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Posted by Jason on October 27, 2003 at 23:44:02:
i have an interesting tuba history. i started off on some trashy Bb that the school had until i learned the basics. than i bought a Conn 3J and learned more and mroe and gotr pretty good. this is when i first really started to use my air to play more than my lips. now i have a MW 2145. i can play it and sound good on it, but it takes SOOOOOOOO much work to get up to the level i want to be at. whereas i am playing my friends piggy and it takes a lot less effort to do everything i want to do (sound, intonation, tuning, articulation, etc.). would it be worth it to go back and look for a good smaller horn?
Follow Ups:
| 136,132
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TITLE: Constructing a space with specified homology
QUESTION [4 upvotes]: I am trying to do as many practise questions as possible for an A.T exam in June, which I am currently studying independently for.
I am stuck on the following;
I am wanting to construct a space X where the ith homology group is given by G_i for i = 1,...,n , where the G_i are some finitely generated abelian groups.
I have made little progress, aside from writing the G_i as direct sums using the fundamental theorem of finitely generated abelian groups, so any help at all would be greatly appreciated.
Thank you
REPLY [5 votes]: Let's do this for $H_n = G$ and zero in all other dimensions - this is known as the Moore space and is denoted by $M(G, n)$. Since $G$ is finitely generated abelian, write it as the direct sum $\Bbb Z \bigoplus_{p} \Bbb Z/p^k\Bbb Z$.
$M(\Bbb Z, n)$ is $S^n$, and $M(\Bbb Z/m\Bbb Z, n)$ can be constructed by gluing an $(n+1)$-cell to $S^n$ by the degree $m$ map $S^n \to S^n$ (this indeed does have $H_{n+1} = 0$ - write down the long exact sequence of $(X, S^n)$ and note that the snake map coming out of the $n+1$-th term is injective)
By Dan Rust's hint write $M(\Bbb Z, n) \bigvee_p M(\Bbb Z/p^k \Bbb Z, n)$. That's the desired $M(G, n)$.
For given $G_1, \cdots, G_n$, look at $M(G_1, 1) \vee M(G_2, 2) \vee \cdots \vee M(G_n, n)$. This space has $H_i \cong G_i$ for each $i = 1, \cdots, n$ as desired.
| 149,525
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| 395,244
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There are 80 of them in total. Even the 52 I met lifted my mood, as I stood by the thinly stacked Eton Manor Olympic cycle park. Their co-ordinated orange tops and orange bikes are a magnificent statement of cycling confidence. Could it be that the example we need to get biking around the Olympics – midway through our year of going Dutch – comes from the Dutch themselves?
Though the bike blog has reported on the limpness of the Olympic Delivery Authority's (ODA) cycling provision several times, it is still disappointing to have to say that so far, visitors cycling to the Olympic Park and nearby venues have been little more than a tenth of the 4,400 cycling visitors per day the ODA was originally aiming for.
The orange-clad cyclists are guests of Dutch firm Pon, and they are so determined to cycle between venues that they have brought a truckload of Dutch bikes with them. British opinion would tend to dismiss such enthusiastic cyclists as a convention of lettuce-eaters, and get back to complaining about the traffic. But Pon is not a niche environmental outfit. It is an international business services company and an official supplier of the Dutch Olympic team, with extensive links to the automotive industry. The visitors are senior executives and their clients, and their visit a lesson in how totally mainstream cycling is for the Dutch.
For Hans van der Valk, Pon's senior vice president for passenger cars, bringing bikes was simply a pragmatic choice. "We studied the logistics of moving around London in a large group, and decided that cycling was the best way," he says. Pon has provided a few cars as well, but only those who are not physically able to ride have chosen the lift. "We are very happy to choose bikes. Most people like biking," he says.
Despite its involvement in the automotive sector, Pon also invests in cycling. It acquired Dutch bike company Gazelle last year. Van der Valk sees the future of the business in increased use of bikes alongside more efficient use of cars. "Cycling is a growth area," he says. It is also something enjoyed by every level of society in the Netherlands. Van der Valk assures me that it is completely normal for a senior Dutch executive to cycle around town, though it is difficult to imagine British board-level executives – let alone automotive chiefs – doing the same.
Van der Valk is diplomatic about his clients' impression of London cycling. "We have had a very positive experience," he says. "People have been very friendly, and the traffic is not too bad."
Unsurprisingly, the condition of the infrastructure disappoints. "The cycle paths are very narrow, and we have to remember to ride on the left," he says. In the Netherlands there are usually separate tracks for each direction of travel, which means cyclists can ride abreast and chat.
Alec Lemaire of the London Bicycle Tour Company, who is project managing the visit, sees the Dutch group's experiences in sharper relief. The respect accorded to cyclists in the Netherlands is not necessarily useful preparation for London traffic. "Dutch cyclists have a sense of entitlement on the road," he says, "which can lead to problems ..."
Lemaire is optimistic that the quality of cycling experience for Olympic visitors can be good, though he has had to choose the routes between venues carefully, using his local knowledge. "It is extremely challenging to get to the Olympic Park venues by bike. Cyclists visiting the Olympics should avoid the most direct route, and try to use quiet back roads," he says. It's not great for independent cyclists new to the area.
Bike parking at the transport hubs is secure, with a bike mechanic providing free check-ups and a Met officer offering security marking. And Olympic staff are friendly. "They are keen to have us," says Lemaire. That may be because cycle parking assistants are getting lonely. At Eton Manor, the 52 Dutch bikes were perhaps a fifth of the total parked. At Victoria Park, the second of the three Olympic Park cycle parks, the Dutch bikes were virtually the only ones there. There's no avoiding the scale of the missed opportunity.
The Dutch show us how easy and relaxed cycling should be, even without the extensive network of traffic-free routes many cyclists were hoping for. The ODA could have attracted more cyclists with some volunteer cycling guides and a bike hire service for visitors. It has provided limousine drivers and public transport stewards by the thousand. But the small-scale, bolt-on bike provision just reminds us again that despite progress on many fronts, cycling is not yet considered a serious, mainstream form of transport. Friday's heavy-handed Critical Mass arrests only confirm the perceived marginalisation.
In Sunday's women's road race, Marianne Vos of the Netherlands pipped Britain's Lizzie Armitstead to the finishing post. If only we were that close behind the Dutch when it comes to putting the bike at the centre of civilised urban life.
| 98,975
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TITLE: G/M/1 Queue with Arrivals Following an Erlang-2 Distribution
QUESTION [1 upvotes]: I look at the following queueing system:
"We have two identical M/M/1 queueing systems, S1 and S2. Jobs arrive according to a Poisson process with rate 2 per hour, and service times are exponential with mean $1/4$ hour. The arrival stream is split into two (stchastically equal) arrival streams, which form the arrival processes for S1 and S2.
Suppose that the spliting is done such that the jobs are alternatingly routed to S1 and S2; as a result te interarrival times at S1 have an Erlang-2 distribution."
Firstly, I want to compute $\rho_1$, the fraction of time the server (at S1) is busy. I know that $\rho_1=\lambda_1 \cdot \mathbb{E}[B]$, where I denote with $\lambda_1$ the arrival rate at S1.
I also know that the interarrival time of the whole system has mean $\mathbb{E}[A]=\frac{1}{\lambda}$, which would give $\mathbb{E}[A]=\frac{1}{2}$. So my idea was, knowing that the mean of the Erlang-2 distribution is $\frac{2}{\lambda_1}$ to compute $\lambda_1$ with
$$\mathbb{E}[A]=\frac{1}{2}=\frac{2}{\lambda_1}+\frac{2}{\lambda_2},$$
where $\lambda_2$ belongs to S2. Now since the situation in S2 is symmetric to the one in S1, I have $\lambda_1=\lambda_2$. That would give me $\lambda=8$ and consequently an unstable system because $\rho=4 \cdot \frac{1}{4}$.
Can somebody help me understand the system? How do I derive the correct arrival rate?
REPLY [1 votes]: We can consider each station as a $G/M/1$ queue with arrival distribution $\mathrm{Erlang}(2,\lambda)$, that is, the interarrival times have density
$$
f_A(t)=\lambda(\lambda t)e^{-\lambda t}\mathsf 1_{(0,\infty)}(t).
$$
Let $X_0=0$ and $X_n$ be the number of customers in the system just before the $n^{\mathrm{th}}$ arrival. Then $\{X_n:n=0,1,2,\ldots\}$ is the embedded Markov chain on the nonnegative integers with transition probabilities
$$
\mathbb P(X_{n+1}=j\mid X_n=i) = \begin{cases}
p_{i,0},& j=0\\
\beta_{i-j+1},& j>0.
\end{cases}
$$
Here $\beta_j$ is the probability of serving $j$ customers during an interarrival time given that the server remains busy during this interval. Let $T$ be the interarrival time, then conditioned on $\{T=t\}$ the number of customers served $N$ has Poisson distribution with parameter $\mu t$. Thus
\begin{align}
\beta_j &= \int_0^\infty \mathbb P(N=j\mid T=t)f_A(t)\ \mathsf dt\\
&= \int_0^\infty \frac{(\mu t)^j}{j!} e^{-\mu t}\lambda(\lambda t)e^{-\lambda t}\ \mathsf dt\\
&= (j+1)\left(\frac\lambda\mu\right)^2\left(\frac\mu{\lambda+\mu}\right)^{j+2},\ j=0,1,2,\ldots.
\end{align}
Since the rows of the transition matrix should sum to one, it follows that
\begin{align}
p_{i,0} &= 1 - \sum_{j=0}^i \beta_j\\
&= \sum_{i=j+1}^\infty \beta_j\\
&= \sum_{i=j+1}^\infty (i+1)\left(\frac\lambda\mu\right)^2\left(\frac\mu{\lambda+\mu}\right)^{i+2}\\
&= \frac{\mu(\mu+\lambda(i+2))}{(\lambda+\mu)^2}\left(\frac\mu{\lambda+\mu}\right)^i,\ i=0,1,2,\ldots
\end{align}
The stationary distribution $\pi$ satisfies the balance equations
\begin{align}
\pi_0 &= \sum_{i=0}^\infty p_{i,0}\pi_i\\
\pi_n &= \sum_{i=0}^\infty \beta_i\pi_{n-i+1},\ n=1,2,\ldots\tag1.
\end{align}
It is known in the literature that $\pi$ has a geometric distribution, i.e. $\pi_n = (1-\sigma)\sigma^n$ for some $\sigma\in(0,1)$. Substituting this into $(1)$ and dividing by the common factor $\sigma^{n-1}$ yields
\begin{align}
\sigma &= \sum_{i=0}^\infty \sigma^i\beta_i\\
&= \sum_{i=0}^\infty \sigma^i (i+1)\left(\frac\lambda\mu\right)^2\left(\frac\mu{\lambda+\mu}\right)^{i+2}\\
&= \frac{\lambda ^2}{(\lambda + \mu(1-\sigma))^2}.
\end{align}
Solving for $\sigma$ yields the roots
\begin{align}
\sigma &= \frac{\mu ^{3/2} \left(-\sqrt{4 \lambda +\mu }\right)+2 \lambda \mu +\mu ^2}{2 \mu ^2}\tag2\\\
\sigma &= \frac{\mu ^{3/2} \sqrt{4 \lambda +\mu }+2 \lambda \mu +\mu ^2}{2 \mu ^2}\tag3.
\end{align}
Since $\sum_{i=0}^\infty\pi_i=1$, it turns out that $(2)$ is the correct root. So we have
$$
\pi_n = \left(\frac{\sqrt{\mu } \sqrt{4 \lambda +\mu }-2 \lambda +\mu }{2 \mu }\right) \left(\frac{2 \lambda +\mu -\sqrt{\mu(1+4\lambda)} }{2\mu }\right)^n,\ n=0,1,2,\ldots
$$
The fraction of time that the server is busy is given by
$$
1 - \pi_0 = \frac{2 \lambda +\mu -\sqrt{\mu(1+4\lambda)} }{2\mu }.
$$
| 178,595
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TITLE: Show that $\omega\wedge f(v_1,\ldots,v_m)=f(v)\operatorname{vol} (v_1,\ldots,v_m)$
QUESTION [0 upvotes]: Show for every $\omega\in \wedge^{m-1}(\mathbb R^m)$ exist one only vector $v$ such that $$\omega\wedge f(v_1,\ldots,v_m)=f(v)\operatorname{vol}(v_1,\ldots,v_m)$$
for all $f\in\wedge^1(\Bbb R^m)$ and for all $v_1,\ldots,v_m$ in $ \Bbb R^m$. Show that exist one isomorphism between $\wedge^{m-1}(\Bbb R^m)$ and $\Bbb R^m$.
My attempt is to try to compute the leftside, I mean
$$\omega\wedge f = \frac{(m-1+1)! }{(m-1)!1!}\operatorname{Alt}(\omega \otimes f)=m\operatorname{Alt}(\omega \otimes f),$$
the other side
$$\operatorname{Alt}(\omega \otimes f)=\frac1{m!} \sum_{\sigma\in S_m} (\operatorname{sign}\sigma ) \omega\otimes f.$$ Then
$$\begin{aligned}\omega\wedge f(v_1\ldots,v_n)&= \frac{m}{m!}\sum_{\sigma\in S_m} (\operatorname{sign }\sigma )\omega(v_{\sigma(1)},\ldots,v_{\sigma(m-1)}).f(v_{\sigma(m)})\\&=\frac1{(m-1)!}\sum_{\sigma\in S_m}(\operatorname{sign} \sigma)\omega(v_{\sigma(1)},\ldots,v_{\sigma(m-1)}).f(v_{\sigma(m)}).\end{aligned}$$
How continued this? And $\frac1{(m-1)!}\sum_{\sigma\in S_m}(\operatorname{sign} \sigma)\omega(v_{\sigma(1)},\ldots,v_{\sigma(m-1)})$ is the volumen element on dimension $\Bbb R^{m-1}$?
Please can somebody give me one hint? Thank you so much.
REPLY [2 votes]: This is a result which I think is much clearer from an abstract perspective. Suppose $V$ is an $m$-dimensional real vector space along with a given volume form $\text{vol}\in \bigwedge^m(V)\setminus\{0\}$. Now, fix an $\omega\in \bigwedge^{m-1}(V)$ and define the following maps:
$A_{\omega}: \bigwedge^1(V)=V^* \to \bigwedge^m(V)$ given by $f\mapsto \omega \wedge f$. i.e this is the "wedging map" (this is easily verified to be a linear map).
Define $\Phi:\Bbb{R}\to \bigwedge^m(V)$ by $\Phi(c) = c \cdot \text{vol}$. This is clearly injective, and since both vector spaces are $1$-dimensional, this is also an isomorphism (all we're doing is mapping the basis element $1\in \Bbb{R}$ to the basis element $\text{vol}\in \bigwedge^m(V)$).
Finally, let $\iota:V \to V^{**}$ be the canonical isomorphism, where for all $v\in V$ and all $f\in V^*$, $[\iota(v)](f):= f(v)$ (i.e it's the evaluation map).
Now, note that $\Phi^{-1} \circ A_{\omega}$ is a linear map $V^* \to \Bbb{R}$, in other words, $\Phi^{-1}\circ A_{\omega} \in V^{**}$ is an element of the second dual space, which is isomorphic to $V$. Now, define $v:= \iota^{-1}[\Phi^{-1}\circ A_{\omega}]\in V$.
This is the unique $v$ which you seek (for the initially given $\omega$), and I leave it as an exercise :) for you to unwind the definitions and prove that $\omega \wedge f = A_{\omega}(f) = f(v) \cdot \text{vol}$.
In this regard you may be interested to read up about the Hodge Dual operation which (on a $m$-dimensional, oriented, (pseudo-)inner product space) provides an isomorphism between $\bigwedge^{k}(V)$ and $\bigwedge^{m - k}(V)$. For example, pages 319-320 of Loomis and Sternberg's Advanced Calculus.
| 202,112
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\begin{document}
\title{Distortion Bounds for Source Broadcast Problem}
\author{Lei Yu, Houqiang Li, \textit{Senior} \textit{Member, IEEE, }and Weiping
Li, \textit{Fellow, IEEE} \thanks{The authors are all with the Department of Electronic Engineering
and Information Science, University of Science and Technology of China,
Hefei 230027, China (e-mail: yulei@ustc.edu.cn, lihq@ustc.edu.cn,
wpli@ustc.edu.cn). The material in this paper was presented in part
at IEEE ISIT 2016 \cite{Yu2016}.}}
\maketitle
\begin{abstract}
This paper investigates the joint source-channel coding problem of
sending a memoryless source over a memoryless broadcast channel. An
inner bound and several outer bounds on the achievable distortion
region are derived, which respectively generalize and unify several
existing bounds. As a consequence, we also obtain an inner bound and
an outer bound for degraded broadcast channel case. When specialized
to Gaussian source broadcast or binary source broadcast, the inner
bound and outer bound not only recover the best known inner bound
and outer bound in the literature, but also are used to generate some
new results. Besides, we also extend the inner bound and outer bounds
to Wyner-Ziv source broadcast problem, i.e., source broadcast with
side information available at decoders. Some new bounds are obtained
when specialized to Wyner-Ziv Gaussian case and Wyner-Ziv binary case.
In addition, when specialized to lossless transmission of a source
with independent components, the bounds for source broadcast problem
(without side information) are also used to achieve an inner bound
and an outer bound on capacity region of general broadcast channel
with common messages, which respectively generalize Marton's inner
bound and Nair-El Gamal outer bound to $K$-user broadcast channel
case.
\end{abstract}
\begin{IEEEkeywords}
Joint source-channel coding (JSCC), hybrid coding, hybrid digital-analog
(HDA), broadcast, Wyner-Ziv, side information, multivariate covering/packing,
network information theory.
\end{IEEEkeywords}
\section{Introduction}
As stated in Shannon's source-channel separation theorem \cite{Shannon48},
cascading source coding and channel coding does not lose the optimality
for the point-to-point communication systems. This separation theorem
does not only suggest a simple system architecture in which source
coding and channel coding are separated by a universal digital interface,
but also guarantees that such architecture does not incur any asymptotic
performance loss. Consequently, it forms the basis of the architecture
of today's communication systems. However, for many multi-user communication
systems, the optimality of such a separation does not hold any more
\cite{Goblick65,Gastpar03}. Therefore, an increasing amount of literature
focus on joint source-channel coding (JSCC) in multi-user setting.
One of the most classical problems in this area is JSCC of transmitting
a Gaussian source over average power constrained $K$-user Gaussian
broadcast channel. Goblick \cite{Goblick65} observed that when the
source bandwidth and the channel bandwidth are matched (i.e., one
channel use per source sample) linear uncoded transmission (symbol-by-symbol
mapping) is optimal. However, the optimality of such a simple linear
scheme cannot be extended to the case of the bandwidth mismatch. One
way to approximately characterize the achievable distortion region
is finding its inner bound and outer bound. For inner bound, analog
coding schemes or hybrid coding schemes have been studied in a vast
body of literature \cite{Gastpar03,Shannon49,Shamai98,Prabhakaran11}.
For 2-user Gaussian broadcast communication, Prabhakaran \emph{et
al.} \cite{Prabhakaran11} gave the tightest inner bound so far, which
is achieved by hybrid digital-analog (HDA) scheme. On the other hand,
Reznic \emph{et al.} \cite{Reznic06} derived a nontrivial outer bound
for 2-user Gaussian broadcast problem with bandwidth expansion (i.e.,
more than one channel uses per source sample) by introducing an auxiliary
random variable (or remote source). Tian \emph{et al.} \cite{Tian11}
extended this outer bound to $K$-user case by introducing more than
one auxiliary random variables. Similar to the results of Reznic \emph{et
al.}, the outer bound given by Tian \emph{et al.} is also nontrivial
only for bandwidth expansion case \cite{Yu2015comments}. Beyond broadcast
communication, Minero \emph{et al.} \cite{Minero} considered sending
memoryless correlated source transmitted over memoryless multi-access
channel and derived an inner bound using a unified framework of hybrid
coding, and also Lee \emph{et al.} \cite{Lee} derived a unified achievability
result for memoryless network communication.
Besides, in \cite{Shamai98,Nayak,Gao} Wyner-Ziv source communication
problem was investigated, in which side information of the source
is available at decoder(s). Shamai \emph{et al.} \cite{Shamai98}
studied the problem of sending Wyner-Ziv source over point-to-point
channel, and proved that for such communication system separate coding
(which combines Wyner-Ziv coding with channel coding) does not incur
any loss of optimality. Nayak \emph{et al.} \cite{Nayak} and Gao
\emph{et al.} \cite{Gao} investigated Wyner-Ziv source broadcast
problem, and obtained an outer bound by simply applying cut-set bound
(the minimum distortion achieved in point-to-point setting) for each
receiver.
In this paper, we consider JSCC of transmitting a memoryless source
over $K$-user memoryless broadcast channel, and give an inner bound
and three outer bounds on the achievable distortion region. The inner
bound is derived by a unified framework of hybrid coding inspired
by \cite{Minero}, and the outer bounds are derived by introducing
auxiliary random variables at sender side or at receiver sides. The
proof method of introducing auxiliary random variables (or remote
sources) at sender side could be found in \cite{Reznic06} and \cite{Tian11}.
However, to the best of our knowledge, it is the first time to prove
outer bounds by introducing auxiliary random variables (or remote
channels) at receiver sides. Our bounds are generalizations and unifications
of several existing bounds in the literature. Besides, as a consequence,
we also obtain an inner bound and an outer bound for degraded broadcast
channel case. Owing to the generalization of our results, when specialized
to Gaussian source broadcast and binary source broadcast, our inner
bound could recover best known performance achieved by hybrid coding,
and our outer bound could recover the best known outer bounds given
by Tian \emph{et al.} \cite{Tian11} and Khezeli \emph{et al. }\cite{Khezeli14}.
Moreover, for these cases, our bounds can also be used to generate
some new results. Besides, we also extend the inner bound and outer
bounds to Wyner-Ziv source broadcast problem, i.e., source broadcast
with side information at decoders. When specialized to Wyner-Ziv Gaussian
case and Wyner-Ziv binary case, our bounds reduce to some new bounds.
In addition, when specialized to lossless transmission of a source
with independent components, the bounds for source broadcast problem
(without side information) is also used to achieve an inner bound
and an outer bound on capacity region of general broadcast channel
with common messages, which respectively generalize Marton's inner
bound and Nair-El Gamal outer bound to $K$-user broadcast channel
case.
The rest of this paper is organized as follows. Section II summarizes
basic notations, definitions and preliminaries, and formulates the
problem. Section III gives the main results for source broadcast problem,
including general, degraded, Gaussian and binary cases. Section IV
extends the results to Wyner-Ziv source broadcast problem. Finally,
Section V gives the concluding remarks.
\section{Problem Formulation and Preliminaries}
\subsection{Notation}
Throughout this paper, we follow the notation in \cite{El Gamal}.
For example, for discrete random variable $X\sim p_{X}$ on alphabet
$\mathcal{X}$ and $\epsilon\in\left(0,1\right)$, the set of $\epsilon$-typical
$n$-sequences $x^{n}$ (or the typical set in short) is defined as
$\mathcal{T}_{\epsilon}^{\left(n\right)}\left(X\right)=\left\{ x^{n}:\left||\{i:x_{i}=x\}|/n-p_{X}(x)\right|\leq\epsilon p_{X}(x)\textrm{ for all }x\in\mathcal{X}\right\} $.
When it is clear from the context, we will use $\mathcal{T}_{\epsilon}^{\left(n\right)}$
instead of $\mathcal{T}_{\epsilon}^{\left(n\right)}\left(X\right)$.
In addition, we use $X_{\mathcal{A}}$ to denote the vector $(X_{j}:j\in\mathcal{A})$,
use $[i:j]$ to denote the set $\left\{ \left\lfloor i\right\rfloor ,\left\lfloor i\right\rfloor +1,\cdots,\left\lfloor j\right\rfloor \right\} $,
and use $\mathbf{1}$ to denote an all-one vector (similarly, use
$\mathbf{2}$ to denote an all-2 vector). We say vector $m_{[1:N]}$
is smaller than vector $m'_{[1:N]}$ if $m_{j}=m'_{j},k<j\leq K$
and $m_{k}<m'_{k}$ for some $k$. For two vectors $m_{\mathcal{I}}$
and $m_{\mathcal{I}}^{\prime}$, we say $m_{\mathcal{I}}$ is component-wise
unequal to $m_{\mathcal{I}}^{\prime}$, if $m_{i}\neq m_{i}^{\prime}$
for all $i\in\mathcal{I}$, and denote it as $m_{\mathcal{I}}\nLeftrightarrow m_{\mathcal{I}}^{\prime}$.
Besides, we use $1\left\{ \mathcal{A}\right\} $ to denote indicator
function of event $\mathcal{A}$, i.e.,
\[
1\left\{ \mathcal{A}\right\} =\begin{cases}
1, & \textrm{if }\mathcal{A}\textrm{ is true};\\
0, & \textrm{if }\mathcal{A}\textrm{ is false}.
\end{cases}
\]
\subsection{Problem Formulation}
Consider the source broadcast system shown in Fig. \ref{fig:broadcast communication system }.
The discrete memoryless source (DMS) $S^{n}$ is first coded into
$X^{n}$ using a source-channel code, then transmitted to $K$ receivers
through a discrete memoryless broadcast channel (DM-BC) $p_{Y_{[1:K]}|X}$,
and finally, the receiver $k$ produces source reconstruction $\hat{S}_{k}^{n}$
from the received signal $Y_{k}^{n}$.
\begin{figure}[t]
\centering\includegraphics[width=0.6\textwidth]{Broadcast} \protect\protect\protect\protect\protect\caption{\label{fig:broadcast communication system }Source broadcast system. }
\end{figure}
\begin{defn}[Source]
\label{def:source} A discrete memoryless source (DMS) is specified
by a probability mass function (pmf) $p_{S}$ on a finite alphabet
${\mathcal{S}}$. The DMS $p_{S}$ generates an i.i.d. random process
$\left\{ S_{i}\right\} $ with $S_{i}\sim p_{S}$.
\end{defn}
\begin{defn}[Broadcast Channel]
\label{def:DBC}A $K$-user discrete memoryless broadcast channel
(DM-BC) is specified by a collection of conditional pmfs $p_{Y_{[1:K]}|X}$
on finite output alphabet ${\mathcal{Y}_{1}\times\cdots\times\mathcal{Y}_{K}}$
for each $x$ in finite input alphabet $\mathcal{X}$.
\end{defn}
\begin{defn}[Degraded Broadcast Channel]
A DM-BC $p_{Y_{[1:K]}|X}$ is stochastically degraded (or simply
degraded) if there exist a random vector $\tilde{Y}_{[1:K]}$ such
that $\tilde{Y}_{k}|\left\{ X=x\right\} \sim p_{Y_{k}|X}(\tilde{y}_{k}|x),1\leq k\leq K$,
i.e., $\tilde{Y}_{[1:K]}$ has the same conditional marginal pmfs
as $Y_{[1:K]}$ (given $X$), and $X\rightarrow\tilde{Y}_{K}\rightarrow\tilde{Y}_{K-1}\rightarrow\cdots\rightarrow\tilde{Y}_{1}$\footnote{To simplify notation, the Markov chain is assumed in this direction.
Note that this differs from that in the conference version \cite{Yu2016}.} form a Markov chain. In addition, as a special case, if $X\rightarrow Y_{K}\rightarrow Y_{K-1}\rightarrow\cdots\rightarrow Y_{1}$,
i.e., $\tilde{Y}_{k}=Y_{k},1\le k\le K$, then $p_{Y_{[1:K]}|X}$
is physically degraded.
\end{defn}
\begin{defn}
An $n$-length source-channel code is defined by the encoding function
$x^{n}:{\mathcal{S}}^{n}\mapsto{\mathcal{X}}^{n}$ and a sequence
of decoding functions $\hat{s}_{k}:\mathcal{Y}_{k}^{n}\mapsto\hat{\mathcal{S}_{k}^{n}},1\leq k\leq K$,
where $\hat{{\mathcal{S}_{k}}}$ is the alphabet of source reconstruction
at receiver $k$.
\end{defn}
For any $n$-length source-channel code, the induced distortion is
defined as
\begin{equation}
\mathbb{E}d_{k}\left(S^{n},\hat{S}_{k}^{n}\right)=\frac{1}{n}\sum_{t=1}^{n}\mathbb{E}d_{k}\left(S_{t},\hat{S}_{k,t}\right),
\end{equation}
for $1\le k\le K$, where $d_{k}\left(s,\hat{s}_{k}\right):{\mathcal{S}}\times\hat{{\mathcal{S}_{k}}}\mapsto\left[0,+\infty\right]$
is a distortion measure function for receiver $k$.
\begin{defn}
For transmitting source $S$ over channel $p_{Y_{[1:K]}|X}$, if there
exists a sequence of source-channel codes such that
\begin{equation}
\mathop{\limsup}\limits _{n\to\infty}\mathbb{E}d_{k}\left(S^{n},\hat{S}_{k}^{n}\right)\le D_{k},
\end{equation}
then we say that the distortion tuple $D_{[1:K]}$ is achievable.
\end{defn}
\begin{defn}
For transmitting source $S$ over channel $p_{Y_{[1:K]}|X}$, the
admissible distortion region is defined as
\begin{align}
\mathcal{R}\triangleq & \left\{ D_{[1:K]}:D_{[1:K]}\textrm{ is achievable}\right\} .
\end{align}
\end{defn}
The admissible distortion region $\mathcal{R}$ only depends on the
marginal distributions of $p_{Y_{[1:K]}|X}$, hence for source broadcast
over stochastically degraded channel it suffices to only consider
the broadcast over physically degraded channel.
In addition, Shannon's source-channel separation theorem shows that
the minimum distortion for transmitting source over point-to-point
channel satisfies
\begin{equation}
R_{k}\left(D_{k}\right)=C_{k},
\end{equation}
where $R_{k}\left(\cdotp\right)$ is the rate-distortion function
of the source with distortion measure $d_{k}$, and $C_{k}$ is the
capacity of the channel of the receiver $k$. Therefore, the optimal
distortion (Shannon limit) is
\begin{equation}
D_{k}^{*}=R_{k}^{-1}\left(C_{k}\right).\label{eq:-28-1}
\end{equation}
Obviously,
\begin{equation}
\mathcal{R}\subseteq\mathcal{R}^{*}\triangleq\left\{ D_{[1:K]}:D_{k}\geq D_{k}^{*},1\le k\le K\right\} ,\label{eq:trivialouterbound}
\end{equation}
where $\mathcal{R}^{*}$ is named \emph{trivial outer bound}.
In the system above, source bandwidth and channel bandwidth are matched.
In this paper, we also consider the communication system with bandwidth
mismatch, whereby $m$ samples of a DMS are transmitted through $n$
uses of a DM-BC. For this case, bandwidth mismatch factor is defined
as $b=\frac{n}{m}$.
\subsection{Multivariate Covering/Packing Lemma}
Two important results we need to prove the achievability part in this
work are the following lemmas, both of which are generalized versions
of the existing covering/packing lemmas.
Let $(U,V_{[0:k]})\sim p_{U,V_{[0:k]}}$, and let $\left(U^{n},V_{0}^{n}\right)\sim p_{U^{n},V_{0}^{n}}$
be a random vector sequence. For each $j\in[1:k]$, let $\mathcal{A}_{j}\subseteq[1:j-1]$.
Assume $\mathcal{A}_{j}$ satisfies if $i\in\mathcal{A}_{j}$, then
$\mathcal{A}_{i}\subseteq\mathcal{A}_{j}$. For each $j\in[1:k]$
and each $m_{\mathcal{A}_{j}}\in\prod_{i\in\mathcal{A}_{j}}[1:2^{nr_{i}}]$,
let $V_{j}^{n}(m_{\mathcal{A}_{j}},m_{j}),m_{j}\in[1:2^{nr_{j}}],$
be pairwise conditionally independent random sequences, each distributed
according to $\prod_{i=1}^{n}p_{V_{j}|V_{\mathcal{A}_{j}},V_{0}}(v_{j,i}|v_{\mathcal{A}_{j},i}(m_{\mathcal{A}_{j}}),v_{0,i})$.
Hence for each $j\in[1:k]$, $\mathcal{A}_{j}\cup\left\{ 0\right\} $
denotes the index set of the random variables on which the codeword
$V_{j}^{n}$ is superposed. Based on the notations above, we have
the following generalized Multivariate Covering Lemma and generalized
Multivariate Packing Lemma.
\begin{lem}[Multivariate Covering Lemma]
\label{lem:Covering} Let $\epsilon'<\epsilon$. If $\lim_{n\rightarrow\text{\ensuremath{\infty}}}\mathbb{P}\left(\left(U^{n},V_{0}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\right)=1$,
then there exists $\delta(\epsilon)$ that tends to zero as $\epsilon\rightarrow0$
such that
\begin{equation}
\lim_{n\rightarrow\text{\ensuremath{\infty}}}\mathbb{P}\left((U^{n},V_{0}^{n},V_{[1:k]}^{n}(m_{[1:k]}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}\textrm{ for some }m_{[1:k]}\right)=1,
\end{equation}
if $\sum_{j\in\mathcal{J}}r_{j}>\sum_{j\in\mathcal{J}}H\left(V_{j}|V_{\mathcal{A}_{j}}V_{0}\right)-H\left(V_{\mathcal{J}}|V_{0}U\right)+\delta(\epsilon)$
for all $\mathcal{J}\subseteq[1:k]$ such that $\mathcal{J}\neq\emptyset$
and if $j\in\mathcal{J}$, then $\mathcal{A}_{j}\subseteq\mathcal{J}$.
\end{lem}
\begin{lem}[Multivariate Packing Lemma]
\label{lem:Packing} There exists $\delta(\epsilon)$ that tends
to zero as $\epsilon\rightarrow0$ such that
\begin{equation}
\lim_{n\rightarrow\text{\ensuremath{\infty}}}\mathbb{P}\left((U^{n},V_{0}^{n},V_{[1:k]}^{n}(m_{[1:k]}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}\textrm{ for some }m_{[1:k]}\right)=0,
\end{equation}
if $\sum_{j\in\mathcal{J}}r_{j}<\sum_{j\in\mathcal{J}}H\left(V_{j}|V_{\mathcal{A}_{j}}V_{0}\right)-H\left(V_{\mathcal{J}}|V_{0}U\right)-\delta(\epsilon)$
for some $\mathcal{J}\subseteq[1:k]$ such that $\mathcal{J}\neq\emptyset$
and if $j\in\mathcal{J}$, then $\mathcal{A}_{j}\subseteq\mathcal{J}$.
\end{lem}
Note that all the existing covering and packing lemmas such as \cite[Lem. 8.2]{El Gamal}
and \cite[Lem. 4]{Shayevitz}, only involve single-layer codebook.
Our Multivariate Covering and Packing Lemmas generalize them to the
case of multilayer codebook, and certainly our Covering/Packing Lemmas
could recover all of them.
\section{Source Broadcast }
\subsection{Source Broadcast}
Now, we bound the distortion region for source broadcast communication.
To write the inner bound, we first introduce an auxiliary random variable
$V_{j},1\leq j\leq N\triangleq2^{K}-1$ for each of the $2^{K}-1$
nonempty subsets $\mathcal{G}_{j}\subseteq[1:K]$, and let $V_{j}$
denote a common message transmitted from sender to all the receivers
in $\mathcal{G}_{j}$. The $V_{j}$ corresponds to a subset $\mathcal{G}_{j}$
by the following one-to-one mapping.
Sort all the nonempty subsets $\mathcal{G}\subseteq[1:K]$ in the
decreasing order\footnote{We say a set $\mathcal{G}$ is larger than another $\mathcal{H}$
if $|\mathcal{G}|>|\mathcal{H}|$, or $|\mathcal{G}|=|\mathcal{H}|$
and there exists some $1\leq i\leq|\mathcal{G}|$ such that $\mathcal{G}\left[i\right]>\mathcal{H}\left[i\right]$
and $\mathcal{G}\left[l\right]=\mathcal{H}\left[l\right]$ for all
$1\leq l\leq i-1$, where $\mathcal{G}\left[i\right]$ (or $\mathcal{H}\left[i\right]$)
denotes the $i$th largest element in $\mathcal{G}$ (or $\mathcal{H}$). }. Map the $j$th subset in the resulting sequence to $j$. Obviously
this mapping is one-to-one corresponding. For example, if $K=3$,
then $\mathcal{G}_{1}=\left\{ 1,2,3\right\} ,\mathcal{G}_{2}=\left\{ 2,3\right\} ,\mathcal{G}_{3}=\left\{ 1,3\right\} ,\mathcal{G}_{4}=\left\{ 1,2\right\} ,\mathcal{G}_{5}=\left\{ 3\right\} ,\mathcal{G}_{6}=\left\{ 2\right\} ,\mathcal{G}_{7}=\left\{ 1\right\} $.
Besides, let
\begin{align}
\mathcal{A}_{j} & \triangleq\left\{ i\in[1:N]:\mathcal{G}_{j}\subsetneqq\mathcal{G}_{i}\right\} ,1\leq j\leq N,\\
\mathcal{D}_{k} & \triangleq\left\{ i\in[1:N]:k\in\mathcal{G}_{i}\right\} ,1\leq k\leq K.
\end{align}
Later we will show that they respectively correspond to the index
set of the random variables on which the codeword $V_{j}^{n}$ is
superposed, and the index set of decodable codewords $V_{j}^{n}$'s
for receiver $k$ in the proposed hybrid coding scheme; see Appendix
\ref{sub:Inner-Bound}. Decoder $k$ is able to recover correctly
the $V_{j}^{n}$, designated by the encoder with probability approaching
1 as $n\rightarrow\infty$ if $j\in\mathcal{D}_{k}$. In addition,
it is easy to verify that if $j\in\mathcal{D}_{k},\textrm{ then }\mathcal{A}_{j}\subseteq\mathcal{D}_{k}$.
It means that the proposed codebook satisfies that if information
$V_{j}^{n}$ can be recovered correctly by receiver $k$, then $V_{\mathcal{A}_{j}}^{n}$
can also be recovered correctly by it.
Based on the notations above, we define distortion region (inner bound)
\begin{align}
\mathcal{R}^{(i)}= & \Bigl\{ D_{[1:K]}:\textrm{There exist some pmf }p_{V_{[1:N]}|S},\textrm{ vector }r_{[1:N]},\nonumber \\
& \text{and functions }x\left(v_{[1:N]},s\right),\hat{s}_{k}\left(v_{\mathcal{D}_{k}},y_{k}\right),1\le k\le K\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},1\le k\le K,\nonumber \\
& \sum_{j\in\mathcal{J}}r_{j}>\sum_{j\in\mathcal{J}}H\left(V_{j}|V_{\mathcal{A}_{j}}\right)-H\left(V_{\mathcal{J}}|S\right)\nonumber \\
& \textrm{for all }\mathcal{J}\subseteq[1:N]\textrm{ such that }\mathcal{J}\neq\emptyset\textrm{ and if }j\in\mathcal{J},\textrm{ then }\mathcal{A}_{j}\subseteq\mathcal{J},\nonumber \\
& \sum_{j\in\mathcal{J}^{c}}r_{j}<\sum_{j\in\mathcal{J}^{c}}H\left(V_{j}|V_{\mathcal{A}_{j}}\right)-H\left(V_{\mathcal{J}^{c}}|Y_{k}V_{\mathcal{J}}\right)\nonumber \\
& \textrm{for all }1\le k\le K\textrm{ and for all }\mathcal{J}\subseteq\mathcal{D}_{k}\textrm{ such that }\mathcal{J}^{c}\triangleq\mathcal{D}_{k}\backslash\mathcal{J}\neq\emptyset\textrm{ and if }j\in\mathcal{J},\textrm{ then }\mathcal{A}_{j}\subseteq\mathcal{J}\Bigr\}.\label{eq:-28}
\end{align}
Besides, define two distortion region (outer bounds, achieved by introducing
auxiliary random variables at sender)
\begin{align}
\mathcal{R}_{1}^{(o)}= & \Bigl\{ D_{[1:K]}:\textrm{There exists some pmf }p_{\hat{S}_{[1:K]}|S}\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},1\le k\le K,\nonumber \\
& \text{and for any pmf }p_{U_{[1:L]}|S},\text{ one can find }p_{\tilde{U}_{[1:L]},X^{n}}\text{satisfying}\nonumber \\
& I\left(\hat{S}_{\mathcal{A}};U_{\mathcal{B}}|U_{\mathcal{C}}\right)\leq\frac{1}{n}I\left(Y_{\mathcal{A}}^{n};\tilde{U}_{\mathcal{B}}|\tilde{U}_{\mathcal{C}}\right)\textrm{ for any }\mathcal{A}\subseteq\left[1:K\right],\mathcal{B},\mathcal{C}\subseteq\left[1:L\right]\Bigr\},
\end{align}
\begin{align}
\mathcal{R}_{1}^{\prime(o)}= & \Bigl\{ D_{[1:K]}:\textrm{There exists some pmf }p_{\hat{S}_{[1:K]}|S}\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},1\le k\le K,\nonumber \\
& \text{and for any pmf }p_{U_{[1:L]}|S},\text{ one can find }p_{X,\tilde{U}_{[1:L]},W_{[1:K]},W_{[1:K]}^{\prime}}\text{satisfying}\nonumber \\
& \sum_{i=1}^{m}I\left(\hat{S}_{\mathcal{A}_{i}};U_{\mathcal{B}_{i}}|U_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\right)\leq\sum_{i=1}^{m}I\left(Y_{\mathcal{A}_{i}};\tilde{U}_{\mathcal{B}_{i}}\tilde{W}_{\mathcal{A}_{i+1}}|\tilde{U}_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\tilde{W}_{\mathcal{A}_{i}}\tilde{W}_{\mathcal{A}_{i-1}}\right),\nonumber \\
& \textrm{for any }m\geq1,\mathcal{A}_{i}\subseteq\left[1:K\right],\mathcal{B}_{i}\subseteq\left[1:L\right],0\le i\le m,\mathcal{A}_{0},\mathcal{A}_{m+1}\triangleq\emptyset,\nonumber \\
& \textrm{and }\tilde{W}_{\mathcal{A}_{i}}\triangleq W_{\mathcal{A}_{i}},\textrm{if }i\textrm{ is odd};W'_{\mathcal{A}_{i}},\textrm{otherwise, or }\tilde{W}_{\mathcal{A}_{i}}\triangleq W'_{\mathcal{A}_{i}},\textrm{if }i\textrm{ is odd};W{}_{\mathcal{A}_{i}},\textrm{otherwise}\Bigr\},
\end{align}
and another distortion region (outer bound, achieved by introducing
auxiliary random variables at receivers)
\begin{align}
\mathcal{R}_{2}^{(o)}= & \Bigl\{ D_{[1:K]}:\textrm{There exists some pmf }p_{X}\textrm{ and some functions }\hat{s}_{k}^{n}\left(\tilde{y}_{k}\right),1\le k\le K\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},1\le k\le K,\nonumber \\
& \text{and for any pmf }p_{U_{[1:L]}|Y_{[1:K]}},\text{ one can find }p_{\tilde{Y}_{[1:K]}|S}p_{\tilde{U}_{[1:L]}|\tilde{Y}_{[1:K]}}\text{satisfying}\nonumber \\
& I\left(S;\tilde{Y}_{\mathcal{B}}\tilde{U}_{\mathcal{B}'}|\tilde{Y}_{\mathcal{C}}\tilde{U}_{\mathcal{C}'}\right)\leq I\left(X;Y_{\mathcal{B}}U_{\mathcal{B}'}|Y_{\mathcal{C}}U_{\mathcal{C}'}\right)\textrm{ for any }\mathcal{B},\mathcal{C}\subseteq\left[1:K\right],\mathcal{B}',\mathcal{C}'\subseteq\left[1:L\right]\Bigr\}.
\end{align}
Then we have the following theorem. The proof is given in Appendix
\ref{sec:broadcast}.
\begin{thm}
\label{thm:AdmissibleRegion-GBC} For transmitting source $S$ over
general broadcast channel $p_{Y_{[1:K]}|X}$,
\begin{equation}
\mathcal{R}^{(i)}\subseteq\mathcal{R}\subseteq\mathcal{R}_{1}^{(o)}\cap\mathcal{R}_{2}^{(o)}\subseteq\mathcal{R}_{1}^{\prime(o)}.
\end{equation}
\end{thm}
\begin{rem}
The inner bound of Theorem \ref{thm:AdmissibleRegion-GBC} can be
easily extended to Gaussian or any other well-behaved continuous-alphabet
source-channel pair by standard discretization method \cite[Thm. 3.3]{El Gamal},
and moreover for this case the outer bounds still hold. Theorem \ref{thm:AdmissibleRegion-GBC}
can be also extended to the case of source-channel bandwidth mismatch,
where $m$ samples of a DMS are transmitted through $n$ uses of a
DM-BC. This can be accomplished by replacing the source and channel
symbols in Theorem \ref{thm:AdmissibleRegion-GBC} by supersymbols
of lengths $m$ and $n$, respectively. Besides, Theorem \ref{thm:AdmissibleRegion-GBC}
could be also extended to the problem of broadcasting correlated sources
(by modifying the distortion measure) or source broadcast with channel
input cost (by adding channel input constraint).
\end{rem}
\begin{figure}[t]
\centering\includegraphics[width=1\columnwidth]{hybridcoding_GBC}
\caption{\label{fig:hybridcoding}A unified hybrid coding used to prove the
inner bound of Theorem \ref{thm:AdmissibleRegion-GBC}. }
\end{figure}
The inner bound $\mathcal{R}^{(i)}$ in Theorem \ref{thm:AdmissibleRegion-GBC}
is achieved by a unified hybrid coding scheme depicted in Fig. \ref{fig:hybridcoding}.
In this scheme, the codebook has a layered (or superposition) structure,
and consists of randomly and independently generated codewords $V_{[1:N]}^{n}(m_{[1:N]})$,
$m_{[1:N]}\in\prod_{i=1}^{N}\left[1:2^{nr_{i}}\right]$, where $r_{[1:N]}$
satisfies \eqref{eq:-28}. At encoder side, upon source sequence $S^{n}$,
the encoder produces digital messages $M_{[1:N]}$ with $M_{i}$ meant
for all the receivers $k$ satisfying $i\in\mathcal{D}_{k}$. Then,
the codeword $V_{[1:N]}^{n}(M_{[1:N]})$ and the source sequence $S^{n}$
are used to generate channel input $X^{n}$ by symbol-by-symbol mapping
$x(v_{[1:N]},s)$. At decoder sides, upon received signal $Y_{k}^{n}$,
decoder $k$ could reconstruct $M_{\mathcal{D}_{k}}$ (and also $V_{\mathcal{D}_{k}}^{n}(M_{\mathcal{D}_{k}})$)
losslessly, and then $\hat{S}_{k}^{n}$ is produced by symbol-by-symbol
mapping $\hat{s}_{k}(v_{\mathcal{D}_{k}},y_{k})$. Such a scheme could
achieve any $D_{[1:K]}$ in the inner bound $\mathcal{R}^{(i)}$.
To reveal essence of such hybrid coding, the digital transmission
part of this hybrid coding can be roughly understood as cascade of
a $K$-user Gray-Wyner source-coding and a $K$-user Marton's broadcast
channel-coding, which share a common codebook. According to \cite[Thm. 13.3]{El Gamal},
the encoding operation of Gray-Wyner source-coding with rates $r_{[1:N]}$
is successful if $\sum_{j\in\mathcal{J}}r_{j}>\sum_{j\in\mathcal{J}}H\left(V_{j}|V_{\mathcal{A}_{j}}\right)-H\left(V_{\mathcal{J}}|S\right)\textrm{ for all }\mathcal{J}\subseteq[1:N]\textrm{ such that }\mathcal{J}\neq\emptyset\textrm{ and if }j\in\mathcal{J},\textrm{ then }\mathcal{A}_{j}\subseteq\mathcal{J}$,
and according to \cite[Thm 5.2]{El Gamal} the decoding operation
of Marton's broadcast channel-coding with rates $r_{[1:N]}$ is successful
if $\sum_{j\in\mathcal{J}^{c}}r_{j}<\sum_{j\in\mathcal{J}^{c}}H\left(V_{j}|V_{\mathcal{A}_{j}}\right)-H\left(V_{\mathcal{J}^{c}}|Y_{k}V_{\mathcal{J}}\right)\textrm{for all }1\le k\le K\textrm{ and for all }\mathcal{J}\subseteq\mathcal{D}_{k}\textrm{ such that }\mathcal{J}^{c}\neq\emptyset\textrm{ and if }j\in\mathcal{J},\textrm{ then }\mathcal{A}_{j}\subseteq\mathcal{J}$.
Note that in Marton's broadcast channel-coding, $r_{[1:N]}$ does
not correspond to the regular broadcast-rates (i.e., the rates of
subcodebooks in multicoding), but is the rates of the whole codebook.
Since the proposed hybrid coding satisfies the two sufficient conditions
above, $V_{\mathcal{D}_{k}}^{n}(M_{\mathcal{D}_{k}})$ could be losslessly
transmitted to receiver $k$. Note that such informal understanding
is inaccurate owing to the use of symbol-by-symbol mapping, but it
provides the rationale for our scheme. Besides, the design of such
unified hybrid coding is inspired by the hybrid coding scheme for
sending correlated sources over multi-access channel in \cite{Minero}.
The outer bounds $\mathcal{R}_{1}^{(o)}$, $\mathcal{R}_{1}^{\prime(o)}$
and $\mathcal{R}_{2}^{(o)}$ of Theorem \ref{thm:AdmissibleRegion-GBC}
are derived by introducing auxiliary random variables $U_{[1:L]}^{n}$
at sender side or at receiver sides. The proof method of introducing
auxiliary random variables (or remote sources) at sender side could
be found in \cite{Tian11}, \cite[Thm. 2]{Khezeli14} and \cite[Lem. 1]{Khezeli}.
However, to the best of our knowledge, it is the first time to prove
outer bounds by introducing auxiliary random variables (or remote
channels) at receiver sides. In \cite{Tian11} it is used to derive
the outer bound for Gaussian source broadcast, and in \cite[Thm. 2]{Khezeli14}
and \cite[Lem. 1]{Khezeli} it is used to derive the outer bounds
for sending source over 2-user general broadcast channel. This proof
method generalizes the one used to derive trivial outer bound, but
it does not always result in a tighter outer bound than the trivial
one \cite{Yu2015comments}. A deeper understanding of these proof
methods has been given by Khezeli \emph{et al. }in \cite{Khezeli}.
$p_{\hat{S}{}_{[1:K]}|S}$ can be considered as a virtual broadcast
channel realized over physical broadcast channel $p_{Y_{[1:K]}|X}$,
and hence certain measurements based on $p_{\hat{S}{}_{[1:K]}|S}$
are less than or equal to those based on $p_{Y_{[1:K]}|X}$. This
leads to the necessary conditions on the communication. Besides, the
necessary conditions can be also understood from the perspective of
virtual source. $X$ and $Y_{[1:K]}$ can be considered as a virtual
source and $K$ virtual reconstructions. Then the physical source
$S$ and the physical reconstructions $\hat{S}{}_{[1:K]}$ are correlated
through the virtual source and virtual reconstructions. Hence the
physical source should be more ``tractable'' than the virtual one,
and certain measurements based on physical source and reconstructions
should be less than or equal to those based on the virtual source
and reconstructions. The analysis above gives the reasons why $\mathcal{R}_{1}^{(o)}$
and $\mathcal{R}_{1}^{\prime(o)}$ are expressed in form of comparison
of the ``capacity regions'' of virtual broadcast channel and physical
broadcast channel, while $\mathcal{R}_{2}^{(o)}$ is expressed in
form of comparison of the ``source-coding rate regions'' of virtual
source and physical source.
For 2-user broadcast case, the inner bound of Theorem \ref{thm:AdmissibleRegion-GBC}
reduces to
\begin{align}
\mathcal{R}^{(i)}= & \Bigl\{\left(D_{1},D_{2}\right):\textrm{There exist some pmf }p_{V_{0},V_{1},V_{2}|S},\nonumber \\
& \text{and functions }x\left(v_{0},v_{1},v_{2},s\right),\hat{s}_{k}\left(v_{0},v_{k},y_{k}\right),k=1,2,\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},\nonumber \\
& I(V_{0}V_{k};S)<I(V_{0}V_{k};Y_{k}),k=1,2,\nonumber \\
& I(V_{0}V_{1}V_{2};S)+I(V_{1};V_{2}|V_{0})<\min\left\{ I(V_{0};Y_{1}),I(V_{0};Y_{2})\right\} +I(V_{1};Y_{1}|V_{0})+I(V_{2};Y_{2}|V_{0}),\nonumber \\
& I(V_{0}V_{1};S)+I(V_{0}V_{2};S)+I(V_{1};V_{2}|V_{0}S)<I(V_{0}V_{1};Y_{1})+I(V_{0}V_{2};Y_{2})\Bigr\}.
\end{align}
This inner bound was first given in by Yassaee \emph{et. al} \cite{Yassaee}.
On the other hand, for 2-user broadcast case, letting $L=1$ for $\mathcal{R}_{1}^{(o)}$
and $\mathcal{R}_{2}^{(o)}$, and $L=3$ for $\mathcal{R}_{1}^{\prime(o)}$,
the outer bounds of Theorem \ref{thm:AdmissibleRegion-GBC} reduces
to
\begin{align*}
\mathcal{R}_{1}^{(o)}= & \Bigl\{\left(D_{1},D_{2}\right):\textrm{There exists some pmf }p_{\hat{S}_{1}\hat{S}_{2}|S}\text{ such that}\\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},k=1,2,\\
& \text{and for any pmf }p_{U|S},\text{ one can find }p_{\tilde{U},X^{n}}\text{satisfying}\\
& I\left(\hat{S}_{1};U\right)\leq\frac{1}{n}I\left(Y_{1}^{n};\tilde{U}\right),\\
& I\left(\hat{S}_{2};U\right)\leq\frac{1}{n}I\left(Y_{2}^{n};\tilde{U}\right),\\
& I\left(\hat{S}_{1}\hat{S}_{2};U\right)\leq\frac{1}{n}I\left(Y_{1}^{n}Y_{2}^{n};\tilde{U}\right),\\
& I\left(\hat{S}_{1};S|U\right)\leq\frac{1}{n}I\left(Y_{1}^{n};X^{n}|\tilde{U}\right),\\
& I\left(\hat{S}_{2};S|U\right)\leq\frac{1}{n}I\left(Y_{2}^{n};X^{n}|\tilde{U}\right),\\
& I\left(\hat{S}_{1}\hat{S}_{2};S|U\right)\leq\frac{1}{n}I\left(Y_{1}^{n}Y_{2}^{n};X^{n}|\tilde{U}\right)\Bigr\},
\end{align*}
\begin{align}
\mathcal{R}_{1}^{\prime(o)}= & \Bigl\{\left(D_{1},D_{2}\right):\textrm{There exists some pmf }p_{\hat{S}_{1},\hat{S}_{2}|S}\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},k=1,2,\nonumber \\
& \text{and for any pmf }p_{U_{1},U_{2},U_{3}|S},\text{ one can find }p_{X,\tilde{U}_{1},\tilde{U}_{2},\tilde{U}_{3},W_{1},W_{2},W'_{1},W'_{2}}\text{satisfying}\nonumber \\
& I\left(\hat{S}_{\mathcal{A}_{1}};U_{\mathcal{B}_{1}}|U_{\mathcal{B}_{0}}\right)\leq I\left(Y_{\mathcal{A}_{1}};\tilde{U}_{\mathcal{B}_{1}}|\tilde{U}_{\mathcal{B}_{0}}\tilde{W}_{\mathcal{A}_{1}}\right),\nonumber \\
& I\left(\hat{S}_{\mathcal{A}_{1}};U_{\mathcal{B}_{1}}|U_{\mathcal{B}_{0}}\right)+I\left(\hat{S}_{\mathcal{A}_{2}};U_{\mathcal{B}_{2}}|U_{\mathcal{B}_{0}}U_{\mathcal{B}_{1}}\right)\leq I\left(Y_{\mathcal{A}_{1}};\tilde{U}_{\mathcal{B}_{1}}\tilde{W}_{\mathcal{A}_{2}}|\tilde{U}_{\mathcal{B}_{0}}\tilde{W}_{\mathcal{A}_{1}}\right)+I\left(Y_{\mathcal{A}_{2}};\tilde{U}_{\mathcal{B}_{2}}|\tilde{U}_{\mathcal{B}_{0}}\tilde{U}_{\mathcal{B}_{1}}\tilde{W}_{\mathcal{A}_{1}}\tilde{W}_{\mathcal{A}_{2}}\right),\nonumber \\
& I\left(\hat{S}_{\mathcal{A}_{1}};U_{\mathcal{B}_{1}}|U_{\mathcal{B}_{0}}\right)+I\left(\hat{S}_{\mathcal{A}_{2}};U_{\mathcal{B}_{2}}|U_{\mathcal{B}_{0}}U_{\mathcal{B}_{1}}\right)+I\left(\hat{S}_{\mathcal{A}_{3}};U_{\mathcal{B}_{3}}|U_{\mathcal{B}_{0}}U_{\mathcal{B}_{1}}U_{\mathcal{B}_{2}}\right)\leq I\left(Y_{\mathcal{A}_{1}};\tilde{U}_{\mathcal{B}_{1}}\tilde{W}_{\mathcal{A}_{2}}|\tilde{U}_{\mathcal{B}_{0}}\tilde{W}_{\mathcal{A}_{1}}\right)\nonumber \\
& \qquad+I\left(Y_{\mathcal{A}_{2}};\tilde{U}_{\mathcal{B}_{2}}\tilde{W}_{\mathcal{A}_{3}}|\tilde{U}_{\mathcal{B}_{0}}\tilde{U}_{\mathcal{B}_{1}}\tilde{W}_{\mathcal{A}_{1}}\tilde{W}_{\mathcal{A}_{2}}\right)+I\left(Y_{\mathcal{A}_{3}};\tilde{U}_{\mathcal{B}_{3}}|\tilde{U}_{\mathcal{B}_{0}}\tilde{U}_{\mathcal{B}_{1}}\tilde{U}_{\mathcal{B}_{2}}\tilde{W}_{\mathcal{A}_{2}}\tilde{W}_{\mathcal{A}_{3}}\right),\nonumber \\
& \textrm{for any }\mathcal{A}_{i}\subseteq\left[1:2\right],\mathcal{B}_{i}\subseteq\left[1:3\right],0\le i\le3,\nonumber \\
& \textrm{and }\tilde{W}_{\mathcal{A}_{i}}\triangleq W_{\mathcal{A}_{i}},\textrm{if }i\textrm{ is odd};W'_{\mathcal{A}_{i}},\textrm{otherwise, or }\tilde{W}_{\mathcal{A}_{i}}\triangleq W'_{\mathcal{A}_{i}},\textrm{if }i\textrm{ is odd};W{}_{\mathcal{A}_{i}},\textrm{otherwise}\Bigr\},
\end{align}
and
\begin{align*}
\mathcal{R}_{2}^{(o)}= & \Bigl\{\left(D_{1},D_{2}\right):\textrm{There exists some pmf }p_{X}\textrm{ and some functions }\hat{s}_{k}\left(\tilde{y}_{k}\right),k=1,2\text{ such that}\\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},k=1,2,\\
& \text{and for any pmf }p_{U|Y_{1}Y_{2}},\text{ one can find }p_{\tilde{Y}_{1}\tilde{Y}_{2}|S}p_{\tilde{U}|\tilde{Y}_{1}\tilde{Y}_{2}}\text{satisfying}\\
& I\left(S;\tilde{U}\right)\leq I\left(X;U\right),\\
& I\left(S;\tilde{Y}_{1}|\tilde{U}\right)\leq I\left(X;Y_{1}|U\right),\\
& I\left(S;\tilde{Y}_{2}|\tilde{U}\right)\leq I\left(X;Y_{2}|U\right),\\
& I\left(S;\tilde{Y}_{1}|\tilde{Y}_{2}\tilde{U}\right)\leq I\left(X;Y_{1}|Y_{2}U\right),\\
& I\left(S;\tilde{Y}_{2}|\tilde{Y}_{1}\tilde{U}\right)\leq I\left(X;Y_{2}|Y_{1}U\right),\\
& I\left(S;\tilde{Y}_{1}\tilde{Y}_{2}|\tilde{U}\right)\leq I\left(X;Y_{1}Y_{2}|U\right)\Bigr\}.
\end{align*}
The outer bound $\mathcal{R}_{1}^{\prime(o)}$ is as tight as if not
tighter than that given by Khezeli \emph{et al. }in \cite[Thm. 1]{Khezeli}.
This is because the outer bound in \cite[Lem 1]{Khezeli} is just
$\mathcal{R}_{1}^{\prime(o)}$ with $\mathcal{A}_{i}$ restricted
to be an element (not a subset) of $\left[1:2\right]$. In addition,
note that introducing nondegenerate auxiliary random variable(s) in
the outer bounds $\mathcal{R}_{1}^{(o)},\mathcal{R}_{1}^{\prime(o)}$
and $\mathcal{R}_{2}^{(o)}$ are not trivial in general. The necessity
of introducing nondegenerate variable(s) for $\mathcal{R}_{1}^{(o)}$
and $\mathcal{R}_{1}^{\prime(o)}$ can be concluded from some special
cases, e.g., source broadcast over degraded channel, quadratic Gaussian
source broadcast or Hamming binary source broadcast (see the details
in the subsequent three subsections). To show the necessity of introducing
nondegenerate variable for $\mathcal{R}_{2}^{(o)}$, we only consider
the first three inequalities on mutual information in $\mathcal{R}_{2}^{(o)}$.
Next we show that the necessary conditions
\begin{align}
& I\left(S;\tilde{U}\right)\leq I\left(X;U\right),\label{eq:-29}\\
& I\left(S;\tilde{Y}_{1}|\tilde{U}\right)\leq I\left(X;Y_{1}|U\right),\\
& I\left(S;\tilde{Y}_{2}|\tilde{U}\right)\leq I\left(X;Y_{2}|U\right),\label{eq:-31}
\end{align}
with nondegenerate $U$ results in a tighter bound than that with
degenerate $U$ (for the latter case, the necessary conditions reduce
to the trivial bound).
Suppose the broadcast channel $P_{Y_{1}Y_{2}|X}$ satisfies $Y_{1}=(Y_{0},Y_{1}'),Y_{2}=(Y_{0},Y_{2}')$
for some $Y_{0},Y_{1}',Y_{2}'$. Suppose lossless transmission case
(Hamming distortion measure and $D_{1}=D_{2}=0$): $S=(S_{1},S_{2}),\hat{S}_{1}=S_{1},\hat{S}_{2}=S_{2}$,
and $H(S_{1})=C_{1}$ and $H(S_{2})=C_{2}$. Hence the trivial outer
bound implies $(S_{1},S_{2})$ can be transmitted losslessly. Now
we show that the outer bound $\mathcal{R}_{2}^{(o)}$ implies $(S_{1},S_{2})$
can not be transmitted losslessly for some cases. Set $U=Y_{0}$ in
$\mathcal{R}_{2}^{(o)}$. Then it is easy to obtain the following
inequalities from \eqref{eq:-29}-\eqref{eq:-31}.
\begin{align}
& I(S_{1}S_{2};V)\leq I(X;Y_{0}),\label{eq:-29-1}\\
& H(S_{1}|V)\leq I(X;Y_{1}|Y_{0}),\\
& H(S_{2}|V)\leq I(X;Y_{2}|Y_{0}),\label{eq:-31-1}
\end{align}
for some $p_{V|S_{1}S_{2}}$. Therefore, we further have
\begin{equation}
H(S_{1})\leq H(S_{1})+I(S_{2};V|S_{1})=I(S_{1}S_{2};V)+H(S_{1}|V)\leq I(X;Y_{0})+I(X;Y_{1}|Y_{0})=I(X;Y_{1})\leq C_{1}.
\end{equation}
On the other hand, since $H(S_{1})=C_{1}$ and $H(S_{2})=C_{2}$,
the equalities hold in the inequalities above, which implies $I(X;Y_{1})=C_{1}$
(i.e., $P_{X}$ is the capacity-achieving distribution), $I(S_{1}S_{2};V)=I(X;Y_{0})$
, and $I(S_{2};V|S_{1})=0$, i.e., $S_{2}\rightarrow S_{1}\rightarrow V$.
Similarly, we have $S_{1}\rightarrow S_{2}\rightarrow V$. In addition,
the Gács-K{\"o}rner common information \cite{Gacs,Ahlswede} can
be expressed as
\begin{equation}
C_{GK}(S_{1};S_{2})=\sup_{P_{V|S_{1}S_{2}}:S_{2}\rightarrow S_{1}\rightarrow V,S_{1}\rightarrow S_{2}\rightarrow V}I(S_{1}S_{2};V).
\end{equation}
Hence there exists $p_{V|S_{1}S_{2}}$ such that $S_{2}\rightarrow S_{1}\rightarrow V,S_{1}\rightarrow S_{2}\rightarrow V$
and $I(S_{1}S_{2};V)=I(X;Y_{0})$ , only if $C_{GK}(S_{1};S_{2})\geq I(X;Y_{0})>0$
(suppose the channel $P_{Y_{0}|X}$ satisfy $I(X;Y_{0})>0$ for the
capacity-achieving distribution $P_{X}$). However GK common information
does not always exist for any source pair $(S_{1},S_{2})$ , e.g.,
$C_{GK}(S_{1};S_{2})=0$ for doubly symmetric binary source. This
implies the outer bound is tighter than the trivial one, which in
turn implies the necessity of introducing nondegenerate variable for
$\mathcal{R}_{2}^{(o)}.$
When consider lossless transmission of independent source, Theorem
\ref{thm:AdmissibleRegion-GBC} can be used to achieve bounds on capacity
region of general broadcast channel with common messages. In this
case, $S=\left(M_{j}:1\leq j\leq N\right)$ and all $M_{j}$'s are
independent with each other. For each $1\leq j\leq N$, let $R_{j}$
denote the rate of the common message $M_{j}$ that is to be transmitted
losslessly from sender to all the receivers in $\mathcal{G}_{j}\subseteq[1:K]$.
The correspondence between $M_{j}$ and $\mathcal{G}_{j}$ is kept
same to that between $V_{j}$ and $\mathcal{G}_{j}$; see the beginning
of this section. Then such achievable rates $R_{[1:N]}$ constitute
the capacity region $\mathcal{C}$.
Now, define rate region
\begin{align}
\mathcal{C}^{(i)}= & \Bigl\{ R_{[1:N]}:\textrm{There exist some pmf }p_{V_{[1:N]}|S},\textrm{ vector }r_{[1:N]}\nonumber \\
& \text{and function }x\left(v_{[1:N]}\right)\text{ such that}\nonumber \\
& \sum_{j\in\mathcal{J}}r_{j}>\sum_{j\in\mathcal{J}}R_{j}+\sum_{j\in\mathcal{J}}H\left(V_{j}|V_{\mathcal{A}_{j}}\right)-H\left(V_{\mathcal{J}}|S\right)\nonumber \\
& \textrm{for all }\mathcal{J}\subseteq[1:N]\textrm{ such that }\mathcal{J}\neq\emptyset\textrm{ and if }j\in\mathcal{J},\textrm{ then }\mathcal{A}_{j}\subseteq\mathcal{J},\nonumber \\
& \sum_{j\in\mathcal{J}^{c}}r_{j}<\sum_{j\in\mathcal{J}^{c}}H\left(V_{j}|V_{\mathcal{A}_{j}}\right)-H\left(V_{\mathcal{J}^{c}}|Y_{k}V_{\mathcal{J}}\right)\nonumber \\
& \textrm{for all }1\le k\le K\textrm{ and for all }\mathcal{J}\subseteq\mathcal{D}_{k}\textrm{such that }\mathcal{J}^{c}\triangleq\mathcal{D}_{k}\backslash\mathcal{J}\neq\emptyset\textrm{ and if }j\in\mathcal{J},\textrm{then }\mathcal{A}_{j}\subseteq\mathcal{J}\Bigr\},
\end{align}
and another rate region
\begin{align}
\mathcal{C}^{(o)}= & \Bigl\{ R_{[1:N]}:\textrm{There exists some pmf }\prod_{i=1}^{N}p_{\tilde{U}_{i}}p_{W_{[1:K]},W_{[1:K]}^{\prime}|\tilde{U}_{[1:N]}}\text{and function }x\left(\tilde{u}_{[1:N]}\right)\text{ such that}\nonumber \\
& \sum_{i=1}^{m}\sum_{j\in\left(\bigcup_{k\in\mathcal{A}_{i}}\mathcal{D}_{k}\right)\cap\mathcal{B}_{i}}R_{j}\leq\sum_{i=1}^{m}I\left(Y_{\mathcal{A}_{i}};\tilde{U}_{\mathcal{B}_{i}}\tilde{W}_{\mathcal{A}_{i+1}}|\tilde{U}_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\tilde{W}_{\mathcal{A}_{i}}\tilde{W}_{\mathcal{A}_{i-1}}\right),\nonumber \\
& \textrm{for any }m\geq1,\mathcal{A}_{i}\subseteq\left[1:K\right],\mathcal{B}_{i}\subseteq\left[1:N\right],0\le i\le m,\mathcal{A}_{0},\mathcal{A}_{m+1}\triangleq\emptyset,\nonumber \\
& \textrm{and }\tilde{W}_{\mathcal{A}_{i}}\triangleq W_{\mathcal{A}_{i}},\textrm{if }i\textrm{ is odd};W'_{\mathcal{A}_{i}},\textrm{otherwise, or }\tilde{W}_{\mathcal{A}_{i}}\triangleq W'_{\mathcal{A}_{i}},\textrm{if }i\textrm{ is odd};W{}_{\mathcal{A}_{i}},\textrm{otherwise}\Bigr\}.
\end{align}
Then as a consequence of Theorem \ref{thm:AdmissibleRegion-GBC},
we can establish the following bounds on the capacity region of general
broadcast channel. The proof is omitted.
\begin{thm}
\label{thm:CapacityRegion-GBC-2} For general broadcast channel $p_{Y_{[1:K]}|X}$,
the capacity region $\mathcal{C}$ with common messages satisfies
\begin{equation}
\mathcal{C}^{(i)}\subseteq\mathcal{C}\subseteq\mathcal{C}^{(o)}.
\end{equation}
\end{thm}
\begin{rem}
The inner bound and the outer bound of Theorem \ref{thm:CapacityRegion-GBC-2}
respectively follow from $\mathcal{R}^{(i)}$ and $\mathcal{R}_{1}^{\prime(o)}$
of Theorem \ref{thm:AdmissibleRegion-GBC}, and respectively generalize
Marton's inner bound and Nair-El Gamal outer bound to the case of
$K$-user broadcast channel.
\end{rem}
\subsection{Source Broadcast over Degraded Channel}
If the channel is degraded, define
\begin{align}
\mathcal{R}_{DBC}^{(i)}= & \Bigl\{ D_{[1:K]}:\textrm{There exist some pmf }p_{V_{K}|S}p_{V_{K-1}|V_{K}}\cdots p_{V_{1}|V_{2}},\nonumber \\
& \text{and functions }x\left(v_{K},s\right),\hat{s}_{k}\left(v_{k},y_{k}\right),1\le k\le K\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},\nonumber \\
& I\left(S;V_{k}\right)\leq\sum_{j=1}^{k}I\left(Y_{j};V_{j}|V_{j-1}\right),1\le k\le K,\text{ where }V_{0}\triangleq\emptyset\Bigr\},
\end{align}
and
\begin{align}
\mathcal{R}_{DBC}^{(o)}= & \Bigl\{ D_{[1:K]}:\textrm{There exists some pmfs }p_{\hat{S}_{[1:K]}|S},p_{X}\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},1\le k\le K,\nonumber \\
& \left(I\left(\hat{S}_{[1:k]};U_{k}|U_{k-1}\right):k\in[1:K]\right)\in\mathcal{B}_{DBC}\left(p_{X}p_{Y_{[1:K]}|X}\right)\nonumber \\
& \text{for any pmf }p_{U_{K-1}|S}p_{U_{K-2}|U_{K-1}}\cdots p_{U_{1}|U_{2}},U_{0}\triangleq\emptyset,U_{K}\triangleq S,\nonumber \\
& \textrm{and }\left(I\left(S;\hat{S}_{[1:k]}\right):k\in[1:K]\right)\in\mathcal{B}_{SRC}\left(p_{X}p_{Y_{[1:K]}|X}\right)\Bigr\},
\end{align}
where
\begin{align}
\mathcal{B}_{DBC}\left(p_{X}p_{Y_{[1:K]}|X}\right)= & \Bigl\{ R_{[1:K]}:\textrm{There exists some pmf }p_{V_{K-1}|X}p_{V_{K-2}|V_{K-1}}\cdots p_{V_{1}|V_{2}}\textrm{\text{such that }}\nonumber \\
& R_{k}\geq0,\sum_{j=1}^{k}R_{j}\leq\sum_{j=1}^{k}I\left(Y_{j};V_{j}|V_{j-1}\right),1\le k\le K,\text{ where }V_{0}\triangleq\emptyset,V_{K}\triangleq X\Bigr\}\label{eq:DBCcapacity}
\end{align}
denotes the capacity of degraded broadcast channel $p_{Y_{[1:K]}|X}$
with input $X$ following $p_{X}$, and
\begin{align}
\mathcal{B}_{SRC}\left(p_{X}p_{Y_{[1:K]}|X}\right)= & \Bigl\{ R_{[1:K]}:R_{k}\geq0,\sum_{j=1}^{k}R_{j}\geq I\left(X;Y_{[1:k]}\right),1\le k\le K\Bigr\}\label{eq:DBCcapacity2}
\end{align}
denotes the successive refinement coding rate region of source $X$
with reconstructions $Y_{[1:K]}$ following $p_{Y_{[1:K]}|X}$.
Then as a consequence of Theorem \ref{thm:AdmissibleRegion-GBC},
the following theorem holds. The proof is given in Appendix \ref{sec:broadcast-1}.
\begin{thm}
\label{thm:AdmissibleRegion-DBC} For transmitting source $S$ over
degraded broadcast channel $p_{Y_{[1:K]}|X}$,
\begin{equation}
\mathcal{R}_{DBC}^{(i)}\subseteq\mathcal{R}\subseteq\mathcal{R}_{DBC}^{(o)}.
\end{equation}
\end{thm}
\begin{rem}
$\mathcal{R}_{DBC}^{(o)}$ can be also expressed as
\begin{align}
\mathcal{R}_{DBC}^{(o)}= & \Bigl\{ D_{[1:K]}:\textrm{There exists some pmf }p_{\hat{S}_{[1:K]}|S},p_{X}\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},1\le k\le K,\nonumber \\
& \mathcal{B}_{DBC}\left(p_{S}p_{\hat{S}_{[1:K]}^{\prime}|S}\right)\subseteq\mathcal{B}_{DBC}\left(p_{X}p_{Y_{[1:K]}|X}\right),\nonumber \\
& \mathcal{B}_{SRC}\left(p_{S}p_{\hat{S}_{[1:K]}^{\prime}|S}\right)\supseteq\mathcal{B}_{SRC}\left(p_{X}p_{Y_{[1:K]}|X}\right),\hat{S}_{k}^{\prime}\triangleq\hat{S}_{[1:k]},1\le k\le K\Bigr\}.
\end{align}
From the last constraint of $\mathcal{R}_{DBC}^{(o)}$, one can obtain
an interesting result: the trivial outer bound $D_{[1:K]}^{*}$ can
be achieved for source broadcast over degraded channel only if the
source is successively refinable.
\end{rem}
Note that the last constraint of $\mathcal{R}_{DBC}^{(i)}$ can be
understood as the intersection between the rate region of successive
refinement coding for source $S$ and reconstructions $V_{[1:K]}$
and the capacity of degraded broadcast channel $p_{Y_{[1:K]}|X}$
with input $X$ and auxiliary random variables $V_{[1:K]}$, is not
empty. The second constraint of $\mathcal{R}_{DBC}^{(o)}$ can be
understood as the capacity of virtual degraded broadcast channel $p_{\hat{S}_{[1:K]}^{\prime}|S}$
with input $S$ is included in the capacity of physical degraded broadcast
channel $p_{Y_{[1:K]}|X}$ with input $X$. The last constraint of
$\mathcal{R}_{DBC}^{(o)}$ can be understood as the rate region of
successive refinement coding for the physical source $S$ and reconstructions
$\hat{S}_{[1:K]}^{\prime}$ includes the rate region of successive
refinement coding for the virtual source $X$ and reconstructions
$Y_{[1:K]}$. Similar to the general broadcast channel case, these
necessary conditions is also consistent with the intuition. From the
perspective of channel, the virtual broadcast channel is realized
over the physical broadcast channel, hence the physical channel should
be more ``capable'' than the virtual one. On the other hand, from
the perspective of source, the physical source is correlated with
the reconstructions through the virtual source and virtual reconstructions,
hence the physical source should be more ``tractable'' than the
virtual one.
\subsection{Quadratic Gaussian Source Broadcast}
Consider sending Gaussian source $S\sim\mathcal{N}\left(0,N_{S}\right)$
with quadratic distortion measure $d_{k}(s,\hat{s})=d(s,\hat{s})\triangleq(s-\hat{s})^{2}$
over power-constrained Gaussian broadcast channel $Y_{k}=X+W_{k},1\le k\le K$
with $\mathbb{E}\left(X^{2}\right)\leq P$ and$W_{k}\sim\mathcal{N}\left(0,N_{k}\right),N_{1}\geq N_{2}\geq\cdots\geq N_{K}$.
Assume bandwidth mismatch factor is $b$. Then the inner bound $\mathcal{R}_{DBC}^{(i)}$
in Theorem \ref{thm:AdmissibleRegion-DBC} could recover the best
known inner bound so far \cite[Thm. 5]{Prabhakaran11} by setting
suitable random variables and symbol-by-symbol mappings.
\begin{cor}
\cite[Thm. 5]{Prabhakaran11}\label{thm:InnerBoundGG} For transmitting
Gaussian source $S$ with quadratic distortion measure over 2-user
Gaussian broadcast channel with bandwidth mismatch factor $b$, $\mathcal{R}_{DBC}^{(i)}\subseteq\mathcal{R}$.
\begin{itemize}
\item For $b<1$ (bandwidth compression)
\begin{equation}
\mathcal{R}_{DBC}^{(i)}=\left\{ \left(D_{1}(\lambda,\gamma),D_{2}(\lambda,\gamma)\right):0\leq\lambda\leq1,0\leq\gamma\leq1\right\} ,
\end{equation}
where
\begin{align}
D_{1}(\lambda,\gamma) & =\frac{bN_{S}}{\frac{\lambda P+N_{1}}{\lambda\gamma P+N_{1}}}+\frac{(1-b)N_{S}}{\left(\frac{P+N_{1}}{\lambda P+N_{1}}\right)^{\frac{b}{1-b}}},\\
D_{2}(\lambda,\gamma) & =\frac{bN_{S}}{\frac{\lambda P+N_{1}}{\lambda\gamma P+N_{1}}}+\frac{(1-b)N_{S}}{\left(\frac{P+N_{1}}{\lambda P+N_{1}}\frac{\lambda\gamma P+N_{2}}{N_{2}}\right)^{\frac{b}{1-b}}}.
\end{align}
\item For $b>1$ (bandwidth expansion)
\begin{equation}
\mathcal{R}_{DBC}^{(i)}=\left\{ \left(D_{1}(\lambda,\gamma),D_{2}(\lambda,\gamma)\right):0\leq\lambda\leq1,0\leq\gamma\leq1\right\} ,
\end{equation}
where
\begin{align}
D_{1}(\lambda,\gamma) & =\frac{N_{S}}{\left(\frac{\frac{b(1-\gamma)}{b-1}P+N_{1}}{\lambda\frac{b(1-\gamma)}{b-1}P+N_{1}}\right)^{b-1}\left(\frac{b\gamma P+N_{1}}{N_{1}}\right)},\\
D_{2}(\lambda,\gamma) & =\frac{N_{S}}{\left(\frac{\frac{b(1-\gamma)}{b-1}P+N_{1}}{\lambda\frac{b(1-\gamma)}{b-1}P+N_{1}}\right)^{b-1}\left(\frac{b\gamma P+N_{2}}{N_{2}}\right)\left(\frac{\lambda\frac{b(1-\gamma)}{b-1}P+N_{2}}{N_{2}}\right)^{b-1}}.
\end{align}
\end{itemize}
\end{cor}
\begin{IEEEproof}
For $b=\frac{n}{m}<1$ (bandwidth compression), the source $S^{m}=\left(S^{n},S_{n+1}^{m}\right)$
is transmitted over channel $Y_{k}^{n}=X^{n}+W_{k}^{n},k=1,2$. Define
a set of random variables $\left(U_{1}^{m-n},E_{1}^{m-n},U_{2}^{m-n},E_{2}^{m-n},X_{1}^{n},X{}_{2}^{n},X'{}_{2}^{n},V_{1},V_{2}\right)$
such that
\begin{align}
S_{n+1}^{m} & =U_{1}^{m-n}+E_{1}^{m-n},\\
E_{1}^{m-n} & =U_{2}^{m-n}+E_{2}^{m-n},\\
X_{2}^{n} & =X_{2}^{\prime n}+\beta\alpha S^{n},\\
V_{1} & =(U_{1}^{m-n},X_{1}^{n}),\\
V_{2} & =(V_{1},U_{2}^{m-n},X{}_{2}^{n}),
\end{align}
where $U_{1}^{m-n},U_{2}^{m-n},E_{2}^{m-n}$ are mutually independent
Gaussian variables, $X_{2}^{\prime n}$ and $X_{1}^{n}$ are Gaussian
variables independent of all the other variables, $\textrm{Var}(X'_{2})=\lambda\gamma P,\textrm{Var}(X_{1})=\left(1-\lambda\right)P,\textrm{Var}(E_{1})=\frac{N_{S}}{\left(\frac{P+N_{1}}{\lambda P+N_{1}}\right)^{\frac{b}{1-b}}},$
$\textrm{Var}(E_{2})=\frac{\textrm{Var}(E_{1})}{\left(\frac{\lambda\gamma P+N_{2}}{N_{2}}\right)^{\frac{b}{1-b}}},$
and $\alpha=\sqrt{\frac{\lambda\left(1-\gamma\right)P}{N_{S}}},\beta=\frac{\lambda\gamma P}{\lambda\gamma P+N_{2}}$.
Define a set of functions
\begin{align}
& x^{n}\left(v_{2},s^{m}\right)=x_{1}^{n}+\alpha s^{n}+x{}_{2}^{n}-\beta\alpha s^{n}=x_{1}^{n}+\alpha s^{n}+x'{}_{2}^{n},\\
& \hat{s}_{1}^{m}\left(v_{1},y_{1}^{n}\right)=\left(\frac{\alpha N_{S}}{\alpha^{2}N_{S}+N_{1}}\left(y_{1}^{n}-x_{1}^{n}\right),u_{1}^{m-n}\right),\\
& \hat{s}_{2}^{m}\left(v_{2},y_{2}^{n}\right)=\left(\frac{\alpha N_{S}}{\alpha^{2}N_{S}+N_{2}}\left(y_{2}^{n}-x_{1}^{n}\right),u_{1}^{m-n}+u_{2}^{m-n}\right).
\end{align}
Substitute these variables and functions into the inner bound $\mathcal{R}_{DBC}^{(i)}$
in Theorem \ref{thm:AdmissibleRegion-DBC}, then the $b<1$ case in
Corollary \ref{thm:InnerBoundGG} is recovered.
For $b=\frac{n}{m}>1$ (bandwidth expansion), the source $S^{n}$
is transmitted over channel $\left(Y_{k}^{m},Y_{k,m+1}^{n}\right)=\left(X^{m},X_{m+1}^{n}\right)+\left(W_{k}^{m},W_{k,m+1}^{n}\right),k=1,2$.
Define a set of random variables $\left(U_{1}^{m},E_{1}^{m},U_{2}^{m},E_{2}^{m},X{}_{1}^{n-m},X{}_{2}^{n-m},V_{1},V_{2}\right)$
such that
\begin{align}
S^{m} & =U_{1}^{m}+E_{1}^{m},\\
E_{1}^{m} & =U_{2}^{m}+E_{2}^{m},\\
V_{1} & =(U_{1}^{m},X{}_{1}^{n-m}),\\
V_{2} & =(V_{1},U_{2}^{m},X{}_{2}^{n-m}),
\end{align}
where $U_{1}^{m},U_{2}^{m},E_{2}^{m}$ are mutually independent Gaussian
variables, $X_{1}^{n-m}$ and $X_{2}^{n-m}$ are two Gaussian variables
independent of all the other random variables, $\textrm{Var}(X_{1})=\frac{\left(1-\lambda\right)\left(1-\gamma\right)bP}{b-1},$
$\textrm{Var}(X_{2})=\frac{\lambda\left(1-\gamma\right)bP}{b-1}$,
$\textrm{Var}(E_{1})=\frac{N_{S}}{\left(\frac{\frac{b(1-\gamma)}{b-1}P+N_{1}}{\lambda\frac{b(1-\gamma)}{b-1}P+N_{1}}\right)^{b-1}}$
and $\textrm{Var}(E_{2})=\frac{\textrm{Var}(E_{1})}{\left(\frac{b\gamma P+N_{2}}{N_{2}}\right)\left(\frac{\lambda\frac{b(1-\gamma)}{b-1}P+N_{2}}{N_{2}}\right)^{b-1}-\frac{b\gamma P}{N_{2}}}$.
Define a set of functions
\begin{align}
& x^{n}\left(v_{2},s^{m}\right)=\left(\alpha\left(s^{m}-u_{1}^{m}\right),x{}_{1}^{n-m}+x{}_{2}^{n-m}\right)=\left(\alpha e_{1}^{m},x{}_{1}^{n-m}+x{}_{2}^{n-m}\right),\\
& \hat{s}_{1}^{m}\left(v_{1},y_{1}^{n}\right)=u_{1}^{m}+\frac{\alpha\textrm{Var}(E_{1})}{\alpha^{2}\textrm{Var}(E_{1})+N_{1}}y_{1}^{m},\\
& \hat{s}_{2}^{m}\left(v_{2},y_{2}^{n}\right)=u_{1}^{m}+\frac{N_{2}}{\alpha^{2}\textrm{Var}(E_{2})+N_{2}}u_{2}^{m}+\frac{\alpha\textrm{Var}(E_{2})}{\alpha^{2}\textrm{Var}(E_{2})+N_{2}}y_{2}^{m},
\end{align}
where $\alpha=\sqrt{\frac{\gamma bP}{\textrm{Var}(E_{1})}}$. Substitute
these variables and functions into the inner bound $\mathcal{R}_{DBC}^{(i)}$
in Theorem \ref{thm:AdmissibleRegion-DBC}, then the $b>1$ case in
Corollary \ref{thm:InnerBoundGG} is also recovered.
\end{IEEEproof}
On the other hand, setting $U_{[1:K-1]}$ to be jointly Gaussian with
$S$, the outer bound $\mathcal{R}_{DBC}^{(o)}$ in Theorem \ref{thm:AdmissibleRegion-DBC}
could recover the best known outer bound \cite[Thm. 2]{Tian11}.
\begin{thm}
\cite[Thm. 2]{Tian11}\label{thm:OuterBoundGG} For transmitting Gaussian
source $S$ with quadratic distortion measure over $K$-user Gaussian
broadcast channel with bandwidth mismatch factor $b$,
\begin{align}
\mathcal{R}\subseteq\mathcal{R}_{DBC}^{(o)}\triangleq & \Bigl\{ D_{[1:K]}:\textrm{For any variables }+\infty=\tau_{0}\geq\tau_{1}\geq\cdots\geq\tau_{K}=0,\nonumber \\
& \frac{1}{b}\left(\frac{1}{2}\log\frac{\left(N_{S}+\tau_{k}\right)\left(D_{k}+\tau_{k-1}\right)}{\left(D_{k}+\tau_{k}\right)\left(N_{S}+\tau_{k-1}\right)}:k\in[1:K]\right)\in\mathcal{C}_{GBC}\Bigr\},
\end{align}
where $\mathcal{C}_{GBC}$ denotes the capacity of Gaussian broadcast
channel given by
\begin{align}
\mathcal{C}_{GBC}= & \Bigl\{ R_{[1:K]}:R_{k}\geq0,1\le k\le K,N_{K+1}=0,\sum_{k=1}^{K}(N_{k}-N_{k+1})\exp\left(2\sum_{j=1}^{k}R_{j}\right)\leq P+N_{1}\Bigr\}.\label{eq:Gaussiancapacity}
\end{align}
\end{thm}
To compare $\mathcal{R}_{DBC}^{(o)}$ with the trivial distortion
bound, we consider a set of Gaussian test channels $p_{\hat{S}_{k}^{*}|S},1\le k\le K$
that achieves the optimal point-to-point distortion for each receiver.
Assume $p_{\hat{S}_{[1:K]}^{*}|S}$ is the backward Gaussian broadcast
channel consisting of subchannels $p_{\hat{S}_{k}^{*}|S},1\le k\le K$.
Since any backward Gaussian channel can be transformed into a forward
Gaussian channel with the same probability distribution, assume $p_{V_{[1:K]}|U}$
is the forward Gaussian broadcast channel of $p_{\hat{S}_{[1:K]}^{*}|S}$.
Then for Gaussian source broadcast, the outer bound is in form of
the comparison of capacity regions for two Gaussian broadcast channels
$p_{V_{[1:K]}|U}$ and $p_{Y_{[1:K]}|X}$. Note that $p_{Y_{[1:K]}|X}$
and $p_{V_{[1:K]}|U}$ have different bandwidth (the bandwidth ratio
is $b$) but the same point-to-point capacity for each receiver. It
can be proved that the Gaussian broadcast capacity region shrinks
as the bandwidth increases. Additionally, $\mathcal{C}_{DBC}\left(p_{V_{[1:K]}|U}\right)=\mathcal{C}_{DBC}\left(p_{Y_{[1:K]}|X}\right)$
when bandwidth matched. Hence $\mathcal{C}_{DBC}\left(p_{V_{[1:K]}|U}\right)\subseteq\mathcal{C}_{DBC}\left(p_{Y_{[1:K]}|X}\right)$
always holds for bandwidth compression case. This is the reason why
the outer bound in \cite{Tian11} is nontrivial only for bandwidth
expansion. The details can be found in \cite{Yu2015comments}.
The bounds in Corollary \ref{thm:InnerBoundGG} and Theorem \ref{thm:OuterBoundGG}
are illustrated in Fig. \ref{fig:HDAcoding-GG}.
\begin{figure}[t]
\centering\includegraphics[width=0.6\textwidth]{Gaussian_NEW} \protect\caption{\label{fig:HDAcoding-GG}Distortion bounds for sending Gaussian source
over Gaussian broadcast channel with $b=2,N_{S}=1,P=50,N_{1}=10,N_{2}=1$.
Outer Bounds 1 and 2 and Inner Bounds 1 and 2 respectively correspond
to the outer bound of Theorem \ref{thm:OuterBoundGG}, the outer bound
of Theorem \ref{thm:AdmissibleRegionGGSI}, the inner bound in Corollary
\ref{thm:InnerBoundGG}, and the inner bound achieved by Wyner-Ziv
separate coding (uncoded systematic code) \cite[Lem. 3]{Nayak}. Trivial
Outer Bound corresponds to the trivial outer bound \eqref{eq:trivialouterbound}.
Besides, Outer Bound 2 and Inner Bound 2 could be considered as the
outer bound and inner bound for Wyner-Ziv source broadcast problem
with $b=1,\beta_{1}=\frac{N_{S}N_{1}}{P+N_{1}},\beta_{2}=\frac{N_{S}N_{2}}{P+N_{2}}$,
and in this case Trivial Outer Bound corresponds to the Wyner-Ziv
outer bound \eqref{eq:wzouterbound}.}
\end{figure}
\subsection{Hamming Binary Source Broadcast}
Consider sending binary source $S\sim\textrm{Bern}\left(\frac{1}{2}\right)$
with Hamming distortion measure $d_{k}(s,\hat{s})=d(s,\hat{s})\triangleq0,\textrm{ if }s=\hat{s};1,\textrm{ otherwise}$,
over binary broadcast channel $Y_{k}=X\oplus W_{k},1\le k\le K$ with
$W_{k}\sim\textrm{Bern}\left(p_{k}\right),\frac{1}{2}\geq p_{1}\geq p_{2}\geq\cdots\geq p_{K}\geq0$.
Assume bandwidth mismatch factor is $b$.
We first consider the inner bound. For bandwidth expansion ($b>1$),
as a special case of hybrid coding systematic source-channel coding
(Uncoded Systematic Coding) has been first investigated in \cite{Shamai98}.
For any point-to-point lossless communication system, such systematic
coding does not loss the optimality; however, for some lossy cases
such as Hamming binary source communication, it is not optimal any
more. To retain the optimality, we can first quantize the source $S$,
and then transmit the quantized signal using Uncoded Systematic Coding.
The performance of this code could be obtained directly from Theorem
\ref{thm:AdmissibleRegion-DBC}.
Specifically, let $U_{2}=S\oplus E_{2}$, $U_{1}=U_{2}\oplus E_{1}$
with $E_{2}\sim\textrm{Bern}(D_{2}),E_{1}\sim\textrm{Bern}(d_{1})$.
Let $V_{2}=\left(U_{2},X^{b-1}\right),V_{1}=\left(U_{1},X_{1}^{b-1}\right),X_{1}^{b-1}=X^{b-1}\oplus B^{b-1}$,
where $X_{1}^{b-1}$ and $X^{b-1}$ are independent of $U_{2}$ and
$U_{1}$, and $X^{b-1}$ and $B^{b-1}$ follow $b-1$ dimensional
$\textrm{Bern}(\frac{1}{2})$ and $\textrm{Bern}(\theta)$, respectively.
Let $x^{b}(v_{2},s)=\left(u_{2},x^{b-1}\right)$, $\hat{s}_{2}\left(v_{2},y_{2}^{b}\right)=u_{2}$
and $\hat{s}_{1}\left(v_{1},y_{1}^{b}\right)=u_{1},\textrm{if }d_{1}<p_{1};y_{1},\textrm{otherwise.}$
Substitute these variables and functions into the inner bound $\mathcal{R}_{DBC}^{(i)}$
in Theorem \ref{thm:AdmissibleRegion-DBC}, then we get the following
corollary.
\begin{cor}[Coded Systematic Coding]
\label{cor:Coded}For transmitting binary source $S$ with Hamming
distortion measure over $2$-user binary broadcast channel with bandwidth
mismatch factor $b$,
\begin{align}
\mathcal{R}\supseteq\mathcal{R}_{CSC}^{(i)}\triangleq\textrm{convexhull} & \Bigl\{\left(D_{1},D_{2}\right):0\leq\theta,d_{1}\leq\frac{1}{2},\nonumber \\
& D_{1}\geq\min\left\{ d_{1}\star D_{2},p_{1}\star D_{2}\right\} ,\nonumber \\
& r_{1}=1-H_{2}(d_{1}\star p_{1})+\left(b-1\right)[1-H_{2}(\theta\star p_{1})],\nonumber \\
& r_{2}=H_{2}(d_{1}\star p_{2})-H_{2}(p_{2})+\left(b-1\right)[H_{2}(\theta\star p_{2})-H_{2}(p_{2})],\nonumber \\
& 1-H_{2}(d_{1}\star D_{2})\leq r_{1},\nonumber \\
& 1-H_{2}(D_{2})\leq r_{1}+r_{2}\Bigr\},
\end{align}
where $\star$ denotes the binary convolution, i.e.,
\begin{equation}
x\star y=(1-x)y+x(1-y),\label{eq:star}
\end{equation}
and $H_{2}$ denotes the binary entropy function, i.e.,
\begin{equation}
H_{2}(p)=-p\log p-(1-p)\log(1-p).\label{eq:binaryentropy}
\end{equation}
\end{cor}
\begin{rem}
Coded Systematic Coding without timesharing does not always lead to
a convex distortion region, hence a timesharing mechanism is needed
to improve performance and achieve $\mathcal{R}_{CSC}^{(i)}$. It
is equivalent to adding a timesharing variable $Q$ into $V_{2}$
and $V_{1}$, before substitute them into the inner bound $\mathcal{R}_{DBC}^{(i)}$.
Besides, note that unlike Uncoded Systematic Coding, the Coded Systematic
Coding could always achieve the optimal distortion for at least one
of the receivers. Moreover, unlike separate coding the Coded Systematic
Coding could weaken the cliff effect, and result in slope-cliff effect.
\end{rem}
In addition, the outer bound of Theorem \ref{thm:AdmissibleRegion-DBC}
reduces to the following outer bound for Hamming binary source broadcast
problem. This outer bound was first given in \cite[Eqn (41)]{Khezeli14}
for 2-user case. The proof is similar to that of \cite[Eqn (41)]{Khezeli14},
hence it is omitted here.
\begin{thm}
\label{thm:AdmissibleRegionBB} For transmitting binary source $S$
with Hamming distortion measure over $K$-user binary broadcast channel
with bandwidth mismatch factor $b$,
\begin{align}
\mathcal{R}\subseteq\mathcal{R}_{DBC}^{(o)}\triangleq & \Bigl\{ D_{[1:K]}:\textrm{For any variables }\frac{1}{2}=\tau_{0}\geq\tau_{1}\geq\tau_{2}\geq\cdots\geq\tau_{K}=0,\nonumber \\
& \frac{1}{b}\left(H_{2}\left(\tau_{k-1}\star D_{k}\right)-H_{2}\left(\tau_{k}\star D_{k}\right):k\in[1:K]\right)\in\mathcal{C}_{BBC}\Bigr\},
\end{align}
where $\mathcal{C}_{BBC}$ denotes the capacity of binary broadcast
channel given by
\begin{align}
\mathcal{C}_{BBC}= & \Bigl\{ R_{[1:K]}:\textrm{There exist some variables }\frac{1}{2}=\theta_{0}\geq\theta_{1}\geq\theta_{2}\geq\cdots\geq\theta_{K}=0\textrm{ \text{such that }}\nonumber \\
& 0\leq R_{k}\leq H_{2}\left(\theta_{k-1}\star p_{k}\right)-H_{2}\left(\theta_{k}\star p_{k}\right),1\le k\le K\Bigr\}.\label{eq:binarycapacity}
\end{align}
\end{thm}
The bounds in Corollary \ref{cor:Coded} and Theorem \ref{thm:AdmissibleRegionBB}
are illustrated in Fig. \ref{fig:HDAcoding-1}.
\begin{figure}[t]
\centering\includegraphics[width=0.6\textwidth]{Binary_NEW} \protect\caption{\label{fig:HDAcoding-1}Distortion bounds for sending binary source
over binary broadcast channel with $b=2,p_{1}=0.18,p_{2}=0.12$. Outer
Bounds 1 and 2 respectively correspond to the outer bound of Theorem
\ref{thm:AdmissibleRegionBB} and the outer bound of Theorem \ref{thm:AdmissibleRegionBBSI}.
Separate Coding, Uncoded Systematic Coding and Coded Systematic Coding
respectively correspond to the separate scheme combining successive-refinement
code \cite[Example 13.3]{El Gamal} with superposition code \cite[Example 5.3]{El Gamal},
the inner bound in Corollary \ref{thm:bandwidthmismatch}, and the
inner bound in Corollary \ref{cor:Coded}. Trivial Outer Bounds 1
and 2 correspond to the trivial outer bound \eqref{eq:trivialouterbound}
and the Wyner-Ziv outer bound \eqref{eq:wzouterbound}, respectively.
Besides, Trivial Outer Bound 2, Outer Bound 2 and Uncoded Systematic
Coding could be considered as the outer bounds and inner bound for
Wyner-Ziv source broadcast problem with $b=1,\beta_{1}=p_{1},\beta_{2}=p_{2}$.}
\end{figure}
\section{Wyner-Ziv Source Broadcast: Source Broadcast with Side Information}
We now extend the problem by allowing decoders to access side information
correlated with the source. As depicted in Fig. \ref{fig:WZbroadcast communication system},
receiver $k$ observes memoryless side information $Z_{k}^{n}$, and
it produces source reconstruction $\hat{S}_{k}^{n}$ from received
signal $Y_{k}^{n}$ and side information $Z_{k}^{n}$.
\begin{figure}[t]
\centering\includegraphics[width=0.6\textwidth]{WynerZivBroadcast}\caption{\label{fig:WZbroadcast communication system}Wyner-Ziv source broadcast
system: broadcast communication system with side information at decoders. }
\end{figure}
\begin{defn}
An $n$-length Wyner-Ziv source-channel code is defined by the encoding
function $x^{n}:{\mathcal{S}}^{n}\mapsto{\mathcal{X}}^{n}$ and a
sequence of decoding functions $\hat{s}_{k}^{n}:\mathcal{Y}_{k}^{n}\times\mathcal{Z}_{k}^{n}\mapsto\hat{\mathcal{S}}_{k}^{n},1\leq k\leq K$.
\end{defn}
\begin{defn}
If there exists a sequence of Wyner-Ziv source-channel codes satisfying
\begin{equation}
\mathop{\limsup}\limits _{n\to\infty}\mathbb{E}d_{k}\left(S^{n},\hat{S}_{k}^{n}\right)\le D_{k},
\end{equation}
then we say that the distortion tuple $D_{[1:K]}$ is achievable.
\end{defn}
\begin{defn}
The admissible distortion region for Wyner-Ziv broadcast problem is
defined as {\small{}{}{}{} }
\begin{align}
\mathcal{R}_{\textrm{SI}}\triangleq & \left\{ D_{[1:K]}:D_{[1:K]}\textrm{ is achievable}\right\} .
\end{align}
\end{defn}
In addition, Shamai \emph{et al.} \cite[Thm. 2.1]{Shamai98} showed
that for transmitting source over point-to-point channel $p_{Y_{k}|X}$
with side information $Z_{k}$ available at decoder, the minimum achievable
distortion satisfies $R_{S|Z_{k}}\left(D_{k}\right)=C_{k}$, where
$R_{S|Z_{k}}\left(\cdotp\right)$ is the Wyner-Ziv rate-distortion
function of the source $S$ given that the decoder observes $Z_{k}$
\cite{El Gamal}. Therefore, the optimal distortion is $D_{\textrm{SI},k}^{*}=R_{S|Z_{k}}^{-1}\left(C_{k}\right).$
Obviously,{\footnotesize{}{}{} }
\begin{equation}
\mathcal{R}_{\textrm{SI}}\subseteq\mathcal{R}_{\textrm{SI}}^{*}\triangleq\left\{ D_{[1:K]}:D_{k}\geq D_{\textrm{SI},k}^{*},1\le k\le K\right\} ,\label{eq:wzouterbound}
\end{equation}
where $\mathcal{R}_{\textrm{SI}}^{*}$ is named \emph{Wyner-Ziv outer
bound}.
Besides, we also consider the communication system with bandwidth
mismatch, whereby $m$ samples of a DMS are transmitted through $n$
uses of a DM-BC with $l$ samples of side information available at
each decoder. For simplicity, we let $m=l$, and for this case, bandwidth
mismatch factor is defined as $b=\frac{n}{m}$.
\subsection{Wyner-Ziv Source Broadcast}
If consider $Z_{[1:K]}$ to be transmitted from sender to the receivers
over a virtual broadcast channel $p_{Z_{[1:K]}|S}$, and define $X'=(S,X)$
and $Y'_{k}=(Z_{k},Y_{k}),1\le k\le K$, then the Wyner-Ziv source
broadcast problem is equivalent to the problem of sending $p_{S}$
over $p_{Y'_{[1:K]}|X'}$ with $S$ restricted to be the input of
subchannel $p_{Z_{[1:K]}|S}$. Hence Wyner-Ziv source broadcast problem
could be considered as the problem of (uncoded) systematic source-channel
coding. If set $x'\left(v_{[1:N]},s\right)=\left(s,x\left(v_{[1:N]},s\right)\right)$,
then from Theorem \ref{thm:AdmissibleRegion-GBC}, we obtain the following
inner bound for such systematic source-channel coding problem. It
is therefore also an inner bound for the Wyner-Ziv source broadcast
problem.
\begin{align}
\mathcal{R}_{\textrm{SI}}^{(i)}= & \Bigl\{ D_{[1:K]}:\textrm{There exist some pmf }p_{V_{[1:N]}|S},\textrm{ vector }r_{[1:N]},\nonumber \\
& \text{and functions }x\left(v_{[1:N]},s\right),\hat{s}_{k}\left(v_{\mathcal{D}_{k}},y_{k},z_{k}\right),1\le k\le K\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},1\le k\le K,\nonumber \\
& \sum_{j\in\mathcal{J}}r_{j}>\sum_{j\in\mathcal{J}}H\left(V_{j}|V_{\mathcal{A}_{j}}\right)-H\left(V_{\mathcal{J}}|S\right)\nonumber \\
& \textrm{for all }\mathcal{J}\subseteq[1:N]\textrm{ such that }\mathcal{J}\neq\emptyset\textrm{ and if }j\in\mathcal{J},\textrm{ then }\mathcal{A}_{j}\subseteq\mathcal{J},\nonumber \\
& \sum_{j\in\mathcal{J}^{c}}r_{j}<\sum_{j\in\mathcal{J}^{c}}H\left(V_{j}|V_{\mathcal{A}_{j}}\right)-H\left(V_{\mathcal{J}^{c}}|Y_{k}Z_{k}V_{\mathcal{J}}\right)\nonumber \\
& \textrm{for all }1\le k\le K\textrm{ and for all }\mathcal{J}\subseteq\mathcal{D}_{k}\textrm{ such that }\mathcal{J}^{c}\triangleq\mathcal{D}_{k}\backslash\mathcal{J}\neq\emptyset\textrm{ and if }j\in\mathcal{J},\textrm{ then }\mathcal{A}_{j}\subseteq\mathcal{J}\Bigr\}.\label{eq:-28-2}
\end{align}
In addition, regard $\left(U_{[1:L]},Z_{[1:K]}\right)$ as auxiliary
random variables following $p_{U_{[1:L]}|S}p_{Z_{[1:K]}|S}$, then
following similar steps to the proof of the outer bounds $\mathcal{R}_{1}^{(o)}$
and $\mathcal{R}_{1}^{\prime(o)}$ of Theorem \ref{thm:AdmissibleRegion-GBC},
we can achieve the following two outer bounds on $\mathcal{R}_{\textrm{SI}}$.
\begin{align}
\mathcal{R}_{\textrm{SI},1}^{(o)}= & \Bigl\{ D_{[1:K]}:\textrm{There exists some pmf }p_{\hat{S}_{[1:K]}|S,Z_{[1:K]}}\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},1\le k\le K,\nonumber \\
& \text{and for any pmf }p_{U_{[1:L]}|S},\text{ one can find }p_{\tilde{U}_{[1:L]},\tilde{Z}_{[1:K]},X^{n}}\text{satisfying}\nonumber \\
& I\left(\hat{S}_{\mathcal{A}};U_{\mathcal{B}}|U_{\mathcal{C}}Z_{\mathcal{A}}\right)\leq\frac{1}{n}I\left(Y_{\mathcal{A}}^{n};\tilde{U}_{\mathcal{B}}|\tilde{U}_{\mathcal{C}}\tilde{Z}_{\mathcal{A}}\right)\textrm{ for any }\mathcal{A}\subseteq\left[1:K\right],\mathcal{B},\mathcal{C}\subseteq\left[1:L\right]\Bigr\},
\end{align}
and
\begin{align}
\mathcal{R}_{\textrm{SI},1}^{\prime(o)}= & \Bigl\{ D_{[1:K]}:\textrm{There exists some pmf }p_{\hat{S}_{[1:K]}|S,Z_{[1:K]}}\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},1\le k\le K,\nonumber \\
& \text{and for any pmf }p_{U_{[1:L]}|S},\text{ one can find }p_{X,\tilde{U}_{[1:L]},\tilde{Z}_{[1:K]},W_{[1:K]},W_{[1:K]}^{\prime}}\text{satisfying}\nonumber \\
& \sum_{i=1}^{m}I\left(\hat{S}_{\mathcal{A}_{i}};U_{\mathcal{B}_{i}}Z_{\mathcal{A}_{i+1}}|U_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}Z_{\cup_{j=1}^{i}\mathcal{A}_{j}}\right)\leq\sum_{i=1}^{m}I\left(Y_{\mathcal{A}_{i}};\tilde{U}_{\mathcal{B}_{i}}\tilde{Z}_{\mathcal{A}_{i+1}}\tilde{W}_{\mathcal{A}_{i+1}}|\tilde{U}_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\tilde{Z}_{\cup_{j=1}^{i}\mathcal{A}_{j}}\tilde{W}_{\mathcal{A}_{i}}\tilde{W}_{\mathcal{A}_{i-1}}\right),\nonumber \\
& \textrm{ for any }m\geq1,\mathcal{A}_{i}\subseteq\left[1:K\right],\mathcal{B}_{i}\subseteq\left[1:L\right],0\le i\le m,\mathcal{A}_{0},\mathcal{A}_{m+1}\triangleq\emptyset,\nonumber \\
& \text{and }\tilde{W}_{\mathcal{A}_{i}}\triangleq W_{\mathcal{A}_{i}},\textrm{if }i\textrm{ is odd};W'_{\mathcal{A}_{i}},\textrm{otherwise, or }\tilde{W}_{\mathcal{A}_{i}}\triangleq W'_{\mathcal{A}_{i}},\textrm{if }i\textrm{ is odd};W{}_{\mathcal{A}_{i}},\textrm{otherwise}\Bigr\}.
\end{align}
Following similar steps to the proof of the outer bound $\mathcal{R}_{2}^{(o)}$
of Theorem \ref{thm:AdmissibleRegion-GBC}, we can also prove the
following outer bound on $\mathcal{R}_{\textrm{SI}}$.
\begin{align}
\mathcal{R}_{\textrm{SI},2}^{(o)}= & \Bigl\{ D_{[1:K]}:\textrm{There exists some pmf }p_{X}\textrm{ and some functions }\hat{s}_{k}^{n}\left(\tilde{y}_{k},z_{k}^{n}\right),1\le k\le K\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},1\le k\le K,\nonumber \\
& \text{and for any pmf }p_{U_{[1:L]}|Y_{[1:K]}},\text{ one can find }p_{\tilde{Y}_{[1:K]}|S}p_{\tilde{U}_{[1:L]}|\tilde{Y}_{[1:K]}}\text{satisfying}\nonumber \\
& I\left(S^{n};\tilde{Y}_{\mathcal{B}}\tilde{U}_{\mathcal{B}'}|\tilde{Y}_{\mathcal{C}}\tilde{U}_{\mathcal{C}'}\right)\leq I\left(X;Y_{\mathcal{B}}U_{\mathcal{B}'}|Y_{\mathcal{C}}U_{\mathcal{C}'}\right)\textrm{ for any }\mathcal{B},\mathcal{C}\subseteq\left[1:K\right],\mathcal{B}',\mathcal{C}'\subseteq\left[1:L\right]\Bigr\}.
\end{align}
Therefore, the following theorem holds. The proof is omitted.
\begin{thm}
\label{thm:AdmissibleRegionSI-GBC} For transmitting source $S$ over
broadcast channel $p_{Y_{[1:K]}|X}$ with side information $Z_{k}$
at decoder $k$,
\begin{equation}
\mathcal{R}_{\textrm{SI}}^{(i)}\subseteq\mathcal{R}_{\textrm{SI}}\subseteq\mathcal{R}_{\textrm{SI},1}^{(o)}\cap\mathcal{R}_{\textrm{SI},2}^{(o)}\subseteq\mathcal{R}_{\textrm{SI},1}^{\prime(o)}.
\end{equation}
\end{thm}
\begin{rem}
Similar to Theorem \ref{thm:AdmissibleRegion-GBC}, Theorem \ref{thm:AdmissibleRegionSI-GBC}
could also be extended to Gaussian or any other well-behaved continuous-alphabet
source-channel pair, the problem of broadcasting Wyner-Ziv correlated
sources, or Wyner-Ziv source broadcast with channel input cost.
\end{rem}
\subsection{Wyner-Ziv Source Broadcast over Degraded Channel with Degraded Side
Information}
Theorem \ref{thm:AdmissibleRegionSI-GBC} can be used to derive the
inner bound and outer bound for the case of degraded channel and degraded
side information. Define
\begin{align}
\mathcal{R}_{\textrm{SI-D}}^{(i)}= & \Bigl\{ D_{[1:K]}:\textrm{There exists some pmf }p_{V_{K}|S}p_{V_{K-1}|V_{K}}\cdots p_{V_{1}|V_{2}},\nonumber \\
& \text{and functions }x\left(v_{K},s\right),\hat{s}_{k}\left(v_{k},y_{k},z_{k}\right),1\le k\le K\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},\nonumber \\
& I\left(S;V_{k}\right)\leq\sum_{j=1}^{k}I\left(Y_{j}Z_{j};V_{j}|V_{j-1}\right),1\le k\le K,\text{ where }V_{0}\triangleq\emptyset\Bigr\},
\end{align}
and where
\begin{align}
\mathcal{R}_{\textrm{SI-D}}^{(o)}= & \Bigl\{ D_{[1:K]}:\textrm{There exists some pmfs }p_{V_{K}|S}p_{V_{K-1}|V_{K}}\cdots p_{V_{1}|V_{2}},p_{X}\nonumber \\
& \text{and functions }\hat{s}_{k}\left(v_{k},z_{k}\right),1\le k\le K\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},\nonumber \\
& \left(I\left(V_{k};U_{k}|U_{k-1}Z_{k}\right):k\in[1:K]\right)\in\mathcal{B}_{DBC}\left(p_{X}p_{Y_{[1:K]}|X}\right)\nonumber \\
& \text{for any pmf }p_{U_{K-1}|S}p_{U_{K-2}|U_{K-1}}\cdots p_{U_{1}|U_{2}},U_{0}\triangleq\emptyset,U_{K}\triangleq S,\nonumber \\
& \textrm{and }\left(I\left(V_{k};S|Z_{k}\right):k\in[1:K]\right)\in\mathcal{B}_{SRC}\left(p_{X}p_{Y_{[1:K]}|X}\right)\Bigr\},
\end{align}
where $\mathcal{B}_{DBC}\left(p_{X}p_{Y_{[1:K]}|X}\right)$ and $\mathcal{B}_{SRC}\left(p_{X}p_{Y_{[1:K]}|X}\right)$
are given in \eqref{eq:DBCcapacity} and \eqref{eq:DBCcapacity2},
respectively. Then we have the following theorem. The proof is analogous
to that of Theorem \ref{thm:AdmissibleRegion-DBC}, and is therefore
omitted.
\begin{thm}
\label{thm:AdmissibleRegionSI-DBC} For transmitting source $S$ over
degraded broadcast channel $p_{Y_{[1:K]}|X}$ ($X\rightarrow Y_{K}\rightarrow Y_{K-1}\rightarrow\cdots\rightarrow Y_{1}$)
with degraded side information $Z_{k}$ ($S\rightarrow Z_{K}\rightarrow Z_{K-1}\rightarrow\cdots\rightarrow Z_{1}$)
at decoder $k$,
\begin{equation}
\mathcal{R}_{\textrm{SI-D}}^{(i)}\subseteq\mathcal{R}_{\textrm{SI}}\subseteq\mathcal{R}_{\textrm{SI-D}}^{(o)}.
\end{equation}
\end{thm}
\begin{rem}
$\mathcal{R}_{\textrm{SI-D}}^{(o)}$ can be also expressed as
\begin{align}
\mathcal{R}_{\textrm{SI-D}}^{(o)}= & \Bigl\{ D_{[1:K]}:\textrm{There exists some pmf }p_{V_{K}|S}p_{V_{K-1}|V_{K}}\cdots p_{V_{1}|V_{2}}\nonumber \\
& \text{and functions }\hat{s}_{k}\left(v_{k},z_{k}\right),1\le k\le K\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},\nonumber \\
& \mathcal{B}_{DBC-SI}\left(p_{S}p_{Z_{[1:K]}|S}p_{V_{[1:K]}|S,Z_{[1:K]}}\right)\subseteq\mathcal{B}_{DBC}\left(p_{X}p_{Y_{[1:K]}|X}\right),\nonumber \\
& \mathcal{B}_{SRC-SI}\left(p_{S}p_{Z_{[1:K]}|S}p_{V_{[1:K]}|S,Z_{[1:K]}}\right)\supseteq\mathcal{B}_{SRC}\left(p_{X}p_{Y_{[1:K]}|X}\right)\Bigr\},
\end{align}
where
\begin{align}
\mathcal{B}_{DBC-SI}\left(p_{X}p_{Z_{[1:K]}|X}p_{Y_{[1:K]}|X,Z_{[1:K]}}\right)= & \Bigl\{ R_{[1:K]}:\textrm{There exists some pmf }p_{V_{K-1}|X}p_{V_{K-2}|V_{K-1}}\cdots p_{V_{1}|V_{2}}\textrm{\text{such that }}\nonumber \\
& R_{k}\geq0,\sum_{j=1}^{k}R_{j}\leq\sum_{j=1}^{k}I\left(Y_{j};V_{j}|V_{j-1}Z_{j}\right),1\le k\le K,\text{ where }V_{0}\triangleq\emptyset,V_{K}\triangleq X\Bigr\},\label{eq:DBCcapacity-3-2}
\end{align}
and
\begin{align}
\mathcal{B}_{SRC-SI}\left(p_{X}p_{Z_{[1:K]}|X}p_{Y_{[1:K]}|X,Z_{[1:K]}}\right)= & \Bigl\{ R_{[1:K]}:R_{k}\geq0,\sum_{j=1}^{k}R_{j}\geq I\left(X;Y_{k}|Z_{k}\right),1\le k\le K\Bigr\}.
\end{align}
\end{rem}
\subsection{Wyner-Ziv Gaussian Source Broadcast}
Consider sending Gaussian source $S\sim\mathcal{N}\left(0,N_{S}\right)$
with quadratic distortion measure $d_{k}(s,\hat{s})=d(s,\hat{s})\triangleq(s-\hat{s})^{2}$
over power-constrained Gaussian broadcast channel $Y_{k}=X+W_{k},1\le k\le K$
with $\mathbb{E}\left(X^{2}\right)\leq P$ and$W_{k}\sim\mathcal{N}\left(0,N_{k}\right),N_{1}\geq N_{2}\geq\cdots\geq N_{K}$.
Assume the side information $Z_{k}$ observed by receiver $k$ satisfies
$S=Z_{k}+B_{k}$ with independent Gaussian variables $Z_{k}\sim\mathcal{N}\left(0,N_{S}-\beta_{k}\right)$
and $B_{k}\sim\mathcal{N}\left(0,\beta_{k}\right)$. Assume bandwidth
mismatch factor is $b$. Then the inner bound of Theorem \ref{thm:AdmissibleRegionSI-DBC}
could recover the existing results in the literature \cite{Nayak,Gao},
and the outer bound of Theorem \ref{thm:AdmissibleRegionSI-DBC} could
be used to prove the following outer bound for Wyner-Ziv Gaussian
source broadcast problem. The proof is given in Appendix \ref{sec:broadcast-GaussianSI}.
\begin{thm}
\label{thm:AdmissibleRegionGGSI}For transmitting Gaussian source
$S$ over Gaussian broadcast channel with degraded side information
$Z_{k}$ ($\beta_{1}\geq\beta_{2}\geq\cdots\geq\beta_{K}$) at decoder
$k$, {\footnotesize{}{}{}{} }
\begin{align}
\mathcal{R}_{\textrm{SI}}\subseteq\mathcal{R}_{\textrm{SI-D}}^{(o)}\triangleq & \Bigl\{ D_{[1:K]}:\textrm{For any variables }+\infty=\tau_{0}\geq\tau_{1}\geq\cdots\geq\tau_{K}=0,\nonumber \\
& \frac{1}{b}\left(\frac{1}{2}\log\frac{\left(\beta_{k}+\tau_{k}\right)\left(D_{k}+\tau_{k-1}\right)}{\left(D_{k}+\tau_{k}\right)\left(\beta_{k}+\tau_{k-1}\right)}:k\in[1:K]\right)\in\mathcal{C}_{GBC}\Bigr\},
\end{align}
where $\mathcal{C}_{GBC}$ denotes the capacity region of the Gaussian
broadcast channel given in \eqref{eq:Gaussiancapacity}.
\end{thm}
The bound of Theorem \ref{thm:AdmissibleRegionGGSI} is shown in Fig.
\ref{fig:HDAcoding-GG}.
\subsection{\label{sub:Wyner-Ziv-Binary-Source}Wyner-Ziv Binary Source Broadcast}
Consider sending binary source $S\sim\textrm{Bern}\left(\frac{1}{2}\right)$
with Hamming distortion measure $d_{k}(s,\hat{s})=d(s,\hat{s})\triangleq0,\textrm{ if }s=\hat{s};1,\textrm{ otherwise},$
over binary broadcast channel $Y_{k}=X\oplus W_{k},1\le k\le K$ with
$W_{k}\sim\textrm{Bern}\left(p_{k}\right),\frac{1}{2}\geq p_{1}\geq p_{2}\geq\cdots\geq p_{K}\geq0$.
Assume the side information $Z_{k}$ observed by receiver $k$ satisfies
$S=Z_{k}\oplus B_{k}$ with independent variables $Z_{k}\sim\textrm{Bern}\left(\frac{1}{2}\right)$
and $B_{k}\sim\textrm{Bern}\left(\beta_{k}\right)$. Assume bandwidth
mismatch factor is $b$.
Let $V_{1}=\left(U_{1},X_{1}^{b}\right),V_{2}=\left(U_{2},X^{b}\right),V_{3}=\emptyset$,
and $U_{1}$ and $U_{2}$ are independent of $X_{1}^{b}$ and $X^{b}$.
$S$, $U_{2}$ and $U_{1}$ satisfy the distribution $p_{U_{2}|S}p_{U_{1}|U_{2}}$,
where
\begin{align}
& \begin{array}{c}
\qquad\qquad\qquad0\qquad1\qquad2\\
p_{U_{2}|S}=\begin{array}{c}
0\\
1
\end{array}\left(\begin{array}{ccc}
q_{2}\bar{\alpha}_{2} & q_{2}\alpha_{2} & \bar{q}_{2}\\
q_{2}\alpha_{2} & q_{2}\bar{\alpha}_{2} & \bar{q}_{2}
\end{array}\right)
\end{array},\\
& \begin{array}{c}
\qquad\qquad\qquad0\qquad1\qquad2\\
p_{U_{1}|U_{2}}=\begin{array}{c}
0\\
1\\
2
\end{array}\left(\begin{array}{ccc}
q'_{1}\bar{\alpha}'_{1} & q'_{1}\alpha'_{1} & \bar{q}'_{1}\\
q'_{1}\alpha'_{1} & q'_{1}\bar{\alpha}'_{1} & \bar{q}'_{1}\\
0 & 0 & 1
\end{array}\right)
\end{array}.
\end{align}
with $0\leq q_{2},q'_{1}\leq1,0\leq\alpha_{2},\alpha'_{1}\leq\frac{1}{2}.$
$X^{b}$ and $X_{1}^{b}$ satisfy $X_{1}^{b}=X^{b}\oplus B^{b},$
$X^{b}\sim b\textrm{ dimensional Bern}(\frac{1}{2}),$ and $B^{b}\sim b\textrm{ dimensional Bern}(\theta)$
with $0\leq\theta\leq\frac{1}{2}.$ Denote $\alpha_{1}=\alpha_{2}\star\alpha'_{1},q{}_{1}=q{}_{2}q'_{1}$,
and set $x^{b}(v_{2},s)=x^{b}$ and for $i=1,2$,
\begin{equation}
\hat{s}_{i}\left(v_{i},y_{i}^{b},z_{i}\right)=\begin{cases}
z_{i}, & \textrm{if }\alpha_{i}\geq\beta_{i}\textrm{ or }\alpha_{i}<\beta_{i},u_{i}=2;\\
u_{i}, & \textrm{if }\alpha_{i}<\beta_{i},u_{i}=0,1.
\end{cases}
\end{equation}
Substitute these random variables and functions into $\mathcal{R}_{\textrm{SI}}^{(i)}$
in Theorem \ref{thm:AdmissibleRegionSI-GBC}, then we get the following
performance (the hybrid coding reduces to a layered digital coding),
which is tighter than that of the Layered Description Scheme (LDS)
\cite[Lem. 4]{Nayak}.
\begin{cor}[Layered Digital Coding]
\label{thm:bandwidthmismatch} For transmitting binary source $S$
with Hamming distortion measure over $2$-user binary broadcast channel
with side information $Z_{k}$ at decoder $k$,
\begin{align}
\mathcal{R}_{\textrm{SI}}\supseteq\mathcal{R}_{LDC}^{(i)}\triangleq & \Bigl\{\left(D_{1},D_{2}\right):0\leq q_{1}\leq q_{2}\leq1,0\leq\alpha_{2}\leq\alpha_{1}\leq\frac{1}{2},0\leq\theta\leq\frac{1}{2},\nonumber \\
& q_{1}r(\alpha_{1},\beta_{1})\leq b\left(1-H_{2}(\theta\star p_{1})\right),\nonumber \\
& q_{1}r(\alpha_{1},\beta_{2})\leq b\left(1-H_{2}(\theta\star p_{2})\right),\nonumber \\
& q_{2}r(\alpha_{2},\beta_{2})\leq b\left(1-H_{2}(p_{2})\right),\nonumber \\
& q_{1}r(\alpha_{1},\beta_{1})+\left(q_{2}r(\alpha_{2},\beta_{2})-q_{1}r(\alpha_{1},\beta_{2})\right)\leq b\left(1-H_{2}(\theta\star p_{1})\right)+b\left(H_{2}(\theta\star p_{2})-H_{2}(p_{2})\right),\nonumber \\
& D_{i}\leq q_{i}\min\left\{ \alpha_{i},\beta_{i}\right\} +(1-q_{i})\beta_{i},i=1,2\Bigr\},
\end{align}
where
\begin{equation}
r(\alpha,\beta)=H_{2}(\alpha\star\beta)-H_{2}(\alpha),
\end{equation}
$\star$ denotes the binary convolution given in \eqref{eq:star},
and $H_{2}$ denotes the binary entropy function given in \eqref{eq:binaryentropy}.
\end{cor}
In addition, the outer bound of Theorem \ref{thm:AdmissibleRegionSI-DBC}
reduces to the following one for Wyner-Ziv binary case. The proof
is given in Appendix \ref{sec:broadcast-BernoulliSI}.
\begin{thm}
\label{thm:AdmissibleRegionBBSI} For transmitting binary source $S$
with Hamming distortion measure over $K$-user binary broadcast channel
with degraded side information $Z_{k}$ ($\frac{1}{2}\geq\beta_{1}\geq\beta_{2}\geq\cdots\geq\beta_{K}\geq0$)
at decoder $k$,
\begin{align}
\mathcal{R}_{\textrm{SI}}\subseteq\mathcal{R}_{\textrm{SI-D}}^{(o)}\triangleq & \Bigl\{ D_{[1:K]}:\textrm{There exists some variables }0\leq\alpha_{1},\alpha_{2},\cdots,\alpha_{K}\leq\frac{1}{2}\nonumber \\
& \text{such that }\alpha_{k}\leq D'_{k}\triangleq\min\left\{ D_{k},\beta_{k}\right\} ,1\le k\le K,\nonumber \\
& \textrm{and for any variables }\frac{1}{2}=\tau_{0}\geq\tau_{1}\geq\tau_{2}\geq\cdots\geq\tau_{K}=0,\nonumber \\
& \frac{1}{b}\Biggl(\eta_{k}\left(H_{2}\left(\beta_{k}\star\tau_{k}\right)-H_{2}\left(\beta_{k}\star\tau_{k-1}\right)-\left(H_{4}\left(\alpha_{k},\beta_{k},\tau_{k}\right)-H_{4}\left(\alpha_{k},\beta_{k},\tau_{k-1}\right)\right)\right):k\in[1:K]\Biggr)\in\mathcal{C}_{BBC}\Bigr\},
\end{align}
where $\mathcal{C}_{BBC}$ denotes the capacity region of the binary
broadcast channel given in \eqref{eq:binarycapacity}, \textup{
\begin{equation}
\eta_{k}\triangleq\begin{cases}
\frac{\beta_{k}-D'_{k}}{\beta_{k}-\alpha_{k}}, & \textrm{if }\alpha_{k}<\beta_{k},\\
0, & \textrm{otherwise,}
\end{cases}
\end{equation}
\begin{align}
H_{4}\left(x,y,z\right)\triangleq & -\left(xyz+\overline{x}\overline{y}\overline{z}\right)\log\left(xyz+\overline{x}\overline{y}\overline{z}\right)-\left(x\overline{y}z+\overline{x}y\overline{z}\right)\log\left(x\overline{y}z+\overline{x}y\overline{z}\right)\nonumber \\
& -\left(xy\overline{z}+\overline{x}\overline{y}z\right)\log\left(xy\overline{z}+\overline{x}\overline{y}z\right)-\left(x\overline{y}\overline{z}+\overline{x}yz\right)\log\left(x\overline{y}\overline{z}+\overline{x}yz\right),\label{eq:H4}
\end{align}
}and $\overline{x}\triangleq1-x.$.
\end{thm}
The bounds in Corollary \ref{thm:bandwidthmismatch} and Theorem \ref{thm:AdmissibleRegionBBSI}
are shown in Fig. \ref{fig:HDAcoding-1}.
\section{Concluding Remarks}
In this paper, we focused on the joint source-channel coding problem
of sending a memoryless source over memoryless broadcast channel,
and developed an inner bound and several outer bounds for this problem.
The inner bound is achieved by a unified hybrid coding scheme, and
it can recover the best known performance of existing hybrid coding.
Similarly, our outer bounds can also recover the best known outer
bound in the literature. Besides, we also extend the results to Wyner-Ziv
source broadcast problem. All these bounds are also used to generate
some new results, including the bounds on capacity region of general
broadcast channel with common messages which respectively generalize
Marton's inner bound and Nair-El Gamal outer bound.
The inner bound achieved by proposed hybrid coding is established
by using generalized Multivariate Covering Lemma and generalized Multivariate
Packing Lemma, and the outer bounds are derived by introducing auxiliary
random variables (at sender side or receiver sides) and exploiting
Csiszár sum identity as in \cite{Khezeli}. These lemmas and tools
are expected to be exploited to derive more and stronger achievability
and converse results for network information theory.
\appendices{}
\section{Proof of Lemma \ref{lem:Covering}}
We follow similar steps to the proof of mutual covering lemma \cite{Minero}.
Let
\begin{equation}
\mathcal{B}=\left\{ m_{[1:k]}\in\prod_{i=1}^{k}[1:2^{nr_{i}}]:(U^{n},V_{0}^{n},V_{[1:k]}^{n}(m_{[1:k]}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}\right\} .
\end{equation}
Then we only need to show $\lim_{n\rightarrow\infty}\mathbb{P}\left(|\mathcal{B}|=0\right)=0$.
On the other hand,
\begin{align}
\lim_{n\rightarrow\infty}\mathbb{P}\left(|\mathcal{B}|=0\right) & =\lim_{n\rightarrow\infty}\sum_{u^{n},v_{0}^{n}}p_{U^{n},V_{0}^{n}}\left(u^{n},v_{0}^{n}\right)\mathbb{P}\left(|\mathcal{B}|=0|u^{n},v_{0}^{n}\right)\\
& \leq\lim_{n\rightarrow\infty}\sum_{(u^{n},v_{0}^{n})\in\mathcal{T}_{\epsilon'}^{\left(n\right)}}p_{U^{n},V_{0}^{n}}\left(u^{n},v_{0}^{n}\right)\mathbb{P}\left(|\mathcal{B}|=0|u^{n},v_{0}^{n}\right)+\lim_{n\rightarrow\infty}\mathbb{P}\left((u^{n},v_{0}^{n})\notin\mathcal{T}_{\epsilon'}^{\left(n\right)}\right)\\
& =\lim_{n\rightarrow\infty}\sum_{(u^{n},v_{0}^{n})\in\mathcal{T}_{\epsilon'}^{\left(n\right)}}p_{U^{n},V_{0}^{n}}\left(u^{n},v_{0}^{n}\right)\mathbb{P}\left(|\mathcal{B}|=0|u^{n},v_{0}^{n}\right)\label{eq:}
\end{align}
To prove $\lim_{n\rightarrow\infty}\mathbb{P}\left(|\mathcal{B}|=0\right)=0$,
it is sufficient to show $\lim_{n\rightarrow\infty}\mathbb{P}\left(|\mathcal{B}|=0|u^{n},v_{0}^{n}\right)=0$
for any $(u^{n},v_{0}^{n})\in\mathcal{T}_{\epsilon'}^{\left(n\right)}$.
Utilizing Chebyshev lemma \cite[App. B]{El Gamal}, we can bound the
probability as
\begin{equation}
\mathbb{P}\left(|\mathcal{B}|=0|u^{n},v_{0}^{n}\right)\leq\mathbb{P}\left((|\mathcal{B}|-\mathbb{E}|\mathcal{B}|)^{2}\geq(E|\mathcal{B}|)^{2}|u^{n},v_{0}^{n}\right)\leq\frac{\textrm{Var}(|\mathcal{B}||u^{n},v_{0}^{n})}{(\mathbb{E}(|\mathcal{B}||u^{n},v_{0}^{n}))^{2}}.\label{eq:-8}
\end{equation}
Next we prove the upper bound $\frac{\textrm{Var}(|\mathcal{B}||u^{n},v_{0}^{n})}{(\mathbb{E}(|\mathcal{B}||u^{n},v_{0}^{n}))^{2}}$
tends to zero as $n\rightarrow\infty$. Define
\begin{equation}
E\left(m_{[1:k]}\right)\triangleq\begin{cases}
1, & \textrm{if }(u^{n},v_{0}^{n},V_{[1:k]}^{n}(m_{[1:k]}))\in\mathcal{T}_{\epsilon}^{\left(n\right)};\\
0, & \textrm{otherwise},
\end{cases}
\end{equation}
for each $m_{[1:k]}\in\prod_{i=1}^{k}[1:2^{nr_{i}}]$, then $|\mathcal{B}|$
can be expressed as
\begin{equation}
|\mathcal{B}|=\sum_{m_{[1:k]}\in\prod_{i=1}^{k}[1:2^{nr_{i}}]}E\left(m_{[1:k]}\right).
\end{equation}
Denote
\begin{align}
p_{0} & =\mathbb{P}\left((u^{n},v_{0}^{n},V_{[1:k]}^{n}(m_{[1:k]}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}|u^{n},v_{0}^{n}\right),\\
p_{\mathcal{I}} & =\mathbb{P}\left((u^{n},v_{0}^{n},V_{[1:k]}^{n}(m_{[1:k]}))\in\mathcal{T}_{\epsilon}^{\left(n\right)},(u^{n},v_{0}^{n},V_{[1:k]}^{n}(m{}_{\mathcal{I}},m'_{\mathcal{I}^{c}}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}|u^{n},v_{0}^{n}\right),
\end{align}
for $m_{[1:k]}=\boldsymbol{1}$, and $m_{[1:k]}^{\prime}=\boldsymbol{2}$.
Obviously, $p_{[1:k]}=p_{0}$. Then
\begin{equation}
\mathbb{E}(|\mathcal{B}||u^{n},v_{0}^{n})=\sum_{m_{[1:k]}}\mathbb{P}\left((u^{n},v_{0}^{n},V_{[1:k]}^{n}(m_{[1:k]}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}|u^{n},v_{0}^{n}\right)=2^{n\sum_{j=1}^{k}r_{j}}p_{0},
\end{equation}
and
\begin{align}
& \mathbb{E}(|\mathcal{B}|^{2}|u^{n},v_{0}^{n})\nonumber \\
& =\sum_{\mathcal{I}\subseteq[1:k]}\sum_{m_{[1:k]}}\sum_{m'_{\mathcal{I}^{c}}:m'_{\mathcal{I}^{c}}\nLeftrightarrow m{}_{\mathcal{I}^{c}}}\mathbb{P}\left((u^{n},v_{0}^{n},V_{[1:k]}^{n}(m_{[1:k]}))\in\mathcal{T}_{\epsilon}^{\left(n\right)},(u^{n},v_{0}^{n},V_{[1:k]}^{n}(m{}_{\mathcal{I}},m'_{\mathcal{I}^{c}}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}|u^{n},v_{0}^{n}\right)\\
& =2^{n\sum_{j=1}^{k}r_{j}}p_{0}+\sum_{\mathcal{I}\subsetneqq[1:k]}\sum_{m_{[1:k]}}\sum_{m'_{\mathcal{I}^{c}}:m'_{\mathcal{I}^{c}}\nLeftrightarrow m{}_{\mathcal{I}^{c}}}\mathbb{P}\left((u^{n},v_{0}^{n},V_{[1:k]}^{n}(m_{[1:k]}))\in\mathcal{T}_{\epsilon}^{\left(n\right)},(u^{n},v_{0}^{n},V_{[1:k]}^{n}(m{}_{\mathcal{I}},m'_{\mathcal{I}^{c}}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}|u^{n},v_{0}^{n}\right).
\end{align}
Define
\begin{equation}
\mathbb{J}\triangleq\left\{ \mathcal{J}\subsetneqq[1:k]:\textrm{ if }j\in\mathcal{J},\textrm{ then }\mathcal{A}_{j}\subseteq\mathcal{J}\right\} .
\end{equation}
Then any set $\mathcal{I}\subsetneqq[1:k]$ can transform into a $\mathcal{J}\left(\mathcal{I}\right)\in\mathbb{J}$
by removing all the elements $j$'s such that $\mathcal{A}_{j}\nsubseteq\mathcal{I}$.
According to generation of random codebook, we can observe that $p_{\mathcal{I}}=p_{\mathcal{J}\left(\mathcal{I}\right)}$.
Therefore,
\begin{align}
& \sum_{\mathcal{I}\subsetneqq[1:k]}\sum_{m_{[1:k]}}\sum_{m'_{\mathcal{I}^{c}}:m'_{\mathcal{I}^{c}}\nLeftrightarrow m{}_{\mathcal{I}^{c}}}\mathbb{P}\left((u^{n},v_{0}^{n},V_{[1:k]}^{n}(m_{[1:k]}))\in\mathcal{T}_{\epsilon}^{\left(n\right)},(u^{n},v_{0}^{n},V_{[1:k]}^{n}(m{}_{\mathcal{I}},m'_{\mathcal{I}^{c}}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}|u^{n},v_{0}^{n}\right)\nonumber \\
& =\sum_{\mathcal{I}\subsetneqq[1:k]}\sum_{m_{[1:k]}}\sum_{m'_{\mathcal{I}^{c}}:m'_{\mathcal{I}^{c}}\nLeftrightarrow m{}_{\mathcal{I}^{c}}}p_{\mathcal{J}\left(\mathcal{I}\right)}\\
& \leq\sum_{\mathcal{I}\subsetneqq[1:k]}2^{n\left(\sum_{j=1}^{k}r_{j}+\sum_{j\in\mathcal{I}^{c}}r_{j}\right)}p_{\mathcal{J}\left(\mathcal{I}\right)}\\
& \leq\sum_{\mathcal{I}\subsetneqq[1:k]}2^{n\left(\sum_{j=1}^{k}r_{j}+\sum_{j\in\left(\mathcal{J}\left(\mathcal{I}\right)\right)^{c}}r_{j}\right)}p_{\mathcal{J}\left(\mathcal{I}\right)}\label{eq:-56}\\
& \leq\sum_{\mathcal{J}\in\mathbb{J}}2^{k-|\mathcal{J}|}2^{n\left(\sum_{j=1}^{k}r_{j}+\sum_{j\in\mathcal{J}^{c}}r_{j}\right)}p_{\mathcal{J}}\label{eq:-58}\\
& \leq\sum_{\mathcal{J}\in\mathbb{J}}2^{n\left(\sum_{j=1}^{k}r_{j}+\sum_{j\in\mathcal{J}^{c}}r_{j}+o(1)\right)}p_{\mathcal{J}},
\end{align}
where \eqref{eq:-56} follows from $\mathcal{J}\left(\mathcal{I}\right)\subseteq\mathcal{I}$,
\eqref{eq:-58} follows from that for each $\mathcal{J}\subseteq\mathbb{J}$,
there are at most $2^{k-|\mathcal{J}|}$ of $\mathcal{I}$'s that
could transform into $\mathcal{J}$, and $o(1)$ denotes a term that
vanishes as $n\rightarrow\infty$. Hence
\begin{align}
\textrm{Var}(|\mathcal{B}||u^{n},v_{0}^{n}) & \leq\mathbb{E}(|\mathcal{B}|^{2}|u^{n},v_{0}^{n})\\
& \leq2^{n\sum_{j=1}^{k}r_{j}}p_{0}+\sum_{\mathcal{J}\in\mathbb{J}}2^{n\left(\sum_{j=1}^{k}r_{j}+\sum_{j\in\mathcal{J}^{c}}r_{j}+o(1)\right)}p_{\mathcal{J}}.
\end{align}
Furthermore we have
\begin{align}
\frac{\textrm{Var}(|\mathcal{B}||u^{n},v_{0}^{n})}{(\mathbb{E}(|\mathcal{B}||u^{n},v_{0}^{n}))^{2}} & \leq\frac{2^{n\sum_{j=1}^{k}r_{j}}p_{0}+\sum_{\mathcal{J}\in\mathbb{J}}2^{n\left(\sum_{j=1}^{k}r_{j}+\sum_{j\in\mathcal{J}^{c}}r_{j}+o(1)\right)}p_{\mathcal{J}}}{\left(2^{n\sum_{j=1}^{k}r_{j}}p_{0}\right)^{2}}\\
& =2^{-n\sum_{j=1}^{k}r_{j}}\frac{1}{p_{0}}+\sum_{\mathcal{J}\in\mathbb{J}}2^{n\left(-\sum_{j\in\mathcal{J}}r_{j}+o(1)\right)}\frac{p_{\mathcal{J}}}{p_{0}^{2}}.\label{eq:-9}
\end{align}
According to generation process of random codebook, we can observe
that
\begin{align}
p_{0} & =\sum_{v_{[1:k]}^{n}:(u^{n},v_{0}^{n},v_{[1:k]}^{n})\in\mathcal{T}_{\epsilon}^{\left(n\right)}}\mathbb{P}\left(V_{[1:k]}^{n}(m_{[1:k]})=v_{[1:k]}^{n}|u^{n},v_{0}^{n}\right)\\
& \geq2^{-n\left(\sum_{j=1}^{k}H\left(V_{j}|V_{\mathcal{A}_{j}}V_{0}\right)-H\left(V_{[1:k]}|UV_{0}\right)+2\delta\left(\epsilon\right)\right)},\label{eq:-60}
\end{align}
where \eqref{eq:-60} follows from that for any $(u^{n},v_{0}^{n},v_{[1:k]}^{n})\in\mathcal{T}_{\epsilon}^{\left(n\right)}$,
\begin{equation}
\mathbb{P}\left(V_{[1:k]}^{n}(m_{[1:k]})=v_{[1:k]}^{n}|u^{n},v_{0}^{n}\right)\geq2^{-n\left(\sum_{j=1}^{k}H\left(V_{j}|V_{\mathcal{A}_{j}}V_{0}\right)+\delta\left(\epsilon\right)\right)},
\end{equation}
and for any $\left(u^{n},v_{0}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}$,
\begin{equation}
\left|\left\{ v_{[1:k]}^{n}:(u^{n},v_{0}^{n},v_{[1:k]}^{n})\in\mathcal{T}_{\epsilon}^{\left(n\right)}\right\} \right|\geq2^{n\left(H\left(V_{[1:k]}|UV_{0}\right)+\delta\left(\epsilon\right)\right)}.
\end{equation}
Similarly, we also can get
\begin{align}
p_{\mathcal{J}} & \leq2^{-n\left(\sum_{j=1}^{k}H\left(V_{j}|V_{\mathcal{A}_{j}}V_{0}\right)+\sum_{j\in\mathcal{J}^{c}}H\left(V_{j}|V_{\mathcal{A}_{j}}V_{0}\right)-H\left(V_{[1:k]}|UV_{0}\right)-H\left(V_{\mathcal{J}^{c}}|UV_{0}V_{\mathcal{J}}\right)-4\delta\left(\epsilon\right)\right)}.\label{eq:-61}
\end{align}
Substitute \eqref{eq:-60} and \eqref{eq:-61} into \eqref{eq:-9},
then we have
\begin{align}
\frac{\textrm{Var}(|\mathcal{B}||u^{n},v_{0}^{n})}{(\mathbb{E}(|\mathcal{B}||u^{n},v_{0}^{n}))^{2}} & \leq2^{-n\left(\sum_{j=1}^{k}r_{j}-\left(\sum_{j=1}^{k}H\left(V_{j}|V_{\mathcal{A}_{j}}V_{0}\right)-H\left(V_{[1:k]}|UV_{0}\right)+2\delta\left(\epsilon\right)\right)\right)}\nonumber \\
& +\sum_{\mathcal{J}\in\mathbb{J}}2^{-n\left(\sum_{j\in\mathcal{J}}r_{j}-\left(\sum_{j\in\mathcal{J}}H\left(V_{j}|V_{\mathcal{A}_{j}}V_{0}\right)-H\left(V_{\mathcal{J}}|UV_{0}\right)+6\delta\left(\epsilon\right)+o(1)\right)\right)}.\label{eq:-10}
\end{align}
\eqref{eq:-10} tends to zero if
\begin{align}
& \sum_{j=1}^{k}r_{j}>\sum_{j=1}^{k}H\left(V_{j}|V_{\mathcal{A}_{j}}V_{0}\right)-H\left(V_{[1:k]}|UV_{0}\right)+2\delta\left(\epsilon\right)\\
& \sum_{j\in\mathcal{J}}r_{j}>\sum_{j\in\mathcal{J}}H\left(V_{j}|V_{\mathcal{A}_{j}}V_{0}\right)-H\left(V_{\mathcal{J}}|UV_{0}\right)+6\delta\left(\epsilon\right)+o(1),
\end{align}
i.e., $\sum_{j\in\mathcal{J}}r_{j}>\sum_{j\in\mathcal{J}}H\left(V_{j}|V_{\mathcal{A}_{j}}V_{0}\right)-H\left(V_{\mathcal{J}}|UV_{0}\right)+\delta^{\prime}\left(\epsilon\right)$
for some $\delta^{\prime}\left(\epsilon\right)$ that tends to zero
as $\epsilon\rightarrow0$. This completes the proof.
\section{Proof of Lemma \ref{lem:Packing}}
For any $\mathcal{J}$ such that $\mathcal{J}\neq\emptyset$ and if
$j\in\mathcal{J}$ then $\mathcal{A}_{j}\subseteq\mathcal{J}$,
\begin{align}
& \mathbb{P}\left((U^{n},V_{0}^{n},V_{[1:k]}^{n}(m_{[1:k]}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}\textrm{ for some }m_{[1:k]}\right)\nonumber \\
& \leq\mathbb{P}\left((U^{n},V_{0}^{n},V_{\mathcal{J}}^{n}(m_{\mathcal{J}}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}\textrm{ for some }m_{\mathcal{J}}\right)\\
& =\sum_{u^{n},v_{0}^{n}}p_{U^{n},V_{0}^{n}}\left(u^{n},v_{0}^{n}\right)\mathbb{P}\left((u^{n},v_{0}^{n},V_{\mathcal{J}}^{n}(m_{\mathcal{J}}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}\textrm{ for some }m_{\mathcal{J}}|u^{n},v_{0}^{n}\right)\\
& \leq\sum_{u^{n},v_{0}^{n}}p_{U^{n},V_{0}^{n}}\left(u^{n},v_{0}^{n}\right)\sum_{m_{\mathcal{J}}}\mathbb{P}\left((u^{n},v_{0}^{n},V_{\mathcal{J}}^{n}(m_{\mathcal{J}}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}|u^{n},v_{0}^{n}\right).\label{eq:-59}
\end{align}
Similar to \eqref{eq:-60}, we can obtain that
\begin{align}
& \mathbb{P}\left((u^{n},v_{0}^{n},V_{\mathcal{J}}^{n}(m_{\mathcal{J}}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}|u^{n},v_{0}^{n}\right)\leq2^{-n\left(\sum_{j\in\mathcal{J}}H\left(V_{j}|V_{\mathcal{A}_{j}}V_{0}\right)-H\left(V_{\mathcal{J}}|UV_{0}\right)-2\delta\left(\epsilon\right)\right)}.
\end{align}
Substitute it into \eqref{eq:-59}, then we have
\begin{align}
& \mathbb{P}\left((U^{n},V_{0}^{n},V_{[1:k]}^{n}(m_{[1:k]}))\in\mathcal{T}_{\epsilon}^{\left(n\right)}\textrm{ for some }m_{[1:k]}\right)\nonumber \\
& \leq2^{n\left(\sum_{j\in\mathcal{J}}r_{j}-\left(\sum_{j\in\mathcal{J}}H\left(V_{j}|V_{\mathcal{A}_{j}}V_{0}\right)-H\left(V_{\mathcal{J}}|UV_{0}\right)-2\delta\left(\epsilon\right)\right)\right)}.\label{eq:-11}
\end{align}
\eqref{eq:-11} tends to zero if
\begin{align}
& \sum_{j\in\mathcal{J}}r_{j}<\sum_{j\in\mathcal{J}}H\left(V_{j}|V_{\mathcal{A}_{j}}V_{0}\right)-H\left(V_{\mathcal{J}}|UV_{0}\right)-2\delta\left(\epsilon\right).
\end{align}
This completes the proof.
\section{Proof of Theorem \ref{thm:AdmissibleRegion-GBC}\label{sec:broadcast}}
\subsection{\label{sub:Inner-Bound}Inner Bound}
Actually the inner bound can be seen as a corollary to \cite[Thm. 1]{Lee}
by choosing proper network topology, transit probability and symbol-by-symbol
functions. For completeness and clarity, next we provide a direct
description of the proposed hybrid coding scheme and a direct proof
for it.
\emph{Codebook Generation}: Fix conditional pmf $p_{V_{[1:N]}|S}$,
vector $r_{[1:N]}$, encoding function $x\left(v_{[1:N]},s\right)$
and decoding functions $\hat{s}_{k}\left(v_{\mathcal{D}_{k}},y_{k}\right)$
that satisfy
\begin{align}
\mathbb{E}d_{k}\left(S,\hat{S}_{k}\right) & \le D_{k},1\le k\le K,\\
\sum_{j\in\mathcal{J}}r_{j} & >\sum_{j\in\mathcal{J}}H\left(V_{j}|V_{\mathcal{A}_{j}}\right)-H\left(V_{\mathcal{J}}|S\right)\nonumber \\
& \qquad\textrm{for all }\mathcal{J}\subseteq[1:N]\textrm{ such that }\mathcal{J}\neq\emptyset\textrm{ and if }j\in\mathcal{J},\textrm{ then }\mathcal{A}_{j}\subseteq\mathcal{J},\label{eq:-14}\\
\sum_{j\in\mathcal{J}^{c}}r_{j} & <\sum_{j\in\mathcal{J}^{c}}H\left(V_{j}|V_{\mathcal{A}_{j}}\right)-H\left(V_{\mathcal{J}^{c}}|Y_{k}V_{\mathcal{J}}\right)\nonumber \\
& \qquad\textrm{for all }1\le k\le K\textrm{ and for all }\mathcal{J}\subseteq\mathcal{D}_{k}\textrm{ such that }\mathcal{J}^{c}\triangleq\mathcal{D}_{k}\backslash\mathcal{J}\neq\emptyset\textrm{ and if }j\in\mathcal{J},\textrm{ then }\mathcal{A}_{j}\subseteq\mathcal{J}.\label{eq:-16}
\end{align}
For each $j\in[1:N]$ and each $m_{\mathcal{A}_{j}}\in\prod_{i\in\mathcal{A}_{j}}[1:2^{nr_{i}}]$,
randomly and independently generate a set of sequences $v_{j}^{n}(m_{\mathcal{A}_{j}},m_{j}),m_{j}\in[1:2^{nr_{j}}],$
with each distributed according to $\prod_{i=1}^{n}p_{V_{j}|V_{\mathcal{A}_{j}}}(v_{j,i}|v_{\mathcal{A}_{j},i}(m_{\mathcal{A}_{j}}))$.
The codebook
\begin{equation}
\mathcal{C}=\left\{ \begin{array}{c}
v_{[1:N]}^{n}\left(m_{[1:N]}\right):m_{[1:N]}\in\prod_{i=1}^{N}[1:2^{nr_{i}}]\end{array}\right\} .
\end{equation}
is revealed to the encoder and all the decoders.
\emph{Encoding}: We use joint typicality encoding. Given $s^{n}$,
encoder finds the smallest index vector $m_{[1:N]}$ such that $\left(s^{n},v_{[1:N]}^{n}\left(m_{[1:N]}\right)\right)\in\mathcal{T}_{\epsilon}^{\left(n\right)}$.
If there is no such index vector, let $m_{[1:N]}=\mathbf{1}$. Then
the encoder transmits the signal
\begin{equation}
x_{i}=x\left(v_{[1:N],i}\left(m_{[1:N]}\right),s_{i}\right),1\leq i\leq n.
\end{equation}
\emph{Decoding}: We use joint typicality decoding. Let $\epsilon'>\epsilon$.
Upon receiving signal $y_{k}^{n}$, the decoder of the receiver $k$
finds the smallest index vector $\hat{m}_{\mathcal{D}_{k}}^{(k)}$
such that
\begin{equation}
(v_{\mathcal{D}_{k}}^{n}(\hat{m}_{\mathcal{D}_{k}}^{(k)}),y_{k}^{n})\in\mathcal{T}_{\epsilon'}^{\left(n\right)}.
\end{equation}
If there is no such index vector, let $\hat{m}_{\mathcal{D}_{k}}^{(k)}=\mathbf{1}$.
The decoder reconstructs the source as
\begin{equation}
\hat{s}_{k,i}=\hat{s}_{k}(v_{\mathcal{D}_{k},i}(\hat{m}_{\mathcal{D}_{k}}^{(k)}),y_{k,i}),1\leq i\leq n.
\end{equation}
\emph{Analysis of Expected Distortion}: We bound the distortion averaged
over $S^{n}$, and the random choice of the codebook $\mathcal{C}$.
Define the ``error\textquotedblright{} event
\begin{align}
\mathcal{E} & =\mathcal{E}_{1}\cup\left(\bigcup_{k}\mathcal{E}_{2,k}\right)\cup\left(\bigcup_{k}\mathcal{E}{}_{3,k}\right),
\end{align}
where
\begin{align}
\mathcal{E}_{1} & =\left\{ \left(S^{n},V_{[1:N]}^{n}\left(m_{[1:N]}\right)\right)\notin\mathcal{T}_{\epsilon}^{\left(n\right)}\textrm{ for all }m_{[1:N]}\right\} ,\\
\mathcal{E}_{2,k} & =\left\{ \left(S^{n},V_{[1:N]}^{n}\left(M_{[1:N]}\right),Y_{k}^{n}\right)\notin\mathcal{T}_{\epsilon'}^{\left(n\right)}\right\} ,\\
\mathcal{E}{}_{3,k} & =\Bigl\{\left(V_{\mathcal{D}_{k}}^{n}(m'_{\mathcal{D}_{k}}),Y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\textrm{ for some }m'_{\mathcal{D}_{k}},m'_{\mathcal{D}_{k}}\neq M{}_{\mathcal{D}_{k}}\Bigr\},
\end{align}
for $1\le k\le K.$ Using union bound, we have
\begin{align}
\mathbb{P}\left(\mathcal{E}\right) & \leq\mathbb{P}\left(\mathcal{E}_{1}\right)+\sum_{k=1}^{K}\mathbb{P}\left(\mathcal{E}_{1}^{c}\bigcap\mathcal{E}_{2,k}\right)+\sum_{k=1}^{K}\mathbb{P}\left(\mathcal{E}{}_{3,k}\right).\label{eq:proberror}
\end{align}
Now we claim that if \eqref{eq:-14} and \eqref{eq:-16} hold, then
$\mathbb{P}\left(\mathcal{E}\right)$ tends to zero as $n\to\infty$.
Before proving it, we show that this claim implies the inner bound
of Theorem \ref{thm:AdmissibleRegion-GBC}.
Define
\begin{equation}
\mathcal{E}_{4,k}=\left\{ \left(S^{n},V_{\mathcal{D}_{k}}^{n}(\hat{M}_{\mathcal{D}_{k}}^{(k)}),Y_{k}^{n}\right)\notin\mathcal{T}_{\epsilon'}^{\left(n\right)}\right\} .
\end{equation}
then we have $\mathcal{E}^{c}\subseteq\mathcal{E}_{4,k}^{c}$, i.e.,
$\mathcal{E}_{4,k}\subseteq\mathcal{E}$. This implies that $\mathbb{P}\left(\mathcal{E}_{4,k}\right)\leq\mathbb{P}\left(\mathcal{E}\right)\rightarrow0$
as $n\to\infty$. Then utilizing typical average lemma \cite{El Gamal},
we have
\begin{align}
& \limsup_{n\to\infty}\mathbb{E}d_{k}\left(S^{n},\hat{S}_{k}^{n}\right)\nonumber \\
& =\limsup_{n\to\infty}\left(\mathbb{P}\left(\mathcal{E}_{4,k}\right)\mathbb{E}\left[d_{k}\left(S^{n},\hat{S}_{k}^{n}\right)|\mathcal{E}_{4,k}\right]+\mathbb{P}\left(\mathcal{E}_{4,k}^{c}\right)\mathbb{E}\left[d_{k}\left(S^{n},\hat{S}_{k}^{n}\right)|\mathcal{E}_{4,k}^{c}\right]\right)\\
& =\limsup_{n\to\infty}\mathbb{E}\left[d_{k}\left(S^{n},\hat{S}_{k}^{n}\right)|\mathcal{E}_{4,k}^{c}\right]\\
& \le\left(1+\epsilon'\right)\mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\\
& \le\left(1+\epsilon'\right)D_{k}.
\end{align}
Therefore, the desired distortions are achieved for sufficiently small
$\epsilon'$.
Next we turn back to prove the claim above. Following from Multivariate
Covering Lemma (Lemma \ref{lem:Covering}), the first term of \eqref{eq:proberror},
$\mathbb{P}\left(\mathcal{E}_{1}\right)$, vanishes as $n\to\infty$,
and according to conditional typicality lemma \cite[Sec. 3.7]{El Gamal},
the second item tends to zero as $n\to\infty$.
Now we focus on the third term of \eqref{eq:proberror}. $\mathcal{E}{}_{3,k}$
can be writen as
\begin{align}
\mathcal{E}{}_{3,k} & =\bigcup_{\mathcal{I}\subseteq\mathcal{D}_{k}}\mathcal{E}{}_{3,k}^{\mathcal{I}},
\end{align}
where
\begin{align}
\mathcal{E}{}_{3,k}^{\mathcal{I}} & =\Bigl\{\left(V_{\mathcal{D}_{k}}^{n}\left(M_{\mathcal{I}},m'_{\mathcal{I}^{c}}\right),Y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\textrm{ for some }m'_{\mathcal{I}^{c}},m'_{\mathcal{I}^{c}}\nLeftrightarrow M_{\mathcal{I}^{c}}\Bigr\},
\end{align}
with $\mathcal{I}^{c}\triangleq\mathcal{D}_{k}\backslash\mathcal{I}.$
Using union bound we have
\begin{align}
\mathbb{P}\left(\mathcal{E}{}_{3,k}\right) & \leq\sum_{\mathcal{I}\subseteq\mathcal{D}_{k}}\mathbb{P}\left(\mathcal{E}{}_{3,k}^{\mathcal{I}}\right).\label{eq:proberror-1}
\end{align}
Each $\mathcal{D}_{k}$ has finite number of subsets, hence we only
need to show for each $\mathcal{I}\subseteq\mathcal{D}_{k}$, $\mathbb{P}\left(\mathcal{E}{}_{3,k}^{\mathcal{I}}\right)$
vanishes as $n\to\infty$. To show this, it is needed to analyze the
correlation between coding index $M_{[1:N]}$ and nonchosen codewords.
Specifically, $M_{[1:N]}$ depends on source sequence and the entire
codebook, and hence standard packing lemma cannot be applied directly.
This problem has been resolved by the technique developed in \cite{Minero,Lee}.
\begin{align}
& \mathbb{P}\left(\left(V_{\mathcal{D}_{k}}^{n}\left(M_{\mathcal{I}},m'_{\mathcal{I}^{c}}\right),Y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\textrm{ for some }m'_{\mathcal{I}^{c}},m'_{\mathcal{I}^{c}}\nLeftrightarrow M_{\mathcal{I}^{c}}\right)\nonumber \\
= & \sum_{m_{[1:N]}}\sum_{y_{k}^{n}}\mathbb{P}\left(M_{[1:N]}=m_{[1:N]},Y_{k}^{n}=y_{k}^{n},\left(V_{\mathcal{D}_{k}}^{n}\left(m_{\mathcal{I}},m'_{\mathcal{I}^{c}}\right),y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\textrm{ for some }m'_{\mathcal{I}^{c}},m'_{\mathcal{I}^{c}}\nLeftrightarrow m_{\mathcal{I}^{c}}\right)\\
\leq & \sum_{m_{[1:N]}}\sum_{y_{k}^{n}}\sum_{m'_{\mathcal{I}^{c}}:m'_{\mathcal{I}^{c}}\nLeftrightarrow m_{\mathcal{I}^{c}}}\mathbb{P}\left(M_{[1:N]}=m_{[1:N]},Y_{k}^{n}=y_{k}^{n},\left(V_{\mathcal{D}_{k}}^{n}\left(m_{\mathcal{I}},m'_{\mathcal{I}^{c}}\right),y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\right)\label{eq:-5}
\end{align}
{\small{}{}{}}where \eqref{eq:-5} follows from the union bound.
Define a sub-codebook as
\begin{align}
& \mathcal{C}_{(m_{\mathcal{I}},m_{\mathcal{I}^{c}}^{\prime})}=\Bigl\{ V_{[1:N]}^{n}\left(m_{\mathcal{I}},m_{\mathcal{I}^{c}}^{\prime\prime},m_{\mathcal{D}_{k}^{c}}^{\prime\prime}\right):\forall\left(m_{\mathcal{I}^{c}}^{\prime\prime},m_{\mathcal{D}_{k}^{c}}^{\prime\prime}\right),m_{\mathcal{I}^{c}}^{\prime\prime}\nLeftrightarrow m_{\mathcal{I}^{c}}^{\prime}\Bigr\}.
\end{align}
{\small{}{}{}}Define another coding index as $\tilde{M}_{[1:N]}$
which is generated by performing the same coding process as $M_{[1:N]}$
but on codebook $\mathcal{C}_{(m_{\mathcal{I}},m_{\mathcal{I}^{c}}^{\prime})}$,
i.e., given source sequence $s^{n}$, encoder finds the smallest index
vector $\tilde{m}_{[1:N]}$ such that $\left(s^{n},v_{[1:N]}^{n}\left(\tilde{m}_{[1:N]}\right)\right)\in\mathcal{T}_{\epsilon}^{\left(n\right)}$;
if there is no such index vector, let $\tilde{m}_{[1:N]}=\mathbf{1}$.
Then according to the generation process of $M_{[1:N]}$ and $\tilde{M}_{[1:N]}$,
we have if $M_{[1:N]}=m_{[1:N]}$, then $\tilde{M}_{[1:N]}=m_{[1:N]}$.
Now continuing with \eqref{eq:-5}, we have
\begin{align}
& \mathbb{P}\left(M_{[1:N]}=m_{[1:N]},Y_{k}^{n}=y_{k}^{n},\left(V_{\mathcal{D}_{k}}^{n}\left(m_{\mathcal{I}},m'_{\mathcal{I}^{c}}\right),y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\right)\nonumber \\
& =\sum_{v_{\mathcal{D}_{k}}^{\prime n},c,s^{n}}\mathbb{P}\left(M_{[1:N]}=m_{[1:N]},\mathcal{C}_{(m_{\mathcal{I}},m_{\mathcal{I}^{c}}^{\prime})}=c,S^{n}=s^{n},V_{\mathcal{D}_{k}}^{n}\left(m_{\mathcal{I}},m'_{\mathcal{I}^{c}}\right)=v_{\mathcal{D}_{k}}^{\prime n}\right)\nonumber \\
& \qquad\prod_{i=1}^{n}p_{Y_{k}|X}\left(y_{k,i}|x\left(v_{[1:N],i}(m_{[1:N]}),s_{i}\right)\right)1\left\{ \left(v_{\mathcal{D}_{k}}^{\prime n},y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\right\} \\
& =\sum_{v_{\mathcal{D}_{k}}^{\prime n},c,s^{n}}\mathbb{P}\left(M_{[1:N]}=m_{[1:N]},\tilde{M}_{[1:N]}=m_{[1:N]},\mathcal{C}_{(m_{\mathcal{I}},m_{\mathcal{I}^{c}}^{\prime})}=c,S^{n}=s^{n},V_{\mathcal{D}_{k}}^{n}\left(m_{\mathcal{I}},m'_{\mathcal{I}^{c}}\right)=v_{\mathcal{D}_{k}}^{\prime n}\right)\nonumber \\
& \qquad\prod_{i=1}^{n}p_{Y_{k}|X}\left(y_{k,i}|x\left(v_{[1:N],i}(m_{[1:N]}),s_{i}\right)\right)1\left\{ \left(v_{\mathcal{D}_{k}}^{\prime n},y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\right\} \\
& \leq\sum_{v_{\mathcal{D}_{k}}^{\prime n},c,s^{n}}\mathbb{P}\left(\tilde{M}_{[1:N]}=m_{[1:N]},\mathcal{C}_{(m_{\mathcal{I}},m_{\mathcal{I}^{c}}^{\prime})}=c,S^{n}=s^{n},V_{\mathcal{D}_{k}}^{n}\left(m_{\mathcal{I}},m'_{\mathcal{I}^{c}}\right)=v_{\mathcal{D}_{k}}^{\prime n}\right)\nonumber \\
& \qquad\prod_{i=1}^{n}p_{Y_{k}|X}\left(y_{k,i}|x\left(v_{[1:N],i}(m_{[1:N]}),s_{i}\right)\right)1\left\{ \left(v_{\mathcal{D}_{k}}^{\prime n},y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\right\} \\
& =\sum_{v_{\mathcal{D}_{k}}^{\prime n},c,s^{n}}\mathbb{P}\left(\tilde{M}_{[1:N]}=m_{[1:N]},\mathcal{C}_{(m_{\mathcal{I}},m_{\mathcal{I}^{c}}^{\prime})}=c,S^{n}=s^{n}\right)\prod_{i=1}^{n}p_{Y_{k}|X}\left(y_{k,i}|x\left(v_{[1:N],i}(m_{[1:N]}),s_{i}\right)\right)\nonumber \\
& \qquad\mathbb{P}\left(V_{\mathcal{D}_{k}}^{n}\left(m_{\mathcal{I}},m'_{\mathcal{I}^{c}}\right)=v_{\mathcal{D}_{k}}^{\prime n}|\mathcal{C}_{(m_{\mathcal{I}},m_{\mathcal{I}^{c}}^{\prime})}=c\right)1\left\{ \left(v_{\mathcal{D}_{k}}^{\prime n},y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\right\} \label{eq:-7}
\end{align}
where $c=\Bigl\{ v_{[1:N]}^{n}\left(m_{\mathcal{I}},m_{\mathcal{I}^{c}}^{\prime\prime},m_{\mathcal{D}_{k}^{c}}^{\prime\prime}\right):\forall\left(m_{\mathcal{I}^{c}}^{\prime\prime},m_{\mathcal{D}_{k}^{c}}^{\prime\prime}\right),m_{\mathcal{I}^{c}}^{\prime\prime}\nLeftrightarrow m_{\mathcal{I}^{c}}^{\prime}\Bigr\}$,
and \eqref{eq:-7} follows from the fact that $V_{\mathcal{D}_{k}}^{n}\left(m_{\mathcal{I}},m'_{\mathcal{I}^{c}}\right)\rightarrow\mathcal{C}_{(m_{\mathcal{I}},m_{\mathcal{I}^{c}}^{\prime})}\rightarrow\left(S^{n},\tilde{M}_{[1:N]}\right)$
forms a Markov chain.
Define
\begin{equation}
\mathbb{J}\triangleq\left\{ \mathcal{J}\subseteq\mathcal{D}_{k}:\textrm{if }j\in\mathcal{J},\textrm{then }\mathcal{A}_{j}\subseteq\mathcal{J}\right\} .
\end{equation}
Then any set $\mathcal{I}\subseteq\mathcal{D}_{k}$ can transform
into a $\mathcal{J}\left(\mathcal{I}\right)\in\mathbb{J}$ by removing
all the elements $j$'s such that $\mathcal{A}_{j}\nsubseteq\mathcal{I}$.
Denote $\mathcal{J}^{c}\triangleq\mathcal{D}_{k}\backslash\mathcal{J}.$
Then according to the generation process of the codebook, continuing
with \eqref{eq:-7}, we have
\begin{align}
& \sum_{v_{\mathcal{D}_{k}}^{\prime n}}\mathbb{P}\left(V_{\mathcal{D}_{k}}^{n}\left(m_{\mathcal{I}},m'_{\mathcal{I}^{c}}\right)=v_{\mathcal{D}_{k}}^{\prime n}|\mathcal{C}_{(m_{\mathcal{I}},m_{\mathcal{I}^{c}}^{\prime})}=c\right)1\left\{ \left(v_{\mathcal{D}_{k}}^{\prime n},y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\right\} \nonumber \\
& =\sum_{v_{\mathcal{J}^{c}}^{\prime n}}\mathbb{P}\left(V_{\mathcal{J}^{c}}^{n}\left(m_{\mathcal{I}},m'_{\mathcal{I}^{c}}\right)=v_{\mathcal{J}^{c}}^{\prime n}|\mathcal{C}_{(m_{\mathcal{I}},m_{\mathcal{I}^{c}}^{\prime})}=c\right)1\left\{ \left(v_{\mathcal{J}}^{n}\left(m_{\mathcal{J}}\right),v_{\mathcal{J}^{c}}^{\prime n},y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\right\} \\
& =\sum_{v_{\mathcal{J}^{c}}^{\prime n}}\prod_{j\in\mathcal{J}^{c}}\prod_{i=1}^{n}p_{V_{j}|V_{\mathcal{A}_{j}}}\left(v_{j,i}^{\prime}|v_{\mathcal{A}_{j}\cap\mathcal{J},i}\left(m_{\mathcal{J}}\right),v_{\mathcal{A}_{j}\cap\mathcal{J}^{c},i}^{\prime}\right)1\left\{ \left(v_{\mathcal{J}}^{n}\left(m_{\mathcal{J}}\right),v_{\mathcal{J}^{c}}^{\prime n},y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\right\} \label{eq:-17}\\
& \leq2^{n\left(H\left(V_{\mathcal{J}^{c}}|Y_{k}V_{\mathcal{J}}\right)-\sum_{j\in\mathcal{J}^{c}}H\left(V_{j}|V_{\mathcal{A}_{j}}\right)+\left(|\mathcal{J}^{c}|+1\right)\delta\left(\epsilon'\right)\right)},\label{eq:-22}
\end{align}
where $\delta\left(\epsilon'\right)$ is a term that tends to zero
as $\epsilon'\rightarrow0$, and \eqref{eq:-22} follows from the
fact that $\prod_{i=1}^{n}p_{V_{j}|V_{\mathcal{A}_{j}}}\left(v_{j,i}|v_{\mathcal{A}_{j},i}\right)\leq2^{-n\left(H\left(V_{j}|V_{\mathcal{A}_{j}}\right)-\delta\left(\epsilon'\right)\right)}$
for any $\left(v_{j}^{n},v_{\mathcal{A}_{j}}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}$
and $\left|\left\{ v_{\mathcal{J}^{c}}^{\prime n}:\left(v_{\mathcal{J}}^{n},v_{\mathcal{J}^{c}}^{\prime n},y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\right\} \right|\leq2^{n\left(H\left(V_{\mathcal{J}^{c}}|Y_{k}V_{\mathcal{J}}\right)+\delta\left(\epsilon'\right)\right)}$
for any $\left(y_{k}^{n},v_{\mathcal{J}}^{n}\right)$.
Combining \eqref{eq:-5}, \eqref{eq:-7} and \eqref{eq:-22} gives
\begin{align}
& \mathbb{P}\left(\left(V_{\mathcal{D}_{k}}^{n}\left(M_{\mathcal{I}},m'_{\mathcal{I}^{c}}\right),Y_{k}^{n}\right)\in\mathcal{T}_{\epsilon'}^{\left(n\right)}\textrm{ for some }m'_{\mathcal{I}^{c}},m'_{\mathcal{I}^{c}}\nLeftrightarrow M_{\mathcal{I}^{c}}\right)\nonumber \\
& \leq2^{n\left(\sum_{j\in\mathcal{J}^{c}}r_{j}-\left(\sum_{j\in\mathcal{J}^{c}}H\left(V_{j}|V_{\mathcal{A}_{j}}\right)-H\left(V_{\mathcal{J}^{c}}|Y_{k}V_{\mathcal{J}}\right)-\left(|\mathcal{J}^{c}|+1\right)\delta\left(\epsilon'\right)\right)\right)}.
\end{align}
Hence if $\sum_{j\in\mathcal{J}^{c}}r_{j}<\sum_{j\in\mathcal{J}^{c}}H\left(V_{j}|V_{\mathcal{A}_{j}}\right)-H\left(V_{\mathcal{J}^{c}}|Y_{k}V_{\mathcal{J}}\right)-\left(|\mathcal{J}^{c}|+1\right)\delta\left(\epsilon'\right)$
for all $\mathcal{J}\in\mathbb{J}$, then the third term of \eqref{eq:proberror}
tends to zero as $n\to\infty$. Letting $\epsilon'$ small enough,
this completes the proof of the inner bound.
Besides, it is worth noting that although Multivariate Packing Lemma
(Lemma \ref{lem:Packing}) has not been employed directly in the proof,
the derivation after \eqref{eq:-17} is essentially the same as that
of Multivariate Packing Lemma.
\subsection{Outer Bound $\mathcal{R}_{1}^{(o)}$}
For fixed $p_{U_{[1:L]}|S}$, we first introduce a set of auxiliary
random variables $U_{[1:L]}^{n}$ that follow distribution $\prod_{i=1}^{n}p_{U_{[1:L]}|S}\left(u_{[1:L],i}|s_{i}\right)$.
Hence the Markov chains $U_{[1:L]}^{n}\rightarrow S^{n}\rightarrow X^{n}\rightarrow Y_{k}^{n}\rightarrow\hat{S}_{k}^{n},1\leq k\leq K$
hold. Consider that
\begin{align}
& I\left(Y_{\mathcal{A}}^{n};U_{\mathcal{B}}^{n}|U_{\mathcal{C}}^{n}\right)\nonumber \\
& =\sum_{t=1}^{n}I\left(Y_{\mathcal{A}}^{n};U_{\mathcal{B},t}|U_{\mathcal{C}}^{n}U_{\mathcal{B}}^{t-1}\right)\label{eq:-13}\\
& =\sum_{t=1}^{n}H\left(U_{\mathcal{B},t}|U_{\mathcal{C}}^{n}U_{\mathcal{B}}^{t-1}\right)-H\left(U_{\mathcal{B},t}|U_{\mathcal{C}}^{n}U_{\mathcal{B}}^{t-1}Y_{\mathcal{A}}^{n}\right)\label{eq:-14-3-1-1}\\
& =\sum_{t=1}^{n}H\left(U_{\mathcal{B},t}|U_{\mathcal{C},t}\right)-H\left(U_{\mathcal{B},t}|U_{\mathcal{C}}^{n}U_{\mathcal{B}}^{t-1}Y_{\mathcal{A}}^{n}\right)\label{eq:-3-1-1}\\
& =\sum_{t=1}^{n}I\left(U_{\mathcal{B},t};U_{\mathcal{C}}^{n}U_{\mathcal{B}}^{t-1}Y_{\mathcal{A}}^{n}|U_{\mathcal{C},t}\right)\\
& \geq\sum_{t=1}^{n}I\left(U_{\mathcal{B},t};\hat{S}_{\mathcal{A},t}|U_{\mathcal{C},t}\right)\\
& =nI\left(U_{\mathcal{B},Q};\hat{S}_{\mathcal{A},Q}|U_{\mathcal{C},Q}Q\right)\label{eq:-2-1-1}\\
& =nI\left(U_{\mathcal{B},Q};\hat{S}_{\mathcal{A},Q}Q|U_{\mathcal{C},Q}\right)\\
& \geq nI\left(U_{\mathcal{B},Q};\hat{S}_{\mathcal{A},Q}|U_{\mathcal{C},Q}\right)\\
& =nI\left(U_{\mathcal{B}};\hat{S}_{\mathcal{A}}|U_{\mathcal{C}}\right),\label{eq:-4-1-1}
\end{align}
where the time-sharing random variable $Q$ is defined to be uniformly
distributed $\left[1:n\right]$ and independent of all other random
variables, and in \eqref{eq:-4-1-1}, $U_{l}\triangleq U_{l,Q},\hat{S}_{k}\triangleq\hat{S}_{k,Q},1\leq l\leq L,1\leq k\leq K$.
Set $\tilde{U}_{l}\triangleq U_{l}^{n},1\leq l\leq L$, then \eqref{eq:-4-1-1}
implies the outer bound $\mathcal{R}_{1}^{(o)}$ holds.
\subsection{Outer Bound $\mathcal{R}_{1}^{\prime(o)}$}
To prove the outer bound $\mathcal{R}_{1}^{\prime(o)}$, we only need
to show $\mathcal{R}_{1}^{(o)}\subseteq\mathcal{R}_{1}^{\prime(o)}$.
Substituting $\mathcal{A}=\mathcal{A}_{i},\mathcal{B}=\mathcal{B}_{i},\mathcal{C}=\cup_{j=0}^{i-1}\mathcal{B}_{j}$
into $\mathcal{R}_{1}^{(o)}$ gives us $I\left(\hat{S}_{\mathcal{A}_{i}};U_{\mathcal{B}_{i}}|U_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\right)\leq\frac{1}{n}I\left(Y_{\mathcal{A}_{i}}^{n};\tilde{U}_{\mathcal{B}_{i}}|\tilde{U}_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\right)$.
Summate both sides of this inequality through $i=1$ to $m$, then
we get
\begin{equation}
\frac{1}{n}\sum_{i=1}^{m}I\left(Y_{\mathcal{A}_{i}}^{n};\tilde{U}_{\mathcal{B}_{i}}|\tilde{U}_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\right)\geq\sum_{i=1}^{m}I\left(\hat{S}_{\mathcal{A}_{i}};U_{\mathcal{B}_{i}}|U_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\right).\label{eq:-19}
\end{equation}
Now, we turn to upper-bounding $\frac{1}{n}\sum_{i=1}^{m}I\left(Y_{\mathcal{A}_{i}}^{n};\tilde{U}_{\mathcal{B}_{i}}|\tilde{U}_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\right)$.
We will show for any $m\geq1$,
\begin{equation}
\sum_{i=1}^{m}I\left(Y_{\mathcal{A}_{i}}^{n};\tilde{U}_{\mathcal{B}_{i}}|\tilde{U}_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\right)\leq\sum_{i=1}^{m}\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{i},t};\tilde{U}_{\mathcal{B}_{i}}\tilde{Y}_{\mathcal{A}_{i+1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{i}}^{t-1}\tilde{Y}_{\mathcal{A}_{i-1}}^{t-1}\right)-\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{m},t};\tilde{Y}_{\mathcal{A}_{m+1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\right)\label{eq:-30}
\end{equation}
by induction method, where
\begin{equation}
\tilde{Y}_{\mathcal{A}_{i}}^{t-1}\triangleq\begin{cases}
Y_{\mathcal{A}_{i}}^{t-1}, & \textrm{if }i\textrm{ is odd};\\
Y_{\mathcal{A}_{i},t+1}^{n}, & \textrm{if }i\textrm{ is even.}
\end{cases}\label{eq:YB}
\end{equation}
For $m=1$,
\begin{align}
& I\left(Y_{\mathcal{A}_{1}}^{n};\tilde{U}_{\mathcal{B}_{1}}|\tilde{U}_{\mathcal{B}_{0}}\right)\nonumber \\
& =\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{1},t};\tilde{U}_{\mathcal{B}_{1}}|\tilde{U}_{\mathcal{B}_{0}}\tilde{Y}_{\mathcal{A}_{1}}^{t-1}\right)\\
& =\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{1},t};\tilde{U}_{\mathcal{B}_{1}}\tilde{Y}_{\mathcal{A}_{2}}^{t-1}|\tilde{U}_{\mathcal{B}_{0}}\tilde{Y}_{\mathcal{A}_{1}}^{t-1}\right)-I\left(Y_{\mathcal{A}_{1},t};\tilde{Y}_{\mathcal{A}_{2}}^{t-1}|\tilde{U}_{\cup_{j=0}^{1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{1}}^{t-1}\right).
\end{align}
Hence \eqref{eq:-30} holds for $m=1$.
Assume \eqref{eq:-30} holds for $m-1$, then we have
\begin{align}
& \sum_{i=1}^{m}I\left(Y_{\mathcal{A}_{i}}^{n};\tilde{U}_{\mathcal{B}_{i}}|\tilde{U}_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\right)\nonumber \\
& \leq I\left(Y_{\mathcal{A}_{m}}^{n};\tilde{U}_{\mathcal{B}_{m}}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\right)+\sum_{i=1}^{m-1}\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{i},t};\tilde{U}_{\mathcal{B}_{i}}^{n}\tilde{Y}_{\mathcal{A}_{i+1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{i}}^{t-1}\tilde{Y}_{\mathcal{A}_{i-1}}^{t-1}\right)\nonumber \\
& \qquad-\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{m-1},t};\tilde{Y}_{\mathcal{A}_{m}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}\right).\label{eq:-32}
\end{align}
Considering the first term of \eqref{eq:-32}, we have
\begin{align}
& I\left(Y_{\mathcal{A}_{m}}^{n};\tilde{U}_{\mathcal{B}_{m}}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\right)\nonumber \\
& =\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{m},t};\tilde{U}_{\mathcal{B}_{m}}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\right)\\
& =\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{m},t};\tilde{U}_{\mathcal{B}_{m}}\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\right)-I\left(Y_{\mathcal{A}_{m},t};\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\right)\\
& =\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{m},t};\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\right)+I\left(Y_{\mathcal{A}_{m},t};\tilde{U}_{\mathcal{B}_{m}}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}\right)\nonumber \\
& \qquad-I\left(Y_{\mathcal{A}_{m},t};\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\right)\\
& =\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{m},t};\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\right)+I\left(Y_{\mathcal{A}_{m},t};\tilde{U}_{\mathcal{B}_{m}}\tilde{Y}_{\mathcal{A}_{m+1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}\right)\nonumber \\
& \qquad-I\left(Y_{\mathcal{A}_{m},t};\tilde{Y}_{\mathcal{A}_{m+1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}\right)-I\left(Y_{\mathcal{A}_{m},t};\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\right)\\
& =\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{m},t};\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\right)+I\left(Y_{\mathcal{A}_{m},t};\tilde{U}_{\mathcal{B}_{m}}\tilde{Y}_{\mathcal{A}_{m+1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}\right)\nonumber \\
& \qquad-I\left(Y_{\mathcal{A}_{m},t};\tilde{Y}_{\mathcal{A}_{m+1}}^{t-1}\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\right)\\
& \leq\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{m},t};\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\right)+I\left(Y_{\mathcal{A}_{m},t};\tilde{U}_{\mathcal{B}_{m}}\tilde{Y}_{\mathcal{A}_{m+1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}\right)\nonumber \\
& \qquad-I\left(Y_{\mathcal{A}_{m},t};\tilde{Y}_{\mathcal{A}_{m+1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\right).\label{eq:-35}
\end{align}
Combine \eqref{eq:-32} and \eqref{eq:-35}, and utilize the following
identity
\begin{equation}
\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{m},t};\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m}}^{t-1}\right)=\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{m-1},t};\tilde{Y}_{\mathcal{A}_{m}}^{t-1}|\tilde{U}_{\cup_{j=0}^{m-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{m-1}}^{t-1}\right),
\end{equation}
which follows from Csiszár sum identity \cite[p. 25]{El Gamal}, then
we have \eqref{eq:-30} holds for $m$. Hence \eqref{eq:-30} holds
for any $m\geq1$.
From \eqref{eq:-30}, we have
\begin{align}
\sum_{i=1}^{m}I\left(Y_{\mathcal{A}_{i}}^{n};\tilde{U}_{\mathcal{B}_{i}}|\tilde{U}_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\right) & \leq\sum_{i=1}^{m}\sum_{t=1}^{n}I\left(Y_{\mathcal{A}_{i},t};\tilde{U}_{\mathcal{B}_{i}}\tilde{Y}_{\mathcal{A}_{i+1}}^{t-1}|\tilde{U}_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{i}}^{t-1}\tilde{Y}_{\mathcal{A}_{i-1}}^{t-1}\right)\label{eq:-30-1}\\
& =n\sum_{i=1}^{m}I\left(Y_{\mathcal{A}_{i},Q};\tilde{U}_{\mathcal{B}_{i}}\tilde{Y}_{\mathcal{A}_{i+1}}^{Q-1}|\tilde{U}_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\tilde{Y}_{\mathcal{A}_{i}}^{Q-1}\tilde{Y}_{\mathcal{A}_{i-1}}^{Q-1}Q\right)\\
& =n\sum_{i=1}^{m}I\left(Y_{\mathcal{A}_{i}};\tilde{U}_{\mathcal{B}_{i}}\tilde{W}_{\mathcal{A}_{i+1}}|\tilde{U}_{\cup_{j=0}^{i-1}\mathcal{B}_{j}}\tilde{W}_{\mathcal{A}_{i}}\tilde{W}_{\mathcal{A}_{i-1}}\right),\label{eq:-57}
\end{align}
where the time-sharing random variable $Q$ is defined above, and
$Y_{k}\triangleq Y_{k,Q},W_{k}\triangleq\left(Y_{k}^{Q-1},Q\right),W_{k}^{\prime}\triangleq\left(Y_{k,Q+1}^{n},Q\right),1\leq k\leq K$,
and $\tilde{W}_{\mathcal{A}_{i}}\triangleq W_{\mathcal{A}_{i}}$,
if $i$ is odd; $W'_{\mathcal{A}_{i}}$, otherwise.
If we redefine \eqref{eq:YB} as
\begin{equation}
\tilde{Y}_{\mathcal{A}_{i}}^{t-1}\triangleq\begin{cases}
Y_{\mathcal{A}_{i},t+1}^{n}, & \textrm{if }i\textrm{ is odd};\\
Y_{\mathcal{A}_{i}}^{t-1}, & \textrm{if }i\textrm{ is even},
\end{cases}\label{eq:YB-1}
\end{equation}
then \eqref{eq:-57} still holds for $\tilde{W}_{\mathcal{A}_{i}}\triangleq W'_{\mathcal{A}_{i}}$,
if $i$ is odd; $W_{\mathcal{A}_{i}}$, otherwise.
Combine bounds \eqref{eq:-19} and \eqref{eq:-57}, then the outer
bound $\mathcal{R}_{1}^{\prime(o)}$ holds.
\subsection{Outer Bound $\mathcal{R}_{2}^{(o)}$}
For fixed $p_{U_{[1:L]}|Y_{[1:K]}}$, we first introduce a set of
auxiliary random variables $U_{[1:L]}^{n}$ that follow distribution
$\prod_{i=1}^{n}p_{U_{[1:L]}|Y_{[1:K]}}\left(u_{[1:L],i}|y_{[1:K],i}\right)$.
Hence the Markov chains $S^{n}\rightarrow X^{n}\rightarrow Y_{[1:K]}^{n}\rightarrow U_{[1:L]}^{n}$
hold. Note that different from the proof of $\mathcal{R}_{1}^{(o)}$,
the auxiliary random variables $U_{[1:L]}^{n}$ here is introduced
at receiver sides, and $p_{Y_{[1:K]}U_{[1:L]}|X}=p_{U_{[1:L]}|Y_{[1:K]}}p_{Y_{[1:K]}|X}$
forms a new memoryless broadcast channel. Consider that
\begin{align}
& I\left(S^{n};Y_{\mathcal{B}}^{n}U_{\mathcal{B}'}^{n}|Y_{\mathcal{C}}^{n}U_{\mathcal{C}'}^{n}\right)\nonumber \\
& \leq I\left(X^{n};Y_{\mathcal{B}}^{n}U_{\mathcal{B}'}^{n}|Y_{\mathcal{C}}^{n}U_{\mathcal{C}'}^{n}\right)\\
& =\sum_{t=1}^{n}I\left(X^{n};Y_{\mathcal{B},t}U_{\mathcal{B}',t}|Y_{\mathcal{C}}^{n}U_{\mathcal{C}'}^{n}Y_{\mathcal{B}}^{t-1}U_{\mathcal{B}'}^{t-1}\right)\\
& =\sum_{t=1}^{n}H\left(Y_{\mathcal{B},t}U_{\mathcal{B}',t}|Y_{\mathcal{C}}^{n}U_{\mathcal{C}'}^{n}Y_{\mathcal{B}}^{t-1}U_{\mathcal{B}'}^{t-1}\right)-H\left(Y_{\mathcal{B},t}U_{\mathcal{B}',t}|Y_{\mathcal{C}}^{n}U_{\mathcal{C}'}^{n}Y_{\mathcal{B}}^{t-1}U_{\mathcal{B}'}^{t-1}X^{n}\right)\label{eq:-14-3-1-1-1}\\
& \leq\sum_{t=1}^{n}H\left(Y_{\mathcal{B},t}U_{\mathcal{B}',t}|Y_{\mathcal{C},t}U_{\mathcal{C}',t}\right)-H\left(Y_{\mathcal{B},t}U_{\mathcal{B}',t}|Y_{\mathcal{C},t}U_{\mathcal{C}',t}X_{t}\right)\label{eq:-3-1-1-1}\\
& =\sum_{t=1}^{n}I\left(Y_{\mathcal{B},t}U_{\mathcal{B}',t};X_{t}|Y_{\mathcal{C},t}U_{\mathcal{C}',t}\right)\\
& =nI\left(Y_{\mathcal{B},Q}U_{\mathcal{B}',Q};X_{Q}|Y_{\mathcal{C},Q}U_{\mathcal{C}',Q}Q\right)\label{eq:-2-1-1-1}\\
& =nH\left(Y_{\mathcal{B},Q}U_{\mathcal{B}',Q}|Y_{\mathcal{C},Q}U_{\mathcal{C}',Q}Q\right)-nH\left(Y_{\mathcal{B},Q}U_{\mathcal{B}',Q}|Y_{\mathcal{C},Q}U_{\mathcal{C}',Q}X_{Q}Q\right)\\
& \leq nH\left(Y_{\mathcal{B},Q}U_{\mathcal{B}',Q}|Y_{\mathcal{C},Q}U_{\mathcal{C}',Q}\right)-nH\left(Y_{\mathcal{B},Q}U_{\mathcal{B}',Q}|Y_{\mathcal{C},Q}U_{\mathcal{C}',Q}X_{Q}\right)\label{eq:-12}\\
& =nI\left(Y_{\mathcal{B},Q}U_{\mathcal{B}',Q};X_{Q}|Y_{\mathcal{C},Q}U_{\mathcal{C}',Q}\right)\\
& =nI\left(X;Y_{\mathcal{B}}U_{\mathcal{B}'}|Y_{\mathcal{C}}U_{\mathcal{C}'}\right),\label{eq:-4-1-1-1}
\end{align}
where \eqref{eq:-3-1-1-1} follows from Markov chain $U_{[1:L]}^{t-1}U_{[1:L],t+1}^{n}Y_{[1:K]}^{t-1}Y_{[1:K],t+1}^{n}X^{t-1}X_{t+1}^{n}\rightarrow X_{t}\rightarrow Y_{[1:K],t}U_{[1:L],t}$
and the fact conditioning reduces entropy, \eqref{eq:-12} follows
from Markov chain $Q\rightarrow X_{Q}\rightarrow Y_{[1:K],Q}U_{[1:L],Q}$
and the fact conditioning reduces entropy, the time-sharing random
variable $Q$ is defined to be uniformly distributed $\left[1:n\right]$
and independent of all other random variables, and in \eqref{eq:-4-1-1-1},
$U_{l}\triangleq U_{l,Q},Y_{k}\triangleq Y_{k,Q},X\triangleq X_{Q},1\leq l\leq L,1\leq k\leq K$.
On the other hand,
\begin{align}
& I\left(S^{n};Y_{\mathcal{B}}^{n}U_{\mathcal{B}'}^{n}|Y_{\mathcal{C}}^{n}U_{\mathcal{C}'}^{n}\right)\nonumber \\
& =\sum_{t=1}^{n}I\left(S_{t};Y_{\mathcal{B}}^{n}U_{\mathcal{B}'}^{n}|Y_{\mathcal{C}}^{n}U_{\mathcal{C}'}^{n}S^{t-1}\right)\\
& =nI\left(S_{Q};Y_{\mathcal{B}}^{n}U_{\mathcal{B}'}^{n}|Y_{\mathcal{C}}^{n}U_{\mathcal{C}'}^{n}S^{Q-1}Q\right)\label{eq:-2-1-1-1-1}\\
& =nI\left(S;\tilde{Y}_{\mathcal{B}}\tilde{U}_{\mathcal{B}'}|\tilde{Y}_{\mathcal{C}}\tilde{U}_{\mathcal{C}'}\right),\label{eq:-4-1-1-1-1}
\end{align}
Set $S\triangleq S_{Q},\tilde{U}_{l}\triangleq U_{l}^{n}S^{Q-1}Q,\tilde{Y}_{k}\triangleq Y_{k}^{n}S^{Q-1}Q,1\leq l\leq L,1\leq k\leq K$,
then combining \eqref{eq:-4-1-1-1} and \eqref{eq:-4-1-1-1-1} gives
us the outer bound $\mathcal{R}_{2}^{(o)}$.
\section{Proof of Theorem \ref{thm:AdmissibleRegion-DBC}\label{sec:broadcast-1}}
\subsection{Inner Bound}
For the inner bound $\mathcal{R}^{(i)}$ of Theorem \ref{thm:AdmissibleRegion-GBC},
retain all the random variables $V_{i}$'s corresponding to the sets
$\mathcal{G}_{i}=[1:K],[2:K],\cdots,\left\{ K\right\} $, rename them
and corresponding rates $r_{i}$'s to $V_{1}^{\prime},V_{2}^{\prime},\cdots,V_{K}^{\prime}$
and $r_{1}^{\prime},r_{2}^{\prime},\cdots,r_{K}^{\prime}$, respectively,
and set all the other random variables to empty, then $\mathcal{R}^{(i)}$
reduces to
\begin{align}
\mathcal{R}^{(i)}= & \Bigl\{ D_{[1:K]}:\textrm{There exist some pmf }p_{V_{[1:K]}^{\prime}|S},\textrm{ vector }r_{[1:K]}^{\prime},\nonumber \\
& \text{and functions }x\left(v_{[1:K]}^{\prime},s\right),\hat{s}_{k}\left(v_{[1:k]}^{\prime},y_{k}\right),1\le k\le K\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},1\le k\le K,\nonumber \\
& \sum_{j=1}^{k}r_{j}^{\prime}>I\left(V_{[1:k]}^{\prime};S\right),1\le k\le K,\nonumber \\
& r_{k}^{\prime}<I\left(V_{k}^{\prime};Y_{k}|V_{[1:k-1]}^{\prime}\right),1\le k\le K\Bigr\},\label{eq:-28-3}
\end{align}
and the coding scheme in the proof of Theorem \ref{thm:AdmissibleRegion-GBC}
reduces to a superposition coding scheme. Define a set of random variables
$V_{k}\triangleq V_{[1:k]}^{\prime},1\le k\le K$. Substitute these
into \eqref{eq:-28-3}, then the inner bound of Theorem \ref{thm:AdmissibleRegion-DBC}
is recovered.
\subsection{Outer Bound}
For the outer bound, we provide two proofs. The first follows from
Theorem \ref{thm:AdmissibleRegion-GBC}, and the second is a more
simple and direct proof that does not utilize the Csiszár sum identity.
\subsubsection*{Proof method 1}
Set $L=K,\mathcal{A}=\mathcal{B}=[1:k],\mathcal{C}=[1:k-1]$ for $1\le k\le m$,
and $U_{0}\triangleq\emptyset,U_{K}\triangleq S,p_{U_{[1:K-1]}|S}=p_{U_{K-1}|S}p_{U_{K-2}|U_{K-1}}\cdots p_{U_{1}|U_{2}}$.
Substitute these into the outer bound $\mathcal{R}_{1}^{(o)}$ of
Theorem \ref{thm:AdmissibleRegion-GBC}, and utilize the degradation
of the channel, then we get $I\left(\hat{S}_{k}^{\prime};U_{k}|U_{k-1}\right)\leq\frac{1}{n}I\left(Y_{[1:k]}^{n};\tilde{U}_{[1:k]}|\tilde{U}_{[1:k-1]}\right),\hat{S}_{k}^{\prime}\triangleq\hat{S}_{[1:k]},1\le k\le K$.
Hence $I\left(\hat{S}_{k}^{\prime};U_{k}|U_{k-1}\right)\in\frac{1}{n}\mathcal{B}_{DBC}\left(p_{X^{n}}p_{Y_{[1:K]}^{n}|X^{n}}\right)\subseteq\mathcal{B}_{DBC}\left(p_{X}p_{Y_{[1:K]}|X}\right)$
for some $p_{\hat{S}_{[1:K]}|S}$ and $p_{X}$. The last equality
follows from that for the memoryless degraded broadcast channel, the
capacity region will not expand after extending the channel to $n$
lettered one.
\begin{align}
\mathcal{R}_{2}^{(o)}= & \Bigl\{ D_{[1:K]}:\textrm{There exists some pmf }p_{X}\textrm{ and some functions }\hat{s}_{k}^{n}\left(\tilde{y}_{k}\right),1\le k\le K\text{ such that}\nonumber \\
& \mathbb{E}d_{k}\left(S,\hat{S}_{k}\right)\le D_{k},1\le k\le K,\nonumber \\
& \text{and for any pmf }p_{U_{[1:L]}|Y_{[1:K]}},\text{ one can find }p_{\tilde{Y}_{[1:K]}|S^{n}}p_{\tilde{U}_{[1:L]}|\tilde{Y}_{[1:K]}}\text{satisfying}\nonumber \\
& \frac{1}{n}I\left(S^{n};\tilde{Y}_{\mathcal{B}}\tilde{U}_{\mathcal{B}'}|\tilde{Y}_{\mathcal{C}}\tilde{U}_{\mathcal{C}'}\right)\leq I\left(X;Y_{\mathcal{B}}U_{\mathcal{B}'}|Y_{\mathcal{C}}U_{\mathcal{C}'}\right)\textrm{ for any }\mathcal{B},\mathcal{C}\subseteq\left[1:K\right],\mathcal{B}',\mathcal{C}'\subseteq\left[1:L\right]\Bigr\}.
\end{align}
In addition, set $\mathcal{B}=[1:k],\mathcal{C}=[1:k-1]$ for $1\le k\le K$
and $\mathcal{B}'=\mathcal{C}'=\emptyset$. Substitute these into
the outer bound $\mathcal{R}_{2}^{(o)}$ of Theorem \ref{thm:AdmissibleRegion-GBC},
and utilize the degradation of the channel, then we get
\begin{equation}
\frac{1}{n}I\left(S^{n};\tilde{Y}_{[1:k]}|\tilde{Y}_{[1:k-1]}\right)\leq I\left(X;Y_{k}|Y_{k-1}\right).
\end{equation}
Summate both sides of this inequality through $k=1$ to $K$, then
we get
\begin{equation}
\frac{1}{n}I\left(S^{n};\tilde{Y}_{[1:k]}\right)\leq I\left(X;Y_{k}\right).\label{eq:-53}
\end{equation}
In addition,
\begin{align}
\frac{1}{n}I\left(S^{n};\tilde{Y}_{[1:k]}\right) & \geq\frac{1}{n}I\left(S^{n};\hat{S}_{[1:k]}^{n}\right)\\
& =\frac{1}{n}\sum_{t=1}^{n}I\left(S_{t};\hat{S}_{[1:k]}^{n}|S^{t-1}\right)\\
& =\frac{1}{n}\sum_{t=1}^{n}I\left(S_{t};\hat{S}_{[1:k]}^{n}S^{t-1}\right)\\
& \geq\frac{1}{n}\sum_{t=1}^{n}I\left(S_{t};\hat{S}_{[1:k],t}\right)\\
& =I\left(S_{Q};\hat{S}_{[1:k],Q}|Q\right)\\
& =I\left(S_{Q};\hat{S}_{[1:k],Q}Q\right)\\
& \geq I\left(S_{Q};\hat{S}_{[1:k],Q}\right)\\
& =I\left(S;\hat{S}_{[1:k]}\right)\\
& =I\left(S;\hat{S}_{k}^{\prime}\right),\label{eq:-62}
\end{align}
where the time-sharing random variable $Q$ is defined to be uniformly
distributed $\left[1:n\right]$ and independent of all other random
variables, and $S\triangleq S_{Q},\hat{S}_{k}\triangleq\hat{S}_{k,Q},\hat{S}_{k}^{\prime}\triangleq\hat{S}_{[1:k]},1\leq k\leq K$.
Combining \eqref{eq:-53} and \eqref{eq:-62} gives us $I\left(S;\hat{S}_{k}^{\prime}\right)\leq I\left(X;Y_{k}\right)$.
Hence the outer bound $\mathcal{R}^{(o)}$ of Theorem \ref{thm:AdmissibleRegion-DBC}
holds.
\subsubsection*{Proof method 2}
For fixed $p_{U_{K-1}|S}p_{U_{K-2}|U_{K-1}}\cdots p_{U_{1}|U_{2}}$,
we first introduce a set of auxiliary random variables $U_{[1:K-1]}^{n}$
that follow distribution $\prod_{i=1}^{n}p_{U_{K-1}|S}\left(u_{K-1,i}|s_{i}\right)p_{U_{K-2}|U_{K-1}}\left(u_{K-2,i}|u_{K-1,i}\right)\cdots p_{U_{1}|U_{2}}\left(u_{1,i}|u_{2,i}\right)$.
Then $U_{1}^{n}\rightarrow U_{2}^{n}\rightarrow\cdots\rightarrow U_{K-1}^{n}\rightarrow S^{n}\rightarrow X^{n}\rightarrow Y_{K}^{n}\rightarrow Y_{K-1}^{n}\rightarrow\cdots\rightarrow Y_{1}^{n}$
follows a Markov chain. We first derive a lower bound for $I\left(Y_{k}^{n};U_{k}^{n}|U_{k-1}^{n}\right)$.
\begin{align}
& I\left(Y_{k}^{n};U_{k}^{n}|U_{k-1}^{n}\right)\nonumber \\
& =I\left(Y_{[1:k]}^{n};U_{k}^{n}|U_{k-1}^{n}\right)\\
& =\sum_{i=1}^{n}I\left(Y_{[1:k]}^{n};U_{k,i}|U_{k+1}^{n}U_{k}^{i-1}\right)\\
& =\sum_{i=1}^{n}H\left(U_{k,i}|U_{k-1}^{n}U_{k}^{i-1}\right)-H\left(U_{k,i}|U_{k-1}^{n}U_{k}^{i-1}Y_{[1:k]}^{n}\right)\label{eq:-14-3}\\
& =\sum_{i=1}^{n}H\left(U_{k,i}|U_{k-1,i}\right)-H\left(U_{k,i}|U_{k-1}^{n}U_{k}^{i-1}Y_{[1:k]}^{n}\right)\label{eq:-3}\\
& =\sum_{i=1}^{n}I\left(U_{k,i};U_{k-1}^{n}U_{k}^{i-1}Y_{[1:k]}^{n}|U_{k-1,i}\right)\\
& \geq\sum_{i=1}^{n}I\left(U_{k,i};\hat{S}_{[1:k],i}|U_{k-1,i}\right)\\
& =nI\left(U_{k,Q};\hat{S}_{[1:k],Q}|U_{k-1,Q}Q\right)\label{eq:-2}\\
& =nI\left(U_{k,Q};\hat{S}_{[1:k],Q}Q|U_{k-1,Q}\right)\\
& \geq nI\left(U_{k,Q};\hat{S}_{[1:k],Q}|U_{k-1,Q}\right)\\
& =nI\left(U_{k};\hat{S}_{[1:k]}|U_{k-1}\right),\label{eq:-4}
\end{align}
where the time-sharing random variable $Q$ is defined to be uniformly
distributed $\left[1:n\right]$ and independent of all other random
variables, and in \eqref{eq:-4}, $\hat{S}_{k}\triangleq\hat{S}_{k,Q},U_{k}\triangleq U_{k,Q},1\leq k\leq K$.
Next, we turn to upper-bounding $I\left(Y_{k}^{n};U_{k}^{n}|U_{k-1}^{n}\right)$,
and write the following:
\begin{align}
& I\left(Y_{k}^{n};U_{k}^{n}|U_{k-1}^{n}\right)\nonumber \\
& =\sum_{i=1}^{n}I\left(Y_{k,i};U_{k}^{n}|U_{k-1}^{n}Y_{k}^{i-1}\right)\\
& \leq\sum_{i=1}^{n}I\left(Y_{k,i};U_{k}^{n}Y_{k+1}^{i-1}|U_{k-1}^{n}Y_{k}^{i-1}\right)\label{eq:-14-2-2}\\
& =n\sum_{i=1}^{n}I\left(Y_{k,Q};U_{k}^{n}Y_{k+1}^{Q-1}|U_{k-1}^{n}Y_{k}^{Q-1}Q\right)\\
& =nI\left(Y_{k};V'_{k}|V'_{k-1}\right),\label{eq:-1}
\end{align}
where the time-sharing random variable $Q$ is defined above, and
$Y_{K+1}^{Q-1}\triangleq X^{Q-1},V'_{k}\triangleq\left(U_{k}^{n},Y_{k+1}^{Q-1},Q\right),Y_{k}\triangleq Y_{k,Q},1\leq k\leq K$
.
Obviously, $V'_{1}\rightarrow V'_{2}\rightarrow\cdots\rightarrow V'_{K}\rightarrow X\rightarrow Y_{k}$
forms a Markov chain, hence
\begin{equation}
\left(I\left(Y_{k};V'_{k}|V'_{k-1}\right):k\in[1:K]\right)\in\mathcal{B}_{DBC}\left(p_{X}p_{Y_{[1:K]}|X}\right).
\end{equation}
Combining this with \eqref{eq:-4} and \eqref{eq:-1}, we have
\begin{equation}
\left(I\left(U_{k};\hat{S}_{[1:k]}|U_{k-1}\right):k\in[1:K]\right)\in\mathcal{B}_{DBC}\left(p_{X}p_{Y_{[1:K]}|X}\right).\label{eq:-54}
\end{equation}
On the other hand, it can be proved that $nI\left(\hat{S}_{[1:k]};S\right)\leq I\left(\hat{S}_{[1:k]}^{n};S^{n}\right)\leq I\left(Y_{k}^{n};X^{n}\right)\leq nI\left(Y_{k};X\right)$,
i.e., $I\left(S;\hat{S}_{k}^{\prime}\right)\leq I\left(X;Y_{k}\right)$.
Combining it with \eqref{eq:-54} completes the proof.
\section{Proof of Theorem \ref{thm:AdmissibleRegionGGSI}\label{sec:broadcast-GaussianSI}}
For Wyner-Ziv Gaussian broadcast with bandwidth mismatch case (bandwidth
mismatch factor $b$), Theorem \ref{thm:AdmissibleRegionSI-DBC} states
that if $D_{[1:K]}$ is achievable, then there exists some pmf $p_{V_{K}|S}p_{V_{K-1}|V_{K}}\cdots p_{V_{1}|V_{2}}$
and functions $\hat{s}_{k}\left(v_{k},z_{k}\right),1\le k\le K$ such
that
\begin{equation}
\mathbb{E}d\left(S,\hat{S}_{k}\right)\le D_{k},\label{eq:-11-1}
\end{equation}
and for any pmf $p_{U_{K-1}|S}p_{U_{K-2}|U_{K-1}}\cdots p_{U_{1}|U_{2}}$,
\begin{equation}
\frac{1}{b}\left(I\left(V_{k};U_{k}|U_{k-1}Z_{k}\right):k\in[1:K]\right)\in\mathcal{C}_{GBC}\label{eq:-12-1}
\end{equation}
holds, where the capacity of Gaussian broadcast channel $\mathcal{C}_{GBC}$
is given in \eqref{eq:Gaussiancapacity}.
Choose $U_{K-1}=S+E'_{K-1}$ and $U_{k}=U_{k+1}+E'_{k},1\le k\le K-2$,
where $E'_{k}\sim\mathcal{N}\left(0,\tau'_{k}\right)$ is independent
of all the other random variables. Define $E_{k}=\sum_{j=k}^{K-1}E'_{j}\sim\mathcal{N}\left(0,\tau{}_{k}\right)$
with $\tau_{k}=\sum_{j=k}^{K-1}\tau'_{j}$. Then
\begin{align}
I\left(V_{1};U_{1}|Z_{1}\right) & \geq I\left(\hat{S}_{1};U_{1}|Z_{1}\right)\\
& =h\left(U_{1}|Z_{1}\right)-h\left(U_{1}|\hat{S}_{1}Z_{1}\right)\\
& =h\left(U_{1}|Z_{1}\right)-h\left(U_{1}-\hat{S}_{1}|\hat{S}_{1}Z_{1}\right)\\
& \geq h\left(U_{1}|Z_{1}\right)-h\left(U_{1}-\hat{S}_{1}\right)\\
& \geq\frac{1}{2}\log\left(2\pi e\left(\beta_{1}+\tau_{1}\right)\right)-\frac{1}{2}\log\left(2\pi e\left(D_{1}+\tau_{1}\right)\right)\label{eq:-15}\\
& =\frac{1}{2}\log\frac{\beta_{1}+\tau_{1}}{D_{1}+\tau_{1}},\label{eq:-24}
\end{align}
where \eqref{eq:-15} follows from Gaussian distribution maximizes
the differential entropy for a given second moment.
On the other hand,
\begin{align}
I\left(V_{k};U_{k}|U_{k-1}Z_{k}\right) & \geq I\left(\hat{S}_{k};U_{k}|U_{k-1}Z_{k}\right)\\
& =I\left(\hat{S}_{k};U_{k}|Z_{k}\right)-I\left(\hat{S}_{k};U_{k-1}|Z_{k}\right)\\
& =h\left(U_{k}|Z_{k}\right)-h\left(U_{k-1}|Z_{k}\right)+h\left(U_{k-1}|Z_{k}\hat{S}_{k}\right)-h\left(U_{k}|Z_{k}\hat{S}_{k}\right).\label{eq:-18}
\end{align}
The first two terms of \eqref{eq:-18}
\begin{equation}
h\left(U_{k}|Z_{k}\right)-h\left(U_{k-1}|Z_{k}\right)=\frac{1}{2}\log\frac{\beta_{k}+\tau_{k}}{\beta_{k}+\tau_{k-1}}.\label{eq:-21}
\end{equation}
The last two terms of \eqref{eq:-18}
\begin{align}
h\left(U_{k-1}|Z_{k}\hat{S}_{k}\right)-h\left(U_{k}|Z_{k}\hat{S}_{k}\right) & =h\left(U_{k-1}|Z_{k}\hat{S}_{k}\right)-h\left(U_{k}|Z_{k}\hat{S}_{k}E'_{k-1}\right)\\
& =h\left(U_{k-1}|Z_{k}\hat{S}_{k}\right)-h\left(U_{k-1}|Z_{k}\hat{S}_{k}E'_{k-1}\right)\\
& =I\left(U_{k-1};E'_{k-1}|Z_{k}\hat{S}_{k}\right)\\
& =h\left(E'_{k-1}\right)-h\left(E'_{k-1}|Z_{k}\hat{S}_{k}U_{k-1}\right)\\
& =h\left(E'_{k-1}\right)-h\left(E'_{k-1}|Z_{k},\hat{S}_{k},U_{k-1}-\hat{S}_{k}\right)\\
& \geq h\left(E'_{k-1}\right)-h\left(E'_{k-1}|U_{k-1}-\hat{S}_{k}\right)\\
& =I\left(E'_{k-1};S-\hat{S}_{k}+E_{k}+E'_{k-1}\right)\\
& \geq\frac{1}{2}\log\frac{D_{k}+\tau_{k-1}}{D_{k}+\tau_{k}},\label{eq:-20}
\end{align}
where \eqref{eq:-20} is by applying the mutual information game result
that Gaussian noise is the worst additive noise under a variance constraint
\cite[p. 298, Problem 9.21]{Cover91} and taking $E'_{k-1}$ as channel
input.
Combining \eqref{eq:-18}, \eqref{eq:-21} and \eqref{eq:-20}, we
have
\begin{align}
I\left(V_{k};U_{k}|U_{k-1}Z_{k}\right) & \geq\frac{1}{2}\log\frac{\left(\beta_{k}+\tau_{k}\right)\left(D_{k}+\tau_{k-1}\right)}{\left(\beta_{k}+\tau_{k-1}\right)\left(D_{k}+\tau_{k}\right)}.\label{eq:-23}
\end{align}
\eqref{eq:-12-1}, \eqref{eq:-24} and \eqref{eq:-23} imply Theorem
\ref{thm:AdmissibleRegionGGSI} holds.
\section{Proof of Theorem \ref{thm:AdmissibleRegionBBSI}\label{sec:broadcast-BernoulliSI}}
Observe that if there is no information transmitted over the channel,
receiver $k$ could produce a reconstruction within distortion $\beta_{k}$.
Hence we only need consider the case of $D_{[1:K]}$ with
\begin{equation}
D_{k}\leq\beta_{k},1\le k\le K.\label{eq:-55}
\end{equation}
For Wyner-Ziv binary broadcast with bandwidth mismatch case (bandwidth
mismatch factor $b$), Theorem \ref{thm:AdmissibleRegionSI-DBC} states
that if $D_{[1:K]}$ is achievable, then there exists some pmf $p_{V_{K}|S}p_{V_{K-1}|V_{K}}\cdots p_{V_{1}|V_{2}}$
and functions $\hat{s}_{k}\left(v_{k},z_{k}\right),1\le k\le K$ such
that
\begin{equation}
\mathbb{E}d\left(S,\hat{S}_{k}\right)=\mathbb{P}\left(\hat{S}_{k}\oplus S=1\right)\le D_{k},\label{eq:-11-2}
\end{equation}
and for any pmf $p_{U_{K-1}|S}p_{U_{K-2}|U_{K-1}}\cdots p_{U_{1}|U_{2}}$,
\begin{equation}
\frac{1}{b}\left(I\left(V_{k};U_{k}|U_{k-1}Z_{k}\right):k\in[1:K]\right)\in\mathcal{C}_{BBC}\label{eq:-12-2}
\end{equation}
holds, where the capacity of binary broadcast channel $\mathcal{C}_{BBC}$
is given in \eqref{eq:binarycapacity} \cite{Bergmans}.
Define the sets
\begin{equation}
\mathcal{A}_{k}=\left\{ v_{k}:\hat{s}_{k}\left(v_{k},0\right)=\hat{s}_{k}\left(v_{k},1\right)\right\} ,1\le k\le K,
\end{equation}
so that their complements
\begin{equation}
\mathcal{A}_{k}^{c}=\left\{ v_{k}:\hat{s}_{k}\left(v_{k},0\right)\neq\hat{s}_{k}\left(v_{k},1\right)\right\} ,1\le k\le K.
\end{equation}
By hypothesis,
\begin{align}
\mathbb{E}d\left(S,\hat{S}_{k}\right) & =\mathbb{P}(V_{k}\in\mathcal{A}_{k})\mathbb{E}\left[d\left(S,\hat{S}_{k}\right)|V_{k}\in\mathcal{A}_{k}\right]+\mathbb{P}(V_{k}\in\mathcal{A}_{k}^{c})\mathbb{E}\left[d\left(S,\hat{S}_{k}\right)|V_{k}\in\mathcal{A}_{k}^{c}\right]\\
& \le D_{k}.\label{eq:-34}
\end{align}
We first show that
\begin{equation}
\mathbb{E}\left[d\left(S,\hat{S}_{k}\right)|V_{k}\in\mathcal{A}_{k}^{c}\right]\geq\beta_{k}.\label{eq:-26}
\end{equation}
To do this, we write
\begin{equation}
\mathbb{E}\left[d\left(S,\hat{S}_{k}\right)|V_{k}\in\mathcal{A}_{k}^{c}\right]=\sum_{v_{k}\in\mathcal{A}_{k}^{c}}\frac{\mathbb{P}(V_{k}=v_{k})}{\mathbb{P}(V_{k}\in\mathcal{A}_{k}^{c})}\mathbb{E}\left[d\left(S,\hat{S}_{k}\right)|V_{k}=v_{k}\right].
\end{equation}
If $v_{k}\in\mathcal{A}_{k}^{c}$ and $\hat{s}_{k}\left(v_{k},0\right)=0$
then $\hat{s}_{k}\left(v_{k},1\right)=1$. Therefore, for such $v_{k}$,
\begin{align}
\mathbb{E}\left[d\left(S,\hat{S}_{k}\right)|V_{k}=v_{k}\right] & =\mathbb{P}\left(Z_{k}=0,S=1|V_{k}=v_{k}\right)+\mathbb{P}\left(Z_{k}=1,S=0|V_{k}=v_{k}\right)\\
& =\mathbb{P}\left(S=1|V_{k}=v_{k}\right)\mathbb{P}\left(Z_{k}=0|S=1\right)+\mathbb{P}\left(S=0|V_{k}=v_{k}\right)\mathbb{P}\left(Z_{k}=1|S=0\right)\label{eq:-25}\\
& =\beta_{k},\label{eq:-27}
\end{align}
where \eqref{eq:-25} follows that $Z_{k}\rightarrow S\rightarrow V_{k}$
forms a Markov chain. If $v_{k}\in\mathcal{A}_{k}^{c}$ but $\hat{s}_{k}\left(v_{k},0\right)=1$,
then for such $v_{k}$,
\begin{align}
\mathbb{E}\left[d\left(S,\hat{S}_{k}\right)|V_{k}=v_{k}\right] & =1-\beta_{k}\geq\beta_{k},\label{eq:-33}
\end{align}
since $\beta_{k}\leq\frac{1}{2}$. Therefore, \eqref{eq:-26} follows
from \eqref{eq:-27} and \eqref{eq:-33}.
Now we write
\begin{equation}
\mathbb{E}\left[d\left(S,\hat{S}_{k}\right)|V_{k}\in\mathcal{A}_{k}\right]=\sum_{v_{k}\in\mathcal{A}_{k}}\frac{\mathbb{P}(V_{k}=v_{k})}{\mathbb{P}(V_{k}\in\mathcal{A}_{k})}\mathbb{E}\left[d\left(S,\hat{S}_{k}\right)|V_{k}=v_{k}\right],
\end{equation}
and define $g_{k}\left(v_{k}\right)\triangleq\hat{s}_{k}\left(v_{k},0\right),\lambda_{v_{k}}\triangleq\frac{\mathbb{P}(V_{k}=v_{k})}{\mathbb{P}(V_{k}\in\mathcal{A}_{k})},\mu_{k}\triangleq\mathbb{P}(V_{k}\in\mathcal{A}_{k})$,
\begin{equation}
d_{v_{k}}\triangleq\mathbb{E}\left[d\left(S,\hat{S}_{k}\right)|V_{k}=v_{k}\right]=\mathbb{P}\left(S\neq g_{k}\left(v_{k}\right)|V_{k}=v_{k}\right),
\end{equation}
then utilizing \eqref{eq:-34} and \eqref{eq:-26}, we have
\begin{equation}
d'_{k}\triangleq\mu_{k}\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}d_{v_{k}}+\left(1-\mu_{k}\right)\beta_{k}\le D_{k}.\label{eq:-42}
\end{equation}
Next we will show
\begin{align}
I\left(V_{k};U_{k}|Z_{k}\right) & \geq\frac{\beta_{k}-D_{k}}{\beta_{k}-\alpha_{k}}\left(H_{2}\left(\beta_{k}\star\tau_{k}\right)-\left(H_{4}\left(\alpha_{k},\beta_{k},\tau_{k}\right)-H_{2}\left(\alpha_{k}\star\beta_{k}\right)\right)\right).\label{eq:-6}
\end{align}
Choose $U_{K-1}=S\oplus E'_{K-1}$ and $U_{k}=U_{k+1}\oplus E'_{k},1\le k\le K-2$,
where $E'_{k}\sim\textrm{Bern}\left(\tau'_{k}\right)$ is independent
of all the other random variables. Define $E_{k}=E'_{K-1}\oplus E'_{K-2}\oplus\cdots\oplus E'_{k}\sim\textrm{Bern}\left(\tau_{k}\right)$
with $\tau_{k}=\tau'_{K-1}\star\tau'_{K-2}\star\cdots\star\tau'_{k}$.
Then
\begin{align}
I\left(V_{k};U_{k}|Z_{k}\right)= & H\left(U_{k}|Z_{k}\right)-H\left(U_{k}|V_{k},Z_{k}\right)\\
= & H_{2}\left(\beta_{k}\star\tau_{k}\right)-\mu_{k}\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}H\left(U_{k}|Z_{k},V_{k}=v_{k}\right)\nonumber \\
& -\left(1-\mu_{k}\right)\sum_{v_{k}\in\mathcal{A}_{k}^{c}}\frac{\mathbb{P}(V_{k}=v_{k})}{\mathbb{P}(V_{k}\in\mathcal{A}_{k}^{c})}H\left(U_{k}|Z_{k},V_{k}=v_{k}\right).\label{eq:-36}
\end{align}
For fixed $v_{k}$, define a set of random variables $\left(V'_{k},S',U'_{k},Z'_{k}\right)\sim1\left\{ v'_{k}=v_{k}\right\} p_{SU_{k}Z_{k}|V_{k}}\left(s',u'_{k},z'_{k}|v'_{k}\right)$,
then $H\left(U'_{k}Z'_{k}|V'_{k}\right)=H\left(U_{k}Z_{k}|V_{k}=v_{k}\right)$
and $H\left(Z'_{k}|V'_{k}\right)=H\left(Z_{k}|V_{k}=v_{k}\right)$.
Since $p_{SU_{k}Z_{k}|V_{k}}$ satisfies
\begin{equation}
p_{SU_{k}Z_{k}|V_{k}}\left(s',u'_{k},z'_{k}|v'_{k}\right)=p_{S|V_{k}}\left(s'|v'_{k}\right)p_{Z_{k}|S}\left(z'_{k}|s'\right)p_{U_{k}|S}\left(u'_{k}|s'\right),
\end{equation}
it holds that $Z'_{k}=S'\oplus B{}_{k},U'_{k}=S'\oplus E{}_{k}$.
Hence $Z'_{k}\oplus U'_{k}=B{}_{k}\oplus E{}_{k}$.
For fixed $v_{k}$, consider
\begin{align}
H\left(U_{k}|Z_{k},V_{k}=v_{k}\right) & =H\left(U_{k}Z_{k}|V_{k}=v_{k}\right)-H\left(Z_{k}|V_{k}=v_{k}\right)\\
& =H\left(U'_{k}Z'_{k}|V'_{k}\right)-H\left(Z'_{k}|V'_{k}\right)\\
& =H\left(U'_{k}|Z'_{k}V'_{k}\right)\\
& =H\left(U'_{k}\oplus Z'_{k}|Z'_{k}V'_{k}\right)\\
& =H\left(B_{k}\oplus E{}_{k}|Z'_{k}V'_{k}\right)\\
& \leq H\left(B_{k}\oplus E{}_{k}\right)\\
& =H_{2}\left(\beta_{k}\star\tau_{k}\right).\label{eq:-37}
\end{align}
Combine \eqref{eq:-36} and \eqref{eq:-37}, then it holds that
\begin{align}
I\left(V_{k};U_{k}|Z_{k}\right) & \geq H_{2}\left(\beta_{k}\star\tau_{k}\right)-\mu_{k}\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}H\left(U_{k}|Z_{k},V_{k}=v_{k}\right)-\left(1-\mu_{k}\right)H_{2}\left(\beta_{k}\star\tau_{k}\right)\\
& =\mu_{k}H_{2}\left(\beta_{k}\star\tau_{k}\right)-\mu_{k}\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}H\left(U_{k}|Z_{k},V_{k}=v_{k}\right).\label{eq:-38}
\end{align}
Now we consider the second term of \eqref{eq:-38}.
\begin{align}
\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}H\left(U_{k}|Z_{k},V_{k}=v_{k}\right) & =\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}\left(H\left(U_{k}Z_{k}|V_{k}=v_{k}\right)-H\left(Z_{k}|V_{k}=v_{k}\right)\right)\\
& =\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}\left(H_{4}\left(d_{v_{k}},\beta_{k},\tau_{k}\right)-H_{2}\left(d_{v_{k}}\star\beta_{k}\right)\right)\label{eq:-39}\\
& =\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}G_{1}\left(d_{v_{k}},\beta_{k},\tau_{k}\right),\label{eq:-41}
\end{align}
where the function $H_{4}\left(x,y,z\right)$ is defined in \eqref{eq:H4}
and
\begin{equation}
G_{1}\left(x,y,z\right)\triangleq H_{4}\left(x,y,z\right)-H_{2}\left(x\star y\right).
\end{equation}
Equality \eqref{eq:-39} follows from calculating the entropies according
to the definition.
Now we show that $G_{1}\left(x,y,z\right)$ is concave in $x$. To
do this, we consider
\begin{align}
& \frac{\partial^{2}}{\partial x^{2}}G_{1}\left(x,y,z\right)\nonumber \\
& =-\frac{\left(yz-\overline{y}\overline{z}\right)^{2}}{xyz+\overline{x}\overline{y}\overline{z}}-\frac{\left(\overline{y}z-y\overline{z}\right)^{2}}{x\overline{y}z+\overline{x}y\overline{z}}-\frac{\left(y\overline{z}-\overline{y}z\right)^{2}}{xy\overline{z}+\overline{x}\overline{y}z}-\frac{\left(\overline{y}\overline{z}-yz\right)^{2}}{x\overline{y}\overline{z}+\overline{x}yz}+\frac{\left(y-\overline{y}\right)^{2}}{x\overline{y}+\overline{x}y}+\frac{\left(y-\overline{y}\right)^{2}}{xy+\overline{x}\overline{y}}\\
& =-\left(\frac{\left(yz-\overline{y}\overline{z}\right)^{2}}{xyz+\overline{x}\overline{y}\overline{z}}+\frac{\left(y\overline{z}-\overline{y}z\right)^{2}}{xy\overline{z}+\overline{x}\overline{y}z}-\frac{\left(y-\overline{y}\right)^{2}}{xy+\overline{x}\overline{y}}\right)-\left(\frac{\left(\overline{y}z-y\overline{z}\right)^{2}}{x\overline{y}z+\overline{x}y\overline{z}}+\frac{\left(\overline{y}\overline{z}-yz\right)^{2}}{x\overline{y}\overline{z}+\overline{x}yz}-\frac{\left(y-\overline{y}\right)^{2}}{x\overline{y}+\overline{x}y}\right)\\
& \leq0,\label{eq:-40}
\end{align}
where \eqref{eq:-40} follows from the following inequality
\begin{align}
\frac{a_{1}^{2}}{b_{1}}+\frac{a_{2}^{2}}{b_{2}} & =\frac{1}{b_{1}+b_{2}}\left(b_{1}+b_{2}\right)\left(\frac{a_{1}^{2}}{b_{1}}+\frac{a_{2}^{2}}{b_{2}}\right)\\
& =\frac{1}{b_{1}+b_{2}}\left(a_{1}^{2}+a_{2}^{2}+\frac{b_{2}a_{1}^{2}}{b_{1}}+\frac{b_{1}a_{2}^{2}}{b_{2}}\right)\\
& \geq\frac{1}{b_{1}+b_{2}}\left(a_{1}^{2}+a_{2}^{2}+2a_{1}a_{2}\right)\\
& =\frac{\left(a_{1}+a_{2}\right)^{2}}{b_{1}+b_{2}},
\end{align}
for $b_{1},b_{2}>0$ and arbitrary real numbers $a_{1},a_{2}$. \eqref{eq:-40}
implies $G_{1}\left(x,y,z\right)$ is concave in $x$.
Then combining the concavity of $G_{1}\left(x,y,z\right)$ with \eqref{eq:-38}
and \eqref{eq:-41}, we have
\begin{align}
I\left(V_{k};U_{k}|Z_{k}\right) & \geq\mu_{k}\left(H_{2}\left(\beta_{k}\star\tau_{k}\right)-G_{1}\left(\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}d_{v_{k}},\beta_{k},\tau_{k}\right)\right)\\
& =\mu_{k}\left(H_{2}\left(\beta_{k}\star\tau_{k}\right)-G_{1}\left(\alpha_{k},\beta_{k},\tau_{k}\right)\right)
\end{align}
where
\begin{equation}
\alpha_{k}\triangleq\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}d_{v_{k}}.\label{eq:alphak}
\end{equation}
From \eqref{eq:-42}, $\alpha_{k}$ satisfies
\begin{equation}
\mu_{k}\alpha_{k}+\left(1-\mu_{k}\right)\beta_{k}\le D_{k}.\label{eq:-43}
\end{equation}
Combine \eqref{eq:-43} with $D_{k}\leq\beta_{k}$ (i.e., \eqref{eq:-55}),
then we have
\begin{equation}
0\leq\alpha_{k}\le D_{k}\leq\beta_{k}.\label{eq:-51}
\end{equation}
Therefore,
\begin{align}
I\left(V_{k};U_{k}|Z_{k}\right) & \geq\frac{\beta_{k}-D_{k}}{\beta_{k}-\alpha_{k}}\left(H_{2}\left(\beta_{k}\star\tau_{k}\right)-G_{1}\left(\alpha_{k},\beta_{k},\tau_{k}\right)\right)\\
& =\frac{\beta_{k}-D_{k}}{\beta_{k}-\alpha_{k}}\left(H_{2}\left(\beta_{k}\star\tau_{k}\right)-\left(H_{4}\left(\alpha_{k},\beta_{k},\tau_{k}\right)-H_{2}\left(\alpha_{k}\star\beta_{k}\right)\right)\right),\label{eq:-53-1}
\end{align}
i.e., \eqref{eq:-6} holds.
Next we will show
\begin{align}
I\left(V_{k};U_{k}|U_{k-1}Z_{k}\right) & \geq\frac{\beta_{k}-D_{k}}{\beta_{k}-\alpha_{k}}\left(H_{2}\left(\beta_{k}\star\tau_{k}\right)-H_{2}\left(\beta_{k}\star\tau_{k-1}\right)-\left(H_{4}\left(\alpha_{k},\beta_{k},\tau_{k}\right)-H_{4}\left(\alpha_{k},\beta_{k},\tau_{k-1}\right)\right)\right).\label{eq:secinequity}
\end{align}
Consider
\begin{align}
I\left(V_{k};U_{k}|U_{k-1}Z_{k}\right) & =H\left(U_{k}|U_{k-1}Z_{k}\right)-H\left(U_{k}|U_{k-1}Z_{k}V_{k}\right)\\
& =H\left(U_{k-1}|U_{k}\right)+H\left(U_{k}|Z_{k}\right)-H\left(U_{k-1}|Z_{k}\right)-H\left(U_{k}|U_{k-1}Z_{k}V_{k}\right)\\
& =H_{2}\left(\tau'_{k-1}\right)+H_{2}\left(\beta_{k}\star\tau_{k}\right)-H_{2}\left(\beta_{k}\star\tau_{k-1}\right)-H\left(U_{k}|U_{k-1}Z_{k}V_{k}\right).\label{eq:-46}
\end{align}
Write the last term as
\begin{align}
& H\left(U_{k}|U_{k-1}Z_{k}V_{k}\right)\nonumber \\
& =-\left(1-\mu_{k}\right)\sum_{v_{k}\in\mathcal{A}_{k}^{c}}\frac{\mathbb{P}(V_{k}=v_{k})}{\mathbb{P}(V_{k}\in\mathcal{A}_{k}^{c})}H\left(U_{k}|U_{k-1},Z_{k},V_{k}=v_{k}\right)-\mu_{k}\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}H\left(U_{k}|U_{k-1},Z_{k},V_{k}=v_{k}\right).\label{eq:-47}
\end{align}
For fixed $v_{k}$, define $\left(V'_{k},S',U'_{k},U'_{k-1},Z'_{k}\right)\sim1\left\{ v'_{k}=v_{k}\right\} p_{SU_{k}U_{k-1}Z_{k}|V_{k}}\left(s',u'_{k},u'_{k-1},z'_{k}|v'_{k}\right)$.
Since
\begin{equation}
p_{SU_{k}U_{k-1}Z_{k}|V_{k}}\left(s',u'_{k},u'_{k-1},z'_{k}|v'_{k}\right)=p_{S|V_{k}}\left(s'|v'_{k}\right)p_{Z_{k}|S}\left(z'_{k}|s'\right)p_{U_{k}|S}\left(u'_{k}|s'\right)p_{U_{k-1}|U_{k}}\left(u'_{k-1}|u'_{k}\right),
\end{equation}
we have $Z'_{k}=S'\oplus B{}_{k},U'_{k}=S'\oplus E{}_{k},U'_{k-1}=U'_{k}\oplus E'{}_{k-1}$.
Hence $Z'_{k}\oplus U'_{k}=B{}_{k}\oplus E{}_{k},Z'_{k}\oplus U'_{k-1}=B{}_{k}\oplus E{}_{k-1}$.
Similar to the derivation for $H\left(U_{k}|U_{k-1},V_{k}=v_{k}\right)$,
we can write
\begin{align}
H\left(U_{k}|U_{k-1},Z_{k},V_{k}=v_{k}\right) & =H\left(U'_{k}|U'_{k-1}Z'_{k}V'_{k}\right)\\
& =H\left(U'_{k}\oplus Z'_{k}|U'_{k-1}\oplus Z'_{k},Z'_{k},V'_{k}\right)\\
& \leq H\left(U'_{k}\oplus Z'_{k}|U'_{k-1}\oplus Z'_{k}\right)\\
& =H\left(B_{k}\oplus E{}_{k}|B{}_{k}\oplus E{}_{k-1}\right)\\
& =H\left(B_{k}\oplus E{}_{k}\right)+H\left(B_{k}\oplus E{}_{k-1}|B{}_{k}\oplus E{}_{k}\right)-H\left(B_{k}\oplus E{}_{k-1}\right)\\
& =H_{2}\left(\tau'_{k-1}\right)+H_{2}\left(\beta_{k}\star\tau_{k}\right)-H_{2}\left(\beta_{k}\star\tau_{k-1}\right).\label{eq:-45}
\end{align}
Combine \eqref{eq:-46}, \eqref{eq:-47} and \eqref{eq:-45}, then
we have
\begin{align}
I\left(V_{k};U_{k}|U_{k-1}Z_{k}\right) & \geq\mu_{k}\left(H_{2}\left(\tau'_{k-1}\right)+H_{2}\left(\beta_{k}\star\tau_{k}\right)-H_{2}\left(\beta_{k}\star\tau_{k-1}\right)\right)-\mu_{k}\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}H\left(U_{k}|U_{k-1},Z_{k},V_{k}=v_{k}\right).\label{eq:-48}
\end{align}
Consider the last term of \eqref{eq:-48},
\begin{align}
& \sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}H\left(U_{k}|U_{k-1},Z_{k},V_{k}=v_{k}\right)\nonumber \\
& =\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}\left(H\left(U_{k}|Z_{k},V_{k}=v_{k}\right)+H\left(U_{k-1}|U_{k},Z_{k},V_{k}=v_{k}\right)-H\left(U_{k-1}|Z_{k},V_{k}=v_{k}\right)\right)\\
& =\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}\left(H\left(U_{k},Z_{k}|V_{k}=v_{k}\right)+H_{2}\left(\tau'_{k-1}\right)-H\left(U_{k-1},Z_{k}|V_{k}=v_{k}\right)\right)\\
& =H_{2}\left(\tau'_{k-1}\right)+\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}\left(H_{4}\left(d_{v_{k}},\beta_{k},\tau_{k}\right)-H_{4}\left(d_{v_{k}},\beta_{k},\tau_{k-1}\right)\right)\label{eq:-49}\\
& =H_{2}\left(\tau'_{k-1}\right)+\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}G_{2}\left(d_{v_{k}},\beta_{k},\tau_{k},\tau_{k-1}\right),\label{eq:-50}
\end{align}
where \eqref{eq:-49} is by directly calculating the entropies according
to the definition, and
\begin{equation}
G_{2}\left(x,y,z,t\right)\triangleq H_{4}\left(x,y,z\right)-H_{4}\left(x,y,t\right).
\end{equation}
Note that function $G_{1}\left(x,y,z\right)$ is a special case of
function $G_{2}\left(x,y,z,t\right)$ given $t=\frac{1}{2}$, i.e.,
\begin{equation}
G_{1}\left(x,y,z\right)=G_{2}\left(x,y,z,\frac{1}{2}\right).
\end{equation}
Now we show that $G_{2}\left(x,y,z,t\right)$ is concave in $x$ when
$0\leq z\leq t\leq\frac{1}{2}$, which generalizes the concavity of
$G_{1}\left(x,y,z\right)$. To do this, we consider
\begin{align}
& \frac{\partial^{2}}{\partial x^{2}}G_{2}\left(x,y,z,t\right)\nonumber \\
& =-\frac{\left(yz-\overline{y}\overline{z}\right)^{2}}{xyz+\overline{x}\overline{y}\overline{z}}-\frac{\left(\overline{y}z-y\overline{z}\right)^{2}}{x\overline{y}z+\overline{x}y\overline{z}}-\frac{\left(y\overline{z}-\overline{y}z\right)^{2}}{xy\overline{z}+\overline{x}\overline{y}z}-\frac{\left(\overline{y}\overline{z}-yz\right)^{2}}{x\overline{y}\overline{z}+\overline{x}yz}+\frac{\left(yt-\overline{y}\overline{t}\right)^{2}}{xyt+\overline{x}\overline{y}\overline{t}}+\frac{\left(\overline{y}t-y\overline{t}\right)^{2}}{x\overline{y}t+\overline{x}y\overline{t}}+\frac{\left(y\overline{t}-\overline{y}t\right)^{2}}{xy\overline{t}+\overline{x}\overline{y}t}+\frac{\left(\overline{y}\overline{t}-yt\right)^{2}}{x\overline{y}\overline{t}+\overline{x}yt},
\end{align}
and
\begin{align}
\frac{\partial}{\partial t}\left(\frac{\partial^{2}}{\partial x^{2}}G_{2}\left(x,y,z,t\right)\right) & =\frac{\partial}{\partial t}\left(\frac{\left(yt-\overline{y}\overline{t}\right)^{2}}{xyt+\overline{x}\overline{y}\overline{t}}+\frac{\left(y\overline{t}-\overline{y}t\right)^{2}}{xy\overline{t}+\overline{x}\overline{y}t}\right)+\frac{\partial}{\partial t}\left(\frac{\left(\overline{y}t-y\overline{t}\right)^{2}}{x\overline{y}t+\overline{x}y\overline{t}}+\frac{\left(\overline{y}\overline{t}-yt\right)^{2}}{x\overline{y}\overline{t}+\overline{x}yt}\right)\\
& =\frac{-y^{2}\cdot\overline{y}^{2}\cdot\left(xy+\overline{x}\overline{y}\right)\cdot\left(1-2t\right)}{\left(xyt+\overline{x}\overline{y}\overline{t}\right)^{2}\left(xy\overline{t}+\overline{x}\overline{y}t\right)^{2}}+\frac{-y^{2}\cdot\overline{y}^{2}\cdot\left(x\overline{y}+\overline{x}y\right)\cdot\left(1-2t\right)}{\left(x\overline{y}t+\overline{x}y\overline{t}\right)^{2}\left(x\overline{y}\overline{t}+\overline{x}yt\right)^{2}}.
\end{align}
Hence for $0\leq t\leq\frac{1}{2}$,
\begin{align}
\frac{\partial}{\partial t}\left(\frac{\partial^{2}}{\partial x^{2}}G_{2}\left(x,y,z,t\right)\right) & \leq0,
\end{align}
i.e., $\frac{\partial^{2}}{\partial x^{2}}G_{2}\left(x,y,z,t\right)$
is decreasing in $t$. Then we have for $0\leq z\leq t\leq\frac{1}{2}$,
\begin{equation}
\frac{\partial^{2}}{\partial x^{2}}G_{2}\left(x,y,z,t\right)\leq\frac{\partial^{2}}{\partial x^{2}}G_{2}\left(x,y,z,z\right)=0.
\end{equation}
It implies $G_{2}\left(x,y,z,t\right)$ is concave in $x$ when $0\leq z\leq t\leq\frac{1}{2}$.
Combining \eqref{eq:-48} and \eqref{eq:-50}, and utilizing the concavity
of $G_{2}\left(x,y,z,t\right)$, we have
\begin{align}
I\left(V_{k};U_{k}|U_{k-1}Z_{k}\right) & \geq\mu_{k}\left(H_{2}\left(\beta_{k}\star\tau_{k}\right)-H_{2}\left(\beta_{k}\star\tau_{k-1}\right)-G_{2}\left(\sum_{v_{k}\in\mathcal{A}_{k}}\lambda_{v_{k}}d_{v_{k}},\beta_{k},\tau_{k},\tau_{k-1}\right)\right)\\
& =\mu_{k}\left(H_{2}\left(\beta_{k}\star\tau_{k}\right)-H_{2}\left(\beta_{k}\star\tau_{k-1}\right)-G_{2}\left(\alpha_{k},\beta_{k},\tau_{k},\tau_{k-1}\right)\right)
\end{align}
where $\alpha_{k}$ is given by \eqref{eq:alphak} and satisfies \eqref{eq:-43}
and \eqref{eq:-51}. Therefore,
\begin{align}
I\left(V_{k};U_{k}|U_{k-1}Z_{k}\right) & \geq\frac{\beta_{k}-D_{k}}{\beta_{k}-\alpha_{k}}\left(H_{2}\left(\beta_{k}\star\tau_{k}\right)-H_{2}\left(\beta_{k}\star\tau_{k-1}\right)-G_{2}\left(\alpha_{k},\beta_{k},\tau_{k},\tau_{k-1}\right)\right)\\
& =\frac{\beta_{k}-D_{k}}{\beta_{k}-\alpha_{k}}\left(H_{2}\left(\beta_{k}\star\tau_{k}\right)-H_{2}\left(\beta_{k}\star\tau_{k-1}\right)-\left(H_{4}\left(\alpha_{k},\beta_{k},\tau_{k}\right)-H_{4}\left(\alpha_{k},\beta_{k},\tau_{k-1}\right)\right)\right),\label{eq:-52}
\end{align}
i.e., \eqref{eq:secinequity} holds.
Combining \eqref{eq:-12-2}, \eqref{eq:-6} and \eqref{eq:secinequity}
gives Theorem \ref{thm:AdmissibleRegionBBSI}.
| 197,252
|
\begin{document}
\maketitle
\renewcommand{\thefootnote}{}
\footnotetext[1]{This work was supported in part by the DARPA InPho
program
under Contract No. HR0011-10-C-0159, and by AFOSR under Grant
No.~FA9550-11-1-0183. Y. Kochman was also supported in part by Israel Science Foundation under Grant No.~956/12.
Y. Kochman is with the School of Computer Science and Engineering, Hebrew
University of Jerusalem, Israel; he was with the Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, MA,
USA (e-mail: yuvalko@cs.huji.ac.il).
L. Wang and G.W. Wornell are with the Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, MA,
USA (e-mail: wlg@mit.edu; gww@mit.edu).}
\renewcommand{\thefootnote}{\arabic{footnote}}
\begin{abstract}
This work considers the distribution of a secret key over an optical (bosonic) channel in the regime of high photon efficiency, i.e., when the number of secret key bits generated per detected photon is high. While in principle the photon efficiency is unbounded, there is an inherent tradeoff between this efficiency and the key generation rate (with respect to the channel bandwidth).
We derive asymptotic expressions for the optimal generation rates in the photon-efficient limit, and propose schemes that approach these limits up to certain approximations. The schemes are practical, in the sense that they use coherent or temporally-entangled optical states and direct photodetection, all of which are reasonably easy to realize in practice, in conjunction with off-the-shelf classical codes.
\end{abstract}
\begin{IEEEkeywords}
Information-theoretic security, key distribution, optical communication, wiretap channel.
\end{IEEEkeywords}
\section{Introduction}
\IEEEPARstart{I}{nformation-theoretic} key distribution \cite{ahlswedecsiszar93,maurer93} involves the generation of a sequence between the participating terminals, such that the mutual information between this sequence and any data obtained by other terminals is close to zero in an appropriate sense. Unlike secure communication through the wiretap channel \cite{wyner75}, the sequence need not be known \emph{a priori} to any of the terminals. Like the latter, however, the information-theoretic approach to key distribution hinges on knowledge of the channel through which an adversarial terminal listens to the communication, as opposed to computational approaches where the assumption is the inability of the adversary to perform certain computations in reasonable time. The computational hardness assumption may no longer be valid when future technology, e.g., quantum computers, becomes available, causing the computational approaches to fail. But the information-theoretic approach also has its drawback: the information obtained by the legitimate terminals cannot prove or disprove the channel assumption on which the key-distribution protocol is based, inhibiting security in a realistic setting.
The situation is much different when a quantum channel is employed \cite{bennettbrassard84,ekert91}. Loosely speaking, the ``no-cloning'' theorem \cite{wootterszurek82} guarantees that information ``stolen'' by an eavesdropper will not reach the legitimate terminal, thus the situation where the adversary is stronger than initially assumed can be detected. In fact, even eavesdroppers that can actively transmit into the quantum channel can be detected, at the cost of key-rate loss, using measurements based on local randomness. We shall come back to these issues in the discussion at the end of the paper. For the main part of the paper, we rely on the existence of good detection methods to assume that the eavesdropper is passive, and that the complete statistical characterization of the eavesdropper's channel is known to the legitimate terminals.
Two-terminal quantum key distribution (QKD) protocols can be roughly divided into two classes. In ``prepare and measure'' protocols, one legitimate terminal (Alice) prepares quantum states that are sent via a quantum channel to the other terminal (Bob) and to the eavesdropper (Eve). In contrast, in entanglement-based protocols, a quantum source emits entangled states, which are observed by all terminals via quantum channels. These two classes are parallel to the ``C'' (channel) and ``S'' (source) models of \cite{ahlswedecsiszar93}; in this work we shall use the C/S notation. In either approach, the quantum stage is followed by the use of a classical communication channel. This channel is assumed to be public, i.e., all information sent is received by Eve; however, it is assumed that Eve cannot transmit into this public channel. The performance of a QKD scheme is measured in terms of the size of the secret key normalized by the quantum-channel resources used. The classical channel is thus ``free'', although its use is limited by the assumption that Eve has full access to this channel.
A quantum channel most often encountered in practice is the optical channel, which is modeled in quantum mechanics as a bosonic channel. When used for communicating classical data at low average input power, it is asymptotically optimal to use a direct-detection receiver, which ignores the phase of the optical signal. This results in an equivalent classical channel where the output has a Poisson distribution whose mean is proportional to the channel's input \cite{shapiro09}. Some of the first important works on this channel model are in \cite{bardavid69,kabanov78,davis80}.
The low-input-power regime can be thought of as a ``photon-efficient regime''. This is because, in the limit of low average photon number per channel use, the communication rate per photon is unbounded.
In this work we consider QKD over the bosonic channel in the photon-efficient regime. We consider both C and S models, and show that in both, as happens in communication, the photon efficiency is unbounded and direct-detection receivers are asymptotically optimal. We further consider specific QKD protocols. We discuss the difficulty of finding code constructions that allow us to approach the theoretical performance limits, since in the photon-efficient regime they have to operate over highly-skewed sequences. We present protocols that overcome this difficulty: in the C model we use pulse-position modulation (PPM), while in the S model we parse the sequence of detections into frames. In both cases, coding over frames is an easier task than coding directly over the detection sequence.
The rest of the paper is organized as follows. We introduce our notation in Section~\ref{sec:notation}. In Section~\ref{sec:setting} we formally describe the problem setting. Then in Section~\ref{sec:communication} we discuss, as a point of reference, photon-efficient communication. Sections~\ref{sec:channel_case} and \ref{sec:source_case} include our main results for key distribution, regarding the C and S models, respectively. We conclude this paper in Section~\ref{sec:discussion} by discussing the gap between our results and fully quantum security proofs.
\section{Notation}\label{sec:notation}
We use a font like $\mathbb{A}$ to denote a Hilbert space. Throughout this paper we shall focus on bosonic Hilbert spaces. We adopt Dirac's notation to use $|\psi\ket$ to denote a unit vector in a Hilbert space, which can describe a pure quantum state, and use $\bra \psi|$ to denote the conjugate of $|\psi\ket$. We follow most of the physics literature to slightly abuse our notation: we shall not make typographical distinctions between number states and coherent states. Hence $|n\ket$, $n\in\Integers_0^+$, (usually) denotes the number state that contains $n$ photons; while $|\alpha\ket$, $\alpha\in\Complex$, (almost everywhere) denotes a coherent state, whose exact characterization is given later. This abuse of notation will not cause confusion within the scope of this paper. We use a Greek letter like $\rho$ to denote a density operator (i.e., a trace-one semidefinite operator) on a Hilbert space, which can describe a pure or mixed quantum state. Note that the density-operator description of a pure state $|\psi\ket$ is $|\psi\ket\bra\psi|$. When considering a system such as a beamsplitter, we reserve the letters $|\psi\ket$ and $\rho$ for input states, and $|\phi\ket$ and $\sigma$ for output states. Sometimes, to be explicit, we add a superscript to a state to indicate its Hilbert space so it looks like $|\psi\ket^\mathbb{A}$ or $\sigma^\mathbb{B}$. We use the notation $\hat{a}$ to denote the annihilation operator on $\mathbb{A}$ (so $\hat{a}^\dag$ is the creation operator on $\mathbb{A}$); similarly, $\hat{b}$ denotes the annihilation operator on $\mathbb{B}$, etc.
For a quantum state $\sigma^\mathbb{AB}$ on the Hilbert spaces $\mathbb{A}$ and $\mathbb{B}$, we use $H(\sigma^\mathbb{A})$, $H(\sigma^\mathbb{A}|\sigma^\mathbb{B})$, and $I(\sigma^\mathbb{A};\sigma^\mathbb{B})$ to denote the corresponding entropy, conditional entropy, and mutual information, respectively. These quantities are defined as follows (see \cite{nielsenchuang00} for more details):
\begin{IEEEeqnarray}{rCl}
H(\sigma^\mathbb{A}) & \triangleq & - \textnormal{tr} \left\{ \sigma^\mathbb{A} \log \sigma^\mathbb{A} \right\}\\
H(\sigma^\mathbb{A}| \sigma^\mathbb{B}) & \triangleq & H(\sigma^{\mathbb{AB}} ) - H(\sigma^\mathbb{B})\\
I(\sigma^\mathbb{A};\sigma^\mathbb{B}) & \triangleq & H(\sigma^\mathbb{A})+ H(\sigma^\mathbb{B})- H(\sigma^{\mathbb{AB}}).
\end{IEEEeqnarray}
For classical or mixed classical-quantum states, we simply replace the density operator by the classical random variable for the classical part in these expressions, so they look like, e.g., $H(X)$, $H(X|\sigma^\mathbb{B})$, and $I(\sigma^\mathbb{A}; Y)$. Sometimes, to be more precise, we also write the mutual information as $I(\mathbb{A};\mathbb{B})\vert_\sigma$, indicating that it is the mutual information between space $\mathbb{A}$ and $\mathbb{B}$ evaluated for the joint state $\sigma$.
Throughout this paper, we use natural logarithms, and measure information in nats, though sometimes we do talk about ``bits'' and ``binary representation''.
We use the usual notation $O(\cdot)$ and $o(\cdot)$ to describe behaviors of functions of $\ave$ in the limit where $\ave$ approaches zero with other variables, if any, held fixed. Specifically, given a reference function $f(\cdot)$ (which might be the constant $1$), a function denoted as $O(f(\ave))$ satisfies
\begin{equation}
\varlimsup_{\ave\downarrow 0} \left|\frac{O(f(\ave))}{f(\ave)} \right|< \infty,
\end{equation}
while a function denoted as $o(f(\ave))$ satisfies
\begin{equation}
\lim_{\ave\downarrow 0} \frac{o(f(\ave))}{f(\ave)} = 0.
\end{equation}
\section{Problem Setting}\label{sec:setting}
In this section we
describe our setups for optical communication and key distribution. To do so, we first recall some basic results in quantum optics.
\subsection{Beamsplitting and Direct Detection}
We briefly describe how \emph{number (Fock) states} and \emph{coherent states} evolve when passed through a beamsplitter, and what outcomes they induce when fed into a direct-detection receiver, i.e., a photon counter. We refer to \cite{mandelwolf95} for more details. For some background in quantum physics and in quantum information theory, we refer to \cite{nielsenchuang00}.
Let $\mathbb{A}$ and $\mathbb{V}$ be the two input spaces to a single-mode beamsplitter, and $\mathbb{B}$ and $\mathbb{E}$ be the two output spaces. Let the beamsplitter's transmissivity from $\mathbb{A}$ to $\mathbb{B}$ be $\eta\in[0,1]$. Then this beamsplitter is characterized in the Heisenberg picture by
\begin{subequations}\label{eq:beamsplitter}
\begin{IEEEeqnarray}{rCl}
\hat{b} & = & \sqrt{\eta}\,\hat{a} + \sqrt{1-\eta}\,\hat{v} \label{eq:beamsplitter1}\\
\hat{e} & = & \sqrt{1-\eta}\,\hat{a} - \sqrt{\eta}\,\hat{v}.
\end{IEEEeqnarray}
\end{subequations}
Throughout this paper we shall only consider situations where the second input space $\mathbb{V}$ (the ``noise mode'') is in its vacuum state $|0\ket$.
Ideal direct detection (i.e., photon counting) measures an optical state in the number-state basis. For direct detection on $\mathbb{A}$, the observable is the Hermitian operator $\hat{a}^\dagger \hat{a}$. On state $\rho$, a photon counter gives outcome $n\in\Integers_0^+$ with probability $\bra n|\rho|n\ket$.
Obviously, when a number state $|n\ket$, $n\in\Integers_0^+$, is fed into an ideal photon counter, the outcome is $n$ with probability one. But passing $|n\ket$ through a beamsplitter is more complicated: if space $\mathbb{A}$ in \eqref{eq:beamsplitter} is in state $|n\ket$, then the output state is an entangled state on $\mathbb{B}$ and $\mathbb{E}$:
\begin{equation}
|\phi\ket^{\mathbb{BE}} = \sum_{i=0}^n \sqrt{\binom{n}{i}} \,\eta^{i/2}(1-\eta)^{(n-i)/2} |i\ket^\mathbb{B} |n-i\ket^\mathbb{E}.
\end{equation}
This implies that performing direct detection on the output of this beamsplitter will yield a binomial distribution on the outcome: the probability of detecting $m$ photons on space $\mathbb{B}$ is
\begin{equation} \label{eq:number_direct}
\bra m | \sigma^\mathbb{B} |m \ket = \binom{n}{m}\eta^m (1-\eta)^{n-m}
\end{equation}
for $0\leq m \leq n$, and is zero otherwise. It also implies that, if direct detection is performed both on $\mathbb{B}$ and on $\mathbb{E}$, then with probability one the sum of the two outcomes is equal to $n$.
A coherent state $|\alpha\ket$, $\alpha\in\Complex$, can be written in the number-state basis as
\begin{equation}\label{eq:coherent_state}
|\alpha\ket = \e^{-|\alpha|^2/2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\ket.
\end{equation}
Thus, when fed into a photon counter, the probability of $n$ photons being observed in $|\alpha\ket$ is
\begin{equation}
\bra n|\alpha\ket\bra \alpha |n\ket = |\bra n|\alpha\ket|^2 = \e^{-|\alpha|^2} \frac{|\alpha|^{2n}}{n!}.
\end{equation}
Namely, the number of photons in $|\alpha\ket$ has a Poisson distribution of mean $|\alpha|^2$.
Coherent states have the nice property that, when passed through a beamsplitter, the outcomes remain in coherent states. If $|\alpha\ket$ is fed into the beamsplitter~\eqref{eq:beamsplitter}, the output state is
\begin{equation}
|\phi\ket^{\mathbb{BE}} = |\sqrt{\eta}\,\alpha\ket^{\mathbb{B}}\otimes |\sqrt{1-\eta}\,\alpha\ket^{\mathbb{E}}.
\end{equation}
Therefore, if direct detection is performed both on $\mathbb{B}$ and on $\mathbb{E}$, the outcomes will be two \emph{independent} Poisson random variables of means $\eta|\alpha|^2$ and $(1-\eta)|\alpha|^2$, respectively.
\subsection{Optical Communication} \label{sec:comm_background}
A single-mode pure-loss optical (i.e., bosonic) channel can be described using the beamsplitter~\eqref{eq:beamsplitter1}, where we ignore the output space $\mathbb{E}$ and assume the noise space $\mathbb{V}$ to be in its vacuum state. In this formula, $\mathbb{A}$ is the input space controlled by the transmitter which, in consistency with the key-distribution part, we call Alice; $\mathbb{B}$ is the output space obtained by the receiver, Bob; and $\eta$ is the transmissivity of the channel. Equivalently, the channel may be described in the Schr\"odinger picture as a completely-positive trace-preserving (CPTP) map from the input state $\rho^\mathbb{A}$ to the output state $\sigma^\mathbb{B}$:
\begin{equation}\label{eq:channel_map}
\sigma^{\mathbb{B}} = \mathcal{C}(\rho^{\mathbb{A}}).
\end{equation}
The explicit characterization of $\mathcal{C}$ is complicated and omitted.
We denote the blocklength of a channel code by $k$. Alice has a message of $kR$ nats\footnote{We ignore the fact that the number of values that the message can take is not an integer.} to convey to Bob. In order to do this, she prepares a state $\rho^k$ over $\mathbb{A}^k$, subject to an average-photon-number constraint $\ave$ per channel use:
\begin{equation}\label{eq:ave_block}
\textnormal{tr}\left\{ \left(\sum_{i=1}^k \hat{a}_i^\dag \hat{a}_i \right) \rho^k \right\} \le k \ave
\end{equation}
where $\hat{a}_i$ is the annihilation operator on the input space of the $i$th channel use. The channel is assumed to be memoryless, so the output is given by
\begin{equation}\label{general_channel}
\sigma^k = \mathcal{C}^{\otimes k} (\rho^k).
\end{equation}
Bob may perform any positive-operator valued measure (POVM) on $\sigma^k$ to reconstruct the message. As usual, the capacity of the channel is defined as the supremum of rates for which there exist sequences of schemes with increasing blocklengths and with the error probabilities approaching zero.
We define the \emph{photon efficiency} of transmission as the rate normalized by the expected number of photons that Bob receives per channel use:\footnote{We adopt this definition rather than normalizing by transmitted photons, because this allows us to derive expressions which are less influenced by the transmissivity of the channel.}
\begin{equation} \label{eq:efficiency}
r (\eta,\ave) \triangleq \frac{R(\eta,\ave)}{\eta \ave}.
\end{equation}
This quantity is upper-bounded by the channel's capacity divided by $\eta \ave$.
\subsection{Key Distribution Using an Optical Channel (Model C)} \label{sec:key_channel_setup}
We next consider the problem where Alice and Bob use the channel of \eqref{eq:beamsplitter} to generate a secret key between them. The channel from Alice to Bob is still characterized by \eqref{eq:beamsplitter1} or by the CPTP~\eqref{eq:channel_map}, but we now assume that an eavesdropper, Eve, obtains the Hilbert space $\mathbb{E}$. Note that this is a worst-case assumption in the sense that Eve obtains the whole \emph{ancilla} system of the channel. Also note that we assume Eve to be passive, so she cannot interfere with the communication; she can only try to distill useful information from her observations. This setting can be seen as a special case of the quantum version of ``Model C'' discussed in \cite{ahlswedecsiszar93}.
The aim of Alice and Bob is to use this channel, together with a two-way, public, but authentic classical channel, to generate a secret key. Let $k$ denote the total number of uses of the optical channel. We impose the same average-photon-number constraint \eqref{eq:ave_block} on Alice's inputs. We assume the public channel is free so we can use it to transmit as many bits as needed, though all these bits will be known to Eve. By the end of a key-distribution protocol, Alice should be able to compute a bit string $S_A$ and Bob should be able to compute $S_B$ such that
\begin{itemize}
\item The probability that $S_A=S_B$ tends to one as $k$ tends to infinity;
\item The key $S_A$ (or $S_B$) is almost uniformly distributed and independent of Eve's observations, in the sense that $$\frac{H(S_A| \rho_{\textnormal{Eve}})}{\log|\mathcal{S}|}$$ tends to one as $k$ tends to infinity, where $\rho_\textnormal{Eve}$ summarizes all of Eve's observations, and where $\mathcal{S}$ denotes the alphabet for $S_A$ and $S_B$.
\end{itemize}
We define the \emph{secret-key rate} of a scheme to be
\begin{equation}
R(\eta,\ave) \triangleq \frac{\log |\mathcal{S}|}{k}
\end{equation}
nats per use of the optical channel. The parameter $\ave$ is the average photon number in~\eqref{eq:ave_block}.
A typical (and rather general) protocol to accomplish this task consists of the following steps:
\textbf{Step 1:} Alice generates random variables $X_1,X_2,\ldots$ which are known to neither Bob nor Eve. She then prepares an optical state $\rho^k$ on $\mathbb{A}^k$ based on $\vect{X}$ and sends the state into the channel, spread over $k$ channel uses.
\textbf{Step 2:} Bob makes measurements on his output state to obtain a sequence $Y_1,Y_2,\ldots$.\footnote{We do not consider feedback from Bob to Alice during the first two steps. As in channel coding, feedback cannot increase the maximum key rate.}
\textbf{Step 3:} (Information Reconciliation) Alice and Bob exchange messages $M_1,M_2,\ldots$
using the public channel. Then Alice computes her raw key $K_A$ as a
function of $(\vect{X},\vect{M})$, and Bob computes his raw key $K_B$
as a function of $(\vect{Y},\vect{M})$. They try to ensure that $K_A=K_B$ with high probability, but Eve might have partial information about the raw key.
\textbf{Step 4:} (Privacy Amplification) Alice and Bob randomly pick one from a set of universal hashing functions. They apply the chosen function to their raw keys $K_A$ and $K_B$ to obtain the secret keys $S_A$ and $S_B$, respectively.
Privacy amplification has been extensively studied in literature. Denote the quantum state that Eve obtained in Step~1 from the optical channel by $\sigma^{\mathbb{E}^k}$. It is shown in \cite{rennerkonig05} that, provided $K_A=K_B$ with probability close to one, the privacy amplification step (i.e., Step~4) can be accomplished successfully with high probability, and the length of the secret key in nats, i.e., $\log|\mathcal{S}|$, can be made arbitrarily close to\footnote{To be precise, to achieve \eqref{eq:rennerkonig}, Alice and Bob should repeat Steps~1 to~3
many times, and then do Step~4 on all the raw keys together.}
\begin{equation}\label{eq:rennerkonig}
H(K_A|\vect{M},\sigma^{\mathbb{E}^k}).
\end{equation}
Hence, in this paper, we shall not discuss how to accomplish Step~4. As we shall see, in some cases Step~4 can be omitted. If not, then we shall concentrate on Steps~1 to~3, try to maximize~\eqref{eq:rennerkonig}, and compute the secret-key rate as
\begin{equation}\label{eq:key_rate}
R(\eta,\ave) = \frac{H(K_A|\vect{M},\sigma^{\mathbb{E}^k})}{k}.
\end{equation}
As mentioned previously, in Step~1, we impose the same average-photon-number constraint on Alice \eqref{eq:ave_block} as in the communications case. Consequently, we define the photon efficiency (of key distribution) $r(\eta,\ave)$ in the same way as in communications, namely, as in \eqref{eq:efficiency}, except that now $R(\eta,\ave)$ is the secret-key rate.
\subsection{Key Distribution Using a Photon Source (Model S)}\label{sec:key_source_setup}
In some key-distribution protocols, Alice and Bob make use of a random source, rather than Alice preparing states, to generate a secret key, as in the ``Model S'' discussed in~\cite{ahlswedecsiszar93}. In optical applications one can, for instance, generate a uniform stream of random, temporally-entangled photon pairs, which are very useful for key distribution; see, e.g., \cite{zhongwong12}.
An accurate model for such temporally-entangled photon sources divides the timeline into very fine temporal modes, where each temporal mode is in a pure, entangled state on its two output Hilbert spaces, with the number of photon pairs having a geometric (Bose-Einstein) distribution of a very small mean. Such a model, however, would be intractable for precise key-rate analyses. We hence choose a simplified model as follows. Let the timeline be divided into slots, where each slot can be thought of as one ``use'' of the source. Each slot contains many, e.g., a thousand, temporal modes. This results in the number of photon pairs in each slot having a Poisson distribution, whose mean $\ave$ equals the total number of temporal modes times the mean photon number in each mode. We ignore the fine structures inside each slot and describe it with only two Hilbert spaces, $\mathbb{C}$ and $\mathbb{D}$. We also ignore the entanglement between the two spaces and simplify the optical state to a mixed one with classical correlation only. The optical state emitted by the source in every source use is thus given by
\begin{equation}\label{eq:source_model}
\rho^{\mathbb{CD}} = \sum_{i=0}^\infty \frac{\ave^i \e^{-\ave}}{i!} |i\ket \bra i|^\mathbb{C} \otimes |i\ket\bra i|^\mathbb{D}.
\end{equation}
To justify the simplification we make, note the following.
\begin{itemize}
\item{Discretization:} In our schemes, Alice and Bob will never measure the arrival time of a photon with higher accuracy than the duration of one slot. In this case, it is easy to show that Eve cannot have any advantage by making finer measurements.
\item{Classical correlation:} When Alice and Bob only make direct detection with the given time accuracy and Eve is listening passively via a beamsplitter, entanglement does not play any role. Note that this would not be the case if we were interested in a secrecy proof against a general (possibly active) Eve; see Section~\ref{sec:discussion}.
\end{itemize}
We assume that the source is collocated with Alice, who keeps $\mathbb{C}$; while the photons in $\mathbb{D}$ are sent to Bob through a lossy optical channel. To account for coupling losses, we can assume that Alice also only has access to a lossy version of $\mathbb{C}$. Specifically, $\rho^\mathbb{C}$ is passed through a beamsplitter, like the one in \eqref{eq:beamsplitter}, of transmissivity $\eta_A$ before it reaches Alice:
\begin{subequations}\label{eq:beamsplitterA}
\begin{IEEEeqnarray}{rCl}
\hat{a} & = & \sqrt{\eta_A}\,\hat{c} + \sqrt{1-\eta_A}\,\hat{v}\\
\hat{f} & = & \sqrt{1-\eta_A}\,\hat{c} - \sqrt{\eta_A}\,\hat{v}.
\end{IEEEeqnarray}
\end{subequations}
But, except for Section~\ref{sec:etaA}, we shall ignore coupling losses and take $\eta_A=1$.
Similarly $\rho^\mathbb{D}$ is passed through a beamsplitter of transmissivity $\eta_B$ before it reaches Bob:
\begin{subequations}\label{eq:beamsplitterB}
\begin{IEEEeqnarray}{rCl}
\hat{b} & = & \sqrt{\eta_B}\,\hat{d} + \sqrt{1-\eta_B}\,\hat{u} \\
\hat{e} & = & \sqrt{1-\eta_B}\,\hat{d} - \sqrt{\eta_B}\,\hat{u}.
\end{IEEEeqnarray}
\end{subequations}
We assume $\eta_B<1$ throughout.
Both noise modes $\mathbb{V}$ and $\mathbb{U}$ are assumed to be in their vacuum states. Note that the two beamsplitters behave independently of each other.
Since the source is collocated with Alice, we know the photons that are lost from $\mathbb{C}$ to $\mathbb{A}$ (in case $\eta_A<1$) should \emph{not} reach Eve; Eve only has access to the Hilbert space $\mathbb{E}$.
It is useful to describe the output states when $\rho^\mathbb{D}$ passes through the beamsplitter on Bob's side. We first write down $\rho^\mathbb{D}$ by taking partial trace of \eqref{eq:source_model}:
\begin{equation}
\rho^\mathbb{D} = \sum_{i=0}^\infty \frac{\ave^i \e^{-\ave}}{i !} |i\ket\bra i|,
\end{equation}
which can be equivalently written as
\begin{equation}\label{eq:rhoD_coherent}
\rho^\mathbb{D} = \frac{1}{2\pi} \int_0^{2\pi} \d \theta |\alpha(\theta)\ket\bra\alpha(\theta)|
\end{equation}
where
\begin{equation}
\alpha(\theta) = \sqrt{\ave} \e^{i\theta}.
\end{equation}
From \eqref{eq:rhoD_coherent} and \eqref{eq:beamsplitterB} it is straightforward to obtain the output optical state on $\mathbb{BE}$:
\begin{IEEEeqnarray}{rCl}
\sigma^{\mathbb{BE}}
& = & \frac{1}{2\pi} \int_0^{2\pi} \d \theta |\sqrt{\eta}\, \alpha(\theta)\ket \bra \sqrt{\eta}\, \alpha(\theta) |^\mathbb{B} \nonumber\\
& & ~~~~~~~~\otimes |\sqrt{1-\eta}\, \alpha(\theta)\ket \bra \sqrt{1-\eta}\, \alpha(\theta)|^\mathbb{E}.\label{eq:sigmaBE}
\end{IEEEeqnarray}
By taking partial traces of \eqref{eq:sigmaBE} we obtain Bob's and Eve's states:
\begin{IEEEeqnarray}{rCl}
\sigma^\mathbb{B} & = & \frac{1}{2\pi} \int_0^{2\pi} \d \theta |\sqrt{\eta}\, \alpha(\theta)\ket \bra \sqrt{\eta}\, \alpha(\theta) | \\
& = & \sum_{i=0}^\infty \frac{(\eta \ave)^i \e^{-\eta \ave}}{i!} |i\ket\bra i|\\
\sigma^\mathbb{E} & = & \frac{1}{2\pi} \int_0^{2\pi} \d \theta |\sqrt{1-\eta} \alpha(\theta)\ket \bra \sqrt{1-\eta} \alpha(\theta) | \\
& = & \sum_{i=0}^\infty \frac{((1-\eta) \ave)^i \e^{-(1-\eta)\ave}}{i!} |i\ket\bra i|. \label{eq:sigmaE}
\end{IEEEeqnarray}
The joint state $\sigma^{\mathbb{BE}}$ is not a tensor state, i.e., $\mathbb{B}$ and $\mathbb{E}$ are not independent. However, if direct detection---namely, projective measurement in the number-state basis---is performed on $\mathbb{B}$ (or on $\mathbb{E}$), the post-measurement state on $\mathbb{E}$ (or on $\mathbb{B}$) is independent of the measurement outcome; in particular, the photon numbers in $\mathbb{B}$ and in $\mathbb{E}$ are independent. Indeed, conditional on the measurement outcome on $\mathbb{B}$ being $i$, the post-measurement state on $\mathbb{E}$ is
\begin{IEEEeqnarray}{rCl}
\lefteqn{\frac{\textnormal{tr}_\mathbb{B} \left\{ |i\ket\bra i|^\mathbb{B} \sigma^{\mathbb{BE}}\right\}}{\textnormal{tr}\left\{|i\ket\bra i|^\mathbb{B} \sigma^\mathbb{B} \right\}}}\nonumber\\
&= & \frac{\displaystyle \frac{1}{2\pi} \int_0^{2\pi} \d \theta |\bra i | \sqrt{\eta}\, \alpha(\theta)\ket|^2 |\sqrt{1-\eta}\, \alpha(\theta)\ket \bra \sqrt{1-\eta}\, \alpha(\theta)|} {\displaystyle \frac{1}{2\pi} \int_0^{2\pi} \d \theta |\bra i | \sqrt{\eta}\, \alpha(\theta)\ket|^2}\nonumber\\ {} \\
& = & \frac{1}{2\pi} \int_0^{2\pi} \d \theta |\sqrt{1-\eta}\, \alpha(\theta)\ket \bra \sqrt{1-\eta}\, \alpha(\theta) | \label{eq:indep_theta}\\
& = & \sigma^\mathbb{E}
\end{IEEEeqnarray}
where \eqref{eq:indep_theta} follows because
\begin{equation}
|\bra i | \sqrt{\eta}\, \alpha(\theta)\ket|^2 = \frac{(\eta \ave)^i \e^{-\eta \ave}}{i!}
\end{equation}
does not depend on $\theta$.
We now describe a scheme (which is again rather general) for Alice and Bob to use this source $k$ times to generate a secret key. In this scheme, Steps~3 and~4 are exactly the same as in Section~\ref{sec:key_channel_setup}, but Steps~1 and~2 are now replaced by:
\textbf{Step 1':} Alice makes measurements on her state $\sigma^{\mathbb{A}^k}$ to obtain the sequence $X_1,X_2,\ldots$.
\textbf{Step 2':} Bob makes measurements on his state $\sigma^{\mathbb{B}^k}$ to obtain the sequence $Y_1,Y_2,\ldots$.
As in Section~\ref{sec:key_channel_setup}, we shall concentrate on Steps~1', 2', and~3. The secret-key rate, denoted by $R(\eta_A,\eta_B,\ave)$, is again given by the right-hand side of \eqref{eq:key_rate}, with unit ``nats per source use''. But the photon efficiency in this setting is defined as
\begin{equation}\label{eq:PE_etaAB}
r(\eta_A,\eta_B,\ave) \triangleq \frac{R(\eta_A,\eta_B,\ave)}{\eta_A \eta_B \ave}.
\end{equation}
We choose this definition because $\eta_A \eta_B \ave$ is the expected number of photon pairs in each source use that reach both Alice and Bob,\footnote{We interpret this quantity in a semi-classical way: each photon pair reaches Alice with probability $\eta_A$, and reaches Bob with probability $\eta_B$ independently of whether it reaches Alice or not, hence the fraction of photon pairs that reach both Alice and Bob is $\eta_A \eta_B$. We do not know if there exists a physical observable, i.e., a Hermitian operator that corresponds to this value.} and because these photon pairs are those that contain correlated information that can be used to generate the secret key. When $\eta_A=1$, we omit the subscript in $\eta_B$, and denote the secret-key rate and photon efficiency simply by $R(\eta,\ave)$ and $r(\eta,\ave)$, respectively. Obviously, they are again related by \eqref{eq:efficiency}.
\section{Background: Photon-Efficient Communication using Pulse-Position Modulation}\label{sec:communication}
Before we address key distribution, we give some results regarding communications over the bosonic channel described in Section~\ref{sec:comm_background}.
These results serve as a point of reference, and the derivation provides tools later used in key distribution. See also \cite{kochmanwornell12,erkmen12,wangwornell14}.
The capacity of a quantum channel is characterized by the formula found by Holevo \cite{holevo98} and by Schumacher and Westmoreland \cite{schumacherwestmoreland97}. For the pure-loss bosonic channel \eqref{eq:beamsplitter1} under constraint~\eqref{eq:ave_block}, this capacity is $g(\eta \ave)$ nats per channel use \cite{giovannettiguha04}, where
\begin{equation} \label{eq:g_x} g(x) \triangleq (x+1) \log (x+1) - x \log x,\quad x>0. \end{equation}
This immediately implies that the photon efficiency \eqref{eq:efficiency} satisfies:
\begin{equation}\label{holevo_eff} r_\textrm{quantum}(\eta,\ave) = \frac{g(\eta \ave)}{\eta \ave} = \log{\frac{1}{\eta \ave}} + 1 + o(1).
\end{equation}
Note that the efficiency is unbounded, that is,
\begin{equation} \lim_{\ave \downarrow 0} r_\textnormal{quantum}(\eta,\ave) = \infty. \end{equation}
Hence, in terms of \cite{gallager87,verdu90}, the capacity per unit cost $\sup_{\ave} r(\eta,\ave)$ of the channel \eqref{eq:channel_map} is infinite.
The capacity $g(\eta \ave)$ is achievable by Alice using product (i.e., nonentangled), pure input states
\begin{equation}\label{eq:indep_input}
|\psi^k \ket = |\psi_1 \ket \otimes |\psi_2 \ket \otimes \cdots \otimes |\psi_k \ket .
\end{equation}
Indeed, in this paper we limit our attention to such mode of operation, where
the average-photon-number constraint \eqref{eq:ave_block} becomes
\begin{equation}\label{eq:ave}
\frac{1}{k} \sum_{i=1}^k \bra {\psi_i}|\hat{a}_i^\dag\hat{a}_i| \psi_i \ket \le \ave.
\end{equation}
For the degenerate case $\eta=1$, a simple capacity-achieving codebook consists only
of number states, where the photon numbers' empirical distribution is independent and identically distributed (i.i.d.) geometric (i.e., Bose-Einstein). Bob's optimal measurement for this codebook is simply per-channel-use direct detection. We shall see in Section~\ref{sec:comm_number} that, in the photon-efficient regime, this code construction can be further simplified and can be used also when $\eta<1$, without sacrificing much photon efficiency.
For the general case where $\eta$ may not be one, the capacity can be achieved if
Alice's codebook consists of coherent states
\begin{equation}
|\psi^k\ket = |\alpha_1\ket \otimes |\alpha_2\ket \otimes \cdots \otimes |\alpha_k\ket,
\end{equation}
and if Bob performs a general (not per-channel-use) POVM on the output state, which is
\begin{equation}
|\phi^k\ket = |\sqrt{\eta}\,\alpha_1\ket \otimes |\sqrt{\eta}\,\alpha_2\ket \otimes \cdots \otimes |\sqrt{\eta}\,\alpha_k\ket.
\end{equation}
In this case, the average-photon-number constraint \eqref{eq:ave_block} becomes
\begin{equation}\label{eq:ave_coherent}
\sum_{i=1}^k |\alpha_i|^2 \le k\ave.
\end{equation}
It is known that capacity-achieving codebooks of coherent states should have empirical distributions that resemble i.i.d. complex-Gaussian with mean zero and variance $\ave$ \cite{giovannettiguha04}. The main problem with such a code is that Bob's POVM is almost impossible to implement using today's technology. Hence we are interested in ``practical'' schemes, in particular, in schemes where Bob uses per-channel-use direct detection while Alice sends coherent states. As we shall see in Section~\ref{sec:comm_coherent}, this restriction induces a second-order-term loss in photon efficiency.
\subsection{Alice Sends Binary Number States}\label{sec:comm_number}
Consider the case where the sequence of states sent by Alice consists only of the number states $|0\ket$ and $|1\ket$, and where Bob uses direct detection. Recalling \eqref{eq:number_direct}, for input $|0\ket$ Bob will always detect no photon, while for input $|1\ket$ Bob detects one photon with probability $\eta$, and detects no photon otherwise. Thus the scheme induces a classical Z channel. The maximum achievable rate is, according to the classical channel coding theorem \cite{shannon48}, the maximum mutual information over this channel.
Let
\begin{equation} \label{I_Z} I_\textnormal{Z}(q,\mu) \triangleq H_2(q \mu) - q H_2(\mu) \end{equation} be the mutual information over a Z channel with input probability $P_X(1)=q$ and transition probability $P_{Y|X}(1|1)=\mu$, where $H_2(\cdot)$ is the binary entropy function
\begin{equation}
H_2 (x) \triangleq x\log\frac{1}{x}+(1-x)\log\frac{1}{1-x},\quad 0<x<1.
\end{equation}
Due to the photon-number constraint, the input distribution must satisfy $q \leq \ave$.\footnote{The expected number of photons translates to a per-codeword constraint via a standard expurgation argument.} It is easy to see that $I_\textnormal{Z}(q,\mu)$ is monotonically increasing in $q$ for small enough $q$, and hence, in the regime of interest, we should choose $q=\ave$, achieving rate $I_\textnormal{Z}(\ave,\eta)$. The resulting photon efficiency can be readily shown to satisfy:
\begin{equation}\label{eq:numZ}
r_\textnormal{num,Z} (\eta,\ave) = \frac{I_\textnormal{Z}(\ave,\eta)}{\eta \ave} = r_\textrm{quantum}(\eta,\ave) - \frac{H_2(\eta)}{\eta} + o(1), \end{equation} reflecting a constant efficiency loss with respect to the optimum \eqref{holevo_eff}.
For the scheme described above, the task of (classical) coding is difficult: one needs mutual-information-approaching codes for a Z channel with a highly skewed input. We can solve this problem by replacing the i.i.d. binary codebook by PPM: the input sequence consists of ``frames'' of length $\lceil\nicefrac{1}{\ave}\rceil$, where each frame includes exactly one photon, whose position is uniformly chosen inside the frame. (If the blocklength is not divisible by $\lceil\nicefrac{1}{\ave}\rceil$, then we ignore the remainder.) This scheme converts the channel to a $\lceil\nicefrac{1}{\ave}\rceil$-ary erasure channel. By computing the capacity of this erasure channel, we easily see that the photon efficiency of the PPM scheme is:
\begin{equation} r_\textnormal{num,PPM}(\eta, \ave) = \log {\frac{1}{\ave}} + o(1), \end{equation} which again reflects only a constant loss compared to the optimal efficiency \eqref{holevo_eff}. The large-alphabet erasure channel is much like a packet-erasure channel encountered in internet applications, and good off-the-shelf codes are available.
\subsection{Alice Sends Binary Coherent States}\label{sec:comm_coherent}
Generating the number state $|1\ket$ is hard in practice. We hence turn to coherent states, which are a good model for light coming out of laser sources \cite{shapiro09}.
We consider a simple binary-coherent-state scheme. In this scheme, Alice first generates a classical binary codebook where the probability of $1$ is $q$. She then maps $0$ and $1$ to the coherent states $|0\ket$ and $\left|\nicefrac{\ave}{q}\right\ket$, respectively. Note that doing this satisfies the average-power constraint \eqref{eq:ave_coherent}. Bob uses direct detection that is not photon-number resolving (PNR), i.e., he views a measurement with no photon as a logical $0$, and views any measurement with at least one photon as a logical~$1$. (Such a detector is easier to build than a PNR detector, which outputs the exact number of detected photons.) This results again in a classical Z channel, with
\begin{equation} P_{Y|X}(1|1) = \mu_\textnormal{coh} (q,\ave) \triangleq 1 - \exp\left(-\frac{\eta \ave}{q}\right). \end{equation}
We can thus achieve $I_\textnormal{Z}(q,\mu_\textnormal{coh})$ nats per channel use, where $q$ should be chosen to maximize $I_\textnormal{Z}(q,\mu_\textnormal{coh})$. The exact analytical optimization is complicated, but in the photon-efficient regime the approximate optimum (which yields the best rate up to the approximation of interest) is given by
\begin{equation}\label{p_star} q^*(\ave) = \frac{\eta \ave}{2}\log\frac{1}{\ave}. \end{equation}
The resulting photon efficiency is given by:
\begin{IEEEeqnarray}{rCl}
\label{Z_eff1} r_\textnormal{coh,Z}(\eta,\ave) & = & \frac{I_Z\bigl(q^*(\ave),\mu_\textnormal{coh}(q^*(\ave),\ave)\bigr)}{\eta \ave} \\
& =& \log{\frac{1}{\eta \ave}} - \log \log{\frac{1}{\ave}} + \log 2 -1 + o(1). \label{Z_eff} \IEEEeqnarraynumspace
\end{IEEEeqnarray}
Comparing to the quantum limit \eqref{holevo_eff}, we see that the efficiency loss of the coherent-state-and-direct-detection scheme with respect to the optimal performance grows as $\log \log \nicefrac{1}{\ave}$ as $\ave$ decreases in the photon-efficient regime. This loss is inherent to any ``classical'' transmission scheme, even if general (non-binary) coherent states are sent \cite{wangwornell14}, or if the receiver is allowed to use feedback between measurements \cite{chungguhazheng11}.
Similarly to the case of Alice sending number states, we can alleviate the difficulty of coding by replacing the i.i.d. codebooks with PPM frames, an idea already exploited in \cite{pierce78,massey81}. Indeed, using PPM frames of length $b$ with the optimum (to the approximation order) choice of~\eqref{p_star} and $b=\lceil \nicefrac{1}{q^*(\ave)} \rceil$, this efficiency is
\begin{equation}\label{PPM_eff}
r_\textnormal{coh,PPM}(\ave)= \frac{\mu_\textnormal{coh} \bigl(q^*(\ave), \ave\bigr) \log b}{\eta b \ave},
\end{equation}
and has the same expression as on the right-hand side of \eqref{Z_eff}, i.e., the further efficiency loss incurred by restricting to PPM is $o(1)$.
\begin{figure}[t]
\centering
\psfrag{E}[cc]{\small $\mathcal{E}$}
\psfrag{rquantum}[Bl][Bl]{\small $r_\textnormal{quantum}$}
\psfrag{rz}[Bl][Bl]{\small $r_\textnormal{coh,Z}$}
\psfrag{rppm}[Bl][Bl]{\small $r_\textnormal{coh,PPM}$}
\psfrag{Photon efficiency (nats/photon)}[Bl][Bl]{\footnotesize Photon efficiency (nats/photon)}
\includegraphics[width=0.5\textwidth]{COMM.eps}
\caption{Photon efficiency in the different cases discussed in Section~\ref{sec:communication}. Efficiency in the quantum case $r_\textnormal{quantum}$ is computed from \eqref{holevo_eff}; efficiency for coherent-state inputs and Z-channel model $r_\textnormal{coh,Z}$ from \eqref{Z_eff1}; and efficiency for coherent-state inputs and PPM $r_\textnormal{coh,PPM}$ from \eqref{PPM_eff}. For all three we let the channel be lossless, i.e., we choose $\eta=1$.}
\label{fig_eff}
\end{figure}
Figure~\ref{fig_eff} depicts the photon efficiency in the different cases discussed in this section. It can be appreciated that, while the loss of using coherent states with direct detection is large, the further loss of PPM is small. As we shall see, similar phenomena are also observed in key-distribution scenarios.
\section{Key Distribution in Model C}\label{sec:channel_case}
In this section we study the key-distribution problem in Model C, which we set up in Section~\ref{sec:key_channel_setup}.
To the best of our knowledge, the maximum secret-key rate, and hence also the maximum photon efficiency, in this setting are not yet known. However, in the photon-efficient regime we have the following asymptotic upper bound. (Later we show that this upper bound is tight within a constant term).
\begin{proposition}\label{prp:C_max}
The maximum photon efficiency for key distribution in Model~C as described in Section~\ref{sec:key_channel_setup} satisfies
\begin{equation}\label{eq:PE_channel_max}
r_\textnormal{max} (\ave) \le \log{\frac{1}{\eta \ave}} + 1 + o(1).
\end{equation}
\end{proposition}
\begin{proof} We use the fact that the maximum secret-key rate over a quantum channel cannot exceed the communication capacity of the same channel. This follows, e.g., from \cite[Chapter I, Theorem 5.1]{wilmink03}. Recalling \eqref{holevo_eff}, the proof is completed. \end{proof}
As in the communication setting, we shall mostly focus on key-distribution schemes in which Bob only employs direct detection. As we shall see in Section~\ref{sec:key_channel_number}, if Alice can send number states---even only binary number states---the photon-efficiency loss of direct detection is at most a constant term in the photon-efficient regime. However, in Section~\ref{sec:key_channel_coherent} we show that if Alice can only send coherent states, then the loss in photon efficiency scales like $\log\log\nicefrac{1}{\ave}$. These results are similar to their optical-communication counterparts. Also similar to the communication scenario is the fact that PPM is nearly optimal in terms of photon efficiency; in the context of key distribution, PPM allows us to greatly simplify the coding task in the information-reconciliation step.
\subsection{Alice Sends Binary Number States}\label{sec:key_channel_number}
Consider the following key-distribution scheme.
\setcounter{sc}{3}
\begin{scheme} \label{Scheme_C-1}
\mbox{}
\begin{enumerate}
\item Let $b\triangleq \lceil \nicefrac{1}{\ave} \rceil$. We divide
the whole block of $k$ channel uses into frames each
consisting of $b$ consecutive uses (and ignore the remainder).
\item Alice generates a sequence of integers
$\tilde{X}_1,\tilde{X}_2,\ldots$ i.i.d.
uniformly in $\{1,\ldots,b\}$. These are the
``pulse positions''. Within the $i$th frame,
$i\in\{1,2,\ldots\}$, she sends the
number state $|1\ket$ in the $\tilde{X}_i$th channel use, and
sends $|0\ket$ in all other channel uses.
\item Bob makes direct detection on every channel output. Since Alice sends one
photon per frame, Bob will either detect a single photon or no photon per frame.
Let the set of frames where Bob had a detection be denoted as $\{i_1,i_2,\ldots\}$, and denote the detection positions inside
these bins by $\tilde{Y}_{i_1},\tilde{Y}_{i_2},\ldots$. Bob tells Alice
the values of $i_1,i_2,\ldots$ using the public channel.
\item Alice generates the secret key from
$\tilde{X}_{i_1},\tilde{X}_{i_2},\ldots$, and Bob generates the
secret key from $\tilde{Y}_{i_1},\tilde{Y}_{i_2},\ldots$, both by
directly taking the binary representation of these integers.
\end{enumerate}
\end{scheme}
The average-photon-number constraint \eqref{eq:ave_block} is clearly satisfied. Scheme~\ref{Scheme_C-1} is rather simple in the sense that
\begin{itemize}
\item Alice's input states are either $|0\ket$ or $|1\ket$;
\item Bob's detector can be non-PNR;
\item The information-reconciliation step is uncoded, and only involves one-way communication from Bob to Alice;
\item There is no privacy-amplification step.
\end{itemize}
As the next proposition shows, this simple scheme performs very well in the photon-efficient regime: it is at most a constant term away from optimum. Compared to the communication case~\eqref{holevo_eff}, this proposition also shows that the loss in photon efficiency due to the secrecy requirement is at most a constant term.
\begin{proposition}\label{prp:C-1}
Scheme~\ref{Scheme_C-1} generates a secret key between Alice and Bob, and its photon efficiency is
\begin{equation}
r_{\textnormal{C-1}}(\eta,\ave) = \log\frac{1}{\ave}+o(1)
\end{equation}
for all $\eta\in(0,1]$.
\end{proposition}
\begin{proof}
We first verify that Scheme~\ref{Scheme_C-1} indeed
generates a secret key.
To this end, first note that
$\tilde{X}_{i_j}=\tilde{Y}_{i_j}$ for all $j\in\{1,2,\ldots\}$. This
is because Alice sends only one non-vacuum state in each frame, and
because Bob cannot detect any photon in a channel use where Alice
sends $|0\ket$. Hence the keys obtained by Alice and by Bob are the same. Second, by the way Alice chooses~$\tilde{\vect{X}}$,
every $\tilde{X}_{i_j}$ (or,
equivalently, $\tilde{Y}_{i_j}$) is uniformly distributed in
$\{1,\ldots,b\}$, independently of $\tilde{X}_{i_{j'}}$ where $j'\neq
j$. This shows that the key is uniformly distributed. It now remains to verify that the key is dependent neither on Eve's
output states from the optical channel nor on the messages which Bob
sends to Alice. It is independent of Eve's optical states because,
in every selected frame, Bob detects the only photon that Alice
transmits, so Eve's post-measurement state in this frame is the all-vacuum
state. It is independent of Bob's messages because Bob only sends the labels of the selected frames to Alice, and because Alice chooses the
pulse positions independently of the frame labels.
We next compute the photon efficiency achieved by Scheme~\ref{Scheme_C-1}. Let $N(k)$ be the total number of frames selected by Bob within $k$ channel uses. Since each frame is selected when Bob detects a photon in that frame, which happens with probability $\eta$, we have from the Law of Large Numbers that
\begin{equation}
\lim_{k\to\infty} \frac{N(k)}{k} = \lim_{k\to\infty}\frac{\eta \lfloor\nicefrac{k}{b}\rfloor}{k} = \eta \ave
\end{equation}
with probability one. Each detected photon (or, equivalently, each selected frame)
provides $\log b$ nats of secret key. So, as $k$ tends to infinity, the achieved photon efficiency tends to
\begin{equation}
\lim_{k\to\infty} \frac{N(k) \log b}{k \eta \ave } = \log b =
\log\frac{1}{\ave}+o(1).
\end{equation}
\end{proof}
\subsection{Alice Sends Coherent States}\label{sec:key_channel_coherent}
We now restrict Alice to sending coherent states since, as discussed previously, generating the number state $|1\ket$ is hard in practice. Under this restriction, Alice generates a sequence of complex numbers $\alpha_1,\alpha_2,\ldots,\alpha_k$ satisfying \eqref{eq:ave_coherent}, prepares the coherent states
$|\alpha_1\ket,|\alpha_2\ket,\ldots,|\alpha_k\ket$, and sends them over the
channel. As the next proposition shows, this restriction induces a loss of $\log\log\nicefrac{1}{\ave}$ in the photon efficiency, even if the scheme employed is more sophisticated than Scheme~\ref{Scheme_C-1}.
\begin{proposition}\label{prp:coherent_converse}
The maximum photon efficiency in Model~C when Alice sends only
coherent states and when Bob uses only direct detection satisfies
\begin{equation}\label{eq:coherent}
r_{\textnormal{coh}} (\eta,\ave) \le \log\frac{1}{\ave} -
\log\log\frac{1}{\ave} + O(1)
\end{equation}
for all $\eta\in(0,1]$.
\end{proposition}
\begin{proof}
We note that, when Alice sends the coherent state $|\alpha\ket$, Bob's measurement
outcome $Y$ has a Poisson distribution of mean
$\eta|\alpha|^2$. We can bound the achievable secret-key rate as
\begin{IEEEeqnarray}{rCl}
R_\textnormal{coh} (\eta,\ave) & \le & \max_{\E{|X|^2} \le \ave} I(X;Y) \label{eq:coherent_2}\\
& = & \max_{\E{|X|^2} \le \ave} I(|X|^2;Y), \label{eq:coherent_1}
\end{IEEEeqnarray}
where \eqref{eq:coherent_2} follows because the secret-key rate over a channel cannot be larger than the communication capacity of the channel (see, e.g., \cite{ahlswedecsiszar93}); and where \eqref{eq:coherent_1} follows because $|X|^2$ is a deterministic function of $X$, and because $X\markov |X|^2 \markov Y$
forms a Markov chain. Finally, the right-hand side of \eqref{eq:coherent_1}, which is the maximum mutual information over a Poisson channel under an average-photon-number constraint, is shown in \cite{wangwornell14} to satisfy
\begin{equation} \label{eq:from_seminal}
\max_{\E{|X|^2} \le \ave} I(|X|^2;Y) \le \eta \ave \left\{ \log\frac{1}{\ave} -
\log\log\frac{1}{\ave} +O(1) \right\}.
\end{equation}
\end{proof}
We do not specify the $O(1)$ term, as the derivation of \eqref{eq:from_seminal} in\cite{wangwornell14} yields expressions that are rather involved. In the sequel we show that the bound~\eqref{eq:coherent} is tight within a constant term.
As in Section~\ref{sec:comm_coherent}, to simplify the coding task for the information-reconciliation step, Alice and Bob can use a PPM-based scheme. We choose the PPM frame-length to be:
\begin{equation}\label{eq:blocklog}
b\triangleq \left\lceil \frac{1}{\ave\log\nicefrac{1}{\ave}}
\right\rceil.
\end{equation}
This choice is optimal up to the order of approximation of interest. Note that $b$ in \eqref{eq:blocklog} is half the frame-length chosen for the communication setting, where the latter is $\lceil\nicefrac{1}{q^*(\ave)}\rceil$ with $q^*(\ave)$ given in~\eqref{p_star}.
\begin{scheme}\label{Scheme_C-2}
\mbox{}
\begin{enumerate}
\item
We divide
the whole block of $k$ channel uses into frames each
consisting of $b$ consecutive uses (and ignore the remainder).
\item Alice generates a sequence of integers
$\tilde{X}_1,\tilde{X}_2,\ldots$ i.i.d.
uniformly in $\{1,\ldots,b\}$. Within the $i$th frame,
$i\in\{1,2,\ldots\}$, she sends the
coherent state $|\sqrt{b \ave}\ket$ in the $\tilde{X}_i$th
channel use,
and sends the vacuum state
$|0\ket$ in all other channel uses.
\item Bob makes direct detection on every channel-output. Since all channel input-states but one are in vacuum state, he will have detections in at most one output. Let the set of frames where Bob had a detection be denoted as $\{i_1,i_2,\ldots\}$, and denote the detection positions inside
these bins by $\tilde{Y}_{i_1},\tilde{Y}_{i_2},\ldots$. He tells Alice
the values of $i_1,i_2,\ldots$ using the public channel.
\item Alice generates the raw key $K_A$ from
$\tilde{X}_{i_1},\tilde{X}_{i_2},\ldots$, and Bob generates the
raw key $K_B$ from $\tilde{Y}_{i_1},\tilde{Y}_{i_2},\ldots$, both by
directly taking the binary representation of these integers.
\item Alice and Bob perform privacy amplification on their raw keys to
obtain the secret keys.
\end{enumerate}
\end{scheme}
The average-photon-number constraint \eqref{eq:ave_block} or \eqref{eq:ave_coherent} is clearly satisfied. Also note that, in this scheme,
\begin{itemize}
\item Alice's input states are binary: either $|0\ket$ or $|\sqrt{b \ave}\ket$;
\item Bob's detector can be non-PNR;
\item The information-reconciliation step is uncoded, and only involves one-way
communication from Bob to Alice.
\end{itemize}
In contrast to the restriction on Alice to sending only coherent states, which results in a loss of $\log\log \nicefrac{1}{\ave}$ in photon efficiency, the further simplifications employed in Scheme~\ref{Scheme_C-2} induce at most a constant-term loss.
\begin{proposition}\label{prp:C-2}
Scheme~\ref{Scheme_C-2} achieves photon efficiency
\begin{equation}
r_{\textnormal{C-2}} (\eta,\ave) \ge \log\frac{1}{\ave} - \log\log\frac{1}{\ave} - (1-\eta) + o(1)
\end{equation}
for all $\eta\in(0,1]$.
\end{proposition}
The proof, which appears in Appendix~\ref{app:C-2}, is more involved than that of Scheme~\ref{Scheme_C-1}, since in the case of coherent states, the raw key depends upon Eve's optical states (since, if Bob and Eve both see detections in some frame, then they must be in the same location). However, we bound the information leakage and show that it leads to at most a constant key-efficiency loss.
\section{Key Distribution in Model S}\label{sec:source_case}
In this section we study the key-distribution problem in Model S, which we set up in Section~\ref{sec:key_source_setup}. Apart from Section~\ref{sec:etaA}, we shall focus on the case where $\eta_A=1$. In this case, we omit the subscript of $\eta_B$ to denote it simply as~$\eta$.
\begin{proposition}\label{prp:IAB}
The maximum photon efficiency achievable in Model~S satisfies
\begin{equation} \label{eq:efficiency_Squantum}
r_\textnormal{quantum}(\eta,\ave) \le \log{\frac{1}{\eta \ave}} + 1 + o(1).
\end{equation}
\end{proposition}
\begin{proof}
We note that, without further constraints, the secret-key rate and hence the photon efficiency achievable in Model~S cannot exceed those achievable in Model~C. This is because any measurement Alice performs in Step~1' in Model~S, which is described in Section~\ref{sec:key_source_setup}, can be simulated in Model~C in the following way. Alice first generates random numbers that have the same statistics as the outcomes of the measurement that she would perform in Model~S. Then, for each number, she generates the corresponding post-measurement state on $\mathbb{D}$ and sends it to Bob. Doing these will generate the same correlation between Alice, Bob, and Eve as the corresponding strategy in Model~S would do. The claim now follows immediately from Proposition~\ref{prp:C_max}.
\end{proof}
\emph{Note:} The above proof says that, when Alice and Bob can both use fully quantum devices, there is no advantage in Model~S over Model~C. However, as we later show, this need not be the case when Alice and Bob are restricted, e.g, to direct detection.
For practicality, for the rest of this section we restrict both Alice and Bob to using only direct detection on their quantum states. In fact, Alice and Bob will only use non-PNR direct detection. In contrast, we do not impose any constraint on Eve's measurement, thus our schemes are secure against a fully-quantum (though passive) Eve.
\subsection{Direct Detection Combined with Optimal Binary Slepian-Wolf Codes}
After Alice and Bob perform direct detection on their optical states, each of them has a binary sequence where $1$ indicates photons are detected in the corresponding source use. Denote their sequences by $\vect{A}$ and $\vect{B}$, respectively. Due to our source model, $\vect{A}$ and $\vect{B}$ are distributed i.i.d. in time, while each pair $(A,B)$ has joint distribution according to a Z channel with
\begin{subequations}\label{eq:PAB}
\begin{IEEEeqnarray}{rCl}
q &\triangleq& P_A(1) = 1 - \e^{-\ave}\\
\mu & \triangleq& P_{B|A}(1|1) = \frac{1-\e^{-\eta \ave}}{1-\e^{-\ave}}.
\end{IEEEeqnarray}
\end{subequations}
Bob can help Alice to know $\vect{B}$ by sending her a Slepian-Wolf code \cite{slepianwolf73}. For the moment, we assume that Alice and Bob have an optimal Slepian-Wolf code for the joint distribution $P_{AB}$ (Later we drop this assumption to find more realistic code constructions.) Then they can use the following key-distribution scheme.
\setcounter{sc}{19}
\setcounter{scheme}{0}
\begin{scheme}\label{Scheme_S-1}
\mbox{}
\begin{enumerate}
\item Alice and Bob perform non-PNR direct detection to obtain binary sequences $\mathbf{A}$ and $\mathbf{B}$, respectively.
\item Bob sends Alice an optimal Slepian-Wolf code so that Alice knows $\vect{B}$ with high probability. They use $\vect{B}$ as the raw key.
\item Alice and Bob perform privacy amplification on $\vect{B}$ to obtain the secret key.
\end{enumerate}
\end{scheme}
The key rate and photon efficiency of Scheme~\ref{Scheme_S-1} satisfy the following.
\begin{proposition}\label{prp:S-1}
Scheme~\ref{Scheme_S-1} achieves the key rate
\begin{equation}\label{eq:S-1_IAB}
R_\textnormal{S-1}(\eta,\ave) = I(A;B)
\end{equation}
where the mutual information is computed on the joint distribution $P_{AB}$ given by \eqref{eq:PAB}.
Furthermore, for all $\eta\in(0,1]$, the photon efficiency of Scheme~\ref{Scheme_S-1} satisfies
\begin{equation}\label{eq:S-1_efficiency}
r_\textnormal{S-1}(\eta,\ave) = \log\frac{1}{\eta \ave} + 1 - \frac{H_2(\eta)}{\eta} + o(1).
\end{equation}
\end{proposition}
\begin{proof}
We first prove \eqref{eq:S-1_IAB}. Its converse part follows immediately from \cite[Chapter I, Theorem 5.3]{wilmink03}, which states that the secret-key rate cannot exceed $I(A;B)$ even if Eve possesses no quantum state that is correlated to $A$ and $B$. Its achievability part follows from \cite[Chapter III, Theorem 2.2]{wilmink03}: when we eliminate the ``helper subalgebra'', the theorem says that the forward key capacity (i.e., the maximum key rate achievable when Alice does not communicate to Bob) is lower-bounded by $I(A;B) - I(B;\mathbb{E})$ evaluated for the joint state consisting of Alice's and Bob's measurement outcomes and Eve's post-measurement state. As shown in Section~\ref{sec:key_source_setup}, Bob's measurement outcome is independent of Eve's post-measurement state, so $I(B;\mathbb{E})=0$.\footnote{In Section~\ref{sec:key_source_setup} we consider the case where Bob performs a complete projective measurement in the number-state basis, whereas here Bob's non-PNR detection only distinguishes between zero and positive photon numbers. But extending our claim for the former case to the latter is straightforward.}
We next prove \eqref{eq:S-1_efficiency}. Direct evaluation for the Z-channel mutual information \eqref{I_Z} for the channel parameters $q$ and $\mu$ of \eqref{eq:PAB} gives:
\begin{IEEEeqnarray}{rCl}
I(A;B) & = & I_Z(q,\mu) \\
& = & H_2(\e^{-\eta \ave}) - \left(1-\e^{-\ave}\right) H_2\left(\frac{1-\e^{-\eta \ave}}{1-\e^{-\ave}}\right) \IEEEeqnarraynumspace\\
& = & \eta \ave\log\frac{1}{\eta \ave} + \eta \ave-\ave H_2(\eta)+o(\ave). \label{eq:S-1_last}
\end{IEEEeqnarray}
Substituting in \eqref{eq:S-1_IAB} and dividing by $\eta \ave$ yields \eqref{eq:S-1_efficiency}.
\end{proof}
Hence the conceptually simple Scheme~\ref{Scheme_S-1}, which only uses non-PNR direct detection both at Alice and at Bob, is at most a constant term away from the optimal quantum efficiency whose upper bound is given in \eqref{eq:efficiency_Squantum} . Comparing this with \eqref{holevo_eff} and \eqref{eq:PE_channel_max} we see that the differences between the optimal photon efficiencies in communication, in Model~C, and in Model~S are at most constants. Interestingly, $r_\textnormal{S-1}(\eta,\ave)$ is asymptotically the same as the photon efficiency in the communication scenario where Alice sends binary number states \eqref{eq:numZ}.
The problem with Scheme~\ref{Scheme_S-1} is, though, that the source distribution $P_{AB}$ is highly skewed, which makes it difficult to find a good Slepian-Wolf code, much like the difficulty to obtain a channel code in the communication setting of Section~\ref{sec:communication}. While in communication and in Model~C Alice can use PPM to simplify code design, in Model~S this is no longer possible, as the sequences $\vect{A}$ and $\vect{B}$ are governed by the source, over which neither Alice nor Bob have control. Nevertheless, Alice and Bob can use a PPM-like scheme by \emph{parsing} the sequences into frames, as we next propose.
\subsection{Simple Frame-Parsing}
In a simple PPM-like scheme, Alice and Bob parse the source uses into frames, and only use the frames where each of them has exactly one detection to generate the key.
\begin{scheme}\label{Scheme_S-2}
\mbox{}
\begin{enumerate}
\item Alice and Bob perform non-PNR direct detection to obtain binary sequences $\mathbf{A}$ and $\mathbf{B}$, respectively.
\item Let $b$ be as in \eqref{eq:blocklog}. We divide the whole block of $k$ source uses into frames each consisting of $b$ consecutive uses (and ignore the remainder).
\item Bob selects all the frames in which he detects at least one photon ($B=1$ for at least one source use). Denote the labels of these frames by $\{i_1,i_2,\ldots\}$. He tells Alice the values of $i_1,i_2,\ldots$ using the public channel.
\item Alice selects the frames among $i_1,i_2,\ldots$ in which $A=1$ for \emph{exactly one} source use. Denote the labels of these frames by $\{i_{j_1},i_{j_2}\ldots\}$, Alice's detection positions within these frames by $\{Y_{i_{j_1}},Y_{i_{j_2}},\ldots\}$, and Bob's (unique) detection positions within these frames by $\{X_{i_{j_1}},X_{i_{j_2}},\ldots\}$. She tells Bob the values of $j_1,j_2,\ldots$ using the public channel.
\item Alice and Bob generate the raw key by taking the binary representations of $\{X_{i_{j_1}},X_{i_{j_2}},\ldots\}$ and of $\{Y_{i_{j_1}},Y_{i_{j_2}},\ldots\}$, respectively.
\item Alice and Bob perform privacy amplification on the raw key to obtain the secret key.
\end{enumerate}
\end{scheme}
As in Schemes~\ref{Scheme_C-1} and \ref{Scheme_C-2}, the information-reconciliation step in Scheme~\ref{Scheme_S-2} is uncoded and hence very simple. The performance of Scheme~\ref{Scheme_S-2} is similar to that of Scheme~\ref{Scheme_C-2} where Alice sends coherent states, in the sense that it loses a $\log\log\nicefrac{1}{\ave}$ term in photon efficiency compared to the optimum \eqref{eq:efficiency_Squantum}. Interestingly, here the loss does not come from the input states used, as they are identical to those in Scheme~\ref{Scheme_C-1}, but rather from the parsing process.
\begin{proposition}\label{prp:S-2}
The photon efficiency of Scheme~\ref{Scheme_S-2} satisfies
\begin{equation}\label{eq:S-2}
r_{\textnormal{S-2}}(\eta,\ave) = \log\frac{1}{\ave} - \log\log\frac{1}{\ave} - 1 + o(1).
\end{equation}
\end{proposition}
The scheme has some information leakage, since Eve can use her knowledge about the frames which were selected for key generation (obtained by listening to the public channel), in conjunction with the measurements she performs on the same frames. The proof, which appears in Appendix~\ref{app:S-2}, shows that this leakage is vanishing in the photon-efficient limit.
\emph{Note:} If Alice uses PNR direct detection (which is technically more difficult than non-PNR), then Scheme~\ref{Scheme_S-2} can be simplified so that it does not contain an privacy-amplification step. Indeed, Alice can select those frames in which she detects \emph{only one photon}. In this case, since Bob also detects photons (in fact, only one photon) in every such frame, we know that Eve's post-measurement states in these frames are all vacuum. Hence Eve has no information about $\tilde{X}$, and taking the binary representation of $\tilde{X}$ already gives Alice and Bob a secret key.
The information loss of Scheme~\ref{Scheme_S-2} compared to Scheme~{\ref{Scheme_S-1}} comes from two sources. First, the sequence $\{i_1,i_2,\ldots\}$ itself contains useful information that can be used to generate secret bits, but is not exploited in Scheme~\ref{Scheme_S-2}. Second, frames in which Alice detects photons in two or more source uses are discarded. As it turns out, the first source of information loss is dominant in the photon-efficient regime; we next show how this loss can be recovered. (Loss from the second source can also be partially recovered, e.g., by varying the frame-lengths~\cite{zhouwornell13}.)
\subsection{Enhanced Frame-Parsing}
Our idea of enhancing the frame-parsing scheme~\ref{Scheme_S-2} is to extract secret-key bits also from the sequence $\{i_1,i_2,\ldots\}$, which indicates the positions of frames selected by Bob. To this end, instead of sending this sequence uncoded, Bob uses a binary Slepian-Wolf code to send this information to Alice. Note that such a code is much easier to construct than the one in Scheme~\ref{Scheme_S-1}, as the zeros (frames not selected by Bob) and ones (frames selected by Bob) are much more balanced than in the original binary sequence $\vect{B}$; recall \eqref{eq:blocklog}. Assuming that an optimal Slepian-Wolf can be found, we can completely recover the $\log\log\nicefrac{1}{\ave}$ term and reduce the loss in photon efficiency to a constant term.
\begin{scheme}\label{Scheme_S-3}
\mbox{}
\begin{enumerate}
\item Alice and Bob use non-PNR direct detection to obtain binary sequences $\mathbf{A}$ and $\mathbf{B}$, respectively.
\item Let $b$ be as in \eqref{eq:blocklog}. We divide the whole block of $k$ source uses into frames each consisting of $b$ consecutive uses (and ignore the remainder).
\item Let $\tilde{B}_i$ be the indicator that Bob detects at least one photon within the $i$th frame, and let $\tilde{A}_i$ be the same indicator for Alice. Bob sends a Slepian-Wolf code to Alice using the public channel, so that Alice can recover $\tilde{\vect{B}}$ based on the codeword together with $\vect{\tilde{A}}$ with high probability.
\item Corresponding to every $i$ such that $\tilde{B}_i=1$, Alice sends a binary symbol $C_i$ to Bob: $C_i=1$ if within the $i$th frame there is \emph{exactly} one source use where $A=1$, and $C_i=0$ otherwise. Note that since Alice knows $\vect{\tilde{B}}$ with high probability, she can send $C_i$s simply as a bitstream in an increasing order in $i$ (and skip the $i$s for which $\tilde{B}_i=0$).
\item Alice and Bob perform privacy amplification on $\vect{\tilde{B}}$ to obtain the first part of the secret key.
\item For every $i$ such that $\tilde{B}_i=C_i=1$, let $X_i$ be the position where $A=1$, and let $Y_i$ be the (unique) position where $B=1$. Alice and Bob generate the second part of the secret key by taking the binary representations of $X_i$ and of $Y_i$, respectively, for all such $i$s, and by then performing privacy amplification.
\end{enumerate}
\end{scheme}
\begin{proposition}\label{prp:S-3}
Scheme~\ref{Scheme_S-3} achieves photon efficiency
\begin{equation}\label{eq:S-3}
r_\textnormal{S-3} (\eta,\ave) \ge \log\frac{1}{\ave}-\frac{H_2(\eta)}{\eta}+o(1)
\end{equation}
for all $\eta\in(0,1]$.
\end{proposition}
The proof, which appears in Appendix~\ref{app:S-3}, evaluates the key rate that Step 5) adds over the rate of Scheme~\ref{Scheme_S-2}. This part of the key consists of frame labels, thus it is obviously correlated with the messages sent over the public channel. However, we show that in the photon-efficient limit Eve must ``lose synchronization'' with the frame locations, thus the leakage is vanishing.
\subsection{Extension to the Case $\eta_A<1$}\label{sec:etaA}
The results for the case where $\eta_A$ in \eqref{eq:beamsplitterA} is equal to one can be extended to the case where $\eta_A<1$, though the expressions become considerably more cumbersome. We hence only give some heuristic explanations how our schemes should be modified, and how they perform. Note that for the following discussions the photon efficiency is defined in \eqref{eq:PE_etaAB}. Also recall that we assume that the source is co-located with Alice, such that the photons lost do not reach Eve.
\emph{Quantum Limit:} Proposition~\ref{prp:IAB} holds but with a different constant term. The same proof ideas apply.
\emph{Direct Detection:} Scheme~\ref{Scheme_S-1} can be directly applied to the case where $\eta_A<1$ without modification, and its photon efficiency is different from the right-hand side of \eqref{eq:S-1_efficiency} by a constant term, i.e., it is again at most a constant away from the quantum limit.
\emph{Simple Frame-Parsing:} Scheme~\ref{Scheme_S-2} needs some modifications in order to work when $\eta_A<1$. First, in Step 2) Bob should select only those frames in which there is \emph{exactly} one source use where $B=1$. This is because there can be frames in which Bob has more detections than Alice, due to the loss to Alice. Second, after Step~3) Bob needs to send Alice a $b$-ary Slepian-Wolf code on his detection positions inside the selected frames, so that Alice will know these positions with high probability. (This is a large-alphabet code for symmetric errors, and is relatively easy to construct.) This step is needed because, since both Alice and Bob only observe lossy versions of the source, their detection positions inside the selected frames might be different. Indeed, the two positions are equal if they come from the same source photon-pair, and are independent of each other if they come from two different source photon-pairs. Finally, for Step~5) (privacy amplification), Eve's side information needs to be examined more carefully compared to the case where $\eta_A=1$. After these modifications, one can show that the photon efficiency is the same as the right-hand side of \eqref{eq:S-2} up to the second term, i.e., the loss in photon efficiency scales like $\log\log\nicefrac{1}{\ave}$.
\emph{Enhanced Frame-Parsing:} If we incorporate the aforementioned modifications for Scheme~\ref{Scheme_S-2} to Scheme~\ref{Scheme_S-3}, then Scheme~\ref{Scheme_S-3} also works for the case $\eta_A<1$, and its photon efficiency is different from the right-hand side of \eqref{eq:S-3} by a constant.
We finally note that, for all three cases in which we restrict Alice and Bob to using direct detection, we can also take \emph{detector dark counts} into account. Statistically, a dark count at Alice can be treated as a source photon-pair that reaches Alice but not Bob; similarly for a dark count at Bob. For example, when the dark-count rates at Alice and at Bob are $\lambda_A$ and $\lambda_B$ counts per slot, respectively, we can model the system by replacing $\eta_A$, $\eta_B$, and $\ave$ with $\eta_A'$, $\eta_B'$, and $\ave'$ that are solved from
\begin{subequations}
\begin{IEEEeqnarray}{rCl}
\eta_A' \ave' & = & \eta_A \ave + \lambda_A\\
\eta_B' \ave' & = & \eta_B \ave + \lambda_B\\
\eta_A' \eta_B' \ave' & = & \eta_A \eta_B \ave
\end{IEEEeqnarray}
\end{subequations}
without introducing any new elements to the model. Note that this replacement of parameters yields the desired correlation between Alice's and Bob's photon counts, but does \emph{not} yield the correct form for Eve's optical states after Alice's and Bob's measurements. However, as our proofs show, information in Eve's optical states does not affect the dominant terms in secret-key rate in the regime of interest.
This observation combined with our results shows that dark counts only affect the constant term in photon efficiency, which is again similar to the previous results in optical communications~\cite{wangwornell14}.
\section{Discussion: Toward Secrecy with a General Adversary}\label{sec:discussion}
In this work we have presented schemes that approach the optimal key rate in the photon-efficient limit, up to a constant efficiency loss. Moreover, these schemes are practical, both in the physical sense (utilizing realizable transmissions and measurements) and in the algorithmic sense (using simple protocols and off-the-shelf codes). However, throughout the work we have assumed that Eve is limited to passive eavesdropping through a beamsplitter channel. We now comment on the problems that may arise when this model does not hold, and point out ways to overcome them.
First, suppose that Eve is still passive, but is free to change the beamsplitter transmissivity $\eta$ as a function of time, as long as it satisfies some average constraint $\bar \eta$. We now distinguish between two strategies that Eve can use:
\begin{enumerate}
\item Pre-scheduled transmissivity. Take, for example, Scheme~\ref{Scheme_C-2}, and imagine that for each PPM frame, Eve uses $\eta=0$ for half the block, and $\eta=1$ for the other half. Then she knows that the key pertaining to this frame must correspond to the part where $\eta=1$, gaining one bit per detected photon (thus reducing the key efficiency by $\log 2$). This kind of attack can go undetected, provided that Eve randomizes the schedule. However, it is plausible that the efficiency loss is bounded by a constant for any schedule.
\item Measurement-dependent transmissivity. In principle, Eve can change $\eta$ in a causal manner, based upon her measurement outcomes. However, we believe that the gain from using measurements can be shown to vanish in the photon-efficient limit, by the same techniques used to show that the information leakage is small.
\end{enumerate}
It however still remains to be investigated whether our intuitions above are correct, i.e., whether Eve indeed cannot gain from changing the beamsplitter transmissivity.
If Eve is allowed to transmit as well, other types of attacks are possible. A very simple and efficient one is ``intercept and resend'': Eve uses direct detection on the channel meant for Bob, and then upon detection of a photon, transmits a substitute one to Bob. This way Eve can obtain information about Bob's sequence of detections, and if she uses much higher bandwidth than Bob, the delay will not be detected.
In fact, all QKD protocols face this problem. For example, in the BB84 protocol \cite{bennettbrassard84}, the key is generated using the polarization of a photon; Eve can make a measurement, then transmit to Bob a photon with the same polarization. The solution for BB84 is that Alice and Bob measure in either of two \emph{mutually unbiased} bases, according to local randomness. Only if they happened to measure in the same basis, the measurement results are used, inflicting a rate loss of factor~$2$. By sacrificing rate, they can now \emph{a posteriori} find out whether they used the same basis, and compare the correlation of the polarizations to the expected statistics, thus authenticating the received photons.
Extending this idea to schemes based on photon arrival times involves an extension of the concept of mutually unbiased bases to continuous variables; see \cite{weigertwilkinson08}.
Specifically, in Model C the modulation and measurements can be performed either in the time or in the frequency domain with the help of dispersive optics; see \cite{mowerzhang12}. Alternatively, in Model S, one can use interferometry to verify that the photons received by Alice and Bob are indeed entangled; see \cite{mowerwong11}.
\appendix
\subsection{Proof of Proposition~\ref{prp:C-2}}\label{app:C-2}
By the same argument as in the proof of
Proposition~\ref{prp:C-1}, we know that the raw keys generated by Alice and Bob (before privacy amplification) are the same, and are independent of Bob's messages in the information-reconciliation step. It is, however, dependent on Eve's optical states. We thus need to determine how much secret key can be distilled from the raw key.
The quantum states in different frames are mutually independent, so we need only to analyze one frame that is selected by Bob. We note that, when
Alice sends the coherent state $|\sqrt{b \ave}\ket$,
Eve's output
state is $|\sqrt{(1-\eta)b \ave}\ket$, and is independent of
Bob's measurement outcome conditional on Alice's input. Thus, using~\eqref{eq:rennerkonig}, we know
that the number of secret nats we can obtain in each selected frame can
be arbitrarily close to $H(\tilde{X}| \rho^{\mathbb{E}^b})$, where $\tilde{X}$ is uniformly distributed over $\{1,\ldots,b\}$, and where
$\rho^{\mathbb{E}^b}$ is a $b$-mode bosonic state described as follows: conditional on
$\tilde{X}=i$, $i\in\{1,\ldots,b\}$, $\rho^{\mathbb{E}^b}$
has the
coherent state $|\sqrt{(1-\eta)b \ave}\ket$ in the $i$th
mode and has the vacuum state $|0\ket$ in all other modes. Note
that
the total number of photons in $\rho^{\mathbb{E}^b}$ is $(1-\eta)
b \ave$, so
\begin{IEEEeqnarray}{rCl}
H(\rho^{\mathbb{E}^b}) & \le & b \big\{\bigl(1+(1-\eta)\ave\bigr)
\log \bigl(1+(1-\eta)\ave\bigr) \nonumber\\
& & ~~~{}- (1-\eta)\ave\log
\bigl((1-\eta)\ave\bigr)\bigr\}\label{eq:coherent_22}\\
& = & \left\lceil \frac{1}{\ave\log\nicefrac{1}{\ave}}
\right\rceil \left\{(1-\eta)\ave\log\frac{1}{\ave} + O(\ave) \right\}
\\
& = & (1-\eta)+o(1).
\end{IEEEeqnarray}
Here, \eqref{eq:coherent_22} follows from the well-known fact that
the maximum entropy
of a $b$-mode bosonic state with a certain average photon number is achieved by the state consisting of $b$ i.i.d. thermal states
\cite{holevosohmahirota99}. Now the number of secret nats per
selected frame satisfies
\begin{IEEEeqnarray}{rCl}
H(\tilde{X}| \rho^{\mathbb{E}^b}) & = & H(\tilde{X}) - I(\tilde{X};\rho^{\mathbb{E}^b}) \\
& \ge & H(\tilde{X}) - H(\rho^{\mathbb{E}^b}) \\
& = & \log b - H(\rho^{\mathbb{E}^b})\\
& = & \log\frac{1}{\ave} - \log\log\frac{1}{\ave} - (1-\eta) + o(1).
\end{IEEEeqnarray}
We next consider the number of frames per $k$ channel uses that will be selected by Bob, which we denote by $N(k)$. When Alice sends $|\sqrt{b \ave}\ket$, Bob's output has
a Poisson distribution of mean $\eta b \ave$, so the
probability that Bob detects at least one photon is $1-\e^{-\eta
b \ave}$. Hence, by the Law of Large Numbers,
\begin{equation}\label{eq:C-2_1}
\lim_{k\to\infty} \frac{N(k)}{k} =\lim_{k\to\infty} \frac{(1-\e^{-\eta b \ave})\lceil \nicefrac{k}{b}\rceil }{k} = \frac{1-\e^{-\eta b \ave}}{b}
\end{equation}
with probability one. Using
\begin{equation}
\e^{-x} \le 1 - x + \frac{x^2}{2},\quad x\ge 0,
\end{equation}
the right-hand side of \eqref{eq:C-2_1} can be lower-bounded as
\begin{equation}
\frac{1-\e^{-\eta b \ave}}{b} \ge \eta \ave \left(1 -
\frac{\eta b \ave}{2}\right).
\end{equation}
The photon efficiency of the proposed scheme can now be
lower-bounded as
\begin{IEEEeqnarray}{rCl}
r_{\textnormal{C-2}} & = & \frac{1}{\eta \ave} \cdot \frac{1-\e^{-\eta b \ave}}{b} \cdot H(\tilde{X}| \rho^{\mathbb{E}^b})\\
& \ge & \left(1 - \frac{\eta b \ave}{2}\right)
\left\{\log\frac{1}{\ave} - \log\log\frac{1}{\ave}
- (1-\eta) + o(1)\right\} \nonumber\\ {} \\
& = & \left(1-\frac{\eta \left\lceil \frac{1}{\ave \log\nicefrac{1}{\ave}} \right\rceil \ave}{2} \right) \nonumber\\
& & {}\cdot \left\{\log\frac{1}{\ave} - \log\log\frac{1}{\ave}
- (1-\eta) + o(1)\right\} \\
& = & \log\frac{1}{\ave} - \log\log\frac{1}{\ave} - (1-\eta) + o(1),
\end{IEEEeqnarray}
which is as claimed.
\subsection{Proof of Proposition~\ref{prp:S-2}}\label{app:S-2}
We first observe that, in every selected frame, the detection positions of Alice and Bob must be the same. This is because, due to \eqref{eq:PAB}, $B=1$ can happen only if $A=1$, and because by our choice each selected frame contains only one source use where $A=1$. We thus know that Alice's and Bob's raw keys are the same with probability one.
To obtain the secret-key rate, we need to compute the entropy of the raw key conditional on Eve's observations. Note that the quantum states inside different frames are mutually independent. We consider one frame that is selected by Alice and Bob. Denote the detection position in the frame by $\tilde{X}$. It is clear that $\tilde{X}$ is uniformly distributed over $\{1,\ldots,b\}$ and is independent of the label of this frame. All Eve's information about $\tilde{X}$ is in her optical state from the $b$ source uses that form this frame: if the source use where $A=B=1$ contains more than one photons, then Eve could also detect a photon in this source use, hence knowing Alice's and Bob's detection position. But, as we next show, this information leakage is small. To this end, we first note that in source uses where $A=B=0$, Eve's optical state is vacuum. Indeed, according to our source model, the number of photons in Alice's state equals the sum of the numbers of photons in Bob's and Eve's states with probability one. Therefore, when both Alice and Bob make direct detections on a source use and observe no photon, Eve's post-measurement state in the same source use becomes the vacuum state. In the (unique) source use where $A=B=1$, Eve's post-measurement state is the same as her state \emph{without} the condition $A=B=1$ given in \eqref{eq:sigmaE}. This is because $A=B=1$ means nothing but that Bob's photon number is positive, but Eve's post-measurement state is independent of Bob's photon number, as shown in Section~\ref{sec:key_source_setup}. Denote Eve's state over the whole frame by $\sigma^{\mathbb{E}^b}$. We now know that it consists of $b-1$ vacuum states and one state of the form \eqref{eq:sigmaE} whose position inside the frame is random. The expected number of photons in $\sigma^{\mathbb{E}^b}$ is $(1-\eta)\ave$, so the entropy of $\sigma^{\mathbb{E}^b}$ is upper-bounded by \cite{holevosohmahirota99}
\begin{equation}
H(\sigma^{\mathbb{E}^b}) \le b\cdot g\left(\frac{(1-\eta)\ave}{b}\right) = o(1).
\end{equation}
Thus the amount of secret information extractable from one selected frame is lower-bounded by
\begin{IEEEeqnarray}{rCl}
H(\tilde{X}| \sigma^{\mathbb{E}^b}) & = & H(\tilde{X}) - I(\tilde{X};\sigma^{\mathbb{E}^b})\\
& \ge & H(\tilde{X}) - H(\sigma^{\mathbb{E}^b})\\
& \ge & \log \left\lceil \frac{1}{\ave\log\nicefrac{1}{\ave}} \right\rceil +o(1)\\
& = & \log\frac{1}{\ave} - \log\log\frac{1}{\ave} + o(1).
\end{IEEEeqnarray}
It now remains to compute the probability that a specific frame will be selected by Alice and Bob. A simple lower bound on the probability of a frame being selected is the following: suppose both Bob and Eve make PNR direct detections on their states, then a frame is selected by Alice and Bob if (but not only if) Bob detects exactly one photon in the frame while Eve detects no photon. Bob's photon number has a Poisson distribution of mean $\eta b \ave$, while Eve's photon number has a Poisson distribution of mean $b(1-\eta)\ave$, and the two photon numbers are independent. Hence the probability a frame being selected is lower-bounded by
\begin{equation}\label{eq:probselect}
\left(\eta b \ave \e^{-\eta b \ave}\right)\cdot \left(\e^{-b(1-\eta)\ave}\right) = \eta b \ave - \eta b^2 \ave^2 + o\left(\frac{1}{\log\nicefrac{1}{\ave}}\right).
\end{equation}
Multiplying \eqref{eq:probselect} with $H(\tilde{X}| \sigma^{\mathbb{E}^b})$ gives us the secret-key nats per frame, where we count both selected and unselected frames. Simple normalization then yields the photon efficiency
\begin{IEEEeqnarray}{rCl}
r_\textnormal{S-2}(\eta,\ave)
& \ge & \frac{1}{{\eta b \ave}} \cdot \left(\eta b \ave - \eta b^2 \ave^2 + o\left(\frac{1}{\log\nicefrac{1}{\ave}}\right)\right) \nonumber\\
& & {} \cdot \left(\log\frac{1}{\ave} - \log\log\frac{1}{\ave} + o(1)\right)\\
& = & \log\frac{1}{\ave} - \log\log\frac{1}{\ave} - 1 + o(1).
\end{IEEEeqnarray}
\subsection{Proof of Proposition~\ref{prp:S-3}}\label{app:S-3}
The second part of the secret key, which is generated in Step 5) in Scheme~\ref{Scheme_S-3}, is exactly the (whole) secret key generated by Scheme~\ref{Scheme_S-2}, and hence contributes to the total photon efficiency by the right-hand side of~\eqref{eq:S-2}. It is clear that this is independent of the first part of the key, as the latter only contains information of the frame labels. We thus only need to evaluate the contribution to the photon efficiency from the first part of the key which is generated in Step~5).
Consider a block of $\ell$ length-$b$ frames. To compute the length of the first part of the key that can be obtained from these frames, we first consider the information leakage due to Bob's message to Alice. Note that $(\tilde{A}^\ell,\tilde{B}^\ell)$ is distributed i.i.d. in time, where
each pair $(A,B)$ has joint distribution according to a Z channel with
\begin{subequations}\label{eq:PABtilde}
\begin{IEEEeqnarray}{rCl}
\tilde q &\triangleq& P_A(1) = 1 - \e^{-b \ave} \\
\tilde \mu & \triangleq& P_{B|A}(1|1) = \frac{1-\e^{-\eta b \ave}}{1-\e^{-b \ave}}.
\end{IEEEeqnarray}
\end{subequations}
The optimal Slepian-Wolf code for Bob to convey $\tilde{B}^{\ell}$ to Alice should contain, asymptotically, $H(\tilde{B}|\tilde{A})$ nats per frame \cite{slepianwolf73}. Let $M_B$ be the message which Bob sends to Alice, then
\begin{equation}
H(M_B) = \ell H(\tilde{B}|\tilde{A}) + \ell \epsilon
\end{equation}
where $\epsilon$ tends to zero as $\ell$ tends to infinity.
We next bound the information leakage due to the message which Alice sends to Bob. A simple upper bound is: for each frame where $\tilde{B}=1$, Alice needs to send Bob at most one bit. From \eqref{eq:PABtilde} we can obtain
\begin{equation}\label{eq:S-3_1}
P_{\tilde{B}}(1) = 1-\e^{-\eta b \ave}.
\end{equation}
Let $M_A$ be the message which Bob sends to Alice for $\ell$ frames, then
\begin{equation}\label{eq:S-3_2}
H(M_A) \le \ell \left(1-\e^{-\eta b \ave}\right) + \ell \epsilon.
\end{equation}
We finally consider Eve's quantum state from the optical channel. Denote this state over $\ell$ frames by $\rho^{\mathbb{E}^{b\ell}}$. Since, as shown in Section~\ref{sec:key_source_setup}, it is independent of Bob's measurement (direct detection) outcomes, and since $\tilde{B}$ is a function of Bob's measurement outcomes, we know that
\begin{equation}\label{eq:S-3_3}
I\left(\tilde{B}^{\ell}; \rho^{\mathbb{E}^{b\ell}} \right) = 0.
\end{equation}
We now use \eqref{eq:S-3_1}, \eqref{eq:S-3_2} and \eqref{eq:S-3_3} to bound the length of the first part of the key for $\ell$ frames which, according to \eqref{eq:rennerkonig}, is given by
\begin{IEEEeqnarray}{rCl}
\lefteqn{H(\tilde{B}^{\ell}| M_A,M_B, \rho^{\mathbb{E}^{b\ell}})}~~\nonumber\\
& = & H(\tilde{B}^{\ell}) - \underbrace{I(\tilde{B}^{\ell}; \rho^{\mathbb{E}^{b\ell}})}_{=0} - \underbrace{I(M_A,M_B; \tilde{B}^{b\ell}| \rho^{\mathbb{E}^{b\ell}})}_{\le H(M_A) + H(M_B)} \IEEEeqnarraynumspace\\
& \ge & \underbrace{H(\tilde{B}^\ell)}_{=\ell H(\tilde{B})} - \underbrace{H(M_A)}_{\le \ell(1-\e^{-\eta b \ave}) + \ell \epsilon} - \underbrace{H(M_B)}_{=\ell H(\tilde{B}|\tilde{A}) + \ell \epsilon}\\
& \ge & \ell H(\tilde{B}) - \ell \left(1-\e^{-\eta b \ave}\right) - \ell H(\tilde{B}|\tilde{A}) - 2 \ell \epsilon\\
& = & \ell I(\tilde{A};\tilde{B}) + \ell \eta b \ave - 2 \ell \epsilon + o(\ave).
\end{IEEEeqnarray}
Hence, for large enough $\ell$, the length of the first part of the key \emph{per frame} is given by
\begin{equation}\label{eq:S-3_4}
I(\tilde{A};\tilde{B}) + \ell \eta b \ave + o(\ave).
\end{equation}
We next evaluate $I(\tilde{A};\tilde{B})$. Comparing the parameters \eqref{eq:PABtilde} to \eqref{eq:PAB}, we see that $I(\tilde{A};\tilde{B})$ is the same as $I(A;B)$ \eqref{eq:S-1_last}, replacing $\ave$ with $b \ave$, where, recalling \eqref{eq:blocklog},
\begin{equation}
b \ave = \frac{1}{\log\nicefrac{1}{\ave}} + o(\ave).
\end{equation}
Thus,
\begin{IEEEeqnarray}{rCl}
I(\tilde{A};\tilde{B}) & = & H_2(\e^{-\eta b \ave}) - \left(1-\e^{-b \ave}\right) H_2\left(\frac{1-\e^{-\eta b \ave}}{1-\e^{-b \ave}}\right) \IEEEeqnarraynumspace\\
& = & \eta b \ave \log \log \frac{1}{\ave} + \eta b \ave-b \ave H_2(\eta)\nonumber\\
& & {} +o\left(\frac{1}{\log\nicefrac{1}{\ave}}\right).\label{eq:S-3_5}
\end{IEEEeqnarray}
We can now compute the photon efficiency coming from the first part of the secret key in Scheme~\ref{Scheme_S-3} by dividing \eqref{eq:S-3_4} by $\eta b \ave$ (the average number of photons Bob detects per frame), and by using \eqref{eq:S-3_5}. This photon efficiency is at least
\begin{equation} \label{eq:S-3_6}
\log\log\frac{1}{\ave} + 1 - \frac{H_2(\eta)}{\eta} + o(1).
\end{equation}
Adding \eqref{eq:S-3_6} to the right-hand side of \eqref{eq:S-2}, i.e., to the photon efficiency coming from the second part of the secret key, we conclude that
\begin{equation}
r_\textnormal{S-3} (\eta,\ave) \ge \log\frac{1}{\ave} - \frac{H_2(\eta)}{\eta} + o(1).
\end{equation}
\section*{Acknowledgments}
The authors thank Nivedita Chandrasekaran and Jeffrey Shapiro for helpful discussions, and the anonimous reviewers for useful comments.
\bibliographystyle{IEEEtran}
\bibliography{/Volumes/Data/wang/Library/texmf/tex/bibtex/header_short,/Volumes/Data/wang/Library/texmf/tex/bibtex/bibliofile}
\end{document}
| 120,024
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\begin{document}
\title[Properties of Strongly Balanced Tilings by Convex Polygons]{Properties of
Strongly Balanced Tilings by Convex Polygons}
\address{The Interdisciplinary Institute of Science, Technology and Art\\
Suzukidaini-building 211, 2-5-28 Kitahara, Asaka-shi, Saitama, 351-0036,
Japan}
\email{ismsugi@gmail.com}
\author{Teruhisa Sugimoto}
\maketitle
\begin{abstract}
Every normal periodic tiling is a strongly balanced
tiling. The properties of periodic tilings by convex polygons are rearranged
from the knowledge of strongly balanced tilings. From the results, we show
the properties of representative periodic tilings by a convex pentagonal
tile.
\end{abstract}
\section{Introduction}
\label{section1}
A collection of sets (the `tiles') is a\textit{ tiling} (or tessellation) of the plane if
their union covers the whole plane but the interiors of different tiles are
disjoint. If all the tiles in a tiling are of the same size and shape, then
the tiling is \textit{monohedral}.
In this study, a polygon that admits a monohedral tiling is a \textit{polygonal tile}.
A tiling by convex polygons is \textit{edge-to-edge} if any two convex polygons
in a tiling are either disjoint or share one vertex (or an entire edge) in common.
A tiling is periodic if it coincides with its translation by a nonzero vector. The
unit that can generate a periodic tiling by translation only is known as the
\textit{fundamental region}~\cite{G_and_S_1987, Sugimoto_2012a, Sugimoto_2015,
Sugimoto_NoteTP, Sugimoto_APTCP}.
In the classification problem of convex polygonal tiles, only the pentagonal
case is open. At present, fifteen types of convex pentagon tiles are known
(see Figure~\ref{fig1}) but it is not known whether this list is
complete~\cite{Gardner_1975a, Gardner_1975b, G_and_S_1987, Hallard_1991,
Hirshh_1985, Kershner_1968, Klamkin_1980, Mann_2015, Reinhardt_1918,
Schatt_1978, Stein_1985, Sugimoto_2012a, Sugimoto_2015, Sugi_Ogawa_2006}.
However, it has been proved that a convex pentagonal tile that can generate
an edge-to-edge tiling belongs to at least one of the eight known
types~\cite{Bagina_2011, Sugimoto_2012b, Sugimoto_NoteTP, Sugimoto_2016}.
We are interested in the problem of convex pentagonal tiling (i.e., the complete
list of types of convex pentagonal tile, regardless of edge-to-edge and
non-edge-to-edge tilings). However, the solution of the problem is not easy.
Therefore, we will first treat only convex pentagonal tiles that admit at
least one periodic tiling\footnote{ We know as a fact that the 15 types of
convex pentagonal tile admit at least one periodic tiling. From this, we find
that the convex pentagonal tiles that can generate an edge-to-edge tiling
admit at least one periodic tiling~\cite{G_and_S_1987, Mann_2015, Sugimoto_2012a,
Sugimoto_2012b, Sugimoto_2015, Sugimoto_NoteTP, Sugimoto_APTCP, Sugimoto_2016}.
On the other hand, there is no proof that they admit at least one periodic
tiling without using this fact. That is, there is no assurance yet that all
convex pentagonal tiles admit at least one periodic tiling. In the solution
of the problem of convex pentagonal tiling, it is necessary to consider whether
there is a convex polygonal tile that admits infinitely many tilings of the
plane, none of which is periodic.}. As such, we consider that the properties
of the periodic tilings by convex polygons should be rearranged. From
statement 3.4.8 (``every normal periodic tiling is strongly balanced'') in
\cite{G_and_S_1987}, we see that periodic tilings by convex polygonal tiles
(i.e., monohedral periodic tilings by convex polygons) are contained in
the strongly balanced tilings. The definitions of normal and strongly balanced
tilings are given in Section~\ref{section2}. In this paper, the properties of
strongly balanced tilings by convex polygons are presented from the
knowledge of strongly balanced tilings in general. That is, the properties
correspond to those of periodic tilings by a convex polygonal tile.
\renewcommand{\figurename}{{\small Figure.}}
\begin{figure}[htbp]
\centering\includegraphics[width=13cm,clip]{PSBTCP_fig1.png}
\caption{{\small
Fifteen types of convex pentagonal tiles. If a convex pentagon can generate
a monohedral tiling and is not a new type, it belongs to at least one of types 1--15.
Each of the convex pentagonal tiles is defined by some conditions between
the lengths of the edges and the magnitudes of the angles, but some degrees
of freedom remain. For example, a convex pentagonal tile belonging to type 1
satisfies that the sum of three consecutive angles is equal to $360^\circ$.
This condition for type 1 is expressed as $A + B + C = 360^\circ$ in this figure.
The pentagonal tiles of types 14 and 15 have one degree of freedom, that of
size. For example, the value of $C$ of the pentagonal tile of type 14 is $\cos
^{ - 1}((3\sqrt {57} - 17) / 16) \approx 1.2099 \; \mbox{rad} \approx 69.32^\circ$.
}
\label{fig1}
}
\end{figure}
\renewcommand{\figurename}{{\small Figure.}}
\begin{figure}[htbp]
\centering\includegraphics[width=11cm,clip]{PSBTCP_fig2.png}
\caption{{\small
The differences between corners and vertices, sides and edges,
adjacents, and neighbors. The points $A$, $B$, $C$, $E$, $F$, and $G$ are corners
of the tile $T$; but $A$, $C$, $D$, $E$, and $G$ are vertices of the tiling (we note
that the valence of vertices $A$ and $G$ is four, and the valence of vertices
$C$, $D$, and $E$ is three). The line segments \textit{AB}, \textit{BC}, \textit{CE},
\textit{EF}, \textit{FG,} and \textit{GA} are sides of $T$, while \textit{AC},
\textit{CD}, \textit{DE}, \textit{EG}, and \textit{GA} are edges of the tiling.
The tiles $T_{1}$, $T_{3}$, $T_{4}$, $T_{5}$, and $T_{6}$ are adjacents (and
neighbors) of $T$, whereas tiles $T_{2}$ and $T_{7}$ are neighbors (but not
adjacents) of $T$ \cite{G_and_S_1987}.
}
\label{fig2}
}
\end{figure}
\section{Preparation}
\label{section2}
Definitions and terms of this section quote from \cite{G_and_S_1987}.
Terms ``vertices'' and ``edges'' are used by both polygons and tilings. In order
not to cause confusion, \textit{corners} and \textit{sides} are referred to instead
of vertices and the edges of polygons, respectively. At a vertex of a polygonal tiling,
corners of two or more polygons meet and the number of polygons meeting at
the vertex is called the \textit{valence} of the vertex , and is at least three (see
Figure~\ref{fig2}). Therefore, an edge-to-edge tiling by polygons is such that the
corners and sides of the polygons in a tiling coincide with the vertices and edges
of the tiling.
Two tiles are called \textit{adjacent} if they have an edge in common, and then
each is called an adjacent of the other. On the other hand, two tiles are called
\textit{neighbors} if their intersection is nonempty (see Figure~\ref{fig2}).
There exist positive numbers $u$ and $U$ such that any tile contains a certain
disk of radius $u$ and is contained in a certain disk of radius $U$ in which case
we say the tiles in tiling are \textit{uniformly bounded}.
A tiling $\Im$ is called \textit{normal} if it satisfies following conditions:
(i) every tiles of $\Im $ is a topological disk; (ii) the intersection of every two
tiles of $\Im$ is a connected set, that is, it does not consist of two (or
more) distinct and disjoint parts; (iii) the tiles of $\Im$ are uniformly
bounded.
Let $D(r,M)$ be a closed circular disk of radius $r$, centered at any point
$M$ of the plane. Let us place $D(r,M)$ on a tiling, and let $F_{1}$ and $F_{2}$
denote the set of tiles contained in $D(r,M)$ and the set of meeting
boundary of $D(r,M)$ but not contained in $D(r,M)$, respectively. In
addition, let $F_{3}$ denote the set of tiles surrounded by these in
$F_{2}$ but not belonging to $F_{2}$. The set $F_1 \cup F_2 \cup F_3 $ of
tiles is called the patch $A(r,M)$ of tiles generated by $D(r,M)$.
For a given tiling $\Im$, we denote by $v(r,M)$, $e(r,M)$, and $t(r,M)$ the
numbers of vertices, edges, and tiles in $A(r,M)$, respectively. The tiling
$\Im$ is called \textit{balanced} if it is normal and satisfies the following
condition: the limits
\[
\mathop {\lim }\limits_{r \to \infty } \frac{v(r,M)}{t(r,M)}\quad
\mbox{and}\quad \mathop {\lim }\limits_{r \to \infty }
\frac{e(r,M)}{t(r,M)}
\]
\noindent
exist. Note that $v(r,M) - e(r,M) + t(r,M) = 1$ is called Euler's Theorem
for Planar Maps.
\begin{sta}[Statement 3.3.13 in \cite{G_and_S_1987}]
\label{sta1}
Every normal periodic tiling is balanced.
\end{sta}
\begin{nameth}[Statement 3.3.3 in \cite{G_and_S_1987}]
\label{EulerThm}
For any normal tiling $\Im$, if one of the limits
$v(\Im ) = \mathop {\lim }\limits_{r \to \infty } \frac{v(r,M)}{t(r,M)}$ or
$e(\Im ) = \mathop {\lim }\limits_{r \to \infty } \frac{e(r,M)}{t(r,M)}$ exists
and is finite, then so does the other. Thus the tiling is balanced and,
moreover,
\begin{equation}
\label{equqtion1}
v(\Im ) = e(\Im ) - 1.
\end{equation}
\end{nameth}
For a given tiling $\Im$, we write $t_h (r,M)$ for the number of tiles with
$h$ adjacents in $A(r,M)$, and $v_j (r,M)$ for the numbers of $j$-valent
vertices in $A(r,M)$. Then the tiling $\Im$ is called \textit{strongly balanced}
if it is normal and satisfies the following condition: all the limits
\[
t_h (\Im ) = \mathop {\lim }\limits_{r \to \infty } \frac{t_h
(r,M)}{t(r,M)}\quad \mbox{and}\quad v_j (\Im ) = \mathop {\lim }\limits_{r
\to \infty } \frac{v_j (r,M)}{t(r,M)}
\]
\noindent
exist. Then,
\begin{equation}
\label{equqtion2}
\sum\limits_{h \ge 3} {t_h (\Im ) = 1} \quad \mbox{and}\quad v(\Im )
= \sum\limits_{j \ge 3} {v_j (\Im )}
\end{equation}
\noindent
hold. Therefore, every strongly balanced tiling is necessarily balanced.
When $\Im$ is strongly balanced, we have
\begin{equation}
\label{equqtion3}
2e(\Im ) = \sum\limits_{j \ge 3} {j \cdot v_j (\Im )}
= \sum\limits_{h \ge 3} {h \cdot t_h (\Im )} .
\end{equation}
In addition, as for strongly balanced tiling, following properties are
known.
\begin{sta}[Statement 3.4.8 in \cite{G_and_S_1987}]
\label{sta2}
Every normal periodic tiling is strongly balanced.
\end{sta}
\begin{sta}[Statement 3.5.13 in \cite{G_and_S_1987}]
\label{sta3}
For each strongly balanced tiling $\Im $ we have
\begin{equation}
\label{equqtion4}
\frac{1}{\sum\limits_{j \ge 3} {j \cdot w_j (\Im )} } +
\frac{1}{\sum\limits_{h \ge 3} {h \cdot t_h (\Im )} } = \frac{1}{2}
\end{equation}
\noindent
where
\[
w_j (\Im ) = \frac{v_j (\Im )}{v(\Im )}.
\]
\end{sta}
Thus $w_j (\Im )$ can be interpreted as that fraction of the total number of
vertices in $\Im$ which have valence $j$, and $\sum\limits_{j \ge 3} {j \cdot
w_j (\Im )} $ is the \textit{average} valence taken over all the vertices. Since
$\sum\limits_{h \ge 3} {t_h (\Im ) = 1} $ there is a similar interpretation
of $\sum\limits_{h \ge 3} {h \cdot t_h (\Im )} $: it is the \textit{average}
number of adjacents of the tiles, taken over all the tiles in $\Im$.
Since the valence of the vertex is at least three,
\begin{equation}
\label{equqtion5}
\sum\limits_{j \ge 3} {j \cdot w_j (\Im )} \ge 3.
\end{equation}
\begin{sta}[Statement 3.5.6 in \cite{G_and_S_1987}]
\label{sta4}
In every strongly balanced tiling $\Im$ we have
\[
2\sum\limits_{j \ge 3} {(j - 3) \cdot v_j (\Im )} + \sum\limits_{h \ge 3}
{(h - 6) \cdot t_h (\Im ) = 0} ,
\]
\[
\sum\limits_{j \ge 3} {(j - 4) \cdot v_j (\Im )} + \sum\limits_{h \ge 3} {(h
- 4) \cdot t_h (\Im ) = 0} ,
\]
\[
\sum\limits_{j \ge 3} {(j - 6) \cdot v_j (\Im )} + 2\sum\limits_{h \ge 3}
{(h - 3) \cdot t_h (\Im ) = 0} .
\]
\end{sta}
\section{Consideration and Discussion}
\label{section3}
A polygon with $n$ sides and $n$ corners is referred to as an \textit{n-gon}.
For the discussion below, note that $n$-gons in a strongly balanced tiling do
not need to be congruent (i.e., the tiling does not need to be monohedral).
\subsection{Case of convex $n$-gons}
\label{subsection3_1}
Let $\Im _n^{sb} $ be a strongly balanced tiling by convex $n$-gons.
\begin{prop}\label{prop1}
$\sum\limits_{h \ge 3} {h \cdot t_h (\Im _n^{sb} )} \le 6$.
\end{prop}
\noindent
\textbf{\textit{Proof}.}
From (\ref{equqtion4}), we have that
\[
\frac{2\sum\limits_{h \ge 3} {h \cdot t{ }_h(\Im _n^{sb} )} }{\sum\limits_{h
\ge 3} {h \cdot t_h (\Im _n^{sb} )} - 2}
= \sum\limits_{j \ge 3} {j \cdot w_j (\Im _n^{sb} )} .
\]
\noindent
Since the valence of the vertex is at least three, i.e.,
$\sum\limits_{j \ge 3} {j \cdot w_j (\Im _n^{sb} )} \ge 3$,
\[
\frac{2\sum\limits_{h \ge 3} {h \cdot t{ }_h(\Im _n^{sb} )} }{\sum\limits_{h
\ge 3} {h \cdot t_h (\Im _n^{sb} )} - 2} \ge 3 \quad .
\]
\noindent
Therefore, we obtain Proposition~\ref{prop1}.
\hspace{7cm} $\square$
\bigskip
From Proposition~\ref{prop1}, there is no strongly balanced tiling that is formed by
convex $n$-gons for $n \ge 7$, since the average number of adjacents is greater
than six. Note that the number of sides of all convex polygons in $\Im _n^{sb}$
does not have the same necessity. For example, there is no strongly balanced
tiling by convex 6-gons and convex 8-gons, and there is no strongly balanced
tiling by only convex 5-gons whose number of adjacents is seven or more.
Note that Proposition~\ref{prop1} is not a proof that there is no convex
polygonal tile with seven or more sides. If there were a proof that all convex
polygonal tiles admit at least one periodic tiling, it could be used to
prove that there is no convex polygonal tile with seven or more sides from
Proposition~\ref{prop1}.
\begin{prop}\label{prop2}
$3 \le \sum\limits_{j \ge 3} {j \cdot w_j (\Im _n^{sb} )} \le \frac{2n}{n - 2}$.
\end{prop}
\noindent
\textbf{\textit{Proof}.}
From (\ref{equqtion4}) and $\sum\limits_{h \ge n} {h \cdot t_h (\Im _n^{sb} )} \ge n$,
\[
\frac{2\sum\limits_{j \ge 3} {j \cdot w{ }_j(\Im _n^{sb} )} }{\sum\limits_{j
\ge 3} {j \cdot w_j (\Im _n^{sb} )} - 2} = \sum\limits_{h \ge n} {h \cdot
t_h (\Im _n^{sb} )} \ge n.
\]
\noindent
Therefore, from the above inequality and (\ref{equqtion5}), we obtain
Proposition~\ref{prop2}.
\hspace{1.6cm} $\square$
\bigskip
Let $\Im _n^{sbe} $ be a strongly balanced edge-to-edge tiling by convex
$n$-gons.
\begin{prop}\label{prop3}
$\sum\limits_{j \ge 3} {j \cdot w_j (\Im _n^{sbe} )} = \frac{2n}{n - 2}$.
\end{prop}
\noindent
\textbf{\textit{Proof}.}
The number of adjacents of all convex $n$-gons in $\Im _n^{sb} $ is equal to $n$.
That is, $\sum\limits_{h \ge n} {h \cdot t_h (\Im _n^{sbe} )} = n$. Then,
(\ref{equqtion4}) is $\frac{1}{\sum\limits_{j \ge 3} {j \cdot w_j (\Im _n^{sbe} )} }
+ \frac{1}{n} = \frac{1}{2}$. Therefore, we obtain Proposition~\ref{prop3}.
\hspace{11.4cm} $\square$
\subsection{Case of convex hexagons}
\label{subsection3_2}
As for $\Im _6^{sb}$ (i.e., a strongly balanced tiling by convex hexagons
(6-gon)), the number of adjacents of each convex hexagon should be greater
than or equal to six (i.e., $h \ge 6)$. Therefore,
\begin{equation}
\label{equqtion6}
\sum\limits_{h \ge 6} {t_h (\Im _6^{sb} ) = t_6 (\Im _6^{sb} ) +
\sum\limits_{h \ge 7} {t_h (\Im _6^{sb} )} } = 1.
\end{equation}
\noindent
On the other hand, from Proposition~\ref{prop1}, we have
\begin{equation}
\label{equqtion7}
\sum\limits_{h \ge 6} {h \cdot t_h (\Im _6^{sb} )} = 6 \cdot t_6 (\Im
_6^{sb} ) + \sum\limits_{h \ge 7} {h \cdot t_h (\Im _6^{sb} )} \le 6.
\end{equation}
\begin{prop}\label{prop4}
$\sum\limits_{j \ge 3} {j \cdot w_j (\Im _6^{sb} )} = 3$.
\end{prop}
\noindent
\textbf{\textit{Proof}.}
From (\ref{equqtion6}) and (\ref{equqtion7}),
\[
6\,\left( {1 - \sum\limits_{h \ge 7} {t_h (\Im _6^{sb} )} } \right) +
\sum\limits_{h \ge 7} {h \cdot t_h (\Im _6^{sb} )} \le 6.
\]
\noindent
Hence, we obtain $\sum\limits_{h \ge 7} {(h - 6) \cdot t_h (\Im _6^{sb} )} \le 0$.
On the other hand, $\sum\limits_{h \ge 7} {(h - 6) \cdot t_h (\Im _6^{sb} )} \ge 0$
holds because $t_h (\Im _6^{sb} ) = \mathop {\lim }\limits_{r \to \infty }
\frac{t_h (r,M)}{t(r,M)} \ge 0$. From these inequalities,
\[
\sum\limits_{h \ge 7} {(h - 6) \cdot t_h (\Im _6^{sb} )} = 0.
\]
\noindent
Therefore, $t_h (\Im _6^{sb} ) = 0$ for $h \ne 6$. That is, we have
\[
t_6 (\Im _6^{sb} ) = 1,
\quad
\sum\limits_{h \ge 6} {h \cdot t_h (\Im _6^{sb} )} = 6
\quad \mbox{and}\quad
\sum\limits_{h
\ge 7} {t_h (\Im _6^{sb} )} = \sum\limits_{h \ge 7} {h \cdot t_h (\Im _6^{sb} )} = 0.
\]
\noindent
Thus, from these relationships and (\ref{equqtion4}), we arrive at
Proposition~\ref{prop4}.
\hspace{2.4cm} $\square$
\bigskip
From Statement~\ref{sta2}, a monohedral periodic tiling by a convex hexagon
is strongly balanced. If a fundamental region in a monohedral periodic tiling
by a convex hexagon has vertices with valences of four or more,
$\sum\limits_{j \ge 3} {j \cdot w_j (\Im _6^{sb} )} > 3$ and
$\sum\limits_{h \ge 6} {h \cdot t_h (\Im _6^{sb} )} < 6$ from (\ref{equqtion4}),
which is a contradiction of Proposition~\ref{prop4}. Thus, we have the
following corollary.
\begin{cor}\label{cor1}
A monohedral periodic tiling by a convex hexagon is an edge-to-edge tiling
with only $3$-valent vertices.
\end{cor}
It is well known that convex hexagonal tiles (i.e., convex hexagons that
admit a monohedral tiling) belong to at least one of the three types shown
in Figure~\ref{fig3}. That is, convex hexagonal tiles admit at least one periodic
edge-to-edge tiling whose valence is three at all vertices. In fact, the
representative tilings of the three types in Figure~\ref{fig3} are periodic
edge-to-edge tilings whose valence is three at all vertices. From Corollary
1 and the fact that the valence of vertices is at least three, it might be
considered that the valence of all vertices in monohedral tilings by convex
hexagons is three; however, that is not true. For example, as shown Figure~\ref{fig4},
there are monohedral tilings by convex hexagons with vertices of valence
equal to four (note that a monohedral tiling is not always a periodic
tiling). However, it is clear that the convex hexagonal tiles of Figure~\ref{fig4} can
generate a periodic edge-to-edge tiling in which the valence of all vertices
is three.
\renewcommand{\figurename}{{\small Figure.}}
\begin{figure}[htbp]
\centering\includegraphics[width=13cm,clip]{PSBTCP_fig3.png}
\caption{{\small
Three types of convex hexagonal tiles. If a convex hexagon can
generate a monohedral tiling, it belongs to at least one of types 1--3.
The pale gray hexagons in each tiling indicate the fundamental region.
}
\label{fig3}
}
\end{figure}
\renewcommand{\figurename}{{\small Figure.}}
\begin{figure}[htbp]
\centering\includegraphics[width=10cm,clip]{PSBTCP_fig4.png}
\caption{{\small
Monohedral tilings by convex hexagons with vertices of valence equal
to four.}
\label{fig4}
}
\end{figure}
\subsection{Case of convex pentagons}
\label{subsection3_3}
As for $\Im _5^{sb}$ (i.e., a strongly balanced tiling by convex pentagons
(5-gons)), the number of adjacents of each convex pentagon should be greater
than or equal to five. From Proposition~\ref{prop1}, we obtain the following.
\begin{prop}\label{prop5}
$5 \le \sum\limits_{h \ge 5} {h \cdot t_h (\Im _5^{sb} )} \le 6$.
\end{prop}
\noindent
Since $\sum\limits_{h \ge 5} {h \cdot t_h (\Im _5^{sb} )}$ is the average
number of adjacents of the convex pentagons in $\Im _5^{sb}$, we obtain
the following theorem~\cite{Sugimoto_2014}.
\begin{thm}\label{thm1}
A tiling $\Im _5^{sb}$ contains a convex pentagon whose number of
adjacents is five or six.
\end{thm}
\noindent
Then, we obtain the following propositions.
\begin{prop}\label{prop6}
$\frac{5}{2} \le e(\Im _5^{sb} ) \le 3$.
\end{prop}
\begin{prop}\label{prop7}
$\frac{3}{2} \le v(\Im _5^{sb} ) \le 2$.
\end{prop}
\noindent
\textbf{\textit{Proof of Propositions $\ref{prop6}$ and $\ref{prop7}$}.}
From (\ref{equqtion3}) and Proposition~\ref{prop5}, we have that
$5 \le 2e(\Im _5^{sb} ) \le 6$. Since each strongly balanced tiling is
necessarily balanced, $v(\Im _5^{sb} ) = e(\Im _5^{sb} ) - 1$ holds from
Euler's Theorem for Tilings (see P.\pageref{EulerThm}). Therefore, we
have that $5 \le 2(v(\Im _5^{sb} ) +1) \le 6$. Thus, we obtain
Propositions $\ref{prop6}$ and $\ref{prop7}$.\hspace{3.2cm} $\square$
\bigskip
\begin{prop}\label{prop8}
$3 \le \sum\limits_{j \ge 3} {j \cdot w_j (\Im _5^{sb} )} \le \frac{10}{3}$.
\end{prop}
\noindent
\textbf{\textit{Proof}.}
It is clear from Proposition~\ref{prop2}.
\hspace{6.6cm} $\square$
\bigskip
\begin{prop}\label{prop9}
$0 \le \sum\limits_{h \ge 7} {(h - 6) \cdot t_h (\Im _5^{sb} )} \le t_5 (\Im _5^{sb} ) \le 1$.
\end{prop}
\noindent
\textbf{\textit{Proof}.}
From Proposition~\ref{prop5},
\begin{equation}
\label{equqtion8}
5 \le \sum\limits_{h \ge 5} {h \cdot t_h (\Im _5^{sb} )} = 5t_5 (\Im _5^{sb}
) + 6t_6 (\Im _5^{sb} ) + \sum\limits_{h \ge 7} {h \cdot t_h (\Im _5^{sb} )}
\le 6.
\end{equation}
\noindent
From (\ref{equqtion2}),
\[
\sum\limits_{h \ge 3} {t_h (\Im _5^{sb} )} = t_5 (\Im _5^{sb} ) + t_6 (\Im _5^{sb} )
+ \sum\limits_{h \ge 7} {t_h (\Im _5^{sb} )} = 1,
\]
\begin{equation}
\label{equqtion9}
t_6 (\Im _5^{sb} ) = 1 - t_5 (\Im _5^{sb} ) - \sum\limits_{h \ge 7} {t_h (\Im _5^{sb} )} .
\end{equation}
\noindent
From (\ref{equqtion8}), (\ref{equqtion9}) and $0 \le t_h (\Im _5^{sb} ) \le 1$, we obtain
the inequality of Proposition~\ref{prop9}.
\hspace{0.6cm} $\square$
\bigskip
From Proposition~\ref{prop9}, we obtain the following theorem~\cite{Sugimoto_2014}.
\begin{thm}\label{thm2}
A tiling $\Im _5^{sb}$ that satisfies $\sum\limits_{h \ge 7} {t_h (\Im _5^{sb} )} > 0$
contains a convex pentagon whose number of adjacents is five.
\end{thm}
As for $\Im _5^{sbe} $ (i.e., a strongly balanced edge-to-edge tiling by
convex pentagons), we have the following proposition.
\begin{prop}\label{prop10}
$t_5 (\Im _5^{sbe} ) = 1$, $v(\Im _5^{sbe} ) = \frac{3}{2}$,
$e(\Im _5^{sbe} ) = \frac{5}{2}$, and
$\sum\limits_{j \ge 3} {j \cdot w_j (\Im _5^{sbe} )} = \frac{10}{3}$.
\end{prop}
\noindent
\textbf{\textit{Proof}.}
It is clear from Proposition~\ref{prop3}, Euler's Theorem for Tilings
(see P.\pageref{EulerThm}), and $t_h (\Im _5^{sbe} ) = 0$ for $h \ne 5$.
\hspace{9cm} $\square$
\bigskip
Next, we have other propositions as follows.
\begin{prop}\label{prop11}
$v_3 (\Im _5^{sb} ) = 2 + \sum\limits_{j \ge 4} {(2 - j) \cdot v_j (\Im _5^{sb} )} $.
\end{prop}
\noindent
\textbf{\textit{Proof}.}
From (\ref{equqtion1}) and the definition of $v(\Im )$, we have
\begin{equation}
\label{equqtion10}
e(\Im _5^{sb} ) = v(\Im _5^{sb} ) + 1
= \sum\limits_{j \ge 3} {v_j (\Im _5^{sb} ) + 1 }
= v_3 (\Im _5^{sb} ) + \sum\limits_{j \ge 4} {v_j (\Im _5^{sb} )} + 1.
\end{equation}
\noindent
On the other hand, from (\ref{equqtion3}),
\begin{equation}
\label{equqtion11}
2e(\Im _5^{sb} ) = \sum\limits_{j \ge 3} {j \cdot v_j (\Im _5^{sb} ) =
3v_3 (\Im _5^{sb} ) + \sum\limits_{j \ge 4} {j \cdot v_j (\Im _5^{sb} )} }
\end{equation}
\noindent
holds. Therefore, from (\ref{equqtion10}) and (\ref{equqtion11}), we have
\[
2\,\left( {v_3 (\Im _5^{sb} ) + \sum\limits_{j \ge 4} {v_j (\Im _5^{sb} ) +
1} } \right) = 3v_3 (\Im _5^{sb} ) + \sum\limits_{j \ge 4} {j \cdot v_j (\Im
_5^{sb} )} .
\]
\noindent
Thus, we obtain Proposition~\ref{prop11}.
\hspace{7.7cm} $\square$
\bigskip
\begin{prop}\label{prop12}
$v_3 (\Im _5^{sbe} ) = \sum\limits_{j \ge 4} {(3j - 10)} \cdot v_j (\Im _5^{sbe} )$.
\end{prop}
\noindent
\textbf{\textit{Proof}.}
From Proposition~\ref{prop10} and the definitions of
$w_j \left( \Im \right)$ and $v\left( \Im \right)$,
\[
\sum\limits_{j \ge 3} {j \cdot w_j (\Im _5^{sbe} ) = } \frac{\sum\limits_{j
\ge 3} {j \cdot v_j (\Im _5^{sbe} )} }{v(\Im _5^{sbe} )} = \frac{3v_3 (\Im
_5^{sbe} ) + \sum\limits_{j \ge 4} {j \cdot v_j (\Im _5^{sbe} )} }{v_3 (\Im
_5^{sbe} ) + \sum\limits_{j \ge 4} {v_j (\Im _5^{sbe} )} } = \frac{10}{3}.
\]
\noindent
Thus, we obtain Proposition~\ref{prop12}.
\hspace{7.7cm} $\square$
\bigskip
As for $v(\Im _5^{sb} )$, we have the following propositions.
\begin{prop}\label{prop13}
$v(\Im _5^{sb} ) = \sum\limits_{j \ge 3} {v_j (\Im _5^{sb} )} =
\frac{1}{2}\sum\limits_{h \ge 5} {\left( {h - 2} \right)} \cdot t_h (\Im _5^{sb} )$.
\end{prop}
\begin{prop}\label{prop14}
$v(\Im _5^{sb} ) = \frac{1}{2} +
\frac{1}{2}\sum\limits_{h \ge 5} {\left( {h - 3} \right)} \cdot t_h (\Im _5^{sb} )$.
\end{prop}
\begin{prop}\label{prop15}
$v(\Im _5^{sb} ) = 2 + \frac{1}{2}\sum\limits_{h \ge 5} {(h - 6) \cdot t_h (\Im _5^{sb} )}$.
\end{prop}
\begin{prop}\label{prop16}
$v(\Im _5^{sb} ) = 2 - \sum\limits_{j \ge 4} {(j - 3) \cdot v_j (\Im _5^{sb} )} $.
\end{prop}
\noindent
\textbf{\textit{Proof of Propositions $\ref{prop13}, \ref{prop14}, \ref{prop15},$ and
$\ref{prop16}$}.} From the first equation in Statement~\ref{sta4},
\[
2\sum\limits_{j \ge 4} {(j - 3) \cdot v_j (\Im _5^{sb} )} +
\sum\limits_{h \ge 5} {(h - 6) \cdot t_h (\Im _5^{sb} )} = 0.
\]
\noindent
Note that $t_3 (\Im _5^{sb} ) = t_4 (\Im _5^{sb} ) = 0$, since $\Im _5^{sb}$
is $h \ge 5$. The above equation is rearranged as
\begin{equation}
\label{equqtion12}
\sum\limits_{j \ge 4} {(j - 3) \cdot v_j (\Im _5^{sb} )}
= - \frac{1}{2}\sum\limits_{h \ge 5} {(h - 6) \cdot t_h (\Im _5^{sb} )}.
\end{equation}
\noindent
From the second equation in Statement~\ref{sta4},
\[
- v_3 (\Im _5^{sb} ) + \sum\limits_{j \ge 5} {(j - 4) \cdot v_j (\Im _5^{sb} )}
+ \sum\limits_{h \ge 5} {(h - 4) \cdot t_h (\Im _5^{sb} )} = 0.
\]
\noindent
The above equation is rearranged as
\begin{equation}
\label{equqtion13}
\sum\limits_{j \ge 5} {(j - 4) \cdot v_j (\Im _5^{sb} )} =
v_3 (\Im _5^{sb} ) - \sum\limits_{h \ge 5} {(h - 4) \cdot t_h (\Im _5^{sb} )} .
\end{equation}
\noindent
Then,
\begin{equation}
\label{equqtion14}
\begin{array}{l}
\sum\limits_{j \ge 4} {(j - 3) \cdot v_j (\Im _5^{sb} )} = v_4 (\Im _5^{sb} )
+ \sum\limits_{j \ge 5} {(j - 4 + 1) \cdot v(\Im _5^{sb} )} \\
\quad \quad \quad \quad \quad \quad \quad \quad \:\:\: = v_4 (\Im _5^{sb} ) +
\sum\limits_{j \ge 5} {(j - 4) \cdot v(\Im _5^{sb} ) + \sum\limits_{j \ge 5}
{v_j (\Im _5^{sb} )} }. \\
\end{array}
\end{equation}
\noindent
By replacing $\sum\limits_{j \ge 4} {(j - 3) \cdot v_j (\Im _5^{sb} )} $ of
(\ref{equqtion12}) and $\sum\limits_{j \ge 5} {(j - 4) \cdot v_j (\Im _5^{sb} )} $
of (\ref{equqtion13}) in (\ref{equqtion14}), the latter becomes
\[
- \frac{1}{2}\sum\limits_{h \ge 5} {(h - 6) \cdot t_h (\Im _5^{sb} )} =
v_4 (\Im _5^{sb} ) + v_3 (\Im _5^{sb} ) - \sum\limits_{h \ge 5} {(h - 4) \cdot
t_h (\Im _5^{sb} )} + \sum\limits_{j \ge 5} {v_j (\Im _5^{sb} )} .
\]
\noindent
Simplifying both sides,
\[
\sum\limits_{j \ge 3} {v_j (\Im _5^{sb} ) = } \frac{1}{2}\sum\limits_{h \ge 5}
{\left( { - h + 6 + 2h - 8} \right) \cdot t_h (\Im _5^{sb} ) = }
\frac{1}{2}\sum\limits_{h \ge 5} {\left( {h - 2} \right) \cdot t_h (\Im _5^{sb} )} .
\]
\noindent
Thus, we obtain Proposition~\ref{prop13}.
Next, from $\sum\limits_{j \ge 4} {(j - 3) \cdot v_j (\Im _5^{sb} )} $ and
Proposition~\ref{prop11},
\[
\begin{array}{l}
\sum\limits_{j \ge 4} {(j - 3) \cdot v_j (\Im _5^{sb} )} = - \sum\limits_{j \ge 4} {(2 - j)
\cdot v_j (\Im _5^{sb} )} - \sum\limits_{j \ge 4} {v_j (\Im_5^{sb} )} \\
\quad \quad \quad \quad \quad \quad \quad \quad \:\:\:
= 2 - v_3 (\Im _5^{sb} ) - \sum\limits_{j \ge 4} {v_j (\Im _5^{sb} )} \\
\quad \quad \quad \quad \quad \quad \quad \quad \:\:\:
= 2 - \sum\limits_{j \ge 3} {v_j (\Im _5^{sb} )}. \\
\end{array}
\]
\noindent
Thus, we obtain Proposition~\ref{prop16}.
From (\ref{equqtion12}) and Proposition~\ref{prop16},
\[
2 - \sum\limits_{j \ge 3} {v_j (\Im _5^{sb} )} =
- \frac{1}{2}\sum\limits_{h \ge 5} {(h - 6) \cdot t_h (\Im _5^{sb} )} .
\]
\noindent
Thus, we obtain Proposition~\ref{prop15}.
From the third equation in Statement~\ref{sta4},
\[
- 3v_3 (\Im _5^{sb} ) + \sum\limits_{j \ge 4} {(j - 6) \cdot v_j (\Im _5^{sb} )}
+ 2\sum\limits_{h \ge 5} {(h - 3) \cdot t_h (\Im _5^{sb} ) = 0}.
\]
\noindent
By replacing $v_3 (\Im _5^{sb} )$ of Proposition~\ref{prop11} in the above equation,
it becomes
\begin{equation}
\label{equqtion15}
4\sum\limits_{j \ge 4} {(j - 3) \cdot v_j (\Im _5^{sb} )} =
6 - 2\sum\limits_{h \ge 5} {(h - 3) \cdot t_h (\Im _5^{sb} )}.
\end{equation}
\noindent
Form Proposition~\ref{prop11} and (\ref{equqtion15}),
\[
\begin{array}{l}
v_3 (\Im _5^{sb} ) = 2 - \sum\limits_{j \ge 4} {v_j (\Im _5^{sb} )} -
\sum\limits_{j \ge 4} {(j - 3) \cdot v_j (\Im _5^{sb} )}\\ \quad \quad \quad \;\;
= \frac{1}{2} - \sum\limits_{j \ge 4} {v_j (\Im _5^{sb} )}
+ \frac{1}{2}\sum\limits_{h \ge 5} {(h - 3) \cdot t_h (\Im _5^{sb} )}.
\end{array}
\]
\noindent
Simplifying,
\[
v_3 (\Im _5^{sb} ) + \sum\limits_{j \ge 4} {v_j (\Im _5^{sb} )} =
\frac{1}{2} + \frac{1}{2}\sum\limits_{h \ge 5} {(h - 3) \cdot t_h (\Im
_5^{sb} )} .
\]
\noindent
Thus, we obtain Proposition~\ref{prop14}.
\hspace{7.7cm} $\square$
\bigskip
Here, we consider the case of $v(\Im _5^{sb} ) = \frac{3}{2}$ (i.e., the
minimum case of $v(\Im _5^{sb} ))$. From Proposition~\ref{prop15}, we have
\[
2 + \frac{1}{2}\sum\limits_{h \ge 5} {(h - 6) \cdot t_h (\Im _5^{sb} )}
= 2 - \frac{1}{2}t_5 ( {\Im _5^{sb} } ) +
\frac{1}{2}\sum\limits_{h \ge 6} {(h - 6) \cdot t_h (\Im _5^{sb} )} = \frac{3}{2}.
\]
\noindent
Simplifying this equation,
\begin{equation}
\label{equqtion16}
t_5 ( {\Im _5^{sb} } ) = 1 + \sum\limits_{h \ge 6} {(h - 6) \cdot
t_h (\Im _5^{sb} )} .
\end{equation}
\noindent
Since $\sum\limits_{h \ge 5} {t_h (\Im _5^{sb} ) = t_5 \left( {\Im _5^{sb} } \right) + }
\sum\limits_{h \ge 6} {t_h (\Im _5^{sb} ) = 1} $,
$\sum\limits_{h \ge 6} {(h - 6) \cdot t_h (\Im _5^{sb} )} $ in (\ref{equqtion16}) is equal
to zero. That is, in the case of $v(\Im _5^{sb} ) = \frac{3}{2}$,
$\sum\limits_{h \ge 6} {t_h (\Im _5^{sb} )} = 0$. Therefore, $v(\Im _5^{sb}
) = \frac{3}{2}$ if and only if $\Im _5^{sb} $ is $\Im _5^{sbe} $.
\subsection{Properties of representative periodic tilings by a
convex pentagonal tile}
\label{subsection3_4}
Let $\Im _5^{r(x)} $ be a representative periodic tiling by a convex
pentagonal tile of type $x$. That is, $\Im _5^{r(x)} $ for $x = 1,\,\ldots ,\,15$
is a strongly balanced tiling by a convex pentagonal tile.
Representative tilings of types 1 or 2 are generally non-edge-to-edge, as
shown in Figure~\ref{fig1}. However, in special cases, the convex pentagonal tiles of
types 1 or 2 can generate edge-to-edge tilings, as shown in Figure~\ref{fig5}. Here,
the convex pentagonal tiles of (a) and (b) in Figure~\ref{fig5} are referred to as
those of types 1e and 2e, respectively. Then, let $\Im _5^{r(1e)} $ and
$\Im _5^{r(2e)} $ be representative edge-to-edge periodic tilings by a convex
pentagonal tile of types 1e and 2e, respectively.
The properties of each tiling $\Im _5^{r(x)} $ are obtained from the results
of Section~\ref{subsection3_3}, etc. Table 1 summarizes the results~\cite{Sugimoto_2014}.
We can check that the representative periodic tilings of each type of convex
pentagonal tile that can generate an edge-to-edge tiling satisfy
Proposition~\ref{prop10}.
\renewcommand{\figurename}{{\small Figure.}}
\begin{figure}[htbp]
\centering\includegraphics[width=13cm,clip]{PSBTCP_fig5.png}
\caption{{\small
Examples of edge-to-edge tilings by convex pentagonal tiles that
belong to types 1 or 2. The pale gray pentagons in each tiling indicate the
fundamental region.}
\label{fig5}
}
\end{figure}
\section{Conclusions}
In this paper, although it is accepted as a fact, it is proved that the only
convex polygonal tiles that admit at least one periodic tiling are
triangles, quadrangles, pentagons, and hexagons. As for the fact (proof)
that the only convex polygonal tiles are triangles, quadrangles, pentagons,
and hexagons, note that it is necessary to take except periodic tilings also
into consideration.
Although a solution to the problem of classifying the types of convex
pentagonal tile is not yet in sight, we suggest that the properties that
could lead to such a solution are those that are shown in Section~\ref{section3}.
\begin{landscape}
\begin{table}[htbp]
{\small
\caption[Table 1.]{Properties of $\Im _5^{r(x)} $}
\label{table1}
}
\
\renewcommand{\arraystretch}{2.1}
\begin{tabular}
{p{90pt}|p{105pt}p{90pt}p{50pt}p{90pt}p{75pt}}
\hline
\hfil \hfil \raisebox{0ex}[0cm][0cm]{$x$ of $\Im _5^{r(x)} $}
&
\hfil \hfil \raisebox{0.5ex}[0cm][0cm]{$t_h \Bigl(\Im _5^{r(x)} \Bigr)$}
&
\hfil \hfil \raisebox{0.5ex}[0cm][0cm]{$v_j \Big(\Im _5^{r(x)} \Bigr)$}
&
\hfil \hfil \raisebox{0.5ex}[0cm][0cm]{$2e \Big(\Im _5^{r(x)} \Bigr)$}
&
\hfil \raisebox{0.5ex}[0cm][0cm]{$w_j \Big(\Im _5^{r(x)} \Bigr)$}
&
\hfil \raisebox{1ex}[0cm][0cm]{$\sum\limits_{j \ge 3} {j \cdot w_j \Big(\Im _5^{r(x)} \Bigr)} $} \\
\hline
\hfil \raisebox{-1.50ex}[0cm][0cm]{1e, 2e, 4, 6, 7, 8, 9}
&
$t_5 = 1,$ \par $t_h = 0$ for $h \ne 5$
&
$v_3 = 1, v_4 = \frac{1}{2},$ \par $v_j = 0$ for $j \ne 3\;\mbox{or}\;4$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{5}
&
$w_3 = \frac{2}{3}, w_4 = \frac{1}{3},$ \par $w_j = 0$ for $j \ne 3\;\mbox{or}\;4$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{$\frac{10}{3} = 3.\dot {3}$} \\
\hline
\hfil \raisebox{-1.50ex}[0cm][0cm]{5}
&
$t_5 = 1,$ \par $t_h = 0$ for $h \ne 5$
&
$v_3 = \frac{4}{3}, v_6 = \frac{1}{6},$ \par $v_j = 0$ for $j \ne 3\;\mbox{or}\;6$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{5}
&
$w_3 = \frac{8}{9}, w_6 = \frac{1}{9},$ \par $w_j = 0$ for $j \ne 3\;\mbox{or}\;6$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{$\frac{10}{3} = 3.\dot {3}$} \\
\hline
\hfil \raisebox{-1.50ex}[0cm][0cm]{1, 2, 3, 12}
&
$t_6 = 1,$ \par $t_h = 0$ for $h \ne 6$
&
$v_3 = 2,$ \par $v_j = 0$ for $j \ne 3$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{6}
&
$w_3 = 1,$ \par $w_j = 0$ for $j \ne 3$&
\hfil \raisebox{-1.50ex}[0cm][0cm]{3} \\
\hline
\hfil \raisebox{-1.50ex}[0cm][0cm]{10}
&
$t_5 = \frac{2}{3},t_7 = \frac{1}{3},$ \par $t_h = 0$ for $h \ne 5\;\mbox{or}\;7$
&
$v_3 = \frac{5}{3},v_4 = \frac{1}{6},$ \par $v_j = 0$ for $j \ne 3\;\mbox{or}\;4$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{$\frac{17}{3} = 5.\dot {6}$}
&
$w_3 = \frac{10}{11},w_4 = \frac{1}{11},$ \par $w_j = 0$ for $j \ne 3\;\mbox{or}\;4$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{$\frac{34}{11} = 3.\dot {0}\dot {9}$} \\
\hline
\hfil \raisebox{-1.50ex}[0cm][0cm]{11}
&
$t_5 = t_7 = \frac{1}{2},$ \par $t_h = 0$ for $h \ne 5\;\mbox{or}\;7$
&
$v_3 = 2,$ \par $v_j = 0$ for $j \ne 3$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{6}
&
$w_3 = 1,$ \par $w_j = 0$ for $j \ne 3$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{3} \\
\hline
\hfil \raisebox{-1.50ex}[0cm][0cm]{13}
&
$t_5 = t_6 = \frac{1}{2},$ \par $t_h = 0$ for $h \ne 5\;\mbox{or}\;6$
&
$v_3 = \frac{3}{2},v_4 = \frac{1}{4},$ \par $v_j = 0$ for $j \ne 3\;\mbox{or}\;4$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{$\frac{11}{2} = 5.5$}
&
$w_3 = \frac{6}{7},w_4 = \frac{1}{7},$ \par $w_j = 0$ for $j \ne 3\;\mbox{or}\;4$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{$\frac{22}{7} \approx 3.142...$} \\
\hline
\hfil \raisebox{-1.50ex}[0cm][0cm]{14}
&
$t_5 = t_6 = t_7 = \frac{1}{3},$ \par $t_h = 0$ for $h \ne 5, 6, \;\mbox{or}\;7$
&
$v_3 = 2$ \par $v_j = 0$ for $j \ne 3$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{6}
&
$w_3 = 1,$ \par $w_j = 0$ for $j \ne 3$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{3} \\
\hline
\hfil \raisebox{-1.50ex}[0cm][0cm]{15}
&
$t_5 = \frac{2}{3},t_6 = \frac{1}{3},$ \par $t_h = 0$ for $h \ne 5\;\mbox{or}\;6$
&
$v_3 = \frac{4}{3},v_4 = \frac{1}{3},$ \par $v_j = 0$ for $j \ne 3\;\mbox{or}\;4$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{$\frac{16}{3} = 5.\dot {3}$}
&
$w_3 = \frac{4}{5},w_4 = \frac{1}{5},$ \par $w_j = 0$ for $j \ne 3\;\mbox{or}\;4$
&
\hfil \raisebox{-1.50ex}[0cm][0cm]{$\frac{16}{5} = 3.2$} \\
\hline
\end{tabular}
\label{tab1}
\end{table}
\end{landscape}
| 134,778
|
TITLE: What is the condition that two maps need to meet in order for their compositive map to exist?
QUESTION [0 upvotes]: In the book that I'm currently reading it states that:
"Two maps can be applied one after the other provided the co-domain of the first map is the same as the domain of the second. This process is called composition of maps"
I'm just curious as to know whether this statement is fully accurate? Shouldn't it actually be something more around the lines of:
"Two maps can be applied one after the other provided the image set of the first map (say, Im(f) where the map is f) is a subset of the domain of the second"
REPLY [1 votes]: Both statements are sufficient for the composite function to be well defined.
All we need is that for every $x$ in the domain of $f$, we can define $g(f(x))$ so $f(x)$ should be in the domain of $g$
| 218,727
|
I have something very interesting for you all today: an international giveaway+ book review of A World Without States. Read on please…..
About the book:
“However, in the end it all comes all down to one thing: the desire for power. The question arises: should world politics still merely revolve around this? It might have been logical in the time of Pharaohs, but today? Back then, ages before Christ, there were only a few states. But how many are there today? Since the Second World War colonies have freed themselves from oppression. The United Nation has 193 members. Therefore there are 193 states that have nuclear weapons at their disposal. I believe there are 193 states to many. If we ever want to end this desire for power we can do nothing but eliminating the system of states. It needs to be replaced by a global governance that puts an end to all militarism. Militarism does not belong to this day and age. Is has become an anachronism. Merely a new world order states can ensure civilization,” a very powerfull thought of Dago Steenis from the book A World Without States.
Steenis has an accessible writing style. I loved his book very much. I thought that books about history were boring but Steenis shows us with his book that history is anything but boring. The book is very good. He teaches the readers history in a very cool way. This is a very philosophic book. In 133 pages you get to know his very intelligent vision about the world.
This book has seven chapters. The first chapter is dedicated to ‘The age in which we live.’ Steenis tells about the current polital system, the growth of the humanitarism, violence is never the answer, empathy: us and the other and the existence: from nature to culture. The second chapter is dedicated the The emergence of the state. The third chapter is dedicated to The final crisis of the transitium. The last four chapters tells us about The United States enforces its standards on the World, Terror and Anti-terror, Counter-moment: growth of humanity and the Evolution and progress.
This is one of my favourite books about the history of mankind. This books shows us history in a different way ( positive). In a very interesting way. Steenis tells us about the First World War, gladiators, slavery, states, nationalism, sovereignty, globalization, inhumanity, Second World War etc. It’s a very addictive book, once you start to read it you cannot stop reading. I would definitely recommend this book to everyone. This is definitely one of the best books I have ever read.
Dago Steenis (1925) wrote a remarkably positive book something that is actually more than a book, it is a thought that can be passed on. Not like a pamphlet, but based on his speech. According to Steenis history cannot be a linear concept, not a static process, without progress. On the contrary. History precisely demonstrates that we have a reason to be optimistic. The developments of mankind are evolutionary, and an increasing form of human growth nd potency. Since the emergence of culture man gradually replaced natural evolution by cultural evolution.
Order this book at Uitgeverij Aspekt. Also available in Dutch and many more languages.
_________________________________________________________________
The giveaway:
What do you have to do to win this book?
-Follow our instagrampage or like or facebookpage.
-comment below that you are entering this giveaway ( please let us know where you follow us or like us)
Optional:
-Share the giveaway on instagram, facebook, twitter etc.
– there are 4 books available for the giveaway this means 1 book per person.
-We will mail the winners*( make sure your e-mail address is right on the form).
*There is no possibility to discuss about the giveaway we will raffle the names of the contestants in an raffle machine.
-This giveaway is valid untill 1 October 2016.
________________________________________________________________
We would love to thank the Publisher: Uitgeverij Aspekt!
Good luck all!!!
_________________________________________________________________
Pssst you can also enter the Dutch giveaway here
0 thoughts on “Review: A World Without States + international giveaway 4x (closed)”
Ik zou graag kans willen maken op de Engelse versie. Ik heb geliked op facebook.
Veel succes.
Great giveaway. I am following your instagram page. I also shared it on twitter.
Good luck.
I am also in. I already liked and followed everything.
Good luck.
Another giveaway love it. I like the Facebook page and shared it.
Good luck.
Super tof dit. Ik doe mee met insta ?
Veel succes!
Nice international giveaway. I am from Germany. Nice blog. I would love to have this book.I am following your page on instagram.
Good luck!
Thank you for your book review. I am very curious about this book. Please include me in the raffle. I gave you a like on facebook.
Good luck!
I like the book review. I am entering this giveaway by following the instagram page.
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You wrote a very good bookreview. I hope I can win this book. I did everything also the optional steps.
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I am also collaborating. I already liked your page and shared it with my friends <3
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I am now following you on instagram. I also shared the pic.
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Cool hoor deze winactie. Ik likte jullie facebook al. Heb wel even gedeeld.
Veel succes.
I think this is an interesting book see my instagram follow plzzz.
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Wow I want to win. Nice book. I like your facebook.
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Dit klinkt interessant! Ik lees eigenlijk altijd in het Engels en het onderwerp spreekt mij aan, dus ik doe mee 🙂 Ik volg je nu op Instagram en heb je Facebookpagina geliked.
Veel succes!
Sounds like a great read!
Florals&Smiles
Thank you! Are you also entering the giveaway?
Ik lees liever in het Engels. Dolgraag zou ik dit boek in mijn nieuwe boekenkast willen plaatsen. Hij is nog leeg. Ik denk dat dit een leuk boek is.
Veel succes.
Great giveaway! Shared this on Twitter and Instagram.
Good luck!
[…] staten van Dago Steenis. De Engelse versie heb ik al verslonden met veel plezier zie hier mijn recensie. Nu de Nederlandse versie nog. Een stukje over het boek: De mensheid leeft nog in de overgang […]
Als ik de recensie moet geloven is dit een goed boek. Ik doe mee. Zie je Facebook.
Dit zou een leuk cadeautje zijn voor mijn vriend. Hij houd van Engelse boeken. Ik doe hier ook aan mee.
[…] –Een wereld zonder staten Engels […]
| 130,837
|
Shanah Tovah forgiveness for any misdeeds; and that, the Shofar are blown to awaken the hearers from their spiritual slumber, to make them aware of their actions and their repercussions.
And let me also use this opportunity to seek your kind forgiveness for any wrong act of omission or commission, which I might have done knowingly or unknowingly, and which could have caused any pain or loss to you.
And of course, most importantly, I wish you “tizkee vetihyee ve’orekh yamim / tizkeh v’tihyeh ve’orech yamim “
| 141,226
|
Ghirardelli Drum Holiday Gift Basket
by Wine Country Gift Baskets | Item #: 569950 |
(1)
|
What an amazing way to make the Ghirardelli lover in your life smile. Laden with six delectable Ghirardelli treats, this holiday gift is tough to beat.
Details
Assembled Country
USA
Assembled Size
Gift size: 8.5" x 8.5" x 13.75"
Component Country
USA And Imported
Warranty InformationThis product is covered by the Sam's Club Member Satisfaction Guarantee.
There is no additional information available for this item.
| 132,895
|
In case you find your Read more [...]
Vasilis Grillhouse
Place Category: Feature, Food, Restaurants, and Taverns to ensure maximum quality from the very first step of the cooking process, they themselves produce some of the ingredients used in meal preparation, including most of the groceries, eggs, goat cheese, rabbit meat, and of course milk and olive oil.
Vasilis’ cuisine is primarily focused on traditional Greek dishes such as rabbit stew, lamb “Youvetsi” (lamb with orzo pasta), goose “Pastitsada” (goose with pasta), and Rotisserie grill. Fresh sea food is also on offer, ready to be picked from a fish tank, prominently featuring the famed lobsters of Paxos Island. The restaurant offers a vast range of options from appetizers to dessert, surely able to satisfy your every need.
In the cool shade of a towering plum tree, in one of the picturesque crossroads amidst the narrow streets of Gaios, about 50m from the town square, Vasilis Grill-Restaurant is ready to provide you with the serenity needed to enjoy a meal alone or in the company of a few friends. Traditionally decorated, its walls are oozing with elements of the sea and maritime life, characteristics deeply etched in the culture of Ionian folk. It is safe to say that Vasilis’ restaurant is among the top choices available on Paxos Island when it comes to food. The restaurant is open year round and also takes orders for delivery.
| 86,228
|
TITLE: How to take the square root of a dual number ($\sqrt{a + b \cdot \varepsilon}$ with $a, ~b \in \mathbb{R}$ and $\varepsilon^{2} = 0$)?
QUESTION [0 upvotes]: How to take the square root of a dual number:
$\sqrt{\Xi}$ with $\begin{align*}
a, ~b &\in \mathbb{R}\\
\varepsilon^{2} &= 0\\
\varepsilon &\ne 0\\
\\
\Xi &:= a + b \cdot \varepsilon\\
\end{align*}$
My first attempt worked (at least I think so), but the second one came to nothing, but it seems to me that taking the square root of binary numbers can also be derived in a different way, which is why I am here under the question in the question and the answers to show all possibilities to the question. (I wrote my first attempt as an answer under the question.)
Surely the question sounds strange, because why should one calculate that at all, but let's just ask ourselves how that would work?
My attempt $2$
My attempt $2$ was "try and hope that it works":
$$
\begin{align*}
\Xi^{2} &= \left( a + b \cdot \varepsilon \right)^{2}\\
\Xi^{2} &= a^{2} + 2 \cdot a \cdot + b \cdot \varepsilon + \left( b \cdot \varepsilon \right)^{2}\\
\Xi^{2} &= a^{2} + 2 \cdot a \cdot b \cdot \varepsilon + b^{2} \cdot \varepsilon^{2}\\
\Xi^{2} &= a^{2} + 2 \cdot a \cdot b \cdot \varepsilon + b^{2} \cdot 0\\
\Xi^{2} &= a^{2} + 2 \cdot a \cdot b \cdot \varepsilon + 0\\
\Xi^{2} &= a^{2} + 2 \cdot a \cdot b \cdot \varepsilon \\
\Xi &= \sqrt{a^{2} + 2 \cdot a \cdot b \cdot \varepsilon}\\
a + b \cdot \varepsilon &= \sqrt{a^{2} + 2 \cdot a \cdot b \cdot \varepsilon}\\
c := a^{2} &\wedge d := 2 \cdot a \cdot b \cdot \varepsilon\\
a = \sqrt{c} &\wedge b = \frac{d}{2 \cdot a} \cdot \varepsilon^{-1}\\
a = \sqrt{c} &\wedge b = \frac{d}{2 \cdot \sqrt{c}} \cdot \varepsilon^{-1}\\
\\
\Xi &= \sqrt{a^{2} + 2 \cdot a \cdot b \cdot \varepsilon}\\
a + b \cdot \varepsilon &= \sqrt{c + d \cdot \varepsilon} \quad\mid\quad a = \sqrt{c} \wedge b = \frac{d}{2 \cdot \sqrt{c}} \cdot \varepsilon^{-1}\\
\sqrt{c} + \frac{d}{2 \cdot \sqrt{c}} \cdot \varepsilon^{-1} \cdot \varepsilon &= \sqrt{c + d \cdot \varepsilon}\\
\sqrt{c} + \frac{d}{2 \cdot \sqrt{c}} \cdot 1 &= \sqrt{c + d \cdot \varepsilon}\\
\sqrt{c} + \frac{d}{2 \cdot \sqrt{c}} &= \sqrt{c + d \cdot \varepsilon}\\
\sqrt{c + d \cdot \varepsilon} &= \sqrt{c} + \frac{d}{2 \cdot \sqrt{c}}\\
\sqrt{a + b \cdot \varepsilon} &= \sqrt{a} + \frac{b}{2 \cdot \sqrt{a}} \quad\mid\quad \left( ~~ \right)^{2}\\
a + b \cdot \varepsilon &= \left( \sqrt{a} + \frac{b}{2 \cdot \sqrt{a}} \right)^{2}\\
a + b \cdot \varepsilon &= a + 2 \cdot \sqrt{a} \cdot \frac{b}{2 \cdot \sqrt{a}} + \frac{b^{2}}{4 \cdot a}\\
a + b \cdot \varepsilon &= a + b + \frac{b^{2}}{4 \cdot a}\\
b \cdot \varepsilon &= b + \frac{b^{2}}{4 \cdot a}\\
\end{align*}
$$
But that makes no sense...
REPLY [1 votes]: From calculus, $\sqrt{1+x} = 1 + \frac{x}{2} + O(x^2)$.
With the obvious caveats on the sign of $a$, we have
$$ \sqrt{a + b\epsilon} = \pm \sqrt{a} \sqrt{1 + \frac{b\epsilon}{a}} = \pm \sqrt{a} (1 + \frac{b\epsilon}{2a} + O(\epsilon^2)). $$
| 171,839
|
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| 154,438
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\begin{document}
\title[Twisted vector bundles on pointed nodal curves]
{Twisted vector bundles on pointed nodal curves}
\author[Ivan Kausz]{Ivan Kausz}
\thanks{Partially supported by the DFG}
\date{January 12, 2005}
\address{NWF I - Mathematik, Universit\"{a}t Regensburg, 93040 Regensburg,
Germany}
\email{ivan.kausz@mathematik.uni-regensburg.de}
\begin{abstract}
Motivated by the quest for a good compactification
of the moduli space of $G$-bundles on a nodal curve
we establish a striking relationship between Abramovich's and
Vistoli's twisted bundles and Gieseker vector bundles.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
This paper grew out of an attempt to understand a recent
draft of Seshadri (\cite{Seshadri-notes})
and is meant as a contribution in the quest for a good compactification
of the moduli space (or stack) of $G$-bundles on a nodal curve.
We are led by the idea that such a compactification should behave well in
families and also under partial normalization of nodal curves.
This statement may be reformulated by saying that we are looking for
an object which has the right to be called
the moduli stack of stable maps into the classifying stack $BG$
of a reductive group $G$.
For finite groups $G$ the stack of stable maps into $BG$ has been
recently constructed by means of so called twisted bundles
by D. Abramovich and A. Vistoli
(\cite{AV}, \cite{ACV}).
On the other hand, as shown in
\cite{stable maps}, the notion of Gieseker vector bundles leads to
the construction of the stack of stable maps into $B\GL_r$.
In this note we establish a connection between the straightforward
generalization of the notion of twisted bundles to the case of the non-finite
reductive group $\GL_r$ and Gieseker vector bundles.
My hope is that this relationship - whose observation is entirely due
to Seshadri, and which in my mind is really striking -
may help to find the right notion for more general reductive
groups $G$.
I would like to thank Seshadri for generously imparting his ideas.
This paper owes very much to long discussions which I had with
Nagaraj in November and December 2002. I would like to thank the Institute
of Mathematical Sciences in Chennai, whose hospitality made these
discussions possible.
\vspace{5mm}
\section{Twisted $G$-bundles}
Throughout this section $k$ denotes an algebraically closed field
and $G$ a reductive group over $k$.
A twisted $G$-bundle is a twisted object in the sense of
\cite{AV}, \S 3 where the target stack $\M$ is taken
to be the classifying stack $BG$.
For convenience we recall the necessary definitions from loc. cit.
\begin{definition}
1.
An {\em $n$-marked curve} consists of data $(U\to S,\Sigma_i)$
where $\pi:U\to S$ is a nodal curve and
$\Sigma_1,\dots,\Sigma_n\subset U$ are pairwise disjoint closed
subschemes whose supports do not intersect the singular locus
$U_{\sing}$ of $\pi$ and are such that the projections $\Sigma_i\to S$ are
\'etale.
2.
A {\em morphism between two $n$-marked curves
$(U\to S, \Sigma^U_i)$ and $(V\to S, \Sigma^V_i)$}
is an $S$-morphism $f:U\to V$ such that
$f(\Sigma^U_i)\subseteq \Sigma^V_i$ for each $i$.
Such a morphism is called {\em strict}, if for each $i$ the support of
$f^{-1}(\Sigma^V_i)$ coincides with the support of $\Sigma^U_i$
and if furthermore
the support of $f^{-1}(V_{\sing})$ coincides with the one of $U_{\sing}$.
3.
The {\em pull back} of an $n$-marked curve
$(U\to S, \Sigma_i)$ by a morphism $S'\to S$ is the $n$-marked curve
$(U\times_SS', \Sigma_i\times_SS')$.
4.
An {\em $n$-pointed nodal curve} is an $n$-marked curve where
the projections $\Sigma_i\to S$ are isomorphisms.
5.
Let $(U\to S,\Sigma_i)$ be an $n$-marked curve.
The complement (inside $U$) of the union of the singular
locus $U_{\sing}$ and the markings $\Sigma_i$ is called
the {\em generic locus} of $U$ and is denoted by $U_{\gen}$.
\end{definition}
\begin{definition}
1.
An {\em action of a finite group $\Gamma$} on an
$n$-marked nodal curve $(U\to S, \Sigma_i)$ is an action
of $\Gamma$ on $U$ as an $S$-scheme which leaves the $\Sigma_i$ invariant.
Such an action is called {\em tame}, if for each geometric point
$u$ of $U$ the stabilizer $\Gamma_u\subseteq\Gamma$ of $u$ has order prime to
the characteristic of $u$.
2.
Let $S$ be a $k$-scheme.
Let $(U\to S, \Sigma_i)$ be an $n$-marked nodal curve and let
$\eta$ be a principal $G$-bundle on $U$.
A {\em essential action of a finite group $\Gamma$ on $(\eta, U)$}
is a pair of actions of $\Gamma$ on $\eta$ and on $(U\to S, \Sigma_i)$
such that
\begin{enumerate}
\def\theenumi{\roman{enumi}}
\def\labelenumi{(\theenumi)}
\item
the actions of $\Gamma$ on $\eta$ and on $U$ are compatible, i. e.
if $\pi: \eta\to U$ denotes the projection, then
$\pi\comp\gamma = \gamma\comp\pi$ for each $\gamma\in\Gamma$.
\item
if $\gamma\in\Gamma$ is an element different from the identity and
$u$ is a geometric point of $U$ fixed by $\gamma$, then
the automorphism of the fiber $\eta_u$ induced by $\gamma$ is not trivial.
\end{enumerate}
3.
An essential action of a finite group $\Gamma$ on $(\eta, U)$ is
called {\em tame}, if the action of $\Gamma$ on $(U\to S, \Sigma_i)$
is tame.
\end{definition}
\begin{definition}
Let $S$ be a $k$-scheme.
Let $C\to S$ be an $n$-pointed nodal curve and let $\xi$ be a principal
$G$-bundle over $C_{\gen}$.
A {\em chart $(U,\eta,\Gamma)$ for $\xi$} consists of the following
data
\begin{enumerate}
\item
An $n$-marked curve $U\to S$ and a strict morphism $\phi: U\to C$,
\item
A principal $G$-bundle $\eta$ on $U$.
\item
An isomorphism $\eta\times_UU_{\gen}\isomto \xi\times_CU_{\gen}$
of $G$-bundles on $U_{\gen}$.
\item
A finite group $\Gamma$.
\item
A tame, essential action of $\Gamma$ on $(\eta, U)$.
\end{enumerate}
These data are required to satisfy the following conditions
\begin{enumerate}
\def\theenumi{\roman{enumi}}
\def\labelenumi{(\theenumi)}
\item
The action of $\Gamma$ leaves the morphisms $U\to C$ and
$\eta\times_UU_{\gen}\isomto \xi\times_CU_{\gen}$ invariant.
\item
The induced morphism $U/\Gamma\to C$ is \'etale.
\end{enumerate}
\end{definition}
\begin{proposition}
\label{cyclic}
(cf. \cite{AV}, Prop 3.2.3)
Let $C\to S$ be an $n$-pointed nodal curve over a $k$-scheme $S$
and let $\xi$ be a principal $G$-bundle on $C_{\gen}$.
Let $(U,\eta,\Gamma)$ be a chart for $\xi$. Then the following
holds.
\begin{enumerate}
\item
The action of $\Gamma$ on $U_{\gen}$ is {\em free}.
\end{enumerate}
Let $s$ be a geometric point of $S$ and let $u$ be a closed
point of the curve $U_s$. Let $\Gamma_u\subseteq \Gamma$ be the
stabilizer of $u$. Then $\Gamma_u$ is a cyclic group.
Let $e$ be its order and let $\gamma_u$ be a generator of $\Gamma_u$.
Then
\begin{enumerate}
\setcounter{enumi}{1}
\item
if $u$ is a regular point, the action of $\gamma_u$ on the tangent
space of $U_s$ at $u$ is via multiplication by a primitive $e$-th
root of unity.
\item
if $u$ is a singular point, $\Gamma_u$
leaves each of the two branches of $U_s$ at $u$ invariant.
The action of $\gamma_u$ on the tangent space of each of the branches is
via multiplication with a primitive $e$-th root of unity.
\end{enumerate}
\end{proposition}
\begin{definition}
Let $C\to S$ be an $n$-pointed nodal curve over a $k$-scheme $S$ and
let $\xi$ be a principal $G$-bundle on $C_{\gen}$.
A chart $(U,\eta,\Gamma)$ for $\xi$ is called {\em balanced},
if for each geometric fiber of $U\to S$ and each singular point $u$ on
it the action of $\gamma_u$ on the tangent spaces of the two branches
is via multiplication with primitive roots of unity which are inverse
to each other.
\end{definition}
\begin{definition}
Let $C\to S$ be an $n$-pointed nodal curve over a $k$-scheme $S$ and
let $\xi$ be a principal $G$-bundle on $C_{\gen}$.
Two charts $(U_1, \eta_1, \Gamma_1)$
and $(U_2, \eta_2, \Gamma_2)$ of $\xi$ are called {\em compatible},
if for each pair of $u_1$, $u_2$ of geometric points of $U_1$, $U_2$
lying above the same geometric point $u$ of $C$ the following holds:
\begin{inmargins}{8mm}
Let $C^{\sh}$ denote the strict henselization of $C$ at $u$.
For $j=1,2$
let $\Gamma'_j\subseteq\Gamma_j$ denote the stabilizer
subgroup of the point $u_j$,
let $U_j^{\sh}$ denote the strict henselization
of $U_j$ at $u_j$, and let $\eta_j^{\sh}:=\eta_j\times_{U_j}U_j^{\sh}$.
Then there exists an isomorphism $\theta: \Gamma'_1\to\Gamma'_2$,
a $\theta$-equivariant isomorphism $\phi:U_1^{\sh}\isomto U_2^{\sh}$
of $C^{\sh}$-schemes, and a $\theta$-equivariant isomorphism
$\eta_1^{\sh}\isomto\phi^*\eta_2^{\sh}$ of $G$-bundles.
\end{inmargins}
\end{definition}
\begin{definition}
Let $g$ and $n$ be two non-negative integers.
An {\em $n$-pointed twisted $G$-bundle of genus $g$} is a triple
$(\xi, C\to S, \Ac)$ where
\begin{enumerate}
\item
$S$ is a $k$-scheme,
\item
$C\to S$ is proper $n$-pointed nodal curve of finite presentation
with geometrically connected fibers of genus $g$,
\item
$\xi$ is a principal $G$-bundle on $C_{\gen}$,
\item
$\Ac=\{(U_\alpha,\eta_\alpha,\Gamma_\alpha)\}$ is a balanced atlas, i.e.
a collection of
mutually compatible balanced charts for $\xi$, such that the images of the
$U_\alpha$ cover $C$.
\end{enumerate}
\end{definition}
\begin{definition}
Let $(\xi, C\to S, \Ac)$ be an $n$-pointed twisted $G$-bundle
of genus $g$. A morphism of $k$-schemes $S'\to S$ induces
a triple $(\xi', C'\to S', \Ac')$ as follows:
\begin{itemize}
\item
The $n$-pointed nodal curve $C'\to S'$ is the pull back of
$C\to S$ by $S'\to S$.
\item
Thus we have a morphism $C'_{\gen}\to C_{\gen}$, and the $G$-bundle
$\xi'$ is the pull back of $\xi$ by this morphism.
\item
Let $\{U_\alpha,\eta_\alpha,\Gamma_\alpha)\}$ be the set of charts
which make up the atlas $\Ac$. Then
$\Ac'=\{U'_\alpha,\eta'_\alpha,\Gamma_\alpha)\}$, where
$U_{\alpha}'\to S'$ is the pull back of the $n$-marked curve
$U_{\alpha}\to S$,
and $\eta'_\alpha$ is the pull back of $\eta_\alpha$ by the morphism
$U'_{\alpha}\to U_{\alpha}$.
Since the $(U'_\alpha, \eta'_\alpha, \Gamma_\alpha)$ are charts for
$\xi'$ which are balanced and mutually compatible (cf. \cite{AV},
Prop. 3.4.3), $\Ac'$ is a balanced atlas.
\end{itemize}
Thus the triple $(\xi', C'\to S', \Ac')$ is an
$n$-pointed twisted $G$-bundle of genus $g$. It is called the
{\em pull back} of $(\xi, C\to S, \Ac)$ by the morphism $S'\to S$.
\end{definition}
\begin{definition}
A {\em morphism} between two $n$-pointed twisted $G$-bundles
$(\xi', C'\to S', \Ac')$ and $(\xi, C\to S, \Ac)$ consists
of a Cartesian diagram
$$
\xymatrix@R=2ex{
C' \ar[r] \ar[d] &
C \ar[d] \\
S' \ar[r] &
S}
$$
and an isomorphism $\xi'\isomto\xi\times_{C_{\gen}}C'_{\gen}$
such that the pull-back of the charts in $\Ac$
(considered as charts for $\xi'$) are compatible
with all the charts in $\Ac'$.
\end{definition}
\vspace{5mm}
\section{Review of Gieseker vector bundles}
\label{review}
In this section I will recall some definitions from my earlier papers
\cite{kgl} and \cite{degeneration}.
Let $k$ be an algebraically closed field.
Let $n\geq 1$ be an integer and let $R_1,\dots,R_n$ be $n$
copies of the projective line $\Pp^1$. On each $R_i$ we choose
two distinct points $x_i$ and $y_i$.
Let $R$ be the nodal curve over $k$ constructed from $R_1,\dots, R_n$
by identifying $y_i$ with $x_{i+1}$ for $i=1,\dots,n-1$.
We call such a curve $R$ a {\em chain of projective lines} of length $n$ with components
$R_1,\dots,R_n$. On the extremal components $R_1$ and $R_n$ we have the
two points $x_1$ and $y_n$ respectively, which are smooth points of $R$.
\begin{definition}
\label{admissible}
A vector bundle $E$ of rank $r$ on $R$ is called {\em admissible},
if
\begin{enumerate}
\item
for each $i\in[1,n]$ the restriction of $E$ on the component $R_i$ is
of the form
$$
d_i\Oo_{R_i}(1)\oplus(r-d_i)\Oo_{R_i}
$$
for some integer $d_i\geq 1$ and
\item
there exists no nonvanishing global section of $E$ over $R$ which
vanishes in the two points $x_1$ and $y_n$.
\end{enumerate}
\end{definition}
Let $C$ be an irreducible curve with exactly one double point $p$.
Let $\Ct\to C$ be the normalization of $C$ and let $p_1, p_2\in \Ct$
be the two points lying above $p$.
Let $C_0:=C$.
For $n\geq 1$ we let $C_n$ denote reducible nodal curve
which is constructed from $\Ct$ and a chain $R=R_1\cup\dots\cup R_n$
of projective lines by identifying the points $p_1, x_1$ and
$p_2, y_n$ respectively.
\begin{definition}
\label{GVB}
A {\em Gieseker vector bundle} on $C$ is a pair $(C'\to C,\F)$ where
$C'=C_n$ for some $n\geq 0$, the morphism $C'\to C$ is the one which
contracts the chain of projective lines to the point $p$ and $\F$ is
a vector bundle on $C'$ whose restriction to the chain of projective
lines is admissible in the sense of \ref{admissible}.
\end{definition}
\begin{definition}
\label{GVBD}
A {\em Gieseker vector bundle datum} on the two-pointed curve
$(\Ct,p_1,p_2)$ is a triple $(C'\to C, F, p')$, where
$(C'\to C,\F)$ is a Gieseker vector bundle on $C$ and $p'$ is
a singular point in $C'$.
\end{definition}
Let $V$ and $W$ be two $r$-dimensional $k$-vector spaces.
In \cite{kgl} I have constructed a certain compactification
$\KGL(V,W)$
of the space $\Isom(V,W)$ of linear isomorphisms from $V$ to $W$
which has properties similar to De Concinis and Procesis so called
wonderfull compactification of adjoint linear groups.
We need the following fact about $\KGL(V,W)$ whose proof can
be found in \cite{kgl}, \S 9:
The variety $\KGL(V,W)$
is the disjoint union of strata $\bO_{I,J}\subset\KGL(V,W)$ indexed
by pairs of subsets $I,J\in[0,r-1]$ such that $\min(I)+\min(J)\geq r$.
Let $I,J$ be such a pair.
Let us write $I=\{i_1,\dots,i_{n_1}\}$ and $J=\{j_1,\dots,j_{n_2}\}$
where $i_1<\dots<i_{n_1}<i_{n_1+1}:=r$ and $j_1<\dots<j_{n_2}<j_{n_2+1}:=r$.
A ($k$-valued) point in $\bO_{I,J}$ is given by
the data
$$
\Phi=
(F_{\bullet}(V),F_{\bullet}(W),\varphib_1,\dots,\varphib_{n_1},\psib_1,\dots,\psib_{n_2},\Phi')
$$
where
\begin{enumerate}
\item
$F_{\bullet}(V)$ denotes a flag
$$
0=F_0(V)\subsetneq F_1(V)\subsetneq\dots\subsetneq F_{n_2}(V)\subseteq F_{n_2+1}(V)
\subsetneq\dots\subsetneq F_{n_1+n_2+1}(V)=V
$$
whith
$\dim F_{\nu}(V)=r-j_{n_2+1-\nu}$ for $\nu\in [0,n_2]$ and
$\dim F_{\nu}(V)=i_{\nu-n_2}$ for $\nu\in [n_2+1,n_1+n_2+1]$,
\item
$F_{\bullet}(W)$ denotes a flag
$$
0=F_0(W)\subsetneq F_1(W)\subsetneq\dots\subsetneq F_{n_1}(W)\subseteq F_{n_1+1}(W)
\subsetneq\dots\subsetneq F_{n_1+n_2+1}(W)=W
$$
where
$\dim F_{\nu}(W)=r-i_{n_1+1-\nu}$ for $\nu\in [0,n_1]$ and
$\dim F_{\nu}(W)=i_{\nu-l}$ for $\nu\in [n_1+1,n_1+n_2+1]$,
\item
the symbol
$\varphib_{\nu}$ denotes the homothety class of an isomorphism
from the subquotient $F_{n_1-\nu+1}(W)/F_{n_1-\nu}(W)$ of $W$ to the subquotient
$F_{n_2+\nu+1}(V)/F_{n_2+\nu}(V)$ of $V$,
\item
the symbol
$\psib_{\nu}$ denotes the homothety class of an isomorphism
from the subquotient $F_{n_2-\nu+1}(V)/F_{n_2-\nu}(V)$ of $V$ to the subquotient
$F_{n_1+\nu+1}(W)/F_{n_1+\nu}(W)$ of $W$,
\item
the symbol
$\Phi'$ denotes an isomorphism from the subquotient
$F_{n_2+1}(V)/F_{n_2}(V)$ of $V$ to the subquotient $F_{n_1+1}(W)/F_{n_1}(W)$ of $W$.
\end{enumerate}
The relationship between Gieseker vector bundles and the compactification $KGL(V,W)$ is
given by the following
\begin{theorem}
\label{GVBD->KGL}
(Cf. \cite{degeneration}, Theorem 9.5)
There exists a natural
bijection from the set of all Gieseker vector bundle data on $(\Ct,p_1,p_2)$
to the set of all pairs $(\E,\Phi)$, where $\E$ is a vector bundle on
$\Ct$ and $\Phi$ is a $k$-valued point in
$
\KGL(\E[p_1],\E[p_2])
$.
More precisely,
let $(C'\to C,\F,p')$ be a Gieseker vector bundle datum on $(\Ct,p_1,p_2)$.
Let $R=R_1\cup\dots\cup R_n$ be the chain of projective lines in $C'$.
Let $y_0:=p_1$ and $x_{n+1}:=p_2$. Let $n_1+n_2=n$ be such that
the singular point $p'\in C'$ comes from identifying the points $y_{n_1}$ and
$x_{n_1+1}$.
Let $d_i$ be the degree of $\F$ restricted to $R_i$.
Let $(\E,\Phi)$ be the pair associated to the given Gieseker vector bundle datum
$(C'\to C, \F, p')$.
Then $\Phi$ is in fact a point in the stratum
$\bO_{I,J}$, where $I=\{i_1,\dots,i_{n_1}\}$, $J=\{j_1,\dots,j_{n_2}\}$ and the
$i_{\nu}$, $j_{\nu}$ are defined by
$$
i_{\nu} = r-\sum_{i=\nu}^{n_1}d_i
\qquad,\qquad
j_{\nu} = r-\sum_{i=n_1+1}^{n-\nu+1}d_i
\quad.
$$
The special case $n=0$ is included here in the sense that then $I=J=\emptyset$
and $\Phi\in\bO_{\emptyset,\emptyset}=\Isom(\E[p_1],\E[p_2])$.
\end{theorem}
\vspace{5mm}
\section{Twisted $\GL_r$-bundles on a fixed curve}
Throughout this section $k$ denotes an algebraically closed field
and $r$ a positive integer.
Let $(C,p_i)$ be an $n$-pointed nodal curve over $k$.
Let $\TVB_r(C,p_i)$ be the set of isomorphism classes of
$n$-pointed twisted $\GL_r$-bundles of the form
$$
(\xi, C\to\Spec(k), \Ac)
\quad.
$$
\vspace{3mm}
\noindent
{\bf The case of a one-pointed smooth curve.}
Assume that $C$ is smooth and that $n=1$, i.e. $(C,p_i)=(C,p)$ is a
one-pointed smooth curve. Let $\PB_r(C,p)$ be the set of isomorphism
classes of vector bundles $E$ of rank $r$ on $C$ together with a
flag in the fiber at $p$.
\begin{theorem}
\label{thm1}
There is a natural surjection
$$
\TVB_r(C,p)\to\PB_r(C,p)
\quad.
$$
\end{theorem}
We skip the proof of Theorem \ref{thm1}, since on the one hand the result
is well known (cf. \cite{MS}, \cite{Biswas}) and on the other hand
there is a proof analogous to (and easier than) the proof of Theorem
\ref{thm2} which we give in detail below.
\vspace{3mm}
\noindent
{\bf The case of a nodal curve with one singularity.}
Assume now that $n=0$ and $C$ has exactly one double point.
Let $\GVB_r(C)$ be the set of isomorphism classes of
Gieseker vector bundles of rank $r$ on $C$.
\begin{theorem}
\label{thm2}
There is a natural surjection
$$
\TVB_r(C)\to\GVB_r(C)
\quad.
$$
\end{theorem}
The rest of the paper is concerned with the proof of Theorem \ref{thm2}.
\vspace{5mm}
\section{Construction}
Let $C$ be a nodal curve over $\Spec(k)$ with one singular point $p$.
Let $(\xi, C\to\Spec(k), \Ac)$ be an object of $\TVB_r(C)$.
Let $(U,\eta,\Gamma)$ be a chart belonging to $\Ac$ such that
there is a point $q\in U$ which is mapped to $p$.
We denote by $\widehat{\Oo}_{p}$ and $\widehat{\Oo}_{q}$
the completion of the local rings $\Oo_{C,p}$ and $\Oo_{U,q}$
respectively.
Let $\Gamma_q\subseteq\Gamma$ be the subgroup consisting of those
elements, which leave $q$ invariant.
$\Gamma_q$ acts on $\widehat{\Oo}_q$, and $\widehat{\Oo}_p$
may be identified with the set of invariants under that action.
By proposition \ref{cyclic} the group $\Gamma_q$ is cyclic of
some order $e$ (which is prime to $\chara(k)$ by the tameness
assumption).
Let $\gamma$ be a generator of $\Gamma_q$.
We choose an isomorphism
\begin{eqnarray}
\label{1}
\widehat{\Oo}_p\isomto k[[s,t]]/(s\cdot t)
\quad.
\end{eqnarray}
It follows from \ref{cyclic}.(3) that there exists an isomorphism
\begin{eqnarray}
\label{2}
\widehat{\Oo}_q\isomto k[[u,v]]/(u\cdot v)
\end{eqnarray}
and a primitive $e$-th root of unity $\zeta$ such that the
diagrams
$$
\xymatrix{
\text{$\widehat{\Oo}_q$} \ar[r]^(0.3){\isomorph}
\ar[d]^{\gamma}
&
k[[u,v]]/(u\cdot v) \ar[d] &
u \ar@{|->}[d] & v \ar@{|->}[d]
\\
\text{$\widehat{\Oo}_q$} \ar[r]^(0.3){\isomorph}
&
k[[u,v]]/(u\cdot v) &
\zeta u & \zeta^{-1}v
}
$$
and
$$
\xymatrix{
\text{$\widehat{\Oo}_q$} \ar[r]^(0.3){\isomorph}
&
k[[u,v]]/(u\cdot v) &
u^e & v^e
\\
\text{$\widehat{\Oo}_p$} \ar[r]^(0.3){\isomorph}
\ar@{^(->}[u]
&
k[[s,t]]/(s\cdot t) \ar@{^(->}[u] &
s \ar@{|->}[u] & t \ar@{|->}[u]
}
$$
are commutative.
Let $\widehat{K}_p$ be the total quotient ring of $\widehat{\Oo}_p$.
Then we have
$
\Spec(\widehat{K}_p)=\Spec(\widehat{\Oo}_p)\times_CC_{\gen}
$
and the isomorphism (\ref{1}) induces an isomorphism
$
\widehat{K}_p \isomto k((s))\times k((t))
$.
We choose an isomorphism
\begin{eqnarray}
\label{3}
\xi\times_{C_{\gen}}\Spec(\widehat{K}_p)\isomto
\GL_r\times\Spec(\widehat{K}_p)
\quad.
\end{eqnarray}
The group $\Gamma_q$ acts on $\eta\times_U\Spec(\widehat{\Oo}_q)$
(since it acts compatibly on $\eta$, $U$, $\Spec(\widehat{\Oo}_q)$).
To analyse this action we need the following
\begin{lemma}
\label{action}
Let $k$ be an algebraically closed field.
Let $(R,\m)$ be a local $k$-algebra with residue field $R/\m=k$.
Let $\Gamma$ be a cyclic group of order $e$ prime to the
characteristic of $k$ and let $\gamma\in\Gamma$ be a generator.
Assume that $\Gamma$ acts on $R$ such that the induced action
on $k$ is trivial.
Let $M$ be a trivial $R$-module of rank $r$ on which $\Gamma$
acts such that $\gamma(ax)=\gamma(a)\gamma(x)$ for all
$a\in R$, $x\in M$.
Then there is a basis $x_1,\dots,x_r$ of $M$ such that
$\gamma(x_i)=\zeta_ix_i$ for some $e$-th roots of unity $\zeta_i$.
\end{lemma}
\begin{proof}
Let $e_1,\dots,e_r$ be an arbitrary basis of $M$.
Let $a=(a_{i,j})\in\GL_r(R)$ be defined by
$
\gamma(e_j)=\sum_{i=1}^ra_{i,j}e_i
\quad.
$
Since $\gamma$ is of order $e$, it follows that
$$
\prod_{j=0}^{e-1}\gamma^j(a)=1
\quad.
$$
We have to show that there is a matrix $b\in\GL_r(R)$
such that
$$
a\cdot\gamma(b)=b\cdot z
$$
for some diagonal matrix $z\in\GL_r(k)$ with $z^e=1$.
Representation theory of finite groups tells us
that there is a matrix $c\in\GL_r(k)$ and a diagonal matrix
$z\in\GL_r(k)$ with $z^e=1$ such that
$a\cdot c\equiv c\cdot z$ modulo $\m$.
Let $a':=c^{-1}\cdot a\cdot c$ and let $b'$ be the matrix
$$
b':=\sum_{i=0}^{e-1}\left(\prod_{j=0}^{i-1}\gamma^j(a')\right)z^{-i}
\quad.
$$
Since $b\equiv e\cdot 1$ modulo $\m$, it follows that $b'\in\GL_r(R)$.
Using the fact that $\prod_{i=0}^{e-1}\gamma^i(a')=1$ a simple
calculation shows that
$$
\gamma(b')=(a')^{-1}\cdot b'\cdot z
\quad.
$$
Therefore, if we set $b:=c\cdot b'$, we get the desired equality.
\end{proof}
\begin{corollary}
\label{alpha}
There exists an isomorphism
\begin{eqnarray}
\label{4}
\eta\times_U\Spec(\widehat{\Oo}_q)\isomto
\GL_r\times\Spec(\widehat{\Oo}_q)
\end{eqnarray}
of principal $\GL_r$-bundles on $\Spec(\widehat{\Oo}_q)$,
and elements $\alpha_1,\dots\alpha_r\in\Z/e\Z$
such that the following diagram commutes:
$$
\xymatrix{
\text{$\eta\times_U\Spec(\widehat{\Oo}_q)$}
\ar[r]^{\isomorph}
\ar[d]^{\gamma}
&
\GL_r\times\Spec(\widehat{\Oo}_q)
\ar[d]^{\diag(\zeta^{\alpha_1},\dots,\zeta^{\alpha_r})\times\gamma}
\\
\text{$\eta\times_U\Spec(\widehat{\Oo}_q)$}
\ar[r]^{\isomorph}
&
\GL_r\times\Spec(\widehat{\Oo}_q)
}
$$
where the morphism
$\diag(\zeta^{\alpha_1},\dots,\zeta^{\alpha_r}): \GL_r\to \GL_r$
is multiplication from the left with the matrix whose only
non-zero entries are the values $\zeta^{\alpha_1},\dots,\zeta^{\alpha_r}$
on the diagonal.
\end{corollary}
\begin{proof}
This is immediate from lemma \ref{action}.
\end{proof}
Let $\widehat{K}_q$ be the total quotient ring of
$\widehat{\Oo}_q$.
The $\Gamma$-equivariant isomorphism
$
\eta\times_UU_{\gen} \isomto \xi\times_{C_{\gen}}U_{\gen}
$,
which is part of the data of the chart $(U,\eta,\Gamma)$,
induces a $\Gamma_q$-equivariant isomorphism
\begin{eqnarray}
\label{5}
\eta\times_U\Spec(\widehat{K}_q)
\isomto
\xi\times_{C_{\gen}}\Spec(\widehat{K}_q)
\end{eqnarray}
of principal $\GL_r$-bundles over $\Spec(\widehat{K}_q)$.
Via the isomorphisms (\ref{3}) and (\ref{4}) such an isomorphism
is given by a matrix
$
F \in \GL_r(\widehat{K}_q)
$
such that
$$
\gamma(F) =
F \cdot \diag(\zeta^{\alpha_1},\dots,\zeta^{\alpha_r})
$$
The isomorphism (\ref{2}) induces an isomorphism
$
\GL_r(\widehat{K}_q)\isomto \GL_r(k((u)))\times\GL_r(k((v)))
$
and we denote by $(F^1(u), F^2(v))$ the image of $F$ under this isomorphism.
The above condition on $F$ translates into the condition
\begin{eqnarray}
\label{6}
F^1_{i,j}(\zeta u)
&=&
\zeta^{\alpha_j} F^1_{i,j}(u)
\\
\label{7}
F^2_{i,j}(\zeta^{-1} v)
&=&
\zeta^{\alpha_j} F^2_{i,j}(v)
\end{eqnarray}
for the entries
$F^1_{i,j}(u)\in k((u))$ and $F^2_{i,j}(v)\in k((v))$
of the matrices $F^1(u)$ and $F^2(v)$.
After possibly changing the isomorphism (\ref{4}) by a permutation
matrix, we can choose integers $a_1,\dots,a_r$ with
\begin{eqnarray}
\label{8}
0\leq a_1\leq a_2\leq \dots \leq a_r < e
\quad
\text{and $a_i\congruent\alpha_i$ mod $e\Z$.}
\end{eqnarray}
Conditions (\ref{6}), (\ref{7}) imply that there are matrices
$H^1(s)$ and $H^2(t)$ with entries
$H^1_{i,j}(s)\in k((s))$ and $H^2_{i,j}(t)\in k((t))$
such that
\begin{eqnarray}
\label{9}
F^1_{i,j}(u) &=& u^{a_j} H^1_{i,j}(u^e)
\\
\label{10}
F^2_{i,j}(v) &=& v^{-a_j} H^2_{i,j}(v^e)
\end{eqnarray}
We will now use the $\GL_r$-bundle $\xi$ over $C_{\gen}$,
the isomorphisms (\ref{1}) and (\ref{3}), the numbers
$a_1,\dots,a_r$ and the matrices $H^1(s)$ and $H^2(t)$,
to construct a Gieseker vector bundle of rank $r$ on the curve $C$.
Let $p_1$ and $p_2$ denote the closed point of
$\Spec(k[[s]])$ and $\Spec(k[[t]])$ respectively.
Let $\V$ be the trivial vector bundle $\Oo^{[1,r]}$
on the disjoint union
$\Spec(k[[s]])\sqcup\Spec(k[[t]])$
(the normalization of $\Spec(k[[s,t]]/(s\cdot t))$),
and let $V$ and $W$ be its fiber at $p_1$ and $p_2$ respectively.
Of course, both $V$ and $W$ are naturally identified with $k^{[1,r]}$.
The numbers $a_1,\dots,a_r$ define a partition
$$
[1,r]=D_1\sqcup D_2\sqcup\dots\sqcup D_m
\quad
$$
characterized by the following properties:
\begin{enumerate}
\item
$D_1$ is the (possibly empty)
set of all indices $i$ such that $a_i=0$.
\item
For $\nu\geq 2$ the set $D_\nu$ is non-empty.
\item
If $1\leq\nu<\nu'\leq m$, $i\in D_\nu$ and $j\in D_{\nu'}$
then $a_i<a_j$.
\item
For all $\nu\in[1,m]$ and $i,j\in D_\nu$ we have $a_i=a_j$.
\end{enumerate}
We define filtrations
\begin{eqnarray*}
0= & F_0(V)\subseteq F_1(V)\subsetneq F_2(V)\subsetneq\dots
\subsetneq F_{m-1}(V)
\subsetneq F_{m}(V) & =V
\\
0=& F_0(W)\subsetneq F_1(W)\subsetneq F_2(W)\subsetneq\dots
\subsetneq F_{m-1}(W)
\subseteq F_{m}(W) & =W
\end{eqnarray*}
by setting
$$
F_i(V) := k^{D_{1}\sqcup\dots\sqcup D_i}
\qquad\text{and}\qquad
F_i(W) := k^{D_{m-i+1}\sqcup\dots\sqcup D_m}
$$
for $i=0,\dots,m$.
For $i=1,\dots,m-1$ let
$$
\varphi_i: F_{m-i}(W)/F_{m-i-1}(W)=k^{D_{i+1}}
\ \Isomto\
k^{D_{i+1}}=F_{i+1}(V)/F_{i}(V)
$$
be the identity morphism on $k^{D_{i+1}}$ and let
$\varphib_i$ be the homothety class of $\varphi_i$.
Finally let
$$
\Phi': F_1(V)/F_{0}(V)= k^{D_1} \ \Isomto\ k^{D_1}=F_m(W)/F_{m-1}(W)
$$
be the identity morphism on $k^{D_1}$.
By \cite{kgl} 9.3 the data
$$
((F_{\bullet}(V),F_{\bullet}(W)),\varphib_1,\dots,\varphib_{m-1},\Phi')
$$
define a $k$-valued point of $KGL(V,W)$, i.e.
a generalized isomorphism $\Phi$ from $V$ to $W$.
Let $\Ct\to C$ be the normalization of the curve $C$.
By a slight abuse of notation we denote also by $p_1$, $p_2$ the
two points of $\Ct$ which lie above the singular point $p$ of $C$.
Let $\E_{\xi}$ be the rank $r$ vector bundle on
$C_{\gen}=\Ct\setminus\{p_1,p_2\}$
associated to the principal $\GL_r$-bundle $\xi$.
We use the isomorphism
$$
\xymatrix{
(k((s))\times k((t)))^{[1,r]}
\ar[r]^{(H^1,H^2)}
&
(k((s))\times k((t)))^{[1,r]}
\\
\V\tensor_{k[[s]]\times k[[t]]} (k((s))\times k((t)))
\ar@{=}[u]
&
\E_{\xi}\tensor_{\Oo_{\Ct}}\widehat{K}_p
\ar[u]^{\isomorph}_{\text{(\ref{1}), (\ref{3})}}
}
$$
as a glueing datum to define a vector bundle $\E$
on $\Ct$, whose fibers at the points $p_1$ and $p_2$ are naturally
identified with $V$ and $W$ respectively.
By \ref{GVBD->KGL} the pair $(\E,\Phi)$ induces a
Gieseker vector bundle datum $(C'\to C,\F,p')$ on $(\Ct,p_1,p_2)$ which
in turn induces a Gieseker vector bundle $(C'\to C, \F)$ on $C$.
For the convenience of the reader I will now describe the Gieseker
vector bundle $(C'\to C, \F)$ explicitely.
Let $R_0:=\Spec(k[[s]])$, $R_m:=\Spec(k[[t]])$.
If $m=1$, we set $R=\Spec(k[[s,t]]/(s\cdot t))$, which is nothing
else but the nodal curve which arrises from $R_0\sqcup R_m$ by
identifying the points $p_1$ and $p_2$.
If $m\geq 2$,
let $R_1,\dots,R_{m-1}$ be $m-1$ copies of the projective line
$\Pp^1$ and let $x_i, y_i$ be two distinct points
in $R_i$. Let $R$ be the nodal curve which arrises from the
union
$$
R_0\sqcup R_1\sqcup\dots\sqcup R_{m-1}\sqcup R_m
$$
by identifying $p_1\in R_0$ and $p_2\in R_m$ with
$x_1\in R_1$ and $y_{m-1}\in R_{m-1}$ respectively
and by identifying $y_i\in R_i$ with $x_{i+1}\in R_{i+1}$ for $i\in[1, m-2]$:
\vspace{1mm}
\begin{center}
\parbox{8cm}{
\xy <0mm,-4mm>; <.9mm,-4mm> :
(3,1);(3,-7)**@{-};
(3,-10) *{\text{$p_1=x_1$}};
(3,4) *{\text{$R_0$}};
(0,-4);(30,4)**@{-}; (17,4) *{\text{$R_1$}};
(27.5,0) *{\text{$y_1=x_2$}};
(25,4);(55,-4)**@{-}; (40,4) *{\text{$R_2$}};
(52.5,-6) *{\text{$y_2=x_3$}};
(50,-4);(80,4)**@{-}; (65,4) *{\text{$R_3$}};
(85,0)*{\cdot};
(90,0)*{\cdot};
(95,0)*{\cdot};
(100,4);(140,-4) **@{-}; (120,4) *{\text{$R_{m-2}$}};
(132.5,-6) *{\text{$y_{m-2}=x_{m-1}$}};
(125,-4);(160,4) **@{-}; (145,4) *{\text{$R_{m-1}$}};
(157,-1);(157,7)**@{-};
(157,10) *{\text{$R_m$}};
(157,-4) *{\text{$y_{m-1}=p_2$}};
\endxy
}
\end{center}
\vspace{3mm}
Let $\Oo_{R_i}(1)$ be the defining bundle on $R_i=\Pp^1$
together with isomorphisms
\begin{eqnarray}
\label{triv O(1)}
\Oo_{R_i}(1)[x_i]\isomto k
\quad\text{and}\quad
\Oo_{R_i}(1)[y_i]\isomto k
\quad.
\end{eqnarray}
We define the rank $r$ vector bundles
$$
E_i:=\Oo_{R_i}^{D_1\sqcup\dots\sqcup D_{i}} \oplus
\Oo_{R_i}(1)^{D_{i+1}} \oplus
\Oo_{R_i}^{D_{i+2}\sqcup\dots\sqcup D_{m}}
$$
on $R_i$ together with the isomorphisms
\begin{eqnarray}
\label{triv E}
E_i[x_i]\isomto k^{[1,r]}
\quad\text{and}\quad
E_i[y_i]\isomto k^{[1,r]}
\end{eqnarray}
induced by (\ref{triv O(1)}).
The maximal ideal $sk[[s]]$ of $k[[s]]$ is a free module of rank one
and as such defines a line bundle $\Oo_{R_0}(-1)$
on $R_0=\Spec(k[[s]])$.
We consider this line bundle together with the isomorphism
\begin{eqnarray}
\label{triv O(-1)}
\Oo_{R_0}(-1)[p_2]\isomto k
\end{eqnarray}
given by $sk[[s]]/s^2k[[s]]\to k$, $s\mapsto 1$.
The generic fiber of $\Oo_{R_0}(-1)$ is identified with $k((s))$ via
the inclusion $sk[[s]]\injto k[[s]]$.
Then we have the rank $r$ vector bundles
$$
E_0:=\Oo_{R_0}^{D_1}\oplus\Oo_{R_0}(-1)^{D_2\sqcup\dots\sqcup D_m}
\qquad\text{and}\qquad
E_m:=\Oo_{R_m}^{[1,r]}
$$
on $R_0$ and $R_m$ together with isomorphisms
\begin{eqnarray}
\label{triv E_0}
E_0[p_1]\isomto k^{[1,r]}
\quad\text{and}\quad
E_m[p_2]\isomto k^{[1,r]}
\end{eqnarray}
(the first one being induced by (\ref{triv O(-1)})) and isomorphisms
\begin{eqnarray}
\label{triv E_0 gen}
E_0\tensor_{\Oo_{R_0}}k((s))\isomto k((s))^{[1,r]}
\quad\text{and}\quad
E_m\tensor_{\Oo_{R_m}}k((t))\isomto k((t))^{[1,r]}
\quad.
\end{eqnarray}
The vector bundles $E_0,\dots,E_m$ glue together via the isomorphisms
(\ref{triv E}) and (\ref{triv E_0}) to form a rank $r$
vector bundle $E$ on $R$.
Let $C'\to C$ be the modification of $C$ obtained by glueing together
$R$ and $C_{\gen}$ along the isomorphism
$$
\xymatrix@R=2ex{
\Spec(k((s)))\sqcup\Spec(k((t))) \ar[r]^(.7){(\ref{1})} \ar@{^(->}[d]
&
\Spec(\widehat{K}_p) \ar@{^(->}[d]
\\
R & C_{\gen}
}
$$
and let $\F$ be the rank $r$ vector bundle on $C'$ obtained by
glueing together $E$ and $\E_{\gen}$ via the isomorphism
$$
\xymatrix{
(k((s))\times k((t)))^{[1,r]}
\ar[r]^{(H^1,H^2)}
&
(k((s))\times k((t)))^{[1,r]}
\\
E\tensor_{\Oo_{R}} (k((s))\times k((t)))
\ar[u]^{\isomorph}_{\text{(\ref{triv E_0 gen})}}
&
\E_{\xi}\tensor_{\Oo_{\Ct}}\widehat{K}_p
\ar[u]^{\isomorph}_{\text{(\ref{1}), (\ref{3})}}
}
$$
It is easy to check that $(C'\to C, \F)$ is indeed a Gieseker
vector bundle on $C$.
\vspace{3mm}
It remains to be shown that the association
$$
(\xi, C\to\Spec(k), \Ac)
\ \mapsto\
(C'\to C, \F)
$$
is surjective and
does not depend on the choices
(\ref{1}), (\ref{2}), (\ref{3}), (\ref{4})
which we made during the construction.
This will be done in the next sections.
\vspace{5mm}
\section{Independence of the isomorphisms (\ref{1}) and (\ref{2})}
Let
$$
\widehat{\Oo}_p\isomto k[[s,t]]/(s\cdot t)
\eqno(\ref{1}')
$$
be another isomorphism and let
$$
\widehat{\Oo}_q\isomto k[[u,v]]/(u\cdot v)
\eqno(\ref{2}')
$$
be an isomorphism with the required property with respect to
(\ref{1}').
For the moment we make the following assumption:
$$
\text{
The images of the
two minimal ideals of $\widehat{\Oo}_p$
under (\ref{1}) and (\ref{1}')
are the same.
}
\eqno(*)
$$
Then there are units $\sigma(s),\pi(s)\in k[[s]]^\times$ and
$\tau(t),\omega(t)\in k[[t]]^\times$ such that the following diagrams commute:
$$
\xymatrix{
k[[s,t]]/(s\cdot t)
\ar[rr]^{s \mapsto s\sigma(s)}_{t\mapsto t\tau(t)}
& &
k[[s,t]]/(s\cdot t)
\\
&
\text{$\widehat{\Oo}_p$}
\ar[lu]^{(1)}
\ar[ru]_{(1')}
&
}
$$
$$
\xymatrix{
k[[u,v]]/(u\cdot v)
\ar[rr]^{u \mapsto u\pi(u^e)}_{v\mapsto v\omega(v^e)}
& &
k[[u,v]]/(u\cdot v)
\\
&
\text{$\widehat{\Oo}_p$}
\ar[lu]^{(2)}
\ar[ru]_{(2')}
&
}
$$
Furthermore we have $\pi^e=\sigma$ and $\omega^e=\tau$.
It should be noticed that the $e$-th root of unity $\zeta$ is independent
of whether we choose (\ref{1}) or (\ref{1}'),
since it is the eigenvalue of $\gamma$ operating on the tangent
space of one of the branches of $\Spec(\widehat{\Oo}_q)$
and by assumption $(*)$ both the isomorphisms
(\ref{1}) and (\ref{1}') map that branch
$\Spec(\widehat{\Oo}_q)$ to the same branch $\{v=0\}$ of
$\Spec(k[[u,v]]/(u\cdot v))$.
Therefore the elements $\alpha_1,\dots,\alpha_r\in\Z/e\Z$ and
the numbers $a_1,\dots,a_r$ are independent
of whether we choose (\ref{1}) or (\ref{1}').
Let $(\tF_1(u),\tF_2(v))$ be the image of $F$ under the isomorphism
$
\GL_r(\widehat{K}_q)\isomto\GL_r(k[[u]])\times\GL_r(k[[v]])
$
induced by (\ref{2}'). Then we have $\tF^1(u)=F^1(u\pi(u^e))$ and
$\tF^2(v)=F^2(v\omega(v^e))$ and it follows that
\begin{eqnarray*}
\tF^1_{i,j}(u) &=& u^{a_j}\cdot\tH^1_{i,j}(u^e)
\quad,
\\
\tF^2_{i,j}(v) &=& v^{-a_j}\cdot\tH^2_{i,j}(v^e)
\quad,
\end{eqnarray*}
where
\begin{eqnarray*}
\tH^1_{i,j}(s) &=& \pi^{a_j}H^1_{i,j}(s\sigma)
\quad,
\\
\tH^2_{i,j}(t) &=& \omega^{-a_j}H^2_{i,j}(t\tau)
\quad.
\end{eqnarray*}
Therefore the following diagram commutes:
$$
\xymatrix{
(k((s))\times k((t)))^{[1,r]}
\ar[rr]^{(H^1,H^2)}
\ar[dd]
& &
(k((s))\times k((t)))^{[1,r]}
\ar[dd]_{s\mapsto s\sigma}^{t\mapsto t\tau}
\\
&
\E_{\xi}\tensor_{\Oo_{\Ct}}\widehat{K}_p
\ar[ur]^{\isomorph}_{\text{(\ref{1}), (\ref{3})}}
\ar[dr]_{\isomorph}^{\text{(\ref{1}'), (\ref{3})}}
&
\\
(k((s))\times k((t)))^{[1,r]}
\ar[rr]^{\text{$(\tH^1,\tH^2)$}}
& &
(k((s))\times k((t)))^{[1,r]}
}
$$
where the left vertical arrow maps an element $(x(s),y(t))$
to the element
$$
(\diag(\pi^{a_1},\dots,\pi^{a_r}) x(s\sigma),
\diag(\omega^{-a_1},\dots,\omega^{-a_r}) y(t\tau))
\quad.
$$
Let $\tilde{\E}$ be the vector bundle on $\Ct$ obtained by the glueing datum
$(\tH^1,\tH^2)$. Then the above diagram shows that there is an
isomorphism $\E\isomto\tilde{\E}$ which induces the isomorphisms
$$
\xymatrix@R=1ex{
\E[p_1]=k^{[1,r]}
\ar[rrrr]^{\diag(\pi(0)^{-a_1},\dots,\pi(0)^{-a_r})}
& & & &
k^{[1,r]}=\tilde{\E}[p_1]
\\
\E[p_2]=k^{[1,r]}
\ar[rrrr]^{\diag(\omega(0)^{a_1},\dots,\omega(0)^{a_r})}
& & & &
k^{[1,r]}=\tilde{\E}[p_2]
}
$$
between the fibers at $p_1$ and $p_2$ respectively.
Thus it maps the generalized isomorphism
$
\Phi\in\KGL(k^{[1,r]},k^{[1,r]})=\KGL(\E[p_1],\E[p_2])
$
to the generalized isomorphism
$
\Phi\in\KGL(k^{[1,r]},k^{[1,r]})=\KGL(\tilde{\E}[p_1],\tilde{\E}[p_2])
$.
This shows that the pairs $(\E,\Phi)$
and $(\tilde{\E},\Phi)$ are isomorphic. Consequently
this is also true for the associated Gieseker vector bundles.
To get rid of the assumption $(*)$ we investigate now what happens
if we change the isomorphisms
(\ref{1}), (\ref{2})
by composing them with the automorphisms
$$
\xymatrix@R=1ex{
k[[s,t]]/(s\cdot t) \ar[r]^{s\mapsto t}_{t\mapsto s}
&
k[[s,t]]/(s\cdot t)
\\
k[[u,v]]/(u\cdot v) \ar[r]^{u\mapsto v}_{v\mapsto u}
&
k[[u,v]]/(u\cdot v)
}
$$
respectively.
This means that $\zeta^{-1}$ takes the role of $\zeta$ and consequently
the set $\{\alpha_1,\dots,\alpha_r\}\subseteq \Z/e\Z$ from
\ref{alpha} is replaced by the set $\{-\alpha_1,\dots,-\alpha_r\}$.
It follows that in (\ref{8}) we would choose integers
$
\tilde{a}_1,\dots,\tilde{a}_r
$
instead of
$
a_1,\dots,a_r
$,
where
$$
\tilde{a}_i=
\left\{
\begin{array}{ll}
a_i \quad & \text{for $i\in[1,i_1]=D_1$} \\
e-a_{r+i_1+1-i} \quad & \text{for $i\in[i_1+1,r]$}
\end{array}
\right.
$$
Then the matrix $F$ is replaced by the matrix
$\tilde{F}=F\cdot\Lambda$, where
$$
\Lambda =
\left[
\vcenter{
\hbox{
\xymatrix@C.5ex@R.5ex{
\eins_{i_1} & \ar@{-}'[dddd] & & 0 & \\
\ar@{-}'[rrrr] & & & & \\
& & & & 1 \ar@{.}[ddll] \\
0 & & & & \\
& & 1 & &
}
}}
\right]
$$
is the permutation matrix
belonging to the permutation $\lambda\in S_r$, where
$$
\lambda(i)=
\left\{
\begin{array}{ll}
i \quad & \text{for $i\in[1,i_1]$} \\
r+i_1+1-i \quad & \text{for $i\in[i_1+1,r]$}
\end{array}
\right.
\quad,
$$
and the matrices $H^1(s)$ and $H^2(t)$ are replaced by the matrices
$\tilde{H}^1(s)$ and $\tilde{H}^2(t)$ respectively, where
$$
\tilde{H}^1(s) = H^2(s)\cdot\Lambda\cdot
\left[
\vcenter{
\hbox{
\xymatrix@C.5ex@R.5ex{
\eins_{i_1} & 0 \\
0 & s^{-1}\eins_{r-i_1}
}
}}
\right]
\qquad\text{and}\qquad
\tilde{H}^2(t) = H^1(t)\cdot\Lambda\cdot
\left[
\vcenter{
\hbox{
\xymatrix@C.5ex@R.5ex{
\eins_{i_1} & 0 \\
0 & t\eins_{r-i_1}
}
}}
\right]
\quad.
$$
The numbers $\tilde{a}_1,\dots,\tilde{a}_r$ define the partition
$$
[1,r]=\tilde{D}_1\sqcup\tilde{D}_2\sqcup\dots\sqcup\tilde{D}_m
$$
where $\tilde{D}_1=D_1$ and $\tilde{D}_i=\lambda(D_{m+2-i})$
for $i\in[2,m]$.
Let
$(\tilde{R}=
\tilde{R}_0\cup\dots\cup\tilde{R}_m,\tilde{E})$
be the nodal curve associated to this partition,
together with isomorphisms
$\tilde{E}\tensor_{\Oo_{\tilde{R}}}k((s))\isomto k((s))^{[1,r]}$
and
$\tilde{E}\tensor_{\Oo_{\tilde{R}}}k((t))\isomto k((t))^{[1,r]}$
as in (\ref{triv E_0 gen}).
Now one checks easily that there is an isomorphism
$$
\rho:(R,E)\isomto(\tilde{R},\tilde{E})
$$
which sends the component $R_i$ to $\tilde{R}_{m-i}$ ($i=0,\dots,m$),
such that the following diagram commutes:
$$
\xymatrix{
E\tensor_{\Oo_R}(k((s))\times k((t)))
\ar[r]^{\isomorph}
\ar[d]^{\rho}
&
(k((s))\times k((t)))^{[1,r]}
\ar[d]_{\rho'}
\\
\text{$\tilde{E}\tensor_{\Oo_{\tilde{R}}}(k((s))\times k((t)))$}
\ar[r]^{\isomorph}
&
(k((s))\times k((t)))^{[1,r]}
}
$$
where the morphism $\rho'$ is given by
$$
(x(s),y(t))\mapsto
\left(
\Lambda\cdot
\left[
\vcenter{
\hbox{
\xymatrix@C.5ex@R.5ex{
\eins_{i_1} & 0 \\
0 & s^{-1}\eins_{r-i_1}
}
}}
\right]\cdot
y(s)
\ ,\
\Lambda\cdot
\left[
\vcenter{
\hbox{
\xymatrix@C.5ex@R.5ex{
\eins_{i_1} & 0 \\
0 & t\eins_{r-i_1}
}
}}
\right]\cdot
x(t)
\right)
$$
From the commutativity of the diagram
$$
\xymatrix{
(k((s))\times k((t)))^{[1,r]}
\ar[rr]^{(H^1(s),H^2(t))}
\ar[d]^{\rho'}
& &
(k((s))\times k((t)))^{[1,r]}
\ar[d]^{s\mapsto t}_{t\mapsto s}
&
\E_{\xi}\tensor_{\Oo_{\tilde{C}}}\widehat{K}_p
\ar[l]_(.4){(1),(3)}
\ar@{=}[d]
\\
(k((s))\times k((t)))^{[1,r]}
\ar[rr]^{(\tilde{H}^1(s),\tilde{H}^2(t))}
& &
(k((s))\times k((t)))^{[1,r]}
&
\E_{\xi}\tensor_{\Oo_{\tilde{C}}}\widehat{K}_p
\ar[l]_(.4){(1'),(3)}
}
$$
it finally follows that the Gieseker vector bundle
$(\tilde{C}'\to C,\tilde{\F})$
constructed from the data $\xi$, (\ref{1}'),
(\ref{3}), $(\tilde{a}_1,\dots,\tilde{a}_r)$,
$\tilde{H}^1(s)$, $\tilde{H}^2(t)$
is isomorphic to the Gieseker vector bundle
$(C'\to C,\F)$ constructed from the data $\xi$, (\ref{1}),
(\ref{3}), $(a_1,\dots,a_r)$, $H^1(s)$, $H^2(t)$.
\vspace{5mm}
\section{Independence of the isomorphisms (\ref{3}) and (\ref{4})}
Independence of (\ref{3}) is immediate, since if we change it by
an automorphism of $\GL_r\times\Spec(\widehat{K}_p)$
(which can be written as an element in $\GL_r(k((s)))\times\GL_r(k((t)))$),
then $(H^1(s),H^2(t))$ is changed by that same matrix.
Two isomorphisms (\ref{4}) differ by a matrix
$
A=(A_{i,j}) \in \GL_r(\widehat{\Oo}_q)
$
such that
\begin{eqnarray}
\label{cond on A}
A = \diag(\zeta^{-\alpha_1},\dots,\zeta^{-\alpha_r})\cdot
\gamma(A)\cdot
\diag(\zeta^{\alpha_1},\dots,\zeta^{\alpha_r})
\quad.
\end{eqnarray}
After identifying $\widehat{\Oo}_q$ with the ring
$k[[u,v]]/(u\cdot v)$ via the isomorphism (\ref{1}), we can write
$$
A=A^0 + u\cdot A^1(u) + v\cdot A^2(v)
$$
with uniquely determined matrices
$A^0\in\GL_r(k)$,
$A^1(u)\in M(r\times r, k[[u]])$ and
$A^2(v)\in M(r\times r, k[[v]])$.
Condition (\ref{cond on A}) implies that $A^0$ is a block matrix
of the form
\begin{equation}
\label{A^0}
A^0 =
\left[
\vcenter{
\hbox{
\xymatrix@C.5ex@R.5ex{
A^0_1 \ar@{.}[dr] & 0 \\
0 & A^0_m
}
}}
\right]
\end{equation}
where the $A^0_i$ are blocks of size $\card D_i$ for $i=1,\dots,m$.
Condition (\ref{cond on A}) implies furthermore that there
are matrices
$B^1(s)=(B^1_{i,j}(s))\in\GL_r(k[[s]])$ and
$B^2(t)=(B^2_{i,j}(t))\in\GL_r(k[[t]])$
such that
\begin{eqnarray*}
A^1(u) &=& u^{-1}\diag(u^{a_1},\dots,u^{a_r})
\cdot B^1(u^e)\cdot
\diag(u^{-a_1},\dots,u^{-a_r})
\quad,
\\
A^2(v) &=& v^{-1}\diag(v^{-a_1},\dots,v^{-a_r})
\cdot B^2(v^e)\cdot
\diag(v^{a_1},\dots,v^{a_r})
\end{eqnarray*}
and such that
\begin{equation}
\label{B(0)}
\begin{array}{ll}
B^1_{i,j}(0) &= 0
\quad\text{for $a_i-a_j\leq 0$}
\quad,
\\
B^2_{i,j}(0) &= 0
\quad\text{for $a_j-a_i\leq 0$}
\quad.
\end{array}
\end{equation}
The change of (\ref{4}) by the matrix $A$ means that we have
to replace $F$ by the matrix
$$
\tilde{F}=F\cdot A
$$
and that consequently we have to replace
the matrices $H^1(s)$ and $H^2(t)$ by the matrices
\begin{eqnarray*}
\tilde{H}^1(s) &=& H^1(s)\cdot(A^0 + B^1(s))
\qquad\text{and}
\\
\tilde{H}^2(t) &=& H^2(t)\cdot(A^0 + B^2(t))
\end{eqnarray*}
respectively.
The pair of matrices
$(A^0 + B^1(s)$, $A^0 + B^2(t))$
defines an automorphism of $\V$ which induces the automorphisms
$A^0 + B^1(0)$ and $A^0 + B^2(0)$ on the special fibers
$V$ and $W$ respectively.
From (\ref{A^0}) and (\ref{B(0)}) it follows that the induced automorphism
of $\KGL(V,W)$ maps the generalized isomorphism $\Phi$ to itself.
It follows that the pair $(\tilde{\E},\tilde{\Phi})$ obtained by the
glueing datum $(\tilde{H}^1, \tilde{H}^2)$ is isomorphic to the pair
$(\E,\Phi)$ obtained by the glueing datum $(H^1, H^2)$. Therefore
also the induced Gieseker vector bundles are isomorphic.
\vspace{5mm}
\section{Surjectivity}
Let $(C'\to C,\F)$ be a Gieseker vector bundle on $C$.
By definition, $C'$ is either isomorphic to $C$, or it is
the union of the normalization $\Ct$ of $C$ and a chain
$R$ of projective lines which intersects $\Ct$ in the two
points $p_1$ and $p_2$ lying above the singularity $p\in C$.
In the first case we let $p':=p$, in the second case
we let $p'=p_2$.
Then the tripel
$$
(\Ct'\to\Ct, \Ft', p')
$$
is a Gieseker vector bundle datum in the sense of \ref{GVBD}.
By \ref{GVBD->KGL} such a datum induces a vector bundle
$\E$ on the curve $\Ct$ together with a generalized isomorphism
$\Phi$ from $V:=\E[p_1]$ to $W:=\E[p_2]$.
More precisely, $\Phi$ is a $k$-valued point of $\KGL(V,W)$ which
lies in the stratum $\bO_{I,J}$ for some $I\subseteq[0,r-1]$ and $J=\emptyset$.
As we have recalled in \S \ref{review}, such a point is given by a tupel
$$
((F_{\bullet}(V),F_{\bullet}(W)), \varphib_1,\dots,\varphib_{m-1},\Phi')
\quad,
$$
where $m:=|I|+1$,
\begin{eqnarray*}
0= & F_0(V)\subseteq F_1(V)\subsetneq F_2(V)\subsetneq\dots
\subsetneq F_{m-1}(V)
\subsetneq F_{m}(V) & =V
\\
0=& F_0(W)\subsetneq F_1(W)\subsetneq F_2(W)\subsetneq\dots
\subsetneq F_{m-1}(W)
\subseteq F_{m}(W) & =W
\end{eqnarray*}
are flags in $V$ and $W$ respectively,
$\varphib_i$ is the homothety class of an isomorphism
$$
\varphi_{\nu}: F_{m-\nu}(W)/F_{m-\nu-1}(W)\isomto F_{\nu+1}(V)/F_{\nu}(V)
$$
and $\Phi'$ denotes an isomorphism
$F_1(V)/F_0(V)\isomto F_m(W)/F_{m-1}(W)$.
There is a basis $v_1,\dots,v_r$ of $V$ and $w_1,\dots,w_r$ of $W$
and a partition
$$
[1,r]=D_1\sqcup D_2\sqcup \dots \sqcup D_m
$$
with the property that
\begin{enumerate}
\item
$i\in D_{\nu}$, $j\in D_{\nu'}$ and $i<j$ implies $\nu\leq \nu'$,
\item
$F_{\nu}(V)$ is generated by
$\{v_i\ |\ i\in D_1\sqcup\dots\sqcup D_{\nu}\}$
and
$F_{\nu}(W)$ is generated by
$\{w_i\ |\ i\in D_{m-\nu+1}\sqcup\dots\sqcup D_{m}\}$,
\item
for $i\in D_{\nu+1}$
the isomorphism $\varphi_{\nu}$ sends the residue class of $w_i$
mod $F_{m-\nu-1}(W)$
to the residue class of $v_i$ mod $F_{\nu}(V)$,
\item
for $i\in D_1$ the isomorphism $\Phi'$ sends
$v_i$ to the residue class of $w_i$ mod $F_{m-1}(W)$.
\end{enumerate}
For $i=1,2$
we choose an isomorphism
\begin{equation}
\label{E at p_i}
\E\tensor_{\Oo_{\Ct}}\Oh_{p_i}\isomto \Oh_{p_i}^r
\end{equation}
which induces the isomorphism
$V\to k^r$, $v_i\to e_i$
and
$W\to k^r$, $w_i\to e_i$
from the fibres at $p_1$ and $p_2$ respectively,
where $e_1,\dots,e_r$ is the canonical basis of $k^r$.
Let $\xi$ be the prinicpal $\GL_r$-bundle on $C_{\gen}$ of local frames of
the restriction of the vector bundle $\E$ to
$C_{\gen}=\Ct\setminus\{p_1,p_2\}$.
Then the isomorphisms \ref{E at p_i} induce the isomorphism
\begin{equation}
\label{xi at K_p}
\xi\times_{C_{\gen}}\Spec(\Kh_p)
\isomto
\GL_r\times\Spec(\Kh_p)
\end{equation}
\begin{lemma}
There is a morphism $f:U\to C$, an integer
$e\geq m$ prime to the characteristic of $k$
and an operation of $\Gamma:=\Z/e\Z$
on $U$ such that
\begin{enumerate}
\item
$\Gamma$ leaves $f$ invariant and the induced morphism
$U/\Gamma\to C$ is etale,
\item
$U$ has exactly one singular point $q$ and $f^{-1}(p)=\{q\}$,
\item
the action of $\Gamma$ on $f^{-1}(C_{\gen})$ is free.
\end{enumerate}
\end{lemma}
\begin{proof}
Since $p$ is an ordinary double point of $C$, there exists a diagram
of pointed schemes and \'etale morphisms as follows:
$$
\xymatrix{
(C,p)
&
(U_0,q_0)
\ar[l]_{\text{\'etale}}
\ar[r]^(.24){\text{\'etale}}
&
(V_0,y_0):=((\Spec(k[s,t]/(s\cdot t)),(s,t))
}
\quad.
$$
After removing from $U_0$
the points $\neq q_0$ in the fiber of $U_0\to C$
we may assume that $q_0$ is the only point lying above $p$.
Choose $e\in\Z$ prime to $\chara(k)$ with $e\geq m$.
Let
$$(V,y):=(\Spec(k[u,v]/(u\cdot v)),(u,v))$$
and let $(V,y)\to(V_0,y_0)$ be defined by $s\mapsto u^e$, $t\mapsto v^e$.
Let $\gamma$ be a generator of $\Gamma=\Z/e\Z$ and let $\zeta\in k$ be a
primitive $e$-th root of unity. We define an action of $\Gamma$ on $(V,y)$
by letting $\gamma(u)=\zeta u$ and $\gamma(v)=\zeta^{-1} v$.
Now we set
$$
U:=U_0\times_{V_0}V
$$
and let $f:U\to C$ be the composition $U\to U_0\to C$.
From $V$ the scheme $U$ inherits an action of the group $\Gamma$.
Since $V/\Gamma=V_0$ and $U_0\to V_0$ is flat we have
$U/\Gamma=U_0$ which by construction is \'etale over $C$.
The only point in the fibre of $f$ over $p$ is the point
$q=(q_0,y)\in U$. Since $U_0\to V_0$ is \'etale
and $V\to V_0$ is smooth outside the point $y$,
it follows that the fibre product $U=U_0\times_{V_0}V$ is regular
outside $q$.
Furthermore, since the action of $\Gamma$ on $V\setminus\{y\}$ is
free the same holds for the action of $\Gamma$ on $U\setminus \{q\}$.
\end{proof}
In what follows we will construct a chart $(U,\eta,\Gamma)$ for $\xi$
where $U\to C$ and $\Gamma$ are chosen as in the lemma and the
$\GL_r$-bundle $\eta$ with $\Gamma$-operation
is glued together from an object $\eta_{\gen}$
over $U_{\gen}$ and an object $\etah_q$ over
the completion of $U$ at the singular point $q$.
To fix notation,
let $\Oh_q$ be the completion of the local ring $\Oo_{U,q}$ and
let $\gamma$ be a generator of $\Gamma$.
There exists an isomorphism
\begin{equation}
\Oh_q\isomto k[[u,v]]/(u\cdot v)
\end{equation}
and a primitive $e$-th root of unity $\zeta$ such that
the automorphism $\gamma:\Oh_q\isomto\Oh_q$ translates
into the automorphism $u\mapsto \zeta u$, $v\mapsto \zeta^{-1} v$
of $k[[u,v]]/(u\cdot v)$ (cf. \cite{ACV}, 2.1.2).
Let $a_i\in[0,e-1]$ ($i\in[1,r]$) be chosen such that:
\begin{eqnarray*}
a_i=0 &\text{for}& i\in D_1, \\
a_i<a_j &\text{for}& i\in D_{\nu},\ j\in D_{\nu'},\ \nu<\nu', \\
a_i=a_j &\text{for}& i,j\in D_{\nu},\ \nu \in[1,m].
\end{eqnarray*}
Let $\etah_q:=\GL_r\times\Spec\Oh_q$ together with the $\Gamma$-operation
defined by
$$
\diag(\zeta^{a_1},\dots,\zeta^{a_r})\times\gamma:
\GL_r\times\Spec\Oh_q\isomto\GL_r\times\Spec\Oh_q
\quad.
$$
Let $\eta_{\gen}:=\xi\times_{C_{\gen}}U_{\gen}$ together with the
$\Gamma$-operation given by
$$
\id\times\gamma:\xi\times_{C_{\gen}}U_{\gen}\isomto
\xi\times_{C_{\gen}}U_{\gen}
\quad.
$$
Now we glue together $\etah_q$ and $\eta_{\gen}$ along
$\Spec\Kh_q\isomorph k((u))\times k((v))$ via the
isomorphism
$$
\xymatrix{
\text{$\etah_q\times_{\Oh_q}\Spec(\Kh_q)$}
\ar[r]^{(\ref{xi at K_p})}
&
\text{$\GL_r\times\Spec(\Kh_q)$}
\ar[r]^(0.35){F^1\times F^2}
&
\text{$\GL_r\times\Spec(\Kh_q)$}
=
\text{$\eta_{\gen}\times_{U_{\gen}}\Spec(\Kh_q)$}
}
$$
where
$$
F^1=\diag(u^{a_1},\dots,u^{a_r})
\quad\text{and}\quad
F^2=\diag(v^{-a_1},\dots,v^{-a_r})
\quad.
$$
This gives a principal $\GL_r$-bundle $\eta$ on $U$.
From the commutativity of the diagram
$$
\xymatrix{
\text{$\etah_q\times_{\Oh_q}\Spec(\Kh_q)$}
\ar[r]^{\isomorph}
\ar[d]_{\diag(\zeta^{a_1},\dots,\zeta^{a_r})\times\gamma}
&
\text{$\eta_{\gen}\times_{U_{\gen}}\Spec(\Kh_q)$}
\ar[d]^{\id\times\gamma}
\\
\text{$\etah_q\times_{\Oh_q}\Spec(\Kh_q)$}
\ar[r]^{\isomorph}
&
\text{$\eta_{\gen}\times_{U_{\gen}}\Spec(\Kh_q)$}
}
$$
it follows that the $\Gamma$-operation on $\etah_q$ and $\eta_{\gen}$
induces a $\Gamma$-operation on $\eta$.
It is clear from the construction that the triple $(U,\eta,\Gamma)$
forms a chart for $\xi$.
There is a chart $(U_1,\eta_1,\Gamma_1)$ for $\xi$,
where $U_1:=C_{\gen}$, $\eta:=\xi$,
$\Gamma:=(1)$.
This chart together with the chart $(U,\eta,\Gamma)$ make up
a balanced atlas $\Ac$ for $\xi$.
It is clear by construction that the twisted $G$-bundle
$(\xi,C\to\Spec(k),\Ac)$ is mapped to the Gieseker
vector bundle $(C'\to C,\F)$.
\section{Further directions}
The relationship between twisted $GL_r$-bundles and
Gieseker vector bundles should be further investigated since
it might lead to a clue what the right notion of stable maps
to the classifying stack of a reductive group are.
The next step would be to try to extend the mapping given in
\ref{thm2} so that it works for families.
For example let $A:=\C[[t]]$, let $S:=\Spec A$
and let $C\to S$ be a
stable curve over $S$. Let
$$
\left(
\vcenter{
\xymatrix@R=1ex{
C' \ar[rr] \ar[dr] & &
C \ar[dl] \\
& S &
}},
\F
\right)
$$
be a Gieseker vector bundle of rank $r$ on $C$.
Assume in paticular that the generic fiber of $C\to S$ is smooth
and that its special fiber is irreducible with one double point $p$.
Then it can be shown that there is a twisted $GL_r$-bundle
$(\xi,C\to S,\Ac)$ such that if we apply the mappings from theorems
\ref{thm1} and \ref{thm2} to the isomorphism class of the
generic and special fiber of $(\xi,C\to S,\Ac)$,
then we obtain the generic and special fiber of the Gieseker vector
bundle
$(C'\to C,\F)$ respectively.
Indeed, in the neighbourhood of $p$ one may chose a chart
$(U,\eta,\Gamma)$ for $\xi$, where $U\to C$ \'etale locally looks
like
$$
\xymatrix@R=1ex{
\Spec A[u,v]/(uv-t)\ar[r] &
\Spec A[x,y]/(xy-t^e) \\
u^e &
x \ar@{|->}[l] \\
v^e &
y \ar@{|->}[l]
}
$$
On the other hand, assume $C=C_0\times S$, where $C_0$ is an irreducible
curve with one ordinary double point $p$, and assume that
$C'\to C$ induces an isomorphism of the generic fibers and the
morphism $C_1\to C_0$ on the special fibers.
For simplicity let us assume furthermore that the rank $r$ of
the Gieseker bundle is one.
In this situation it would be interesting to
know, whether there is a twisted $\GL_1$-bundle
$(\xi,C\to S,\Ac)$ such that the map of theorem \ref{thm2}
maps the generic and the special fiber of $(\xi,C\to S,\Ac)$
to the generic and special fiber of $(C'\to C,\F)$ respectively.
| 24,752
|
TITLE: If $f:\Bbb R \to\Bbb R$ is continuous and $f^2$ is uniformly continuous, does that necessarily imply that $f$ is uniformly continuous?
QUESTION [0 upvotes]: Actually, by given criterion, if $\{x_n\}$ and $\{y_n\}$ be any two sequences such that $|x_n - y_n|$ goes to $0$, then $|f^2(x_n) - f^2(y_n)|$ also goes to $0$, now if $|f(x_n) + f(y_n)|$ is bounded, then f is also uniformly continuous.
But, I think it has a counter-example, I was thinking about a function $f(x)= \sqrt{\ln(1+x)} ; x\geq 0$, it can be shown by the mean value theorem that $f^2$ is uniformly continuous, but I can't think it about $f$.
One fact is obvious that as $f'$ is unbounded, then $f$ is not Lipschitz.
REPLY [6 votes]: First, note that the composition of two uniformly continuous functions is uniformly continuous; and $g(x) = \sqrt{x}$ is uniformly continuous on $[0, \infty)$. Therefore, if $f^2$ is uniformly continuous, then $g \circ f^2 = |f|$ is also uniformly continuous.
It remains to show that $f$ continuous and $|f|$ uniformly continuous implies that $f$ is uniformly continuous. To see this, for $\epsilon > 0$ choose $\delta > 0$ such that $|x-y| < \delta$ implies $|\,|f(x)| - |f(y)|\,| < \frac{\epsilon}{2}$. Then whenever $|x-y| < \delta$, if $f(x)$ and $f(y)$ have the same sign, then $|f(x) - f(y)| = |\,|f(x)| - |f(y)|\,| < \frac{\epsilon}{2} < \epsilon$. On the other hand, if $f(x)$ and $f(y)$ have opposite signs, then by the intermediate value theorem there exists $z$ between $x$ and $y$ such that $f(z) = 0$. Then $|x-z| < \delta$ and $|z-y| < \delta$; so $$|f(x) - f(y)| \le |f(x) - f(z)| + |f(z) - f(y)| = \\ |\,|f(x)| - |f(z)|\,| + |\,|f(z)| - |f(y)|\,| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.$$
| 94,391
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TITLE: Integral closure of p-adic integers in maximal unramified extension
QUESTION [10 upvotes]: Let $\mathbb Q_p$ be the field of p-adic numbers, and let $\mathbb Q_p^{\text{unr}}$ be maximal unramified extension in some algebraic closure of $\mathbb Q_p$. My understanding is that $\mathbb Q_p^{\text{unr}}$ has a fairly explicit description:
$$ \mathbb Q_p^{\text{unr}} = \mathbb Q_p \left(\bigcup_{(n,p)=1} \mu_n \right)$$
where $\mu_n$ is a primitive $n$th root of unity, i.e. we adjoin all $n$th roots of unity with $n$ relatively prime to $p$.
My question is: Does the integral closure of $\mathbb Z_p$ in $\mathbb Q_p^{\text{unr}}$ have a similarly explicit description? For example, does it equal:
$$ \mathbb Z_p \left[\bigcup_{(n,p)=1} \mu_n \right] $$
perhaps?
REPLY [11 votes]: Yes, this is true. Since the integral closure of a directed union is the union of the integral closures, it suffices to establish this at every finite level: that is, for $n$ prime to $p$, the ring of integers in $\mathbb{Q}_p(\zeta_n)$ is $\mathbb{Z}_p[\zeta_n]$.
Here are two methods of proof:
First Proof (Local): This follows from the structure theory of unramified extensions of local fields. For instance, you can apply Proposition 4 of these notes on local fields to $\overline{f}$, the minimal polynomial over $\mathbb{F}_p$ of a primitive $n$th root of unity.
Second Proof (Global): Show that the discriminant of the order $\mathcal{O} = \mathbb{Z}[\zeta_n]$ -- or, in plainer terms, of $(1,\zeta_n,\ldots,\zeta_n^{\varphi(n)-1})$ -- is prime to $p$. Therefore the localized order $\mathcal{O} \otimes \mathbb{Z}_p$ is maximal.
| 211,301
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TITLE: local minimum of $|f|$
QUESTION [1 upvotes]: Suppose $f \in H(\Omega)$, where $\Omega\subset\mathbb C$ is an open set. Under what condition can $|f|$ have a local minimum?
Here $|f| = u^2 +v^2 = g$ say. We assumed $f(x,y)= u(x,y) +i v(x,y)$.
Then $g$ has local minimum if $g_{xx} > 0$ and $g_{xx}= 2[u_x^2 +uu_{xx} +v_x ^2 +vv_{xx}]$. So as square terms are positive always, the required condition is $uu_{xx}+vv_{xx} >0$.
I am asking if this a correct answer; if not then please guide me in the right way.
Thanks in advance.
REPLY [0 votes]: If $f$ is a constant, $|f|$ has a local minimum at any point of $\Omega.$
If $f$ is not a constant and there exists $z\in\Omega$ s.t. $f(z)=0,$ then the minimum occurs at those points $z\in\Omega.$
If $f$ is not a constant and there isn't any $z\in\Omega$ s.t. $f(z)=0,$ then $\frac{1}{f}$ doesn't have singularities on $\Omega.$ So, $f\in h(\Omega)$ and $\frac{1}{f}\in h(\Omega).$ Let suppose that $|f|$ occurs minimum at $a\in\Omega.$ Then, by the Principle of Maximum Modulus for $\frac{1}{f}$ we have that $\vert\frac{1}{f(a)}\vert<\{\vert\frac{1}{f(z)}\vert\vert z\in\partial D\}$ for any neighborhood $\bar{D}\subset\Omega$ of $a,$ i.e. $|f(a)|>|f(z)|$ for a $z\in\partial D\subset\Omega$ witch contradicts the supposition that $a$ was a local minimum. This means that under there conditions, |f| does not occur a local minimum on $\Omega.$
| 159,686
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Solutions In Background Check - Some Thoughts
Migration Background Examine - Free Document Look for United States Citizenship
Anybody that may be putting on end up being an U.S. person is needed to undergo an immigration background examination. Given that the Usa already has enough concerns taking care of its own criminals, the last point the country wants is to permit one more countries criminals into the united state borders. To stop this from occurring the U.S. Citizenship as well as Immigration Solutions will examine the criminal background records of any sort of applicant for UNITED STATE citizenship with the authorities responsible for keeping that info in the candidate's house country. In addition, the USCIS will additionally inquire companies like Interpol to make sure that the visitor obtaining citizenship has no outstanding rap sheet.
Because the taking place of the 9/11 bombings, these migration record searches have boosted significantly as it had actually appeared that some individuals had slid with the system and also still find more info were eligible to get in into the confines of the Usa with questionable documents.
The criminal and terrorist document search is growing in appeal and is a should activity for any type of firm that is working with workers from an additional country. Any kind of company that is intending on employing a worker from another country will certainly be required to have that worker undergo a record search before the issuance of their work visa into the Usa.
The immigration background check is an indicate of avoiding any sort of unwanted site visitor from foreign soil into the United States. While the united state is open to immigration from virtually any kind of nation on the planet, there is a distinction between a prospective candidate that will certainly be a law abiding UNITED STATE citizen as well as someone who is more likely to dedicate a criminal activity within the U.S. perimeters.
These rap sheet searches are right here to remain since the 9/11 Globe Field Center Bombings and also will just enhance over the next few years as the U.S. government undergoes and check every visitor who has secured U.S. citizenship to make sure that he or she had not slid via the cracks and also provided citizenship despite a doubtful criminal past history in their home nation.
| 207,766
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100 ft. Steel Metric Long Tape Measure, Brown
- Item# 4CP84
- Mfr. Model# HW226ME
- Catalog Page# N/A
- UNSPSC# 27111801
Using 360° Viewing:
- Rotate: Use top-to-bottom, side-to-side by use of mouse arrow.
- Zoom In: Double click on image.
- Zoom Out/Reset: Put photo at full zoom & then double click.
- Shipping Weight 1.65 lbs.
Product Details
Technical Specs
- Tape Measure Product Grouping Tape Measures
- Item Long Tape Measure
- Measurement Type Metric
- Blade Length 100 ft.
- Blade Width 3/8"
- Graduation Type mm/cm
- Graduations 1/8" Top, 1mm Bottom
- Blade Material Steel
- Tip Style Folding Hook
- Case Type Closed
- Case Material ABS
- Belt Clip No
- Blade Color Yellow
- Case Color Brown
- Blade Coating Clear
- Lock Type No Lock
- Rewind Type Manual
- Standout Not Rated
- Stud Markings No
- Features Large Easy Action Winding Drum and Friction Washers, Roller Guide, Bold Graduations with Black and Red Numbers
Compliance and Restrictions
None
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Hi Purseholic Team,
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For the Spring 2009, Coach launched a Bonnie Cashin collection, paying tribute to its late former designer, Ms Bonnie Cashin. Cashin was the one who launched the women accesories division for Coach back to 1962 and created a few iconic products for the brand especially leather products in colors such as pink, green and yellow [...]
Men’s bags commonly come in a messenger, briefcase or tote shape so this satchel from Coach is something different and i am loving it. Let me start wit the material. The leather of the bag is rugged and rough, the way a man’s bag should be. I also like that the bag is slouchy without [...]
This is not a typical Coach bag and it is not a bad thing at all. Gone is the signature CC pattern which has become tired after a while, replaced by soft leather in palatable color such as this shade of pink. Do not get me wrong, I have always been a fan of Coach [...]
Buying a gift for a man is supposed to be simple but could be tricky if you do not know the basic rules. First, unless you make decisions on his wardrobe and know his sizes by heart - shirts, belts or shoes are risky choices. Second, unless specifically requested or needed, avoid electronic gadgets such as ipod or mouse as [...]
The Holiday season is around the corner and one of the problems most people face annually is to find the perfect gifts under budget. Especially in harder times like now, each of our Dollar must stretch more than ever, don’t you agree? Besides budget, always remember that you will never go wrong to go with a nice handbag or wrislet [...]
Coach hit it BIG with the latest Sabrina bags (under Madison handbags). I like the shape, the functionality, the look, the horse and carriage logo emblem (Reed Krakoff pays tributes to the brand’s heritage back to the 1960s) and of course the price point, typically what Coach bags are loved for. Another beauty about this [...]
| 363,257
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Title and writing inspiration: bit . ly/1vxyDvX
Character pics on the blog: bit . ly/1qpjRat
Thanks be to MrsSpaceCowboy and lellabeth for pre-reading.
Thanks be to beegurl13 for the lovely banner.
The moment Bella reaches her bedroom, the adrenaline kicks in. She grabs her already-packed suitcase, makes sure it includes everything on her list, and dumps out her purse onto the bed. She grabs what she needs from that pile and begins placing it into her empty backpack. She ignores the aftershave and Old Spice deodorant on the bathroom counter as she reaches for her toothbrush inside the medicine cabinet and her travel bag of shower supplies underneath the sink. She places them both inside her backpack, checks her list again, grabs his black hoodie from behind the bedroom door, and makes her way downstairs with everything.
As she's taking them to her car, she calls Charlie.
"Hey, Dad."
"My beautiful pain in the ass," he laughs. "How are ya, kiddo?"
"Are you working tonight?" she asks, completely bypassing his question.
"No. I worked the day shift. Why?"
"I decided to get a jump on this trip to Renee's, so I'm leaving tonight. If traffic isn't a problem, I should be there by six."
"Everything okay?" Charlie asks, his voice instantly going from playful father to concerned cop. "I thought you guys were starting tomorrow morning."
"I've been better," she admits. "But I'll see you tonight?"
"Sure."
"Okay."
"Drive safe, kiddo. Love you."
"Love you, too, Dad."
After hanging up, Bella heads back up the porch steps to lock the house. She stares at the open letter on the kitchen table for a minute before she sets the alarm and pulls the door shut.
She's in her car and pulling out of the driveway before she can change her mind.
Once she's just outside Seattle, she calls Angela, frustrated when her voicemail answers.
"It's me. I, uh…" she begins, clearing her throat. "I left."
She's quiet for a bit before she begins again.
"I need to think, Angie. There's so much…"
Before her emotions get the best of her, Bella takes a deep breath.
"I'll call you when I can. I love you."
Bella ends the call, turns up the radio, and hums along to whatever's on, needing to drown out her thoughts just for a little while.
When she arrives in Port Angeles a couple of hours later, she stops at a gas station to top off, buys a large bottle of water, and texts Charlie her ETA. As soon as she pulls out of the station, her phone rings, and she answers without checking the caller ID. She already knows who it is.
"Hey."
"Hey," Angela replies.
Bella immediately begins to tear up.
"Do you want to talk about it?"
Angela's voice is full of concern, though it's without surprise.
"No."
"Is there anything I can do?"
"Not right now."
"Okay. Just check in every once in a while, so I know you're okay."
"Okay," Bella whispers. "I love you, Ella."
"I love you, too, La La."
Bella's about to hang up before that pulling sensation in her chest gets the best of her.
"Wait! Ella?"
"I'm here."
"Does he… Has he…"
"I don't think he's home yet."
With that, Bella says her goodbye once more and hangs up.
She spends the next hour thinking about what she doesn't want to think about. She turns the volume up as high as it'll go in the hopes that it'll drown out the chaos that refuses to leave her.
It's not until she's almost to Forks that she feels her phone vibrate with a text message.
Be home in half an hour. Stopping to grab a pizza. Feel like watching something? Your pick. Love you.
Bella pulls over to the side of the road and stares at the message. Nothing about it is out of the ordinary. It's a message she's received many times before. But it's the ordinary that makes her realize how extraordinary it really is, and that thought is more than she can stand.
The moment she feels her emotions slipping, she shuts off her screen, effectively ridding herself of Edward's text for the time being. She collects herself and pulls back onto the road, grateful when the Welcome to Forks sign comes into view.
It's not long before she arrives at Charlie's. Her childhood home hasn't changed much ‒ if at all ‒ since she was a kid, and she's thankful for that.
With all of the changes going on around her, she needs something familiar.
Before she's even opened her door, Charlie's walking out of the house still in his uniform. His handsome face is graced by a smile she knows she's the cause of. It reminds her of another smile she knows she's also the cause of, and her heart beats harder within her chest. But she lets go of that to focus on her dad.
"Hey, old man," Bella greets, a teasing smile on her face.
Charlie keeps walking toward her as he quickly glances behind him.
"Who are you talking to?" he asks with a laugh before pulling her into his arms for a hug, the type that have always reminded Bella that everything would be okay.
"How are you? How was the drive?"
He grabs her bag for her, closes her car door, and ushers her toward the house.
"Both good. You? How was work?"
"The usual. It's Forks. You know how that goes."
Walking through the front door and into the kitchen, Bella smiles at the well-known features of the room and takes in the silence.
"Where is everyone?"
"Leah's out with Rachel. She's been running around crazy this past week over some big date she's got planned. I don't know what's gotten into her. "
"She didn't tell you?"
"Apparently not."
Charlie raises one of the only two take-out menus Forks has to offer, and Bella opts for pizza.
"Leah's asking Rachel to move in with her."
"No shit?"
"She called me last week for tips on how to make it romantic," she laughs. "I told her I had no idea and gave the phone to‒"
Before she can stop it, the memory of Edward sitting at their kitchen table and passing on suggestions to her stepsister flood her mind. She pictures his smile, sincere in the happiness he wanted for Leah, and remembers his words about doing something that's distinctly them, not cliché.
Noticing Charlie's confusion over her sudden pause, Bella clears her throat.
"She talked to Edward. So," she begins, needing to quickly change the subject. "Jake? Sue?"
Charlie gives his order to the person on the other end of the line before he hangs up and answers her.
"He's probably at the shop with Sam."
"He still freaking out over the baby?"
"Not really," he replies. "They have their first appointment with the doctor this week, so it's safe to say he's accepted it. Actually, I get the feeling he's happy about becoming a dad."
Charlie grabs a beer from the fridge and follows Bella into the living room. He takes his seat in his favorite recliner, the bane of Sue's existence, while Bella plops down on the couch.
"And my wicked stepmother?" she asks, smiling.
"As evil as ever," Charlie laughs. "She's at some last-minute women's retreat the tribe wanted her to oversee. She gets back on Sunday. It's not soon enough, if you ask me."
As usual, Charlie's face lights up when he talks about Sue, the love he has for her plain as day. It immediately makes her think of Edward and how she can always tell what he's feeling just by looking at him.
It makes her wonder what she's doing, why she's running this time, and her smile fades.
Charlie and Bella exchange random chit-chat as they eat dinner. He keeps glancing over at her, and she knows he's going to ask where Edward is and why they're not leaving for Arizona together. Bella's not ready to voice what's plaguing her, so she hopes Charlie will give her some more time before the inquisition begins.
Sure enough, the questions start as soon as he clears the table.
"So, where's Edward?"
"Can we not, Dad? I really don't wanna talk about it."
"Bella, I deserve an explanation at the very least. I was already worried about you taking this road trip to see Renee, and now you show up early and alone. Where's Edward?"
Bella sighs and runs her hands over her face.
"He's probably at the house. I don't know," she says, shaking her head, irritated. "Seriously. I don't wanna talk about this."
Once Bella calms down and admits to herself that Charlie's only asking because he loves her and is concerned, she gives him what she can.
"Nothing's changed. I'm still driving to Phoenix. I'm still seeing Renee. I just need to do this alone."
"You need to see Renee alone because it's part of this closure you've been talking about, or you need to see Renee alone because you don't wanna do this with Edward?"
Bella exhales loudly but doesn't respond.
"Does he at least know? Did you tell him you were leaving?"
"No," Bella whispers, hanging her head.
Rather than push her for more information, Charlie simply pulls her out of her seat and draws her in for a hug. Bella manages to keep the tears at bay and is thankful that he's not pushing it.
"I hope you sort out whatever it is you're going through, kid. You know I'm here if you want to talk."
And for a few minutes, Bella holds on to her dad. When she hears the rain start to fall outside, she pulls away.
"I should get going."
"Why don't you stay the night and leave in the morning? I don't like the idea of you driving in the middle of nowhere alone, especially in the rain."
Bella quietly shakes her head before she grabs her bag from the floor next to the front door.
"It's better if I go now. I'll get a motel in Portland. Promise."
Bella gives him another hug before kissing his cheek and opening the front door.
"Keep your cell on. Call if you need anything."
"Of course. Thanks for the four-star meal," she jokes.
"Any time, kiddo. Be safe out there."
Bella knows it's not the out there so much as the inside of her mind – her thoughts ‒ she needs to safeguard from, but she nods just the same.
"Bye, Dad."
As always, I give no warnings regarding content, but I will say that nobody dies in this story. lol
I was going to wait until I had a majority of the story written before posting, but the nervous energy was becoming overwhelming. I don't know how many chapters it'll take, but this is a small, simple story. It won't be long at all.
I'm not sure when I'll have chapter 2 up, but I'm hoping it'll be in the next two weeks. After that, it'll post every Saturday until it's complete.
See you soon.
P.S. I missed you guys. :)
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Topaz Lake
Topaz Lake
At the northern end of the Antelope Valley, where California greets Nevada, you can find “Topaz Lake … Home of the Trophy Trout.”
It’s easy to feel like a winner at Topaz Lake simply by spending a day fishing, sailing or exploring the lake and its 25 miles of shoreline. Or you can hit the Topaz Lodge Resort and Casino overlooking the lake and try your luck there.
Besides the option for gambling, there are a couple other things that make Topaz Lake unique. The manmade lake is in both California and Nevada, making a fishing license from either state valid. The other is that Topaz Lake is actually at its best in the winter and early spring, when most bodies of water in the Eastern Sierra are frozen or buried under snow. Topaz Lodge even runs an annual winter fishing derby, offering anglers a nice outdoor fix in the midst of the chilly winter months.
For those who prefer drier activities, Northern Mono County is also known for offering some of the best ATV riding in the Golden State. The hiking and horseback riding near Topaz Lake are also terrific.
Directions: Topaz Lake is located along Highway 395 at the California/Nevada border. It is located about halfway between Reno and Mammoth Lakes.
Camping at Topaz Lake
The Topaz Lake County Park, part of the Topaz Lake Recreation Area, offers full RV hook ups, partial hook ups and over 40 dry campsites at Topaz Lake. For more information or to make reservations, please call 775-266-3343 or 775-782-9828. The Topaz Lake Lodge and RV Park also offers a large variety of camping options, for more information or reservations, please call 775-226-3337
Directions: Topaz Lake is located along Highway 395 at the California/Nevada border. It is located about halfway between Reno and Mammoth Lakes.
Fishing at Topaz Lake casting can, however, also be effective, especially along the great access the western edge of the lake provides. Unlike many lakes in the region that tend to fish best in summer and fall, winter and spring are usually the best times to fish Topaz Lake. The annual fishing derby put on by Topaz Lake Lodge runs from New Years Day into April, when the general fishing season in the Eastern Sierra is closed. So if you’re looking for so good off-season trout fishing, the lake along the state border, where the edge of the Eastern Sierra meets the Great Basin, is an excellent option.
Directions: Topaz Lake is located along Highway 395 at the California/Nevada border. It is located about halfway between Reno and Mammoth Lakes.
| 360,333
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TITLE: Sum of Odd Triangular numbers (closed form)
QUESTION [0 upvotes]: I am looking for the sum of consecutive odd triangular numbers. I am trying to relate the number of $k \times k$ rhombi in an $n \times n \times n$ equilateral triangle. While I have figured out an answer to the problem as it relates to sums of triangular numbers, in particular the sum of odd triangular numbers when n is even and the sum of even triangular numbers when $n$ is odd, I am having trouble finding a closed form for the expression. For example, in a $6 \times 6 \times 6$ equilateral triangle composed of unit triangles there are $66$ total rhombi; 1 $3 \times 3$, 6 $2 \times 2$, and 15 $1 \times 1$, and we have that $T_1=1$, $T_3=6$, and $T_5=15$.
$1+6+15 = 22$. And there are three types of rhombi; left, right, and vertical, so $22 \times 3$ is $66$. The same works for $n=7$ but it is the sum of $T_2 + T_4 + T_6$.
So i know how to answer the question of total rhombi, but I can not get a closed form of the summation of odd or even triangular numbers.
REPLY [0 votes]: An automatic solution with GP:
T(n)=n*(n+1)/2
sumformal(T(2*n+1))
Alternately: odd triangular numbers are represented by a quadratic polynomial, and so their sum is a cubic polynomial. Work out the first four sums then interpolate.
| 89,452
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If excess gum tissue is affecting your smile, Dr. Hernickson may recommend gum lifting in Maplewood, Minnesota. Also known as crown lengthening, this procedure involves removing excess gum to improve the appearance of your smile. Our dentists can consult with you about whether this treatment is right for you. To schedule your consultation, call the office of Henrickson Dental today at 651-777-8900!
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\begin{document}
\tolerance = 9999
\maketitle
\markboth{Tatsuro Kawakami and Masaru Nagaoka}{Du Val del Pezzo surfaces in positive characteristic}
\begin{abstract}
In this paper, we study pathologies of Du Val del Pezzo surfaces defined over an algebraically closed field of positive characteristic by relating them to their non-liftability to the ring of Witt vectors.
More precisely,
we investigate
the condition (NB): all the anti-canonical divisors are singular,
(ND): there are no Du Val del Pezzo surfaces over the field of complex numbers with the same Dynkin type, Picard rank, and anti-canonical degree,
(NK): there exists an ample $\Z$-divisor which violates the Kodaira vanishing theorem for $\Z$-divisors, and
(NL): the pair $(Y, E)$ does not lift to the ring of Witt vectors, where $Y$ is the minimal resolution and $E$ is its reduced exceptional divisor.
As a result, for each of these conditions, we determine all the Du Val del Pezzo surfaces which satisfy the given one.
\end{abstract}
\tableofcontents
\section{Introduction}
We say that $X$ is a Du Val del Pezzo surface if $X$ is a normal projective surface whose anti-canonical divisor is ample and which has at worst Du Val singularities, i.e., 2-dimensional canonical singularities.
By \textit{the Dynkin type of $X$}, we mean the corresponding Dynkin diagrams of singularities on $X$.
For example, we say that $X$ is of type $3A_1+D_4$ if $X$ has three $A_1$-singularities and one $D_4$-singularity. In this case, we also write $\Dyn(X) = 3A_1+D_4$ and $X=X(3A_1+D_4)$.
In positive characteristic, it has become clear that many pathological phenomena occur on Du Val del Pezzo surfaces.
For example, Keel-M\textsuperscript{c}Kernan \cite[end of Section 9]{KM} constructed a Du Val del Pezzo surface $X(7A_1)$ of Picard rank one and degree $K_X^2=2$ in characteristic two. This Dynkin type does not appear in characteristic zero (see \cite[Theorem 2, Table (II)]{Fur} or \cite[Theorem 1.1]{Bel}).
Furthermore, Cascini-Tanaka \cite[Proposition 4.3 (iii)]{CT18} pointed out that anti-canonical members of $X(7A_1)$ are all singular. Since the complete linear system of the anti-canonical divisor of $X(7A_1)$ is base point free, this gives a counterexample to Bertini's theorem in positive characteristic.
Cascini-Tanaka \cite[Theorem 4.2 (6)]{CT19} also showed that there exists an ample $\Z$-divisor $A$ on $X(7A_1)$ such that $H^1(X, \sO_X(-A))\neq 0$, which gives a counterexample to the Kodaira vanishing theorem for ample $\Z$-divisors on klt surfaces.
On the other hand, the question of whether a variety admits a lifting to the ring of Witt vectors $W(k)$ is often related to pathological phenomena in positive characteristic.
Since all Du Val del Pezzo surfaces considered by themselves are always liftable (see Remark \ref{lift remark}), it is more useful to consider the notion of log liftability.
\begin{defn}[\textup{cf.~Definition \ref{loglift:Def} and Lemma \ref{equiv}}]\label{Intro:loglift:Def}
We say that a Du Val del Pezzo surface $X$ is \textit{log liftable over the ring of Witt vectors $W(k)$} if for the minimal resolution $\pi \colon Y \to X$, the pair of $Y$ and its reduced exceptional divisor $E_\pi$ lifts to $W(k)$.
\end{defn}
Indeed, it is known that the Keel-M\textsuperscript{c}Kernan's surface $X(7A_1)$ is not log liftable over $W_2(k)$ (see \cite[Proposition 11.1]{Langer19} and \cite[Proposition 4.1]{Lan}).
The aim of this paper is to investigate the relationship between these pathological phenomena and non-log liftability of Du Val del Pezzo surfaces over $W(k)$.
For simplicity of notation, we define the following conditions.
\begin{defn}\label{notation}
For a Du Val del Pezzo surface $X$ over an algebraically closed field $k$ of characteristic $p>0$, we say that $X$ satisfies:
\begin{itemize}
\item (ND) if there does not exist any Du Val del Pezzo surface $X_{\C}$ over the field of complex numbers $\C$ with the same Dynkin type, the same Picard rank, and the same degree as $X$.
\item (NB) if anti-canonical members of $X$ are all singular.
\item (NK) if $H^1(X, \sO_X(-A))\neq0$ for some ample $\Z$-divisor $A$ on $X$.
\item (NL) if $X$ is not log liftable over $W(k)$.
\end{itemize}
\end{defn}
Our main results consist of three theorems.
One is the following, which shows that (NK) $\Rightarrow$ (NL) and (ND) $\Rightarrow$ (NL) $\Rightarrow$ (NB).
\begin{thm}\label{smooth, Intro}
Let $X$ be a Du Val del Pezzo surface over an algebraically closed field $k$ of characteristic $p>0$.
Then the following hold.
\begin{enumerate}\renewcommand{\labelenumi}{$($\textup{\arabic{enumi}}$)$}
\item{If a general anti-canonical member is smooth, then $X$ is log liftable over $W(k)$.}
\item{If $X$ is log liftable over $W(k)$, then there exists a Du Val del Pezzo surface over $\C$ with the same Dynkin type, the same Picard rank, and the same degree as $X$.}
\item{If $X$ is log liftable over $W(k)$, then $H^1(X, \sO_X(-A))=0$ for every ample $\Z$-divisor $A$.}
\end{enumerate}
\end{thm}
The second main theorem is Theorem \ref{sing}, which classifies Du Val del Pezzo surfaces satisfying (NB), the weakest condition among four pathological phenomena in Definition \ref{notation}.
\begin{thm}\label{sing}
Let $X$ be a Du Val del Pezzo surface over an algebraically closed field $k$ of characteristic $p>0$.
Suppose that anti-canonical members of $X$ are all singular.
Then the following hold.
\begin{enumerate}
\item[\textup{(0)}] $K_X^2 \leq 2$ and $p=2$ or $3$.
\item[\textup{(1)}] When $K_X^2=1$ and $p=2$ (resp.~$p=3$), the Dynkin type of $X$ is $E_8$, $D_8$, $A_1+E_7$, $2D_4$, $2A_1+D_6$, $4A_1+D_4$, or $8A_1$ (resp.~$E_8$, $A_2+E_6$, or $4A_2$). In particular, the Picard rank of $X$ is equal to one.
\item[\textup{(2)}] When $K_X^2=2$, we have $p=2$ and the Dynkin type of $X$ is $E_7$, $A_1+D_6$, $3A_1+D_4$, or $7A_1$.
Furthermore, the anti-canonical morphism $\phi_{|-K_X|}\colon X\to \PP_k^2$ is purely inseparable and hence $X$ is homeomorphic to $\PP^2_k$. In particular, the Picard rank of $X$ is equal to one.
\item[\textup{(3)}]
The isomorphism class of $X$ is uniquely determined by its Dynkin type except when the Dynkin type is $2D_4, 4A_1+D_4$, or $8A_1$.
In these cases, the isomorphism classes of del Pezzo surfaces of type $8A_1$ (resp.~each of types $2D_4$ and $4A_1+D_4$) correspond to the closed points of $\mathcal{D}_n / \PGL(n+1, \FF_2)$ with $n=2$ (resp.~$n=1$), where $\mathcal{D}_n \subset \PP^n_k$ is the complement of the union of all the hyperplane sections defined over the prime field $\FF_2$ of $k$.
\end{enumerate}
Summarizing these statements, we obtain Table \ref{table:sing}.
\begin{table}[htbp]
\caption{}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{3}{|c||}{Degree} & \multicolumn{5}{|c|}{$K_X^2=1$} \\
\hline
\multicolumn{3}{|c||}{Dynkin type} &$E_8$ & $A_2+E_6$ & \multicolumn{1}{c|}{$4A_2$} & $D_8$ & $A_1+E_7$ \\ \hline
\multicolumn{3}{|c||}{Characteristic} &$p=2, 3$& \multicolumn{2}{|c|}{$p=3$} &\multicolumn{2}{|c|}{$p=2$} \\ \hline
\multicolumn{3}{|c||}{No. of isomorphism classes} & $1$ & $1$ & \multicolumn{1}{c|}{$1$} & $1$ & $1$ \\ \hline \cline{1-8}
\multicolumn{4}{|c||}{$K_X^2=1$} & \multicolumn{4}{|c|}{$K_X^2=2$}\\\cline{1-8}
$2D_4$ & $2A_1+D_6$ & $4A_1+D_4$& \multicolumn{1}{c||}{$8A_1$} & $E_7$& $A_1+D_6$ & $3A_1+D_4$& $7A_1$ \\ \cline{1-8}
\multicolumn{4}{|c||}{$p=2$} & \multicolumn{4}{|c|}{$p=2$} \\ \cline{1-8}
$\infty$ & $1$ &$\infty$ & \multicolumn{1}{c||}{$\infty$ } & $1$ & $1$ & $1$ & $1$ \\ \cline{1-8}
\end{tabular}
\label{table:sing}
\end{table}
\end{thm}
\begin{rem}
The list of root bases in the lattice $\mathbb{E}_{9-d}$ gives the list of possible Dynkin types of Du Val del Pezzo surfaces of degree $d$.
When $k= \C$ and $d=2$ (resp.~$d=1$), it is also shown that any root bases are realized as the Dynkin type of a Du Val del Pezzo surfaces of degree $d$ except $7A_1$ (resp.~$4A_1+D_4, 8A_1$, and $7A_1$) (see \cite[Chapter 8]{Dol} and the references given there for more details).
Theorems \ref{smooth, Intro} and \ref{sing} show that $7A_1$ (resp.~each of $4A_1+D_4$ and $8A_1$) is realized as the Dynkin type of a Du Val del Pezzo surface of degree two (resp.~one) only in characteristic two.
On the other hand, $7A_1$ cannot be realized as the Dynkin type of a Du Val del Pezzo surface of degree one in any characteristic.
\end{rem}
\begin{rem}
Theorem \ref{sing} describes isomorphism classes of several rational quasi-elliptic surfaces and, combining Ito's results \cite{Ito1, Ito2}, we obtain the complete classification of isomorphism classes of rational quasi-elliptic surfaces (see Corollary \ref{Itorem} for more details). In the process of the proof of Theorem \ref{sing}, we also describe automorphism group scheme structures of all rational quasi-elliptic surfaces (see Corollary \ref{cor:q-ellaut} and Remark \ref{autorem}).
\end{rem}
The last main theorem is Theorem \ref{pathologies}. It clarifies which Du Val del Pezzo surfaces satisfying (NB) also satisfy additional pathological phenomena.
As a consequence, we conclude that (NK) $\Rightarrow$ (ND) $\Rightarrow$ (NL) $\Rightarrow$ (NB) and none of the opposite directions hold.
\begin{thm}\label{pathologies}
Let $X$ be a Du Val del Pezzo surface over an algebraically closed field $k$ of characteristic $p>0$.
Then the following hold.
\begin{enumerate}
\item [\textup{(1)}] The surface $X$ is not log liftable over $W(k)$ if and only if $(p, \Dyn(X))=(3, 4A_2)$, $(2, 4A_1+D_4)$, $(2, 8A_1)$, or $(2, 7A_1)$.
\item [\textup{(2)}] There exists no Du Val del Pezzo surface $X_{\C}$ over $\C$ with the same Dynkin type, the same Picard rank, and the same degree as $X$ if and only if $(p, \Dyn(X))=(2, 4A_1+D_4)$, $(2, 8A_1)$, or $(2, 7A_1)$.
\item [\textup{(3)}] There exists an ample $\Z$-divisor on $X$ such that $H^1(X, \sO_X(-A))\neq0$ if and only if $(p, \Dyn(X))=(2, 8A_1)$ or $(2, 7A_1)$.
\end{enumerate}
\end{thm}
\begin{rem}
The vanishing of the second cohomology of the tangent sheaf is important because this implies that there are no local-to-global obstructions to deformations (cf.~\cite[Theorem 4.13]{LN}).
Theorem \ref{pathologies} asserts that there exists a Du Val del Pezzo surface $X$ with $H^2(X, T_X) \neq 0$ when $p=2$ or $3$. We remark that this does not happen when $p>3$ since a general anti-canonical member is smooth in this case (see the proof of Proposition \ref{NBtoNL} for more details).
\end{rem}
\subsection{Structure of the paper}
This paper is structured as follows.
In Section \ref{sec:pre}, we recall some facts on liftability of pairs, Du Val del Pezzo surfaces, and rational quasi-elliptic surfaces.
In Section \ref{sec:smooth}, we prove Theorem \ref{smooth, Intro}.
The main idea of the proof is to utilize a smooth anti-canonical member of a Du Val del Pezzo surface $X$ to show the vanishing of $H^2(X, T_X)$,
which is the obstruction of log lifting over $W(k)$.
After that, we construct the desired surface over $\C$ from the generic fiber of the log lifting.
Theorem \ref{smooth, Intro} (3) is an easy consequence of Hara's vanishing theorem \cite[Corollary 3.8]{Hara98}.
In Sections \ref{sec:sing} and \ref{sec:singisom}, we prove Theorem \ref{sing} by using the description of the configuration of negative rational curves on rational quasi-elliptic surfaces by Ito \cite{Ito1, Ito2}. In these sections, we also determine the automorphism groups of Du Val del Pezzo surfaces satisfying (NB).
In the rest of the paper, we prove Theorem \ref{pathologies} as follows.
We only have to consider Du Val del Pezzo surfaces satisfying (NB) due to Theorem \ref{smooth, Intro}.
Fix such a surface $X$.
We note that the assertion (2) now follows from the list of Du Val del Pezzo surfaces over $\C$ of Picard rank one \cite[Theorem 2, Table (II)]{Fur}.
In Section \ref{sec:singliftable}, we show the assertion (1).
The main difficulty of the proof is to show non-log liftability of $X(4A_2)$ over $W(k)$, which does not satisfy (ND).
For the proof, we observe that log liftability of $X(4A_2)$ over $W(k)$ would give the extremal rational elliptic surface over $\C$ constructed from the Hesse pencil $\Lambda=\{s(x^3+y^3+z^3)+t(xyz)\}$.
Then we obtain a contradiction by showing that $W(k)$ must contain the cube root of unity, which appears as the singular member locus of $\Lambda$.
In Section \ref{sec:KV}, we show the assertion (3).
By \cite[Theorem 4.8]{Kaw2} and Theorem \ref{smooth, Intro} (3), it suffices to consider the case where $X$ is of type $7A_1$, $8A_1$, or $4A_1+D_4$.
\cite[Theorem 4.2 (6)]{CT19} shows that $X(7A_1)$ satisfies (NK).
Choosing an ample $\Z$-divisor based on [\textit{ibid}.], we show that $X(8A_1)$ also satisfies (NK).
On the other hand, we utilize a birational map between $X(4A_1+D_4)$ and the Du Val del Pezzo surface of type $7A_1$ to conclude that $X(4A_1+D_4)$ does not satisfy (NK).
We have thus proved Theorem \ref{pathologies}.
\subsection{Related results}
Log liftability was originally considered by Cascini-Tanaka-Witaszek \cite{CTW}, in which they proved that surfaces of del Pezzo type are either log liftable over $W(k)$ or globally $F$-regular in large characteristic. After that Lacini \cite{Lac} classified klt del Pezzo surfaces of Picard rank one in characteristic $p>3$, from which he deduced that klt del Pezzo surfaces of Picard rank one in $p>5$ are log liftable over $W(k)$. Arvidsson-Bernasconi-Lacini \cite{ABL} generalized his result to the case of klt projective surfaces $X$ with Iitaka dimension $\kappa(X, K_X)=-\infty$. The first author \cite{Kaw21} showed that normal projective surfaces $X$ in large characteristic with Iitaka dimension $\kappa(X, K_X)\leq0$ are log liftable to characteristic zero.
The second author \cite{Nag21} determined klt del Pezzo surfaces of Picard rank one in $p=5$ which are not log liftable over $W(k)$.
\subsubsection{Classification of del Pezzo surfaces in positive characteristic}
Lacini \cite{Lac} classified klt del Pezzo surfaces of Picard rank one in characteristic $p>3$. Inspired by his work, the authors \cite{KN20} determined Du Val del Pezzo surfaces of Picard rank one in $p=2$ or $3$.
Martin and Stadlmayr \cite{M-S} classified smooth weak del Pezzo surfaces with non-zero global vector fields in any characteristic.
\subsection{Notation}
We work over an algebraically closed field $k$ of characteristic $p>0$.
A \textit{variety} means an integral separated scheme of finite type over $k$.
A \textit{curve} (resp.~a \textit{surface}) means
a variety of dimension one (resp.~two).
We call two-dimensional canonical singularities \textit{Du Val singularities}.
We always require quasi-elliptic surfaces to be relatively minimal.
Throughout this paper, we also use the following notation:
\begin{itemize}
\item $\FF_q$: the finite field of order $q$.
\item $W(k)$ (resp.~$W_n(k)$): the ring of Witt vectors (resp.~the ring of Witt vectors of length $n$).
\item $E_f$: the reduced exceptional divisor of a birational morphism $f$.
\item $\rho(X)$: the Picard rank of a projective variety $X$.
\item $T_X\coloneqq \mathcal{H}om_{\sO_X}(\Omega_X, \sO_X)$: the tangent sheaf of a normal variety $X$
\item $T_Y(-\log E) \coloneqq \mathcal{H}om_{\sO_Y}(\Omega_Y(\log E), \sO_Y)$: the logarithmic tangent bundle of a smooth variety $Y$ and a simple normal crossing divisor $E$ on $Y$.
\item $\Aut X$: the automorphism group of a variety $X$.
\item $\MW(Z)$: the Mordell-Weil group of a genus one fibration $f \colon Z \to \PP^1_k$ defined by $|-K_Z|$.
\item $\Dyn(X)$ : the Dynkin type of $X$.
\end{itemize}
\section{Preliminaries}
\label{sec:pre}
\subsection{Liftability of pairs}
In this subsection, we review a general theory of liftability to Noetherian irreducible schemes.
\begin{defn}
Let $T$ be a Noetherian irreducible scheme, $Y$ be a smooth separated scheme over $T$, and $E \coloneqq \sum_{i=1}^r E_i$ be a reduced divisor on $Y$, where each $E_i$ is an irreducible component of $E$. We say that $E$ is \textit{simple normal crossing over $T$} if, for any subset $J \subseteq \{1, \ldots, r\}$ such that $\bigcap_{i \in J} E_i\neq \emptyset$, the scheme-theoretic intersection $\bigcap_{i \in J} E_i$ is smooth over $T$ of relative dimension $\dim Y_{\eta}-|J|$, where $Y_{\eta}$ is the generic fiber of $Y\to T$.
\end{defn}
\begin{defn}\label{d-liftable}
Let $\alpha \colon S \to T$ be a morphism between Noetherian irreducible schemes.
Let $Y$ be a smooth projective scheme over $S$ and $E$ a simple normal crossing divisor over $S$ on $Y$ with $E = \sum_{i=1}^r E_i$ the irreducible decomposition.
We say that the pair $(Y,E)$ \textit{lifts to $T$ via $\alpha$} if
there exist
\begin{itemize}
\item a smooth and projective morphism $\mathcal{Y} \to T$ and
\item effective divisors $\mathcal{E}_1, \dots, \mathcal E_r$ on $\mathcal{Y}$ such that $\sum _{i=1}^r \mathcal{E}_i$ is simple normal crossing over $T$
\end{itemize}
such that the base change of the schemes $\mathcal{Y}, \mathcal{E}_1,\cdots,\mathcal{E}_r$ by $\alpha \colon S \to T$ are isomorphic to $Y, E_1,\cdots, E_r$ respectively.
When $T$ is the spectrum of a local ring $(R, m)$ and $\alpha$ is induced by $R/m \cong k$, we also say that $(Y, E)$ \textit{lifts to $R$} for short.
\end{defn}
The following theorem is a log version of \cite[Theorem 8.5.9]{FAG}. It seems to be well-known for experts, but we include the sketch of the proof for the convenience of the reader.
\begin{thm}\label{loglift:criterion 1}
Let $Y$ be a smooth projective variety
and $E$ a simple normal crossing divisor on $Y$. If $H^2(Y, T_{Y}(-\log\,E))=H^2(Y, \sO_Y)=0$, then $(Y ,E)$ lifts to every Noetherian complete local ring $(R, m)$ with the residue field $k$.
\end{thm}
\begin{proof}
We denote $R/m^n$ by $R_n$.
Let $(Y^n, E^n)$ be a lifting of $(Y, E)$ over $\Spec\,R_n$. We first see that $(Y^n, E^n)$ is liftable to $\Spec\,R_{n+1}$. Since $E^n$ is simple normal crossing over $\Spec\,R_n$, we can take an affine open covering $\{U_i\}$ of $Y^n$ such that $(U_i, E|_{U_i})$ lifts to $\Spec\,R_{n+1}$.
Then for each $i$ and any open subset $U$ of $U_{i}$, the set of equivalence classes of such liftings is a torsor under the action of $\mathcal{H}om(\Omega_{U}(\log E), m^{n-1}\sO_{U})$. We refer to the arguments of \cite[Section 8]{EV} for the details. Then by a similar argument as in \cite[Theorem 8.5.9 (b)]{FAG},
the obstruction for the lifting of $(Y^n, E^n)$ over $\Spec\,R_{n+1}$ is contained in $H^2(Y, T_{Y}(-\log\,E))\otimes m^{n}/m^{n+1}$.
Thus the vanishing of $H^2(Y, T_{Y}(-\log\,E))$ gives a lifting of $Y$ and $E_{i}$ over $\Spec R$ as formal schemes.
Since $H^2(Y, \sO_Y)=0$, they are algebraizable and we get a projective scheme $\mathcal{Y}$ over $\Spec\,R$ and a closed subscheme $\mathcal{E} \coloneqq \sum_{i=1}^r \mathcal{E}_i$ on $\mathcal{Y}$ such that
$\mathcal{Y}\otimes_{R} R_n =Y^n$ and $\mathcal{E}_i\otimes_{R} R_n=E^n_i$ for each $n$ and $i$ by \cite[Corollary 8.5.6 and Corollary 8.4.5]{FAG}.
We take a subset $J \subseteq \{1, \ldots, r\}$. Since $(\bigcap_{i \in J} \mathcal{E}_i)\otimes_{R} R_n=\bigcap_{i \in J} {E}^n_i$ is smooth over $\Spec\,R_n$ for all $n>0$ and $\mathcal{Y}$ is projective over $\Spec\,R$, \cite[Chapitre 0, Proposition (10.2.6)]{EGAIII} and \cite[Th\'eor\`eme 12.2.4 (iii)]{EGAIV3} show that $\bigcap_{i \in J} \mathcal{E}_i$ is smooth of relative dimension $\dim \mathcal{Y}_{\eta}-|J|$ except when $\bigcap_{i \in J} \mathcal{E}_i = \emptyset$, where $\mathcal{Y}_{\eta}$ is the generic fiber. Therefore $(\mathcal{Y}, \mathcal{E}=\sum_{i=1}^r \mathcal{E}_i)$ is a lifting of $(Y, E)$ over $\Spec\,R$.
\end{proof}
We use the following Hara's vanishing theorem in Propositions \ref{prop:W2lift} and \ref{lem:KV-3}.
\begin{thm}[\textup{cf.~\cite[Corollary 3.8]{Hara98}}]\label{loglift vanishing}
Let $Y$ be a smooth projective variety and $E$ a simple normal crossing divisor on $Y$ such that $(Y, E)$ lifts to $W_2(k)$.
Let $A$ be an ample $\Q$-divisor the support of whose fractional part is contained in $E$.
Suppose that $p\geq \dim\,Y$.
Then $H^{j}(Y, \Omega_Y^{i}(\log \, E)\otimes\sO_Y(-\lceil A \rceil))=0$ if $i+j<\dim\, Y$.
\end{thm}
\begin{proof}
When $p>\dim\,Y$, the assertion follows from \cite[Corollary 3.8]{Hara98} and the Serre duality.
We remark that even when $p=\dim\,Y$, the assertion holds. This is because, in the proof of \cite[Corollary 3.8]{Hara98}, the assumption that $p>\dim\,Y$ is only used for the quasi-isomorphism
$\bigoplus_{i}\Omega_Y^i(\log\, E)[-i]\simeq F_{*}\Omega_Y^{\bullet}(\log E)$, which holds even in $p=\dim\,Y$ as in \cite[10.19 Proposition]{EV}.
\end{proof}
As we will see in Remark \ref{lift remark}, all Du Val del Pezzo surfaces lift to every Noetherian complete local ring with the residue field $k$.
For this reason, we will mainly consider the following notion of liftability.
\begin{defn}\label{loglift:Def}
Let $X$ be a normal projective surface. Fix a Noetherian irreducible scheme $T$ and a morphism $\alpha \colon \Spec k \to T$.
We say that $X$ is \textit{log liftable over $T$ via $\alpha$} (or \textit{log liftable over $R$ via $\alpha$} when $T= \Spec R$) if the pair $(Z, E_{f})$ lifts to $T$ via $\alpha$ for some log resolution $f \colon Z \to X$.
When $T$ is the spectrum of a local ring $(R, m)$ and $\alpha$ is induced by $R/m \cong k$,
We also say that $X$ is \textit{log liftable over $R$} for short.
\end{defn}
\begin{rem}
We remark that log liftability over a scheme smooth and separated over $\Z$ is equivalent to log liftability over $W(k)$ by \cite[Proposition 2.5]{ABL}.
\end{rem}
Let us see conditions equivalent to log liftability.
We note that the minimal resolution of a Du Val singularity is a log resolution.
\begin{lem}\label{equiv}
Let $X$ be a normal projective surface and $R$ a Noetherian complete local ring with the residue field $k$.
Let $\pi \colon Y\to X$ be a log resolution and $f \colon Z \to X$ a log resolution which factors through $\pi$.
Then the following holds.
\begin{enumerate}
\item[\textup{(1)}] Suppose that $(Z, E_{f})$ lifts to $R$.
Then $(Y, E_{\pi})$ lifts to $R$ as a formal scheme.
If $H^2(Z, \sO_Z)=0$ in addition, then $(Y, E_{\pi})$ lifts to $R$.
\item[\textup{(2)}] Suppose that $(Y, E_{\pi})$ lifts to $R$ and $R$ is regular.
Then $(Z, E_{f})$ lifts to $R$.
\end{enumerate}
\end{lem}
\begin{proof}
The assertion (1) is \cite[Proposition 4.3 (1)]{AZ}.
We show (2). Since the morphism $Z\to Y$ is a birational morphism of smooth projective surfaces, this is a composition of blow-ups at a smooth point.
Since $R$ is complete and regular, the essentially same argument as \cite[Proposition 2.9]{ABL} shows the liftability of $(Z, E_{f})$.
\end{proof}
In particular, Definition \ref{Intro:loglift:Def} agrees with Definition \ref{loglift:Def}.
\begin{prop}\label{prop:W2lift}
Let $X$ be a normal projective surface.
Suppose that one of the following conditions holds.
\begin{enumerate}
\item[\textup{(1)}] $-K_X$ is ample $\Q$-Cartier, the minimal resolution $\pi\colon Y\to X$ is a log resolution, and there exists a log resolution $f\colon Z\to X$ such that $(Z, E_f)$ lifts to $W_2(k)$.
\item[\textup{(2)}] $H^2(X, T_X)=0$ and $H^2(X, \sO_X)=0$.
\end{enumerate}
Then, for every log resolution $f'\colon Z'\to X$, the pair $(Z',E_{f'})$ lifts to every Noetherian complete local ring with residue field $k$.
\end{prop}
\begin{proof}
We first show (1).
We take a $\pi$-exceptional effective $\Q$-divisor $F$ such that $\pi^{*}(-K_X)-F$ is ample and $\lceil \pi^{*}(-K_X)-F\rceil=\lceil \pi^{*}(-K_X)\rceil$.
Since $\pi$ is minimal, we have $-K_Y \geq \lceil \pi^{*}(-K_X) \rceil=\lceil \pi^{*}(-K_X)-F\rceil$.
Let $f'\colon Z'\to X$ be a log resolution.
Then $f'$ decomposes into $g\colon Z'\to Y$ and the minimal resolution $\pi\colon Y\to X$.
We have the injective morphism
\begin{align*}
g_{*}(\Omega_{Z'}(\log\,E_{f'})\otimes \sO_{Z'}(K_{Z'}))\hookrightarrow&
(g_{*}(\Omega_{Z'}(\log\,E_{f'})\otimes \sO_{Z'}(K_{Z'})))^{**}\\
=&\Omega_Y(\log\,E_{\pi})\otimes\sO_Y(K_Y)
\end{align*}
and then the Serre duality yields
\begin{align*}
H^2(Z', T_{Z'}(-\log\, E_{f'}))\cong&H^0(Z', \Omega_{Z'}(\log\,E_{f'})\otimes \sO_{Z'}(K_{Z'}))\\
\hookrightarrow&H^0(Y, \Omega_Y(\log\,E_{\pi})\otimes \sO_Y(K_Y))\\
\hookrightarrow&H^0(Y, \Omega_Y(\log\,E_{\pi})\otimes \sO_Y(-\lceil \pi^{*}(-K_X)-F\rceil)).
\end{align*}
Since $(Z, E_f)$ lifts to $W_2(k)$ by assumption, so does $(Y, E_{\pi})$ by Lemma \ref{equiv} (1), and hence the last cohomology vanishes by Theorem \ref{loglift vanishing}.
Together with
\[
H^2(Z', \sO_{Z'})\cong H^0(Z', \sO_{Z'}(K_{Z'}))\hookrightarrow H^0(X, \sO_X(K_X))=0,\]
we obtain the liftability of $(Z',E_{f'})$ by Theorem \ref{loglift:criterion 1}.
In the case of (2), we have
\begin{align*}
H^2(Z', T_{Z'}(-\log\, E_{f'}))\cong&H^0(Z', \Omega_{Z'}(\log\,E_{f'})\otimes \sO_{Z'}(K_{Z'}))\\
\hookrightarrow&H^0(X, (\Omega_X\otimes \sO_X(K_X))^{**})\\
\cong&H^2(X, T_X)=0,
\end{align*}
and the rest proof is similar to (1).
\end{proof}
As a consequence of Proposition \ref{prop:W2lift} (1), log liftability of a del Pezzo surface $X$ with rational singularities over $W(k)$ is equivalent to log liftability of $X$ over $W_2(k)$, and one of the conditions implies log liftability of $X$ over every Noetherian complete local ring with the residue field $k$.
\subsection{Du Val del Pezzo surfaces}
In this subsection, we gather some basic results of Du Val del Pezzo surfaces.
\begin{defn}\label{GdelPezzo}
Let $X$ be a normal projective surface.
We say that $X$ is a \textit{Du Val del Pezzo surface} if $-K_X$ is ample and $X$ has only Du Val singularities.
\end{defn}
\begin{rem}\label{lift remark}
Let $X$ be a normal projective surface with only rational singularities with Iitaka dimension $\kappa(\tilde{X}, K_{\tilde{X}})=-\infty$, where $\tilde{X}\to X$ is a resolution. Let us see that $X$ lifts to every Noetherian complete local ring $R$ with the residue field $k$.
First, we recall that $\tilde{X}$ lifts to $R$ (see \cite[8.5.26]{FAG}). Then $X$ is formally liftable to $R$ by \cite[Proposition 4.3(1)]{AZ}, and the formal lifting is algebraizable since $H^2(X, \sO_X)=0$.
In particular, all Du Val del Pezzo surfaces lift to $R$.
\end{rem}
\begin{lem}\label{basic}
Let $X$ be a Du Val del Pezzo surface of degree $d \coloneqq K_X^2$.
Then the following hold.
\begin{enumerate}\renewcommand{\labelenumi}{$($\textup{\arabic{enumi}}$)$}
\item{$\dim|-K_X|=d$.}
\item{$|-K_X|$ has no fixed part.}
\item{A general anti-canonical member is a locally complete intersection curve with arithmetic genus one. Moreover, if $p>3$, then a general anti-canonical member is smooth.}
\item{If $d\geq3$, then $|-K_X|$ is very ample. }
\item{If $d\geq2$, then $|-K_X|$ is base point free.}
\end{enumerate}
\end{lem}
\begin{proof}
We refer to \cite[Propositions 2.10, 2.12, and 2.14]{BT} and \cite[Proposition 4.6]{Kaw2} for the proof.
\end{proof}
\subsection{Quasi-elliptic surfaces}
In this subsection, we compile the results on rational quasi-elliptic surfaces by Ito \cite{Ito1, Ito2}, which we will use in Sections \ref{sec:sing} and \ref{sec:singisom}.
\begin{thm}[{\cite[Theorems 3.1--3.3]{Ito1}}]\label{thm:q-ell3}
Suppose $p=3$.
Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] The configurations of reducible fibers of rational quasi-elliptic surfaces and their Mordell-Weil groups are listed in Table \ref{q-ell3}, where we use Kodaira's notation.
\item[\textup{(2)}] Rational quasi-elliptic surfaces of each type (1), (2), and (3) uniquely exist.
\item[\textup{(3)}] Sections on rational quasi-elliptic surfaces are disjoint from each other.
Moreover, the dual graphs of negative rational curves in rational quasi-elliptic surfaces are as in Figure \ref{fig:dual(1)-(3)} and Table \ref{tab:type(3)}, where black nodes (resp.\ white nodes) correspond to $(-1)$-curves (resp.\ $(-2)$-curves).
\end{enumerate}
\end{thm}
\begin{table}[htbp]
\centering
\caption{}
\begin{tabular}{|c|c|c|} \hline
Type & Reducible fibers &$\mathrm{MW}(Z)$ \\ \hline \hline
\textup{(1)} & $\textup{II}^*$ & $\{1\}$ \\ \hline
\textup{(2)} & $\textup{IV}^*, \textup{IV}$ & $\ZZ/3\ZZ$ \\ \hline
\textup{(3)} & four $\textup{IV}$ & $(\ZZ/3\ZZ)^{2}$ \\ \hline
\end{tabular}
\label{q-ell3}
\end{table}
\begin{figure}[htbp]
\captionsetup[subfigure]{labelformat=empty}
\begin{tabular}{cc}
\multirow{2}{*}{
\begin{minipage}[t]{0.2\hsize}
\centering
{\label{fig:figA}
\begin{tikzpicture}
\draw[thin ](-0.2,-0)--(-0.8,0);
\draw[thin ](-1.2,-0)--(-1.8,0);
\draw[thin ](-2,-0.2)--(-2,-0.8);
\draw[thin ](-2,-1.2)--(-2,-1.8);
\draw[thin ](-2,-2.2)--(-2,-2.8);
\draw[thin ](-2,-3.2)--(-2,-3.8);
\draw[thin ](-2,-4.2)--(-2,-4.8);
\draw[thin ](-2,-5.2)--(-2,-5.8);
\draw[thin ](-2.2,-4)--(-2.8,-4);
\fill (0,0) circle (2pt);
\draw[very thick ] (-1,0)circle(2pt);
\draw[very thick] (-2,0)circle(2pt);
\draw[very thick] (-2,-1)circle(2pt);
\draw[very thick] (-2,-2)circle(2pt);
\draw[very thick] (-2,-3)circle(2pt);
\draw[very thick] (-2,-4)circle(2pt);
\draw[very thick] (-2,-5)circle(2pt);
\draw[very thick] (-2,-6)circle(2pt);
\draw[] (-3,-4)circle(2pt);
\node(a)at(0,0.3){$O$};
\node(a)at(-1,0.3){$\Theta_{\infty, 0}$};
\node(a)at(-2,0.3){$\Theta_{\infty, 1}$};
\node(a)at(-1.4,-1){$\Theta_{\infty, 2}$};
\node(a)at(-1.4,-2){$\Theta_{\infty, 3}$};
\node(a)at(-1.4,-3){$\Theta_{\infty, 4}$};
\node(a)at(-1.4,-4){$\Theta_{\infty, 5}$};
\node(a)at(-1.4,-5){$\Theta_{\infty, 6}$};
\node(a)at(-1.4,-6){$\Theta_{\infty, 7}$};
\node(a)at(-3,-3.7){$\Theta_{\infty, 8}$};
\end{tikzpicture}
} \subcaption{Type (1)}
\label{composite}
\end{minipage}
}
&
\begin{minipage}[t]{0.7\hsize}
\centering
{\label{fig:figB}
\begin{tikzpicture}
\draw[thin ](0.2,1)--(0.8,1);
\draw[thin ](0.2,0)--(0.8,0);
\draw[thin ](0.2,-1)--(0.8,-1);
\draw[thin ](1.2,1)--(1.8,1);
\draw[thin ](1.2,0)--(1.8,0);
\draw[thin ](1.2,-1)--(1.8,-1);
\draw[thin ](2.1,0.9)--(2.9,0.1);
\draw[thin ](2.2,0)--(2.8,0);
\draw[thin ](2.1,-0.9)--(2.9,-0.1);
\draw[thin ](-1.8,1)--(-0.2,1);
\draw[thin ](-0.8,0)--(-0.2,0);
\draw[thin ](-1.8,-1)--(-0.2,-1);
\draw[thin ](-2,0.8)--(-2,-0.8);
\draw[thin ](-1.9,0.9)--(-1.1,0.1);
\draw[thin ](-1.9,-0.9)--(-1.1,-0.1);
\fill (0,1) circle (2pt);
\fill (0,0) circle (2pt);
\fill (0,-1) circle (2pt);
\draw[very thick] (1,-1)circle(2pt);
\draw[very thick] (1,0)circle(2pt);
\draw[very thick] (1,1)circle(2pt);
\draw[very thick] (2,-1)circle(2pt);
\draw[very thick] (2,0)circle(2pt);
\draw[very thick] (2,1)circle(2pt);
\draw[] (3,0)circle(2pt);
\draw[] (-1,0)circle(2pt);
\draw[] (-2,1)circle(2pt);
\draw[] (-2,-1)circle(2pt);
\node(a)at(0,1.3){$O$};
\node(a)at(0,0.3){$P$};
\node(a)at(0,-0.7){$2P$};
\node(a)at(1,1.3){$\Theta_{\infty, 2}$};
\node(a)at(1,0.3){$\Theta_{\infty, 4}$};
\node(a)at(1,-0.7){$\Theta_{\infty, 6}$};
\node(a)at(2,1.3){$\Theta_{\infty, 1}$};
\node(a)at(2,0.3){$\Theta_{\infty, 3}$};
\node(a)at(2.6,-1){$\Theta_{\infty, 5}$};
\node(a)at(-2.5,1){$\Theta_{0,0}$};
\node(a)at(-1.5,0){$\Theta_{0,1}$};
\node(a)at(-2.5,-1){$\Theta_{0,2}$};
\node(a)at(3.5,0){$\Theta_{\infty, 0}$};
\end{tikzpicture}
}
\subcaption{Type (2)}
\label{Gradation}
\end{minipage} \\
\begin{minipage}[t]{0.2\hsize}
\centering
\label{fill}
\end{minipage} &
\begin{minipage}[t]{0.7\hsize}
\centering
{\label{fig:figC}
\begin{tikzpicture}
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\draw (-0.1,1.94) parabola bend (-1,1.94) (-1.9,1.1);
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\fill (0,0.5) circle (2pt);
\fill (0,1) circle (2pt);
\fill (0,1.5) circle (2pt);
\fill (0,2) circle (2pt);
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\draw[] (-2.5,-1)circle(2pt);
\draw[] (-3,-1)circle(2pt);
\draw[] (-4,-1)circle(2pt);
\draw[] (-4.5,-1)circle(2pt);
\draw[] (-5.5,-1)circle(2pt);
\draw[] (-6,-1)circle(2pt);
\draw[] (-7,-1)circle(2pt);
\draw[] (-2,1)circle(2pt);
\draw[] (-3.5,1)circle(2pt);
\draw[] (-5,1)circle(2pt);
\draw[] (-6.5,1)circle(2pt);
\node(a)at(0.3,2){$O$};
\node(a)at(0.3,1.5){$P$};
\node(a)at(0.4,1){$2P$};
\node(a)at(0.3,0.5){$Q$};
\node(a)at(0.4,0){$2Q$};
\node(a)at(0.6,-0.5){$P+Q$};
\node(a)at(0.7,-1){$2P+Q$};
\node(a)at(0.7,-1.5){$P+2Q$};
\node(a)at(0.8,-2){$2P+2Q$};
\node(a)at(-7,1){$\Theta_{0,0}$};
\node(a)at(-5.6,1){$\Theta_{-1,0}$};
\node(a)at(-4,1){$\Theta_{1,0}$};
\node(a)at(-2.5,1){$\Theta_{\infty,0}$};
\node(a)at(-7,-1.3){$\Theta_{0,1}$};
\node(a)at(-6.2,-1.3){$\Theta_{0,2}$};
\node(a)at(-5.4,-1.3){$\Theta_{-1,1}$};
\node(a)at(-4.5,-1.3){$\Theta_{-1,2}$};
\node(a)at(-3.7,-1.3){$\Theta_{1,1}$};
\node(a)at(-3.05,-1.3){$\Theta_{1,2}$};
\node(a)at(-2.2,-1.3){$\Theta_{\infty,1}$};
\node(a)at(-1.4,-1.3){$\Theta_{\infty,2}$};
\end{tikzpicture}
}
\vspace{-3em}
\subcaption{Type (3)}
\label{transform}
\end{minipage}
\end{tabular}
\caption{Dual graphs of negative rational curves in rational quasi-elliptic surfaces of types (1)--(3)}
\label{fig:dual(1)-(3)}
\end{figure}
\begin{table}[htbp]
\caption{}
\begin{tabular}{|l|l|l|l|l|} \hline
& $\beta=0$ & $\beta=-1$ & $\beta=1$ & $\beta=\infty$ \\ \hline
Sections adjacent & $O, 2P+Q,$&$O, Q,$ & $O, P,$ & $O, P+Q,$ \\
to $\Theta_{\beta, 0}$ & $P+2Q$ &$2Q$ & $2P$ & $2P+2Q$ \\ \hline
Sections adjacent & $P, Q,$ &$2P, 2P+Q,$ & $2Q, P+2Q,$ & $P, 2Q$ \\
to $\Theta_{\beta, 1}$ & $2P+2Q$ &$2P+2Q$ & $2P+2Q$ & $2P+Q$ \\ \hline
Sections adjacent & $2P, 2Q,$ &$P, P+Q,$ & $Q, P+Q$ & $Q, 2P,$ \\
to $\Theta_{\beta, 2}$ & $P+Q$ &$P+2Q$ & $2P+Q$ & $P+2Q$ \\ \hline
\end{tabular}
\label{tab:type(3)}
\end{table}
\begin{thm}[{\cite[\S 5]{Ito2}}]\label{q-ell}
Suppose $p=2$. Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] The configurations of reducible fibers of rational quasi-elliptic surfaces and their Mordell-Weil groups are listed in Table \ref{q-ell2}, where we use Kodaira's notation.
\item[\textup{(2)}] Rational quasi-elliptic surfaces of each type (a)--(c) and (e) uniquely exist.
\item[\textup{(3)}] For each rational quasi-elliptic surface of one of the types (a)--(e), sections are disjoint from each other.
Moreover, the dual graphs of negative rational curves in rational quasi-elliptic surfaces of types (a)--(e) are as in Figure \ref{fig:dual(a)-(e)}, where black nodes (resp.\ white nodes) correspond to $(-1)$-curves (resp.\ $(-2)$-curves).
\item[\textup{(4)}] For each rational quasi-elliptic surface of type (f), sections are disjoint from each other.
There is an element $a \in k \setminus \{0\}$ such that the reducible fiber of type $\textup{I}^*_0$ lies over $t=1$ and reducible fibers of type \textup{III} lie over the points $t=0, \infty, \alpha_1, \alpha_2$ of the base curve $\PP^1_k$, where $\alpha_1$ and $\alpha_2$ are two solutions of the equation $t^2+at+1=0$.
Moreover, Figure \ref{fig:dual(f)} and Table \ref{typef} describe the dual graph of the configuration of negative rational curves.
\item[\textup{(5)}] For each rational quasi-elliptic surface of type (g), there are eight pairs of two sections intersecting with each other transversally and not intersecting with any other sections.
There are no irreducible components of reducible fibers intersecting with two sections in a pair.
Figure \ref{fig:dual(g)} describes the above situation.
\end{enumerate}
\end{thm}
\begin{table}[htbp]
\caption{}
\begin{tabular}{|c|c|c||c|c|c|} \hline
Type &Reducible fibers &$\mathrm{MW}(Z)$ & Type & Reducible fibers &$\mathrm{MW}(Z)$ \\ \hline \hline
\textup{(a)} &$\textup{II}^*$ &$\{1\}$ & \textup{(e)} &$\textup{I}^*_2, \textup{III}, \textup{III}$ & $(\ZZ/2\ZZ)^2$ \\ \hline
\textup{(b)} &$\textup{I}^*_4$ &$\ZZ/2\ZZ$ & \textup{(f)} &$\textup{I}^*_0$ and four $\textup{III}$ & $(\ZZ/2\ZZ)^3$ \\ \hline
\textup{(c)} &$\textup{III}^*, \textup{III}$ &$\ZZ/2\ZZ$ & \textup{(g)} &eight $\textup{III}$ & $(\ZZ/2\ZZ)^4$ \\ \hline
\textup{(d)} &$\textup{I}^*_0, \textup{I}^*_0$ &$(\ZZ/2\ZZ)^2$ & & & \\ \hline
\end{tabular}
\label{q-ell2}
\end{table}
\begin{rem}
\begin{enumerate}
\item Table 2 of \cite{Ito2} contains misprints.
By substituting $t=1$ to the equations of $P_2$, $P_3$, $Q_1$, and $R_1$ in the bottom of p.\ 246 of [\textit{ibid}], we see at once that $Q_1$ and $P_2$ in the bottom table should be interchanged with each other.
We also have to replace $R_3$ by $R_2$.
\item In Lemma \ref{lem:typeg}, we will clarify that Figure \ref{matrix(g)} is the intersection matrix of negative rational curves in a rational quasi-elliptic surface of type (g).
\item In Corollary \ref{Itorem}, we will give the parametrizing spaces of the isomorphism classes of rational quasi-elliptic surfaces of type (d), (f), or (g).
\end{enumerate}
\end{rem}
\begin{figure}[htbp]
\captionsetup[subfigure]{labelformat=empty}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\begin{tikzpicture}
\draw[thin ](-0.2,-0)--(-0.8,0);
\draw[thin ](-1.2,-0)--(-1.8,0);
\draw[thin ](-2,-0.2)--(-2,-0.8);
\draw[thin ](-2,-1.2)--(-2,-1.8);
\draw[thin ](-2,-2.2)--(-2,-2.8);
\draw[thin ](-2,-3.2)--(-2,-3.8);
\draw[thin ](-2,-4.2)--(-2,-4.8);
\draw[thin ](-2,-5.2)--(-2,-5.8);
\draw[thin ](-2.2,-4)--(-2.8,-4);
\fill (0,0) circle (2pt);
\draw[very thick ] (-1,0)circle(2pt);
\draw[very thick ] (-2,0)circle(2pt);
\draw[very thick ] (-2,-1)circle(2pt);
\draw[very thick ] (-2,-2)circle(2pt);
\draw[very thick ] (-2,-3)circle(2pt);
\draw[very thick ] (-2,-4)circle(2pt);
\draw[very thick ] (-2,-5)circle(2pt);
\draw[very thick ] (-2,-6)circle(2pt);
\draw[] (-3,-4)circle(2pt);
\node(a)at(0,0.3){$O$};
\node(a)at(-1,0.3){$\Theta_{\infty, 0}$};
\node(a)at(-3,-3.7){$\Theta_{\infty, 8}$};
\end{tikzpicture}
\caption{Type (a) }
\label{fig:dual(a)}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.3\textwidth}
\centering
\begin{tikzpicture}
\draw[thin ](-0.8,3)--(-0.2,3);
\draw[thin ](-0.8,-3)--(-0.2,-3);
\draw[thin ](-1.1,2.9)--(-1.9,2.1);
\draw[thin ](-1.1,-2.9)--(-1.9,-2.1);
\draw[thin ](-2.1,2.1)--(-2.9,2.9);
\draw[thin ](-2.1,-2.1)--(-2.9,-2.9);
\draw[thin ](-2,1.2)--(-2,1.8);
\draw[thin ](-2,0.2)--(-2,0.8);
\draw[thin ](-2,-0.2)--(-2,-0.8);
\draw[thin ](-2,-1.2)--(-2,-1.8);
\fill (0,-3) circle (2pt);
\fill (0,3) circle (2pt);
\draw[very thick ] (-1,3)circle(2pt);
\draw[very thick ] (-1,-3)circle(2pt);
\draw[very thick ] (-2,2)circle(2pt);
\draw[] (-2,1)circle(2pt);
\draw[very thick ] (-2,0)circle(2pt);
\draw[very thick ] (-2,-1)circle(2pt);
\draw[very thick ] (-2,-2)circle(2pt);
\draw[very thick ] (-3,3)circle(2pt);
\draw[] (-3,-3)circle(2pt);
\node(a)at(0,3.3){$O$};
\node(a)at(0,-2.7){$P$};
\node(a)at(-1,3.3){$\Theta_{\infty, 0}$};
\node(a)at(-3,3.3){$\Theta_{\infty, 1}$};
\node(a)at(-2.5,2){$\Theta_{\infty, 2}$};
\node(a)at(-2.5,1){$\Theta_{\infty, 3}$};
\node(a)at(-2.5,0){$\Theta_{\infty, 4}$};
\node(a)at(-2.5,-1){$\Theta_{\infty, 5}$};
\node(a)at(-2.5,-1.9){$\Theta_{\infty, 6}$};
\node(a)at(-1,-2.4){$\Theta_{\infty, 7}$};
\node(a)at(-3,-2.4){$\Theta_{\infty, 8}$};
\end{tikzpicture}
\caption{Type (b) }
\label{fig:dual(b)}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.3\textwidth}
\centering
\begin{tikzpicture}
\draw[thin ](-0.8,3)--(-0.2,3);
\draw[thin ](-0.8,-3)--(-0.2,-3);
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\draw[thin ](-1,0.2)--(-1,0.8);
\draw[thin ](-1,-0.2)--(-1,-0.8);
\draw[thin ](-1,-1.2)--(-1,-1.8);
\draw[thin ](-1.8,0)--(-1.2,0);
\draw[thin ](0.1,2.9)--(1,1.2);
\draw[thin ](0.1,-2.9)--(1,-1.2);
\draw[thin ](0.95,0.8)--(0.95,-0.8);
\draw[thin ](1.05,0.8)--(1.05,-0.8);
\fill (0,-3) circle (2pt);
\fill (0,3) circle (2pt);
\draw[very thick ] (-1,3)circle(2pt);
\draw[very thick ] (-1,2)circle(2pt);
\draw[] (-1,1)circle(2pt);
\draw[very thick ] (-1,0)circle(2pt);
\draw[very thick ] (-1,-1)circle(2pt);
\draw[very thick ] (-1,-2)circle(2pt);
\draw[very thick ] (-1,-3)circle(2pt);
\draw[very thick ] (-2,0)circle(2pt);
\draw[] (1,1)circle(2pt);
\draw[] (1,-1)circle(2pt);
\node(a)at(0,3.3){$O$};
\node(a)at(0,-2.7){$P$};
\node(a)at(0.5,0.9){$\Theta_{\infty, 0}$};
\node(a)at(0.5,-1.1){$\Theta_{\infty, 1}$};
\node(a)at(-1,3.3){$\Theta_{0, 0}$};
\node(a)at(-1.5,0.9){$\Theta_{0, 2}$};
\end{tikzpicture}
\caption{Type (c) }
\label{fig:dual(c)}
\end{subfigure} \\
\vspace{1em}
\begin{subfigure}[b]{0.4\textwidth}
\centering
\begin{tikzpicture}
\draw[thin ](0.2,-1.5)--(0.8,-1.5);
\draw[thin ](0.2,-0.5)--(0.8,-0.5);
\draw[thin ](0.2,0.5)--(0.8,0.5);
\draw[thin ](0.2,1.5)--(0.8,1.5);
\draw[thin ](-0.2,-1.5)--(-0.8,-1.5);
\draw[thin ](-0.2,-0.5)--(-0.8,-0.5);
\draw[thin ](-0.2,0.5)--(-0.8,0.5);
\draw[thin ](-0.2,1.5)--(-0.8,1.5);
\draw (1.2,1.5) parabola bend (1.5,1.5) (2,0.2);
\draw (1.2,0.5) parabola bend (1.5,0.5) (1.9,0.1);
\draw (1.2,-0.5) parabola bend (1.5,-0.5) (1.9,-0.1);
\draw (1.2,-1.5) parabola bend (1.5,-1.5) (2,-0.2);
\draw (-1.2,1.5) parabola bend (-1.5,1.5) (-2,0.2);
\draw (-1.2,0.5) parabola bend (-1.5,0.5) (-1.9,0.1);
\draw (-1.2,-0.5) parabola bend (-1.5,-0.5) (-1.9,-0.1);
\draw (-1.2,-1.5) parabola bend (-1.5,-1.5) (-2,-0.2);
\fill (0,-1.5) circle (2pt);
\fill (0,-0.5) circle (2pt);
\fill (0,0.5) circle (2pt);
\fill (0,1.5) circle (2pt);
\draw[very thick ] (1,-1.5)circle(2pt);
\draw[very thick ] (1,-0.5)circle(2pt);
\draw[very thick ] (1,0.5)circle(2pt);
\draw[] (1,1.5)circle(2pt);
\draw[] (-1,-1.5)circle(2pt);
\draw[] (-1,-0.5)circle(2pt);
\draw[] (-1,0.5)circle(2pt);
\draw[very thick ] (-1,1.5)circle(2pt);
\draw[] (2,0)circle(2pt);
\draw[very thick ] (-2,0)circle(2pt);
\node(a)at(0,1.8){$O$};
\node(a)at(0,0.8){$P_1$};
\node(a)at(0,-0.2){$P_2$};
\node(a)at(0,-1.2){$P_3$};
\node(a)at(-1,1.8){$\Theta_{0,0}$};
\node(a)at(-1,0.8){$\Theta_{0,1}$};
\node(a)at(-1,-0.2){$\Theta_{0,2}$};
\node(a)at(-1,-1.2){$\Theta_{0,3}$};
\node(a)at(-2.5,0){$\Theta_{0,4}$};
\node(a)at(1,1.8){$\Theta_{\infty,0}$};
\node(a)at(1,0.8){$\Theta_{\infty,1}$};
\node(a)at(1,-0.2){$\Theta_{\infty,2}$};
\node(a)at(1,-1.2){$\Theta_{\infty,3}$};
\node(a)at(2.5,0){$\Theta_{\infty,4}$};
\end{tikzpicture}
\caption{Type (d) }
\label{fig:dual(d)}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.5\textwidth}
\centering
\begin{tikzpicture}
\draw[thin ](0.2,-1.5)--(0.8,-1.5);
\draw[thin ](0.2,-0.5)--(0.8,-0.5);
\draw[thin ](0.2,0.5)--(0.8,0.5);
\draw[thin ](0.2,1.5)--(0.8,1.5);
\draw (1.2,1.5) parabola bend (3,1.5) (4,0.2);
\draw (1.2,0.5) parabola bend (1.5,0.5) (2,0.2);
\draw (1.2,-0.5) parabola bend (1.5,-0.5) (2,-0.2);
\draw (1.2,-1.5) parabola bend (3,-1.5) (4,-0.2);
\draw[thin ](2.2,0)--(2.8,0);
\draw[thin ](3.2,0)--(3.8,0);
\draw[thin ](-0.95,0.8)--(-0.95,-0.8);
\draw[thin ](-1.05,0.8)--(-1.05,-0.8);
\draw[thin ](-1.95,0.8)--(-1.95,-0.8);
\draw[thin ](-2.05,0.8)--(-2.05,-0.8);
\draw (-0.2,1.45) parabola bend (-0.7,1.45) (-1,1.2);
\draw (-0.2,-1.45) parabola bend (-0.7,-1.45) (-1,-1.2);
\draw (-0.2,0.55) parabola bend (-0.7,0.55) (-0.9,0.9);
\draw (-0.2,-0.55) parabola bend (-0.7,-0.55) (-0.9,-0.9);
\draw (-0.2,1.55) parabola bend (-1.5,1.55) (-2,1.2);
\draw (-0.2,-1.55) parabola bend (-1.5,-1.55) (-2,-1.2);
\draw (-0.2,0.45) parabola bend (-1,0.45) (-1.9,-0.9);
\draw (-0.2,-0.45) parabola bend (-1,-0.45) (-1.9,0.9);
\fill (0,-1.5) circle (2pt);
\fill (0,-0.5) circle (2pt);
\fill (0,0.5) circle (2pt);
\fill (0,1.5) circle (2pt);
\draw[] (1,-1.5)circle(2pt);
\draw[] (1,-0.5)circle(2pt);
\draw[] (1,0.5)circle(2pt);
\draw[] (1,1.5)circle(2pt);
\draw[] (2,0)circle(2pt);
\draw[] (3,0)circle(2pt);
\draw[] (4,0)circle(2pt);
\draw[] (-1,-1)circle(2pt);
\draw[] (-1,1)circle(2pt);
\draw[] (-2,-1)circle(2pt);
\draw[] (-2,1)circle(2pt);
\node(a)at(0,1.8){$O$};
\node(a)at(0,0.8){$Q$};
\node(a)at(0,-0.2){$R$};
\node(a)at(0,-1.2){$P$};
\node(a)at(-2.4,1.1) {$\Theta_{0,0}$};
\node(a)at(-2.4,-1.1) {$\Theta_{0,1}$};
\node(a)at(-0.5,1.1) {$\Theta_{\infty,0}$};
\node(a)at(-0.5,-1.1) {$\Theta_{\infty,1}$};
\node(a)at(3,0.3) {$\Theta_{1,0}$};
\node(a)at(1,0.8) {$\Theta_{1,1}$};
\node(a)at(1,1.8) {$\Theta_{1,2}$};
\node(a)at(4.4,0.3) {$\Theta_{1,3}$};
\node(a)at(1,-1.2) {$\Theta_{1,4}$};
\end{tikzpicture}
\caption{Type (e) }
\label{fig:dual(e)}
\end{subfigure}
\caption{Dual graphs of negative rational curves in rational quasi-elliptic surfaces of types (a)--(e)}
\label{fig:dual(a)-(e)}
\end{figure}
\begin{figure}[htbp]
\centering
\begin{tikzpicture}
\draw[thin ](1.2,0)--(1.8,0);
\draw[thin ](2.2,0)--(2.8,0);
\draw[thin ](2,0.2)--(2,0.8);
\draw[thin ](2,-0.2)--(2,-0.8);
\draw[thin ](-0.95,0.8)--(-0.95,-0.8);
\draw[thin ](-1.05,0.8)--(-1.05,-0.8);
\draw[thin ](-1.95,0.8)--(-1.95,-0.8);
\draw[thin ](-2.05,0.8)--(-2.05,-0.8);
\draw[thin ](-2.95,0.8)--(-2.95,-0.8);
\draw[thin ](-3.05,0.8)--(-3.05,-0.8);
\draw[thin ](-3.95,0.8)--(-3.95,-0.8);
\draw[thin ](-4.05,0.8)--(-4.05,-0.8);
\draw[thin ] (0.9,0.1)--(0.1,0.9);
\draw[thin ] (0.9,-0.1)--(0.1,-0.9);
\draw (0,1.1) parabola bend (-2,2.2) (-4,1.1);
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\draw (0,-1.1) parabola bend (-2,-2.2) (-4,-1.1);
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\draw[thin ](1.9,1.1)--(1.6,1.4);
\draw[thin ](2.1,1.1)--(2.4,1.4);
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\draw[thin ](2.1,-1.1)--(2.4,-1.4);
\draw[thin ](3.1,0.1)--(3.4,0.4);
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\draw[thin ](1.56,1.6)--(1.56,1.9);
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\draw[thin ](1.48,1.6)--(1.48,1.9);
\draw[thin ](1.44,1.6)--(1.44,1.9);
\node(a)at(1.5,2.3){$\vdots$};
\draw[thin ](2.56,1.6)--(2.56,1.9);
\draw[thin ](2.52,1.6)--(2.52,1.9);
\draw[thin ](2.48,1.6)--(2.48,1.9);
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\node(a)at(2.5,2.3){$\vdots$};
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\node(a)at(1.5,-2.1){$\vdots$};
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\draw[thin ](2.52,-1.6)--(2.52,-1.9);
\draw[thin ](2.48,-1.6)--(2.48,-1.9);
\draw[thin ](2.44,-1.6)--(2.44,-1.9);
\node(a)at(2.5,-2.1){$\vdots$};
\draw[thin ](3.6,0.56)--(3.9,0.56);
\draw[thin ](3.6,0.52)--(3.9,0.52);
\draw[thin ](3.6,0.48)--(3.9,0.48);
\draw[thin ](3.6,0.44)--(3.9,0.44);
\node(a)at(4.2,0.5){$\cdots$};
\draw[thin ](3.6,-0.56)--(3.9,-0.56);
\draw[thin ](3.6,-0.52)--(3.9,-0.52);
\draw[thin ](3.6,-0.48)--(3.9,-0.48);
\draw[thin ](3.6,-0.44)--(3.9,-0.44);
\node(a)at(4.2,-0.5){$\cdots$};
\fill (0,-1) circle (2pt);
\fill (0,1) circle (2pt);
\fill (3.5,-0.5) circle (2pt);
\fill (3.5,0.5) circle (2pt);
\fill (1.5,1.5) circle (2pt);
\fill (2.5,1.5) circle (2pt);
\fill (1.5,-1.5) circle (2pt);
\fill (2.5,-1.5) circle (2pt);
\draw[] (2,0)circle(2pt);
\draw[] (1,0)circle(2pt);
\draw[] (3,0)circle(2pt);
\draw[] (2,1)circle(2pt);
\draw[] (2,-1)circle(2pt);
\draw[] (-1,-1)circle(2pt);
\draw[] (-1,1)circle(2pt);
\draw[] (-2,-1)circle(2pt);
\draw[] (-2,1)circle(2pt);
\draw[] (-3,-1)circle(2pt);
\draw[] (-3,1)circle(2pt);
\draw[] (-4,-1)circle(2pt);
\draw[] (-4,1)circle(2pt);
\begin{scriptsize}
\node(a)at(-4.3,1.2){$\Theta_{0,0}$};
\node(a)at(-4.3,-1.2){$\Theta_{0,1}$};
\node(a)at(-3.3,1.2){$\Theta_{\infty,0}$};
\node(a)at(-3.3,-1.2){$\Theta_{\infty,1}$};
\node(a)at(-2.3,1.2){$\Theta_{\alpha_1,0}$};
\node(a)at(-2.3,-1.2){$\Theta_{\alpha_1,1}$};
\node(a)at(-1.3,1.2){$\Theta_{\alpha_2,0}$};
\node(a)at(-1.3,-1.2){$\Theta_{\alpha_2,1}$};
\node(a)at(0.3,1.2){$O$};
\node(a)at(0.3,-1.2){$P_1$};
\node(a)at(1.2,-1.5){$Q_2$};
\node(a)at(2.8,-1.5){$R_2$};
\node(a)at(3.5,-0.8){$Q_1$};
\node(a)at(3.5,0.8){$R_1$};
\node(a)at(1.2,1.5){$P_2$};
\node(a)at(2.8,1.5){$P_3$};
\node(a)at(1.3,0.2){$\Theta_{1, 0}$};
\node(a)at(2.4,-0.9){$\Theta_{1, 1}$};
\node(a)at(3.5,0.0){$\Theta_{1, 2}$};
\node(a)at(2.4,0.9){$\Theta_{1, 3}$};
\node(a)at(2.4,0.2){$\Theta_{1, 4}$};
\end{scriptsize}
\end{tikzpicture}
\caption{Dual graph of negative rational curves in a rational quasi-elliptic surface of type (f)}
\label{fig:dual(f)}
\end{figure}
\begin{table}[htbp]
\caption{}
\begin{tabular}{|l|c|c|c|c|} \hline
& $\beta=0$ & $\beta=\infty$ & $\beta=\alpha_1$ & $\beta=\alpha_2$ \\ \hline
Section intersecting & $O, R_2$ &$O, Q_2$ & $O, Q_2$ & $O, R_2$ \\
with $\Theta_{\beta, 0}$ & $R_1, P_3$ &$P_3, Q_1$ & $R_1, P_2$ & $P_2, Q_1$ \\ \hline
Section intersecting & $P_1, Q_2$ &$P_1, R_2$ & $P_1, R_2$ & $P_1, Q_2$ \\
with $\Theta_{\beta, 1}$ & $P_2, Q_1$ &$R_1, P_2$ & $P_3, Q_1$ & $R_1, P_3$ \\ \hline
\end{tabular}
\vspace{1em}
\begin{tabular}{|l|c|} \hline
& $\gamma=1$ \\ \hline
Section intersecting with $\Theta_{\gamma, 0}$ & $O, P_1$ \\ \hline
Section intersecting with $\Theta_{\gamma, 1}$ & $Q_2, R_2$ \\ \hline
Section intersecting with $\Theta_{\gamma, 2}$ & $Q_1, R_1$ \\ \hline
Section intersecting with $\Theta_{\gamma, 3}$ & $P_2, P_3$ \\ \hline
\end{tabular}
\label{typef}
\end{table}
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[xscale=0.75]
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\begin{tiny}
\node(a)at(0.4,0.9){$A_{0,1}$};
\node(a)at(0.4,-0.9){$A_{0,2}$};
\node(a)at(0.4,-1.15){\rotatebox{90}{$\;\;=\;\;$}};
\node(a)at(0.4,-1.4){$O$};
\node(a)at(1.4,0.9){$A_{1,1}$};
\node(a)at(1.4,-0.9){$A_{1,2}$};
\node(a)at(2.4,0.9){$A_{2,1}$};
\node(a)at(2.4,-0.9){$A_{2,2}$};
\node(a)at(3.4,0.9){$A_{3,1}$};
\node(a)at(3.4,-0.9){$A_{3,2}$};
\node(a)at(4.4,0.9){$A_{4,1}$};
\node(a)at(4.4,-0.9){$A_{4,2}$};
\node(a)at(5.4,0.9){$A_{5,1}$};
\node(a)at(5.4,-0.9){$A_{5,2}$};
\node(a)at(6.4,0.9){$A_{6,1}$};
\node(a)at(6.4,-0.9){$A_{6,2}$};
\node(a)at(7.4,0.9){$A_{7,1}$};
\node(a)at(7.4,-0.9){$A_{7,2}$};
\node(a)at(-1.4,0.9){$\Theta_{0,1}$};
\node(a)at(-1.4,-0.9){$\Theta_{0,2}$};
\node(a)at(-2.4,0.9){$\Theta_{1,1}$};
\node(a)at(-2.4,-0.9){$\Theta_{1,2}$};
\node(a)at(-3.4,0.9){$\Theta_{2,1}$};
\node(a)at(-3.4,-0.9){$\Theta_{2,2}$};
\node(a)at(-4.4,0.9){$\Theta_{3,1}$};
\node(a)at(-4.4,-0.9){$\Theta_{3,2}$};
\node(a)at(-5.4,0.9){$\Theta_{4,1}$};
\node(a)at(-5.4,-0.9){$\Theta_{4,2}$};
\node(a)at(-6.4,0.9){$\Theta_{5,1}$};
\node(a)at(-6.4,-0.9){$\Theta_{5,2}$};
\node(a)at(-7.4,0.9){$\Theta_{6,1}$};
\node(a)at(-7.4,-0.9){$\Theta_{6,2}$};
\node(a)at(-8.4,0.9){$\Theta_{7,1}$};
\node(a)at(-8.4,-0.9){$\Theta_{7,2}$};
\end{tiny}
\end{tikzpicture}
\caption{Dual graph of negative rational curves in a rational quasi-elliptic surface of type (g)}
\label{fig:dual(g)}
\end{figure}
By \cite[Theorem 3.1]{LPS}, each cuspidal cubic curve in $\PP^2_{k, [x:y:z]}$ with an inflexion point is projectively equivalent to $C=\{x^3+y^2z=0\}$.
Moreover, since the automorphism $[x:y:z] \mapsto [ax:y:a^3z]$ of $\PP^2_k$ with $a \in k^*$ fixes $C$, the pair of $C$ and a point $p \in C$ is projectively equivalent to the pair of $C$ and $[1:1:-1]$ unless $p=[0:0:1]$ or $[0:1:0]$.
From these facts, we can interpret \cite[Example 3.8]{Ito1} and \cite[Remark 4]{Ito2} as follows.
\begin{lem}[{\cite[Example 3.8]{Ito1}, \cite[Remark 4]{Ito2}}]\label{lem:Ito2Rem4}
Let $Z$ be a quasi-elliptic surface of type one of (1)--(3) in characteristic three or one of (a)--(d) in characteristic two.
When $Z$ is of type (1), (a), or (b), we choose a general fiber $F$ in addition.
Then, contracting all curves corresponding to bold white node or black node in Figure \ref{fig:dual(1)-(3)} and Types (a)--(d) of Figure \ref{fig:dual(a)-(e)},
we obtain a morphism $h \colon Z \to \PP^2_k$.
Moreover, there are coordinates $[x:y:z]$ of $\PP^2_k$ such that the images of $F$ and negative rational curves by $h$ are written as follows.
If $Z$ is of type (1), then
\begin{align*}
&h(F) = \{x^3+y^2z=0\},
&&h(\Theta_{\infty, 8}) = \{z=0\},
&&h(O)=[0:1:0].
\end{align*}
If $Z$ is of type (2), then
\begin{align*}
&h(\Theta_{\infty, 0}) = \{x=0\},
&&h(\Theta_{0,0})=\{y=0\},
&&h(\Theta_{0,1})=\{y=z\},
&&h(\Theta_{0,2})=\{z=0\}, \\
&h(O)=[0:0:1],
&&h(P)=[0:1:1],
&&h(2P)=[0:1:0].
\end{align*}
If $Z$ is of type (3), then
\begin{align*}
&h(\Theta_{0,0})=\{x=0\},
&&h(\Theta_{0,1})=\{x=z\},
&&h(\Theta_{0,2})=\{x=-z\}, \\
&h(\Theta_{-1,0})=\{y=0\},
&&h(\Theta_{-1,1})=\{y=z\},
&&h(\Theta_{-1,2})=\{y=-z\}, \\
&h(\Theta_{1,0})=\{x+y=0\},
&&h(\Theta_{1,1})=\{x+y=-z\},
&&h(\Theta_{1,2})=\{x+y=z\}, \\
&h(\Theta_{\infty,0})=\{x-y=0\},
&&h(\Theta_{\infty,1})=\{x-y=-z\},
&&h(\Theta_{\infty,2})=\{x-y=z\},\\
&h(O)=[0:0:1],
&&h(P)=[1:-1:1],
&&h(2P)=[-1:1:1],\\
&h(Q)=[1:0:1],
&&h(2Q)=[-1:0:1],
&&h(P+Q)=[-1:-1:1],\\
&h(2P+Q)=[0:1:1],
&&h(P+2Q)=[0:-1:1],
&&h(2P+2Q)=[1:1:1].
\end{align*}
If $Z$ is of type (a), then
\begin{align*}
&h(F) = \{x^3+y^2z=0\},
&&h(\Theta_{\infty, 8}) = \{z=0\},
&&h(O)=[0:1:0].
\end{align*}
If $Z$ is of type (b), then
\begin{align*}
&h(F) = \{x^3+y^2z=0\},
&&h(\Theta_{\infty, 8})=\{z=0\},
&&h(\Theta_{\infty, 3})=\{x+z=0\}, \\
&h(P)=[0:1:0],
&&h(O)=[1:1:1].
\end{align*}
If $Z$ is of type (c), then
\begin{align*}
&h(\Theta_{0,2})=\{z=0\},
&&h(\Theta_{\infty, 0})=\{x=0\},
&&h(\Theta_{\infty, 1})=\{xz+y^2=0\}, \\
&h(O)=[0:1:0],
&&h(P)=[1:0:0].
\end{align*}
If $Z$ is of type (d), then
\begin{align*}
&h(\Theta_{\infty, 4}) = \{x=0\},
&&h(\Theta_{0, 1})=\{y=0\},
&&h(\Theta_{0, 2})=\{y+z=0\},
&&h(\Theta_{0, 3})=\{z=0\}, \\
&h(\Theta_{0,4})=[1:0:0],
&&h(\Theta_{\infty, 1})=[0:0:1],
&&h(\Theta_{\infty, 2})=[0:1:1],
&&h(\Theta_{\infty, 3})=[0:1:0].
\end{align*}
\end{lem}
Rational quasi-elliptic surfaces are naturally endowed with the action of the Mordell-Weil groups.
The next lemma shows that these surfaces may have other automorphisms.
\begin{lem}\label{lem:type(d)auto}
A rational quasi-elliptic surface $Z$ of type (d) has an involution which sends $\Theta_{0, i}$ in Type (d) of Figure \ref{fig:dual(a)-(e)} to $\Theta_{\infty, i}$ for $0 \leq i \leq 4$.
\end{lem}
\begin{proof}
Let $\phi \colon Z \to \PP^1_k \times \PP^1_k$ be the contraction of $O$, $P_1, P_2, P_3$, $\Theta_{0, 0}$, $\Theta_{0, 1}$,$\Theta_{\infty, 2}$, and $\Theta_{\infty, 3}$.
Then we can choose coordinates $([x:y],[s:t])$ of $\PP^1_k \times \PP^1_k$ such that $\phi(\Theta_{0,4})=\{x=0\}$ and $\phi(\Theta_{\infty,4})=\{y=0\}$.
Hence the involution $([x:y],[s:t]) \mapsto ([y:x],[s:t])$ induces the desired involution on $Z$.
\end{proof}
The next lemma clarifies the whole configuration of negative rational curves in a quasi-elliptic surface of type (g).
\begin{lem}\label{lem:typeg}
Figure \ref{matrix(g)} is the intersection matrix of negative rational curves on a rational quasi-elliptic surface of type (g).
\end{lem}
\begin{figure}[htbp]
\centering
\begin{align*}
{\fontsize{7pt}{4}\selectfont
\left(\!\!\!\!
\vcenter{
\xymatrix@=-1.8ex{
&&&&&&&&&\ar@{-}[ddddddddddddddddddddddddddddddddddddd]&&&&&&&&& \ar@{-}[ddddddddddddddddddddddddddddddddddddd] &&&&&&&&& \ar@{-}[ddddddddddddddddddddddddddddddddddddd] &&&&&&&&\\
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&\\
&-1&0 &0 &0 &0 &0 &0 &0 &&1 &0 &0 &0 &0 &0 &0 &0 &&1 &1 &1 &1 &1 &1 &1 &1 &&0 &0 &0 &0 &0 &0 &0 &0 &\\
& &-1&0 &0 &0 &0 &0 &0 &&0 &1 &0 &0 &0 &0 &0 &0 &&1 &1 &1 &1 &0 &0 &0 &0 &&0 &0 &0 &0 &1 &1 &1 &1 &\\
& & &-1&0 &0 &0 &0 &0 &&0 &0 &1 &0 &0 &0 &0 &0 &&1 &1 &0 &0 &1 &1 &0 &0 &&0 &0 &1 &1 &0 &0 &1 &1 &\\
& & & &-1&0 &0 &0 &0 &&0 &0 &0 &1 &0 &0 &0 &0 &&1 &0 &1 &0 &1 &0 &1 &0 &&0 &1 &0 &1 &0 &1 &0 &1 &\\
& & & & &-1&0 &0 &0 &&0 &0 &0 &0 &1 &0 &0 &0 &&1 &1 &0 &0 &0 &0 &1 &1 &&0 &0 &1 &1 &1 &1 &0 &0 &\\
& & & & & &-1&0 &0 &&0 &0 &0 &0 &0 &1 &0 &0 &&1 &0 &0 &1 &1 &0 &0 &1 &&0 &1 &1 &0 &0 &1 &1 &0 &\\
& & & & & & &-1&0 &&0 &0 &0 &0 &0 &0 &1 &0 &&1 &0 &1 &0 &0 &1 &0 &1 &&0 &1 &0 &1 &1 &0 &1 &0 &\\
& & & & & & & &-1&&0 &0 &0 &0 &0 &0 &0 &1 &&1 &0 &0 &1 &0 &1 &1 &0 &&0 &1 &1 &0 &1 &0 &0 &1 &\\
& \ar@{-}[rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr]&&&&&&&&& &&&&&&&&& &&&&&&&&&& &&&&&&&&\\
& & & & & & & & &&-1&0 &0 &0 &0 &0 &0 &0 &&0 &0 &0 &0 &0 &0 &0 &0 &&1 &1 &1 &1 &1 &1 &1 &1 &\\
& & & & & & & & && &-1&0 &0 &0 &0 &0 &0 &&0 &0 &0 &0 &1 &1 &1 &1 &&1 &1 &1 &1 &0 &0 &0 &0 &\\
& & & & & & & & && & &-1&0 &0 &0 &0 &0 &&0 &0 &1 &1 &0 &0 &1 &1 &&1 &1 &0 &0 &1 &1 &0 &0 &\\
& & & & & & & & && & & &-1&0 &0 &0 &0 &&0 &1 &0 &1 &0 &1 &0 &1 &&1 &0 &1 &0 &1 &0 &1 &0 &\\
& & & & & & & & && & & & &-1&0 &0 &0 &&0 &0 &1 &1 &1 &1 &0 &0 &&1 &1 &0 &0 &0 &0 &1 &1 &\\
& & & & & & & & && & & & & &-1&0 &0 &&0 &1 &1 &0 &0 &1 &1 &0 &&1 &0 &0 &1 &1 &0 &0 &1 &\\
& & & & & & & & && & & & & & &-1&0 &&0 &1 &0 &1 &1 &0 &1 &0 &&1 &0 &1 &0 &0 &1 &0 &1 &\\
& & & & & & & & && & & & & & & &-1&&0 &1 &1 &0 &1 &0 &0 &1 &&1 &0 &0 &1 &0 &1 &1 &0 &\\
& \ar@{-}[rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr]&&&&&&&&& &&&&&&&&& &&&&&&&&&& &&&&&&&&\\
& & & & & & & & && & & & & & & & &&-2&0 &0 &0 &0 &0 &0 &0 &&2 &0 &0 &0 &0 &0 &0 &0 &\\
& & & & & & & & && & & & & & & & && &-2&0 &0 &0 &0 &0 &0 &&0 &2 &0 &0 &0 &0 &0 &0 &\\
& & & & & & & & && & & & & & & & && & &-2&0 &0 &0 &0 &0 &&0 &0 &2 &0 &0 &0 &0 &0 &\\
& & & & & & & & && & & & & & & & && & & &-2&0 &0 &0 &0 &&0 &0 &0 &2 &0 &0 &0 &0 &\\
& & & & & & & & && & & & & & & & && & & & &-2&0 &0 &0 &&0 &0 &0 &0 &2 &0 &0 &0 &\\
& & & & & & & & && & & & & & & & && & & & & &-2&0 &0 &&0 &0 &0 &0 &0 &2 &0 &0 &\\
& & & & & & & & && & & & & & & & && & & & & & &-2&0 &&0 &0 &0 &0 &0 &0 &2 &0 &\\
& & & & & & & & && & & & & & & & && & & & & & & &-2&&0 &0 &0 &0 &0 &0 &0 &2 &\\
& \ar@{-}[rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr]&&&&&&&&& &&&&&&&&& &&&&&&&&&& &&&&&&&&\\
& & & & & & & & && & & & & & & & && & & & & & & & &&-2&0 &0 &0 &0 &0 &0 &0 &\\
& & & & & & & & && & & & & & & & && & & & & & & & && &-2&0 &0 &0 &0 &0 &0 &\\
& & & & & & & & && & & & & & & & && & & & & & & & && & &-2&0 &0 &0 &0 &0 &\\
& & & & & & & & && & & & & & & & && & & & & & & & && & & &-2&0 &0 &0 &0 &\\
& & & & & & & & && & & & & & & & && & & & & & & & && & & & &-2&0 &0 &0 &\\
& & & & & & & & && & & & & & & & && & & & & & & & && & & & & &-2&0 &0 &\\
& & & & & & & & && & & & & & & & && & & & & & & & && & & & & & &-2&0 &\\
& & & & & & & & && & & & & & & & && & & & & & & & && & & & & & & &-2&\\
& & & & & & & & && & & & & & & & && & & & & & & & && & & & & & & & &
}
}\!\!\!\!\right)
}
\end{align*}
\caption{The intersection matrix of
$A_{0, 1}, \ldots, A_{7, 1}$, $A_{0, 2}, \ldots, A_{7,2}$, $\Theta_{0,1}, \ldots, \Theta_{7,1}$, $\Theta_{0,2}, \ldots, \Theta_{7,2}$ in this order
in a rational quasi-elliptic surface of type (g).
}
\label{matrix(g)}
\end{figure}
\begin{proof}
Let $Z$ be a rational quasi-elliptic surface of type (g).
Then there are exactly sixteen $(-2)$-curves $\{\Theta_{i, j}\}_{0 \leq i \leq 7, 1 \leq j \leq 2}$ on $Z$, which satisfies that
\begin{align*}
(\Theta_{i,j}, \Theta_{i',j'}) >0 \iff (\Theta_{i,j}, \Theta_{i',j'}) = 2 \iff i=i' \text{ and } j \neq j'.
\end{align*}
On the other hand, as we described in Theorem \ref{q-ell} (5), there is exactly sixteen sections $\{A_{k, l}\}_{0 \leq k \leq 7, 1 \leq l \leq 2}$ on $Z$, which satisfies that
\begin{align*}
(A_{k,l}, A_{k',l'}) >0 \iff (A_{k,l}, A_{k',l'}) = 1 \iff k=k' \text{ and } l \neq l'.
\end{align*}
By Theorem \ref{q-ell} (5), we may assume that
\begin{align*}
(\Theta_{i,j}, A_{0,2})\neq 0 &\iff (\Theta_{i,j}, A_{0,2})=1 \iff j=2, \text{ and }\\
(\Theta_{0,2}, A_{k,l})\neq 0 &\iff (\Theta_{0,2}, A_{k,l})=1 \iff l=2.
\end{align*}
By contracting $A_{0,2}, \Theta_{0,2}$ and $A_{i,1}$ for $1 \leq i \leq 7$, we get a birational morphism $h \colon Z \to \PP^2_k$.
Let $t=h(A_{0,2} \cup \Theta_{0,2})$, $t_i=h(A_{i,1})$, and $D_i=h_*\Theta_{i,1}$ for $1 \leq i \leq 7$.
To show the assertion, we prepare some claims.
\begin{cln}\label{cl:typeg-1}
$h_*\Theta_{i,j} \sim \sO_{\PP^2_k}(j)$ for each $1 \leq i \leq 7$ and $1 \leq j \leq 2$.
\end{cln}
\begin{clproof}
We need only consider the case where $i=1$ by symmetry and the case where $j=1$ since $h_*(\Theta_{1, 1}+\Theta_{1,2}) \sim h_*(-K_Y) \sim \sO_{\PP^2_k}(3)$.
Suppose by contradiction that $h_*\Theta_{1,1} \sim \sO_{\PP^2_k}(2)$.
Then exactly six of $A_{1,1}, A_{2, 1}, \ldots, A_{7,1}$ intersect with $\Theta_{1,1}$ since $(h_*\Theta_{1,1})^2-\Theta_{1,1}^2=6$.
We may assume that $(A_{1,1}, \Theta_{1,1})=0$.
Assume that $h_*\Theta_{i,1} \sim \sO_{\PP^2_k}(2)$ for some $2 \leq i \leq 7$.
Then $(h_*\Theta_{1,1}, h_*\Theta_{i,1}) = 4$.
However, at least five of $A_{1,1}$, $A_{2, 1}$, $\ldots$, $A_{7,1}$ intersect with both $\Theta_{1,1}$ and $\Theta_{i,1}$, which implies that $(h_*\Theta_{1,1}, h_*\Theta_{i,1}) \geq 5$, a contradiction.
Hence $h_*\Theta_{i,1} \sim \sO_{\PP^2_k}(1)$ and $(h_*\Theta_{1,1}, h_*\Theta_{i,1}) = 2$ for each $2 \leq i \leq 7$.
For such an $i$, exactly three of $A_{1,1}, A_{2, 1}, \ldots, A_{7,1}$ intersect with $\Theta_{i,1}$ since $(h_*\Theta_{i,1})^2-\Theta_{i,1}^2=3$.
Moreover, $A_{1,1}$ intersects with $\Theta_{i,1}$ since otherwise we would obtain $(h_*\Theta_{1,1}, h_*\Theta_{i,1}) \geq 3$.
On the other hand, assume that $A_{k, 1}$ intersects with both $\Theta_{i_1, 1}$ and $\Theta_{i_2, 1}$ for some $2 \leq k \leq 7$ and $2 \leq i_1 < i_2 \leq 7$.
Then $(h_*\Theta_{i_1,1}, h_*\Theta_{i_2,1}) = 1$ since they are lines.
However, $A_{1, 1}$ also intersects with both $\Theta_{i_1, 1}$ and $\Theta_{i_2, 1}$, which implies that $(h_*\Theta_{i_1,1}, h_*\Theta_{i_2,1}) \geq 2$, a contradiction.
Hence we may assume that $\Theta_{i,1}$ intersects with $A_{1,1}, A_{2i-2,1}, A_{2i-1,1}$ for $2 \leq i \leq 4$.
However, it implies that $(h_*\Theta_{5,1}, h_*\Theta_{i,1}) \geq 2$ for some $2 \leq i \leq 4$, a contradiction.
Therefore $h_*(\Theta_{1,1}) \sim \sO_{\PP^2_k}(1)$.
\hfill $\blacksquare$
\end{clproof}
\begin{cln}\label{cl:typeg-2}
There are coordinates of $\PP^2_k$ such that $\{t_i\}_{1 \leq i \leq 7}$ is the set of $\FF_2$-rational points and $\{D_i\}_{1 \leq i \leq 7}$ is the set of lines defined over $\FF_2$.
\end{cln}
\begin{clproof}
By Claim \ref{cl:typeg-1}, $\{D_i\}_{1 \leq i \leq 7}$ is a set of lines passing through exactly three of $\{t_i\}_{1 \leq i \leq 7}$.
Hence the set $\Sigma \coloneqq \{(i,j) \mid D_i \text{ passes through } t_j\}$ consists of $21$ elements.
On the other hand, distinct two lines cannot share two points.
Combining this fact and $\sharp \Sigma=21$, we conclude that $\{t_i\}_{1 \leq i \leq 7}$ is a set of points contained in exactly three of $\{D_i\}_{1 \leq i \leq 7}$.
Next, let us show that $\{t_i\}_{1 \leq i \leq 7}$ contains four points in general position.
Changing the indices of $\{D_i\}_{1 \leq i \leq 7}$ and $\{t_i\}_{1 \leq i \leq 7}$, we may assume that $D_1$ (resp.\,$D_2$) passes through $t_1$ and $t_2$ (resp.\,$t_1$ and $t_3$).
Since three of $\{D_i\}_{1 \leq i \leq 7}$ passes through $t_2$, it contains the line spanned by $t_2$ and $t_3$, say $D_4$.
Then there is a unique point, say $t_7$, in $\{t_i\}_{1 \leq i \leq 7}$ disjoint from $D_1 \cup D_2 \cup D_4$.
Hence $t_1, t_2, t_3$, and $t_7$ are in general position.
Then there are coordinates of $\PP^2_k$ such that $t_1=[1:0:0]$, $t_2=[0:1:0]$, $t_3=[0:0:1]$ and $t_7=[1:1:1]$.
Then we may assume that $t_4=[1:1:0], t_5=[0:1:1]$ and $t_6=[1:0:1]$.
Since each of $D_i$ is a span of two $\FF_2$-rational point, it is also defined over $\mathbb{F}_2$.
\hfill $\blacksquare$
\end{clproof}
Fix coordinates of $\PP^2_k$ as above.
By construction, $t$ is not contained in $D_i$ for $1 \leq i \leq 7$.
Now define $\zeta \colon \{1, \ldots, 7\} \to I= \{(1,2,4)$, $(1,3,6)$, $(1,5,7)$, $(2,3,5)$, $(2,6,7)$, $(3,4,7)$, $(4,5,6)\}$ which maps $1$, $2$, $3$, $4$, $5$, $6$, and $7$ to $(1,2,4)$, $(1,3,6)$, $(1,5,7)$, $(2,3,5)$, $(2,6,7)$, $(3,4,7)$, and $(4,5,6)$ respectively.
By Claims \ref{cl:typeg-1} and \ref{cl:typeg-2}, we may assume that $D_i$ contains $t_l$ for all $l \in \zeta(i)$ and $1 \leq i \leq 7$.
Then the following hold for $1 \leq i \leq 7$.
\begin{itemize}
\item $\Theta_{i,1}$ is the strict transform by $h$ of the line passing through $t_l$ for all $l \in \zeta(i)$.
\item $\Theta_{i,2}$ is the strict transform by $h$ of the conic passing through $t$ and $t_l$ for all $l \in \{1, \ldots, 7\} \setminus \zeta(i)$.
\item $A_{i, 1}$ is the exceptional divisor over $t_i$.
\item $A_{i, 2}$ is the strict transform by $h$ of the line passing through $t$ and $t_i$.
\item $\Theta_{0,1}$ is the strict transform by $h$ of the cuspidal cubic passing through $t, t_1, \ldots, t_7$ which has a cusp at $t$.
\item The tangent line of $h(\Theta_{i,2})$ at $t$ is independent of the choice of $i$, and $A_{0,1}$ is the strict transform of this line by $h$.
\item $Z$ is obtained by blowing up $\PP^2_k$ at $t_j$ once for $1 \leq j \leq 7$ and at $t$ twice along $h(A_{0,1})$, and the $h$-exceptional divisor over $t$ consists of $A_{0,2}$ and $\Theta_{0,2}$.
\end{itemize}
From these facts, it is easy to check that Figure \ref{matrix(g)} is the intersection matrix of $A_{0, 1}, \ldots, A_{7, 1}$, $A_{0, 2}, \ldots, A_{7,2}$, $\Theta_{0,1}, \ldots, \Theta_{7,1}$, $\Theta_{0,2}, \ldots, \Theta_{7,2}$ in this order.
\end{proof}
\section{Proof of Theorem \ref{smooth, Intro}}
\label{sec:smooth}
This section is devoted to proving Theorem \ref{smooth, Intro}.
First, we show that (NL) $\Rightarrow$ (NB).
\begin{prop}\label{NBtoNL}
Let $X$ be a Du Val del Pezzo surface whose general anti-canonical member is smooth. Then $X$ is log liftable over every Noetherian complete local ring with the residue field $k$.
\end{prop}
\begin{proof}
Let $\pi\colon Y\to X$ be the minimal resolution.
By Proposition \ref{prop:W2lift} (2), it suffices to show that $H^2(X, T_X)=H^2(X, \sO_X)=0$.
Since $-K_X$ is ample, it follows that $H^2(X, \sO_X)\cong H^0(X, \sO_X(K_X))=0$.
Now we show that $H^2(X, T_X)=0$.
By the Serre duality, it follows that
\begin{align*}
H^2(X, T_X)\cong&\mathrm{Hom}_{\sO_X}(T_X, \sO_X(K_X))\\
\cong&\mathrm{Hom}_{\sO_X}(\sO_X(-K_X), \Omega_X^{[1]}),
\end{align*}
where $\Omega_X^{[1]}$ denotes the double dual of $\Omega_X$.
Suppose by contradiction that there exists an injective $\sO_X$-module homomorphism $s\colon \sO_X(-K_X)\hookrightarrow \Omega_X^{[1]}$.
Let $C\in |-K_X|$ be a general member and $s|_C\colon\sO_C(-K_X)\rightarrow \Omega^{[1]}_{X}|_{C}$ be the restriction of $s$ on $C$.
By Lemma \ref{basic} (2), we may assume that $C$ is not contained in the zero locus of $s$. In particular, $s|_C$ is injective.
By assumption, we also may assume that $C$ is a smooth Cartier divisor.
In particular, $X$ is smooth along $C$ and hence $\Omega^{[1]}_{X}|_{C}=\Omega_{X}|_{C}$.
Let $t \colon \sO_C(-K_X) \to \Omega_C$ be the composition of $s|_C \colon \sO_C(-K_X)\hookrightarrow \Omega_{X}|_{C}$ and the canonical map
$\Omega_{X}|_{C} \to \Omega_C$.
By the conormal exact sequence, we obtain the following diagram.
\begin{equation*}
\xymatrix{ & & \sO_C(-K_X) \ar@{.>}[ld] \ar[d]^{s|_C} \ar[rd]^{t} &\\
0\ar[r] &\sO_C(-C) \ar[r] & \Omega_X|_{C} \ar[r] & \Omega_C \ar[r] & 0.}
\end{equation*}
Then $t$ is the zero map since $\sO_C(-K_X)$ is ample and $\Omega_C=\sO_C$.
Hence the above diagram induces an injective $\sO_C$-module homomorphism $\sO_C(-K_X) \hookrightarrow \sO_C(-C)$, but this is a contradiction because $\sO_C(-C)=\sO_C(K_X)$ is anti-ample.
Therefore we obtain the assertion.
\end{proof}
Next, we prove that (ND) $\Rightarrow$ (NL).
\begin{prop}\label{NDtoNL}
Let $X$ be a Du Val del Pezzo surface. Let $R$ be a Noetherian integral domain of characteristic zero with a surjective homomorphism $R \to k$.
If $X$ is log liftable over $R$ via the associated morphism $\alpha \colon \Spec k \to \Spec R$,
then there exists a Du Val del Pezzo surface over $\C$ which has the same Dynkin type, the same Picard rank, and the same degree as $X$.
\end{prop}
\begin{proof}
Let $m$ be the kernel of the homomorphism $R \to k$.
Replacing $R$ with the completion of $R_m$, we may assume that $R$ is a Noetherian complete local ring with residue field $k$.
Thus, by Lemma \ref{equiv} (1), the pair $(Y, E_{\pi})$ lifts to $R$, where $\pi \colon Y \to X$ is the minimal resolution.
We denote by $E_{\pi}\coloneqq\sum_{i=1}^{r} E_i$ the irreducible decomposition.
Let $(\mathcal{Y}, \mathcal{E} \coloneqq \sum_{i=1}^{r} \mathcal{E}_i)$ be an $R$-lifting of $(Y, E_{\pi})$.
We take a subfield $K$ of the field of fractions of $R$ such that $K$ is of finite transcendence degree over $\Q$, and the generic fiber of $\mathcal{Y}$ and that of each $\mathcal{E}_i$ are defined over $K$.
Fix an inclusion $K \subset \C$ and take $\overline{K} \subset \C$ as the algebraic closure of $K$.
For a field extension $K \subset F$, we use the notation $Y_{F} \coloneqq \mathcal{Y}\otimes_{R} F$ and $E_{i,F} \coloneqq \mathcal{E}_i\otimes_{R} F$ for each $i$.
Since the geometrical connectedness are open property by \cite[Th\'eor\`eme 12.2.4 (viii)]{EGAIV3}, $Y_{\C}$ and $E_{i,\C}$ are smooth varieties.
Since $E_{\C} \coloneqq \sum_{i=1}^{r} E_{i,\C}$ has the same intersection matrix as $E_{\pi}$, we have a contraction $\pi_{\C} \colon Y_{\C} \to X_{\C}$ of $E_{\C}$ and $X_{\C}$ has the same Dynkin type as $X$.
By the crepantness of $\pi$ and $\pi_{\C}$, we obtain $K_X^2=K_Y^2=K_{Y_{\C}}^2=K_{X_{\C}}^2$.
Next, we prove that ${X_{\C}}$ is a Du Val del Pezzo surface.
For the sake of contradiction, we assume that $-K_{X_{\C}}$ is not ample. Since $K^2_{X_{\overline{K}}}=K_{X_{\C}}^2>0$, there exists an integral curve $C_0\subset Y_{\C}$ defined over $\overline{K}$ such that $C_0$ is not contained in $E_{\C}$ and $(-K_{Y_{\C}} \cdot C_0) \leq 0$.
We take a finite Galois extension field $L$ of $K$ such that $C_{0}$ is defined over $L$
and write $C \coloneqq \sum_{\sigma\in \mathrm{Gal}(L/K)} \sigma(C_{0})$, which is defined over $K$.
By the choice of $C_0$, there are no components contained in both $C$ and $E_{K}$.
We denote by $\overline{C}$ the closure of $C$ in $\mathcal{Y}$ and define an effective divisor $C_{k} \coloneqq \overline{C}\otimes_{R}k$.
Now, assume that $\Supp{C_{k}}\subset E_{\pi}$.
Then we can write $C_{k}=\sum_{i=1}^r a_iE_{i}$ for some $a_i \geq 0$.
Since $C$ and $E_{K}$ have no common components, we have $C_{k}^2=(C \cdot \sum_{i=1}^r a_i E_{i,K}) \geq 0$.
By the negative definiteness of $E_{\pi}$, we obtain $a_i=0$ for $1 \leq i \leq r$, a contradiction.
Thus there exists an integral curve $C'_{k}\subset C_{k}$ such that $C'_{k}$ is not contained in $E_{\pi}$.
Since $-K_{Y}$ is nef, we have
$0 \leq (-K_Y \cdot C'_k) \leq (-K_{Y}\cdot C_{k}) = (-K_{Y_{K}}\cdot C) = |\mathrm{Gal}(L/K)| (-K_{Y_{\C}}\cdot C_{0}) \leq 0$.
Hence $(-K_Y \cdot C'_{k}) = (-K_X\cdot \pi_{*}(C'_{k})) = 0$, a contradiction with
the ampleness of $-K_{X}$.
Therefore, $X_{\C}$ is a Du Val del Pezzo surface.
Finally, we show that $\rho(X)=\rho(X_{\C})$.
Since $Y$ and $Y_{\C}$ are smooth rational surfaces, we have $\rho(Y_{\C})=10-K_{Y_{\C}}^2=10-K_Y^2=\rho(Y)$. Then we obtain $\rho(X)=\rho(X_{\C})$ because $\pi_{\C}$ contracts the same number of $(-2)$-curves as $\pi$.
\end{proof}
Finally, we prove that (NK) $\Rightarrow$ (NL).
\begin{lem}\label{lem:KV-2}
Let $f\colon Z\to X$ be a birational morphism of normal projective klt surfaces and $A$ an ample $\Z$-divisor on $X$.
Suppose that $(Z, \lceil f^{*}A\rceil-f^{*}A)$ is klt.
Then $H^i(X, \sO_X(-A))=H^i(Z, \sO_Z(-\lceil f^{*}A\rceil))$ for $i\geq 0$.
\end{lem}
\begin{proof}
By \cite[Theorem 2.12]{Tan15}, it follows that $R^if_{*}\sO_Z(K_Z+\lceil f^{*}A\rceil))=0$ for $i \geq 1$.
Then the Leray spectral sequence
\[
E_2^{p,q}=H^{p}(X, R^{q}f_{*}\sO_{Z}(K_{Z}+\lceil f^{*}A \rceil))\Rightarrow E^{p+q}=H^{p+q}(Z,\sO_{Z}(K_{Z}+\lceil f^{*}A \rceil))
\]
gives $H^i(X, f_{*}\sO_Z(K_Z+\lceil f^{*}A \rceil))\cong H^i(Z, \sO_{Z}(K_{Z}+\lceil f^{*}A \rceil))$.
Since $X$ is klt, we obtain
\begin{align*}
K_{Z}+\lceil f^{*}A \rceil&=\lceil K_Z-f^{*}K_X+f^{*}(K_X+A) \rceil\\
&=\lfloor f^{*}(K_X+A) \rfloor +F
\end{align*}
for some effective $f$-exceptional $\Z$-divisor $F$.
Then $H^i(X, f_{*}\sO_Z(K_Z+\lceil f^{*}A \rceil))=H^i(X, \sO_X(K_X+A))$ by the projection formula.
Hence the assertion follows from the Serre duality for Cohen-Macaulay sheaves \cite[Theorem 5.71]{KM98}.
\end{proof}
\begin{prop}\label{lem:KV-3}
Let $X$ be a normal projective surface and $A$ an ample $\Q$-Cartier $\Z$-divisor.
Suppose that there exists a log resolution $f\colon Z\to X$ such that $(Z, E_{f})$ lifts to $W_2(k)$.
Then $H^1(X, \sO_X(-A))=0$.
\end{prop}
\begin{proof}
By Lemma \ref{lem:KV-2}, it follows that $H^1(X, \sO_X(-A))=H^1(Z, \sO_Z(-\lceil f^{*}A\rceil))$.
Take an $f$-exceptional effective $\Q$-divisor $F$ such that $\lceil f^{*}A-F\rceil=\lceil f^{*}A\rceil$ and $f^{*}A-F$ is ample. Since $\Supp(\lceil f^{*}A-F\rceil-(f^{*}A-F))$ is contained in $E_{f}$,
Theorem \ref{loglift vanishing} shows that $H^1(Z, \sO_Z(-\lceil f^{*}A-F\rceil))=0$. Hence we get the assertion.
\end{proof}
Now we can prove Theorem \ref{smooth, Intro}.
\begin{proof}[Proof of Theorem \ref{smooth, Intro}]
The assertions (1), (2), and (3) follow from Propositions \ref{NBtoNL}, \ref{NDtoNL}, and \ref{lem:KV-3} respectively.
\end{proof}
\section{Dynkin types}\label{sec:sing}
In this section, we determine the Dynkin types of Du Val del Pezzo surfaces satisfying (NB).
By Lemma \ref{basic}, such a del Pezzo surface is of degree at most two, and $p=2$ or $3$.
First, we treat the case where the degree is one.
\begin{prop}\label{prop:deg=1}
Let $X$ be a Du Val del Pezzo surface with $K_X^2=1$ and $\pi \colon Y \to X$ the minimal resolution.
Take $g \colon Z \to Y$ as the blow-up at the base point of $|-K_Y|$ and $f \colon Z \to \PP^1_k$ the genus one fibration defined by $|-K_Z|$.
\[
\xymatrix{
Z \ar[r]^{g}\ar[rrd]^f & Y \ar[r]^{\pi} & X \ar@{..>}[d]^{\phi_{|-K_X|}} \\
& & \PP^1_k \\
}
\]
Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] $X$ satisfies (NB) if and only if $f$ is a quasi-elliptic fibration.
\item[\textup{(2)}] For another Du Val del Pezzo surface $X'$ of degree one, take $\pi' \colon Y' \to X'$ and $g' \colon Z' \to Y'$ as above.
Then $X \cong X'$ if and only if $Z \cong Z'$.
\item[\textup{(3)}] Suppose that $p=3$ and $Z$ is a rational quasi-elliptic surface.
Then $Z$ is of type (1) (resp.\ (2), (3)) if and only if $\Dyn(X)=E_8$ (resp.\ $A_2+E_6, 4A_2$).
\item[\textup{(4)}] Suppose that $p=2$ and $Z$ is a rational quasi-elliptic surface.
Then $Z$ is of type (a) (resp.\ (b), (c), (d), (e), (f), (g)) if and only if $\Dyn(X)=E_8$ (resp.\ $D_8, A_1+E_7, 2D_4, 2A_1+D_6, 4A_1+D_4, 8A_1$)
\end{enumerate}
\end{prop}
\begin{proof}
\noindent (1): A general member of $|-K_Y|$ is isomorphic to its image by $\pi$ since it is disjoint from the exceptional divisor $E_\pi$ of $\pi$.
On the other hand, the base locus of $|-K_{Y}|$ consists of one point, say $y$.
Since any two members of $|-K_{Y}|$ intersect transversely with each other at $y$, each $f$-fiber is isomorphic to its image on $Y$.
Hence a general $f$-fiber is isomorphic to its image on $X$, and the assertion holds.
\noindent (2):
Take $f' \colon Z' \to \PP^1_k$ as the morphism given by $|-K_{Z'}|$.
Suppose that there is an isomorphism $X \cong X'$.
Then it ascends to an isomorphism $Z \cong Z'$ since the construction of $\pi$, $\pi'$, $g$, and $g'$ are canonical.
On the other hand, suppose that there is an isomorphism $\sigma \colon Z \cong Z'$.
Then this isomorphism is compatible with the genus one fibration structures since $-K_Z \sim \sigma^*(-K_Z')$.
In particular, $\sigma$ maps each $f$-section to an $f'$-section.
Since $E_g$ (resp.\ $E_{g'}$) is an $f$-section (resp.\ an $f'$-section) and the Mordell-Weil group $\MW(Z)$ acts on the set of $f$-sections transitively, we may assume that $\sigma$ maps $E_g$ to $E_{g'}$.
Hence it descends to an isomorphism $\sigma_Y \colon Y \cong Y'$.
Since both $\pi$ (resp.\ $\pi'$) is the contraction of all the $(-2)$-curves on $Y$ (resp.\ $Y'$), $\sigma_Y$ also descends to the desired isomorphism $\sigma_X \colon X \cong X'$.
\noindent (3) and (4): Since $\MW(Z)$ acts on the set of $f$-sections transitively, we may assume that $E_g$ is the section $O$ in Figures \ref{fig:dual(1)-(3)}--\ref{fig:dual(g)}.
Hence the assertions follows from Figures \ref{fig:dual(1)-(3)}--\ref{matrix(g)} and Tables \ref{tab:type(3)} and \ref{typef}.
\end{proof}
We will use the following proposition in Section \ref{sec:singliftable}.
\begin{prop}\label{4A_2NB}
Let $X$ be a Du Val del Pezzo surface with $\Dyn(X)=4A_2$. Then $X$ satisfies (NB) if and only if $p=3$.
\end{prop}
\begin{proof}
The only if part follows from Proposition \ref{prop:deg=1}.
To show the other direction, we suppose that $p=3$.
Let us take $Y$ and $Z$ as in Proposition \ref{prop:deg=1}.
Suppose by contradiction that a general anti-canonical member of $X$ is smooth.
Then $Z$ is an extremal rational elliptic surface with four singular fibers.
By \cite[Theorem 2.1]{Lang1}, its singular fibers are $(\textup{I}_8, \textup{I}_2, \textup{I}_1, \textup{I}_1)$, $(\textup{I}_5, \textup{I}_5, \textup{I}_1, \textup{I}_1)$, or $(\textup{I}_4, \textup{I}_4, \textup{I}_2, \textup{I}_2)$.
However, this implies that $\Dyn(X)=A_1+A_7, 2A_4$, or $2A_1+2A_3$, a contradiction.
\end{proof}
Next, we treat the case where the degree is two.
The following proposition claims that the anti-canonical double covering must be purely inseparable.
\begin{prop}\label{sep}
Let $X$ be a Du Val del Pezzo surface with $K_X^2=2$.
Suppose that the anti-canonical double covering $\phi_{|-K_X|}\colon X \to \PP_k^2$ is separable.
Then a general anti-canonical member is smooth.
\end{prop}
\begin{proof}
Take the minimal resolution $\pi \colon Y \to X$.
Let $t \in \PP^2_k$ be a general point and $V \subset |-K_Y|$ the pullback of the pencil of lines in $\PP^2_k$ passing through $t$.
Then the base locus of $V$ consists of two points, say $y_1$ and $y_2$, such that there are no $(-1)$-curves passing through $y_1$ or $y_2$ because $t$ is general and there exist only finitely many $(-1)$-curves on $Y$.
Let $g \colon Z \to Y$ be the blow-up at $y_1$ and $y_2$, and $E_i$ the $g$-exceptional divisor over $y_i$ for $i \in \{1, 2\}$.
Then $g$ gives a resolution $f\colon Z \to \PP^1_k$ of the indeterminacy of the pencil $\phi_V \colon Y \dashrightarrow \PP^1_k$. Since any two members of $V$ intersect transversely at $y_1$ and $y_2$,
a general $f$-fiber is isomorphic to its image on $Y$.
\[
\xymatrix{
Z \ar[r]^{g}\ar[rrd]^f & Y \ar[r]^{\pi} & X \ar[r]^{\phi_{|-K_X|}}\ar@{..>}[d]^{\phi_{V}} & \PP^2_k \\
& & \PP^1_k & \\
}
\]
Now let us show that a general member of $|-K_X|$ is smooth.
Suppose by contradiction that members of $|-K_X|$ are all singular.
Then $Z$ is a quasi-elliptic surface, and $E_1$ and $E_2$ are $f$-sections by the same arguments as in Proposition \ref{prop:deg=1}.
Since there are no $(-1)$-curves on $Y$ which pass through $y_1$ or $y_2$,
each $(-2)$-curves in $Z$ either intersects with both $E_1$ and $E_2$ or is disjoint from both $E_1$ and $E_2$.
However, there is no such a choice of two sections by Figures \ref{fig:dual(1)-(3)}--\ref{matrix(g)} and Tables \ref{tab:type(3)} and \ref{typef}, a contradiction.
\end{proof}
\begin{prop}\label{insep}
Let $X$ be a Du Val del Pezzo surface with $K_X^2=2$ satisfying (NB).
Then $p=2$ and $\Dyn(X)=E_7$, $A_1+D_6$, $3A_1+D_4$, or $7A_1$.
\end{prop}
\begin{proof}
By Proposition \ref{sep}, the anti-canonical double covering $\phi_{|-K_X|}\colon X \to \PP^2_k$ is purely inseparable.
In particular, we have $p=2$.
Take $\pi, t$ and $V$ as in Proposition \ref{sep}.
By the generality of $t$, the base locus of $V$ consists of one point, say $y$, and no $(-1)$-curves pass through $y$.
For general two members $C_1$ and $C_2$ of $V$, they intersect with each other at $y$ with multiplicity two since $\phi_{|-K_X|}$ is a homeomorphism.
Moreover, one of them is smooth at $y$ since otherwise $2=K_Y^2=(C_1 \cdot C_2) \geq 4$.
Thus general members of $V$ are smooth at $y$, and have the same tangent direction at $y$.
Hence there is a point $y'$ infinitely near $y$ such that the blow-up $g \colon Z \to Y$ at $y$ and $y'$ gives a resolution $f\colon Z \to \PP^1_k$ of indeterminacy of the pencil $\phi_V \colon Y \dashrightarrow \PP^1_k$.
Since a general member of $V$ is smooth at $y$, a general $f$-fiber is isomorphic to its image on $X$.
In particular, $Z$ is a quasi-elliptic surface.
By construction, $E_g$ consists of a $(-1)$-curve $E_1$ and a $(-2)$-curve $E_2$.
In particular, $E_1$ is an $f$-section and $E_2$ is contained in a reducible $f$-fiber.
Suppose that the $f$-fiber containing $E_2$ has simple normal crossing support.
Then there is another $(-2)$-curve $C$ intersecting with $E_2$.
Since $C$ and $E_2$ are contained in the same $f$-fiber, $E_1$ is disjoint from $C$.
This implies, however, $g_*C$ is a $(-1)$-curve passing through $y$, a contradiction with the choice of $y$.
Hence $E_2$ is contained in a reducible $f$-fiber whose support is not simple normal crossing.
Theorem \ref{q-ell} now shows that $E_2$ is contained in a reducible $f$-fiber of type III, where we use Kodaira's notation, and $Y$ is one of the types (c), (e), (f), and (g) in Table \ref{q-ell2}.
By Figures \ref{fig:dual(a)-(e)}--\ref{matrix(g)} and Table \ref{typef}, we conclude that $\Dyn (X) = E_7$, $A_1+D_6$, $3A_1+D_4$, or $7A_1$.
\end{proof}
Finally, let us show that there are several constructions of Du Val del Pezzo surfaces of degree two satisfying (NB).
\begin{lem}\label{lem:deg=2pre}
Let $X$ be a del Pezzo surface satisfying (NB) such that $\Dyn(X)=E_7, A_1+D_6, 3A_1+D_4$, or $7A_1$.
Let $Y$ be the minimal resolution of $X$.
Then the following hold.
\begin{enumerate}
\item[\textup{(0)}] $K_X^2=2$ and $p=2$.
\item[\textup{(1)}] For each point $t \in Y$ not contained in any negative rational curves, there is a rational quasi-elliptic surface $Z$, an irreducible component $T$ of reducible fiber of type III, and a section $S$ of $Z$ intersecting with $T$ such that $Y$ is given from $Z$ by contracting $S \cup T$ to $t$.
\item[\textup{(2)}] If $\Dyn(X)=E_7$ (resp.\ $A_1+D_6, 3A_1+D_4, 7A_1$), then $Z$ as in the assertion (1) is of type (c) (resp.\ (e), (f), (g)).
\item[\textup{(3)}] If $\Dyn(X)=E_7, A_1+D_6$, or $3A_1+D_4$, then the union of the negative rational curves on $Y$ is a simple normal crossing divisor.
Moreover, Figure \ref{fig:dualE_7-3A_1D_4} is the dual graph of the configuration of the negative rational curve, where black nodes (resp. white nodes) corresponds to a $(-1)$-curve (resp.\ a $(-2)$-curve).
\item[\textup{(4)}] If $\Dyn(X)=7A_1$, then there are exactly seven $(-1)$-curves and seven $(-2)$-curves whose intersection matrix is as in Figure \ref{matrix7A1}.
\item[\textup{(5)}] If $\Dyn(X)=E_7$, then $Y$ is also obtained from the rational quasi-elliptic surface of type (a) by blowing down $O$ and $\Theta_{\infty, 0}$ in Type (a) of Figure \ref{fig:dual(a)-(e)}.
\item[\textup{(6)}] If $\Dyn(X)=A_1+D_6$, then $Y$ is also obtained from the rational quasi-ellptic surface of type (b) (resp.\ (c)) by blowing down $O$ and $\Theta_{\infty, 0}$ (resp.\ $O$ and $\Theta_{0, 0}$) in Type (b) (resp.\ (c)) of Figure \ref{fig:dual(a)-(e)}.
\item[\textup{(7)}] If $\Dyn(X)=3A_1+D_4$, then $Y$ is also obtained from a rational quasi-ellptic surface of type (d) (resp.\ (e)) by blowing down $O$ and $\Theta_{0, 0}$ (resp.\ $O$ and $\Theta_{1, 2}$) in Type (d) (resp.\ (e)) of Figure \ref{fig:dual(a)-(e)}.
\item[\textup{(8)}] If $\Dyn(X)=7A_1$, then $Y$ is also obtained from a rational quasi-ellptic surface of type (f) by blowing down $O$ and $\Theta_{1, 0}$ in Figure \ref{fig:dual(f)}.
\end{enumerate}
\end{lem}
\begin{figure}[htbp]
\captionsetup[subfigure]{labelformat=empty}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\label{fig:dualE7}
\begin{tikzpicture}
\draw[thin ](0,-0.2)--(0,-0.8);
\draw[thin ](0,-1.2)--(0,-1.8);
\draw[thin ](0,-2.2)--(0,-2.8);
\draw[thin ](0,-3.2)--(0,-3.8);
\draw[thin ](0,-4.2)--(0,-4.8);
\draw[thin ](0,-5.2)--(0,-5.8);
\draw[thin ](-0.2,-4)--(-0.8,-4);
\fill (0,0) circle (2pt);
\draw[] (0,-1)circle(2pt);
\draw[] (0,-2)circle(2pt);
\draw[] (0,-3)circle(2pt);
\draw[] (0,-4)circle(2pt);
\draw[] (0,-5)circle(2pt);
\draw[] (0,-6)circle(2pt);
\draw[] (-1,-4)circle(2pt);
\end{tikzpicture}
\caption{Type $E_7$}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.3\textwidth}
\centering
\label{fig:dualA1D6}
\begin{tikzpicture}
\draw[thin ](-0.8,-3)--(-0.2,-3);
\draw[thin ](-1.1,-2.9)--(-1.9,-2.1);
\draw[thin ](-2.1,2.1)--(-2.9,2.9);
\draw[thin ](-2.1,-2.1)--(-2.9,-2.9);
\draw[thin ](-2,1.2)--(-2,1.8);
\draw[thin ](-2,0.2)--(-2,0.8);
\draw[thin ](-2,-0.2)--(-2,-0.8);
\draw[thin ](-2,-1.2)--(-2,-1.8);
\fill (-2,2) circle (2pt);
\fill (0,-3) circle (2pt);
\draw[] (-1,-3)circle(2pt);
\draw[] (-2,-2)circle(2pt);
\draw[] (-2,1)circle(2pt);
\draw[] (-2,0)circle(2pt);
\draw[] (-2,-1)circle(2pt);
\draw[] (-3,3)circle(2pt);
\draw[] (-3,-3)circle(2pt);
\end{tikzpicture}
\caption{Type $A_1+D_6$}
\end{subfigure}
\quad
\begin{subfigure}[b]{0.3\textwidth}
\centering
\label{fig:dual3A1D4}
\begin{tikzpicture}
\draw[thin ](-0.1,1.9)--(-0.9,1.1);
\draw[thin ](0,1.8)--(0,1.2);
\draw[thin ](0.1,1.9)--(0.9,1.1);
\draw[thin ](-1,0.8)--(-1,0.2);
\draw[thin ](0,0.8)--(0,0.2);
\draw[thin ](1,0.8)--(1,0.2);
\draw[thin ](-1,-0.2)--(-1,-0.8);
\draw[thin ](0,-0.2)--(0,-0.8);
\draw[thin ](1,-0.2)--(1,-0.8);
\draw[thin ](-0.1,-1.9)--(-0.9,-1.1);
\draw[thin ](0,-1.8)--(0,-1.2);
\draw[thin ](0.1,-1.9)--(0.9,-1.1);
\fill (1,0) circle (2pt);
\fill (0,0) circle (2pt);
\fill (-1,0) circle (2pt);
\fill (0,-2) circle (2pt);
\draw[] (1,1)circle(2pt);
\draw[] (0,1)circle(2pt);
\draw[] (-1,1)circle(2pt);
\draw[] (0,2)circle(2pt);
\draw[] (1,-1)circle(2pt);
\draw[] (0,-1)circle(2pt);
\draw[] (-1,-1)circle(2pt);
\node(a)at(0,-3){\phantom{$O$}};
\end{tikzpicture}
\caption{Type $3A_1+D_4$}
\end{subfigure}
\caption{Dual graphs of negative rational curves in a Du Val del Pezzo surface of type $E_7, A_1+D_6$, or $3A_1+D_4$ satisfying (NB)}
\label{fig:dualE_7-3A_1D_4}
\end{figure}
\begin{figure}[htbp]
\centering
\begin{align*}
{\fontsize{7pt}{4}\selectfont
\left(\!\!\!\!
\vcenter{
\xymatrix@=-1.8ex{
&&&&&&&&\ar@{-}[ddddddddddddddddd]&&&&&&&& \\
& & & & & & & && & & & & & & &\\
&-1&0 &0 &0 &0 &0 &0 &&1 &1 &1 &0 &0 &0 &0 &\\
& &-1&0 &0 &0 &0 &0 &&1 &0 &0 &1 &1 &0 &0 &\\
& & &-1&0 &0 &0 &0 &&0 &1 &0 &1 &0 &1 &0 &\\
& & & &-1&0 &0 &0 &&1 &0 &0 &0 &0 &1 &1 &\\
& & & & &-1&0 &0 &&0 &0 &1 &1 &0 &0 &1 &\\
& & & & & &-1&0 &&0 &1 &0 &0 &1 &0 &1 &\\
& & & & & & &-1&&0 &0 &1 &0 &1 &1 &0 &\\
\ar@{-}[rrrrrrrrrrrrrrrr]& &&&&&&&&& &&&&&&&&&\\
& & & & & & & &&-2&0 &0 &0 &0 &0 &0 &\\
& & & & & & & && &-2&0 &0 &0 &0 &0 &\\
& & & & & & & && & &-2&0 &0 &0 &0 &\\
& & & & & & & && & & &-2&0 &0 &0 &\\
& & & & & & & && & & & &-2&0 &0 &\\
& & & & & & & && & & & & &-2&0 &\\
& & & & & & & && & & & & & &-2&\\
& & & & & & & && & & & & & & &
}
}\!\!\!\!\right)
}
\end{align*}
\caption{The intersection matrix of negative rational curves in a Du Val del Pezzo surface of type $7A_1$
}
\label{matrix7A1}
\end{figure}
\begin{proof}
The assertion (0) follows from Lemma \ref{basic} and Propositions \ref{prop:deg=1} and \ref{insep}.
The essentially same proof as in that of Proposition \ref{insep} shows the assertions (1) and (2).
We see at once that the contraction of $O$ and $\Theta_{\infty, 0}$ in Types (c) and (e) of Figure \ref{fig:dual(a)-(e)} and Figure \ref{fig:dual(f)} gives the dual graph as in Figure \ref{fig:dualE_7-3A_1D_4}, and the assertion (3) holds.
Suppose that $\Dyn(X)=7A_1$ and we follow the notation of the proof of Lemma \ref{lem:typeg}.
By contracting $A_{0,2}$ and $\Theta_{0,2}$ in Figure \ref{fig:dual(g)}, $A_{i, 1}$, $A_{i, 2}$, $\Theta_{i,1}$, $\Theta_{i,2}$ $A_{0, 1}$, and $\Theta_{0,1}$ become a $(-1)$-curve, a $(0)$-curve, a $(-2)$-curve, a $(0)$-curve, a $(1)$-curve, and a cuspidal curve of self intersection number two respectively for $1 \leq i \leq 7$.
Hence the assertion (4) holds.
Finally, let us show the assertions (5)--(8).
Let $E$ be a $(-1)$-curve in $Y$ and $t \in E$ a point not contained in any $(-2)$-curve.
Then the blow-up $Y_t$ of $Y$ at $t$ is a weak del Pezzo surface whose anti-canonical members are all singular.
Hence $Y_t$ is the blow-down of a section in a rational quasi-elliptic surface $Z_t$.
Now suppose that $\Dyn(X)=E_7$ and let $E$ correspond the black node in Type $E_7$ of Figure \ref{fig:dualE_7-3A_1D_4}.
Then $Y_t$ contains eight $(-2)$-curves whose configuration is the Dynkin diagram $E_8$.
By Proposition \ref{prop:deg=1} (4), $Z_t$ is of type (a), and hence the assertion (5) holds.
Similarly, if $\Dyn(X)=A_1+D_6$ (resp.\ $3A_1+D_4$), then by Type $A_1+D_6$ (resp.\ $3A_1+D_4$) of Figure \ref{fig:dualE_7-3A_1D_4}, there are two possibility of the number of $(-2)$-curves intersecting with $E$, and $Y_t$ contains eight $(-2)$-curves whose configuration is the Dynkin diagram $D_8$ or $A_1+E_7$ (resp.\ $2D_4$ or $2A_1+D_6$).
On the other hand, if $\Dyn(X)=7A_1$, then Figure \ref{matrix7A1} shows that $E$ is unique up to symmetry, and $Y_t$ contains eight $(-2)$-curves whose configuration is the Dynkin diagram $D_4+4A_1$.
Therefore Proposition \ref{prop:deg=1} (4) shows assertions (6)--(8).
\end{proof}
\section{Isomorphism classes and automorphism groups}
\label{sec:singisom}
In this section, we determine the isomorphism classes and the automorphism groups of Du Val del Pezzo surfaces satisfying (NB).
\subsection{Characteristic three}
In this subsection, we treat the case where $p=3$.
\begin{prop}\label{prop:p=3isom}
Let $X$ be a Du Val del Pezzo surface satisfying (NB) in $p=3$ and $\pi \colon Y \to X$ be the minimal resolution.
Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] $K_X^2=1$.
\item[\textup{(2)}] $\Dyn(X) =E_8$, $A_2+E_6$, or $4A_2$.
Moreover, $X$ is uniquely determined up to isomorphism by $\Dyn(X)$.
\item[\textup{(3)}] If $\Dyn(X)=E_8$, then $Y$ is constructed from $\PP^2_{k, [x:y:z]}$ by blowing up at $[0:1:0]$ eight times along $\{x^3+y^2z=0\}$.
Moreover, each negative rational curve is either exceptional over $\PP^2_{k}$ or the strict transform of $\{z=0\}$.
\item[\textup{(4)}] If $\Dyn(X)=A_2+E_6$, then $Y$ is constructed from $\PP^2_{k, [x:y:z]}$ by blowing up at $[0:0:1]$ twice along $\{y=0\}$, at $[0:1:1]$ three times along $\{y=z\}$, and at $[0:1:0]$ three times along $\{z=0\}$.
Moreover, each negative rational curve is either exceptional over $\PP^2_{k}$ or the strict transform of $\{y=0\}$, $\{y=z\}$, $\{z=0\}$, or $\{x=0\}$.
\item[\textup{(5)}] If $\Dyn(X)=4A_2$, then $Y$ is constructed from $\PP^2_{k, [x:y:z]}$ by blowing up all the $\FF_3$-rational points on $\{z \neq 0\}$ except $[0:0:1]$.
Moreover, each negative rational curve on $Y$ is either exceptional over $\PP^2_{k}$ or the strict transform of lines passing through two of the eight points as above.
\item[\textup{(6)}] $Y$ and each negative rational curve on $Y$ are defined over $\FF_3$.
\end{enumerate}
\end{prop}
\begin{proof}
\noindent (1): The assertion follows from Lemma \ref{basic} and Proposition \ref{insep}.
\noindent (2): The assertion follows from Proposition \ref{prop:deg=1} and Theorem \ref{thm:q-ell3}.
\noindent (3)--(5): Let $g \colon Z \to Y$ be the blow-up at the base point of $|-K_Y|$.
By Proposition \ref{prop:deg=1} (3), $Z$ is a rational quasi-elliptic surface of type (1), (2), and (3) when $\Dyn(X) = E_8, A_2+E_6$, and $4A_2$ respectively.
Since the $\MW(Z)$-action on the set of sections of $Z$ is transitive, we may assume that $g$ is the contraction of the section $O$ in Figure \ref{fig:dual(1)-(3)}.
Take $h \colon Z \to \PP^2_k$ as in Lemma \ref{lem:Ito2Rem4}.
Then the assertion follows from the description of the induced morphism $h' \colon Y \to \PP^2_k$ and the image of negative rational curves on $Z$ via $h$.
\noindent (6): The assertions directly follow from the assertions (3)--(5).
\end{proof}
\begin{cor}\label{cor:deg=1p=3auto}
Let $X$ be a Du Val del Pezzo surface satisfying (NB) in $p=3$.
When $\Dyn(X)=E_8$ (resp.\ $A_2+E_6$, $4A_2$), $\Aut X$ is isomorphic to
\begin{align*}
\left\{
\begin{psmallmatrix}
a&0&c\\
0&1&0\\
0&0&a^3
\end{psmallmatrix}
\in \PGL(3, k) \middle| a\in k^*, c\in k
\right\} \text{ (resp.\ } k^* \times \ZZ/2\ZZ, \mathrm{GL}(2, \FF_3)\text{)}.
\end{align*}
\end{cor}
\begin{proof}
We follow the notation of the proof of Proposition \ref{prop:p=3isom}.
Since every morphism from $Y$ to $X$ factors through the minimal resolution $\pi$, we have a canonical homomorphism $\varphi \colon \Aut X \to \Aut Y$ such that $\sigma \circ \pi =\pi \circ \varphi(\sigma)$ for all $\sigma \in \Aut X$.
On the other hand, $\pi$ is the contraction of all the $(-2)$-curves on $Y$.
Since each automorphism of $Y$ fixes the union of $(-2)$-curves, we also have a canonical homomorphism $\psi \colon \Aut Y \to \Aut X$, which is the inverse of $\varphi$.
Hence $\Aut X \cong \Aut Y$.
First, suppose that $\Dyn(X)=E_8$.
By Type (1) of Figure \ref{fig:dual(1)-(3)}, each negative rational curve on $Y$ is $g(\Theta_{\infty, i})$ for some $0 \leq i \leq 8$.
The $\Aut Y$-action on $Y$ fixes the unique $(-1)$-curve $g(\Theta_{\infty, 0})$.
It also fixes $g(\Theta_{\infty, 1})$, which is the unique $(-2)$-curve intersecting with $g(\Theta_{\infty, 0})$.
By a similar argument, it fixes each negative rational curve.
Hence the $\Aut Y$-action descends to $\PP^2_k$ via $h'$.
In particular, $\Aut Y$ is contained in the subgroup $G$ of $\PGL(3, k) \cong \Aut \PP^2_k$ fixing $h'_*|-K_Y|$.
On the other hand, since $h_*|-K_Z|=h'_*|-K_Y|$ and $|-K_Z|$ are base point free, $Z$ is the minimal resolution of indeterminacy of $h'_*|-K_Y|$.
In particular, the $G$-action on $\PP^2_k$ ascends to $Z$.
Since $Z$ has a unique section, it descends to $Y$.
Therefore $\Aut Y \cong G$.
Since $h(F)=\{x^3+y^2z=0\}$ and $h(\Theta_{\infty, 8})=\{z=0\}$, we have $h'_*|-K_Y| = \{sz^3+t(x^3+y^2z)=0 \mid [s:t] \in \PP^1_k\}$.
Let
\begin{align*}
A=
\begin{psmallmatrix}
a&b&c\\
d&e&f\\
g&h&i
\end{psmallmatrix}
\end{align*}
be an element of $G$.
Since $\Aut Y$ fixes $h'(g(\Theta_{\infty, 8}))= h(\Theta_{\infty, 8})=\{z=0\}$, we have $g=h=0$ and $i \neq 0$.
On the other hand, we have
\begin{align*}
A \cdot (x^3+y^2z) = (a^3x^3+b^3y^3+c^3z^3)+(dx+ey+fz)^2(iz) \in h'_*|-K_Y|.
\end{align*}
Since the coefficients of $y^3$, $x^2z$, and $yz^2$ must be zero, we have $b=0$, $d=0$, $e \neq 0$, and $f=0$.
Since the coefficient of $x^3$ must coincide with that of $y^2z$, we have $a^3=e^2i$.
Fixing $e=1$, we obtain the assertion.
Next, suppose that $\Dyn(X)=A_2+E_6$.
By Type (2) of Figure \ref{fig:dual(1)-(3)}, each negative rational curve on $Y$ is $g(P)$, $g(2P)$, $g(\Theta_{0, i})$ for some $0 \leq i \leq 2$, or $g(\Theta_{\infty, i})$ for some $0 \leq i \leq 6$.
The $\Aut Y$-action on $Y$ fixes $g(\Theta_{0, 0})$, which is the unique $(-1)$-curve intersecting with one $(-1)$-curve and two $(-2)$-curves.
Then it also fixes $g(\Theta_{\infty, 2})$, $g(\Theta_{\infty, 1})$, and $g(\Theta_{\infty, 0})$.
On the other hand, there are exactly two $(-1)$-curves on $Y$ intersecting with no other $(-1)$-curves, which are $g(P)$ and $g(2P)$.
Then the $\Aut Y$-action on $Y$ fixes $g(P) \cup g(2P)$.
Similarly, it fixes $g(\Theta_{\infty, 4}) \cup g(\Theta_{\infty, 6})$, $g(\Theta_{\infty, 3}) \cup g(\Theta_{\infty, 5})$, and $g(\Theta_{0, 1}) \cup g(\Theta_{0, 2})$.
Hence the $\Aut Y$-action descends to $\PP^2_k$ via $h'$.
In particular, the $\Aut Y$-action on $\PP^2_k$ fixes $h(O)=[0:0:1]$, $h(\Theta_{\infty,0})=\{x=0\}$, and $h(\Theta_{0,1}) \cup h(\Theta_{0,2})=\{z(y+z)=0\}$.
In particular, it fixes $h(\Theta_{0,1}) \cap h(\Theta_{0,2})=[1:0:0]$ and $h(\Theta_{\infty,0}) \cap (h(\Theta_{0,1}) \cup h(\Theta_{0,2}))=\{[0:1:0], [0:1:1]\}$.
On the other hand, by construction, every automorphism on $\PP^2_k$ ascends to $Y$ via $h'$ if they fix $[0:0:1]$, $[1:0:0]$, and $\{[0:1:0], [0:1:1]\}$.
Hence $\Aut Y$ is isomorphic to
\begin{align*}
\left\{
\begin{psmallmatrix}
a&0&0\\
0&1&0\\
0&h&i
\end{psmallmatrix}
\in \PGL(3, k) \middle| a \in k^*, (h,i)=(0,1) \text{ or } (1, -1)
\right\} \cong k^* \times \ZZ/2\ZZ.
\end{align*}
Finally, suppose that $\Dyn(X)=4A_2$.
By Type (3) of Figure \ref{fig:dual(1)-(3)}, each $(-1)$-curve on $Y$ is either $g(\Theta_{i, 0})$ for some $i=0, -1, 1, \infty$, or the image of a section.
The former intersects with another $(-1)$-curve at $g(O)$ and the latter intersects with no other $(-1)$-curves.
Hence the $\Aut Y$-action on $Y$ fixes $g(O)$ and $E_{h'}$.
In particular, it descends to $\PP^2_k$ via $h'$ and fixes $h(O)=[0:0:1]$ and $h(E_h)$, which are $\FF_3$-rational points not contained in $\{z=0\}$.
On the other hand, by construction, every automorphism on $\PP^2_k$ ascends to $Y$ via $h'$ if they fix $[0:0:1]$ and $h(E_h)$.
Hence $\Aut Y$ is isomorphic to the subgroup of $\PGL(3, \FF_3)$ which fixes $\{z=0\}$ and $[0:0:1]$, which is $\mathrm{GL}(2, \FF_3)$.
Combining these arguments, we complete the proof.
\end{proof}
\subsection{Characteristic two}
In this subsection, we always assume that $p=2$.
First let us show that, when the degrees are two, Dynkin types determine the isomorphism classes of Du Val del Pezzo surfaces satisfying (NB).
\begin{prop}\label{E_7}
The minimal resolution of each del Pezzo surface of type $E_7$ satisfying (NB) is constructed from $\PP^2_{k, [x:y:z]}$ by blowing up at $[0:1:0]$ seven times along $\{x^3+y^2z=0\}$.
In particular, there is a unique del Pezzo surface of type $E_7$ satisfying (NB).
\end{prop}
\begin{proof}
We follow the notation of Type (a) of Figure \ref{fig:dual(a)-(e)}.
Let $Z$ be the rational quasi-elliptic surface of type (a) and $F$ a general fiber.
Let $g \colon Z \to Y$ be the contraction of $O$ and $\Theta_{\infty, 0}$ and $\pi \colon Y \to X$ the contraction of all $(-2)$-curves.
Then the desired del Pezzo surface must be $X$ by Lemma \ref{lem:deg=2pre} (5).
Take $h \colon Z \to \PP^2_k$ and coordinates of $\PP^2_k$ as in Lemma \ref{lem:Ito2Rem4}.
Let $h' \colon Y \to \PP^2_k$ be the morphism induced by $h$.
Then $h'$ is the blow-up at $h(O)=[0:1:0]$ seven times along $h(F)=\{x^3+y^2z=0\}$.
Hence it suffices to show that $X$ satisfies (NB).
Since $\pi$ and $h'$ is an isomorphism around a general member of $|-K_X|$, we are reduced to proving that $h'_*|-K_Y|$ has only a singular member.
By construction, $h'_*|-K_Y|$ consists of cubic curves intersecting with $h(F)=\{x^3+y^2z=0\}$ at $h(O)=[0:1:0]$ with multiplicity seven.
Then it is generated by $\{x^3+y^2z=0\}, \{z^3=0\}$, and $\{xz^2=0\}$.
The Jacobian criterion now shows that $h'_*|-K_{Y'}|$ has only a singular member, and the assertion holds.
\end{proof}
\begin{cor}\label{E_7aut}
Let $X$ be the del Pezzo surface of type $E_7$ satisfying (NB) and $\pi \colon Y \to X$ the minimal resolution.
Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] $Y$ and each negative rational curve on $Y$ are defined over $\FF_2$.
\item[\textup{(2)}] $\Aut X$ is isomorphic to
\begin{align*}
\left\{
\begin{psmallmatrix}
a&0&d^2a\\
d&1&f\\
0&0&a^3
\end{psmallmatrix}
\in \PGL(3, k) \middle| a \in k^*, d \in k, f \in k
\right\}.
\end{align*}
\end{enumerate}
\end{cor}
\begin{proof}
We follow the notation of the proof of Proposition \ref{E_7}.
\noindent (1):
By the construction of $h'$, $Y$ and each irreducible component of the exceptional divisor $E_{h'}$ of $h'$ are defined over $\FF_2$.
Since $Z$ is of type (a), Lemma \ref{lem:Ito2Rem4} shows that a negative rational curve on $Y$ is either a component of $E_{h'}$ or the strict transform of $h(\Theta_{\infty, 8}) = \{z=0\}$.
Hence the assertion holds.
\noindent (2): As in the proof of Corollary \ref{cor:deg=1p=3auto}, we have $\Aut X \cong \Aut Y$.
By Type $E_7$ of Figure \ref{fig:dualE_7-3A_1D_4}, the $\Aut Y$-action on $Y$ fixes the $(-1)$-curve and each $(-2)$-curve.
In particular, the $\Aut Y$-action naturally descends to $\PP^2_k$ via $h'$.
Hence $\Aut Y$ is contained the subgroup $G$ of $\PGL(3, k)$ which fixes the net $h'_*|-K_Y|=\{sz^3+t(xz^2)+u(x^3+y^2z)=0 \mid [s:t:u] \in \PP^2_k\}$.
On the other hand, $|-K_Y|$ is basepoint free by Lemma \ref{basic} (5).
Since $(-1)$-curves on $Y$ are of $(-K_Y)$-degree one, every blow-down of $Y$ collapses the basepoint freeness of $|-K_Y|$.
Hence $Y$ is the minimal resolution of indeterminacy of $h'_*|-K_Y|$.
In particular, we obtain $G \subset \Aut Y$.
Therefore $\Aut Y \cong G$.
Let
\begin{align*}
A=
\begin{psmallmatrix}
a&b&c\\
d&e&f\\
g&h&i
\end{psmallmatrix}
\end{align*}
be an element of $G$.
Then
\begin{align*}
&A \cdot z^3 \\
= &i^3z^3+gi^2xz^2+(g^3x^3+h^2iy^2z)+h^3y^3+g^2hx^2y+g^2ix^2z+h^2gxy^2+hi^2yz^2.
\end{align*}
Since the coefficient of $y^3$ must be zero, we have $h=0$.
Now the coefficient of $x^3$ also must be zero. Hence $g=0$ and $i \neq 0$.
Similarly,
\begin{align*}
A \cdot xz^2 = ci^2z^3+ai^2xz^2+bi^2yz^2
\end{align*}
implies that $b=0$ and
\begin{align*}
A \cdot (x^3+y^2z) = (c^3+f^2i)z^3+ac^2xz^2+(a^3x^3+e^2iy^2z)+(a^2c+d^2i)x^2z
\end{align*}
implies that $a^3=e^2i$ and $a^2c=d^2i$.
Fixing $e=1$, we obtain the assertion.
\end{proof}
\begin{prop}\label{A_1D_6}
The minimal resolution of each del Pezzo surface of type $A_1+D_6$ satisfying (NB) is constructed from $\PP^2_{k, [x:y:z]}$ by blowing up at $[0:1:0]$ five times along $\{x^3+y^2z=0\}$ and at $[1:1:1]$ twice along $\{x^3+y^2z=0\}$.
In particular, there is a unique del Pezzo surface of type $A_1+D_6$ satisfying (NB).
\end{prop}
\begin{proof}
We follow the notation of Type (b) of Figure \ref{fig:dual(a)-(e)}.
Let $Z$ be the rational quasi-elliptic surface of type (b) and $F$ a general fiber.
Let $g \colon Z \to Y$ be the contraction of $O$ and $\Theta_{\infty, 0}$ and $\pi \colon Y \to X$ the contraction of all $(-2)$-curves.
Then the desired del Pezzo surface must be $X$ by Lemma \ref{lem:deg=2pre} (6).
Take $h \colon Z \to \PP^2_k$ and coordinates of $\PP^2_k$ as in Lemma \ref{lem:Ito2Rem4}.
Let $h' \colon Y \to \PP^2_k$ be the morphism induced by $h$.
Then $h'$ is the composition of the blow-ups at $h(P)=[0:1:0]$ five times along $h(F)=\{x^3+y^2z=0\}$ and at $h(O)=[1:1:1]$ twice along $\{x^3+y^2z=0\}$.
Hence it suffices to show that $X$ satisfies (NB).
Since $\pi$ and $h'$ is an isomorphism around a general member of $|-K_X|$, it suffices to show that $h'_*|-K_Y|$ has only a singular member.
By construction, $h'_*|-K_Y|$ consists of cubic curves intersecting with $h(F)=\{x^3+y^2z=0\}$ at $h(P)=[0:1:0]$ five times and at $h(O)=[1:1:1]$ twice.
Then it is generated by $\{x^3+y^2z=0\}, \{(x+z)z^2=0\}$, and $\{(x+z)^2z=0\}$.
The Jacobian criterion now shows that $h'_*|-K_{Y}|$ has only a singular member, and the assertion holds.
\end{proof}
\begin{cor}\label{A_1D_6aut}
Let $X$ be the del Pezzo surface of type $A_1+D_6$ satisfying (NB) and $Y$ the minimal resolution of $X$.
Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] $Y$ and each negative rational curve on $Y$ are defined over $\FF_2$.
\item[\textup{(2)}] $\Aut X$ is isomorphic to
\begin{align*}
\left\{
\begin{psmallmatrix}
a&0&a^3+a\\
d&1&a^3+d+1\\
0&0&a^3
\end{psmallmatrix}
\in \PGL(3, k) \middle| a \in k^*, d \in k
\right\}.
\end{align*}
\item[\textup{(3)}] There is a birational morphism $h'_1 \colon Y \to \PP^1_{k, [x:y]} \times \PP^1_{k, [s:t]}$ such that each negative rational curve on $Y$ is either $h'_1$-exceptional or the strict transform of $\{x=0\}$, $\{y=0\}$, or $\{s=0\}$.
Moreover, $h'_1$ is decomposed into six blow-ups at $\FF_2$-rational points.
\end{enumerate}
\end{cor}
\begin{proof}
We follow the notation of the proof of Proposition \ref{A_1D_6}.
\noindent (1):
By the construction of $h'$, $Y$ and each irreducible component of $E_{h'}$ are defined over $\FF_2$.
Since $Z$ is of type (b), Lemma \ref{lem:Ito2Rem4} shows that a negative rational curve on $Y$ is either a component of $E_{h'}$ or the strict transform of $h(\Theta_{\infty, 8})=\{z=0\}$ or $h(\Theta_{\infty, 3})=\{x+z=0\}$.
Hence the assertion holds.
\noindent (2): Analysis similar to that in the proof of Corollary \ref{E_7aut} shows that $\Aut X \cong \Aut Y$ is the subgroup of $\PGL(3, k)$ which fixes $h'_*|-K_Y|=\{s((x+z)z^2)+t((x+z)^2z)+u(x^3+y^2z)=0 \mid [s:t:u] \in \PP^2_k\}$, and the assertion holds.
\noindent (3): Take $h_1 \colon Z \to \PP^1_{k} \times \PP^1_{k}$ as the contraction of $O$, $P$, and $\Theta_{\infty, i}$ for $i=0, 2, 3, 5, 6$, and $7$.
The induced morphism $h'_1 \colon Y \to \PP^1_{k} \times \PP^1_{k}$ satisfies the first assertion.
The second assertion follows from (1).
\end{proof}
\begin{prop}\label{3A_1D_4}
The minimal resolution of each del Pezzo surface of type $3A_1+D_4$ satisfying (NB) is constructed from $\PP^2_{k, [x:y:z]}$ by blowing up at $[1:0:0]$ once,
at $[0:0:1]$ twice along $\{y=0\}$,
at $[0:1:1]$ twice along $\{y+z=0\}$,
and at $[0:1:0]$ twice along $\{z=0\}$.
In particular, there is a unique del Pezzo surface of type $3A_1+D_4$ satisfying (NB).
\end{prop}
\begin{proof}
We follow the notation of Type (d) of Figure \ref{fig:dual(a)-(e)}.
Let $Z$ be a rational quasi-elliptic surface of type (d).
Let $g \colon Z \to Y$ be the contraction of $O$ and $\Theta_{0, 0}$ and $\pi \colon Y \to X$ the contraction of all $(-2)$-curves.
Then the desired del Pezzo surface must be $X$ by suitable choice of $Z$ by Lemma \ref{lem:deg=2pre} (7).
Hence it suffices to show that $X$ is independent of the choice of $Z$ and satisfies (NB).
Take $h \colon Z \to \PP^2_k$ and coordinates of $\PP^2_k$ as in Lemma \ref{lem:Ito2Rem4}.
Let $h' \colon Y \to \PP^2_k$ be the morphism induced by $h$.
Then $h'$ is the composition of the blow-ups at $h(\Theta_{0,4})=[1:0:0]$ once,
at $h(\Theta_{\infty, 1})=[0:0:1]$ twice along $h(\Theta_{0, 1})=\{y=0\}$,
at $h(\Theta_{\infty, 2})=[0:1:1]$ twice along $h(\Theta_{0, 2})=\{y+z=0\}$,
and at $h(\Theta_{\infty, 3})=[0:1:0]$ twice along $h(\Theta_{0, 3})=\{z=0\}$.
Hence it suffices to show that $X$ satisfies (NB).
Since $\pi$ and $h'$ is an isomorphism around a general member of $|-K_X|$, we are reduced to proving that $h'_*|-K_Y|$ has only a singular member.
By construction, $h'_*|-K_{Y}|$ consists of cubic curves intersecting with $h(\Theta_{0, i})$ at $h(\Theta_{\infty, i})$ with multiplicity two for $1 \leq i \leq 3$ and passing through $h(\Theta_{0,4})$.
Then it is generated by $\{x^2y=0\}, \{x^2z=0\}$, and $\{yz(y+z)=0\}$.
The Jacobian criterion now shows that $h'_*|-K_{Y}|$ has only a singular member, and the assertion holds.
\end{proof}
\begin{cor}\label{3A_1D_4aut}
Let $X$ be the del Pezzo surface of type $3A_1+D_4$ satisfying (NB) and $Y$ the minimal resolution of $X$.
Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] $Y$ and each negative rational curve on $Y$ are defined over $\FF_2$.
\item[\textup{(2)}] $\Aut X \cong k^* \times \PGL(2, \FF_2)$.
\end{enumerate}
\end{cor}
\begin{proof}
We follow the notation of the proof of Proposition \ref{3A_1D_4}.
\noindent (1):
By the construction of $h'$, $Y$ and every irreducible component of $E_{h'}$ are defined over $\FF_2$.
Since $Z$ is of type (d), Lemma \ref{lem:Ito2Rem4} shows that
a negative rational curve on $Y$ is either $h'$-exceptional or the strict transform of one of $\{y=0\}$, $\{y+z=0\}$, $\{z=0\}$, or $h(\Theta_{\infty, 4}) = \{x=0\}$.
Hence the assertion holds.
\noindent (2): By the symmetry of Type $3A_1+D_4$ of Figure \ref{fig:dualE_7-3A_1D_4}, the $\Aut X \cong \Aut Y$-action on $Y$ naturally descends to $\PP^2_k$ via $h'$.
Hence $\Aut Y$ is isomorphic to
the subgroup of $\Aut \PP^2_k$ generated by automorphisms fixing $\{[0:1:0], [0:1:1], [0:0:1]\}$ and $[1:0:0]$, which is
\begin{align*}
\left\{
\begin{psmallmatrix}
a&0&0\\
0&e&f\\
0&h&i
\end{psmallmatrix}
\in \PGL(3, k) \middle| a \in k^*,
\begin{psmallmatrix}
e&f\\
h&i
\end{psmallmatrix}
\in \PGL(2, \FF_2)
\right\} \cong k^* \times \PGL(2, \FF_2).
\end{align*}
\end{proof}
\begin{prop}\label{7A_1}
The minimal resolution of each del Pezzo surface of type $7A_1$ is constructed from $\PP^2_{k, [x:y:z]}$ by blowing up all the $\FF_2$-rational points.
In particular, there is a unique del Pezzo surface of type $7A_1$.
\end{prop}
\begin{proof}
We follow the notation of the proof of Lemma \ref{lem:typeg}.
Since \cite{Ye} shows that the desired surface satisfies (ND), it also satisfies (NB) by Theorem \ref{smooth, Intro}.
Let $Z$ be a rational quasi-elliptic surface of type (g).
Let $g \colon Z \to Y$ be the contraction of $A_{0,2}$ and $\Theta_{0,2}$ and $\pi \colon Y \to X$ the contraction of all $(-2)$-curves.
Then the desired del Pezzo surface must be $X$ by a suitable choice of $Z$ by Lemma \ref{lem:deg=2pre} (1) and (2).
Claim \ref{cl:typeg-2} in the proof of Lemma \ref{lem:typeg} now shows that the morphism $h' \colon Y \to \PP^2_k$ induced by $h \colon Z \to \PP^2_k$ is the blow-up of all the points in $\PP^2_k$ defined over $\FF_2$.
\end{proof}
\begin{rem}
Cascini-Tanaka \cite[Proposition 6.4]{CT18} proved that some del Pezzo surfaces constructed by Keel-M\textsuperscript{c}Kernan \cite[end of section 9]{KM} are isomorphic to the del Pezzo surface constructed by Langer \cite[Example 8.2]{Lan}.
Proposition \ref{7A_1} gives another proof of this fact.
Moreover, Proposition \ref{7A_1} says that this surface is also isomorphic to a counterexample to the Akizuki-Nakano vanishing theorem in \cite[Proposition 11.1 (1)]{Gra} with $p=n=2$.
\end{rem}
\begin{cor}\label{7A_1aut}
Let $X$ be the del Pezzo surface of type $7A_1$ and $Y$ the minimal resolution of $X$.
Let $h' \colon Y \to \PP^2_k$ be the blow-up of all the $\FF_2$-rational points.
Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] $(-1)$-curves (resp.\ $(-2)$-curves) on $Y$ are $h'$-exceptional (resp.\ the strict transform of lines in $\PP^2_k$ are defined over $\FF_2$).
In particular, $Y$ and every negative rational curve on $Y$ are defined over $\FF_2$.
\item[\textup{(2)}] The class divisor group of $Y$ is generated by the seven $(-1)$-curves and any one of $(-2)$-curves.
\item[\textup{(3)}] $\Aut X \cong \Aut Y \cong \PGL(3, \FF_2)$.
\item[\textup{(4)}] $\Aut Y$ acts on both the set of $(-1)$-curves on $Y$ and that of $(-2)$-curves transitively.
\item[\textup{(5)}] For each $(-1)$-curve $E$ on $Y$, the stabilizer subgroup of $\Aut Y$ with respect to $E$ is isomorphic to $\FF_2^2 \rtimes \PGL(2, \FF_2)$.
The first (resp.\ second) factor acts on $E$ trivially (resp.\ as $\Aut \PP^1_{\FF_2}$).
\end{enumerate}
\end{cor}
\begin{proof}
\noindent (1): There are seven $h'$-exceptional curves and the strict transform of lines in $\PP^2_k$ defined over $\FF_2$, which are $(-1)$-curves and $(-2)$-curves respectively.
On the other hand, Lemma \ref{lem:deg=2pre} (4) shows that $Y$ contains exactly seven $(-1)$-curves and seven $(-2)$-curves.
Hence the assertion holds.
\noindent (2): The assertion is obvious from the assertion (1).
\noindent (3): By the assertion (1), the $\Aut Y$-action on $Y$ fixes $E_{h'}$ and descends to $\PP^2_k$ via $h'$.
Hence $\Aut Y$ equals the stabilizer subgroup of $\PGL(3, k)$ with respect to the set of $\FF_2$-rational points on $\PP^2_k$, which is $\PGL(3, \FF_2)$.
\noindent (4): The assertion is obvious from the assertion (3).
\noindent(5): Fix coordinates $[x:y:z]$ of $\PP^2_k$.
By the assertion (4), we may assume that $E$ is the strict transform of $\{x=0\} \subset \PP^2_k$.
Then the stabilizer subgroup of $\Aut Y$ with respect to $E$ is
\begin{align*}
\left\{
\begin{psmallmatrix}
1&0&0\\
d&e&f\\
g&h&i
\end{psmallmatrix}
\in \PGL(3, \FF_2)
\right\}
&\cong \left\{
\begin{psmallmatrix}
1&0&0\\
d&1&0\\
g&0&1
\end{psmallmatrix}
\in \PGL(3, \FF_2)
\right\}
\rtimes
\left\{
\begin{psmallmatrix}
1&0&0\\
0&e&f\\
0&h&i
\end{psmallmatrix}
\in \PGL(3, \FF_2)
\right\} \\
&\cong \FF_2^2 \rtimes \PGL(2, \FF_2),
\end{align*}
and the assertion holds.
\end{proof}
Next, we treat the case where the degree is one.
\begin{prop}\label{prop:deg1isom}
Let $X$ be a Du Val del Pezzo surface satisfying (NB).
Suppose that $p=2$ and $\Dyn(X)=E_8, D_8, A_1+E_7$, or $2A_1+D_6$.
Then the isomorphism class of $X$ is uniquely determined by $\Dyn(X)$.
\end{prop}
\begin{proof}
By Proposition \ref{prop:deg=1} (4), the minimal resolution of $X$ is obtained from the rational quasi-elliptic surface $Z$ of type (a), (b), (c), or (e) by contracting a section.
Since $Z$ is unique up to isomorphism for each types by Theorem \ref{q-ell}, the assertion follows from Proposition \ref{prop:deg=1} (2).
\end{proof}
\begin{lem}\label{lem:deg=1p=2auto}
Let $X$ be a Du Val del Pezzo surface satisfying (NB) and $\pi \colon Y \to X$ be the minimal resolution.
Suppose that $p=2$ and $\Dyn(X)=E_8$, $D_8$, or $A_1+E_7$ in addition.
Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] $Y$ and every negative rational curve on $Y$ are defined over $\FF_2$.
\item[\textup{(2)}] $\Aut X$ is isomorphic to
\begin{align*}
\left\{
\begin{psmallmatrix}
a&0&0\\
0&1&f\\
0&0&a^3
\end{psmallmatrix}
\in \PGL(3, k) \middle| a\in k^*, f\in k
\right\}
\end{align*}
when $\Dyn(X)=E_8$,
\begin{align*}
\left\{
\begin{psmallmatrix}
1&0&0\\
d&1&d\\
0&0&1
\end{psmallmatrix}
\in \PGL(3, k) \middle| d\in k
\right\} \cong k
\end{align*}
when $\Dyn(X)=D_8$, and
\begin{align*}
\left\{
\begin{psmallmatrix}
1&0&0\\
0&e&0\\
0&0&e^2
\end{psmallmatrix}
\in \PGL(3, k) \middle| e\in k^*
\right\}
\cong k^*
\end{align*}
when $\Dyn(X)=A_1+E_7$.
\end{enumerate}
\end{lem}
\begin{proof}
Let $g \colon Z \to Y$ be the blow-up at the base point of $|-K_Y|$.
When $\Dyn(X)=E_8$ (resp.~$D_8$, $A_1+E_7$), $Z$ is the rational quasi-elliptic surface of type (a) (resp.\ (b), (c)) by Proposition \ref{prop:deg=1} (4).
We may assume that $g$ is the contraction of $O$ in Types (a)--(c) of Figure \ref{fig:dual(a)-(e)} by virtue of the $\MW(Z)$-action on $Y$.
From now on, we follow the notation of Lemma \ref{lem:Ito2Rem4}.
Then $h \colon Z \to \PP^2_k$ induces a morphism $h' \colon Y \to \PP^2_k$.
\noindent (1):
First, suppose that $\Dyn(X)=E_8$.
Then $h'$ is the blow-up of $\PP^2_k$ at $h(O)=[0:1:0]$ eight times along $h(F) = \{x^3+y^2z=0\}$.
Hence $Y$ and each irreducible component of $E_{h'}$ are defined over $\FF_2$.
Since each negative rational curve on $Y$ is either a component of $E_h$ or the strict transform of $h(\Theta_{\infty, 8}) = \{z=0\}$, the assertion holds.
Next, suppose that $\Dyn(X)=D_8$.
Then $h'$ is the composition of the blow-up of $\PP^2_k$ at $h(P)=[0:1:0]$ five times along $h(F) = \{x^3+y^2z=0\}$ and the blow-up at $h(O)=[1:1:1]$ three times along $\{x^3+y^2z=0\}$.
Hence $Y$ and each irreducible component of $E_{h'}$ are defined over $\FF_2$.
Since each negative rational curve on $Y$ is either a component of $E_{h'}$ or the strict transform of $h(\Theta_{\infty, 8})=\{z=0\}$ or $h(\Theta_{\infty, 3})=\{x+z=0\}$, the assertion holds.
Finally, suppose that $\Dyn(X)=A_1+E_7$.
Then $h'$ is the composition of the blow-up of $\PP^2_k$ at $h(P)=[1:0:0]$ six times along $h(\Theta_{\infty, 1})=\{xz+y^2=0\}$ and the blow-up at $h(O)=[0:1:0]$ twice along $h(\Theta_{\infty, 0})=\{x=0\}$.
Hence $Y$ and each irreducible component of $E_{h'}$ are defined over $\FF_2$.
Since each negative rational curve on $Y$ is either a component of $E_{h'}$ or the strict transform of $\{xz+y^2=0\}$, $\{x=0\}$, or $h(\Theta_{0,2})=\{z=0\}$, the assertion holds.
\noindent (2): From Types (a)--(c) of Figure \ref{fig:dual(a)-(e)}, it is easily seen that the $\Aut Y$-action on $Y$ fixes each negative rational curve.
In particular, the $\Aut Y$-action naturally descends to $\PP^2_k$ via $h'$.
Hence $\Aut Y$ is contained in the subgroup $G$ of $\PGL(3, k)$ which fixes the net $h'_*|-K_Y|$.
On the other hand, we have $h'_*|-K_Y| = h_*|-K_Z|$.
Since $|-K_Z|$ is base point free, $Z$ is the minimal resolution of indeterminacy of $h'_*|-K_Y|$.
Hence the $G$-action on $\PP^2_k$ ascends to $Z$.
When $\Dyn(X)=E_8$, it descends to $Y$ since there is a unique section on $Z$.
On the other hand, when $\Dyn(X)=D_8$ or $A_1+E_7$, it also descends to $Y$ by the asymmetry of $E_{h}$.
Therefore $\Aut Y \cong G$.
By the choice of coordinates $[x:y:z]$ of $\PP^2_k$, $h'_*|-K_Y|$ is generated by $\{x^3+y^2z=0\}$ and $\{z^3=0\}$ (resp.\ $\{x^3+y^2z=0\}$ and $\{(x+z)^2z=0\}$, $\{(xz+y^2)x=0\}$ and $\{z^3=0\}$) when $\Dyn(X)=E_8$ (resp.\ $D_8$, $A_1+E_7$).
Hence an easy computation as in the proof of Corollary \ref{E_7aut} gives the assertion.
\end{proof}
\begin{lem}\label{2A_1D_6aut}
Let $X$ be the Du Val del Pezzo surface of type $2A_1+D_6$ satisfying (NB) and $\pi \colon Y \to X$ be the minimal resolution.
Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] $Y$ and every negative rational curve on $Y$ are defined over $\FF_2$.
\item[\textup{(2)}] $\Aut X$ is isomorphic to
\begin{align*}
\left\{
\begin{psmallmatrix}
1&0&0\\
0&1&0\\
0&0&1
\end{psmallmatrix}
,
\begin{psmallmatrix}
1&0&0\\
0&1&1\\
0&0&1
\end{psmallmatrix}
\in \PGL(3, k)
\right\}
\cong \ZZ/2\ZZ.
\end{align*}
\end{enumerate}
\end{lem}
\begin{proof}
Since $\pi$ is the minimal resolution, we have $\Aut Y \cong \Aut X$.
Let $g \colon Z \to Y$ be the blow-up at the base point of $|-K_Y|$.
Then $Z$ be the rational quasi-elliptic surface of type (e) by Proposition \ref{prop:deg=1} (4).
In what follows, we use the notation of Type (e) of Figure \ref{fig:dual(a)-(e)}.
Then we may assume that $g$ is the contraction of $O$.
By the shape of Type (e) of Figure \ref{fig:dual(a)-(e)}, the $\Aut Y$-action on $Y$ fixes $g(\Theta_{1,2})$.
By Lemma \ref{lem:deg=2pre} (7), the contraction of $g(\Theta_{1,2})$ gives a morphism $h \colon Y \to W$ to the minimal resolution of the Du Val del Pezzo surface of type $3A_1+D_4$ satisfying (NB).
Hence $\Aut Y$ is isomorphic to the stabilizer subgroup of $\Aut W \cong k^* \times \PGL(2, \FF_2)$ with respect to $t=h \circ g(\Theta_{1,2})$.
\noindent (2): By Type (e) of Figure \ref{fig:dual(a)-(e)}, $E=h \circ g(\Theta_{1, 3})$ is the unique negative rational curve containing $t$.
Hence the $\Aut Y$-action on $W$ fixes $E$.
Moreover $E$ is a $(-1)$-curve intersecting with exactly two $(-2)$-curves, which are $E_1 = h \circ g(\Theta_{1, 0})$ and $E_2 = h \circ g(\Theta_{1, 4})$.
We have seen in the proof of Corollary \ref{3A_1D_4aut} that the first factor
(resp.\ the second factor) of $\Aut W \cong k^* \times \PGL(2, \FF_2)$ acts on $E \setminus (E \cap (E_1 \cup E_2)) \cong k^*$ freely and transitively
(resp.\ acts as a permutation of the third nodes from the top in Type $3A_1+D_4$ of Figure \ref{fig:dualE_7-3A_1D_4}).
Hence the assertion holds.
\noindent (1): By Corollary \ref{3A_1D_4aut}, $W$ and each negative rational curve on $W$ are defined over $\FF_2$.
By virtue of the $k^*$-action on $W$, we may assume that $t$ is an $\FF_2$-rational point.
Hence $Y$ and each negative rational curve on $Y$ except $g(\Theta_{0,0})$ and $g(\Theta_{\infty,0})$ are defined over $\FF_2$.
On the other hand, $g(\Theta_{0,0})$ and $g(\Theta_{\infty,0})$ are defined over $\FF_{2^m}$ for some $m>0$ since $Y$ is defined over $\overline{\FF_2}$, and are the unique $(-1)$-curves on $Y$ intersecting with $g(\Theta_{0,1})$ and $g(\Theta_{\infty,1})$ twice respectively.
Since the field extension $\FF_{2^m}/\FF_2$ is Galois, they are also defined over $\FF_2$.
Hence the assertion holds.
\end{proof}
To determine the isomorphism classes of Du Val del Pezzo surfaces of one of the types $2D_4$, $4A_1+D_4$, and $8A_1$ satisfying (NB), we need the following notation and auxiliary lemmas.
\begin{defn}
For coordinates of $\PP^n_k$, let $\mathcal{D}_n \subset \PP^n_k$ denote the complement of all the hyperplane sections defined over $\FF_2$.
Note that $\PGL (n+1, \FF_2)$ naturally acts on $\mathcal{D}_n$.
\end{defn}
\begin{lem}\label{lem:D1aut}
Let $\Sigma_t$ be the stabilizer subgroup of $\PGL(2, \FF_2)$ with respect to $t \in \mathcal{D}_1$.
Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] $\Sigma_t$ is trivial unless $t$ is an $\FF_4$-rational point.
\item[\textup{(2)}] The $\PGL(2, \FF_2)$-action on the set $\mathcal{D}_1(\FF_4)$ of $\FF_4$-rational points on $\mathcal{D}_1$ is transitive.
\item[\textup{(3)}] $\Sigma_t = \ZZ/3\ZZ$ if $t$ is an $\FF_4$-rational point.
\item[\textup{(4)}] $\mathcal{D}_1 / \PGL(2, \FF_2) \cong \mathbb{A}^1_k$ with a distinct point which corresponds to $\mathcal{D}_1(\FF_4)$.
\end{enumerate}
\end{lem}
\begin{proof}
\noindent (1):
Suppose that there is a non-trivial element $A \in \Sigma_t$.
Since $\PGL(2, \FF_2)$ is isomorphic to the symmetric group of three letters, it has exactly three conjugacy classes.
Since $A_1=
\begin{psmallmatrix}
0&1\\
1&0\\
\end{psmallmatrix}
$ and $A_2=
\begin{psmallmatrix}
0&1\\
1&1\\
\end{psmallmatrix}
$ are non-trivial and have different minimal polynomials, $A$ is conjugate to $A_i$ for some $i$.
Then $A_i$ also fixes some point in $\mathcal{D}_1$.
In $\PP^1_{k, [x:y]}$, the fixed point locus of $A_1$ (resp.\ $A_2$) equals $\{[1:1]\}$ (resp.\ $\{[1:s] \mid s^2+s+1=0\}$).
Hence $i=2$ and $t \in \mathcal{D}_1(\FF_4)$.
\noindent (2):
Since $A_1$ interchanges two points in $\mathcal{D}_1(\FF_4)$ with each other, the assertion holds.
\noindent (3):
Since the order of $\Sigma_t$ equals $|\PGL(2, \FF_2)|/|\mathcal{D}_1(\FF_4)|=3$, we obtain $\Sigma_t = \ZZ/3\ZZ$.
\noindent (4):
$\mathcal{D}_1 / \PGL(2, \FF_2)$ is naturally embedded into $\PP^1_k / \PGL(2, \FF_2) \cong \PP^1_k$.
The complement is a point since $\PGL(2, \FF_2)$ acts on $\PP^1_k(\FF_2)$ transitively.
\end{proof}
\begin{lem}\label{lem:D2aut}
Let $\Sigma_t$ be the stabilizer subgroup of $\PGL(3, \FF_2)$ with respect to $t \in \mathcal{D}_2$.
Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] $\Sigma_t$ is trivial unless $t$ is an $\FF_8$-rational point.
\item[\textup{(2)}] The $\PGL(3, \FF_2)$-action on the set $\mathcal{D}_2(\FF_8)$ of $\FF_8$-rational points on $\mathcal{D}_2$ is transitive.
\item[\textup{(3)}] $\Sigma_t = \ZZ/7\ZZ$ if $t$ is an $\FF_8$-rational point.
\item[\textup{(4)}] $\mathcal{D}_2 / \PGL(3, \FF_2)$ is a surface with a unique singular point, which corresponds to $\mathcal{D}_2(\FF_8)$.
\end{enumerate}
\end{lem}
\begin{proof}
\noindent (1): Suppose that there is a non-trivial element $A \in \Sigma_t$.
By \cite[27.1 Lemma]{J-L}, $\PGL(3, \FF_2) \cong \mathrm{PSL}(2, \FF_7)$ has exactly six conjugacy classes.
Since
\begin{align*}
A_1=
\begin{psmallmatrix}
1&1&0\\
0&1&0\\
0&0&1
\end{psmallmatrix},
A_2=
\begin{psmallmatrix}
1&1&0\\
0&1&1\\
0&0&1
\end{psmallmatrix},
A_3=
\begin{psmallmatrix}
1&0&0\\
0&0&1\\
0&1&1
\end{psmallmatrix},
A_4=
\begin{psmallmatrix}
0&1&0\\
0&0&1\\
1&1&0
\end{psmallmatrix},
\text{ and }
A_5=
\begin{psmallmatrix}
0&1&0\\
0&0&1\\
1&0&1
\end{psmallmatrix}
\end{align*}
are non-trivial and have different minimal polynomials to each other, $A$ is conjugate to $A_i$ for some $1 \leq i \leq 5$.
Then $A_i$ also fixes some point in $\mathcal{D}_2$.
In $\PP^2_{k, [x:y:z]}$, the fixed point locus of $A_i$ equals
\begin{align*}
\begin{cases}
\{y=0\} & (i=1) \\
\{[1:0:0]\} & (i=2)\\
\{[1:0:0]\} \cup \{[0:1:s] \mid s^2+s+1=0\} & (i=3) \\
\{[1:s:s^2] \mid s^3+s+1=0\} & (i=4)\\
\{[1:s:s^2] \mid s^3+s^2+1=0\} & (i=5)\\
\end{cases}
\end{align*}
Hence $i=4$ or $5$, and $t \in \mathcal{D}_2(\FF_8)$.
We have proved more, namely that $A$ is of order seven and fixes exactly three points in $\mathcal{D}_2(\FF_8)$.
By \cite[27.1 Lemma]{J-L}, the size of its conjugacy class is 24.
\noindent (2): $\PP^1_k(\FF_8)$ (resp.\ $\PP^2_k(\FF_8)$) consists of nine (resp.\ 73) points.
Since $\PP^2_k \setminus \mathcal{D}_2$ is the union of seven $\PP^1_k$'s passing through three $\FF_2$-rational points, $\mathcal{D}_2(\FF_8)$ consists of $73-7 \cdot 9 + (3-1) \cdot 7 = 24$ points.
The Burnside lemma now shows that the number of the $\PGL(3, \FF_2)$-orbits is
\begin{align*}
|\mathcal{D}_2(\FF_8)/\PGL(3, \FF_2)|=\frac{1}{168}(24 \cdot 3+24 \cdot 3+1 \cdot 24 + (168-24-24-1) \cdot 0)=1.
\end{align*}
Hence $\PGL(3, \FF_2)$ acts on $\mathcal{D}_2(\FF_8)$ transitively.
\noindent (3): Since the order of $\Sigma_t$ equals $|\PGL(3, \FF_2)|/|\mathcal{D}_2(\FF_8)|=7$, we obtain $\Sigma_t = \ZZ/7\ZZ$.
\noindent (4): The quotient morphism $\mathcal{D}_2 \to \mathcal{D}_2 / \PGL(3, \FF_2)$ is \'{e}tale outside the image of $\mathcal{D}_2(\FF_8)$.
Hence the assertion holds.
\end{proof}
Finally, let us investigate Du Val del Pezzo surfaces of one of types $2D_4$, $4A_1+D_4$, and $8A_1$ satisfying (NB).
\begin{prop}\label{2D_4isom}
Let $W$ be the minimal resolution of the Du Val del Pezzo surface of type $3A_1+D_4$ satisfying (NB) and $E$ the $(-1)$-curve intersecting with three $(-2)$-curves.
Note that $E$ is unique by Lemma \ref{lem:deg=2pre} (3) and $W$ and $E$ are defined over $\FF_2$ by Corollary \ref{3A_1D_4aut}.
Then the following holds.
\begin{enumerate}
\item[\textup{(1)}] The minimal resolution of each Du Val del Pezzo surface of type $2D_4$ satisfying (NB) is obtained from $W$ by blowing up a point in $E \setminus E(\FF_2) \cong \mathcal{D}_1$.
\item[\textup{(2)}] Let $h_t \colon Y_{t} \to W$ be the blow-up at $t \in E \setminus E(\FF_2)$.
Then $Y_t$ is the minimal resolution of a Du Val del Pezzo surface of type $2D_4$ satisfying (NB).
Moreover, for $t' \in E \setminus E(\FF_2)$, $Y_{t} \cong Y_{t'}$ if and only if $t'$ is contained in the $\PGL(2, \FF_2)$-orbit of $t$.
\end{enumerate}
As a result, there is a one-to-one correspondence between the isomorphism classes of del Pezzo surfaces of type $2D_4$ satisfying (NB) and the closed points of $\mathcal{D}_1 /\PGL (2, \FF_2)$.
\end{prop}
\begin{proof}
We follow the notation of Type (d) of Figure \ref{fig:dual(a)-(e)}.
\noindent(1): Let $Z$ be a rational quasi-elliptic surface of type (d).
Let $g \colon Z \to Y$ be the contraction of $O$.
Then the minimal resolution of each del Pezzo surface of type $2D_4$ satisfying (NB) is isomorphic to $Y$ by suitable choice of $Z$ by Proposition \ref{prop:deg=1} (4).
On the other hand, by Lemma \ref{lem:deg=2pre} (7), the contraction of $g(\Theta_{0,0})$ gives a morphism $h \colon Y \to W$.
We check at once that $E=h \circ g(\Theta_{0, 4})$ and it contains the point $t=h \circ g(\Theta_{0,0})$, which is not contained in any $(-2)$-curves.
By Corollary \ref{3A_1D_4aut}, the set of $\FF_2$-rational points on $E$ is the intersection of $E$ and all $(-2)$-curves.
Therefore $Y$ is the blow-up of $W$ at $t \in E \setminus E(\FF_2)$.
\noindent (2):
By Lemma \ref{lem:deg=2pre} (7), $Y_t$ is obtained from a rational quasi-elliptic surface of type (d) by contracting a section.
Hence the former assertion follows from Proposition \ref{prop:deg=1} (4).
We have seen in the proof of Corollary \ref{3A_1D_4aut} that first (resp.\ second) factor of $\Aut W=k^* \times \PGL(2, \FF_2)$ acts on $h(\Theta_{\infty, 4})$ trivially (resp.\ as $\Aut \PP^1_{\FF_2}$).
The same conclusion can be drawn for $E$ by the choice of its coordinates.
In particular, $t' \in E \setminus E(\FF_2)$ is contained in the $\PGL(2, \FF_2)$-orbit of $t$ if and only if it is contained in the $\Aut W$-orbit of $t$ in $W$.
On the other hand, $Y_t \cong Y_{t'}$ if $t'$ is contained in the $\Aut W$-orbit of $t$.
Hence it remains to prove that $t'$ is contained in the $\Aut W$-orbit of $t$ if $Y_t \cong Y_{t'}$.
Suppose that there is an isomorphism $\sigma \colon Y_t \cong Y_{t'}$.
Since the involution as in Lemma \ref{lem:type(d)auto} fixes the section $O$, it descends to an involution $\tau \in \Aut Y_t$.
Then, replacing $\sigma$ with $\sigma \circ \tau$ if necessary, we may assume that $\sigma(E_{h_t}) = E_{h_{t'}}$.
Hence $\sigma$ descends to an isomorphism $\overline \sigma \in \Aut W$ such that $\overline \sigma(t)=t'$, and the latter assertion holds.
\end{proof}
\begin{cor}\label{2D_4aut}
Let $X_t$ be the contraction of all $(-2)$-curves in $Y_t$ as in Proposition \ref{2D_4isom}.
Then $\Aut X_t \cong \Aut Y_t \cong (k^* \times \ZZ/3\ZZ) \rtimes \ZZ/2\ZZ$ when $t$ is an $\FF_4$-rational point of $\mathcal{D}_1$ and $k^* \rtimes \ZZ/2\ZZ$ otherwise.
In particular, there is a unique Du Val del Pezzo surface $X(2D_4)$ satisfying (NB) such that $\Aut X \cong (k^* \times \ZZ/3\ZZ) \rtimes \ZZ/2\ZZ$.
\end{cor}
\begin{proof}
We follow the notation of Proposition \ref{2D_4isom}.
Let $\Sigma$ be the stabilizer subgroup of $\Aut W$ with respect to $t$.
Then $\Sigma = k^* \times \Sigma'$ for some $\Sigma' \subset \PGL(2, \FF_2)$ since $k^*$ acts on $E$ trivially.
By Lemma \ref{lem:D1aut}, $\Sigma = k^* \times \ZZ/3\ZZ$ if $t \in \mathcal{D}_1(\FF_4)$ and $\Sigma = k^*$ otherwise.
On the other hand, we can identify $\Sigma$ with the stabilizer subgroup of $\Aut Y_t$ with respect to $E_{h_t}$.
For $\eta \in \Aut Y_t$, either $\eta$ or $\eta \circ \tau$ belongs to $\Sigma$.
Hence $\Aut Y_t \cong \Sigma \rtimes \ZZ/2\ZZ$, where the last factor is generated by $\tau$, and the first assertion holds.
Since $\PGL(2, \FF_2)$ acts on $\mathcal{D}_1(\FF_4)$ transitively, the second assertion follows from Proposition \ref{2D_4isom} (2).
\end{proof}
\begin{prop}\label{4A_1D_4isom}
Let $W$ be the minimal resolution of the Du Val del Pezzo surface of type $7A_1$ and $E$ a $(-1)$-curve.
Note that $W$ and $E$ are defined over $\FF_2$ and $E$ is unique up to the $\Aut W$-action on $W$ by Corollary \ref{7A_1aut} (1) and (4).
Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] The minimal resolution of each Du Val del Pezzo surface of type $4A_1+D_4$ is obtained from $W$ by blowing up a point in $E \setminus E(\FF_2) \cong \mathcal{D}_1$.
\item[\textup{(2)}] Let $h_t \colon Y_{t} \to W$ be the blow-up at $t \in E \setminus E(\FF_2)$.
Then $Y_t$ is the minimal resolution of a Du Val del Pezzo surface of type $4A_1+D_4$.
Moreover, for $t' \in E \setminus E(\FF_2)$, $Y_{t} \cong Y_{t'}$ if and only if $t'$ is contained in the $\PGL(2, \FF_2)$-orbit of $t$.
\end{enumerate}
As a result, there is a one-to-one correspondence between the isomorphism classes of del Pezzo surfaces of type $4A_1+D_4$ and the closed points of $\mathcal{D}_1 /\PGL (2, \FF_2)$.
\end{prop}
\begin{proof}
We follow the notation of Figure \ref{fig:dual(f)}.
Note that, since \cite{Ye} shows that $4A_1+D_4$ is not feasible over $\C$, every Du Val del Pezzo surface of type $4A_1+D_4$ satisfies (NB) by Theorem \ref{smooth, Intro}.
\noindent(1): Let $Z$ be a rational quasi-elliptic surface of type (f).
Let $g \colon Z \to Y$
be the contraction of $O$.
Then the minimal resolution of each del Pezzo surface of type $4A_1+D_4$ is isomorphic to $Y$ by suitable choice of $Z$ by Proposition \ref{prop:deg=1} (4).
On the other hand, by Lemma \ref{lem:deg=2pre} (8), the contraction of $g(\Theta_{1,0})$ gives a morphism $h \colon Y \to W$.
We may assume that $E=h \circ g(\Theta_{1, 4})$.
Then $E$ contains the point $t=h \circ g(\Theta_{1,0})$, which is not contained in any $(-2)$-curves.
By Corollary \ref{7A_1aut} (1), the set of $\FF_2$-rational points on $E$ is the intersection of $E$ and all $(-2)$-curves.
Therefore $Y$ is the blow-up of $W$ at $t \in E \setminus E(\FF_2)$.
\noindent (2):
By Lemma \ref{lem:deg=2pre} (8), $Y_t$ is obtained from a rational quasi-elliptic surface of type (f) by contracting a section.
Hence the former assertion follows from Proposition \ref{prop:deg=1} (4).
Let $C_t$ be the strict transform of $E$ in $Y_t$, which is a $(-2)$-curve.
By Figure \ref{matrix7A1}, $C_t$ intersects with three $(-2)$-curves in $Y_t$.
Hence $C_t$ is the central curve of the Dynkin diagram $D_4$.
In particular, every automorphism of $Y_t$ fixes $C_t$.
By Corollary \ref{7A_1aut} (5), $t' \in E \setminus E(\FF_2)$ is contained in the $\PGL(2, \FF_2)$-orbit of $t$ if and only if it is contained in the $\Aut W=\FF_2^2 \rtimes \PGL(2, \FF_2)$-orbit of $t$ in $W$.
On the other hand, $Y_t \cong Y_{t'}$ if $t'$ is contained in the $\Aut W$-orbit of $t$.
Hence it remains to prove that $t'$ is contained in the $\Aut W$-orbit of $t$ if $Y_t \cong Y_{t'}$.
Suppose that there is an isomorphism $\sigma \colon Y_t \cong Y_{t'}$.
Then $\sigma(C_t) = C_{t'}$.
By Figure \ref{fig:dual(f)}, $E_{h_t}$ is the unique $(-1)$-curve intersecting with $C_t$.
Hence $\sigma(E_{h_t})=E_{h_{t'}}$ and $\sigma$ descends to an isomorphism $\overline{\sigma} \in \Aut W$ such that $\overline{\sigma}(t)=t'$, and the latter assertion holds.
\end{proof}
\begin{cor}\label{4A_1D_4aut}
Let $X_t$ be the contraction of all $(-2)$-curves in $Y_t$ as in Proposition \ref{4A_1D_4isom}.
Then $\Aut X_t \cong \Aut Y_t \cong (\ZZ/2\ZZ)^2\rtimes \ZZ/3\ZZ$ when $t$ is an $\FF_4$-rational point and $(\ZZ/2\ZZ)^2$ otherwise.
In particular, there is a unique Du Val del Pezzo surface $X(4A_1+D_4)$ such that $\Aut X \cong (\ZZ/2\ZZ)^2\rtimes \ZZ/3\ZZ$.
\end{cor}
\begin{proof}
We follow the notation of Proposition \ref{4A_1D_4isom}.
Since each automorphism of $Y_t$ fixes $E_{h_t}$, the group $\Aut Y_t$ equals the stabilizer subgroup $\Sigma$ of $\Aut W = \FF_2^2 \rtimes \PGL(2, \FF_2)$ with respect to $t$.
Then $(\ZZ/2\ZZ)^2 \cong \FF_2^2 \subset \Aut Y_t$ since $\FF_2^2$ acts on $E$ trivially.
The rest of the proof runs as in Corollary \ref{2D_4aut}.
\end{proof}
\begin{prop}\label{8A_1isom}
Let $W$ be the minimal resolution of the Du Val del Pezzo surface of type $7A_1$ and $B$ the union of all negative rational curves on $W$.
Note that $W$ and $B$ are defined over $\FF_2$ and $W \setminus B \cong \mathcal{D}_2$ by Corollary \ref{7A_1aut} (1).
Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] The minimal resolution of each Du Val del Pezzo surface of type $8A_1$ is obtained from $W$ by blowing up a point in $W \setminus B$.
\item[\textup{(2)}] Let $h_t \colon Y_{t} \to W$ be the blow-up at $t \in W \setminus B$.
Then $Y_t$ is the minimal resolution of a Du Val del Pezzo surface of type $8A_1$.
\item[\textup{(3)}] For $t \in W \setminus B$, Figure \ref{matrix} is the intersection matrix of negative rational curves on $Y_t$.
Moreover, there is a $(-2)$-curve $C_t$ such that $E_{h_t}$ is a unique $(-1)$-curve intersecting with $C_t$ twice.
\item[\textup{(4)}] For $t \in W \setminus B$, $\Aut Y_t$ is contained in the affine linear group $\FF_2^3 \rtimes \mathrm{GL}(3, \FF_2)$ and contains its normal subgroup $\FF_2^3$, which acts on the set of $(-2)$-curves transitively.
\item[\textup{(5)}] For $t$ and $t' \in W \setminus B$, $Y_{t} \cong Y_{t'}$ if and only if $t'$ is contained in the $\Aut W \cong \PGL(3, \FF_2)$-orbit of $t$.
\end{enumerate}
As a result, there is a one-to-one correspondence between the isomorphism classes of del Pezzo surfaces of type $8A_1$ and the closed points of $\mathcal{D}_2 /\PGL (3, \FF_2)$.
\end{prop}
\begin{figure}[htbp]
\centering
\begin{align*}
{\fontsize{7pt}{4}\selectfont
\left(
\vcenter{
\xymatrix@=-1.8ex{
&&&&&&&&\ar@{-}[ddddddddddddddddddddddddddddddddddd]&&&&&&&& \ar@{-}[ddddddddddddddddddddddddddddddddddd] &&&&&&&&& \ar@{-}[ddddddddddddddddddddddddddddddddddd] &&&&&&&&\\
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&\\
&-1&0 &0 &0 &0 &0 &0 & &1 &0 &0 &0 &0 &0 &0 &&1 &1 &1 &1 &0 &0 &0 &0 &&0 &0 &0 &0 &1 &1 &1 &1 &\\
& &-1&0 &0 &0 &0 &0 & &0 &1 &0 &0 &0 &0 &0 &&1 &1 &0 &0 &1 &1 &0 &0 &&0 &0 &1 &1 &0 &0 &1 &1 &\\
& & &-1&0 &0 &0 &0 & &0 &0 &1 &0 &0 &0 &0 &&1 &0 &1 &0 &1 &0 &1 &0 &&0 &1 &0 &1 &0 &1 &0 &1 &\\
& & & &-1&0 &0 &0 & &0 &0 &0 &1 &0 &0 &0 &&1 &1 &0 &0 &0 &0 &1 &1 &&0 &0 &1 &1 &1 &1 &0 &0 &\\
& & & & &-1&0 &0 & &0 &0 &0 &0 &1 &0 &0 &&1 &0 &0 &1 &1 &0 &0 &1 &&0 &1 &1 &0 &0 &1 &1 &0 &\\
& & & & & &-1&0 & &0 &0 &0 &0 &0 &1 &0 &&1 &0 &1 &0 &0 &1 &0 &1 &&0 &1 &0 &1 &1 &0 &1 &0 &\\
& & & & & & &-1& &0 &0 &0 &0 &0 &0 &1 &&1 &0 &0 &1 &0 &1 &1 &0 &&0 &1 &1 &0 &1 &0 &0 &1 &\\
\ar@{-}[rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr]&&&&&&&& &&&&&&&& &&&&&&&&&& &&&&&&&&\\
& & & & & & & & &-1&0 &0 &0 &0 &0 &0 &&0 &0 &0 &0 &1 &1 &1 &1 &&1 &1 &1 &1 &0 &0 &0 &0 &\\
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& & & & & & & & & & & & & &-1&0 &&0 &1 &0 &1 &1 &0 &1 &0 &&1 &0 &1 &0 &0 &1 &0 &1 &\\
& & & & & & & & & & & & & & &-1&&0 &1 &1 &0 &1 &0 &0 &1 &&1 &0 &0 &1 &0 &1 &1 &0 &\\
\ar@{-}[rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr]&&&&&&&& &&&&&&&& &&&&&&&&&& &&&&&&&&\\
& & & & & & & & & & & & & & & &&-2&0 &0 &0 &0 &0 &0 &0 &&2 &0 &0 &0 &0 &0 &0 &0 &\\
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\ar@{-}[rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr]&&&&&&&& &&&&&&&& &&&&&&&&&& &&&&&&&&\\
& & & & & & & & & & & & & & & && & & & & & & & &&-1&1 &1 &1 &1 &1 &1 &1 &\\
& & & & & & & & & & & & & & & && & & & & & & & && &-1&1 &1 &1 &1 &1 &1 &\\
& & & & & & & & & & & & & & & && & & & & & & & && & &-1&1 &1 &1 &1 &1 &\\
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&&&&&&&& &&&&&&&& &&&&&&&&& &&&&&&&&&
}
}\right)
}.
\end{align*}
\caption{The intersection matrix of the $(-1)$-curves and $(-2)$-curves in a del Pezzo surface of type $8A_1$}
\label{matrix}
\end{figure}
\begin{proof}
We follow the notation of the proof of Lemma \ref{lem:typeg}.
Note that, since \cite{Ye} shows that $8A_1$ is not feasible over $\C$, every Du Val del Pezzo surface of type $8A_1$ satisfies (NB) by Theorem \ref{smooth, Intro}.
\noindent(1): Let $Z$ be a rational quasi-elliptic surface of type (g).
Let $g \colon Z \to Y$ be the contraction of $A_{0,2}$.
Then the minimal resolution of each del Pezzo surface of type $8A_1$ is isomorphic to $Y$ by suitable choice of $Z$ by Proposition \ref{prop:deg=1} (4).
On the other hand, by Lemma \ref{lem:deg=2pre} (1) and (2), the contraction of $g(\Theta_{0,2})$ gives a morphism $h \colon Y \to W$ such that $t = h \circ g(\Theta_{0,2}) \in W \setminus B$.
Therefore $Y$ is the blow-up of $W$ at $t \in W \setminus B$.
\noindent (2):
By Lemma \ref{lem:deg=2pre} (1) and (2), $Y_t$ is obtained from a rational quasi-elliptic surface $Z_t$ of type (g) by contracting a section.
Hence the assertion follows from Proposition \ref{prop:deg=1} (4).
\noindent (3):
By Lemma \ref{lem:typeg}, $Z_t$ has exactly sixteen $(-1)$-curves $A_{0, 1}, \ldots, A_{7,1}$, $A_{0,2}, \ldots, A_{7, 2}$ and exactly sixteen $(-2)$-curves $\Theta_{0,1}, \ldots, \Theta_{7,1}$, $\Theta_{0,2}, \ldots, \Theta_{7,2}$, whose intersection matrix is Figure \ref{matrix(g)}.
We may assume that the contraction of $A_{0,2}$ gives a morphism $g_t \colon Z_t \to Y_t$ and $E_{h_t}=g_t(\Theta_{0,2})$.
Then $g_t(A_{0,1})$ is a $(0)$-curve and $A'_{i,j} \coloneqq g_t(A_{i,j})$ is a $(-1)$-curve for $1 \leq i \leq 7$ and $j=1,2$.
Moreover, $\Theta'_{i,1} \coloneqq g_t(\Theta_{i,1})$ and $\Theta'_{i,2} \coloneqq g_t(\Theta_{i,2})$ is a $(-2)$-curve and a $(-1)$-curve respectively for $0 \leq i \leq 7$.
Hence Figure \ref{matrix} is the intersection matrix of $A'_{1, 1}, \ldots, A'_{7,1}$, $A'_{1,2}, \ldots, A'_{7, 2}$, $\Theta'_{0,1}, \ldots, \Theta'_{7,1}$, $\Theta'_{0,2}, \ldots, \Theta'_{7,2}$ in this order.
Moreover, $E_{h_t}=\Theta'_{0,2}$ is a unique $(-1)$-curve intersecting with $C_t = \Theta'_{0,1}$ twice.
\noindent (4):
Suppose that an automorphism of $Y_t$ fixes each $(-1)$-curve and each $(-2)$-curve.
Then it fixes $A'_{1, 1}, \ldots, A'_{7,1}$, and $\Theta'_{0,2}$.
By Claim \ref{cl:typeg-2} in the proof of Lemma \ref{lem:typeg}, it descends to an automorphism of $\PP^2_k$ fixing all the $\FF_2$-rational points, which is the identity.
Thus an automorphism of $Y_t$ is determined by the image of all $(-1)$-curves and $(-2)$-curves.
Let $S_8$ be the permutation group of $\{0, 1, \ldots, 7\}$.
By Figure \ref{matrix}, the images of all $(-1)$-curves are determined by those of $(-2)$-curves $\Theta'_{0,1}, \ldots, \Theta'_{7,1}$.
Hence there is an injection $\iota \colon \Aut Y_t \to S_8$ which sends $\eta \in \Aut Y_t$ to $\sigma \in S_8$ such that $\eta(\Theta'_{i,1})=\Theta'_{\sigma(i),1}$ for $0 \leq i \leq 7$.
Moreover, $(0^8)$, $(1^8)$, and fourteen rows in the $(1, 3)$ block or $(2,3)$ block of Figure \ref{matrix} form the $[8,4,4]$ extended Hamming code, which is also the Reed-Muller code $R(1,3)$.
Hence $\iota$ factors through the automorphism group of $R(1,3)$, which is the affine linear group $\FF_2^3 \rtimes \mathrm{GL}(3, \FF_2) \subset S_8$ by \cite[Chapter 13, \S 9, Theorem 24]{Mac-Sloane}.
Since the normal group $\FF_2^3 \subset S_8$ is generated by $(01)(23)(45)(67)$, $(02)(13)(46)(57)$ and $(04)(15)(26)(37)$, it acts on $\{0, 1, \ldots, 7\}$ transitively.
Hence it suffices to show that $\FF_2^3 \subset \Aut Y_t$.
We show only the existence of $\eta \in \Aut Y_t$ such that $\iota(\eta) = (01)(23)(45)(67)$; the same proof works for $(02)(13)(46)(57)$ and $(04)(15)(26)(37)$.
Let $\varphi \colon Y_t \to \overline{Y}$ be the contraction of $A'_{1,1}$, $A'_{2,1}$, and $A'_{4,1}$.
Set $s_i=\varphi(A'_{i,1})$ for $i=1,2,$ and $4$.
Then $\overline{Y}$ is a smooth del Pezzo surface of degree four since each $(-2)$-curve in $Y_t$ intersects with $A'_{1,1}$, $A'_{2,1}$, or $A'_{4,1}$.
Generally speaking, a smooth del Pezzo surface of degree four contains sixteen $(-1)$-curves, and each $(-1)$-curve intersects with five $(-1)$-curves.
In the present case, $\overline{A}_{i,1}=\varphi(A'_{i,1})$, $\overline{A}_{i,2}=\varphi(A'_{i,2})$, $\overline{\Theta}_{j,1}=\varphi(\Theta'_{j,1})$, and $\overline{\Theta}_{k,1}=\varphi(\Theta'_{k,1})$ are $(-1)$-curves on $Y$ for $i=3,5,6$, or $7$, $2 \leq j \leq 7$, and $k=0,1$.
Figure \ref{matrixdeg5} is the intersection matrix of $\overline{A}_{3,1}, \ldots, \overline{A}_{7,1}$, $\overline{A}_{3,2}, \ldots, \overline{A}_{7,2}$, $\overline{\Theta}_{2,1}, \ldots, \overline{\Theta}_{7,1}$, $\overline{\Theta}_{0,2}$, and $\overline{\Theta}_{1,2}$ in this order.
In particular, $\mathcal{M}=(\overline{\Theta}_{1,2}, \overline{A}_{3,2}, \overline{A}_{5,2}, \overline{A}_{6,2}, \overline{A}_{7,2})$ and $\mathcal{M'}=(\overline{\Theta}_{0,2}, \overline{A}_{3,1}, \overline{A}_{5,1}, \overline{A}_{6,1}, \overline{A}_{7,1})$ are the 5-tuples of $(-1)$-curves intersecting with $\overline{\Theta}_{0,2}$ and $\overline{\Theta}_{1,2}$ respectively.
Since $\mathcal{M}$ and $\mathcal{M}'$ satisfy the condition (1) of \cite[Theorem 2.1]{Hos}, there is an automorphism $\overline{\eta}$ of $\overline{Y}$ which interchanges $\mathcal{M}$ with $\mathcal{M'}$.
By Figure \ref{matrixdeg5}, $\overline{\eta}$ also interchanges $\overline{\Theta}_{2,1}$ with $\overline{\Theta}_{3,1}$, $\overline{\Theta}_{4,1}$ with $\overline{\Theta}_{5,1}$, and $\overline{\Theta}_{6,1}$ with $\overline{\Theta}_{7,1}$.
Then $\overline{\eta}$ fixes $s_1 = \overline{\Theta}_{2,1} \cap \overline{\Theta}_{3,1}$, $s_2 = \overline{\Theta}_{4,1} \cap \overline{\Theta}_{5,1}$, and $s_4 = \overline{\Theta}_{6,1} \cap \overline{\Theta}_{7,1}$.
Hence $\overline{\eta}$ induces an automorphism $\eta \in \Aut Y_t$, which interchanges $\Theta'_{0,2}$ with $\Theta'_{1,2}$, $\Theta'_{2,1}$ with $\Theta'_{3,1}$, $\Theta'_{4,1}$ with $\Theta'_{5,1}$, and $\Theta'_{6,1}$ with $\Theta'_{7,1}$.
Since $\Theta'_{0,1}$ (resp.\ $\Theta'_{1,1}$) is the unique $(-2)$-curve which intersects with $\Theta'_{0,2}$ (resp.\ $\Theta'_{1,2}$) twice, $\eta$ also interchanges $\Theta'_{0,1}$ with $\Theta'_{1,1}$.
Hence $\iota(\eta) = (01)(23)(45)(67)$, and the assertion holds.
\begin{figure}[t]
\centering
\begin{align*}
{\fontsize{7pt}{4}\selectfont
\left(
\vcenter{
\xymatrix@=-1.8ex{
&&&&&\ar@{-}[ddddddddddddddddddddd]&&&&& \ar@{-}[ddddddddddddddddddddd] &&&&&&& \ar@{-}[ddddddddddddddddddddd]&&\\
&&&&& &&&&& &&&&&&&& &&\\
&-1&0 &0 &0 &&1 &0 &0 &0 &&1 &0 &1 &0 &1 &0 &&0 &1 &\\
& &-1&0 &0 &&0 &1 &0 &0 &&0 &1 &1 &0 &0 &1 &&0 &1 &\\
& & &-1&0 &&0 &0 &1 &0 &&1 &0 &0 &1 &0 &1 &&0 &1 &\\
& & & &-1&&0 &0 &0 &1 &&0 &1 &0 &1 &1 &0 &&0 &1 &\\
\ar@{-}[rrrrrrrrrrrrrrrrrrrr]&&&&& &&&&& &&&&&&&& &&\\
& & & & &&-1&0 &0 &0 &&0 &1 &0 &1 &0 &1 &&1 &0 &\\
& & & & && &-1&0 &0 &&1 &0 &0 &1 &1 &0 &&1 &0 &\\
& & & & && & &-1&0 &&0 &1 &1 &0 &1 &0 &&1 &0 &\\
& & & & && & & &-1&&1 &0 &1 &0 &0 &1 &&1 &0 &\\
\ar@{-}[rrrrrrrrrrrrrrrrrrrr]&&&&& &&&&& &&&&&&&& &&\\
& & & & && & & & &&-1&1 &0 &0 &0 &0 &&0 &0 &\\
& & & & && & & & && &-1&0 &0 &0 &0 &&0 &0 &\\
& & & & && & & & && & &-1&1 &0 &0 &&0 &0 &\\
& & & & && & & & && & & &-1&0 &0 &&0 &0 &\\
& & & & && & & & && & & & &-1&1 &&0 &0 &\\
& & & & && & & & && & & & & &-1&&0 &0 &\\
\ar@{-}[rrrrrrrrrrrrrrrrrrrr]&&&&& &&&&& &&&&&&&& &&\\
& & & & && & & & && & & & & & &&-1&1 &\\
& & & & && & & & && & & & & & && &-1&\\
&&&&& &&&&& &&&&&&&& &&\\
}
}\right)
}.
\end{align*}
\caption{The intersection matrix of the $(-1)$-curves in $\overline{Y}$}
\label{matrixdeg5}
\end{figure}
\noindent (5):
If some automorphism of $W$ sends $t$ to $t'$, then it ascends to an isomorphism $Y_t \cong Y_{t'}$.
On the other hand, suppose that there is an isomorphism $Y_{t} \cong Y_{t'}$.
By the assertion (4), we may assume that this isomorphism sends $C_t$ to $C_{t'}$.
Then by the assertion (3), it also sends $E_{h_t}$ to $E_{h_{t'}}$.
Hence it descends to an isomorphism of $W$, which sends $t$ to $t'$.
\end{proof}
\begin{cor}\label{8A_1aut}
Let $X_t$ be the contraction of all $(-2)$-curves in $Y_t$ as in Proposition \ref{8A_1isom}.
Then $\Aut X_t \cong \Aut Y_t \cong (\ZZ/2\ZZ)^3 \rtimes \ZZ/7\ZZ$ when $t$ is an $\FF_8$-rational point and $(\ZZ/2\ZZ)^3$ otherwise.
In particular, there is a unique Du Val del Pezzo surface $X(8A_1)$ such that $\Aut X \cong (\ZZ/2\ZZ)^3 \rtimes \ZZ/7\ZZ$.
\end{cor}
\begin{proof}
We follow the notation of Proposition \ref{8A_1isom}.
Let $\Sigma \subset \Aut Y_t$ be the stabilizer subgroup with respect to $C_t$.
Since $\FF_2^3 \cong (\ZZ/2\ZZ)^3$ is a normal subgroup of $\Aut Y_t$ which acts on the set of $(-2)$-curves in $Y_t$ transitively, we obtain $\Aut Y_t \cong (\ZZ/2\ZZ)^3 \rtimes \Sigma$.
By Proposition \ref{8A_1isom} (3), $\Sigma$ is the same as the stabilizer subgroup of $\PGL(3, \FF_2)$ with respect to $t \in \mathcal{D}_2$.
Now the first assertion follows from Lemma \ref{lem:D2aut}.
Since $\PGL(3, \FF_2)$ acts on $\mathcal{D}_2(\FF_8)$ transitively, the second assertion follows from Proposition \ref{8A_1isom} (5).
\end{proof}
\begin{cor}\label{Itorem}
There are one-to-one correspondences between the isomorphism classes of rational quasi-elliptic surfaces of type (d), (f), and (g), and the closed points of $\mathcal{D}_1 /\PGL (2, \FF_2)$, $\mathcal{D}_1 /\PGL (2, \FF_2)$, and $\mathcal{D}_2 /\PGL (3, \FF_2)$ respectively.
\end{cor}
\begin{proof}
By Proposition \ref{prop:deg=1}, there is one-to-one correspondence between isomorphism classes of del Pezzo surfaces of type $2D_4$ satisfying (NB) (resp.\ type $4A_1+D_4$, type $8A_1$) and those of rational quasi-elliptic surfaces of type (d) (resp.\ (f), (g)).
Hence the assertion follows from Propositions \ref{2D_4isom}, \ref{4A_1D_4isom} and \ref{8A_1isom}.
\end{proof}
Now we can prove Theorem \ref{sing}.
\begin{proof}[Proof of Theorem \ref{sing}]
The assertions (0), (1), and (2) follow from Lemma \ref{basic}, Proposition \ref{prop:deg=1}, and Propositions \ref{sep} and \ref{insep} respectively.
The assertion (3) follows from Propositions \ref{prop:p=3isom}, \ref{E_7}, \ref{A_1D_6}, \ref{3A_1D_4}, \ref{7A_1}, \ref{prop:deg1isom}, \ref{2D_4isom}, \ref{4A_1D_4isom}, and \ref{8A_1isom}.
\end{proof}
\subsection{List of automorphism groups}
As a consequence, we obtain the list of automorphisms of Du Val del Pezzo surfaces satisfying (NB) and rational quasi-elliptic surfaces as follows.
\begin{thm}\label{auto}
Let $X$ be a Du Val del Pezzo surface satisfying (NB).
Then $\Aut X$ is described in Table \ref{table:auto}.
Furthermore, suppose that $p=2$.
Then for each of types $2D_4$, $4A_1+D_4$, and $8A_1$, there is a unique del Pezzo surface of the given type such that the group $G$ in Table \ref{table:auto} is non-trivial.
\begin{table}[htbp]
\caption{}
\begin{tabular}{|c|c|c|} \hline
$\Dyn(X)$ & Characteristic & Automorphism groups \\ \hline \hline
\multirow{2}{*}{$E_8$} &$p=2$ &
$\left\{
\begin{psmallmatrix}
a&0&0\\
0&1&f\\
0&0&a^3
\end{psmallmatrix}
\in \PGL(3, k) \middle| a\in k^*, f\in k
\right\}$
\\ \cline{2-3}
&\multirow{3}{*}{$p=3$} &
$\left\{
\begin{psmallmatrix}
a&0&c\\
0&1&0\\
0&0&a^3
\end{psmallmatrix}
\in \PGL(3, k) \middle| a\in k^*, c\in k
\right\}$
\\ \cline{1-1} \cline{3-3}
$A_2+E_6$ & &$k^* \times \ZZ/2\ZZ$ \\ \cline{1-1} \cline{3-3}
$4A_2$ & &$\mathrm{GL}(2, \FF_3)$ \\ \hline
$D_8$ &\multirow{10}{*}{$p=2$} &$k$ \\ \cline{1-1} \cline{3-3}
$A_1+E_7$ & &$k^*$\\ \cline{1-1} \cline{3-3}
$2D_4$ & &$(k^* \times G) \rtimes \ZZ/2\ZZ$ with $G=\{1\}$ or $\ZZ/3\ZZ$ \\ \cline{1-1} \cline{3-3}
$2A_1+D_6$ & &$\ZZ/2\ZZ$ \\ \cline{1-1} \cline{3-3}
$4A_1+D_4$ & &$(\ZZ/2\ZZ)^2 \rtimes G$ with $G=\{1\}$ or $\ZZ/3\ZZ$ \\ \cline{1-1} \cline{3-3}
$8A_1$ & &$(\ZZ/2\ZZ)^3 \rtimes G$ with $G=\{1\}$ or $\ZZ/7\ZZ$ \\ \cline{1-1} \cline{3-3}
$E_7$ & &$\left\{
\begin{psmallmatrix}
a&0&d^2a\\
d&1&f\\
0&0&a^3
\end{psmallmatrix}
\in \PGL(3, k) \middle| a \in k^*, d \in k, f \in k
\right\}$ \\ \cline{1-1} \cline{3-3}
$A_1+D_6$ & &
$ \left\{
\begin{psmallmatrix}
a&0&a^3+a\\
d&1&a^3+d+1\\
0&0&a^3
\end{psmallmatrix}
\in \PGL(3, k) \middle| a \in k^*, d \in k
\right\}$\\ \cline{1-1} \cline{3-3}
$3A_1+D_4$ & &$k^* \times \PGL(2, \FF_2)$ \\ \cline{1-1} \cline{3-3}
$7A_1$ & &$\PGL(3, \FF_2)$ \\ \hline
\end{tabular}
\label{table:auto}
\end{table}
\end{thm}
\begin{proof}
The assertion follows from Corollaries \ref{cor:deg=1p=3auto}, \ref{E_7aut}, \ref{A_1D_6aut}, \ref{3A_1D_4aut}, \ref{7A_1aut}, Lemmas \ref{lem:deg=1p=2auto}, \ref{2A_1D_6aut}, Corollaries \ref{2D_4aut}, \ref{4A_1D_4aut}, and \ref{8A_1aut}.
\end{proof}
\begin{cor}\label{cor:q-ellaut}
Let $Z$ be a rational quasi-elliptic surface and $O \subset Z$ a section.
Take $g \colon Z \to Y$ as the contraction of $O$ and $\pi \colon Y \to X$ the contraction of all the $(-2)$-curves.
Then $\Aut Z \cong \MW (Z) \cdot \Aut X$.
In particular, $\Aut Z \cong (\ZZ/p\ZZ)^n \cdot H$ for some $0 \leq n \leq 4$ and for some group $H$ listed in Table \ref{table:auto}.
\end{cor}
\begin{proof}
Note that $X$ is a Du Val del Pezzo surface satisfying (NB) by Proposition \ref{prop:deg=1} (1) and $\Aut Y \cong \Aut X$ since $\pi$ is the minimal resolution.
Since $h(O)$ is the base point of $|-K_Y|$, $h$ induces an isomorphism between $\Aut Y$ and the stabilizer subgroup of $\Aut Z$ with respect to $O$.
Hence the first assertion follows from the transitivity of the $\MW (Z)$-action on the set of sections on $Z$.
The second assertion follows from Theorems \ref{thm:q-ell3}, \ref{q-ell}, and \ref{auto}.
\end{proof}
\begin{rem}\label{autorem}
We follow the notation in Corollary \ref{cor:q-ellaut}.
We have described the reduced scheme structure of $\Aut Y$ and $\Aut Z$.
We can also describe the scheme structure of them by virtue of \cite[Main Theorem]{M-S}, which calculates the identity component of $\Aut Y$ as a scheme.
On the other hand, what is still lacking is the determination of the scheme structure of $\Aut X$ since the contraction of $(-2)$-curves may thicken the scheme structures of the automorphism groups.
For example, smooth K3 surfaces in characteristic $p>0$ admit no non-trivial $\mu_p$-actions but RDP K3 surfaces may admit such actions (see \cite[Remark 2.3]{Mat17}).
\end{rem}
\section{Log liftability}
\label{sec:singliftable}
In this section, we determine all the Du Val del Pezzo surfaces which are not log liftable over $W(k)$.
Note that by Theorem \ref{smooth, Intro} (1), it suffices to consider Du Val del Pezzo surfaces satisfying (NB).
\begin{prop}\label{p=3liftable}
Let $X$ be a Du Val del Pezzo surface satisfying (NB) and $\pi \colon Y \to X$ the minimal resolution.
Suppose that $p=3$ and $\Dyn(X)=E_8$ or $A_2+E_6$.
Then the pair of $(Y, E_\pi)$ lifts to $\Spec \Z$ via $\Spec \FF_3 \to \Spec \Z$.
As a result, $X$ is log liftable both over $\Z$ via $\Spec \FF_3 \to \Spec \Z$ and over $W(k)$.
\end{prop}
\begin{proof}
Note that $Y$ and each $(-2)$-curve on $Y$ are defined over $\FF_3$ by Proposition \ref{prop:p=3isom} (6).
Suppose that $\Dyn(X)=E_8$.
Take a birational morphism $h'_{\ZZ} \colon Y_{\ZZ} \to \PP^2_{\ZZ}$ as the blow-up at $[0:1:0]$ eight times along $\{x^3+y^2z=0\}$.
By Proposition \ref{prop:p=3isom} (3), we have $Y \cong Y_{\ZZ} \otimes_\ZZ \FF_3$ and each negative rational curve on $Y$ is the specialization of either an $h'_{\ZZ}$-exceptional curve or the strict transform of $\{z=0\} \subset \PP^2_{\ZZ}$ via $h'_{\ZZ}$.
Hence we obtain the desired lift.
The proof for the case where $\Dyn(X)=A_2+E_6$ is similar by virtue of Proposition \ref{prop:p=3isom} (4).
\end{proof}
\begin{prop}\label{p=2liftable}
Let $X$ be a Du Val del Pezzo surface satisfying (NB) and $\pi \colon Y \to X$ the minimal resolution.
Suppose that $p=2$.
Then the following hold.
\begin{enumerate}
\item[\textup{(1)}] Suppose that $\Dyn(X)=E_7$, $A_1+D_6$, or $3A_1+D_4$.
Then the log smooth pair of $Y$ and the union $B$ of negative rational curves lifts to $\Spec \Z$ via $\Spec \FF_2 \to \Spec \Z$.
\item[\textup{(2)}] Suppose that $\Dyn(X)=E_8$, $D_8$, $A_1+E_7$, or $2A_1+D_6$.
Then the pair $(Y, E_\pi)$ lifts to $\Spec \Z$ via $\Spec \FF_2 \to \Spec \Z$.
\end{enumerate}
As a result, $X$ is log liftable both over $\Z$ via $\Spec \FF_2 \to \Spec \Z$ and over $W(k)$.
\end{prop}
\begin{proof}
By Theorem \ref{sing}, $X$ is uniquely determined up to isomorphism by $\Dyn(X)$.
Moreover, we have shown in \S \ref{sec:singisom} that $Y$ and each negative rational curve on $Y$ are defined over $\FF_2$.
\noindent (1):
By Lemma \ref{lem:deg=2pre} (3), the pair $(Y, B)$ is log smooth.
Now suppose that $\Dyn(X)=E_7$.
Take a birational morphism $h'_{\ZZ} \colon Y_{\ZZ} \to \PP^2_{\ZZ}$ as the blow-up at $[0:1:0]$ seven times along $\{x^3+y^2z=0\}$.
By Proposition \ref{E_7}, we have $Y \cong Y_{\ZZ} \otimes_\ZZ \FF_2$ and each negative rational curve on $Y$ is the specialization of either an $h'_{\ZZ}$-exceptional curve or the strict transform of $\{z=0\} \subset \PP^2_{\ZZ}$ via $h'_{\ZZ}$.
Hence we obtain the desired lift.
The proof for the cases where $\Dyn(X)=A_1+D_6$ and $3A_1+D_4$ is similar by virtue of Corollary \ref{A_1D_6aut} (3) and Proposition \ref{3A_1D_4} respectively.
\noindent (2):
By Proposition \ref{prop:deg=1} (4) and Lemma \ref{lem:deg=2pre}
(5)--(7), for some Du Val del Pezzo surface of type $E_7$, $A_1+D_6$, or $3A_1+D_4$ satisfying (NB) and its minimal resolution $W$, there exist a $(-1)$-curve $E \subset W$ and an $\FF_2$-rational point $t \in E$ not contained in any $(-2)$-curves such that $Y$ is the blow-up of $W$ at $t$.
Hence the assertion follows from the assertion (1) and \cite[Proposition 2.9]{ABL}.
\end{proof}
By Proposition \ref{2D_4isom}, there are infinitely many Du Val del Pezzo surfaces of type $2D_4$ satisfying (NB).
In particular, they are not defined over $\FF_2$ in general.
On the other hand, we can show their log liftability over $W(k)$ as follows.
\begin{prop}
Let $X$ be a Du Val del Pezzo surface of type $2D_4$ satisfying (NB) in $p=2$.
Take $R$ as a Noetherian irreducible ring with surjective ring homomorphism $f \colon R \rightarrow k$.
Then $X$ is log liftable over $R$ via the induced morphism $\Spec k \to \Spec R$.
\end{prop}
\begin{proof}
Let $\pi \colon Y \to X$ be the minimal resolution.
By Proposition \ref{2D_4isom} (1), on the minimal resolution $W$ of the Du Val del Pezzo surface of type $3A_1+D_4$ satisfying (NB), there are the $(-1)$-curve $E \subset W$ intersecting with exactly three $(-2)$-curves and a closed point $t \in E$ not contained in any $(-2)$-curves such that $Y$ is the blow-up of $W$ at $t$.
Let $D$ be the union of the $(-2)$-curves in $W$.
By Proposition \ref{p=2liftable} (1), the log smooth pair $(W, D \cup E)$ lifts to $\Spec \ZZ$ via $\Spec \FF_2 \to \Spec \ZZ$.
Take $(\mathcal{W}, \mathcal{D} \cup \mathcal{E})$ as the base change of such a lifting by the natural homomorphism $\ZZ \to R$.
Fix coordinates $[x:y]$ of $\mathcal{E} \cong \PP^1_R$ and choose $a, b \in k$ so that
$t=[a:b] \in \PP^1_{k, [x:y]}$.
Since $f\colon R\to k$ is surjective, we can take a lifting $\tilde{a}$ (resp.~$\tilde{b}$) of $a$ (resp.~$b$).
Then $\tilde{t}= [\tilde{a}:\tilde{b}]\in \mathcal{E} \cong \PP^1_R$ is a lifting of $t$.
Let $\Phi\colon \mathcal{Y} \to \mathcal{W}$ be the blow-up along $\tilde{t}$.
Then $(\mathcal{Y}, \Phi^{-1}_{*}(\mathcal{D} \cup \mathcal{E}))$ is the desired lift.
\end{proof}
\begin{prop}\label{ND}
Let $X$ be a Du Val del Pezzo surface with $\Dyn(X)=4A_1+D_4$, $8A_1$, or $7A_1$.
Then $X$ is not log liftable over any Noetherian integral domain $R$ of characteristic zero via any morphism $\Spec k \to \Spec R$ induced by a surjective homomorphism $R \to k$.
\end{prop}
\begin{proof}
By \cite[Theorem 1.2]{Ye}, the surface $X$ satisfies (ND).
Hence the assertion follows from Proposition \ref{NDtoNL}.
\end{proof}
\begin{prop}\label{4A_2lift}
Let $X$ be a Du Val del Pezzo surface of type $4A_2$ in $p=3$.
Then $X$ is not log liftable over $W(k)$.
\end{prop}
\begin{proof}
We note that $X$ satisfies (NB) by Proposition \ref{4A_2NB}.
Suppose by contradiction that $X$ is log liftable over $W(k)$.
Take $\pi \colon Y \to X$ as the minimal resolution and $(\mathcal{Y}, \mathcal{E})$ as a $W(k)$-lifting of $(Y, E_{\pi})$.
We follow the notation used in the proof of Proposition \ref{NDtoNL}.
Then the blow-up $Z_K \to Y_K$ at the base point of $|-K_{Y_K}|$ gives the anti-canonical morphism $f_K \colon Z_K \to \PP^1_K$.
Let $G$ be the strict transform of $E_K=\sum_{i=1}^8 E_{i, K}$ in $Z_K$.
Then $f_K(G)$ consists of four $K$-rational points.
We fix coordinates of $\PP^1_K$ such that $f(G)=\{0, 1, \infty, \alpha\}$ for some $\alpha \in \PP^1_K \setminus \{0,1,\infty\}$.
On the other hand, by Proposition \ref{NDtoNL}, $X_{\C}$ is the del Pezzo surface of degree one of type $4A_2$.
By \cite[Table 4.1]{Ye}, the blow-up $Z_{\C} \to Y_{\C}$ at the base point of $|-K_{Y_{\C}}|$ gives an elliptic fibration $f_{\C} \colon Z_{\C} \to \PP^1_{\C}$ with four singular fibers of type $\textup{I}_3$.
Since $f_K(G) \subset \PP^1_{\C}$ is the singular fiber locus of $f_{\C}$, \cite[Th\'{e}or\`{e}me]{Bea} now yields the existence of $\sigma \in \Aut \PP^1_{\C}$ which sends $f_K(G)$ to $\{1, \omega, \omega^2, \infty\}$, where $\omega$ is a primitive cube root of unity.
An easy computation shows that $\alpha=-\omega$ and hence $\omega \in K$.
However, by the Eisenstein criterion and the Gauss lemma, the cyclotomic polynomial $t^2+t+1$ is irreducible in $K[t]$, a contradiction.
Therefore $(Y, E_{\pi})$ does not lift to $W(k)$.
\end{proof}
\begin{rem}
One question still unanswered is whether $X(4A_2)$ in $p=3$ is log liftable over any Noetherian integral domain of characteristic zero.
\end{rem}
\begin{rem}\label{remliftable}
As we saw in the proof of Propositions \ref{7A_1}, \ref{4A_1D_4isom}, and \ref{8A_1isom} (resp.\ Proposition \ref{prop:p=3isom} (4)), the surfaces as in Proposition \ref{ND} (resp.\ Proposition \ref{4A_2lift}) are obtained from the configuration of all the lines in $\PP^2_k$ defined over $\FF_p$, which is not realizable in $\PP_{\C}^2$ by the Hirzebruch inequality for line arrangements (see \cite{Hir} and \cite[Example 3.2.2]{Miy}).
This is the reason why we cannot apply the proof of Proposition \ref{p=3liftable} for such surfaces.
\end{rem}
\section{Kodaira type vanishing theorem}\label{sec:KV}
In this section, we determine all the Du Val del Pezzo surfaces which violate the Kodaira vanishing theorem for ample $\Z$-divisors. Note that by Theorem \ref{smooth, Intro} (3), it suffices to consider Du Val del Pezzo surfaces satisfying (NL).
\begin{lem}[\textup{cf.~\cite[Theorem 4.8]{Kaw2}}]\label{lem:KV-1}
Let $X$ be a Du Val del Pezzo surface and $A$ an ample $\Z$-divisor on $X$. If $H^1(X, \sO_X(-A))\neq 0$, then $p=2$ and $(-K_X \cdot A)=1$.
\end{lem}
\begin{proof}
We refer to the proof of \cite[Theorem 4.8]{Kaw2} for the details.
\end{proof}
\begin{prop}\label{KV:8A_1}
Let $X$ be a del Pezzo surface of type $8A_1$.
Then there is an ample $\Z$-divisor $A$ such that $H^1(X, \sO_X(-A)) \neq 0$.
\end{prop}
\begin{proof}
We follow the notation of the proof of Proposition \ref{8A_1isom}.
Let $\pi \colon Y \to X$ be the minimal resolution and
$A \coloneqq \pi_*(A'_{1,1}+A'_{2,1}-A'_{4,1})$.
Then $A$ is ample since $\rho(X)=1$ and $(-K_X \cdot A)=1$.
By Figure \ref{matrix}, we have
\begin{align*}
&\lceil \pi^* A \rceil\\
=&\lceil A'_{1,1}+\frac12(\Theta'_{0,1}+\Theta'_{1,1}+\Theta'_{2,1}+\Theta'_{3,1})+A'_{2,1}+\frac12(\Theta'_{0,1}+\Theta'_{1,1}+\Theta'_{4,1}+\Theta'_{5,1})\\
&-A'_{4,1}-\frac12(\Theta'_{0,1}+\Theta'_{1,1}+\Theta'_{6,1}+\Theta'_{7,1}) \rceil \\
=&A'_{1,1} + A'_{2,1} - A'_{4,1} + \Theta'_{0,1} + \Theta'_{1,1} + \Theta'_{2,1} + \Theta'_{3,1} + \Theta'_{4,1} + \Theta'_{5,1}
\end{align*}
In particular, $\lceil \pi^* A \rceil^2=-3$ and $(-K_Y \cdot \lceil \pi^* A \rceil)=1$. Lemma \ref{lem:KV-2} now yields $H^{i}(Y, \sO_Y(-\lceil \pi^* A \rceil))=H^{i}(X, \sO_X(-A))$ for $i \geq 0$.
Since $\lceil \pi^* A \rceil$ is big, we have $H^0(Y, \sO_Y(-\lceil \pi^* A \rceil))=0$.
Next assume that $H^2(Y, \sO_Y(-\lceil \pi^* A \rceil)) \neq 0$.
Then there is an effective divisor $C \sim K_Y+\lceil \pi^* A \rceil$ by the Serre duality.
Since $(-K_Y \cdot C)=0$, the curve $C$ is a sum of $(-2)$-curves.
Since $(-2)$-curves in $Y$ are disjoint from each other, we have $(C \cdot \Theta'_{0,1}) \in 2 \ZZ$.
However, $(C \cdot \Theta'_{0,1})=(\lceil \pi^* A \rceil \cdot \Theta'_{0,1})=-1$ by Figure \ref{matrix}, a contradiction.
Combining these results and the Riemann-Roch theorem, we conclude that
\begin{align*}
\dim_k H^1(X, \sO_X(-A))
&=\dim_k H^1(Y, \sO_Y(-\lceil \pi^* A \rceil))\\
&=-\chi(Y, \sO_Y(-\lceil \pi^* A \rceil))\\
&=-(\chi(Y, \sO_Y)+\frac12((-\lceil \pi^* A \rceil)^2+(-K_Y \cdot -\lceil \pi^* A \rceil)))=1.
\end{align*}
Therefore $H^1(X, \sO_X(-A)) \neq 0$.
\end{proof}
\begin{prop}\label{KV:4A_1+D_4}
Let $X$ be a del Pezzo surface of type $4A_1+D_4$.
Then $H^1(X, \sO_X(-A))=0$ for any ample $\Z$-divisor $A$.
\end{prop}
\begin{proof}
Let $\pi \colon Y \to X$ be the minimal resolution.
By Proposition \ref{prop:deg=1} (4), there exist a rational quasi-elliptic surface $Z$ of type (f) and a section $O$ such that the contraction of $O$ gives a birational morphism $g \colon Z \to Y$.
In what follows, we use the notation of Figure \ref{fig:dual(f)}.
For a birational morphism $Z \to S$ and a curve $C \subset Z$, we denote $(C)_S$ the strict transform of $C$ in $S$.
By Lemma \ref{lem:deg=2pre} (8), the contraction of $(\Theta_{1,0})_Y$ gives a morphism $h \colon Y \to W$ to the minimal resolution of the Du Val del Pezzo surface $V$ of type $7A_1$.
Let $\xi \colon W \to V$ be the contraction of all the $(-2)$-curves and $\nu = \xi \circ h$.
By Corollary \ref{7A_1aut} (2), the class divisor group of $W$ is generated by $(\Theta_{1, 4})_W$, $(R_2)_W$, $(Q_2)_W$, $(R_1)_W$, $(Q_1)_W$, $(P_3)_W$, $(P_2)_W$, and any one of $(-2)$-curves.
Since the point $h((\Theta_{1,0})_Y)$ lies on $(\Theta_{1, 4})_W$ and $\pi$ contracts all the $(-2)$-curves, the class divisor group of $X$ is generated by
$(R_2)_X$, $(Q_2)_X$, $(R_1)_X$, $(Q_1)_X$, $(P_3)_X$, $(P_2)_X$, and $(\Theta_{1,0})_X$, whose anti-canonical degrees are one.
Then an easy computation shows the following.
\begin{align*}
\pi^* (\Theta_{1,0})_X
&=&&(\Theta_{1, 0})_Y + (\Theta_{1, 1})_Y + (\Theta_{1, 2})_Y + (\Theta_{1, 3})_Y + 2(\Theta_{1, 4})_Y\\
&=&&2((\Theta_{1, 0})_Y + \frac12(\Theta_{1, 1})_Y + \frac12(\Theta_{1, 2})_Y + \frac12(\Theta_{1, 3})_Y + (\Theta_{1, 4})_Y)-(\Theta_{1, 0})_Y\\
&=&&2\nu^*(\Theta_{1,4})_V - (\Theta_{1, 0})_Y,\\
\pi^* (Q_1)_X
&=&&(Q_1)_Y + \frac12 (\Theta_{0, 1})_Y + \frac12 (\Theta_{\alpha_1, 1})_Y\\
&&&+ \frac12 (\Theta_{1, 1})_Y + (\Theta_{1, 2})_Y + \frac12 (\Theta_{1, 3})_Y + (\Theta_{1, 4})_Y\\
&=&&((Q_1)_Y + \frac12 (\Theta_{0, 1})_Y + \frac12 (\Theta_{\alpha_1, 1})_Y + \frac12 (\Theta_{1, 2})_Y)\\
&&&+ ((\Theta_{1, 0})_Y + \frac12 (\Theta_{1, 1})_Y + \frac12 (\Theta_{1, 2})_Y + \frac12 (\Theta_{1, 3})_Y + (\Theta_{1, 4})_Y) - (\Theta_{1, 0})_Y\\
&=&&\nu^*(Q_1)_V + \nu^*(\Theta_{1,4})_V - (\Theta_{1, 0})_Y,\\
\pi^* (R_1)_X&=&&\nu^*(R_1)_V + \nu^*(\Theta_{1,4})_V - (\Theta_{1, 0})_Y,\\
\pi^* (Q_2)_X&=&&\nu^*(Q_2)_V + \nu^*(\Theta_{1,4})_V - (\Theta_{1, 0})_Y,\\
\pi^* (R_2)_X&=&&\nu^*(R_2)_V + \nu^*(\Theta_{1,4})_V - (\Theta_{1, 0})_Y,\\
\pi^* (P_2)_X&=&&\nu^*(P_2)_V + \nu^*(\Theta_{1,4})_V - (\Theta_{1, 0})_Y,\\
\pi^* (P_3)_X&=&&\nu^*(P_3)_V + \nu^*(\Theta_{1,4})_V - (\Theta_{1, 0})_Y.
\end{align*}
Now let us show the assertion.
Let $A$ be an ample $\Z$-divisor on $X$.
By Lemma \ref{lem:KV-2}, we only have to show that $H^1(Y, \sO_Y(-\lceil \pi^*A \rceil))=0$.
By Lemma \ref{lem:KV-1}, we may assume that $(-K_X \cdot A)=1$.
Then $A \sim n_1 (R_2)_X + n_2 (Q_2)_X + n_3 (R_1)_X + n_4 (Q_1)_X + n_5 (P_3)_X + n_6 (P_2)_X + n_7 (\Theta_{1,0})_X$ with $n_1+ \cdots +n_7=1$.
Set $B= n_1 (R_2)_V + n_2 (Q_2)_V + n_3 (R_1)_V + n_4 (Q_1)_V + n_5 (P_3)_V + n_6 (P_2)_V + n_7 (\Theta_{1,4})_V$.
Then we obtain $\pi^* A =\nu^*(B + (\Theta_{1,4})_V) - (\Theta_{1, 0})_Y$.
Since $\nu$ sends $E_h = (\Theta_{1, 0})_Y$ to a smooth point of $V$, the support of $\lceil \nu^*(B + (\Theta_{1,4})_V) \rceil - \nu^*(B + (\Theta_{1,4})_V)$ is contained in $E_\xi$.
Since $E_h$ is disjoint from $E_\xi$ in $Y$, we obtain
$(E_h \cdot \lceil \pi^*A \rceil)
= (E_h \cdot \nu^*(B + (\Theta_{1,4})_V) -E_h) =1$.
Hence we have an exact sequence
\begin{align*}
\xymatrix@C=10pt{
0 \ar[r] &\sO_Y(-\lceil \nu^*(B + (\Theta_{1,4})_V) \rceil ) \ar[r] & \sO_Y(-\lceil \pi^*A \rceil) \ar[r] & \sO_{E_h}(-1) \ar[r] & 0.
}
\end{align*}
Thus $H^1(Y, \sO_Y(-\lceil \pi^*A \rceil)) \cong H^1(Y, \sO_Y(-\lceil \nu^*(B + (\Theta_{1,4})_V) \rceil ))$.
Since $(Y, E_\nu)$ is a log smooth pair, Lemma \ref{lem:KV-2} yields $H^1(Y, \sO_Y(-\lceil \nu^*(B + (\Theta_{1,4})_V) \rceil )) \cong H^1(V, \sO_V(-(B + (\Theta_{1,4})_V))).$
Since $(-K_V \cdot B+(\Theta_{1,4})_V)=2$, Lemma \ref{lem:KV-1} yields $H^1(V, \sO_V(-(B + (\Theta_{1,4})_V))) \cong 0$.
Hence the assertion holds.
\end{proof}
Now we can prove Theorem \ref{pathologies}.
\begin{proof}[Proof of Theorem \ref{pathologies}]
By Theorem \ref{smooth, Intro}, it suffices to show the assertions when $X$ satisfies (NB), i.e., $X$ is listed in Table \ref{table:sing}.
Then the assertions (1) and (2) follow from Propositions \ref{p=3liftable}--\ref{4A_2lift} and \cite[Theorem 2, Table (II)]{Fur} respectively.
Finally, we show the assertion (3).
Suppose that $X$ satisfies (NK).
Then $p=2$ and $X$ satisfies (NL) by Lemma \ref{lem:KV-1} and Theorem \ref{smooth, Intro} (3) respectively.
The assertion (1) now shows that $\Dyn(X)=7A_1$, $8A_1$, or $4A_1+D_4$.
If $\Dyn(X)=7A_1$, then $X$ satisfies (NK) by \cite[Theorem 4.2 (6)]{CT19} with $(d, q_1, q_2)=(3, 1, 2)$.
If $\Dyn(X)=8A_1$, then $X$ satisfies (NK) by Proposition \ref{KV:8A_1}.
If $\Dyn(X)=4A_1+D_4$, then $X$ does not satisfy (NK) by Proposition \ref{KV:4A_1+D_4}.
Hence we get the assertion (3).
\end{proof}
\section*{Acknowledgements}
The authors would like to thank Professor Keiji Oguiso and Professor Shunsuke Takagi for their helpful advice and comments.
They are indebted to Professor Hiroyuki Ito for helpful advice on rational quasi-elliptic surfaces.
They would like to thank the referee for valuable advice which improved the paper.
Discussions with Teppei Takamatsu, Yuya Matsumoto, and Takeru Fukuoka on the automorphism groups of Du Val del Pezzo surfaces have been insightful.
They are grateful to Jakub Witaszek for letting them know about Remark \ref{lift remark}.
The authors would like to thank Fabio Bernasconi for telling them the paper \cite{AZ}.
They also wish to express their gratitude to Shou Yoshikawa, Yohsuke Matsuzawa, and Naoki Koseki for helpful discussions and comments.
The authors are supported by JSPS KAKENHI Grant Number JP19J21085 and JP19J14397.
\input{list.bbl}
\end{document}
| 1,947
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Police have widened their probe into the Andrew Mitchell "plebgate" row amid allegations that a police officer tried to "blacken" the politician's name.
Some 30 officers are now working on the investigation and will look at claims the officer posed as a member of the public and falsely claimed to have witnessed the argument.
Scotland Yard confirmed it will also be examining whether there was any sort of "conspiracy" to smear the then Tory Chief Whip as part of a "large scale and complex investigation".
The row was revived after claims the officer wrote an email to his local MP giving details of of Mr Mitchell's behaviour when he was prevented from cycling through the Downing Street gates.
There are also fresh questions after CCTV footage of the altercation on September 19 emerged and appeared to conflict with the official police version of events.
David Cameron said at PMQs: Independent Police Complaints Commission will be supervising the investigation and I think we should allow them to get to the truth."
Mr Mitchell, who eventually quit in October after a month under intense pressure, has claimed he was the victim of a "stitch-up" and is demanding a full inquiry.
In an earlier statement, Number 10 described allegations that an officer pretending to be a bystander and had fabricated evidence as "exceptionally serious".
Scotland Yard has vowed to establish the truth "as quickly as possible" but warned "the investigation will not be short".
Mr Mitchell was thrust to the centre of a political storm three months ago when a police report about his rant at the Downing Street officer was leaked to the press.
It claimed the senior Tory had warned the policeman: "Best you learn your f****** place. You don't run this f****** government. You're f****** plebs."
The politician has always denied using the word "plebs", although he did admit swearing and getting angry. Instead, he claims he said: "I thought you guys were supposed to f****** help us."
But the "pleb" claim was seized on by the Police Federation and Labour who demanded that he stand down.
The email, now known to be from a fellow police officer, allegedly helped fuel the row and keep up the momentum that eventually cost Mr Mitchell his job.
The policeman wrote to his MP John Randall, apparently not disclosing his job and describing how he had been walking past Downing Street with his nephew when the spat happened.
It suggested Mr Mitchell had sworn repeatedly and called the officers "plebs", as well as claiming passers-by near the gates had been shocked.
The account closely matched the official police log's version of events, which was eventually leaked and published in full by the press.
Mr Cameron summoned his Chief Whip after being told about the email and suggested he had been "caught bang to rights", according to an investigation by Channel 4 News.
When Mr Mitchell flatly denied key parts, the Prime Minister ordered an investigation but this failed to establish who sent the email.
It emerged only when the officer was arrested on suspicion of misconduct in public office last week.
Contacted by Channel 4 News, the individual seemed to admit that he had never been present when the row happened.
Mr Cameron was said to be "furious" when he found out.
Previously unreleased CCTV footage of the clash also showed no evidence of passers-by who could be a man with his nephew.
The video, which has no sound, shows Mr Mitchell talking to three officers by the main gate for around 20 seconds before wheeling his bicycle to the side gate and leaving.
Clips from other cameras suggest there were few members of the public close by at the time - apparently contradicting the police log.
Mr Mitchell said: "‘Three phrases were hung around my neck for 28 days and used to destroy my political career and toxify the Conservative Party.
'They are completely untrue - I never said them. I have never called someone a f****** pleb and never would.
"I always knew that the emails were false, although extremely convincing. It has shaken my lifelong support and confidence in the police.
"I believe now there should be a full inquiry so we can get to the bottom of this."
Met Police Chief Bernard Hogan-Howe, speaking before Channel 4 broadcast its programme, said he did not think the new revelations "affected the original account of officers at the scene".
But Mayor of London Boris Johnson said: "These are very serious allegations that must be investigated with all possible urgency.
"An allegation that a serving police officer posed as a member of the public whilst fabricating evidence is a matter of the utmost gravity."
Keith Vaz, the chairman of the Commons Home Affairs Committee, suggested the police watchdog or the HM Inspectorate of Police should investigate the affair instead of Scotland Yard.
"There is clearly a need for a robust, transparent and comprehensive investigation," he said.
John Tully, chairman of the Metropolitan Police Federation (MPF), said: "The serious allegations aired in the Channel 4 News report are of concern to the MPF.
"However, as this is an ongoing investigation, we are unable to make further comment, other than to say we support a full and thorough investigation to establish the truth."
Former Tory leader Michael Howard said he was "appalled" by the claims and hoped Mr Mitchell would be back in Government "at the earliest opportunity".
The Prime Minister's spokesman said: "The Prime Minister's view remains that he hopes in time Mr Mitchell will be able to return to public life."
| 211,937
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Halloween is just around the corner and if you like to decorate and garden, here is a idea for vexing good fun in the fall bed. Design a dark and sinister Halloween garden. When we think of plants and flowers green, yellow, white and other bright colors usually come to mind. Even so, black can make a terrific haunted holiday project that draws the eye and stimulates wonder.
Believe it or not, a plethora of dark diversity exists for manipulation. Here are several suggestions: Diablo ninebark, Voodoo lily, Smokebush, Black pearl(ornamental pepper), Queen of the night tulip, Black elephant ears, Hellebores, and the Black lace elder flower. Adding some Purple majesty millet will attract fall birds for extra delight and do not forget to don some ghostly and ghoulish decor, just as you would gnomes and stepping stones in a regular bed.
For more information visit: or
| 310,002
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Mishap averted as two planes come close at Delhi airport
New Delhi, Dec 27: A mishap was averted after two aircraft Indigo and SpiceJet came face to face at the runway of Delhi's IGI Airport on Tuesday morning due to an ATC miscommunication, sources said. The IndiGo flight had reportedly just arrived from Lucknow and the Hyderabad-bound SpiceJet aircraft carrying 187 passengers flight was about to take off when the two aircraft came close.
Mishap averted after two aircraft (Indigo and SpiceJet) came face to face at Delhi's IGI Airport; reported to DGCA. Probe underway. pic.twitter.com/djyEOeCuHS— ANI (@ANI_news) December 27, 2016
The matter has been reported to the Directorate General of Civil Aviation (DGCA) and an investigation is underway.
Meanwhile,, it said adding, "at no stage the safety of passengers and crew was compromised. All concerned authorities were immediately informed," SpiceJet spokesperson said in the statement. However, the IndiGo spokesperson was not available for comments.
Earlier today, a Mumbai- bound Jet Airways aircraft with 161 passengers including seven crew members onboard skidded off the runway while aligning to takeoff from Dabolim Airport.
OneIndia News (with IANS inputs)
| 384,249
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TITLE: Providing a grading for the polynomial ring over a commutative unital graded ring
QUESTION [1 upvotes]: Let $R$ be a commutative unital $G$-graded ring , where $G$ is a monoid ; then does there exist a $G$-grading on $R[X]$ such that whenever we have a commutative unital $G$-graded ring $S$ , $a \in S$ and a graded homomorphism $\phi : R \to S$ , then there exists a graded homomorphism $\bar \phi : R[X]\to S$ such that $\bar \phi |_{R}=\phi$ and $\bar \phi (X)=a$ ? If this cannot always be done , then what if we took $a\in S$ to be a homogenous element ? If even this cannot always be done ; then is there any condition on the monoid $G$ such that this type of grading can be done for any $G$-graded ring ?
REPLY [0 votes]: This answer is only about the case where $G$ is a (commutative) group. I have not thought about more general monoids yet.
As observed by Remy in his comment, one has to choose $X$ to be homogeneous of some fixed degree and $a$ to be homogeneous as well. Then, this is indeed possible for polynomial algebras with an arbitrary set of indeterminates, or even more generally for algebras of monoids. For details you may have a look at paragraph 2.1.6 in this article (or here for free).
| 199,006
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View frequently asked questions on topics including the environment, traffic management, economic considerations and construction.babyliss hair curler 271ce hair curler qoo10 4 Browsing with Live Assistance We aim to offer you the best possible help and advice in the context of your visit to our page. Discover the best new music courtesy of The List back to top Products & services Add an event Add a place Email newsletters Edinburgh Festival Guides & publications Magazine subscriptions Jobs The List About us Contact us Advertise with us Terms & conditions Follow us Twitter Facebook Google+ ©.
This is going to earn the honour of being written into my recipe folder so will be used over and over. FeedbackContact usInterpreter servicesDisclaimer & copyrightFreedom of information & privacyAccessibilityEmail WebmanagerSite map Login ©.babyliss pro nano-titanium and ceramic curling iron review 99 In Stock New Rock Rings The Cocktagon lll 3 Pack 1 Review(s) Everything you need in a cock ring collection is here with the Rock Rings The Cocktagon lll, varying sizes for versatile play £ 14.how to curl your hair with the babyliss pro original
My paintings are made with Oil or Acrylic on canvas or in the case of gouache I paint on heavy weight textured paper. Build on that in the same way as a ballet dancer analyses her movements in the dance room mirror to perfect her step next time. with Sir Roy StrongA special In Conversation event for Bath in Fashion 2016Tuesday 19 April, 6 7pm£12.
Although the more I work and network within the industry I'd suggest that as a 18 19 old designer I would have learned a hell of a lot more working 1 year at an agency than three years studying.babyliss secret curl or perfect curl zlava If you answer yes to any of the above then Windows hosting is the option for you but otherwise you may find our Linux web hosting plans easier to manage, maintain and of course cheaper. FIND OUT MORE ABOUT TRADE ADVERTISING Value your classic When you're looking to sell your classic car setting the right price is possibly one of the most difficult parts of the process: price it too high and you'll find it difficult to find a buyer.
| 315,707
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TITLE: How to calculate some conditional probabilities
QUESTION [3 upvotes]: A red die, a blue die, and a yellow die (all six sided) are rolled. Given that no two of the dice land on the same number, what is the conditional probability that blue is less than yellow which is less than red?
The Answer is a sixth. I have absolutely no idea how to do this though.
REPLY [2 votes]: Imagine recording the outcomes as $(a,b,c)$, where $a$ is the number on the red, $b$ the number on the blue, and $c$ the number on the yellow. Fix a particular set of distinct numbers, such as $\{1,4,5\}$. All ordered triples (permutations) of these three numbers are equally likely. There are $3!$ such permutations. In exactly $1$ of these permutations, we have $\text{blue}\lt \text{yellow}\lt \text{red}$. Thus the required probability is $\frac{1}{3!}$.
REPLY [2 votes]: Since the three die rolls are unequal, every set of possible rolls $\{x,y,z\}$, with $x < y < z$, can be obtained in exactly six ways:
$$
\begin{array}{|c|c|c|}
\hline
\text{blue} & \text{yellow} & \text{red} \\
\hline
x & y & z \\
y & z & x \\
z & x & y \\
x & z & y \\
y & x & z \\
z & y & x \\
\hline
\end{array}
$$
Note that out of these six equally likely possibilities, the only one where blue < yellow < red is $x, y, z$ (since we assumed $x < y < z$).
| 78,422
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TITLE: Is there a way to find out how many distinct roots a polynomial has?
QUESTION [1 upvotes]: Let say we have an arbitrary polynomial over the reals, and we do not know whether it is separable or not. Is there some algorithm to find the number of roots it has in the complex number?
REPLY [7 votes]: If $f(z) = \prod (z-z_i)^{n_i}$, with $z_i$ distinct, then $GCD(f(z), f'(z)) = \prod (z-z_i)^{n_i-1}$, so the number of distinct complex roots is $\deg f - \deg GCD(f, f')$. If $f$ has rational coefficients, or in some other sense can be computed with exactly, this is a practical method; polynomial GCD can be computed by the Euclidean algorithm. If the coefficients of $f$ are only known approximately, then you need to think about what you mean by number of roots, since perturbing $x^2$ (one root) can give $x^2-\epsilon$ (two roots).
Also, if you meant to ask how many real roots there are, see Sturm's theorem.
REPLY [3 votes]: Yes, there is. The key is the square-free factorization which is an algorithm for factoring a polynomial $f$ into
$f=f_1^{e^1}\cdots f_r^{e_r}$, where all the $f_i$ are square free.
In your case, the number of complex roots would then be the degree of $f_1\cdots f_r$.
Look for example in von zur Gathen, J. and Gerhard, J. Modern Computer Algebra. Cambridge, England: Cambridge University Press, pp. 601-606, 1999.
| 212,752
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\begin{document}
\title{Quantum sphere $\mathbb{S}^4$ as a non-Levi conjugacy class}
\author{A.~Mudrov \\
\small Department of Mathematics, \\
\small University of Leicester,
\\
\small University Road,
Leicester, LE1 7RH, UK.\\
\small e-mail: am405@le.ac.uk\\
}
\date{ }
\maketitle
\begin{abstract}
We construct a $U_\hbar\bigl(\s\p(4)\bigr)$-equivariant quantization of the four-dimensional
complex sphere $\mathbb{S}^4$ regarded as a conjugacy class, $Sp(4)/Sp(2)\times Sp(2)$,
of a simple complex group with non-Levi isotropy subgroup,
through an operator realization of the quantum polynomial
algebra $\C_\hbar[\mathbb{S}^4]$ on a highest weight module of $U_\hbar\bigl(\s\p(4)\bigr)$.
\end{abstract}
{\small \underline{Key words}: quantum groups, quantization, Verma modules.}
\\
{\small \underline{AMS classification codes}: 17B10, 17B37, 53D55.}
\section{Introduction}
There are two types of closed conjugacy classes in a simple complex algebraic group $G$.
One type consists of classes that are isomorphic to orbits in the adjoint representation
on the Lie algebra $\g$. They are homogeneous spaces of $G$ whose stabilizer of the initial point
is a Levi subgroup in $G$.
Our concern is equivariant quantization of classes of second type, i. e. whose isotropy subgroup {\em is not} Levi.
Regarding the classical series, such classes are present only in the orthogonal and symplectic groups.
The group $G$ supports a (Drinfeld-Sklyanin) Poisson bivector field $\pi_0\in \Lambda^2(G)$ associated with a solution of the classical Yang-Baxter
equation. This structure makes $G$ a Poisson group, whose multiplication $G\times G\to G$ is a Poisson map
(here $G\times G$ is equipped with the Poisson structure of Cartesian product).
The Drinfeld-Sklyanin bracket gives rise to the quantum group $U_\hbar(\g)$, which is a deformation,
along the parameter $\hbar$, of the universal enveloping algebra $U(\g)$ in the class of Hopf algebras, \cite{D}.
There is a Poisson structure $\pi_1\in \Lambda^2(G)$ compatible
with the conjugacy action of the Poisson group on itself, \cite{S}.
It means that action map from the Cartesian product of $(G,\pi_0)$ and $(G,\pi_1)$
to $(G,\pi_1)$ is Poisson. Then $G$ is said to be a Poisson space
over the Poisson group $G$, under the conjugacy action
The Poisson bivector field $\pi_1$ restricts to every closed conjugacy class making it a Poisson $G$-variety, \cite{AM}. In this sense,
the group $G$ is analogous to $\g\simeq \g^*$ equipped with the canonical $G$-invariant bracket.
Quantization of conjugacy classes with Levi isotropy subgroups has been constructed in various settings, namely,
as a star product and in terms of generators and relations, \cite{EEM,M2}.
Both approaches rely upon the representation theory of the quantum group $U_\hbar(\g)$ and make use of
the following
facts: a) the universal enveloping algebra $U(\l)$ of the isotropy subgroup is quantized to a Hopf subalgebra
$U_\hbar(\l)\subset U_\hbar(\g)$,
b) there is a triangular factorization of $U(\g)$ relative to $U(\l)$, which amounts
to a factorization of quantum groups and facilitates parabolic induction.
In particular, quantum conjugacy classes of the Levi type have been realized by operators on scalar parabolic Verma modules
in \cite{M2}.
The above mentioned conditions are violated for non-Levi conjugacy classes, which makes the conventional
methods of quantization inapplicable in this case. In this paper, we show how to overcome these obstructions for the
simplest non-Levi conjugacy class $Sp(4)/Sp(2)\times Sp(2)$. This is the class of symplectic invertible
$4\times 4$-matrices
with eigenvalues $\pm 1$, each of multiplicity $2$. As an affine variety, it
is isomorphic to the four-dimensional complex sphere $\mathbb{S}^4$. Although the quantization of $\mathbb{S}^4$
can be obtained by other methods, e. g. as in \cite{FRT}, we are interested in $\mathbb{S}^4$
as an illustration of our approach to a general non-Levi class.
The idea is to find a suitable highest weight $U_\hbar(\g)$-module where the quantum sphere could be
represented by linear operators. We consider an auxiliary parabolic Verma module $\hat M_\la$ as a starting point. For a
special value of
weight $\la$, the module $\hat M_\la$ has a singular vector generating a submodule in $\hat M_\la$. The quotient $ M_\la$
of $\hat M_\la$ over that submodule is irreducible. The deformation of the polynomial algebra
$\C[\mathbb{S}^4]$ is realized by a $U_\hbar(\g)$-invariant subalgebra in $\End(M_\la)$.
This also allows us to describe the quantized polynomial algebra $\C_\hbar[\mathbb{S}^4]$ in terms of generators and relations.
Irreducibility of $M_\la$ implies non-degeneracy of the Shapovalov form on it. In the simple case of
$Sp(4)/Sp(2)\times Sp(2)$ this form can be calculated explicitly. This provides a bi-differential
operator relating the multiplication in $\C_\hbar[\mathbb{S}^4]$ to the multiplication in the dual Hopf algebra
$U_\hbar^*(\g)$, as explained in \cite{KST}.
We start from description of the classical conjugacy class $Sp(4)/Sp(2)\times Sp(2)$ and the Poisson structure on it.
Next we collect the necessary facts about the quantum group $U_\hbar(\g)$. Further we describe the quantization of
the polynomial algebra $\C_\hbar[G]$ and its properties. After that we construct the module $M_\la$
and analyze the submodule structure of the tensor product $\C^4\tp M_\la$. This allows us to realize
$\C_\hbar[\mathbb{S}^4]$
by operators on $M_\la$ and describe it in generators and relations. In conclusion, we calculate the
invariant pairing between $M_\la$ and its dual and discuss the star product on $\C_\hbar[\mathbb{S}^4]$.
\section{The classical conjugacy class $Sp(4)/Sp(2)\times Sp(2)$}
Let $Sp(4)$ denote the complex algebraic group of matrices preserving
the antisymmetric skew-diagonal bilinear form $C_{ij}=\epsilon_{i}\delta_{ij'}$, where $i'=5-i$,
$(\epsilon_1,\epsilon_2,\epsilon_3,\epsilon_4)=(1,1,-1,-1)$, and $\dt_{ij}$ is the Kronecker symbol.
We are interested in the conjugacy class of symplectic matrices with eigenvalues
$\pm1$ each of multiplicity $2$. It is an $Sp(4)$-orbit with respect to the conjugation action
on itself.
The initial point $A_o$ of the class
and its isotropy subgroup can be taken as
$$
A_o=
\left(
\begin{array}{cccc}
-1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&-1\\
\end{array}
\right),
\quad
Sp(2)\times Sp(2)=
\left(
\begin{array}{cccc}
*&0&0&*\\
0&*&*&0\\
0&*&*&0\\
*&0&0&*\\
\end{array}
\right)\subset Sp(4).
$$
This conjugacy class is a subvariety in $Sp(4)$
defined by the system of equations
\be
ACA^t-C=0, \quad \Tr(A)=0, \quad A^2-1=0,
\label{ideal_class}
\ee
where $1$ in the third equality is the matrix unit.
This is a system of polynomial equations on the matrix coefficients $A_{ij}$, which can be written in
an alternative way:
$$
A^t+CAC=0, \quad \Tr(A)=0, \quad A^2-1=0.
$$
The first two equations are linear and allow for the following non-zero entries:
$$
A=
\left(
\begin{array}{cccc}
a&b&y&0\\
c&-a&0&-y\\
z&0&-a&b\\
0&-z&c&a\\
\end{array}
\right).
$$
The quadratic equation is then equivalent to
\be
\label{sphere}
a^2+bc+yz-1=0.
\ee
Thus, the conjugacy class of $A_o$ is isomorphic to the complex sphere
$\mathbb{S}^4$. The ideal generated by the entries of the matrix equations
(\ref{ideal_class}) along with
the zero trace condition is, in fact, generated by a single irreducible
polynomial and is the defining ideal of the class.
Consider the r-matrix
$$
r=\sum_{i=1}^4(e_{ii}\tp e_{ii}-e_{ii}\tp e_{i'i'})
+2\sum_{i,j=1\atop i>j}^4(e_{ij}\tp e_{ji}-\epsilon_{i}\epsilon_{j}e_{ij}\tp e_{i'j'})\in \s\p(4)\tp \s\p(4)
$$
solving the classical Yang-Baxter equation, \cite{D}. It induces a Drinfeld-Sklyanin bivector field $\pi_0$ on $Sp(4)$ making it a Poisson group, \cite{D}.
We are concerned with the following Poisson structure, $\pi_1$, on $Sp(4)/Sp(2)\times Sp(2)\simeq \mathbb{S}^4$:
\be
\{A_1,A_2\}=\frac{1}{2}(A_2r_{21}A_1-A_1rA_2+A_2A_1r-r_{21}A_1A_2).
\label{poisson_br}
\ee
This equation is understood in $\End(\C^4)\tp \End(\C^4)\tp \C[\mathbb{S}^4]$ and is a shorthand matrix form
of the system of $n^2\times n^2$ identities defining the Poisson brackets $\{A_{ij}, A_{kl}\}$ of the coordinate
functions. The subscripts indicate the
copy of $\End(\C^4)$ in the tensor square, as usual in the quantum group literature.
Explicitly, the brackets of the generators $a,b,c,y,z\in \C[\mathbb{S}^4]$ read
$$
\{a,b\}=ab,\quad \{a,c\}=-ac, \quad\{a,y\}=ay,\quad\{a,z\}=-az,
$$
$$
\{b,y\}=by,\quad\{b,z\}=-bz,\quad
\{c,y\}=cy,\quad\{c,z\}=-cz,
$$
$$
\{y,z\}=2a^2+2bc,\quad\{b,c\}=2a^2.
$$
This Poisson structure restricts from $Sp(4)$ and makes
$\mathbb{S}^4$ a Poisson manifold under the conjugacy action of the Poisson group $Sp(4)$, \cite{S}.
In can be shown that such a Poisson structure on $\mathbb{S}^4$ is unique.
\section{Quantum group $U_\hbar(\s\p(4)$)}
\label{QG}
Throughout the paper, $\g$ stands for the Lie algebra $\s\p(4)$.
We are looking for quantization of the polynomial algebra $\C[\mathbb{S}^4]$
along the Poisson bracket (\ref{poisson_br}) that is invariant
under an action of the quantized universal enveloping algebra $U_\hbar(\g)$.
In this section we recall the definition of $U_\hbar(\g)$, following \cite{D}.
The root system of $\g$ is generated by
the simple positive roots $\al$, $\bt$, which are defined in the orthogonal
basis $\ve_1$, $\ve_2$ as
$$\al=\ve_1-\ve_2,\quad \bt=2\ve_2.$$
The other positive roots are
$\gm=\al+\bt$ and $\dt=2\al+\bt$.
Root vectors and Cartan elements are represented by the matrices
\be
\begin{array}{c}
e_\al=e_{12}-e_{34},\quad e_\bt=e_{23},\quad e_\gm=e_{13}+e_{24},\quad e_\dt=e_{14},
\\
f_\al=e_{21}-e_{43},\quad f_\bt=e_{32},\quad f_\gm=e_{31}+f_{42},\quad f_\dt=e_{41},
\\
h_\al=e_{11}-e_{22}+e_{33}-e_{44},\quad h_\bt=2e_{22}-2e_{33},
\end{array}
\label{C4rep}
\ee
where $\{e_{ij}\}$ is the standard matrix basis.
The quantized universal enveloping algebra (quantum group) $U_\hbar(\g)$
is a $\C[[\hbar]]$-algebra generated by the elements
$e_\al,e_\bt,f_\al, f_\bt, h_\al,h_\bt$
subject
to the commutator relations
$$
[h_{\al},e_{\al}]=2 e_{\al},
\quad
[h_{\al},f_{\al}]=-2 f_{\al},
\quad
[h_{\bt},e_{\bt}]=4 e_{\bt},
\quad
[h_{\bt},f_{\bt}]=-4 f_{\bt},
$$
$$
[h_{\al},e_{\bt}]=-2 e_{\bt},
\quad
[h_{\al},f_{\bt}]= 2 f_{\bt},
\quad
[h_{\bt},e_{\al}]=- 2 f_{\al},
\quad
[h_{\bt},f_{\al}]= 2 f_{\al},
$$
$$
[e_{\al},f_{\al}]=\frac{q^{h_{\al}}-q^{-h_{\al}}}{q-q^{-1}},
\quad
[e_{\al},f_{\bt}]=0=[e_{\bt},f_{\al}],
\quad
[e_{\bt},f_{\bt}]=\frac{q^{h_{\bt}}-q^{-h_{\bt}}}{q^2-q^{-2}},
$$
plus the Serre relations
$$
e_{\al}^{3}
e_{\bt}
-
(q^2+1+q^{-2})e_{\al}^{2}
e_{\bt}e_{\al}
+
(q^2+1+q^{-2})e_{\al}
e_{\bt}e_{ \al}^{2}
-
e_{\bt}e_{\al}^{3}
=0
,
$$
$$
e_{\bt}^{2}
e_{\al}
-
(q^2+q^{-2})
e_{ \bt}e_{ \al}e_{\bt}
+
e_{\al}e_{\bt}^{2}
=0,
$$
and similar relations for $f_\al$, $f_\bt$.
Here and further on $q=e^\hbar$.
The comultiplication $\Delta$ and antipode $\gm$ are defined on the generators by
$$
\Delta(h)=h\tp 1+1\tp h, \quad \gm(h)=-h, \quad h\in \h,
$$
$$
\Delta(e_\mu)=e_\mu\tp 1+q^{h_\mu}\tp e_\mu, \quad \gm(e_\mu)=-q^{-h_\mu}e_\mu,
\quad \mu=\al,\bt,
$$
$$
\Delta(f_\mu)=f_\mu\tp q^{-h_\mu}+1\tp f_\mu, \quad \gm(f_\mu)=-f_\mu q^{h_\mu},
\quad \mu=\al,\bt.
$$
The counit homomorphism $\ve\colon U_\hbar(\g)\to \C[[\hbar]]$ is nil on the generators.
\begin{remark}
The quantum group $U_\hbar(\g)$ is regarded as a $\C[[\hbar]]$-algebra, bearing in mind its application to
deformation quantization.
Accordingly, all its modules are understood as free $\C[[\hbar]]$-modules. However, we will suppress the
reference to $\C[[\hbar]]$ in order to
simplify the formulas. For instance, the vector representation of $U_\hbar(\g)$ will be denoted simply as $\C^4$.
The tensor products and linear maps are also understood over $\C[[\hbar]]$.
\end{remark}
Let us introduce higher root vectors $e_\gm, f_\gm, e_\dt, f_\dt \in U_\hbar(\g)$ (the coincidence in the notation for the weight and the antipode should not cause a confusion) by
$$
f_{\gm}=f_{\bt}f_{\al}-q^{-2} f_{\al}f_{\bt},
,\quad
f_{\dt}=f_{\gm}f_{\al}-q^{2}f_{\al}f_{\gm},
$$
$$
e_{\gm}=e_\al e_\bt-q^{2}e_\bt e_\al, \quad
e_{\dt}=e_\al e_\gm-q^{-2}e_\gm e_\al.
$$
Our definition of $e_\dt, f_\dt$ is different from the usual definition
$e_\dt=[e_\al,e_\gm]$, $f_\dt=[f_\gm,f_\al]$, corresponding to $(\al,\gm)=0$, \cite{ChP}. The reason for that
will be clear later on.
The elements $h_\al, h_\bt$ span the Cartan subalgebra $\h$
and generate the Hopf subalgebra $U_\hbar(\h)\subset U_\hbar(\g)$.
The vectors $e_\al,e_\bt$ along with $\h$ generate
the positive Borel subalgebra $U_\hbar(\b^+)$
in $U_\hbar(\g)$. Similarly, $f_\al,f_\bt$, and $\h$ generate
the negative Borel subalgebra $U_\hbar(\b^-)$. They are
Hopf subalgebras of $U_\hbar(\g)$.
\begin{lemma}
The root vectors satisfy the relations
$$
e_{ \gm}e_{ \bt}
-
q^{-2}e_{ \bt}e_{ \gm}
=0
, \quad
[e_{ \al},e_{ \dt}]
=0
,\quad [e_{\bt},e_{\dt}]=0
,\quad
[e_{ \gm},e_{ \dt}]
=0,
$$
$$
f_{\bt}f_{\gm}-q^{2}f_{\gm}f_{\bt}
=0
,\quad
[f_{\al},f_{\dt}]=0
,\quad
[f_{\bt},f_{\dt}]=0
,\quad
[f_{\gm},f_{\dt}]=0.
$$
\label{Borel}
\end{lemma}
\begin{proof}
The first two equalities in both lines are simply a rephrase of the Serre relations
in the new terms. The last equalities follow from the second and third.
Let us check the third equality, say, in the first line:
\be
e_{\bt}e_{\dt}&=&
e_{\bt}(e_{\al}e_{\gm}-q^{-2} e_{\gm} e_{\al})=e_{\bt}e_{\al}e_{\gm}- e_{\gm}e_{\bt} e_{\al}
\nn\\
&=&
q^{-2}e_{\al}e_{\bt}e_{\gm}-q^{-2}e_{\gm}^2- q^{-2}e_{\gm} e_{\al}e_{\bt}+ q^{-2}e_{\gm}^2
=
q^{-2}e_{\al}e_{\bt}e_{\gm}- q^{-2}e_{\gm} e_{\al}e_{\bt}
\nn\\
&=&
e_{\al}e_{\gm}e_{\bt}- q^{-2}e_{\gm} e_{\al}e_{\bt}=e_{\dt}e_{\bt},
\nn
\ee
as required.
\end{proof}
Denote by $U_\hbar(\n^\pm_0)$ the subalgebras generated by, respectively, positive and
negative Chevalley generators. The Borel subalgebras $\U_\hbar(\b^\pm)$ are freely generated by $\U_\hbar(\n^\pm_0)$ over $\U_\hbar(\h)$ with respect to right or left multiplication.
These equalities facilitate the following
\begin{corollary}
The positive (respectively, negative) root vectors generate a Poincar\`{e}-Birkgoff-Witt
basis in $\U_\hbar(\n^\pm_0)$.
\label{PBW}
\end{corollary}
\begin{proof}
The presence of PBW basis in the quantum group is a well known fact. However,
we use a non-standard definition of the root vectors $e_\dt$, $f_\dt$, therefore the lemma is substantial.
To prove it, say, for $U_\hbar(\n^-_0)$ one should check that the system of monomials $f_\al^a f_\gm^c f_\dt^d f_\bt^b=f_\al^a f_\dt^d f_\gm^c f_\bt^b$, where
$a$, $b$, $c$, and $d$ are non-negative integers, is linearly independent and complete in $\U_\hbar(\n^-)$.
The ordered sequence of the elements $f_\al,f_\dt'=[f_\gm,f_\al],f_\gm,f_\bt$ does generate a PBW
basis, \cite{ChP}. Using this fact along with Lemma \ref{Borel} relations, one can easily check the statement
via the substitution
$f'_\dt=f_\dt+(q^{2}-1)f_\al f_\gm$.
\end{proof}
\section{The algebra of quantized polynomials on $Sp(4)$}
\label{S_qG}
We adopt the convention throughout the paper that $G$ stands for the
complex algebraic group $Sp(4)$.
The conjugacy class of our interest is a closed affine variety
in $G$, and its polynomial ring is a quotient
of the polynomial ring $\C[G]$ by a certain ideal.
Our goal is to obtain an analogous description of the quantum conjugacy class.
To that end, we need to describe the quantum analog of the algebra $\C[G]$ first.
Recall from \cite{J,B} that the image of the universal R-matrix of the quantum group $U_\hbar(\g)$ in the
vector representation is equal, up to
a scalar factor, to
$$
R=\sum_{i,j=1 }^4 q^{\delta_{ij}-\delta_{ij'}}e_{ii}\tp e_{jj}
+
(q-q^{-1})\sum_{i,j=1 \atop i>j}^4(e_{ij}\tp e_{ji}
- q^{\rho_i-\rho_j}\epsilon_i\epsilon_j
e_{ij}\tp e_{i'j'}),
$$
where $(\rho_1,\rho_2,\rho_3,\rho_4)=(2,1,-1,-2)$.
Denote by $S$ the $U_\hbar(\g)$-invariant operator $PR\in \End(\C^4)\tp \End(\C^4)$,
where $P$ is the ordinary flip of $\C^4\tp \C^4$.
This operator has three invariant projectors to its eigenspaces,
among which there is a one-dimensional projector $\sim
\sum_{i,j=1}^4q^{\rho_i-\rho_j}\epsilon_i\epsilon_j e_{i'j}\tp e_{ij'}
$
to the
trivial $U_\hbar(\g)$-submodule, call it $\kappa$.
Denote by $\C_\hbar[G]$ the associative algebra generated by
the entries of the matrix $K=||k_{ij}||_{i,j=1}^4\in \End(\C^4)\tp \C_\hbar[G]$
modulo the relations
\be
S_{12}K_2S_{12}K_2=K_2S_{12}K_2S_{12}
,\quad K_2S_{12}K_2\kappa=- q^{-5}\kappa=\kappa K_2S_{12}K_2.
\label{ideal_quant_linear}
\ee
These relations are understood in $\End(\C^4)\tp \End(\C^4)\tp \C_\hbar[G]$,
and the indices distinguish the two copies of $\End(\C^4)$, as usual.
The algebra $\C_\hbar[G]$ is an equivariant quantization of $\C[G]$, \cite{RS,FRT}, which is different
from the $RTT$-quantization and is not a Hopf algebra. It carries a $U_\hbar(\g)$-action, which is a deformation of the conjugation $U(\g)$-action
on $\C[G]$. It admits a $U_\hbar(\g)$-equivariant algebra
monomorphism to $U_\hbar(\g)$, where the latter is regarded as the adjoint module.
The monomorphism is implemented by the assignment
$$
K\mapsto (\phi\tp \id)(\Ru_{21}\Ru)=\Q\in \End(\C^4)\tp U_\hbar(\g),
$$
where $\phi\colon U_\hbar(\g)\to \End(\C^4)$ is the vector representation and $\Ru$ is the
universal R-matrix of $U_\hbar(\g)$.
The matrix $\Q$ is important for our presentation, and the reader is referred
to \cite{M2} for detailed explanation of its role in quantization and
for its basic characteristics.
\section{The generalized Verma module $M_\la$}
Denote by $\l$ the Levi subalgebra in $\g=\s\p(4)$ spanned by
$
e_\bt,f_\bt, h_\bt, h_\al.
$
It is a Lie subalgebra of maximal rank, and its semisimple part is
isomorphic to $\s\l(2)\simeq \s\p(2)$. The universal enveloping algebra
$U(\l)$ is quantized as a Hopf subalgebra in $U_\hbar(\g)$.
Denote by $\n^+$ and $\n^-$ the nilpotent subalgebras in $\g$ spanned,
respectively, by $\{e_\al, e_\gm, e_\dt\}$ and $\{f_\al,f_\gm, f_\dt\}$.
The sum $\l+\n^\pm$ is a parabolic subalgebra $\p^\pm\subset \g$ whose universal
enveloping algebra
is quantized to a Hopf subalgebra in $U_\hbar(\p^\pm)\subset U_\hbar(\g)$.
Let $U_\hbar(\n^\pm)$ be the subalgebras in $U_\hbar(\g)$ generated by
the quantum root vectors $\{e_\al, e_\gm, e_\dt\}$ and $\{f_\al,f_\gm, f_\dt\}$, respectively. The quantum group $U_\hbar(\g)$
is a free $U_\hbar(\n^-)-U_\hbar(\n^+)$-bimodule generated by
$U_\hbar(\l)$:
\be
\label{triangular_fact}
U_\hbar(\p^-)=U_\hbar(\n^-)U_\hbar(\l),
\quad
U_\hbar(\g)=U_\hbar(\n^-)U_\hbar(\l)U_\hbar(\n^+),
\quad
U_\hbar(\p^+)=U_\hbar(\l)U_\hbar(\n^+).
\ee
The factorizations of $U_\hbar(\p^\pm)$ have the structure of smash product.
Fix a weight $\la\in \h^*$ orthogonal to $\bt$. It can be regarded as a
one-dimensional representation of $U_\hbar(\l)$,
$$
\la\colon e_\bt, f_\bt, h_\bt\mapsto 0, \quad \la \colon h_\al\mapsto (\al,\la),
$$
which can be extended to a representation of $U_\hbar(\p^+)$ by
$\la\colon e_\al\mapsto 0$. Let $\C_\la$ denote the one-dimensional vector space supporting this representation.
Consider the scalar parabolic Verma module $\hat M_\la$ induced from $\C_\la$,
$$
\hat M_\la= U_\hbar(\g)\tp_{U_\hbar(\p^+)}\C_\la.
$$
As a module over $U_\hbar(\n^-)$, it is freely generated by its highest weight
vector $v_\la$. As a module over the Cartan subalgebra, it isomorphic
to $U_\hbar(\n^-)\tp \C_\la$, where $U_\hbar(\n^-)$ is the natural module over
$U_\hbar(\h)$.
The $U_\hbar(\g)$-module $\hat M_\la$ is irreducible except for special values of
$\la$, when $\hat M_\la$ may contain singular vectors. Recall that a weight vector is called
singular if it is annihilated by the positive Chevalley generators. Such vectors generate
submodules in $\hat M_\la$, where they carry the highest weight.
We are looking for such $\la$ that $\hat M_\la$ admits a singular
vector of weight $\la-\dt$. Quotienting out the corresponding submodule yields a module that supports
quantization of $\C[\mathbb{S}^4]$.
\begin{propn}
The module $\hat M_\la$ admits a singular vector of weight $\la-\dt$
if and only if
$
q^{2(\al,\la)}=-q^{-2}.
$ Then $f_\dt v_\la$ is the singular vector.
\end{propn}
\begin{proof}
The general expression for the vector of
weight $\la-\dt$ in $M_\la$ is
$$
\bigl(f_{\al}^2f_{\bt}-(a+b)f_{\al}f_{\bt}f_{\al}+abf_{\bt}f_{\al}^2\bigr)v_\la=
\bigl(-(a+b)f_{\al}f_{\bt}f_{\al}+abf_{\bt}f_{\al}^2\bigr)v_\la,
$$
where $a,b$ are some scalars. For this vector being singular, we have a system
of two equations on $a,b$ resulted from the action of $e_\bt$ and $e_\al$:
$$
\left\{
\begin{array}{rrr}
\bigl(-(a+b)f_{\al}[e_{\bt},f_{\bt}]f_{\al}+ab[e_{\bt},f_{\bt}]f_{\al}^2\bigr)v_\la&=&0,
\\
\bigl(-(a+b)[e_{\al},f_{\al}]f_{\bt}f_{\al}
+abf_{\bt}[e_{\al},f_{\al}]f_{\al}
+abf_{\bt}f_{\al}[e_{\al},f_{\al}]\bigr)v_\la&=&0.
\end{array}
\right.
$$
The non-zero solution of this system is unique (up to permutation $a \leftrightarrow b$)
and equal to
$$
q^{2(\al,\la)}=-q^{-2}
,\quad a=q^2,\quad b=q^{-2},
$$
as required.
Finally, notice that
$
f_{\dt}=f_{\al}^2f_{\bt}-(q^2+q^{-2}) f_{\al}f_{\bt}f_{\al}+f_{\bt}f_{\al}^2.
$
This completes the proof.
\end{proof}
Denote by $M_\la$ the quotient of $\hat M_\la$
by the submodule $U_\hbar(\g)f_{\dt}v_\la$.
By Corollary \ref{PBW}, the vectors $f_\al^k f_\gm^l f_\dt^mv_\la$ for all non-negative integer $k,l,m$
form a basis in $\hat M_\la$.
Therefore, $M_\la$
is spanned by $f_{\al}^kf_{\gm}^l v_\la$, $k,l\geqslant 0$.
\begin{propn}
The module $M_\la$ is irreducible.
\end{propn}
\begin{proof}
Irreducibility follows from non-degeneracy of the invariant bilinear pairing
of $M_\la$ with its dual, see Section \ref{Shapovalov}. One can also verify
that $M_\la$ has no singular vector. Omitting the details,
the action of the positive Chevalley generators on $M_\la$ is given by
\be
e_\al f_{\al}^k f_{\gm}^m v_\la&=&
q^{(\al,\la)+1}\frac{q^{2k}-q^{-2k}}{(q-q^{-1})^2} f_{\al}^{k-1} f_{\gm}^m v_\la,
\nn
\\
e_\bt f_{\al}^kf_{\gm}^m v_\la&=& \frac{q^{2m}-q^{-2m}}{q^{2}-q^{-2}}f_{\al}^{k+1} f_{\gm}^{m-1} v_\la.
\nn
\ee
Here we assume that $k>0$ in the first line and $m>0$ in the second; otherwise the right hand side is nil.
This immediately implies the absence of singular vectors in $M_\la$.
\end{proof}
\section{The $U_\hbar(\g)$-module $\C^4\tp M_\la$}
The tautological assignment (\ref{C4rep}) defines the four-dimensional irreducible representation of $U(\g)$.
Similar assignment on the quantum Chevalley generators and Cartan elements defines a representation of $U_\hbar(\g)$.
Our next object of interest is the $U_\hbar(\g)$-module $\C^4\tp M_\la$.
In particular, we shall study the decomposition of $\C^4\tp M_\la$ into direct sum of irreducible submodules.
Choose the standard basis $\{w_i\}_{i=1}^4\subset \C^4$ of columns with the only nonzero entry
$1$ in the $i$-TtH place from the top. Their weights are $\ve_1, \ve_2,-\ve_2,-\ve_1$, respectively.
As a $U_\hbar(\l)$-module, $\C^4$ splits into the sum of two one-dimensional blocks of weights
$
\pm\ve_1
$
and one two-dimensional block of highest weights
$
\ve_2
$.
The parabolic Verma module contains three blocks
of highest weights
$
\ve_1+\la, \ve_2+\la, -\ve_1+\la,
$
which we denote by $\hat V_{\ve_1+\la}$, $\hat V_{\ve_2+\la}$, $\hat V_{-\ve_1+\la}$.
For generic $\la$ these submodules are
irreducible, and
\be
\C^4\tp \hat M_\la =\hat V_{\ve_1+\la}\oplus \hat V_{\ve_2+\la}\oplus \hat V_{-\ve_1+\la}.
\label{Verma_split}
\ee
All these blocks are parabolic Verma modules corresponding to the $U_\hbar(\l)$-submodules of $\C^4$.
Clearly $\la+\ve_1$ is the highest weight of $\C^4\tp \hat M_\la$ and $w_1\tp v_\la$ is the
highest weight vector.
The other singular vectors in $\C^4\tp \hat M_\la$ are given next.
\begin{lemma}
The vectors
\be
u_{\ve_1}&=& w_1\tp v_\la,\nn\\
u_{\ve_2}&=&w_1\tp f_\al v_\la-q\frac{q^{(\al,\la)}-q^{-(\al,\la)}}{q-q^{-1}} w_2\tp v_\la,\nn\nn\\
u_{-\ve_1}&=& f_\dt w_1\tp v_\la +
(q^{(\la,\al)+1}+q^{-(\la,\al)-1})\times\nn\\
&\times&
\Bigl(q w_2\tp f_\bt f_\al v_\la-q^3 w_3\tp f_\al v_\la-q^4\frac{q^{(\la,\al)}-q^{-(\la,\al)}}{q-q^{-1}}w_4\tp v_\la\Bigr)
\nn
\ee
are singular and generate the submodules
$\hat V_{\ve_1+\la}$, $\hat V_{\ve_2+\la}$, $\hat V_{-\ve_1+\la}$, respectively.
\end{lemma}
\begin{proof}
One should check that $u_{\ve_1}$, $u_{\ve_2}$, $u_{-\ve_1}$ are annihilated by $e_\al$ and $e_\bt$. That is obvious for $u_{\ve_1}$ and relatively easy for $u_{\ve_2}$. The case of $u_{-\ve_1}$ requires
bulky but straightforward calculation, which is omitted here.
\end{proof}
We denote by $V_{\ve_1+\la}$, $V_{\ve_2+\la}$, $V_{-\ve_1+\la}$ the images of
$\hat V_{\ve_1+\la}$, $\hat V_{\ve_2+\la}$, $\hat V_{-\ve_1+\la}$ under the projection
$\C^4\tp \hat M_\la\to \C^4\tp M_\la$, assuming $q^{2(\al,\la)}=-q^{-2}$.
An important fact is that for $q^{2(\al,\la)}=-q^{-2}$ the
singular vector $u_{-\ve_1}$ turns into $w_1\tp f_\dt v_\la$ and thus
disappears from $\C^4\tp M_\la$. The submodule $\hat V_{-\ve_1+\la}$ is killed
by the projection $\C^4\tp \hat M_\la\to \C^4\tp M_\la$, so $V_{-\ve_1+\la}=\{0\}$.
\begin{propn}
\label{dir_sum}
The module
$\C^4\tp M_\la$ is a direct sum of the submodules $V_{\ve_1+\la}$ and $V_{\ve_2+\la}$.
\end{propn}
\begin{proof}
The modules $V_{\ve_1+\la}$ and $V_{\ve_2+\la}$ have zero intersection, as they carry different eigenvalues of
the invariant matrix $\Q$, see below.
We must show that the sum $V_{\ve_1+\la}\oplus V_{\ve_2+\la}$ exhausts all of
$\C^4\tp M_\la$. To that end,
it is sufficient to show that $\C^4\tp v_\la$ lies in $V=V_{\ve_1+\la}\oplus V_{\ve_2+\la}$.
Indeed, then for all $u\in U_\hbar(\g)$ and $w\in \C^4$,
$$
w\tp uv_\la=\Delta(u^{(2)})\bigl(\gm^{-1}(u^{(1)})w\tp v\bigr)\in V,
$$
as required.
In what follows $\equiv $ will mean equality modulo $V$.
Obviously, $w_1\tp v_\la\equiv 0$.
Applying $f_{\al}$ to $w_1\tp v_\la$ gives $w_1\tp f_{\al}v_\la+ q^{-(\al,\la)}w_2\tp v_\la\equiv 0$.
Comparing this with $u_{\ve_2+\la}\in V$ we conclude that $w_2\tp v_\la\equiv 0$.
Applying $f_{\bt}$ to $w_2\tp v_\la$ gives $w_3\tp v_\la \equiv 0 $.
Thus, we are left to check that $w_4\tp v\in V$. We have
$$
0\equiv f_{\al}(w_1\tp v_\la)\equiv w_1 \tp f_\al v_\la ,
\quad
0\equiv f_{\al}( w_2\tp v_\la)= w_2 \tp f_\al v_\la,
$$
$$
0\equiv f_{\al}^2(w_1\tp v_\la)\equiv f_\al(w_1\tp f_\al v_\la)
=w_1\tp f_\al^2 v_\la+q^{-2-(\al,\la)}w_2\tp f_\al v_\la \equiv w_1\tp f_\al^2 v_\la,
$$
\be
0\equiv f_\bt f_{\al}^2(w_1\tp v_\la)\equiv f_\bt(w_1\tp f_\al^2 v_\la)= w_1\tp f_\bt f_\al^2 v_\la.
\label{aux1}
\ee
Further,
$$
0\equiv f_\bt f_{\al}(w_1\tp v_\la)\equiv f_\bt( w_1\tp f_\al v_\la)=
w_1\tp f_\bt f_\al v_\la,
$$
\be
0\equiv f_\al f_\bt f_{\al}(w_1\tp v_\la)\equiv
f_\al (w_1\tp f_\bt f_\al v_\la)=
w_1\tp f_\al f_\bt f_\al v_\la
+q^{-(\al,\la)}w_2\tp f_\bt f_\al v_\la.
\label{aux2}
\ee
Combining (\ref{aux1}) and (\ref{aux2}), we calculate $f_\dt (w_1\tp v_\la)\in V$:
$$
0\equiv
\bigl(-(q^2+q^{-2}) f_{\al}f_{\bt}f_{\al}+f_{\bt}f_{\al}^2\bigr)(w_1\tp v_\la)
\equiv w_1\tp f_\dt v_\la-(q^2+q^{-2})q^{-(\al,\la)}w_2\tp f_\bt f_\al v_\la.
$$
The first equality takes place because $f_\bt(w_1\tp v_\la)=0$.
Therefore $w_2\tp f_\bt f_\al v_\la\equiv 0$, and
$$
0\equiv f_{\bt}(w_2\tp f_\al v_\la)=
w_2\tp f_{\bt}f_\al v_\la+q^2w_3\tp f_\al v_\la\equiv q^2 w_3\tp f_\al v_\la.
$$
Finally,
$$
0\equiv f_\al (w_3\tp v_\la)=w_3\tp f_\al v_\la
-
q^{-(\al,\la)} w_4\tp v_\la\equiv
-
q^{-(\al,\la)}w_4\tp v_\la,
$$
as required.
\end{proof}
Now consider the action of the matrix $\Q$ on $\C^4\tp \hat M_\la$.
It satisfies a cubic polynomial equation, and its eigenvalues in $\C^4\tp \hat M_\la$
can be found in \cite{M2}:
$$
q^{2(\la,\ve_1)}=-q^{-2},
$$
$$
q^{2(\la+\rho,\ve_2)-2(\rho,\nu)}=q^{2(\rho,\ve_2)-2(\rho,\nu)}=q^{-2},
$$
$$
q^{2(\la+\rho,-\ve_1)-2(\rho,\nu)}=q^{-2(\la,\ve_1)-4(\rho,\nu)}=-q^{-6}.
$$
The operator $\Q$ is semisimple on $\C^4\tp \hat M_\la$ for generic $\la$.
Due to Proposition \ref{dir_sum}, it is semisimple
on $\C^4\tp M_\la$ and has eigenvalues $\pm q^{-2}$.
\section{Quantization of $\mathbb{S}^4$}
Let $\phi$ denote the representation homomorphism $U_\hbar(\g)\to \End(\C^4)$.
The $q$-trace of $\Q$ is a weighted trace $\Tr_q(\Q)=\Tr(D\Q)$,
where $D$ is the diagonal matrix $\diag(q^4,q^2,q^{-2},q^{-4})$. It belongs
to the center of $U_\hbar(\g)$ and hence the center of
$\C_\hbar[G]\subset U_\hbar(\g)$.
A module of highest weight $\la$ defines a central character $\chi_\la$ of the
algebra $\C_\hbar[G]$, which returns zero on $\Tr_q(\Q)$:
\be
\chi_\la\bigl(\Tr_q(\Q)\bigr)&=&
\Tr\bigl(\phi(q^{h_\la+h_\rho})\bigr)=q^{2(\la+\rho,\ve_1)}
+q^{2(\la+\rho,\ve_2)}
+q^{2(\la+\rho,-\ve_2)}
+q^{2(\la+\rho,-\ve_1)}
\nn\\
&=&q^{2(\la,\ve_1)+4}
+q^{2}
+q^{-2}
+q^{-2(\la,\ve_1)-4}
=
-q^{2}
+q^{2}
+q^{-2}
-q^{-2}
=0,
\nn
\ee
cf. \cite{M2}.
Thus, the $q$-trace of the matrix $\Q$ vanishes in $M_\la$.
Also, the entries of the matrix
$
\Q^2-q^{-4}
$
are annihilated in
$\End(M_{\la})$, as discussed in the previous section.
\begin{propn}
The image of $\C_\hbar[G]$ in $\End(M_{\la})$ is a quantization of $\C_\hbar[\mathbb{S}^4]$.
It is isomorphic to the subalgebra in $U_\hbar(\g)$ generated by the entries of the matrix $\Q=(\phi\tp \id)(\Ru_{21}\Ru)$, modulo
the relations
\be
\Q^2=q^{-4}, \quad \Tr_q(\Q)=0.
\label{q-rel}
\ee
\end{propn}
\begin{proof}
The center of $\C_\hbar[G]$ is formed by $U_\hbar(\g)$-invariants, which are also central in $U_\hbar(\g)$. Therefore, $\ker \chi_\la$ lies in the kernel of the
representation $\C_\hbar[G]\to \End(M_{\la})$. The quotient of $\C_\hbar[G]$ by the ideal generated by $\ker \chi_\la$
is free over $\C[[\hbar]]$ and is a direct sum of isotypical $U_\hbar(\g)$-components of finite multiplicities, \cite{M1}.
Therefore, the
image of $\C_\hbar[G]$ in $\End(M_{\la})$ is a direct sum of isotypical $U_\hbar(\g)$-components
which are free and finite over $\C[[\hbar]]$.
The ideal in $\C_\hbar[G]$ generated by (\ref{q-rel}) lies in the kernel of
the homomorphism $\phi\colon\C_\hbar[G]\to \End(M_\la)$
and turns into the defining ideal of $\mathbb{S}^4$ modulo $\hbar$.
Therefore this ideal coincides with $\ker \phi$, and the quotient of $\C_\hbar[G]$
by this ideal is a quantization of $\C[\mathbb{S}^4]$, see \cite{M2} for details.
\end{proof}
We will give a
more explicit description of $\C_\hbar[\mathbb{S}^4]$.
The matrix $\Q$ is the image of the matrix $K$ from Section \ref{S_qG} under the embedding $\C_\hbar[G]\to U_\hbar(\g)$.
The algebra $\C_\hbar[\mathbb{S}^4]$ is generated by
elements $a,b,c,y,z$ arranged in the matrix
$$
\left(
\begin{array}{cccc}
a&b&y&0\\
c&-q^2a&0&-y\\
z&0&-q^2a&q^2b\\
0&-z&q^2c&q^4a\\
\end{array}
\right).
$$
This matrix is obtained from $K$
by imposing the linear relations on its entries
derived from (\ref{ideal_quant_linear}) by the substitution $K^2=q^{-4}$, $\Tr_q(K)=0$.
The generators of $\C_\hbar[\mathbb{S}^4]$ obey the relations
$$
ab=q^2ba,\quad ac=q^{-2}ca, \quad ay=q^2ya,\quad az=q^{-2}za,
$$
$$
by=q^2yb,\quad bz=q^{-2}zb,\quad
cy=q^{2}yc,\quad cz=q^{-2}zc,
$$
$$
[b,c]=(q^4-1)a^2,
\quad [y,z]=(q^4-1)a^2+(q^4-1)bc,
$$
plus
$$
a^2+bc+yz=q^{-4},
$$
which is a deformation of (\ref{sphere}).
Remark that $\C_\hbar[\mathbb{S}^4]$ has a 1-dimensional representation $a,b,c\mapsto 0$, $y,z\mapsto q^{-2}$.
Therefore it can be realized as a subalgebra in the Hopf algebra dual to $U_\hbar(\g)$, as explained in \cite{DM}.
\section{On invariant star product on $\mathbb{S}^4$}
\label{Shapovalov}
It follows from \cite{KST} that the star product on the conjugacy class $Sp(4)/Sp(2)\times Sp(2)$
can be calculated by means of the invariant pairing between the
modules $M_{-\la}^-$ and $M^+_\la$, where $M^+_\la=M_\la$ and
$M_{-\la}^-$ is its restricted dual. The module $M_{-\la}^-$ is the quotient
of the lower parabolic Verma module
$
\hat M^-_{-\la}= U_\hbar(\g)\tp_{ U_\hbar(\p^-)}\C_{-\la}
$
by the submodule $U_\hbar(\g)e_\dt v_{-\la}$.
Explicitly, the pairing is given by the assignment
$$
xv_{-\la}\tp yv_{\la}\mapsto \langle xv_{-\la},yv_{\la}\rangle= \la([\gm(x)y]_{\l}).
$$
Here $x\mapsto [x]_{\l}$ is the projection $U_\hbar(\g)\to U_\hbar(\l)$
along $U_\hbar'(\n^-)U_\hbar(\g)+U_\hbar(\g)U_\hbar'(\n^+)$, where the prime designates the kernel of the counit.
This projection is facilitated by the triangular factorization (\ref{triangular_fact}) and it
is a homomorphism of $U_\hbar(\l)$-bimodules.
The modules $M_{\pm\la}^\pm$ are irreducible if and only if the pairing is non-degenerate, \cite{Ja}.
Our next goal is to calculate it explicitly.
Put
$$
x_1=e_\al, \quad x_2=e_{\gm}
,\quad
\tilde x_1=e_\al, \quad \tilde x_2=q^4e_{\bt}e_{\al}-q^{2} e_{\al}e_{\bt},
\quad y_1=f_\al, \quad y_2=f_{\gm}.
$$
The twiddled root vectors are related to non-twiddled via the antipode:
$$
\gm(\tilde e_{\gm})= q^4 q^{-h_\bt}e_\bt q^{-h_\al}e_\al-q^{2}q^{-h_\al}e_\al q^{-h_\bt}e_\bt=-q^{-h_\gm}e_\gm,
$$
with a similar relation for the root $\al$.
The following system of monomials constitute
bases in $M_{-\la}^-$ and $M^+_\la$:
$$
\bigl(y^k_1 y^m_2v_\la\bigr)_{k,m=0}^\infty \subset M^+_\la
,\quad
\bigl(\tilde x^k_1 \tilde x^m_2v_\la\bigr)_{k,m=0}^\infty \subset M_{-\la}^-.
$$
Further we need the identities
\be
[e_\al, f_{\gm}]=-(q+q^{-1})q^{-2}f_{\bt}q^{-h_{\al}},\quad [e_\bt, f_{\gm}]=f_{\al}q^{h_{\bt}},
\\[1pt]
[e_\gm, f_\al]=-(q+q^{-1})e_\bt q^{h_\al},
\quad
[e_\gm, f_\bt]=q^2e_\bt q^{-h_\bt},
\ee
which can be proved directly from the defining relations $U_\hbar(\g)$ and the definition of
$e_\gm$ and $f_\gm$. Also, one can check that
\be
[e_{\gm}, f_{\gm}]=\frac{q^{h_{\gm}}-q^{-h_{\gm}}}{q-q^{-1}}.
\ee
Thus, for $\nu=\al, \gm$ and any positive integer $k$ we have
\be
[e_{\nu}, f_{\nu}^k]= q^{h_\nu+1}\frac{1-q^{-2k}}{(q-q^{-1})^2}+q^{-h_\nu-1}\frac{1-q^{2k}}{(q-q^{-1})^2}
.
\ee
\begin{lemma}
The matrix coefficient
$
\langle \tilde x_1^i\tilde x_2^jv_{-\la}, y_1^k,y_2^mv_{\la}\rangle
$
is zero unless $i=k$, $j=m$.
\end{lemma}
\begin{proof}
It follows that $[e_\al,f_\gm^k]$ belongs to the
left ideal $U_\hbar(\g)f_\bt$, hence $x_1 y_2^kv_\la=0$.
We have
$$
x_1 y_1^ky_2^m v_\la =e_\al f_\al^kf_\gm^m v_\la\sim
f_\al^{k-1}f_\gm^m v_\la+f_\al^{k-1}[e_\al,f_\gm^m] v_\la=f_\al^{k-1}f_\gm^m v_\la=y_1^{k-1}y_2^m v_\la.
$$
Using this, we find
$$
\langle \tilde x_1^i\tilde x_2^jv_{-\la}, y_1^k,y_2^mv_{\la}\rangle
=
\langle v_{-\la}, x_2^j x_1^iy_1^ky_2^mv_{\la}\rangle
\sim
\langle v_{-\la}, x_2^j x_1^{i-k}y_2^mv_{\la}\rangle =0,
$$
assuming $i>k$.
If $i<k$, then $y_1$ is pulled to the left in a similar way, so
the matrix coefficient can be non-zero only if $i=k$.
Suppose $i=k$ but $j>m$.
Then
$$
\langle \tilde x_1^k\tilde x_2^jv_{-\la}, y_1^k,y_2^mv_{\la}\rangle
\sim
\langle v_{-\la}, x_2^k x_1^iy_1^ky_2^mv_{\la}\rangle
\sim
\langle v_{-\la}, x_2^j y_2^mv_{\la}\rangle \sim
\langle v_{-\la}, x_2^{j-m}v_{\la}\rangle =0.
$$
The case $i=k$, $j<m$ is verified similarly.
\end{proof}
Define $[2l]_q!!=\prod_{i=1}^{l}[2l]_q$ for all positive integer $l$ and put $[0]_q!!=1$.
\begin{propn}
The matrix coefficients of the invariant pairing are given by the formula
$$
\langle \tilde x_1^k\tilde x_2^mv_{-\la}, y_1^k y_2^mv_{\la}\rangle=
q^{m(m-2)+k(k-2)}
\Bigl(\frac{1}{q-q^{-1}}\Bigr)^{k+m}[2k]_q!![2m]_q!!,
$$
for all $k,m=0,1,\ldots$.
\end{propn}
\begin{proof}
The matrix coefficient $\langle \tilde x_1^k\tilde x_2^mv_{-\la}, y_1^k y_2^mv_{\la}\rangle$ is equal to
$$
\langle v_{-\la},(-q^{-h_\gm}e_\gm)^m (-q^{-h_\al}e_\al)^kf_\al^k,f_\gm^mv_{\la}\rangle=
c\langle v_{-\la},e_\gm^m e_\al^kf_\al^kf_\gm^mv_{\la}\rangle,
$$
where $c=(-1)^{k+m}q^{(k+m)(\al,\la)+m(m-1)+k(k-1)}$.
Further,
$
\langle v_{-\la}, e_\gm^m e_\al^k f_\al^k f_\gm^m v_{\la}\rangle
$
is found to be
$$
\langle v_{-\la}, e_\gm^m e_\al^{k-1}
f^{k-1}_{\al}\Bigl( q^{h_\al+1}\frac{1-q^{-2k}}{(q-q^{-1})^2}+q^{-h_\al-1}\frac{1-q^{2k}}{(q-q^{-1})^2}\Bigr) f_\gm^m v_{\la}\rangle+\ldots
$$
The omitted term is zero, as it involves $e_\al f_\gm^m v_{\la}=0$.
We continue in this way and get
$$
[k]_q!\prod_{i=1}^k[(\al,\la)+1-i]_q[m]_q!\prod_{i=1}^m[(\al,\la)+1-i]_q
=[2k]_q!![2m]_q!!
\Bigl(\frac{q^{(\al,\la)}q}{q-q^{-1}}\Bigr)^{k+m},
$$
since
$
[(\al,\la)+1-i]_q=\frac{q^{(\al,\la)+1-i}-q^{-(\al,\la)-1+i}}{q-q^{-1}}
=q^{(\al,\la)+1}\frac{q^{-i}+q^{i}}{q-q^{-1}}
=\frac{q^{(\al,\la)+1}[2i]_q}{(q-q^{-1})[i]_q}
$.
Combining this result with the multiplier $c$ calculated earlier and taking into account $q^{2(\al,\la)}=-q^{-2}$
we prove the statement.
\end{proof}
Let $\C_\hbar[G_{DS}]$ denote the affine coordinate ring on the quantum
group $Sp_q(4)$, i.e. the quantization of the algebra of polynomial functions
on $G$ along the Drinfeld-Sklyanin bracket. It is the Hopf dual to the quantized universal enveloping algebra
$U_\hbar(\g)$, and the reader should not confuse it with $\C_\hbar[G]$ defined
in Section \ref{S_qG}. It is known that the multiplication in $\C_\hbar[G]$, call it
$\cdot_\hbar\>$,
is a star product, \cite{T}.
Denote by $\C_\hbar[G_{DS}]^\l$ the subalgebra
of $U_\hbar(\l)$-invariants in $\C_\hbar[G]$ under the left co-regular action.
It is a natural right $U_\hbar(\g)$-module algebra under the right co-regular action and is
also a quantization a closed conjugacy class (quotient by the Levi subgroup).
The Shapovalov form on $\hat M_\la$ is invertible for non-special $\la$ and its inverse
provides an associative multiplication on $\C_\hbar[G_{DS}]^\l$. For the special value of
$\la$, it has a pole, while its regular part still defines an associative multiplication
on a certain subspace of $\C_\hbar[G_{DS}]$, as argued in \cite{KST}. Description of this subspace
boils down to description of the "stabilizer" of the quantum conjugacy class. It should
be an appropriate deformation of $U(\k)\supset U(\l)$ within $U_\hbar\bigl(\s\p(4)\bigr)$,
where $\k=\s\p(2)\oplus \s\p(2)$. The algebra $U_\hbar(\k)$ is unknown to us at present and will be a subject of our future research.
The stabilizer $U_\hbar(\k)$ will determine the subspace $\C_\hbar[G_{DS}]^\k$
of its invariants that supports the associative multiplication
\be
f\tp g\mapsto f*_\hbar g=\sum_{k,m=0}^\infty
q^{-m(m-2)-k(k-2)}\frac{(q-q^{-1})^{k+m}}{[2k]_q!![2m]_q!!}
\bigl(\tilde x_1^{k}\tilde x_{2}^{m}f\bigr)\cdot_\hbar\bigl(y_1^{k} y_{2}^{m}g\bigr),
\label{star}
\ee
for $f,g\in \C_\hbar[G_{DS}]^\k$. This multiplication will be a star-product
on $\C[\mathbb{S}^4]$.
Remark that product (\ref{star}) is not perfectly explicit
because it is expressed through $\cdot_\hbar$,
whose explicit expression through the classical multiplication in $\C[G]$ is unknown.
Also, the new multiplication should be isomorphic to $\cdot_\hbar$,
because $\mathbb{S}^4$ has a unique structure of
Poisson manifold over the Poisson group $G$.
Thus, neither (\ref{star}) nor (\ref{q-rel}) are particularly new with regard to {\em abstract} quantization.
For instance, one can apply the method of characters (which is doable in this special case)
and realize the quantized polynomial algebra on $\mathbb{S}^4$ both as a quotient of $\C_\hbar[G]$
and a subalgebra in $\C_\hbar[G_{DS}]$, \cite{DM}. Alternatively, one can quantize $\mathbb{S}^4$
through the quantum plane, along the lines of \cite{FRT}.
The novelty of the present work is a realization of $\C[\mathbb{S}^4]$ by operators on a highest weight module.
This approach admits far reaching generalization that unites the
non-Levi conjugacy classes and the classes
with Levi isotropy subgroups in a common quantization context.
The general approach to quantization of non-Levi conjugacy classes of simple
complex matrix groups will be the subject of our future research.
\vspace{20pt}
{\bf Acknowledgements}.
This research is supported in part by the RFBR grant 09-01-00504. The author is grateful
to the Max-Plank Institute for Mathematics for hospitality.
| 78,261
|
TITLE: Prove that the cardinality of the reals and all binary funcions is not equal
QUESTION [0 upvotes]: Let $S$ be the the set of all real functions that bring back only two values: 0 and 1 (Binary functions).
If $f\in S$ then $f:\mathbb{R}\rightarrow \left\{0,1\right\}$.
Prove that $|\mathbb{R}| \neq |S| $.
I tried to start with a proof by contradiction that there's no one to to correspondence but I got stuck. I also assume that by the end of the proof we show that $|S|=\aleph_0 \ne C=|\mathbb{R}|$.
Thanks in advance.
REPLY [3 votes]: If you’ve not seen Cantor’s theorem before, this is a fairly hard problem. Suppose that $\varphi:\Bbb R\to S$ is an injection (one-to-one function). I claim that $\varphi$ cannot be a surjection (onto function). If true, this means that there is no bijection from $\Bbb R$ to $S$ and hence that $|\Bbb R|\ne|S|$.
For each $r\in\Bbb R$ let $f_r=\varphi(r)$; $f_r$ is a function from $\Bbb R$ to $\{0,1\}$. To show that $\varphi$ is not surjective, we’ll find a function $g:\Bbb R\to\{0,1\}$ that is different from $f_r$ for every $r\in\Bbb R$. Or rather, I’ll tell you how to construct it and let you finish the details, including the verification that $g\ne f_r$ for every $r\in\Bbb R$. This will show that $g$ is not in the range of $\varphi$ and hence that $\varphi$ is not a surjection.
The idea is simple but powerful: for each $r\in\Bbb R$ choose $g(r)\in\{0,1\}$ so that $g(r)\ne f_r(r)$. That doesn’t give you much choice, since $\{0,1\}$ has only two elements; in fact, it completely defines the function $g$.
| 185,466
|
Download Kaspersky Internet Security 2013 Technical Preview for Windows 8
I have already written about Internet Security suites that work with Windows 8 and the Windows 8 itself includes a basic antivirus called Windows Defender. Yes, Microsoft Security Essentials has been merged into Windows Defender so that it will protect the system from viruses and spyware.
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Download Kaspersky Internet Security 2013 technical preview
Installing Kaspersky Internet Security 2013
When I installed the security suite on Windows 8, 64-bit edition, it took about 15 minutes to install which is a huge amount of time. Kaspersky needs to look into this and reduce it to less than 5 minutes.
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The following protection modules are included in KIS 2013:
- File antivirus
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You also get additional tools useful for troubleshooting infected computers like create rescue disk, Microsoft Windows troubleshooting tool, privacy cleaner and browser configuration analysis tool.
If the computer is used by multiple people, you can setup parental controls to limit the usage of computer by different individuals.
Overall KIS 2013 is a great security suite which although takes a long time to install, does not seem to affect the computer performance a lot. Please note that you should never use technical preview or beta products on production systems.
| 135,052
|
Internationalisation- what is it all about?
One of the most widely discussed issues in high-flying universities right now is internationalisation. Susan Bassnett FRSL, Pro Vice-Chancellor and Professor in the Centre for Translation and Comparative Cultural Studies, looks at what it actually means and how it might differ from what has gone before.
An international campus
As anyone walking around the Warwick campus can see, the population today is indeed international. Every year thousands of students from around the world apply to Warwick programmes. The lucky few find themselves living in the lovely Warwickshire countryside, on a campus that has grown beyond all recognition in the last two decades. Indeed, from being a small radical institution in the 1960s, Warwick has been transformed into one of the UK’s top ten universities. Its population has increased beyond the wildest dreams of the pioneering academics and students who first ventured onto what was then little more than a building site in a field in the early years.
Unprecedented movement
The increased international population of academics and students at Warwick reflects broader social trends. Millions of people move around the planet today on an unprecedented scale. Some are driven by wars, famine or persecution to leave their homelands and seek a new life elsewhere, many move for improved opportunities and better jobs, or to acquire an international education. The English-speaking world has enjoyed a great advantage in such a climate: since English has become such an important language globally, studying in English is an aspiration for many students, whether the variety of English is British, North American or Antipodean. During the 1990s, following the collapse of the Berlin Wall and the opening up of former Communist countries, plus the open door policy pursued by the expanding economic power that is China, English-speaking universities could rest assured that their international numbers would grow steadily.
Double advantage
Today however, we have to rethink what we understand by internationalisation. Although the demand to learn English is high, it carries its own disadvantage. The strength of learning another language is that you are able to access another culture, to understand other ways of thinking. This has great advantages in the job market. Those overseas students who come to study in the UK are doubly advantaged: they leave with a UK degree and with an excellent command of English. In contrast, home students, unless they have studied a language, leave as they arrived, that is as monolinguals, and perhaps even more significantly, without any first-hand awareness of cultural diversity.
Catastrophic decline
At least one leading UK university is reintroducing a compulsory foreign language entry requirement, but with the catastrophic decline of language teaching in state schools such a decision is bound to be controversial. What is needed is a much broader strategy, an internationalisation plan that goes out beyond the purely linguistic. Although it is valuable to learn another language, it is possible to learn a great deal about cultural differences in other ways.
Cultural awareness
International business has long recognised the importance of cultural awareness training: differences in business practices, in timekeeping, in politeness, in the managing of formal and informal meetings, in the importance of hospitality, in dress and body language codes that are vitally significant and can all be taught. Students in a genuinely internationalised university should be undertaking such training, which will be invaluable when they start to compete in the international job market.
Information flow
What can also be taught, even without knowledge of another language, is how complex the process of transferring information across languages can be. My Arts and Humanities Research Council-funded project on global news translation, for example, studied the ways in which the news we read about or listen to has been constructed across language frontiers, passing through all kinds of processes of translation, editing and rewriting. Students need to be made aware of the processes that affect information flows in our global world, so alongside cultural awareness training, the study of translation as a shaping force in world affairs should also be taught as part of a basic university education.
Internationalisation in education, business and trade is unstoppable; what we in leading universities need to do is equip our students for the future. Joining a cosmopolitan academic community is one step in the right direction; what we need to do now is find ways of equipping students with the right tools that will enhance the quality of their Warwick degrees even further.
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TITLE: How can this IBVP with regularity boundary conditions be solved?
QUESTION [2 upvotes]: I have a radial Schrödinger equation for a particle in Coulomb potential:
$$i\partial_t f(r,t)=-\frac1{r^2}\partial_r\left(r^2\partial_r f(r,t)\right)-\frac2rf(r,t)$$
with initial condition
$$f(r,0)=e^{-r^2}$$
and boundary conditions
$$\begin{cases}
|f(0,t)|<\infty\\
|f(\infty,t)|<\infty.
\end{cases}$$
Trying to solve it, I couldn't come up to any analytical solution, so I tried to solve it numerically.
Simplest what I thought of was finite differences method. But while I can limit domain to make it finite, imposing zero Dirichlet condition at $r=A$ for some $A$, I would need to somehow impose the condition of regularity at $r=0$, where $f$ doesn't have to be neither zero, nor non-zero in general. I don't know how to do this with finite differences.
Another approach I tried was to expand the initial condition in eigenfunctions of the Hamiltonian operator (the RHS of the PDE) and then approximate the solution taking finite number of eigenstates into account. But the expansion appeared to converge extremely slowly, and after taking 200 bound eigenstates (simple because of closed-form solution in terms of Laguerre polynomials) and 70 eigenstates of continuous spectrum (taking those of them which vanish at $r=A$, so as to impose zero boundary condition at $A$; had $A=10$), I still got nothing similar to initial function.
So, the question is: is there any explicit solution for this IBVP (be it closed-form one or in some sort of series, but necessarily explicit and fast convergent)? If no, how can it be solved numerically?
REPLY [0 votes]: This may be a wild goose chase, and at the very least it will be a mess, but here are my thoughts. The time-independent Schrodinger equation (TISE) with Coulomb potential can be solved exactly -- the negative-energy solutions are in any quantum mechanics book and the positive-energy solutions are confluent hypergeometric functions of some sort (I believe they are given in Schiff, for instance). So you can write a general solution of the time-dependent equation (TDSE) as a linear combination of all solutions of the TISE each multiplied by exp(-iEt); the coefficients are determined by the initial condition. (You only need the l=0 solutions, of course, since your initial condition is rotationally invariant.) I have not mentioned boundary conditions, and indeed I can only say that I think the above solutions of the TISE are well-behaved at the origin. The positive-energy solutions surely don't go to zero as r->infinity but I think the sum must do so.
| 93,840
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Walkinside 6.1
Old versionsSee all
Walkinside automatically creates a complete 3D virtual model of facilities, merging native files from diverse data sources and proprietary formats without creating an additional data base. Multi-vendor 3D CAD data, together with associated engineering databases are typically imported including:
- laser scanning output
- aerial photography
- digital terrain models (DTM)
Share your experience:Write a review about this program
Info updated on:
| 193,956
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We retired Tax Justice Blog in April 2017. For new content on issues related to tax justice, go to
In recent years, Georgia has been a hotbed for regressive proposals to eliminate or lower the state’s
reliance on income taxes and replace that revenue with higher sales taxes. So far each of these proposals has been rejected, though late last year voters did cap the state’s top personal income tax rate—a change that could lead to financial problems down the road and may prevent future Georgians from making needed investments.
But hope springs eternal as there are indications that during the upcoming legislative session lawmakers are interested in tax reform yet again. While one of the most serious proposals on table is a familiar sort of regressive tax shift, the Georgia Budget and Policy Institute (GBPI) has released a new report explaining that the state has a variety of tax reform options at hand that would actually improve the fairness of the state’s tax code. In “A Tax Blueprint to Strengthen Georgia,” GBPI prescribes a tax plan that provides:
“a targeted tax cut to Georgians climbing the ladder toward the middle class, while protecting the state’s most critical investments. The plan consists of three core tenets: cut income taxes from the bottom up; modernize the sales tax to fit today’s online commerce and make special tax deductions less generous.”
An Institute on Taxation and Economic Policy (ITEP) analysis of the GBPI plan found that the overall fairness of Georgia’s tax structure would be improved under the proposal and the middle 20 percent of Georgians would see an average tax cut of $206. This blueprint for Georgia tax reform should be required reading for Georgia lawmakers. Once the debate heats up let’s hope they also heed the words of Wesley Tharpe from GBPI who opined in the Atlanta Journal Constitution, “One thing is for sure: A drastic shift from income to sales taxes is a flawed approach to reform. Georgia can do better.”
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Published: March 2017 | Report Code: LS10824 | Available Format: PDF | Pages: 179
The research offers historical market size of the global wearable injectors market for the period 2012 - 2015 and market forecast for the period 2016 – 2024.
GLOBAL WEARABLE INJECTORS MARKET
GLOBAL WEARABLE INJECTORS MARKET, BY GEOGRAPHY
North America Wearable Injectors Market
Europe Wearable Injectors Market
APAC Wearable Injectors Market
Latin America Wearable Injectors Market
MEA Wearable Inject
| 213,893
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Raktu (Boyar) seeds are related to the more popular (and increasingly harder to source) phoenix-eye / bodhi seed from Namo Buddha area in Nepal. Malas made from seeds are said to be good for any kind of mantra practice. This is a good "starter mala" if you are new to the use of malas for your practice.
This is simple prayer bead mala features 108 Raktu seed beads that are hand strung on durable knotting cord. It is the perfect mala for meditation practice and every day wear. With meaning similar to the bodhi seed, many practitioners use raktu malas for their essence and inherent teachings of enlightenment within each seed.
These mala beads are great if you like a natural seed bead in a smaller size. Seed malas are some of the most common malas used in Tibet and Nepal. These seeds are sustainable and are harvested each year then drilled and made into mala beads.
That is how your Raktu Mala will looks after 8 years use.
| 382,182
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Karachi:
International Cricket Council's promise to pay them the hosting fees notwithstanding, Pakistan Cricket Board stands to lose at least USD 2.75 million after Champions Trophy was relocated from the country last month.
Sources in the board said the USD three million it has been promised by the ICC is actually participating fees and PCB is actually set to lose its hosting fees.
"It is a wrong impression that Pakistan could get USD six million hosting fees. Pakistan is guaranteed, like every other participating country, USD three million as participation fees, another USD 750,000 for gate money receipts and USD two million as the hosting fees," the source said.
But the ICC has confirmed only the USD three million participation fees for the Pakistan board and no understanding or agreement had been signed on the remaining USD 2.75 million, he claimed.
The ICC executive board decided last month that due to security conditions the Champions Trophy would be relocated from Pakistan and a new venue would be announced in March/April for the eight-nation event.
The ICC had also said that because Pakistan was being deprived of the tournament it would still get the hosting fees even if the tournament was held elsewhere.
| 172,584
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Learn more at the Research Data Management Website
For personalized assistance with research data management or to arrange a workshop, please email research.data@ubc.ca
| 318,523
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Overview
America is a Premier destination for international students owing to its strong economy, vibrant campus life and the variety, flexibility and quality academic options it offers. From fabulous cities to beautiful natural parks, you are sure to have a spell bound life in the US.
Quick Facts
Capital: Washington, D.C.
Population : 32.57 million
GDP : $18.57 trillion
Number of universities : 4140
Foreign Student Enrolment 2012/2013 : 425,260
Currency : Dollar
Official Languages : English
Why Study in USA ?
1. Ranked no 1 for overall quality of education.
2. Numerous Study Choices.
3. Internationally recognised.
4. Flexible Education system.
5. Merit based university assistance provided.
6. Scholarships and fellowships awarded by departmen.
Education System in USA
The higher education imparting bodies in the USA are
1. USA education system comprises of 12 complete years of primary and secondary education prior to university or graduate college.
2. On completion of high school students move on to attend undergraduate school followed by graduate school, post-graduation and PhD study.
Required Tests
The test required are;
1. IELTS or TOEFL scores or any equivalent as a proof of English Proficiency.
2. Diploma programs: Overall band score of 6.0.
3. Bachelor's Degree: Overall band score of 7.0.
4. Masters Programs: Overall band score of 7.0 – 7.5
Intake
January /August.
Cost Of Education
Average tuition fees per year range from $15,000 to $25,000 (Private Institutions) and $10,000 to $20,000 (State Institutions)*.
Cost of Living
1. Estimated living expenses per year – $10,000-$16,000*.
Work Rights & Stay Back
1. 12 months optional practical training 17 months' extension for STEM Majors.
2. Authorization to work 20 hours per week (Only on Campus).
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Bullfrogs, Toads and Grey Tree Frogs in Massachusetts
The story continues with Jack Stearns, a scientist and meteorologist in Massachusetts, who had rescued a Bullfrog (Bartholomew) last Winter, updates us on his progress along with a discovery of Grey Tree Frogs in the area.
Bart must be happy back in his pond as my wife hears deep croaking when she walks by the pond at lunch time. It has to be Bart! Also seen has been a big bullfrog near the spot where we let him go and he makes quite a splash when he jumps into the pond.
Another story involves a Gray Tree Frog. My wife works as a receptionist and is located in a huge lobby which has a big indoor garden, complete with trees and many plants. A couple of years ago a Gray Tree Frog got in and took up residence and proceeded to serenade the guards at night. It took them months to figure out what the noise was since the chirping resonates in the big lobby. He only hibernated for two months and came out in February to start singing again. It was weird to see snow falling and hearing this frog chirp. In fact, that was his name, Chirp.
He disappeared in the spring and we figure he got out the same way he got in, under the door that is right by the indoor garden.
Well it looks like history is repeating itself. Above is a picture of a very young tree frog who got into the lobby. After this picture was taken he proceeded to scurry up the wall behind him into the indoor garden. Apparently a few others have been seen entering as well, especially at night. No noise yet, but I figure that by early spring there will be another chorus of tree frogs in the solarium. There is plenty for them to eat as they have been observed close to the outdoor window, snagging bugs that land there.
The frog population may be declining but not around where my wife works. The underground garage has lots of toads in the summer season who know that bugs are attracted by the lights and the toads come in for a quick meal. Everyone is careful of the toads when walking around the garage and there have been very few fatalities.
— Jack Stearns
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Your Ministry’s Financial Security: It’s a Big DealJohn Gilman August 5, 2016
A friend of mine, an administrative pastor of a large church told me about a time he had a group of retired special forces officers drive him to the bank. His church collected over $600,000 on one Sunday during a capital campaign, and one of these guys thought they should take extra precautions. He was… Continue Reading »
| 96,984
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Create me a Unique Handwritten Font
125
I'd like a unique - as in: no one else will ever own it, Font. Just for my business. I would need truetype and illustrator vectorized versions. I'd also like to get an unlimited font license, and an agreement that it wouldn't be used by anyone else. Of course I would credit you, and you could showcase it as needed, but I'd like something that is completely unique to me, my brand and my business.
I might be crazy. If I am explain that to me. Tell me why this is not possible, etc.
I'm also flexible on price. Tell me what it would take to make it happen.
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\begin{document}
\maketitle
\begin{abstract}
In a recent paper with Thomas Hoffmann-Ostenhof, we proved that the number of critical points $\nu_k$ in the boundary set of a $k$-minimal partition
tends to $+\infty$ as $k\ar +\infty$. In this note, we show that $\nu_k$ increases linearly with $k$ as suggested by a hexagonal conjecture about the asymptotic behavior of the energy of these minimal partitions. As the original proof by Pleijel,
this involves Faber-Krahn's inequality and Weyl's formula, but this time, due to the magnetic characterization of the minimal partitions, we have to establish
a Weyl's formula for Aharonov-Bohm operator controlled with respect to a $k$-dependent number of poles.
\end{abstract}
\section{Introduction}
We consider the Dirichlet Laplacian in a bounded
regular domain $\Omega\subset \mathbb R^2$.
In \cite{HHOT} we have analyzed the elations between the nodal domains
of the real-valued eigenfunctions of this Laplacian and the partitions of
$\Omega$
by $k$ disjoint
open sets $ D_i$ which are minimal in the sense that the
maximum over the $
D_i$'s of the ground state energy (or smallest eigenvalue) of the Dirichlet
realization
of the Laplacian in $ D_i$ is minimal.
We denote by $ (\lambda_j(\Omega))_{j\in \mathbb N}$ the
non decreasing sequence of its eigenvalues and by $\phi_j$ some associated
orthonormal basis of real-valued eigenfunctions.
The groundstate $ \phi_1$ can be chosen to be
strictly positive in $ \Om$, but the other eigenfunctions
$ \phi_j$ ($j>1$) must have non empty zeroset in $\Omega$.
By the zero-set of a real-valued continuous function $u$ on
$ \overline\Om $, we mean
$
N(u)=\overline{\{x\in \Om\:\big|\: u(x)=0\}}
$
and call the components of $ \Om\setminus N(u)$ the nodal
domains of $u$.
The number of
nodal domains of $ u$ is called $ \mu(u)$. These
$\mu(u)$ nodal
domains define a $k$-partition of $ \Omega$, with $k=\mu(u)$.
We recall that the Courant nodal
Theorem \cite{Cou} says that, for $k\geq 1$, and if $ E(\lambda_k)$ denotes the eigenspace associated with $\lambda_k\,$, then, for all real-valued $ u\in E(\lambda_k)\setminus \{0\}\;,\;
\mu (u)\le k\,.
$
A theorem due to Pleijel \cite{Pl} in
1956 says
that this cannot be true when the dimension (here we consider the
$2D$-case) is larger than one. In the next section, we describe the link of these results with the question of spectral minimal partitions which were introduced by Helffer--Hoffmann-Ostenhof--Terracini \cite{HHOT}.
\section{Minimal spectral partitions}
We now introduce for $k\in \mathbb N$ ($k\geq 1$),
the notion of $k$-partition. We
call {\bf $ k$-partition} of $ \Omega$ a family
$ \mathcal D=\{D_i\}_{i=1}^k$ of mutually disjoint sets in $\Omega$.
We denote by $ \mathfrak O_k(\Omega)$ the set of open connected
partitions of $\Omega$.
We now introduce the notion of energy of the partition $\mathcal D$ by
\begin{equation}\label{LaD}
\Lambda(\mathcal D)=\max_{i}\la(D_i)\,.
\end{equation}
Then we define for any $k$ the minimal energy in $\Omega$ by
\begin{equation}\label{frakL}
\mathfrak L_{k}(\Omega)=\inf_{\mathcal D\in \mathfrak O_k}\:\Lambda(\mathcal D).
\end{equation}
and call $ \mathcal D\in \mathfrak O_k$ a minimal $k$-partition if
$ \mathfrak L_{k}=\Lambda(\mathcal D)$.
We associate with a partition its {\bf boundary set}:
\begin{equation}\label{assclset}
N(\mathcal D) = \overline{ \cup_i \left( \partial D_i \cap \Omega
\right)}\;.
\end{equation}
The properties of the boundary of a minimal partition are quite close to the properties of nodal sets can be described in the following way:
\begin{enumerate}
\item
Except for finitely many distinct $ X_i\in \Om\cap N$
in the neighborhood of which $ N$ is the union of $\nu_i= \nu(X_i)$
smooth curves ($ \nu_i\geq 3$) with one end at $ X_i$,
$ N$ is locally diffeomorphic to a regular
curve.\\
\item
$ \pa\Om\cap N$ consists of a (possibly empty) finite set
of points $ Y_i$. Moreover
$N$ is near $ Y_i$ the union
of $ \rho_i$ distinct smooth half-curves which hit
$ Y_i$.\\
\item $ N$ has the {\bf equal angle
meeting
property}\footnote{ The half curves meet with equal angle at each critical
point of $ N$ and also at the boundary together with the
tangent to the boundary.}
\end{enumerate}
The $X_i$ are called the critical points and define the set
$X(N)$. A particular role is played by $X^{odd}(N)$ corresponding to the critical points for which $\nu_i$ is odd.
It has been proved by Conti-Terracini-Verzini (existence)
and Helffer--Hoffmann-Ostenhof--Terracini (regularity) (see \cite{HHOT} and references therein) that for any $ k$, there exists a minimal regular $
k$-partition, and moreover that
any minimal $ k$-partition has a regular
representative\footnote{possibly
after a modification of the open sets of the partition by capacity $0$ subsets.}.
In a recent paper with Thomas Hoffmann-Ostenhof \cite{HHO}, we proved that the number of odd critical points of a minimal $k$-partition $\mathcal D_k$
\begin{equation}\label{weak}
\nu_k:=\# X^{odd} (N(\mathcal D_k))
\end{equation}
tends to $+\infty$ as $k \ar + \infty$. \\
In this note, we will show that it increases linearly with $k$ as suggested by the hexagonal conjecture as discussed in \cite{BHV,CL, BBO,HHO}. This conjecture says that
\begin{equation}\label{hexa}
A(\Omega) \lim_{k\ar +\infty} \frac{\mathfrak L_k(\Omega)}{k} = \lambda({\rm Hexa_1})\,,
\end{equation}
where ${\rm Hexa_1}$ denotes the regular hexagon of area $1$ and $A(\Omega)$ denotes the area of $\Omega$.\\
Behind this conjecture, there is the idea that $k$-minimal partitions will look (except at the boundary where one can imagine that pentagons will appear) as the intersection with $\Omega$ of a tiling by hexagons of area $\frac{1}{k} A(\Omega)$.\\
The proof presented here gives not only a better result but is at the end simpler, although based on the deep magnetic characterization of minimal partitions of \cite{HHmag} which will be recalled in the next section.
\section{ Aharonov-Bohm operators and magnetic characterization.}
Let us recall some definitions about the Aharonov-Bohm
Hamiltonian in an open set $\Omega$ (for short $ {{\bf A}{\bf B}}X$-Hamiltonian) with a singularity at $ X\in \Omega$ as considered in \cite{HHOO,AFT}. We denote by $ X=(x_{0},y_{0})$ the coordinates of the pole and
consider the magnetic potential with flux at $ X$:
$ \Phi = \pi $, defined in $\dot{\Omega_X}= \Omega \setminus \{X\}$:
\begin{equation}
{{\bf A}^X}(x,y) = (A_1^X(x,y),A_2^X(x,y))=\frac 12\, \left( -\frac{y-y_{0}}{r^2}, \frac{x-x_{0}}{r^2}\right)\,.
\end{equation}
The ${{\bf A}{\bf B}}X$-Hamiltonian is defined by considering the Friedrichs
extension starting from $ C_0^\infty(\dot \Omega_{X})$
and the associated differential operator is
\begin{equation}
-\Delta_{{\bf A}^X} := (D_x - A_1^X)^2 + (D_y-A_2^X)^2\,\mbox{with }D_x =-i\pa_x\mbox{ and }D_y=-i\pa_y.
\end{equation}
Let $ K_{X}$ be the antilinear operator
$ K_{X} = e^{i \theta_{X}} \; \Gamma\,$,
with $ (x-x_0)+ i (y-y_0) = \sqrt{|x-x_0|^2+|y-y_0|^2}\, e^{i\theta_{X}}\,$, $\theta_X$ such that $d\theta_X= 2 {\bf A}^X\,$,
and where $ \Gamma$ is the complex conjugation operator
$ \Gamma u = \bar u\,$. A function $ u$ is called $ K_{X}$-real, if
$ K_{X} u =u\,.$ The operator $ -\Delta_{{\bf A}^X}$ is preserving the
$ K_{X}$-real functions and we can consider a
basis of $K_{X}$-real eigenfunctions.
Hence we only analyze the
restriction of the ${{\bf A}{\bf B}}X$-Hamiltonian
to the $ K_{X}$-real space $ L^2_{K_{X}}$ where
$$
L^2_{K_{X}}(\dot{\Omega}_{X})=\{u\in L^2(\dot{\Omega}_{X}) \;,\; K_{X}\,u =u\,\}\,.
$$
This construction can be extended to
the case of a
configuration with $ \ell$ distinct points $ X_1,\dots, X_\ell$ (putting a flux $ \pi$ at each
of these points). We just take as magnetic potential
$$
{\bf A}^X = \sum_{j=1}^\ell {\bf A}^{X_j}\,, \mbox{ where } X=(X_1,\dots,X_\ell)\,.$$
We can also construct the antilinear
operator $ K_X$, where $ \theta_X$ is replaced by a
multivalued-function $ \phi_X$ such that $ d\phi_X = 2 {\bf A}^{X}$. We can then consider the
real subspace of the $ K_X$-real
functions in $ L^2_{K_{X}}(\dot{\Omega}_{X})$.
It was shown in \cite{HHOO} and \cite{AFT} that the $ K_{X}$-real eigenfunctions have a regular nodal set
(like the eigenfunctions of the Dirichlet Laplacian) with the
exception that at each singular point $ X_j$ ($ j=1,\dots,\ell$)
an odd number of half-lines meet.\\
The next theorem which
is the most interesting part of the magnetic characterization of the minimal partitions given in \cite{HHmag} will play a basic role in the proof of our main theorem.
\begin{theorem}\label{thchar}[Helffer--Hoffmann-Ostenhof]~\\
Let $ \Omega$ be simply connected. If $ \mathcal D$ is a $k$-minimal partition of $\Omega$, then, by choosing $ (X_1,\dots, X_\ell)= X^{odd}(N(\mathcal D))$, $ \mathcal D$ is the nodal partition
of some $ k$-th $ K_{X}$-real eigenfunction of the Aharonov-Bohm Laplacian associated with $ \dot{\Omega}_X$.
\end{theorem}
\section{Analysis of the critical sets in the large limit case}
We can now state our main theorem, which improves \eqref{weak} as proved in \cite{HHO}.
\begin{theorem}[Main theorem]~\\
Let $(\mathcal D_k)_{k\in \mathbb N} $ be a sequence of regular minimal $k$-partitions.
Then there exists $c_0 >0$ and $k_0$ such that for $k\geq k_0$,
$$\nu_k:= \# X^{odd}(N(\mathcal D_k ))\geq c_0 k\,.
$$
\end{theorem}
{\bf Proof}\\
The proof is inspired by the proof of Pleijel's theorem, with the particularity that the operator, which is now the Aharonov-Bohm operator will depend on $k$. For each $\mathcal D_k$, we consider the corresponding Aharonov-Bohm operator as constructed in Theorem \ref{thchar}.\\
We come back to the proof of the lower bound of the Weyl's formula but we will make a partition in squares depending on
$$\lambda =\mathfrak L_k\,.$$
We introduce a square $Q_p$ of size $t/\sqrt{\lambda}$ with $t\geq 1$ which will be chosen large enough (independently of $k$) and will be determined later.
Having in mind the standard proof of the Weyl's formula (see for example \cite{CH}), we recall the following proposition
\begin{proposition}\label{Corw}~\\ If $\mathcal D$ is a partition of $\Omega$, then
\begin{equation}\label{mmw}
\sum_i n(\lambda, D_i) \leq n(\lambda, \Omega)\;.
\end{equation}
\end{proposition}
Here $n(\lambda,\Omega)$ is the counting function of the eigenvalues $< \lambda$ of $H(\Omega)$.\\ This proposition is actually present in the proofs of
the asymptotics of the counting function. We will apply this proposition in the case of Aharonov-Bohm operators $H:= -\Delta_{{\bf A}^X }$ restricted to $K_X$-real $L^2$ spaces. $H(D)$ means the Dirichlet realization (obtained via the Friedrichs extension theorem) of $-\Delta_{{\bf A}^X}$ in an open set $D\subset \Omega$.\\
\begin{remark}\label{remutile}
Note that if no pole belongs to $D$ (a pole on $\partial D$ is permitted) and if $D$ is simply connected, then $H(D)$ is unitary equivalent (the magnetic potential can be gauged away) to the Dirichlet Laplacian in $D$.
We refer to \cite{AFT,Len,NT} for a careful analysis of the domains of the involved operators.
\end{remark}
We now consider a maximal partition of $\Omega$ with squares $Q_p$ of size $t/\sqrt{\lambda}$ with the additional rule that the squares should not contain the odd critical points of $\mathcal D_k$.\\
The area $A(\Omega_{k,t,\lambda})$, where $\Omega_{k,t,\lambda}$ is the union of these squares, satisfies
$$
A( \Omega_{k,t,\lambda} ) \geq A(\Omega) - \ell t^2
/\lambda - C (t,\Omega)\frac{1}{\sqrt{k}}\,.
$$
The second term on the right hand side estimates from above the area of the squares containing a critical point and the last term takes account of the effect of the boundary.\\
Note that this lower bound of $A(\Omega_{k,t,\lambda})$ leads to the estimate of the cardinal of the squares using
$
\# \{ Q_p \} = A( \Omega_{k,t,\lambda} ) \frac{\lambda}{t^2}\,.
$
In each of the squares, because (as recalled in Remark \ref{remutile}) the magnetic Laplacian is isospectral to the usual Laplacian,
we have (after a dilation argument) :
$$ n( \lambda , Q_p) = n\left(t, (0,1)^2\right)\,.$$
Hence we need
to find a lower bound of $n(t):=n(t, (0,1)^2)$, the number of eigenvalues less than $t^2$ for the standard Dirichlet Laplacian in the fixed unit square. \\
We know, that for any $\epsilon >0$ there exists $t$ such that
\begin{equation} \label{wq}
n(t) \geq (1-\epsilon) \frac{1}{4\pi} t^2 \,.
\end{equation}
This leads, using Proposition \ref{Corw} for $H= - \Delta_{{\bf A}^X}$ (remember that $X$ is given by the magnetic characterization of $\mathcal D_k$) and applying \eqref{wq} in each square, to the lower bound as $k\ar +\infty$,
\begin{equation}\label{z1}
k= n(\mathfrak L_k,\Omega) \geq \left( \frac{1}{4\pi} (1-\epsilon)t^2\right)\, \left( A(\Omega) - \ell t^2 /\mathfrak L_k + o(1) \right) \, \left(\frac{\mathfrak L_k}{t^2}\right) \,
\end{equation}
Let us recall from \cite{HHOT} the following consequence of Faber-Krahn's inequality
\begin{equation}\label{z2}
A(\Omega) \frac{\mathfrak L_k(\Omega)}{k} \geq \pi {\bf j}^2\,,
\end{equation}
where ${\bf j}\sim 2.405 $ is the first zero of the first Bessel function.
Dividing \eqref{z1} by $k$ and using \eqref{z2},
we get, as $k\ar +\infty$
$$
1 \geq \frac{{\bf j}^2}{4} (1-\epsilon) (1 - \frac{\ell}{k} t^2 \pi^{-1} {\bf j}^{-2}) (1 + o(1)).
$$
If we assume that the number $\ell$ of critical points satisfies
$$
\ell \leq \alpha k\,, \, \mbox{for some }\alpha >0\,,
$$
we get
\begin{equation}\label{contra}
1 \geq \frac{{\bf j}^2}{4} (1-\epsilon) (1 - \alpha t^2 \pi^{-1} {\bf j}^{-2}) (1 + o(1)).
\end{equation}
We see that if $\epsilon$ is small enough (this determines $t =t(\epsilon)$) and $\alpha t^2$ is small enough such that
$$
\frac{{\bf j}^2}{4} (1-\epsilon) (1 - \alpha t^2 \pi^{-1} {\bf j}^{-2}) >1
$$
(this gives the condition on $\alpha$),
we will get a contradiction for $k$ large.\\
As recalled in \cite{HHO}, Euler's formula implies that for a minimal $k$-partition
$\mathcal D$ of a simply connected domain $\Omega$
the cardinal of $X^{odd}(N(\mathcal D))$ satisfies
\begin{equation}
\# X^{odd}(N(\mathcal D) ) \leq 2k -4 \,.
\end{equation}
This estimate seems optimal and is compatible with the hexagonal conjecture, which, for critical points, will read
\begin{conjecture}
\begin{equation}
\lim_{k\ar +\infty} \frac{\# X^{odd}(N(\mathcal D_k) )}{k}=2 \,.
\end{equation}
\end{conjecture}
\section{Explicit lower bounds}
Looking at the proof of the main theorem, the contradiction is obtained if \eqref{contra} is satisfied.
Using the universal lower bound for $n(t)$ (see for example \cite{Pl}), we have, if $t\geq 2$
\begin{equation}
n(t) > \frac {1}{4\pi} t^2 - \frac{2}{\pi^2} t + \frac{1}{\pi^2} \,.
\end{equation}
We look for $t=t(\epsilon) \geq 2$ such that
\begin{equation*}
\frac {1}{4\pi} t^2 - \frac{2}{\pi^2} t + \frac{1}{\pi^2} \geq (1-\epsilon) \frac {1}{4\pi} t^2 \,,
\end{equation*}
which leads to the condition
\begin{equation}\label{conde}
\epsilon \frac {1}{4\pi} t^2 - \frac{2}{\pi^2} t + \frac{1}{\pi^2}\geq 0\,.
\end{equation}
We can choose
$t(\epsilon) = \max (2, \frac{ 8}{\epsilon \pi })\,$.
We then get a condition on $\alpha$ through \eqref{contra}. For some admissible $\epsilon$, i.e satisfying: $$ \frac{{\bf j}^2}{4} (1-\epsilon) >1 \,,
$$
the proof works if
$ \alpha < c_0(\epsilon) $, with $c_0(\epsilon) $ solution of
\begin{equation} \label{zc0}
1 = \frac{{\bf j}^2}{4} (1-\epsilon) (1 - c_0(\epsilon) t(\epsilon)^2 \pi^{-1} {\bf j}^{-2}) \,.
\end{equation}
Hence the $c_0$ announced in the theorem can be chosen as
$$
c_0:= \sup_{\epsilon \in (0, 1-4/{\bf j}^2)} c_0(\epsilon)\,,
$$
It remains to determine this $\sup$.
Note that $1-4/{\bf j}^2 \sim 0,36\,$.
Hence we can assume $t(\epsilon)= \frac{8}{\epsilon \pi}$ and get for $c_0(\epsilon)$ the equation
\begin{equation}
c_0(\epsilon)= \epsilon^2 2^{-6} \pi^3 {\bf j}^{2} \left(1 - \frac{4}{{\bf j}^2(1-\epsilon)}\right)\,.
\end{equation}
But $c_0(\epsilon)$ being $0$ at the ends of the interval $(0, 1-4/{\bf j}^2)) $, the maximum is obtained inside by looking at the zero of the derivative with respect to $\epsilon$.
We get
\begin{equation}
\epsilon _{max}= (1 -{\bf j}^{-2} ) - \sqrt{(1- {\bf j}^{-2})^2 - (1- 4{\bf j}^{-2}) } = (1-{\bf j}^{-2}) - {\bf j}^{-2} \sqrt{1 + 2 {\bf j}^2}\,.
\end{equation}
and
\begin{equation}\label{forc0}
c_0 = 2^{-6}{\bf j}^{-2} \pi^3 \left( ({\bf j}^4 + 10 {\bf j}^2 - 2 ) -2 (2 {\bf j}^2+1) \sqrt{1 + 2 {\bf j}^2} \right) \,.
\end{equation}
Numerics with ${\bf j}$ replaced by its approximation gives
$
c_0 \sim 0.014\,$.
This is extremely small and very far from from the conjectured value $2$ !\\
\begin{remark}
One can actually in \eqref{forc0} replace ${\bf j}^2$ by $\frac{A(\Omega)}{\pi} \lim\inf \frac{\mathfrak L_k}{k}$. The constant ${\bf j}^2$ appears indeed only through \eqref{z2}. Because of the monotonicity of $c_0$
as a function of ${\bf j}^2$ which results of the definition of $c_0$ as a sup. any improvement of a lower bound for $\frac{A(\Omega)}{\pi} \lim\inf \frac{\mathfrak L_k}{k}$ will lead to a corresponding improvment of $c_0$.
\end{remark}
{\bf Acknowledgements.\\} I would like to thank T. Hoffmann-Ostenhof for former discussions on the subject, V. Felli and L. Abatangelo for inviting me to give a course on the subject in Milano (February 2015) and the Isaac Newton Institute where the final version of this note was completed, the author being Simons Foundation visiting Fellow there.\\
| 99,030
|
TITLE: Some field extensions with coprime degrees
QUESTION [1 upvotes]: Let $L/K$ be a finite field extension with degree $m$. And let $n\in \mathbb N$ such that $m$ and $n$ are coprime. Show the following:
If there is a $a\in \mathbb K$ such that an $n$-th root of $a$ lies in $L$ then we have already $a\in \mathbb K$.
My attempt:
The field extension $K(\sqrt[n]{a})/K$ has degree smaller $n$. The minimal polynomial of
$\sqrt[n]{a}$ namely $m_{\sqrt[n]{a}}(X)\in K[X]$ divides $X^n-a$. I.e. let $k$ be the degree of the minimal polynomial, then $k|n$.
But because of the formula $[L:K]=[L:K(\sqrt[n]{a}]\cdot [K(\sqrt[n]{a}):K]$ $k|m$, hence $k=1$ and hence our conclusion follows because $[K(\sqrt[n]{a}):K]=1$ yields $\sqrt[n]{a}\in K$ .
Can someone go through it? Thanks.
REPLY [0 votes]: The non-separable case is actually not the problem: Let $p$ be the characteristic.
If $X^n-a$ is not separable, we have $p|n$ and we can write $X^n-a=(X^d-\sqrt[p^r]{a})^{p^r}$ with the inner polynomial being separable. By assumption $\sqrt[n]{a} \in L$, hence $\sqrt[p^r]{a} \in L$. But $\sqrt[p^r]{a}$ is purely inseparable over $K$ and seprable over $K$, because it lies in the separable extension $L/K$. This extension is separable because $m$ and $p$ must be co-prime. Thus $\sqrt[p^r]{a} \in K$. So we are reduced to the separable case with the polynomial $X^d-\sqrt[p^r]{a} \in K[X]$, since $d$ and $m$ are co-prime.
| 66,908
|
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HP OfficeJet Pro 8600 Scanner Software
05-27-2012 10:57 AM
I am running Windows Vista and just purchased the HP OfficeJet Pro 8600 printer. I am having a hell of a time with this printer and am seriously considering returning it if I can't get this figured out asap.
When I open the HP OfficeJet Pro 8600 software, the main panel opens I double click on the "Scanner Actions" button, and then the "Scan Document or Photo" button, nothing happens. I can not get the scanner software to open so that I can control my scan options from the computer.
As well, from the printer control pad, when I go to scan, I am only able to "Scan to Network Folder" and am unable to scan to computer. The message that I am getting is "Computer is Unavailable".
I am able to "Scan to Network Folder" so I am not overly concerned with that issue however, I need to be able to access the scanner software so that I can adjust my scan settings etc on my computer.
I purchase the unit for the wireless. I could not access the software while connected with wireless so I attempted to connect via usb. No difference what so ever.
I have uninstalled and reinstalled the software 4 times so far. 2 times from the CD and 2 times from the software available on the HP site.
This question was solved. View Solution.
Re: HP OfficeJet Pro 8600 Scanner Software
05-31-2012 11:36 AM
Hey dleman. Sorry to hear your having issues and I appreciate your detailed messages. I would try using the HP Print & Scan Doctor. This tool will try to resolve the issue for you and provide a specific code if it can not resolve the issue.
Let me know how it goes and if it does not solve the issue we can go from there.
Thanks
SeanS
Re: HP OfficeJet Pro 8600 Scanner Software
08-12-2012 08:54 AM
Worked for me. Apparently it was a WIA issue. Got it running now. Thank you!
Re: HP OfficeJet Pro 8600 Scanner Software
08-13-2012 10:11 AM
Awesome! Glad to hear it was able to resolve the issue. If you every have any other issues dont hesitate to post them.
Take care!
Sean
Re: HP OfficeJet Pro 8600 Scanner Software
10-17-2012 04:01 PM
HP OfficeJet Pro 8600 Scanner Software
10-19-2012 12:19 PM
Re: HP OfficeJet Pro 8600 Scanner Software
10-19-2012 12:35 PM
The HP print and scan doctor came up with no errors. Still could not connect.
Eventually I found something some where where it told me that it will not dedicate unless you have a windows password set. I did not have a windows password set and therefore it would not allow wireless network connectivity.
When doing a clean install of windows 7, I made sure to set it up with a password. I then reinstalled the HP drivers and when I connected to the printer... No problems what so ever. Every feature works. Don't know what will happen though when it comes time for me to change my password.
Re: HP OfficeJet Pro 8600 Scanner Software
10-26-2012 05:11 AM
My hp officejet pro 8600 the scanner doesn't work please help me to resolve my problem even the webscan doesn't work too.
Re: HP OfficeJet Pro 8600 Scanner Software
07-30-2013 09:09 PM
Sorry to hear about your problem. But is your printer conencted to the network?
Can you try the Print and Scan doctor at the follwoing link?
Let us know if it works.
Click on "Kudos" if the solution works.
Re: HP OfficeJet Pro 8600 Scanner Software
11-30-2013 01:49 AM
Same problems as others seem to be having with scanning. The HP Scanner Doctor does not help as I get a message to say that the program has stopped working. ??
| 56,024
|
Israel Will Return Home 14 1The Lord will show mercy to the people of Jacob, and he will again choose the people of Israel. He will settle them in their own land. Then non-Israelite people will join the Israelites and will become a part of the family of Jacob. 2Nations will take the Israelites back to their land. Then those men and women from the other nations will become slaves to Israel in the Lord’s land. In the past the Israelites were their slaves, but now the Israelites will defeat those nations and rule over them. The King of Babylon Will Fall 3The Lord will take away the Israelites’ hard work and will comfort them. They will no longer have to work hard as slaves. 4On that day Israel will sing this song about the king of Babylon: The cruel king who ruled us is finished; his angry rule is finished! 5The Lord has broken the scepter of evil rulers and taken away their power. 6The king of Babylon struck people in anger again and again. He ruled nations in anger and continued to hurt them. 7But now, the whole world rests and is quiet. Now the people begin to sing. 8Even the pine trees are happy, and the cedar trees of Lebanon rejoice. They say, “The king has fallen, so no one will ever cut us down again.” 9The place of the dead is excited to meet you when you come. It wakes the spirits of the dead, the leaders of the world. It makes kings of all nations stand up from their thrones to greet you. 10All these leaders will make fun of you and will say, “Now you are weak, as we are. Now you are just like us.” 11Your pride has been sent down to the place of the dead. The music from your harps goes with it. Flies are spread out like your bed beneath you, and worms cover your body like a blanket. 12King of Babylon, morning star, you have fallen from heaven, even though you were as bright as the rising sun! In the past all the nations on earth bowed down before you, but now you have been cut down. 13You told yourself, “I will go up to heaven. I will put my throne above God’s stars. I will sit on the mountain of the gods, on the slopes of the sacred mountain. 14I will go up above the tops of the clouds. I will be like God Most High.” 15But you were brought down to the grave, to the deep places where the dead are. 16Those who see you stare at you. They think about what has happened to you and say, “Is this the same man who caused great fear on earth, who shook the kingdoms, 17who turned the world into a desert, who destroyed its cities, who captured people in war and would not let them go home?” 18Every king of the earth has been buried with honor, each in his own grave. 19But you are thrown out of your grave, like an unwanted branch. You are covered by bodies that died in battle, by bodies to be buried in a rocky pit. You are like a dead body other soldiers walk on. 20You will not be buried with those bodies, because you ruined your own country and killed your own people. The children of evil people will never be mentioned again. 21Prepare to kill his children, because their father is guilty. They will never again take control of the earth; they will never again fill the world with their cities. 22The Lord All-Powerful says this: “I will fight against those people; I will destroy Babylon and its people, its children and their descendants,” says the Lord. 23“I will make Babylon fit only for owls and for swamps. I will sweep Babylon as with a broom of destruction,” says the Lord All-Powerful. God Will Punish Assyria 24The Lord All-Powerful has made this promise: “These things will happen exactly as I planned them; they will happen exactly as I set them up. 25I will destroy the king of Assyria in my country; I will trample him on my mountains. He placed a heavy load on my people, but that weight will be removed. 26“This is what I plan to do for all the earth. And this is the hand that I have raised over all nations.” 27When the Lord All-Powerful makes a plan, no one can stop it. When the Lord raises his hand to punish people, no one can stop it. God’s Message to Philistia 28This message was given in the year that King Ahaz died: 29Country of Philistia, don’t be happy that the king who struck you is now dead. He is like a snake that will give birth to another dangerous snake. The new king will be like a quick, dangerous snake to bite you. 30Even the poorest of my people will be able to eat safely, and people in need will be able to lie down in safety. But I will kill your family with hunger, and all your people who are left will die. 31People near the city gates, cry out! Philistines, be frightened, because a cloud of dust comes from the north. It is an army, full of men ready to fight. 32What shall we tell the messengers from Philistia? Say that the Lord has made Jerusalem strong and that his poor people will go there for safety.
| 297,966
|
\begin{document}
\maketitle
\begin{abstract} Let $K$ be a finite extension of $\mathbb{Q}_{p}$ with ring of integers $\mathfrak{o}$, and let $H_{0}$ be a formal $\mathfrak{o}$-module of finite height over a separable closure of the residue class field of $K$. The Lubin-Tate moduli space $X_{m}$ classifies deformations of $H_{0}$ equipped with level-$m$-structure. In this article, we study a particular type of $p$-adic representations originating from the action of $\text{Aut}(H_{0})$ on certain equivariant vector bundles over the rigid analytic generic fibre of $X_{m}$. We show that, for arbitrary level $m$, the Fr\'{e}chet space of the global sections of these vector bundles is dual to a locally $K$-analytic representation of $\text{Aut}(H_{0})$ generalizing the case $K=\mathbb{Q}_{p}$ and $m=0$ of \cite{kohliwamo}, Theorem 3.5. As a first step towards a better understanding of these representations, we compute their locally finite vectors. Essentially, all locally finite vectors arise from the global sections over the projective space via pullback along the Gross-Hopkins period map, and are in fact locally algebraic vectors.
\end{abstract}
\tableofcontents
\section*{Introduction}
\vspace{2mm}
\noindent Let $p$ be a prime number and $K$ be a finite extension of $\mathbb{Q}_{p}$ with ring of integers $\mathfrak{o}$, uniformizer $\varpi$ and residue class field $k$. Let $\breve{K}$ denote the completion of the maximal unramified extension of $K$ and $\breve{\mathfrak{o}}$ denote its ring of integers. We fix a one dimensional formal $\mathfrak{o}$-module $H_{0}$ of height $h$ over a separable closure $k^{\textnormal{sep}}$ of $k$ and consider a problem of its deformations to local $\breve{\mathfrak{o}}$-algebras with residue class field $k^{\textnormal{sep}}$. Extending the work of Lubin-Tate, Drinfeld showed that the functor of deformations together with level-$m$-structure is representable by a smooth affine formal $\breve{\mathfrak{o}}$-scheme $X_{m}:=\textnormal{Spf}(R_{m})$. Further, the formal scheme $X_{0}$ commonly known as \emph{the Lubin-Tate moduli space} is non-canonically isomorphic to the formal spectrum $\text{Spf}(\breve{\mathfrak{o}}[[u_{1},\ldots, u_{h-1}]])$. The Lubin-Tate moduli space $X_{0}$ plays a pivotal role in understanding the arithmetic of the local field $K$. For $h=1$, it realizes the main theorem of local class field theory by explicitly constructing the totally ramified part of the maximal abelian extension of $K$.
\vspace{2.6mm}\\
\noindent The generic fibre $X^{\textnormal{rig}}_{m}$ of $X_{m}$ has a structure of a rigid analytic variety over $\breve{K}$, and is a finite \'{e}tale covering of the $(h-1)$-dimensional open unit polydisc $X^{\text{rig}}_{0}$ over $\breve{K}$. The \emph{Lubin-Tate tower} $\varprojlim_{m}X^{\text{rig}}_{m}$ carries actions of important groups which is what makes it an interesting object to study. For every $m\geq 0$, there are natural commuting actions of the groups $\Gamma:=\text{Aut}(H_{0})$ and $G_{0}:=GL_{h}(\mathfrak{o})$ on the universal deformation ring $R_{m}$, where $\Gamma$ is isomorphic to the group $\mathfrak{o}_{B_{h}}^{\times}$ of units in the maximal order of the central $K$-division algebra $B_{h}$ of invariant $1/h$. By functoriality, these group actions pass on to the rigid spaces $X^{\text{rig}}_{m}$ and their global sections $R^{\text{rig}}_{m}:=\mathcal{O}_{X^{\text{rig}}_{m}}(X^{\text{rig}}_{m})$. The $G_{0}$-action on $X^{\text{rig}}_{m}$ factors through a quotient by the $m$-th principal congruence subgroup $G_{m}:=1+\varpi^{m}M_{h}(\mathfrak{o})$ making $X^{\text{rig}}_{m}\longrightarrow X^{\text{rig}}_{0}$ a Galois covering with Galois group $G_{0}/G_{m}.$ Allowing deformations of $H_{0}$ with quasi-isogenies of arbitrary heights in the moduli problem, the generalized Lubin-Tate tower admits an action of the triple product group $GL_{h}(K)\times B_{h}^{\times}\times W_{K}$, where $W_{K}$ is the Weil group of $K$, and its $l$-adic \'{e}tale cohomology realizes both the Jacquet-Langlands correspondence and the local Langlands correspondence for $GL_{h}(K)$ as shown by Harris and Taylor (cf. \cite{ht}). On the other hand, the action of $\Gamma$ on $R_{0}\simeq\breve{\mathfrak{o}}[[u_{1},\ldots, u_{h-1}]]$ is also shown to be related to important problems in stable homotopy theory by Devinatz and Hopkins (cf. \cite{dh95}). Despite its general interest, the $\Gamma$-action on $X^{\text{rig}}_{m}$ is poorly understood. The aim of this article is to study this $\Gamma$-action in terms of interesting $p$-adic representations it gives rise to on the global sections of certain equivariant vector bundles on $X^{\text{rig}}_{m}$.\vspace{2.6mm}\\
\noindent The Lie algebra $\textnormal{Lie}(\mathbb{H}^{(m)})$ of the universal formal $\mathfrak{o}$-module $\mathbb{H}^{(m)}$ over $R_{m}$ is a free $R_{m}$-module of rank 1 and is equipped with a semilinear $\Gamma$-action. For any integer $s$, the $s$-fold tensor power $\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ of $\textnormal{Lie}(\mathbb{H}^{(m)})$ gives rise to a $\Gamma$-equivariant line bundle $(\mathcal{M}^{s}_{m})^{\textnormal{rig}}$ on $X^{\textnormal{rig}}_{m}$ with global sections $M^{s}_{m}$ isomorphic to $R^{\textnormal{rig}}_{m}\otimes_{R_{m}}\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$. The functoriality induces a $\Gamma$-action on $M^{s}_{m}$ such that the isomorphism $M^{s}_{m}\simeq R^{\textnormal{rig}}_{m}\otimes_{R_{m}}\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ is $\Gamma$-equivariant for the diagonal $\Gamma$-action on the right. The algebra $R^{\text{rig}}_{m}$ has a natural structure of a $\breve{K}$-Fr\'{e}chet space and endows $M^{s}_{m}$ with the same type of topology. Note that $\Gamma\simeq\mathfrak{o}_{B_{h}}^{\times}$ is a compact locally $K$-analytic group, i.e. a compact Lie group over a $p$-adic field $K$. The $\breve{K}$-linear $\Gamma$-representation $M^{s}_{m}$ is then an example of a representation of a $p$-adic analytic group on a $p$-adic locally convex vector space. Such representations have been systematically studied by Schneider and Teitelbaum in a series of articles including \cite{stladist} and \cite{stadmrep}. Motivated by the questions in the $p$-adic Langlands programme, they introduce a category of \emph{locally analytic representations}. This is a subcategory of continuous representations of $\Gamma$ on locally convex $\breve{K}$-vector spaces large enough to contain smooth and finite dimensional algebraic representations, and which can be algebraized.\vspace{2.6mm}\\
\noindent A locally analytic representation $V$ is defined by the property that, for each $v\in V$, the orbit map $\Gamma\longrightarrow V$, $\gamma\mapsto\gamma(v)$ is locally on $\Gamma$ given by a convergent power series with coefficients in $V$. These representations can be analysed by viewing them as modules over the \emph{distribution algebra} $D(\Gamma,\breve{K})$ of $\breve{K}$-valued locally analytic distributions on $\Gamma$. A fundamental theorem \cite{stladist}, Corollary 3.4 of locally analytic representation theory says that the category of locally analytic $\Gamma$-representations on $\breve{K}$-vector spaces of compact type is anti-equivalent to the category of continuous $D(\Gamma,\breve{K})$-modules on nuclear $\breve{K}$-Fr\'{e}chet spaces via the duality functor. We now state our first main result regarding the $\Gamma$-representations $M^{s}_{m}$ (cf. Theorem \ref{genlaK0} and Theorem \ref{genlaKm}):
\begin{theorema}\label{theorema}
For all $m\geq 0$ and $s\in\mathbb{Z}$, the action of $\Gamma$ on the nuclear $\breve{K}$-Fr\'{e}chet space $M^{s}_{m}$ \linebreak extends to a continuous action of the distribution algebra $D(\Gamma,\breve{K})$. Therefore, its strong \linebreak topological $\breve{K}$-linear dual $(M^{s}_{m})'_{b}$ is a locally $K$-analytic representation of $\Gamma$.
\end{theorema}
\noindent We point out that the continuity and the differentiability of the $\Gamma$-action on $R^{\textnormal{rig}}_{0}=M^{0}_{0}$ was already known by the work of Gross and Hopkins (cf. \cite{gh}, Proposition 19.2 and Proposition 24.2). In \cite{kohliwamo}, Theorem 3.5, Kohlhaase first proves the local analyticity of the $\Gamma$-action on $M^{s}_{0}$ when $K=\mathbb{Q}_{p}$ making previous results of Gross and Hopkins more precise. Our result generalizes Kohlhaase's theorem to any finite base extension $K$ of $\mathbb{Q}_{p}$ and also extends it to the higher levels $m$. The proof of the above theorem essentially follows the same approach as the one of Kohlhaase. Using continuity of the $\Gamma$-action on $\text{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ obtained in Theorem \ref{ctsthm2}, one shows that it remains continuous after passing to rigidification (cf. Proposition \ref{mainprop} and Proposition \ref{mainprop2}). Together with the structure theory of the distribution algebra $D(\Gamma,\breve{K})$, this yields local $\mathbb{Q}_{p}$-analyticity of the $\Gamma$-action on $M^{s}_{m}$. To show that it is indeed locally $K$-analytic, for level $m=0$, we make use of the explicit locally $K$-analytic action of $\Gamma$ on the sections $M^{s}_{D}$ of our line bundle over the \emph{Gross-Hopkins fundamental domain $D$}. The local $K$-analyticity at level $0$ then implies the local $K$-analyticity at higher levels because the covering morphisms are \'{e}tale. \\\\
\noindent The Gross-Hopkins' \emph{$p$-adic period map} $\Phi:X^{\text{rig}}_{0}\longrightarrow\mathbb{P}^{h-1}_{\breve{K}}$ constructed in \cite{gh} turns out to be a crucial tool to understand the action of $\Gamma$ on the Lubin-Tate moduli space. The period map $\Phi$ is an \'{e}tale surjective morphism of rigid analytic spaces which is also $\Gamma$-equivariant for an explicitly known linear action of $\Gamma$ on the projective space. It is constructed in such a way that the line bundle $(\mathcal{M}^{s}_{m})^{\text{rig}}$ is a pullback of $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)$ along the composition $\Phi_{m}:X^{\text{rig}}_{m}\xrightarrow[\text{map}]{\text{covering}}X^{\text{rig}}_{0}\xrightarrow{\Phi}\mathbb{P}^{h-1}_{\breve{K}}$ (cf. Remark \ref{our_line_bundle_is_a_pullback_of_O(s)*}). The fundamental domain $D$ is a certain affinoid subdomain of $X^{\text{rig}}_{0}$ on which $\Phi$ is injective. This allows us to obtain explicit formulae for the action of $\Gamma$ on $M^{s}_{D}$ and also for the action of the Lie algebra $\mathfrak{g}:=\text{Lie}(\Gamma)$ on $M^{s}_{D}$ (cf. Proposition \ref{dh}, Remark \ref{our_line_bundle_is_a_pullback_of_O(s)} and Lemma \ref{explicit_g-action_lemma}).
\vspace{2.6mm}\\
\noindent Using the Lie algebra action, we prove very first results concerning the structure of the global sections $M^{s}_{m}$ of equivariant vector bundles as $\breve{K}$-linear representations of $\Gamma$. A \emph{locally finite vector} in $M^{s}_{m}$ is a vector contained in a finite dimensional sub-$H$-representation of $M^{s}_{m}$ for some open subgroup $H\subseteq\Gamma$. The set $(M^{s}_{m})_{\text{lf}}$ of all locally finite vectors is again a $\Gamma$-representation and is equal to the union of all finite dimensional subrepresentations of $M^{s}_{m}$ due to compactness of $\Gamma$. Consider the $h$-dimensional $\breve{K}$-linear representation $B_{h}\otimes_{K_{h}}\breve{K}$ of $\Gamma$ on which $\Gamma\simeq\mathfrak{o}_{B_{h}}^{\times}$ acts by the left multiplication, and let $\breve{K}_{m}$ denote the $m$-th Lubin-Tate extension of $\breve{K}$ equipped with the $\Gamma$-action via $\mathfrak{o}_{B_{h}}^{\times}\xrightarrow{\textnormal{Nrd}}\mathfrak{o}^{\times}\twoheadrightarrow(\mathfrak{o}/\varpi^{m}\mathfrak{o})^{\times}\simeq\text{Gal}(\breve{K}_{m}/\breve{K})$. Here Nrd denotes the reduced norm. Then our next main result concerning the locally finite vectors in $M^{s}_{m}$ shows that they all come from the projective space via pullback (cf. Remark \ref{Vs_is_sym-s-part_of_V1}).
\begin{theoremb}
For all $m\geq 0$ and $s\in\mathbb{Z}$, we have
\begin{equation*}
(M^{s}_{m})_{\textnormal{lf}}\simeq
\breve{K}_{m}\otimes_{\breve{K}}\textnormal{Sym}^{s}(B_{h}\otimes_{K_{h}}\breve{K})\simeq\breve{K}_{m}\otimes_{\breve{K}}\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(\mathbb{P}^{h-1}_{\breve{K}})
\end{equation*} as $\breve{K}[\Gamma]$-modules with the diagonal $\Gamma$-action on the tensor product. Thus, $(M^{s}_{m})_{\textnormal{lf}}$ is zero if $s<0$ and is a finite dimensional semi-simple locally algebraic $\Gamma$-representation if $s\geq 0$.
\end{theoremb}
\noindent We prove the above theorem in several parts (cf. Corollary \ref{lf1}, Theorem \ref{lf2}, Theorem \ref{lf3}, Remark \ref{la_rmk}). The action of the covering group $G_{0}$ on the Lubin-Tate tower commuting with the $\Gamma$-action plays a prominent role in the proof. The case $m=0$ is solved using the formulae of the Lie algebra action obtained in Lemma \ref{explicit_g-action_lemma}, whereas the case $s\leq 0$ is solved using the property of \emph{generic flatness} of the line bundle induced by the Lie algebra of the universal additive extension $\mathbb{E}^{(m)}$ (cf. Remark \ref{Lie(Ems)_generically_flat} and Theorem \ref{lf2}). Implicitly, we also make use of Strauch's computation of the geometrically connected components of $X^{\text{rig}}_{m}$ (cf. \cite{strgeo}). Namely, it implies that $(M^{0}_{m})_{\text{lf}}=(R^{\text{rig}}_{m})_{\text{lf}}=\breve{K}_{m}$ (cf. Theorem \ref{lfinRm}). For $m>0$, $s>0$, we make use of the action of a bigger group $G^{0}\times B_{h}^{\times}$ on the Lubin-Tate tower, where $G^{0}:=\lbrace g\in GL_{h}(K)|\hspace*{.1cm}\textnormal{det}(g)\in\mathfrak{o}^{\times}\rbrace$. Let $D_{m}\subset X^{\text{rig}}_{m}$ denote the inverse image of $D$ under the covering map and $\Pi$ be a uniformizer of $B_{h}$, then it is known that the ``$g$-translates'' of $D_{m}$ with $g\in G^{0}$ cover $\Phi^{-1}_{m}(\Phi(D))$, and the ``$\Pi$-translates'' of $\Phi^{-1}_{m}(\Phi(D))$ cover $X^{\text{rig}}_{m}$. Using complete reducibility of finite dimensional $\mathfrak{sl}_{h}(K_{h})$-representations and highest weight theory, the problem of computation of locally finite vectors then reduces to computing invariants of the sections over the aforementioned covering under the upper nilpotent Lie algebra action (cf. Proposition \ref{ninvariantsequalginvariants} and Proposition \ref{keyprop}). \vspace{2.6mm}\\
\noindent It is noteworthy to mention that the first, well-studied example of $p$-adic representations stemming from equivariant vector bundles on $p$-adic analytic spaces did not concern the Lubin-Tate spaces. Rather, the geometric object here was \emph{Drinfeld's upper half space} $\mathcal{X}$ obtained by deleting all $K$-rational hyperplanes from the projective space $\mathbb{P}^{h-1}_{K}$. The natural action of $GL_{h}(K)$ on the projective space stabilizes $\mathcal{X}$. Restricting any $GL_{h}(K)$-equivariant vector bundle $\mathcal{F}$ on $\mathbb{P}^{h-1}_{K}$ to $\mathcal{X}$ gives rise to a locally analytic $GL_{h}(K)$-representation on the strong dual of the nuclear Fr\'{e}chet space $\mathcal{F}(\mathcal{X})$ of its global sections (cf. \cite{orl}). By the work of Orlik-Strauch, the Jordan-H\"{o}lder series of these locally analytic representations is explicitly known. The Jordan-H\"{o}lder constituents are subrepresentations of parabolic inductions of certain locally algebraic representations (cf. \cite{orlstr}). Though our results are not as precise, we hope that they lay the groundwork for further study of locally analytic representations arising from the Lubin-Tate moduli spaces. We also mention a closely related work of Chi Yu Lo showing anlyticity of a certain rigid analytic group on a particular closed disc of $X_{0}$ in the case of height $h=2$ (cf. \cite{lo15}). \vspace{2.6mm}\\
\noindent We conclude this introduction by discussing a few important questions that we were unable to answer.
\begin{itemize}[leftmargin=*]
\item A major open question concerning the locally analytic $\Gamma$-representations $(M^{s}_{m})'_{b}$ is whether they are \emph{admissible} or not in the sense of \cite{stadmrep}, § 6. In the language of § \ref{step2} and § \ref{step3}, this would require proving that the $\Lambda(\Gamma_{*})_{\breve{K},l}$-modules $M^{s}_{m,l}$ are finitely generated and the natural maps $\Lambda(\Gamma_{*})_{\breve{K},l}\otimes_{\Lambda(\Gamma_{*})_{\breve{K},l+1}}M^{s}_{m,l+1}\longrightarrow M^{s}_{m,l}$ are isomorphisms for all $l$. By \cite{stadmrep}, Lemma 3.8, this would imply the \emph{coadmissibility} of $D(\Gamma,\breve{K})$-modules $M^{s}_{m}$.
\item Similar to the case of Drinfeld's upper half space, one would like to know the Jordan-H\"{o}lder series of the representations $(M^{s}_{m})'_{b}$. In Theorem \ref{top_finiteness_of_MsD_thm}, for $s\geq 0$, we construct a filtration $M^{s}_{D}\supset V_{s}\supset \lbrace 0\rbrace$ on the sections of $(\mathcal{M}^{s}_{0})^{\text{rig}}$ over $D$ by closed $\Gamma$-stable subspaces with topologically irreducible quotients, where $V_{s}\simeq\textnormal{Sym}^{s}(B_{h}\otimes_{K_{h}}\breve{K})$. This gives rise by duality to a filtration \begin{equation*}
\lbrace 0\rbrace\subset (M^{s}_{0}/V_{s})'_{b}\subset(M^{s}_{0})'_{b}
\end{equation*} by locally analytic $\Gamma$-representations on the strong dual of the global sections. The quotient $\frac{(M^{s}_{0})'_{b}}{(M^{s}_{0}/V_{s})'_{b}}\simeq (V_{s})'_{b}$ is irreducible as a $\Gamma$-representation. However, the irreducibility of $(M^{s}_{0}/V_{s})'_{b}$ is not yet clear.
\end{itemize}
\vspace{2.5mm}
\textit{Organization of the content}: The first section provides a brief overview of several important notions and results in non-archimedean functional analysis. These include locally convex vector spaces of compact type, locally analytic distribution algebra and Schneider-Teitelbaum's fundamental theorem in locally analytic representation theory. In order to explain the Fr\'{e}chet structure of the distribution algebra, a short introduction to uniform pro-$p$-groups is given in § \ref{Structure of the distribution algebra}.\vspace{2.6mm}\\
\noindent In § 2, we present the Lubin-Tate deformation problem with Drinfeld's level-$m$-structures and define the actions of the groups $\Gamma$ and $G_{0}$ on the universal deformation rings $R_{m}$. Although we are mainly interested in the $\Gamma$-action, the action of the covering group $G_{0}$ is needed later in § 4 to compute locally finite vectors. The main results of this section concerning the continuity of the $\Gamma$-action on $R_{m}$ and on $\text{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ are Theorem \ref{ctsthm} and Theorem \ref{ctsthm2} which generalize \cite{kohliwamo} Theorem 2.4 and \cite{kohliwamo} Theorem 2.6 to higher levels $m>0$ respectively. This continuity is used crucially later to prove local analyticity. In § \ref{Rigidification and the equivariant vector bundles}, we pass on to the rigidifications of the formal schemes $X_{m}=\text{Spf}(R_{m})$ and introduce our main example $M^{s}_{m}$ of $\breve{K}$-linear $\Gamma$-representations which arise as global sections of rigid equivariant vector bundles on $X^{\text{rig}}_{m}$. \vspace{2.6mm}\\
\noindent The § 3 is devoted to proving that the strong dual of $M^{s}_{m}$ is a locally analytic $\Gamma$-representation for all $s$ and $m$ (Theorem A). The proof is divided into 3 steps which form § \ref{step1}, § \ref{step2} and § \ref{step3} respectively. The key ingredients of the proof are Proposition \ref{mainprop} and Proposition \ref{mainprop2} exhibiting continuity of the $\Gamma$-action on $R^{\text{rig}}_{m}$. The subsection § \ref{Phi&D} summarizes Gross-Hopkins' construction of the period morphism $\Phi$ and of the fundamental domain $D$. The interpretation of the $\Gamma$-equivariant line bundles $(\mathcal{M}^{s}_{m})^{\text{rig}}$ as the pullback of invertible sheaves $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)$ on the projective space along the covering morphism $\Phi_{m}$ is explained in Remark \ref{our_line_bundle_is_a_pullback_of_O(s)*} at the end.
\vspace{2.6mm}\\
\noindent The final section deals with the computation of $\Gamma$-locally finite vectors in $M^{s}_{m}$ (Theorem B). The Remark \ref{emerton_defn} regarding continuity of the representations makes these computations easier. Our explicit results on the locally finite vectors allow us to deduce that the subspace $(M^{s}_{m})_{\text{lf}}$ of locally finite vectors is a finite dimensional semi-simple locally algebraic representation (cf. Theorem \ref{lf3} and Remark \ref{la_rmk}).
\vspace{4mm}\\
\textit{Acknowledgements}: The results presented in this article form an integral part of the author's Ph.D. thesis under the supervision of Jan Kohlhaase. The author is extremely grateful to his supervisor for suggesting valuable ideas, for pointing out the errors, and for many very helpful discussions.
\vspace{4mm}\\
\textit{Notation and conventions}: $\mathbb{N}$ and $\mathbb{N}_{0}$ denote the set of positive integers and the set of non-negative integers respectively. If $\alpha=(\alpha_{1},\ldots ,\alpha_{r})\in\mathbb{N}^{r}_{0}$ is an $r$-tuple of non-negative integers and $T=(T_{1},\ldots ,T_{r})$ is a family of indeterminates for some $r\in\mathbb{N}$, then we set $|\alpha|:=\alpha_{1}+\ldots+\alpha_{r}$, and $T^{\alpha}:=T_{1}^{\alpha_{1}}\cdots T_{r}^{\alpha_{r}}$.\vspace{2.6mm}\\
\noindent Unless stated otherwise, all rings are considered to be commutative with identity. A ring extension $A\subseteq B$ will be denoted by $B|A$, and its degree by $[B:A]$ if it is finite and free. \vspace{2.6mm}\\
\noindent Let $p$ be a fixed prime number and let $K$ be a finite field extension of $\mathbb{Q}_{p}$ with the valuation ring $\mathfrak{o}$. We fix a uniformizer $\varpi$ of $K$ and let $k:=\mathfrak{o}/\varpi\mathfrak{o}$ denote its residue class field of characteristic $p$ and cardinality $q$. The absolute value $\vert\cdot\vert$ of $K$ is assumed to be normalized through $\vert p\vert =p^{-1}$.\newpage
\noindent We denote by $\breve{K}$ the completion of the maximal unramified extension of $K$, and by $\breve{\mathfrak{o}}$ its valuation ring. We denote by $\sigma$ the Frobenius automorphism of a separable closure $k^{\text{sep}}$ of $k$, as well as its unique lift to a ring automorphism of $\breve{\mathfrak{o}}$ and the induced field automorphism of $\breve{K}$. We also fix an algebraic closure $\overline{\breve{K}}$ of $\breve{K}$ and denote its valuation ring by $\overline{\breve{\mathfrak{o}}}$. The absolute value $|\cdot|$ on $K$ extends uniquely to $\breve{K}$, and to $\overline{\breve{K}}$.\vspace{2.6mm}\\
\noindent For a positive integer $h$, let $K_{h}$ be the unramified extension of $K$ of degree $h$, $\mathfrak{o}_{h}$ be its valuation ring, and $B_{h}$ be the central $K$-division algebra of invariant $1/h$. We fix an embedding $K_{h} \hookrightarrow B_{h}$ and a uniformizer $\Pi$ of $B_{h}$, satisfying $\Pi^{h}=\varpi$. Let $\text{Nrd}:B_{h}\longrightarrow K$ denote the reduced norm of $B_{h}$ over $K$.\vspace{2.6mm}\\
\noindent The symbol $\mathbb{P}^{h-1}_{\breve{K}}$ always denotes the $(h-1)$-dimensional rigid analytic projective space over $\breve{K}$.
\section{Preliminaries on locally analytic representation theory}
\noindent Let $L|K$ be a field extension such that $L$ is spherically complete with respect to a non-archimedean absolute value $|\cdot|$ extending the one on $K$. We let $\mathfrak{o}_{L}$ denote the ring of integers $L$. For us, the field $K$ is the ``base field'', while $L$ is the ``coefficient field", i.e. our analytic groups are defined over $K$, and their representations have coefficients in $L$. From § 2 onwards, the role of $L$ will be played by $\breve{K}$.
\subsection{Locally convex vector spaces}
\noindent Let $V$ be an $L$-vector space. A \emph{lattice} $\Lambda$ in $V$ is an $\mathfrak{o}_{L}$-submodule satisfying the condition that for any vector $v\in V$ there is a non-zero scalar $a\in L^{\times}$ such that $av\in \Lambda$. The natural map $L\otimes_{\mathfrak{o}_{L}}\Lambda\longrightarrow V$, $(a\otimes v\longmapsto av)$ is bijective (cf. \cite{schnfa}, top of page 10). Let $(\Lambda_{j})_{j\in J}$ be a non-empty family of lattices in $V$ satisfying the following two conditions:
\begin{itemize}
\item for any $j\in J$ and $a\in L^{\times}$ there exists a $k\in J$ such that $\Lambda_{k}\subseteq a\Lambda_{j}$
\item for any two $i,j\in J$ there exists a $k\in J$ such that $\Lambda_{k}\subseteq \Lambda_{i}\cap \Lambda_{j}$.
\end{itemize}
Then the subsets $v+\Lambda_{j}$ of $V$ with $v\in V$ and $j\in J$ form the basis of a topology on $V$ which is called the \emph{locally convex topology on $V$ defined by the family $(\Lambda_{j})_{j\in J}$}.
\begin{definition} A \emph{locally convex $L$-vector space} is an $L$-vector space equipped with a locally convex topology.
\end{definition}
\noindent The notion of a locally convex topology is equivalent to the notion of a topology defined by a family of seminorms (cf. \cite{schnfa}, Proposition 4.3 and Proposition 4.4). Thus any normed $L$-vector space (in particular $L$-Banach space) is a locally convex $L$-vector space.
\begin{definition}
A locally convex $L$-vector space is called an \emph{$L$-Fr\'{e}chet space} if it is metrizable and complete.
\end{definition}
\noindent Any countable projective limit of Banach spaces with the projective limit topology is a Fr\'{e}chet space (cf. \cite{schnfa}, Proposition 8.1).\vspace{2.6mm}\\
\noindent A subset $B$ of a locally convex $L$-vector space $V$ is said to be \emph{compactoid} if for any open lattice $\Lambda\subseteq V$, there are finitely many vectors $v_{1},\dots,v_{m}$ such that $B\subseteq\Lambda+\mathfrak{o}_{L}v_{1}+\dots+\mathfrak{o}_{L}v_{m}$. A compactoid and complete $\mathfrak{o}_{L}$-submodule of $V$ is called \emph{c-compact}. A continuous linear map $f:V\longrightarrow W$ between two locally convex $L$-vector spaces is called \emph{compact} if there is an open lattice $\Lambda\subseteq V$ such that $f(\Lambda)$ is c-compact in $W$. Now consider a sequence $V_{1}\longrightarrow\ldots\longrightarrow V_{n}\xrightarrow{i_{n}} V_{n+1}\longrightarrow\ldots$ of locally convex $L$-vector spaces $V_{n}$ with continuous linear transition maps $i_{n}$. The vector space inductive limit $V:=\varinjlim V_{n}$ equipped with the finest locally convex topology such that all the natural maps $j_{n}:V_{n}\longrightarrow V$ are continuous is called the \emph{locally convex inductive limit} of this sequence.
\begin{definition} A locally convex $L$-vector space is called \emph{vector space of compact type} if it is a locally convex inductive limit of a sequence of $L$-Banach spaces with injective and compact transition maps.
\end{definition}
\noindent We say a locally convex $L$-vector space $V$ is \emph{barrelled} if every closed lattice in $V$ is open. Fr\'{e}chet spaces and vector spaces of compact type are examples of barrelled locally convex vector spaces (cf. \cite{schnfa}, Examples on page 40). \vspace{2.6mm}\\
\noindent For locally convex $L$-vector spaces $V$ and $W$, the set $\mathcal{L}(V,W)$ of all continuous linear maps from $V$ to $W$ can be equipped with several locally convex topologies (cf. \cite{schnfa}, Lemma 6.4). We write $V'_{s}$ and $V'_{b}$ to denote the dual vector space of $V$ equipped with the \emph{weak} and \emph{strong} topology respectively (cf. \cite{schnfa}, Examples on page 35). A Hausdorff locally convex $L$-vector space $V$ is called \emph{reflexive} if the duality map $\delta:V\longrightarrow (V'_{b})'_{b}$ $((\delta(v))(l):=l(v))$ is a topological isomorphism.
\begin{theorem}
The duality functor $V\longmapsto V'_{b}$ is an anti-equivalence between the category of vector spaces of compact type and the category of nuclear Fr\'{e}chet spaces.
\end{theorem}
\begin{proof}
See \cite{stladist} Corollary 1.4.
\end{proof}
\noindent For the notion of a nuclear locally convex $L$-vector space, we refer the reader to \cite{schnfa}, § 19.
\subsection{Locally analytic functions and distributions}
\noindent Let $(V,\|\cdot\|)$ be an $L$-Banach space. For any $\varepsilon >0$ the power series $f(X)=\sum_{\alpha\in\mathbb{N}_{0}^{r}}v_{\alpha}X^{\alpha}$ in $r$ variables $X=(X_{1},\ldots, X_{r})$ with $v_{\alpha}\in V$ is called \emph{$\varepsilon$-convergent} if $\lim_{|\alpha|\to \infty}\|v_{\alpha}\|\varepsilon^{|\alpha|}=0$. We denote by $\mathcal{F}_{\varepsilon}(K^{r},V)$ the $L$-vector space of all $\varepsilon$-convergent power series in $r$ variables with coefficients from $V$. This is a Banach space with respect to the norm $\|f\|_{\varepsilon}:=\max_{\alpha}\|v_{\alpha}\|\varepsilon^{|\alpha|}$.
\begin{definition} Let $U\subseteq K^{r}$ be an open subset. A function $f:U\longrightarrow V$ is called \emph{locally $K$-analytic}, if for any point $x\in U$ there exists a closed ball $B_{\varepsilon}(x)\subseteq U$ of radius $\varepsilon>0$ around $x$, and a power series $F\in\mathcal{F}_{\varepsilon}(K^{r},V)$ such that $f(y)=F(x-y)$ for all $y\in B_{\varepsilon}(x)$.
\end{definition}
\noindent Let $M$ be a Hausdorff topological space. A chart $(U,\varphi)$ of dimension $n$ for $M$ is an open subset $U\subseteq M$ together with a map $\varphi:U\longrightarrow K^{n}$ such that $\varphi(U)$ is open in $K^{n}$ and $\varphi:U\simeq\varphi(U)$ is a homeomorphism. Two charts $(U_{1},\varphi_{1})$ and $(U_{2},\varphi_{2})$ are said to be compatible if the maps $\varphi_{2}\circ\varphi_{1}^{-1}:\varphi_{1}(U_{1}\cap U_{2})\longrightarrow \varphi_{2}(U_{1}\cap U_{2})$ and $\varphi_{1}\circ\varphi_{2}^{-1}:\varphi_{2}(U_{1}\cap U_{2})\longrightarrow \varphi_{1}(U_{1}\cap U_{2})$ are locally $K$-analytic. Two compatible charts with non-empty intersection have the same dimension. An atlas for $M$ is a collection of compatible charts that cover $M$. Given an atlas, one can enlarge it by adding all charts compatible with every chart in the given atlas, giving a maximal atlas. The topological space $M$ with such a maximal atlas is called a \emph{locally $K$-analytic manifold}. In this case, $M$ is said to have dimension $n$ if all charts in its atlas have dimension $n$.
\begin{definition} Let $M$ be a locally $K$-analytic manifold and $V$ be an $L$-Banach space. A function $f:M\to V$ is called \emph{locally $K$-analytic} if $f\circ\varphi^{-1}:\varphi(U)\to V$ is locally $K$-analytic for any chart $(U,\varphi)$ for $M$.
\end{definition}
\noindent A map $f:M\longrightarrow N$ between two locally $K$-analytic manifolds is said to be locally $K$-analytic if for any point $x\in M$ there exists a chart $(U,\varphi)$ for $M$ around $x$ and a chart $(V,\psi)$ for $N$ around $f(x)$ such that $f(U)\subseteq V$ and the map $\psi\circ f\circ\varphi^{-1}:\varphi(U)\longrightarrow K^{n}$ is locally $K$-analytic.\vspace{2.6mm}\\
\noindent At several places later, we shall use the following lemma on restriction of scalars.
\begin{lemma}\label{laLlaK} Let $K'|K$ be a field extension of finite degree $d$. Let $M$ and $N$ be locally $K'$-analytic manifolds of dimensions $m$ and $n$ respectively, and $g:M\longrightarrow N$ be a locally $K'$-analytic map. Then $M$ and $N$ are locally $K$-analytic manifolds of dimensions $md$ and $nd$ respectively, and $g$ is a locally $K$-analytic map too.
\end{lemma}
\begin{proof} Let $\mathcal{A}'=\lbrace (U_{i},\phi_{i}, K'^{m})\rbrace_{i\in I}$ be an atlas for $M$ over $K'$. We claim that \linebreak $\mathcal{A}=\lbrace (U_{i},\phi_{i}, K^{md})\rbrace_{i\in I}$ is an atlas for $M$ over $K$.
Since $K'^{m}\simeq K^{md}$ is a finite dimensional $K$-vector space, any two vector space norms on it are equivalent. So the topology induced by the supremum norm $\|\cdot\|_{K'^{m}}$ of $K'^{m}$, and that induced by the supremum norm $\|\cdot\|_{K^{md}}$ of $K^{md}$ are the same. Hence, $(U_{i},\phi_{i}, K^{md})$ is a chart for every ${i\in I}$. \vspace{2.6mm}\\
\noindent Now, for any two $i,j \in I$, the map $\phi_{j}\circ\phi_{i}^{-1}:\phi_{i}(U_{i}\cap U_{j})\longrightarrow K'^{m}$ is locally $K'$-analytic, i.e. for every $x\in \phi_{i}(U_{i}\cap U_{j})$, there exists $\varepsilon_{x} > 0$ and $s_{x}(X)=\sum_{\beta'\in\mathbb{N}_{0}^{m}}(s_{x})_{\beta'}X^{\beta'}\in\mathcal{F}_{\varepsilon_{x}}(K'^{m},K'^{m})$ such that for all $y\in B_{\varepsilon_{x}}^{K'^{m}}(x)$, $\phi_{j}\circ\phi_{i}^{-1}(y)=s_{x}(y-x)$. Let $\{a_{1},a_{2},\dots ,a_{d}\}$ be a basis of $K'/K$, and let $m_{0}:=\textnormal{max}\lbrace \vert a_{1}\vert,\dots ,\vert a_{d}\vert\rbrace$. Then $B_{\varepsilon_{x} /m_{0}}^{K^{md}}(x)\subseteq B_{\varepsilon_{x}}^{K'^{m}}(x)$. For $y\in B_{\varepsilon_{x} /m_{0}}^{K^{md}}(x)\subseteq B_{\varepsilon_{x}}^{K'^{m}}(x)$, we have \begin{equation*}
s_{x}(y-x)=\sum_{\beta'\in\mathbb{N}_{0}^{m}}(s_{x})_{\beta'}(y_{1}-x_{1})^{\beta'_{1}}\dots (y_{m}-x_{m})^{\beta'_{m}} .
\end{equation*} Writing $z$ for $(y-x)$, we have
\begin{align*}
s_{x}(z)&=\sum_{\beta'\in\mathbb{N}_{0}^{m}}(s_{x})_{\beta'}z_{1}^{\beta'_{1}}\dots z_{m}^{\beta'_{m}}\\&=\sum_{\beta'\in\mathbb{N}_{0}^{m}}(s_{x})_{\beta'}(z_{11}a_{1}+\dots +z_{1d}a_{d})^{\beta'_{1}}\dots (z_{m1}a_{1}+\dots +z_{md}a_{d})^{\beta'_{m}}\\&=\sum_{\beta\in\mathbb{N}_{0}^{md}}(t_{x})_{\beta}z_{11}^{\beta_{1}}\dots z_{1d}^{\beta_{d}}z_{21}^{\beta_{d+1}}\dots z_{m(d-1)}^{\beta_{md-1}}z_{md}^{\beta_{md}}.
\end{align*} Given $\beta\in\mathbb{N}_{0}^{md}$, the monomial $z_{11}^{\beta_{1}}\dots z_{md}^{\beta_{md}}$ in the previous expression appears only in\linebreak $(z_{11}a_{1}+\dots +z_{1d}a_{d})^{(\beta_{1}+\dots+\beta_{d})}\dots (z_{m1}a_{1}+\dots +z_{md}a_{d})^{(\beta_{(m-1)d+1}+\dots+\beta_{md})}$ in the expression above it. By comparing coefficients, we get \begin{equation*}
(t_{x})_{\beta}=n_{\beta}a_{1}^{(\beta_{1}+\beta_{d+1}+\dots +\beta_{(m-1)d+1})}\dots a_{d}^{(\beta_{d}+\beta_{2d}+\dots +\beta_{md})}(s_{x})_{\beta'}
\end{equation*} for some integer $n_{\beta}$ and $\beta'=((\beta_{1}+\dots+\beta_{d}),\dots ,(\beta_{(m-1)d+1}+\dots+\beta_{md}))$. Thus, \begin{equation*}
\|(t_{x})_{\beta}\|_{K^{md}}\leq C\| (t_{x})_{\beta}\|_{K'^{m}}\leq C\| (s_{x})_{\beta'}\|_{K'^{m}}m_{0}^{\vert\beta\vert}
\end{equation*} for some constant $C$. Therefore,
\begin{equation*}
\| (t_{x})_{\beta}\|_{K^{md}}\Big(\frac{\varepsilon_{x}}{m_{0}}\Big)^{\vert\beta\vert}\leq C\| (s_{x})_{\beta'}\|_{K'^{m}}\varepsilon_{x}^{\vert\beta\vert}=C\| (s_{x})_{\beta'}\|_{K'^{m}}\varepsilon_{x}^{\vert\beta'\vert} .
\end{equation*} As $\vert\beta\vert\to\infty$, $\vert\beta'\vert\to\infty$ and the right hand side of the above inequality tends to 0. So the left hand side also tends to 0 as $\vert\beta\vert\to\infty$. Hence, $\sum_{\beta\in\mathbb{N}_{0}^{md}}(t_{x})_{\beta}Y^{\beta}\in\mathcal{F}_{\varepsilon_{x} /m_{0}}(K^{md},K^{md})$, and the map $\phi_{j}\circ\phi_{i}^{-1}:\phi_{i}(U_{i}\cap U_{j})\longrightarrow K^{md}$ is locally $K$-analytic for any $x\in\phi_{i}(U_{i}\cap U_{j})$. This implies that $\mathcal{A}$ is an atlas for $M$ over $K$, and $M$ is a locally $K$-analytic manifold of dimension $md$.\vspace{2.6mm}\\
\noindent A similar argument as above can be used to show the local analyticity of the map $g$ over $K$.
\end{proof}
\noindent Let $M$ be a strictly paracompact $m$-dimensional locally $K$-analytic manifold. This means that any open covering of $M$ can be refined into a covering by pairwise disjoint open subsets. Let $V$ be a Hausdorff locally convex $L$-vector space. In this situation the locally convex $L$-vector space $C^{an}(M,V)$ of all $V$-valued locally $K$-analytic functions on $M$ can be defined as follows. A $V$-index $\mathcal{I}$ on $M$ is a family of triples $\lbrace (D_{i},\phi_{i},V_{i})\rbrace_{i\in I}$ where the $D_{i}$ are pairwise disjoint open subsets of $M$ which cover $M$, each $\phi_{i}:D_{i}\longrightarrow K^{m}$ is a chart for $M$ such that $\varphi_{i}(D_{i})=B_{\varepsilon_{i}}(x_{i})$ and $V_{i}\hookrightarrow V$ is a continuous linear injection from an $L$-Banach space $V_{i}$ into $V$. Let $\mathcal{F}_{\phi_{i}}(V_{i})$ denote the $L$-Banach space of all functions $f:D_{i}\longrightarrow V_{i}$ such that $f\circ\phi_{i}^{-1}(x)=F_{i}(x-x_{i})$ for some $F_{i}\in\mathcal{F}_{\varepsilon_{i}}(K^{m},V_{i})$. Then we form the locally convex direct product $\mathcal{F}_{\mathcal{I}}(V):=\prod_{i\in I}\mathcal{F}_{\phi_{i}}(V_{i})$ and we define $C^{an}(M,V):=\varinjlim_{\mathcal{I}}\mathcal{F}_{\mathcal{I}}(V)$ (cf. \cite{schplie}, § 11).
\begin{definition} The elements of the strong dual $D(M,V):=C^{an}(M,V)'_{b}$ of $C^{an}(M,V)$ are called \emph{$V$-valued locally $K$-analytic distributions on $M$}.
\end{definition}
\noindent Denote by $\delta_{x}\in D(M,L)$ the Dirac distributions defined by $\delta_{x}(f):=f(x)$. Then there is a unique natural $L$-linear map $\int:C^{an}(M,V)\longrightarrow \mathcal{L}(D(M,L),V)$ such that for all $x\in M$, $\int(f)(\delta_{x})=f(x)$ which can be considered as the integration map (cf. \cite{stladist}, Theorem 2.2).\begin{definition} A \emph{locally $K$-analytic group G} is a locally $K$-analytic manifold which also carries a group structure such that the multiplication map $G\times G\longrightarrow G$ $((g,h)\longmapsto gh)$ is locally $K$-analytic.
\end{definition}
\noindent We note that by \cite{schplie}, Proposition 13.6 the inversion $G\longrightarrow G$, $(g\mapsto g^{-1})$, is automatically locally $K$-analytic.\vspace{2.6mm}\\
\noindent If $G$ is a locally $K$-analytic group, then $D(G,L)$ is a unital, associative $L$-algebra with separately continuous multiplication such that the inclusion $L[G]\hookrightarrow D(G,L)$, $(\sum_{g\in G}a_{g}g\mapsto\sum_{g\in G}a_{g}\delta_{g})$, is a homomorphism of $L$-algebras (cf. \cite{stladist}, § 2). One method to explicitly construct elements in $D(G,L)$ is through the Lie algebra $\mathfrak{g}$ of $G$. From the discussion after \cite{stladist}, Proposition 2.3, one has the inclusion of algebras $U(\mathfrak{g})\otimes_{K}L\hookrightarrow D(G,L)$ where $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$ over $K$.
\subsection{Structure of the distribution algebra}
\label{Structure of the distribution algebra}
\noindent If $G$ is a compact locally $K$-analytic group then $D(G,L)$ is an $L$-Fr\'{e}chet algebra (cf. \cite{stladist}, Proposition 2.3). Further, if $G$ is a \emph{uniform pro $p$-group} then this Fr\'{e}chet structure can be made very explicit. This is what we will explain next.\vspace{2.6mm}\\
\noindent Let $p$ be a prime number. A \emph{pro $p$-group} is a profinite group in which every open normal subgroup has index equal to some power of $p$. A topological group $G$ is a pro $p$-group if and only if $G$ is topologically isomorphic to an inverse limit of finite $p$-groups (cf. \cite{ddms}, Proposition 1.12).\vspace{2.6mm}\\
\noindent Let $G$ be a pro $p$-group. Set $P_{1}(G):=G$ and $P_{i+1}(G):=\overline{P_{i}(G)^{p}[P_{i}(G),G]}$ for $i\geq 1$. Here $P_{i}(G)^{p}[P_{i}(G),G]$ denotes the subgroup of $G$ generated by the $p$th powers of elements of $P_{i}(G)$ and by all commutators $[a,b]$ with $a\in P_{i}(G)$ and $b\in G$; $\overline{X}$ denotes the topological closure of a subset $X$ of $G$. If $G$ is topologically finitely generated then $P_{i}(G)$ is open in $G$ for each $i$ and the set $\lbrace P_{i}(G)|i\geq 1\rbrace$ is a base for the neighbourhoods of identity of $G$ (cf. \cite{ddms}, Proposition 1.16). A pro $p$-group is called \emph{powerful} if $p$ is odd and $G/\overline{G^{p}}$ is abelian or if $p=2$ and $G/\overline{G^{4}}$ is abelian.
\begin{definition} A pro $p$-group $G$ is called \emph{uniform} if it is topologically finitely generated, powerful and if $|P_{i}(G):P_{i+1}(G)|=|G:P_{2}(G)|$ for all $i\geq 1$.
\end{definition}
\noindent We call the minimal cardinality of a topological generating set of a uniform pro $p$-group $G$ the \emph{rank} of $G$. The following easy lemma will be handy later.
\begin{lemma}\label{pseries} If $G$ is a uniform pro $p$-group of rank $r$ then $P_{i}(G)$ is also a uniform pro $p$-group of rank $r$ for all $i\geq 1$.
\end{lemma}
\begin{proof}
Since $P_{i}(G)$ is open and closed in $G$, it follows from \cite{ddms}, Proposition 1.7 and Proposition 1.11 (i) that $P_{i}(G)$ is topologically finitely generated pro $p$-group for all $i\geq 1$. Now \cite{ddms}, Theorem 3.6 (i) tells us that $P_{i}(G)$ is powerful and $P_{i}(P_{j}(G))=P_{i+j-1}(G)$ implies that it is uniform for all $i\geq 1$. The rank statement follows from \cite{ddms}, Proposition 4.4.
\end{proof}
\noindent Locally $\mathbb{Q}_{p}$-analytic groups have a following group-theoretic characterization due to Lazard:
\begin{theorem}\label{Lazard's_characterization}
A topological group $G$ is locally $\mathbb{Q}_{p}$-analytic if and only if it contains an open subgroup which is a uniform pro-$p$ group.
\end{theorem}
\begin{proof}
See \cite{ddms}, Theorem 8.32.
\end{proof}
\noindent Let $G$ be a uniform pro $p$-group of rank $r$ and $\lbrace a_{1},\ldots,a_{r}\rbrace$ be an (ordered) minimal topological generating set for $G$. Then the mapping $(\lambda_{1},\ldots,\lambda_{r})\mapsto a_{1}^{\lambda_{1}}\ldots a_{r}^{\lambda_{r}}$ from $\mathbb{Z}_{p}^{r}$ to $G$ is a well-defined homeomorphism (cf. \cite{ddms}, Theorem 4.9). Of course, unless $G$ is commutative, this is not an isomorphism of groups. Using the inverse as a global chart for $G$, we may identify the locally convex $L$-vector spaces of locally $\mathbb{Q}_{p}$-analytic functions \begin{equation*}
C^{an}(\mathbb{Z}_{p}^{r},L)\simeq C^{an}(G,L),
\end{equation*} and also the locally convex $L$-vector spaces of locally analytic $\mathbb{Q}_{p}$-distributions \begin{equation*}
D(\mathbb{Z}_{p}^{r},L)\simeq D(G,L)
\end{equation*} after dualizing. By Amice's theorem, $f\in C^{an}(\mathbb{Z}_{p}^{r},L)$ if and only if the coefficients $c_{\alpha}\in L$ of its Mahler expansion satisfy $\lim_{|\alpha|\to\infty}|c_{\alpha}|r^{|\alpha|}=0$ for some real number $r>1$ (cf. \cite{stadmrep}, page 16). Set $b_{i}:=a_{i}-1\in\mathfrak{o}_{L}[G]\subset D(G,L)$ for all $1\leq i\leq r$, and $b^{\alpha}:=b_{1}^{\alpha_{1}}\cdots b_{r}^{\alpha_{r}}$ for $\alpha\in\mathbb{N}^{r}_{0}$. Then it follows that any distribution $\mu\in D(G,L)$ has a unique convergent expansion of the form \begin{equation*}
\mu=\sum_{\alpha\in\mathbb{N}^{r}_{0}}d_{\alpha}b^{\alpha}
\end{equation*} with $d_{\alpha}\in L$ such that $\lim_{|\alpha|\to\infty}|d_{\alpha}|r^{|\alpha|}=0$ for all $0<r<1$. The family of norms defined by \begin{equation*}
\big\|\sum_{\alpha\in\mathbb{N}^{r}_{0}}d_{\alpha}b^{\alpha}\big\|_{r}:=\sup_{\alpha\in\mathbb{N}^{r}_{0}}|d_{\alpha}|r^{|\alpha|}
\end{equation*} with $1/p\leq r<1$ endow $D(G,L)$ a structure of an $L$-Fr\'{e}chet algebra (cf. \cite{stadmrep}, Proposition 4.2). In fact, the norms $\|\cdot\|_{r}$ for $1/p\leq r<1$ are submultiplicative in the sense that $\|\mu\mu'\|_{r}\leq\|\mu\|_{r}\|\mu'\|_{r}$ for all $\mu,\mu'\in D(G,L)$.
\subsection{Locally analytic representations}
\noindent Let $G$ be a finite dimensional locally $K$-analytic group.
\begin{definition}\label{lardefn} A \emph{locally $K$-analytic representation} of $G$ (over $L$) is a barrelled Hausdorff locally convex $L$-vector space equipped with a $G$-action by continuous linear endomorphisms such that for each $v\in V$, the orbit map $\rho_{v}=(g\mapsto g(v))$ is a $V$-valued locally $K$-analytic function on $G$.
\end{definition}
\noindent Let $V$ a be locally $K$-analytic representation of $G$. Then the Lie algebra $\mathfrak{g}$ of $G$ acts on $V$ by continuous linear endomorphisms as follows : if $\mathfrak{x}\in\mathfrak{g}$ and $v\in V$ then
\begin{equation}\label{g_action_eqn}
\mathfrak{x}(v):=\frac{d}{dt}\exp(t\mathfrak{x})(v)\big\vert_{t=0}=\lim_{t\to 0}\frac{\exp(t\mathfrak{x})(v)-v}{t}
\end{equation}
where $\textnormal{exp}:\mathfrak{g}\longrightarrow G$ is the exponential map defined locally around zero on $\mathfrak{g}$. This $\mathfrak{g}$-action extends to an action of the universal enveloping algebra $U(\mathfrak{g})$ on $V$ by continuous linear endomorphisms. Using the integration map, we obtain a $D(G,L)$-module structure on $V$ via $\mu(v):=\int(\rho_{v})(\mu)$, which is separately continuous and extends the action of $U(\mathfrak{g})$ on $V$ (cf. \cite{stladist}, Proposition 3.2).
\begin{theorem}[Schneider-Teitelbaum]\label{stantiequivalence}
If $G$ is compact then the duality functor $V\longmapsto V'_{b}$ gives an anti-equivalence between the following categories
\end{theorem}\begin{center}
\begin{tabular}{p{6.2cm} p{.5cm} p{6.5cm}}
locally $K$-analytic representations of $G$ on $L$-vector spaces of compact type with continuous linear $G$-maps &\vspace{.2cm} $\longrightarrow$ & continuous $D(G,L)$-modules on nuclear $L$-Fr\'{e}chet spaces with continuous $D(G,L)$-module maps
\end{tabular}
\end{center}
\begin{proof}
See \cite{stladist}, Corollary 3.4.
\end{proof}
\section{The Lubin-Tate moduli spaces and the group actions}
\noindent The Lubin-Tate moduli spaces are deformation spaces that parametrize the deformations of formal $\mathfrak{o}$-modules together with level structures. Before explaining this deformation problem precisely, we first recall some basic definitions and results on formal $\mathfrak{o}$-modules.
\subsection{Deformations of formal $\mathfrak{o}$-modules with level structures}
\begin{definition} Let $A$ be a commutative unital $\mathfrak{o}$-algebra with the structure map $i:\mathfrak{o}\longrightarrow A$. A (one-dimensional) \emph{formal $\mathfrak{o}$-module $F$ over $A$} is a one-dimensional commutative formal group law $F(X,Y)\in A[[X,Y]]$ together with a ring homomorphism $[\hspace{.1cm}\cdot\hspace{.1cm}]_{F}:\mathfrak{o}\longrightarrow \textnormal{End}(F)$ such that $[a]_{F}(X)=i(a)X\mod{\textnormal{deg}\hspace{.1cm}2}$ for all $a\in\mathfrak{o}$. Here $\textnormal{End}(F)$ denotes the ring of endomorphisms of the formal group law $F$ over $A$.
\end{definition}
\noindent Higher dimensional formal $\mathfrak{o}$-modules are defined similarly, i.e. a formal $\mathfrak{o}$-module $F$ over $A$ of dimension $d$ for some positive integer $d$ is a $d$-dimensional commutative formal group law $F(X,Y)=(F_{j}(X_{1},\ldots,X_{d},Y_{1},\ldots,Y_{d}))_{1\leq j\leq d}$ together with a ring homomorphism $\mathfrak{o}\longrightarrow \textnormal{End}(F)$, $a\mapsto[a]_{F}(X)=(([a]_{F})_{j}(X_{1},\ldots,X_{d}))_{1\leq j\leq d}$ satisfying $([a]_{F})_{j}(X_{1},\ldots,X_{d})=i(a)X_{j}\mod{\textnormal{deg}\hspace{.1cm}2}$ for all $a\in\mathfrak{o}$ and for all $1\leq j\leq d$. However, as we are mostly concerned with the one dimensional formal $\mathfrak{o}$-modules in this article, we restrict ourselves to the one-dimensional case in this exposition.
\begin{example}\label{examples_of_formal_modules}
\begin{enumerate}
\item[]
\item The additive formal group law $\mathbb{G}_{a}(X,Y)=X+Y$ is a formal $\mathfrak{o}$-module over any $\mathfrak{o}$-algebra $A$ with the $\mathfrak{o}$-multiplication given by $[a]_{\mathbb{G}_{a}}(X)=i(a)X$.
\item The multiplicative formal group law $\mathbb{G}_{m}(X,Y)=(1+X)(1+Y)-1$ becomes a formal $\mathbb{Z}_{p}$-module over $\mathbb{Z}_{p}$ for the $\mathbb{Z}_{p}$-multiplication given by $[a]_{\mathbb{G}_{m}}(X)=\sum_{n=1}^{\infty}\binom{a}{n}X^{n}$. Here $\binom{a}{n}:=\frac{a(a-1)\cdots(a-n+1)}{n!}\in\mathbb{Z}_{p}$ for all $a\in\mathbb{Z}_{p}$.
\end{enumerate}
\end{example}
\noindent Given a homomorphism of $\mathfrak{o}$-algebras $f:A\longrightarrow B$ and a formal $\mathfrak{o}$-module $F$ over $A$, its base change $F\otimes_{A}B$ (or pushforward $f_{*}F$) is a formal $\mathfrak{o}$-module over $B$ obtained by applying $f$ to the coefficients of $F(X,Y)$. The $\mathfrak{o}$-action $[\hspace{.1cm}\cdot\hspace{.1cm}]_{F\otimes_{A}B}$ on $F\otimes_{A}B$ is given by the composition of the induced map $\text{End}(F)\xrightarrow[\text{to}\hspace{.1cm}\text{coeff}]{\text{apply}\hspace{.1cm}f}\text{End}(F\otimes_{A}B)$ with $[\hspace{.1cm}\cdot\hspace{.1cm}]_{F}$.\vspace{2.6mm} \\
\noindent A homomorphism between formal $\mathfrak{o}$-modules is a homomorphism of formal group laws which commutes with $\mathfrak{o}$-multiplications. By abuse of notation, we denote the endomorphism ring of a formal $\mathfrak{o}$-module $F$ over $A$ again by $\textnormal{End}(F)$ or by $\textnormal{End}_{A}(F)$ to emphasize the ring $A$ over which the endomorphisms are defined.
\begin{definition} Let $A[[X]]dX$ be the free $A[[X]]$-module of rank 1 generated by $dX$. For a formal $\mathfrak{o}$-module $F$ over $A$, the $A$-module $\omega(F)$ of \emph{invariant differentials on $F$} is an $A$-submodule of $A[[X]]dX$ consisting of differentials $f(X)dX$ that satisfy $f(F(X,Y))d(F(X,Y))=f(X)dX+f(Y)dY$ and $f([a]_{F}(X))d([a]_{F}(X))=i(a)f(X)dX$ for all $a\in\mathfrak{o}$.
\end{definition}
\noindent It follows from \cite{gh}, Proposition 2.2 that $\omega(F)$ is free of rank 1 over $A$, and generated by $F_{X}(0,X)^{-1}dX$, where $F_{X}(X,Y)$ denotes the formal partial derivative of $F(X,Y)$ with respect to $X$. The dual notion to $\omega(F)$ is the module of \emph{$F$-invariant $A$-linear derivations of $A[[X]]$}, which is a free $A$-module of rank 1 spanned by the derivation $F_{X}(0,X)\frac{d}{dX}$ (cf. \cite{gh}, § 2).
\begin{definition} The \emph{Lie algebra} Lie$(F)$ of a formal $\mathfrak{o}$-module $F$ over $A$ is the tangent space $\text{Hom}_{A}((X)/(X)^{2},A)$ of its coordinate ring $A[[X]]$ (equipped with the trivial Lie bracket). By definition of formal modules, the endomorphism $[a]_{F}(X)$ with $a\in\mathfrak{o}$ acts on Lie$(F)$ by the multiplication by $i(a)$.
\end{definition}
\noindent Clearly, Lie$(F)$ is a free $A$-module of rank 1 with a basis given by the formal derivative $\frac{d}{dX}$. The map $a\frac{d}{dX}\longmapsto aF_{X}(0,X)\frac{d}{dX}$ gives a canonical isomorphism from Lie$(F)$ to the module of $F$-invariant $A$-linear derivations of $A[[X]]$. Thus, the Lie algebra Lie$(F)$ of $F$ can be interpreted as the $A$-linear dual of the free $A$-module $\omega(F)$ of invariant differentials on $F$.\vspace{2.6mm}\\
\noindent A homomorphism of $\varphi:F\longrightarrow F'$ of formal $\mathfrak{o}$-modules over $A$ induces a natural $A$-linear map $\text{Lie}(\varphi):\text{Lie}(F)\longrightarrow\text{Lie}(F')$ between their Lie algebras mapping the basis element $\frac{d}{dX}$ to $\varphi'(0)\frac{d}{dX'}$, where $\varphi'(X)$ is the formal derivative of the formal power series $\varphi(X)\in A[[X]]$. It then follows readily that $\text{Lie}(\varphi\circ\psi)=\text{Lie}(\varphi)\circ\text{Lie}(\psi)$ for any two homomorphisms $\psi:F\longrightarrow F'$ and $\varphi:F'\longrightarrow F''$ of formal $\mathfrak{o}$-modules over $A$.\vspace{2.6mm}\\
\noindent A homomorphism $\varphi:F\longrightarrow\mathbb{G}_{a}$ of formal $\mathfrak{o}$-modules over $A$ gives rise to an invariant differential $d\varphi=\varphi'(X)dX\in\omega(F)$. If $A$ is a flat $\mathfrak{o}$-algebra and $\omega\in\omega(F)$ is a basis, then there exists a unique isomorphism $\varphi:F\iso\mathbb{G}_{a}$ over $A\otimes_{\mathfrak{o}}K$ such that $d\varphi=\omega$ (cf. \cite{gh}, Proposition 3.2). The power series $\varphi(X)\in(A\otimes_{\mathfrak{o}}K)[[X]]$ is called a \emph{logarithm of $F$}, and can be constructed by formally integrating $\omega$.
\begin{definition}
For a formal $\mathfrak{o}$-module $F$ over $k^{\textnormal{sep}}$, either $[\varpi]_{F}(X)=0$ or there exists a unique positive integer $h$ such that $[\varpi]_{F}(X)=f(X^{q^{h}})$ with $f'(0)\neq 0$ (cf. \cite{gh}, Lemma 4.1). In the latter case, we say $F$ has \emph{height} $h$. For example, the multiplicative formal $\mathbb{Z}_{p}$-module $\mathbb{G}_{m}\otimes_{\mathbb{Z}_{p}}\overline{\mathbb{F}}_{p}$ (cf. Example \ref{examples_of_formal_modules} (2)) over $\overline{\mathbb{F}}_{p}$ has height 1.
\end{definition}
\noindent We now fix a one-dimensional formal $\mathfrak{o}$-module $H_{0}$ of finite height $h$ over $k^{\text{sep}}$ which is defined over $k$. The formal module $H_{0}$ is unique up to isomorphism, and we have \begin{equation}\label{endH0}
\text{End}(H_{0})\simeq\mathfrak{o}_{B_{h}}
\end{equation} where $\mathfrak{o}_{B_{h}}$ is the valuation ring of the central $K$-division algebra $B_{h}$ of invariant $1/h$ (cf. \cite{dr} Proposition 1.6 and 1.7). Let $\mathcal{C}$ be the category of commutative unital complete Noetherian local $\breve{\mathfrak{o}}$-algebras $R=(R,\mathfrak{m}_{R})$ with residue class field $k^{\textnormal{sep}}$. We wish to consider the liftings of $H_{0}$ to the objects of $\mathcal{C}$ together with certain additional data defined below.
\begin{definition} Let $R\in\text{Ob}(\mathcal{C})$ and $H$ be a formal $\mathfrak{o}$-module over $R$.
\begin{itemize}
\item A pair $(H,\rho)$, where $\rho:H_{0}\iso H\otimes_{R}k^{\text{sep}}$ is an isomorphism of formal $\mathfrak{o}$-modules over $k^{\text{sep}}$, is called \emph{a deformation of $H_{0}$ to $R$}.
\item Denote by $(\mathfrak{m}_{R},+_{H})$ the abstract $\mathfrak{o}$-module $\mathfrak{m}_{R}$ in which addition and $\mathfrak{o}$-multiplication are defined as $x+_{H}y:=H(x,y)$ and $ax:=[a]_{H}(x)$ respectively for all $x,y\in\mathfrak{m}_{R}$, $a\in\mathfrak{o}$. For a non-negative integer $m$, a \emph{Drinfeld level-$m$-structure on $H$} is a homomorphism $\phi : \big(\frac{\varpi^{-m}\mathfrak{o}}{\mathfrak{o}}\big)^{h}\longrightarrow (\mathfrak{m}_{R},+_{H})$ of abstract $\mathfrak{o}$-modules such that $\prod_{\alpha\in(\frac{\varpi^{-m}\mathfrak{o}}{\mathfrak{o}})^{h}}(X-\phi(\alpha))$ divides $[\varpi^{m}]_{H}(X)$ in $R[[X]]$.
\item We call the triple $(H,\rho,\phi)$ \emph{a deformation of $H_{0}$ to $R$ with level-m-structure} if $(H,\rho)$ is a deformation of $H_{0}$ to $R$ and $\phi$ is a Drinfeld level-$m$-structure on $H$.
\end{itemize}
\end{definition}
\noindent Two deformations $(H,\rho,\phi)$ and $(H',\rho',\phi')$ of $H_{0}$ to $R$ with level-$m$-structures are isomorphic if there is an isomorphism $f:H\iso H'$ of formal $\mathfrak{o}$-modules over $R$ making the following diagrams commutative ($f\mod\mathfrak{m}_{R}$ is the isomorphism obtained by reducing the coefficients of $f$ modulo $\mathfrak{m}_{R}$):
\\\\
\xymatrix{
&&& H\otimes_{R}k^{\text{sep}}\ar[dd]^{f\mod\mathfrak{m}_{R}} \\
&& H_{0} \ar[ur]^{\rho}\ar[dr]_{\rho'}& \\
&&& H'\otimes_{R}k^{\text{sep}}}
\hspace{2cm}\xymatrix{
& (\mathfrak{m}_{R},+_{H}) \ar[dd]^{f} \\
\big(\frac{\varpi^{-m}\mathfrak{o}}{\mathfrak{o}}\big)^{h} \ar[ur]^{\phi}\ar[dr]_{\phi'}& \\
&(\mathfrak{m}_{R},+_{H'}) }
\\ \\
\noindent For any integer $m\geq 0$, consider the set valued functor Def$_{H_{0}}^{m}:\mathcal{C}\longrightarrow \textnormal{Set}$, which associates to an object $R$ of $\mathcal{C}$ the set of isomorphism classes of deformations of $H_{0}$ to $R$ with level-$m$-structures. For a morphism $\varphi:R\longrightarrow R'$ in $\mathcal{C}$, Def$_{H_{0}}^{m}(\varphi)$ is defined by sending a class $[(H, \rho, \phi)]$ to the class $[(H\otimes_{R}R',\rho,\varphi\circ\phi)]$. Notice that $\rho:H_{0}\iso H\otimes_{R}k^{\text{sep}}\simeq (H\otimes_{R}R')\otimes_{R'}k^{\text{sep}}$. We denote the triple $(H\otimes_{R}R',\rho,\varphi\circ\phi)$ by $\varphi_{*}(H,\rho,\phi)$ for simplicity.
\begin{theorem}[Lubin-Tate, Drinfeld]\label{fundthm}\hfill
\begin{enumerate}
\item The functor $\textnormal{Def}^{m}_{H_{0}}$ is representable by a regular local ring $R_{m}$ of dimension $h$ for all $m\geq 0$.
\item For any two integers $0\leq m\leq m'$, the natural transformation $\textnormal{Def}^{m'}_{H_{0}}\longrightarrow\textnormal{Def}^{m}_{H_{0}}$ of functors defined by sending a class $[(H,\rho,\phi)]$ to $[(H,\rho,\phi\vert_{(\frac{\varpi^{-m}\mathfrak{o}}{\mathfrak{o}})^{h}})]$ induces a homomorphism of local rings $R_{m}\longrightarrow R_{m'}$ which is finite and flat.
\item The ring $R_{0}$ is non-canonically isomorphic to the ring $\breve{\mathfrak{o}}[[u_{1},\ldots,u_{h-1}]]$ of formal power series in $h-1$ indeterminates over $\breve{\mathfrak{o}}$.
\end{enumerate}
\end{theorem}
\begin{proof}
See \cite{dr}, Proposition 4.2 and 4.3.
\end{proof}
\begin{definition} The smooth affine formal $\breve{\mathfrak{o}}$-scheme $X_{m}:=\textnormal{Spf}(R_{m})$ is called \emph{the Lubin-Tate moduli space of finite level $m$}. In the literature, often $X_{0}$ is referred as \emph{the Lubin-Tate moduli space} because the above theorem was initially proven by Lubin and Tate for the case $m=0$ and $\mathfrak{o}=\mathbb{Z}_{p}$.
\end{definition}
\noindent Let us denote the universal deformation of $H_{0}$ to $R_{m}$ with level-$m$-structure by the triple $(\mathbb{H}^{(m)},\rho^{(m)},\phi^{(m)})$. Here $\mathbb{H}^{(m)}=\mathbb{H}^{(0)}\otimes_{R_{0}} R_{m}$ i.e. the universal deformation $\mathbb{H}^{(m)}$ with level-$m$-structure is given by the base change of the universal deformation $\mathbb{H}^{(0)}$ without level structure under the map $R_{0}\longrightarrow R_{m}$ induced by Part 2, Theorem \ref{fundthm}. We note that, since $R_{0}\simeq\breve{\mathfrak{o}}[[u_{1},\ldots,u_{h-1}]]$ is an integral domain, the flatness of the map $R_{0}\longrightarrow R_{m}$ implies $R_{0}\hookrightarrow R_{m}$ for all $m\geq 0$. \vspace{2.6mm}\\
\noindent By the universal property, given an object $R$ of $\mathcal{C}$ and a deformation $(H,\rho,\phi)$ of $H_{0}$ to $R$ with level-$m$-structure, there is a unique $\breve{\mathfrak{o}}$-linear local ring homomorphism $\varphi : R_{m}\longrightarrow R$ such that $\text{Def}^{m}_{H_{0}}(\varphi)([(\mathbb{H}^{(m)},\rho^{(m)},\phi^{(m)})])=[\varphi_{*}(\mathbb{H}^{(m)},\rho^{(m)},\phi^{(m)})]=[(H,\rho,\phi)]$. The unique isomorphism between the deformations $\varphi_{*}(\mathbb{H}^{(m)},\rho^{(m)},\phi^{(m)})$ and $(H,\rho,\phi)$ over $R$ will be denoted by $[\varphi]:\varphi_{*}(\mathbb{H}^{(m)},\rho^{(m)},\phi^{(m)})\iso (H,\rho,\phi)$ .
\subsection{The group actions}
\label{the_group_actions}
\noindent For all $m\geq 0$, the functor $\text{Def}^{m}_{H_{0}}$ admits natural commuting left actions of the groups $\Gamma:=\textnormal{Aut}(H_{0})$ and $G_{0}:=GL_{h}(\mathfrak{o})$ for which the morphisms $\textnormal{Def}^{m'}_{H_{0}}\longrightarrow\textnormal{Def}^{m}_{H_{0}}$ of functors mentioned in Part 2, Theorem \ref{fundthm} are equivariant. On $R$-valued points, they are given by
\begin{equation}
[(H,\rho,\phi)]\longmapsto[(H,\rho\circ\gamma^{-1},\phi)]\hspace{.2cm}\text{and}\hspace{.2cm}[(H,\rho,\phi)]\longmapsto[(H,\rho,\phi\circ g^{-1})]\hspace{.2cm}\text{for}\hspace{.2cm}\gamma\in\Gamma,g\in G_{0}.
\end{equation}
Because of the representability, these actions give rise to commuting left actions of $\Gamma$ and $G_{0}$ on the universal deformation rings $R_{m}$. Indeed, given $\gamma\in\Gamma$, $g\in G_{0}$, as explained earlier, there are unique $\breve{\mathfrak{o}}$-linear local ring endomorphisms of $R_{m}$ denoted by the same letters $\gamma$ and $g$, and unique isomorphisms \begin{align*}
&[\gamma]:\gamma_{*}(\mathbb{H}^{(m)},\rho^{(m)},\phi^{(m)})\iso (\mathbb{H}^{(m)},\rho^{(m)}\circ\gamma^{-1},\phi^{(m)}) \\&[g]:g_{*}(\mathbb{H}^{(m)},\rho^{(m)},\phi^{(m)})\iso (\mathbb{H}^{(m)},\rho^{(m)},\phi^{(m)}\circ g^{-1})
\end{align*} of deformations over $R_{m}$. Here $g^{-1}\in GL_{h}(\mathfrak{o})$ acts on the free $(\frac{\mathfrak{o}}{\varpi^{m}\mathfrak{o}})$-module $(\frac{\varpi^{-m}\mathfrak{o}}{\mathfrak{o}})^{h}$ by considering it as an $\mathfrak{o}$-module via the natural reduction map $\mathfrak{o}\twoheadrightarrow\frac{\mathfrak{o}}{\varpi^{m}\mathfrak{o}}$. It follows from the uniqueness that the resulting endomorphisms of $R_{m}$ are in fact automorphisms.
\begin{remark}\label{Galois_action_on_LT_tower} For $m\geq 1$, denote by $G_{m}:=1+\varpi^{m}M_{h}(\mathfrak{o})$ the $m$-th principal congruence subgroup of $G_{0}$. If $g\in G_{m}$, then by writing $g^{-1}=1+\varpi^{m}g'$, we see that for all $\alpha\in\big(\frac{\varpi^{-m}\mathfrak{o}}{\mathfrak{o}}\big)^{h}$, $\phi^{(m)}\circ g^{-1}(\alpha)=\phi^{(m)}(1+\varpi^{m}g'(\alpha))=\phi^{(m)}(\alpha+g'(\varpi^{m}\alpha))=\phi^{(m)}(\alpha)$. As a result, the $G_{0}$-action on $R_{m}$ factors through a quotient by $G_{m}$. For $m'\geq m\geq 0$, the induced action of $G_{m}/G_{m'}$ makes $R_{m'}[\frac{1}{\varpi}]$ \'{e}tale and Galois over $R_{m}[\frac{1}{\varpi}]$ with Galois group $G_{m}/G_{m'}$ (cf. Theorem 2.1.2 (ii), \cite{strdeform}).
\end{remark}
\noindent The actions of $\Gamma$ and $G_{0}$ on $R_{m}$ induce semilinear actions of $\Gamma$ and $G_{0}$ on the Lie algebra $\text{Lie}(\mathbb{H}^{(m)})$ of the universal deformation $\mathbb{H}^{(m)}$. We now describe the $\Gamma$-action on $\text{Lie}(\mathbb{H}^{(m)})$; the $G_{0}$-action is defined in a same way. Given $\gamma\in\Gamma$, extend the ring automorphism $\gamma$ of $R_{m}$ to $R_{m}[[X]]$ by sending $X$ to itself. This induces a homomorphism \begin{equation*}
\gamma_{*}:\textnormal{Lie}(\mathbb{H}^{(m)})\longrightarrow\textnormal{Lie}(\gamma_{*}\mathbb{H}^{(m)})
\end{equation*} of additive groups. The isomorphism $[\gamma]:\gamma_{*}\mathbb{H}^{(m)}\iso\mathbb{H}^{(m)}$ also induces a natural $R_{m}$-linear map \begin{equation*}
\textnormal{Lie}([\gamma]):\textnormal{Lie}(\gamma_{*}\mathbb{H}^{(m)})\longrightarrow\textnormal{Lie}(\mathbb{H}^{(m)}).
\end{equation*} We define $\gamma:\textnormal{Lie}(\mathbb{H}^{(m)})\longrightarrow\textnormal{Lie}(\mathbb{H}^{(m)})$ as the composite of these two maps i.e. $\gamma:=\textnormal{Lie}([\gamma])\circ\gamma_{*}$.\vspace{2.6mm}\\
\noindent Given another element $\gamma'\in\Gamma$, let $\gamma'_{*}[\gamma]:\gamma'_{*}(\gamma_{*}\mathbb{H}^{(m)})\iso\gamma'_{*}\mathbb{H}^{(m)}$ be the isomorphism obtained by applying $\gamma'$ to the coefficients of $[\gamma]$. Then $[\gamma']\circ\gamma'_{*}[\gamma]$ is an isomorphism between the formal $\mathfrak{o}$-modules $(\gamma'\gamma)_{*}\mathbb{H}^{(m)}$ and $\mathbb{H}^{(m)}$ over $R_{m}$. Therefore by uniqueness, we have $[\gamma'\gamma]=[\gamma']\circ\gamma'_{*}[\gamma]$. One also checks easily that the following diagram commutes.
$$
\xymatrixcolsep{4pc}
\xymatrix{
\text{Lie}(\gamma_{*}\mathbb{H}^{(m)})\ar[d]_{\gamma'_{*}}\ar[r]^{\text{Lie}([\gamma])}&\text{Lie}(\mathbb{H}^{(m)})\ar[d]^{\gamma'_{*}}\\
\text{Lie}(\gamma'_{*}(\gamma_{*}\mathbb{H}^{(m)}))\ar[r]_{\text{Lie}(\gamma'_{*}[\gamma])}&\text{Lie}(\gamma'_{*}\mathbb{H}^{(m)})
}$$ Then it follows that \begin{align}\label{gamma_action_on_lie_algebra}
\text{Lie}([\gamma'\gamma])\circ(\gamma'\gamma)_{*}&=\text{Lie}([\gamma'])\circ\text{Lie}(\gamma'_{*}[\gamma])\circ\gamma'_{*}\circ\gamma_{*}\\&=\text{Lie}([\gamma'])\circ(\gamma'_{*}\circ\text{Lie}([\gamma])\circ(\gamma'_{*})^{-1})\circ\gamma'_{*}\circ\gamma_{*}\nonumber\\&=\text{Lie}([\gamma'])\circ\gamma'_{*}\circ\text{Lie}([\gamma])\circ\gamma_{*}\nonumber.
\end{align}
\noindent Thus we obtain an action of $\Gamma$ (and of $G_{0}$) on the additive group $\textnormal{Lie}(\mathbb{H}^{(m)})$ which is semilinear for the action of $\Gamma$ (and of $G_{0}$ respectively) on $R_{m}$ because $\gamma_{*}$ is semilinear. Given a positive integer $s$, we denote by $\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ the $s$-fold tensor product of $\textnormal{Lie}(\mathbb{H}^{(m)})$ over $R_{m}$ with itself. This is a free $R_{m}$-module of rank 1 with a semi-linear action of $\Gamma$ defined by $\gamma(\delta_{1}\otimes\cdots\otimes\delta_{s}):=\gamma(\delta_{1})\otimes\cdots\otimes\gamma(\delta_{s})$. Set $\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes 0}:=R_{m}$ and $\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}:=\textnormal{Hom}_{R_{m}}(\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes (-s)},R_{m})$ if $s$ is a negative integer. In the latter case, a semi-linear action of $\Gamma$ is defined by $\gamma(\varphi)(\delta_{1}\otimes\cdots\otimes\delta_{-s}):=\gamma(\varphi(\gamma^{-1}(\delta_{1})\otimes\cdots\otimes\gamma^{-1}(\delta_{-s})))$. The semi-linear actions of $G_{0}$ on the $s$-fold tensor products are defined similarly. As before, for all $s\in\mathbb{Z}$, the $G_{0}$-action on $\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ factors through $G_{0}/G_{m}$.
\vspace{1mm}
\begin{remark}\label{Gamma_and_G0_actions_commute}
Using that the group actions of $\Gamma$ and $G_{0}$ on $R_{m}$ commute, one can show that they commute on $\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ as follows: It suffices to show the commutativity for $s=1$. Since the $G_{0}$-action is defined likewise, we may use (\ref{gamma_action_on_lie_algebra}) for $\gamma\in\Gamma$ and $g\in G_{0}$. As a result, we get \begin{align*}
\text{Lie}([g])\circ g_{*}\circ\text{Lie}([\gamma])\circ\gamma_{*}=\text{Lie}([g\gamma])\circ (g\gamma)_{*}=\text{Lie}([\gamma g])\circ(\gamma g)_{*}=\text{Lie}([\gamma])\circ\gamma_{*}\circ\text{Lie}([g])\circ g_{*}.
\end{align*}
\end{remark} \vspace{1.5mm}
\noindent We are primarily interested in the $\Gamma$-action on $\text{Lie}(\mathbb{H}^{(m)})^{\otimes s}$. It follows from (\ref{endH0}) that $\Gamma\simeq\mathfrak{o}_{B_{h}}^{\times}$. Recall that the division algebra $B_{h}$ is a $K_{h}$-vector space of dimension $h$ with basis $\lbrace\Pi^{i}\rbrace_{0\leq i\leq h-1}$ whose multiplication is determined by the relations $\Pi^{h}=\varpi$ and $\Pi\lambda=\lambda^{\sigma}\Pi$ for all $\lambda\in K_{h}$ ($\lambda^{\sigma}$ denotes the image of $\lambda$ under the Frobenius automorphism $\sigma$). Thus, any $\gamma\in\Gamma=\mathfrak{o}_{B_{h}}^{\times}$ can be uniquely written as \begin{equation*}
\text{$\gamma = \sum_{i=0}^{h-1}\lambda_{i}\Pi^{i}$}
\end{equation*}
with $\lambda_{0}\in\mathfrak{o}_{h}^{\times}$ and $\lambda_{1},\dots ,\lambda_{h-1}\in \mathfrak{o}_{h}$. The map \begin{align}\label{chart_for_Gamma}
&\hspace{.6cm}\psi:\Gamma\longrightarrow K^{h}_{h}\\&\sum_{i=0}^{h-1}\lambda_{i}\Pi^{i}\longmapsto(\lambda_{0},\lambda_{1}\dots \lambda_{h-1})\nonumber
\end{align} identifies $\Gamma$ with a compact open subset $\mathfrak{o}_{h}^{\times}\times\mathfrak{o}_{h}^{h-1}$ of $K_{h}^{h}$ making it into a compact open locally $K_{h}$-analytic submanifold of $K_{h}^{h}$. The composition map \begin{equation*}
\psi(\Gamma)\times\psi(\Gamma)\xrightarrow{\psi^{-1}\times\psi^{-1}}\Gamma\times\Gamma\xrightarrow{\textnormal{multiplication}}\Gamma\xrightarrow{\psi}\psi(\Gamma)
\end{equation*} from an open subset in $K_{h}^{2h}$ to $K_{h}^{h}$ can be easily seen to be locally $K$-analytic since each component of this map is a composition of a polynomial and a $K$-linear Frobenius automorphism $\sigma$, both being locally $K$-analytic. Hence, $\Gamma$ is a locally $K$-analytic group. Notice that $\Gamma$ is not locally $K_{h}$-analytic group because $\sigma:\mathfrak{o}_{h}^{\times}\longrightarrow\mathfrak{o}_{h}^{\times}$ is not locally $K_{h}$-analytic unless $h=1$.\vspace{2.6mm}\\
\noindent Alternatively, if $\mathbb{B}_{h}^{\times}$ denotes the algebraic group over $K$ defined by $\mathbb{B}_{h}^{\times}(A):=(B_{h}\otimes_{K}A)^{\times}$ for any $K$-algebra $A$, then the local $K$-analyticity of the group $\Gamma$ also follows from the fact that the group of $K$-valued points $\mathbb{B}_{h}^{\times}(K)=B_{h}^{\times}$ is a locally $K$-analytic group and $\mathfrak{o}_{B_{h}}^{\times}$ is open in $B^{\times}_{h}$.
\noindent Now, being a compact and a totally disconnected Hausdorff topological group, $\Gamma$ is a profinite topological group. A basis of neighbourhoods of the identity is given by the normal subgroups $\Gamma_{i}:=1+\varpi^{i}\mathfrak{o}_{B_{h}}=1+\varpi^{i}\textnormal{End}(H_{0})$, $i\geq 1$ of finite index. We put $\Gamma_{0}:=\Gamma$. \vspace{2.6mm}\\
\noindent Our next aim is to show that the $\Gamma$-action on $\text{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ is continuous, i.e. the action map $\Gamma\times\text{Lie}(\mathbb{H}^{(m)})^{\otimes s}\longrightarrow\text{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ is continuous for the $\mathfrak{m}_{R_{m}}$-adic topology on $\text{Lie}(\mathbb{H}^{(m)})^{\otimes s}$, and for the product of profinite and $\mathfrak{m}_{R_{m}}$-adic topology on the left hand side. But, first we need a couple of lemmas. For any two non-negative integers $n$ and $m$, set $\mathbb{H}_{n}^{(m)}:=\mathbb{H}^{(m)}\otimes_{R_{m}}(R_{m}/\mathfrak{m}_{R_{m}}^{n+1})$. We have $H_{0}\iso\mathbb{H}_{0}^{(m)}$ via $\rho^{(m)}$ for all $m\geq 0$.
\begin{lemma} If $n$ and $m$ are non-negative integers then the homomorphism of $\mathfrak{o}$-algebras $\textnormal{End}(\mathbb{H}_{n+1}^{(m)})\longrightarrow \textnormal{End}(\mathbb{H}_{n}^{(m)})$, induced by reduction modulo $\mathfrak{m}_{R_{m}}^{n+1}$, is injective.
\end{lemma}
\begin{proof}
Let $m\geq 0$ be arbitrary. We show by induction on $n$ that the ring homomorphism $i_{n}:\textnormal{End}(\mathbb{H}_{n}^{(m)})\longrightarrow\textnormal{End}(\mathbb{H}_{0}^{(m)})$, induced by reduction modulo the maximal ideal, is injective for every $n\in\mathbb{N}_{0}$. The case $n=0$ is trivial. Let $n\geq 1$ and assume that $i_{n-1}$ is injective. Since $H_{0}$ is of height $h$, we have $[\varpi]_{\mathbb{H}_{0}^{(m)}}(X)\equiv\overline{u}X^{q^{h}}\mod{\textnormal{deg}\hspace{.1cm} q^{h}+1}$ for some $u\in R_{m}^{\times}$. Then $[\varpi]_{\mathbb{H}_{0}^{(m)}}=i_{n}([\varpi]_{\mathbb{H}_{n}^{(m)}})$ implies that \begin{equation*}
[\varpi]_{\mathbb{H}_{n}^{(m)}}(X)\equiv\overline{\varpi}X+\overline{b_{2}}X^{2}+\cdots+\overline{b_{q^{h}-1}}X^{q^{h}-1}+\overline{u}X^{q^{h}}\mod{\textnormal{deg}\hspace{.1cm} q^{h}+1} \end{equation*} for some $b_{2},\cdots,b_{q^{h}-1}\in\mathfrak{m}_{R_{m}}$.\vspace{2.6mm}\\
\noindent Now let $f(X)=\sum_{i=1}^{\infty}\overline{a_{i}}X^{i}\in\textnormal{End}(\mathbb{H}_{n}^{(m)})$ such that $i_{n}(f)=0$ i.e. $a_{i}\in\mathfrak{m}_{R_{m}}$ for all $i\geq 1$. We need to show that $a_{i}\in\mathfrak{m}_{R_{m}}^{n+1}$ for all $i\geq 1$. However the induction hypothesis implies that $a_{i}\in\mathfrak{m}_{R_{m}}^{n}$. Thus $[\varpi]_{\mathbb{H}_{n}^{(m)}}\circ f=0$. Since $[\varpi]_{\mathbb{H}_{n}^{(m)}}\circ f=f\circ[\varpi]_{\mathbb{H}_{n}^{(m)}}$, we get $a_{i}u^{i}\in\mathfrak{m}_{R_{m}}^{n+1}$ by induction $i$ and hence $a_{i}\in\mathfrak{m}_{R_{m}}^{n+1}$ for all $i\geq 1$.
\end{proof}
\noindent The above lemma allows us to consider all the $\mathfrak{o}$-algebras $\text{End}(\mathbb{H}^{(m)}_{n})$ as subalgebras of $\text{End}(\mathbb{H}^{(m)}_{0})$.
\begin{proposition}\label{subringprop} For all $n\geq 0$, $m\geq 0$, the subalgebra $\textnormal{End}(\mathbb{H}_{n}^{(m)})$ of $\textnormal{End}(\mathbb{H}_{0}^{(m)})$ contains $\varpi^{n}\textnormal{End}(\mathbb{H}_{0}^{(m)})$.
\end{proposition}
\begin{proof}
Let $m\geq 0$ be arbitrary. We proceed by induction on $n$, the case $n=0$ being trivial. Let $n\geq 1$ and assume the assertion to be true for $n-1$. Let $\varphi\in\varpi^{n}\textnormal{End}(\mathbb{H}_{0}^{(m)})$. By induction hypothesis, we have $\varphi\in\varpi\text{End}(\mathbb{H}^{(m)}_{n-1})$. Now for any $\psi\in\text{End}(\mathbb{H}^{(m)}_{n-1})$, choose a power series $\tilde{\psi}\in(R_{m}/\mathfrak{m}_{R_{m}}^{n+1})[[X]]$ with trivial constant term such that $\tilde{\psi}\mod{\mathfrak{m}_{R_{m}}^{n}}=\psi$. The power series $\varpi\tilde{\psi}=[\varpi]_{\mathbb{H}_{n}^{(m)}}\circ\tilde{\psi}$ is a lift of $\varpi\psi=[\varpi]_{\mathbb{H}_{n-1}^{(m)}}\circ\psi$. We claim that $\varpi\tilde{\psi}\in\text{End}(\mathbb{H}^{(m)}_{n})$ and $(\varpi\psi\longmapsto\varpi\tilde{\psi}):\varpi\text{End}(\mathbb{H}^{(m)}_{n-1})\longrightarrow\text{End}(\mathbb{H}^{(m)}_{n})$ is a well-defined injective map. The proposition then follows from the claim. \vspace{2.6mm}\\
\noindent First, let us see why $\varpi\tilde{\psi}$ defines an endomorphism of $\mathbb{H}^{(m)}_{n}$. Since $\psi\in\text{End}(\mathbb{H}^{(m)}_{n-1})$, we have\begin{equation*}
0=\psi(X+_{\mathbb{H}^{(m)}_{n-1}}Y)-_{\mathbb{H}^{(m)}_{n-1}}\psi(X)-_{\mathbb{H}^{(m)}_{n-1}}\psi(Y)=(\tilde{\psi}(X+_{\mathbb{H}^{(m)}_{n}}Y)-_{\mathbb{H}^{(m)}_{n}}\tilde{\psi}(X)-_{\mathbb{H}^{(m)}_{n}}\tilde{\psi}(Y))\mod{\mathfrak{m}_{R_{m}}^{n}}.
\end{equation*} Thus all the coefficients of the power series $(\tilde{\psi}(X+_{\mathbb{H}^{(m)}_{n}}Y)-_{\mathbb{H}^{(m)}_{n}}\tilde{\psi}(X)-_{\mathbb{H}^{(m)}_{n}}\tilde{\psi}(Y))$ lie in $\mathfrak{m}^{n}_{R_{m}}/\mathfrak{m}^{n+1}_{R_{m}}$. Since $\varpi\in\mathfrak{m}_{R_{m}}$ and $(\mathfrak{m}_{R_{m}}^{n})^{k}\subseteq \mathfrak{m}_{R_{m}}^{n+1}$ for all integers $k>1$, we get $[\varpi]_{\mathbb{H}_{n}^{(m)}}\circ(\tilde{\psi}(X+_{\mathbb{H}^{(m)}_{n}}Y)-_{\mathbb{H}^{(m)}_{n}}\tilde{\psi}(X)-_{\mathbb{H}^{(m)}_{n}}\tilde{\psi}(Y))=0$. Consequently, $\varpi\tilde{\psi}(X+_{\mathbb{H}^{(m)}_{n}}Y)=\varpi\tilde{\psi}(X)+_{\mathbb{H}^{(m)}_{n}}\varpi\tilde{\psi}(Y)$. Similarly one shows that \begin{equation*}
0=[\varpi]_{\mathbb{H}_{n}^{(m)}}\circ([a]_{\mathbb{H}^{(m)}_{n}}\circ\tilde{\psi}-_{\mathbb{H}^{(m)}_{n}}\tilde{\psi}\circ[a]_{\mathbb{H}^{(m)}_{n}})=[a]_{\mathbb{H}^{(m)}_{n}}\circ\varpi\tilde{\psi}-_{\mathbb{H}^{(m)}_{n}}\varpi\tilde{\psi}\circ[a]_{\mathbb{H}^{(m)}_{n}}
\end{equation*} for all $a\in\mathfrak{o}$. Therefore $\varpi\tilde{\psi}\in\text{End}(\mathbb{H}^{(m)}_{n})$.\vspace{2.6mm}\\
\noindent To see that the above map is well-defined, take another lift $\tilde{\psi}'$ of $\psi$ with trivial constant terms. Then $(\tilde{\psi}'-_{\mathbb{H}_{n}^{(m)}}\tilde{\psi})\mod{\mathfrak{m}_{R_{m}}^{n}}=\psi-_{\mathbb{H}_{n-1}^{(m)}}\psi=0$. Thus $[\varpi]_{\mathbb{H}_{n}^{(m)}}\circ(\tilde{\varphi}'-_{\mathbb{H}_{n}^{(m)}}\tilde{\varphi})=0$ as above. Hence $\varpi\tilde{\psi}'=\varpi\tilde{\psi}$.\vspace{2.6mm}\\
\noindent Injectivity is clear because $\varpi\tilde{\psi_{1}}=\varpi\tilde{\psi_{2}}$ implies $\varpi\psi_{1}=\varpi\psi_{2}$ after reduction modulo $\mathfrak{m}_{R_{m}}^{n}$.
\end{proof}
\begin{theorem}\label{ctsthm} For all $n\geq 0$, $m\geq 0$, the induced action of $\Gamma_{n+m}$ on $R_{m}/\mathfrak{m}_{R_{m}}^{n+1}$ is trivial. Thus the map $((\gamma,f)\mapsto \gamma(f)):\Gamma\times R_{m}\longrightarrow R_{m}$ is continuous where the left hand side carries the product topology.
\end{theorem}
\begin{proof}
Let $n$ and $m$ be arbitrary non-negative integers. Let $\gamma\in\Gamma_{n+m}$ and pr$^{(m)}_{n}:R_{m}\longrightarrow R_{m}/\mathfrak{m}_{R_{m}}^{n+1}$ denote the natural projection. Consider the level-$m$-structure $\phi^{(m)}_{n}:=\textnormal{pr}^{(m)}_{n}\circ\phi^{(m)}$ on $\mathbb{H}_{n}^{(m)}$ and consider the deformation $(\mathbb{H}_{n}^{(m)},\rho^{(m)}\circ\gamma^{-1},\phi^{(m)}_{n})$ of $H_{0}$ to $R_{m}/\mathfrak{m}_{R_{m}}^{n+1}$ with this level-$m$-structure. Let $\gamma_{n}^{(m)}:R_{m}\longrightarrow R_{m}/\mathfrak{m}_{R_{m}}^{n+1}$ denote the unique ring homomorphism for which there exists an isomorphism $[\gamma_{n}^{(m)}]:(\gamma^{(m)}_{n})_{*}(\mathbb{H}^{(m)},\rho^{(m)},\phi^{(m)})\iso(\mathbb{H}_{n}^{(m)},\rho^{(m)}\circ\gamma^{-1},\phi^{(m)}_{n})$. Note that also the ring homomorphism pr$^{(m)}_{n}\circ\gamma:R_{m}\longrightarrow R_{m}/\mathfrak{m}_{m}^{n+1}$ admits an isomorphism of deformations \begin{equation*}
(\text{pr}^{(m)}_{n}\circ\gamma)_{*}(\mathbb{H}^{(m)},\rho^{(m)},\phi^{(m)})=(\text{pr}^{(m)}_{n})_{*}(\gamma_{*}(\mathbb{H}^{(m)},\rho^{(m)},\phi^{(m)}))\iso(\mathbb{H}_{n}^{(m)},\rho^{(m)}\circ\gamma^{-1},\phi^{(m)}_{n}).
\end{equation*} Therefore by uniqueness, we have pr$^{(m)}_{n}\circ\gamma=\gamma^{(m)}_{n}$ and $[\gamma^{(m)}_{n}]=[\gamma]\mod{\mathfrak{m}_{R_{m}}^{n+1}}$.\vspace{2.6mm}\\
\noindent Since the map $(\sigma\mapsto\rho^{(m)}\circ\sigma\circ(\rho^{(m)})^{-1})$ is an isomorphism End$(H_{0})\iso\textnormal{End}(\mathbb{H}^{(m)}_{0})$ of $\mathfrak{o}$-algebras, Proposition~\ref{subringprop} shows that $\rho^{(m)}\circ\gamma^{-1}\circ(\rho^{(m)})^{-1}\in 1+\varpi^{m}\textnormal{End}(\mathbb{H}^{(m)}_{n})\subseteq\textnormal{Aut}(\mathbb{H}^{(m)}_{n})$. We claim that $(\rho^{(m)}\circ\gamma^{-1}\circ(\rho^{(m)})^{-1})\circ\phi^{(m)}_{n}=\phi^{(m)}_{n}$ : Write $\rho^{(m)}\circ\gamma^{-1}\circ(\rho^{(m)})^{-1}=1+\varepsilon\varpi^{m}$ for some $\varepsilon\in\textnormal{End}(\mathbb{H}^{(m)}_{n})$ and let $\alpha\in(\frac{\varpi^{-m}\mathfrak{o}}{\mathfrak{o}})^{h}$ be arbitrary. Then \begin{align*}
(\rho^{(m)}\circ\gamma^{-1}\circ(\rho^{(m)})^{-1})(\phi^{(m)}_{n}(\alpha))=(1+\varepsilon\varpi^{m})(\phi^{(m)}_{n}(\alpha))&=\phi^{(m)}_{n}(\alpha)+_{\mathbb{H}^{(m)}_{n}}\varepsilon(\varpi^{m}(\phi^{(m)}_{n}(\alpha)))\\&=\phi^{(m)}_{n}(\alpha)+_{\mathbb{H}^{(m)}_{n}}\varepsilon(\phi^{(m)}_{n}(\varpi^{m}\alpha))\\&=\phi^{(m)}_{n}(\alpha)+_{\mathbb{H}^{(m)}_{n}}\varepsilon(\phi^{(m)}_{n}(0))\\&=\phi^{(m)}_{n}(\alpha)
\end{align*} Therefore, the automorphism $\rho^{(m)}\circ\gamma^{-1}\circ(\rho^{(m)})^{-1}$ of $\mathbb{H}^{(m)}_{n}$ defines an isomorphism of deformations \begin{align*}
(\mathbb{H}^{(m)}_{n},\rho^{(m)},\phi^{(m)}_{n})&\simeq(\mathbb{H}^{(m)}_{n},(\rho^{(m)}\circ\gamma^{-1}\circ(\rho^{(m)})^{-1})\circ\rho^{(m)},(\rho^{(m)}\circ\gamma^{-1}\circ(\rho^{(m)})^{-1})\circ\phi^{(m)}_{n})\\&=(\mathbb{H}_{n}^{(m)},\rho^{(m)}\circ\gamma^{-1},\phi^{(m)}_{n}).
\end{align*} However; $(\mathbb{H}^{(m)}_{n},\rho^{(m)},\phi^{(m)}_{n})=\textnormal{(pr}^{(m)}_{n})_{*}(\mathbb{H}^{(m)},\rho^{(m)},\phi^{(m)})$. By uniqueness again, we have pr$^{(m)}_{n}=\textnormal{pr}^{(m)}_{n}\circ\gamma=\gamma^{(m)}_{n}$. This implies that $\Gamma_{n+m}$ acts trivially on $R_{m}/\mathfrak{m}_{R_{m}}^{n+1}$ and $[\gamma]\mod{\mathfrak{m}_{R_{m}}^{n+1}}=\rho^{(m)}\circ\gamma^{-1}\circ(\rho^{(m)})^{-1}$.\\
\end{proof}
\noindent The $R_{m}$-module $\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ is complete and Hausdorff for the $\mathfrak{m}_{R_{m}}$-adic topology because it is free of finite rank. By the semi-linearity of the $\Gamma$-action, the $R_{m}$-submodules $\mathfrak{m}_{R_{m}}^{n}\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ are $\Gamma$-stable for any non-negative integer $n$.
\begin{theorem}\label{ctsthm2}
Let $s$, $n$, $m$ be integers with $n\geq 0$ and $m\geq 0$. The induced action of $\Gamma_{2n+m+1}$ on $\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}/\mathfrak{m}_{R_{m}}^{n+1}\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ is trivial. Thus the map $((\gamma,\delta)\mapsto \gamma(\delta)):\Gamma\times \textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}\longrightarrow \textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ is continuous where the left hand side carries the product topology.
\end{theorem}
\begin{proof}
If we assume the assertion to be true for $s=1$, then by the definition of the action, it is easy to see that it holds for all positive $s$. On the other hand, let \linebreak $\overline{\varphi}\in\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes -1}/\mathfrak{m}_{R_{m}}^{n+1}\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes -1}=\text{Hom}_{R_{m}}(\text{Lie}(\mathbb{H}^{(m)}),R_{m})/\mathfrak{m}_{R_{m}}^{n+1}\text{Hom}_{R_{m}}(\text{Lie}(\mathbb{H}^{(m)}),R_{m})$, and let $\delta\in\text{Lie}(\mathbb{H}^{(m)})$ and $\gamma\in\Gamma_{2n+m+1}$. Then by assumption, $\gamma(\delta)-\delta\in\mathfrak{m}_{R_{m}}^{n+1}\text{Lie}(\mathbb{H}^{(m)})$. Write $\gamma^{-1}(\delta)=\delta+\sum_{i=1}^{r}\alpha_{i}\eta_{i}$ with $\alpha_{i}\in\mathfrak{m}_{R_{m}}^{n+1}$ and $\eta_{i}\in\text{Lie}(\mathbb{H}^{(m)})$. Then \begin{align*}
(\varphi-\gamma(\varphi))(\delta)=\varphi(\delta)-\gamma(\varphi)(\delta)=\varphi(\delta)-\gamma(\varphi(\gamma^{-1}(\delta)))&=\varphi(\delta)-\gamma(\varphi(\delta+\sum_{i=1}^{r}\alpha_{i}\eta_{i}))\\&=\varphi(\delta)-\gamma(\varphi(\delta))-\sum_{i=1}^{r}\gamma(\alpha_{i})\gamma(\varphi(\eta_{i})).
\end{align*} Since $2n+m+1\geq n+m$, by Theorem \ref{ctsthm}, we have $\varphi(\delta)-\gamma(\varphi(\delta))\in\mathfrak{m}_{R_{m}}^{n+1}$. Also $\gamma(\alpha_{i})\in\mathfrak{m}_{R_{m}}^{n+1}$. Therefore $
(\varphi-\gamma(\varphi))(\delta)\in\mathfrak{m}_{R_{m}}^{n+1}$. If $\delta_{0}$ is a basis of $\text{Lie}(\mathbb{H}^{(m)})$ over $R_{m}$, and $\psi\in\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes -1}$ is defined by $\psi(\delta_{0})=1$, then $\varphi-\gamma(\varphi)=(\varphi-\gamma(\varphi))(\delta_{0})\psi\in\mathfrak{m}_{R_{m}}^{n+1}\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes -1}$. Thus $\overline{\varphi}=\gamma(\overline{\varphi})$. A similar argument like this can be used to show that the assertion is true for all higher negative $s$. Hence it is sufficient to prove the theorem for $s=1$. \vspace{2.6mm}\\
\noindent Let $\gamma\in\Gamma_{2n+m+1}$. By identifying $\textnormal{Lie}(\mathbb{H}^{(m)})/\mathfrak{m}_{R_{m}}^{n+1}\textnormal{Lie}(\mathbb{H}^{(m)})=\textnormal{Lie}(\mathbb{H}^{(m)}_{n})$, Theorem~\ref{ctsthm} and its proof show that the map $\gamma\mod{\mathfrak{m}_{R_{m}}^{n+1}}:\textnormal{Lie}(\mathbb{H}^{(m)}_{n})\longrightarrow\textnormal{Lie}(\mathbb{H}^{(m)}_{n})$ is given by $\textnormal{Lie}(\rho^{(m)}\circ\gamma^{-1}\circ(\rho^{(m)})^{-1})$ where $\rho^{(m)}\circ\gamma^{-1}\circ(\rho^{(m)})^{-1}\in 1+\varpi^{2n+m+1}\textnormal{End}(\mathbb{H}^{(m)}_{0})\subseteq 1+\varpi^{n+m+1}\textnormal{End}(\mathbb{H}^{(m)}_{n})$. Therefore it suffices to show that the natural action of $1+\varpi^{n+m+1}\textnormal{End}(\mathbb{H}^{(m)}_{n})\subset\textnormal{End}(\mathbb{H}^{(m)}_{n})$ on $\textnormal{Lie}(\mathbb{H}_{n}^{(m)})$ is trivial. However if $\varphi\in\textnormal{End}(\mathbb{H}^{(m)}_{n})$ and $\delta\in\textnormal{Lie}(\mathbb{H}^{(m)}_{n})$, then \begin{align*}
(\textnormal{Lie}(1+\varpi^{n+m+1}\varphi)(\delta))(\overline{X})=\delta(\overline{(1+\varpi^{n+m+1}\varphi)(X)})&=\delta(\overline{X+_{\mathbb{H}^{(m)}_{n}}\varpi^{n+m+1}\varphi(X)})\\&=\delta(\overline{X+\varpi^{n+m+1}\varphi(X)})=\delta(\overline{X})\end{align*} because $\varpi^{n+m+1}\in\mathfrak{m}_{R_{m}}^{n+1}$.
\end{proof}
\begin{remark}\label{iwasawa_alg_action} The $\Gamma$-action on $\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ gives rise to an action of the group ring $\breve{\mathfrak{o}}[\Gamma]$ on $\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$. By Theorem \ref{ctsthm2}, the induced action of $\breve{\mathfrak{o}}[\Gamma]$ on $\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}/\mathfrak{m}_{R_{m}}^{n+1}\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ factors through $(\breve{\mathfrak{o}}/\varpi^{n+1}\breve{\mathfrak{o}})[\Gamma/\Gamma_{2n+m+1}]$ such that the following diagram with the horizontal maps given by the action and the vertical maps given by reduction commutes for all $n$.
\vspace{2.6mm}\\
\xymatrix{
(\breve{\mathfrak{o}}/\varpi^{n+1}\breve{\mathfrak{o}})[\Gamma/\Gamma_{2n+m+1}]\times\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}/\mathfrak{m}_{R_{m}}^{n+1}\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}\ar[d]\ar[r]&\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}/\mathfrak{m}_{R_{m}}^{n+1}\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}\ar[d]\\
(\breve{\mathfrak{o}}/\varpi^{n}\breve{\mathfrak{o}})[\Gamma/\Gamma_{2(n-1)+m+1}]\times\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}/\mathfrak{m}_{R_{m}}^{n}\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}\ar[r]&\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}/\mathfrak{m}_{R_{m}}^{n}\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}
}
\vspace{2.6mm}\\
Taking projective limits over $n$, we obtain an action of the Iwasawa algebra $\breve{\mathfrak{o}}[[\Gamma]]$ on $\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ that extends the action of $\Gamma$.
\end{remark}
\subsection{Rigidification and the equivariant vector bundles}
\label{Rigidification and the equivariant vector bundles}
\noindent Berthelot's rigidification functor associates to every locally Noetherian adic formal scheme over Spf$(\breve{\mathfrak{o}})$ whose reduction is a scheme locally of finite type over Spec$(k^{\text{sep}})$, a rigid analytic space over $\breve{K}$ (cf. \cite{jong}, § 7). For an affine formal $\breve{\mathfrak{o}}$-scheme $\text{Spf}(A)$, there is a bijection between the closed points of its generic fibre $\text{Spec}(A\otimes_{\breve{\mathfrak{o}}}\breve{K})$ and the points of the associated rigid analytic space (cf. \cite{jong}, Lemma 7.1.9). Let us denote by $X^{\textnormal{rig}}_{m}$ the rigidification of the affine formal $\breve{\mathfrak{o}}$-scheme $X_{m}=\textnormal{Spf}(R_{m})$ under Berthelot's functor, and by $R^{\textnormal{rig}}_{m}:=\mathcal{O}_{X^{\textnormal{rig}}_{m}}(X^{\textnormal{rig}}_{m})$ the $\breve{K}$-algebra of the global rigid analytic functions on $X^{\textnormal{rig}}_{m}$.\vspace{2.6mm}\\
\noindent It follows from the isomorphism $R_{0}\simeq\breve{\mathfrak{o}}[[u_{1},\ldots,u_{h-1}]]$ that $X^{\textnormal{rig}}_{0}$ is isomorphic to the $(h-1)$-dimensional rigid analytic open unit polydisc over $\breve{K}$, and the isomorphism $R_{0}\simeq\breve{\mathfrak{o}}[[u_{1},\ldots,u_{h-1}]]$ extends to an isomorphism \begin{equation}
\text{$R^{\textnormal{rig}}_{0}\simeq\Big\lbrace\sum_{\alpha\in\mathbb{N}_{0}^{h-1}}c_{\alpha}u^{\alpha}\mid c_{\alpha}\in\breve{K}$ and $\lim_{|\alpha|\to\infty}|c_{\alpha}|r^{|\alpha|}=0$ for all $0<r<1$\Big\rbrace}
\end{equation} of $\breve{K}$-algebras. This allows us to view $R^{\textnormal{rig}}_{0}$ as a topological $\breve{K}$-Fr\'{e}chet algebra whose topology is defined by the family of norms $\|\cdot\|_{l}$, given by \begin{equation*}
\Big\|\sum_{\alpha\in\mathbb{N}_{0}^{h-1}}c_{\alpha}u^{\alpha}\Big\|_{l}:=\sup_{\alpha\in\mathbb{N}_{0}^{h-1}}\lbrace\vert c_{\alpha}\vert\vert \varpi\vert^{\vert \alpha\vert /l}\rbrace
\end{equation*} for any positive integer $l$. Let $R^{\textnormal{rig}}_{0,l}$ be the completion of $R^{\textnormal{rig}}_{0}$ with respect to the norm $\|\cdot\|_{l}$. Then \begin{equation*}
R^{\text{rig}}_{0,l}\simeq\Big\lbrace\sum_{\alpha\in\mathbb{N}_{0}^{h-1}}c_{\alpha}u^{\alpha}\big|c_{\alpha}\in\breve{K},\lim_{|\alpha|\to\infty}|c_{\alpha}||\varpi|^{|\alpha|/l}=0\Big\rbrace
\end{equation*} is the $\breve{K}$-Banach algebra of rigid analytic functions on the affinoid subdomain \begin{equation*}
\text{$\mathbb{B}_{l}:=\lbrace x\in X^{\textnormal{rig}}_{0}\mid |u_{i}(x)|\leq |\varpi|^{1/l}$ for all $1\leq i\leq h-1\rbrace$}
\end{equation*} of $X^{\textnormal{rig}}_{0}$. Further, $R^{\textnormal{rig}}_{0}\simeq\varprojlim_{l}R^{\textnormal{rig}}_{0,l}$ is the topological projective limit of the $\breve{K}$-Banach algebras $R^{\textnormal{rig}}_{0,l}$.\vspace{2.6mm}\\
\noindent By functoriality, $X^{\text{rig}}_{m}$ and $R^{\textnormal{rig}}_{m}$ carry commuting (left) actions of $\Gamma$ and $G_{0}$, and the $G_{0}$-action factors through $G_{0}/G_{m}$ (cf. Remark \ref{Galois_action_on_LT_tower}). For $m'\geq m\geq 0$, let \begin{equation*}
\pi_{m',m}:X^{\text{rig}}_{m'}\longrightarrow X^{\text{rig}}_{m}
\end{equation*} denote the morphism of rigid analytic spaces induced by Part 2, Theorem \ref{fundthm} and by functoriality. It follows from the properties of the rigidification functor that the morphism $\pi_{m',m}$ is a finite \'{e}tale Galois covering with Galois group $G_{m}/G_{m'}$ (cf. \cite{jong}, § 7). Consequently, the ring extension $R^{\text{rig}}_{m'}\big|R^{\text{rig}}_{m}$ is finite Galois with Galois group $G_{m}/G_{m'}$. Since the group actions commute, we have an action of the product group $\Gamma\times G_{0}$ on $X^{\text{rig}}_{m}$ (and on $R^{\text{rig}}_{m}$) by setting $(\gamma,g)(x):=\gamma(g(x))=g(\gamma(x))$ for $(\gamma,g)\in\Gamma\times G_{0}$ and $x\in X^{\text{rig}}_{m}$. We note that all covering morphisms are $(\Gamma\times G_{0})$-equivariant.\vspace{2.6mm}\\
\noindent Since $R_{0}$ is a local ring, the finite flat $R_{0}$-module $R_{m}$ is free by \cite{matcrt}, Theorem 7.10, and has rank $r:=|G_{0}/G_{m}|$ (cf. Remark \ref{Galois_action_on_LT_tower}). Let $R^{\text{rig}}_{m,l}$ denote the affinoid $\breve{K}$-algebra of the rigid analytic functions on the affinoid subdomain $\mathbb{B}_{m,l}:=\pi_{m,0}^{-1}(\mathbb{B}_{l})$ of $X^{\text{rig}}_{m}$. Then by \cite{jong}, Lemma 7.2.2, we have \begin{equation}\label{rigidification@l_is_a_basechange}
R^{\text{rig}}_{m,l}\simeq R_{m}\otimes_{R_{0}}R^{\text{rig}}_{0,l}
\end{equation} as $R_{m}|R_{0}$ is finite. Let us fix a basis $\lbrace e_{1},\cdots, e_{r}\rbrace$ of $R_{m}$ over $R_{0}$ and view it as an $R^{\textnormal{rig}}_{0, l}$-basis of $R^{\textnormal{rig}}_{m,l}=R_{m}\otimes_{R_{0}}R^{\textnormal{rig}}_{0,l}$. The next lemma shows that $R^{\textnormal{rig}}_{m,l}$ is a $\breve{K}$-Banach algebra with respect to the norm $\|(f_{1}e_{1}+\cdots +f_{r}e_{r})\|_{l}:=\max_{1\leq i\leq r}\lbrace\|f_{i}\|_{l}\rbrace$, where $f_{i}\in R^{\textnormal{rig}}_{0,l}$ for all $i$, by showing that it is indeed an algebra norm.
\begin{lemma}\label{lnorm_is_algebra_norm} Let $f=f_{1}e_{1}+\cdots +f_{r}e_{r}$, $g=g_{1}e_{1}+\cdots +g_{r}e_{r}\in R^{\textnormal{rig}}_{m,l}$. Then $\| fg\|_{l}\leq\| f\|_{l}\| g\|_{l}$.
\end{lemma}
\begin{proof}
Let $e_{i}e_{j}=\sum_{k=1}^{r}a_{ijk}e_{k}$ for all $1\leq i,j \leq r$. Note that $a_{ijk}\in R_{0}=\breve{\mathfrak{o}}[[u_{1},\dots ,u_{h-1}]]$ and thus $\| a_{ijk}\|_{l}\leq 1$ for all $1\leq i,j,k \leq r$. Also $\|\cdot\|_{l}$ is multiplicative on $R^{\textnormal{rig}}_{0,l}$. Therefore \begin{align*}
\| fg\|_{l}=\max_{1\leq k\leq r}\Big\lbrace\Big\|\sum_{1\leq i,j\leq r}f_{i}g_{j}a_{ijk}\Big\|_{l}\Big\rbrace &\leq\max_{1\leq k\leq r}\Big\lbrace\max_{1\leq i,j\leq r}\| f_{i}g_{j}a_{ijk}\|_{l}\Big\rbrace\\&\leq\max_{1\leq i,j\leq r}\lbrace\| f_{i}\|_{l}\| g_{j}\|_{l}\rbrace\leq\| f\|_{l}\| g\|_{l}.
\end{align*}
\end{proof}
\noindent It then follows from \cite{bgr}, (6.1.3), Proposition 2 that the affinoid topology on $R^{\text{rig}}_{m,l}$ coincides with Banach topology given by the aforementioned norm $\|\cdot\|_{l}$. The natural maps $R^{\textnormal{rig}}_{m,l+1}\longrightarrow R^{\textnormal{rig}}_{m,l}$ induced from $R^{\textnormal{rig}}_{0,l+1}\longrightarrow R^{\textnormal{rig}}_{0,l}$ endow the projective limit \begin{equation*}
R^{\textnormal{rig}}_{m}\simeq\varprojlim_{l}R^{\textnormal{rig}}_{m,l}
\end{equation*} with the structure of a $\breve{K}$-Fr\'{e}chet algebra. Indeed, this projective limit is isomorphic to the $\breve{K}$-algebra of global rigid analytic functions on $X^{\textnormal{rig}}_{m}$ by \cite{jong}, Lemma 7.2.2. Thus we have \begin{equation}\label{rigidification_is_a_basechange}
R^{\textnormal{rig}}_{m}\simeq R_{m}\otimes_{R_{0}}R^{\textnormal{rig}}_{0}
\end{equation} as $R_{m}$ is a finite free $R_{0}$-module, and $R^{\textnormal{rig}}_{m,l}$ can be viewed as the Banach completion of $R^{\textnormal{rig}}_{m}$ with respect to the norm $\|\cdot\|_{l}$ defined as before by $\|(f_{1}e_{1}+\cdots +f_{r}e_{r})\|_{l}:=\max_{1\leq i\leq r}\lbrace\|f_{i}\|_{l}\rbrace$, with $f_{i}\in R^{\textnormal{rig}}_{0}$. \vspace{2.6mm}\\
\noindent The $\Gamma$-action on $X^{\text{rig}}_{0}$ stabilizes the affinoid subdomains $\mathbb{B}_{l}$ for all positive integers $l$: Since $\gamma(u_{i})$ belongs to the maximal ideal $(\varpi,u_{1},\dots ,u_{h-1})$ of $R_{0}$, $\|\gamma(u_{i})\|_{l}\leq|\varpi|^{1/l}$ for all $1\leq i\leq h-1$. This implies that $\|\gamma(f)\|_{l}\leq\|f\|_{l}$ for all $f\in R^{\text{rig}}_{0}$. Thus the $\Gamma$-action on $R^{\text{rig}}_{0}$ extends to its completion $R^{\text{rig}}_{0,l}$ for all positive integers $l$. As a result the affinoid subdomains $\mathbb{B}_{m,l}$ of $X^{\text{rig}}_{m}$ are stable under the $(\Gamma\times G_{0})$-action, and the isomorphism (\ref{rigidification@l_is_a_basechange}) is $(\Gamma\times G_{0})$-equivariant for the diagonal $(\Gamma\times G_{0})$-action on the right. Therefore the isomorphism (\ref{rigidification_is_a_basechange}) is also $(\Gamma\times G_{0})$-equivariant for the diagonal $(\Gamma\times G_{0})$-action on the right.\vspace{2.6mm}\\
\noindent By \cite{bgr}, (6.1.3), Theorem 1, the $\breve{K}$-algebra automorphism of an affinoid $\breve{K}$-algebra $R^{\text{rig}}_{m,l}$ corresponding to $(\gamma,g)\in\Gamma\times G_{0}$ is automatically continuous for its $\breve{K}$-Banach topology defined by the norm $\|\cdot\|_{l}$. Since the $\breve{K}$-Fr\'{e}chet topology of $R^{\text{rig}}_{m}$ is given by the family of norms $\|\cdot\|_{l}$, $l\in\mathbb{N}$, the group $\Gamma$ acts on $R^{\text{rig}}_{m}$ by continuous $\breve{K}$-algebra automorphisms for all $m\geq 0$.
\begin{remark}\label{mLText} A \emph{Lubin-Tate formal $\mathfrak{o}$-module} is a formal $\mathfrak{o}$-module $F$ over $\mathfrak{o}$ such that $[\varpi]_{F}(X)\equiv X^{q}\mod{\varpi}$, i.e. $F\otimes_{\mathfrak{o}}k^{\text{sep}}$ has height 1. By Theorem \ref{fundthm}, over $\breve{\mathfrak{o}}$, there exists a unique such formal $\mathfrak{o}$-module up to isomorphism. Let $\breve{K}_{m}$ denote the finite Galois extension of $\breve{K}$ obtained by adjoining to it the $\varpi^{m}$-torsion points of any Lubin-Tate formal $\mathfrak{o}$-module $F$ over $\mathfrak{o}$, i.e. \begin{equation*}
\breve{K}_{m}=\breve{K}(\lbrace\alpha\in\mathfrak{m}_{\overline{\breve{\mathfrak{o}}}}\hspace{.2cm}|\hspace{.2cm}[\varpi^{m}]_{F}(\alpha)=0\rbrace).
\end{equation*} This is a totally ramified extension of $\breve{K}$ with $\textnormal{Gal}(\breve{K}_{m}/\breve{K})\simeq(\mathfrak{o}/\pi^{m}\mathfrak{o})^{\times}$ and plays a crucial role in local class field theory. It is a non-trivial result of M. Strauch (cf. \cite{strgeo}, Corollary 3.4 (ii)) that $\breve{K}_{m}\subset R^{\textnormal{rig}}_{m}$. In fact, $\breve{K}_{m}$ is stable under the actions of $G_{0}/G_{m}$ and $\Gamma$ on $ R^{\textnormal{rig}}_{m}$. For $g\in G_{0}/G_{m}$, $\gamma\in\Gamma$ and $\alpha\in\breve{K}_{m}$, these actions are given by $g(\alpha)=\textnormal{det}(g)^{-1}(\alpha)$ and $\gamma(\alpha)=\textnormal{Nrd}(\gamma)(\alpha)$ viewing $\breve{K}$ as a left $\mathfrak{o}^{\times}$-module via the map $\mathfrak{o}^{\times}\twoheadrightarrow(\mathfrak{o}/\varpi^{m}\mathfrak{o})^{\times}\simeq\text{Gal}(\breve{K}_{m}/\breve{K})$ (cf. \cite{strgeo}, Theorem 4.4). If the height $h$ of $H_{0}$ equals 1 then we have the equality $\breve{K}_{m}=R^{\textnormal{rig}}_{m}$.
\end{remark}
\noindent Following \cite{gh}, we now recall the following definition:
\begin{definition} A \emph{$\Gamma$-equivariant vector bundle $\mathcal{M}$} on the formal scheme $X_{m}$ is a locally free $\mathcal{O}_{X_{m}}$-module $\mathcal{M}$ of finite rank equipped with a (left) $\Gamma$-action that is compatible with the $\Gamma$-action on $X_{m}$. In other words, $\gamma(fs)=\gamma(f)\gamma(s)$ for $\gamma\in\Gamma$ and for the sections $f$ and $s$ of $\mathcal{O}_{X_{m}}$ and $\mathcal{M}$ respectively over some $\Gamma$-stable open subset of $X_{m}$. The notion of a $G_{0}$-equivariant vector bundle can be defined similarly. A homomorphism $f:\mathcal{M}\longrightarrow\mathcal{N}$ of $\Gamma$-equivariant vector bundles on $X_{m}$ is a homomorphism of $\mathcal{O}_{X_{m}}$-modules which is $\Gamma$-equivariant.
\end{definition}
\noindent Since $X_{m}$ is formally affine, a $\Gamma$-equivariant vector bundle $\mathcal{M}$ on $X_{m}$ is completely determined by its global sections $\mathcal{M}(X_{m})$. Hence, for all $s\in\mathbb{Z}$ and $m\geq 0$, the free $R_{m}$-module $\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}$ of rank 1 equipped with a semilinear $\Gamma$-action gives rise to a $\Gamma$-equivariant line bundle \begin{equation*}
\mathcal{M}^{s}_{m}:=\mathcal{L}\textnormal{ie}(\mathbb{H}^{(m)})^{\otimes s}
\end{equation*} on $X_{m}$. Its rigidification $(\mathcal{M}^{s}_{m})^{\textnormal{rig}}$ is a locally free $\mathcal{O}_{X^{\textnormal{rig}}_{m}}$-module of rank 1 by \cite{jong}, 7.1.11. Let \begin{equation*}
M^{s}_{m}:=(\mathcal{M}^{s}_{m})^{\textnormal{rig}}(X^{\textnormal{rig}}_{m})
\end{equation*} denote its global sections. Because of the fact that $X_{m}$ is affine, the natural map \begin{equation}\label{G_eqv_iso_eqn1}
R^{\textnormal{rig}}_{m}\otimes_{R_{m}}\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}\longrightarrow M^{s}_{m}
\end{equation} is an isomorphism.
By functoriality, $\Gamma$ acts on $(\mathcal{M}^{s}_{m})^{\textnormal{rig}}$ in such a way that the map (\ref{G_eqv_iso_eqn1}) is $\Gamma$-equivariant for the diagonal $\Gamma$-action on the left and for the $\Gamma$-action induced by functoriality on the right. In particular, the $\Gamma$-action on $M^{s}_{m}$ is semilinear for its action on $R^{\text{rig}}_{m}$, and thus $(\mathcal{M}^{s}_{m})^{\text{rig}}$ is a \emph{rigid $\Gamma$-equivariant line bundle} on $X^{\text{rig}}_{m}$. In a similar fashion, it can be seen that $(\mathcal{M}^{s}_{m})^{\text{rig}}$ is a rigid $G_{0}$-equivariant line bundle on $X^{\text{rig}}_{m}$, and the actions of $\Gamma$ and $G_{0}$ commute (cf. Remark \ref{Gamma_and_G0_actions_commute}). By functoriality again, the $G_{0}$-action on $(\mathcal{M}^{s}_{m})^{\text{rig}}$ factors through the quotient group $G_{0}/G_{m}$ (cf. Remark \ref{Galois_action_on_LT_tower}).\vspace{2.6mm}\\
\noindent For all $s,m$ and $l$, set $M^{s}_{m,l}:=(\mathcal{M}^{s}_{m})^{\text{rig}}(\mathbb{B}_{m,l})$. Then $M^{s}_{m,l}$ is a free $R^{\textnormal{rig}}_{m,l}$-module of rank 1 for which the natural $R^{\textnormal{rig}}_{m,l}$-linear map \begin{equation}\label{G_eqv_iso_eqn3}
R^{\textnormal{rig}}_{m,l}\otimes_{R_{m}}\textnormal{Lie}(\mathbb{H}^{(m)})^{\otimes s}\longrightarrow M^{s}_{m,l}
\end{equation} is an isomorphism (cf. \cite{jong}, 7.1.11), and is $(\Gamma\times G_{0})$-equivariant for the diagonal $(\Gamma\times G_{0})$-action on the left. Endowing $M^{s}_{m}$ and $M^{s}_{m,l}$ with the natural topologies of finitely generated modules over $R^{\textnormal{rig}}_{m}$ and $R^{\textnormal{rig}}_{m,l}$ respectively, makes them a $\breve{K}$-Fr\'{e}chet space and a $\breve{K}$-Banach space respectively. One then has a topological isomorphism \begin{equation*}
M^{s}_{m}\simeq\varprojlim_{l}M^{s}_{m,l}
\end{equation*} for the projective limit topology on the right, and the group $\Gamma\times G_{0}$ acts on $M^{s}_{m}$ by continuous $\breve{K}$-vector space automorphisms for all $s$ and $m$.\vspace{2.6mm}\\
\noindent The $\breve{K}$-linear $\Gamma$-representations $M^{s}_{m}$ so obtained will be the central topic of study in the subsequent sections.
\section{Locally analytic representations arising from the Lubin-Tate spaces}
\noindent As seen in the previous section, the group $\Gamma=\text{Aut}(H_{0})$ acts on the global sections $M^{s}_{m}$ of the rigid $\Gamma$-equivariant line bundle $(\mathcal{M}^{s}_{m})^{\text{rig}}$ by continuous vector space automorphisms. The goal of this section is to show that the strong topological $\breve{K}$-linear dual $(M^{s}_{m})'_{b}$ with the induced $\Gamma$-action is a locally $K$-analytic representation of $\Gamma$ in the sense of Definition \ref{lardefn} for all $s\in\mathbb{Z}$ and levels $m\geq 0$. We outline a strategy of the proof:
\begin{itemize}
\item[Step 1.] Show that the explicitly known $\Gamma$-action on the sections $M^{s}_{D}$ over \emph{the Gross-Hopkins fundamental domain} $D$ is locally $K$-analytic by direct computations.
\item[Step 2.] Using continuity of the $\Gamma$-action on $M^{s}_{0}$, show that it is locally $\mathbb{Q}_{p}$-analytic. Use Step 1 and the $\mathfrak{g}_{\mathbb{Q}_{p}}$-equivariant $\breve{K}$-linear embedding $M^{s}_{0}\hookrightarrow M^{s}_{D}$ to deduce that the $\Gamma$-action on $(M^{s}_{0})'_{b}$ is not only locally $\mathbb{Q}_{p}$-analytic but also locally $K$-analytic.
\item[Step 3.] For level $m>0$, first prove that the $\Gamma$-action on $(M^{s}_{m})'_{b}$ is locally $\mathbb{Q}_{p}$-analytic as done in Step 2. Then using the local $K$-analyticity of the $\Gamma$-action at level 0 and the \'{e}taleness of the extension $R^{\text{rig}}_{m}\big|R^{\text{rig}}_{0}$, deduce that the $\Gamma$-action on $(M^{s}_{m})'_{b}$ is locally $K$-analytic.
\end{itemize}
\subsection{The period morphism and the Gross-Hopkins fundamental domain}
\label{Phi&D}
\noindent We first briefly review Gross-Hopkins' construction of the period morphism $\Phi$, and recall the definition of the fundamental domain $D$ as given in \cite{gh}, § 23.
\begin{definition} A sequence $0\longrightarrow F'\xrightarrow{\hspace{.1cm}\alpha\hspace{.1cm}}E\xrightarrow{\hspace{.1cm}\beta\hspace{.1cm}}F\longrightarrow 0$ of formal $\mathfrak{o}$-modules over an $\mathfrak{o}$-algebra $A$ is said to be \emph{exact} if the associated sequence $0\longrightarrow\text{Lie}(F')\xrightarrow{\text{Lie}(\alpha)}\text{Lie}(E)\xrightarrow{\text{Lie}(\beta)}\text{Lie}(F)\longrightarrow 0$ of free $A$-modules is exact. A formal $\mathfrak{o}$-module $E$ is then said to be \emph{an extension of $F$ by $F'$}. With an obvious notion of an isomorphism of extensions, we denote by $\text{Ext}(F,F')$ the set of isomorphism classes of extensions of $F$ by $F'$.
\end{definition}
\noindent It follows from \cite{gh}, Proposition 9.8 that $\text{Ext}(\mathbb{H}^{(m)},\mathbb{G}_{a})$ is a free $R_{m}$-module of rank $h-1$ for all $m\geq 0$. Let $\text{Ext}(\mathbb{H}^{(m)},\mathbb{G}_{a})^{*}:=\text{Hom}_{R_{m}}(\text{Ext}(\mathbb{H}^{(m)},\mathbb{G}_{a}),R_{m})$ denote the $R_{m}$-linear dual of $\text{Ext}(\mathbb{H}^{(m)},\mathbb{G}_{a})$, and $(\mathbb{H}^{(m)})':=\mathbb{G}_{a}\otimes_{R_{m}}\text{Ext}(\mathbb{H}^{(m)},\mathbb{G}_{a})^{*}$ be the associated additive formal $\mathfrak{o}$-module of dimension $h-1$. Then \begin{equation*}
\text{Ext}(\mathbb{H}^{(m)},(\mathbb{H}^{(m)})')=\text{Ext}(\mathbb{H}^{(m)},\mathbb{G}_{a})\otimes_{R_{m}}\text{Ext}(\mathbb{H}^{(m)},\mathbb{G}_{a})^{*}=\text{End}_{R_{m}}(\text{Ext}(\mathbb{H}^{(m)},\mathbb{G}_{a})).
\end{equation*} Let $0\longrightarrow (\mathbb{H}^{(m)})'\xrightarrow{\alpha^{(m)}}\mathbb{E}^{(m)}\xrightarrow{\beta^{(m)}}\mathbb{H}^{(m)}\longrightarrow 0$ be an extension in the class corresponding to the identity homomorphism in $\text{End}_{R_{m}}(\text{Ext}(\mathbb{H}^{(m)},\mathbb{G}_{a}))$. Then the extension $\mathbb{E}^{(m)}$ is unique up to a unique isomorphism, and is a \emph{universal additive extension of $\mathbb{H}^{(m)}$} in the sense that given any other additive extension $E'$ of $\mathbb{H}^{(m)}$ by an additive formal $\mathfrak{o}$-module $F'$, there exists unique homomorphisms $(\mathbb{H}^{(m)})'\longrightarrow F'$ and $\mathbb{E}^{(m)}\longrightarrow E'$ of formal $\mathfrak{o}$-modules over $R_{m}$ making all the relevant diagrams commute (cf. \cite{gh}, Proposition 11.3).
\begin{remark}\label{Group_actions_on_Lie(E)} Let us recall from \cite{gh}, § 16 how $\mathcal{L}\text{ie}(\mathbb{E}^{(m)})$ and its tensor powers can be viewed as $\Gamma$-equivariant vector bundles on $X_{m}$. Given $\gamma\in\Gamma$, note that $\gamma_{*}\mathbb{E}^{(m)}$ is an extension of $\gamma_{*}\mathbb{H}^{(m)}$ by $\gamma_{*}(\mathbb{H}^{(m)})'$ obtained by applying $\gamma\in\text{Aut}(R_{m})$ to the coefficients of $\mathbb{E}^{(m)}$. The isomorphism $[\gamma]:\gamma_{*}\mathbb{H}^{(m)}\iso\mathbb{H}^{(m)}$ and the universal property of $\mathbb{E}^{(m)}$ gives us a map $\text{Lie}([\gamma]_{\mathbb{E}^{(m)}}):\text{Lie}(\gamma_{*}\mathbb{E}^{(m)})\longrightarrow\text{Lie}(\mathbb{E}^{(m)})$ which is an isomorphism of free $R_{m}$-modules of rank $h$. As before (cf. § \ref{the_group_actions}), we have a map $\gamma_{*}:\textnormal{Lie}(\mathbb{E}^{(m)})\longrightarrow\textnormal{Lie}(\gamma_{*}\mathbb{E}^{(m)})
$ of additive groups induced by the base change $\gamma:R_{m}[[X_{1},\ldots ,X_{h}]]\longrightarrow R_{m}[[X_{1},\ldots ,X_{h}]]$, ($a\mapsto\gamma(a)$, $a\in R_{m}$ and $X_{i}\mapsto X_{i}$ for all $1\leq i\leq h$). As a result, we obtain a semilinear $\Gamma$-action on $\text{Lie}(\mathbb{E}^{(m)})$ given by $\gamma=\text{Lie}([\gamma]_{\mathbb{E}^{(m)}})\circ\gamma_{*}$. This $\Gamma$-action is extended to $\text{Lie}(\mathbb{E}^{(m)})^{\otimes s}$ as before for all $s\in\mathbb{Z}$.\vspace{1.5mm}\\
\noindent A semilinear $G_{0}$-action on $\text{Lie}(\mathbb{E}^{(m)})^{\otimes s}$ is defined similarly with the subgroup $G_{m}\subseteq G_{0}$ acting trivially. The actions of $\Gamma$ and $G_{0}$ on $\text{Lie}(\mathbb{E}^{(m)})^{\otimes s}$ commute. It follows directly from the definitions of the actions that the maps $\text{Lie}(\alpha^{(m)})$ and $\text{Lie}(\beta^{(m)})$ are $(\Gamma\times G_{0})$-equivariant.
\end{remark}
\vspace{2mm}
\noindent By rigidification, the exact sequence \begin{equation*}
0\longrightarrow \text{Lie}((\mathbb{H}^{(m)})')^{\otimes s}\xrightarrow{\text{Lie}(\alpha^{(m)})^{\otimes s}}\text{Lie}(\mathbb{E}^{(m)})^{\otimes s}\xrightarrow{\text{Lie}(\beta^{(m)})^{\otimes s}}\text{Lie}(\mathbb{H}^{(m)})^{\otimes s}\longrightarrow 0
\end{equation*} gives rise to an exact sequence \begin{equation*}
0\longrightarrow(\mathcal{L}\text{ie}((\mathbb{H}^{(m)})')^{\otimes s})^{\text{rig}}\longrightarrow(\mathcal{L}\text{ie}(\mathbb{E}^{(m)})^{\otimes s})^{\text{rig}}\longrightarrow(\mathcal{M}^{s}_{m})^{\text{rig}}\longrightarrow 0
\end{equation*} of rigid $(\Gamma\times G_{0})$-equivariant line bundles on $X^{\text{rig}}_{m}$ for all non-negative $s$, and for negative $s$ in the opposite direction. Since $X_{m}$ is an affine formal scheme, by taking global sections, we get an exact sequence \begin{equation}\label{exact_seq_of_global_sections_of_eqv_bundles}
0\longrightarrow R^{\text{rig}}_{m}\otimes_{R_{m}}\text{Lie}((\mathbb{H}^{(m)})')^{\otimes s}\longrightarrow R^{\text{rig}}_{m}\otimes_{R_{m}}\text{Lie}(\mathbb{E}^{(m)})^{\otimes s}\longrightarrow M^{s}_{m}\longrightarrow 0
\end{equation} of $\breve{K}$-linear $(\Gamma\times G_{0})$-representations for $s\geq 0$, and in the opposite direction for $s<0$. \vspace{2.6mm}\\
\noindent The following proposition constitutes a key ingredient in the construction of the period morphism.
\begin{proposition}\label{Lie(E0)_is_generically_flat} The $\Gamma$-equivariant line bundle $\mathcal{L}\textnormal{ie}(\mathbb{E}^{(0)})$ is generically flat, i.e. there exists a basis $\lbrace c_{0},c_{1},\ldots ,c_{h-1}\rbrace$ of $R^{\textnormal{rig}}_{0}\otimes_{R_{0}}\textnormal{Lie}(\mathbb{E}^{(0)})$ over $R^{\textnormal{rig}}_{0}$ such that the $\breve{K}$-subspace of $R^{\textnormal{rig}}_{0}\otimes_{R_{0}}\textnormal{Lie}(\mathbb{E}^{(0)})$ spanned by $c_{i}$'s is $\Gamma$-stable. Let $B_{h}\otimes_{K_{h}}\breve{K}$ be the $h$-dimensional $\breve{K}$-linear $\Gamma$-representation where the action of $\Gamma\simeq\mathfrak{o}_{B_{h}}^{\times}$ is given by left multiplication. Then we have an isomorphism
\begin{equation}\label{Lie(E0)_is_generically_flat_iso}
R^{\textnormal{rig}}_{0}\otimes_{R_{0}}\textnormal{Lie}(\mathbb{E}^{(0)})\simeq R^{\textnormal{rig}}_{0}\otimes_{\breve{K}}(B_{h}\otimes_{K_{h}}\breve{K})
\end{equation} of $R^{\textnormal{rig}}_{0}[\Gamma]$-modules with $\Gamma$ acting diagonally on both sides.
\end{proposition}
\begin{proof}
The construction of the basis $\lbrace c_{i}\rbrace_{0\leq i\leq h-1}$ is given in \cite{gh}, § 21. The isomorphism $
R^{\text{rig}}_{0}\otimes_{R_{0}}\text{Lie}(\mathbb{E}^{(0)})\simeq R^{\text{rig}}_{0}\otimes_{\breve{K}}(B_{h}\otimes_{K_{h}}\breve{K})$ is proved in \cite{gh}, Proposition 22.4.
\end{proof}
\begin{remark}\label{Lie(Ems)_generically_flat}
Since $\mathbb{H}^{(m)}=\mathbb{H}^{(0)}\otimes_{R_{0}}R_{m}$, it follows from the universality of $\mathbb{E}^{(m)}$ that $\mathbb{E}^{(m)}=\mathbb{E}^{(0)}\otimes_{R_{0}}R_{m}$. The isomorphism $\text{Lie}(\mathbb{E}^{(m)})\simeq R_{m}\otimes_{R_{0}}\text{Lie}(\mathbb{E}^{(0)})$ of $R_{m}[\Gamma\times G_{0}]$-modules gives rise to an isomorphism $R^{\text{rig}}_{m}\otimes_{R_{m}}\text{Lie}(\mathbb{E}^{(m)})\simeq R^{\text{rig}}_{m}\otimes_{R_{0}}\text{Lie}(\mathbb{E}^{(0)})$ of $R^{\text{rig}}_{m}[\Gamma\times G_{0}]$-modules. Then using (\ref{rigidification_is_a_basechange}) and (\ref{Lie(E0)_is_generically_flat_iso}), we have an isomorphism \begin{equation}\label{Lie(Em)_is_generically_flat_iso}
R^{\textnormal{rig}}_{m}\otimes_{R_{m}}\textnormal{Lie}(\mathbb{E}^{(m)})\simeq R^{\textnormal{rig}}_{m}\otimes_{\breve{K}}(B_{h}\otimes_{K_{h}}\breve{K})
\end{equation} of $R^{\text{rig}}_{m}[\Gamma\times G_{0}]$-modules, where $\Gamma$ and $G_{0}$ act diagonally on both sides. The action of $G_{0}$ on $B_{h}\otimes_{K_{h}}\breve{K}$ by convention is trivial.
\vspace{1mm}\\
\noindent For $s>0$, the $s$-fold tensor product $(B_{h}\otimes_{K_{h}}\breve{K})^{\otimes s}$ of $B_{h}\otimes_{K_{h}}\breve{K}$ over $\breve{K}$ with itself is a $\Gamma$-representation with the diagonal $\Gamma$-action. Set $(B_{h}\otimes_{K_{h}}\breve{K})^{\otimes 0}:=\breve{K}$ with the trivial $\Gamma$-action, and $(B_{h}\otimes_{K_{h}}\breve{K})^{\otimes s}:=\textnormal{Hom}_{\breve{K}}((B_{h}\otimes_{K_{h}}\breve{K})^{\otimes -s},\breve{K})$ for $s<0$ equipped with the contragredient $\Gamma$-action, i.e. if $\gamma\in\Gamma$, $\varphi\in\textnormal{Hom}_{\breve{K}}((B_{h}\otimes_{K_{h}}\breve{K})^{\otimes -s},\breve{K})$, and $v\in(B_{h}\otimes_{K_{h}}\breve{K})^{\otimes -s}$, then $(\gamma(\varphi))(v)=\gamma(\varphi(\gamma^{-1}(v)))=\varphi(\gamma^{-1}(v))$.
Since \begin{align*}
R^{\textnormal{rig}}_{m}\otimes_{R_{m}}\text{Hom}_{R_{m}}(\textnormal{Lie}(\mathbb{E}^{(m)}),R_{m})&\simeq\text{Hom}_{R^{\text{rig}}_{m}}(R^{\text{rig}}_{m}\otimes_{R_{m}}\text{Lie}(\mathbb{E}^{(m)}),R^{\text{rig}}_{m})\\&\simeq\text{Hom}_{R^{\text{rig}}_{m}}(R^{\text{rig}}_{m}\otimes_{\breve{K}}(B_{h}\otimes_{K_{h}}\breve{K}),R^{\text{rig}}_{m})\\&\simeq R^{\textnormal{rig}}_{m}\otimes_{\breve{K}}\text{Hom}_{\breve{K}}(B_{h}\otimes_{K_{h}}\breve{K},\breve{K}),
\end{align*}
the isomorphism (\ref{Lie(Em)_is_generically_flat_iso}) extends to the isomorphism
\begin{equation}\label{Lie(Ems)_is_generically_flat_iso}
R^{\textnormal{rig}}_{m}\otimes_{R_{m}}\textnormal{Lie}(\mathbb{E}^{(m)})^{\otimes s}\simeq R^{\textnormal{rig}}_{m}\otimes_{\breve{K}}(B_{h}\otimes_{K_{h}}\breve{K})^{\otimes s}
\end{equation} of $R^{\text{rig}}_{m}[\Gamma\times G_{0}]$-modules for all $s\in\mathbb{Z}$. To put it differently, the $(\Gamma\times G_{0})$-equivariant line bundle $\mathcal{L}\textnormal{ie}(\mathbb{E}^{(m)})^{\otimes s}$ is \emph{generically flat} for all $m\geq 0$ and $s\in\mathbb{Z}$.
\end{remark}
\vspace{2mm}
\noindent Let $v_{i}$ denote the images of the basis elements $c_{i}$ under the second map in the short exact sequence (\ref{exact_seq_of_global_sections_of_eqv_bundles}) for $s=1$, $m=0$. According to \cite{gh}, Proposition 23.2, the global sections $\lbrace v_{i}\rbrace_{0\leq i\leq h-1}$ of the line bundle $(\mathcal{M}^{1}_{0})^{\text{rig}}$ have no common zeros on $X^{\text{rig}}_{0}$, and are linearly independent over $\breve{K}$. If $\mathbb{V}$ denotes the $\breve{K}$-subspace of $M^{1}_{0}$ spanned by them, then $\mathbb{V}$ is $\Gamma$-stable, and is isomorphic to $B_{h}\otimes_{K_{h}}\breve{K}$ as a $\Gamma$-representation. Let $\mathbb{P}(\mathbb{V})$ be the projective space of all hyperplanes in $\mathbb{V}$, then the map \begin{align*}
\Phi:X&^{\text{rig}}_{0}\longrightarrow\mathbb{P}(\mathbb{V})\\&x\longmapsto\lbrace v\in \mathbb{V}|v(x)=0\rbrace
\end{align*} is an \'{e}tale surjective morphism of rigid analytic spaces, if $\mathbb{P}(\mathbb{V})$ is identified with the $(h-1)$-dimensional rigid analytic projective space $\mathbb{P}^{h-1}_{\breve{K}}$ (cf. \cite{gh}, Proposition 23.5). The morphism $\Phi:X^{\text{rig}}_{0}\longrightarrow\mathbb{P}^{h-1}_{\breve{K}}$ is called \emph{the period morphism}. In homogeneous projective coordinates, it is given by $\Phi(x)=[\varphi_{0}(x):\ldots:\varphi_{h-1}(x)]$ where $\varphi_{0},\ldots,\varphi_{h-1}\in R^{\textnormal{rig}}_{0}$ are certain global rigid analytic functions without any common zero. These functions can be constructed from the logarithm $g_{0}(X)=\sum_{n\geq 0}a_{n}X^{q^{n}}$ of the universal formal $\mathfrak{o}$-module $\mathbb{H}^{(0)}$ over $R_{0}$ as the limits \begin{align}\label{varphi_i_formulae}
&\varphi_{0}:=\lim_{n\to\infty}\varpi^{n}a_{nh}\\
&\varphi_{i}:=\lim_{n\to\infty}\varpi^{n+1}a_{nh+i}, \hspace{.2cm}\text{if}\hspace{.2cm}1\leq i\leq h-1\nonumber
\end{align} in the Fr\'{e}chet topology of $R^{\text{rig}}_{0}$ (cf. \cite{gh}, (21.6) and (21.13)).\vspace{2.6mm}\\
\noindent An important property of the period morphism $\Phi$ is that it is $\Gamma$-equivariant for the $\Gamma$-action on $\mathbb{P}^{h-1}_{\breve{K}}$ by fractional linear transformations via following inclusion of groups (cf. \cite{kohliwath}, Remark 1.4): \begin{align}\label{Gamma_as_a_subgroup_of_GLh}
&\hspace{2.5cm} j:\Gamma\hookrightarrow GL_{h}(K_{h})\nonumber \\
&\sum_{i=0}^{h-1}\lambda_{i}\Pi^{i}\longmapsto \begin{pmatrix}
\lambda_{0} & \varpi\lambda_{1} & \varpi\lambda_{2} & \cdots & \cdots & \varpi\lambda_{h-1} \\
\lambda_{h-1}^{\sigma} & \lambda_{0}^{\sigma} & \lambda_{1}^{\sigma} & \cdots & \cdots & \lambda_{h-2}^{\sigma}\\
\lambda_{h-2}^{\sigma^{2}} & \varpi\lambda_{h-1}^{\sigma^{2}} & \lambda_{0}^{\sigma^{2}} & \cdots & \cdots & \lambda_{h-3}^{\sigma^{2}}\\
\vdots & \vdots & \ddots & \ddots & &\vdots \\
\vdots & \vdots & &\ddots & \ddots & \vdots \\
\lambda_{1}^{\sigma^{h-1}} & \varpi\lambda_{2}^{\sigma^{h-1}} & \cdots & \cdots & \varpi\lambda_{h-1}^{\sigma^{h-1}} & \lambda_{0}^{\sigma^{h-1}}
\end{pmatrix}
\end{align}
The map $j$ is the group homomorphism coming from the $\Gamma$-action via left multiplication on the right $K_{h}$-vector space $B_{h}$ with basis $\lbrace 1,\Pi^{h-1},\Pi^{h-2},\ldots,\Pi\rbrace$.
In a sense, the period morphism ``linearises'' the complicated $\Gamma$-action on $X^{\text{rig}}_{0}$. \vspace{2.6mm}\\
\noindent The Gross-Hopkins fundamental domain $D$ is the affinoid subdomain of $X^{\textnormal{rig}}_{0}$ defined as follows: \begin{equation}\label{DefnD}
D:=\left\lbrace \text{$x\in X^{\textnormal{rig}}_{0} \hspace{.1cm}\Big\vert\hspace{.1cm} |u_{i}(x)|\leq |\varpi|^{(1-\frac{i}{h})}$ for all $1\leq i \leq h-1$} \right\rbrace \end{equation}
According to \cite{gh}, Lemma 23.14, the function $\varphi_{0}$ does not have any zeroes on $D$, hence is a unit in $\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)$. Setting $w_{i}:=\frac{\varphi_{i}}{\varphi_{0}}$ for $1\leq i\leq h-1$, \cite{gh}, Lemma 23.14 implies that the affinoid $\breve{K}$-algebra $\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)$ is isomorphic to the generalized Tate algebra. \begin{align}\label{expression_of_O(D)}
\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)&\simeq \breve{K} \langle\varpi^{-(1-\frac{1}{h})}w_{1},\dots ,\varpi^{-(1-\frac{h-1}{h})}w_{h-1}\rangle\\&:=\Big\{\sum_{\alpha\in\mathbb{N}_{0}^{h-1}}c_{\alpha}w^{\alpha}\in\breve{K}[[w_{1},\dots ,w_{h-1}]]\hspace{.1cm}\Big\vert\hspace{.1cm}\lim_{\vert \alpha\vert\to\infty}\vert c_{\alpha}\vert\vert\varpi\vert^{\sum_{i=1}^{h-1}\alpha_{i}(1-\frac{i}{h})}=0\Big\}\nonumber
\end{align}
\noindent Note that over a field extension $L$ of $\breve{K}$ containing an $h$-th root of $\varpi$, this is isomorphic to the Tate algebra $L\langle T_{1},\ldots,T_{h-1}\rangle$.\vspace{2.6mm}\\
\noindent It follows from \cite{fgl}, Remarque I.3.2 that $D$ is stable under the $\Gamma$-action on $X^{\text{rig}}_{0}$. Also, the $\Gamma$-equivariant period morphism $\Phi$ restricts to an isomorphism $\Phi:D\iso\Phi(D)$ over $D$ (cf. \cite{gh}, Corollary 23.15). As a result, we have an explicit formula for the $\Gamma$-action on $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$ similar to the one of Devinatz-Hopkins (cf. \cite{kohliwath}, Proposition 1.3.):
\begin{proposition}\label{dh} Fix $i$ with $1\leq i\leq h-1$, and let $\gamma=\sum_{j=0}^{h-1}\lambda_{j}\Pi^{j}\in\Gamma$. Then \begin{equation}\label{dheq}
\gamma(w_{i})=\frac{\sum_{j=1}^{i}\lambda_{i-j}^{\sigma^{j}}w_{j}+\sum_{j=i+1}^{h}\varpi\lambda_{h+i-j}^{\sigma^{j}}w_{j}}{\lambda_{0}+\sum_{j=1}^{h-1}\lambda_{h-j}^{\sigma^{j}}w_{j}}.
\end{equation}
The group $\Gamma$ acts on $\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)$ by continuous $\breve{K}$-algebra endomorphisms extending the action on $R^{\textnormal{rig}}_{0}$.
\end{proposition}
\begin{proof} This is straightforward since $\gamma$ acts on the projective homogeneous coordinates \linebreak $[\varphi_{0}:\ldots:\varphi_{h-1}]$ through right multiplication with the matrix $j(\gamma)$ in (\ref{Gamma_as_a_subgroup_of_GLh}). By \cite{bgr}, (6.1.3), Theorem 1, the induced $\breve{K}$-algebra endomorphism $\gamma$ of the affinoid $\breve{K}$-algerba $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$ is automatically continuous.
\end{proof}
\begin{remark}\label{our_line_bundle_is_a_pullback_of_O(s)}
\emph{A rigidified extension} $(E,s)$ of $\mathbb{H}^{(0)}$ by $\mathbb{G}_{a}$ is an extension $E$ of $\mathbb{H}^{(0)}$ by $\mathbb{G}_{a}$ together with a section $s:\text{Lie}(\mathbb{H}^{(0)})\longrightarrow\text{Lie}(E)$. The set $\text{RigExt}(\mathbb{H}^{(0)},\mathbb{G}_{a})$ of isomorphism classes of rigidified extensions of $\mathbb{H}^{(0)}$ by $\mathbb{G}_{a}$ is a free $R_{0}$-module of rank $h$, and has a basis $\lbrace g_{0},g_{1},\ldots, g_{h-1}\rbrace$ where $g_{0}\in R_{0}[[X]]$ is the logarithm of $\mathbb{H}^{(0)}$, and $g_{i}:=\frac{\partial g_{0}}{\partial u_{i}}$ for $1\leq i\leq h-1$ (cf. \cite{gh}, Proposition 9.8). Moreover, the $R_{0}$-module $\omega(\mathbb{E}^{(0)})$ of invariant differentials on the universal additive extension is isomorphic to $\text{RigExt}(\mathbb{H}^{(0)},\mathbb{G}_{a})$ (cf. \cite{gh}, (11.4)). Thus, $R^{\textnormal{rig}}_{0}\otimes_{R_{0}}\textnormal{Lie}(\mathbb{E}^{(0)})\simeq\text{Hom}_{R_{0}}(\text{RigExt}(\mathbb{H}^{(0)},\mathbb{G}_{a}),R^{\textnormal{rig}}_{0})$. The functions $\varphi_{i}$ in (\ref{varphi_i_formulae}) are precisely $c_{i}(g_{0})$, and the basis $dg_{0}$ of $\omega(\mathbb{H}^{(0)})$ is mapped to $g_{0}$ under the natural map $\omega(\mathbb{H}^{(0)})\longrightarrow\omega(\mathbb{E}^{(0)})$. As a result, the global sections $v_{i}$ and $v_{j}$ of the line bundle $(\mathcal{M}^{1}_{0})^{\text{rig}}$ (see paragraph after Remark \ref{Lie(Ems)_generically_flat}) are related by the relation $\varphi_{j}v_{i}=\varphi_{i}v_{j}$ for all $0\leq i,j\leq h-1$. Consequently, we have $\varphi_{j}^{s}v_{i}^{s}=\varphi_{i}^{s}v_{j}^{s}$ in $M^{s}_{0}$. Let $U_{i}\subset X^{\text{rig}}_{0}$ be the non-vanishing locus of $\varphi_{i}$, then on $U_{i}\cap U_{j}$, we get $v_{i}^{s}=\frac{\varphi_{i}^{s}}{\varphi_{j}^{s}} v_{j}^{s}$ and $v_{j}^{s}=\frac{\varphi_{j}^{s}}{\varphi_{i}^{s}} v_{i}^{s}$. The $U_{i}$'s cover $X^{\text{rig}}_{0}$ as the functions $\varphi_{i}$'s do not vanish simultaneously at any point on $X^{\text{rig}}_{0}$. The data $\big\lbrace U_{i},\varphi_{i}^{s}\in\mathcal{O}^{\times
}_{U_{i}},\frac{\varphi_{i}^{s}}{\varphi_{j}^{s}}\in\mathcal{O}^{\times
}_{U_{i}\cap U_{j}}\big\rbrace_{0\leq i,j\leq h-1}$ represent an element in $H^{1}(X^{\text{rig}}_{0},\mathcal{O}^{\times
}_{X^{\text{rig}}_{0}})$ corresponding to the line bundle $(\mathcal{M}^{s}_{0})^{\text{rig}}$. This means that $(\mathcal{M}^{s}_{0})^{\text{rig}}\big|_{U_{i}}\simeq\mathcal{O}_{U_{i}}\varphi_{i}=\mathcal{O}_{X^{\text{rig}}_{0}}\big|_{U_{i}}\varphi_{i}$ for all $0\leq i\leq h-1$. In particular, for $i=0$, we have an isomorphism $M^{s}_{D}:=(\mathcal{M}^{s}_{0})^{\text{rig}}(D)\simeq\mathcal{O}_{X^{\text{rig}}_{0}}(D)\varphi_{0}^{s}$ of $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$-modules which is also $\Gamma$-equivariant. The $\Gamma$-action on $M^{s}_{D}$ is semilinear for its action on $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$, i.e. $\gamma(f\varphi_{0}^{s})=\gamma(f)\gamma(\varphi_{0}^{s})=\gamma(f)\gamma(\varphi_{0})^{s}$ for $\gamma\in\Gamma$ and $f\in\mathcal{O}_{X^{\text{rig}}_{0}}(D)$.
\end{remark}
\begin{remark}\label{our_line_bundle_is_a_pullback_of_O(s)*} The discussion in Remark \ref{our_line_bundle_is_a_pullback_of_O(s)} shows that the generating global sections $v_{i}$'s of the line bundle $(\mathcal{M}^{1}_{0})^{\text{rig}}$ are the pullbacks $\Phi^{*}(\varphi_{i})$ of $\varphi_{i}$'s along the period morphism $\Phi$ for all $0\leq i\leq h-1$. As a consequence, it follows that \begin{equation*}
(\mathcal{M}^{1}_{0})^{\text{rig}}\simeq\Phi^{*}\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(1).
\end{equation*} By the general properties of the inverse image functor, we then have \begin{equation*}
(\mathcal{M}^{s}_{0})^{\text{rig}}\simeq\Phi^{*}\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)
\end{equation*} for all $s\in\mathbb{Z}$.
\end{remark}
\subsection{Local analyticity of the $\Gamma$-action on $M^{s}_{D}$}
\label{step1}
\noindent In this subsection, we show that the orbit map $(\gamma\mapsto\gamma(f\varphi_{0}^{s})):\Gamma\longrightarrow M^{s}_{D}$ explicitly given by Proposition \ref{dh} and Remark \ref{our_line_bundle_is_a_pullback_of_O(s)} is locally $K$-analytic for all $f\varphi_{0}^{s}\in M^{s}_{D}$.\vspace{2.6mm}\\
\noindent Let $M_{h}(K_{h})$ denote the ring of $h\times h$ matrices with entries from $K_{h}$. It carries an induced topology from the identification with $K_{h}^{h^{2}}$, which endows it with a structure of a locally analytic $K_{h}$-manifold. The subset $GL_{h}(K_{h})$ of invertible matrices is open and forms a locally $K_{h}$-analytic group.
Consider the subgroup $P$ of $GL_{h}(K_{h})$ defined as follows. It is conjugate to a standard Iwahori subgroup of $GL_{h}(K_{h})$.
\begin{equation*}
\textnormal{$P:=\big\lbrace a=(a_{ij})_{0\leq i,j\leq h-1}\in GL_{h}(\mathfrak{o}_{h})$ $\vert$ $ a_{ij},a_{0k}\in \varpi\mathfrak{o}_{h}$ for all $1\leq i,j,k \leq h-1$ with $i>j\big\rbrace$}
\end{equation*}
The conditions on the entries of a matrix in $P$ force all of its diagonal entries to lie in $\mathfrak{o}_{h}^{\times}$. Since $B_{\vert\varpi^{2} \vert}(a)\subseteq P$ for any $a\in P$, $P$ is open in $GL_{h}(K_{h})$. Thus, $P$ is a locally $K_{h}$-analytic subgroup of $GL_{h}(K_{h})$.
The inclusion map $j:\Gamma\hookrightarrow GL_{h}(K_{h})$ mentioned in (\ref{Gamma_as_a_subgroup_of_GLh}) has image in $P$.
\begin{lemma}\label{j_is_locKan} The inclusion map $j:\Gamma\hookrightarrow P$
in (\ref{Gamma_as_a_subgroup_of_GLh}) is locally $K$-analytic.
\end{lemma}
\begin{proof} The global chart for $P$ induced from that for $M_{h}(K_{h})$ sends $a$ in $P$ to
\begin{equation*}
(a_{00},a_{01},\dots a_{0(h-1)},a_{10},a_{11},\dots ,a_{(h-1)(h-2)},a_{(h-1)(h-1)})
\end{equation*} in $K_{h}^{h^{2}}$. Recall the global chart $\psi$ for $\Gamma$ from (\ref{chart_for_Gamma}). Using the global charts for both groups, it is easy to see that the corresponding map from the open subset $\psi(\Gamma)$ in $K_{h}^{h}$ to $K_{h}^{h^{2}}$ is locally $K$-analytic since each component of this map is either a linear polynomial or a $K$-linear Frobenius automorphism $\sigma$ or a composition of both, all being locally $K$-analytic. As before, we remark that $j$ is generally not locally $K_{h}$-analytic because $\sigma:\mathfrak{o}_{h}^{\times}\longrightarrow\mathfrak{o}_{h}^{\times}$ is not locally $K_{h}$-analytic unless $h=1$.
\end{proof}
\noindent The algebra $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$ is a $\breve{K}$-Banach algebra with respect to the multiplicative norm $\|\cdot\|_{D}$defined as follows: for $f=\sum_{\alpha\in\mathbb{N}_{0}^{h-1}}c_{\alpha}w^{\alpha}\in\mathcal{ O}_{X^{\text{rig}}_{0}}(D)$, $\|f\|_{D}:=\sup_{\alpha\in\mathbb{N}_{0}^{h-1}}\vert c_{\alpha}\vert \vert\varpi\vert^{\sum_{i=1}^{h-1}\alpha_{i}(1-\frac{i}{h})}$ (cf. \cite{bgr}, § 6.1.5, Proposition 1 and 2). Let $P$ act on $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$ by $\breve{K}$-linear ring automorphisms by defining \begin{equation}\label{actI}
a(w_{i}):=\frac{a_{0i}+\sum_{j=1}^{h-1}a_{ji}w_{j}}{a_{00}+\sum_{j=1}^{h-1}a_{j0}w_{j}}
\end{equation}
for $a\in P$ and for $1\leq i\leq h-1$. This gives an action of $P$ on $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$ by continuous $\breve{K}$-linear ring automorphisms which, when restricted to $\Gamma$ via $j$, coincides with the $\Gamma$-action on $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$ (cf. Proposition \ref{dh}). Indeed, note that $a_{00}+\sum_{j=1}^{h-1}a_{j0}w_{j}=a_{00}(1+\sum_{j=1}^{h-1}a_{00}^{-1}a_{j0}w_{j})\in(\mathcal{O}_{X^{\text{rig}}_{0}}(D))^{\times}$ is a unit of norm 1, and $\big\|a_{0i}+\sum_{j=1}^{h-1}a_{ji}w_{j}\big\|_{D}=\|w_{i}\|_{D}$ by the strict triangle inequality. Altogether, $\|a(w_{i})\|_{D}=\|w_{i}\|_{D}$ which ensures that $P$ acts on $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$ via $\sum_{\alpha\in\mathbb{N}_{0}^{h-1}}c_{\alpha}w^{\alpha}\longmapsto\sum_{\alpha\in\mathbb{N}_{0}^{h-1}}c_{\alpha}a(w_{1})^{\alpha_{1}}\ldots a(w_{h-1})^{\alpha_{h-1}}$ in a well-defined way. \vspace{2.6mm}\\
\noindent We now show that the above action is locally $K$-analytic.
\begin{lemma}\label{invla}
The map $\iota:P\longrightarrow P$, $(a_{ij})_{0\leq i,j\leq h-1}\longmapsto(\iota(a)_{ij})_{0\leq i,j\leq h-1}$ defined by \begin{equation*}
\iota(a)_{ij}=
\begin{cases}
a_{ij}^{-1}, &\text{if $i=j=0$;}\\
a_{ij}, &\text{otherwise}
\end{cases}
\end{equation*} is locally $K_{h}$-analytic, and thus locally $K$-analytic.
\end{lemma}
\begin{proof}
This follows from \cite{schplie}, Proposition 13.6, and the fact that $K_{h}^{\times}$ is a locally $K_{h}$-analytic group.
\end{proof}
\begin{proposition}\label{laofI} The action of $P$ on $\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)$ is locally $K_{h}$-analytic, and thus locally $K$-analytic.
\end{proposition}
\begin{proof}
By Lemma \ref{invla}, it is enough to show that, for each $f\in\mathcal{O}_{X^{\text{rig}}_{0}}(D)$, the map $\iota(a)\longmapsto a(f)$ from $P$ to $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$ is locally $K_{h}$-analytic. Consider the open neighbourhood $U$ of $0$ in $K_{h}^{h^{2}}$ defined as follows:\begin{equation*}
U:=\lbrace x=(x_{1},x_{2},\dots ,x_{h^{2}})\in \mathfrak{o}_{h}^{h^{2}}\mid\text{$x_{i}, x_{qh+r}\in\varpi\mathfrak{o}_{h}$ $\forall$ $2\leq i\leq h$ and $\forall$ $q\geq r$ with $q,r>1$ }\rbrace
\end{equation*} Let $T=(T_{1},T_{2},\dots ,T_{h^{2}})$, and let $\mathcal{F}_{U}(K_{h}^{h^{2}},\mathcal{O}_{X^{\text{rig}}_{0}}(D))$ denote the set of power series in $T$ with coefficients from $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$ which converge on $U$, i.e. $\mathcal{F}_{U}(K_{h}^{h^{2}},\mathcal{O}_{X^{\text{rig}}_{0}}(D)):=$ \begin{equation*}
\left\lbrace\sum_{\alpha\in\mathbb{N}_{0}^{h^{2}}}f_{\alpha}T^{\alpha}\in\mathcal{O}_{X^{\text{rig}}_{0}}(D)[[T]]\hspace*{.1cm}\Big\vert\hspace*{.1cm}\lim_{\vert \alpha\vert\to\infty}\| f_{\alpha}\|_{D}\vert\varpi\vert^{(\alpha_{2}+\alpha_{3}+\dots +\alpha_{h}+\alpha_{2h+2}+\alpha_{3h+2}+\alpha_{3h+3}+\dots +\alpha_{(h-1)h+h-1})}=0\right\rbrace
\end{equation*} Like $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$, $\mathcal{F}_{U}(K_{h}^{h^{2}},\mathcal{O}_{X^{\text{rig}}_{0}}(D))$ is also a $\breve{K}$-Banach algebra with respect to the following multiplicative norm (cf. \cite{schplie}, Proposition 5.3):\begin{equation*}
\Big\|\sum_{\alpha\in\mathbb{N}_{0}^{h^{2}}}f_{\alpha}T^{\alpha}\Big\|_{U}:=\sup_{\alpha\in\mathbb{N}_{0}^{h^{2}}}\| f_{\alpha}\|_{D}\vert\varpi\vert^{(\alpha_{2}+\alpha_{3}+\dots +\alpha_{h}+\alpha_{2h+2}+\alpha_{3h+2}+\alpha_{3h+3}+\dots +\alpha_{(h-1)h+h-1})}
\end{equation*}
Under the global chart of $P$ in the proof of \ref{j_is_locKan}, we now show that, for a monomial $w^{\alpha}\in\mathcal{O}_{X^{\text{rig}}_{0}}(D)$, the map $\iota(a)\longmapsto a(w^{\alpha})$ belongs to $\mathcal{F}_{U}(K_{h}^{h^{2}},\mathcal{O}_{X^{\text{rig}}_{0}}(D))$ for every $\alpha\in \mathbb{N}_{0}^{h-1}$.\vspace{2.6mm}\\
\noindent By (\ref{actI}), we have \begin{align}\label{3bra}
a(w^{\alpha})&\nonumber=a(w_{1})^{\alpha_{1}}\dots a(w_{h-1})^{\alpha_{h-1}}\\\nonumber&=\Bigg(\frac{a_{01}+\sum_{j=1}^{h-1}a_{j1}w_{j}}{a_{00}+\sum_{j=1}^{h-1}a_{j0}w_{j}}\Bigg)^{\alpha_{1}}\dots\Bigg(\frac{a_{0(h-1)}+\sum_{j=1}^{h-1}a_{j(h-1)}w_{j}}{a_{00}+\sum_{j=1}^{h-1}a_{j0}w_{j}}\Bigg)^{\alpha_{h-1}}
\\\nonumber&=\Bigg(\prod_{i=1}^{h-1}\Big(a_{0i}+\sum_{j=1}^{h-1}a_{ji}w_{j}\Big)^{\alpha_{i}}\Bigg)(a_{00}^{-1})^{\vert \alpha\vert}\Big(1+a_{00}^{-1}\sum_{j=1}^{h-1}a_{j0}w_{j}\Big)^{-\vert \alpha\vert}\\&=\Bigg(\prod_{i=1}^{h-1}\Big(a_{0i}+\sum_{j=1}^{h-1}a_{ji}w_{j}\Big)^{\alpha_{i}}\Bigg)(a_{00}^{-1})^{\vert \alpha\vert}\Bigg(\sum_{l=0}^{\infty}\Big(-a_{00}^{-1}\sum_{j=1}^{h-1}a_{j0}w_{j}\Big)^{l}\Bigg)^{\vert \alpha\vert}
\end{align}
\noindent Thus the expression of $a(w^{\alpha})$ is a product of $(a_{00}^{-1})^{\vert \alpha\vert}$ and two big brackets. The first big bracket in (\ref{3bra}) is a product of polynomials in $a_{ij}$'s with coefficients from $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$, and hence is the evaluation at $\iota(a)$ of an element in $\mathcal{F}_{U}(K_{h}^{h^{2}},\mathcal{O}_{X^{\text{rig}}_{0}}(D))$. Similarly, $(a_{00}^{-1})^{\vert \alpha\vert}$ is the evaluation at $\iota(a)$ of the monomial $T_{1}^{\vert \alpha\vert}$ which belongs to $\mathcal{F}_{U}(K_{h}^{h^{2}},\mathcal{O}_{X^{\text{rig}}_{0}}(D))$. The second big bracket is the $\vert \alpha\vert$-th power of a certain geometric series. The $l$-th term in that series is the evaluation of the polynomial $\big(-T_{1}\sum_{j=1}^{h-1}T_{jh+1}w_{j}\big)^{l}$ at $\iota(a)$, and \begin{equation*}
\Big\|\Big(-T_{1}\sum_{j=1}^{h-1}T_{jh+1}w_{j}\Big)^{l}\Big\|_{U}=\Big(\Big\|-T_{1}\sum_{j=1}^{h-1}T_{jh+1}w_{j}\Big\|_{U}\Big)^{l}=\vert\varpi\vert^{\frac{l}{h}}
\end{equation*} Hence, the series $\sum_{l=0}^{\infty}(-T_{1}\sum_{j=1}^{h-1}T_{jh+1}w_{j})^{l}$ converges in $\mathcal{F}_{U}(K_{h}^{h^{2}},\mathcal{O}_{X^{\text{rig}}_{0}}(D))$, and the map $\iota(a)\longmapsto a(w^{\alpha})\in\mathcal{F}_{U}(K_{h}^{h^{2}},\mathcal{O}_{X^{\text{rig}}_{0}}(D))$ for every $\alpha\in \mathbb{N}_{0}^{h-1}$.\vspace{2.6mm}\\
\noindent Let us calculate the norms $\|\cdot\|_{U}$ of the above power series corresponding to the terms in the expression (\ref{3bra}) or find an upper bound for them. First, $\| T_{1}^{|\alpha|}\|_{U}=1$. Since \begin{equation*}
\Big\|\sum_{l=0}^{\infty}\Big(-T_{1}\sum_{j=1}^{h-1}T_{jh+1}w_{j}\Big)^{l}\Big\|_{U}\leq\sup_{l\geq 0}\Big\|\Big(-T_{1}\sum_{j=1}^{h-1}T_{jh+1}w_{j}\Big)^{l}\Big\|_{U}=\sup_{l\geq 0}\vert \varpi\vert^{\frac{l}{h}}=1,
\end{equation*} the power series corresponding to the second big bracket in (\ref{3bra}) has the norm $\leq$ 1. The first big bracket is obtained by evaluating $\prod_{i=1}^{h-1}\Big(T_{i+1}+\sum_{j=1}^{h-1}T_{jh+i+1}w_{j}\Big)^{\alpha_{i}}$ at $\iota(a)$, and \begin{equation*}
\Big\|\prod_{i=1}^{h-1}\Big(T_{i+1}+\sum_{j=1}^{h-1}T_{jh+i+1}w_{j}\Big)^{\alpha_{i}}\Big\|_{U}=\prod_{i=1}^{h-1}\Big\|\Big(T_{i+1}+\sum_{j=1}^{h-1}T_{jh+i+1}w_{j}\Big)\Big\|_{U}^{\alpha_{i}}=\prod_{i=1}^{h-1}\vert\varpi\vert^{\alpha_{i}(1-\frac{i}{h})}
\end{equation*} Therefore, the power series corresponding to the first big bracket has the norm $\vert\varpi\vert^{\sum_{i=1}^{h-1}\alpha_{i}(1-\frac{i}{h})}$. So, for every $\alpha\in \mathbb{N}_{0}^{h-1}$, the map $\iota(a)\longmapsto a(w^{\alpha})$ is given by an element in $\mathcal{F}_{U}(K_{h}^{h^{2}},\mathcal{O}_{X^{\text{rig}}_{0}}(D))$ whose norm is bounded above by $\vert\varpi\vert^{\sum_{i=1}^{h-1}\alpha_{i}(1-\frac{i}{h})}$.\vspace{2.6mm}\\
\noindent Now for every $f=\sum_{\alpha\in\mathbb{N}_{0}^{h-1}}c_{\alpha}w^{\alpha}\in\mathcal{O}_{X^{\text{rig}}_{0}}(D)$, we have $a(f)=\sum_{\alpha\in\mathbb{N}_{0}^{h-1}}c_{\alpha}(a (w^{\alpha}))$, and for every $\alpha\in \mathbb{N}_{0}^{h-1}$, $c_{\alpha}(a (w^{\alpha}))$ is represented by a power series in $\mathcal{F}_{U}(K_{h}^{h^{2}},\mathcal{O}_{X^{\text{rig}}_{0}}(D))$ having norm $\leq\vert c_{\alpha}\vert\vert\varpi\vert^{\sum_{i=1}^{h-1}\alpha_{i}(1-\frac{i}{h})}$. Since, $\lim_{\vert \alpha\vert\to\infty}\vert c_{\alpha}\vert\vert\varpi\vert^{\sum_{i=1}^{h-1}\alpha_{i}(1-\frac{i}{h})}=0$, we see that the map $\iota(a)\longmapsto a(f)$ from $P$ to $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$ is given by a convergent power series in $\mathcal{F}_{U}(K_{h}^{h^{2}},\mathcal{O}_{X^{\text{rig}}_{0}}(D))$. As $a$ is arbitrary, this implies that the action of $P$ on $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$ is locally $K_{h}$-analytic, and thus locally $K$-analytic by Lemma \ref{laLlaK}.
\end{proof}
\begin{proposition}\label{laofG} The $\breve{K}$-vector space $\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)$ is a locally $K$-analytic representation of $\Gamma$.\end{proposition}
\begin{proof}
This follows from the Lemma \ref{j_is_locKan} and Proposition \ref{laofI}.
\end{proof}
\noindent The above proposition can be generalized as follows completing the Step 1 of the strategy mentioned in the beginning:
\begin{theorem}\label{laofGonMD}
Let $s$ be any integer, then the $\breve{K}$-vector space $M^{s}_{D}$ is a locally $K$-analytic representation of $\Gamma$.
\end{theorem}
\begin{proof}
According to Remark \ref{our_line_bundle_is_a_pullback_of_O(s)}, we have a $\Gamma$-equivariant, $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$-linear isomorphism $M^{s}_{D}\simeq\mathcal{O}_{X^{\text{rig}}_{0}}(D).\varphi_{0}^{s}$ of free $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$-modules of rank 1. Then $M^{s}_{D}$ obtains a structure of a $\breve{K}$-Banach space with respect to the norm defined as $\| f\varphi^{s}_{0}\|_{M^{s}_{D}}:=\| f\|_{D}$. Since the $\Gamma$-action on $M^{s}_{D}$ is semilinear for its action on $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$, we have $\gamma(f\varphi^{s}_{0})=\gamma(f)\gamma(\varphi^{s}_{0})$ for all $\gamma\in\Gamma$ and $f\in\mathcal{O}_{X^{\text{rig}}_{0}}(D)$. Now, as mentioned in the proof of Proposition \ref{dh}, $\gamma(\varphi^{s}_{0})=\gamma(\varphi_{0})^{s}=(\lambda_{0}\varphi_{0}+\lambda^{\sigma}_{h-1}\varphi_{1}+\dots +\lambda^{\sigma^{h-1}}_{1}\varphi_{h-1})^{s}=(\lambda_{0}+\lambda^{\sigma}_{h-1}w_{1}+\dots +\lambda^{\sigma^{h-1}}_{1}w_{h-1})^{s}\varphi_{0}^{s}$. So the orbit map from $\Gamma$ to $M^{s}_{D}$ is given by sending $\gamma$ to $\gamma(f\varphi^{s}_{0})=\gamma(f)(\lambda_{0}+\lambda^{\sigma}_{h-1}w_{1}+\dots +\lambda^{\sigma^{h-1}}_{1}w_{h-1})^{s}\varphi^{s}_{0}$. The map $\gamma\mapsto\gamma(f)$ is locally $K$-analytic by Proposition \ref{laofG}, and the map $\gamma\mapsto(\lambda_{0}+\lambda^{\sigma}_{h-1}w_{1}+\dots +\lambda^{\sigma^{h-1}}_{1}w_{h-1})^{s}\varphi^{s}_{0}$ is also locally $K$-analytic since it is given by a linear polynomial in the coordinates of $\gamma$. Thus, the orbit map, being a product of these two maps, is locally $K$-analytic. Therefore, $M^{s}_{D}$ is a locally analytic $\Gamma$-representation for all integers $s$.
\end{proof}
\subsection{Local analyticity of the $\Gamma$-action on $M^{s}_{0}$}
\label{step2}
\noindent Recall that the free $R^{\text{rig}}_{0}$-module $M^{s}_{0}$ of rank 1 is a $\breve{K}$-Fr\'{e}chet space, and the group $\Gamma$ acts on it by continuous $\breve{K}$-linear automorphisms. This induces an action of $\Gamma$ on the strong topological $\breve{K}$-linear dual $(M^{s}_{0})'_{b}$ of $M^{s}_{0}$ given by \begin{align*}
\gamma:(M^{s}_{0})&'_{b}\longrightarrow (M^{s}_{0})'_{b}\\
& l\longmapsto(\delta\mapsto l(\gamma^{-1}(\delta))).
\end{align*} The goal of this subsection is to show that the above action of $\Gamma$ on $(M^{s}_{0})'_{b}$ is locally $K$-analytic by showing that its dual $M^{s}_{0}$ is a continuous $D(\Gamma,\breve{K})$-module. \vspace{2.6mm}\\
\noindent The continuity of the $\Gamma$-action on the universal deformation ring $R_{0}$ (cf. Theorem \ref{ctsthm}) leads to a continuous $\Gamma$-action on $R^{\text{rig}}_{0}$, i.e. the action map $\Gamma\times R^{\text{rig}}_{0}\longrightarrow R^{\text{rig}}_{0}$ is continuous for the Fr\'{e}chet topology on $R^{\text{rig}}_{0}$ and for the product of profinite and Fr\'{e}chet topology on the left. This is implied by the next proposition which also forms the main ingredient of the Step 2 of our strategy.
\begin{proposition}\label{mainprop} Let $n$ and $l$ be integers with $n\geq 0$ and $l\geq 1$. If $\gamma\in\Gamma_{n}$, and if $f\in R^{\textnormal{rig}}_{0}$, then $\|\gamma(f)-f\|_{l}\leq\vert \varpi\vert^{n/l}\| f\|_{l}$.
\end{proposition}
\begin{proof}
Note that $R^{\text{rig}}_{0,l}$ is a generalized Tate algebra over $\breve{K}$ in the variables $(\varpi^{-1/l}u_{i})_{1\leq i\leq h-1}$. Then by \cite{bgr}, (6.1.5), Proposition 5, we have \begin{equation*}
\| g\|_{l}=\sup\big\lbrace\vert g(x)\vert\hspace{.1cm}\big\vert\hspace{.1cm} x\in\mathbb{B}_{l}\big(\overline{\breve{K}}\big)\big\rbrace\hspace{.2cm}\text{for any}\hspace{.2cm} g\in R^{\textnormal{rig}}_{0,l},
\end{equation*} where \begin{equation*}
\mathbb{B}_{l}\big(\overline{\breve{K}}\big)=\big\lbrace x\in\big(\overline{\breve{K}}\big)^{h-1}\hspace{.1cm}\big\vert\hspace{.1cm}\text{$\vert x_{i}\vert\leq\vert\varpi\vert^{1/l}$ for all $1\leq i \leq h-1$}\big\rbrace.
\end{equation*} Let us first prove the assertion for $f=u_{i}$ for some $1\leq i\leq h-1$. If $x\in\mathbb{B}_{l}\big(\overline{\breve{K}}\big)$ and $y=(y_{j}):=(\gamma(u_{j})(x))$, then we need to show that $\vert x_{i}-y_{i}\vert\leq\vert\varpi\vert^{(n+1)/l}$ because $\|u_{i}\|_{l}=|\varpi|^{1/l}$. Consider the commutating diagram \begin{displaymath}
\xymatrix{
R_{0}\ar[dr]_{f\mapsto f(y)}\ar[rr]^{\gamma} & & R_{0}\ar[dl]^{f\mapsto f(x)} \\
& \overline{\breve{\mathfrak{o}}} & }
\end{displaymath}
of homomorphisms of $\breve{\mathfrak{o}}$-algebras. Choosing $z\in\overline{\breve{\mathfrak{o}}}$ with $\vert z\vert = \vert\varpi\vert^{1/l}$, we have $x_{j}\in z\overline{\breve{\mathfrak{o}}}$ for all $j$. Further, $\varpi\in z\overline{\breve{\mathfrak{o}}}$ because $l\geq 1$. As a consequence, the right oblique arrow of the above diagram maps $\mathfrak{m}_{R_{0}}=(\varpi,u_{1},\ldots,u_{h-1})$ to $z\overline{\breve{\mathfrak{o}}}$. Note that $\gamma(u_{j})\in\mathfrak{m}_{R_{0}}$ so we obtain $y_{j}\in z\overline{\breve{\mathfrak{o}}}$ as well. Therefore, also the left oblique arrow maps $\mathfrak{m}_{R_{0}}$ to $z\overline{\breve{\mathfrak{o}}}$. Now consider the induced diagram
\begin{displaymath}
\xymatrix{
R_{0}/\mathfrak{m}_{R_{0}}^{n+1}\ar[dr]\ar[rr]^{\gamma} & & R_{0}/\mathfrak{m}_{R_{0}}^{n+1}\ar[dl] \\
& \overline{\breve{\mathfrak{o}}}/(z^{n+1}) & }
\end{displaymath}
According to Theorem \ref{ctsthm}, the upper horizontal arrow is the identity. It follows that $x_{i}-y_{i}\in z^{n+1}\overline{\breve{\mathfrak{o}}}$, i.e. $\vert x_{i}-y_{i}\vert\leq\vert\varpi\vert^{(n+1)/l}$.
\vspace{2.6mm}\\
\noindent We now prove the assertion for $f=u^{\alpha}$ by induction on $|\alpha|$. The case $|\alpha|=0$ is trivial. Let $|\alpha|>0$. Choose an index $i$ with $\alpha_{i}>0$. Define $\beta_{j}:=\alpha_{j}$ if $j\neq i$, and $\beta_{i}:=\alpha_{i}-1$. Then for $x\in\mathbb{B}_{l}\big(\overline{\breve{K}}\big)$,\begin{align*}
\vert \gamma(u^{\alpha})(x)-u^{\alpha}(x)\vert&=\vert y^{\alpha}-x^{\alpha}\vert=\vert y_{i}y^{\beta}-x_{i}x^{\beta}\vert\\&\leq\max\lbrace\vert y_{i}\vert\vert y^{\beta}-x^{\beta}\vert,\vert y_{i}-x_{i}\vert\vert x^{\beta}\vert\rbrace.
\end{align*} Now $\vert y_{i}\vert\vert y^{\beta}-x^{\beta}\vert\leq\vert\varpi\vert^{1/l}\|\gamma(u^{\beta})-u^{\beta}\|_{l}\leq\vert\varpi\vert^{(n+1)/l}\| u^{\beta}\|_{l}=\vert\varpi\vert^{n/l}\| u^{\alpha}\|_{l}$ by the induction hypothesis and $\vert y_{i}-x_{i}\vert\vert x^{\beta}\vert\leq\vert\varpi\vert^{(n+1)/l}\vert\varpi\vert^{\vert \beta\vert/l}=\vert\varpi\vert^{n/l}\| u^{\alpha}\|_{l}$ as seen above. Thus we obtain $\vert \gamma(u^{\alpha})(x)-u^{\alpha}(x)\vert\leq\vert\varpi\vert^{n/l}\|u^{\alpha}\|_{l}$ for all $x\in\mathbb{B}_{l}\big(\overline{\breve{K}}\big)$ as required. \vspace{2.6mm}\\
\noindent Therefore if $f=\sum_{\alpha\in\mathbb{N}^{h-1}_{0}}c_{\alpha}u^{\alpha}\in R^{\textnormal{rig}}_{0}$, then by continuity of $\gamma$, we get\begin{align*}
\|\gamma(f)-f\|_{l}=\Big\|\sum_{\alpha\in\mathbb{N}^{h-1}_{0}}c_{\alpha}(\gamma(u^{\alpha})-u^{\alpha})\Big\|_{l}&\leq\sup_{\alpha\in\mathbb{N}^{h-1}_{0}}|c_{\alpha}|\|\gamma(u^{\alpha})-u^{\alpha}\|_{l}\\&\leq\sup_{\alpha\in\mathbb{N}^{h-1}_{0}}|c_{\alpha}||\varpi|^{n/l}\|u^{\alpha}\|_{l}=|\varpi|^{n/l}\|f\|_{l}.
\end{align*}
\end{proof}
\noindent We write $\Gamma_{\mathbb{Q}_{p}}$ for $\Gamma$ when viewed as a locally $\mathbb{Q}_{p}$-analytic group, and $\mathfrak{g}_{\mathbb{Q}_{p}}$ for its Lie algebra $\mathfrak{g}$ when considered as a $\mathbb{Q}_{p}$-vector space. Let $d:=[K:\mathbb{Q}_{p}]$. Since $\Gamma_{\mathbb{Q}_{p}}$ is a compact locally $\mathbb{Q}_{p}$-analytic group of dimension $t:=dh^{2}$, it contains an open subgroup $\Gamma_{o}$ which is a uniform pro-$p$ group of rank $t$ (cf. Theorem \ref{Lazard's_characterization}). The subgroups in its lower $p$-series $P_{i}(\Gamma_{o})$ $(i\geq 1)$ form a basis of open neighbourhoods of the identity in $\Gamma_{o}$ and are also uniform pro-$p$ groups of rank $t$ by Lemma \ref{pseries}. Let $n$ be a positive integer such that $\Gamma_{n}\subseteq\Gamma_{o}$. As $\Gamma_{n}$ is open in $\Gamma_{o}$, it contains $\Gamma_{*}:=P_{i}(\Gamma_{o})$ for some $i\geq 1$. In what follows, we view $\Gamma_{*}$ as a locally $\mathbb{Q}_{p}$-analytic group.\vspace{2.6mm}\\
\noindent Let us denote by $\Lambda(\Gamma_{*}):=\breve{\mathfrak{o}}[[\Gamma_{*}]]$ the Iwasawa algebra of $\Gamma_{*}$ over $\breve{\mathfrak{o}}$. Set $b_{i}:=\gamma_{i}-1\in\Lambda(\Gamma_{*})$ and $b^{\alpha}:=b_{1}^{\alpha_{1}}\cdots b_{t}^{\alpha_{t}}$ for any $\alpha\in\mathbb{N}_{0}^{t}$ where $\lbrace\gamma_{1},\dots ,\gamma_{t}\rbrace$ is a minimal topological generating set of $\Gamma_{*}$. By \cite{ddms}, Theorem 7.20, any element $\mu\in\Lambda(\Gamma_{*})$ admits a unique expansion of the form \begin{equation*}
\text{$\mu =\sum_{\alpha\in\mathbb{N}_{0}^{t}}d_{\alpha}b^{\alpha}$ with $d_{\alpha}\in\breve{\mathfrak{o}}$ $\forall$ $\alpha\in\mathbb{N}_{0}^{t}$}
\end{equation*}
For any $l\geq 1$, this allows us to define the $\breve{K}$-norm $\|\cdot\|_{l}$ on the algebra $\Lambda(\Gamma_{*})_{\breve{K}}:=\Lambda(\Gamma_{*})\otimes_{\breve{\mathfrak{o}}}\breve{K}$ through \begin{equation}\label{lnorm}
\Big\|\sum_{\alpha\in\mathbb{N}_{0}^{t}}d_{\alpha}b^{\alpha}\Big\|_{l}:=\sup_{\alpha\in\mathbb{N}_{0}^{t}}\lbrace\vert d_{\alpha}\vert\vert \varpi\vert^{\vert \alpha\vert /l}\rbrace
\end{equation} By \cite{stadmrep}, Proposition 4.2, the norm $\|\cdot\|_{l}$ on $\Lambda(\Gamma_{*})_{\breve{K}}$ is submultiplicative. As a consequence, the completion \begin{equation*}
\Lambda(\Gamma_{*})_{\breve{K},l}=\Big\lbrace\sum_{\alpha\in\mathbb{N}_{0}^{t}}d_{\alpha}b^{\alpha}\mid d_{\alpha}\in\breve{K} , \lim_{\vert \alpha\vert\to\infty}\vert d_{\alpha}\vert\vert \varpi\vert^{\vert \alpha\vert /l}=0\Big\rbrace
\end{equation*} of $\Lambda(\Gamma_{*})_{\breve{K}}$ with respect to $\|\cdot\|_{l}$ is a $\breve{K}$-Banach algebra. The natural inclusions $\Lambda(\Gamma_{*})_{\breve{K},l+1}\hookrightarrow\Lambda(\Gamma_{*})_{\breve{K},l}$ endow the projective limit \begin{equation*}D(\Gamma_{*},\breve{K})=\varprojlim_{l}\Lambda(\Gamma_{*})_{\breve{K},l}\end{equation*} with the structure of a $\breve{K}$-Fr\'{e}chet algebra. As explained in § \ref{Structure of the distribution algebra}, the above projective limit is indeed equal to the algebra of $\breve{K}$-valued locally $\mathbb{Q}_{p}$-analytic distributions on $\Gamma_{*}$.
By fixing coset representatives $\lbrace\gamma'_{1}=1,\gamma'_{2},\ldots,\gamma'_{s}\rbrace$ of $\Gamma_{*}$ in $\Gamma_{\mathbb{Q}_{p}}$, the natural topological isomorphism $C^{an}(\Gamma_{\mathbb{Q}_{p}},\breve{K})\simeq\prod_{i=1}^{s}C^{an}(\gamma'_{i}\Gamma_{*},\breve{K})$ of locally convex $\breve{K}$-vector spaces induces a topological isomorphism \begin{equation}\label{top_iso_of_distribution_alg}
D(\Gamma_{\mathbb{Q}_{p}},\breve{K})\simeq\bigoplus_{i=1}^{s}\delta_{\gamma'_{i}} D(\Gamma_{*},\breve{K})\hspace{1.5cm}\text{($\delta_{\gamma'_{i}}$'s are Dirac distributions)}
\end{equation} by dualizing (cf. \cite{fe99}, Korollar 2.2.4). This defines a $\breve{K}$-Fr\'{e}chet algebra structure on $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$ given by the family of norms $\|\delta_{\gamma'_{1}}\mu_{1}+\cdots+\delta_{\gamma'_{s}}\mu_{s}\|_{l}:=\max_{i=1}^{s}\lbrace\|\mu_{i}\|_{l}\rbrace$ with $l\geq 1$ (cf. \cite{stadmrep}, Theorem 5.1).\vspace{2.6mm}\\
\noindent Note that $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$ is not the same as the distribution algebra $D(\Gamma,\breve{K})$ of $\breve{K}$-valued locally $K$-analytic distributions on $\Gamma$. In fact, the natural embedding $C^{an}(\Gamma,\breve{K})\hookrightarrow C^{an}(\Gamma_{\mathbb{Q}_{p}},\breve{K})$ induces a map $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})\longrightarrow D(\Gamma,\breve{K})$ which is a strict surjection and a homomorphism of $\breve{K}$-algebras by \cite{kohlinvdist}, Lemma 1.3.1. The kernel $I$ of the sujection $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})\twoheadrightarrow D(\Gamma,\breve{K})$ has the following explicit description due to \cite{kohlinvdist}, Lemma 1.3.2 and Lemma 1.3.3: \begin{align}\label{description_of_I}
&\text{If $i:\mathfrak{g}_{\mathbb{Q}_{p}}\hookrightarrow D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$ denotes the natural inclusion, then $I$ is the closure of the ideal } \\ & \text{generated by all elements of the form $i(t\mathfrak{x})-ti(\mathfrak{x})$ with $\mathfrak{x}\in\mathfrak{g}_{\mathbb{Q}_{p}}$ and $t\in K$.\nonumber}
\end{align} The quotient topology on $D(\Gamma,\breve{K})$ is the $\breve{K}$-Fr\'{e}chet topology mentioned in the beginning of § \ref{Structure of the distribution algebra}.
\begin{theorem}\label{laK0} The action of $\Gamma_{\mathbb{Q}_{p}}$ on $R^{\textnormal{rig}}_{0}$ extends to a continuous action of the $\breve{K}$-Fr\'{e}chet algebra $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$, which then factors through a continuous action of $D(\Gamma,\breve{K})$ on $R^{\textnormal{rig}}_{0}$. Hence the action of $\Gamma$ on the strong continuous $\breve{K}$-linear dual $(R^{\textnormal{rig}}_{0})'_{b}$ of $R^{\textnormal{rig}}_{0}$ is locally $K$-analytic.
\end{theorem}
\begin{proof}
First, we show that $R^{\textnormal{rig}}_{0,l}$ is a topological Banach module over the $\breve{K}$-Banach algebra $\Lambda(\Gamma_{*})_{\breve{K},l}$ for all $l\geq 1$. To show this, let us prove by induction on $\vert \alpha\vert$ that $\| b^{\alpha}(f)\|_{l}\leq\| b^{\alpha}\|_{l}\| f\|_{l}$ for any $f\in R^{\textnormal{rig}}_{0}$. This is clear if $\vert \alpha\vert=0$. Let $|\alpha|>0$ and let $i$ be the minimal index such that $\alpha_{i}>0$. Define $\beta_{j}:=\alpha_{j}$ if $j\neq i$, and $\beta_{i}:=\alpha_{i}-1$. Since $\Gamma_{*}\subseteq \Gamma_{n}$, Proposition \ref{mainprop} and the induction hypothesis imply \begin{align*}
\| b^{\alpha}(f)\|_{l}&=\| ((\gamma_{i}-1)b^{\beta})(f)\|_{l}=\| (\gamma_{i}-1)(b^{\beta}(f))\|_{l}\leq\vert\varpi\vert^{n/l}\| b^{\beta}(f)\|_{l}\\&\leq\vert\varpi\vert^{1/l}\| b^{\beta}\|_{l}\| f\|_{l}\leq\vert\varpi\vert^{1/l}\vert\varpi\vert^{\vert \beta\vert /l}\| f\|_{l}=\vert\varpi\vert^{(\vert \beta\vert+1) /l}\| f\|_{l}=\| b^{\alpha}\|_{l}\| f\|_{l}
\end{align*} as required. By Remark \ref{iwasawa_alg_action}, this immediately gives $\|\mu(f)\|_{l}\leq\|\mu\|_{l}\| f\|_{l}$ for all $\mu\in\Lambda(\Gamma_{*})_{\breve{K}}$ and $f\in R_{0}[\frac{1}{\varpi}]=R_{0}\otimes_{\breve{\mathfrak{o}}}\breve{K}$. Hence the map $\Lambda(\Gamma_{*})_{\breve{K}}\times R_{0}[\frac{1}{\varpi}]\longrightarrow R_{0}[\frac{1}{\varpi}]$ $((\mu,f)\mapsto\mu(f))$ is continuous if $\Lambda(\Gamma_{*})_{\breve{K}}$ and $R_{0}[\frac{1}{\varpi}]$ are endowed with the respective $\|\cdot\|_{l}$-topologies, and if the left hand side carries the product topology. Since $R_{0}[\frac{1}{\varpi}]$ is dense in $R^{\textnormal{rig}}_{0,l}$, we obtain a map $\Lambda(\Gamma_{*})_{\breve{K},l}\times R^{\textnormal{rig}}_{0,l}\longrightarrow R^{\textnormal{rig}}_{0,l}$ by passing to completions. By continuity, it gives $R^{\textnormal{rig}}_{0,l}$ the structure of a topological Banach module over the $\breve{K}$-Banach algebra $\Lambda(\Gamma_{*})_{\breve{K},l}$.\vspace{2.6mm}\\
\noindent Taking projective limits over $l$, we obtain a continuous map $D(\Gamma_{*},\breve{K})\times R^{\textnormal{rig}}_{0}\longrightarrow R^{\textnormal{rig}}_{0}$ giving $R^{\textnormal{rig}}_{0}$ the structure of a continuous module over $D(\Gamma_{*},\breve{K})$. Because of the topological isomorphism (\ref{top_iso_of_distribution_alg}), $R^{\textnormal{rig}}_{0}$ becomes a continuous module over $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$. To see that the $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$-action on $R^{\textnormal{rig}}_{0}$ factors through a continuous action of $D(\Gamma,\breve{K})$, it suffices from (\ref{description_of_I}) to check that $i(t\mathfrak{x})(f)=ti(\mathfrak{x})(f)$ for all $t\in K$, $\mathfrak{x}\in\mathfrak{g}_{\mathbb{Q}_{p}}\subseteq D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$ and $f\in R^{\textnormal{rig}}_{0}$. However by Theorem \ref{laofGonMD}, this holds for all $f\in \mathcal{O}_{X^{\text{rig}}_{0}}(D)$ because $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$, being a locally $K$-analytic $\Gamma$-representation, carries an action of the Lie algebra $\mathfrak{g}$. As the $K$-linear inclusion $R^{\text{rig}}_{0}\hookrightarrow \mathcal{O}_{X^{\text{rig}}_{0}}(D)$ is continuous, it is $\mathfrak{g}_{\mathbb{Q}_{p}}$-equivariant. Hence the equality $i(t\mathfrak{x})(f)=ti(\mathfrak{x})(f)$ is true for all $f\in R^{\text{rig}}_{0}$. \vspace{2.3mm}\\
\noindent Now it follows from \cite{schnfa}, Proposition 19.9 and the arguments proving the claim on page 98, that the $\breve{K}$-Fr\'{e}chet space $R^{\textnormal{rig}}_{0}$ is nuclear. Therefore, Theorem \ref{stantiequivalence} implies that the locally convex $\breve{K}$-vector space $(R^{\textnormal{rig}}_{0})'_{b}$ is of compact type and that the action of $\Gamma$ obtained by dualizing is locally $K$-analytic.
\end{proof}
\noindent The preceding theorem can be generalized as follows. Let $\Gamma_{*}$ be a uniform pro-$p$ group contained in $\Gamma_{2n+1}$ for some positive integer $n$.
\begin{theorem}\label{genlaK0}
The action of $\Gamma_{\mathbb{Q}_{p}}$ on $M^{s}_{0}$ extends to a continuous action of the $\breve{K}$-Fr\'{e}chet algebra $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$, which then factors through a continuous action of $D(\Gamma,\breve{K})$ on $M^{s}_{0}$. Hence the action of $\Gamma$ on the strong continuous $\breve{K}$-linear dual $(M^{s}_{0})'_{b}$ of $M^{s}_{0}$ is locally $K$-analytic for any $s\in\mathbb{Z}$.
\end{theorem}
\begin{proof}
Choose a generator $\delta$ of the $R_{0}$-module $\textnormal{Lie}(\mathbb{H}^{(0)})^{\otimes s}$. Then by (\ref{G_eqv_iso_eqn1}) and (\ref{G_eqv_iso_eqn3}), $M^{s}_{0}=R^{\textnormal{rig}}_{0}\delta$ and $M^{s}_{0,l}=R^{\textnormal{rig}}_{0,l}\delta$. The topology on $M^{s}_{0,l}$ is defined by the norm $\| f\delta\|_{l}:=\|f\|_{l}$. Let $\gamma(\delta)=f_{0}\delta$ then by $\Gamma$-equivariance, we have $\gamma(f\delta)=\gamma(f)\gamma(\delta)=\gamma(f)f_{0}\delta$ for all $f\delta\in M^{s}_{0}$. Hence \begin{equation}\label{simplifying_eqn1}
\gamma(f\delta)-f\delta=(\gamma(f)f_{0}-f)\delta=(\gamma(f)f_{0}-ff_{0}+ff_{0}-f)\delta=((\gamma(f)-f)f_{0}+f(f_{0}-1))\delta
\end{equation} Now if $\gamma\in\Gamma_{*}\subseteq\Gamma_{2n+1}$ and if $f\delta\in M^{s}_{0}$, then $\|\gamma(f)-f\|_{l}\leq |\varpi|^{\frac{2n+1}{l}}\|f\|_{l}$ by Proposition \ref{mainprop} and $\gamma(\delta)-\delta=(f_{0}-1)\delta\in\mathfrak{m}^{n+1}_{R_{0}}\textnormal{Lie}(\mathbb{H}^{(0)})^{\otimes s}$ by Theorem \ref{ctsthm2} i.e. $f_{0}-1\in\mathfrak{m}_{R_{0}}^{n+1}$. Since $\| y\|_{l}\leq\vert\varpi\vert^{1/l}$ for any $y\in\mathfrak{m}_{R_{0}}=(\varpi,u_{1},\cdots,u_{h-1})$, $\|f_{0}-1\|_{l}\leq|\varpi|^{\frac{n+1}{l}}$ and $\|f_{0}\|_{l}\leq\max\lbrace\|f_{0}-1\|_{l},1\rbrace=1$.
Thus by the multiplicativity of the norm $\|\cdot\|_{l}$ on $R^{\textnormal{rig}}_{0}$ and by (\ref{simplifying_eqn1}), we have \begin{align*}
\|\gamma(f\delta)-f\delta\|_{l}=\|(\gamma(f)-f)f_{0}+f(f_{0}-1)\|_{l}&\leq\max\lbrace\|(\gamma(f)-f)\|_{l}\|f_{0}\|_{l},\|f\|_{l}\|(f_{0}-1)\|_{l}\rbrace\\&\leq\max\lbrace|\varpi|^{\frac{2n+1}{l}}\|f\|_{l},|\varpi|^{\frac{n+1}{l}}\|f\|_{l}\rbrace\\&=|\varpi|^{\frac{n+1}{l}}\|f\|_{l}=|\varpi|^{\frac{n+1}{l}}\|f\delta\|_{l}
\end{align*}
The rest of the proof now proceeds along the same lines as for Theorem \ref{laK0}.
\end{proof}
\subsection{Local analyticity of the $\Gamma$-action on $M^{s}_{m}$ with $m>0$}
\label{step3}
\noindent As the title indicates, this subsection extends the theorems of the previous subsection to higher levels $m>0$. The following observation together with the continuity of the $\Gamma$-action on $R_{m}$ and on $R^{\text{rig}}_{0}$ (cf. Theorem \ref{ctsthm} and Proposition \ref{mainprop} respectively) allows us to show the continuity of the $\Gamma$-action on $R^{\text{rig}}_{m}$ for $m>0$.
\begin{lemma}\label{km} For every $m\geq 0$, there exists a positive integer $k_{m}$ such that $\mathfrak{m}_{R_{m}}^{n}\subseteq\mathfrak{m}_{R_{0}}R_{m}$ for all $n\geq k_{m}$.
\end{lemma}
\begin{proof}
Since $R_{m}$ is a finite free module over $R_{0}$, $R_{m}/\mathfrak{m}_{R_{0}}R_{m}$ is a finite dimensional vector space over $R_{0}/\mathfrak{m}_{R_{0}}=k^{\textnormal{sep}}$. Moreover, $R_{m}/\mathfrak{m}_{R_{0}}R_{m}$ is still a Noetherian local ring with the maximal ideal $\mathfrak{m}_{R_{m}}/\mathfrak{m}_{R_{0}}R_{m}$. The powers $(\mathfrak{m}_{R_{m}}/\mathfrak{m}_{R_{0}}R_{m})^{n}$, $n\in\mathbb{N}$, of the ideal $\mathfrak{m}_{R_{m}}/\mathfrak{m}_{R_{0}}R_{m}$ form a descending sequence of finite dimensional subspaces which eventually must become stationary. Let $k_{m}$ be a positive integer such that $(\mathfrak{m}_{R_{m}}/\mathfrak{m}_{R_{0}}R_{m})^{n+1}=(\mathfrak{m}_{R_{m}}/\mathfrak{m}_{R_{0}}R_{m})^{n}$ for all $n\geq k_{m}$. Then by Nakayama's lemma, for all $n\geq k_{m}$, $(\mathfrak{m}_{R_{m}}/\mathfrak{m}_{R_{0}}R_{m})^{n}=0$, in other words, $\mathfrak{m}_{R_{m}}^{n}\subseteq\mathfrak{m}_{R_{0}}R_{m}$.
\end{proof}
\begin{proposition}\label{mainprop2} Let $m$, $n$ and $l$ be integers with $m\geq 1$, $l\geq 1$ and $n\geq k_{m}-1$ where $k_{m}$ is as stated in Lemma \ref{km}. If $\gamma\in\Gamma_{n+m}$ and if $f\in R^{\textnormal{rig}}_{m}$, then $\|\gamma(f)-f\|_{l}\leq\vert \varpi\vert^{1/l}\| f\|_{l}$.
\end{proposition}
\begin{proof} Write $f=f_{1}e_{1}+\cdots +f_{r}e_{r}$ where $\lbrace e_{1},\cdots, e_{r}\rbrace$ is a basis of $R_{m}$ over $R_{0}$ and $f_{i}\in R^{\textnormal{rig}}_{0}$ for all $1\leq i\leq r$. Let $x_{i}:=\gamma(e_{i})-e_{i}$. Then $x_{i}\in\mathfrak{m}_{R_{m}}^{n+1}$ for all $1\leq i\leq r$ by Theorem \ref{ctsthm} and thus by Lemma \ref{km}, $x_{i}\in\mathfrak{m}_{R_{0}}R_{m}$ for all $1\leq i\leq r$. Since $\| y\|_{l}\leq\vert\varpi\vert^{1/l}$ for any $y\in\mathfrak{m}_{R_{0}}=(\varpi,u_{1},\cdots,u_{h-1})$, $\| x_{i}\|_{l}\leq\vert\varpi\vert^{1/l}$ for all $1\leq i\leq r$. Now note that
\begin{align*}
\|\gamma(f)-f\|_{l}&\leq\max_{1\leq i\leq r}\lbrace\|\gamma(f_{i}e_{i})-f_{i}e_{i}\|_{l}\rbrace =\max_{1\leq i\leq r}\lbrace\|\gamma(f_{i})\gamma(e_{i})-f_{i}e_{i}\|_{l}\rbrace\\&=\max_{1\leq i\leq r}\lbrace\|\gamma(f_{i})\gamma(e_{i})-f_{i}\gamma(e_{i})+f_{i}\gamma(e_{i})-f_{i}e_{i}\|_{l}\rbrace \\&=\max_{1\leq i\leq r}\lbrace\|(\gamma(f_{i})-f_{i})\gamma(e_{i})+(\gamma(e_{i})-e_{i})f_{i}\|_{l}\rbrace\\& =\max_{1\leq i\leq r}\lbrace\|(\gamma(f_{i})-f_{i})(e_{i}+x_{i})+x_{i}f_{i}\|_{l}\rbrace
\end{align*}
Then Lemma \ref{lnorm_is_algebra_norm} and Proposition \ref{mainprop} imply that for every $1\leq i\leq r$,
\begin{align*}\|(\gamma(f_{i})-f_{i})(e_{i}+x_{i})+x_{i}f_{i}\|_{l}&\leq\max\lbrace\|(\gamma(f_{i})-f_{i})(e_{i}+x_{i})\|_{l},\| x_{i}f_{i}\|_{l}\rbrace\\&\leq\max\lbrace\|(\gamma(f_{i})-f_{i})\|_{l},\| x_{i}\|_{l}\| f_{i}\|_{l}\rbrace\\&\leq\max\lbrace\vert\varpi\vert^{(n+m)/l}\| f_{i}\|_{l},\vert\varpi\vert^{1/l}\| f_{i}\|_{l}\rbrace=\vert\varpi\vert^{1/l}\| f_{i}\|_{l}\end{align*} where we use that $e_{i}+x_{i}=\gamma(e_{i})\in R_{m}$ has $\|\cdot\|_{l}$-norm less than or equal to 1.
Therefore,
$\|\gamma(f)-f\|_{l}\leq\max_{1\leq i\leq r}\lbrace\vert\varpi\vert^{1/l}\| f_{i}\|_{l}\rbrace=\vert \varpi\vert^{1/l}\| f\|_{l}$.
\end{proof} \noindent We now fix a level $m\geq 1$. As before, we have a uniform pro-$p$ group $\Gamma_{o}$ of rank $t$ as an open subgroup of $\Gamma_{\mathbb{Q}_{p}}$. We also fix a positive integer $n\geq k_{m}-1$ such that $\Gamma_{n+m}\subseteq\Gamma_{o}$. Then $\Gamma_{n+m}$ contains $\Gamma_{*}:=P_{i}(\Gamma_{o})$ for some $i\geq 1$ which is also a uniform pro-$p$ group of rank $t$. \vspace{2.6mm}\\
\noindent Let $\lbrace\gamma_{1},\dots ,\gamma_{t}\rbrace$ be an ordered basis of $\Gamma_{*}$ and let $b_{i}:=\gamma_{i}-1\in\Lambda(\Gamma_{*})$. Then as before, we equip the $\breve{K}$-algebra $\Lambda(\Gamma_{*})_{\breve{K}}$ with the sub-multiplicative norm $\|\cdot\|_{l}$ defined in (\ref{lnorm}) for every positive integer $l$. The natural inclusions $\Lambda(\Gamma_{*})_{\breve{K},l+1}\hookrightarrow\Lambda(\Gamma_{*})_{\breve{K},l}$ of $\breve{K}$-Banach completions endow the projective limit $D(\Gamma_{*},\breve{K})=\varprojlim_{l}\Lambda(\Gamma_{*})_{\breve{K},l}$ with the structure of a $\breve{K}$-Fr\'{e}chet algebra which is equal to the algebra of $\breve{K}$-valued locally $K$-analytic distributions on $\Gamma_{*}$.
\begin{proposition}\label{lam} For any integer $l\geq 1$, the action of $\Gamma_{*}$ on $R^{\textnormal{rig}}_{m}$ extends to $R^{\textnormal{rig}}_{m,l}$ and makes $R^{\textnormal{rig}}_{m,l}$ a topological Banach module over the $\breve{K}$-Banach algebra $\Lambda(\Gamma_{*})_{\breve{K},l}$. The action of $\Gamma_{\mathbb{Q}_{p}}$ on $R^{\textnormal{rig}}_{m}$ extends to a continuous action of the $\breve{K}$-Fr\'{e}chet algebra $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$.
\end{proposition}
\begin{proof} The proof is similar to that of Theorem \ref{laK0}. First, we prove by induction on $\vert \alpha\vert$ that $\| b^{\alpha}(f)\|_{l}\leq\| b^{\alpha}\|_{l}\| f\|_{l}$ for any $f\in R^{\textnormal{rig}}_{m}$. This is clear if $\vert \alpha\vert=0$. Let $|\alpha|>0$ and let $i$ be the minimal index such that $\alpha_{i}>0$. Define $\beta_{j}:=\alpha_{j}$ if $j\neq i$ and $\beta_{i}:=\alpha_{i}-1$. Then Proposition \ref{mainprop2} and the induction hypothesis imply \begin{align*}
\| b^{\alpha}(f)\|_{l}&=\| ((\gamma_{i}-1)b^{\beta})(f)\|_{l}=\| (\gamma_{i}-1)(b^{\beta}(f))\|_{l}\leq\vert\varpi\vert^{1/l}\| b^{\beta}(f)\|_{l}\\&\leq\vert\varpi\vert^{1/l}\| b^{\beta}\|_{l}\| f\|_{l}\leq\vert\varpi\vert^{1/l}\vert\varpi\vert^{\vert \beta\vert /l}\| f\|_{l}=\vert\varpi\vert^{(\vert \beta\vert+1) /l}\| f\|_{l}=\| b^{\alpha}\|_{l}\| f\|_{l}
\end{align*} as required. By Remark \ref{iwasawa_alg_action}, this immediately gives $\|\mu(f)\|_{l}\leq\|\mu\|_{l}\| f\|_{l}$ for all $\mu\in\Lambda(\Gamma_{*})_{\breve{K}}$ and $f\in R_{m}[\frac{1}{\varpi}]=R_{m}\otimes_{\breve{\mathfrak{o}}}\breve{K}$. Hence the map $\Lambda(\Gamma_{*})_{\breve{K}}\times R_{m}[\frac{1}{\varpi}]\longrightarrow R_{m}[\frac{1}{\varpi}]$ $((\mu,f)\mapsto\mu(f))$ is continuous if $\Lambda(\Gamma_{*})_{\breve{K}}$ and $R_{m}[\frac{1}{\varpi}]$ are endowed with the respective $\|\cdot\|_{l}$-topologies and if the left hand side carries the product topology. Since $R_{m}[\frac{1}{\varpi}]$ is dense in $R^{\textnormal{rig}}_{m,l}$, we obtain a map $\Lambda(\Gamma_{*})_{\breve{K},l}\times R^{\textnormal{rig}}_{m,l}\longrightarrow R^{\textnormal{rig}}_{m,l}$ by passing to completions. By continuity, it gives $R^{\textnormal{rig}}_{m,l}$ the structure of a topological Banach module over the $\breve{K}$-Banach algebra $\Lambda(\Gamma_{*})_{\breve{K},l}$.\vspace{2.6mm}\\
\noindent Taking projective limits over $l$, we obtain a continuous map $D(\Gamma_{*},\breve{K})\times R^{\textnormal{rig}}_{m}\longrightarrow R^{\textnormal{rig}}_{m}$ giving $R^{\textnormal{rig}}_{m}$ the structure of a continuous module over $D(\Gamma_{*},\breve{K})$. Since $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$ is topologically isomorphic to the locally convex direct sum $\bigoplus_{\gamma\Gamma_{*}\in \Gamma_{\mathbb{Q}_{p}}/\Gamma_{*}}\delta_{\gamma} D(\Gamma_{*},\breve{K})$, $R^{\textnormal{rig}}_{m}$ is a continuous module over $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$.
\end{proof}
\noindent We want to show that the $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$-action on $R^{\text{rig}}_{m}$ factors through a continuous $D(\Gamma,\breve{K})$-action. As mentioned in the Step 3, the idea is to use the local $K$-analyticity of the $\Gamma$-action on $R^{\text{rig}}_{0}$ obtained in Theorem \ref{laK0} and the property of \'{e}taleness of the extension $R^{\text{rig}}_{m}|R^{\text{rig}}_{0}$.\vspace{2.6mm}\\
\noindent For a ring homomorphism $A\to B$ of commutative unital rings, let $\textnormal{Der}_{A}(B,B)$ denote the $B$-module of $A$-linear derivations from $B$ to $B$, and let $\Omega_{B/A}$ denote the $B$-module of differentials of $B$ over $A$. Recall that $B$ is said to be \emph{formally smooth} over $A$ if for any $A$-algebra $C$, for any ideal $I\subset C$ satisfying $I^{2}=0$ and for any $A$-algebra homomorphism $u:B\to C/I$, there exists a lifting $v:B\to C$ of $u$ making the following diagram commutative.
$$
\xymatrix{
B \ar[r]^{u} \ar@{-->}[rd]^{v} & C/I \\
A \ar[r] \ar[u] & C \ar[u]
}
$$
It is said to be \emph{formally unramified} if there exists atmost one such lifting, and \emph{formally \'{e}tale} if it is both formally smooth and formally unramified, i.e. if there exists a unique such lifting (cf. \cite[Tag 00UQ]{stacks-project}).
\begin{lemma}\label{formallyetale} Let $A\to B\to C$ be ring homomorphisms such that $B\to C$ is formally \'{e}tale. Then $\Omega_{B/A}\otimes_{B}C\simeq\Omega_{C/A}$.
\end{lemma}
\begin{proof}
Since $B\to C$ is formally smooth, we have $\Omega_{C/A}\simeq\Omega_{C/B}\oplus(\Omega_{B/A}\otimes_{B}C)$ by \cite[Tag 031K]{stacks-project}. However, $\Omega_{C/B}=0$ by \cite[Tag 00UO]{stacks-project} because $B\to C$ is formally unramified.
\end{proof}
\noindent Since $R_{m}[\frac{1}{\varpi}]$ is \'{e}tale over $R_{0}[\frac{1}{\varpi}]$ (cf. Remark \ref{Galois_action_on_LT_tower}), $R^{\textnormal{rig}}_{m}\simeq R_{m}[\frac{1}{\varpi}]\otimes_{R_{0}[\frac{1}{\varpi}]}R^{\textnormal{rig}}_{0}$ is \'{e}tale over $R^{\textnormal{rig}}_{0}$ by \cite[Tag 00U0]{stacks-project}, and so is formally \'{e}tale by \cite[Tag 00UR]{stacks-project}. Then by applying Lemma \ref{formallyetale} to the ring homomorphisms $\breve{K}\hookrightarrow R^{\textnormal{rig}}_{0}\hookrightarrow R^{\textnormal{rig}}_{m}$, we get $\Omega_{R^{\textnormal{rig}}_{m}/\breve{K}}\simeq\Omega_{R^{\textnormal{rig}}_{0}/\breve{K}}\otimes_{R^{\textnormal{rig}}_{0}}R^{\textnormal{rig}}_{m}$. Therefore \begin{align*}
\textnormal{Der}_{\breve{K}}(R^{\textnormal{rig}}_{0},R^{\textnormal{rig}}_{0})&\simeq\textnormal{Hom}_{R^{\textnormal{rig}}_{0}}(\Omega_{R^{\textnormal{rig}}_{0}/\breve{K}},R^{\textnormal{rig}}_{0})\\&\hookrightarrow\textnormal{Hom}_{R^{\textnormal{rig}}_{0}}(\Omega_{R^{\textnormal{rig}}_{0}/\breve{K}},R^{\textnormal{rig}}_{m})\\&\simeq\textnormal{Hom}_{R^{\textnormal{rig}}_{m}}(\Omega_{R^{\textnormal{rig}}_{0}/\breve{K}}\otimes_{R^{\textnormal{rig}}_{0}}R^{\textnormal{rig}}_{m},R^{\textnormal{rig}}_{m})\\&\simeq\textnormal{Hom}_{R^{\textnormal{rig}}_{m}}(\Omega_{R^{\textnormal{rig}}_{m}/\breve{K}},R^{\textnormal{rig}}_{m})\simeq\textnormal{Der}_{\breve{K}}(R^{\textnormal{rig}}_{m},R^{\textnormal{rig}}_{m}).
\end{align*} In other words,
\begin{lemma}\label{derivationextensionlemma} Any $\breve{K}$-linear derivation from $R^{\textnormal{rig}}_{0}$ to $R^{\textnormal{rig}}_{0}$ extends uniquely to a $\breve{K}$-linear derivation from $R^{\textnormal{rig}}_{m}$ to $R^{\textnormal{rig}}_{m}$.
\hspace{10.85cm}\qedsymbol
\end{lemma}
\begin{theorem}\label{laKm}
The action of $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$ on $R^{\textnormal{rig}}_{m}$ factors through a continuous action of $D(\Gamma,\breve{K})$ on $R^{\textnormal{rig}}_{m}$. Hence the action of $\Gamma$ on the strong continuous $\breve{K}$-linear dual $(R^{\textnormal{rig}}_{m})'_{b}$ of $R^{\textnormal{rig}}_{m}$ is locally $K$-analytic.
\end{theorem}
\begin{proof}
Recall the inclusion map $i:\mathfrak{g}_{\mathbb{Q}_{p}}\hookrightarrow D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$ from (\ref{description_of_I}). Because of Theorem \ref{lam}, for every $\mathfrak{x}\in\mathfrak{g}_{\mathbb{Q}_{p}}$, $i(\mathfrak{x})\in D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$ acts on $R^{\textnormal{rig}}_{m}$ by $\breve{K}$-linear vector space endomorphism and for any $f, g\in R^{\textnormal{rig}}_{m}$, we have \begin{align*}
i(\mathfrak{x})(fg)=\mathfrak{x}(fg)&=\frac{d}{dt}\exp(t\mathfrak{x})(fg)\big\vert_{t=0}\\&=\lim_{t\to 0}\frac{\exp(t\mathfrak{x})(fg) -fg}{t}\\&=\lim_{t\to 0}\frac{\exp(t\mathfrak{x})(f)\exp(t\mathfrak{x})(g) -fg}{t}\\&=\lim_{t\to 0}\frac{\exp(t\mathfrak{x})(f)\exp(t\mathfrak{x})(g)-g\exp(t\mathfrak{x})(f)+g\exp(t\mathfrak{x})(f)-fg}{t}\\&=\lim_{t\to 0}\exp(t\mathfrak{x})(f)\Big(\frac{\exp(t\mathfrak{x})(g)-g}{t}\Big)+\lim_{t\to 0}g\Big(\frac{\exp(t\mathfrak{x})(f)-f}{t}\Big)\\&=f\mathfrak{x}(g)+g\mathfrak{x}(f)=fi(\mathfrak{x})(g)+gi(\mathfrak{x})(f)
\end{align*} Therefore we have a natural map $\partial:i(\mathfrak{g}_{\mathbb{Q}_{p}})\longrightarrow\textnormal{Der}_{\breve{K}}(R^{\textnormal{rig}}_{m},R^{\textnormal{rig}}_{m})$. For $t\in K$ and $\mathfrak{x}\in\mathfrak{g}_{\mathbb{Q}_{p}}$, consider the derivation $\partial(i(t\mathfrak{x})-ti(\mathfrak{x}))\in\textnormal{Der}_{\breve{K}}(R^{\textnormal{rig}}_{m},R^{\textnormal{rig}}_{m})$. It is zero on $R^{\textnormal{rig}}_{0}$ by Theorem \ref{laK0} and thus by Lemma \ref{derivationextensionlemma}, it is zero on $R^{\textnormal{rig}}_{m}$ i.e. we have \begin{equation*}
0=\partial(i(t\mathfrak{x})-ti(\mathfrak{x}))(f)=(i(t\mathfrak{x})-ti(\mathfrak{x}))(f)
\end{equation*} for all $t\in K$, $\mathfrak{x}\in\mathfrak{g}_{\mathbb{Q}_{p}}$ and $f\in R^{\textnormal{rig}}_{m}$. This means that the action of $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$ on $R^{\textnormal{rig}}_{m}$ factors through a continuous action of $D(\Gamma,\breve{K})$ on $R^{\textnormal{rig}}_{m}$ by (\ref{description_of_I}).\vspace{2.6mm}\\
\noindent As the $\breve{K}$-Fr\'{e}chet space $R^{\textnormal{rig}}_{m}$ is topologically isomorphic to $\bigoplus_{i=1}^{r} R^{\textnormal{rig}}_{0}$, it follows from \cite{schnfa}, Proposition 19.7, that $R^{\textnormal{rig}}_{m}$ is nuclear. Therefore, Theorem \ref{stantiequivalence} implies that the locally convex $\breve{K}$-vector space $(R^{\textnormal{rig}}_{m})'_{b}$ is of compact type and that the action of $\Gamma$ obtained by dualizing is locally $K$-analytic.
\end{proof}
\noindent Similar to the before, Theorem \ref{laKm} can be generalized as follows. Fix $m\geq 1$, $n\geq k_{m}-1$ and a uniform pro-$p$ group $\Gamma_{*}\subseteq\Gamma_{2n+m+1}$ with $k_{m}$ as in Lemma \ref{km}.
\begin{theorem}\label{genlaKm}
The action of $\Gamma_{\mathbb{Q}_{p}}$ on $M^{s}_{m}$ extends to a continuous action of the $\breve{K}$-Fr\'{e}chet algebra $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$, which then factors through a continuous action of $D(\Gamma,\breve{K})$ on $M^{s}_{m}$. Hence the action of $\Gamma$ on the strong continuous $\breve{K}$-linear dual $(M^{s}_{m})'_{b}$ of $M^{s}_{m}$ is locally $K$-analytic for any $s\in\mathbb{Z}$.
\end{theorem}
\begin{proof}
Using Theorem \ref{ctsthm2}, Lemma \ref{km} and Proposition \ref{mainprop2}, the proof of the first part of the assertion is similar to that of Theorem \ref{genlaK0}.\vspace{2.6mm}\\
\noindent Observe that the isomorphism $\text{Lie}(\mathbb{H}^{(m)})\simeq\text{Lie}(\mathbb{H}^{(0)})\otimes_{R_{0}}R_{m}$ is $\Gamma$ -equivariant for the diagonal $\Gamma$-action on the right. Therefore, we have the following $\Gamma$-equivariant isomorphisms by (\ref{G_eqv_iso_eqn1})
\begin{align*}
M^{s}_{m}&\simeq R^{\text{rig}}_{m}\otimes_{R_{m}}\textnormal{Lie}(\mathbb{H}^{(m)})\\&\simeq R^{\text{rig}}_{m}\otimes_{R_{0}}\textnormal{Lie}(\mathbb{H}^{(0)})\\&\simeq R^{\text{rig}}_{m}\otimes_{R^{\text{rig}}_{0}}R^{\text{rig}}_{0}\otimes_{R_{0}}\textnormal{Lie}(\mathbb{H}^{(0)})\\&\simeq R^{\text{rig}}_{m}\otimes_{R^{\text{rig}}_{0}}M^{s}_{0}
\end{align*} with $\Gamma$-acting diagonally on all the tensor products. As a consequence, the $\mathfrak{g}_{\mathbb{Q}_{p}}$-action on $f\otimes\delta\in M^{s}_{m}$ is given by $\mathfrak{x}(f\otimes\delta)=\mathfrak{x}(f)\otimes\delta+f\otimes\mathfrak{x}(\delta)$. However, by Theorem \ref{genlaK0} and Theorem \ref{laKm}, $M^{s}_{0}$ and $R^{\text{rig}}_{m}$ are not only $\mathfrak{g}_{\mathbb{Q}_{p}}$-modules but also $\mathfrak{g}$-modules. Thus, it follows from (\ref{description_of_I}) that the $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$-action on $M^{s}_{m}$ factors through the continuous $D(\Gamma,\breve{K})$-action as required.
\end{proof}
\begin{remark} By Theorem \ref{genlaK0} and Theorem \ref{genlaKm}, we have a continuous action of $D(\Gamma_{\mathbb{Q}_{p}},\breve{K})$ on $M^{s}_{m,l}$ via the natural map $D(\Gamma_{*},\breve{K})\longrightarrow\Lambda(\Gamma_{*})_{\breve{K},l}$ for all $s\in\mathbb{Z}$, $m\geq 0$, $l\geq 1$. This action factors through a continuous action of $D(\Gamma,\breve{K})$ by the similar arguments as in the proofs of Theorem \ref{genlaK0} and Theorem \ref{genlaKm}. Therefore, the $\breve{K}$-Banach space $M^{s}_{m,l}$ is a locally $K$-analytic $\Gamma$-representation for all $s\in\mathbb{Z}$, $m\geq 0$, $l\geq 1$ by \cite{stladist}, discussion before Corollary 3.3.
\end{remark}
\section{Locally finite vectors in the global sections of equivariant vector bundles}
\noindent In this section, we study representation-theoretic aspects of our $\Gamma$-representations $M^{s}_{m}$. Our results include a complete description of the $\Gamma$-locally finite vectors in $M^{s}_{m}$ for all $s\in\mathbb{Z}$ and $m\geq 0$.
\subsection{Locally finite vectors in the $\Gamma$-representations $M^{s}_{0}$}
\begin{definition} Let $G$ be a topological group and $V$ be a vector space over a field $F$ equipped with an $F$-linear $G$-action. We say that a vector $v\in V$ is \emph{locally finite} (or \emph{$G$-locally finite}) if there is an open subgroup $H$ of $G$ and a finite dimensional $H$-stable subspace $W$ of $V$ containing $v$.
It follows easily that the set $V_{\textnormal{lf}}$ of all locally finite vectors of $V$ forms a $G$-stable subspace. $V$ is called a locally finite representation of $G$ if $V_{\textnormal{lf}}=V$. If $V$ and $W$ are $F$-linear $G$-representations, and if $f:V\longrightarrow W$ is an $F$-linear $G$-equivariant map, then clearly $f(V_{\textnormal{lf}})\subseteq W_{\textnormal{lf}}$.
\end{definition}
\begin{remark}\label{emerton_defn} In \cite{eme04}, Proposition-Definition 4.1.8, the notion of a locally finite vector is defined for the vector spaces over a complete non-archimedean field $F$, and requires locally finite vector $v$ to be contained in a \emph{continuous} finite dimensional $H$-representation $W$ for its natural Hausdorff topology as a finite dimensional $F$-vector space. Since all the $\Gamma$-representations we are concerned with in this section are continuous representations on $\breve{K}$-Fr\'{e}chet spaces, the continuity condition is automatically satisfied. Thus, the above definition without the continuity condition coincides with the one in \cite{eme04} for our representations.
\end{remark}
\noindent To calculate locally finite vectors, we make extensive use of the Lie algebra action. Let $\mathfrak{g}$ be the Lie algebra of the Lie group $\Gamma$ over $K$ and $U(\mathfrak{g})$ be its universal enveloping algebra over $K$. Note that $\mathfrak{g}$ is isomorphic to the Lie algebra associated with the associative $K$-algebra $B_{h}$. Thus, $\mathfrak{g}\hookrightarrow\mathfrak{g}\otimes_{K}K_{h}\simeq\mathfrak{gl}_{h}(K_{h})$. We denote by $\mathfrak{x}_{ij}\in\mathfrak{gl}_{h}(K_{h})$ the matrix with entry 1 at the place $(i,j)$ and zero everywhere else. By Theorem \ref{laofGonMD}, the $\Gamma$-representation $M^{s}_{D}\simeq\mathcal{O}_{X^{\text{rig}}_{0}}(D)\varphi_{0}^{s}$ carries a continuous linear action of $U(\mathfrak{g})\otimes_{K}\breve{K}\simeq U(\mathfrak{gl}_{h}(K_{h}))\otimes_{K_{h}}\breve{K}\hookrightarrow D(\Gamma,\breve{K})$. Since $GL_{h}(K_{h})$ acts on the projective coordinates $\varphi_{0},\ldots,\varphi_{h-1}$ by fractional linear transformations, one can explicitly determine the Lie algebra action directly using the definition (\ref{g_action_eqn}). Here we use the description of $\mathcal{O}_{X^{\text{rig}}_{0}}(D)$ in terms of power series in $w$ as in (\ref{expression_of_O(D)}).
\begin{lemma}\label{explicit_g-action_lemma} Let $i$, $j$ and $s$ be integers with $0\leq i,j\leq h-1$. Put $w_{0}:=1$. If $f\in\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)$ then
\begin{equation}\label{explicit_g-action_formulae}
\mathfrak{x}_{ij}(f\varphi^{s}_{0})=
\begin{cases}
w_{i}\frac{\partial f}{\partial w_{j}}\varphi^{s}_{0}, &\text{if $j\neq 0$;}\\
(sf-\sum_{l=1}^{h-1}w_{l}\frac{\partial f}{\partial w_{l}})\varphi^{s}_{0}, &\text{if $i=j=0$;}\\
w_{i}(sf-\sum_{l=1}^{h-1}w_{l}\frac{\partial f}{\partial w_{l}})\varphi^{s}_{0}, &\text{if $i>j=0$.}
\end{cases}
\end{equation}
\end{lemma}
\begin{proof}
This is exactly same as \cite{kohliwamo}, Lemma 4.1, which treats the case $K=\mathbb{Q}_{p}$.
\end{proof}
\noindent Given a Lie subalgebra $\mathfrak{h}\subseteq\mathfrak{gl}_{h}(K_{h})$, and a $\breve{K}$-linear $\mathfrak{gl}_{h}(K_{h})$-representation $W$, the $\breve{K}$-subspace of \emph{$\mathfrak{h}$-invariants of $W$} is the subspace \textnormal{$\lbrace w\in W|\mathfrak{x}(w)=0$ for all $\mathfrak{x}\in\mathfrak{h}\rbrace$} of $W$. Let us denote by $\mathfrak{n}$ the Lie subalgebra of $\mathfrak{gl}_{h}(K_{h})$ consisting of strictly upper triangular matrices. For later use, we calculate the $\mathfrak{g}$-invariants and the $\mathfrak{n}$-invariants of $\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)$ in the next lemma using the formulae (\ref{explicit_g-action_formulae}).\vspace{2mm}
\begin{lemma}\label{gninvO(D)}
$\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)^{\mathfrak{g}=0}=\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)^{\mathfrak{n}=0}=\breve{K}$.
\end{lemma}
\begin{proof}
Since $\breve{K}\subseteq\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)^{\mathfrak{g}=0}\subseteq\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)^{\mathfrak{n}=0}$, it suffices to show that the latter is $\breve{K}$. Now if $f\in\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)^{\mathfrak{n}=0}$ then applying the formulae (\ref{explicit_g-action_formulae}), we get $\mathfrak{x}_{0j}(f)=\frac{\partial f}{\partial w_{j}}=0$ for all $1\leq j\leq h-1$. Therefore, $f$ must be a constant power series.
\end{proof}
\vspace{4mm}
\noindent We now compute the space $(M^{s}_{D})_{\text{lf}}$ of locally finite vectors in the $\Gamma$-representation $M^{s}_{D}$. The key step is the following lemma based on Lemma \ref{explicit_g-action_lemma}:
\begin{lemma}\label{polysubspace_contained_in_homopolygensubspace} The subspace $\breve{K}[w_{1},\dots ,w_{h-1}]\varphi^{s}_{0}$ of $M^{s}_{D}$ is contained in $(U(\mathfrak{g})\otimes_{K}\breve{K})(f\varphi_{0}^{s})$ for any non-zero homogeneous polynomial $f\in\breve{K}[w_{1},\dots ,w_{h-1}]$ of total degree $d>s$.
\end{lemma}
\begin{proof} Using (\ref{explicit_g-action_formulae}), we have for all $0<i,j\leq h-1$ \begin{align*}
&\mathfrak{x}_{0j}(f\varphi^{s}_{0})=\frac{\partial f}{\partial w_{j}}\varphi^{s}_{0}\\
&\mathfrak{x}_{i0}(f\varphi^{s}_{0})=(s-d)w_{i}f\varphi_{0}^{s}.
\end{align*}
To obtain $g\varphi^{s}_{0}$ with a monomial $g$ of total degree $\leq s$, first reduce $f\varphi^{s}_{0}$ to $\varphi^{s}_{0}$ by applying suitable $\mathfrak{x}_{0j}$ $(j\neq 0)$ to it iteratively and then apply appropriate $\mathfrak{x}_{i0}$ $(i>0)$ to $\varphi^{s}_{0}$ to get the desired element $g\varphi^{s}_{0}$. To obtain $g\varphi^{s}_{0}$ with a monomial $g$ of total degree $>s$, reverse the procedure i.e. first apply appropriate $\mathfrak{x}_{i0}$ $(i>0)$ to $f\varphi^{s}_{0}$ and then reduce the result to $g\varphi^{s}_{0}$ by applying suitable $\mathfrak{x}_{0j}$ $(j\neq 0)$ to it.
\end{proof}
\noindent Recall that, if $V$ is a topological vector space over a field $F$ with an $F$-linear action of a group $G$, then $V$ is called \textit{topologically irreducible} if it is non-zero and if it does not have a closed, non-trivial ($\neq \lbrace 0\rbrace, V$) $G$-stable subspace. For any $s\in\mathbb{Z}$, we define the $\breve{K}$-subspace of $M^{s}_{D}$ by \begin{equation*}
V_{s}:=\sum_{\vert \alpha\vert\leq s}\breve{K}w^{\alpha}\varphi^{s}_{0}.
\end{equation*} Note that we have $V_{s}=0$ if $s<0$.
\vspace{2mm}
\begin{theorem}\label{top_finiteness_of_MsD_thm} The $\Gamma$-representation $M^{s}_{D}$ is topologically irreducible if $s<0$, and if $s\geq 0$ then $V_{s}$ is a topologically irreducible sub-representation of $M^{s}_{D}$ with topologically irreducible quotient $M^{s}_{D}/V_{s}$.
\end{theorem}
\begin{proof}
\underline{Case $s<0$}: Let $V$ be a non-zero closed $\Gamma$-stable subspace of $M^{s}_{D}$. Let $f_{0}\varphi^{s}_{0}\in V$ where $f_{0}=\sum_{\alpha\in\mathbb{N}_{0}^{h-1}}c_{\alpha}w^{\alpha}\neq 0$ and $d$ $(\geq 0)$ be the smallest natural number such that $c_{\alpha}\neq 0$ for some $\alpha\in\mathbb{N}_{0}^{h-1}$ with $\vert \alpha\vert=d$. Thus, \begin{equation*}
f_{0}\varphi^{s}_{0}=\Bigg(\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}+\sum_{i=1}^{\infty}\sum_{\vert \alpha\vert =d+i}c_{\alpha}w^{\alpha}\Bigg)\varphi^{s}_{0}\in V
\end{equation*} where $\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}\neq 0$. For $n>0$, define a sequence of elements of $M^{s}_{D}$ inductively as follows: \begin{equation*}
f_{n}\varphi^{s}_{0}:=\frac{1}{n}\Big((d+n-s)f_{n-1}\varphi^{s}_{0}+\mathfrak{x}_{00}(f_{n-1}\varphi^{s}_{0})\Big).\end{equation*} Since $V$ is closed and $\Gamma$-stable, $V$ is stable under the action of the Lie algebra (\ref{g_action_eqn}) and thus $f_{n}\in V$ for all $n\in\mathbb{N}_{0}$.\vspace{2.6mm}\\
\noindent We prove by induction on $n$ that \begin{equation}\label{simplifying_eqn2}
f_{n}\varphi^{s}_{0}=\Bigg(\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}+\sum_{i=1}^{\infty}\sum_{\vert \alpha\vert =d+i}(-1)^{n}\binom{i-1}{n}c_{\alpha}w^{\alpha}\Bigg)\varphi^{s}_{0}
\end{equation} Here the generalized binomial coefficients are defined by $\binom{x}{n}:=\frac{x(x-1)\dots (x-n+1)}{n!}$ for any $x\in\mathbb{Z}$ and $n\in\mathbb{N}_{0}$. The case $n=0$ is true by definition. Assuming that (\ref{simplifying_eqn2}) holds for $n-1$, we compute using Lemma \ref{explicit_g-action_lemma} that
\begin{align*}
f_{n}\varphi^{s}_{0}&=\frac{1}{n}\Big((d+n-s)f_{n-1}\varphi^{s}_{0}+\mathfrak{x}_{00}(f_{n-1}\varphi^{s}_{0})\Big)\\&=\frac{1}{n}\Bigg((d+n-s)\Big(\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}+\sum_{i=1}^{\infty}\sum_{\vert \alpha\vert =d+i}(-1)^{n-1}\binom{i-1}{n-1}c_{\alpha}w^{\alpha}\Big)\varphi^{s}_{0}\\&\ \ \ \ \ \ \ \ +\mathfrak{x}_{00}\Bigg(\Big(\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}+\sum_{i=1}^{\infty}\sum_{\vert \alpha\vert =d+i}(-1)^{n-1}\binom{i-1}{n-1}c_{\alpha}w^{\alpha}\Big)\varphi^{s}_{0}\Bigg)\Bigg)\\&=\frac{1}{n}\Bigg((d+n-s)\Big(\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}+\sum_{i=1}^{\infty}\sum_{\vert \alpha\vert =d+i}(-1)^{n-1}\binom{i-1}{n-1}c_{\alpha}w^{\alpha}\Big)\varphi^{s}_{0}\\&\ \ \ \ \ \ \ \ +\Bigg(\sum_{\vert \alpha\vert =d}(s-d)c_{\alpha}w^{\alpha}+\sum_{i=1}^{\infty}\sum_{\vert \alpha\vert =d+i}(s-(d+i))(-1)^{n-1}\binom{i-1}{n-1}c_{\alpha}w^{\alpha}\Bigg)\varphi^{s}_{0}\Bigg)\\&=\Bigg(\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}+\sum_{i=1}^{\infty}\sum_{\vert \alpha\vert =d+i}(n-i)(-1)^{n-1}\frac{1}{n}\binom{i-1}{n-1}c_{\alpha}w^{\alpha}\Bigg)\varphi^{s}_{0}\\&=\Bigg(\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}+\sum_{i=1}^{\infty}\sum_{\vert \alpha\vert =d+i}(-1)^{n}\binom{i-1}{n}c_{\alpha}w^{\alpha}\Bigg)\varphi^{s}_{0}
\end{align*}
We now claim that, as $n\to\infty$, the sequence $f_{n}\varphi^{s}_{0}$ converges to $\Big(\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}\Big)\varphi^{s}_{0}$ with respect to the norm $\|\cdot\|_{M^{s}_{D}}$ defined in the proof of Theorem \ref{laofGonMD}: As $f_{0}\in\mathcal{O}(D)$, we know that given $\varepsilon >0$, there exists $N_{\varepsilon}\in\mathbb{N}$ such that for all $\alpha\in\mathbb{N}_{0}^{h-1}$ with $\vert \alpha\vert > N_{\varepsilon}$, we have $\vert c_{\alpha}\vert \vert\varpi\vert^{\sum_{i=1}^{h-1}\alpha_{i}(1-\frac{i}{h})}<\varepsilon$. Therefore, $\sup_{\vert \alpha\vert >N_{\varepsilon}}\vert c_{\alpha}\vert \vert\varpi\vert^{\sum_{i=1}^{h-1}\alpha_{i}(1-\frac{i}{h})}<\varepsilon$. Now for all $n>N_{\varepsilon}-d$, using (\ref{simplifying_eqn2}), we have \begin{align*}\Big\| f_{n}\varphi^{s}_{0}-\Big(\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}\Big)\varphi^{s}_{0}\Big\|_{M^{s}_{D}}&=\Big\|\Big(\sum_{i=1}^{\infty}\sum_{\vert \alpha\vert =d+i}(-1)^{n}\binom{i-1}{n}c_{\alpha}w^{\alpha}\Big)\varphi^{s}_{0}\Big\|_{M^{s}_{D}}\\&=\Big\|\sum_{i=1}^{\infty}\sum_{\vert \alpha\vert =d+i}(-1)^{n}\binom{i-1}{n}c_{\alpha}w^{\alpha}\Big\|_{D}\\&=\Big\|\sum_{i=n+1}^{\infty}\sum_{\vert \alpha\vert =d+i}(-1)^{n}\binom{i-1}{n}c_{\alpha}w^{\alpha}\Big\|_{D}\\&\leq\Big\|\sum_{\vert \alpha\vert >d+n}c_{\alpha}w^{\alpha}\Big\|_{D}\\&=\sup_{\vert \alpha\vert >d+n}\vert c_{\alpha}\vert \vert\varpi\vert^{\sum_{i=1}^{h-1}\alpha_{i}(1-\frac{i}{h})}\\&\leq\sup_{\vert \alpha\vert >N_{\varepsilon}}\vert c_{\alpha}\vert \vert\varpi\vert^{\sum_{i=1}^{h-1}\alpha_{i}(1-\frac{i}{h})}<\varepsilon
\end{align*} Hence, $f_{n}\varphi^{s}_{0}$ converges to $\Big(\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}\Big)\varphi^{s}_{0}$ as $n\to\infty$ and $\Big(\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}\Big)\varphi^{s}_{0}\in V$ because $V$ is closed. Since $\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}$ is a non-zero homogeneous polynomial of total degree $d\geq 0>s$, it follows from Lemma \ref{polysubspace_contained_in_homopolygensubspace} that $\breve{K}[w_{1},\dots ,w_{h-1}]\varphi^{s}_{0}\subseteq V$. Since $\breve{K}[w_{1},\dots ,w_{h-1}]\varphi^{s}_{0}$ is dense in $M^{s}_{D}=\mathcal{O}_{X^{\text{rig}}_{0}}(D)\varphi_{0}^{s}$ and $V$ is closed, it follows that $V=M^{s}_{D}$. Hence, $M^{s}_{D}$ is topologically irreducible for all $s<0$.\vspace{2.6mm}\\
\underline{Case $s\geq 0$}:
First we show that the finite dimensional subspace $V_{s}$ of $M^{s}_{D}$ is stable under the action of $\Gamma$:
It is sufficient to prove that $\gamma(w^{\alpha}\varphi^{s}_{0})\in V_{s}$ for any $w^{\alpha}\varphi^{s}_{0}$ with $\vert \alpha\vert\leq s$ and for any $\gamma=\sum_{i=0}^{h-1}\lambda_{i}\Pi^{i}\in\Gamma$. In this case, using the action of the matrix (\ref{Gamma_as_a_subgroup_of_GLh}) on the projective coordinates $[\varphi_{0}:\ldots:\varphi_{h-1}]$, we find \begin{align*} &\gamma(w^{\alpha}\varphi^{s}_{0})=\gamma(w_{1}^{\alpha_{1}}\dots w_{h-1}^{\alpha_{h-1}}\varphi^{s}_{0})=\gamma(\varphi_{1}^{\alpha_{1}}\dots \varphi_{h-1}^{\alpha_{h-1}}\varphi^{s-\vert \alpha\vert}_{0})=\gamma(\varphi_{1})^{\alpha_{1}}\dots\gamma(\varphi_{h-1})^{\alpha_{h-1}}\gamma(\varphi_{0})^{s-\vert \alpha\vert}\\&=(\varpi\lambda_{1}\varphi_{0}+\dots +\varpi\lambda_{2}^{\sigma^{h-1}}\varphi_{h-1})^{\alpha_{1}}\dots(\varpi\lambda_{h-1}\varphi_{0}+\dots +\lambda_{0}^{\sigma^{h-1}}\varphi_{h-1})^{\alpha_{h-1}}(\lambda_{0}\varphi_{0}+\dots +\lambda_{1}^{\sigma^{h-1}}\varphi_{h-1})^{s-\vert \alpha\vert}\\&=(\varpi\lambda_{1}+\dots +\varpi\lambda_{2}^{\sigma^{h-1}}w_{h-1})^{\alpha_{1}}\dots(\varpi\lambda_{h-1}+\dots +\lambda_{0}^{\sigma^{h-1}}w_{h-1})^{\alpha_{h-1}}(\lambda_{0}+\dots +\lambda_{1}^{\sigma^{h-1}}w_{h-1})^{s-\vert \alpha\vert}\varphi^{s}_{0}\\&\in V_{s}.
\end{align*}
\noindent Let $V$ be a non-zero closed $\Gamma$-stable subspace of $V_{s}$. Then $V$ is stable under the action of the Lie algebra and thus it becomes a module over $U(\mathfrak{g})\otimes_{K}\breve{K}$. As mentioned in the proof of Lemma \ref{polysubspace_contained_in_homopolygensubspace}, any non-zero element $f\varphi^{s}_{0}$ of $V$ can be reduced to $\varphi^{s}_{0}$ by applying suitable $\mathfrak{x}_{0j}$ $(j\neq 0)$ to it iteratively and then $\varphi^{s}_{0}$ can be converted into any monomial of total degree $\leq s$ multiplied with $\varphi^{s}_{0}$ by applying appropriate $\mathfrak{x}_{i0}$ $(i>0)$ to it. Therefore $V=V_{s}$ and $V_{s}$ is topologically irreducible.\vspace{2.6mm}\\
\noindent Now let $\phi :M^{s}_{D}\longrightarrow M^{s}_{D}/V_{s}$ be the canonical surjective map and let $W\subset M^{s}_{D}/V_{s}$ be a non-zero, closed $\Gamma$-stable subspace. Then $\phi^{-1}(W)$ is a non-zero, closed $\Gamma$-stable subspace of $M^{s}_{D}$ not equal to $V_{s}$. \vspace{2.6mm}\\
\noindent Let $\Big(\sum_{\alpha\in\mathbb{N}_{0}^{h-1}}c_{\alpha}w^{\alpha}\Big)\varphi^{s}_{0}+V_{s}$ be a non-zero element of $W$. Then $f_{0}\varphi^{s}_{0}:=\Big(\sum_{\vert \alpha\vert>s}c_{\alpha}w^{\alpha}\Big)\varphi^{s}_{0}\neq 0\in\phi^{-1}(W)$. Let $d>s$ be the smallest natural number such that $c_{\alpha}\neq 0$ for some $\alpha\in\mathbb{N}_{0}^{h-1}$ with $\vert \alpha\vert=d$. Thus, \begin{equation*}
f_{0}\varphi^{s}_{0}=\Bigg(\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}+\sum_{i=1}^{\infty}\sum_{\vert \alpha\vert =d+i}c_{\alpha}w^{\alpha}\Bigg)\varphi^{s}_{0}\in \phi^{-1}(W)
\end{equation*} where $\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}\neq 0$.\vspace{2.6mm}\\
\noindent As in the case of $s<0$, we define a sequence of elements in $\phi^{-1}(W)$ inductively for $n>0$ as follows: \begin{equation*}
f_{n}\varphi^{s}_{0}:=\frac{1}{n}\Big((d+n-s)f_{n-1}\varphi^{s}_{0}+\mathfrak{x}_{00}(f_{n-1}\varphi^{s}_{0})\Big).\end{equation*} Using exactly the same proof in the previous case of $s<0$, it can be shown that $f_{n}\varphi^{s}_{0}$ converges to $\Big(\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}\Big)\varphi^{s}_{0}$ as $n\to\infty$ and $\Big(\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}\Big)\varphi^{s}_{0}\in \phi^{-1}(W)$ because $\phi^{-1}(W)$ is closed. Since $\sum_{\vert \alpha\vert =d}c_{\alpha}w^{\alpha}$ is a non-zero homogeneous polynomial of total degree $d>s$, it follows from Lemma \ref{polysubspace_contained_in_homopolygensubspace} that $\breve{K}[w_{1},\dots ,w_{h-1}].\varphi^{s}_{0}\subseteq \phi^{-1}(W)$. Since $\breve{K}[w_{1},\dots ,w_{h-1}]\varphi^{s}_{0}$ is dense in $M^{s}_{D}=\mathcal{O}_{X^{\text{rig}}_{0}}(D)\varphi_{0}^{s}$ and $\phi^{-1}(W)$ is closed, it follows that $\phi^{-1}(W)=M^{s}_{D}$. Hence $W=\phi(M^{s}_{D})=M^{s}_{D}/V_{s}$. Thus $M^{s}_{D}/V_{s}$ is topologically irreducible.
\end{proof}
\begin{corollary}\label{lf0} For all $s\in\mathbb{Z}$, we have \begin{equation*}
(M^{s}_{D})_{\textnormal{lf}}=V_{s}.
\end{equation*} Thus, $(M^{s}_{D})_{\textnormal{lf}}$ is zero if $s<0$ and is a finite dimensional irreducible $\Gamma$-representation if $s\geq 0$.
\end{corollary}
\begin{proof} Let $s<0$ and $v$ be a non-zero locally finite vector in the $\breve{K}$-vector space $M^{s}_{D}$. Thus $v$ is contained in a finite dimensional $H$-subrepresentation $W$ of $M^{s}_{D}$ for some open subgroup $H$ of $\Gamma$. Being finite dimensional, $W$ is complete and hence closed $H$-stable subspace. Therefore $W$ is stable under the action of $\mathfrak{g}=\textnormal{Lie}(H)$. Then it follows from the proof of Theorem \ref{top_finiteness_of_MsD_thm} (case $s<0$) that $W=M^{s}_{D}$ which is a contradiction since $M^{s}_{D}$ is not finite dimensional.\vspace{2.6mm}\\
\noindent Let $s\geq 0$. It is clear that $V_{s}\subseteq(M^{s}_{D})_{\textnormal{lf}}$. Suppose there exists a locally finite vector $v$ not contained in $V_{s}$. Then $v$ is contained in a finite dimensional $H$-subrepresentation $W$ of $M^{s}_{D}$ for some open subgroup $H$ of $\Gamma$. Now $W/(W\cap V_{s})$ is a non-zero finite dimensional $H$-stable subspace of $M^{s}_{D}/V_{s}$ and thus it is closed in $M^{s}_{D}/V_{s}$. Then it follows from the proof of Theorem \ref{top_finiteness_of_MsD_thm} (case $s\geq 0$) that $W/(W\cap V_{s})=M^{s}_{D}/V_{s}$ which is again a contradiction due to dimensionality argument.
\end{proof}
\noindent A few remarks are in order.
\begin{remark} In the context of Theorem \ref{top_finiteness_of_MsD_thm}, one might say that the $\Gamma$-representation $M^{s}_{D}$ is \emph{topologically} of length 1 or 2. However; in this very general situation, the notion \emph{topological length} is problematic since the image of a closed subspace of $M^{s}_{D}$ in $M^{s}_{D}/V_{s}$ under the quotient map is generally not closed for the quotient topology on $M^{s}_{D}/V_{s}$. For this, one would have to work in the category of admissible Banach space representations.
\end{remark}
\begin{remark}\label{highest_wt_of_Vs} For $s\geq 0$, the finite-dimensional $\Gamma$-representation $V_{s}$ is also a $\mathfrak{gl}(K_{h})$-module. Let $\mathfrak{t}\subset\mathfrak{sl}_{h}(K_{h})$ be the Cartan subalgebra of $\mathfrak{sl}_{h}(K_{h})$ consisting of diagonal matrices, and let $\lbrace\varepsilon_{1},\ldots,\varepsilon_{h-1}\rbrace$ be the basis of the root system $(\mathfrak{sl}_{h}(K_{h}),\mathfrak{t})$ given by $\varepsilon_{i}(\text{diag}(t_{0},\ldots,t_{h-1})):=t_{i-1}-t_{i}$. Define the fundamental dominant weight $\chi_{0}:=\frac{1}{h}\sum_{i=1}^{h-1}(h-i)\varepsilon_{i}\in\mathfrak{t}^{*}$. Then, by the same proof as in \cite{kohliwamo}, Proposition 4.3, it follows that $V_{s}$ is an irreducible $\mathfrak{sl}_{h}(K_{h})$-representation of highest weight $s\chi_{0}$. Although this is stronger than saying that $V_{s}$ is an irreducible $\Gamma$-representation, our result (Theorem \ref{top_finiteness_of_MsD_thm}) also gives information about the $\Gamma$-representation $M^{s}_{D}$ when $s<0$.
\end{remark}
\vspace{1mm}
\noindent The Corollary \ref{lf0} leads us to calculate locally finite vectors in the global sections $M^{s}_{0}$ over $X^{\text{rig}}_{0}$. Recall from \cite{bgr}, (9.3.4), Example 3, that the rigid analytic projective space $\mathbb{P}^{h-1}_{\breve{K}}$ has a finite admissible covering by the $(h-1)$-dimensional closed unit polydiscs $V_{i}:=\text{Sp}(\breve{K}\langle\frac{\varphi_{0}}{\varphi_{i}},\ldots,\frac{\varphi_{h-1}}{\varphi_{i}}\rangle)$, $0\leq i\leq h-1$. If $V_{ij}:=\text{Sp}(\breve{K}\langle\frac{\varphi_{0}}{\varphi_{i}},\ldots,\frac{\varphi_{h-1}}{\varphi_{i}},(\frac{\varphi_{j}}{\varphi_{i}})^{-1}\rangle)$ for $0\leq i,j\leq h-1$, then gluing the $V_{i}$'s along the identification $V_{ij}\simeq V_{ji}$ of affinoid subdomains via \begin{equation*}
\breve{K}\Big\langle\frac{\varphi_{0}}{\varphi_{i}},\ldots,\frac{\varphi_{h-1}}{\varphi_{i}},\Big(\frac{\varphi_{j}}{\varphi_{i}}\Big)^{-1}\Big\rangle=\breve{K}\Big\langle\frac{\varphi_{0}}{\varphi_{j}},\ldots,\frac{\varphi_{h-1}}{\varphi_{j}},\Big(\frac{\varphi_{i}}{\varphi_{j}}\Big)^{-1}\Big\rangle
\end{equation*} gives the rigid analytic projective space $\mathbb{P}^{h-1}_{\breve{K}}$. The affinoid covering $\lbrace V_{i}\rbrace_{0\leq i\leq h-1}$ allows us to describe the construction of the line bundles $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)$ on the rigid analytic projective space in a way analogous to the classical construction. For $s\geq 0$, define its sections over the affinoid space $V_{i}$
\begin{equation*}
\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(V_{i}):=\breve{K}\Big\langle\frac{\varphi_{0}}{\varphi_{i}},\ldots,\frac{\varphi_{h-1}}{\varphi_{i}}\Big\rangle\varphi_{i}^{s}
\end{equation*} to be a free module of rank 1 generated by $\varphi_{i}^{s}$ over $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(V_{i})=\breve{K}\langle\frac{\varphi_{0}}{\varphi_{i}},\ldots,\frac{\varphi_{h-1}}{\varphi_{i}}\rangle$, and the transition functions $\psi_{ij}:V_{ij}\iso V_{ji}$ induced by the homomorphisms of affinoid $\breve{K}$-algebras \begin{equation*}
\psi_{ij}^{*}:\breve{K}\Big\langle\frac{\varphi_{0}}{\varphi_{j}},\ldots,\frac{\varphi_{h-1}}{\varphi_{j}},\Big(\frac{\varphi_{i}}{\varphi_{j}}\Big)^{-1}\Big\rangle\varphi_{j}^{s}\xrightarrow{\textnormal{multiply by}\hspace{.1cm}\frac{\varphi_{i}^{s}}{\varphi_{j}^{s}}}\breve{K}\Big\langle\frac{\varphi_{0}}{\varphi_{i}},\ldots,\frac{\varphi_{h-1}}{\varphi_{i}},\Big(\frac{\varphi_{j}}{\varphi_{i}}\Big)^{-1}\Big\rangle\varphi_{i}^{s}
\end{equation*} for all $0\leq i,j\leq h-1$. The above datum gives rise to a locally free $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{k}}}$-module $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)$ of rank 1. For $s<0$, $\mathcal{O}_{\mathbb{P}_{\breve{K}}^{h-1}}(s)$
turns out to be the $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{k}}}$-linear dual of $\mathcal{O}_{\mathbb{P}_{\breve{K}}^{h-1}}(-s)$. It then follows easily from the above description that the global sections of $\mathcal{O}_{\mathbb{P}_{\breve{K}}^{h-1}}(s)$ are the $\breve{K}$-vector space of homogeneous polynomials of degree $s$ in the variables $\varphi_{i}$'s if $s\geq 0$, and are 0 otherwise. The line bundles $\mathcal{O}_{\mathbb{P}_{\breve{K}}^{h-1}}(s)$ carry a canonical action of $\Gamma$ induced by its action on the projective space $\mathbb{P}_{\breve{K}}^{h-1}$. \vspace{2.6mm}\\
\noindent Now for any $\mathcal{O}_{X^{\textnormal{rig}}_{0}}$-module $\mathcal{F}$ and $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}-$module $\mathcal{G}$, there is a bijection between the sets \linebreak $\textnormal{Hom}_{\mathcal{O}_{X^{\textnormal{rig}}_{0}}-\textnormal{mod}}(\Phi^{*}\mathcal{G},\mathcal{F})=\textnormal{Hom}_{\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}-\textnormal{mod}}(\mathcal{G},\Phi_{*}\mathcal{F})$, where $\Phi:X^{\textnormal{rig}}_{0}\longrightarrow\mathbb{P}^{h-1}_{\breve{K}}$ is the Gross-Hopkins' period morphism. The morphism id$_{\Phi^{*}\mathcal{G}}$ corresponds to the adjunction morphism $\textnormal{ad}:\mathcal{G}\longrightarrow\Phi_{*}\Phi^{*}\mathcal{G}$. Let $\mathcal{G}=\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)$ with $s\in\mathbb{Z}$. The period morphism $\Phi$ is constructed in such a way that $\Phi^{*}\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)\simeq(\mathcal{M}^{s}_{0})^{\textnormal{rig}}$ (cf. Remark \ref{our_line_bundle_is_a_pullback_of_O(s)*}). This gives us a map $\textnormal{ad}:\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)\longrightarrow \Phi_{*}(\mathcal{M}^{s}_{0})^{\textnormal{rig}}$ of $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}$-modules. Taking global sections, we get a homomorphism of $\Gamma$-representations \begin{equation*}
\textnormal{ad}_{\mathbb{P}^{h-1}_{\breve{K}}}:\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(\mathbb{P}^{h-1}_{\breve{K}})\longrightarrow \Phi_{*}(\mathcal{M}^{s}_{0})^{\textnormal{rig}}(\mathbb{P}^{h-1}_{\breve{K}})=(\mathcal{M}^{s}_{0})^{\textnormal{rig}}(\Phi^{-1}(\mathbb{P}^{h-1}_{\breve{K}}))=(\mathcal{M}^{s}_{0})^{\textnormal{rig}}(X^{\textnormal{rig}}_{0})=M^{s}_{0}. \end{equation*}
\begin{lemma} The map $
\textnormal{ad}_{\mathbb{P}^{h-1}_{\breve{K}}}$ is injective.
\end{lemma}
\begin{proof}
The period morphism $\Phi$, when restricted to the affinoid subdomain $D$, is an isomorphism. Thus $(\mathcal{M}^{s}_{0})^{\textnormal{rig}}(D)\simeq\Phi^{*}\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(D)\simeq\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(\Phi(D))$. Also we have $(\mathcal{M}^{s}_{0})^{\textnormal{rig}}(D)\simeq\mathcal{O}_{X^{\text{rig}}_{0}}(D)\varphi^{s}_{0}\simeq\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(\Phi(D))\varphi^{s}_{0}$. As a result, it follows from the preceding discussion on the line bundles that $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(\mathbb{P}^{h-1}_{\breve{K}})$ maps bijectively onto $V_{s}\subset\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(\Phi(D))$ under the restriction map. The lemma now follows from the following commutative diagram with vertical restriction maps.
$$
\xymatrixcolsep{11pc}
\xymatrixrowsep{5pc}
\xymatrix{
\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(\mathbb{P}^{h-1}_{\breve{K}}) \ar[r]^{\textnormal{ad}_{\mathbb{P}^{h-1}_{\breve{K}}}} \ar@{^{(}->}[d]
& \Phi_{*}(\mathcal{M}^{s}_{0})^{\textnormal{rig}}(\mathbb{P}^{h-1}_{\breve{K}})=M^{s}_{0} \ar[d]\\
\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(\Phi(D)) \ar[r]_{\textnormal{ad}_{\Phi(D)}}^{\simeq} & \Phi_{*}(\mathcal{M}^{s}_{0})^{\textnormal{rig}}(\Phi(D))=(\mathcal{M}^{s}_{0})^{\textnormal{rig}}(D)=M^{s}_{D}
}
$$
\end{proof}
\noindent In fact, also the right vertical arrow in the commutative diagram above is injective. Namely, the inclusion $R^{\text{rig}}_{0}\hookrightarrow\mathcal{O}_{X^{\text{rig}}_{0}}(D)$ gives rise to a $\Gamma$-equivariant inclusion $M^{s}_{0}\hookrightarrow M^{s}_{D}\simeq\mathcal{O}_{X^{\text{rig}}_{0}}(D)\otimes_{R_{0}}\text{Lie}(\mathbb{H}^{(0)})^{\otimes s}$ using (\ref{G_eqv_iso_eqn1}) and the freeness of $\text{Lie}(\mathbb{H}^{(0)})^{\otimes s}$ as an $R_{0}$-module. As $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(\mathbb{P}^{h-1}_{\breve{K}})$ is a finite dimensional $\breve{K}$-vector space, we have for $s\geq 0$, $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(\mathbb{P}^{h-1}_{\breve{K}})\subseteq (M^{s}_{0})_{\textnormal{lf}}\subseteq (M^{s}_{D})_{\textnormal{lf}}=V_{s}$, where the first and the last $\breve{K}$-vector spaces are isomorphic as mentioned in the proof of the previous lemma. Hence,
\begin{corollary}\label{lf1}
For all $s\in\mathbb{Z}$, we have an isomorphism of $\Gamma$-representations \begin{equation*}
(M^{s}_{0})_{\textnormal{lf}}=\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(\mathbb{P}^{h-1}_{\breve{K}})\simeq V_{s}.
\end{equation*} Thus, $(M^{s}_{0})_{\textnormal{lf}}$ is zero if $s<0$ and is a finite dimensional irreducible $\Gamma$-representation if $s\geq 0$.\hspace{.3cm}\qedsymbol
\end{corollary}
\begin{remark}\label{Vs_is_sym-s-part_of_V1} From now on, we identify the subrepresentation $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(\mathbb{P}^{h-1}_{\breve{K}})$ of $M^{s}_{0}$ with $V_{s}$. For $s=1$, the $\Gamma$-locally finite subrepresentation $V_{1}$ of $M^{1}_{0}$ is the representation $\mathbb{V}$ mentioned in the construction of the period morphism $\Phi$ (cf. the paragraph after Remark \ref{Lie(Ems)_generically_flat}), and thus is isomorphic to the $h$-dimensional $\Gamma$-representation $B_{h}\otimes_{K_{h}}\breve{K}$. Since the $\Gamma$-representations $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(\mathbb{P}^{h-1}_{\breve{K}})$ and $V_{s}$ are isomorphic, and $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(s)(\mathbb{P}^{h-1}_{\breve{K}})$ is same as the $s$-th symmetric power $\text{Sym}^{s}(\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(1)(\mathbb{P}^{h-1}_{\breve{K}}))$ of $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(1)(\mathbb{P}^{h-1}_{\breve{K}})$, we obtain the isomorphism, \begin{equation*}
(M^{s}_{0})_{\textnormal{lf}}= V_{s}\simeq \textnormal{Sym}^{s}(B_{h}\otimes_{K_{h}}\breve{K})
\end{equation*} of $\Gamma$-representations for all $s\geq 0$. This makes the shape of these representations completely explicit.
\end{remark}
\subsection{Locally finite vectors in the $\Gamma$-representations $M^{s}_{m}$ with $m>0$}
\noindent We compute the locally finite vectors in two parts: $s\leq0$ and $s>0$. The idea here is to use the commuting actions of $\Gamma$ and the finite group $G_{0}/G_{m}$ on $M^{s}_{m}$.
\vspace{2mm}
\begin{center}
\textbf{Part I : $s\leq0$}
\end{center}
\begin{lemma}\label{finiteextlemma} Let $G$ be a finite group acting on an integral domain $R$ by ring automorphisms such that the subring of $G$-invariants $R^{G}$ is a perfect field $F$. Then $R$ is a field and the extension $R/F$ is finite.
\end{lemma}
\begin{proof}
If $\alpha\in R$ then $\prod_{\sigma\in G}(t-\sigma(\alpha))$ is a monic polynomial of degree $|G|$ with coefficients in $R^{G}=F$, and has $\alpha$ as a root. This implies that every nonzero $\alpha$ has a unique inverse, since $R$ is an integral domain. The second assertion now follows from \cite{langalgebra}, Chapter VI, Lemma 1.7.
\end{proof}
\noindent The following result relies on the Strauch's computation of the geometrically connected components of the Lubin-Tate tower (cf. \cite{strgeo} and Remark \ref{mLText}).
\begin{theorem}\label{lfinRm} For all $m\geq0$, $(M^{0}_{m})_{\textnormal{lf}}=(R^{\textnormal{rig}}_{m})_{\textnormal{lf}}=\breve{K}_{m}$.
\end{theorem}
\begin{proof}
The kernel of the composition map $\Gamma\xrightarrow{\textnormal{Nrd}}\mathfrak{o}^{\times}\longrightarrow(\mathfrak{o}/\varpi^{m}\mathfrak{o})^{\times}$ is an open subgroup of $\Gamma$ which acts trivially on $\breve{K}_{m}$ (cf. Remark \ref{mLText}). Thus $\breve{K}_{m}\subseteq (R^{\textnormal{rig}}_{m})_{\textnormal{lf}}$. Notice that $(R^{\textnormal{rig}}_{m})_{\textnormal{lf}}$ is a subring of $R^{\textnormal{rig}}_{m}$. Indeed, if $f_{1}, f_{2}\in (R^{\textnormal{rig}}_{m})_{\textnormal{lf}}$ then there exist open subgroups $H_{1}$ and $H_{2}$ of $\Gamma$ and finite dimensional sub-representations $V_{1}$ and $V_{2}$ of $H_{1}$ and $H_{2}$ in $R^{\textnormal{rig}}_{m}$ respectively such that $f_{1}\in V_{1}$ and $f_{2}\in V_{2}$. Let $V_{1}+V_{2}=\lbrace v_{1}+v_{2}|v_{1}\in V_{1}, v_{2}\in V_{2}\rbrace$ and $V_{1}V_{2}$ be the $\breve{K}$-vector space generated by $\lbrace v_{1}v_{2}\rbrace_{v_{1}\in V_{1}, v_{2}\in V_{2}}$. Then $V_{1}+V_{2}$ and $V_{1}V_{2}$ are finite dimensional sub-representations of the open subgroup $H_{1}\cap H_{2}$ containing $f_{1}+f_{2}$ and $f_{1}f_{2}$ respectively.\vspace{2.6mm}\\
\noindent The $\breve{K}$-algebra $R^{\textnormal{rig}}_{m}$ carries commuting actions of the groups $\Gamma$ and $G_{0}/G_{m}$ with $(R^{\textnormal{rig}}_{m})^{G_{0}/G_{m}}=R^{\textnormal{rig}}_{0}$ (cf. \cite{Koh11}, Theorem 1.4 (i)). Now let $f\in (R^{\textnormal{rig}}_{m})_{\textnormal{lf}}$ and $V$ be a finite dimensional $H$-subrepresentation of $R^{\textnormal{rig}}_{m}$ containing $f$ for some open subgroup $H$ of $\Gamma$. Let $g\in G_{0}/G_{m}$. Then the $\breve{K}$-vector space $gV$ is $H$-stable since the actions of $H$ and $G_{0}/G_{m}$ on $V$ commute. Thus $gV$ is a finite dimensional $H$-subrepresentation of $R^{\textnormal{rig}}_{m}$ containing $gf$ implying that $gf$ is locally finite. Hence $(R^{\textnormal{rig}}_{m})_{\textnormal{lf}}$ is stable under the action of $G_{0}/G_{m}$ with the ring of invariants $(R^{\textnormal{rig}}_{m})_{\textnormal{lf}}^{G_{0}/G_{m}}=((R^{\textnormal{rig}}_{m})^{G_{0}/G_{m}})_{\textnormal{lf}}=(R^{\textnormal{rig}}_{0})_{\textnormal{lf}}=\breve{K}$ by Corollary \ref{lf1}. As $G_{0}/G_{m}$ is finite, $(R^{\textnormal{rig}}_{m})_{\textnormal{lf}}$ is a finite field extension of $\breve{K}$ by Lemma \ref{finiteextlemma}. So it is also finite over $\breve{K}_{m}$. However it follows from the proof of \cite{Koh11}, Theorem 1.4 that $\breve{K}_{m}$ is separably closed in $R^{\textnormal{rig}}_{m}$. Therefore $(R^{\textnormal{rig}}_{m})_{\textnormal{lf}}=\breve{K}_{m}$.
\end{proof}
\begin{remark}\label{ginvinRm} By Theorem \ref{laKm}, we have a $\mathfrak{g}$-action on $R^{\textnormal{rig}}_{m}$. The subspace of $\mathfrak{g}$-invariants $(R^{\textnormal{rig}}_{m})^{\mathfrak{g}=0}$ forms a subring of $R^{\textnormal{rig}}_{m}$, and is stable under the action of $G_{0}/G_{m}$ because the $G_{0}/G_{m}$-action on $R^{\textnormal{rig}}_{m}$ is continuous and commutes with that of $\Gamma$. As said in the proof of Theorem \ref{lfinRm}, the kernel of the composition map $\Gamma\xrightarrow{\textnormal{Nrd}}\mathfrak{o}^{\times}\longrightarrow(\mathfrak{o}/\varpi^{m}\mathfrak{o})^{\times}$ is an open subgroup of $\Gamma$ which acts trivially on $\breve{K}_{m}$. Thus, $\breve{K}_{m}\subseteq (R^{\textnormal{rig}}_{m})^{\mathfrak{g}=0}$. Proceeding similarly as above, we have $((R^{\textnormal{rig}}_{m})^{\mathfrak{g}=0})^{G_{0}/G_{m}}=((R^{\textnormal{rig}}_{m})^{G_{0}/G_{m}})^{\mathfrak{g}=0}=(R^{\textnormal{rig}}_{0})^{\mathfrak{g}=0}=\breve{K}$ (cf. Lemma \ref{gninvO(D)}). Then by the same arguments as above, we get $(R^{\textnormal{rig}}_{m})^{\mathfrak{g}=0}=\breve{K}_{m}$.
\end{remark}
\noindent For all integers $s$, the $\Gamma$-equivariant isomorphism $M^{s}_{m}\simeq R^{\text{rig}}_{m}\otimes_{R^{\text{rig}}_{0}}M^{s}_{0}$ (cf. proof of Theorem \ref{genlaKm}) and the freeness of the $R^{\text{rig}}_{0}$-module $M^{s}_{0}$ give rise to a $\Gamma$-equivariant inclusion $M^{s}_{0}\subset M^{s}_{m}$ of $\breve{K}$-vector spaces. Consequently, we have $(M^{s}_{0})_{\textnormal{lf}}\subseteq (M^{s}_{m})_{\textnormal{lf}}$. Using the above theorem, we see that $(M^{s}_{m})_{\textnormal{lf}}$ is a module over $(R^{\textnormal{rig}}_{m})_{\textnormal{lf}}=\breve{K}_{m}$, and thus we obtain a natural map \begin{equation*}
\breve{K}_{m}\otimes_{\breve{K}}(M^{s}_{0})_{\textnormal{lf}}\longrightarrow (M^{s}_{m})_{\textnormal{lf}}
\end{equation*} of $\breve{K}$-vector spaces. Our objective is to show that this map is an isomorphism of $\breve{K}[\Gamma]$-modules for all integers $s$.
\begin{lemma}\label{lflemma} Suppose $V$ and $W$ are two representations of a topological group $G$ over a field $F$ such that one of them, say $W$, is finite dimensional. Consider the representation $V\otimes_{F}W$ with diagonal $G$-action. Then $(V\otimes_{F}W)_{\textnormal{lf}}=V_{\textnormal{lf}}\otimes_{F}W$.
\end{lemma}
\begin{proof}
We omit the subscript $F$ in $\otimes_{F}$, as all the tensor products are over $F$. The inclusion $V_{\textnormal{lf}}\otimes W\subseteq (V\otimes W)_{\textnormal{lf}}$ is clear. Let $W^{*}$ be the $F$-linear dual of $W$ equipped with the contragredient $G$-action i.e. $(gf)(w)=f(g^{-1}w)$ for all $g\in G, w\in W$ and $f\in W^{*}$. Choose an $F$-basis $\lbrace w_{1},\ldots ,w_{d}\rbrace$ of $W$, and let $\lbrace f_{1},\ldots ,f_{d}\rbrace$ be the dual basis of $W^{*}$ (i.e. $f_{i}(w_{j})=\delta_{ij}$). Then the natural evaluation map $W\otimes W^{*}\longrightarrow F$ $(w\otimes f\mapsto f(w))$ is $G$-equivariant for the diagonal $G$-action on the left and for the trivial $G$-action on the right. Tensoring both sides with $V$, we get a $G$-equivariant map $\phi:V\otimes W\otimes W^{*}\longrightarrow V$ for the diagonal $G$-action on the left, sending $v\otimes w\otimes f$ to $f(w)v$. Because of its $G$-equivariance, $\phi$ maps locally finite vectors to locally finite vectors. Now let $x\in (V\otimes W)_{\textnormal{lf}}$. Then $x$ can be uniquely written as $x=\sum_{i=1}^{d}x_{i}\otimes w_{i}$ for some $x_{1},\ldots ,x_{d}\in V$. Since $W^{*}$ is finite dimensional, $x\otimes f_{i}\in (V\otimes W)_{\textnormal{lf}}\otimes (W^{*})_{\textnormal{lf}}\subseteq (V\otimes W\otimes W^{*})_{\textnormal{lf}}$ for all $1\leq i\leq d$. Hence, $\phi(x\otimes f_{i})=x_{i}\in V_{\textnormal{lf}}$ for all $1\leq i\leq d$. Therefore, $x\in V_{\textnormal{lf}}\otimes W$.
\end{proof}
\noindent The following theorem is based on the property of generic flatness of the line bundles $\mathcal{L}\textnormal{ie}(\mathbb{E}^{(m)})^{\otimes s}$ obtained in Remark \ref{Lie(Ems)_generically_flat}.
\begin{theorem}\label{lf2} For all $s<0$ and for all $m\geq 0$, $(M^{s}_{m})_{\textnormal{lf}}\simeq\breve{K}_{m}\otimes_{\breve{K}}(M^{s}_{0})_{\textnormal{lf}}=0$.
\end{theorem}
\begin{proof} Recall from (\ref{Lie(Ems)_is_generically_flat_iso}) that we have an isomorphism \begin{equation*}
R^{\textnormal{rig}}_{m}\otimes_{R_{m}}\textnormal{Lie}(\mathbb{E}^{(m)})^{\otimes s}\simeq R^{\textnormal{rig}}_{m}\otimes_{\breve{K}}(B_{h}\otimes_{K_{h}}\breve{K})^{\otimes s}
\end{equation*} of $\Gamma$-representations. As a result, using Lemma \ref{lflemma} together with Theorem \ref{lfinRm}, we obtain locally finite vectors in the global sections of $\mathcal{L}\text{ie}(\mathbb{E}^{(m)})^{\otimes s}$, i.e. \begin{equation*}
(R^{\textnormal{rig}}_{m}\otimes_{R_{m}}\textnormal{Lie}(\mathbb{E}^{(m)})^{\otimes s})_{\textnormal{lf}}\simeq \breve{K}_{m}\otimes_{\breve{K}}(B_{h}\otimes_{K_{h}}\breve{K})^{\otimes s}.
\end{equation*}
Then, since $s<0$, the $(\Gamma\times(G_{0}/G_{m}))$-equivariant inclusion \begin{equation*}
M^{s}_{m}\subset R^{\textnormal{rig}}_{m}\otimes_{R_{m}}\textnormal{Lie}(\mathbb{E}^{(m)})^{\otimes s}
\end{equation*} from (\ref{exact_seq_of_global_sections_of_eqv_bundles}) gives rise to a $(\Gamma\times (G_{0}/G_{m}))$-equivariant inclusion \begin{equation*}
(M^{s}_{m})_{\textnormal{lf}}\subseteq\breve{K}_{m}\otimes_{\breve{K}}(B_{h}\otimes_{K_{h}}\breve{K})^{\otimes s}
\end{equation*} of $\breve{K}$-vector spaces. As the action of $SL_{h}(\mathfrak{o}/\varpi^{m}\mathfrak{o})\subset G_{0}/G_{m}$ on the right hand side above is trivial (cf. Remark \ref{mLText}), we get $(M^{s}_{m})_{\textnormal{lf}}=\big(M^{s}_{m}\big)_{\textnormal{lf}}^{SL_{h}(\mathfrak{o}/\varpi^{m}\mathfrak{o})}=\Big(\big(M^{s}_{m}\big)^{SL_{h}(\mathfrak{o}/\varpi^{m}\mathfrak{o})}\Big)_{\textnormal{lf}}$, where the latter equality is due to the fact that the both group actions on $M^{s}_{m}$ commute. Therefore,
\begin{align*}(M^{s}_{m})_{\textnormal{lf}}=\Big(\big(M^{s}_{m}\big)^{SL_{h}(\mathfrak{o}/\varpi^{m}\mathfrak{o})}\Big)_{\textnormal{lf}}&\simeq\Big(\big(R^{\textnormal{rig}}_{m}\otimes_{R^{\textnormal{rig}}_{0}}M^{s}_{0}\big)^{SL_{h}(\mathfrak{o}/\varpi^{m}\mathfrak{o})}\Big)_{\textnormal{lf}}\\&\simeq\Big(\big(R^{\textnormal{rig}}_{m}\big)^{SL_{h}(\mathfrak{o}/\varpi^{m}\mathfrak{o})}\otimes_{R^{\textnormal{rig}}_{0}}M^{s}_{0}\Big)_{\textnormal{lf}}\\&\simeq\big((\breve{K}_{m}\otimes_{\breve{K}}R^{\textnormal{rig}}_{0})\otimes_{R^{\textnormal{rig}}_{0}}M^{s}_{0}\big)_{\textnormal{lf}}\\&\simeq\big(\breve{K}_{m}\otimes_{\breve{K}}M^{s}_{0}\big)_{\textnormal{lf}}=\breve{K}_{m}\otimes_{\breve{K}}(M^{s}_{0})_{\textnormal{lf}}=0
\end{align*}
where the second isomorphism holds because $M^{s}_{0}$ is free over $R^{\text{rig}}_{0}$ with trivial $G_{0}/G_{m}$-action, and the third isomorphism holds because $\big(R^{\textnormal{rig}}_{m}\big)^{SL_{h}(\mathfrak{o}/\varpi^{m}\mathfrak{o})}$ is Galois over $R^{\textnormal{rig}}_{0}$ with the Galois group isomorphic to $\frac{G_{0}/G_{m}}{SL_{h}(\mathfrak{o}/\varpi^{m}\mathfrak{o})}\simeq(\mathfrak{o}/\varpi^{m}\mathfrak{o})^{\times}\simeq\textnormal{Gal}(\breve{K}_{m}/\breve{K})$ and $\breve{K}_{m}\subseteq R^{\text{rig}}_{m}$. For the second last equality in the above, we use Lemma \ref{lflemma} again. The final result then follows from Corollary \ref{lf1}.
\end{proof}
\begin{center}
\textbf{Part II : $s>0$}
\end{center}
\vspace{2.6mm}
To compute the locally finite vectors in $M^{s}_{m}$ for $s>0$, we make use of the action of the group $G^{0}:=\lbrace g\in GL_{h}(K)|\hspace*{.1cm}\textnormal{det}(g)\in\mathfrak{o}^{\times}\rbrace$ on \textit{the Lubin-Tate tower} $\big(X_{m}^{\textnormal{rig}}\big)_{m\in\mathbb{N}_{0}}$ described in \cite{strdeform}, § 2.2.2.\vspace{2.6mm}\\
\noindent Given $g\in G^{0}$ and $m\geq 0$, for every $m'\geq m$ sufficiently large (depending on $g$), there is a morphism $g_{m',m}:X_{m'}^{\textnormal{rig}}\longrightarrow X_{m}^{\textnormal{rig}}$ of rigid analytic spaces satisfying the following properties:
\begin{enumerate}
\item[(i)] For all $g\in G^{0}$ and for all $n\geq m''\geq m'\geq m$, we have $g_{n,m}=\pi_{m',m}\circ g_{m'',m'} \circ \pi_{n,m''}$, where recall that $\pi_{m',m}:X_{m'}^{\textnormal{rig}}\longrightarrow X_{m}^{\textnormal{rig}}$ denotes the covering morphism. In particular, if $g=e$, and if $m=m'=m''$, then we get $e_{n,m}=\pi_{n,m}$ for all $n\geq m$ because $e_{m,m}=\text{id}_{ X_{m}^{\textnormal{rig}}}$ by definition (cf. \cite{strdeform}, § 2.2.2).
\item[(ii)] $(gh)_{m'',m}=g_{m',m}\circ h_{m'',m}$ for all $g,h\in G^{0}$ and for all $m''\geq m' \geq m$.
\item[(iii)] Set $\Phi_{m}:=\Phi\circ\pi_{m,0}:X^{\textnormal{rig}}_{m}\longrightarrow\mathbb{P}^{h-1}_{\breve{K}}$. Then $\Phi_{m'}=\Phi_{m}\circ g_{m',m}$ for all $g\in G^{0}$, $m'\geq m$. In other words, the Gross-Hopkins period morphism is $G^{0}$-equivariant for the trivial $G^{0}$-action on $\mathbb{P}^{h-1}_{\breve{K}}$.
\item[(iv)] All $g_{m',m}$ are $\Gamma$-equivariant morphisms.
\item[(v)] For $g\in GL_{h}(\mathfrak{o})$ and $m\geq 0$, $g_{m,m}$ is defined. The gives an action of $GL_{h}(\mathfrak{o})$ on $X_{m}^{\textnormal{rig}}$ which factors through $GL_{h}(\mathfrak{o}/\varpi^{m}\mathfrak{o})=G_{0}/G_{m}$. The induced $G_{0}/G_{m}$-action coincides with the $G_{0}/G_{m}$-action introduced in § \ref{the_group_actions}.
\end{enumerate}
In the above and hereafter, $m'\geq m$ means that $m'$ is sufficiently larger than or equal to $m$ so that $g_{m',m}$ is defined.\vspace{2.6mm}\\
\noindent Let $D_{m}:=\pi_{m,0}^{-1}(D)$ where $D$ is the Gross-Hopkins fundamental domain $D$ in $X^{\textnormal{rig}}_{0}$. The admissible open $D_{m}$ is a $\Gamma$-stable affinoid subdomain because $\pi_{m,0}$ is a finite, $\Gamma$-equivariant morphism, and $D$ is $\Gamma$-stable. For every $g\in G^{0}$ and $m\geq 0$, we define a $g$-translate of $D_{m}$ \hspace{0cm} as $gD_{m}:=g_{m',m}(D_{m'})$ by choosing $m'\geq m$ large enough. Note that this definition is independent of the choice of $m'$, since by property (i), for $m''\geq m'\geq m$, \begin{align*}
g_{m'',m}(D_{m''})=g_{m',m}(\pi_{m'',m'}(D_{m''}))&=g_{m',m}(\pi_{m'',m'}(\pi_{m'',0}^{-1}(D)))\\&=g_{m',m}(\pi_{m'',m'}(\pi_{m'',m'}^{-1}(\pi_{m',0}^{-1}(D))))=g_{m',m}(D_{m'}),\end{align*} using that $\pi_{m'',m'}$ is surjective.
\begin{proposition}\label{gDmisacovering} The set $\lbrace gD_{m}\rbrace_{g\in G^{0}}$ forms an admissible affinoid covering of $\Phi_{m}^{-1}(\Phi(D))$ consisting of $\Gamma$-stable affinoid subdomains.
\end{proposition}
\begin{proof}
This is a part of the cellular decomposition of the Lubin-Tate tower in \cite{fgl}, Proposition I.7.1 relying on \cite{gh}, Corollary 23.26. The $\Gamma$-stability of $gD_{m}$ follows from (iv) and that of $D_{m'}$.
\end{proof}
\begin{lemma}\label{inclusionofaffisubd} For all $g\in G^{0}$ and $m'\geq m$, the maps $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(\Phi(D))\longrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})\longrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})$ of affinoid $\breve{K}$-algebras induced by the morphisms $D_{m'}\xrightarrow{g_{m',m}}gD_{m}\xrightarrow{\Phi_{m}}\Phi(D)$ are injective.
\end{lemma}
\begin{proof}
By property (iii), the composition $\Phi_{m}\circ g_{m',m}=\Phi_{m'}=\Phi\circ\pi_{m',0}$ is flat because $\Phi$ and $\pi_{m',0}$ are flat. Hence the composition map $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(\Phi(D))\longrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})$ of affinoid $\breve{K}$-algebras is flat. However, $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(\Phi(D))\simeq\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)$ is an integral domain (cf. \cite{bgr}, (6.1.5), Proposition 2). So the map $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(\Phi(D))\longrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})$ is injective: the multiplication by a non-zero element $f$ on $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(\Phi(D))$ is injective, which remains injective after flat base change. In particular, the image of $f$ in $\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})$ cannot be zero.\vspace{2.6mm}\\
\noindent To show that the other map $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})\longrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})$ is injective, choose $m''\geq m'$ large enough so that $g^{-1}_{m'',m'}:X^{\textnormal{rig}}_{m''}\longrightarrow X^{\textnormal{rig}}_{m'}$ is defined. Then using properties (i) and (ii), we see that $g^{-1}_{m'',m'}(gD_{m''})=g^{-1}_{m'',m'}(g_{n,m''}(D_{n}))=e_{n,m'}(D_{n})=\pi_{n,m'}(D_{n})=D_{m'}$ and thus $gD_{m}=g_{m',m}(D_{m'})=g_{m',m}(g^{-1}_{m'',m'}(gD_{m''}))=e_{m'',m}(gD_{m''})=\pi_{m'',m}(gD_{m''})$. In other words, \begin{equation*}
\big(g_{m',m}\circ g^{-1}_{m'',m'}\big)\big|_{gD_{m''}}=\pi_{m'',m}.\end{equation*}
Hence the induced composition $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})\longrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})\longrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m''}}(gD_{m''})$ of the maps of affinoid $\breve{K}$-algebras is flat. Now it is not clear if the algebra $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})$ is an integral domain. However we can decompose $gD_{m}$ into its finitely many disjoint connected components $gD_{m}=\bigsqcup_{i=1}^{r} U_{i}$ so that each $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(U_{i})$ is an integral domain (cf. discussion after \cite{bgr}, (9.1.4), Proposition 8 as well as \cite{con}, Lemma 2.1.4). This decomposition also gives a decomposition $gD_{m''}=\bigsqcup_{i=1}^{r}(\pi_{m'',m}|_{gD_{m''}})^{-1}(U_{i})$ of $gD_{m''}$ into disjoint admissible open subsets. By the same argument as in the first paragraph, each map $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(U_{i})\longrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m''}}((\pi_{m'',m}|_{gD_{m''}})^{-1}(U_{i}))$ is injective. As a consequence, the composition $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})\longrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m''}}(gD_{m''})$ is injective since it is the finite direct product of all these maps.
\end{proof}
\begin{remark}\label{O(gDm)isDmod_part1}
The affinoid subdomain $D_{m}$, by definition, is the same as the fibre product $X^{\textnormal{rig}}_{m}\times_{X^{\textnormal{rig}}_{0}}D$ for the maps $\pi_{m,0}:X^{\textnormal{rig}}_{m}\longrightarrow X^{\textnormal{rig}}_{0}$ and $D\hookrightarrow X^{\textnormal{rig}}_{0}$. Thus, we have an isomorphism $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(D_{m})\simeq R^{\textnormal{rig}}_{m}\otimes_{R^{\textnormal{rig}}_{0}}\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D) $ because $R^{\textnormal{rig}}_{m}\big|R^{\textnormal{rig}}_{0}$ is finite. The Galois group $G_{0}/G_{m}=\textnormal{Gal}(R^{\textnormal{rig}}_{m}\big|R^{\textnormal{rig}}_{0})$ acts on $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(D_{m})$ via $\sum_{i=1}^{r}f_{i}\otimes f'_{i}\mapsto\sum_{i=1}^{r}g(f_{i})\otimes f'_{i}$ for $g\in G_{0}/G_{m}$, which gives an action on $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(D_{m})$ by $\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)$-linear automorphisms. Hence the extension $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(D_{m})|\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)$ is finite Galois with the Galois group $G_{0}/G_{m}$. Consequently, for all $m\geq 0$, the extension of coordinate rings $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(D_{m})|\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(\Phi(D))$ induced by the map $\Phi_{m}$ is finite Galois with the same Galois group.
\end{remark}
\begin{remark}\label{O(gDm)isDmod} As both $R^{\textnormal{rig}}_{m}$ and $\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)$ are $\mathfrak{g}$-modules (cf. Proposition \ref{laofG}, Theorem \ref{laKm}), we have a $\mathfrak{g}$-action on $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(D_{m})\simeq R^{\textnormal{rig}}_{m}\otimes_{R^{\textnormal{rig}}_{0}}\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)$. Namely, if $\mathfrak{x}\in\mathfrak{g}$ then on simple tensors, $\mathfrak{x}(f\otimes f')=\mathfrak{x}(f)\otimes f'+f\otimes\mathfrak{x}(f')$. The $\mathfrak{g}$-action on $\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})$ restricts to the subalgebra $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})$, because by Remark \ref{O(gDm)isDmod_part1}, $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})$ is a $\Gamma$-stable submodule of the finitely generated $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(\Phi(D))$-module $\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})$ and hence is closed in $\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})$ by \cite{bgr}, (3.7.3), Proposition 1. Denoting by $\text{Ad}_{\gamma}$ the adjoint automorphism of $\mathfrak{g}$ corresponding to $\gamma\in\Gamma$, we remark that the actions of $\Gamma$ and $\mathfrak{g}$ on $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})$ are compatible in the sense that $\gamma(\mathfrak{x}(f))=\text{Ad}_{\gamma}(\mathfrak{x})(\gamma(f))$, since the Lie algebra action comes from the action of the distribution algebra $D(\Gamma,\breve{K})$ on $R^{\text{rig}}_{m}$ and $\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)$. Using the isomorphism $(\mathcal{M}^{s}_{m})^{\text{rig}}(gD_{m})\simeq\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})\otimes_{R^{\text{rig}}_{m}}M^{s}_{m}$ and Theorem \ref{genlaKm}, one obtains that $(\mathcal{M}^{s}_{m})^{\text{rig}}(gD_{m})$ carries compatible actions of $\Gamma$ and $\mathfrak{g}$ for all $s\in\mathbb{Z}$.
\end{remark}
\begin{proposition}\label{ninvariantsequalginvariants} For all $g\in G^{0}$ and for all $m\geq 0$, $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})_{\textnormal{lf}}=\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})^{\mathfrak{g}=0}=\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})^{\mathfrak{n}=0}$. All these $\breve{K}$-vector spaces are finite dimensional.
\end{proposition}
\begin{proof}
\indent Let $g\in G^{0}, m\geq 0$ be arbitrary, and $m'\geq m$ so that $g_{m',m}$ is defined. As seen in the proof of Lemma \ref{inclusionofaffisubd}, the composition $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(\Phi(D))\hookrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})\hookrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})$ is induced by $\Phi_{m'}$. The $\Gamma$-equivariance of $g_{m',m}$ and of $\Phi_{m}$ yields the inclusions $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(\Phi(D))_{\textnormal{lf}}\hookrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})_{\textnormal{lf}}\hookrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})_{\textnormal{lf}}$ of $\breve{K}$-algebras. The Galois action on $\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})$ commutes with the $\Gamma$-action. As a result, $\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})_{\textnormal{lf}}$ is stable under the Galois action, and $\big(\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})_{\textnormal{lf}}\big)^{G_{0}/G_{m'}}$=\linebreak$\big(\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})^{G_{0}/G_{m'}}\big)_{\textnormal{lf}}=\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(\Phi(D))_{\textnormal{lf}}=\mathcal{O}_{X^{\textnormal{rig}}_{0}}(D)_{\textnormal{lf}}=\breve{K}$ (cf. Corollary \ref{lf0}). Since $G_{0}/G_{m'}$ is finite, $\mathcal{O}_{X^{\textnormal{rig}}_{m'}}(D_{m'})_{\textnormal{lf}}$ is integral over $\breve{K}$, and thus $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})_{\textnormal{lf}}$ is integral over $\breve{K}$. As before, we write $gD_{m}=\bigsqcup_{i=1}^{r} U_{i}$ where $U_{i}$s are the connected components of $gD_{m}$. Let $\Gamma_{i}$ be the stabilizer of $U_{i}$ in $\Gamma$; then each $\Gamma_{i}$ has a finite index in $\Gamma$. Let $\Gamma_{o}\subseteq\Gamma$ be an open subgroup which is a uniform pro-$p$ group. Then for every $i$, the intersection $\Gamma_{i}\cap\Gamma_{o}$ has a finite index in $\Gamma_{o}$, and thus is open in $\Gamma_{o}$ by \cite{ddms}, Theorem 1.17. As a result, $\Gamma_{i}\cap\Gamma_{o}$ is open in $\Gamma$, and \begin{equation*}
\Gamma_{i}=\bigcup_{\bar{\gamma}\in\Gamma_{i}/\Gamma_{i}\cap\Gamma_{o}}\gamma(\Gamma_{i}\cap\Gamma_{o})
\end{equation*} implies that $\Gamma_{i}$ is open in $\Gamma$ for all $i$. Therefore their intersection $\Gamma':=\bigcap_{i=1}^{r}\Gamma_{i}$ is again an open subgroup of $\Gamma$. \vspace{2.6mm}\\
\noindent Now the decomposition $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})\simeq\prod_{i=1}^{r}\mathcal{O}_{X^{\textnormal{rig}}_{m}}(U_{i})$ of $\breve{K}$-algebras is $\Gamma'$-equivariant for the componentwise $\Gamma'$-action on the right. Thus the compactness of $\Gamma$ gives the decomposition $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})_{\textnormal{lf}}=\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})_{\Gamma'-\textnormal{lf}}\simeq\prod_{i=1}^{r}\mathcal{O}_{X^{\textnormal{rig}}_{m}}(U_{i})_{\Gamma'-\textnormal{lf}}$ of locally finite vectors. Denote by $K_{i}$ the integral closure of $\breve{K}$ in the integral domain $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(U_{i})$ for each $i$. It then follows that $K_{i}$ is a field extension of $\breve{K}$. Since every projection $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})_{\textnormal{lf}}\longrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m}}(U_{i})_{\Gamma'-\textnormal{lf}}$ is a surjective $\breve{K}$-algebra homomorphism, the integrality of $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})_{\textnormal{lf}}$ over $\breve{K}$ implies that $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(U_{i})_{\Gamma'-\textnormal{lf}}$ is integral over $\breve{K}$ for all $i$. Therefore, $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(U_{i})_{\Gamma'-\textnormal{lf}}\subseteq K_{i}$ for all $i$. On the other hand, for each $i$, $K_{i}$ is $\Gamma'$-stable as $\Gamma'$ acts $\breve{K}$-linearly on $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(U_{i})$. Now for any classical point $x\in U_{i}$, the composition map $K_{i}\hookrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m}}(U_{i})\twoheadrightarrow\kappa(x)$ is injective, and because $\kappa(x)|\breve{K}$ is finite, $K_{i}|\breve{K}$ is a finite extension. This gives the other inclusion $K_{i}\subseteq\mathcal{O}_{X^{\textnormal{rig}}_{m}}(U_{i})_{\Gamma'-\textnormal{lf}} $ for all $i$. Thus we have $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})_{\textnormal{lf}}=\prod_{i=1}^{r}K_{i}$ with each $K_{i}$ a finite field extension of $\breve{K}$.\vspace{2.6mm}\\
\noindent We now claim that $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})^{\mathfrak{g}=0}=\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})^{\mathfrak{n}=0}=\prod_{i=1}^{r}K_{i}$. Note that $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(U_{i})$ is $\mathfrak{g}$-stable for all $i$ because the projection map $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})\twoheadrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m}}(U_{i})$ of affinoid $\breve{K}$-algebras is surjective, continuous and $\Gamma'$-equivariant. Then all arguments in the last two paragraphs carry over to these cases since $\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(\Phi(D))^{\mathfrak{g}=0}=\mathcal{O}_{\mathbb{P}^{h-1}_{\breve{K}}}(\Phi(D))^{\mathfrak{n}=0}=\breve{K}$ (cf. Lemma \ref{gninvO(D)}). The only thing that remains to be shown is $K_{i}\subseteq\mathcal{O}_{X^{\textnormal{rig}}_{m}}(U_{i})^{\mathfrak{g}=0}$ for all $i$ : Write $K_{i}=\breve{K}[\alpha_{i}]$. Then the set $\lbrace\gamma(\alpha_{i})\rbrace_{\gamma\in\Gamma'}$ is finite as $\Gamma'$ takes $\alpha_{i}$ to its conjugates. Therefore, the stabilizer $\Gamma'_{i}$ of $\alpha_{i}$ in $\Gamma'$ has a finite index in $\Gamma'$, and thus we obtain the open subgroup $\Gamma'_{i}$ of $\Gamma$ which acts trivially on $K_{i}$.
\end{proof}
\noindent The embedding $\Gamma\hookrightarrow GL_{h}(K_{h})$ in (\ref{Gamma_as_a_subgroup_of_GLh}) extends to an embedding $B_{h}^{\times}\hookrightarrow GL_{h}(K_{h})$ of locally $K$-analytic groups via the same map. This yields an action of $B_{h}^{\times}$ on $\mathbb{P}_{\breve{K}}^{h-1}$. Similarly, the deformation spaces $X^{\textnormal{rig}}_{m}$ carry an action of the full group $B_{h}^{\times}$ that extends the $\Gamma$-action, and for which the maps $\Phi_{m}$ are $B_{h}^{\times}$-equivariant (cf. \cite{car}, page 20-21).
\begin{lemma}\label{coveringofXm} The set $\lbrace\Pi^{i}\Phi(D)\rbrace_{0\leq i\leq h-1}$ forms an admissible covering of $\mathbb{P}_{\breve{K}}^{h-1}$. Thus, $X^{\textnormal{rig}}_{m}$ has an admissible covering $\lbrace \Pi^{i}\Phi_{m}^{-1}(\Phi(D))\rbrace_{0\leq i\leq h-1}$ for all $m\geq 0$.
\end{lemma}
\begin{proof} This is proved as a part of \cite{gh}, Corollary 23.21.
\end{proof}
\noindent For $0\leq i\leq h-1$, $s\geq 0$, $m\geq 0$, define $N^{s}_{m}(i):=(\mathcal{M}^{s}_{m})^{\textnormal{rig}}(\Pi^{i}\Phi^{-1}_{m}(\Phi(D)))$ and $A_{m}(i):=N^{0}_{m}(i)=\mathcal{O}_{X^{\textnormal{rig}}_{m}}(\Pi^{i}\Phi^{-1}_{m}(\Phi(D)))$. Note that each $\Pi^{i}\Phi_{m}^{-1}(\Phi(D))$ is $\Gamma$-stable because the conjugation by $\Pi^{-i}$ $(\gamma\mapsto\Pi^{-i}\gamma\Pi^{i})$ is an automorphism of $\Gamma$. Therefore, all $A_{m}(i)$ and $N^{s}_{m}(i)$ are $\Gamma$-representations.\vspace{2.6mm}\\
\noindent Because of Proposition \ref{gDmisacovering}, we have the exact diagram
\[
\xymatrix{
A_{m}(0)\ar[r]^-{r}&\prod_{g\in G^{0}}\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})\ar@<-.6ex>[r]_-{r_{2}}\ar@<.6ex>[r]^-{r_{1}}&\prod_{g,g'\in G^{0}}\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m}\cap g'D_{m})
}
\]
with the maps given by $r(f)=\big(f\big|_{gD_{m}}\big)_{g\in G^{0}}$, $r_{1}((f_{g})_{g\in G^{0}})=\big(f_{g}\big|_{gD_{m}\cap g'D_{m}}\big)_{g,g'\in G^{0}}$, and \linebreak $r_{2}((f_{g})_{g\in G^{0}})=\big(f_{g'}\big|_{gD_{m}\cap g'D_{m}}\big)_{g,g'\in G^{0}}$. The continuity of the restriction maps $\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})\longrightarrow\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m}\cap g'D_{m})$ between affinoid $\breve{K}$-algebras implies that the maps $r_{1}$ and $r_{2}$ are continuous for the product topology on their source and target. The Remark \ref{O(gDm)isDmod} allows us to view $\prod_{g\in G^{0}}\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})$ as a $\mathfrak{g}$-module with the componentwise $\mathfrak{g}$-action. Now, $A_{m}(0)$ can be identified with the kernel of the continuous map
\begin{align*}
r_{1}-r_{2}:\prod_{g\in G^{0}}\mathcal{O}&_{X^{\textnormal{rig}}_{m}}(gD_{m})\longrightarrow\prod_{g,g'\in G^{0}}\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m}\cap g'D_{m})\\&(f_{g})_{g\in G^{0}}\longmapsto r_{1}((f_{g})_{g\in G^{0}})-r_{2}((f_{g})_{g\in G^{0}}).
\end{align*} Hence, $A_{m}(0)$ is a closed $\Gamma$-stable subspace of $\prod_{g\in G^{0}}\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})$ as $r$ is $\Gamma$-equivariant. Consequently, $A_{m}(0)$ is stable under the induced $\mathfrak{g}$-action. \vspace{2.6mm}\\
\noindent Observe that the isomorphism $(\mathcal{M}^{s}_{m})^{\text{rig}}(gD_{m}\cap g'D_{m})\simeq\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m}\cap g'D_{m})\otimes_{R^{\text{rig}}_{m}}M^{s}_{m}$ yields a $\mathfrak{g}$-action on $(\mathcal{M}^{s}_{m})^{\text{rig}}(gD_{m}\cap g'D_{m})$ (cf. Theorem \ref{genlaKm} and Remark \ref{O(gDm)isDmod}). The restriction maps $(\mathcal{M}^{s}_{m})^{\text{rig}}(gD_{m})\longrightarrow(\mathcal{M}^{s}_{m})^{\text{rig}}(gD_{m}\cap g'D_{m})$ are continuous for the topology of finitely generated Banach modules. Then by the similar argument as in the last paragraph, $N^{s}_{m}(0)$ carries a $\mathfrak{g}$-module structure. The $\mathfrak{g}$-action and the $\Gamma$-action on $A_{m}(0)$ and on $N^{s}_{m}(0)$ are compatible with each other (cf. Remark \ref{O(gDm)isDmod}). \vspace{2.6mm}\\
\noindent Now since $M^{s}_{m}$ is generated over $R^{\textnormal{rig}}_{m}$ by $V_{s}$ (cf. (\ref{exact_seq_of_global_sections_of_eqv_bundles}), Proposition \ref{Lie(E0)_is_generically_flat}), $N^{s}_{m}(i)$ is generated by $V_{s}$ as an $A_{m}(i)$-module for all $0\leq i\leq h-1$. Let $A_{m}(i)^{\mathfrak{g}=0}V_{s}$ and $A_{m}(i)^{\mathfrak{g}=0}\varphi_{0}^{s}$ denote the $A_{m}(i)^{\mathfrak{g}=0}$-submodules of $N^{s}_{m}(i)$ generated by $V_{s}$ and $\varphi_{0}^{s}$ respectively.
\begin{proposition}\label{keyprop} For all $0\leq i\leq h-1$, $s\geq 0$, $m\geq 0$, we have $N^{s}_{m}(i)_{\textnormal{lf}}\subseteq A_{m}(i)^{\mathfrak{g}=0}V_{s}$ and $(N^{s}_{m}(i)_{\textnormal{lf}})^{\mathfrak{n}=0}\subseteq A_{m}(i)^{\mathfrak{g}=0}\varphi_{0}^{s}$. \end{proposition}
\begin{proof}
We first show that $N^{s}_{m}(0)_{\textnormal{lf}}\subseteq A_{m}(0)^{\mathfrak{g}=0}V_{s}$. Note that $\varphi_{0}\in\mathcal{O}_{\mathbb{P}_{\breve{K}}^{h-1}}(\Phi(D))^{\times}\hookrightarrow A_{m}(0)^{\times}$. This implies that $\varphi^{s}_{0}$ alone generates $N_{m}^{s}(0)$ as a free $A_{m}(0)$-module of rank one. Now let $W\subseteq N^{s}_{m}(0)$ be a finite dimensional $\Gamma$-stable subspace. As an $\mathfrak{sl}_{h}(K_{h})$-representation, $W$ decomposes as a direct sum of simple $\mathfrak{sl}_{h}(K_{h})$-modules by Weyl's complete reducibility theorem. From highest weight theory, we know that each simple module in the decomposition is generated by an element annihilated by the subalgebra $\mathfrak{n}$ of strictly upper triangular matrices. Now $W^{\mathfrak{n}=0}\subseteq N^{s}_{m}(0)^{\mathfrak{n}=0}=(A_{m}(0)\varphi^{s}_{0})^{\mathfrak{n}=0}=A_{m}(0)^{\mathfrak{n}=0}\varphi^{s}_{0}$ because $\mathfrak{n}\varphi^{s}_{0}=0$ (cf. Lemma \ref{explicit_g-action_lemma}). Let $f\in A_{m}(0)^{\mathfrak{n}=0}$, then $f\big|_{gD_{m}}\in \mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})^{\mathfrak{n}=0}=\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})^{\mathfrak{g}=0}$ for all $g\in G^{0}$ by Proposition \ref{ninvariantsequalginvariants}. The $\mathfrak{g}$-linear injection $A_{m}(0)\hookrightarrow\prod_{g\in G^{0}}\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})$ of $\breve{K}$-algebras induces an equality \begin{equation*}
A_{m}(0)^{\mathfrak{g}=0}=A_{m}(0)\cap\prod_{g\in G^{0}}\mathcal{O}_{X^{\textnormal{rig}}_{m}}(gD_{m})^{\mathfrak{g}=0}.
\end{equation*} Therefore, $f\in A_{m}(0)^{\mathfrak{g}=0}$, and hence $A_{m}(0)^{\mathfrak{g}=0}=A_{m}(0)^{\mathfrak{n}=0}$. \vspace{2.7mm}\\
\noindent Since the $\mathfrak{sl}_{h}(K_{h})$-simple modules in the decomposition of $W$ are also irreducible as $\Gamma$ \linebreak-representations, $W$ is generated over $\Gamma$ by its $\mathfrak{n}$-invariants and thus we have \begin{equation*}
W=\Gamma .W^{\mathfrak{n}=0}\subseteq \Gamma .(A_{m}(0)^{\mathfrak{g}=0}\varphi^{s}_{0})=A_{m}(0)^{\mathfrak{g}=0}V_{s}.
\end{equation*} The last equality follows from the observation that $A_{m}(0)^{\mathfrak{g}=0}$ is $\Gamma$-stable and $\Gamma .\varphi^{s}_{0}=V_{s}$. This proves the desired inclusion $N^{s}_{m}(0)_{\textnormal{lf}}\subseteq A_{m}(0)^{\mathfrak{g}=0}V_{s}$. On the way, we have also seen that $(N^{s}_{m}(0)_{\textnormal{lf}})^{\mathfrak{n}=0}\subseteq A_{m}(0)^{\mathfrak{g}=0}\varphi_{0}^{s}$.
\noindent If the $\Gamma$-action on $\Phi_{m}^{-1}(\Phi(D))$ is changed via the automorphism $\gamma\mapsto\Pi^{-i}\gamma\Pi^{i}$, then the map $\Pi^{i}:\Phi_{m}^{-1}(\Phi(D))\iso\Pi^{i}\Phi_{m}^{-1}(\Phi(D))$ is a $\Gamma$-equivariant isomorphism. We note that the new $\Gamma$-action does not change the locally finite vectors in $N^{s}_{m}(0)$. Writing $\varphi_{h}:=\varphi_{0}$ formally, we have an induced isomorphism $\Pi^{i}:(N^{s}_{m}(i))_{\textnormal{lf}}\iso (N^{s}_{m}(0))_{\textnormal{lf}}$ mapping $\varphi_{0}^{s}$ to $\varphi_{h-i}^{s}$, and the $\mathfrak{n}$-invariants onto the $\mathfrak{n}_{i}:=\textnormal{Ad}_{\Pi^{i}}(\mathfrak{n})$-invariants for all $0\leq i\leq h-1$. Therefore,
\begin{align*}
(N^{s}_{m}(i)_{\textnormal{lf}})^{\mathfrak{n}=0}=(\Pi^{i})^{-1}((N^{s}_{m}(0)_{\textnormal{lf}})^{\mathfrak{n}_{i}=0})&\subseteq (\Pi^{i})^{-1}((A_{m}(0)^{\mathfrak{g}=0}V_{s})^{\mathfrak{n}_{i}=0})\\&=(\Pi^{i})^{-1}(A_{m}(0)^{\mathfrak{g}=0}V_{s}^{\mathfrak{n}_{i}=0})\\&=(\Pi^{i})^{-1}(A_{m}(0)^{\mathfrak{g}=0}\varphi^{s}_{h-i})\\&=A_{m}(i)^{\mathfrak{g}=0}\varphi_{0}^{s}.
\end{align*}
\noindent As before, this also implies $N^{s}_{m}(i)_{\textnormal{lf}}\subseteq A_{m}(i)^{\mathfrak{g}=0}V_{s}$ for all $0\leq i\leq h-1$.
\end{proof}
\begin{theorem}\label{lf3} For all $s\geq 0$, $m\geq 0$, we have an isomorphism
\begin{equation*}
(M^{s}_{m})_{\textnormal{lf}}\simeq\breve{K}_{m}\otimes_{\breve{K}}V_{s}\simeq\breve{K}_{m}\otimes_{\breve{K}}\mathcal{O}_{\mathbb{P}_{\breve{K}}^{h-1}}(s)(\mathbb{P}_{\breve{K}}^{h-1})\simeq\breve{K}_{m}\otimes_{\breve{K}}\textnormal{Sym}^{s}(B_{h}\otimes_{K_{h}}\breve{K})
\end{equation*} of $\Gamma$-representations for the diagonal $\Gamma$-action on the tensor products. The representation $(M^{s}_{m})_{\textnormal{lf}}$ is a finite dimensional semi-simple representation of $\Gamma$.
\end{theorem}
\begin{proof}
As before, $(M^{s}_{m})_{\textnormal{lf}}$ is generated over $\Gamma$ by its $\mathfrak{n}$-invariants. Let $x\in((M^{s}_{m})_{\textnormal{lf}})^{\mathfrak{n}=0}$. Then, using the preceding proposition, $x\big|_{\Pi^{i}\Phi_{m}^{-1}(\Phi(D))}\in(N^{s}_{m}(i)_{\textnormal{lf}})^{\mathfrak{n}=0}\subseteq A_{m}(i)^{\mathfrak{g}=0}\varphi_{0}^{s}$ for all $0\leq i\leq h-1$. Let $Y_{i}:=\Pi^{i}\Phi_{m}^{-1}(\Phi(D))$, and write $x\big|_{Y_{i}}=f_{i}\varphi^{s}_{0}$ with $f_{i}\in A_{m}(i)^{\mathfrak{g}=0}$. \vspace{2.6mm}\\
\noindent For all $0\leq i,j\leq h-1$, we have $\big(f_{i}\big|_{Y_{i}\cap Y_{j}}-f_{j}\big|_{Y_{i}\cap Y_{j}}\big)\varphi^{s}_{0}=x\big|_{Y_{i}\cap Y_{j}}-x\big|_{Y_{i}\cap Y_{j}}=0$. Now $M^{s}_{m}$ is free over the integral domain $R^{\textnormal{rig}}_{m}$, and contains $\varphi^{s}_{0}\neq 0$. Hence the map $(r\mapsto r\varphi^{s}_{0})$ from $R^{\textnormal{rig}}_{m}$ to $M^{s}_{m}$ is injective and remains injective after any flat base change. In particular, the map $(r\mapsto r\varphi^{s}_{0}):\mathcal{O}_{X^{\textnormal{rig}}_{m}}(Y_{i}\cap Y_{j})\longrightarrow(\mathcal{M}^{s}_{m})^{\textnormal{rig}}(Y_{i}\cap Y_{j})$ is injective, and thus $f_{i}\big|_{Y_{i}\cap Y_{j}}=f_{j}\big|_{Y_{i}\cap Y_{j}}$ for all $0\leq i,j\leq h-1$. Therefore, by the sheaf axioms, the functions $(f_{i})_{i}$ glue together to a global section $f\in R^{\textnormal{rig}}_{m}$ and $x=f\varphi^{s}_{0}$. Since $f\big|_{Y_{i}}=f_{i}\in A_{m}(i)^{\mathfrak{g}=0}$ for all $i$, and the map $R^{\textnormal{rig}}_{m}\hookrightarrow\prod_{i=0}^{h-1}A_{m}(i)$ is $\mathfrak{g}$-equivariant, $f\in(R^{\textnormal{rig}}_{m})^{\mathfrak{g}=0}=\breve{K}_{m}$ (cf. Remark \ref{ginvinRm}). Hence $x\in\breve{K}_{m}\varphi^{s}_{0}$. As a result, $(M^{s}_{m})_{\textnormal{lf}}\subseteq\Gamma.(\breve{K}_{m}\varphi^{s}_{0})=\breve{K}_{m}V_{s}$. The other inclusion $\breve{K}_{m}V_{s}\subseteq (M^{s}_{m})_{\textnormal{lf}}$ is easy to see as $(M^{s}_{m})_{\textnormal{lf}}$ is a module over $(R^{\textnormal{rig}}_{m})_{\textnormal{lf}}=\breve{K}_{m}$, and $V_{s}=(M^{s}_{0})_{\textnormal{lf}}\subseteq (M^{s}_{m})_{\textnormal{lf}}$.\vspace{2.6mm}\\
\noindent Now to justify the isomorphism $\breve{K}_{m}\otimes_{\breve{K}}V_{s}\simeq\breve{K}_{m}V_{s}$, it is enough to show that the natural map
\begin{align*}
& \hspace{1.6cm} \breve{K}_{m}\otimes_{\breve{K}}V_{s}\longrightarrow \breve{K}_{m}V_{s}\\&
\sum_{0\leq|\alpha|\leq s}c_{\alpha}(1\otimes w^{\alpha}\varphi_{0}^{s})\longmapsto\sum_{0\leq|\alpha|\leq s}c_{\alpha} w^{\alpha}\varphi_{0}^{s}
\end{align*}
is injective. Here the set $\lbrace 1\otimes w^{\alpha}\varphi_{0}^{s}\rbrace_{0\leq|\alpha|\leq s}$ forms a $\breve{K}_{m}$-basis of $\breve{K}_{m}\otimes_{\breve{K}}V_{s}$. By Lemma \ref{explicit_g-action_lemma}, we have $\mathfrak{x}_{00}(w^{\alpha}\varphi_{0}^{s})=(s-|\alpha|)w^{\alpha}\varphi_{0}^{s}$ and $\mathfrak{x}_{ii}(w^{\alpha}\varphi_{0}^{s})=\alpha_{i}w^{\alpha}\varphi_{0}^{s}$ for all $1\leq i\leq h-1$. Since $\mathfrak{g}$ annihilates $\breve{K}_{m}$, if $\sum_{0\leq|\alpha|\leq s}c_{\alpha} w^{\alpha}\varphi_{0}^{s}=0$, one can use the above actions of the diagonal matrices iteratively to deduce that each summand $c_{\alpha} w^{\alpha}\varphi_{0}^{s}$ is zero, and therefore $c_{\alpha}=0$ for all $0\leq |\alpha|\leq s$.\vspace{2.6mm}\\
\noindent Unlike $(M^{s}_{0})_{\text{lf}}=V_{s}$, the space of locally finite vectors $(M^{s}_{m})_{\text{lf}}\simeq\breve{K}_{m}\otimes_{\breve{K}}V_{s}$ at level $m>0$ is not an irreducible $\Gamma$-representation as it properly contains the representation $V_{s}$. However; it is semi-simple and this can be seen as follows: the action of $\Gamma$ on $\breve{K}_{m}$ factors through a finite group (cf. Remark \ref{mLText}). As a result, $\breve{K}_{m}$ decomposes into a direct sum $\breve{K}_{m}\simeq\bigoplus_{i=1}^{n}W_{i}$ of irreducible representations. This gives us a decomposition \begin{equation}\label{ss_decomp}
\breve{K}_{m}\otimes_{\breve{K}}V_{s}\simeq \bigoplus_{i=1}^{n}(W_{i}\otimes_{\breve{K}}V_{s}).
\end{equation} Now we note that $V_{s}\simeq\textnormal{Sym}^{s}(B_{h}\otimes_{K_{h}}\breve{K})$ is an irreducible algebraic representation of $\Gamma\simeq\mathfrak{o}_{B_{h}}^{\times}$ (cf. Theorem \ref{top_finiteness_of_MsD_thm}, \cite{kohliwamo}, Remark 4.4), and $\breve{K}_{m}$ is a smooth representation of $\Gamma$ by Remark \ref{ginvinRm}. Thus every direct summand in (\ref{ss_decomp}) is a tensor product of a smooth irreducible representation and an irreducible algebraic representation of $\Gamma$. Such a representation is irreducible by \cite{stug}, Appendix by Dipendra Prasad, Theorem 1. As a consequence, $(M^{s}_{m})_{\text{lf}}$ is a semi-simple representation of $\Gamma$.
\end{proof}
\begin{remark}\label{la_rmk}
We recall from \cite{eme04} § 4.2 that a vector $v\in M^{s}_{m}$ is said to be \emph{locally algebraic} if there exists a finite dimensional algebraic representation $W$ of $\mathbb{B}_{h}^{\times}$ and an $H$-equivariant homomorphism $\phi:W^{n}\longrightarrow M^{s}_{m}$ with $v\in\phi(W^{n})$ for some open subgroup $H\subseteq\Gamma$ and $n\in\mathbb{N}$. The set of all locally algebraic vectors $(M^{s}_{m})_{\text{lalg}}$ forms a $\Gamma$-stable subspace of $M^{s}_{m}$ (cf. \cite{eme04}, Proposition-Definition 4.2.6), and it is clear that $(M^{s}_{m})_{\text{lalg}}\subseteq (M^{s}_{m})_{\text{lf}}$. However; Theorem 1 in \cite{stug}, Appendix by Dipendra Prasad asserts that the direct summands in the decomposition (\ref{ss_decomp}) of $\breve{K}_{m}\otimes_{\breve{K}}V_{s}$ are in fact irreducible locally algebraic representations of $\Gamma$. Therefore, our explicit results show that the space of locally finite vectors $(M^{s}_{m})_{\textnormal{lf}}=(M^{s}_{m})_{\text{lalg}}\simeq\breve{K}_{m}\otimes_{\breve{K}}V_{s}$ is actually a locally algebraic representation of $\Gamma$.
\end{remark}
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Picture it: the year is 5,000 B.C., and domesticated wheat and barley have been introduced to your neighborhood. You’re hanging out around the fire with your buddies one night, chewing the fat, making some stone tools, and getting ready to make some bread.
Stop for a second and take a look around. What do you see?
Mud huts? Yes. Dogs and sheep? Yes. Pottery? Yes. Bread maker? Uhhhh….no. Not a bread maker in sight.
And there you have it: historical proof that you don’t need a bread machine to make bread. Now, you don’t need dogs or sheep or pottery to make bread either, but you definitely don’t need a bread machine. In fact the only “bread machine” you need looks a little something like this:
If you have a pair of these, along with some simple ingredients, you’re all set to make bread Stone Age style.
Now … a few caveats. You’re going to need a couple things that our Stone Age buddies didn’t have. Like a stove, for instance, unless you want to build a fire. And yeast, too, which didn’t show up on the scene until maybe the 12th century B.C. So there’s that. But other than that, the main thing you’re going to need is something the Neos DID have, which maybe you have not so much of: time.
Which is not to say that surviving in the Stone Age was a 9-to-5 job, of course. It likely wasn’t. But in addition to surviving and making bread, you’ve probably got a lot of other things to do, like getting to work, driving the kids around, cleaning the hut — that sort of thing. So in recognition of that, we’re going to break it down like so:
We’ve included a simple, basic bread recipe below. The first iteration is for “Weekend Bread,” because it’s a longer process, and you’re going to need to come back to it a few times. It’s a great recipe when you’re hanging around the hut anyway, and you’ve got a day to cook. The second iteration is for “Weekday Bread.” Same recipe, but the process is truncated so you can start it in the morning, leave for the entire day, come back home, and put it in the oven. It makes a denser bread, but it’s still delicious, and it’s one less thing you have to think about when you’re out there hunting and gathering.
WEEKEND OAT BREAD
3 cups water
1 cup rolled oats
4 cups unbleached white flour (You can play around with this a bit and substitute some whole wheat, rye, or spelt flour for some of the white flour. However, we generally substitute no more than one cup because this ratio helps to maintain the bread’s lighter texture)
1 tsp salt
1 tsp yeast
olive oil
1. Boil 2 cups of water, add oats, reduce heat, and stir occasionally for 2-3 minutes. Once the oats are cooked, cover and let sit for about 5 minutes more. Remove the cover and allow oats to cool.
2. Once the oats are cool, transfer the oats into a bowl, and mix in 1 cup of cold water.
3. To the oats and water, add yeast, flour, add salt, and stir with a wooden spoon until well combined.
4. Knead the dough into a ball (this doesn’t take much–only about a dozen or so turns).
5. Coat the dough lightly in olive oil. It should look something like this:
6. Cover the dough with a dish towel or cloth, and allow to sit until it has doubled in volume. How long this takes is going to depend on a number of things, including how warm the room is, the amount of moisture in the air, the peppiness of the yeast, etc. But it should look noticeably softer, puffier and bigger. After it has doubled in size, the dough shown above looked like this:
See how it’s all puffed up?
7. Take the risen dough, and fold it over onto itself maybe a dozen times. Then gather it back into a ball, and it should look something like this:
8. Cover and let it rise again until it has doubled in size. You know what that looks like:
8. Fold it over maybe a dozen times, then gather the dough into a ball and cut the ball in half.
9. Shape the balls into two baguette-shaped loaves, roll them in flour, and place them in a lightly oiled loaf pan or on a baking sheet. A baking sheet will work just fine here, but if you plan on making bread more than occasionally, pick up an Italian loaf pan the next time you’re in a kitchen store, or ask someone to buy it for you for your birthday! It’s a neat tool, easy to clean, and it helps the loaves maintain their shape. This helps the bread to cook more evenly.
10. Score the loaves across the top gently with a sharp knife if you like. This makes the bread looks very official.
11. Let the loaves rise. How much? Well, let’s say double in volume, but at this point, depending on the mood your bread is in, the loaves may not rise as much as they did last time, or as much as you expect, and they may not double in size. That’s totally fine. Just give it a chance to rise a bit, and then call it. It should look something like this:
12. Preheat the oven to 500 degrees. Once it hits 500, turn it down to 450 and put the bread in. Cook the bread at 450 for 10 minutes, and then turn the oven down to 400 degrees and bake the bread for 20-25 minutes more.
And that’s it! Fresh baked bread made with your very own hands! Now for the shortened version:
WEEKDAY OAT BREAD
– Follow Steps 1 – 5 as written above. However, for this version you want to make the oats as cold as you can, so when you toss in that cup of cold water in Step 2, throw a couple ice cubes in the cup that you’re measuring the water in. (Don’t add them after you add the water, or once they melt, you will have more than a cup of total liquid). When you’re stirring all the ingredients together in Step 3, just make sure the ice has melted before adding the flour.
– Skip steps 6 and 7. Instead, oil the dough, the cover the bowl with plastic wrap or the lid from a large stock pot that will entirely cover the bowl, and just walk away! That’s right — walk away. Take all day, if you like. You want to go shopping? Cool. Catch a movie? Fine with us. Take a nap? You deserve it. Head over to the bakery? No, no — you’ve got bread doing its thing at home. And whenever you’re ready to return to it…
– Follow Steps 9 – 11 above. You can let the loaves rise in the pan while you’re making dinner or watching TV or going through the mail. It doesn’t need your attention, and it will only take 30-35 minutes to cook. And when it’s done, you’ll have some delicious homemade bread to snack on. Rock on!
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\appendix
\section{Adapted Weyl type estimates}
\begin{framed}
\begin{lem}
\label{lem:weyl_law_compact}Let $a_{\hbar}\in S_{\mu}\left(\langle x\rangle^{-2}\langle\xi\rangle^{-2}\right)$
with $0\leq\mu<\frac{1}{2}$ be a real compactly supported symbol.
$\forall\hbar>0,$ $\hat{A}:=\mbox{\emph{Op}}_{\hbar}^{w}(a_{\hbar})$
is self-adjoint and trace class on $L^{2}\left(\mathbb{R}\right)$
and for any $\epsilon>0$, as $\hbar\rightarrow0$ :
\begin{equation}
\left(2\pi\hbar\right)\sharp\left\{ \lambda_{i}^{\hbar}\in\sigma\left(\hat{A}\right)\,|\,\left|\lambda_{i}^{\hbar}\right|\geq\epsilon\right\} \leq C_{1}\mathrm{Leb}\left\{ (x,\xi)\,;\,|a|>0\right\} +C_{2}\hbar\label{eq:loi_weyl}
\end{equation}
where $C_{1}$ and $C_{2}$ depend only on $\mu$ and $\epsilon$.\end{lem}
\end{framed}
\begin{proof}
As $a_{\hbar}$ is compactly supported $\hat{A}$ is trace class for
every $\hbar$ (see theorem C.17 \cite{zworski-03}). Consequently
also $\frac{1}{\epsilon^{2}}\hat{A}^{2}$ is trace class and its trace
is given by Lidskii's theorem by $Tr(\frac{1}{\epsilon^{2}}\hat{A}^{2})=\sum{}_{i}\left(\frac{{\lambda_{i}^{\hbar}}}{\epsilon}\right)^{2}$.
As $\hat{A}$ is self adjoint all $\lambda_{i}^{\hbar}$ are real
and one clearly has
\[
\sharp\left\{ \lambda_{i}^{\hbar}\in\sigma\left(\hat{A}\right)\,|\,\left|\lambda_{i}^{\hbar}\right|\geq\epsilon\right\} \leq Tr\left(\frac{1}{\epsilon^{2}}\hat{A}^{2}\right).
\]
If we denote by $b_{\hbar}(x,\xi)$ the complete symbol of $\hat{A^{2}}$
we can calculate the trace by the following exact formula
\[
Tr(\hat{A}^{2})=\frac{1}{2\pi\hbar}\int b_{\hbar}(x,\xi)dxd\xi
\]
According to the theorem of composition of PDOs $b_{\hbar}$ can be
written as $b_{\hbar}=b_{\hbar}^{(1)}+\hbar^{1}b_{\hbar}^{(2)}$ where
$\mathrm{supp}b_{\hbar}^{(1)}=\mathrm{supp}a_{\hbar}$ and $b_{\hbar}^{(2)}\in\text{ \ensuremath{S_{\mu}\left(\langle x\rangle^{-2}\langle\xi\rangle^{-2}\right)}}$(note
that this decomposition depends on $\mu$. Thus
\[
\frac{1}{\epsilon\text{\texttwosuperior\ }}Tr(\hat{A}^{2})=\frac{1}{2\pi\hbar\epsilon^{2}}\left(\int b_{\hbar}^{(1)}(x,\xi)dxd\xi+\hbar^{1}\int b_{\hbar}^{(1)}(x,\xi)dxd\xi\right)\leq\left(\frac{1}{2\pi\hbar}C_{1}\textup{Leb}(\textup{supp}(a_{\hbar}))+C_{2}\hbar\right)
\]
\end{proof}
\section{General lemmas on singular values of compact operators}
Let $\left(P_{n}\right)_{n\in\mathbb{N}}$ be a family of compact
operators on some Hilbert space. For every $n\in\mathbb{N}$ let $(\lambda_{j,n})_{j\in\mathbb{N}}\in\mathbb{C}$
be the sequence of eigenvalues of $P_{n}$ counted with multiplicity
and ordered decreasingly:
\[
|\lambda_{0,n}|\geq|\lambda_{1,n}|\geq...
\]
In the same manner, define $(\mu_{j,n})_{j\in\mathbb{N}}\in\mathbb{R}^{+}$,
the decreasing sequence of singular values of $P_{n}$, i.e. the eigenvalues
of $\sqrt{P_{n}^{*}P_{n}}$ .
\begin{framed}
\begin{lem}
\label{lem:``Singular-and-eigenvalues} Suppose there exits a map
$N:\mathbb{N}\rightarrow\mathbb{N}$ s.t. $N(n)\underset{n\rightarrow\infty}{\rightarrow}\infty$
and \textup{$\mu_{N(n),n}\underset{n\rightarrow\infty}{\rightarrow}0$,
}then $\forall C>1,$ $|\lambda_{\left[C\cdot N(n)\right],n}|\underset{n\rightarrow\infty}{\rightarrow}0$
where $\left[.\right]$ stands for the integer part. \end{lem}
\begin{cor}
\label{cor:singular} Suppose there exits a map $N:\mathbb{N}\rightarrow\mathbb{N}$
s.t. $\forall\varepsilon>0,$ $\exists A_{\varepsilon}\geq0$ s.t.
$\forall n\geq A_{\varepsilon}$,
\[
\#\left\{ j\in\mathbb{N}\;\mbox{s.t.}\;\mu_{j,n}>\varepsilon\right\} <N\left(n\right),
\]
then $\forall C>1,$ $\forall\varepsilon>0$, $\exists B_{C,\varepsilon}\geq0$
s.t. $\forall n\geq B_{C,\varepsilon}$
\begin{equation}
\#\left\{ j\in\mathbb{N}\;\mbox{s.t.}\;|\lambda_{j,n}|>\varepsilon\right\} \leq C\cdot N\left(n\right).\label{eq:cor2}
\end{equation}
\end{cor}
\end{framed}
\begin{proof}
\emph{(Of corollary \ref{cor:singular})}. Suppose that for any $\varepsilon>0$,
there exists $A_{\varepsilon}$ s.t. for all $n\geq A_{\varepsilon}$,
$\#\left\{ j\in\mathbb{N}\;\mbox{s.t.}\;\mu_{j,n}>\varepsilon\right\} <N\left(n\right).$
Then $\mu_{N(n),n}\rightarrow_{n\rightarrow\infty}0$ and from Lemma
\ref{lem:``Singular-and-eigenvalues}, $\forall C>1$, $|\lambda_{[C\cdot N(n)],n}|\rightarrow_{n\rightarrow\infty}0,$
which can be directly restated as (\ref{eq:cor2}).
\end{proof}
\medskip{}
\begin{proof}
\emph{(Of lemma \ref{lem:``Singular-and-eigenvalues}).}Let $m_{j,n}:=-\log\mu_{j,n}$
and $l_{j,n}:=-\log|\lambda_{j,n}|$, $M_{k,n}:=\sum_{j=0}^{k}m_{j,n}$
and $L_{k,n}:=\sum_{j=0}^{k}l_{j,n}$. Weyl inequalities relate singular
values and eigenvalues by (see \cite{gohberg-00} p. 50 for a proof)
:
\begin{equation}
\prod_{j=1}^{k}\mu_{j,n}\leq\prod_{j=1}^{k}|\lambda_{j,n}|,\;\forall k\geq1.\label{eq:weyl_inequalities}
\end{equation}
This rewrites:
\begin{equation}
M_{k,n}\leq L_{k,n},\quad\forall k,n\label{eq:Weyl_ineq_2}
\end{equation}
The sequence $\left(l_{j,n}\right)_{j\geq0}$ is increasing in $j$
so, $\forall n,\forall k$ we have
\begin{equation}
k\cdot l_{k,n}\geq L_{k,n}.\label{eq:16}
\end{equation}
Suppose that $\mu_{N(n),n}\rightarrow0$ as $n\rightarrow\infty$
hence
\begin{equation}
m_{N(n),n}\underset{n\rightarrow\infty}{\rightarrow}\infty\label{eq:17}
\end{equation}
Let $C>1$. The sequence $\left(m_{j,n}\right)_{j\geq0}$ is increasing
in $j$ hence
\begin{equation}
M_{[C\cdot N(n)],n}\geq\left([C\cdot N(n)]-N(n)\right)\cdot m_{N(n),n},\label{eq:S_vs_m}
\end{equation}
hence
\begin{eqnarray*}
l_{[C\cdot N(n)],n} & \underset{(\ref{eq:16})}{\geq} & \frac{1}{[CN(n)]}\cdot L_{[C\cdot N(n)],n}\underset{(\ref{eq:Weyl_ineq_2})}{\geq}\frac{1}{[C\cdot N(n)]}M_{[C\cdot N(n)],n}\\
& \underset{(\ref{eq:S_vs_m})}{\geq} & \frac{[C\cdot N(n)]-N(n)}{[C\cdot N(n)]}\cdot m_{N(n),n}\underset{(\ref{eq:17})}{\longrightarrow}\infty
\end{eqnarray*}
Thus $l_{[C\cdot N(n)],n}\underset{n\rightarrow\infty}{\rightarrow}\infty$
and $|\lambda_{\left[C\cdot N(n)\right],n}|\underset{n\rightarrow\infty}{\rightarrow}0$.
\end{proof}
\section{Symbol classes of local $\hbar$-order\label{sec:Appendix-Symbol-classes}}
In this Appendix we will first repeat the definitions of the standard
symbol classes which are used in this article as well as their well
known quantization rules. Then we will introduce a new symbol class
which allows $\hbar$-dependent order functions and will prove some
of the classical results which are known in the usual case for these
new symbol classes.
\subsection{Standard semiclassical Symbol classes and their quantization}
The standard symbol classes (see e.g. \cite{zworski-03} chapter 4
or \cite{dimassi-99} ch 7) of $\hbar$PDO's are defined with respect
to an order function $f(x,\xi)$. This order function is required
to be a smooth positive valued function on $\mathbb{R}^{2n}$ such
that there are constants $C_{0}$ and $N_{0}$ fullfilling
\begin{equation}
f(x,\xi)\leq C_{0}\langle(x,\xi)-(x',\xi')\rangle^{N_{0}}f(x',\xi').\label{eq:OrderFunctionClassical}
\end{equation}
An important example of such an order function is given by $f(x,\xi)=\langle\xi\rangle^{m}$
with $k\in\mathbb{R}$.
\begin{framed}
\begin{defn}
\label{def:SymbolClass_S_mu}For $0\leq\mu\leq\frac{1}{2}$ the symbol
classes $\hbar^{k}S_{\mu}(m)$ contain all families of functions $a_{\hbar}(x,\xi)\in C^{\infty}(\mathbb{R}^{2n})$
parametrized by a parameter $\hbar\in]0,\hbar_{0}]$ such that
\[
|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a_{\hbar}(x,\xi)|\leq C\hbar^{k-\mu(|\alpha|+|\beta|)}f(x,\xi)
\]
where $C$ depends only on $\alpha$ and $\beta$.\end{defn}
\end{framed}
Unless we want to emphasize the dependence of the symbol $a_{\hbar}$
on $\hbar$ we will drop the index in the following. For the special
case of order function $f(x,\xi)=\langle\xi\rangle^{m}$ we also write
$S_{\mu}^{m}=S_{\mu}(\langle\xi\rangle^{m})$, if $\mu=0$ we write
$S(f):=S_{0}(f)$.
As quantization we use two different quantization rules in this article
which are called standard quantization respectively Weyl quantization.
\begin{framed}
\begin{defn}
\label{def:Quantization}Let $a_{\hbar}\in S_{\mu}\left(f\right)$
the Weyl quantization is a family of operators $\mbox{Op}_{\hbar}^{w}(a):\mathcal{S}\left(\mathbb{R}^{n}\right)\rightarrow\mathcal{S}\left(\mathbb{R}^{n}\right)$,
defined by
\begin{equation}
\left(\mbox{Op}_{\hbar}^{w}(a_{\hbar})\varphi\right)(x)=\left(2\pi\hbar\right)^{-n}\int e^{\frac{i}{\hbar}\xi(x-y)}a_{\hbar}\left(\frac{x+y}{2},\xi\right)\varphi(y)dyd\xi,\qquad\varphi\in\mathcal{S}\left(\mathbb{R}^{n}\right).\label{eq:weyl_quant}
\end{equation}
while the standard quantization $\mbox{Op}_{\hbar}(a):\mathcal{S}\left(\mathbb{R}^{n}\right)\rightarrow\mathcal{S}\left(\mathbb{R}^{n}\right)$
is given by
\begin{equation}
\left(\mbox{Op}_{\hbar}(a_{\hbar})\varphi\right)(x)=\left(2\pi\hbar\right)^{-n}\int e^{\frac{i}{\hbar}\xi(x-y)}a_{\hbar}\left(x,\xi\right)\varphi(y)dyd\xi,\qquad\varphi\in\mathcal{S}\left(\mathbb{R}^{n}\right).\label{eq:stand_quant}
\end{equation}
\end{defn}
\end{framed}
Both quantization extend continuously to operators on $\mathcal{S}^{\prime}(\mathbb{R}^{n})$.
While the standard quantization is slightly easier to define, the
Weyl quantization has the advantage, that real symbols are mapped
to formally self adjoint operators.
\subsection{Definition of the Symbol classes $S_{\mu}(A_{\hbar})$}
In this standard $\hbar$-PDO calculus the symbols are ordered by
there asymptotic behavior for $\hbar\to0$. If we take for example
a symbol $a\in\hbar^{k}S_{\mu}(f)$ then $a(x,\xi)$ is of order $\hbar^{k}$
for all $(x,\xi)\in\mathbb{R}^{2n}$. The symbol classes that we will
now introduce will also allow $\hbar-$dependent order function which
will allow to control the $\hbar$-order of a symbol locally, i.e.
in dependence of $(x,\xi)$. First we define these $\hbar$-dependent
order functions:
\begin{framed}
\begin{defn}
\label{def:h-OrderFunction}Let $f$ be an order function on $\mathbb{R}^{2n}$
and $0\leq\mu\leq\frac{1}{2}$. Let $A_{\hbar}\in S_{\mu}(f)$ a (possibly
$\hbar$-dependent) positive symbol such that for some $c\geq0$ there
is a constant $C$ that fulfills
\begin{equation}
A_{\hbar}(x,\xi)\geq C\hbar^{c}f(x,\xi)\label{eq:AlowerBd}
\end{equation}
\end{defn}
and that for all multiindices $\alpha,\beta\in\mathbb{N}^{n}$:
\begin{equation}
\left|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}A_{\hbar}(x,\xi)\right|\leq C_{\alpha,\beta}\hbar^{-\mu(|\alpha|+|\beta|)}A_{\hbar}(x,\xi)\label{eq:dxxiEstimate_for_OF}
\end{equation}
holds. Then we call $A_{\hbar}$an $\hbar$-dependent order function
and say $A_{\hbar}\in\mathcal{OF}^{c}(f)$\end{framed}
\begin{framed}
\begin{defn}
\label{def:hSymbolClassNew}The symbol class $S_{\mu}(A_{\hbar})$
is then defined to be the space of smooth functions $a_{\hbar}(x,\xi)$
defined on $R^{2n}$ and parametrized by $\hbar>0$ such that
\begin{equation}
\left|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a_{\hbar}(x,\xi)\right|\leq C_{\alpha,\beta}\hbar^{-\mu(|\alpha|+|\beta|)}A_{\hbar}(x,\xi)\label{eq:dxxiEstimate}
\end{equation}
By $\hbar^{k}S_{\mu}(A_{\hbar})$ we will as usual denote the symbols
$a_{\hbar}$ for which $\hbar^{-k}a_{\hbar}\in S_{\mu}(A_{\hbar})$\end{defn}
\end{framed}
As $A_{h}(x,\xi)\leq C_{0}f(x,\xi)$ and from (\ref{eq:dxxiEstimate_for_OF})
it is obvious, that
\begin{equation}
S_{\mu}(A_{\hbar})\subset S_{\mu}(f)\label{eq:IncSymbClass}
\end{equation}
and via this inclusion for $a_{\hbar}\in S_{\mu}(A_{\hbar})$ the
standard Quantization $Op_{\hbar}(a)$ and the Weyl quantization $Op_{\hbar}^{w}(a_{\hbar})$
are well defined and give continuous operators on $\mathcal{S}(\mathbb{R}^{n})$
respectively on $\mathcal{S}'(\mathbb{R}^{n})$. Furthermore equation
(\ref{eq:AlowerBd}) gives us a second inclusion
\begin{equation}
S_{\mu}(f)\subset\hbar^{-c}S_{\mu}(A_{\hbar})\label{eq:IncSymbClass2}
\end{equation}
thus combining these two inclusions we have:
\[
\hbar^{c}S_{\mu}(f)\subset S_{\mu}(A_{\hbar})\subset S_{\mu}(f)
\]
As for standard $\hbar-PDO$ symbol we can define asymptotic expansions:
\begin{framed}
\begin{defn}
Let $a_{j}\in S_{\mu}(A_{\hbar})$ for $j=0,1,\dots$ then we call
$\sum\limits _{j}\hbar^{j}a_{j}$ an asymptotic expansion of $a\in S_{\mu}(A_{\hbar})$
(writing $a\sim\sum\limits _{j}\hbar^{j}a_{j}$) if and only if:
\[
a-\sum\limits _{j<N}\hbar^{j}a_{j}\in\hbar^{N}S_{\mu}(A_{\hbar})
\]
\end{defn}
\end{framed}As in for the standard $\hbar$-PDOs we have some sort of Borel's
theorem also for symbols in $S_{\mu}(A_{\hbar})$
\begin{framed}
\begin{prop}
Let $a_{i}\in S_{\mu}(A_{\hbar})$ then there is a symbol $a\in S_{\mu}(A_{\hbar})$
such that
\begin{equation}
a-\sum\limits _{j<k}\hbar^{j}a_{j}\in\hbar^{k}S_{\mu}(A_{\hbar})\label{eq:AssExp1}
\end{equation}
\end{prop}
\end{framed}
\begin{proof}
Once more we can use the inclusion (\ref{eq:IncSymbClass}) into the
standard $h-PDO$ classes and obtain the existence of a symbol $a\in S_{\mu}(f)$
such that
\begin{equation}
a-\sum\limits _{j<k}\hbar^{j}a_{j}\in\hbar^{k}S_{\mu}(f)\label{eq:AssExp2}
\end{equation}
and we will show that this symbol belongs to $S_{\mu}(A_{\hbar})$
and that (\ref{eq:AssExp1}) holds: For the first statement we write
\[
a=\underbrace{a-\sum\limits _{j<c}\hbar^{j}a_{j}}_{\in\hbar^{c}S_{\mu}(f)}+\underbrace{\sum\limits _{j<c}\hbar^{j}a_{j}}_{\in S_{\mu}(A_{\hbar})}
\]
and use the inverse inclusion (\ref{eq:IncSymbClass2}).
In order to prove (\ref{eq:AssExp1}) write
\[
a-\sum\limits _{j<k}\hbar^{j}a_{j}=\underbrace{a-\sum\limits _{j<k+c}\hbar^{j}a_{j}}_{\in\hbar^{c+k}S_{\mu}(f)}+\underbrace{\sum\limits _{j=k}^{k+c-1}\hbar^{j}a_{j}}_{\in\hbar^{k}S_{\mu}(A_{\hbar})}
\]
and use once more (\ref{eq:IncSymbClass2}).
\end{proof}
The advantage of this new symbol class is, that the order function
$A_{\hbar}(x,\xi)$ itself can depend on $\hbar$ and thus the control
in $\hbar$ can be localized. A simple example for such an order function
would be $A_{\hbar}=\hbar^{m\mu}\langle\frac{\xi}{\hbar^{\mu}}\rangle^{m}\in\mathcal{OF}^{c}(\langle\xi\rangle^{m})$.
For $\xi\neq0$ this function is of order $\hbar^{0}$ whereas for
$\xi=0$ it is of order $\hbar^{m\mu}$. Thus also all symbols in
$S_{\mu}(A_{\hbar})$ have to show this behavior.
\subsection{Composition of symbols}
By using the inclusion (\ref{eq:IncSymbClass}) we will show a result
for the composition of Symbols absolutely analogous to the one in
the standard case Theorem 4.18 in \cite{zworski-03}. We first note
that for $A_{\hbar}\in\mathcal{OF}^{c_{A}}(f{}_{A})$ and $B_{\hbar}\in\mathcal{OF}^{c_{B}}(f_{B})$
the product formula for derivative yields that $A_{\hbar}B_{\hbar}\in\mathcal{OF}^{c_{A}+c_{B}}(f_{A}f_{B})$
and can now formulate the following theorem:
\begin{framed}
\begin{thm}
\label{thm:Comp} Let $A_{\hbar}\in\mathcal{OF}^{c_{A}}(f_{A})$ and
$B_{\hbar}\in\mathcal{OF}^{c_{B}}(f_{B})$ be two $\hbar$-dependent
order functions and $a\in S_{\mu}(A_{\hbar})$ and $b\in S_{\mu}(B_{\hbar})$
two $\hbar$-local symbols. Then there is a symbol
\[
a\#b\in S_{\mu}(A_{\hbar}B_{\hbar})
\]
such that
\begin{equation}
Op_{\hbar}^{w}(a)Op_{\hbar}^{w}(b)=Op_{\hbar}^{w}(a\#b)\label{eq:CompOp}
\end{equation}
as operators on $\mathcal{S}$ and the at first order we have
\begin{equation}
a\#b-ab\in\hbar^{1-2\mu}S_{\mu}(A_{\hbar}B_{\hbar})\label{eq:CompFirstOrder}
\end{equation}
\end{thm}
\end{framed}
\begin{proof}
The standard theorem of composition of $\hbar$-PDOs (see e.g. Th
4.18 in \cite{zworski-03}) together with the inclusion of symbol-classes
(\ref{eq:IncSymbClass}) provides us a symbol $a\#b\subset S_{\mu}(f_{A}\cdot f_{B})$
that fulfills equation (\ref{eq:CompOp}). Furthermore it provides
us with a complete asymptotic expansion for $a\#b$:
\begin{equation}
a\#b-\sum\limits _{k=0}^{N-1}\left(\frac{1}{k!}\left[\frac{i\hbar(\langle D_{x},D_{\eta}\rangle-\langle D_{y},D_{\xi}\rangle)}{2}\right]^{k}a(x,\xi)b(y,\eta)\right)_{|y=x,\eta=\xi}\in\hbar^{N(1-2\mu)}S_{\mu}(f_{A}\cdot f_{B})\label{eq:CompAsymptExp}
\end{equation}
In order to prove our theorem it thus only rests to show, that $a\#b\in S_{\mu}(A_{\hbar}B_{\hbar})$
and that equation (\ref{eq:CompFirstOrder}) holds. We start with
the second one. First let $N\in\mathbb{N}$ be such that $(N-1)(1-2\mu)\geq c_{A}+c_{B}$,
then equation (\ref{eq:CompAsymptExp}) and inclusion (\ref{eq:IncSymbClass2})
assure that the remainder term in (\ref{eq:CompAsymptExp}) is in
$\hbar^{1-2\mu}S_{\mu}(A_{\hbar}B_{\hbar})$. For $0\leq k\leq N-1$
each term in (\ref{eq:CompAsymptExp}) can be written as a sum of
finitely many terms of the form
\[
\frac{(i\hbar)^{k}}{2^{k}k!}\left(D_{x}^{\alpha}D_{\xi}^{\beta}a(x,\xi)\right)\cdot\left(D_{x}^{\gamma}D_{\xi}^{\delta}b(x,\xi)\right)
\]
where $\alpha,\beta,\gamma,\delta\in\mathbb{N}^{n}$ are multiindices
fulfilling $|\alpha|+|\beta|+|\gamma|+|\delta|=2k$. Via the product
formula one easily checks, that these terms are all in $\hbar^{k(1-2\mu)}S_{\mu}(A_{\hbar}B_{\hbar})$
which proves that $a\#b\in S_{\mu}(A_{\hbar}B_{\hbar})$.
\end{proof}
\subsection{Ellipticity and inverses}
In this section we will define ellipticity for our new symbol classes
and will prove a result on $L^{2}$-invertibility.
\begin{framed}
\begin{defn}
We call a symbol $a\in S_{\mu}(A_{\hbar})$ elliptic if there is a
constant $C$ such that:
\begin{equation}
|a(x,\xi)|\geq CA_{\hbar}(x,\xi)\label{eq:ElipCond}
\end{equation}
\end{defn}
\end{framed}For an $\hbar$-dependent order function $A_{\hbar}\in\mathcal{OF}^{c}(f)$,
from (\ref{eq:dxxiEstimate_for_OF}) and (\ref{eq:AlowerBd}) it follows,
that $\hbar^{c}A_{\hbar}^{-1}\in\mathcal{OF}^{c}(f^{-1})$ is again
a $\hbar$- dependent order function and we can formulate the following
proposition:
\begin{framed}
\begin{prop}
If $a\in S_{\mu}(A_{\hbar})$ is elliptic then $a^{-1}\in\hbar^{-c}S_{\mu}(\hbar^{c}A_{\hbar}^{-1})$\end{prop}
\end{framed}
\begin{proof}
We have to show, that $|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a^{-1}(x,\xi)|\leq C\hbar^{-\mu(|\alpha|+|\beta|)}A_{\hbar}^{-1}(x,\xi)$
uniformly in $\hbar,x$ and $\xi$. For some first derivative (i.e.
for $\alpha\in\mathbb{N}^{2n},|\alpha|=1$) we have
\[
|\partial_{x,\xi}^{\alpha}a^{-1}|=\frac{|\partial_{x,\xi}^{\alpha}a|}{|a^{2}|}\leq C\frac{\hbar^{-\mu}A_{\hbar}}{A_{\hbar}^{2}}=C\hbar^{-\mu}A_{\hbar}^{-1}
\]
where the inequality is obtained by (\ref{eq:dxxiEstimate_for_OF})
and (\ref{eq:ElipCond}). The estimates of higher order derivatives
can be obtained by induction.
\end{proof}
As for standard $\hbar$-PDOs this notion of ellipticity implies that
the corresponding operators are invertible for sufficiently small
$\hbar$.
\begin{framed}
\begin{prop}
\label{prop:L2inverse}Let $A_{\hbar}\in\mathcal{OF}^{c}(1)$ and
$a\in S_{\mu}(A_{\hbar})$ be an elliptic symbol, then $Op_{\hbar}^{w}(a):L^{2}(\mathbb{R}^{n})\to L^{2}(\mathbb{R}^{n})$
is a bounded operator. Furthermore there exists $\hbar_{0}>0$ such
that $Op_{\hbar}^{w}(a)$ is invertible for all $\hbar\in]0,\hbar_{0}]$.
Its inverse is again bounded and a pseudodifferential operator $Op_{\hbar}^{w}(b)$
with symbol $b\in S_{\mu}(A_{\hbar}^{-1})$. At leading order its
symbol is given by
\[
b-a^{-1}\in\hbar^{1-2\mu}S_{\mu}(A_{\hbar}^{-1})
\]
\end{prop}
\end{framed}
\begin{proof}
As $a\in S_{\mu}(A_{\hbar})\subset S_{\mu}(1)$ the boundedness of
$Op_{\hbar}^{w}(a)$ follows from theorem 4.23 in \cite{zworski-03}.
By theorem \ref{thm:Comp} we calculate
\[
Op_{\hbar}^{w}(a)Op_{\hbar}^{w}(a^{-1})=Id+R
\]
where $R=Op_{\hbar}^{w}(r)$ is a PDO with symbol $r\in\hbar^{1-2\mu}S_{\mu}(1)$.
Again from theorem 4.23 in \cite{zworski-03} we obtain $\|R\|_{L^{2}}\leq C\hbar^{1-2\mu}$
thus there is $\hbar_{0}$ such that $\|R\|_{L^{2}}<1$ for $\hbar\in]0,\hbar_{0}]$.
According to theorem C.3 in \cite{zworski-03} we can conclude that
$Op_{\hbar}^{w}(a)$ is invertible and that the inverse is given by
$Op_{\hbar}^{w}(a^{-1})(Id+R)^{-1}$. The semiclassical version of
Beals theorem allows us to conclude that $(Id+R)^{-1}=\sum\limits _{k=0}^{\infty}(-R)^{k}$
is a PDO with symbol in $S_{\mu}(1)$ (cf. theorem 8.3 and the following
remarks in \cite{zworski-03}). The representation of $(Id-R)^{-1}$
as a series finally gives us the symbol of the inverse operator at
leading order.
\end{proof}
\subsection{Egorov's theorem for diffeomorphisms}
In this section we will study the behavior of symbols $a\in S_{\mu}(A_{\hbar})$
under variable changes. Let $\gamma:\mathbb{R}^{n}\to\mathbb{R}^{n}$
be a diffeomorphism that equals identity outside some bounded set
then the pullback with this coordinate change acts as a continuous
operator on $\mathcal{S}(\mathbb{R}^{n})$ by:
\[
(\gamma^{*}u)(x):=u(\gamma(x))
\]
Which can be extended by its adjoint to a continuous operator $\gamma^{*}:\mathcal{S}'(\mathbb{R}^{n})\to\mathcal{S}'(\mathbb{R}^{n})$.
By a variable change of an operator we understand its conjugation
by $\gamma$ and we are interested for which $a\in S_{\mu}(A_{\hbar})$
the conjugated operator $(\gamma^{*})^{-1}Op_{\hbar}(a)\gamma^{*}$
is again a $\hbar$-PDO with symbol $a_{\gamma}$. At leading order
this symbol will be the composition of the original symbol with the
so called canonical transformation
\[
T:\mathbb{R}^{2n}\to\mathbb{R}^{2n},(x,\xi)\mapsto(\gamma^{-1}(x),(\partial\gamma(\gamma^{-1}(x)))^{T}\xi)
\]
and the symbol class of $a_{\gamma}$ will be $S_{\mu}(A_{\hbar}\circ T)$.
For the $A_{\hbar}\in\mathcal{OF}^{c}(f)$ defined in Definition \ref{def:h-OrderFunction}
the composition $A_{\hbar}\circ T$ will in general however not be
a $\hbar$-dependent order function itself because the derivatives
in $x$ create a supplementary $\xi$ factor which has to be compensated
(cf. discussion in chapter 9.3 in \cite{zworski-03}). We therefore
demand in this section that our order function $A_{\hbar}$ satisfies:
\[
\left|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}A_{\hbar}(x,\xi)\right|\leq C_{\alpha,\beta}\hbar^{\mu(|\alpha|+|\beta|)}\langle\xi\rangle^{-|\beta|}A_{\hbar}(x,\xi)
\]
A straightforward calculation shows then, that $A_{\hbar}\circ T\in\mathcal{OF}^{c}(f\circ T)$
is again a $\hbar$- dependent order function. The same condition
has to be fulfilled by the symbol of the conjugated operator:
\begin{equation}
|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\leq\hbar^{-\mu(|\alpha|+|\beta|)}\langle\xi\rangle^{-|\beta|}A_{\hbar}(x,\xi)\label{eq:EgoDecayXiCondition}
\end{equation}
\begin{framed}
\begin{thm}
\label{thm:Ego}Let $a\in S_{\mu}(A_{\hbar})$ be an symbol which
fulfills (\ref{eq:EgoDecayXiCondition}) and has compact support in
$x$ (i.e. $\overline{\{x\in\mathbb{R}^{n}|\exists\xi\in\mathbb{R}^{n}:a(x,\xi)\neq0\}}$
is compact) and let $\gamma:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a
diffeomorphism. Then there is a symbol $a_{\gamma}\in S_{\mu}(A_{\hbar}\circ T)$
such that
\begin{equation}
(Op_{\hbar}(a_{\gamma})u)(\gamma(x))=(Op_{\hbar}(a)(u\circ\gamma))(x)\label{eq:EgoOpIdentity}
\end{equation}
for all $u\in\mathcal{S}'(\mathbb{R}^{n})$. Furthermore $a_{\gamma}$
has the following asymptotic expansion.
\begin{equation}
a_{\gamma}(\gamma(x),\eta)\sim\sum\limits _{n=0}^{k-n}\frac{1}{\nu!}\langle i\frac{\hbar}{\langle\eta\rangle}D_{y},D_{\xi}\rangle^{\nu}e^{\frac{i}{\hbar}\langle\rho_{x}(y),\eta\rangle}a(x,\xi)_{\big|y=0,\xi=(\partial\gamma(x))^{T}\eta}\label{eq:EgoAssExp}
\end{equation}
where $\rho_{x}(y)=\gamma(y+x)-\gamma(x)-\gamma'(x)y$. The terms
of the series are in $\hbar^{\frac{\nu(1-2\mu)}{2}}S_{\mu}(\langle\eta\rangle^{\frac{\nu}{2}}A_{\hbar}\circ T(\gamma(x),\eta))$.\end{thm}
\end{framed}We will prove this theorem similar to theorem 18.1.17 in \cite{hormander_3}
by using a parameter dependent stationary phase approximation (Thm7.7.7
in \cite{hormander_1}) as well as the following proposition which
forms the analog to Proposition 18.1.4 of \cite{hormander_3} for
our symbol classes and which we will prove first.
\begin{framed}
\begin{prop}
\label{prop:AssExp}Let $a(x,\xi;\hbar)\in C^{\infty}(\mathbb{R}^{2n})$
a family of smooth functions that fulfills
\begin{equation}
|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\leq C\hbar^{-l}\langle\xi\rangle^{l}f(x,\xi)\label{eq:AssCond1}
\end{equation}
where $C$ and $l$ may depend on $\alpha$ and $\beta$. Let $a_{j}\in S_{\mu}(A_{\hbar})$,
$j=0,1,\dots$ be a sequence of symbols such that
\begin{equation}
|a(x,\xi)-\sum\limits _{j<k}\hbar^{j}a_{j}(x,\xi)|\leq C\hbar^{\tau k}\langle\xi\rangle^{-\tau k}f(x,\xi)\label{eq:AssCond2}
\end{equation}
where $\tau>0$. Then $a\in S_{\mu}(A_{\hbar})$ and $a\sim\sum\hbar^{j}a_{j}$.\end{prop}
\end{framed}
\begin{proof}
We have to show that for all $k\geq0$ and $g_{k}(x,\xi):=a(x,\xi)-\sum\limits _{j<k}\hbar^{j}a_{j}(x,\xi)$
we have $|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}g_{k}|\leq C\hbar^{k-\mu(|\alpha|+|\beta|)}A_{\hbar}$.
This result can be obtained by iterating the following argument for
the first derivative in $x_{1}$:
Let $e_{1}\in\mathbb{R}^{n}$ be the first eigenvector and $0<\varepsilon<1$.
For arbitrary $j\in\mathbb{N}$ we can write by Taylor's Formula
\[
|g_{j}(x+\varepsilon e_{1},\xi)-g_{j}(x,\xi)-\partial_{x_{1}}g_{j}(x,\xi)\varepsilon)|\leq C\varepsilon^{2}\sup_{t\in[0,\varepsilon]}|\partial_{x_{1}}^{2}g_{j}(x+te_{1},\xi)|
\]
From (\ref{eq:AssCond1}) and the property, that all $a_{j}$ are
in $S_{\mu}(A_{\hbar})$ we get
\[
\sup_{t\in[0,\varepsilon]}|\partial_{x_{1}}^{2}g_{j}(x+te_{1},\xi)|\leq C\hbar^{-l}\langle\xi\rangle^{l}f(x,\xi)
\]
for some $l\in\mathbb{R}$ and get
\[
|\partial_{x_{1}}g_{j}(x,\xi)|\leq C\varepsilon\hbar^{-l}\langle\xi\rangle^{l}m(x,\xi)+\frac{|g_{j}(x+\varepsilon e_{1},\xi)-g_{j}(x,\xi)|}{\varepsilon}
\]
which turns for $j>\frac{2k+2c+l}{\tau}$ and $\varepsilon=\hbar^{k+l+c}\langle\xi\rangle^{-(k+l+c)}$
into:
\[
|\partial_{x_{1}}g_{j}(x,\xi)|\leq C\hbar^{c+k}\langle\xi\rangle^{-(c+k)}f(x,\xi)\leq C\hbar^{k}A_{\hbar}(x,\xi)
\]
where we used (\ref{eq:IncSymbClass2}) in the second equation. Thus
\[
|\partial_{x_{1}}g_{k}(x,\xi)|\leq C\hbar^{k}A_{\hbar}(x,\xi)+|\sum\limits _{i=k}^{j}\hbar^{i}\partial_{x_{1}}a_{i}(x,\xi)|\leq C\hbar^{k-\mu}A_{\hbar}(x,\xi)
\]
which finishes the proof.
\end{proof}
After having proven this proposition we can start with the proof of
theorem \ref{thm:Ego}:
\begin{proof}
If we define
\begin{equation}
a_{\gamma}(\gamma(x),\eta):=e^{-\frac{i}{\hbar}\gamma(x)\eta}Op_{\hbar}(a)e^{\frac{i}{\hbar}\gamma(\cdot)\eta}\label{eq:EgoDefaGamma}
\end{equation}
then equation (\ref{eq:EgoOpIdentity}) holds for all $e^{\frac{i}{\hbar}x\eta}$
which form a dense subset of $\mathcal{S}'(\mathbb{R}^{n})$. We thus
have to show that $a_{\gamma}$ defined in (\ref{eq:EgoDefaGamma})
is in $S_{\mu}(A_{\hbar})$ and that (\ref{eq:EgoAssExp}) holds.
We will first write $a_{\gamma}$ as an oscillating integral in order
to apply the stationary phase theorem. By definition of $Op_{\hbar}(a)$
one obtains
\[
a_{\gamma}(\gamma(x),\eta)=\frac{1}{(2\pi\hbar)^{n}}\iint a(x,\tilde{\xi})e^{\frac{i}{\hbar}((x-\tilde{y})\tilde{\xi}+(\gamma(\tilde{y})-\gamma(x))\eta)}d\tilde{y}d\tilde{\xi}
\]
which we can transform by a variable transformation $\tilde{\xi}=\langle\eta\rangle\xi$
and $\tilde{y}=y+x$ into
\[
a_{\gamma}(\gamma(x),\eta)=\frac{1}{(2\pi\tilde{\hbar})^{n}}\iint a(x,\langle\eta\rangle\xi)e^{\frac{i}{\tilde{\hbar}}(-y\xi+(\gamma(y+x)-\gamma(x))\frac{\eta}{\langle\eta\rangle})}dyd\xi
\]
where $\tilde{\hbar}=\frac{\hbar}{\langle\eta\rangle}$.
The critical points of the phase function are given by
\[
y=0\textup{ and }\xi=(\partial\gamma(x))^{T}\frac{\eta}{\langle\eta\rangle}
\]
Let $\chi\in C_{c}^{\infty}([-2,2]^{n})$ such that $\chi=1$ on $[-1,1]^{n}$
then we can write
\[
a_{\gamma}(\gamma(x),\eta)=I_{1}(\tilde{\hbar})+I_{2}(\tilde{\hbar})
\]
with
\[
I_{1}(\tilde{\hbar})=\frac{1}{(2\pi\tilde{\hbar})^{n}}\iint\chi\left(y\right)\chi\left(\xi-(\partial\gamma(x))^{T}\frac{\eta}{\langle\eta\rangle}\right)a(x,\langle\eta\rangle\xi)e^{\frac{i}{\tilde{\hbar}}(-y\xi+(\gamma(y+x)-\gamma(x))\frac{\eta}{\langle\eta\rangle})}dyd\xi
\]
and
\[
I_{2}(\tilde{\hbar})=\frac{1}{(2\pi\tilde{\hbar})^{n}}\iint\left(1-\chi\left(y\right)\chi\left(\xi-(\partial\gamma(x))^{T}\frac{\eta}{\langle\eta\rangle}\right)\right)a(x,\langle\eta\rangle\xi)e^{\frac{i}{\tilde{\hbar}}(-y\xi+(\gamma(y+x)-\gamma(x))\frac{\eta}{\langle\eta\rangle})}dyd\xi.
\]
While $I_{1}(\hbar)$ still contains critical points, for $I_{2}(\hbar)$
there are no critical points in the support of the integrand anymore.
$I_{1}$ is of the form studied in theorem 7.7.7 in \cite{hormander_1}.
Here the role of $x$ and $y$ is interchanged and there is an additional
parameter $\frac{\eta}{\langle\eta\rangle}$. We thus get from this
stationary phase theorem
\begin{equation}
\begin{array}{lc}
\left|I_{1}(\tilde{\hbar})-\sum\limits _{\nu=0}^{k-n}\frac{1}{\nu!}\langle i\tilde{\hbar}D_{y},D_{\xi}\rangle^{\nu}e^{\frac{i}{\tilde{\hbar}}\langle\rho_{x}(y),\frac{\eta}{\langle\eta\rangle}\rangle}u(x,\xi,y,\eta)_{\big|y=0,\xi=(\partial\gamma(x))^{T}\frac{\eta}{\langle\eta\rangle}}\right|\\
\leq C\tilde{\hbar}^{\frac{k+n}{2}}\sum\limits _{|\alpha|\leq2k}\sup_{y,\xi}|D_{y,\xi}^{\alpha}u(x,\xi,y,\eta)|
\end{array}\label{eq:EgoExp1}
\end{equation}
where $u(x,\xi,y,\eta)=\chi\left(y\right)\chi\left(\xi-(\partial\gamma(x))^{T}\frac{\eta}{\langle\eta\rangle}\right)a(x,\langle\eta\rangle\xi)$.
Because of (\ref{eq:EgoDecayXiCondition}) and (\ref{eq:OrderFunctionClassical})
we can estimate
\[
\sup_{y,\xi}|D_{y,\xi}^{\alpha}u(x,\xi,y,\eta)|\leq C\hbar^{-\mu|\alpha|}f(x,(\partial\gamma(x))^{T}\eta)=C\hbar^{-\mu|\alpha|}f\circ T(\gamma(x),\eta)
\]
Thus transforming the expansion (\ref{eq:EgoExp1}) back to an expansion
in $\hbar$ we get
\[
\begin{array}{lc}
\left|I_{1}(\hbar)-\sum\limits _{\nu=0}^{k-n}\frac{1}{\nu!}\langle i\frac{\hbar}{\langle\eta\rangle}D_{y},D_{\xi}\rangle^{\nu}e^{\frac{i}{\hbar}\langle\rho_{x}(y),\eta\rangle}u(x,\xi,y,\eta)_{\big|y=0,\xi=(\partial\gamma(x))^{T}\frac{\eta}{\langle\eta\rangle}}\right|\\
\leq C\hbar^{\frac{k(1-2\mu)+n}{2}}\langle\eta\rangle^{-\frac{k+n}{2}}f\circ T(\gamma(x),\eta)
\end{array}
\]
As the stationary points for $I_{2}$ are not contained in the support
of the integrand we get by the non stationary phase theorem:
\[
|I_{2}(\hbar)|\leq C\left(\frac{\hbar}{\langle\eta\rangle}\right)^{N}f\circ T(\gamma(x),\eta)
\]
for all $N\in\mathbb{N}$. Thus we finally get
\begin{equation}
\begin{array}{lc}
\left|a_{\gamma}(\gamma(x),\eta)-\sum\limits _{\nu=0}^{k-n}\frac{1}{\nu!}\langle i\frac{\hbar}{\langle\eta\rangle}D_{y},D_{\xi}\rangle^{\nu}e^{\frac{i}{\hbar}\langle\rho_{x}(y),\eta\rangle}u(x,\xi,y,\eta)_{\big|y=0,\xi=(\partial\gamma(x))^{T}\frac{\eta}{\langle\eta\rangle}}\right|\\
\leq C\hbar^{\frac{k(1-2\mu)+n}{2}}\langle\eta\rangle^{-\frac{k+n}{2}}f\circ T(\gamma(x),\eta)
\end{array}\label{eq:EgoExp2}
\end{equation}
If we show that the elements of the series are in $\hbar^{\frac{\nu(1-2\mu)}{2}}S_{\mu}(\langle\eta\rangle^{\frac{\nu}{2}}A_{\hbar}\circ T(\gamma(x),\eta))$
then this equation is of the form (\ref{eq:AssCond2}). The terms
of order $\nu$ in the series are of the form
\[
\left.\left(\frac{i\hbar}{\langle\eta\rangle}\right)^{\nu}\partial_{y}^{\alpha}e^{\frac{i}{\hbar}\langle\rho_{x}(y),\eta\rangle}(\partial_{\xi}^{\alpha}a)(x,(\partial\gamma(x))^{T}\eta)\langle\eta\rangle^{\nu}\right._{\big|y=0}
\]
Where $\alpha\in\mathbb{N}^{n}$ with $|\alpha|=\nu$. The second
factor $(\partial_{\xi}^{\alpha}a)(x,(\partial\gamma(x))^{T}\eta)\langle\eta\rangle^{\nu}$
is in $\hbar^{-\mu\nu}S_{\mu}(A_{\hbar}\circ T(\gamma(x),\eta))$
as we demanded the condition (\ref{eq:EgoDecayXiCondition}) on our
symbol $a$. Thus it remains to show that the other factor is of order
$\left(\frac{\hbar}{\langle\eta\rangle}\right)^{\frac{\nu}{2}}$ on
the support of $a$. This is the case because $\rho_{x}(y)$ vanishes
at second order in $y=0$. Each derivative of $e^{\frac{i}{\hbar}\langle\rho_{x}(y),\eta\rangle}$
produces a factor $\frac{i}{\hbar}\langle\partial_{y_{i}}\rho_{x}(0),\eta\rangle$.
But as $\partial_{y_{i}}\rho_{x}(0)$ vanishes we need a second derivative,
now acting on $\partial_{y_{i}}\rho_{x}(y)$, in order to get a contribution.
Thus in the worst case $\partial_{y}^{\alpha}e^{\frac{i}{\hbar}\langle\rho_{x}(y),\eta\rangle}$
is of order $\left(\frac{\hbar}{\langle\eta\rangle}\right)^{-\frac{\nu}{2}}$.
Thus we have shown that (\ref{eq:EgoExp2}) is of the form (\ref{eq:AssCond2}).
The last thing that we have to show is thus, that $a_{\gamma}$ fulfills
(\ref{eq:AssCond1}). If we consider the definition (\ref{eq:EgoDefaGamma})
of $a_{\gamma}$ we see that $\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a_{\gamma}(\gamma(x),\eta)$
can be written as a sum of terms of the form $\frac{P(\eta)}{\hbar^{k}}e^{-\frac{i}{\hbar}\gamma(x)\eta}Op_{\hbar}(b)e^{\frac{i}{\hbar}\gamma(\cdot)\eta}$
where $b\in S_{\mu}(A_{\hbar}\langle\xi\rangle^{j})$ and $P(\eta)$
is a polynomial in $\eta$. The constants $j,k$ and the degree of
$P(\eta)$ depend on $\alpha$ and $\beta$. Thus writing these terms
as oscillating integrals and applying the same arguments as above
one gets (\ref{eq:AssCond1}).
We have thus shown that all the conditions for proposition \ref{prop:AssExp}
are fulfilled and can conclude that $a_{\gamma}$ belongs to $S_{\mu}(A_{\hbar})$
and that (\ref{eq:EgoExp2}) is also an asymptotic expansion w.r.t.
the order function $A_{\hbar}$.\end{proof}
| 2,225
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Pastor Didi Panzo in the Democratic Republic of the Congo
By Pastor Didi Panzo
When a farmer plants seeds in the ground, he does not know what to expect: the seeds might sprout to produce a crop, or they may die. During the visit of the Rev. Dr. Paul Gossman, World Mission Prayer League (WMPL) executive director, and Gloria Suck, our colleague missionary in Kenya, we visited Stanley Baoba and the Port of Boma.
At the port, we were well received by Christian workers and they invited us to visit one of their offices. We went to learn about what it is like to be a Christian working at the port, where no salary is paid for over 10 months. As the officer asked us to pray, Paul prayed that the place where we sat may turn into a place where people or colleagues of the company may be strengthened as they seek God’s comfort in their suffering.
Now the offices are a place of prayer for many Christians who work at the port. Some even come to discover and receive Jesus as their Savior. Now, they are growing in number as the port director gave them a larger room where they may spend their prayer time, which they call “One Hour for Jesus.” Also, the group wants to explore the possibility of beginning a seafarer ministry at that port. If someone has experience with seafarer ministries, please share with us for the benefit of the group and for God’s glory.
Suicide
Azimack was a young, married farmer with three children. He used to feed his family from his farming income. A few weeks back, Azimack changed his behavior and became a heavy drinker. He didn’t care about his family and argued with his wife every day. Azimack told no one of what was going on in his life. On Saturday, March 23, Azimack again changed his way of living. He became happier, receptive and began talking again with his wife, kids and others. It turns out he had a plan to commit suicide. He was found dead that evening inside the bedroom of his wife’s parents. Please pray for his wife and kids.
Less Production for Farmers
The city of Boma and its surrounding areas suffered a hot season of 40 degrees Celsius (104 degrees Fahrenheit) during December and January. All the plants withered and died. Farmers are complaining that they are unable to harvest enough to feed their families and to generate income. We are encouraging them to rely on God, the One who can answer their “Why?” Please pray for them.
Troubling Road
During the month of March, it rained a lot in the two territories of Lukula and Tshela. That rain caused the deterioration of the road Boma-Tshela. It now takes two days to travel a distance of 120 kilometers (72 miles). This is the road I use every week to go teach at the theological school in Tshela. Two times we were stopped and could not pass because of a car that was stuck in the mud. Traveling to Tshela requires dedication and sacrifice. Pray for the Democratic Republic of the Congo (DRC) government to organize the infrastructure of the country.
Women’s Convention
The women of Confessional Lutheran Churches in Congo (CELCCO) and the Evangelical Lutheran Church in Congo (EELCO) gathered for a convention organized by the women’s regional office of CELCCO. The main theme of the convention was based on the book of Esther, chapter five. They encouraged all women to play the role of intercessor — asking for God’s favor for the situation that the DRC is facing. There also was a day that women spoke out against domestic violence, sexual abuse and injustice. Mrs. Elisee, who faced domestic violence for many years, recovered from trauma through our counseling program and was honored during the convention as a warrior woman. At the end of the convention, we celebrated her marriage to Mr. Jose. Let us wish them peace and blessings.
Prayer Requests
+ Pray for the port ministry.
+ Pray for Azmack’s family (wife and three children).
+ Pray for the DRC government and peace in the country.
+ Pray for God’s provision for mission and outreach.
+ Pray that farmers may harvest well next season.
+ Praise God for the life of Elisee and her husband Mayela.
Pastor Didi Panzo is an NALC Global Worker in the Democratic Republic of the Congo. You may support his ministry though World Mission Prayer League (WMPL). Gifts may be given online at wmpl.org or sent to World Mission Prayer League; 232 Clifton Ave.; Minneapolis, MN 55403 or World Mission Prayer League; 5408 49th Ave.; Camrose, AB T4V-0N7.
| 393,990
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Results 1 to 4 of 4
SC - mom caught on video stuffing cotton down baby's throat
MBP maybe???
Chevonne Deandra Younginer
17-year-old Summerville mother was arrested after she stuffed a piece of cotton down her child's throat, covered her with a blanket and walked away at the children's hospital last Friday.
the reason the baby was under video surveillance was because the child had suffered from "numerous, life-threatening episodes.."
-
Yep, sounds like classic MSBP to me.
RIP McStays
-
January 2014:.
-
Thank you Rayemonde, for following up on so many crimes against children cases to update status for all of us. I appreciate it so much and I know others do as well. It is very sad that there are so many cases and new ones coming in every day.
Websleuths now on Facebook
Welcome to all new members. Thank you for joining the conversation. Please take a moment to become familiar with the TOS and rules, etiquette and information.
mni wiconi I Stand With Standing Rock
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Chloe Sevigny: Miu Miu's Newest Face!
Check out Chloe Sevigny in this new campaign shot for her latest venture with the fashion brand Miu Miu!
The 37-year-old actress is the label’s newest face for the Fall 2012 ad campaign.
PHOTOS: Check out the latest pics of Chloe Sevigny
You may remember, Chloe was also the face of Miu Miu‘s campaign back in 1996!!!
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Debates
The Rome Debate: Rebuilding Trust In The Misinformation Era
*Registration is now closed. If you still wish to attend, please contact us directly ([email protected]).
We are excited to announce the very first event of our new EACD Debate series and delighted to invite you to join The Rome Debate.
The Rome Debate: Rebuilding trust in the misinformation era
Date:. Prior to joining the OECD, Anthony spent 13 years with the European Commission, serving as the EU’s Trade Spokesman and Special Adviser to the EU’s Chief Trade Negotiator, Pascal Lamy. He also headed the Commission’s media and public diplomacy operations in the US and the UK. He has been an EU Visiting Fellow to the University of Southern California (USC) and a Fellow of the USC Center for Public Diplomacy.
Ryan O’Keeffe, Director of Communications at ENEL
Ryan O’Keeffe oversees Enel’s brand strategy, global advertising, internal communications, media relations and digital communications at Group level. Ryan joined UBS Investment Bank in its South African office before moving to London where he worked in the M&A and Leveraged Finance teams. In 2006 he joined Finsbury, the global strategic communications consultancy, culminating in leading Finsbury’s 2011 merger with US-based Robinson, Lerer & Montgomery. Ryan Jyrki.
Paolo Messa, Member of Board of Directors at RAI Radio Televisione Italiana
Paolo Messa is Director at the Center for American Studies and founder of Formiche, a publishing project that includes a monthly magazine, an online newspaper ANSA
He has been the editor in chief of the Italian news agency ANSA since June 2009. He started his career working for the economic newspaper Ore 12 before joining the finance editorial department of ANSA. ([email protected]).
| 9,213
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Home > HOME > Vol. 2 > Iss. 2 (2016)
Tennessee Medicine E-Journal
Article Title
Myeloid Sarcoma: solidification of liquid cells.
Abstract
Myeloid sarcoma (MS) is a solid tumor of immature myeloid cells occurring in an extra-medullary site. MS can occur in de novo, as initial manifestation or concurrently with acute chronic myeloid leukemia (3-8%) and less frequently with other types of myeloproliferative disorders or MDS. Majority of the cases of MS involve the skin, lymph nodes, bone, or gastrointestinal tract. MS can however occur at virtually any extra-medullary site. We present a case of myeloid sarcoma of the neck in a patient with history of essential thrombocythemia that later transformed into Acute myeloid leukemia (AML).
Recommended Citation
Siddiqui, Badar F. MD; Tawadro, Fady MD; Nguyen, Kim MD; Hamati, Agnes MD; and Cook, Emilie
(2016)
"Myeloid Sarcoma: solidification of liquid cells.,"
Tennessee Medicine E-Journal: Vol. 2
:
Iss.
2
, Article 2.
Available at:
Included in
Medicine and Health Sciences Commons
| 292,026
|
\begin{document}
\keywords{}
\subjclass[2010]{}
\begin{abstract} We study some properties of the solutions of the functional equation $$f(x)+f(a_1x)+\cdots+f(a_Nx)=0,$$ which was introduced in the literature by Mora, Cherruault and Ziadi in 1999, for the case $a_k=k+1$, $k=1,2,\cdots,N$ \cite{mora_1} and studied by Mora \cite{mora_2} and Mora and Sepulcre \cite{mora_3, mora_4}. \end{abstract}
\maketitle
\markboth{J. M. Almira, Kh. F. Abu-Helaiel}{On solutions of $f(x)+f(a_1x)+\cdots+f(a_Nx)=0$}
\section{Motivation}
The functional equation \begin{equation}\label{mora}
f(x)+f(a_1x)+\cdots+f(a_Nx)=0,\end{equation}
was introduced in the literature by Mora, Cherruault and Ziadi in 1999, for the case $a_k=k+1$, $k=1,2,\cdots,N-1$ \cite{mora_1} and studied by Mora \cite{mora_2} and Mora and Sepulcre \cite{mora_3,mora_4}. Concretely, the equations $f(x)+f(2x)=0$ and $f(x)+f(2x)+f(3x)=0$ were used for modeling certain processes related to combustion of hydrogen in a car engine \cite{mora_1, mora_2} and, later, the most general equation
\begin{equation} \label{ecumora}
f(x)+f(2x)+\cdots+f(Nx)=0
\end{equation}
was studied from a more theoretical point of view \cite{mora_2,mora_3,mora_4}. Concretely, by imposing a solution of the form $z^\alpha$, these authors have shown that there exists a strong connection between the continuous solutions of $(\ref{ecumora})$ and the zeroes of the exponential functions $G_N(z)=1+2^z+\cdots+N^z$, and they have developed a very interesting theory with some deep results concerning the distribution of the zeroes of $G_N(z)$. Note that the functions $H_N(z)=G_N(-z)$ are the partial sums of the Riemann zeta function $\zeta(z)=\sum_{n=1}^\infty n^{-z}$, so that the study of the zeroes of $G_N(z)$ is an important problem, connected with the well known Riemann's conjecture.
In \cite{mora_2} the author proved that if $f(z)$ is a solution of $(\ref{ecumora})$ and $f(z)\neq 0$, then $f(z)$ cannot be analytic at $z=0$. Furthermore, he also proved that the set of solutions of $(\ref{ecumora})$ that are analytic on $\Omega=\mathbb{C}\setminus (-\infty,0]$ is an infinite dimensional vector space. In this paper we study the more general equation $(\ref{mora})$. In Section 2 we prove that if $f(x)$ is a solution of $(\ref{mora})$, with $0<a_1<\cdots<a_N$ and $a_k\neq 1$ for $k=1,2,\cdots,N$, there exists a positive natural number $m=m(a_1,\cdots,a_N)$ such that, for any $\delta >0$, if $f\in \mathbf{C}^{(m)}[0,\delta]$, then $f_{|[0,\delta]}=0$ and, if if $f\in \mathbf{C}^{(m)}[-\delta,0]$, then $f_{|[-\delta,0]}=0$. In particular, if
$f\in \mathbf{C}^{(k)}(\mathbb{R})$ is a solution of $(\ref{mora})$ and $k\geq m(a_1,\cdots,a_N)$, then $f=0$. We also give upper and lower bounds for $m(a_1,\cdots,a_N)$. In Section 3 we concentrate on the study of the equation $(\ref{mora})$ with the additional restriction $x>0$. We transform this equation into the easier one
\begin{equation} \label{nueva}
g(w)+g(w+b_1)+\cdots+g(w+b_N)=0 \ \ (w\in\mathbb{R}),
\end{equation}
and we give a new elegant argument to prove that the set of continuous solutions of both equations is an infinite dimensional vector space. Furthermore, we also study the existence of continuous periodic solutions for $(\ref{nueva})$.
\section{A regularity result for $f(x)+f(a_1x)+\cdots+f(a_Nx)=0$}
We study, for $N\geq 1$, the functional equation:
$$
f(x)+f(a_1x)+\cdots+f(a_Nx)=0,
$$
where $0<a_1<a_2<\cdots<a_N$ are positive real numbers, $a_k\neq 1$ for all $k\in\{1,\cdots,N\}$, and $x$ is a real variable. Moreover, in all what follows we set $a_0=1$.
The first thing we observe is that we can assume that $1<a_1$ since otherwise, making the change of variable $y=a_1x$, the equation $(\ref{mora})$ is transformed into the equation
\begin{equation*}
f(y)+f(b_1y)+\cdots+ \cdots +f(b_Ny)=0 \ \ (x\in (0,\infty)),
\end{equation*}
where
\[
(b_1,b_2,\cdots,b_N)==\left\{
\begin{array}{llll}
(\frac{a_2}{a_1},\cdots, \frac{a_k}{a_1},\frac{1}{a_1},\frac{a_{k+1}}{a_1},\cdots ,\frac{a_N}{a_1}) & \ & \text{if} & a_k< 1< a_{k+1} \\
(\frac{a_2}{a_1},\cdots, \frac{a_N}{a_1},\frac{1}{a_1}) & \ & \text{if} & a_N < 1 \\
\end{array}
\right.
.
\]
Hence, we impose the conditions $0<a_0=1<a_1<\cdots<a_N$ through this paper. Furthermore, we use the notation $\mathbf{a}=(a_1,a_2,\cdots,a_N)$.
\begin{definition} Given $I\subseteq \mathbb{R}$ an interval, we say that a function $f:I\to\mathbb{R}$ satisfies $(\ref{mora})$ (or that $f$ is a solution of $(\ref{mora})$ on $I$) if $f(x)+f(a_1x)+\cdots+f(a_Nx)=0$ whenever $\{x,a_1x,\cdots,a_Nx\}\subseteq I$.
\end{definition}
\begin{definition} Let $N\geq 1$ and $\mathbf{a}=(a_1,a_2,\cdots,a_N)$. We define the natural number
\[
m(\mathbf{a})=\min\left\{m\in\mathbb{N}: \sum_{k=0}^{N-1}(\frac{a_k}{a_N})^{m}<1\right\}.
\]
\end{definition}
Obviously, $m(\mathbf{a})$ is well defined because all fractions $\frac{a_k}{a_N}$ appearing under the summation symbol satisfy $0<\frac{a_k}{a_N}<1$, and the number $N$ of summands is fixed.
\begin{theorem}\label{teoregularidad} Let $\delta>0$ be a positive real number. Then:
\begin{itemize}
\item[$(a)$] If $f$ is a solution of $(\ref{mora})$ on $[0,\delta]$ and $f\in \mathbf{C}^{(m(\mathbf{a}))}[0,\delta]$, then $f_{|[0,\delta]}=0$.
\item[$(b)$] If $f$ is a solution of $(\ref{mora})$ on $[-\delta,0]$ and $f\in \mathbf{C}^{(m(\mathbf{a}))}[-\delta,0]$, then $f_{|[-\delta,0]}=0$.
\end{itemize}
\end{theorem}
\noindent \textbf{Proof. } We only prove part $(a)$ of the theorem, since part $(b)$ follows with the very same arguments. Assume that $f\in \mathbf{C}^{(m(\mathbf{a}))}[0,\delta]$ is a solution of $(\ref{mora})$ over $[0,\delta]$. Let us set $x=h/a_N$, so that equation $(\ref{mora})$ is transformed into
\begin{equation}\label{mora_transf}
f(h/a_N)+f(a_1h/a_N)+\cdots+f(a_{N-1}h/a_N)+f(h)=0.\end{equation}
Taking derivatives $m(\mathbf{a})$ times at $(\ref{mora_transf})$ and defining $\varphi(h)=f^{(m(\mathbf{a}))}(h)$, we get
\begin{equation*}
\varphi(h)=(-1)\left[(\frac{1}{a_N})^{m(\mathbf{a})}\varphi(\frac{h}{a_N})+ (\frac{a_1}{a_N})^{m(\mathbf{a})}\varphi(\frac{a_1}{a_N}h)+\cdots+ (\frac{a_{N-1}}{a_N})^{m(\mathbf{a})}\varphi(\frac{a_{N-1}}{a_N}h)\right].\end{equation*}
It follows that
\[
\|\varphi\|_{[0,\delta]}\leq \|\varphi\|_{[0,\delta]} \sum_{k=0}^{N-1}(\frac{a_k}{a_N})^{m(\mathbf{a})},
\]
so that $\|\varphi\|_{[0,\delta]}=0$ (since $\sum_{k=0}^{N-1}(\frac{a_k}{a_N})^{m(\mathbf{a})}<1$). This implies that $f_{|[0,\delta]}$ is a polynomial of degree $\leq m(\mathbf{a})-1$. In particular, $f$ is real analytic on $[0,\delta]$ and, given any natural number $m\in \mathbb{N}$, we have that
\begin{equation*}
f^{(m)}(h)=(-1)\left[(\frac{1}{a_N})^{m}f^{(m)}(\frac{h}{a_N})+ (\frac{a_1}{a_N})^{m}f^{(m)}(\frac{a_1}{a_N}h)+\cdots+ (\frac{a_{N-1}}{a_N})^{m}f^{(m)}(\frac{a_{N-1}}{a_N}h)\right], \end{equation*}
so that
\begin{equation*}
f^{(m)}(0)\left[1+ (\frac{1}{a_N})^{m}+ (\frac{a_1}{a_N})^{m}+\cdots+ (\frac{a_{N-1}}{a_N})^{m}\right]=0.\end{equation*}
Hence $f^{(m)}(0)=0$ for all $m$. This means that $f_{|[0,\delta]}=0$. {\hfill $\Box$}
\begin{corollary} If $f\in \mathbf{C}^{(k)}(\mathbb{R})$ is a solution of $(\ref{mora})$ and $k\geq m(\mathbf{a})$, then $f=0$. In particular, $f=0$ is the unique solution of $(\ref{mora})$ which admits infinitely many derivatives in all the real line.
\end{corollary}
\begin{remark}It is important to note that, for all $N\geq 2$, the function $f:[0,\infty)\to\mathbb{R}$ given by $f(x)=0$ can be extended in infinitely many ways to a solution $\widetilde{f}$ of equation $(\ref{ecumora})$ such that $\widetilde{f}\in \mathbf{C}(\mathbb{R})\cap \mathbf{C}^{(\infty)}(\mathbb{R}\setminus \{0\})$ and $\widetilde{f}$ does not vanish identically.
To prove this, we use that, for $N\geq 2$, the function $G_N(z)=1+2^z+\cdots+N^z$ has infinitely many zeros in the complex plane (see \cite[Proposition 1]{mora_2} for a proof of this fact). Thus, let $\alpha=a+\mathbf{i}b\in\mathbb{C}$ be a zero of $G_N(z)$ and let us define $\widetilde{f}_{\alpha}(x)=\mathbf{Re}(|x|^{\alpha})$ for $x<0$ and $\widetilde{f}(x)=0$ for $x\geq 0$. Then $\widetilde{f}_{\alpha}$ is clearly a solution of equation $(\ref{ecumora})$ in $[0,\infty)$, and, for $x<0$ we have that
\[
\sum_{k=1}^N\widetilde{f}_{\alpha}(kx)=\mathbf{Re}\left(|x|^{\alpha}\sum_{k=1}^N k^{\alpha}\right) = 0,
\]
so that $\widetilde{f}_{\alpha}$ is also a solution of equation $(\ref{ecumora})$ in $\mathbb{R}$. \end{remark}
\begin{remark} The natural number $m(\mathbf{a})$ appearing in Theorem \ref{teoregularidad} is not optimal. For example, it is clear that, for $N=1$, $a_1=2$, $m(\mathbf{a})=1$. Now, let us assume that $f$ is a solution of $f(x)+f(2x)=0$ and $f(h_0)\neq 0$ for some $h_0>0$. Then $f(\frac{1}{2}h_0)=(-1)f(h_0)$ and, for all $k\in\mathbb{N}$ we have that $f(\frac{1}{2^k}h_0)=(-1)^kf(h_0)$. This obviously implies that $f(x)$ cannot be continuous at $x=0$.
\end{remark}
The following result gives a simple estimation of the number $m(\mathbf{a})$:
\begin{proposition} $$\frac{1}{2}\frac{a_N}{\max_{0\leq k<N}(a_{k+1}-a_k)}-1\leq m(\mathbf{a})\leq \frac{a_N}{\min_{0\leq k<N}(a_{k+1}-a_k)}$$
\end{proposition}
\noindent \textbf{Proof. } The result follows directly from the interpretation of the sum $$\sum_{k=0}^{N-1}\left(\frac{a_k}{a_N}\right)^m\left(\frac{a_{k+1}}{a_N}-\frac{a_k}{a_N}\right)$$
as a lower Riemann sum for the integral $\int_0^1x^mdx$ and the sum $$\sum_{k=0}^{N}\left(\frac{a_k}{a_N}\right)^m\left(\frac{a_{k}}{a_N}-\frac{a_{k-1}}{a_N}\right)$$
as an upper Riemann sum for the same integral (here we impose $a_{-1}=0$). Concretely, we have that
\[
\sum_{k=0}^{N-1}\left(\frac{a_k}{a_N}\right)^m\left(\frac{a_{k+1}}{a_N}-\frac{a_k}{a_N}\right)\leq \int_0^1x^mdx=\frac{1}{m+1},
\]
so that, for all $m\geq \frac{a_N}{\min_{0\leq k<N}(a_{k+1}-a_k)}$,
\begin{eqnarray*}
\sum_{k=0}^{N-1}\left(\frac{a_k}{a_N}\right)^m &\leq & \sum_{k=0}^{N-1}\left(\frac{a_k}{a_N}\right)^m\left(\frac{a_{k+1}}{a_N}-\frac{a_k}{a_N}\right) \frac{a_N}{\min_{0\leq k<N}(a_{k+1}-a_k)}\\
& \leq & \frac{1}{m+1}\frac{a_N}{\min_{0\leq k<N}(a_{k+1}-a_k)}<1,
\end{eqnarray*}
which implies that $m(\mathbf{a})\leq \frac{a_N}{\min_{0\leq k<N}(a_{k+1}-a_k)}$.
On the other hand,
\begin{eqnarray*}
\sum_{k=0}^{N}\left(\frac{a_k}{a_N}\right)^m &\geq & \sum_{k=0}^{N}\left(\frac{a_k}{a_N}\right)^m\left(\frac{a_{k}}{a_N}-\frac{a_{k-1}}{a_N}\right)\frac{a_N}{\max_{0\leq k<N}(a_{k+1}-a_k)}\\
& \geq & (\int_0^1x^mdx) \frac{a_N}{\max_{0\leq k<N}(a_{k+1}-a_k)} \\
&=& \frac{1}{m+1} \frac{a_N}{\max_{0\leq k<N}(a_{k+1}-a_k)},
\end{eqnarray*}
so that, for all $m < \frac{1}{2}\frac{a_N}{\max_{0\leq k<N}(a_{k+1}-a_k)}-1$,
\[
\sum_{k=0}^{N-1}\left(\frac{a_k}{a_N}\right)^m \geq \frac{1}{m+1}\frac{a_N}{\max_{0\leq k<N}(a_{k+1}-a_k)}-1>1,
\]
which implies that $m(\mathbf{a})\geq \frac{1}{2}\frac{a_N}{\max_{0\leq k<N}(a_{k+1}-a_k)}-1$. {\hfill $\Box$}
Note that the best bounds for $m(\mathbf{a})$ appear when the points $a_k$ are equidistributed (i.e., when $a_k=1+k(a_1-1)$ for all $k\in\{0,1,\cdots,N\}$) since it is precisely in this case when $\max_{0\leq k<N}(a_{k+1}-a_k)$ attains its minimum and $\min_{0\leq k<N}(a_{k+1}-a_k)$ attains its maximum, and both coincide with $d=a_1-1$. Hence, in this case we get the following bounds: $$\frac{1}{2}N-1\leq m((1+d,1+2d,\cdots,1+Nd))\leq N,$$ which are independent of the separation $d$.
\section{A related functional equation}
Let us consider the equation
\begin{equation}\label{morageneral}
f(x)+f(a_1x)+\cdots +f(a_Nx)=0 \ \ (x\in (0,\infty)),
\end{equation}
where $a_0=1<a_1<a_2<\cdots<a_N$ are real numbers.
If we set $x=e^w$ and $g(w)=f(e^w)$ then, taking into account that $f(a_kx)=f(a_ke^w)=f(e^{w+\ln a_k})=g(w+\ln a_k)$, the equation $(\ref{morageneral})$ can be written as
\begin{equation}\label{aditiva}
g(w)+g(w+b_1)+\cdots+g(w+b_N)=0 \ \ (w\in\mathbb{R}),
\end{equation}
where $0<b_k=\ln a_k <\ln a_{k+1}< b_{k+1}$, $k=1,\cdots,N-1$.
\begin{lemma} \label{auxiliar} Let us assume that $g:[0,b_N]\to \mathbb{R}$ is a continuous function which satisfies $(\ref{aditiva})$. Then there exists a unique $\widetilde{g}\in\mathbf{C}(\mathbb{R})$ such that $\widetilde{g}$ is a solution of $(\ref{aditiva})$ on $\mathbb{R}$ and $\widetilde{g}_{|[0,b_N]}=g$.
\end{lemma}
\noindent \textbf{Proof. } First of all, we note that $g:[0,b_N]\to \mathbb{R}$ satisfies $(\ref{aditiva})$ if and only if it satisfies the interpolation condition:
\[
g(0)+g(b_1)+\cdots+g(b_N)=0.
\]
Moreover, if $y,w$ denote two real numbers satisfying the relation $y=w+b_N$ and $\widetilde{g}$ is any solution of $(\ref{aditiva})$ on $\mathbb{R}$, then a simple substitution shows that $\widetilde{g}$ satisfies
\[
\widetilde{g}(y)+\widetilde{g}(y-b_N)+\widetilde{g}(y-(b_N-b_1))+\cdots+\widetilde{g}(y-(b_N-b_{N-1}))=0 \ \ (y\in\mathbb{R}).
\]
We will use this relation to (uniquely) define on $[0, \infty)$ the solution $\widetilde{g}$ such that $\widetilde{g}_{|[0,b_N]}=g$. Furthermore, we will use the original equation $(\ref{aditiva})$ to (uniquely) extend the solution $\widetilde{g}$ over the negative part of the real axis.
Let us set $I_0=[0,b_N]$, $\widetilde{g}_0=g$, and define, for $h\in I_1=[b_N, b_N+(b_{N}-b_{N-1})]$, the function
\[
\widetilde{g}_1(y)= (-1)\left[\widetilde{g}_0(y-b_N)+\widetilde{g}_0(y-(b_N-b_1))+\widetilde{g}_0(y-(b_N-b_2))+\cdots+\widetilde{g}_0(y-(b_N-b_{N-1}))\right].
\]
Obviously, $\widetilde{g}_1$ is well defined, since $t\in I_1$ implies that
$$0\leq y-b_N\leq y-(b_N-b_{1})\leq y-(b_N-b_{2})\leq \cdots \leq y-(b_N-b_{N-1})\leq b_N.$$
Moreover, $\widetilde{g}_1\in \mathbf{C}(I_1)$.
For $k\geq 2$, we set $I_{k}=[b_N+(k-1)(b_{N}-b_{N-1}), b_N+k(b_{N}-b_{N-1})]$ and
\[
\widetilde{g}_k(y)= (-1)\left[\widetilde{g}_{k-1}(y-b_N)+\widetilde{g}_{k-1}(y-(b_N-b_1))+\cdots+\widetilde{g}_{k-1}(y-(b_N-b_{N-1}))\right] \ \ (y\in I_{k}).
\]
Let us now consider the negative part of the real axis. Set $I_{-1}=[-b_1,0]$ and
\[
\widetilde{g}_{-1}(x)= (-1)\left[\widetilde{g}_{0}(x+b_1)+\widetilde{g}_{0}(x+b_2)+\cdots+\widetilde{g}_{0}(x+b_N)\right] \ \ (x\in I_{-1}).
\]
For $k\leq -2$, we set $I_{k}=[kb_{1}, (k+1)b_1]$ and
\[
\widetilde{g}_{k}(x)= (-1)\left[\widetilde{g}_{k+1}(x+b_1)+\widetilde{g}_{k+1}(x+b_2)+\cdots+\widetilde{g}_{k+1}(x+b_N)\right] \ \ (x\in I_{k}).
\]
Clearly, $\bigcup_{k\in\mathbb{Z}} I_k=\mathbb{R}$ and $\widetilde{g}(x)=\widetilde{g}_k(x)$ ($x\in I_k$, $k\in\mathbb{Z}$) is the function we were looking for. The uniqueness is guaranteed by the construction we have used for the definition of $\widetilde{g}$.
{\hfill $\Box$}
For the proof of the following theorem, we need firstly to recall the concept of exponential polynomial which is of common use for people working on functional equations.
\begin{definition} We say that $f(x)\in\mathbf{C}(\mathbb{R})$ is a (real) exponential polynomial if $f(x)$ is the solution of some ordinary homogeneous linear differential equation with constant coefficients, $y^{(n)}+a_1y^{(n-1)}+\cdots+a_ny=0$. These functions are completely characterized as finite $\mathbb{R}$-linear combinations of the real and imaginary parts of functions of the form $m(x)=x^k e^{\lambda x}$, where $k\leq n-1$ is a natural number and $\lambda$ is a complex number. Furthermore, they can also be characterized as the continuous solutions of certain functional equations which do not involve the use of derivatives, such as Popoviciu's equation (see, for example, \cite{roscau}, \cite{rado}):
\[
\det \left[
\begin{array}{cccccc}
f(x) & f(x+h) & \cdots & f(x+nh) \\
f(x+h) & f(x+2h) & \cdots & f(x+(n+1)h) \\
f(x+2h) & f(x+3h) & \cdots & f(x+(n+2)h) \\
\vdots & & \ddots & \vdots & \\
f(x+nh) & f(x+(n+1)h) & \cdots & f(x+2nh)
\end{array}
\right] = 0.
\]
We say that $f(x)\in\mathbf{C}(\mathbb{R},\mathbb{C}):=\{f:\mathbb{R}\to\mathbb{C},\ f\text{ is continuous}\}$ is a (complex) exponential polynomial if $f(x)$ is a finite $\mathbb{C}$-linear combination of functions of the form $m(x)=x^k e^{\lambda x}$, where $k$ is a natural number and $\lambda$ is a complex number.
\end{definition}
\begin{theorem} \label{dimension} Let $\mathbf{S}=\{g\in\mathbf{C}(\mathbb{R}):g \text{ is a solution of } (\ref{aditiva})\}$. Then $\mathbf{S}$ is an infinite dimensional vector space. As a consequence, the space of continuous solutions of $(\ref{morageneral})$ is also an infinite dimensional vector space.
\end{theorem}
\noindent \textbf{Proof. } It is well known (see \cite{anselone}, \cite{engert}) that, if $V$ is a finite dimensional subspace of $\mathbf{C}(\mathbb{R},\mathbb{C})$ and $V$ is invariant by translations (i.e., $f(x)\in V$ implies $g_L(x)=f(x-L)\in V$ for all $L\in\mathbb{R}$), then all elements of $V$ are (complex) exponential polynomials. In particular, all elements of $V$ are analytic functions. On the other hand, the space $\mathbf{S}$ is obviously invariant by translations and can be considered in a natural way as a subspace of $\mathbf{C}(\mathbb{R},\mathbb{C})$. Moreover, Lemma \ref{auxiliar} implies that $\mathbf{S}$ is nonempty. Hence, the proof will end as soon as we find a continuous solution of (\ref{aditiva}) which is not an exponential polynomial.
Let us define
$$
g(x)= \left\{
\begin{array}{lll}
1 & \text{ if } 0\leq x \leq b_{N-1} \\
\frac{b_N+Nb_{N-1}-(N+1)x}{b_N-b_{N-1}} & \text{ if } b_{N-1}<x\leq b_N \end{array}
\right.
$$
Obviously, $g$ satisfies $(\ref{aditiva})$ in $[0,b_N]$, so that Lemma \ref{auxiliar} implies that there exists $\widetilde{g}\in\mathbf{S}$ such that $\widetilde{g}_{|[0,b_N]}=g$. In particular, $\widetilde{g}$ is not an exponential polynomial, since $g$ is not differentiable. {\hfill $\Box$}
It is interesting to note that, in some cases, the function $\widetilde{g}$ we get in the construction shown at the proof of Theorem \ref{dimension} is periodic. For example, if we impose $b_k=k$, $k=1,2,\cdots,N$ and we follow all steps of the proof, we get that $\widetilde{g}(x)$ is the $(N+1)$-periodic extension of the function :
$$
g(x)= \left\{
\begin{array}{lll}
1 & \text{ if } 0\leq x \leq N-1 \\
-(N+1)x + N^2 & \text{ if } N-1<x\leq N \\
(N+1)x -N(N+2) & \text{ if } N<x\leq N+1 \\
\end{array}
\right.
$$
Thus, an interesting question is, under which conditions on $(b_1,\cdots,b_N)$ can we guarantee that equation $(\ref{aditiva})$ admits periodic solutions? The following theorem partially solves this question:
\begin{theorem} The equation $(\ref{aditiva})$ admits a continuous periodic solution $g\neq 0$ if and only if there exists $\alpha\in\mathbb{R}$ such that
\begin{equation}\label{condiciongeneral}
\left\{ \begin{array}{lll}
1+\sum_{k=1}^N\cos(\alpha b_k) & = & 0 \\
\sum_{k=1}^N\sin(\alpha b_k) & =& 0 \\
\end{array}
\right.
\end{equation}
Furthermore, in such a case, there are trigonometric polynomials satisfying equation $(\ref{aditiva})$ which are periodic of fundamental period equal to $2\pi/\alpha$. Finally, the equation $(\ref{aditiva})$ admits continuous periodic solutions for $(b_1,\cdots,,b_N)$ if and only if it admits continuous periodic solutions for $(d b_1,\cdots, d b_N)$ for all $d> 0$.
\end{theorem}
\noindent \textbf{Proof. } Assume $g$ is a continuous $T$-periodic solution of $(\ref{aditiva})$ and set $\theta=2\pi/T$. Let
\[
g(x)=\sum_{k=1}^{\infty}(a_k(g)\cos(k\theta t)+b_k(g)\sin(k\theta t)) +\frac{a_0(g)}{2}
\]
be the Fourier series expansion of $g$ (this expansion exists because, being $g$ continuous, its Fourier coefficients are well defined). Then $h(x)=g(x)+\sum_{k=1}^Ng(x+b_k)$ is also continuous and $T$-periodic, so that its Fourier coefficients are well defined. Indeed, a simple computation gives $a_0(h)=a_0(g)$ and
\[
\left(
\begin{array}{lll}
a_k(h) \\
b_k(h) \\
\end{array}
\right)
= \left(
\begin{array}{lll}
1+\sum_{k=1}^N\cos(k\theta b_k) & \sum_{k=1}^N\sin(k\theta b_k) \\
-\sum_{k=1}^N\sin(k\theta b_k) & 1+\sum_{k=1}^N\cos(k\theta b_k) \\
\end{array}
\right) \left(
\begin{array}{lll}
a_k(g) \\
b_k(g) \\
\end{array}
\right) \text{ for all } k\geq 1.
\]
Now, the function $h$ vanishes identically if and only if all its Fourier coefficients are zero. Thus, if $g$ is a solution of $(\ref{aditiva})$ then $a_0(g)=0$ and
\begin{equation} \label{matriz}
\left(
\begin{array}{lll}
1+\sum_{k=1}^N\cos(k\theta b_k) & \sum_{k=1}^N\sin(k\theta b_k) \\
-\sum_{k=1}^N\sin(k\theta b_k) & 1+\sum_{k=1}^N\cos(k\theta b_k) \\
\end{array}
\right)
\left(
\begin{array}{lll}
a_k(g) \\
b_k(g) \\
\end{array}
\right) = \left(
\begin{array}{lll}
0 \\
0 \\
\end{array}
\right)
\end{equation}
for all $k\geq 1$. Let us denote by $A_k$ the matrix appearing in $(\ref{matriz})$. If $g$ is not the zero function, then there exists $k\geq 1$ such that $(a_k(g),b_k(g))\neq (0,0)$ and, for this concrete value of $k$ we should have
\[
\det(A_k)=\left(1+\sum_{k=1}^N\cos(k\theta b_k) \right)^2 + \left(\sum_{k=1}^N\sin(k\theta b_k) \right)^2 = 0.
\]
In other words, the system of equations $(\ref{condiciongeneral})$ is satisfied for $\alpha=k\theta$. What is more: as soon as $\det(A_k)=0$ we have that $A_k=0$ is the null matrix, which implies that for all $(a_k,b_k)\in\mathbb{R}^2$ the trigonometric polynomial
$$g(x) = a_k\cos(k\theta t)+b_k\sin(k\theta t)$$
satisfies $(\ref{aditiva})$ and is a periodic function with fundamental period $T=\frac{2\pi}{k\theta}$. The last claim of the theorem follows from the fact that $\alpha$ is a solution of $(\ref{condiciongeneral})$ for $(b_1,\cdots,b_N)$ if and only if $\frac{\alpha}{d}$ if a solution of $(\ref{condiciongeneral})$ for $(d b_1,\cdots,d b_N)$.
{\hfill $\Box$}
\begin{corollary} Let $d>0$ and set $b_k=k d$, $k=1,2,\cdots N$ and let $m\in\mathbb{Z}\setminus (N+1)\mathbb{Z}$. Then equation $(\ref{aditiva})$ admits continuous periodic solutions of period $T=\frac{N+1}{m}$. \end{corollary}
\noindent \textbf{Proof. } It is only necessary to make the proof for $d=1$. Assume that $b_k=k$ for $k=1,2,\cdots,N$. Then $(\ref{condiciongeneral})$ becomes:
\begin{equation} \label{condiciongeneralcaso}
\left\{ \begin{array}{lll}
\frac{\sin\left( \frac{N+1}{2}\alpha \right)}{\sin\left( \frac{\alpha}{2}\right)} \cos\left( \frac{N}{2}\alpha \right) = & 0 \\
\frac{\sin\left( \frac{N+1}{2}\alpha \right)}{\sin\left( \frac{\alpha}{2}\right)} \sin\left( \frac{N}{2}\alpha \right) = & 0 \\
\end{array}
\right. .
\end{equation}
Hence, in this case, to find $\alpha\in\mathbb{R}$ which solves $(\ref{condiciongeneral})$ is equivalent to find a real solution $\alpha$ of
\[
\left(\frac{\sin\left( \frac{N+1}{2}\alpha \right)}{\sin\left( \frac{\alpha}{2}\right)}\right)^2 = \left[\frac{\sin\left( \frac{N+1}{2}\alpha \right)}{\sin\left( \frac{\alpha}{2}\right)} \cos\left( \frac{N}{2}\alpha \right) \right]^2+ \left[\frac{\sin\left( \frac{N+1}{2}\alpha \right)}{\sin\left( \frac{\alpha}{2}\right)} \sin\left( \frac{N}{2}\alpha \right) \right]^2=0.
\]
Equivalently, we are looking for the real solutions of
\[
\frac{\sin\left( \frac{N+1}{2}\alpha \right)}{\sin\left( \frac{\alpha}{2}\right)}=0,
\]
which exist and are given by $\alpha = \frac{2m\pi}{N+1}$, $m\in\mathbb{Z}\setminus (N+1)\mathbb{Z}$. This ends the proof. {\hfill $\Box$}
\begin{corollary} Let us assume that $b> 0$. The equation \begin{equation}\label{dos} g(x)+g(x+a)+g(x+b)=0\ \ (x\in\mathbb{R})\end{equation} admits a continuous periodic solution $g\neq 0$ if and only if
\begin{equation} \label{condiciondos}
\frac{a}{b} \in\{\frac{2+3k}{1+3m}, \frac{1+3m}{2+3k}: (m,k)\in\mathbb{Z}^2\}.
\end{equation}
In particular, if $a/b\in\mathbb{R}\setminus\mathbb{Q}$ then $(\ref{dos})$ admits no continuous periodic solutions. Furthermore, there are infinitely many rational numbers $p/q$ such that $a/b=p/q$ implies that $(\ref{dos})$ admits no continuous periodic solutions. \end{corollary}
\noindent \textbf{Proof. } In this case, the equations we must study are given by: \begin{equation}\label{condiciongeneraltres}
\left\{ \begin{array}{lll}
1+ cos(\alpha a) +cos(\alpha b) & = & 0 \\
sin(\alpha a) +sin(\alpha b) & = & 0 \\
\end{array}
\right.
\end{equation}
Solving the second equation in the system we get that $\alpha b= -\alpha a + 2k\pi$ or $\alpha b= \alpha a + (2k+1)\pi$ for a certain $k\in\mathbb{Z}$. We consider both cases separately:
\noindent \textbf{Case 1: $\alpha b= \alpha a + (2k+1)\pi$. } Introducing the corresponding values into the first equation in the system, we get
\[
1+ cos(\alpha a) +cos(\alpha a + (2k+1)\pi) =1\neq 0.
\]
Hence, in this case we get no solutions of $(\ref{condiciongeneraltres})$.
\noindent \textbf{Case 2: $\alpha b= -\alpha a + 2k\pi$. } Introducing the corresponding values into the first equation in the system, we get
\[
1+ cos(\alpha a) +cos(-\alpha a + 2k\pi) =1 +2 cos(\alpha a) = 0 \Leftrightarrow
\alpha a \in \{2\pi/3+2m\pi, \pi/3+(2m+1)\pi\}_{m\in\mathbb{Z}}.
\]
Taking into account that $\alpha b= -\alpha a + 2k\pi$, we conclude that $\alpha$ is a solution of the system of equations $(\ref{condiciongeneraltres})$ if and only if it is a solution of at least one of the following two systems:
\begin{equation}\label{I}
\left\{ \begin{array}{lll}
\alpha a & = & 2\pi/3+2m\pi \\
\alpha b & = & -(2\pi/3+2m\pi) +2k\pi \\
\end{array}
\right. \text{ for some } (m,k)\in\mathbb{Z}^2.
\end{equation}
or
\begin{equation}\label{II}
\left\{ \begin{array}{lll}
\alpha a & = & \pi/3+(2m+1)\pi\\
\alpha b & = & -(\pi/3+(2m+1)\pi) +2k\pi \\
\end{array}
\right. \text{ for some } (m,k)\in\mathbb{Z}^2.
\end{equation}
Let us study the equations given in $(\ref{I})$. Clearly, we should have $a,b\neq 0$. Furthermore,
\[
\alpha = \frac{1}{a} \left(2\pi/3+2m\pi\right) = \frac{1}{b} \left(-(2\pi/3+2m\pi) +2k\pi\right)
\]
In particular, the system has real solutions if and only if
\[
\frac{b}{a} \in \{\frac{-1+3(k-m)}{1+3m}: (m,k)\in\mathbb{Z}^2\} =\{\frac{2+3k}{1+3m}: (m,k)\in\mathbb{Z}^2\}.
\]
As the parameters $a,b$ are interchangeable in all the argument above, we conclude that
\[
\frac{b}{a} \in \{\frac{2+3k}{1+3m}, \frac{1+3m}{2+3k}: (m,k)\in\mathbb{Z}^2\}.
\]
Finally, it is clear that the system $(\ref{II})$ is a particular case of $(\ref{I})$.
{\hfill $\Box$}
\begin{remark} The condition $(\ref{condiciondos})$ can be studied for any particular instance of $a,b$. For example, if $a=b$ then $(\ref{condiciondos})$ becomes
\[
1 \in\{\frac{2+3m}{1+3k}, \frac{1+3k}{2+3m}: (k,m)\in\mathbb{Z}^2\}
\]
which is clearly impossible. It follows that there are no continuous periodic functions $g\neq 0$ satisfying $g(x)+2g(x+a)=0$.
We give here a direct proof of this fact: Assume, on the contrary, that $g(x)+2g(x+a)=0$ and $g(x)$ is continuous and $T$-periodic with $T>0$ a fundamental period. Let $s_0\in [0,T]$ be such that $|g(s_0)|\neq 0$. There are two possibilities:
\begin{itemize}
\item[$(a)$] The numbers $\{a,T\}$ are linear dependent when dealing $\mathbb{R}$ as a $\mathbb{Q}$-vector space.
In this case, there are natural numbers $k,m$ such that $kT=ma$. Then
\[
0<|g(s_0)|=|g(s_0+kT)|=|g(s_0+ma)|=\frac{1}{2^m}|g(s_0)|<|g(s_0)|,
\]
which is a contradiction.
\item[$(b)$] $\dim \mathbf{span}_{\mathbb{Q}}\{a,T\}=2$. Taking into account that $g$ is continuous and periodic, it is uniformly continuous. Hence, given $\varepsilon>0$ there exists $\delta>0$ such that $|g(x)-g(y)|<\varepsilon$ whenever $|x-y|<\delta$. Now, our hypothesis on $\{a,T\}$ implies that for any $\delta>0$ there exists $n,m\in\mathbb{Z}$ such that $|nT+ma|<\delta$ (indeed, we may assume $m>0$). Hence, if $d=|g(s_0)|>0$ and $\varepsilon<d/2$,
\[
|g(s_0+nT+ma)-g(s_0)|<\varepsilon
\]
which implies $|g(s_0+nT+ma)|>d/2$. On the other hand,
\[
|g(s_0+nT+ma)|=|g(s_0+ma)|=\frac{1}{2^m}|g(s_0)| \leq \frac{1}{2}d,
\]
a contradiction.
\end{itemize}
Of course, a direct proof of existence (or nonexistence) of continuous periodic solutions of equation $(\ref{dos})$ for each instance of the parameters $a,b$, would be a difficult task. Instead of that, checking if condition $(\ref{condiciondos})$ is (or it is not) satisfied is always easy.
\end{remark}
\section{Acknowledgements}
We are very grateful to the referee for reading our paper carefully
and thoroughly, and making many helpful suggestions.
\bibliographystyle{amsplain}
| 8,851
|
Bluegrass Reunion Deluxe Edition
- CD
Did you know that Grisman got his first gig as a bluegrass picker back in ’65 with Red Allen and the Kentuckians? They reunited for this 1992 LP featuring guitar and vocals by Jerry Garcia, who was thrilled to meet and jam with Allen, one of his bluegrass heroes. This was Allen’s last studio recording before his death: "She’s No Angel," "Ashes of Love," "Little Maggie," "On and On," "Is This My Destiny?," "Is It Too Late Now?," "Will You Miss Me When I’m Gone?" and more. This edition adds two unreleased bonus tracks featuring Garcia on lead vocals! Acoustic Disc.
To play the media you will need to either update your browser to a recent version or update your version of Flash Player.
Get ADOBE® FLASH® Player.
| 394,338
|
Monday, March 3, 2014
NJOY Gets $70M
Scottsdale, Arizona-based electronics cigarette maker NJOY announced this morning that it has raised over $70M in a funding round, which was led by Brookside Capital. The company said other investors in the funding also included Morgan Stanley Investment Management, GAM Technology Strategy, as well as other institutional investors. The company said the funds will go towards marketing, international expansion, and research and development. More information »
| 196,923
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TITLE: Graphical regular representation - specific choice of generators
QUESTION [2 upvotes]: From all I know the finite groups admitting a GRR are known completely. I currently try to use Godsil's results for some own ideas. That raises the following question:
If $G$ is a given group that admits a GRR, is it possible to choose a generating set $S$ of $G$ such that the set $\left( \bigcup_{s \in S} \langle s \rangle \right) \backslash \{1\}$ is a generating set which provides a GRR of $G$?
Or more generally: For a given group $G$ that admits a GRR, are there some subgroups $U_i \leq G$ such that the set $\left( \bigcup_i U_i \right) \backslash \{1\}$ provides a GRR?
REPLY [1 votes]: I think the answer is no. Take $G$ to be the dihedral group of order 14, with presentation $\langle a,b|a^7=b^2=(ab)^2=1\rangle$ and suppose that $S$ is a generating set for $G$ of the type you require and such that $Cay(G,S)$ is a GRR.
If $S$ intersects $\langle a\rangle\cong C_7$ then $S$ must contain all the non-identity elements of $\langle a\rangle$. Taking graph complements if necessary, this implies that $G$ has a GRR with connection set containing only involutions.
Unless I made a mistake, I think this does not happen. (I checked, with some help from a computer.)
| 217,695
|
Extract
As society at large becomes increasingly concerned with the issues surrounding global climate change, so the pressure on the scientific community to produce models and predictions of climate variability increases. In many respects, however, that branch of science concerned with climate change is in its infancy. While recent meteorological and oceanographic studies have shed light on the processes and mechanisms of atmospheric and oceanic circulation, this has produced only a ‘snapshot’ perspective of global change, limited by the range of instrumental or historical records. On the other hand, palaeoclimatic and palaeoceanographic studies have been mainly on coarser (millennial) timescales that have a more academic and less immediate appeal. The palaeorecords which have the required temporal (interannual/ decadal) resolution are limited to tree rings, ice cores, coral records and laminated marine or lacustrine sediments. This volume is concerned with the wide-ranging application of lacustrine and marine laminated sediments as palaeo-indicators.
Environments of lamina formation and preservation The two fundamental requirements for development of a laminated sediment sequence are: (1) variation in input/chemical conditions/biological activity that will result in compositional changes in the sediment; and (2) environmental conditions that will preserve the laminated sediment fabric from bioturbation. Within lakes, strong seasonal signals are dominant while preservation is effected by bottom water/sediment anoxia resulting from stratification, high salinities or high sedimentation rates. In the marine environment, the dominant control on lamina preservation is reduced oxygen in anoxic silled basins (e.g. California Borderland basins) or marginal seas (e.g. Black Sea), or beneath regions of high
- © The Geological Society 1996
| 178,880
|
- 1 night
1 Room, 1 Adult
- Update
Allegro Hoi An - A Little Boutique Lux Hotel & Spa
4.7/5Outstanding
4.7/5Outstanding
- 29.5km
- 29.8km
- 1 night
1 Room, 1 Adult
Hotel Policy
- Check-in from 14:00
Hotel Description
- Opened: 2017
- Renovated: 2019
- Number of Rooms: 94
Services & Amenities
- Wi-Fi in designated areasFree
- Airport pickup service
- Airport shuttle service(Additional fee)
- Shuttle service to nearby landmarks(Additional fee)
- Car rental
- Public transport tickets
- Bicycle rental
- Bellmen
- Luggage storage
- Front desk safe
- Valet parkingFree
- Concierge service
- Attraction ticket service
- Express check-in/check-out
- Locker
- Currency exchange
- Front desk (24 hours)
- Doorman
- Private check-in/check-out
- Restaurant
- Café
- Room service
- Children's meals
- Bar
- Elevator
- Designated smoking area
- ATM
- Accessible rooms
- Newspaper in lobby
- Non-smoking floor
- CCTV in public areas
- Security alarm
- Fire extinguisher
- Smoke alarm
- Conference hall(Additional fee)
- Business center
- Dry cleaning
- Laundry room
- Ironing service(Additional fee)
- Room cleaned: daily
- Laundry service(Additional fee)
- Laundry service (off-site)
- Outdoor swimming pool
- Pool with view
- Children's pool
- Private beach
- Beach towels
- Beach/pool sun umbrellas
- Beach
- Hiking
- Tennis
- Pool bar
- Swimming pool toys
- Barbecue
- Mini shopping mall
- Lobby seating area
- Gift shop
- Library
- Garden
- Outdoor furniture
- Gym
- Sauna
- SPA(Additional fee)
- Massage
- Beauty salon
- Facials
- Pedicures
- Manicures
- Body treatments
- Hair removal/waxing
- Exfoliation
- Slimming and toning
- Light therapy
- Childcare service
-
| 14,605
|
9.7% Bodyfat?!?!
Damn I need to kick up my HIIT Cardio! haha
But at least my muscle quality is through the roof!
The Skulpt is awesome. I measured my bodyfat in less than 30 seconds.
If you haven't yet, enter the raffle to win a free Apple Watch + Skulpt Aim!
Three are up for grabs and I was assured at least one winner will be from our community so your chances are looking good Nation! :-D
Here is the link to enter the raffle!
If you don't want to wait to win, use my link to order one today!
| 255,274
|
Eric Clapton has threatened to cancel performances at U.K. venues that require audience members to be vaccinated to attend his concerts.
Clapton’s message, which was posted on the Telegram page of Robin Monotti, an anti-vaccine advocate, was in response to U.K. Prime Minister Boris Johnson‘s recent announcement that COVID-19 vaccine passes would be required to attend events at nightclubs and other venues.
“Following the PM’s announcement on Monday…I feel honour bound to make an announcement of my own,” Slowhand writes. “I will not perform on any stage where there is a discriminated audience present. Unless there is provision made for all people to attend, I reserve the right to cancel the show.”
Accompanying the message is a link to Clapton’s rendition of the 2020 Van Morrison song “Stand and Deliver,” which criticizes the U.K. government’s pandemic-related restrictions on live performances.
In May, another message from Clapton was posted on Monotti’s Telegram page in which he revealed he experienced a severe reaction after receiving the AstraZeneca vaccine.
“[M]y hands and feet were either frozen, numb or burning, and pretty much useless for two weeks, I feared I would never play again,” Eric wrote. “[I] should never have gone near the needle.”
In a statement to Rolling Stone in May, a spokesperson for the U.K. government agency overseeing the vaccine maintained that “over 56 million doses of vaccines against COVID-19 have now been administered in the UK, saving thousands of lives through the biggest vaccination programme that has ever taken place in this country.”
The rep added, “Our advice remains that the benefits of the COVID-19 Vaccine AstraZeneca outweigh the risks in the majority of people.”
Clapton’s next U.K. concerts are scheduled for May 2022. He begins a run of U.S. dates in September 2021.
| 210,539
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TITLE: Show $S_3$ is not cyclic and find the subgroups of order $2$
QUESTION [2 upvotes]: Consider the symmetric group $S_3=\{i,\alpha,\beta,\rho,\sigma,\tau\}$ of all permutations on {1,2,3}. It's operation table is
$$\begin{matrix}
\ &i&\alpha&\beta&\rho&\sigma&\tau\\
\ &\underline{\quad}&\underline{\quad}&\underline{\quad}&\underline{\quad}&\underline{\quad}&\underline{\quad}\\
i\vert&i&\alpha&\beta&\rho&\sigma&\tau\\
\alpha\vert&\alpha&\beta&i&\sigma&\tau&\rho\\
\beta\vert&\beta&i&\alpha&\tau&\rho&\sigma\\
\rho\vert&\rho&\tau&\sigma&i&\beta&\alpha\\
\sigma\vert&\sigma&\rho&\tau&\alpha&i&\beta\\
\tau\vert&\tau&\sigma&\rho&\beta&\alpha&i
\end{matrix}$$
(I hope the table makes sense).
Show that $S_3$ is not cyclic.
Write down a complete list of all subgroups of order $2$ in $S_3$.
Show that none of the above subgroups is normal in $S_3$.
Solution
We use the table to check the cycle of each element, to see if any are a viable generator.
Starting with $\alpha$
$$\alpha\alpha=\beta,\ \alpha\beta=i,\ \alpha i=\alpha$$
$\alpha$ is not a viable generator as not all elements of $S_3$ have been found.
Doing the same for the rest
$$\beta\beta=\alpha,\ \beta\alpha=i,\ \beta i=\beta$$
$$\rho\rho=i,\ \rho i=\rho$$
$$\sigma\sigma=i,\ \sigma i=\sigma$$
$$\tau\tau=i,\ \tau i=\tau$$
we can clearly see that none of the elements are a suitable generator of $S_3$, hence the group is not cyclic.
All subgroups of order $2$ were found above, these are $\{i,\rho\},\{i,\sigma\},\{i,\tau\}.$
I have no idea how to do this, any hints are appreciated.
REPLY [1 votes]: It is easy to prove that a cyclic group of order $n$ has precisely one subgroup of each order $d$ which is a factor of $n$.
In particular, a cyclic group of even order has precisely one subgroup of order $2$. Since each element of order $2$ generates a distinct subgroup of order $2$, a cyclic group of even order has exactly one element of order $2$.
Your group table shows more than one by easy observation (look a the leading diagonal). Actually exhibiting two distinct elements of order $2$ would be enough.
I mention this because it links the first and second parts of your question.
For the third part, if an element $a$ of order $2$ in a group and an element $b$ of order $3$ commute with each other, their product $c=ab$ has order $6$. Since the group has order $6$ and is not cyclic, no element of order $2$ can commute with any element of order $3$.
This means $ab\neq ba$ or $bab^{-1}\neq a$ for all pairs of elements of order $2$ and order $3$ respectively. Also $bab^{-1} \neq 1$. So if $A$ is the subgroup of order $2$ generated by $a$ you have $bA\neq Ab$ or $bAb^{-1} \neq A$
It is, of course, possible to do everything by calculating explicitly with the actual elements of the group - and there is much to be said for getting a feel for groups with such hands-on work. This answer is intended to indicate some of the patterns which start to appear, and to indicate the kinds of arguments which emerge once such patterns are known.
| 60,060
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TITLE: Dimension of a quotient vector space of meromorphic functions
QUESTION [4 upvotes]: Let $U$ be an open set of the Riemann sphere, $z_i$ be $n$ distinct points of $U$, and $E$ the vector space of meromorphic functions on $U$ with poles of order no more than 2.
Let $F$ be the subspace of $E$ whose elements are holomorphic in a neighborhood of the $z_i$.
Does $E/F$ have finite dimension ? If so, what is it ?
It is clear that it has dimension at least $2n$, since the $\frac{1}{(z-z_i)^k}$, $k=1,2$, form a free family. However, I couldn't determine if there was more (intuition suggests not).
REPLY [3 votes]: If $f$ is an element in $E$, you can subtract all principal parts of $f$ at $z_i$, then you'll get a function which is holomorphic near every $z_i$. Since every principal part is a linear combination of $\frac{1}{z-z_i}$ and $\frac{1}{(z-z_i)^2}$, we can deduce that $E/F$ is a complex vector space of dimension $2n$.
| 61,004
|
> more.
Virtually all Hindus practice rituals of daily worship, or puja. A particularly devout Hindu may visit a temple every day, but most daily worship takes place in the home, where contact is direct between the worshipper and deity. The rituals that take place there are the foundation of all family actions and decisions.
> more.
Each day, members of the family perform personal rituals at their household shrine, which may be a large and impressive room or a tiny niche in a wall. Through worship, the energy of a God or Goddess is summoned to enter the image in the shrine. The deity is then present within the home, protecting the family and engendering a positive future. Shrines contain images relating to the beliefs of each person living in a household, and some homes have separate shrines dedicated to more than one deity.
| 411,410
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TITLE: plotting Keplerian orbit from implicit radial and angular relationship
QUESTION [1 upvotes]: Recently I have been reading about Keplerian orbit and I came across the implicit relationship between $r$ and $\theta$ as:
$\theta = \theta_{0} + L \int_{r_{0}}^{r} \frac{dr}{r^2} \frac{1}{\sqrt{2(E+1/r-L^2/2r^2)}}$
It's the equation of the path, that means by plotting this equation I should get an ellipse. If I have been provided with initial conditions on $r$, $\theta$, $\dot{r}$ and $\dot{\theta}$, then is it possible to derive all the parameters needed to integrate and plot an orbit numerically?
REPLY [1 votes]: The orbit equation for gravity can actually be solved explicitly. It results in an expression for $r$ in terms of $\theta$ which can be written as
$r(\theta) = \frac{p}{1+\epsilon\cos(\theta+C)}$
where $p = \frac{L^2}{m\alpha}$ and $\epsilon = \sqrt{1+\frac{2L^2E}{m\alpha^2}}$
$\alpha$ can be seen as the strength of the gravitational field, and $L$ and $E$ are the angular momentum and energy respectively. $C$ here is the constant of integration and will depend on the initial position and initial velocity of the particle.
| 129,597
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TITLE: Expected value calculation.
QUESTION [0 upvotes]: Let $E(h,k,i,j)$ , $i \leq h$,$j\leq k$ be the expected number of the common numbers of two independently chosen subsets of a set of $h$ elements and $k$ elements respectively, where the first and second subsets respectively contain $i$ and $j$ elements. How to calculate $E(h,k,i,j)$ for two possible case here $h\leq k$ and $k\leq h$? Here is one example shown in figure .
In this scenario how can I calculate expected number of overlap or common numbers which are indicated by T,T' in the figure? Here T is chosen from (5*9 = 45) grid and T' is chosen (7*13=91) grid.
REPLY [0 votes]: If we assume that $H=\{1,2,\dots,h\}$ and $K=\{1,2,\dots,k\}$, and if $i$ elements are chosen (uniformly and without replacement) from $H$ and $j$ elements are chosen (uniformly and without replacement) from $K$, and if $E(h,k,i,j)$ is the expected number of elements common to both samples, then $E(h,k,i,j)=\dfrac{ij\min(h,k)}{hk}=\dfrac{ij}{\max(h,k)}$; this is obtained by adding, for each $n\in\{1,2,\dots,\min(h,k)\}$, the probability that $n$ is chosen in both samples.
Is that what you meant?
| 191,517
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Vij: Alone In A Church With A Dead Russian Witch — embedded YouTube video
Absolutely breathtaking. This is a work of art.
The IMDB has it listed as Viy with a 1967 date. 1967! Forty years ago and what I see puts today’s CGI-infested blockbusters to shame.
Look at that! The sense of composition, the color, the lighting. It’s like a painting come to life. And this was done before computers, when the people involved had to have talent and know their shit firsthand. No relying on “We’ll fix it in post with the computer” or “We’ll have the colorist touch it up later.”
Hollywood, with its fractions-of-a-billion-dollars budgets hasn’t produced anything as beautiful or as frightening.
Update: TrueGore has it on DVD with English subtitles for US$10.00.
| 348,419
|
The gym is a pricey, but effective tool in transforming your physique into tip-top shape. The various equipment, classes, personal trainers, and pool (if you’re lucky) make health clubs the perfect setting to shred the fat – for some. For others, the gym is not an option, nor is it ideal. Maybe monthly gym fees, or the yearlong contractual commitment is a bit much for you. For some, joining a gym may be inconvenient due to its distance from their home or job – or maybe they’re adverse to the whole gym scene. Fortunately gym memberships aren’t a necessity in getting in great physical shape. You can get an effective workout at home by implementing the following steps:
Create Realistic Goals:
Whether it’s a long-term, or a series of short-term goals, setting clear expectations at the onset of a fitness program really helps to kick motivation into high gear. Are you looking to drop a specific amount of weight, target & tone, or enhance your vitality? Regardless of the end game, Fitness gurus always warn us to take it slow and ease into our routine – hence the term ‘realistic.’ Too many make the mistake of expecting too much too soon. To reach the goals you’ve set, be easy on yourself, as well as consistent. Slow and steady yields results.
Methodology:
Now that you’ve created your goals, you can develop an exercise schedule to reach them. The process of getting physically fit is hard work, but it doesn’t mean that you can’t enjoy the process of attaining it. What works for you? Dance, weights, swimming, Zumba, kickboxing, Pilates, interval training, yoga? There are tons of ways to get fit within the comfort of your own home, so feel free to explore. There are DVD’s a plenty, or workouts you can follow on YouTube, Hulu or various fitness sites. If you want to start by using a blend of exercises of your choosing, software such as FitnessBliss allows you to create, print, track and chart your own fitness routine online. Just turn up your fitness playlist and get to work.
Duration & Intensity:
Fitness experts recommend beginners start by walking, jogging, cycling, light weight workouts or beginner’s level fitness classes. It’s best to acclimate your body by beginning your quest with 20-30 minutes of exercise 3-4 times per week. After a few weeks, you can gradually step up the frequency, duration and intensity of your workouts. From personal experience, after having baby number 2, I was absolutely abs-obsessed, but I eventually extended my workouts, adding full body & cardio work into the mix once my body adjusted to the regular exertion. Listen to your body – not your mind. It will let you know when you’re ready to take it to the next level.
Timing:
Some of us have hectic schedules that limit activities of the fitness variety to very specific times of day. However, if you have any kind of flexibility at all in your schedule, try to figure in exercise when your energy is at its peak. Perhaps it’s your lunch break, right after work or first thing in the morning. Whenever the magic hour, you’re most likely to get the most out of your session when your energy is at its peak.
Fueling Up Right:
Exercise is a vital way to get and stay in shape, but without a healthy, balanced diet you’re not likely to see the transformation to a sleeker sexier physique. It’s counterproductive to spend 30 minutes 4 times a week sweating it out just to fuel your body up with junk food. Weight loss diets are a dime a dozen. Ultimately it takes a few simple steps to detoxify your diet and eliminate unnecessary calories and fat. Cut out junk food, fried foods, white flour, excessive sugar & carbs and swap soda and other unhealthy drinks for as much water as possible. If you haven’t done so, gradually introduce your palate to more fruits & veggies, lean proteins like turkey and fish, and whole grains. A shift towards the healthful will not only give you more energy for exercise, but will assist you in getting the results you desire.
Tracking Your Progress:
It’s your fitness plan so how you choose to monitor your progress is entirely up to you. Many trainers recommend tracking the measurement of your target areas, be it your abs, hips, etc., from the jump, observing the change in measurements over time. Tracking by weight may be ideal for those specifically interested in weight loss, as well as keeping a record that includes photos for the use of comparison as you advance in your training. Others prefer to use their own increased strength and vitality as a gauge for success – that and perhaps those favorite pair of jeans they’ve been dying to get back into. No matter your technique, it’s important to be patient, consistent and give your body time to transform.
Room to Move:
If you’re like me, and face challenges in setting up the proper space for a proper work out, there are three main factors to consider. First, make sure you have enough space for the type of work out you plan to engage in. This may sound like a no-brainer, but if you live in close quarters, it’s easy to underestimate the amount of space needed to move about vigorously and accidentally come into contact with an immovable piece of furniture, for example. Secondly, work out in front of a mirror large enough to check for proper form and last but not least, make sure the space has the proper ventilation and temperature while you exert yourself.
| 414,935
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Support PDF,DOC,DOCX,TXT,XLS,WPD,HTM,HTML fils up to 5MB
Company name
Organization TypeLaw Firm
Job Type
Years of Experience
Location
Date Last Verified
Posted on
ProfileBusiness / Corporate Litigator The candidates should be a Partner and senior counsel-level position available for business / corporate litigation attorney with portable business. Litigation clients include publicly held corporations, privately owned companies and high net worth individuals in the general business, real estate and development, and intellectual property fields. California Bar admission, excellent academic credentials are required.
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| 147,506
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Gangster John Gilligan checks out of hospital and head straight for ferry.
A Spanish based Irish gangster ordered the hit on gangland boss John Gilligan, according to police sources.
The Irish Independent’s crime reporter Paul Williams has said that the Gilligan hit followed demands for cash from underworld figures.
The paper says Gilligan has attempted to ‘tap’ Ireland’s most notorious criminals for money since his release from Portlaoise Prison last October.
Associates of a notorious Irish criminal, now based in Spain, were tapped by Gilligan.
The Spanish based criminal regards Gilligan as ‘a weak link,’ attracting police attention to their bids to set up profitable drug deals.
Police sources told the paper that it is not yet clear whether the overseas criminal played any direct role in the failed assassination attempt on Gilligan, who was behind the murder of journalist Veronica Guerin. favor because of his standing before he was sent to prison.”
Source have also described Saturday’s attack in a Dublin house as ‘amateurish’ – two gunmen fired six shots with Gilligan who was hit at least three times in the stomach, hip and leg.
Police say the gun used in the attack was similar to the weapon abandoned by gunmen after a previous planned shooting of Gilligan last December.
Gilligan was reported to be in a stable condition at Connolly Hospital in Blanchardstown, Co Dublin following surgery.
| 156,823
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How To Play Music With Lyrics On Androidadmin January 18, 2019 0 COMMENTS
>>IMAGE may have the habit of singing along with the music. You will search for the lyrics on the Internet if you can’t understand the music.
Stock Android music player plays the music well but lacks to play lyrics. Since it doesn’t provide the benefits of displaying lyrics, you need to look for the best one. If the music player can scroll lyrics, you can too sing along with the artist.
Music can please you depending upon your mood. You may shake your head while listening to the beats. Now, you can sing too by looking at your mobile screen. In this post, you will see how to play music with lyrics on Android.
Requirements
- Android mobile.
- MusixMatch App.
- Internet connection.
Why MusixMatch App?
- MusixMatch has the largest collection of lyrics for the songs.
- It synchronizes with the songs without delay in time.
- As the name implies, it matches with the exact lyrics for the specific song.
- Option to translate the song in many languages.
- Floating lyrics widget to track the lyrics.
- Android TV support.
- You can cast music and lyrics to your smart television. For this, you will need Chromecast app as additional.
Now, let’s see how to play lyrics along with the music. Follow the simple steps given below:
Steps To Play Music With Lyrics on Android
- Download the MusixMatch – Lyrics & Music from Play store and launch the app.
- Login with Facebook or Google account to explore more features. If you need only lyrics for music, skip the login option.
- On the next screen, you’ll be asked to enable the option to fetch lyrics with music. Tap on Enable Now
- On the next screen, you’ll be asked to Allow Notification access for Musixmatch. Tap on Allow
- This is the final option for granting permission. On this screen, Tap on Allow Access to give permission to MusixMatch to scan your music library.
Once you’ve finished the steps, search for your favorite song in this app.
Play the app and check for the lyrics.
Have you seen that? The lyrics are scrolling down automatically. It synchronizes and accurately fetches with the song.
You can see an option to play Translation along with the Live lyrics. If you play a regional music, you will see the original lyric language and translated lyrics. It supports around 56 languages for translation.
To translate the lyrics into the available language, you need to signup with this app.
To explore additional options, tap on 3 vertical dots on the top right side on the lyrics screen. Make sure, you’re playing a music.
There you will see the options like Adding music to Favorites, create lyrics card, sleep timer and Equalizer.
One of the best features we like in this app is that you can sing with lyrics when you locked the mobile screen. Tap the Power button once but don’t unlock your device. The lyrics will show for the music.
The screen will turn off as per the sleep settings made on your mobile.
To Sum up
MusixMatch is a free app which shows ads. If you don’t want to see ads, buy the Premium version. This app requires an Internet connection to play lyrics in the free version. To download lyrics for offline use, you may prefer the paid version. Some of the alternatives to MusixMatch are SongtexteMania and QuickLyric. You could get it for free from the Play Store.
About the Author:
Dinesh is a technology geek who loves to write a lot on Android guides, mobile apps and software tips on his site blogtom.com. His greatest pleasure is to share innovative ideas to inspire others.
| 93,464
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TITLE: What's the inverse of $x^5 +3x^3 + 2x + 1$?
QUESTION [3 upvotes]: Let $f$ be a one-to-one function whose inverse is given by:
$f^{-1}(x)=x^5+3x^3+2x+1$
Compute $f^{−1}(1)$.
My attempt at this yielded a very straightforward answer:
$f^{-1}(1)=1^5+3(1)^3+2(1)+1\\
f^{-1}(1)= 7$
Compute $f(1)$.
I have searched around on this Stack Exchanged and discovered that finding the inverse of a degree 5 polynomial is not viable using only algebra level math.
I would greatly appreciate it if someone was able to point me in the right direction. Thanks in advance.
REPLY [1 votes]: Inverse here means the inverse function.
$$f^{-1}(x)=x^5+3x^3+2x+1$$
1.)
Your calculation is correct. For $f^{-1}(1)$, we have to set $x=1$ and get $f^{-1}(1)=7$ from the above equation.
$\ $
2.)
The preconditions of the task say i.a. that $f$ is a one-to-one function and $f^{-1}$ is its inverse. That means, the inverse of $f^{-1}$ does exist and is $f$.
If $f$ and $f^{-1}$ are inverses of each other, $f(f^{-1}(x))=f^{-1}(f(x))=x$ holds.
If the inverse of a function $f\colon x\mapsto f(x)$ given in closed form does exist, we therefore can try to find the function term of $f^{-1}$ with $y=f^{-1}(x)$ by solving $f(y)=x$ for $y$.
Now $f^{-1}$ is given, and we are looking for its inverse $f$ with $y=f(x)$:
$$f^{-1}(y)=x,$$
$$y^5+3y^3+2y+1=x.$$
The solution $y$ cannot be represented as explicit algebraic function of $x$. But the task needs $f$ only at one point $x=1$.
We set $x=1$ therefore:
$$y^5+3y^3+2y+1=1,$$
$$y^5+3y^3+2y=0,$$
and get $$y_{1,2,3,4,5}={0,-i,i,-i\sqrt{2},i\sqrt{2}}.$$
Because, according to the preconditions, $f$ is a one-to-one function, only one of these values can be $f^{-1}(1)$. To ensure that the task is given correctly, the range of $f^{-1}$ or the domain of $f$ should be specified.
| 182,010
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Discount Hotel in Los Angeles City
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- TV with remote control
- Cable/satellite TV
- All-news channel
- In-room movies
- Newspaper delivered (Mon-Fri)
- Minibar
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- On-site Restaurant
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- Safe deposit box at front desk
- Rental car desk: Hertz
Restaurants & Lounges :
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| 131,775
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\section{Proposed discretisation of the discontinuous Galerkin method on equidistant and scattered points}
\label{sec:proposed}
In this section, we propose a stable discretisation for (high-order) DG methods on equidistant and scattered points.
Therefor, we utilise the techniques discussed in \S \ref{sec:DLS}, involving DLS approximations, LS-QRs, and bases of
DOPs.
Starting point of our discretisation is the analytical DG method \eqref{eq:anal-DG}.
Transformed to the reference element $\Omega_{\mathrm{ref}} = (-1,1)$, the task is to find
$u^i \in \mathbb{P}_K$ such that
\begin{equation}\label{eq:trans_an_DG}
\frac{\Delta x_i}{2} \int_{-1}^1 \dot{u}^i v \intd \xi
- \int_{-1}^1 f(u^i) v' \intd \xi
+ \fnum_R v(1) - \fnum_L v(-1)
= 0
\end{equation}
for all $v \in \mathbb{P}_k$.
Here, $u^i$ denotes the transformation of the numerical solution $u$, consisting of piecewise polynomials, on the
element $\Omega_i$ to the reference element.
Further, $\dot{u}^i$ denotes the temporal derivative $\partial_t u^i$ and $v'$ the spatial derivative $\partial_x v$.
We start our discretisation by replacing $f(u^i)$ in \eqref{eq:trans_an_DG} with a polynomial $f^i \in \mathbb{P}_K$
given by a DLS approximation
\begin{equation}\label{eq:DLS-approx}
f(u^i) \approx f^i := \sum_{k=0}^K \hat{f}^i_{k,N} \varphi_k,
\end{equation}
where $\{ \varphi_k \}_{k=0}^K$ is a basis of $\mathbb{P}_K$ and the modal coefficients are given by a simple finite
sum
\begin{equation}\label{eq:coeffs}
\hat{f}^i_{k,N}
= \scp{f(u^i)}{\varphi_k}_{\vec{\omega}}
= \sum_{n=0}^N \omega_n f(u^i(\xi_n)) \varphi_k(\xi_n).
\end{equation}
For $N=K$, this is the usual polynomial interpolation.
Yet, in our discretisation, $N$ might be chosen greater than $K$.
Next, the involved integrals are replaced by LS-QRs
\begin{equation}\label{eq:DLS-QR}
Q_N[g]
= \sum_{n=0}^N \omega_n^* g(\xi_n)
\approx \int_{-1}^1 g(\xi) \intd \xi
\end{equation}
as discussed in \S \ref{sub:QRs}.
Utilising the discrete inner product $\scp{\cdot}{\cdot}_{\vec{\omega}^*}$, this results in the discretisation
\begin{equation}\label{eq:DG-DLS}
\frac{\Delta x_i}{2} \scp{\dot{u}^i}{v}_{\vec{\omega}^*}
= \scp{f^i}{v'}_{\vec{\omega}^*}
- \left[ \fnum_R v(1) - \fnum_L v(-1) \right].
\end{equation}
$N \in \N$ and $\vec{\omega}^* \in \R^{N+1}$ are chosen such that the resulting LS-QR is stable with
$\kappa(\vec{\omega}^*) = 1$ and provides order of exactness $2K$.
This will be crucial to prove conservation and linear stability of the resulting discretisation.
Further, we follow the idea of collocation and match the quadrature points with the points at which the nodal values of
$f(u^i)$ are used to construct the DLS approximation $f^i \in \mathbb{P}_K$.
This results in a more efficient implementation of the proposed discretisation.
For sake of simplicity, we also use the same weights $\vec{\omega}^*$ for the LS-QR \eqref{eq:DLS-QR} and the DLS
approximation \eqref{eq:DLS-approx}.
Different choices are possible but will not be investigated here.
Finally, we include bases of DOPs.
Note that we can avoid computing a mass matrix on the left hand side of \eqref{eq:DG-DLS} by utilising such a basis of
DOPs w.\,r.\,t.\ the discrete inner product $\scp{\cdot}{\cdot}_{\vec{\omega}^*}$.
Since, when choosing $v = \varphi_l$, we have
\begin{equation}\label{eq:mass_matrix}
\scp{\dot{u}^i}{v}_{\vec{\omega}^*}
= \sum_{k=0}^K \frac{\d}{\d t} \hat{u}_k^i \scp{\varphi_k}{\varphi_l}_{\vec{\omega}^*}
= \frac{\d}{\d t} \hat{u}^i_l
\end{equation}
and \eqref{eq:DG-DLS} becomes a system of $K+1$ ODEs
\begin{equation}\label{eq:DG-DLS-ODE}
\frac{\Delta x_i}{2} \frac{\d}{\d t} \hat{u}^i_l
= \scp{f^i}{\varphi_l'}_{\vec{\omega}^*}
- \left[ \fnum_R \varphi_l(1) - \fnum_L \varphi_l(-1) \right]
\end{equation}
for $l=0,\dots,K$.
The discrete inner product on the right hand side of \eqref{eq:DG-DLS-ODE} is given by
\begin{equation}\label{eq:v-term}
\scp{f^i}{\varphi_l'}_{\vec{\omega}^*}
= \sum_{k=0}^K \hat{f}^i_{k,N} \scp{\varphi_k}{\varphi_l'}_{\vec{\omega}^*}.
\end{equation}
For the sake of brevity, we will refer to the DLS based discretisation \eqref{eq:DG-DLS-ODE} of the DG method as the
\textit{discontinuous Galerkin discrete least squares (DGDLS) method}.
A main part of the DGDLS method is to initially determine a suitable vector of weights $\vec{\omega}^*$ and
a corresponding basis of DOPs.
Figure \ref{fig:flowchart} provides a flowchart which summarises this procedure.
\begin{figure}[!htb]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{
flowchart}
\end{subfigure}
\caption{Flowchart describing the initial construction of suitable weights $\vec{\omega}^*$ and a DOP-basis $\{
\varphi_k \}_{k=0}^K$}
\label{fig:flowchart}
\end{figure}
\begin{remark}
Computing the derivative $\frac{\d}{\d t} \hat{u}^i_l$ for a fixed $l \in \{ 0,\dots,K \}$ in the DGDLS method has
the following complexity:
First note that the inner products $\scp{\varphi_k}{\varphi_l'}_{\vec{\omega}^*}$ and
$\scp{\varphi_k}{\varphi_l'}_{\vec{\omega}^*}$ respectively in \eqref{eq:v-term} and \eqref{eq:coeffs} are computed
once-for-all a priori.
Thus, the computation of a flux coefficients $\hat{f}^i_{k,N}$ is performed in $\mathcal{O}(N)$ operations and the
whole set $\{\hat{f}^i_{k,N}\}_{k=0}^K$, corresponding to a fixed element $\Omega_i$, is computed in $\mathcal{O}(NK)$
operations.
This also yields $\frac{\d}{\d t} \hat{u}^i_l$ in the DGDLS method \eqref{eq:DG-DLS-ODE} to be computed in
$\mathcal{O}(NK)$ operations.
In the subsequent numerical tests, we found the choice $N=2K$ to be sufficient for a stable computation on
equidistant points by the DGDLS method.
From point of complexity, this compares to a DG method using over-integration with $2K$ Gauss--Lobatto or
Gauss--Legendre points and is by a factor $2$ less efficient than a collocation-type DGSEM method using $K+1$
Gauss--Lobatto or Gauss--Legendre points.
\end{remark}
\begin{remark}
In many methods it is desirable to avoid mass matrices and, in particular, their inversion, see
\cite{abgrall2017high, abgrall2016avoid}.
In the proposed discretisation of the DG method this was possible by using bases of DOPs.
Continuous Galerkin methods might benefit even more from the combination of DLS approximations and QRs with
bases of DOPs.
Of course, the restriction to a continuous approximation space has to be regarded.
Nevertheless, we believe that the application of DOPs could have a positive impact on these schemes.
Future work will investigate this possibility.
\end{remark}
| 31,935
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This Week’s Top Story
Chicago interior designer Kara Mann was recently named interior designer and interior architect of an upcoming 74-story condo tower at 1000 South Michigan Avenue.
Mann founded Kara Mann Design (20 W. Hubbard St., 2nd Fl., karamann.com) in 2005. Known for her designs inspired by her background in fashion and art, she made a name for herself by designing high-end single-family homes, but she has been moving towards the hospitality and commercial spaces for some time.
Mann and her team will be responsible for the South Loop building’s 323 individual residences and nearly 40,000 square feet of indoor and outdoor amenity space. Buyers will have four different interior finish packages to choose from when selecting their condos. Mann will also provide residents on the upper floors with the option to have their outdoor spaces staged and decorated.
Just last month, Mann’s designs were front and center at the reopening of Gold Coast’s Talbott Hotel. Built in 1927, the Talbott Hotel received the luxury treatment as Mann cut her teeth on her first hotel project.
Interior Intel
Acclaimed NFL player? Check. Actor? Check. Furniture designer? Check again. Terry Crews presented his new furniture collection at NeoCon, the annual trade show for commercial designers at the Merchandise Mart (222 W. Merchandise Mart Plaza, Ste. 470, themart.com). Crews’s line for Bernhardt Design takes inspiration from ancient Egyptian culture.
Does your office need new furniture? West Elm (1000 W. North Ave., westelm.com) can help. West Elm Workspace with Inscape took home three Best of Show awards at NeoCon, including gold in the furniture systems category for its Conduit system. Created in collaboration with Gensler, this new system is designed to offer flexibility and choice with reconfigurable furniture systems.
Sales
Unison (1911 W. Division St., unisonhome.com) will be closing its doors on June 25, but will remain online. On Friday, the store will launch a major clearance sale. Shoppers can grab up to 80 percent off clearance items such as bedding, throw pillows, tabletop, and linens.
Mitchell Gold + Bob Williams (1555 N. Halsted St., mgbwhome.com) is offering 10 percent off its entire collection through June 18. Decor and home accessories will also ship for free until June 18.
| 150,114
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I love working with children. My loving nature and patience is what allows me to succeed as a nanny. I have raised three children of my own, now 33, 30, and 27. I am very proud of my childrens' success and happy to say that I raised them.
frances nanny Gender:Female Age:61 Preferred Wage ($/Wk): 400 - 500 Ref ID:417696
Position Duration:Permanent / Part-time,Temporary / Summer
Would prefer to:Live-out Preferred States:PA
My past childcare experience: I have worked for the same family for the past 11 years. I raised the oldest child from 5 months old and the second was born 10 months later I felt like part of their family and I loved the children like my own. The oldest boy was in my daughter's wedding. I even took care of their grandfather until his passing. He reminded me of my own father. Ive raised three of my own children. I have 2 married children with five grandchildren , my youngest daughter has graduated from Penn state and is now a math teacher.
| 207,798
|
NORDISK SOMMERUNIVERSITET
Nordic Summer University
Ad-hoc symposium 27.-29. April 2012
Roskilde University, Denmark
Call for Papers - Deadline 15th March 2012
Nordic Summer University
Ad-hoc symposium 27.-29. April 2012
Roskilde University, Denmark
Call for Papers - Deadline 15th March 2012
Universities in Europe, have in the last decade, been through a period of structural and organizational changes. There is a growing political focus on results from research activities, but at the same time an increasing pressure to do more teaching. In the process of controlling and directing the academic work, new financial and bureaucratic procedures are introduced and exercised on both the university as system and the researcher/lecturer as individual. This calls for an open and creative discussion about the university both as idea and institution, and its possible ‘futures’.
Questions which concern the current development include: Is the present development therefore an important departure from the Humboldtian system of research-based university education? Will the policies from the Bologna process undermine the tradition of the freedom of enquiry in the universities? Will this development increase the quality of the academic work or rather lead to the homogenisation and loss of diversity which still exist between universities in Europe? Do those political attempts, such as getting direct control of the research activities and the curriculum in the university education, threaten the university as a free institution in the Humboldt tradition? Will they change the university's role as the basis for a democratic society? What kind of future for the universities is that current higher education policies are forming for us? What are the possibilities for the universities? Should the university as idea be protected as a world Cultural Heritage?
The symposium, drawing on the research and experiences of the participants, will discuss above important matters. It aims to have some understanding of the overall situation, as well as identify the areas where further research and discussions are needed. We would also like to discuss different scenarios for possible university ‘futures' and to create directions for the university's role in future society.
The ad-hoc symposium, will as part of discussions, prepare a proposal (a manchet) for a coming three year NSU study circle on the theme: “The futures of the University” to be presented at the NSU summer session 2012.
Abstract Submission
Word limit: 350 words with a short biography.
Send it to: nsu-symp-papers [at] ruc.dk
Submission deadline: the 15th March 2012. Authors will be notified by 20th March 2012.
Nordic as well as international participants are welcome. Participants without presentations are welcome too. Presentations may be made either in English or in one of the Scandinavian languages.
More information:
Symposium fee: 67€ (paid on arrival) including 2-night accommodation (shared with 2 or 4 persons), symposium dinner, lunches, coffees and teas. Single accommodation can also be arranged with different fees.
Preliminary program can be found: from the 20th March 2012 on where you can also find more information about NSU and sign up for the newsletter.
Registration: from 15th March 2012. (Further notice on registration at from 15th March)
Coordinator: Jesper Jorgensen E-mail: jesperjo [at] ruc.dk
Ingen kommentarer:
Send en kommentar
| 43,366
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\begin{document}
\title[Vector-valued singular integrals]
{Foundations of vector-valued singular integrals revisited---with random dyadic cubes}
\author[T.~P.\ Hyt\"onen]{Tuomas P.\ Hyt\"onen}
\address{Department of Mathematics and Statistics, University of Helsinki, Gustaf H\"all\-str\"omin katu 2b, FI-00014 Helsinki, Finland}
\email{tuomas.hytonen@helsinki.fi}
\date{\today}
\keywords{Calder\'on--Zygmund operator, martingale transform, random dyadic cubes, UMD space, operations on the Haar basis}
\subjclass[2000]{42B20, 60G46}
\maketitle
\begin{abstract}
The vector-valued $T(1)$ theorem due to Figiel, and a certain square function estimate of Bourgain for translations of functions with a limited frequency spectrum, are two cornerstones of harmonic analysis in UMD spaces. In this paper, a simplified approach to these results is presented, exploiting Nazarov, Treil and Volberg's method of random dyadic cubes, which allows to circumvent the most subtle parts of the original arguments.
\end{abstract}
\section{Introduction}
The investigation, during the 1980's \cite{Bourgain:86,Burkholder:83,Figiel:89,McConnell:84}, of the interrelation between the boundedness properties of vector-valued singular integral operators, and geometric or probabilistic properties of the underlying Banach space, culminated in the end of the decade in the proof of the full $T(1)$ theorem for UMD space -valued functions by Figiel \cite{Figiel:90}. This result remains a heavy piece of hard analysis, whose proof depends on subtle combinatorial arguments related to the rearrangements of dyadic cubes~\cite{Figiel:89}. An alternative Fourier-analytic proof of Weis and the author \cite{HytWei:06}, in turn, relies on a delicate square function estimate of Bourgain \cite{Bourgain:86} concerning the action of translation operators on functions of bounded frequency spectrum, whose proof, which predates and foreshadows the rearrangements of Figiel, is similar in spirit and on the same level of difficulty.
Interestingly, some of the complications in Figiel's proof are related to essentially similar configurations of cubes (a smaller cube close to the boundary of a much larger one), which were termed ``bad'' by Nazarov, Treil and Volberg in their $T(1)$ theorem for non-homogeneous spaces \cite{NTV:97,NTV:03}, and which they overcame by a probabilistic argument using a random choice of a dyadic system instead of a fixed one. This suggests the possibility of simplifying Figiel's proof with the help of the random dyadic systems, and indeed this idea works out surprisingly nicely in this paper.
Since there exists already a reasonably streamlined vector-valued argument for the paraproduct operators (see Figiel and Wojtaszczyk \cite{FigWoj:01}, who attribute it to Bourgain), the present article concentrates on a short proof of the following special $T(1)$ theorem, whose quantitative form seems also new as such:
\begin{theorem}[Figiel \cite{Figiel:90}]\label{thm:Figiel}
Let $X$ be a UMD space, $p\in(1,\infty)$, and $\beta_{p,X}$ be the unconditionality constant of martingale differences in $L^p(\R^n;X)$.
Let $T$ be a Calder\'on--Zygmund operator on $\R^n$ which satisfies the standard kernel estimates, the weak boundedness property $\abs{\pair{1_I}{T1_I}}\leq C\abs{I}$ for all cubes $I$, and the vanishing paraproduct conditions $T(1)=T^*(1)=0$.
Then $T$ extends to a bounded linear operator on $L^p(\R^n;X)$, and more precisely
\begin{equation*}
\Norm{T}{\bddlin(L^p(\R^n;X))}
\leq C\beta_{p,X}^2,
\end{equation*}
where $C$ depends only on the dimension $n$ and the constants in the standard estimates and the weak boundedness property.
\end{theorem}
It would be natural to conjecture that the correct estimate should be $C\beta_{p,X}$, but the quadratic bound in terms of $\beta_{p,X}$ is the best that is known in general UMD spaces even for the Hilbert transform. In the scalar-valued case, both singular integral and martingale transform norms in $L^p$ grow like $\max\{p,(p-1)^{-1}\}$, and a similar behaviour in noncommutative $L^p$ spaces has been verified by Randrianantoanina \cite{Randria:02} for martingale transforms, and recently by Parcet \cite{Parcet:09} for singular integrals, so the conjecture is true in these special cases.
In abstract UMD spaces, a linear bound has only been shown for some special classes of operators with an even kernel (see \cite{GMS:10,Hyt:07}). The largest class of operators for which the quadratic bound was previously known seems to consist of the Fourier multiplier operators with symbol $\sigma$ satisfying $\abs{\partial^{\alpha}\sigma(\xi)}\leq C\abs{\xi}^{-\abs{\alpha}}$ for all multi-indices $\abs{\alpha}\leq n+1$. This estimate can be extracted out of the proof of McConnell~\cite{McConnell:84}, although it is not explicitly formulated in his paper. The above multiplier condition, which is stronger than the usual Mihlin or H\"ormander conditions, implies the standard estimates for the corresponding convolution kernel, so these operators also fall under the scope of the above theorem. For more general multiplier classes with fewer derivatives, the known estimates give higher powers of~$\beta_{p,X}$.
As another application of random dyadic systems to vector-valued analysis, I also provide a simpler proof of the mentioned square function estimate of Bourgain (see Theorem~\ref{thm:Bourgain}). Besides the Fourier-analytic approach to the vector-valued $T(1)$ theorem, Bourgain's inequality is also central to various other results in harmonic analysis in UMD spaces, so it seems useful to make it more approachable.
It should be emphasized that this paper is largely expository in character, and even the simplified proofs borrow their general structure and much of the details from the original arguments of Bourgain \cite{Bourgain:86} and Figiel \cite{Figiel:89,Figiel:90}. For Theorem~\ref{thm:Figiel}, a completely selfcontained proof will be provided in order to convince the reader that it can indeed be done in just about five pages. (The proof has not yet started, and it will be finished on page~\pageref{p:endFigiel}.) In the case of Bourgain's inequality, a couple of simpler lemmas (which are important on their own, and quite well known among experts in vector-valued harmonic analysis) from his original paper \cite{Bourgain:86} are taken for granted here. The principal novelty in both proofs consists of avoiding the more subtle points with the help of the probabilistic argument of Nazarov, Treil and Volberg; everything else is basically due to the original authors, even when this is not indicated explicitly at every step.
\subsection*{Acknowledgement}
Support by the Academy of Finland, grants 130166, 133264 and 218148, is gratefully acknowledged.
Some of the ideas in this paper grew out of my discussions with Michael Lacey.
\section{Random dyadic systems; good and bad cubes}
The following construction of random dyadic systems is, up to some details, from Nazarov, Treil and Volberg \cite{NTV:97,NTV:03}, these details being as in \cite{Hyt:A2}. Only one random system, rather than two, will be used here; this is akin to \cite{Hyt:A2,NTV:97} but in contrast to the probably best-known appearance of this kind of constructions in \cite{NTV:03}.
Let $\mathscr{D}^0:=\bigcup_{j\in\Z}\mathscr{D}^0_j$, $\mathscr{D}^0_j:=\{2^{-j}([0,1)^n+m):m\in\Z^n\}$ be the standard system of dyadic cubes. For every $\beta=(\beta_j)_{j\in\Z}\in(\{0,1\}^n)^{\Z}$, consider the dyadic system $\mathscr{D}^{\beta}:=\bigcup_{j\in\Z}\mathscr{D}^{\beta}_j$, where $\mathscr{D}^{\beta}_j:=\mathscr{D}^0_j+\sum_{i>j}2^{-i}\beta_i$. It is also convenient to define the shift of an individual cube $I\in\mathscr{D}^0$ by the formal shift parameter $\beta$ by using the same truncation procedure, $I+\beta:=I+\sum_{i:2^{-i}<\ell(I)}2^{-i}\beta_i$, so that $\mathscr{D}^{\beta}=\{I+\beta:I\in\mathscr{D}^0\}$.
On the space $(\{0,1\}^n)^{\Z}$, consider the natural probability $\prob_{\beta}$, which makes the coordinates $\beta_j$ independent and uniformly distributed over the set $\{0,1\}^n$. This induces a probability on the family of all dyadic systems $\mathscr{D}^{\beta}$ as defined above.
Consider for the moment a fixed dyadic system $\mathscr{D}=\mathscr{D}^{\beta}$ for some $\beta$. A cube $I\in\mathscr{D}$ is called \emph{bad} (with parameters $r\in\Z_+$ and $\gamma\in(0,1)$) if there holds
\begin{equation*}
\dist(I,J^c)\leq \ell(I)^{\gamma}\ell(J)^{1-\gamma}\qquad
\text{for some }J=I^{(k)},\quad k\geq r,
\end{equation*}
where $\ell(I)$ denotes the sidelength and $I^{(k)}$ the $k$th dyadic ancestor of $I$. Otherwise, $I$ is said to be \emph{good}.
Fixing a $I\in\mathscr{D}^0$, consider the random event that its shift $I+\beta$ is bad in $\mathscr{D}^{\beta}$. The badness only depends on the relative position of $I+\beta$ inside the bigger cubes in $\mathscr{D}^{\beta}$, which is determined by the coordinates $\beta_j$ with $2^{-j}\geq\ell(I)$. On the other hand, the absolute position of $I+\beta$ only depends on the coordinates $\beta_j$ with $2^{-j}<\ell(I)$, and hence the badness and position of $I+\beta$ are independent under the random choice of $\beta$. For reasons of symmetry it is obvious that the probability $\prob_{\beta}(I+\beta\text{ is bad})$ is independent of the cube $I$, and we denote it by $\pi_{\bad}$; similarly one defines $\pi_{\good}=1-\pi_{\bad}$. It is easy to see that the probability of $(I+\beta)^{(k)}$ making $I+\beta$ bad for a fixed $k$ (an event which depends on $\sum_{j:\ell(I)\leq 2^{-j}<2^k\ell(I)}\beta_j 2^{-j}$) is
\begin{equation*}
\big(1-2\cdot 2^{-k}\ceil{2^{k(1-\gamma)}}\big)^n\leq
2n\cdot 2^{-k}(2^{k(1-\gamma)}+1)\leq 4n 2^{-k\gamma},
\end{equation*}
and hence
\begin{equation*}
\pi_{\bad}
\leq\sum_{k=r}^{\infty}4n 2^{-k\gamma}
=\frac{4n 2^{-r\gamma}}{1-2^{-\gamma}}.
\end{equation*}
The only thing that is needed about this number in the present paper, as in \cite{Hyt:A2}, is that $\pi_{\bad}<1$, and hence $\pi_{\good}>0$, as soon as $r$ is chosen sufficiently large. We henceforth consider the parameters $\gamma$ and $r$ being fixed in such a way.
\section{The dyadic representation of an operator}
For $\mathscr{D}=\mathscr{D}^{\beta}$, let $\Exp_j$ denote the conditional expectation with respect to $\mathscr{D}_j$, and $\D_j:=\Exp_{j+1}-\Exp_j$. These operators can be conveniently represented in terms of the Haar functions $h_I^{\theta}$, $\theta\in\{0,1\}^n$, defined as follows: For $n=1$,
\begin{equation*}
h^0_I:=\abs{I}^{-1/2}1_I,\qquad
h^1_I:=\abs{I}^{-1/2}(1_{I_+}-1_{I_-}),
\end{equation*}
where $I_+$ and $I_-$ are the left and right halves of $I$, and in general,
\begin{equation*}
h^{\theta}_I(x)=h^{(\theta_1,\ldots,\theta_n)}_{I_1\times\cdots\times I_n}(x_1,\ldots,x_n)
:=\prod_{i=1}^n h^{\theta_i}_{I_i}(x_i).
\end{equation*}
Then
\begin{equation*}
\Exp_j f=\sum_{I\in\mathscr{D}_j}h^0_I\pair{h^0_I}{f},\qquad
\D_j f=\sum_{I\in\mathscr{D}_j}\sum_{\theta\in\{0,1\}^n\setminus\{0\}}
h^{\theta}_I\pair{h^{\theta}_I}{f}.
\end{equation*}
A frequently occurring object is the translation of a dyadic cube $I$ by $m\in\Z^n$ times its sidelength $\ell(I)$; this will be abbreviated as $I\dot+m:=I+\ell(I)m$.
The convergence of $\Exp_j f$ to $f$ as $j\to\infty$ and to $0$ as $j\to-\infty$ (both a.e. and in $L^p(\R^n)$ for $p\in(1,\infty)$) leads to Figiel's representation of an operator $T$ as the telescopic series
\begin{align*}
\pair{g}{Tf}
&=\sum_{j\in\Z}\big(\pair{\Exp_{j+1} g}{T\Exp_{j+1} f}
-\pair{\Exp_j g}{T\Exp_j f}\big) \\
&=\sum_{j\in\Z}\big(
\pair{\D_j g}{T\D_j f}
+\pair{\Exp_j g}{T\D_j f}+\pair{\D_j g}{T\Exp_j f}\big)
=: A+B+C,
\end{align*}
where, upon expanding in terms of the Haar functions,
\begin{align*}
A &=\sum_{m\in\Z^n}\sum_{I\in\mathscr{D}}
\sum_{\eta,\theta\in\{0,1\}^n\setminus\{0\}}
\pair{g}{h_{I\dot+m}^{\eta}}\pair{h_{I\dot+m}^{\eta}}{Th_I^{\theta}}
\pair{h_I^{\theta}}{f}, \\
B &=\sum_{m\in\Z^n}\sum_{I\in\mathscr{D}}
\sum_{\theta\in\{0,1\}^n\setminus\{0\}}
\pair{g}{h_{I\dot+m}^{0}}\pair{h_{I\dot+m}^{0}}{Th_I^{\theta}}
\pair{h_I^{\theta}}{f} \\
&=\sum_{m\in\Z^n}\sum_{I\in\mathscr{D}}
\sum_{\theta\in\{0,1\}^n\setminus\{0\}}
\pair{g}{h_{I\dot+m}^{0}-h_I^0}\pair{h_{I\dot+m}^{0}}{Th_I^{\theta}}
\pair{h_I^{\theta}}{f} \\
&\qquad+\sum_{I\in\mathscr{D}}\sum_{\theta\in\{0,1\}^n\setminus\{0\}}
\ave{g}_I\pair{T^*1}{h_I^{\theta}}\pair{h_I^{\theta}}{f}=:B^0+P,
\end{align*}
and the term $C$ is essentially dual to $B$ and can be treated similarly by splitting into a cancellative part $C^0$ and a paraproduct part $Q$. It is quite explicit in the above formula that $P$ vanishes under the condition that $T^*1=0$, and similarly $Q$ is zero if $T1=0$.
If $T$ is a Calder\'on--Zygmund singular integral
\begin{equation*}
Tf(x)=\int_{\R^n}K(x,y)f(y)\ud y,\qquad x\notin\supp f,
\end{equation*}
which satisfies the standard estimates $\abs{K(x,y)}\leq C\abs{x-y}^{-n}$ and
\begin{equation*}
\abs{K(x+h,y)-K(x,y)}+\abs{K(x,y+h)-K(x,y)}
\leq\frac{C\abs{h}^{\alpha}}{\abs{x-y}^{n+\alpha}}
\end{equation*}
for $\abs{x-y}>2\abs{h}$, as well as the weak boundedness property $\abs{\pair{1_I}{T1_I}}\leq C\abs{I}$ for all cubes $I$, then the Haar coefficients of $T$ satisfy
\begin{align*}
\abs{\pair{h_{I\dot+m}^{\eta}}{Th_I^{\theta}}}
\lesssim (1+\abs{m})^{-n-\alpha},
\qquad(\eta,\theta)\in\{0,1\}^{2n}\setminus\{(0,0)\}.
\end{align*}
(Here and below, the notation $\lesssim$ indicates an inequality of the type ``$\leq C\times\ldots$'', where $C$ may depend at most on the dimension $n$ and the constants appearing in the Calder\'on--Zygmund conditions.) The above estimate was observed by Figiel, and it follows by elementary computations, using the H\"older estimate for the kernel when $m\notin\{-1,0,1\}^n$, the pointwise bound when $m\in\{-1,0,1\}^n\setminus\{0\}$, and the pointwise bound in combination with the weak boundedness property for $m=0$. Then it is easy to check that all the above expansions converge absolutely for example for $f\in C_c^1(\R^n;X)$ and $g\in C_c^1(\R^n;X^*)$.
Now the above expansions of $\pair{g}{Tf}$ are valid with any dyadic system $\mathscr{D}=\mathscr{D}^{\beta}$. Hence they are also valid if we take the average over a random choice of the dyadic system. For the manipulation of such averages, it is convenient to organise the summations over the fixed reference system of dyadic cubes $\mathscr{D}^0$, so that the summation condition does not depend on any random variable.
A basic observation is the following. Let
\begin{align*}
\pi_{\good}:=\prob_{\beta}(I+\beta\text{ is good})=\Exp_{\beta}1_{\good}(I+\beta),
\end{align*}
recalling that this number is independent of the particular cube $I$. For any function $\phi(I)$ defined on the collection of all cubes $I$, we then have
\begin{align*}
\pi_{\good}\Exp_{\beta}\sum_{I\in\mathscr{D}^{\beta}}\phi(I)
&=\sum_{I\in\mathscr{D}^{0}}
\Exp_{\beta}1_{\good}(I+\beta)\Exp_{\beta}\phi(I+\beta) \\
&=\sum_{I\in\mathscr{D}^{0}}
\Exp_{\beta}\big(1_{\good}(I+\beta)\phi(I+\beta)\big)
=\Exp_{\beta}\sum_{I\in\mathscr{D}^{\beta}_{\good}}\phi(I+\beta).
\end{align*}
The second equality used the crucial fact that the event that $I+\beta$ is good is independent of the position of the cube $I+\beta$, and hence of the function $\phi(I+\beta)$. The conclusion of this computation is this: inside the average over the random choice of our dyadic system, any summation over the dyadic cubes may be restricted to the good ones, and the final result is only changed by the absolute multiplicative factor $\pi_{\good}$. This gives rise to the final form of our dyadic representation,
\begin{align*}
\pair{g}{Tf}
=\frac{1}{\pi_{\good}}\Exp_{\beta}(A_{\good}+B^0_{\good}+C^0_{\good})
+\Exp_{\beta}(P+Q),
\end{align*}
where e.g.
\begin{align*}
A_{\good}=A_{\good}^{\beta}
=\sum_{m\in\Z^n}\sum_{I\in\mathscr{D}_{\good}^{\beta}}
\sum_{\eta,\theta\in\{0,1\}^n\setminus\{0\}}
\pair{g}{h_{I\dot+m}^{\eta}}\pair{h_{I\dot+m}^{\eta}}{Th_I^{\theta}}
\pair{h_I^{\theta}}{f},
\end{align*}
and the terms $B^0_{\good}$ and $C^0_{\good}$ are defined in an analogous manner by restricting the summations over $I\in\mathscr{D}$ appearing in $B^0$ and $C^0$ to $I\in\mathscr{D}_{\good}$ only. This could have been done for the paraproduct terms as well, but the known arguments for handling them do not seem to gain any particular simplification from this reduction.
\section{Estimating the expansions as martingale transforms}
The estimation of the operator norm of $T$ via the size of the pairings $\pair{g}{Tf}$ has now been reduced to the estimation of the cancellative parts $A_{\good}$, $B^0_{\good}$ and $C^0_{\good}$ as well as, in general, the paraproducts $P$ and $Q$ which are assumed to be zero here. The randomisation over the choice of the dyadic system was already fully exploited in making this reduction, and the remaining estimates will be carried out uniformly for any fixed choice of $\mathscr{D}=\mathscr{D}^{\beta}$.
Figiel's key idea for the estimation of $A$ is the interpretation of the shifted functions $h^{\eta}_{I\dot+m}$ as martingale transforms of the $h^{\theta}_I$, for each $m\in\Z^n$.
For a single $I\in\mathscr{D}$, this is easily achieved by defining the two-element martingale difference sequence
\begin{align*}
d_{I,m,u}^{\eta,\theta}
:=\frac{1}{2}\big(h_I^{\eta}+(-1)^u h_{I\dot+m}^{\theta}\big),\qquad u=0,1,
\end{align*}
so that
\begin{align*}
h_I^{\eta}=d_{I,m,0}^{\eta,\theta}+d_{I,m,1}^{\eta,\theta},\qquad
h_{I\dot+m}^{\theta}=d_{I,m,0}^{\eta,\theta}-d_{I,m,1}^{\eta,\theta}.
\end{align*}
A little more tricky is to do this in such a way that the $d_{I,m,u}^{\eta,\theta}$ still form a martingale difference sequence when also the cube $I$ is allowed to vary. This is not true for all $I\in\mathscr{D}_{\good}$, but can be achieved for appropriate subcollections which partition $\mathscr{D}_{\good}$.
For each $m$, let $M=M(m):=\max\{r,\ceil{(1-\gamma)^{-1}\log_2^+\abs{m}}\}$. Let then $a(I):=\log_2\ell(I)\mod M+1$, and define $b(I)$ to be alternatingly $0$ and $1$ along each orbit of the permutation $I\mapsto I\dot+m$ of $\mathscr{D}$. We claim that if $(a(I),b(I))=(a(J),b(J))$ for two different cubes $I,J\in\mathscr{D}_{\good}$, then the cubes satisfy the following \emph{$m$-compatibility} condition: either the sets $I\cup(I\dot+m)$ and $J\cup(J\dot+m)$ are disjoint, or one of them, say $I\cup(I\dot+m)$, is contained in a dyadic subcube of $J$ or $J\dot+m$.
Suppose first that $\ell(I)=\ell(J)$. Then $b(I)=b(J)$ ensures that $I\dot+m\neq J$ and $J\dot+m\neq I$, so the disjointness condition holds. Let then for example $\ell(I)<\ell(J)$. Then $a(I)=a(J)$ implies that in fact $\ell(J)\geq 2^{M+1}\ell(I)$. If $I$ intersects $J\cup(J\dot+m)$, then it is contained in a dyadic subcube $K$ of $J$ or $J\dot+m$ of sidelength $\ell(K)=2^{-1}\ell(J)\geq 2^M\ell(I)\geq 2^r\ell(I)$. Since $I$ is good,
\begin{align*}
\dist(I,K^c)>
\ell(I)^{\gamma}\ell(K)^{1-\gamma}
\geq 2^{M(1-\gamma)}\ell(I)\geq\abs{m}\ell(I),
\end{align*}
and hence $\dist(I\dot+m,K^c)\geq\dist(I,K^c)-\abs{m}\ell(I)>0$. This means that also $I\dot+m$ is contained in $K$, as we wanted.
We can hence decompose $\mathscr{D}_{\good}$ into collections of pairwise $m$-compatible cubes by setting
\begin{align*}
\mathscr{D}^m_{k,v}
:=\{I\in\mathscr{D}_{\good}:a(I)=i,b(I)=v\},\qquad
i=0,\ldots,M(m),\quad v=0,1.
\end{align*}
The total number of these collections is $2(1+M(m))\lesssim(1+\log^+\abs{m})$.
The estimate for $A_{\good}$ now finally begins with
\begin{align*}
\abs{A_{\good}}
&\leq\sum_{m\in\Z^n}\sum_{I\in\mathscr{D}_{\good}}
\sum_{\eta,\theta\in\{0,1\}^n\setminus\{0\}}
\abs{\pair{g}{h^{\eta}_{I\dot+m}}\pair{h^{\eta}_{I\dot+m}}{Th^{\theta}_I}
\pair{h^{\theta}_I}{f}} \\
&\leq\Norm{g}{p'} \sum_{m\in\Z^n}\sum_{\eta,\theta}
\sum_{k,v}\BNorm{\sum_{I\in\mathscr{D}_{k,v}^m}
\zeta^{\theta,\eta}_{I,m}h^{\eta}_{I\dot+m}
\pair{h^{\eta}_{I\dot+m}}{Th^{\theta}_I}
\pair{h^{\theta}_I}{f}}{p},
\end{align*}
where the $\zeta^{\theta,\eta}_{I,m}$ are angular factors of the quantities inside the absolute values on the previous line.
For a fixed $\mathscr{D}^m_{k,v}$, the pairwise $m$-compatibility of its cubes ensures that the $d^{\eta,\theta}_{I,m,k}$ defined above, for $I\in\mathscr{D}_{k,v}^m$, form a martingale difference sequence with respect to their generated filtration, when ordered primarily according to decreasing $\ell(I)$, arbitrarily among the intervals $I$ of the same sidelength, and secondarily according to the parameter $u=0,1$. Then the support of any given $d^{\eta,\theta}_{I,m,0}$ is entirely contained in a set where all the previous members of the sequence are constant, so that its vanishing integral ensures that it is indeed a legitimate next member of a martingale difference sequence. And clearly $d^{\eta,\theta}_{I,m,1}$ has a vanishing integral separately on all the sets where $d^{\eta,\theta}_{I,m,0}$ (or the previous members of the sequence) take a given value.
Hence
\begin{align*}
\sum_{I\in\mathscr{D}^m_{k,v}}\zeta^{\theta,\eta}_{I,m}h^{\eta}_{I\dot+m}
\pair{h^{\eta}_{I\dot+m}}{Th^{\theta}_I}\pair{h^{\theta}_I}{f}
\end{align*}
is a martingale transforms of
\begin{align*}
\sum_{I\in\mathscr{D}^m_{k,v}}h^{\theta}_{I}\pair{h^{\theta}_I}{f}
\quad\text{by the multiplying numbers}\quad
\pm\zeta^{\theta,\eta}_{I,m}\pair{h^{\eta}_{I\dot+m}}{Th^{\theta}_I}
\end{align*}
all of which are bounded by $(1+\abs{m})^{-n-\alpha}$.
By a direct application of the defining property of UMD spaces, it then follows that
\begin{align*}
\abs{A_{\good}}
&\lesssim\Norm{g}{p'} \sum_{m\in\Z^n}\sum_{\eta,\theta}
\sum_{k,v}(1+\abs{m})^{-n-\alpha}\beta_{p,X}\BNorm{\sum_{I\in\mathscr{D}^m_{k,v}}
h^{\theta}_I\pair{h^{\theta}_I}{f}}{p},
\end{align*}
Another application of UMD with the transforming sequence of zeros and ones gives
\begin{align*}
\BNorm{\sum_{I\in\mathscr{D}^m_{k,v}}h^{\theta}_I\pair{h^{\theta}_I}{f}}{p}
\lesssim\beta_{p,X}\BNorm{\sum_{I\in\mathscr{D}}
\sum_{\eta\in\{0,1\}^n\setminus\{0\}}
h^{\eta}_I\pair{h^{\eta}_I}{f} }{p}
=\beta_{p,X}\Norm{f}{p},
\end{align*}
and hence
\begin{align*}
\abs{A_{\good}}
&\lesssim\Norm{g}{p'} \sum_{m\in\Z^n}\sum_{\eta,\theta}
\sum_{k,v}(1+\abs{m})^{-n-\alpha}\beta_{p,X}^2\Norm{f}{p} \\
&\lesssim\Norm{g}{p'} \sum_{m\in\Z^n}(1+\log\abs{m})
(1+\abs{m})^{-n-\alpha}\beta_{p,X}^2\Norm{f}{p}
\lesssim\beta_{p,X}^2\Norm{g}{p'}\Norm{f}{p},
\end{align*}
and this completes the estimate for $A_{\good}$.
The considerations for $B^0_{\good}$ are almost the same, we only need to realise the $h^0_{I\dot+m}-h^0_I$ as martingale transforms of the corresponding $h^{\theta}_I$. This is achieved by setting
\begin{align*}
d_{I,m,0}^{0,\theta}
:=\frac{1}{3}\big(h^0_{I\dot+m}+(h^{\theta}_I)^+\big)
-(h^{\theta}_I)^-,\qquad
d_{I,m,1}^{0,\theta}
:=\frac{1}{3}\big(-h^0_{I\dot+m}+2(h^{\theta}_I)^+\big),
\end{align*}
where $h^{\theta}_I=(h^{\theta}_I)^+-(h^{\theta}_I)^-$ is the splitting into positive and negative parts, so that, observing the identity $h^{0}_I=(h^{\theta}_I)^+ +(h^{\theta}_I)^-$,
\begin{align*}
h^{\theta}_I
=d_{I,m,0}^{0,\theta}+d_{I,m,1}^{0,\theta},\qquad
h^0_{I\dot+m}-h^0_I
=d_{I,m,0}^{0,\theta}-2d_{I,m,1}^{0,\theta},
\end{align*}
and hence
\begin{align*}
\sum_{I\in\mathscr{D}_{i,j}}\zeta^{\theta,0}_{I,m}(h^{0}_{I\dot+m}-h^0_I)
\pair{h^{0}_{I\dot+m}}{Th^{\theta}_I}\pair{h^{\theta}_I}{f}
\end{align*}
is a martingale transforms of
\begin{align*}
\sum_{I\in\mathscr{D}_{i,j}}h^{\theta}_{I}\pair{h^{\theta}_I}{f}
\quad\text{by the multiplying numbers}\quad
\{1,-2\}\cdot\zeta^{\theta,0}_{I,m}\pair{h^{0}_{I\dot+m}}{Th^{\theta}_I}
\end{align*}
all of which are bounded by $(1+\abs{m})^{-n-\alpha}$.
With obvious changes the same computation as for $A_{\good}$ then gives
\begin{align*}
\abs{B^0_{\good}}
\lesssim\beta_{p,X}^2\Norm{g}{p'}\Norm{f}{p}
\end{align*}
and the argument for $C^0_{\good}$, as mentioned, is dual to this. This completes the proof of Theorem~\ref{thm:Figiel}.\label{p:endFigiel}
An inspection of the argument gives the following slight generalisation:
\begin{corollary}[Figiel \cite{Figiel:90}]
Let $X$ be a UMD space and $p\in(1,\infty)$.
Let $T$ be a linear operator which satisfies the vanishing paraproduct conditions $T(1)=T^*(1)=0$ and
\begin{align*}
\sum_{m\in\Z^n}(1+\log^+\abs{m})\sup_I\abs{\pair{h_{I\dot+m}^{\eta}}{Th_I^{\theta}}}
\leq C, \qquad(\eta,\theta)\in\{0,1\}^{2n}\setminus\{(0,0)\},
\end{align*}
where the supremum is taken over all cubes $I$ in $\R^n$.
Then $T$ extends to a bounded linear operator on $L^p(\R^n;X)$, and more precisely
\begin{equation*}
\Norm{T}{\bddlin(L^p(\R^n;X))}
\leq C\beta_{p,X}^2,
\end{equation*}
where $C$ depends only on the dimension $n$ and the constant $C$ in the bound for the Haar coefficients.
\end{corollary}
Indeed, the bound $\abs{\pair{h_{I\dot+m}^{\eta}}{Th_I^{\theta}}}\leq C(1+\abs{m})^{-n-\alpha}$ was only used to ensure the summability of the series as in the statement of the corollary.
\section{Bourgain's inequality for translations}
This final section deals with the following result of Bourgain, which has become a cornerstone of Fourier analysis in UMD spaces. Besides Bourgain's original application \cite{Bourgain:86} to vector-valued singular integrals of convolution type, it is a key ingredient of the results in \cite{GirWei:03,HytPor:08,HytWei:06} concerning vector-valued Fourier multipliers, pseudodifferential operators, and the $T(1)$ theorem, respectively, and also in other papers. Let $\hat{f}=\mathscr{F}f$ denote the Fourier transform of $f$.
\begin{theorem}[Bourgain \cite{Bourgain:86}]\label{thm:Bourgain}
Let $X$ be a UMD space and $p\in(1,\infty)$. Let $f_j\in L^p(\R^n;X)$ be functions with $\supp\hat{f}_j\subseteq B(0,2^{-j})$. Then
\begin{equation*}
\BNorm{\sum_j\radem_j f_j(\cdot-2^j y_j)}{p}
\leq C\alpha_{p,X}^n\beta_{p,X}^2
(1+\sup_j\log^+\abs{y_j})\BNorm{\sum_j\radem_j f_j}{p},
\end{equation*}
where the $\radem_j$ are independent symmetric random signs on a probability space $\Omega$, the norms are those of the space $L^p(\Omega\times\R^n;X)$, the constant $\alpha_{p,X}$ is the norm of the Hilbert transform on $L^p(\R;X)$, and $C$ is a constant depending only on the dimension $n$.
\end{theorem}
This was originally proven by Bourgain for periodic functions $f_j\in L^p(\T;X)$. It was transfered to $L^p(\R^n;X)$ by Girardi and Weis \cite{GirWei:03}, but only under the additional condition that the $y_j$ lie on the same line through the origin (which of course is no restriction when $n=1$). However, in the abovementioned applications one needs the case when $y_j\equiv y$, so the restricted statement (which can be deduced from the one-dimensional version as explained by Girardi and Weis, and only requires the constant $\alpha_{p,X}$ in place of $\alpha_{p,X}^n$) is more than sufficient.
Of course $\alpha_{p,X}\leq C\beta_{p,X}^2$ by Theorem~\ref{thm:Figiel}, and it is known by other methods (in fact, Burkholder's original proof \cite{Burkholder:83}) that one can take $C=1$ here, but as the precise connection between $\alpha_{p,X}$ and $\beta_{p,X}$ remains unknown (see \cite{GMS:10} for an up-to-date discussion), it seems better to use both constants in the estimate in the way in which they naturally appear from the proof.
We begin by deriving a dyadic analogue of this inequality, whose proof is greatly simplified by restricting ourselves to good cubes only. The estimate is the following:
\begin{equation*}
\BNorm{\sum_j\radem_j\sum_{I\in\mathscr{D}^{\good}_j}
h^0_{I\dot+m_j}\pair{h^0_I}{f_j}}{p}
\lesssim\beta_{p,X}^2(1+\sup_j\log^+\abs{m_j})\BNorm{\sum_j\radem_j f_j}{p},
\end{equation*}
where $m_j\in\Z^n$ for each $j$, and $\mathscr{D}^{\good}_j:=\mathscr{D}_{\good}\cap\mathscr{D}_j$.
As before, we decompose $\mathscr{D}_{\good}$ into collections of pairwise compatible cubes. The notion of compatibility needs only slight tuning due to the fact that the relevant shifts of the cubes, $\psi:I\mapsto\psi(I):= I\dot+m_j$, $j=-\log_2\ell(I)$, now depend on the the sidelength of the cube in a more general way than before. It is now required that $I\cup\psi(I)$ and $J\cup\psi(J)$ are either disjoint or one of them, say $I\cup\psi(I)$, is contained in a dyadic subcube of either $J$ or $\psi(J)$. We let $M(\psi)$ be defined like $M(m)$ above, using $\sup_j\abs{m_j}$ in place of $\abs{m}$, and then the function $a(I)$ has exactly the same definition as before, and $b(I)$ is obviously defined by using the orbits of $\psi$ now. This provides us with a partition of $\mathscr{D}_{\good}$ into the $O(1+\sup_j\log^+\abs{m_j})$ subcollections $\mathscr{D}_{k,v}$ with $k=0,\ldots,M(\psi)$, $v=0,1$. We write $\mathscr{D}^{k,v}_j:=\mathscr{D}_{k,v}\cap\mathscr{D}_j$.
We then turn to the martingale differences, which are now functions on the product space $\Omega\times\R^n$. For each $I\in\mathscr{D}^{\good}_j$, let
\begin{equation*}
d_{I,u}:=\radem_j(h^0_I+(-1)^uh^0_{I\dot+m_j}),\qquad u=0,1,
\end{equation*}
so that
\begin{equation*}
\radem_j h^0_I=d_{I,0}+d_{I,1},\qquad
\radem_j h^0_{I\dot+m_j}=d_{I,0}-d_{I,1},
\end{equation*}
and hence
\begin{equation*}
\sum_j\radem_j\sum_{I\in\mathscr{D}^{k,v}_j}
h^0_{I\dot+m_j}\pair{h^0_I}{f_j}
\end{equation*}
is a martingale transform of
\begin{equation*}
\sum_j\radem_j\sum_{I\in\mathscr{D}^{k,v}_j}
h^0_I\pair{h^0_I}{f_j}\quad
\text{by the multiplying numbers}\quad\pm 1.
\end{equation*}
It follows that
\begin{align*}
&\BNorm{\sum_j\radem_j\sum_{I\in\mathscr{D}^{\good}_j}
h^0_{I\dot+m_j}\pair{h^0_I}{f_j}}{p} \\
&\leq\sum_{k,v}\BNorm{\sum_j\radem_j\sum_{I\in\mathscr{D}^{k,v}_j}
h^0_{I\dot+m_j}\pair{h^0_I}{f_j}}{p} \\
&\leq\sum_{k,v}\beta_{p,X}\BNorm{\sum_j\radem_j\sum_{I\in\mathscr{D}^{k,v}_j}
h^0_{I}\pair{h^0_I}{f_j}}{p} \\
&\lesssim(1+\sup_j\log^+\abs{m_j})\beta_{p,X}
\BNorm{\sum_j\radem_j\sum_{I\in\mathscr{D}_j}
h^0_{I}\pair{h^0_I}{f_j}}{p},
\end{align*}
where the last estimate also used the contraction property of the $\radem_j$-randomised sums (pointwise in $x\in\R^n$) to return back to the full collection $\mathscr{D}_j$. Here
\begin{align*}
\BNorm{\sum_j\radem_j\sum_{I\in\mathscr{D}_j}
h^0_{I}\pair{h^0_I}{f_j}}{p}
=\BNorm{\sum_j\radem_j\Exp_j f_j}{p}
\leq\beta_{p,X}\BNorm{\sum_j\radem_j f_j}{p}.
\end{align*}
The last estimate is the vector-valued Stein inequality, which is also due to Bourgain~\cite{Bourgain:86}. It can be proven in a couple of lines directly from the definition of UMD (see~\cite{FigWoj:01}), so no difficulties of the proof are hidden into this estimate. The dyadic analogue of Bourgain's inequality has now been proven.
The next task is to compute the average
\begin{equation*}
\Exp_{\beta}\sum_j\radem_j\sum_{I\in\mathscr{D}^{\beta}_{\good,j}}
h^0_{I\dot+m_j}\pair{h^0_I}{f_j}
=\pi_{\good}\cdot\Exp_{\beta}\sum_j\radem_j\sum_{I\in\mathscr{D}^{\beta}_{j}}
h^0_{I\dot+m_j}\pair{h^0_I}{f_j}
\end{equation*}
over the random choice of $\mathscr{D}^{\beta}$. Of course, this will also satisfy the same norm bound, which all these expressions with a fixed $\beta$ have.
Recalling that $\mathscr{D}_j^{\beta}=\mathscr{D}_j^{0}+\sum_{i>j}2^{-i}\beta_i$, where the last binary expansion is uniformly distributed over $[0,2^{-j})^n$ under the random choice of $\beta$, it follows that
\begin{align*}
&\Exp_{\beta}\sum_{I\in\mathscr{D}^{\beta}_{j}}
h^0_{I\dot+m_j}\pair{h^0_I}{f_j} \\
&=\int_{[0,1)^n}\sum_{k\in\Z^n}
h^0_{2^{-j}([0,1)^n+k+u+m_j)}\pair{h^0_{2^{-j}([0,1)^n+k+u)}}{f_j}\ud u \\
&=\int_{\R^n}
2^{jn}h^0(2^j\cdot-u-m_j)
\pair{h^0(2^j\cdot-u)}{f_j}\ud u\qquad\big(h^0:=h^0_{[0,1)^n}\big) \\
&=(\varphi_{2^j}*f_j)(\cdot-2^{-j}m_j),\qquad
\varphi(x):=\int_{\R^n}h^0(x+u)h^0(u)\ud u.
\end{align*}
Hence, as the second intermediate estimate towards Theorem~\ref{thm:Bourgain}, we obtain
\begin{align*}
\BNorm{\sum_j\radem_j(\varphi_{2^j}*f_j)(\cdot-2^{-j}m_j)}{p}
&\lesssim\Exp_{\beta}\BNorm{\sum_j\radem_j
\sum_{I\in\mathscr{D}^{\beta}_{\good,j}}h^0_{I\dot+m_j}\pair{h^0_I}{f_j}}{p} \\
&\lesssim\beta_{p,X}^2(1+\sup_j\log^+\abs{m_j})\BNorm{\sum_j\radem_j f_j}{p}.
\end{align*}
This still deviates from the final goal in two respects: there is additional smoothing on the left, and the shifts $m_j$ are restricted to integer values. Both these drawbacks may be corrected simultaneously by a Fourier multiplier technique going back to Bourgain's original proof.
By a simple change of variable, we may assume that $\supp\hat{f}_j\subseteq B(0,2^{-j-1})$ rather than just $B(0,2^{-j})$. Let $m_j$ be the integer point nearest to $y_j$, so that $z_j:=y_j-m_j\in[2^{-1},2^{-1})^n$. The quantity to be estimated has the Fourier transform
\begin{align*}
\sum_j\radem_j \exp(-i2\pi 2^j y_j\cdot\xi)\hat{f}_j(\xi),
\end{align*}
whereas the one appearing in the intermediate estimate has transform
\begin{align*}
\sum_j\radem_j \exp(-i2\pi 2^j m_j\cdot\xi)\hat{\varphi}(2^j\xi)\hat{f}_j(\xi),
\end{align*}
where $\hat\varphi$ is immediately computed as
\begin{align*}
\hat\varphi(\xi)=\prod_{i=1}^n\sinc^2(\pi\xi_i),\qquad
\sinc u:=\frac{\sin u}{u}.
\end{align*}
This function is smooth and bounded away from zero in $B(0,2^{-1})$, so that it is easy to find a function $\chi\in C_c^{\infty}(B(0,1))$ so that $\chi\hat{\varphi}\equiv 1$ in $B(0,2^{-1})$. By the support property of $\hat{f}_j$, this implies $(\chi\hat{\varphi})(2^j\xi)\hat{f}_j(\xi)\equiv\hat{f}_j(\xi)$, and hence
\begin{align*}
&\exp(-i2\pi 2^j y_j\cdot\xi)\hat{f}_j(\xi) \\
&=\exp(-i2\pi 2^j z_j\cdot\xi)\chi(2^j\xi)
\times\exp(-i2\pi 2^j m_j\cdot\xi)\hat{\varphi}(2^j\xi)\hat{f}_j(\xi),
\end{align*}
The multipliers $\sigma_j(\xi):=\exp(-i2\pi z_j\cdot\xi)\chi(\xi)$, $z_j\in[-2^{-1},2^{-1})^n$, and hence their dilations $\sigma_j(2^j\xi)$ appearing above, have uniformly bounded variation in the sense that
\begin{align*}
\int_{\R^n}\abs{\partial_1\cdots\partial_n\sigma_j(\xi)}\ud\xi\leq C,
\end{align*}
and hence the corresponding Fourier multiplier operators $T_j:=\mathscr{F}^{-1}\sigma_j(2^j\cdot)\mathscr{F}$ satisfy the estimate
\begin{align*}
\BNorm{\sum_j\radem_j T_j g_j}{p}
\lesssim\alpha_{p,X}^n\BNorm{\sum_j\radem_j g_j}{p}
\end{align*}
for all $g_j\in L^p(\R^n;X)$. This is another lemma of Bourgain \cite{Bourgain:86}, but not a particurly difficult one; it is based on representation of such operators as convex combinations of frequency modulations of $n$-fold products of Hilbert transforms in all coordinate directions.
The proof is hence completed by combining the estimate
\begin{align*}
\BNorm{\sum_j\radem_j f_j(\cdot-2^{-j}y_j)}{p}
&=\BNorm{\sum_j\radem_j T_j[\varphi_{2^j}*f_j(\cdot-2^{-j}m_j)]}{p} \\
&\lesssim\alpha_{p,X}^n\BNorm{\sum_j\radem_j \varphi_{2^j}*f_j(\cdot-2^{-j}m_j)}{p}
\end{align*}
with the intermediate inequality already established.
\bibliography{umd-literature}
\bibliographystyle{plain}
\end{document}
| 149,653
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\begin{document}
\title{ Singular sectors of the 1-layer Benney and dToda systems and their
interrelations. }
\author{B. Konopelchenko $^{1}$, L. Mart\'{\i}nez Alonso$^{2}$ and E. Medina$^{3}$
\\
\emph{ $^1$ Dipartimento di Fisica, Universit\'a del Salento and Sezione INFN}
\\ {\emph 73100 Lecce, Italy}\\
\emph{$^2$ Departamento de F\'{\i}sica Te\'orica II, Universidad
Complutense}\\
\emph{E28040 Madrid, Spain}\\
\emph{$^3$ Departamento de Matem\'aticas, Universidad de C\'adiz}\\
\emph{E11510 Puerto Real, C\'adiz, Spain}
}
\maketitle
\begin{abstract}
Complete description of the singular sectors of the 1-layer Benney system ( classical long wave equation)
and dToda system is presented. Associated Euler-Poisson-Darboux
equations E(1/2,1/2) and E(-1/2,-1/2) are the main tool in the
analysis. A complete list of solutions of the 1-layer Benney
system depending on two parameters and belonging to the singular
sector is given. Relation between Euler-Poisson-Darboux equations
$E(\varepsilon,\varepsilon)$ with opposite sign of $\varepsilon$
is discussed.
\end{abstract}
\maketitle
\section{Introduction}
The 1-layer Benney system (classical long wave equation)
\begin{equation}\label{ben}
\begin{cases}
u_t\,+\,u\,u_x\,+\,v_x\,=\,0,\\ \\
v_t\,+\,(u\,v)_x\,=\,0
\end{cases}
\end{equation}
and dToda equation $v_{xx}=(log v)_{tt}$ or equivalently the
system
\begin{equation}\label{toda2}
\begin{cases}
u_t\,+\,v_x\,=\,0,\\ \\
v_t\,+v u_x\,=\,0
\end{cases}
\end{equation}
are the two distinguished integrable systems of hydrodynamical
type (see e.g. \cite{dub,ben}). 1-layer Benney system describes
long waves in shallow water with free surface in gravitational
field. It is the dispersionless limit of the nonlinear
Schr\"{o}dinger equation \cite{zak}. Recently, the 1-layer Benney
($B(1)$) system became a crucial ingredient in the analysis of the
universality of critical behavior for nonlinear equations
\cite{dubgr}. The dToda equation is the 1+1-dimensional version of
the Boyer-Finley equation from the general relativity
\cite{boyer}. It shows up in various problems of fluid mechanics (
see e.g. \cite{mineev}). It is known also (see e.g.\cite{dsl})
that the hodograph equations of the dToda hierarchy determine the
large $N$-limit of the Hermitian model in random matrix theory.
In general, these two systems are an excellent laboratory for
studying properties of integrable hydrodynamical type systems.
In the present work we analyze the structures of the set of hodograph equations of the $B(1)$
hierarchy and dToda hierarchy in terms of its Riemann invariants. These hodograph solutions describe the critical
points
\begin{equation}\label{crit}
\dfrac{\partial
W}{\partial\beta_i}=0,\quad i=1,2,
\end{equation}
of a function $W=W(\bt,\beta_1,\beta_2)$ which depend linearly on
the coordinates $\bt$, where $\bt$ denotes the flow parameters of
the Benney hierarchy and dToda hierarchy respectively and obey an
Euler-Poisson-Darboux equations $E(\varepsilon,\varepsilon)$
\cite{dar}
\begin{equation}\label{edpa}
(\beta_1\,-\,\beta_2)\, \dfrac{\partial^2
W}{\partial\beta_1\,\partial\beta_2}\,=\varepsilon\Big(\,\dfrac{\partial
W}{\partial\beta_1}\,-\,\dfrac{\partial W}{\partial\beta_2}\Big).
\end{equation}
where for the Benney system one has $\varepsilon=1/2$ and for the
dToda system $\varepsilon=-1/2$. The equation \eqref{edpa} and
its multidimensional version are well known for a long time in
classical geometry \cite{dar}.
Its relevance to the theory of Whitham equations has been demonstrated recently in the papers \cite{kud}-\cite{tian2}.
Here we will use classical notation $E(\varepsilon,\mu)$ for the
Euler-Poisson-Darboux equation proposed in \cite{dar} where such
equations with different $\varepsilon$ and $\mu$ have been studied
too.
If we denote by $\mathcal{M}$ the set of solutions $(\bt,\bb)\, (\beta_1\neq \beta_2)$ of the hodograph equations \eqref{crit}, we may distinguish a regular and a singular sector in $\mathcal{M}$
\[
\mathcal{M}={\mathcal{M}}^{\mbox{reg}}\cup{\mathcal{M}}^{\mbox{sing}},
\]
such that given $(\bt,\bb)\in \mathcal{M}$
\[
\mbox{$(\bt,\bb)\in\mathcal{M}^{\mbox{reg}}$ if $\det \Big(\dfrac{\partial^2\,W(\bt,\bb)}{\partial\,\beta_i\,\partial\,\beta_j}\Big)\neq 0$},\quad
\mbox{$(\bt,\bb)\in\mathcal{M}^{\mbox{sing}}$ if $\det \Big(\dfrac{\partial^2\,W(\bt,\bb)}{\partial\,\beta_i\,\partial\,\beta_j}\Big)=0$}.
\]
The elements of $\mathcal{M}^{\mbox{reg}}$, correspond to the case when the system \eqref{crit} is uniquely solvable and hence, it defines a unique solution $\bb(\bt)$.
The singular class $\mathcal{M}^{\mbox{sing}}$ represents degenerate critical points of the function $W$ and are the points on which the implicit solutions $\bb(\bt)$ of the hodograph equations exhibit ``gradient catastrophe''
behaviour. As we will see in this paper, the Euler-Poisson-Darboux equation is of great help to analyze the structure of $\mathcal{M}^{\mbox{sing}}$.
As the
illustration of the general result a complete list of solutions of the 1-layer Benney hierarchy
from $\mathcal{M}^{\mbox{sing}}$ depending on two parameters is
presented.
We also discuss the relation between Euler-Poisson-Darboux equations with opposite $a$ and Euler-Poisson-Darboux
equations for symmetries and densities of integrals of motion for integrable hydrodynamical type systems.
\section{1-layer Benney hierarchy and its singular sector}
The $B(1)$ system \eqref{ben} is a member of a dispersionless integrable hierarchy of deformations of the curve
(see e.g. \cite{kod,km}).
\begin{equation}\label{curve}
p^2=(\lambda-\beta_1)\,(\lambda-\beta_2).
\end{equation}
where $u=-(\beta _{1}+\beta _{2}),v=\frac{1}{4}(\beta _{1}-\beta
_{2})^{2}$. The flows $\bb(\bt)$ are characterized by the
following condition: There exists a family of functions
$S(\lambda,\bt,\bb)$ satisfying
\begin{equation}\label{kdV}
\partial_{t_n}\, S(\lambda,\bt,\bb(\bt))=\Omega_n(\lambda,\bb(\bt)),
\quad n\geq 1.
\end{equation}
where
\begin{equation}
\Omega_n(\lambda,\bb)=
\Big(\dfrac{\lambda^n}{\sqrt{(\lambda-\beta_1)\,(\lambda-\beta_2)}}\Big)_{\oplus}\,\sqrt{(\lambda-\beta_1)\,(\lambda-\beta_2)}.
\end{equation}
where $\oplus$ denotes the standard projection on the positive
powers of $\lambda$.Functions $S$ which satisfy \eqref{kdV} are
referred to as \emph{action functions} in the theory of
dispersionless integrable systems (see e.g. \cite{kri}). Notice
that for $n=1$ \eqref{kdV} reads
\[
p=\dfrac{\partial S}{\partial x},\quad x:=t_1,
\]
so that the sytem \eqref{kdV} is equivalent to
\begin{equation}\label{kdvp}
\partial_{t_n} p=\,\partial_x\,\Omega_n,
\end{equation}
and, in terms of Riemann invariants $\bb$, it can be rewritten in
the hydrodynamical form
\begin{equation}\label{eqbeta}
\partial_{t_n}\,\beta_i=\Big(\dfrac{\lambda^n}{\sqrt{(\lambda-\beta_1)\,(\lambda-\beta_2)}}\Big)_{\oplus}\,\Big|_{\lambda=\beta_i}\,\partial_x\,\beta_i,\quad i=1,2.
\end{equation}
The $t_2$-flow of this hierarchy is the $B(1)$ system \eqref{ben}
($t:=t_2$)
\begin{equation}\label{benr}
\begin{cases}
\partial_{t}\,\beta_1=\dfrac{1}{2}\,(3\,\beta_1+\beta_2)\,\beta_{1\,x},\\ \\
\partial_{t}\,\beta_2=\dfrac{1}{2}\,(3\,\beta_2+\beta_1)\,\beta_{2\,x}.
\end{cases}
\end{equation}
For $v>0$ the $B(1)$ system is hyperbolic while for $v<0$ it is elliptic.
\vspace{0.3cm}
It was proved in \cite{kmm} that the system \eqref{crit} for the critical points of the function
\begin{equation}\label{w1}
W(\bt,\bb)\,:=\,\oint_{\gamma}\dfrac{\d \lambda}{2\,i\,\pi}\,\dfrac{V(\lambda,\bt)}{\sqrt{(\lambda-\beta_1)\,(\lambda-\beta_2)}},
\end{equation}
where $V(\lambda,\bt)=\sum_{n\geq 1}t_n\,\lambda^n$, is a system of hodograph equations for the Benney hierarchy. Moreover, the action function for the corresponding solutions is given by
\begin{equation}\label{sol}
S(\lambda,\bt,\bb)=\sum_{n\geq 1}\, t_n\,\Omega_n(\lambda,\bb)=h(\lambda,\bt,\bb)\,\sqrt{(\lambda-\beta_1)(\lambda-\beta_2)}.
\end{equation}
where
\[
h(\lambda,\bt,\bb):=\Big(\dfrac{V(\lambda,\bt)}{\sqrt{(\lambda-\beta_1)(\lambda-\beta_2)}}\Big)_{\oplus}.
\]
Obviously, the function $W$ satisfies the Euler-Poisson-Darboux equation $E(1/2, 1/2)$.
Written explicitly, $W$ represents itself the series
\begin{align}\label{Wexp}
W&=\,\dfrac{x}{2}(\beta_1+\beta_2)+\dfrac{t_2}{8}(3\beta_1^2+2\beta_1\beta_2+3\beta_2^2)+\dfrac{t_3}{16}
\left(5\beta_1^3+
3\beta_1^2\beta_2+3\beta_1\beta_2^2+5\beta_2^3\right) \nonumber \\
&+
\dfrac{t_4}{128}(35\beta_1^4+20\beta_1^3\beta_2+18\beta_1^2\beta_2^2+20\beta_1\beta_2^3+35\beta_2^4)+\cdots.
\end{align}
The hodograph equations \eqref{crit} with $t_n\,=\,0$ for $n\,\geq\,5$ take the form
\begin{equation}\label{h2}
\everymath{\displaystyle}
\begin{cases}
8x+4t_2(3\beta_1+\beta_2)+3t_3\left(5\beta_1^2+
2\beta_1\beta_2+\beta_2^2\right)+
\dfrac{t_4}{8}(140\beta_1^3+60\beta_1^2\beta_2+18\beta_1\beta_2^2+20\beta_2^3)=0,\\\\
8x+4t_2(\beta_1+3\beta_2)+3t_3\left(\beta_1^2+
2\beta_1\beta_2+5\beta_2^2\right)
+\dfrac{t_4}{8}(140\beta_2^3+60\beta_2^2\beta_1+18\beta_2\beta_1^2+20\beta_1^3)=0.
\end{cases}
\end{equation}
Detailed analysis of equations \eqref{h2} will be performed in section 3.
Here, we would like to make two observations. First, one is that the formulae \eqref{h2} point out on the possible
alternative interpretation of the times $t_2$, $t_3$, $t_4$,... of the $B(1)$ hierarchy. Namely, taking $t_2\,=\,0$ in the
formulae \eqref{h2}, we see that $t_3$ and $t_4$ are parameters appearing in the initial data $\beta_1(x,t_2=0)$ and
$\beta_2(x,t_2=0)$. Thus, one can view hodograph equations \eqref{crit} (in particular, equations \eqref{h2}) as
equations describing the time evolution of the family of initial data for the $B(1)$ system , parametrized by the
variables $t_3$, $t_4$, $t_5$,...
Second observation concerns with the elliptic version of the $B(1)$ system. In this case
$\beta_2\,=\,\overline{\beta}_1$ and the system \eqref{benr} reduces to the single equation
\begin{equation}\label{redben}
\partial_{t}\,\beta\,=\,\frac{1}{2}\,(3\,\beta\,+\,\overline{\beta})\,\beta_x,\quad t:=t_2,\quad \beta:=\beta_1.
\end{equation}
This equation is equivalent to the nonlinear Beltrami equation
\begin{equation}\label{bel}
\beta_{\bar{z}}\,=\,\frac{2\,i\,-\,3\,\beta\,-\,\overline{\beta}}{2\,i\,+\,3\,\beta\,+\,\overline{\beta}}\,\beta_z,
\end{equation}
where $z\,=\,x\,+\,i\,t$. This fact indicates that the theory of quasi-conformal mappings (see e.g. \cite{alp})
can be relevant for the analysis of properties of the elliptic $B(1)$ system \eqref{ben} $(v\,<\,0)$. Hence,
since the elliptic $B(1)$ system is the quasiclassical limit \cite{zak} of the focusing nonlinear Schr\"{o}dinger (NLS) equation
\[
i\,\epsilon\,\psi_t\,+\,\dfrac{\epsilon^2}{2}\,\psi_{xx}\,+\,|\psi|^2
\,\psi\,=\,0,
\]
with $\psi\,=\,A\,\exp{\Big(\dfrac{i}{\epsilon}\,S\Big)}$,
$u\,=\,\dfrac{\epsilon}{2\,i}\Big(\dfrac{\psi_x}{\psi}-\dfrac{\overline{\psi}_x}{\overline{\psi}}\Big)$, $v\,=\,-\,|\psi|^2$,
$\epsilon\,\rightarrow\,0$, the quasiconformal mapping can be useful also in the study of the small dispersion limit of the focusing
NLS equation (compare with \cite{dubgr}).
In order to analyse singular sector of the 1-layer Benney
hierarchy we first observe that due to the Euler-Poisson-Darboux
equation for given $(\bt,\bb)\in \mathcal{M}$, as a consequence
of \eqref{edpa} one has
\begin{equation}\label{nondiag}
\frac{\partial^2\,W}{\partial\,\beta_1\,\partial\,\beta_2}\,=\,0.
\end{equation}
Consequently
\begin{equation}\label{det}
\det\Big(\frac{\partial^2\,W}{\partial\,\beta_i\,\partial\,\beta_j}\Big)=
\frac{\partial^2\,W}{\partial\,\beta_1^2}\cdot\frac{\partial^2\,W}{\partial\,\beta_2^2}.
\end{equation}
Thus, we have
\begin{pro}
Given $(\bt,\bb)\in \mathcal{M}$ then
\begin{enumerate}
\item $(\bt,\bb)\,\in\,\mathcal{M}^{\mbox{reg}}$ if and only if
$\everymath{\displaystyle}
\frac{\partial^2\,W}{\partial\,\beta_1^2}\,\neq\,0\, \mbox{and} \,
\frac{\partial^2\,W}{\partial\,\beta_2^2}\,\neq\,0.
$
\item $(\bt,\bb)\,\in\,\mathcal{M}^{\mbox{sing}}$ if and only at least one of the
derivatives
$\everymath{\displaystyle}
\frac{\partial^2\,W}{\partial\,\beta_1^2},\, \frac{\partial^2\,W}{\partial\,\beta_2^2},
$
vanishes.
\end{enumerate}
\end{pro}
Furthermore, using \eqref{edpa} it follows easily that at any point $(\bt,\bb)\in {\mathcal{M}}$ all mixed derivatives $\partial_{\beta_1}^i\partial_{\beta_2}^j W$ can be expressed in terms of linear combination of derivatives $\partial_{\beta_1}^nW$ and $\partial_{\beta_2}^m W$. Hence if we define ${\mathcal{M}}^{\mbox{sing}}_{n_1,n_2}$ as the set of points $(\bt,\bb)\in {\mathcal{M}}$ such that
\begin{equation}\label{m12}
\dfrac{\partial^{n_i+2}W}{\partial\beta_i^{n_i+2}}\neq 0, \quad
\dfrac{\partial^k W}{\partial \beta_i^k}=0,\quad \forall 1\leq k\leq n_i+1,\quad (i=1,2),
\end{equation}
it follows that
$$
{\mathcal{M}}^{\mbox{sing}}=\bigcup _{n_1+n_2 \geq 1}{\mathcal{M}}^{\mbox{sing}}_{n_1,n_2},
$$
where
$$
{\mathcal{M}}^{\mbox{sing}}_{n_1,n_2}\bigcap{\mathcal{M}}^{\mbox{sing}}_{n'_1,n'_2}=\emptyset,\;
\mbox{ for $(n_1,n_2)\neq (n'_1,n'_2)$}
$$
We may characterize the classes $\mathcal{M}^{\mbox{sing}}_{n_1,n_2}$ of the singular sector in terms of the behaviour of $S(\lambda)$
at $\lambda=\beta_i\, (i=1,2)$ . Indeed the derivative $\partial_{\beta_1}^{k+1}W$ with $k\geq 1$ is proportional to the integral
\begin{equation*}\everymath{\displaystyle}
\oint_{\gamma}\dfrac{\d \lambda}{2\,i\,\pi}
\,\dfrac{V(\lambda,\bt)}{
(\lambda-\beta_1)^{k+1}\,\sqrt{(\lambda-\beta_1)\,(\lambda-\beta_2)}}=\oint_{\gamma}\dfrac{\d \lambda}{2\,i\,\pi}\
\dfrac{h(\lambda)}{
(\lambda-\beta_1)^{k+1}}
=\dfrac{1}{k!}\,\Big(\partial_{\lambda}^{k} \,h(\lambda)\Big)\Big|_{\lambda=\beta_1},
\end{equation*}
and a similar result follows for the derivatives $\partial_{\beta_2}^{k+1}W$ with $k\geq 2$. As a consequence we have
\begin{pro}
A point $(\bt,\bb)\in\mathcal{M}$ belongs to the singularity class $\mathcal{M}^{\mbox{sing}}_{n_1,n_2}$ if and only if
\begin{equation}
\everymath{\displaystyle}
S(\lambda,\bt,\bb)\sim (\lambda-\beta_i)^{\frac{2n_i+3}{2}}\quad \mbox{as $\lambda\rightarrow \beta_i,\quad (i=1,2)$}
\end{equation}
\end{pro}
\noindent
\section{Explicit determination of singular sectors}
It is easy to see that the singular classes ${\mathcal{M}}^{\mbox{sing}}_{n_1,n_2}$ can be determined by means of a system of $n_1+n_2$ constraints for the coordinates $\bt$. Indeed, the points $(\bt,\bb)$ of $ {\mathcal{M}}^{\mbox{sing}}_{n_1,n_2}$ are characterized by the equations
\begin{equation}\label{m12b}
\dfrac{\partial^k W}{\partial \beta_i^k}=0,\quad \forall 1\leq k\leq n_i+1,\quad i=1,2,
\end{equation}
and
\begin{equation}\label{m12c}
\dfrac{\partial^{n_i+2}W}{\partial\beta_i^{n_i+2}}\neq 0, \quad i=1,2.
\end{equation}
Now the observation is that the jacobian matrix of the the system of two equations
\begin{equation}\label{m12d}
\dfrac{\partial^{n_i+1}W}{\partial\beta_i^{n_i+1}}=0, \quad i=1,2
\end{equation}
is not singular as
\begin{equation}\label{Delta23}\everymath{\displaystyle}
\Delta\,:=\,\left|\begin{array}{cc}
\frac{\partial^{n_1+2}\,W}{\partial\,\beta_1^{n_1+2}} & \frac{\partial^{n_2+2}\,W}{\partial\, \beta_1\partial\,\beta_2^{n_2+1}}
\\ \\
\frac{\partial^{n_1+2}\,W}{\partial\,\beta_1^{n_1+1}\,\partial\,\beta_2} & \frac{\partial^{n_2+2}\,W}{\partial\,\beta_2^{n_2+2}}
\end{array}\right|\,\neq\,0.
\end{equation}
Indeed, we notice that as a consequence of \eqref{edpa} the derivatives outside the diagonal of $\Delta$ are linear combinations of the derivatives $\{\partial_{\beta_i}^k\,W, \; 1\leq k\leq n_i+1,\; i=1,2\}$, so that from \eqref{m12b}-\eqref{m12c} we have
\[
\Delta=\frac{\partial^{n_1+2}\,W}{\partial\,\beta_1^{n_1+2}}\cdot \frac{\partial^{n_2+2}\,W}{\partial\,\beta_2^{n_2+2}}\neq 0.
\]
Therefore, one can solve \eqref{m12d} and get a solution $\bb(\bt)$. Substituting this solution in the remaining equations
\eqref{m12b} gives $n_1+n_2$ constraints of the form
\[
f_k(\bt)=0,\quad k=1,\ldots,n_1+n_2.
\]
It is not difficult to determine the solutions of \eqref{m12b}-\eqref{m12c} in two simple cases: with one parameter $t_3$ ($t_4=t_5=\cdots=0$),
and with two parameters $t_3$, $t_4$ ($t_5=t_6=\cdots=0$). We have that in this case
\[
{\mathcal{M}}^{\mbox{sing}}\,=\,{\mathcal{M}}^{\mbox{sing}}_{10}\,\cup\,{\mathcal{M}}^{\mbox{sing}}_{01}
\]
with ${\mathcal{M}}^{\mbox{sing}}_{10}$ defined by
$$\everymath{\displaystyle}\begin{array}{ll}
\textbf{1.} &
x\,=\,\frac{-45t_4 t_3^3+180 t_2 t_4^2
t_3+\sqrt{15}(8t_2t_4-3t_3^2)\sqrt{t_4^2 \left(3 t_3^2-8
t_2 t_4\right)}}{360 t_4^3},\\ \\
&\beta_1\,=\,-\frac{5 t_3 t_4+\sqrt{15} \sqrt{t_4^2 \left(3
t_3^2-8 t_2 t_4\right)}}{20t_4^2},\qquad \beta_2\,=\,\frac{-3 t_3 t_4+\sqrt{15} \sqrt{t_3^2 \left(3 t_3^2-8 t_2
t_4\right)}}{12 t_4^2},\\ \\ \\
\textbf{2.} & x\,=\,\frac{-45t_4 t_3^3+180 t_2 t_3^2
t_3-\sqrt{15}(8t_2t_4-3t_3^2)\sqrt{t_4^2 \left(3 t_3^2-8
t_2 t_4\right)}}{360 t_4^3},\\ \\
&\beta_1\,=\,\frac{-5 t_3 t_4+\sqrt{15} \sqrt{t_4^2 \left(3
t_3^2-8 t_2 t_4\right)}}{20t_4^2},\qquad \beta_2\,=\,-\frac{3 t_3 t_4+\sqrt{15} \sqrt{t_4^2 \left(3 t_3^2-8 t_2
t_4\right)}}{12 t_4^2},
\end{array}$$
and ${\mathcal{M}}^{\mbox{sing}}_{01}$ by
$$\everymath{\displaystyle}\begin{array}{ll}
\textbf{3.} & x\,=\,\frac{-45t_4 t_3^3+180 t_2 t_4^2
t_3-\sqrt{15}(8t_2t_4-3t_3^2)\sqrt{t_4^2 \left(3 t_3^2-8
t_2t_4\right)}}{360 t_4^3},\\ \\
&\beta_1\,=\,-\frac{3 t_3 t_4+\sqrt{15} \sqrt{t_4^2 \left(3 t_3^2-8 t_2
t_4\right)}}{12 t_4^2},\qquad \beta_2\,=\,\frac{-5 t_3 t_4+\sqrt{15} \sqrt{t_4^2 \left(3
t_3^2-8 t_2 t_4\right)}}{20t_4^2},\\ \\ \\
\textbf{4.} & x\,=\,\frac{-45t_4 t_3^3+180 t_2 t_4^2
t_3+\sqrt{15}(8t_2t_4-3t_3^2)\sqrt{t_4^2 \left(3 t_3^2-8
t_2 t_4\right)}}{360 t_4^3},\\ \\
&\beta_1\,=\,\frac{-3 t_3 t_4+\sqrt{15} \sqrt{t_4^2 \left(3 t_3^2-8 t_2
t_4\right)}}{12 t_4^2},\qquad \beta_2\,=\,-\frac{5 t_3 t_4+\sqrt{15} \sqrt{t_4^2 \left(3
t_3^2-8 t_2 t_4\right)}}{20t_4^2}.
\end{array}$$
\section{Singular sector of the elliptic $B(1)$ system}
Now, we will consider the elliptic $B(1)$ system \eqref{ben}. Singular sector ${\mathcal{M}}^{\mbox{sing}}$ has
in this case a structure which is quite different from that of the hyperbolic system. Indeed, since $\beta_1\,=\,\overline{\beta}_2$,
the function $W$ for real $x,\,t_2,\,t_3,\,\dots$ is the real valued function
\[
\overline{W(\bt,\beta,\overline{\beta})}\,=\,W(\bt,\beta,\overline{\beta}),
\]
and the hodograph equation \eqref{crit} has the form of the Cauchy-Riemann condition $(\beta\,=\,\beta_1)$
\begin{equation}\label{CR}
\frac{\partial\,W}{\partial\,\overline{\beta}}\,=\,0.
\end{equation}
Regular and singular sectors $\mathcal{M}^{\mbox{reg}}$ and $\mathcal{M}^{\mbox{sing}}$ are defined as the sets
$(\bt,\beta,\overline{\beta})$ of solutions of equation \eqref{CR} such that the hermitian form
$$\d^2\,W\,=\,\frac{\partial^2\,W}{\partial\,{\beta}^2}\,\d\beta^2\,+\,
2\,\frac{\partial^2\,W}{\partial\,\beta\,\partial\,\overline{\beta}}\,\d\beta\,\d\,\overline{\beta}\,+\,
\frac{\partial^2\,W}{\partial\,\overline{\beta}^2}\,\d\,\overline{\beta}^2,$$
is nondegenerate or degenerate, respectively. For unreduced solutions $(\beta\,\neq\,\overline{\beta})$, the corresponding
Euler-Poisson-Darboux equation implies that
$$\frac{\partial^2\,W}{\partial\,\beta\,\partial\,\overline{\beta}}\,=\,0,$$
and, hence
\begin{equation}\label{det}
\everymath{\displaystyle}\left|
\begin{array}{cc}
\frac{\partial^2\,W}{\partial\,{\beta}^2} &
\frac{\partial^2\,W}{\partial\,\beta\,\partial\,\overline{\beta}}\\ \\
\frac{\partial^2\,W}{\partial\,\beta\,\partial\,\overline{\beta}} &
\frac{\partial^2\,W}{\partial\,\overline{\beta}^2}
\end{array}\right|\,=\,
\left|\frac{\partial^2\,W}{\partial\,\overline{\beta}^2}\right|^2.
\end{equation}
Thus, one has
\begin{pro}
For unreduced solutions of the elliptic $B(1)$ system, the regular sector $\mathcal{M}^{\mbox{reg}}$ is defined by the condition
\begin{equation}\label{regconel}
\frac{\partial\,W}{\partial\,\overline{\beta}}\,=\,0,\quad \frac{\partial^2\,W}{\partial\,\overline{\beta}^2}\,\neq\,0.
\end{equation}
\end{pro}
A similar analysis to that of the hyperbolic case readily leads to
\begin{pro}
Singular sector $\mathcal{M}^{\mbox{sing}}$ of the elliptic $B(1)$ system \eqref{ben} is the union of the subspaces
$\mathcal{M}_n^{\mbox{sing}}$, $(n\,=\,1,2,3,\dots)$ defined as
\begin{equation}\label{singellip}
\mathcal{M}_n^{\mbox{sing}}\,=\,\left\{(\bt,\beta,\overline{\beta})\,\in\,\mathcal{M}^{\mbox{sing}}:\;
\frac{\partial^k\,W}{\partial\,\overline{\beta}^k}\,=\,0,\;k\,=\,1,\dots,n+1;\;
\frac{\partial^{(n+2)}\,W}{\partial\,\overline{\beta}^{(n+2)}}\,\neq\,0\right\}
\end{equation}
Solutions belonging to $\mathcal{M}_n^{\mbox{sing}}$ are defined on a subspace of codimension $2\,n$ in the space of parameters
$x$, $t_2$, $t_3$, ... .
\end{pro}
So, in the elliptic case, gradient catastrophe happens in the point $(x,t)$ at fixed parameters $t_2$, $t_3$,...
Similar to the hyperbolic case, the subspace $\mathcal{M}_n^{\mbox{sing}}$ is not empty if at least $n$ parameters
$t_2$, $t_3$,...,$t_{n+1}$ are different from zero in the formula \eqref{Wexp}.
It is instructive to rewrite the formula \eqref{Wexp} for the function $W$ in terms of the real and imaginary part of $\beta_1$. i.e.
$\beta_1\,=\,U\,+\,i\,V$:
\begin{equation}\label{Wexpri}\everymath{\displaystyle}\begin{array}{lll}
W&=&x\,U\,+\,t_2\,(U^2\,-\,\frac{1}{2}V^2)\,+\,t_3\,(U^3\,-\,\frac{3}{2}\,U\,V^2)\,+\,
t_4\,(U^4\,-\,3\,U^2\,V^2\,+\,\frac{3}{8}\,V^4)\\ \\
& &\,+\,t_5\,(U^5\,-\,5\,U^3\,V^2\,+\,\frac{15}{8}\,U\,V^4)\,+\,\cdots.
\end{array}\end{equation}
This formula explicitly shows the character of elliptic singularities exhibited for the function $W$ for various values of
parameters $t_2$, $t_3$, ....
Basic equations \eqref{CR}, \eqref{Wexp} and also conditions \eqref{singellip} defining subspaces $\mathcal{M}_n^{\mbox{sing}}$
can be easily rewritten in terms of the original variables $u$ and $v$. Since
$$\frac{\partial\,W}{\partial\,\beta}\,=\,-\,\frac{\partial\,W}{\partial\,u}\,+\,i\,\sqrt{-\,v}\,\frac{\partial\,W}{\partial\,v},$$
the hodograph equation \eqref{CR} becomes (for $v\,\neq\,0$)
\begin{equation}\label{conuv}
\frac{\partial\,W}{\partial\,u}\,=\,0,\quad \frac{\partial\,W}{\partial\,v}\,=\,0,
\end{equation}
while the Euler-Poisson-Darboux equation and equation \eqref{conuv} take the form
\begin{equation}\label{epdellip}
\frac{\partial^2\,W}{\partial\,u^2}\,-\,v\,\frac{\partial^2\,W}{\partial\,v^2}\,=\,0.
\end{equation}
For the subspace $\mathcal{M}_1^{\mbox{sing}}$ conditions \eqref{singellip} are
\begin{equation}\label{s1e}
\frac{\partial\,W}{\partial\,\beta}\,=\,0,\quad \frac{\partial^2\,W}{\partial\,\beta^2}\,=\,0,\quad
\frac{\partial^3\,W}{\partial\,\beta^3}\,\neq\,0.
\end{equation}
Since
$$\frac{\partial^2\,W}{\partial\,\beta^2}\,=\,\frac{\partial^2\,W}{\partial\,u^2}\,-\,2\,i\,\sqrt{-\,v}\,
\frac{\partial^2\,W}{\partial\,u\,\partial\,v}\,+\,v\,\frac{\partial^2\,W}{\partial\,v^2}\,+\,
\frac{1}{2}\,\frac{\partial\,W}{\partial\,v},$$
one concludes taking into account equation \eqref{conuv} and \eqref{epdellip} that the second condition \eqref{s1e}
is satisfied if and only if
\begin{equation}\label{ss}
\frac{\partial^2\,W}{\partial\,u^2}\,=\,0,\quad \frac{\partial^2\,W}{\partial\,v^2}\,=\,0,\quad
\frac{\partial^2\,W}{\partial\,u\,\partial\,v}\,=\,0.
\end{equation}
Thus, the subspace $\mathcal{M}_1^{\mbox{sing}}$ is characterized by the conditions \eqref{conuv}, \eqref{ss} and by requirement of
nonvanishing third order derivatives of $W$.
In order to compare these conditions with those of paper \cite{dubgr}, we first observe that the $B(1)$ system \eqref{ben}
is converted into the system (1.8) by the substitution $u\,\rightarrow\,v$, $v\,\rightarrow\,-\,u$. Then, with the choice
$$W\,=\,f(u,v)\,+\,x\,v\,-\,u\,v\,t,$$
the hodograph equations \eqref{conuv} become equations (2.4) of \cite{dubgr} and equation \eqref{epdellip} is reduced to
their equation (2.5). Finally, with such a choice, the conditions \eqref{ss} are converted to the condition (2.12) from
the paper \cite{dubgr}.
Finally, we note that according to the proposition 4 for the subspace $\mathcal{M}_1^{\mbox{sing}}$, the codimension of the
corresponding subspace of $(x,t_2,t_3,\dots)$ is equal to two and the function $W$ with $t_n\,=\,0$, $n\,\geq\,4$ i.e.
$$W\,=\,x\,U\,+\,t_2\,(U^2\,-\,\frac{1}{8}V^2)\,+\,t_3\,(U^3\,-\,\frac{3}{2}\,U\,V^2),$$
exhibits the elliptic umbilic singularity according to Thom's classification \cite{thom} (see also \cite{alp}-\cite{arn2}).
These results reproduce those originally obtained in the paper \cite{dubgr} (formula (4.2))
\section{5. dToda hierarchy.}
Now let us consider the function
\begin{equation}
W_{T}(x,\beta _{1,}\beta _{2})=\int \frac{d\lambda }{2\pi i}\,V_{T}(x,\lambda )
\sqrt{(1-\frac{\beta _{1}}{\lambda })\,(1-\frac{\beta _{2}}{\lambda })}
\end{equation}
where $V_{T}(x,\lambda )=\sum_{n\geq 0}\lambda ^{n}x_{n}.$ Critical points for this
function are defined by the equations
\begin{equation}
\frac{\partial W_{T}}{\partial \beta _{1}}=0,\quad \frac{\partial
W_{T}}{\partial \beta _{2}}=0.
\end{equation}
It is a simple check to see that $W_{T}$ obeys the Euler-Poisson-Darboux equation
of the type $E(-1/2,-1/2)$
\begin{equation}
2(\beta _{1}-\beta _{2})\frac{\partial ^{2}W_{T}}{\partial \beta
_{1}\partial \beta _{2}}=-(\frac{\partial W_{T}}{\partial \beta _{1}}-\frac{
\partial W_{T}}{\partial \beta _{2}}).
\end{equation}
Written explicitly the function $W_{T}$ is the series
\begin{equation}
W_{T}=-\frac{1}{2}x_{0}(\beta _{1}+\beta _{2})-\frac{1}{8}x_{1}(\beta
_{1}-\beta _{2})^{2}-\frac{1}{16}x_{2}(\beta _{1}+\beta _{2})(\beta
_{1}-\beta _{2})^{2}-\frac{1}{128}x_{3}(5\beta _{1}^{2}+6\beta _{1}\beta
_{2}+5\beta _{1}^{2})(\beta _{1}-\beta _{2})^{2}+...
\end{equation}
while the hodograph equations take the form
\begin{align*}
& x_{0}+\frac{1}{2}x_{1}(\beta _{1}-\beta _{2})+\frac{1}{8}x_{2}(3\beta
_{1}^{2}-2\beta _{1}\beta _{2}-\beta _{2}^{2})+\ldots=0, \\
&x_{0}-\frac{1}{2}x_{1}(\beta _{1}-\beta
_{2})+\frac{1}{8}x_{2}(3\beta _{2}^{2}-2\beta _{1}\beta _{2}-\beta
_{1}^{2})+\ldots=0.
\end{align*}
These hodograph equations provide us with the solutions of the system
\begin{equation}
\frac{\partial \beta _{1}}{\partial x_{1}}=\frac{1}{2}(\beta _{1}-\beta _{2})
\frac{\partial \beta _{1}}{\partial x_{0}},\quad \frac{\partial \beta _{2}}{
\partial x_{1}}=-\frac{1}{2}(\beta _{1}-\beta _{2})\frac{\partial \beta _{2}
}{\partial x_{0}}.
\end{equation}
In terms of the variables $u=-(\beta _{1}+\beta _{2}),v=\frac{1}{4}(\beta
_{1}-\beta _{2})^{2}$ one has the dToda system (2). \ Considering the higher
times $x_{2},x_{3},...$ .one gets the whole dToda hierarchy.
Similar to the Benney case \ the function $W_{T}$ is the generating function
for classical singularities for functions of two variables. Indeed,
in the variables $X=\frac{1}{2}(\beta _{1}+\beta _{2}),Y=\frac{1}{2}(\beta
_{1}-\beta _{2})$ it is of the form
\begin{equation}
W_{T}=-x_{0}X-\frac{1}{2}x_{1}Y^{2}-\frac{1}{2}x_{2}XY^{2}-\frac{1}{8}x_3
(4X^{2}+Y^{2})Y^{2}+...
\end{equation}
The third term here represents the parabolic umbilic singularity both for
hyperbolic and elliptic cases.
The formulas for the dToda hierarchy presented here coincide with
those given in the paper \cite{konopel} after the identification
\begin{equation}
V_{T}(x,\lambda )=-2T\lambda +\lambda V_{H}^{\prime }(t,\lambda ).
\end{equation}
i.e. $x_{0}=-2T,x_{n}=nt_{n},n=1,2,3,...$.
It is obvious that the descriptions of the regular and singular sectors of
the dToda hierarchy completely coincide with those of 1-layer Benney
hierarchy.
\bigskip
\section{6. Interrelations between the Euler-Poisson-Darboux equations with different indices
and those for function W and densities of integrals of motion.}
\bigskip
1-layer Benney hierarchy and dToda hierarchy are two examples of
hydrodynamical type systems for which functions $W$ obey the
Euler-Poisson-Darboux equations
\begin{equation}
L_{\varepsilon }W_{\varepsilon }:= \Big[\frac{\partial ^{2}}{\partial
x\partial y}-\dfrac{\varepsilon}{x-y} \Big(\frac{\partial }{\partial x}-\frac{\partial }{
\partial y}\Big)\Big]W_{\varepsilon }=0,
\end{equation}
with different indexes $\varepsilon $. \ Such linear equations \ \ are well
studied ( see e.g. \cite{dar}). The operators $L_{\varepsilon }$ have a number
of remarkable properties. One of them ( probably missed before) is given by the identity
\begin{equation}
L_{\varepsilon +1}L_{\mu }=L_{\mu +1}L_{\varepsilon }
\end{equation}
for arbitrary indices $\varepsilon$ and $\mu$ . This identity
implies, for instance, that for any solution $W_{\varepsilon}$ the function $L_{\mu}W_{\varepsilon}$ with arbitrary
$\mu$ obeys the
Euler-Poisson-Darboux equation with index $\varepsilon +1$, more precisely $
L_{\mu }W_{\varepsilon }=\varepsilon (\varepsilon -\mu )W_{\varepsilon +1}$
. In particular, at $\varepsilon =-\frac{1}{2}$ and $\mu =0$ one has $L_{
\frac{1}{2}}L_{0}=L_{1}L_{-\frac{1}{2}}.$ In terms of the operators $
\widetilde{L}_{\varepsilon }$ defined as $\widetilde{L}_{\varepsilon
}=(x-y)L_{\varepsilon}$ the last relation takes the form
\begin{equation}
\partial _{x}\partial _{y}\widetilde{L}_{-\frac{1}{2}}=\widetilde{L}_{\frac{1
}{2}}\partial _{x}\partial _{y}.
\end{equation}
This identity clearly demonstrates the duality between the
Euler-Poisson-Darboux equations with indices $\frac{1}{2}$ and -$\frac{1}{2}$
and consequently between 1-layer Benney \ and dToda hierarchies.
Duality between the functions W and densities of integrals of motions is the
another type of duality typical for the so-called $\varepsilon$ integrable
hydrodynamical type systems. Indeed, due to the Tsarev's result \cite{tsa} , a
symmetry $w_{i}$ of a semi-Hamiltonian hydrodynamical system
\begin{equation}\label{44}
\frac{\partial \beta _{i}}{\partial t}=\lambda _{i}(\beta )\frac{\partial
\beta _{i}}{\partial x},\quad i=1,...,n,
\end{equation}
i.e. a solution of the system
\begin{equation}
\frac{\partial \beta _{i}}{\partial \tau }=w_{i}(\beta )\frac{\partial \beta
_{i}}{\partial x},\quad i=1,...,n
\end{equation}
which commutes with the system \eqref{44}, are defined by the system
\begin{equation}\everymath{\displaystyle}
\frac{\frac{\partial w_{k}}{\partial \beta _{i}}}{w_{i}-w_{k}}=\frac{\frac{
\partial \lambda _{k}}{\partial \beta _{i}}}{\lambda _{i}-\lambda _{k}},\quad i\neq k.
\end{equation}
Such $w_{i}$ provide us with the solutions of the systems \eqref{44} via
the hodograph equations
\begin{equation}
\Omega _{i}:= -x+\lambda _{i}(\beta )t+w_{i}=0,i=1,...,n.
\end{equation}
For such system \eqref{44} densities P of integrals of motion obey the
equations \cite{tsa}
\begin{equation}
\frac{\partial ^{2}P}{\partial \beta _{i}\partial \beta _{k}}=\frac{\frac{
\partial \lambda _{i}}{\partial \beta _{k}}}{\lambda _{i}-\lambda _{k}}\frac{
\partial P}{\partial \beta _{i}}-\frac{\frac{\partial \lambda _{k}}{\partial
\beta _{i}}}{\lambda _{i}-\lambda _{k}}\frac{\partial P}{\partial \beta _{k}}
,\quad i\neq k.
\end{equation}
Let us define $\varepsilon $-systems as those ( for particular class of such
systems see e.g. \cite{pav2003}) for which
\begin{equation}
\frac{\frac{\partial \lambda _{i}}{\partial \beta _{k}}}{\lambda
_{i}-\lambda _{k}}=\frac{\frac{\partial \lambda _{k}}{\partial \beta _{i}}}{
\lambda _{i}-\lambda _{k}}=\frac{\varepsilon }{\beta _{i}-\beta _{k}}
\end{equation}
For such systems densities of integrals obey
Euler-Poisson-Darboux equations
\begin{equation}
\frac{\partial ^{2}P}{\partial \beta _{i}\partial \beta _{k}}=\frac{
\varepsilon }{\beta _{i}-\beta _{k}}\frac{\partial P}{\partial \beta _{i}}-
\frac{\varepsilon }{\beta _{i}-\beta _{k}}\frac{\partial P}{\partial \beta
_{k}},\quad i\neq k.
\end{equation}
At the same time the equations for $w_{i}$ become
\begin{equation}
\frac{\partial w_{k}}{\partial \beta _{i}}=-\varepsilon \frac{w_{i}-w_{k}}{
\beta _{i}-\beta _{k}},\quad i\neq k
\end{equation}
Symmetry of these equations with respect to the transposition of indices $i$ and
$k$ implies that
$\frac{\partial w_{k}}{\partial \beta _{i}}=\frac{\partial
w_{i}}{\partial \beta k}.$ Hence
$$
w_{i}=\frac{\partial \widetilde{W}}{
\partial \beta _{i}},\quad i=1,...,n,
$$
for a certain function $\widetilde{W}$.
Thus, equations (51) are the Euler-Poisson-Darboux equations of the type $
E(-\varepsilon ,-\varepsilon )$ for the function $\widetilde{W}$
\begin{equation}
\frac{\partial ^{2}\widetilde{W}}{\partial \beta _{i}\partial \beta _{k}}=-\Big(
\frac{\varepsilon }{\beta _{i}-\beta _{k}}\frac{\partial \widetilde{W}}{
\partial \beta _{i}}-\frac{\varepsilon }{\beta _{i}-\beta _{k}}\frac{
\partial \widetilde{W}}{\partial \beta _{k}}\Big),\quad i\neq k.
\end{equation}
The fact that the generating function for symmetries of the Whitham
equations and some other integrable hydrodynamical systems obey the
Euler-Poisson-Darboux equations has been observed earlier in the papers \cite{kud,pav,pav2003}.
Also the duality between the Euler-Poisson-Darboux equations for the densities of integrals
of motions and generating functions of symmetries has been noted before too. However the
demonstration presented above seems to be different from those discussed earlier.
In addition one can note that equations (49) imply that for $\varepsilon $
-systems also $\lambda _{i}=\frac{\partial g}{\partial \beta _{i}}$ with
some function $g$. \ As the result the hodograph equations (47) for the $
\varepsilon $-systems take the form
\begin{equation}
\Omega _{i}=-x+t\frac{\partial g}{\partial \beta _{i}}+\frac{\partial
\widetilde{W}}{\partial \beta _{i}}=\frac{\partial W}{\partial \beta _{i}}
=0,i,...,n
\end{equation}
where $W=-x(\beta _{1}+\beta _{2})+gt+\widetilde{W.}$ Thus,
hodograph equations for the integrable hydrodynamical type systems
\ are nothing but the equations defining the critical points of
the function $W$. \ It seems \ that this fact has been missing in
the previous publications. Moreover, due to the equations (49) the
function $g$ also obeys the $E(-\varepsilon ,-\varepsilon )$
Euler-Poisson-Darboux equation and , hence, the function $W$ does
the same. Note that particular class of $\varepsilon $-systems for
which $\lambda _{i}$ are linear functions of $\beta_i$ has been
discussed in \cite{pav2003}.
So, for integrable hydrodynamical systems of the
$\varepsilon $ type, the densities of integrals and the functions
$W$ ( as well as the functions $\widetilde{W}$ generating
symmetries) play a dual role
obeying the Euler-Poisson-Darboux equations with opposite sign of the index $
\varepsilon $ . This property resembles a lot the well-known duality between
the generating functions of integrals of motion and symmetries for the
dispersionful integrable equations.
\vspace{0.5cm}
\subsection* { Acknowledgements}
\vspace{0.3cm} The authors wish to thank the Spanish Ministerio de
Educaci\'on y Ciencia (research project FIS2008-00200/FIS).
| 48,009
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The Japan Times reports forecasts that the population of the Prefecture of Tokyo, the central jurisdiction of the metropolitan area, could decline by nearly 50 percent (chart) between 2010 and 2100 (Note). Yet, while the overall population is dropping in half, the elderly population would increase by more than 20 percent. The resulting far less favorable ratio of elderly to the working population would present unprecedented social and economic challenges.
The article provides no information on the population of the entire urban area in 2100. The Prefecture of Tokyo constitutes somewhat over one third of the present population of the urban area.
During the last census period (between 2005 2010) the four prefecture Tokyo metropolitan area (Tokyo, Kanagawa, Saitama and Chiba), gained approximately 1,100,000 new residents, while the balance of the country was losing 1,400,000 residents. Japan is forecast to suffer substantial population losses in the decades to come. The United Nations forecasts that its population will decline from approximately 125 million in 2010 to 90 million in 2100. This is the optimistic scenario. The National Institute of Population and Social Security Research forecasts a drop to under 50 million, a more than 60 percent population reduction.
There are serious concerns about the projected population decline. According to the Japan Times, the researchers said that " ... it will be crucial to take measures to turn around the falling birthrate and enhance social security measures for the elderly," A professor the National Graduate Institute for Policy Studies, expressed concern that "If the economies of developing countries continue growing, the international competitiveness of major companies in Tokyo will dive."
----
Note: the Prefecture of Tokyo government is called the Tokyo Metropolitan Government. This term can mislead, because the prefecture itself is not the metropolitan area, but only part of the four prefecture metropolitan area. The pre-– amalgamation predecessor of the current city of Toronto was called the Municipality of Metropolitan Toronto. Like the Prefecture of Tokyo, the Municipality of Metropolitan Toronto comprised only part of the Toronto metropolitan area. Confusion over these terms not only resulted in incorrect press reports, but even misled some academic researchers to treat these sub-metropolitan jurisdictions as metropolitan
| 348,896
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+++
Several architectural and design firms were involved in the restoration and adaptive reuse of Temple Court. Although the interior is not a designated New York City landmark, the lead firm, Gerner, Kronick + Varcel Architects, restored many aspects of the original interior, including the historic cast-iron balconies, the grand skylight, the atrium, and the wood millwork doors and windows surrounding the atrium.
+++
“Temple Court was the first “fireproof” building in New York. Because of modern fire code regulations, which prohibit an atrium that physically connects multiple floors, a smoke curtain system was put in place along the perimeter of the restored atrium. Detectors on each floor activate the smoke curtains, which fall and seal off the atrium. With the modern smoke curtains in place, the atrium, in effect, functions much like a fireplace chimney, directing smoke upward and out via ducts located at the base of the historic skylight.
Today, the two turrets function as penthouse hotel suites.
+++
+++
In conjunction with GKV Architects, EverGreene Architectural Arts‘ craftsmen meticulously removed the original floor tiles, cleaned them, replaced those that were broken and reinstalled them. EverGreene artists also restored the plaster, wood and metal elements of the atrium, including as cast iron railings and plaster arches.
The basement was transformed into an event space and offices.
These before photos show how badly the building had deteriorated.
But the life inside the building today proves what a successful restoration and rehabilitation project this was.
All photos taken by James and Karla Murray exclusively for 6sqft. Photos are not to be reproduced without written permission from 6sqft.
James and Karla Murray are husband-and-wife New York based professional : 123 Nassau Street, 5 Beekman Street, James and Karla Murray, Temple Court, the Beekman
Neighborhoods : Financial District
| 248,982
|
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| 362,175
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TITLE: Solving recurrence relation without initial condition
QUESTION [2 upvotes]: Any idea on how I can approach this recurrence relation? It is very different to other questions I have encountered where there is only one term of $T(n)$ on the RHS, and the initial condition isn't given as well.
$$T(n)=\frac2n\big(T(0)+T(1)+\ldots+T(n-1)\big)+5n$$
for $n\ge 1$.
REPLY [0 votes]: Use generating functions. Define $t(z) = \sum_{n \ge 0} T(n) z^n$, shift the recurrence and multiply out to get rid of fractions; multiply by $z^n$, sum over $n \ge 0$ and recognize resulting sums:
$\begin{align*}
(n + 1) T(n + 1)
&= \sum_{0 \le k \le n} T(k) + 5 (n + 1)^2 \\
\sum_{n \ge 0} (n + 1) T(n + 1) z^n
&= \sum_{n \ge 0} z^n \sum_{0 \le k \le n} T(k)
+ 5 \sum_{n \ge 0} (n + 1)^2 z^n \\
\frac{1}{z} \sum_{n \ge 0} n T(n) z^n
&= \sum_{n \ge 0} z^n \sum_{0 \le k \le n} T(k)
+ 5 \frac{1 + z}{(1 - z)^3} \\
\frac{1}{z} \cdot z t'(z)
&= \frac{t(z)}{1 - z} + 5 \frac{1 + z}{(1 - z)^3}
\end{align*}$
No need for any initial values here. Solve for $t(z)$:
$\begin{align*}
t(z)
&= \frac{10}{(1 - z)^2} + 5 \frac{1}{1 - z} \ln \frac{1}{1 - z} + \frac{c}{1 - z}
\end{align*}$
Using the generalized binomial theorem and the generating function of the harmonic numbers:
$\begin{align*}
(1 - z)^{-m}
&= \sum_{n \ge 0} (-1)^n \binom{-m}{n} z^n \\
&= \sum_{n \ge 0} \binom{n + m - 1}{m - 1} z^n \\
H_n
&= \sum_{1 \le k \le n} \frac{1}{n} \\
\sum_{n \ge 0} H_n z^n
&= \frac{1}{1 - z} \ln \frac{1}{1 - z}
\end{align*}$
we see that:
$\begin{align*}
T(n)
&= [z^n] t(n) \\
&= 10 \binom{n + 2 - 1}{2 - 1} + 5 H_n + c \\
&= 10 n + 5 H_n + c'
\end{align*}$
Here $c' = T(0)$.
| 148,093
|
Respond to customers' storage buying habits
Act on your customers storage buying habits. Contributor Jerome Wendt offers three key takeaways from a recent Storage magazine purchasing intentions survey.
As users buy more storage capacity with the money that they do spend, being more receptive to technologiesCSI while still questioning the value proposition of storage management software, here are some thoughts on how VARs and systems integrators like yourself should act on this information.
1. Users are more educated than they were just a few years ago on how storage works, so you should play to sensitive areas. The survey did uncover that large companies were more concerned about support, and small and midsize businesses were more sensitive to price, so vendors should construct their bids to reflect those concerns.
2. As iSCSI gains acceptance and momentum among end users, you should add the appropriate products and skill sets to your portfolio. Expect high initial interest from small and midsize businesses. However, this interest will trickle up to larger enterprises over the next two years as they look for a low cost way to connect their Windows and Linux farms with a storage area network (SAN). Also keep in mind that while iSCSI costs less up front, the same management challenges you'd find with Fibre Channel SANs -- LUN security, data migration and performance tuning -- remain.
3. Recognize that an opportunity to sell storage management and reporting tools to users exists -- if you offer the right product at the right price. For instance, you may want to consider TeraCloud's TSF Lite, which allows either you or your client to measure storage environments with up to 20 TBs of storage for $395 per month. whether you use it only once or over an ongoing basis.
An additional service TeraCloud offers is the ability to pipe the information back to TeraCloud, so the vendor can then produce a set of analytic reports complete with measurements against best practices based on the facts you feed them. A tool like this allows you to offer your clients the ability to perform a low-risk, low-cost assessment of their storage environment.
End-user appetites for storage are only increasing. However, periods of belt tightening or panic will surely come as users realize their storage infrastructures are growing out of control. Whether you're standing by to watch it happen or you're called upon to help bail them out, introducing the right combination of storage products, SAN infrastructure and storage software can go a long way in preventing this scenario or helping customers handle it when it does. Storage Buying Guide Part 1: What are your customers buying? Part 2: Increased SATA hard drive adoption Part 3: Hard drive replacement Part 4: Production iSCSI SAN interest rises Part 5: Less data storage management spending Part+
Content
Find more PRO+ content and other member only offers, here.
| 244,718
|
\section{Moment Estimation}\label{sec:moments1}
We will now work towards proving our result for Poincare distributions. A key ingredient in our algorithm will be estimating the moment tensor of a mixture $\mcl{M}$ i.e. for an unknown mixture
\[
\mcl{M} = w_1\mcl{D}(\mu_1) + \dots + w_k \mcl{D}(\mu_k)
\]
we would like to estimate the tensor
\[
w_1 \mu_1^{\otimes t} + \dots + w_k \mu_k^{\otimes t}
\]
for various values of $t$ using samples from $\mcl{M}$. Naturally, it suffices to consider the case where we are given samples from $\mcl{D}(\mu)$ for some unknown $\mu$ and our goal is to estimate the tensor $\mu^{\otimes t}$. For our full algorithm, we will need to go up to $t \sim \log k/ \log \log k$ but of course for such $t$, our estimate has to be implicit because we cannot write down the full tensor in polynomial time. In this section, we address this task by constructing an unbiased estimator with bounded variance that can be easily manipulated implicitly.
\\\\
We make the following definition to simplify notation later on.
\begin{definition}
For integers $t$ and a distribution $\mcl{D}$, we define the tensor
\[
D_{t, \mcl{D}} = \E_{z \sim \mcl{D}}[ z^{\otimes t}] \,.
\]
We will drop the subscript $\mcl{D}$ when it is clear from context.
\end{definition}
\subsection{Adjusted Polynomials}
First, we just construct an unbiased estimator for $\mu^{\otimes t}$ (without worrying about making it implicit). This estimator is given in the definition below.
\begin{definition}\label{def:adjusted-polynomial}
Let $\mcl{D}$ be a distribution on $\R^d$. For $x \in \R^d$, define the polynomials $P_{t, \mcl{D}}(x)$ for positive integers $t$ as follows. $P_{0, \mcl{D}}(x) = 1$ and for $t \geq 1$,
\begin{equation}\label{eq:recursive-def}
P_{t, \mcl{D}}(x) = x^{\otimes t} - \sum_{sym} D_{1, \mcl{D}} \otimes P_{t-1, \mcl{D}}(x) - \sum_{sym} D_{2, \mcl{D}} \otimes P_{t-2, \mcl{D}}(x) - \dots - D_{t, \mcl{D}} \,.
\end{equation}
We call $P_{t, \mcl{D}}$ the $\mcl{D}$-adjusted polynomials and will sometimes drop the subscript $\mcl{D}$ when it is clear from context.
\end{definition}
We now prove that the $\mcl{D}$-adjusted polynomials give an unbiased estimator for $\mu^{\otimes t}$ when given samples from $\mcl{D}(\mu)$.
\begin{claim}\label{claim:adjusted-polynomial}
For any $\mu \in \R^d$,
\[
\E_{z \sim \mcl{D}(\mu)} [P_{t, \mcl{D}}(z)] = \mu^{\otimes t} \,.
\]
\end{claim}
\begin{proof}
To simplify notation, we will drop all of the $\mcl{D}$ from the subscripts as there will be no ambiguity. We prove the claim by induction on $t$. The base case for $t = 1$ is clear. Now for the inductive step, note that
\begin{align*}
\E_{z \sim \mcl{D}(\mu)} [P_t(z)] &= \E_{z \sim \mcl{D}(\mu)} \left[ z^{\otimes t} - \sum_{sym} D_1 \otimes P_{t-1}(z) - \sum_{sym} D_2 \otimes P_{t-2}(z) - \dots - D_t \right] \\ &= \E_{x \sim \mcl{D}}[ (x + \mu)^{\otimes t}] - \E_{z \sim \mcl{D}(\mu)} \left[ \sum_{sym} D_1 \otimes P_{t-1}(z) + \sum_{sym} D_2 \otimes P_{t-2}(z) + \dots + D_t \right] \\ & = \E_{x \sim \mcl{D}} \left[ \mu^{\otimes t} - \sum_{sym} (D_1 - x^{\otimes 1}) \otimes \mu^{t - 1} - \sum_{sym} (D_2 - x^{\otimes 2}) \otimes \mu^{t - 2} - \dots - (D_t - x^{\otimes t}) \right] \\ & = \mu^{\otimes t} \,.
\end{align*}
where we used the induction hypothesis and then the definition of $D_t$ in the last two steps.
\end{proof}
\subsection{Variance Bounds for Poincare Distributions}
In the previous section, we showed that the $\mcl{D}$-adjusted polynomials give an unbiased estimator for $\mu^{\otimes t}$. We now show that they also have bounded variance when $\mcl{D}$ is Poincare. This will rely on the following claim which shows that the $\mcl{D}$-adjusted polynomials recurse under differentiation.
\begin{claim}\label{claim:derivative-recurrence}
Let $\mcl{D}$ be a distribution on $\R^d$. Then
\[
\frac{\partial P_{t, \mcl{D}}(x)}{\partial x_i} = \sum_{\sym } e_i \otimes P_{t - 1, \mcl{D}}(x)
\]
where we imagine $x = (x_1, \dots , x_d)$ so $x_i$ is the $i$\ts{th} coordinate of $x$ and
\[
e_i = ( \underbrace{0, \dots , 1}_i , \dots, 0 )
\]
denotes the $i$\ts{th} coordinate basis vector.
\end{claim}
\begin{proof}
We will prove this by induction on $t$. The base case for $t = 1$ is clear. In the proceeding computations, we drop the $\mcl{D}$ from all subscripts as there will be no ambiguity. Differentiating the definition of $P_{t, \mcl{D}}$ and using the induction hypothesis, we get
\begin{align*}
\frac{\partial P_{t}(x)}{\partial x_i} &= \sum_{sym} e_i \otimes x^{\otimes t - 1} - \sum_{sym} D_{1} \otimes e_i \otimes P_{t-2}(x) - \dots - \sum_{sym} D_{t - 2} \otimes e_i \otimes P_1(x) - \sum_{sym} D_{t - 1} \otimes e_i \\ &= \sum_{sym} e_i \otimes \left( x^{\otimes t-1} - \sum_{sym} D_1 \otimes P_{t-2}(x) - \dots - D_{t-1} \right) \\ & = \sum_{sym} e_i \otimes P_{t-1}(x)
\end{align*}
where in the last step we again used (\ref{eq:recursive-def}), the recursive definition of $P_{t-1}$. This completes the proof.
\end{proof}
Now, by using the Poincare property, we can prove a bound on the variance of the estimator $P_{t, \mcl{D}}(x)$.
\begin{claim}\label{claim:variance-bound}
Let $\mcl{D}$ be a distribution on $\R^d$ that is $1$-Poincare. Let $v \in \R^{d^t}$ be a vector. Then
\[
\E_{z \sim \mcl{D}(\mu)} [ \left( v \cdot \flatten( P_{t, \mcl{D}}(z)) \right)^2 ] \leq ( \norm{\mu}^2 + t^2)^t \norm{v}^2 \,.
\]
\end{claim}
\begin{proof}
We will prove the claim by induction on $t$. The base case for $t = 1$ follows because
\begin{align*}
\E_{z \sim \mcl{D}(\mu)} [ \left( v \cdot \flatten( P_{1, \mcl{D}}(z)) \right)^2 ] = (v \cdot \mu)^2 + \Var_{z \sim \mcl{D}(\mu)} \left( v \cdot \flatten( P_{1, \mcl{D}}(z)) \right) \leq (v \cdot \mu)^2 + \norm{v}^2 \\ \leq ( \norm{\mu}^2 + 1) \norm{v}^2
\end{align*}
where we used the fact that $\mcl{D}$ is $1$-Poincare. Now for the inductive step, we have
\begin{align*}
\E_{z \sim \mcl{D}(\mu)} [ \left( v \cdot \flatten( P_{t, \mcl{D}}(z)) \right)^2 ] &= (v \cdot \flatten( \mu^{\otimes t}) )^2 + \Var_{z \sim \mcl{D}(\mu)} \left( v \cdot \flatten( P_{t, \mcl{D}}(z)) \right) \\ & \leq \norm{\mu}^{2t}\norm{v}^2 + \E_{z \sim \mcl{D}(\mu)} \left[ \sum_{i = 1}^d \left(v \cdot \frac{\partial P_{t, \mcl{D}}(x)}{\partial x_i} \right)^2 \right] \\ & = \norm{\mu}^{2t}\norm{v}^2 + \E_{z \sim \mcl{D}(\mu)} \left[ \sum_{i = 1}^d \left( \sum_{j = 1}^t v_{\eta_j = i} \cdot P_{t-1, \mcl{D}}(z)\right)^2 \right] \\ & \leq \norm{\mu}^{2t}\norm{v}^2 + \E_{z \sim \mcl{D}(\mu)} \left[ t \sum_{j = 1}^t \sum_{i = 1}^d \left( v_{\eta_j = i} \cdot P_{t-1, \mcl{D}}(z)\right)^2 \right] \\ & \leq \norm{\mu}^{2t} \norm{v}^2 + t \sum_{j = 1}^t \sum_{i = 1}^d ( \norm{\mu}^2 + (t-1)^2)^{t-1} \norm{v_{\eta_j = i}}^2 \\ & = \norm{\mu}^{2t} \norm{v}^2+ t^2 ( \norm{\mu}^2 + (t-1)^2)^{t-1} \norm{v}^2 \\ & \leq ( \norm{\mu}^2 + t^2)^t \norm{v}^2
\end{align*}
where in the above manipulations, we first used Claim \ref{claim:adjusted-polynomial}, then the fact that $\mcl{D}$ is $1$-Poincare, then Claim \ref{claim:derivative-recurrence}, then Cauchy Schwarz, then the inductive hypothesis, and finally some direct manipulation. This completes the inductive step and we are done.
\end{proof}
\subsection{Efficient Implicit Representation} \label{sec:implicit-moments1}
In the previous section, we showed that for $x \sim \mcl{D}(\mu)$ for unknown $\mu$, $P_{t, \mcl{D}}(x)$ gives us an unbiased estimator of $\mu^{\otimes t}$ with bounded variance. Still, it is not feasible to actually compute $P_{t, \mcl{D}}(x)$ in polynomial time because we cannot write down all of its entries and there is no nice way to implicitly work with terms such as $D_{t, \mcl{D}}$ that appear in $P_{t, \mcl{D}}(x)$. In this section, we construct a modified estimator that is closely related to $P_{t, \mcl{D}}(x)$ but is also easy to work with implicitly because all of the terms will be rank-$1$ i.e. of the form $v_1 \otimes \dots \otimes v_t$ for some vectors $v_1, \dots , v_t \in \R^d$. Throughout this section, we will assume that the distribution $\mcl{D}$ that we are working with is fixed and we will drop it from all subscripts as there will be no ambiguity.
Roughly, the way that we construct this modified estimator is that we take multiple variables $x_1, \dots , x_t \in \R^d$. We start with $P_t(x_1)$. We then add various products
\[
P_{a_1}(x_1) \otimes \dots \otimes P_{a_t}(x_t)
\]
to it in a way that when expanded as monomials, only the leading terms, which are rank-$1$ since they are a direct product of the form $x_1^{\otimes a_1} \otimes \dots \otimes x_t^{\otimes a_t}$, remain. If we then take $x_1 \sim \mcl{D}(\mu)$ and $x_2, \dots, x_t \sim \mcl{D}$, then Claim \ref{claim:adjusted-polynomial} will immediately imply that the expectation is $\mu^{\otimes t}$. The key properties are stated formally in Corollary \ref{coro:rank1-identity-p2}, Corollary \ref{coro:rank1-estimator-mean} and Corollary \ref{coro:rank1-estimator-variance}.
\\\\\
First, we write out an explicit formula for $P_t(x)$.
\begin{claim}\label{claim:explicit-formula}
We have
\[
P_{t}(x) = \sum_{ S_0 \subseteq [t] } \left( x^{\otimes S_0 } \right) \otimes \left( \sum_{\{S_1, \dots , S_t \} \in Z_t( [t] \backslash S_0)} (-1)^{\mcl{C} \{S_1, \dots , S_t \} } (\mcl{C} \{S_1, \dots , S_t \} )! (D_{|S_1|})^{(S_1)} \otimes \dots \otimes (D_{|S_t|})^{(S_t)} \right)
\]
\end{claim}
\begin{proof}
We use induction. The base case is trivial. Now, it suffices to compute the coefficient of some ``monomial" in $P_{t}$. Note that the coefficient of the monomial $x^{\otimes t}$ is clearly $1$ which matches the desired formula. Otherwise, consider a monomial
\[
A = \left( x^{\otimes S_0} \right)\otimes (D_{|S_1|})^{(S_1)} \otimes \dots \otimes (D_{|S_a|})^{(S_a)}
\]
where $S_1, \dots , S_a$ are nonempty for some $1 \leq a \leq t$. Recall the recursive definition of $P_t$ in (\ref{eq:recursive-def}). There are exactly $a$ terms on the RHS of (\ref{eq:recursive-def}) that can produce the monomial $A$. These terms are
\[
-D_{|S_1|}^{(S_1)} \otimes P_{t - |S_1|}(x)^{([t]\backslash S_1)}, \dots , -D_{|S_a|}^{(S_a)} \otimes P_{t - |S_a|}(x)^{([t]\backslash S_a)} \,.
\]
However, by the inductive hypothesis, the coefficient of $A$ produced by each of these terms is exactly $(-1)^a ( a-1)!$. Thus, combining over all $a$ terms, the resulting coefficient is $(-1)^a a!$ which matches the desired formula. This completes the proof.
\end{proof}
Now, we are ready to write out our estimator that can be written as a sum of few rank-$1$ terms. The key identity is below.
\begin{definition}
For $x_1, \dots , x_t \in \R^d$, define the polynomial
\begin{equation}\label{eq:rank1}
Q_t(x_1, \dots , x_t) = \sum_{\substack{S_1 \cup \dots \cup S_t = [t] \\ |S_i \cap S_j| = 0}} \frac{(-1)^{\mcl{C} \{ S_1, \dots , S_t \}}}{\binom{t - 1}{\mcl{C} \{ S_1, \dots , S_t \} - 1 } } \left( P_{|S_1|}(x_1) \right)^{(S_1)} \otimes \dots \otimes \left( P_{|S_t|}(x_t) \right)^{(S_t)} \,.
\end{equation}
\end{definition}
\begin{lemma}\label{lem:rank1-identity-p1}
All nonzero monomials in $Q_t$ either have total degree $t$ in the variables $x_1, \dots , x_t$ or are constant.
\end{lemma}
\begin{proof}
Consider substituting the formula in Claim \ref{claim:explicit-formula} into the RHS for all occurrences of $P$. Consider a monomial
\[
A = \left( x_{i_1}^{\otimes U_1} \right) \otimes \dots \otimes \left( x_{i_a}^{\otimes U_a} \right) \otimes (D_{|V_1|})^{(V_1)} \otimes \dots \otimes (D_{|V_b|})^{(V_b)}
\]
where $U_1, \dots , U_a, V_1, \dots , V_b$ are nonempty. Note that when we expand out the RHS, all monomials are of this form. It now suffices to compute the coefficient of this monomial $A$ in the expansion of the RHS.
\\\\
The terms in the sum on the RHS are of the form
\[
\frac{(-1)^c}{\binom{t-1}{c-1}} P_{|S_{j_1}|}( x_{j_1})^{(S_{j_1})} \otimes \dots \otimes P_{|S_{j_c}|}( x_{j_c})^{(S_{j_c})}
\]
where $S_{j_1}, \dots , S_{j_c}$ are nonempty. We now consider summing over all such terms that can produce the monomial $A$ and sum the corresponding coefficient to get the overall coefficient of $A$.
\\\\
In order for the monomial $A$ to appear in the expansion of this term, we need $\{i_1, \dots , i_a \} \subseteq \{ j_1, \dots , j_c \}$. Once the indices $j_1, \dots , j_c$ are fixed, it remains to assign each of the terms
\[
(D_{|V_1|})^{(V_1)} , \dots , (D_{|V_b|})^{(V_b)}
\]
to one of the variables $x_{j_1}, \dots , x_{j_c}$. This will correspond to which of the polynomials
\[
P_{|S_{j_1}|}( x_{j_1})^{(S_{j_1})}, \dots , P_{|S_{j_c}|}( x_{j_c})^{(S_{j_c})}
\]
that the term came from. Specifically, for each integer $f $ with $1 \leq f \leq c$, let $B_f \subset [b]$ be the indices of the terms that are assigned to the variable $x_{j_f}$. The sets $B_1, \dots , B_c$ uniquely determine the sets $S_{j_1}, \dots , S_{j_c}$ but also need to satisfy the constraint that if $j_f \notin \{i_1, \dots , i_a \}$ then $B_f \neq \emptyset $. Once these sets are all fixed, by Claim \ref{claim:explicit-formula}, the desired coefficient is simply
\[
\frac{(-1)^c}{\binom{t-1}{c-1}} (-1)^{|B_1| + \dots +|B_c|} |B_1|! \cdots |B_c|! = \frac{(-1)^c}{\binom{t-1}{c-1}} (-1)^{b} |B_1|! \cdots |B_c|! \,.
\]
Now overall, the desired coefficient is
\begin{equation}\label{eq:binomsum}
\sum_{c = a}^{a + b} \frac{(-1)^c}{\binom{t-1}{c-1}} \binom{t - a}{c - a} (-1)^{b} \sum_{\substack{B_1 \cup \dots \cup B_c = [b] \\ B_i \cap B_j = \emptyset \\ B_{a+1}, \dots , B_c \neq \emptyset} } |B_1|! \cdots |B_c|! \,.
\end{equation}
This is because there are $\binom{t - a}{c - a}$ to choose the set $\{ j_1, \dots , j_c \}$ (since it must contain $ \{ i_1, \dots , i_a \}$ ). Once we have chosen this set, WLOG we can label $j_1 = i_1, \dots, j_a = i_a$ so that $j_{a+1}, \dots , j_c$ are the elements that are not contained in $ \{ i_1, \dots , i_a \}$ and thus the constraint on $B_1, \dots , B_c$ is simply that $B_{a+1}, \dots , B_c$ are nonempty.
\\\\
To evaluate (\ref{eq:binomsum}), we will evaluate the inner sum differently. Imagine first choosing the sizes $s_1 = |B_1|, \dots , s_c = |B_c|$ and then choosing the sets $B_1, \dots , B_c$ to satisfy these size constraints. The inner sum can then be rewritten as
\[
\sum_{ \substack{s_1 + \dots + s_c = b \\ s_{a+1}, \dots , s_c > 0}} \binom{b}{s_1, \dots , s_c } s_1! \cdots s_c ! = b! \sum_{ \substack{s_1 + \dots + s_c = b \\ s_{a+1}, \dots , s_c > 0}} 1 = b! \binom{b + a - 1}{c - 1}
\]
where the last equality follows from counting using stars and bars. Now we can plug back into (\ref{eq:binomsum}). Assuming that $a,b \geq 1$, the coefficient of the monomial $A$ is
\begin{align*}
\sum_{c = a}^{a + b} \frac{(-1)^c}{\binom{t-1}{c-1}} \binom{t - a}{c - a} (-1)^{b} b! \binom{b + a - 1}{c - 1} &= (-1)^b b! \sum_{c = a}^{a + b} \frac{(-1)^c(t-a)! (b+a - 1)!}{(c-a)! (t-1)! (b+a - c)!} \\ &= (-1)^b \frac{(t-a)! (b+a - 1)!}{ (t-1)! } \sum_{c = a}^{a + b} (-1)^c \binom{b}{c - a} \\ & = 0 \,.
\end{align*}
Thus, the only monomials that have nonzero coefficient either have $a = 0$ (meaning they are constant) or $b = 0$ (meaning they have degree $t$). This completes the proof.
\end{proof}
Now, we can easily eliminate the constant term by subtracting off $Q(x_{t+1}, \dots , x_{2t})$ for some additional variables $x_{t+1}, \dots , x_{2t}$ and we will be left with only degree-$t$ terms. It will be immediate that the degree-$t$ terms are all rank-$1$ and this will give us an estimator that can be efficiently manipulated implicitly.
\begin{definition}\label{def:key-polynomial}
For $x_1, \dots , x_{2t} \in \R^d$, define the polynomial
\[
R_t(x_1, \dots , x_{2t}) = - Q_t(x_1, \dots , x_t) + Q_t(x_{t+1}, \dots , x_{2t}) \,.
\]
\end{definition}
\begin{corollary}\label{coro:rank1-identity-p2}
We have the identity
\[
R_t(x_1, \dots , x_{2t}) = \sum_{\substack{S_1 \cup \dots \cup S_t = [t] \\ |S_i \cap S_j| = 0}} \frac{(-1)^{\mcl{C} \{ S_1, \dots , S_t \} -1 }}{\binom{t - 1}{\mcl{C} \{ S_1, \dots , S_t \} - 1 } } \left( x_1^{\otimes S_1} \otimes \dots \otimes x_t^{\otimes S_t} - x_{t+1}^{\otimes S_1} \otimes \dots \otimes x_{2t}^{\otimes S_t} \right)\,.
\]
\end{corollary}
\begin{proof}
This follows immediately from Lemma \ref{lem:rank1-identity-p1} and the definition of $R_t(x_1, \dots , x_{2t})$ because the constant terms cancel out and the degree-$t$ terms clearly match the RHS of the desired expression.
\end{proof}
Corollary \ref{coro:rank1-identity-p2} gives us a convenient representation for working implicitly with $R_t(x_1, \dots , x_{2t})$. We now show why this polynomial is actually useful. In particular, we show that for $x_1 \sim \mcl{D}(\mu)$ and $x_2, \dots , x_{2t} \sim \mcl{D}$, $R_{t}(x_1, \dots , x_{2t})$ is an unbiased estimator of $\mu^{\otimes t}$ and furthermore that its variance is bounded. These properties will follow directly from the definitions of $R_t, Q_t$ combined with Claim \ref{claim:adjusted-polynomial} and Claim \ref{claim:variance-bound}.
\begin{corollary}\label{coro:rank1-estimator-mean}
We have
\[
\E_{z_1 \sim \mcl{D}(\mu), z_2, \dots , z_{2t} \sim \mcl{D}}[ R_t(z_1, \dots , z_{2t}) ] = \mu^{\otimes t}
\]
and for fixed $z_1$, we have
\[
\E_{ z_2, \dots , z_{2t} \sim \mcl{D}}[ R_t(z_1, \dots , z_{2t}) ] = P_t(z_1) \,.
\]
\end{corollary}
\begin{proof}
Using the definition of $Q_t$ in (\ref{eq:rank1}) and Claim \ref{claim:adjusted-polynomial}, the expectations of all of the terms are $0$ except for the leading term $P_t(z_1)$. Thus,
\[
\E_{ z_2, \dots , z_{2t} \sim \mcl{D}}[ R_t(z_1, \dots , z_{2t}) ] = P_t(z_1) \,.
\]
Also by Claim \ref{claim:adjusted-polynomial},
\[
\E_{z_1 \sim \mcl{D}(\mu)}[ P_{t}(z_1)] = \mu^{\otimes t}
\]
and this gives us the two desired identities.
\end{proof}
\begin{corollary}\label{coro:rank1-estimator-variance}
Let $\mcl{D}$ be a distribution that is $1$-Poincare. We have
\[
\E_{z_1 \sim \mcl{D}(\mu), z_2, \dots , z_{2t} \sim \mcl{D}}\left[ \flatten(R_t(z_1, \dots , z_{2t}))^{\otimes 2} \right] \preceq (20t)^{2t} (\norm{\mu}^{2t} + 1)I_{d^t} \,.
\]
where recall $I_{d^t}$ denotes the $d^t$-dimensional identity matrix.
\end{corollary}
\begin{proof}
Using the definition of $R_t$ and $Q_t$ and Cauchy Schwarz, we have
\begin{align*}
& \E_{z_1 \sim \mcl{D}(\mu), z_2, \dots , z_{2t} \sim \mcl{D}}\left[ \flatten(R_t(z_1, \dots , z_{2t}))^{\otimes 2} \right] \\ & \preceq 2\E_{z_1 \sim \mcl{D}(\mu), z_2, \dots , z_{t} \sim \mcl{D}} \left[ \flatten \left( Q_t(z_1, \dots , z_t)\right)^{\otimes 2} \right] + 2 \E_{z_{t+1}, \dots , z_{2t} \sim \mcl{D}} \left[ \flatten \left( Q_t(z_{t+1}, \dots , z_{2t})\right)^{\otimes 2} \right] \,.
\end{align*}
Now by Cauchy Schwarz again,
\begin{align*}
&\E_{z_1 \sim \mcl{D}(\mu), z_2, \dots , z_{t} \sim \mcl{D}} \left[ \flatten \left( Q_t(z_1, \dots , z_t)\right)^{\otimes 2} \right] \preceq \left( \sum_{\substack{S_1 \cup \dots \cup S_t = [t] \\ |S_i \cap S_j| = 0}} |S_1|! \cdots |S_t|!\right) \\ & \quad \cdot \left( \sum_{\substack{S_1 \cup \dots \cup S_t = [t] \\ |S_i \cap S_j| = 0}} \frac{\E_{z_{1} \sim \mcl{D}(\mu), z_2 \dots , z_{t} \sim \mcl{D}} \left[ \flatten \left( \left( P_{|S_1|}(z_{1}) \right)^{(S_1)} \otimes \dots \otimes \left( P_{|S_t|}(z_{t}) \right)^{(S_t)} \right)^{\otimes 2} \right] }{|S_1|! \cdots |S_t|!} \right) \,.
\end{align*}
Let the first term above be $C_1$ and the second term be $C_2$. We can rearrange the sum over partitions of $[t]$ as follows. We can first choose the sizes $s_1 = |S_1|, \dots , s_t = |S_t|$ and then choose the partition according to these constraints. We get
\[
C_1 = \sum_{s_1 + \dots + s_t = t} \binom{t}{s_1, \dots , s_t} s_1 ! \cdots s_t ! = t! \binom{2t - 1}{t} \leq (2t)^t
\]
and similarly (and also using Claim \ref{claim:variance-bound})
\begin{align*}
C_2 &\preceq \left(\sum_{\substack{S_1 \cup \dots \cup S_t = [t] \\ |S_i \cap S_j| = 0}} \frac{(\norm{\mu}^2 + |S_1|^2)^{|S_1|} |S_2|^{2|S_2|} \cdots |S_t|^{2|S_t|} }{|S_1|! \cdots |S_t|! } \right) I_{d^t} \\ &\preceq \left(20^t \sum_{\substack{S_1 \cup \dots \cup S_t = [t] \\ |S_i \cap S_j| = 0}} \frac{\left( \norm{\mu}^{2|S_1|} + (|S_1|!)^2 \right) (|S_2|!)^2 \cdots (|S_t|!)^2 }{|S_1|! \cdots |S_t|! } \right) I_{d^t} \\ & \preceq \left( 20^t (\norm{\mu}^{2t} + 1) \sum_{s_1 + \dots + s_t = t} \binom{t}{s_1, \dots , s_t} s_1 ! s_2! \cdots s_t! \right) I_{d^t} \\ &\preceq (40t)^t( \norm{\mu}^{2t} + 1) I_{d^t} \,.
\end{align*}
Note that in the first step above, we also used the fact that $z_1, \dots , z_t$ are drawn independently. Thus, overall we have shown
\[
\E_{z_1 \sim \mcl{D}(\mu), z_2, \dots , z_{t} \sim \mcl{D}} \left[ \flatten \left( Q_t(z_1, \dots , z_t)\right)^{\otimes 2} \right] \preceq (10t)^{2t} (\norm{\mu}^{2t} + 1) I_{d^t} \,.
\]
Similarly, we have
\[
\E_{z_{t+1} , \dots , z_{2t} \sim \mcl{D}} \left[ \flatten \left( Q_t(z_1, \dots , z_t)\right)^{\otimes 2} \right] \preceq (10t)^{2t} I_{d^t}
\]
and putting everything together, we conclude
\[
\E_{z_1 \sim \mcl{D}(\mu), z_2, \dots , z_{2t} \sim \mcl{D}}\left[ \flatten(R_t(z_1, \dots , z_{2t}))^{\otimes 2} \right] \preceq (20t)^{2t} (\norm{\mu}^{2t} + 1) I_{d^t}
\]
and we are done.
\end{proof}
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