s^j} k_{s^{m}} ds = \int P_6^{2m+2\ell}\left( k\right) ds\mbox{,}$$
where no higher derivative than $k_{s^{m+\ell}}$ appears on the right hand side.\\
It remains now to consider the summation term in \eqref{E:ksl2}. For $j=0$ this term has the same form as that dealt with above, so assume $1\leq j \leq \ell$. To handle the boundary terms arising from integration by parts, it is going to be easiest to do $j$ integrations by parts first without substituting in the form of $F$. The first integration by parts gives
$$\int k_{s^\ell} \partial_{s^j} \left( k\, k_{s^{\ell-j}} F\right) ds = \left[ k_{s^\ell} \partial_{s^{j-1}} \left( k\, k_{s^{\ell-j}} F\right) \right]_{\partial \gamma} - \int k_{s^{\ell+1}} \partial_{s^{j-1}} \left( k\, k_{s^{\ell-j}} F\right) ds \mbox{.}$$
If $\ell$ is odd, then the above boundary term is clearly equal to zero. If $\ell$ is even then we consider two cases. If $j$ is also even then the other factor in the boundary term has the form $\left( e\, e\, o\right)_o$, that by Lemma \ref{T:BD2} always contains an odd derivative of $k$ or of $F$. If $k$ is odd then the same factor instead has the form $\left( e\, e\, e\right)_o$ that by Lemma \ref{T:BD2} also always contains an odd derivative of $k$ or of $F$. We conclude by Lemma \ref{T:BCs} that the boundary term in all cases is equal to zero.
Integrating by parts a second time (for $j>1$) we have
$$\int k_{s^\ell} \partial_{s^j} \left( k\, k_{s^{\ell-j}} F\right) ds = - \left[ k_{s^\ell+1} \partial_{s^{j-2}} \left( k\, k_{s^{\ell-j}} F\right) \right]_{\partial \gamma} - \int k_{s^{\ell+2}} \partial_{s^{j-2}} \left( k\, k_{s^{\ell-j}} F\right) ds \mbox{.}$$
Here the boundary term is clearly equal to zero if $\ell$ is even, while if $\ell$, two cases depending on $j$ are handled as above.
Continuing like this we have for each $j=1, \ldots, \ell$,
\begin{multline} \label{E:intFin}
\int k_{s^\ell} \partial_{s^j} \left( k\, k_{s^{\ell-j}} F\right) ds = \left( -1\right)^j \int k_{s^{\ell+j}} k\, k_{s^{\ell-j}} F ds\\
= \left( -1\right)^j \int k_{s^{\ell+j}} k\, k_{s^{\ell-j}} \left[ \left( -1\right)^m k_{s^{2m+2}} - \sum_{i=1}^m k\, k_{s^{m+i}} k_{s^{m-i}} - \frac{1}{2} k\, k_{s^m}^2 \right] ds \mbox{.}
\end{multline}
Let us now deal with each of these terms separately, remembering that we want factors of the form $ \int P_4^{2m+2\ell+2}\left( k \right) ds$ with no higher derivative than $k_{s^{m+ \ell + 1}}$ and $\int P_6^{2m+2\ell} \left( k \right) ds$ with no higher derivative than $k_{s^{m+\ell}}$. The last term has the correct form for $j\leq m$ but if $j>m$ needs $m-j$ integrations by parts. In this case we have
$$\int k^2 k_{s^m}^2 k_{s^{\ell-j}} k_{s^{\ell+j}} ds = \left[ k^2 k_{s^m}^2 k_{s^{\ell-j}} k_{s^{\ell+j-1}} \right]_{\partial \gamma} - \int k_{s^{\ell + j-1}} \left( k^2 k_{s^m}^2 k_{s^{\ell-j}} \right)_s ds \mbox{;}$$
the boundary term clearly contains an odd derivative of $k$ so is equal to zero by Lemma \ref{T:BCs}. Integrating by parts again (if $j>m+1$),
$$\int k^2 k_{s^m}^2 k_{s^{\ell-j}} k_{s^{\ell+j}} ds = -\left[ \left( k^2 k_{s^m}^2 k_{s^{\ell-j}} \right)_s k_{s^{\ell+j-2}} \right]_{\partial \gamma} - \int k_{s^{\ell + j-2}} \left( k^2 k_{s^m}^2 k_{s^{\ell-j}} \right)_{ss} ds \mbox{.}$$
Now, if $\ell+j$ is odd then the boundary term is equal to zero via Lemma \ref{T:BCs}. If $\ell+j$ is even then so is $\ell-j$ and the other factor in the boundary term has the form $\left( \mbox{square } \times \mbox{ square }\times e \right)_o$. As we saw before, such terms always contain an odd derivative factor, so are equal to zero by Lemma \ref{T:BCs}. Integrating by parts once again (if $j>m+2$),
$$\int k^2 k_{s^m}^2 k_{s^{\ell-j}} k_{s^{\ell+j}} ds = \left[ \left( k^2 k_{s^m}^2 k_{s^{\ell-j}} \right)_{ss} k_{s^{\ell+j-3}} \right]_{\partial \gamma} - \int k_{s^{\ell + j-3}} \left( k^2 k_{s^m}^2 k_{s^{\ell-j}} \right)_{s^3} ds \mbox{.}$$
Here, if $\ell+j$ is even then the boundary term is again equal to zero. If $\ell+j$ is odd then the other factor in the boundary term has the form $\left( \mbox{square } \times \mbox{ square }\times o \right)_e$. Expanding out such a term using the binomial theorem, all terms with $e$ derivatives of $o$ are fine so we need only consider the terms with $o$ derivatives of $o$. For such terms one square factor has an $o$ derivative, so, as before, this will generate an $o$ and again the boundary term is equal to zero by Lemma \ref{T:BCs}. Continuing in this way, we can write for each integer $j\in \left( m, \ell\, \right]$,
$$\int k^2 k_{s^m}^2 k_{s^{\ell-j}} k_{s^{\ell+j}} ds = \left( -1\right)^{j-m} \int \left( k^2 k_{s^m}^2 k_{s^{\ell-j}} \right)_{s^{j-m}} k_{s^{\ell-\left( j-m\right)}} ds = \int P_6^{2m+2\ell}\left( k\right) ds \mbox{,}$$
with no higher derivative appearing than $k_{s^{\ell+m}}$ appearing on the right hand side.
Turning now to the first term of \eqref{E:intFin}, if $j\leq m+1$ then this term already has the form $\int P_4^{2m + 2\ell+2}\left( k\right) ds$ with no higher derivative than $k_{s^{m+\ell +1}}$ appearing. On the other hand,
if $j>m+1$ then we will need to perform $j-\left( m+1\right)$ integrations by parts. We have
$$\int k_{s^{\ell+j}} k\, k_{s^{\ell-j}} k_{s^{2m+2}} ds = \left[ k\, k_{s^{\ell-j}} k_{s^{2m+2}} k_{s^{\ell+j-1}} \right]_{\partial \gamma} - \int \left( k\, k_{s^{\ell-j}} k_{s^{2m+2}}\right)_s k_{s^{\ell+j-1}} ds \mbox{.}$$
The boundary term here clearly has an odd derivative of $k$ so is equal to zero by Lemma \ref{T:BCs}. Integrating by parts again, if necessary,
$$\int k_{s^{\ell+j}} k\, k_{s^{\ell-j}} k_{s^{2m+2}} ds = - \left[ \left( k\, k_{s^{\ell-j}} k_{s^{2m+2}}\right)_s k_{s^{\ell+j-2}} \right]_{\partial \gamma} + \int \left( k\, k_{s^{\ell-j}} k_{s^{2m+2}}\right)_{ss} k_{s^{\ell+j-2}} ds \mbox{.}$$
Here, if $\ell+j$ is odd then the boundary term is clearly equal to zero. If $\ell+j$ is even, then so is $\ell-j$ and the other factor in the boundary term has the form $\left( e\, e\, e\right)_o$; Lemma \ref{T:BD2} and Lemma \ref{T:BCs} then give that the boundary term is again equal to zero. Integrating by parts once more, if necessary,
$$\int k_{s^{\ell+j}} k\, k_{s^{\ell-j}} k_{s^{2m+2}} ds = \left[ \left( k\, k_{s^{\ell-j}} k_{s^{2m+2}}\right)_{ss} k_{s^{\ell+j-3}} \right]_{\partial \gamma} - \int \left( k\, k_{s^{\ell-j}} k_{s^{2m+2}}\right)_{s^3} k_{s^{\ell+j-3}} ds \mbox{.}$$
In this case the boundary term is clearly equal to zero if $\ell+j$ is even, while if $\ell +j $ is odd then the other factor has the form $\left( e\, o\, e\right)_e$; again Lemma \ref{T:BD2} and \ref{T:BCs} give that the boundary term is equal to zero. Continuing in this way, we have for each $j>m+1$,
$$\int k_{s^{\ell+j}} k\, k_{s^{\ell-j}} k_{s^{2m+2}} ds = \left( -1\right)^{j-\left( m+1\right)} \int \left( k\, k_{s^{\ell-j}} k_{s^{2m+2}}\right)_{s^{j-\left( m+1\right)}} k_{s^{\ell+m+1}} ds = \int P_4^{2m+2\ell+2}\left( k\right) ds \mbox{,}$$
where the highest derivative of $k$ that appears on the right is $k_{s^{m+\ell +1}}$.
Finally for each $j$ we look at the middle summation term of \eqref{E:intFin}. Since $i \leq m < \ell$, the $k_{s^{m+i}}$ factors are no problem. However, regardless of $i$, there will be a derivative factor of too high an order if $j>m$, so we need to do $j-m$ integrations by parts. We have
$$\int k^2 k_{s^{m-i}} k_{s^{m+i}} k_{s^{\ell-j}} k_{s^{\ell+j}} ds = \left[ k^2 k_{s^{m-i}} k_{s^{m+i}} k_{s^{\ell-j}} k_{s^{\ell+j-1}} \right]_{\partial \gamma} - \int \left( k^2 k_{s^{m-i}} k_{s^{m+i}} k_{s^{\ell-j}} \right)_s k_{s^{\ell+j-1}} ds \mbox{.}$$
Clearly the boundary term here is equal to zero. Integrating by parts a second time, if necessary, we have
\begin{multline*}
\int k^2 k_{s^{m-i}} k_{s^{m+i}} k_{s^{\ell-j}} k_{s^{\ell+j}} ds\\
= - \left[ \left( k^2 k_{s^{m-i}} k_{s^{m+i}} k_{s^{\ell-j}} \right)_s k_{s^{\ell+j-2}} \right]_{\partial \gamma} + \int \left( k^2 k_{s^{m-i}} k_{s^{m+i}} k_{s^{\ell-j}} \right)_{ss} k_{s^{\ell+j-2}} ds \mbox{.}
\end{multline*}
If $\ell +j$ is odd then the boundary term is clearly equal to zero. If $\ell + j$ is even then, as usual, we examine the other factor in the boundary term. In this case it has the form $\left( e\, e\, o\, o\, e\right)_s$ or $\left( e\, e\, e\, e\, e\right)_s$. More generally, with subsequent integrations by parts we are going to have $\left( e\, e\, o\, o\, e\right)_o$ or $\left( e\, e\, e\, e\, e\right)_o$. Working similarly as in the proof of Lemma \ref{T:BD2} shows that such factors always have an $o$ factor, so these boundary terms are always equal to zero. Another integration by parts, if necessary, gives
\begin{multline*}
\int k^2 k_{s^{m-i}} k_{s^{m+i}} k_{s^{\ell-j}} k_{s^{\ell+j}} ds\\
= \left[ \left( k^2 k_{s^{m-i}} k_{s^{m+i}} k_{s^{\ell-j}} \right)_{ss} k_{s^{\ell+j-3}} \right]_{\partial \gamma} - \int \left( k^2 k_{s^{m-i}} k_{s^{m+i}} k_{s^{\ell-j}} \right)_{s^3} k_{s^{\ell+j-3}} ds \mbox{.}
\end{multline*}
Similar arguments as before give that the boundary term is again equal to zero.
In general, we have for each $i$,
$$ \int k^2 k_{s^{m-i}} k_{s^{m+i}} k_{s^{\ell-j}} k_{s^{\ell+j}} ds = \left( -1\right)^{j-m} \int \left( k^2 k_{s^{m-i}} k_{s^{m+i}} k_{s^{\ell-j}} \right)_{s^{j-m}} k_{s^{\ell+m}} ds = \int P_6^{2m + 2\ell}\left( k\right) ds \mbox{,}$$
with no higher derivative than $k_{s^{m+\ell}}$ appearing on the right hand side.\\
Using all these results in \eqref{E:ksl2} we have for each $\ell>m$,
$$\frac{d}{dt} \int k_{s^\ell}^2 ds = -2 \int k_{s^{m+\ell +2}}^2 ds + \int P_4^{2m+2\ell+2}\left( k\right) ds + \int P_6^{2m+2\ell}\left( k\right) ds \mbox{,}$$
where no higher derivative that $k_{s^{m+\ell+1}}$ appears in the $\int P_4^{2m+2\ell+2}\left( k\right) ds$ term and no higher derivative than $k_{s^{m+\ell}}$ appears in $\int P_6^{2m+2\ell}\left( k\right) ds$. On the second term we use \eqref{E:P4} while for the last we estimate using Proposition \ref{T:int} to estimate
$$\int P_6^{2m + 2\ell}\left( k \right) ds \leq \varepsilon \int k_{s^{m+\ell+2}}^2 ds + c\left( \int k^2 ds\right)^{m+\ell+5} \mbox{,}$$
where to have the second statement of Proposition \ref{T:int}, it is necessary to use that $P_6^{2m+2\ell}\left( k\right)$ contains derivatives of $k$ of order \emph{at most} $m+\ell +1$ (which is bigger than $m+\ell$, the highest order that actually occurs). Therefore we obtain
$$\frac{d}{dt} \int k_{s^\ell}^2 ds \leq - \left( \frac{\pi^2}{L_0^2}\right)^{m+2} \int k_{s^\ell}^2 ds + c\left( \int k^2 ds\right)^{m + \ell +5}$$
from which the result follows since the second term above is bounded via Lemma \ref{T:PSW} and Lemma \ref{T:Length}. \hspace*{\fill}$\Box$
\noindent \textbf{Remark:} As with the polyharmonic curve diffusion, we can establish the similar curvature blow up rate in the case that the maximal existence time $T$ is finite. In this case,
$$\frac{d}{dt} \int k^2 ds = -2 \int k_{s^{m +2}}^2 ds + \int P_4^{2m+2}\left( k\right) ds + \int P_6^{2m}\left( k\right) ds \mbox{;}$$
estimating the $P$ terms as above we get
$$\frac{d}{dt} \int k^2 ds \leq c \left( \int k^2 ds\right)^{2m+5}$$
from which the blow up rate again follows.\\[8pt]
\begin{kexp} \label{T:kexp}
If the initial curve $\gamma_0$ has sufficiently small scale-invariant energy
$$L^{2m+1} \int k_{s^m}^2 ds \leq c\left( m\right)$$
then, under \eqref{E:theflow} with normal speed \eqref{E:speed}, $\int k^2 ds$ decays exponentially.
\end{kexp}
\noindent \textbf{Proof:} We have using Lemma \ref{T:evlneqns}, the boundary conditions and \eqref{E:speed}
\begin{equation} \label{E:k22}
\frac{d}{dt} \int k^2 ds = 2\int \left( k_{ss} + \frac{1}{2} k^3 \right) \left[ \left( -1\right)^{m+1} k_{s^{2m+2}} + \sum_{j=1}^m \left( -1\right)^{j+1} k\, k_{s^{m+j}} k_{s^{m-j}} - \frac{1}{2} k\, k_{s^m}^2 \right] ds \mbox{.}
\end{equation}
We will deal with each of these terms separately. For the first term in \eqref{E:k22} we have by $m$ integrations by parts and using Lemma \ref{T:BCs}
$$\int k_{ss} k_{s^{2m+2}} ds = \left( -1\right)^m \int k_{s^{m+2}}^2 ds \mbox{.}$$
The next summation of terms in \eqref{E:k22} is
$$\sum_{j=1}^m\left( -1\right)^{j+1} \int k_{ss} k\, k_{s^{m+j}} k_{s^{m-j}} ds = \int P_4^{2m+2}\left( k \right) ds \mbox{.}$$
We want to verify that this can be rewritten such that the highest order derivative of $k$ is $k_{s^{m+1}}$. For the $j$th term in the sum, this will require integrating by parts $j-1$ times. For each integration by parts, the boundary term always contains a factor of an odd derivative of $k$, so the boundary terms are always equal to zero. We obtain for each $j$
$$\int k_{ss} k\, k_{s^{m-j}} k_{s^{m+j}} ds = \left( -1\right)^{j-1} \int \left( k_{ss} k\, k_{s^{m-j}}\right)_{s^{j-1}} k_{s^{m+1}} ds = \int P_4^{2m+2}\left( k\right) ds \mbox{,}$$
where the highest order derivative of $k$ is $k_{s^{m+1}}$.
For the third term in \eqref{E:k22} we have
$$\int k_{ss} k\, k_{s^m}^2 ds = \int P_4^{2m+2}\left( k\right) ds$$
and the highest order derivative is $k_{s^m}$.
For the fourth term in \eqref{E:k22} we have similarly as the first by $m$ integrations by parts
$$\int k^3 k_{s^{2m+2}} ds = \left( -1\right)^m \int \left( k^3\right)_{s^{m+1}} k_{s^{m+1}} ds = \int P_4^{2m+2}\left( k\right) ds$$
with the highest order derivative of $k$ being $k_{s^{m+1}}$.
For the fifth summation term in \eqref{E:k22} we integrate the $j$th term by parts $j$ times observing again that every boundary term that arises contains an odd derivative of $k$ so is equal to zero. We obtain
$$\int k^4 k_{s^{m-j}} k_{s^{m+j}} ds = \left( -1\right)^j \int \left( k^4 k_{s^{m-j}} \right)_{s^j} k_{s^m} ds = \int P_6^{2m}\left( k\right) ds$$
with highest order derivative of $k$ begin $k_{s^m}$. The last term in \eqref{E:k22} already has the form $\int P_6^{2m}\left( k\right) ds$.
Therefore we have two kinds of term to estimate using Lemma \ref{T:int}. We first use again \eqref{E:P4} together with Lemma \ref{T:PSW}:
$$\int P_4^{2m+2}\left( k \right) ds \leq c\, L \int k^2 ds \int k_{s^{m+2}}^2 ds \leq c\, L^{2m+1} \int k_{s^m}^2 ds \int k_{s^{m+2}}^2 ds \mbox{.}$$
Using the same interpolation inequalities we also have
$$\int P_6^{2m}\left( k \right) ds \leq c \left( \int k^2 ds\right)^2 \int k_{s^{m+1}}^2 ds \leq c\, L^{4m+2} \left( \int k_{s^m}^2 ds \right)^2 \int k_{s^{m+2}}^2 ds \mbox{.}$$
Substituting all these into \eqref{E:k22} we obtain
$$\frac{d}{dt} \int k^2 ds \leq \left[ -2 + c\, L^{2m+1} \int k_{s^m}^2 ds + c\, L^{4m+2} \left( \int k_{s^m}^2 ds \right)^2 \right] \int k_{s^{m+2}}^2 ds \mbox{.}$$
From Lemma \ref{T:Length} we know for sufficiently small initial energy that $L$ does not increase. Also the energy itself is nonicreasing so if the above coefficient on the right hand side is initially less than $-\delta$ say, for some $\delta >0$, this will remain so and we have
$$\frac{d}{dt} \int k^2 ds \leq -\delta \int k_{s^{m+2}}^2 ds \leq -\tilde \delta\left( m, L_0\right) \int k^2 ds$$
hence the result.\hspace*{\fill}$\Box$
We may now prove similarly as for Corollary \ref{T:PHCDL3} exponential decay of curvature derivatives.
\begin{dexp} \label{T:dexp}
If the initial curve $\gamma_0$ has sufficiently small scale-invariant energy
$$L^{2m+1} \int k_{s^m}^2 ds \leq c\left( m\right)$$
then, under \eqref{E:theflow} with normal speed \eqref{E:speed}, $\int k_{s^{\ell}}^2 ds$ and $\left\| k_{s^\ell} \right\|_\infty$ decay exponentially for all $\ell$.
\end{dexp}
With these estimates in hand the exponential convergence of $\gamma$ to a unique horizontal straight segment now follows by exactly the same argument as in the previous section. This completes the proof of Theorem \ref{T:main2}. \hspace*{\fill}$\Box$\\[8pt]
\end{document}
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\subsection{Choosing the ``Multimodal" States, $X_{t,s}$}
\label{chooseKunimod}
Corollary \ref{ldssvec} gives a unimodality condition that needs to be verified separately for each particle and each $Y_t$ at each $t$. An exact algorithm to do this would be to begin by checking at each $t$, for each $i$, if Theorem \ref{unimodthm} holds with $K=0$. Keep increasing $K$ and doing this until find a $K$ for which Corollary \ref{ldssvec} holds conditioned on $X_{t,1:K}^i$. This can be done efficiently only if $\Delta^*$ can be computed analytically or using some efficient numerical techniques. That will be the focus of future research. But, as discussed earlier, PF-EIS works even if unimodality of $\pss(X_{t,r})$ holds for most particles at most times, i.e. it holds w.h.p.
We use the temperature tracking problem of Example \ref{temptrack} to explain how to choose $X_{t,s}$. For a given $K$, we would like to choose $X_{t,s}=v_{t,s}$ that makes it most likely for $\pss(v_{t,r})$ to be unimodal.
Given $X_{t-1}^i$, $v_{t,s}^i$, $v_{t,r}$ is a linear function of $C_t$. If $v_{t,r}$ were also a one-to-one function of $C_t$, then one could equivalently find conditions for unimodality of $\pss(C_{t})$, which is easier to analyze. For an approximate analysis, we make it a one-to-one function of $C_t$ by adding a very small variance (compared to that of any $\nu_{t,p}$) noise, $n_{t,s}$, along $B_s$, i.e. given $X_{t-1}^i,v_{t,s}^i$, set $C_t = C_{t-1}^i+B_s v_{t,s}^i + B_r v_{t,r} + B_s n_{t,s}$. Now, $C_t$ is a one-to-one and linear function of $[v_{t,r},n_{t,s}]$. This also makes $\pss(C_{t})$ a non-degenerate pdf.
First consider the case where w.h.p. OL can be multimodal as a function of temperature at only one node $p_0$, for e.g., $h_p(C_{t,p}) = C_{t,p}$, $\forall p \neq p_0$, $\alpha_p^j = 0$, $\forall p \neq p_0$, and either $\alpha_{p_0}^j > 0$ or $h_{p_0}(C_{t,p_0})$ is many-to-one. Then,
\bea
\pss(C_{t}) \se \overbrace{\zeta p(Y_{t,p_0}|C_{t,p_0}) p(C_{t,p_0}|X_{t-1}^i,v_{t,s}^i)}^{\pss(C_{t,p_0})} \times \nn \\
&& [\prod_{p \neq p_0} p(Y_{t,p}|C_{t,p})] p(C_{t}|C_{t,p_0},X_{t-1}^i,v_{t,s}^i) \ \ \
\label{pssct}
\eea
and the last two terms above are Gaussian (and hence strongly log-concave) as a function of $C_{t,p}, \ p \neq p_0$. If $\pss(C_{t,p_0})$ is also strongly log-concave then $\pss(C_{t})$ (and hence $\pss(v_{t,r})$) will be strongly log-concave, and hence unimodal. Now, $\pss(C_{t,p_0})$ will be strongly log-concave if $\exists \ \eps_0 >0$ such that $\Delta_{C,p_0} = Var[p(C_{t,p_0}|X_{t-1}^i,v_{t,s}^i)] < \inf_{\{ C_{t}: \nabla_{C_{t,p_0}}^2 E_{Y_t}(C_t) < 0 \}} \frac{1}{|\nabla_{C_{t,p_0}}^2 E_{Y_t}(C_t)|+\eps_0}$. This bound can only be computed on the fly. A-priori, $\pss(C_{t,p_0})$ will be most likely to be log-concave if $v_{t,s}$ is chosen to ensure that $\Delta_{C,p_0}$ is smallest.
Let $v_{t,s} = v_{t,k_0}$ where the set $k_0$ contains $K$ elements out of $[1,\dots M]$ and $K$ is fixed. Then, $\Delta_{C,p_0} = \sum_{k \notin k_0} B_{p_0,k}^2 \Delta_{\nu,k}$. This ignores the variance of $n_{t,s}$ (valid since the variance is assumed very small compared to all $\Delta_{\nu,p}$'s). Thus, $\Delta_{C,p_0}$ will be smallest if $v_{t,s}$ is chosen as
\bea
v_{t,s} = v_{t,k_s}, \ k_s \defn \arg \min_{k_0} \sum_{k \notin k_0} B_{p_0,j}^2 \Delta_{\nu,k}
\label{choosevts}
\eea
When $K=1$, this is equivalent to choosing $k_s = \arg \max_k B_{p_0,k}^2 \Delta_{\nu,k}$. Based on the above discussion, we have the following heuristics.
\begin{heuristic}
If OL can be multimodal as a function of temperature at only a single node, $p_0$, and is unimodal as a function of temperature at other nodes, select $v_{t,s}$ using (\ref{choosevts}).
\label{mm1}
\end{heuristic}
\begin{heuristic}
If OL is much more likely to be multimodal as a function of $C_{t,p_0}$, compared to temperature at any other node (e.g. if a sensor at $p_0$ is old so that its failure probability is much larger than the rest), apply Heuristic \ref{mm1} to that $p_0$.
\label{mmmost}
\end{heuristic}
\begin{heuristic}
When $p_0$ is a set (not a single index), Heuristic \ref{mm1} can be extended to select $k_s$ to minimize the spectral radius (maximum eigenvalue) of the matrix, $\sum_{k \notin k_0} B_{p_0,k}B_{p_0,k}^T \Delta_{\nu,k}$.
\label{mm2}
\end{heuristic}
\begin{heuristic}
If OL is equally likely to be multimodal as a function of any $C_{t,p}$ (e.g. if all sensors have equal failure probability), then $p_0 = [1, \dots M]$. Applying Heuristic \ref{mm2}, one would select the $K$ largest variance directions of STP as $v_{t,s}$.
\label{equal}
\end{heuristic}
\begin{heuristic}
If the probability of OL being multimodal is itself very small, then $K=0$ can be used. In Example \ref{temptrack} with all linear sensors, this probability is roughly $1 - \prod_{p,j} (1-\alpha_p^j)$.
\label{h2}
\end{heuristic}
\begin{heuristic}
For $J=2$ and all linear sensors, $p_0$ may be chosen on-the-fly as $\arg \max_p [(Y_{t,p}^{(1)}-Y_{t,p}^{(2)})^2/\sigma_{obs,p}^2]$ (larger the difference, the more likely it is for OL to be multimodal at that $p$). If the maximum itself is small, set $K=0$.
\label{onfly}
\end{heuristic}
We show an example now. Consider Example \ref{example1} with $\alpha^{(1)} = \alpha^{(2)} = [0.4,0.01,0.01]$, $\Sigma_\nu = diag([10,5,5])$, $B= [0.95, 0.21, 0.21]'; [-0.21,0.98,-0.05]'; [-0.22, 0, 0.98]'$ (using MATLAB notation). By Heuristic \ref{h2}, the probability of OL being multimodal is about 0.65 which is not small. So we choose $K>0$ ($K=1$). By Heuristic \ref{mmmost}, we choose $p_0=1$ since OL is multimodal as a function of
$C_{t,1}$ with probability 0.64, while that for $C_{t,2}$ or $C_{t,3}$ together is $0.02$ (much smaller).
Applying (\ref{choosevts}) for $p_0=1$, we get $v_{t,s} = v_{t,1}$.
| 106,699
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Matzek Taking Control Of Chance July 16, 2014 by Jack Etkin DENVER—Lefthander Tyler Matzek took advantage of an unexpected opportunity and made his major league debut on June 11. Largely because of injuries, the Rockies have been forced to use 13 […]... Click here to log in and read the full article.
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Tips. Attaching your key fob to a bigger item like your purse or a wallet makes it more difficult to lose. If you carry a laptop or briefcase, consider putting your keys in the bag. Just walk up to your car, wave your hand over the unlock button and open the door—your keys don't have to leave your bag until you're in the car.
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Nitty Gritty WAV-DiSCOVER25 February 2016
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\begin{document}
\newtheorem{theo}{Theorem}[section]
\newtheorem{definition}[theo]{Definition}
\newtheorem{lem}[theo]{Lemma}
\newtheorem{prop}[theo]{Proposition}
\newtheorem{coro}[theo]{Corollary}
\newtheorem{exam}[theo]{Example}
\newtheorem{rema}[theo]{Remark}
\newtheorem{example}[theo]{Example}
\newtheorem{principle}[theo]{Principle}
\newcommand{\ninv}{\mathord{\sim}}
\newtheorem{axiom}[theo]{Axiom}
\title{A Quantum Version of The Spectral Decomposition Theorem of Dynamical Systems, Quantum Chaos Hierarchy: Ergodic, Mixing and Exact}
\author{{\sc Ignacio G\'{o}mez}$^1$ {\sc and} \ {\sc Mario Castagnino}$^2$}
\maketitle
\begin{small}
\begin{center}
1- Instituto de F\'{\i}sica de Rosario (IFIR-CONICET), Rosario, Argentina\\
2- Instituto de F\'{\i}sica de Rosario (IFIR-CONICET) and \\
Instituto de Astronom\'{\i}a y F\'{\i}sica del Espacio, \\
Casilla de Correos 67, Sucursal 28, 1428 Buenos Aires, Argentina.\\
\end{center}
\end{small}
\vspace{1cm}
\begin{abstract}
In this paper we study the Spectral Decomposition Theorem \cite{LM}
and translate it to the quantum language through the Wigner
Transform. We obtain a Quantum Version of The Spectral Decomposition
Theorem that allows us to three things: First, to rank the levels of
the \emph{Quantum Ergodic Hierarchy} \cite{0}. Second, to analyze
the classical limit in quantum ergodic systems and quantum mixing
systems. And third, and maybe the most important feature, to find a
relevant and simple connection between the first three levels of the
quantum ergodic hierarchy (ergodic, exact and mixing) and the type
of spectrum of the quantum systems in the classical limit.
\end{abstract}
\begin{small}
\centerline{\em Key words: QSDT-Quantum Ergodic Hierarchy-classical
limit-weak limit-Wigner transform}
\end{small}
\bigskip
\noindent
\bibliography{pom}
\section{Introduction}
As in previous works \cite{0},\cite{FOP},\cite{NACHOSKY MARIO} in
this paper we study the quantum chaos from the view point of the
Berry's definition: ``\emph{a quantum system is chaotic if its
classical limit exhibits chaos}". This particular approach of
quantum chaos deviates from the standard approaches such as the
complexity (see \cite{MANTICA1}, \cite{MANTICA2}), the exponential
divergence trajectories (see \cite{SCHUSTER}, \cite{BATTERMAN}), the
threatment of chaos based on the introduction of non-linear terms in
the Schrodinger equation (see \cite{WEINBERG}) and the non-unitary
evolution of a quantum system as an indicator of quantum chaos (see
\cite{GHIRARDI}). However, as mentioned in (\cite{0} pag. 247, 248)
the study of chaos based on the quantum ergodic hierarchy takes into
account the real conflict that the classical limit of a system does
not exhibit a chaotic behavior, which would be a threat to the
correspondence principle. The actual quantum version of the Spectral
Decomposition Theorem gives a direct connection with the
\emph{Quantum Ergodic Hierarchy} \cite{0} and the classical limit,
and also the degree of generality of it presents could be used as an
attempt towards a general theory of quantum chaos. All these aspects
in accordance with the conceptual foundations in the context of
Belot-Earman program's \cite{BELOT}. The main goal of this paper is
to obtain the first two levels of the quantum ergodic hierarchy from
a Quantum Version of the Spectral Decomposition Theorem \cite{LM}
and establish a simple connection between these ergodic levels and
the type of spectrum that presents a quantum closed system in the
classical limit. In section 2 we begin introducing a brief review of
the minimal notions of the theory of densities necessary for the
development of the following sections.
\section{Theory of densities and Markov Operators}
Historically, the concept of density has only recently appeared in
order to unify the descriptions of the phenomena of stadistical
nature. Clear examples of this were the Maxwell velocity
distribution and the quantum mechanics both as attempts to unify the
theory of gases and to justify the derivation of the Planck
distribution of radiation from the black body respectively. Further
development of modern physics demonstrated the usefulness of the
densities for the description of large systems with a large number
of degrees of freedom which have an uncertainty by ignorance. In
this section we introduce a brief review of the concepts of the
theory of densities and the Markov Operators based on the formalism
of dynamical systems \cite{LM}.
\subsection{Densities functions and Dynamical Systems}
We start by recalling the fundamental mathematical elements of the
theory of dynamical systems (\cite{3M},\cite{LM}). Given a set $X$,
$\Sigma$ is a $\sigma$\emph{-algebra} of subsets of $X$ if it
satisfies:
\begin{enumerate}
\item[$(I)$] $X \in \Sigma$
\item[$(II)$] $A,B \in \Sigma \Longrightarrow A\backslash B \in \Sigma$
\item[$(III)$] $(B_i)\in \Sigma\Longrightarrow \cup_{i}B_i \in \Sigma$
\end{enumerate}
\nd A function $\mu$ on $\Sigma$ is a \emph{probability measure} if
it satisfies:
\begin{enumerate}
\item[$(I)$] $\mu:\Sigma \rightarrow [0,1]$ and $\mu(X)=1$
\item[$(II)$] For all family of pairwise disjoint subsets $(B_i)\in \Sigma \Longrightarrow \mu(\cup_{i}B_i )= \sum_{i}\mu(B_i
)$
\end{enumerate}
\nd A \emph{measure space} is any terna of the form
$(X,\Sigma,\mu)$. Given a measure space $(X,\Sigma,\mu)$, a
\emph{measurable preserving transformation or automorphism} $T$ is a
biyective function $T:X\rightarrow X$ which satisfies:
\begin{equation}\label{automorphism}
\forall A\in \Sigma: \mu(T^{-1}A)=\mu(A)
\end{equation}
\nd We say then that the family of automorphisms
$\tau:=\{T_t\}_{t\in I}$ is a group of measure-preserving
automorphisms and we call it a \emph{dynamical law} $\tau$. With
these definitions, we say that the quaternary $(X,\Sigma,\mu,\tau)$
is a \emph{dynamical system} The central notion of the dynamic
system concept is the definition of density: given a dynamical
system $(X,\Sigma,\mu,\tau)$ and $D(X,\Sigma ,\mu )=\{f\in
L^{1}(X,\Sigma ,\mu ):f\geq 0\,\ ;\,\ \Vert f\Vert =1\}$ then any
function $f \in D(X,\Sigma,\mu,\tau)$ is called a \emph{density}.
For a dynamical system $S$ in the context of classical mechanics it
is usual to take $X=\mathcal{M}$ the phase space,
$\Sigma=\mathcal{P}(\mathcal{M})$ the subsets of the phase space,
$\mu$ the Lebesgue measure and $T_t$ the temporal evolution. This
context will be clarified in the next sections. Besides the concept
of dynamical system, the other fundamental component in order to
describe the temporal evolution of a classical density is the notion
of Markov operator. This important class of operators is presented
below.
\subsection{Markov Operators}
Given a classical system S with an initial state given by a density
$f_0$ we know that its temporal evolution will be determined by the
Liouville equation. Except in simple cases we know that this
equation has no exact solution and therefore we are forced to use
another strategy to study the evolution of the system. In this
context Markov operator are very useful because their properties
allow us to know the asymptotic behavior of the densities and
general behavior of the densities can be well developed in both
dynamical systems and stochastic systems. Markov operators contain
global information of the densities when $t\rightarrow \infty$ and
under certain hypotheses on these gives certain conditions for the
existence of an equilibrium density $f_\ast$ which physically
corresponds to the arrival of the system at the equilibrium. This
approach at the equilibrium when $t\rightarrow \infty$ through the
global properties of Markov operators will be the connection between
the classical limit and the \emph{Quantum Spectral Decomposition
Theorem}. This will be considered in the section 4. We present a
brief review of the concepts necessary for the development of the
present paper beginning with the following definition (see
\cite{LM}, pag. 32).
\begin{definition}(\emph{Markov Operator})\label{markov operator}
Given a measure space $(X,\Sigma,\mu)$, a linear operator
$P:L^{1}\rightarrow L^{1} $ is called a \emph{Markov operator} if it
satisfies:
\begin{enumerate}
\item[$(a)$] $Pf \geq 0$
\item[$(b)$] $\|Pf\| = \|f\|$
for all $f\in L^{1} $, $f\geq 0$
\end{enumerate}
\end{definition}
\nd From the condition (b) of the definition of Markov operator
follows that $P$ is \emph{monotonic}, that is if $f,g \in L^{1}$
with $f \geq g$ then $Pf \geq Pg$.
Markov operators satisfy the following important properties that
will be crucial in order to obtain the quantum version of the
spectral decomposition theorem (see \cite{LM}, pag. 33):
\begin{theorem}\label{markov properties}
Let $(X,\Sigma,\mu)$ be a $\sigma$-algebra and let $f \in L^{1}$. If
P is a Markov operator then:
\begin{enumerate}
\item[$(I)$] $\|Pf\| \leq \|f\|$ (\emph{contractive} property)
\item[$(II)$] $|Pf(x)| \geq P|f(x)|$
\end{enumerate}
\end{theorem}
\nd The following concept of a fixed point of a Markov operator $P$
is crucial for establishing the arrival of a density $f$ at the
equilibrium (see \cite{LM}, pag. 35).
\begin{definition}(\emph{Fixed Point})\label{fixed point}
Let $P$ be a Markov operator. If $f \in L^{1} $ with $Pf=f$ then $f$
is called a \emph{fixed point} of P. In a more general way, any $f
\in D(X,\Sigma ,\mu)$ that satisfies $Pf = f$ is called a
\emph{stationary density} of P.
\end{definition}
A family of automorphisms $\{T_t\}_{t\in I}$ which represent the
temporal evolution of any dynamical system are a special class of
Markov operators called \emph{Frobenius-Perron operators}. They are
defined as follows (see \cite{LM}, pag. 36).
\begin{definition}(\emph{Frobenius-Perron Operator})\label{perron operator}
Given a measure space $(X,\Sigma,\mu)$ and $T:X\rightarrow X $ a
\emph{non singular automorphism} (e.g. $\mu(T^{-1}(A))=0$ for all $A
\in \Sigma$ such that $\mu(A)=0$) the unique operator
$P:L^{1}\rightarrow L^{1}$ defined for all $A \in \Sigma$ by the
equation
\begin{equation}
\int_{A}Pf(x)\mu(dx)=\int_{T^{-1}(A)}f(x)\mu(dx)
\end{equation}
is called the \emph{Frobenius-Perron operator} corresponding to T.
\end{definition}
From the equation (2) it follows that the Frobenius-Perron operator
is lineal and satisfies the following properties (see \cite{LM},
pag. 37):
\begin{theorem}\label{perron properties}
Let T be an automorphism. If $P$ and $P_n$ are the Frobenius-Perron
operators corresponding to $T$ and $T^{n}$ respectively. Then we
have that
\begin{enumerate}
\item[$(I)$] $\int_{X}Pf(x)\mu(dx)=\int_{X}f(x)\mu(dx)$
\item[$(II)$] $P_n=P^{n}$
\end{enumerate}
\end{theorem}
\nd The adjoint operator to the Frobenius-Perron operator is defined
as follows (see \cite{LM}, pag. 42).
\begin{definition}(\emph{Koopman Operator})\label{perron operator}
Given a measure space $(X,\Sigma,\mu)$ and $T:X\rightarrow X $ a
\emph{non singular automorphism}, the unique operator
$U:L^{\infty}\rightarrow L^{\infty}$ defined for all $f \in
L^{\infty}$ by the equation
\begin{equation}\label{koopman}
Uf(x)=f(T(x))
\end{equation}
which is called the \emph{Koopman operator} corresponding to T.
\end{definition}
From the equation (3) it follows that the Koopman operator is
lineal, and satisfies the following properties(see \cite{LM}, pag.
43):
\begin{theorem}\label{Koopman properties}
Let T be an automorphism. If $U$ and $U_n$ are the Koopman operators
corresponding to $T$ and $T^{n}$ respectively. Then
\begin{enumerate}
\item[$(I)$] $\|Uf\|_{L^{\infty}}\leq \|f\|_{L^{\infty}}$
\item[$(II)$] $U_n=U^{n}$
\item[$(III)$] $\langle P_n f, g\rangle = \langle f, U_n g\rangle$ \,\,\,\,\,\ for every $f \in L^{1}, g \in L^{\infty}$, $n \in \mathbb{N}_0$
\end{enumerate}
where $\langle f,g\rangle = \int_{X} f(x)g(x)d(\mu x)$ for all $f
\in L^{1}, g \in L^{\infty}$.
\end{theorem}
\nd We now recall the \emph{Ergodic Hierarchy} for dynamical systems
(see \cite{LM}, pag. 68):
\begin{theorem}(\emph{Ergodic, Mixing and Exact})\label{EH}
Let $(X,\Sigma,\mu)$ be a normalized measure space, $T:X \rightarrow
X$ an automorphism and $P$, $U$ the Frobenius-Perron and Koopman
operators corresponding to $T$. Then:
\begin{enumerate}
\item[$(a)$] $T$ is \emph{ergodic} $\Leftrightarrow$ $lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=0}^{n}\langle P^{k} f, g\rangle
= lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=0}^{n} \langle f, U_k
g\rangle = \langle f,1\rangle \langle 1,g \rangle$
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ for all $f \in L^{1}, g \in
L^{\infty}$
\item[$(b)$] $T$ is \emph{mixing} $\Leftrightarrow$ $lim_{n\rightarrow \infty} \langle P^{n} f, g\rangle
= lim_{n\rightarrow \infty} \langle f, U_n g\rangle = \langle
f,1\rangle \langle 1,g \rangle$
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\
for all $f \in L^{1}, g \in L^{\infty}$
\item[$(b)$] $T$ is \emph{exact} $\Leftrightarrow lim_{n\rightarrow \infty} \| \langle P^{n} f - \langle f,1\rangle \|= 0$ \,\,\,\,\,\ for all $f \in
L^{1}$
\end{enumerate}
\end{theorem}
\nd From these definitions it follows that
\begin{equation}
MIXING \,\,\,\ \subseteq \,\,\,\ EXACT \,\,\,\ \subseteq \,\,\,\
ERGODIC
\end{equation}
\nd these are the inclusions of the Ergodic Hierarchy levels, and
these inclusions are strict.
To complete this section we introduce the notion of constrictive
operator allowing us to guarantee the existence of an equilibrium
density (see \cite{LM}, pag. 87).
\begin{definition}(\emph{constrictive operator})\label{constrictive}
A Markov operator P will be called \emph{constrictive} if there
exist a precompact set $\mathcal{F}\subseteq L^{1}$ such that for
all $f \in D(X,\Sigma ,\mu)$:
\begin{equation}
lim_{n\rightarrow \infty} d(P^{n}f,\mathcal{F}) = lim_{n\rightarrow
\infty} inf _{g \in \mathcal{F}}\|P^{n}f - g \| = 0
\end{equation}
\end{definition}
\nd An important result is that every Markov constrictive operator
has an equilibrium density (see \cite{LM}, pag. 87).
\begin{theorem}\label{constrictive MO}
Let $(X,\Sigma,\mu)$ be a normalized measure space, and
$P:L^{1}\rightarrow L^{1}$ a constrictive Markov operator. Then P
has a stationary density, i.e there is a $f_{\ast}\in L^{1}$ such
that $Pf_{\ast} = f_{\ast}$.
\end{theorem}
The existence of a equilibrium density $f_{\ast}$ is fundamental in
order to obtain a framework for quantum chaos and it can be
translated to quantum language through the Wigner transform in a way
that the equilibrium state of any quantum closed system of
continuous spectrum is represented by the weak limit $\rho_{\ast}$
(see \cite{MARIO OLIMPIA}, pag. 889 eq. (3.28)).
\section{The Spectral Decomposition Theorem of Dynamical Systems}
With all the mathematical background of the previous sections we are
able to present one of the main results of the theory of dynamical
systems which is called the \emph{Spectral Decomposition Theorem}
(see \cite{LM}, pag. 88):
\begin{theorem}(\emph{The Spectral Decomposition Theorem}\emph{(version I)})\label{SDT2}
Let P be a Markov constrictive operator. Then there is an integer r,
two sequences of nonnegative functions $g_i \in D(X,\Sigma ,\mu)$,
$k_i \in L^{\infty}$, $i=1,...,r$, and an operator
$Q:L^{1}\rightarrow L^{1}$ such that for all $f \in L^{1}$, $Pf$ may
be written as
\begin{equation}\label{SDT1 decomposition}
Pf(x)=\sum_{i=1}^{r}\lambda_i(f)g_i(x) + Qf(x)
\end{equation}
\nd where
\begin{equation}\label{lambda}
\lambda_i(f)= \int_{X} f(x)k_i(x)\mu(dx)=\langle f(x),k_i(x)\rangle
\end{equation}
\nd The functions $g_i$ and the operator $Q$ have the following
properties:
\begin{enumerate}
\item[$(I)$] $g_i(x)g_j(x)=0$ for all $i\neq j$, so that functions $g_i$ have disjoint supports.
\item[$(II)$] For each integer $i$ exists a unique integer $\alpha (i)$ such that $Pg_i=g_{\alpha (i)}$. Further $\alpha (i) \neq \alpha (j)$ for $i \neq j$ and thus the operator $P$ just allows to permute the functions $g_i$.
\item[$(III)$] $\|P^{n}Qf\| \rightarrow 0$ \,\,\, as \,\,\, $n \rightarrow \infty$ for every $f \in L^{1}$.
\end{enumerate}
\end{theorem}
Basically, this theorem describes the temporal evolution of any
density in a simple manner with an initial term which oscillates
between each of the densities $g_i$ under the assumption that this
evolution has a constrictive Perron-Frobenius operator. And with a
remaining term $Qf$ goes to zero, which is the expression of a
process of relaxation to equilibrium which we will analize in the
section 5. By the property (2) of the theorem, it follows that
\begin{equation}
P^{n}f(x)=\sum_{i=1}^{r}\lambda_i(f)g_{\alpha^{n}(i)}(x) +
P^{n-1}Qf(x)=\sum_{i=1}^{r}\lambda_{\alpha^{-n}(i)}(f)g_i(x) +
Q_{n}f(x)
\end{equation}
\nd where $Q_{n}f(x)=P^{n-1}Qf(x)$ and $\{\alpha^{-n}(i)\}$ is the
inverse permutation of $\{\alpha^{n}(i)\}$.
In the case that the measurable space is normalized and the Markov
operator P has a constant stationary density $f_{\ast}$ (e.g. if P
is an Frobenius-Perron operator this is equivalent to $\mu_f$ being
invariant (see \cite{LM}, pag. 46) the Spectral Decomposition
Theorem takes the following compact form (see \cite{LM}, pag. 90):
\begin{theorem}(\emph{The Spectral Decomposition Theorem}\emph{(version II)})\label{SDT}
Let $(X,\Sigma ,\mu)$ be a normalized measure space and
$P:L^{1}\rightarrow L^{1}$ a constrictive Markov operator. If P has
an stationary density, then the representation of $P^{n}f$ takes the
simple form for all $f \in L^{1}$
\begin{equation}\label{SDT2 decomposition}
P^{n}f(x)=\sum_{i=1}^{r}\lambda_{\alpha^{-n}(i)}(f)\overline{1}_{A_i}(x)
+ Q_{n}f(x)
\end{equation}
\nd where
\begin{equation}\label{characteristic1}
\overline{1}_{A_i}(x)=[1/\mu(A_i)]1_{A_i}
\end{equation}
\begin{equation}\label{characteristic1bis}
\bigcup_{i}A_i=X \,\,\,\ with \,\,\,\ A_i \cap A_j = \emptyset
\,\,\,\ for \,\,\,\ i\neq j
\end{equation}
\nd and
\begin{equation}
\mu(A_i) = \mu(A_j) \,\,\,\ if \,\,\,\ j=\alpha^{n}(i) \,\,\,\ for
\,\,\,\ some\,\,\,\ n.
\end{equation}
\end{theorem}
\nd The last version of the Spectral Decomposition Theorem allows to
characterize the Ergodic Hierarchy through the permutation
$\{\alpha^{n}(i)\}$ (see \cite{LM}, pag. 92,93 and 94).
\begin{theorem}(\emph{The Spectral Decomposition Theorem and The Ergodic Hierarchy})\label{SDT and EH}
Let $(X,\Sigma ,\mu)$ be a normalized measure space and
$P:L^{1}\rightarrow L^{1}$ a constrictive Markov operator. Then
\begin{enumerate}
\item[$(I)$] $P$ is \emph{ergodic} $\Longleftrightarrow$ the permutation $\{\alpha(1),...,\alpha(r)\}$ of the secuence $\{1,...,r\}$ is \emph{cyclical} (that is, for which there no is invariant subset).
\item[$(II)$] If $r=1$ in the representation of equation $(6)$ $\Longrightarrow$ $P$ is \emph{exact}.
\item[$(III)$] If $P$ is \emph{mixing} $\Longrightarrow$ $r=1$ in the representation of equation $(6)$.
\end{enumerate}
\end{theorem}
The version 2 of the \emph{Spectral Decomposition Theorem} says that
if the classical Hamiltonian is such that Frobenius-Perron operator
$P$ associated to the temporal evolution $T$ admits an equilibrium
density $f_{\ast}$ then for very long times ($n \rightarrow \infty$)
the state of the system $U_nf(x)=f(T_n(x))$ will oscillate between
the characteristic functions $\overline{1}_{A_i}(x)$ with a remanent
term $Q_{n}f(x)$ going to zero. In the next section we will see that
this decomposition in two terms contains relevant information with
each term associated to the type of spectrum. We have seen that the
hypothesis of constrictiveness of the Markov operator $P$ and
normalization of the measure space are sufficient to ensure the
existence of a stationary density and to obtain a representation of
the temporal evolution of any density through The Spectral
Decomposition Theorem.
\section{The Quantum Version of The Spectral Decomposition Theorem (QSDT)}
The aim of this paper is to obtain a quantum version of \emph{The
Spectral Decomposition Theorem} that can be useful to study quantum
systems in the classical limit and can give a general framework to a
theory describing the quantum chaos according to the Berry's
definition. In this section we begin by defining the mathematical
elements necessary for this purpose using the observables as central
object of quantum treatment and functional states as living in the
dual space of those. Let $\widehat{\mathcal{A}}$ be the
characteristic algebra of the quantum system, that is,
$\widehat{\mathcal{A}}$ is an algebra whose self-adjoint elements
$\hat{O}=\hat{O}^{\dag}$ are the observables belonging to the space
$\mathcal{O}$. The space of states is the positive cone
\begin{equation}
\mathcal{N}=\{\hat{\rho} \in
\mathcal{O}^{\prime}:\hat{\rho}(\mathbb{I})=1, \,\,\
\hat{\rho}^{\dag}=\hat{\rho}, \,\,\
\hat{\rho}(\hat{a}.\hat{a}^{\dag})\geq 0 \,\,\ for \,\,\ all \,\,\
\hat{a} \in \mathcal{O}\}
\end{equation}
\nd where the action $\hat{\rho}(\hat{O})$ of the functional
$\hat{\rho} \in \mathcal{O}^{\prime}$ on the observable $\hat{O} \in
\mathcal{O}$ is denoted by $(\hat{\rho}|\hat{O})$, and in the case
that $\hat{O}=\mathbb{I}$ this action is the \emph{trace} of
$\hat{\rho}$ equal to
$tr(\hat{\rho})=\hat{\rho}(\mathbb{I})=(\hat{\rho}|\mathbb{I})=1$.
In this aproach the state $\hat{\rho}$ is unknown and we study the
expectation values $(\hat{\rho}(t)|\hat{O})$ when $t \rightarrow
\infty$. If there exists a unique $\hat{\rho}_{\ast} \in
\mathcal{O}^{\prime}$ for all $\hat{\rho} \in \mathcal{O}^{\prime}$
such that
\begin{equation}
lim_{t \rightarrow \infty} (\hat{\rho}(t)|\hat{O}) = lim_{t
\rightarrow \infty} (\hat{U}_t \hat{\rho}\hat{U}_t^{\dag}|\hat{O}) =
(\hat{\rho}_{\ast}|\hat{O})
\end{equation}
\nd we say that the evolution $\hat{U_t}$ has \emph{weak-limit}
$\hat{\rho}_{\ast}$ (see \cite{0} pag. 248). This functional
$\hat{\rho}_{\ast}$ is interpreted as the average value that would
result if the state $\hat{\rho}(t)$ had a limit $\hat{\rho}_{\ast}$
for large times, i.e it is a weak limit and it is not a limit in the
(strong) usual sense. In other words, $\hat{\rho}_{\ast}$ is the
equilibrium state in the weak sense that the system reaches in the
relaxation.
In this framework the fundamental element to realize the
\emph{classical limit} of a quantum system, which mathematically
consists in to obtain a classical algebra from a quantum algebra via
an algebra ``deformation" in the limit $\hbar \rightarrow 0$, is the
\emph{Wigner transform}. This transform allows us to obtain a
function $f(\phi)$ defined over the phase space $\Gamma$ from a
quantum state $\hat{\rho}$ where the function $f(\phi)$ can be
interpreted such a distribution probability analogous to the
statistic mechanics density $\rho(p,q)$ governed by the Liouville
equation. Now we present a brief review of the more important Wigner
transform properties.
Let $\Gamma=\mathcal{M}_{2(N+1)}\equiv \mathbb{R}^{2(N+1)}$ be the
phase space. The Wigner transformation
$symb:\widehat{\mathcal{A}}\rightarrow \mathcal{A}_q$ that sends the
quantum algebra $\widehat{\mathcal{A}}$ to ``classical like"
$\mathcal{A}_q$ algebra is given by (see \cite{HILLERY},
\cite{DITO}, \cite{GADELLA}).
\begin{equation}
symb(\hat{f})=f(\phi)=\int \langle q+\Delta|\hat{f}|q-\Delta\rangle
e^{i\frac{p\Delta}{\hbar}}d^{\Delta+1}
\end{equation}
\nd where $f(\phi)\in \mathcal{A}_q$ are the functions over the
space $\Gamma$ phase with coordinates
$\phi=(q^{1},...,q^{N+1},p_{q}^{1},...,p_{q}^{N+1})$.
The star product between two operators
$\hat{f},\widehat{g}\in\hat{\mathcal{A}}$ is given by (see [39])
\begin{equation}\label{star product}
symb(\hat{f}.\hat{g})=symb(\hat{f})\ast symb(\hat{g})=(f\ast
g)(\phi)
\end{equation}
\nd and the \emph{Moyal bracket} is
\begin{equation}\label{moyal}
\{f,g\}_{mb}=\frac{1}{i\hbar}(symb(\hat{f})\ast
symb(\hat{g})-symb(\hat{g})\ast
symb(\hat{f}))=symb(\frac{1}{i\hbar}[\hat{f},\hat{g}])
\end{equation}
\nd Two important properties are (see [36])
\begin{equation}
(f\ast g)(\phi)=f(\phi)g(\phi)+0(\hbar) \,\,\,\,\,\,\, ,
\,\,\,\,\,\,\, \{f,g\}_{mb}=\{f,g\}_{pb}+0(\hbar^{2})
\end{equation}
\nd The \emph{symmetrical} or Weyl ordering prescription is used to
define the inverse \emph{symb}$^{-1}$, that is
\begin{equation}
symb^{-1}[q^{i}(\phi),p^{j}(\phi)]=\frac{1}{2}(\hat{q}^{i}\hat{p}^{j}+\hat{p}^{j}\hat{q}^{i})
\end{equation}
\nd Therefore, with \emph{symb} and \emph{symb}$^{-1}$ is defined an
isomorphism between the algebras $\widehat{\mathcal{A}}$ and
$\mathcal{A}_q$,
\begin{equation}
symb:\widehat{\mathcal{A}}\rightarrow \mathcal{A}_q \,\,\,\,\,\,\, ,
\,\,\,\,\,\,\, symb^{-1}:\mathcal{A}_q\rightarrow
\widehat{\mathcal{A}}
\end{equation}
\nd On the other hand the Wigner transformation for states is
\begin{equation}
\rho(\phi)=(2\pi\hbar)^{-(N+1)}symb(\hat{\rho})
\end{equation}
\nd and the fundamental property of the Wigner transformation used
in this work is the preservation of the inner product between states
$\hat{\rho}\in \mathcal{N}$ and observables
$\hat{O}\in\widehat{\mathcal{A}}$ which physically represents the
preservation of the expectation values that gives the same result to
be calculated in both $\widehat{\mathcal{A}}$ and $\mathcal{A}_q$,
that is,
\begin{equation}\label{wigner}
\hat{\rho}(\hat{O})=\langle
\hat{O}\rangle_{\hat{\rho}}=(\hat{\rho}|\hat{O})=(symb(\hat{\rho})|symb(\hat{O}))=\langle\rho(\phi),O(\phi)\rangle=\int
d\phi^{2(N+1)}\rho(\phi)O(\phi)
\end{equation}
\subsection{The Quantum Spectral Decomposition Theorem (QSDT)}
In the previous section we have establish the framework based on the
classical limit and the Wigner transform. Now we can write the
spectral decomposition theorem in quantum language. First, we assume
that
\begin{itemize}
\item In the classical limit $\hbar\rightarrow0$ the quantum system
has a classical evolution $T$\footnote{Of course, we make the
natural choice of $T$ as $T:\Gamma\rightarrow \Gamma$ with
$T(\phi=(q,p))=\phi(1)=(q(1),p(1))$, that is, $T$ is the classical
evolution operator determined by the Hamilton equations.} defined
over the phase space $\Gamma$ with an associated Frobenius-Perron
operator $P$ constrictive.
\item There exists a stationary density $f_{\ast}$, that is,
$Pf_{\ast}=f_{\ast}$.
\end{itemize}
Under these hypothesis we have the following quantum version of the
Spectral Decomposition Theorem.
\begin{theorem}(\emph{The Quantum Spectral Decomposition Theorem}\emph{(QSDT)})\label{QSDT}
Let $\hat{\rho} \in \mathcal{N}$ and let $\hat{O}$ be an observable.
Then there exists pure states
$\hat{\rho}_1,\hat{\rho}_2,...,\hat{\rho}_r$; observables
$\hat{O}_1,\hat{O}_2,...,\hat{O}_r$ ; a permutation
$\alpha:\{1,...,r\}\longrightarrow \{1,...,r\}$ and
$\widetilde{\rho}_0 \in \mathcal{O}^{\prime}$ such that
\begin{equation}\label{QSDT1}
(\hat{\rho}(n)|\hat{O})=\sum_{i=1}^{r}\lambda_{\alpha^{-n}(i)}(\hat{\rho}_i|\hat{O})
+ (\widetilde{\rho}_0(n-1)|\hat{O})
\end{equation}
\nd where
\begin{equation}\label{QSDT2}
\lambda_i(\hat{\rho})= (\hat{\rho}|\hat{O}_i)
\end{equation}
\nd The states $\hat{\rho}_i$ and $\widetilde{\rho}_0$ have the
following properties:
\begin{enumerate}
\item[$(I)$] $\hat{\rho}_i\hat{\rho}_j=0(\hbar)$ for all $i\neq j$ and $\hat{\rho}_i^{2}=\hat{\rho}_i+0(\hbar)$. So that the states $\hat{\rho}_i$ are projectors in the classical limit $(\hbar\rightarrow
0)$. Moreover, we have a descomposition of the identity:
\begin{equation}\label{QSDT3}
\hat{1} = \sum_{i}\alpha_i\hat{\rho}_i \,\,\,\,\,\,\,\ with
\,\,\,\,\,\,\,\ \alpha_i\geq0 \,\,\,,\,\,\sum_{i}\alpha_i=1
\end{equation}
\item[$(II)$] For each integer $i$ exists a unique integer $\alpha (i)$ such that $(\hat{U}\hat{\rho}_i\hat{U}^{\dag}|\hat{O})=(\hat{\rho}_{\alpha (i)}|\hat{O})$. Further $\alpha (i) \neq \alpha (j)$ for $i \neq j$ an thus the evolution operator $\hat{U}=e^{-\frac{i}{\hbar}\hat{H}}$ (that is, $\hat{U}$ is $\hat{U}_t=e^{-\frac{i}{\hbar}\hat{H}t}$ with $t=1$) is used to permute the states $\hat{\rho}_i$.
\item[$(III)$] $(\widetilde{\rho}_{\hat{0}(n-1)}|\hat{O}) \longrightarrow 0$ \,\,\, as \,\,\, $n \longrightarrow \infty$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $\hat{\rho} \in \mathcal{N}$ and let $\hat{O}$ be an observable.
If we define $f=symb(\hat{\rho})$ and $g=symb(\hat{O})$ then
multiplying the equation 9 by $g$ and integrating over all space we
have
\begin{equation}
\int_{X}P^{n}f(x)g(x)dx=\sum_{i=1}^{r}\lambda_{\alpha^{-n}(i)}(f)\int_{X}\overline{1}_{A_i}(x)g(x)dx
+ \int_{X}Q_{n}f(x)g(x)dx
\end{equation}
\nd Equivalently,
\begin{equation}
\langle P^{n}f,g\rangle =\sum_{i=1}^{r}\lambda_{\alpha^{-n}(i)}(f)
\langle\overline{1}_{A_i},g\rangle + \langle P^{n-1}Qf,g\rangle
\end{equation}
\nd Since $\langle P^{n}f,g\rangle = \langle f,U^{n}g\rangle$ and
$\langle P^{n-1}Qf,g\rangle = \langle Qf,U^{n-1}g\rangle$ (the
Koopman operator is the dual operator of the Frobenius-Perron
operator) and thus
\begin{equation}
\langle f,U^{n}g\rangle =\sum_{i=1}^{r}\lambda_{\alpha^{-n}(i)}(f)
\langle\overline{1}_{A_i},g\rangle + \langle Qf,U^{n-1}g\rangle
\end{equation}
\nd Now if we call $\hat{\rho}_i=sym^{-1}(\overline{1}_{A_i})$,
$\widetilde{\rho}_0=symb^{-1}(Qf)$ and we use the fact that
$U^{n}g=g(n)=symb(\hat{O}(n))$,$U^{n-1}g=g(n-1)=symb(\hat{O}(n-1))$
(see equation \eqref{koopman}) then
\begin{equation}\label{symb1}
\langle symb(\hat{\rho}),symb(\hat{O}(n))\rangle
=\sum_{i=1}^{r}\lambda_{\alpha^{-n}(i)}(f) \langle
symb(\hat{\rho}_i),symb(\hat{O})\rangle + \langle
symb(\widetilde{\rho}_0),symb(\hat{O}(n-1))\rangle
\end{equation}
\nd If we call $k_i=symb(\hat{O}_i)$ and using equation
\eqref{lambda} then the coefficient $\lambda_{\alpha^{-n}(i)}(f)$
can be written as
\begin{equation}
\lambda_{\alpha^{-n}(i)}(f)=\int_{X}
f(x)k_{\alpha^{-n}(i)}(x)dx=\langle
f,k_{\alpha^{-n}(i)}\rangle=\langle
symb(\hat{\rho}),symb(\hat{\rho}_{\alpha^{-n}(i)})\rangle=\lambda_{\alpha^{-n}(i)}(\hat{\rho})
\end{equation}
\nd Therefore, the equation \eqref{symb1} reads as
\begin{equation}\label{symb2}
\langle symb(\hat{\rho}),
symb(\hat{O}(n))\rangle=\sum_{i=1}^{r}\lambda_{\alpha^{-n}(i)}(\hat{\rho})\langle
symb(\hat{\rho}_i), symb(\hat{O})\rangle + \langle
symb(\widetilde{\rho}_0), symb(\hat{O}(n-1))\rangle
\end{equation}
\nd Finally we can use the property of the Weyl symbol (see
\cite{MARIO OLIMPIA}, pag. 251 eq.(24)), that is
\begin{equation}\label{WEYL}
\forall \hat{O}\in \widehat{\mathcal{A}},\forall\hat{\rho}\in
\widehat{\mathcal{A}}^{\prime}:(\hat{\rho}|\hat{O})=\langle
symb(\hat{\rho}), symb(\hat{O})\rangle=\int \rho(\phi)O(\phi)\phi
\end{equation}
\nd Using this property the equation \eqref{symb2} can be expressed
in quantum language as
\begin{equation}\label{symb3}
(\hat{\rho}|\hat{O}(n))=\sum_{i=1}^{r}\lambda_{\alpha^{-n}(i)}(\hat{\rho})(\hat{\rho}_i|\hat{O})
+ (\widetilde{\rho}_0|\hat{O}(n-1))
\end{equation}
\nd We know that $(\hat{\rho}|\hat{O}(n))$ and
$(\widetilde{\rho}_0|\hat{O}(n-1))$ are equal to
$(\hat{\rho}(n)|\hat{O})$ and $(\widetilde{\rho}_0(n-1)|\hat{O})$
respectively, and then equation \eqref{symb3} reads as
\begin{equation}
(\hat{\rho}(n)|\hat{O})=\sum_{i=1}^{r}\lambda_{\alpha^{-n}(i)}(\hat{\rho})(\hat{\rho}_i|\hat{O})
+ (\widetilde{\rho}_0(n-1)|\hat{O})
\end{equation}
\nd and therefore we have proved the equation \eqref{QSDT1}.
(I): On one hand we have that
\begin{equation}
\begin{split}
&tr(\hat{\rho}_i)=(\hat{\rho}_i|\hat{1})=\langle symb(\hat{\rho}_i), symb(\hat{1})\rangle=\langle \overline{1}_{A_i},1_{X}\rangle=\int_{X}\overline{1}_{A_i}dx=\int_{X}[1/\mu(A_i)]1_{A_i}dx=\\
&=[1/\mu(A_i)]\int_{X}1_{A_i}dx=[1/\mu(A_i)]\int_{A_i}dx=[1/\mu(A_i)]\mu(A_i)=1
\end{split}
\end{equation}
\nd where we have used the definition of $\overline{1}_{A_i}$
(equation \eqref{characteristic1}), the property of the Weyl's
symbol (equation \eqref{WEYL}) and $symb(\hat{1})=1_X$. Therefore
$tr(\hat{\rho}_i)=1$, that is, $\hat{\rho}_i \in \mathcal{N}$ for
all $i$. On the other hand if we use equations \eqref{star product}
and \eqref{moyal} applied to $f=\overline{1}_{A_i}$ and
$g=\overline{1}_{A_j}$ we have
\begin{equation}\label{symb4}
\begin{split}
& symb(\hat{\rho}_i\hat{\rho}_j)=\overline{1}_{A_i}(x)\overline{1}_{A_j}(x)+0(\hbar)=\\
&=0(\hbar) \,\,\,\ if \,\ i\neq j\\
&=\overline{1}_{A_i}(x)+0(\hbar)\,\,\,\ if \,\ i=j
\end{split}
\end{equation}
\nd Now applying the inverse $symb^{-1}$ to both sides of the
equation \eqref{symb4} we have
\begin{equation}
\begin{split}
& \hat{\rho}_i\hat{\rho}_j=0(\hbar) \,\,\,\ if \,\ i\neq j\\
&=\hat{\rho}_i+0(\hbar)\,\,\,\ if \,\ i=j
\end{split}
\end{equation}
On the other hand from the equations
\eqref{characteristic1},\eqref{characteristic1bis} we have that
$1_X=\sum_i \mu(A_i)\overline{1}_{A_i}$ and therefore
$symb^{-1}(1_X)=symb^{-1}(\sum_i \mu(A_i)\overline{1}_{A_i})=\sum_i
\mu(A_i)symb^{-1}(\overline{1}_{A_i})$ where we have used that
$symb^{-1}$ is a linear map.
Now if we call $\alpha_i=\mu(A_i)$
since $symb^{-1}(1_X)=\hat{1}$ and
$\hat{\rho}_i=symb^{-1}(\overline{1}_{A_i})$ then we deduce the
equation \eqref{QSDT3}.
(II): Due to part (II) of the Spectral Decomposition Theorem version
I and considering that we are in the hypothesis of version II of the
Spectral Decomposition Theorem we have
\begin{equation}\label{characteristic2}
P\overline{1}_{A_i}=\overline{1}_{A_\alpha(i)}
\end{equation}
\nd where $\alpha:\{1,...,r\}\longrightarrow \{1,...,r\}$ is a
permutation which satisfies $\alpha (i) \neq \alpha (j)$ for $i \neq
j$ an thus the operator $P$ is used to permute the functions
$\overline{1}_{A_i}$. Let $\hat{O}$ be an observable and
$g=symb(\hat{O})$. Then from equation \eqref{characteristic2} it
follows that
\begin{equation}\label{characteristic3}
\langle P\overline{1}_{A_i},g
\rangle=\langle\overline{1}_{A_\alpha(i)},g\rangle
\end{equation}
\nd and noting that
\begin{equation}
\begin{split}
& \langle P\overline{1}_{A_i},g \rangle=\langle \overline{1}_{A_i},Ug \rangle=\langle symb(\hat{\rho}_i),symb(\hat{O}(1)) \rangle=(\hat{\rho}_i|\hat{O}(1))=(\hat{U}\hat{\rho}_i\hat{U}^{\dag}|\hat{O})\\
& \langle\overline{1}_{A_\alpha(i)},g\rangle=\langle
symb(\hat{\rho}_{\alpha(i)}),symb(\hat{O})
\rangle=(\hat{\rho}_{\alpha(i)}|\hat{O})
\end{split}
\end{equation}
\nd then from \eqref{characteristic2} we have that
$(\hat{U}\rho_i\hat{U}^{\dag}|\hat{O})=(\hat{\rho}_{\alpha(i)}|\hat{O})$.
(III): Let $\hat{O}$ be an observable and $\varepsilon > 0$. Then by
the condition (III) of the Spectral Decomposition Theorem version I
we have
\begin{equation}
\|P^{n-1}Qf\|=\|Q_nf\|< \frac{\varepsilon}{max_{\phi \in
X}|O(\phi)|}=\frac{\varepsilon}{\|O\|_{\infty}}
\,\,\,\,\,\,\,\,\,\,\ with \,\,\ O=symb(\hat{O})
\end{equation}
\nd Then
\begin{equation}\label{symb5}
(\widetilde{\rho}_0(n-1)|\hat{O})=\langle
symb(\widetilde{\rho}_0(n-1)), symb(\hat{O})\rangle=\langle Q_n f,
O\rangle\leq \|Q_nf\|\|O\|_{\infty}<\varepsilon
\end{equation}
\nd Therefore, from the equation \eqref{symb5} it follows that
$(\widetilde{\rho}_0(n-1)|\hat{O}) \longrightarrow 0$.
\end{proof}
\subsection{The Levels of The Quantum Ergodic Hierarchy: Ergodic, Mixing and Exact}
The theorem 8 and the Spectral Decomposition Theorem (theorem 7) can
be used simultaneously to determine the ergodicity, the mixing or
the exactness just looking the terms that there are in the sum of
\eqref{SDT1 decomposition} or \eqref{SDT2 decomposition}. And this
also holds for the \emph{Quantum Spectral Decomposition Theorem}
(QSDT, theorem 9) simply because this is a quantum language
translation of the spectral theorem for dynamical systems. Therefore
we apply the theorem 8 and QSDT to obtain the following theorem
which classifies the ergodic, exact and mixing levels of the Quantum
Ergodic Hierarchy (see \cite{0} theorems 1,2 pags. 261, 263).
\begin{theorem}(\emph{The levels of The Quantum Ergodic Hierarchy: Ergodic, Exact and Mixing})\label{QSDT AND HE}
Let $S$ be a quantum system. Let $\hat{\rho} \in \mathcal{N}$ and
let $\hat{O}$ be an observable. Then
\begin{enumerate}
\item[$(I)$] $S$ is \emph{ergodic} $\Longleftrightarrow$ the permutation $\{\alpha(1),...,\alpha(r)\}$ of the secuence $\{1,...,r\}$ is \emph{cyclical} (that is, for which there no is invariant subset).
\item[$(II)$] If $r=1$ in the representation of \eqref{QSDT1} $\Longrightarrow$ $S$ is \emph{exact}.
\item[$(III)$] If $S$ is \emph{mixing} \emph{(see \cite{0} theorem 1 pag. 261)} $\Longrightarrow$ $r=1$ in the representation of \eqref{QSDT1}.
\end{enumerate}
\end{theorem}
\begin{proof}
It is enough to apply the theorem 8 and the theorem 9.
\end{proof}
In the following subsections we examine the ergodic hierarchy
established by the theorem 10 and their consequences in more detail.
\subsubsection{A consequence of QSDT: homogenization of the mixing level}
If the system is \emph{mixing} then from theorem 10 and the equation
\eqref{QSDT1} it follows that
\begin{equation}\label{QSDT consequence mixing}
(\hat{\rho}(n)|\hat{O})=(\hat{\rho}|\hat{O}_1)(\hat{\rho}_1|\hat{O})
+ (\widetilde{\rho}_0(n-1)|\hat{O})
\end{equation}
where $\hat{\rho}_1$ is a pure state (in the classical limit) and
$\hat{O}_1$ is an observable which does not depend on the observable
$\hat{O}$. Further, since the system is mixing then it has a weak
limit $\hat{\rho}_{\ast}$ such that $lim_{n\rightarrow
\infty}(\hat{\rho}(n)|\hat{O})=(\hat{\rho}_{\ast}|\hat{O})$. From
this limit and the equation \eqref{QSDT consequence mixing} we
obtain
\begin{equation}\label{QSDT consequence mixing2}
(\hat{\rho}_{\ast}|\hat{O})=lim_{n\rightarrow
\infty}(\hat{\rho}|\hat{O}_1)(\hat{\rho}_1|\hat{O}) +
lim_{n\rightarrow
\infty}(\widetilde{\rho}_0(n-1)|\hat{O})=(\hat{\rho}|\hat{O}_1)(\hat{\rho}_1|\hat{O})
\end{equation}
since $lim_{n\rightarrow \infty}(\widetilde{\rho}_0(n-1)|\hat{O})=0$
(see Theorem 9 (III) subsection 4.1). Now if we make $O=1$ in the
equation \eqref{QSDT consequence mixing2} since
$(\hat{\rho}_{\ast}|\hat{1})=tr(\hat{\rho}_{\ast})=1$ and
$(\hat{\rho}_1|\hat{1})=tr(\hat{\rho}_1)=1$ we conclude that
$(\hat{\rho}|\hat{O}_1)=1$ then
$(\hat{\rho}_{\ast}|\hat{O})=(\hat{\rho}_1|\hat{O})$ for all
$\hat{O}$ observable and therefore $\hat{\rho}_1=\hat{\rho}_{\ast}$.
That is, physically, the mixing level is responsible for
``\emph{homogenize}" the initial state $\hat{\rho}$ to take it to
the weak limit $\hat{\rho}_1$ which is a pure state. In this sense
QSDT gives a physical interpretation of the mixing level.
\subsubsection{The exact level}
Let $\hat{\rho}$ be an state and $\hat{O}$ an observable. Then by
QSDT we have the representation of $(\hat{\rho}|\hat{O})$ given by
the equation \eqref{QSDT1}. If $r=1$ then by the the theorem 10 the
system is \emph{exact}. That is, for the exact case the theorem 10
only gives a sufficient condition.
\subsubsection{The ergodic level: Oscillation of the mean values}
From theorem 10 we can deduce a necessary and sufficient condition
for ergodicity. The system is \emph{ergodic} if and only if the
permutation $\alpha$ of
\begin{equation}\label{QSDT consequence ergodic1}
(\hat{\rho}(n)|\hat{O})=\sum_{i=1}^{r}(\hat{\rho}|\hat{O}_{\alpha^{-n}(i)})(\hat{\rho}_i|\hat{O})
+ (\widetilde{\rho}_0(n-1)|\hat{O})
\end{equation}
is cyclical and $(\widetilde{\rho}_0(n-1)|\hat{O})\rightarrow 0$.
Since $\alpha$ is cyclical there is an integer $N>0$ such that
$\alpha^{N}(i)=i$ and $\alpha^{-N}(i)=i$ for $i=1,...,r$. That is,
the inverse permutation $\alpha^{-1}$ operates on the indices
$i=1,...,r$ as
\begin{equation}\label{QSDT consequence ergodic2}
(1,...,r)\longrightarrow^{\alpha^{-1}}(2,3,...,r-2,r-1,r,1)\longrightarrow^{\alpha^{-2}}(3,4,...,r-1,r,1,2)...\longrightarrow^{\alpha^{-k}}...\longrightarrow^{\alpha^{-N}}(1,...,r)
\end{equation}
After $N$ successive steps we are back to the original cycle
$(1,...,r)$. This behavior indicates that the sum of \eqref{QSDT
consequence ergodic1} will also return to its original value after
$N$ successive time instants. Then the sum of \eqref{QSDT
consequence ergodic1} is periodic with a period equal to $N$, i.e.
with the same period as the cycle ${\alpha^{-1}}$. Indeed, we have
\begin{equation}\label{QSDT consequence ergodic3}
\begin{split}
&(\hat{\rho}(0)|\hat{O})=(\hat{\rho}|\hat{O}_{1})(\hat{\rho}_1|\hat{O})+(\hat{\rho}|\hat{O}_{2})(\hat{\rho}_2|\hat{O})+...+(\hat{\rho}|\hat{O}_{r-1})(\hat{\rho}_{r-1}|\hat{O})+(\hat{\rho}|\hat{O}_r)(\hat{\rho}_{r}|\hat{O})+(\widetilde{\rho}_0(-1)|\hat{O}) \\
&(\hat{\rho}(1)|\hat{O})=(\hat{\rho}|\hat{O}_{2})(\hat{\rho}_1|\hat{O})+(\hat{\rho}|\hat{O}_{3})(\hat{\rho}_2|\hat{O})+...+(\hat{\rho}|\hat{O}_{r})(\hat{\rho}_{r-1}|\hat{O})+(\hat{\rho}|\hat{O}_1)(\hat{\rho}_{r}|\hat{O})+(\widetilde{\rho}_0(0)|\hat{O}) \\
&(\hat{\rho}(2)|\hat{O})=(\hat{\rho}|\hat{O}_{3})(\hat{\rho}_1|\hat{O})+(\hat{\rho}|\hat{O}_{4})(\hat{\rho}_2|\hat{O})+...+(\hat{\rho}|\hat{O}_{1})(\hat{\rho}_{r-1}|\hat{O})+(\hat{\rho}|\hat{O}_2)(\hat{\rho}_{r}|\hat{O})+(\widetilde{\rho}_0(1)|\hat{O}) \\
&.\\
&.\\
&.\\
&(\hat{\rho}(N)|\hat{O})=(\hat{\rho}|\hat{O}_{1})(\hat{\rho}_1|\hat{O})+(\hat{\rho}|\hat{O}_{2})(\hat{\rho}_2|\hat{O})+...+(\hat{\rho}|\hat{O}_{r-1})(\hat{\rho}_{r-1}|\hat{O})+(\hat{\rho}|\hat{O}_r)(\hat{\rho}_{r}|\hat{O})+(\widetilde{\rho}_0(N-1)|\hat{O}) \\
&(\hat{\rho}(N+1)|\hat{O})=(\hat{\rho}|\hat{O}_{2})(\hat{\rho}_1|\hat{O})+(\hat{\rho}|\hat{O}_{3})(\hat{\rho}_2|\hat{O})+...+(\hat{\rho}|\hat{O}_{r})(\hat{\rho}_{r-1}|\hat{O})+(\hat{\rho}|\hat{O}_1)(\hat{\rho}_{r}|\hat{O})+(\widetilde{\rho}_0(N)|\hat{O}) \\
\end{split}
\end{equation}
Then
$\sum_{i=1}^{r}(\hat{\rho}|\hat{O}_{\alpha^{-(N+1)}(i)})(\hat{\rho}_i|\hat{O})=\sum_{i=1}^{r}(\hat{\rho}|\hat{O}_{\alpha^{-1}(i)})(\hat{\rho}_i|\hat{O})$.
That is, if we call
$F(n)=\sum_{i=1}^{r}(\hat{\rho}|\hat{O}_{\alpha^{-n}(i)})(\hat{\rho}_i|\hat{O})$
then
\begin{equation}
F(N+1)=F(1)
\end{equation}
and therefore this sum $F(n)$ is periodic with a period equal to
$N$. Therefore, we see that in the ergodic case the mean values
consist of an oscillating part plus a term which tends to zero for
large times ($n$ goes to $\infty$). The physical interpretation of
this behavior is studied in the next section and it is the key to
find the type of spectrum of the ergodic systems.
\section{QSDT and the quantum spectrum}
The theorem QSDT besides characterizing the mean values of the
ergodic hierarchy also gives us a connection between ergodic
hierarchy levels and the spectra. In this section we extend this
connection to the cases of the discrete spectra, the continuous
spectra and for a more general case where both spectra are
simultaneously present.
\subsection{Discrete spectrum}
Let $\hat{\rho}\in\mathcal{N}$ an state and let $\hat{O}$ be an
observable. We assume that the spectrum is discrete and $E_1,
E_2,...$ are the energies of the system with
$\omega_1=\frac{E_1}{\hbar}, \omega_2=\frac{E_2}{\hbar},...$ the
natural frequencies of each energy level. Then the mean value of
$\hat{O}$ in the state $\hat{\rho}$ in the instant $t=n$ is
\begin{equation}\label{DISCRETE SPECTRUM1}
(\hat{\rho}(n)|\hat{O})=\sum_{j}(\hat{\xi}_j|\hat{O})e^{-j\frac{\omega_j}{\hbar}n}
\end{equation}
where $\hat{\xi}_j=|j\rangle\langle j|\in\mathcal{N}$ and
$|j\rangle$ is the eigenstate with energy $E_j$. Furthermore, by the
QSDT theorem we have
\begin{equation}\label{DISCRETE SPECTRUM2}
(\hat{\rho}(n)|\hat{O})=\sum_{i=1}^{r}\lambda_{\alpha^{-n}(i)}(\hat{\rho}_i|\hat{O})
+ (\widetilde{\rho}_0(n-1)|\hat{O})
\end{equation}
Then,
\begin{equation}\label{DISCRETE SPECTRUM3}
\sum_{j}(\hat{\xi}_j|\hat{O})e^{-j\frac{\omega_j}{\hbar}n}=\sum_{i=1}^{r}\lambda_{\alpha^{-n}(i)}(\hat{\rho}_i|\hat{O})
+ (\widetilde{\rho}_0(n-1)|\hat{O})
\end{equation}
Since $\sum_{j}(\hat{\xi}_j|\hat{O})e^{-j\frac{\omega_j}{\hbar}n}$
and $\sum_{i=1}^{r}\lambda_{\alpha^{-n}(i)}(\hat{\rho}_i|\hat{O})$
are periodic functions and
$(\widetilde{\rho}_0(n-1)|\hat{O})\rightarrow 0$ when $n\rightarrow
\infty$ it follows that $\widetilde{\rho}_0$ is equal to zero. Then
we conclude that the discrete spectrum corresponds to the ergodic
level. That is, from the QSDT theorem it follows that in the
classical limit the quantum systems of spectrum discrete correspond
to the ergodic level.
\subsection{Continuous spectrum}
Now we assume that the spectrum is continuous with $\omega\in
[0,\infty)$ and $|\omega\rangle$ which are the energies and the
eigenvectors of the system. Let $\hat{\rho}\in\mathcal{N}$ an state
and let $\hat{O}$ be an observable. In order to obtain an
equilibrium arrival of the system we restrict the space of
observables only considering the \emph{Van Hove observables}. This
restriction does not lose generality because the observables that do
not belong to Van Hove space are not experimentally accessible (see
\cite{MARIO OLIMPIA2} for a complete argument). The components of a
Van Hove observable $\hat{O}_R$ are
$O_R(\omega,\omega^{\prime})=O(\omega)\delta(\omega-\omega^{\prime})+O(\omega,\omega^{\prime})$.
Then we can expand $\hat{O}$ in the basis
$\{|\omega\rangle\langle\omega|,|\omega\rangle\langle\omega^{\prime}|\}$
as
\begin{equation}\label{CONTINUOUS SPECTRA1}
\hat{O}=\int_{0}^{\infty}O(\omega)|\omega\rangle\langle\omega|d\omega+\int_{0}^{\infty}\int_{0}^{\infty}O(\omega,\omega^{\prime})|\omega\rangle\langle\omega^{\prime}|
\end{equation}
Therefore, the mean value of $\hat{O}$ in the state $\hat{\rho}$ for
the instant $t=n$ is
\begin{equation}\label{CONTINUOUS SPECTRA3}
\begin{split}
&(\hat{\rho}(n)|\hat{O})=\int_{0}^{\infty}\rho(\omega)^{\ast}O(\omega)d\omega+\int_{0}^{\infty}\int_{0}^{\infty}\rho(\omega,\omega^{\prime})^{\ast}O(\omega,\omega^{\prime})e^{-i\frac{(\omega-\omega^{\prime})}{\hbar}n}d\omega d\omega^{\prime}\\
\end{split}
\end{equation}
On the other hand, by QSDT theorem we have
\begin{equation}\label{CONTINUOUS SPECTRA4}
\begin{split}
&(\hat{\rho}(n)|\hat{O})=\sum_{i=1}^{r}(\hat{\rho}|\hat{O}_{\alpha^{-n}(i)})(\hat{\rho}_i|\hat{O}) + (\widetilde{\rho}_0(n-1)|\hat{O})=\\
&=\int_{0}^{\infty}\rho(\omega)^{\ast}O(\omega)d\omega+\int_{0}^{\infty}\int_{0}^{\infty}\rho(\omega,\omega^{\prime})^{\ast}O(\omega,\omega^{\prime})e^{-i\frac{(\omega-\omega^{\prime})}{\hbar}n}d\omega d\omega^{\prime}\\
\end{split}
\end{equation}
If we suppose that
$\rho(\omega,\omega^{\prime})^{\ast}O(\omega,\omega^{\prime})\in
L^{1}([0,\infty)\times[0,\infty))$ then by the Riemann-Lebesgue
Lemma is
\begin{equation}\label{CONTINUOUS SPECTRA5}
\int_{0}^{\infty}\int_{0}^{\infty}\rho(\omega,\omega^{\prime})^{\ast}O(\omega,\omega^{\prime})e^{-i\frac{(\omega-\omega^{\prime})}{\hbar}n}d\omega
d\omega^{\prime}\longrightarrow0
\end{equation}
when $n\longrightarrow\infty$. That is, under the restriction of the
observable space (Van Hove algebra) and under the assumption that
$\rho(\omega,\omega^{\prime})^{\ast}O(\omega,\omega^{\prime})\in
L^{1}([0,\infty)\times[0,\infty))$ then the system is \emph{mixing}.
Moreover, from \eqref{CONTINUOUS SPECTRA3} and \eqref{CONTINUOUS
SPECTRA5} it follows that
$(\hat{\rho}(n)|\hat{O})\longrightarrow(\hat{\rho}_{\ast}|\hat{O})=\int_{0}^{\infty}\rho(\omega)^{\ast}O(\omega)d\omega$
as $n\longrightarrow\infty$ with
$\hat{\rho}_{\ast}=\int_{0}^{\infty}\rho(\omega)^{\ast}|\omega\rangle\langle\omega|d\omega$.
Therefore, $\hat{\rho}$ has weak limit equal to $\hat{\rho}_{\ast}$.
Now since the system is mixing then by the theorem 10 it follows
that $r=1$ in the sum of \eqref{CONTINUOUS SPECTRA4}. Then we have
\begin{equation}\label{CONTINUOUS SPECTRA6}
\begin{split}
&(\hat{\rho}(n)|\hat{O})=(\hat{\rho}|\hat{O}_1)(\hat{\rho}_1|\hat{O}) + (\widetilde{\rho}_0(n-1)|\hat{O})=\\
&=\int_{0}^{\infty}\rho(\omega)^{\ast}O(\omega)d\omega+\int_{0}^{\infty}\int_{0}^{\infty}\rho(\omega,\omega^{\prime})^{\ast}O(\omega,\omega^{\prime})e^{-i\frac{(\omega-\omega^{\prime})}{\hbar}n}d\omega d\omega^{\prime}\\
\end{split}
\end{equation}
Since $(\hat{\rho}|\hat{O}_1)(\hat{\rho}_1|\hat{O})$ and
$\int_{0}^{\infty}\rho(\omega)^{\ast}O(\omega)d\omega$ are constants
and
$\int_{0}^{\infty}\int_{0}^{\infty}\rho(\omega,\omega^{\prime})^{\ast}O(\omega,\omega^{\prime})e^{-i\frac{(\omega-\omega^{\prime})}{\hbar}n}d\omega
d\omega^{\prime},(\widetilde{\rho}_0(n-1)|\hat{O})\longrightarrow0$
then by \eqref{CONTINUOUS SPECTRA6} we have that
\begin{equation}\label{CONTINUOUS SPECTRA7}
\begin{split}
&(\hat{\rho}|\hat{O}_1)(\hat{\rho}_1|\hat{O})=\int_{0}^{\infty}\rho(\omega)^{\ast}O(\omega)d\omega\\
&(\widetilde{\rho}_0(n-1)|\hat{O})=\int_{0}^{\infty}\int_{0}^{\infty}\rho(\omega,\omega^{\prime})^{\ast}O(\omega,\omega^{\prime})e^{-i\frac{(\omega-\omega^{\prime})}{\hbar}n}d\omega d\omega^{\prime}\\
\end{split}
\end{equation}
From \eqref{CONTINUOUS SPECTRA7} we see the physical interpretation
of the term $(\widetilde{\rho}_0(n-1)|\hat{O})$ in the decomposition
\eqref{QSDT1}, that is, the term $(\widetilde{\rho}_0(n-1)|\hat{O})$
is the manifestation of the Riemann-Lebesgue Lemma in closed quantum
systems of continuous spectrum \cite{MARIO OLIMPIA}. Therefore, a
consequence of QSDT is that the mixing systems which have continuous
spectrum are those with only one term
$(\hat{\rho}|\hat{O}_1)(\hat{\rho}_1|\hat{O})$ in the decomposition
of the mean value $(\hat{\rho}(n)|\hat{O})$ given by the equation
\eqref{QSDT1}, and with the Riemann-Lebesgue Lemma contained in the
term $(\widetilde{\rho}_0(n-1)|\hat{O})$ that goes to zero as
$n\rightarrow\infty$.
\subsection{The General Case: Discrete and Continuous Spectrum}
If the discrete and continuous spectrum are simultaneously present
in accordance with \eqref{DISCRETE SPECTRUM1} and \eqref{CONTINUOUS
SPECTRA3} we have
\begin{equation}\label{GENERAL CASE1}
(\hat{\rho}(n)|\hat{O})=\sum_{j}(\hat{\xi_j}|\hat{O})e^{-j\frac{\omega_j}{\hbar}n}+\int_{0}^{\infty}\rho(\omega)^{\ast}O(\omega)d\omega+\int_{0}^{\infty}\int_{0}^{\infty}\rho(\omega,\omega^{\prime})^{\ast}O(\omega,\omega^{\prime})e^{-i\frac{(\omega-\omega^{\prime})}{\hbar}n}d\omega
d\omega^{\prime}
\end{equation}
where the summa is the contribution to the mean value of the
discrete spectrum and integrals are the contributions from the
continuum. Now if we call $A=max_{1\leq n\leq
N}\{\sum_{i=1}^{r}(\hat{\rho}|\hat{O}_{\alpha^{-n}(i)})(\hat{\rho}_i|\hat{O})\}$
and $B=min_{1\leq n\leq
N}\{\sum_{i=1}^{r}(\hat{\rho}|\hat{O}_{\alpha^{-n}(i)})(\hat{\rho}_i|\hat{O})\}$
we can express the decomposition given by the equation \eqref{QSDT1}
as
\begin{equation}\label{GENERAL CASE2}
(\hat{\rho}(n)|\hat{O})=\{\sum_{i=1}^{r}(\hat{\rho}|\hat{O}_{\alpha^{-n}(i)})(\hat{\rho}_i|\hat{O})-\frac{A+B}{2}\}+\frac{A+B}{2}+(\widetilde{\rho}_0(n-1)|\hat{O})
\end{equation}
where we have used the equation \eqref{QSDT2} for the coefficients
$\lambda_{\alpha^{-n}(i)}$. We see the usefulness of this rewriting
of $(\hat{\rho}(n)|\hat{O})$ below. Now if we compare the three
terms of \eqref{GENERAL CASE1} and \eqref{GENERAL CASE2} we conclude
that
\begin{equation}\label{GENERAL CASE3}
\begin{split}
&\sum_{j}(\hat{\xi_j}|\hat{O})e^{-j\frac{\omega_j}{\hbar}n}=\{\sum_{i=1}^{r}(\hat{\rho}|\hat{O}_{\alpha^{-n}(i)})(\hat{\rho}_i|\hat{O})-\frac{A+B}{2}\}\\
&\int_{0}^{\infty}\rho(\omega)^{\ast}O(\omega)d\omega=\frac{A+B}{2}\\
&\int_{0}^{\infty}\int_{0}^{\infty}\rho(\omega,\omega^{\prime})^{\ast}O(\omega,\omega^{\prime})e^{-i\frac{(\omega-\omega^{\prime})}{\hbar}n}d\omega d\omega^{\prime}=(\widetilde{\rho}_0(n-1)|\hat{O})\\
\end{split}
\end{equation}
Therefore, from \eqref{GENERAL CASE3} we see that the QSDT theorem
can express the mean value of any observable and also at the same
time it indicates to which level of the ergodic hierarchy it
belongs. This relation that QSDT establishes between the type of
spectrum and the ergodic hierarchy levels provides a formal
framework for the study of the possible connections between the
classical limit and the quantum chaos.
\section{An Application of QSDT: Two-Level System}
We conclude this paper by applying QSDT on the two-level system to
make clear the consequences described in section 4.2.3. Despite
being one of the simplest quantum models, the two-level system is
one of the most used for testing formalisms, for teaching purposes,
etc. The Hamiltonian of the two-level system is given by
\begin{equation}\label{TWO LEVEL2}
\hat{H}=E_1|1\rangle\langle1|+E_2|2\rangle\langle2|
\end{equation}
where $|1\rangle$, $|2\rangle$ are the eigenstates whose energies
are $E_1$ and $E_2$ respectively. We consider an observable
$\hat{O}$ and a state $\hat{\rho}$ given by
\begin{equation}\label{TWO LEVEL3}
\hat{\rho}=\rho_{11}|1\rangle\langle1|+\rho_{22}|2\rangle\langle2|+\rho_{12}|1\rangle\langle2|+\rho_{21}|2\rangle\langle1|
\end{equation}
where $\rho_{ij}=\langle i|\widehat{\rho}|j\rangle$ and
$\rho_{ii}\geq0$\,,\,$\rho_{ji}=\rho_{ij}^{\ast}$ with $i,j=1,2$.
The state $\rho$ in the instant $t=n$ is
\begin{equation}\label{TWO LEVEL4}
\hat{\rho}(n)=\rho_{11}|1\rangle\langle1|+\rho_{22}|2\rangle\langle2|+\rho_{12}e^{-\frac{i}{\hbar}(E_1-E_2)n}|1\rangle\langle2|+\rho_{21}
e^{-\frac{i}{\hbar}(E_2-E_1)n}|2\rangle\langle1|
\end{equation}
Then the mean value of $\hat{O}$ in the instant $t=n$ is
\begin{equation}\label{TWO LEVEL5}
\begin{split}
&(\hat{\rho}(n)|\hat{O})=tr(\hat{\rho}(n)\hat{O})=\langle1|\hat{\rho}(n)\hat{O}|1\rangle+\langle2|\hat{\rho}(n)\hat{O}|2\rangle=\\
&=\rho_{11}O_{11}+\rho_{22}O_{22}+\rho_{12}e^{-\frac{i}{\hbar}(E_1-E_2)n}O_{12}+\rho_{21}
e^{-\frac{i}{\hbar}(E_2-E_1)n}O_{21}
\end{split}
\end{equation}
where $\langle j|\hat{O}|i\rangle=O_{ij}$ with $i=1,2$. Now by QSDT
we have
\begin{equation}\label{TWO LEVEL6}
\begin{split}
&\sum_{i=1}^{r}(\hat{\rho}_{\psi}|\hat{O}_{\alpha^{-n}(i)})(\hat{\rho}_i|\hat{O}) + (\widetilde{\rho}_0(n-1)|\hat{O})=\\
&=\rho_{11}O_{11}+\rho_{22}O_{22}+\rho_{12}e^{-\frac{i}{\hbar}(E_1-E_2)n}O_{12}+\rho_{21}
e^{-\frac{i}{\hbar}(E_2-E_1)n}O_{21}
\end{split}
\end{equation}
Since the right member of \eqref{TWO LEVEL6} is oscillatory because
there are imaginary exponentials, then the left member of \eqref{TWO
LEVEL6} must be oscillatory. From this fact it follows that
$\widetilde{\rho}_0=0$ and the permutation $\alpha$ is cyclical.
Therefore, from the theorem 10 (I) it follows that the two-level
system is \emph{ergodic}. Moreover, we can obtain the C\'{e}saro
limit (see \cite{LM} corollary 4.4.1. (a)):
\begin{equation}\label{TWO LEVEL7}
\begin{split}
&lim_{M\rightarrow\infty}\frac{1}{M}\sum_{n=0}^{M-1}(\hat{\rho}(n)|\hat{O})=\\
&=\rho_{11}O_{11}+\rho_{22}O_{22}+\rho_{12}O_{12}\{lim_{M\rightarrow\infty}\frac{1}{M}\sum_{n=0}^{M-1}e^{-\frac{i}{\hbar}(E_1-E_2)n}\}+\rho_{21} O_{21}\{lim_{M\rightarrow\infty}\frac{1}{M}\sum_{n=0}^{M-1}e^{-\frac{i}{\hbar}(E_2-E_1)n}\}=\\
&=\rho_{11}O_{11}+\rho_{22}O_{22}+\rho_{12}O_{12}\{lim_{M\rightarrow\infty}\sigma_M\}+\rho_{21} O_{21}\{lim_{M\rightarrow\infty}\sigma_M\}^{\ast}=\\
&=\rho_{11}O_{11}+\rho_{22}O_{22}+\rho_{12}O_{12}(\frac{1}{1-z})+\rho_{21} O_{21}(\frac{1}{1-z})^{\ast}\\
\end{split}
\end{equation}
with
$\sigma_M=\frac{1}{M}\sum_{n=0}^{M-1}e^{-\frac{i}{\hbar}(E_1-E_2)n}$
and $z=e^{-\frac{i}{\hbar}(E_1-E_2)}$. From \eqref{TWO LEVEL7} we
have that the C\'{e}saro limit $\hat{\rho}_{c}$ of $\hat{\rho}$ is
\begin{equation}\label{TWO LEVEL8}
\hat{\rho}_{c}=\rho_{11}|1\rangle\langle1|+\rho_{22}|2\rangle\langle2|+\rho_{12}(\frac{1}{1-z})|1\rangle\langle2|+\rho_{21}(\frac{1}{1-z})^{\ast}|2\rangle\langle1|
\end{equation}
This is so because
\begin{equation}\label{TWO LEVEL9}
\begin{split}
&(\hat{\rho}_{c}|\hat{O})=\langle1|\hat{\rho}_{c}\hat{O}|1\rangle+\langle2|\hat{\rho}_{c}\hat{O}|2\rangle=\\
&=\rho_{11}O_{11}+\rho_{22}O_{22}+\rho_{12}O_{12}(\frac{1}{1-z})+\rho_{21}O_{21}(\frac{1}{1-z})^{\ast}\\
\end{split}
\end{equation}
and therefore, from \eqref{TWO LEVEL7} and \eqref{TWO LEVEL9} it
follows that
\begin{equation}\label{TWO LEVEL10}
lim_{M\rightarrow\infty}\frac{1}{M}\sum_{n=0}^{M-1}(\hat{\rho}(n)|\hat{O})=(\hat{\rho}_{c}|\hat{O})
\end{equation}
\section{Conclusions}
Under the assumption that in the classical limit the quantum system
has an analogue classical system whose evolution $T$ has an
associated Frobenius-Perron operator $P$ constrictive we have been
able to translate The Spectral Decomposition Theorem of dynamical
systems (theorems 6,7) to quantum language (QSDT) which gives a
representation of the expectation value of an observable $\hat{O}$
(equation \eqref{QSDT1}) in a state $\hat{\rho}$.
The relevant feature of the QSDT representation is that it can
express in a simple way in the classical limit ($\hbar\rightarrow0$)
the oscillatory nature of the expectation values in the case of
discrete spectrum (equation \eqref{QSDT consequence ergodic3}) and
the manifestation of the Riemann-Lebesgue lemma for the Van Hove
observables in the case of continuous spectrum (equation
\eqref{CONTINUOUS SPECTRA7}).
QSDT theorem also connects the lower levels of the hierarchy ergodic
with spectral type (equations \eqref{DISCRETE
SPECTRUM3},\eqref{CONTINUOUS SPECTRA7}) obtaining that the discrete
spectrum corresponds to the ergodic level. For the mixing case, QSDT
provides a physical interpretation of the ``homogenization"
(equations \eqref{QSDT consequence mixing} and \eqref{QSDT
consequence mixing2}) that is represented by the presence of the
pure state $\hat{\rho}_1$ as the weak limit for any initial state
$\hat{\rho}$.
When both spectra are present QSDT also gives a representation of
the expectation value where the sum represents the discrete spectrum
and the constant term plus the term that goes to zero represents the
continuous spectrum (equation \eqref{GENERAL CASE3}).
In Section 6 the application of QSDT allowed us to classify the
two-level system in the ergodic level and this was verified by
performing the corresponding Cesaro limit (equations \eqref{TWO
LEVEL7}, \eqref{TWO LEVEL8} and \eqref{TWO LEVEL9}). We hope that
all the features and connections provided by the QSDT among the
classical limit, the Quantum Ergodic Hierarchy and the type of
spectrum can be extended in future studies through more examples or
theoretical essays.
| 108,245
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TITLE: Why does the parameter of a logistic map influence its behaviour?
QUESTION [4 upvotes]: I am looking at choatic behaviour in discrete dynamical systems. As an example, I have taken the iteration map $x_{n+1} = rx_{n}(1-x_{n})$, where $r$ is the parameter.
By plotting a bifurcation map, I can see for what value of $r$ the system becomes chaotic (the 'shaded' parts):
My question is, why do these values of $r$ yield chaos: what is special about them? So far I have only found texts which deal with describing the phenomenon, but which do not actually explain the mathematics behind it.
REPLY [9 votes]: The key issue that you need to understand is the relationship between slope and dynamical stability. To be clear, let's suppose that $f:\mathbb R \to \mathbb R$ is a function and that $x_0$ is a fixed point of that function, i.e. $f(x_0) = x_0$. Then we say that
$x_0$ is attractive if $|f'(x_0)| < 1$,
$x_0$ is repulsive if $|f'(x_0)| > 1$, and
$x_0$ is neutral if $|f'(x_0)| = 1$.
You might also see the term super-attractive if $f'(x_0)=0$, though that's not particularly important for your specific question. This classification is justified by theorems such as
If $x_0$ is an attractive fixed point of $f$, then there is a neighborhood of $x_0$ such that the orbit of any seed in that neighborhood will converge to $x_0$ under iteration of $f$.
By contrast, repulsive fixed points are, well, repulsive and there are examples showing that orbits might converge or not for neutral fixed points.
These types of theorems can be proven with the mean value theorem but we can get a very simple intuitive understanding by examining the function $\ell_r(x) = rx$. Note that the derivative is simply
$\ell'(x) = r$ and the $n^\text{th}$ iterate of $\ell$ is $\ell^n(x)=r^nx$. Thus, it's easy to see that the the fixed point zero behaves exactly as described by the classification.
Often, this is illustrated with a cobweb plot:
In this picture, the blue line is the line $y=x$ and the yellow line is the graph of $\ell_r$. If a point $(x_i,x_i)$ on the line $y=x$ and moves vertically to the graph of the function, it arrives at the point
$$(x_i,f(x_i)) = (x_i,x_{i+1}).$$
If it then moves horizontally back to the line, it arrives at $(x_{i+1},x_{i+1})$. As a result, the process of moving vertically from the line $y=x$ to the graph of $y=f(x)$ and then horizontally back to the line is a geometric representation of one iteration of $f$. Furthermore, you can see from the picture exactly what attraction and repulsion look like.
Let's now apply these ideas to the logistic function:
$$f_r(x) = rx(1-x).$$
As you point out in your comment,
at 2.9 there is 1 steady value, at 3.4 there are 2 steady values, and at 3.6 there are a lot of steady values
More precisely, I would say that for $r$ a little less than 3, there is a single, attractive fixed point, and for $r$ a little more than 3, there is an attractive orbit of period two. At $r=3$ the fixed point is neutral; the family undergoes a bifurcation as $r$ increases through 3. To see why, take a look at the cobweb plots:
Note that for $r=2.9$, the slope of the graph at the fixed point appears to be close to $-1$ but a little less in absolute value. For $r=3.1$ it appears that the slope is larger than $-1$ in absolute value. In fact, you can show by direct computation that for $r=3$ exactly, $x_0=2/3$ is a fixed point and that $f_3'(2/3) = -1$.
That explains why the fixed point disappeared but, perhaps, not why the new attractive orbit appeared. To understand this next point it helps to add the graph of
$$f_r^2(x) = r^2 (1-x) x (1-r (1-x) x)$$
to the picture because the points in an attractive orbit of period 2 are fixed points of $f_r^2$.
Note that, after the bifurcation occurs, the graph of $f_r^2$ crosses the line $y=x$ twice with small slope.
As $r$ increases further, the same thing happens to the slope of $f_r^2$. You can prove that when $r=1+\sqrt{6}\approx3.44949$, $f_r^2$ has a neutral fixed point and another bifurcation occurs as $r$ passes through this value. Near that point, the picture looks something like so:
| 41,171
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\begin{document}
\maketitle
\begin{abstract}
A stochastic queueing network model with parameter-dependent service times and
routing mechanism, and its related performance measures are considered. An
estimate of performance measure gradient is proposed, and rather general
sufficient conditions for the estimate to be unbiased are given. A gradient
estimation algorithm is also presented, and its validity is briefly discussed.
\\
\textit{Key-Words:} queueing networks, parameter-dependent routing, performance measure gradient, unbiased estimate.
\end{abstract}
\section{Introduction}
The evaluation of performance measure gradient presents one of the main issues
of analysis of queueing network performance. Except in a few particular
models, there are generally no closed-form representations as functions of
network parameters available for performance measures and their gradients.
In this situation, one normally applies the Monte Carlo approach to estimate
gradient of network performance measures.
In the last decade, infinitesimal perturbation analysis (IPA)
\cite{HoYC83,HoYC87,Suri89} has received wide acceptance in queueing system
performance evaluation as an efficient technique underlying the calculation of
gradient estimates as well as the examination of their unbiasedness.
Specifically, this technique was employed in \cite{CaoX91} to calculate
gradient estimates in closed networks with an ordinary probabilistic routing
mechanism and general service time distributions. An extension of IPA,
smoothed perturbation analysis, has been applied in \cite{Gong92} to the
development of asymptotically unbiased gradient estimates in queueing networks
with parameter-dependent routing.
Another approach based on the analysis of algebraic representation of queueing
system dynamics and their performance has been implemented in
\cite{Kriv90a,Kriv90b,Kriv93}. This approach offers a convenient and unified
way of analytical study of gradient estimates, and it leads to computational
procedures closely similar to those of IPA. In this paper, based on this
approach, a rather general queueing network model with parameter-dependent
service times and routing mechanism is presented. For the performance
measures which one normally chooses in analysis of network performance, we
propose a gradient estimate, and give sufficient conditions for the estimate
to be unbiased. These conditions are rather general and normally met in
analysis of queueing network performance. Finally, an algorithm of estimating
gradient of a particular performance measure is presented, and its validity is
briefly discussed.
\section{The Underlying Network Model}
We consider a generalized model of a queueing network consisting of $ N $
nodes, with customers of a single class. As is customary in queueing network
models, customers are assumed to circulate through the network to receive
service at appropriate nodes. We do not restrict ourselves to a particular
type of nodes, it is suggested that any node may have a single server as well
as several servers operating either in parallel or in tandem.
Furthermore, there is a buffer with infinite capacity in each node, in which
customers are placed at their arrival to wait for service if it cannot be
initiated immediately. We assume the queue discipline underlying the operation
of any node to be first-come, first-served. Upon his service completion at one
node, each customer goes to another node chosen according to some routing
procedure described below. We suppose that the transition of any customer
between nodes requires no time, and he therefore arrives immediately into the
next node. Finally, we assume that the network starts operating at time zero;
at the initial time, the server at any node $ n $ is free, whereas its
buffer contains $ K_{n} $ customers, $ 0 \leq K_{n} \leq \infty $,
$ n=1,\ldots,N $.
We now turn to the formal description of the network dynamics from an
algebraic viewpoint, and then introduce randomness into the network model.
\subsection{Algebraic Description of Node Dynamics}
In a general sense, each node can be regarded as a processor which produces an
output sequence of departure times of customers from another two, an input and
control sequences formed respectively by the arrival times and the service
times of customers. Let us denote for every node $ n $,
$ 1 \leq n \leq N $, the $k$th arrival epoch to the node by $ A_{n}^{k} $,
and the $k$th departure epoch from the node by $ D_{n}^{k} $. Notice,
because the transition of customers from one node to another is immediate,
each $ A_{n}^{k} $ coincides with some $ D_{i}^{j} $ with the
exception of $ A_{n}^{k} = 0 $ for all $ k \leq K_{n} $. Finally, we
denote the service time associated with the $k$th service initiation in node
$ n $, by $ \tau_{n}^{k} $. The set of all service times
$ \mbox{\boldmath $T$}=\{\tau_{n}^{k} | n=1,\ldots,N; k=1,2,\ldots \} $
is assumed to be given.
The usual way to represent the operation of a node is based on recursive
equations describing evolution of $ D_{n}^{k} $ as a state variable
\cite{Chen90,HuJQ92,Kriv94b}. Note that these recursive equations are often
rather difficult to resolve. Below are given two equations which describe
dynamics of nodes operating as the $ G/G/1 $ and $ G/G/2 $ queueing
systems. Other examples may be found in \cite{Chen90,HuJQ92,Kriv94a,Kriv94b}.
\subsubsection{The $G/G/1$ queue.}
Suppose first that node $ n $ is represented as the $ G/G/1 $ queue.
Its associated recursive equation may be written as \cite{Chen90}
$$
D_{n}^{k} = (A_{n}^{k} \vee D_{n}^{k-1}) + \tau_{n}^{k},
$$
where $ \vee $ denotes the maximum operator, and
$ D_{n}^{k} \equiv 0 $ for all $ k < 0 $. It is easy to see that the
solution of the equation in terms of arrival and service times has the form
$$
D_{n}^{k}
= \bigvee_{i=1}^{k} \left(A_{n}^{i} + \sum_{j=i}^{k} \tau_{n}^{j} \right).
$$
\subsubsection{The $G/G/2$ queue.}
The equation which describes the dynamics of a node operating as the
$ G/G/2 $ queue may be considered as rather difficult to handle. For node
$ n $, it is written as \cite{Kriv94a}
$$
D_{n}^{k} =
\bigvee_{i=1}^{k} \left((A_{n}^{i}\vee D_{n}^{i-2}) + \tau_{n}^{i} \right)
\wedge \left((A_{n}^{k+1}\vee D_{n}^{k-1}) + \tau_{n}^{k+1} \right),
$$
where $ \wedge $ stands for the minimum operator. Although there are no
closed-form solutions of the equation, known to the author, it is clear that
it exists.
\subsection{Routing Mechanism and Interaction of Nodes}
The routing mechanism inherent in the network is defined by the sequences
$ \mbox{\boldmath $R$}_{n} = \{\rho_{n}^{1}, \rho_{n}^{2}, \ldots\} $
given for each node $ n $, where $ \rho_{n}^{k} $ represents the next
node to be visited by the customer who is the $k$th to depart from node
$ n $, $ \rho_{n}^{k} \in \{1, \ldots, N\} $, $ k=1,2, \ldots $. The
matrix
$$
\mbox{\boldmath $R$} =
\left( \begin{array}{ccccc}
\rho_{1}^{1} & \rho_{1}^{2} & \ldots & \rho_{1}^{k} & \ldots \\
\rho_{2}^{1} & \rho_{2}^{2} & \ldots & \rho_{2}^{k} & \ldots \\
\vdots & \vdots & & \vdots & \\
\rho_{N}^{1} & \rho_{N}^{2} & \ldots & \rho_{N}^{k} & \ldots \\
\end{array}
\right)
$$
is referred to as the routing table of the network.
In order to describe the dynamics of the network completely, it remains to
define formally interactions between nodes. In fact, a relationship between
arrival and departure times of distinct nodes is to be established. To this
end, for each node $ n $, let us introduce the set
\begin{equation} \label{D-def}
\mbox{\boldmath $D$}_{n}
= \{ D_{i}^{j} | \rho_{i}^{j} = n; i=1,\ldots,N; j=1,2,\ldots \}
\end{equation}
which is constituted by the departure times of the customers who have to go to
node $ n $. Furthermore, we denote by $ {\cal A}_{n}^{k} $ the arrival
time of the customer which is the $k$th to arrive into node $ n $ after
his service at any node of the network. In other words, the symbol
$ {\cal A}_{n}^{k} $ differs from $ A_{n}^{k} $ in that it refers only
to the customers really arriving into node $ n $, and does not to those
occurring in this node at the initial time.
It has been shown in \cite{Kriv90b,Kriv93} that it holds
\begin{equation} \label{A1-def}
{\cal A}_{n}^{k} =
\bigwedge_{\{D_{1},\ldots,D_{k}\} \subset \bm{D}_{n}}
(D_{1} \vee \cdots \vee D_{k}),
\end{equation}
where minimum is taken over all $k$-subsets of the set
$ \mbox{\boldmath $D$}_{n} $. The times $ A_{n}^{k} $ and
$ {\cal A}_{n}^{k} $ are related by the equality
\begin{equation} \label{A-def}
A_{n}^{k} = \left\{\begin{array}{ll}
0, & \mbox{if $ k \leq K_{n} $} \\
{\cal A}_{n}^{k-K_{n}}, & \mbox{otherwise}
\end{array}
\right..
\end{equation}
Clearly, if deterministic routing with an integer matrix $ R $ as the
routing table is adopted in the model, each set
$ \mbox{\boldmath $D$}_{n} $, $ n=1,\ldots,N $, is determined uniquely
from (\ref{D-def}), and then a straightforward algebraic representation of
$ A_{n}^{k} $ may be obtained using (\ref{A1-def}-\ref{A-def}). In this
case, starting from the above representations of node dynamics, one may
eventually arrive at algebraic expressions for any arrival time
$ A_{n}^{k} $ and departure time $ D_{n}^{k} $, which are written in
terms of service times $ \tau \in \mbox{\boldmath $T$} $, and involve only
the operations of maximum, minimum, and addition.
\subsection{Representation of Network Performance}
One of the features of the formal network model described above is that it
offers the potential for representing network performance criteria in a rather
simple and convenient way. Suppose that we observe the network until the $K$th
service completion at node $ n $, $ 1 \leq n \leq N $. As performance
criteria for node $ n $ in the observation period, one normally chooses
the following average quantities \cite{Chen90,Kriv90a,Kriv90b,Kriv93,Kriv94b}:
$$\begin{array}{ll}
\mbox{System time of one customer,} &
S_{n}^{K} = \sum_{k=1}^{K} (D_{n}^{k}-A_{n}^{k})/K, \\ \\
\mbox{Waiting time of one customer,} &
W_{n}^{K} = \sum_{k=1}^{K} (D_{n}^{k}-A_{n}^{k}-\tau_{n}^{k})/K, \\ \\
\mbox{Throughput rate of the node,} &
T_{n}^{K} = K/D_{n}^{K}, \\ \\
\mbox{Utilization of the server,} &
U_{n}^{K} = \sum_{k=1}^{K} \tau_{n}^{k}/D_{n}^{K}, \\ \\
\mbox{Number of customers,} &
J_{n}^{K} = \sum_{k=1}^{K} (D_{n}^{k}-A_{n}^{k})/D_{n}^{K}, \\ \\
\mbox{Queue length at the node,} &
Q_{n}^{K} = \sum_{k=1}^{K} (D_{n}^{k}-A_{n}^{k}-\tau_{n}^{k})/D_{n}^{K}.
\end{array}$$
It is easy to see that with the routing mechanism determined by an integer
matrix, all these criteria may be represented only in terms of service times
in closed form.
\subsection{Stochastic Aspect and Performance Evaluation}
Let us suppose that for all $ n=1,\ldots,N $, and $ k=1,2,\ldots $, the
service times are defined as random variables
$ \tau_{n}^{k} = \tau_{n}^{k}(\theta,\omega) $, where
$ \theta \in \Theta \subset \mathbb{R} $ is a decision parameter,
$ \omega $ is a random vector which represents the random effects involved
in network behaviour. First we assume the routing table
$ \mbox{\boldmath $R$} = R $ to be an integer matrix. Since deterministic
routing leads to algebraic expressions in terms of the random variables
$ \tau \in \mbox{\boldmath $T$} $ for the performance criteria introduced
above, one can conclude that these criteria also present random variables.
Let $ F=F(\theta,\omega) $ be a random performance criterion of the
network. As is customary, we define the performance measure associated with
$ F $ as the expected value
\begin{equation} \label{E-def}
\mbox{\boldmath $F$}(\theta) = E_{\omega}[F(\theta,\omega)].
\end{equation}
Although we may express $ F $ in closed form, it is often very difficult
or impossible to obtain analytically the performance measure
$ \mbox{\boldmath $F$} $. In this situation, one generally applies a
simulation technique which allows of obtaining values of
$ F(\theta,\omega) $, and then estimate the network performance by using the
Monte Carlo approach.
We now turn to the networks with parameter-dependent probabilistic routing. We
assume $ \rho_{n}^{k} = \rho_{n}^{k}(\theta,\omega) $ to be a discrete
random variable ranging over the set $ \{1,\ldots,N\} $. The routing
mechanism of the network is now defined by the random matrix
$ \mbox{\boldmath $R$} = \mbox{\boldmath $R$}(\theta,\omega) $ with
particular routing tables as its values. We denote the set of all possible
routing tables $ R $ by $ {\cal R} $.
Obviously, the expression of $ A_{n}^{k} $ defined by
(\ref{A1-def}-\ref{A-def}) may change from one shape into another, depending
on particular routing tables. To take this into account, we now define the
random performance criteria in (\ref{E-def}) as \cite{Kriv90b,Kriv93}
\begin{equation} \label{F-def}
F(\theta,\omega,\mbox{\boldmath $R$}(\theta,\omega)) =
\sum_{R\in{\cal R}} {\bf 1}_{\{\bm{R}(\theta,\omega) = R\}}
F_{R}(\theta,\omega),
\end{equation}
where $ {\bf 1}_{\{\bm{R}(\theta,\omega)=R\}} $ is the indicator
function of the event $ \{\mbox{\boldmath $R$}(\theta,\omega) = R\} $, and
$ F_{R}(\theta,\omega) = F(\theta,\omega,R) $ is the performance criterion
evaluated under the condition that the network operates according to the
deterministic routing mechanism defined by the routing table
$ R \in {\cal R} $.
\section{Performance Measure Gradient Estimation}
Since there are generally no explicit representations as functions of system
parameters $ \theta $, available for the performance measure
$ \mbox{\boldmath $F$} $, one may evaluate its gradient
$ \partial\mbox{\boldmath $F$}(\theta)/\partial\theta $ by no way other
than through the use of either finite difference estimates
\cite{CaoX85,HoYC87} or the estimate
\begin{equation} \label{g-def}
g(\theta,\omega_{1},\ldots,\omega_{M})
= \frac{1}{M} \sum_{i=1}^{M} \frac{\partial}{\partial\theta}
F(\theta,\omega_{i}),
\end{equation}
where $ \omega_{i} $, $ i=1,\ldots,M $, are independent realization of
$ \omega $, provided that the derivative
$ \partial F(\theta,\omega) / \partial\theta $ exists.
Very efficient procedures of obtaining gradient estimates may be designed
using the IPA technique \cite{HoYC83,HoYC87,Suri89}. Such a procedure can
yield the exact values of the derivative
$ \partial F(\theta,\omega) / \partial\theta $ by performing only one
simulation run. Furthermore, in the case of a vector parameter
$ \theta \in \mathbb{R}^{d} $, the IPA procedures provide all partial
derivatives $ \partial F(\theta,\omega) / \partial \theta_{i} $,
$ i=1,\ldots,d $, simultaneously, and take an additional computational cost
which is normally very small compared with that required for the simulation
run alone. Finally, it can be easily shown \cite{CaoX85,Suri89} that if the
IPA estimate of the derivative in (\ref{g-def}) is unbiased, the mean square
error of $ g $ has the order which is significantly less than those of any
finite difference estimates based on the same number of simulation runs.
A sufficient condition for the estimate (\ref{g-def}) to be unbiased at some
$ \theta \in \Theta $ requires \cite{CaoX85,HoYC87,Suri89}
\begin{equation} \label{C-def}
\frac{\partial}{\partial\theta} E[F(\theta,\omega)]
= E\left[ \frac{\partial}{\partial\theta} F(\theta,\omega) \right].
\end{equation}
A usual way of examining the interchange in (\ref{C-def}) involves the
application of the Lebesgue dominated convergence theorem \cite{Loev60}
\begin{theorem} \label{L-t}
Let $ (\Omega,{\cal F},P) $ be a probability space,
$ \Theta \subset \mathbb{R}^{d} $, and
$ F: \Theta \times \Omega \rightarrow \mathbb{R} $ be a
${\cal F}$-measurable function for any $ \theta \in \Theta $ and such that
the following conditions hold:
\begin{description}
\item{{\rm (i)}} for every $ \theta \in \Theta $, there exists
$ \partial F(\theta,\omega) / \partial\theta $ at $ \theta $ w.p.~1,
\item{{\rm (ii)}} for all $ \theta_{1}, \theta_{2} \in \Theta $, there is
a random variable $ \lambda(\omega) $ with $ E\lambda < \infty $ and
such that
\begin{equation} \label{L-def}
| F(\theta_{1},\omega)-F(\theta_{2},\omega) |
\leq \lambda(\omega) \parallel \theta_{1}-\theta_{2} \parallel \;\;\;
\mbox{w.p.~1.}
\end{equation}
\end{description}
Then equation {\rm (\ref{C-def})} holds on $ \Theta $.
\end{theorem}
In \cite{Kriv90a,Kriv90b,Kriv93} the approach based on the implementation of
Theorem~\ref{L-t} has been applied to analyze estimates of performance
gradient in the networks models with the parameter-independent probabilistic
routing mechanism determined by a random routing table
$ \mbox{\boldmath $R$} = \mbox{\boldmath $R$}(\omega) $. Specifically,
starting from the representations of network dynamics, discussed in Section~2,
it has been shown that
\begin{description}
\item{{\rm (i)}} if each service time $ \tau \in \mbox{\boldmath $T$} $
satisfies the conditions of Theorem~\ref{L-t}, and for every
$ \theta \in \Theta $, all $ \tau \in \mbox{\boldmath $T$} $ present
continuous and independent random variables, then the average total time
$ S_{n}^{K} $ and waiting time $ W_{n}^{K} $ satisfy the conditions of
Theorem~\ref{L-t};
\item{{\rm (ii)}} if in addition to previous assumptions, there exist random
variables $ \mu, \nu > 0 $ such that for all
$ \tau \in \mbox{\boldmath $T$} $ it holds $ \nu \leq |\tau| \leq\mu $
w.p.~1 for all $ \theta \in \Theta $, and the condition
$ E[\mu\lambda/\nu^{2}] < \infty $ is fulfilled, where $ \lambda $ is
the random variable providing $ \tau $ with (\ref{L-def}), then the
average throughput rate $ T_{n}^{K} $, utilization $ U_{n}^{K} $, number
of customers $ J_{n}^{K} $, and queue length $ Q_{n}^{K} $ satisfy the
conditions of Theorem~\ref{L-t}.
\end{description}
Note that the above conditions do not involve independence at each
$ \theta $ between the random variables $ \tau(\theta,\omega) $ and
$ \rho(\omega) $ in the probabilistic sense.
\section{An Unbiased Gradient Estimate for Networks with
Parameter-Dependent Routing}
We start the section with an example which exhibits difficulties arising in
gradient estimation when there is a parameter dependence involved in the
routing mechanism of the network, and then present our main result offering an
unbiased estimate of performance measure gradient in networks with
parameter-dependent routing.
\subsection{Preliminary Analysis}
Suppose that there is a parameter dependence of the routing mechanism in the
network, that is
$ \mbox{\boldmath $R$} = \mbox{\boldmath $R$}(\theta,\omega) $. In this
case, the random performance criterion $ F $ generally violates condition
(\ref{L-def}). As an illustration, one can consider the following example.
Let $ \Theta = [0,1] $, $ \Omega_{1}=\Omega_{2}=[0,1] $, and
$ (\Omega,{\cal F},P) $ be a probability space, where $ {\cal F} $ is
the $\sigma$-field of Borel sets of
$ \Omega = \Omega_{1} \times \Omega_{2} $, $ P $ is the Lebesgue measure
on $ \Omega $. Denote $ \omega = (\omega_{1},\omega_{2})^{T} $, and define
the function
$$
F(\theta,\omega,\rho(\theta,\omega))
= \left\{\begin{array}{ll}
\tau_{1}(\theta,\omega), & \mbox{if $ \rho(\theta,\omega)=1 $} \\
\tau_{2}(\theta,\omega), & \mbox{if $ \rho(\theta,\omega)=2 $}
\end{array},
\right.
$$
where
$$
\tau_{1}(\theta,\omega) = \theta+\omega_{1}+1, \;\;\;
\tau_{2}(\theta,\omega) = \theta+\omega_{1},
$$
and
$$
\rho(\theta,\omega)
= \left\{\begin{array}{ll}
1, & \mbox{if $ \omega_{2} \leq \theta $} \\
2, & \mbox{if $ \omega_{2} > \theta $}
\end{array}
\right..
$$
We may treat $ \tau_{1} $ and $ \tau_{2} $ as the service time of a
customer respectively at node $ 1 $ and $ 2 $. The function $ F $ is
then assumed to be the service time of the customer which may arrive into
either node $ 1 $ or $ 2 $, according to one of the two possible values
of $ \rho $.
Clearly, $ \tau_{1} $ and $ \tau_{2} $ satisfy the conditions of
Theorem~\ref{L-t}, whereas the function $ F $ now represented as
$$
F(\theta,\omega)
= \left\{\begin{array}{ll}
\theta+\omega_{1}+1, & \mbox{if $ \omega_{2} \leq \theta $} \\
\theta+\omega_{1}, & \mbox{if $ \omega_{2} > \theta $}
\end{array}
\right.,
$$
is differentiable w.p.~1 at any $ \theta \in \Theta $, and
$ \partial F(\theta,\omega) / \partial\theta = 1 $ w.p.~1. However, for
any $ \theta_{1},\theta_{2} \in \Theta $ such that
$ \theta_{1} \geq \omega_{2} $ and $ \theta_{2} < \omega_{2} $, it holds
$$
| F(\theta_{1},\omega)-F(\theta_{2},\omega) | \geq 1,
$$
and therefore, condition (\ref{L-def}) is violated.
On the other hand, it is easy to verify that
$$
\begin{array}{lcl}
E[F(\theta,\omega)] = 2\theta+\frac{1}{2}, & &
\frac{\partial}{\partial\theta} E[F(\theta,\omega)] = 2 \\ \\
\frac{\partial}{\partial\theta} F(\theta,\omega) = 1 \;\; \mbox{w.p.~1}, & &
E\left[\frac{\partial}{\partial\theta} F(\theta,\omega) \right] = 1.
\end{array}
$$
In other words, equation (\ref{C-def}) proves to be not valid, and we finally
conclude that estimate (\ref{g-def}) will be biased.
\subsection{The Main Result}
To suppress the bias in estimates of the gradient
\begin{equation} \label{gF-def}
\frac{\partial}{\partial\theta}\mbox{\boldmath $F$}(\theta)
= \frac{\partial}{\partial\theta}
E[F(\theta,\omega,\mbox{\boldmath $R$}(\theta,\omega))],
\end{equation}
let us replace (\ref{g-def}) by the estimate
\begin{equation} \label{h-def}
{\widetilde g}(\theta,\omega_{1},\ldots,\omega_{M})
= \frac{1}{M} \sum_{i=1}^{M} G(\theta,\omega_{i}),
\end{equation}
where $ G(\theta,\omega) $ will be defined in the next theorem.
\begin{theorem} \label{M-t}
Suppose that a random performance criterion $ F $ is represented in
form {\rm (\ref{F-def})}, and for each $ R \in {\cal R} $ the following
conditions hold:
\begin{description}
\item{{\rm (i)}} $ F_{R} $ satisfies the conditions of Theorem~\ref{L-t};
\item{{\rm (ii)}} for any $ \theta \in \Theta $, the random variable
$ F_{R}(\theta,\omega) $ and the random matrix
$ \mbox{\boldmath $R$}(\theta,\omega) $ are independent;
\item{{\rm (iii)}} for any $ \theta \in \Theta $, the function
$ \Phi(\theta,R)
= \mbox{\rm Pr}\{\mbox{\boldmath $R$}(\theta,\omega)=R\} $ is continuously
differentiable at $ \theta $, and $ \Phi(\theta,R) > 0 $.
\end{description}
Then for any $ \theta_{0} \in \Theta $, estimate {\rm (\ref{h-def})} with
$$
G(\theta_{0},\omega) = \left. \frac{\partial}{\partial\theta}
F(\theta,\omega,\mbox{\boldmath $R$}(\theta_{0},\omega))
\right|_{\theta=\theta_{0}}
+ F(\theta_{0},\omega,\mbox{\boldmath $R$}(\theta_{0},\omega))
\Psi(\theta_{0},\mbox{\boldmath $R$}(\theta_{0},\omega)),
$$
where $ \Psi(\theta,R) = \frac{\partial}{\partial\theta}\ln\Phi(\theta,R) $,
is unbiased.
\end{theorem}
\begin{proof}
Clearly, it is sufficient to show that the equation
$$
\frac{\partial}{\partial\theta}
E[F(\theta,\omega,\mbox{\boldmath $R$}(\theta,\omega))]
= E[G(\theta,\omega)]
$$
holds for any $ \theta \in \Theta $.
To verify this equation, let us first represent $ F $ in form
(\ref{F-def}), and consider its expected value
$$
E[F(\theta,\omega,\mbox{\boldmath $R$}(\theta,\omega))]
= \sum_{R\in{\cal R}} E\left[{\bf 1}_{\{\bm{R}(\theta,\omega) = R\}}
F(\theta,\omega,R)\right].
$$
Since $ E[{\bf 1}_{\{\bm{R}(\theta,\omega) = R\}}]
= \mbox{\rm Pr}\{\mbox{\boldmath $R$}(\theta,\omega) = R\}
= \Phi(\theta,R) $, it follows from condition~(ii) of the theorem that
\begin{eqnarray*}
\lefteqn{E[F(\theta,\omega,\mbox{\boldmath $R$}(\theta,\omega))]}
\;\;\;\;\;\;\;\;\;\; \\
& = & \sum_{R\in{\cal R}} E[F(\theta,\omega,R)]
\mbox{\rm Pr}\{\mbox{\boldmath $R$}(\theta,\omega) = R\}
= \sum_{R\in{\cal R}} E[F(\theta,\omega,R)] \Phi(\theta,R).
\end{eqnarray*}
For any $ \theta_{0} \in \Theta $, under conditions (i) and (iii), we
successively get
\begin{eqnarray*}
\lefteqn{\left. \frac{\partial}{\partial\theta}
E[F(\theta,\omega,\mbox{\boldmath $R$}(\theta,\omega))]
\right|_{\theta=\theta_{0}}} \\
& = & \sum_{R\in{\cal R}} \left( \left. \frac{\partial}{\partial\theta}
E[F(\theta,\omega,R)]\right|_{\theta=\theta_{0}} \Phi(\theta_{0},R)
\right. \\
& & \left. \mbox{} + E[F(\theta_{0},\omega,R)] \left.
\frac{\partial}{\partial\theta} \Phi(\theta,R)
\right|_{\theta=\theta_{0}} \right) \\
& = & \sum_{R\in{\cal R}} \left( E\left[\left. \frac{\partial}{\partial\theta}
F(\theta,\omega,R)]\right|_{\theta=\theta_{0}} \right] \Phi(\theta_{0},R)
\right. \\
& & \left. \mbox{} + E[F(\theta_{0},\omega,R)]
\frac{\left. \frac{\partial}{\partial\theta} \Phi(\theta,R)
\right|_{\theta=\theta_{0}}}
{\Phi(\theta_{0},R)} \Phi(\theta_{0},R) \right) \\
& = & \sum_{R\in{\cal R}} \Biggl( E\left[ \left. \frac{\partial}{\partial\theta}
F(\theta,\omega,R)\right|_{\theta=\theta_{0}} \right]
+ E[F(\theta_{0},\omega,R)] \Psi(\theta_{0},R) \Biggr)
\Phi(\theta_{0},R) \\
& = & \sum_{R\in{\cal R}} E\Biggl[ \left. \frac{\partial}{\partial\theta}
(\theta,\omega,R)\right|_{\theta=\theta_{0}}
+ F(\theta_{0},\omega,R) \Psi(\theta_{0},R) \Biggr]
\mbox{\rm Pr}\{\mbox{\boldmath $R$}(\theta_{0},\omega) = R\} \\
& = & E \Biggl[ \left. \frac{\partial}{\partial\theta}
F(\theta,\omega,\mbox{\boldmath $R$}(\theta_{0},\omega))
\right|_{\theta=\theta_{0}}
+ F(\theta_{0},\omega,\mbox{\boldmath $R$}(\theta_{0},\omega))
\Psi(\theta_{0},\mbox{\boldmath $R$}(\theta_{0},\omega)) \Biggr] \\
& = & E[G(\theta_{0},\omega)].
\qedhere
\end{eqnarray*}
\end{proof}
It is not difficult to obtain the conditions for estimate (\ref{h-def}) to be
unbiased for gradient of particular performance measures. They can be stated
by combining the conditions in Section~3, related to particular
performance criteria in networks with parameter-independent routing, with
those of Theorem~\ref{M-t}. Note that these conditions are rather general, and
normally met in analysis of queueing networks.
Let us now return to the example presented in the previous subsection. First,
we have
$$ \Phi(\theta,1) = \mbox{\rm Pr}\{\rho(\theta,\omega)=1\}=\theta, $$
$$ \Phi(\theta,2) = \mbox{\rm Pr}\{\rho(\theta,\omega)=2\}=1-\theta, $$
and then
$$ \Psi(\theta,1) = \frac{d}{d\theta}\ln\theta=\frac{1}{\theta}, $$
$$ \Psi(\theta,2) = \frac{d}{d\theta}\ln(1-\theta)=\frac{1}{1-\theta}. $$
In this case, the function $ G $ is defined as
$$
G(\theta,\omega)
= \left\{\begin{array}{ll}
1+\frac{\theta+\omega_{1}+1}{\theta}, &
\mbox{if $ \omega_{2} \leq \theta $} \\ \\
1+\frac{\theta+\omega_{1}}{\theta-1}, &
\mbox{if $ \omega_{2} > \theta $}
\end{array}
\right.
$$
for any $ \theta \in (0,1) $. Finally, evaluation of its expected value
gives
$$
E[G(\theta,\omega)]
= \frac{\partial}{\partial\theta}
E\left[F(\theta,\omega,\rho(\theta,\omega))\right] = 2.
$$
\section{Application to Network Simulation}
Consider a network with $ N $ single-server nodes, and assume that the
$K$th service completion at a fixed node $ n $, $ 1 \leq n \leq N $, comes
with probability one after a finite number of service completions in the
network. In this case, to observe evolution of the network until the $K$th
completion at the node, it will suffice to take into consideration only finite
routes defined by a right truncated routing table with integer
($N\times L$)-matrices
$$
R = \left( \begin{array}{cccc}
r_{1}^{1} & r_{1}^{2} & \ldots & r_{1}^{L} \\
r_{2}^{1} & r_{2}^{2} & \ldots & r_{2}^{L} \\
\vdots & \vdots & & \vdots \\
r_{N}^{1} & r_{N}^{2} & \ldots & r_{N}^{L}
\end{array}
\right)
$$
as its values, with some $ L \geq K $.
Furthermore, we assume, as is customary in network simulation, that for each
$ \theta \in \Theta $, the random variables
$ \rho_{n}^{k}(\theta,\omega) $ are independent for all
$ n=1, \ldots, N $, and $ k=1, \ldots, L $. With this condition and the
notation $ \varphi_{n}^{k}(\theta,r) =
\mbox{\rm Pr}\{\rho_{n}^{k}(\theta,\omega) = r\} $, we may represent the
function $ \Phi $ introduced in Theorem~\ref{M-t}, as
$$
\Phi(\theta,R) = \mbox{\rm Pr}\{\mbox{\boldmath $R$}(\theta,\omega) = R\}
= \prod_{n=1}^{N} \prod_{k=1}^{L} \varphi_{n}^{k}(\theta,r_{n}^{k}),
$$
and then get the function $ \Psi $ in the form
$$
\Psi(\theta,R) = \frac{\partial}{\partial\theta}\ln\Phi(\theta,R)
= \sum_{n=1}^{N} \sum_{k=1}^{L}
\frac{\partial}{\partial\theta}\ln\varphi_{n}^{k}(\theta,r_{n}^{k}).
$$
Suppose now that we have to estimate the gradient of a performance measure,
say $ \mbox{\boldmath $U$}_{n}^{K}(\theta) = E[U_{n}^{K}(\theta,\omega)] $,
the expected value of the average utilization of the server at node $ n $.
It results from Theorem~\ref{M-t} that, as an unbiased estimate, the function
\begin{equation} \label{G-def}
G(\theta,\omega) = \frac{\partial}{\partial\theta} F(\theta,\omega,R)
+ F(\theta,\omega,R) \Psi(\theta,R),
\end{equation}
may be applied with $ R = \mbox{\boldmath $R$}(\theta,\omega) $, and
$$
F(\theta,\omega,R)
= \sum_{k=1}^{K} \tau_{n}^{k}(\theta,\omega) / D_{n}^{K}(\theta,\omega,R).
$$
It is not difficult to construct the next algorithm which produces the value
of $ G(\theta,\omega) $ for fixed
$ \theta \in \Theta \subset \mathbb{R} $, and $ \omega \in \Omega $,
provided that there is a network simulation procedure into which the algorithm
may be incorporated. It actually combines an IPA algorithm \cite{HoYC83} for
obtaining the derivative $ \partial F(\theta,\omega,R)/\partial\theta $
with additional computations according to (\ref{G-def}).
\begin{algorithm}{5.1}
{\mdseries\itshape Initialization:} \\
for $ i=1,\ldots, N $ do $ g_{i} \longleftarrow 0 $; \\
$ s,t,t^{\prime} \longleftarrow 0 $; \\
$ R \longleftarrow \mbox{\boldmath $R$}(\theta,\omega) $; \\ \\
{\mdseries\itshape Upon the $k$th service completion at node $ i $, perform the instructions:} \\
$ g_{i} \longleftarrow g_{i}
+ \frac{\partial}{\partial\theta} \tau_{i}^{k}(\theta,\omega) $; \\
if \= $ i = n $ then \=
$ t \longleftarrow t + \tau_{i}^{k}(\theta,\omega) $; \\
\> \> $ t^{\prime} \longleftarrow t^{\prime}
+ \frac{\partial}{\partial\theta} \tau_{i}^{k}(\theta,\omega) $; \\
\> \> if $ k = K $ then \=
$ d \longleftarrow D_{n}^{K}(\theta,\omega,R) $; \\
\> \> \> stop; \\
$ r \longleftarrow r_{i}^{k} $; \\
$ s \longleftarrow s
+ \frac{\partial}{\partial\theta}\ln\varphi_{i}^{k}(\theta,r) $; \\
if {\mdseries\itshape the server of node $ r $ is free} then
$ g_{r} \longleftarrow g_{i} $.
\end{algorithm}
On completion of the algorithm, it remains to compute
$ (t^{\prime}d - t g_{n})/d^{2} + t s/d $ as the value of $ G $.
Note, in conclusion, that estimate (\ref{g-def}) with the function $ G $
evaluated using the algorithm will not be unbiased in general. For the
estimate to be unbiased, the function $ \Psi(\theta,R) $ in (\ref{G-def})
must be calculated as the sum with the same number of summands
$ \partial \ln\varphi_{n}^{k}(\theta,r)/\partial\theta $ for any of
simulation runs. However, during the simulation runs with distinct
realizations of $ \omega $, there may be different numbers of service
completions encountered at nodes $ i \neq n $, and then considered in
evaluation of $ \Psi $. This normally involves an insignificant error in
estimating the gradient, which becomes inessential as $ K $ increases.
\section{Acknowledgements}
The research described in this publication was made possible in part by Grant
\# NWA000 from the International Science Foundation.
\bibliographystyle{utphys}
\bibliography{Unbiased_gradient_estimation_in_queueing_networks_with_parameter-dependent_routing}
\end{document}
| 128,748
|
TITLE: variational problem with constraints
QUESTION [1 upvotes]: Let me bring to your attention the following problem.
Suppose we have the functional
$$ F = \int\limits_{a}^{b} f(y(x))\cdot\frac{dy}{dx} dx .$$
It is easy to see that that the Euler-Lagrange equation vanishes identically, in other words, every function $y (x)$ is an extremal of the functional. If we write the Euler-Lagrange equation we can see, that it is an identity.
Еvery Lagrangian of the form "function multiplied by the derivative of the 1st order" has this property (so called zero Lagrangian). However, the question arises whether the extremum is minimum or maximum. The second variation for it is identically zero, IMHO. Weierstrass function too. The question is how to determine whether the extremal is minimum or maximum?
Let us complicate the problem. There is a functional
$$ F = \int\limits_{a}^{b} n(y(x))\cdot f(y(x))\cdot\frac{dy}{dx} dx $$
with differential constraint
$ (\frac{dy}{dx} / n(y(x))) - 1 = 0 $ - DE.
We write the Lagrangian for this problem
$$
L1 = n(y(x))\cdot f(y(x))\cdot\frac{dy}{dx} + \lambda(x) \cdot ((\frac{dy}{dx} / n(y(x))) - 1 )$$
It is easy to see that this is "zero Lagrangian" and the EL equation becomes an identity. Can we know whether it is a minimum or maximum? How to find the Lagrange multiplier in such case?
If we now use the constraint (DE) and replace in the integral n(y) on the derivative, we obtain the Lagrangian which is quadratic in the derivative
$$
L2 = f(y(x))\cdot(\frac{dy}{dx})^2 + \lambda(x) \cdot ((\frac{dy}{dx} / n(y(x))) - 1 )$$
EL equation is no longer an identity:
$$
\frac{d( f(y(x))\cdot (\dot{y})^2)}{dx} + \dot{\lambda} =0 $$ or
$$
f(y(x))\cdot (\dot{y})^2 + {\lambda(x)} =0 .$$ Very strange EL equation. From my point of view, it means the following. Lagrange multiplier is a function only of the argument $x$, then it (and therefore the Lagrangian, as can be seen from equation) can be represented as a derivative with respect to $x$ of a function $p$. And this is in accordance with the well-known theorem sufficient criterion of "zero Lagrangian" (EL equation is an identity). The Weierstrass function also is non-zero.
Both of these problems (before the substitution of the equation in integral and after) are, I think, the same problem because, well, result can not depend on the arbitrary recording method. This is the apparent paradox.
Thank you for your help.
REPLY [1 votes]: You are aware that if $F$ is an anti-derivative of $f$, $F'(y)=f(y)$ then by the substitution rule, which is the counter-part of the chain rule
$$
I = \int\limits_{a}^{b} f(y(x))\cdot\frac{dy}{dx} dx =\int_a^b\frac{d}{dx}F(y(x))\,dx=F(y(b))-F(y(a)) .
$$
so that the independence to the curve under constant end-points $y(a)=y_a$, $y(b)=y_b$, is rather trival.
In the same way, if $H(y)$ is an anti-derivative to $n(y)\cdot f(y)$ then
$$
I = \int\limits_{a}^{b} n(y(x))\cdot f(y(x))\cdot\frac{dy}{dx} dx=H(y(b))-H(y(a))
$$
i.e., this case is not really different from the first one.
Forcing additionally $y'(x)=n(y(x))$ via Lagrange-multipliers makes the existence of admissible functions an accident, the IVP $y(a)=y_a$ rarely satisfies $y(b)=y_b$.
The further transformations do not, IMO, change this problem.
| 174,307
|
\vspace{-.2in}
\subsection{Last-Iterate Convergence of EG with Arbitrary Convex Constraints}\label{sec:last-iterate EG}
We establish the last-iterate convergence rate of the EG algorithm in the constrained setting in this section. The plan is similar to the one in Section~\ref{sec:warm up}. First, we use the assistance of SOS programming to prove the monotonicity of the \projham (Theorem~\ref{thm:monotone hamiltonian}), then combine it with the best-iterate convergence guarantee from Lemma~\ref{lem:best iterate hamiltonian} to derive the last-iterate convergence rate (Theorem~\ref{thm:EG last-iterate constrained}).
The crux is proving the monotonicity of the \projhamnospace. Due to the constraints introduced by the domain $\Z$, our approach in the unconstrained case no longer applies.
More specifically, the coordinates are now entangled due to projection, ruining the two key properties -- Property~\ref{property:unconstrained one} and~\ref{property:unconstrained two} that we rely on in the unconstrained setting. We show how to reduce \emph{both the number of constraints and the dimension by exploiting the low dimensionality of the EG update rule}. Equipped with the reduction, we convert the problem of proving the monotonicity of the \projham to solving a small constant size SOS program.
\paragraph{Reducing the Number of Constraints.}
Suppose we are not given the description of $\Z$, and we only observe one iteration of the EG algorithm. In other words, we know $z_k$, $z_{k+\half}$, and $z_{k+1}$, as well as $F(z_k)$, $F(z_{k+\half})$, and $F(z_{k+1})$. To compute the squared \projham at $z_k$, let us also assume that the unit vector $-a_k\in \unitnormal(z_k)$ satisfies $r^{tan}_{(F,\Z)}(z_k)^2=\InNorms{F(z_k)}^2-\InAngles{F(z_k),-a_k}^2$. From this limited information, what can we learn about $\Z$? We can conclude that $\Z$ must lie in the intersection of the following halfspaces: (a) $\InAngles{a_k,z}\geq b_k$, where $b_k=\InAngles{a_k,z_k}$. This is true because $-a_k\in \unitnormal(z_k)$. (b) $\InAngles{a_{k+\half},z}\geq b_{k+\half}$, where $a_{k+\half}=\frac{z_{k+\half}-z_k+\eta F(z_k)}{\InNorms{z_{k+\half}-z_k+\eta F(z_k)}}$ and $b_{k+\half}=\InAngles{a_{k+\half},z_{k+\half}}$. This is true because $z_{k+\half}=\Pi_\Z(z_k-\eta F(z_k))$, so $\InAngles{z_{k+\half}-z_k+\eta F(z_k), z-z_{k+\half}}\geq 0$ for all $z\in \Z$. (c) $\InAngles{a_{k+1},z}\geq b_{k+1}$, where $a_{k+1}=\frac{z_{k+1}-z_k+\eta F(z_{k+\half})}{\InNorms{z_{k+1}-z_k+\eta F(z_{k+\half})}}$ and $b_{k+1}=\InAngles{a_{k+1},z_{k+1}}$. This is true because $z_{k+1}=\Pi_\Z(z_k-\eta F(z_{k+\half}))$, so $\InAngles{z_{k+1}-z_k+\eta F(z_{k+\half}), z-z_{k+1}}\geq 0$ for all $z\in \Z$. See Figure~\ref{fig:geometry of EG}
for illustration.
\begin{figure}[ht]
\centering
\includegraphics[width=0.75\textwidth]{COLT/eg_update.pdf}
\caption{We illustrate how to reduce the number of constraints. Let $\Z' = \left\{z:\InAngles{a_{k+\frac{i}{2}},z}\geq b_{k+\frac{i}{2}}, \text{ for $i\in \{0,1,2\}$} \right\}$. Pictorially $-F(z_{k+1})\in \normal_\Z (z_{k+1})$,
while $-F(z_{k+1})\notin \normal_{\Z'} (z_{k+1})$
which further implies $\Ham_{(F,\Z)}(z_{k+1})=0$ and $\Ham_{(F,\Z')}(z_{k+1})> 0$.}
\label{fig:geometry of EG}
\end{figure}
The ``hardest instance'' of $\Z$ that is consistent with our knowledge of $z_k$, $z_{k+\half}$, and $z_{k+1}$ is when $\Z$ is exactly the intersection of these three halfspaces. In such case, the squared \projham of $z_{k+1}$ is $\InNorms{F(z_{k+1})}^2-\InAngles{F(z_{k+1}),a_{k+1}}^2\cdot \ind[\InAngles{F(z_{k+1}),a_{k+1}}\geq 0]$, and it is an upper bound of $r^{tan}_{(F,\Z)}(z_{k+1})^2$ for any other consistent $\Z$. Our goal is to prove the \projham is non-increasing even in the "hardest case", that is, to prove the non-negativity of
\vspace{-.1in}
\begin{align}\label{eq:hardest instance}
\InNorms{F(z_k)}^2-\InAngles{F(z_k),a_k}^2-\left(\InNorms{F(z_{k+1})}^2-\InAngles{F(z_{k+1}),a_{k+1}}^2\cdot \ind[\InAngles{F(z_{k+1}),a_{k+1}}\geq 0]\right)
\end{align}
\vspace{-.2in}
\paragraph{Low-dimensionality of an EG Update.} As there are only three hyperplanes $\InAngles{a_i,z}\geq b_i$ for $i\in\{k,k+\half,k+1\}$ involved, we can choose a new basis, so that $a_{k+1}=(1,0,\ldots, 0)$, $a_{k+\half}=(\theta_1,\theta_2,0,\ldots, 0)$, and $a_{k}=(\sigma_1,\sigma_2,\sigma_3,0,\ldots, 0)$. As $a_{k+\half}$ and $z_{k+\half}-z_k+\eta F(z_k)$ are co-directed, and $a_{k+1}$ and $z_{k+1}-z_k+\eta F(z_{k+\half})$ are co-directed, \emph{an important property of this change of basis is that the EG update from $z_k$ to $z_{k+1}$ is unconstrained in all coordinates $\ell\geq 4$.} More specifically,
\vspace{-.1in}
$$z_{k+\frac{1}{2}}[\ell]- z_k[\ell]+\eta F(z_{k})[\ell]=0,\quad z_{k+1}[\ell] - z_k[\ell] + \eta F(z_{k+\frac{1}{2}})[\ell]=0,~\forall \ell\geq 4.$$
\vspace{-.1in}
Hence, we can represent all of the coordinates $\ell\geq 4$ with one coordinate in the SOS program similar to the unconstrained case. We still need to keep the first three dimensions, but now we only face a problem in dimension $4$ rather than in dimension $n$, and we can form a constant size SOS program to search for a certificate of non-negativity for Expression~\eqref{eq:hardest instance}.
\notshow{
In Lemma~\ref{lem:reduce to cone} we prove that if for two consecutive iterations of the EG method the projected Hamiltonian increases, then it guarantees the existence of a finite number of vectors $\bx=\{\bx_i\}_{i\in[n]}$ that satisfy a set of polynomial inequalities $\{g_i(\bx)\leq 0\}_{i\in[m]}$ and certify that a polynomial $p(\bx) < 0$ can be strictly negative.
Then in Theorem \ref{thm:monotone hamiltonian} we show that the polynomial $p(\bx)$ is always non-negative over the semi-algebraic group $\mathcal{S}=\{g_i(x) \leq 0\}_{i\in[m]}$, which leads to a contradiction.
Vectors $\bx=\{\bx_i\}_{i\in[n]}$ that are guaranteed to exist are directly inspired by the original instance where the projected Hamiltonian in increasing between two consecutive iterations and corresponds to a low-dimensional instance of a VI problem,
where the projected Hamiltonian across two carefully selected consecutive iterations is also increasing.
Polynomial $p(\cdot)$ and $\{g_i(\cdot)\}_{i\in[m]}$ do not depend on the original instance and encode properties that the original instance satisfies.
More specifically polynomial $p(\cdot)$ encodes that the projected Hamiltonian among the two consecutive iterations increases and polynomials $\{g_i(\cdot)\}_{i\in[m]}$ encode a subset of properties of the EG method (e.g. that operator $F(\cdot)$ is Lipschitz etc.).
Before stating Lemma~\ref{lem:reduce to cone},
we would like to provide intuition behind the defined vectors and imposed constraints.
In a high-level idea,
we construct a new instance of the VI problem on operator $\bF(\cdot)$ over a cone $C=\{\bz:\cap_{i\in\{k,k+\half,k+1\}} \InAngles {\ba_i,\bz}\geq 0\}$ defined as the intersection of three half-spaces,
and we perform one iteration of the EG method starting at point $\bz_k\in C$.
Vectors $\bz_{k+\half}$ and $\bz_{k+1}$ are the projected points according to Equation~\eqref{def:k+1/2-th step} and Equation~\eqref{def:k+1-th step} when we apply the EG method on operator $\bF(\cdot)$ on cone $C$ starting at $\bz_k$.
Vectors $\bF(\bz_k)$, $\bF(\bz_{k+\half})$, $\bF(\bz_{k+1})$ correspond to the value of operator $\bF(\cdot)$ at points $\bz_k$, $\bz_{k+\half}$ and $\bz_{k+1}$.
Under this intuition,
observe that Property~3 implies that $\bz_k$,
$\bz_{k+\half}$ and $\bz_{k+1}$ all lie in cone $C$.
Moreover it implies that $\bz_{k+\half}$ lies in hyperplane $\InAngles{\ba_{k+\half},\bz}=0$ and is the projection of $\bz_k - \eta\bF(\bz_k)$ on cone $C$, e.g. $\bz_{k+\half}=\prod_C\InParentheses{\bz_k - \eta \bF(\bz_k)}=\prod_{\bz:\InAngles{\ba_{k+\half},\bz}\geq 0}\InParentheses{\bz_k - \eta \bF(\bz_k)}$.
Similar reasoning holds for point $\bz_{k+1}$.
Equation~\eqref{eq:reduced Lipschitz} and Equation~\eqref{eq:reduced monotone} corresponds to the fact that the reduced operator should be $L$-Lipschitz and monotone.
Finally, combining Equation~\eqref{eq: reduced gradient plane} with $\InAngles{\ba_k,\bz_k}=\InAngles{\ba_{k+1},\bz_{k+1}}=0$,
the RHS of the equation of Property~2 can be thought of as an upper bound of $H_{\bF,C}(z_k)-H_{\bF,C}(z_{k+1})$ and Property~2 further implies that this upper bound is strictly negative.
Property~1 guarantees that the normals of the three half-spaces that define cone $C$, $\ba_k$,$\ba_{k+\half}$ and $\ba_{k+1}$ are low-dimension,
which is crucial to show the impossibility of the promised inequalities from Lemma~\ref{lem:reduce to cone},
regardless of the dimension $\Z$.
In Remark~\ref{rem:primitive cone} we show the primitive instance for the reduction,
that is we show what kind of MVI instances coincide with the reduced VI instances (up to rotation of the vectors).
}
In Lemma~\ref{lem:reduce to cone}, we further simplify the instance that we need to consider. In particular, we argue that it is w.l.o.g. to assume that (1) $a_{k+1}$, $a_{k+\half}$, and $a_{k}$ are linear independent and (2) the intersection of the three halfspaces forms a cone, i.e., $b_k=b_{k+\half}=b_{k+1}=0$. Both assumption (1) and (2) reduce the number of variables we need to consider in the SOS program, so a low degree SOS proof is more likely to exist.
To maximally reduce the number of variables, we only included the minimal number of constraints that suffice to derive an SOS proof.
The proof of Lemma~\ref{lem:reduce to cone} is postponed to Appendix~\ref{appx:last iterate EG}.
\vspace{-.1in}
\begin{lemma}[Simplification Procedure]\label{lem:reduce to cone}
Let $\mathcal{I}$ be a variational inequality problem for a closed convex set $\Z\subseteq \R^n$ and a monotone and $L$-Lipschitz operator $F:\Z\rightarrow{\R}^n$.
Suppose the EG algorithm has a constant step size $\eta$. Let $z_k$ be the $k$-th iteration of the EG algorithm, $z_{k+\half}$ be the $(k+\half)$-th iteration as defined in~\eqref{def:k+1/2-th step}, and $z_{k+1}$ be the $(k+1)$-th iteration as defined in~\eqref{def:k+1-th step}.
Then either $r^{tan}_{(F,\Z)}(z_k) \geq r^{tan}_{(F,\Z)}(z_{k+1})$, or there exist vectors $\ba_k,\ba_{k+\half},\ba_{k+1}$,$\bz_k$,$\bz_{k+\half}$,$\bz_{k+1}$,
$\bF(\bz_k)$, $\bF(\bz_{k+\half})$, $\bF(\bz_{k+1})\in \R^{N}$ with $N\leq n+5$ that satisfy the following conditions.
\begin{enumerate}
\item $\ba_k=(\beta_1,\beta_2,1,0,\ldots,0),\ba_{k+\half}=(\alpha, 1, 0,\ldots,0)$, and $\ba_{k+1}=(1,0,\ldots,0)$ for some $\alpha,\beta_1,\beta_2\in \R$.
\item
$\InNorms{\bF(\bz_k)-\frac{\InAngles{F(\bz_k),\ba_k}\cdot \ba_k}{\InNorms{\ba_k}^2}}^2
-
\InNorms{
\bF(\bz_{k+1})-\frac{\InAngles{\bF(\bz_{k+1}),\ba_{k+1}}\cdot\ba_{k+1}}{\InNorms{\ba_{k+1}}^2}\mathbbm{1}[\InAngles{\bF(\bz_{k+1}),\ba_{k+1}}\ge 0]}^2<0$.
\item Additionally, $\langle \ba_i,\bz_j \rangle \geq 0$ and $\langle \ba_i,\bz_i \rangle = 0$ for all $i,j \in\{k,k+\half,k+1\}$.
$\ba_{k+\half}$ and $\bz_{k+\half} -\bz_k +\eta \bF(\bz_k)$ are co-directed, i.e., they are colinear and have the same direction,
and
$\ba_{k+1}$ and $\bz_{k+1}-\bz_k + \eta \bF(\bz_{k+\half})$ are co-directed.
\item \begin{align}
\InNorms{\bF(\bz_{k+1}) - \bF(\bz_{k+\half})}^2 \leq& L^2\InNorms{\bz_{k+1} - \bz_{k+\half}}^2\label{eq:reduced Lipschitz}\\
\InAngles{\bF(\bz_{k+1}) - \bF(\bz_k),\bz_{k+1}-\bz_k} \geq&0 \label{eq:reduced monotone}\\
\InAngles{\ba_k, \bF(\bz_k) }\geq& 0. \label{eq: reduced gradient plane}
\end{align}
\end{enumerate}
\end{lemma}
\vspace{-.1in}
\begin{comment}
The following remark characterizes the ''primitive'' instance that we consider.
High-levelly speaking, in a ''primitive'' instance,
the convex set is a cone defined over three hyperplanes that are linear independent,
and when we apply the EG,
the midpoint and the endpoint are projected onto two of the three hyperplanes.
\begin{remark}\label{rem:primitive cone}
Observe that if in the instance $\mathcal{I}$ that is fed on Lemma~\ref{lem:reduce to cone}:
\begin{enumerate}
\item The closed convex set $\Z=\{z:\InAngles{a_k,z}\geq 0,\InAngles{a_{k+\half},z}\geq 0,\InAngles{a_{k+1},z}\geq 0\}$ is a cone defined over hyperplanes $a_k$, $a_{k+\half}$, and $a_{k+1}$.
\item $z_{k+\half}=\Pi_{\left\{z:\InAngles{a_{k+1/2},z}\geq 0\right\}}\InParentheses{z_k - \eta F(z_k)}$ and $z_{k+1}=\Pi_{\left\{z:\InAngles{a_{k+1},z}\geq 0\right\}}\InParentheses{z_k - \eta F(z_{k+\half})}$
\item $H(z_k)= \InNorms{F(z_k)}^2 - \InAngles{a_k,F(z_k)}^2$
\item $a_k$, $a_{k+\half}$ and $a_{k+1}$ are linearly independent.
\end{enumerate}
Then the reduced instance coincides with a rotation of $\mathcal{I}$,
e.g. there exists orthonormal matrix $Q$ s.t. $\bz_k=Q\cdot z_k$, $\bz_{k+\half}=Q\cdot z_{k+\half}$, $\bz_{k+1}=Q\cdot z_{k+1}$, $\bF(\bz_k)=Q\cdot F(z_k)$, $\bF(\bz_{k+\half})=Q\cdot F(z_{k+\half})$ and $\bF(\bz_{k+1})=Q\cdot F(z_{k+1})$.
\end{remark}
\end{comment}
\notshow{
\begin{proof}
Recall that the $k$-th update of EG is as follows:
\begin{align}
z_{k+\frac{1}{2}} &= \Pi_\Z\left[z_k-\eta F(z_k)\right]=\arg \min_{z\in \Z} \| z-\left(z_k-\eta F(z_k)\right)\|\\
z_{k+1} & = \Pi_\Z\left[z_k-\eta F(z_{k+\frac{1}{2}})\right]= \arg \min_{z\in \Z} \left \| z-\left(z_k-\eta F(z_{k+\frac{1}{2}})\right)\right \|
\end{align}
We define the following vectors
\begin{align}
-a_k &\in \argmin_{\substack{a\in \unitnormal_\Z(z_k),\\ \InAngles{F(z_k),a}\le 0}}
\|F(z_k) - \InAngles{F(z_k),a}a\|^2 \label{dfn:a_k} \\
a_{k+\half} &= z_{k+\half} - z_k + \eta F(z_k) \label{dfn:a_k+1/2}\\
a_{k+1} &= z_{k+1} - z_k + \eta F(z_{k+\half}).\label{dfn:a_k+1}
\end{align}
For now, let us assume that $a_k$, $a_{k+\frac{1}{2}}$, and $a_{k+1}$ are linear independent and have the form in the statement of the lemma and that $\InAngles{a_k,z_k}$, $\InAngles{a_{k+\half},z_{k+\half}}$, $\InAngles{a_{k+1},z_{k+1}}$ are all zero.
Then the instance $\mathcal{I}'$ coincides with instance $\mathcal{I}$,
that is we set $\ha_k=a_k,\ha_{k+\half}=\ha_{k+\half},\ha_{k+1}=a_{k+1}$,
$\hz_k=z_k,\hz_{k+\half}=\hz_{k+\half},\hz_{k+1}=z_{k+1}$, $\hF(\cdot) = F(\cdot)$ and $\hZ = \Z$.
It is obvious that $\widehat{F}$ is monotone and $L$-Lipschitz.
We verify the other properties of $a_k$, $a_{k+\frac{1}{2}}$, $a_{k+1}$ as follows.
By Equation (\ref{dfn:a_k}), we know that $\InAngles{F(z_k),a_k} \ge 0$. We also know that $-a_k \in \unitnormal_\Z(z_k)$ and thus
\begin{align}
\InAngles{a_k ,z }\geq& \InAngles{a_k ,z_k } \quad \forall z \in \Z.
\end{align}
According to the update of EG (\ref{alg:EG}), Equation (\ref{dfn:a_k+1/2}), and Equation (\ref{dfn:a_k+1}), we know that for all $z \in \Z$,
\begin{align}
&\InAngles{a_{k+\half} ,z - z_{k+\half} } = \InAngles{ z_{k+\half} - z_k + \eta F(z_k), z-z_{k+\half}} \ge 0, \\
&\InAngles{a_{k+1} ,z - z_{k+1} } = \InAngles{ z_{k+1} - z_k + \eta F(z_{k+\half}), z-z_{k+1}} \ge 0.
\end{align}
Thus $-a_{k+\half} \in \unitnormal(z_{k+\half})$ and $-a_{k+1} \in \unitnormal(z_{k+1})$. Additionally, we have
\begin{align}
\InAngles{a_{k+\frac{1}{2}},z_k-\eta F(z_k) - z_{k+\half}} &\le 0,\\
\InAngles{a_{k+1},z_k-\eta F(z_{k+\half}) - z_{k+1}} &\le 0.
\end{align}
By the definition of $a_{k+\frac{1}{2}}$ and $a_{k+1}$, we know that for any $a_{k+\half}^\perp$ and $a_{k+1}^{\perp}$ such that $\InAngles{a_{k+\half}^\perp, a_{k+\half} }=\InAngles{a_{k+1}^\perp, a_{k+1} }=0$:
\begin{align*}
&\InAngles{a_{k+\half}^\perp ,z_k - F(z_k) - z_{k+\half}} = 0,\\
&\InAngles{a_{k+1}^\perp ,z_k - F(z_{k+\half}) - z_{k+1}} = 0.
\end{align*}
According to Definition \ref{def:projected hamiltonian} and Equation (\ref{dfn:a_k}), we know
\begin{align}
H(z_k) = \|F(z_k) - \InAngles{F(z_k),-a_k} (-a_k)\|^2 = \|F(z_k) - \InAngles{F(z_k),a_k} a_k\|^2.
\end{align}
Note that $-a_{k+1} \in \unitnormal(z_{k+1})$. If $\InAngles{F(z_{k+1}),a_{k+1}}\ge 0$, then
\begin{align}
&\|F(z_{k+1})-\InAngles{F(z_{k+1}),a_{k+1}}a_{k+1}\mathbbm{1}[\InAngles{F(z_{k+1}),a_{k+1}}\ge 0]\|^2 \\
&= \|F(z_{k+1})-\InAngles{F(z_{k+1}),a_{k+1}}a_{k+1}\|^2 \\
&\ge \min_{\substack{a\in \unitnormal_\Z(z_{k+1}),\\ \InAngles{F(z_{k+1}),a}\le 0}}
\|F(z_{k+1}) - \InAngles{F(z_{k+1}),a}a\|^2\\
& = H(z_{k+1}).
\end{align}
Otherwise $\InAngles{F(z_{k+1}),a_{k+1}}< 0$, then
\begin{align}
&\|F(z_{k+1})-\InAngles{F(z_{k+1}),a_{k+1}}a_{k+1}\mathbbm{1}[\InAngles{F(z_{k+1}),a_{k+1}}\ge 0]\|^2 \\
&= \|F(z_{k+1})\|^2 \\
& \ge H(z_{k+1}).
\end{align}
Thus we know
\begin{align*}
&H(z_k) - H(z_{k+1}) \\
&\geq
\|F(z_k)-\InAngles{F(z_k),a_k}a_k\|^2 - \|F(z_{k+1})-\InAngles{F(z_{k+1}),a_{k+1}}a_{k+1}\mathbbm{1}[\InAngles{F(z_{k+1}),a_{k+1}}\ge 0]\|^2.
\end{align*}
This completes the proof for the case where $a_k$, $a_{k+\frac{1}{2}}$, $a_{k+1}$ are linear independent and $\InAngles{\ha_k,\hz_k}$,$\InAngles{\ha_{k+\half}, \hz_{k+\half}}$, $\InAngles{\ha_{k+1},\hz_{k+1}}$ are all equal to $0$.
In a high-level idea, we introduce four dummy dimensions,
the purpose of the dummy dimension is to ensure that the updated equations $\InAngles{\ha_k,\hz_k}$,$\InAngles{\ha_{k+\half}, \hz_{k+\half}}$, $\InAngles{\ha_{k+1},\hz_{k+1}}$ are all equal to zero and the remaining three dummy dimensions ensure that our vectors are linearly independent.
Now we return to the case where $a_k$, $a_{k+\frac{1}{2}}$, $a_{k+1}$ are linear dependent or when $\InAngles{\ha_k,\hz_k},\InAngles{\ha_{k+\half},\hz_{k+\half}},\InAngles{\ha_{k+1},\hz_{k+1}}$ are not all equal to $0$. The idea here is to introduce four more dimensions. Specifically, we define $\hZ \subseteq \R^{n+4}$ and $\widehat{F}: \hZ \rightarrow \R^{n+4}$ as follows
\begin{align}
&\hZ := \{(-1,0,0,0,z): z\in \Z \}.\\
&\widehat{F}((-1,0,0,0,z)) :=
\begin{cases}
(\InAngles{a_k,z_k},1,0,0, F(z_k)), &\text{ if $z=z_k$ } \\
(\InAngles{a_{k+\half},z_{k+\half}},0,1,0, F(z_{k+\half})), &\text{ if $z=z_{k+\half}$ } \\
(\InAngles{a_{k+1},z_{k+1}},0,0,1, F(z_{k+1})), &\text{ if $z=z_{k+1}$ } \\
(0,0,0,0, F(z)), &\forall z\in \Z\backslash\{z_k,z_{k+\half},z_{k+1}\}.
\\
\end{cases}
\end{align}
Since the first four coordinates of each point in $\hZ$ are fixed,
it is easy to see that $\widehat{F}$ over $\hZ$ is monotone and $L$-Lipschitz.
\yangnote{We define $\hz_k$ to be $(-1,0,0,0,z_k)$. It is not hard to see by executing EG on $\hZ$ for $\hF$, $\hz_{k+\half}=(-1,0,0,0,z_{k+\half})$ and $\hz_{k+1}=(-1,0,0,0,z_{k+1})$.}
We define $\ha_k$, $\ha_{k+\frac{1}{2}}$, $\ha_{k+1}$ as follows:
\begin{align}
\ha_k &= (\InAngles{a_k,z_k},1,0,0,a_k),\\
\ha_{k+\half} &= (\InAngles{a_{k+\half},z_{k+\half}},0,1,0,a_{k+\half}),\\
\ha_{k+1} &= (\InAngles{a_{k+1},z_{k+1}},0,0,1,a_{k+1}).
\end{align}
Clearly, $\ha_k$, $\ha_{k+\frac{1}{2}}$, $\ha_{k+1}$ are linear independent.
Note that $\forall \hz=(-1,0,0,0,z)\in \hZ$
\begin{align}
\langle \ha_i,\hz \rangle =& \langle a_i, z \rangle - \langle a_i, z_i \rangle \geq 0
\qquad &\forall i\in\{k,k+\half,k+1\},\\
\langle \ha_i,\hz_i \rangle =& 0
\qquad &\forall i\in\{k,k+\half,k+1\},\\
\InAngles{\ha_k, \hF(\hz_k)} =& \InAngles{a_k, F(z_k)}\geq 0
\\
\ha_i =& \hz_i - \hz_k -\hF(\hz_{i-\half})
&\forall i\in\{k+\half,k+1\}
\end{align} Hence, all the properties are satisfied by $\hz_k,\hz_{k+\half},\hz_1$ and $\ha_k,\ha_{k+\half},\ha_{k+1}$.
Now we perform a change of base so that vectors $\ha_k,\ha_{k+\half}$ and $\ha_{k+1}$ only depend on the first three coordinates.
We use Gram–Schmidt process to generate a basis,
where vectors $\ha_k,\ha_{k+\half},\ha_{k+1}$ all lie in the span of the first three vector of the new basis.
More formally,
let $\{b_i\}_{i\in[N]}$ be a sequence of orthonormal vectors produced by the Gram-Schmidt process on ordered input $\ha_{k+1},\ha_{k+\half},\ha_k$ and $\{e_i\}_{i\in[N]}$.
By carefully choosing the directions of vectors $\{b_1,b_2,b_3\}$,
we also have that $\InAngles{b_1,\ha_k}\geq 0$,
$\InAngles{b_2,\ha_{k+\half}}\geq 0$
and $\InAngles{b_3,\ha_{k+1}}\geq 0$.
Let $Q$ be the $N\times N$ matrix,
where the $i$-th row of $Q$ is vector $b_i$.
Since $Q$ is orthonormal,
the inverse matrix of $Q$ is $Q^T$.
Observe that a vector $z\in \R^N$ written in base $\{e_i\}_{i\in[N]}$ can be written into base $\{b_i\}_{i\in [N]}$ with coefficients $Q\cdot z$.
We consider the set $\bZ = \{Q\cdot z: z\in \hZ\}$ and operator $\bF(\bz)=Q \cdot \hF(Q^T\cdot \bz):\bZ\rightarrow \R$.
Set $\bZ$ contains the vectors in $\Z$ written in base $\{b_i\}_{i\in[N]}$ and operator $\bF(\bz)$ is the outcome of $\hF(\hz)$ written in base $\{b_i\}_{i\in[N]}$,
where $\hz$ satisfies $\bz=Q\cdot \hz$,
that is $\bz$ is $\hz$ written in base $\{b_i\}_{i\in[N]}$.
Since $Q$ is orthonormal, function $f(\hz)=Q\cdot \hz:\hZ\rightarrow \bZ$ is a bijection mapping between set $\hZ$ and set $\bZ$ with inverse mapping $f^{-1}(\bz)=Q^T\cdot \bz:\bZ\rightarrow \hZ$.
Note that the only difference between using operator $\hF(\hz)$ on $\hz\in \hZ$ is the same as using $\bF(\bz)$ on $\bz\in \bZ$ just written in different base, which implies that we can couple the execution of EG on operator $\bF(\cdot)$ on set $\bZ$ with the execution of EG on operator $\hF(\cdot)$ on set $\hZ$.
For completeness reasons, we include a proof that operator $\bF(\cdot)$ is monotone and that the updates when using operator $\bF(\cdot)$ are the same as using operator $\hF(\cdot)$ just written in different bases.
Let $\ba_k = Q \cdot \ha_k$, $\ba_{k+\half} = Q \cdot \ha_{k+\half}$,
$\ba_{k+1} = Q \cdot \ha_{k+1}$,
$\bz_k = Q \cdot \hz_k$, $\bz_{k+\half} = Q \cdot \hz_{k+\half}$ and
$\bz_{k+1} = Q \cdot \hz_{k+1}$.
Vector $\bz_k$ is vector $\hz_k$ written in base $\{b_i\}_{i\in[N]}$,
similar reasoning holds for the rest of the defined vectors.
Clearly, for any $\hz,\hz'\in \hZ$ and their corresponding points $\bz=Q\cdot \hz,\bz'=Q\cdot \hz\in \bZ$:
\begin{align*}
\langle \bz,\bz' \rangle = \langle \hz,\hz' \rangle \label{eq:equal after rotation}
\end{align*}
Combining Equation~\eqref{eq:equal after rotation} and that properties of the Lemma hold for points $\ha_k,\ha_{k+\half},\ha_{k+1},\hz_k,\hz_{k+\half},\hz_{k+1}$, set $\hZ$ and operator $\hF(\cdot)$,
we conclude that the desired properties also hold for $\ba_k,\ba_{k+\half},\ba_{k+1},\bz_k,\bz_{k+\half},\bz_{k+1}$, set $\bZ$ and operator $\bF(\cdot)$.
Moreover, by properties of the Gram-Schmidt process and the order of the vector in its input,
since vectors $\ha_k,\ha_{k+\half}$ and $\ha_{k+1}$ are linearly independent,
$\ha_{k+1}\in Span(b_1),\ha_{k+\half}\in Span(b_1,b_2)$ and $\ha_k\in Span(b_1,b_2,b_3)$ and
$\InAngles{\ha_{k+1},b_1}>0$, $\InAngles{\ha_{k+\half},b_2}>0$,
and $\InAngles{\ha_k,b_3}>0$.
Thus $\ba_k=Q\ha_k=(\beta_1,\beta_2,b,0,\ldots,0)$,
$\ba_{k+\half}=(\alpha,c,0,\ldots,0)$
and $\ba_k=(d,0,\ldots,0)$,
where $\beta_1,\beta_2, \alpha\in \R$ and $b,c,d > 0$.
If $b\neq 1$ or $c\neq 1$ or $d \neq 1$,
then by properly scaling vectors $\ba_k$,$\ba_{k+\half}$ and $\ba_{k+1}$ we can finish the proof.
For completeness reasons, we verify that that operator $\bF(\cdot)$ is monotone and that the updates of EG with operator $\bF(\cdot)$ starting from $\bz_k$ coincide with the updates of EG on $\hF(\cdot)$ on set $\hZ$ written in different bases.
First we show that $\bF(\cdot)$ is a monotone operator.
For any $\bz,\bz'\in \bZ$:
\begin{align*}
\InAngles{\bF(\bz) - \bF(\bz'), \bz-\bz'}
=&\InAngles{Q\cdot \left(\hF(Q^T\cdot\bz+z^*) - \hF(Q^T\cdot\bz'+z^*)\right), Q\cdot \left(Q^T\left(\bz+z^*-(\bz'+z^*)\right)\right)} \\
=&\InAngles{\hF(Q^T\bz+z^*) - \hF(Q\bz'+z^*),\left(Q^T\bz+z^*\right)-\left(Q^T\bz'+z^*\right)}\geq 0
\end{align*}
In the first equality we used that the since $Q$ is orthonormal,
its inverse matrix is $Q^T$ and in the second equation we used the fact that $F$ is a monotone operator.
Consider the points $\bz_k = Q(\hz_k-z^*)\in \bZ$ and let $\bz_{k+\half}$ and $\bz_{k+1}$ be the points according to the EG update when starting at $\bz_k$ with operator $\bF(\cdot)$ and convex set $\bZ$.
We are going to show that $\bz_{k+\half}=Q\left( z_{k+\half}- \bz^*\right)$ and $\bz_{k+1}=Q\left( z_{k+1}- \bz^*\right)$.
According to EG update rule Equation~\eqref{alg:EG} we have that
\begin{align*}
\bz_{k+\half} =
\arg \min_{\bz\in \bZ} \| \bz-\left(\bz_k-\eta \bF(\bz_k)\right)\|
= &
\arg \min_{\bz\in \bZ} \| \bz-\left(Q(\hz_k-z^*)-\eta Q\cdot \hF(Q^T\cdot\bz_k+z^*\right)\| \\
=& Q\left( \arg \min_{\hz\in \hZ} \| Q\cdot\left(\hz-z^*\right)-\left(Q(\hz_k-z^*)-\eta Q\cdot \hF(\hz_k)\right)\| - z^*\right) \\
=& Q\left( \arg \min_{\hz\in \hZ} \| Q\cdot\left(\hz - \hz_k- \hF(\hz_k\right)\| - z^*\right) \\
=& Q\left( \arg \min_{\hz\in \hZ} \| \hz - \hz_k- \hF(\hz_k)\| - z^* \right)\\
=& Q\left( \hz_{k+\half} - z^*\right)
\end{align*}
In the second equality we substitute $\bz\in \bZ$ with $Q(\hz-\hz^*)$ and minimized over $\hz\in\hZ$.
In the fourth equation we used that $Q$ is orthonormal matrix and in the final equality we used that $\hz_{k+\half} =
\arg \min_{\hz\in \hZ} \| \hz-\left(\hz_k-\eta \hF(\hz_k)\right)\|$.
We can similarly show that $\bz_{k+1}= Q\left(\hz_{k+1} - z^* \right)$.
\end{proof}
}
\begin{comment}
In Theorem~\ref{thm:monotone hamiltonian} we show that the \projham is non-increasing between two consecutive iterations of the EG method.
To show that,
we assume towards a contradiction that the projected Hamiltonian increases,
in which case Lemma~\ref{lem:reduce to cone} implies that there exist vectors $\ba_k,\ba_{k+\half},\ba_{k+1}$ that lie in a three-dimensional space, parameterized by numbers $\alpha$,$\beta_1$,$\beta_2\in \R$ and vectors $\bz_k$, $\bz_{k+\half}$, $\bz_{k+1}$,
$\bF(\bz_k)$, $\bF(\bz_{k+\half})$, $\bF(\bz_{k+1})$ that possibly lie in a high dimensional space that satisfy a set of constraints.
Our goal is to consider vectors $\bz_k$,$\bz_{k+\half}$,$\bz_{k+1}$,
$\bF(\bz_k)$,$\bF(\bz_{k+\half})$,$\bF(\bz_{k+1})$ and numbers $\alpha$,$\beta_1$,$\beta_2$ as variables and show that the system of non-linear polynomial inequalities defined by the constrains of Lemma~\ref{lem:reduce to cone} is infeasible.
Showing infeasibility for a system of non-linear polynomial inequalities is notoriously hard and undecidable when we want an integer solution.
However our problem is structured and allows for an efficient and verifiable witness for the infeasibility.
More specifically our verifiable witness start with term $\InNorms{\bF(\bz_k)-\frac{\InAngles{F(\bz_k),\ba_k}\cdot \ba_k}{\InNorms{\ba_k}^2}}^2
-
\InNorms{
\bF(\bz_{k+1})-\frac{\InAngles{\bF(\bz_{k+1}),\ba_{k+1}}\cdot\ba_{k+1}}{\InNorms{\ba_{k+1}}^2}\mathbbm{1}[\InAngles{\bF(\bz_{k+1}),\ba_{k+1}}\ge 0]}^2$, which by Lemma~\ref{lem:reduce to cone} is strictly negative and we add several non-positive terms, and show that the summation of these terms is a sum of squares,
which is clearly non-negative, reaching a contradiction.
The non-negativeness of the added terms is directly implied by the constraints of Lemma~\ref{lem:reduce to cone}.
To construct our witness,
we formulate the set of constraints as a SOS program and we search for a feasible solution.
We exploit the low-dimensionality of vectors $\ba_k$,$\ba_{k+\half}$, $\ba_{k+1}$ and using a technique similar to the one described in Section~\ref{sec:warm up} and Figure~\ref{fig:sos program unconstrained} we can formulate the polynomial inequalities as a SOS program with a constant number of variables, regardless of the original dimension of the vectors.
A general template of the SOS program that we solve to construct the witness is located in Figure~\ref{fig:construct witness sos}.
Observe that a solution to the SOS program in Figure~\ref{fig:construct witness sos} is sufficient to conclude that the set of polynomial inequalities is infeasible,
but not necessary.
In Theorem~\ref{thm:monotone hamiltonian} we present an identity that certifies that the promised inequalities from Lemma~\ref{lem:reduce to cone} are infeasible.
More specifically,
the constraints we use are combinations of the inequalities by Lemma~\ref{lem:reduce to cone} and the coefficients of the constraints are sum of squares polynomials of at most degree $6$.
Due to limited computational power, we used heuristics to identify the polynomial and coefficients that we used in the proof,
which we describe next.
To identify the set of polynomial that we are going to use,
we considered the primitive instance that we describe in Remark~\ref{rem:primitive cone} and wrote down all the quantities up to degree two that are non-negative or zero.
Then we sample the values of $\alpha$,$\beta_1$ and $\beta_2$ and identify the minimal set of constraints that are essential to construct a witness.
Once we were confident about what constraints to use,
we wanted to figure out their coefficients which could possibly depend on the values $\alpha,\beta_1,\beta_2$ that we had sampled in the previous step.
We figured out the dependency of the coefficients on $\alpha$ and $\beta_1$,$\beta_2$ separately.
To find the dependence of the coefficients on $\alpha$,
we sampled the values of $\beta_1$ and $\beta_2$ and we allowed the coefficients to be polynomials of $\alpha$ of degree two that are sum of squares for the terms that are non-negative and polynomials of $\alpha$ of degree two (not necessarily that are sum of squares) for terms that are zero.
Observing the feasible solution, we were able to figure out the dependence of the coefficients on $\alpha$.
In a similar way,
by sampling $\alpha$ and allowing the coefficients to be polynomials that depend only on $\beta_1$ and $\beta_2$ we were able to figure out the dependence of the coefficients on $\beta_1$ and $\beta_2$.
We provide a high-level overview of the identity we discovered in the sketch of the proof of Theorem~\ref{thm:monotone hamiltonian}.
\end{comment}
\begin{comment}
\begin{figure}[h!]
\colorbox{MyGray}{
\begin{minipage}{0.97\textwidth} {
\noindent\textbf{Feasibility of set of polynomial Inequalities:}
\begin{itemize}
\item $v=\{v_i\}_{i\in[N]}$ are the independent variables.
\end{itemize}
\begin{equation*}
\begin{array}{ll@{}ll}
\text{find} & \displaystyle v \\
\text{s.t.} & \displaystyle f_0(v) < 0\\
&f_i(v) \geq 0 \qquad\qquad \forall i \in [M]\\
&h_i(v) = 0 \qquad\qquad \forall i \in [M']\\
\end{array}
\end{equation*}
\textbf{Constructing a Witness through a SOS Program:}
\begin{itemize}
\item $d\in \N$, denotes the maximum degree of a coefficient.
\item $v=\{v_i\}_{i\in[N]}$ are the independent variables.
\end{itemize}
\begin{equation*}
\begin{array}{ll@{}ll}
\text{find} & \displaystyle g_0(v),\{g_i(v)\}_{i\in[M]},\{g'_i(v)\}_{i\in[N]}\\
\text{s.t.} & \displaystyle (g_0(v)+1)\cdot f_0(v) + \sum_{i\in[M]}g_i(v)\cdot f_i(v) + \sum_{i\in[M']}g'_i(v) \cdot h_i(v) \text{ is sum of squares.} \\
&\displaystyle g_0(v) \text{ is sum of squares of degree $d$ $\forall i \in[M]$}\\
&\displaystyle g_i(v) \text{ is sum of squares of degree $d$ $\forall i \in[M]$}\\
&\displaystyle g'_i(v) \text{ is polynomial of degree $d$ $\forall i \in[M']$}\\
\end{array}
\end{equation*}}
\end{minipage}} \caption{Constructing a witness for infeasibility of a set of polynomial Inequalities.}\label{fig:construct witness sos}
\end{figure}
\end{comment}
\notshow{
\begin{figure}[h!]
\colorbox{MyGray}{
\begin{minipage}{0.97\textwidth} {
\noindent\textbf{Feasibility of set of polynomial Inequalities:}
\begin{itemize}
\item $v=\{v_i\}_{i\in[N]}$ are the independent variables.
\end{itemize}
\begin{equation*}
\begin{array}{ll@{}ll}
\text{find} & \displaystyle v \\
\text{s.t.} & \displaystyle f_0(v) < 0\\
&f_i(v) \geq 0 \qquad\qquad \forall i \in [M]\\
&h_i(v) = 0 \qquad\qquad \forall i \in [M']\\
\end{array}
\end{equation*}
\textbf{Constructing a Witness through a SOS Program:}
\begin{itemize}
\item $v=\{v_i\}_{i\in[N]}$ are the independent variables.
\end{itemize}
\begin{equation*}
\begin{array}{ll@{}ll}
\text{find} & \displaystyle \{d_i\}_{i\in[M]},\{d'_i\}_{i\in[N]}\\
\text{s.t.} & \displaystyle f_0(v) + \sum_{i\in[M]}d_i\cdot f_i(v) + \sum_{i\in[M']}d'_i \cdot h_i(v) \text{ is sum of squares.} \\
&\displaystyle d_i\geq 0 \qquad \forall i \in[M]\\
&\displaystyle d_i\in\R \qquad \forall i \in[M']
\end{array}
\end{equation*}}
\end{minipage}} \caption{Constructing a witness for infeasibility of a set of polynomial Inequalities.}\label{fig:construct witness sos}
\end{figure}
}
In Theorem~\ref{thm:monotone hamiltonian}, we establish the monotonicity of the \projhamnospace. Our proof is based on the solution to the degree-8 SOS program concerning polynomials in $27$ variables (Figure~\ref{fig:sos constrainedprogram}). We include a proof sketch of Theorem~\ref{thm:monotone hamiltonian} in the main body, and the full proof in Appendix~\ref{appx:last iterate EG}.
\vspace{-.1in}
\begin{theorem}\label{thm:monotone hamiltonian}
Let $\Z \subseteq \R^n$ be a closed convex set and $F:\Z \rightarrow \R^n$ be a monotone and $L$-Lipschitz operator. For any step size $\eta \in (0, \frac{1}{L}$) and any $z_k \in \Z$, the EG method update satisfies $r^{tan}_{(F,\Z)}(z_k) \ge r^{tan}_{(F,\Z)}(z_{k+1})$.
\end{theorem}
\noindent{\bf Proof Sketch }
Assume towards contradiction that $\Ham(z_k) < \Ham(z_{k+1})$,
using Lemma~\ref{lem:reduce to cone} there exist numbers $\alpha, \beta_1, \beta_2\in \R$ and vectors $\ba_k$, $ \ba_{k+\half}$, $\ba_{k+1}$, $\bz_k$, $\bz_{k+\half}$, $\bz_{k+1}$, $\bF(\bz_k)$, $\bF(\bz_{k+\half})$, $\bF(\bz_{k+1})\in \R^N$ where $N=n+5$ that satisfy the properties in the statement of Lemma~\ref{lem:reduce to cone} and
\begin{align*}
0&>\InNorms{\bF(\bz_k)-\frac{\InAngles{F(\bz_k),\ba_k}}{\InNorms{\ba_k}^2}\ba_k}^2
- \InNorms{ \bF(\bz_{k+1})-\frac{\InAngles{\bF(\bz_{k+1}),\ba_{k+1}}\ba_{k+1}}{\InNorms{\ba_{k+1}}^2}\mathbbm{1}[\InAngles{\bF(\bz_{k+1}),\ba_{k+1}}\ge 0]}^2.
\end{align*}
We use $\tar$ to denote the RHS above.
We first present 8 inequalities/equations, which directly follow from Lemma \ref{lem:reduce to cone}. We verify their correctness in the complete proof in Appendix~\ref{appx:last iterate EG}.
\vspace{-0.2in}
\begin{align}
\InAngles{\eta \bF(\bz_{k+1}) - \eta\bF(\bz_k),\bz_k-\bz_{k+1}} &\leq0,\label{eq:sketch_mononotone} \\
\InNorms{\eta \bF(\bz_{k+1}) - \eta \bF(\bz_{k+\half})}^2- \InNorms{\bz_{k+1} - \bz_{k+\half}}^2 &\le 0,\label{eq:sketch_lipsitz}\\
\bz_{k+\half}[1]\InParentheses{\InParentheses{\bz_{k+\half} -\bz_k +\eta \bF(\bz_k)}[1] - \alpha \InParentheses{\bz_{k+\half} -\bz_k +\eta \bF(\bz_k)}[2]} &= 0,\label{eq:sketch_cons1}\\
\bz_{k+1}[2]\InParentheses{\InParentheses{\bz_{k+\half} -\bz_k +\eta \bF(\bz_k)}[1] - \alpha \InParentheses{\bz_{k+\half} -\bz_k +\eta \bF(\bz_k)}[2]} &= 0,\label{eq:sketch_cons2}\\
\InParentheses{\alpha\InParentheses{\bz_{k}-\eta\bF(\bz_{k})}[1] + \InParentheses{\bz_{k}-\eta\bF(\bz_{k})}[2]} \InParentheses{\alpha\bz_{k+1}[1]+\bz_{k+1}[2]} &\le 0,\label{eq:sketch_cons3} \\
-\eta\InParentheses{\beta_1\bF(\bz_k)[1] + \beta_2\bF(\bz_k)[2] + \bF(\bz_k)[3]} \InParentheses{\beta_1\bz_{k+\half}[1] + \beta_2 \bz_{k+\half}[2] + \bz_{k+\half}[3]} &\le 0,\label{eq:sketch_cons4} \\
\bz_k[1]\InParentheses{\bz_k[1] - \eta \bF(\bz_{k+\half})[1]} &\le 0,\label{eq:sketch_cons5} \\
-\eta\bF(\bz_{k+1})[1] \mathbbm{1}\inteval{\bF(\bz_{k+1})[1]\le 0 }\InParentheses{\bz_k[1] - \eta \bF(\bz_{k+\half})[1]} &\le 0.\label{eq:sketch_cons6}
\end{align}
We first add several non-positive terms to $\tar$ to derive Expression~\ref{eq:proof sketch Hamiltonian monotonicity LHS}.
\begin{align}\label{eq:proof sketch Hamiltonian monotonicity LHS}
&\eta^2\cdot \tar+ 2\cdot \LHSI~\eqref{eq:sketch_mononotone}+\LHSI~\eqref{eq:sketch_lipsitz}+2\cdot \LHSE~\eqref{eq:sketch_cons1}\notag\\
&+ \frac{2\alpha}{1+\alpha^2} \cdot \LHSE~\eqref{eq:sketch_cons2} \quad+ \frac{2}{1+\alpha^2} \cdot\LHSI~\eqref{eq:sketch_cons3}+ \frac{2}{1+\beta_1^2+\beta_2^2} \cdot \LHSI~\eqref{eq:sketch_cons4} \notag \\
&+ 2\cdot \LHSI~\eqref{eq:sketch_cons5}+ 2\cdot \LHSI~\eqref{eq:sketch_cons6}
\end{align}
After substituting the following six variables $\bz_{k}[3], \bz_{k+\half}[2], \bz_{k+\half}[3],\bz_{k+1}[1],\bz_{k+1}[2], \bz_{k+1}[3]$ using Equation~\eqref{substitute-1} to~\eqref{substitute-6}, Expression~\eqref{eq:proof sketch Hamiltonian monotonicity LHS} equals to the following polynomial, which is clearly non-negative. We verify the validity of the substitutions (Equation~\eqref{substitute-1} to~\eqref{substitute-6}) in the full proof.
\notshow{
\begin{align*}
& \InParentheses{\bz_k[1]-\eta\bF(\bz_{k+\half})[1] + \eta\bF(\bz_{k+1})[1] \cdot\ind[\bF(\bz_{k+1})[1]\geq 0]}^2\\
& + \frac{\left(\bz_k[2] -\eta\bF(\bz_{k})[2] + \alpha \bz_{k+\half}[1]\right)^2}{1+\beta_2^2}\\
& + \frac{\left( \eta\bF(\bz_k)[3] + \beta_1\bz_k[1] + \beta_2 \bz_{k}[2] + \InParentheses{\alpha\beta_2-\beta_1}\bz_{k+\half}[1]\right)^2}{1+\beta_1^2+\beta_2^2} \\
& + \frac{\left(\InParentheses{1+\beta_2^2}\InParentheses{\eta\bF(\bz_k)[1]- \bz_k[1]} -\beta_1\beta_2 \InParentheses{\eta\bF(\bz_k)[2] - \bz_{k}[2]} + \InParentheses{1+\beta_2^2 + \alpha\beta_1\beta_2}\bz_{k+\half}[1]\right)^2 }{\InParentheses{1+\beta_2^2}\InParentheses{1+\beta_1^2+\beta_2^2}}
\end{align*}
}
\begin{align*}
&\InParentheses{\bz_k[1]-\eta\bF(\bz_{k+\half})[1] + \eta\bF(\bz_{k+1})[1] \cdot\ind[\bF(\bz_{k+1})[1]\geq 0]}^2 \\
& + \frac{\left(\bz_k[1] - \eta\bF(\bz_k)[1] -\bz_{k+\half}[1]\right)^2}{1+\beta_1^2+\beta_2^2} \\
& + \frac{\left( \eta\bF(\bz_k)[3] + \beta_1\bz_k[1] + \beta_2 \bz_{k}[2] + \InParentheses{\alpha\beta_2-\beta_1}\bz_{k+\half}[1]\right)^2}{1+\beta_1^2+\beta_2^2} \\
& + \frac{\left(\bz_k[2] -\eta\bF(\bz_{k})[2] + \alpha \bz_{k+\half}[1]\right)^2}{1+\beta_1^2+\beta_2^2} \\
& + \frac{\left(\beta_1\InParentheses{\bz_k[2] -\eta\bF(\bz_{k})[2] + \alpha \bz_{k+\half}[1]} - \beta_2 \InParentheses{\bz_k[1] - \eta\bF(\bz_k)[1] -\bz_{k+\half}[1]}\right)^2}{1+\beta_1^2+\beta_2^2}
\end{align*}
$\hfill \blacksquare$
\vspace{-0.15in}
\begin{theorem}\label{thm:EG last-iterate constrained}
Let $\Z \subseteq \R^n$ be a closed convex set, $F(\cdot):\Z\rightarrow \R^n$ be a monotone and $L$-Lipschitz operator and $z^*\in \Z$ be the solution to the variational inequality. Then for any $T \ge 1$, $z_T$ produced by EG with any constant step size $\eta \in (0,\frac{1}{L})$ satisfies $\gap(z_T) \le \frac{1}{\sqrt{T}}\frac{3D||z_0-z^*||}{\eta\sqrt{1-(\eta L)^2}}$.
\end{theorem}
Theorem~\ref{thm:EG last-iterate constrained} is implied by combing~\Cref{lem:hamiltonian to gap}, \Cref{lem:best iterate hamiltonian}, \Cref{thm:monotone hamiltonian} and the fact that $\eta\in \InParentheses{0,\frac{1}{L}}$. Choosing $\eta$ to be $\frac{1}{2L}$ and $D=O(\InNorms{z_0-z^*})$, then $\gap(z_T)= O(\frac{D^2L}{\sqrt{T}})$ matching the $\Omega(\frac{D^2L}{\sqrt{T}})$ lower bound for EG, OGDA, and more generally all p-SCLI algorithms~\citep{golowich_last_2020,golowich_tight_2020} in terms of the dependence on $D$, $L$, and $T$.
\notshow{
\begin{proof}
Unless $H(z_{k+1})\geq H(z_k)$,
using Lemma~\ref{lem:reduce to cone} there exist vectors $\ba_k,\ba_{k+\half},\ba_{k+1}$,$\bz_k$,$\bz_{k+\half}$,$\bz_{k+1}$,
$\bF(\bz_k)$,$\bF(\bz_{k+\half})$, $\bF(\bz_{k+1})\in \R^N$ where $N=n+4$ that satisfy the properties in the statement of Lemma~\ref{lem:reduce to cone} and:
\begin{align}
&2\eta^2\InParentheses{ H_{(F,\Z)}(z_k) - H_{(F,\Z)}(z_{k+1})} \nonumber\\
&\geq
\InNorms{\eta\bF(\bz_k)-\frac{\InAngles{\eta \bF(\bz_k),\ba_k}}{\InNorms{\ba_k}^2}\ba_k}^2
-
\InNorms{
\eta\bF(\bz_{k+1})-\frac{\InAngles{\eta\bF(\bz_{k+1}),\ba_{k+1}}\ba_{k+1}}{\InNorms{\ba_{k+1}}^2}\mathbbm{1}[\InAngles{\bF(\bz_{k+1}),\ba_{k+1}}\ge 0]}^2 \label{eq:obj}
\end{align}
Our goal is to show that Term~\eqref{eq:obj} is non-negative.
To prove that our objective is non-negative,
we add some non-positive terms that we inferred using the SoS program and show that the new term can be factored into sum of squares,
which implies non-negativity of our objective.
According to Lemma \ref{lem:reduce to cone} and using the fact $\InParentheses{\eta L}^2 \leq \frac{1}{2}$ and $\eta>0$, the following two terms are non-positive:
\begin{align}
\InAngles{\eta \bF(\bz_{k+1}) - \eta\bF(\bz_k),\bz_k-\bz_{k+1}} &\leq0,\label{eq:mononotone} \\
\InNorms{\eta \bF(\bz_{k+1}) - \eta \bF(\bz_{k+\half})}^2- \InNorms{\bz_{k+1} - \bz_{k+\half}}^2 &\le 0\label{eq:lipsitz},
\end{align}
Observe that $\eqref{eq:obj}\geq\eqref{eq:obj} - 2\cdot \eqref{eq:mononotone} - \eqref{eq:lipsitz}$.
We consider the following partition of a vector in $u=(u_1,\ldots,u_N)\in\R^N$,
we denote by $u^{(\leq 3)}=(u_1,u_2,u_3)$ and by $u^{(> 3)}=(u_4,\ldots,u_N)$.
The reason behind our decomposition is that $\ba_k^{(>3)}=\ba_{k+\half}^{(>3)}=\ba_{k+1}^{(>3)}=(0\ldots,0)$,
which will help us \todo{add....}.
Since $\ba_k^{(>3)}=\ba_{k+\half}^{(>3)}=\ba_{k+1}^{(>3)}=(0\ldots,0)$,
it is enough to prove that the following two terms are non-positive:
\begin{align}
\InNorms{\eta\bF^{(>3)}(\bz_k)}^2
-
\InNorms{
\eta\bF^{(>3)}(\bz_{k+1})}^2
-\InAngles{\eta \bF^{(>3)}(\bz_{k+1}) - \eta\bF^{(>3)}(\bz_k),\bz^{(>3)}_k-\bz^{(>3)}_{k+1}} \nonumber\\
\qquad\qquad +\InNorms{\bz^{(>3)}_{k+1} - \bz^{(>3)}_{k+\half}}^2 - \InNorms{\eta \bF^{(>3)}(\bz_{k+1}) - \eta \bF^{(>3)}(\bz_{k+\half})}^2- \geq 0 \label{eq:unconstrained obj}
\end{align}
\begin{align}
&\InNorms{\eta\bF^{(\leq3)}(\bz_k)-\frac{\InAngles{\eta \bF^{(\leq3)}(\bz_k),\ba_k}}{\InNorms{\ba_k}^2}\ba_k}^2
-
\InNorms{
\eta\bF^{(\leq3)}(\bz_{k+1})-\frac{\InAngles{\eta\bF^{(\leq3)}(\bz_{k+1}),\ba_{k+1}}\ba_{k+1}}{\InNorms{\ba_{k+1}}^2}\mathbbm{1}[\InAngles{\bF^{(\leq3)}(\bz_{k+1}),\ba_{k+1}}\ge 0]}^2 \nonumber\\
&\qquad\qquad-\InAngles{\eta \bF^{(\leq3)}(\bz_{k+1}) - \eta\bF^{(\leq3)}(\bz_k),\bz^{(\leq3)}_k-\bz^{(\leq3)}_{k+1}} \nonumber\\
&\qquad\qquad\qquad +\InNorms{\bz^{(\leq3)}_{k+1} - \bz^{(\leq3)}_{k+\half}}^2 - \InNorms{\eta \bF^{(\leq3)}(\bz_{k+1}) - \eta \bF^{(\leq3)}(\bz_{k+\half})}^2 \geq 0 \label{eq:constrained obj}
\end{align}
First we show Equation~\eqref{eq:unconstrained obj}.
According to Lemma \ref{lem:reduce to cone}, $\ba_{k+\half}$ has the same direction as $\bz_{k+\half} -\bz_k +\eta \bF(\bz_k)$.
For any $i\in[4,N]$, $\InAngles{e_i,\bz_{k+\half}-\bz_k + \eta \bF(\bz_k)}=\InAngles{e_i,\ba_{k+\half}}=0$ which implies that $\bz^{(>3)}_{k+\half} = \bz^{(>3)}_k -\eta \bF^{(>3)}(\bz_k)$.
We can similarly show that $\bz^{(>3)}_{k+1} = \bz^{(>3)}_k -\eta \bF^{(>3)}(\bz_{k+\half})$.
According to Lemma \ref{lem:reduce to cone}, to prove the monotonicity of the \projham $\Ham(z_k) \ge \Ham(z_{k+1})$, it suffices to show the non-negativeness of the following summation.
\begin{align}\label{eq:general potential}
& \InNorms{\bF(\bz_k)-\frac{\InAngles{F(\bz_k),\ba_k}}{\InNorms{\ba_k}^2}\ba_k}^2
-
\InNorms{
\bF(\bz_{k+1})-\frac{\InAngles{\bF(\bz_{k+1}),\ba_{k+1}}\ba_{k+1}}{\InNorms{\ba_{k+1}}^2}\mathbbm{1}[\InAngles{\bF(\bz_{k+1}),\ba_{k+1}}\ge 0]}^2\notag\\
& = \sum_{i=1}^N (\bF(\bz_k)[i]^2 - \bF(\bz_{k+1})[i]^2) - (\beta_1 \bF(\bz_k)[1] + \beta_2 \bF(\bz_k)[2] + \bF(\bz_k)[3])^2\notag\\
&\quad + \bF(\bz_{k+1})[1]^2 \mathbbm{1}[\bF(\bz_{k+1})[1]\ge 0].
\end{align}
We do a partition of $[N]$. We define $p_1 = \{1,2,3\}$ and $p_2 = \{4,5,\cdots,N\}$. For any vector $z \in \R^N$ and $j \in \{1,2\}$, we define $p_j(z) \in \R^n$ such that $p_j(z)[i] = z[i]$ for $i \in p_i$ and $p_j(z)[i] = 0$ otherwise.
According to Lemma \ref{lem:reduce to cone}, the following two terms are non-positive:
\begin{align}
\InAngles{\bF(\bz_{k+1}) - \bF(\bz_k),\bz_k-\bz_{k+1}} &\leq0,\label{eq:general-1} \\
\eta^2\InNorms{\bF(\bz_{k+1}) - \bF(\bz_{k+\half})}^2- \InNorms{\bz_{k+1} - \bz_{k+\half}}^2 &\le 0\label{eq:general-2},
\end{align}
where in the second equality we use the fact that $\eta^2 L^2 < \half$. Then in order to show (\ref{eq:general potential}) is non-negative, it suffices to show the following is non-negative
\notshow{\begin{align}
&\eta^2\times(\ref{eq:general potential}) +2\eta\times (\ref{eq:general-1}) + (\ref{eq:general-2})\notag \\
& = \sum_{j=1}^2 \left(\eta^2 \InNorms{p_j(\bF(\bz_k))-p_j(\bF(\bz_{k+1}))}^2 + 2\eta \InAngles{p_j(\bF(\bz_{k+1})) - p_j(\bF(\bz_k)) ,p_j(\bz_k)-p_j(\bz_{k+1})}.\right \notag\\
&\left.\quad + \eta^2\InNorms{p_j(\bF(\bz_{k+1})) - p_j(\bF(\bz_{k+\half}))}^2- \InNorms{p_j(\bz_{k+1}) - p_j(\bz_{k+\half})}^2 \right)\notag\\
&\quad - \eta^2(\beta_1 \bF(\bz_k)[1] + \beta_2 \bF(\bz_k)[2] + \bF(\bz_k)[3])^2 +\eta^2 \bF(\bz_{k+1})[1]^2 \mathbbm{1}[\bF(\bz_{k+1})[1]\ge 0] \label{eq:general-reduced potential}.
\end{align}}
According to Lemma \ref{lem:reduce to cone}, $\ba_{k+\half}$ has the same direction as $\bz_{k+\half} -\bz_k +\eta \bF(\bz_k)$
and
$\ba_{k+1}$ has the same direction as $\bz_{k+1}-\bz_k + \eta \bF(\bz_{k+\half})$. Since $\ba_{k+\frac{1}{2}}[i] = \ba_{k+1}[i] = 0$ for any $i \in p_2$, we know for any $i \in p_2$,
\begin{align*}
\bz_{k+\half} &= \bz_k -\eta \bF(\bz_k), \\
\bz_{k+1} &= \bz_k - \eta \bF(\bz_{k+\half}).
\end{align*}
Therefore, the summation of $p_2$ in (\ref{eq:general-reduced potential}) is
\begin{align*}
&\eta^2 \InNorms{p_2(\bF(\bz_k))-p_2(\bF(\bz_{k+1}))}^2 + 2\eta \InAngles{p_2(\bF(\bz_{k+1})) - p_2(\bF(\bz_k)) ,p_2(\bz_k)-p_2(\bz_{k+1})} \\
&\quad + \eta^2\InNorms{p_2(\bF(\bz_{k+1})) - p_2(\bF(\bz_{k+\half}))}^2- \InNorms{p_2(\bz_{k+1}) - p_2(\bz_{k+\half})}^2 \\
& = \eta^2 \InNorms{p_2(\bF(\bz_k))-p_2(\bF(\bz_{k+1}))}^2 + -2\eta^2 \InAngles{p_2(\bF(\bz_{k+1})) - p_2(\bF(\bz_k)) ,p_2(\bF(\bz_{k+\half}))} \\
&\quad + \eta^2\InNorms{p_2(\bF(\bz_{k+1})) - p_2(\bF(\bz_{k+\half}))}^2- \eta^2\InNorms{p_2(\bF(\bz_k) - p_2(\bF(\bz_{k+\half}))}^2 \\
& = 0.
\end{align*}
Then it remains to show that the summation of $p_1 = \{1,2,3\}$ in (\ref{eq:general-reduced potential}) is non-negative
\begin{align}
&\sum_{i=1}^3 \eta^2(\bF(\bz_k)[i]^2 - \bF(\bz_{k+1})[i]^2- \eta^2(\beta_1 \bF(\bz_k)[1] + \beta_2 \bF(\bz_k)[2] + \bF(\bz_k)[3])^2 \notag\\ &\quad +\eta^2 \bF(\bz_{k+1})[1]^2 \mathbbm{1}[\bF(\bz_{k+1})[1]\ge 0].\label{eq:general-final-potential}
\end{align}
The idea is to introduce five non-positive terms, adding them to (\ref{eq:general-final-potential}) and show the summation is a sum of squares which is non-negative.
According to Lemma \ref{lem:reduce to cone},
$\ba_{k+\half}$ is in the same direction as $\bz_{k+\half} -\bz_k +\eta \bF(\bz_k)$. Since $(1,-\alpha,0,\cdots,0)$ is orthogonal to $\ba_{k+\frac{1}{2}}$, we know
\begin{align*}
&\InAngles{(1,-\alpha,0,\cdots,0) ,\bz_{k+\half} -\bz_k +\eta \bF(\bz_k)} = 0\\
\Leftrightarrow & (\bz_{k+\half} -\bz_k +\eta \bF(\bz_k))[1] - \alpha (\bz_{k+\half} -\bz_k +\eta \bF(\bz_k))[2] = 0
\end{align*}
The first two non-positive terms are
\begin{align}
-\bz_{k+\half}[1]\left((\bz_{k+\half} -\bz_k +\eta \bF(\bz_k))[1] - \alpha (\bz_{k+\half} -\bz_k +\eta \bF(\bz_k))[2]\right) &= 0,\label{eq:cons1}\\
-\bz_{k+1}[2]\left((\bz_{k+\half} -\bz_k +\eta \bF(\bz_k))[1] - \alpha (\bz_{k+\half} -\bz_k +\eta \bF(\bz_k))[2]\right) &= 0.\label{eq:cons2}
\end{align}
By Lemma \ref{lem:reduce to cone}, we have $\ba_{k+\half}$ is in the same direction as $\bz_{k+\half} -\bz_k +\eta \bF(\bz_k)$, $\InAngles{\ba_{k+\half},\bz_{k+\half}} = 0$, and $\InAngles{\ba_{k+\half},\bz_{k+1}} \ge 0$. Thus we have
\begin{align*}
\InParentheses{\alpha\InParentheses{\bz_{k}}-\eta\bF(\bz_{k})}[1] + \InParentheses{\alpha\InParentheses{\bz_{k}}-\eta\bF(\bz_{k})}[2] = \InAngles{\ba_{k+\half},\bz_{k} - \eta \bF(\bz_k)} = \InAngles{\ba_{k+\half},\bz_{k} - \eta \bF(\bz_k)- \bz_{k+\half}} &\le 0,\\
\alpha \bz_{k+1}[1] + \bz_{k+1}[2] = \InAngles{\ba_{k+\half},\bz_{k+1}} &\ge 0.
\end{align*}
The third non-positive term is
\begin{align}\label{eq:cons3}
\InParentheses{\InParentheses{\alpha\InParentheses{\bz_{k}}-\eta\bF(\bz_{k})}[1] + \InParentheses{\alpha\InParentheses{\bz_{k}}-\eta\bF(\bz_{k})}[2]} \InParentheses{\alpha\bz_{k+1}[1]+\bz_{k+1}[2]} \le 0.
\end{align}
By Lemma \ref{lem:reduce to cone}, we have $\InAngles{\ba_k,\bF(\ba_k)} \ge 0$ and $\InAngles{\ba_k,\bF(\ba_{k+\half})} \ge 0$. The fourth non-positive term is
\begin{align}\label{eq:cons4}
&-\InAngles{\ba_k,\bF(\ba_k)}\InAngles{\ba_k,\bF(\ba_{k+\half})}\notag\\
&= -\InParentheses{\beta_1\bF(\bz_k)[1] + \beta_2\bF(\bz_k)[2] + \bF(\bz_k)[3]} \InParentheses{\beta_1\bz_{k+\half}[1] + \beta_2 \bz_{k+\half}[2] + \bz_{k+\half}[3]} \le 0.
\end{align}
By Lemma \ref{lem:reduce to cone}, we have $\ba_{k+1}$ is in the same direction as $\bz_{k+1}-\bz_k + \eta \bF(\bz_{k+\half})$, $\InAngles{\ba_{k+1},\bz_{k+1}} = 0$, $\InAngles{\ba_{k+1},\bz_k} \ge 0$. Thus
\begin{align*}
\bz_k[1] - \eta \bF(\bz_{k+\half})[1] = \InAngles{a_{k+1},\bz_k - \eta \bF(\bz_{k+\half})} = \InAngles{a_{k+1},\bz_k - \eta \bF(\bz_{k+\half})-\bz_{k+1}} &\le 0 \\
\bz_k[1] = \InAngles{\ba_{k+1},\bz_{k}} &\ge 0.
\end{align*}
The fifth non-positive term is
\begin{align}\label{eq:cons5-1}
\bz_k[1]\InParentheses{\bz_k[1] - \eta \bF(\bz_{k+\half})[1]} \le 0.
\end{align}
Let us first consider the case when $\bF(\bz_{k+1})[1] \le 0$.
\paragraph{Case 1: $\bF(\bz_{k+1})[1] \ge 0$. } Then (\ref{eq:general-final-potential}) can be written as follows:
\begin{align}
&\sum_{i=1}^3 \eta^2\InParentheses{\bF(\bz_k)[i]^2 - \bF(\bz_{k+1})[i]^2}- \eta^2(\beta_1 \bF(\bz_k)[1] + \beta_2 \bF(\bz_k)[2] + \bF(\bz_k)[3])^2 +\eta^2 \bF(\bz_{k+1})[1]^2 .\label{eq:general-final-potential-1}
\end{align}
The following equality holds
\begin{align*}
&(\ref{eq:general-final-potential-1}) + 2\times \InParentheses{ (\ref{eq:cons1})+ (\ref{eq:cons5-1})} + \frac{2\alpha}{1+\alpha^2} \times (\ref{eq:cons2}) + \frac{2}{1+\alpha^2} \times (\ref{eq:cons3}) + \frac{2}{1+\beta_1^2+\beta_2^2} \times (\ref{eq:cons4}) \\
& = \InParentheses{\bz_k[1]-\bF(\bz_{k+\half})[1] + \bF(\bz_{k+1})[1]}^2 + \frac{1}{1+\beta_2^2} \InParentheses{\bz_k[2] - \bF(\bz_{k})[2] + \alpha \bz_{k}[1]}^2 \\
& \quad + \frac{1}{1+\beta_1^2+\beta_2^2} \left(\InParentheses{\bF(\bz_k)[3]^2 + \beta_1\bz_k[1]^2 + \beta_2 \bz_{k}[2] + \InParentheses{\alpha\beta_2-\beta_1}\bz_{k+\half}[1]}^2 \right.\\&\left.\quad + \frac{1}{1+\beta_2^2} \InParentheses{\InParentheses{1+\beta_2^2}\InParentheses{\bF(\bz_k)[1]- \bz_k[1]} -\beta_1\beta_2 \InNorms{\bF(\bz_k[2]) - \bz_{k}[2]} + \InParentheses{1+\beta_2^2 + \alpha\beta_1\beta_2}\bz_{k+\half}[2]}^2 \right)\\
&\ge 0.
\end{align*}
Thus (\ref{eq:general-final-potential}) is non-negative.
\paragraph{Case 1: $\bF(\bz_{k+1})[1] \le 0$. } Then (\ref{eq:general-final-potential}) can be written as follows:
\begin{align}
&\sum_{i=1}^3 \eta^2\InParentheses{\bF(\bz_k)[i]^2 - \bF(\bz_{k+1})[i]^2}- \eta^2(\beta_1 \bF(\bz_k)[1] + \beta_2 \bF(\bz_k)[2] + \bF(\bz_k)[3])^2 .\label{eq:general-final-potential-2}
\end{align}
We add one more non-positive term
\begin{align}\label{eq:cons6}
\bF(\bz_{k+1})[1] \InParentheses{\bz_k[1] - \eta \bF(\bz_{k+\half})[1]} \le 0.
\end{align}
The following equality holds
\begin{align*}
&(\ref{eq:general-final-potential-2}) + 2\times \InParentheses{ (\ref{eq:cons1})+ (\ref{eq:cons5-1})+(\ref{eq:cons6})}+ \frac{2\alpha}{1+\alpha^2} \times (\ref{eq:cons2}) + \frac{2}{1+\alpha^2} \times (\ref{eq:cons3}) + \frac{2}{1+\beta_1^2+\beta_2^2} \times (\ref{eq:cons4}) \\
& = \InParentheses{\bz_k[1]-\bF(\bz_{k+\half})[1] }^2 + \frac{1}{1+\beta_2^2} \InParentheses{\bz_k[2] - \bF(\bz_{k})[2] + \alpha \bz_{k}[1]}^2 \\
& \quad + \frac{1}{1+\beta_1^2+\beta_2^2} \left(\InParentheses{\bF(\bz_k)[3]^2 + \beta_1\bz_k[1]^2 + \beta_2 \bz_{k}[2] + \InParentheses{\alpha\beta_2-\beta_1}\bz_{k+\half}[1]}^2 \right.\\&\left.\quad + \frac{1}{1+\beta_2^2} \InParentheses{\InParentheses{1+\beta_2^2}\InParentheses{\bF(\bz_k)[1]- \bz_k[1]} -\beta_1\beta_2 \InNorms{\bF(\bz_k[2]) - \bz_{k}[2]} + \InParentheses{1+\beta_2^2 + \alpha\beta_1\beta_2}\bz_{k+\half}[2]}^2 \right)
\end{align*}
Thus (\ref{eq:general-final-potential}) is non-negative. This also completes the proof.
\end{proof}
We make the following assumptions. We denote $w = (x,y,z)$ and $F(x_i) = F(w_i)_x$ for $i \in [3]$. We consider the following cone:
\begin{align}
\begin{pmatrix}
\beta_1 & \beta_2 & 1\\
\alpha & 1 & 0 \\
1 & 0 & 0
\end{pmatrix}
w \ge 0.
\end{align}
We assume that $w_1$ lies on $(\beta_1,\beta_2,1)^\top w = 0$, $w_2$ lies on $(\alpha,1,0)^\top w = 0$ and $w_3$ lies on $(1,0,0)^\top w = 0$. Recall the update rule of EG:
\begin{align}
&w_2 = \Pi[w_1 - F(w_1)], \\
&w_3 = \Pi[w_2 - F(w_2)].
\end{align}
We assume $w_1 - F(w_1)$ violates $(\alpha,1,0)^\top w \ge 0$ and $w_1 - F(w_2)$ violates $(1,0,0)^\top w \ge 0$. We consider the following two cases based on the sign of $F(x_3)$.
\paragraph{Case 1: $F(x_3) \le 0.$}
The following terms are negative:
\begin{align*}
(F(x_2)-F(x_3))^2 + (F(y_2)-F(y_3))^2 + (F(z_2)-F(z_3))^2 - (x_2-x_3)^2 - (y_2 - y_3)^2 - (z_2 - z_3)^2 & \le 0 \tag{1} \\
(F(x_1) - F(x_3)) (x_3-x_1) + (F(y_1) -F(y_3))(y_3 - y_1) + (F(z_1) - F(z_3))(z_3 - z_1) & \le 0 \tag{2} \\
(x_1 - F(x_3))(x_1 - F(x_2)) & \le 0 \tag{2}\\
(\alpha x_3 + y_3)(\alpha (x_1 - F(x_1))+(y_1 - F(y_1))) & \le 0 \tag{$\frac{2}{1+\alpha^2}$} \\
- (\beta_1 F(x_1) + \beta_2 F(y_1) + F(z_1)) (\beta_1 x_2 + \beta_2 y_2 + z_2) & \le 0 \tag{$\frac{2}{(1+\beta_1^2+\beta_2^2)}$}
\end{align*}
The first inequality is due to the Lipschitzness of $F$ between $w_2$ and $w_3$. The second inequality is due to the monotonicity of $F$ between $w_1$ and $w_3$. In the third inequality, we use $(1,0,0)^\top w_1 \ge 0$, $F(x_3) \le 0$ and $(1,0,0)^\top (w_1 - F(w_2)) \le 0$. In the fourth inequality, we use $(\alpha,1,0)^\top w_3 \ge 0$ and $(\alpha,1,0)^\top (w_1 - F(w_1)) \le 0$. In the last inequality, we use $(\beta_1,\beta_2,1)^\top F(w_1) \le 0$ and $(\beta_1,\beta_2,1)^\top w_2 \ge 0$.
We assume that $w_1 - F(w_1) - w_2$ is orthogonal to $(\alpha, 1, 0)^\top w = 0$. Thus
\begin{align}
(1, -\alpha, 0)^\top w = (x_1 - F(x_1) - x_2) - \alpha(y_1 - F(y_1) - y_2) = 0.
\end{align}
Therefore, the following terms are zero:
\begin{align*}
x_2((x_1 - F(x_1) - x_2) - \alpha(y_1 - F(y_1) - y_2)) &= 0 \tag{-2} \\
y_3((x_1 - F(x_1) - x_2) - \alpha(y_1 - F(y_1) - y_2)) &= 0 \tag{$\frac{-2\alpha}{1+\alpha^2}$} \\
\end{align*}
Adding the above seven terms with corresponding coefficients to the objective, we get a sum of squares polynomial which is non-negative.
\begin{align*}
&F(x_1)^2 + F(y_1)^2 + F(z_1)^2 - (\beta_1 F(x_1) + \beta_2 F(y_1) + F(z_1))^2/(\beta_1^2 + \beta_2^2 + 1) \\
& \quad - F(x_3)^2 - F(y_3)^2 - F(z_3)^2.
\end{align*}
The sum of squares polynomial is the following:
\begin{align}
&(x_1 - F(x_2))^2\\
&+ \frac{1}{1+\beta_2^2}(y_1-F(y_1) + \alpha x_2)^2 \\
&+\frac{1}{1+\beta_1^2+\beta_2^2}(F(z_1) + \beta_1 x_1 + \beta_2 y_1 + (\alpha\beta_2-\beta_1)x_2)^2\\
& +\frac{1}{(1+\beta_2^2)(1+\beta_1^2+\beta_2^2)}((1+\beta_2^2)(F(x_1)-x_1)- \beta_1\beta_2 (F(y_1)-y_1) + (1 + \beta_2^2 + \alpha\beta_1\beta_2)x_2 )^2
\end{align}
\paragraph{Case 2: $F(x_3) \ge 0$.}
Then we do not count $F(x_3)^2$ in $H(w_3)$.
The only difference to Case 1 is that we use $x_1(x_1 - F(x_2))$ instead of $(x_1-F(x_3)(x_1 -F(x_2))$.
\notshow{
Therefore, we have the following five negative terms:
\begin{align*}
(F(x_2)-F(x_3))^2 + (F(y_2)-F(y_3))^2 + (F(z_2)-F(z_3))^2 - (x_2-x_3)^2 - (y_2 - y_3)^2 - (z_2 - z_3)^2 & \le 0 \tag{1} \\
(F(x_1) - F(x_3)) (x_3-x_1) + (F(y_1) -F(y_3))(y_3 - y_1) + (F(z_1) - F(z_3))(z_3 - z_1) & \le 0 \tag{2} \\
x_1(x_1 - F(x_2)) & \le 0 \tag{2}\\
(\alpha x_3 + y_3)(\alpha (x_1 - F(x_1))+(y_1 - F(y_1))) & \le 0 \tag{$\frac{2}{1+\alpha^2}$} \\
- (\beta_1 F(x_1) + \beta_2 F(y_1) + F(z_1)) (\beta_1 x_2 + \beta_2 y_2 + z_2) & \le 0 \tag{$\frac{2}{(1+\beta_1^2+\beta_2^2)}$}
\end{align*}
The following terms are zero:
\begin{align*}
x_2((x_1 - F(x_1) - x_2) - \alpha(y_1 - F(y_1) - y_2)) &= 0 \tag{-2} \\
y_3((x_1 - F(x_1) - x_2) - \alpha(y_1 - F(y_1) - y_2)) &= 0 \tag{$\frac{-2\alpha}{1+\alpha^2}$} \\
\end{align*}
}
By adding the seven terms with corresponding coefficients to the following objective, we get a sum of squares polynomial which is non-negative.
\begin{align*}
&F(x_1)^2 + F(y_1)^2 + F(z_1)^2 - (\beta_1 F(x_1) + \beta_2 F(y_1) + F(z_1))^2/(\beta_1^2 + \beta_2^2 + 1) \\
& \quad - F(y_3)^2 - F(z_3)^2.
\end{align*}
The sum of squares polynomial is the following:
\begin{align}
&(x_1 - F(x_2) + F(x_3))^2\\
&+ \frac{1}{1+\beta_2^2}(y_1-F(y_1) + \alpha x_2)^2 \\
&+\frac{1}{1+\beta_1^2+\beta_2^2}(F(z_1) + \beta_1 x_1 + \beta_2 y_1 + (\alpha\beta_2-\beta_1)x_2)^2\\
& +\frac{1}{(1+\beta_2^2)(1+\beta_1^2+\beta_2^2)}((1+\beta_2^2)(F(x_1)-x_1)- \beta_1\beta_2 (F(y_1)-y_1) + (1 + \beta_2^2 + \alpha\beta_1\beta_2)x_2 )^2
\end{align} }
| 20,869
|
Dell Inspiron 15 3521(Pentium 2nd Gen/2GB/500GB/Linux) is a budget laptop with a very good battery life when compared with other Dell laptops. This laptop comes with black matte textured finish which attracts many users who wants their laptop to look simple and at the same time presentable. The laptop comes with Linux which makes it more easy to use. Though multi-tasking makes the system little slow because of pretty basic processor. Apart from this, it is one of good laptops priced so reasonably..About my lap ! It will be the cheap and best lap you get from dell for daily use.Major features!...The quality of dell that stand first, most of the lap under 26k are normally poor quality, in this case its dell, Its high quality body give it a Contemporary look which have black matte finish, textured lid and large palmrest.Processor Pentium Dual Core (2nd Generation), Variant 997, As if now i don't feel any speed issue, most of my software and games are working well with my 2 GB DDR3 Ram, and 500 GB hard. Keyboard Its have a Standard Keyboard which is too nice and perfect for my job ! V - good Touch-pad. Good Audio !DVD writer with Dual Layer SupportPorts Two USB 3.0, Two USB 2.0, HDMI 1.4, RJ45 Ethernet ports, card reader, and Headphone port.ports are working perfectly.Most important feature Power If the battery is in fully charged - Am getting Battery Backup Almost more than 6 hr if i adjust my display, and working normally ! Without adjusting and play video and game it will long up to 5.15 hours max. In ubuntu It will be more than 5.20 hours with adjusting, Without adjusting less than 5 hr It will take almost 3.30 hr to get full charge from 10% to 100 %some con....The Stereo Speakers are placed below . (But i don't care) Ram 2GB and 1 Unused Slot (Hmmm????? ) Dell have only 1 year warranty, and their extendable warranty cost too high ! (if its running without any prob. am lucky) This lap is a basic model of dell and its condition is suitable for some normal day-today works, so its not a multitask lap with many feature or a gaming material !According to me its a profitable gadget and money which i spend is really cheap and so if you are looking for a cheap Dell lap ill recommend this !
Faster Processor:
2411 vs 1525
Lesser weight:
2.36 Kg vs 2.37 Kg
More thickness:
34.5 Millimeter vs 33 Millimeter
Faster Processor:
3997 vs 1525
Lower battery life:
4 Hrs vs 5 Hrs
Common Features
Lesser thickness:
30.3 Millimeter vs 33 Millimeter
Higher battery life:
7 Hrs vs 5 Hrs
More hard-disk capacity:
1024 GB vs 500 GB
Lesser thickness:
25.4 Millimeter vs 33 Millimeter
Lesser weight:
2 Kg vs 2.37 Kg
Fewer USB 2.0 ports:
2 vs 3
Higher display resolution:
1600 x 900 Pixels vs 1366 x 768 Pixels
Lesser thickness:
25.3 Millimeter vs 33 Millimeter
Higher battery life:
8.25 Hrs vs 5 Hrs
Higher battery life:
6 Hrs vs 5 Hrs
Faster Processor:
2411 vs 1525
Lesser weight:
2.15 Kg vs 2.37 Kg
Sorry, no question asked yet.
| 90,041
|
The video that relates to this news can be viewed in our group VKontakte, on Facebook or on YOUTUBE at this link: In the LIGHT OF LIGHT in the tourist area of Tianyang in Sanya City, Hainan there are two amazing coconut palms that intersect at a right angle, and then vertically grow upwards, from the side it looks like two spouses who stand with their hands-you can see it on the video. You can see fresh video or photos related to this news and previous news, and ask all the questions you are interested in our group on Facebook: or to our forum on Hainan Island:
| 136,656
|
.
8 Gifts for the Moms in Your Life
Mother’s Day is May 8th, which means there’s no time like the present to purchase your gift. Here are Favor’s curated picks for thoughtful, personalized Mother’s Day gifts that are guaranteed to please.
Meet Our Newest Executive Hires
Welcoming The Pill Club's Chief Financial Officer, VP of Product Management, and Chief Compliance Officer.
Introducing The Pill Club's New CEO, Liz Meyerdirk
Leadership Highlight: Liz Meyerdirk, CEO
Get to know The Pill Club CEO Liz Meyerdirk!
3 Questions with Janeen “Neen” Kennamore, Lead Pharmacy Technician
3 Questions with Kristine Fernandez, Social Media Supervisor
Employee Spotlight: Pamon Forouhar
3 Questions with Kristine Fernandez, Lifecycle Marketing Associate
Now Delivering Birth Control in Arkansas
Employee Spotlight: Story Sylwester, Operations
3 Questions with Silvia Leo, Patient Care Advocate
Care About Equality? Take Action for Telehealth Awareness Week
3 Questions with Tresa Wallace, Nurse Practitioner
3 Questions with Michael Easley, Social Media Care Coordinator
Meet Our Head of People
3 Questions with LaKeisha Shade, Staff Pharmacist
Meet Our Head of CA Pharmacy Ops
| 4,502
|
mountain berry
see cloudberry
From The Food Lover's Companion, Fourth edition by Sharon Tyler Herbst and Ron Herbst. Copyright © 2007, 2001, 1995, 1990 by Barron's Educational Series, Inc.
Next Up
Rosé Strawberries and Raspberries Now Exist
We’ll give you a moment to drink that fact in.
Alex Eats: Strawberries
There are so many things you can do with strawberries during the season, both sweet and savory. Below, find my recipe for Strawberry tarts.
Win Driscoll's Berries!
We're giving away $50 worth of Driscoll's berries; just tell us how you like to eat your strawberries, blueberries, raspberries or blackberries.
In Season: Raspberries
Raspberries have a sweet-tart flavor and are full of vitamin C and fiber -- a perfect summer berry to enjoy.
Strawberries in July
I went to the farmers' market to get strawberries. I thought I might have missed their short season, but they were in fact there. And then, as if I were somewhere I might never visit again, I suddenly needed everything else there, too.
In Season: Blueberries
Blueberries are a definitely healthy powerhouse -- full of vitamin k, vitamin c, the mineral manganese and the mega-antioxidants, anthocyanidins. Here’s are some recipes to show some blueberry love, especially during National Blueberry Month!.
Blueberries 5 Ways
Despite their small size, these berries are a nutrition powerhouse. Packed with vitamins, minerals and antioxidants, serve them up in these five fun recipes.
In Season: Strawberries
I adore these luscious red berries for more than just their sweet flavor. They're full of vitamin C and have some great, heart-healthy benefits, too. Learn more.
| 172,954
|
TITLE: Current knowledge of Higgs
QUESTION [5 upvotes]: What is the current knowledge about the Higgs field after its discovery in the LHC? Does it exactly mimic the standard model Higgs? Does this knowledge rules out the possibility of other Higgs particles beyond the standard model like triplet higgs model etc?
REPLY [2 votes]: Your question is extremely vast. Here is a quick answer:
Does it exactly mimic the standard model Higgs?
Within experimental errors, the answer is yes (so far). The couplings tested so far (directly or via quantum loops) show that they are proportional to the mass of the fermions as expected by the standard model (SM). Only heavy fermions have reasonably been tested: $b,t,\tau$. The couplings to gauge bosons $Z,W,\gamma$ are also compatible with the SM.
The spin and CP numbers are compatible with SM expectation even if statistic accumulated during the run1 (which lasted 2 years) is not large enough to prove it with a $5 \sigma$ confidence level.
One important thing to be tested with run2 data (started reasonably in May-June at 13 TeV): the self coupling of the higgs boson (to itself). We estimate that by the end of run2 (~2023), the trilinear coupling will be tested: $H \to HH$. The quadrilinear coupling $H\to HHH$ (the famous $\lambda$ in the Higgs scale potential) is probably beyond the reach of LHC run2.
Does this knowledge rules out the possibility of other Higgs particles beyond the standard model like triplet higgs model etc?
The minimal supersymmetric model (MSSM) is now severely constrained. The whole phase space is however not totally ruled out. With run2 data, one might discover some heavy super-particles or deviation of the SM coupling to low mass fermions for instance: $H \to \mu\mu$. But if so, scenarios beyond MSSM are probable.
| 129,392
|
\section{Strict Ordering Preserved under Cancellability in Totally Ordered Semigroup}
Tags: Ordered Semigroups
\begin{theorem}
Let $\left({S, \circ, \preceq}\right)$ be a [[Definition:Totally Ordered Semigroup|totally ordered semigroup]].
If either:
: $x \circ z \prec y \circ z$
or
: $z \circ x \prec z \circ y$
then $x \prec y$.
\end{theorem}
\begin{proof}
Let $\preceq$ be a [[Definition:Total Ordering|total ordering]].
Let $x \circ z \prec y \circ z$.
But we have, by hypothesis:
:$x \succeq y \implies x \circ z \succeq y \circ z$
which contradicts $x \circ z \prec y \circ z$.
So $x \prec y$.
Similarly for $z \circ x \prec z \circ y$.
{{qed}}
\end{proof}
| 57,987
|
TITLE: Local trivialization for cotangent space of algebraic variety
QUESTION [1 upvotes]: I'm trying to understand why around any non-singular point in an $n$-dimensional variety we can find $n$ regular functions $\{f_i\}$ such that $\{df_i\}$ generate the cotangent space at every point in the neighborhood.
Given a basis for $T_p^{\ast}V$, I know that Nakayama's lemma tells us that there exist
$n$ functions $\{f_i\}$ such that $\{df_i\}$ is the basis we started out with. What's not clear is why these differentials of these $n$ functions also generate the cotangent space in a neighborhood of $p$. Is the property of $\{f_i\}$ generating a basis an open property, i.e. given by the non-vanishing of some polynomials? If not, how does one show this?
I'd appreciate any help on this problem. Thanks!
REPLY [1 votes]: I found an answer here, in the proof of lemma 1.6. I'll sketch out a proof here as well if the link ever dies.
Without loss of generality, we can assume $X$ is an affine non-singular variety after
passing to an affine open neighborhood. Our first claim is that we can do this in a
manner such that the first $i$ coordinate functions $t_i$ when restricted to some open set $U$ around $p$ are the functions $f_i$. This can be seen by extending the collection $\{f_i\}$ to a generating set of $\mathcal{O}_X$.
That means the collection of the differentials $\{dt_i\}$ generate the cotangent space
at each point. But at $p$, the differentials $df_i$ also generate the cotangent space
at $p$. That means at $p$, the change of basis matrix from $dt_i$ to $df_i$ which is given by $\left( \frac{\partial f_i}{\partial t_i} \right)$ is invertible, i.e. its
determinant is non-zero, but that's an open condition, which means its determinant
is non-zero in a neighborhood, and in that neighborhood $\{df_i\}$ spans the cotangent
space.
| 128,507
|
\begin{document}
\maketitle
\begin{abstract}
By making use of the variational tricomplex, a covariant procedure is proposed for deriving the classical BRST charge of the BFV formalism from a given BV master action.
\end{abstract}
\section{Introduction}
The BRST theory provides the most powerful approach to the quantization of gauge systems \cite{HT}. It includes the Batalin-Vilkovisky (BV) formalism for Lagrangian gauge systems and its Hamiltonian counterpart known as the
Batalin-Fradkin-Vilkovisky (BFV) formalism. Usually, the two formalisms are developed in parallel starting,
respectively, from the classical action or the first-class
constraints on the phase space of the system.
In either case one applies the homological
perturbation theory (hpt) to obtain the master action or the
classical BRST charge at the output. A relationship between both the pictures of gauge dynamics is
established through the Dirac-Bergmann (DB) algorithm, which allows
one to generate the complete set of first-class constraints by
the classical action. All these can be displayed diagrammatically as
follows:
$$
\xymatrix@R=10pt{*+[F]\txt{\Small \;Lagrangian gauge theory\;\;\\\Small with action $S_0$} \ar[rr]^-{\txt{\Small \textit{hpt}}}\ar[dd]|{\txt{\Small \textit{DB algorithm}}} &&*+[F]\txt{\Small Master action \\\Small \;\; $S=S_0+\cdots\;\;$}\ar@{.>}[dd]^{?}\\&&\\
*+[F]\txt{\Small Hamiltonian theory with\\\Small the $1$-st class constraints $T_a$} \ar[rr]^-{\txt{\Small \textit{hpt}}}&&*+[F]\txt{\Small BRST charge \\\Small $\Omega=C^aT_a+\cdots$} }
$$
Looking at this picture it is natural to ask about the dotted arrow making the diagram commute. The arrow symbolizes a hypothetical map or construction connecting the BV and BFV formalisms at the level of generating functionals. As we show below such a map really exists. By making use of the variational tricomplex \cite{Sh1}, we propose a direct construction of the classical BRST charge from the BV master action. The construction is explicitly covariant (even though we pass to the Hamiltonian picture) and generates the full spectrum of BFV ghosts immediately from that of the BV theory. We also derive a covariant Poisson bracket on the extended phase space of the theory, with respect to which the classical BRST charge obeys the master equation. The construction of the covariant Poisson bracket is similar to that presented in \cite{Dickey}, except that our Poisson bracket is defined off shell.
Finally, it should be noted that the first variational tricomplex for gauge systems was introduced in \cite{BH} as the Koszul-Tate resolution of the usual variational bicomplex for partial differential equations. Using this tricomplex, the authors of \cite{BH} were able to relate various Lie algebras associated with the global symmetries and conservation laws of a classical gauge system. Our tricomplex is similar in nature but involves the full BRST differential, and not its Koszul-Tate part.
\section{Variational tricomplex of a local gauge system}
In modern language the classical fields are just the sections of a locally trivial, fiber bundle $\pi : E\rightarrow M$ over an $n$-dimensional space-time manifold $M$. The typical fiber $F$ of $E$ is called the \textit{target space of fields}. In case the bundle is trivial, i.e., $E=M\times F$, the fields are merely the smooth mappings from $M$ to $F$. For the sake of simplicity, we restrict ourselves to fields associated with vector bundles. In this case the space of fields $\Gamma (E)$ has the structure of a real vector space.
Bearing in mind gauge theories as well as field theories with fermions, we assume $\pi: E\rightarrow M$ to be a $\mathbb{Z}$-graded supervector bundle over the ordinary (non-graded) smooth manifold $M$. The Grassmann parity and the $\mathbb{Z}$-grading of a homogeneous object $A$ will be denoted by $\epsilon(A)$ and $\deg A$, respectively. It should be emphasized that in the presence of fermionic fields there is no natural correlation between the Grassmann parity and the $\mathbb{Z}\,$-grading. Since throughout the paper we work exclusively in the category of $ \mathbb{Z}\,$-graded supermanifolds, we omit the boring prefixes ``super'' and ``graded'' whenever possible. For a quick introduction to the graded differential geometry and some of its applications we refer the reader to \cite{V1}, \cite{CatSch}, \cite{V2}.
In the local field theory, the dynamics of fields are governed by partial differential equations.
The best way to account for the local structure of fields is to introduce the variational bicomplex $\Lambda^{\ast,\ast}(J^\infty E; d, \delta)$ on the infinite jet bundle $J^\infty E$ associated with the vector bundle $\pi: E\rightarrow M$. Here $d$ and $\delta$ denote the horizontal and vertical differentials in the bigraded space $\Lambda^{\ast,\ast}(J^\infty E)=\bigoplus \Lambda^{p,q}(J^\infty E)$ of differential forms on $J^\infty E$, where $p$ and $q$ refer to the vertical and horizontal degrees, respectively. A brief account of the concept of a vatiational bicomplex can be found in \cite{Anderson}, \cite{Dickey}.
The free variational bicomplex represents thus a natural kinematical basis for defining local field theories. In order to specify dynamics two more geometrical ingredients are needed. These are the classical BRST differential and the BRST-invariant (pre)symplectic structure on $J^\infty E$. Let us give the corresponding definitions.
\subsection{Presymplectic structure}\label{PrSt} By a \textit{presymplectic} $(2,m)$-form on $J^\infty E$ we understand an
element $\omega\in \widetilde{\Lambda}^{2,m}(J^\infty E)$ satisfying
\footnote{By abuse of notation, we denote by $\omega$ an element of the quotient space $\widetilde{\Lambda}^{2,m}=\Lambda^{2,m}/d\Lambda^{2,m-1}$ and its representative in $\Lambda^{2,m}$. The sign $\simeq$ means equality modulo $d\Lambda^{\ast,\ast}$.}
\begin{equation}\label{dom}
\delta \omega \simeq 0\,.
\end{equation}
The form $\omega$ is assumed to be homogeneous, so that we can speak of an odd or even presymplectic structure of definite $\mathbb{Z}$-degree. The triviality of the relative ``$\delta$ modulo $d$'' cohomology in positive vertical degree (see \cite[Sec. 19.3.9]{Dickey}) implies that any presymplectic $(2,m)$-form is exact, namely, there exists a homogeneous $(1,m)$-form $\theta$ such that $\omega\simeq\delta\theta$. The form $\theta$ is called the \textit{presymplectic potential} for $\omega$. Clearly, the presymplectic potential is not unique. If $\theta_0$ is one of the presymplectic potentials for $\omega$, then setting $\omega_0=\delta \theta_0$ we get
$$
\delta \omega_0=0\,,\qquad \omega_0\simeq \omega\,.
$$
In other words, any presymplectic form has a $\delta$-closed representative.
Denote by $\ker\omega$ the space of all evolutionary vector fields $X$ on $J^\infty E$ that fulfill the relation\footnote{Recall that a vertical vector field $X$ is called \textit{evolutionary } if $i_X d+(-1)^{\epsilon({X})}di_{X}=0$, where $i_X$ is the operation of contraction of $X$ with differential forms.}
$$
i_X\omega \simeq 0\,.
$$
A presymplectic form $\omega$ is called non-degenerate if $\ker \omega=0$, in which case we refer to it as a \textit{symplectic form}.
An evolutionary vector field $X$ is called \textit{Hamiltonian} with respect to $\omega$ if it preserves the presymplectic form, that is,
\begin{equation}\label{Xom}
L_X\omega\simeq 0\,.
\end{equation}
Obviously, the Hamiltonian vector fields form a subalgebra in the Lie algebra of all evolutionary vector fields. Eq. (\ref{Xom}) is equivalent to
$$
\delta i_X\omega\simeq 0\,.
$$
Again, because of the triviality of the relative $\delta$-cohomology, we can write
\begin{equation}\label{HVF}
i_X\omega \simeq \delta H
\end{equation}
for some $H\in \widetilde{\Lambda}^{0,m}(J^\infty E)$. We refer to $H$ as a \textit{Hamiltonian form} (or \textit{Hamiltonian}) associated with $X$. Sometimes, to indicate the relationship between the Hamiltonian vector fields and forms, we will write $X_H$ for $X$. In general, the relationship is far from being one-to-one.
The space ${\Lambda}_\omega^{0,m}(J^\infty E)$ of all Hamiltonian $m$-forms can be endowed with the structure of a Lie algebra. The corresponding Lie bracket is defined as follows: If $X_A$ and $X_B$ are two Hamiltonian vector fields associated with the Hamiltonian forms $A$ and $B$, then
\begin{equation}\label{PB}
\{A,B\}=(-1)^{\epsilon(X_A)}i_{X_A}i_{X_B}\omega\,.
\end{equation}
The next proposition shows that the bracket is well defined and possesses all the required properties.
\begin{prop}[\cite{Sh1}]\label{2.1}
The bracket (\ref{PB}) is bilinear over reals, maps the Hamiltonian forms to Hamiltonian ones, enjoys the symmetry property
\begin{equation}\label{sym}
\{A,B\}\simeq -(-1)^{(\epsilon(A)+\epsilon(\omega))(\epsilon(B)+\epsilon(\omega))}\{B,A\}\,,
\end{equation}
and obeys the Jacobi identity
\begin{equation}\label{jac}
\{C,\{A,B\}\}\simeq \{\{C,A\},B\}+(-1)^{(\epsilon(C)+\epsilon(\omega))(\epsilon(A)+\epsilon(\omega))}\{A,\{C,B\}\}\,.
\end{equation}
\end{prop}
\subsection{Classical BRST differential}\label{brst}
An odd evolutionary vector field $Q$ on $J^\infty E$ is called \textit{homological} if
\begin{equation}\label{QQ}
[Q,Q]=2Q^2=0\,, \qquad \deg\,Q=1\,.
\end{equation}
The Lie derivative along the homological vector field $Q$ will be denoted by $\delta_Q$. It follows from the definition that $\delta_Q^2=0$. Hence, $\delta_Q$ is a differential of the algebra $\Lambda^{\ast,\ast}(J^\infty E)$ increasing the $\mathbb{Z}$-degree by 1. Moreover, the operator $\delta_Q$ anticommutes with the coboundary operators $d$ and $\delta$:
$$
\delta_Q d+d\delta_Q=0\,,\qquad \delta_Q \delta+\delta\delta_Q=0\,.
$$
This allows us to speak of the tricomplex $\Lambda^{\ast,\ast,\ast}(J^\infty E; d, \delta, \delta_Q)$, where
$$
\delta_Q: \Lambda^{p,q,r}(J^\infty E)\rightarrow \Lambda^{p,q,r+1}(J^\infty E)\,.
$$
In the physical literature the homological vector field $Q$ is known as the \textit{classical BRST differential} and the $\mathbb{Z}$-grading is called the \textit{ghost number}. These are the two main ingredients of all modern approaches to the covariant quantization of gauge theories. In the BV formalism, for example, the BRST differential carries all the information about equations of motions, their gauge symmetries and identities, and the space of physical observables is naturally identified with the group $H^{0,{n},0}(J^\infty E; \delta_Q/d) $ of ``$\delta_Q$ modulo $d$'' cohomology in ghost number zero. For general non-Lagrangian gauge theories the classical BRST differential was systematically defined in \cite{LS0}, \cite{KazLS}.
The equations of motion of a gauge theory can be recovered by considering the zero locus of the homological vector field $Q$. In terms of adapted coordinates $(x^i, \phi^a_I)$ on $J^\infty E$ the vector field $Q$, being evolutionary, assumes the form\footnote{We use the multi-index notation according to which the multi-index $I=i_1i_2\cdots i_k$ represents the set of symmetric covariant indices and $\partial_I=\partial_{i_1}\cdots\partial_{i_k}$. The \textit{order} of the multi-index is given by $|I|=k$.}
$$
Q=\partial_I Q^a\frac{\partial}{\partial \phi_I^a}\,.
$$
Then there exists an integer $l$ such that the equations
$$
\partial_I Q^a=0\,,\qquad |I|=k\,,
$$
define a submanifold $\Sigma^k\subset J^{l+k}E$. The standard regularity condition implies that $\Sigma^{k+1}$ fibers over $\Sigma^k$ for each $k$. This gives the infinite sequence of projections
$$
\xymatrix{\cdots\ar[r]& \Sigma^{l+3}\ar[r]&\Sigma^{l+2}\ar[r]&\Sigma^{l+1}\ar[r]&\Sigma^l}\rightarrow M\,,
$$
which enables us to define the zero locus of $Q$ as the inverse limit
$$
\Sigma^\infty =\lim_{\longleftarrow}\Sigma^k\,.
$$
In physics, the submanifold $\Sigma^\infty\subset J^\infty E$ is usually referred to as the \textit{shell}. The terminology is justified by the fact that the classical field equations as well as their differential consequences can be written as
$$
(j^{\infty}\phi)^\ast (\partial_I Q^a)=0\,.
$$
In other words, the field $\phi\in \Gamma(E)$ satisfies the classical equations of motion iff $j^\infty \phi \in \Sigma^\infty$.
\subsection{$Q$-invariant presymplectic structure and its descendants}
By a \textit{gauge system} on $J^\infty E$ we will mean a pair $(Q, \omega)$ consisting of a homological vector field $Q$ and a $Q$-invariant presymplectic $(2,m)$-form $\omega$. In other words, the vector field $Q$ is supposed to be Hamiltonian with respect to $\omega$, so that $\delta_Q\omega\simeq 0$. The last relation implies the existence of forms $\omega_1$, $H$, and $\theta_1$ such that
\begin{equation}\label{des}
\delta_Q \omega=d\omega_1\,, \qquad i_Q\omega =\delta H +d\theta_1\,.
\end{equation}
As was mentioned in Sec.\ref{PrSt}, we can always assume that $\omega =\delta\theta$ for some presymplectic potential $\theta$, so that $\delta\omega=0$. Then applying $\delta$ to the second equality in (\ref{des}) and using the first one, we find $d(\omega_1-\delta\theta_1)=0$. On account of the exactness of the variational bicomplex, the last relation is equivalent to
$$
\omega_1\simeq \delta\theta_1\,.
$$
Thus, $\omega_1$ is a presymplectic $(2,m-1)$-form on $J^\infty E$ coming from the presymplectic potential $\theta_1$. Furthermore, the form $\omega_1$ is $Q$-invariant as one can easily see by applying $\delta_Q$ to the first equality in (\ref{des}) and using once again the fact of exactness of the variational bicomplex. Let $H_1$ denote the Hamiltonian for $Q$ with respect to $\omega_1$, i.e.,
$$
i_Q\omega_1\simeq \delta H_1\,, \qquad H_1\in \widetilde{\Lambda}^{0,m-1}(J^\infty E)\,.
$$
Given the pair $(Q,\omega)$, we call $\omega_1$ the \textit{descendent presymplectic structure} on $J^\infty E$ and refer to $(Q,\omega_1)$ as the \textit{descendent gauge system}.
The next proposition provides an alternative definition for the descendent Hamiltonian of the homological vector field.
\begin{prop}[\cite{Sh1}]\label{p2}
Let $\omega$ be a $\delta$-closed representative of a presymplectic $(2,m)$-form on $J^\infty E$ and $\mathrm{deg} H_1\neq 0$, then
\begin{equation}\label{HH}
dH_1=-\frac12\{H,H\}\,.
\end{equation}
\end{prop}
\begin{cor}\label{c1}
$H$ is a Maurer-Cartan element of the Lie algebra $\Lambda^{0,m}_\omega(J^\infty E)$, that is, $$\{H,H\}\simeq 0\,.$$
\end{cor}
\begin{cor}\label{CL}
The Hamiltonian form $H_1$ is $d$-closed on-shell. In particular, for $m=n$ it defines a conservation law.
\end{cor}
\begin{prop}[\cite{Sh1}]\label{p5}
Suppose that the $Q$-invariant presymplectic form $\omega$ of top horizontal degree has the structure
\begin{equation}\label{ff}
\omega=P_{ab}\wedge\delta\phi^a\wedge\delta\phi^b\,,\qquad P_{ab}\in \Lambda^{0,n}(J^\infty E)\,,
\end{equation}
and $H$ is the Hamiltonian of $Q$ with respect to $\omega$. Then the presymplectic potential for the descendent presymplectic (2,n-1)-form $\omega_1\simeq \delta\theta_1$ is defined by the equation
\begin{equation}\label{ht1}
\delta H=\delta\phi^a\wedge \frac{\delta H}{\delta\phi^a}-d\theta_1 \,.
\end{equation}
\end{prop}
The above construction of the descendent gauge system $(Q,\omega_1)$ can be iterated producing a sequence of gauge systems $(Q, \omega_k)$, where the $k$-th presymlectic form $\omega_k\in {\Lambda}^{2,m-k}(J^\infty E)$ is the descendant of $\omega_{k-1}$. The minimal $k$ for which $\omega_k \simeq 0$ gives a numerical invariant of the original gauge system $(Q,\omega)$.
\section{BFV from BV}\label{BV-BFV}
In this section, we apply the construction of the variational tricomplex for establishing a direct correspondence between the BV formalism of Lagrangian gauge systems and its Hamiltonian counterpart known as the BFV formalism. We start from a very brief account of both the formalisms in a form suitable for our purposes. For a systematic exposition of the subject we refer the reader to \cite{HT}.
\subsection{BV formalism} The starting point of the BV formalism is an infinite-dimensional manifold $\mathcal{M}_0$ of gauge fields that live on an $n$-dimensional space-time $M$. Depending on a particular structure of gauge symmetry the manifold $\mathcal{M}_0$ is extended to an $\mathbb{N}$-graded manifold $\mathcal{M}$ containing $\mathcal{M}_0$ as its body. The new fields of positive $\mathbb{N}$-degree are called the \textit{ghosts} and the $\mathbb{N}$-grading is referred to as the \textit{ghost number}. Let us collectively denote all the original fields and ghosts by $\Phi^A$ and refer to them as fields. At the next step the space of fields $\mathcal{M}$ is further extended by introducing the odd cotangent bundle $\Pi T^\ast[-1]\mathcal{M}$. The fiber coordinates, called \textit{antifields}, are denoted by $\Phi_A^\ast$. These are assigned with the following ghost numbers and Grassmann parities:
$$
\mathrm{gh} (\Phi^\ast_A)=-\mathrm{gh} (\Phi^A)-1\,,\qquad \epsilon (\Phi^\ast_A)=\epsilon (\Phi^A)+1 \quad (\mbox{mod}\, 2)\,.
$$
Thus, the total space of the odd cotangent bundle $\Pi T^\ast[-1]\mathcal{M}$ becomes a $\mathbb{Z}$-graded supermanifold. The canonical Poisson structure on $\Pi T^\ast[-1]\mathcal{M}$ is determined by the following odd Poisson bracket in the space of functionals of $\Phi$ and $\Phi^\ast$:
\begin{equation}\label{abr}
(A,B)=\int_M \left(\frac{\delta_r A}{\delta \Phi^A}\frac{\delta_l B}{\delta \Phi^\ast_A}-\frac{\delta_r A}{\delta \Phi^\ast_A}\frac{\delta_l B}{\delta \Phi^A}\right)d^nx\,.
\end{equation}
Here $d^nx$ is a volume form on $M$ and the subscripts $l$ and $r$ refer to the standard left and right functional derivatives.
In the physical literature the above bracket is usually called the \textit{antibracket} or the \textit{BV bracket}.
The functionals of the form
$$
A=\int_M (j^\infty \phi)^\ast(a)\,,
$$
where $\phi=(\Phi, \Phi^\ast)$ and $a\in \widetilde{\Lambda}^{0,n}(J^\infty E)$, are called \textit{local}. Under suitable boundary conditions for $\phi$'s the map $a \mapsto A$ defines an isomorphism of vector spaces, which gives rise to a pulled-back Poisson bracket on $\widetilde{\Lambda}^{0,n}(J^\infty E)$. This last bracket is determined by the symplectic structure
\begin{equation}\label{ops}
\omega= \delta \Phi_A^\ast\wedge \delta\Phi^A\wedge d^nx
\end{equation}
according to (\ref{PB}).
By definition, $\mathrm{gh} (\omega)= - 1$ and $\epsilon (\omega)=1$.
The central goal of the BV formalism is the construction of a \textit{master action} $S$ on the space of fields and antifields. This is defined as a proper solution to the \textit{classical master equation}
\begin{equation}\label{BV_MEq}
(S,S)=0\,.
\end{equation}
The local functional $S$ is required to be of ghost number zero and start with the action $S_0$ of the original fields to which one couples vertices involving antifields. All these vertices can be found systematically from the master equation (\ref{BV_MEq}) by means of the so-called \textit{homological perturbation theory} \cite{HT}.
The classical BRST differential on the space of fields and antifields is canonically generated by the master action through the antibracket:
\begin{equation}\label{CLBRSTD}
Q=(S\,,\,\cdot\,)\,.
\end{equation}
Because of the master equation for $S$ and the Jacobi identity for the antibracket (\ref{abr}), the operator $Q$ squares to zero in the space of smooth functionals. The physical quantities are then identified with the cohomology classes of $Q$ in ghost number zero. When restricted to the subspace of local functionals the classical BRST differential (\ref{CLBRSTD}) induces a homological vector field on the total space of the jet bundle $J^\infty E$.
\subsection{BFV formalism} The Hamiltonian formulation of the same gauge dynamics implies a prior splitting $M=N\times \mathbb{R}$ of the original space-time into space and time; the factor $N$ can be viewed as the physical space at a given instant of time. The initial values of the original fields are then considered to form an infinite-dimensional manifold $\mathcal{N}_0$. To allow for possible constraints on the initial data of fields the manifold $\mathcal{N}_0$ is extended to an $\mathbb{N}$-graded supermanifold $\mathcal{N}$ by adding new fields, called ghosts, of positive $\mathbb{N}$-degree. Then the space of original fields and ghosts is doubled by introducing the cotangent bundle $T^\ast \mathcal{N}$ endowed with the canonical symplectic structure. If we denote the local coordinates on $\mathcal{N}$ by $\Phi^a$ and the linear coordinates in the cotangent spaces by $\bar{\Phi}_a$, then the canonical Poisson bracket in the space of functionals of $\Phi^a$ and $\bar{\Phi}_a$ reads
\begin{equation}\label{epb}
\{A,B\}=\int_N \left(\frac{\delta_r A}{\delta \Phi^a}\frac{\delta_l B}{\delta \bar{\Phi}_a}-(-1)^{\epsilon(\Phi_a)}\frac{\delta_r A}{\delta \bar{\Phi}_a}\frac{\delta_l B}{\delta \Phi^a}\right)d^{n-1}x\,.
\end{equation}
Here $d^{n-1}x$ stands for a volume form on $N$. By the definition of the cotangent bundle of a graded manifold
$$
{\mathrm{gh}} ( \bar{\Phi}_a) =-\mathrm{gh}({\Phi}^a)\,,\qquad \epsilon(\bar{\Phi}_a) =\epsilon({\Phi}^a)\,,
$$
Again, the space of local functionals, i.e., functionals of the form
$$
B=\int_N j^\infty(\phi)^\ast (b)\,, \qquad \phi=(\Phi,\bar\Phi)\,,\qquad b\in \tilde{\Lambda}^{0,n-1}(J^\infty E)\,,
$$
appears to be closed w.r.t. the even Poisson bracket (\ref{epb}) and the map $b\mapsto B$ induces an even Poisson bracket on $\widetilde{\Lambda}^{0,n-1}(J^\infty E)$. The latter is determined by the even symplectic form
$$
\omega_1=\delta \bar{\Phi}_a\wedge \delta \Phi^a \wedge d^{n-1}x
$$
of ghost number zero.
The gauge structure of the original dynamics is encoded by the \textit{classical $BRST$ charge} $\Omega$. This is given by an odd, local functional of ghost number $1$ satisfying the classical master equation
$$
\{\Omega,\Omega\}=0\,.
$$
The classical BRST differential in the
extended space of fields and momenta is given now by the Hamiltonian action of the BRST charge:
\begin{equation}\label{BDIF}
Q=\{\Omega\,,\,\cdot\,\}\,.
\end{equation}
It is clear that $Q^2=0$. The group of $Q$-cohomology in ghost number zero is then naturally identified with the space of physical observables. Upon restriction to the space of local functionals the variational vector field (\ref{BDIF}) induces a homological vector field on the total space of the infinite jet bundle.
\subsection{From BV to BFV} It must be clear from the discussion above that any gauge system in the BFV formalism may be viewed as the descendant of the same system in the BV formalism. More precisely, we can define the even presymplectic structure $\omega_1$ on the phase space of a gauge theory as the descendant of the odd symplectic structure (\ref{ops}):
$$
d\omega_1 =\delta_Q(\delta \Phi^\ast_A\wedge \delta \Phi^A\wedge d^nx)=\delta \left( \delta\Phi^A\wedge \frac{\delta S}{\delta \Phi^A}+\delta\Phi^\ast_A\wedge \frac{\delta S}{\delta \Phi^\ast_A}\right)\,.
$$
The corresponding classical BRST charge is given by
$$
\Omega_N=\int_N (j^{\infty}\phi)^\ast(J)\,,
$$
where $N\subset M$ is a space-like, Cauchy hypersurface and $J\in \Lambda^{0,n-1}_{\omega_1}(J^\infty E)$ is the Hamiltonian of the classical BRST differential $Q=(S,\,\cdot\,)$ w.r.t. the descendent presymplectic form $\omega_1$, i.e.,
\begin{equation}\label{J}
\delta J\simeq i_Q\omega_1\,.
\end{equation}
It is clear that $\mathrm{gh}(\Omega)=1$. In virtue of Corollary \ref{c1}, the functional $\Omega$ obeys the classical master equation $\{\Omega,\Omega\}=0$ with respect to the even Poisson bracket associated with $\omega_1$. According to Corollary \ref{CL} the form $J$ represents a conserved current, the BRST current. Formally, this means that the ``value'' of the odd charge $\Omega_N$ does not depend on the choice of $N$ provided that $j^\infty\phi\in \Sigma^\infty$.
Since the canonical symplectic structure (\ref{ops}) on the space of fields and antifields is $\delta$-exact, we can give an equivalent definition for $J$ in terms of the antibracket (\ref{abr}). For this end, consider the dynamics of fields in a domain $D\subset M$ bounded by two Cauchy hypersurfaces $N_1$ and $N_2$. The fields and antifields are assumed to vanish on space infinity together with their derivatives. By Proposition \ref{p2},
$$
-\frac12(S,S)=\int_D (j^\infty \phi)^\ast (d J)=\int_D d [(j^\infty \phi)^\ast (J)]=\Omega_{N_2}-\Omega_{N_1}\,.
$$
Let us illustrate the general construction by a particular example of gauge theory.
\subsection{Maxwell's electrodynamics} In the BV formalism, the free electromagnetic field in $4$-dimensional Minkowski space is described by the master action
\begin{equation}\label{MED}
S=\int L \,,\qquad L=-\Big(\frac14 F_{\mu\nu}F^{\mu\nu}+C\partial^\mu A_\mu^\ast\Big)d^4x\,.
\end{equation}
Here
$$
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu
$$
is the strength tensor of the electromagnetic field, $A^\ast_\mu$ is the antifield to the electromagnetic potential $A_\mu$, and $C$ is the ghost field associated with the standard gauge transformation
$
\delta_\varepsilon A_\mu=\partial_\mu \varepsilon
$.
Since the gauge symmetry is abelian, the master action (\ref{MED}) does not involve the ghost antifield $C^\ast$. The odd symplectic structure (\ref{ops}) on the space of fields and antifields assumes the form
$$
\omega=(\delta A^\ast_\mu\wedge \delta A^\mu+\delta C^\ast\wedge \delta C)\wedge d^4 x\,,\qquad d^4x=dx^0\wedge dx^1\wedge dx^2\wedge dx^3\,,
$$
and the action of the classical BRST differential is given by
\begin{equation}\label{brst-t}
\delta_Q A_\mu =\partial_\mu C\,,\qquad \delta_Q A^\ast_\mu =\partial^\nu F_{\nu\mu}\,,\qquad \delta_Q C=0\,, \qquad \delta_Q C^\ast=\partial^\mu A^\ast_\mu\,.
\end{equation}
The variation of the Lagrangian density reads
$$
\delta L=(\partial^\mu F_{\mu\nu}\delta A^\nu + \partial^\mu A^\ast_\mu\delta C +\partial^\mu C\delta A^\ast_\mu -\partial^\mu \theta_\mu)\wedge d^4x\,,\qquad \theta_\mu =
F_{\mu\nu}\delta A^\nu +C \delta A_\mu^\ast\,.
$$
One can easily check that $i_Q \omega \simeq \delta L$. By Proposition \ref{p5} the form
$$
\theta_1=-\theta_\mu\wedge d^3x^\mu\,,\qquad d^3x^\mu=\eta^{\mu\nu}i_{\frac{\partial}{\partial x^\nu}}d^4x\,,$$
defines the potential for the descendent presymplectic form
\begin{equation}\label{WC}
\omega_1=\delta\theta_1= -(\delta F_{\nu\mu}\wedge \delta A^\mu+\delta C\wedge \delta A^\ast_\nu)\wedge d^3x^\nu\,.
\end{equation}
(Of course, one could arrive at this expression by considering the BRST variation $\delta_Q\omega = d\omega_1$ of the original symplectic structure.)
Applying the BRST differential to the form $\omega_1$ yields one more descendent presymplectic form
$$
\omega_2= \delta C\wedge \delta F_{\mu\nu}\wedge d^2x^{\mu\nu}\,, \qquad d^2x^{\mu\nu}=\eta^{\mu\alpha}i_{\frac{\partial}{\partial x^\alpha}}d^3x^\nu\,.
$$
This last form, being ``absolutely'' invariant under the BRST transformations (\ref{brst-t}), leaves no further descendants.
The $3$-form of the conserved BRST current $J$ associated to the BRST symmetry transformations (\ref{brst-t}) is determined by Eq. (\ref{J}). We find
$$
J=J_\nu d^3x^\nu\simeq-C\partial^\mu F_{\mu\nu} d^3x^\nu\,.
$$
Once we identify $x^0$ with time in the Hamiltonian formalism, the antifield $A^\ast_0$ plays the role of ghost momentum canonically conjugate to $C$ with respect to the presymplectic structure (\ref{WC}). The on-shell conservation of the corresponding BRST charge $\Omega=\int_{\mathbb{R}^3} J_0d^3x$ expresses nothing but the Gauss law $\partial^i F_{i0}=0$.
\vspace{0.2 cm}
\noindent \textbf{Acknowledgements.} The work was partially supported by the
RFBR grant 13-02-00551.
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Saturday, April 24, 2010
Doors, Window, Doors
"Doors, Windows, Doors" is the title of this page spread in my altered book on dreamscapes. As I worked on this spread, I was struck by the lightness of all the doors and windows. They are clean and washed and bright and inviting.
The most obvious symbolism here is opportunity, the invitation to walk through an opening and into something else. Less obvious, but more meaningful is the idea of freedom, safety, and choosing. In the process of creating this spread, I realized that these are safe doors and windows. They allow me to look out into the new landscape. They give me permission to go out and to come in. I can leave and return as I choose. They offer the security of a safe return but encourage exploration beyond their safety.
As in my "Dry, Empty, Barren" spread, the dream weaving is here to capture what is good from this dreamscape. It has snagged for me the freedom of a door and a window. They are invitations to the unknown. They swing inwardly or outwardly depending upon their design. They close up and provide safety. They open wide and offer risk. I choose. That is my freedom.
These are God's doors and windows. They are perfectly placed, christened, and carved by the Carpenter's hands.
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TITLE: Show that if $f:[a,b]\rightarrow \mathbb{R}$ is continuous and $[a,b]\subseteq f([a,b])$ then there exists a fixed point $f(x)=x$
QUESTION [1 upvotes]: I have tried to prove the statement but there is a part where I am not sure if I can argue like that, I will mark it by setting it in bold
We know that $[a,b]\subseteq f([a,b])$ and that $f$ is continuous
$$\Longrightarrow \exists_{x,y\in [a,b]}f(x)=a\wedge f(y)=b$$
Without loss of generality assume $x<y$
$$\Longrightarrow[x,y]\subseteq[a,b]$$
The function $f_{|[x,y]}$ is continuous and for all $\gamma\in [f(x),f(y)]$ there is a $z\in[x,y]$ such that $f(z)= \gamma$
Here is the part where I am not quite sure. Also if you have an idea how to prove it differently please share it with me:
Assume there exists finitely many points $z\in [x,y]$ such that $f(z)\neq z$.
Then we are already done because that means there must exist infinitely many Points for which the opposite is true, since the interval consists of infinitely many Points.
Therefore assume there exists infinitely many Points such that $z\in [x,y] \wedge f(z)\neq z$
Then there must also exist infinitely many points for which $f(z)>z$ and $f(z')<z'$
Without loss of generality assume there are only finitely many Points for which $z<f(z)$. Then there exists a maximal element of this set and one can split the interval $[x,y]$ into $[x,z]$ and $(z,y]$. If there exists a $w\in (z,y]$ such that $f(w) = w$ we are already done. Therefore assume $\forall_{w\in (z,y]} f(w)>w$. This would invoke a contradiction that $f$ is continuous in $z$.
Therefore we have showed there exists infinitely many $z,z'\in [x,y]$ such that $f(z)>z$ and $f(z')<z'$
One can pick now $z,z'$ such that $f(z)>z$ and $f(z')<z'$
Without loss of generality $z<z'$
Then $[z,z']$ is a closed interval and due to the Intermediate-value-theorem
$$\forall_{\gamma\in[f(z),f(z')]}\exists_{x\in[z,z']}f(x)=\gamma$$
If we repeat this process by Always choosing an element $l$ for the left end of the interval for which we have $f(l)>l$ and for the Right end an element for which we have $f(r)<r$. Then we get a nested interval which must converge to a fixed Point.
Is my thought process consistent? One Thing that this idea lacks is probably that there must always exist Points $l$ and $r$ like described above otherwise the proof does not work. Is it possible to prove this somehow?
REPLY [0 votes]: There is no reason to split your argument into counting finitely/infinitely many things. The moment you have a single fixed point, you're done. Or if you don't have any fixed points, then you will have at least one(infinite or not doesn't matter) $z$ and one $z'$ (guaranteed by the assumption that $[a,b]$ is contained in the image) such that
$f(z)>z$
and $f(z′)<z′$
Once you have this, you can just apply intermediate value theorem once as in the standard proof (see below).
PS I cannot understand what you wrote starting from the line
Without loss of generality assume there are only finitely many Points for which $z<f(z)$.
The standard proof. Without loss $[a,b]= [0,1]$. $f$ is continuous, so by extreme value theorem, it attains its min $m\le 0$ and max $M\ge 1$ at the points $x_0$ and $x_1$ respectively. Define for $t\in[0,1]$, $x_t = tx_0 + (1-t)x_1$ and consider $F(t) = f(x_t)-x_t$. We have $F(0) = m - x_0 ≤ -x_0\le 0$, and $F(1) = M - x_1 ≥ 1-x_1 \ge 0.$ By intermediate value theorem, there is a point $s$ in $[0,1]$ such that $F(s) = 0$. This corresponds to a point $x_s \in [0,1]$ such that $f(x_s) = x_s$, as needed.
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Robbie Amell Skips The Unemployment Line
Photo Credit: Jason Merritt/Getty Images for CW
A collective cry was heard around the nation as news broke that the CW show The Tomorrow People would not be renewed for a second season. Our hearts broke for Robbie Amell, out newest CW obsession. We grew so fond of the young actor in just one, short season. We were, and still are, so disappointed. But we have found new hope… Robbie Amell ain’t going nowhere!
Photo Credit:
That’s right Tomorrow Fans, Robbie skipped right over that unemployment line and landed himself a great gig. He has been casted in the new CBS film called The DUFF. TheWrap reported that The DUFF is the story of a 17 year-old girl named Bianca who believes she is the DUFF or “designated ugly fat friend.” Robbie is set to play Wesley, a jock who Bianca enlists in her quest to overthrow the school’s evil queen Madison. Although the film sounds like a complete 180 from his work on The Tomorrow People, we are so excited to see him back in action and working on a comedy.
We aren’t the only ones excited about the new role. Robbie, himself, has taken to his socials to introduce us to his co-stars and producers. Take a look↓
Photo Credit:
“Regram. #theDuff cast. Missing @maewhitman…@nickeversman @skyler_samuels @bellathorne@biancaalexasantos“
Photo Credit:
“Uh oh. @bellathorne is going against @itsmaryviola‘s donut on a stick. #theDUFF”
As we mourn the loss of The Tomorrow People, we look to the future with great hope. We can’t wait to see the film and we know only good things lie ahead for Robbie Amell :)
Vanessa Rao | CW44 Tampa Bay
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\begin{document}
\title[The Ellis semigroup of bijective substitutions]{The Ellis semigroup of bijective substitutions}
\author{Johannes Kellendonk}
\address{
Institut Camille Jordan, Universit\'{e} Lyon-1, France
}
\email{kellendonk@math.univ-lyon1.fr}
\author{Reem Yassawi}
\address{ School of Mathematics and Statistics, The Open University, U.K. and Institut Camille Jordan, Universit\'{e} Lyon-1, France
}
\email{reem.yassawi@open.ac.uk}
\subjclass[2010]{ 37B15, 54H20, 20M10}
\begin{abstract}
For topological dynamical systems $(X,T,\sigma)$ with abelian group $T$ which admit an equicontinuous factor
$\pi:(X,T,\sigma)\to (Y,T,\delta)$ the Ellis semigroup $E(X)$ is an extension of $Y$ by its subsemigroup $ \Ef(X)$ of elements which preserve the fibres of $\pi$.
We establish methods to compute $\Ef(X)$ and use them to determine the Ellis semigroup
of dynamical systems arising from primitive aperiodic bijective substitutions.
As an application we show that for these substitution shifts, the virtual automorphism group is isomorphic to the classical automorphism group.
\end{abstract}
\maketitle
\section{Introduction}
Consider $(X,T,\sigma)$, the action $\sigma$ of an abelian group (or semigroup) $T$ by homeomorphisms on a compact metrisable space $X$.
The {\em Ellis semigroup} $E(X)$ of $(X,T,\sigma)$ is {the} compactification of the group action in the topology of pointwise convergence on $X^X$.
The study of its topological and algebraic structure, which was initiated by Ellis \cite{Ellis-1960}, reveals dynamical properties of $(X,T,\sigma)$ and is
consequently an area of active study.
One topological property which has recently incited a lot of interest is tameness: $(X,T,\sigma)$ is {\em tame} if $E(X)$ is the sequential compactification of the action \cite{Huang-2006, glasnerCM}, that is, each element of $E(X)$ is a limit of a sequence (as opposed to a limit of a net, or generalised sequence) of homeomorphisms coming from the group action.
This can be expressed purely using cardinality: $E(X)$ is tame if and only if its cardinality
is at most that of the continuum
\cite{glasnerCM}.
Tameness implies, for instance, the following dynamical property \cite{Huang-2006,glasner2018structure,fuhrmann2018irregular}:
If a compact metrisable minimal system which admits an invariant measure is tame, then it is a {\em $\mu$-almost one to one} extension of its maximal equicontinuous factor. Here $\mu$ is the unique ergodic probability measure on the maximal equicontinuous factor of $(X,T,\sigma)$, and $\mu$-almost one-to-one means that the set of points in the maximal equicontinuous factor which have a unique pre-image under the factor map has full $\mu$-measure. As soon as all fibres of the maximal equicontinuous factor map contain more than one point, the system is thus not tame.
Systematic investigations focussing on the algebraic structure of $E(X)$ are to our knowledge, restricted to the question of when $E(X)$ is a group, when it has a single minimal left ideal, or, in the case of $T=\Z^+$, when its adherence subsemigroup is left simple. $E(X)$ is a group if and only if $(X,T,\sigma)$ is {\em distal} (proximality is trivial),
$E(X)$ has a single minimal left ideal if and only if proximality is transitive (see, for instance, \cite{Auslander}), and the adherence subsemigroup is left simple if and only if forward proximality implies forward asymptoticity \cite{Blanchard}.
Recently, a detailed computation of
the Ellis semigroups of the dynamical systems arising from almost canonical projection method tilings \cite{Aujogue,Aujogue-Barge-Kellendonk-Lenz} has exhibited another algebraic structure which seems worthwhile investigating, namely
the semigroups are all disjoint unions of groups. Semigroups which are disjoint unions of groups are precisely those which are {\em completely regular}, which means that every element admits a generalised inverse with which it commutes. Ellis semigroups associated to almost canonical projection method tilings are tame \cite{Aujogue}.
For the most part, good descriptions of Ellis semigroups are only currently available for tame systems.
The present paper arose from a desire to obtain explicit algebraic descriptions of Ellis semigroups for a class of dynamical systems which are not tame. We study the Ellis semigroup of systems $(X_\theta, \Z,\sigma)$ arising from bijective substitutions
$\theta$. The fibres of the maximal equicontinuous factor map of such systems are never singletons and so the resulting semigroup is not tame. They also enjoy two properties which we harness. The first is that for these $\Z$-actions, forward and backward proximality are non-trivial and equal to forward and backward asymptoticity. We describe systems $(X,\Z,\sigma)$ with this property in Section \ref{Section:comp reg}, and show that their Ellis group $E(X)$ is the disjoint union of the acting group $\Z$ with its {\em kernel} $\M(X)$, that is, the smallest bilateral ideal of $E(X)$; in particular, $E(X)$ is completely regular.
This reduces the task to the study of $\M(X)$.
The kernel $\M(X)$ of a compact sub-semigroup of $X^X$ is always {\em completely simple} and therefore can be described by the Rees-Suskevitch theorem and its topological extensions (Theorems \ref{Rees-theorem}, \ref{thm-Rees-str-1}, \ref{thm-Rees-str-2}).
This theorem characterises a completely simple semigroup as a {\em matrix semigroup} $M[G;I,\Lambda;A]$, where $G$ is the so-called {\em structure group}, where $I$ and $\Lambda$ index the right and left ideals respectively, and where $A$ is a matrix through which the semigroup operation is defined. Its entries are elements of $G$ which specify the idempotents.
The second property that bijective substitution systems enjoy is that they are {\em unique singular orbit systems}. This means that they have exactly one orbit of singular fibres
(fibres of the factor map on which proximality is non-trivial) over an equicontinuous factor. We study these systems in Section~\ref{fibre-preserving} in a way which can be summarised as follows.
Given an equicontinuous factor $\pi:(X,T,\sigma)\to (Y,T,\delta)$ we obtain a
short exact sequence of right-topological semigroups for the Ellis semigroup which restricts to a short exact sequence of its kernel
\begin{equation}\label{eqn:exact}
\Ef(X) \hookrightarrow E(X) \stackrel{\tilde\pi} \twoheadrightarrow E(Y)\cong Y \quad \mathrm{and}\quad \M^{fib}(X) \hookrightarrow \M(X) \stackrel{\tilde\pi} \twoheadrightarrow \M(Y)\cong Y.
\end{equation}
Here $\Ef(X)$ is the subsemigroup of functions which preserve the fibres of the factor map $\pi$, and $\M^{fib}(X)$ is the kernel of $\Ef(X)$. The two matrix semigroups associated to $\M^{fib}(X)$ and $\M(X)$ via the Rees-Suskevitch theorem are related (when properly normalised): they share the same $I$, $\Lambda$ and $A$, and their corresponding structure groups form an exact sequence
\begin{equation}\label{eqn:exact-1}
\RSTfp \hookrightarrow \RSTp \stackrel{\tilde\pi} \twoheadrightarrow Y,
\end{equation}
derived from the above (\ref{eqn:exact}). Finally we make one further reduction: we restrict $\M^{fib}(X)$ to a singular fibre and obtain again a completely simple semigroup to which we can apply the Rees-Suskevitch theorem. If $Y$ contains a single $T$-orbit, say that of $y_0$, such that $\pi^{-1}(y_0)$ is singular, then the restriction of $\M^{fib}(X)$ to $\pi^{-1}(y_0)$, denoted $\M^{fib}_{y_0}(X)$, has a matrix form which shares the same $I$, $\Lambda$ and $A$ as the other two matrix semigroups above. Furthermore,
we show in
Corollary~\ref{cor:reproducing M} that, if the singular fibres are finite and the idempotents generate $\M^{fib}_{y_0}(X)$, then the structure group $\mathcal G^{fib}$ equals the infinite Cartesian product $G_\pi^{Y/T}$, where $G_\pi$ is the structure group of
$\M^{fib}_{y_0}(X)$, and $Y/T$ is the space of $T$-orbits of $Y$. We thus obtain a description of $\M(X)$ through the finite semigroup $\M^{fib}_{y_0}(X)$ and the extension (\ref{eqn:exact-1}). We prove that the extension is algebraically split so that $\RSTp$ is a semidirect product of $\RSTfp$ with $Y$.
While $\M^{fib}(X)$ is topologically isomorphic its matrix semigroup representation, $\M(X)$ is only algebraically isomorphic to it.
The dynamical system $(X_\theta,\Z, \sigma)$ associated to a primitive aperiodic bijective substitution of length $\ell$ has a natural equicontinuous factor, namely the adding machine $(\Z_\ell,(+1))$, and only the orbit of $0\in\Z_\ell$ has singular fibres. We use the hierarchical symmetry defined by the substitution $\theta$ to compute the matrix form of
$\M^{fib}_{0}(X_\theta)$ in Theorem~\ref{thm-RMG},
\[ E_0^{fib}(X_\theta)\backslash \{\Id\} = \M^{fib}_{0}(X_\theta)\cong
M[G_\theta;I_\theta,\Lambda;A].\]
$M[G_\theta;I_\theta,\Lambda;A]$ is a finite semigroup, to which we refer also as the {\em structural semigroup} of the substitution. The structure group $G_\theta$ has already appeared in work by Lemanczyk and Mentzen in \cite{L-M} who identify it as the object whose centraliser completely encodes the {\em essential centraliser} of $(X_\theta,\sigma)$.
Provided that the smallest normal subgroup of $G_\theta$ which contains the group generated by the entries of $A$, which we denote by $\Glstr$, is all of $G_\theta$, Theorem~\ref{thm-main2} gives a complete description of $E(X_\theta)$ from $E_0^{fib}(X_\theta)$. In particular, $E(X_\theta)\backslash \Z$ is completely simple and there is a semigroup isomorphism
\begin{equation}\label{eq:one}E(X_\theta)\backslash \Z = \M(X_\theta)\cong M[\Gstr^{\Z_\ell/\Z}\rtimes \Z_\ell;I_\theta,\Lambda;A].\end{equation}
On the way to achieving this we also show that $\Ef(X_\theta)$ is topologically isomorphic to
$$\Ef(X_\theta) \cong ( M[G_\theta;I_\theta,\Lambda;A] \cup \{\Id\}) \:\:\times \prod_{\stackrel{[z]\in\Z_\ell /\Z}{\scriptscriptstyle{[z]\neq [0]}}}\Gstr ,$$
and this isomorphism makes clear where the non-tameness comes from.
In general, $\Glstr$ can be a proper subgroup of $\Gstr$, but the quotient group $\Gstr/\Glstr$ is always a cyclic group. We call its order $h$ the {\em generalised height} of the substitution. $h$ is at least as large as the classical height of a constant length substitution, and we give in Section \ref{Examples} examples where it is strictly larger. It is related to the topological spectrum of the dynamical system which is given by the action of $\Z$ on a minimal left ideal of $E(X_\theta)$, and $E(X_\theta)$ factors onto $\Z/h\Z$. In other words, $E(X_\theta)$ is a graded semigroup and its calculation can be reduced to its elements of degree $0$. In the case of nontrivial generalised height our result is Theorem~\ref{thm-main4}. Here, with the assumption that the generalised height equals the classical height, we are able to describe $E(X_\theta)$ algebraically in a similar way as in the trivial height case, but with the structure group $G_\theta$ replaced with $\bar\Gamma_\theta$. However, when the generalised height is strictly larger than the classical height, the extension problem (\ref{eqn:exact-1}) remains unsolved.
In Section \ref{Ellis-group}, we apply our machinery to partly answer a recent question of Auslander and Glasner in \cite{AG-2019}. They
define the notion of a {\em semi-regular} dynamical system, and ask whether a minimal, point distal shift which is not distal can be semi-regular. They show that the Thue-Morse shift is semi-regular. We extend their result, by showing in Corollary \ref{Vag=Aut}, that
the shift generated by a primitive aperiodic bijective substitution is semi-regular. Implicitly, we relate the structure group $G_\theta$ to the {\em virtual automorphism group} that Auslander and Glasner define.
Our work is related to recent work of Staynova \cite{Staynova}, in which she computes the minimal idempotents of the Ellis semigroup for dynamical systems of bijective substitutions $\theta$ that are an {\em AI extension} of their maximal equicontinuous factor. In other words, $(X_\theta, \sigma)$ is an isometric extension, via $f:X_\theta\rightarrow X_\phi$, of a constant length substitution shift $(X_\phi, \sigma)$, which is in turn an almost one-to-one extension, via $\pmax : X_\phi\rightarrow \Xmax$, of its maximal equicontinuous factor. Martin \cite{martin} characterises the bijective substitutions that are AI extensions of their maximal equicontinuous factor using a combinatorial property on the set of two-letter words allowed for $\theta$, namely that they are {\em partitioned} into sets according to what indices they appear at, as we scan all fixed points. Staynova uses the functoriality of the Ellis semigroup construction, namely that a map between dynamical systems induces a semigroup morphism between their Ellis semigroups,
and the fact that the Ellis semigroup of an equicontinuous system is a group, thus having exactly one idempotent. Using Martin's combinatorial condition, she first computes the preimages of that idempotent in $E(X_\phi)$. Apart from the identity map, all pre-images are minimal idempotents and live in the unique minimal left ideal. She then pulls this information up through the factor map $f$ to find that each of these minimal idempotents has two preimages, one for each minimal left ideal in $E(X_\theta)$.
Our work goes beyond the results of Staynova in several respects. First, our techniques apply to all bijective substitutions. Indeed it is easy to define substitutions that do not satisfy Martin's criterion, so that their dynamical systems are not AI extensions of their maximal equicontinuous factor (see Example \ref{Martin}). Second, we do not only determine the idempotents, but the complete algebraic structure of $E(X_\theta)$, at least if generalised height is not larger than classical height.
\bigskip
This paper is organised as follows. In Section \ref{preliminaries} we provide the necessary background on semigroups and the Ellis semigroup of a dynamical system, and study $\Z$-actions for
which forward and backward proximality implies forward and backward asymptoticity, respectively.
In Section \ref{fibre-preserving} we study the Ellis semigroup for dynamical systems which have a single orbit of singular fibres under an equicontinuous factor map. In Section \ref{bijective}, we study in detail the Ellis semigroup of a bijective substitution dynamical system, and give an algorithm that computes its structural semigroup. In Section \ref{Ellis-group} we apply our results to investigate the virtual automorphism group of bijective substitution shifts. We end in Section \ref{Examples} with some examples.
\section{Preliminaries}\label{preliminaries}
The literature on the algebraic aspects of semigroups is vast and, although our work is partly based on now classical results from the the forties we provide some background to the reader, {who may not be familiar with the basic material.} This can all be found in \cite{howie1995fundamentals}. We then recall the basic definitions and results on the Ellis semigroup of topological dynamical systems. These can mostly be found in \cite{Auslander} or \cite{hindman}.
\subsection{Semigroups, basic algebraic notions}
A {\em semigroup} is a set $S$ with an associative binary operation, which we denote multiplicatively. {Some of the semigroups in this paper have an identity element, but some do not. However they will never have a $0$ element.}
A {\em normal inverse} to $s\in S$ is an element $t\in S$ such that $sts = s$, $tst = t$ and $st=ts$. A general element in a general semigroup need not admit a normal inverse, but if it exists, it is unique. We may therefore denote it by $s^{-1}$.
A semigroup is called {\em completely regular} if every element admits a normal inverse. Completely regular semigroups have been studied in great detail \cite{petrich1999completely}. They are exactly the semigroups which may be written as disjoint unions of groups, i.e.\
$S=\bigsqcup_{i} \Gg_i$ such that multiplication restricted to $\Gg_i$ defines a group structure \cite[Theorem~II.1.4]{petrich1999completely}. The normal inverse of $s\in \Gg_i$ is then its group inverse in $\Gg_i$.
Of particular importance in the analysis of a semigroup are its
idempotents and its ideals.
An idempotent of a semigroup $S$ is an element $p\in S$ satisfying $pp=p$.
The set of idempotents of $S$ is partially ordered via
$p\leq q$ if $p = pq = qp$.
An idempotent is called {\em minimal} if it is minimal w.r.t.\ the above order.
In general, we cannot expect to have minimal idempotents.
A (left, right, or bilateral) ideal of a semigroup $S$ is a {nonempty} subset $I\subseteq S$ satisfying
$SI\subseteq I$, $IS\subseteq I$, {or} $SI\cup IS\subseteq I$ {respectively}. The different kind of ideals will play different roles below. When we simply say ideal we always mean bilateral ideal.
A semigroup is called {\em simple} if it does not have any proper ideal, and left simple if it does not have any proper left ideal. Note that a left simple semigroup is simple.
(Left, right, or bilateral)
ideals are ordered by inclusion. A {\em minimal} (left, right, or bilateral) ideal
is a minimal element w.r.t.\ this order, that is, a (left, right, or bilateral) ideal is minimal if it does not properly contain another (left, right, or bilateral) ideal. In general, we cannot expect to have minimal ideals, but their existence in our specific context will be guaranteed for by compactness, see below.
Whereas the intersection of two left ideals may be empty, this is not the case for the intersection of two bilateral ideals, or the intersection of a left ideal with a bilateral ideal.
Therefore the intersection of all bilateral ideals of a semigroup $S$ is either the unique minimal ideal of $S$, also called the {\em kernel} of $S$, or the intersection is empty, in which case $S$ does not admit a minimal ideal.
The kernel of a semigroup without zero element is always simple \cite{howie1995fundamentals}.
Related to left and right ideals are the so-called Green's equivalence relations. Two elements $x,y\in S$ are $\mathcal L$-related if they generate the same left ideal, that is, there are $s,s'\in S$ such that $x=sy$ and $y = s'x$. Likewise $x,y\in S$ are $\mathcal R$-related if they generate the same right ideal.
The intersection of the $\mathcal L$-relation with the $\mathcal R$-relation is called the $\mathcal H$-relation. The relation generated by the $\mathcal L$-relation and the $\mathcal R$-relation, that is the join of $\mathcal L$ and $\mathcal R$, is called the $\mathcal D$-relation. The relations $\mathcal L$ and $\mathcal R$ commute, so $x$ and $y$ are $\mathcal D$-related if there is a $z$ such that $x$ and $z$ are $\mathcal L$-related and $z$ and $y$ are $\mathcal R$-related. Two results are of importance for what follows: First, an $\mathcal H$-class of $S$ which contains an idempotent is a subgroup of $S$ whose neutral element is the idempotent \cite[Corollary~2.26]{howie1995fundamentals}, and second, two $\mathcal H$-classes containing idempotents and which belong to a common $\mathcal D$-class must be isomorphic as groups \cite[Proposition~2.3.6]{howie1995fundamentals}.
\subsection{Simple semigroups and the Rees matrix form}\label{matrix-form}
Let $G$ be a group, let $I$ and $\Lambda$ be non-empty sets, and let
$A = (a_{\lambda i})_{\lambda\in \Lambda,i\in I}$ be a $\Lambda\times I$ matrix with entries from $G$. Then the {\em matrix semigroup $M[G;I,\Lambda;A]$}
is the set $I\times G \times \Lambda$ together with the multiplication
\begin{equation*} (i,g,\lambda)(j,h,\mu) = (i, g a_{\lambda j} h,\mu).\end{equation*}
The matrix $A$ is called the {\em sandwich matrix} and the group $G$ is called the {\em structure group}.
It is an easy exercise to determine the idempotents and the left and the right ideals of $M[G;I,\Lambda;A]$. Indeed, an idempotent is of the form
\begin{equation*}\label{idempotent-form}(i,a^{-1}_{\lambda i},\lambda),\end{equation*}
the left ideals are the sets $I\times G\times\Lambda'$, $\Lambda'\subset \Lambda$, and the right ideals are $I'\times G\times \Lambda$, $I'\subset I$. In particular, a completely simple semigroup has minimal left and minimal right ideals, namely those for which $\Lambda'$ or $I'$ contain a single element. These minimal left and right ideals are also the $\mathcal L$ and the $\mathcal R$ classes, and so the $\mathcal H$-classes are of the form
$\{i\}\times G\times\{\lambda\}$. $\{i\}\times G\times\{\lambda\}$
is a subsemigroup of $M[G;I,\Lambda;A]$ which is a group.
The identity element of this group is the idempotent $(i,a^{-1}_{\lambda i},\lambda)$. It is isomorphic to $G$ via the isomorphism $(i,g,\lambda)\mapsto a_{\lambda i}g$.
The normal inverse of
$(i,g,\lambda)$ is $(i,a_{\lambda i}^{-1}g^{-1} a_{\lambda i}^{-1},\lambda)$.
In particular, a matrix semigroup as defined above is completely regular.
A {\em completely simple} semigroup is a simple semigroup which has minimal
idempotents.
We have the following characterisation of completely simple semigroups.\footnote{Recall that we excluded the case that $S$ has a $0$-element.
For semigroups with $0$-element there is an analogous but slightly different characterisation \cite{howie1995fundamentals}.}
{\begin{thm}[Rees-Suskevitch]\label{Rees-theorem}
A semigroup is completely simple if and only if it is isomorphic to a matrix semigroup $M[G;I,\Lambda;A]$ for some group $G$.
\end{thm}}
A proof of this theorem can be found in almost any textbook on semigroups. Since this result will be important in what follows we give a partial sketch of how to construct a Rees matrix from for a completely simple semigroup $S$. Proofs can be found in \cite{howie1995fundamentals}.
$S$ can be partitioned into its $\mathcal R$-classes, which we index by a set $I$. It can also be partitioned into its $\mathcal L$-classes, which we index by $\Lambda$. These partitions intersect yielding a partition into $\mathcal H$-classes. It can be shown that if $S$ is simple and contains an idempotent, then it consists of a single $\mathcal D$-class and that all its $\mathcal H$-classes contain an idempotent. In particular, $S$ is a disjoint union of groups which are all isomorphic. Moreover, each $\mathcal R$-class is a minimal right ideal and each $\mathcal L$-class is a minimal left ideal so that each $\mathcal H$-class is the intersection of a minimal right with a minimal left ideal. Up to here, everything is canonical. But now we choose a minimal right ideal $R_{i_0}$ and a minimal left ideal $L_{\lambda_0}$ and set
$$G := H_{i_0\lambda_0}$$
where we use the notation $H_{i\lambda} = R_{i}\cap L_{\lambda}$.
As mentioned above, all other $\mathcal H$-classes are isomorphic to $G$, and indeed, given any $r_i \in H_{i\lambda_0}$ and $q_\lambda \in H_{i_0\lambda}$
\begin{eqnarray}\label{eq-iso1}
&H_{i_0\lambda_0} \ni x \mapsto r_i x \in H_{i\lambda_0}& \\
\label{eq-iso2}
&H_{i_0\lambda_0} \ni x \mapsto xq_\lambda \in H_{i_0\lambda}&
\end{eqnarray}
are bijections which are group isomorphisms if $r_i$ and $q_\lambda$ are idempotents. Now the isomorphism between $G$ and the other $H_{i\lambda}$ will follow from the fact that $\mathcal L$ commutes with $\mathcal R$.
Taking into account these choices define the matrix $A=(a_{\lambda i})$
through
$$ a_{\lambda i} = q_\lambda r_i.$$
Then a direct calculation shows that
$$ M(G;I,\Lambda;A)\ni (i,g,\lambda) \mapsto r_i g q_\lambda \in S$$
yields the desired isomorphism.
We must ask how the Rees matrix form of a completely simple semigroup depends on the choices. The first choice is that of the right and left ideals indexed $i_0$ and $\lambda_0$, it defines the structure group $G=\mathcal H_{i_0 \lambda_0}$.
A different choice will lead to a different but isomorphic structure group. An isomorphism can always be constructed using (\ref{eq-iso1},\,\ref{eq-iso2}). The second choice is that of the elements $r_i$ and $q_\lambda$. It affects the sandwich matrix. Indeed, one has the freedom to multiply any row of $A$ from the left and, independently, any column of $A$ from the right by an element of $G$ to obtain a sandwich matrix which defines an isomorphic semigroup. It is therefore possible to normalise $A$ in such a way that one of its rows and one of its columns contains only the identity element of $G$. More precisely, having chosen
the right and left ideals indexed by $i_0$ and $\lambda_0$ we can always bring $A$ into its so-called {\em normalised} form by taking $r_i$ to be the unique idempotent of $\mathcal H_{i\lambda_0}$ and $q_\lambda$ to be the unique idempotent of $\mathcal H_{i_0\lambda}$ \cite[Theorem~3.4.2]{howie1995fundamentals}. Up to the choice of $i_0$ and $\lambda_0$ this {\em normalised} Rees matrix form is then unique. Since any pair $(i,\lambda)\in I\times \Lambda$ determines a unique idempotent of $S$ we can also formulate this as follows: once we have chosen an idempotent of $S$, typically denoted $e$, we obtain a unique normalised Rees matrix form for $S$. To be precise we call this {\em the normalised Rees matrix form for $S$ w.r.t.\ $e$.} In what follows the use of $e$ refers to this chosen minimal idempotent.
Given a normalised matrix semigroup $M(G;I,\Lambda;A)$ w.r.t.\ $e$,
we call the subgroup $\Gamma$ of $G$ generated by the coefficients
$a_{\lambda i}$ of $A$ the {\em little structure group}.
\begin{lem} \label{lem-lsgroup}
Consider a normalised matrix semigroup $M(G;I,\Lambda;A)$ w.r.t.\ $e=(i_0,1,\lambda_0)$.
The subsemigroup of $M(G;I,\Lambda;A)$ which is generated by the idempotents is equal to $M(\Gamma;I,\Lambda;A)$.
\end{lem}
\begin{proof} Let $K$ be the subsemigroup of $M(G;I,\Lambda;A)$ which is generated by the idempotents. By definition of the little structure group, $(i,G,\lambda)\cap K \subset (i,\Gamma,\lambda)$.
Normalisation implies $a_{\lambda i}=1$ provided $i=i_0$ or $\lambda=\lambda_0$. Given $a_{\lambda i}$ we know that $(i,a_{\lambda i}^{-1},\lambda)$ is an idempotent. Hence
$$(i_0,a_{\lambda i},\lambda_0)=(i_0,1,\lambda)(i,a_{\lambda i}^{-1},\lambda)(i,1,\lambda_0)\in (i_0,G,\lambda_0)\cap K.$$
This shows that $(i_0,\Gamma,\lambda_0)\subset (i_0,G,\lambda_0)\cap K$. Hence also
$$(i,\Gamma,\lambda) = (i,1,\lambda_0) (i_0,\Gamma,\lambda_0)(i_0,1,\lambda)
\subset (i,G,\lambda)\cap K.$$
This shows that $M(G;I,\Lambda;A)\cap K = M(\Gamma;I,\Lambda;A)$.
\end{proof}
\subsubsection{Example }\label{ex:matrix-semigroup-ex}
We consider a class
of matrix semigroups $M[G;I,\{\pm\};A]$ which will play a major role later. For this family, $G$ is a finite group with neutral element $\one$
and $I\subseteq G$ is a subset which generates $G$. Fix $g_0\in I$. Let $\Lambda = \{+,-\}$ be a set of two elements. Define the $\Lambda\times I$ matrix $A=(a_{\lambda i})_{\lambda i}$
\begin{equation}\label{eq:matrix} a_{+\,g} = \one \qquad a_{-\,g} = g_0 g^{-1} \end{equation}
Then $ M[G;I,\{\pm\};A] $ has $2|I||G|$ elements
of which $2|I|$ are idempotents.
Note that $M[G;I,\{\pm\};A]$ is normalised w.r.t.\ the idempotent $e=(g_0,\one,+)$.
\begin{lem}\label{lem-JSG} With the notation above, the little structure group of
$M[G;I,\{\pm\};A]$ is the group generated by $g h^{-1}$, $g,h\in I$.
\end{lem}
\begin{proof} This follows directly from $g h^{-1} = a_{-g}^{-1} a_{-h}$.
\end{proof}
\subsection{Compact semigroups}\label{compact semigroups}
A {\em topological} semigroup is a semigroup $S$ equipped with a topology in which the multiplication map $S\times S\to S$ is (jointly) continuous. A semigroup (equipped with a topology) is called {\em right-topological} if, for any $s\in S$ {\em right multiplication} $\rho_s: S\to S$, $\rho_s(t) := ts$ is continuous. Note that this is equivalent to multiplication $S\times S\to S$ being continuous in the {\em left} variable which is why
the term left-topological is also sometimes employed. We follow here the terminology of \cite{hindman}. A topological semigroup is right-topological and left-topological (with the obvious definition), but the converse need not be true.
Let $X$ be a topological space. The set $F(X)$ of functions $X\to X$ with the topology of pointwise convergence is
the same as the infinite Cartesian product $X^X$ with product topology. It is perhaps
the simplest example of a right-topological semigroup, the semigroup product being composition of functions. If $X$ is compact then $F(X)$ is compact. Only if $X$ is discrete is $F(X)$ a topological semigroup.
Let $\pi:X\to Y$ be a continuous surjection. We call the preimage $\pi^{-1}(y)$ the $\pi$-fibre of $y$. Let $F^{fib}(X)\subset F(X)$ be the subsemigroup of all functions $X\to X$ which preserve the $\pi$-fibres. Since fibres are closed subspaces of $X$, $F^{fib}(X)$ is a closed subsemigroup of $F(X)$.
We can view $f\in F^{fib}(X)$ as a function $\tilde f$ on $Y$,
\begin{equation} \label{eq:definition-f-tilde}
\tilde f : y
\mapsto f|_{\pi^{-1}(y)}\end{equation}
which, evaluated at $y$ is the restriction of $f$ to $\pi^{-1}(y)$, $\tilde f(y)(x) = f(x)$ for $x\in \pi^{-1}(y)$. This identification $f \mapsto \tilde f$
yields a {\em topological isomorphism}, i.e. a homeomorphism which is also a semigroup isomorphism, between
$F^{fib}(X)$ and the direct product $\prod_{y\in Y} F(\pi^{-1}(y))$
where the semigroup multiplication in the latter space is $\tilde f_1\tilde f_2(y) = \tilde f_1(y)\circ \tilde f_2(y)$ and
we equip it with the product topology, $F(\pi^{-1}(y))$ still carrying the topology of pointwise convergence. Recall that $F(\pi^{-1}(y))$ is a topological semigroup if the fibre of $y$ is finite. By definition of the product topology we therefore get that $\prod_{y\in Y} F(\pi^{-1}(y))$ is a topological semigroup provided all fibres are finite.
For compact semigroups one has the following results concerning their kernels and corresponding Rees matrix form.
\begin{thm}\label{thm-Rees-str-1}
Let $S$ be a compact right-topological semigroup. Then $S$ admits a
kernel $\M(S)$ which contains all minimal idempotents, so that $\M(S)$ is isomorphic to a matrix semigroup. Furthermore, all minimal left ideals are compact and homeomorphic, and two $\mathcal H$-classes of $\M(S)$ which belong to the same minimal right ideal are topologically isomorphic.
\end{thm}
This theorem is discussed in \cite[Corollary~2.6 and Theorem~2.11]{hindman}.
The essential input from the compact topology is the existence of an idempotent and the continuity of the map (\ref{eq-iso2}).
We mention that in general, minimal right ideals are not closed, nor are $\mathcal H$-classes closed, nor are two $\mathcal H$-classes topologically isomorphic which do not belong to the same minimal right ideal. $\M(S)$ is then not topologically isomorphic to a matrix semigroup.
One consequence of Theorem \ref{thm-Rees-str-1} will be particularly important below, namely that for any minimal idempotent $p$ of a compact right-topological semigroup $S$, $pSp$ is a group. Indeed, the chain of inclusions
\begin{equation} \label{pEp-group} p\M(S) p\subset pSp = pSp p \subset p\M(S) p. \end{equation}
shows that $pSp$ is isomorphic to the structure group of the kernel $\M(S)$.
If the multiplication of $S$ is jointly continuous then the topological aspects of
Theorem \ref{thm-Rees-str-1} can be strengthened.
We can equip the normalised Rees matrix form $M[G;I,\Lambda;A]$ (w.r.t.\ $e=(i_0,1,\lambda_0)$) of $\M(S)$ with the following topology: We identify $G$ with $H_{i_0\lambda_0}=eSe$, $I$ with the set of idempotents of $L_{\lambda_0}$, $\Lambda$ with the set of idempotents of $R_{i_0}$, and we equip all these subsets of $S$ with the relative topology, and finally $I\times G\times \Lambda$ with the product topology.
Under the assumption that $S$ is a compact topological semigroup, we have that $G$ is a compact topological group, $I$ and $\Lambda$ compact subsets and the semigroup product on $M[G;I,\Lambda;A]$ is jointly continuous.
\begin{thm}\label{thm-Rees-str-2}
Let $S$ be a compact topological semigroup. Then $\M(S)$ is topologically isomorphic to the normalised matrix semigroup $M[G;I,\Lambda;A]$.
\end{thm}
A proof of this theorem can be found in \cite[Theorem~3.21]{carruth1983theory}.
As now also the map (\ref{eq-iso1}) is continuous, all $\mathcal H$-classes of $M(S)$ are closed and topologically isomorphic.
\subsection{Extensions of groups by completely simple semigroups}
We now use the above description of completely simple semigroups to study extensions
$$ K\hookrightarrow S \stackrel{\pi} \twoheadrightarrow Y $$
where $Y$ is a group with neutral element $y_0$,
$S$ a semigroup, $\pi$ a semigroup epimorphism and $K$ the kernel of $\pi$,
$$K = \{s\in S: \pi(s)=y_0\}.$$
$K$ is a subsemigroup of $S$ which is closed if $S$ and $Y$ are right-topological and $\pi$ continuous.
If $e\in S$ is an idempotent then $\pi(e)$ must be an idempotent, hence equal to $y_0$ so that we obtain a restricted extension
$ eKe\hookrightarrow eSe \stackrel{\pi_e} \twoheadrightarrow Y $ where $\pi_e$ is the restriction of $\pi$ to $eSe$.
If moreover $e$ is an idempotent in the kernel of $S$ then by \eqref{pEp-group},$eSe$ is a group, as is $eKe$,
so that
the restricted extension is an extension of groups.
A semigroup $S$ is {\em regular} if for any $s\in S$ there exists $t\in S$ such that $s=sts$.
Clearly, any completely regular semigroup is regular.
\begin{prop}\label{extension-semigroup}
Consider an extension $ K\hookrightarrow S \stackrel{\pi} \twoheadrightarrow Y $ of a group $Y$ by a completely simple semigroup $K$, where $S$ is regular. Then $S$ is completely simple. If $K$ has normalised Rees matrix form $M[G;I,\Lambda;A]$ w.r.t.\ an idempotent $e$ then $S$ has normalised Rees matrix form $M[\mathcal G;I,\Lambda;A]$
w.r.t.\ $e$, where $\mathcal G=eSe$ is the extension of $Y$ by $G=eKe$
determined by the exact sequence of groups $eKe \hookrightarrow eSe \stackrel{\pi_e}\twoheadrightarrow Y$.
\end{prop}
\begin{proof} We first show that $S$ must be completely simple. Let $M\subset S$ be an ideal. Then $M\cap K$ is an ideal of $K$. As $K$ is simple, $M\cap K = K$. Thus $M$ contains all idempotents of $S$. Let $s\in S$ and $t\in S$ such that $s=sts$. Then $ts$ is an idempotent and so we see that $S\subset SK\subset SM\subset M$. Therefore $S$ is completely simple.
Let $M[\mathcal G;I,\Lambda;A]$ be the normalised Rees matrix form of $S$ w.r.t.\ $e$, in particular $\mathcal G=eSe$. Since $K$ contains the subsemigroup generated by the idempotents of $S$,
by Lemma~\ref{lem-lsgroup}, the coefficients of $A$ belong to $G:= eKe = \ker \pi_e$. Hence
$M[G;I,\Lambda;A]$ is well-defined and, with $e=(i_0,1,\lambda_0)$ and
$(i_0,1,\lambda_0)(i,g,\lambda)(i_0,1,\lambda_0)=(i_0, g ,\lambda_0)$
we obtain
$$\pi(i,g,\lambda) = \pi(i_0,g,\lambda_0) = \pi_e(g)$$
so that $K= M[\ker\pi_e;I,\Lambda;A]$.
Since the normalised Rees matrix form w.r.t.\ $e$ is unique we see that $I,\Lambda$ and $A$ are completely determined by $K$.
\end{proof}
We will apply Proposition \ref{extension-semigroup} to a situation in which $Y$ is a topological group and $K$ is a topological semigroup, but $S$ is only a right-topological semigroup. Therefore we cannot conclude that $S$ is topologically isomorphic to $M[\mathcal G;I,\Lambda;A]$. Indeed, the group $\mathcal G=eSe$ is only closed if the right ideal containing $e$ is closed, which we cannot expect. The interest in the above construction is therefore only algebraic.
It is particularly useful if the group extension is split so that $\mathcal G$ is a semidirect product of $G$ with $Y$.
\subsection{Ellis semigroup of a dynamical system}
Given a dynamical system $(X,T,\sigma)$ the family of homeomorphisms
$\{\sigma^t | t\in T\}$ is a subsemigroup of $F(X)$.
Its closure, denoted $E(X,T, \sigma)$, or simply $E(X)$ if the rest is understood, is still a semigroup, called the {\em Ellis semigroup} (or {\em enveloping semigroup}) of the dynamical system.
Since $X$ is compact the set of all functions $X\to X$ is compact in the topology of pointwise convergence and so $E(X,T, \sigma)$ is a compact \rst\ semigroup, by construction.
The Ellis semigroup is closely related to the {\em proximality} relation. Given a metric $d$ on $X$ which generates the topology, a pair of points $x,x'$ are {\em proximal} if $\inf_{t\in T} d(\sigma^t(x),\sigma^t(x')) = 0$. The proximal relation does not depend on the choice of metric (which generates the topology). Its relation with the Ellis semigroup is the following:
\begin{thm}\cite[Chapter 3, Proposition 8]{Auslander}
Let $E(X)$ be the Ellis semigroup of a dynamical system $(X,T, \sigma)$. Two points $x$ and $y$ are proximal if and only if there exists $f\in E(X)$ such that $f(x)=f(y)$.
\end{thm}
In particular we see that, given any idempotent $p\in E(X)$ and $x\in X$, the points $p(x)$ and $x$ are proximal.
\subsection{Complete regularity for $\Z$-actions}\label{Section:comp reg}
In this section we provide a criterion for complete regularity of the Ellis semigroup for $\Z$ actions. We will see below that it is satisfied by the dynamical systems defined by bijective substitutions.
Since the union of the closure of two sets is the closure of their union we can decompose
\begin{equation}\label{eq-fb}
E(X) = E(X,\Z^+) \cup E(X,\Z^-)
\end{equation}
where $E(X,\Z^\pm)$ is the closure of $\{\sigma^t | t\in \Z^\pm\}$. This allows us to compute the elements of $E(X)$ by looking independently, forward in ``time", and backwards in ``time".
We say that two points $x,x'\in X$ are {\em forward proximal} if
$$\inf_{t\in\Z^+ } d(\sigma^t(x),\sigma^t(x')) = 0$$
We say that two points $x,x'\in X$ are
{\em forward asymptotic} if $$\lim_{t\to +\infty} d(\sigma^t(x),\sigma^t(x')) = 0$$
Similarly, we define {\em backward} proximality and asymptoticity using $\sigma^{-1}$ in place of $\sigma$.
Clearly sequences which are forward asymptotic are forward proximal. The following lemma is related to the work of \cite{Blanchard} in which the adherence semigroup of a $\Z^+$-action is defined and analysed.
\begin{lem}\label{lem:prox=asym}
Let $(X,\sigma)$ be a dynamical system for which forward proximality agrees with forward asymptoticity. Then $E(X,\Z^+)$ has a unique minimal left ideal $\M(X,\Z^+)$ and contains besides this ideal only $\Z^+$.
\end{lem}
\begin{proof}
An element $f\in E(X,\Z^+)\backslash \Z^+$ is the limit of a generalised sequence $(\sigma^{t_\nu})_{\nu}$, $t_\nu\in\Z^+$ which is not in $\Z^+$. Hence the generalised sequence $(t_\nu)_\nu$ has the property that for any finite $N\in\Z^+$ there exists $\nu_0$ such that
$t_{\nu}\geq N$ for all $\nu >\nu_0$. In particular, if $x$ and $y$ are forward asymptotic points then $\lim_\nu d(\sigma^\nu(x),\sigma^\nu(y))=0$, and hence $f(x) = f(y)$.
$E(X,\Z^+)$ is also a compact \rst\ semigroup and hence has minimal left ideals and minimal idempotents. Furthermore, $x,y\in X$ are forward proximal if and only if there exists $f\in E(X,\Z^+)$ such that $f(x) = f(y)$.
Let $p\in E(X,\Z^+)$ be any idempotent. For any $x\in X$, $p(x)$ is forward proximal to $x$, and by our assumption therefore forward asymptotic to $x$. This implies that if $f\in E(X,\Z^+)\backslash \Z^+$, then $f(p(x))=f(x)$. Since $x$ was arbitrary we find $f=fp$.
This identity
shows that any $f\in E(X,\Z^+)\backslash \Z^+$ lies in the ideal generated by the idempotent $p$. If $p$ is minimal then this ideal is a minimal left ideal. Since $p$ can be any minimal idempotent there can only be one minimal left ideal.
\end{proof}
Note that the unique minimal left ideal $\M(X,\Z^+)$ of the previous lemma is the kernel of $E(X,\Z^+)$.
\begin{cor}\label{cor:prox=asym}
Let $(X,\sigma)$ be a dynamical system for which forward proximality agrees with forward asymptoticity and backward proximality agrees with backward asymptoticity. Then $E(X)$ is completely regular. If moreover the forward and the backward proximality relations are non-trivial (not diagonal) then $E(X)$ is the disjoint union of its kernel $\M(X)$ with the acting group $\Z$.\end{cor}
\begin{proof}
Minimal left ideals which contain idempotents are completely simple and hence, by the Rees structure theorem, disjoint unions of groups.
Therefore Lemma \ref{lem:prox=asym}
implies that $E(X,\Z^+)\backslash \Z^+$ is completely regular. $E(X)$ is thus a union of completely regular sub-semigroups. Hence any element of $E(X)$ has an inverse with which it commutes.
To proof the second statement we first show that $\M(X,\Z^+)$ is a minimal left ideal in $E(X)$.
Let $f\in E(X)$, $g\in E(X,\Z^+)\backslash \Z^+$. So $f = \lim \sigma^{n_\nu}$ and $g = \lim \sigma^{m_\mu}$, however with $m_\mu\to+\infty$. Then $fg = \lim_\nu \sigma^{n_\nu} g$.
Since $\sigma^{n_\nu} g = \lim_{\mu} \sigma^{n_\nu+m_\mu} \in E(X,\Z^+)$ and $E(X,\Z^+)$ is closed
we have $fg\in E(X,\Z^+)$. Suppose that $fg=\sigma^n$ for some $n\in \Z$. As $g\in \M(X,\Z^+)$ we have $gp = g$ for some idempotent $p\in \M(X,\Z^+)$. It follows that
$1=\sigma^{-n}fg= \sigma^{-n}fgp = p$. This implies that $E(X,\Z^+)$ is a group and thus contradicts the assumption that the forward proximality relation is non-trivial \cite{Auslander}.
Hence $fg\notin \Z$ so that by Lemma~\ref{lem:prox=asym},
$E(X)\M(X,\Z^+)\subset \M(X,\Z^+)$ and moreover $\M(X,\Z^+)\cap \Z=\emptyset$.
To show minimality of $\M(X,\Z^+)$ it suffices to show that all idempotents of $\M(X,\Z^+)$ are minimal in $E(X)$.
Let $q\in \M(X,\Z^+)$ and $p\in E(X)$ be idempotents such that $p\leq q$. This means that $pq = qp = p$. As we just showed, $p=pq\in \M(X,\Z^{+})$.
But then $p=q$ as $q$ is minimal in $E(\M,\Z^+)$.
As the kernel of a semigroup is the union of its minimal left ideals we now have shown that
$\M(X)=\M(X,\Z^+)\cup \M(X,\Z^-)$ and that $\M(X)\cap \Z=\emptyset$.
\end{proof}
The arguments in the second part of the proof were adapted from \cite{BargeKellendonk} where it is also shown that, for minimal $\Z$-actions on totally disconnected compact metric spaces, the condition that forward proximality agrees with forward asymptoticity and backward proximality agrees with backward asymptoticity is also necessary for complete regularity.
\subsection{Equicontinuous factors and the structure of $E(X)$}\label{semigroup-of-factor}
In this section $T$ is an abelian group. When we have a factor map between two dynamical systems, the acting group is the same. A dynamical system $(X, \sigma)$ is called {\em equicontinous} if the family of homeomorphisms
$\{\sigma^t, t\in T\}$ is equicontinuous. If the action is transitive then this is the case if and only if, for any choice of $x_0\in X$ there is an abelian group structure on $X$ (denoted additively) such that $x_0$ is the identity element and $\sigma^t(x) = x+ \sigma^t(x_0)-x_0$. This group structure is topological.
Moreover, for a minimal equicontinuous system and w.r.t.\ the above group structure on $X$,
$ev_{x_0} : E(X) \to X$ is an isomorphism of topological groups, where $ev_{x_0}$ is evaluation at the point $x_0\in X$, $ev_{x_0}(f) = f(x_0)$ \cite[Chap.~3, Theorem~6]{Auslander}.
An {\em equicontinuous} factor is a factor $\pi:(X, \sigma)\to (Y,\delta)$ such that $(Y,\delta)$ is equicontinuous. As with any factor map, $\pi$ induces a continuous semigroup morphism
$\pi_*: E(X) \to E(Y)$ via $\pi_*(f)(y) = \pi(f(x))$ where $x$ is any pre-image of $y$ under $\pi$. As $(Y,\delta)$ is equicontinuous $ev_{y_0}:E(Y) \to Y$ is a semigroup isomorphism where $y_0$ is the identity element in $Y$. We denote by $\tilde \pi:E(X)\to Y$ the composition $ev_{y_0}\circ \pi_*$, which is also a continuous surjective semigroup morphism.
\begin{definition}\label{def:Efib}
Let $\pi:(X,\sigma)\to (Y,\delta)$ be an equicontinuous factor.
Define $\Ef(X)$ to be the subsemigroup of $E(X)$ which consists of those elements which preserve the $\pi$-fibres $\pi^{-1}(y)$, $y\in Y$.
\end{definition}
In other words, $\Ef(X)$ is the kernel of the continuous semigroup morphism $\tilde\pi$ and therefore a closed subsemigroup. We summarize this situation with the exact sequence of right-topological semigroups
\begin{equation}\label{eq-ext}
\Ef(X) \hookrightarrow E(X) \stackrel{\tilde\pi} \twoheadrightarrow Y
\end{equation}
in which the involved maps are continuous semigroup morphisms. While $E(X)$ is only right-topological, $Y$ is topological. As we will see below, under certain circumstances, $\Ef(X)$ is also a topological semigroup.
\section{The fibre-preserving part $\Ef(X)$}\label{fibre-preserving}
In this section we investigate the fibre-preserving part $\Ef(X)$ of $E(X)$ for dynamical systems which factor onto an equicontinuous system, $\pi:X\to Y$.
We call a point $y\in Y$ is {\em regular} (for $\pi$)
if the proximal relation restricted to $\pi^{-1}(y)$ is trivial.
Otherwise we call the point {\em singular} (for $\pi$).
\begin{prop}
Suppose that $\pi:(X,\sigma)\to (Y,\delta)$ is an equicontinuous factor map whose fibres $\pi^{-1}(y)$ are all finite. Then $\Ef(X)$ is a compact topological semigroup.
\end{prop}
\begin{proof}
$\Ef(X)$ is a compact subsemigroup of $F^{fib}(X)$, defined in Section \ref{compact semigroups}. By assumption, all $F(\pi^{-1}(y))$ are (trivially) topological semigroups. Therefore the semigroup multiplication of $\prod_{y\in Y} F(\pi^{-1}(y))$ is jointly continuous.
As $\Ef(X)$ is a closed subsemigroup of $F^{fib}(X)$ its product is also jointly continuous.
\end{proof}
We can now apply Theorem \ref{thm-Rees-str-2} to conclude the following.
\begin{cor}\label{cor-ff}
Suppose that $\pi:(X,\sigma)\to (Y,\delta)$ is an equicontinuous factor map whose fibres $\pi^{-1}(y)$ are all finite. Then the kernel $\M^{fib}(X)$ of $\Ef(X)$ is topologically isomorphic to its normalised Rees matrix form.
\end{cor}
We now consider more closely the algebraic structure of $\Ef(X)$.
To simplify the notation we drop the reference to $X$ and denote it by $\Ef$.
When identifying $\Ef$ with a subsemigroup of
$F^{fib}(X)\cong \prod_{y\in Y} F(\pi^{-1}(y))$
we observe that it belongs actually to the smaller semigroup
$\prod_{y\in Y} \Ef_y$ where
\begin{equation*}\label{eq:Efib restriction}
\Ef_y := \Ef|_{\pi^{-1}(y)},
\end{equation*}
the restriction of $\Ef$ to the fibre $\pi^{-1}(y)$. Indeed, any $f\in \Ef$ corresponds to a function $\tilde f$ on $Y$ whose value $\tilde f(y)$ belongs to $\Ef_y$. $\Ef_y$ is a compact subsemigroup of $F(\pi^{-1}(y))$.
Moreover, since the elements of $E$ commute with the action $\sigma$ of $T$ the functions $\tilde f$ have to be {\em covariant} in the sense that
\begin{equation*}\label{eq-cov}
\tilde f(\delta^t(y)) = \sigma^t \tilde f(y)\sigma^{-t}
\end{equation*}
for all $y\in Y$ and $t\in T$. In other words, $\Ef$ is a subsemigroup of
\begin{equation*}\label{eq:definiton-of-Cf} \Cf := \{\tilde f\in \prod_{y\in Y} \Ef_y
: \tilde f \;\mbox{is covariant}\},\end{equation*}
again equipped with the pointwise semigroup multiplication
$(\tilde f_1\tilde f_2)(y) = \tilde f_1(y)\circ \tilde f_2(y)$. Equipped with the product topology,
$\Cf$ is compact and the inclusion $\Ef\subset \Cf$
is continuous.
Recall {by \eqref{pEp-group}} that if $p$ is any minimal idempotent then $pEp$ is a group.
We fix any such minimal idempotent $e$, recalling that different minimal idempotents define isomorphic groups.
As $\tilde\pi: E \to Y$ is onto, and $\tilde \pi (efe)= \tilde \pi (f)$, the restriction
$\tilde\pi: eEe \to Y$ is also onto.
A {\em lift under $\tilde\pi$} is a right inverse $s:Y\to eE e$ to $\tilde\pi:e E e\to Y$, i.e.\ it satisfies $\tilde \pi \circ s = \Id$. A lift always exists by the axiom of choice.
We do not demand that it is continuous, nor, for the time being, that it preserves the group structure. But we can and do demand that it satisfies
$s(\delta^t(y)) = \sigma^t s(y)$ for all $t\in T$, and also that $s(y)^{-1}=s(-y)$ for each $y\in Y$. We impose the latter condition now although we will not use it until
Proposition \ref{prop-shomo}.
Given a lift $s:Y\to e E e$, we define $\Phi_{y_1}^{y_2}:\Ef\to \Ef$ by
\begin{equation}\label{eq-conj}
\Phi_{y_1}^{y_2} (f) = s(y_2-y_1) f \, s(y_2-y_1)^{-1}
\end{equation}
where $s(z)^{-1}$ is the group inverse to $s(z)$. Although we do not include this in our notation, it must be kept in mind that $\Phi_{y_1}^{y_2}$ depends on the choice of lift.
Since $s(\delta^t(y)) = \sigma^t s(y)$ we have
$\Phi_{y_1}^{\delta^t(y_2)} (f) = \sigma^t \Phi_{y_1}^{y_2} (f) \sigma^{-t}$.
Note that $\Phi_{y_1}^{y_2}$ also defines a map from $\Ef_{y_1}$ to $\Ef_{y_2} $, namely if $\varphi\in \Ef_{y_1} $ and $f$ is an element of $\Ef$ which restricts to $\varphi$ on $\pi^{-1}(y_1)$, that is $\varphi = \tilde f(y_1)$ in the notation above,
then $\Phi_{y_1}^{y_2} ( \varphi )$ is defined to be the restriction of
$\Phi_{y_1}^{y_2} (f) $ to $\pi^{-1}(y_2)$. This does not depend on the choice of $f$, as $s(y_2-y_1)^{-1}$ maps $\pi^{-1}(y_2)$ to $\pi^{-1}(y_1)$.
\begin{lem}\label{structure}
Let $e\in E(X)$ be a minimal idempotent.
\begin{enumerate}
\item $e\Ef_y e$ is a group.
\item If $y$ is regular then $e\Ef_y e = \Ef_y$.
\item For any $y_1, y_2$, the restriction $\Phi_{y_1}^{y_2}:e\Ef_{y_1} e\to e\Ef_{y_2} e$
is a group isomorphism.
\end{enumerate}
\end{lem}
\begin{proof}
$e\Ef_y e$ is entirely determined by the action of $e\Ef e$ on $e\pi^{-1}(y)$. It is hence the homomorphic image of a group.
Idempotents must act like the identity on a regular fibre, as can be seen as follows:
The points $e(x)$ and $x$ are proximal. In a regular fibre this can only be the case if $e(x)=x$. Hence $e\Ef_y e = \Ef_y$ if $y$ is regular.
Let $s:Y\to e E e$ be a right inverse to $\tilde\pi:eE e\to Y$;
$s(z)$ restricts to a map $e\pi^{-1}(y)\to e\pi^{-1}(y+z)$ whose inverse is the restriction of $s(z)^{-1}$, as $s(z)^{-1}s(z) = s(z)s(z)^{-1} = e$. Hence
$\Phi_{y_1}^{y_2}$ is conjugation with a bijection.
\end{proof}
\begin{definition}\label{def:structure group} We call the group determined up to isomorphism by Lemma~\ref{structure}
the {\em structure group} of the factor system $(X,\sigma)\stackrel{\pi}\to (Y,\delta)$ and denote it by $G_\pi$.
\end{definition}
\begin{definition} {We say that the system $(X,\sigma)$ is a {\em unique singular orbit system} if it admits an equicontinuous factor which has a single orbit of singular points.}
\end{definition}
We now specialize to the context of unique singular orbit systems and fix a singular point $y_0\in Y$. Define $\Et\subset \Ef$ to be
\begin{equation}\label{eq:def T}
\Et=\{ f\in \Ef: f(x)= x \mbox{ for all $x$ in a regular fibre}\}.
\end{equation}
Since idempotents can only project proximal points, and regular fibres contain no proximal pairs, so idempotents belong to $\Et$.
However $\Et$ may be larger.
Given the minimal idempotent $e$, $e\Et e$ is a subsemigroup of $e\Ef e$. We claim that it is even a normal subgroup. Indeed, if $f\in e\Et e$ then its inverse in $e\Ef e$ also acts trivially on regular fibres and so belongs to $e\Et e$. Furthermore, an element $g\in e\Ef e$ acts bijectively on regular fibres and hence $gfg^{-1}$ acts as $g g^{-1}=\Id$ on them.
Let $\Et_{y_0}$ be the restriction of $\Et $ to $\pi^{-1}(y_0)$; it is a subsemigroup of $\Ef_{y_0}$.
Then $e\Et_{y_0} e$ is the restriction of $e\Et e$ to $e\pi^{-1}(y_0)$; it is a normal subgroup of $e\Ef_{y_0} e$. We now use the maps $\Phi_{y_0}^{y}$ from (\ref{eq-conj}) to transport the group $e\Et_{y_0} e$ along $Y$ and define the subsemigroup of $\Cf$
\begin{equation} \label{eq:def Ct}
\Ct :=
\{ \tilde f\in \Cf : \tilde f (y_0) \in \Et_{y_0}, \mbox{ and } \tilde f (y) \in \Phi_{y_0}^{y}(e\Et_{y_0} e) \mbox{ for all $y$ regular} \}.
\end{equation}
Although the map $\Phi_{y_0}^{y}$ depends on the choice of a lift $s:Y\to eEe$ for $\tilde \pi$, the space $\Ct$ does not.
Indeed, if we take another lift to obtain a map ${\Phi'}_{y_0}^{y}$ then $\Phi_{y_0}^{y}(\tilde f)(y_0)$ will differ from ${\Phi'}_{y_0}^{y}(\tilde f)(y_0)$ by a conjugation with an element $h\in e \Ef_{y_0} e$, which does not matter as $e\Et_{y_0} e$ is a normal subgroup of
$e\Ef_{y_0} e$.
By covariance $\Ct$ does not depend on the choice of $y_0$ in the unique orbit of singular points.
Finally, the dependence of $\Ct$ on the choice of minimal idempotent $e$ can be controlled with the isomorphisms (\ref{eq-iso1},\ref{eq-iso2}). If the singular fibre $\pi^{-1}(y_0)$ is finite then the isomorphisms are bicontinuous by Theorem ~\ref{thm-Rees-str-2} and Corollary ~\ref{cor-ff}.
\begin{thm} \label{thm-main}
Let $(X,\sigma)$ be a minimal unique singular orbit system. $\Ct$ is a subsemigroup of $\Ef$.
\end{thm}
\begin{proof}
Let $g \in \Et$ and $y\in Y$. Then $f:= \Phi_{y_0}^{y} (g)$ belongs to
$\Ct\cap \Ef$. Indeed, $\Phi_{y_0}^{y} (g)=\Phi_{y_0}^{y} (ege)\in \Phi_{y_0}^{y} (e\Et_{y_0}e)$.
By definition $g$ acts non-trivially only on the fibres of the $T$-orbit of $y_0$. Hence
$f:= \Phi_{y_0}^{y} (g)$ acts non-trivially only on the fibres of the $T$-orbit of $y$.
As $(\Phi_{y_0}^{y})^{-1}(\tilde f(y))$ is the restriction of $ege$ to $e\pi^{-1}(y_0)$
we find that, given any regular point $y$ and any $g\in e\Et_{y_0} e$, $\Ct\cap \Ef$ contains the function $f$ which satisfies $(\Phi_{y_0}^{y})^{-1}(\tilde f(y)) = g$ and $\tilde f(y') = \Id$ for any point $y'$ in another orbit. By taking finite products of such functions we see that
$\Ct\cap \Ef$ contains, for any choice of $k$ points $y_1,\cdots,y_k$ in distinct regular orbits
and any choice of $k+1$ elements $g_i\in \Et_{y_0}$, $i=0,\cdots,k$ a function $f$ such that
$(\Phi_{y_0}^{y_i})^{-1}(\tilde f(y_i)) = eg_ie$, $i\geq 1$, $\tilde f(y_0) = g_0$, and $\tilde f(y') = \Id$ for a point $y'$ in another orbit.
By definition of the topology of pointwise convergence and since covariance is a closed relation, the set of these elements is dense in $\Ct$. Since $\Ef$ is the kernel of a continuous map it is closed; it hence contains $\Ct$.
\end{proof}
\begin{cor} \label{cor-main}
Let $(X,\sigma)$ be a minimal unique singular orbit system. If $\Ef_{y_0}=\Et_{y_0}$ then $\Ef$ is topologically isomorphic to $\Cf$.
\end{cor}
\begin{proof}
If $\Ef_{y_0}=\Et_{y_0}$ then $\Phi_{y_0}^{y}(e\Et_{y_0}e)=\Ef_y$, for regular $y$, so that the condition $\tilde f(y)\in \Phi_{y_0}^{y}(e\Et_{y_0}e)$ is trivially satisfied as is $\tilde f(y_0)\in \Et_{y_0}$. Hence $\Ct=\Cf$. {Thus by Theorem \ref{thm-main} we have $\Cf \subseteq \Ef$. On the other hand,
we saw that $\Ef$ is a subsemigroup of $\Cf$ and that the inclusion is continuous.
Since $\Cf$ is compact this gives the result.}
\end{proof}
We end this section by establishing a criterion which implies the condition of the last corollary, namely that $\Ef_{y_0}=\Et_{y_0}$.
\begin{definition}\label{coincidence-definition}
Let $\pi:(X,\sigma)\to (Y,\delta)$ be an equicontinuous factor.
The {\em minimal rank} $r_\pi$ of the factor $\pi$ is the smallest possible cardinality $|\pi^{-1}(y)|$ of a fibre, $y\in Y$.
The {\em coincidence rank} $cr_\pi(y)$ of the fibre $y\in Y$ is the largest possible cardinality a subset of $\pi^{-1}(y)$ can have, which contains only pairwise non-proximal elements. \end{definition}
If the system $(X,\sigma)$ is minimal, then the coincidence rank of
an equicontinuous factor can be shown to be independent of $y$ and so $cr_\pi=cr_\pi(y)$ is the coincidence rank of the factor $\pi:(X,\sigma)\to (Y,\delta)$. If the factor is not specified then the coincidence rank is meant to be the coincidence rank of the maximal equicontinuous factor.
See \cite{Aujogue-Barge-Kellendonk-Lenz} for details and a context.
Not every system contains regular fibres. It can be shown that for minimal systems with finite coincidence rank for the maximal equicontinuous factor, the maximal equicontinuous factor contains a regular fibre
if and only if the system is {\em point distal } i.e.\ contains a point $x$ that is proximal only to itself \cite{Aujogue-Barge-Kellendonk-Lenz}, and if that is the case, any other equicontinuous factor must also contain regular fibres.
(Since this is a side remark we don't include a proof.)
\begin{lem} If the minimal rank $r_\pi$ of
the equicontinuous factor $Y$ of a minimal system is finite and the factor contains some regular fibre then $y\in Y$ is regular if and only if $|\pi^{-1}(y)|=r_\pi$.
\end{lem}
\begin{proof}
Let $y_0$ be a regular point. Then $cr_\pi=|\pi^{-1}(y_0)|$.
It follows that $cr_\pi \geq r_\pi$. On the other hand, since $r_\pi$ is finite there exists a point $y_1$ for which $r_\pi=|\pi^{-1}(y_1)|$. Clearly $cr_\pi(y_1)\leq |\pi^{-1}(y_1)|$. Hence
$cr_\pi=r_\pi$. Thus all points of a regular fibre must be pairwise non-proximal, and moreover, a fibre cannot contain more than $r_\pi$ pairwise non-proximal points.
\end{proof}
\begin{lem} \label{lem-idp} Let $\pi:(X,\sigma)\to (Y,\delta)$ be an equicontinuous factor with finite minimal rank.
Let $f\in \Ef$ be an element which acts on the singular fibres as an idempotent.
Then for some $N$, $f^N = f$ on the singular fibres and $f^N=\Id$ on the regular fibres.
\end{lem}
\begin{proof} Since regular fibres contain only distal points, and only finitely many, any element of $f\in \Ef$ must act on a regular fibre as a bijection. Since regular fibres have $r_\pi$ elements, then if $N=r_\pi!\,$,
$f^N$ acts like the identity on a regular fibre. If $f$ acts like an idempotent on the singular fibre then $f^N$ acts like $f$ on the singular fibres.
\end{proof}
We denote by $Y/T$ the space of $T$-orbits of $Y$ and its elements by $[y]$.
\begin{cor} \label{cor-main1}
Consider a unique singular orbit system with finite minimal rank. Let $y_0\in Y$ be singular.
The restriction $\Et_{y_0}$ of $\Et $ to $\pi^{-1}(y_0)$ contains all idempotents of $\Ef_{y_0}$. In particular, if $\Ef_{y_0}$ is generated by its idempotents then $\Ef_{y_0}=\Et_{y_0}$ and consequently,
$$\Ef = \Cf \cong \Ef_{y_0}\times \prod_{\stackrel{[y]\in Y/T}{y\neq y_0}} G_\pi.$$
This is a topological isomorphism if we equip the r.h.s.\ with the product topology.
\end{cor}
\begin{proof}
Any idempotent of $\Ef_{y_0}$ is the restriction of an element $f\in \Ef$ which, by Lemma~\ref{lem-idp}, may be assumed to act trivially on all regular fibres. Hence any idempotent of $\Ef_{y_0}$ is the restriction of an element $f\in \Et$.
Under the assumption of finite minimal rank the structure group $G_\pi$ must be finite and thus topologically isomorphic to $\Ef_y$ for regular $y$. Covariance allows us to factor out the action of $T$ and thus describe $\Cf$ as a direct product over the space of orbits $Y/T$.
\end{proof}
\subsection{Recovering $E(X)$ from $E^{fib}(X)$ and $Y$}
Although we now have a pretty good description of $E^{fib}(X)$ and $Y$ for unique singular orbit systems with finite minimal rank, it is not obvious how this describes $E(X)$. As our interest lies in minimal systems which have a singular fibre, their Ellis semigroup must contain two non-commuting idempotents. This implies that $E(X)$ cannot be
left-topological\footnote{Since $T$ is abelian, if left multiplication is continuous then $\lim_\nu \sigma^{t_\nu} \lim_\mu \sigma^{s_\mu} = \lim_\mu \lim_\nu \sigma^{t_\nu+s_\mu} = \lim_\mu \sigma^{s_\mu} \lim_\nu \sigma^{t_\nu}$, hence all elements of $E(X)$ commute.},
even when $E^{fib}(X)$ and $Y$ are topological. This is a sign that we cannot expect a semidirect product construction, paralleling that of groups, which describes $E(X)$ with its topology through $E^{fib}(X)$ and $Y$. However, on the purely algebraic side, we will see that Proposition \ref{extension-semigroup} turns out to be useful in this regard.
\begin{notation}\label{not:structure groups}
We let
$\M(X)$ denote the kernel of $E(X)$ and $\M^{fib}(X)$ denote the kernel of $\Ef(X)$. Recall that these kernels are completely simple. Picking a minimal idempotent $e$, we let
$\RSTp=eE(X)e$ and $\RSTfp=e\Ef(X) e$ denote the Rees structure group of $\M(X)$ and $\M^{fib}(X)$ respectively.
\end{notation}
As $Y=\tilde\pi(E(X)) = \tilde\pi(eE(X)) \subset \tilde\pi(\M(X))$,
\eqref{eq-ext} gives rise to the exact sequence
\begin{equation}\label{eq-kernel-ext}\nonumber
\Ef(X)\cap \M(X) \hookrightarrow \M(X) \stackrel{\tilde \pi} \twoheadrightarrow Y .
\end{equation}
$\Ef(X)\cap \M(X)$ contains all idempotents of $\M(X)$. Moreover, it is simple, as can be seen as follows:
As $\M(X)$ is completely simple, given $x,y\in \Ef(X)\cap \M(X)$ there is an idempotent $z \in \M(X)$ such that $x,z$ belong to the same minimal left, while $z,y$ belong to the same minimal right ideal of $\M(X)$. Since $z$ is an idempotent we have $z\in \Ef(X)$.
Since $x,z\in \Ef(X)\cap \M(X)$ belong to the same minimal left ideal of $\M(X)$ then there is $a\in \M(X)$ such that $x=az$. It follows that $\tilde \pi(a) = 0$, thus $a\in \Ef(X)\cap \M(X)$. Similarly, since $z,y\in \Ef(X)\cap \M(X)$ belong to the same minimal right ideal of $\M(X)$ then there is $b\in \Ef(X)\cap \M(X)$ such that $z=yb$. Hence $x=ayb$ for $a,b\in\Ef(X)\cap \M(X)$. This proves that $\Ef(X)\cap \M(X)$ is simple
and therefore equal to the kernel $\M^{fib}(X)$ of $\Ef(X)$.
A further restriction of \eqref{eq-ext} to $e \M(X) e$ leads to the exact sequence of groups
\begin{equation}\label{eq-kernel-ext1}\nonumber
\RSTfp \hookrightarrow \RSTp \stackrel{\tilde \pi} \twoheadrightarrow Y .
\end{equation}
We show now that for systems which satisfy the conclusion of Corollary~\ref{cor-main} the above sequence has a split section, so that the structure group $\Gg$ is the semi-direct product of $\RSTfp$ with the group $Y$.
\begin{prop} \label{prop-shomo}
Let $(X, \sigma)$ be a minimal
system with an equicontinuous factor $\pi:(X,\sigma)\to (Y,\delta)$ such that $\Ef=\Cf$. Let $e$ be a minimal idempotent of $E$ and
$s:Y\to eEe$ be a lift of $\tilde \pi$ which satisfies $s(\delta^t(y)) = \sigma^t s(y)$ and
$s(-y)=s(y)^{-1}$. Define, for $y\in Y$ the map $\hat s(y):eX\to eX$ by
$$\hat s(y)(x) = s(\pi(x)+y) s(\pi(x))^{-1} (x).$$
Then $\hat s:Y\to eEe$ is a right inverse to $\tilde\pi$ which
satisfies $\hat s(\delta^t(y)) = \sigma^t \hat s(y)$ and is a group homomorphism.
\end{prop}
\begin{proof} Let $y\in Y$.
By definition
$$\hat s(y)(x) = s(y) g(\pi(x)) (x)$$
where $g(z) = s(y)^{-1} s(z+y) s(z)^{-1}$.
We see that $g(z)(x)\in e\pi^{-1}(z)$ for all $x\in e\pi^{-1}(z)$,
hence $g(z):e\pi^{-1}(z)\to e\pi^{-1}(z)$ is an is element of $e\Ef_{z} e$.
Using $s(\delta^t(y)) = \sigma^t s(y)$ we obtain
$g(\delta^t(\pi(x)))(\sigma^t x) = \sigma^t g(\pi(x))(x)$. Thus
$Y\ni z\mapsto g(z)\in e\Ef_{z} e$ is covariant along the orbit of $y$ and hence an element of $e \Cf e$. By assumption $e \Cf e=e \Ef e$.
Thus $\hat s(y) = \tilde s(y)g$ is an element of $eEe$.
We show that $\hat s(y)$ is a right inverse to $\tilde\pi$. Let $x\in \pi^{-1}(0)$. We have
$$\tilde\pi \hat s(y) = \pi_*(\hat s(y))(0) = \pi(\hat s(y)(x)) = \pi(s(y)(x))=y+\pi(x) = y$$
where we have used $s(0)^{-1}=e$ in the third equality.
The identity $\hat s(\delta^t(y)) = \sigma^t \hat s(y)$ follows readily.
It remains to show that $\hat s$ is multiplicative:
\begin{eqnarray*}
\hat s(y_1+y_2)(x) &=& s(\pi(x)+y_1+y_2) s(\pi(x))^{-1} (x)\\
&=& s(\pi(x)+y_1+y_2)s(\pi(x)+y_2)^{-1} s(\pi(x)+y_2)s(\pi(x))^{-1} (x)\\
&=& \hat s(y_1)\hat s(y_2)(x).
\end{eqnarray*}
\end{proof}
\begin{cor}\label{cor:reproducing M}
Let $(X,T,\sigma)$ be a minimal
system with an equicontinuous factor $\pi:(X,\sigma)\to (Y,\delta)$ such that $\Ef=\Cf$.
Then the structure group $\RSTfp $ of $\M^{fib}(X)$ is isomorphic to $G_\pi^{Y/T}$. Moreover, the structure group of $\M(X)$ is $\RSTp=\RSTfp\rtimes Y$. Furthermore
if $M[\RSTfp;I,\Lambda;A]$ is the normalised Rees matrix form for $\M^{fib}(X)$ w.r.t.\ $e$, then $\M(X)$ is algebraically isomorphic to $M[\RSTfp\rtimes Y;I,\Lambda;A]$.
\end{cor}
\begin{proof} Apply Proposition~\ref{extension-semigroup}, {Corollary \ref{cor-ff}} and Proposition~\ref{prop-shomo}.
\end{proof}
For unique singular orbit systems with finite minimal rank and for which $\Ef_{y_0}(X)$ is generated by its idempotents, we have now reduced the calculation of the kernel of their Ellis semigroup to the calculation of the kernel of $\Ef_{y_0}(X)$ which we denote $\M^{fib}_{y_0}(X)$. Indeed, if $M[G;I,\Lambda;A]$ is the normalised matrix form of $\M^{fib}_{y_0}(X)$ w.r.t.\ $e$, then $G=G_\pi$ which we may identify with the subgroup $G_\pi\times \prod_{[y_0]\neq [y]\in Y/ T}\{\one\}$ of $G_\pi^{Y/T}$. It follows that
\begin{equation}\label{eq-alter}
\M^{fib}(X) \cong M[G_\pi;I,\Lambda;A]\times \prod_{[y_0]\neq [y]\in Y/ T}G_\pi \cong M[G_\pi^{Y/T};I,\Lambda;A]
\end{equation}
where the second topological isomorphism is given by the map $((i,g,\lambda),f)\mapsto
(i,(g,f),\lambda)$.
We will see in the next section how to compute $G_\pi$, $I$, $\Lambda$, and $A$
for systems arising from bijective substitutions.
\section{Bijective substitutions and their Ellis semigroup}\label{bijective}
In this section we discuss the Ellis semigroup of a family of minimal $\Z$-actions which are both unique singular orbit systems, and also systems for which forward/backward proximality is non-trivial and agrees with forward/backward asymptoticity. This is the family of bijective constant length substitution shifts.
For these systems, Corollary \ref{cor:prox=asym} tells us that $E(X)$ is the disjoint union of its kernel $\M(X)$ with the acting group $\Z$, so that a description of
$\M(X)$ suffices to completely describe the Ellis semigroup. Next,
for most of these systems, Corollary \ref{cor:reproducing M} will apply, so that we are on the way to describing $\M(X)$ once we know its restriction to a singular fibre which we call below the structural semigroup. This is the content of Theorem \ref{thm-RMG}. We consolidate to get a global statement in Theorem \ref{thm-main2}.
Finally, we identify the substitution shifts to which we cannot apply Corollary \ref{cor:reproducing M}, and we replace it with Theorem \ref{thm-main4}.
\subsection{Generalities}\label{generalities} We briefly summarise the notation and results concerning substitutions that we will need; for an extensive background see \cite{Baake-Grimm} or \cite{Pytheas-Fogg}.
A {\em substitution} is a map from a finite set $\mathcal A$, the alphabet, to the set of
nonempty finite words (finite sequences) on $\mathcal A$. We extend $\theta$ to a map on finite words by concatenation:
\begin{equation}\label{eq-concat}\theta(a_1\cdots a_k) = \theta(a_1)\cdots \theta(a_k),
\end{equation}
and to bi-infinite sequences $\cdots u_{-2} u_{-1} u_0 u_1 \cdots$ as
\[\theta (\cdots u_{-2} u_{-1} u_0 u_1 \cdots ) := \cdots \theta (u_{-2}) \theta ( u_{-1} ) \theta (u_{-1})\cdot \theta ( u_{0}) \theta ( u_{1}) \cdots \, .\]
Here the $\cdot$ indicates the position between the negative indices and the nonnegative indices.
We say that $\theta$ is {\em primitive} if there is some
$k\in \N$ such that for any $a,a'\in \mathcal A$,
the word $\theta^k(a)$ contains at least one occurrence of $a'$.
We say that a finite word is {\em allowed} for $\theta$ if it appears somewhere in $\theta^k(a)$ for some $a\in \Aa$ and some $k\in\N$.
The {\em substitution shift} $( X_\theta, \sigma)$ is the dynamical system where the space $X_\theta$ consists of all bi-infinite sequences
all of whose subwords are allowed for $\theta$. If $\theta$ is primitive, $X_\theta=X_{\theta^n}$ for each $n\in \N$.
We equip $X_\theta$ with the subspace topology of the product topology on $\Aa^\Z$, making the left shift map $\sigma$ a continuous $\Z$-action. Primitivity of $\theta$ implies that $(X_\theta,\sigma)$ is minimal.
We say that a primitive substitution is {\em aperiodic} if $X_\theta$ does not contain any $\sigma$-periodic sequences. This is the case if and only if $X_\theta$ is an infinite space.
The substitution $\theta$ has
{\em (constant) length~$\ell$} if for each $a\in \mathcal A$,
$\theta (a)$ is a word of length $\ell$. In this case one can describe the substitution with $\ell$ maps $\theta_i:\mathcal A \rightarrow \mathcal A$, $0\leq i \leq \ell-1$, such that
\begin{equation}\label{eq-as-perm}\nonumber
\theta(a) = \theta_0(a)\cdots \theta_{\ell-1}(a)
\end{equation}
for all $a\in\Aa$.
A substitution $\theta$ is {\em bijective} if it has constant length and each of the maps $\theta_i$ is a bijection.
If $\theta$ is bijective, then $X_\theta$ is the disjoint union of finitely many primitive bijective substitution shifts, and consequently its Ellis semigroup is also the disjoint union of finitely many Ellis semigroups of primitive substitution shifts. Henceforth we assume that $\theta$ is primitive but this comment means that all our results have analogous statements for non-primitive bijective substitutions.
We say that {the bijective} $\theta$ is {\em simplified} if
\begin{enumerate}
\item every $\theta$-periodic point is a fixed point of $\theta$, so that in particular $\theta_0=\theta_{\ell-1} = \one$, and
\item each word $\theta(a)$ contains all letters from $\mathcal A$.
\end{enumerate}
Given any bijective substitution $\theta$, both properties will be satisfied by a large enough power $\theta^n$ of $\theta$. Indeed, if $M$ is the lowest common multiple of the least periods of the periodic points, then each periodic point is a fixed point under $\theta^M$. Since for any $n\in \N$, $X_\theta= X_{\theta^n}$, there will be no loss in generality in assuming that $\theta$ is simplified and this is henceforth a standing assumption.
\subsection{An equicontinuous factor with a unique orbit of singular fibres}
Let $\theta$ be an aperiodic primitive substitution of length $\ell$. Define
$B^{(n)}:=\theta^n(X_\theta),$
which is a clopen subset of $X_\theta$. Then
$\sigma^i(B^{(n)})= \sigma^j(B^{(n)})$ if $i-j=0\: \mbox{\rm mod }\ell^n$ whereas
otherwise $\sigma^i(B^{(n)})\cap \sigma^j(B^{(n)})=\emptyset$ \cite[Lemma II.7]{dekking}. In other words
\[ \mathcal P_n = \{ \sigma^k( B^{(n)}) : 0\leq k\leq \ell^n-1 \} \]
is a $\sigma^{\ell^n}$-cyclic partition of $X_\theta$ of size $\ell^n$
For $n\geq1$, define
$ \pi_n : X_\theta \to \Z/\ell^n\Z$ by
$$\pi_n(x) = i \quad\mbox{\rm if }x\in \sigma^{i} (B^{(n)}).$$
The map $\pi_n$ can be described as follows. Using the partition $\mathcal P_1$, any bi-infinite sequence $x=(x_i)_{i\in\Z}\in X_\theta$ can be uniquely decomposed into blocks of length $\ell$ such that
\begin{itemize}
\item[{(i)}] The $i$-th block is a substitution word $\theta(a_i)$, for some $a_i\in \Aa$. Here we say that the $0$-th block is the one which contains $x_0$, and
\item[{(ii)}] The sequence $(a_i)_{i\in\Z}$ is an element of $X_\theta$.
\end{itemize}
Now set $\pi_1(x):=i$ if the $0$-th block starts at index $-i$ (if we shift that block $i$ units to the right then its first letter has index $0$).
This procedure can be performed with $\mathcal P_n$ and $\theta^n$ yielding an analogous definition for
$\pi_n(x)$. In particular, the $\pi_n$ are pattern equivariant (or local) and hence continuous.
Note that if $\pi_n(x) = i$, then $\pi_{n+1}(x) \equiv i \mod \ell^{n}$. Therefore, the collection of these maps $\pi_n$ defines a continuous map
\begin{equation}\label{ef-map} \pi : X_\theta \to Y:=\lim_{\leftarrow} \Z/\ell^n\Z \end{equation}
onto the inverse limit $\lim_{\leftarrow} \Z/\ell^n\Z$ defined by the canonical projections $\Z/\ell^{n+1}\Z \twoheadrightarrow \Z/\ell^n\Z$. The inverse limit space can be identified with
the space of left-sided sequences $(y_i)_{i<0} = \cdots y_{-2}\, y_{-1}$, $0\leq y_i<\ell$, and then $\pi(x)= (y_i)_{i<0}$ is such that for each positive integer
$n$, $\pi_{n}(x)=\sum_{i=-n}^{-1} \ell^{-i-1}y_{i}$.
It then follows that $\pi\circ\sigma = \add\circ \pi$ where $\add$ is addition of $1 = \cdots 00 1$ (only the last digit is not $0$) with carry over.
Its additive inverse is addition of $-1= \cdots \ell\!-\!1\, \ell\!-\!1\, \ell\!-\!1\,$.
In other words $(X_\theta,\sigma)$ factors onto the odometer with $\ell$ digits (adding machine).
This is the equicontinuous factor map with which we work.
As the space is the space of $\ell$-adic integers, we will denote it using the notation $\Z_\ell$.
\begin{prop}\label{prop-sf}
Let $\theta$ be a primitive aperiodic bijective (and simplified)
substitution of length $\ell$ and $\pi:X_\theta\to\Zl$ be defined by \eqref{ef-map}.
The fibre $\pi^{-1}(0)$ contains exactly the $\theta$-fixed
points. These are in one-to-one correspondence with the allowed two letter words.
\end{prop}
\begin{proof}
It is quickly seen that $\pi\circ \theta = (\times \ell)\circ \pi$ where $(\times \ell)$ is multiplication by $\ell$ in $\Zl$ and corresponds to left shift with adjoining a $0$:
$(\times\ell)(\cdots y_{-2}\, y_{-1}) = \cdots y_{-2}\, y_{-1} 0$. Hence any $\theta$-fixed point is mapped by $\pi$ to a $(\times \ell)$-fixed point in $\Z_\ell$, and the only such one is $0$.
Thus all $\theta$-fixed points belong to $\pi^{-1}(0)$.
It also follows that $\theta$ must preserve $\pi^{-1}(0)$.
Since the maps $\theta_i$ are bijections of $\Aa$, $\theta$ is injective on $X_\theta$. Hence it is injective on $\pi^{-1}(0)$.
We claim that $\pi^{-1}(0)$ must be finite. To prove the claim
let $x,x'\in X_\theta$. If $x\neq x'$ there exists $n\in\N$ such that $x_{[-\ell^n,\ell^n-1]} \neq x'_{[-\ell^n,\ell^n-1]}$
(here $x_{[n,m]}$ is the word $x_n x_{n+1}\cdots x_m$). It follows that $\theta^{-n}(x)_{[-1,0]} \neq \theta^{-n}(x')_{[-1,0]}$.
Since there are only finitely many words of length two
$\pi^{-1}(0)$ must be finite. It follows that the restriction of $\theta$ to $\pi^{-1}(0)$ is bijective and thus $\pi^{-1}(0)$ must be a union of periodic orbits under $\theta$.
As $\theta$ is bijective and simplified we have $\theta(x)_{[-1,0]} = x_{[-1,0]}$ for any $x\in X_\theta$. It follows that $\theta$-periodic points are $\theta$-fixed points and that they are in
one-to-one correspondence with the allowed two letter words.
\end{proof}
\begin{prop}\label{one-singular-orbit}
Let $\theta$ be a primitive aperiodic bijective substitution of length $\ell$ and let
$\pi:X_\theta\to\Zl$ be defined by \eqref{ef-map}.
Then the orbit of $\pi^{-1}(0)$ is the only singular fibre orbit. The minimal rank is
$r_\pi = s$ where $s$ is the size of the alphabet.
\end{prop}
\begin{proof}
Suppose that $y=\ldots y_{-2} y_{-1}$ does not belong to the $\Z$-orbit of $0$.
This is the case precisely if for infinitely many $n$, $y_{-n}\neq 0$ and, for infinitely many $n$, $y_{-n}\neq \ell-1$. Now if we take $x\in \pi^{-1}(y)$ and decompose it into substitution words $\theta^n(a)$ of level $n$ (as described above),
then the substitution word $\theta^n(a_0)$ which covers index $0$ must be
$\theta^n(a_0) = x_{[k_n,k_n+\ell^n-1]}$ where $k_n = -\sum_{i=-n}^{-1}\ell^{-i-1}y_{i}$.
Since $y_{-n}\neq 0$ for infinitely many $n$ we have $k_n\stackrel{n\to \infty} \longrightarrow -\infty$, and
since $y_{-n}\neq \ell-1$ for infinitely many $n$ we have $k_n + \ell^n-1\stackrel{n\to \infty} \longrightarrow=+\infty$.
Furthermore,
by bijectivity of $\theta$, $a_0$ is uniquely determined by $x_0$.
It follows that $x$ is uniquely determined by $y$ and $x_0$. Since there are exactly $s$ choices for $x_0$ we see that $\pi^{-1}(y)$ contains $s$ elements.
We now show that $\pi^{-1}(y)$ is a regular fibre if $y$ does not belong to the orbit of $\Z$.
Suppose that $x,x'$ were proximal. Then there exists $n\in\Z$ such that $x_n=x'_n$. In other words $\sigma^n(x)_0 = \sigma^n(x')_0$. Also $y+n$ does not belong to the $\Z$-orbit of $0$ and since $\sigma^n(x),\sigma^n(x')\in \pi^{-1}(y+n)$ we conclude from the above that $x=x'$. Hence all points of $\pi^{-1}(y)$ are pairwise non-proximal.
We have seen above that $\pi^{-1}(0)$ has $s^{(2)}$ elements where $s^{(2)}$ is the number of allowed two letter words. Given that $\theta$ is aperiodic we must have $s^{(2)}>s$. Thus
$\pi^{-1}(0)$ cannot be a regular fibre.
\end{proof}
Since $\theta$ is simplified its fixed points are precisely those of the form $\theta^{\infty}(a)\cdot\theta^{\infty}(b)$, where $ab$ is an allowed word for $\theta$. Such a fixed point is uniquely determined by $ab$.
We will use the notation $a\cdot b$ to denote it.
\begin{cor}
Let $\theta$ be a primitive bijective aperiodic substitution of constant length. If two points $x,x'\in X_\theta$ are forward (or backward) proximal then they are forward (or backward) asymptotic. Furthermore, forward and backward proximality are non-trivial.
In particular the Ellis semigroup $E(X_\theta)$ is completely regular and the disjoint union of its kernel $\M(X_\theta)$ with $\Z$.
\end{cor}
\begin{proof}
Suppose that $x,x'\in X_\theta$ are forward proximal. Then they are proximal and so by Propositions ~\ref{one-singular-orbit} and \ref{prop-sf} there is $n\in\Z$ such that $\sigma^n(x)$ and $\sigma^n(x')$ are fixed points of the (simplified) substitution. Since they are forward proximal and $\sigma$ is left shift this means that there are allowed two-letter words $ba,b'a$ such that $\sigma^n(x)=b\cdot a$ and $\sigma^n(x')=b'\cdot a$. In particular the two sequences agree to the right and hence are forward asymptotic. If forward proximality were trivial then every two letter word would be determined by its right letter. This cannot be the case as the substitution is aperiodic. Hence forward proximality is non-trivial. For the backward motion we argue in a similar way. The result now follows from Corollary \ref{cor:prox=asym}.
\end{proof}
\subsection{The kernel $\M^{fib}_0(X_\theta)$}\label{structural-section}
The restriction $E^{fib}_0(X_\theta)$ of $\Ef(X_\theta)$ to the singular fibre $\pi^{-1}(0)$ of the factor map $\pi:X_\theta\rightarrow \Z_\ell$ contains besides the identity map only its kernel $\M^{fib}_0(X_\theta)$. This kernel is a finite semigroup which we now compute. We also call it the {\em structural semigroup} of $\theta$. It completely determines $\Ef(X_\theta)$.
\bigskip
Recall that since $\theta$ has length $\ell$ there are maps $\theta_i:\mathcal A \rightarrow \mathcal A$ such that
$
\theta(a) = \theta_0(a)\cdots \theta_{\ell-1}(a)
$
for all $a\in\Aa$.
$\theta$ is thus uniquely determined by what we call its {\em expansion}, namely its representation as a concatenation of $\ell$ maps, which we write as
$$\theta = \theta_0|\theta_1|\cdots |\theta_{\ell-1}.$$
It follows from (\ref{eq-concat}) that
the composition of two substitutions $\theta$, $\theta'$ of length $\ell$ and $\ell'$ over the same alphabet (which we simply denote by $\theta \theta'$) has then an expansion into
$\ell \ell'$ maps
\begin{equation}\nonumber
\theta \theta' = \theta_0\theta'_0 |\cdots |\theta_{\ell-1}\theta'_0|\theta_0\theta'_1| \cdots |
\theta_{\ell-1}\theta'_{\ell'-1}
\end{equation}
where the product $\theta_i\theta'_j$ is that of permutations, that is, composition of bijections.
In particular, the expansion of $\theta^2$ is given by
\begin{equation}\label{expansion}
(\theta^2)_0|\cdots |(\theta^2)_{\ell^2-1} = \theta_0\theta_0 | \cdots |\theta_{\ell - 1}\theta_{0}| \theta_0\theta_1 | \cdots |\theta_{\ell-1}\theta_{\ell-1}
\end{equation}
and iteratively we find, for any given $n$ the $\ell^n$ bijections $(\theta^n)_i$ corresponding to the expansion of $\theta^n$.
\begin{definition}\label{def:structure group substitution} Given a bijective substitution $\theta$, we define the
{\em structure group} $\Gstr$ of $\theta$ to be the group generated by all the bijections
$(\theta^n)_i$, $n\in\N$, $i=0,\cdots,\ell^n-1$, and its {\em R-set} by
$$\rset:= \{(\theta^n)_{i}(\theta^n)_{i-1}^{\,-1}\in G_\theta : n\in\N,i=1,\cdots,\ell^{n}-1\}.$$
\end{definition}
Note that $\rset$ is the collection of bijections we need to apply (from the left) to go from some element $(\theta^n)_{i-1}$ in the expansion of some power of the substitution to its successor $(\theta^n)_{i}$. The name $R$-set is motivated by the fact that $\rset$ will label the right ideals of $\Sfib$.
\begin{lem}\label{I-is-everything}
If $\theta$ is simplified, then $\Gstr$ is generated by $\rset$ and
$$\rset = \{\theta_{i}\theta_{i-1}^{\,-1}\in G_\theta : i=1,\cdots,\ell-1\}.$$
\end{lem}
\begin{proof}
The first statement follows recursively as $\theta_i = \theta_i \theta_{i-1}^{-1} \theta_{i-1}$ and $\theta_0=\one$.
We prove the second statement for $n=2$
as the general statement then follows by induction. Let $(\theta^2)_{i-1}(\theta^2)_{i}$ be two consecutive bijections in the expansion of $\theta^2$. We consider two cases, the first if
$(\theta^2)_{i-1}(\theta^2)_{i}$ appears as two consecutive columns in a single substitution word, the second if it lies on the boundary, across two substitution words.
In the first case, $(\theta^2)_{i-1} | (\theta^2)_{i} = \theta_{j-1} \theta_k |\theta_{j} \theta_k $ for some $j\leq \ell - 1$ and some $0\leq k \leq \ell -1$, as in Equation \eqref{expansion}. But then
\[ (\theta^2)_{i} (\theta^2)_{i-1}^{\,-1} = \theta_{j} \theta_k (\theta_{j-1} \theta_k)^{-1} = \theta_{j} (\theta_{j-1})^{-1}, \]
and we are done as this last expression belongs to $I$.
In the second case, $(\theta^2)_{i-1} | (\theta^2)_{i} = \theta_{\ell-1} \theta_k |\theta_{0} \theta_{k+1}$ for some $k<\ell -1$. Since $\theta$ is simplified, $\theta_0=\theta_{\ell -1}= \one$, and here also we are done.
\end{proof}
In this section we will prove Theorem \ref{thm-RMG}; in its statement the semigroup has sandwich matrix as defined in \eqref{eq:matrix}.
\begin{thm}\label{thm-RMG}
Let $\theta$ be a primitive aperiodic bijective substitution.
The \Sstr\ $\Sfib$ of $\theta$ is isomorphic to the matrix semigroup $M[ \Gstr, \rset, \{ \pm\}; A]$ where
$\Gstr$ is the structure group and
$\rset$ is the R-set of $\theta$.
\end{thm}
We will first give a description of $\Sfib$ as a subsemigroup of $F(\pi^{-1}(0))$, the set of functions from $\pi^{-1}(0)$ to itself, and then compute its Rees matrix form.
We
recall that $\pi^{-1}(0)$ is the set of fixed points of $\theta$ which we denote $a\cdot b$ where $ab$ is an allowed two-letter word of $\Aa$.
To describe the action of $E^{fib}(X_\theta)$ on such a fixed point $a\cdot b$ we consider the set $\tc$ of all possible pairs of consecutive permutations $(\theta^n)_{i-1},(\theta^n)_{i}$, occurring in $\theta^n$, $n\in\N$, $i=1,\cdots \ell^n-1$. We write them with a dot
$(\theta^n)_{i-1}\cdot (\theta^n)_{i}$, or abstractly $L \cdot R$. We note that the R-set is related to $\tc$, namely
$$ \rset = \{RL^{-1}: L \cdot R\in \tc\}.$$
Notice also that $\tc$ is the same for any power of the substitution. \begin{lem}\label{lem-inv}
$\tc$ is invariant under the right
diagonal $G_\theta$-action $$(L \cdot R)g = (L g \cdot R g).$$ \end{lem}
\begin{proof}
Suppose that $L \cdot R\in G^{(2)}$, then for some $k\in \N$ and $0<i< \ell^k$ it appears as $L \cdot R= (\theta^k)_{i-1}\cdot(\theta^k)_{i}$. Let $g\in G_\theta$, so that it appears as $g=(\theta^n)_j$ for some $n\in \N $ and some $0\leq j\leq \ell^n -1$. Then, using the expansion of $\theta^n$ obtained in
Equation \eqref{expansion},
we find that $L g\cdot R g$
appears as two consecutive columns in the expansion of $ \theta^{n+k}$ and hence belongs to $\tc$
for each $g\in G_\theta$.
\end{proof}
As we assume that $\theta$ is simplified, we have
$(\theta^k)_0 = (\theta^k)_{\ell^k -1} = \one$ for each $k\geq 1$. Then
$$\theta^k(L|R) = (\theta^k)_0L |\cdots |(\theta^k)_{\ell^k-2} L | L|R | (\theta^k)_1 R|\cdots|
(\theta^k)_{\ell^k-1} R$$
where we have used that $(\theta^k)_{\ell^k -1}L=L$ and $(\theta^k)_{0}R = R$.
Hence if $L \cdot R = (\theta^n)_{i-1}\cdot(\theta^n)_{i}$ then the expansion of
$\theta^k\theta^n$ contains $\theta^k(\theta^n)_{i-1}| \theta^k(\theta^n)_{i}$ at positions
$[\ell^{k}(i-2),\ell^{k}i-1]$.
\begin{prop} \label{prop-hin}
Let $L \cdot R\in\tc$. Then
$E^{fib}$ contains an element
$f_{[L \cdot R;+]}$
which acts on the singular fibre $\pi^{-1}(0)$ as
$$f_{[L \cdot R;+]}(a\cdot b) = L(b)\cdot R(b),$$
and it contains an element $ f_{[L \cdot R;-]}$ which acts on this fibre as
$$\quad f_{[L \cdot R;-]}(a\cdot b) = L(a)\cdot R(a).$$
\end{prop}
\begin{proof}
Recall we assume that $\theta$ is simplified, so
that $\theta_0=\theta_{\ell-1}=\one$.
Let $n$ be such that $L \cdot R = (\theta^n)_{\nu-1}\cdot(\theta^n)_\nu$, for some $1\leq \nu\leq \ell^n-1$.
Let
$a\cdot b$ be a fixed point.
Then the two-letter word $\sigma^{\nu}(a\cdot b)_{[-1,0]}$ is $(\theta^n)_{\nu-1}(b)(\theta^n)_\nu(b)$. Furthermore
the expansion of
$\theta^k\theta^n$ contains $\theta^k(\theta^n)_{i-1}| \theta^k(\theta^n)_{i}$ at positions
$[\ell^{k}(i-2),\ell^{k}i-1]$.
Hence
$$\sigma^{\nu\ell^k}(a\cdot b)_{[-\ell^k,\ell^k-1]} = \theta^k L(b)\,\theta^kR(b).$$
It follows that
$$\sigma^{\nu \ell^k}(a\cdot b) \stackrel{k\to +\infty}\longrightarrow L(b)\cdot R(b)$$
in the topology of $X_\theta$.
By compactness there exists $f_{[L \cdot R;+]}\in E(X_\theta)$ which agrees with the map
$a\cdot b\mapsto L(b)\cdot R(b)$ on the singular fibre. It follows from the exact sequence (\ref{eq-ext}) that an element of $E(X_\theta)$ either preserves all $\pi$-fibres or none. Hence
$f_{[L \cdot R;+]}\in E(X_\theta)^{fib}$.
To construct elements in $E(X_\theta)$ which acts like
$a\cdot b\mapsto L(a)\cdot R(a)$ on the singular fibre we take
$\nu' = \nu-\ell^{n}$ with $n$ and $\nu$ as above. Then the two-letter word $\sigma^{\nu'}(a\cdot b)_{[-1,0]}$ is $L(a)\,R(a)$ and, similarly we find
$$\sigma^{\nu' \ell^k}(a\cdot b) \stackrel{k\to +\infty}\longrightarrow L(a)\cdot R(a).$$
By compactness we find the required map $ f_{[L \cdot R;-]}$.
\end{proof}
Let us denote the restriction of $f_{[L \cdot R;\epsilon]}$ to $\pi^{-1}(0)$ by $[L \cdot R;\epsilon]$ and set
$$\tc_\pm := \{[L\cdot R;\epsilon]: L\cdot R\in \tc, \epsilon \in\{\pm\}\}$$
It is easily checked that different elements of $\tc_\pm$ define different maps on $\pi^{-1}(0)$.
\begin{prop}\label{prop-rueck}
For any $\varphi\in \Sfib$ there exists $L \cdot R\in\tc$ and $\epsilon=\pm$ such that $\varphi = [L \cdot R;\epsilon]$.
\end{prop}
\begin{proof} Any $\varphi\in \Sfib\subset \Ef_0(X_\theta)$ is the restriction of a function $f\in \Ef(X_\theta)$ to $\pi^{-1}(0)$. We consider first the case that this function belongs to $E(X,\Z^+)$,
that is, $f$ is a pointwise limit of a generalised sequence $(\sigma^{\nu_i})_i$ with $\nu_i> 0$ (the inequality is strict as $f\neq \Id$). Note that $\varphi(a\cdot b)_{[-1,0]}$ is an open neighbourhood of $\varphi(a\cdot b)$. Thus given $a\cdot b\in\pi^{-1}(0)$ there exists a
$i_0$ such that $\varphi(a\cdot b)_{[-1,0]} = \sigma^{\nu_i}(a\cdot b)_{[-1,0]}$ for $i\geq i_0$. As $\pi^{-1}(0)$ is finite there exists a
$\nu>0$ such that $\varphi(a\cdot b)_{[-1,0]} = \sigma^{\nu}(a\cdot b)_{[-1,0]}$ for all $a\cdot b\in\pi^{-1}(0)$. Let
$L=(\theta^n)_{\nu-1}$ and $R=(\theta^n)_{\nu}$ where $\ell^n\geq \nu$. Then, for all $a\cdot b\in\pi^{-1}(0)$ we have $\varphi(a\cdot b)_{[-1,0]} = L(b)\cdot R(b)_{[-1,0]}$. Since the fixed points are uniquely determined by their two-letter word on $[-1,0]$ the map $\varphi$ is given by $a\cdot b\mapsto L(b)\cdot R(b)$. It is thus the restriction of $f_{[L\cdot R;+]}$ to $\pi^{-1}(0)$.
If $f \in E^-(X_\theta)$ we argue similarly:
there exists a $\nu<0$ such that, for all $a\cdot b\in\pi^{-1}(0)$ we have $\varphi(a\cdot b)_{[-1,0]} = \sigma^{\nu}(a\cdot b)_{[-1,0]}$. Then we take $L=(\theta^n)_{\ell^n-\nu-1}$ and $R=(\theta^n)_{\ell^n-\nu}$ where $\ell^n\geq \nu$. This leads to $\varphi(a\cdot b)_{[-1,0]} = L(a)\cdot R(a)_{[-1,0]}$, for all $a\cdot b\in\pi^{-1}(0)$ and we conclude that $\varphi$ is the restriction of $f_{[L\cdot R;-]}$ to $\pi^{-1}(0)$.
\end{proof}
We can compute the compositions of elements of $\tc_\pm$, for example
\begin{eqnarray*}
{[L \cdot R;+]} {{[L'\cdot R';+]}}(a\cdot b) &= & {[L \cdot R;+]}(L'(b)\cdot R'(b))\\
& =& LR'(b)\cdot RR'(b) \\
& = & {[L{R'},R{R'},+]}(a\cdot b)
\end{eqnarray*}
and likewise
\begin{eqnarray*}
{[L \cdot R;+]} {{[L'\cdot R';-]}}(a\cdot b)
&= & {[L \cdot R;+]}(L'(a)\cdot R'(a))\\
& =& LR'(a)\cdot RR'(a)\\
& = & {[LR' \cdot R{R'};-]}(a\cdot b).
\end{eqnarray*}
In this way we get
\begin{cor}\label{cor-G2}
$\Sfib = \tc_\pm$
with product given by
\begin{eqnarray*}
{[L \cdot R;+]} {[L'\cdot R';+]} &=& {[LR' \cdot RR';+]} \\
{[L \cdot R;-]} {[L'\cdot R';-]} &=& {[LL'\cdot RL';-]} \\
{[L \cdot R;+]} {[L'\cdot R';-]} &=& {[LR' \cdot RR';-]} \\
{[L \cdot R;-]} {[L'\cdot R';+]} &=& {[LL'\cdot R{L'};+]} .
\end{eqnarray*}
\end{cor}
\begin{proof}
{Combine Propositions~\ref{prop-hin} and ~\ref{prop-rueck}
together with the fact that all $[L \cdot R;\epsilon]$ act differently
to see that there is a one-to-one correspondence between the elements of $\Sfib$ and $\tc_\pm$. The form of the product is a direct calculation along the lines above.}
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm-RMG}]
Given the result of Corollary~\ref{cor-G2} it remains to show that $\tc_\pm$ is isomorphic to $M[ \Gstr, \rset, \{ \pm\}; A]$ for a (any) fixed choice of $g_0=R_0L_0^{-1}\in I_\theta$. Recall that, as a set, $M[ \Gstr, \rset, \{ \pm\}; A]= \rset\times\Gstr\times \{\pm\}$.
Consider the map
\begin{equation}\label{eq-bij}
\tc_\pm\ni [L \cdot R;\epsilon] \mapsto
\begin{cases}
(RL^{-1} , RL_0R_0^{-1},\epsilon) \ & \text{if $\epsilon = -$ } \\
(RL^{-1} , R,\epsilon) & \text{if $\epsilon=+$}.
\end{cases}
\end{equation}
Its injectivity is clear and its surjectivity is equivalent to Lemma~\ref{lem-inv}.
A direct calculation shows that it preserves the product structures.
\end{proof}
The Rees structure group of $M[ \Gstr, \rset, \{ \pm\}; A]$ is $G_\theta$, the structure group of $\theta$. It is related to the structure group $G_\pi$ of the factor
$\pi:(X_\theta,\sigma)\to (\Z_\ell,(+1))$ as follows: The choice of $g_0\in \rset$ to define the normalised matrix semigroup corresponds to a choice of
a minimal idempotent $e$ of $\Ef_0(X_\theta,\Z^+)$. Then we may view $G_\pi=e \Ef_0 e$. $G_\pi$ is thus a permutation of $e\pi^{-1}(0)$ and if we identify $e\pi^{-1}(0)$ via
$a\cdot b\mapsto b$ with $\Aa$ then we obtain $G_\theta$.
We remark that an idempotent $e\in \Ef_0(X_\theta,\Z^+)$ has the form $e=(i,\one,+)$, $i\in\rset$,
so that $e \Sfib e = \{i\}\times G_\theta \times \{+\}$ which is isomorphic to $G_\theta$ via the projection onto the middle component.
\begin{example} \label{Martin}
Consider the substitution $\theta$ given by
\[
\begin{array}{c} a\\ b\\ c \end{array}
\mapsto
\begin{array}{c} a\\ b\\ c \end{array}
\!\!\!\!\!\!{\begin{array}{c} b\\ a\\ c \end{array}}
\!\!\!\!\!\!{\begin{array}{c} a\\ c\\ b \end{array}}
\!\!\!\!\!\!{\begin{array}{c} a\\ b\\ c \end{array}}
\]
We use the notation $\begin{pmatrix}
\alpha \\
\beta \\
\gamma
\end{pmatrix} $
to denote the bijection that sends $a$ to $\alpha$, $b$ to $\beta$ and $c$ to $\gamma$.
The expansion of $\theta$ is $\theta_0|\theta_1|\theta_2|\theta_3$ with $\theta_0=\theta_3 = \one = \begin{pmatrix} a\\b\\c \end{pmatrix}$, while $\theta_1 = \begin{pmatrix} b\\a\\c \end{pmatrix}$ and $\theta_2 = \begin{pmatrix} a\\c\\b \end{pmatrix}$ are two transpositions.
We quickly find that
\[ \rset = \left\{ \theta_1\theta^{-1}_0 = \begin{pmatrix}
b \\
a \\
c
\end{pmatrix},
\theta_2\theta^{-1}_1 = \begin{pmatrix} c\\a\\b \end{pmatrix} ,
\theta_3\theta^{-1}_2 =
\begin{pmatrix} a\\c\\b \end{pmatrix} \right\}.
\]
Clearly $\rset$ generates $\Gstr=S_3$. The normalised sandwich matrix is
$$ A \begin{pmatrix} \one & \one & \one \\ \one & \tau_1 & \tau_2 \end{pmatrix} $$
where $\tau_1 = \theta_1 \theta^{-1}_0\theta_1 \theta_2^{-1}=\theta_2$ and $\tau_2= \theta_2 \theta^{-1}_1\theta_2 \theta_3^{-1}=\theta_2 \theta_1\theta_2$ are transpositions.
$\Sfib=M[S_3, \rset, \{ \pm\}; A]$ has 2 minimal left ideals each of which contains 18 elements, and 3 minimal right ideals each of which contains $12$ elements.
We note that $(X_\theta,\sigma)$ is not an {\em AI-extension} of $(\Z_4,+1)$; this can be
seen by applying Martin's criterion, which we do not describe here (see \cite{martin} or \cite{Staynova} for definitions and details); it suffices to note that $\theta$ admits seven two-letter words, has height one, and the set of two-letter words cannot be partitioned into sets of size three, creating an obstruction. Thus the techniques of \cite{Staynova} do not apply.
\end{example}
\subsection{The full semigroup $E(X_\theta)$}
We now combine Theorem~\ref{thm-main} with
our results from Section~\ref{fibre-preserving} to determine $\Ef(X_\theta)$ and, as far as possible, $E(X_\theta)$.
We consider first the simpler case in which $\Sfib$ is generated by its idempotents.
\begin{thm}\label{thm-main2}
Let $\theta$ be a primitive aperiodic bijective substitution and suppose that its structural semigroup $\Sfib$ is generated by its idempotents.
Then $\Ef(X_\theta)$ is topologically isomorphic to
$$\Ef(X_\theta) \cong (\Sfib \cup \{\Id\}) \:\:\times \prod_{\stackrel{[z]\in\Z_\ell /\Z}{\scriptscriptstyle{[z]\neq [0]}}}\Gstr .$$
Using the sets $\rset$, $\{\pm\}$ and the sandwich matrix $A$ from
normalised Rees matrix form $M[\Gstr;\rset,\{\pm\};A]$ for $\Sfib$ we can also express this as follows:
$\Ef(X_\theta)\backslash \{\Id\}$ is completely simple and topologically isomorphic to
$$\Ef(X_\theta)\backslash \{\Id\} \cong M[\RSTf;I_\theta,\{\pm\};A],\quad \RSTf=\Gstr^{\Z_\ell/\Z}.$$
Here an entry $a_{\lambda,g}$ of $A$ is identified with the function $\tilde f\in \RSTf$ which satisfies $\tilde f(0)=a_{\lambda,g}$ and $\tilde f(z)=\one$ for regular $z$.
Furthermore $E(X_\theta)\backslash \Z$ is algebraically isomorphic to
$$E(X_\theta)\backslash \Z \cong M[\RST;I_\theta,\{\pm\};A],\quad \RST = \RSTf\rtimes \Z_\ell$$
where $A$ is understood to take values in the subgroup $\Gstr\times \{\one\}^{\Z_\ell/\Z\backslash \{[0]\}}\rtimes \{0\}$ of $\Gstr^{\Z_\ell/\Z}\rtimes \Z_\ell$.
\end{thm}
\begin{proof}
As $\Ef(X_\theta)_0 = \Sfib\cup \{\Id\}$ it is generated by its idempotents.
Furthermore, the minimal rank $r_\pi$ is equal to $|\Aa|$.
The first expression for $\Ef(X_\theta)$ is thus a direct application of Corollary~\ref{cor-main1}.
The second expression corresponds to (\ref{eq-alter}). The last statement follows from Corollary~\ref{cor:reproducing M}.
\end{proof}
While the isomorphism between $E(X_\theta)\backslash \Z$ and $ M[\Gstr^{\Z_\ell/\Z}\rtimes \Z_\ell;I_\theta,\{\pm\};A]$ is not continuous when $\Gstr^{\Z_\ell/\Z}\rtimes \Z_\ell$ is equipped with the product topology, Theorem \ref{thm-main2} makes clear where the non-tameness comes from: whereas the restrictions of $E^{fib}(X_\theta)$ to individual fibres are finite semigroups, it is the fact that the structure group of its kernel
consists of {\em all possible} functions over the orbit space which implies that
the cardinality of $ E^{fib}(X_\theta)$, and hence of $E(X_\theta)$, is larger than that of the continuum.
\subsection{Height}\label{generalised-height}
The assumption of the last theorem, namely that $\Sfib$ is generated by its idempotents, is not always satisfied. To treat the other cases we introduce a new notion of height.
Let $\rset$ be the R-set of a bijective substitution $\theta$ and $\Gamma_\theta$ be the group generated by $\{gh^{-1}:g,h\in I_\theta\}$. We have seen in Lemma~\ref{lem-JSG} that $\Gamma_\theta$ is the Rees structure group of the subsemigroup generated by the idempotents of $\Sfib$.
If $\rset$ contains $\id$ then $\Gamma_\theta$ contains $\rset$ and therefore coincides with $\Gstr$, and consequently $\Sfib$ is generated by its idempotents. However, $\Gamma_\theta$ need not even be a normal subgroup of $\Gstr$; see Section~\ref{ex462} for an example.
\begin{lem}
Let $\tilde\Gamma_\theta$ be a normal subgroup of $\Gstr$ which contains the little structure group $\Gamma_\theta$.
Denote by $\phi:\Gstr\to \Gstr/\tilde\Gamma_\theta$ the canonical projection. Then
$\phi(g_1) = \phi(g_2)$ for any two elements of $\rset$. In particular, $\Gstr/\tilde\Gamma_\theta$ is a finite cyclic group.
\end{lem}
\begin{proof}
If $\phi(g_1)\neq \phi(g_2)$ for two elements of $\rset$ then $\phi(g_1 g_2^{-1})\neq 1$. But $g_1g_2^{-1}\in\Gamma_\theta\subseteq \ker\phi$. Since $\rset$ generates $\Gstr$,
the class of $\rset$ is a generator of $\Gstr/\tilde\Gamma_\theta$.
\end{proof}
We denote the order of $\Gstr/\tilde\Gamma_\theta$ by $\tilde h$.
Note that $\tilde h$ must devide any $\nu>0$ for which $(\theta^n)_\nu \in\tilde\Gamma_\theta$ (here $n$ is large enough such that $\nu\leq \ell^n$). Indeed, $(\theta^n)_\nu {(\theta^n)_0}^{-1}$ is a product of $\nu$ elements of $\rset$ and so its image in $\Gstr/\tilde\Gamma_\theta$ is $\nu$ times the generator of $\Gstr/\tilde\Gamma_\theta$. In particular, $\tilde h$ devides $\ell-1$.
\subsubsection{Classical height}
To better understand the meaning of the quantity $\tilde h$ we recall the notion of height which occurs in the context of constant length substitutions.
\newcommand{\hc}{h_{cl}}
The equicontinuous factor $\Zl$ which we have described above for a primitive aperiodic substitution $\theta$ of constant length is not always the {\em maximal} equicontinuous factor, i.e.\ there might be an intermediate equicontinuous factor $(\Xmax, +g)$ between $(X_\theta,\sigma)$ and
$(\Z_\ell, +1)$. The relevant quantity which governs this is the height of the substitution. In view of what follows we refer to it here as its {\em classical height}.
This is a natural number arising as the height of a tower
comprising a Kakutani-Rohlin model for the dynamical system and shows up also in the spectral analysis. It can be computed as follows. Consider a
one-sided fixed point $u=u_0u_1\cdots $ of $\theta$.
The (classical) height $\hc$ of $\theta$ is defined as
\begin{equation} \label{height}
\hc(\theta):= \max \{n\geq 1: \gcd(n,\ell)=1, n | \gcd\{k: u_k=u_0 \} \}\, .
\end{equation}
For primitive substitutions
it turns out to be independent of the choice of $u$. The following result was shown by Dekking \cite{dekking}, with partial
results by Kamae \cite{kamae} and Martin \cite{martin}.
\begin{thm} \label{thm:dekking}
Let $\theta$ be a primitive aperiodic substitution of length $\ell$ and classical height $\hc$. Then
the maximal equicontinuous factor of $(X_\theta, \sigma)$ is
$(\Z_{\ell} \times \Z/\hc\Z , \add\times\add)$.
\end{thm}
The theorem says that $\frac1\hc$ corresponds to a topological eigenvalue of the dynamical system $(X_\theta,\sigma)$ which does not already occur in the spectrum of $(\Z_l,+1)$. It moreover implies that the surjection $\tilde\pi$ in (\ref{eq-ext})
factors through $\Z/\hc \Z\times\Z_\ell$,
$$E(X_\theta)\twoheadrightarrow \Z/\hc \Z\times\Z_\ell \twoheadrightarrow \Z_\ell $$ and therefore leads to a continuous surjective semigroup morphism
\begin{equation*} \label{classical grading} E^{fib}(X_\theta)\twoheadrightarrow \Z/\hc \Z
\end{equation*}
Stated differently, $E^{fib}(X_\theta)$ is a $\Z/\hc \Z$-graded right topological semigroup.
A more detailed analysis yields the following picture.
Let $u$ be any one-sided fixed point of $\theta$.
For $a\in \mathcal A$, let $i_u(a) = \min\{k:u_k=a\}$. We claim that the set $\{n\in\N: u_n=a\}$ of occurences of $a$ in $u$ is contained in $i_u(a)+\hc\N$ where $\hc$ is as in \eqref{height}. For, say that $a$ occurs at indices $i$ and $j$ in $u$. Let $v$ be the one-sided fixed point of $\theta$ that starts with $a$. By minimality there exists $i_0\in \N_0$ such that we see $a$ in $v$ at the indices $i_0+i$ and $i_0+j$, $a=v_{i_0+i}=v_{i_0+j}$.
Recall that the height $h$ can be defined using any fixed point of $\theta$. Taking $v$ in place of $u$ in Definition \eqref{height} we see that
all indices at which we see the letter $a$ in $v$ are multiples of $h$. Thus $h$ divides $i-j$, and our claim follows.
In other words, we can partition the alphabet into subsets $\mathcal A_k:=\{a\in \mathcal A: i_u(a)\equiv k\bmod h \}$ and $\sigma(A_k) = A_{k+1}$.
Note also that if $\theta$ is simplified then $\{k: u_k=u_0 \}$ contains $\ell-1$ and hence the height must divide $\ell-1$. {In Lemma \ref{general-height-vs-height} and Theorem \ref{thm-grading}, we extend this partition to $\Gstr$ and $E(X_\theta)$.}
\begin{lem}\label{general-height-vs-height}
Let $\theta$ be a primitive aperiodic bijective substitution with structure group $\Gstr$. If $\theta$ has classical height $h_{cl}$, then there is a surjective group homomorphism $\phi_{cl}:\Gstr\to \Z/h_{cl}\Z$ such that for all $g\in \rset$
we have $\phi_{cl}(g) = 1$.
\end{lem}
\begin{proof} We may assume that $\theta$ is simplified and hence the height $h_{cl}$ divides $\ell-1$.
Fix an arbitrary one-sided fixed point $u=u_0u_1\cdots$ of $\theta$.
For $a\in \mathcal A$, let $i_u(a) = \min\{k:u_k=a\}$; we have seen that $\{n\in\N: u_n=a\}$ is contained in $i_u(a)+\hc\N$.
We now understand $k$ and $i_u(a)$ as an index modulo $\hc$. As
$\theta_j(u_k) = u_{\ell k+j}$ we see that $i_u(\theta_j(a)) - i_u(a) \equiv (\ell-1) i_u(a) + j $. Since the height must divide $\ell-1$ we find $(\ell-1) i_u(a) + j \equiv j$. Hence $i_u(\theta_j(a)) - i_u(a)$ does not depend on $a$ and so
$\phi_{cl}(\theta_j) := i_u(\theta_j(a)) - i_u(a)$ is a well defined map from $\{\theta_j,j\geq 0\}$ to $\Z/\hc\Z$. We compute $i_u(\theta_{j'}\theta_j(a)) \equiv j'+\ell(j+\ell i_u(a)) \equiv j'+j+i_u(a)$ and thus see that $\phi_{cl}$ is multiplicative. It hence
induces a surjective group homomorphism $\phi_{cl} :\Gstr \to \Z/h_{cl}\Z$. Clearly $\phi_{cl}(\theta_j\theta_{j-1}^{-1}) = 1$.
\end{proof}
\subsubsection{Generalised height}\label{sec:generalised height}
To discuss generalised height we introduce the maps $\evo$ and $\evo^z$, where
$$\evo:X_\theta\to\Aa,\quad \evo(x) := x_0$$
reads the letter on $0$ of $x$ and
$\evo^z:= \evo|_{e\pi^{-1}(z)}$ is the restriction of $\evo$ to the subset $e\pi^{-1}(z)$ of the fibre over $z$. Here again, $e$ is a chosen minimal idempotent, but we take it from $E(X_\theta,\Z^+)$. Clearly, if $z$ is regular then $e\pi^{-1}(z)=\pi^{-1}(z)$. Note that if $z\in \Z^+$ then $\evo(x) = \evo(e(x))$ because $e$ does not affect the right infinite part of a fixed point. $\evo^z$
is a bijection which we will use below to identify $e\pi^{-1}(z)$ with $\Aa$. As we already mentioned,
conjugation with $\evo^0$ identifies $G_\pi$ with $G_\theta$ and $\Gamma_\pi$ with $\Gamma_\theta$.
\begin{lem} \label{lem-fz}
Let $f\in E(X_\theta,\Z^+)$ and $z\in\Z_\ell\backslash \Z^-$.
There exists $f_z\in \Gstr$ such that for all $x\in \pi^{-1}(z)$ we have $\evo(f(x)) = f_z(x_0)$.
\end{lem}
\begin{proof}
We first show the result for $z=n\in \Z^+$ and $f=\sigma^m$, $m\geq 0$. Then $x = \sigma^n(a.b)$ for some fixed point $a.b$ and $n\geq 0$. It follows that $\sigma^m(x) = (\theta^{k})_{n+m} (\theta^{k})_{n}^{-1}(x_0)$ for $x\in \pi^{-1}(z)$ and all $k$ with
$\ell^k>n+m$. Thus for $z=n$ we can take ${\sigma^m}_z = (\theta^{k})_{n+m} (\theta^{k})_{n}^{-1}$.
Next suppose $z$ is regular and $f=\sigma^m$.
Since $(X,\sigma)$ is forward minimal there is $h\in E(X_\theta,\Z^+)$ with $\tilde\pi(h)=z$, where
$h$ is the limit of some generalised sequence $\sigma^{n_\nu}$ and $\pi^{-1}(z) = h(\pi^{-1}(0))$.
Let $x\in \pi^{-1}(z)$, i.e.\ $x=h(a.b)$ for some fixed point $a.b$. By continuity of $\sigma$ and $\evo$ we have
$$\evo\circ \sigma^m\circ h (a.b) = \lim_{\nu} \evo\circ \sigma^{m+n_\nu}(a.b)$$ and since $\pi^{-1}(0)$ is finite there exists $\nu_0$ such that for all $\nu\geq \nu_0$ and all $a.b\in\pi^{-1}(z)$,
$\lim_{\nu'} \evo\circ \sigma^{m+n_{\nu'}}(a.b) = \evo\circ \sigma^{m+n_\nu}(a.b)$.
Hence
$${\sigma^m}_z = {\sigma^{m}}_{n_\nu}$$
once $\nu\geq \nu_0$.
Now let $f\in E(X_\theta,\Z^+)$ and $z\in\Z_\ell\backslash \Z^-$. $f$ is the limit of some generalised sequence $\sigma^{n_\mu}$. We have
$$\evo\circ f (x) = \lim_{\mu} \evo\circ \sigma^{n_\mu}(x)$$
As $\pi^{-1}(z)$ is finite there exists $\mu_0$ such that for all $\mu\geq \mu_0$ and all $x\in \pi^{-1}(z)$,
$\lim_{\mu'} \evo\circ \sigma^{n_{\mu'}}(x) = \evo\circ \sigma^{n_\mu}(x)$.
Hence $f_z = {\sigma^{n_\mu}}_z$ once $\mu\geq \mu_0$.
\end{proof}
\begin{thm}\label{thm-grading} Let $\tilde\Gamma_\theta$ be a normal subgroup of $\Gstr$ which contains the little structure group $\Gamma_\theta$.
There exists a continuous surjective semigroup morphism
$$\eta: E(X_\theta,\Z^+)\to \Gstr/\tilde\Gamma_\theta\cong \Z/\tilde h\Z$$
such $\eta(\sigma f) = \eta(f)+1$. In other words, $\frac1{\tilde h}$ is a topological eigenvalue of the dynamical system $(E(X_\theta,\Z^+),\lambda_\sigma)$.
\end{thm}
\begin{proof}
By Lemma~\ref{lem-fz}, given $z\in\Z_{\ell}\backslash\Z^-$ we can assign to any $f\in E(X_\theta,\Z^+)$ a map $f_z\in I_\theta$. It depends on $z$ but since two elements of $\rset$ differ by an element of $\Gamma_\theta$ the class $f_z \tilde\Gamma_\theta$ is independent of $z$.
We can therefore define $$\eta(f) := f_z\tilde\Gamma_\theta.$$
We show that $\eta$ is locally constant.
For that we consider the following neighbourhood of $f$:
given $z\notin\Z^-$ let $$V = \{f'\in E(X_\theta,\Z^+): \evo(f(x)) = \evo(f'(x))\, \forall x\in\pi^{-1}(z) \}.$$ Then, if $f'\in V$ we must have $f_z = f'_{z}$ and hence $\eta(f) =\eta(f')$. Hence $\eta$ is locally constant, so continuous. It is immediate that $\eta(\sigma f) = \eta(f) + 1$.
By continuity we obtain $\eta(f_1 f_2) = \eta(f_1) + \eta(f_2)$.
Finally $e^{2\pi i\eta}$ is a continuous eigenfunction for the eigenvalue $\frac1{\tilde h}$.
\end{proof}
Let us compute the degree of the map $f_{[L\cdot R,+]}$ where $L \cdot R = (\theta^n)_{\nu-1}\cdot (\theta^n)_{\nu}$, $\nu\geq 0$.
We have seen that $f_{[(\theta^n)_{\nu-1}\cdot (\theta^n)_{\nu},+]}$ is the limit of a subnet of $(\sigma^{\nu \ell^k})_k$. Furthermore $\tilde h$ devides $\ell-1$. Hence $\nu$ and $\nu\ell^k$ agree modulo $\tilde h$ showing that
$\eta({\sigma^{\nu \ell^k}})$ is independent of $k$ and equal to $\nu$ modulo $\tilde h$. Thus
$$\eta(f_{[(\theta^n)_{\nu-1}\cdot (\theta^n)_{\nu},+]}) = \nu\: \mbox{modulo}\: \tilde h.$$
The above calculation shows also that the degree of an element of
$\M^{fib}(X_\theta,\Z^+)$ depends only on its restriction to the singular fibre $\pi^{-1}(0)$.
It is hence determined by a grading on $\M^{fib}_0(X_\theta,\Z^+)\cong \rset \times \Gstr\times\{+\}$. The latter can be easily obtained using the bijection (\ref{eq-bij}) between
$G^{(2)}_+$ and $\rset \times \Gstr\times\{+\}$; it is given by
$\eta:\rset \times \Gstr\times\{+\} \to \Gstr/\tilde \Gamma_\theta $,
$$ \eta(i,g,+) = g \tilde \Gamma_\theta.$$
In particular, as it should be, idempotents have degree $0$.
\bigskip
As an aside we remark that the grading $\eta$ can be extended to all of $E(X_\theta)$. We can repeat the above with $E(X_\theta,\Z^-)$ while exchanging $\sigma$ with $\sigma^{-1}$ and $\rset$ with $\rset^{-1}$. One then finds that
$\eta(f_{[(\theta^n)_{\nu-1}\cdot (\theta^n)_{\nu},-]}) = \nu -1$ modulo $\tilde h$ and
$\eta(i,g,-) = g \tilde \Gamma_\theta$.
\bigskip
The grading of $E(X_\theta,\Z^+)$ restricts to a grading of the group $\RST = eE(X_\theta,\Z^+) e$.
We denote the elements of degree $k$ by ${\RST}_k := \eta^{-1}(\{k\})\cap \RST$. Thus $ \RST = \bigsqcup_{k\in \Z/\tilde h\Z} {\RST}_k$ with ${\RST}_k {\RST}_l = {\RST}_{k+l}$ and
${\RST}_k = \fg^k {\RST}_0$
for any choice of element $\fg\in {\RST}_1$.
Similarly, the grading restricts to $\RSTf$, the structure group of the kernel of $\Ef$.
Recall the definition \eqref{eq:def T} of $\Et$ as the subsemigroup of elements of $\Ef$ which act trivially on all regular fibres. Since we have only one orbit of singular fibres, its restriction $\Et_0$ to the fibre at $0\in\Z_\ell$ is faithful and $e\Et e$ isomorphic to $e\Et_0 e$.
We identify $e\pi^{-1}(z)$ with $\Aa$ through the restricted evaluation map $\evo^z:e\pi^{-1}(z) \to \Aa$. Define $\Et_\theta:= \evo^0 e\Et_0 e (\evo)^{-1}$, a subgroup of the group of bijections of $\Aa$.
We now consider the situation in which $\tilde\Gamma_\theta$ is a subgroup of $\Et_\theta=\evo^0 e\Et_0 e (\evo)^{-1}$
and define
\begin{equation*}\label{def: CfGamma}\Cf(\tilde\Gamma_\theta) :=\{\tilde f\in e\,\Cf\, e\, : (\Phi^z_{0})^{-1}(\tilde f(z))\in (\evo^0)^{-1} \tilde\Gamma_\theta \evo^0 \:\forall z\in\Z_{\ell}\} . \end{equation*}
This group is independent of the choice of lift to define $\Phi^z_{0}$, as $\tilde\Gamma_\theta$ is normal in $\Gstr$.
\begin{prop}\label{fibre-structure-height1} Let $\tilde\Gamma_\theta$ be a normal subgroup of $\Gstr$ which contains $\Gamma_\theta$ and is contained in $\Et_\theta$. Then, w.r.t.\ the grading defined by $\tilde\Gamma_\theta$, we have the inclusion of groups
$$\RSTf_0 \subset \Cf(\tilde\Gamma_\theta) . $$
\end{prop}
\begin{proof}
Given $f\in \RSTf$ we have $\eta(f) =0$ if and only if, for all $z\in\Z_\ell\backslash \Z^-$ we have ${f}_z\in \tilde\Gamma_\theta$. Note that, with $\tilde f$ defined as in \eqref{eq:definition-f-tilde}, $f_z\circ \evo^z = \evo^z\circ \tilde f(z)$.
Moreover, if $h\in E(\Z^+)$ then $h_z\circ \evo^z = \evo^{z+\pi_*(h)}\circ h\left|_{\pi^{-1}(z)}\right.$.
It follows from Lemma~\ref{lem-fz} that $\evo^{z+\pi_*(h)}\circ h\left|_{\pi^{-1}(z)}\right.\circ (\evo^z)^{-1}\in G_\theta$. We apply this to $h=s(z)$, {the lift of $z$}, to see that
$g:=\evo^{z}\circ s(z)\left|_{\pi^{-1}(0)}\right.(\evo^0)^{-1}\in G_\theta$. Now
\begin{align*}(\Phi^z_{0})^{-1}(\tilde f(z)) &= s(z)^{-1} \tilde f(z) s(z) = s(z)^{-1} \circ(\evo^z)^{-1}\circ f_z\circ \evo^z\circ s(z)\\& \in (\evo^0)^{-1}\circ g^{-1}\tilde\Gamma_\theta g \circ \evo^0 = (\evo^0)^{-1}\circ \tilde\Gamma_\theta \circ \evo^0\end{align*}
as $\tilde \Gamma_\theta$ is normal in $G_\theta$. Thus
$f\in \Cf(\tilde\Gamma_\theta) $. \end{proof}
\begin{definition} The normal completion $\Glstr$ of the little structure group $\Gamma_\theta$ is the smallest normal subgroup of $\Gstr$ which contains $\Gamma_\theta$.
The generalised height $h$ of a primitive aperiodic bijective substitution is the order of
$\Gstr/\Glstr$.
\end{definition}
\begin{prop} Generalised height must be at least as large as classical height.
\end{prop}
\begin{proof}
Recall the quotient map $\phi_{cl}:\Gstr\to \Z/h_{cl}\Z$ from Lemma~\ref{general-height-vs-height}.
It satisfies $\phi_{cl}(\Gamma_\theta) = 0$. As $\Glstr$ is generated by elements of the form $g h g^{-1}$ with $h\in \Gamma_\theta$ and $g\in \Gstr$ we have $\phi_{cl}(\Glstr) = 0$
and hence $\Gstr/\Glstr$ factors onto $\Z/h_{cl}\Z$.
\end{proof}
In Section~\ref{ex462} we provide an example with trivial classical height but non-trivial generalised height.
\begin{prop}\label{fibre-structure-height2}
$\Glstr= \Et_\theta$ and
$$\RSTf_0 = \Cf(\Glstr)\cong \Glstr^{\Z_\ell/\Z} .$$
\end{prop}
\begin{proof} By Theorem~\ref{thm-main} we have the inclusion of groups $ \Cf(\Et_\theta) = e\Ct e \subseteq e\Ef e=\RSTf$. Hence, by Proposition ~\ref{fibre-structure-height1},
$ \Cf(\Et_\theta)$ is a subgroup of $\bigsqcup_{k\in \Z/\tilde h\Z} \mathfrak f^k \Cf(\Glstr)$
where $\mathfrak f$ is some element from $\RSTf_1$.
Since $s(z)$ and $s(z)^{-1}$ have opposite degree, $(\Phi_0^z)^{-1}(\tilde{\mathfrak f}(z)) \Glstr\in \Gstr/\Glstr$ must be the generator of $\Gstr/\Glstr$ for all $z$. Hence, for any $f\in \Cf(\Et_\theta)$, $(\Phi_0^z)^{-1}(\tilde{f}(z)) \Glstr\in \Gstr/\bar\Gamma_\theta$ is constant in $z$.
This is possible only if $\Et_\theta$ is a subgroup of $\Glstr$.
If that is the case then $\Cf(\Et_\theta) = \RSTf_0$, again by Proposition~\ref{fibre-structure-height1}.
The covariance condition says that a function $\tilde f\in \Cf$ is determined on the $\Z$-orbit of a point $z\in\Z_\ell$ by its value on $z$. We may hence chose for each $\Z$-orbit $[z]\in\Z_\ell/\Z$ a representative $z$ and then the obtain a bijection from $\Cf(\Glstr)$ to $\Glstr$-valued functions over the orbit space $\Z_\ell/\Z$ by restricting $\tilde f$ to the chosen representatives. This bijection is, of course, not canonical as it involves an uncountable choice of representatives, but it is a topological isomorphism of semigroups.
\end{proof}
We end this section with the generalisation of Theorem~\ref{thm-main2} to substitutions which may have non-trivial height.
\begin{thm}\label{thm-main4}
Let $\theta$ be a primitive aperiodic bijective substitution with generalised height $h$.
Using the sets $\rset$, $\{\pm\}$ and the sandwich matrix $A$ from
normalised Rees matrix form $M[\Gstr;\rset,\{\pm\};A]$ for $\Sfib$,
$\Ef(X_\theta)\backslash \{\Id\}$ is
topologically isomorphic to
$$\Ef(X_\theta)\backslash \{\Id\} \cong M[\RSTf;I_\theta,\{\pm\};A].$$
where $\RSTf$ is a $\Z/h\Z$-graded group and
$$\RSTf_k = \fg^k \Cf(\Glstr) $$
for any element $\fg$ of degree $1$ of $\RSTf$.
Here an entry $a_{\lambda,g}$ of $A$ is identified with the function $\tilde f\in \RSTf_0$ which satisfies $\tilde f(0)=a_{\lambda,g}$ and $\tilde f(z)=\one$ for regular $z$.
If $\Gstr$ contains an element of order $h$ then $\RSTf$ is a semidirect product
$$\RSTf \cong \Cf(\Glstr)\rtimes \Z/h\Z.$$
Furthermore $E(X_\theta)\backslash \Z$ is algebraically isomorphic to
$$E(X_\theta)\backslash \Z \cong M[\RST;I_\theta,\{\pm\};A]$$
where $\RST$ is the extension determined by $\RSTf\hookrightarrow \RST \twoheadrightarrow \Z_\ell$ and $A$ is understood to take values in the subgroup $\RSTf$.
If moreover, generalised height is equal to classical height then
there is a split section $s:\Z_\ell \to \RST$ whose image belongs to ${\RST}_0$. In particular we have the algebraic isomorphism
$$\RST\cong \RSTf\rtimes \Z_{\ell}.$$
\end{thm}
\begin{proof}
The first part follows from Prop.~\ref{fibre-structure-height2}.
It remains to show that $\RSTf \cong \Cf(\Glstr)\rtimes \Z/h\Z$ provided
$\Gstr$ contains an element of order $h$. An element $\fg\in \RSTf$ with $\eta(\fg) = 1$ can be constructed as follows. We saw that any $(i_0,i,+)\in M[G_\theta;\rset,\{\pm\},A]$ with $i\in \rset$ has degree $1$. Pick $i\in \Gstr$ and
let $\fg$ be the function which satisfies
$$ \fg(z):=\Phi_0^z((\evo^0)^{-1}(i_0,i,+)\evo^0). $$
Then $\fg$ is an element of degree $1$ in $\Cf(\Gstr)$.
As $\Phi_0^z$ restricts to a group isomorphism from $e\Ef_0e$ to $\Ef_z$ it preserves the order of an element and therefore, if $i^h=\one$, then $\fg$ has order $h$. In that case, $\Z/h\Z \ni 1 \mapsto \fg \in \RSTf$ induces a split section for the exact sequence
$\RSTf_0\hookrightarrow \RSTf\stackrel{\eta}\twoheadrightarrow \Z/h\Z$.
The result for $E(X_\theta)$ follows from Proposition~\ref{extension-semigroup}.
We have seen that, if $h$ is equal to the classical height, then $\tilde\pi$ factors through the maximal equicontinuous factor which is
$\Z/h\Z\times\Z_\ell$. By the axiom of choice we can construct a covariant lift
$s:\Z_\ell \to \RST$ which factors through $\{0\}\times\Z_\ell\subset \Z/h\Z\times\Z_\ell$. It hence takes values in ${\RST}_0$. It follows that the split homomorphism $\hat s$ constructed in Proposition~\ref{prop-shomo} also takes values in ${\RST}_0$. Thus
${\RST}_0 \cong \RSTf_0\rtimes \Z_\ell$. This implies the last statement.
\end{proof}
\section{The virtual automorphism group of unique singular orbit systems}\label{Ellis-group}
In this section we investigate the relationship between the automorphism group and the virtual automorphism group of bijective substitutional systems, as defined by
Auslander and Glasner \cite{AG-2019}. They show that an almost automorphic system is semi-regular iff it is equicontinuous. They also show that the Thue-Morse shift is semi-regular. Using our tools, in this section we will extend their result to show that bijective substitution shifts are semi-regular.
We start a little bit more generally considering minimal systems $(X,T, \sigma)$ where $T$
is an abelian group, unless we are talking about a substitution, in which case $T=\Z$.
\subsection{Automorphism groups of bijective substitutions}
The {\em automorphism group} $\Aut(X)$ of a dynamical system $(X,\sigma)$ is the group, under composition, of all homeomorphisms of $X$ which commute with $\sigma$.
Since the elements of $E(X)$ are limits of generalised sequences of powers of $\sigma$, the automorphism group, viewed as a subset of $X^X$, lies in the commutant of $E(X)$.
Similar to the situation of Ellis semigroups described in Section~\ref{semigroup-of-factor},
if $(X,\sigma)$ is minimal and $\pi: X \rightarrow Y$ is an equicontinuous factor map,
then the map $\pi_*$ from Section~\ref{semigroup-of-factor} is well-defined on automorphisms of $X$
inducing a group morphism $\pi_*: \Aut(X)\rightarrow \Aut(Y)$; see \cite{coven-quas-yassawi}. We also have $\Aut(Y)\cong Y$ for a minimal equicontinuous system. But $\pi_*$ is usually not surjective, although its image always contains $T\subset Y$.
We can therefore analogously define $\Aut^{fib}(X)$ as the kernel of $\pi_*$, that is, the subgroup of automorphisms which preserve the $\pi$-fibres, and then determine $\Aut(X)$ through the extension $\Aut^{fib}(X) \hookrightarrow \Aut(X) \twoheadrightarrow \pi_*(\Aut(Y))$. Contrary to most elements of $E^{fib}(X)$, the elements of $\Aut^{fib}(X)$ are always continuous and therefore, again by minimality, $\Aut^{fib}(X)$ is determined by its restriction to a fibre $\pi^{-1}(y_0)$. Furthermore, $\Aut^{fib}(X)$ must commute with $\RSTfp=eE^{fib}(X)e$ and therefore its restriction to $\pi^{-1}(y_0)$ is
contained in the centraliser of the structure group $G_\pi=eE^{fib}_{y_0}(X)e$ in the permutation group $S_{e\pi^{-1}(y_0)}$ of the factor
\cite[Theorem 33]{M-Y}. For primitive bijective substitution shifts it can be shown that $\Aut^{fib}(X)$ exhausts that centraliser and that
$\pi_*: \Aut(X)\rightarrow \Aut(Y)$ is as small as possible:
\begin{thm} \cite[Theorem 5]{L-M}\label{automorphism-theorem}
Let $\theta$ be a primitive aperiodic bijective substitution over the alphabet $\Aa$. Then $\Aut^{fib}(X_\theta)$ is isomorphic to the centraliser $C_{S_{\mathcal A}} (G_\theta) $ and
$$\Aut(X_\theta)\cong C_{S_{\mathcal A}} (G_\theta) \times \Z.$$
\end{thm}
\subsection{Virtual automorphism groups}
We start with the two algebraic lemmas. Given a set $X$, $x\in X$ and a group $G\subseteq S_X$, the permutation group of $X$, let $\Stab_G(x)$ denote the stabilizer of $x$ in $G$, let $N_G(\Stab_G(x))$
denote the normaliser of $\Stab_G(x)$ in $G$, and let $C_{S_X}(G)$ denote the centraliser of $G$ in $S_X$.
The following lemma from group theory is well-known but we include a proof.
Recall that a group G acts {\em transitively} on $X$ if for each pair $x, y$ in $X$ there exists $g\in G$ such that $g(x) = y$.
\begin{lem}\label{algebraic-lemma}
Let $G$ be a subgroup of the permutation group $S_X$ which acts transitively on $X$, and let $x\in X$. Then $$N_G(\Stab_G(x)) / \Stab_G(x) \cong C_{S_X}(G).$$
\end{lem}
\begin{proof}
Let $h\in N_G(\Stab_G(x))$. We first claim that if two $f, f'\in G$ satisfy $f(x)=f'(x)$, then $fh^{-1}(x)= f'h^{-1}(x)$. Indeed, $f(x)=f'(x)$ implies
$f' = f s$ for some $s\in \Stab_G(x)$. Furthermore $sh^{-1} = h^{-1}s'$ with $s'$ in $\Stab_G(x)$, as $h$ normalises $\Stab_G(x)$. Hence
\[fh^{-1}(x) = fh^{-1}(s'x) = f s h^{-1}(x)= f'h^{-1}(x).\]
Therefore the map
$ F: N_G(\Stab_G(x)) / \Stab_G(x) \to S_X$ defined by
\begin{equation*} F(h)(y) := fh^{-1}(x),\quad \mbox{for any $f\in G$ such that $f(x)=y$} \label{definition-phi} \end{equation*}
is well-defined. We show $F$ preserves the group multiplication. Let $h_1,h_2\in N_G(\Stab_G(x)) /\Stab_G(x)$. Then
\begin{equation}\label{eq-F-homo}
F(h_1)(F(h_2)(y)) = f_1h_1^{-1}(x) ,\quad \mbox{for any $f_1\in G$ such that
$f_1(x)=F(h_2)(y)$}
\end{equation}
and $F(h_2)(y) = f_2h_2^{-1}(x)$ for any $f_2\in G$ such that $f_2(x) = y$. In particular, $f_1(x)=f_2h_2^{-1}(x)$ so that we may take $f_1=f_2h_2^{-1}$ in (\ref{eq-F-homo}).
Thus $F(h_1)(F(h_2)(y))=f_2h_2^{-1}h_1^{-1}(x)$. As $f_2(x) = y$ this is also equal to
$F(h_1h_2)(y)$.
Note that if $F(h) = \Id$ then $fh^{-1}(x) = f(x)$ for all $f$, which implies that $F$ is injective.
Furthermore, using our first claim again, it can be checked that $F(h)$ commutes with the elements of $\Stab_G(x)$ and so the image of $F$ is contained in
$C_{S_X}(G)$.
To see that the image of $F$ is all of $C_{S_X}(G)$ let $\psi\in C_{S_X}(G)$. Since $G$ acts transitively there exists $h\in G$ such that $\psi(x) = h^{-1}(x)$. Then, for $f\in G$,
\begin{equation}\psi(f(x)) = f\psi(x) = fh^{-1}(x).\label{surjective}\end{equation}
If we apply this formula to $f\in \Stab_G(x)$ we get $h^{-1}(x) = \psi (x) = \psi(f(x)) = fh^{-1}(x)$ which implies that $h^{-1}$ normalises $\Stab_G(x)$. Now \eqref{surjective} shows that $\psi=F([h])$ where $[h]$ is the class of $h$ in $N_G(\Stab_G(x)) /\Stab_G(x)$.
\end{proof}
Let $G_1$, $G_2$ be two groups acting on $X_1$ and $X_2$, resp..
Let $G\subset G_1\times G_2$ a subgroup such that the projection $p_1:G\to G_1$ is surjective. $G_1\times G_2$ acts on $X_1$ via the projection $p_1$ hence $G$ also acts on $X_1$: if $f\in G$ then $f(x):= f_1(x)$ where $f_1=p_1(f)$.
\begin{lem} \label{lem-exercice}
In the above context, let $x\in X_1$. Then $p_1$ induces an isomorphism
$$N_G(\Stab_G(x))/\Stab_G(x) \to N_{G_1}(\Stab_{G_1}(x))/\Stab_{G_1}(x).$$
\end{lem}
\begin{proof}
$f\in G$ belongs to $\Stab_G(x)$ iff $f_1(x) = x$. Hence
$p_1^{-1}(\Stab_{G_1}(x))=\Stab_G(x)$. In particular $p_1$ descends to the quotients.
$h\in G$ belongs to $N_G(\Stab_G(x))$ iff $\forall f\in \Stab_G(x)$ we have $fh^{-1}(x) = h^{-1}(x)$. Since $p_1^{-1}(\Stab_{G_1}(x))=\Stab_G(x)$ the latter is equivalent to
$\forall f_1\in \Stab_{G_1}(x)$ we have $f_1h_1^{-1}(x) = h_1^{-1}(x)$. Hence
$N_{G_1}(\Stab_{G_1}(x))=p_1(N_G(\Stab_G(x)))$. Hence the map induced by $p_1$ is surjective.
$p_1^{-1}(\Stab_{G_1}(x))=\Stab_G(x)$ shows that $p_1(h) \in \Stab_{G_1}(x)$ implies
$h\in \Stab_G(x)$ and so the map induced by $p_1$ is injective.
\end{proof}
\subsubsection{Definition of the virtual automorphism group}
Let $(X,\sigma,T)$ be a minimal system with abelian group $T$. We denote its Ellis semigroup here simply by $E$.
Let $e\in E$ be a minimal idempotent and
recall that $\RSTp =eEe$ is a group. Pick $x_0\in e(X)$. Applying Lemma \ref{algebraic-lemma} to the space $e(X)$ and the group $G=\RSTp$, we get
\[ N_{\RSTp } (\Stab_{\RSTp }(x_0) ) / (\Stab_{\RSTp }(x_0)) \cong C_{S_{e(X)}}(\RSTp ). \]
\begin{definition}\label{vag-definition}
Let $(X,\sigma,T)$ be minimal and $e\in E$ be a minimal idempotent in its Ellis semigroup.
The {\em virtual automorphism group $V(X)$ of $(X,\sigma)$} is defined to be $C_{S_{e(X)}}(\RSTp )$.
\end{definition}
While this definition depends on the choice of the minimal idempotent $e$ it does so only up to isomorphism, as different choices of idempotents lead to isomorphic groups.
We remark that the restriction map
$$\Aut(X)\ni \Phi\mapsto\Phi|_{e(X)}\in C_{S_{e(X)}}(\RSTp )$$ is well-defined, as automorphisms commute with the elements of $E(X)$. Furthermore, if $e(X)$ is dense in $X$ then this restriction map is an injective group homomorphism, as automorphisms are
continuous. A condition guaranteeing that $e(X)$ is dense in $X$ is point distality of
$(X, \sigma,T)$, as minimal idempotents fix distal points. So for a point distal minimal system $(X, \sigma,T)$, $\Aut(X)$ is a subgroup of $V(X)$.
\begin{definition}\label{semi-regular-definition}
The minimal dynamical system $(X,\sigma,T)$ is called {\em semi-regular}
if the restriction map $\Phi\mapsto\Phi|_{e(X)}$ is an isomorphism between
the automorphism group $\Aut(X)$ and the virtual automorphism group $V(X)$. \end{definition}
Note that our definition is slightly different to Auslander and Glasner's, who simply require that the map $\Aut(X)\rightarrow V(X)$ be onto. However for point distal systems, and the systems we study here are point distal, the definitions coincide.
\subsubsection{Unique singular fibre systems}
We investigate the virtual automorphism group $V(X)$ for minimal systems which have an equicontinuous factor with a unique orbit of singular points.
Recall the definition (\ref{eq:def T}) of $\Et$ as the subsemigroup of elements of $\Ef$ which act trivially on regular fibres and that it restriction $\Et_{y_0} $ to $\pi^{-1}(y_0)$ is faithful.
Recall the notation $\RSTfp=e\Ef e$.
\begin{lem} \label{lem-aus1}
Let $(X,\sigma,T)$ be a minimal unique singular orbit system.
Let $y_0$ be a singular point in its maximal equicontinuous factor and $x_0\in e\pi^{-1}(y_0)$.
If $e\Et_{y_0} e$ acts effectively\footnote{For any $x\in e\pi^{-1}(y_0)$ there exists $\gamma\in e\Et_{y_0} e$ such that $\gamma(x)\neq x$.}
on $e\pi^{-1}(y_0)$, then
$$N_{\RSTp }(\Stab_{\RSTp }(x_0)) = N_{\RSTfp }(\Stab_{\RSTfp }(x_0)) \times T.$$
\end{lem}
\begin{proof}
By Theorem~\ref{thm-main} $\Ef$ contains $\Ct$ as defined in (\ref{eq:def Ct}) and therefore
\begin{equation}\label{eq-aus1}
\Stab_{\RSTp }(x_0)\supset \{\tilde f\in e\Ct e :\tilde f (y_0)(x_0)=x_0\}.
\end{equation}
Let $h\in N_{\RSTp }(\Stab_{\RSTp }(x_0))$. Then $f (h(x_0))=h(x_0)$ for all $f\in \Stab_{\RSTp }(x_0)$. Using $y:=\tilde\pi(h)+y_0$ this means that $\tilde f(y)(h(x_0)) = h(x_0)$. Suppose that $y-y_0 \notin T\subset Y$ ($T$ seen as a subgroup of the maximal equicontinuous factor $Y$).
Since $e\Et_{y_0} e$ acts effectively there is $\gamma\in e\Et_{y_0} e$ such that
$\Phi^{y}_{y_0}(\gamma)(h(x_0)) \neq h(x_0)$.
By (\ref{eq-aus1}) there exists $f\in \Stab_{\RSTp }(x_0)$ such that $\tilde f(y) = \Phi^{y}_{y_0}(\gamma)$. For this element we have
$f(h(x_0)) =\Phi^{y}_{y_0}(\gamma)(h(x_0))\neq h(x_0)$. This is a contradiction and thus all $h\in N_{\RSTp }(\Stab_{\RSTp }(x_0))$ must satisfy $\tilde\pi(h)\in T$. Clearly $\sigma\in N_{\RSTp }(\Stab_{\RSTp }(x_0))$ and so the above shows that
the exact sequence $ {\RSTfp }\hookrightarrow {\RSTp }\twoheadrightarrow Y $
restricts to the exact sequence
$$ N_{\RSTfp }(\Stab_{\RSTfp }(x_0))\hookrightarrow N_{\RSTp }(\Stab_{\RSTp }(x_0)) \twoheadrightarrow T .$$
We can lift the subgroup $T\subseteq Y$ with the lift $s:T\to E$ given by
$s(t) = \sigma^t$ to see that $N_{\RSTp }(\Stab_{\RSTp }(x_0))$ is a semi-direct product, which is in fact direct, as $\sigma^t$ commutes with $E$, since $T$ is abelian.
\end{proof}
\begin{lem} \label{lem-vag-structure}
Let $(X,\sigma,T)$ be a minimal system with equicontinuous factor $\pi:X\to Y$.
Let $y_0\in Y$, $x_0\in e\pi^{-1}(y_0)$ where $e$ is a minimal idempotent of $E(X)$.
We have
$$N_{\RSTfp }(\Stab_{\RSTfp }(x_0))\cong C_{S_{e\pi^{-1}(y_0)}}(G_\pi) .$$
\end{lem}
\begin{proof}
$\RSTfp $ is a subgroup of $e\Cf e=\prod_{y\in Y_0} e\Ef_y e$ where $Y_0\subset Y$ contains exactly one representative for each orbit and we suppose that $y_0\in Y_0$. Thus $e\Cf e$ has the form $G_1\times G_2$ with $G_1 = G_\pi = e\Ef_{y_0} e$ and the projection $p_1$ is surjective. We thus can apply Lemma~\ref{lem-exercice} and then Lemma~\ref{algebraic-lemma} to see that
$$N_{\RSTfp }(\Stab_{\RSTfp }(x_0))\cong N_{e\Ef_{y_0} e}(\Stab_{e\Ef_{y_0} e}(x_0))\cong C_{S_{e\pi^{-1}(y_0)}}(G_\pi).$$
\end{proof}
\begin{cor} \label{vag-structure}
Let $(X,\sigma)$ be a minimal unique singular orbit system. Let $y_0$ be a singular point of its maximal equicontinuous factor. If
$e\Et_{y_0}e$ acts effectively then the virtual automorphism group is given by
$$V(X)\cong C_{S_{e\pi^{-1}(y_0)}}(G_\pi) \times T.$$
\end{cor}
\begin{proof}
Combine Lemmata~\ref{lem-aus1} and \ref{lem-vag-structure}. \end{proof}
We provide a criterion for effectiveness of the action of $e\Et_{y_0} e$.
\begin{lem}\label{effective-lemma}
Let $X$ be a set and
$\Gamma$ be a non-trivial normal subgroup of a subgroup $G\subseteq S_X$ which acts transitively on $X$. Then $\Gamma$ acts effectively on $X$.
\end{lem}
\begin{proof}
Let $F_\Gamma=\{ a\in X:\Gamma(a)=\{a\} \}$.
If $h\in N_{S_X}( \Gamma)$ then for $a\in F_\Gamma$
\[ \Gamma h(a) = \{ \gamma h(a): \gamma\in \Gamma \} = \{ h\gamma' (a): \gamma'\in \Gamma \} = \{ h(a)\}, \]
so that $h(a)\in F_\Gamma$.
By assumption, $G$ lies in $N_{S_X}( \Gamma)$, so that $G(F_{\Gamma})\subseteq F_{\Gamma}$. Since $G$ is transitive we have either $F_\Gamma=X$ or $F_\Gamma=\emptyset$. In the first case, $\Gamma$ can consist only of the identity, and in the second $\Gamma$ acts effectively.
\end{proof}
We thus see that $e\Et_{y_0} e$ acts effectively if it is non-trivial and
$e\Ef_{y_0} e$ acts transitively on $e\pi^{-1}(y_0)$.
\subsubsection{Bijective substitutions}
We now focus again on the dynamical systems of primitive aperiodic bijective substitutions.
Recall that in this case we can identify $e\pi^{-1}(0)$, the image of the singular fibre at $0\in\Z_\ell$ under the chosen minimal idempotent $e\in E(X_\theta,\Z^+)$, with the alphabet $\Aa$ using the map $\evo^0$. Under this isomorphism
$G_\pi = e \Ef_0 e \cong G_\theta$ and $e \Et_0 e \cong \Glstr$. Since $\theta$ is assumed primitive, the structure group $\Gstr$ must act transitively on $\Aa$. Aperiodicity of the substitution implies that $\rset$ must consist of at least $2$ elements.
Hence $\Gamma_\theta$ is non-trivial, so by Lemma \ref{effective-lemma}, $\Glstr$ acts effectively on $\mathcal A$.
\begin{cor} \label{vag-structure-height}
Let $\theta$ be a primitive aperiodic bijective substitution.
The virtual automorphism group is given by
$$V(X_\theta)\cong C_{S_\Aa}(\Gstr) \times \Z.$$
\end{cor}
\begin{proof} All hypothesis of Corollary~\ref{vag-structure} are satisfied. \end{proof}
\begin{cor}\label{Vag=Aut} The dynamical system of a primitive aperiodic bijective substitution is semi-regular.
\end{cor}
\begin{proof}
We see from Theorems ~\ref{automorphism-theorem} and Corollary~\ref{vag-structure-height}
that the virtual automorphism group is isomorphic to the automorphism group.
Furthermore, their fibre preserving parts are isomorphic. Since these are finite groups and the automorphism group is included in the virtual automorphism group, the map from Definition~\ref{semi-regular-definition} must be an isomorphism.
\end{proof}
Since the virtual automorphism group $V(X_\theta)$ can be expressed by means of the Ellis semigroup $E(X_\theta)$, as we saw above, Corollary~\ref{Vag=Aut} describes the relation between $E(X_\theta)$ and $\Aut(X_\theta)$.
Note that no non-trivial element of $\Aut^{fib}(X_\theta)$ can be an element of $E^{fib}(X_\theta)$ as
$E^{fib}(X_\theta)\backslash \{ \Id\}$ is a proper ideal and therefore cannot contain an invertible element.
\section{Examples}\label{Examples}
We provide here a list of examples of Ellis semigroups of dynamical systems defined by a primitive, aperiodic, bijective substitution $\theta$ of constant length $\ell$ over a finite alphabet $\Aa$. For the benefit of the reader we summarise results and recall calculation of $E(X_\theta)$.
$E(X_\theta)$ is the disjoint union of its kernel $\M(X_\theta)$ with the acting group $\Z$.
It depends only on the $\Rr$-set $\rset\subset S_\Aa$, which can easily be computed as in Lemma \ref{I-is-everything}.
But to describe its associated Rees matrix form we make a choice of minimal idempotent $e\in E(X_\theta,\Z^+)$ which amounts to a choice of element $g_0\in \rset$. Different choices for $e$ lead to isomorphic expressions, and, as far as the fibre preserving parts are concerned, even homeomorphic ones.
The first result is that the structural semigroup $\Sfib$ is isomorphic to the normalised matrix semigroup
$$\Sfib \cong M[\Gstr;I_\theta,\{\pm\};A]$$
where the structure group $\Gstr$ is the group generated by $\rset$, the $+$ entries of $A$ equal 1, and the $-$ entries of $A$ equal $g_0^{-1} g$, $g\in \rset$. They generate a subgroup of $\Gstr$ which we call the little structure group $\Gamma_\theta$.
Everything is finite at this level and so topologically trivial.
Next, the fibre preserving part $\M^{fib}(X_\theta)$ is topologically isomorphic the normalised matrix semigroup
$$\M^{fib}(X_\theta)\cong M[\RSTf;I_\theta,\{\pm\};A]$$
where $\rset$ is the same as above. The quotient $\Gstr/\Glstr$ of $\Gstr$ by the normal completion $\Glstr$ of the little structure group must be a cyclic group, its order $h$ is the {generalised} height of the substitution.
The structure group of $\M^{fib}(X_\theta)$ is $\Z/h\Z$-graded and its subgroup of elements of degree $0$ is
$$\RSTf_0 \cong \Glstr^{\Z_\ell/\Z}.$$
If $\Gstr$ contains an element of order $h$ then
$$\RSTf \cong \Glstr^{\Z_\ell/\Z}\rtimes \Z/h\Z,$$
a semidirect product whose explicit expression depends on the choice of
an element $\mathfrak f$ of $\RSTf$ of degree $1$.
The sandwich matrix $A$ is the same as that for $\Sfib$, because we view $\Glstr$ as a subgroup of $\Glstr^{\Z_\ell/\Z}$: an element $g\in \Glstr$ maps to the function of
$\Glstr^{\Z_\ell/\Z}$ which takes value $g$ on $[0]$ and $\one$ otherwise.
If generalised height is trivial then we can rewrite the above
$$\M^{fib}(X_\theta)\cong \Sfib\times \prod_{[z]\in\Z_\ell/\Z\atop [z]\neq [0]} \Gstr$$
again a topological isomorphism.
Finally, the kernel $\M(X_\theta)$ of the full Ellis semigroup has Rees-matrix form
\begin{equation}\label{eq-Rees-for-M}\nonumber
\M(X_\theta) \cong M[\RST;I_\theta,\{\pm\};A],
\end{equation}
where $\RST$ is an extension of the equicontinuous factor $\Z_\ell$ by $\RSTf$.
This isomorphism is only algebraic. Again $\rset$ and the sandwich matrix $A$ are the same, as $\RSTf$ is a subgroup of $\RST$. The extension is algebraically split if the height is equal to the classical height $h=h_{cl}$ of the substitution. In this case
$$\RST \cong \RSTf \rtimes \Z_\ell,$$
algebraically.
\bigskip
Besides the details for the Ellis semigroup we provide below also $C_\Aa(\Gstr)$, the centraliser of $\Gstr$ in the group of permutations of the alphabet, which is also
$\mbox{\rm Aut}^{fib}(X_\theta)$.
For arbitrary size of the alphabet we can say the following. There is no aperiodic bijective substitution with $|\rset|=1$.
If $\rset$ contains two elements then $\Gamma_\theta$ must be a cyclic subgroup of $\Gstr$.
\subsection{Two-letter alphabet} To be compatible with primitivity and aperiodicity we must have $\rset=S_2$. Hence $\Gstr=S_2=\Z/2\Z$ and $\Gamma_\theta=\Glstr=S_2$. Thus all primitive, aperiodic, bijective substitutions on a two-letter alphabet have the same structural semigroup, namely $M[ S_2; S_2, \{ \pm\}; A]$.
The sandwich matrix is $A=\begin{pmatrix} \one & \one \\ \one & \omega\end{pmatrix}$ where $\omega$ interchanges $a$ with $b$.
The generalised height is trivial for these substitutions. We thus have
$$\RSTf \cong S_2^{\Z_\ell/\Z},\quad \mathrm{and}\quad
\RST \cong S_2^{\Z_\ell/\Z} \rtimes \Z_\ell$$
where $\ell$ is the length of the substitution.
$M[S_2; S_2; \{ \pm\}; A]$ is perhaps the simplest non-orthodox semigroup.
Since $S_2$ is abelian, we have $C_\Aa(\Gstr)= \Gstr$. Thus all these substitutions have $\mbox{\rm Aut}^{fib}(X_\theta) = S_2$, generated by the map $\omega$. The simplest example of this type is the (simplified) Thue-Morse substitution,
$\theta(a) = abba$, $\theta(b) = baab$, where the above result has been obtained by Marcy Barge in a direct calculation \cite{Barge}.
\subsection{Three-letter alphabet}\label{ex462}
If $\Gstr$ is a subgroup of $S_2\subset S_3$ then we reproduce the above results for the semigroup, but these can never be realised by a primitive substitution on three letters, as one letter would stay fixed. So we consider the two possible other cases, $\Gstr=S_3$ and $\Gstr=A_3\cong \Z/3\Z$. For
$\Gstr=S_3$, we give below examples where $\Gamma_\theta \cong \Z/2\Z$ or $\Gamma_\theta\cong\Z/3\Z$. In the first case, $\Gamma_\theta \cong \Z/2\Z$ has normal completion $\Gstr$ and so the height is trivial. In the second case $\Gamma_\theta\cong\Z/3\Z$ is normal in $\Gstr$ and the height equal to $2$. The example we provide for this case has classical height $1$. We also give an example where $\Gstr\cong \Z/3\Z$, which, for aperiodic $\theta$, forces $\Gamma_\theta\cong\Z/3\Z$ so that we have again trivial height.
\begin{enumerate}
\item
Consider the substitution $\theta$
\[
\begin{array}{c} a\\ b\\ c \end{array}
\mapsto
\begin{array}{c} a\\ b\\ c \end{array}
\!\!\!\!\!\!{\begin{array}{c} b\\ a\\ c \end{array}}
\!\!\!\!\!\!{\begin{array}{c} c\\ b\\ a \end{array}}
\!\!\!\!\!\!{\begin{array}{c} c\\ a\\ b \end{array}}
\!\!\!\!\!\!{\begin{array}{c} a\\ b\\ c \end{array}}
\]
Then it can be verified that $I_{\theta}=\left\{ \begin{pmatrix} b\\a\\c \end{pmatrix}, \begin{pmatrix} b\\c\\a \end{pmatrix} \right\}$ which generates $\Gstr = S_3$.
The structural semigroup is $M[S_3; I_\theta, \{ \pm\}; A]$, whose normalised sandwich matrix $A=\begin{pmatrix} \one & \one \\
\one &\omega \end{pmatrix}$, where $\omega$ exchanges $b$ with $c$.
One finds $\Gamma_\theta=\left\{ \one, \begin{pmatrix} c\\b\\a \end{pmatrix} \right\}\cong \Z/2\Z$, which is not normal in $S_3$, and $ \Glstr = G_\theta= S_3$. Thus $\theta$ has trivial generalised height. Hence
$$\RSTf \cong S_3^{\Z_5/\Z},\quad \mathrm{and}\quad
\RST \cong S_3^{\Z_5/\Z} \rtimes \Z_5$$
Also $C_\Aa(\Gstr) = \mbox{\rm Aut}^{fib}(X_\theta)$ is trivial.
\item
Consider the substitution $\theta$
\[
\begin{array}{c} a\\ b\\ c \end{array}
\mapsto
\begin{array}{c} a\\ b\\ c \end{array}
\!\!\!\!\!\!{\begin{array}{c} b\\ a\\ c \end{array}}
\!\!\!\!\!\!{\begin{array}{c} a\\ b\\ c \end{array}}
\!\!\!\!\!\!{\begin{array}{c} c\\ b\\ a \end{array}}
\!\!\!\!\!\!{\begin{array}{c} a\\ b\\ c \end{array}}
\!\!\!\!\!\!{\begin{array}{c} a\\ c\\ b \end{array}}
\!\!\!\!\!\!{\begin{array}{c} a\\ b\\ c \end{array}}
\]
It has $\theta_0=\theta_2=\theta_4=\theta_6 = \id$ and the other three are the transpositions of $S_3$,
$\theta_1 = \begin{pmatrix} b\\a\\c\end{pmatrix}$,
$\theta_3 = \begin{pmatrix}c\\b\\a \end{pmatrix}$, and
$\theta_5 =\begin{pmatrix}a\\c\\b \end{pmatrix}$.
Hence $\rset = \{\theta_1,\theta_3,\theta_5\}$ and $\Gstr=S_3$. The structural semigroup is $M[S_3; I_\theta, \{ \pm\}; A]$ has normalised sandwich matrix
$\begin{pmatrix} \one & \one & \one \\
\one &\omega&\omega^2 \end{pmatrix}$, where
$\omega = \begin{pmatrix} b\\c\\a\end{pmatrix}$, a cyclic permutation.
Every element in $\Gamma_\theta$ is an even permutation, thus $\Gamma_\theta =
\Glstr=A_3$. It follows that $\Gstr/\Gamma_\theta \cong \Z/2\Z$ and $\theta$ has generalised height equal to 2 and therefore
$$\RSTf_0 \cong {A_3}^{\Z_7/\Z},\quad \RSTf \cong {A_3}^{\Z_7/\Z}\rtimes \Z/2\Z,$$
as $\theta_1\in \rset$ has order $2$.
Note that the substitution has trivial classical height as any fixed point must contain the word $aa$. We do therefore not know whether the extension ${A_3}^{\Z_7/\Z}\rtimes \Z/2\Z\hookrightarrow \RST \stackrel{\tilde\pi}\twoheadrightarrow \Z_7$ defining the structure group $\RST$ of $\M(X_\theta)$ splits.
Again, $C_\Aa(\Gstr)=\mbox{\rm Aut}^{fib}(X_\theta)$ is trivial.
This example has a natural generalisation to alphabets of any size $s$ with $\Gstr=S_s$ and $\Gamma_\theta=A_s$, the alternating group on $s$ elements.
\item
Consider the substitution $\theta$
$$
\begin{array}{c c l}
a & & abc \\
b & \mapsto & bca \\
c & & cab
\end{array}
$$
whose third power is simplified. We find $\rset = \{\one,\omega,\omega^{2}\}\cong \Z/3\Z$ where $\omega =\begin{pmatrix} b\\c\\a \end{pmatrix}$
is a cyclic permutation. It follows that $\Gstr=\Gamma_\theta=\Glstr\cong\Z/3\Z$. The structural semigroup is $M[\Z/3\Z; I_\theta, \{ \pm\}; A]$ where
$A= \begin{pmatrix} \one & \one & \one \\
\one&\omega&\omega^{2} \end{pmatrix}$.
$$\RSTf \cong (\Z/3\Z)^{\Z_{3}/\Z},\quad \mathrm{and}\quad
\RST \cong (\Z/3\Z)^{\Z_{3}/\Z} \rtimes \Z_3.$$
Finally, $C_\Aa(\Gstr)=\mbox{\rm Aut}^{fib}(X_\theta)\cong\Z/3\Z$, generated by $\omega$.
There is an obvious generalisation of this example to alphabets of any size $s\geq 2$, the case $s=2$ corresponding again to the Thue-Morse substitution.
\end{enumerate}
\subsection{Four-letter alphabet}
Our last example is related to the dihedral group $D_4$. Consider the substitution
$\theta$ of length $7$
$$
\begin{array}{c c l}
a & & abadcba \\
b & \mapsto & badcbab \\
c & & cdcbadc\\
d & & dcbadcd
\end{array}
$$
which has classical height 2. If we identify the letters with the edges of a square whose center is $0$ in such a way that
$a$ corresponds to the lower right corner and we order the edges counterclockwise, then
$\theta_1$ and $\theta_3$ amount to the reflection along the $x$-axis and the $y$-axis, resp., while $\theta_2$ and $\theta_4$ amount to the reflection along the diagonal with slope $-1$ and $+1$, resp.. Finally $\theta_5=\theta_1$ and $\theta_6=\theta_0=\one$.
Thus $\rset = \{\theta_1,\rho\}$ where $\rho=\theta_1\theta_2$ is the rotation by $\frac\pi2$ to the right. It follows that $G_\theta = D_4$ is the dihedral group of order $4$ and that $\Gamma_\theta$ the group of order $2$ generated by the reflection $\theta_2$. Its normal completion is thus the group generated by the reflections $\theta_2$ and $\theta_4$, which commute, so $\Glstr\cong \Z/2\Z\times\Z/2\Z$ showing that height is equal to the classical height, namely 2. Moreover, the element $\theta_1$ of $\rset$ has order $2$.
The structural semigroup is
$\Sfib=M[D_4; I_\theta, \{ \pm\}; A]$ with $A=\begin{pmatrix} \one & \one \\
\one &\theta_4 \end{pmatrix}$.
Furthermore,
$$\RSTf \cong (\Z/2\Z\times\Z/2\Z)^{\Z_7/\Z}\rtimes\Z/2\Z,\quad \mathrm{and}\quad
\RST \cong (\Z/2\Z\times\Z/2\Z)^{\Z_7/\Z}\rtimes\Z/2\Z \rtimes \Z_7$$
Finally
$C_\Aa(\Gstr) = \mbox{\rm Aut}^{fib}(X_\theta)= \{ \id,\rho^2 \}$.
\section*{Acknowledgement}
The authors are very grateful to the referee, whose numerous suggestions and questions led not only to an improvement in the exposition, but also to an extension of the results, namely the recovering of the full Ellis semigroup from its fibre preserving part in
Corollary~\ref{cor:reproducing M}, and Theorems~\ref{thm-main2} and
\ref{thm-main4}.
{\footnotesize
\bibliographystyle{abbrv}
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by1pt\rlap{\hbox to\wd0{\hss\raise\dimen0
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Meetings start at 4:00 p.m.
Meetings are held on the 4th Thursday of the month, at 4 p.m. at the North Texas Municipal Water District offices in Wylie, Texas. Call 972-442-5405 for more information on meeting dates and times.
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263,737
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Time Preparing: 20 mins
Time Cooking: 50 mins
Makes: 12-18
Difficulty: 3/5
Season: Summer, Autumn & Winter
Ingredients
– 350g Beetroot
– 150ml Sunflower Oil
– 150g Self Raising Flour
– 250g Golden Caster Sugar
– 4 Eggs
– 200g White Chocolate
– 40g Cocoa Powder
– 3tsp Vanilla Essence
– 1/2tsp Salt
– 1tsp Baking Powder
– 150g Butter
– 300g Icing Sugar
– 1tbsp Milk
Method
1. Preheat oven to 180 degrees
2. Wrap beetroot in kitchen foil, then cook for 35 minutes. Once cooked, peel skin off beetroot
3. Blend the beetroot, 2tsp vanilla essence and sunflower oil until smooth
4. Sift flour, cocoa powder, baking powder and salt into and bowl. In a seperate bowl, whisk caster sugar and eggs together
5. Whisk the beetroot mix into the sugar and eggs until smooth, then fold into the flour mix
6. Break white chocolate into small chunks and add them to the mix
7. Carefully spoon into cupcake holders, then cook in the middle of the oven for 15 minutes
8. To make the icing, mix together the remaining ingredients, and spoon over cupcakes once they’re cool
| 342,796
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TITLE: When are particles distinguishable?
QUESTION [0 upvotes]: I'm just revisting some basics from statisitical mechanics for an exam. One of the exercises asks the reader to calculate the canoncial partition functions of $N$ harmonic oscillators. How should I know wheather these oscillators are distinguishable or not? Because the result will be different and arguably harder to obtain if they're not. The solution distinguishes between the oscillators, so that for e.g the Eigenvalue $E=\hbar\omega(2+1/2)$ is degenerated (e.g $(2,0...,0)\neq(0,2,0...0)$).
Should I just regard all physical particles (e.g electrons) to be indistinguishable and just take the exercisese with distinguishable particles as artificial questions? In some cases distinguishing between particles leads to non-extensive entropys after all.
REPLY [0 votes]: Particles (specifically bosons) are distinguishable if any of their degrees of freedom are different. The degrees of freedom include the spatial and temporal degrees of freedom (e.g. spatial and temporal frequencies), and their internal degrees of freedom (spin). For identical (indistinguishable) particles (bosons), all these degrees of freedom must be identical.
Fermions (such as electrons) are always distinguishable. The Pauli exclusion principle (which follows from Fermi-Dirac statistics) does not allow two fermions to be identical.
| 180,780
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\begin{document}
\begin{frontmatter}
\title{Fluid limits for overloaded multiclass FIFO~single-server
queues with general abandonment}
\runtitle{Overloaded FIFO queues with general abandonment}
\author[a]{\fnms{Otis B.} \snm{Jennings}\ead[label=e1]{otisj@alumni.princeton.edu}}
\and
\author[b]{\fnms{Amber L.} \snm{Puha}\corref{}\ead[label=e2]{apuha@csusm.edu}\thanksref{t2}}
\thankstext{t2}{Research supported in part by UCLA-IPAM}
\address[a]{Otis B.\ Jennings\\
323 E.\ Chapel Hill St\\
Unit 1571\\
Durham NC, 27702-2469\\
\printead{e1}}
\address[b]{Amber L.\ Puha\\
Department of Mathematics\\
California State University San Marcos\\
333 S.\ Twin Oaks Valley Road\\
San Marcos, CA 92096-0001\\
\printead{e2}}
\affiliation{California State University San Marcos}
\runauthor{O.B. Jennings and A.L. Puha}
\begin{abstract}
We consider an overloaded multiclass nonidling first-in-first-out
single-server queue with abandonment. The interarrival times, service
times, and deadline times are sequences of independent and identically,
but generally distributed random variables. In prior work, Jennings and
Reed studied the workload process associated with this queue. Under
mild conditions, they establish both a functional law of large numbers
and a functional central limit theorem for this process. We build on that
work here. For this, we consider a more detailed description of the
system state given by $K$ finite, nonnegative Borel measures on the
nonnegative quadrant, one for each job class. For each time and job
class, the associated measure has a unit atom associated with each job
of that class in the system at the coordinates determined by what are
referred to as the residual virtual sojourn time and residual patience
time of that job. Under mild conditions, we prove a functional law of
large numbers for this measure-valued state descriptor. This yields
approximations for related processes such as the queue lengths and
abandoning queue lengths. An interesting characteristic of these
approximations is that they depend on the deadline distributions in
their entirety.
\end{abstract}
\begin{keyword}[class=AMS]
\kwd[Primary ]{60B12}
\kwd{60F17}
\kwd{60K25}
\kwd[; secondary ]{68M20}
\kwd{90B22}.
\end{keyword}
\begin{keyword}
\kwd{Overloaded queue}
\kwd{abandonment}
\kwd{first-in-first-out}
\kwd{multiclass queue}
\kwd{measure-valued state descriptor}
\kwd{queue-length vector}
\kwd{fluid limits}
\kwd{fluid model}
\kwd{invariant states}.
\end{keyword}
\received{\smonth{10} \syear{2012}}
\end{frontmatter}
\section{Introduction}
Here we consider a single-server queue fed by $K$ arrival streams, each
corresponding to
a distinct job class. Upon arrival, each job declares its service time
and deadline requirements.
If a job doesn't enter service within the deadline time of its arrival,
it abandons the queue before
initiating service. Otherwise it remains in queue until it receives its
full service
time requirement. Nonabandoning jobs are served in a nonidling,
first-in-first-out (FIFO)
fashion. The queue is assumed to be overloaded, i.e., the offered load
exceeds one. In addition, it is assumed that the interarrival times,
service times,
and deadline times are mutually independent sequences of independent and
identically, but generally distributed random variables. One aim of
this work is to provide
approximations for functionals such as the queue-length process.
Queueing systems with abandonment are observed in many applications.
Hence, it is a natural phenomenon to
study. For FIFO single-server queues with abandonment, the earliest
analysis focused on models with exponential
abandonment~\cite{ref:AG}, which is not an ideal modeling assumption.
Thereafter, general abandonment
distributions were considered in~\cite{ref:BH}, which restricted to
Markovian service and arrival processes. Shortly thereafter,
stability conditions were determined in~\cite{ref:BBH} without the
Markovian restriction. In the last decade, there has
been considerable progress with analyzing the critically loaded FIFO
single-server queue with general abandonment
\cite{ref:GlW, ref:RW}.
Here we focus on general abandonment for the multiclass overloaded FIFO
single-server model described above.
In~\cite{ref:JR}, Jennings and Reed study the workload process
associated with this
queue under the assumption that the abandonment distributions are
continuous. Note that
the workload process is well defined since each job declares its
deadline upon arrival. In
particular, the value of the workload process is increased by a job's
service time at its arrival
time if and only if the value of the workload process immediately
before the job's arrival is strictly less
than the job's deadline time. As usual, the value of the workload
decreases at
rate one while the workload is positive. Under mild conditions, they
establish both a functional
law of large numbers and a functional central limit theorem for this process.
In this paper, we build on that work. One outcome
is to provide a fluid approximation for the vector-valued
queue-length process. Because the abandonment distributions are not necessarily
exponential, the queue-length vector together with the vector of
residual interarrival times,
class in service, and residual service time does not provide a
Markovian description of the
system state. Additional information about the residual deadline times
is also needed.
We track this information using a measure-valued state descriptor,
which is defined precisely in Section~\ref{sec:sm}. We give an informal
description here.
The state $\z(t)$ of the
system at time $t$ consists of $K$ finite, nonnegative Borel measures
on $\Rp^2$, one for each
class, where $\Rp$ denotes the nonnegative real numbers. Each measure
consists entirely of
unit atoms, one corresponding to each job of that class in the system.
The coordinates of each
atom are determine by two quantities associated with the job. The first
coordinate is the residual virtual sojourn time of that job, which is the
amount of work in the system
(the cumulative residual service times of jobs that don't abandon)
associated with this job and the jobs that arrived ahead of it.
The second coordinate is the residual patience time. This is given by
the job's deadline minus the time in system if it will abandon before
entering service, and is given by
the job's deadline plus its service time minus the time in system
otherwise. Note that the atoms associated
with jobs that will eventually be served
are initially located in $\Rp^2$ above the line of slope one
intersecting the origin, while the atoms
associated with
jobs that won't be served are initially placed on or below this line.
With this description, new jobs
arrive and the corresponding unit atoms are added at the appropriate
coordinates of the system state.
Further, each coordinate of each atom decreases at rate one until the
unit atom reaches one of the
coordinate axes and exits the system. Jobs associated with atoms that
hit the vertical coordinate
axis exit due to service completion. Jobs associated with atoms that
ultimately hit the horizontal
coordinate axis exit due to abandonment (see Figure~\ref{fig:state}).
Our choice of state descriptor is reminiscent of the one used by
Gromoll, Robert, and Zwart~\cite{ref:GRZ}
to analyze an overloaded processor sharing (PS) queue with abandonment.
One distinction is that their work
is for a single-class queue, and therefore has one coordinate rather
than $K$. Another is that the first coordinate
for the unit atom associated with a given job in the PS system is
simply the job's residual service time. We use the residual virtual
sojourn time to determine the first coordinate since it captures the
order of arrivals, which is needed for FIFO.
The evolution of the state descriptor in~\cite{ref:GRZ} is slightly
more complicated than the one used here.
In both cases, the residual patience times decrease at rate one, but in
\cite{ref:GRZ} the service times decrease at rate one
over the number of jobs in the queue. Hence, the nice relationship with
the line of slope one intersecting the origin
present in the FIFO model is not present in the PS model. However, in
both models it is true that hitting a coordinate
axis corresponds to exiting the system with the vertical axis being
associated with service completion and the horizontal
axis being associated with abandonment. So this work provides an
example of how the modeling framework
developed for analyzing PS with abandonment can be adapted to yield an
analysis in another abandonment setting.
Measure valued processes have been used rather extensively for modeling
many server queues with and without abandonment
(see~\cite{ref:KangPang,ref:KangR2010,ref:KangR2012,ref:KaspiR},
and~\cite{ref:PW}).
An important distinction between single-server and many server
first-come-first-serve queues is that the later is not
first-in-first-out. Therefore the measure valued descriptor and
analysis given here is quite different from that found in the many
server queues literature.
We develop a fluid approximation for this measure-valued state
descriptor, which yields
fluid approximations for the queue-length vector and other functionals
of interest.
Similarly to the approximations derived in~\cite{ref:JR} and~\cite{ref:W2006},
these approximations depend on the entire abandonment distribution of
each job class.
We begin by introducing an associated fluid model, which can be viewed
as a formal law of large numbers
limit of the stochastic system. Hence fluid model solutions are
$K$-dimensional measure-valued
functions with each coordinate taking values the space of finite,
nonnegative Borel measures
on $\Rp^2$ that satisfy an appropriate fluid model equation (see
\eqref{fdevoeq2}). We analyze the behavior of fluid
model solutions. Through this analysis we identify a nonlinear mapping
from the fluid workload
to the fluid queue-length vector (see \eqref{eq:fq}). Using the nature
of fluid model solutions,
this mapping is refined to yield the approximations for the number of
jobs of each class in queue
that will and will not abandon (see \eqref{eq:fn} and \eqref{eq:fa}).
In a similar spirit,
we obtain approximations for the number of jobs of each class in system
of a certain age or older
(see \eqref{eq:age1} and \eqref{eq:age2}). In addition, we characterize
the invariant states for this fluid model
(Theorem~\ref{thrm:is}).
Next we justify interpreting fluid model solutions and functionals
derived from them
as first order approximations of prelimit functionals in the stochastic
model by proving a fluid limit theorem
(Theorem~\ref{thrm:flt}). It states that under mild assumptions the
fluid scaled state
descriptors for the stochastic system converge in distribution to
measure-valued functions
that are almost surely fluid model solutions.
A basic element of our fluid limit result is the nature of the scaling employed.
Consistent with the framework in~\cite{ref:JR}, we accelerate both the
arrival process and the service rates, while leaving the abandonment
times unchanged.
One should think of a concomitant speeding up of the server's
processing rate to accommodate the increased customer demand; the
content of the work for any particular customer is the same.
We assume any given customer is unaffected by the increase in the
number of fellow customers, but rather it is the time
in queue that triggers abandonment.
Hence, abandonment propensity does not require adjusting as the other
system parameters increase.
The paper is organized as follows. We conclude this section with a
listing of our notation.
The formal stochastic and fluid models are given Section~\ref{sec:m}.
Then, in Section~\ref{sec:main}, we present the main results (Theorems
\ref{thrm:eu},~\ref{thrm:flt} and~\ref{thrm:is})
and several approximations derived from our fluid model. Theorems~\ref
{thrm:eu} and~\ref{thrm:is} are proved
in Section~\ref{sec:prfthrmeu}.
In the final two sections, we provide the proof
of Theorem~\ref{thrm:flt}. The proof of tightness is presented in
Section~\ref{sec:tightness}, while the
characterization of fluid limit points as fluid model solutions is
presented in Section~\ref{sec:char}.
For this, we prove a functional law of large numbers for a sequence of
measure-valued processes
that we refer to as residual deadline related processes (see Lemma \ref
{lem:ResDeadFLLN}).
Since the residual deadline related processes don't include information
about the service discipline,
the result in Lemma~\ref{lem:ResDeadFLLN} may be of more general
interest for analyzing queues with abandonment.
\subsection{Notation}
The following notation will be used throughout the paper.
Let $\N$ denote the set of strictly positive integers and
let $\R$ denote the set of real numbers.
For $x, y\in\R$, $x\vee y=\max(x, y)$, $x\wedge y=\min(x, y)$, and
$x^+=x\vee0$.
For $x\in\R$, $\lfloor x\rfloor$ denotes the largest integer less than
or equal to $x$ and
$\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$.
For $x\in\R$, $\Vert x \Vert=\vert x\vert$ and for $x\in\R^2$,
$\Vert
x \Vert=\sqrt{x_1^2+x_2^2}$.
The nonnegative real numbers $[0, \infty)$ will be denoted by $\Rp$.
Let $\CB_1$ denote the Borel subsets of $\Rp$
and $\CB_2$ denote the Borel subsets of $\Rp^2$.
On occasion we will use the notation $\Rp^1$ in place of $\Rp$.
For a Borel set $B\in\CB_i$, $i=1, 2$, we denote the indicator function
of the set $B$ by $1_{B}$.
For $i=1, 2$, $B\in\CB_i$ and $x\in\Rp^i$, let $B_x$ denote the
$x$-shift of the set $B$, which is given by
\begin{equation}\label{def:xshift}
B_x=\{ y \in\Rp^i : y-x\in B\}.
\end{equation}
Given $x\in\Rp$ and $B\in\CB_2$, we adopt the following shorthand notation:
\[
B_x=B_{(x, x)}.
\]
For $i=1, 2$, $B\in\CB_i$, and $\kappa>0$,
let $B^{\kappa}$ denote the $\kappa$-enlargement of $B$, which is
given by
\begin{equation}\label{def:kappa}
B^{\kappa}=\left\{ x\in\Rp^i : \inf_{y\in B} \Vert x-y\Vert
<\kappa\right
\}.
\end{equation}
Notice that given $i=1, 2$ and $B\in\CB_i$, $(B^{\kappa})_x\subseteq
(B_x)^{\kappa}$,
but $(B^{\kappa})_x$ and $(B_x)^{\kappa}$ are not necessarily the
same set.
In particular, the set resulting from shifting before enlarging may
contain additional points.
We adopt the convention that $B_x^{\kappa}=(B_x)^{\kappa}$, i.e.,
$B_x^{\kappa}$
is the larger of the two sets.
For $i=1, 2$, let $\C_b(\Rp^i)$ denote the set of bounded continuous
functions from $\Rp^i$ to $\R$. Given a finite, nonnegative Borel
measure $\zeta$ on $\Rp^i$, let $\bL(\zeta)$ denote the set of Borel
measurable functions from $\Rp^i$ to $\R$ that are integrable
with respect to $\zeta$. For $g\in\bL(\zeta)$, we define
$\langle g, \zeta\rangle =\int_{{\mathbb R}_+^i}g \der\zeta$ and
adopt the shorthand notation $\zeta(B)=\langle 1_B, \zeta\rangle $ for
$B\in
\CB_i$.
In addition, let $\chi:\Rp\to\R$ be the identify map $\chi(x)=x$ for
all $x\in\Rp$.
Given a finite, nonnegative Borel measure $\zeta$ on $\Rp$, if $\chi
\in
\bL(\zeta)$,
i.e., if $\langle \chi, \zeta\rangle <\infty$, we say that $\zeta$
has a finite
first moment.
For $i=1, 2$, let $\M_i$ denote the set of finite, nonnegative Borel
measures on
$\Rp^i$. The zero measure in $\M_i$ is denoted by ${\bf0}$. For
$x\in
\Rp$,
$\delta_x\in\M_1$ is the measure that puts one unit of mass
at $x$. Similarly, for $x, y\in\Rp$, $\delta_{(x, y)}\in\M_2$ is the
measure that puts one unit of mass at $(x, y)$.
The set $\M_i$ is endowed with the weak topology, that is, for
$\zeta^n, \zeta\in\M_i$, $n\in\N$, we have $\zeta^n\wk\zeta$ as
$n\rightarrow\infty$ if and only if $\langle g, \zeta^n\rangle\rightarrow \langle g, \zeta\rangle $
as $n\rightarrow\infty$, for all $g\in\C_b(\Rp^i)$.
With this topology, $\M_i$ is a Polish space~\cite{ref:P}. Also define
\[
\M_i^K=\{ (\zeta_1, \dots, \zeta_K) : \zeta_k\in\M_i\hbox{ for
}1\le k\le
K\}.
\]
Then $\M_i^K$ endowed with the product topology is also a Polish space.
Given $\zeta\in\M_i^K$ and $g\in\cap_{k=1}^K\bL(\zeta_k)$, we define
the shorthand notation
\[
\left\langle g, \zeta\right\rangle =(\left\langle g, \zeta_1\right\rangle , \dots, \left\langle g,
\zeta_K\right\rangle ).
\]
Given $i=1, 2$ and $\zeta\in\M_i^K$, it will be handy to introduce the
notation $\zeta_+\in\M_i$
for the superposition measure, which is given by
\begin{equation}\label{def:sp}
\zeta_+(B)=
\sum_{k=1}^K \zeta_k(B), \qquad\hbox{for all }B\in\CB_i.
\end{equation}
Let $\M_{1, 0}$ denote the subset of $\M_1$ containing those measures
that assign zero
measure to the set $\{0\}$. Similarly, let $\M_{2, 0}$ denote the subset
of $\M_2$ containing
those measures that assign measure zero to the set
\begin{equation}\label{def:C}
C=\Rp\times\{0\}\cup\{0\}\times\Rp.
\end{equation}
For $x, y\in\Rp$, let $\delta_x^+\in\M_{1, 0}$ and $\delta
_{(x, y)}^+\in\M
_{2, 0}$ respectively denote the measures that put a unit mass at the
point $x$ if $x>0$ and $(x, y)$ if $x, y>0$, and are
the zero measure otherwise.
For $i=1, 2$, the set $\M_{i, 0}^K$ is
defined analogously to $\M_i^K$ except that $\zeta_k\in\M_{i, 0}$ for
$1\le k\le K$.
In addition, let $\CB_{1, 0}$ denote those sets in $\CB_1$
that do not contain zero.
Similarly, let $\CB_{2, 0}$ denote those sets in $\CB_2$
that do not meet the set $C$.
Finally, we will use ``$\Rightarrow$'' to denote convergence in
distribution of random
elements of a metric space. Following Billingsley~\cite{ref:B}, we will
use $\bP$ and $\E$
respectively to denote the probability measure and expectation operator
associated with
whatever space the relevant random element is defined on. All
stochastic processes used
in this paper will be assumed to have paths that are right continuous
with finite left limits (r.c.l.l.).
For a Polish space $\mathcal{S}$, we denote by $\bD([0, \infty),
\mathcal{S})$
the space of r.c.l.l.\ functions
from $[0, \infty)$ into~$\mathcal{S}$, and we endow this space with the
usual Skorohod $J_1$-topology
(cf.~\cite{ref:EK}). There are six Polish spaces that will be
considered in this paper: $\R$, $\Rp$,
$\M_1$, $\M_1^K$, $\M_2$, and $\M_2^K$.
\section{The Stochastic and Fluid Models}\label{sec:m}
\subsection{The Stochastic Model}\label{sec:sm}
In this section, we define the model of the GI/GI/1
+ GI queue serving $K$ distinct customer classes, which
will be used for the remainder of the paper.
\paragraph{Initial condition and associated system dynamics}
The initial condition specifies the number $Z_+(0)$ of jobs in the
queue at time zero, as well as the initial virtual sojourn time, initial
patience time and class of each job. Assume that $Z_+(0)$ is
nonnegative integer-valued random variable that is finite
almost surely. The initial virtual sojourn times, initial
patience times
and classes are the first, second, and third coordinates respectively
of the first $Z_+(0)$ elements of the random sequence
$\{(\tilde w_j, \tilde p_j, k_j)\}_{j\in\N}\subset\Rp\times\Rp
\times\{
1, \dots, K\}$. For $1\le j\le Z_+(0)$,
the initial job with initial virtual sojourn time $\tilde w_j$, initial
patience time $\tilde p_j$, and class $k_j$ is called job $j$.
We assume that the elements of the sequence $\{\tilde w_j\}_{j\in\N}$
are finite, positive, and
nondecreasing, which reflects the fact that the service discipline is
first-in-first-out. In particular, the job with the smallest index is
regarded as having arrived to the system before all
other jobs, and therefore has the smallest initial virtual sojourn time.
This is the job currently in service and $\tilde w_1$ represents the time
until its service is completed.
The remaining jobs are waiting in the
queue in order. For $2\le j \le Z_+(0)$, $\tilde w_j$ represents the
amount of time that job $j$
will remain in the system, provided that it doesn't abandon before
entering service.
We assume that the elements of the sequence $\{\tilde p_j\}_{j\in\N}$ are
finite and positive, which reflects the fact that none of the jobs
would be
regarded as having abandoned the queue by time zero.
For $1\le j\le Z_+(0)$ such that $\tilde p_j>\tilde w_j$,
job $j$ is sufficiently patient to wait in the queue until service completion.
For $1\le j\le Z_+(0)$ such that $\tilde p_j\le\tilde w_j$,
job $j$ abandons the queue at time $\tilde p_j$ before entering service.
We assume that $\tilde w_1<\tilde p_1$, which reflects the fact that
job 1
is presently in service and is therefore patient enough to stay in the queue
until service completion. For $2\le j\le Z_+(0)$, we assume that
$\tilde p_j\le\tilde w_j$ if and only if $\tilde w_j=\tilde w_{j-1}$.
When $j$ is such that $\tilde p_j>\tilde w_j$,
one regards $\tilde w_j-\tilde w_{j-1}$ as the service time of job $j$
and this
service time is included in the initial virtual sojourn time of all
jobs $\ell$ such that $j\le\ell\le Z_+(0)$. When $j$ is such that
$\tilde p_j\le\tilde w_j$, a service time for job $j$ is not included
in any of the initial virtual sojourn times.
Since the queue is nonidling and first-in-first-out, all $Z_+(0)$ jobs
in the system at time zero will either abandon or be served by time
$W(0)=\tilde w_{Z_+(0)}$. We
refer to $W(0)$ as the initial workload.\vadjust{\eject}
A convenient way to express the initial condition is to define an
initial random measure $\z(0)\in\M_2^K$. For this, we will find it
convenient to separate the sequence $\{ (\tilde w_j, \tilde p_j, k_j)\}
_{j=1}^{Z_+(0)}$ into $K$ separate
sequences $\{ (\tilde w_{k, j}, \tilde p_{k, j})\}_{j=1}^{Z_k(0)}$, one
for each class. For $1\le k\le K$,
let $Z_k(0)$ denote the number of class $k$ initial jobs. Given
$1\le k \le K$, for $1\le j\le Z_k(0)$, let $i(k, j)$ be the $j$th
smallest index
such that $k_{i(k, j)}=k$ and set $\tilde w_{k, j}=\tilde w_{i(k, j)}$ and
$\tilde p_{k, j}=\tilde p_{i(k, j)}$.
Then, for $1\le k\le K$, let $\z_k(0)\in\M_2$ be given by
\[
\z_k(0)=\sum_{j=1}^{Z_k(0)} \delta_{(\tilde w_{k, j}, \tilde p_{k, j})}^+,
\]
which equals $\mathbf{0}$ if $Z_k(0)=0$. Then let
\[
\z(0)=(\z_1(0), \dots, \z_K(0)).
\]
Our assumptions imply that
$\z(0)$ satisfies
\begin{equation}\label{eq:InitialConditionFinite}
\mathbf{P}\left(\max_{1\le k\le K}\left\langle 1, \z_k(0)\right\rangle \vee
W(0)<\infty
\right)=1.
\end{equation}
Furthermore,
\[
\mathbf{P}\left(\max_{1\le k\le K}\z_k(0)(C)=0\right)=1.
\]
In particular, $\z(0)\in\M_{2, 0}^K$ almost surely.
\paragraph{Stochastic primitives and associated system dynamics}
The stochastic primitives consist of $K$ exogenous arrival processes
$E_k(\cdot)$, $1\le k\le K$, $K$ sequences of service times $\{
v_{k, i}\}
_{i\in\N}$, $1\le k\le K$,
and $K$ sequences of deadlines $\{d_{k, i}\}_{i\in\N}$, $1\le k\le K$.
We assume that the exogenous arrival processes, the
sequences of service times, and the sequences of deadlines are all
independent of one another.
For a given $1\le k\le K$, the class $k$ arrival process $E_k(\cdot)$
is a rate
$\lambda_k\in(0, \infty)$ renewal process. For $t\in\ptime$, $E_k(t)$
represents the number of class $k$ jobs that arrive to the queue
during the time interval $(0, t]$. We assume that the interarrival times
are strictly positive and denote the sequence of interarrival
times by $\{ \xi_{k, i}\}_{i\in\N}$. Class $k$ jobs arriving after
time zero
are indexed by integers $j>Z_k(0)$. For $t\in\ptime$, let
\begin{equation}\label{def:A}
A_k(t)=Z_k(0)+E_k(t).
\end{equation}
Then class $k$ job $j$ arrives at time $t_{k, j}=\inf\{t\in\ptime
: A_k(t)\ge j\}$.
Hence, for $j^\prime<j$, $t_{k, j^\prime}\le t_{k, j}$ and we say that
class $k$ job $j^\prime$
arrives before class $k$ job $j$. The inequality is strict for indices
$j>Z_k(0)$. Moreover, for each $j\le Z_k(0)$, $t_{k, j} = 0$.
Given $1\le k\le K$, for each $i\in\N$, the random variable $v_{k, i}$
represents the service time of the $(Z_k(0)+i)$th class $k$ job. That is,
class $k$ job $j>Z_k(0)$ has service time $v_{k, j-Z_k(0)}$. Assume that
the random variables $\{v_{k, i}\}_{i\in\N}$ are strictly positive and
form an independent and identically distributed sequence with finite
positive mean
$1/\mu_k$. Define the class $k$ offered load to be $\rho_k=\lambda
_k/\mu_k$.
We assume that the queue is overloaded. In particular, we assume that
$\rho=\sum_{k=1}^K\rho_k>1$.
Given $1\le k\le K$, for each $i\in\N$, the random variable $d_{k, i}$
represents the deadline of the $(Z_k(0)+i)$th class $k$ job. That is,
class $k$
job $j>Z_k(0)$ has deadline $d_{k, j-Z_k(0)}$. Assume that the random
variables $\{d_{k, i}\}_{i\in\N}$ are strictly positive and form an
independent
and identically distributed sequence of random variables with common
continuous distribution $\Gamma_k$. Assume that the mean $1/\gamma_k$
is finite. Let $F_k(\cdot)$ denote the cumulative distribution function
associated
with $\Gamma_k$, i.e., $F_k(x)=\langle 1_{[0, x]}, \Gamma_k\rangle $
for all $x\in\Rp$.
Denote its complement by $G_k(\cdot)=1-F_k(\cdot)$.
As is the case for jobs in the system at time zero, jobs arriving after
time zero are served in a first-in-first-out, nonidling fashion.
A class $k$ job $j$ arriving to the system after time zero immediately
enters service if the server is available. Otherwise, for class $k$ job $j$
to be served, it must wait until all other jobs currently in the queue
exit via
service completion or abandonment. If class $k$ job $j$ has not entered
service before time $t_{k, j}+d_{k, j-Z_k(0)}$, class $k$ job $j$
abandons the
queue at this time. Otherwise, when class $k$ job $j$ enters service,
it is
served for $v_{k, j-Z_k(0)}$ time units.
It will be convenient to combine the exogenous arrival
process and deadlines into a single measure-valued deadline process.
\begin{definition}\label{def:mvdp} For $1\le k\le K$, the class $k$ deadline
process is given by
\[
\dead_k(t)=\sum_{i=1}^{E_k(t)}\delta_{d_{k, i}}, \quad t\in\ptime.
\]
Then the deadline process is given by
\[
\dead(t)=(\dead_1(t), \dots, \dead_K(t)), \quad t\in\ptime.
\]
\end{definition}
Note that $\dead_k(\cdot)\in\bD(\ptime, \M_1)$ for $1\le k\le K$ and
$\dead(\cdot)\in\bD(\ptime, \M_1^K)$.
\paragraph{The workload process}
The workload process $W(\cdot)\in\bD(\ptime, \Rp)$ tracks as a function
of time, the amount of time needed for
the server to process all jobs currently in the system that will not
abandon. If no additional jobs were to arrive
after time $t$, the system would be empty $W(t)$ time units in the
future. This quantity also records the amount
of time that a newly arriving job would have to wait before being
served, if the job were sufficiently patient.
In particular, as in~\cite{ref:JR}, $W(\cdot)$ almost surely satisfies,
for all $t\in\ptime$,
\begin{eqnarray}
W(t)&=&W(0)+\sum_{k=1}^K\int_{(0, t]} v_{k, E_k(s)} 1_{\{
d_{k, E_k(s)}>W(s-)\}}\der E_k(s)-B(t), \label{eq:vwp}\\
B(t)&=&\int_0^t 1_{\{ W(s)>0\}}\der s.\label{eq:btp}
\end{eqnarray}
Here $B(\cdot)$ denotes the busy time process. Since the server works
at rate one during any
busy period, the amount of work served by time $t$ is equal to $B(t)$.
Occasionally, it will be convenient to refer to the idle time process,
which is given by
$I(t)=t-B(t)$ for $t\in\ptime$.
For a given time $t$,
the first and second terms in \eqref{eq:vwp} add up all of the work
that enters
the system by time $t$ and doesn't abandon before entering service.
Note that because of the indicator in the integrand, a particular job's
service time is added to the workload if and only if that job will not
abandon before it enters service. Fluid and diffusion limit results for
this process were proved in~\cite{ref:JR},
where it was referred to as the virtual waiting time process. We will
leverage that fluid limit result to carry out the analysis here.
\paragraph{Evolution of the virtual sojourn times and patience times}
For $1\le k\le K$, let
\begin{eqnarray}
w_{k, j}&=&
\begin{cases}
\tilde w_{k, j}, &1\le j\le Z_k(0), \\
W(t_{k, j}), &j>Z_k(0),
\end{cases}
\nonumber\\
p_{k, j}&=&
\begin{cases}
\tilde p_{k, j}, &1\le j\le Z_k(0),
\\ d_{k, j-Z_k(0)} + v_{k, j-Z_k(0)}1_{\{d_{k, j-Z_k(0)}>W(t_{k, j}-)\}
}, &j>Z_k(0).
\end{cases}
\nonumber
\end{eqnarray}
Note that for each $1\le k\le K$ and $j>Z_k(0)$,
\[
W(t_{k, j})=W(t_{k, j}-) + v_{k, j-Z_k(0)} 1_{\{
d_{k, j-Z_k(0)}>W(t_{k, j}-)\}}.
\]
Hence, if class $k$ job $j$ is served, $w_{k, j}$ includes
the job's service time in addition to the time spent waiting for
service to begin.
So, if class $k$ job $j$ is served, it stays in the system $w_{k, j}$
time units.
Therefore, $w_{k, j}$ is referred to as the virtual sojourn time of
class $k$ job $j$.
Further, $p_{k, j}$ represents the initial patience of class $k$ job $j$.
If a class $k$ job $j$ arriving after time zero will enter service before
time $t_{k, j}+d_{k, j-Z_k(0)}$, the job's patience time is taken to be
$d_{k, j-Z_k(0)}+v_{k, j-Z_k(0)}$ to account for the fact that it will
not abandon
$d_{k, j-Z_k(0)}$ time units after arrival. Instead, it will stay until time
$t_{k, j}+w_{k, j}$ when it receives its full service time requirement.
Otherwise, if class $k$ job $j$ won't enter service before the deadline
expires, the job will abandon at time $t_{k, j}+d_{k, j-Z_k(0)}$, and
$p_{k, j}=d_{k, j-Z_k(0)}$.
Then for each $1\le k\le K$ and $j\in\N$, the sojourn time $s_{k, j}$ of
class $k$
job $j$ is given by
\[
s_{k, j}=\min(w_{k, j}, p_{k, j}).
\]
This quantity indicates precisely how long class $k$ job $j$ will
reside in the system.
For all $t\in\ptime$, $1\le k\le K$, and $1\le j\le A_k(t)$, define
\begin{eqnarray}
w_{k, j}(t) &=&(w_{k, j}-(t-t_{k, j}))^+, \label{eq:rst}\\
p_{k, j}(t)&=&(p_{k, j}-(t-t_{k, j}))^+.\label{eq:rpt}
\end{eqnarray}
For $1\le k\le K$, $j\in\N$ and $t\ge t_{k, j}$, $w_{k, j}(t)$ and
$p_{k, j}(t)$
respectively represent the residual virtual sojourn time and residual
patience time of
class $k$ job $j$ at time $t$. Then the residual sojourn time $s_{k, j}(t)$
for class $k$ job $j$ at time $t\ge t_{k, j}$ is given by
\[
s_{k, j}(t)=\min(w_{k, j}(t), p_{k, j}(t)).
\]
\paragraph{Measure-valued state descriptor}
For $1\le k\le K$, define the class $k$ state descriptor
by
\begin{equation}\label{eq:zk}
\z_k(t)=\sum_{j=1}^{A_k(t)}\delta^+_{(w_{k, j}(t), p_{k, j}(t))},
\qquad
t\in\ptime.
\end{equation}
The state descriptor is defined as
\begin{equation}\label{eq:z}
\z(t)=(\z_1(t), \dots, \z_K(t)), \qquad t\in\ptime.
\end{equation}
For each $1\le k\le K$, $\z_k(\cdot)\in\bD(\ptime, \M_2)$, and
$\z(\cdot)\in\bD(\ptime, \M_2^K)$.
Figure~\ref{fig:state} depicts one component of a hypothetical
system state at a fixed time. The points in the figure correspond
to unit atoms of the measure.
All points move to the left and
down at rate one. The dotted diagonal line $p=w$ separates the points
in the figure into two groups. The jobs associated with points
on or below the line will eventually abandon; the points above
the line represent jobs that will be served.
Once a point reaches one of the coordinate axes it immediately
leaves the system and so is no longer included in the system state.
Among all of the components of the system state, there is a unique point
that has the smallest residual virtual sojourn time coordinate
and lies above the diagonal. This point corresponds to the job
currently in service.
Notice that in Figure~\ref{fig:state} there are three sets of points
that are
aligned vertically.
In each set, at most one of these points is above the
dotted line and the corresponding job will be served.
For each residual virtual sojourn time assumed by some job that is in
the system
at this fixed time, there is exactly one of these jobs for which the location
of the corresponding point in the component of the state descriptor associated
with that job's class lies above the diagonal. This job will be served
and it
arrived before any other job with the same residual virtual sojourn time.
These later arriving jobs aren't patient enough to remain in queue
until service begins.
Instead each will abandon. Therefore the corresponding
points in the components of the state descriptor associated
with those jobs' classes lie on or below the diagonal.
Although their sequence of arrivals relative to one another is not captured
by the state descriptor, the relative residual patience times
reveal the order in which they will abandon.
\begin{figure}[t]
\begin{center}
\begin{pspicture}(-1, -1)(8, 8)
\psline[linestyle=dashed, dash=3pt 2pt, linecolor=gray](0, 0)(7, 7)
{\psset{linewidth=3\pslinewidth}
\psline{-}(0.0, 0.0)(2.0, 0.0)
\psline{->}(5.0, 0.0)(7.0, 0.0)
\psline{-}(0.0, 0.0)(0.0, 1.5)
\psline{->}(0.0, 5.5)(0.0, 7.0)}
\rput(3.5, 0.0){Abandonment}
\rput{90}(0.0, 3.5){Service Completion}
\psline{*->}(6.0, 6.5)(5.6, 6.1)
\psline{*->}(3.5, 6.8)(3.1, 6.4)
\psline{*->}(3.5, 3.2)(3.1, 2.8)
\psline{*->}(4.5, 4.3)(4.1, 3.9)
\psline{*->}(4.5, 3.0)(4.1, 2.6)
\psline{*->}(4.5, 0.7)(4.1, 0.3)
\psline{*->}(2.3, 5.0)(1.9, 4.6)
\psline{*->}(2.3, 1.7)(1.9, 1.3)
\psline{*->}(1.7, 3.0)(1.3, 2.6)
\psline{*->}(0.9, 1.5)(0.5, 1.1)
\rput(0, 7.5){Patience Time}
\rput(7, -0.5){Virtual Sojourn Time}
\end{pspicture}
\caption{One coordinate of a hypothetical system state at a fixed time
with depicted transitions.} \label{fig:state}
\end{center}
\end{figure}
The state descriptor satisfies the following system of dynamic equations.
For each $1\le k\le K$ and for all $B\in\CB_{2, 0}$ and $t\in\ptime$,
\begin{equation} \label{eq:dynamics1}
\z_k(t)(B)
=
\sum_{j=1}^{A_k(t)}
1_{B_{t-t_{k, j}}}(w_{k, j}, p_{k, j}).
\end{equation}
To see this, simply note that for each $B\in\CB_{2, 0}$, $x\in\Rp$, and
$w, p>0$,
\[
((w-x)^+, (p-x)^+)\in B
\quad\Leftrightarrow\quad
(w-x, p-x)\in B
\quad\Leftrightarrow\quad
(w, p)\in B_x.
\]
The dynamic equation \eqref{eq:dynamics1} is equivalent to
\begin{equation}\label{eq:dynamics1prime}
\z_k(t)(B)
=
\z_k(0)(B_t) +
\sum_{j=1+Z_k(0)}^{A_k(t)}
1_{B_{t-t_{k, j}}}(w_{k, j}, p_{k, j}).
\end{equation}
Given $B\in\CB_{2, 0}$ and $x, y\in\Rp$, notice that
\[
(B_x)_y=B_{x+y}.
\]
This together with \eqref{eq:dynamics1} implies that,
for each $1\le k\le K$ and for all $B\in\CB_{2, 0}$ and $h, t\in
\ptime$,
\begin{equation} \label{eq:dynamics2}
\z_k(t+h)(B)
=
\z_k(t)(B_h)
+
\sum_{j=A_k(t)+1}^{A_k(t+h)}
1_{B_{t+h-t_{k, j}}}(w_{k, j}, p_{k, j}).
\end{equation}
\subsection{The Fluid Model}\label{sec:fms}
In this section, we define the fluid model associated with this
GI/GI/1+GI queue with $K$ distinct job classes. The primitive data for
this fluid model consists of the
vector $\lambda$ of arrival rates, the vector $\mu$ of service rates,
and the vector $\Gamma$ of deadline distributions. The triple
$(\lambda
, \mu, \Gamma)$ is referred to as supercritical data since $\rho>1$.
We begin by summarizing the results in~\cite{ref:JR} that suggest a
workload fluid model. Then, we develop
a full measure-valued fluid model. For this, we first need to define
the scaling regimes that yield the desired limiting dynamics.
\paragraph{The Sequence of Time Accelerated Systems}
We consider a sequence of systems indexed by $n \in\N$
in which the arrival rates and mean service times in the $n$th
system are sped up by a factor of $n$. We use a superscript
$n$ to denote all processes and parameters associated with
the $n$th system.
Specifically, the interarrival times $\{\xi^n_{k, i}\}_{i \in\N}$
for class $k$ in the $n$th system are given by
$\xi^n_{k, i} = \xi_{k, i} / n$ for $i\in\N$. Then $E_k^n(\cdot)$
denotes the class $k$ exogenous arrival processes in
the $n$th system. Hence, for $1\le k\le K$, $E_k^n(t)=E_k(nt)$
for $t\in\ptime$ and $\lambda_k^n=n\lambda_k$.
Similarly, the class $k$ service times $\{v^n_{k, i}\}_{i \in\N}$ in the
$n$th system are given by $v^n_{k, i} = v_{k, i} / n$ for $i\in\N$. Then
$\mu_k^n=n\mu_k$ and $\rho_k^n=\rho_k$. Hence, $\rho^n=\rho$.
The deadline sequence is unscaled. In particular, for each $1\le k\le
K$ and
$n, i\in\N$, $d_{k, i}^n=d_{k, i}$. Hence, we omit the superscript
when referring
to the class $k$ deadlines, distribution $\Gamma_k$, mean $1/\gamma
_k$, and
cumulative distribution function $F_k$ or its complement $G_k$.
Then the class $k$ deadline process for the $n$th system is given by
\[
\dead_k^n(t)=\sum_{i=1}^{E_k^n(t)} \delta_{d_{k, i}}, \qquad t\in
\ptime,
\]
and $\dead^n(\cdot)=(\dead_1^n(\cdot), \dots, \dead_K^n(\cdot))$.
For each $n\in\N$, there is an initial condition $Z^n(0)$ and
$\{(\tilde w_j^n, \tilde p_j^n, k_j^n)\}_{j\in\N}$ satisfying the conditions
indicated above. The initial conditions may vary with $n$.
Then for $n\in\N$, $1\le k\le K$, and $t\in\ptime$,
$A_k^n(t) = Z_k^n(0) + E_k^n(t)$.
If we suppose for simplicity that $Z_+(0)=0$ in the unscaled system,
then the job arrival times $\{t^n_{k, j}\}_{j \in\N}$
for class $k$ jobs in the $n$th system would be given by
\[
t^n_{k, j} =
\begin{cases}
0, &1\le j\le Z_k^n(0), \\
\frac{t_{k, j-Z_k^n(0)}}{n}, &j>Z_k^n(0).
\end{cases}
\]
The workload $W^n(\cdot)$, busy time $B^n(\cdot)$, and idle time
$I^n(\cdot)$ processes for the $n$th system satisfy equations analogous
to \eqref{eq:vwp} and \eqref{eq:btp}.
The residual virtual sojourn $w_{k, j}^n(\cdot)$ and residual patience
$p_{k, j}^n(\cdot)$ times, $1\le k\le K$ and $j\in\N$, for the $n$th
system are defined as in \eqref{eq:rst} and \eqref{eq:rpt}. The class
$k$ state descriptor $\z_k^n(\cdot)$ and the state descriptor $\z
^n(\cdot)$ for the $n$th system are defined as in \eqref{eq:zk} and
\eqref{eq:z}. Analogs of \eqref{eq:dynamics1} and \eqref{eq:dynamics2}
hold for the $n$th system as well.
\paragraph{The Workload Fluid Model}
For the time accelerated scaling regime, the authors of~\cite{ref:JR}
identify what one might
refer to as a workload fluid model. Namely, define a \textit{workload
fluid model
solution} to be a function $w:\ptime\to\Rp$ satisfying
\begin{equation}\label{eq:flwp}
w(t)=w(0)+\sum_{k=1}^K\rho_k \int_0^tG_k(w(s))\der s -t, \qquad t\in
\ptime.
\end{equation}
Equation \eqref{eq:flwp} can be interpreted as a fluid analog of
\eqref
{eq:vwp} and \eqref{eq:btp}.
Notice that any solution to \eqref{eq:flwp} is necessarily Lipschitz
continuous with Lipschitz constant
$\rho-1$, and therefore is almost everywhere differentiable. In \cite
{ref:JR}, it is asserted that for each $w_0\in\Rp$
a unique workload fluid model solution $w(\cdot)$ with $w(0)=w_0$
exists. They further show that workload fluid model
solutions satisfy a nice monotonicity property \cite[Theorem
1]{ref:JR}. To describe this, let
\begin{eqnarray*}
w_l&=&\sup\left\{ u\in\Rp: \sum_{k=1}^K\rho_k G_k(u) < 1\right\}
, \\
w_u&=&\sup\left\{ u\in\Rp: \sum_{k=1}^K\rho_k G_k(u) \le1\right
\}.
\end{eqnarray*}
Note that by continuity of $G_k(\cdot)$ for each $k$ and the fact that
$\rho>1$, $0<w_l\le w_u<\infty$.
They show that for a workload fluid model solution $w(\cdot)$ such that
$w(0)\not\in[w_l, w_u]$,
$w(\cdot)$ is strictly monotone and
\begin{equation}\label{eq:fdwl}
\lim_{t\to\infty}w(t)
=
\begin{cases}
w_l, &\hbox{if } w(0)<w_l, \\
w_u, &\hbox{if } w(0)>w_u.
\end{cases}
\end{equation}
Otherwise, if $w(0)\in[w_l, w_u]$, then $w(t)=w(0)$ for all $t\in
\ptime$.
Workload fluid model solutions also satisfy a useful relative ordering property.
In particular, given two workload fluid model solutions $w_1(\cdot)$
and $w_2(\cdot)$
such that $w_1(0)\le w_2(0)$ it follows that for all $t\in\ptime$,
\begin{equation}\label{eq:RelOrd}
w_1(t)\le w_2(t).
\end{equation}
To see this note that if $w_1(0)\le w_u$ and $w_l\le w_2(0)$, then
\eqref{eq:RelOrd}
follows immediately from the monotoncity property of workload fluid
model solutions.
Otherwise, $w_2(0)<w_l$ or $w_1(0)>w_u$. If $w_1(0)>w_u$, then by
continuity and
\eqref{eq:fdwl} there exists $t\in\ptime$ such that $w_2(t)=w_1(0)$.
For $s\in\ptime$,
set $w(s)=w_2(t+s)$. Then, it is straightforward to verify that
$w(\cdot
)$ is a workload
fluid model solution with $w(0)=w_1(0)$. Hence,
by uniqueness of fluid model solutions, $w(\cdot)=w_1(\cdot)$ so that
$w_2(\cdot+t)=w_1(\cdot)$. Then for all
$s\in\ptime$, $w_1(s)=w_2(s+t)\le w_2(s)$. An analogous argument demonstrates
\eqref{eq:RelOrd} when $w_2(0)<w_l$.
In~\cite{ref:JR}, the authors justify interpreting the fluid model as a
first order approximation
of the stochastic system by showing that if $W^n(0)\to w_0$ almost
surely as $n\to\infty$,
where $w_0$ is a finite nonnegative constant, then
$(W^n, I^n)\Rightarrow(w(\cdot), 0)$,
as $n\to\infty$, where $w(\cdot)$ is the unique workload fluid model
solution such that $w(0)=w_0$
\cite[Theorem~1]{ref:JR}. Here, we wish to cite a slightly modified
version of \cite[Theorem~1]{ref:JR},
which we state next. Before this, we make note of a representation for
the workload process developed
in~\cite{ref:JR} that will be of use here. By \cite[(19) and arguments
presented in the proof of Theorem~1]{ref:JR},
for all $n\in\N$ and $0\le s\le t<\infty$,
\begin{eqnarray}\label{eq:WorkRep}
W^n(t)&=&W^n(s) + I^n(t)-I^n(s) - (t-s) \\
&&+X^n(t)-X^n(s) +\sum_{k=1}^K \rho_k \int_s^t G_k(W^n(u))\der
u, \nonumber
\end{eqnarray}
where $\{X^n(\cdot)\}_{n\in\N}\subset\bD(\ptime, \R)$ is such
that as
$n\to\infty$,
\begin{equation}\label{eq:Xn}
X^n(\cdot)\Rightarrow0.
\end{equation}
\begin{thrm}\label{thrm:JR}
If $W^n(0)\Rightarrow W_0$ as $n\to\infty$, then as $n\to\infty$,
\begin{equation}\label{eq:JR}
(W^n, I^n)\Rightarrow(W^*(\cdot), 0),
\end{equation}
where $W^*(0)$ is equal in distribution to $W_0$. Furthermore,
$W^*(\cdot)$ is almost surely a workload fluid model solution.
\end{thrm}
\begin{proof}[Summary of proof]
Let $\iota:\ptime\to\ptime$ be given by $\iota(t)=t$ for all $t\in
\ptime$.
Since the limit in \eqref{eq:Xn} is deterministic, it follows that as
$n\to\infty$,
\[
W^n(0)-\iota(\cdot) + X^n(\cdot)\Rightarrow W_0-\iota(\cdot).
\]
Using tools developed in~\cite{ref:RW}, the authors further show that
for all $n\in\N$,
\begin{equation}\label{eq:CMT}
(W^n(\cdot), I^n(\cdot))=(\varphi_G, \psi_G)\left(W^n(0)-\iota
(\cdot
)+X^n(\cdot)\right),
\end{equation}
where $(\varphi_G, \psi_G):\bD(\ptime, \R)\to\bD(\ptime, \Rp^2)$
is a Lipschitz
continuous function in the topology of uniform convergence on compact sets.
Then, by the continuous mapping theorem, it follows that as $n\to
\infty$,
\begin{equation}\label{eq:CMT2}
(W^n(\cdot), I^n(\cdot))\Rightarrow(\varphi_G, \psi_G)\left
(W_0-\iota
(\cdot)\right).
\end{equation}
The authors of~\cite{ref:JR} go on to show that for
$w_0\in\Rp$ $(\varphi_G, \psi_G)(w_0-\iota(\cdot))=(w(\cdot),
0)$, where
$w(\cdot)$ is the unique workload fluid model solution such that $w(0)=w_0$.
\end{proof}
\paragraph{The Measure-Valued Fluid Model}
To develop a measure-valued fluid model, we add spatial scaling to the
sequence of
time accelerated systems. In particular in order to obtain a fluid
scaled system, we divide
space in the $n$th time accelerated system by a factor of $n$. Then,
for $n\in\N$,
$1 \le k \le K$, and $t\in\ptime$,
\[
\Ebn_k(t)=\frac{E^n_k(t)}{n}, \qquad\CDbn_k(t)=\frac{\CD^n_k(t)}{n},
\qquad\hbox{and}\qquad\bzn_k(t)=\frac{\z^n_k(t)}{n}.
\]
The $n$th fluid scaled state descriptor in the sequence is given by
\[
\bzn(\cdot)
=
(\bzn_1(\cdot), \ldots, \bzn_K(\cdot)).
\]
We are interested the the limiting behavior as $n$ approaches infinity.
One can speculate as to how the dynamics of the system will behave
in the limit. Class $k$ fluid should arrive to the system at rate
$\lambda_k$
and be distributed over the positive quadrant as it enters. Class $k$
fluid arriving at
time $t$ should have virtual sojourn time $w(t)$, where $w(\cdot)$ is
an appropriate fluid
workload solution, and patience time distributed according to $\Gamma
_k$. Then the residual virtual sojourn and residual patience times
should each decrease at rate one
until at least one is zero, at which time the fluid would exit from the
system. In particular,
at time $t+h$, fluid that arrived to $(w, p)$ at time
$t$ should be at $(w-h, p-h)$ if $(w-h)\wedge(p-h)>0$ and should have
exited the system otherwise.
Such departures would be associated with abandonment if $p-h\le0\wedge
(w-h)$ and service
completion if $w-h\le0$ and $w-h < p-h$.
We wish to capture these dynamics by defining an appropriate fluid model.
The initial measure will need to satisfy various natural conditions.
To define these, first recall the definition of $C$ given in \eqref{def:C}.
Then, given $\vartheta\in\M_2^K$, recall that $\vartheta_+\in\M_2$
denotes the superposition
measure (see \eqref{def:sp}). Let
\[
w_{\vartheta}=\sup\{ x\in\Rp: \vartheta_+([x, \infty)\times\Rp
)>0\}.
\]
Define $\I\subset\M_2^K$ to be the collection of $\vartheta\in\M_2^K$
such that
\begin{enumerate}
\item[(I.1)] $\vartheta_+(C_x)=0\hbox{ for all }x\in\Rp^2$,
\item[(I.2)] $w_{\vartheta}<\infty$,
\item[(I.3)] $\max_{1\le k\le K} G_k(w_{\vartheta
}-\varepsilon)>0\hbox{ for all }\varepsilon>0$.
\end{enumerate}
We refer to the collection ${\mathcal C}=\{ C_x : x\in\Rp^2\}$ as the
corner sets. Then
condition (I.1) is that each coordinate of the initial measure doesn't
charge corner sets,
which is equivalent to the condition that each coordinate of the
initial measure doesn't charge vertical or horizontal lines. A
motivation for requiring (I.1) is to
prevent exiting mass from resulting in discontinuities. Since (I.1) it
is preserved by the
fluid model dynamics (see Property~2 of Theorem~\ref{thrm:eu}), it is a
natural to require it of the initial condition.
Given $\vartheta\in\M_2^K$ that satisfies (I.1)
and $\varepsilon>0$ there exists $\kappa>0$ such that
\begin{equation}\label{eq:InitialCorner}
\max_{1\le k\le K} \sup_{x\in\Rp^2} \vartheta_k(C_x^{\kappa
})<\varepsilon.
\end{equation}
To see this, given $x\in\Rp^2$, let $p_i(x)=x_i$ for $i=1, 2$.
Then, given $\nu\in\M_2$ and $i=1, 2$, let $\pi_i(\nu)\in\M_1$ be
the projection measure such that for all $f\in\C_b(\Rp^1)$
\begin{equation}\label{def:pi}
\left\langle f, \pi_i(\nu)\right\rangle =\left\langle f\circ p_i, \nu\right\rangle .
\end{equation}
Then, if $\vartheta\in\M_2^K$ satisfies (I.1), it follows that for
$i=1, 2$, $\pi_i(\vartheta_+)$
doesn't charge points. Hence, for every $\varepsilon>0$ there exists
$\kappa>0$ such that
\[
\max_{i=1, 2}\sup_{y\in\Rp} \pi_i(\vartheta_+)([(y-\kappa
)^+, y+\kappa
])<\frac{\varepsilon}{2},
\]
(cf.\ \cite[Lemma A.1]{ref:GPW}).
Since for each $1\le k\le K$ and $x\in\Rp^2$,
\[
\vartheta_k(C_x^{\kappa})
\le\vartheta_+(C_x^{\kappa})
\le\pi_1(\vartheta_+)([(x_1-\kappa)^+, x_1+\kappa])+ \pi
_2(\vartheta
_+)([(x_2-\kappa)^+, x_2+\kappa]),
\]
\eqref{eq:InitialCorner} follows.
Condition (I.2) dictates that the fluid analog of the initial workload
is finite and
condition (I.3) requires that the initial fluid workload does not
exceed the maximal deadline.
In particular, let
\[
d_{\max}=\max_{1\le k\le K}\sup\{ x\in\Rp: G_k(x)>0\}.
\]
Then, by (I.2), $w_{\vartheta}<d_{\max}$ if $d_{\max}=\infty$.
Further, by (I.3), $w_{\vartheta}\le d_{\max}$ if $d_{\max}<\infty$,
which is natural. Indeed in the stochastic system
the workload can only exceed the maximal deadline by at most
one service time, i.e., once the workload has jumped above the
maximal deadline, all incoming jobs necessarily abandon until
such time as the maximal deadline exceeds the workload.
This discrepancy vanishes in the fluid limit.
The reader will see that it is needed mathematically when
$d_{\max}<\infty$ to prevent fluid from building up on a moving
vertical line during $(0, w_{\vartheta}-d_{\max}]$, which
would prevent (I.1) from being preserved.
A \textit{fluid model solution} for the initial measure $\vartheta\in\I$
is a function
$\zeta:\ptime\to\M_2^K$ such that $\zeta(0)=\vartheta$ and
for each $1\le k\le K$, $B\in\CB_2$, and $t\in\ptime$, $\zeta_k$ satisfies
\begin{equation}\label{fdevoeq2}
\zeta_k(t)(B)
=
\zeta_k(0)(B_t)
+
\lambda_k \int_0^t \left(\delta_{w(s)}^+\times\Gamma_k\right)(B_{t-s})
\der s,
\end{equation}
where $w(\cdot)$ denotes the unique workload fluid model solution with
$w(0)=w_\vartheta$.
Notice that \eqref{fdevoeq2} is a fluid analog of \eqref{eq:dynamics1}.
\section{Main Results and Approximation Formulas} \label{sec:main}
Here we state the two main theorems proved in this paper (Theorems \ref
{thrm:eu} and~\ref{thrm:flt}),
which together validate approximating various performance processes via
fluid model solutions.
Then, we derive some specific approximation formulas. Lastly, we
identify the set of invariant
states for the fluid model.
\begin{thrm}\label{thrm:eu} Let $\vartheta\in\I$. Then there exists a
unique fluid model solution
$\zeta(\cdot)$ for the data $(\lambda, \mu, \Gamma)$ such that
$\zeta
(0)=\vartheta$.
In addition,
\begin{enumerate}
\item$w_{\zeta(t)}=w(t)$ for all $t\in\ptime$,
where $w(\cdot)$ is the unique workload fluid model solution such that
$w(0)=w_{\vartheta}$;
\item$\zeta_+(t)(C_x)=0$ for all $x\in\Rp^2$ and $t\in\ptime$;
\item$\zeta(\cdot)$ is continuous.
\end{enumerate}
\end{thrm}
The proof of this theorem is given in Section~\ref{sec:prfthrmeu}.
Property~1 is that the right edge of the support
of the fluid model solution superposition measure is equal to the
workload fluid model solution for all time,
which is true by definition at time zero. It implies that (I.2) holds
for all time. Then by the monotoncity
properties of workload fluid model solutions, (I.3) holds for all time
as well. The proof of Property~1 is fairly
straightforward, as the reader will see in Section~\ref{sec:prfthrmeu}.
Property~2 is that the fluid model solution superposition
measure doesn't charge corner sets for all time, i.e., that (I.1) holds
for all time, and its proof is more involved. For this,
we first prove Lemma~\ref{prop:NearBoundary}, from which Property~2
follows by letting $\varepsilon$
decrease to zero. Lemma~\ref{prop:NearBoundary} is then used together
with some additional arguments
to verify Property~3, continuity.
The next result justifies regarding fluid model solutions as
first order approximations for the measure-valued state descriptor of
the original stochastic system.
\begin{thrm}\label{thrm:flt}
Suppose that $\z_0^*=(\z_{0, 1}^*, \dots, \z_{0, K}^*)$ is a random measure
in $\M_2^K$ with superposition measure $\z_{0, +}^*$ such that
\begin{itemize}
\item[(A.1)] $\bP( \z_0^*\in\I)=1$,
\item[(A.2)] $\bE[ \langle p_1+p_2, \z_{0, +}^*\rangle ]<\infty$,
\item[(A.3)] $\bE[ \z_{0, +}^*(\Rp^2)]<\infty$.
\end{itemize}
Let $W_0^*=w_{\z_0^*}$. Also suppose that, as $n\to\infty$,
\begin{eqnarray}\label{eq:IC}
&&\left(\bzn(0), \left\langle p_1, \bzn(0)\right\rangle , \left\langle p_2, \bzn
(0)\right\rangle
, W^n(0)\right) \\
&&\qquad\Rightarrow\left(\z_0^*, \left\langle p_1, \z_0^*\right\rangle , \left
<p_2, \z
_0^*\right\rangle
, W_0^*\right).\nonumber
\end{eqnarray}
Then, as $n\to\infty$,
\[
\bzn(\cdot)\Rightarrow\z^*(\cdot),
\]
where $\z^*(0)$ is equal in distribution to $\z_0^*$.
Furthermore, $\z^*(\cdot)$ is almost surely a fluid model solution.
\end{thrm}
This result is proved via verifying tightness and then proving that any
limit point is almost surely a fluid model solution, which are done in
Sections~\ref{sec:tightness} and~\ref{sec:char} respectively.
Next, we use the fluid model to derive various approximation formulas.
These include approximation formulas that demonstrate the nonlinear
nature of
state space collapse that takes place for this model. That is to say,
one can recover
various $K$ dimensional quantities such as the fluid approximation for
the queue-length
vector from the fluid approximation for the workload process. However,
this mapping
depends on the entirety of the deadline distributions and is therefore
nonlinear in its
dependence on the workload process. See \eqref{eq:fq}, \eqref{eq:fn},
and \eqref{eq:fa}
below. In addition, we are able to approximate various age related
processes such
as the number of jobs in the system that are within a specific age range.
To describe these, we will need to introduce the following functional.
Given $\vartheta\in\I$,
let $\zeta(\cdot)$ denote the unique fluid model solution such that
\mbox{$\zeta(0)=\vartheta$}.
Let $w(\cdot)$ denote the unique workload fluid model solution such
that $w(0)=w_{\vartheta}$.
Fluid that arrives at time $s>0$ has fluid workload $w(s)$. Some of
this fluid remains in the system
at time $t\ge s$ only if
\[
w(s)-(t-s)>0.
\]
Recall that $w(\cdot)$ is continuous. Furthermore, it is strictly
decreasing if $w(0)>w_u$
and bounded above by $w_u$ otherwise. Additionally, since
$\rho>1$, $w_u<d_{\max}$. Finally, by (I.3), $w(0)\le d_{\max}$.
Hence, $w(s)<d_{\max}$ for all $s>0$. Therefore, $w'(s)=\sum_{k=1}^K
\rho_kG_k(w(s))-1>-1$
for all $s>0$. Then, as a function of $s\in\ptime$, $w(s)+s$ is
continuous and strictly increasing.
Furthermore, for $s\in[0, t]$, the values range from $w(0)$ to
$w(t)+t\ge t$ at time $t$. Hence,
$\inf\{ 0\le s\le t : w(s)+s\ge t\}$ is well defined, finite, and can
be interpreted as the time at which fluid
departing at time $t$ via service completion arrived to the system. For
$t\in\ptime$, let
\begin{equation}\label{eq:tau}
\tau(t)=
\inf\{ 0\le s\le t : w(s)+s\ge t\}.
\end{equation}
Then $\tau(t)=0$ for all $t\in[0, w(0)]$, so that $w(\tau(t))+\tau(t)>
t$ for all $t\in[0, w(0))$.
Further, for $t\ge w(0)$,
\begin{equation}\label{eq:tauprop}
w(\tau(t))+\tau(t)=t.
\end{equation}
For $t>w(0)$, all fluid in the system at time $t$ arrived during the
time interval $(\tau(t), t]$.
\paragraph{Nonlinear State Space Collapse}
Given $\vartheta\in\I$, let $\zeta(\cdot)$ denote the unique fluid
model solution with $\zeta(0)=\vartheta$
and $w(\cdot)$ denote the unique workload fluid model solution with
$w(0)=w_{\vartheta}$.
\textit{Queue-Length Vector Fluid Approximation.}
For $1\le k\le K$ and $t\in\ptime$,
\begin{equation}\label{eq:fq}
z_k(t)
\equiv
\zeta_k(t)(\Rp^2)
=
\begin{cases}
\zeta_k(0)((\Rp^2)_t)
+
\lambda_k \int_0^{t} G_k(a)\der a
, & t< w(0), \\
\lambda_k \int_0^{w(\tau(t))} G_k(a)\der a
, & t\ge w(0).
\end{cases}
\end{equation}
We have chosen $a$ as the variable of integration to suggest the
interpretation age, or time in system.
This provides a simple interpretation of this formula. At time
$t<w(0)$, none of the fluid that entered the
system after time zero has been fully processed. So the integral term
only needs to address departures
resulting from expiring deadlines. Then, fluid arriving in $(0, t]$
remains in the system at time $t$
if and only if the initial deadline exceeds the current age.
Hence the simple form of the integral.
At time $t\ge w(0)$, all fluid initially in the system has departed,
and $w(\tau(t))$ can be interpreted as the age of the fluid departing
the system via service completion at time $t$. So, for $t\ge\tau(t)$,
$w(\tau(t))$ can be
thought of as the age of the oldest fluid in the system at time $t$.
\textit{Nonabandoning Jobs Queue-Length Vector Fluid Approximation.}
Let $U=\{ (w, p)\in\Rp^2 : w<p\}$.
Then the mass present in $U$ at time $t$ is associated with fluid that
doesn't abandon.
For $1\le k\le K$ and $t\in\ptime$,
\begin{equation}\label{eq:fn}
n_k(t)
\equiv
\zeta_k(t)(U)
=
\begin{cases}
\zeta_k(0)(U_t)
+
\lambda_k \int_0^{t}G_k(w(v)) \der v
, & t<w(0), \\
\lambda_k \int_{\tau(t)}^t G_k(w(v)) \der v
, & t\ge w(0).
\end{cases}
\end{equation}
Here the variable of integration should be regarded as time, and the
formula has the following
interpretation. Fluid that has not departed the system by time $t$ will
not abandon prior to service
completion if the initial deadline exceeds the fluid workload at the
time of arrival.
For $t\ge w(0)$, we can rewrite this result in an alternative form that
highlights the independence
of the service time distributions to the deadline and interarrival time
distributions. For $1\le k\le K$ and
$t\ge w(0)$, let
\[
w_k(t)=\rho_k \int_{\tau(t)}^t G_k(w(v)) \der v.
\]
Then $w_k(t)$ denotes the amount of fluid workload in the system at
time $t$ due to class $k$ fluid.
For each $1\le k\le K$, we have for all $t\ge w(0)$,
\[
n_k(t)=\mu_kw_k(t).
\]
In this form, the formula is reminiscent of linear state space collapse,
but the dependence of $w_k(\cdot)$ on $w(\cdot)$ is nonlinear.
\textit{Abandoning Jobs Queue-Length Vector Fluid Approximation.}
Let $L=\{ (w, p)\in\Rp^2 : p\le w\}$.
Then the mass present in $L$ at time $t$ is associated with fluid that abandons.
For $1\le k\le K$ and $t\in\ptime$, let $a_k(t)=\zeta_k(t)(L)$.
Then, for
$1\le k\le K$ and $t\in\ptime$,
\begin{equation}\label{eq:fa}
a_k(t)
=
\begin{cases}
\zeta_k(0)(L_t)
+
\lambda_k \int_0^{t} \left(G_k(t-v)-G_k(w(v))\right) \der v
, & t<w(0), \\
\lambda_k \int_{\tau(t)}^t \left(G_k(t-v)-G_k(w(v))\right)\der v
, & t\ge w(0).
\end{cases}
\end{equation}
Note that it is easy to verify that $z_k(t)=a_k(t)+n_k(t)$ for $1\le
k\le K$ and $t\in\ptime$.
\paragraph{Age Related Fluid Approximations}
Given $\vartheta\in\I$, let $\zeta(\cdot)$ denote the unique fluid
model solution with $\zeta(0)=\vartheta$
and let $w(\cdot)$ denote the unique workload fluid model solution with
$w(0)=w_{\vartheta}$.
Fluid in the system at time $t>0$ that arrived at time $s\in(0, t]$ is
$t-s$ units old at time $t$.
Hence, for fluid in the system at time $t>0$ that arrives in $(0, t]$ to
be of age at least
$u$, it must have arrived by time $t-u$. Then $s\in(0, t-u]$. This fluid
has residual offered
waiting time $(w(s)-(t-s))^+$. Since it is in the system at
time $t$, $w(s)-(t-s)>0$,
i.e., $w(s)+s>t$ so that $s\in(\tau(t), t-u]$. For $t\ge w(0)$, this is
an empty time interval if
$w(\tau(t))\le u$. Otherwise, for $t\ge w(0)$ and $s\in(\tau(t), t-u]$,
$w(s)-(t-s)\in(0, w(t-u)-u]$.
For $t\ge w(0)$ and $0\le u\le t$, let
\[
H(t, u)=[0, \left(w(t-u)-u\right)^+]\times\Rp.
\]
This is the line $\{0\}\times\Rp$ if $u\ge w(\tau(t))$ and is a
vertical stripe otherwise.
Then, for $1\le k\le K$, $t\ge w(0)$, and $0\le u\le t$, we can interpret
\[
z_k(t, u)\equiv\zeta_k(t)(H(t, u)),
\]
to be the amount of class $k$ fluid in the system at time $t$ of age at
least $u$.
If $t<w(0)$, some initial fluid may remain in the system at time $t$.
We regard that fluid as being $t$
units old. At time $t$, the residual offered waiting time of such fluid
lies in $[0, w(0)-t]$. If we consider
a time $0\le u\le t<w(0)$ and ask for the total amount of fluid of age
at least $u$, this would also include
fluid that arrived after time zero and by time $t-u$. By \eqref
{eq:flwp}, $w(t-u)-u>w(0)-t$. Hence the
definition of $H(t, u)$ and interpretation of $z_k(t, u)$ naturally
extend to all $0\le u\le t<w(0)$.
Then, for $1\le k\le K$ we have the following: for $0\le u\le t<w(0)$,
\begin{align}\label{eq:age1}
z_k(t, u)&=
\zeta_k(0)(H(t, u)_t)
+
\lambda_k \int_u^t G_k(v)\der v, \\
\intertext{and for $t\ge w(0)$, }
z_k(t, u)&=
\begin{cases}
\lambda_k \int_u^{w(\tau(t))} G_k(v)\der v
, & 0\le u< w(\tau(t)), \\
0, & w(\tau(t))\le u\le t.\label{eq:age2}
\end{cases}
\end{align}
One can obtain similar approximations for the abandoning and
nonabandoning queue length of a certain age or older by
computing the measure of an appropriately chosen set.
\paragraph{Invariant States}
An \textit{invariant state} is a measure $\theta\in\I$ such that the
unique fluid model solution $\zeta(\cdot)$
with initial measure $\theta$ satisfies $\zeta(t)=\theta$ for all
$t\in
\ptime$. Here we identify the collection of invariant states.
For this, we begin by recalling some measure theoretic background.
Details can be found in \cite[Chapter~1]{ref:F}. Let
\[
{\mathcal R}=\left\{ [a, b)\times[c, d) \subset\Rp^2 : a, c\in\Rp,
a\le
b\le\infty\hbox{ and } c\le d\le\infty\right\}.
\]
Then ${\mathcal R}$ is an elementary family (a collection of sets that
contains the emptyset, is closed under
pairwise intersection, and such that complements of members of the collection
can be written as finite unions of members of the collection).
Further note that $\sigma({\mathcal R})=\CB_2$. Let
\[
{\mathcal R}'=\left\{\cup_{l=1}^{m} R_l : m\in\N\hbox{ and }R_l\in
{\mathcal R}\right\}.
\]
Then ${\mathcal R}'$ is an algebra (a collection of sets that contains
the emptyset and is closed under
pairwise union and relative complementation). In fact, ${\mathcal R}'$
is the algebra generated by ${\mathcal R}$. It is easy to see that any
finite pre-measure
(additive $\Rp$ valued function that assigns value zero to the
emptyset) on ${\mathcal R}$ extends to a finite pre-measure on
${\mathcal R}'$.
Then, by the Carath\'{e}odory extension theorem \cite[Theorem
1.14]{ref:F}, any finite pre-measure defined on ${\mathcal R}'$
uniquely extends to a finite Borel measure on $\Rp^2$. Thus, in order
to uniquely specify a finite Borel measure on $\Rp^2$, it suffices to
specify a finite pre-measure on~${\mathcal R}$.
Given $w_l\le w\le w_u$, let $\theta^w\in\I$ be the unique
finite Borel measure on $\Rp^2$ that for $1\le k\le K$ satisfies
\[
\theta_k^w([w, \infty)\times\Rp)=0,
\]
and for $0\le a<b\le w$ and $0\le c<d\le\infty\in\Rp$,
\[
\theta_k^w([a, b)\times[c, d))
=
\lambda_k \int_{w-b}^{w-a}\Gamma_k([c+u, d+u))\der u.
\]
It is easy to verify that these relationships determine a finite
pre-measure on ${\mathcal R}$.
Indeed, since $[w, \infty)\times\Rp$ has measure zero, it suffices to
specify each
$\theta_k^w$ on sets in ${\mathcal R}$ that don't meet $[w, \infty
)\times
\Rp$.
Define
\[
\J=\{ \theta^w : w_l\le w\le w_u\}.
\]
\begin{thrm}\label{thrm:is} The set of invariant states is given by
$\J$.
\end{thrm}
The proof of Theorem~\ref{thrm:is} is given in Section~\ref{sec:prfthrmeu}.
Note that for $w_l\le w\le w_u$ and $1\le k\le K$,
the fluid approximations for queue length $z_k^w$ and nonabandoning
queue length
$n_k^w$ take the form
\begin{eqnarray*}
z_k^w&\equiv&\theta_k^w(\Rp^2)=\lambda_k\int_0^w G_k(u)\der u, \\
n_k^w&\equiv&\theta_k^w(U)=\lambda_k w G_k(w).\\
\end{eqnarray*}
\section{Properties of Fluid Model Solutions}\label{sec:prfthrmeu}
In this section, we prove Theorems~\ref{thrm:eu} and~\ref{thrm:is}.
The proof of existence and uniqueness for Theorem~\ref{thrm:eu}
is relatively straightforward. Indeed, since the right hand side of
\eqref{fdevoeq2} only depends on $(\lambda, \mu, \Gamma)$ and
$\vartheta$, \eqref{fdevoeq2} can be regarded as a definition,
provided that the integral term is a well defined function taking
values in $\M_2^K$. In this regard, note that $\vartheta_k\in\M_2$
and $\delta_{w(s)}^+\times\Gamma_k\in\M_2$ for all $s\in\ptime$
and $1\le k\le K$. Hence, the main issue is to show that the integral
is well defined on all of ${\mathcal B}_2$, which is demonstrated
here using the Carath\'{e}odory extension theorem and
Dynkin's $\pi\lambda$-theorem.
\begin{proof}[Proof of Existence and Uniqueness for Theorem~\ref{thrm:eu}]
Fix \mbox{$\vartheta\in\I$}.
First we verify existence of a fluid model solution with initial
measure $\vartheta$.
For this, fix $1\le k\le K$ and $0<t<\infty$.
Given $B\in{\mathcal R}$, we have that $B=[a, b)\times[c, d)$ for some
$a, c\in\Rp$, $a\le b\le\infty$, and $c\le d\le\infty$. Define
$f:[0, t]\to[0, 1]$ by
\[
f(s)=\left(\delta_{w(s)}^+\times\Gamma_k\right)(B_{t-s}).
\]
By \eqref{eq:fdwl}, the fact that $w_l>0$ and monotonicity properties
of $w(\cdot)$, $w(s)>0$ for all $s>0$.
Then, since $t>0$, we have that for $s\in(0, t]$,
\[
f(s)=\left(\delta_{w(s)}\times\Gamma_k\right)(B_{t-s}).
\]
Therefore, for $s\in[0, t]$,
\[
f(s)
=
\begin{cases}
G_k(c+t-s)-G_k(d+t-s), &\hbox{if }a+t-s\le w(s)< b+t-s, \\
0, &\hbox{otherwise.}
\end{cases}
\]
We see that $a+t-s\le w(s)< b+t-s$ if and only if $a+t\le w(s)+s< b+t$
if and only if
$\tau(a+t)\le s<\tau(b+t)$, where $\tau(\infty)=\infty$. Hence, for
$s\in[0, t]$,
\[
f(s)
=
\begin{cases}
G_k(c+t-s)-G_k(d+t-s), &\hbox{if } \tau(a+t) \le s <\tau(b+t), \\
0, &\hbox{otherwise.}
\end{cases}
\]
Then, since $G_k(\cdot)$ and $\tau(\cdot)$ are continuous, $f$ is Borel
measurable and\break
$\int_0^tf(s)ds$ is well defined. Further, since $\tau(\cdot)$ is
monotone increasing,
\[
\int_0^t f(s)\der s
=
\int_{t\wedge\tau(a+t)}^{t\wedge\tau(b+t)} \left(
G_k(c+t-s)-G_k(d+t-s)\right)\der s.
\]
For $B\in{\mathcal R}$, define
\[
\gamma_k(t)(B)=\vartheta_k(B_t)+\lambda_k\int_0^t \left(\delta
_{w(s)}^+\times\Gamma_k\right)(B_{t-s})\der s.
\]
It is clear that $\gamma_k(t)(\emptyset)=0$.
Further, $\gamma_k(t)(\Rp^2)\le\vartheta_k(\Rp^2)+\lambda_k
t<\infty$.
So, in order to verify that $\gamma_k(t)$ is a finite premeasure on
${\mathcal R}$, it suffices to verify countable additivity.
This amounts to demonstrating that the summation and integral can be
interchanged, which is an immediate consequence
of the monotone convergence theorem. Then, by the Carath\'{e}odory
extension theorem, $\gamma_k(t)$
extends to a finite Borel measure $\gamma_k^*(t)$ on ${\mathcal B}_2$.
We must verify that $\gamma_k^*(t)$ satisfies \eqref{fdevoeq2} for all
$B\in{\mathcal B}_2$. For this, we
use Dynkin's $\pi\lambda$-theorem \cite[Chapter~1
Theorem~3.3]{ref:B}. Let
\begin{equation}\label{def:Pi}
{\mathcal P}=\{ [a, \infty)\times[c, \infty) : 0\le a, c<\infty\}.
\end{equation}
This is a $\pi$-system since it is closed under intersection. Let
\[
{\mathcal L}=\left\{ B\in{\mathcal B}_2 :
\gamma_k^*(t)(B)=\vartheta_k(B_t)+\lambda_k\int_0^t \left(\delta
_{w(s)}\times\Gamma_k\right)(B_{t-s})\der s
\right\}.
\]
Then ${\mathcal L}$ is a $\lambda$-system since $\Rp^2\in{\mathcal L}$,
${\mathcal L}$
is closed under countable unions (by the monotone convergence theorem), and
$A\setminus B\in{\mathcal L}$ whenever $B, A\in{\mathcal L}$ and
$B\subset A$. Further,
\[
{\mathcal P}\subset{\mathcal R}\subset{\mathcal L}\subset{\mathcal B}_2.
\]
Then, since the $\sigma$-algebra generated by ${\mathcal P}$ is
${\mathcal B}_2$,
it follows from Dynkin's $\pi\lambda$-theorem that ${\mathcal
L}={\mathcal B}_2$.
Since $1\le k\le K$ and $t>0$ were arbitrary, it follows that $\gamma
^*:\ptime\to\M_2^K$,
which is given by
\[
\gamma^*(t)=(\gamma_1^*(t), \dots, \gamma_K^*(t)),
\]
is a fluid model solution. Uniqueness is immediate since any fluid
model solution $\zeta$
such that $\zeta(0)=\vartheta$ satisfies $\zeta_k(t)(B)=\gamma
_k^*(t)(B)$ for all
$B\in{\mathcal R}$ (see \cite[Theorem~1.14]{ref:F}).
\end{proof}
\begin{proof}[Proof of Property~1 for Theorem~\ref{thrm:eu}]
Fix $\vartheta\in\I$. Let $w(\cdot)$ be the unique workload fluid model
solution with $w(0)=w_{\vartheta}$
and $\zeta(\cdot)$ be the unique fluid model solution with $\zeta
(0)=\vartheta$.
Since $\zeta(0)=\vartheta$ and $w(0)=w_{\vartheta}$, $w_{\zeta(0)}=w(0)$.
Fix $t\in(0, \infty)$ and $\varepsilon\in(0, w(t))$.
Set $B=[w(t)-\varepsilon, \infty)\times\Rp$. Then $B_{(2\varepsilon
, 0)}=[w(t)+\varepsilon, \infty)\times\Rp$.
We wish to show that $\zeta_+(t)(B)>0$ and $\zeta
_+(t)(B_{(2\varepsilon, 0)})=0$.
Then, since $\varepsilon\in(0, w(t))$ is arbitrary, it follows that
$w_{\zeta(t)}=w(t)$.
Suppose that $\zeta_+(t)(B_{(2\varepsilon, 0)})>0$. Then by \eqref{fdevoeq2},
there exists $s\in[0, t]$ such that $w(t)+\varepsilon+t-s\le w(s)$, i.e.,
$w(t)+t+\varepsilon\le w(s)+s$. But $w(\cdot)+\iota(\cdot)$ is
strictly increasing,
so that $w(t)+t+\varepsilon> w(s)+s$, which is a contradiction. Thus,
$\zeta_+(t)(B_{(2\varepsilon, 0)})=0$.
Since $w(\cdot)+\iota(\cdot)$ is continuous and strictly increasing,
there exists
$s_1\in[0, t)$ such that $w(t)+t-\varepsilon\le w(s_1)+s_1$. Then, for
all $s\in(s_1, t]$,
$w(t)+t-\varepsilon\le w(s)+s$. Let $s_2=(t-d_{\max})^+$. Then
$s_2\in[0, t)$
and $t-s<d_{\max}$ for all $s\in(s_2, t]$. Let $s^*= s_1 \vee s_2$.
Then $s^*<t$ and
\[
\sum_{k=1}^K \int_{0}^t \left( \delta_{w(s)}^+ \times\Gamma_k
\right
)(B_{t-s})\der s
\ge
\sum_{k=1}^K \int_{s_1}^t G_k(t-s)\der s
\ge
\sum_{k=1}^K \int_{s^*}^t G_k(t-s)\der s
>
0.
\]
Hence, by \eqref{fdevoeq2}, $\zeta_+(t)(B)>0$.
\end{proof}
The next goal is to verify Properties~2 and~3 for Theorem~\ref{thrm:eu}.
For this, we state and prove the following lemma. Property~2 follows
immediately from Lemma~\ref{prop:NearBoundary} by letting $\varepsilon$
decrease to zero. To verify Property~3, we must show that fluid model
solutions change very little over
short time intervals. Note that there are three mechanisms that cause
the measure-valued function
to change: fluid arriving, fluid departing, and the measure evolving. Verifying
that the fluid departs in a smooth way is the main issue that needs to
be addressed. We will
use the result in the following lemma to assist with this as well.
\begin{lemma}\label{prop:NearBoundary}
Given $\vartheta\in\I$, let $\zeta(\cdot)$ be the unique fluid
model solution
such that $\zeta(0)=\vartheta$. For every $T, \varepsilon>0$, there exists
$\kappa>0$ such that
\[
\max_{1\le k\le K}\sup_{t\in[0, T]}\sup_{x, y\in\Rp} \zeta
_k(t)(C_{(x, y)}^{\kappa})<\varepsilon.
\]
\end{lemma}
To ease the reader's efforts to follow the proof of Lemma \ref
{prop:NearBoundary}
given below, we outline the basic strategy. Having a good understanding
of this deterministic
argument will assist the reader in following the stochastic
generalization used to prove
Lemma~\ref{lem:BndReg}. The basic idea in this case is to use \eqref
{fdevoeq2} on a given
$\kappa$ enlargement of a corner set. Then \eqref{eq:InitialCorner} can
be used to bound
the contribution from mass present in the system at time zero. Next one
determines
the time interval during which mass must arrive in order to contribute
to the vertical portion of the
enlarged corner set. As the reader will see, the end points of this
time interval can be expressed
in terms of the function $\tau(\cdot)$. Then one obtains a bound on the
mass that can be present
in the horizontal portion of the enlarged corner set. This second part
turns out to be fairly easy to
bound, as the reader will see. Hence the main difficultly is to bound
the amount of mass that falls
in the vertical portion. This relies on bounding the length of the
aforementioned time interval, which
can be done by demonstrating that the function $w(\cdot)+\iota(\cdot)$
increases sufficiently quickly.
But $w(\cdot)+\iota(\cdot)$ actually increases quite slowly at times
when the workload fluid model
solution is near $d_{\max}$, i.e., possibly at small times. So one must
wait a short amount
of time (until time $\delta$ in the proof) before implementing this
strategy during which just a small amount of mass enters the entire system.
Once that small amount of time has elapsed, \eqref{eq:flwp} can be used
to obtain the desired bound.
The mathematical details are given next.
\begin{proof}[Proof of Lemma~\ref{prop:NearBoundary}] Fix $T,
\varepsilon>0$.
Set $\lambda_+=\sum_{k=1}^K\lambda_k$, $\delta=\varepsilon
/(4\lambda_+)$,
$M=w_u\vee w(\delta)$, and $c=\sum_{k=1}^K\rho_k G_k(M)$. Note that
$c>0$ since $M<d_{\max}$.
Further, by monotonicity properties of workload fluid model solutions,
$w(u)\le M$ for all $u\ge\delta$.
For $t\in[0, T]$, $1\le k\le K$, $x, y\in\Rp$, and $\kappa>0$, by
\eqref
{fdevoeq2},
\[
\zeta_k(t)\left(C_{(x, y)}^{\kappa}\right)
=
\zeta_k(0)\left(\left(C_{(x, y)}^{\kappa}\right)_t\right)
+
\lambda_k\int_0^t \left(\delta_{w(s)}^+\times\Gamma_k\right
)\left(\left
(C_{(x, y)}^{\kappa}\right)_{t-s}\right)\der s.
\]
By \eqref{eq:InitialCorner} there exists $\kappa_0$ such that for all
$0<\kappa<\kappa_0$,
\[
\max_{1\le k\le K} \sup_{t\in[0, T]} \sup_{x, y\in\Rp}\zeta
_k(0)\left
(\left(C_{(x, y)}^{\kappa}\right)_t\right)
<\frac{\varepsilon}{4}.
\]
Hence, for $t\in[0, T]$, $1\le k\le K$, $x, y\in\Rp$, and $0<\kappa
<\kappa_0$,
\begin{equation}\label{eq:fdevoeqk}
\zeta_k(t)\left(C_{(x, y)}^{\kappa}\right)
<
\frac{\varepsilon}{4}
+
\lambda_k\int_0^t \left(\delta_{w(s)}^+\times\Gamma_k\right
)\left(\left
(C_{(x, y)}^{\kappa}\right)_{t-s}\right)\der s.
\end{equation}
We must show that there exists $\kappa^*\le\kappa_0$ such that for all
$t\in[0, T]$, $1\le k\le K$, $x, y\in\Rp$, and $0<\kappa<\kappa^*$,
the integral term is bounded above by $3\varepsilon/4$.
Fix $t\in[0, T]$, $1\le k\le K$, and $x, y\in\Rp$.
For $s\in[0, T]$ and $\kappa>0$, let
\[
h_k(s, \kappa)=G_k((y-\kappa)^++t-s)-G_k(y+\kappa+t-s).
\]
For $0<\kappa<\kappa_0$, the integrand in \eqref{eq:fdevoeqk}
at time $s\in[0, t]$ is bounded above by
\[
\begin{cases}
0, &\hbox{if } w(s)+s<(x-\kappa)^++t, \\
1, &\hbox{if } (x-\kappa)^+ + t\le w(s)+s \le x+\kappa+t, \\
h_k(s, \kappa), &\hbox{if }x+\kappa+t<w(s)+s.
\end{cases}
\]
Equivalently, for $0<\kappa<\kappa_0$,
using continuity and monotonicity properties of $w(\cdot)+\iota(\cdot)$,
the integrand in \eqref{eq:fdevoeqk} at time $s\in[0, t]$ is bounded
above by
\begin{equation}\label{eq:bnds}
\begin{cases}
0, &\hbox{if } 0\le s<\tau((x-\kappa)^++t), \\
1, &\hbox{if } \tau((x-\kappa)^++t)\le s \le\tau(x+\kappa+t), \\
h_k(s, \kappa), &\hbox{if }\tau(x+\kappa+t)<s\le t.
\end{cases}
\end{equation}
This together with \eqref{eq:fdevoeqk} yields that for $0<\kappa
<\kappa_0$,
\begin{eqnarray}
\zeta_k(t)\left(C_{(x, y)}^{\kappa}\right)
&<&
\frac{\varepsilon}{4}
+
\lambda_k\int_{\tau((x-\kappa)^++t)\wedge t}^{\tau(x+\kappa
+t)\wedge
t}\der s
+\lambda_k\int_{\tau(x+\kappa+t)\wedge t}^t h_k(s, \kappa)\der
s\nonumber\\
&\le&
\frac{\varepsilon}{4}
+
\lambda_k\left( \tau(x+\kappa+t)\wedge t-\tau((x-\kappa
)^++t)\wedge t
\right)\nonumber\\
&&+\
\lambda_k \int_{(y-\kappa)^+}^{y+\kappa} G_k(u) \der u\nonumber\\
&\le&
\frac{\varepsilon}{4}
+
\lambda_k\left( \tau(x+\kappa+t)\wedge t-\tau((x-\kappa
)^++t)\wedge t
\right)
+
2\lambda_k\kappa.\label{eq:taumin}
\end{eqnarray}
For $0<\kappa<\kappa_0$, let
\[
\Delta(\kappa)= \tau(x+\kappa+t)\wedge t-\tau((x-\kappa
)^++t)\wedge t.
\]
Fix $0<\kappa<\kappa_0$.
If $\tau((x-\kappa)^++t)\ge t$, $\Delta(\kappa)=0$.
Also, if $x+\kappa+t\le w(0)$, then $\Delta(\kappa)=0$.
Henceforth, we assume that $\tau((x-\kappa)^++t)<t$ and $x+\kappa+t>w(0)$.
If $ \tau(x+\kappa+t)\wedge t\le\delta$, $\Delta(\kappa)\le
\delta$.
Otherwise, $ \tau(x+\kappa+t)\wedge t> \delta$, and there are two cases
to consider.
First consider the case where $\tau((x-\kappa)^++t)\ge\delta$.
Then $(x-\kappa)^++t> w(0)$ since $\tau((x-\kappa)^++t)>0$.
Hence, by \eqref{eq:tauprop} and \eqref{eq:flwp}, since $\tau
((x-\kappa
)^++t)\ge\delta$,
\begin{eqnarray*}
2\kappa
&=& x+\kappa+t-(x-\kappa+t)\\
&\ge& x+\kappa+t-((x-\kappa)^++t)\\
&=&w(\tau(x+\kappa+t))+\tau(x+\kappa+t)\\
&&- w(\tau((x-\kappa)^++t))-\tau((x-\kappa)^++t)\\
&=& \sum_{k=1}^{K} \rho_k\int_{\tau((x-\kappa)^++t)}^{\tau
(x+\kappa+t)}
G_k(w(u))\der u\\
&\ge& \sum_{k=1}^{K} \rho_k\int_{\tau((x-\kappa)^++t)}^{\tau
(x+\kappa
+t)} G_k(M)\der u\\
&\ge& c \Delta(\kappa).
\end{eqnarray*}
In particular, $\Delta(\kappa)\le2\kappa/c$.
The other case to consider is $\tau((x-\kappa)^++t)<\delta$. Then
\[
\Delta(\kappa)
\le\tau(x+\kappa+t)\wedge t
=\left(\tau(x+\kappa+t)\wedge t-\delta\right)+\delta.
\]
Further, by \eqref{eq:tauprop} and monotoncity of $w(\cdot)+\iota
(\cdot)$,
\begin{eqnarray*}
x-\kappa+t
&\le& (x-\kappa)^++t
\le w(\tau((x-\kappa)^++t))+\tau((x-\kappa)^++t)\\
&\le& w(\delta)+\delta
\le w(\tau(x+\kappa+t))+\tau(x+\kappa+t)
= x+\kappa+ t.
\end{eqnarray*}
Then, by \eqref{eq:tauprop} and \eqref{eq:flwp},
\begin{eqnarray*}
2\kappa
&\ge& x+\kappa+t- w(\delta)-\delta\\
&=&w(\tau(x+\kappa+t))+\tau(x+\kappa+t)- w(\delta)-\delta\\
&=& \sum_{k=1}^{K} \rho_k\int_{\delta}^{\tau(x+\kappa+t)}
G_k(w(u))\der
u\\
&\ge& \sum_{k=1}^{K} \rho_k\int_{\delta}^{\tau(x+\kappa+t)}
G_k(M)\der
u\\
&=& c\left(\tau(x+\kappa+t)-\delta\right).
\end{eqnarray*}
In particular,
\begin{equation}\label{eq:Delta}
\Delta(\kappa)\le\frac{2\kappa}{c}+ \delta,
\end{equation}
which is the largest of the four upper bounds.
By combining \eqref{eq:taumin}, \eqref{eq:Delta}, and the definition of
$\delta$
it follows that for $0<\kappa<\kappa_0$,
\begin{eqnarray*}
\zeta_k(t)\left(C_{(x, y)}^{\kappa}\right)
&<&
\frac{\varepsilon}{4}
+
\frac{2\lambda_k\kappa}{c}
+
\lambda_k\delta
+
2\lambda_k\kappa
\le
\frac{\varepsilon}{2}
+
\frac{2\lambda_k\kappa}{c}
+
2\lambda_k\kappa.
\end{eqnarray*}
Let $0<\kappa^*\le\kappa_0$ be such that for all $0<\kappa<\kappa^*$
\[
\max_{1\le k\le K}2\lambda_k\left(\frac{1}{c}+1\right)\kappa
<\frac
{\varepsilon}{2}.
\]
Then, for all $0<\kappa<\kappa^*$,
$\zeta_k(t)(C_{(x, y)}^{\kappa})<\varepsilon$.
Since $1\le k\le K$, $t\in[0, T]$, and $x, y, \in\Rp$ were chosen
arbitrarily, the result follows.
\end{proof}
Next, Lemma~\ref{prop:NearBoundary} is used to prove continuity for
Theorem~\ref{thrm:eu}.
For this, we need to specify a metric on $\M_2$ that induces the
topology of weak convergence.
For efficiency, we will specify such a metric on $\M_i$ for $i=1, 2$.
Given $i=1, 2$ and
$\zeta$, $\zeta'\in\M_i$, let ${\bf d}[\zeta, \zeta']$ denote the Prohorov
distance between
$\zeta$ and $\zeta'$. Specifically,
\begin{eqnarray*}
{\bf d}[\zeta, \zeta']
&=&
\inf\{ \varepsilon>0 : \zeta(B)\le\zeta'(B^{\varepsilon})
+\varepsilon
\hbox{ and }\zeta'(B)\le\zeta(B^{\varepsilon}) +\varepsilon\\
&&\qquad\hbox{ for all closed }B\in\CB_i\}.
\end{eqnarray*}
Then the following is a natural metric on $\M_i^K$. Given $i=1, 2$ and
$\zeta, \zeta'\in\M_i^K$,
define
\begin{equation}\label{eq:metric}
{\bf d}_K[\zeta, \zeta'] =\max_{1\le k\le K} {\bf d}[\zeta_k, \zeta'_k].
\end{equation}
In the following proof, we show that for each fluid model solution
$\zeta(\cdot)$,
$\lim_{h\to0} {\bf d}_K[\zeta(t+h), \zeta(t)]=0$ for all $t\in
\ptime$.
\begin{proof}[Proof of Property~3 for Theorem~\ref{thrm:eu}]
Let $\vartheta\in\I$ and let $\zeta(\cdot)$ be the unique fluid model
solution such that
$\zeta(0)=\vartheta$. It suffices to prove continuity of $\zeta
_k(\cdot
)$ on $\ptime$ for
each $1\le k\le K$. Fix $1\le k\le K$. We prove continuity of $\zeta
_k(\cdot)$
on $[0, T]$ for each $T>0$. Fix $T, \varepsilon>0$. By Lemma \ref
{prop:NearBoundary}, there exists
$\kappa$ such that for all $0<h<\kappa$,
\[
\sup_{s\in[0, T]}\zeta_k(s)(C^h) <\varepsilon.
\]
Let $0\le s < t \le T$ be such that $t-s<\min(\kappa, \varepsilon
/\max
(2, \lambda_k))$.
Note that $( \delta_{w(u)}^+\times\Gamma_k)(\Rp^2)=1$
for all $u\in\ptime$. Let $B\in\CB_2$ be closed and set $h=t-s>0$.
Note that $B_h\subset B^{2h}$.
By subtracting \eqref{fdevoeq2} applied to $B_h$ at time $s$ from
\eqref{fdevoeq2}
applied to $B$ at time $t$, we obtain
\begin{eqnarray*}
\zeta_k(t)(B)
&=&
\zeta_k(s)(B_h)+\lambda_k\int_s^t\left( \delta_{w(u)}^+\times
\Gamma
_k\right)(B_{t-u})\der u\\
&\le&
\zeta_k(s)(B^{2h})+\lambda_k h\\
& < &
\zeta_k(s)(B^{\varepsilon})+\varepsilon.
\end{eqnarray*}
Consider \eqref{fdevoeq2} applied to $B$ at time $s$, to $B^{2h}$ at
time $t$, and to $C^h$ at time $s$. For $u\in[0, s]$, we see that
$(w, p)\in B_u$ implies that
$(w-u, p-u)\in B$, which implies that either $(w-u, p-u)\in B\cap
C^h\subset C^h$
or $(w-u, p-u)\in B\setminus C^h$. If $(w-u, p-u)\in C^h$, then $(w,
p)\in
(C^h)_u$.
If $(w-u, p-u)\in B\setminus C^h$,
then $(w-u-h, p-u-h)\in B^{2h}$ and so $(w, p)\in(B^{2h})_{u+h}$
and $u+h=u+t-s\le t$. Therefore, for $u\in[0, s]$, $B_{s-u}\subset
(C^h)_{s-u}\cup
(B^{2h})_{t-v}$, where $v=t-(s-u+h)\in[0, t]$. Hence,
\begin{eqnarray*}
\zeta_k(s)(B)\le\zeta_k(t)(B^{2h})+ \zeta_k(s)(C^h)
<
\zeta_k(t)(B^{\varepsilon})+ \varepsilon.
\end{eqnarray*}
Then for all $0\le s<t\le T$ such that $0<t-s<\min(\kappa,
\varepsilon
/\max(2, \lambda_k))$,
\[
{\bf d}[\zeta_k(t), \zeta_k(s)]<\varepsilon.
\]
Hence, $\zeta_k(\cdot)$ is continuous on $[0, T]$. Since $T>0$ was
arbitrary, $\zeta_k(\cdot)$
is continuous on $\ptime$. Since $1\le k\le K$ was arbitrary, $\zeta
(\cdot)$ is continuous on $\ptime$.
\end{proof}
Now that each statement in Theorem~\ref{thrm:eu} has been verified, we
prove Theorem~\ref{thrm:is}.
\begin{proof}[Proof of Theorem~\ref{thrm:is}]
First suppose that $\vartheta\in\I$ is an invariant state. We must show
that $\vartheta\in\J$, i.e.,
we must show that for some $w\in[w_l, w_u]$, $\vartheta_k(B)=\theta
_k^w(B)$ for all
$B\in\CB_2$ and $1\le k\le K$.
Since $\vartheta$ is an invariant state, Theorem~\ref{thrm:eu} Property
1 implies that the unique workload
fluid model solution such that $w(0)=w_{\vartheta}$ is constant, i.e.,
$w(t)=w_{\vartheta}$ for all $t\in\ptime$.
Then, by the monotonicity properties of workload fluid model solutions,
$w_l\le w_{\vartheta}\le w_u$.
Let $w=w_{\vartheta}$. Fix $1\le k\le K$, $B\in{\mathcal P}$,
and $t>w$. Then $B=[a, \infty)\times[c, \infty)$ for some $a, c\in
\Rp$.
Further, $a+t-w>0$ and $(w-a)^+<t$. Hence, by \eqref{fdevoeq2}, we have
\begin{eqnarray*}
\vartheta_k(B)
&=& \lambda_k\int_0^t\left(\delta_w^+\times\Gamma_k\right
)(B_{t-s})\der
s\\
&=& \lambda_k\int_{(a+t-w)\wedge t}^t G_k(c+t-s)\der s\\
&=& \lambda_k\int_0^{(w-a)^+} G_k(c+u)\der u\\
&=& \theta_k^w(B).
\end{eqnarray*}
Since $B\in\CP$ was arbitrary, $\vartheta_k$ and $\theta_k^w$ agree on
$\CP$.
Since $\vartheta_k$ and $\theta_k^w$ agree on ${\mathcal P}$, they
agree on ${\mathcal R}$
and therefore they agree on $\CB_2$. So $\vartheta_k=\theta_k^w$. Since
$1\le k\le K$ was arbirary, $\vartheta=\theta^w\in\J$.
Next, fix $w_l\le w\le w_u$. Then the unique fluid workload solution
$w(\cdot)$ with $w(0)=w$
is constant, i.e., $w(t)=w$ for all $t\in\ptime$. Set $\zeta
(t)=\theta
^w$ for all $t\in\ptime$.
Then, in order to show that $\zeta(\cdot)$ is a fluid model solution,
it suffices to verify that
$\zeta(\cdot)$ satisfies \eqref{fdevoeq2}. For this, it suffices to
verify that
for all $B\in{\mathcal P}$, $1\le k\le K$, and $t>0$,
\[
\theta_k^w(B) = \theta_k^w(B_t)+\lambda_k\int_0^t\left(\delta
_w^+\times
\Gamma_k\right)(B_{t-s})\der s.
\]
For this fix $B\in{\mathcal P}$, $1\le k\le K$, and $t>0$. We have
\begin{eqnarray*}
\theta_k^w(B_t)&+&\lambda_k\int_0^t\left(\delta_w^+\times\Gamma
_k\right
)(B_{t-s})\der s\\
&=&
\theta_k^w(B_t)+\lambda_k\int_ {(a+t-w)^+\wedge t}^t G_k(c+t-s)\der s.
\end{eqnarray*}
\noindent\textit{Case~1}: Suppose that $a\ge w$.
Then $(a+t-w)^+\wedge t= t$ and it follows that
\[
\theta_k^w(B_t)+\lambda_k\int_0^t\left(\delta_w^+\times\Gamma
_k\right
)(B_{t-s})\der s
=0
=\theta_k^w(B).
\]
\noindent\textit{Case~2}:
Suppose that $a<w\le a+t$.
Then $(a+t-w)^+\wedge t=a+t-w$ and
\begin{eqnarray*}
\theta_k^w(B_t)&+&\lambda_k\int_0^t\left(\delta_w^+\times\Gamma
_k\right
)(B_{t-s})\der s\\
&=&
0+\lambda_k \int_ {a+t-w} ^ t G_k(c+t-s) \der s\\
&=&
\lambda_k \int_ {0} ^ {w-a}G_k(c+u)\der u\\
&=&
\theta_k^w(B).
\end{eqnarray*}
\noindent\textit{Case~3}:
If $a+t<w$, then $(a+t-w)^+\wedge t=0$ and
\begin{eqnarray*}
\theta_k^w(B_t)&+&\lambda_k\int_0^t\left(\delta_w^+\times\Gamma
_k\right
)(B_{t-s})\der s\\
&=&
\theta_k^w([a+t, w)\times[c+t, \infty))
+
\lambda_k \int_ 0^t G_k(c+t-s)\der s\\
&=&
\lambda_k\int_0^{w-a-t}G_k(c+t+u)\der u
+
\lambda_k \int_ 0^t G_k(c+u)\der u\\
&=&
\lambda_k\int_t^{w-a}G_k(c+u)\der u
+
\lambda_k \int_ 0^t G_k(c+u)\der u\\
&=&
\lambda_k \int_ 0^{w-a} G_k(c+u)\der u\\
&=&
\theta_k^w(B).
\end{eqnarray*}
Hence $\theta^w$ is an invariant state.
\end{proof}
\section{Proof of Tightness} \label{sec:tightness}
In this section, we prove that the sequence $\{\bzn(\cdot)\}_{n\in\N}$
is relatively
compact under the standing assumption \eqref{eq:IC}.
For this, we apply \cite[Corollary~3.7.4]{ref:EK}. In particular, it
suffices to
prove compact containment and an oscillation inequality. This is done in
Lemmas~\ref{lem:compact containment} and~\ref{lem:osc} below. Throughout,
\[
\lambda_+=\sum_{k=1}^K\lambda_k\qquad\hbox{and}\qquad g_+=\sum
_{k=1}^K\frac{1}{\gamma_k}.
\]
\subsection{Preliminaries}
For $1\le k\le K$ and $t\in\ptime$, let $\lambda_k^*(t)=\lambda_kt$.
For $1\le k\le K$, let $\dead_k^*(\cdot)=\lambda_k^* (\cdot) \Gamma_k$.
Define $\lambda^*(\cdot)=(\lambda_1^*(\cdot), \dots, \lambda
_K^*(\cdot))$
and $\dead^*(\cdot)=(\dead_1^*(\cdot)\dots, \dead_K^*(\cdot))$.
By a functional law of large numbers for renewal processes, as $n\to
\infty$,
\begin{equation}\label{eq:EFLLN}
\Ebn(\cdot)\Rightarrow\lambda^*(\cdot).
\end{equation}
Further, as a special case of Lemma~\ref{lem:deadFLLN} stated below,
the following
functional law of large numbers for the deadline process holds:
as $n\to\infty$,
\begin{equation}\label{eq:deadFLLN}
\left(\CDbn(\cdot), \left\langle \chi, \CDbn(\cdot)\right\rangle \right
)\Rightarrow
\left(\dead
^*(\cdot), \left\langle \chi, \dead^*(\cdot)\right\rangle \right).
\end{equation}
To state Lemma~\ref{lem:deadFLLN}, we need to introduce some
additional notation
and asymptotic assumptions.
\paragraph{Asymptotic Assumptions (AA)}
For each $n\in\N$ suppose that we have $K$ independent sequences of
strictly positive
independent and identically distributed random variables that are
independent of the
exogenous arrival process. For each $n\in\N$, denote the $k$th
sequence by
$\{g_{k, i}^n\}_{i\in\N}$, and for each $n\in\N$ and $1\le k\le K$,
denote the distribution of
$g_{k, 1}^n$ by $\Gamma_k^n$. We assume that $\Gamma_k^n$ has a
finite mean
$1/\gamma_k^n$, but we do not necessarily assume that $\Gamma_k^n$ is
continuous.
We further assume that for each $1\le k\le K$,
\begin{equation}\label{eq:ui}
\lim_{M\to\infty} \sup_{n\in\N} \left\langle \chi1_{(M, \infty)},
\Gamma
_k^n\right\rangle =0,
\end{equation}
and, as $n\to\infty$,
\begin{equation}\label{eq:weak}
\Gamma_k^n\wk\Gamma_k.
\end{equation}
One interpretation of (AA) is that for large $n$ and $1\le k\le K$,
$\Gamma_k^n$ approximates $\Gamma_k$.
So, for $n\in\N$ and $1\le k\le K$, one can regard $\{g_{k, i}^n\}
_{i\in
\N}$
as a collection of approximate patience times.
Here we are primarily interested in two particular choices of
approximate patience times. These are introduced below, shortly after
the statement of Lemma~\ref{lem:ResDeadFLLN}.
Note that \eqref{eq:ui} is a uniform integrability condition and,
together with \eqref{eq:weak},
it implies that for each $1\le k\le K$, as $n\to\infty$,
\[
1/\gamma_k^n\to1/\gamma_k.
\]
For each $n\in\N$, let $\Gamma^n=\Gamma_1^n\times\dots\times
\Gamma_K^n$
and let
$\Gamma=\Gamma_1\times\dots\times\Gamma_K$. For $n\in\N$, $1\le
k\le
K$, and
$t\in\ptime$, define
\[
{\mathcal G}_k^n(t)=\sum_{i=1}^{E_k^n(t)} \delta_{g_{k, i}^n}^+
\qquad\hbox{and}\quad
\bar{\mathcal G}_k^n(t)=\frac{1}{n}{\mathcal G}_k^n(t).
\]
For $n\in\N$, let ${\mathcal G}^n(\cdot)=({\mathcal
G}_1^n(\cdot
), \dots, {\mathcal G}_K^n(\cdot))$
and $\bar{\mathcal G}^n(\cdot)=(\bar{\mathcal G}_1^n(\cdot
), \dots,
\bar{\mathcal G}_K^n(\cdot))$.
So then, for each $n\in\N$, ${\mathcal G}^n(\cdot)\in\bD
(\ptime, \M_1^K)$. We refer to
${\mathcal G}^n(\cdot)$, $n\in\N$, as a deadline related process.
\begin{lemma}\label{lem:deadFLLN} Suppose that (AA) holds. Then, as
$n\to\infty$,
\[
\left(\bar{\mathcal G}^n(\cdot), \left\langle \chi, \bar{\mathcal
G}^n(\cdot
)\right\rangle
\right)
\Rightarrow
\left( \CD^*(\cdot), \left\langle \chi, \CD^*(\cdot)\right\rangle \right).
\]
\end{lemma}
Lemma~\ref{lem:deadFLLN} holds by \eqref{eq:EFLLN} and \cite[Theorem
5.1]{ref:GW}. To see this, note the following.
Respectively, the number of classes and deadline distributions play the
same role for the deadline related processes defined
here as the number of routes and service time distributions do for the
load process defined in~\cite{ref:GW}.
Then \cite[Theorem 5.1]{ref:GW} holds under \cite[Assumption
(A)]{ref:GW}, and the conditions in \cite[Assumption (A)]{ref:GW}
relevant to \cite[Theorem 5.1]{ref:GW} are \cite[(4.8)--(4.13)]{ref:GW}.
Conditions \cite[(4.8)--(4.9)]{ref:GW} hold here because the
deadlines are assumed to be strictly positive and have finite means.
Condition \cite[(4.10)]{ref:GW} corresponds to \eqref{eq:EFLLN}
above.
Conditions \cite[(4.11)--(4.13))]{ref:GW} hold here due to \eqref{eq:ui}
and \eqref{eq:weak} above.
For each $n\in\N$ and $1 \le k \le K$, define the fluid scaled
increments as follows:
for $0\le s\le t<\infty$,
\[
\Ebn_k(s, t) = \Ebn_k(t) - \Ebn_k(s)
\qquad\hbox{and}\qquad
\lambda^*_k(s, t) = \lambda^*_k(t)-\lambda^*_k(s),
\]
and
\[
\CGbn_k(s, t) = \CGbn_k(t) - \CGbn_k(s)
\qquad\hbox{and}\qquad
\dead^*_k(s, t) = \dead^*_k(t)-\dead^*_k(s).
\]
Then, by \eqref{eq:EFLLN}, given $T, \varepsilon, \eta>0$,
\begin{equation}\label{eq:EincFLLN}
\liminf_{n\to\infty} \bP
\left(
\max_{1\le k \le K} \sup_{0\le s \le t \le T}
\left| \Ebn_k(s, t) - \lambda^*_k(s, t) \right| \le\varepsilon
\right) \ge1-\eta.
\end{equation}
Similarly, the following corollary holds as a consequence of
Lemma~\ref{lem:deadFLLN} and continuity of the measures
$\Gamma_k$, $1 \le k \le K$.
\begin{cor}\label{cor:deadlineLimit}
Let $T, \varepsilon, \eta> 0$ and $0\le a\le b< \infty$. Suppose that
(AA) holds. Then
\[
\liminf_{n\to\infty} \bP
\left(
\max_{1\le k \le K} \sup_{0\le s \le t \le T}
\left| \CGbn_k(s, t)((a, b)) - \dead^*_k(s, t)((a, b)) \right| \le
\varepsilon
\right) \ge1-\eta.
\]
\end{cor}
We also wish to consider a version of a deadline related process
involving residual approximate patience times.
We refer to such a process as a residual deadline related process.
To define these, let $\{g_{k, i}^n\}_{i\in\N}$, $1\le k\le K$, and
$n\in
\N$,
satisfy (AA). For each $n\in\N$, $1\le k\le K$ and $t\in\ptime$, let
\[
g_{k, i}^n(t)=\left( g_{k, i}^n-\left(t-t_{k, Z_k^n(0)+i}^n\right
)^+\right)^+,
\]
and set
\[
{\mathcal R}_k^n(t)=\sum_{i=1}^{E_k^n(t)} \delta_{g_{k, i}^n(t)}^+
\qquad\hbox{and}\qquad
\bar{\mathcal R}_k^n(t)=\frac{{\mathcal R}_k^n(t)}{n}.
\]
Then, for each $n\in\N$ and $1\le k\le K$, ${\mathcal R}_k^n(\cdot
)\in\bD(\ptime, \M_1)$.
For $n\in\N$, set ${\mathcal R}^n(\cdot)=({\mathcal R}_1^n(\cdot),
\dots, {\mathcal R}_K^n(\cdot))$
and $\bar{\mathcal R}^n(\cdot)=(\bar{\mathcal R}_1^n(\cdot), \dots
, \bar{\mathcal R}_K^n(\cdot))$.
For each $1\le k\le K$ and $t\in\ptime$, let ${\mathcal R}_k^*(t)\in
\M
_1$ be the measure that is
absolutely continuous with respect to Lebesgue measure with density
$\lambda_k(G_k(\cdot)-\break G_k(\cdot+t))$.
Set ${\mathcal R}^*(\cdot)=({\mathcal R}_1^*(\cdot), \dots
, {\mathcal R}_K^*(\cdot))$.
\begin{lemma}\label{lem:ResDeadFLLN} Suppose that (AA) holds. Then, as
$n\to\infty$,
\[
\bar{\mathcal R}^n(\cdot)\Rightarrow{\mathcal R}^*(\cdot).
\]
\end{lemma}
In order to prove Lemma~\ref{lem:ResDeadFLLN}, we must first prove that
$\{ \bar{\mathcal R}^n(\cdot)\}_{n\in\N}$ is tight. This is done using
the same
general approach as that outlined for proving tightness of $\{\bzn
(\cdot
)\}_{n\in\N}$.
To illustrate similarity of proof techniques, we consecutively execute each
step for proving tightness of both processes in Sections~\ref{sec:cc}
through~\ref{sec:osc}
below. Then in Section~\ref{sec:char},
we complete the proof of Lemma~\ref{lem:ResDeadFLLN} by uniquely characterizing
the limit points.
There are two specific choices of $\{g_{k, i}^n\}_{i\in\N}$, $1\le
k\le
K$ and $n\in\N$, that will
be of particular interest for proving tightness of $\{\bzn(\cdot)\}
_{n\in\N}$ and Theorem~\ref{thrm:flt}.
\noindent\textit{Special Case 1:} For $n\in\N$, $1\le k\le K$, and
$i\in\N
$, let $g_{k, i}^n=d_{k, i}$.
This choice clearly satisfies (AA). Hence, Lemma~\ref{lem:deadFLLN}
implies \eqref{eq:deadFLLN}.
Further, for $n\in\N$, $1\le k\le K$, $i\in\N$, and $t\in\ptime$,
let
\begin{equation}\label{eq:a}
a_{k, i}^n(t)=\left( d_{k, i} - \left( t - t_{k, Z_k^n(0)+i}^n\right
)^+\right)^+.
\end{equation}
Then, for $n\in\N$ and $t\in\ptime$, set
\[
\CA_k^n(t)=\sum_{i=1}^{E_k^n(t)} \delta_{a_{k, i}^n(t)}^+
\qquad\hbox{and}\qquad
\CAbn_k(t)=\frac{1}{n}\CA^n_k(t).
\]
By Lemma~\ref{lem:ResDeadFLLN}, as $n\to\infty$,
\begin{equation}\label{eq:A}
\CAbn(\cdot)\Rightarrow{\mathcal R}^*(\cdot).
\end{equation}
\noindent\textit{Special Case 2:} For $n\in\N$, $1\le k\le K$, and
$i\in\N
$, let $g_{k, i}^n=d_{k, i}+v_{k, i}^n$.
This choice satisfies (AA) as well. For $n\in\N$, $1\le k\le K$,
$i\in\N
$, and $t\in\ptime$,
let
\begin{equation}\label{eq:v}
v_{k, i}^n(t)=\left( d_{k, i}+v_{k, i}^n - \left( t - t_{k,
Z_k^n(0)+i}^n\right)^+\right)^+.
\end{equation}
Then, for $n\in\N$ and $t\in\ptime$, set
\[
\CV_k^n(t)=\sum_{i=1}^{E_k^n(t)} \delta_{v_{k, i}^n(t)}^+
\qquad\hbox{and}\qquad
\CVbn_k(t)=\frac{1}{n}\CV_k^n(t).
\]
By Lemma~\ref{lem:ResDeadFLLN}, as $n\to\infty$,
\begin{equation}\label{eq:V}
\CVbn(\cdot)\Rightarrow{\mathcal R}^*(\cdot).
\end{equation}
Even though the steps for proving tightness of $\{ \bar{\mathcal
R}^n(\cdot)\}_{n\in\N}$
and $\{\bzn(\cdot)\}_{n\in\N}$ are executed consecutively, it is worth
noting that
Lemma~\ref{lem:ResDeadFLLN} is actually used to prove tightness of
$\{\bzn(\cdot)\}_{n\in\N}$ through the application of \eqref{eq:A}
and~\eqref{eq:V}.
\subsection{Compact Containment}\label{sec:cc}
In this section, we demonstrate the compact containment properties
needed to prove
tightness of $\{\CRbn(\cdot)\}_{n\in\N}$ and of $\{\bzn(\cdot)\}
_{n\in\N}$.
In both cases, we will utilize \cite[Lemma 15.7.5]{ref:K}. The
application turns out to be
simpler for $\{\CRbn(\cdot)\}_{n\in\N}$ since these are measures in~$\M_1^K$.
\paragraph{Compact Containment in $\M_1^K$}
Let $K_m=[0, m]$ and let $K_m^c$ denote its complement.
Then $\{K_m\}_{m\in\N}$ forms a sequence of sets that increases to
$\Rp$.
By \cite[Lemma 15.7.5]{ref:K}, $\K'\subset\M_1^K$ is relatively compact
if and only if there exists a positive constant $\check z$
and a sequence of positive constants $\{b_m\}_{m\in\N}$ tending to zero
such that for each $\zeta\in\K'$
\[
\max_{1\le k\le K} \zeta_k(\Rp) \le\check z
\qquad\mbox{and}\qquad
\max_{1 \le k \le K} \zeta_k(K_m^c) \le b_m, \ \forall\ m \in\N.
\]
Note that for $1\le k\le K$ and $m\in\N$, $\zeta_k(K_m^c)\le\langle\chi, \zeta_k\rangle /m$.
Hence it suffices to show
that there exist positive constants $\check z$ and $\check w$
such that for each $\zeta\in\K'$
\begin{equation}\label{eq:Kcompact1}
\max_{1\le k\le K} \zeta_k(\Rp) \le\check z
\qquad\mbox{and}\qquad
\max_{1 \le k \le K} \left\langle \chi, \zeta_k\right\rangle \le\check w.
\end{equation}
We will verify \eqref{eq:Kcompact1} to prove Lemma~\ref{lem:cc1}.
\begin{lemma} \label{lem:cc1} Suppose that (AA) holds.
Let $T, \eta>0$.
There exists a compact set $\K\subset\M^K_1$ such that
\[
\liminf_{n \to\infty}
\bP(\CRbn(t) \in\K\mbox{ for all } t \in[0, T]) \ge1- \eta.
\]
\end{lemma}
\begin{proof} Fix $T, \eta>0$. For all $1\le k\le K$ and $t\in[0, T]$,
\begin{equation}\label{eq:UBs}
\CRbn_k(t)(\Rp)\le\bar E_k^n(T)
\qquad\hbox{and}\qquad
\left\langle \chi, \CRbn_k(t) \right\rangle \le\left\langle \chi, \CGbn_k(T) \right\rangle .
\end{equation}
Define
\[
\Omega_1^n=\left\{\max_{1\le k\le K} \bar E_k^n(T) \le2\lambda_+
T\right\}
\quad\hbox{and}\quad
\Omega_2^n=\left\{\max_{1\le k\le K} \left\langle \chi, \CGbn_k(T)
\right\rangle \le
2g_+T\right\}.
\]
Set $\Omega_0^n=\Omega_1^n\cap\Omega_2^n$. By \eqref{eq:EincFLLN} and
Lemma~\ref{lem:deadFLLN},
\begin{equation}\label{eq:HighProb}
\liminf_{n \to\infty}\bP\left( \Omega^n_0 \right)\ge1 - \eta.
\end{equation}
The result follows by combining \eqref{eq:Kcompact1}, \eqref{eq:UBs}
and \eqref{eq:HighProb}.
\end{proof}
\paragraph{Compact Containment in $\M_2^K$}
For each $m \in\N$, let $K_m = [0, m] \times[0, m]$ and let $K_m^c$
denote its complement.
Then $\{K_m\}_{m\in\N}$ forms a sequence of sets that increases to
$\Rp^2$.
By \cite[Lemma 15.7.5]{ref:K}, $\K'\subset\M_2^K$ is relatively compact
if and only if there exists a positive constant $\check z$
and a sequence of positive constants $\{b_m\}_{m\in\N}$ tending to zero
as $m$ tends to infinity
such that for each $\zeta\in\K'$
\begin{equation}\label{eq:Kcompact}
\max_{1\le k\le K} \zeta_k(\Rp^2) \le\check z
\qquad\mbox{and}\qquad
\max_{1 \le k \le K} \zeta_k(K_m^c) \le b_m, \ \forall\ m \in\N.
\end{equation}
We will verify \eqref{eq:Kcompact} to prove Lemma~\ref{lem:compact
containment}.
\begin{lemma} \label{lem:compact containment}
Let $T, \eta>0$.
There exists a compact set $\K\subset\M^K_2$ such that
\[
\liminf_{n \to\infty}
\bP(\bzn(t) \in\K\mbox{ for all } t \in[0, T]) \ge1- \eta.
\]
\end{lemma}
\begin{proof}
Fix $T, \eta> 0$.
First we identify a sequence of sets $\{\Omega_0^n\}_{n\in\N}$ for which
we will show that for each $n\in\N$ on $\Omega_0^n$, $\bzn(t)$ remains
in a particular relatively compact set for all $t\in[0, T]$.
By \eqref{eq:IC}, there exists a compact set $\K_0$ such that
\[
\liminf_{n\to\infty}\bP(\bzn(0)\in\K_0)\ge1-\frac{\eta}{4}.
\]
Since $\K_0$ is compact, there exists a positive constant $\check z_0$
and a sequence of positive constants $\{ a_m\}_{m\in\N}$
tending to zero as $m$ tends to infinity such that
\[
\K_0\subset\left\{ \zeta\in\M^K_2 :
\max_{1\le k\le K} \zeta_k(\Rp^2) \le\check z_0
\mbox{ and }
\max_{1 \le k \le K} \zeta_k(K_m^c) \le a_m, \ \forall\ m \in\N
\right\}.
\]
For $n\in\N$, let
\[
\Omega^n_1
=
\left\{
\max_{1\le k\le K} \bzn_k(0)(\Rp^2) \le\check z_0
\hbox{ and }
\max_{1 \le k \le K} \bzn_k(0)(K_m^c)\le a_m, \ \forall\ m \in\N
\right\}.
\]
Then
\[
\liminf_{n \to\infty}
\bP\left(\Omega^n_1 \right)
\ge1- \frac{\eta}{4}.
\]
For $n\in\N$, let
\[
\Omega^n_2
=
\left\{
\max_{1\le k\le K} \Ebn_k(T)
\le
2\lambda_+ T
\right\}.
\]
By \eqref{eq:EincFLLN},
\[
\liminf_{n \to\infty}
\bP\left( \Omega^n_2 \right)
\ge
1-\frac{\eta}{4}.
\]
Note that, for all $n\in\N$,
\[
\sup_{t\in[0, T]}W^n(t)\le W^n(0) + \frac{1}{n}\sum_{k=1}^K \sum
_{i=1}^{E_k^n(T)} v_{k, i}.
\]
Then, by \eqref{eq:IC} and a functional strong law of large numbers,
there exists a positive
constant $\check{w}$ such that
\[
\liminf_{n\to\infty} \bP\left( \sup_{t\in[0, T]} W^n(t) \le
\check
{w}\right)\ge1-\frac{\eta}{4}.
\]
For $n\in\N$, let
\[
\Omega^n_3
=
\left\{\sup_{t\in[0, T]} W^n(t) \le\check{w} \right\}.
\]
Notice that for each $n, m\in\N$, $1\le k\le K$, and $t \in[0, T]$,
\begin{equation}\label{eq:TailBnd}
\bar\dead^n_k(t)((m, \infty))
\le
\bar\dead^n_k(T)((m, \infty))
\le
\frac{\left\langle \chi, \bar\dead^n_k(T) \right\rangle }{m}.
\end{equation}
Set $c = 2 \lambda_+g_+ T$.
For each $m \in\N$, let $c_m=c/m$.
Then \eqref{eq:deadFLLN} and \eqref{eq:TailBnd} together imply that
\[
\liminf_{n \to\infty}
\bP\left(
\max_{1 \le k \le K}\sup_{t\in[0, T]}
\bar\dead^n_k(t)((m, \infty))
\le
c_m, \
\forall\ m \in\N
\right)
\ge
1-\frac{\eta}{4}.
\]
For $n\in\N$, let
\[
\Omega^n_4
=
\left\{
\max_{1 \le k \le K}\sup_{t\in[0, T]}
\bar\dead^n_k(t)((m, \infty))
\le
c_m, \ \forall\ m \in\N
\right\}.
\]
Finally, for each $n\in\N$ set
\[
\Omega^n_0 = \Omega^n_1 \cap\Omega^n_2 \cap\Omega^n_3 \cap\Omega^n_4.
\]
It follows that
\begin{equation} \label{eq:moreNiceOmegas}
\liminf_{n \to\infty}\bP\left( \Omega^n_0 \right)\ge1 - \eta.
\end{equation}
Next we identify the relatively compact set $\K'$.
For this, let $\check z=\check z_0 + 2\lambda_+T$ and for $m\in\N$, let
\[
b_m=
\begin{cases}
\check z, & 1\le m\le\check w, \\
a_m+c_{m-\check w}, & m>\check w.
\end{cases}
\]
Then $\{b_m\}_{m\in\N}$ is a sequence of postive numbers tending
to zero as $m$ tends to infinity. Let
\[
\K'=\left\{\zeta\in\M^K_2:
\max_{1 \le k \le K} \zeta_k(\Rp^2) \le\check z
\mbox{ and }
\max_{1 \le k \le K} \zeta_k(K_m^c) \le b_m, \forall m \in\N
\right\}.
\]
Then, by \eqref{eq:Kcompact}, $\K'$ is relatively compact.
It suffices to show that for each $n\in\N$, on $\Omega^n_0$,
$\bzn(t) \in\K'$ for all $t \in[0, T]$. Fix $n\in\N$.
On $\Omega^n_1 \cap\Omega^n_2$ we have that, for all $1 \le k \le K$
and $t \in[0, T]$,
\begin{equation}\label{eq:TM}
\bzn_k(t)(\Rp^2)
\le
\bzn_k(0)(\Rp^2) + \bar{E}^n_k(T)
\le
\check z_0 + 2\lambda_+T=\check z.
\end{equation}
Next, by \eqref{eq:dynamics1} under fluid scaling and the fact that for
any $x\in\Rp$ and $m\in\N$,
$(K_m^c)_x\subseteq K_m^c$, we have, for each $m\in\N$, $1 \le k \le
K$, and $t\in[0, T]$,
\begin{eqnarray*}
\bzn_k(t) (K^c_m)
&=&
\frac{1}{n}
\sum_{j = 1}^{A^n_k(t)}
1_{ \left(K^c_m\right)_{t-t_{k, j}^n} }(w_{k, j}^n, p_{k, j}^n) \\
&\le&
\frac{1}{n}
\sum_{j = 1}^{A^n_k(t)}
1_{K^c_m }(w_{k, j}^n, p_{k, j}^n) \\
&\le&
\bzn_k(0)(K_m^c)+
\frac{1}{n}
\sum_{j =Z^n_k(0)+1}^{A^n_k(t)}\left(
1_{\left\{w^n_{k, j} > m\right\}}
+
1_{\left\{p^n_{k, j} > m\right\}}\right).
\end{eqnarray*}
Recall that on $\Omega^n_1$, $\max_{1\le k\le K}\bzn_k(0)(K^c_m) \le
a_m$ for all $m\in\N$.
Further, on $\Omega^n_3$, for $m > \check w$, $1_{\{w^n_{k, j} >m\}}=0$
for each $1\le k\le K$ and $Z^n_k(0)+1\le j \le A^n_k(T)$.
Additionally, on $\Omega^n_3$,
for each $1\le k\le K$ and $Z^n_k(0)+1\le j \le A^n_k(T)$,
\[
p^n_{k, j}\le d_{k, j-Z^n_k(0)} +W^n(t_{k, j}^n)\le d_{k, j-Z^n_k(0)} +
\check{w}.
\]
Hence, on $\Omega^n_3 \cap\Omega^n_4$, for $m>\check w$ and $1\le
k\le K$,
\[
\frac{1}{n}
\sum_{j = Z^n_k(0)+1}^{A^n_k(t)}
1_{\left\{p^n_{k, j} > m\right\}}
\le
\bar\dead^n_k(t)((m-\check w, \infty))
\le
c_{m-\check w}.
\]
Then on $\Omega_1^n\cap\Omega_3^n\cap\Omega_4^n$ for $m>\check w$,
\begin{equation}\label{eq:TailM}
\bzn_k(t) (K^c_m)\le a_m+c_{m-\check w}.
\end{equation}
By \eqref{eq:TM} and \eqref{eq:TailM}, it follows that on $\Omega^n_0$,
$\bzn_k(t) (K^c_m)\le b_m$ for all $m\in\N$. This together with
\eqref{eq:TM}
implies that on $\Omega^n_0$, $\bzn(t) \in\K'$ for each $t \in[0, T]$.
\end{proof}
\subsection{Asymptotic Regularity}
This section contains results that are preparatory for proving the
oscillation bounds.
For the sequences of fluid scaled residual deadline related processes
and state descriptors,
the sudden arrival or departure of a
large amount of mass may result in a large oscillation. The focus here
is showing
that it is very unlikely that large
oscillations take place due to departing mass.
\paragraph{Asymptotic Regularity in $\M_1^K$}
Consider the sequence $\{ \CRbn(\cdot)\}_{n\in\N}$ of fluid scaled
residual deadline related processes.
Given $x\in\Rp$ and $\kappa>0$, let
\[
I_x^{\kappa}=\left( (x-\kappa)^+, x+\kappa\right).
\]
Note that for $t\in\ptime$, $x\in\Rp$, and $\kappa>0$, the mass in
$I_{x}^{\kappa}$ at time $t$ will all depart the system
during the time interval $(t+(x-\kappa)^+, t+x+\kappa)$. This time
interval is small if $\kappa$ is small.
Hence, in order to avoid an abrupt departure of a large amount of mass,
one needs to show that asymptotically
such sets contain arbitrarily small mass. This is stated precisely in
the following lemma.
\begin{lemma}\label{lem:BndReg1} Suppose that (AA) holds.
Let $T, \varepsilon, \eta>0$. Then there exists $\kappa>0$ such that
\[
\liminf_{n\to\infty}\bP\left(
\max_{1 \le k \le K}
\sup_{t\in[0, T]} \sup_{x\in\Rp} \CRbn_k(t)(I_x^{\kappa})\le
\varepsilon
\right)\ge1-\eta.
\]
\end{lemma}
\begin{proof}
Fix $T, \varepsilon, \eta> 0$.
Set $\kappa= \frac{\varepsilon}{8 \lambda_+}$ and $M_t = \lceil
t/\kappa\rceil$ for $0< t \le T$.
For each integer $m \ge3$, $n\in\N$, $1\le k\le K$, and $t \in[0, T]$,
\[
\CGbn_k(t)(((m-2) \kappa, \infty))
\le
\CGbn_k(T)(((m-2)\kappa, \infty))
\le
\frac{\left\langle \chi, \CGbn_k(T) \right\rangle }{(m-2) \kappa}.
\]
It follows by Lemma~\ref{lem:deadFLLN} that for some $m \ge3$,
\[
\liminf_{n \to\infty}
\bP\left(
\max_{1 \le k \le K}\sup_{t\in[0, T]}
\CGbn_k(t)(((m - 2) \kappa, \infty))
\le
\varepsilon
\right)
\ge
1-\frac{\eta}{2}.
\]
Fix such an $m$ and for $n\in\N$, let
\[
\Omega^n_1
=
\left\{
\max_{1 \le k \le K}\sup_{t\in[0, T]}
\CGbn_k(t)(((m- 2) \kappa, \infty))
\le
\varepsilon
\right\}.
\]
Also, for $n\in\N$, define
\begin{eqnarray*}
\Omega^n_2
&=&
\left\{
\max_{1 \le k \le K}
\sup_{0\le s\le t\le T}
\max_{0\le j \le m+M_T}
\left|
\CGbn_k(s, t)( (j\kappa, (j+4)\kappa))\right.\right.\\
&&\qquad\left. \left. -
\dead^*_k(s, t)( (j\kappa, (j+4)\kappa))
\right|
\le\frac{\varepsilon}{2 M_T}
\right\}.
\end{eqnarray*}
By Corollary~\ref{cor:deadlineLimit},
\[
\liminf_{n \to\infty}\bP\left(\Omega^n_2\right) \ge1-\frac
{\eta}{2}.
\]
For $n\in\N$, set $\Omega_0^n = \Omega^n_1 \cap\Omega^n_2$. Then, we
have that
\[
\liminf_{n \to\infty}\bP\left(\Omega_0^n \right) \ge1-\eta.
\]
Fix $n\in\N$, $1 \le k \le K, x\in\Rp, $ and $0 < t \le T$.
First, note that for any $1 \le i \le E^n_k(t)$, we have
$t^n_{k, i+Z^n_k(0)} \le t$,
so that $(t-t^n_{k, i+Z^n_k(0)})^+ = t-t^n_{k, i+Z^n_k(0)}$.
It follows that
\[
\CRbn_k(t)(I_x^{\kappa})
=
\frac{1}{n}
\sum_{i=1}^{E^n_k(t)}
1_{I^\kappa_x}(g^n_{k, i}(t)) \\
\le
\frac{1}{n}
\sum_{i=1}^{E^n_k(t)}
1_{I^\kappa_{x+t-t^n_{k, i+Z^n_k(0)}}}(g^n_{k, i}).
\]
Then, for $x\ge m\kappa$ and $1 \le i \le E^n_k(t)$,
\[
\left(x+t-t^n_{k, i+Z^n_k(0)}-\kappa\right)^+\ge x-\kappa\ge
(m-1)\kappa
>(m-2)\kappa.
\]
Hence, on $\Omega_1^n$, for $x\ge m\kappa$,
\[
\CRbn_k(t)(I_x^{\kappa})\le\frac{1}{n}
\sum_{i=1}^{E^n_k(t)}
1_{ ((m-2)\kappa, \infty) }(g^n_{k, i})=\CGbn_k(t)\left
(((m-2)\kappa
, \infty)\right)\le\varepsilon.
\]
Otherwise, $x< m\kappa$. Then
\begin{eqnarray*}
\CRbn_k(t)(I_x^{\kappa})
&\le&
\frac{1}{n}\left(
\sum_{j=0}^{M_t-2}
\sum_{i=E^n_k(j \kappa) +1}^{E^n_k((j+1)\kappa)}
1_{I^\kappa_{x+t-t^n_{k, i+Z^n_k(0)}}}(g^n_{k, i})\right.\\
&&+
\left.\sum_{i=E^n_k((M_t-1) \kappa) +1}^{E^n_k(t)}
1_{I^\kappa_{x+t-t^n_{k, i+Z^n_k(0)}}}(g^n_{k, i})\right).
\end{eqnarray*}
Noting that for $0 \le j \le M_t-1$ and $E^n_k(j\kappa) < i \le
E^n_k((j+1)\kappa\vee t)$ we have that
\[
x+t-\kappa- t^n_{k, i+Z^n_k(0)}
\ge
x+t - (j+2) \kappa
\ge\kappa\left( \left\lfloor\frac{x+t}{\kappa} \right\rfloor-
j -
2\right)
\]
and
\[
x+t +\kappa- t^n_{k, i+Z^n_k(0)}
\le
x+t -( j-1) \kappa
\le\kappa\left( \left\lfloor\frac{x+t}{\kappa} \right\rfloor-
j +
2\right).
\]
Then it follows that
\begin{eqnarray*}
\CRbn_k(t)(I_x^{\kappa})
&\le&
\frac{1}{n}
\sum_{j=0}^{M_t-2}
\sum_{i=E^n_k(j \kappa) +1}^{E^n_k((j+1)\kappa)}
1_{I^{2 \kappa}_{\kappa\left( \left\lfloor\frac{x+t}{\kappa}
\right
\rfloor-j \right)}}(g^n_{k, i})\\
&&+\frac{1}{n}
\sum_{i=E^n_k((M_t-1) \kappa) +1}^{E^n_k(t)}
1_{I^{2 \kappa}_{\kappa\left( \left\lfloor\frac{x+t}{\kappa}
\right
\rfloor-(M_t-1) \right)}}(g^n_{k, i}) \\
&=&
\sum_{j=0}^{M_t-2}
\bar{\mathcal{G}}^n_k(j \kappa, (j+1)\kappa)
\left(I^{2 \kappa}_{\kappa\left( \left\lfloor\frac{x+t}{\kappa}
\right\rfloor-j \right)}\right)\\
&&+ \bar{\mathcal{G}}^n_k((M_t-1)\kappa, t )
\left(I^{2 \kappa}_{\kappa\left( \left\lfloor\frac{x+t}{\kappa}
\right\rfloor-(M_t-1) \right)}\right).
\end{eqnarray*}
Since $x<m\kappa$, for all $1\le j\le M_t-1$, the left end point of
$I^{2 \kappa}_{\kappa(\lfloor\frac{x+t}{\kappa}
\rfloor-j)}$ satisfies
\[
0
\le
\left( \kappa\left( \left\lfloor\frac{x+t}{\kappa} \right
\rfloor-j
\right)-2\kappa\right)^+
\le
x+t < m\kappa+M_t\kappa\le(m+M_T)\kappa.
\]
Also note that $0<t-(M_t-1)\kappa\le t- (t/\kappa-1)\kappa=\kappa$.
Then, on $\Omega^n_2$,
since $x< m \kappa$,
\begin{eqnarray*}
\CRbn_k(t)(I_x^{\kappa})
&\le&
\sum_{j=0}^{M_t-2}
{\mathcal{D}}^*_k(j \kappa, (j+1)\kappa)
\left(I^{2 \kappa}_{\kappa\left( \left\lfloor\frac{x+t}{\kappa}
\right\rfloor-j \right)} \right)\\
&&+ {\mathcal{D}}^*_k((M_t-1)\kappa, t)
\left(I^{2 \kappa}_{\kappa\left( \left\lfloor\frac{x+t}{\kappa}
\right\rfloor-(M_t-1) \right)} \right)
+ \frac{\varepsilon}{2}\\
&\le&
\sum_{j=0}^{M_t-1}
\lambda_k \kappa\left[
F_k \left(\kappa\left( \left\lfloor\frac{x+t}{\kappa} \right
\rfloor
-j + 2 \right)^+\right) \right. \\
&&\left.\qquad\qquad\qquad-
F_k \left(\kappa\left( \left\lfloor\frac{x+t}{\kappa} \right
\rfloor
-j - 2 \right)^+\right)
\right] + \frac{\varepsilon}{2} \\
&\le&
\lambda_k \kappa
\sum_{j=-2}^\infty\left[
F_k((j+4)\kappa) - F_k(j\kappa)
\right]
+\frac{\varepsilon}{2}.
\end{eqnarray*}
Note that $F_k(\cdot)$ is a cumulative distribution function and that
each point in $\Rp$
is included in at most four intervals of the form $(j\kappa
, (j+4)\kappa
]$, $j=-2, -1, 0, 1, 2, \dots$. Thus,
on $\Omega^n_2$, since $x<m\kappa$,
\[
\CRbn_k(t)(I_x^{\kappa}) \le4 \lambda_k \kappa+ \frac{\varepsilon}{2}
\le\varepsilon.
\]
Since, $n\in\N$, $1\le k\le K$, $x\in\Rp$, and $t\in[0, T]$ were chosen
arbitrarily, this concludes the proof.
\end{proof}
\paragraph{Asymptotic Regularity in $\M_2^K$}
We need to prove an analog of Lemma~\ref{lem:BndReg1} for $\{\bzn
(\cdot
)\}_{n\in\N}$.
In particular, we wish to prove the following
prelimit version of Lemma~\ref{prop:NearBoundary}.
\begin{lemma}\label{lem:BndReg}
Let $T, \varepsilon, \eta>0$. Then there exists $\kappa>0$ such that
\[
\liminf_{n\to\infty}\bP\left(
\max_{1 \le k \le K}
\sup_{x\in\Rp^2} \sup_{t\in[0, T]} \bzn_k(t)(C_x^{\kappa})\le
\varepsilon
\right)\ge1-\eta.
\]
\end{lemma}
Before proving Lemma~\ref{lem:BndReg}, we verify the following
regularity result for the initial state,
which is the stochastic analog of \eqref{eq:InitialCorner}.
\begin{lemma}\label{lem:BndReg0} Let $\varepsilon, \eta>0$. Then there
exists $\kappa>0$ such that
\[
\liminf_{n\to\infty}\bP\left(
\max_{1\le k\le K}
\sup_{x\in\Rp^2} \bzn_k(0)(C_x^{\kappa})\le\varepsilon\right
)\ge1-\eta.
\]
\end{lemma}
\begin{proof} Fix $\varepsilon, \eta>0$. Given $i=1, 2$, recall
definition \eqref{def:pi} of the projection
mapping $\pi_i:\M_2\to\M_1$. We apply the argument given in
\cite[Pages 835--836]{ref:GPW} for measures in $\M_1$ to the
projection mappings
applied to $\bzn_+(0)$, $n\in\N$, to verify that there exists
$\kappa>0$
such that
\begin{equation}\label{eq:d1}
\liminf_{n\to\infty}
\bP\left( \max_{i=1, 2}\sup_{x\in\Rp} \left\langle 1_{[(x-\kappa)^+,
x+\kappa]},
\pi_i\left({\bzn_+(0)}\right)\right\rangle
<
\frac{\varepsilon}{2}\right)\ge1-\eta.
\end{equation}
The desired result follows from \eqref{eq:d1} since for all $n\in\N$,
$1\le k\le K$, $x=(x_1, x_2)\in\Rp^2$ and
$\kappa>0$,
\[
\bzn_k(0)(C_x^{\kappa})
\le
\left\langle 1_{[(x_1-\kappa)^+, x_1+\kappa]}, \pi_1\left(\bzn
_+(0)\right
)\right\rangle
+
\left\langle 1_{[(x_2-\kappa)^+, x_2+\kappa]}, \pi_2\left(\bzn
_+(0)\right
)\right\rangle .
\]
In order to verify \eqref{eq:d1}, we must verify that suitable
$K$-dimensional analogs of
\cite[(3.19)--(3.22)]{ref:GPW} hold. For this, for $\zeta\in\M_2^K$
and $i=1, 2$ we adopt
the shorthand notation
\[
\pi_i(\zeta)=\left( \pi_i(\zeta_1), \dots, \pi_i(\zeta_K)\right
)\hbox{
and }
\left\langle \chi, \pi_i(\zeta)\right\rangle = \left(\left\langle \chi, \pi_i(\zeta
_1)\right\rangle ,
\dots, \left\langle \chi
, \pi_i(\zeta_K)\right\rangle
\right).
\]
Note that for $\nu\in\M_2$ and $i=1, 2$, $\langle \chi, \pi_i(\nu)
\rangle
=\langle \chi
\circ p_i, \nu\rangle =\langle p_i, \nu\rangle $.
Then, by \eqref{eq:IC}, as $n\to\infty$,
\begin{eqnarray*}
&&\left( \pi_1(\bzn(0)), \pi_2(\bzn(0)), \left\langle \chi, \pi_1(\bzn
(0))\right\rangle , \left\langle
\chi, \pi_2(\bzn(0))\right\rangle \right)\\
&&\qquad\Rightarrow
\left( \pi_1(\z_0^*), \pi_1(\z_0^*), \left\langle \chi, \pi_1(\z_0^*)
\right\rangle ,
\left\langle \chi
, \pi_1(\z_0^*) \right\rangle \right),
\end{eqnarray*}
which is the $K$-dimensional analog of \cite[(3.19)]{ref:GPW}. Using
(A.3) we obtain the following $K$-dimensional
analog of \cite[(3.20)]{ref:GPW}:
\[
\max_{1\le k\le K}\max_{i=1, 2} \bE\left[ \left\langle 1, \pi_i(\z
_{0, k}^*)\right\rangle
\right]
\le
\bE\left[ \z_{0, +}^*(\Rp^2) \right]<\infty.
\]
Using (A.2) we obtain the following $K$-dimensional analog of \cite
[(3.21)]{ref:GPW}:
\[
\max_{1\le k\le K}\max_{i=1, 2}\bE\left[\left\langle \chi, \pi_i(\z
_{0, k}^*)\right\rangle
\right]
\le
\bE\left[ \left\langle p_1+p_2, \z_{0, +}^*\right\rangle \right]<\infty.
\]
Using (A.1) (and in particular (I.1)) gives the following
$K$-dimensional analog of \cite[(3.22)]{ref:GPW}:
\[
\bP\left( \max_{1\le k\le K}\max_{i=1, 2}\sup_{x\in\Rp} \left
<1_{\{
x\}},
\pi_i(\z_{0, k}^*)\right\rangle =0\right)=1.\qedhere
\]
\end{proof}
Before moving on to prove Lemma~\ref{lem:BndReg}, we obtain an almost sure
upper bound on the mass in $C_{(x, y)}^\kappa$ in an arbitrary
coordinate of the $n$th
system at time $t$ for each $\kappa>0$, $x, y \in\Rp$, and $t\in
\ptime$.
To this end,
let $n\in\N$, $1\le k\le K$, $t\in\ptime$, $x, y \in\Rp$, and
$\kappa
>0$. The $n$th
system analog of \eqref{eq:dynamics2} for the set $C_{(x, y)}^\kappa$ is
\begin{eqnarray*}
\zn_k(t)(C_{(x, y)}^\kappa)
&\le&
\zn_k(0)\left( \left(C_{(x, y)}^{\kappa}\right)_t\right)
+
\sum_{j=Z^n_k(0)+1}^{A^n_k(t)}
1_{\left(C_{(x, y)}^{\kappa}\right)_{t-t_{k, j}^n}} (w_{k, j}^n,
p_{k, j}^n)
\\
&=&
\zn_k(0)\left( \left(C_{(x, y)}^{\kappa}\right)_t\right)
+
\sum_{j=Z^n_k(0)+1}^{A^n_k(t)}
1_{C_{(x, y)}^{\kappa}} (w_{k, j}^n(t), p_{k, j}^n(t)).
\end{eqnarray*}
But then, since the $(x, y)$-shift of a set followed by the $\kappa
$-enlargement
contains the $\kappa$-enlargement followed by the $(x, y)$-shift,
\begin{equation}\label{eq:jobsInCorner1}
\zn_k(t)(C_{(x, y)}^{\kappa})
\le
\zn_k(0)\left(C_{(x+t, y+t)}^{\kappa}\right)
+
\sum_{j=Z^n_k(0)+1}^{A^n_k(t)}
1_{ C_{(x, y)}^{\kappa} } (w_{k, j}^n(t), p_{k, j}^n(t)).
\end{equation}
We simplify the summation term by focusing on each coordinate separately.
For this, we classify jobs by those whose residual patience time causes the
associated unit atom to lie in a certain horizontal band and those
whose residual
virtual sojourn time causes the associated unit atom to lie in a certain
vertical band. Then, for each $n\in\N$, $1\le k\le K$, $t\in\ptime$,
$x, y \in\Rp$, and
$\kappa> 0$,
\begin{eqnarray} \label{eq:jobsInCorner}
\zn_k(t)(C_{(x, y)}^{\kappa})
&\le&
\zn_k(0)(C_{(x+t, y+t)}^{\kappa})
+
\sum_{j=Z^n_k(0)+1}^{A^n_k(t)}
1_{I^\kappa_{x}} (w^n_{k, j}(t))\\
&&+
\sum_{j=Z^n_k(0)+1}^{A^n_k(t)}
1_{I^\kappa_{y}} (p^n_{k, j}(t)).\nonumber
\end{eqnarray}
We are prepared to prove Lemma~\ref{lem:BndReg}. The proof given below
can be
regarded as a stochastic version of the proof of Lemma~\ref{prop:NearBoundary}.
\begin{proof}[Proof of Lemma~\ref{lem:BndReg}]
Fix $\varepsilon$, $\eta>0$ and $T>\frac{\varepsilon}{24 \lambda_+}$.
We begin by defining a sequence $\{\Omega_0^n\}_{n\in\N}$ of events
on which the prelimit processes satisfy properties analogous to those
exhibited by fluid model solutions and used in the proof
of Lemma~\ref{prop:NearBoundary}.
By Lemma~\ref{lem:BndReg0}, there exists $\kappa_0>0$ such that
\[
\liminf_{n\to\infty} \bP\left( \max_{1\le k\le K} \sup_{x, y \in
\Rp}
\bzn_k(0)\left(C_{(x, y)}^{\kappa_0}\right)\le\frac{\varepsilon
}{4}\right)\ge1-\frac{\eta}{6}.
\]
For $n\in\N$, let
\[
\Omega^n_1
=
\left\{\max_{1 \le k \le K}
\sup_{x, y \in\Rp} \bzn_k(0)\left(C_{(x, y)}^{\kappa_0}\right
)\le
\frac
{\varepsilon}{4}\right\}.
\]
By \eqref{eq:A} and \eqref{eq:V}, there exists a $\kappa_1 > 0$ such that
\[
\liminf_{n\to\infty} \bP\left( \max_{1\le k\le K} \sup_{y\in\Rp
} \sup
_{t \in[0, T]}
\bar{\mathcal{A}}^n_k(t)(I_y^{\kappa_1})\le\frac{\varepsilon
}{4}\right
)\ge1-\frac{\eta}{6}
\]
and
\[
\liminf_{n\to\infty} \bP\left( \max_{1\le k\le K} \sup_{y \in
\Rp} \sup
_{t \in[0, T]}
\bar{\mathcal{V}}^n_k(t)(I_y^{\kappa_1})\le\frac{\varepsilon
}{4}\right
)\ge1-\frac{\eta}{6}.
\]
For $n\in\N$, let
\[
\Omega^n_2
=
\left\{\max_{1 \le k \le K}
\sup_{y \in\Rp}
\sup_{t \in[0, T]}
\bar{\mathcal{A}}^n_k(t)(I_y^{\kappa_1})\le\frac{\varepsilon
}{4}\right
\}
\]
and
\[
\Omega^n_3
=
\left\{\max_{1 \le k \le K}
\sup_{y \in\Rp}
\sup_{t \in[0, T]}
\bar{\mathcal{V}}^n_k(t)(I_y^{\kappa_1})\le\frac{\varepsilon
}{4}\right
\}.
\]
For $n\in\N$, let
\begin{eqnarray*}
\Omega^n_4
&=&
\left\{\max_{1\le k\le K} \sup_{0\le s\le t\le T}
\bar{A}^n_k(t) - \bar{A}^n_k(s) \le2 \lambda_+ (t-s)
\right\}.
\end{eqnarray*}
Since for all $n\in\N$, $1\le k\le K$, and $0\le s\le t\le T$, $\bar
{A}^n_k(t) - \bar{A}^n_k(s)=\bar{E}^n_k(t) - \bar{E}^n_k(s)$,
\eqref{eq:EincFLLN} implies that
\[
\liminf_{n\to\infty} \bP\left(\Omega_4^n \right)
\ge1-\frac{\eta}{6}.
\]
Next we identify positive constants $\delta$ and $M$, analogous to the
constants $\delta$
and $M$ defined in the proof of Lemma~\ref{prop:NearBoundary}.
Let $\delta= \frac{\varepsilon}{24 \lambda_+}$. Then $\delta<T$.
In order to define $M$, there are two
cases to consider, based on the nature of the abandonment distributions.
\noindent\textit{Case 1}: First suppose that $d_{\max}< \infty$. Let
$w_{\max}(\cdot)$ denote the maximal
workload fluid model solution, i.e., the workload fluid model solution
such that $w_{\max}(0)=d_{\max}$.
By the relative ordering property of workload fluid model solutions, it
follows that for any workload fluid
model solution $w(\cdot)$, $w(t)\le w_{\max}(t)$ for all $t\in\ptime$.
In particular, for any workload fluid
model solution $w(\cdot)$ for all $t\ge\delta$,
\begin{equation}\label{eq:AfterDelta}
w(t)\le w_{\max}(\delta).
\end{equation}
Set $M_1=(w_{\max}(\delta)+d_{\max})/2$.
By monotonicity properties of workload fluid model
solutions, $w_u < w_{\max}(\delta)<M_1<d_{\max}$.
It follows from \eqref{eq:IC}, (A.1), Theorem~\ref{thrm:JR},
and \eqref{eq:AfterDelta} that
\[
\bP\left( \sup_{t \in[\delta, T]} W^*(t) < M_1 \right) = 1.
\]
\noindent\textit{Case 2}: Next consider the case $d_{\max}=\infty$. By
\eqref{eq:IC} and (A.1),
there exists $M_2 > w_u$ such that
\[
\bP\left( W^*(0) < M_2 \right) \ge1 - \frac{\eta}{6}.
\]
Since $M_2 > w_u$, \eqref{eq:IC}, (A.1), Theorem~\ref{thrm:JR} and
monotonicity properties of workload fluid model solutions imply that
\[
\bP\left( \sup_{t \in[0, T]} W^*(t) < M_2 \right)
=
\bP\left( W^*(0) < M_2 \right)
\ge
1 - \frac{\eta}{6}.
\]
Set
\[
M=
\begin{cases} M_1, &\hbox{if }d_{\max}<\infty, \\ M_2, &\hbox{if
}d_{\max
}=\infty.
\end{cases}
\]
Now we proceed to bound the prelimt processes by $M$ with probability
asymptotically
close to one. For $n\in\N$, let
\[
\Omega_5^n = \left\{ \sup_{t\in[\delta, T]} W^n(t)<M\right\}.
\]
Note that by \eqref{eq:IC}, (A.1), Theorem~\ref{thrm:JR} and that fact
that workload fluid model
solutions are continuous, the convergence in distribution in \eqref
{eq:JR} takes place with
respect to the topology of uniform convergence on compact sets.
Furthermore, the set
$\{ f\in\bD(\ptime, \Rp) : \sup_{t\in[\delta, T]} f(t)<M\}$ is
open with
respect to this topology.
Then by the Portmanteau theorem
\[
\liminf_{n \to\infty}\bP\left( \Omega_5^n\right)\ge1-\frac
{\eta}{6}.
\]
Set
\[
c=\sum_{k=1}^K\rho_k G_k(M)
\qquad\hbox{and}\qquad
\kappa= \min\left( \kappa_0, \kappa_1, \frac{\varepsilon}{24
\lambda
_+}, \frac{\varepsilon c}{72\lambda_+}\right).
\]
Note that $c>0$ since $M<d_{\max}$, and so $\kappa>0$. Finally, for
$n\in\N$, let
\[
\Omega_6^n=\left\{ \sup_{t\in[0, T]} \left| X^n(t)\right| \le
\frac
{\kappa}{2} \right\}.
\]
Then, by \eqref{eq:Xn},
\[
\liminf_{n \to\infty}\bP\left( \Omega_6^n\right)\ge1-\frac
{\eta}{6}.
\]
For $n\in\N$, set
\[
\Omega^n_0 = \Omega^n_1 \cap\Omega^n_2 \cap\Omega^n_3 \cap\Omega
^n_4\cap\Omega_5^n\cap\Omega_6^n.
\]
Then
\begin{equation} \label{eq:niceOmegas}
\liminf_{n \to\infty}
\bP(\Omega^n_0) \ge1 - \eta.
\end{equation}
Fix $n\in\N$ such that $1/n\le\varepsilon/12$, $1\le k\le K$, $x, y
\in
\Rp$, and $t\in[0, T]$.
The fluid scaled analog of (\ref{eq:jobsInCorner}) is
\begin{eqnarray} \label{eq:lem:BndReg1}
\bzn_k(t)\left(C_{(x, y)}^{\kappa}\right)
&\le&
\bzn_k(0)\left(C_{(x+t, y+t)}^{\kappa}\right)
+
\frac{1}{n}
\sum_{j=Z^n_k(0)+1}^{A^n_k(t)}
1_{I^\kappa_{x}} \left(w^n_{k, j}(t)\right) \\
&&+ \nonumber
\frac{1}{n}
\sum_{j=Z^n_k(0)+1}^{A^n_k(t)}
1_{I^\kappa_{y}} \left(p^n_{k, j}(t)\right).
\end{eqnarray}
We will show that on $\Omega_0^n$, the right hand side of \eqref
{eq:lem:BndReg1} is
less than or equal to~$\varepsilon$.
Since $\kappa\le\kappa_0$, we have $C_{(x+t, y+t)}^{\kappa}
\subseteq
C_{(x+t, y+t)}^{\kappa_0}$.
Then, on $\Omega^n_1$,
\begin{equation} \label{eq:lem:BndReg1.1}
\bzn_k(0)\left(C_{(x+t, y+t)}^{\kappa}\right) \le\frac
{\varepsilon}{4}.
\end{equation}
Also, for all $Z_k^n(0)+1\le j\le A_k^n(t)$, the quantity
$p^n_{k, j}(t)$ is either
$a^n_{k, j-Z^n_k(0)}(t)$ of
$v^n_{k, j-Z^n_k(0)}(t)$ (recall \eqref{eq:a} and \eqref{eq:v}). It
follows that, on
$\Omega^n_2 \cap\Omega^n_3$,
\begin{equation} \label{eq:lem:BndReg1.3}
\frac{1}{n}
\sum_{j=Z^n_k(0)+1}^{A^n_k(t)}
1_{I_y^\kappa}
\left(p^n_{k, j}(t)\right)
\le
\bar{\mathcal{A}}^n_k(t)\left(I_y^\kappa\right)
+
\bar{\mathcal{V}}^n_k(t)\left(I_y^\kappa\right)
\le
\frac{\varepsilon}{2}.
\end{equation}
Combining \eqref{eq:lem:BndReg1}, \eqref{eq:lem:BndReg1.1}, and
\eqref
{eq:lem:BndReg1.3}, we see that on $\Omega_0^n$,
\begin{equation} \label{eq:BndReg2}
\bzn_k(t)\left(C_{(x, y)}^{\kappa}\right)
\le
\frac{3\varepsilon}{4}
+
\frac{1}{n}\sum_{j=Z^n_k(0)+1}^{A^n_k(t)}
1_{I^\kappa_x} \left(w^n_{k, j}(t)\right).
\end{equation}
Finally, we bound the second term on the right side of \eqref{eq:BndReg2}.
To this end, define the prelimit version $\tau^n(\cdot)$ of $\tau
(\cdot
)$ as follows:
\[
\tau^n(s)=\inf\left\{u \ge0: W^n(u) + u \ge s\right\}, \qquad s
\in
\ptime.
\]
Notice that for all $s\in\ptime$, $W^n(s) + s\ge s$, so that $\tau^n(s)
\in[0, s]$.
During busy periods, $W^n(\cdot) + \iota(\cdot)$ jumps up exactly when
jobs arrive
that contribute to the workload, and remains constant otherwise. During
idle periods,
$W^n(\cdot) + \iota(\cdot)$ increases at rate one. In particular,
$W^n(\cdot) + \iota(\cdot)$
is right continuous and nondecreasing. Then,
\begin{eqnarray}
W^n(\tau^n(s))+\tau^n(s) &\ge& s, \qquad\hbox{for }s\in\ptime
, \label
{eq:taunprop1}\\
W^n(\tau^n(s)-)+\tau^n(s)&\le&s, \qquad\hbox{for }s\in
(W^n(0), \infty
).\label{eq:taunprop2}
\end{eqnarray}
The function $\tau^n(\cdot)$ will be used to determine which
arrivals may contribute to the sum in \eqref{eq:BndReg2}.
First we show that
\begin{eqnarray} \label{lem:BndReg2.1}
\frac{1}{n}
\sum_{j=Z^n_k(0)+1}^{A^n_k(t)}
1_{I^\kappa_x} \left(w^n_{k, j}(t)\right)
&\le&
\bar{A}^n_k\left( \tau^n(x+ \kappa+t) \wedge t \right)\\
&&-
\bar{A}^n_k\left( \tau^n( (x-\kappa)^++t) \wedge t \right)+\frac
{1}{n}.\nonumber
\end{eqnarray}
To see this, let $Z^n_k(0)+1\le j\le A^n_k(t)$. Then $t_{k, j}^n\le t$ and
\[
\left( x-\kappa\right)^+ < w_{k, j}^n(t) < x+\kappa
\qquad\Leftrightarrow\qquad
\left( x-\kappa\right)^+ + t < w_{k, j}^n + t_{k, j}^n < x+\kappa+t.
\]
But $w_{k, j}^n =W^n( t_{k, j}^n)$. Then, since $W^n(\cdot)+\iota
(\cdot)$
is nondecreasing,
$t_{k, j}^n\le t$ and $w_{k, j}^n(t)\in I_{x}^{\kappa}$ imply that
\[
\tau^n\left(\left( x-\kappa\right)^++t\right)\wedge t \le
t_{k, j}^n\le
\tau^n(x+\kappa+t)\wedge t.
\]
Therefore, \eqref{lem:BndReg2.1} holds.
Next, we proceed to bound the right hand side of \eqref{lem:BndReg2.1}
on $\Omega_0^n$.
By \eqref{eq:BndReg2}, \eqref{lem:BndReg2.1}, the definition of
$\Omega_4^n$,
and the fact that $n$ is such that $1/n\le\varepsilon/12$,
it suffices to show that on $\Omega_0^n$,
\begin{eqnarray}\label{eq:TimeChange}
2\lambda_+\left( \tau^n(x+\kappa+t) \wedge t
-
\tau^n( (x-\kappa)^++t) \wedge t \right)\le\frac{\varepsilon}{6}.
\end{eqnarray}
If $\tau^n( (x-\kappa)^++t)\ge t$, then the left side of \eqref
{eq:TimeChange} is zero,
and so \eqref{eq:TimeChange} holds. Henceforth, we assume that $\tau^n(
(x-\kappa)^++t)< t$.
If $ \tau^n(x+\kappa+t) \wedge t\le\delta$, then \eqref
{eq:TimeChange} holds
since $\delta=\varepsilon/24\lambda_+$, which implies that the left
hand side of
\eqref{eq:TimeChange} is no larger than $\varepsilon/12$.
Otherwise, $\delta< \tau^n(x+\kappa+t) \wedge t$
(so that $\tau^n(x+\kappa+t)>0$ and then $x+\kappa+t>W^n(0)$).
First consider the case where $\delta\le\tau^n( (x-\kappa)^++t)$. Then,
by \eqref{eq:taunprop1}, \eqref{eq:taunprop2}, and the nondecreasing
nature of $W^n(\cdot)+\iota(\cdot)$,
\begin{eqnarray*}
2\kappa
&=&x+\kappa+t-(x-\kappa+t)\ge x+\kappa+t-((x-\kappa)^++t)\\
&\ge& W^n(\tau^n(x+\kappa+t)-)+\tau^n(x+\kappa+t)\\
&&-W^n(\tau^n((x-\kappa)^++t))-\tau^n((x-\kappa)^++t)\\
&\ge& W^n(\tau^n(x+\kappa+t)\wedge t-)+\tau^n(x+\kappa+t)\wedge t\\
&&-W^n(\tau^n((x-\kappa)^++t))-\tau^n((x-\kappa)^++t).
\end{eqnarray*}
Using \eqref{eq:WorkRep} and the nondecreasing nature of the idle time
process, we obtain
\begin{eqnarray*}
2\kappa
&\ge& X^n(\tau^n(x+\kappa+t)\wedge t-)-X^n(\tau^n((x-\kappa
)^++t))\\
&& + \sum_{k=1}^K\rho_k\int_{\tau^n((x-\kappa)^++t)}^{\tau
^n(x+\kappa
+t)\wedge t}G_k(W^n(u))\der u.
\end{eqnarray*}
Since $\delta\le\tau^n( (x-\kappa)^++t)$, on $\Omega_5^n\cap
\Omega_6^n$,
\[
2\kappa\ge-\kappa+ c\left(\tau^n(x+\kappa+t)\wedge t-\tau
^n((x-\kappa
)^++t)\right).
\]
By definition of $\kappa$, $3\kappa/c\le\varepsilon/24\lambda_+$,
which implies that the left side of
\eqref{eq:TimeChange} is no larger than $\varepsilon/12$ on $\Omega
_5^n\cap\Omega_6^n$, and
so \eqref{eq:TimeChange} holds.
The last case to consider is the case $\tau^n( (x-\kappa)^++t)<\delta<
\tau^n(x+ \kappa+t) \wedge t$.
Then, by definition of~$\delta$, on $\Omega_4^n$,
\begin{eqnarray}\label{eq:deltabetween}
&&2\lambda_+\left( \tau^n(x+ \kappa+t) \wedge t
-
\tau^n( (x-\kappa)^++t) \wedge t \right)\\
&&\qquad\le
\frac{\varepsilon}{12}+2\lambda_+\left( \tau^n(x+\kappa+t) \wedge
t-\delta\right).\nonumber
\end{eqnarray}
By \eqref{eq:taunprop1}, \eqref{eq:taunprop2}, and monotonicity of
$W^n(\cdot)+\iota(\cdot)$,
\begin{eqnarray*}
x-\kappa+t &\le& (x-\kappa)^++t\\
&\le& W^n(\tau^n((x-\kappa)^++t))+\tau^n((x-\kappa)^++t)\\
&\le&
W^n(\delta)+\delta\\
&\le&
W^n(\tau^n(x+\kappa+t)\wedge t-)+\tau^n(x+\kappa+t)\wedge t
\le
x+\kappa+t.
\end{eqnarray*}
This together with \eqref{eq:WorkRep} and the nondecreasing nature of
the idle time process
implies that, on $\Omega_5^n\cap\Omega_6^n$,
\begin{eqnarray*}
2\kappa
&\ge& W^n(\tau^n(x+\kappa+t)\wedge t-)+\tau^n(x+\kappa+t)\wedge t-
W^n(\delta)-\delta\\
&\ge& X^n(\tau^n(x+\kappa+t)\wedge t-)-X^n(\delta) \\
&& + \sum_{k=1}^K\rho_k\int_{\delta}^{\tau^n(x+\kappa+t)\wedge t}
G_k(W^n(u))\der u\\
&\ge& -\kappa+ c\left(\tau^n(x+\kappa+t) \wedge t-\delta\right).
\end{eqnarray*}
Using the definition of $\kappa$ and \eqref{eq:deltabetween} implies
\eqref{eq:TimeChange}.
\end{proof}
\subsection{Oscillation Bounds}\label{sec:osc}
This section establishes the second main ingredient for proving
tightness of the fluid scaled residual deadline related processes and state
descriptors, controlled oscillations (see Lemmas~\ref{lem:osc1} and
\ref
{lem:osc}). Both proofs are similar in spirit. The one for the residual
deadline processes is slightly simpler, so it is presented first. For
this, recall the definition of ${\bf d}_K$ given in \eqref{eq:metric}.
Then the modulus of continuity is defined as follows.
\begin{definition} Let $i=1, 2$. For each $\zeta(\cdot)\in\bD
(\ptime
, \M_i^K)$
and each $T>\delta>0$, define the modulus of continuity on $[0, T]$
by
\[
\bw_T(\zeta(\cdot), \delta)=\sup_{t \in[0, T-\delta]}\sup_{h\in
[0, \delta
]} {\bf d}_K[ \zeta(t+h), \zeta(t)].
\]
\end{definition}
\begin{lemma}\label{lem:osc1} Suppose that (AA) holds.
For all $T>0$ and $\varepsilon, \eta\in(0, 1)$, there exists $\delta
\in(0, T)$
such that
\[
\liminf_{n\to\infty} \bP( \bw_T(\CRbn(\cdot), \delta)\le
\varepsilon
)\ge1-\eta.
\]
\end{lemma}
\begin{proof}
Fix $T>0$ and $\varepsilon, \eta\in(0, 1)$.
By \eqref{eq:EFLLN} and Lemma~\ref{lem:BndReg1}, there exists a
$\kappa\in(0, \varepsilon)$ such that for any $\delta\in(0, T)$,
the events
\begin{eqnarray*}
\Omega^n_1
&=&
\left\{ \max_{1\le k \le K}
\sup_{t \in[0, T]} \CRbn_k(t)([0, \kappa]) \le\varepsilon\right\}
, \\
\Omega^n_2
&=&
\left\{\max_{1 \le k \le K}
\sup_{t\in[0, T-\delta]}\sup_{h\in[0, \delta]} \left[\Ebn_k(t+h)
- \Ebn
_k(t)\right] \le2 \lambda_+ \delta\right\}, \\
\Omega^n_0&=&\Omega^n_1 \cap\Omega^n_2,
\end{eqnarray*}
satisfy
\begin{equation} \label{eq:osc0}
\liminf_{n \to\infty} \bP(\Omega^n_0) \ge1-\eta.
\end{equation}
Fix such a $\kappa$ and set $\delta= \min(\kappa, \frac
{\varepsilon}{2 \lambda_+})$.
Fix $n\in\N$, $1\le k\le K$, and a closed set $B\in\CB_1$.
Let $t\in[0, T-\delta]$, $h\in(0, \delta]$, and $1\le i\le E_k^n(t)$.
If $g_{k, i}^n(t)>h$ or $g_{k, i}^n(t+h)>0$, then
\[
g_{k, i}^n(t)=g_{k, i}^n(t+h)+h.
\]
Then, for $h\in(0, \delta]$, since $h\le\delta\le\kappa
<\varepsilon$,
\begin{eqnarray*}
\CRbn_k(t)(B)&\le& \CRbn_k(t+h)(B^{\varepsilon})+\CRbn
(t)([0, \kappa])\\
\CRbn_k(t+h)(B)&\le& \CRbn_k(t)(B^{\varepsilon})+\Ebn_k(t+h)-\Ebn_k(t).
\end{eqnarray*}
Then, on $\Omega_0^n$, for all $t\in[0, T-\delta]$ and $h\in
[0, \delta]$,
\[
{\bf d}(\CRbn_k(t+h), \CRbn_k(t))<\varepsilon.
\]
This together with the fact that $n\in\N$, $1\le k\le K$, and the
closed set $B\in\CB_2$
were arbitrary and \eqref{eq:osc0} implies the result.
\end{proof}
Next we generalize the preceding argument to prove the following
analogous lemma
for the fluid scaled state descriptors. There are two main
distinctions. One is that
$\kappa<\varepsilon/2$ since in $h>0$ time units a unit atom moves
along a diagonal
path a distance of $\sqrt{2}h$. The other is that \eqref{eq:dynamics2}
has been established.
\begin{lemma}\label{lem:osc}
For all $T>0$ and $\varepsilon, \eta\in(0, 1)$, there exists $\delta
\in(0, T)$
such that
\[
\liminf_{n\to\infty} \bP( \bw_T(\bzn(\cdot), \delta)\le
\varepsilon)\ge
1-\eta.
\]
\end{lemma}
\begin{proof}
Fix $T>0$ and $\varepsilon, \eta\in(0, 1)$.
For each $\kappa>0$, let $\bar{C}^\kappa$ be the closure of
$C^\kappa
$, or
\[
\bar{C}^\kappa= [0, \kappa] \times\Rp\cup\Rp\times[0, \kappa].
\]
By \eqref{eq:EFLLN} and Lemma~\ref{lem:BndReg}, there exists a
$\kappa\in(0, \varepsilon/2)$ such that for any $\delta\in(0, T)$,
the events
\begin{eqnarray*}
\Omega^n_1
&=&
\left\{ \max_{1\le k \le K}
\sup_{t \in[0, T]} \bzn_k(t)(\bar{C}^\kappa) \le\varepsilon
\right
\}
, \\
\Omega^n_2
&=&
\left\{\max_{1 \le k \le K}
\sup_{t\in[0, T-\delta]}\sup_{h\in[0, \delta]} \left[\Ebn_k(t+h)
- \Ebn
_k(t)\right] \le2 \lambda_+ \delta\right\}, \\
\Omega^n_0&=&\Omega^n_1 \cap\Omega^n_2,
\end{eqnarray*}
satisfy
\begin{equation} \label{eq:lem:osc0}
\liminf_{n \to\infty} \bP(\Omega^n_0) \ge1-\eta.
\end{equation}
Fix such a $\kappa$ and set $\delta= \min(\kappa, \frac
{\varepsilon}{2 \lambda_+})$.
We begin by noting two basic facts that will be used in the proof.
Firstly, for all $B\in\CB_2$ and $h\in[0, \delta]$,
\begin{equation} \label{eq:lem:osc1.1}
B \subseteq(B^\varepsilon)_h \cup\bar{C}^\kappa.
\end{equation}
To see this, take some $B\in\CB_2$, $h\in[0, \delta]$, and $(w, p)
\in B
\backslash\bar{C}^\kappa$.
By the construction of $\bar{C}^\kappa$, we have $w, p > \kappa\ge
\delta\ge h$.
Because $h \le\delta\le\kappa< \varepsilon/2$, it follows that
$(w-h, p-h) \in B^\varepsilon$
and $(w, p) \in(B^\varepsilon)_h$. In addition, since $\delta
<\varepsilon/2$, we have that
for all $B\in\CB_2$ and $h\in[0, \delta]$,
\begin{equation} \label{eq:lem:osc1.2}
B_h \subseteq B^\varepsilon.
\end{equation}
Fix $n\in\N$, $1\le k\le K$, and a closed set $B\in\CB_2$. Let
$\check
B=B\setminus C$.
Note that $\check B\in\CB_{2, 0}$ and by \eqref{eq:zk}, $\bzn
_k(t)(B)=\bzn_k(t)(\check B)$
for all $t\in[0, T]$. Then, we can then conclude from \eqref{eq:zk},
\eqref{eq:dynamics2}, and \eqref{eq:lem:osc1.1} that, on $\Omega^n_0$,
for all
$t\in[0, T-\delta]$ and $h\in[0, \delta]$,
\begin{eqnarray} \nonumber
\bzn_k(t)(B) &=&\bzn_k(t)(\check B)\\
&\le&\nonumber
\bzn_k(t)((\check B^\varepsilon)_h) + \bzn_k(t)( \bar{C}^\kappa) \\
&\le& \nonumber
\bzn_k(t+h)(\check B^\varepsilon) + \varepsilon\\
&\le& \label{eq:lem:osc1}
\bzn_k(t+h)(B^\varepsilon) + \varepsilon.
\end{eqnarray}
Also, by \eqref{eq:zk}, \eqref{eq:dynamics2}, \eqref{eq:lem:osc1.2},
and the fact that $\delta< \frac{\varepsilon}{2\lambda_+}$,
it is true that on $\Omega^n_0$, for all $t\in[0, T-\delta]$ and
$h\in
[0, \delta]$,
\begin{eqnarray} \nonumber
\bzn_k(t+h) (B) &=& \bzn_k(t+h) (\check B)\\
&\le& \nonumber
\bzn_k(t)(\check B_h) + \bar{E}^n_k(t+h) - \bar{E}^n_k(t) \\
&\le& \nonumber
\bzn_k(t)(B_h) + \varepsilon\\
&\le& \label{eq:lem:osc2}
\bzn_k(t)(B^\varepsilon) + \varepsilon.
\end{eqnarray}
Because $k$ and $B$ were chosen arbitrarily, \eqref{eq:lem:osc1} and
\eqref{eq:lem:osc2}
imply that on $\Omega_0^n$
\[
\bw_T(\bzn(\cdot), \delta)\le\varepsilon.
\]
The result follows from this and \eqref{eq:lem:osc0}.
\end{proof}
\section{Characterization of Limit Points} \label{sec:char}
The main goal of this section is to prove Theorem~\ref{thrm:flt}.
First we prove Lemma~\ref{lem:ResDeadFLLN},
which is then used in the proof of Theorem~\ref{thrm:flt}.
\begin{proof}[Proof of Lemma~\ref{lem:ResDeadFLLN}]
Throughout, we assume that (AA) holds.
Together Lemmas~\ref{lem:cc1} and~\ref{lem:osc1} imply tightness of
$\{\CRbn(\cdot)\}_{n\in\N}$. Let $\bbM\subset\N$ be a strictly increasing
subsequence tending to infinity and $\CRb(\cdot)$ a process such that
as $m\to\infty$,
\[
\CRbm(\cdot)\Rightarrow\CRb(\cdot).
\]
By Lemma~\ref{lem:BndReg1}, $\CRb(t)$ doesn't charge points for all
$t\in\ptime$ almost surely.
By \eqref{eq:EFLLN} and Lemma~\ref{lem:deadFLLN} and the deterministic
nature of those limiting processes,
as $m\to\infty$,
\begin{equation}\label{eq:JC1}
\left(\CRbm(\cdot), \bar E^m(\cdot), \bar{\mathcal G}^m(\cdot),
\left\langle
\chi
, \bar{\mathcal G}^m(\cdot)\right\rangle \right)
\Rightarrow
\left(\CRb(\cdot), E^*(\cdot), \CD^*(\cdot), \left\langle \chi, \CD
^*(\cdot
)\right\rangle \right).
\end{equation}
Using the Skorohod representation we may assume without loss of
generally that all
random elements are defined on a common probability space $(\Omega
, {\mathcal F}, \bP)$
such that the joint convergence in \eqref{eq:JC1} is almost sure.
Fix $\omega\in\Omega$ such that $\CRb(t)(\omega)$ doesn't charge points
for all $t\in\ptime$
and as $m\to\infty$,
\begin{eqnarray}\label{eq:as1}
&&
\left(\CRbm(\cdot)(\omega), \bar E^m(\cdot)(\omega), \bar
{\mathcal
G}^m(\cdot)(\omega), \left\langle \chi, \bar{\mathcal G}^m(\cdot)\right
>(\omega
)\right
)\\
&&\qquad\rightarrow
\left(\CRb(\cdot)(\omega), E^*(\cdot), \CD^*(\cdot), \left\langle \chi
, \CD
^*(\cdot
)\right\rangle \right).\nonumber
\end{eqnarray}
Henceforth, all random variables are evaluated at this $\omega$. It
suffices to show that
$\CRb(\cdot)=\CR^*(\cdot)$.
For this, it suffices to show that for all $1\le k\le K$, $t\in\ptime$
and $x\in\Rp$,
\begin{equation}\label{eq:R*}
\CRb_k(t)(x, \infty)=\lambda_k\int_0^t G_k(x+t-s)\der s.
\end{equation}
To see this, note that for all $1\le k\le K$, $t\in\ptime$, and $x\in
\Rp$
\begin{eqnarray*}
\CR_k^*(t)(x, \infty)
&=&
\lambda_k\int_x^{\infty} \left[G_k(y)-G_k(y+t)\right]\der y\\
&=&
\lambda_k\int_x^{x+t} G_k(y)\der y
=
\lambda_k\int_0^t G_k(x+t-s)\der s.
\end{eqnarray*}
Next we verify \eqref{eq:R*}. For this, fixed $1\le k\le K$, $t\in
\ptime
$, and $x\in\Rp$.
Given $L\in\N$, let $\kappa=t/L$ and set
\[
t_{\ell} = \ell\kappa\qquad\hbox{and}\qquad x_{\ell}=x+t-t_{\ell
}, \qquad\hbox{for }\ell=0, \dots, L.
\]
Then, given $L\in\N$, $1\le i\le E_k^m(t)$ if and only if there exists
$\ell\in\{0, \dots, L-1\}$ such that
$t_{k, Z_k^m(0)+i}^m\in(t_{\ell}, t_{\ell+1}]$. Further, given $L\in
\N$,
$1\le i\le E_k^m(t)$ and $\ell\in\{0, \dots, L-1\}$
such that $t_{k, Z_k^m(0)+i}^m\in(t_{\ell}, t_{\ell+1}]$, $x_{\ell
}<g_{k, i}^m$ implies that $x<g_{k, i}^m(t)$. Similarly,
given $L\in\N$, $1\le i\le E_k^m(t)$ and $\ell\in\{0, \dots, L-1\}
$ such
that $t_{k, Z_k^m(0)+i}^m\in(t_{\ell}, t_{\ell+1}]$,
$x<g_{k, i}^m(t)$ implies that $x_{\ell+1}<g_{k, i}^m$. Hence, given
$L\in
\N$,
\[
\sum_{\ell=0}^{L-1} \CGbm_k\left(t_{\ell}, t_{\ell+1}\right
)\left
(x_{\ell
}, \infty\right)
\le
\CRbm_k(t)(x, \infty)
\le
\sum_{\ell=0}^{L-1} \CGbm_k\left(t_{\ell}, t_{\ell+1}\right
)\left
(x_{\ell
+1}, \infty\right).
\]
By \eqref{eq:as1}, given $L\in\N$ and $\varepsilon>0$ there exists
$M\in
\N$
such that for all $m\ge M$
\begin{eqnarray*}
\max_{0\le\ell\le L-1} \left|
\CGbm_k\left(t_{\ell}, t_{\ell+1}\right)\left(x_{\ell}, \infty
\right)
-
\CD_k^*\left(t_{\ell}, t_{\ell+1}\right)\left(x_{\ell}, \infty
\right)
\right|
&<&\frac{\varepsilon}{2L}, \\
\max_{0\le\ell\le L-1} \left|
\CGbm_k\left(t_{\ell}, t_{\ell+1}\right)\left(x_{\ell+1}, \infty
\right)
-
\CD^*_k\left(t_{\ell}, t_{\ell+1}\right)\left(x_{\ell+1}, \infty
\right)
\right|
&<&\frac{\varepsilon}{2L}.
\end{eqnarray*}
Hence, given $L\in\N$ and $\varepsilon>0$ there exists $M\in\N$
such that for all $m\ge M$,
\[
\sum_{\ell=0}^{L-1} \CD_k^*\left(t_{\ell}, t_{\ell+1}\right
)\left
(x_{\ell
}, \infty\right)-\frac{\varepsilon}{2}
\le
\CRbm_k(t)(x, \infty)
\le
\sum_{\ell=0}^{L-1} \CD_k^*\left(t_{\ell}, t_{\ell+1}\right
)\left
(x_{\ell
+1}, \infty\right)+\frac{\varepsilon}{2}.
\]
Given $L\in\N$, we have that
\begin{eqnarray*}
\sum_{\ell=0}^{L-1} \CD_k^*\left(t_{\ell}, t_{\ell+1}\right
)\left
(x_{\ell
}, \infty\right)
&=&
\sum_{\ell=0}^{L-1}\lambda_k\kappa G_k\left(x_{\ell}\right)\\
\sum_{\ell=0}^{L-1} \CD_k^*\left(t_{\ell}, t_{\ell+1}\right
)\left
(x_{\ell
+1}, \infty\right)
&=&
\sum_{\ell=0}^{L-1}\lambda_k\kappa G_k\left(x_{\ell+1}\right).
\end{eqnarray*}
Respectively these are upper and lower Riemann sums, and since
$G_k(\cdot)$ is continuous, they both converge to
$\lambda_k\int_0^t G_k(x+t-s)\der s$ as $L\to\infty$. Given
$\varepsilon
>0$, let $\hat L\in\N$ be such that
\[
\left| \sum_{\ell=0}^{\hat L-1}\lambda_k\kappa G_k\left(x_{\ell
+1}\right
)- \sum_{\ell=0}^{\hat L-1}\lambda_k\kappa G_k\left(x_{\ell}\right
)\right|
<\frac{\varepsilon}{2}.
\]
Hence, given $\varepsilon>0$, there exists $\hat M\in\N$ such that for
all $m\ge\hat M$,
\[
\left|\CRbm_k(t)(x, \infty)-\lambda_k\int_0^t G_k(x+t-s)\der
s\right
|<\varepsilon.
\]
Thus, \eqref{eq:R*} holds, as desired.
\end{proof}
Having proved Lemma~\ref{lem:ResDeadFLLN}, we are now ready to prove
Theorem~\ref{thrm:flt}.
\begin{proof}[Proof of Theorem~\ref{thrm:flt}]
Together Lemmas~\ref{lem:compact containment} and~\ref{lem:osc} imply
tightness of
$\{\bzn(\cdot)\}_{n\in\N}$. Let $\bbM\subset\N$ be a strictly
increasing subsequence
tending to infinity and $\z^*(\cdot)$ a process such that as $m\to
\infty$,
\begin{equation}\label{eq:SubSeq}
\bzm(\cdot)\Rightarrow\z^*(\cdot).
\end{equation}
Note that by (A.1), $\z^*(\cdot)\in\I$ almost surely. Hence, in order
to prove Theorem~\ref{thrm:flt},
it suffices to show that $\z^*(\cdot)$ satisfies \eqref{fdevoeq2}
almost surely. Indeed, once this
is verified, it follows by the uniqueness asserted in Theorem \ref
{thrm:eu} that the law of the limit
point $\z^*(\cdot)$ is unique, and so $\bzn(\cdot)\Rightarrow\z
^*(\cdot
)$ as $n\to\infty$.
By \eqref{eq:IC}, (A.1), and Theorem~\ref{thrm:JR}, as $m\to\infty$,
\begin{equation}\label{eq:JR2}
W^m(\cdot)\Rightarrow W^*(\cdot),
\end{equation}
where $W^*(\cdot)$ is almost surely a workload fluid model solution
such that $W^*(0)$
is equal in distribution to $W_0^*$.
We would like to argue that this convergence is joint with \eqref{eq:SubSeq}.
By \eqref{eq:IC}, \eqref{eq:Xn}, \eqref{eq:A}, \eqref{eq:V},
and the fact that the limit in \eqref{eq:Xn} and $\CR^*(\cdot)$ are
deterministic,
as $m\to\infty$,
\[
\left(\bzm(\cdot), W^m(0), X^m(\cdot), \bar{\mathcal A}^m(\cdot
), \bar
{\mathcal V}^m(\cdot)\right)
\Rightarrow
\left(\z^*(\cdot), W_0, 0, \CR^*(\cdot), \CR^*(\cdot)\right).
\]
This together with \eqref{eq:CMT}, \eqref{eq:CMT2}, and \eqref{eq:JR2}
implies that
as $m\to\infty$,
\begin{eqnarray}\label{eq:JointConvergence}
&&\left(\bzm(\cdot), W^m(\cdot), X^m(\cdot), \bar{\mathcal
A}^m(\cdot), \bar
{\mathcal V}^m(\cdot)\right)\\
&&\qquad\Rightarrow
\left(\z^*(\cdot), W^*(\cdot), 0, \CR^*(\cdot), \CR^*(\cdot
)\right
).\nonumber
\end{eqnarray}
Using the Skorohod representation we may assume without loss of
generally that all
random elements are defined on a common probability space $(\Omega
, {\mathcal F}, \bP)$
such that the joint convergence in \eqref{eq:JointConvergence} is
almost sure.
By \eqref{eq:IC}, $\z^*(0)$ satisfies $w_{\z^*(0)}=W^*(0)$, (A.1),
(A.2), and (A.3) almost surely.
Furthermore, by Lemma~\ref{lem:osc}, $\z^*(\cdot)$ is continuous
almost surely.
In addition, by Lemma~\ref{lem:BndReg},
\begin{equation}\label{eq:NoCornerMass}
\bP\left(\z_+^*(t)(C_x)=0\hbox{ for all }t\in\ptime\hbox{ and
}x\in\Rp
^2\right)=1.
\end{equation}
(cf.\ \cite[Lemma 6.2] {ref:GRZ}).
Fix $\omega\in\Omega$ such that $W^*(\cdot)(\omega)$ is a workload
fluid model solution
and $\z^*(\cdot)(\omega)$ is continuous and
satisfies \eqref{eq:NoCornerMass}, $w_{\z^*(0)(\omega
)}=W^*(0)(\omega
)$, (A.1), (A.2), (A.3)
and, as $m\to\infty$,
\begin{eqnarray*}
&&\left(\bzm(\cdot)(\omega), W^m(\cdot)(\omega), \bar{\mathcal
A}^m(\cdot
)(\omega), \bar{\mathcal V}^m(\cdot)(\omega)\right)\\
&&\qquad\rightarrow\left(\z^*(\cdot)(\omega), W^*(\cdot)(\omega
), \CR
^*(\cdot), \CR^*(\cdot)\right).
\end{eqnarray*}
For all $t\in\ptime$ and $m\in\bbM$, set
\begin{eqnarray*}
\zeta^m(t)=\bzm(t)(\omega)&\qquad\hbox{and}\qquad&
w^m(t)=W^m(t)(\omega
), \\
\zeta(t)=\z^*(t)(\omega)&\qquad\hbox{and}\qquad&
w(t)=W^*(t)(\omega).
\end{eqnarray*}
By \eqref{eq:IC}, $w(0)=w_{\vartheta}$ where $\vartheta=\zeta(0)$.
In order to prove Theorem~\ref{thrm:flt}, it suffices to show that
$\zeta(\cdot)$ is a fluid model solution for the supercritical data
$(\lambda, \mu, \Gamma)$ and initial measure $\vartheta$. In particular,
we must show that $\zeta(\cdot)$ satisfies \eqref{fdevoeq2}. In this
regard, recall that ${\mathcal P}$ is a $\pi$-system (see \eqref
{def:Pi}). As in the proof of Theorem~\ref{thrm:eu} given in Section
\ref{sec:prfthrmeu}, it is enough to show that $\zeta(\cdot)$ satisfies
\eqref{fdevoeq2} for all $B\in{\mathcal P}$.
Fix $B\in{\mathcal P}$. Then $B=[a, \infty)\times[c, \infty)$ for some
$0\le a, c<\infty$.
Fix $t\in\ptime$. In what follows, all random elements are evaluated at
the specific $\omega$
fixed in the preceding paragraph.
Since $\zeta(\cdot)$ is continuous, $\zeta^n(s)\wk\zeta(s)$ for all
$s\in[0, t]$.
By \eqref{eq:NoCornerMass}, $\zeta_+(0)(C_{(a+t, c+t)})=0$ and $\zeta
_+(t)(C_{(a, c)})=0$. Then,
for each $1\le k\le K$, we have
\[
\lim_{m\to\infty}\zeta_k^m(0)(B_t)=\zeta_k(0)(B_t)
\qquad\hbox{and}\qquad
\lim_{m\to\infty}\zeta_k^m(t)(B)=\zeta_k(t)(B).
\]
However, by the fluid scaled version of \eqref{eq:dynamics2} with $h=t$
and $t=0$,
for each $m\in\bbM$ and $1\le k\le K$,
\begin{eqnarray*}
\zeta_k^m(t)(B)-\zeta_k^m(0)(B_t)
&=&
\frac{1}{m}\sum_{j=Z_k^m(0)+1}^{A_k^m(t)} 1_{B_{t-t_{k, j}^m}}\left
(w_{k, j}^m, p_{k, j}^m\right)\\
&=&
\frac{1}{m}\int_0^{t} 1_{B_{t-s}}\left(w^m(s), p_{k,
A_k^m(s)}^m\right)
\der E_k^m(s).
\end{eqnarray*}
Hence, in order to verify that \eqref{fdevoeq2} holds, it suffices to
show that for each $1\le k\le K$,
\[
\lim_{m\to\infty}\frac{1}{m}\int_0^{t} 1_{B_{t-s}}\left
(w^m(s), p_{k, A_k^m(s)}^m\right)
\der E_k^m(s)
= \lambda_k\int_0^t\left( \delta_{w(s)}^+\times\Gamma_k\right
)(B_{t-s})\der s.
\]
Note that for $1\le k\le K$,
\begin{eqnarray*}
\lambda_k\int_0^t\left( \delta_{w(s)}^+\times\Gamma_k\right
)(B_{t-s})\der s
&=& \lambda_k
\int_0^t
1_{\{w(s) \ge a + t-s\}} G_k(c+t-s)\der s \\
&=& \lambda_k
\int_{\tau(a+t)\wedge t}^t
G_k(c+t-s)\der s \\
&=& \lambda_k
\int_c^{c+t-\tau(a+t)\wedge t}
G_k(u)\der u.
\end{eqnarray*}
Hence, in order to verify that \eqref{fdevoeq2} holds, it suffices to
show that for each $1\le k\le K$,
\begin{eqnarray}\label{eq:MainPart}
&& \lim_{m\to\infty}\frac{1}{m}\int_0^{t} 1_{B_{t-s}}\left
(w^m(s), p_{k, A_k^m(s)}^m\right) \der E_k^m(s)\\
&&\qquad=\lambda_k
\int_c^{c+t-\tau(a+t)\wedge t}
G_k(u)\der u.\nonumber
\end{eqnarray}
In order to verify \eqref{eq:MainPart}, fix $1\le k\le K$.
Note that for all $m\in\bbM$ and $1\le i \le E_k^m(t)$,
\[
d_{k, i}
\le
p_{k, Z_k^m(0)+i}^m
\le
d_{k, i}+v_{k, i}^m.
\]
Then
\begin{eqnarray*}
&&\int_0^{t} 1_{B_{t-s}}\left(w^m(s), d_{k, E_k^m(s)}\right) \der
E_k^m(s)\\
&&\qquad\le
\int_0^{t} 1_{B_{t-s}}\left(w^m(s), p_{k, A_k^m(s)}^m\right) \der
E_k^m(s)\\
&&\qquad\le
\int_0^{t} 1_{B_{t-s}}\left
(w^m(s), d_{k, E_k^m(s)}+v_{k, E_k^m(s)}^m\right
) \der E_k^m(s).
\end{eqnarray*}
Let $0<\delta<t$. Take $M$ such that for all $m\ge M$, $\sup_{0\le
s\le t}|w^m(s)-w(s)|<\delta$.
Then, for all $m\ge M$,
\begin{eqnarray}
&&\int_0^{t} 1_{B_{t-s}}\left(w(s)-\delta, d_{k, E_k^m(s)}\right)
\der
E_k^m(s)\nonumber\\
&&\qquad\le
\int_0^{t} 1_{B_{t-s}}\left(w^m(s), p_{k, A_k^m(s)}^m\right) \der
E_k^m(s)\label{eq:center}\\
&&\qquad\le
\int_0^{t} 1_{B_{t-s}}\left(w(s)+\delta
, d_{k, E_k^m(s)}+v_{k, E_k^m(s)}^m\right) \der E_k^m(s).\nonumber
\end{eqnarray}
For $s\in\ptime$, $w(s)-\delta\ge a+t-s$ if and only if $s\ge\tau
(a+t+\delta)$. Then
\begin{eqnarray*}
&&\int_0^{t} 1_{B_{t-s}}\left(w(s)-\delta, d_{k, E_k^m(s)}\right)
\der
E_k^m(s)\\
&&\quad=
\int_{\tau(a+t+\delta)\wedge t} ^{t} 1_{[c+t-s, \infty)}\left
(d_{k, E_k^m(s)}\right) \der E_k^m(s)\\
&&\quad=
\int_{\tau(a+t+\delta)\wedge t} ^{t} 1_{[c, \infty)}\left(d_{k, E_k^m(s)}
- (t-s)\right) \der E_k^m(s)\\
&&\quad=
\int_{0} ^{t} 1_{[c, \infty)}\left(a_{k, E_k^m(s)}^m(t) \right)
\der
E_k^m(s)\\
&&\qquad-
\int_0^{\tau(a+t+\delta)\wedge t} 1_{[c, \infty)}\left
(a_{k, E_k^m(s)}^m(t)\right) \der E_k^m(s)\\
&&\quad=
{\mathcal A}_k^m(t)([c, \infty))
-{\mathcal A}_k^m(\tau(a+t+\delta)\wedge t)([c+t-\tau(a+t+\delta
)\wedge
t, \infty)).
\end{eqnarray*}
Similarly,
\begin{eqnarray*}
&&\int_0^{t} 1_{B_{t-s}}\left(w(s)+\delta
, d_{k, E_k^m(s)}+v_{k, E_k^m(s)}^m\right) \der E_k^m(s)\\
&&\quad=
{\mathcal V}_k^m(t)([c, \infty))
-{\mathcal V}_k^m(\tau(a+t-\delta)\wedge t)([c+t-\tau(a+t-\delta
)\wedge
t, \infty)).
\end{eqnarray*}
Then, by \eqref{eq:A} and \eqref{eq:center},
\begin{eqnarray*}
&&\liminf_{m\to\infty}\frac{1}{m}\int_0^{t} 1_{B_{t-s}}\left
(w^m(s), p_{k, A_k^m(s)}^m\right) \der E_k^m(s)\\
&&\quad\ge
\CR_k^*(t)([c, \infty))-\CR_k^*(\tau(a+t+\delta)\wedge
t)([c+t-\tau
(a+t+\delta)\wedge t, \infty)).
\end{eqnarray*}
But,
\begin{eqnarray*}
&&\CR_k^*(t)([c, \infty))
-
\CR_k^*(\tau(a+t+\delta)\wedge t )([c+t-\tau(a+t+\delta)\wedge
t, \infty
))\\
&&\quad= \lambda_k \int_c^{\infty}\left( G_k(u)-G_k(u+t)\right
)\der
u\\
&&\qquad-\lambda_k \int_{ c+t-\tau(a+t+\delta)\wedge
t}^{\infty
}\left( G_k(u)-G_k(u+\tau(a+t)\wedge t )\right)\der u\\
&& \quad= \lambda_k
\int_c^{c+t-\tau(a+t+\delta)\wedge t}
G_k(u)\der u.
\end{eqnarray*}
So then, since $\delta\in(0, t)$ is arbitrary, letting $\delta
\searrow
0$ and using continuity of $\tau(\cdot)$
yields that
\[
\liminf_{m\to\infty}\frac{1}{m}\int_0^{t} 1_{B_{t-s}}\left
(w^m(s), p_{k, A_k^m(s)}^m\right) \der E_k^m(s)
\ge
\lambda_k
\int_c^{c+t-\tau(a+t)\wedge t}
G_k(u)\der u.
\]
Similarly, using \eqref{eq:V} in place of \eqref{eq:A},
\[
\limsup_{m\to\infty} \frac{1}{m}\int_0^{t} 1_{B_{t-s}}\left
(w^m(s), p_{k, A_k^m(s)}^m\right) \der E_k^m(s)
\le
\lambda_k \int_c^{c+t-\tau(a+t)\wedge t} G_k(u)\der u.
\]
Hence \eqref{eq:MainPart} holds, as desired.
\end{proof}
\bibliographystyle{acmtrans-ims}
| 49,785
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Shilpa Shetty provides health and fitness tips on the launch of her web site theshilpashetty.com. sourceRead More »
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| 381,094
|
TITLE: Find all $n$ for which $504$ doesn't divide $n^8-n^2$
QUESTION [2 upvotes]: I have seen a similar problem where someone showed that it does divide $n^9-n^3$ but here one has to show that it isn't the case for $n^8-n^2$ for all $n$.
REPLY [2 votes]: By the Chinese remainder theorem, $n^8-n^2$ is divisible by $504$ if and only if it is divisible by $8$, $7$ and $9$.
Mod $8$, $n^2\equiv 0$ or $1$, except if $n\equiv \pm2\mod8$, hence $n^8-n^2=n^2(n^6-1)\equiv 0$, except if $n\equiv \pm2\mod8$, in which case $n^8-n^2\equiv 4$.
Mod $7$, Lil' Fermat says $n^7\equiv n\mod 7$, hence $n^8-n^2\equiv n^2-n^2=0\mod7$.
Mod $9$, $n^2\equiv 0$ if $n\equiv 0,\pm3$, and $n^6\equiv 1$ if $n\equiv \pm1,\pm 2, 4$, so $n^8-n^2\equiv 0\mod9$ for all $n$.
As a conclusion, $n^8-n^2\equiv 0\mod 504$, except if $n\equiv\pm2$ (i.e. $n\equiv 2,6$) $\bmod8$. As observed by @lhf, the simplifies to $n\equiv 2\mod4$.
| 203,556
|
TITLE: Categorification of the (co-)induced topology
QUESTION [3 upvotes]: In second semester analysis we learned about the product topology which is quite easy to categorify using limits. However, we also learned about the coinduced topology $\mathfrak{V}$ induced by $f: X → Y$ and $(X, \mathfrak{U})$ on $Y$. It is the strongest topology on Y such that $f: (X, \mathfrak{U}) → (Y, \mathfrak{V})$ is continuous.
I would love to to express this coinduced topology using the categories Top and Set. Sadly I can't realy figure out how to do it as I have few experience with Category Theory.
Could someone please explain this to me?
REPLY [3 votes]: The final topology $\mathfrak{V}$ induced by a map $f : X \to Y$ satisfies the following universal property: for all spaces $(Z, \mathfrak{W})$ and all continuous maps $h : (X, \mathfrak{U}) \to (Z, \mathfrak{W})$, if $h : X \to Z$ factors through $f : X \to Y$ for some $g : Y \to Z$, then the map $g : (Y, \mathfrak{V}) \to (Z, \mathfrak{W})$ is continuous.
Another way to think of the final topology is as a colimit. If I'm not mistaken, it is the colimit of a diagram induced by the coslice category $(f \downarrow ((X, \mathfrak{U}) \downarrow \textbf{Top}))$, but I don't think this is helpful...
| 133,699
|
Bob
Agramonte of La Conner, WA asks:
Have you sighted other vessels?
Yes, one only, range lights and radar.
How is email set up?
We use the Iridium SAT phone and Microsoft Outlook (stratosnet)
Regular time? Sea State?
11:00a.m./4:00 p.m. local time. Sea state is 20 ft.
Fuel sight gauge calibrated? How?
Yes, fuel sight gauge is calibrated - in gallons
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| 183,408
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The American businesswoman being investigated over her ties to British leader Boris Johnson went on live TV Monday — and played coy over reports of their alleged affair.
Former model Jennifer Arcuri, 34, is in the spotlight in the UK amid allegations she received money and perks from London coffers while Johnson was mayor of the British capital.
But she would only say on “Good Morning Britain” that she and Johnson met in 2011 and “share a very close bond,” while refusing to confirm or deny if they were romantically involved.
“Would you like me to ask about your sex life?” she asked the show’s hosts, Piers Morgan and Susanna Reid, after one of the nine times they asked her about the affair reports, according to The Guardian.
Arcuri admitted that her answer would “weaponize” the discussion even as she was asked on “Good Morning Britain” to give a simple denial if the reports were untrue.
“It’s really categorically no one’s business what private life we had or didn’t have,” she said, complaining about being painted as an “objectified ex-model pole dancer.”
Revealing that Johnson is listed as “Alex the Great” in her phone contacts, she admitted the then-married mayor had made a “handful” of visits to her London home, joking about the stripper pole she has there.
She denied ever using the pole for him — but refused to rule out him taking a twirl.
“I’m never going to tell you that,” she smiled. “Can you imagine Boris Johnson on a pole?”
Asked if she loved Johnson, she said, “I care about him deeply as a friend and we do share a very close bond.”
“I wish him well. I want him to be happy,” she added.
Arcuri was, however, quick to deny allegations that he helped her businesses, Innotech and Hacker House, get public money and privileged access to foreign trade missions led by Johnson.
“Boris never, ever gave me favoritism,” she insisted. “Categorically, Boris has nothing to do with all of my other achievements.”
Johnson, who is being investigated for allegedly breaking the Greater London Assembly’s code of conduct in the matter, was asked about Arcuri’s interview later Monday.
“I’ve said everything I’m going to say on that matter,” he told ITV.
Johnson’s office has previously said, “This allegation is untrue.”
| 348,158
|
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